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# Where have all the solar-like stars gone? Rotation period detectability at various inclinations and metallicities Timo Reinhold Max-Planck-Institut für Sonnensystemforschung Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany Alexander I. Shapiro Max- Planck-Institut für Sonnensystemforschung Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany Veronika Witzke Max-Planck- Institut für Sonnensystemforschung Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany Nina-E. Nèmec Max-Planck- Institut für Sonnensystemforschung Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany Institut für Astrophysik, Georg-August-Universität Göttingen, Friedrich-Hund-Platz 1, D-37077 Göttingen, Germany Emre Işık Dept. of Computer Science, Turkish-German University Şahinkaya Cd. 108, Beykoz, 34820 Istanbul, Turkey Sami K. Solanki Max-Planck- Institut für Sonnensystemforschung Justus-von-Liebig-Weg 3, 37077 Göttingen, Germany School of Space Research, Kyung Hee University, Yongin, Gyeonggi, 446-701, Korea ###### Abstract The plethora of photometric data collected by the Kepler space telescope has promoted the detection of tens of thousands of stellar rotation periods. However, these periods are not found to an equal extent among different spectral types. Interestingly, early G-type stars with near-solar rotation periods are strongly underrepresented among those stars with known rotation periods. In this study we investigate whether the small number of such stars can be explained by difficulties in the period determination from photometric time series. For that purpose, we generate model light curves of early G-type stars with solar rotation periods for different inclination angles, metallicities and (magnitude-dependent) noise levels. We find that the detectability is determined by the predominant type of activity (i.e. spot or faculae domination) on the surface, which defines the degree of irregularity of the light curve, and further depends on the level of photometric noise. These two effects significantly complicate the period detection and explain the lack of solar-like stars with known near-solar rotation periods. We conclude that the rotation periods of the majority of solar-like stars with near-solar rotation periods remain undetected to date. Finally, we promote the use of new techniques to recover more periods of near-solar rotators. ††journal: ApJL ## 1 Introduction Stellar brightness variations at the timescale of stellar rotation are caused by transits of magnetic features (such as dark spots or bright faculae) rotating across the visible disk. These variations have routinely been observed by transit photometry missions. In particular, the Kepler telescope obtained light curves of roughly 150,000 main-sequence stars. Some of these light curves exhibit clear signatures of stellar rotation, which can be extracted by standard frequency analysis tools such as Lomb-Scargle periodograms, auto-correlation functions, or wavelet transforms. The biggest survey of rotation periods based on the Kepler data has been published by McQuillan et al. 2014 (hereafter MMA14), who detected rotation periods for 34,030 presumably main-sequence stars. However, for the majority of the main-sequence stars the light curves were either too noisy or too irregular for the rotation period to be determined. MMA14 found that the fraction of stars with detectable periods strongly depends on the effective temperature. Interestingly, this fraction appeared to be lowest for stars with near-solar effective temperature (between 5500–6000 K, hereafter referred to as early G-type stars), reaching only 16% (see Table 3 in MMA14). On the contrary, van Saders et al. (2019) used Galactic evolution models to predict that $\sim$59% of the early G-type stars should have detectable rotation periods. The lack of stars with known rotation period becomes even more severe for early G-type stars of near-solar age. Recently, Reinhold et al. (2020) showed that only a dozen Kepler stars with near-solar fundamental parameters and rotation periods between 20–30 days (i.e. encompassing the solar rotation period of $\sim$25 days) exhibit rotational variability levels similar to that of the Sun. In contrast, the majority of these stars are substantially more variable than the Sun and also show more regular light curves patterns. It has been proposed that such a conspicuous difference between the Sun and other solar-like stars can be explained by a detection bias towards more active stars in bulk rotation period measurements (see the discussion in Amazo-Gómez et al. 2020), thus missing the majority of early G-type stars with near-solar rotation periods and small variabilities. In this Letter we address the question whether the ”missing” solar-like stars (i.e. stars with solar fundamental parameters and rotation periods) do not exist or simply go undetected. Our approach is based on the solar paradigm, i.e. we build on the comprehensive understanding of solar brightness variability (see, e.g., reviews by Ermolli et al. 2013; Solanki et al. 2013), and extend solar models to solar-like stars. Namely, we combine two recently developed physics-based models by Witzke et al. (2020) and Nèmec et al. (2020). This allows calculating light curves of stars with a solar distribution of active regions and solar effective temperature, but various metallicities and observed at arbitrary inclination angles. These light curves are used to identify obstacles in the period determination of solar-like stars, and to discuss possible limitations of period measurements in real data sets. We further compare the number of actual period measurements in Kepler data to predictions from Galactic evolution models using the detection rate obtained from the model light curves. ## 2 Methods ### 2.1 The curious case of the Sun The morphology of the solar light curve (as it would be observed in the Total Solar Irradiance or in the broad-band spectral passband like those of CoRoT, Kepler, or TESS) changes significantly depending on the phase of the solar activity cycle. While it appears to be quite regular at 11-year cycle minima when activity is low, the regularity disappears at periods of intermediate and high solar activity (Lanza & Shkolnik, 2014; Aigrain et al., 2015; He et al., 2015). In particular, Amazo-Gómez et al. (2020) showed that if the Sun were observed by Kepler, the standard frequency analysis tools would most probably fail to detect the correct rotation period (unless observations are done during epochs of low solar activity). The causes for this inability are manifold: solar rotational variability is mainly brought about by spots (see, e.g., Shapiro et al. 2016). The relatively short lifetimes of sunspots from days to weeks (see, e.g., Solanki 2003) implies that most of the spots transit the visible solar disk only once, which leads to irregularities in the solar light curve, and hampers the detection of the solar rotation period. Furthermore, the brightness changes of dark spots and bright faculae partly compensate each other, which decreases the amplitude of the rotational signal, further hindering the period determination (Shapiro et al., 2017; Witzke et al., 2020; Nèmec et al., 2020). The exception from this general tendency are epochs of low solar activity with a small number of active regions. At these times the rotational variability is attributed to long-lived facular features and the light curve pattern becomes more periodic. ### 2.2 The model While the irregularity of the solar light curve is quite well understood, the situation gets more complicated for other early G-type stars. Their light curves look different, partly because the stars are observed at various inclinations. For example, faculae appear brighter at the limb and therefore contribute more strongly to the variability when the star is observed out of the ecliptic plane (Nèmec et al., 2020). Additionally, stellar metallicity $\rm[Fe/H]$ affects facular (and to a smaller degree spot) contrasts (Witzke et al., 2018), which eventually has an impact on the period detectability (Witzke et al., 2020). To synthesize the light curves of solar-like stars, we built on recent calculations by Nèmec et al. (2020) and by Witzke et al. (2018). Nèmec et al. (2020) utilized a semi-empirical sunspot-group record by Jiang et al. (2011) and the Surface Flux Transport Model by Cameron et al. (2010) to reconstruct the distribution of active regions on the solar surface from the year 2010 back to 1700 with a daily cadence. By applying an appropriate geometrical transformation, Nèmec et al. (2020) calculated the distribution of active regions on the solar disk as it would be observed at arbitrary inclinations. Witzke et al. (2018) calculated the brightness contrasts of faculae and spots relative to the quiet Sun (i.e. free from any apparent manifestations of magnetic activity) as a function of wavelength and position on the visible disk for stars with different metallicities and solar effective temperature. All in all, by combining the reconstructed disk distribution of active regions with their brightness contrasts, we generated light curves with a time span of 310 years as they would be seen in the passband of the Kepler telescope. The light curves were calculated for ten inclination angles $0^{\circ}\leq i\leq 90^{\circ}$ (with a step of $10^{\circ}$), and nine different metallicities $\rm-0.4\leq[Fe/H]\leq 0.4$ dex (with a step of 0.1 dex). The solar record from 1700–2010 covers epochs of both low solar activity (like the Dalton minimum around 1790–1830), and very high solar activity (like the modern grand maximum around 1950–2000, see Solanki et al. 2004; Usoskin et al. 2007), which allows studying rotation period detectability during activity cycles of very different strengths. We note that, by assuming a solar disk distribution of active regions, we only account for the metallicity effect on the contrasts of magnetic features. A change in metallicity also affects the depth of the convective zone, which in turn could influence the stellar dynamo, in particular the length of the stellar activity cycle or the emergence latitudes of magnetic bipoles (Schuessler & Solanki, 1992). However, this effects is rather weak, e.g. doubling the metallicity of a star with solar temperature will deepen the convective zone by only ca. 8% (van Saders & Pinsonneault, 2012; Karoff et al., 2018). Therefore, we expect these effects to be relatively small. Studying them is beyond the scope of the present paper, but would be an interesting future exercise. ### 2.3 Monte Carlo approach We take a Monte Carlo approach to analyze light curves with different realizations of inclination angles and metallicities. The distribution of inclination angles is uniform in $\cos{i}$, where $i=0^{\circ}$ denotes a pole-on view and $i=90^{\circ}$ an equator-on view. The input distribution of metallicities was adapted for solar-like stars in the Kepler field (see Fig. A.1 and Reinhold et al. 2020). For each (random) parameter combination $(i,[{\rm Fe/H}])$, we chose the model light curve from the grid with the closest parameters in metallicity and inclination angle. Following the observing strategy of the Kepler telescope, we pick a random 4-year segment of the full time series (see top row of Fig. 1 for example). This light curve is then cut into 90-day segments (i.e. similar to the Kepler observing quarters), where each 90-day chunk is normalized by its median, and appended to the previous one, to form a 4-year time series. These Keplerized light curves (Fig. 1, middle row) will be analyzed for rotation in the next step. The detrending is necessary because it filters out brightness variations on the activity-cycle timescale, and renders the light curves comparable to detrended Kepler data. In addition to the various inclination and metallicity combinations, we study the impact of noise on the period detectability. The model light curves are by definition noise-free. In real observations, the visual stellar magnitude defines the noise level. We use the distribution of Kepler magnitudes $Kp$ of solar-like stars to compute different noise realizations $\sigma$ (see Reinhold et al. 2020 for details). A noise time series with zero mean and standard deviation $\sigma$ is then added to the time series in the Monte- Carlo simulation. In total, we conducted 50,000 Monte Carlo runs, both for the noise-free and the noisy cases to study them separately. ## 3 Results ### 3.1 Period detection From among the various period detection methods, we chose the auto-correlation function (ACF) to search for periodicity in the time series (i.e. the same method as employed by MMA14). The ACF returns peaks of different power as a measure of the periodicity in the light curve. To quantify the strength of the periodicity, we adapt the measure of MMA14, where the local peak height (LPH) is computed as the difference between the highest peak and the mean of the two troughs on either side (see Fig. 1, bottom row). We only search for peaks at periods less than 70 days, consistent with MMA14. If the highest ACF peak lies between 24–30 days and $LPH>0.1$, we count it as a detection. If the peak lies outside this period range or is smaller ($LPH<0.1$), it is counted as a false or non-detection. Fig. 1 illustrates the difficulty of detecting the correct rotation period of the Sun from the photometric time series obtained in the Kepler passband. The top row of Fig. 1 shows the modeled light curve computed for a star with solar metallicity, $\rm[Fe/H]=0$, as it would be observed outside of its equatorial plane at $i=40^{\circ}$. The bottom row gives the ACF and the computed LPH for three different 4-year segments of the same light curve. Depending on the selected segment, the ACF shows the highest peak at different periods. The first panel shows a peak with a moderate LPH but outside the range of 24–30 days (red dashed lines), i.e. a false detection. The second panel shows a peak close to the model rotation period of 27 days, although with a rather low LPH. The last panel shows a clear peak within the range 24–30 days, although this period is found to be the first harmonic of the highest peak at twice the correct rotation period (so that this panel corresponds to a false detection again). The light curve segment shown in this panel corresponds to an epoch of relatively low magnetic activity when the rotational variability of the Sun becomes faculae-dominated. Since faculae have significantly longer lifetimes than spots, this segment shows a more stable periodicity but even in such cases the correct rotation period is not necessarily associated with the highest ACF peak. We now consider how the apparent magnitude of a star affects the period detection. For that purpose, Fig. 2 shows the same light curve as Fig. 1, but with different noise levels to simulate the star as observed at different magnitudes. To demonstrate the effect of noise on the ACF, we chose a segment during solar minimum111The chosen segment slightly differs from the one chosen in the third panel Fig. 1 to demonstrate the effect of noise on the LPH. where the correct period was detected, and added Poisson noise to the light curve, representative of a solar-like star at 11th, 13th, and 15th Kepler magnitude (see Reinhold et al. 2020 for details). In all cases, the correct period was detected. While from 11th to 13th magnitude the LPH only slightly decreases, it decreases by more than half at 15th magnitude. Figure 1: Top row: Model light curve (black) with an inclination of $i=40^{\circ}$ and solar metallicity $\rm[Fe/H]=0$, with three randomly chosen 4-year segments (green). Middle row: Keplerized light curves of the chosen segment from the top row (see Sect. 2.3 for details). Bottom row: auto- correlation function (ACF) of the selected 4-year segment. The measured period is indicated by the red asterisk, and the local peak height (LPH) is shown as the vertical gray line between the peak and the two troughs on either side. The vertical dashed red lines indicate the period detection window from 24 to 30 days. Figure 2: Same as Fig. 1 but choosing the same light curve segment for three different noise realizations corresponding to 11th (left), 13th (middle), and 15th (right) Kepler magnitude. The LPH decreases towards fainter stars. ### 3.2 LPH dependence on spot area The two examples in Sect. 3.1 illustrated how the period detection is affected by the activity level (Fig. 1) and the amount of observational noise (Fig. 2). Fig. 3 combines both of these effects by showing the LPH when the highest peak was found between 24–30 days, as a function of the sunspot coverage on the visible solar disc, averaged over the 4-year segment (in ppm), for different magnitudes. The red diamonds show the median LPH values for the selected spot area bins to better illustrate the LPH dependence. The upper left panel shows only stars brighter than 12th Kepler magnitude (i.e. with small noise levels). The LPH increases with decreasing spot area. As mentioned above, small spot coverages are typically found during activity minima when variability becomes faculae-dominated. Consequently, the light curves become more regular. A similar trend is found for stars between 12th and 13th magnitude but with larger scatter (upper right panel). Between 13th to 14th magnitude (lower left panel), the noise level becomes comparable to the variability amplitude during epochs of small spot coverages. As a result, the LPH drops for small coverages and the increase of the LPH with decreasing spot area can only be identified down to spot areas of 100 ppm. For the faintest stars down to 15th magnitude (lower right panel), the larger noise further decreases the LPH for small spot areas, and for spot areas above 100 ppm no trend can be identified any longer. Figure 3: Local peak height (LPH) vs. spot area fraction for different Kepler magnitudes. The red diamonds show the median LPH values for the selected spot area bins. The very few peaks with $\rm LPH<0.01$ were excluded from the analysis. ### 3.3 Period distribution As shown in the previous section, the position of the highest peak and the associated LPH determine the period detection. The percentages of correct and false detections are given in Table 1. The period distribution is shown in Fig. 4 for different LPH constraints for the noise-free (black) and the noisy (red) case. From the upper left to the lower right panel, the LPH threshold increases from 0.1 to 0.4. Consequently, the detection fraction decreases, but also the number of false detections drops. The decrease of detections is even stronger for the noisy case. We note that also the false detections decrease more strongly for the noisy case (see Table 1). This is caused by the fact that a peak is more easily found in the noise-free case, but the associated period lies outside the range of 24–30 days, and is therefore counted as false detection. When measuring rotation periods in real data, the period is a priori unknown, and one has to assign a certain LPH threshold, for which periods are considered as significant. The upper left panel in Fig. 4 shows that even for small values ($0.1<LPH<0.2$), most detections are found at the correct (model) rotation period of 27 days. However, the number of false detections is also quite high (see Table 1). Further increasing the LPH threshold significantly decreases the number of false detections but also lowers the number of correct detections. Finding an optimal LPH threshold that compromises between discarding correct detections and not having too many false periods is non- trivial. We stress that MMA14 required $LPH>0.3$ to count the period as a real detection. As seen in the lower left panel, this threshold eliminates almost all false detections but strongly decreases the number of real detections (see discussion below). Figure 4: Rotation period distribution for different local peak heights of the noise-free (black) and noisy (red) cases. The blue dashed-dotted line indicates the model rotation period of 27 days. LPH \| Detection \| False detection ---\|---\|--- \| Noise-free \| Noisy \| Noise-free \| Noisy $>$0.1 \| 23.8% \| 17.3% \| 35.2% \| 27.4% $>$0.2 \| 15.5% \| 7.4% \| 9.0% \| 4.5% $>$0.3 \| 9.5% \| 2.9% \| 1.9% \| 0.5% $>$0.4 \| 6.1% \| 1.0% \| 0.4% \| 0.0% Table 1: Detections and false detections for different LPH for both the noise- free and noisy cases. ### 3.4 Detection rate We now turn to the question how the period detectability is affected by stellar inclination and metallicity. For that purpose, we define the detection rate as the number of detections divided by the number of different Monte Carlo runs at a given parameter. In Fig. 5 we show the detection rate as a function of the inclination of the rotation axis of the model star (integrated over all metallicities) for different LPH values. The error bars indicate the square root of the number of detections divided by the number of models. As before, we consider the noise-free (left panel) and the noisy (right panel) cases separately. The noise-free case qualitatively displays the same behavior for all LPH thresholds. As expected, the detection rate is zero for the pole-on view. However, when increasing the inclination angle the detection rate steeply increases to the maximum at an inclination near $20^{\circ}$, and gradually decreases towards the equator-on view at $90^{\circ}$. This result might be surprising at first glance but can be explained by the dominant contribution of faculae to brightness variability for stars with near-equatorial activity belts (similar to those on the Sun) observed close to the pole-on view (Shapiro et al., 2016). In such stars each facular feature spends roughly half a rotation period on the far-side of a star and the remaining half of the time near the limb on the visible disk. Faculae appear especially bright near the limb, and usually last for several stellar rotations. Consequently, the light curves of such stars appear more regular, leading to higher LPH values. We note that the calculations are performed assuming a solar latitudinal distribution of active regions. A change of the distribution would affect the visibility of the active regions, and consequently, the inclination angle corresponding to the maximum of the detection rate. However, we expect that a solar distribution is typical for stars with solar rotation period and temperature (see Sect. 2.2). In the noisy case (right panel) the detection rates are generally smaller (cf. Table 1 and Fig. 4). While the curves have a similar shape to the noise-free cases, their maxima are shifted to higher inclinations. Such a shift is caused by the decrease of the amplitude of brightness variability with decreasing inclination (Nèmec et al., 2020), and consequently a decrease of the signal- to-noise ratio. Consequently, the detection peak near $20^{\circ}$ is suppressed, leaving a residual peak near $40^{\circ}$. As already shown in Figs. 3 and 4, the noise decreases the LPH such that only a few cases with $LPH>0.3$ remain. Figure 5: The detection rate as a function of inclination angle, integrated over all metallicities, for different LPH thresholds. The error bars show the square root of the number of detections divided by the number of models at a given inclination. In Fig. 6 we show the detection rate as a function of metallicity (integrated over all inclinations) for different LPH values. We note that the qualitative shape of the $LPH>0.1$ curve is consistent with the one found in Witzke et al. (2020), who considered the noise-free case (see Sect. C in the appendix for the difference between the calculations in this study and those employed in Witzke et al. 2020). Fig. 6 shows that for both the noise-free (left panel) and the noisy (right panel) case, the detection rate increases with metallicity. This is caused by the stronger contribution of the faculae to the overall variability. Only for the cases $LPH>0.1$ and $LPH>0.2$, does the detection rate show a minimum at $\rm[Fe/H]=-0.3$ dex or $-0.2$ dex (noisy), increasing again towards smaller metallicity values. We expect that this trend continues towards even smaller metallicities. Figure 6: The detection rate as a function of metallicity, integrated over all inclinations, for different LPH thresholds. The error bars show the square root of the number of detections divided by the number of models at a given metallicity. ### 3.5 Comparison with observations We now compare our detection rates (see Table 1) to period detections of solar-like stars in the MMA14 sample. As mentioned above, MMA14 used a relatively conservative detection threshold of $LPH>0.3$. The bottom left panel of Fig. 4 and Table 1 indicate that this threshold represents only the tip of the iceberg: for $LPH>0.3$ (noisy case), the rotation periods can be correctly detected for only 2.9% of our modeled light curves. Galactic evolution models (van Saders et al., 2019) predict that 16% of the (dwarf) stars in the Kepler field with effective temperatures $5500<T_{\rm eff}<6000$ K should have rotation periods between 24–30 days (van Saders, private communications). Using the latest Kepler parameter catalog (Mathur et al., 2017), we select stars in this temperature range, with surface gravities $\log g>4.2$ to exclude more evolved stars, and brighter than 15th Kepler magnitude (following the selection criteria used in Reinhold et al. 2020). We further restrict the catalog metallicities to $\rm-0.45<[Fe/H]<0.45$ dex, which corresponds to the range of simulated metallicities (see Fig. A.1). Selecting such stars from tables 1 and 2 in MMA14 yields $N=16890$ stars. Among those, only 16% will have periods between 24–30 days, and according to our analysis, only 2.9% of these stars will have detectable periods. Thus, we estimate that $N_{det}=N0.160.029=78$ stars should have detectable periods. MMA14 found 455 stars in this parameter range with periods $24\leq P_{\rm rot}\leq 30$ days. However, the vast majority of these stars exhibits variability levels much higher than the Sun, and represent a regime of variability very different from that of the Sun (Reinhold et al., 2020; Zhang et al., 2020; Isık et al., 2020). Consequently, the light curves of these stars cannot be accounted for by our model. To correct for such stars, we followed the approach of Witzke et al. (2020) and selected the stars with variability (regressed to solar values of effective temperature, metallicity, and rotation period, see Fig. S8 and its detailed discussion in Reinhold et al. 2020) below 0.18%, which corresponds to the maximum variability of the Sun over the last 140 years. All in all, only 73 out of the 455 stars satisfied such a criterion. This number is gratifyingly close to our estimate of 78 stars. ## 4 Conclusions In this study we identified biases in the period determination of stars with solar-like variability. The detection rates among these stars are lower than for stars of other spectral types. In particular, only 2.9% of them would be detectable using the thresholds set in MMA14. This is mainly caused by the small variability amplitudes of the rotational tracers and their relatively short lifetimes compared to the rotation period. The very low detection rate explains the large discrepancy between the number of measured rotation periods (MMA14), and those predicted by Galactic evolution models (van Saders et al., 2019). The predicted number of stars with detectable periods (78), and that for which rotation periods have actually been measured (73), is remarkably similar. Fig. 4 shows that many more rotation periods of solar-like stars may be measured when lowering the thresholds in the automated period surveys. However, this will also add a number of false periods, depending on how the thresholds are set. Our study revealed that the rotation periods of most solar-like stars will go undetected using standard frequency analysis tools. Thus, we emphasize the importance of alternative methods for period detection such as the GPS method (Shapiro et al., 2020; Amazo-Gómez et al., 2020) or new approaches based on Gaussian process regression (Foreman-Mackey et al., 2017; Angus et al., 2018; Kosiarek & Crossfield, 2020). We would like to thank Jennifer van Saders for providing model numbers and for helpful discussion. This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 715947). This work has been partially supported by the BK21 plus programme through the National Research Foundation (NRF) funded by the Ministry of Education of Korea. We would like to thank the International Space Science Institute, Bern, for their support of science team 446 and the resulting helpful discussions. ## Appendix A Monte Carlo input distribution Fig. A.1 shows the distributions of input parameters used in the Monte Carlo simulation. The first panel shows the distribution of inclination angles. It can be shown that isotropic inclination angles $i$ exhibit a uniform distribution in $\cos i$ (see e.g. http://keatonb.github.io/archivers/uniforminclination for a detailed derivation). The last bin of the distribution ($85-90^{\circ}$) only contains half the number of realizations because no inclination angles greater than $90^{\circ}$ exist. The same argument applies to the first bin from $0-5^{\circ}$. The middle panel shows the distribution of metallicities of the solar-like stars in the Kepler field. The catalog values were adapted from Mathur et al. (2017) and the selection of solar-like stars can be found in Reinhold et al. (2020). We note that the Sun ($\rm[Fe/H]=0$) is slightly more metal-rich than the peak of the distribution. The last panel shows the apparent magnitudes of the stars in the Kepler field. It is obvious that the majority of stars is very faint. Since the stellar magnitudes define the noise in the light curves, it is crucial to adapt this distribution for the noise model (see Reinhold et al. 2020) to make realistic predictions about stars in the Kepler field. Figure A.1: Input distributions of inclination angles (left) and metallicities (middle), and Kepler magnitudes (right) of the Monte Carlo simulation. ## Appendix B Generating light curves The total spectral flux at a certain time is composed of fluxes emerging from surface areas with different levels of magnetic activity. Following the detailed description in Shapiro et al. (2014), we decompose the spectral flux from a star, $F$, into contributions from the quiet stellar region ($F_{\rm Q}$), faculae ($F_{\rm fac}$), and spots ($F_{\rm spot}$): $F(\lambda)=F_{\rm Q}(\lambda)+F_{\rm fac}(\lambda)+F_{\rm spot}(\lambda),$ (B1) where $\lambda$ is the wavelength. For the quiet stellar region, the disc- integrated flux $\rm F_{Q}(\lambda)$ is obtained by integrating $F_{Q}(\lambda)=\int_{0}^{1}I_{Q}(\lambda,\mu)\omega(\mu)d\mu,$ (B2) where $\omega(\mu)=2\pi\mu(r_{star}/d_{star})^{2}$ is a weighting function with the stellar radius, $r_{star}$, and the distance between the star and the observer, $d_{star}$. The emergent intensity, $I_{Q}(\lambda,\mu)$, also depends on $\mu$, which is the cosine of the angle between the observer’s direction and the local stellar radius. In this formulation the stellar disc center is associated with $\mu=1$ and the limb with $\mu=0$. Both faculae and spots are taken into account through their contrast with respect to the quiet regions. Therefore, the contribution of faculae is defined as $F_{Fac}(\lambda)=\int_{0}^{1}\alpha_{Fac}(\mu)\left[I_{Fac}(\lambda,\mu)-I_{Q}(\lambda,\mu)\right]\omega(\mu)d\mu,$ (B3) where the fractional coverage of the ring corresponding to a given $\mu$ by faculae is given by the function $\alpha_{Fac}(\mu)$. The contribution from spots consists of those from spot umbrae and spot penumbrae: $\displaystyle F_{\rm spot}(\lambda)$ $\displaystyle=$ $\displaystyle\int_{0}^{1}\alpha_{\rm pen}(\mu)\left(I_{\rm pen}(\lambda,\mu)-I_{Q}(\lambda,\mu)\right)\omega(\mu)d\mu$ (B4) $\displaystyle+$ $\displaystyle\int_{0}^{1}\alpha_{\rm umb}(\mu)\left(I_{\rm umb}(\lambda,\mu)-I_{Q}(\lambda,\mu)\right)\omega(\mu)d\mu,$ where $I_{\rm umb}$ and $I_{\rm pen}$ are the emergent intensities from the spot umbrae and spot penumbrae, respectively, and the $\alpha_{\rm umb}$ and $\alpha_{\rm pen}$ denote the corresponding surface coverages. The surface coverages for the magnetic features (i.e. $\alpha_{\rm Fac}$, $\alpha_{\rm umb}$ and $\alpha_{\rm pen}$) used in this work are taking from Nèmec et al. (2020). Furthermore, calculations of the emergent intensities for all stellar regions follow the approach used in Witzke et al. (2020), but with a small modification which is explained in Appendix C. ## Appendix C Calculating emergent intensities The model atmospheres and corresponding emergent spectra computed by Unruh et al. (1999) for the solar faculae, spots, and quiet regions (hereafter, original models) proved to be very successful in reproducing the solar brightness variations with high accuracy (Krivova et al., 2003; Solanki et al., 2013; Ermolli et al., 2013). Here, we extend the intensities for different surface components to different metallicities following the approach outlined in Witzke et al. (2018, 2020), but with slight modifications to cover a broader metallicity range. In our modeling approach, we aim to match the intensity contrasts for the solar metallicity as closely as possible to the original models. Thus, in the first step we searched for the model parameters (input parameters for calculating stellar atmospheres with ATLAS9), such as convection settings, surface gravity and continuum opacity sources for the quiet Sun, spot-umbra and spot-penumbra, to match the original models and spectra by Unruh et al. (1999). The closest match is achieved by the parameters listed in Table 2. Then for the generation of the faculae model we assumed that the temperature difference, $\Delta\text{T}$, and pressure difference, $\Delta P$, as a function of column mass between the original facular and quiet Sun models are the same as between our new facular and quiet Sun models. Finally, to calculate atmospheric models for different metallicity values, we first generated atmosphere models for the quiet regions and the spots assuming radiative equilibrium. Then we followed up on the Witzke et al. (2018) approach and assumed that a change of the metallicity value has the same effect on the temperature and pressure structures of the quiet Sun and faculae. Using the quiet stellar atmosphere models for different metallicities, we applied the solar $\Delta\text{T}$ and $\Delta P$ with column mass to calculate the facular models. Using these atmospheric models for the quiet regions and magnetic features, we generated the emergent intensities $I_{\lambda,\mu}$ for each metallicity value using the MPS-ATLAS code (Witzke et al., in prep.). model \| effective \| surface \| mixing \| over ---\|---\|---\|---\|--- \| temperature [K] \| gravity \| -length \| -shoot Quiet region \| 5777 \| 4.43777 \| 1.25 \| on Spot umbra \| 4500 \| 4.0 \| 1.25 \| on Spot penumbra \| 5450 \| 4.0 \| 1.25 \| on Table 2: Input parameters for model atmospheres in radiative equilibrium. ## References * Aigrain et al. 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# One Model to Serve All: Star Topology Adaptive Recommender for Multi-Domain CTR Prediction Xiang-Rong Sheng, Liqin Zhao and Guorui Zhou, Xinyao Ding, Binding Dai, Qiang Luo Siran Yang, Jingshan Lv, Chi Zhang, Hongbo Deng, Xiaoqiang Zhu Alibaba GroupBeijing, China (2021) ###### Abstract. Traditional industry recommendation systems usually use data in a single domain to train models and then serve the domain. However, a large-scale commercial platform often contains multiple domains, and its recommendation system often needs to make click-through rate (CTR) predictions for multiple domains. Generally, different domains may share some common user groups and items, and each domain may have its own unique user groups and items. Moreover, even the same user may have different behaviors in different domains. In order to leverage all the data from different domains, a single model can be trained to serve all domains. However, it is difficult for a single model to capture the characteristics of various domains and serve all domains well. On the other hand, training an individual model for each domain separately does not fully use the data from all domains. In this paper, we propose the Star Topology Adaptive Recommender (STAR) model to train a single model to serve all domains by leveraging data from all domains simultaneously, capturing the characteristics of each domain, and modeling the commonalities between different domains. Essentially, the network of each domain consists of two factorized networks: one centered network shared by all domains and the domain-specific network tailored for each domain. For each domain, we combine these two factorized networks and generate a unified network by element-wise multiplying the weights of the shared network and those of the domain-specific network, although these two factorized networks can be combined using other functions, which is open for further research. Most importantly, STAR can learn the shared network from all the data and adapt domain-specific parameters according to the characteristics of each domain. The experimental results from production data validate the superiority of the proposed STAR model. Since late 2020, STAR has been deployed in the display advertising system of Alibaba, obtaining 8.0% improvement on CTR and 6.0% increase on RPM (Revenue Per Mille). Multi-Domain Learning, Recommender System, Display Advertising ††copyright: acmcopyright††journalyear: 2021††copyright: acmcopyright††conference: Proceedings of the 30th ACM International Conference on Information and Knowledge Management; November 1–5, 2021; Virtual Event, QLD, Australia††booktitle: Proceedings of the 30th ACM International Conference on Information and Knowledge Management (CIKM ’21), November 1–5, 2021, Virtual Event, QLD, Australia††price: 15.00††doi: 10.1145/3459637.3481941††isbn: 978-1-4503-8446-9/21/11††ccs: Information systems Information retrieval Figure 1. Two representative business domains, Banner and Guess What You Like on Taobao mobile app home. A business domain is referred to as a specific spot that items are presented to users in the mobile app and PC websites. ## 1\. Introduction Traditional CTR prediction models (Zhou et al., 2018, 2019; Rendle, 2010; Guo et al., 2017b; Cheng et al., 2016) focus on single-domain prediction, where the CTR model serves for a single business domain after trained with examples collected from this domain. Each business domain is a specific spot that items are presented to users on the mobile app or PC websites. At large commercial companies like Alibaba and Amazon, there are often many business domains that need CTR prediction to enhance user satisfaction and improve business revenue. For example, in Alibaba, the business domains range from Guess What You Like in Taobao App homepage, Banner of Taobao App homepage to other domains (Zhu et al., 2017). Figure 1 shows two representative business domains in Alibaba. * • Banner: In banner, the items to be recommended appears in the top banner of the Taobao home page. The item can be a single commodity, a store, or a brand. * • Guess What You Like: In Guess What You Like, items are all single commodities and displayed to users in the left or right column. Since different business domains have overlapping user groups and items, there exist commonalities among these domains. Enabling information sharing is beneficial for learning the CTR model of each domain. However, the specific user group may be different and the users’ behaviors also change in various domains. These distinctions result in domain-specific data distributions. Simply mixing all the data and training a single shared CTR model can not work well on all domains. Besides mixing data and training a shared model, another simple solution is to build a separate model per business domain. This strategy also has some downsides: (1) some business domains have much less data than other domains. Splitting the data neglects the domain commonalities and causes much less training data, making the models hard to learn. (2) Maintaining multiple models cause a tremendous amount of resource consumption and require much more human cost. It is unduly burdensome when the number of business domains is up to hundreds. Figure 2. (a): Single shared model for all domains, square nodes indicate the shared model. (b): One model per domain where each model is learned separately. The circle node indicates the domain-specific model. (c): The proposed Star Topology Adaptive Recommender (STAR), where each domain has specific parameters and also shares a common centered model. The edges mean the combination of the center shared parameters with the domain-specific parameters. This paper aims to learn an effective and efficient CTR model to handle multiple domains simultaneously. We formulate multi-domain CTR prediction as the problem that the recommender needs to make CTR prediction for $M$ business domains $D_{1},D_{2},\dots,D_{M}$ simultaneously. The model takes input as $({\bf x},y,p)$, where ${\bf x}$ is the common feature used by multiple business domains like historical user behavior, user profile feature, item feature, and context feature. $y\in\\{0,1\\}$ is the clicked label, and $p$ is the domain indicator that indicates which domain this sample is collected. Note that $({\bf x},y)$ is drawn from the domain-specific distribution $D_{p}$, and the distribution varies with different domains. Multi-domain CTR prediction aims to construct an effective and efficient model that gives accurate CTR prediction for each domain and at a trivial cost on resource consumption. To achieve this goal, the model should make full use of the domain commonalities and capture the domain distinction. One possible strategy to improve learning with multiple domains is multi-task learning (Ruder, 2017; Caruana, 1998; Ma et al., 2018b). As shown in Figure 3, the difference between multi-domain CTR prediction and multi-task learning is that multi-domain CTR prediction solves the same task, i.e., CTR prediction, across different domains, in which the label spaces of different domains are the same and the data distribution is different. By contrast, most multi-task learning approaches (Ma et al., 2018b; Misra et al., 2016; Ma et al., 2018a, 2019; Tang et al., 2020) address various tasks in the same domain, where the label space might be different, e.g., jointly estimate CTR and conversion rate (CVR) (Ma et al., 2018a; Wen et al., 2020). Due to the heterogeneity of tasks, existing multi-task learning approaches focus on sharing information in the bottom layers but keeping separate task-specific output layers (Ruder, 2017). Directly adapting multi-task approaches to multi-domain CTR prediction can not sufficiently exploit the domain relationship in the label space and neglect the distinct data distribution of different domains. Figure 3. Comparison of multi-task learning with multi-domain learning. Most multi-task learning approaches focus on tackling different tasks within a single domain. In contrast, multi-domain learning makes predictions for multiple domains addressing the same task, e.g., CTR prediction, where the label spaces are of the same. Directly adapting multi-task approaches to multi-domain CTR prediction can not sufficiently exploit the domain relationship in the label space and neglects the distinct data distribution of different domains. To fully exploit the domain relationship, we propose Star Topology Adaptive Recommender (STAR) for multi-domain CTR prediction. The proposed STAR model has the star topology, as illustrated in Figure 4. STAR consists of shared centered parameters and multiple sets of domain-specific parameters. The final model of each domain is obtained by combining the shared centered parameters and the domain-specific parameters. The centered parameters are used to learn general behaviors among all domains, in which the common knowledge can be learned and transferred among all domains. The domain-specific parameters capture specific behaviors in different domains to facilitate more refined CTR prediction. The star topology facilitates effective information transformation across multiple domains to learn domain commonalities and distinctions. This paper implements the STAR model with the element-wise product of weights in each layer as the combination strategy. Since embedding layers contribute most parameters in industrial recommender, the added domain-specific parameters are negligible to the total amount of parameters. Thus, using the STAR model to serve multiple domains only adds little computational and memory costs while yielding much better performance. The main contributions of this work can be summarized as follows: * • We propose Star Topology Adaptive Recommender (STAR) to tackle multi-domain CTR prediction. The star topology facilitates effective information transformation across multiple domains to learn domain commonalities while capturing domain distinction. * • Different domains have different data distributions, this leads to inaccurate statistics when using batch normalization. We propose Partitioned Normalization (PN) that privatizes normalization for examples from different domains to address this issue. PN can lead to more accurate moments within the domain, which improves model performance. * • In multi-domain CTR prediction, features that depict the domain information are of importance. We propose an auxiliary network that treats the domain indicator directly as the input and learns its embeddings to depict the domain. The embeddings are then fed to the auxiliary network, which is much simpler than the original network. This makes the domain indicator influence the final prediction in a direct manner. * • We evaluate STAR on the industrial production dataset and deploy it in the display advertising system of Alibaba in 2020. The consistent superiority validates the efficacy of STAR. Up to now, the deployment of STAR brings 6% CTR and 8% RPM lift. We believe the lessons learned in our deployment generalize to other setups and are thus of interest to researchers and industrial practitioners. ## 2\. Related Work Our work is closely related to traditional single-domain CTR prediction, where the recommender is trained on a single business domain and then serve for this business domain. Besides, our work is also related to multi-task learning and multi-domain learning. In this section, we give a brief introduction. ### 2.1. Single-Domain CTR Prediction Inspired by the success within deep learning, recent CTR prediction model has made the transition from traditional shallow approaches (Friedman, 2001; Rendle, 2010; Koren, 2008; Koren et al., 2009; Zhou et al., 2008) to modern deep approaches (Guo et al., 2017b; Cheng et al., 2016; Qu et al., 2016; Zhou et al., 2018, 2019; Pi et al., 2019). Most deep CTR models follow the embedding and MLP paradigm. Wide & Deep (Cheng et al., 2016) and deepFM (Guo et al., 2017b) combine low-order and high-order features to improve the expression power of the model. PNN (Qu et al., 2016) introduces a product layer to capture interactive patterns between inter-field categories. In these models, the user’s history behaviors are transformed into low-dimensional vectors after the embedding and pooling. DIN (Zhou et al., 2018) employs the mechanism of attention to activate historical behaviors locally w.r.t. the given the target item, and successfully captures the diversity characteristic of user interest. DIEN (Zhou et al., 2019) further proposes an auxiliary loss to capture latent interest from historical behaviors. Additionally, DIEN integrates the attention mechanism with GRU to model the dynamic evolution of user interest. MIND (Li et al., 2019) and DMIN (Xiao et al., 2020) argue that a single vector might be insufficient to capture complicated pattern lying in the user and items. Capsule network and the dynamic routing mechanism are introduced in MIND to learn multiple representations to aggregate raw features. Moreover, inspired by the success of the self-attention architecture in the tasks of sequence to sequence learning (Vaswani et al., 2017), Transformer is introduced in (Feng et al., 2019) for feature aggregation. MIMN (Pi et al., 2019) proposes a memory-based architecture to aggregate features and tackle the challenge of long-term user interest modeling. SIM (Pi et al., 2020) extracts user interests with two cascaded search units, which achieves better ability to model lifelong sequential behavior data in both scalability and accuracy. ### 2.2. Multi-Task Learning Multi-task learning (Caruana, 1998; Ruder, 2017) aims to improve generalization by sharing knowledge across multiple related tasks. The shared knowledge and task-specific knowledge are explored to facilitate the learning of each task. Multi-task learning has been used successfully on multiple application domains, ranging from natural language processing (Collobert and Weston, 2008), speech recognition (Deng et al., 2013), recommender system (Yuan et al., 2020) to computer vision (Kendall et al., 2018). In early literature on MTL for linear models, Argyriou et al. (2008) propose a method to learn sparse representations shared across multiple tasks. In the context of deep learning, multi-task learning is typically done with parameter sharing of hidden layers (Caruana, 1998; Ma et al., 2018a). Misra et al. (2016) propose cross-stitch units to learn unique combinations of task-specific hidden-layers for each task. Ma et al. (2018b) proposes Multi-gate Mixture-of- Experts (MMoE) to model task relationships by sharing the expert sub-models across all tasks, while also having a gating network trained to optimize each task. Kendall et al. (2018) propose a principled approach to multi-task deep learning which weighs multiple loss functions by considering the homoscedastic uncertainty of each task. In multi-task learning, different tasks may conflict, necessitating a trade-off, optimize a proxy objective that minimizes a weighted linear combination of per-task losses may not be optimal. To address this issue, Sener and Koltun (2018) explicitly cast multi-task learning as multi-objective optimization, with the overall objective of finding a Pareto optimal solution. Note that (Kendall et al., 2018; Sener and Koltun, 2018) are complementary to this work and could be potentially combined to achieve better performance. ### 2.3. Multi-Domain Learning In real-world applications, it is oftentimes that the data are collected from multiple domains (Dredze et al., 2010; Joshi et al., 2012; Li et al., 2020). Multi-domain learning enables knowledge transfer between domains to improve learning. As such, it contrasts with the domain adaptation (DA) problem (Bickel et al., 2007; Ben-David et al., 2010), where knowledge transfer is only one way, i.e., from the source domain to the target domain. Wang et al. (2019) propose Transferable Normalization in place of existing normalization techniques for domain adaptation and reveals that BN (Ioffe and Szegedy, 2015) is the constraint of transferability. Multi-domain CTR prediction can be seen as a special kind of multi-domain learning problem, in which each domain corresponds to a business domain and the task is the CTR prediction. Compared with traditional multi-domain learning, our work focuses on CTR prediction. The proposed model makes full use of the domain indicator that is directly fed as the ID feature and learning its semantic embeddings to facilitates the model learning, which is neglected by previous literature. The difference between multi-domain learning and multi-task learning is that multi-domain learning makes prediction for multiple domains addressing the same problem, e.g., CTR prediction, where the label spaces are of the same. In contrast, multi-task learning focuses on tackling different problems (Yang and Hospedales, 2015). For example, in the field of video recommendation, a multi-task learning problem can be as simultaneously predicting CTR and expected watch time of videos for a single business domain and multi-domain CTR prediction makes CTR predictions for multiple business domains, e.g., multiple video platforms. ## 3\. The Proposed Approach Figure 4. Comparison of model for single-domain CTR prediction and the Star Topology Adaptive Recommender (STAR) for multi-domain CTR prediction. In STAR, the partitioned normalization (PN) privatizes normalization for examples from different domains. The normalized features are then fed as input to the following star topology fully-connected neural network (star topology FCN). The star topology FCN consists of shared centered FCN and multiple domain- specific FCNs. The final combined model of each domain is obtained by the element-wise product of weights in each layer. In this section, we first give a brief introduction about the background of multi-domain CTR prediction. Next is the architecture overview of the proposed method, star topology adaptive recommender (STAR) for multi-domain CTR prediction. Then we introduce STAR in detail, including the proposed star topology network, partitioned normalization, and auxiliary network. ### 3.1. Multi-Domain CTR Prediction In sequential recommender systems, the model takes input as the user historical behavior, user profile feature, target item feature, and other features like context feature. The predicted CTR $\hat{y}$ of a user $u$ clicking on an item $m$ is calculated via: $\hat{y}=\mathrm{f}(E(u_{1}),\dots,E(u_{i});E(m_{1}),\dots,E(m_{j});E(c_{1}),\dots,E(c_{k})),$ where $\\{u_{1},\dots,u_{i}\\}$ is the set of user features including user historical behavior and user profile feature. $\\{m_{1},\dots,m_{j}\\}$ is the set of target item feature and $\\{c_{1},\dots,c_{k}\\}$ is the set of other features. The $E(\cdot)\in\mathbb{R}^{d}$ means the embedding layer which maps the sparse IDs into learnable dense vectors. After mapping the raw features to low-dimensional embeddings, the common practice is to aggregate these embeddings to obtain fixed-length vectors. Different kinds of aggregation methods like (Zhou et al., 2018, 2019) can be employed to aggregate these embeddings to extract user interest and get the fixed-length representation. The obtained representation is then fed into the following deep neural network, e.g., a multi-layer fully-connected network, to get the final CTR prediction. Traditional CTR models (Guo et al., 2017b; Lian et al., 2018; Cheng et al., 2016; Zhou et al., 2018, 2019) are usually trained on data from a single business domain. However, real-world recommender often has to deal with multiple business domains. Concretely, the recommender needs to make CTR prediction for $M$ domains $D_{1},D_{2},\dots,D_{M}$ simultaneously. The model takes input as $({\bf x},y,p)$, where ${\bf x}$ is the common feature used by multiple domains like user historical behavior and user profile feature, target item feature as mentioned above. $y\in\\{0,1\\}$ is the clicked label and $p\in\\{1,2,\dots,M\\}$ is the domain indicator that indicates which domain this sample is collected. Note that $({\bf x},y)$ is drawn from the domain-specific distribution $D_{p}$ and the distribution varies for different domains. The goal of multi-domain CTR prediction is to construct a single CTR model that can give accurate CTR prediction to serve all domains at low resource consumption and human cost. ### 3.2. Architecture Overview As mentioned above, ignoring domain indicator $p$ and learning a single shared CTR model neglect the domain differences. This leads to inferior model performance. On the other hand, training separate models for each domain performs much worse since splitting the domains provides much less data for each model. Besides, it is infeasible to maintain each domain a separate model in production due to the resource consumption and human cost. To this end, we propose Star Topology Adaptive Recommender (STAR) for multi- domain CTR prediction to better utilize the similarity among different domains while capturing the domain distinction. As shown in Figure 4, STAR consists of three main components: (1) the partitioned normalization (PN) which privatizes normalization for examples from different domains, (2) the star topology fully-connected neural network (star topology FCN), (3) the auxiliary network that treats the domain indicator directly as the input feature and learns its semantic embeddings to capture the domain distinction. During training, a domain indicator $p$ is first sampled and then a mini-batch of $B$ instances $({\bf x}_{1},p),({\bf x}_{2},p),\dots,({\bf x}_{B},p)$ is sampled from this domain. STAR first embeds these input features as low- dimensional vectors by an embedding layer. In industrial recommender, the model is often trained with billions of features (Jiang et al., 2019) and the parameters of embedding are usually much more than other parts of the model. This makes it difficult for different domains to learn domain-specific embeddings with limited data. For example, for models used in our daily tasks, the embeddings parameters are 10,000 times more than the parameters of fully- connected layers (Jiang et al., 2019). Thus, in the proposed STAR model, we let all business domains share the same embedding layer, i.e., the same ID features in different domains share the same embedding. Sharing embedding layer across multiple domains can significantly reduce the computational and memory cost. The embeddings are then pooled and concatenated to obtain $B$ fixed-length representations. After that, the $B$ extracted representations are processed by the proposed partitioned normalization (PN) layer that privatizes normalization statistics for different domains. The normalized vectors are then fed as input to the proposed star topology FCN to get the output. The star topology FCN consists of shared centered FCN and multiple domain-specific FCNs. The final model of each domain is obtained by combining the shared centered FCN and domain-specific FCN. In multi-domain CTR prediction, features that depict the domain information is of importance. In the STAR model, the auxiliary network treats the domain indicator as input and fed with other features depicting the domain to the auxiliary network. The output of the auxiliary network is added with the output of the star topology FCN to get the final prediction. We make the auxiliary network much simpler than the star topology FCN to let the model capture the domain distinction in a direct and easy manner. In what follows we will describe these components in detail. ### 3.3. Partitioned Normalization As mentioned above, the raw features are first transformed into low- dimensional embeddings and then pooled and aggregated to get the intermediate representation. Denote the intermediate representation of an instance as ${\bf z}$, to train deep networks fast and stably, a standard practice is applying normalization layer to the intermediate representation ${\bf z}$. Among all normalization methods, batch normalization (BN) (Ioffe and Szegedy, 2015) is a representative method that is proved to be crucial to the successful training of very deep neural networks (Ioffe and Szegedy, 2015; Radford et al., 2016). BN uses a global normalization for all examples, which accumulates normalization moments and learns shared parameters across all samples. Concretely, the normalization of BN in training is given as (1) ${\bf z}^{\prime}=\gamma\frac{{\bf z}-\mu}{\sqrt{\sigma^{2}+\epsilon}}+\beta,$ where ${\bf z}^{\prime}$ is the output, $\gamma,\beta$ are the learnable scale and bias parameters, $\mu,\sigma^{2}$ are mean and variances of current mini- batch. During testing, moving averaged statistics of mean $E$ and variance $Var$ across all samples are used instead (2) ${\bf z}^{\prime}=\gamma\frac{{\bf z}-E}{\sqrt{Var+\epsilon}}+\beta.$ In other words, BN assumes all samples are i.i.d. and use the shared statistics across all training samples. However, in multi-domain CTR prediction, samples are only assumed to be locally i.i.d. within a specific domain. Thus, data from different domains have different normalization moments. Sharing global moments and parameters of BN layers during testing will obscure domain differences and lead to degraded model performance. To capture the unique data characteristic of each domain, we propose partitioned normalization (PN) which privatizes normalization statistics and parameters for different domains. Concretely, during training, suppose the current mini-batch is sampled from the $p$-th domain, we compute the mean and variances of the current mini-batch and normalize the feature as: (3) $z^{\prime}=(\gamma\gamma_{p})\frac{z-\mu}{\sqrt{\sigma^{2}+\epsilon}}+(\beta+\beta_{p}),$ where $\gamma,\beta$ are the global scale and bias, and $\gamma_{p},\beta_{p}$ are the domain-specific scale and bias parameters. For each mini-batch, it receives the final scale by element-wise multiplying the shared $\gamma$ with the domain-specific $\gamma_{p}$, i.e., PN adaptively scales the representation according to the domain indicator. Similarly, the bias of PN is also adaptive conditioned on the domain, which is implemented by the addition of global bias $\beta$ and domain-specific bias $\beta_{p}$. Note that compared with BN, PN also uses the moments of the current mini-batch during training, but PN introduces domain-specific scale and bias $\gamma_{p},\beta_{p}$ to capture the domain distinction. Besides the modification of the scale and bias, PN also let different domains to accumulate the domain-specific moving average of mean $E_{p}$ and variance $Var_{p}$. During testing, PN transforms instance ${\bf z}$ from the $p$-th domain as: (4) $\displaystyle{\bf z}^{\prime}=(\gamma\gamma_{p})\frac{{\bf z}-E_{p}}{\sqrt{Var_{p}+\epsilon}}+(\beta+\beta_{p}).$ From Equation 4, we can see that PN uses the domain-specific mean $E_{p}$ and variance $Var_{p}$ to normalize the intermediate representation ${\bf z}$. Thus PN adaptively alters the intermediate representation conditioned on the domain indicator to capture the distinctive characteristics. ### 3.4. Star Topology FCN Figure 5. An illustration on how STAR generates the parameters of fully- connected network (FCN) for different domains. STAR consists of a shared centered FCN and independent FCNs per domain. For each domain, the final weights of a neural network layer are obtained by element-wise multiplying the weights of the shared FCN and the domain-specific FCN. The shared parameters are updated through the gradient of all examples, while the domain-specific parameters are only updated through examples within this domain. After the PN layer, the representation ${\bf z}^{\prime}$ is fed as input to the following star topology multi-layer fully-connected neural network (star topology FCN). As depicted in Figure 5, the proposed star topology FCN consists of a shared centered FCN and independent FCNs per domain, thus the total number of FCN is $M+1$. The final model of $p$-th domain is obtained by combining the shared centered FCN and domain-specific FCN, in which the centered parameters learn general behaviors among all domains, and the domain- specific parameters capture specific behaviors in different domains to facilitate more refined CTR prediction. Specifically, for the shared FCN, let $W$ be the weights and $b$ be the bias in a neural network layer respectively. For the specific FCN of the $p$-th domain, let $W_{p}$ be the weights and $b_{p}$ be the bias in the corresponding layer. Denote the input dimension as $c$ and the output dimension as $d$, i.e, $W,W_{p}\in\mathbb{R}^{c\times d},b,b_{p}\in\mathbb{R}^{d}$. The final weights $W^{\star}_{i}$ and bias $b^{\star}_{i}$ for the $p$-th domain is obtained by: (5) $W^{\star}_{p}=W_{p}\otimes W,b^{\star}_{p}=b_{p}+b,$ where $\otimes$ denotes the element-wise multiplication. Let $in_{p}\in\mathbb{R}^{c\times 1}$ denote the input of this neural network layer from the $p$-th domain, the final output $out_{p}\in\mathbb{R}^{d}\times 1$ is given by: (6) $out_{p}=\phi((W^{\star}_{p})^{\top}in_{p}+b^{\star}_{p}),$ where $\phi$ denotes the activation function of this layer. The combination of shared parameters of domain-specific parameters is employed in all layers. By this means, STAR can modulate its parameters conditioned on the domain. Note that we implement the combination strategy of the shared centered FCN and domain-specific FCN by element-wise product of between weights and addition of bias in each layer, other strategies can also be investigated for better performance. The shared parameters are updated through the gradient of all examples while the domain-specific parameters are only updated through examples within this domain. This helps captures the domain differences for more refined CTR prediction while learning the domain commonality through the shared centered parameters. As mentioned above, most of the parameters in industrial recommenders are contributed by the embedding layer, the increased $M$ FCNs is negligible to the total amount of parameters. Thus STAR uses one model to effectively serve all business domains in a parameter efficient and memory friendly manner. ### 3.5. Auxiliary Network In the traditional way of CTR modeling, all features are treated equally and fed to the complicated model. In multi-domain CTR prediction, however, it may be hard for the model to automatically learn the domain distinction. We argue that a good multi-domain CTR model should have the following characteristic: (1) have informative features regarding the domain characteristic (2) make these features easily and directly influence the final CTR prediction. The intuition behind is that features that depict the information of domains are of importance since it can reduce the difficulty for the model to capture the distinction among domains. To this end, we propose an auxiliary network to learn the domain distinction. To augment informative features regarding the domain characteristic, we treat the domain indicator directly as the ID feature input. The domain indicator is first mapped into embedding vector and concatenated with other features. The auxiliary network then computes forward pass with respect to the concatenated features to gets the one-dimensional output. Denote the one-dimensional output of star topology FCN as $s_{m}$ and the output of the auxiliary network as $s_{a}$. $s_{m}$ and $s_{a}$ are added to get the final logit. Sigmoid is then applied to get the CTR prediction: (7) $\textrm{Sigmoid}(s_{m}+s_{a}).$ In our implementation, the auxiliary network is much simpler than the main network, which is a two-layer fully connected neural network. The simple architecture makes the domain features directly influence the final prediction. Denote $\hat{y}^{p}_{i}$ the predicted probability for the $i$-th instance in the $p$-th domain and $y^{p}_{i}\in\\{0,1\\}$ the ground truth. We minimize the cross entropy loss function between the $\hat{y}^{p}_{i}$ and label $y^{p}_{i}$ in all domains as: (8) $\displaystyle\min\sum_{p=1}^{M}\sum_{i=1}^{N_{p}}-y^{p}_{i}\mathrm{log}(\hat{y}^{p}_{i})-(1-y^{p}_{i})\mathrm{log}(1-\hat{y}^{p}_{i}).$ ## 4\. Experiments Table 1. The example percentage and average click-through rate (CTR) of each domain. \| 1 \| 2 \| 3 \| 4 \| 5 \| 6 \| 7 \| 8 \| 9 \| 10 \| 11 \| 12 \| 13 \| 14 \| 15 \| 16 \| 17 \| 18 \| 19 ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Percentage \| 0.99% \| 1.61% \| 3.40% \| 3.85% \| 2.79% \| 0.56% \| 4.27% \| 16.76% \| 10.00% \| 12.16% \| 0.76% \| 1.31% \| 3.34% \| 28.76% \| 1.17% \| 0.46% \| 1.05% \| 0.91% \| 5.85% CTR \| 2.14% \| 2.69% \| 2.97% \| 3.63% \| 2.77% \| 3.45% \| 3.59% \| 3.24% \| 3.23% \| 2.08% \| 12.05% \| 3.52% \| 1.27% \| 3.75% \| 12.03% \| 4.02% \| 1.63% \| 4.64% \| 1.42% We evaluate the efficacy of STAR in this section. We begin by introducing the setup including the used production dataset, compared methods and implementation details in Sec. 4.1. The results and discussion are elaborated in Sec. 4.2. We also perform in-depth ablation studies in Sec. 4.3. Experimental results on production environment are shown in Sec. 4.4. ### 4.1. Experimental Settings Dataset. Due to the lack of public dataset on multi-domain CTR prediction, we use Alibaba production data regarding user click behavior on 19 business domains to perform the offline evaluation. The training data is collected from traffic logs of the online display advertising system of Alibaba. Data of one day from 19 business domains are used for training and the data of the following day is used for testing. The training dataset consists of billions of examples. Table 1 shows the example percentage and average CTR (# Click/# Impression, i.e., ratio of positive examples) of each domain in the training set. As shown in Table1, different domains have different domain-specific data distribution, which can be reflected from the different CTR. It can be seen that domain with the highest CTR (domain #15) is 12.03% while domain with the lowest CTR (domain #13) is only 1.27%. In this dataset, the majority of items are available in most of the business domains while only some of users are overlapping, e.g., domain #1 and domain #2 have the same set of items but only have 8.52% overlapping users. Compared models. To verify the effectiveness of the proposed approach, we compare STAR with the following models: * • Base. We refer to Base as the model composed of embedding layer, pooling & concatenation layer, batch normalization, and a 7-layer fully-connected network. Specifically, the pooling & concatenation layer is based on DIEN (Zhou et al., 2019), which extracts user interest after the embedding layer. We mix all samples from different domains and train the base model. * • Shared Bottom. The Shared Bottom model is a multi-task model that shares the parameters of the bottom layers. In our implementation, we let the Shared Bottom share the embedding layer. Each domain will also have a specific 7-layer fully-connected network that is not shared. * • MulANN. MulANN (Sebag et al., 2019) adds domain discriminator module to the Base model. The domain discriminator classifies which domain the examples are from. MulANN adopts a adversarial loss to let the domain discriminator indiscriminates with respect to the shift between the domains. * • MMoE. MMoE (Ma et al., 2018b) implicitly models task relationships for multi- task learning, where different tasks may have different label spaces. Here we adapt MMoE for multi-domain CTR prediction, where each expert is a 7-layer fully-connected network. The number of experts is equal to the number of domains. Besides, MMoE also learns gating networks per domain that takes the input features and outputs softmax gates assembling the experts with different weights. * • Cross-Stitch. Cross-Stitch (Misra et al., 2016) uses linear cross-stitch units to learn an optimal combination of task-specific representations. In the cross-stitch method, each domain have a 7-layer fully-connected network and the cross-stitch units are added in each hidden layer to learn task-specific representations. To give a fair comparison, all compared methods and the STAR model are trained with the proposed auxiliary network in Sec. 4.2. The ablation study about the auxiliary network is performed in Sec. 4.3. Implementation details. All models are trained with Adam (Kingma and Ba, 2015), the learning rate is set to 0.001 and the batch size is 2000. We minimize the cross-entropy loss for samples from all domains to train the model. Metrics. Area under the ROC curve (AUC) is the common metric used to evaluate the performance of CTR prediction. An variation of user weighted AUC (Zhou et al., 2018) measures the goodness of intra-user order by averaging AUC over users and is shown to be more relevant to online performance in recommender system. It is calculated as follows: (9) $\textrm{AUC}=\frac{\sum_{i}^{n}\\#\textrm{impression}_{i}\times\textrm{AUC}_{i}}{\sum_{i}^{n}\\#\textrm{impression}_{i}},$ where $n$ is the number of users, $\textrm{impression}_{i}$ and $\textrm{AUC}_{i}$ are the number of impressions and AUC of the $i$-th user, respectively. We use this weighted AUC as the evaluation metric and still refer it to as AUC for simplicity. Concretely, we use the AUC of each domain and overall AUC (mixing samples from all domains to calculate the overall AUC) as the metrics. ### 4.2. Results We evaluate all approaches on the Alibaba production dataset. To give a fair comparison, all compared methods and STAR model are trained with the proposed auxiliary network. As illustrated in Table 2, the consistent improvement validates the efficacy of STAR. Note that the performance of MulANN is worse than the Base model, which proves obscuring domain difference hurts the modeling of multi-domain CTR prediction. Besides, the shared Bottom model, MMoE, Cross-Stitch and STAR all achieve better overall performance than the Base model. This demonstrates the importance of exploiting domain relationship and capturing domain distinction to enhance the prediction performance. Although the Shared Bottom, MMoE, and Cross-Stitch achieve better overall performance than the Base model, it is notable that in some domains, the AUCs of Shared Bottom, MMoE, and Cross-Stitch are worse than the Base model, e.g., domain # 5, #6, and #16. We hypothesize this is because the learning of these models conflicts in different domains. In contrast, STAR avoids this issue by its star topology, where the the domain-specific parameters are only updated through examples within this domain. The proposed STAR model exhibits superior performance across all domains compared with the Base model. STAR also achieves consistent improvement over the Shared Bottom, which demonstrates the importance of information sharing on top specific layers for multi-domain learning, where all domains share the same label space. STAR also outperforms MMoE and Cross-Stitch, which shows the superiority of explicitly modeling domain relationships compared with implicitly modeling domain relationships by the gate networks or cross-stitch units. Table 2. Results of different approaches on offline Alibaba production dataset. \| Base \| Shared Bottom \| MulANN \| MMoE \| Cross-Stitch \| STAR ---\|---\|---\|---\|---\|---\|--- #1 \| 0.6134 \| 0.6186 \| 0.6143 \| 0.6143 \| 0.6183 \| 0.6306 #2 \| 0.6321 \| 0.6320 \| 0.6321 \| 0.6355 \| 0.6337 \| 0.6417 #3 \| 0.6281 \| 0.6293 \| 0.6282 \| 0.6311 \| 0.6307 \| 0.6372 #4 \| 0.6326 \| 0.6361 \| 0.6333 \| 0.6373 \| 0.6372 \| 0.6451 #5 \| 0.6308 \| 0.6292 \| 0.6302 \| 0.6336 \| 0.6322 \| 0.6388 #6 \| 0.6378 \| 0.6383 \| 0.6336 \| 0.6412 \| 0.6368 \| 0.6494 #7 \| 0.6305 \| 0.6329 \| 0.6310 \| 0.6340 \| 0.6352 \| 0.6410 #8 \| 0.6297 \| 0.6278 \| 0.6297 \| 0.6330 \| 0.6328 \| 0.6411 #9 \| 0.6264 \| 0.6283 \| 0.6258 \| 0.6292 \| 0.6278 \| 0.6368 #10 \| 0.6392 \| 0.6434 \| 0.6375 \| 0.6431 \| 0.6278 \| 0.6577 #11 \| 0.6469 \| 0.6529 \| 0.6445 \| 0.6508 \| 0.6548 \| 0.6719 #12 \| 0.6506 \| 0.6575 \| 0.6498 \| 0.6518 \| 0.6570 \| 0.6676 #13 \| 0.6558 \| 0.6612 \| 0.6538 \| 0.6603 \| 0.6637 \| 0.6739 #14 \| 0.6362 \| 0.6405 \| 0.6371 \| 0.6412 \| 0.6411 \| 0.6486 #15 \| 0.6745 \| 0.6888 \| 0.6710 \| 0.6787 \| 0.6819 \| 0.7021 #16 \| 0.6638 \| 0.6627 \| 0.6517 \| 0.6634 \| 0.6727 \| 0.6901 #17 \| 0.6524 \| 0.6658 \| 0.6499 \| 0.6519 \| 0.6575 \| 0.6715 #18 \| 0.6493 \| 0.6480 \| 0.6375 \| 0.6500 \| 0.6610 \| 0.6754 #19 \| 0.6330 \| 0.6375 \| 0.6306 \| 0.6374 \| 0.6381 \| 0.6476 Overall AUC \| 0.6364 \| 0.6398 \| 0.6353 \| 0.6403 \| 0.6415 \| 0.6506 ### 4.3. Ablation Study To investigate the effect of each component, we conduct several ablation studies. #### 4.3.1. STAR Topology FCN and PN Table 3. Ablation study of partitioned normalization (PN) and star topology fully-connected neural networks (STAR FCN). All models are trained with the proposed auxiliary network. \| Base (BN) \| Base (PN) \| STAR FCN (BN) \| STAR FCN (PN) ---\|---\|---\|---\|--- Overall AUC \| 0.6364 \| 0.6485 \| 0.6455 \| 0.6506 We analyze the influence of different components of STAR. Concretely, the separate effects of star topology FCN and PN are investigated. We compare (a) the Base model trained with BN, (b) Base model trained with PN, (c) STAR FCN with BN and (d) STAR model (STAR FCN + PN). The result is reported in Table 3. We observe that using star topology FCN and PN separately can outperform the Base model. Bring them together can further boost performance. The result validates the effect of both star topology FCN and PN. #### 4.3.2. Normalization Normalization methods are very effective components in deep learning, which have been shown by many practices to ease optimization and enable very deep networks to converge. We analyze the effect of different normalization methods including Batch Normalization (BN) (Ioffe and Szegedy, 2015), Layer Normalization (LN) (Ba et al., 2016) and the proposed Partitioned Normalization (PN) on multi-domain CTR prediction. BN accumulates global statistics and learns global parameters for samples from all domains. LN is a representative instance-based normalization method, which operates along the channel dimension and avoids mixing statistics for samples from different domains. The result is shown in Table 4. Our first observation is that both LN and PN outperforms BN. This observation validates that data from different domains have distinct distribution and need specific normalization. Using global normalization obscures domain differences, which will hurt performance for multi-domain CTR prediction. We also observe that PN outperforms LN, which validates that domain-specific normalization is better than the instance- specific normalization, since PN leads to more accurate moments within the domain. Table 4. Ablation study of normalization methods for multi-domain CTR prediction. STAR FCN is trained BN, LN, and PN respectively. \| STAR FCN (BN) \| STAR FCN (LN) \| STAR FCN (PN) ---\|---\|---\|--- Overall AUC \| 0.6455 \| 0.6463 \| 0.6506 #### 4.3.3. Auxiliary network We conduct experiment to assess the effect of the auxiliary network for different models. All methods are trained with and without the proposed auxiliary network. The result is illustrated in Figure 6. We observe that the auxiliary network improves all methods consistently. The result validates the importance of making full utilization of domain features and using it to capture the domain distinction. We also observe the improvement of the auxiliary network for MulANN is slightly weaker than the other methods. The reason may due to the fact that the adversarial loss for obscuring domain differences contradicts with the domain feature to capture the domain differences. Figure 6. The performance of different methods trained with (w/) and without (w/o) the auxiliary network. #### 4.3.4. Ability to Capture Domain Distinction Figure 7. Predicted CTR over CTR (PCOC) of the Base model and STAR in all domains. Each circle means PCOC of a specific domain. Cost-per-click (CPC) is a widely used performance-dependent payment model in display advertising, where advertisers bid for clicks. In CPC, the display systems compute the effective cost per mille (eCPM) as the product of bid times its CTR. The systems allocate impressions according to the descending order of the eCPM. In CPC, the CTR model needs to be well-calibrated (Guo et al., 2017a) in order to achieve a competitive advertising system, i.e., the predicted CTR should be as close as to the actual CTR. We show that STAR is more well-calibrated and is capable of capturing domain distinctions. We compute the predicted CTR over CTR (PCOC) in each domain. Note that the closer PCOC is to 1.0, the more accurate the CTR prediction is. For the simplicity of illustration, we show the PCOCs of the Base model and STAR in Figure 7. We can see that the PCOCs of STAR in different domains are more compact and concentrated around 1.0 compared with the Base model. The result validates the ability of STAR to capture the domain distinction. ### 4.4. Production Online serving and challenges. One of the challenges in industrial recommender is that the distribution of features and CTR exhibits large shifts over time. To capture the dynamic change of data in real-time, it is important to use real-time examples to update the CTR models continuously to prevent them from becoming stale. However, for multi-domain CTR prediction, the percentage of examples of each domain changes over time. For example, some business domains have traffic spike in the morning while some business domains have traffic spike in the evening. If we train the model directly in the chronological order, the changes in data percentage over time will cause the instability of model learning. To address this issue, we redesign the data pipeline and maintain a buffer that stores a sliding window of history samples to avoid the sudden change of example percentage. Specifically, samples in the buffer are shuffled firstly and then sampled to construct a mini-batch. After fed to the model, this mini-batch of samples are removed from the buffer and new arriving data is added to this buffer. We empirically found this training manner is more stable than the traditional way of online updates. Note that during serving, the weights of FCN for each domain are pre-computed to achieve faster inferences. By this means, the computational time of STAR equals the Shared Bottom model. The systematical optimization makes STAR capable of serving main traffic of multiple business domains stably. Since 2020, STAR is deployed and serves more than 60 business domains on the display advertising system of Alibaba. We compute the overall improvements of all domains. Table 5 shows the improvement of STAR over the previous production model, the Base model. The introduction of STAR brings +8.0% overall CTR lift and +6.0% overall RPM lift in our online A / B test. Table 5. CTR and RPM gains in online display advertising system of Alibaba. \| CTR \| RPM ---\|---\|--- Overall \| +8.0% \| +6.0% ## 5\. Conclusion In this paper, we propose the star topology adaptive recommender to address the problem of multi-domain CTR prediction. Instead of keeping unique models for different domains or simply mixing all samples and maintaining a shared model, STAR has the star topology, which consists of shared centered parameters and domain-specific parameters. The shared parameters learn commonalities, which is updated through all examples. The domain-specific parameters capture domain distinction for more refined prediction, which is learned using examples within the specific domain. By this means, STAR can adaptively modulate its parameters conditioned on the domain for more refined prediction. The experiments demonstrate that the superiority of STAR on multi- domain CTR prediction. Since 2020, STAR is deployed in the advertising system of Alibaba, obtaining 8.0% improvement on CTR and 6.0% on RPM. ## References * (1) * Argyriou et al. 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# TSQA: Tabular Scenario Based Question Answering Xiao Li, Yawei Sun, Gong Cheng ###### Abstract Scenario-based question answering (SQA) has attracted an increasing research interest. Compared with the well-studied machine reading comprehension (MRC), SQA is a more challenging task: a scenario may contain not only a textual passage to read but also structured data like tables, i.e., tabular scenario based question answering (TSQA). AI applications of TSQA such as answering multiple-choice questions in high-school exams require synthesizing data in multiple cells and combining tables with texts and domain knowledge to infer answers. To support the study of this task, we construct GeoTSQA. This dataset contains 1k real questions contextualized by tabular scenarios in the geography domain. To solve the task, we extend state-of-the-art MRC methods with TTGen, a novel table-to-text generator. It generates sentences from variously synthesized tabular data and feeds the downstream MRC method with the most useful sentences. Its sentence ranking model fuses the information in the scenario, question, and domain knowledge. Our approach outperforms a variety of strong baseline methods on GeoTSQA. ## 1 Introduction Scenario-based question answering (SQA) is to answer questions contextualized by scenarios (Lally et al. 2017). Compared with the well-studied task of machine reading comprehension (MRC) which requires reading a passage to extract or infer an answer (Rajpurkar et al. 2016; Lai et al. 2017), a SQA task requires reading a scenario which commonly contains both a textual passage and a set of structured data. One such prominent AI application of SQA is answering multiple-choice questions in high-school geography exams (Ding et al. 2018; Huang et al. 2019). Those questions are contextualized by scenarios containing tables and diagrams, where the rich information cannot be captured by current MRC methods but have to be manually interpreted using natural language. Thus, one natural research question arises: can we solve SQA in a fully automated manner? Figure 1: Left: an example question contextualized by a tabular scenario in GeoTSQA. Right: an overview of our approach. #### Task and Challenges. Specifically, we focus on questions contextualized by a scenario consisting of a textual passage and a set of tables. We refer to this branch of SQA as TSQA, short for _Tabular Scenario based Question Answering_. To support the study of this task, we construct a dataset named GeoTSQA. It contains 1k real questions contextualized by tabular scenarios in the geography domain, collected from China’s high-school exams. Compared with existing datasets for table-based question answering like WikiTableQuestions (Pasupat and Liang 2015), GeoTSQA requires fundamentally different reading and reasoning skills, and poses new research challenges. For instance, Figure 1 shows a question in GeoTSQA. To answer it, tabular data needs to be synthesized via a complex operation: identifying a monotonic increase in ELP over the interval 2000–2003. Focusing on this particular interval rather than many other intervals is implicitly suggested in the question: after year 2000. Moreover, the passage in the scenario helps to link ELP with educational level, and the retrieved domain knowledge bridges the gap between educational level and rural labor which is the correct answer. To conclude, TSQA methods need to _properly manipulate tabular data_ , and _comprehend fused textual information_. #### Our Approach. To meet the challenges, considering that text reading has been extensively studied in MRC research, we propose to extend state-of-the-art MRC methods with a novel table-to-text generator named TTGen to specifically handle tabular data. The basic idea is straightforward: feeding a MRC model with sentences generated from tables _using templates that encapsulate many and various predefined operations for manipulating tabular data_. However, the potentially large number (e.g., hundreds) of generated sentences may easily exceed the capacity of typical MRC models, and produce much noise information influencing the accuracy of reading comprehension. To address this problem, TTGen incorporates a sentence ranking model that fuses the information in the scenario, question, and domain knowledge to effectively _select sentences that are most useful for answering the question_. It outperforms a variety of strong baseline methods in extensive experiments on GeoTSQA. We summarize our contributions in the paper as follows. * • We construct and publish GeoTSQA, the first dataset dedicated to TSQA. It requires reading and reasoning with tables, texts, and domain knowledge at high school level. * • We extend MRC methods with TTGen to solve TSQA. TTGen performs question and knowledge aware ranking of sentences generated from synthesized tabular data. #### Outline. The remainder of the paper is organized as follows. We discuss and compare with related work in Section 2. We formally define the TSQA task and describe the construction of the GeoTSQA dataset in Section 3. We introduce our approach in Section 4. We present experiment settings in Section 5 and report experiment results in Section 6. Finally we conclude the paper in Section 7. Our code and data are available on Github.111https://github.com/nju- websoft/TSQA ## 2 Related Work ### 2.1 SQA SQA is an emerging AI task and has found application in many domains. The pioneering WatsonPaths system provides recommendations for diagnosis and treatment based on a medical scenario about a patient (Lally et al. 2017). In the legal domain, SQA supports judgment prediction based on the fact description of a legal case (Ye et al. 2018; Zhong et al. 2018; Yang et al. 2019b). We focus on TSQA where a scenario contains both textual and tabular data. Such questions are common in, for example, China’s high-school geography and history exams where a scenario describes a concrete fact or event to contextualize a set of questions. Previous efforts in this domain either ignore tables (Cheng et al. 2016) or manually transform tables into triple- structured knowledge (Ding et al. 2018) or natural language descriptions for machine reading (Huang et al. 2019). In contrast, we aim at _solving TSQA in a fully automated manner by generating texts from tables_. ### 2.2 Table-to-Text Generation Table-to-text generation has been studied for decades. Early methods rely on handcrafted rules to generate texts for specific domains such as stock market summaries (Kukich 1983) and weather forecasts (Goldberg, Driedger, and Kittredge 1994). They typically implement a pipeline of modules including content planning, sentence planning, and surface realization. Today, it is feasible to train neural generation models in an end-to-end fashion, thanks to the availability of effective pre-trained language models (Devlin et al. 2019; Radford et al. 2019) and large datasets (Lebret, Grangier, and Auli 2016; Wiseman, Shieber, and Rush 2017; Dusek, Novikova, and Rieser 2019). Current models often adopt an encoder-decoder architecture with a copy mechanism (Wiseman, Shieber, and Rush 2017; Puduppully, Dong, and Lapata 2019a). Moreover, they can be enhanced with entity representations (Puduppully, Dong, and Lapata 2019b) and external background knowledge (Chen et al. 2019). The above methods are targeted on surface-level description of tabular data, which is insufficient for our task where data in multiple cells needs to be _synthesized using various operations_ (e.g., extremum, monotonicity, trend). Generating such natural language statements that are logically entailed from tabular data, rather than superficial restatements, has recently attracted research attention (Chen et al. 2020a, d). However, they are primarily focused on high-fidelity generation, i.e., the generated text should be faithful to the tabular data. Fidelity is necessary but insufficient for our task where the generated text also needs to be useful for answering the question. It is thus essential to _select the proper operation and data from a potentially very large space_. To this end, our proposed generator TTGen features a sentence ranking model that fuses the information in the scenario, question, and domain knowledge. ### 2.3 Table-Based Question Answering Similar to TSQA, there has been a line of research of answering questions over tabular data (Pasupat and Liang 2015; Jauhar, Turney, and Hovy 2016; Yin et al. 2016; Yu et al. 2020). Like our constructed dataset GeoTSQA, these datasets also require performing various operations over multiple cells. Differently, their questions can be answered solely on the basis of tabular data, whereas the questions in GeoTSQA are more naturally contextualized by a scenario containing _both_ a set of tables and a textual passage which are equally important and are _dependent on each other_. From this angle, the most similar dataset to GeoTSQA is HybridQA (Chen et al. 2020c), where table cells are linked with Wikipedia pages. However, GeoTSQA has its _unique challenges_ due to the source of questions—high-school geography exams. For example, table cells mainly contain non-linkable numeric values; more complex operations (e.g., monotonicity) are needed; it would be helpful to incorporate domain knowledge into question answering. ## 3 Task and Dataset We firstly define the task of TSQA, and then we construct the GeoTSQA dataset to support the study of TSQA. ### 3.1 Task Definition A TSQA task consists of a scenario $\langle P,T\rangle$, a question $Q$, and a set of options $O$ as candidate answers of which only one is correct. The scenario contains a passage $P$ and a set of tables $T$. Each table in $T$ has a header row, a header column, and a set of content cells. The goal is to select an option from $O$ as the answer to $Q$ contextualized by $\langle P,T\rangle$. ### 3.2 Dataset Construction We constructed GeoTSQA. To the best of our knowledge, it is the first dataset dedicated to the TSQA task. #### Collecting Questions. We collected multiple-choice questions contextualized by tabular scenarios in the geography domain from China’s high-school exams. A related dataset is GeoSQA (Huang et al. 2019). We not only collected all the questions from GeoSQA but also reused the code for constructing GeoSQA to crawl much more questions from the Web to expand our dataset. However, many collected scenarios are not tabular. Indeed, each scenario is associated with a set of image files. Each image file depicts either a table or another kind of diagram such as a map or a histogram. Therefore, we need to identify images depicting tables or table-like diagrams. #### Identifying Tables. We looked for tables, or charts that can be straightforwardly converted to tables (e.g., histograms, line charts). We manually identified 200 such image files as positive examples and another 200 image files as negative examples. We used them to train an image classifier (Szegedy et al. 2016) to classify all the remaining image files. Finally, for all the image files that were classified as positive, we manually checked them for classification errors. #### Extracting Tables. We recruited 15 undergraduate students from a university in China as annotators. For image files depicting tables, we used Baidu’s OCR tool to extract tabular data. OCR errors were manually corrected by annotators. For image files depicting charts, annotators manually extracted tabular data, assisted with a tool we developed. The annotator used that tool to easily click key points in the image, e.g., the origin, coordinate axes, data points. The tool then automatically converted data points to data tables. Annotators manually checked each extracted table and filtered out irregular tables (e.g., with multi-level headers). #### Filtering Questions. Last but not least, annotators filtered out questions that can be answered without using any table. Therefore, every question in GeoTSQA is contextualized by a tabular scenario, and it is essential to employ the information in the given tables to answer the question. ### 3.3 Dataset Statistics Scenarios \| 556 \| ---\|---\|--- Chinese characters per passage \| $52.42$ \| $\pm 32.99$ Tables per scenario \| $1.58$ \| $\pm 0.93$ Cells per table \| $26.98$ \| $\pm 17.51$ Questions \| 1,012 \| Chinese characters per question \| $44.02$ \| $\pm 15.89$ Table 1: Statistics about GeoTSQA. GeoTSQA contains 556 scenarios and 1,012 multiple-choice questions. Each question has four options. More statistics about the dataset are shown in Table 1. Out of the 878 tables in GeoTSQA, 96% only contain numeric content cells. It differs from HybridQA (Chen et al. 2020c) where content cells are often entities linked with Wikipedia pages, thereby providing extra background knowledge for answering questions. For GeoTSQA, to obtain information that is not explicitly given in the scenario but critical for answering questions, it is essential to entail from tabular data via operations over multiple cells. ## 4 Approach We propose a two-step approach to solve TSQA. As illustrated in Figure 1, the first step (Section 4.2) is a table-to-text generator named TTGen. From the tables $T$ in a scenario $\langle P,T\rangle$, TTGen generates top-$k$ sentences $S$ that are most useful for answering the question $Q$. The second step (Section 4.1) is a MRC method based on K-BERT (Liu et al. 2020), a state- of-the-art knowledge-enabled language model. It fuses the information in the passage $P$, generated sentences $S$, question $Q$, and domain knowledge $K$ to rank the options in $O$. ### 4.1 MRC with Domain Knowledge Our MRC method is based on K-BERT (Liu et al. 2020). This state-of-the-art language model extends BERT (Devlin et al. 2019) with the capability to utilize external knowledge such as domain knowledge. #### MRC with K-BERT. For each option $o_{i}\in O$, we concatenate the passage $P$, top-$k$ sentences $S=\\{s_{1},\ldots,s_{k}\\}$ generated from the tables $T$, question $Q$, and $o_{i}$ in a standard way, starting with a [CLS] token and separating with [SEP]: $I^{\text{MRC}}_{i}=\text{[CLS] $P$ $s_{1}\cdots s_{k}$ $Q$ [SEP] $o_{i}$ [SEP] \emph{NUMS}${}_{i}$ [SEP]}\,,$ (1) where _NUMS_ i is a concatenation of all the numeric tokens in $P$, $S$, $Q$, and $o_{i}$. Each numeric token in the original position is replaced by a special token [NUM]. We use K-BERT to obtain a vector representation for each token in $I^{\text{MRC}}_{i}$ to capture its semantic features: $\langle\mathbf{h}^{\text{MRC}}_{i1},\mathbf{h}^{\text{MRC}}_{i2},\ldots\rangle=\text{K-BERT}(I^{\text{MRC}}_{i},~{}K)\,,$ (2) where $K$ is an external knowledge base we will explain later. The vector representation for the [CLS] token, i.e., $\mathbf{h}^{\text{MRC}}_{i1}$, is used as an aggregate representation for $I^{\text{MRC}}_{i}$. It is fed into two dense layers followed by a softmax layer to obtain a correctness score $\hat{\omega}_{i}$ for each option $o_{i}\in O$: $\begin{split}\omega_{i}&=\mathbf{w}_{2}^{\intercal}\tanh(\mathbf{W}_{1}\mathbf{h}^{\text{MRC}}_{i1}+\mathbf{b}_{1})+b_{2}\,,\\\ \mathbf{\Omega}&=[\hat{\omega}_{1};\hat{\omega}_{2};\ldots]=\mathtt{softmax}([\omega_{1};\omega_{2};\ldots])\,,\end{split}$ (3) where $\mathbf{W}_{1}$ is a trainable matrix, $\mathbf{w}_{2}$ and $\mathbf{b}_{1}$ are trainable vectors, and $b_{2}$ is a trainable parameter. In the training phase, we minimize the negative log-likelihood loss which measures the difference between $\mathbf{\Omega}$ and the binary correctness label on each option (we will detail in Section 5.1). In the test phase, we choose the option in $O$ with the highest correctness score $\hat{\omega}$ as the answer. K-BERT extends BERT with an external knowledge base $K$. It helps to fuse the information in $P$, $S$, $Q$, $O$, and $K$. We refer the reader to Liu et al. (2020) for a detailed description of K-BERT. Briefly, each entry in $K$ is a pair $\langle\text{entity},~{}\text{fact sentence}\rangle$, or a triple $\langle\text{entity},~{}\text{property},~{}\text{value}\rangle$ which can be converted into a pair by concatenating the property and the value into a fact sentence. K-BERT employs $K$ to expand the input sequence into a tree of tokens: fact sentences about an entity are retrieved from $K$ and inserted as branches after each mention of the entity in the input sequence. In our implementation, for each entity, we retrieve top-$\epsilon$ fact sentences that are most relevant to the input sequence. The relevance of a fact sentence to the input sequence is measured by the cosine similarity between their average pre-trained BERT embedding vectors. #### Domain Knowledge. For the external knowledge base $K$, for our experiments we use domain knowledge since all the questions in GeoTSQA are in the geography domain. We obtain domain knowledge from two sources. First, we import all the triples in Clinga (Hu et al. 2016), a large Chinese geographical knowledge base. Second, we reuse the corpus in (Huang et al. 2019). The corpus contains a geography textbook providing a set of entity descriptions. We pair each entity with each sentence in its description as a fact sentence. The corpus also contains a subset of Chinese Wikipedia. We treat the title of each page as an entity and pair it with each sentence in the page as a fact sentence. ### 4.2 Table-to-Text Generation (TTGen) Below we describe the generation of sentences from tables to be fed into our MRC method. We rely on templates that encapsulate predefined operations for manipulating tabular data. It enables us to perform complex operations that are needed for answering hard questions such as those in GeoTSQA. We generate sentences from tables using all the applicable templates. However, it is infeasible for a MRC model like K-BERT to jointly encode a large number (e.g., hundreds) of sentences. Therefore, we rank the generated sentences and select $k$ top-ranked sentences that are most useful for answering the question. By filtering the generated sentences, we can also reduce noise information that may influence the accuracy of reading comprehension. #### Sentence Generation. By significantly extending the operations considered in Chen et al. (2020a, b), we define six table-to-text templates that encapsulate different powerful operations for synthesizing numeric tabular data. As we will show in the experiments, these templates have covered most needs about tables in GeoTSQA. One can easily add new templates to accommodate other applications. * • Extremum. This template reports the maximum or minimum value of a row or column. An example sentence generated from the table in Figure 1 is: _ELP reaches a maximum of 2.504 at Year 2000._ * • Special values. This template reports or compares with a special value (e.g., under a column header that is mentioned in the question), e.g., _ELP at Year 2000 is 2.504._ * • Comparison with average. This template reports a maximal sequence of cells where all the values are above or below the average of the entire row or column, e.g., _ELP is relatively large between Year 2000 and 2002._ * • Monotonicity. This template reports a monotonic increase or decrease over a maximal sequence of cells, e.g., _ELP decreases between Year 2000 and 2003._ * • Trend. This template reports the overall trend of a row or column, e.g., _ELP generally increases and then decreases._ * • Range comparison. This template reports a comparison between two maximal corresponding sequences of cells from different rows or columns. For non-numeric tabular data, we simply concatenate each row header, each column header, and the corresponding content cell into a sentence. #### Sentence Ranking. Let $\hat{S}$ be the set of sentences generated from the tables $T$ using all the applicable templates. We compute a usefulness score for each sentence $s_{j}\in\hat{S}$, and choose $k$ top-ranked sentences $S\subseteq\hat{S}$. To select sentences that are most useful for answering the question, our ranking model employs K-BERT to fuse the information in the passage $P$, question $Q$, and domain knowledge $K$ to perform question and knowledge aware ranking. Figure 2 presents an overview of the model. It integrates two complementary rankers: sentence-level ranking directly assesses the usefulness of each individual sentence; template-level ranking infers useful templates purely from the passage and question. Figure 2: Sentence ranking model in TTGen. For sentence-level ranking, we concatenate the passage $P$, question $Q$, and sentence $s_{j}$ in a standard way: $I^{\text{SR}}_{j}=\text{[CLS] $P$ $Q$ [SEP] $s_{j}$ [SEP] \emph{NUMS}${}_{j}$ [SEP]}\,,$ (4) where _NUMS_ j is a concatenation of all the numeric tokens in $P$, $Q$, and $s_{j}$. Each numeric token in the original position is replaced by a special token [NUM]. We use K-BERT to obtain a vector representation for each token in $I^{\text{SR}}_{j}$: $\langle\mathbf{h}^{\text{SR}}_{j1},\mathbf{h}^{\text{SR}}_{j2},\ldots\rangle=\text{K-BERT}(I^{\text{SR}}_{j},~{}K)\,.$ (5) The vector representation for the [CLS] token, i.e., $\mathbf{h}^{\text{SR}}_{j1}$, is fed into two dense layers followed by a softmax layer to obtain a usefulness score $\hat{\phi}_{j}$ for each sentence $s_{j}\in\hat{S}$: $\begin{split}\phi_{j}&=\mathbf{w}_{4}^{\intercal}\tanh(\mathbf{W}_{3}\mathbf{h}^{\text{SR}}_{j1}+\mathbf{b}_{3})+b_{4}\,,\\\ \mathbf{\Phi}&=[\hat{\phi}_{1};\hat{\phi}_{2};\ldots]=\mathtt{softmax}([\phi_{1};\phi_{2};\ldots])\,,\end{split}$ (6) where $\mathbf{W}_{3}$ is a trainable matrix, $\mathbf{w}_{4}$ and $\mathbf{b}_{3}$ are trainable vectors, and $b_{4}$ is a trainable parameter. In the training phase, we minimize the negative log-likelihood loss which measures the difference between $\mathbf{\Phi}$ and the binary usefulness label on each generated sentence (we will detail in Section 5.1). For template-level ranking, we concatenate the passage $P$ and question $Q$ in a standard way: $I^{\text{TR}}=\text{[CLS] $P$ $Q$ [SEP]}\,.$ (7) We use K-BERT to obtain a vector representation for each token in $I^{\text{TR}}$: $\langle\mathbf{h}^{\text{TR}}_{1},\mathbf{h}^{\text{TR}}_{2},\ldots\rangle=\text{K-BERT}(I^{\text{TR}},~{}K)\,.$ (8) The vector representation for the [CLS] token, i.e., $\mathbf{h}^{\text{TR}}_{1}$, is fed into two dense layers followed by a sigmoid layer to obtain a usefulness score $\hat{\psi}$ for each of the six templates: $\begin{split}[\psi_{1};\ldots;\psi_{6}]&=\mathbf{W}_{6}\tanh(\mathbf{W}_{5}\mathbf{h}^{\text{TR}}_{1}+\mathbf{b}_{5})+\mathbf{b}_{6}\,,\\\ \mathbf{\Psi}&=[\hat{\psi}_{1};\ldots;\hat{\psi}_{6}]=\mathtt{sigmoid}([\psi_{1};\ldots;\psi_{6}])\,,\end{split}$ (9) where $\mathbf{W}_{5}$ and $\mathbf{W}_{6}$ are trainable matrices, $\mathbf{b}_{5}$ and $\mathbf{b}_{6}$ are trainable vectors. Let sentence $s_{j}$ be generated by the $\tau_{j}$-th template. We derive usefulness labels on templates for training from usefulness labels on generated sentences: a template is labeled useful if and only if at least one sentence it generates is labeled useful. Multiple sentences and hence multiple templates may be labeled useful for answering a question. Therefore, in the training phase, we formulate a multi-label binary classification task, and we minimize the binary cross-entropy loss which measures the difference between $\mathbf{\Psi}$ and the binary usefulness label on each template. Finally, in the test phase, we compute: $\text{usefulness score of }s_{j}=\hat{\phi}_{j}\cdot\hat{\psi}_{\tau_{j}}\,.$ (10) ## 5 Experiment Setup We compared our approach with a variety of strong baseline methods for TSQA. We also evaluated our sentence ranking model, which is the core component of our approach. ### 5.1 Labeled Data #### Correctness Labels on Options. For each question, from its known correct answer, we derived a label for each of the four options indicating whether it is the correct answer. These binary correctness labels were used to train and evaluate TSQA methods. #### Usefulness Labels on Generated Sentences. The number of all the sentences $\hat{S}$ generated by our templates for a question is in the range of 2–176, with a mean of 41.58 and a median of 38. For each question, we asked an annotator (recruited in Section 3.2) to read $\hat{S}$ and assign a label to each sentence indicating whether it is useful for answering the question. These binary usefulness labels were used to train and evaluate sentence ranking models. #### Gold-Standard Sentences. Furthermore, the annotator manually summarized the tables in one sentence describing necessary information for answering the question. This gold- standard sentence was used for comparison. We randomly sampled 100 questions from GeoTSQA. For 92 questions, $\hat{S}$ fully covers the information in the gold-standard sentence. For 6 questions, $\hat{S}$ partially covers that information. Therefore, our six templates show good coverage of the various operations required by GeoTSQA. ### 5.2 Baselines Our approach extends MRC methods. It is not our focus to compare existing MRC methods. Instead, table-to-text generation is our major technical contribution. Therefore, in the experiments we consistently used the MRC method based on K-BERT described in Section 4.1, but fed it with sentences generated from tables by the following different methods. #### Supervised Methods. Output of linearization for the table in Figure 1: --- … ELP at Year 1998 is 2.465. ELP at Year 1999 is 2.476. ELP at Year 2000 is 2.504. ELP at Year 2001 is 2.490. ELP at Year 2002 is 2.482. ELP at Year 2003 is 2.473. Table 2: Example output of Linearization. Firstly, we compared with three table-to-text generators that achieved state- of-the-art results on the recent LogicNLG dataset (Chen et al. 2020a) which, similar to our GeoTSQA, requires synthesizing data in multiple cells. These generators are open source. Field-Infusing employs LSTM to encode each table into a sequence of vectors and then applies Transformer to generate text. GPT- Linearization linearizes each table as a paragraph by horizontally scanning the table and concatenating each content cell with its row header and column header into a sentence. Table 2 illustrates such a paragraph. The resulting paragraph is then fed into GPT-2 to generate a new text. Coarse-to-Fine is an enhanced version of GPT-Linearization. It adopts a two-step text generation process: generating a template and then filling it. Furthermore, we implemented an enhanced version of GPT-Linearization and Coarse-to-Fine, referred to as GPT-Linearization+ and Coarse-to-Fine+, respectively. At the beginning of the paragraph fed into GPT-2, we inserted the scenario passage and question to enable GPT-2 to perform question-aware text generation. All the above supervised table-to-text generators were trained based on sentences with positive usefulness labels. #### Unsupervised Methods. We also compared with two naive table-to-text generators. Recall that GPT-Linearization generates a paragraph from tables and then feeds it into GPT-2 to generate a new text. We implemented Linearization. It directly outputs the generated paragraph without feeding it into GPT-2. Output of templation for the table in Figure 1: --- … ELP at Year 2000 is 2.504. … ELP decreases between Year 2000 and 2003. … ELP generally increases and then decreases. … ELP reaches a maximum of 2.504 at Year 2000. … ELP is relatively large between Year 2000 and 2002. … Table 3: Example output of Templation. Besides, we implemented Templation. It generates a paragraph consisting of all the sentences $\hat{S}$ generated by our templates. Sentences are sorted in ascending order of length so that if the paragraph has to be truncated by the maximum sequence length of K-BERT, the largest number of sentences can be retained. Table 3 illustrates such a paragraph. #### Gold-Standard Sentence. Last but not least, we used manually annotated gold-standard sentence as a reference. ### 5.3 Implementation Details We performed 5-fold cross-validation. For each fold, we split GeoTSQA into 80% for training and 20% for test. For model selection, we relied on an inner holdout 80%/20% training/development split. We ran all the experiments on TITAN RTX GPUs. For K-BERT, we used BERT-wwm-ext (Cui et al. 2019), a pre-trained Chinese language model as the underlying language model. We set $\text{maximum sequence length}=256$, $\text{self-attention layer}=12$, $\text{hidden units}=768$, $\text{epochs}=15$ for MRC and template-level ranking, $\text{epochs}=5$ for sentence-level ranking, $\text{batch size}=8$ for MRC, $\text{batch size}=16$ for template-level ranking and sentence-level ranking, $\text{learning rate}=1e\text{--}5$, and $\text{attention heads}=12$. For knowledge base retrieval we set $\epsilon=2$. Inspired by Jin et al. (2020), for the K-BERT model in our MRC method (but not the one in TTGen), we coarse- tuned it on $\text{C}^{3}$ (Sun et al. 2020), a Chinese MRC dataset. For GPT-2, we used $\text{CDialGPT2}_{\text{LCCC-base}}$ (Wang et al. 2020), a pre-trained Chinese GPT-2 model. For $\text{CDialGPT2}_{\text{LCCC-base}}$, and for LSTM and Transformer in Field-Infusing, we followed the recommended hyperparameter settings in their original implementation. For our TTGen, by default we set $k=2$ to only select the top-2 generated sentences for MRC. We will report a comparison in different settings of $k$. ### 5.4 Evaluation Metrics To evaluate TSQA, we measured accuracy, i.e., the proportion of correctly answered questions. To evaluate sentence ranking, we measured the quality of the whole ranked list of all the sentences $\hat{S}$ generated by our templates. We used two standard information retrieval evaluation metrics: Mean Average Precision (MAP) and Mean Reciprocal Rank (MRR). ## 6 Experiment Results We report average results on the test sets over all the folds. ### 6.1 Results on TSQA #### Comparison with Baselines. \| Accuracy ---\|--- Field-Infusing \| 0.353 ∙ GPT-Linearization \| 0.370 Coarse-to-Fine \| 0.367 GPT-Linearization+ \| 0.348 ∙ Coarse-to-Fine+ \| 0.359 ∘ Linearization \| 0.235 ∙ Templation \| 0.243 ∙ TTGen \| 0.397 Gold-Standard Sentence \| 0.418 Table 4: Accuracy of TSQA. We mark the results of baselines that are significantly lower than TTGen under $p<0.01$ $(^{\bullet})$ or $p<0.05$ $(^{\circ})$. Table 4 shows the accuracy of TSQA achieved by each method. Our TTGen outperforms all the baselines by 2.7–16.2 percent of accuracy. TTGen exceeds three state-of-the-art table-to-text generators, i.e., Field- Infusing, GPT-Linearization, and Coarse-to-Fine, by 2.7–4.4 percent of accuracy. The enhanced version of these generators that we implemented, i.e., GPT- Linearization+ and Coarse-to-Fine+, exhibit surprisingly worse performance than their original version. Their generation methods are significantly inferior to our TTGen by 3.8–5.1 percent of accuracy. The two naive generators, i.e., Linearization and Templation, produce much noise information for MRC and achieve accuracy even lower than random guess (i.e., 0.25). It demonstrates the necessity of ranking and selecting generated sentences. The accuracy of using gold-standard sentence is 0.418. On the one hand, compared with the accuracy 0.397 of our TTGen, it suggests that there is still room for improving our templates and/or our sentence ranking model. On the other hand, the achieved accuracy is not satisfying. To improve the overall performance of our approach, we need to combine our TTGen with novel MRC methods that are more powerful than K-BERT to meet the unique challenges raised by the GeoTSQA dataset. This will be our future work. #### Varying $k$. \| $k=1$ \| $k=2$ \| $k=3$ \| $k=4$ \| $k=5$ ---\|---\|---\|---\|---\|--- Accuracy \| 0.390 \| 0.397 \| 0.352 \| 0.343 \| 0.330 Table 5: Accuracy of TSQA by varying $k$ in TTGen. Table 5 shows the accuracy of TSQA achieved by our approach under different settings of $k$. Increasing $k$ from 1 to 2 (the default value), the accuracy remains stable. Further increasing $k$ to 3 or larger, the accuracy drops substantially, probably influenced by the extra noise information. It is thus important to rank generated sentences and only select those useful for answering the question. #### Ablation Study. \| Accuracy ---\|--- TTGen \| 0.397 TTGen w/o tabular data \| 0.372 TTGen w/o domain knowledge \| 0.380 Table 6: Accuracy of TSQA (ablation study). To analyze the usefulness of tabular data and domain knowledge in TSQA, we implemented two variants of our approach. The first variant ignored tabular data. The second variant ignored domain knowledge. Table 6 shows the accuracy of TSQA achieved by each variant. Compared with the full version of our approach, the accuracy of both variants decrease, by 2.5 percent of accuracy without tabular data and by 1.7 percent of accuracy without domain knowledge. The results reveal the usefulness of tabular data and of domain knowledge. ### 6.2 Results on Sentence Ranking We compared our sentence ranking model with a strong baseline method: RE2 (Yang et al. 2019a). This state-of-the-art text matcher is open source. We employed it to compute the semantic relevance of each generated sentence in $\hat{S}$ to the question. Specifically, we used RE2 as a text pair classifier to predict a ranking score for each generated sentence conditioned on (i.e., paired with) a concatenation of the scenario passage and question. We followed the recommended hyperparameter setting in its original implementation. Table 7 shows the quality of sentence ranking computed by each method. Our TTGen exceeds RE2 by 5.2 percent of MAP and by 6.0 percent of MRR. Paired t-tests show that all these differences are statistically significant under $p<0.01$. ### 6.3 Error Analysis We randomly sampled 100 questions to which our approach provided incorrect answers. We analyzed the question answering process and identified the following three main causes of errors. Multiple causes could apply to a question. #### Knowledge Base. For 76% of the errors, there is a lack of necessary domain or commonsense knowledge for answering the question, such as the location of a particular lake. It suggests expanding our knowledge base. However, this is orthogonal to our technical contribution. #### Reasoning Capabilities. For 62% of the errors, more advanced reasoning skills are needed. For example, some questions require multi-hop math calculations over a group of related domain concepts. K-BERT as a language model cannot calculate. It is also impracticable to encapsulate such extremely complex operations with predefined templates. Therefore, it suggests incorporating specific calculators and powerful reasoners into MRC models. #### Sentence Ranking. For 54% of the errors, our sentence ranking model chooses a sentence that is not useful for answering the question. Indeed, some templates and their generated sentences are linguistically similar though logically different, e.g., _is relatively large_ , _reaches maximum_ , and _increases_. This sometimes challenges our sentence ranking model as well as our MRC method. We will focus on this problem in the future work. \| MAP \| MRR ---\|---\|--- RE2 \| 0.434 ∙ \| 0.461 ∙ TTGen \| 0.486 \| 0.521 Table 7: Quality of sentence ranking. We mark the results of baselines that are significantly lower than TTGen under $p<0.01$ $(^{\bullet})$. ## 7 Conclusion Our study aims at solving TSQA in a fully automated manner to avoid manually interpreting tabular data using natural language descriptions as done in previous research. To support this study, we constructed and published the first dataset GeoTSQA that is dedicated to the TSQA task. With only six templates encapsulating predefined operations for synthesizing tabular data in various ways, we covered most needs about tables in GeoTSQA but then, the problem turned into selecting, among a large number of sentences generated from templates, the most useful ones for answering the question. Our proposed model effectively integrates sentence-level and template-level ranking, and exploits the scenario passage, question, and domain knowledge by fusing their information with K-BERT. Our approach has the potential to be adapted to other AI applications that require table comprehension and explanation. Although our approach outperformed a variety of strong baselines in the experiments, its accuracy is still not satisfying. Following the results of our error analysis, for the future work, we plan to enhance our sentence ranking model with more powerful semantic matching techniques. 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# An Explainable CNN Approach for Medical Codes Prediction from Clinical Text Hu S.Y.ShuYuan Hu Teng F.Fei Teng Leeds joint School, SWJTU, Cheng Du, China The school of information science and technology, SWJTU, Cheng Du, China ###### Abstract Background Clinical notes are unstructured text documents generated by clinicians during patient encounters, generally are annotated with ICD codes, which give formatted information about the diagnosis and treatment. ICD code has shown its potentials in many fields, but manual coding is labor-intensive and error-prone, lead to researches of automatic coding. Two specific challenges of this task are 1) given an annotated clinical notes, the reasons behind specific diagnoses and treatments are lost; 2) explainability is important for practical automatic coding method, the method should not only explain its prediction output but also have explainable internal mechanics. This study aims to develop an explainable CNN approach to address these two challenges. Method We develop CNN-based methods for automatic ICD coding based on clinical text from the intensive care unit (ICU) stays. We come up with the Shallow and Wide Attention convolutional Mechanism (SWAM), which allows our model to learn local and low-level features for each label. The key idea behind our model design is to look for the presence of informative snippets in the clinical text that correlated with each code, and we infer that there exists a correspondence between “informative snippet” and convolution filter. Results We evaluate our approach on MIMIC-III, an open-access dataset of ICU medical records. Our approach substantially outperforms previous results on top-50 medical code prediction on MIMIC-III dataset. We attribute this improvement to SWAM, by which the wide architecture gives the model ability to more extensively learn the unique features of different codes, and we prove it by ablation experiment. Besides, we perform manual analysis of the performance imbalance between different codes, and preliminary conclude the characteristics that determine the difficulty of learning specific codes. Conclusions We present an explainable CNN approach for multi-label document classification, which employs a wide convolution layer to learn local and low- level features for each label, yields strong improvements over previous metrics on the ICD-9 code prediction task while providing satisfactory explanations for its internal mechanics. ICD coding, Machine learning, Attentional Convolution for NLP, ###### keywords: Research ## Background Clinical notes are written by clinicians during patient encounters, they are usually unstructured text narratives and accompanied by a set of metadata codes from the International Classification of Diseases (ICD), which present a standardized way of indicating diagnoses and procedures that were performed during the encounter. There is much research that demonstrates the practical application with ICD codes [7, 26, 3]. For example in work by Choi et al. [5], they proposed the Doctor AI system based on the presence of ICD codes to predict future patient states from learning patient representation from a large dataset of patient records. But manual coding is time-consuming and error-prone, so much research on automatic coding has been done in the past three decades, some recent works are Zhang et al. [40]; Kavuluru et al. [12] and Avati et al. [3]. And automatic coding is considered a multi-label classification task. Two domain- specific challenges are facing this task. First, a reasonable guess is that for a certain code prediction task, most of the text is not informative, only a few snippets are related to the code. However given the annotated text, the connections between code and its corresponding informative snippets are lost, in other words, the model has to learn the reasons behind specific diagnoses and treatments; second, interpretability is a crucial obstacle for practical automatic coding in both perspective of inferring and internal mechanics, the method is supposed to explain its prediction as well as have an explainable internal mechanics. To address these two specific challenges together, in this paper, we develop CNN-based methods for automatic ICD code assignment based on text discharge summaries from intensive care unit (ICU) stays, we come up with Shallow and Wide Attention convolutional Mechanism (SWAM), which allows our model to learn local and low-level features for each label. Our model design is motivated by the way human clinicians manual label the clinical notes, which is to look for informative snippets that are relevant to each code. We consider the “informative snippets” as local and low-level features. SWAM address the two mentioned challenges in automatic coding: first, by transferring the base representation (i.e. clinical notes in the word-embedding form) to the convolution representation which represents the presence of “informative snippets”, the model could filter out the irrelevant information in the text, and through the attention mechanism the model could learn the correlation between “informative snippets” and labels; second, SWAM gives “informative snippets” extracted from clinical notes as explanations of its prediction result, and provides a new perspective for understanding the internal mechanics of the machine learning method. We evaluate our approach on the MIMIC-III dataset [10], an open dataset of ICU medical records. With the Shallow and Wide Attention CNN mechanism, the model can learn non-generic features associated with specific labels that are not informative for other labels, which the narrow one are failed to learn. With the performance improvement gained from these specific labels, our approach outperforms previous results on medical code prediction on MIMIC-III dataset. ### Related work #### Automatic ICD coding ICD coding has been a long-established task in the medical informatics community for decades, from the perspective of data, the current approaches of this task can be divided into two factions: much recent research focuses on unstructured text data [34, 12], while the other incorporates structured data as well [30]. We develop our methods on unstructured text data from the MIMIC3. Also, from the perspective of the code set, many approaches [36, 24] evaluate on a subset of the full ICD label space, while there are also methods [21] developed on the full code set. We develop our methods on the top-50 code set because the advantage of SWAM is learning specific features associated with specific labels that are not generic feature for other labels, so instead of carrying out a surprisingly large network to learn all non-generic features on the full code set, using ensemble method to cover the whole code set is preferred, which is discussed in later part. A tendency in recent years is developing Neural Network-based methods for this task. Shi et al. [32] applied attentional LSTMs to form a soft matching between sentence representations from discharge summaries and the top 50 codes. Prakash et al. [24] generated predictions of the top 50 codes by memory networks built from discharge summaries and Wikipedia. Mullenbach et al. [21] applied a per-label mechanism to extract the most important snippet for each code from discharge summaries. SWAM is compared with the published result from all these papers, and it achieves state-of-the-art results across many indicators. We attribute these improvements to the ability to learn non- generic features associated with specific labels that are not informative for other labels, which bring significant performance improvements on these specific labels. #### Attentional convolution for NLP and explainable text classification Combing convolution with attention has been proved is efficient in different tasks among NLP [2, 39, 29, 38, 4]. Yang et al. [37] and Mullenbach et al. [21] utilize attentional convolution to select the most relevant parts of the clinical text of each code. We refer to the per-label attention mechanism from those of Mullenbach et al. [21], in which per-label parameter vectors are used to compute attention over specific locations in the text. Our work differs in that SWAM establishes the correspondence between the “informative snippet” and convolution filter, which makes the network a wider one comparing to its of Mullenbach et al. [21] and is better tuned to our goal of learning low-level feature, a.k.a. informative snippet with explainable internal mechanics. Figure 1: Internal mechanism of SWAM. Attentional Convolution has also been applied to make explainable text classification. Some prior works like Rushet al. [28] and Rocktäschel et al. [27] employ attention to highlight salient features of the text. The per-label attention mechanism [21] we referred extract snippet from the text as automatically generated explanation of the prediction in the same medical codes prediction task, and the informativeness of explanations are rated by a physician. Their work illustrates that the neural network work in an explainable way for this task: the model will try to find parts of the text that are most relevant to each code. Our work differs in that instead of making the model explainable by explaining its prediction, we take a further step forward to make the internal mechanics of the method explainable by opening the black box of the neural network to establish the correspondence between the “informative snippet” and convolution filter. We also bring out a preliminary analysis of the imbalance performance between the labels, provide a rational explanation of why the model performs terribly on certain codes. #### Neural network architecture design for text classification work Hoa et al. [17] compared the deep CNN and shallow CNN under text classification task, a practical rule is summarized that deep models do not seem to bring a significant advantage over shallow networks for text classification, another observation they made is that a global max-pooling [6], which retrieves the most influential feature could already be good enough for the text classification task. The authors believe one possible reason may be related to these facts that images are represented as real and dense values, as opposed to the discrete, artificial, and sparse representation of text. Their work indicates that local and low-level features extracted by shallow CNN work well for text classification tasks and inspires us to explore the underlying correspondence between local and low-level features and snippets in the text. Gong and Ji [8] find that in CNN for the text classification task, the convolution filters have learned division of labor. More than half of the kernels have a preference for one specific label. Their work inspires us to associate the width of the network with the learning of features of specific labels that are not generic for other labels. Table 1: Table of Notations Notation \| Description ---\|--- $\mathcal{L}$ \| The set of ICD-9 codes. $y_{i,\ell}\in\ {0,\ 1}$ \| The true value of the label task for instance i and $\ell\in\mathcal{L}$, 1 indicates the label is true for instance i. $d_{e}$ \| The size of the input embedding $d_{c}$ \| The size of the convolution output, a.k.a. the number of convolution filters $\mathbf{X}=[\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N}]$ \| The matrix of a document instance, where $\mathbf{N}$ is the length of the document and $\mathbf{x}_{i}$ is the vector representation of the word. $\mathbf{W}_{c}\in\mathbb{R}^{k\times d_{e}\times d_{c}}$ \| Convolution filters, where k is the width of filter window. $\mathbf{H}\in\mathbb{R}^{d_{c}\times N}$ \| Convolutional representation of the document. $\ast$ \| Convolution operator. $g$ \| An element-wise nonlinear transformation. $\mathbf{b}_{c}\in\mathbb{R}^{d_{c}}$ \| The bias in convolutional operation. $\mathbf{u}_{\ell}\in\mathbb{R}^{d_{c}}$ \| Attention parameter vector for label. $\ell$ $\boldsymbol{\alpha}_{\ell}\in\mathbb{R}^{N}$ \| Attention result vector for label. $\ell$ $b_{\ell}$ \| Scalar offset in linear layer for label. $\ell$ $\boldsymbol{\beta}_{\ell}\in\mathbb{R}^{d_{c}}$ \| Vector of prediction weights. $\sigma$ \| Sigmoid function. $\operatorname{SoftMax}()$ \| $\operatorname{SoftMax}(\mathbf{x})=\frac{\operatorname{\exp}(\mathbf{x})}{\sum_{i}{\operatorname{\exp}(x_{i})}}$ , where $\operatorname{\exp}(\mathbf{x})$ is the element-wise exponentiation of the vector $\mathbf{x}$. ## Methods In this paper, we use notations shown in Table 1. We present SWAM, a CNN-based method for automatic ICD coding from the clinical text, which provides a good explanation of its internal mechanics. SWAM is motivated by the way human clinicians manual label the clinical notes, to help the reader understand the method, firstly here is a brief introduction of the way human clinicians manual label the clinical notes. Normally, human clinicians will look for informative snippets that are relevant to each code. For example, as shown in Figure 1, given code 96.04 in the figure, a human clinician will look for the presence of relevant snippets in the clinical notes. In this case, the relevant snippets are “intubation” and “endotracheal intubation”, if the human clinician finds the relevant snippets, he/she will give a positive prediction of code 96.04. SWAM refers to the same idea of manual coding. As shown in Figure 1, the first step, through the convolutional layer the model will extract informative snippets that could be relevant to any code. In the second step, the attention layer will assign importance weight to snippets to select the relevant snippets of each code, and in the final step the model summary the weighted score of all relevant snippets of each code to give the predictions of the presence of each code. Figure 2: The architecture of the model with per-label mechanism. ### The correspondence between “informative snippet” and convolution filter Our explanation of the internal mechanics of SWAM builds on the correspondence between “informative snippet” and convolution filter. Firstly, we classify the ”informative snippet” into two categories: ”generic snippet” and ”non-generic snippet”. ”generic snippet” refers to snippets that are informative for multiple labels, for example, in our task, ”experience fever” is likely to be a ”generic snippet” since it is the symptom correlated with multiple diagnoses. ”non-generic snippet” refers to snippets that are only considered as informative to a specific or a few labels, for example, in top-50 code task, “endotracheal intubation” will be considered as a ”non-generic snippet” since it brings little information gain to the other 49 labels than it brings to the code 96.04 “Insertion of endotracheal tube”. Then we infer that there exists a correspondence between “informative snippet” and convolution filter, which means one convolution filter can only generate a high activation value for a specific “informative snippet”. Given that in the CNN context, the “informative snippet” can be considered as a set of word embedding sequences that are close in the embedding space. For example, ”large mucus plug” and ”big mucus plug” are the same ”informative snippet” since they have similar meanings and therefore are close in the embedding space. It is most likely that for different ”informative snippets”, they will have very little chance to be close in the embedding space. For each filter, it will be ”highly activated” ouput exceeds threshold when the snippet in its window is close to its parameters in embedding space, and this snippet can be considered as the ”informative snippet” corresponds to this filter. Based on the above inference, an obvious conclusion is that the choice of the width of the convolution layers, a.k.a. the number of convolution filters should depend on the total numbers of ”informative snippets” in the task, more accurately, the numbers of ”non-generic snippet” since it will much larger than the numbers of ”generic snippet” in the large-scale coding task. Besides, empirical guidance in our architecture design is that there could be multiple ”non-generic snippets” for each code [8]. Therefore we develop Shallow and Wide Attention CNN for this task: the presence of the informative snippet of each code could be considered as a local, low-level feature learned by the shallow CNN, and we also need the convolutional layer to be wide since the model needs to learn the ”non-generic snippets” of all codes. The mechanism behind Shallow and Wide Attention CNN is general for a set of similar text classification tasks that informative snippets relevant to each label scattered at random locations in the input document. So SWAM can be regarded as a general architecture with the following three characteristics, and implement details can be varied (e.g. the attention layer in the model can be either per-label attention mechanism [21] or the full connected layer in textCNN [14]). 1\. The convolutional layer should be sufficiently wide, a.k.a. enough convolution filters to not only extract all generic snippets that are informative for multiple labels, but also all non-generic features that are correlated to specific label and not informative to other labels, the certain number of filters depends on task context, a.k.a. the total number of generic features and non-generic features in the task. 2\. The network architecture should be shallow, this model is designed to extract snippets of text, which can be considered as local, low-level features, so a deeper network is unnecessary since informative snippets relevant to each label scattered at random locations in the input document, it is not likely that we can earn any benefit from the global, high-level features by combining the adjacent snippets. 3\. Attention mechanism should be introduced to learn the correlations between important/informative snippets and each code. #### Word embedding The word embedding model used in this paper is the word2vec CBOW method by Mikolov et al. [20], we pre-train word embedding of size $d_{e}=100$ on the preprocessed text from all discharge summaries in MIMIC3, which is the same dataset for training our model. Details about the dataset can be found in Dataset. We treat ICD code prediction as a multilabel text classification problem [19]. For clinical note instance i, we want to determine $y_{i,\ell}\in\ {0,\ 1}$ for all $\ell\in\mathcal{L}$. We train a neural network which passes text through a convolutional layer to compute a base representation of the text of each document [14], and makes $\ \|\mathcal{L}\|$ binary classification decisions. #### Convolutional Layer The input of convolutional layer is the clinical notes in form of pre-trained embeddings representing by the matrix $\mathbf{X}=[\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{N}]$. The convolution of adjacent embeddings are computed with a convolution filter $W_{c}\in\mathbb{R}^{k\times d_{e}\times d_{c}}$. At step n, we compute $\mathbf{h}_{n}=g(\mathbf{W}_{c}\ast\mathbf{x}_{n:n+k-1}+\mathbf{b}_{c}),$ The input is padded on both sides with zeros so the base representation $\mathbf{H}$ keeps the same length as $\mathbf{X}$. #### Attention layer Nowadays attention mechanism has been generalized and has been employed in many different forms [35]. The core idea of the attention mechanism can be regarded as ”giving weight to different parts of the input, to select the part in the input that is more important for the current task”. So the full connected layer in textCNN [14] can also be regarded as a kind of ”attention” since it weighs input separately for each label. As we mention in The correspondence between “informative snippet” and convolution filter, SWAM can be regarded as a general CNN architecture, and implement details can be varied. We adopt two different implements of the attention layer in our model for different consider considerations. The first one is the per-label attention mechanism by Mullenbach et al. [21], we adapt it because it can extract snippets from the clinical text as explanations of the model prediction, which can be used to verify our conjecture about the correspondence between “informative snippet” and convolution filter. The second one is the common full connected layer in textCNN [14], we adapt it since the textCNN is the basis of many works so it can prove the versatility of SWAM. For the per-label attention mechanism (the implement shown in Figure 2), the idea is to calculate the per-label representation of the document and use an attention vector $\mathbf{\alpha}_{\ell}$ to represents importance distribution over locations in the document. To obtain the per-label representation of the document, formally a vector parameter $\mathbf{u}_{\ell}\in\mathbb{R}^{d_{c}}$ is used to compute the matrix-vector product, $\mathbf{H}^{\top}\mathbf{u}_{\ell}$, which can be taken as that the base representation $\mathbf{H}$ is weighted for label $\ell$. The resulting vector is then normalized using a SoftMax operation, obtaining $\mathbf{\alpha}_{\ell}$, the attention vector, the value of the element in the attention vector is the weighted sum of convolutional features from all kernels in the same place of the document. $\mathbf{\alpha}_{\ell}=\operatorname{SoftMax}(\mathbf{H}^{\top}\mathbf{u}_{\ell})$ $\mathbf{\alpha}_{\ell}$ is also taken as the location indicator of the most important snippet for label $\ell$, every element in $\mathbf{\alpha}_{\ell}$ is corresponding to a location in the document, the value of the element is seen as the importance of the corresponding location for label $\ell$. The highest element value in $\mathbf{\alpha}_{\ell}$ means the snippet in this location is most important (a.k.a. most informative) for the prediction of label$\ell$. Therefore we obtain an explanation of the prediction in the form of extracted snippets from the document. $\alpha_{\ell,n}\mathbf{h}_{n}$, the element-wise vector product is then computed, applies the attention vectors on the base representation to get the vector document representations $\mathbf{v}_{\ell}$ for label $\ell$, $\mathbf{v}_{\ell}=\sum_{n=1}^{N}{\alpha_{\ell,n}\mathbf{h}_{n}}$ For full connected attention, we instead use max-pooling to filter the base representation down to a vector $\mathbf{v}\in d_{c}$ where every element in $\mathbf{v}$ corresponds to the highest action value of a convolution filter in the text, $v_{j}=\max_{n}h_{n,j}$ #### Classification Given the vector representation $v_{\ell}$, the likelihood for label $\ell$ is computed using a linear layer and a non-linear function sigmoid: $\hat{y}_{\ell}=\sigma\left(\boldsymbol{\beta}_{\ell}^{\top}\boldsymbol{v}_{\ell}+b_{\ell}\right)$ #### Loss function The training procedure use BCE (binary cross-entropy) as the loss function, the optimization goal is to minimize the loss. $L_{\mathrm{BCE}}(X,y)=-\sum_{\ell=1}^{\mathcal{L}}y_{\ell}\log\left(\hat{y}_{\ell}\right)+\left(1-y_{\ell}\right)\log\left(1-\hat{y}_{\ell}\right)$ ### Why Shallow and Wide Attention CNN SWAM can learn a large scale of local and low-level features, it is suitable for the multi-label text classification task that informative snippets relevant to each label are not shared. Also, SWAM addresses the challenges of interpretability: it provides a satisfactory explanation of the internal mechanics of the deep learning method by establishing the correspondence between “informative snippet” and convolution filter. ## Results ### Dataset MIMIC-III [10] is an open-access dataset comprising health data in the form of text and structured records of ICU admissions. Since MIMIC was built, it has become the basis of many works on multi-label classification [36, 25]. Following previous works, we train our model on discharge summaries in MIMIC, which summary records about one stay into a single document. We focus on the raw text of the data and ignore the attached features like admission time. Every discharge summary is corresponding to an admission, and each admission is annotated with a set of ICD-9 codes, describing both diagnoses and treatments that occurred during the patient’s stay. We train and evaluate SWAM on a label set consisting of the 50 most frequent labels. We filter the dataset down to the instances that have at least one of the top 50 most frequent codes, after the filter, there are 8,067 summaries for training, 1,574 for validation, and 1,730 for testing. Other detailed statistics for the setting are summarized in Table 2. Table 2: Descriptive statistics for MIMIC3 \| MIMIC-III full \| MIMIC-III 50 ---\|---\|--- Training Documents \| 47,724 \| 8,067 Vocabulary Size \| 51,917 \| 51,917 Mean Tokens per Document \| 1,485 \| 1,530 Mean Labels per Document \| 15.9 \| 5.7 Total Labels \| 8922 \| 50 #### Preprocessing We remove the tokens that contain no alphabetic characters (e.g., removing ‘100’ but keeping ‘100ml’ ). For those tokens that appear too few times to make their semantics difficult to learn, a threshold that only remains tokens that appear in no fewer than 3 training documents is setting, and all tokens that failed the threshold are replaced with an ‘UNK’ token. The distribution of discharge summaries conforms to the long-tailed distribution, 90% of discharge summaries are short than 2500 tokens, so we truncated discharge summaries to a maximum length of 2500 tokens. Table 3: Results on MIMIC-III, 50 labels. Model \| AUC \| F1 \| P@5 ---\|---\|---\|--- Macro \| Micro \| Macro \| Micro \| C-MemNN [24] \| 0.833 \| - \| - \| - \| 0.42 Shi et al.(2017) [32] \| - \| 0.900 \| - \| 0.532 \| - CAML [21] \| 0.875 \| 0.909 \| 0.532 \| 0.614 \| 0.609 Logistic regression \| 0.828 \| 0.862 \| 0.477 \| 0.530 \| 0.545 SWAM-CAML \| 0.900* \| 0.924* \| 0.593 \| 0.648 \| 0.625* SWAM-textCNN \| 0.892 \| 0.919 \| 0.603* \| 0.652* \| 0.620 ### Baselines As mentioned in The correspondence between “informative snippet” and convolution filter, SWAM can be regarded as a general CNN architecture and implement details can be varied. In model part Attention layer two different implements of attention layer are adapt for different considerations, we name those two implements as ”SWAM-textCNN”[14] and ”SWAM-CAML” [21] separately to indicate the attention approaches they refers. The baseline we compare against is a bag-of-words logistic regression model, we also compare SWAM-CAML with the origin implement of CAML [21] at the same setting. For SWAM-textCNN and SWAM-CAML we initialize the embedding weights using the same pre-trained word2vec vectors. The logistic regression model consists of $\ \|\mathcal{L}\|$ binary one-vs-rest classifiers acting on unigram bag-of- words features. #### Parameter tuning We tune the hyper-parameters of our SWAM-models using grid search.We sample parameter values for the learning rate $\eta$ ,as well as filter size $k$ , number of filters $d_{c}$, and dropout probability $q$ as shown in Table 4. We also adapt hyper-parameter tuning in the previous works as empirical guidance [14, 21, 1]. We use a fixed batch size of 16, and train the model with early stopping, in the case that the f1-macro does not improve for 10 epochs the training will terminate. Table 4: Hyper-parameter tuning ranges and optimal values for SWAM model \| Range \| Optimal Value ---\|---\|--- $\eta$ \| 0.0001,0.0003, \| 0.001 (learning rate) \| 0.001,0.003 \| $k$ \| 1-10 \| 4 (filter size) \| \| $d_{c}$ \| 50-500 \| 500 (number of filters) \| \| $q$ \| 0.2-0.8 \| 0.2 (dropout probability) \| \| ### Evaluation Metrics We focus on two metrics: Macro-averaged F1 and precision at n (denoted as ’P@n’), which is the fraction of the $n$ highest-scored labels that are present in the ground truth. The reason we focus on Macro-averaged F1 is that it pays attention to per-label performance, which can reflect the average performance of the model on different label tasks. As for P@n, we choose it because it reflects the performance of the model as a practical decision support system which presents a fixed number of predicted codes to help user annotated the clinical text. To facilitate comparison with both future and prior work, we also report a variety of metrics includes the area under the ROC curve (AUC) and micro-averaged F1. For recall, Macro-averaged values are calculated by averaging metrics computed per-label. Micro-averaged values are calculated by treating each (document, code) pair as a separate prediction. ### Results on quantitative evaluation Our main quantitative evaluation involves predicting the 50-code set of ICD-9 codes based on the text of the MIMIC-III discharge summaries. These results are shown in Table 3. We adopt two different implements of attention layer, named ”SWAM-textCNN” and ”SWAM-CAML” The SWAM models give the strongest results on all metrics, especially on F1-Macro, which emphasis average performance over different labels. We attribute this improvement to SWAM, by which the wide architecture gives the model ability to more extensively learn the unique features of different codes. ### Ablation experiment on width of network Figure 3: Comparing per-label performances of SWAM-CAML with CAML [21],the only difference between two models are the width of the network. SWAM has 500 filters, while CAML has 50 filters. According to our inference about the correspondence between the informative snippet and convolution filter, since each ”non-generic snippet” has to correspond to a convolution filter, if the network is too narrow, the model will fail to learn the ”non-generic snippets” of some labels. Therefore the impact of the width of the network can be observed from the perspective of per-label performance. We carry out an ablation experiment on the width of the network, we consider the original implement of CAML [21] as a narrow variant of SWAM, the only difference between SWAM-CAML and CAML are the width of the network, the former has 500 convolution filters and the latter has 50. The experiment results are in line with our expectations. For the narrow model(CAML), the performance of 5 labels is 0, while on the opposite, the wide model(SWAM-CAML) make significant performance improvement that 4 of 5 labels that with a 0 precision in the narrow model now have an average precision of 0.53, at the same time the overall performance of the model is improved. As for the only ICD-9 code 285.9 ”Anemia, unspecified” that has 0 precision in both models, we make a manual analysis in Section Analysis of the reason behind bad performance code. ### Secondary evaluation #### Comparing informative snippets extracted by narrow and wide models To verify our inference about The correspondence between “informative snippet” and convolution filter, we also compare the informative snippets extracted by both the narrow (CAML) and the wide (SWAM-CAML) model. In order to make the cases representative, from the five labels that has 0 precision in the narrow model, we pick up a label (276.1: Hyposmolality and/or hyponatremia) that has a improved precision in the wide model, and the only label(285.9 Anemia, unspecified) that has 0 precision in both models for analyse. Table 5 shows the informative snippets extracted by the SWAM-CAML and the CAML model during the prediction of code 276.1 in two random selected documents. Through a simple analysis, it can be found that the word ”hyponatremia” extracted by the SWAM model that appears in both the document and the code description plays an important role in the prediction. While on the opposite, the snippet extracted by the CAML model is not informative since it has 0 precision on this code. The word ”hyponatremia”, as a local and low-level feature can be learned by a single convolution filter according to our inference The correspondence between “informative snippet” and convolution filter. Since the only difference between the SWAM-CAML model and the CAML model is the number of filters in the convolutional layer. Table 5 proves that the performance difference between the narrow and the wide model comes from the learning of non-generic ”informative snippet”. Table 5: informative snippets extracted by the wide (SWAM-CAML) and narrow (CAML) models for prediction of ICD code 276.1 ICD code 276.1: ”Hyposmolality and/or hyponatremia” SWAM-CAML …dehydration and increased abd… SWAM-CAML …hyponatremia and possible initiation of chemotherapy… CAML …peritonitis renal failure and ileus on the floor the patient was followed by… CAML …renal failure and small bowel obstruction of note the provided information on… Table 6: ICD Code with 0 precision in both the wide (SWAM-CAML) and narrow (CAML) models before and after data shuffle Before shuffle \| ICD Code with 0 precision ---\|--- SWAM-CAML \| 285.9 ”Anemia, unspecified” CAML \| 285.9 ”Anemia, unspecified” \| V15.82 ”History of tobacco use” \| 276.1: ”Hyposmolality and/or hyponatremia” \| 305.1 ”Tobacco use disorder” \| 311 ”Depressive disorder, not elsewhere classified” After shuffle \| ICD Code with 0 precision SWAM-CAML \| 285.9 ”Anemia, unspecified” CAML \| 285.9 ”Anemia, unspecified” \| V15.82 ”History of tobacco use” \| 272.0 ”Pure hypercholesterolemia” \| V45.81 ”Postsurgical aortocoronary bypass status” #### Factors determine which ”non-generic snippet” will fail to be learned by narrow model According to our inference, a narrow network will fail to learn the ”non- generic snippet” of a part of labels, which naturally raises a question: what factors determine which ”non-generic snippet” will not be learned? The essence of failing to learn different ”non-generic snippet” is that the model converges to different local optimal parameters. The convergence result of the model is related to the distribution of data during training, in other words, the order that ”non-generic snippets” appear during training: once a filter learns a specific ”non-generic snippet” for some labels, the loss function will encourage it to keep its parameters unchanged, and after all the filters have learned corresponding ”informative snippet”, it’s difficult for the model to leave the current local optimal solution and learn new ”informative snippet”.Therefore we shuffle the training dataset with a different random seed and re-train the models with the shuffled dataset. The results are shown in Table 6. The results in Table 6 are in line with our expectations, after shuffle and re-training, the 0 precision labels in the narrow model (CAML) change. On the opposite, the only 0 precision label 285.9 in the wide model (SWAM-CAML) still can not be learned. The distribution of data during training is a factor that determines which ”non-generic snippet” will fail to be learned by the narrow model. And the reason behind the bad performance ICD code 285.9 ”Anemia, unspecified” is something beyond the local optimum. #### Analysis of the reason behind bad performance code The ICD-9 code 285.9 ”Anemia, unspecified” is failed to be predicted by both narrow (CAML) and wide (SWAM) models. Through manual analyzing, we found there are more than 50 codes in the ICD-9 that are in form of “Anemia + specific reason”, which means the snippets related to anemia cannot are necessary but not sufficient for prediction of code 285.9. The prediction of 285.9 Anemia, unspecified is not only based on the presence of snippet related to ‘Anemia’, it is also based on the information that all possible reasons are absent. This is a blind spot in all current machine learning models. It is difficult for models to learn inferences based on missing information. ## Discussion For future work, we are considering several different directions. From the application perspective, since our approach does work well on 50 labels task, the next step is to apply the approach to the full code set. A major challenge is for full code set, we may need tens of thousands of convolution filters, as the number of filters in the network increases, unnecessary overlap in the features captured by the network’s filters will also increase [23]. We plan to address this challenge by adapting the ensemble method, we plan to cluster the ICD codes and train a classifier for each clustered subset. From the linguistic perspective, we plan to explore reasons behind hard-to-learn codes such as ICD-9 code 285.9 ”Anemia, unspecified”, and leverage the hierarchy of ICD codes to improve performance on these codes. ## Conclusion We present SWAM, an explainable CNN approach for multi-label document classification, which employs a wide convolution layer to learn local and low- level features for each label, yields strong improvements over previous metrics on the ICD-9 code prediction task, while providing satisfactory explanations for its internal mechanics. ## Abbreviations ICD:International Classification of Diseases;CNN:Convolutional Neural Network;ICU:Intensive care unit;SWAM:Shallow and Wide Attention convolutional Mechanism;MIMIC-III:Medical Information Mart for Intensive Care III;NLP:Natural Language;CBOW:Continuous Bag-of-Words;BCE:Binary cross- entropy;CAML:Convolutional Attention for Multi-Label classification;F1:F-measure ## Availability of data and materials The datasets is available from https://mimic.physionet.org/. ## Competing interests The authors declare that they have no competing interests. ## Author’s contributions Hu S.Y. conceived of the presented idea, developed the theory and carry out the experiments. Teng F. encouraged Hu S.Y. to investigate per-label performance of the model and supervised the findings of this work. All authors discussed the results. Hu S.Y. wrote the manuscript under supervising from Teng F., all authors have read and approved the final manuscript. ## Funding No funding to declare. ## Ethics approval and consent to participate Not applicable. ## Consent for publication Not applicable. ## Acknowledgements None. ## References * [1] Ahmad Aghaebrahimian and Mark Cieliebak. Hyperparameter tuning for deep learning in natural language processing. In 4th Swiss Text Analytics Conference (SwissText 2019), Winterthur, June 18-19 2019. Swisstext, 2019. * [2] Miltiadis Allamanis, Hao Peng, and Charles Sutton. A convolutional attention network for extreme summarization of source code. In International conference on machine learning, pages 2091–2100, 2016. * [3] Anand Avati, Kenneth Jung, Stephanie Harman, Lance Downing, Andrew Ng, and Nigam H Shah. 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labelinglabel # Analysis of basic emotions in texts based on bert vector representation A. Artemov, A. Veselovskiy, I. Khasenevich, I. Bolokhov Kognitivnie Sistemi <EMAIL_ADDRESS> ###### Abstract In the following paper the authors present a GAN-type model and the most important stages of its development for the task of emotion recognition in text. In particular, we propose an approach for generating a synthetic dataset of all possible emotions combinations based on manually labelled incomplete data. Keywords: 7-dimensional emotional model, Pytorch, GAN, deep learning, cosine similarity, Ekman, BERT, NLP, BRAIN2NLP, emotions, collisions, Big Data ## Introduction There are several approaches to classification of human emotions. Authors consider the methodology according to P.Ekman to be the most reasonable [4]. In accordance with it, the following seven human emotions can be distinguished: fear, sadness, anger, disgust, calm, happiness, surprise. These emotions were classified according to the analysis of facial expressions of a person during events of different semantic meaning. The unambiguous perception of facial expressions with different people and nations was the proof of the sufficiency and validity of the seven basic emotions. At the same time, although there is a number of solutions for emotional recognition, a standard for the number of basic emotions for text data has not yet been created. For example, in Rodrigo Masaru Ohashi’s work [7] the model of bidirectional LSTM with a convolutional neural network layer was successfully trained to identify 4 emotions: joy, fear, sadness and anger. Chew-Yean Yam’s article “Emotion Detection and Recognition from Text Using Deep Learning [2]” describes the neural network with 3 hidden layers of 125, 25 and 5 neurons for determining 5 emotions: anger, sadness, fear, happiness, excitement. Jangwon Park [8] created a Huggingface Transformers based model that predicts 7 emotions, the F1-measure of the model is 59.81 and 61.48 for the train and test dataset, respectively. This model is based on the GoEmotions [3] dataset with 27 emotion classes (see Table 5). Let us assume, as a hypothesis, that the 7 basic emotions by Ekman are applicable to texts. That is, the labelling of texts for this set of emotion classes by a person is complete. Based on the possibility of obtaining such a dataset, the task is to train a model that can predict the vector of seven emotions for an arbitrary text. The result of this work is described in the article. A positive solution to this problem opens up prospects for combining different data channels (audio and video) to determine emotions when machines are communicating with humans. Moreover, of particular interest is the description of information spread through means of mass communication (social networks), taking into account the phenomenon of emotions social exchange. ## 1 Dataset vectorization Initially, it is necessary to form a dataset for training the model. The dataset should contain data about seven basic emotions for each certain piece of text - a paragraph or a sentence. The authors couldn’t find a publicly available dataset with this or similar data. Therefore, it was decided to form it automatically using data on the emotional reaction of people, expressed in texts. Such data were emoticons in texts from Twitter, YouTube and other social networks. The original texts were divided into sentences, emoticons were extracted from each sentence. In order to vectorize emotions, a special dictionary was compiled containing emoticons divided into emotion types according to uniquely interpreted classes: The classification of the texts’ emotions was carried out using the created dictionary. By counting the number of encounters of each emotion in the text, vectors of 7 emotions were generated for each sentence. Text vectorization was carried out using our own NLP framework - BRAIN2NLP111Russian cognitive processor for automated labelling of natural text. The main functionality of BRAIN2NLP includes language detection, stem extraction, morphological and syntactic labelling of text, extraction of nominal entities and definition of a semantic vector of words.. The text vectorization was based on the BERT language model trained by Google. Thus, at the output, pairs of the following form were obtained: 512-dimensional vector BERT to seven-dimensional vector of emotion. In total, more than 100,000 such pairs were constructed. In the process, it turned out that the resulting dataset contains a lot of conflicting data (collisions). Collisions are examples of data, where the same properties of an object lead to different classes or to a significant number of classes. The automation of the collision search process required the development of a small script. The algorithm of its work is given below. * 1. Determine the allowed fraction of the k-number of classes to which objects in the same cluster can belong. * 2. Classify dataset objects by the vector of their properties (features). * 3. Assign the same ID to dataset objects belonging to the same cluster. * 4. Add up the values by class for objects with the same ID. * 5. Estimate for each ID the mathematical expectation of the vector of values by class. * 6. Calculate the number of Z values of the class vector that is greater than or equal to the expected value. * 7. Evaluate each cluster, and where $Z>k$ mark it as a collision. * 8. Assign a collision mark to all objects in the cluster. Algorithm 1 Finding collisions in a dataset For the purpose of "human" marking, 10,000 texts were selected, among which there were only 6913 examples without collisions with a length of no more than two sentences. The experts were asked to assign no more than 2 emotions to each of these texts. But the dataset formed in this way also contained minor examples of collisions, which were also eliminated [1]. As a result, a golden dataset with 2813 examples was obtained. ## 2 Search for Architecture BERT-vectors are successfully used for classification tasks of texts [9] and their fragments [5]. Considering this fact, we have chosen these vectors as a basic description of the space of text properties, which can be used to predict the emotion vector. During the search for the optimal solution for the classification of emotions, different variants of embedding length BERT (lengths 768 and 512) and different sizes of datasets were tested. Also, we have tested a certain number of negative samples, various dictionaries for the extraction of emoticons, normalization of the dataset, different model architectures, loss functions and activation functions. As a result of this work, the following hypothesis was formulated and confirmed in practice: It is possible to improve the quality of classification by abandoning the attempt to normalize the unbalanced initial data - the number of examples for a set of classes, replacing it with uniformly generated BERT vectors for each set of classes. Among all tested neural network architectures, Generative adversarial network (GAN) [6] showed the best results - an architecture consisting of a generative network and a discriminative network configured to work against each other. Discriminative models learn the boundary between the classes; generative model generates new data instances. A collision free dataset (Dataset_1) was used for training the generative network and for final testing of the discriminative network. In the formulation of the study, the vector of emotions has 7 dimensions, we denote it as V1. Table 1 shows a list of combinations of vector V1 values found in Dataset_1. Table 1: Combinations of emotion vectors Dataset_1 # \| fear \| sadness \| anger \| disgust \| calm \| happiness \| surprise ---\|---\|---\|---\|---\|---\|---\|--- 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 2 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 3 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 4 \| 0 \| 0 \| 0 \| 0 \| 1 \| 0 \| 1 5 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 0 6 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 \| 1 7 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 8 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 \| 1 9 \| 0 \| 0 \| 0 \| 1 \| 1 \| 0 \| 0 10 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 11 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 \| 1 12 \| 0 \| 0 \| 1 \| 0 \| 0 \| 1 \| 0 13 \| 0 \| 0 \| 1 \| 0 \| 0 \| 1 \| 1 14 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 0 15 \| 0 \| 0 \| 1 \| 0 \| 1 \| 0 \| 1 16 \| 0 \| 0 \| 1 \| 0 \| 1 \| 1 \| 0 17 \| 0 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 18 \| 0 \| 0 \| 1 \| 1 \| 0 \| 0 \| 1 19 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 20 \| 0 \| 1 \| 0 \| 0 \| 0 \| 0 \| 1 21 \| 0 \| 1 \| 0 \| 0 \| 0 \| 1 \| 0 22 \| 0 \| 1 \| 0 \| 0 \| 1 \| 0 \| 0 23 \| 0 \| 1 \| 0 \| 0 \| 1 \| 0 \| 1 24 \| 0 \| 1 \| 0 \| 0 \| 1 \| 1 \| 0 25 \| 0 \| 1 \| 0 \| 1 \| 0 \| 0 \| 0 26 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 27 \| 0 \| 1 \| 1 \| 0 \| 0 \| 0 \| 1 28 \| 0 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 29 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 0 30 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 \| 1 31 \| 1 \| 0 \| 0 \| 0 \| 0 \| 1 \| 1 32 \| 1 \| 0 \| 0 \| 0 \| 1 \| 0 \| 0 33 \| 1 \| 0 \| 0 \| 0 \| 1 \| 0 \| 1 34 \| 1 \| 0 \| 0 \| 1 \| 0 \| 0 \| 0 35 \| 1 \| 0 \| 1 \| 0 \| 0 \| 0 \| 1 36 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 \| 0 37 \| 1 \| 1 \| 1 \| 0 \| 0 \| 0 \| 0 Since 37 combinations222Not entirely confident in ”Les 36 situations dramatiques” concept, we decided to generate all possible combinations of emotions. do not make up a complete list of 128 combinations, the missing ones had to be generated. For this purpose, based on Dataset_1, we have trained a generator model for known combinations of emotions. Then the combinations V1 were initiated, where the initial - has the values [0, 0, 0, 0, 0, 0, 0], and the final one is [1, 1, 1, 1, 1, 1, 1]. The total number of vectors V1 was 128. Using the generator, 128 BERT - V2 vectors were obtained corresponding to various V1 combinations. Thus, a new dataset (Dataset_2) was formed, where each vector V1 corresponded to the generated vector V2 (Figure 1). Dataset_2 was used for fine-tuning the Generator model and Discriminator training, predicting the emotion vector from the BERT vector of the text. Figure 1 shows the process of sequential training of the Generator, and Figure 2 shows the Discriminator. Figure 1: The process of generator training with the formation of Dataset_2 Figure 2: The process of joint training of the Generator and the Discriminator Figure 3 shows graphs of the "descent rate" when training the Generator and the Discriminator. The large value of the generator error is explained by the fact that a simple product of vectors is used, and the error is considered by MSE. The opposite situation is with the discriminator, where not only the output value is normalized, but also the error is calculated by the cosine measure. a) b) Figure 3: Gradient descent of a) Generator model; b) Discriminator model In order to move to a non-negative real number, the output vector of the discriminator value was normalized as follows $M_{ij}^{\ast}=\dfrac{M_{ij}-min\|M_{ij}\|}{\sum_{M_{ij}\in V}(M_{ij}-min\|M_{ij}\|)}$ (1) where V is a vector of forecast values. ## 3 Results Model training was performed on 70% of the data with testing on the remaining 30%. The evaluation of the quality of the model is illustrated in Figure 4. Comparative results of the model training process are presented in Table 2. Table 2: Accuracy rating Emotion \| Accuracy \| Figure 4: The matrix of the predictive distribution and the real value of the Discriminator based on Dataset_1 ---\|---\|--- fear \| 0.62 sadness \| 0.70 anger \| 0.63 disgust \| 0.76 calm \| 0.64 happiness \| 0.68 surprise \| 0.54 MEAN \| 0.65 Table 3: The result of the model training process Network \| Dataset \| Architecture \| Learn ---\|---\|---\|--- \| Collisions --- Train \| Test \| \| 1-st layer --- \| 2-nd layer --- \| Weight Initialization --- \| Activation Function --- Error \| \| Gradient descent --- Epochs \| \| Train accuracy --- \| Test accuracy --- \| Generator --- Yes \| 4855 \| 2082 \| 7x128 \| 128x512 \| Random \| No \| MSE \| ADAM \| 10 \| 0.99 \| 0.99 \| Discriminator --- Yes \| 89 \| 39 \| 512x7 \| No \| \| Metod --- FM* \| Cosine --- similarity \| Cosine --- Loss ADAM \| 50 \| 0.75 \| 0.81 \| Generator --- No \| 1969 \| 844 \| 7x128 \| 128x512 \| Random \| No \| MSE \| ADAM \| 10 \| 0.96 \| 0.9 \| Discriminator --- No \| 89 \| 39 \| 512x7 \| No \| \| Metod --- FM* \| Cosinus --- similarity \| Cosine --- Loss ADAM \| 50 \| 0.74 \| 0.87 FM Method (frequency-matrix) is a method for forming a frequency matrix based on pairs of property and class vectors. The original dataset of the emotion vector and BERT vector pairs of text was divided into two matrices of emotion vectors and BERT vectors. Then, by multiplying the obtained matrices, a frequency matrix is formed. Based on the input data, the model makes a forecast in the form of a 7-dimensional vector of emotions. To be able to compare it with gold dataset, it was converted into 2-dimensional vector by selecting emotion classes with maximum values. If at least one of the two emotions coincided with the gold labelling, then it was considered that the emotion was predicted correctly. The accuracy of the forecast for such an assessment varies depending on the class of emotion (see Table 3). The average accuracy of emotion predictions is 0.65. The results of the model for different words with different emotional sentiment are shown in Table 4. Table 4: An example of model’s evaluation the emotional sentiment of a sentence - \| Example of sentence \| Expert’s Emotion \| Predicted emotion by the Model ---\|---\|---\|--- fear \| sadness \| anger \| disgust \| calm \| happiness \| surprise 1 \| \| There is danger all around, --- it’s scary to go outside. \| fear, --- sadness 0.49 \| 0.12 \| 0.17 \| 0.13 \| 0.00 \| 0.00 \| 0.09 2 \| \| It is very sad that nothing --- can be done. The city burned down and everyone died \| sadness, --- fear 0.18 \| 0.27 \| 0.21 \| 0.27 \| 0.00 \| 0.00 \| 0.07 3 \| \| I am super angry, --- I’m in a fury! You’re dead, bastards! anger \| 0.21 \| 0.15 \| 0.22 \| 0.19 \| 0.00 \| 0.12 \| 0.11 4 \| \| The soup was terrible, --- I’ve never tasted such disgusting meal. I’ve been sick of it for an hour. \| disgust, --- anger 0.17 \| 0.26 \| 0.15 \| 0.30 \| 0.00 \| 0.07 \| 0.06 5 \| \| Relax and listen to nature, feel this --- light summer wind. The silence calms better than a thousand words. \| calm, --- happiness 0.03 \| 0.20 \| 0.19 \| 0.14 \| 0.14 \| 0.30 \| 0.00 6 \| \| What could be better than --- watching the sunset together with close friends! Just wonderful! \| happiness, --- calm 0.01 \| 0.05 \| 0.10 \| 0.03 \| 0.00 \| 0.63 \| 0.18 7 \| \| John won more than --- $1 million in the lottery, that was a pleasant surprise for him. \| surprise, --- happiness 0.00 \| 0.04 \| 0.03 \| 0.15 \| 0.07 \| 0.47 \| 0.25 Based on the study of the data on model’s performance, the following conclusions were made: - in all cases, the model correctly identified at least one of the dominant emotions, a little less often – two of the two. * - the model does not define the calm emotion well, in most cases its value is the lowest. * - Intends to show more emotions than necessary (see examples in the first five sentences of Table 4). That is, the model "gives a chance" to emotions that were not defined in the gold dataset. We consider this to be a special property of the model rather than a bug, since the model was trained on a larger variety (of generated data) than those observed by experts. This is a very important consequence that needs to be taken into account when reconstructing complete data based on expert assessments. ## 4 Conclusion Among all the architectures that were tested, GAN performed better than the others (see Table 5). When compared with the models of emotions of some other authors, our model showed the best result. The F1-measure value is 0.025 higher than the "goemotions-pytorch" model, and the accuracy value is 0.0053 higher than the "Emotion Detection and Recognition from Text Using Deep Learning" model, but 0.19 lower than the "Bidirectional LSTM with a Convolutional Neural Network"model. In the latter case, it is worth saying that in our model the number of defined emotions is seven against four in "Bidirectional LSTM with a Convolutional Neural Network". Table 5: Comparing the quality of different emotion models Model \| Emotions \| F1 macro \| Accuracy ---\|---\|---\|--- \| «Bidirectional LSTM with a Convolutional Neural --- Network» by Rodrigo Masaru Ohashi (2019) 4 \| - \| 0.84 \| «Emotion Detection and Recognition from Text Using --- Deep Learning» by Chew-Yean Yam (2015) 5 \| - \| 0.64 «GoEmotions-pytorch» by Jangwon Park (2020) \| 7 \| 0.61 \| - GAN BERT-EMO model (2020) \| 7 \| 0.64 \| 0.65 The hypothesis about the possibility of improving the quality of classification by refusing to normalize the class distribution of a real sample and replacing it with a sample obtained by generating all possible combinations of classes (BERT vectors) for a finite number of variables was fully justified. The authors have not encountered such an approach in other publications. At the same time, it opens up new perspectives for machine learning with minimizing human involvement by automatically generating "big data" for training neural networks on "small" incomplete data. ## References * [1] A.Artemov, & A. Veselovskiy. CLASSIFICATION-OF-BASIC-EMOTIONS-USING-BERT. 2021. https://github.com/cogsyscompany/datasets * [2] Chew-Yean. Emotion Detection and Recognition from Text Using Deep Learning. November 29, 2015. https://devblogs.microsoft.com/cse/2015/11/29/emotion-detection-and-recognition-from-text-using-deep-learning/ * [3] D. Demszky, D. Movshovitz-Attias, Jeongwoo Ko, A. Cowen, G. Nemade, & Sujith Ravi. GoEmotions: A Dataset of Fine-Grained Emotions. 2020. https://arxiv.org/abs/2005.00547 * [4] P. E.Ekman, , & R. J.Davidson, (1994). The Nature of Emotion Fundamental Questions. Oxford Oxford University Press. * [5] Y. Feng, Y. Wang, & Hang Li. A Sequence-to-Sequence Approach to Dialogue State Tracking. 2020. https://arxiv.org/pdf/2011.09553.pdf * [6] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, Yoshua Bengio. Generative Adversarial Networks. 2014. https://arxiv.org/abs/1406.2661 * [7] R. M. Ohashi. From Sentiment Analysis to Emotion Recognition: A NLP story. 2019. https://medium.com/neuronio/from-sentiment-analysis-to-emotion-recognition-a-nlp-story-bcc9d6ff61ae * [8] Jangwon Park. GoEmotions Pytorch. 2020. https://github.com/monologg/GoEmotions-pytorch * [9] M. Tan, & J. Jiang. A BERT-based Dual Embedding Model for Chinese Idiom Prediction. 2020. https://arxiv.org/abs/2011.02378
11institutetext: Instituto de Astrofísica de Canarias, Vía Láctea S/N, La Laguna 38205, Tenerife, Spain 11email<EMAIL_ADDRESS><EMAIL_ADDRESS>22institutetext: Departamento de Astrofísica, Universidad de La Laguna, 38205, Tenerife, Spain # Mapping the Sun’s upper photosphere with artificial neural networks H. Socas-Navarro 1122 A. Asensio Ramos 1122 (Received ; accepted ) We have developed an inversion procedure designed for high-resolution solar spectro-polarimeters, such as Hinode/SP or DKIST/ViSP. The procedure is based on artificial neural networks trained with profiles generated from random atmospheric stratifications for a high generalization capability. When applied to Hinode data we find a hot fine-scale network structure whose morphology changes with height. In the middle layers this network resembles what is observed in G-band filtergrams but it is not identical. Surprisingly, the temperature enhancements in the middle and upper photosphere have a reversed pattern. Hot pixels in the middle photosphere, possibly associated to small- scale magnetic elements, appear cool at the $\log\tau_{500}$=$-3$ and $-4$ level, and viceversa. Finally, we find hot arcs on the limb side of magnetic pores, which we interpret as the first direct observational evidence of the ”hot wall” effect in temperature. ###### Key Words.: Sun: photosphere – Sun: faculae, plages – Sun: magnetic fields – Methods: numerical – Methods: data analysis ## 1 Introduction Inversion techniques allow us to retrieve information encoded in spectral lines about the atmospheres where they form. A wide variety of strategies have been employed for decades in solar physics to interpret spectroscopic and spectropolarimetric observations (see e.g. the reviews by del Toro Iniesta & Ruiz Cobo 2016; Bellot Rubio 2006; Socas-Navarro 2001). Most applications are based on the least-squares fitting of the observed spectral lines with synthetic profiles, which are computed from model atmospheres whose parameters are iteratively adjusted until a satisfactory fit is attained. However, advances in instrumentation are driving an increasing interest in the exploration of alternative methods. Two-dimensional spectropolarimetry is now very common and fast growing data rates motivate the exploration of new algorithms that have the potential of being faster and/or more robust for systematic application. Artificial neural networks (ANNs) offer a promising new approach for many purposes where profile fitting is inadequate because one needs a faster or a more robust performance. The first applications of ANNs in solar physics are almost 20 years old, dating back to Carroll & Staude (2001) and Socas-Navarro (2002). However, while those first efforts produced encouraging results, ANN inversions were not immediately adopted by the community, for two reasons mainly. First, disentangling the magnetic filling factor from the intrinsic field strength has proven extremely challenging, as noted since those early works (Socas-Navarro 2003). The magnetic field tends to exhibit small-scale structures in the solar photosphere. In arc-second resolution observations, it is common to find pixels where the magnetic field occupies less than 10% of the resolution element. This area fraction is referred to as the filling factor and it introduces an important complication for ANN inversions. Second, a more practical issue is the complexity involved in the coding of algorithms for the training of an ANN model. Those early problems have been largely resolved in recent years, leading to a renewed interest in ANNs (e.g., Liu et al. 2020; Guo et al. 2020; Díaz Baso & Asensio Ramos 2018; Asensio Ramos & Díaz Baso 2019; Felipe & Asensio Ramos 2019; Milić & Gafeira 2020). Current (and upcoming) instrumentation are delivering very high resolution observations, mitigating the filling factor problem. Furthermore, there has been a tremendous development in the field of deep learning and many sophisticated tools have been made publicly available to simplify the problem of building and training ANNs (e.g., Abadi et al. 2015; Paszke et al. 2019). In this paper we use a relatively simple ANN model and take a different approach to previous work for the training strategy. Instead of using a simulation snapshot as the starting point for a training set, as in Asensio Ramos & Díaz Baso (2019) or Milić & Gafeira (2020), we create a database of profiles from random stratifications of the relevant parameters. This provides a wide coverage of the parameter space and guarantees that the ANN is not specialized on any particular scenario. Unlike Asensio Ramos & Díaz Baso (2019), whose ANN performs a full inversion of the entire 2D field at once, this ANN works on each pixel independently. In that regard, it is more similar to a traditional inversion technique. We created two different ANNs, one to invert photospheric observations from DKIST/ViSP (Daniel K Inouye Solar Telescope/Visible Spectro-Polarimeter, see Rimmele et al. 2020) and the other one for the Hinode satellite’s SOT/SP (Solar Optical Telescope/Spectro-Polarimeter). DKIST/ViSP data are not yet available so we focus here on the analysis of the Hinode inversions. After testing the procedure with synthetic data and previous inversions of real observations, we applied it to Hinode observations of active regions. In this manner we obtained datacubes with a fairly unique combination of high spatial resolution, large field of view and depth-dependent temperatures. These maps show a fine hot network in active regions, particularly around sunspots and pores. We find some surprising results in this application, such as an anticorrelation between hot pixels in middle and upper layers. Also, the inversions reveal a series of hot arcs running along the limb side of pores in the observed regions. We interpret these arcs as the first direct observation of the ”hot wall” effect, a prediction of fluxtube models since the work of Spruit (1976) which had not been directly observed thus far. ## 2 The ANN model and training set All the calculations presented in this paper were produced with relatively standard computer hardware. We employed a Linux workstation powered by eight 3·GHz Intel Xeon cores. The system is equipped with a GTX 1080 GPU that handles most of the ANN-related processing. Our ANN model and codes are publicly available in a repository111https://github.com/hsocasnavarro/Paper_SNAR21. The ANN is created and trained using PyTorch (Paszke et al. 2019). It is a simple multilayer perceptron with six hidden layers between the input and output layers. Each hidden layer comprises 300 neurons. The input layer has a number of neurons that matches the number of spectral pixels in a given profile (175 for DKIST/ViSP and 112 for Hinode SOT/SP). The output layer has 9 neurons, which correspond to the output parameters that we wish to retrieve. These parameters are: 5 temperatures at different heights, 3 components of the magnetic field vector and a single-valued line-of-sight velocity. In all cases, the activation function chosen is a leaky ReLU (Maas et al. 2013). The entire training procedure takes a few hours on our hardware described above. For the training and validation sets we compute one million synthetic profiles from randomized model atmospheres. A thousand models and profiles are taken as the validation set and the rest are used for training. These models are obtained as random variations from four different reference atmospheres, namely: HSRA (Gingerich et al. 1971), VAL-C (Vernazza et al. 1981), FAL-C (Fontenla et al. 1993) and the sunspot model M of Maltby et al. (1986). We provide here a description of the randomization procedure in some detail because the construction of this database is critical for the ability of the ANN to perform adequately when faced with real observations and to exhibit good generalization properties. For each relevant parameter, we take the stratification in the reference atmosphere and add a depth-dependent perturbation to it. The perturbation is constructed by assigning values to certain layers and then interpolating in depth. In the case of temperature, the parameter to which spectral lines are most sensitive, we start by creating a perturbation at four layers. These layers are not necessarily the same heights that the ANN will retrieve. They are equispaced in the logarithm of continuum optical depth at 500 nm ($\log\tau_{500}$) and their actual location is different for the four reference atmospheres. The perturbations at these four points are drawn from a Gaussian distribution with a 1,500 K standard deviation. From these four values, the depth-dependent perturbation is interpolated to the entire grid and added to the reference model. With the new thermal stratification, the model is set in hydrostatic equilibrium and the equation of state is solved to compute plasma densities, ionization fractions, relevant molecules and electron densities. For the magnetic field, we have that $B_{z}$ (the line-of-sight component) is linear in $\log\tau_{500}$, whereas $B_{x}$ and $B_{y}$ (the transverse components) are constant with height. $B_{z}$ is defined by $B_{z}(0)$, its value at $\log\tau_{500}$ $=0$, and its gradient. We construct three possible scenarios with weak, strong and extreme fields, having probabilities of 45%, 45% and 10% respectively. The field strength $B_{z}(0)$ takes values from a uniform distribution with a width of 500, 2000 and 6000 $G$ for the weak, strong and extreme fields, respectively. The sign for each field component is randomly set to $\pm 1$ except for $B_{x}$, which is always taken as positive. Since the Zeeman effect has a 180-degree ambiguity in the transverse component of the field, we restrict our solutions to the subspace with positive $B_{x}$. The $B_{z}$ gradient is set to either 0 or a random value, with a 50% probability. The random value is taken from a uniform probability distribution between -150 and 150 $G$ per unit in $\log\tau_{500}$. The filling factor ($\alpha$) is set to 1 in 50% of the models. The rest have a uniform distribution between 0.1 and 1. In addition to the filling factor, we consider a fixed amount of stray light in the instrument by adding an average quiet Sun profile to Stokes I. The amount of stray light is fixed to 10%, which is a typical value for spectrographs. This training set was built with the purpose of covering a sufficiently wide range of profiles for the ANN to work with all possible observations of the solar photosphere in the 630 nm spectral window observed by Hinode. The statistical distribution of our random atmospheres is not necessarily optimal. We relied on past experience and numerical experimentation to determine a suitable set. A systematic analysis is beyond the scope of this work. We used NICOLE (Socas-Navarro et al. 2015) to compute synthetic Stokes profiles for the entire set of one million random models in the database. The synthesis parameters for one of the training sets were defined to mimic Hinode/SP observations. DKIST/ViSP will also feature a preset mode to observe the same 630 nm window so we produced another similar set of profiles simulating those observations in anticipation of its science operations. Both training sets are publicly available in the repository mentioned above. Figure 1: Comparison of inversions of a Hinode map performed with NICOLE (left) and our ANN (right). The Stokes profiles are fed as inputs to the ANN. For the outputs, we extract a set of 9 parameters from the random model atmospheres in the database. These parameters are: five temperatures ($T_{0}$, …, $T_{4}$), extracted at optical depths $\log\tau_{500}$= $0$,…,$-4$, a bulk Doppler velocity ($v_{z}$) and the three components of the pixel-averaged magnetic field ($F_{x}$, $F_{y}$, $F_{z}$, where $F_{i}=\alpha B_{i}$ for $i=x,y,z$). We do not aim here at disentangling the filling factor $\alpha$ from the intrinsic magnetic field strength ($B_{i}$) in the magnetic element. We seek to retrieve the magnetic flux density ($F$) in the resolution element, which simplifies the problem. Figure 2: Comparison of inversions of a Hinode map performed with NICOLE (left) and our ANN (right). Positive (negative) velocities are directed downards (upwards). Positive (negative) magnetic polarity represents fields pointing up (down) from the solar surface. ## 3 Comparisons with other inversions After successfully training the ANN and observing a good recovery of the validation set (see Figure 3), we tested it with real observations. As noted in previous work (Socas-Navarro 2005), a good performance with the validation set composed of synthetic observations does not guarantee a good operation with real data. Figure 3: Tests of the ANN performance with the validation set. Abscissas are the ”true” values and ordinates are the ANN outputs. Each plot shows the average (median) error in the recovery of that parameter. Ideally, one would like to have inversions of Hinode/SP data to compare with our ANN. Unfortunately, there are very few inversions of Hinode/SP maps that yield the height stratification. The standard pipeline includes an inversion carried out by the instrument team with the MERLIN code (Lites et al. 2007), which is based on the Milne-Eddington approximation and therefore does not provide information on the height dependence of any physical quantities. One of the few inversions with the height stratification existing in the literature is that of Socas-Navarro (2011) using NICOLE (later refined in Socas-Navarro 2015). We took the same Hinode/SP observations used for the NICOLE inversions and processed them with our ANN. The NICOLE inversions took about 5 hours on a dedicated parallel run over the eight cores of our workstation. The ANN inversion was completed in half a second. Figure 4: Comparison of inversions of a Hinode map (map1) performed with MERLIN (left column) and our ANN (center). Scatter plots are shown in the right column. Positive (negative) magnetic polarity represents fields pointing up (down) from the solar surface. Figure 5: Comparison of inversions of a Hinode map (map2) performed with MERLIN (left column) and our ANN (center). Scatter plots are shown in the right column. Positive (negative) magnetic polarity represents fields pointing up (down) from the solar surface. An additional postprocessing renormalization was applied on each ANN temperature output, so that the average value would match that of the NICOLE inversions. This is done to remove (at least to first order) some small residuals that arise in the application to real data. We do not find these residuals in the validation tests so they must be due to systematic differences between our synthetic training set and the real observations, such as observation artifacts, differences in the PSF or an inaccurate estimate of the stray light used in the synthesis. A detailed analysis of these residuals is beyond the scope of this paper but for our purposes here, this simple renormalization (the same for all observations) resolves the issue. The normalization factors for $T_{0}$,…,$T_{4}$ are 0.90,1.26,1.29,1.44 and 1.60, respectively. The growing trend of these factors indicates that the ANN produces models that are, on average, steeper than those obtained with NICOLE. The synthesis tests presented below demonstrate that the model atmospheres obtained in this manner produce spectral profiles very similar to the observations. For the magnetic field (see below), this calibration yields a factor of 0.7 in all three components. We incorporate this normalization factor in all subsequent inversions. A comparison of the maps produced by the ANN and those from NICOLE (the 2015 version) is presented in Figs 1 and 2. The similarity between the spatial structures in the images obtained with both techniques is remarkable. The NICOLE inversions are much more noisy, especially in the higher layers. ANNs are known to have good noise filtering properties. In this case, most of the noise in the NICOLE data is ”inversion noise” produced by the specific $\chi^{2}$ fitting procedure that seeks the best fit to the entire line profile. The upper layers are probed only by the core of the spectral lines. Since the core occupies very few pixels in the spectral profile, there is very little information about those upper layers. For very similar profiles, the $\chi^{2}$ minimization might reach slightly different solution where the core is fitted with more or less accuracy, perhaps compensating it with a better fit to other spectral regions. The end result is a pixel-to-pixel variation that becomes more important in those layers where the profile is less sensitive. This problem could be mitigated by fine-tuning the weights, giving more weight to the pixels that carry the relevant information. However, different layers would require a different optimization. The ANN, on the other hand, ”learns” what are the optimal spectral points that it needs to focus on for each layer. There is a direct, deterministic mapping between the observations and the inversion result. For that reason the ANN maps (right column in the figures) look cleaner. We can even see some of the residual defects in the data reduction that are still present in the observations, as they propagate directly into the results. The similarity between both sets of images confirms that NICOLE and the ANN are giving consistent results. This test should not be viewed as NICOLE giving the ”correct” answer and our ANN being an approximation. Both techniques are approximations and the difference between them is the sum of their respective errors. The accuracy of the ANN in recovering the magnetic field is not relevant for the purposes of this paper. Nevertheless, we show here similar comparisons for the sake of completeness. We processed two active region maps observed with Hinode/SP (more details in section 4.2 below). These maps are 384$\times$384 (map1) and 871$\times$512 (map2) spatial pixels. The ANN inversions took 4 and 11 seconds, respectively. Standard inversions with the MERLIN code are available for these maps. Figures 4 and 5 show the comparison of the magnetic flux inferred by MERLIN and our ANN. For consistency with the training set, the 180-degree ambiguity is resolved by choosing the solution that has a positive component along the $x$ axis. ### 3.1 Reconstruction fits A common problem with ANN-based inversions is that they are not based on fitting the observations, unlike $\chi^{2}$ fitting methods. The quality of a fit is usually a good indicator to assess the validity of the results. Our approach suffers from this limitation too but it does provide enough information to reconstruct a model atmosphere and, from there, synthesize spectral profiles that can then be compared to the observations. It does not provide the same information as a fit because the reconstruction of the atmosphere implies additional approximations. Nevertheless, it is still useful information. We take the parameters from the ANN inversion and compute model atmospheres by interpolating them in optical depth. For the temperature stratification, we perform a cubic interpolation of the five temperatures between $\log\tau_{500}$ $=0$ and $-4$. Above $\log\tau_{500}$$=-4$ we impose that the stratification becomes flat. In the deeper layers below $\log\tau_{500}$ $=0$, the temperature gradient usually becomes steeper. After some experimentation, we concluded that a gradient that is 30% larger than between $\log\tau_{500}$ $=0$ and $-1$ works best in reproducing the observations. Hydrostatic equilibrium is imposed and the plasma equation of state is solved numerically to determine gas and electron densities, ionization stages and relevant molecules. The magnetic field and bulk Doppler velocities are taken as constant with height from the ANN inversion. We computed the synthetic profiles from the models reconstructed from the ANN outputs, obtaining the results shown in Fig 6. These ”reconstruction fits” support the notion that the temperature stratification retrieved by the ANN is consistent with the observations. The first panel shows the average profiles over the entire region. The other three are selected representative samples of profiles having a value of $\chi^{2}$ equal to the median over the region, and the median plus/minus a standard deviation of all $\chi^{2}$ values. Figure 6: Upper left: Average observed and synthetic profiles in the region. Upper right: Sample profiles where the $\chi^{2}$ is equal to the median in the region. Lower left: Profiles representative of a good reconstructed fit, where the $\chi^{2}$ is equal to the median minus one standard deviation. Lower right: Profiles representative of a poor reconstructed fit, where the $\chi^{2}$ is equal to the median plus one standard deviation. In all cases, the profile in blue is the observation and orange is the synthesis from the reconstructed model atmosphere (reconstructed fit). ## 4 Results We employed the ANN-based inversions described in the previous sections to explore the thermal stratification of the solar photosphere. The maps discussed above are in agreement with previous works in showing a rich thermal structure, rapidly changing with height. In this paper, we compare the spatial distribution to what is observed in the Ca II or G-Band filtergrams. ### 4.1 Quiet Sun We start by considering the quiet Sun map inverted in the tests of section 3. The maps are 200$\times$200 pixels but the field of view is not exactly square because the slit stepping, which establishes the sampling in the $x$-direction, does not necessarily match the pixel size. In this case the sampling reported in the file headers is 0.15$\times$0.16 arc-seconds per pixel. The first recognizable pattern that stands out is the similarity of the mid- photosphere temperature map to the Doppler velocity distribution. This is the well known reversed granulation effect, a natural consequence of convective motions. The tightest correlation in our dataset, shown in Figure 7, is between temperature at $\log\tau_{500}$ $=-2$ and the Doppler velocity ($v_{z}$), with a Pearson’s correlation coefficient of 0.45 (recall that, as noted above, the velocities retrieved by our procedure are at the base of the photosphere). Reversed granulation is characterized also by an anticorrelation with the $\log\tau_{500}$ $=0$ map, which in our data is of $-0.37$. Figure 7: Scatter plot of the mid-photospheric temperatures and the Doppler velocities retrieved by the ANN. The scatter plot exhibits some vertical features. These are the result of the ANN assigning nearly the same value of $v_{z}$ to many different profiles instead of smearing them over the uncertainty range of that parameter. In least-squares inversions, the solutions for similar profiles tend to be spread over the error bar for that parameter because each inversion has followed a different path on the $\chi^{2}$ hypersurface. However, an ANN might end up assigning a specific value for a parameter (or a narrow range of values) as a ”sticky solution” for a range of input profiles. This means that the resulting maps will usually be less noisy but the noise level should not be considered an indication of the uncertainties. Figure 8: Comparison of temperatures at different heights, as retrieved by the ANN inversion, to Ca II filtergram (bottom right panel) in a quiet Sun region. The temperature maps retrieved in the ANN inversions show a different network structure at each atmospheric height. Bright photospheric networks have been observed in the wings of the Ca II lines and in the G-Band, which are also accessible to Hinode’s narrow band instrument (Hinode/NB). It is then of interest to investigate whether these structures are related to those, both in the quiet Sun and active regions. Figure 8 shows a comparison of the temperature maps to a simultaneous Ca II image from Hinode/NB. Alignment of SP and NB observations is not straightforward. We made use of the pointing information stored in the data headers to bring both datasets to a common reference frame. It should be noted, however, that the alignment is only accurate to a few arc-seconds 222https://hesperia.gsfc.nasa.gov/ssw/hinode/sot/doc/guide/SAGv3.3.pdf. The figure shows no obvious similarities between the Ca II intensity and the photospheric temperatures retrieved at any of the heights. The very weak magnetic fields in this quiet region do not appear to be correlated to the Ca II filtergram, either. ### 4.2 Active regions In this section we analyze two large-field active-region maps for which there exists simultaneous G-Band and Ca II imaging. The datasets were acquired on January 11, 2010 around 18:30 UT (map1) and January 22, 2012 around 06:30 UT (map2). The spatial sampling is coarser than in the quiet Sun observations (0.30 and 0.32 arc-seconds in the $x$ and $y$ directions, respectively) to encompass a larger field of view. Map1 consists of 384$\times$384 spatial pixels, while map2 is 871$\times$512\. The full maps are shown in Figs 9 to 10, with the various panels displaying temperatures at various heights, along with the narrow-band images (the magnetic field was already introduced in Figs. 4 and 5). Figure 9: ANN inversion of map1 (active region) at different heights (except $\log\tau_{500}$= 0) alongside G-band and Ca II filtergrams. The arrow points in the direction to the nearest solar limb. Figure 10: ANN inversion of map2 (active region) at different heights (except $\log\tau_{500}$= 0) alongside G-band and Ca II filtergrams. The arrow points in the direction to the nearest solar limb. The data shows a fine network of hot pixels that roughly follows, in the mid- photosphere, the magnetic field distribution ($\log\tau_{500}$ $=-1$ and $-2$). Higher up the structure is more patchy and does not follow the magnetic field maps. Each layer exhibits a different structure and, more importantly, they also differ from both the G-Band and Ca II H images. We discuss these differences below. A reversal of hot and cool areas between the middle and the upper photosphere is also apparent. For instance, the lower left corner of both maps (Figs. 9 and 10) is cool at $\log\tau_{500}$ $=-1$ but hot at $\log\tau_{500}$ $=-4$ (upper left and upper right panels in both figures). The same anticorrelation is apparent in the upper left and lower right corners of map2 (Fig. 10). In fact, most of the region left of $x=200$ arc-sec in Fig. 10 has a reversed appearance. The hot network at $\log\tau_{500}$ $=-1$ (upper left panel) is seen as a dark shadow at $\log\tau_{500}$ $=-4$ (upper right panel). The same is true about the area left of $x=-240$ in map1 (Fig. 9). We quantified this by selecting only the pixels that are hot in either layer ($T>$5400 at $\log\tau_{500}$ $=-1$ or $T>$5000 at $\log\tau_{500}$ $=-4$) and computing the Pearson’s correlation coefficient. The values obtained are $-0.77$ in map1 and $-0.78$ in map2. The temperature maps do not match the narrow-band images. There is some similarity between the temperature at $\log\tau_{500}$ $=-1$ and the G-Band image in the overall distribution of the hot network. However, a closer look shows important differences (see discussion of Fig. 11 below). The comparison with the Ca II H images is even more puzzling. The Ca emission follows the pattern of the hot pixels in the mid-photosphere at $\log\tau_{500}$ $=-1$ instead of the upper photosphere, as one would have expected (recall that, as discussed above, the distribution of hot pixels at $\log\tau_{500}$ $=-4$ is anticorrelated with that at $\log\tau_{500}$ $=-1$). However, the Ca images show small-scale filamentary structure in the network, as opposed to the chains of dots that appear in the temperature maps. The appearance of filaments would suggest that we are seeing higher layers but the brightness distribution follows that of the mid-photosphere. We speculate that the most plausible explanation is that the this spectral band has contributions to its response function from both the low photosphere and the (low) chromosphere. Detailed radiative transfer modeling would be necessary to confirm this point but it would require some knowledge of the chromospheric conditions, which is not available from these data. Figure 11 shows a zoom on two regions containing several pores in both maps. The magnification is different in each figure because the area with pores is larger in map2. Even though the G-Band bright points extend over the same area as the hot network at $\log\tau_{500}$ $=-1$, they do not exhibit the same features when seen at high resolution. Figure 11: Enlargement of of the areas with pores in the active regions map1 (upper panels) and map2 (lower panels). The left panels show ANN temperature reconstructions at $\log\tau_{500}$=-1 and the right panels, the respective G-band filtergrams. Notice the bright arcs around the limb-side (indicated by the orange arrow) of the pores in the left panels. A very remarkable feature in these images is the presence of a bright arc around the edge of pores, tracing the limb side (the arrows indicate the direction to the closest limb). These arcs are visible around virtually every pore in both datasets. They have a width of one or two pixels, suggesting that they are not fully resolved in the observations, and their temperature is always between 5,600 and 5,700 K. By contrast, the pores have temperatures mostly of 4,600 to 4,800 K but in some cases, particularly the larger ones in map2, they may go down to 3,600 K, such as in the feature at coordinates (165,390) of map2. The bright arcs are probably the pores’ ”hot walls”. The idea of a hot wall seen in perspective was introduced in early fluxtube models to explain the center-to-limb variation of faculae and G-band points (Spruit 1976; Knoelker & Schuessler 1988; Topka et al. 1997). Spruit’s original work considered unresolved fluxtubes and small pores of up to 1,000 km. The pores in our observations are significantly larger, starting from roughly 3,500 km, but there is no reason why the same effect should not take place in these. ## 5 Conclusions ANN-based inversions are enabling the analysis of large spectroscopic (and spectropolarimetric) datasets. One such application is presented in this paper. The training strategy appears to be sufficiently robust for application to real observations in various situations. It is puzzling to find such a clear anticorrelation in the location of hot points in the middle and upper photosphere. This is counterintuitive and warrants further work to confirm it, since it appears to challenge the generally accepted idea that small magnetic elements act as channels to propagate energy into the upper atmosphere (e.g., Jefferies et al. 2006; Rajaguru et al. 2019). One possibility is that the energy dissipation and associated heating might occur at higher layers than we observe here. That would explain the presence of hot points at intermediate heights that do not exhibit a temperature enhancement in the upper photosphere. However, this would not explain the patches with hot points in the upper layers that appear as quiet lower down. In our ANN approach, each pixel is inverted independently of the rest. Therefore, the spatial distributions obtained cannot be artifacts of the procedure, they must be present in the data somehow. A possible mundane explanation for the pixels that are hot in middle layers and quiet at the top could be that the ANN is not properly trained for such situations and the closest models in the training set that reproduce the lower and mi layers are quiet in the upper layers. However, that would not explain the opposite scenario in the anticorrelation, i.e. the patches with quiet lower and middle photosphere having enhanced temperature in the upper layers. Another important result presented in this paper is the first observation of the ”hot wall” effect, which has been a model prediction since the 1970 (Spruit 1976) and explains the bright appearance of small magnetic elements. The original theoretical models considered smaller pores of up to 1,000 km but we have detected it here in structures of at least 3,500 km. Hot walls are believed to be responsible for the brightness of faculae and small magnetic flux elements (e.g., Topka et al. 1997). This view, which is now the community consensus, is strongly reinforced by our data. Our results open the possibility of a future application to chromospheric lines. The chromosphere is much more complicated to simulate due to NLTE radiative transfer, much faster and vigorous dynamics, faster wave phase velocities, etc. As a result, MHD numerical models are not yet sufficiently realistic that they can be used to match the observations and this limits the ability to train an ANN with chromospheric simulated profiles. However, our training strategy does not require a full numerical model. In principle one could apply a similar training using semiempirical 1D models with random perturbations to produce a large number of NLTE profiles. The computational effort involved in the database generation would be significantly higher than here but still feasible. An area of improvement that we have found with this technique is the ”sticky solution”, where the ANN returns basically the same values within a broader uncertainty range, creating the vertical features seen in the scatter plot of Fig 7. Finally, we would like to mention a negative result. 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# ’Controlling by Showing: i-Mimic’: A Video-based Method to Control Robotic Arms Debarati B. Chakraborty<EMAIL_ADDRESS>Mukesh Sharma <EMAIL_ADDRESS>Bhaskar Vijay<EMAIL_ADDRESS>Dept. of Computer Science and Engineering, Indian Institute of Technology, Jodhpur, India Department of Mechanical Engineering, Indian Institute of Technology, Jodhpur, India ###### Abstract A novel concept of vision-based intelligent control of robotic arms is developed here in this work. This work enables the controlling of robotic arms motion only with visual inputs, that is, controlling by showing the videos of correct movements. This work can broadly be sub-divided into two segments. The first part of this work is to develop an unsupervised vision-based method to control robotic arm in 2-D plane, and the second one is with deep CNN in the same task in 3-D plane. The first method is unsupervised, where our aim is to perform mimicking of human arm motion in real-time by a manipulator. Mimicking, here involves a series of steps, namely, tracking the motion of the arm in videos, estimating motion parameters, and replicating the motion parameters in the robot. We developed a network, namely the vision-to-motion optical network (DON), where the input should be a video stream containing hand movements of human, the the output would be out the velocity and torque information of the hand movements shown in the videos. The output information of the DON is then fed to the robotic arm by enabling it to generate motion according to the real hand videos. The method has been tested with both live- stream video feed as well as on recorded video obtained from a monocular camera. Besides, the DON also enables a method to behave intelligently by predicting the trajectory of human hand even if the hand gets occluded for some time with varying degree of occlusion. This is why the mimicry of the arm incorporates some intelligence to it and becomes intelligent mimic (i- mimic). Alongside the unsupervised method another method has also been developed deploying the deep neural network technique with CNN (Convolutional Neural Network) to perform the mimicking, where labelled datasets are used for training. The same dataset, as used in the unsupervised DON-based method, is used in the deep CNN method, after manual annotations. Both the proposed methods are validated with off-line as well as with on-line video datasets in real-time, enabling the robotic arm in ’i- mimic’ and thereby showing the effectiveness of the proposed method. The entire methodology is validated with real-time 1-link and simulated n-link manipulators alongwith suitable comparisons. ###### keywords: Visio-based robot control, video processing , deep network , convolutional neural network , robotic arm control ## 1 Introduction Robotic arms are used to perform mechanical tasks in industries over decades. It was mainly used for performing repetitive tasks in the industries to cut down the labor cost [1, 2]. Normally, robotic arms are quite complex with five or more degree of freedom as it aims to perform human tasks. Recently, the application of robotic arms in conducting the domestic work has drawn attention. Controlling of these robotic arms to perform different tasks is still a major issue to be addressed. Here in this work we have defined a new concept of controlling robotic arms only with visual information, that is, the motion of different parts of the robotic arms could be controlled by providing according human hand movement videos as the input to those arms. The entire method could be subdivided into two parts. In the first part of the this work we aim to find out a simple solution for unsupervised controlling the robotic arm only with visual information. Here we aim to deal with the issues of i) unavailability of sufficient training datasets, ii) domain adaptation and iii) economic cost. On the way to search for a solution of the two initial issues we concluded that the teaching/ training part should be removed. But how could it be controlled then? ’Mimic’ is the solution that stroked in our mind. The controlling could only be achieved by showing the arm the desired movement and making it enabled to follow it. Visual mimic-based controlling of 1-link and n-linked robotic arms is the primary contribution of this work. The mechanism of this set-up is quite simple and the 1-link manipulator is developed only by the authors. Either the recorded or real-time video could be shown to the arm to achieve the control. It should be noted that the real-time testing of this ’mimic’ with robotic arm is in a very primary level where the arm is a simple 1-link robotic manipulator with a degree of freedom of 120 degrees. The rest of the tests are conducted in the form of simulation, where the simulated arm is able to mimic the motion of a single joint (solder or elbow) or real hand. Another contribution in this part of the work is development of the vision-to- motion optical network (DON) to process the optical flow information of the input video and to convert it into the physical force, to be fed the robotic arm. The proposed DON is different from the existing deep networks in the following manners: i) it does not require any labeled data set or manual intervention, ii) it does neither require background estimation or a large number of input frames for training, iii) the functioning of the intermediate layers are simple which enables computational gain iv) all the intermediate layers are not active simultaneously; some layer gets activated depending on the values of the outputs of its previous layer v) the single network can perform both estimation and prediction and vi) it produces torque and angular velocity as the output. In the second part of the work we focus on implementation rather than definition. We implemented deep neural network with CNN and Refiner Network for the purpose of i-mimic. To deal with unavailability of large set of labelled data, we used a CNN network whose output are further fed into the refinement network that smoothens the final output and enables to interpolate to a larger range. The deep CNN is proven to be less effective with 2-D n-link manipulators, but, it performs better in 3-D plane with n-link manipulators. The rest of the article is organised as follows. Sec. 2 presents the background research, Sec. 3 describes the layer-wise formulation of vision-to- motion optical network (DON). The architecture of CNN and Refinement Network, customized loss function, dataset, training are described in Sec. 4. The experimental set-ups with four different variations in experimental studies are described in Sec. 5. The real-time experimental results tested under different scenarios like without occlusion, with low occlusion, with high occlusion, and with multiple joints are given in Sec. 6 along with parameter section and comparative study. The overall conclusion of this work with its future scope are discussed in Sec. 7. ## 2 Background Research Vision-based robotics to serve domestic purpose has drawn the attention of researchers in several areas. Most of the approaches developed so far for this purpose implied training the system through labeled data, i.e., with supervised or semi-supervised learning. Automated driving [3], grasping [4] and block stacking [5] are among few the applications where this kind of learning were used. But gathering adequate amount of labeled data for training is a challenging issue for this type of approaches. Semi-supervised learning or reinforcement learning has appeared to be the substitute of supervised learning in recent literature’s where the training is carried out with less amount of labeled data or weakly labeled data. Rusu _et al._ [6] used learning with progressive network for Jaco robot gripping to have a faster algorithm with less amount of training data. KUKA IIWA robot grasping with deep network and domain adaptation was developed by Bousmalis _et al._ [7]. A method of training with weakly labeled images with adaptation from real world to simulation using a PR2 robot was proposed by Tzeng _et al._ [8]. Zuo _et al._ [9] came up with a solution of semi supervised method of 3D pose estimation where the training was carried out in a virtual environment. Its real world implementation was done after domain adaptation. Domain adaptation is another challenge while semi-supervised/ reinforcement learning is carried out. There are many rich works carried out so far to deal with this problem. Domain adaptation with back-propagation by inducing an ’inverse- gradient layer’ to the deep network was formulated by Ganin and Lempitsky [10]. In another work Ganin _et al._ [11] came up with a solution of carrying out the training and testing of the network with the features that are non- discriminative and domain invariant for training and test data. Bousmalis _et al._ [12] came up with another solution of identifying the unique feature of each domain to extract out the common features in the domains. They have recently developed another way of domain adaptation with simultaneous simulation [7]. Sing _et al._ [13] demonstrated that passively collected data can be paired with interaction data to learn visual representations for end- to-end control policies that generalize substantially better to unseen environments. However, less amount of labeled data or synthetically labeled data are always required in all of the aforementioned approaches. Economic cost of a robotic arm controller is another major issue to be dealt with to make the robotic arms be implementable for domestic purpose. The controlling of the robotic arms are normally carried out with multiple sensors which make it more costly. The robotic arms like PR2 [8], Jaco [6] or KUKA IIWA [7] costs around USD 20,000/- to USD 50,000/-. A pocket friendly robotic arm is developed recently [9] but but it still have the issues of synthetic labeled data and domain adaptation. ## 3 DON: Vision-to-Motion Optical Network Here we developed a network based on the information of optical flow from frame-to- frame of a the input hand movement video sequence. The major challenges that are to be addressed in the task of ’i-mimic’ are: unavailability of sufficient number of labelled data, computation time and the lag between input-video to robotic-arm gestures. The proposed deep flow network is able to minimize all these parameters simultaneously. First of all, no labelled data is required here and the process is fully automatic. The computation complexity of this method is quite low and there the lag is as less as around ten frames here. The vision-to-motion optical network is a layer-wise network with multiple layers between the input and output layer. Different feature of the optical flow information is process in different layers of this network. The layer- wise architecture is shown in Fig. 1. It can be noticed from the diagram that the videos are fed in the input layer of the network, whereas we get velocity and torque values that generates the physical motion at the output layer. That is why it is named as a ’vision-to-motion optical network network’. The output of the previous layer is the input to the next layer. All the layers of the network may not be active at a time. Rather, the activation of some layer of the network is dependent on the outputs of its previous layer. The layer-wise working principles of this network are described in details in the following sections. Figure 1: Architecture of Deep Flow Network ### 3.1 Layer 1: Input Layer The video sequence is the input that is fed to the network. But it is not the entire sequence that is given as the input at a time since a on-line processing is going on here with the input video frames. As we have already stated that the proposed method is unsupervised, therefore the output is to be produced only by automated processing of input data. Here, the frame that gets generated on the current instant, say, at the instant $t$ is fed to the network along-with $N$-number of previous frames that gets generated earlier to the current frame. Let the current frame be denoted as $f_{t}$here. The frames generated in the earlier instances be denoted by $f_{t-1},f_{t-2},...,f_{t-N}$. Therefore, the input layer contains the frames: $f_{t},f_{t-1},...,f_{t-N}$. ### 3.2 Layer 2: Frame Difference Computation Layer The network is supposed to deal with the optical flow information. In case of the video sequences that we are dealing with is captured by static cameras. Therefore the changed information from frame to frame reflects the optical flow of the sequence. Here two types of differences are computed here in this layer. That is why two different colored of nodes (DO1 and DO2) are shown there in Fig. 1. There are total $N+N=2N$ number of nodes present there in layer 2. The difference operation carried out in DO1 the difference between consecutive frames ($\delta 1$) given in Eqn (1). The difference between the current to all its previous frames ($\delta 2$) is carried out in Eqn (2). $\delta 1_{p}=\|f_{t-p}-f_{t-(p-1)}\|:p=0,...,N-1$ (1) $\delta 2_{p}=\|f_{t}-f_{t-p}\|:p=1,...,N$ (2) Therefore $N$ number of binarized $\delta 1$ and $\delta 2$ frames, i.e., $2N$ number of difference frames in total are the output of this layer. $\delta 1_{p}$ are the binarized outputs from the nodes of type DO1, whereas, $\delta 2_{p}$ are the output from DO2 type of nodes. ### 3.3 Layer 3: Object Identification Layer The third layer of this network is developed to find out the locations and the shape of the moving hand in all the $N$-number of previous frames. As it can be observed from Fig. 1 that this layer contains $N$ number of nodes, labeled as $ObjectFinderF1,...,ObjectFinderFN$. The input fed to a certain node $ObjectFinderFp$ are: ${\delta 2_{p}}:p=1,...,N$ and $\delta 2_{p}$. That is, all the DO2- difference frames and only $p^{th}$ DO1 difference frame are the input to the said node of layer 3. Let the location of the moving object segment in the $p^{th}$-frame be represented by $l_{p}$. The operation that is carried out in each node of the third layer is given by the Eqn. (3). $l_{p}=(\cup\ _{p=1}^{N}\delta 2_{p})\cap\delta 1_{p}$ (3) Please note that the union of $\delta 2_{p}\forall p=1,...,N$ is taken here to have the entire moving obeject region as a subset of that union and intersection of it to that of $\delta 1_{p}$ is carried out to extract out the obvious moving region in the $p^{th}$-frame. The pixels those belong to the set $l_{p}$ are in the region that definitely belong to the moving object in the $p^{th}$ frame. For the sake of simplicity, here in this work we consider only the skeleton and the locations of the corner pixels of the $l_{p}$ (moving hand) to be the output from each node of this layer as we need to find out the angular velocity and torque from the hand movement video. ### 3.4 Layer 4: Estimation Layer This layer contains two nodes and the operations and functioning of these two nodes are different from each other. The location of the object in $N$-number of frames are the input to this layer. Two types of estimations are performed simultaneously in this layer with the two nodes. The path estimation node gives the probable trajectory of the moving object as the output whereas the trust factor estimator node computes the reliability of the estimated path. The output of trust factor estimator node determines the activation of the next layer, i.e., prediction layer. The working principles of the two nodes in the forth layer are described below. #### 3.4.1 Path Estimator As discussed before the prediction of probable trajectory of the object is carried out here. This is done by computing the optical velocity and acceleration of the moving object from frame-to-frame displacement. Let $\varsigma_{p}$ be the location $l_{p}$ (see Eqn. (3)) in the $p^{th}$-frame. Then the velocity ($v_{p}$) and acceleration ($a_{p}$) of that object are computed according to Eqn. (4). The velocity and acceleration values for all the $N$ frames are stored in the sets $V$ and $A$ respectively. $\displaystyle{V}$ $\displaystyle=\\{v_{p}:v_{p}=\varsigma_{p}-\varsigma_{p-1}\forall p=1,...,N\\}$ (4) $\displaystyle A$ $\displaystyle=\\{a_{p}:a_{p}=v_{p}-v_{p-1}\forall p=2,...,N\\}$ (5) Please note that signed difference between the locations and velocity are taken while computing $v_{p}$ and $a_{p}$ in Eqn. (4). It is known from Sec. 3.3 that the input $\varsigma_{p}$ could be a scalar or vector component based on the type of object representation. But the two components $v_{p}$ and $a_{p}$ should always be a vector since these components contain both magnitude and signs. Consideration of those signs helps in the incorporation of the information of the direction and the change in the direction of the moving hand. Here the robotic arm with revolute joint is supposed to mimic the movement of the arm shown in real time or recorded video. Therefore, the movement of the arm is always supposed to be circular in nature with respect to any joint (e.g. elbow) with maximum 180 degree of freedom. This phenomenon is kept in mind and the determination of the radius w.r.t the angular motion is computed by measuring the length of the skeleton of the arm. Let the skeleton of the moving part of the arm be of length $r$. The angular velocity ($\omega_{p}$) and torque ($\theta_{p}$) are then computed as: $\omega_{p}=\frac{v_{p}}{r}$ and $\theta_{p}=I\alpha_{p}$ where $I$ is mass moment of inertia of the manipulator arm and $\alpha_{p}$ is the angular acceleration computed as: $\alpha_{p}=\frac{a_{p}}{r}$ For any given one-link manipulator the algorithm computes $\alpha_{p}$ and having $I$ of the manipulator one can compute the torque required to be applied to at the joint. #### 3.4.2 Trust Factor Computation The working principle of this particular node is different from any other nodes present there in a network. It takes input from the previous layer but does not transmit its output to the next layer. Instead the output from this layer determines which layer should be the fifth layer of this network. That is the which path should be followed by the output information from the path estimator node is decided with the output of this node. Since, only motion of the moving hand of a static person is considered here, it can be assumed that the size of the moving object will remain almost the same throughout the sequence. This assumption is applied during formulation of the trust factor. Let there be $M_{p}$ be the region of $l_{p}$ (see Eqn. (3)) in the $p^{th}$-frame. Let, the set $\\{S\\}$ the regions of the object in all the $N$ frames and the set $\\{S_{d}\\}$ contains the values of change in regions. Those are computed according to Eqn. (3.4.2). $\displaystyle{S}$ $\displaystyle=\\{M_{p}:p=1,...,N\\}$ $\displaystyle S_{d}$ $\displaystyle=\\{c_{p}:c_{p}=\|M_{1}-M_{p}\|\forall p=2,...,N\\}$ (6) The trust factor ($\eta$) is computed as: $\eta=1-\frac{max(S_{d})}{max(S)}$ (7) In Eqn. (7) max(.) represents the element with maximum magnitude present in a set. Physically, the effectiveness of measuring the $\eta$ is in determining the amount of occlusion took place over the moving object. If huge amount of occlusion is present there for some frames, then the estimation with those frame may lead to a wrong trajectory. Therefore, prediction should be carried out from the previous set of information and ignoring the wrong (occluded) visual information. That is why the activation of the prediction layer is necessary in this scenario. The path leading to prediction layer gets only activated if the value of $\eta$ is low. ### 3.5 Layer 5: Prediction Layer There is only one node in this layer. But, the input fed to this layer is not only from the previous layer, but the output of layer 4 of the previous execution of the network is also an input here. Please note that this layer can not be active in the first execution of the network, but from the second execution onward it could get activated any time. Here the velocity and acceleration values from the frames without occlusion, or with minimal occlusion are considered. There inputs that are provided to this node are: i) the velocity and acceleration information (sets $V$ and $A$) from the previous execution, ii) the velocity and acceleration information ($V$ and $A$ from Eqn. (4)) from the previous layer and iii) Object regions and change in the regions ($S$ and $S_{d}$ from Eqn. (3.4.2)). Let the velocity and acceleration from the previous be denoted here as $\tilde{V}$ and $\tilde{A}$. We only consider the information of the frames for with $c_{p}<0.05Xmax(S)$ ($C_{p}$ is as defined in Eqn. (3.4.2)). That is, frames maximum with $5\%$ change in object size will be taken into account. Let $k$ number of frames out of the $N$ frames failed to satisfy the criterion. Then only $N-k$ elements from the sets $V$ and $A$ will be taken by merging it with the sets $\tilde{V}$ and $\tilde{A}$ respectively. Therefore, the new sets will be $V^{k}=\\{\tilde{V}\|V(1:N-k)\\}$ and $A^{k}=\\{\tilde{A}\|A(1:N-k)\\}$ with $N+N-k=2N-k$ number of elements in each set. Now we need to predict the information from the $(N-k+1)^{th}$ frame onward. As it is known, the consecutive difference between the elements of $V^{k}$ forms $A^{k}$, i.e., $A^{k}$ could be said the first order derivative of $V^{k}$. We can similarly compute the second order derivative of $V^{k}$ or the first order derivative of $A^{k}$ and represent it by $A^{k^{\prime}}$. Now, the $(2N-k+1)^{th}$ element of the sets $V^{k}$ and $A^{k}$ will be approximated as: $\displaystyle v_{2N-k+1}=v_{2N-k}+a_{2N-k-1}$ $\displaystyle a_{2N-k+1}=a_{2N-k}+a^{\prime}_{2N-k-1}$ (8) In Eqn. 3.5 the symbols $V_{p}$, $a_{p}$ and $a^{\prime}_{p}$ represents the $p^{th}$ element of the sets $V^{k}$, $A^{k}$ and $A^{k^{\prime}}$ respectively. The element will get inserted to the sets $V^{k}$ and $A^{k}$ as the $(2N-k+1)^{th}$ elements of them. The process will be repeated and the next element will be approximated. The process will continue until the set is going to have $2N$ number of elements. Once it is done, the last $N$ elements of $V^{k}$ and $A^{k}$ will be stored in the sets $V$ and $A$ respectively and will be given as the output to the output layer. The experimentation those are carried out with this proposed DON are described in the following section. Please note that, one additional layer, namely object regression layer to this network is introduced while working with multiple joints. It is shown in Fig. 9. ## 4 DNN:Deep Neural Network In our second approach, we designed Convolutional Neural Network(CNN) for which the input corresponds to the image(frames of video stream) and labels correspond to coordinates$(x,y)$ of joints(shoulder, elbow and wrist joints). We have used two networks for our purpose to get the joints coordinates from which further joint angles, joint velocities and and joint torque can be obtained in terms of pixel coordinates which are mapped to real value using the mapping function or mapping factor. In [14] after using DNN based regression a DNN based refiner network was added, which takes cropped image around prediction as input to improve the prediction but we used a different approach by mapping a simple neural network to refine the predictions of the DNN based regression network. Our approach reduces the training time of the network with similar accuracy for our dataset. ### 4.1 Convolutional Neural Network(CNN) The architecture of CNN is shown in Fig. 2. The input to this network is the images(from a stream of video feed) and labels as the pixel coordinates$(x,y)$ of the shoulder joint, elbow joint and wrist joint. The network has six hidden layers wherein there are 2 Convolutional layers followed by max pooling and flatten. Activation functions for all the layers other than the last layer are ReLu activation. Final layer has a Linear Activation function. This network uses Mean Square Error(MSE) Loss function. Figure 2: CNN Network ### 4.2 Refiner Network The architecture of Refiner Network is shown in Fig. 3. This network is a simple neural network with input corresponding to the output of CNN Network and labels as the pixel coordinates of the shoulder joint, elbow joint and wrist joint respectively. Activation function for the last layer is Linear and for rest is ReLu Activation. This network uses customized loss function that includes MSE and simple error in link length. Figure 3: Refiner Network ### 4.3 Customized Loss Function For the error in coordinates of joint Mean Square Error(MSE) has been used and another error for link length has also been taken into consideration. Let $d$ denote error in link length and $p$ denote the MSE in position. For real joint coordinates $(x_{s},y_{s})$, $(x_{e},y_{e})$, $(x_{w},y_{w})$, and predicted coordinates $(x_{sp},y_{sp})$, $(x_{ep},y_{ep})$, $(x_{wp},y_{wp})$ where subscripts $s$, $e$ and $p$ represents shoulder, elbow and wrist joints respectively and subscript $s$, $e$, $w$ followed by $p$ represents corresponding predicted coordinates respectively. The respective error are computed following the Eqns. (9)-(11). $d=(\sqrt{(x_{s}-x_{e})^{2}-(y_{s}-y_{e})^{2}}-\sqrt{(x_{sp}-x_{ep})^{2}-(y_{sp}-y_{ep})^{2}})^{2}+\\\ (\sqrt{(x_{e}-x_{w})^{2}-(y_{e}-y_{w})^{2}}-\sqrt{(x_{ep}-x_{wp})^{2}-(y_{ep}-y_{wp})^{2}})^{2}$ (9) $p=(x_{s}-x_{sp})^{2}+(y_{s}-y_{sp})^{2}+(x_{e}-x_{ep})^{2}+(y_{e}-y_{ep})^{2}+(x_{w}-x_{wp})^{2}+(y_{w}-y_{wp})^{2}$ (10) $loss=d+p$ (11) ### 4.4 Dataset For the purpose of training the deep CNN, we created our own dataset of 300 images and manually labelled the pixel coordinates of the joints for each of the images. From the total dataset, 224 images were used for training and rest were used for testing. ### 4.5 Training Our second approach requires training for which training and validation dataset has been used as specified under sub-heading 4.4. For the optimization purpose Adam optimizer has been used, with batch size of 14 images. The training has been done on the sample dataset for the case of one-link and two- link cases( forearm and arm) both on CNN network and Refinement network and the combined network. CNN network has been trained on 10 epoch and Combined Network on 100 epoch. ## 5 Experiment The proposed method is tested with both real-time and recorded video sequences. But the processing of both type of the sequences are carried out in real-time since the ’mimic’ of robotic arm is a real-time task. The experiments adressing four different challenges viz., i)presence of different level of background noise, ii) variation in distance between arm and camera, iii) variation in speed of hand movement, and iv) different number of links are performed to test the proposed methodology and setups are accordingly made. The experiments has been performed at frame rate of 30 fps and video resolution of 240x320 pixels. ### 5.1 Experiment-1 In the first experiment, the recorded video of hand motion is given as input to the network and the motion of the arm is mimicked by one-link manipulator in virtual environment of PyBullet. This experiment requires preparation of virtual environment and no physical setup is required. For the recorded video even noisy data was used to test the method. The recorded data used for testing is of [15] (lossy compressed AVI format devel-1). Additional setups were not made for getting the recorded video. Videos used are available here as ( Exp-1 Video-1, Exp-1 Video-2) ### 5.2 Experiment-2 In the second experiment, the motion of actual one-link manipulator against stationary background is mimicked in simulation environment. This required preparation of physical setup of the manipulator and camera. The experimental setup consists of a camera(Model: HP HD 4310 H2W19AA) is mounted(fixed) at a height of 32 cm(can be varied) above the manipulator. The manipulator’s link length is 10.5 cm which is made up of paper to reduce the weight of the link mounted on the servo motor(specifications: Model: SM-S2309S, Size: $22.9\times 12.3\times 22.2$mm, Weight: 9.9g, Rotation angle $\equiv 120$ , Micro analog servo, 4 plastic gears + 1 metal gear). An Arduino UNO board has been used as a controller to provide signal to the servo motor for the motion (Fig.2(a)). The videos of experiment is available here as ( Exp-2 Video-1, Exp-2 Video-2) (a) (b) (c) Figure 4: (a) Experimental setup for experiment-2, (b) Experimental setup for experiment-3, (c) Manipulator for experiment-4 ### 5.3 Experiment-3 In the third experiment, the proposed method is used to mimic the forearm motion of a standing person by an actual one-link manipulator. The one-link manipulator used in Experiment-2 has been used here except the positioning has been changed as shown in Fig.2(b). Here, the distance between the forearm and the laptop(HP laptop AU030WM Pavilion) camera is varied. whereas the camera was mounted at fixed distance from the manipulator in earlier setup. Also, the background here is not stationary as noise is introduced while moving forearm other body parts too move slightly. ### 5.4 Experiment-4 In the fourth experiment, we extend our method to n-link planar manipulator(four-link, one-link is fixed). Here in case of human arm, we considered forearm, thumb and index finger having three joints overall. One additional layer to DON is added for the sake of experimentation here. It is shown in Fig. 9. Please note no physical setup is prepared for this experiment and the testing is done in simulation. The manipulator used in Pybullet is shown in Fig.2(c) ### 5.5 Parameter Tuning The actuator of our experimental setup enabled us to test the method using position control. However, velocity control and torque control techniques can also be used with the values obtained from the algorithm with the actuators that enables velocity control and torque control. For the uniformity, we use position control in simulator as well as on actual manipulator. The joint angle computed in optical flow and the actual joint angle remains the same. Same is not the case with angular velocity, angular acceleration, torque and the link length. The term aspect ratio(ratio of value in optical flow and actual value) has been introduced for mapping optical flow value to actual value. These values shall be experimentally determined and is dependent on the experimental setup. Since the algorithm is completely unsupervised, the requirement of labelled data and domain adaptation is not required. The issue of cost is also dealt since the algorithm can easily run on low computation powered devices such as mobile handset, laptops, computers, etc. There is only lag time between the input of the image frame from video and the output signal to the manipulator which is the processing time. (a) (b) Figure 5: (a) Computation time vs No of frames considered, (b) Average area captured vs intensity level This processing time is a function of number of input frames (Fig.3(a)) in Input Layer, filters used, filter sizes, etc. The lag time can be decreased by using optimal number of input frames in Input Layer and appropriate filter with optimal filter size. For the case of experimentation the number of input frames is $10$ with Gaussian filter size $15$x$15$ which has been determined as optimal values through an iterative process. Further the performance of Object Identification Layer is also dependent on intensity of light and background noise(movement of other objects). The algorithm works well for the intensity of light above 50%(determined using experimental setup in Fig.2(a)) shown in Fig.3(b) and small background noise. The algorithm captures the major motion in the video. So, the performance of the algorithm is not affected until the major motion is of the arm. ## 6 Results and Discussions ### 6.1 Results of DON The Experiment-1 performed on hand motion data set could hardly track the arm joint angle due to extremely random and fast hand movement and very large noise due to movement of other body part. This experiment was performed on both the RGB video and depth video. The results are available here RGB videos(Video-1 at 10fps, Video-2 at 30fps) and Depth videos (Video-1 at 10fps, Video-2 at 30fps) The Experiment-2 performed for mimicking actual one-link manipulator motion in simulation result is shown in Fig.4. In addition we used [16] to generate depth images from RGB image and tested our algorithm. In both the cases our result obtained is the same. (a) (b) Figure 6: Plot of (a)Angular velocity vs Time, (b) Error vs Time Figure 7: Visual _i_ -mimic in Real-time: (a) Without occlusion ($0.9<\eta<1$), (b) Low occlusion to background and human hand ($0.7<\eta<0.5$) and (c) High occlusion ($0.4<\eta<0.3$) The Experiment-3 is performed with variation of trust factor ($\eta$ in Eqn. (7)) that is with various degrees of occlusion. Example frames with arm-mimic for three different type of occlusions are shown in Fig.9, where there is no occlusion present there in Fig. 9(a), low amount of occlusion is present there in Fig. 9(b) and the amount of occlusion is quite high in Fig. 9(c). It is also observed that the proposed method works well for moving hand. The videos corresponding to the experimental results are: no occlusion, minor occlusion and major occlusion. Figure 8: Adjusted portion of Architecture of Deep Flow Network In Experiment-4, since there are three rotating link, three-joints angles are to be estimated. To accommodate multiple links, in the architecture of DON, an additional layer is added between Object Identification Layer and Estimation Layer to fit straight lines(Object Regression Layer) on objects as shown in Fig.6. Rest of the network remains the same. The joint angle between forearm and fixed link, thumb and forearm, index-finger and forearm are estimated accordingly. The results obtained after Object Regression Layer is shown in Fig.7. The video expressing the obtained result is here. Figure 9: Snapshot image after Object Regression Layer In addition to, the algorithm has been tested on videos of different resolutions at 30 fps. The average algorithm run time per loop execution for video input of different resolution before giving signal to manipulator for is presented in Table 1. This experiment has been performed on DELL Laptop with 8 GB RAM and Intel core i5, the algorithm run time will vary depending upon the computation power of hardware used for testing. Table 1: Average loop run time for videos of different resolutions. Resolution \| Average loop run time(in milliseconds) ---\|--- 240x320 \| 20 480x640 \| 45 720x960 \| 68 960x1280 \| 124 ### 6.2 Results of DNN The CNN network and Refiner network has been tested on single link and two link cases. The loss occurred during the training are shown in Fig. 10 and Fig. 11. Likewise, the scattered plots of actual coordinate and predicted coordinates by combined Networks of one-link, shoulders joint and wrist joint obtained are also shown in Fig. 10 and Fig. 11. Figure 10: Result for single link arm Figure 11: Results for two link arm ### 6.3 Discussions As we stated earlier, four different experiments under different circumstances, joints are performed here. In case of extremely random and very fast hand movement, the method is found to be ineffective as in Experiment-1 while in Experiment-2 where the background noise, arm speed are limited, the method performance i.e. mimicking is near to perfect. There is negligible lag time because the arm control command is given to simulator which runs nearly at 240Hz in Pybullet. In Experiment-3, the mimicking is performed with small lag time. This lag time(as seen in video) has been caused due to hardware limitation of the manipulator and setup. This can be reduced with good enough hardware, since the algorithm has been fine-tuned with optimal parameter values. The algorithm extended to n-link planar manipulator in Experiment-4 is able to estimate the three-joint angles between lines accurately as depicted in Fig.6. The algorithm can be extended to n-link planar manipulator just by introducing Object Regression Layer and the outcome would be as desired. Further, experimented with videos of different resolution shows the algorithm run time increases with increase in video resolution. In addition, our algorithm tracks/detects the objects on the basis of motion and not probabilistic color distribution or the object features. This enables our algorithm to run on both RGB and depth images independently and give the same result. In case of our second method, for the one-link(forearm) case the Predicted Joint positions improved after adding a second network. The second network has acted as a smoothing/refining network which refines the outcome of the first network. For the two link cases, the performance of the combined network is better compared to just use of CNN Network. ### 6.4 Comparative Study Please note that, first approach’s application that we proposed here is new to literature. Therefore, no similar method is available to compare with. Therefore, we focus on comparing the proposed tracking algorithm. We could not conduct any direct comparative study for our tracking method too since no other tracking algorithm, formulated so far, gives torque and velocity as the output. For example, we carried out the same experimentation with two other robust and popular unsupervised tracking algorithms, namely, MoG2 and CAMSHIFT for the sake of comparison. In this study we verified that the proposed algorithm tracks down the object within about 30 ms, whereas the time consumed by CAMSHIF and MoG2 are about 149 ms and 100ms respectively on HP laptop AU030WM Pavilion. Besides, these methods is not robust enough since they loose the object trajectory even to stationary background, therefore, its performance get reduced with reduction of trust factor. Above all, none of the algorithm enables us to compute values of kinematic and dynamic parameters of motion like our algorithm for mimicking and hence failed to mimic. This is the main cause why we failed to carry out suitable comparative study for this application. For the second approach of deep learning, we compare our results to that of [14] using Percentage correct parts(PCP) at link length threshold of 0.5(PCP 0.5), for upper arm and lower arm. PCP 0.5 was calculated on our model using our dataset on 20 images and also on 40 images. The comparison results are shown in Table 2 and 3 respectively. The data taken from [14] is generalised on large dataset that has been used, for our case the method has not been generalised and results shown are obtained on our dataset. Table 2: Comparison PCP(0.5) on 20 images Model \| Upper Arm \| Lower Arm ---\|---\|--- Deep Pose 1st \| 0.5 \| 0.27 Deep Pose 2nd \| 0.56 \| 0.35 Deep Pose 3rd \| 0.56 \| 0.35 CNN Network \| 0.40 \| 0.1 CNN + Refinement Network \| 0.7 \| 0.35 Table 3: Comparison PCP(0.5) on 40 images Model \| Upper Arm \| Lower Arm ---\|---\|--- Deep Pose 1st \| 0.5 \| 0.27 Deep Pose 2nd \| 0.56 \| 0.35 Deep Pose 3rd \| 0.56 \| 0.35 CNN Network \| 0.25 \| 0.125 CNN + Refinement Network \| 0.566 \| 0.275 From the above table, it is clear that our combined network comprising of CNN and Refiner Network performs better compared to just CNN network and Deep Pose Network for Upper Arm. However for lower the results are as good as that of Deep pose for the case of 20 images. On increasing the number of images to 40, performance decreases but still is better than that of Deep Pose and CNN Network for Upper Arm. However, for lower arm combined is poor. ## 7 Conclusions and Future Work In the proposed work we aimed to develop a method with which robotic arms could be controlled only by showing video sequences in real-time. It is tested both with unsupervised DON network and deep CNN with manually labelled training samples. This approach is proven to be successful with adequate amount of demonstration shown here. The unsupervised DON based method is proved to be effective in achieving control over n-link manipulator in 2-D plane. Control over n-link manipulator plane can be achieved better with hybridization of CNN and refinement network as shown here in the study. The proposed technique performs well both with real-arm manipulator and synthetic arms. The mimic-based control therefore could be implemented to control robotic arm in different tasks. The approach is in its entry level as of now and more complex scenarios could be addressed in future with mimicking the motion of more than one joint on manipulator in spatial 3D environment. ## References * [1] R. Basu, S. Padage, Development of 5 dof robot arm-gripper for sorting and investigating rtm concepts, Materials Today 4 (2) (2017) 1634–1643. * [2] M. H. Ali, K. Aizat, K. Yerkhan, T. Zhandos, O. Anuar, Vision-based robot manipulator for industrial applications, Procedia Computer Science, Elsevier 133 (2) (2018) 205–212. * [3] A. Geiger, P. Lenz, C. Stiller, R. Urtasun, Vision meets robotics: The kitti dataset, The International Journal of Robotics Research 32 (11) (2013) 1231–1237. * [4] S. Levine, P. Pastor, A. Krizhevsky, J. Ibarz, D. Quillen, Learning hand-eye coordination for robotic grasping with deep learning and large-scale data collection, The International Journal of Robotics Research 37 (4-5) (2018) 421–436. * [5] A. Hundt, V. Jain, C. Paxton, G. D. 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2.5cm2.5cm2.5cm2.5cm # Iwahori-Hecke model for mod $p$ representations of ${\rm GL}_{2}(F)$ U. K. Anandavardhanan and Arindam Jana Department of Mathematics, Indian Institute of Technology Bombay, Mumbai - 400076, India<EMAIL_ADDRESS>Department of Mathematics, Indian Institute of Technology Bombay, Mumbai - 400076, India<EMAIL_ADDRESS> ###### Abstract. For a $p$-adic field $F$, the space of pro-$p$-Iwahori invariants of a universal supersingular mod $p$ representation $\tau$ of ${\rm GL}_{2}(F)$ is determined in the works of Breuil, Schein, and Hendel. The representation $\tau$ is introduced by Barthel and Livné and this is defined in terms of the spherical Hecke operator. In [AB13, AB15], an Iwahori-Hecke approach was introduced to study these universal supersingular representations in which they can be characterized via the Iwahori-Hecke operators. In this paper, we construct a certain quotient $\pi$ of $\tau$, making use of the Iwahori-Hecke operators. When $F$ is not totally ramified over $\mathbb{Q}_{p}$, the representation $\pi$ is a non-trivial quotient of $\tau$. We determine a basis for the space of invariants of $\pi$ under the pro-p Iwahori subgroup. A pleasant feature of this ”new” representation $\pi$ is that its space of pro-$p$-Iwahori invariants admits a more uniform description vis-à-vis the description of the space of pro-$p$-Iwahori invariants of $\tau$. ###### 1991 Mathematics Subject Classification: Primary 20G05; Secondary 22E50, 11F70 ## 1\. Introduction For a $p$-adic field $F$, the study of irreducible smooth mod $p$ representations of $\rm GL_{2}(F)$ started with the famous work of Barthel and Livné [BL94]. They showed that there exist irreducible smooth representations, called supersingular representations, which cannot be obtained as a subquotient of a parabolically induced representation. It is shown in [BL94] that a supersingular representation can be realized as the quotient of a universal module constructed as follows. Let $G=\rm GL_{2}(F)$ and let $K$ be its standard maximal compact subgroup. Let $Z$ denote the center of $G$. For an irreducible representation $\sigma$ of $KZ$, let ind${}_{KZ}^{G}\sigma$ be the representation of $G$ compactly induced from $\sigma$. Its endomorphism algebra is a polynomial algebra in one variable: ${\rm End}_{G}\left({\rm ind}_{KZ}^{G}\sigma\right)\simeq\overline{\mathbb{F}}_{p}[T],$ where $T$ is the standard spherical Hecke operator and $\overline{\mathbb{F}}_{p}$ denotes an algebraic closure of the finite field $\mathbb{F}_{p}$ of $p$ elements [BL94, Proposition 8]. The universal module in consideration is $\tau=\frac{{\rm ind}_{KZ}^{G}\sigma}{(T)}$ and a supersingular representation of $G$ is an irreducible quotient of the universal module for some $\sigma$ of $KZ$ up to a twist by a character [BL94]. Explicitly constructing a supersingular representation of $\rm GL_{2}(F)$ is a challenging problem when $F\neq\mathbb{Q}_{p}$ [BP12]. When $F=\mathbb{Q}_{p}$, Breuil proved that the universal representation $\tau$ itself is irreducible [Bre03, Theorem 1.1]. The key step in Breuil’s proof of the irreducibility of $\tau$ is the explicit computation of its $I(1)$-invariant space, which is of dimension $2$, where $I(1)$ is the pro-$p$-Iwahori subgroup of $K$ [Bre03, Theorem 3.2.4]. The space of $I(1)$-invariants of $\tau$ is infinite dimensional when $F\neq\mathbb{Q}_{p}$. An explicit basis for this infinite dimensional space is computed by Schein when $F$ is totally ramified over $\mathbb{Q}_{p}$ [Sch11, §2] and by Hendel more generally for any $p$-adic field $F$ [Hen19, Theorem 1.2]. One can also construct a universal module from the perspective of the Iwahori- Hecke operators instead of the spherical Hecke operator $T$ [AB13, AB15]. For this, instead of doing compact induction from an irreducible representation of $KZ$, we start with a regular character $\chi$ of $IZ$, where $I$ is the Iwahori subgroup $K$, and consider the compactly induced representation ind${}_{IZ}^{G}\chi$. Its endomorphism algebra is [BL94, Proposition 13]: ${\rm End}_{G}({\rm ind}_{IZ}^{G}\chi)\simeq\frac{\overline{\mathbb{F}}_{p}[T_{-1,0,},T_{1,2}]}{(T_{-1,0}T_{1,2},T_{1,2}T_{-1,0})},$ where $T_{-1,0}$ and $T_{1,2}$ are the Iwahori-Hecke operators. When $F$ is a totally ramified extension of $\mathbb{Q}_{p}$, it is proved in [AB15, Proposition 3.1 & Remark 1] that the image of one of these operators is equal to the kernel of the other; i.e., (1) ${\rm Im}~{}T_{-1,0}={\rm Ker}~{}T_{1,2}~{}\&~{}{\rm Im}~{}T_{1,2}={\rm Ker}~{}T_{-1,0}.$ Let $\mathbb{F}_{q}$ be the residue field of $F$ where $q=p^{f}$. Assume $0<r<q-1$ and write $r=r_{0}+r_{1}p+\dots+r_{f-1}p^{f-1}$ with $0\leq r_{i}\leq p-1$ for $0\leq i\leq f-1$. Let $\sigma_{r}={\rm Sym}^{r_{0}}\overline{\mathbb{F}}_{p}^{2}\otimes{\rm Sym}^{r_{1}}\overline{\mathbb{F}}_{p}^{2}\circ{\rm Frob}\otimes\dots\otimes{\rm Sym}^{r_{f-1}}\overline{\mathbb{F}}_{p}^{2}\circ{\rm Frob}^{f-1}$ be an irreducible representation of $\rm GL_{2}(\mathbb{F}_{q})$, where Frob is the Frobenius morphism. We continue to denote the corresponding irreducible representation of $K$, obtained via inflation, by $\sigma_{r}$. Similarly, let $\chi_{r}$ be the character of $I$, valued in $\overline{\mathbb{F}}_{p}^{\times}$, obtained via the character of the Borel subgroup of $\rm GL_{2}(\mathbb{F}_{q})$ defined by $\left(\begin{array}[]{cc}a&b\\\ \ 0&d\end{array}\right)\mapsto d^{r}.$ We fix a uniformizing element $\varpi$ of the ring of integers $\mathcal{O}$ of $F$. The representation $\sigma_{r}$ is treated as a representation of $KZ$ by making diag$(\varpi,\varpi)$ acting trivially and similarly the character $\chi_{r}$ is treated as a character of $IZ$. For $g\in G$ and $v\in\sigma_{r}$, let $g\otimes v$ be the function in ${\rm ind}_{KZ}^{G}\sigma_{r}$ supported on $KZg^{-1}$ that sends $g^{-1}$ to $\sigma_{r}(k)v$. Similarly, for $g\in G$, by $[g,1]$ we define the function in ${\rm ind}_{IZ}^{G}\chi_{r}$ which is supported on $IZg^{-1}$ and sending $g^{-1}$ to $1$. It can be seen that every element of ${\rm ind}_{IZ}^{G}\chi_{r}$ (resp. ${\rm ind}_{KZ}^{G}\sigma_{r}$) is a finite sum of these type of functions $[g,1]$ (resp. $g\otimes v$). Now [AB15, Theorem 1.1] takes the form: ###### Theorem 1.1. Let $F$ be a finite extension of $\mathbb{Q}_{p}$ with residue field $\mathbb{F}_{q}$ and residue degree $f$. Let $0<r<q-1$ and $r=r_{0}+r_{1}p+\dots+r_{f-1}p^{f-1}$ with $0\leq r_{i}\leq p-1$. Then $\tau_{r}=\frac{{\rm ind}_{KZ}^{G}\sigma_{r}}{(T)}\simeq\frac{{\rm ind}_{IZ}^{G}\chi_{r}}{({\rm Im}~{}T_{1,2},{\rm Ker}~{}T_{1,2})}.$ Moreover, this isomorphism is determined by ${\rm Id}\otimes\bigotimes_{j=0}^{f-1}x_{j}^{r_{j}}\mod T\mapsto[\beta,1]\mod({\rm Im}~{}T_{1,2},{\rm Ker}~{}T_{1,2}).$ ###### Remark 1. Theorem 1.1 is stated and proved in [AB15, Theorem 4.1] when $F$ is a totally ramified extension of $\mathbb{Q}_{p}$ (see [AB15, Remark 3]) and exactly the same proof goes through in the general case as well. ###### Remark 2. As mentioned earlier, the space of $I(1)$-invariants of $\tau_{r}$ is computed by Hendel [Hen19, Theorem 1.2]. Stating an explicit basis for this space involves four cases; (i) $e=1,f=1$, (ii) $e>1,f=1$, (iii) $e=1,f>1$, and (iv) $e>1,f>1$. In this paper, we study a new universal representation given by $\pi_{r}=\frac{{\rm ind}_{IZ}^{G}\chi_{r}}{({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})}$ which is a further quotient of $\tau_{r}$. Note that this representation equals $\tau_{r}$ when $F$ is totally ramified over $\mathbb{Q}_{p}$ by (1). We show that when $F$ is not totally ramified over $\mathbb{Q}_{p}$, we have strict containments (2) ${\rm Im}~{}T_{-1,0}\subsetneq{\rm Ker}~{}T_{1,2}~{}\&~{}{\rm Im}~{}T_{1,2}\subsetneq{\rm Ker}~{}T_{-1,0}.$ and thus we have a new representation to investigate for its properties (cf. Remark 4). At this stage, we also note that the representation $\pi_{r}$ is indeed non-trivial (cf. Lemma 3.4). The main result of this paper gives an explicit basis for the space of $I(1)$-invariants of $\pi_{r}$. This space turns out to be infinite dimensional as well as in the case of [Hen19, Theorem 1.2]. However, in this case the basis can be written in a uniform manner whenever $F\neq\mathbb{Q}_{p}$. Thus, the statement involves only two cases; (i) $F=\mathbb{Q}_{p}$ and (ii) $F\neq\mathbb{Q}_{p}$. It is interesting to compare our result with that of Hendel in this aspect (cf. Remark 2). In order to state the theorem, we introduce a few more notations. Set $I_{0}=\\{0\\}$, and for $n\in\mathbb{N}$, let $I_{n}=\left\\{[{\mu}_{0}]+[{\mu}_{1}]\varpi+\dots+[\mu_{n-1}]{\varpi}^{n-1}\mid{\mu}_{i}\in\mathbb{F}_{q}\right\\}\subset\mathcal{O},$ where, for $x\in\mathbb{F}_{q}$, we denote its multiplicative representative in $\mathcal{O}$ by $[x]$. If $0\leq m\leq n,$ let $[\cdot]_{m}:I_{n}\rightarrow I_{m}$ be the truncation map defined by $\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{n-1}[\lambda_{i}]\varpi^{i}\mapsto\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{m-1}[\lambda_{i}]\varpi^{i}.$ Let us denote $\alpha=\left(\begin{array}[]{cc}1&0\\\ 0&\varpi\end{array}\right),~{}\beta=\left(\begin{array}[]{cc}0&1\\\ \varpi&0\end{array}\right),~{}w=\left(\begin{array}[]{cc}0&1\\\ 1&0\end{array}\right),$ and observe that $\beta=\alpha w$ normalizes $I(1)$. For any $n\in\mathbb{N}$, we denote $\displaystyle s_{n}^{k}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\mu_{n-1}^{k}\left[\left(\begin{array}[]{cc}\varpi^{n}&\mu\\\ 0&1\end{array}\right),1\right],$ $\displaystyle t_{n}^{s}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\mu_{n-1}^{s}\left[\left(\begin{array}[]{cc}\varpi^{n-1}&[\mu]_{n-1}\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right],$ where $0\leq k,s\leq q-1$. For $0\leq l\leq f-1$ and $m\geq 1$, we define the following sets $\displaystyle\mathcal{S}_{m}^{l}$ $\displaystyle=\\{s_{n}^{q-1-r+p^{l}}\\}_{n\geq m}\cup\\{\beta s_{n}^{q-1-r+p^{l}}\\}_{n\geq m}$ $\displaystyle\mathcal{S}_{m}$ $\displaystyle=\bigcup_{l=0}^{f-1}\mathcal{S}_{m}^{l},$ $\displaystyle\mathcal{T}_{m}^{l}$ $\displaystyle=\\{t_{n}^{r+p^{l}}\\}_{n\geq m}\cup\\{\beta t_{n}^{r+p^{l}}\\}_{n\geq m},$ $\displaystyle\mathcal{T}_{m}$ $\displaystyle=\bigcup_{l=0}^{f-1}T_{m}^{l}.$ Now we state the main theorem of this paper. ###### Theorem 1.2. Let $F$ be a finite extension of $\mathbb{Q}_{p}$ with ramification index $e$. Let $\mathbb{F}_{q}$ be the residue field of $F$ with $q=p^{f}.$ Let $0<r<q-1$ and $r=r_{0}+r_{1}p+\dots+r_{f-1}p^{f-1}$ with $0<r_{j}<p-1$ for all $0\leq j\leq f-1$. When $f=1$, we assume $2<r<p-3$. Then a basis of the space of $I(1)$-invariants of the representation $\pi_{r}=\frac{{\rm ind}_{IZ}^{G}\chi_{r}}{({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})}$ as an $\overline{\mathbb{F}}_{p}$-vector space is given by the images of the following sets in $\pi_{r}$: 1. (1) $\left\\{\left[{\rm Id},1\right],\left[\beta,1\right]\right\\}$ when $F=\mathbb{Q}_{p}$ 2. (2) $\mathcal{S}_{2}\bigcup\left\\{\left[{\rm Id},1\right],\left[\beta,1\right]\right\\}\bigcup\mathcal{T}_{2}~{}$ when $F\neq\mathbb{Q}_{p}$. ###### Remark 3. The representation $\pi_{r}$ that we construct and investigate in this paper is a quotient of the representation $\tau_{r}$ considered in [BL94, Bre03, Sch11, Hen19]; $0\rightarrow\frac{{\rm Ker}~{}T_{-1,0}}{{\rm Im~{}}T_{1,2}}\rightarrow\tau_{r}\rightarrow\pi_{r}\rightarrow 0.$ When $F$ is totally ramified over $\mathbb{Q}_{p}$, the representations $\tau_{r}$ and $\pi_{r}$ are isomorphic by Theorem 1.1 together with the equality of spaces in (1). However, $\pi_{r}$ is a “new” representation when $F$ is not totally ramified over $\mathbb{Q}_{p}$. That there is no isomorphism between $\tau_{r}$ and $\pi_{r}$ can be checked, for instance, from the characterization of the space of $I(1)$-invariants of $\pi_{r}$ in Theorem 1.2 and that of $\tau_{r}$ in [Hen19, Theorem 1.2]. We give more details in §4.5. Following the argument in [Hen19, Conclusion 3.10] word to word, we get the following corollary to Theorem 1.2. ###### Corollary 1.3. The representation $\pi_{r}$ is indecomposable; i.e., ${\rm End}_{G}(\pi_{r})\simeq\overline{\mathbb{F}}_{p}.$ The plan of the paper is as follows. We collect many results about the Iwahori-Hecke operators in Section 3. Several of these results are contained in some form in [AB13, AB15]. Theorem 1.2 and the key ideas in its proof are inspired by the work of Hendel [Hen19], though the Iwahori-Hecke approach which is employed in this paper as in [AB13, AB15] seems to be more amenable to carrying out the necessary calculations. We take up the proof in Section 4. ## 2\. Two basic results As in the work of Hendel [Hen19], we will need to frequently make use of the following two results in our computations. The first one is the classical result in modular combinatorics due to Lucas which gives a condition for a binomial coefficient ${n\choose r}$ to be zero modulo $p$. ###### Theorem 2.1 (Lucas). Let $n,r\in\mathbb{N}$ be such that $n=\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{k}n_{i}p^{i}$ and $r=\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{k}r_{i}p^{i},$ where $0\leq n_{i}\leq p-1$ and $0\leq r_{i}\leq p-1.$ Then ${n\choose r}\equiv\prod_{i=0}^{k}{n_{i}\choose r_{i}}\mod p.$ ###### Corollary 2.2. Let $n,r\in\mathbb{N}.$ Then $p$ divides ${n\choose r}$ if and only if $n_{i}<r_{i}$ for some $0\leq i\leq k.$ The next result gives a formula for adding multiplicative representatives in $\mathcal{O}$ [Hen19, Lemma 1.7]. As in [Hen19], this formula will play a crucial role in the calculations to follow. ###### Lemma 2.3. Let $x,y\in\mathbb{F}_{q}$ with $q=p^{f}$. Then $[x]+[y]\equiv[x+y]+\varpi^{e}[P_{0}(x,y)]\mod\varpi^{e+1},$ where $P_{0}(x,y)=\frac{x^{q^{e}}+y^{q^{e}}-(x+y)^{q^{e}}}{\varpi^{e}}.$ ## 3\. Preliminaries on the Iwahori-Hecke operators For $n\in\mathbb{N}\cup\\{0\\}$ and $\lambda\in I_{n}$, define $g_{n,\lambda}^{0}=\left(\begin{array}[]{cc}\varpi^{n}&\lambda\\\ 0&1\end{array}\right)~{}~{}\&~{}~{}g_{n,\lambda}^{1}=\left(\begin{array}[]{cc}1&0\\\ \varpi\lambda&\varpi^{n+1}\end{array}\right).$ We have the relations $g_{0,0}^{0}={\rm Id},~{}g_{0,0}^{1}=\alpha,\beta g_{n,\lambda}^{0}=g_{n,\lambda}^{1}w.$ Now $G$ acts transitively on the Bruhat-Tits tree of ${\rm SL}_{2}(F)$, whose vertices are in a $G$-equivariant bijection with the cosets $G/{KZ}$ and whose oriented edges are in a $G$-equivariant bijection with the cosets $G/{IZ}$. We have the explicit Cartan decomposition given by $G=\underset{\begin{subarray}{c}i\in\\{0,1\\}\\\ n\geq 0,~{}\lambda\in I_{n}\end{subarray}}{\coprod}g_{n,\lambda}^{i}KZ$ and an explicit set of coset representatives of $G/IZ$ is given by (3) $\left\\{g_{n,\lambda}^{0},g_{n,\lambda}^{0}\left(\begin{array}[]{cc}1&\mu\\\ 0&1\end{array}\right)w,g_{n,\lambda}^{1}w,g_{n,\lambda}^{1}w\left(\begin{array}[]{cc}1&\mu\\\ 0&1\end{array}\right)w\right\\}_{n\geq 0,\lambda\in I_{n}},$ where $\mu\in I_{1}.$ Now we recall a few details about the Iwahori-Hecke algebra [BL94, §3.2]. By definition, this algebra, denoted by $\mathcal{H}(IZ,\chi_{r})$, is the endomorphism algebra of the compactly induced representation ${\rm ind}_{IZ}^{G}\chi_{r}$. For $n\in\mathbb{Z}$, let $\phi_{n,n+1}$ denote the convolution map supported on $IZ{\alpha}^{-n}I$ such that $\phi_{n,n+1}(\alpha^{-n})=1$ [BL94, Lemma 9]. We denote by $T_{n,n+1}$ the corresponding element in $\mathcal{H}(IZ,\chi_{r})$. By [BL94, Proposition 13], for $0<r<q-1$, we have: $\mathcal{H}(IZ,\chi_{r})\simeq\frac{\overline{\mathbb{F}}_{p}[T_{-1,0,},T_{1,2}]}{(T_{-1,0}T_{1,2},T_{1,2}T_{-1,0})}.$ Substituting $n=1$ in [BL94, (16), (17)], we have the following explicit formulas for $T_{-1,0}$ and $T_{1,2}$: (4) $T_{-1,0}(\left[g,1\right])=\sum\limits_{\begin{subarray}{c}\lambda\in I_{1}\end{subarray}}\left[gg_{1,\lambda}^{0},1\right],$ (5) $T_{1,2}(\left[g,1\right])=\sum\limits_{\begin{subarray}{c}\lambda\in I_{1}\end{subarray}}\left[g\beta\left(\begin{array}[]{cc}1&\lambda\\\ 0&1\end{array}\right)w,1\right].$ The following proposition characterizes the kernel of the Iwahori-Hecke operators $T_{-1,0}$ and $T_{1,2}$ [AB13, AB15]. ###### Proposition 3.1. We have: 1. (1) ${\rm Ker~{}}T_{-1,0}$ is generated as a $G$-module by the vectors 1. (a) $(-1)^{q-1-r}s_{0}^{0}+t_{1}^{r}$, 2. (b) $t_{1}^{s}$ where $0\leq s\leq r-1$, 3. (c) $t_{1}^{s}$ where $s>r$ and ${q-1-r\choose q-1-s}\equiv 0\mod p$. 2. (2) ${\rm Ker~{}}T_{1,2}$ is generated as a $G$-module by the vectors 1. (a) $t_{0}^{0}+s_{1}^{q-1-r}$, 2. (b) $s_{1}^{k}$ where $0\leq k\leq q-2-r$, 3. (c) $s_{1}^{k}$ where $k>q-1-r$ and ${r\choose q-1-k}\equiv 0\mod p$. ###### Proof. We indicate the proof for Ker $T_{1,2}$, with the other case being similar. An arbitrary vector in ind${}_{IZ}^{G}\chi_{r}$ is an $\overline{\mathbb{F}}_{p}$-linear combination of vectors $[g,1]$, where $g$ is in the set of coset representatives (3) of $G/IZ$. Arguing as in the proof of [AB15, Proposition 3.1], we can restrict our attention to the vectors $\left\\{[{\rm Id},1],[\beta,1],[g_{1,\mu}^{0},1],\left[\left(\begin{array}[]{cc}1&\mu\\\ 0&1\end{array}\right)w,1\right]\right\\}$ for $\mu\in I_{1}$. Now the proof boils down to elementary linear algebra as in [AB15, Lemma 3.2], where one is led to analyse the indices $i$ for which $\sum_{\mu\in\mathbb{F}_{q}}\mu^{i}(\mu-\lambda)^{r}=0,$ for $\lambda\in\mathbb{F}_{q}$. Alternatively, this last step can be deduced directly from the explicit formulas for the Iwahori-Hecke operators in [AB13, p. 63-64]. ∎ ###### Remark 4. We remarked in (2) in Section 1 that we have strict containments (6) ${\rm Im}~{}T_{-1,0}\subsetneq{\rm Ker}~{}T_{1,2}~{}\&~{}{\rm Im}~{}T_{1,2}\subsetneq{\rm Ker}~{}T_{-1,0}.$ when $F$ is not a totally ramified extension of $\mathbb{Q}_{p}$. The reason for this is that the third type of vectors in both (1) and (2) in Proposition 3.1 do not belong to the images of the Iwahori-Hecke operators. Note that such vectors do not exist when $f=1$; i.e., when $q=p$. By the argument in [AB15, Lemma 3.2], it can be shown that the first two types of vectors are indeed in the image of the relevant Iwahori-Hecke operator. ###### Corollary 3.2. A basis of the space of $I(1)$-invariants of ${\rm Ker~{}}T_{-1,0}$ is given by $\\{t_{n}^{0},\beta t_{n}^{0}\\}_{n\geq 1}$ and that of ${\rm Ker~{}}T_{1,2}$ is given by $\\{s_{n}^{0},\beta s_{n}^{0}\\}_{n\geq 1}$. Moreover, the action of $I$ is given by $\left(\begin{array}[]{cc}a&b\\\ \varpi c&d\end{array}\right)\cdot v=\begin{cases}a^{r}v&\text{$v=t_{n}^{0}$ or $\beta s_{n}^{0}$,}\\\ d^{r}v&\text{$v=s_{n}^{0}$ or $\beta t_{n}^{0}$.}\end{cases}$ ###### Proof. The first part of Proposition 3.1 together with the observation that the space of $I(1)$-invariants of the full induced representation is given by $\left({\rm ind}_{IZ}^{G}\chi_{r}\right)^{I(1)}=\langle s_{n}^{0},t_{n}^{0},\beta s_{n}^{0},\beta t_{n}^{0}\rangle_{n\geq 0}.$ For the second part, observe that since $I/{I(1)}=\left\\{\left(\begin{array}[]{cc}a&0\\\ 0&d\end{array}\right)\mid a,d\in{\mathbb{F}_{q}}^{\times}\right\\},$ it follows that $\left(\begin{array}[]{cc}a&b\\\ \varpi c&d\end{array}\right)~{}\&~{}\left(\begin{array}[]{cc}a&0\\\ 0&d\end{array}\right)$ have the same action on any $I(1)$-invariant vector. Now, for any $k\geq 0$, we have $\displaystyle\left(\begin{array}[]{cc}a&0\\\ 0&d\end{array}\right)s_{n}^{k}$ $\displaystyle=\left(\begin{array}[]{cc}a&0\\\ 0&d\end{array}\right)\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\mu_{n-1}^{k}\left[\left(\begin{array}[]{cc}\varpi^{n}&\mu\\\ 0&1\end{array}\right),1\right]$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\mu_{n-1}^{k}\left[\left(\begin{array}[]{cc}\varpi^{n}&ad^{-1}\mu\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}a&0\\\ 0&d\end{array}\right),1\right]$ $\displaystyle=d^{r}(da^{-1})^{k}s_{n}^{k}.$ A similar computation gives $\displaystyle\left(\begin{array}[]{cc}a&0\\\ 0&d\end{array}\right)t_{n}^{s}$ $\displaystyle=a^{r}(da^{-1})^{s}t_{n}^{s}.$ Similarly, we can check the action on $\beta s_{n}^{k}$ and $\beta t_{n}^{k}$. ∎ Next, we recall [AB15, Proposition 3.3], whose proof in [loc. cit.] is valid for any $q$. ###### Proposition 3.3. We have ${\rm Ker}~{}T_{-1,0}\cap{\rm Ker}~{}T_{1,2}=\\{0\\}.$ As a corollary to Proposition 3.3, we have the following lemma. ###### Lemma 3.4. For the iwahori-Hecke operators $T_{-1,0}$ and $T_{1,2}$, we have ${\rm ind}_{IZ}^{G}\chi_{r}\neq{\rm Ker}~{}T_{-1,0}\oplus{\rm Ker}~{}T_{1,2}.$ ###### Proof. If possible, let ${\rm ind}_{IZ}^{G}\chi_{r}={\rm Ker}~{}T_{-1,0}\oplus{\rm Ker}~{}T_{1,2}.$ Then we get $[{\rm Id},1]=v_{1}+v_{2}$ for some $v_{1}\in{\rm Ker}~{}T_{-1,0}$ and $v_{2}\in{\rm Ker}~{}T_{1,2}$. Then, for an element $g\in I$, $g(v_{1}+v_{2})=\left(\begin{array}[]{cc}a&b\\\ \varpi c&d\end{array}\right)(v_{1}+v_{2})=d^{r}[{\rm Id},1]=d^{r}(v_{1}+v_{2})$ and this implies $gv_{1}-d^{r}v_{1}=-gv_{2}+d^{r}v_{2}=0,$ by Proposition 3.3. In particular, both $v_{1}$ and $v_{2}$ are $I(1)$-invariant. By Corollary 3.2, $v_{1}$ is a linear combination of vectors of the form $\\{\beta t_{n}^{0}\\}_{n\geq 1}$ and $v_{2}$ is a linear combination of vectors of the form $\\{s_{n}^{0}\\}_{n\geq 1}$. But $[{\rm Id},1]$ cannot be written as a linear combination of these types of vectors. ∎ We end this section with two more results which immediately follow from considerations similar to Proposition 3.1. We state these in a ready to use format here (see also [AB13, p. 63-64]). ###### Lemma 3.5. Let $0\leq i_{j}\leq q-1$ for $0\leq j\leq n-1$ and $\mu=[\mu_{0}]+[\mu_{1}]\varpi+\dots+[\mu_{n-1}]\varpi^{n-1}\in I_{n}.$ Write $i_{n-1}=i_{n-1,0}+i_{n-1,1}p+\dots+i_{n-1,f-1}p^{f-1}.$ Then * (1) $\sum\limits_{\begin{subarray}{c}\mu_{0}\end{subarray}}\dots\sum\limits_{\begin{subarray}{c}\mu_{n-1}\end{subarray}}\mu_{0}^{i_{0}}\dots\mu_{n-1}^{i_{n-1}}\left[g^{0}_{n,\mu},1\right]\in{\rm Ker}~{}T_{1,2}$ if and only if $0\leq i_{n-1}\leq q-2-r$ or $i_{n-1}>q-1-r$ such that $i_{n-1,j}<p-1-r_{j}$ for some $0\leq j\leq f-2$. * (2) $\sum\limits_{\begin{subarray}{c}\mu_{0}\end{subarray}}\dots\sum\limits_{\begin{subarray}{c}\mu_{n-1}\end{subarray}}\mu_{0}^{i_{0}}\dots\mu_{n-1}^{i_{n-1}}\left[g^{0}_{n-1,[\mu]_{n-1}}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right]\in{\rm Ker}~{}T_{-1,0}$ if and only if $0\leq i_{n-1}\leq r-1$ or $i_{n-1}>r$ such that $i_{n-1,j}<r_{j}$ for some $0\leq j\leq f-2$. ###### Remark 5. Note that in Lemma 3.5, the range for $j$ is $0\leq j\leq f-2$ because $i_{n-1}>q-1-r\implies i_{n-1,f-1}\geq p-1-r_{f-1}.$ ###### Remark 6. Note that the condition $i_{n-1}>q-1-r~{}\&~{}i_{n-1,j}<p-1-r_{j}\mbox{~{}for~{}some~{}}0\leq j\leq f-2$ in Lemma 3.5 (1) is precisely what gives, by Theorem 2.1, ${r\choose q-1-i_{n-1}}\equiv 0\mod p$ which is related to the condition in (c) of Proposition 3.1 (2). Similarly, the condition $i_{n-1}>r~{}\&~{}i_{n-1,j}<r_{j}\mbox{~{}for~{}some~{}}0\leq j\leq f-2$ in Lemma 3.5 (2) is related to (c) of Proposition 3.1 (1). The following lemma is [AB13, Lemma 3.1]. We note that its proof in [loc. cit.] is valid for any $q$. ###### Lemma 3.6. Let $\mu=[\mu_{0}]+\dots+[\mu_{n-1}]\varpi^{n-1}\in I_{n}.$ Then modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$, we have the identities 1. (1) $\sum\limits_{\begin{subarray}{c}\mu_{n-1}\in I_{1}\end{subarray}}\mu_{n-1}^{q-1-r}\left[g^{0}_{n,\mu},1\right]=-\left[g^{0}_{n-2,[\mu]_{n-2}}\left(\begin{array}[]{cc}1&[\mu_{n-2}]\\\ 0&1\end{array}\right)w,1\right],$ 2. (2) $\sum\limits_{\begin{subarray}{c}\mu_{n-1}\in I_{1}\end{subarray}}\mu_{n-1}^{r}\left[g^{0}_{n-1,[\mu]_{n-1}}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right]=(-1)^{r-1}\left[g^{0}_{n-1,[\mu]_{n-1}},1\right].$ ###### Remark 7. In fact, (1) is true modulo ${\rm Ker}~{}T_{1,2}$ and (2) is true modulo ${\rm Ker}~{}T_{-1,0}$ (cf. [AB13, (4) & (5) on p. 62]). ## 4\. Proof of Theorem 1.2 In this section we take up the proof of Theorem 1.2. As mentioned in Section 1, several of the ideas of the proof here are already there in [Hen19]. ### 4.1. A set of $I(1)$-invariants First we make the following observation [Hen19, §2.1]. For $a,b,c\in\mathcal{O}$, any matrix in $I(1)$ can be written as $\left(\begin{array}[]{cc}1+\varpi a&b\\\ \varpi c&1+\varpi d\end{array}\right)=\left(\begin{array}[]{cc}1&(1+\varpi d)^{-1}b\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1&0\\\ \varpi ct^{-1}&1\end{array}\right)\left(\begin{array}[]{cc}t&0\\\ 0&1+\varpi d\end{array}\right),$ where $t=1+\varpi(a-bc(1+\varpi d)^{-1}).$ Hence to prove that a certain vector is $I(1)$-invariant modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$, it is enough to check for invariance under $\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right),\left(\begin{array}[]{cc}1&0\\\ \varpi c&1\end{array}\right),\left(\begin{array}[]{cc}1+\varpi a&0\\\ 0&1\end{array}\right),$ where $a,b,c\in\mathcal{O}.$ We first prove that the set of vectors $\mathcal{S}_{2}$ and $\mathcal{T}_{2}$ are $I(1)$-invariants when considered as vectors in $\pi_{r}$; i.e., when we consider the images of these vectors modulo ${\rm Ker}~{}T_{-1,0}\oplus{\rm Ker}~{}T_{1,2}$. The first step in achieving this is an inductive argument which reduces the general case to the case $n=2$. ###### Lemma 4.1. If $s_{n-1}^{k}$ (resp. $t_{n-1}^{s}$) is $I(1)$-invariant modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$, then, for all $n\geq 2$, the vector $s_{n}^{k}$ (resp. $t_{n}^{s}$) is also $I(1)$-invariant modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$. ###### Proof. We prove the case of $s_{n}^{k}$ and the case of $t_{n}^{s}$ is similar. Assume that $s_{n-1}^{k}$ is $I(1)$-invariant modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Now, $\displaystyle\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{n}^{k}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\mu_{n-1}^{k}\left[\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi^{n}&\mu\\\ 0&1\end{array}\right),1\right]$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\mu_{n-1}^{k}\left[\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi^{n-1}&\displaystyle{\sum_{i=1}^{n-1}}[\mu_{i}]\varpi^{i-1}\\\ 0&1\end{array}\right),1\right]$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\mu_{n-1}^{k}\left[\left(\begin{array}[]{cc}\varpi&[\mu_{0}+b_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1&B(\mu_{0},b)\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi^{n-1}&\displaystyle{\sum_{i=1}^{n-1}}[\mu_{i}]\varpi^{i-1},\\\ 0&1\end{array}\right),1\right]$ where (7) $B(\mu_{0},b)=\varpi^{e-1}[P_{0}(\mu_{0},b_{0})]+[b_{1}]+[b_{2}]\varpi+\dots$ and (8) $P_{0}(\mu_{0},b_{0})=\frac{\mu_{0}^{q^{e}}+b_{0}^{q^{e}}-(\mu_{0}+b_{0})^{q^{e}}}{\varpi^{e}}$ is obtained from the formula in Lemma 2.3. Let $\mu^{\prime}=[\mu_{1}]+[\mu_{2}]\varpi+\dots+[\mu_{n-1}]\varpi^{n-2}.$ We continue by making the substitution $\mu_{0}\rightarrow\mu_{0}-b_{0}$. Thus, $\displaystyle\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{n}^{k}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu_{0}\in I_{1}\end{subarray}}\left(\begin{array}[]{cc}\varpi&[\mu_{0}]\\\ 0&1\end{array}\right)\left\\{\left(\begin{array}[]{cc}1&B(\mu_{0}-b_{0},b)\\\ 0&1\end{array}\right)\sum\limits_{\begin{subarray}{c}\mu^{\prime}\in I_{n-1}\end{subarray}}{\mu^{\prime}}_{n-2}^{k}\left[\left(\begin{array}[]{cc}\varpi^{n-1}&\mu^{\prime}\\\ 0&1\end{array}\right),1\right]\right\\}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu_{0}\in I_{1}\end{subarray}}\left(\begin{array}[]{cc}\varpi&[\mu_{0}]\\\ 0&1\end{array}\right)\left\\{\sum\limits_{\begin{subarray}{c}\mu^{\prime}\in I_{n-1}\end{subarray}}{\mu^{\prime}}_{n-2}^{k}\left[\left(\begin{array}[]{cc}\varpi^{n-1}&\mu^{\prime}\\\ 0&1\end{array}\right),1\right]+x_{\mu_{0}}\right\\},$ by our assumption, where $x_{\mu_{0}}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$. Thus, we get $\displaystyle\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{n}^{k}$ $\displaystyle=s_{n}^{k}+\sum\limits_{\begin{subarray}{c}\mu_{0}\in I_{1}\end{subarray}}\left(\begin{array}[]{cc}\varpi&[\mu_{0}]\\\ 0&1\end{array}\right)x_{\mu_{0}},$ and hence $\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{n}^{k}-s_{n}^{k}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Checking for invariance under $\left(\begin{array}[]{cc}1&0\\\ \varpi c&1\end{array}\right)~{}\&~{}\left(\begin{array}[]{cc}1+\varpi a&0\\\ 0&1\end{array}\right)$ is even easier which we skip. ∎ Now we take the case $n=2$. Recall that $e$ (resp. $f$) is the ramification index (resp. residue degree) of $F$ over $\mathbb{Q}_{p}$. We write $r=r_{0}+r_{1}p+\dots+r_{f-1}p^{f-1}$ where $0\leq r_{j}\leq p-1$ for $0\leq j\leq f-1$. We first observe that for $a,b,c\in\mathcal{O}$, we have (9) $\left(\begin{array}[]{cc}1+\varpi a&b\\\ c\varpi&1+d\varpi\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu]\\\ 0&1\end{array}\right)=\left(\begin{array}[]{cc}\varpi&[\mu+b_{0}]\\\ 0&1\end{array}\right)k,$ for $k\in I(1)$. Indeed, $\displaystyle{\rm LHS}$ $\displaystyle=\left(\begin{array}[]{cc}\varpi(1+\varpi a)&[\mu+b_{0}]+\varpi()\\\ c\varpi^{2}&1+\varpi(\Delta)\end{array}\right)$ $\displaystyle=\left(\begin{array}[]{cc}\varpi&[\mu+b_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1+a\varpi-[\mu+b_{0}]c\varpi&()-(\mu+b_{0})\Delta\\\ c\varpi^{2}&1+\varpi\Delta\end{array}\right),$ where $,\Delta\in\mathcal{O}$. Similarly, one can show that (10) $\left(\begin{array}[]{cc}1+\varpi a&b\\\ c\varpi&1+d\varpi\end{array}\right)\left(\begin{array}[]{cc}1&[\mu]\\\ 0&1\end{array}\right)w=\left(\begin{array}[]{cc}1&[\mu+b_{0}]\\\ 0&1\end{array}\right)wk^{\prime}$ for some $k^{\prime}\in I(1)$. ###### Lemma 4.2. Assume $0<r_{j}<p-1$, and if $f=1$, assume further that $2<r<p-3$. Then when $(e,f)\neq(1,1)$, we have $gs_{2}^{q-1-r+p^{l}}-s_{2}^{q-1-r+p^{l}}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ and $gt_{2}^{r+p^{l}}-t_{2}^{r+p^{l}}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ for all $g\in I(1)$ and $0\leq l\leq f-1.$ ###### Proof. We have $\displaystyle\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{2}^{q-1-r+p^{l}}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi^{2}&[\mu_{0}]+[\mu_{1}]\varpi\\\ 0&1\end{array}\right),1\right]$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{1}]\\\ 0&1\end{array}\right),1\right]$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}\varpi&[\mu_{0}+b_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1&B(\mu_{0},b)\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{1}]\\\ 0&1\end{array}\right),1\right],$ where $B(\mu_{0},b)$ is given by (7) in the proof of Lemma 4.1. Now write $B(\mu_{0},b)=[b_{1}+Z]+()\varpi$ where $Z=0$ for $e>1$ and $Z=P_{0}(\mu_{0},b_{0})$ for $e=1$. To continue, the above expression equals $\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}\varpi&[\mu_{0}+b_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1&[b_{1}+Z]+()\varpi\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{1}]\\\ 0&1\end{array}\right),1\right]$ which equals $\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}\varpi&[\mu_{0}+b_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{1}+b_{1}+Z]\\\ 0&1\end{array}\right)k,1\right]$ for $k\in I(1)$, by (9). We continue by making the change of variables $\mu_{1}\rightarrow\mu_{1}-b_{1}-Z~{}\&~{}\mu_{0}\rightarrow\mu_{0}-b_{0},$ and we get $\displaystyle\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}(\mu_{1}-b_{1}-Z)^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}\varpi^{2}&\mu\\\ 0&1\end{array}\right),1\right]$ $\displaystyle=s_{2}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{q-1-r+p^{l}-1}{q-1-r+p^{l}\choose i}(-b_{1}-Z)^{q-1-r+p^{l}-i}\mu_{1}^{i}\left[\left(\begin{array}[]{cc}\varpi^{2}&\mu\\\ 0&1\end{array}\right),1\right].$ Now we read the above expression modulo ${\rm Ker}~{}T_{1,2}$. We claim that only the term corresponding to $i=q-1-r$ remains amongst the $q-1-r+p^{l}$ terms in the inner summation in the above expression. By Lemma 3.6 (1), we know that $\sum_{\mu\in I_{2}}\mu_{1}^{i}\left[\left(\begin{array}[]{cc}\varpi^{2}&\mu\\\ 0&1\end{array}\right),1\right]\in{\rm Ker}~{}T_{1,2}$ precisely when $0\leq i\leq q-2-r$ or $i>q-1-r$ such that $i_{j}<p-1-r_{j}$ for some $0\leq j\leq f-2$. Note that if $i>q-1-r$ and $i_{j}\geq p-1-r_{j}$ for all $0\leq j\leq f-1$ (cf. Remark 5) then $i_{j}>p-1-r_{j}$ for some $0\leq j\leq l-1$ (since $i\leq q-1-r+p^{l}-1$). If this is the case then observe that ${q-1-r+p^{l}\choose i}\equiv 0\mod p$ by Corollary 2.2. Thus, modulo ${\rm Ker}~{}T_{1,2}$, we get $\displaystyle\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{2}^{q-1-r+p^{l}}$ $\displaystyle=s_{2}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}{q-1-r+p^{l}\choose q-1-r}(-b_{1}-Z)^{p^{l}}\mu_{1}^{q-1-r}\left[\left(\begin{array}[]{cc}\varpi^{2}&\mu\\\ 0&1\end{array}\right),1\right]$ $\displaystyle=s_{2}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}\mu_{0}\in I_{1}\end{subarray}}(p-r_{l})(-b_{1}-Z)^{p^{l}}\left[\left(\begin{array}[]{cc}1&[\mu_{0}]\\\ 0&1\end{array}\right)w,1\right]$ by Lemma 3.6 (1) and the binomial coefficient here is computed via Theorem 2.1. Now if $e>1$ then we have $Z=0$. Therefore, it follows, by Lemma 3.5 (2), that $\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{2}^{q-1-r+p^{l}}-s_{2}^{q-1-r+p^{l}}\in{\rm Ker}~{}T_{-1,0},$ and thus we have proved $\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{2}^{q-1-r+p^{l}}\equiv s_{2}^{q-1-r+p^{l}}\mod({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ If $e=1$ then $Z=P_{0}(\mu_{0},b_{0})$. As $F$ is unramified over $\mathbb{Q}_{p}$, we have $\varpi=p$. Now by Corollary 2.2, it follows that $\displaystyle Z$ $\displaystyle=\frac{\mu_{0}^{q^{e}}+b_{0}^{q^{e}}-(\mu_{0}+b_{0})^{q^{e}}}{\varpi^{e}}$ $\displaystyle\equiv-\sum\limits_{\begin{subarray}{c}i=1\end{subarray}}^{p-1}\frac{1}{p}{p^{f}\choose ip^{f-1}}b_{0}^{p^{f}-ip^{f-1}}\mu_{0}^{ip^{f-1}}\mod p.$ In this case, if further $f\neq 1$ we have, modulo ${\rm Ker}~{}T_{1,2}$, $\displaystyle\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{2}^{q-1-r+p^{l}}-s_{2}^{q-1-r+p^{l}}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu_{0}\in I_{1}\end{subarray}}(p-r_{l})(-b_{1}-Z)^{p^{l}}\left[\left(\begin{array}[]{cc}1&[\mu_{0}]\\\ 0&1\end{array}\right)w,1\right]$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu_{0}\in I_{1}\end{subarray}}r_{l}(b_{1}^{p^{l}}+Z^{p^{l}})\left[\left(\begin{array}[]{cc}1&[\mu_{0}]\\\ 0&1\end{array}\right)w,1\right].$ Note that both $\sum_{\mu_{0}\in I_{1}}\left[\left(\begin{array}[]{cc}1&[\mu_{0}]\\\ 0&1\end{array}\right)w,1\right]~{}\&~{}\sum_{\mu_{0}\in I_{1}}\mu_{0}^{ip^{l-1}}\left[\left(\begin{array}[]{cc}1&[\mu_{0}]\\\ 0&1\end{array}\right)w,1\right]$ are in ${\rm Ker}~{}T_{-1,0}$, by Lemma 3.5 (2). Thus, once again we have proved $\left(\begin{array}[]{cc}1&b\\\ 0&1\end{array}\right)s_{2}^{q-1-r+p^{l}}\equiv s_{2}^{q-1-r+p^{l}}\mod({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Now we analyze invariance for the lower unipotent representative of $I(1)$. We have $\displaystyle\left(\begin{array}[]{cc}1&0\\\ \varpi c&1\end{array}\right)s_{2}^{q-1-r+p^{l}}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}1&0\\\ \varpi c&1\end{array}\right)\left(\begin{array}[]{cc}\varpi^{2}&[\mu_{0}]+[\mu_{1}]\varpi\\\ 0&1\end{array}\right),1\right]$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}1&0\\\ \varpi c&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{1}]\\\ 0&1\end{array}\right),1\right]$ which we express as $\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}\varpi&[\mu_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1-\varpi c[\mu_{0}]&-[\mu_{0}^{2}]c\\\ \varpi^{2}c&1+\varpi c[\mu_{0}]\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{1}]\\\ 0&1\end{array}\right),1\right]$ and this equals $\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}\mu_{1}^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}\varpi&[\mu_{0}]\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi&[\mu_{1}-c_{0}\mu_{0}^{2}]\\\ 0&1\end{array}\right)k,1\right]$ for $k\in I(1)$ by (9). Changing $\mu_{1}\rightarrow\mu_{1}+c_{0}\mu_{0}^{2}$, we get $\displaystyle\left(\begin{array}[]{cc}1&0\\\ \varpi c&1\end{array}\right)s_{2}^{q-1-r+p^{l}}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}(\mu_{1}+c_{0}\mu_{0}^{2})^{q-1-r+p^{l}}\left[\left(\begin{array}[]{cc}\varpi^{2}&\mu\\\ 0&1\end{array}\right),1\right]$ which we read modulo ${\rm Ker}~{}T_{1,2}$ and get $\displaystyle s_{2}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}\mu\in I_{2}\end{subarray}}{q-1-r+p^{l}\choose q-1-r}(c_{0}\mu_{0}^{2})^{p^{l}}\mu_{1}^{q-1-r}\left[\left(\begin{array}[]{cc}\varpi^{2}&\mu\\\ 0&1\end{array}\right),1\right]$ by Corollary 2.2 together with Lemma 3.5 (1), exactly as we have argued before. Now this equals, modulo ${\rm Ker}~{}T_{1,2}$, $\displaystyle s_{2}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}\mu_{0}\in I_{1}\end{subarray}}(p-r_{l})c_{0}^{p^{l}}\mu_{0}^{2p^{l}}\left[\left(\begin{array}[]{cc}1&[\mu_{0}]\\\ 0&1\end{array}\right)w,1\right]$ by Theorem 2.1 and Lemma 3.6 (1). By Lemma 3.5 (2), this vector belongs to ${\rm Ker}~{}T_{-1,0}$ (with the extra assumption that $3\leq r$ when $f=1$). Thus, we have proved $\left(\begin{array}[]{cc}1&0\\\ \varpi c&1\end{array}\right)s_{2}^{q-1-r+p^{l}}\equiv s_{2}^{q-1-r+p^{l}}\mod({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ The proof for showing that $\left(\begin{array}[]{cc}1+\varpi a&0\\\ 0&1\end{array}\right)s_{2}^{q-1-r+p^{l}}-s_{2}^{q-1-r+p^{l}}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ is similar and therefore we skip it. The argument for $\displaystyle gt_{2}^{r+p^{l}}-t_{2}^{r+p^{l}}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ for all $g\in I(1)$ is similar to the one for $s_{2}^{q-1-r+p^{l}}$. Note that corresponding to the case $3\leq r$ in the totally ramified case for $s_{2}^{q-1-r+p^{l}}$, in the case of $t_{2}^{r+p^{l}}$ we will get $r\leq p-4$. ∎ ### 4.2. Linear independence The following lemma gives the action of the Iwahori subgroup $I$ on the $I(1)$-invariant vectors (cf. [Hen19, Lemma 3.6]). ###### Lemma 4.3. Let $\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\in I.$ Let $s_{n}^{k}$ and $t_{n}^{s}$ be $I(1)$-invariants modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Then they are $I$-eigenvectors and those actions are given by (1) $\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\cdot s_{n}^{k}=d^{r}(da^{-1})^{k}s_{n}^{k},$ * (2) $\left(\begin{array}[]{cc}a&b\\\ c&d\end{array}\right)\cdot t_{n}^{s}=a^{r}(da^{-1})^{s}t_{n}^{s}.$ ###### Proof. The proof is straightforward and we have already done it in the proof of the second part of Corollary 3.2. ∎ ###### Remark 8. Lemmas 4.1, 4.2 and 4.3 remain true for $\beta s_{n}^{k}$ and $\beta t_{n}^{s}$. ###### Proposition 4.4. The set of vectors in $\mathcal{S}_{2}\cup\mathcal{T}_{2}$ of Theorem 1.2 are linearly independent. ###### Proof. Note that the vectors in $\mathcal{S}_{2}\cup\mathcal{T}_{2}$ consist of vectors of the form $s_{n}^{q-1-r+p^{l}},\beta s_{n}^{q-1-r+p^{l}},t_{n}^{r+p^{l}},\beta t_{n}^{r+p^{l}}$ for $n\geq 2$ and $0\leq l\leq f-1$. These are invariant under $I(1)$ modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ except for the case when both $e=1$ and $f=1$ (cf. Lemmas 4.1, 4.2 and Remark 8). For any vector $v\in{\rm ind}_{IZ}^{G}\chi_{r}$, note that $v$ and $\beta v$ cannot cancel each other (pictorially they are on two different sides of the tree of ${\rm SL}_{2}(F)$). Therefore, it is enough to show that the set $\\{s_{n}^{q-1-r+p^{l}},t_{n}^{r+p^{l}}\\}$, for $n\geq 2$ and $0\leq l\leq f-1$, is linearly independent. Since $s_{n}^{q-1-r+p^{l}}$ and $t_{n}^{r+p^{l}}$ have different $I$-eigenvalues, it is enough to show that $\\{s_{n}^{q-1-r+p^{l}}\\}$ and $\\{t_{n}^{r+p^{l}}\\}$, for $n\geq 2$ and $0\leq l\leq f-1$, are linearly independent. We show that the vectors in $\\{s_{n}^{q-1-r+p^{l}}\\}_{n\geq 2,0\leq l\leq f-1}$ are linearly independent, and the proof for $\\{t_{n}^{r+p^{l}}\\}$ is similar. Suppose that $\sum\limits_{\begin{subarray}{c}i=2\end{subarray}}^{n}c_{i}s_{i}^{q-1-r+p^{l}}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ where $c_{i}\in\overline{{\mathbb{F}}}_{p}$ and $n\in\mathbb{N}$. Since no reduction is possible in the above expression and also these vectors obviously cannot be in ${\rm Ker}~{}T_{-1,0}$, it follows that $\sum\limits_{\begin{subarray}{c}i=2\end{subarray}}^{n}c_{i}s_{i}^{q-1-r+p^{l}}\in{\rm Ker}~{}T_{1,2}.$ For $i\neq j$ with $2\leq i,j\leq n$, once again from the formula for $T_{1,2}$, there cannot be any cancellation between $T_{1,2}(c_{i}s_{i}^{q-1-r+p^{l}})$ and $T_{1,2}(c_{j}s_{j}^{q-1-r+p^{l}}),$ so we get $c_{i}s_{i}^{q-1-r+p^{l}}\in{\rm Ker}~{}T_{1,2}$ for all $2\leq i\leq n.$ By Lemma 3.5 (1), it follows that $c_{i}=0$ for all $2\leq i\leq n$. ∎ ###### Remark 9. It follows by eigenvalue considerations as in the proof of Proposition 4.4 that the set $\mathcal{S}_{2}\cup\\{[{\rm Id},1],[\beta,1]\\}\cup\mathcal{T}_{2}$ is linearly independent. ### 4.3. Auxiliary lemmas We will have to make use of the following elementary lemma [Hen19, Lemma 2.8]. ###### Lemma 4.5. Let $n\geq 1$ and $\phi:I_{n}\rightarrow\overline{\mathbb{F}}_{p}$ be any set map. Then there exists a unique polynomial $Q(x_{0},\dots,x_{n-1})\in\overline{\mathbb{F}}_{p}[x_{0},x_{1},\dots,x_{n-1}]$ in which degree of each variable is at most $q-1$ and $\phi(\mu)=Q(\mu_{0},\mu_{1},\dots,\mu_{n-1})$ for all $\mu\in I_{n}.$ The next two lemmas are the first steps towards the proof of Theorem 1.2. ###### Lemma 4.6. Let $\mu=[\mu_{0}]+[\mu_{1}]\varpi+\dots+[\mu_{n-1}]\varpi^{n-1}\in I_{n}$ and $r=r_{0}+r_{1}p+\dots+r_{f-1}p^{f-1}$ with $0<r_{j}<p-1$ for all $0\leq j\leq f-1.$ Let $f_{n}=f_{n}^{\prime}+f_{n}^{\prime\prime}$ be such that $f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}a(\mu_{0},\mu_{1},\dots,\mu_{n-1})\left[\left(\begin{array}[]{cc}\varpi^{n}&\mu\\\ 0&1\end{array}\right),1\right]$ and $f_{n}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}b(\mu_{0},\mu_{1},\dots,\mu_{n-1})\left[\left(\begin{array}[]{cc}\varpi^{n-1}&[\mu]_{n-1}\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right],$ where $a(\mu_{0},\dots,\mu_{n-1})$ and $b(\mu_{0},\dots,\mu_{n-1})$ are polynomials in $\mu_{0},\dots,\mu_{n-1}.$ Suppose $\left(\begin{array}[]{cc}1&-\varpi^{n-1}\\\ 0&1\end{array}\right)f_{n}-f_{n}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Then 1. (1) the possible powers of $\mu_{n-1},$ say $k=k_{0}+k_{1}p+\dots+k_{f-1}p^{f-1}$, in $a(\mu_{0},\dots,\mu_{n-1})$ will satisfy one of the following three conditions: 1. (a) there exists some $0\leq j^{\prime}\leq f-1$ such that $k_{j^{\prime}}<p-1-r_{j^{\prime}},$ 2. (b) $k_{j}=p-1-r_{j}$ for all $0\leq j\leq f-1,$ 3. (c) $k_{j}=p-1-r_{j}$ for $j\neq l$ and $k_{l}=p-r_{l}$ for some $0\leq l\leq f-1.$ 2. (2) the possible powers of $\mu_{n-1},$ say $k=k_{0}+k_{1}p+\dots+k_{f-1}p^{f-1}$, in $b(\mu_{0},\dots,\mu_{n-1})$ will satisfy one of the following three conditions: 1. (a) there exists some $0\leq j^{\prime}\leq f-1$ such that such that $k_{j^{\prime}}<r_{j^{\prime}},$ 2. (b) $k_{j}=r_{j}$ for all $0\leq j\leq f-1,$ 3. (c) $k_{j}=r_{j}$ for $j\neq l$ and $k_{l}=r_{l}+1$ for some $0\leq l\leq f-1.$ ###### Proof of Lemma 4.6. We will prove $(1)$ and the proof of $(2)$ is similar. Suppose $(1)$ does not hold. Then there exists $k$ such that $k_{j}\geq p-1-r_{j}$ for all $0\leq j\leq f-1$ with $k_{j_{0}}>p-1-r_{j_{0}}\mbox{~{}for~{}some~{}}0\leq j_{0}\leq f-1~{}\&~{}k\neq(p-r_{j_{0}})p^{j_{0}}+\sum\limits_{\begin{subarray}{c}j_{0}\neq j=0\end{subarray}}^{f-1}(p-1-r_{j})p^{j}.$ Then either there exists $j_{1}$ with $j_{1}\neq j_{0}$ such that $k_{j_{1}}>p-1-r_{j_{1}}$ or $k=k_{j_{0}}p^{j_{0}}+\sum\limits_{\begin{subarray}{c}j_{0}\neq j=0\end{subarray}}^{f-1}(p-1-r_{j})p^{j}$ with $k_{j_{0}}>p-r_{j_{0}}.$ Choose $k$ with the above property such that there is no other monomial $\mu_{n-1}^{k^{\prime}}$ in $a(\mu_{0},\dots,\mu_{n-1})$ with $k_{j}\leq k^{\prime}_{j}$ for all $0\leq j\leq f-1$. Since a polynomial is of finite degree, such a $k$ exists. Let $g=\left(\begin{array}[]{cc}1&-\varpi^{n-1}\\\ 0&1\end{array}\right).$ We have $gf_{n}-f_{n}=(gf_{n}^{\prime}-f_{n}^{\prime})+(gf_{n}^{\prime\prime}-f_{n}^{\prime\prime})\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Note that, $\displaystyle gf_{n}^{\prime}-f_{n}^{\prime}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\left[a([\mu]_{n-1},\mu_{n-1}+1)-a([\mu]_{n-1},\mu_{n-1})\right]\left[\left(\begin{array}[]{cc}\varpi^{n}&\mu\\\ 0&1\end{array}\right),1\right]$ and $\displaystyle gf_{n}^{\prime\prime}-f_{n}^{\prime\prime}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\left[b([\mu]_{n-1},\mu_{n-1}+1)-b([\mu]_{n-1},\mu_{n-1})\right]\left[\left(\begin{array}[]{cc}\varpi^{n-1}&[\mu]_{n-1}\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right].$ Let $\Delta a=a([\mu]_{n-1},\mu_{n-1}+1)-a([\mu]_{n-1},\mu_{n-1})$ considered as a polynomial in $\mu_{n-1}$ with coefficients in $\overline{\mathbb{F}}_{p}[\mu_{0},\dots,\mu_{n-2}]$. By Theorem 2.1, we have $(\mu_{n-1}+1)^{k}-\mu_{n-1}^{k}\equiv\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{k-1}\prod_{j=0}^{f-1}{k_{j}\choose i_{j}}\mu_{n-1}^{i}\mod p.$ Now if there exists $j_{1}$ with $j_{1}\neq j_{0}$ such that $k_{j_{1}}>p-1-r_{j_{1}},$ take $k^{\prime}=(k_{j_{1}}-1)p^{j_{1}}+\sum\limits_{\begin{subarray}{c}j_{1}\neq j=0\end{subarray}}^{f-1}k_{j}p^{j}.$ The coefficient of $\mu_{n-1}^{k^{\prime}}$ in $\Delta a$ is ${k\choose k^{\prime}}={k_{j_{1}}\choose k_{j_{1}}-1}\not\equiv 0\mod p$ by Theorem 2.1 and Corollary 2.2. Note that the term involving $\mu_{n-1}^{k^{\prime}}$ in $gf_{n}^{\prime}-f_{n}^{\prime}$ cannot get cancelled by any other term in $gf_{n}-f_{n}$. Indeed, it cannot get cancelled with any other term in $gf_{n}^{\prime}-f_{n}^{\prime}$ because of the choice of $k$ and anyway no term in $gf_{n}^{\prime}-f_{n}^{\prime}$ can get cancelled with a term in $gf_{n}^{\prime\prime}-f_{n}^{\prime\prime}$ (pictorially they represent edges of opposite orientation on the tree of ${\rm SL}_{2}(F)$). So this term involving $\mu_{n-1}^{k^{\prime}}$ must be there in $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}),$ but then Lemma 3.5 (1) would imply that there exists some $0\leq l\leq f-1$ such that $k^{\prime}_{l}<p-1-r_{l},$ which contradicts our assumption. So $k$ must be of the form $k=k_{j_{0}}p^{j_{0}}+\sum\limits_{\begin{subarray}{c}j_{0}\neq j=0\end{subarray}}^{f-1}(p-1-r_{j})p^{j}$ with $k_{j_{0}}>p-r_{j_{0}}.$ Taking $k^{\prime}=(k_{j_{0}}-1)p^{j_{0}}+\sum\limits_{\begin{subarray}{c}j_{0}\neq j=0\end{subarray}}^{f-1}(p-1-r_{j})p^{j},$ and using the same argument as in the previous case, we arrive at a contradiction. ∎ ###### Remark 10. The idea of choosing $k$ as in Lemma 4.6 is already employed by Hendel in [Hen19, Lemma 3.13]. Now we state one more lemma whose main idea of proof also comes from [Hen19, Lemma 3.13]. In what follows, $B(t)$ denotes the ball of radius $m$ on the tree of ${\rm SL}_{2}(F)$ with center at the vertex representing the trivial coset $G/KZ$. Explicitly it consists of linear combinations of vectors of the form $B^{0}(t)=\left\\{[g_{n,\mu}^{0},1],\left[g_{n-1,[\mu]_{n-1}}^{0}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right]\right\\}_{n\leq t},$ and $B^{1}(t)=\left\\{[g_{n-1,[\mu]_{n-1}}^{1}w,1],\left[g_{n-2,[\mu]_{n-2}}^{1}w\left(\begin{array}[]{cc}1&[\mu_{n-2}]\\\ 0&1\end{array}\right)w,1\right]\right\\}_{n\leq t},$ where $\mu=[\mu_{0}]+[\mu_{1}]\varpi+\dots+[\mu_{n-1}]\varpi^{n-1}\in I_{n}$. ###### Lemma 4.7. Let $f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}P_{l}([\mu]_{n-1})\mu_{n-1}^{q-1-r+p^{l}}\left[g^{0}_{n,\mu},1\right]$ and $f_{n}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}Q_{l}([\mu]_{n-1})\mu_{n-1}^{r+p^{l}}\left[g^{0}_{n-1,[\mu]_{n-1}}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right],$ where $P_{l}([\mu]_{n-1})$ and $Q_{l}([\mu]_{n-1})$ are polynomials in $\mu_{0},\dots,\mu_{n-2}$. Let $f_{n}=f_{n}^{\prime}+f_{n}^{\prime\prime}$. Let $f=f_{n}+f^{\prime}$ be such that $f^{\prime}\in B(n-1)$ and $\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f-f\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}),$ for all $1\leq m\leq n-1.$ Then we have $f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l}\mu_{n-1}^{q-1-r+p^{l}}\left[g^{0}_{n,\mu},1\right]$ and $f_{n}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l}\mu_{n-1}^{r+p^{l}}\left[g^{0}_{n-1,[\mu]_{n-1}}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right],$ where $a_{l}$ and $b_{l}$ are constants. ###### Proof of Lemma 4.7. We do the proof only for $f_{n}^{\prime}$, as the case of $f_{n}^{\prime\prime}$ is similar. The proof is by induction on $n$. Note that $P_{l}([\mu]_{n-1})$ is independent of $\mu_{n-1}.$ Suppose it is independent of $\mu_{n-1},\dots,\mu_{n-m+1}.$ Then $f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}P_{l}([\mu]_{n-m},\mu_{n-m})\mu_{n-1}^{q-1-r+p^{l}}\left[g^{0}_{n,\mu},1\right]$ We show that it is independent of $\mu_{n-m}$. It is given to us that $\displaystyle\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f-f$ $\displaystyle=\left[\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f_{n}-f_{n}\right]+\left[\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f^{\prime}-f^{\prime}\right]$ $\displaystyle\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Now, $\displaystyle\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)\left(\begin{array}[]{cc}\varpi^{n}&\displaystyle{\sum_{i=0}^{n-1}}[\mu_{i}]\varpi^{i}\\\ 0&1\end{array}\right)$ $\displaystyle=\left(\begin{array}[]{cc}\varpi^{n}&[\mu_{0}]+\dots+[\mu_{n-1}]\varpi^{n-1}-\varpi^{n-m}\\\ 0&1\end{array}\right)$ and this equals $\displaystyle\left(\begin{array}[]{cc}\varpi^{n}&\displaystyle{\sum_{i=0}^{n-m-1}}[\mu_{i}]\varpi^{i}+[\mu_{n-m}-1]\varpi^{n-m}+[\mu^{\prime}_{n-m+1}]\varpi^{n-m+1}+\dots+[\mu^{\prime}_{n-1}]\varpi^{n-1}\\\ 0&1\end{array}\right)$ where $\mu^{\prime}_{k}=\mu_{k}+c_{k}(\mu_{n-m},\dots,\mu_{n-2})$ for $n-m+1\leq k\leq n-1.$ Note that the transformation $\mu^{\prime}_{k}\mapsto\mu_{k}-c_{k}(\mu_{n-m},\dots,\mu_{n-2})$ does not affect the variables $\mu_{k}$ for $n-m+1\leq k\leq n-1$ in $P_{l}([\mu]_{n-1}),$ as it is independent of these variables. This transformation together with $\mu_{n-m}\mapsto\mu_{n-m}+1$ gives $\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}P_{l}([\mu]_{n-m},\mu_{n-m}+1)(\mu_{n-1}-c_{n-1})^{q-1-r+p^{l}}[g_{n,\mu}^{0},1].$ In the above expression, by $c_{n-1}$ we mean $c_{n-1}(\mu_{n-m},\dots,\mu_{n-2})$. Now, $\displaystyle\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f_{n}^{\prime}-f_{n}^{\prime}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}\alpha(\mu,l)[g^{0}_{n,\mu},1],$ where $\alpha(\mu,l)=\left[P_{l}([\mu]_{n-m},\mu_{n-m}+1)(\mu_{n-1}-c_{n-1})^{q-1-r+p^{l}}-P_{l}([\mu]_{n-m},\mu_{n-m})\mu_{n-1}^{q-1-r+p^{l}}\right].$ Thus, $\displaystyle\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f_{n}^{\prime}-f_{n}^{\prime}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}\left[P_{l}([\mu]_{n-m},\mu_{n-m}+1)-P_{l}([\mu]_{n-m},\mu_{n-m})\right]\mu_{n-1}^{q-1-r+p^{l}}[g^{0}_{n,\mu},1]$ $\displaystyle+\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}\sum\limits_{\begin{subarray}{c}i=0\end{subarray}}^{q-1-r+p^{l}-1}\beta(\mu,l,i)[g^{0}_{n,\mu},1],$ where $\beta(\mu,l,i)=P_{l}([\mu]_{n-m},\mu_{n-m}+1)(-1)^{i}{q-1-r+p^{l}\choose i}(-c_{n-1})^{q-1-r+p^{l}-i}\mu_{n-1}^{i}.$ Now we read this modulo ${\rm Ker}~{}T_{1,2}$. Thus, we get $\displaystyle\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f_{n}^{\prime}-f_{n}^{\prime}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}\left[P_{l}([\mu]_{n-m},\mu_{n-m}+1)-P_{l}([\mu]_{n-m},\mu_{n-m})\right]\mu_{n-1}^{q-1-r+p^{l}}[g^{0}_{n,\mu},1]$ $\displaystyle+\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}P_{l}([\mu]_{n-m},\mu_{n-m}+1){q-1-r+p^{l}\choose q-1-r}(-c_{n-1})^{p^{l}}\mu_{n-1}^{q-1-r}[g^{0}_{n,\mu},1],$ by Corollary 2.2 and Lemma 3.5 (1), exactly as we have argued before in the proof of Lemma 4.2. Now by Lemma 3.6 (1), it follows that, modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$, we have $\displaystyle\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f_{n}^{\prime}-f_{n}^{\prime}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}\left[P_{l}([\mu]_{n-m},\mu_{n-m}+1)-P_{l}([\mu]_{n-m},\mu_{n-m})\right]\mu_{n-1}^{q-1-r+p^{l}}[g^{0}_{n,\mu},1]+g_{n-1}$ where $g_{n-1}\in B(n-1)$. As $r_{l}\neq 0,$ by Lemmas 3.5 (1) and 3.6 (1) we have $\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\mu_{n-1}^{q-1-r+p^{l}}\left[g^{0}_{n,\mu},1\right]\notin({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Also the term involving $\mu_{n-1}^{q-1-r+p^{l}}$ cannot get cancelled by any other term in the expression $\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f-f.$ So it follows that $P_{l}([\mu]_{n-m},\mu_{n-m}+1)-P_{l}([\mu]_{n-m},\mu_{n-m})=0.$ Hence $P_{l}([\mu]_{n-1})$ is independent of $\mu_{n-m}.$ Therefore, by induction $P_{l}([\mu]_{n-1})$ is a constant. ∎ ### 4.4. Proof of Theorem 1.2 Clearly the vectors $\left[{\rm Id},1\right]$ and $\left[\beta,1\right]$ are fixed by $I(1).$ By Lemmas 4.1 and 4.2 and Remark 8 the vectors in $\mathcal{S}_{2}$ and $\mathcal{T}_{2}$ are $I(1)$-invariant modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ except for the case when both $e=1$ and $f=1.$ By Remark 9, the set $\mathcal{S}_{2}\cup\\{[{\rm Id},1],[\beta,1]\\}\cup\mathcal{T}_{2}$ is linearly independent. Now let $f\in{\rm ind}_{IZ}^{G}\chi_{r}$ be an $I(1)$-invariant of $\pi_{r}=\frac{{\rm ind}_{IZ}^{G}\chi_{r}}{({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})}.$ We write $f=f^{0}+f^{1}$ where $f^{0}$ (resp. $f^{1}$) is a linear combination of vectors on the zero side (resp. one side) of the tree of ${\rm SL}_{2}(F)$. By this, we mean $f^{0}$ is a linear combination of vectors of the form $[g_{n,\mu}^{0},1],\left[g_{n-1,[\mu]_{n-1}}^{0}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right]$ and $f^{1}$ is a linear combination of vectors of the form $[g_{n-1,[\mu]_{n-1}}^{1}w,1],\left[g_{n-2,[\mu]_{n-2}}^{1}w\left(\begin{array}[]{cc}1&[\mu_{n-2}]\\\ 0&1\end{array}\right)w,1\right].$ Then, $gf^{i}-f^{i}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}),$ for all $i\in\\{0,1\\}$ and $g\in I(1)$. Since $\beta f^{1}$ is a linear combination of vectors on the zero side and $\beta$ normalizes $I(1)$, without loss of generality, we may assume $f=f^{0}$. Write $f=f_{n}+f^{\prime}$ with $f_{n}\neq 0$, $f^{\prime}\in B(n-1)$, for $n$ maximal. Now, $f_{n}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}a_{{\mu}}\left[g^{0}_{n,\mu},1\right]+\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}b_{{\mu}}\left[g^{0}_{n-1,[\mu]_{n-1}}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right],$ where $\mu=[\mu_{0}]+[\mu_{1}]\varpi+\dots+[\mu_{n-1}]\varpi^{n-1}$ and $a_{{\mu}},b_{{\mu}}\in\overline{\mathbb{F}}_{p}.$ By Lemma 4.5, the coefficients $a_{{\mu}}$ and $b_{{\mu}}$ can be replaced by the polynomials $a(\mu_{0},\dots,\mu_{n-1})$ and $b(\mu_{0},\dots,\mu_{n-1})$ respectively, where each $\mu_{i}$ has maximum degree $q-1.$ Write $f_{n}=f_{n}^{\prime}+f_{n}^{\prime\prime},$ where $f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}i\end{subarray}}a(i_{0},i_{1},\dots,i_{n-1})\mu_{0}^{i_{0}}\dots\mu_{n-1}^{i_{n-1}}\left[g^{0}_{n,\mu},1\right],$ and $f_{n}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}j\end{subarray}}b(j_{0},j_{1},\dots,j_{n-1})\mu_{0}^{j_{0}}\dots\mu_{n-1}^{j_{n-1}}\left[g^{0}_{n-1,[\mu]_{n-1}}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right].$ Let $g^{\prime}=\left(\begin{array}[]{cc}1&-\varpi^{n-1}\\\ 0&1\end{array}\right)\in I(1).$ Since $f^{\prime}$ belongs in $B(n-1)$, it is easy to check that $g^{\prime}$ fixes $f^{\prime}$. This gives $g^{\prime}f_{n}-f_{n}\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Now Lemma 3.5 (1) together with Lemma 4.6 (1) gives $\displaystyle f_{n}^{\prime}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}i\end{subarray}}a(i_{0},\dots,i_{n-2},q-1-r)\mu_{0}^{i_{0}}\dots\mu_{n-1}^{q-1-r}\left[g^{0}_{n,\mu},1\right]$ $\displaystyle+\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l}([\mu]_{n-1})\mu_{n-1}^{q-1-r+p^{l}}\left[g^{0}_{n,\mu},1\right],$ which in turn implies that $\displaystyle f_{n}^{\prime}-\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l}([\mu]_{n-1})\mu_{n-1}^{q-1-r+p^{l}}\left[g^{0}_{n,\mu},1\right]$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu_{0},\dots,\mu_{n-1}\end{subarray}}\sum\limits_{\begin{subarray}{c}i_{0},\dots,i_{n-2}\end{subarray}}a(i_{0},\dots,q-1-r)\mu_{0}^{i_{0}}\dots\mu_{n-1}^{q-1-r}\left[g^{0}_{n,\mu},1\right]$ which modulo ${\rm Ker}~{}T_{1,2}$ equals $\displaystyle\sum\limits_{\begin{subarray}{c}\mu_{0},\dots,\mu_{n-2}\end{subarray}}\sum\limits_{\begin{subarray}{c}i_{0},\dots,i_{n-2}\end{subarray}}a(i_{0},\dots,q-1-r)\mu_{0}^{i_{0}}\dots\mu_{n-2}^{i_{n-2}}\left[g^{0}_{n-2,[\mu]_{n-2}}\left(\begin{array}[]{cc}1&[\mu_{n-2}]\\\ 0&1\end{array}\right)w,1\right],$ by Lemma 3.6 (1). This vector belongs to $B(n-1)$ which we call $g_{n-1}^{\prime}$. We get $f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l}([\mu]_{n-1})\mu_{n-1}^{q-1-r+p^{l}}\left[g^{0}_{n,\mu},1\right]+g_{n-1}^{\prime}.$ Similarly, working with $f_{n}^{\prime\prime},$ we get $f_{n}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l}([\mu]_{n-1})\mu_{n-1}^{r+p^{l}}\left[g_{n-1,[\mu]_{n-1}}^{0}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right]+g_{n-1}^{\prime\prime}$ for some $g_{n-1}^{\prime\prime}\in B(n-1)$, by Lemmas 3.5 (2), 4.6 (2) and 3.6 (2). For $1\leq m\leq n-1,$ we note that $\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)\in I(1).$ Using the condition $\left(\begin{array}[]{cc}1&-\varpi^{n-m}\\\ 0&1\end{array}\right)f-f\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}),$ by Lemma 4.7, we have $f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l,n}\mu_{n-1}^{q-1-r+p^{l}}\left[g^{0}_{n,\mu},1\right]+g_{n-1}^{\prime}$ and $f_{n}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l,n}\mu_{n-1}^{r+p^{l}}\left[g_{n-1,[\mu]_{n-1}}^{0}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right]+g_{n-1}^{\prime\prime},$ where $a_{l}$ and $b_{l}$ are constants. Hence $f_{n}$ takes the form $f_{n}=\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l,n}s_{n}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l,n}t_{n}^{r+p^{l}}+g_{n-1},$ where $g_{n-1}=g_{n-1}^{\prime}+g_{n-1}^{\prime\prime}\in B(n-1).$ Thus it follows that $f-\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l,n}s_{n}^{q-1-r+p^{l}}-\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l,n}t_{n}^{r+p^{l}}=g_{n-1}+f^{\prime}$ is an $I(1)$-invariant vector modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ in $B(n-1)$. Applying this argument on vectors in $B(n-1)$ and repeating this process, we get $\displaystyle f=\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l,n}s_{n}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l,n}t_{n}^{r+p^{l}}+\dots+\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l,2}s_{2}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l,2}t_{2}^{r+p^{l}}+f_{1},$ where $f_{1}$ is an $I(1)$-invariant in $B(1)$. Write $f_{1}=f_{1}^{\prime}+f_{1}^{\prime\prime},$ where $f_{1}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{1}\end{subarray}}\sum\limits_{\begin{subarray}{c}i\end{subarray}}a_{i}\mu^{i}\left[g^{0}_{1,\mu},1\right],$ and $f_{1}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{1}\end{subarray}}\sum\limits_{\begin{subarray}{c}j\end{subarray}}b_{j}\mu^{j}\left[\left(\begin{array}[]{cc}1&\mu\\\ 0&1\end{array}\right)w,1\right].$ Using the action of $u=\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right)$ on $f_{1}$, by Lemma 4.6 (1), the possible powers $i$ of $\mu$ in $f_{1}^{\prime}$ will satisfy either $0\leq i\leq q-1-r$ or $i=q-1-r+p^{l}$ for some $0\leq l\leq f-1.$ If $i=q-1-r+p^{l},$ then $\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right)f_{1}^{\prime}-f_{1}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{1}\end{subarray}}\mu^{q-1-r}\left[g^{0}_{1,\mu},1\right]\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ This, by Lemma 3.6 (1), gives $\left[\beta,1\right]\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}),$ which is not possible. So we must have $0\leq i\leq q-1-r.$ Then, by Lemma 3.5(1) and Lemma 3.6 (1), we have $f_{1}^{\prime}=\left[\beta,1\right]\mod({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Similarly, by Lemmas 3.5 (2) and 4.6 (2) and Lemma 3.6 (2) , we can show that $f_{1}^{\prime\prime}=\left[{\rm Id},1\right]\mod({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Thus, we have $\displaystyle f=\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l,n}s_{n}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l,n}t_{n}^{r+p^{l}}+\dots$ $\displaystyle+\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}a_{l,2}s_{2}^{q-1-r+p^{l}}+\sum\limits_{\begin{subarray}{c}l=0\end{subarray}}^{f-1}b_{l,2}t_{2}^{r+p^{l}}+c\left[\beta,1\right]+d\left[{\rm I}d,1\right].$ Now assume $e=1$ and $f=1$. Let $f\in{\rm ind}_{IZ}^{G}\chi_{r}$ be an $I(1)$-invariant vector modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ As in the previous case, we concentrate only on the zero side of the tree and assume that $f=f^{0}.$ We write $f=f_{n}+f^{\prime}$ where $f_{n}\neq 0$ and $f^{\prime}\in B(n-1).$ We further write $f_{n}=f_{n}^{\prime}+f_{n}^{\prime\prime}$ where $f_{n}^{\prime}$ and $f_{n}^{\prime\prime}$ are same as in the previous case. Following the steps in the previous case, we have $f_{n}^{\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}a_{0}\mu_{n-1}^{p-r}\left[g^{0}_{n,\mu},1\right]+g_{n-1}^{\prime},$ and $f_{n}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}b_{0}\mu_{n-1}^{r+1}\left[g_{n-1,[\mu]_{n-1}}^{0}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right]+g_{n-1}^{\prime\prime},$ where $a_{0}$ and $b_{0}$ are constants and $g_{n-1}^{\prime},g_{n-1}^{\prime\prime}\in B(n-1).$ Thus, $f_{n}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}a_{0}\mu_{n-1}^{p-r}\left[g^{0}_{n,\mu},1\right]+\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}b_{0}\mu_{n-1}^{r+1}\left[g_{n-1,[\mu]_{n-1}}^{0}\left(\begin{array}[]{cc}1&[\mu_{n-1}]\\\ 0&1\end{array}\right)w,1\right]+g_{n-1},$ where $g_{n-1}=g_{n-1}^{\prime}+g_{n-1}^{\prime\prime}\in B(n-1).$ Write $f=f_{n}+f_{n-1}+f^{\prime}.$ We get $\displaystyle\left(\begin{array}[]{cc}1&p^{n-2}\\\ 0&1\end{array}\right)f-f$ $\displaystyle=\left[\left(\begin{array}[]{cc}1&p^{n-2}\\\ 0&1\end{array}\right)f_{n}-f_{n}\right]+\left[\left(\begin{array}[]{cc}1&p^{n-2}\\\ 0&1\end{array}\right)f_{n-1}-f_{n-1}\right]$ $\displaystyle\in({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ For $e=1$, we have $\displaystyle\left(\begin{array}[]{cc}1&p^{n-2}\\\ 0&1\end{array}\right)f_{n}^{\prime}-f_{n}^{\prime}$ $\displaystyle=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}a_{0}\left[(\mu_{n-1}-())^{p-r}-\mu_{n-1}^{p-r}\right]\left[g^{0}_{n,\mu},1\right],$ where $()=\sum\limits_{\begin{subarray}{c}s=1\end{subarray}}^{p-1}(-1)^{p-s}\frac{{p\choose s}}{p}\mu_{n-2}^{s}.$ Then, by Lemmas 3.6 (1) and 3.5 (1), modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$, the above expression becomes (11) $-\sum\limits_{\begin{subarray}{c}\mu\in I_{n-1}\end{subarray}}a_{0}{p-r\choose p-1-r}()\left[g^{0}_{n-2,[\mu]_{n-2}}\left(\begin{array}[]{cc}1&[\mu_{n-2}]\\\ 0&1\end{array}\right)w,1\right].$ Writing $f_{n-1}=f_{n-1}^{\prime}+f_{n-1}^{\prime\prime}$, we have, $\left(\begin{array}[]{cc}1&p^{n-2}\\\ 0&1\end{array}\right)f_{n-1}-f_{n-1}=\left[\left(\begin{array}[]{cc}1&p^{n-2}\\\ 0&1\end{array}\right)f_{n-1}^{\prime}-f_{n-1}^{\prime}\right]+\left[\left(\begin{array}[]{cc}1&p^{n-2}\\\ 0&1\end{array}\right)f_{n-1}^{\prime\prime}-f_{n-1}^{\prime\prime}\right].$ No term in the first summand of the above equation can cancel a term in (11). Also, by Lemma 4.6 (2), the possible powers, say $k,$ of $\mu_{n-2}$ in $f_{n-1}^{\prime\prime}$ must satisfy either $0\leq k\leq r$ or $k=r+1.$ As $r<p-1$, we have $\max(r+1)=p-1.$ So the maximum power of $\mu_{n-2}$ in the second summand of the above equation is $p-2.$ In both the cases, the term involving $\mu_{n-2}^{p-1}$ in (11) will not get cancelled. Since there is no reduction, this term must be in ${\rm Ker}~{}T_{-1,0},$ which is not possible by Lemma 3.5 (2). Thus we arrive at a contradiction. So $i_{n-1}$ can not be $p-r.$ Thus one can always modify $f_{n}^{\prime}$ by a vector $g_{n-1}^{\prime}$ in $B(n-1).$ Similarly, working with $f_{n}^{\prime\prime},$ we can modify it by a vector $g_{n-1}^{\prime\prime}$ in $B(n-1).$ Thus $f_{n}$ is congruent to a vector $f_{n-1}$ in $B(n-1)$ modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})$ and hence by induction, $f$ is congruent to a vector $f_{1}$ in $B(1)$ modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}).$ Write $f_{1}=f_{1}^{\prime}+f_{1}^{\prime\prime},$ where $f_{1}^{\prime}=\sum\limits_{\begin{subarray}{c}i\end{subarray}}\sum\limits_{\begin{subarray}{c}\mu\in I_{1}\end{subarray}}a_{i}\mu^{i}\left[g^{0}_{1,\mu},1\right],$ and $f_{1}^{\prime\prime}=\sum\limits_{\begin{subarray}{c}j\end{subarray}}\sum\limits_{\begin{subarray}{c}\mu\in I_{1}\end{subarray}}b_{j}\mu^{j}\left[\left(\begin{array}[]{cc}1&\mu\\\ 0&1\end{array}\right)w,1\right].$ Considering the action of $\left(\begin{array}[]{cc}1&1\\\ 0&1\end{array}\right)$ on $f_{1}$ as in the previous case, we have $0\leq i\leq p-1-r$ and $0\leq j\leq r$, by Lemma 3.6 and Lemma 4.6. Then, by Lemma 3.6 and Lemma 3.5, modulo $({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2}),$ we get $f_{1}^{\prime}=\left[\beta,1\right]$ and $f_{1}^{\prime\prime}=\left[{\rm Id},1\right].$ Thus we can conclude that $f=c\left[{\rm Id},1\right]+d\left[\beta,1\right].$ This finishes the proof of Theorem 1.2. ### 4.5. A remark on $\pi_{r}$ We show that there is no isomorphism between $\tau_{r}=\frac{{\rm ind}_{KZ}^{G}\sigma_{r}}{(T)}$ and $\pi_{r}=\frac{{\rm ind}_{IZ}^{G}\chi_{r}}{({\rm Ker}~{}T_{-1,0},{\rm Ker}~{}T_{1,2})}$ when $f\neq 1$; i.e., $F$ is not a totally ramified extension of $\mathbb{Q}_{p}$ (cf. Remark 3). Note that any $G$-linear isomorphism $\varphi:\pi_{r}\rightarrow\tau_{r}$ must preserve $I(1)$-invariants and the corresponding $I$-eigenvalues. Suppose $e=1,f\neq 1$; i.e., $F/\mathbb{Q}_{p}$ is unramified. In this case, $s_{n}^{q-1-r+p^{l}}$, for $n\geq 2$, is an $I(1)$-invariant in $\pi_{r}$ such that $\left(\begin{array}[]{cc}a&b\\\ \varpi c&d\end{array}\right)\cdot s_{n}^{q-1-r+p^{l}}=a^{r-p^{l}}d^{p^{l}}\cdot s_{n}^{q-1-r+p^{l}}$ by Lemma 4.3. By [Hen19, Theorem 1.2], a basis of the $I(1)$-invariants in $\tau_{r}$ consists of the vectors ${\rm Id}\otimes\bigotimes_{j=0}^{f-1}x_{j}^{r_{j}},\alpha\otimes\bigotimes_{j=0}^{f-1}y_{j}^{r_{j}},c_{n}^{p^{l}(r_{l}+1)},\beta c_{n}^{p^{l}(r_{l}+1)}$ for $n\geq 1$, where $c_{n}^{k}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\left(\begin{array}[]{cc}\varpi^{n}&\mu\\\ 0&1\end{array}\right)\otimes\mu_{n-1}^{k}\bigotimes_{j=0}^{f-1}x_{j}^{r_{j}}.$ By [Hen19, Lemma 3.6], $\left(\begin{array}[]{cc}a&b\\\ \varpi c&d\end{array}\right)\cdot c_{n}^{k}=a^{r-2k}(ad)^{k}\cdot c_{n}^{k},$ and it follows that there is no $I(1)$-invariant vector in $\tau_{r}$ with $I$-eigenvalue $a^{r-p^{l}}d^{p^{l}}$. Thus there is no vector in $\tau_{r}$ where $s_{n}^{q-1-r+p^{l}}$ can be mapped under $\varphi$. This gives a contradiction. Now, suppose $e>1,f>1$. In this case $t_{n}^{r+p^{l}}$, $n\geq 2$, is an $I(1)$-invariant vector in $\pi_{r}$ with $I$-eigenvalue $a^{q-1-p^{l}}d^{r+p^{l}}$, by Lemma 4.3. A basis of the $I(1)$-invariants in $\tau_{r}$ consists of the vectors ${\rm Id}\otimes\bigotimes_{j=0}^{f-1}x_{j}^{r_{j}},\alpha\otimes\bigotimes_{j=0}^{f-1}y_{j}^{r_{j}},c_{n}^{p^{l}(r_{l}+1)},\beta c_{n}^{p^{l}(r_{l}+1)},d_{n}^{l},\beta d_{n}^{l},$ for $n\geq 1$, where $d_{n}^{l}=\sum\limits_{\begin{subarray}{c}\mu\in I_{n}\end{subarray}}\left(\begin{array}[]{cc}\varpi^{n}&\mu\\\ 0&1\end{array}\right)\otimes\bigotimes_{l\neq j=0}^{f-1}x_{j}^{r_{j}}\otimes x_{l}^{r_{l}-1}y_{l},$ by [Hen19, Theorem 1.2]. By [Hen19, Lemma 3.6], $\left(\begin{array}[]{cc}a&b\\\ \varpi c&d\end{array}\right)\cdot d_{n}^{l}=a^{r-2p^{l}}(ad)^{p^{l}}\cdot d_{n}^{l},$ and once again it can be checked that there is no $I(1)$-invariant vector in $\tau_{r}$ with $I$-eigenvalue $a^{q-1-p^{l}}d^{r+p^{l}}$, where $t_{n}^{r+p^{l}}$ can be mapped under $\phi$, giving a contradiction. ## Acknowledgements The second author would like to thank Council of Scientific and Industrial Research, Government of India (CSIR) and Industrial Research and Consultancy Centre, IIT Bombay (IRCC) for financial support. ## References [AB13] U. K. Anandavardhanan and Gautam H. Borisagar, _On the $K(n)$-invariants of a supersingular representation of $GL_{2}(\mathbb{Q}_{p})$_, The legacy of Srinivasa Ramanujan, Ramanujan Math. Soc. Lect. Notes Ser., vol. 20, Ramanujan Math. Soc., Mysore, 2013, pp. 55–75. MR 3221302 * [AB15] by same author, _Iwahori-Hecke model for supersingular representations of ${\rm GL}_{2}(\mathbb{Q}_{p})$_, J. Algebra 423 (2015), 1–27. MR 3283706 * [BL94] L. Barthel and R. Livné, _Irreducible modular representations of ${\rm GL}_{2}$ of a local field_, Duke Math. J. 75 (1994), no. 2, 261–292. MR 1290194 * [BP12] Christophe Breuil and Vytautas Paškūnas, _Towards a modulo $p$ Langlands correspondence for ${\rm GL}_{2}$_, Mem. Amer. Math. Soc. 216 (2012), no. 1016, vi+114. MR 2931521 * [Bre03] Christophe Breuil, _Sur quelques représentations modulaires et $p$-adiques de ${\rm GL}_{2}(\mathbb{Q}_{p})$. I_, Compositio Math. 138 (2003), no. 2, 165–188. MR 2018825 * [Hen19] Yotam I. Hendel, _On the universal ${\rm mod}\,p$ supersingular quotients for ${\rm GL}_{2}(F)$ over $\overline{\mathbb{F}}_{p}$ for a general $F/\mathbb{Q}_{p}$_, J. Algebra 519 (2019), 1–38. MR 3873949 * [Sch11] Michael M. Schein, _An irreducibility criterion for supersingular $\mod\ p$ representations of ${\rm GL}_{2}(F)$ for totally ramified extensions $F$ of $\mathbb{Q}_{p}$_, Trans. Amer. Math. 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###### Abstract In this article we consider the estimation of the log-normalization constant associated to a class of continuous-time filtering models. In particular, we consider ensemble Kalman-Bucy filter based estimates based upon several nonlinear Kalman-Bucy diffusions. Based upon new conditional bias results for the mean of the afore-mentioned methods, we analyze the empirical log-scale normalization constants in terms of their $\mathbb{L}_{n}-$errors and conditional bias. Depending on the type of nonlinear Kalman-Bucy diffusion, we show that these are of order $(t^{1/2}/N^{1/2})+t/N$ or $1/N^{1/2}$ ($\mathbb{L}_{n}-$errors) and of order $[t+t^{1/2}]/N$ or $1/N$ (conditional bias), where $t$ is the time horizon and $N$ is the ensemble size. Finally, we use these results for online static parameter estimation for above filtering models and implement the methodology for both linear and nonlinear models. Keywords: Kalman-Bucy filter, Riccati equations, nonlinear Markov processes. Log-Normalization Constant Estimation using the Ensemble Kalman-Bucy Filter with Application to High-Dimensional Models BY DAN CRISAN1, PIERRE DEL MORAL2, AJAY JASRA3 & HAMZA RUZAYQAT3 1Department of Mathematics, Imperial College London, London, SW7 2AZ, UK. E-Mail<EMAIL_ADDRESS> 2Center INRIA Bordeaux Sud-Ouest & Institut de Mathematiques de Bordeaux, Bordeaux, 33405, FR. E-Mail<EMAIL_ADDRESS> 3Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955, KSA. E-Mail: <EMAIL_ADDRESS><EMAIL_ADDRESS> ## 1 Introduction The filtering problem concerns the recursive estimation of a partially observed Markov process conditioned on a path of observations. It is found in a wide class of real applications, including finance, applied mathematics and engineering; see [2] for instance. This article focusses upon the computation of the normalizing constant for certain classes of filtering problems, to be introduced below. These normalizing constants can be of interest in statistics and engineering for model selection and/or parameter estimation. In this article we study linear and Gaussian models in continuous-time. It is well-known that, for such models, under some minimal assumptions, the filter is Gaussian, with mean satisfying the Kalman-Bucy (KB) equations and the covariance matrix obeying a Riccati equation. In many cases of practical interest, these latter equations may not be computable, for instance if one does not have access to an entire trajectory of data. Instead one can use some suitable approximations obtained via time-discretization methods. However, even then, if the dimension $r_{1}$ of the state-vector is very large, the numerical approximation of the KB and Riccati equations can become computationally prohibitive, often with a computational effort of at least $\mathcal{O}(r_{1}^{2})$ per update. The case where $r_{1}$ is large occurs in many applications including ocean and atmosphere science (e.g. [1]) and oil reservoir simulations (e.g. [17]). A rather elegant and successful solution, in discrete-time, to this problem was developed in [10] in the guise of the ensemble Kalman filter (EnKF), which can reduce the cost to $\mathcal{O}(r_{1})$. The EnKF can be interpreted as a mean-field particle approximation of a conditional McKean-Vlasov diffusion (the K-B diffusion). This latter diffusion shares the same law as the filter associated to the linear and Gaussian model in continuous-time. Hence a possible alternative to recursively solving the K-B and Riccati equations is to generate $N\in\mathbb{N}$ independent copies from the K-B diffusion and use a simple Monte Carlo estimate for expectations with respect to the filter. However, the diffusion process cannot be simulated exactly, but can be approximated in a principled way by allowing the $N$ samples to interact; precise details are given in Section 2. The resulting method, named the Ensemble Kalman-Bucy Filter (EnKBF), is by now rather well- understood with several contributions on its convergence (as $N\rightarrow\infty$) analysis; see for instance [3, 5, 8, 12, 19]. In this work we focus upon using several versions of EnKBF for an online estimate of the normalization constant. In particular the contributions of this work are: 1. 1. New results on the conditional bias of the mean using the EnKBF 2. 2. A derivation of an estimate of the normalization constant using the EnKBF. 3. 3. A proof that the $\mathbb{L}_{n}-$error of the estimate on the log-scale is of $\mathcal{O}\big{(}\sqrt{\frac{t}{N}}+\frac{t}{N}\big{)}$ or $\mathcal{O}\big{(}\tfrac{1}{\sqrt{N}}\big{)}$, depending on the nonlinear Kalman-Bucy diffusion and where $t$ is the time parameter. 4. 4. A proof that the conditional bias of the estimate on the log-scale is of $\mathcal{O}\big{(}\tfrac{t+\sqrt{t}}{N}\big{)}$ or $\mathcal{O}\big{(}\frac{1}{N}\big{)}$, depending on the nonlinear Kalman-Bucy diffusion. 5. 5. A development of a method that uses this estimate to perform online static parameter estimation. The result in 1. is of independent interest, but is used directly in 4.. To the best of our knowledge the estimate in 2. is new and can be computed with a little extra computational cost over applying an EnKBF (under time discretization). Whilst several authors have used the EnKF for normalization constant estimation, e.g. [9], we have not seen the EnKBF version investigated in the literature. In addition to contribution 3. & 4., the results establish the decay of the mean square error (MSE), for instance, of the log-normalizing constant estimate as the time parameter increases. This rate is expected to occur in practice, as we will see in simulations, and parallels other results found in the literature for particle estimates of the normalization constant (e.g. such as particle filters [7]). In relation to 5., if one assumes that the model of interest has several unknown and time-homogeneous parameters, $\theta$ say, then one is interested to estimate such parameters, for instance using likelihood based methods. We show how our estimate can be leveraged for this latter task. In this paper we use the simultaneous stochastic perturbation stochastic approximation (SPSA) method [18] popular in engineering applications. SPSA is based upon a finite difference estimator of the gradient, w.r.t. $\theta$, of the log-normalizing constant. It constructs a stochastic approximation scheme for the estimation of static parameters. We are not aware of this method being used for the EnKBF. As it is a zero-order optimization method, we expect to be computationally less expensive than resorting to using other estimates of the gradient (of the log-normalizing constant). It should be noted that our work focusses upon the standard or ‘vanilla’ EnKBF, the deterministic EnKBF [16] and deterministic transport type EnKBFs [15]; other extensions are possible, for instance, to the feedback particle filter. This article is structured as follows. In Section 2 we provide details on the class of filtering problems that we address as well as details on the ensemble Kalman Bucy filter. The conditional bias of the mean associated to various EnKBFs is also presented there. In Section 3 we discuss how the normalizing constant estimate can be computed, as well as its $\mathbb{L}_{n}-$error and conditional bias. The latter is supported by numerical results checking that the order of convergence rate indeed holds even under naive Euler discretization. In Section 4 we show how the normalizing constant estimate can be used in online static parameter estimation problems. ## 2 Description of the Model and Algorithms ### 2.1 The Kalman-Bucy Filter Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space together with a filtration $(\mathcal{G}_{t})_{t\geq 0}$ which satisfies the usual conditions. On $(\Omega,\mathcal{F},\mathbb{P})$ we consider two $\mathcal{G}_{t}$-adapted processes $X=\\{X_{t},\ t\geq 0\\}$ and $Y=\\{Y_{t},\ t\geq 0\\}$ that form a time homogeneous linear-Gaussian filtering model of the following form $\left\\{\begin{array}[]{rcl}dX_{t}&=&A\leavevmode\nobreak\ X_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ dW_{t}\\\ dY_{t}&=&C\leavevmode\nobreak\ X_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{2}\leavevmode\nobreak\ dV_{t}.\end{array}\right.$ (1) In the above display, $(W_{t},V_{t})$ is an $\mathcal{G}_{t}$-adapted $(r_{1}+r_{2})$-dimensional Brownian motion, $X_{0}$ is an $\mathcal{G}_{0}$-measurable $r_{1}$-valued Gaussian random vector with mean and covariance matrix $(\mathbb{E}(X_{0}),P_{0})$ (independent of $(W_{t},V_{t})$), the symmetric matrices $R^{1/2}_{1}$ and $R^{1/2}_{2}$ are invertible, $A$ is a square $(r_{1}\times r_{1})$-matrix, $C$ is an $(r_{2}\times r_{1})$-matrix, and $Y_{0}=0$. We let ${\cal F}_{t}=\sigma\left(Y_{s},\leavevmode\nobreak\ s\leq t\right)$ be the filtration generated by the observation process. It is well-known that the conditional distribution $\eta_{t}$ of the signal state $X_{t}$ given ${\cal F}_{t}$ is a $r_{1}$-dimensional Gaussian distribution with a a mean and covariance matrix $\widehat{X}_{t}:=\mathbb{E}(X_{t}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t})\quad\mbox{\rm and}\quad P_{t}:=\mathbb{E}\left(\left(X_{t}-\mathbb{E}(X_{t}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t})\right)\left(X_{t}-\mathbb{E}(X_{t}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t})\right)^{\prime}\right)$ given by the Kalman-Bucy and the Riccati equations $\displaystyle d\widehat{X}_{t}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \widehat{X}_{t}\leavevmode\nobreak\ dt+P_{t}\leavevmode\nobreak\ C^{\prime}R^{-1}_{2}\leavevmode\nobreak\ \left(dY_{t}-C\widehat{X}_{t}dt\right)$ (2) $\displaystyle\partial_{t}P_{t}$ $\displaystyle=$ $\displaystyle\mbox{\rm Ricc}(P_{t}).$ (3) with the Riccati drift function from $\SS^{+}_{r_{1}}$ into $\SS_{r_{1}}$ (where $\SS^{+}_{r_{1}}$ (resp. $\SS_{r_{1}}$) is the collection of symmetric and positive definite (resp. semi-definite) $r_{1}\times r_{1}$ matrices defined for any $Q\in\SS^{+}_{r_{1}}$ ) by $\mbox{\rm Ricc}(Q)=AQ+QA^{\prime}-QSQ+R\quad\mbox{\rm with}\quad R=R_{1}\quad\mbox{\rm and}\quad S:=C^{\prime}R^{-1}_{2}C$ (4) ### 2.2 Nonlinear Kalman-Bucy Diffusions We now consider three conditional nonlinear McKean-Vlasov type diffusion processes $\displaystyle d\overline{X}_{t}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \overline{X}_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ d\overline{W}_{t}+{\cal P}_{\eta_{t}}C^{\prime}R^{-1}_{2}\leavevmode\nobreak\ \left[dY_{t}-\left(C\overline{X}_{t}dt+R^{1/2}_{2}\leavevmode\nobreak\ d\overline{V}_{t}\right)\right]$ (5) $\displaystyle d\overline{X}_{t}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \overline{X}_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ d\overline{W}_{t}+{\cal P}_{\eta_{t}}C^{\prime}R^{-1}_{2}\leavevmode\nobreak\ \left[dY_{t}-\left(\frac{1}{2}C\left[\overline{X}_{t}+\eta_{t}(e)\right]dt\right)\right]$ (6) $\displaystyle d\overline{X}_{t}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \overline{X}_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R_{1}{\cal P}_{\eta_{t}}^{-1}\left(\overline{X}_{t}-\eta_{t}(e)\right)\leavevmode\nobreak\ dt+{\cal P}_{\eta_{t}}C^{\prime}R^{-1}_{2}\leavevmode\nobreak\ \left[dY_{t}-\left(\frac{1}{2}C\left[\overline{X}_{t}+\eta_{t}(e)\right]dt\right)\right]$ (7) where $(\overline{W}_{t},\overline{V}_{t},\overline{X}_{0})$ are independent copies of $(W_{t},V_{t},X_{0})$ (thus independent of the signal and the observation path). In the above displayed formula ${\cal P}_{\eta_{t}}$ stands for the covariance matrix ${\cal P}_{\eta_{t}}=\eta_{t}\left[(e-\eta_{t}(e))(e-\eta_{t}(e))^{\prime}\right]\quad\mbox{\rm with}\quad\eta_{t}:=\mbox{\rm Law}(\overline{X}_{t}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t})\quad\mbox{\rm and}\quad e(x):=x$ (8) and we use the notation, $\eta_{t}(f)=\int_{\mathbb{R}^{r_{1}}}f(x)\eta_{t}(dx)$ for $f:\mathbb{R}^{r_{1}}\rightarrow\mathbb{R}^{r}$ that is $\eta_{t}-$integrable (in particular, $r=r_{1}$ in (6) and (7) and $r=r_{1}^{2}$ in (8)). Any of these probabilistic models will be commonly referred to as Kalman-Bucy (nonlinear) diffusion processes. In the following we will denote by ${\mathcal{G}}_{t}$ the augmented filtration generated by $\overline{X}_{0}$ and the triplet of independent Brownian motions $(Y_{t},\overline{W}_{t},\overline{V}_{t})$. The process (5) corresponds to the vanilla type ensemble Kalman-Bucy filter that is typically used in the literature. The process (6) is associated the so-called deterministic ensemble Kalman-Bucy filter [16] and (7) is a deterministic transport-inspired equation [15]. We have the following result that is considered, for instance, in [8]: ###### Lemma 2.1. Let $\overline{X}_{t}$ be a process such that $\mathbb{E}[\left\|\overline{X}_{0}\right\|^{2}]<\infty$ and that it satisfies any one of (5)-(7). Then the conditional mean and the conditional covariance matrix (given ${\mathcal{F}}_{t}$) of any of the nonlinear Kalman-Bucy diffusions (5)-(7) satisfy equations (2) and (3), respectively. This result enables us to approximate the mean and covariance associated to the linear filtering problem in (1) by simulating $N$ i.i.d. samples from one of the processes (5)-(7) exactly and then use the sample mean and sample covariance to approximate the mean and covariance of the filter. Since this exact simulation is seldom possible, we consider how one can generate $N-$particle systems whose sample mean and sample covariance can be used to replace the exact simulation. ###### Remark 2.1. Let us observe that equations (5)-(7) have indeed a unique solution in the class of processes $Z_{t}$ such that $\mathbb{E}[\sup_{t\in[0,T]}\left\|Z_{t}\right\|^{2}]<\infty$. To show existence, one considers the (linear version of the) equations (5)-(7) with the conditional expectations $\eta_{t}(e)$ and $\eta_{t}\left[(e-\eta_{t}(e))(e-\eta_{t}(e))^{\prime}\right]$ replaced by the solution of the equations (2) and (3), respectively, that is, $\displaystyle d\overline{X}_{t}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \overline{X}_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ d\overline{W}_{t}+P_{t}C^{\prime}R^{-1}_{2}\leavevmode\nobreak\ \left[dY_{t}-\left(C\overline{X}_{t}dt+R^{1/2}_{2}\leavevmode\nobreak\ d\overline{V}_{t}\right)\right]$ (9) $\displaystyle d\overline{X}_{t}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \overline{X}_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ d\overline{W}_{t}+P_{t}C^{\prime}R^{-1}_{2}\leavevmode\nobreak\ \left[dY_{t}-\left(\frac{1}{2}C\left[\overline{X}_{t}+\eta_{t}(e)\right]dt\right)\right]$ (10) $\displaystyle d\overline{X}_{t}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \overline{X}_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R_{1}P_{t}^{-1}\left(\overline{X}_{t}-\hat{X}_{t}\right)\leavevmode\nobreak\ dt+P_{t}C^{\prime}R^{-1}_{2}\leavevmode\nobreak\ \left[dY_{t}-\left(\frac{1}{2}C\left[\overline{X}_{t}+\hat{X}_{t}\right]dt\right)\right].$ (11) Equations (9)-(11) have a unique solution (as they are linear) that indeed satisfy the corresponding (nonlinear) equations (5)-(7). To show the uniqueness of the solutions of equations (5)-(7), one observes first that any solution $\eta_{t}$ of the (nonlinear) equations (5)-(7) has its conditional expectations $\eta_{t}(e)$ and $\eta_{t}\left[(e-\eta_{t}(e))(e-\eta_{t}(e))^{\prime}\right]$ uniquely characterized by the equations (2) and (3). Therefore they satisfy the corresponding linear versions of the equations (5)-(7), with the conditional expectations $\eta_{t}(e)$ and $\eta_{t}\left[(e-\eta_{t}(e))(e-\eta_{t}(e))^{\prime}\right]$ replaced by the solution of the equations (2) and (3). In other words, they satisfy equations (9)-(11) and therefore they are unique as the corresponding linear equations (9)-(11) have a unique solution. ###### Remark 2.2. If one modifies (1) to $\left\\{\begin{array}[]{rcl}dX_{t}&=&f(X_{t})\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ dW_{t}\\\ dY_{t}&=&C\leavevmode\nobreak\ X_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{2}\leavevmode\nobreak\ dV_{t}.\end{array}\right.$ for some non-linear function $f:\mathbb{R}^{r_{1}}\rightarrow\mathbb{R}^{r_{1}}$, one can consider a modification of any of (5)-(7). For instance in the case (5) $d\overline{X}_{t}=f(\overline{X}_{t})\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ d\overline{W}_{t}+{\cal P}_{\eta_{t}}C^{\prime}R^{-1}_{2}\leavevmode\nobreak\ \left[dY_{t}-\left(C\overline{X}_{t}dt+R^{1/2}_{2}\leavevmode\nobreak\ d\overline{V}_{t}\right)\right].$ We note however that any approximation of this process, as alluded to above, does not typically provide an approximation of the non-linear filter. Nonetheless it is considered in many works in the field. ### 2.3 Ensemble Kalman-Bucy Filters We now consider three Ensemble Kalman-Bucy filters that correspond to the mean-field particle interpretation of the nonlinear diffusion processes (5)-(7). To be more precise we let $(\overline{W}_{t}^{i},\overline{V}_{t}^{i},\xi_{0}^{i})_{1\leq i\leq N}$ be $N$ independent copies of $(\overline{W}_{t},\overline{V}_{t},\overline{X}_{0})$. In this notation, we have the three McKean-Vlasov type interacting diffusion processes for $i\in\\{1,\dots,N\\}$ $\displaystyle d\xi_{t}^{i}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \xi_{t}^{i}dt+R_{1}^{1/2}d\overline{W}_{t}^{i}+p_{t}C^{\prime}R_{2}^{-1}\left[dY_{t}-\left(C\xi_{t}^{i}dt+R_{2}^{1/2}\leavevmode\nobreak\ d\overline{V}_{t}^{i}\right)\right]$ (12) $\displaystyle d\xi_{t}^{i}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \xi_{t}^{i}dt+R_{1}^{1/2}d\overline{W}_{t}^{i}+p_{t}C^{\prime}R_{2}^{-1}\left[dY_{t}-\left(\frac{1}{2}C\left[\xi_{t}^{i}+\eta_{t}^{N}(e)\right]dt\right)\right]$ (13) $\displaystyle d\xi_{t}^{i}$ $\displaystyle=$ $\displaystyle A\leavevmode\nobreak\ \xi_{t}^{i}dt+R_{1}(p_{t})^{-1}\left(\xi_{t}^{i}-\eta_{t}^{N}(e)\right)\leavevmode\nobreak\ dt+p_{t}C^{\prime}R_{2}^{-1}\left[dY_{t}-\left(\frac{1}{2}C\left[\xi_{t}^{i}+\eta_{t}^{N}(e)\right]dt\right)\right]$ (14) with the rescaled particle covariance matrices $p_{t}:=\left(1-\frac{1}{N}\right)^{-1}\leavevmode\nobreak\ \mathcal{P}_{\eta_{t}^{N}}=\frac{1}{N-1}\sum_{1\leq i\leq N}\left(\xi_{t}^{i}-m_{t}\right)\left(\xi_{t}^{i}-m_{t}\right)^{\prime}$ (15) and the empirical measures $\eta_{t}^{N}:=\frac{1}{N}\sum_{1\leq i\leq N}\delta_{\xi_{t}^{i}}\quad\mbox{\rm and the sample mean}\quad m_{t}:=\frac{1}{N}\sum_{1\leq i\leq N}\xi_{t}^{i}.$ Note that for (14) one needs $N\geq r_{1}$ for the almost sure invertibility of $p_{t}$, but it is also sufficient to use the pseudo inverse instead of the inverse. These processes have been thoroughly studied in the literature and a review can be found in [3]. We have the evolution equations for (12) and (13) $\begin{array}[]{rcl}dm_{t}&=&\displaystyle A\leavevmode\nobreak\ m_{t}dt+p_{t}\leavevmode\nobreak\ C^{\prime}\Sigma^{-1}\leavevmode\nobreak\ \left(dY_{t}-Cm_{t}\leavevmode\nobreak\ dt\right)+\frac{1}{\sqrt{N+1}}\leavevmode\nobreak\ d\overline{M}_{t}\\\ &&\\\ dp_{t}&=&\displaystyle\mbox{\rm Ricc}(p_{t})\leavevmode\nobreak\ dt+\frac{1}{\sqrt{N}}\leavevmode\nobreak\ dM_{t}\end{array}$ with a triplet of pairwise orthogonal martingales $M_{t}$ and $\overline{M}_{t}$ and the innovation martingale $\widehat{M}_{t}:=Y_{t}-C\int_{0}^{t}\widehat{X}_{s}ds.$ Note that for any $t\geq 0$, the absolute moments of $\overline{M}_{t}$ and $M_{t}$ are uniformly, w.r.t. $N$, upper-bounded. We remark that the mathematical theory for (14) in the linear-Gaussian setting reverts to that of the standard Kalman-Bucy filter as detailed in [5]. ###### Remark 2.3. Returning to the context of Remark 2.2, one could, for instance, run the following version of (12) for $i\in\\{1,\dots,N\\}$ $d\xi_{t}^{i}=f(\xi_{t}^{i})dt+R_{1}^{1/2}d\overline{W}_{t}^{i}+p_{t}^{N}C^{\prime}R_{2}^{-1}\left[dY_{t}-\left(C\xi_{t}^{i}dt+R_{2}^{1/2}\leavevmode\nobreak\ d\overline{V}_{t}^{i}\right)\right]$ again, noting that this will typically not produce a consistent approximation of the non-linear filter, as would be the case for (12) with the model (1). Below we denote $\\|\cdot\\|$ as the $L_{2}-$norm for vectors. For a square matrix $B$ say, $B_{\textrm{sym}}=\tfrac{1}{2}(B+B^{\prime})$ and $\mu(B)$ being the largest eigenvalue of $B_{\textrm{sym}}$. In the cases of (12) and (13), [8] consider the following assumption (recall (4)): we have $\mu(S)>0$ with $S=\mu(S)Id$ (16) where $Id$ is the $r_{1}\times r_{1}$ identity matrix (in general we use $Id$ to denote the identity matrix of ‘appropriate dimension’, appropriate depending on the context). [8] prove the following time-uniform convergence theorem for the mean. ###### Theorem 2.1. Consider the cases of (12) and (13). Assume that (16) holds and that $\mu(A)<0$. Then for any $n\geq 1$ and $N$ sufficiently large, we have that $\sup_{t\geq 0}\mathbb{E}[\\|m_{t}-\widehat{X}_{t}\\|^{n}]^{\tfrac{1}{n}}\leq\frac{\mathsf{C}(n)}{\sqrt{N}}$ where $\mathsf{C}(n)<\infty$ is a constant that does not depend upon $N$. We denote by $P_{\infty}$ as the solution to $\mbox{\rm Ricc}(P_{\infty})=0$. Set $\mu(A-P_{\infty}S)$ as the largest eigenvalue of $\tfrac{1}{2}(A-P_{\infty}S+(A-P_{\infty}S)^{\prime})$. For the case of (13), the following result, presented in [3], can be shown. ###### Theorem 2.2. Consider the case of (13). Assume that $\mu(A-P_{\infty}S)<0$ and $S\in\mathbb{S}_{r_{1}}^{+}$. Then for any $n\geq 1$, $N\geq 2$ there exists a $t(n,N)>0$ with $t(n,N)\rightarrow\infty$ as $N\rightarrow\infty$, such that for any $t\in[0,t(n,N)]$ $\mathbb{E}[\\|m_{t}-\widehat{X}_{t}\\|^{n}]^{\tfrac{1}{n}}\leq\frac{\mathsf{C}}{\sqrt{N}}$ where $\mathsf{C}<\infty$ is a constant that does not depend upon $N$. $t(n,N)$ is characterized in [6]. We note that $t(n,N)$ is $\mathcal{O}(\log(N))$; see [6, Theorem 2.1] when $\epsilon=1/\sqrt{N}$ ($\epsilon$ is as [6, Theorem 2.1]). For the case of (14) we have the following result. Below we cite [3, eq. (2.7)] and it should be noted that the notation of that article differs slightly to this one, so we mean of course that the equation holds in the notation of this article. Also note that we use the inverse of $p_{t}^{N}$, but as noted above this need not be the case and then the constraint on $N$ below is not required. ###### Theorem 2.3. Consider the case of (14). Assume that [3, eq. (2.7)] holds. Then for any $n\geq 1$ there exists a $\mathsf{C}<\infty$, $\kappa\in(0,1)$ such that for any $N\geq r_{1}$ and $t\geq 0$ we have that $\mathbb{E}[\\|m_{t}-\widehat{X}_{t}\\|^{n}]^{\tfrac{1}{n}}\leq\frac{\mathsf{C}\kappa^{t}}{\sqrt{N}}.$ ### 2.4 Conditional Bias Result We set $\widehat{m}_{t}:=\mathbb{E}(m_{t}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t})\qquad\widehat{p}_{t}:=\mathbb{E}(p_{t}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}).$ We consider bounds on $\mathbb{E}\left(\\|\widehat{m}_{t}-\widehat{X}_{t}\\|^{n}\right)^{1/n}.$ These bounds do not currently exist in the literature and will allow us to investigate our new estimator for the normalization constant, later on in the article. In the case (14) we have (see e.g. [5]) $\widehat{m}_{t}=\widehat{X}_{t}\quad\mbox{\rm and}\quad\widehat{p}_{t}=P_{t}$ (17) so we focus only on (12) and (13). Using the second order Taylor-type expansions presented in [4] when the pair $(A,R_{1}^{1/2})$ is stabilizable and $(A,S^{1/2})$ is detectable we have the uniform almost sure bias and for any $n\geq 1$, the $\mathbb{L}_{n}$-mean error estimates $0\leq P_{t}-\widehat{p}_{t}\leq\mathsf{C}_{1}/N\quad\mbox{\rm and}\quad\mathbb{E}\left(\\|P_{t}-p_{t}\\|^{n}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)^{1/n}\leq\mathsf{C}_{2}(n)/\sqrt{N}$ (18) as soon as $N$ is chosen sufficiently large, for some deterministic constants $\mathsf{C}_{1},\mathsf{C}_{2}(n)$ that do not depend on the time horizon or $N$ (but $\mathsf{C}_{2}(n)$ does depend upon $n$). Whenever $\mu(A)<0$, Theorem 2.1 yields the rather crude estimate $\mathbb{E}\left(\\|\widehat{m}_{t}-\widehat{X}_{t}\\|^{n}\right)^{1/n}=\mathbb{E}\left(\\|\mathbb{E}\left(m_{t}-\widehat{X}_{t}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)\\|^{n}\right)^{1/n}\leq\mathbb{E}\left(\\|m_{t}-\widehat{X}_{t}\\|^{n}\right)^{1/n}\leq\frac{\mathsf{C}(n)}{\sqrt{N}}.$ (19) Our objective is to improve the estimate (19). We observe that $\begin{array}[]{l}d(m_{t}-\widehat{X}_{t})\\\ \\\ =\displaystyle A\leavevmode\nobreak\ (m_{t}-\widehat{X}_{t})dt+(p_{t}-P_{t})\leavevmode\nobreak\ C^{\prime}\Sigma^{-1}\leavevmode\nobreak\ \left(dY_{t}-Cm_{t}\leavevmode\nobreak\ dt\right)\\\ \\\ \displaystyle\hskip 85.35826pt+P_{t}\leavevmode\nobreak\ C^{\prime}\Sigma^{-1}\leavevmode\nobreak\ \left(dY_{t}-Cm_{t}\leavevmode\nobreak\ dt\right)-P_{t}\leavevmode\nobreak\ C^{\prime}\Sigma^{-1}\leavevmode\nobreak\ \left(dY_{t}-C\widehat{X}_{t}\leavevmode\nobreak\ dt\right)+\frac{1}{\sqrt{N+1}}\leavevmode\nobreak\ d\overline{M}_{t}\end{array}$ (20) which yields $\begin{array}[]{l}d(\widehat{m}_{t}-\widehat{X}_{t})\\\ \\\ =\displaystyle(A-P_{t}S)\leavevmode\nobreak\ (\widehat{m}_{t}-\widehat{X}_{t})dt+(\widehat{p}_{t}-P_{t})\leavevmode\nobreak\ C^{\prime}\Sigma^{-1}\leavevmode\nobreak\ \left(dY_{t}-C\widehat{X}_{t}\leavevmode\nobreak\ dt\right)+\mathbb{E}\left((P_{t}-p_{t})\leavevmode\nobreak\ S(m_{t}-\widehat{X}_{t})\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)\leavevmode\nobreak\ dt.\end{array}$ (21) Equation (21) is deduced by conditioning both terms in (20) with respect to ${\cal F}_{t}$, exchanging the (stochastic) integration with the conditional expectation for all the terms on the right hand side of (20). To justify this Fubini-like exchange one can use, for example, Lemma 3.21 in [2]. One also needs to use the fact that, for arbitrary $0\leq s\leq t$, $\widehat{m}_{s}=\mathbb{E}(m_{s}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{s})=\mathbb{E}(m_{s}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t})$ and $\widehat{p}_{s}:=\mathbb{E}(p_{s}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{s})=\mathbb{E}(p_{s}\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t})$. To justify this, one can use, for example, Proposition 3.15 in [2]. We note that both Proposition 3.15 and Lemma 3.21 hold true when the conditioning is done with respect the equivalent probability measure $\tilde{\mathbb{P}}$ under which $Y$ is a Brownian motion. However since $(\overline{W}_{t},\overline{V}_{t},\overline{X}_{0})$ are independent of $Y$ under $\mathbb{P}$ they remain independent of $Y$ under $\tilde{\mathbb{P}}$ and therefore conditioning the processes $\xi^{i}$ under $\tilde{\mathbb{P}}$ is the same as conditioning them under $\mathbb{P}$. Note that argument this does _not_ apply to the original signal $X$. Let ${\cal E}_{s,t}$ be the exponential semigroup (or the state transition matrix) associated with the smooth flow of matrices $t\mapsto(A-P_{t}S)$ defined for any $s\leq t$ by the forward and backward differential equations, $\partial_{t}\,{\cal E}_{s,t}=(A-P_{t}S)\,{\cal E}_{s,t}\quad\mbox{\rm and}\quad\partial_{s}\,{\cal E}_{s,t}=-{\cal E}_{s,t}\,(A-P_{s}S)$ with ${\cal E}_{s,s}=Id$. Equivalently in terms of the matrices ${\cal E}_{t}:={\cal E}_{0,t}$ we have ${\cal E}_{s,t}={\cal E}_{t}{\cal E}_{s}^{-1}$. Under some observability and controllability conditions, we have that the drift matrices $(A-P_{t}S)$ delivers a uniformly stable time varying linear system in the sense that $\\|{\cal E}_{s,t}\\|\leq c\leavevmode\nobreak\ e^{-\lambda\leavevmode\nobreak\ (t-s)}\quad\mbox{\rm for some}\quad\lambda>0.$ (22) See for instance Eq. (2.13) in the review article [3]. In this notation, recalling that $\widehat{m}_{0}=\widehat{X}_{0}$ we have the bias formulae $\widehat{m}_{t}-\widehat{X}_{t}=\int_{0}^{t}{\cal E}_{s,t}\leavevmode\nobreak\ \underbrace{(\widehat{p}_{s}-P_{s})}_{\in[-c_{1}/N,0]\leavevmode\nobreak\ a.e.}\leavevmode\nobreak\ C^{\prime}\Sigma^{-1}\leavevmode\nobreak\ \left(dY_{s}-C\widehat{X}_{s}\leavevmode\nobreak\ dt\right)+\int_{0}^{t}{\cal E}_{s,t}\leavevmode\nobreak\ \mathbb{E}\left((P_{s}-p_{s})\leavevmode\nobreak\ S(m_{s}-\widehat{X}_{s})\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{s}\right)\leavevmode\nobreak\ ds.$ Note that $(\widehat{p}_{s}-P_{s})\in[-c_{1}/N,0]$ a.e. can be justified via (18). Combining the Burkholder-Davis-Gundy inequality with the almost sure and uniform bias estimate (18) and the exponential decay (22) we obtain the uniform estimate $\begin{array}[]{l}\displaystyle\mathbb{E}\left(\\|\int_{0}^{t}{\cal E}_{s,t}\leavevmode\nobreak\ (\widehat{p}_{s}-P_{s})\leavevmode\nobreak\ C^{\prime}\Sigma^{-1}\leavevmode\nobreak\ \left(dY_{s}-C\widehat{X}_{s}\leavevmode\nobreak\ dt\right)\\|^{n}\right)^{1/n}\displaystyle\leq\frac{\mathsf{C}(n)}{N}\end{array}$ for some deterministic constants $\mathsf{C}(n)$ that do not depend upon $t$ nor $N$. Conversely, combining Hölder’s inequality with the uniform variance estimate (18) we have the almost sure inequality $\mathbb{E}\left(\\|\mathbb{E}\left((P_{s}-p_{s})\leavevmode\nobreak\ S(m_{s}-\widehat{X}_{s})\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{s}\right)\\|^{n}\right)^{1/n}\leq\frac{\mathsf{C}(n)}{N}\leavevmode\nobreak\ $ for some deterministic constants $\mathsf{C}(n)$ that do not depend upon $t$ nor $N$. Applying the generalized Minkowski inequality we have thus proved the following theorem. ###### Theorem 2.4. Consider the cases of (12) and (13). Assume that (16) holds and that $\mu(A)<0$. Then for any $n\geq 1$ and $N$ sufficiently large, we have $\sup_{t\geq 0}\mathbb{E}\left(\\|\widehat{m}_{t}-\widehat{X}_{t}\\|^{n}\right)^{1/n}\leq\frac{\mathsf{C}(n)}{N}$ where $\mathsf{C}(n)<\infty$ is a constant that does not depend upon $N$. ## 3 Computing the Normalizing Constant ### 3.1 Estimation We define $Z_{t}:=\frac{{\mathcal{L}}_{X_{0:t},Y_{0:t}}}{{\mathcal{L}}_{X_{0:t},W_{0:t}}}$ to be the density of ${\mathcal{L}}_{X_{0:t},Y_{0:t}}$, the law of the process $(X,Y)$ and that of ${\mathcal{L}}_{X_{0:t},W_{0:t}}$, the law of the process $(X,W)$. That is, ${\mathbb{E}}[f(X_{0:t})g(Y_{0:t})]={\mathbb{E}}[f(X_{0:t})g(W_{0:t})Z_{t}(X,Y)].$ One can show that (see Exercise 3.14 pp 56 in [2]) $Z_{t}(X,Y)=\exp{\left[\int_{0}^{t}\left[\langle CX_{s},R_{2}^{-1}dY_{s}\rangle-\frac{1}{2}\langle X_{s},SX_{s}\rangle\leavevmode\nobreak\ ds\right]\right]}.$ Following a standard approach (see, e.g., Chapter 3 in [2]), we introduce a probability measure $\tilde{\mathbb{P}}$ by specifying its Radon–Nikodym derivative with respect to $\mathbb{P}$ to be given by $Z_{t}(X,Y)^{-1}$, i.e. $\left.\frac{{\mathrm{d}}\tilde{\mathbb{P}}}{{\mathrm{d}}\mathbb{P}}\right\|_{\mathcal{G}_{t}}=Z_{t}.$ Under $\tilde{\mathbb{P}}$, the observation process $Y$ is a scaled Brownian motion independent of $X$; additionally the law of the signal process $X$ under $\tilde{\mathbb{P}}$ is the same as its law under $\mathbb{P}$111The proof at this statement is an immediate application of Girsanov’s theorem and follows in a similar manner with the proof of Proposition 3.13 in [2].. Moreover, for every $f$ defined on the signal path space, we have the Bayes’ formula $\mathbb{E}\left(f((X_{s})_{s\in[0,t]})\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)=\frac{\tilde{\mathbb{E}}\left(f((X_{s})_{s\in[0,t]})\leavevmode\nobreak\ Z_{t}(X,Y)\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)}{\tilde{\mathbb{E}}\left(Z_{t}(X,Y)\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)}.$ (23) where $\tilde{\mathbb{E}}()$ means expectation with respect to $\tilde{\mathbb{P}}$222 Formula (23) is commonly known as the Kallianpur- Striebel formula. For a proof, see, e.g., Prop 3.16 in [2].. Since, under $\tilde{\mathbb{P}}$, $X$ and $Y$ are independent we can interpret the numerator $\tilde{\mathbb{E}}\left(f((X_{s})_{s\in[0,t]})\leavevmode\nobreak\ Z_{t}(X,Y)\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)$ and $\tilde{\mathbb{E}}\left(Z_{t}(X,Y)\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)$ as integrals with respect to the law of the signal, where the integrant has the observation process path fixed $Y$.333This interpretation can be made rigorous, see e.g. the robust representation formulation of the filtering problem described in Chapter 5 in [2].. We can therefore, let $\overline{Z}_{t}(Y)$ be the likelihood function defined by $\overline{Z}_{t}(Y):=\mathbb{E}_{Y}\left(Z_{t}(X,Y)\right),$ where $\mathbb{E}_{Y}\left(\mbox{\LARGE.}\right)$ stands for the expectation w.r.t. the signal process when the observation is fixed and independent of the signal. In this notation, the Bayes’ formula (23) takes the form $\mathbb{E}\left(f((X_{s})_{s\in[0,t]})\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right)=\frac{\mathbb{E}_{Y}\left(f((X_{s})_{s\in[0,t]})\leavevmode\nobreak\ Z_{t}(X,Y)\right)}{\mathbb{E}_{Y}\left(Z_{t}(X,Y)\right)}.$ We have $\displaystyle dZ_{t}(X,Y)$ $\displaystyle=$ $\displaystyle\left[\langle CX_{t},R_{2}^{-1}dY_{t}\rangle-\frac{1}{2}\langle X_{t},SX_{t}\rangle\leavevmode\nobreak\ dt\right]Z_{t}+\frac{1}{2}\leavevmode\nobreak\ \langle X_{t},SX_{t}\rangle\leavevmode\nobreak\ Z_{t}\leavevmode\nobreak\ dt$ $\displaystyle=$ $\displaystyle Z_{t}(X,Y)\leavevmode\nobreak\ \langle CX_{t},R_{2}^{-1}dY_{t}\rangle$ We check this claim using the fact that $\displaystyle d\langle CX_{t},R_{2}^{-1}dY_{t}\rangle d\langle CX_{t},R_{2}^{-1}dY_{t}\rangle$ $\displaystyle=$ $\displaystyle\sum_{k,l}(CX_{t})(k)(R^{-1/2}_{2}dV_{t})(k)(CX_{t})(l)(R^{-1/2}_{2}dV_{t})(l)$ $\displaystyle=$ $\displaystyle\sum_{k,l,k^{\prime},l^{\prime}}(CX_{t})(k)\leavevmode\nobreak\ R^{-1/2}_{1}(k,k^{\prime})dV_{t}(k^{\prime})(CX_{t})(l)R_{2}^{-1/2}(l,l^{\prime})dV_{t}(l^{\prime})$ $\displaystyle=$ $\displaystyle\sum_{k,l,k^{\prime}}(CX_{t})(k)\leavevmode\nobreak\ R_{2}^{-1}(k,l)(CX_{t})(l)dt=\langle X_{t},SX_{t}\rangle\leavevmode\nobreak\ dt$ This implies that $\overline{Z}_{t}(Y)=1+\int_{0}^{t}\leavevmode\nobreak\ \overline{Z}_{s}(Y)\leavevmode\nobreak\ \overline{Z}_{s}(Y)^{-1}\mathbb{E}_{Y}\left(Z_{s}(X,Y)\leavevmode\nobreak\ \langle CX_{s},R_{2}^{-1}dY_{s}\rangle\right)=1+\int_{0}^{t}\leavevmode\nobreak\ \overline{Z}_{s}(Y)\leavevmode\nobreak\ \langle C\widehat{X}_{s},R_{2}^{-1}dY_{s}\rangle$ from which we conclude that $\overline{Z}_{t}(Y)=\exp{\left[\int_{0}^{t}\left[\leavevmode\nobreak\ \langle C\widehat{X}_{s},R_{2}^{-1}dY_{s}\rangle-\frac{1}{2}\langle\widehat{X}_{s},S\widehat{X}_{s}\rangle\leavevmode\nobreak\ ds\right]\right]}$ or equivalently $d\overline{Z}_{t}(Y)=\overline{Z}_{t}(Y)\leavevmode\nobreak\ \langle C\widehat{X}_{t},R_{2}^{-1}dY_{t}\rangle.$ This suggests the estimator: $\overline{Z}_{t}^{N}(Y)=\exp{\left[\int_{0}^{t}\left[\leavevmode\nobreak\ \langle Cm_{s},R_{2}^{-1}dY_{s}\rangle-\frac{1}{2}\langle m_{s},Sm_{s}\rangle\leavevmode\nobreak\ ds\right]\right]}.$ (24) which satisfies the equation $d\overline{Z}_{t}(Y)=\overline{Z}_{t}(Y)\leavevmode\nobreak\ \langle Cm_{t},R_{2}^{-1}dY_{t}\rangle.$ (25) We remark that any of the three processes (12)-(14) can be used to compute $\overline{Z}_{t}^{N}(Y)$. ### 3.2 Justification of the Estimator We now seek to justify the estimator $\overline{Z}_{t}^{N}(Y)$. We have $\begin{array}[]{l}\overline{Z}_{t}(X,Y)\\\ \\\ :=\overline{Z}_{t}(Y)^{-1}Z_{t}(X,Y)\\\ \\\ =\exp{\left[\int_{0}^{t}\left[\langle C(X_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle-\left[\frac{1}{2}\langle X_{s},SX_{s}\rangle-\frac{1}{2}\langle\widehat{X}_{s},S\widehat{X}_{s}\rangle-\langle X_{s}-\widehat{X}_{s},S\widehat{X}_{s}\rangle\right]\leavevmode\nobreak\ ds\right]\right]}\\\ \\\ =\exp{\left[\int_{0}^{t}\left[\langle C(X_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle-\frac{1}{2}\langle(X_{s}-\widehat{X}_{s}),S(X_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ ds\right]\right]}\end{array}$ from which we conclude that $d\overline{Z}_{t}(X,Y)=\overline{Z}_{t}(X,Y)\leavevmode\nobreak\ \langle C(X_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle.$ This already implies that $\mathbb{E}\left[\overline{Z}_{t}(X,Y)\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right]=1\Longleftrightarrow\mathbb{E}\left[Z_{t}(X,Y)\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t}\right]=\overline{Z}_{t}(Y).$ Now, we have $\begin{array}[]{l}\overline{Z}_{t}^{N}(Y)\overline{Z}_{t}^{-1}(Y)\\\ \\\ =\exp{\left[\int_{0}^{t}\left[\leavevmode\nobreak\ \langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}dY_{s}\rangle-\frac{1}{2}\left[\langle m_{s},Sm_{s}\rangle-\langle\widehat{X}_{s},S\widehat{X}_{s}\rangle\right]\leavevmode\nobreak\ ds\right]\right]}\\\ \\\ =\exp{\left[\int_{0}^{t}\left[\leavevmode\nobreak\ \langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}dY_{s}\rangle-\frac{1}{2}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ ds\right]\right]\nobreak\leavevmode}\\\ \\\ \times\exp{\left[\int_{0}^{t}\left[\leavevmode\nobreak\ \frac{1}{2}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle-\frac{1}{2}\left[\langle m_{s},Sm_{s}\rangle-\langle\widehat{X}_{s},S\widehat{X}_{s}\rangle\leavevmode\nobreak\ ds\right]\right]\right]}\\\ \\\ =\exp{\left[\int_{0}^{t}\left[\leavevmode\nobreak\ \langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}dY_{s}\rangle-\frac{1}{2}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ ds\right]\right]}\times\exp{\left[\int_{0}^{t}\langle\widehat{X}_{s}-m_{s},S\widehat{X}_{s}\rangle ds\right]}\\\ \\\ =\exp{\left[\int_{0}^{t}\left[\leavevmode\nobreak\ \langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}dY_{s}\rangle-\frac{1}{2}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ ds\right]\right]}\times\exp{\left[-\int_{0}^{t}\langle C(m_{s}-\widehat{X}_{s}),R^{-1}C\widehat{X}_{s}\rangle ds\right]}\\\ \\\ =\exp{\left[\int_{0}^{t}\left[\leavevmode\nobreak\ \langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle-\frac{1}{2}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ ds\right]\right]}\end{array}$ Observe that $dY_{s}-C\widehat{X}_{s}ds$ is an ${\cal F}_{s}$-martingale increment. We also have $d(\overline{Z}_{t}^{N}(Y)\overline{Z}_{t}^{-1}(Y))=\overline{Z}_{t}^{N}(Y)\overline{Z}_{t}^{-1}(Y)\leavevmode\nobreak\ \langle C(m_{t}-\widehat{X}_{t}),R_{2}^{-1}(dY_{t}-C\widehat{X}_{t}dt)\rangle.$ (26) Now to conclude, it follows that that $\overline{Z}_{t}^{N}(Y)\overline{Z}_{t}^{-1}(Y)$ is a positive local martingale and therefore a supermartingale, hence $\mathbb{E}(\overline{Z}_{t}^{N}(Y)\overline{Z}_{t}^{-1}(Y))\leq\mathbb{E}(\overline{Z}_{0}^{N}(Y)\overline{Z}_{0}^{-1}(Y))=1$ for all $t\geq 0$. To justify the martingale property of $\overline{Z}_{t}^{N}(Y)\overline{Z}_{t}^{-1}(Y)$ one can use, for example, an argument based on Corollary 5.14 in [13]. As a result of the above calculations we can deduce that $\overline{Z}_{t}^{N}(Y)$ is in some sense well-defined, but a biased estimator of the normalization constant. Therefore, we will focus on the logarithm of the normalization constant, as it is this latter quantity that is used in practical algorithms. We begin by investigating the conditional bias. ### 3.3 Conditional Bias We now consider the estimate of the logarithm of the normalizing constant $U_{t}^{N}(Y):=\log(\overline{Z}_{t}^{N}(Y))$ with the notation $U_{t}(Y)=\log(\overline{Z}_{t}^{N}(Y))$. Set $\widehat{U}_{t}(Y):=\mathbb{E}(U_{t}^{N}(Y)\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{t})$. Then, using (26) we have $\begin{array}[]{l}U_{t}^{N}(Y)-U_{t}(Y)\\\ \\\ =\log(\overline{Z}_{t}^{N}(Y)/\overline{Z}_{t}(Y))\\\ \\\ \displaystyle=\int_{0}^{t}\left[\leavevmode\nobreak\ \langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle-\frac{1}{2}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ ds\right].\end{array}$ This yields the bias formula $\begin{array}[]{l}\widehat{U}_{t}(Y)-U_{t}(Y)\\\ \\\ \displaystyle=\int_{0}^{t}\left[\leavevmode\nobreak\ \langle C(\widehat{m}_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle-\frac{1}{2}\leavevmode\nobreak\ \mathbb{E}\left(\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{s}\right)\leavevmode\nobreak\ ds\right].\end{array}$ (27) Taking the expectation in the above display, when (16) holds and $\mu(A)<0$ and applying Theorem 2.1 we readily check that $0\leq\mathbb{E}\left(U_{t}(Y)\right)-\mathbb{E}\left(\widehat{U}_{t}(Y)\right)=\frac{\mu(S)}{2}\leavevmode\nobreak\ \int_{0}^{t}\mathbb{E}\left(\\|m_{s}-\widehat{X}_{s}\\|^{2}\right)\leavevmode\nobreak\ ds\leq\frac{\mathsf{C}t}{N}$ which is the bias, but is not of particular interest. So we will focus on the conditional bias $\mathbb{E}\left(\left[\widehat{U}_{t}(Y)-U_{t}(Y)\right]^{n}\right)^{1/n}$. In the case (14), recall (17). In this context, using the generalized Minkowski inequality and under the assumptions of Theorem 2.3 we find that $\sup_{t\geq 0}{\mathbb{E}\left(\left\|\widehat{U}_{t}(Y)-U_{t}(Y)\right]^{n}\right\|^{1/n}}\leq\frac{\mathsf{C}(n)}{N}$ where $\mathsf{C}(n)<\infty$ is a constant that does not depend upon $N$. For the cases of (12) and (13), we start with the fact that the conditional bias in (27) is decomposed into two terms $\alpha_{t}:=\int_{0}^{t}\langle C(\widehat{m}_{t}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle\quad\mbox{\rm and}\quad\beta_{t}:=\frac{1}{2}\leavevmode\nobreak\ \int_{0}^{t}\mathbb{E}\left(\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ \|\leavevmode\nobreak\ {\cal F}_{s}\right)\leavevmode\nobreak\ ds.$ Using the uniform estimates presented in Section 2.3 we have $\mathbb{E}\left(\left\|\beta_{t}\right\|^{n}\right)^{1/n}\leq\frac{\mathsf{C}(n)t}{N}$ for some deterministic constants $\mathsf{C}(n)$ that do not depend on the time horizon. On the other hand, combining the Burkholder-Davis-Gundy inequality with the uniform bias estimate (19) we have the rather crude estimate $\mathbb{E}\left(\left\|\alpha_{t}\right\|^{n}\right)\leq\mathsf{C}(n)\left(\frac{t}{N}\right)^{n/2}$ (28) for some deterministic constants $\mathsf{C}(n)$ that do not depend on the time horizon. Arguing as for the proof of Theorem 2.4, we check that $\mathbb{E}\left(\left\|\alpha_{t}\right\|^{n}\right)\leq\mathsf{C}(n)\left(\frac{\sqrt{t}}{N}\right)^{n}.$ Observe that the above improves the crude estimate stated in (28). This yields the following corollary. ###### Corollary 3.1. Consider the cases of (12) and (13). Assume that (16) holds and that $\mu(A)<0$. Then for any $n\geq 1$, $t\geq 0$ and $N$ sufficiently large, we have that $\mathbb{E}\left(\left\|\widehat{U}_{t}(Y)-U_{t}(Y)\right\|^{n}\right)^{1/n}\leq\frac{\mathsf{C}(n)(t+\sqrt{t})}{N}$ where $\mathsf{C}(n)<\infty$ is a constant that does not depend upon $t$ or $N$. ### 3.4 $\mathbb{L}_{n}-$Error We now consider the $\mathbb{L}_{n}-$error of the estimate of the log-of the normalizing constant $U_{t}^{N}(Y)=\log(\overline{Z}_{t}^{N}(Y)).$ In the case of (12) and (13) we have the following. ###### Proposition 3.1. Consider the cases of (12) and (13). Assume that (16) holds and that $\mu(A)<0$. Then for any $n\geq 1$, $t\geq 0$ and $N$ sufficiently large, we have that $\mathbb{E}\left(\left\|U_{t}^{N}(Y)-U_{t}(Y)\right\|^{n}\right)^{1/n}\leq\mathsf{C}(n)\Big{(}\sqrt{\frac{t}{N}}+\frac{t}{N}\Big{)}$ where $\mathsf{C}(n)<\infty$ is a constant that does not depend upon $t$ or $N$. ###### Proof. We have that $U_{t}^{N}(Y)-U_{t}(Y)=\int_{0}^{t}\Big{[}\langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle-\frac{1}{2}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle\leavevmode\nobreak\ ds\Big{]}.$ So one can apply the Minkowski inequality to yield that $\mathbb{E}\left(\left\|U_{t}^{N}(Y)-U_{t}(Y)\right\|^{n}\right)^{1/n}\leq$ $\mathbb{E}\left(\Big{\|}\int_{0}^{t}\langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle ds\Big{\|}^{n}\right)^{1/n}+\frac{1}{2}\leavevmode\nobreak\ \mathbb{E}\left(\Big{\|}\int_{0}^{t}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle ds\Big{\|}^{n}\right)^{1/n}.$ (29) For the left term on the R.H.S. of (29) one can use the Burkholder-Gundy-Davis inequality along with Theorem 2.1 to yield that $\mathbb{E}\left(\Big{\|}\int_{0}^{t}\langle C(m_{s}-\widehat{X}_{s}),R_{2}^{-1}(dY_{s}-C\widehat{X}_{s}ds)\rangle ds\Big{\|}^{n}\right)^{1/n}\leq\frac{\mathsf{C}(n)t^{\frac{1}{2}}}{\sqrt{N}}$ for some $\mathsf{C}<\infty$ a constant that does not depend upon $t$ or $N$. For the right term on the R.H.S. of (29) one can use the generalized Minkowski inequality and Theorem 2.1 to give $\mathbb{E}\left(\Big{\|}\int_{0}^{t}\langle(m_{s}-\widehat{X}_{s}),S(m_{s}-\widehat{X}_{s})\rangle ds\Big{\|}^{n}\right)^{1/n}\leq\frac{\mathsf{C}(n)t}{N}$ for some $\mathsf{C}(n)<\infty$ a constant that does not depend upon $t$ or $N$. The proof is now easily concluded. ∎ In the case of (13) we can refine further: ###### Proposition 3.2. Consider the case of (13). Assume that $\mu(A-P_{\infty}S)<0$ and $S\in\mathbb{S}_{r_{1}}^{+}$. Then for any $N\geq 2$ there exists a $t(N)>0$ with $t(N)\rightarrow\infty$ as $N\rightarrow\infty$, such that for any $t\in[0,t(N)]$ $\mathbb{E}\left(\left\|U_{t}^{N}(Y)-U_{t}(Y)\right\|^{n}\right)^{1/n}\leq\mathsf{C}(n)\Big{(}\sqrt{\frac{t}{N}}+\frac{t}{N}\Big{)}$ where $\mathsf{C}(n)<\infty$ is a constant that does not depend upon $t$ or $N$. ###### Proof. The proof is essentially as Proposition 3.1 except one should use Theorem 2.2 instead of Theorem 2.1. ∎ In the case of (14) we have the following, whose proof is again similar to the above results. ###### Proposition 3.3. Consider the case of (14) . Assume that [3, eq. (2.7)] holds. Then for any $t\geq 0$ and $N\geq r_{1}$, we have that $\mathbb{E}\left(\left\|U_{t}^{N}(Y)-U_{t}(Y)\right\|^{n}\right)^{1/n}\leq\frac{\mathsf{C}(n)}{\sqrt{N}}$ where $\mathsf{C}(n)<\infty$ is a constant that does not depend upon $t$ or $N$. Both Proposition 3.1 and Proposition 3.2 establish that one can estimate the log of the normalization constant using the ensemble Kalman-Bucy type filters, with a mean square error (for instance) that grows at most linearly in time. Proposition 3.3 provides a uniform in time error, mostly following from the deterministic nature of the algorithm and the dependence on standard Kalman- Bucy theory. One can say more, when considering the average estimation error in over a window of time $w$ as we now state. We restrict ourselves to the mean square error below. In the cases of both (12) and (13) we have the time-uniform upper-bound. ###### Corollary 3.2. Consider the case of (12) and (13). Assume that (16) holds and that $\mu(A)<0$. Then for any $t>0$, $0<w<t$ and $N$ sufficiently large, we have that $\mathbb{E}\Bigg{[}\Bigg{(}\frac{1}{w}\Bigg{\\{}\log\Bigg{(}\frac{\overline{Z}_{t}^{N}(Y)}{\overline{Z}_{t-w}^{N}(Y)}\Bigg{)}-\log\Bigg{(}\frac{\overline{Z}_{t}(Y)}{\overline{Z}_{t-w}(Y)}\Bigg{)}\Bigg{\\}}\Bigg{)}^{2}\Bigg{]}\leq\frac{\mathsf{C}}{N}$ where $\mathsf{C}<\infty$ is a constant that does not depend upon $t$, $w$ or $N$. In the case of (13) we refine further and have the following time-uniform upper-bound. ###### Corollary 3.3. Consider the cases of (13). Assume that $\mu(A-P_{\infty}S)<0$ and $S\in\mathbb{S}_{r_{1}}^{+}$. Then for any $N\geq 2$ there exists a $t(N)>0$ with $t(N)\rightarrow\infty$ as $N\rightarrow\infty$, such that for any $t\in(0,t(N)]$, $0<w<t$ $\mathbb{E}\Bigg{[}\Bigg{(}\frac{1}{w}\Bigg{\\{}\log\Bigg{(}\frac{\overline{Z}_{t}^{N}(Y)}{\overline{Z}_{t-w}^{N}(Y)}\Bigg{)}-\log\Bigg{(}\frac{\overline{Z}_{t}(Y)}{\overline{Z}_{t-w}(Y)}\Bigg{)}\Bigg{\\}}\Bigg{)}^{2}\Bigg{]}\leq\frac{\mathsf{C}}{N}$ where $\mathsf{C}<\infty$ is a constant that does not depend upon $t$, $w$ or $N$. Note that, in Corollary 3.3 as we require $t\in(0,t(N)]$ and $t(N)=\mathcal{O}(\log(N))$ this may not be as useful as Corollary 3.2, but the assumption of a stable system in Corollary 3.2 is much stronger than the hypotheses in Corollary 3.3; see [3] for a discussion. ### 3.5 Simulation Results For $(i,k,L)\in\\{1,\cdots,N\\}\times\mathbb{N}_{0}\times\mathbb{N}_{0}$, let $\Delta_{L}=2^{-L}$ and consider the Euler-discretization of (12)-(14): $\begin{array}[]{lccl}(\textbf{F1})&\xi_{(k+1)\Delta_{L}}^{i}&=&\xi_{k\Delta_{L}}^{i}+A\xi_{k\Delta_{L}}^{i}\Delta_{L}+R_{1}^{1/2}\big{\\{}\overline{W}_{(k+1)\Delta_{L}}^{i}-\overline{W}_{k\Delta_{L}}^{i}\big{\\}}+\\\ &&&p_{k\Delta_{L}}C^{\prime}R_{2}^{-1}\Big{(}\big{\\{}Y_{(k+1)\Delta_{L}}-Y_{k\Delta_{L}}\big{\\}}-\Big{[}C\xi_{k\Delta_{L}}^{i}\Delta_{L}+R_{2}^{1/2}\big{\\{}\overline{V}_{(k+1)\Delta_{L}}^{i}-\overline{V}_{k\Delta_{L}}^{i}\big{\\}}\Big{]}\Big{)}\\\ (\textbf{F2})&\xi_{(k+1)\Delta_{L}}^{i}&=&\xi_{k\Delta_{L}}^{i}+A\xi_{k\Delta_{L}}^{i}\Delta_{L}+R_{1}^{1/2}\big{\\{}\overline{W}_{(k+1)\Delta_{L}}^{i}-\overline{W}_{k\Delta_{L}}^{i}\big{\\}}+\\\ &&&p_{k\Delta_{L}}C^{\prime}R_{2}^{-1}\left(\big{\\{}Y_{(k+1)\Delta_{L}}-Y_{k\Delta_{L}}\big{\\}}-C\left(\dfrac{\xi_{k\Delta_{L}}^{i}+m_{k\Delta_{L}}}{2}\right)\Delta_{L}\right)\\\ (\textbf{F3})&\xi_{(k+1)\Delta_{L}}^{i}&=&\xi_{k\Delta_{L}}^{i}+A\xi_{k\Delta_{L}}^{i}\Delta_{L}+R_{1}\left(p_{k\Delta_{L}}\right)^{-1}\left(\xi_{k\Delta_{L}}^{i}-m_{k\Delta_{L}}\right)\Delta_{L}+\\\ &&&p_{k\Delta_{L}}C^{\prime}R_{2}^{-1}\left(\big{\\{}Y_{(k+1)\Delta_{L}}-Y_{k\Delta_{L}}\big{\\}}-C\left(\dfrac{\xi_{k\Delta_{L}}^{i}+m_{k\Delta_{L}}}{2}\right)\Delta_{L}\right)\end{array}$ (30) and the discretization of $\overline{Z}_{T}^{N}(Y)$: $\displaystyle\overline{Z}_{T}^{N,L}(Y)=\exp\left\\{\sum_{k=0}^{T/\Delta_{L}-1}\left\langle m_{k\Delta_{L}},C^{\prime}R_{2}^{-1}\big{[}Y_{(k+1)\Delta_{L}}-Y_{k\Delta_{L}}\big{]}\right\rangle-\frac{1}{2}\left\langle m_{k\Delta_{L}},S\,m_{k\Delta_{L}}\right\rangle\Delta_{L}\right\\}.$ (31) For the purpose of showing that the mean square error of the estimate of the log of the normalization constant in the cases F1 & F2 is of $\mathcal{O}(\frac{t}{N})$ and in the case F3 is of $\mathcal{O}(\frac{1}{N})$, we take $r_{1}=r_{2}=1$, $A=-2$, $R_{1}^{-1/2}=1$, $R_{2}^{-1/2}=2$, $C$ a uniform random number in $(0,1]$ and $\xi_{0}^{i}\overset{i.i.d.}{\sim}\mathcal{N}(0.5,0.2)$ (normal distribution mean 0.5 and variance 0.2). In Tables 1 \- 2 and Figures 1 \- 2, we show that the rate of the MSE of the estimate in (31) for the cases F1 & F2 is of $\mathcal{O}(\frac{t}{N})$, even though we have used a naive time discretization for our results. In Table 3 and Figure 3 we show that the rate in the F3 case is of $\mathcal{O}(1/N)$. \| $N=1000$ \| $N=500$ \| $N=250$ ---\|---\|---\|--- t \| MSE \| MSE$/(t/N)$ \| MSE \| MSE$/(t/N)$ \| MSE \| MSE$/(t/N)$ 50 \| 2.194E-04 \| 4.4E-03 \| 5.404E-04 \| 5.4E-03 \| 8.903E-04 \| 4.5E-03 100 \| 5.483E-04 \| 5.5E-03 \| 1.442E-03 \| 7.2E-03 \| 3.082E-03 \| 7.7E-03 200 \| 1.085E-03 \| 5.4E-03 \| 2.901E-03 \| 7.3E-03 \| 4.442E-03 \| 5.6E-03 400 \| 2.283E-03 \| 5.7E-03 \| 5.047E-03 \| 6.3E-03 \| 1.017E-02 \| 6.4E-03 800 \| 4.894E-03 \| 6.1E-03 \| 8.579E-03 \| 5.4E-03 \| 1.985E-02 \| 6.2E-03 1600 \| 9.716E-03 \| 6.1E-03 \| 2.039E-02 \| 6.4E-03 \| 2.904E-02 \| 4.5E-03 3200 \| 1.974E-02 \| 6.2E-03 \| 3.504E-02 \| 5.5E-03 \| 7.835E-02 \| 6.1E-03 6400 \| 3.571E-02 \| 5.6E-03 \| 7.087E-02 \| 5.5E-03 \| 1.599E-01 \| 6.2E-03 Table 1: The mean square error (MSE) and MSE$/(t/N)$ for $N\in\\{250,500,1000\\}$ in the F1 case. \| $N=1000$ \| $N=500$ \| $N=250$ ---\|---\|---\|--- t \| MSE \| MSE$/(t/N)$ \| MSE \| MSE$/(t/N)$ \| MSE \| MSE$/(t/N)$ 50 \| 1.137E-03 \| 5.7E-03 \| 5.127E-04 \| 5.1E-03 \| 3.317E-04 \| 6.6E-03 100 \| 2.157E-03 \| 5.4E-03 \| 1.128E-03 \| 5.6E-03 \| 8.892E-04 \| 8.9E-03 200 \| 4.601E-03 \| 5.8E-03 \| 2.708E-03 \| 6.8E-03 \| 1.260E-03 \| 6.3E-03 400 \| 9.777E-03 \| 6.1E-03 \| 5.250E-03 \| 6.6E-03 \| 2.436E-03 \| 6.1E-03 800 \| 1.979E-02 \| 6.2E-03 \| 9.949E-03 \| 6.2E-03 \| 4.639E-03 \| 5.8E-03 1600 \| 4.901E-02 \| 7.7E-03 \| 1.595E-02 \| 5.0E-03 \| 8.895E-03 \| 5.6E-03 3200 \| 9.593E-02 \| 7.5E-03 \| 3.001E-02 \| 4.7E-03 \| 1.925E-02 \| 6.0E-03 6400 \| 1.744E-01 \| 6.8E-03 \| 5.415E-02 \| 4.2E-03 \| 3.635E-02 \| 5.7E-03 Table 2: The MSE and MSE$/(t/N)$ for $N\in\\{250,500,1000\\}$ in the F2 case. $N$ \| MSE \| MSE$\times N$ ---\|---\|--- 50 \| 1.073E-05 \| 5.4E-04 100 \| 4.951E-06 \| 5.0E-04 200 \| 2.867E-06 \| 5.7E-04 400 \| 1.492E-06 \| 6.0E-04 800 \| 8.157E-07 \| 6.5E-04 1600 \| 3.119E-07 \| 5.0E-04 3200 \| 1.773E-07 \| 5.7E-04 6400 \| 6.344E-08 \| 4.1E-04 Table 3: The MSE and MSE$\times N$ for $t=100$ in the F3 case. Figure 1: The mean square error associated to the EnKBF in the F1 case. This plots the MSE against the time parameter for $N\in\\{250,500,1000\\}$. MSE $\approx$ 5.9E-03 $\left(\frac{t}{N}\right)$, $\approx$ 6.1E-03 $\left(\frac{t}{N}\right)$, $\approx$ 5.6E-03 $\left(\frac{t}{N}\right)$, for $N=250,500,1000$, respectively. Figure 2: The mean square error associated to the EnKBF in the F2 case. This plots the MSE against the time parameter for $N\in\\{250,500,1000\\}$. MSE $\approx$ 6.4E-03 $\left(\frac{t}{N}\right)$, $\approx$ 5.5E-03 $\left(\frac{t}{N}\right)$, $\approx$ 6.4E-03 $\left(\frac{t}{N}\right)$, for $N=250,500,1000$, respectively. Figure 3: The mean square error associated to the EnKBF in the F3 case. This plots the MSE against the ensemble size $N$ for fixed time $t=100$. MSE $\approx$ 5.4E-04$/N$. ## 4 Application to Static Parameter Estimation ### 4.1 Approach We now assume that there are a collection of unknown parameters, $\theta\in\Theta\subseteq\mathbb{R}^{d_{\theta}}$, associated to the model (1). For instance $\theta$ could consist of some or all of the elements of $A,C,R_{1},R_{2}$. We will then include an additional subscript $\theta$ in each of the mathematical objects that have been introduced in the previous two sections. As an example, we would write $Z_{t,\theta}(X,Y)=\exp{\left[\int_{0}^{t}\left[\langle CX_{s},R_{2}^{-1}dY_{s}\rangle-\frac{1}{2}\langle X_{s},SX_{s}\rangle\right]\leavevmode\nobreak\ ds\right]}.$ For $0<s<t$ we introduce the notation $\overline{Z}_{s,t,\theta}(Y):=\frac{\overline{Z}_{t,\theta}(Y)}{\overline{Z}_{s,\theta}(Y)}$ with the obvious extension to $\overline{Z}_{s,t,\theta}^{N}(Y):=\overline{Z}_{t,\theta}^{N}(Y)/\overline{Z}_{s,\theta}^{N}(Y)$. The basic idea of our approach is to consider the recursive estimation of $\theta$ on the basis of the arriving observations. In particular, for notational convenience, we shall consider a method which will update our current estimate of $\theta$ at unit times. Our objective is to follow a recursive maximum likelihood (RML) method (e.g. [11]) which is based upon the following update at any time $t\in\mathbb{N}$ $\theta_{t}=\theta_{t-1}+\kappa_{t}\nabla_{\theta}\log\Big{(}\overline{Z}_{t-1,t,\theta}(Y)\Big{)}\Big{\|}_{\theta=\theta_{t-1}}$ where $\\{\kappa_{t}\\}_{t\in\mathbb{N}}$ is a sequence of real numbers with $\kappa_{t}>0$, $\sum_{t\in\mathbb{N}}\kappa_{t}=\infty$, $\sum_{t\in\mathbb{N}}\kappa_{t}^{2}<\infty$. Computing the gradient of $\overline{Z}_{t-1,t,\theta}(Y)$ can be computationally expensive, so our approach is to use a type of finite difference estimator via SPSA. We note that a classical finite difference estimator would require $2d_{\theta}$ evaluations of $\overline{Z}_{t-1,t,\theta}(Y)$, whereas the SPSA method keeps this evaluation down to 2; as computational cost is a concern, we prefer this afore-mentioned method. Our approach is the following, noting that we will use the argument $(k)$ to denote the $k^{th}-$element of a vector. 1. 1. Initialization: Set an initial $\theta_{0}\in\Theta$ and choose two step sizes $\\{\kappa_{t}\\}_{t\in\mathbb{N}}$ and $\\{\nu_{t}\\}_{t\in\mathbb{N}}$ such that $\kappa_{t}>0$, $\kappa_{t},\nu_{t}\rightarrow 0$, $\sum_{t\in\mathbb{N}}\kappa_{t}=\infty$, $\sum_{t\in\mathbb{N}}\frac{\kappa_{t}^{2}}{\nu_{t}^{2}}<\infty$. Set $t=1$ and generate i.i.d. the initial ensemble from $\eta_{0,\theta_{0}}$. 2. 2. Iteration: * • For $k\in\\{1,\dots,d_{\theta}\\}$, independently, sample $\Delta_{t}(k)$ from a Bernoulli distribution with success probability $1/2$ and support $\\{-1,1\\}$. * • Set $\theta_{t-1}^{+}=\theta_{t-1}+\nu_{t}\Delta_{t}$, $\theta_{t-1}^{+}=\theta_{t-1}-\nu_{t}\Delta_{t}$. * • Using the EnKBF, with samples simulated under $\theta_{t-1}$, generate estimates $\overline{Z}_{t-1,t,\theta_{t-1}^{+}}^{N}(Y)$ and $\overline{Z}_{t-1,t,\theta_{t-1}^{-}}^{N}(Y)$. * • Set, for $k\in\\{1,\dots,d_{\theta}\\}$, $\theta_{t}(k)=\theta_{t-1}(k)+\kappa_{t}\frac{1}{2\nu_{t}\Delta_{t}(k)}\log\Bigg{(}\frac{\overline{Z}_{t-1,t,\theta_{t-1}^{+}}^{N}(Y)}{\overline{Z}_{t-1,t,\theta_{t-1}^{-}}^{N}(Y)}\Bigg{)}.$ * • Run the EnKBF from $t-1$ (starting with samples simulated under $\theta_{t-1}$) up-to time $t$ using the parameter $\theta_{t}$. Set $t=t+1$ and return to 2.. We note that in practice, one must run a time-discretization of the EnKBF, rather than the EnKBF itself. We remark also that the use of SPSA for parameter estimation associated to hidden Markov models has been considered previously, for instance in [14]. ### 4.2 Simulation Results We use the algorithm described in the previous section along with the Euler- discretization in (30) to estimate the parameters in three different models. In particular, we show that the algorithm works in both linear and non-linear models. In all three models, the data is generated from the true parameters. #### 4.2.1 Linear Gaussian Model In the first example, we take $A=\theta_{1}Id$, $R_{1}^{-1/2}=\theta_{2}R$, where $\displaystyle R=\begin{bmatrix}1&0.5\\\ 0.5&1&0.5\\\ &0.5&\ddots&\ddots\\\ &&\ddots&\ddots&0.5\\\ &&&0.5&1\end{bmatrix}$ $C=\alpha_{1}(r_{1},r_{2})C^{}$, where $C^{}$ is a uniform random matrix and $\alpha_{1}(r_{1},r_{2})$ is a constant, $R_{2}^{-1/2}=\alpha_{2}(r_{2})Id$, where $\alpha_{2}(r_{2})$ is a constant. In Figures 4 \- 9, we show the results for the parameters estimation of $\theta_{1}$ and $\theta_{2}$ in the cases $r_{1}=r_{2}=2$ and $r_{1}=r_{2}=100$. In all cases we take $N=100$ except in the case when $r_{1}=r_{2}=100$ in F3, where we take $N=200$ to assure the invertibility of $p_{t}^{N}$, otherwise, the condition number of $p_{t}^{N}$ will be huge. The discretization level is $L=8$ in all cases. The initial state is $X_{0}\sim\mathcal{N}(4\mathbf{1}_{r_{1}},Id)$, where $\mathbf{1}_{r_{1}}$ is a vector of 1’s in $\mathbb{R}^{r_{1}}$. The results display that in a reasonable case, one can estimate low- dimensional parameters using RML via SPSA. We now consider a few nonlinear models. (a) (b) Figure 4: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2})$ in the case F1 with $r_{1}=r_{2}=2$. The initial values of the parameters are $(-1,2)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{})=(-2,1)$. We take $\nu_{t}=t^{-0.1}$ and $\kappa_{t}=0.09$ when $t\leq 400$ and $\kappa_{t}=3\times t^{-0.601}$ otherwise. (a) (b) Figure 5: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2})$ in the case F1 with $r_{1}=r_{2}=100$. The initial values of the parameters are $(-1,2)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{})=(-2,1)$. We take $\nu_{t}=t^{-0.1}$ and $\kappa_{t}=0.1$ when $t\leq 400$ and $\kappa_{t}=3\times t^{-0.601}$ otherwise. (a) (b) Figure 6: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2})$ in the case F2 with $r_{1}=r_{2}=2$. The initial values of the parameters are $(-1,2)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{})=(-2,1)$. We take $\nu_{t}=t^{-0.1}$ and $\kappa_{t}=0.09$ when $t\leq 300$ and $\kappa_{t}=t^{-0.7}$ otherwise. (a) (b) Figure 7: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2})$ in the case F2 with $r_{1}=r_{2}=100$. The initial values of the parameters are $(-1,2)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{})=(-2,1)$. We take $\nu_{t}=t^{-0.1}$ and $\kappa_{t}=0.08$ when $t\leq 400$ and $\kappa_{t}=3\times t^{-0.64}$ otherwise. (a) (b) Figure 8: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2})$ in the case F3 with $r_{1}=r_{2}=2$. The initial values of the parameters are $(-1,2.2)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{})=(-2,1)$. We take $\nu_{t}=t^{-0.1}$ and $\kappa_{t}=0.09$ when $t\leq 100$ and $\kappa_{t}=t^{-0.8}$ otherwise. (a) (b) Figure 9: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2})$ in the case F3 with $r_{1}=r_{2}=100$. The initial values of the parameters are $(-1,2)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{})=(-2,1)$. We take $\nu_{t}=t^{-0.1}$ and $\kappa_{t}=0.09$ when $t\leq 200$ and $\kappa_{t}=2\times t^{-0.601}$ otherwise. #### 4.2.2 Lorenz’ 63 Model We now consider the following nonlinear model with $r_{1}=r_{2}=3$, which is a simplified mathematical model for atmospheric convection. $\begin{array}[]{rcl}dX_{t}&=&f(X_{t})\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ dW_{t}\\\ dY_{t}&=&C\leavevmode\nobreak\ X_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{2}\leavevmode\nobreak\ dV_{t},\end{array}$ where $\displaystyle f_{1}(X_{t})$ $\displaystyle=\theta_{1}(X_{t}(2)-X_{t}(1)),$ $\displaystyle f_{2}(X_{t})$ $\displaystyle=\theta_{2}X_{t}(1)-X_{t}(2)-X_{t}(1)X_{t}(3),$ $\displaystyle f_{3}(X_{t})$ $\displaystyle=X_{t}(1)X_{t}(2)-\theta_{3}X_{t}(3),$ where $X_{t}(i)$ is the $i^{th}$ component of $X_{t}$. We also have $R_{1}^{1/2}=Id$, $\displaystyle C_{ij}=\left\\{\begin{array}[]{cl}\frac{1}{2}&\text{if }i=j\\\ \frac{1}{2}&\text{if }i=j-1\\\ 0&\text{otherwise }\end{array}\right.,\qquad i,j\in\\{1,2,3\\},$ and $(R_{2}^{1/2})_{ij}=2\,q\left(\frac{2}{5}\min\\{\|i-j\|,r_{2}-\|i-j\|\\}\right)$, $i,j\in\\{1,2,3\\}$, where $\displaystyle q(x)=\left\\{\begin{array}[]{cl}1-\frac{3}{2}x+\frac{1}{2}x^{3}&\text{if }0\leq x\leq 1\\\ 0&\text{otherwise }\end{array}\right..$ In Figures 10 \- 12, we show the results for the parameters estimation of $\theta=(\theta_{1},\theta_{2},\theta_{3})$ in the F1, F2 and F3 cases. The ensemble size is $N=100$ and the discretization level is $L=8$ in all cases. The initial state is $X_{0}\sim\mathcal{N}(\mathbf{1}_{r_{1}},\frac{1}{2}Id)$, where $\mathbf{1}_{r_{1}}$ is a vector of 1’s in $\mathbb{R}^{r_{1}}$. (a) (b) (c) Figure 10: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2},\theta_{3})$ in the case F1. The initial values of the parameters are $(7.5,26.7,6.5)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{},\theta_{3}^{})=(10,28,\frac{8}{3})$. We take $\nu_{t}=t^{-0.2}$ an $\kappa_{t}=0.0314$ when $t\leq 100$ and $\kappa_{t}=t^{-0.71}$ otherwise. (a) (b) (c) Figure 11: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2},\theta_{3})$ in the case F2. The initial values of the parameters are $(7.5,26.7,6.5)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{},\theta_{3}^{})=(10,28,\frac{8}{3})$. We take $\nu_{t}=t^{-0.2}$ and $\kappa_{t}=0.0139$ when $t\leq 300$ and $\kappa_{t}=t^{-0.75}$ otherwise. (a) (b) (c) Figure 12: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $(\theta_{1},\theta_{2},\theta_{3})$ in the case F3. The initial values of the parameters are $(7.5,26.7,6.5)$. The green horizontal lines represent the true parameter values $(\theta_{1}^{},\theta_{2}^{},\theta_{3}^{})=(10,28,\frac{8}{3})$. We take $\nu_{t}=t^{-0.2}$ and $\kappa_{t}=0.0314$ when $t\leq 100$ and $\kappa_{t}=t^{-0.71}$ otherwise. #### 4.2.3 Lorenz’ 96 Model Finally, we consider the following nonlinear model, Lorenz’ 96, with $r_{1}=r_{2}=40$. The solution of this model has a chaotic behavior and it describes the evolution of a scalar quantity on a circle of latitude. $\begin{array}[]{rcl}dX_{t}&=&f(X_{t})\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{1}\leavevmode\nobreak\ dW_{t}\\\ dY_{t}&=&X_{t}\leavevmode\nobreak\ dt\leavevmode\nobreak\ +\leavevmode\nobreak\ R^{1/2}_{2}\leavevmode\nobreak\ dV_{t},\end{array}$ where $\displaystyle f_{i}(X_{t})$ $\displaystyle=(X_{t}(i+1)-X_{t}(i-2))X_{t}(i-1)-X_{t}(i)+\theta$ where $X_{t}(i)$ is the $i^{th}$ component of $X_{t}$, and it is assumed that $X_{t}(-1)=X_{t}(r_{1}-1)$, $X_{t}(0)=X_{t}(r_{1})$ and $X_{t}(r_{1}+1)=X_{t}(1)$. We also have $R_{1}^{1/2}=\sqrt{2}Id$ and $R_{2}^{1/2}=\frac{1}{2}Id$. $\theta$ here represents the external force in the system, while $(X_{t}(i+1)-X_{t}(i-2))X_{t}(i-1)$ is the advection-like term and $-X_{t}(i)$ is the damping term. In Figure 13, we show the results for the parameters estimation of $\theta$ in the F1, F2 and F3 cases. The ensemble size is $N=100$ and the discretization level is $L=8$ in all cases. In F1 & F2 cases, $X_{t}$ is initialized as follows: $X_{0}(1)=8.01$ and $X_{0}(k)=8$ for $1<k\leq 40$. In F3, to avoid having the matrix $p_{0}^{N}$ equal to zero, we take $X_{0}\sim\mathcal{N}(8\mathbf{1}_{r_{1}},0.05Id)$. (a) (b) (c) Figure 13: The blue curves along with their average (in red) are trajectories from the execution of the algorithm in Section 4.1 for the estimation of $\theta$ in the case F1 (left), F2 (middle) and F3 (right). The initial value of $\theta$ is 10. The green horizontal lines represent the true parameter value $\theta^{}=8$. We take $\nu_{t}=t^{-0.1}$ and $\kappa_{t}=0.0314$ when $t\leq 50$ and $\kappa_{t}=t^{-0.75}$ otherwise. #### Acknowldegements AJ & HR were supported by KAUST baseline funding. 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Machine learning with limited data Doctor of Philosophy Biological Sciences iFlyTek-Surrey Joint Research Centre on Artificial Intelligence, UK August 27, 2024 ###### Abstract Abstract Thanks to the availability of powerful computing resources, big data and deep learning algorithms, we have made great progress on computer vision in the last few years. Computer vision systems begin to surpass humans in some tasks, such as object recognition, object detection, face recognition and pose estimation. Lots of computer vision algorithms have been deployed to real world applications and started to improve our life quality. However, big data and labels are not always available. Sometimes we only have very limited labeled data, such as medical images which requires experts to label them. In this paper, we study few shot image classification, in which we only have very few labeled data. Machine learning with little data is a big challenge. To tackle this challenge, we propose two methods and test their effectiveness thoroughly. One method is to augment image features by mixing the style of these images. The second method is applying spatial attention to explore the relations between patches of images. We also find that domain shift is a critical issue in few shot learning when the training domain and testing domain are different. So we propose a more realistic cross-domain few-shot learning with unlabeled data setting, in which some unlabeled data is available in the target domain. We propose two methods in this setting. Our first method transfers the style information of the unlabeled target dataset to the samples in the source dataset and trains a model with stylized images and original images. Our second method proposes a unified framework to fully utilize all the data. Both of our methods surpass the baseline method by a large margin. Key words: few shot learning, image classification, domain shift ###### Contents 1. 0 Introduction 1. 1 Introduction 2. 2 Structure of the Confirmation Report 3. 3 Novel Work Undertaken 2. 1 Literature review 1. 1 Definition 2. 2 Relation with other types of machine learning 3. 3 Overview of few shot learning 3. 2 Methods for Few Shot Learning 1. 1 Style mix for few shot learning 1. 1 Background 2. 2 Approach 3. 3 Experiments 2. 2 Spatial attention for few shot learning 1. 1 Background 2. 2 Approach 3. 3 Experiments 3. 3 Problems of few shot learning methods and remaining challenges 4. 3 Cross-domain Few shot learning with unlabeled data 1. 1 Background 1. 1 Domain shift in few shot learning 2. 2 Cross domain few shot learning 2. 2 Cross domain few shot learning with unlabeled data 1. 1 Definition 2. 2 Related work 3. 3 Style transfer for cross-domain few-shot learning with unlabeled data 1. 1 Background 2. 2 Approach 3. 3 Experiments 4. 4 Discussion 4. 4 Contrastive learning for cross-domain few-shot learning with unlabeled data 1. 1 Background 2. 2 Approach 3. 3 Experiment 4. 4 Discussion 5. 4 Future directions and open challenges ###### List of Figures 1. 1 data augmentation ([10]) 2. 2 style transfer ([27]) 3. 3 TSNE plots of features 4. 4 Attention ([2]) 5. 1 Domain shift in few shot learning ([7]) 6. 2 style transfer ([29]) 7. 3 stylized images ([21]) 8. 4 DomainNet ([37]) ###### List of Tables 1. 1 Results of style mix 2. 2 Result of style mix in different places 3. 3 Hyperparameter tuning for $\alpha$ and p 4. 4 Results of style mix when inserting style mix layers during testing 5. 5 Results of spatial attention 6. 1 Results of style transfer 7. 2 Results of contrastive learning ## Chapter 0 Introduction ### 1 Introduction Nowadays, it’s well known that deep learning is powerful. We use it to solve lots of problems in computer vision, natural language processing, and speech processing. However, we also know that deep learning models have a tremendous number of parameters and consume enormous amount of data. For example, an image recognition model, AlexNet([32]), was trained on ImageNet([11]), which has millions of images. BERT([12]), with 340 million parameters, used BooksCorpus (800M words) (Zhu et al.,2015) and English Wikipedia (2,500M words). GPT-3 ([3]), a recently developed natural language model, has 175 billion parameters. Thus a large number of memory cards, storage drives, and GPUs are consumed. One consequence is that training these large models consumes lots of energy and produces tons of carbon dioxide, which is not environmentally friendly. This poses a huge challenge for us. Then can we learn a model with little data? For example, what if you are asked to recognize dogs and cats with only one image labeled as dog and one image labeled as cat? From a statistical point of view, this is almost impossible. Rules can only be derived from a large amount of samples. But we all know human beings have remarkable ability to learn from little data. For instance, suppose you are given an image of an apple, then you can tell apples of slightly different shape and different color and texture from oranges even though you only see one example of apples. To address this problem, in recent years, some researchers start to study this few shot learning problem, which is learning with only a very limited amount of data. In this setting, we are allowed to transfer knowledge from another additional dataset and test our trained models on a small dataset. Most few- shot learning methods are also meta learning methods, in which the algorithm tries to learn to learn the model. So it’s called learning to learn, which is a bi-level learning. There are also some non meta-learning based methods, one of which is transfer learning. Using transfer learning, we train a model on the additional dataset and then fine tune it on our target dataset which only has limited data. Most of these methods can be categorized as 4 different types of methods: optimization based method, metric learning based methods, data augmentation based methods and parameter generation based methods (or black box / model- based methods). Optimization-based method is the most intuitive meta learning method in which the model tries to optimize the inner optimization problem. The inner optimization tasks are subtasks. By doing this, we hope the model can learn some meta knowledge for all sub-tasks and then generalize to unseen tasks during testing. Metric learning based methods try to find a feature space where all sub-tasks can perform well, i.e., images in the same class are closer in the feature space while images in different classes stay further. Data augmentation based methods deal with the data deficiency by learning to augment the data during meta-training on the meta-training dataset and augmenting the novel dataset during meta-testing. The Parameter generation methods, also called black box or model-based methods, try to generate parameters of sub-tasks from a meta-learner (the learner in the outside level). During training, the model takes samples from a task as input and saves knowledge in the activation states (parameters) in the model. During testing, the model outputs parameters for a task conditioned on the labeled training data and makes predictions using these parameters. We introduce 2 methods from different perspectives. Our first style mix method borrows ideas from the literature of image style transfer and then tries to augment the data in the feature space by mixing the style from different images. Doing so, we hope to mitigate the problem of scarce data in few shot learning. Inspired by ([51]), we also try to use attention mechanisms to address few shot image classification. In the method, we let the query images explicitly attend to important parts of images in the support set. The spatial relations among different parts of image are utilized explicitly, which is not the case in ResNet ([25]) used by most other methods. We carefully design and conduct our experiments and thoroughly verify our ideas above. But the improvement is not impressive enough. We then find out the reason why: most of meta learning methods don’t work well for few shot classification. We do lots of experiments and gather lots of evidence from recently published papers. A paper [Rapid learning or feature reuse] claims that feature reuse is the reason why most optimization-based methods work, by conducting intensive experiments. We also find that pretraining the feature extractor is vital to achieve high accuracy for most methods. Further meta learning only gives 1 or 2 percent of accuracy improvement. Another factor we found is non-episodic training, such as Baseline++ ([7]) and SimpleShot ([52]), can achieve almost the same accuracy as episodic training, which is the common practise in few shot learning. Additionally, deeper models can also achieve better accuracies ([7]). And if there is a domain shift between the training set and the testing set, the accuracy will drop significantly ([7]). All these evidences show that the current meta-learning methods don’t work well for few shot image classification and can’t help much. Later on, after we found this phenomenon, a series of papers, including ([45]; [13]) were published, verifying this idea. All of these show that meta learning reached its bottleneck and we need some fundamental and theoretical breakthroughs. And it may take several decades to see these breakthroughs. We then shift research focus to related problems, cross domain few shot learning and cross-domain few shot learning with unlabeled data. We observe that models trained on base dataset perform badly when there is a big domain shift between the base dataset and the novel dataset. Thus it’s important to study cross-domain few shot learning, in which we have one or multiple datasets from different domains as training datasets and we will test our model on another different domain. But this is a too challenging task since during training we are not able to touch the target testing domain. So a new and more realistic setting for few shot learning is proposed and studied. We provide some unlabeled target data for training. In most cases, unlabeled data are easy to obtain. For example, we can have tons of chest X-ray images collected from hospitals when they do body checks every day for patients. But collecting labels (canner or not) is not easy since they require medical expertise, which is usually only owned by doctors. So we propose a new setting, in which there is some unlabeled data in the target domain. We introduce two methods we propose to tackle this problem. Our first method transfers domain information to the source labeled dataset by style transfer and then trains a model using the stylized images and the images in the original source dataset. We also propose a pseudo-label method to solve this problem: clustering for all unlabeled data and then co-trained the model using both labled source domain data, clustered target domain data and unclustered target data with a contrastive loss. We achieve big accuracy improvements compared to our baseline for these two methods. We also propose another research idea, which combines domain translation based methods and pseudo- label learning methods. We are implementing this and hopefully we get more improvements. In the future,we hope to keep exploring cross-domain few shot learning with or without unlabeled data. Also we plan to study compositional and causal models, which are critical abilities when there are only very little data. We hope that by developing compostial and causal models, we can really get rid of the need of big data and build extremely efficient models. Also compositial and causal learning are the ways to general artificial intelligence, in my opinion. ### 2 Structure of the Confirmation Report In chapter 2, we give a detailed survey about this field. We first give a formal definition of few shot learning. Then we show a taxonomy and an overview of previous methods in literature. Details of important papers in this field are also discussed. In the end, we point out problems and challenges for researchers. In chapter 3, we introduce two algorithms we propose and experiment results and analysis of these two methods. Since we only have very limited data, we first propose a method to increase features. We then show how it works and experiment results of it. Inspired by ([51]),we propose a method which fully exploits the spatial relations between the query image and gallery images. We present the experimental results of this method. In chapter 4, we show the necessities to study cross-domain few shot learning and present a new machine learning setting, cross-domain few shot learning with unlabelled data. We claim this is a more realistic and important setting. Several methods are proposed and show excellent results. We will give some preliminary results and show our future directions. Finally, we discuss open challenges in this field and our future directions in chapter 6. Cross-domain few shot learning needs to be further explored. Models with compositional and causal learning should be proposed to address the data scarcity problem in few-shot learning. ### 3 Novel Work Undertaken We propose 2 new methods for the conventional few shot learning and systematically experimental their effectiveness. We point out the problem in the development of meta learning when no one in the research community noticed this. We propose a new and more realistic cross domain few shot learning setting, which is cross-domain few shot learning with unlabeled data. We develop two methods in this setting and design and conduct experiments for it. Our methods surpass the baseline methods with a large margin. ## Chapter 1 Literature review ### 1 Definition In this chapter we give a detailed introduction to few shot learning. Most of the time, few shot learning means few shot classification. We start from a formal definition. Conventional supervised machine learning In the conventional supervised machine learning, we have a dataset $D=\\{X_{j},Y_{j}\\}_{j=1..N}$, where $\\{X_{j},Y_{j}\\}$ is a pair of a data point and it’s label. In the image classification problem, we $X_{j}$ is an image and $Y_{j}$ is a category. D can be split into a training set $D_{train}$ and a testing set $D_{test}$. The loss function can be cross-entropy loss. Then we optimize the loss with respect to the parameter theta: $\theta^{}=\underset{\theta}{argmin}\;L(D;\theta)$ Few shot learning In few shot learning, our dataset D is very small (less than 30 samples per class in classification problem). In most cases, we only study the case where there are only less than 10 samples per class. This is an ill- posed problem and almost impossible to achieve high performance without utilizing another dataset based on our current understanding of machine learning. So we are allowed to use another dataset in the hope that we can learn some meta-knowledge from it and transfer it to the small dataset. The additional dataset is called the auxiliary dataset or the base dataset. Most methods in literature address few shot learning problems as meta learning problems. Here, let’s formally define meta learning: We have a dataset D, splitted to a training set (called meta-training set or base dataset) $D_{base}$ and testing set (called meta-testing set or novel dataset) $D_{novel}$. We sample lots of few shot learning tasks from D and assume all tasks have a task distribution $p(\tau)$. During training (also called meta- training), we sample some few shot learning tasks [$D_{train}$, $D_{test}$] from $D_{base}$. $D_{train}$ and $D_{test}$ are training and testing sets of a few shot learning tasks. They are also called support sets and query sets which are different from meta-training and meta-testing sets. Our learning goal is to minimize a loss function $L(D_{base},w)$. L measures the performance of the model on a series of tasks. $\omega$ is the parameter of the model, also called meta-parameters which represent meta-knowledge from all sampled tasks: $\omega^{}=\underset{\omega}{argmin}\;E_{\tau\sim p(\tau)}(D;\omega)$ During the meta-testing (also called fine-tuning in some papers) stage, we use the meta-knowledge learned during meta-training to learn a model for each task sampled from $D_{novel}$. $\theta$ is the parameter for each few shot learning task: $\theta^{}=\underset{\theta}{argmax}\;log\;p(\theta\|\omega^{},D^{train(i)}_{novel})$ Finally, we evaluate our algorithm on $D_{novel}^{test(i)}$. Note that $D_{novel}^{train(i)}$ and $D_{novel}^{test(i)}$ are the support and query set of the $i^{th}$ task sampled from the novel dataset. Meta learning tries to learn meta-knowledge from a series of few short tasks sampled from the auxiliary dataset and hope the model can perform well to novel tasks which have different classes. In conventional machine learning, we assume there is a data distribution. In meta learning, we assume there is a task distribution. All tasks are sampled from the same task distribution, so if the model fits the tasks sampled from the auxiliary dataset well, it can also generalize to tasks which have new categories. The new categories are never seen by the model during meta-training, so performing well during meta- testing requires generalization. The above definition is from a task-distribution point of view. There is also another point of view: bi-level optimization point of view ([26]), especially for optimization based methods. This kind of method treats learning to learning as two levels of optimization problem: in the inner loop, you optimize the model for the specific few shot learning problem and get optimal parameters $\theta$ while in the outer loop you optimize the mata-model for all few shot tasks to get the optimal meta-parameters $\omega$. Note that $\theta$ is conditional on the outer-loop parameters $\omega$. In all optimization based papers, the most famous one is MAML([17]). In MAML, meta- parameter $\omega$ is the initial value of $\theta$. $\omega^{}=\underset{\omega}{argmin}\sum_{i=1}^{M}L^{meta}(\theta^{(i)}(\omega)),\omega,D^{test(i)}_{base})$ $s.t.\;\theta^{(i)}(w)=\underset{\theta}{argmin}\;L^{task}(\theta\|\omega,D^{train(i)}_{base})$ Relation between few shot learning and meta-learning Meta-learning is just one way to deal with few shot learning and it treats few shot learning as a ‘data sample’. There are also other ways to tackle few shot learning, such as transfer learning, such as GPT-3([3]). ### 2 Relation with other types of machine learning One/Zero shot learning When the number of samples in each class is one, it’s called one shot learning. When it’s zero, it’s called zero shot learning. One shot learning is always studied in few shot learning while zero shot learning is not the same. In few shot learning, we leverage the auxiliary dataset while in zero shot learning, the model relies on attributes or semantic information of those classes to transfer some supervision signals and make learning feasible ([53]). There is a chance to borrow some methods from zero-shot learning for Few-shot learning. Transfer learning Transfer learning tries to transfer knowledge learned from another dataset to the target dataset. In the current setting of few shot classification, it can be viewed as a special case of transfer learning, we transfer knowledge from base dataset to novel dataset. Multitask learning multitask learning aims to learn several related tasks together. This is a more efficient way of learning so that we can save time and computing resources. In addition, this should also be one feature of general artificial intelligence. Meta learning can also be viewed as a special case of multitask learning in which all tasks are sampled from the same task distribution, which means all tasks share the same statistics. Also during testing, it’s tested on novel tasks. It’s not the case for regular multitask learning. Semi-supervised learning Semi-supervised learning has both labeled data and unlabeled data while in the classic few shot learning, we only have very limited labeled data. The amount of both labeled and unlabeled data, which semi-supervised data can use, is much more than than few shot learning. There is an intersection for these two types of learning: semi-supervised few shot learning. ### 3 Overview of few shot learning In this section, we will discuss the taxonomy of few shot learning methods and methods under each category. There are four types of methods: optimization based methods, metric learning based methods, parameter generation based methods (or black box / model-based methods), and data augmentation based methods. Optimization based methods These methods treat inner-level tasks as optimization problems and try to extract meta-knowledge for inner tasks ([26]). This can also be viewed as learning good global initial points for inner-level tasks so that in the inner loop, after a few steps of gradient descent, the model can quickly adapt to an unseen task. Rapid learning is what we need in mata learning and also avoids overfitting. The most famous one is MAML ([17]), in which in the inner loop, you update the model parameters theta a few steps using $D_{base}^{train}$ and in the outer loop, you update the original model, before being updated by the inner loop, using loss computed on $D_{base}^{test}$. Because MAML differentiates through the optimization process, we need to compute second order derivatives. Then the vanilla MAML algorithm has a great computing cost when there are more than one steps of gradient descent in the inner loop. To mitigate this problem, the author also proposes an approximation method, FOMAML, which drops the second order derivatives and only keeps the first order derivatives. ([40]) further proposes a method which only depends on the solution of the inner loop optimization not the path of it, which enables the inner model to update infinite steps without worrying about the computation cost. ([36]) presents a method, called Reptile, which keeps sampling tasks and then trains the model on them with a few steps of gradient descent and then moves the meta- parameters towards the updated parameters. LSTM meta learner ([41]) models the meta-learner as a LSTM. It takes sampled tasks as input and keeps updating the model for T rounds. Their contribution is representing the optimization of the inner task as evolution of LSTM cell states. MetaOptNet ([33]) uses linear classifiers and SVMs for the base leaner in the inner loop. Since it has a closed form solution, it doesn’t have the high computation cost problem as MAML. Metric learning based methods try to ‘learn to compare’ (compare samples from query set to the samples from the support set) and obtain a metric function under which data samples in the query and support set stay closer if they are from the same class and stay further if they are from different classes. $P_{\theta}(y\|x,S)=\sum_{(x_{i},y_{i})\in S}k_{\theta}(x,x_{i})y_{i}$ The probability of label y of sample x in the query set is the weighted sum of all the labels in the support set and these weights are generated by a kernel function meaning the distance between the sample x and another sample x’ in the support set. ([31]) first use a siamese neural network to deal with this problem. They solve this task as a verification problem in which a pair of images go through the same embedding network and the L1 distance is optimized to be small if they are from the same class. Matching Networks ([50]), shows the kernel function can be an attention kernel which is the normalized softmax of cosine similarities between samples from the query and the support set. The feature embedding functions for the support and query set are LSTMs. Instead of fixed metric functions, such as L1, L2 or cosine distance , Relation Network ([44]) shows that learned metric functions parametrized by neural networks are also possible. Prototypical Networks ([43]) is a simple metric learning method in which the kernel function is the L2 distance between the query sample and the centroids of all features of each class. Model based methods don’t have an explicit form of the base learner. The meta learner gets samples of each task as input and encodes the information in the internal states of the model. Based on these states, the model makes predictions for the query set. ([42]) use external memory for encoding and retrieving task information. Because the model is only exposed to the label of $x_{t}$ at time t+1, the model is forced to predict the label of $x_{t}$ given data before time t. ([34]) uses temporal convolution to gather information for past experience (tasks) and soft attention to pinpoint specific information for the current tasks for prediction. Meta networks ([35]) has meta learner and base learner. The meta-learner is trained to generate parameters (called fast weights in the paper) using LSTM for itself and for the base learner. ([22]) proposes to leverage an attention base classification weight generator to generate weights for those classes in the base learner. The base learner is a cosine distance classifier. Different from other papers, the author claims that their methods not only perform well on novel classes but also on base classes. Imprinted weights ([38]) has the similar idea that the weight of novel classes are generated using the normalized averaged features of samples in the support set. Data augmentation based methods mitigate the data scarcity problem by augmenting the data, which is an intuitive strategy. Sometimes, this method can be integrated with other types of methods, such as metric learning based methods. Some of these methods learn a data generator using the base dataset and generate data for novel classes during meta-testing. ([24]) aims to augment the data using appearance variance in the base dataset. Style transfer can also be leveraged to augment the data for few shot learning ([1]). [54] integrates data generators with meta learning methods directly in one framework. ([56]) uses a pretrained saliency model to segment the foreground and background and then combine these foregrounds and backgrounds to get new images. ([8]) has an image deformation sub-network which takes a pair of images as input and one for keeping the content and another for diversifying the deformations. The output images will be used to train the model. ## Chapter 2 Methods for Few Shot Learning In this chapter we present two methods we propose for few shot learning from different perspectives. And then we show experiments we conduct and conclusions we draw from these experiments. Our first method, style mix, borrowing ideas from style transfer, augments the data in the feature space. Then we present a method exploring the relationship between tasks during training. We apply knowledge distillation techniques for two tasks sharing the same label space and domain adaptation methods for two tasks having disjoint label space. Inspired by the attention mechanism proposed by Non-local neural networks ([51]), we explore this idea in few shot learning: ask the images in the query set to attend to pixels in the images in the support set. We will introduce each of them in the following sections. ### 1 Style mix for few shot learning #### 1 Background Data augmentation is widely used in machine learning and deep learning. When you have only very little data, overfitting is almost unavoidable. To increase the dataset without labeling, the easiest way is to augment the dataset. Common augmentation techniques including geometric and color transformation, such as relation, random cropping, random easing, horizontal and vertical flipping, noise injection, as shown in figure 3.1. These are basic techniques and later researchers also developed more sophisticaled techniques, such as feature space augmentation, adversarial training, and GAN data augmentation. Recently some automatically learned augmentation methods are proposed, such as AutoAugment ([9]) which search optimal combinations of augmentation techniques as hyper-parameter searching. The principle of data augmentation is to preserve the semantic identity (labels) while modifying other aspects using some human known prior information. For example, we know rotating images will not alter the label of the objects in the image. Some methods apply too much transformation and the resulting images make less sense to humans. Figure 1: data augmentation ([10]) Mixup is a regulation technique to overcome overfitting in deep learning, first proposed by ([57]). As shown in the formulas, $\tilde{y}=\lambda y_{i}+(1-\lambda)y_{j},\;where\;y_{i},y_{j}\;are\;raw\;input\;vectors$ (1) $\tilde{x}=\lambda x_{i}+(1-\lambda)x_{j},\;where\;x_{i},x_{j}\;are\;raw\;input\;vectors$ (2) we get the linear combination of a pair of samples and their corresponding labels. Then $\tilde{x}$ and $\tilde{y}$ are used as training samples. This regularization forces models to favor those with linear behaviors of samples. Although it is quite simple, this has been proved to be very effective. Later, in ([49]), instead of using linear combination of samples, it uses linear combinations of hidden representation of samples. They show that this can improve representation by obtaining smoother decision boundaries and fully utilizing hidden representation which contains high level information. Style transfer is a task in image processing in which the artistic styles of images are required to change but the content should be kept, as shown in figure 3.2. ([19]) first studies how to use CNN to deal with this problem. In this work, it shows the feature response of a pretrained convolutional neural network can be used as content and the feature summary statistics contains style information. Then by recombining content and style information, we can create new images with different styles. ([15]) shows that a conditional install normalization layer can be used to transfer the styles: $CIN(x,s)=\gamma^{s}(\frac{x-\mu(x)}{\sigma(x)})+\beta^{s}$ (3) in which the feature x is first normalized by its mean and standard deviation. And then two parameters gamma and beta are learned for the style s. Then ([29]) modifies this, and uses the mean and standard deviation of the style image directly without learning those parameters: $AdaIN(x,y)=\gamma(y)(\frac{x-\mu(x)}{\sigma(x)})+\mu(y)$ (4) in which $\sigma$ and $\mu$ are computed across spatial locations of the style image. With this modification, we can replace the style of image x with the style of image y. This method can produce images of arbitrary styles in real time, so it’s much better compared to the previous method. Figure 2: style transfer ([27]) #### 2 Approach Inspired by Adain ([29]) and manifold mixup ([49]), we propose a method to increase the diversity of the features for few shot learning. Because we have very limited data in shot learning, it’s important to augment the data. We propose to mix styles of different images in the same task and in two different tasks. In the first case, during meta learning, for each image, we randomly select another image. Then we compute the mean and standard deviation for these two images across the spatial locations: $\sigma_{nc}(x)=\sqrt{\frac{1}{HW}\sum_{h=1}^{H}\sum_{w=1}^{W}(x_{nchw}-\mu_{nc}(x))^{2}+\epsilon}$ (5) $\mu_{nc}(x)=\frac{1}{HW}\sum_{h=1}^{H}\sum_{w=1}^{W}x_{nchw}$ (6) in which H, W are the height and width of the feature map x. After we get the mean, $\mu$, and standard deviation, $\sigma$ for the two images x and y, we mix the style (styled information is contained in the mean and standard deviation) of them: $\mu=\lambda\mu_{x}+(1-\lambda)\mu_{y}$ (7) $\sigma=\lambda\sigma_{x}+(1-\lambda)\sigma_{y}$ (8) $\tilde{x}=\sigma(\frac{x-\mu(x)}{\sigma(x)})+\mu$ (9) in which $\lambda$ is the mixing coefficient, sampled from a Beta distribution, which is proposed in ([57]). Notice in the formula 3.9, we first get the normalized feature without style info and then we scale it with a mixed standard deviation and shift it with the mixed mean. After this operation, the feature will have the mixed style of the images. In this way we hope we get features of diverse styles and we can overcome the data scarcity problem in few-shot learning. The algorithm is shown in Algorithm 1. Result: Classification accuracy while _Traning_ do Sample a task T of size n; while _$n_{image}$ <n_ do Sample two images $x$ and $y$ from T; Get feature maps for $x$ and $y$, f(x) and f(y); Obtain $\mu_{x}$ and $\mu_{y}$ using formula 3.5 and 3.6; Obtain the feature map $\tilde{x}$ with mixed style of image x and y using formula 3.7, 3.8 and 3.9; $n_{image}$ = $n_{image}$ +1 Calcuate the few shot classfication loss and accuracy using $\tilde{x}$ ; Update the model Algorithm 1 style mix #### 3 Experiments Dataset We benmark our algorithm on MiniImageNet, which is widely used as a benchmark in few-shot learning. MiniImagenet is a subset of ImageNet ([11]), which has 100 classes in total and 600 images per class. 1000 classes are divided into 64 classes for training, 16 classes for validation and 20 classes for testing. This is a small data but has enough clases and images per class. Thus, it is a perfect testbed for a few-shot learning algorithm. we sample 5-way 1-shot (5 clases and 1 image per class in the query set and 15 images in the support set) tasks and 5-way 5-shot (5 classes and 5 images per class in the query set and 15 images in the support set) tasks for training and testing. During testing ,we average our result over 2000 episodes, while earlier work only use 600 episodes, which has high variance and is not trustable. Backbone We adopt ResNet12 ([25]) here for the feature extractor and Prototypical networks ([43]) for few-shot learning. Earlier work uses Conv4 (Convolutional neural works with only 4 layers) and later on, Resnet12, Resnet18 and wideresnet28-10 are used. Conv4 is too small, has underfit the dataset easily and WideResnet28-10 has too many parameters. We compare Resnet12, Resnet18 and WideResnet and found Resnet12 is enough. Style mix implementation We implement a stylemix layer in which we perform style mixing within each episode or two episodes. This layer can be inserted after each ResNet block and we tested several combinations. Training our training has two steps: pretraining on all 64 clases and episodic training using prototypical networks with style mix. PreTraining is widely used in few-shot learning and has been proven to be vital for the final performance. Result We give the averaged classification accuracy on the testing set over 2000 episodes in table 3.1. method \| 5-way 5-shot accuracy ---\|--- Baseline \| 78.34 ProtoNet \| 79.26 Style mix (after 1st Resnet blcok) \| 78.62 Table 1: Results of style mix In this table, our method achieves comparable results with Protonet. We conduct thorough hyperparameter tuning and detailed analysis trying to figure out why it doesn’t work. Hyperparameter tuning First we study where we should put the style mix layers and how many style mix layers we should put in the feature extractor. There are 3 possible locations to insert the style mix layer and we also try inserting multiple style mix layers. The result is shown in table 3.2. We can see that the deeper and more the style mix layers are, the worse performance we get. method \| 5-way 5-shot accuracy ---\|--- Baseline \| 78.34 ProtoNet \| 79.26 Style mix (after 1st Resnet blcok) \| 78.62 Style mix (after 2st Resnet blcok) \| 78.47 Style mix (after 3st Resnet blcok) \| 77.67 Style mix (after all 4 Resnet blcoks) \| 74.35 Table 2: Result of style mix in different places Then we tune two hyperparameters in our method (table 3.3), $\alpha$ and p. $\alpha$ is a parameter in the Beta distribution, which controls the shape of the distribution. When $\alpha$ gets bigger, we mix the style more. p is the probability of mixing. When p is larger, we mix the styles more. One conclusion can be drawn that no matter how we fine tune these parameters, style mix can not beat the baseline and ProtoNet. \| $\alpha=0.1$ \| $\alpha=0.1$ \| $\alpha=0.1$ \| $\alpha=0.1$ ---\|---\|---\|---\|--- p=0 \| \| 79.26 \| \| p=0.2 \| \| 79.37 \| \| p=0.5 \| 78.87 \| 79.26 \| 79.37 \| 78.60 p=0.8 \| \| 78.97 \| \| Table 3: Hyperparameter tuning for $\alpha$ and p Analysis Then we perform some analysis trying to figure out how our method doesn’t work well as we expect. We insert style mix layers only during testing to see the effect and them. As we can see from table 3.4, the performance gets worse when the style mix layers are inserted deeper. From the below t-SNE embedding of feature maps (figure 3.3), we show that features start to mix together when there is too much style mixing. So the discriminative power is reduced. That explain why the stylization doesn’t work. Insert styleMix during testing ONLY using trained ProtoNet, 5-shot, MiniImageNet --- No styleMix \| After 1st block \| After 2nd block \| After 3rd block \| After 4th block 79.26 \| 74.69 \| 71.46 \| 66.23 \| 60.61 Table 4: Results of style mix when inserting style mix layers during testing Figure 3: TSNE plots of features Fig 1 (right top): no StyleMix layer. Fig 2 (left bottom): 1 StyleMix after 1st ResNet block. Fig 3 (right bottom): 4 styleMix layers after each ResNet block. Mixing probability is 1. ### 2 Spatial attention for few shot learning #### 1 Background Attention is all you need ([48]) first explores attention mechanism in deep learning. Attention mechanism tries to mimic the attention mechanism in bilgraph, in which we only focus on the most important parts. For instance, if we look at the figure 3.4, most of us only focus or visual attention on the dog head. Then BERT ([12]) and GPT ([3]) show it’s superior performance in natural language processing. Non-local neural networks is the first work applying attention mechanism in computer vision, in which a spatial attention on feature maps is designed and shows excellent performance in image and video processing tasks. Then vision transformer ([14]) and i-GPT ([5]) shows transformers with attention can even beat Convolutional neural networks when a large amount of data is available. Figure 4: Attention ([2]) Here we briefly introduce the most basic self-attention mechanism. It tries to improve the representation of each element in a set by aggregating information from other elements in the same set based on the similarity with other elements: $Attention(Q,K,V)=softmax(\frac{QK^{T}}{\sqrt{d_{k}}})V$ (10) The input of a transformer is a set of Q (query), K (key) and V(value). Query, key and value are concepts from retrieval systems. Q is the element we aggregate information for. Key and value are paired. Value is what we look for but it’s often easier to retrieve value by key. In the above formula, we first compute the scaled dot product of Q and K. After applying a softmax function, the attention weight is normalized and it can be viewed as the similarity between the query and the key. based on similarities, we aggregate values. The final output contains information from other elements in the same set so it’s called self attention. And in self-attention, Q=K=V. Our work is also inspired by DeepEMD ([55]) in which image patches are used to computing the matching cost. So the relation between image patches can be utilized. #### 2 Approach We develop several variants of our method applying spatial attention in few- shot learning. We start our introduction from the most basic one. We first apply self attention to feature maps of images in the same class. 5 feature maps of the shape NCHW from the same class are concatenated to the shape NC(5H)W. Then we apply self-attention to it. Each element of the feature maps will aggregate information from other elements of the feature maps. Doing so we hope the output feature map becomes more information and then we get better prototypes (averaged feature maps) for late classification. we also apply self-attention to 5 prototypes so that our final feature maps become more separable. The algorithm is shown in Algorithm 2. Inspired by DeepEMD ([55]), we concatenate the feature maps of each query image and 5 prototypes and then apply self-attention. In this way, we hope to fully utilize the spatial correspondence among all locations of the feature map of the query image and 5 prototypes. This is different from the original prototypical networks ([43]) and here we focus more on local information rather than global information and local information are richer. Result: Classification accuracy while _Traning_ do Sample a s-way 5-shot task T of size; while _$n_{class}$ <5_ do Get five images $x_{1}$, $x_{2}$, … $x_{5}$ for the class $n_{c}lass$ from T; Get feature maps for $x_{1}$, $x_{2}$, … $x_{5}$, $f(x_{1})$, $f(x_{2})$, … $f(x_{5})$; concatenate the feature maps $f(x_{1})$, $f(x_{2})$, … $f(x_{5})$ along the asix H; Applay self attention on the concatenated feature map f(x) using the formula 3.10; Split the feature map after the attention to five feature maps $f(x_{1})^{\prime}$, $f(x_{2})^{\prime}$… $f(x_{5})^{\prime}$; $n_{class}$ = $n_{class}$ +1; Obtain five prototypes by averaging five feature maps in each class; Apply attention on five prototypes; Calcuate the few shot classfication loss and accuracy using augmented prototypes after the attention ; Update the model Algorithm 2 Spatial attention #### 3 Experiments We verify our idea on the same MiniImagenet 5-shot taks. Protonet is our baseline. We use ResNet12 as the feature extractor as it’s large enough to fit the dataset compared to Conv4 and less computationally intensive compared to WideResnet-28-10. We implement an attention layer and insert it into different places of the feature extractor, which are layers after each residual block. The result is shown in table 3.5. We can see that no matter where the attention is inserted, the performance doesn’t change. We will discuss the reason in the following section. method \| 5-way 5-shot accuracy ---\|--- ProtoNet \| 79.97 Spatial attention after the 2nd residual block) \| 79.59 Spatial attention after the 3rd residual block) \| 79.55 Spatial attention after the 4th residual block) \| 79.51 Spatial attention after 4 residual blocks) \| 80.02 Table 5: Results of spatial attention ### 3 Problems of few shot learning methods and remaining challenges Since this paper ([39]) was published, we realize that we have reached a bottleneck for all few-shot learning methods. This paper gives a conclusion based on a series of experiments that MAML works is not due to it can rapidly adapt to new tasks and it’s because it learned good features from the base dataset which is useful during testing for the novel dataset. Baseline++ ([7]) shows that a simple cosine classifier can outperform all the star-of-the-art methods. ([45]) also claims that meta learning doesn’t work well for few shot image classification and all performance boost is from the good feature embedding. ([13]) and SimpleShot([52]) also show the similar phenomena: feature embedding is the key in few shot image classification. All of this mean that we haven’t got enough progress even though lots of fancy meta learning methods are proposed. The area reaches its bottleneck and we won’t get any real progress until there is a theoretical and funmentai breakthrough. This could explain why our earlier methods don’t show significant improvement. It’s hard to get any progress in the area so we transfer our focus to other settings. ## Chapter 3 Cross-domain Few shot learning with unlabeled data In this chapter we introduce the domain shift which exists in few-shot learning and two settings of cross-domain few-shot learning. Domain shift is studied in domain adaptation, domain generalization and it has been observed that the accuracy will drop significantly if there exists a domain shift between the base dataset and the novel dataset. To address this problem, we introduce cross-domain few shot learning. This is a more realistic setting as the domain shift is everywhere in our real life. Also, because we know from the previous section that most meta-learning methods just learn useful features, the accuracy will drop if the training and testing distribution doesn’t match. So cross-domain few shot learning becomes an important research problem. But without touching the target domain where testing happens, the problems become extremely difficult. In most cases, it’s easy to collect some unlabeled data. So cross-domain few shot learning with unlabeled data becomes a more realistic and feasible problem. Studying this problem will have more applications in real life. We will give both the formal definition and related work of cross-domain few shot learning and cross-domain few shot learning with domain shift. Then we will introduce two methods we propose for cross-domain few shot learning with unlabeled data. Detailed and through experiment will be shown in later sections. ### 1 Background #### 1 Domain shift in few shot learning Domain shift means the mismatch between two distributions, which is well studied in domain adaptation and domain generalization. Domain adaptation tries to transfer knowledge from one source domain to another domain but these two domains share the same label space. Domain generalization tries to train a model on some source domains and generalize to another unseen domain. The difference between domain adaptation and domain generalization is that in domain generalization the model never sees the target domain during training. In baseline++ ([7]), the author shows that the classification accuracy will drop if the domain shift between training and testing increases as shown in the figure 4.1. In this paper the author conduct several experiments: evaluating several few-shot learning methods, such as baseline, baseline++, MachingNet, ProtoNet, MAML and RelationNet on CUB->CUB, MiniImageNet, MiniImageNet->CUB. MiniImageNet->CUB means training on MiniImageNet but testing on CUB, which is the cross-domain few shot learning. Domain shift affects the classification accuracy so much and it is so common in real life, so we decide to dive into cross-domain few-shot learning in the next section. Figure 1: Domain shift in few shot learning ([7]) #### 2 Cross domain few shot learning Definition We already discuss domain shift in the above section. Here let’s give a formal definition for cross-domain few shot learning: In cross-domain few shot learning, we have one or several source dataset (base dataset) which has $C1...C_{n}$ classes and one target dataset (novel dataset) which contains $C_{n+1}...C_{n+m}$ classes. So the source dataset does not contain classes from the novel dataset and vice versa. During training only source dataset can be used and during testing the target dataset is used. We can see from the above definition that this setting requires generalization abilities which means that the model needs to perform well not only on the base dataset but also on the novel dataset. Related work There is very little work been done on this topic. Feature-wise transformation ([46]) is one of the first work on this. It tries to augment image features using affine transformation to simulate image distributions from different domains during training. The hyperparameters of the affine transformation are learned using a learning-to-learn approach. In this paper, MiniImageNet, CUB, Cars, Places and Plantae are used as datasets, which are all natural image datasets. This means that the domain shift is not big. Then in ([23]), the author proposes another dataset, which includes non-natural images, such as satellite and chest X-ray images. This makes the benchmark more realistic. The author also proposes some baseline methods in the papers. But this setting is too challenging as it requires the model to get good performance on unseen domain and unseen classes. So instead we put our focus on another new setting where there are some unlabeled data in the target domain. In this case the model can gather information about the target domain and bridge the domain gap between the base dataset and the target dataset. ### 2 Cross domain few shot learning with unlabeled data We propose a new setting which is not studied by other researchers before: Cross-domain few-shot learning with unlabeled data. we add unlabeled data for the target domain for two reasons: unlabeled data are easy to collect in most cases and unlabeled data can bridge the domain gap between the source dataset and the target domain. #### 1 Definition We have a labeled dataset from domain S (base dataset) which has classes $C_{1}...C_{n}$ and another unlabeled dataset from another domain T containing classes $C_{n+1}...C_{n+m}$ for training. Testing data is from domain T but containing classes $C{n+m}...C{m+p}$. Note that the two later datasets are from the same domain and all 3 datasets have disjoint label space. #### 2 Related work Since this setting is proposed by us, there is no work that has been done by others. Related areas include heterogeneous domain adaptation where the source and target domain don’t share the same label space, unsupervised meta learning where the base dataset doesn’t have labels and unsupervised domain adaptation for person/object re-identification. Heterogeneous domain adaptation is also less studied also. ([18]) adopts a deep clustering method for this problem. ([4]) proposes a method using a shared space between the source and target domain. It trains the model with an unsupervised factorisation loss and a graph-based loss. Unsupervised meta learning Recently some researchers start to study this problem. UMTRA ([30]) is an unsupervised meta-learning method with tasks constructed by random sampling and augmentation. CACTUs([28]) uses clustering to automatically construct tasks for unsupervised meta-learning. UDA-Re-ID is called unsupervised domain adaptation for person/object identification. In this setting, they don’t have class labels but have person ids which can be viewed as fine-grained class labels. The person IDs in the target and source domain are disjoint. There are two types of methods in this field, domain translation based methods ,such as using CycleGan ([58]) to translate images from the target domain to the source domain and then train a model with translated images. Another type of methods is pseudo-label based method in which clustering techniques are used to extract pseudo-labels for the unlabeled target data and then unlabeled data with pseudo-labels is used to train a model. ### 3 Style transfer for cross-domain few-shot learning with unlabeled data We propose a method using style transfer. We first feed the images in the unlabeled target dataset as style images and imges in the source dataset as content images to a style transfer model. Then we obtain stylized images which have the same style as the unlabeled target dataset but have same semantic content as the source dataset. We train the stylized images and the original images together. We will show our methods obtain superior results compared to baselines. #### 1 Background Style transfer is introduced in section 3.1.1 so we will not introduce it here. ([21]) shows that Convolutional neural networks trained on ImageNet are heavily biased toward texture instead of shape, which is not preferred because texture information is not the essential feature of a class. If we change the texture of an object in an image, it should belong to the same class. But the shape is different. Then the authors try to remove texture information and obtain a dataset with only shape information using a style transfer model. Then they use the stylized ImageNet to train a model. Results show that the method is more robust to texture variance and environment changes. #### 2 Approach Our method is inspired by ([21]). In our case, we have a different problem: cross-domain few-shot learning with unlabeled data. How to utilize the source dataset and the unlabeled target dataset is a question. We come up with a method which transfers the style of the source dataset to the style of the unlabeled target dataset by feeding images in the unlabeled target dataset as style images and images in source dataset as content images to a style transfer model ([29]). The style transfer model is shown in the figure 4.2. The stylized image looks like images in figure 4.3 if painting images are used as style images and natural images are used as content images. Figure 2: style transfer ([29]) Figure 3: stylized images ([21]) Why is style transfer related to this problem? We find style information carry domain information and lots of style transfer methods share the same ideas as domain adaptation methods. By transferring styles, we actually transfer the domain information and bridge the domain gap between the source dataset and the target dataset. After style transfer we train a ProtoNet ([43]) with both the source dataset images and the stylized images. Testing is conducted on the target testing target dataset. The algorithm is shown in algorithm 3. Result: Classification accuracy Download the style transfer model from ([29]); N = total number of images; while _$n_{img}$ < N_ do Sample a image $x$ from the source dataset and another image $y$ from the unlabeled dataset; Feed $x$ as the content image and $y$ as the style image to the model and obtain a stylized image $z$; $n_{img}$ = $n_{img}$ +1; Train a ProtoNet with all original images and stylized images Algorithm 3 Style transfer #### 3 Experiments Dataset We build our own dataset using DomainNet ([37]). The original dataset contains 6 domains: Real, clipart, painting, sketch, infograph and quickdraw and 345 classes. A quick view is in figure 4.4. Instead of using the whole dataset, we select only a subset of the dataset. Since infograph is too noisy and quickdraw contains too little information, we only use four domains: real, sketch, clipart and sketch. Real is used as the source dataset and 3 others are used as target dataset. Similar to MiniImageNet, we have 64 labeled classes as the base dataset and 16 classes for the unlabeled target dataset and 20 classes as the testing target dataset. Figure 4: DomainNet ([37]) Result We still use the same dataset, MiniImagenet to evaluate our algorithm. ResNet12 is used as the feature extractor, the same as before. we also fine- tune the hyperparameter, style transfer coefficient, which means the degree of style transfer. We find we get the best performance when the style transfer coefficient is 1.0. Our result is shown in table 4.1, in which we have two baselines: ‘Backbone trained on real without unlabeled data’ and ‘ADDA’. ‘Backbone trained on real without unlabeled data’ means we only train a ResNet12 using the labeled source dataset without using the unlabeled target dataset and test it on the testing dataset using mean centroid classifier. ‘ADDA’ is a method in ([47]) and we apply it for this problem. The table shows that our stylization method outperforms all the baseline methods by a large margin, especially on sketch, on which the improvement is 5 percent. \| clipart \| painting \| sketch \| average ---\|---\|---\|---\|--- ADDA \| 59.03 \| 55.48 \| 53.71 \| 56.07 Stylization \| 67.42 \| 60.45 \| 58.62 \| 62.16 Table 1: Results of style transfer #### 4 Discussion Our stylization method has a clear motivation and also shows greater accuracy improvement compared to baseline methods. There is only one defect: the style transfer model used models weights trained on ImageNet, which means that our methods take advantage of another dataset. This is a little bit controversial because we can claim that we only used the style information and the paper, ([21]), also uses the same method. In the future, we can try to avoid using model weights trained on ImageNet. ### 4 Contrastive learning for cross-domain few-shot learning with unlabeled data We also propose a method using contrastive learning, in which we use a clustering method, DBSCAN, to get pseudo-labels and update our model using a loss combining all images in the labeled dataset, clustered images in the unlabeled target dataset and unclustered images. We will show that our method obtains surprising results compared to baseline methods. #### 1 Background Clustering is an unsupervised learning method to group samples, which stay close according to some distance measure, together as a cluster when we don’t have labels for it. Common methods are k-means, DBSCAN ([16]) and Hierarchical clustering. DBSCAN is a density-based method where samples in the high density region are grouped together and points in the low density region are marked as outliers. Contrastive learning is now widely used in self-supervised learning. In contrastive learning we try to find positive and negative samples and force the model to obtain a small distance between a reference sample and its positive samples and big distance between the reference and negative sample. For example, in SimCLR ([6]), augmented images of the same image are treated as positive samples while other images are treated as negative samples. Using this distance supervision signal we can learn meaningful representation even though we may not have labels. Recent self-supervised methods are going to surpass the performance of supervised methods. #### 2 Approach Our method is inspired by ([20]). The original method is designed for object re-identification and we re-purpose it for our setting. First we use the clustering algorithm DBSCAN (k-means can also be applied) to cluster the unlabeled target domain data. Then we have 3 sets of samples: labeled source dataset, clustered unlabeled target dataset and unclustered samples in the unlabeled target dataset. So methods in UDA Re-ID discard those unclustered data but we think it’s a waste of data so we propose a unified framework to fully utilize all the data we have: $L_{f}=-log\frac{exp(<f,z^{+}>/\tau)}{\sum_{k=1}^{n^{s}}exp(<f,w_{k}>/\tau)+\sum_{k=1}^{n_{c}^{t}}exp(<f,c_{k}>/\tau)+\sum_{k=1}^{n_{o}^{t}}exp(<f,v_{k}>/\tau)}$ (1) This is our loss function in which f is a feature vector of an image either from the source dataset, clustered unlabeled target dataset or uncluster outliers. $z^{+}$ indicates a positive prototype corresponding to $f$. $w_{k}$ is the centroid of all features in the class $k$, $c_{k}$ is the centroid of all features in the cluster $k$ and $v_{k}$ is the $k^{th}$ outlier feature. If f is from the source dataset, then $z^{+}=w_{k}$ is the centroid feature vector of class k that f belongs to; if f is from the clustered dataset, then $z^{+}=c_{k}$ is the centroid feature vector of the cluster $k$ that f belongs to and if $f$ is from the unclusted outliers then $z^{+}=v_{k}$ is the outlier feature that f belongs to. Our algorithm alternates between updating the model using the above comparative learning loss and producing pseudo-labels using clustering methods. Detailed description is show in algorithm 4. Result: Classification accuracy Let N be the number of iterations for updating the model after each clustering; while _Traing_ do Obtain pseudo-labels for clustered samples and unclustered samples using DBSCAN algorithm with the unlabeled dataset; $iter$ = 0; while _$iter$ <N_ do $iter$ = $iter$ \+ 1 ; Updating the model using the loss function in formula 1.1 with all labeled samples in the source dataset, clustered samples with their pseudo-labels and unclustered samples; Algorithm 4 Contrastive learning #### 3 Experiment We first benchmark our algorithm on the same dataset as the dataset in 4.3. Resnet12 is used as the feature extractor. we update the model 400 iterations after each step of clustering. As clustering takes a long time (several minutes), we should update our model for more iteration after each clustering. Our result is shown in table 4.2: \| clipart \| painting \| sketch \| average ---\|---\|---\|---\|--- Backbone trained on real without unlabeled data \| 65.17 \| 59.13 \| 53.62 \| 59.31 ADDA \| 59.03 \| 55.48 \| 53.71 \| 56.07 Stylization \| 67.42 \| 60.45 \| 58.62 \| 62.16 Contrastive learning \| 66.18 \| 58.60 \| 56.38 \| 60.39 Contrastive learning (enlarged unlabeled dataset) \| 69.78 \| 60.94 \| 58.76 \| 63.16 Table 2: Results of contrastive learning We can see from the table our method surpasses the baseline on most target dataset but the margin is not big enough. So we suspect that the unlabeled dataset is not big enough. The original unlabeled target dataset has only about 2-3k images, much smaller than the source dataset 20-40 k images. Increase the size of the unlabeled dataset We increase the size of unlabeled dataset for two reasons: First, in most cases, unlabeled dataset are easy to build because no labels are needed and we don’t need experts to label them. Second, most un-supervised learning methods consume more samples compared to supervised learning methods. By increasing the size of the unlabeled dataset, we have more room for these algorithms. Our result is shown in table 4.2 after the size of unlabeled dataset is increased. From the table we can see that our method shows a big improvement on accuracies compared to our baselines, which means our method is very effective for this setting. #### 4 Discussion This work is still ongoing and there are lots of things to be done. For example, the current clustering is still very slow, especially when the unlabeled dataset is very big. We need to find a faster clustering algorithm or reduce the number of clustering. Also we have another idea: combining domain-translation based methods and pseudo-based methods, which will take advantage of both methods and has a great potential to fully use the data and surpass all the methods. ## Chapter 4 Future directions and open challenges In the next future years of my Phd research, there are several directions I want to explore: Cross-domain few shot learning with unlabeled data It is still worth exploring. This is a more realistic setting and we should carefully study this problem and develop more algorithms for it. New research ideas can be drawn from UDA ReID or self-supervised learning. Compositional learning for few-shot learning Current few-shot learning methods still heavily rely on learned features from the base dataset which does not and will not solve the few-shot learning. In my opinion, if we can find primitive features and then compose features from them. The data we need will be very little. The tasks can also be composed of primitive tasks if we can find the relations between different tasks. Causal inference for few shot learning Causal inference is always an interesting topic which tries to find causal factors behind data. But it’s so different from machine learning and it has not been well studied by computer vision research yet. If we can fully utilize causal inference models, then we will be closer to artificial general intelligence and the data we need to train a model will also be much less. Analysis of common practice in few shot learning Lots of common practice are only based on intuitions, which means they may be wrong practice. To move the research in this field forward, we need to carefully verify them by conducting thorough experiments or deriving theorems. For instance, why should we adopt episodic training? Why does pretraining help? 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# Dimerization in quantum spin chains with $O(n)$ symmetry Jakob E. Björnberg Department of Mathematics, Chalmers University of Technology and the University of Gothenburg, Sweden<EMAIL_ADDRESS>, Peter Mühlbacher Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom<EMAIL_ADDRESS>, Bruno Nachtergaele Department of Mathematics and Center for Quantum Mathematics and Physics University of California, Davis Davis, CA 95616, USA<EMAIL_ADDRESS>and Daniel Ueltschi Department of Mathematics, University of Warwick, Coventry, CV4 7AL, United Kingdom <EMAIL_ADDRESS> ###### Abstract. We consider quantum spins with $S\geq 1$, and two-body interactions with $O(2S+1)$ symmetry. We discuss the ground state phase diagram of the one- dimensional system. We give a rigorous proof of dimerization for an open region of the phase diagram, for $S$ sufficiently large. We also prove the existence of a gap for excitations. ###### 1991 Mathematics Subject Classification: 82B10, 82B20, 82B26 © 2021 by the authors. This paper may be reproduced, in its entirety, for non- commercial purposes. ###### Contents 1. 1 Introduction 1. 1.1 A family of quantum spin chains with $O(n)$-invariant interactions 2. 1.2 Ground state phase diagram for general $n\geq 3$ 3. 1.3 The $S=1$ model ($n=3$) 4. 1.4 Our result about dimerization 5. 1.5 Gap for excitations 2. 2 Graphical representation for $O(n)$ models 3. 3 The contour model 1. 3.1 Contours 2. 3.2 Domains and admissibility of contours 3. 3.3 Decomposition of $H(\omega)$ 4. 4 Proof of dimerization 1. 4.1 Setting of the cluster expansion 2. 4.2 Cluster expansion for the partition function 3. 4.3 Dimerization 4. 4.4 Proof of exponential decay of correlations 5. 5 Proof of the spectral gap 6. A The interaction $uT+vP$ when $n$ is even ## 1\. Introduction Over the course of almost a century of studying quantum spin chains, physicists and mathematicians have uncovered a wide variety of interesting physical phenomena and in the process invented an impressive arsenal of new mathematical techniques and structures. Nevertheless, our understanding of these simplest of quantum many-body systems is still far from complete. For many models of interest we have only partial information about the ground state phase diagram, the nature of the phase transitions, and the spectrum of excitations. We consider here a family of spin systems with two-body interactions, where interactions are translation invariant and $O(2S+1)$ invariant. We investigate the ground state phase diagram, looking for ground states that possess less symmetry than the interactions. Our main result is a rigorous proof of dimerization (where translation invariance is broken) in a region of the phase diagram with $S$ large enough (Theorem 1.1). We also prove exponential clustering (Theorem 1.2) and the existence of a gap (Theorem 1.4). The family of models is introduced in Section 1.1; the phase diagram for general $S\geq 1$ is described in Section 1.2; the case $S=1$ has received a lot of attention and we discuss it explicitly in Section 1.3; our result about dimerization is stated in Section 1.4. The $O(n)$ models have a graphical representation which we describe in Section 2. We use it to define a “contour model” in Section 3 where contours are shown to have small weights. This allows to use the method of cluster expansion and prove dimerization in Section 4. ### 1.1. A family of quantum spin chains with $O(n)$-invariant interactions We consider a family of quantum spin chains consisting of $2\ell$ spins of magnitude $S$ defined by a nearest-neighbor Hamiltonian $H_{\ell}$ acting on the Hilbert space $\mathcal{H}_{\ell}=({\mathbb{C}}^{n})^{\otimes 2\ell}$, with $n=2S+1\geq 2$, of the form $H_{\ell}=\sum_{x=-\ell+1}^{\ell-1}h_{x,x+1},$ (1.1) where $h_{x,x+1}$ denotes a copy of $h=h^{}\in M_{n}({\mathbb{C}})\otimes M_{n}({\mathbb{C}})$ acting on the nearest neighbor pair at sites $x$ and $x+1$. We are interested in the family of interactions $h=uT+vQ,\quad u,v\in{\mathbb{R}},$ (1.2) where $T$ is the transposition operator defined by $T(\phi\otimes\varphi)=\varphi\otimes\phi$, for $\phi,\varphi\in{\mathbb{C}}^{n}$, and $Q$ is the orthogonal projection onto the one-dimensional subspace of ${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}$ spanned by a vector of the form $\psi=\frac{1}{\sqrt{n}}\sum_{\alpha=1}^{n}e_{\alpha}\otimes e_{\alpha},$ (1.3) for some orthornormal basis $\\{e_{\alpha}\|\alpha=1,\ldots,n\\}$ of ${\mathbb{C}}^{n}$. The spectrum of $h$ is easy to find. $T$ has the eigenvalues $1$ and $-1$, corresponding to the symmetric and antisymmetric subspaces of ${\mathbb{C}}^{n}\otimes{\mathbb{C}}^{n}$, whose dimensions are $n(n+1)/2$ and $n(n-1)/2$, respectively. Since $\psi$ is symmetric, the eigenvalues of $h$ are $u+v,u,-u$. Let $R$ be a linear transformation represented by an orthogonal matrix in the basis $\\{e_{\alpha}\\}$, meaning $\langle e_{\alpha},RR^{\rm T}e_{\beta}\rangle=\delta_{\alpha\beta}$. This amounts to defining a specific representation of $O(n)$ on the system under consideration. It is then straightforward to check $(R\otimes R)\psi=\psi$. It follows that $R\otimes R$ commutes with $Q=\|\psi\rangle\langle\psi\|$. Since $T$ also commutes with $R\otimes R$, the Hamiltonians with interaction $h$ given in (1.2) have a local $O(n)$ symmetry. This family of models is in fact, up to a trivial additive constant, the most general translation-invariant nearest neighbor Hamiltonian for spins of dimension $n$ and with a translation-invariant local $O(n)$ symmetry. To make contact with previous results in the literature, it is useful to note a couple of equivalent forms of the spin chains we consider. First, for integer values of $S$, that is odd dimensions $n$, consider the orthonormal basis $\\{e_{\alpha}\\}$, relabeled by $\alpha=-S,\ldots,S$, and related to the standard eigenbasis of the third spin matrix $S^{(3)}$, satisfying $S^{(3)}\|\alpha\rangle=\alpha\|\alpha\rangle$, as follows: for $\alpha=0$ take $e_{0}={\rm i}^{S}\|0\rangle$, and for $\alpha>0$ define $e_{\alpha}=\frac{{\rm i}^{S-\alpha}}{\sqrt{2}}\bigl{(}\|\alpha\rangle+\|-\alpha\rangle\bigr{)},\quad e_{-\alpha}=\frac{{\rm i}^{S-\alpha+1}}{\sqrt{2}}\bigl{(}\|\alpha\rangle-\|-\alpha\rangle\bigr{)}.$ (1.4) Then, we have $\psi=\phi:=\frac{1}{\sqrt{n}}\sum_{\alpha=-S}^{S}(-1)^{S-\alpha}\|\alpha,-\alpha\rangle,$ (1.5) which is the $SU(2)$ singlet vector in the standard spin basis. The transposition operator $T$ is of course not affected by any translation- invariant local basis change. Therefore, for odd $n$, and with a simple change of basis, the family of interactions (1.2) is seen to be equivalent to $\tilde{h}=uT+vP,\quad u,v\in{\mathbb{R}},$ (1.6) where $P$ is the orthogonal projection onto the singlet state $\phi$. The case of even $n$ is different. Interactions $h$ and $\tilde{h}$ are not unitarily equivalent. But the model with interaction $\tilde{h}$ is nonetheless interesting and we discuss it in Appendix A. We also prove dimerization and a gap in this case, see Theorem 1.3 and Theorem 1.4. For $n\geq 2$, $u=0$, and $v=-1$, this is the much studied $-P^{(0)}$ spin chain [7, 2, 13, 5, 20, 19, 4]. ### 1.2. Ground state phase diagram for general $n\geq 3$ We start with the phase diagram for arbitrary $n\geq 3$ and discuss the special case $n=3$ in Section 1.3. The ground state phase diagram of the spin chain with nearest-neighbor interactions $h_{x,x+1}=uT_{x,x+1}+vQ_{x,x+1}$ is depicted in Fig. 1. It can be broadly divided into four domains. $u$$v$AA’BB’CReshetikhin$v=-\frac{2n}{n-2}u$ferromagneticdimerizationincommensuratephase correlationsMatrix-product state(s)$v=-2u$ Figure 1. Ground state phase diagram for the chain with nearest-neighbor interactions $uT+vQ$ for $n\geq 3$. Our main result, Theorem 1.1, is a proof of dimerization in an open region around the point B’. The domain formed by the quadrant $u\leq 0,v\geq 0$ (blue region in Fig. 1) is ferromagnetic. There are many ground states and they minimize $h_{x,x+1}$ for all $x$; that is, they are frustration-free. The ground state energy per bond is equal to $u$. Indeed, let $\varphi=\sum_{\alpha}c_{\alpha}e_{\alpha}$ with $\sum_{\alpha}\|c_{\alpha}\|^{2}=1$. It is clear that $\|\varphi\otimes\varphi\rangle$ is eigenstate of $T$ with eigenvalue 1; further, we have $\langle\varphi\otimes\varphi\|Q\|\varphi\otimes\varphi\rangle=\frac{1}{n}\Bigl{\|}\sum_{\alpha}c_{\alpha}^{2}\Bigr{\|}^{2}.$ (1.7) The latter is zero when $\sum_{\alpha}c_{\alpha}^{2}=0$. Since $Q$ is a projector, such a state is eigenstate with eigenvalue 0. Notice that the state $R\varphi$ also satisfies this condition, for all orthogonal transformation $R$. The product state $\otimes_{x=-\ell+1}^{\ell}\varphi$ is then a ground state of $h_{x,x+1}$ with eigenvalue $u$, for all $x$. In addition to these product states, we can obviously take linear combinations. The next domain is the arc-circle between $(u,v)=(-1,0)$ and the “Reshetikhin point” with $v=-\frac{2n}{n-2}u$ (yellow region in Fig. 1), which features dimerization. In order to see that dimerization is plausible as soon as $v<0$, let $\varphi_{x,x+1}=\sqrt{1-{\varepsilon}^{2}}\,\|S,S\rangle+\frac{{\varepsilon}}{\sqrt{n-1}}\sum_{\alpha=-S}^{S-1}\|\alpha,\alpha\rangle.$ (1.8) and consider the (partially) dimerized state $\varphi_{-\ell+1,-\ell+2}\otimes\varphi_{-\ell+3,-\ell+4}\otimes\dots$. For ${\varepsilon}=0$, this is a product state, but for ${\varepsilon}\neq 0$ it is not. Roughly half the edges, namely the edges $x,x+1$ with $x=-\ell+1,-\ell+3,\dotsc$, are dimerized and their energy is $\langle\varphi_{x,x+1}\|uT_{x,x+1}+vQ_{x,x+1}\|\varphi_{x,x+1}\rangle=u+\frac{v}{n}(1+2\sqrt{n-1}{\varepsilon})+O({\varepsilon}^{2}).$ (1.9) The non-dimerized edges contribute $\langle\varphi_{x-1,x}\otimes\varphi_{x+1,x+2}\|uT_{x,x+1}+vQ_{x,x+1}\|\varphi_{x-1,x}\otimes\varphi_{x+1,x+2}\rangle=u+\frac{v}{n}+O({\varepsilon}^{2}).$ (1.10) The average energy per bond of the state $\varphi_{x,x+1}$ is then $u+\frac{v}{n}+v\frac{\sqrt{n-1}}{n}{\varepsilon}$, up to $O({\varepsilon}^{2})$ corrections. When $v<0$ the optimal product states have energy $u+\frac{v}{n}$ (using (1.7) with $\sum_{\alpha}c^{2}_{\alpha}=1$), so the partially dimerized state (1.8) has lower energy when ${\varepsilon}$ is positive and small. Our main result is that dimerization does occur in an open domain around the point B’, provided $n$ is sufficiently large, see Theorem 1.1. This extends the results of [19, 4], valid at the point B’. Then comes the domain formed by the arc-circle between the Reshetikhin point $v=-\frac{2n}{n-2}u$ and $(u,v)=(1,0)$ (red region in Fig. 1). For $n$ odd a unique translation-invariant ground state is expected. This domain contains several interesting special cases. The direction $(u=1,v=0)$ is the the $SU(n)$ generalization of the spin-1/2 Bethe-ansatz solvable Heisenberg model studied by Sutherland and others [24]. The direction $v=-\frac{2n}{n-2}u$ was solved by Reshetikhin [22] (this generalizes the Takhtajan–Babujian model for $n=3$). These models are gapless. The direction $v=-2u$ is a frustration free point and the ground states are given matrix- product states. For odd $n$, these are generalizations of the AKLT model. The ground state for the infinite chain is unique and is in the Haldane phase. For even $n$, there are two matrix-product ground states that break the translation invariance of the chain down to period 2 [28]. The final domain is the quadrant $u,v>0$. The ground states are expected to have slow decaying correlations with incommensurate phase correlations. That is, spin-spin correlations between sites 0 and $x$ are expected to be of the form $\|x\|^{-r}\cos(\omega\|x\|)$ for $\|x\|$ large, and where $r,\omega$ depend on the parameters $u,v$ [11]. It is perhaps worth mentioning that the phase diagram for spatial dimensions other than 1 is quite different. Dimerization is not expected. Instead, the system displays various forms of magnetic long-range orders (ferromagnetic, spin nematic, Néel, …). See [30] for results about magnetic ordering for all $n\geq 2$ and for parameters that correspond to the dimerized phase here. ### 1.3. The $S=1$ model ($n=3$) For $n=3$, the family of models is equivalent to the familiar spin-1 chain with bilinear and biquadratic interactions. The latter is most often parametrized by an angle $\phi$ as follows: $\cos\phi\,\vec{S}_{x}\cdot\vec{S}_{x+1}+\sin\phi\,(\vec{S}_{x}\cdot\vec{S}_{x+1})^{2}=3(\sin\phi-\cos\phi)P+\cos\phi\,T+\sin\phi\,I.$ (1.11) We can apply the change of basis that is the inverse of Eq. (1.4), namely $\|0\rangle=-{\rm i}\,e_{0},\quad\|1\rangle=\tfrac{1}{\sqrt{2}}(e_{1}-{\rm i}\,e_{-1}),\quad\|-1\rangle=\tfrac{1}{\sqrt{2}}(e_{1}+{\rm i}\,e_{-1}).$ (1.12) Then the interaction is given by (1.11) but with the operator $Q$ instead of $P$. $\cos\phi$$\sin\phi$ASutherlandA’BB’CTakhtajan-BabujianAKLT ($\tan\phi=\frac{1}{3}$)ferromagneticdimerizationincommensuratephase correlations Figure 2. Ground state phase diagram for the $S=1$ chain with nearest-neighbor interactions $\cos\phi\vec{S}_{x}\cdot\vec{S}_{x+1}+\sin\phi(\vec{S}_{x}\cdot\vec{S}_{x+1})^{2}$. The domains and the points are the same as those in Fig. 1. The ground state phase diagram with parameter $\phi$ is depicted in Fig. 2. The domains and the points are the same as in Fig. 1. The ferromagnetic domain corresponds to $\phi\in(\frac{\pi}{2},\frac{5\pi}{4})$, and the model is frustration-free in this range. Among the ground states, there is a family of product states that shows that the $O(3)$ symmetry of the Hamiltonian is spontaneously broken. As a consequence, the Goldstone Theorem [15] implies that there are gapless excitations above the ground state in this region. The dimerization domain is $\phi\in(\frac{5\pi}{4},\frac{7\pi}{4})$. The next domain is $\phi\in(-\frac{\pi}{4},\frac{\pi}{4})$ with unique, translation- invariant ground states. Finally, the domain $\phi\in(\frac{\pi}{4},\frac{\pi}{2})$ is expected to display states with slow decay of correlations, with incommensurate phase. There are several points where exact and/or rigorous information is available: (i) $\phi\in[0,\pi/2]$ with $\tan\phi=1/3$, it is the spin-1 AKLT chain [3] with interaction $\tilde{h}$ given by the orthogonal projection on the spin-2 states. In the thermodynamic limit, it has a unique ground state of Matrix Product form with a non-vanishing spectral gap and exact exponential decay of correlations; (ii) the two points with $\tan\phi=1$, A and A’ in Fig. 2, have $SU(3)$ symmetry and are often referred to as the Sutherland model [24]. An exact solution for the ground state at $\phi=-3\pi/4$ is gapless and highly degenerate, while for $\phi=\pi/4$ is believed to be a unique critical state with gapless excitations; (iii) the point $\phi=-\pi/4$ is the Bethe-ansatz solvable Takhtajan–Babujian model [25, 6], which is also gapless; (iv) the point $\phi=-\pi/2$, is the $-P^{(0)}$ spin-1 chain, already mentioned above. Aizenman, Duminil-Copin, and Warzel proved that it has two dimerized (2-periodic) ground states with exponential decay of correlations [4]; all evidence indicates that these states are gapped. Let us briefly comment on higher spatial dimensions. Dimerization is not expected. Various rigorous results about magnetic long-range order have been established: for $\phi=0$ [9]; for $\phi\gtrsim\frac{5\pi}{4}$ [26, 30]; and for $\phi\lesssim 0$ [16]. Recently, the model on the complete graph has been studied by Ryan using methods based on the Brauer algebra [23], which plays a role in the representation theory of the orthogonal groups analogous to that of the symmetric group for the general linear groups. ### 1.4. Our result about dimerization Let us introduce the operators $L^{\alpha,\alpha^{\prime}}$, $1\leq\alpha<\alpha^{\prime}\leq n$, that are generators of the Lie algebra $\mathfrak{o}(n)$: $L^{\alpha,\alpha^{\prime}}=\|\alpha\rangle\langle\alpha^{\prime}\|-\|\alpha^{\prime}\rangle\langle\alpha\|.$ (1.13) And for $x\in\\{-\ell+1,\dots,\ell\\}$, let $L_{x}^{\alpha,\alpha^{\prime}}$ be the operator in $\mathcal{H}_{\ell}$ that acts as $L^{\alpha,\alpha^{\prime}}$ at the site $x$, and as the identity elsewhere. ###### Theorem 1.1. There exist constants $n_{0},u_{0},c>0$ (independent of $\ell$) such that for $n>n_{0}$ and $\|u\|<u_{0}$, we have that for all $1\leq\alpha<\alpha^{\prime}\leq n$, $\begin{split}&\lim_{\beta\to\infty}\Bigl{[}\langle L_{0}^{\alpha,\alpha^{\prime}}L_{1}^{\alpha,\alpha^{\prime}}\rangle_{\ell,\beta,u}-\langle L_{-1}^{\alpha,\alpha^{\prime}}L_{0}^{\alpha,\alpha^{\prime}}\rangle_{\ell,\beta,u}\Bigr{]}>c\qquad\text{for all $\ell$ odd;}\\\ &\lim_{\beta\to\infty}\Bigl{[}\langle L_{0}^{\alpha,\alpha^{\prime}}L_{1}^{\alpha,\alpha^{\prime}}\rangle_{\ell,\beta,u}-\langle L_{-1}^{\alpha,\alpha^{\prime}}L_{0}^{\alpha,\alpha^{\prime}}\rangle_{\ell,\beta,u}\Bigr{]}<-c\qquad\text{for all $\ell$ even.}\end{split}$ Figure 3. Illustration for dimerization. Depending on whether $\ell$ is even or odd, the site $x=0$ is more entangled with its left or its right neighbor. Theorem 1.1 establishes the existence of at least two distinct infinite-volume ground states, close to the point B’ of the phase diagram (see Fig. 3). Notice that the same result holds if we replace the operators $L_{0}^{\alpha,\alpha^{\prime}}L_{1}^{\alpha,\alpha^{\prime}}$ with spin operators $S_{0}^{(3)}S_{1}^{(3)}$, diagonal in the basis $\\{e_{\alpha}\\}$. We expect that there are exactly two extremal ground states, precisely given by limits $\ell\to\infty$ along odd or even integers. We also expect that, if we take the chain to be $\\{-\ell,-\ell+1,\dots,\ell\\}$, the corresponding infinite-volume ground state is equal to the average of the two extremal states. The next result shows that the ground state retains the $O(n)$ symmetry of the system, that there is no magnetic long range order. This is indeed an attribute of dimerisation. ###### Theorem 1.2. There exist constants $n_{0},u_{0},c_{1},c_{2},C>0$ (independent of $\ell$) such that for $n>n_{0}$ and $\|u\|<u_{0}$, we have $\lim_{\beta\to\infty}\bigl{\|}\langle L_{x}^{\alpha,\alpha^{\prime}}\,{\rm e}^{-tH_{\ell}}\,L_{y}^{\alpha,\alpha^{\prime}}\,{\rm e}^{tH_{\ell}}\,\rangle_{\ell,\beta,u}\bigr{\|}\leq C\,{\rm e}^{-c_{1}\|x-y\|-c_{2}\|t\|}\,$ for all $\ell\in{\mathbb{N}}$, all $x,y\in\\{-\ell+1,\dots,\ell\\}$, all $1\leq\alpha<\alpha^{\prime}\leq n$, and all $t\in{\mathbb{R}}$. Dimerization has been established in [19, 4] at the point B’ in the phase diagrams of Figs 1 and 2. The earlier result [19] uses the loop representation of [5] combined with a Peierls argument; it holds for $S\geq 8$ (or $n\geq 17$). The second result, due to Aizenman, Duminil-Copin and Warzel, remarkably holds for all $S\geq 1$ ($n\geq 3$), i.e., for all values of $S$ (or $n$) where dimerization is expected. It uses the loop representation and random cluster representation of [5] as well as recent results for the two- dimensional random cluster model [8, 21]. Away from the point B’ these methods do not apply. In this article we use the loop representation of [30], which combines those of [27, 5], in order to get a contour model; see Theorem 2.1. The loop representation involves a probability measure for $u,v\leq 0$ only; it involves a signed measure otherwise. This is described in Section 2. For large $n$, typical configurations involve many loops, that are short loops located on all the dimerized edges. We define contours to be excitations with respect to this background. It is possible to obtain a contour model with piecewise compatible contours, that is suitable for a cluster expansion (Section 3). This method is robust regarding signs and it allows to intrude in the region with positive parameter $u$. Proving that the expansion converges is difficult, since the cost of excitations is entropic rather than energetic. This is done in Section 4. This allows to establish dimerization in the loop model, see Theorem 4.7. It is equivalent to Theorem 1.1, thus proving our main result. Theorem 1.2 is proved in Subsection 4.4. The interaction that is responsible for dimerization is the operator $Q_{x,x+1}$ and we prove that dimerization is stable under perturbations of this interaction by $uT_{x,x+1}$, with $\|u\|$ sufficiently small. It should be possible to prove stability under more general perturbations that are not necessarily invariant under the group $O(n)$. Since the unperturbed model is not frustration-free, this does not follow from the recent result about the stability of gapped phases with discrete symmetry breaking in [18], which requires the frustration-free property. For translation-invariant perturbations by $O(n)$ invariant next-nearest neighbor or further terms, the methods of this paper should generalize in a straightforward manner. We now discuss the case of the spin chain with Hamiltonian $\tilde{H}_{\ell}=\sum_{x=-\ell+1}^{\ell-1}\bigl{(}uT_{x,x+1}+vP_{x,x+1}\bigr{)},$ (1.14) where $P$ is projection onto the singlet state (recall (1.6)). We have a similar result about dimerization. Let $S^{(i)}$, $i=1,2,3$, be the spin operators that are the generators of the $SU(2)$ symmetry group for $\tilde{H}_{\ell}$. In the basis $\|\alpha\rangle$ where $P$ is the projection onto the vector $\phi$ in (1.5), we can choose $S^{(3)}$ such that $S^{(3)}\|\alpha\rangle=\alpha\|\alpha\rangle$. Let $\langle S_{x}^{(i)}S_{y}^{(i)}\rangle_{\ell,\beta,u}^{\tilde{}}=\frac{1}{{\operatorname{Tr\,}}\,{\rm e}^{-\beta\tilde{H}_{\ell}}\,}{\operatorname{Tr\,}}S_{x}^{(i)}S_{y}^{(i)}\,{\rm e}^{-\beta\tilde{H}_{\ell}}\,.$ (1.15) ###### Theorem 1.3. Let $v=-1$, and $i\in\\{1,2,3\\}$. There exist constants $n_{0},u_{0},c>0$ (independent of $\ell$) such that for $n>n_{0}$ and $\|u\|<u_{0}$, we have $\begin{split}&\lim_{\beta\to\infty}\Bigl{[}\langle S_{0}^{(i)}S_{1}^{(i)}\rangle_{\ell,\beta,u}^{\tilde{}}-\langle S_{-1}^{(i)}S_{0}^{(i)}\rangle_{\ell,\beta,u}^{\tilde{}}\Bigr{]}>c\qquad\text{for all $\ell$ odd;}\\\ &\lim_{\beta\to\infty}\Bigl{[}\langle S_{0}^{(i)}S_{1}^{(i)}\rangle_{\ell,\beta,u}^{\tilde{}}-\langle S_{-1}^{(i)}S_{0}^{(i)}\rangle_{\ell,\beta,u}^{\tilde{}}\Bigr{]}<-c\qquad\text{for all $\ell$ even.}\end{split}$ When $n$ is odd this theorem is equivalent to Theorem 1.1, as the correlations of spin operators are the same as correlations of operators $L_{x,y}^{\alpha,\alpha^{\prime}}$, up to some factors. In the case where $n$ is even, this is no longer the case and the proof needs to be adapted; the modifications are described in Appendix A. ### 1.5. Gap for excitations Let $E_{0}^{(\ell)}<E_{1}^{(\ell)}<\dots$ be the eigenvalues of $H_{\ell}$, and $\tilde{E}_{0}^{(\ell)}<\tilde{E}_{1}^{(\ell)}<\dots$ be the eigenvalues of $\tilde{H}_{\ell}$. The gaps are defined as $\begin{split}\Delta^{(\ell)}=E_{1}^{(\ell)}-E_{0}^{(\ell)},\\\ \tilde{\Delta}^{(\ell)}=\tilde{E}_{1}^{(\ell)}-\tilde{E}_{0}^{(\ell)}.\end{split}$ (1.16) The gaps are obviously positive but the question is whether they are so uniformly in $\ell$. ###### Theorem 1.4. There exist constants $n_{0},u_{0},c>0$ (independent of $\ell$) such that for $n>n_{0}$ and $\|u\|<u_{0}$, we have (a) The multiplicities of $E_{0}^{(\ell)}$ and $\tilde{E}_{0}^{(\ell)}$ are equal to 1. (That is, ground states are unique.) * (b) $\Delta^{(\ell)}\geq c$ and $\tilde{\Delta}^{(\ell)}\geq c$ for all $\ell$. Recall that the chain is $\\{-\ell+1,\dots,\ell\\}$ and it always contains an even number of sites. Our theorem does not cover the chains with odd numbers of sites, although we expect the corresponding Hamiltonians to be gapped as well. The spatial exponential decay proved in Theorem 1.2 is also a consequence of Theorem 1.4, due to the Exponential Clustering Theorem (see the simultaneous articles [10, 17]). Our proof here is motivated by [12]. For the model $H_{\ell}$ it can be found in Section 5. It relies on a loop and contour representation, and on cluster expansions, as for the proof of dimerization. The modifications for $\tilde{H}_{\ell}$ are discussed in the appendix. ## 2\. Graphical representation for $O(n)$ models Consider the one-dimensional graph consisting of the $2\ell$ vertices $V_{\ell}\mathrel{\mathop{:}}=\\{-\ell+1,\dots,\ell\\}$ and the edges $E_{\ell}\mathrel{\mathop{:}}=\big{\\{}(x,x+1):-\ell+1\leq x\leq\ell-1\big{\\}}$. Fix $\beta>0$. To each vertex and edge of this graph we associate a periodic time interval $T_{\beta}=(-\beta,\beta)_{\text{per}}$ to obtain a set of _space-time vertices_ $\overline{V}_{\ell,\beta}\mathrel{\mathop{:}}=V_{\ell}\times T_{\beta}$ as well as a set of _space-time edges_ $\overline{E}_{\ell,\beta}\mathrel{\mathop{:}}=E_{\ell}\times T_{\beta}$. By a _configuration_ $\omega$ we mean a finite subset of $\overline{E}_{\ell,\beta}$, each point of $\omega$ receiving a _mark_ or . The points of $\omega$ will collectively be called _links_ , those marked being referred to as _crosses_ and those marked as _double-bars_. We write $\omega=(\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{dbar.pdf}}}}},\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}})$ and denote the set of all such (link) configurations $\Omega_{\ell,\beta}$. To every configuration $\omega\in\Omega_{\ell,\beta}$ corresponds a set of _loops_ ; see Fig. 4 for an illustration. A loop $l$ is a closed, injective trajectory $\displaystyle[0,L]_{\text{per}}$ $\displaystyle\to\overline{V}_{\ell}$ $\displaystyle t$ $\displaystyle\mapsto l(t)=(v(t),T(t)),$ such that $x(t)$ is piecewise constant and $T^{\prime}(t)\in\\{\pm 1\\}$. We call $L\equiv\|l\|$ the _length_ of $l$, that is the smallest $L>0$ in the above equation. A jump occurs at $t\in[0,L]$ provided that $\\{x(t-),x(t+)\\}\times T(t)$ contains a link. We have $T^{\prime}(t+)=-T^{\prime}(t-)$ in case that link is a double bar and $T^{\prime}(t+)=T^{\prime}(t-)$ in case it is a cross. We identify loops with identical support and we occasionally abuse notation and identify a loop with the set of links it traverses. The number of loops in a configuration $\omega$ is denoted $\mathcal{L}(\omega)$. The number of links in a configuration $\omega$ is denoted by $\\#\omega$. Similarly the number of double bars is denoted by $\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{dbar.pdf}}}}}$ and the number of crosses is denoted by $\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}}$. For $u\in\mathbb{R}$, we define the following _signed measure_ on the set $\Omega_{\ell,\beta}$ of link configurations $\omega$: ${\rm d}\bar{\rho}_{u}(\omega)=u^{\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{cross.pdf}}}}}}{\rm d}^{\otimes\\#\omega}x,$ (2.1) where ${\rm d}x$ is the Lebesgue measure on $\overline{E}_{\ell,\beta}$. We also introduce the following normalized measure $\rho_{u}$, satisfying $\rho_{u}(\Omega_{\ell,\beta})=1$: ${\rm d}\rho_{u}(\omega)=\,{\rm e}^{-(1+u)2\beta\|E_{\ell}\|}\,{\rm d}\bar{\rho}_{u}(\omega)$ (2.2) If $u$ is positive, the measure $\rho_{u}$ is a positive measure and hence a probability measure; in fact, under this measure $\omega$ has the distribution of a Poisson point process with intensity $u$ for crosses and intensity $1$ for double-bars . But we also allow small, negative $u$. Let $Z_{\ell,\beta,n,u}\mathrel{\mathop{:}}=\int_{\Omega_{\ell,\beta}}{\rm d}\rho_{u}(\omega)\,n^{\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}.$ (2.3) This loop model is equivalent to the quantum spin system, and the next result is an instance of this equivalence. The equivalence goes back to Tóth [27] and Aizenman–Nachtergaele [5] for special choices of the parameters; the general case of the interaction (1.2) is due to [30]. Note that it holds for arbitrary finite graphs, not only for chains. We write $x\not\leftrightarrow y$ to characterize the set of configurations $\omega$ where $(x,0)$ and $(y,0)$ belong to distinct loops; $x\overset{+}{\longleftrightarrow}y$ where the top of $(x,0)$ is connected to the bottom of $(y,0)$; and $x\overset{-}{\longleftrightarrow}y$ where the top of $(x,0)$ is connected to the top of $(y,0)$ (see [30, Fig. 2] for an illustration). ###### Theorem 2.1. For the Hamiltonian (1.1) with $h_{x,x+1}=-uT_{x,x+1}-Q_{x,x+1}$, we have that * (a) $\displaystyle{\operatorname{Tr\,}}\,{\rm e}^{-2\beta H_{\ell}}\,=\,{\rm e}^{2\beta(1+u)\|E_{\ell}\|}\,Z_{\ell,\beta,n,u}$. * (b) For all $1\leq\alpha<\alpha^{\prime}\leq n$, we have ${\operatorname{Tr\,}}L_{x}^{\alpha,\alpha^{\prime}}L_{y}^{\alpha,\alpha^{\prime}}\,{\rm e}^{-2\beta H_{\ell}}\,=\tfrac{2}{n}\,{\rm e}^{2\beta(1+u)\|E_{\ell}\|}\,\int_{\Omega_{\ell,\beta}}{\rm d}\rho_{u}(\omega)\,n^{\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}\bigl{(}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}[x\overset{-}{\longleftrightarrow}y]-{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}[x\overset{+}{\longleftrightarrow}y]\bigr{)}.$ The sign of the parameter $u$ in the definition of the interaction has indeed changed; but the theorem holds for arbitrary real (or even complex) parameters. Theorem 2.1 can also be formulated for the interaction $h_{x,x+1}=-uT_{x,x+1}-vQ_{x,x+1}$, by inserting the factor $v^{\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}$ inside the integrals. ###### Proof. The proof of (a) can be found in [30, Theorem 3.2] and (b) is similar, so we only sketch it here. Let $\Sigma(\omega)$ be the set of “space-time spin configurations” that are constant along the loops (so that $\|\Sigma(\omega)\|=n^{\mathcal{L}(\omega)}$). By a standard Feynman-Kac expansion, we get ${\operatorname{Tr\,}}\,{\rm e}^{-2\beta H_{\ell}}\,=\,{\rm e}^{2\beta(1+u)\|E_{\ell}\|}\,\int_{\Omega_{\ell,\beta}}{\rm d}\rho_{u}(\omega)\,n^{-\\#\tilde{\omega}_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}\sum_{\sigma\in\Sigma(\omega)}1.$ (2.4) We recognize the partition function in (2.3), so we get (a). For (b) we need a modified set of space-time spin configurations where the spin value must jump from $\alpha$ to $\alpha^{\prime}$, or from $\alpha^{\prime}$ to $\alpha$, at the points $(x,0)$ and $(y,0)$. Let $\Sigma_{x,y}^{\alpha,\alpha^{\prime}}$ be this set. We then have ${\operatorname{Tr\,}}_{x}^{\alpha,\alpha^{\prime}}L_{y}^{\alpha,\alpha^{\prime}}\,{\rm e}^{-2\beta H_{\ell}}\,=\,{\rm e}^{2\beta(1+u)\|E_{\ell}\|}\,\int_{\Omega_{\ell,\beta}}{\rm d}\rho_{u}(\omega)\,n^{-\\#\tilde{\omega}_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}\\\ \sum_{\sigma\in\Sigma_{x,y}^{\alpha,\alpha^{\prime}}(\omega)}\langle\sigma_{x,0+}\|L_{x}^{\alpha,\alpha^{\prime}}\|\sigma_{x,0-}\rangle\,\langle\sigma_{y,0+}\|L_{y}^{\alpha,\alpha^{\prime}}\|\sigma_{y,0-}\rangle.$ (2.5) It is necessary that $(x,0)$ and $(y,0)$ belong to the same loop in order to get a nonzero contribution. Further, we have $\langle\sigma_{x,0+}\|L_{x}^{\alpha,\alpha^{\prime}}\|\sigma_{x,0-}\rangle\,\langle\sigma_{y,0+}\|L_{y}^{\alpha,\alpha^{\prime}}\|\sigma_{y,0-}\rangle=\begin{cases}-1&\text{if }x\overset{+}{\longleftrightarrow}y,\\\ +1&\text{if }x\overset{-}{\longleftrightarrow}y.\end{cases}$ (2.6) Since $\|\Sigma_{x,y}^{\alpha,\alpha^{\prime}}(\omega)\|=\frac{2}{n}n^{\mathcal{L}(\omega)}$, we get (b). ∎ From now on and to the end of this article we work with the loop model. ###### Remark 2.2 (Intuition). It is helpful to think of $\rho_{u}$ as an a-priori measure on a gas of loops, and rewrite the integrand $n^{\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}$ as $\,{\rm e}^{-(\log n)H(\omega)}\,$, with ‘Hamiltonian’ $-H(\omega)\mathrel{\mathop{:}}=\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{dbar.pdf}}}}},$ (2.7) and inverse temperature $\log n$. Thinking of $n$ as large, the Laplace principle tells us that ‘typical’ configurations should maximise $n^{\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}$. Our goal is to write $Z_{\ell,\beta,n,u}$ as a dominant contribution from such maximizers, and some excitations. We end this section with the following remark about working with a signed measure. Since the (possibly signed) measure $\rho_{u}$ is closely related to the probability measure $\rho_{1}$, it is easy to see that any event $A$ satisfying $\rho_{1}(A)=0$ also has zero measure under $\rho_{u}$. In fact, we have the following slightly stronger property: ###### Lemma 2.3. If $A$ is an event such that $\rho_{1}(A)=0$ and $f:\Omega_{\ell,\beta}\to\mathbb{R}$ is a $\rho_{1}$-integrable function, then for any $u\in\mathbb{R}$ we have that $\int_{A}{\rm d}\rho_{u}(\omega)f(\omega)=0.$ (2.8) ###### Proof. Using (2.1) and (2.2) it is easy to see that $\left\|\int_{A}{\rm d}\rho_{u}(\omega)f(\omega)\right\|\leq C\int_{A}{\rm d}\rho_{1}(\omega)\|f(\omega)\|=0,$ (2.9) for some finite constant $C$ depending only on $u,\ell,\beta$. ∎ As a consequence, we may assume that crosses and double-bars occur at different times, also when $u<0$ and the measure $\rho_{u}$ carries signs. We implicitly used this property when defining loops. ## 3\. The contour model ### 3.1. Contours We classify loops as follows, see Fig. 4. A loop is _contractible_ if it can be continuously deformed to a point and _winding_ otherwise. Not all loops are contractible since our time interval $T_{\beta}$ is periodic. A loop is _long_ if it visits three or more distinct vertices or if it is winding; it is _short_ otherwise. Figure 4. A configuration $\omega$ consisting of three short loops (green, brown, purple), and three long loops (red, blue, orange) two of which are winding loops (blue, orange). We define a _canonical orientation_ of the space-time vertices $\overline{V}_{\ell,\beta}$, using the directions up ($\uparrow$) and down ($\downarrow$), by orienting the leftmost space-time vertex $\\{-\ell+1\\}\times T_{\beta}$ _down_ $\downarrow$ and requiring that neighbouring space-time vertices have opposite orientations; see Fig. 5. We write $V^{\uparrow}_{\ell}\mathrel{\mathop{:}}=\\{x\in V_{\ell}:x+\ell\mbox{ is even}\\}$ for the set of vertices with up-orientation, and $V^{\downarrow}_{\ell}\mathrel{\mathop{:}}=\\{x\in V_{\ell}:x+\ell\mbox{ is odd}\\}$ for the set of vertices with down-orientation, and introduce the following subsets of the edge-set $E_{\ell}$: $\begin{split}E_{\ell}^{+}&\mathrel{\mathop{:}}=\big{\\{}(x,x+1)\in E_{\ell}:x\in V^{\downarrow}_{\ell},x+1\in V^{\uparrow}_{\ell}\big{\\}},\\\ E^{-}_{\ell}&\mathrel{\mathop{:}}=E_{\ell}\setminus E^{+}_{\ell}=\big{\\{}(x,x+1)\in E_{\ell}:x\in V^{\uparrow}_{\ell},x+1\in V^{\downarrow}_{\ell}\big{\\}}.\end{split}$ (3.1) We define $\overline{E}_{\ell,\beta}^{+}$ and $\overline{E}_{\ell,\beta}^{-}$, as well as $\overline{V}^{\uparrow}_{\ell,\beta}$ and $\overline{V}^{\downarrow}_{\ell,\beta}$, analogously. Figure 5. (a) The canonical orientation of $\overline{V}_{\ell,\beta}$ with the set $E^{+}_{\ell}$ highlighted red. (b) a configuration $\omega$ with many short loops; these are positively oriented under the canonical orientation. These definitions are motivated as follows. We expect that ‘typical’ configurations $\omega$ contain many short loops. To maximize the number of short loops one places only double-bars in $\overline{E}^{+}_{\ell}$, as in Fig. 5 (b). The canonical orientation is chosen so that all the short loops in such a configuration are positively oriented (i.e. counter-clockwise). The canonical orientation will be useful in classifying the excitations away from such ‘typical’ $\omega$. Also note that if the origin $0$ belongs to a short, positively oriented loop, then we have $0\leftrightarrow 1$ for $\ell$ odd and $0\leftrightarrow-1$ for $\ell$ even. To prove our main result Theorem 1.1 we will essentially argue that the origin is likely to belong to a short, positively oriented loop. Given a loop $l$ in a configuration $\omega$, we define a _segment_ of $l$ as a trajectory of $l$ between two times $0\leq s_{1}<s_{2}\leq L(l)$ when $l$ passes through height $\beta$. That is to say, $l(s_{1})=(v_{1},\beta),l(s_{2})=(v_{2},\beta)$ for some $v_{i}\in V_{\ell}$, while $l$ does not pass through height $\beta$ in times $t\in(s_{1},s_{2})$. We say that a segment is _spanning_ if for every $t\in T_{\beta}$ there exists a $v=v(t)\in V_{\ell}$ such that the segment traverses $(v,t)$. Note that a spanning segment is not necessarily part of a winding loop. See Fig. 6. Figure 6. The leftmost and rightmost loops are winding loops with one spanning segment each. The loop in the middle is contractible. There are four spanning segments in total. ###### Definition 3.1 (Contours). We say that two loops are _connected_ if they share a link or both are winding. A _contour_ is then a maximally connected set of long loops. ###### Remark 3.2. For later reference, we note here that any cross which is traversed by some loop in a contour is necessarily traversed both ways by the contour; see Fig. 7. Figure 7. A cross is traversed by a contour $\gamma$. If the red loop visits a third vertex, it is a long loop; otherwise it must be a winding loop. In both cases, it is actually part of $\gamma$. A contour which contains at least one winding loop will be called a _winding contour_. See Fig. 8. Figure 8. Two contours: One winding contour, consisting of two winding loops, and one consisting of four long, but contractible loops. We need a notion of _interior_ of a contour, and for this it is useful to regard our configuration $\omega$ as living in the bi-infinite cylinder $C_{\beta}={\mathbb{R}}\times T_{\beta}$. More precisely, given $\omega$ we consider the subset $\overline{\omega}$ of $C_{\beta}$ obtained as the union of (i) $\overline{V}_{\beta}$ embedded in $C_{\beta}$ in the natural way, and (ii) the links of $\omega$ embedded as straight line segments connecting adjacent points of $\overline{V}_{\beta}$. Note that, in the embedding $\overline{\omega}$, crosses and double-bars are embedded in the same way. For a loop $l$ of $\omega$, define its _support_ $S(l)$ as the subset of $\overline{\omega}$ traced out by $l$, meaning the union of the vertical and horizontal line segments of $\overline{\omega}$ corresponding to the intervals of $\overline{V}_{\beta}$ and the links of $\omega$ traversed by $l$. For a contour $\gamma$ of $\omega$ we then make the following definitions. * • The _support_ $S(\gamma)$ is the union of the supports $S(l)$ of the loops $l$ belonging to $\gamma$. Note that $S(\gamma)$ is a closed subset of $C_{\beta}$. * • The _exterior_ $E(\gamma)$ is the union of the unbounded connected components of $C_{\beta}\setminus S(\gamma)$. Note that $E(\gamma)$ is open. * • The _interior_ $I(\gamma)\mathrel{\mathop{:}}={C_{\beta}\setminus\overline{E(\gamma)}}$. Note that $I(\gamma)$ is an open set. * • The _boundary_ $B(\gamma)\mathrel{\mathop{:}}=\overline{E(\gamma)}\setminus E(\gamma)$ which is a closed set. * • The (vertical) length $\|\gamma\|$ of a contour as the sum of the (vertical) lengths of its loops, $\|\gamma\|\mathrel{\mathop{:}}=\sum_{l\in\gamma}\|l\|$. These notions are illustrated in Figs 9–11. Having defined $I(\gamma)$ as a subset of the cylinder $C_{\beta}$, we may also regard $I(\gamma)$ (or more precisely, its closure $\overline{I(\gamma)}$) as a subset of $\overline{E}_{\ell,\beta}$ by identifying a point $(x,x+1)\times\\{t\\}\in\overline{E}_{\ell,\beta}$ with the closed line-segment from $(x,t)$ to $(x+1,t)$ in $C_{\beta}$. Similarly, $S(\gamma)$ and $B(\gamma)$ may be regarded as subsets of $\overline{V}_{\ell,\beta}\cup\omega$. We freely switch between these points of view. Figure 9. A configuration $\omega$ with three contours highlighted green, blue and red. The green contour consists of two winding loops. Figure 10. The corresponding embedding $\overline{\omega}\subseteq C_{\beta}=T_{\beta}\times\mathbb{R}$, with the supports $S(\gamma)$ of the contours highlighted with the corresponding colors. Figure 11. The interiors of the corresponding contours with the boundaries $B(\gamma)$ receiving the canonical orientation. The green and blue contours are of positive type (interiors $I(\gamma)$ on the left) while the red contour is of negative type (interior on the right). Fixing a contour $\gamma$, note that the boundary $B(\gamma)$ consists of a collection of closed curves and horizontal line segments (of length 1). We use the canonical orientation of $\overline{V}_{\ell,\beta}$ to orient each vertical segment of $B(\gamma)$. It is not hard to see that this gives a consistent orientation of all the closed curves constituting $B(\gamma)$. (This follows from Remark 3.2.) Recall the standard notion of a _positively oriented curve_ as one whose interior is always on the left. ###### Definition 3.3 (Type of a contour). We say that the contour $\gamma$ is of _positive type_ if the canonical orientation of $B(\gamma)$ is _positive_ in the sense that $I(\gamma)$ is on the left of each closed curve of $B(\gamma)$. Otherwise we say that $\gamma$ is of _negative type_ (being of negative type is equivalent to the interior being on the right). ###### Remark 3.4. Suppose that $\omega\in\Omega_{\ell,\beta}$ is such that a given point $\bar{v}\in\overline{V}_{\ell,\beta}$ is not on or inside any contour, that is to say $\bar{v}\in\bigcap_{\gamma\in\Gamma(\omega)}E(\gamma),$ (3.2) where $E(\gamma)$ is the exterior of $\gamma$ defined above. Then we have that $\bar{v}$ is on a _positively oriented_ short loop. Indeed, this is related to the fact that all external contours are of positive type, see Lemma 3.6. ### 3.2. Domains and admissibility of contours We now introduce several notations and definitions pertaining to contours and how they relate to each other. First, given $\omega\in\Omega_{\ell,\beta}$ we define $\Gamma(\omega)=\\{\gamma_{1},\dotsc,\gamma_{k}\\}$ as the set of contours in the configuration $\omega$. Here, and in what follows, a contour may be identified with the set of links it traverses. The collection of all possible contours will be denoted $X_{\ell,\beta}=\bigcup_{\omega\in\Omega_{\ell,\beta}}\Gamma(\omega)$, and we write $X^{+}_{\ell,\beta}\subseteq X_{\ell,\beta}$ for the collection of _positive-type_ contours. We write $\mathfrak{X}_{\ell,\beta}=\bigcup_{k\geq 0}\binom{X_{\ell,\beta}}{k}\quad\mbox{and}\quad\mathfrak{X}^{+}_{\ell,\beta}=\bigcup_{k\geq 0}\binom{X^{+}_{\ell,\beta}}{k}$ (3.3) for the set of finite collections of contours, respectively positive-type contours. Elements of $\mathfrak{X}_{\ell,\beta}$ and of $\mathfrak{X}^{+}_{\ell,\beta}$ will usually be denoted by $\Gamma$. It is important to note that far from every such set $\Gamma$ of contours can be obtained as $\Gamma(\omega)$ for some $\omega\in\Omega_{\ell,\beta}$; in fact, we will devote some effort to identifying criteria under which such an $\omega$ does indeed exist. We say that $\Gamma\in\mathfrak{X}_{\ell,\beta}$ is _admissible_ if $\Gamma=\Gamma(\omega)$ for some $\omega\in\Omega_{\ell,\beta}$, and write $\mathfrak{A}_{\ell,\beta}=\Gamma(\Omega_{\ell,\beta})$ for the collection of admissible sets of contours. Recall that the interior $I(\gamma)$ of a contour $\gamma$ is by definition an open subset of the cylinder $C_{\beta}$. Also recall that we regard $\overline{E}_{\ell,\beta}$ as a closed subset of $C_{\beta}$ by identifying a point $(x,x+1)\times\\{t\\}\in\overline{E}_{\ell,\beta}$ with the closed line- segment from $(x,t)$ to $(x+1,t)$. We now define the _(interior) domains_ of $\gamma$ as follows. ###### Definition 3.5. A domain $D$ of $\gamma$ is a subset of $\overline{E}_{\ell,\beta}\cap{I(\gamma)}$ which, when regarded as a subset of $C_{\beta}$ as above, is connected, satisfies $D\cap S(\gamma)=\varnothing$, and is maximal with these properties. We define the _type_ of a domain in a similar way to the type of a contour. Namely, we orient the (topological) boundary of $D$ consistenly with the canonical orientation of $\overline{V}_{\ell,\beta}$ and say that $D$ is of _positive type_ if this is a positive orientation (interior on the left), and of _negative type_ otherwise. See Fig. 12. Figure 12. A contour $\gamma$ of positive type, containing three domains $D_{1},D_{2},D_{3}$. Domains $D_{1}$ and $D_{3}$ are of negative type, while $D_{2}$ is of positive type. Given two contours $\gamma$ and $\gamma^{\prime}$, we say that $\gamma$ is a _descendant_ of $\gamma^{\prime}$, writing $\gamma\prec\gamma^{\prime}$, if $S(\gamma)\subseteq D$ for some domain $D$ of $\gamma^{\prime}$. Given $\Gamma\in\mathfrak{X}_{\ell,\beta}$ and $\gamma,\gamma^{\prime}\in\Gamma$, we say that $\gamma$ is an _immediate descendant of $\gamma^{\prime}$ in $\Gamma$_ if $\gamma\prec\gamma^{\prime}$ and there is no $\overline{\gamma}\in\Gamma$ satisfying both $\gamma\prec\overline{\gamma}$ and $\overline{\gamma}\prec\gamma^{\prime}$. It is important to note that the notion of being an _immediate_ descendant depends not only on the two contours $\gamma$ and $\gamma^{\prime}$ but on the set $\Gamma$; in other words, immediate descendancy cannot be checked in a pairwise manner. If $\gamma\in\Gamma$ is not the descendant of any other contour $\gamma^{\prime}\in\Gamma$ then we say that $\gamma$ is an _external_ contour; this notion is also dependent on the set $\Gamma$. Note that the unique (if it exists) winding contour is always external since a winding loop cannot be in the interior of any contractible loop. ###### Lemma 3.6. Fix $\Gamma\in\mathfrak{X}_{\ell,\beta}$. Then $\Gamma$ is admissible, i.e. $\Gamma\in\mathfrak{A}_{\ell,\beta}$, if and only if the following hold: 1. (1) all external contours in $\Gamma$ are of positive type; 2. (2) for any pair of distinct contours $\gamma,\gamma^{\prime}\in\Gamma$ we have that either $\overline{I(\gamma)}\cap\overline{I(\gamma^{\prime})}=\varnothing$ or $\gamma\prec\gamma^{\prime}$ or $\gamma^{\prime}\prec\gamma$; 3. (3) if $\gamma$ is an immediate descendant of $\gamma^{\prime}$, in a domain $D$ of $\gamma^{\prime}$, then the types of $\gamma$ and of $D$ coincide; 4. (4) there exists at most one winding contour $\gamma\in\Gamma$. ###### Proof. It is easy to see that the four conditions above hold for any admissible $\Gamma=\Gamma(\omega)$. To show the converse, we construct an explicit $\omega\in\Omega_{\ell,\beta}$ with $\Gamma(\omega)=\Gamma$. Starting from the empty configuration $\omega_{0}=\varnothing\in\Omega_{\ell,\beta}$, add all links of all external contours and then place a double bar at height 0, say, on each $e\in E_{\ell}^{+}$ that does not have any link on it. This defines $\omega_{1}$ such that $\Gamma(\omega_{1})$ is precisely the set of external contours of $\Gamma$. Next, add the links of all contours which are immediate descendants of external contours. This does not create any new long loops apart from those in these contours because their types coincide with those of the domains they are in. Iterate this procedure until there are no more contours left to add. ∎ An important prerequisite for applying a cluster expansion is to be able to verify the admissibility of a set of contours in a pairwise manner. As indicated above, and in the light of Lemma 3.6, this is not directly possible since the notion of being an immediate descendant depends on the whole set $\Gamma$. We get around this issue by introducing a notion of _compatibility_ which applies to sets of positive-type contours $\Gamma\in\mathfrak{X}^{+}_{\ell,\beta}$, and which can be checked in a pairwise manner. We then show that there is a bijective correspondence between admissible and compatible sets of contours. The bijective correspondence referred to above involves _shifting_ contours and rests on the simple observation that if $\gamma$ is a negative-type contour, then $\gamma^{\prime}=\gamma+(1,0)$ (i.e. $\gamma$ translated to the right one unit) is a positive-type contour. Figure 13. A contour $\gamma$ and its two appropriately shifted domains (shaded areas). The lower one was not moved since it already was positive type. The upper one was shifted one column to the right. If a $\gamma^{\prime}\in X_{\ell,\beta}$ gets placed inside it, $S^{-1}(\Gamma=(\gamma,\gamma^{\prime}))$ will return an admissible collection of contours. Given a positive-type contour $\gamma\in X^{+}_{\ell,\beta}$ with domains $D_{1}(\gamma),\dotsc,D_{k}(\gamma)\subseteq I(\gamma)$, we define the _appropriately shifted domains_ $D_{i}^{+}(\gamma)$ of $\gamma$ by $D_{i}^{+}(\gamma)=\left\\{\begin{array}[]{ll}D_{i}(\gamma),&\mbox{if }D_{i}(\gamma)\mbox{ is of positive type},\\\ D_{i}(\gamma)+(1,0),&\mbox{otherwise}.\end{array}\right.$ (3.4) Note that while $D_{i}^{+}(\gamma)\subseteq\overline{I(\gamma)}$, a shifted domain may intersect the boundary $B(\gamma)$. See Fig. 13. ###### Definition 3.7. Given two positive-type contours $\gamma,\gamma^{\prime}\in X^{+}_{\ell,\beta}$, we say that $\gamma$ and $\gamma^{\prime}$ are _compatible_ if one of the following hold: 1. (1) $\overline{I(\gamma)}\cap\overline{I(\gamma^{\prime})}=\varnothing$, or 2. (2) $S(\gamma)\subseteq D_{i}^{+}(\gamma^{\prime})$ for some $i$, or 3. (3) $S(\gamma^{\prime})\subseteq D_{i}^{+}(\gamma)$ for some $i$, or 4. (4) at least one of $\gamma$ or $\gamma^{\prime}$ is not a winding contour. We define $\delta(\gamma,\gamma^{\prime})=\begin{cases}1&\text{if $\gamma,\gamma^{\prime}$ are compatible}\\\ 0&\text{otherwise.}\end{cases}$ (3.5) Finally, we let $\mathfrak{C}^{+}_{\ell,\beta}\subseteq\mathfrak{X}^{+}_{\ell,\beta}$ to be the collection of all pairwise compatible sets of positive-type contours; that is, $\Gamma=\\{\gamma_{1},\dotsc,\gamma_{k}\\}\in\mathfrak{X}^{+}_{\ell,\beta}$ belongs to $\mathfrak{C}^{+}_{\ell,\beta}$ if $\prod_{1\leq i<j\leq k}\delta(\gamma_{i},\gamma_{j})=1$. A compatible set $\Gamma$ is generally itself not admissible, since compatible contours may overlap but admissible contours may not. Intuitively, one obtains an admissible set of contours from a compatible set by ‘shifting back’ the appropriately shifted domains $D_{i}^{+}(\gamma)$ and the contours they contain. For nested contours, the shift is performed iteratively; see Fig. 14. Figure 14. (a) A compatible set of contours $\Gamma=\\{\gamma_{1},\gamma_{2},\gamma_{3},\gamma_{4},\gamma_{5}\\}$. We have, for example, $\sigma_{\Gamma}(\gamma_{3})=2$ since $\gamma_{3}$ lies in a shifted domain of both $\gamma_{1}$ and $\gamma_{2}$, while $\sigma_{\Gamma}(\gamma_{4})=1$ since it lies in a shifted domain of $\gamma_{1}$ only. (b) The admissible set $\Sigma(\Gamma)$. More formally, define the _shift_ $\Sigma:\mathfrak{C}^{+}_{\ell,\beta}\to\mathfrak{X}_{\ell,\beta}$ as follows. First, given $\Gamma\in\mathfrak{C}^{+}_{\ell,\beta}$ and $\gamma\in\Gamma$, write $\sigma_{\Gamma}(\gamma)$ for the number of contours $\gamma^{\prime}\in\Gamma\setminus\\{\gamma\\}$ such that $\gamma\subseteq D^{+}_{i}(\gamma^{\prime})\neq D_{i}(\gamma^{\prime})$. This represents the number of times $\gamma$ is shifted to the right in order to obtain the compatible set $\Gamma$ from an admissible set of contours. We define $\Sigma(\Gamma)=\\{\gamma-(\sigma_{\Gamma}(\gamma),0):\gamma\in\Gamma\\}.$ (3.6) ###### Lemma 3.8. The shift $\Sigma$ is a bijection from $\mathfrak{C}^{+}_{\ell,\beta}$, the collection of compatible sets of contours, to $\mathfrak{A}_{\ell,\beta}$, the collection of admissible sets of contours. ###### Proof. It is easy to construct an inverse $\Sigma^{-1}$ of $\Sigma$ on $\mathfrak{A}_{\ell,\beta}$, as follows. Given $\Gamma\in\mathfrak{A}_{\ell,\beta}$, start with an external contour $\gamma$ (which is of positive type by Lemma 3.6) and form its appropriately shifted domains $D_{i}^{+}(\gamma)$. In doing so, shift also the descendants of $\gamma$ along with their domains. Note that all the immediate descendants of $\gamma$ are then mapped to positive type contours. Then iteratively continue this procedure for the (shifted) immediate descendants of $\gamma$. The resulting set $\Sigma^{-1}(\Gamma)$ then satisfies Definition 3.7. It remains to show that $\Sigma(\Gamma)\in\mathfrak{A}_{\ell,\beta}$ for all $\Gamma\in\mathfrak{C}^{+}_{\ell,\beta}$, i.e. that $\Sigma(\Gamma)$ satisfies Lemma 3.6. Compatibility ensures that there is at most one winding contour. It is clear that external contours are of positive type since they are not shifted. For $\gamma,\gamma^{\prime}$ with disjoint interiors, this property is preserved by $\Sigma$; if $\gamma\subseteq D_{i}^{+}(\gamma^{\prime})$ then the shifting ensures that the images of $\gamma,\gamma^{\prime}$ under $\Sigma$ satisfy $\gamma\prec\gamma^{\prime}$, while the relative amounts by which the contours are shifted ensures that the types of immediate descendants in $\Sigma(\Gamma)$ coincide with the types of the relevant domains. ∎ We close this subsection with a simple lemma about counting the amount of ‘available space’ for short loops in a configuration $\omega$, in terms of the lengths of the contours. For $\Gamma\in\mathfrak{A}_{\ell,\beta}$, we define the _free set_ $F(\Gamma)\subseteq\overline{E}_{\ell,\beta}$ as the space-time edges where we can add links without modifying the contours in $\Gamma$ or creating new ones. ###### Lemma 3.9. Let $\Gamma\in\mathfrak{A}_{\ell,\beta}$ be an admissible set of contours. Then $\|F(\Gamma)\|=\|\overline{E}_{\ell,\beta}^{+}\|-\tfrac{1}{2}{\textstyle\sum_{\gamma\in\Gamma}\|\gamma\|.}$ (3.7) ###### Proof. We need to show that $2\|\overline{E}_{\ell,\beta}^{+}\|=2\|F(\Gamma)\|+{\textstyle\sum_{\gamma\in\Gamma}\|\gamma\|.}$ Note that $2\|\overline{E}_{\ell,\beta}^{+}\|=\|\overline{V}_{\ell,\beta}\|$ and that $2\|F(\Gamma)\|$ equals the total length of all the short loops. But any point in $\overline{V}_{\ell,\beta}$ lies either on a contour or on a short loop, thus $\|\overline{V}_{\ell,\beta}\|=2\|F(\Gamma)\|+{\textstyle\sum_{\gamma\in\Gamma}\|\gamma\|},$ as required. ∎ ### 3.3. Decomposition of $H(\omega)$ Recall from (2.7) the quantity $-H(\omega)=\mathcal{L}(\omega)-\|\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{dbar.pdf}}}}}\|$. We now show that $H(\omega)$ can be decomposed as a sum over contours and we prove bounds on the summands. To this end, for a loop $l$ let $\mathcal{T}(l)$ denote the number of _turns_ that $l$ makes; symbolically $\mathcal{T}(l)=\\#\mathchoice{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar- top.pdf}}}}{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar- top.pdf}}}}{\vbox{\hbox{\includegraphics[width=6.00006pt]{dbar- top.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.0pt]{dbar- top.pdf}}}}+\\#\mathchoice{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar- bot.pdf}}}}{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar- bot.pdf}}}}{\vbox{\hbox{\includegraphics[width=6.00006pt]{dbar- bot.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.0pt]{dbar-bot.pdf}}}}$. For a contour $\gamma$, write $\mathcal{T}(\gamma)$ for the total number of U-turns of all loops in $\gamma$. Next define the function $h:X_{\ell,\beta}\to\mathbb{Z}$ by $h(\gamma)=\mathcal{L}(\gamma)-\tfrac{1}{2}\mathcal{T}(\gamma)$ (3.8) where $\mathcal{L}(\gamma)$ denotes the number of loops in the contour $\gamma$. ###### Lemma 3.10. For $\omega\in\Omega_{\ell,\beta}$ with contours $\Gamma=\Gamma(\omega)$ we have $-H(\omega)=\sum_{\gamma\in\Gamma}h(\gamma)$. ###### Proof. Since every double-bar of $\omega$ accounts for exactly two turns (of either one or two loops), we have $-H(\omega)=\sum_{l}\big{(}1-\tfrac{1}{2}\mathcal{T}(l)\big{)}$ (3.9) where the sum is over all loops $l$ in the configuration $\omega$. The result now follows from the observation that short, non-winding loops make exactly two turns. ∎ Write $\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{dbar.pdf}}}}}$ for the number of double-bars visited by $\gamma$ and $\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}}$ for the number of crosses. ###### Lemma 3.11. For contours $\gamma$ without crosses, the function $h:X_{\ell,\beta}\to\mathbb{Z}$ satisfies $h(\gamma)\leq-\tfrac{1}{3}\\#\gamma+2\ell\hbox{\rm 1\kern-2.70004ptI}\\{\gamma\text{ has a spanning segment}\\}.$ (3.10) Note that the constant $-\tfrac{1}{3}$ is tight for the smallest non-winding contours with six double-bars and no crosses, while for larger contours the constant may be taken closer to $-\tfrac{1}{2}$. As to the indicator function, we will see that contours containing spanning segments become very rare asymptotically. ###### Proof. Write $\mathcal{W}(l)$ for the number of winding segments in $l$ and $\mathcal{W}(\gamma)=\sum_{l\in\gamma}\mathcal{W}(l)$. We claim that it suffices to show that $h(\gamma)\leq r(\gamma)$ where $r(\gamma)=-\tfrac{1}{3}\mathcal{T}(\gamma)+\mathcal{W}(\gamma).$ (3.11) Indeed, $r(\gamma)$ is bounded above by the right-hand-side of (3.10) for the following reasons: * • double-bars visited twice by $\gamma$ count twice in $\mathcal{T}(\gamma)$ but only once in $\\#\gamma=\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{dbar.pdf}}}}}$, while those visited once by $\gamma$ count once in both, meaning that $\mathcal{T}(\gamma)\geq\\#\gamma$; * • $\mathcal{W}(\gamma)\leq 2\ell\hbox{\rm 1\kern-2.70004ptI}\\{\gamma\text{ has a spanning segment}\\}$ since each point of the form $(x,0)\in\overline{V}_{\ell,\beta}$ is visited by at most one winding segment. Next, the claimed inequality $h(\gamma)\leq r(\gamma)$ is equivalent to: $\mathcal{T}(\gamma)+6\mathcal{W}(\gamma)\geq 6\mathcal{L}(\gamma).$ (3.12) To establish (3.12), first note that both sides are additive over loops. Thus it suffices to show that any long or winding loop $l$ satisfies $\mathcal{T}(l)+6\mathcal{W}(l)\geq 6.$ (3.13) If $\mathcal{W}(l)\geq 1$ this is clear, hence we may assume that the loop is non-winding. A long, non-winding loop which traverses only double-bars necessarily makes at least 6 turns, see Fig. 15. This proves (3.13) and hence the claim. ∎ ###### Lemma 3.12. For all contours $\gamma\in X_{\ell,\beta}$ we have $h(\gamma)\leq 2\ell\hbox{\rm 1\kern-2.70004ptI}\\{\gamma\text{ has a spanning segment}\\}.$ (3.14) In particular $h(\gamma)\leq 0$ for all non-winding contours. ###### Proof. Note that $h(\gamma)$ is additive over loops $l\in\gamma$. So since there can be at most $2\ell$ spanning segments, it suffices to show for every loop that $1-{1\over 2}(\\#\mathchoice{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar- top.pdf}}}}{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar- top.pdf}}}}{\vbox{\hbox{\includegraphics[width=6.00006pt]{dbar- top.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.0pt]{dbar- top.pdf}}}}+\\#\mathchoice{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar- bot.pdf}}}}{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar- bot.pdf}}}}{\vbox{\hbox{\includegraphics[width=6.00006pt]{dbar- bot.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.0pt]{dbar- bot.pdf}}}})\leq\hbox{\rm 1\kern-2.70004ptI}\\{l\text{ has a spanning segment}\\}$. This is clearly true. ∎ ###### Lemma 3.13. For $\gamma\in X_{\ell,\beta}$, all $u$ with $\|u\|\leq 1$, and all $\kappa>0$, we have $n^{h(\gamma)}\|u\|^{\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{cross.pdf}}}}}}\leq\min(n,\|u\|^{-\kappa/2})^{-(\frac{1}{3}-\kappa)\\#\gamma}n^{2\ell{\scriptsize\hbox{\rm 1\kern-1.89003ptI}}\\{\gamma\text{ has a spanning segment}\\}}.$ (3.15) ###### Proof. If $\gamma$ has no spanning segment and $\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}}\geq{\kappa\over 2}(\frac{1}{3}-\kappa)\\#\gamma$, the claim follows from $h(\gamma)\leq 0$ (Lemma 3.12). Now consider the case where $\gamma$ has no spanning segment and $\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}}\leq{\kappa\over 2}(\frac{1}{3}-\kappa)\\#\gamma$. If $\\#\gamma<({\kappa\over 2}({1\over 3}-\kappa))^{-1}$, then $\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}}=0$ and thus we may apply Lemma 3.11 to get the desired bound in this case. So assume now that $1\leq{\kappa\over 2}({1\over 3}-\kappa)\\#\gamma$ (and still that $\gamma$ has no spanning segment and $\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}}\leq{\kappa\over 2}(\frac{1}{3}-\kappa)\\#\gamma$). Let $\Gamma$ denote the collection of contours and small loops obtained by removing all crosses from $\gamma$, and let $m$ denote the number of small loops in $\Gamma$. Since the removal of a cross can only create at most one more loop, we have that $m\leq\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}}+1\leq{\kappa\over 2}(\frac{1}{3}-\kappa)\\#\gamma+1\leq\kappa(\frac{1}{3}-\kappa)\\#\gamma$, and that $h(\gamma)\leq h(\Gamma)+\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{cross.pdf}}}}}\leq h(\Gamma)+\frac{\kappa}{2}(\tfrac{1}{3}-\kappa)\\#\gamma.$ (3.16) Since every short loop uses at most two double bars, the number of double bars belonging to contours of $\Gamma$ is at least $\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{dbar.pdf}}}}}-2m\geq\bigl{(}1-{\kappa\over 2}(\tfrac{1}{3}-\kappa)\bigr{)}\\#\gamma-2\kappa(\tfrac{1}{3}-\kappa)\\#\gamma=\bigl{(}1-\tfrac{5}{2}\kappa(\tfrac{1}{3}-\kappa)\bigr{)}\\#\gamma.$ (3.17) Applying Lemma 3.11, this allows us to conclude that $h(\Gamma)\leq-\tfrac{1}{3}\big{(}1-\tfrac{5}{2}\kappa(\tfrac{1}{3}-\kappa)\big{)}\\#\gamma.$ (3.18) Combining (3.16) and (3.18) we conclude that $h(\gamma)\leq-\big{[}\tfrac{1}{3}-3\kappa(\tfrac{1}{3}-\kappa)\big{]}\\#\gamma\leq-(\tfrac{1}{3}-\kappa)\\#\gamma.$ (3.19) It remains to show the claim for $\gamma$ with a spanning segment. To this end, consider $\omega$ obtained as follows: Denote by $\omega_{0}$ a configuration of links such that its set of contours $\Gamma(\omega_{0})=\\{\gamma\\}$. Now add $\|E^{+}_{\ell}\|=\ell$ double bars at the same height, exactly one per column in $E^{+}_{\ell}$. Denote this configuration by $\omega$ and note that $\Gamma(\omega)$ does not contain any winding contours. Observe that the number of crosses is unchanged and we added $\ell$ links, hence changing the number of loops by at most $\ell$. Using these observations and Lemma 3.10 we thus get $h(\gamma)=-H(\omega_{0})=-H(\omega)+\mathcal{E}=\sum_{\gamma^{\prime}\in\Gamma(\omega)}h(\gamma^{\prime})+\mathcal{E},$ (3.20) where $\mathcal{E}$ is an error that is bounded by $\|\mathcal{E}\|\leq 2\ell$ and all $\gamma^{\prime}\in\Gamma(\omega)$ are non-winding so that the previous bounds apply. This concludes the proof. ∎ Figure 15. A long, non-winding loop makes at least 6 turns. ## 4\. Proof of dimerization ### 4.1. Setting of the cluster expansion We summarize the main results of the method of cluster expansion as we need it. The following setting and theorem was proposed in [29], extending the results of [14] to the continuous setting and general repulsive interactions. Let $\Gamma$ be a measurable space, $\eta$ a complex measure on $\Gamma$ such that $\|\eta\|(\Gamma)<\infty$, where $\|\eta\|$ is the total variation (absolute value) of $\eta$. Let $\zeta$ be a symmetric function $\Gamma\times\Gamma\to{\mathbb{C}}$ such that $\|1+\zeta(\gamma,\gamma^{\prime})\|\leq 1$ for all $\gamma,\gamma^{\prime}\in\Gamma$. Define the partition function $Z$ by $Z=\sum_{k\geq 0}\frac{1}{k!}\int{\rm d}\eta(\gamma_{1})\dots\int{\rm d}\eta(\gamma_{k})\prod_{1\leq i<j\leq k}\big{(}1+\zeta(\gamma_{i},\gamma_{j})\bigr{)}.$ (4.1) Finally, define the cluster function $\varphi(\gamma_{1},\dots,\gamma_{k})=\begin{cases}1&\text{if }k=1,\\\ \frac{1}{k!}\sum_{G}\prod_{\\{i,j\\}\in G}\zeta(\gamma_{i},\gamma_{j})&\text{otherwise,}\end{cases}$ (4.2) where the sum is over connected graphs of $k$ elements, and the product is over the edges of $G$. Then we have the following expressions and estimates. ###### Theorem 4.1. Assume that there exist functions $a,b:\Gamma\to[0,\infty)$ such that for all $\gamma\in\Gamma$, we have the following Kotecký-Preiss criterion $\int{\rm d}\|\eta\|(\gamma^{\prime})\|\zeta(\gamma,\gamma^{\prime})\|\,{\rm e}^{a(\gamma^{\prime})+b(\gamma^{\prime})}\,\leq a(\gamma).$ (4.3) (Also, assume that $\int{\rm d}\|\eta\|(\gamma)\,{\rm e}^{a(\gamma)+b(\gamma)}\,<\infty$.) Then we have the following. * (a) The partition function is equal to $Z=\exp\biggl{\\{}\sum_{k\geq 1}\int{\rm d}\eta(\gamma_{1})\dots\int{\rm d}\eta(\gamma_{k})\;\varphi(\gamma_{1},\dots,\gamma_{k})\biggr{\\}},$ where the combined sum and integral converges absolutely. * (b) For all $\gamma_{1}\in\Gamma$, $1+\sum_{k\geq 2}k\int{\rm d}\|\eta\|(\gamma_{2})\|\dots\int{\rm d}\|\eta\|(\gamma_{k})\Bigl{(}\sum_{i=1}^{k}\|\zeta(\gamma,\gamma_{i})\|\Bigr{)}\|\varphi(\gamma_{1},\dots,\gamma_{k})\|\,{\rm e}^{b(\gamma_{1})+\dots+b(\gamma_{k})}\,\leq\,{\rm e}^{a(\gamma_{1})}\,.$ * (c) For all $\gamma\in\Gamma$, $\sum_{k\geq 1}\int{\rm d}\|\eta\|(\gamma_{1})\|\dots\int{\rm d}\|\eta\|(\gamma_{k})\Bigl{(}\sum_{i=1}^{k}\|\zeta(\gamma,\gamma_{i})\|\Bigr{)}\|\varphi(\gamma_{1},\dots,\gamma_{k})\|\,{\rm e}^{b(\gamma_{1})+\dots+b(\gamma_{k})}\,\leq a(\gamma).$ This theorem can be found in [29], see Theorems 1 and 3 there, as well as Eqs (18) and (19). Notice that the term $b(\gamma)$ is not usually part of the Kotecký–Preiss criterion and is not needed for convergence of the cluster expansion. But it gives better estimates, see (b) and (c) above, which are most helpful in proving exponential decay. ### 4.2. Cluster expansion for the partition function Let us return to our loop model. We start with the partition function (2.3), namely $Z_{\ell,\beta,n,u}=\,{\rm e}^{-(1+u)\|\overline{E}_{\ell,\beta}\|}\,\int_{\Omega_{\ell,\beta}}{\rm d}\bar{\rho}_{u}(\omega)n^{\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}$ (4.4) where ${\rm d}\bar{\rho}_{u}(\omega)=u^{\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{cross.pdf}}}}}}{\rm d}^{\otimes\\#\omega}x$ is given in (2.1). Since we identify contours $\gamma\in X_{\ell,\beta}$ with the links they are made up of, ${\rm d}\bar{\rho}_{u}(\gamma)$ is also well defined. We define $\tilde{w}(\gamma)\mathrel{\mathop{:}}=\,{\rm e}^{-(1+u)\frac{1}{2}\|\gamma\|}\,n^{h(\gamma)}u^{\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{cross.pdf}}}}}},$ (4.5) where $h(\gamma)$ is defined in (3.8). Let $\mathcal{L}(\Gamma)=\\{l:\exists\gamma\in\Gamma:l\in\gamma\\}$ be the set of loops in a (not necessarily admissible) collection of contours $\Gamma$. Let $Y_{\ell,\beta}\subseteq X_{\ell,\beta}$ the set of contours $\gamma$ (not necessarily admissible) consisting of two adjacent winding loops not traversing any links and let $\mathcal{Y}_{\ell,\beta}\mathrel{\mathop{:}}=\\{\Gamma\subseteq Y_{\ell,\beta}:\gamma\cap\gamma^{\prime}=\emptyset,\;\forall\gamma\neq\gamma^{\prime}\in\Gamma\\}$. Now let $w(\gamma)\mathrel{\mathop{:}}=\sum_{\tilde{\gamma}\in g(\gamma)}\tilde{w}(\tilde{\gamma})(-\,{\rm e}^{-2\beta}\,)^{\\#\gamma\setminus\tilde{\gamma}\over 2},$ (4.6) where $g(\gamma)=\\{\tilde{\gamma}\subseteq\gamma\mid\exists\Gamma^{\prime}\in\mathcal{Y}_{\ell,\beta}:\gamma=\tilde{\gamma}\cup\mathcal{L}(\Gamma^{\prime})\\}$ (4.7) is the set of contours $\tilde{\gamma}$ such that $\gamma$ can be obtained by adding pairs of adjacent, winding loops not traversing any links (those that come from having an “empty good column”) to $\tilde{\gamma}$ and $\\#\gamma\setminus\tilde{\gamma}$ denotes the number of loops that are in $\gamma$, but not in $\tilde{\gamma}$ – necessarily an even number by the definition of $\mathcal{Y}_{\ell,\beta}$. Note that for $\gamma\in X_{\ell,\beta}^{\rm{nw}}$ we have $g(\gamma)=\\{\gamma\\}$, so $w(\gamma)=\tilde{w}(\gamma)$. ###### Proposition 4.2. We have, for any $u\in\mathbb{R}$, $Z_{\ell,\beta,n,u}=\,{\rm e}^{-(1+u)\|\overline{E}_{\ell}^{-}\|}\,\sum_{k\geq 0}\frac{1}{k!}\int_{X^{+}_{\ell}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\dots\int_{X^{+}_{\ell}}{\rm d}\bar{\rho}_{1}(\gamma_{k})\Bigl{(}\prod_{i=1}^{k}w(\gamma_{i})\Bigr{)}\prod_{1\leq i<j\leq k}\delta(\gamma_{i},\gamma_{j}).$ ###### Proof. First note that $d\bar{\rho}_{u}$ factorises, i.e. for $\omega_{1},\omega_{2}\in\Omega_{\ell,\beta}$ sharing no links, we have $d\bar{\rho}_{u}(\omega_{1}\cup\omega_{2})=d\bar{\rho}_{u}(\omega_{1})d\bar{\rho}_{u}(\omega_{2})$. In particular, for any admissible set $\\{\gamma^{1},\dotsc,\gamma^{k}\\}\in\mathfrak{A}_{\ell,\beta}$ of contours, ${\rm d}\bar{\rho}_{u}(\gamma^{1}\cup\dotsc\cup\gamma^{k})=\prod_{i=1}^{k}{\rm d}\bar{\rho}_{u}(\gamma^{i}).$ (4.8) Now let $\Gamma_{0}\in\mathfrak{C}^{+}_{\ell,\beta}$ denote a fixed set of _compatible positive-type_ contours and let $\mathcal{A}(\Gamma_{0})=\\{\omega\in\Omega_{\ell,\beta}:\Gamma(\omega)=\Sigma(\Gamma_{0})\\}$ (4.9) denote the set of link-configurations $\omega$ that induce the set of contours $\Gamma_{0}$ without adding any new contours. By considering the admissible set $\Gamma(\omega)$ and its shift $\Gamma_{0}=\Sigma^{-1}(\Gamma(\omega))=\\{\gamma^{1},\dotsc,\gamma^{k}\\}$, we conclude that $\begin{split}Z_{\ell,\beta,n,u}=\,{\rm e}^{-(1+u)\|\overline{E}_{\ell}\|}\,\sum_{k\geq 0}&\frac{1}{k!}\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma^{1})u^{\\#\gamma^{1}_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{cross.pdf}}}}}}\dots\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma^{k})u^{\\#\gamma^{k}_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{cross.pdf}}}}}}\\\ &\prod_{1\leq i<j\leq k}\delta(\gamma^{i},\gamma^{j})\int_{\mathcal{A}(\Gamma_{0})}{\rm d}\bar{\rho}_{u}\big{(}\omega\setminus\Sigma(\Gamma_{0})\big{)}n^{\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}.\end{split}$ (4.10) We also used Remark 3.2, which tells us that crosses are never ‘shared’ between distinct contours or between a contour and a short loop. (Note that in the last integral, we have the measure $\bar{\rho}_{u}$ rather than $\bar{\rho}_{1}$ as in the other integrals.) Next, applying Lemma 3.10 to write $\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.60002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.20004pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.5pt]{dbar.pdf}}}}}=\sum_{i=1}^{k}h(\gamma^{i})$, we obtain $\int_{\mathcal{A}(\Gamma_{0})}{\rm d}\bar{\rho}_{u}\big{(}\omega\setminus\Sigma(\Gamma_{0})\big{)}n^{\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}=\prod_{i=1}^{k}n^{h(\gamma^{i})}\int_{\mathcal{A}(\Gamma_{0})}{\rm d}\bar{\rho}_{u}\big{(}\omega\setminus\Sigma(\Gamma_{0})\big{)}.$ (4.11) For $\Gamma\in\mathfrak{C}^{+}_{\ell,\beta}$ denote by $\mathfrak{W}(\Gamma)$ the set of columns of $\bar{E}_{\ell,\beta}$ where adding a double bar at any height would not change the set of contours $\Sigma(\Gamma)$. Using Lemma 3.9 and recalling that $\|T_{\beta}\|=2\beta$, we get $\begin{split}\int_{\mathcal{A}(\Gamma_{0})}{\rm d}\bar{\rho}_{u}\big{(}\omega\setminus\Sigma(\Gamma_{0})\big{)}&=\,{\rm e}^{(1+u)\|F(\Gamma(\omega))\|}\,\big{(}\,{\rm e}^{-\|T_{\beta}\|}\,(\,{\rm e}^{\|T_{\beta}\|}\,-1)\big{)}^{\|\mathfrak{W}(\Gamma_{0})\|}\\\ &=\,{\rm e}^{(1+u)\|\overline{E}_{\ell}^{+}\|}\,\Bigl{(}\prod_{\gamma\in\Gamma_{0}}\,{\rm e}^{-(1+u)\frac{1}{2}\|\gamma\|}\,\Bigr{)}(1-\,{\rm e}^{-2\beta}\,)^{\|\mathfrak{W}(\Gamma_{0})\|}.\end{split}$ (4.12) Denote by $\gamma_{w}(\Gamma)$ the unique winding contour in $\Gamma$, if it exists, and $\emptyset$ otherwise. For all $\Gamma\in\mathfrak{C}^{+}_{\ell,\beta}$ we have $\begin{split}(1-\,{\rm e}^{-2\beta}\,)^{\|\mathfrak{W}(\Gamma)\|}&=\sum_{W\subseteq\mathfrak{W}(\Gamma)}\prod_{w\in W}(-\,{\rm e}^{-2\beta}\,)\\\ &=\sum_{\Gamma^{\prime}\in\mathcal{Y}_{\ell,\beta}}\prod_{\gamma^{\prime}\in\Gamma^{\prime}}(-\,{\rm e}^{-2\beta}\,)\times\big{(}\hbox{\rm 1\kern-2.70004ptI}\big{\\{}\gamma^{\prime}_{w}\in\mathcal{A}_{\ell,\beta}\big{\\}}\prod_{\gamma\in\Gamma\setminus\\{\gamma_{w}(\Gamma)\\}}\delta(\gamma,\gamma^{\prime}_{w})\big{)}\\\ &=\int_{\mathfrak{X}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\Gamma^{\prime})\bigg{[}\prod_{\gamma^{\prime}\in\Gamma^{\prime}}(-\,{\rm e}^{-2\beta}\,\hbox{\rm 1\kern-2.70004ptI}\\{\gamma^{\prime}\in Y_{\ell,\beta}\\})\bigg{]}\\\ &\qquad\times\hbox{\rm 1\kern-2.70004ptI}\big{\\{}\gamma^{\prime}_{w}\in\mathcal{A}_{\ell,\beta}\big{\\}}\prod_{\gamma\in\Gamma\setminus\\{\gamma_{w}(\Gamma)\\}}\delta(\gamma,\gamma^{\prime}_{w})\prod_{\gamma,\gamma^{\prime}\in\Gamma^{\prime}}\hbox{\rm 1\kern-2.70004ptI}\\{\gamma\cap\gamma^{\prime}=\emptyset\\},\end{split}$ (4.13) where $\gamma^{\prime}_{w}\equiv\gamma^{\prime}_{w}(\Gamma,\Gamma^{\prime})\mathrel{\mathop{:}}=\gamma_{w}(\Gamma)\cup\mathcal{L}(\Gamma^{\prime})$. For the last equation we used that $\bar{\rho}_{1}$ is just the counting measure on subsets of $Y_{\ell,\beta}$ since these loops do not traverse any links. Intuitively this amounts to summing over “admissible extensions” $\Gamma^{\prime}$ of some given set of contours $\Gamma$ and assigning a different weight to these extensions. But instead of integrating over one set of contours and then another set of contours that are treated differently, we might also integrate over one set of contours and then decide which weight to give to each part of the contour. More rigorously, we combine Eqs (4.10), (4.11), (4.12) and (4.13) to get $\begin{split}Z_{\ell,\beta,n,u}\,{\rm e}^{(1+u)\|\overline{E}_{\ell}^{-}\|}\,&=\int_{\mathfrak{C}^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\Gamma)\bigg{(}\prod_{\gamma\in\Gamma}\tilde{w}(\gamma)\bigg{)}(1-\,{\rm e}^{-2\beta}\,)^{\|\mathfrak{W}(\Gamma)\|}\\\ &=\int_{\mathfrak{C}^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\Gamma)\bigg{(}\prod_{\gamma\in\Gamma}\tilde{w}(\gamma)\bigg{)}\int_{\mathfrak{X}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\Gamma^{\prime})\bigg{[}\prod_{\gamma^{\prime}\in\Gamma^{\prime}}(-\,{\rm e}^{-2\beta}\,\hbox{\rm 1\kern-2.70004ptI}\\{\gamma^{\prime}\in Y_{\ell,\beta}\\})\bigg{]}\\\ &\qquad\times\hbox{\rm 1\kern-2.70004ptI}\big{\\{}\gamma^{\prime}_{w}\in\mathcal{A}_{\ell,\beta}\big{\\}}\prod_{\gamma\in\Gamma\setminus\\{\gamma_{w}(\Gamma)\\}}\delta(\gamma,\gamma^{\prime}_{w})\prod_{\gamma,\gamma^{\prime}\in\Gamma^{\prime}}\hbox{\rm 1\kern-2.70004ptI}\\{\gamma\cap\gamma^{\prime}=\emptyset\\}\\\ &=\int_{\mathfrak{C}^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\Gamma)\prod_{\gamma\in\Gamma}\sum_{\tilde{\gamma}\in g(\gamma)}\tilde{w}(\tilde{\gamma})(-\,{\rm e}^{-2\beta}\,)^{\\#\gamma\setminus\tilde{\gamma}\over 2}.\end{split}$ (4.14) ∎ In what follows we estimate integrals over contours $\gamma$ which intersect a given point or interval. Winding and non-winding contours are treated separately; we will actually see that winding contours play a very limited role for $\beta$ large. We write $X_{\ell,\beta}^{\mathrm{w}}\subseteq X_{\ell,\beta}$ and $X_{\ell,\beta}^{\mathrm{nw}}\subseteq X_{\ell,\beta}$ for the sets of winding and non-winding contours, respectively. We also write $X_{\ell,\beta}^{\mathrm{w}}(k)\subseteq X_{\ell,\beta}$ for the set of winding contours which traverse exactly $k$ links, and we write $X_{\ell,\beta}^{\mathrm{nw}}(\bar{v},k)\subseteq X_{\ell,\beta}$ for the set of non-winding contours $\gamma$, which traverse $k$ links and which visit the point $\bar{v}\in\overline{V}_{\ell}$. Similarly, if $I=[(v,s),(v,t)]\subseteq\overline{V}_{\ell,\beta}$ is an interval we write $X_{\ell,\beta}^{\mathrm{nw}}(I,k)$ for the set of non-winding contours $\gamma$, which traverse $k$ links and which intersect $I$. ###### Lemma 4.3. Fix any $\ell\in{\mathbb{N}},\beta>0$ $c>0$ and points $(v,s),(v,t)\in\overline{V}_{\ell}$ with $s<t$. Write $I=[(v,s),(v,t)]$. Then we have $\displaystyle\int_{X_{\ell,\beta}^{\mathrm{nw}}(I,k)}{\rm d}\bar{\rho}_{1}(\gamma)\,{\rm e}^{-c\|\gamma\|}\,\leq 8^{k-1}c^{-k}\big{(}1+c\|I\|\big{)},$ (4.15) $\displaystyle\int_{X_{\ell,\beta}^{\mathrm{w}}(k)}{\rm d}\bar{\rho}_{1}(\gamma)\,{\rm e}^{-c\|\gamma\|}\,\leq(2\ell+1)^{2\ell+2}(k+1)^{2\ell}8^{k-2}c^{-k}.$ (4.16) ###### Proof. Let us start with the case of non-winding contours and the case when $I=\\{\bar{v}\\}$ contains only one point. Elements $\gamma\in X_{\ell,\beta}^{\mathrm{nw}}(\bar{v},k)$ may be encoded using tuples $(t_{1},\dots,t_{k},l_{1},\dots,l_{k-1})\in{\mathbb{R}}_{+}^{k}\times(\\{\mathchoice{\vbox{\hbox{\includegraphics[width=8.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=8.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=6.00006pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.0pt]{cross.pdf}}}},\mathchoice{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=8.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=6.00006pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=5.0pt]{dbar.pdf}}}}\\}\times\\{\mathtt{L},\mathtt{R}\\}\times\\{\mathtt{1},\mathtt{2}\\})^{k-1}$, as follows. * • Consider a walker started at $\bar{v}$ and travelling upwards until it first encounters the endpoint of a link; store the vertical distance traversed as $t_{1}$. * • This link can go to the left, $\mathtt{L}$, or to the right $\mathtt{R}$; it can be a double bar or a cross ; and it can be traversed by loops in $\gamma$ once, $\mathtt{1}$, or twice, $\mathtt{2}$. Store this information as $l_{1}$. * • Having crossed the link, our walker follows $\gamma$ and records vertical distances until _previously unexplored_ links as $t_{i}$ and information about those links as $l_{i}$, as before. * • If a loop is closed and there are still links that are traversed twice by loops in $\gamma$, but have only been visited once by our walker, the walker continues walking from such a link and recording $t_{i}$ and $l_{i}$ as before (we fix some arbitrary rule for selecting the link and the direction of travel). * • This procedure is iterated until the entire contour has been traversed. See Fig. 16 for an illustration. Figure 16. Illustration of $t_{1},\dots,t_{k},l_{1},\dots,l_{k-1}$. Note, for example, that $t_{5}$ is not merely the distance between the fourth and fifth links, and that $t_{6}$ does not readily admit an interpretation as distance between links at all. Noting that $t_{1}+\dots+t_{k}\leq\|\gamma\|$ and that the number of options for $\\{l_{i}\\}_{i=1}^{k-1}$ is bounded by $8^{k-1}$, we get $\int_{X_{\ell,\beta}^{\mathrm{nw}}(\bar{v},k)}{\rm d}\bar{\rho}_{1}(\gamma)\,{\rm e}^{-c\|\gamma\|}\,\leq 8^{k-1}\int_{0}^{\infty}dt_{1}\dots\int_{0}^{\infty}{\rm d}t_{k}\;\,{\rm e}^{-c(t_{1}+\dots+t_{k})}\,=8^{k-1}c^{-k}.$ (4.17) Next, we may apply a similar argument to obtain that, for ${\varepsilon}>0$ small enough, $\int_{X_{\ell,\beta}^{\mathrm{nw}}((v,t),k)\setminus X_{\ell,\beta}^{\mathrm{nw}}((v,t+{\varepsilon}),k)}{\rm d}\bar{\rho}_{1}(\gamma)\,{\rm e}^{-c\|\gamma\|}\,\leq 8^{k-1}c^{-k+1}\tfrac{1-\,{\rm e}^{-{\varepsilon}c}\,}{c}\leq 8^{k-1}c^{-k+1}{\varepsilon}.$ (4.18) Indeed, for a contour $\gamma$ which visits $(v,t)$ but not $(v,t+{\varepsilon})$, we must have $t_{1}\leq{\varepsilon}$ in the encoding above, and replacing the integral over $t_{1}\in[0,\infty)$ with an integral over $t_{1}\in[0,{\varepsilon}]$ gives the claim. Next, to deduce (4.15) from (4.17) and (4.18), we argue as follows. If $\gamma$ visits $I=[(v,s),(v,t)]$, then either $\gamma$ contains the endpoint $(v,t)$, or there are $r\in(s,t)$ and ${\varepsilon}>0$ such that $(v,r)\in\gamma$ but $(v,r+{\varepsilon})\not\in\gamma$. Using (4.17), the first possibility accounts for the first term $8^{k-1}c^{-k}$ in (4.15). The other possibility accounts for the second term, which one may, for example, see by using a fine dyadic discretization of the interval $I$ and passing to the limit using (4.18) and the monotone convergence theorem. For winding contours $\gamma$, recall that they consist of $r_{1}\leq 2\ell+1$ winding loops with a finite number of contractible long loops attached to at least one of them. In particular there are at most $r\leq r_{1}\leq 2\ell+1$ winding loops that do not share a link and are not connected via a sequence of long, but contractible loops. Let us denote these by $\gamma_{1},\dots,\gamma_{r}$, and the numbers of links they each visit by $k_{1},\dots,k_{r}$, respectively, where $k=\sum_{i=1}^{r}k_{i}$. There are $r$ vertices $v_{1},\dots,v_{r}\in V_{\ell}$ such that $\gamma_{i}$ visits $(v_{i},0)$. Summing over the possibilities for $r$, $v_{1},\dots,v_{r}$, as well as $k_{1},\dots,k_{r}$, and applying the argument for (4.17) to each $\gamma_{i}$, we obtain $\begin{split}\int_{X_{\ell,\beta}^{\mathrm{w}}(k)}&{\rm d}\bar{\rho}_{1}(\gamma)\,{\rm e}^{-c\|\gamma\|}\,\leq\sum_{r=1}^{2\ell+1}{2\ell+1\choose r}\sum_{k_{1},\dots,k_{r}\geq 0\atop k_{1}+\dots+k_{r}=k}8^{k-2}c^{-k}\\\ &\leq(2\ell+1)(2\ell+1)^{2\ell+1}\max_{1\leq r\leq 2\ell+1}\Bigl{\|}\Bigl{\\{}(k_{1},\dots,k_{r})\in{\mathbb{N}}^{r}:\sum_{i}k_{i}=k\Bigr{\\}}\Bigr{\|}8^{k-2}c^{-k}\\\ &\leq(2\ell+1)^{2\ell+2}(k+1)^{2\ell}8^{k-2}c^{-k}.\end{split}$ (4.19) ∎ In order to ensure the convergence of the cluster expansion, we need to check that interactions between contours are small so as to satisfy the Kotecký- Preiss criterion in Eq. (4.3). For $a_{1},a_{2},b_{1},b_{2}\geq 0$, let us introduce $a(\gamma)=a_{1}\|\gamma\|+a_{2}\\#\gamma,\qquad b(\gamma)=b_{1}\|\gamma\|+b_{2}\\#\gamma,$ (4.20) where $\\#\gamma$ denotes the number of links visited by $\gamma$. Then we have the following bound. ###### Lemma 4.4 (Kotecký–Preiss criterion). Let $w(\gamma)$ be as in (4.6). Then there exist $n_{0}$, $u_{0}$, $a_{1}$, $a_{2}$, $b_{1}$, $b_{2}>0$ (independent of $\ell,n,u$), and $\beta_{0}(\ell,n)$, such that for $n>n_{0}$, $\|u\|<u_{0}$, and $\beta>\beta_{0}(\ell,n)$, we have for any $\ell$ and any $\gamma_{0}\in X^{+}_{\ell,\beta}$ that $\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\|w(\gamma)\|\,{\rm e}^{a(\gamma)+b(\gamma)}\,(1-\delta(\gamma,\gamma_{0}))\leq a(\gamma_{0}).$ (4.21) ###### Proof. Let us alleviate the notation by introducing $\bar{w}(\gamma)\mathrel{\mathop{:}}=\|w(\gamma)\|\,{\rm e}^{a(\gamma)+b(\gamma)}\,.$ (4.22) We use Lemma 3.13 with $\kappa$ such that $\frac{1}{3}-\kappa=\frac{1}{4}$ and with $u_{0}$ such that $u_{0}^{-\kappa/2}=n_{0}$, and we set $c_{1}=\tfrac{1-u_{0}}{2}-a_{1}-b_{1},\qquad c_{2}=\tfrac{1}{4}\log n_{0}-a_{2}-b_{2}.$ (4.23) Clearly, $\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma)(1-\delta(\gamma,\gamma_{0}))\leq\int_{X^{\mathrm{w}}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma)+\int_{X^{\mathrm{nw}}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma)(1-\delta(\gamma,\gamma_{0})).$ (4.24) Let us first consider the contribution of winding contours. Notice that $\hbox{\rm 1\kern-2.70004ptI}\\{\gamma\text{ has a spanning segment}\\}\leq\|\gamma\|/(2\beta).$ (4.25) For $\gamma\in X_{\ell,\beta}^{\mathrm{w}}$, note that $\|w(\gamma)\|\leq\|g(\gamma)\|\tilde{w}_{u=-u_{0}}(\gamma)$ where $g(\gamma)$ is given in (4.7). Here $\|g(\gamma)\|\leq 2^{\|E_{\ell}\|}$, which is some constant depending only on $\ell$. Hence, also for $\gamma\in X^{\rm w}_{\ell,\beta}$ we get $\bar{w}(\gamma)\leq\,{\rm e}^{-c_{1}\|\gamma\|}\,\,{\rm e}^{-c_{2}\\#\gamma}\,$ for $\beta\equiv\beta(\ell,n)$ large enough. Using Lemma 4.3 and the fact that a winding contour $\gamma$ satisfies $\|\gamma\|\geq\tfrac{1}{2}\|\gamma\|+\beta$, the first term on the right in (4.24) satisfies $\begin{split}\int_{X^{\mathrm{w}}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma)&\leq\,{\rm e}^{-c_{1}\beta}\,\sum_{k\geq 0}\,{\rm e}^{-c_{2}k}\,\int_{X^{\mathrm{w}}_{\ell,\beta}(k)}{\rm d}\bar{\rho}_{1}(\gamma)\,{\rm e}^{-\tfrac{c_{1}}{2}\|\gamma\|}\,\\\ &\leq(2\ell+1)^{2\ell+2}\,{\rm e}^{-c_{1}\beta}\,\sum_{k\geq 0}(k+1)^{2\ell}\big{(}\tfrac{16}{c_{1}}\,{\rm e}^{-c_{2}}\,\big{)}^{k}\leq c(\ell,n)\,{\rm e}^{-c^{\prime}\beta}\,,\end{split}$ (4.26) with some absolute constant $c^{\prime}>{1\over 4}$ and $c(\ell,n)<\infty$ for $n_{0}$ sufficiently large such that the geometric series converges. In particular (4.26) gets arbitrarily small for $\beta,n_{0}$ large enough. We now turn to non-winding contours. We have $\begin{split}\bar{w}(\gamma)&=\,{\rm e}^{-{1+u\over 2}\|\gamma\|+(a_{1}+b_{1})\|\gamma\|}\,n^{h(\gamma)}\|u\|^{\\#\gamma_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{cross.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{cross.pdf}}}}}}\,{\rm e}^{(a_{2}+b_{2})\\#\gamma}\,\\\ &\leq\,{\rm e}^{-(\frac{1-u_{0}}{2}-a_{1}-b_{1})\|\gamma\|}\,n_{0}^{-\frac{1}{4}\\#\gamma}\,{\rm e}^{(a_{2}+b_{2})\\#\gamma}\,\\\ &=\,{\rm e}^{-c_{1}\|\gamma\|}\,\,{\rm e}^{-c_{2}\\#\gamma}\,.\end{split}$ (4.27) Note that if $\delta(\gamma,\gamma_{0})=0$ then $\gamma$ and $\gamma_{0}$ intersect somewhere on $\overline{V}_{\ell,\beta}$. We may decompose the subset of $\overline{V}_{\ell,\beta}$ visited by $\gamma_{0}$ as a union of closed intervals $I_{1},\dotsc,I_{m}$ where $m\leq\\#\gamma_{0}$ is the number of links of $\gamma_{0}$. Noting also that a non-winding contour $\gamma$ has at least 5 links, we obtain from Lemma 4.3 that $\begin{split}\int_{X^{\mathrm{nw}}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma)(1-\delta(\gamma,\gamma_{0}))&\leq\sum_{j=1}^{m}\sum_{k\geq 5}\,{\rm e}^{-c_{2}k}\,\int_{X^{\mathrm{nw}}_{\ell,\beta}(I_{j},k)}{\rm d}\bar{\rho}_{1}(\gamma)\,{\rm e}^{-c_{1}\|\gamma\|}\,\\\ &\leq\sum_{j=1}^{m}\sum_{k\geq 5}\,{\rm e}^{-c_{2}k}\,8^{k-1}c_{1}^{-k}\big{(}1+c_{1}\|I_{j}\|\big{)}\\\ &\leq\big{(}\tfrac{1}{8}\\#\gamma_{0}+\tfrac{c_{1}}{8}\|\gamma_{0}\|\big{)}\sum_{k\geq 5}\big{(}\tfrac{8}{c_{1}}\,{\rm e}^{-c_{2}}\,\big{)}^{k}.\end{split}$ (4.28) The Lemma 4.4 holds true provided that $\beta$ is large enough, and $\tfrac{1}{8}\sum_{k\geq 5}\big{(}\tfrac{8}{c_{1}}\,{\rm e}^{-c_{2}}\,\big{)}^{k}\leq a_{2},\qquad\tfrac{c_{1}}{8}\sum_{k\geq 5}\big{(}\tfrac{8}{c_{1}}\,{\rm e}^{-c_{2}}\,\big{)}^{k}\leq a_{1}.$ (4.29) Both conditions are fulfilled for $n_{0}$ (and therefore $c_{2}$) large enough. ∎ We will also need an estimate on the integral of contours that contain or surround a given point. ###### Corollary 4.5. For any ${\varepsilon}>0$, there exists $n_{0},u_{0},a_{1},a_{2},b_{1},b_{2}>0$ (independent of $\ell,n,u$) such that for $n>n_{0}$, $\|u\|<u_{0}$, $\beta>\beta_{0}(\ell,n)$, we have $\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\|w(\gamma)\|\,{\rm e}^{a(\gamma)+b(\gamma)}\,{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\\{(0,0)\in\overline{I(\gamma)}\\}\leq{\varepsilon}.$ (4.30) ###### Proof. We proceed as in Eq. (4.24), so that it suffices to bound the contribution from winding and non-winding contours separately. Recall the definition of $\bar{w}$ in (4.22). Using Eq. (4.26), we can make the contribution from winding contours arbitrarily small, say ${\varepsilon}/2$, by choosing $\beta\equiv\beta(\ell,n)$ sufficiently large, i.e. we have $\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma){\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\\{(0,0)\in\overline{I(\gamma)}\\}\leq\int_{X^{\text{nw}}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma)\hbox{\rm 1\kern-2.70004ptI}\\{(0,0)\in\overline{I(\gamma)}\\}+{{\varepsilon}\over 2}.$ (4.31) If $\gamma$ is non-winding and has $k$ links, then it must pass by a site at time 0 at distance less than $k/2$ from $(0,0)$. Thus we have the bound $\int_{X^{\text{nw}}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma){\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\\{(0,0)\in\overline{I(\gamma)}\\}\leq\int_{X^{\text{nw}}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma)\bar{w}(\gamma)\\#\gamma\;{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\\{(0,0)\in\gamma)\\}.$ (4.32) This can be shown to be arbitrarily small, say less than ${\varepsilon}/2$, when $n$ and $\beta$ are large, as in the previous lemma. ∎ Let $\mathcal{C}_{k}$ denote the set of connected (undirected) graphs with vertex set $\\{1,\dotsc,k\\}$ and define $\varphi(\gamma_{1},\dotsc,\gamma_{k})=\left\\{\begin{array}[]{ll}1,&\mbox{if }k=1,\\\ \tfrac{1}{k!}\sum_{G\in\mathcal{C}_{k}}\prod_{ij\in G}(\delta(\gamma_{i},\gamma_{j})-1),&\mbox{if }k\geq 2,\end{array}\right.$ (4.33) where the product in the second line is over the edges of $G$. The following is the main consequence of Theorem 4.1, which holds because of Lemma 4.4. ###### Proposition 4.6 (Cluster expansion of the partition function). For parameters as in Lemma 4.4, the following sum converges absolutely: $\Phi_{\ell,\beta}:=\sum_{m\geq 1}\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\dots\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{k})\Bigl{(}\prod_{i=1}^{k}w(\gamma_{i})\Bigr{)}\varphi(\gamma_{1},\dotsc,\gamma_{m}),$ (4.34) and we have that $\,{\rm e}^{(1+u)\|\overline{E}^{-}_{\ell,\beta}\|}\,Z_{\ell,\beta,n,u}=\exp\big{(}\Phi_{\ell,\beta}\big{)}.$ (4.35) Notice that $\Phi_{\ell,\beta}$ depends on $n$ and $u$ as well. ### 4.3. Dimerization Let us introduce the (signed) measure $\mu_{\ell,\beta,n,u}$ such that the integral of a function $f:\Omega_{\ell,\beta}\to{\mathbb{C}}$ is given by $\mu_{\ell,\beta,n,u}(f)=\frac{1}{Z_{\ell,\beta,n,u}}\int_{\Omega_{\ell,\beta}}{\rm d}\rho_{u}(\omega)\,n^{\mathcal{L}(\omega)-\\#\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}}f(\omega).$ (4.36) The next theorem can be understood as dimerization in the loop model. Together with Theorem 2.1 it implies Theorem 1.1. ###### Theorem 4.7. For any $c>0$, there exist $n_{0},u_{0}>0$ such that for all $n>n_{0}$ and $\|u\|<u_{0}$, we have * (a) For all $\ell$ even: $\begin{split}&\liminf_{\beta\to\infty}\mu_{\ell,\beta,n,u}(0\overset{-}{\longleftrightarrow}-1)>1-c,\quad\text{and}\quad\limsup_{\beta\to\infty}\|\mu_{\ell,\beta,n,u}(0\overset{+}{\longleftrightarrow}-1)\|<c;\\\ &\limsup_{\beta\to\infty}\|\mu_{\ell,\beta,n,u}(0\overset{+}{\longleftrightarrow}1)\|<c,\quad\text{and}\quad\limsup_{\beta\to\infty}\|\mu_{\ell,\beta,n,u}(0\overset{-}{\longleftrightarrow}1)\|<c.\end{split}$ (4.37) * (b) For all $\ell$ odd: $\begin{split}&\liminf_{\beta\to\infty}\mu_{\ell,\beta,n,u}(0\overset{-}{\longleftrightarrow}1)>1-c,\quad\text{and}\quad\limsup_{\beta\to\infty}\|\mu_{\ell,\beta,n,u}(0\overset{+}{\longleftrightarrow}1)\|<c;\\\ &\limsup_{\beta\to\infty}\|\mu_{\ell,\beta,n,u}(0\overset{+}{\longleftrightarrow}-1)\|<c,\quad\text{and}\quad\limsup_{\beta\to\infty}\|\mu_{\ell,\beta,n,u}(0\overset{-}{\longleftrightarrow}-1)\|<c.\end{split}$ (4.38) Notice that the limits $\beta\to\infty$ actually exist; this could be established using the correspondence with the quantum spin system, where convergence is clear. This is less visible in the loop model, though, hence the use of $\limsup$ and $\liminf$ so we do not need to prove it. ###### Proof. Assume without loss of generality that $\ell$ is odd. The case of $\ell$ even works similarly. Let $\mathcal{O}$ (for _outside_) denote the event that $(0,0)$ is not on or inside any contour, that is $\mathcal{O}=\Big{\\{}\omega\in\Omega_{\ell,\beta}:(0,0)\in\bigcap_{\gamma\in\Gamma(\omega)}E(\gamma)\Big{\\}}.$ (4.39) By Remark 3.4 and the fact that $\mu_{\ell,\beta,n,u}(1)=1$, $\mu_{\ell,\beta,n,u}(0\overset{-}{\longleftrightarrow}1)=1-\mu_{\ell,\beta,n,u}({\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\\{0\overset{-}{\longleftrightarrow}1\\}^{\rm c})=1-\mu_{\ell,\beta,n,u}(\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\\{0\overset{-}{\longleftrightarrow}1\\}^{\rm c}).$ (4.40) We also have $\begin{split}&\mu_{\ell,\beta,n,u}(0\overset{+}{\longleftrightarrow}1)=\mu_{\ell,\beta,n,u}(\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\\{0\overset{+}{\longleftrightarrow}1\\}^{\rm c}),\\\ &\mu_{\ell,\beta,n,u}(0\overset{\pm}{\longleftrightarrow}-1)=\mu_{\ell,\beta,n,u}(\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}\\{0\overset{\pm}{\longleftrightarrow}-1\\}^{\rm c}).\end{split}$ (4.41) Then Theorem 4.7 follows from the next lemma (Lemma 4.8), with the function $f$ being an indicator function. ∎ Given a set of compatible contours $\Gamma\in\mathfrak{C}^{+}_{\ell,\beta}$ and its “shifted version” $\Sigma(\Gamma)\in\mathfrak{A}_{\ell,\beta}$, we identify their contours in the natural way; i.e. for every $\gamma\in\Gamma$ there exists a unique $\Sigma(\gamma;\Gamma)\in\Sigma(\Gamma)$ that is obtained by shifting $\gamma$. Every $\Gamma\in\mathfrak{C}^{+}_{\ell,\beta}$ can now be uniquely decomposed into $\Gamma=\Gamma_{0}\dot{\cup}\Gamma\setminus\Gamma_{0}$ with $\Gamma_{0}\mathrel{\mathop{:}}=\\{\gamma\in\Gamma\mid(0,0)\in\overline{I(\Sigma(\gamma;\Gamma))}\\},$ (4.42) where $I(\gamma)$ is the interior of $\gamma$, defined in Section 3. This decomposition will be useful in the proof of the following lemma. ###### Lemma 4.8. Let $g:\Omega_{\ell,\beta}\to{\mathbb{R}}$ be a function that, for every $\omega$, only depends on the contours $\gamma\in\Gamma(\omega)\in\mathfrak{A}_{\ell,\beta}$ that surround or contain $(0,0)$. Assume that $\|g\|\leq 1$. Then for every ${\varepsilon}>0$, there exists $n_{0}\in{\mathbb{N}},u_{0}>0$ such that for all $\ell$, $n>n_{0}$, $\|u\|<u_{0}$, and $\beta\equiv\beta(\ell,n)$ large enough, $\|\mu_{\ell,\beta,n,u}(\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}g)\|<{\varepsilon}.$ (4.43) ###### Proof. We have $\mu_{\ell,\beta,n,u}(\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}g)=\frac{Z_{\ell,\beta,n,u}[\mathcal{O}^{c};g]}{Z_{\ell,\beta,n,u}},$ (4.44) where $\begin{split}Z_{\ell,\beta,n,u}[\mathcal{O}^{c};g]&=\,{\rm e}^{-(1+u)\|\overline{E}_{\ell,\beta}^{-}\|}\,\int_{\mathfrak{C}^{+}_{\ell,\beta}\setminus\\{\emptyset\\}}{\rm d}\bar{\rho}_{1}(\Gamma_{0})g(\Gamma_{0})\Big{(}\prod_{\gamma\in\Gamma_{0}}w(\gamma)\Big{)}\hbox{\rm 1\kern-2.70004ptI}\Bigl{\\{}(0,0)\in\overline{I(\Sigma(\gamma;\Gamma_{0}))}\;\forall\gamma\in\Gamma_{0}\Bigr{\\}}\\\ &\times\sum_{m\geq 0}{1\over m!}\int_{X_{\ell,\beta}^{+}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\dots\int_{X_{\ell,\beta}^{+}}{\rm d}\bar{\rho}_{1}(\gamma_{m})\Bigl{(}\prod_{i=1}^{m}w_{\Gamma_{0}}(\gamma_{i})\Bigr{)}\prod_{1\leq i<j\leq m}\delta(\gamma_{i},\gamma_{j}).\end{split}$ (4.45) Notice that the contour weights in the last line depend on $\Gamma_{0}$ and are defined as $w_{\Gamma_{0}}(\gamma)\mathrel{\mathop{:}}=w(\gamma)\hbox{\rm 1\kern-2.70004ptI}\\{(0,0)\notin\overline{I(\Sigma(\gamma;\Gamma_{0}))}\\}\prod_{\gamma_{0}\in\Gamma_{0}}\delta(\gamma,\gamma_{0}).$ (4.46) Intuitively, we first integrate over all contours surrounding $(0,0)$ (after shifting, they are called $\Gamma_{0}$) and then we integrate out the remaining contours that are compatible with $\Gamma_{0}$. The second line of Eq. (4.45) has the structure of a partition function. Since $\|w_{\Gamma_{0}}(\gamma_{i})\|\leq\|w(\gamma_{i})\|$, Lemma 4.4 holds for the modified weights too, and therefore also the suitable modification of Proposition 4.6. This is then equal to $\exp\big{(}\Phi_{\ell,\beta}(\Gamma_{0})\big{)}$ where $\Phi_{\ell,\beta}(\Gamma_{0})\mathrel{\mathop{:}}=\sum_{m\geq 1}\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\dots\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{m})\Bigl{(}\prod_{i=1}^{m}w_{\Gamma_{0}}(\gamma_{i})\Bigr{)}\varphi(\gamma_{1},\dotsc,\gamma_{m}).$ (4.47) Notice that the sum in $\Phi_{\ell,\beta}(\Gamma_{0})$ converges absolutely. Then $\mu(\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}g)=\int_{\mathfrak{C}^{+}_{\ell,\beta}\setminus\\{\emptyset\\}}{\rm d}\bar{\rho}_{1}(\Gamma_{0})g(\Gamma_{0})\Big{(}\prod_{\gamma\in\Gamma_{0}}w(\gamma)\Big{)}\\\ \hbox{\rm 1\kern-2.70004ptI}\bigl{\\{}(0,0)\in\overline{I(\Sigma(\gamma;\Gamma_{0}))}\;\forall\gamma\in\Gamma_{0}\bigr{\\}}\exp\big{\\{}\Phi_{\ell,\beta}(\Gamma_{0})-\Phi_{\ell,\beta}\big{\\}}$ (4.48) Let $\delta_{0}(\gamma,\Gamma_{0})\mathrel{\mathop{:}}=\hbox{\rm 1\kern-2.70004ptI}\\{(0,0)\notin\overline{I(\Sigma(\gamma;\Gamma_{0}))}\\}\prod_{\gamma_{0}\in\Gamma_{0}}\delta(\gamma,\gamma_{0})$ be the indicator function for $\gamma$ being a contour that is compatible with $\Gamma_{0}$ and should not be part of $\Gamma_{0}$. Then $\Phi_{\ell,\beta}-\Phi_{\ell,\beta}(\Gamma_{0})=\sum_{m\geq 1}\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\dots\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{m})\Big{(}\prod_{i=1}^{m}w(\gamma_{i})\Big{)}\varphi(\gamma_{1},\dots,\gamma_{m})\\\ \hbox{\rm 1\kern-2.70004ptI}\bigl{\\{}\exists i\leq m:\delta_{0}(\gamma_{i},\Gamma_{0})=0\bigr{\\}}.$ (4.49) We bound these “corrections coming from contours not in $\Gamma_{0}$” as follows: $\begin{split}&\|\Phi_{\ell,\beta}-\Phi_{\ell,\beta}(\Gamma_{0})\|\\\ &\leq\sum_{m\geq 1}\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\dots\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{m})\Big{(}\prod_{i=1}^{m}\|w(\gamma_{i})\|\Big{)}\|\varphi(\gamma_{1},\dots,\gamma_{m})\|\hbox{\rm 1\kern-2.70004ptI}\\{\exists i\leq m:\delta_{0}(\gamma_{i},\Gamma_{0})=0\\}\\\ &\leq\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\|w(\gamma_{1})\|(1-\delta_{0}(\gamma_{1},\Gamma_{0}))\biggl{(}1+\sum_{m\geq 2}m\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{2})\dots\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{m})\|\varphi(\gamma_{1},\dots,\gamma_{m})\|\biggr{)}\\\ &\leq\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\|w(\gamma_{1})\|\,{\rm e}^{a(\gamma_{1})}\,(1-\delta_{0}(\gamma_{1},\Gamma_{0})).\end{split}$ (4.50) The last inequality follows from Theorem 4.1 (b). It is easy to see from the definition of $\delta_{0}$ that $1-\delta_{0}(\gamma_{1},\Gamma_{0})\leq\sum_{\gamma_{0}\in\Gamma_{0}}(1-\delta(\gamma_{0},\gamma_{1}))+\hbox{\rm 1\kern-2.70004ptI}\\{(0,0)\in\overline{I(\Sigma(\gamma_{1};\Gamma_{0}))}\\}$. Thus $\|\Phi_{\ell,\beta}-\Phi_{\ell,\beta}(\Gamma_{0})\|\leq\sum_{\gamma_{0}\in\Gamma_{0}}a(\gamma_{0})+\int_{X^{+}_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\|w(\gamma_{1})\|\,{\rm e}^{a(\gamma_{1})}\,\hbox{\rm 1\kern-2.70004ptI}\\{(0,0)\in\overline{I(\gamma_{1})}\\}.$ (4.51) We used Lemma 4.4 for the first summand and translation invariance for the second summand. Using Corollary 4.5, we can bound the second summand by ${\varepsilon}$, arbitrarily (and uniformly in $\ell,n,u$) small, for $n_{0},\beta$ large enough. Plugging these bounds back into (4.48) and using $\|g\|\leq 1$, we get $\begin{split}\big{\|}\mu_{\ell,\beta,n,u}&(\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}g)\big{\|}\leq\,{\rm e}^{{\varepsilon}}\,\int_{\mathfrak{C}^{+}_{\ell,\beta}\setminus\\{\emptyset\\}}{\rm d}\bar{\rho}_{1}(\Gamma_{0})\prod_{\gamma_{0}\in\Gamma_{0}}\big{(}\|w(\gamma_{0})\|\,{\rm e}^{a(\gamma_{0})}\,\big{)}\hbox{\rm 1\kern-2.70004ptI}\\{(0,0)\in\overline{I(\Sigma(\gamma_{0};\Gamma_{0}))}\;\forall\gamma_{0}\in\Gamma_{0}\\}\\\ &\leq\,{\rm e}^{{\varepsilon}}\,\sum_{m\geq 1}\int_{X_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{1})\bar{w}(\gamma_{1})\hbox{\rm 1\kern-2.70004ptI}\\{(0,0)\in\overline{I(\gamma_{1})}\\}\;\dots\int_{X_{\ell,\beta}}{\rm d}\bar{\rho}_{1}(\gamma_{m})\bar{w}(\gamma_{m})\hbox{\rm 1\kern-2.70004ptI}\\{(0,0)\in\overline{I(\gamma_{m})}\\}\\\ &\leq 2\,{\rm e}^{{\varepsilon}}\,\sum_{m\geq 1}{\varepsilon}^{m}=2\,{\rm e}^{{\varepsilon}}\,\bigg{(}{{\varepsilon}\over 1-{\varepsilon}}\bigg{)}.\end{split}$ (4.52) For the second inequality we use the fact that all admissible $\Sigma(\Gamma_{0})\in\mathfrak{A}_{\ell,\beta}$ such that $(0,0)\in\overline{I(\Sigma(\gamma_{0};\Gamma_{0}))}\;\forall\gamma_{0}\in\Gamma_{0}$ can be written uniquely as $\\{\gamma_{1},\dots,\gamma_{m}\\}\in\mathfrak{A}_{\ell,\beta}$ with $(0,0)\in\overline{I(\gamma_{m})}$ and $\gamma_{i}$ surrounding $\gamma_{j}$ whenever $i<j$. In particular all $\gamma_{i}$ must surround $(0,0)$. The last inequality follows from applying Corollary 4.5 for each of the nested integrals, with a factor of $2$ since we integrate over $X_{\ell,\beta}$ instead of $X_{\ell,\beta}^{+}$. Since ${\varepsilon}$ can be made arbitrarily small by taking $n_{0},\beta$ large, we get the lemma. ∎ ### 4.4. Proof of exponential decay of correlations We now turn to exponential decay. Theorem 1.2 is an immediate consequence of the following result about loop correlations. ###### Theorem 4.9. There exists an $n_{0},u_{0},C,c_{1},c_{2}>0$ (independent of $\ell,n,u$) such that for $n>n_{0},\|u\|<u_{0}$, we have $\left.\begin{array}[]{ll}\big{\|}\mu_{\ell,\beta,n,u}\big{(}(x,s)\leftrightarrow(y,t)\big{)}\big{\|}\\\ \big{\|}\mu_{\ell,\beta,n,u}\big{(}(x,s)\ext@arrow 9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{+}(y,t)\big{)}\big{\|}\\\ \big{\|}\mu_{\ell,\beta,n,u}\big{(}(x,s)\ext@arrow 9999{\arrowfill@\leftarrow\relbar\rightarrow}{}{-}(y,t)\big{)}\big{\|}\end{array}\right\\}\leq C\,{\rm e}^{-c_{1}\|x-y\|-c_{2}\|s-t\|}\,.$ (4.53) for all $\ell\in{\mathbb{N}}$, all $x,y\in\\{-\ell+1,\dots,\ell\\}$, and all $s,t\in{\mathbb{R}}$. ###### Proof. All three bounds can be proved in the same way; here we only discuss the first one. We closely follow the proof of Lemma 4.8 and assume $x=s=0$ for notational convenience. It turns out that the proof for $\|x-y\|\leq 1$ present uninformative technical difficulties. On the other hand, exponential decay can be easily proved in the equivalent quantum model by expanding the trace in the basis of eigenvectors of the Hamiltonian and by using the existence of a spectral gap (which is proved in the next section). So it is enough to consider here $\|x-y\|>1$. This allows us to write $\hbox{\rm 1\kern-2.70004ptI}\\{(x,s)\leftrightarrow(y,t)\\}(\omega)=\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}(\omega)g(\omega)$ with a function $g$ such that $\|g\|\leq 1$ and that only depends on the contours $\gamma\in\Gamma(\omega)\in\mathfrak{A}_{\ell,\beta}$ that surround or contain $(x,s)$. (Recall that we assumed $(x,s)$ to be the origin $(0,0)$ — otherwise one would simply have to redefine $\mathcal{O}$ and $\Gamma_{0}$ to depend on $(x,s)$.) We proceed as in Lemma 4.8 to get $\mu_{\ell,\beta,n,u}\big{(}(x,s)\leftrightarrow(y,t)\big{)}=\frac{Z_{\ell,\beta,n,u}[\mathcal{O}^{c};g]}{Z_{\ell,\beta,n,u}},$ (4.54) where $Z_{\ell,\beta,n,u}[\mathcal{O}^{c};g]$ is given as in Eq. (4.45). Proceeding exactly the same way, we get the analogue of Eq. (4.52), namely $\begin{split}&\,{\rm e}^{c_{1}\|x-y\|+c_{2}\|s-t\|}\,\|\mu_{\ell,\beta,n,u}(\hbox{\rm 1\kern-2.70004ptI}_{\mathcal{O}^{c}}g)\|\leq\,{\rm e}^{c_{1}\|x-y\|+c_{2}\|s-t\|}\,\,{\rm e}^{{\varepsilon}}\,\int_{\mathfrak{C}^{+}_{\ell,\beta}\setminus\\{\emptyset\\}}{\rm d}\bar{\rho}_{1}(\Gamma_{0})g(\Gamma_{0})\prod_{\gamma_{0}\in\Gamma_{0}}\big{(}\|w(\gamma_{0})\|\,{\rm e}^{a(\gamma_{0})}\,\big{)}\\\ &\hskip 227.62204pt\times\hbox{\rm 1\kern-2.70004ptI}\\{(x,s)\in\overline{I(\Sigma(\gamma_{0};\Gamma_{0}))}\forall\gamma_{0}\in\Gamma_{0}\\}\\\ &\qquad\leq\,{\rm e}^{{\varepsilon}}\,\int_{\mathfrak{C}^{+}_{\ell,\beta}\setminus\\{\emptyset\\}}{\rm d}\bar{\rho}_{1}(\Gamma_{0})\prod_{\gamma_{0}\in\Gamma_{0}}\big{(}\|w(\gamma_{0})\|\,{\rm e}^{a(\gamma_{0})+b(\gamma_{0})}\,\big{)}\hbox{\rm 1\kern-2.70004ptI}\\{(x,s)\in\overline{I(\Sigma(\gamma_{0};\Gamma_{0}))}\forall\gamma_{0}\in\Gamma_{0}\\}\\\ &\qquad\leq 2\,{\rm e}^{{\varepsilon}}\,\left({{\varepsilon}\over 1-{\varepsilon}}\right).\end{split}$ (4.55) Here we chose ${\varepsilon}\in(0,1)$ to be a constant, independent of all other parameters $n,\beta,\ell,u$. For all $\gamma\in\Gamma_{0}$ such that $g(\Gamma_{0})\neq 0$ (hence $g(\Gamma_{0})=1$) we have $\|\gamma\|\geq 2\|s-t\|$ and $\\#\gamma\geq 2\|x-y\|$. Recall the function $b$ of Lemma 4.4. Choosing $c_{1}=2b_{1},c_{2}=2b_{2}$, we get the second inequality. Corollary 4.5 then allows us to proceed as in Eq. (4.52), which gives the last inequality. ∎ ## 5\. Proof of the spectral gap We follow the method of Kennedy and Tasaki [12] and show that the method of cluster expansion can be used to prove the existence of a positive spectral gap. Indeed, it implies the validity of the following lemma (recall that $Z_{\ell,\beta}={\operatorname{Tr\,}}\,{\rm e}^{-2\beta H_{\ell}}\,$). ###### Lemma 5.1. There exists $n_{0},u_{0},c>0$ (independent of $\ell,\beta,n,u$) and $C_{\ell}$ (independent of $\beta,n,u$) such that for all $n\geq n_{0}$ and $\|u\|\leq u_{0}$, we have for all $\beta\geq\frac{1}{2}$ that $\bigl{\|}E_{0}^{(\ell)}+\tfrac{1}{2\beta}\log Z_{\ell,\beta}\bigr{\|}\leq C_{\ell}\,{\rm e}^{-\beta c}\,.$ ###### Proof. We check that, for all $1\leq\beta<\beta^{\prime}$, we have $\bigl{\|}\tfrac{1}{2\beta}\Phi_{\ell\beta}-\tfrac{1}{2\beta^{\prime}}\Phi_{\ell\beta^{\prime}}\bigr{\|}\leq C_{\ell}\,{\rm e}^{-\beta c}\,.$ (5.1) Since $-\frac{1}{2\beta}\log Z_{\ell,\beta}=2\ell(1+u)-\frac{1}{2\beta}\log\Phi_{\ell,\beta}$, we get the lemma by taking the limit $\beta^{\prime}\to\infty$. Let ${\mathfrak{C}}_{\beta}^{+}$ denote the set of clusters in $V_{\ell}\times T_{\beta}$, i.e. the sequences of contours $\Gamma=(\gamma_{1},\dots,\gamma_{k})$, $\gamma_{i}\in X_{\ell,\beta}^{+}$, such that $\varphi(\Gamma)\neq 0$. For $t\in T_{\beta}$, let ${\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{t}(\Gamma)$ be the indicator that the cluster $\Gamma$ crosses the line $V_{\ell}\times\\{t\\}$. Let ${\mathfrak{L}}(\Gamma)\in[0,2\beta]$ be the vertical length of the cluster $\Gamma$: ${\mathfrak{L}}(\Gamma)=\int_{-\beta}^{\beta}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{t}(\Gamma)\,{\rm d}t.$ (5.2) Then, from the cluster expansion of Proposition 4.6, $\begin{split}\frac{1}{2\beta}\Phi_{\ell,\beta}&=\frac{1}{2\beta}\int_{{\mathfrak{C}}_{\beta}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)w(\Gamma)\varphi(\Gamma)=\frac{1}{2\beta}\int_{{\mathfrak{C}}_{\beta}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)\int_{-\beta}^{\beta}{\rm d}t\frac{{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{t}(\Gamma)}{{\mathfrak{L}}(\Gamma)}w(\Gamma)\varphi(\Gamma)\\\ &=\frac{1}{2\beta}\int_{-\beta}^{\beta}{\rm d}t\int_{{\mathfrak{C}}_{\beta}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)\frac{w(\Gamma)\varphi(\Gamma)}{{\mathfrak{L}}(\Gamma)}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{t}(\Gamma)=\int_{{\mathfrak{C}}_{\beta}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)\frac{w(\Gamma)\varphi(\Gamma)}{{\mathfrak{L}}(\Gamma)}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{0}(\Gamma).\end{split}$ (5.3) We used Fubini’s theorem to exchange the integrals, and time translation invariance in the last step. The last expression is convenient to cancel terms for different $\beta$s; for $\beta<\beta^{\prime}$, we have $\begin{split}\frac{1}{2\beta}\Phi_{\ell,\beta}&-\frac{1}{2\beta^{\prime}}\Phi_{\ell,\beta^{\prime}}=\int_{{\mathfrak{C}}_{\beta}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)\frac{w(\Gamma)\varphi(\Gamma)}{{\mathfrak{L}}(\Gamma)}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{0}(\Gamma)-\int_{{\mathfrak{C}}_{\beta^{\prime}}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)\frac{w(\Gamma)\varphi(\Gamma)}{{\mathfrak{L}}(\Gamma)}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{0}(\Gamma)\\\ &=\int_{{\mathfrak{C}}_{\beta}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)\frac{w(\Gamma)\varphi(\Gamma)}{{\mathfrak{L}}(\Gamma)}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{0}(\Gamma){\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{{\mathfrak{L}}(\Gamma)=2\beta}-\int_{{\mathfrak{C}}_{\beta^{\prime}}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)\frac{w(\Gamma)\varphi(\Gamma)}{{\mathfrak{L}}(\Gamma)}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{0}(\Gamma)1_{{\mathfrak{L}}(\Gamma)\geq 2\beta}.\end{split}$ (5.4) Indeed, the contribution of clusters with ${\mathfrak{L}}(\Gamma)<2\beta$ has precisely canceled. Figure 17. The contour $\gamma^{(\eta)}$ used in the proof of Lemma 5.1. We can use estimates from cluster expansions in order to bound the terms above. For $\eta>0$, let us introduce the contour $\gamma^{(\eta)}$ that surrounds the horizontal axis at time 0, as shown in Fig. 17. Its vertical length goes to 0 as $\eta\to 0$. We have $a(\gamma^{(\eta)})=2(2\ell-1)a_{1}+4(\ell+1)\eta a_{2}$; and if $\eta<2\beta$, we have ${\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{0}(\Gamma)\leq 1-\delta(\gamma^{(\eta)},\Gamma)\leq\sum_{\gamma\in\Gamma}\bigl{(}1-\delta(\gamma^{(\eta)},\gamma)\bigr{)}.$ (5.5) Using $1\leq 2\beta\leq\sum_{\gamma\in\Gamma}\|\gamma\|$, which holds for contours such that $1_{{\mathfrak{L}}(\Gamma)\geq 2\beta}=1$, we have, using Lemma 4.4 and the estimate in Theorem 4.1 (c) that $\begin{split}\,{\rm e}^{2b_{2}\beta}\,\biggl{\|}\int_{{\mathfrak{C}}_{\beta}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)&\frac{w(\Gamma)\varphi(\Gamma)}{{\mathfrak{L}}(\Gamma)}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{0}(\Gamma){\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}_{{\mathfrak{L}}(\Gamma)=2\beta}\biggr{\|}\\\ &\leq\int_{{\mathfrak{C}}_{\beta}^{+}}{\rm d}\bar{\rho}_{1}(\Gamma)\|w(\Gamma)\|\,{\rm e}^{b_{2}\sum_{\gamma\in\Gamma}\|\gamma\|}\,\|\varphi(\Gamma)\|\sum_{\gamma\in\Gamma}\bigl{(}1-\delta(\gamma^{(\eta)},\gamma)\bigr{)}\\\ &\leq 2(2\ell-1)a_{1}+4(\ell+1)\eta a_{2}.\end{split}$ (5.6) The other term in the right side of (5.4) can be estimated in the same way, giving the same bound. This proves Lemma 5.1 with $C_{\ell}=4(2\ell-1)a_{1}$ (we can take $\eta\to 0$) and $c=2b_{2}$. ∎ ###### Proof of Theorem 1.4. With $m_{i}^{(\ell)}$ the multiplicity of the eigenvalue $E_{i}^{(\ell)}$ (satisfying $\sum_{i\geq 0}m_{i}^{(\ell)}=n^{2\ell}$), we have $Z_{\ell,\beta}=\sum_{i\geq 0}m_{i}^{(\ell)}\,{\rm e}^{-2\beta E_{i}^{(\ell)}}\,=m_{0}^{(\ell)}\,{\rm e}^{-2\beta E_{0}^{(\ell)}}\,\biggl{(}1+\sum_{i\geq 1}\tfrac{m_{i}^{(\ell)}}{m_{0}^{(\ell)}}\,\,{\rm e}^{-2\beta(E_{i}^{(\ell)}-E_{0}^{(\ell)})}\,\biggr{)}.$ (5.7) Thus $-\tfrac{1}{2\beta}\log Z_{\ell,\beta}=E_{0}^{(\ell)}-\tfrac{1}{2\beta}\log m_{0}^{(\ell)}-\tfrac{1}{2\beta}R(\ell,\beta),$ (5.8) where $R(\ell,\beta)=\log\biggl{(}1+\sum_{i\geq 1}\tfrac{m_{i}^{(\ell)}}{m_{0}^{(\ell)}}\,\,{\rm e}^{-2\beta(E_{i}^{(\ell)}-E_{0}^{(\ell)})}\,\biggr{)}\geq\log\biggl{(}1+\tfrac{m_{1}^{(\ell)}}{m_{0}^{(\ell)}}\,\,{\rm e}^{-2\beta\Delta^{(\ell)}}\,\biggr{)}\geq\,{\rm e}^{-3\beta\Delta^{(\ell)}}\,,$ (5.9) for $\beta$ large enough (depending on $\ell$). On the other hand, Lemma 5.1 implies that $-\tfrac{1}{2\beta}\log Z_{\ell,\beta}=E_{0}^{(\ell)}+R^{\prime}(\ell,\beta),$ (5.10) where $\|R^{\prime}(\ell,\beta)\|\leq C_{\ell}\,{\rm e}^{-\beta c}\,$. We then have $-\tfrac{1}{2\beta}\log m_{0}^{(\ell)}-\tfrac{1}{2\beta}R(\ell,\beta)=R^{\prime}(\ell,\beta).$ (5.11) Using the bound $R(\ell,\beta)\leq n^{2\ell}\,{\rm e}^{-2\beta\Delta^{(\ell)}}\,$ where $\Delta^{(\ell)}>0$, and looking at the asymptotic $\beta\to\infty$, we see that $m_{0}^{(\ell)}=1$. Next, using (5.9), we get $\tfrac{1}{2\beta}\,{\rm e}^{-3\beta\Delta^{(\ell)}}\,\leq C_{\ell}\,{\rm e}^{-\beta c}\,$ (5.12) for all $\beta$ sufficiently large; this implies that $\Delta^{(\ell)}\geq\frac{1}{3}c$, uniformly in $\ell,n,u$. ∎ ## Appendix A The interaction $uT+vP$ when $n$ is even For $n$ odd, the interactions $uT+vP$ and $uT+vQ$ are related by the unitary transformation of Eq. (1.4). This holds for models defined on arbitrary graphs or lattices. We now discuss the case of $n$ even. As we shall see, we need to restrict ourselves to bipartite graphs (of which the chain is of course an example). We work with the $S^{(3)}$-eigenbasis $e_{\alpha}:=\|\alpha\rangle$ with $\alpha=-S,-S+1,\dots,S$. To begin, we define a unitary $V$ by setting $V\|\alpha\rangle=(-1)^{S-\alpha}\|-\alpha\rangle.$ (A.1) With $\psi$ the vector of (1.3) and $\phi$ the vector of (1.5), we have $\phi=(\hbox{\rm 1\kern-2.70004ptI}\otimes V)\psi.$ (A.2) Therefore, since $P$ is the projection onto $\phi$, $P=(\hbox{\rm 1\kern-2.70004ptI}\otimes V)Q(\hbox{\rm 1\kern-2.70004ptI}\otimes V^{}).$ (A.3) Since $T\psi=\psi$ and $T\phi=-\phi$, we have $TQT=Q$ and $TPT=P$. Using these properties we find $(V\otimes\hbox{\rm 1\kern-2.70004ptI})Q(V^{}\otimes\hbox{\rm 1\kern-2.70004ptI})=(V\otimes\hbox{\rm 1\kern-2.70004ptI})TQT(V^{}\otimes\hbox{\rm 1\kern-2.70004ptI})=T(\hbox{\rm 1\kern-2.70004ptI}\otimes V)Q(\hbox{\rm 1\kern-2.70004ptI}\otimes V^{})T=P.$ (A.4) Both models are translation-invariant although the unitary that relates them is not: $(V\otimes\hbox{\rm 1\kern-2.70004ptI}\otimes V\cdots\otimes\hbox{\rm 1\kern-2.70004ptI})\left[\sum_{x=-\ell+1}^{\ell-1}Q_{x,x+1}\right](V^{}\otimes\hbox{\rm 1\kern-2.70004ptI}\otimes V^{}\cdots\otimes\hbox{\rm 1\kern-2.70004ptI})=\sum_{x=-\ell+1}^{\ell-1}P_{x,x+1}.$ (A.5) Let $\tilde{T}$ be the transformation of the operator $T$. We have $\tilde{T}=(\hbox{\rm 1\kern-2.70004ptI}\otimes V)T(\hbox{\rm 1\kern-2.70004ptI}\otimes V^{})=(\hbox{\rm 1\kern-2.70004ptI}\otimes V)(V^{}\otimes\hbox{\rm 1\kern-2.70004ptI})T=-(V\otimes V)T.$ (A.6) Let us summarize the above considerations by the following proposition. We define the new Hamiltonian $H_{\ell}^{\prime}=\sum_{x=-\ell+1}^{\ell-1}\bigl{(}u\tilde{T}_{x,x+1}+vQ_{x,x+1}\bigr{)}$. ###### Proposition A.1. For $n$ even, the interaction $uT+vP$ is unitarily equivalent with $u\tilde{T}+vQ$. The Hamiltonian $\tilde{H}_{\ell}$ defined in (1.14) is unitarily equivalent to $H_{\ell}^{\prime}$. Notice that, when $u=0$, the $Q$-model and the $P$-model are unitarily equivalent for all $n$. The proposition is stated for chains, but it clearly holds for arbitrary bipartite graphs. Next, we derive a loop representation for the model $H_{\ell}^{\prime}$. ###### Proposition A.2. There exists a function $s(l)$ from the set of loops to $\pm 1$ such that for all $n\geq 2$, * (a) $\displaystyle{\operatorname{Tr\,}}\,{\rm e}^{-2\beta H_{\ell}^{\prime}}\,=\,{\rm e}^{2\beta(1+u)\|E_{\ell}\|}\,\int{\rm d}\rho_{u}(\omega)n^{\mathcal{L}(\omega)-\|\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}\|}\prod_{\text{loop $l$ in $\omega$}}s(l)$. * (b) For $i=1,2,3$, ${\operatorname{Tr\,}}S_{x}^{(i)}S_{y}^{(i)}\,{\rm e}^{-2\beta H_{\ell}^{\prime}}\,=\tfrac{n^{2}-1}{12}\,{\rm e}^{2\beta(1+u)\|E_{\ell}\|}\,\int_{\Omega_{\ell,\beta}}{\rm d}\rho_{u}(\omega)\,n^{\mathcal{L}(\omega)-\|\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}\|}\\\ \times\bigl{(}{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}[x\overset{+}{\longleftrightarrow}y]-{\mathchoice{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.0mul}{\rm 1\mskip-4.5mul}{\rm 1\mskip-5.0mul}}[x\overset{-}{\longleftrightarrow}y]\bigr{)}\prod_{\text{loop $l$ in $\omega$}}s(l).$ Proposition A.2 is stated for chains, but it actually holds for arbitrary bipartite graphs (unlike Theorem 2.1 which holds for all finite graphs). For odd $n$ the signs $s(l)$ are all equal to $+1$. ###### Proof. First, observe that the number of crosses along the trajectory of a loop, is even (here, if a cross is traversed twice in a loop, it counts as two). Indeed, the total number of crosses and double-bars along the trajectory is even because the graph is bipartite; and the number of double-bars is even because the number of changes in vertical direction is even; so the number of crosses is also even. The expansion of the operator $\,{\rm e}^{-2\beta H_{\ell}^{\prime}}\,$ can be made in terms of configurations $\omega$, and of “space-time spin configurations” (see [30]). The space-time spin configurations that are compatible with $\omega$ have the property that their value on a loop is $\pm\alpha$ for some $\alpha=-S,\dots,S$, the changes of signs occurring when traversing crosses (and any such choice results in a possible space-time spin configuration because the number of crosses along a loop trajectory is even). Proceeding as in Theorem 2.1, we find that ${\operatorname{Tr\,}}\,{\rm e}^{-2\beta H_{\ell}^{\prime}}\,=\,{\rm e}^{2\beta(1+u)\|E_{\ell}\|}\,\int{\rm d}\rho_{u}(\omega)n^{\mathcal{L}(\omega)-\|\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}\|}{\boldsymbol{s}}(\omega),$ (A.7) where ${\boldsymbol{s}}(\omega)$ is an overall sign: ${\boldsymbol{s}}(\omega)=\pm 1$. Notice that, since $\tilde{T}$ involves a minus sign, there is no need to change the sign of $u$ in the interaction as in Theorem 2.1. Figure 18. Signs arising when traversing crosses. Left: the crosses are separated by an even number of double bars which yields the factor $(-1)^{S-\alpha}(-1)^{S+\alpha}=-1$. Right: the crosses are separated by an odd number of double bars which yields the factor 1. The signs are due to the action of operators $V$. We can collect the signs for each loop individually. Consider two successive crosses. If the vertical direction is the same (which is the case if there is an even number of double- bars between them), we get the factor $(-1)^{S-\alpha}(-1)^{S+\alpha}=(-1)^{2S}=-1.$ (A.8) If the vertical direction is opposite (which is the case if there is an odd number of double-bars between them), the factor is $(-1)^{S-\alpha}(-1)^{S-\alpha}=1.$ (A.9) This is illustrated in Fig. 18. The value of $s(l)$ is the product of these factors. Notice that the sign does not depend on the value of $\alpha$ in the loop. This proves item (a) of the proposition. The spin correlations are the same for all $i=1,2,3$ by symmetry and it is enough to consider $i=3$. This is identical to [30, Theorem 3.5 (a)] except for the signs (the claim there was restricted to odd $n$ where $s(l)=+1$). Using space-time spin configurations, we have ${\operatorname{Tr\,}}S_{x}^{(i)}S_{y}^{(i)}\,{\rm e}^{-2\beta H_{\ell}^{\prime}}\,=\,{\rm e}^{2\beta(1+u)\|E_{\ell}\|}\,\int_{\Omega_{\ell,\beta}}{\rm d}\rho_{u}(\omega)\,n^{-\|\omega_{\mathchoice{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=4.00002pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=3.00003pt]{dbar.pdf}}}}{\vbox{\hbox{\includegraphics[width=2.5pt]{dbar.pdf}}}}}\|}\Bigl{(}\prod_{\text{loop $l$ in $\omega$}}s(l)\Bigr{)}\sum_{\sigma\in\Sigma(\omega)}\sigma_{x,0}\sigma_{y,0}.$ (A.10) We used the fact that the signs do not depend on the spin values of the loops. If $(x,0)$ and $(y,0)$ do not belong to the same loop, the sum over $\sigma$ is zero. If $(x,0)$ and $(y,0)$ belong to the same loop and the connection is $x\overset{+}{\longleftrightarrow}y$, then $\sigma_{x,0}=\sigma_{y,0}$ and the sum gives $\frac{3}{S}(S+1)n=\frac{1}{12}(n^{2}-1)n$. If the connection is $x\overset{-}{\longleftrightarrow}y$, then $\sigma_{x,0}=-\sigma_{y,0}$ and we get minus the same factor. This gives the identity (b). ∎ We can now prove Theorem 1.3. ###### Proof of Theorem 1.3. By Proposition A.1, the claims of Theorem 1.3 are equivalent to proving dimerization in the model with Hamiltonian $H_{\ell}^{\prime}$. We use the loop representation of Proposition A.2. We can then retrace the steps of the proof of Theorem 4.7. In doing so, note that all short loops $l$ have $s(l)=+1$, while long or winding loops have $s(l)=\pm 1$. We incorporate the latter factors in the weights $w(\gamma)$ of the contours, see (4.6). Therefore, the only difference is that the weights of contours have possibly other signs. All bounds are the same, though, and the cluster expansion gives the same result. ∎ The proof of the gap for $\tilde{H}_{\ell}$ is exactly the same as the proof for $H_{\ell}$ described in Section 5. Acknowledgements: We are grateful to Vojkan Jakšić and the Centre de Recherches Mathématiques of Montreal for hosting us during the thematic semester “Mathematical challenges in many-body physics and quantum information”, with support from the Simons Foundation through the Simons–CRM scholar-in-residence program. We also thank the referees for useful comments. JEB gratefully acknowledges support from _Vetenskapsrådet_ grants 2015-0519 and 2019-04185 as well as _Ruth och Nils-Erik Stenbäcks stiftelse_. BN is supported in part by the National Science Foundation under grant DMS-1813149. ## References * [1] * [2] I. Affleck, _Exact results on the dimerisation transition in $su(n)$ antiferromagnetic chains_, J. Phys.: Condens. Matter 2, 405–415 (1990) * [3] I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Valence bond ground states in isotropic quantum antiferromagnets, Comm. Math. Phys. 115, 477-528 (1988) * [4] M Aizenman, H. Duminil-Copin, S. 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# Hiding Behind Machines: When Blame Is Shifted to Artificial Agents Till Feier TUM School of Governance, TU Munich, Richard-Wagner-Straße 1, 80333 Munich, Germany<EMAIL_ADDRESS>Jan Gogoll Bavarian Institute for Digital Transformation / TU Munich, Gabelsbergerstr. 4, 80333 Munich, Germany, <EMAIL_ADDRESS>Matthias Uhl ZD.B Junior Research Group “Ethics of Digitization”, TUM School of Governance, TU Munich, Richard-Wagner-Straße 1, 80333 Munich, Germany<EMAIL_ADDRESS> ###### Abstract The transfer of tasks with sometimes far-reaching moral implications to autonomous systems raises a number of ethical questions. In addition to fundamental questions about the moral agency of these systems, behavioral issues arise. This article focuses on the responsibility of agents who decide on our behalf. We investigate the empirically accessible question of whether the production of moral outcomes by an agent is systematically judged differently when the agent is artificial and not human. The results of a laboratory experiment suggest that decision-makers can actually rid themselves of guilt more easily by delegating to machines than by delegating to other people. Our results imply that the availability of artificial agents could provide stronger incentives for decision makers to delegate morally sensitive decisions. > “Once the rockets are up, who cares where they come down? That’s not my > department” says Wernher von Braun. > – Tom Lehrer ## 1 Introduction In 2017, the airline AirBerlin, Lufthansa’s biggest competitor for German domestic flights, went bankrupt. Lufthansa passengers soon noticed a significant spike in ticket prices. The resulting backlash was enormous and subsequently led to an investigation by Germany’s Federal Cartel Office (FCO). Lufthansa was quick to blame its automated booking system, which had allegedly responded to a spike in demand. This did little to quell the public outrage and FCO president Andreas Mundt famously said that “companies can’t hide behind algorithms” (Busse, 2017). Our article tests the empirical content of this normative statement. Algorithms have become an integral part of society and are responsible for more tasks than ever before (Manyika et al., 2017). This is not only pushing the frontiers of technology but also challenging our traditional concepts of guilt and responsibility. The diffusion of responsibility between a multitude of actors has long been a trademark of modern institutions. The challenge of locating responsibility within a complex system may even become exponentially more difficult when moral agency is distributed between human operators and autonomous agents (Nissenbaum, 1996). The fact that even the designers of algorithms cannot fully explain their decisions (e.g., black box models in machine learning) adds to this complexity (Rudin, 2019). Normative ethics raise a fundamental question with respect to the increasing use of artificial agents in decision-making. In which sense can artificial agents also be moral agents and therefore be responsible for their actions? Philosophers and engineers started pondering this question decades ago when computers were merely functioning as calculators (Moor, 1979). There is still little consensus on the matter and the idea of “moral machines” remains under debate (Allen and Wallach, 2012). In any case, the philosophical discussion is predominantly concerned with normative aspects and the question of who ought to be held responsible or blamed if a machine brings about moral evil. Although these normative questions are of obvious importance, the behavioral impact that the interaction with artificial agents has on the operator’s conduct is not well understood either. It seems crucial, however, for research on behavioral consequences to feed back into the ethical debate. Human operators might get trapped between the increasing capabilities and autonomy of artificial agents on one hand and our rigid moral understanding of guilt and responsibility on the other. Operators may end up merely filling responsibility gaps in various systems instead of actually being in charge. They might also over-utilize artificial agents even when they are not the ideal choice for a given task, because they wish to use machines as scapegoats (Danaher, 2016). This study investigates whether delegators will be able to successfully shift blame by delegating tasks to artificial agents. Our investigation focuses on whether people judge delegations to human and artificial agents differently in light of given outcomes. We find no differences between judgments toward human and artificial agents in the event of good outcomes. This means that the morally beneficial delegation to an artificial agent was considered neither better nor worse than the beneficial delegation to a human agent. However, if a bad outcome occurred, delegators fared significantly better if a machine agent caused the failure. Interestingly, participants did not seem to anticipate this pattern as we did not find significant differences regarding delegation decisions themselves. In fact, decisions involving human and artificial agents seem driven by the expected utility of the delegation for the affected party. Our article proceeds as follows. In Section 2, we will derive our research question. In section 3, we will outline the experimental setup to test it. We will discuss our results in Section 4 and conclude in Section 5. ## 2 Background and Research Question A substantial amount of literature is available on the parameters that influence automation use in teams of human supervisors and the machines at their disposal (Dzindolet et al., 2002). In recent years, a number of studies investigated how those parameters change once decisions have moral undertones, i.e., an impact on third parties. Goldbach et al. (2019) found people to be hesitant to delegate decisions to algorithms that affect both the decision- maker herself as well as third parties. Similarly, a study by Niszczota and Kaszás (2020) suggests that algorithm aversion extends to the financial sector, and that people especially prefer human over artificial agents when it comes to making financial decisions with moral undertones. In a laboratory study, Gogoll and Uhl (2018) identified a strong aversion against delegating morally relevant tasks to algorithms. It seems that people were less willing to delegate tasks to machines if those decisions imposed monetary externalities on third parties. While their study checked for “perceived utility” of the artificial agent and the trust in that agent, they were unable to determine the exact causes of the profound algorithm aversion. Other studies also support the idea that a lack of trust is unlikely to be the cause of aversion towards machine use. If anything, there seems to be over- reliance and over-trust in machines, even if the lives of people are at stake (Robinette et al., 2016). This suggests that there have to be other causes for machine aversion in decisions with moral implications. We hypothesize that responsibility is a key concept in understanding this phenomenon. Perceived responsibility is already an important research topic in relation to machine use, probably most prominently in the context of automated driving (Hevelke and Nida-Rümelin, 2015). However, little attention has been paid to the question of how the introduction of machine agents might affect the blame and praise that people ascribe to the delegator in light of a given outcome. Understanding how the use of artificial agents is judged would be an important first step in gaining a deeper understanding of how their availability might influence people’s motivation to delegate morally sensible tasks. This is closely linked to the idea that avoiding blame or shirking responsibility can be pivotal factors in delegation decisions. Strategies of blame avoidance and responsibility shifting have long been discussed in the political sciences (Weaver, 1986). Instances of so-called blame games can frequently be observed in political systems with regard to policy making and implementation in the European Union (Heinkelmann-Wild and Zangl, 2020) or between officials from different levels of government in the United States (Maestas et al., 2008). Unsurprisingly, similar strategies can also be observed in the private sector, for instance, when it comes to upholding employment standards within franchise networks (Hardy, 2019). The idea that responsibility shifting can in fact be a pivotal factor in delegation decisions plays an especially prominent role in public choice theory (Fiorina, 1986). Experimental evidence of this phenomenon comes from Fischbacher et al. (2008), who showed that responsibility attribution can sometimes be effectively shifted and that this constitutes a powerful motive for decision-makers. Other experiments provide evidence that this is true even if the agent in question is effectively powerless (Hill, 2015), or the delegation by the principal eliminated the possibility of a fair outcome (Oexl and Grossman, 2013). But does this also hold true for artificial agents? Popular culture places a strong emphasis on human responsibility despite an empirical decline of human control in many areas (Elish and Hwang, 2015). The concept of “algorithmic outrage asymmetry” suggests that people are less morally outraged by algorithmic wrongdoing than by human wrongdoing, for instance, in cases of discrimination by age, race or gender (Bigman et al., 2020). If, however, people seek to attribute blame, some argue that it will not be placed on algorithms but that humans could emerge as “moral crumple zones.” In human-machine teams, humans would then have to take on blame even for accidents outside of their control (Elish, 2019). While some argue that humans will bear all of the responsibility and none of the control when working with machines, others think that machines are perfectly suited to be used as scapegoats. So far very limited empirical evidence has been provided to back either position. Strobel and Kirchkamp (2017) investigated whether choices in a dictator game and perceived guilt change when players can share responsibility with machines. The authors report that perceived responsibility and guilt did not vary significantly when comparing purely human teams with human-machine teams. They write that people tend to make fewer selfish decisions when partnered with machines, though that effect was insignificant. So while responsibility shifting has long been established as an integral part in delegation decisions, empirical evidence of whether the introduction of artificial agents is rendering this motive mute or even more important is lacking. This constitutes a serious research gap, the implications of which exceed academic relevance. A better understanding of responsibility shifting to artificial agents could explain over and underreliance on machines and profoundly influence legal decision-making regarding automation and digitization. For instance, administrators are struggling to provide governance strategies for automated vehicles, because of the ambiguity with respect to liability (Taeihagh and Lim, 2019). A better understanding of how people actually attribute responsibility is likely to help create guidelines that are not only more effective but also more likely to gain consensus. Deeper insights into the phenomenon might also help to shield human operators from unjust recrimination. As mentioned above, our classic understanding of human-machine teams has cemented a focus on human responsibility despite a decline of human control in various areas (Elish and Hwang, 2015). This is especially troubling since responsibility has proven to be be an important factor in understanding and predicting punishment patterns. (for example (Fehr and Schmidt, 1999; Bolton and Ockenfels, 2000)) It seems that human operators are in fact in harms way and might become scapegoats in cases of technical failure. In contrast, machine use might emerge as a strategy for self-exculpation in critical situations. This could have detrimental effects if it leads to an overuse of machines - a bleak prospect given the growing capabilities of algorithms and the potential harm this implies for workers and consumers. In sum, there are plenty of reasons to investigate attributions of blame and praise in the context of automation. To shed light on this problem, we will test the following conjecture in a laboratory experiment. Conjecture: Delegators are rewarded differently for delegating other-regarding tasks to artificial as compared to human agents. On one hand, the delegation of a task that could carry severe consequences for a third party to an unmonitored machine might be considered careless and result in punishment or defamation. On the other hand, principals might be exculpated entirely since people deem the failure of machines to be even more outside of their control than when delegating to another human. Either way, a better understanding of public reservations regarding the introduction of novel technology is likely to prove useful in future moral and legal considerations. The experiment designed to test the above conjecture will be outlined in the following chapter. Additionally, we will explore whether the effect of perceived utility on delegation decisions varies depending on the agent’s artificial or human nature. Based on a definition by Dzindolet et al. (2002), we define the perceived utility of employing an artificial agent as the difference between the perceived reliability of an automated device and the perceived reliability of manual control. Furthermore, we will include risk attitudes into our analysis (O’Donoghue and Somerville, 2018). ## 3 Design The experiment consisted of (1) a logic task, (2) a delegation decision, (3) an evaluation of the delegation decision, (4) a self-assessment of one’s performance and elicitation of risk attitudes. Subjects received a 40 ECU show-up fee increased by the outcome of the delegation (successful or not), plus (minus) their reward (their punishment) for their decision to delegate or not and the bonus for the accuracy of their self-assessment and the lottery as payment for the experiment. An experimental currency unit was used (ECU) with the exchange rate of 10 ECU for 1 EUR. Subjects received their instructions on-screen and were fully informed about the rules of the game. They were randomly matched according to perfect stranger matching, i.e., no two participants interacted more than once during the experiment. The experimental manipulation consisted of changing the nature of the agent to which a task could be delegated: it was either another human participant or an artificial agent (see 3.2). Table 1 gives an overview over the four stages of the experiment. Stage \| Human (Machine) Treatment ---\|--- (1) Solve logic task \| Participants solve a series of logic puzzles (2) Make delegation decision \| Participants decide whether to delegate to another human (a machine) or to have their own work count (3) Evaluate delegation decision \| Participants reward or punish decision to delegate or not (4) Self-assess & choose lotteries \| Participants guess how many errors they made in logic task and reveal their risk attitude by choosing between lotteries Table 1: Overview of experimental stages ### 3.1 Logic Task The main task of the experiment was a logic puzzle. We asked participants to complete ten puzzles within a five-minute time frame. Each puzzle consisted of a sequence of three patterns and a placeholder for the missing fourth pattern. The right answer had to be derived from the given sequence and selected from a set of four alternatives. To identify the right answer, participants always had to focus on the circles, while ignoring differently shaped symbols and any colors (see Figure 1). Because we assumed that delegation decisions would heavily depend on the agent’s perceived capabilities, we chose a task involving visual perception to foster participants’ intuition that the algorithm could err. Participants had to complete ten of the tasks listed above before they could continue to the second stage of the experiment. Figure 1: Example of a logic task ### 3.2 Delegation Decision After the logic task had been completed, participants were randomly assigned to one of two treatments: the “human” or the “machine” treatment. Only at this point did subjects learn about the nature of the agent to whom they could delegate the task. To give participants an idea of their potential delegatee’s capability, they were shown representations of the agent’s performance. In the human treatment, the $n$ participants were shown a histogram displaying the performance of the other $(n-1)$ participants based on the actual results of the running session. In the machine treatment, participants were shown a histogram with information about how often the algorithm failed to give correct answers in $n$ trial runs. The algorithm was programmed such that it mirrored the performance of the human participants in the room. The performance of the artificial agent was therefore as good as that of the participants in the respective session. This process ensured that the delegation decisions were based on the agent’s nature instead of any assumptions about differing capabilities. Participants were then asked to make the delegation decision. They chose whether their own performance or that of their human or artifical agent (depending on the treatment) would determine the third party’s payoff. The delegation decision would thus not affect the payment of the delegator but that of another participant. Conditional on participants’ decisions to delegate or not, one of the ten solutions to the ten puzzles was randomly chosen from the answers of their agent or their own answers. If the selected puzzle was solved correctly, the third party received an additional payoff. If it was solved incorrectly, the third party did not receive an additional payoff. Figure 2: Delegation decision for human and machine treatment ### 3.3 Evaluation of the Delegation Participants were now given the opportunity to punish or reward another participant’s decision to delegate or not. Participants did not know the actual decision of the participant that they evaluated, nor whether this decision had resulted in a good or bad outcome for themselves. They were asked to alter the payment of the participant upwards or downwards by at most 40 ECU. Reward and punishment choices were contingent on the two possible decisions to delegate or not and the two possible outcomes of success or failure. Thus, in any case, four choices had to be made. Only the one that applied to the actual decision of the evaluated participant combined with actual outcome did finally apply (Selten, 1967). Table 2 depicts the table that subjects saw on their screen. Lower or increase subjects payoff given that: \| Amount ---\|--- Subject used own work – outcome: success \| enter amount Subject used own work – outcome: failure \| enter amount Subject delegated (machine/human) – outcome: success \| enter amount Subject delegated (machine/human) – outcome: failure \| enter amount Table 2: Decision to increase or decrease the delegator’s pay-off ### 3.4 Self-Assessment and Risk Attitudes Subsequently, participants were asked to assess their own performance by estimating how many mistakes they had made during the logic task in the experiment’s first part. This estimate was incentivized by an additional payment of 50 ECU if they guess correctly. This procedure allowed us to analyze the effect of self-assessments on delegation decisions. As is standard in incentivized economic experiments, the experiment was concluded by an elicitation of participants’ risk attitudes.111We used the procedure introduced by Holt and Laury (2002). The task is based on ten choices between pairs of lotteries. The potentional payoffs for the safe lottery range from €2.5 to €1.6 and are therefore always less extreme than those for the risky lottery that range from €4.35 to €0.10. For the first pair of lotteries, the probability of the high payoff is equally low in both lotteries but equally increases for both lotteries with each new pair. A participant should switch to the risky lottery once the probability for the high payoff is sufficiently high according to his or her personal risk attitude. The later a participant switches to the risky lottery, the more risk averse he or she is. ## 4 Results The experiment took place in a major German university between February and May 2019. A total of 149 subjects participated in six sessions, 43% were female and the average age was 23.08 years (sd = 3.83). Participants received a show-up fee of €4.00 and could earn additional money in the experiment. A session lasted about 45 minutes and the average payment was about €13.50 per participant. The experiment was programmed in z-Tree (Fischbacher, 2007) and subjects were recruited via ORSEE (Greiner et al., 2004). Data analysis was conducted using Python’s numpy, scipy, and statsmodels.api libraries. The preprocessed data set and the code are available online.222https://doi.org/10.5281/zenodo.4446581 Remember that after the delegation decision, all participants were asked to evaluate the decision of the participant who was responsible for their own pay-off. Responsibility arose from the decision to either delegate (to a human or machine) or rely on one’s own work (in both treatments). Notice again that participants were informed whether they were randomly assigned to the human or machine treatment, but not whether the other participant had actually delegated or not. They were therefore asked to judge the decision with respect to the four possible outcomes according to the so-called strategy method (see Section 3.3). This means that subjects could increase (or deduct) the payment of their responsible participant by an integer between 0 and 40 for the cases of her (1) having brought about a good outcome herself or (2) having brought about a bad outcome herself. Furthermore, they were increasing (or deducting) the payment for the case of her (3) having delegated to an agent causing a good outcome on her behalf or (4) having delegated to an agent causing a bad outcome on her behalf. To test our conjecture that delegators are rewarded differently for delegating other-regarding tasks to artificial as opposed to human agents, we contrast evaluations between both treatments. Let us first consider the good-outcome case. In the human treatment, participants that did not delegate and caused the good outcome themselves were, on average, rewarded 20.28 ECU (sd = 20.85). Participants that delegated to another human who then brought about the good outcome on their behalf were rewarded 20.94 ECU (sd = 21.34). This difference is insignificant (p = 0.720, paired t-test). In the machine treatment, participants that did not delegate and caused the good outcome themselves were on average rewarded 23.78 ECU (sd = 20.14). Participants that delegated to a machine were rewarded 27.00 ECU (sd = 16.71). This difference is again insignificant (p = 0.106, paired t-test). Result 1: Delegators did not lose any recognition for a good outcome if it was caused by either their human or their artificial agent. Let us now consider the bad-outcome case. In the human treatment, participants that did not delegate and caused the bad outcome themselves were on average rewarded 7.29 ECU (sd = 22.59 ECU). Participants that delegated to another human were on average rewarded with 8.26 ECU (sd = 21.56 ECU). This difference is insignificant (p = 0.559, paired t-test). In the machine treatment, participants that did not delegate and caused the bad outcome themselves were on average rewarded 8.53 ECU (sd = 26.52). Participants that delegated to a machine were on average rewarded 12.96 ECU (sd = 24.44). This difference is significant (p = 0.041, paired t-test). In the machine treatment, delegators earned higher rewards if a machine agent caused the failure. This confirms our conjecture stated in Section 2. Principals are rewarded differently for delegating tasks depending on the nature of the agent. More specifically, they fare better if a machine agent caused a bad outcome than if they did so themselves. Result 2: Delegators did not successfully shift blame for a bad outcome if their human agent caused it instead of themselves. They did, however, successfully shift blame for a bad outcome if their artificial agent caused it instead of themselves. Figure 3 illustrates this asymmetry in termsn of rewards between the human and the machine treatment for the bad-outcome case. Figure 3: Reward decisions if the bad outcome prevails. To see whether participants exploited this incentive to delegate to machine agents that does not exist for human agents, we compare the proportions of delegators in both treatments. In the human treatment, 35 out of 76 (46.1%) delegated the task to another human participant, whereas 41 out of of 73 (56.2%) delegated to the machine agent in the machine treatment. The difference in the proportions of delegators is insignificant (p = 0.28?, Chi- Square Test for Independence). The similar proportions of delegators in both treatments suggests that the decision to delegate to the agent is not primarily driven by strategic concerns of avoiding blame in the event of a bad outcome. This is corroborated by the logistic regression reported in Table 3. The propensity to delegate (1 = yes) is positively predicted by participants’ self-assessment of the number of errors they believe they have committed in the logic task (p < 0.001). “Machine” captures whether the agent is human (0) or artificial (1), which does not significantly influence this result. The effect to which self-assessment predicts the delegation decision is robust to controlling for risk attitude, although participants who are more risk averse are also more likely to delegate (p < 0.017). Dep. Variable: \| Del \| No. Observations: \| 149 ---\|---\|---\|--- Model: \| GLM \| Df Residuals: \| 145 Model Family: \| Binomial \| Df Model: \| 3 \| coef \| std err \| z \| P$>$$\|$z$\|$ \| [0.025 \| 0.975] ---\|---\|---\|---\|---\|---\|--- Intercept \| -3.2254 \| 0.795 \| -4.055 \| 0.000 \| -4.784 \| -1.666 Self_Assessment \| 0.4092 \| 0.105 \| 3.906 \| 0.000 \| 0.204 \| 0.615 Machine \| 0.3228 \| 0.365 \| 0.885 \| 0.376 \| -0.392 \| 1.038 Risk \| 0.1806 \| 0.076 \| 2.383 \| 0.017 \| 0.032 \| 0.329 Table 3: Generalized Linear Model Regression Results It thus appears that the expected utility of delegating was the driving factor for participants’ decisions to do so or not regardless of the nature of their agent. Those who were less confident regarding their own performance were more likely to delegate the task. Their estimated number of errors, which they stated in an incentivized self-assessment, had a significant impact on their delegation decision. As Table 4 shows, the self-assessment of participants is also predicted by the actual number of a subject’s errors in the logic task. Whether they have a human or an artificial agent at their avail did not influence their self- assessment. Also, subjects’ risk attitudes have no impact on their self assessment. These findings indicate that participants have a realistic impression about their own performance. Subjects who performed better in the logic task were accordingly less likely to delegate. Dep. Variable: \| Self_Assessment \| R-squared: \| 0.269 ---\|---\|---\|--- Model: \| OLS \| Adj. R-squared: \| 0.253 Method: \| Least Squares \| F-statistic: \| 17.74 No. Observations: \| 149 \| \| \| coef \| std err \| t \| P$>$$\|$t$\|$ \| [0.025 \| 0.975] ---\|---\|---\|---\|---\|---\|--- Intercept \| 3.2012 \| 0.474 \| 6.749 \| 0.000 \| 2.264 \| 4.139 Machine \| 0.4598 \| 0.276 \| 1.666 \| 0.098 \| -0.086 \| 1.005 Error \| 0.3955 \| 0.059 \| 6.741 \| 0.000 \| 0.280 \| 0.512 Risk \| -0.0298 \| 0.055 \| -0.539 \| 0.591 \| -0.139 \| 0.079 Table 4: Linear Regression – Influences on Self-assessment ## 5 Conclusion Our findings indicate that delegators may be judged with more leniency in the event of a bad outcome if they delegate tasks to artificial agents instead of human agents. It does therefore stand to reason that machine agents can be successfully used to avoid ostracism. The FCO president’s statement quoted at the beginning, i.e. that companies cannot hide behind algorithms, might not hold true empirically. Companies might well capitalize the effective shift of responsibility to algorithms if they fail. This is all the more true if they do not suffer any comparable loss of prestige as a result of the delegation in the event of success, as our data also suggest. The fact that the delegator in a between-subjects design receives a discharge if the agent is artificial but not if the agent is human at least suggests that the corresponding judgment is not reflective, but that it is a subtle behavioral tendency. If this is true, the discharge is not ethically desired by the evaluator. Our results might indicate the importance of institutional solutions that hold companies and ultimately individuals liable for the moral wrongs that their artificial agents bring about. These institutions grow even more important if they have to compensate for consumers’ behavioral reluctance to attribute blame in such cases. A deeper inquiry into the reluctance to punish that we observe seems warranted by the idea that extrinsic social motivation is an important factor in moral decision-making (Cappelen et al., 2017). The ability to use artificial agents as a smoke screen might encourage various decision makers to engage in more activities that are considered undesirable by stakeholders. The reassuring fact that delegators in this experiment did not exploit the effective release from blame does not imply that others will not- especially if they see through this behavioral tendency. Our experiment is subject to limitations. One is its rather explorative nature, which is owed to the lack of a well-established theoretical framework: Scientists have only recently started to address the influence of algorithms on human decision-making empirically. To the best of our knowledge, the projects by Gogoll and Uhl (2018) and Strobel and Kirchkamp (2017) cited above are the only experiments dedicated to the issue so far. Furthermore, the stylized setting used in our experiment limits the scope of the conclusion that we can draw. This concern is best explained in relation to internal and external validity. Internal validity means that the experiment is designed in such a way that it warrants conclusions about the behavior of its participants inside of the laboratory, for instance, by keeping relevant factors constant (ceteris paribus) or omitting irrelevant influences (ceteris absentibus). Experiments are externally valid if their design produces findings that are informative about behavior outside of the laboratory (Guala, 2002). Some authors posit an inverse relationship between internal and external validity (Guala et al., 2005; Loewenstein, 1999). Our findings provide a first indication that the nature of the agent has an effect on the reward and punishment that the delegator receives. The phenomenon we observed would have to be replicated in other contexts inside and outside of the laboratory to eliminate the possibility of it being a mere artifact. We believe that our findings represent an early step in understanding delegation decisions in a domain that is gaining relevance, i.e., the use of artificial agents in moral decision-making. More generally, our experimental results illustrate the necessity of investigating the interaction between humans and machines in behavioral settings. It is insufficient to rely on ethicists’ armchair arguments and on surveys that study laymen’s intuitions regarding the ethical implications of algorithms, because ethically relevant implications may arise as unintended results from the interactions between people and machines. 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# Fluid dynamics in the warp drive spacetime geometry Osvaldo L. Santos-Pereira<EMAIL_ADDRESS>Physics Institute, Universidade Federal do Rio de Janeiro, Brazil Everton M. C. Abreu<EMAIL_ADDRESS>Physics Department, Universidade Federal Rural do Rio de Janeiro, Seropédica, Brazil Physics Department, Universidade Federal de Juiz de Fora, Brazil Graduate Program in Applied Physics, Physics Institute, Universidade Federal do Rio de Janeiro, Brazil Marcelo B. Ribeiro<EMAIL_ADDRESS>Physics Institute, Universidade Federal do Rio de Janeiro, Brazil Graduate Program in Applied Physics, Physics Institute, Universidade Federal do Rio de Janeiro, Brazil Valongo Observatory, Universidade Federal do Rio de Janeiro, Brazil (August 27, 2024) ###### Abstract The Alcubierre warp drive metric is a spacetime geometry featuring a spacetime distortion, called warp bubble, where a massive particle inside it acquires global superluminal velocities, or warp speeds. This work presents solutions of the Einstein equations for the Alcubierre metric having fluid matter as gravity source. The energy-momentum tensor considered two fluid contents, the perfect fluid and the parametrized perfect fluid (PPF), a tentative more flexible model whose aim is to explore the possibilities of warp drive solutions with positive matter density content. Santos-Pereira et al. (2020) already showed that the Alcubierre metric having dust as source connects this geometry to the Burgers equation, which describes shock waves moving through an inviscid fluid, but led the solutions back to vacuum. The same happened for two out of four solutions subcases for the perfect fluid. Other solutions for the perfect fluid indicate the possibility of warp drive with positive matter density, but at the cost of a complex solution for the warp drive regulating function. Regarding the PPF, solutions were also obtained indicating that warp speeds could be created with positive matter density. Weak, dominant, strong and null energy conditions were calculated for all studied subcases, being satisfied for the perfect fluid and creating constraints in the PPF quantities such that positive matter density is also possible for creating a warp bubble. Summing up all results, energy-momentum tensors describing more complex forms of matter, or field, distributions generate solutions for the Einstein equations with the warp drive metric where negative matter density might not be a strict precondition for attaining warp speeds. warp drive, cosmic fluid, Burgers equation, shock waves, Alcubierre geometry ###### pacs: 04.20.Gz; 04.90.+e; 47.40.-x ## 1 Introduction It is well known that in general relativity particles can travel globally at superluminal speeds, whereas locally they cannot surpass the light speed. The warp drive spacetime geometry advanced by Alcubierre Alcubierre1994 uses these physical properties to propel material particles at superluminal speeds. It creates a limited spacetime distortion, called warp bubble, such that the spacetime is contracted in front of it and expanded behind the bubble as it moves along a geodesic. This warp drive metric is such that a particle trapped inside this bubble would locally move at subluminal speeds, whereas the bubble with the particle inside acquires global superluminal velocities, or warp speeds. In the seminal paper, Alcubierre also concluded that the warp metric would imply the violation of the energy conditions since it appeared that negative energy density would be required for the creation of the warp bubble. Since this original work many authors have contributed in our understanding of the theoretical details of the Alcubierre warp drive metric and possible feasibility of matter particles acquiring warp speeds. Ford and Roman FordRoman1996 applied quantum inequalities to calculate the amount of negative energy required to transport particles at superluminal speeds. They concluded that such energy requirements would be huge, so the amount of negative energy density necessary for the practical construction of a warp bubble would be impossible to achieve. Pfenning and Ford Pfenning1997 also used quantum inequalities to calculate the necessary bubble parameters and energy for the warp drive viability, reaching at an enormous amount of energy, ten orders of magnitude greater than the mass-energy of the entire visible Universe, also negative. Hiscock hiscock computed the vacuum energy-momentum tensor (EMT) of a reduced two dimensional quantized scalar field of the warp drive spacetime. He showed that in this reduced context that the EMT diverges if the apparent velocity of the bubble is greater than the speed of light. Such divergence is connected to the construction of an horizon in this two dimensional spacetime. Due to the semiclassical effects, the superluminal travel via warp drive might unfeasible. For example, to an observer within the warp drive bubble, the backward and forward walls look like the horizon of a white hole and of a black hole, respectively, resulting with a Hawking radiation. The issue of superluminal speeds of massive particles traveling faster than photons has also been studied by Krasnikov Krasnikov1998 , who argued that this would not be possible if some conjectures for globally hyperbolic spacetimes are made. He described some spacetime topologies and their respective need of the tachyon existence for the occurrence of travel at warp speeds. This author also advanced a peculiar spacetime where superluminal travel would be possible without tachyons, named as the Krasnikov tube by Everett and Roman EveretRoman1997 , who generalized the metric designed by Krasnikov by proposing a tube in the direction of the particle’s path providing a connection between Earth and a distant star. Inside the tube the spacetime is flat and the lightcones are opened out in order to allow the one direction superluminal travel. For the Krasnikov tube to work they showed that huge quantities of negative energy density would also be necessary. Since the tube does not possess closed timelike curves, it would be theoretically possible to design a two way non-overlapping tube system such that it would work as a time machine. In addition, the EMT is positive in some regions. Both the metric and the obtained EMT were thoroughly analyzed in Refs. Lobo2002 ; Lobo2003 . A relevant contribution to warp drive theory was made by van de Broeck Broeck1999 , who demonstrated that a small modification of the original Alcubierre geometry greatly diminishes to a few solar masses the total negative energy necessary for the creation of the warp bubble distortion, a result that led him to hypothesize that other geometrical modifications of this type could further reduce in a dramatic fashion the amount of energy necessary to create a warp drive bubble. Natario Natario2002 designed a new warp drive concept with zero expansion by using spherical coordinates and the $X$-axis as the polar one. Lobo and Visser LoboVisser2004b ; LoboVisser2004 discussed that the center of the warp bubble, as proposed by Alcubierre, need to be massless (see also White2003 ; White2011 ). A linearized model for both approaches was introduced and it was demonstrated that for small speeds the amassed negative energy inside the warp field is a robust fraction of the particle’s mass inside the center of the warp bubble. Lee and Cleaver cleaver1 ; cleaver2 have looked at how external radiation might affect the Alcubierre warp bubble, possibly making it energetically unsustainable, and how a proposed warp field interferometer could not detect spacetime distortions. Mattingly et al. cleaver3 ; cleaver4 discussed curvature invariants in the Natario and Alcubierre warp drives. In a previous paper nos we have considered some of these issues, but from a different angle. Since the Alcubierre metric was not advanced as a solution of the Einstein equations, as it was originally proposed simply as an ad hoc geometry designed to create a spacetime distortion such that a massive particle inside it travels at warp speeds, whereas locally it never exceeds the light speed, the basic question was then the possible types of matter- energy sources capable of creating a warp bubble. To answer this question the Einstein equations have to be solved with some form of EMT as source. The simplest one to start with is incoherent matter. Following this line of investigation, we showed that the dust solutions of the Einstein equations for the warp drive metric implied in vacuum, that is, such distribution is incapable of creating a warp bubble, nevertheless, the Burgers equation appeared as part of the solution of the Einstein equations. In addition, since the Burgers equation describes shock waves moving in an inviscid fluid, it was also found that at these shock waves behave as plane waves Trefethen2001 ; Evans2010 ; Forsyth1906 ; Bateman1915 ; Burgers1948 . In this paper we generalize the results obtained in Ref. nos by following the next logical step, that is, investigating perfect fluid as EMT source for the Alcubierre metric. We also propose a slightly generalized perfect fluid EMT, called here parametrized perfect fluid (PPF), in order to produce a tentatively more flexible model such that the pressure may have different parameter values. The aim is to see if more flexible EMT distributions could relax the original requirement that warp speeds could only be achieved by means of negative matter density. For the perfect fluid EMT solutions we found that two out of four subcases turn out to be the dust solution of Ref. nos where both the matter density and pressure vanish, but the Burgers equation also appears as a result of the solutions of the Einstein equations turcos . Two other subcases, however, indicate that warp speeds are possible with positive matter density, but at the cost of a complex solution for the warp metric regulating function. Weak, dominant, strong and null energy conditions were calculate for both EMTs and all perfect fluid solutions satisfy them. In the case of the PPF, two out of four solutions give rise to a nonlinear equation of state linking various pressures to the matter density. Other solutions produced results where a nonvanishing pressure occurs with a vanishing matter density, condition considered unphysical and then dismissed. The solutions also produced parameters and equations of state related to pressure and inequalities that satisfy all the energy conditions. These results indicate that energy-momentum tensors describing more complex forms of matter distributions generate solutions for the Einstein equations with the warp drive metric where negative matter density might not be a strict precondition. The plan for the paper is as follows. In Section 2 we briefly review the Alcubierre warp drive theory and present the relevant equations and all nonzero components of the Einstein tensor for the warp drive metric. In Section 3 the Einstein equations are then solved and solutions presented for the warp drive metric having a perfect fluid gravity source. In Section 4 the non-zero components of the Einstein tensor in the warp drive geometry are written in terms of the PPF EMT. Solutions for this more flexible EMT are also obtained and studied in all subcases. Section 5 presents the EMT divergence of both the perfect fluid and PPF, whereas Section 6 discusses the energy conditions for the two types of EMTs. Section 7 provides further discussions on the results presented in the previous sections, and Section 8 depicts our conclusions. ## 2 Einstein Equations We shall start this section by brief reviewing the Alcubierre warp drive metric. Subsequently, the nonzero components of the Einstein tensor of this metric will also be explicitly shown. The expressions presented in this section form the basic set of equations required for the next sections. ### 2.1 The Alcubierre warp drive geometry The start up geometry advanced in Ref. Alcubierre1994 may be written as follows, ${ds}^{2}=-\left(\alpha^{2}-\beta_{i}\beta^{i}\right)\,dt^{2}+2\beta_{i}\,dx^{i}\,dt+\gamma_{ij}\,dx^{i}\,dx^{j}\,\,,$ (2.1) where $d\tau$ is the proper time lapse, $\alpha$ is the lapse function, $\beta^{i}$ is the spacelike shift vector and $\gamma_{ij}$ is the spatial metric for the hypersurfaces.111Throughout this paper Greek indices will range from 0 to 3, whereas the Latin ones indicate the spacelike hypersurfaces and will range from 1 to 3. The lapse function $\alpha$ and the shift vector $\beta^{i}$ are functions to be determined, whereas $\gamma_{ij}$ is a positive-definite metric on each of the spacelike hypersurfaces, for all values of time, a feature that makes the spacetime globally hyperbolic Alcubierre2012 ; DeWitt1979 . Alcubierre Alcubierre1994 assumed the following particular parameter choices for Eq. (2.1), $\displaystyle\alpha$ $\displaystyle=1,$ (2.2) $\displaystyle\beta^{1}$ $\displaystyle=-v_{s}(t)f\big{[}r_{s}(t)\big{]},$ (2.3) $\displaystyle\beta^{2}$ $\displaystyle=\beta^{3}=0,$ (2.4) $\displaystyle\gamma_{ij}$ $\displaystyle=\delta_{ij}.$ (2.5) Hence, the Alcubierre warp drive metric is given by, $ds^{2}=-\left[1-v_{s}(t)^{2}f(r_{s})^{2}\right]dt^{2}-v_{s}(t)f(r_{s})\,dx\,dt+dx^{2}+dy^{2}+dz^{2}\,\,,$ (2.6) where $v_{s}(t)$ is the velocity of the center of the bubble moving along the curve $x_{s}(t)$. This is given by the following expression, $v_{s}(t)=\frac{dx_{s}(t)}{dt}\,\,.$ (2.7) The function $f(r_{s})$ is the warp drive regulating function. It describes the shape of the warp bubble, which is given by the following expression Alcubierre1994 , $f(r_{s})=\frac{\tanh\left[\sigma(r_{s}+R)\right]-\tanh\left[\sigma(r_{s}-R)\right]}{2\tanh(\sigma R)}\,\,,$ (2.8) where $\sigma$ and $R$ are parameters to be determined. The variable $r_{s}(t)$ defines the distance from the center of the bubble $[x_{s}(t),0,0]$ to a generic point $(x,y,z)$ on the surface of the bubble, given by the following equation, $r_{s}(t)=\sqrt{\left[x-x_{s}(t)\right]^{2}+y^{2}+z^{2}}.$ (2.9) From the above one can see that the motion is one-dimensional, since the $x$-coordinate is the only one perturbed by the function $x_{s}(t)$. ### 2.2 Einstein tensor components Let us now adopt Alcubierre’s original notation by assuming $\beta=-\beta^{1}=v_{s}(t)f(r_{s})$ (2.10) in Eq. (2.3). Then, the nonzero components of the Einstein tensor for the metric (2.6) are given by the following expressions: $\displaystyle G_{00}$ $\displaystyle=$ $\displaystyle-\frac{1}{4}(1+3\beta^{2})\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}+\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]-\beta\left(\frac{\partial^{2}\beta}{\partial y^{2}}+\frac{\partial^{2}\beta}{\partial z^{2}}\right),$ (2.11) $\displaystyle G_{01}$ $\displaystyle=$ $\displaystyle\frac{3}{4}\beta\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}+\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]+\frac{1}{2}\left(\frac{\partial^{2}\beta}{\partial y^{2}}+\frac{\partial^{2}\beta}{\partial z^{2}}\right),$ (2.12) $\displaystyle G_{02}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\frac{\partial^{2}\beta}{\partial x\partial y}-\frac{\beta}{2}\left(2\frac{\partial\beta}{\partial y}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial x\partial y}+\frac{\partial^{2}\beta}{\partial t\partial y}\right),$ (2.13) $\displaystyle G_{03}$ $\displaystyle=$ $\displaystyle-\frac{1}{2}\frac{\partial^{2}\beta}{\partial x\partial z}-\frac{\beta}{2}\left(2\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial x\partial z}+\frac{\partial^{2}\beta}{\partial t\partial z}\right),$ (2.14) $\displaystyle G_{11}$ $\displaystyle=$ $\displaystyle-\frac{3}{4}\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}+\left(\frac{\partial\beta}{\partial z}\right)^{2}\right],$ (2.15) $\displaystyle G_{12}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(2\frac{\partial\beta}{\partial y}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial x\partial y}+\frac{\partial^{2}\beta}{\partial t\partial y}\right),$ (2.16) $\displaystyle G_{13}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left(2\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial x\partial z}+\frac{\partial^{2}\beta}{\partial t\partial z}\right),$ (2.17) $\displaystyle G_{23}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial y},$ (2.18) $\displaystyle G_{22}$ $\displaystyle=$ $\displaystyle-\left[\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}\right]-\frac{1}{4}\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}-\left(\frac{\partial\beta}{\partial z}\right)^{2}\right],$ (2.19) $\displaystyle G_{33}$ $\displaystyle=$ $\displaystyle-\left[\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}\right]+\frac{1}{4}\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}-\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]\,\,.$ (2.20) ## 3 Perfect fluid Besides incoherent matter, or dust, already studied in Ref. nos , the simplest matter-energy distribution to be considered as gravity source for the possible creation of a warp bubble, and then warp speeds, is the perfect fluid. Hence, this section will discuss matter content solutions of the Einstein equations considering a perfect fluid matter source EMT for the Alcubierre metric. ### 3.1 Perfect fluid content solutions The EMT for a perfect fluid may be written as follows, $T_{\alpha\beta}=\left(\mu+p\right)\,u_{\alpha}u_{\beta}+p\,g_{\alpha\beta},$ (3.1) where $\mu$ is the matter density, $p$ is the fluid pressure, $g_{\alpha\beta}$ is the metric tensor and $u_{\alpha}$ is the 4-velocity of an observer inside the fluid. Perfect fluids have no shear stress, rotation, heat conduction or viscosity, nevertheless this ideal fluid provides a more complex matter content than simple dust Schutz2009 , allowing us to study if a warp bubble can be created with this gravity source and how the respective gravity field equations solutions can be understood. For the metric (2.6) the perfect fluid EMT assumes the following form, $T_{\alpha\beta}=\begin{pmatrix}\mu+\beta^{2}p&-\beta p&0&0\\\ -\beta p&p&0&0\\\ 0&0&p&0\\\ 0&0&0&p\end{pmatrix}\,.$ (3.2) Let us now use Eqs. (2.11) to (2.20) with the EMT above in the Einstein equations. Substituting components $G_{11}=8\pi T_{11}$ and $G_{01}=8\pi T_{01}$ into $G_{00}=8\pi T_{00}$, after some algebra and simplifications we may write the following expression, $T_{00}+2\beta T_{01}+\frac{1}{3}(3\beta^{2}-1)T_{11}=0\,\,.$ (3.3) Substituting the values for the EMT components $T_{00}=\mu+\beta^{2}p$, $T_{01}=-\beta p$, $T_{11}=p$ Eq. (3.3) results in the following expression, $p=3\mu.$ (3.4) This is an equation of state for the Alcubierre metric having a perfect fluid gravity source EMT. The component $G_{23}$ is zero since $T_{23}=0$. This case leads us to either $\partial\beta/\partial y$, or $\partial\beta/\partial z$, or both, equal to zero. Let us now analyze these possibilities and its consequences. Case 1: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial z}=0\right]}$ As $\beta$ does not depend on $z$, the Einstein tensor components $G_{13}$, $G_{23}$ and $G_{03}$ are identically zero. Substituting this case into $G_{11}=8\pi T_{11}$, where $G_{11}$ is given by Eq. (2.15) and $T_{11}=p$, it follows immediately the result below, $\frac{3}{4}\left(\frac{\partial\beta}{\partial y}\right)^{2}=-8\pi p.$ (3.5) Substituting $(\partial\beta/\partial y)^{2}$ above into $G_{01}=8\pi T_{01}$, where $G_{01}$ is given by Eq. (2.12) and $T_{01}=-\beta p$ from Eq. (3.2), as well as $G_{12}=8\pi T_{12}$ and $G_{02}=8\pi T_{02}$, where $T_{12}=0$ and $T_{02}=0$, the Einstein equations are reduced to the following equations, $\displaystyle p=3\mu,$ (3.6) $\displaystyle\left(\frac{\partial\beta}{\partial y}\right)^{2}=-\frac{32}{3}\pi p=-\,32\pi\,\mu,$ (3.7) $\displaystyle\frac{\partial^{2}\beta}{\partial y^{2}}=0,$ (3.8) $\displaystyle\frac{\partial^{2}\beta}{\partial x\partial y}=0,$ (3.9) $\displaystyle 2\frac{\partial\beta}{\partial y}\,\frac{\partial\beta}{\partial x}+\frac{\partial^{2}\beta}{\partial t\partial y}=0,$ (3.10) $\displaystyle\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}=-\frac{64}{3}\pi p\,=\,-\,64\pi\mu,$ (3.11) $\displaystyle\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}=-\frac{128}{3}\pi p\,=\,-\,128\pi\mu\,\,.$ (3.12) Eq. (3.7) implies that $\partial\beta/\partial y$ must be constant, since the pressure $p$ is assumed constant. Eq. (3.8) also shows that $\beta$ must be a linear function of the $y$-coordinate, which means that $\beta$ must have a possible additional dependence of arbitrary functions on $t$ and $x$. Both expressions in Eqs. (3.11) and (3.12) constitute the same homogeneous partial differential equation, but with different inhomogeneous parts, so the solution of the inhomogeneous equation is not unique, unless the pressure $p$ is zero. Then, considering these points and Eqs. (3.7) and (3.10), it follows that, $\frac{\partial\beta}{\partial y}\,\frac{\partial\beta}{\partial x}=0,$ (3.13) which means that either of these partial derivatives, or both, vanish. Let us discuss both possibilities below. Case 1a: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial z}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial x}=0\right]}$ For this case the set of partial differential equations from Eqs. (3.6) to (3.12) simplify to, $\displaystyle p=3\mu,$ (3.14) $\displaystyle\frac{\partial\beta}{\partial y}=\pm\sqrt{-32\pi\mu}.$ (3.15) The above equations mean that the matter density $\mu$ must be negative or zero for a non complex solution of the Einstein equations, and $\beta$ must be a function of both $t$ and $x$ coordinates only. The equation above is readily integrated, yielding $\beta(t,y)=\pm\sqrt{-32\pi\mu}\,y+g(t)\,\,,$ (3.16) where $g(t)$ is a function to be determined by the boundary conditions. Case 1b: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial z}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial y}=0\right]}$ In this case the pressures vanishes, since $\partial\beta/\partial y=0$, and the set of partial differential equations (3.6) to (3.12) simplify to, $\displaystyle p=3\mu=0,$ (3.17) $\displaystyle\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}=0.$ (3.18) Therefore, for $p=0$ the equation of state $p=3\mu$ implies zero matter density as well, which reduces the solution to the dust case and then vacuum. This also leads to the appearance of shock waves as plane waves, since $\beta=\beta(x,t)$ and the field equations are reduced to the Burgers equation, as studied in Ref. nos . This is the case because Eq. (3.18) can be written in the following form, $\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})=h(t).$ (3.19) Here $h(t)$ is a generic function to be determined by boundary conditions. In its homogeneous form, where $h(t)=0$, it takes the conservation form of the inviscid Burgers equation. $\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})=0\,\,.$ (3.20) See Ref. nos for details of the Burgers equation in this context. Case 2: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial y}=0\right]}$ As $\beta$ does not depend on the $y$-coordinate, it is easy to see that $G_{12}$, $G_{23}$ and $G_{02}$ are identically zero. In addition, considering this case into $G_{11}=8\pi T_{11}$, it follows immediately that, $-\frac{3}{4}\left(\frac{\partial\beta}{\partial z}\right)^{2}=8\pi p.$ (3.21) Substituting Eq. (3.21) in $G_{01}=8\pi T_{01}$, where $G_{01}$ is given by Eq. (2.12), $T_{01}=-\beta p$ from Eq. (3.2), and inserting the component $G_{13}=8\pi T_{13}$ into the component $G_{03}=8\pi T_{03}$, where $T_{13}=T_{03}=0$, after some algebra we reach at the following expressions, $\displaystyle p=3\mu,$ (3.22) $\displaystyle\left(\frac{\partial\beta}{\partial z}\right)^{2}=-\frac{32}{3}\pi p,$ (3.23) $\displaystyle\frac{\partial^{2}\beta}{\partial z^{2}}=0,$ (3.24) $\displaystyle\frac{\partial^{2}\beta}{\partial x\partial z}=0,$ (3.25) $\displaystyle 2\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial x}+\frac{\partial^{2}\beta}{\partial t\partial z}=0,$ (3.26) $\displaystyle\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}=-\frac{64}{3}\pi p,$ (3.27) $\displaystyle\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}=-\frac{128}{3}\pi p.$ (3.28) Eq. (3.24) shows that $\beta$ is a linear function with respect to the $z$-coordinate. Eq. (3.23) implies that $\partial\beta/\partial z$ must be constant since the pressure $p$ is assumed to be a constant. This means that all second partial derivatives of $\partial\beta/\partial z$ must vanish. Eqs. (3.27) and (3.28) are the same homogeneous partial differential equation, but both right hand side of theirs are different. Hence, the solution of the non- homogeneous equation is not unique, unless the pressure $p$ is zero. Considering Eq. (3.26), this case also unfolds in two possibilities, since, $\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial x}=0,$ (3.29) and either or both are zero. Let us now analyze each subcase. Case 2a: $\bm{\displaystyle\left[\frac{\partial\beta}{\partial y}=0\ \text{and}\ \frac{\partial\beta}{\partial x}=0\right]}$ For this case Eqs. (3.22) to (3.28) yield, $\displaystyle p=3\mu\,,$ (3.30) $\displaystyle\frac{\partial\beta}{\partial z}=\pm\,\sqrt{-32\pi\mu}\,,$ (3.31) $\displaystyle\frac{\partial\beta}{\partial z}=\pm\,\sqrt{\pm 96\pi\mu}\,.$ (3.32) Eq. (3.31) means that the matter density $\mu$ must be negative or zero for a non complex solution of the Einstein equations. Eq. (3.32) allows for possible positive matter density. In addition, $\beta$ has its dependence reduced to $\beta=\beta(z,t)$. If the matter density $\mu$ is assumed constant, then the above expressions can be integrated, yielding $\displaystyle\beta(z,t)=\pm\sqrt{-32\pi\mu}\,z+\bar{g}(t)\,,$ (3.33) $\displaystyle\beta(z,t)=\pm\sqrt{\pm 96\pi\mu}\,z+\bar{h}(t)\,.$ (3.34) where $\bar{g}(t)$ and $\bar{h}(t)$ are arbitrary functions to be determined by boundary conditions. Case 2b $\bm{\displaystyle\left[\frac{\partial\beta}{\partial y}=0\ \text{and}\ \frac{\partial\beta}{\partial z}=0\right]}$ For this subcase Eq. (3.23) implies zero pressure, and the set of partial differential equations (3.22) to (3.28) simplify to, $\displaystyle p=3\mu=0,$ (3.35) $\displaystyle\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}=0\,\,,$ (3.36) where the last equation is the result of the only nonzero Einstein tensor components $G_{22}$ and $G_{33}$. These results are the same as Case 1b above, that is, the dust solution for the Alcubierre warp drive metric that results in the Burgers equation (3.19) and its inviscid form given by Eq. (3.20), as well as shock waves as plane waves nos . $\square$ Table 1 summarizes all cases and their respective results of the Einstein equations with the Alcubierre warp drive metric having a perfect fluid matter content as gravity source. Case \| Condition \| Results ---\|---\|--- $1)\ \displaystyle{\frac{\partial\beta}{\partial z}=0}$ \| $1a)\ \displaystyle{\frac{\partial\beta}{\partial x}=0}$ \| $\begin{array}[]{ll}p=3\mu\\\\[6.0pt] \beta=\beta(y,t)\\\\[6.0pt] \displaystyle{\frac{\partial\beta}{\partial y}=\pm\sqrt{-32\pi\mu}}\\\\[8.0pt] \end{array}$ $1b)\ \displaystyle{\frac{\partial\beta}{\partial y}=0}$ \| $\begin{array}[]{ll}p=3\mu=0\\\\[6.0pt] \beta=\beta(x,t)\\\\[6.0pt] \displaystyle{\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})=h(t)}\\\\[8.0pt] \rightarrow\mbox{this is the dust solution of Ref.\ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{nos}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\\[4.0pt] \end{array}$ $2)\ \displaystyle{\frac{\partial\beta}{\partial y}=0}$ \| $2a)\ \displaystyle{\frac{\partial\beta}{\partial x}=0}$ \| $\begin{array}[]{ll}p=3\mu\\\\[6.0pt] \beta=\beta(z,t)\\\\[6.0pt] \displaystyle{\frac{\partial\beta}{\partial z}=\pm\,\sqrt{-32\pi\mu}}\\\\[8.0pt] \displaystyle{\frac{\partial\beta}{\partial z}=\,\pm\sqrt{\pm\,96\pi\mu}}\\\\[8.0pt] \end{array}$ $2b)\ \displaystyle{\frac{\partial\beta}{\partial z}=0}$ \| $\begin{array}[]{ll}p=3\mu=0\\\\[6.0pt] \beta=\beta(x,t)\\\\[6.0pt] \displaystyle{\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})=h(t)}\\\\[6.0pt] \rightarrow\mbox{this is the dust solution of Ref.\ \cite[cite]{\@@bibref{Authors Phrase1YearPhrase2}{nos}{\@@citephrase{(}}{\@@citephrase{)}}}}\\\\[4.0pt] \end{array}$ Table 1: Summary of all solutions of the Einstein equations with the Alcubierre warp drive metric having perfect fluid EMT as mass-energy source. ### 3.2 Discussion Cases 1b and 2b are simply the dust case already studied in Ref. nos , apparently being unable to generate a warp bubble since this is a vacuum solution, although it connects the warp metric to the Burgers equation and then to shock waves in the form of plane waves. Cases 1a and 2a share the same equation of state $p=3\mu$, but the coordinate dependencies are different, since $\beta=\beta(y,t)$ and $\beta=\beta(z,t)$, respectively. For $\beta$ to be a real valued function the matter density $\mu$ must be negative in Case 1a. From Eqs. (3.16) and (3.33) we have assumed a constant matter density, which means a straightforward integration, but $\mu$ can be also a function of both $t$ and $y$ coordinates in the Case 1a, or a function of both $t$ and $z$ coordinates in the Case 2a. Inasmuch as the matter density must be negative for real solutions, one could think of defining the total mass-energy density as follows, $\displaystyle\mu(t,x^{j})=\mu^{+}+a(t,x^{j})\mu^{-}\leq 0,$ (3.37) $\displaystyle x^{j}=y,z,$ where $\mu^{+}$ is the positive portion of the matter density of the perfect fluid and $\mu^{-}$ its negative portion that would allow the warp bubble to exist. $a(t,x^{j})$ would be a regulating function that depends on both time $t$ and space $x^{j}$ coordinates, being related to the shape and location of the bubble. Remembering that $x^{j}=y$ for the Case 1a and $x^{j}=z$ for the Case 2a, since $\mu(t,x^{j})$ must be negative there would be a restriction for the positive and negative portions of the matter density in Eq. (3.37). It might be argued that there is no problem in complex solutions for $\beta$, but in the warp drive scenario $\beta=v_{s}(t)f(r_{s})$ determines both the velocity and the shape of the bubble, so either the velocity $v_{s}(t)$ or the regulating function $f(r_{s})$ of the bubble shape must be complex. A complex velocity has no physical meaning, but a complex regulating function could be acceptable if we only consider its real part. Thus, in such situation the formation of a warp bubble capable of generating warp speeds could still be possible in the presence of a perfect fluid positive matter density EMT as gravity source. Nevertheless, caution is required here because if the result of integrating $\beta$ turns out to be purely imaginary it is not clear what the bubble shape being represented by an imaginary function means. Therefore, in principle it seems reasonable to start with $\beta$ being a real function and the warp bubble requiring a perfect fluid with negative mass-energy density for warp speeds to be physically viable. But, it seems to us that this point remains open to debate. Considering the results above it is apparent that a perfect fluid EMT generated a more complex set of solutions of the Einstein equations than the dust one, then it is conceivable that viable warp speeds are also possible in more complex EMTs. One such possibility will be discussed next. ## 4 Parametrized perfect fluid Let us propose a generalization of the perfect fluid EMT having seven quantities, namely the mass-energy density $\mu$, the $\beta$ function and five different pressures $A,B,C,D$ and $p$. The last quantity $D$ is a momentum density parameter. In the perfect fluid of Eq. (3.2) the pressure denoted by $p$ are all the same in the EMT, a constraint that has been relaxed here. Let us call the perfect fluid generalization with the quantities above as the parametrized perfect fluid (PPF). Its respective EMT may be written as below, $T_{\alpha\sigma}=\begin{pmatrix}\mu+\beta^{2}p&-\beta D&0&0\\\ -\beta D&A&0&0\\\ 0&0&B&0\\\ 0&0&0&C\end{pmatrix}\,.$ (4.1) The quantities $A,B,C,D$ and $p$ will not be assumed as constants, but rather as functions of the spacetime coordinates $(t,x,y,z)$. This is clearly a more flexible EMT than the perfect fluid, and it is being proposed here as a tentative model in order to explore the consequences of more complex EMTs in terms of generating possible positive matter solutions of the Einstein equations with the warp drive metric without the caveats of the perfect fluid solutions discussed above. It is a tentative proposal for a more flexible, or toy, model for the possible creation of warp bubbles, and then warp speeds. Section 7.2 provides more details on the physics of this specific fluid proposal. The nonzero components of the Einstein equations for the PPF are given by the following expressions, $\displaystyle-\frac{1}{4}(1+3\beta^{2})\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}+\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]-\beta\left(\frac{\partial^{2}\beta}{\partial y^{2}}+\frac{\partial^{2}\beta}{\partial z^{2}}\right)$ $\displaystyle=$ $\displaystyle 8\pi(\mu+\beta^{2}p),$ (4.2) $\displaystyle\frac{3}{4}\beta\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}+\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]+\frac{1}{2}\left(\frac{\partial^{2}\beta}{\partial y^{2}}+\frac{\partial^{2}\beta}{\partial z^{2}}\right)$ $\displaystyle=$ $\displaystyle-8\pi\beta D,$ (4.3) $\displaystyle-\frac{1}{2}\frac{\partial^{2}\beta}{\partial x\partial y}-\frac{\beta}{2}\left(2\frac{\partial\beta}{\partial y}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial x\partial y}+\frac{\partial^{2}\beta}{\partial t\partial y}\right)$ $\displaystyle=$ $\displaystyle 0,$ (4.4) $\displaystyle-\frac{1}{2}\frac{\partial^{2}\beta}{\partial x\partial z}-\frac{\beta}{2}\left(2\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial x\partial z}+\frac{\partial^{2}\beta}{\partial t\partial z}\right)$ $\displaystyle=$ $\displaystyle 0,$ (4.5) $\displaystyle-\frac{3}{4}\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}+\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]$ $\displaystyle=$ $\displaystyle 8\pi A,$ (4.6) $\displaystyle\frac{1}{2}\left(2\frac{\partial\beta}{\partial y}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial x\partial y}+\frac{\partial^{2}\beta}{\partial t\partial y}\right)$ $\displaystyle=$ $\displaystyle 0,$ (4.7) $\displaystyle\frac{1}{2}\left(2\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial x}+\beta\frac{\partial^{2}\beta}{\partial x\partial z}+\frac{\partial^{2}\beta}{\partial t\partial z}\right)$ $\displaystyle=$ $\displaystyle 0,$ (4.8) $\displaystyle\frac{1}{2}\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (4.9) $\displaystyle-\left[\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}\right]-\frac{1}{4}\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}-\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]$ $\displaystyle=$ $\displaystyle 8\pi B,$ (4.10) $\displaystyle-\left[\frac{\partial^{2}\beta}{\partial t\partial x}+\beta\frac{\partial^{2}\beta}{\partial x^{2}}+\left(\frac{\partial\beta}{\partial x}\right)^{2}\right]+\frac{1}{4}\left[\left(\frac{\partial\beta}{\partial y}\right)^{2}-\left(\frac{\partial\beta}{\partial z}\right)^{2}\right]$ $\displaystyle=$ $\displaystyle 8\pi C.$ (4.11) Substituting Eqs. (4.6) and (4.3) into Eq. (4.2) it follows immediately that, $\mu=\beta^{2}(2D-A-p)+\frac{A}{3}\,\,.$ (4.12) This expression shows that the fluid density not only depends on the pressure components $A$ and $p$, but also on the momentum component $D$ and the warp bubble, since it varies with the shift vector $\beta$ and, hence, the bubble movement. So, the bubble shape modifies the fluid density in a local way, a result that may imply an analogy with classical fluid dynamics and shock waves in fluids with global velocity greater than the speed of sound in that medium. Applying this analogy to the warp drive, it may mean that the warp bubble plays the role of a shock wave in a fluid that moves with apparent velocity greater than the speed of light for an outside observer far away from the bubble, this being a result of the nonlinearity of the Einstein equations. The bubble modifies the fluid density which then causes the bubble motion. This classical relativistic fluid analogy may be a physical argument for a mechanism which accounts for the great amount of energy necessary for the feasibility of the warp drive. After some algebra on the set of Einstein equations above they can be rewritten as below, $\displaystyle\beta^{2}(2D-A-p)+\frac{A}{3}$ $\displaystyle=$ $\displaystyle\mu,$ (4.13) $\displaystyle\frac{\partial^{2}\beta}{\partial x\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (4.14) $\displaystyle\frac{\partial^{2}\beta}{\partial x\partial z}$ $\displaystyle=$ $\displaystyle 0,$ (4.15) $\displaystyle\left(\frac{\partial\beta}{\partial y}\right)^{2}+\left(\frac{\partial\beta}{\partial z}\right)^{2}$ $\displaystyle=$ $\displaystyle-\frac{32}{3}\pi A,$ (4.16) $\displaystyle\left(\frac{\partial\beta}{\partial y}\right)^{2}-\left(\frac{\partial\beta}{\partial z}\right)^{2}$ $\displaystyle=$ $\displaystyle 16\pi(C-B),$ (4.17) $\displaystyle 2\frac{\partial\beta}{\partial y}\,\frac{\partial\beta}{\partial x}+\frac{\partial^{2}\beta}{\partial t\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (4.18) $\displaystyle 2\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial x}+\frac{\partial^{2}\beta}{\partial t\partial z}$ $\displaystyle=$ $\displaystyle 0,$ (4.19) $\displaystyle\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial y}$ $\displaystyle=$ $\displaystyle 0,$ (4.20) $\displaystyle\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]$ $\displaystyle=$ $\displaystyle-32\pi(B+C),$ (4.21) $\displaystyle\frac{\partial^{2}\beta}{\partial y^{2}}+\frac{\partial^{2}\beta}{\partial z^{2}}$ $\displaystyle=$ $\displaystyle 16\pi\beta(A-D).$ (4.22) Eq. (4.20) shows that the solutions for the set of differential equations above have similar alternative cases as in the perfect fluid solutions, that is, either $\partial\beta/\partial z=0$ and/or $\partial\beta/\partial y=0$. Both situations and their respective subcases will be discussed next. Case 1: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial z}=0\right]}$ The set of equations (4.13) to (4.22) simplify to, $\displaystyle\mu=\beta^{2}(2D-A-p)+\frac{A}{3},$ (4.23) $\displaystyle\frac{\partial^{2}\beta}{\partial x\partial y}=0,$ (4.24) $\displaystyle\left(\frac{\partial\beta}{\partial y}\right)^{2}=-\frac{32}{3}\pi A=16\pi(C-B),$ (4.25) $\displaystyle 2\frac{\partial\beta}{\partial y}\,\frac{\partial\beta}{\partial x}+\frac{\partial^{2}\beta}{\partial t\partial y}=0,$ (4.26) $\displaystyle\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-32\pi(B+C).$ (4.27) $\displaystyle\frac{\partial^{2}\beta}{\partial y^{2}}=16\pi\beta(A-D).$ (4.28) From Eq. (4.25) a relation between the pressures $A,B$ and $C$ are straightforward, $B=C+\frac{2}{3}A.$ (4.29) From Eqs. (4.24) and (4.25) it is easy to verify that $A$ and $C-B$ do not depend on the $x$-coordinate. In addition, for real solutions $A$ must be negative, assuming only real values that are equal to or smaller than zero. Differentiating Eq. (4.25) with respect to $x$ yields, $2\frac{\partial\beta}{\partial y}\frac{\partial^{2}\beta}{\partial x\partial y}=0\,\Longrightarrow\frac{\partial\beta}{\partial y}=0\quad\mbox{and/or}\quad\frac{\partial^{2}\beta}{\partial x\partial y}=0,$ (4.30) and inserting the result $\partial\beta/\partial y=0$ into Eq. (4.26) it follows that, $\frac{\partial^{2}\beta}{\partial t\partial y}=0,$ (4.31) The result $\partial\beta/\partial y=0$ means that $A=D$ from Eq. (4.28) and $C=B$ from Eq. (4.25). Hence, the set of equations from Eq. (4.23) to Eq. (4.27) simplify for $\partial\beta/\partial y=0$, $\displaystyle\mu=\beta^{2}(D-p)+\frac{D}{3}\,=\,\beta^{2}(A-p)+\frac{A}{3},$ (4.32) $\displaystyle\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-64\pi B\,=-64\pi C.$ (4.33) From the result $\partial^{2}\beta/\partial x\partial y=0$ of Eq. (4.30) the set of equations (4.23) to (4.28) is recovered. Therefore, Eqs. (4.30) show that this case separates itself in two conditions, either $\partial\beta/\partial x=0$ or $\partial\beta/\partial y=0$. Next we analyze both conditions. Case 1a: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial z}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial x}=0\right]}$ Setting Eq.(4.27) equal to zero then $B=-C$, and the set of equations (4.23) to (4.28) simplify to, $\displaystyle\mu$ $\displaystyle=$ $\displaystyle\beta^{2}(2D-A-p)+\frac{A}{3},$ (4.34) $\displaystyle B$ $\displaystyle=$ $\displaystyle-C=\frac{1}{3}A,$ (4.35) $\displaystyle\left(\frac{\partial\beta}{\partial y}\right)^{2}$ $\displaystyle=$ $\displaystyle 32\pi C,$ (4.36) $\displaystyle\frac{\partial^{2}\beta}{\partial y^{2}}$ $\displaystyle=$ $\displaystyle 16\pi\beta(A-D).$ (4.37) For this case, there is an equation of state given by Eq. (4.34), $\beta$ is a function of time and $y$ coordinates and must be found by solving both Eqs. (4.36) and (4.37) in terms of the pressures $A,C$ and $D$. Note that Eq. (4.35) relates the pressures $A,B$ and $C$. The EMT for this case may be written as follows, $T_{\alpha\sigma}=\begin{pmatrix}\beta^{2}(2D-A)+A/3&-\beta D&0&0\\\ -\beta D&A&0&0\\\ 0&0&A/3&0\\\ 0&0&0&-A/3\end{pmatrix}\,.$ (4.38) One should also note that for the $T_{00}$ component from Eq. (4.38) to be of positive value the following inequality must hold, $\beta^{2}>\frac{A}{3(A-2D)}.$ (4.39) Case 1b: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial z}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial y}=0\right]}$ Since Eq. (4.25) is equal to zero, then $B=C$. Similarly for Eq. (4.28) it is clear that $A=D$. Consequently, the set of equations (4.23) to (4.28) simplify to, $\displaystyle\mu=\beta^{2}(2D-A-p)+\frac{A}{3},$ (4.40) $\displaystyle B=C+\frac{2}{3}A,$ (4.41) $\displaystyle B=C,$ (4.42) $\displaystyle A=D,$ (4.43) $\displaystyle\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-64\pi B.$ (4.44) But, from Eq. (4.25) we have that $A=0$ because $B=C$. So, from Eq. (4.32) we have that $\mu=-\beta^{2}p$, and one is left with a non homogeneous Burgers equation. This case reduces the above set of equations to $\displaystyle\mu=-\beta^{2}p,$ (4.45) $\displaystyle B=C,$ (4.46) $\displaystyle\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-64\pi B.$ (4.47) Eq. (4.45) represents an equation of state between matter density $\mu$ and the pressure $p$. The pressures $B$ and $C$ are functions of the spacetime coordinates $(t,x,y,z)$ and from Eq. (4.46) they are equal. Eq. (4.47) is a non homogeneous Burgers equation, since its right hand side cannot be readily integrated. It might mean that there is no conservation law that can describe the warp drive for the PPF EMT. The only possible way leading to conservation law is for $B$ being a constant, which then allows a straightforward integration. However, since this is not the case, namely, all the pressures from the EMT are not, necessarily, constant functions, it is necessary to determine this functions through the boundary conditions. The EMT for the Case 1b case then yields, $T_{\alpha\sigma}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&B&0\\\ 0&0&0&B\end{pmatrix}\,.$ (4.48) The only non vanishing components of the PPF EMT above are $T_{22}$ and $T_{33}$. This case recovers the perfect fluid Case 1b, that is, the dust EMTs when one chooses $p=B=0$. For $B\neq 0$, one would have to solve Eq. (4.47) to determine how the bubble moves in this type of fluid spacetime. Again, negative matter density emerges from the Einstein equation solutions for this specific choice of EMT. Nevertheless, this is a rather peculiar EMT, since the $T_{00}$ component is zero, but the equation of state (4.45) remains and only two diagonal terms are not zero. In addition, it is contradictory with the perfect fluid solution because the PPF reduces to the perfect fluid under the condition $p=A=B=C=D,$ (4.49) but in this solution $A=0$, but $B\not=0$. So, we discard this case as unphysical. Case 2: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial y}=0\right]}$ The set of equations (4.13) to (4.22) simplify to, $\displaystyle\mu=\beta^{2}(2D-A-p)+\frac{A}{3},$ (4.50) $\displaystyle\frac{\partial^{2}\beta}{\partial x\partial z}=0,$ (4.51) $\displaystyle\left(\frac{\partial\beta}{\partial z}\right)^{2}=-\frac{32}{3}\pi A=-\,16\pi(B-C),$ (4.52) $\displaystyle 2\frac{\partial\beta}{\partial z}\,\frac{\partial\beta}{\partial x}+\frac{\partial^{2}\beta}{\partial t\partial z}=0,$ (4.53) $\displaystyle\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-32\pi(B+C)\,\,,$ (4.54) $\displaystyle\frac{\partial^{2}\beta}{\partial z^{2}}=16\pi\beta(A-D).$ (4.55) Eq. (4.52) straightforwardly implies the following relationship between pressures $A,B$ and $C$, $B=C+\frac{2}{3}A.$ (4.56) Eqs. (4.51) and (4.52) indicate that $A$ and $B-C$ do not depend on the $x$-coordinate. In addition, for real solutions $A$ must be negative or zero. Finally, Eq. (4.51) shows that that this case also unfolds in two subcases: $\partial\beta/\partial z=0$ and/or $\partial\beta/\partial x=0$. Case 2a: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial y}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial x}=0\right]}$ Clearly $B=-C$ as a consequence of Eq. (4.54). Then, the set of equations Eqs. (4.50) to (4.55) simplifies to, $\displaystyle\mu=\beta^{2}(2D-A-p)+\frac{A}{3},$ (4.57) $\displaystyle B=-C=\frac{A}{3},$ (4.58) $\displaystyle\left(\frac{\partial\beta}{\partial z}\right)^{2}=32\pi C,$ (4.59) $\displaystyle\frac{\partial^{2}\beta}{\partial z^{2}}=16\pi\beta(A-D).$ (4.60) The EMT for this case is given by the following expression, $T_{\alpha\sigma}=\begin{pmatrix}\beta^{2}(2D-A)+A/3&-\beta D&0&0\\\ -\beta D&A&0&0\\\ 0&0&-A/3&0\\\ 0&0&0&A/3\end{pmatrix},$ (4.61) which is almost equal to Case 1a (Eq. 4.38), apart from the change of signs in the components $T_{22}$ and $T_{33}$. Both have the same $T_{00}$ component and hence, the same equation of state, and the condition for $T_{00}$ to be of positive is given by, $\beta^{2}>\frac{A}{3(A-2D)}.$ (4.62) Besides, this expressions determines another inequality that $T_{00}$ must follow to be positive, $A-2D>0.$ (4.63) Case 2b: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial y}=0\,\,\,\,\text{and}\,\,\,\,\frac{\partial\beta}{\partial z}=0\right]}$ The set of equations Eqs. (4.50) to (4.55) immediately simplify to, $\displaystyle\mu=\beta^{2}(2D-A-p)+\frac{A}{3},$ (4.64) $\displaystyle B=C-\frac{2}{3}A,$ (4.65) $\displaystyle B=C,$ (4.66) $\displaystyle A=D=0,$ (4.67) $\displaystyle\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-64\pi B\,=-64\pi C\,\,.$ (4.68) Therefore $\mu=-\beta^{2}p$, a non homogeneous Burgers equation is also present, and the expressions above are reduced to $\displaystyle\mu=-\beta^{2}p,$ (4.69) $\displaystyle B=C,$ (4.70) $\displaystyle\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-64\pi B,$ (4.71) which are the same results as obtained in Case 1b, as well as the EMT, that is, $T_{\alpha\sigma}=\begin{pmatrix}0&0&0&0\\\ 0&0&0&0\\\ 0&0&B&0\\\ 0&0&0&B\end{pmatrix}\,.$ (4.72) $\square$ Again, because this solution cannot fulfill the requirement given by Eq. (4.49), in fact it contradicts it since one cannot have both $A=0$ and $B\not=0$, this solution is, similarly to Case 1b, dismissed as unphysical. Table 2 summarizes the Einstein equations solutions obtained as a result of the PPF EMT in the Alcubierre warp drive metric. Case \| Conditions \| Results ---\|---\|--- $1)\ \displaystyle{\frac{\partial\beta}{\partial z}=0}$ \| $1a)\ \displaystyle{\frac{\partial\beta}{\partial x}=0}$ \| $\begin{array}[]{ll}\\\\[-10.0pt] \displaystyle{\mu=\beta^{2}(2D-A-p)+\frac{A}{3}}\\\\[7.0pt] \beta=\beta(t,y)\\\\[7.0pt] \displaystyle{B=-C=\frac{A}{3}}\\\\[7.0pt] \displaystyle{\left(\frac{\partial\beta}{\partial y}\right)^{2}=32\pi C},\\\\[10.0pt] \displaystyle{\frac{\partial^{2}\beta}{\partial y^{2}}=16\pi\beta(A-D)}\\\\[7.0pt] \end{array}$ $1b)\ \displaystyle{\frac{\partial\beta}{\partial y}=0}$ \| $\begin{array}[]{ll}\\\\[-10.0pt] \displaystyle{\mu=-\beta^{2}p}\\\\[7.0pt] \beta=\beta(t,x)\\\\[7.0pt] \displaystyle{B=C}\\\\[5.0pt] \displaystyle{A=D=0}\\\\[7.0pt] \displaystyle{\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-64\pi B}\\\\[8.0pt] \rightarrow\mbox{solution {dismissed} as unphysical}\\\ \end{array}$ $2)\ \displaystyle{\frac{\partial\beta}{\partial y}=0}$ \| $2a)\ \displaystyle{\frac{\partial\beta}{\partial x}=0}$ \| $\begin{array}[]{ll}\\\\[-10.0pt] \displaystyle{\mu=\beta^{2}(2D-A-p)+\frac{A}{3}}\\\\[7.0pt] \beta=\beta(t,z)\\\\[7.0pt] \displaystyle{B=-C=\frac{A}{3}}\\\\[7.0pt] \displaystyle{\left(\frac{\partial\beta}{\partial z}\right)^{2}=32\pi C}\\\\[9.0pt] \displaystyle{\frac{\partial^{2}\beta}{\partial z^{2}}=16\pi\beta(A-D)}\\\\[7.0pt] \end{array}$ $2b)\ \displaystyle{\frac{\partial\beta}{\partial z}=0}$ \| $\begin{array}[]{ll}\\\\[-10.0pt] \displaystyle{\mu=-\beta^{2}p}\\\\[7.0pt] \beta=\beta(t,x)\\\\[7.0pt] \displaystyle{B=C}\\\\[5.0pt] \displaystyle{A=D=0}\\\\[7.0pt] \displaystyle{\frac{\partial}{\partial x}\left[\frac{\partial\beta}{\partial t}+\frac{1}{2}\frac{\partial}{\partial x}(\beta^{2})\right]=-64\pi B}\\\\[8.0pt] \rightarrow\mbox{solution {dismissed} as unphysical}\end{array}$ Table 2: Summary of all solutions of the Einstein equations for the Alcubierre warp drive metric having the parametrized perfect fluid (PPF) EMT. The cases, conditions and their respective designations are the same as in Table 1 ### 4.1 Discussion As in the perfect fluid situations, Cases 1b and 2b are the same for the PPF, producing the same equations of state, coordinate dependency for the $\beta$ function and a non-homogeneous Burgers equation. As stated above, only for a constant pressure $B$ that a conservation law could possibly emerge from the Burgers equation (4.71) that is common to both cases. Otherwise this expression cannot be readily integrated unless further boundary conditions are met (see Section 5.3 below). Nevertheless, because the solutions of these two cases cannot satisfy the requirement established by Eq. (4.49) for reducing the PPF to the perfect fluid, they are dismissed as unphysical. Regarding Cases 1a and 2a, the only physically plausible solutions remaining for the PPF EMT, they must obey the same inequalities (4.62) and (4.63) so that the $T_{00}$ component be positive in both Eqs. (4.38) and (4.61). The function $\beta$ depends on the $y$ coordinate in the former case and the $z$ one in the later, and, respectively, are a result of the integration of the following differential equations, $\beta\frac{\partial\beta}{\partial y}=\frac{C}{A-D},$ (4.73) $\beta\frac{\partial\beta}{\partial z}=\frac{C}{A-D}.$ (4.74) However, the quantities $A$, $C$ and $D$ are function of the coordinates, thus integrating the expressions above require further conditions, such as the ones respectively discussed in Sections 5.1 and 5.2 below. #### 4.1.1 Equations of state The perfect fluid solutions have equations of state given by (Ref. EllisElst, , p. 14), $p=p(\mu)=(\gamma-1)\mu,\;\;\;\dot{\gamma}=\frac{\mathrm{d}\gamma}{\mathrm{d}t}=0,$ (4.75) Ordinary fluids can be approximated with $1\leq\gamma\leq 2$, where incoherent matter, or dust, corresponds to $\gamma=1$, and radiation to $\gamma=\frac{4}{3}$. In the analyzes above we found that for the perfect fluid content solution Cases 1a and 2a for the warp drive metric the equation of state is given by $p=3\mu$ (see Table 1), which corresponds to $\gamma=4$. However, they do not allow a simple expressions for the equation of state similar to Eq. (4.75) for the PPF solutions 1a and 2a (see Table 2). The PPF content solution for the Cases 1b and 2b were dismissed as unphysical, ## 5 EMT Divergence of the perfect fluid and PPF In this section we will investigate the associated conservation laws to the Einstein equations under a warp drive spacetime by means of the usual condition that the EMT divergence must be zero for both the perfect fluid and PPF. We shall start with the EMT for the PPF because from its very definition the perfect fluid can be recovered by setting the equality (4.49). For the fluids discussed here, setting ${T^{\alpha\sigma}}_{;\sigma}=0$ in the EMT (4.1) results in the following expressions, $\displaystyle-\frac{\partial\beta}{\partial x}(D+\mu)-\frac{\partial\mu}{\partial t}-\beta\left[\frac{\partial D}{\partial x}+\frac{\partial\mu}{\partial x}+\frac{\partial\beta}{\partial t}(2p+A-3D)\right]$ $\displaystyle+\beta^{2}\left[\frac{\partial D}{\partial t}-\frac{\partial p}{\partial t}+3\frac{\partial\beta}{\partial x}(D-p)\right]+\beta^{3}\left(\frac{\partial D}{\partial x}-\frac{\partial p}{\partial x}\right)$ $\displaystyle=0,$ (5.1) $\displaystyle\frac{\partial A}{\partial x}+\frac{\partial\beta}{\partial t}(D-A)+\beta\left[3\frac{\partial\beta}{\partial x}(D-A)+\frac{\partial D}{\partial t}-\frac{\partial A}{\partial t}\right]+\beta^{2}\left(\frac{\partial D}{\partial x}-\frac{\partial A}{\partial x}\right)$ $\displaystyle=0,$ (5.2) $\displaystyle\frac{\partial B}{\partial y}+\beta\frac{\partial\beta}{\partial y}(D-A)$ $\displaystyle=0,$ (5.3) $\displaystyle\frac{\partial C}{\partial z}+\beta\frac{\partial\beta}{\partial z}(D-A)$ $\displaystyle=0.$ (5.4) ### 5.1 Case 1a: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial z}=0\right.}$ and $\bm{\left.\displaystyle\frac{\partial\beta}{\partial x}=0\right]}$ Eqs. (5.1) to Eq. (5.4) are reduced to the ones below, $\displaystyle-\frac{\partial\mu}{\partial t}-\beta\left[\frac{\partial D}{\partial x}+\frac{\partial\mu}{\partial x}+\frac{\partial\beta}{\partial t}(2p+A-3D)\right]+\beta^{2}\left(\frac{\partial D}{\partial t}-\frac{\partial p}{\partial t}\right)$ $\displaystyle+\beta^{3}\left(\frac{\partial D}{\partial x}-\frac{\partial p}{\partial x}\right)$ $\displaystyle=0,$ (5.5) $\displaystyle\frac{\partial A}{\partial x}+\frac{\partial\beta}{\partial t}(D-A)+\beta\left(\frac{\partial D}{\partial t}-\frac{\partial A}{\partial t}\right)+\beta^{2}\left(\frac{\partial D}{\partial x}-\frac{\partial A}{\partial x}\right)$ $\displaystyle=0,$ (5.6) $\displaystyle\frac{\partial B}{\partial y}+\beta\frac{\partial\beta}{\partial y}(D-A)$ $\displaystyle=0,$ (5.7) $\displaystyle\frac{\partial C}{\partial z}$ $\displaystyle=0.$ (5.8) The results above concern the PPF. These four equations together with the five ones shown in the respective results of Table 2 means an overdetermined system for the six unknowns $\mu,p,A,B,C$, $D$. The perfect fluid is recovered by setting Eq. (4.49). Hence, Eqs. (5.5) to (5.8) become, $\displaystyle-\frac{\partial\mu}{\partial t}-\beta\left(\frac{\partial p}{\partial x}+\frac{\partial\mu}{\partial x}\right)$ $\displaystyle=0,$ (5.9) $\displaystyle\frac{\partial p}{\partial x}$ $\displaystyle=0,$ (5.10) $\displaystyle\frac{\partial p}{\partial y}$ $\displaystyle=0,$ (5.11) $\displaystyle\frac{\partial p}{\partial z}$ $\displaystyle=0,$ (5.12) and, thus, the pressure does not depend on the spatial coordinates. In addition, Eq. (5.9) reduces to the expression below, $\frac{\partial\mu}{\partial t}+\beta\frac{\partial\mu}{\partial x}=0,$ (5.13) which is the continuity equation, where $\mu$ plays the role of the fluid density and $\beta$ is the flow velocity vector field. It is worth mentioning that for a constant density the fluid has incompressible flow, then all the partial derivatives of $\beta$ in terms of the spatial coordinates vanish and the flow velocity vector field has null divergence, this being a classical fluid dynamics scenario, and the local volume dilation rate is zero. ### 5.2 Case 2a: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial y}=0\right.}$ and $\bm{\left.\displaystyle\frac{\partial\beta}{\partial x}=0\right]}$ Equations (5.1) to (5.4) simplify to the following expressions: $\displaystyle-\frac{\partial\mu}{\partial t}-\beta\left[\frac{\partial D}{\partial x}+\frac{\partial\mu}{\partial x}+\frac{\partial\beta}{\partial t}(2p+A-3D)\right]$ $\displaystyle+\beta^{2}\left(\frac{\partial D}{\partial t}-\frac{\partial p}{\partial t}\right)+\beta^{3}\left(\frac{\partial D}{\partial x}-\frac{\partial p}{\partial x}\right)$ $\displaystyle=0,$ (5.14) $\displaystyle\frac{\partial A}{\partial x}+\frac{\partial\beta}{\partial t}(D-A)+\beta\left(\frac{\partial D}{\partial t}-\frac{\partial A}{\partial t}\right)+\beta^{2}\left(\frac{\partial D}{\partial x}-\frac{\partial A}{\partial x}\right)$ $\displaystyle=0,$ (5.15) $\displaystyle\frac{\partial B}{\partial y}$ $\displaystyle=0,$ (5.16) $\displaystyle\frac{\partial C}{\partial z}+\beta\frac{\partial\beta}{\partial z}(D-A)$ $\displaystyle=0.$ (5.17) Assuming Eq. (4.49) the perfect fluid is recovered, resulting in the already discussed Eqs. (5.9) to (5.12), as well as the continuity equation (5.13). ### 5.3 Cases 1b and 2b: $\bm{\left[\displaystyle\frac{\partial\beta}{\partial y}=0\right.}$ and $\bm{\left.\displaystyle\frac{\partial\beta}{\partial z}=0\right]}$ It follows from Table 1 that conditions 1b and 2b for the perfect fluid imply $\mu=p=0$, which also means $A=B=C=D=0$. Then, Eqs. (5.1) to (5.4) of the null EMT divergence are are immediately satisfied for the perfect fluid. Regarding the PPF, we have already seen that in these cases the solutions were dismissed as unphysical. Nonetheless, it is worth analyzing the resulting expressions to show that they lead to trivial cases or to the dust solution already studied in Ref. nos . Table 2 shows the following solutions for the PPF EMT considering Cases 1b and 2b: $A=D=0$, $B=C$, and $\mu=-\beta^{2}p$. Hence, Eqs. (5.1) to (5.4) are reduced to the following expressions, $\displaystyle-\frac{\partial\mu}{\partial t}-\beta\left(\frac{\partial\mu}{\partial x}+2p\frac{\partial\beta}{\partial t}\right)-\beta^{2}\frac{\partial p}{\partial t}-\beta^{3}\frac{\partial p}{\partial x}$ $\displaystyle=0,$ (5.18) $\displaystyle\frac{\partial B}{\partial y}$ $\displaystyle=0,$ (5.19) $\displaystyle\frac{\partial C}{\partial z}=\frac{\partial B}{\partial z}$ $\displaystyle=0.$ (5.20) Eqs. (5.19) and (5.20) imply that $B$ does not depend on both the $y$ and $z$ coordinates. Inserting the state equation $\mu=-\beta^{2}p$ (see Table 2) into Eq. (5.18) yields $p\beta^{2}\frac{\partial\beta}{\partial x}=0.$ (5.21) This expression can be separated in three subcases: $\beta=0$, $p=0$, $\partial\beta/\partial x=0$. According to Eq. (4.71) the last subcase implies $B=0$ and, hence, there is no Burgers equation. Let us discuss below the consequences of each of these subcases. #### 5.3.1 Subcase $\left[\,\beta=0\,\right]$ This solution reduces the warp drive metric (2.6) into a Minkowski flat spacetime. There is no warp bubble and, so, no warp drive. #### 5.3.2 Subcase $\left[\,B=0\,\right]$ Remembering Eqs. (4.48) and (4.72) this means that all EMT components are zero, even though, the equation of state $\mu=-\beta^{2}p$ remains. But, since there is no EMT both $\mu$ and $p$ must vanish, hence this case reduces itself to the dust solution nos . #### 5.3.3 Subcase $\left[\,p=0\,\right]$ Considering the equation of state $\mu=-\beta^{2}p$ then the matter density is also zero. The EMT for the PPF assumes the form given by Eq. (4.72), which implies in pressure without matter density, an unphysical situation that should either be dismissed or assumed to be just the dust solution. ## 6 Energy conditions The energy conditions are well known inequalities discussed by Hawking and Ellis HawkingEllis1973 that may be applied to the matter content in physical systems as boundary conditions and to test if the energy of such systems follows positive constraint values. This section aims at obtaining these conditions for the Alcubierre warp drive geometry considering the previously discussed EMTs for both the perfect fluid and PPF. Our focus will be on the main classical inequalities, namely, the weak, dominant, strong and null energy conditions. Our analysis starts with the PPF EMT defined in Eq. (4.1) in order to constrain its quantities so that the inequalities are satisfied for each of the four just named conditions. Then the same analysis for the perfect fluid is performed by reducing the results to the choice set by Eq. (4.49), but considering only the physically plausible cases. ### 6.1 Weak Energy Condition (WEC) The WEC requires that at each point of the spacetime the EMT must obey the following inequality, $T_{\alpha\sigma}\,u^{\alpha}u^{\sigma}\geq 0.$ (6.1) This is valid for any timelike vector $\textbf{u}\,(u_{\alpha}u^{\alpha}<0)$ as well as any null vector $\textbf{k}\,(k_{\alpha}k^{\alpha}=0$; see Section 6.4 below). For an observer with a unit tangent vector v at some point of the spacetime, the local energy density measured by any observer is non-negative HawkingEllis1973 . Considering the Eulerian (normal) observers from Alcubierre1994 with 4-velocity given by the expressions below $u^{\alpha}=(1,\beta,0,0)\,,\ \ u_{\alpha}=(-1,0,0,0),$ (6.2) and the EMT, given by Eq. (4.1), computing the expression $T_{\alpha\sigma}\,u^{\alpha}u^{\sigma}$ for the WEC yields, $\displaystyle T_{\alpha\sigma}\,u^{\alpha}u^{\sigma}$ $\displaystyle=T_{00}\,u^{0}u^{0}+2T_{01}\,u^{0}u^{1}+T_{11}\,u^{1}u^{1}$ $\displaystyle=\mu+\beta^{2}(p-2D+A)\,.$ (6.3) Let us now calculate the WEC for all perfect fluid and PPF Einstein equations solutions obtained in the previous sections as referred to in Table 2. #### 6.1.1 Cases 1a and 2a Let us start by considering the PPF. Substituting the equation of state (4.12) into Eq. (6.3) yields, $T_{\alpha\sigma}\,u^{\alpha}u^{\sigma}=\frac{A}{3}\geq 0.$ (6.4) So, the pressure $A$ must positive. Considering that both cases led to Eq. (4.58), the pressure $C$ must be negative. Besides, for the state equation (4.12) the density $\mu$ becomes positive if $A+p\leq 2D\,\,.$ (6.5) This inequality implies that it is possible for the fluid density to account for both negative and positive values in a local way depending on how the momentum and pressure components relate to each other in the spacetime. Remembering Eq. (4.49), the resulting inequality (6.4) indicates that the WEC is satisfied with the perfect fluid EMT for a positive pressure $p$. Considering Eqs. (3.30) and (3.31) that would mean a complex solution for $\beta$, possibility already envisaged in Section 3.2. Note that this is a general result for the perfect fluid with a cross term solution. #### 6.1.2 Cases 1b and 2b It has been shown above that in these two cases the PPF solutions result in $A=D=0$ and $\mu=-\beta^{2}p$, which substituted in Eq. (6.3) yield $T_{\alpha\sigma}\,u^{\alpha}u^{\sigma}=0$. Hence, the WEC is not violated for the PPF. For the perfect fluid, Table 1 shows that in these two cases the perfect fluid solutions reduce to the dust content solution for the warp drive metric, which trivially satisfies the WEC (Ref. nos, , p. 6). ### 6.2 Dominant Energy Condition (DEC) The DEC states that for every timelike vector $u_{a}$ the following inequality must be satisfied, $T^{\alpha\beta}\,u_{\alpha}u_{\beta}\geq 0,\quad\text{and}\quad F^{\alpha}F_{\alpha}\leq 0,$ (6.6) where $F^{\alpha}=T^{\alpha\beta}u_{\beta}$ is a non-spacelike vector. This condition means that for any observer the local energy density is non-negative and the local energy flow vector is non-spacelike. In any orthonormal basis the energy dominates the other components of the EMT, $T^{00}\geq\|T^{ab}\|,\ \text{for each}\ a,b.$ (6.7) Hawking and Ellis HawkingEllis1973 suggested that this condition must hold for all known forms of matter and that it should be the case in all situations. Evaluating this condition for PPF, the first DEC inequality is given by the following expression, $T^{\alpha\beta}\,u_{\alpha}u_{\beta}=T^{00}=\beta^{2}(A-2D+p)+\mu\geq 0,$ (6.8) whereas the second inequality $F^{\alpha}F_{\alpha}\leq 0$ yields, $\displaystyle(T^{\alpha\beta}\,u_{\beta})(T_{\alpha\beta}\,u^{\beta})=$ $\displaystyle-\mu^{2}-A^{2}\beta^{4}-\beta^{4}p^{2}+A^{2}\beta^{2}-(4\beta^{4}-\beta^{2})D^{2}$ $\displaystyle+2(2A\beta^{4}-A\beta^{2})D-2(A\beta^{2}-2\beta^{2}D)\mu$ $\displaystyle-2(A\beta^{4}-2\beta^{4}D+\beta^{2}\mu)p\leq 0.$ (6.9) As before, let us analyze next the solutions of the Einstein equations considering each group of subcases as referred to in Table 2. #### 6.2.1 Cases 1a and 2a The two important results for these two cases are Eqs. (4.57) and (4.58). Substituting them into the two DEC inequalities above result in the following expressions, $\displaystyle T^{\alpha\beta}\,u_{\alpha}u_{\beta}$ $\displaystyle=\frac{A}{3}\geq 0,$ (6.10) $\displaystyle(T^{\alpha\beta}\,u_{\beta})(T_{\alpha\beta}\,u^{\beta})$ $\displaystyle=\left[(A-D)\beta+\frac{A}{3}\right]\left[(A-D)\beta-\frac{A}{3}\right]\leq 0.$ (6.11) Eq. (6.10) is equal to Eq. (6.4), so this dominant energy condition is the same as the weak one. Eq. (6.11) separates in two inequalities, $\displaystyle-\frac{A}{3\left(D-A\right)}\leq\beta\leq\frac{A}{3\left(D-A\right)},$ (6.12) $\displaystyle-\frac{A}{3\left(A-D\right)}\leq\beta\leq\frac{A}{3\left(A-D\right)},$ (6.13) which can be rewritten as $\left\|\beta\right\|\leq\frac{1}{3}\left\|\frac{A}{D-A}\right\|.$ (6.14) In addition to the inequalities above, Eq. (6.11) also provides the following solutions, $\beta=\pm\frac{A}{3(A-D)}.$ (6.15) Equation (6.14) shows that $\beta$ is upper bounded, but since both $A$ and $D$ have no restraints, $\beta$ can be greater than unity. Remembering that $\beta=v_{s}(t)f[r_{s}(t)]$, the dominant energy condition is not violated for cases where the apparent warp bubble speed is greater than the speed of light. This result also depends on the relation between the pressure component $A$ and the momentum component $D$ as can be seen in Eqs. (6.12) and (6.13). As an example suppose that $A=D+1$ in Eq. (6.12), so that $-\frac{D+1}{3}\leq v_{s}(t)f[r_{s}(t)]\leq\frac{D+1}{3}.$ (6.16) The symmetry on the negative and positive values that $\beta$ may assume means the direction in which the warp bubble is moving on the $x$-axis on the spatial hyper surface. Note that, depending on the value of $D$, the warp bubble may assume the speed $v_{s}(t)$ greater than the speed of light. Regarding the perfect fluid, choosing as in Eq. (4.49) the DEC is given by the following expressions, $\displaystyle T^{\alpha\beta}\,u_{\alpha}u_{\beta}=\frac{p}{3}$ $\displaystyle\geq 0,$ (6.17) $\displaystyle(T^{\alpha\beta}\,u_{\beta})(T_{\alpha\beta}\,u^{\beta})=p^{2}$ $\displaystyle\geq 0,$ (6.18) which means that for the DEC to be satisfied a positively valued matter density is enough (see Table 1), although this also means a complex result for $\beta$ (see Section 3.2). #### 6.2.2 Cases 1b and 2b In these cases the PPF solutions, although considered unphysical, produced $B=C$, $A=D=0$ and $\mu=-\beta^{2}p$ (see Table 2). So, as shown by the expressions below the DEC is immediately satisfied, $\displaystyle T^{\alpha\beta}\,u_{\alpha}u_{\beta}=T^{00}=0,$ (6.19) $\displaystyle(T^{\alpha\beta}\,u_{\beta})(T_{\alpha\beta}\,u^{\beta})=0\,\,,$ (6.20) The perfect fluid is the dust solution for both cases (see Table 1), which is a vacuum solution and satisfies the inequalities above trivially. ### 6.3 Strong Energy Condition (SEC) For any timelike vector $u^{\alpha}$ the EMT must obey the following inequality for the SEC be valid, $\left(T_{\alpha\beta}-\frac{1}{2}T\,g_{\alpha\beta}\right)u^{\alpha}u^{\beta}\geq 0,$ (6.21) This requirement is stronger than the WEC and only makes sense in general relativistic framework because this theory is governed by the Einstein equations. These conditions imply that gravity is always attractive. To obtain expression for the SEC let us start with the scalar $T=g^{{}_{\alpha\beta}}T_{\alpha\beta}$, where $g^{\alpha\beta}$ is given by the Alcubierre warp drive metric and $T_{\alpha\beta}$ is the EMT for the PPF (Eq. 4.1). The result is shown below, $T=g^{{}_{\alpha\beta}}T_{\alpha\beta}=-\mu+A+B+C+\beta^{2}(2D-p-A).$ (6.22) Substituting the equation of state (4.12) into the expression above yields, $T=\frac{2}{3}A+B+C.$ (6.23) As shown in Section 6.1 above, Eq. (4.12) implies that $T_{\alpha\sigma}u^{\alpha}u^{\sigma}=A/3$ for $u^{\sigma}=(1,\beta,0,0)$. Inasmuch as $g_{\alpha\sigma}u^{\alpha}u^{\sigma}=-1$, then the SEC are given by the following expression, $\left(T_{\alpha\sigma}-\frac{1}{2}T\,g_{\alpha\sigma}\right)u^{\alpha}u^{\sigma}=\frac{2A}{3}+\frac{(B+C)}{2}\geq 0.$ (6.24) The particular cases, as summarized in Table 2, can now be discussed. #### 6.3.1 Cases 1a and 2a These cases resulted in the constraint $B=-\,C=A/3$. Eq. (6.24) may then be rewritten in the form below, $\left(T_{\alpha\sigma}-\frac{1}{2}T\,g_{\alpha\sigma}\right)u^{\alpha}u^{\sigma}=\frac{2A}{3}=2B\geq 0\,\,.$ (6.25) Clearly, for the SEC be satisfied it is only necessary that the pressure component $A$ must be non negative. Regarding the perfect fluid its respective SEC reads as, $\left(T_{\alpha\sigma}-\frac{1}{2}T\,g_{\alpha\sigma}\right)u^{\alpha}u^{\sigma}=\frac{2p}{3}\geq 0.$ (6.26) So, the SEC can be satisfied by the perfect fluid in these subcases by requiring a positive pressure $p$. #### 6.3.2 Cases 1b and 2b In these cases the PPF solutions stated that $A=D=0$ and $B=C$. Hence, the SEC satisfied under the constraint below, $\left(T_{\alpha\sigma}-\frac{1}{2}T\,g_{\alpha\sigma}\right)u^{\alpha}u^{\sigma}=B\geq 0,$ (6.27) despite providing unphysical solutions. The perfect fluid solutions are just the dust solution that is a vacuum solution nos , so satisfying the SEC trivially. ### 6.4 Null Energy Condition (NEC) The SEC and WEC are satisfied in the limit of the null observers with 4-velocities k. To satisfy the NEC the EMT must follow the inequality below, $T_{\alpha\sigma}\,k^{\alpha}k^{\sigma}\geq 0,\ \text{for any null vector}\ k^{\alpha}.$ (6.28) To calculate the NEC let us suppose the following null vector below, $k^{\alpha}=(a,b,0,0),$ (6.29) where the relation between $a$ and $b$ can be obtained by imposing the condition $k_{\alpha}k^{\alpha}=0,$ (6.30) that leads to $a^{2}\beta^{2}-2ab\beta-a^{2}+b^{2}=0,$ (6.31) whose roots for $a/b$ yield, $\frac{a}{b}=\frac{1}{\beta+1}\ \ \text{and}\ \ \frac{a}{b}=\frac{1}{\beta-1}.$ (6.32) Considering the results above, the NEC may be written as follows, $T_{\alpha\sigma}\,k^{\alpha}k^{\sigma}=T_{00}k^{0}k^{0}+2T_{01}k^{0}k^{1}+T_{11}k^{1}k^{1},$ (6.33) where $T_{\alpha\sigma}$ is the PPF EMT of Eq. (4.1). So, the null condition is given by the following expression, $T_{\alpha\sigma}\,k^{\alpha}k^{\sigma}=a^{2}\beta^{2}p-2ab\beta D+b^{2}A+a^{2}\mu\geq 0,$ (6.34) Substituting the two roots in Eqs. (6.32) into Eq. (6.34) results in expressions below, $\displaystyle\frac{b^{2}}{(\beta+1)^{2}}\,\left[\mu-\beta^{2}(2D-A-p)+2\beta(A-D)+A\right]\geq 0,$ (6.35) $\displaystyle\frac{b^{2}}{(\beta-1)^{2}}\,\left[\mu-\beta^{2}(2D-A-p)-2\beta(A-D)+A\right]\geq 0.$ (6.36) Next, as previously, we shall proceed to investigate first the Cases 1a and 2a and then 1b and 2b ones, as defined in Tables 1 and 2, for the PPF and the perfect fluid. #### 6.4.1 Cases 1a and 2a Considering the PPF first, the equation of state (4.57) is a solution of both cases (see Table 2), so the resulting NEC given by Eqs. (6.35) and (6.36) simplify to the following expressions, $\displaystyle\frac{b^{2}}{(\beta+1)^{2}}\,\left[\frac{4A}{3}+2\beta(A-D)\right]\geq 0,$ (6.37) $\displaystyle\frac{b^{2}}{(\beta-1)^{2}}\,\left[\frac{4A}{3}-2\beta(A-D)\right]\geq 0,$ (6.38) whose solutions for $\beta$ yield the NEC for the PPF EMT, as below, $\displaystyle\beta\geq\frac{2}{3}\frac{A}{D-A}\ \ \text{and}\ \ \beta\neq-1,$ (6.39) $\displaystyle\beta\leq\frac{2}{3}\frac{A}{A-D}\ \ \text{and}\ \ \beta\neq 1.$ (6.40) Remembering that $\beta=v_{s}(t)f(r_{s})$, then Eq. (6.39) states that the warp bubble can not assume the negative sign of the speed of light, but it may assume values above it. To verify that, we have to choose $A=D-1>2/3$ and $\beta>1$. Eq. (6.40) states that $\beta$ cannot assume the exact value of the speed of light and it is bounded by a superior value which can be greater than the speed of light. To have this, one just have to choose for example $A=D+1>2/3$. For the perfect fluid, according to Eq. (4.49) all pressures are equal. Hence, substituting both values for $a/b$ found in Eq. (6.34) into Eqs. (6.35) and (6.36) they then simplify to expressions below, $\displaystyle\frac{b^{2}}{(\beta+1)^{2}}\,\left(\mu+p\right)\geq 0,$ (6.41) $\displaystyle\frac{b^{2}}{(\beta-1)^{2}}\,\left(\mu+p\right)\geq 0.$ (6.42) These results mean that for the NEC is satisfied for the perfect fluid provided that $\mu+p\geq 0.$ (6.43) Remembering that $p=3\mu$ for the perfect fluid in theses two cases, then the NEC is fulfilled if the matter density does not have negative values. #### 6.4.2 Cases 1b and 2b These cases, although set as unphysical, we already concluded that the PPF relates its pressures by the following expressions: $A=D=0$ and $B=C$. Therefore, Eqs. (6.35) and (6.36) yield, $\displaystyle\frac{b^{2}}{(\beta+1)^{2}}\,\left(\mu+\beta^{2}p\right)\geq 0,$ (6.44) $\displaystyle\frac{b^{2}}{(\beta-1)^{2}}\,\left(\mu+\beta^{2}p\right)\geq 0.$ (6.45) So, the NEC establishes that $\mu+\beta^{2}p\geq 0$, which is readily satisfied since in these cases we already concluded that $\mu=-\beta^{2}p$, provided that $\beta\neq-1$ for the former and $\beta\neq 1$ for the latter. Regarding the perfect fluid, these cases reduce to the dust solution studied in nos , which is a vacuum solution and, therefore, immediately satisfies the NEC. Table 3 summarizes the results of all energy conditions for the PPF and perfect fluid EMTs with the Alcubierre warp drive metric. Cases (refer to Tables 1 and 2) \| Energy Conditions \| Results ---\|---\|--- 1a and 2a \| Weak \| $\begin{array}[]{ll}\\\\[-6.0pt] \text{Perfect fluid:}\ \displaystyle{p\geq 0}\\\\[8.0pt] \text{PPF:}\ \displaystyle{A\geq 0}\\\\[8.0pt] \end{array}$ Dominant \| $\begin{array}[]{ll}\\\\[-6.0pt] \text{Perfect Fluid:}\ \displaystyle{p\geq 0}\\\\[8.0pt] \text{PPF:}\ \displaystyle{A\geq 0}\ \ \text{and}\ \ \displaystyle{\left\|\beta\right\|\leq\frac{1}{3}\left\|\frac{A}{D-A}\right\|}\\\\[12.0pt] \end{array}$ Strong \| $\begin{array}[]{ll}\\\\[-6.0pt] \text{Perfect fluid}:\ \displaystyle{p}\geq 0\\\\[8.0pt] \text{PPF}:\ \displaystyle{A}\geq 0\\\\[8.0pt] \end{array}$ Null \| $\begin{array}[]{ll}\\\\[-6.0pt] \text{Perfect fluid}:\ \displaystyle{\mu+p\geq 0}\\\\[8.0pt] \text{PPF}:\ \displaystyle{\beta\geq\frac{2}{3}\frac{A}{D-A}\ \ \text{and}\ \ \beta\neq-1}\\\\[8.0pt] \text{PPF}:\ \displaystyle{\beta\leq\frac{2}{3}\frac{A}{A-D}\ \ \text{and}\ \ \beta\neq 1}\\\\[8.0pt] \end{array}$ 1b and 2b \| Weak \| immediately satisfied for perfect fluid and PPF. Dominant \| immediately satisfied for perfect fluid and PPF. Strong \| $\begin{array}[]{ll}\\\\[-6.0pt] \text{Perfect fluid: immediately satisfied.}\\\\[8.0pt] \text{PPF:}\ B\geq 0\\\\[2.0pt] \end{array}$ Null \| immediately satisfied for perfect fluid and PPF. Table 3: Summary of all energy conditions results for the perfect fluid and PPF EMTs with the Alcubierre warp drive spacetime geometry. ## 7 Further discussions This section discusses some points, and raises others, all related to the physics of the warp drive as suggested by the results presented in the previous sections. It aims at offering some thoughts that may be important in fostering further understanding on how a superluminal travel can be achieved. ### 7.1 Regulating Function and the Burgers Equation The regulating function (2.8) describes the shape of the warp bubble, but it is not uniquely determined. However, the integration the Einstein equations in both the perfect fluid and the PPF led to the appearance of generic functions in the dynamic equations which may end up connected to the regulating function, a situation that adds to its nonuniqueness. This means that physically feasible superluminal speeds will require the specification of these generic functions, possibly by boundary conditions. We shall show below an example of this situation using the dust solution. Considering Eqs. (2.3) and (2.10) the partial derivative of the Burgers equation (3.19) yields, $\frac{\partial\beta}{\partial t}=f\frac{d^{2}x_{s}}{dt^{2}}+v_{s}\frac{\partial f}{\partial r_{s}}\frac{\partial r_{s}}{\partial t},$ (7.1) where the simplified notation $f=f\left[r_{s}(t)\right]$ and $v_{s}=v_{s}(t)$ was adopted. Since, $\frac{\partial\beta}{\partial x}=v_{s}\frac{\partial f}{\partial r_{s}}\frac{\partial r_{s}}{\partial x},$ (7.2) the Burgers equation becomes, $f\frac{d^{2}x_{s}}{dt^{2}}+v_{s}\frac{\partial f}{\partial r_{s}}\frac{\partial r_{s}}{\partial t}+v_{s}^{2}f\frac{\partial f}{\partial r_{s}}\frac{\partial r_{s}}{\partial x}=h(t).$ (7.3) The partial derivative of Eq. (2.8) may be written as follows, $\frac{\partial f}{\partial r_{s}}=\frac{\sigma}{2\tanh{(\sigma R)}}\left\\{\operatorname{sech}^{2}[\sigma(r_{s}+R)]-\operatorname{sech}^{2}[\sigma(r_{s}-R)]\right\\}.$ (7.4) The partial derivatives of $r_{s}$ yield, $\displaystyle\frac{\partial r_{s}}{\partial t}$ $\displaystyle=\pm\frac{dx_{s}}{dt}=\pm v_{s}(t),$ (7.5) $\displaystyle\frac{\partial r_{s}}{\partial x}$ $\displaystyle=\pm 1,$ (7.6) and remembering that in the dust case $\beta=\beta(x,t)$, then $r_{s}(t)$ is given by (see Eq. 2.9), $r_{s}(t)=\sqrt{\left[x-x_{s}(t)\right]^{2}}=\lvert x-x_{s}(t)\rvert.$ (7.7) Considering the expressions above, Eq. (7.3) may be rewritten as follows, $f\frac{d^{2}x_{s}}{dt^{2}}\pm v_{s}^{2}\frac{\sigma F}{2\tanh{(\sigma R)}}\pm v_{s}^{2}f\frac{\sigma F}{2\tanh{(\sigma R)}}=h(t),$ (7.8) where $F(r_{s})\equiv\operatorname{sech}^{2}[\sigma(r_{s}+R)]-\operatorname{sech}^{2}[\sigma(r_{s}-R)].$ (7.9) Remembering that the regulating function $f[r_{s}(t)]$, as defined by Eq. (2.8), can be approximated by a top hat function when $\sigma\gg R$ (see Ref. nos , §2.1), in this limit Eq. (7.8) takes the following form inside the bubble, $\frac{d^{2}x_{s}}{dt^{2}}=h(t),$ (7.10) where $f=1$. Outside the bubble $f=0$ and then $h(t)=0$, which means plane shock waves described by the inviscid Burgers equation nos . Hence, the nonuniqueness of the shift vector $\beta=v_{s}(r_{s})f[r_{s}(t)]$ arises from the fact that not only the function $h(t)$ is arbitrary, but also because the regulating function $f[r_{s}(t)]$ only requires a top hat behavior with null values outside the bubble. So, any well behaved function that respects such constraints may be part of a solution of the Burgers equation. Moreover, from the energy conditions calculated for the PPF, as summarized in Table 3, one can see that $\beta$ plays a fundamental role in Cases 1a and 2a for the null and dominant energy conditions to be satisfied, which adds further physical constraints to its behavior. So, it is clear that the nonlinearity of the Einstein equations imply that the generic functions appearing in their integration become entangled with the regulating function in a non trivial manner. To extend this analysis to the perfect fluid and PPF contents require further constraints on the shift vector, something which at this stage would be done in an entirely arbitrary manner. Perhaps in the future that can be done under more physically plausible reasoning. ### 7.2 Anisotropic Fluids The PPF proposed in Section 4 aimed at offering an alternative EMT for solving the Einstein equations endowed with the warp drive metric. As we shall see below, The PPF can actually be seen as an anisotropic fluid with heat flux CarlEckart1940 . In general relativity the energy momentum tensor $T_{\mu\nu}$ represents the source of energy and momentum, where $T_{00}$ is the flow of energy across a surface of constant time (energy density), $T_{0i}$ is the energy flux across a surface in the $i$ direction (constant $x^{i}$), $T_{i0}$ is the momentum density and $T_{ij}$ is the momentum flux. If we choose a comoving frame of reference that moves with the same velocity as the fluid this means that particles in this fluid will have zero velocity and the flux of energy will be only through the flux of heat, and the momentum flux will be via some sort of dissipative phenomena such as viscosity, thermal radiation or even some sort of electromagnetic type of radiation. The general stress-energy tensor of a relativistic fluid can be written in the form below CarlEckart1940 ; Pimentel2016 , $T^{\alpha\beta}=\mu u^{\alpha}u^{\beta}+ph^{\alpha\beta}+u^{\alpha}q^{\beta}+u^{\beta}q^{\alpha}+\pi^{\alpha\beta},$ (7.11) where $h_{\alpha\beta}=g_{\alpha\beta}+u_{a}\,u_{b}\,,$ (7.12) projects tensors onto hypersurfaces orthogonal to $u^{\alpha}$, $\mu$ is the matter density, $p$ is the fluid static pressure, $q^{\alpha}$ is the heat flux vector and $\pi^{\alpha\beta}$ is the viscous shear tensor. The world lines of the fluid elements are the integral curves of the four-velocity vector $u^{\alpha}$. The heat flux vector and viscous shear tensor are transverse to the world lines, that is, $q_{a}\,u^{a}=0,\;\;\pi_{ab}\,u^{b}=0.$ (7.13) In terms of coordinates we can write the energy momentum tensor for a general fluid as $T_{\alpha\beta}=\begin{pmatrix}\varepsilon&q_{a}\\\ q_{b}&\pi_{ab}\end{pmatrix},$ (7.14) where $q_{a}$ is the three-vector heat flux vector, $\varepsilon$ is a scalar function and $\pi_{ab}$ is a three by three matrix viscous stress tensor, which is symmetric and traceless. Both $q_{a}$ and $\pi_{ab}$ have respectively three and five linearly independent components. Together with the density $\mu$ and the static pressure $p$, this makes a total of ten linearly independent components which is the number of linearly independent components in a four-dimensional symmetric rank two tensor. We noticed that the Einstein tensor components are highly non-linear for the warp drive metric and the off diagonal terms require those free parameters for a non over determined solution. For a non curved metric, i.e., the Minkowski metric $\eta_{\alpha\beta}$, the energy momentum tensor for a perfect fluid with anisotropic pressures can be written as $T_{\alpha\beta}=\begin{pmatrix}\mu&0&0&0\\\ 0&p_{x}&0&0\\\ 0&0&p_{y}&0\\\ 0&0&0&p_{z}\end{pmatrix}.$ (7.15) Isotropic static pressure means that $p_{x}=p_{y}=p_{z}=p$. The perfect fluid has no heat flux or dissipative phenomena, then $(q^{\alpha}=0,\pi^{\alpha\beta}=0)$. This special case with dust content is the well-known EMT, that is, $T^{\alpha\beta}=\mu u^{\alpha}u^{\beta}+ph^{\alpha\beta}=(\mu+p)u^{\alpha}u^{\beta}+pg^{\alpha\beta}\,.$ (7.16) The PPF proposed in Section 4 has the matrix form given by Eq. (4.1), which may be broken down as the sum of a perfect fluid in a warp drive background as given by Eq. (3.2), and a dissipative fluid with the heat flux four-vector given by, $q_{\alpha}=-\frac{1}{2}(q_{0},q_{1},0,0)$. Hence, $q_{\alpha}u_{\beta}+q_{\beta}u_{\alpha}=\begin{pmatrix}q_{0}&q_{1}&0&0\\\ q_{1}&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\end{pmatrix},$ (7.17) since the four-velocity for the moving frame is $u_{\alpha}=(-1,0,0,0)$. The isotropic term is given by, $\pi_{\alpha\beta}=\begin{pmatrix}\pi_{00}&0&0&0\\\ 0&\pi_{01}&0&0\\\ 0&0&\pi_{02}&0\\\ 0&0&0&\pi_{03}\end{pmatrix},$ (7.18) and $\mu+\beta^{2}p\to(\mu+\pi_{00}+q_{0})+\beta^{2}p\,.$ (7.19) So, the four parameters of the PPF are as follows, $A=\pi_{01}+p\,,$ (7.20) $B=\pi_{02}+p\,,$ (7.21) $C=\pi_{03}+p\,,$ (7.22) $D=p-\frac{q_{1}}{\beta}.$ (7.23) The tensor $\pi_{\alpha\beta}$ must be traceless, giving us one more equation to solve for the free parameters (anisotropic pressures) $\pi_{00}+\pi_{01}+\pi_{02}+\pi_{03}=0$ (7.24) From the above one can see that the warp drive metric endowed with the PPF allows the study of fluid anisotropy coupled with possible dissipative effects that could lead to a warp drive bubble. Perfect fluids are well known to be part of solutions for the Einstein equations, this being the case of the standard FLRW cosmological model that accounts for the expansion, isotropy and spatial homogeneity of the universe. On the other hand, anisotropic imperfect fluids offer a more complex source of gravitational effects, presenting dissipative processes, shear and bulk viscous pressures, interaction between fluids, radiation processes, electromagnetic interaction and even collision between particles, charged or not. These fluids are even known to avoid the big bang singularity in cosmological models murphy ; klimek ; msc ; p1 ; p2 . MacCallum MacCallum1979 discussed various ways of generating anisotropy such as the presence of electromagnetic fields, the presence of viscous terms and the anisotropic stresses due to the anisotropic expansion of a cloud of collisionless particles. Another way to account for viscosity, heat and energy flux is the interaction of two or more fluids Letelier1980 ; Bayin1982 . ### 7.3 Other aspects of warp drive physics The points presented above regarding the physical feasibility of superluminal travel with the Alcubierre spacetime geometry by no means exhaust this discussion. Several issues remain open, with each of them deserving separate studies that are beyond the scope of this paper, since in here we focused on the basic properties of the Einstein equations’ solutions of the warp drive metric with fluid content. With respect to the open issues, one can point out the amount of mass-energy density, exotic or not, necessary for the feasibility for the warp drive in the context of both the perfect fluid and the PPF, as well as quantum effects and the question of stability or instability in our solutions. These issues deserve further investigations and are the subject of ongoing research. ## 8 Conclusions In this work we have analyzed the Einstein equations for the Alcubierre warp drive metric having as gravity source two types of energy-momentum tensors (EMT) for fluid, namely the perfect fluid and the parametrized perfect fluid (PPF). The latter is defined by allowing the EMT pressure components of the perfect fluid to be different from one another and dependent on all coordinates. After obtaining the components of the Einstein tensor for the warp drive metric we calculated the dynamic equations for both fluids by solving the respective Einstein equations. Solutions were found in the form of various equations of state, and, by further imposing the null divergence for the EMTs, new constraints were also found for the various variables. The weak, dominant, strong and null energy conditions were also calculated, which implied further constraints upon the free quantities for the EMTs. For the perfect fluid these were on the matter density $\mu$ and pressure $p$. For the PPF that occurred on the pressure components $p,A,B,C$, the matter density for the fluid $\mu$ and the momentum component $D$. We found two main groups of solutions subcases possessing different conditions for each EMT. For the perfect fluid, one solution may be interpreted as requiring that the warp bubble can only be viable with negative matter density. The alternative interpretation is that the warp bubble is possible with positive matter density, but in this case the regulating function $f(r,s)$, which shapes the warp bubble, becomes a complex function. This comes from results allowing the possibility that the function $\beta=v_{s}(t)f(r_{s})$ may have complex solutions once the matter density is positive. Other results in both fluids reinforce our earlier finding in Ref. nos that the warp bubble necessary for generating superluminal velocities, or warp speeds, can be interpreted as a shock wave from classical fluid dynamics theory. We named four cases arising from the solutions of the Einstein equations: 1a, 1b, 2a and 2b. However, the Cases 1a and 2a are very similar, or equal, to each other, the same happening to Cases 1b and 2b. For this reason they were grouped together in the tables that summarize all results. Specifically, Cases 1b and 2b for the perfect fluid reduced the solutions to the ones found for dust content already studied in Ref. nos , that is, a vacuum solution unable to create a warp bubble, but which connects the warp metric to the inviscid Burgers equation, also yielding $\beta$ as a function of $t$ and $x$ coordinates. The null EMT divergence is satisfied and a continuity equation was also found (Eq. 5.13). Cases 1b and 2b for the PPF resulted in a equation of state of the form $\mu=-\beta^{2}p$, coordinate dependency of the $\beta$ function became $\beta=\beta(x,t)$ and a non-homogeneous Burgers equation (4.71) emerged. However, the pressures were constrained in such a way that this PPF EMT solution for the warp drive was dismissed as unphysical. Cases 1a and 2a for the perfect fluid produced an equation of state relating pressure and matter density given by $p=3\mu$. The $\beta$ function dependencies became $\beta=\beta(y,t)$ in the former case and $\beta=\beta(z,t)$ in the latter, and both produced a differential equation for $\beta$ that either requires negative density for $\beta$ to be a real valued function, or a positive matter density which then leads to a complex solution for $\beta$, which in turn leads to a complex regulating function $f(r_{s})$ whose possible real part would then be related to a physically viable warp bubble. Cases 1a and 2a for the PPF resulted in an equation of state relating almost all quantities, in the form $\mu=\beta^{2}(2D-A-p)+A/3$ which is valid for both cases. Coordinate dependency on $\beta$ became, respectively, resulted in $\beta=\beta(y,t)$ and $\beta=\beta(z,t)$. The function $\beta$ is also governed by the first order differential equations (4.73) and (4.74), respectively. The null EMT divergence $T{{}^{\alpha\sigma}}_{;\sigma}=0$ were calculated, producing further sets of very nonlinear differential equations constraining all quantities in the PPF which could be used, in principle, to determine all pressures and matter density in this fluid. For the perfect fluid, Cases 1a and 2a are reduced to a continuity equation (5.13) including the function $\beta$, which is interpreted as playing the role of the flow velocity vector field. Cases 1b and 2b reduced the PPF to either the trivial condition of Minkowski flat spacetime with no warp drive, or an EMT with all components being zero, that is a vacuum case. It has already been seen in Ref. nos that the Burgers equation appears connected to the dust solution of the warp drive metric, which is in fact a vacuum solution. Cases 1b and 2b of the perfect fluid became reduced to the dust solution, and a non-homogeneous form of the Burgers equation appears in these respective cases for the PPF, although the whole solutions in these cases were dismissed as unphysical. The solutions that presented themselves as the most plausible ones for creating warp speeds, 1a and 2a for both the perfect fluid and PPF, do not generate a Burgers equation. The weak, dominant, strong and null energy conditions were also studied in the context of the perfect fluid and PPF energy-momentum tensors for a warp drive metric. The resulting expressions were found to satisfy all conditions in the perfect fluid EMT. Regarding the PPF, specific expressions constraining its EMT quantities were obtained in order to satisfy these energy conditions, but they do not necessarily lead to the conclusion that negative matter density is always necessary for viable warp speeds, particularly because in the PPF the pressure $A$ must assume values equal to zero or positive. Summing up, the results of this paper indicate that warp speeds might be physically viable in the context of positive matter density as some solutions of the Einstein equations for both fluids keep open this possibility. Nevertheless, such a situation creates the additional issue in the perfect fluid context concerning the meaning of a possible complex regulating function in the warp metric, a result that may be interpreted as a caveat, or major stumbling block. Such difficulty does not appear to happen in the PPF scenario, although this fluid was considered here mainly as a hypothetical model whose aim was to investigate whether or not new possibilities arise in the solutions of the Einstein field equations considering more complex energy- momentum tensors having the Alcubierre warp drive metric. On this front it seems then that the initial conclusions about the unphysical nature of warp drive, or the impossibility of generating warp speeds, may not be not as stringent as initially thought, or, perhaps, not valid at all. ## Acknowledgments We are grateful to the referees for useful comments. E.M.C.A. thanks CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico), Brazilian scientific support federal agency, for partial financial support, grants numbers 406894/2018-3 and 302155/2015-5. ## References * (1) M. Alcubierre, The warp drive: hyper-fast travel within general relativity, Class. 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# VOTE400(Voide Of The Elderly 400 Hours): A Speech Dataset to Study Voice Interface for Elderly-Care Minsu Jang1, Sangwon Seo2, Dohyung Kim1, Jaeyeon Lee1 and Jaehong Kim1 and Jun-Hwan Ahn2 1Minsu Jang, Dohyung Kim, Jaeyeon Lee and Jaehong Kim are with Electronics and Telecommunications Research Institute, Daejeon-si, South Korea minsu at etri.re.kr2Sangwon Seo and Jun-Hwan Ahn is with MINDs Lab Inc., Kyungki-do, South Korea asdn9353 at mindslab.ai ###### Abstract This paper introduces a large-scale Korean speech dataset, called VOTE400, that can be used for analyzing and recognizing voices of the elderly people. The dataset includes about 300 hours of continuous dialog speech and 100 hours of read speech, both recorded by the elderly people aged 65 years or over. A preliminary experiment showed that speech recognition system trained with VOTE400 can outperform conventional systems in speech recognition of elderly people’s voice. This work is a multi-organizational effort led by ETRI and MINDs Lab Inc. for the purpose of advancing the speech recognition performance of the elderly-care robots. ## I INTRODUCTION Voice interface is the most intuitive, comfortable and universal interface for interacting with service robots. Recent advancement of commercial cloud-based speech-to-text (STT) services allowed devising a voice interface for service robots a very simple process of integrating a service API for STT into the robot SW system. While these commercial systems work very well with adults in the ages of between 20 and 60, it easily fails with voices from older adults aged 65 years or over. It is known that speech signals from older adults bring about difficulties for automated speech recognition as they tend to be imprecise in consonant pronunciation, include tremors, and have slower articulations [1]. In the need to develop a speech recognition system that are specialized to the speech signals from older adults, we built a speech dataset by collecting large-scale dialogue and read speech from older adults. The result of our effort is 400 hours of Korean speech data which we named as ’VOTE400 (Voice Of The Elderly 400 Hours) and open-sourced for any non-commercial research projects (https://ai4robot.github.io/mindslab-etri-vote400). ## II Dataset Description ### II-A Dataset Collection For recruiting and collecting voice data from older adults, we got assistance from a Korean governmental office, called Dok-Geo-No-In-Jong-Haap-Ji-Won Center (Ji-Won Center: http://www.1661-2129.or.kr/index.html) devoted to the support of older adults living alone. With the support from the Ji-Won Center, we could collect a large-scale dialog speech and read speech from a number of older adults across various regions of South Korea. #### II-A1 Dialog Speech To collect spontaneous speech data from older adults, we could utilize a support program of Ji-Won Center called Saa-Raang-It-Gi where social workers regularly visit elderly people’s homes for consulting on health-related issues and relieving loneliness. After explaining about the data collection experiment and getting consent of participation from the elderly, conversations between a social worker and an elderly were recorded using a smartphone. The recordings from these program sessions were sent to Ji-Won Center and a screening process was performed to remove every dialogue involving sensitive personal information. Then, a quality assurance process was followed to filter out speech segments incomprehensible by human listener due to imprecise pronunciation or significant noise. #### II-A2 Read Speech To amend the relatively low-quality of the dialog speech dataset, we launched another data collection process to acquire read speech from older adults. We built and utilized in the process a dedicated speech collection system, where a tablet-based client program presents a sentence to read to an elderly user; makes a recording and sends it to a server; where the recording is inspected to be accepted or not. In total, the number of unique sentences chosen to be read by participants was 2,250. These sentences were selected by considering how often these could be casually uttered by older adults in daily lives. TABLE I: Raw Data Collection of Dialog Speech Region(R) \| No. Participants \| Len. (hrs) ---\|---\|--- Seoul-si(SE) \| 620 \| 122 Busan-si(PS) \| 242 \| 90 Daegu-si(DG) \| 202 \| 33 Gwangju-si(GJ) \| 179 \| 63 Daejeon-si(DJ) \| 275 \| 66 Ulsan-si(WS) \| 80 \| 28 Goyang-si(GG) \| 335 \| 69 Gangwon-do(GW) \| 178 \| 45 Chungcheongbuk-do(CB) \| 252 \| 92 Chungcheongnam-do(CN) \| 317 \| 46 Jeollanam-do(JN) \| 323 \| 103 Gyeongsangbuk-do(GB) \| 378 \| 116 Total \| 3,381 \| 873 ### II-B Dialog Speech Data The total number of elderly participants is 3,381 and the total length of recordings is 873 hours. This is the result of collective efforts by regional senior citizens welfare institutes collaborating with the Ji-Won Center. Table I shows the regional distributions of all the participants and the length of recordings per region. After the screening and the QA process mentioned in the previous subsection, we finalized 300 hours of dialog speech to be included in the VOTE400 dataset. Transcription for every final speech data was done by human annotators. In VOTE400, we provide for a recording session in a WAV file. The audio format of the WAV file is as shown in table II. Every WAV file is accompanied by a transcription text file encoded in ISO-8859. The transcription does not include audio-text alignment information. TABLE II: VOTE400 Dialog Speech Audio Format Property \| Value ---\|--- Format. \| PCM Format Settings \| Little/Signed Codec ID \| 1 Bit Rate Mode \| Constant Bit Rate. \| 256 Channel(s) \| 1 Sampling Rate \| 16 kHz Bit Depth \| 16 bits The file name of each recording follows the pattern of <P-ID>_<G>_<A>_<R>_<DT>, where P-ID is a unique participant ID; G is a gender value (F for female, M for male), A is a age value, R is a regional code, and DT is the data-time of the recording session. Participants and speech audio statistics for VOTE400 dialog dataset are shown in table III and table IV. TABLE III: Demographics of VOTE400 Dialog Speech Region(R) \| No. Participants \| Age ($\mu/\sigma$) ---\|---\|--- Seoul-si(SE) \| 251(F:210,M:41) \| 78.98/5.13 Daegu-si(DG) \| 108(F:95,M:13) \| 80.33/6.08 Gyoungki-do(GG) \| 110(F:83,M:27) \| 80.17/5.41 Chungcheongnam-do(CN) \| 6(F:6,M:0) \| 77.00/3.69 Jeollanam-do(JN) \| 70(F:56,M:14) \| 80.76/4.90 Busan-si(PS) \| 160(F:137,M:23) \| 78.70/5.51 Daejeon-si(DJ) \| 96(F:72,M:24) \| 78.81/5.24 Gangwon-do(GW) \| 109(F:94,M:15) \| 80.07/5.50 Gyeongsangbuk-do(GB) \| 98(F:95,M:3) \| 80.87/4.48 Gwangju-si(GJ) \| 87(F:70,M:17) \| 79.39/5.77 Chungcheongbuk-do(CB) \| 17(F:17,M:0) \| 80.47/5.51 Ulsan-si(WS) \| 58(F:49,M:9) \| 76.97/4.48 Total \| 1,170(F:984,M:186) \| 79.47/5.37 TABLE IV: Speech Audio Statistics for VOTE400 Dialog Speech Region(R) \| Len.(secs) \| Len.($\mu/\sigma$) ---\|---\|--- Seoul-si(SE) \| 151,010 \| 601.63/239.83 Daegu-si(DG) \| 60,740 \| 562.42/228.14 Gyoungki-do(GG) \| 107,935 \| 981.23/357.19 Chungcheongnam-do(CN) \| 5,193 \| 865.62/293.98 Jeollanam-do(JN) \| 81,767 \| 1,168.10/294.85 Busan-si(PS) \| 200,207 \| 1,251.30/255.85 Gangwon-do(GW) \| 95,420 \| 875.42/158.18 Daejeon-si(DJ) \| 123,138 \| 1,282.70/293.83 Gyeongsangbuk-do(GB) \| 71,175 \| 726.28/308.80 Gwangju-si(GJ) \| 92,699 \| 1,065.52/276.53 Chungcheongbuk-do(CB) \| 20,135 \| 1,184.41/309.54 Ulsan-si(WS) \| 70,754 \| 1,219.90/254.43 Total \| 1,080,179 \| 923.23/380.17 ### II-C Read Speech Data The total number of elderly participants is 104 and the total length of recordings is 100 hours. Table VI shows the statistics of VOTE400 read speech data. Audio format of the VOTE400 read speech data is as shown in V, which is slightly different from the format of the dialog speech data. TABLE V: VOTE400 Read Speech Audio Format Property \| Value ---\|--- Format. \| PCM Format Settings \| Little/Signed Codec ID \| 1 Bit Rate Mode \| Constant Bit Rate. \| 705.6 kb/s Channel(s) \| 1 Sampling Rate \| 44.1 kHz Bit Depth \| 16 bits TABLE VI: Regional Distributions of VOTE400 Read Dataset Region(G) \| No. Persons \| No. Sent. \| Len.($\mu/\sigma$) ---\|---\|---\|--- Gyeongsangnam-do(GB) \| 20 \| 22,575 \| 3.18/1.38 Seoul-si(SE) \| 18 \| 19,220 \| 3.31/1.49 Jeollanam-do(JN) \| 21 \| 21,393 \| 3.36/1.52 Daegu-si(DG) \| 25 \| 26,950 \| 3.60/1.87 Gangwon-do(GW) \| 20 \| 21,676 \| 2.73/1.12 Total \| 104 \| 111,814 \| 3.25/1.54 The file name of read speech data follows the pattern of PID_<P-ID>_<DATE>_<SENTENCE-NO>_<R>, where P-ID is a unique participant ID, DATA is the date of recording, SENTENCE-NO is a serial number put to each of the recorded sentences, and R is the region code as shown in table VI. Each WAV file contains a single sentence, accompanied by a transcription text file encoded in EUC-KR. Though the number of sentences chosen and presented to the participants was originally 2,250, the final total number of unique sentences in VOTE400 read speech data is 7,832, due to mistakes and slight variations in real utterances by older adults. ## III Preliminary Experiment We conducted a preliminary experiment by training a MINDs Lab Inc.’s proprietary baseline speech recognizer(M), which is based on LSTM architecture, and estimating the STT accuracy using VOTE400. After fine-tuning the baseline with 50 hours each of dialog speech data and read speech data of VOTE400, a simple test with 100 sentences from different regions was performed and the results are as shown in table VII, along with the results when the sentences were tested on a commercial cloud-based STT engine(C). TABLE VII: STT Performance Test Results with VOTE400 Region(G) \| Gender \| Acc. M (%) \| Acc. C (%) ---\|---\|---\|--- SE \| M \| 90 \| 90 SE \| F \| 90 \| 80 GW \| M \| 80 \| 90 GW \| F \| 90 \| 80 DG \| M \| 70 \| 80 DG \| F \| 90 \| 80 GN \| M \| 90 \| 80 GN \| F \| 80 \| 80 JN \| M \| 70 \| 50 JN \| F \| 80 \| 60 ## IV Summary We described a Korean speech dataset VOTE400 which is collected entirely from older adults of more than 75 years old. VOTE400 contains 300 hours of dialogue speech data and 100 hours of read speech data, with proficient varieties in gender and regions. To our knowledge, VOTE400 is by far one of the largest voice datasets that is oriented to voices of the elderly. We hope that this dataset will be useful to study older adult’s voice features and realize voice technologies that work sufficiently well in elderly-care robotics. ## ACKNOWLEDGMENT This work was supported by the Institute of Information communi-cations Technology Planning Evaluation(IITP) grant funded by the Koreagovernment(MSIT) (No. 2017-0-00162, Development of Human-care RobotTechnology for Aging Society) ## References * [1] Vacher, M., Aman, F., Rossato, S. and Portet, F., 2015, August. Development of automatic speech recognition techniques for elderly home support: Applications and challenges. In International Conference on Human Aspects of IT for the Aged Population (pp. 341-353). Springer, Cham.
# Improved Wall-Normal Derivative Formulae for Anisotropic Adaptive Simplex- Element Grids Hiroaki Nishikawa National Institute of Aerospace, Hampton, VA 23666, USA Research Fellow (hiro@nianet.org), 100 Exploration Way, Hampton, VA 23666 USA, Associate Fellow AIAA ###### Abstract In this paper, we explore methods for computing wall-normal derivatives used for calculating wall skin friction and heat transfer over a solid wall in unstructured simplex-element (triangular/tetrahedral) grids generated by anisotropic grid adaptation. Simplex-element grids are considered as efficient and suitable for automatic grid generation and adaptation, but present a challenge to accurately predict wall-normal derivatives. For example, wall- normal derivatives computed by a simple finite-difference approximation, as typically done in practical fluid-dynamics simulation codes, are often contaminated with numerical noise. To address this issue, we propose an improved method based on a common step-length for the finite-difference approximation, which is otherwise random due to grid irregularity and thus expected to smooth the wall-normal derivative distribution over a boundary. Also, we consider using least-squares gradients to compute the wall-normal derivatives and discuss their possible improvements. Numerical results show that the improved methods greatly reduce the noise in the wall-normal derivatives for irregular simplex-element grids. ## 1 Introduction Towards fully automated computational fluid dynamics (CFD) simulations, there is an increasing interest in developing robust and accurate discretizations and solvers for unstructured simplex-element grids with anisotropic grid adaptation [WhiteNishikawaBaurle_scitech2020,Kleb_etal_aiaa2019-2948,Nishikawa_scitech2020,nishikawa_centroid:JCP2020,uns_grid_adaptation_aiaa2018-1103,alauzet- loseille-decade-aniso-adapt-cfd]. While fully-automated CFD simulations have recently been successfully demonstrated for practical three-dimensional problems [Kleb_etal_aiaa2019-2948], improvements are still desired in the discretization method especially in terms of accuracy. For example, computations of derivatives such as the viscous/heat fluxes and the vorticity are known to suffer from numerical noise on highly irregular and skewed simplex-element grids [liu_nishikawa_aiaa2016-3969]. Impact of numerical noise may be reduced in integrated quantities (e.g., drag coefficient), but when accuracy is required for a pointwise quantity, e.g., heating rate at a stagnation point, numerical noise can be a serious problem, causing a large error and a great degree of uncertainty in the prediction. In order to develop a reliable automated CFD solver, it is necessary to be able to accurately predict pointwise derivatives, especially at wall boundaries, on irregular anisotropic simplex-element grids. There have been efforts to improve derivative accuracy on unstructured grids: derivative variables introduced in target equations [nishikawa_hyperbolic_poisson:jcp2020,liu_nishikawa_aiaa2016-3969], improved least-squares methods [NovelGradStencil:CF2018,SozerChristophCetin:AIAA2014,ShimaKitamuraHaga_AIAAJ2013], and implicit gradient methods [WangRenPanLi:JCP2017,nishikawa_igg:JCP2018]. All these methods are promising for improving derivative accuracy, but they are not specifically designed to accurately estimate derivatives at a boundary and thus further improvements may be achieved by developing a more specific technique. In this work, we focus on wall-normal derivatives and explore various methods for accurately estimating the wall-normal derivatives relevant to the viscous stresses and heat fluxes at a wall, as a post-processing step for a given numerical solution on adaptive simplex-element grids. A popular technique used in practical CFD codes for computing the wall-normal derivatives is a one-sided finite-difference formula applied in the wall- normal direction with the numerical solution in a nearby node/cell and a boundary condition value. For example, it provides an approximation to $\partial u/\partial n$ that dominates the viscous stress at a wall, where $u$ is a wall-parallel velocity components and $n$ is the coordinate normal to the wall. It can provide an accurate approximation on structured grids, but will introduce numerical noise on irregular simplex-element grids [WhiteNishikawaBaurle_scitech2020]. We conjecture that the numerical noise in the derivative is due to the irregularity of the spacing used in the finite- difference formula. That is, the distance between the center of a cell adjacent to a boundary and a wall-boundary face centroid varies along a boundary in adaptive simplex-element grids. Then, we propose to keep the same distance for the finite-difference formula applied at all wall-boundary faces. Numerical experiments show that the modified finite-difference method greatly improves the skin friction distribution over a flat plate in a laminar flow, where the distance is determined as $\eta=0.5$ at all boundary faces, where $\eta$ is a normalized coordinate in the Blasius boundary-layer solution. In order to extend this technique to more general flows over complex geometries, it is necessary to develop a method for estimating a universal distance for the finite-difference formula. Another post-processing technique can be devised based on gradients computed by a least-squares method at nodes or cells. Following a recent study [WhiteNishikawaBaurle_scitech2020], where heating rate prediction over a flat plate in a hypersonic turbulent flow has been shown to be greatly improved by the face-averaged nodal-gradient (F-ANG) approach applied to the finite-volume discretization [WhiteNishikawaBaurle_scitech2020,NishikawaWhite_FANG:jcp2020], we employ the F-ANG approach in this study, but only consider the post- processing technique for estimating the wall-normal derivatives for a given numerical solution. Discussions on the impact of the gradient approach (used in the discretization) on wall heating prediciton are given in a companion paper [NishikawaWhite_scitech2021-xxxx]. In the F-ANG method, gradients are computed and stored at nodes while numerical solutions are stored at cells. Therefore, the wall-normal derivatives are already available at wall nodes. However, these derivatives can be noisy for irregular anisotropic simplex- element grids. A simple method for reducing the noise would be to average the nodal gradients, and there are two possible techniques: face and cell averages. We compare these averaged gradients for estimating the wall-normal derivatives. Numerical results indicate that the cell-averaged nodal-gradients can greatly reduce the noise in the wall-normal derivatives, but not as much as the improved finite-difference formula. ## 2 Finite-Volume Discretization We consider a cell-centered finite-volume discretization for the Navier-Stokes equations integrated over a computational cell $j$ (see Figure 1) with the midpoint rule: $\displaystyle{\bf Res}_{j}=\sum_{k\in\\{k_{j}\\}}{\bf\Phi}_{jk}A_{jk},$ (2.1) where $\\{k_{j}\\}$ is a set of neighbors of the cell $j$, $A_{jk}$ is the length of the face across $j$ and $k$, and ${\bf\Phi}_{jk}$ is a numerical flux. In this work, the Roe [Roe_JCP_1981] flux is used for the inviscid terms, and the alpha-damping flux [nishikawa:AIAA2010] for the viscous terms. The solution values are stored in the primitive variables ${\bf w}=(\rho,{\bf v},T)$, at cells as point values at the centroid [nishikawa_centroid:JCP2020], where $\rho$ is the density, ${\bf v}$ is the velocity vector, and $T$ is the temperature. Second-order accuracy is achieved by the linearly-exact flux quadrature (i.e., the midpoint rule) and the numerical flux evaluated with linearly reconstructed solutions. In this work, we focus on the F-ANG approach [WhiteNishikawaBaurle_scitech2020,NishikawaWhite_FANG:jcp2020], which has been shown to yield significantly more accurate wall-derivatives than a conventional finite-volume scheme, but still involves mild oscillations in the derivatives on irregular simplex-element grids. In the F-ANG approach, we compute the solution gradients at nodes using solution values stored at cells by a linear least-squares method. Then, the nodal gradients are used in the linear reconstruction, averaged over the nodes of a face: $\displaystyle{\bf w}_{L}={\bf w}_{j}+\nabla{\bf w}_{f}\cdot\Delta{\bf x}_{jm},\quad{\bf w}_{R}={\bf w}_{k}+\nabla{\bf w}_{f}\cdot\Delta{\bf x}_{km},$ (2.2) where $\nabla{\bf w}_{f}$ denotes the face-averaged gradient: $\displaystyle\nabla{\bf w}_{f}=\frac{1}{2}\left(\nabla{\bf w}_{jk}^{\ell}+\nabla{\bf w}_{jk}^{r}\right),$ (2.3) where the superscripts $\ell$ and $r$ denote the left and right nodes of the face $[j,k]$ (see Figure 1). This approach has various advantages especially for simplex-element grids: e.g., no interpartition is required for gradients, the residual stencil is greatly reduced. See [WhiteNishikawaBaurle_scitech2020,NishikawaWhite_FANG:jcp2020] for further details. Figure 1: Stencil for cell-centered finite-volume discretization. ## 3 Methods for Computing Wall-Normal Derivatives In this section, we describe methods for computing a wall-normal derivative, taking the velocity derivative $\partial u/\partial n$ as an example, where $u$ is the wall-parallel component of the velocity and $n$ denotes the coordinate normal to the wall. We assume that the value at a wall $u_{b}$ is known, e.g., $u_{b}=0$ for a no-slip condition. The wall-normal derivative will be computed at a boundary node or at a boundary face, depending on the method. Figure 2: Geometry of an irregular triangular grid over a boundary. Black filled circles are boundary nodes; smaller circles are the centroids of triangles adjacent to the boundary. ### 3.1 Nodal gradients The wall-normal derivative is computed at a boundary node $i$ by projecting the nodal gradient $\nabla u_{i}$ in the direction of wall normal (see Figure 2): $\displaystyle\left.\frac{\partial u}{\partial n}\right\|_{i}=\nabla u_{i}\cdot\hat{\bf n}_{i},$ (3.1) where $\hat{\bf n}_{i}$ denotes a unit vector at the node $i$ normal to the wall pointing towards the interior domain. This is perhaps the simplest method since the nodal gradients are already available in the F-ANG method. ### 3.2 Face-averaged nodal gradients The wall-normal derivative is computed at a face $j$ with a face-averaged gradient: $\displaystyle\left.\frac{\partial u}{\partial n}\right\|_{j}=\overline{\nabla}u_{j}^{face}\cdot\hat{\bf n}_{j},$ (3.2) where $\overline{\nabla}u_{j}^{face}$ is the average of the nodal gradients over the boundary face $j$. In Figure 2, $\overline{\nabla}u_{j}^{face}$ is obtained by averaging the nodal gradients at the left and right nodes of the boundary face $j$. ### 3.3 Cell-averaged nodal gradients The wall-normal derivative is computed at a face $j$ with a cell-averaged gradient: $\displaystyle\left.\frac{\partial u}{\partial n}\right\|_{j}=\overline{\nabla}u_{j}^{cell}\cdot\hat{\bf n}_{j},$ (3.3) where $\overline{\nabla}u_{j}^{cell}$ is the average of the nodal gradients over a cell adjacent to the face $j$. In Figure 2, $\overline{\nabla}u_{j}^{cell}$ is obtained by averaging the nodal gradients at the three nodes of the triangular cell $c$. (a) Finite-difference I. (b) Finite-difference II. (c) Finite-difference III. Figure 3: Geometries for the finite-difference wall-normal derivative formulae. ### 3.4 Finite-difference I The wall-normal derivative can be computed at a face $j$ by the following finite-difference formula: $\displaystyle\left.\frac{\partial u}{\partial n}\right\|_{j}=\frac{u_{c}-u_{b}}{({\bf x}_{c}-{\bf x}_{j})\cdot\hat{\bf n}_{j}},$ (3.4) where $u_{c}$ is a solution value stored at the cell $c$ adjacent to the boundary face $j$, $u_{b}$ is a known boundary value (e.g., from a boundary condition), ${\bf x}_{c}$ is the centroid position of the cell $c$, ${\bf x}_{j}$ is the centroid of the boundary face $j$. See Figure 3(a). This formula estimates the derivative at a point where the cell-center ${\bf x}_{c}$ is projected on the boundary, not at ${\bf x}_{j}$. ### 3.5 Finite-difference II The wall-normal derivative can be estimated at the face centroid by $\displaystyle\left.\frac{\partial u}{\partial n}\right\|_{j}=\frac{u_{c^{\prime}}-u_{b}}{({\bf x}_{c}-{\bf x}_{j})\cdot\hat{\bf n}_{j}},$ (3.5) where $u_{c^{\prime}}$ is a solution extrapolated from the cell center ${\bf x}_{c}$ to the point right above the face centroid (see Figure 3(b)): $\displaystyle u_{c^{\prime}}=u_{c}+\nabla u_{c}\cdot({\bf x}_{c^{\prime}}-{\bf x}_{c}),\quad{\bf x}_{c^{\prime}}={\bf x}_{j}+[({\bf x}_{c}-{\bf x}_{j})\cdot\hat{\bf n}_{j}]\hat{\bf n}_{j}.$ (3.6) This approximation is exact for linear functions at the face centroid. ### 3.6 Finite-difference III In the previous methods, the finite-difference formula is applied with varying step-length as the distance between the cell center and the face center in the normal direction changes randomly on irregular simplex-element grids. Hence, the methods are strongly dependent on the grid geometry, which is not suitable for adaptive simplex-element grids. To minimize the dependence on the grid geometry, we generalize the previous formula as $\displaystyle\left.\frac{\partial u}{\partial n}\right\|_{j}=\frac{u_{p}-u_{b}}{\overline{h}},$ (3.7) where $u_{p}$ is a solution extrapolated from the cell center ${\bf x}_{c}$ to the point at a certain distance $\overline{h}$ from the face center in the wall normal direction (see Figure 3(c)): $\displaystyle u_{p}=u_{c}+\nabla u_{c}\cdot({\bf x}_{p}-{\bf x}_{c}),\quad{\bf x}_{p}={\bf x}_{j}+\overline{h}\,\hat{\bf n}_{j},$ (3.8) where $\overline{h}$ is a global constant or varies along a wall but constant in a transformed normal coordinate. This approximation is also exact for linear functions at the face centroid. This formula does not depend strongly on the cell centroids of the adjacent cells. Whether in the physical or transformed coordinate, the above formula keeps the same step-length of the finite-difference formula over a wall and thus is expected to produce a smoother derivative (and its error) distribution. In this work, we investigate the effects of the use of a common step-length $\overline{h}$ and explore estimation techniques of $\overline{h}$ for various viscous-flow problems on adaptive grids. Here, we provide preliminary results for a laminar flat-plate computation with the height $\overline{h}$ determined as described below. ### 3.7 Finite-difference with face-area-weighted centroids Finite-difference III requires a common length to be defined. For a laminar flow over a flat plate, a normalized wall-normal distance, as we will describe later, has been found to serve well for subsonic flows. However, such an estimate is not applicable to high-speed flows and practical turbulent flows over complex geometries. As a practical technique, we consider the face-area- weighted centroid [nishikawa_centroid:JCP2020], which could reduce the centroid-height variation along a boundary. The resulting method is similar to the finite-difference II but ${\bf x}_{c^{\prime}}$ is replaced by the face- area-weighted centroid: $\displaystyle{\bf x}_{c^{\prime}}=\frac{\displaystyle\sum_{k\in\\{k_{j}\\}}\hat{A}_{jk}^{p}{\bf x}_{jk}}{\displaystyle\sum_{k\in\\{k_{j}\\}}\hat{A}_{jk}^{p}},\quad\hat{A}_{jk}=\frac{{A}_{jk}}{\displaystyle\max_{k\in\\{k_{j}\\}}{A}_{jk}},$ (3.9) where${A}_{jk}$ is the area (length in two dimensions) of the face across $j$ and $k$, ${\bf x}_{jk}$ is the face centroid, and $p=2$. ## 4 Numerical Results ### 4.1 Two-dimensional laminar flat plate at $M_{\infty}=0.15$, $Re_{\infty}=10^{6}$ For a laminar flat plate, we consider the normalized wall-normal coordinate $\eta$: $\displaystyle\eta=\frac{y}{x}\sqrt{Re_{x}},\quad Re_{x}=\frac{\rho_{\infty}U_{\infty}x}{\mu_{\infty}},$ (4.1) where $x$ is the distance from the leading edge, $\rho_{\infty}$, $U_{\infty}$, $\mu_{\infty}$ are free stream density, flow speed, and viscosity. For the finite-difference III, we fix the height to be $\eta=0.5$, which is translated into the $y$-coordinate as $\displaystyle\overline{h}=\eta\sqrt{\frac{x_{j}}{Re_{\infty}}},$ (4.2) where $\eta=0.5$, $x_{j}$ is the $x$-coordinate of the face centroid, and $Re_{\infty}$ is the free stream Reynolds number per unit length. Note that we compute the wall-normal velocity derivative at boundary face centroids and thus $x_{j}$ will not be zero. As can be seen from the formula, the height $\overline{h}$ changes in the physical coordinate parabolically but remains constant in the normalized wall-normal coordinate $\eta$. In the results, this method is denoted by FD-III($\eta=0.5$). For a more general flow problem, this approach may not be suitable. Therefore, we also consider using a fixed length over the wall in the physical space: $\displaystyle\overline{h}=\eta\sqrt{{Re_{\infty}}},$ (4.3) which corresponds to using the estimated distance with $\eta=0.5$ at $x_{j}=1.0$ everywhere. In this case, $\overline{h}$ is a global constant. An adaptive grid was generated by the open-source tool refine [park_darmofal:AIAA2008-917] developed by Mike Park at NASA Langley Research Center and available at https://github.com/nasa/refine. Adaptation was performed with the multiscale metric technique to control interpolation error in L2-norm [ugawg-aiaa-verification] with Mach Hessian computed at nodes (by a least-squares method) from the numerical solution obtained at cells with the F-ANG finite-volume solver for the Navier-Stokes equations. The resulting adaptive grid is shown in Figures 4(a) and 4(b). As can be seen, it is a fully irregular grid towards the flat plate located at the bottom in $x\in[0,2]$. Note that we already have a numerical solution on this grid as shown by Mach contours in Figure 4(b). Then, the skin friction is computed at the flat plate as $\displaystyle C_{fx}=(\tau_{xx},\tau_{xy})\cdot\hat{\bf n}\frac{2}{M_{\infty}^{2}}=\mu\left(\frac{2}{3}\frac{\partial u}{\partial x}-\frac{1}{3}\frac{\partial v}{\partial y},\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\cdot\hat{\bf n}\frac{2}{M_{\infty}^{2}}=\mu\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\frac{2}{M_{\infty}^{2}},$ (4.4) where $u$ and $v$ are the $x$\- and $y$-components of the velocity, the factor $1/M_{\infty}^{2}$ comes from the nondimensionalization (i.e., the velocity is nondimensionaliezd by the free stream speed of sound), and $\hat{\bf n}$ is either $\hat{\bf n}_{i}$ or $\hat{\bf n}_{j}$, depending on where the gradients are computed. The term $\partial v/\partial x$ is ignored in the finite-difference methods, but retained in the LSQ gradient methods. Results are shown in Figures 4(c)-4(i), where numerical skin-friction distributions are compared with that of the Blasius solution. All numerical skin-friction distributions are close to the theoretical distribution on average, but significant differences are observed in the noise level. First, the FD-I and FD-II methods produce noisy skin-friction distributions as shown in Figures 4(c) and 4(d). On the other hand, as expected, the proposed methods produce much smoother distributions as shown in Figures 4(e) and 4(f). Note that the choice of $\overline{h}$ has no major impact on the skin-friction distribution: Figure 4(e) for $\eta=0.5$, and Figure 4(f) for a global constant corresponding to $\eta=0.5$ and $x=1$. These results indicate that it is the uniformness of the step-length, whether in $\eta$ or $y$, that affects the smoothness of the skin friction distribution. This is encouraging since the method can be extended to more general problems by developing a method for estimating a single value of the step-length $\overline{h}$ in a given physical problem. For the LSQ gradient methods, the direct use of the nodal gradients leads to a noisy skin-friction distribution as shown in Figure 4(g). A slightly smoother distribution is obtained with the face-averaged nodal gradients as shown in Figure 4(h). The smoothest distribution is obtained with the cell-averaged nodal gradients as shown in Figure 4(i). The results indicate that averaging with more nodal gradients leads to a smoother distribution, but not as smooth as those obtained with the finite-difference III. Note that the gradients are linearly exact no mater how they are averaged, and the averaging operator acts as a kind of filtering to remove high-frequency contents from the underlying gradient approximation. (a) Adaptive grid (3457 nodes). (b) Zoomed-in view of the grid (with a scaled $y$-coordinate) and Mach contours. (c) FD-I. (d) FD-II. (e) FD-III ($\eta=0.5$). (f) FD-III ($\eta=0.5$, $x=1$). (g) Nodal gradients. (h) Face-averaged gradients. (i) Cell-averaged gradients. Figure 4: Results for a laminar flow over a flat plate at $M_{\infty}=0.15$ and $Re_{\infty}=10^{6}$ on an adaptive triangular grid. The flat plate is located at $y=0$ in $x\in[0,2]$. The skin friction coefficient $C_{fx}$ is plotted at wall face centers, except for the figure (g). ## 5 Concluding Remarks We have proposed improved wall-normal derivative formulae for adaptive anisotropic simplex-element grids, based on the simple idea of unifying the finite-difference distance over a wall. The improved formulae have been demonstrated for a laminar flow over a flat plate to reduce the amount of numerical noise in the skin friction distribution. In the future work, these formulae will have to be extended to three dimensions and to turbulent flows. ## Acknowledgments This work was supported by the Hypersonic Technology Project, through the Hypersonic Airbreathing Propulsion Branch of the NASA Langley Research Center, under Contract No. 80LARC17C0004. The author would like to thank Mike Park (NASA Langley Geometry Laboratory) for his assistance with the use of refine. ## References * [1] White, J., Nishikawa, H., and Baurle, R., “A 3-D Nodal-Averaged Gradient Approach for Unstructured-grid Cell-centered Finite-volume Methods for Application to Turbulent Hypersonic Flow,” SciTech 2020 Forum, AIAA Paper 2020-0652, Orlando, FL, 2020. * [2] Kleb, W. L., Park, M. A., Wood, W. A., Bibb, K. L., Thompson, K. B., and Gomez, R. 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# Utilizing uncertainty estimation in deep learning segmentation of fluorescence microscopy images with missing markers ###### Abstract Fluorescence microscopy images contain several channels, each indicating a marker staining the sample. Since many different marker combinations are utilized in practice, it has been challenging to apply deep learning based segmentation models, which expect a predefined channel combination for all training samples as well as at inference for future application. Recent work circumvents this problem using a modality attention approach to be effective across any possible marker combination. However, for combinations that do not exist in a labeled training dataset, one cannot have any estimation of potential segmentation quality if that combination is encountered during inference. Without this, not only one lacks quality assurance but one also does not know where to put any additional imaging and labeling effort. We herein propose a method to estimate segmentation quality on unlabeled images by ($i$) estimating both aleatoric and epistemic uncertainties of convolutional neural networks for image segmentation, and ($ii$) training a Random Forest model for the interpretation of uncertainty features via regression to their corresponding segmentation metrics. Additionally, we demonstrate that including these uncertainty measures during training can provide an improvement on segmentation performance. ## 1 Introduction Recent progress in Convolutional Neural Networks (CNNs) for image segmentation has enabled many possibilities in biomedical tissues analysis using fluorescence microscopy (FM) [1], which images fluorescent dyes (called _markers_) that target different cells or anatomical networks in biological specimens [2]. Different markers are combined and registered as image channels, which are for our purposes equivalent to modalities in other imaging techniques such as with different sequences in MRI. However, the maximum number of markers in a FM sample is limited due to their spectral overlap, thus requiring different combinations for each biological study. Furthermore, the lack of immunostaining consistency due to sample preparation difficulties, together with limited sample availability, often leads to datasets of images with missing markers, which have been challenging for traditional CNNs. To address this, a modality sampling and attention approach named _Marker Sampling & Marker Excite_ (_MS-ME_) was proposed in [3], which allows for training and inference with a single model on datasets with heterogeneous marker combinations. Although _MS-ME_ permits predictions on marker combinations unseen during training, one does not know in advance what segmentation quality to expect for such unseen combinations without any existing labeled data. Such quality estimators would be very valuable not only in predicting potential future performance, but also in deciding where to invest additional imaging and labeling effort. Uncertainty estimation in deep neural networks with different Bayesian approximations was shown to improve model predictions, either by explaining this within the loss function or by aggregating predictions from ensembles [4]. Such estimations have been successfully applied to medical image segmentation [5]. _Epistemic_ uncertainty accounts for lack of confidence in the parameters of a model, i.e. uncertainty that can be reduced with additional labeled data. This can be estimated using so-called Dropout layers [6] both at training and inference time, known as Monte Carlo (MC) dropouts [7, 8]. _Aleatoric_ uncertainty captures the inherent noise in the observations, i.e. uncertainty that cannot be reduced with additional data; and it can be estimated with the inclusion of a stochastic loss as proposed in [4]. We herein study the use of the above uncertainty estimation tools to design a method for the estimation of segmentation quality in the above- described problem setting of heterogeneous FM marker combinations. ## 2 Methods As illustrated in Fig. 1, we extend _MS-ME_ to also estimate aleatoric and epistemic uncertainties, using the summary statistics of which we train a regression Random Forest (RF) to predict quantitative segmentation outcomes. Fig. 1: Illustration of our proposed framework for predicting segmentation quality with missing markers. ### 2.1 Learning from images with missing markers FM images are formed by different combinations of markers $m_{G}$, with $G\subseteq\\{1,\dots,K\\}$, and $K$ the number of possible markers represented as channels in an image $x\in\mathbb{R}^{h\times w\times\|G\|}$. We denote the combination as the successive indexes of the markers it contains, e.g. $m_{24}$ is a combination of markers 2 and 4. The challenge of different marker combinations in FM images for both in training and testing was addressed in [3] using _MS-ME_. Marker sampling (MS) refers to MC dropout of modalities at training time, to make testing generalizable to different availability of markers. For Marker Excite (ME), a feature-wise attention module that has 2 fully-connected layers and a one-hot encoded input of available modalities for the sample is added at different layers of a UNet [1]. For a detailed network structure and implementation details, see [3]. ### 2.2 Uncertainty estimation in CNN-based segmentation Different uncertainties are estimated following the frameworks proposed in [4, 5] and included within a _MS-ME_ model $f$ that is applied on an input image $x$ to predict its corresponding segmentation $\hat{y}=\textit{softmax}(z)$, where $z$ are the logits resulting from the model: $z=f(x)$. To estimate epistemic uncertainty $u_{\mathrm{e}}$ of $x$, MC Dropout is employed at different layers of a CNN $f_{\mathrm{e}(p)}$ with probability $p$ both at training and inference. Since the output of the network is stochastic, $\hat{y}$ and $u_{\mathrm{e}}$ are estimated respectively as the mean and standard deviation (SD) of $T$ samples from $f_{\mathrm{e}(p)}(x)$. When explicitly stated, we add MC Dropout only after the last layer as proposed in [7]. Aleatoric uncertainty $u_{\mathrm{a}}$ is calculated by explicitly adding a predictive variance $u_{\mathrm{a}}$ to our model output, i.e. $[z,\,u_{a}]=f_{\mathrm{a}}(x)$ . This model $f_{\mathrm{a}}$ is trained with a stochastic cross-entropy loss that adds a noise component $\epsilon\sim\mathcal{N}(0,\,I)$ to multiple ($T$) model predictions, so that the loss evaluates their mean: $\hat{y}^{\text{training}}=\frac{1}{T}\sum_{t}^{T}\textit{softmax}(z+u_{a}\epsilon_{t})$ At inference, both $\hat{y}$ and $u_{a}$ can be obtained without sampling. Both techniques are simultaneously included in a single _combined_ model $f_{\mathrm{e+a}(p)}$ that is employed to separately estimate both $u_{e}$ and $u_{a}$. The number of prediction samples is set to $T=50$ in our experiments. ### 2.3 Predicting segmentation quality from uncertainty Different uncertainty measures only provide visual cues about prediction errors. We herein, however, aim to obtain a quantitative estimation of the segmentation quality $q$. To this end, we propose different regression models $g$ to obtain quality predictions $\hat{q}=g(u)$ from uncertainties $u$ obtained from $f(x)$ as described above. We train $g$ on $u$ extracted from all possible marker combinations within the validation set of the segmentation task, and compare $\hat{q}$ with the ground truth $q$ extracted from their corresponding segmentations. For comparison, herein we use the $F1$-score for binary classification, i.e.: $q=\frac{2\left\|y\cap\operatorname{argmax}(\hat{y})\right\|}{\|y\|+\|\operatorname{argmax}(\hat{y})\|}\ .$ We subsequently evaluate $\hat{q}$ on the test set using a Root Mean Squared Error (RMSE) metric with respect to $q$ across all possible marker combinations. We employ 4-fold cross-validation on the same data split as the segmentation task. A first approach tested is to train an additional CNN $g_{\mathrm{CNN}}$ to predict $\hat{q}=g_{\mathrm{CNN}}(\\{u_{e},u_{a}\\})$. We design a simple regression architecture with a very small number of parameters to avoid overfitting to the limited size of the validation set. By denoting 2D convolutional layers with $l$ nodes and $3\times 3$ kernels as $\text{C}(l)$, 2D max pooling layers with $3\times 3$ kernel as MP, and fully connected layers with $l$ nodes as $\text{FC}(l)$, we use the following CNN: $\text{C}(4)\shortrightarrow\\!\text{MP}\\!\shortrightarrow\\!\text{C}(8)\shortrightarrow\\!\text{MP}\\!\shortrightarrow\\!\text{C}(16)\shortrightarrow\\!\text{MP}\\!\shortrightarrow\\!\text{C}(32)\shortrightarrow\\!\text{MP}\\!\shortrightarrow\\!\text{C}(64)\shortrightarrow\\!\text{FC}(128)\\!\shortrightarrow\\!\text{FC}(1)$. We add a ReLu activation after every convolutional or fully connected layer, and use a batch size of 2. We train for 100 epochs with $L2$ loss and Adam optimizer [9] with a learning rate of $10^{-3}$. Since different uncertainty maps may contain error-related information that qualitatively correlate with $q$, we hypothesize that traditional machine- learning models that have far lower degrees-of-freedom by utilizing hand- crafted globally informative features are potentially more robust to overfitting to the limited data available for this task. To this end, we alternatively train a RF model $g_{\mathrm{RF}}$ with 128 trees, using mean squared error as split criterion. As features we use: uncertainty map percentiles (99 values from 1st to 99th), its cumulative histogram (in 13 bins from values 0.05 to 0.65), its first four statistical moments, and a one-hot vector indicating which among all possible marker combinations is used. We compare three approaches: $g_{\mathrm{RF}}(u_{e})$ with only epistemic features, $g_{\mathrm{RF}}(u_{a})$ with only aleatoric, and $g_{\mathrm{RF}}(\\{u_{e},u_{a}\\})$ with both. ## 3 Results and Discussion ### 3.1 Dataset and use of markers We employ the FM dataset of bone marrow vasculature described in [10] with the experimental settings in [2]. The dataset contains 8 samples decomposed into different 2D patches (a total of 230), each with $K=5$ markers. We use the annotated class _sinusoids_ , which amounts to 11.41% of the pixels, the rest being considered background. The samples are divided into 5 for training, 1 for validation, and 2 for testing. Since the segmentation quality vastly differs across marker combinations, we evaluate a relative segmentation $F1$-score with respect to a reference model on the test set. This score compares the pair-wise $F1$-scores across 4 cross-validation steps and all possible combinations (31) of the 5 available markers. Since it is not feasible to study all possible marker combinations in the training set, 7 training scenarios are proposed. In the first one, all 5 markers are available in all 5 training samples. In the rest, test cases #1 to #6 given in [2], where different markers are artificially ablated on each of the samples, are adopted herein for comparability with the baseline. ### 3.2 Utilizing uncertainties in segmentation We herein study how the segmentation $F1$-score is affected in our missing markers framework when accounting for the presented uncertainty methods in the _MS-ME_ architecture. We compare the performance to a reference _MS-ME_ without uncertainty estimation across all 31 possible marker combinations, 4 cross-validation steps, and 7 missing marker scenarios. The results in Fig. 2(a) show that in the estimation of $u_{e}$, it is best to use MC dropout in all convolutional layers with $p=0.2$ ($f_{\mathrm{e}}(p=0.2)$), which is also superior to baseline _MS-ME_. We assess that the $F1$-score superiority is not merely due to the use of Dropout by comparing it in Fig. 2(b) to a _MS-ME_ model with standard Dropout layers (not sampling at inference) with the same probability, which is significantly worse than _MS-ME_ and $f_{\mathrm{e}}(p=0.2)$. Fig. 2: Segmentation performance of different proposed models $f$ evaluated w.r.t. baseline _MS-ME_ (n=868). (a) Effect of different $p$ in $f_{\mathrm{e}}(p)$. (b) Comparison with conventional Dropout at training. (c) Comparison of the proposed uncertainty-based models. Symmetrical logarithmic scale (linear between -2 and 2, logarithmic elsewhere) is used. Significance is shown between different models (0pt -.4ex 0pt .5ex$\|$—–0pt -.4ex 0pt .5ex$\|$), or w.r.t. the reference _MS-ME_ (—–), in red when a proposed model performs worse. Using the stochastic loss for the estimation of $u_{a}$ with $f_{\mathrm{a}}$ leads to an inferior $F1$-score as seen in Fig. 2(c). But as discussed in the next part, such information is desirable regardless of the negative results. Thus, we employ $f_{\mathrm{e+a}}(p=0.2)$ that allows for prediction of both $u_{e}$ and $u_{e}$ while providing the best $F1$-score together with $f_{\mathrm{e}}(p=0.2)$. ### 3.3 Predicting segmentation quality for the unseen In addition to the superior $F1$-score achieved with the use of $f_{\mathrm{e+a}}(p=0.2)$, the simultaneous estimation of $u_{e}$ and $u_{a}$ allows to visually inspect potential mistakes in the CNN prediction. Here, we use such uncertainty maps to estimate the $F1$-score $q$ of segmented images. We evaluate our proposed methods on the training setting denominated case #6, for which patches have marker combinations $m_{135}$, $m_{124}$, $m_{35}$, $m_{23}$, or $m_{45}$ depending on which of the 5 training samples they belong to. This setting contains a variety of markers for each sample that depicts scenarios usually found in practise, and it allows to evaluate the predicted $q$ in markers not available for any of its training samples. We show in Fig. 3(a) the RMSE results for the $F1$-score prediction with the methods presented above. Using $g_{\mathrm{CNN}}$ leads to worse RMSE results than any of the $g_{\mathrm{RF}}$ methods, which can be explained by the tendency of CNNs to overfit in such a small training set, and that the simplicity of the task asks for the efficient use of predefined explanatory features where RF methods excel. $g_{\mathrm{RF}}(\\{u_{e},u_{a}\\})$ is seen to be superior in RMSE to both $g_{\mathrm{RF}}(u_{e})$ and $g_{\mathrm{RF}}(u_{a})$. This observation can be attributed to the fact that $u_{e}$ and $u_{a}$ capture information about distinct segmentation errors that both help in the regression of a more accurate $F1$-score value. Although further improvements could be achieved with deep ensembles, which have been reported to estimate more accurate uncertainties than MC dropout [11], they are computationally very expensive, which would annul a main advantage of our framework of training a single model. We subsequently employ $g_{\mathrm{RF}}(\\{u_{e},u_{a}\\})$ to estimate $F1$-score for both seen and unseen marker combinations in our training scenario, which is observed in Fig. 3(b) to closely follow ($R^{2}$$=$$0.98$) the ground-truth predictions on average across cross-validation steps. Thus, despite potentially high standard deviations (and high RMSE) per individual patch predictions, we can still make accurate overall predictions about the expected quality of a marker combination, including those that are unseen. Fig. 3: Prediction of $F1$-score from uncertainty maps on the test set. (a) RMSE of different models (n=124). (b) Comparison of predicted $F1$-score using $g_{\mathrm{RF}}(\\{u_{e},u_{a}\\})$ to ground truth for each of the 31 marker combinations shown as point (mean) and bars (SD) (n=4). We also visually assess the individual effects of $u_{e}$ and $u_{a}$ in Fig. 4. $u_{a}$ is observed to focus on the edges of the vessels, where discrepancies in annotations often exist and thus fit the definition of aleatoric uncertainty of capturing noise inherent in the observations. Meanwhile, $u_{e}$ captures mistakenly segmented vessels which appear in $m_{4}$ but are not sinusoids. Such errors are related to lack of information in the model, and may be corrected by adding labeled training data containing patterns similar to those highlighted by $u_{e}$. Fig. 4: Visual example of a prediction and their uncertainties for the test set marker combination $m_{34}$ in training case #6. The error image shows both false positives and negatives. The $F1$-score is predicted using $g_{\mathrm{RF}}(\\{u_{e},u_{a}\\})$. ## 4 Conclusion With the proposed framework for inspection of segmentation uncertainties with missing FM markers, the advantages are twofold: First, accounting for epistemic and aleatoric uncertainties simultaneously produces segmentation results superior to the state-of-the-art _MS-ME_ model. Second, with a uncertainty feature based regressor, we can estimate segmentation quality for any possible marker combination, whether it was seen during training or not. Thus, in practise we can quantitatively evaluate how suitable a trained model is for future samples stained with a previously unseen marker combination, i.e. to discard them or preferentially annotate them, e.g., in an active learning framework. Furthermore, comparison of different uncertainties may indicate whether to focus more on labeled data or on improving image acquisition quality. Note that the proposed methods are applicable also to other multi-modality frameworks, such as MRI. ## 5 Compliance with Ethical Standards This research was conducted on a dataset described in [10, 3], where the ethical compliance for sample preparation was described. ## 6 Acknowledgments Funding was provided by Hasler Foundation, Swiss National Science Foundation (SNSF), Swiss Cancer Research Foundation, and Julius-Müller Foundation. ## References * [1] Thorsten Falk et al., “U-net: deep learning for cell counting, detection, and morphometry,” Nature methods, vol. 16, no. 1, pp. 67–70, 2019. * [2] Alvaro Gomariz et al., “Imaging and spatial analysis of hematopoietic stem cell niches,” Annals of the New York Academy of Sciences, vol. 1466, no. 1, pp. 5–16, 2020. * [3] Alvaro Gomariz et al., “Modality attention and sampling enables deep learning with heterogeneous marker combinations in fluorescence microscopy,” arXiv preprint arXiv:2008.12380, 2020. * [4] Alex Kendall and Yarin Gal, “What uncertainties do we need in bayesian deep learning for computer vision?,” in NeurIPS, 2017. * [5] Yuta Hiasa et al., “Automated muscle segmentation from clinical ct using bayesian u-net for personalized musculoskeletal modeling,” IEEE Transactions on Medical Imaging, vol. 39, no. 4, pp. 1030–1040, 2019. * [6] Nitish Srivastava et al., “Dropout: a simple way to prevent neural networks from overfitting,” The journal of machine learning research, vol. 15, no. 1, pp. 1929–1958, 2014. * [7] Yarin Gal and Zoubin Ghahramani, “Dropout as a bayesian approximation: Representing model uncertainty in deep learning,” in ICML, 2016. * [8] Yarin Gal and Zoubin Ghahramani, “Bayesian convolutional neural networks with bernoulli approximate variational inference,” in ICLR workshop track, 2016. * [9] Diederik P Kingma and Jimmy Ba, “Adam: A method for stochastic optimization,” in ICLR, 2014. * [10] Alvaro Gomariz et al., “Quantitative spatial analysis of haematopoiesis-regulating stromal cells in the bone marrow microenvironment by 3d microscopy,” Nature communications, vol. 9, no. 1, pp. 1–15, 2018. * [11] Stanislav Fort, Huiyi Hu, and Balaji Lakshminarayanan, “Deep ensembles: A loss landscape perspective,” 2020.
# Deep Neural Networks for Active Wave Breaking Classification Caio E. Stringari France Energies Marines, Plouzané, 29280, France <EMAIL_ADDRESS>Pedro V. Guimarães France Energies Marines, Plouzané, 29280, France PPGOceano, Federal University of Santa Catarina, Florianópolis, 88040-900, Brazil these authors contributed equally to this work Jean-François Filipot France Energies Marines, Plouzané, 29280, France these authors contributed equally to this work Fabien Leckler France Energies Marines, Plouzané, 29280, France Rui Duarte France Energies Marines, Plouzané, 29280, France ###### Abstract Wave breaking is an important process for energy dissipation in the open ocean and coastal seas. It drives beach morphodynamics, controls air-sea interactions, determines when ship and offshore structure operations can occur safely, and influences on the retrieval of ocean properties from satellites. Still, wave breaking lacks a proper physical understanding mainly due to scarce observational field data. Consequently, new methods and data are required to improve our current understanding of this process. In this paper we present a novel machine learning method to detect active wave breaking, that is, waves that are actively generating visible bubble entrainment in video imagery data. The present method is based on classical machine learning and deep learning techniques and is made freely available to the community alongside this publication. The results indicate that our best performing model had a balanced classification accuracy score of $\approx$ 90% when classifying active wave breaking in the test dataset. An example of a direct application of the method includes a statistical description of geometrical and kinematic properties of breaking waves. We expect that the present method and the associated dataset will be crucial for future research related to wave breaking in several areas of research, which include but are not limited to: improving operational forecast models, developing risk assessment and coastal management tools, and refining the retrieval of remotely sensed ocean properties. ## Introduction Wave breaking is one of the most challenging water wave phenomena to investigate. Despite nearly two centuries of research, the current practical description of the process is still mostly empirical [1, 2, 3, 4]. Precise wave breaking modelling requires explicit knowledge of the wave phase speed and the fluid velocity distribution on the wave crest which can only be done by numerically solving the Navier-Stokes equations in a framework that currently is too computationally expensive for practical applications[5]. Consequently, current large-scale state-of-the-art wave models rely on statistical approaches (mostly the spectral approach) to represent the waves [6, 7, 8]. This type of models parameterizes wave breaking as a function of known parameters such as the wind speed [6], the local wave height to water depth ratio [7] or semi-empirical breaking wave height probability distributions [2, 9]. Due to a lack of observed data, the constants involved in these models have been derived from limited datasets that may not adequately represent the natural environment. For example, a recent study has shown that popular surf zone parametric wave breaking models incorrectly represented the fraction of broken waves in their formulations with errors >50%, despite these models being able to adequately represent surf zone energy dissipation possibly due to parameter tuning [10]. This paper aims to start addressing the data unavailability issue by providing a reliable and reproducible method to detect and track waves that are actively breaking in video imagery data. Wave breaking directly affects several environmental phenomena. For example, wave breaking is considered as the main driver for air-sea exchanges[11], being often described as a function of Phillips’ [12] $\Lambda(c)dc$ parameter. This parameter is defined as the average total length per unit surface area of breaking fronts that have velocities in the range $c$ to $c+dc$ and its moments are assumed to correspond to different phenomena such as whitecap coverage (second moment), rate of air entrainment (third moment), and momentum flux (fourth moment). As previously identified [13], different interpretations of how data is processed to obtain $\Lambda(c)dc$ resulted in differences of 175% in the first moment and 300% in the fifth moment of $\Lambda(c)$ [14, 13]. This issue does not seem to limit the applicability of Phillips’ framework. A recent study, for example, used a wave breaking parameterization fully based on $\Lambda(c)dc$ to derived whitecap coverage off the coast of California[15]. To the authors’ knowledge, however, no study has assessed the impact that errors in wave breaking detection has on measured $\Lambda(c)$ distributions and how such errors would impact on the conclusions that have been drawn from these models (for example, the assumption that $\Lambda(c)$ is a simple function of wind speed[11]). More importantly, in the case which new observations diverge from Phillips’ original theory (as we will later show in this paper), new and improved wave breaking models will certainly follow. Further, breaking waves have a direct impact on the remote sensing of the ocean surface from satellites, aircraft, or platform mounted instruments. Wave breaking has been observed to significantly modulate measurements of backscatter off radars with large increases in backscatter being directly correlated with breaking waves[16]. Given that part of the data that will be used here (see Methods for details) coincides with the location of previous wave backscatter studies[16], data derived from our method could to lead to explicit formulations correlating wave breaking and radar backscatter. Such formulations could, in the future, lead to more reliable satellite-derived scatterometer data, which currently ignores [17] or relies on empirical wave breaking models[18]. The foam generated by breaking waves at the sea surface also influences on the measurement of microwave brightness temperature signatures for moderate to high wind speeds [19, 20]. To date, no study has, however, systematically quantified such temperature signatures for large field datasets and most of our knowledge is empirical. Such limitation could be addressed by extending wave breaking detection and tracking methods to infrared wavelengths, which is directly correlated to water temperature[21]. Advancements in this regard could lead to improvements in global climate models given that the inclusion of wave-generated heat transfer has shown to reduce cold biases in such models for non-breaking waves[22]. Breaking waves are expected to transfer more heat than non-breaking waves[23] and, when considered, could improve climate models even further. With the recent explosion in the usage of machine learning, particularly deep neural networks, it was only a matter of time before these techniques were adapted to wave research. Recent studies have shown that deep neural networks can accurately be used to classify different types of surf zone breakers [24], obtain wave heights from video data [25], track waves in the surf zone[26] and the shoreline position [27] and, when applied to pressure transducer data, distinguish between broken and unbroken waves [10]. In this paper, we describe a robust and extensible method to identify and track breaking waves in video imagery data using a combination of classic machine learning algorithms and deep learning. More importantly, we make the present dataset and code base fully available for future researchers who can either use it directly, re- train the models adding more training data, or expand the present method to other cases (for example, object segmentation). The development and standardization of a general framework to detect wave breaking in video imagery data should help to provide the wave breaking statistics database that is currently needed to assess, or further develop, our current understanding of several processes related to breaking waves. Ultimately, the efforts initiated here aim to lead to improvements in several areas of research such as wave modelling and forecasting, wave energy tracking and harvesting [28], ocean-atmosphere interaction [15], remote sensing of the ocean[16], and safety at sea and coasts[29]. ## Method ### Model Definition In this study, we have developed a novel method to detect active wave breaking in video imagery data. Similarly to previous studies, we exploit the fact that breaking wave crests generate characteristic white foam patches from bubble entrainment [30, 31]. Differently from the vast majority of previous methods, here we make a clear distinction between active wave breaking (that is, visible foam being actively generated by wave breaking) from passive foam and to all other non-wave breaking instances. To the authors’ knowledge, only the methods of Mironov & Dulov[30] and Kleiss & Melville[32] have previously attempted to distinguish between active wave breaking and passive foam. Both their methods start similarly to ours (see next paragraph) by identifying bright pixels in a given series of images via thresholding. Next, interconnected pixels are connected in space and time using a nearest neighbor search (Mironov & Dulov[30]) or by extracting connected contours (Kleiss & Melville[32]). In both previous methods, the separation between active and passive wave breaking is then done by empirically defining foam expansion direction and velocity limits. The present method aims to remove empirical steps in active wave breaking detection by using data-driven methods and deep neural networks instead of experiment-specific parameters. Our method starts by naïvely identifying wave breaking candidates in a given input image. This was done by identifying bright (that is, white) pixels in the image using the local thresholding algorithm available from the OpenCV[33] library with its default parameters. Interconnected regions of bright pixels in the image were then clustered using DBSCAN [34] and a minimum enclosing ellipse [35] was fitted to each identified cluster. The DBSCAN step requires the user to define a minimum number of pixels to be clustered (default value is 10 for all data used here) and a maximum distance allowed between pixels (default value of 5 for all data used here). This step resulted in large quantities of bright pixel patches being detected. At this stage, it was not possible to determine whether a given identified cluster of pixels represented active wave breaking, passive foam, or any other instance of bright pixels (for example, birds, boats, coastal structures, sun glints or white clouds). Generically, this step can be thought of a feature extraction step and can be replaced by any other equivalent approach (for example, image feature extractors). To avoid exponential memory consumption growth generated by DBSCAN, the input image may be subdivided into blocks of regular size. This step was easily parallelized with performance increasing linearly with number of computing threads. The second step of the method consisted of training a deep convolutional neural network to distinguish between active wave breaking and passive foam (and all other non-desired occurrences). This step can be understood as a binary classification step. Figure 1 shows a schematic representation of the present method. From the original images, randomly selected subsets of size 256x256 pixels centered on the fitted ellipses were extracted and manually labelled as either active wave breaking (label 1, or positive) or otherwise (label 0, or negative). The present training dataset was generated using raw video imagery data from Guimaraẽs et al. [36] and consisted of 19000 training and 1300 testing images (see Table 1). The training dataset is further split into 80% training and 20% validation data[37]. The validation dataset is reserved for future hyper-parameter fine-tuning. Note that the present dataset has a class imbalance of $\approx$90% towards the negative label, that is, for each active wave breaking sample (label 1, or positive) there are nine instances that were not active wave breaking (label 0, or negative). Data augmentation [38] (rotation, vertical and horizontal flips, and zoom) was employed during training to increase the variety of samples of the positive label. Five state-of-the-art neural network architectures, or backbones (VGG16 [39], ResNet50V2 [40], InceptionResNetV2 [41], MobileNetV2 [42] and EfficientNetB5 [43]), were implemented and can be chosen by end users (see our Github code repository https://github.com/caiostringari/deepwaves for usage guidance). All of these backbones make use of convolutional layers to gradually extract information from the input images. The main differences between them are how many layers they have, the size of the convolution windows, how data is normalized in each layer and how each layer connects to each other (or to previous layers in the stack). For example, VGG16 is 16 layers deep, uses 3x3 convolution windows and has no normalization. InceptionResNetV2 is 164 layers deep, uses a mix of 5x5, 3x3 and 1x1 convolution windows and uses batch normalization[44] to help avoiding overfitting. Residual Networks (namely, ResNet50V2) not only connect adjacent layers but also take into account errors (residuals) from previous layers. In general, more modern architectures (namely, EfficientNet) are wider (mainly by having parallel convolution windows) as well as much deeper than older architectures (namely, VGG16). The final top layers of the network were fixed for all backbones and consisted of flattening the last convolutional layer, two fully-connected layers with 50% dropout [45], and a final classification layer with sigmoid activation. The optimization step (training) was done using the Adam [46] implementation of the stochastic gradient descent method [47] and binary cross-entropy was used as the loss function. Note that this step must be computed using a graphics processing unit (GPU) in order to achieve feasible computation times. The models took from three (VGG16) to twelve (EffientNetB5) hours to train using a NVIDIA GTX 1080 GPU with a batch size of sixty-four images. After the neural networks were sufficiently trained, the best performing network (VGG16, see below) was used to classify all the naïvely identified wave breaking candidates and only the events classified as active wave breaking were kept for further analyses. Although VGG16 was chosen for presenting the results, the performance of the other architectures is nearly identical to VGG16 on real-world applications. Finally, note that user-tunable parameters for the neural networks (for example, learning rate and neuron activation thresholds) were kept unchanged from their default values in the TensorFlow library[48]. This implicates that more aggressive parameter optimization could improve the results presented here even further. For sake of brevity, we refer the reader to the project’s code repository for guidance on how to select hyper- parameters. Figure 1: Schematic representation of a deep convolutional neural network. The input layer consists of an active wave breaking candidate and has shape 256x256x3 (image height, image width and number of RGB channels). In the case of a grey-scale image, the single grey channel was triplicated. The red dashed rectangle in the analyzed image shows the region of interest which, in this particular case, was equal to the stereo video reconstruction area. The intermediary convolutional layers (or backbones) are interchangeable with different options available (see text for details). The function of the convolutional layers is to extract features from the input image by using convolutions (3x3 in this example) and max pooling[49] (that is, selecting the brightest pixel in a given window). The last convolutional layer is flattened (that is, turned into a one-dimensional vector) and is connected to two fully- connected (that is, multi-layer perceptron-like) layers and one final classification layer. A 50% dropout (that is, random selection of neurons in a layer) is applied after each fully connected layer. The final classification layer has one unit and uses sigmoid activation with a threshold of 0.5. Table 1: Data characterization summary table. $f$ is the sampling frequency in Hertz, $D$ is the duration of the experiment in seconds, $H_{s}$ is the significant wave height computed from the wave spectrum, $T_{p1}$ and $T_{p2}$ are, respectively, the peak wave period of the first and second spectral partitions computed following Hanson & Jensen[50], $D_{p}$ is the peak wave direction, and $U_{10}$ is the wind speed measured or converted (denoted by the $$) to a height of 10m above the sea surface using Large & Pond’s[51] formula. Tr. S. and Ts. S. are the sample size of the train and test datasets, respectively. Location \| Date and Time \| $f$ $[Hz]$ \| D. $[min]$ \| $H_{s}$ [m] \| $T_{p1}$ $[s]$ \| $T_{p2}$ $[s]$ \| $U_{10}$ $[ms^{-1}]$ \| $D_{p}$ \| Tr. S. \| Ts. S. ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- Black Sea \| 2011/10/01 14:18 \| 12 \| 07 \| 0.3 \| 6.20 \| 3.10 \| 10.7 \| WSW \| 1000 \| 100 Black Sea \| 2011/10/04 09:38 \| 12 \| 20 \| 0.36 \| 6.10 \| 2.63 \| 10.1* \| WSW \| 1000 \| 100 Black Sea \| 2011/10/04 11:07 \| 12 \| 30 \| 0.45 \| 6.10 \| 3.16 \| 12.2* \| WSW \| 1000 \| 100 Black Sea \| 2011/10/04 13:30 \| 12 \| 30 \| 0.55 \| 6.60 \| 3.71 \| 12.9* \| WSW \| 1000 \| 100 Black Sea \| 2013/09/22 13:00 \| 10 \| 15 \| 0.66 \| 4.30 \| 30 \| 8.7* \| E \| 1000 \| 100 Black Sea \| 2013/09/25 12:15 \| 12 \| 15 \| 0.41 \| 4.10 \| 1.20 \| 6.1* \| N \| 1000 \| 100 Black Sea \| 2013/09/30 10:20 \| 12 \| 15 \| 0.65 \| 5.70 \| 1.80 \| 15.2* \| N \| 1000 \| 100 Adriatic Sea \| 2014/03/27 09:10 \| 12 \| 60 \| 1.36 \| 5.02 \| - \| 9.9 \| ENE \| 2000 \| 100 Adriatic Sea \| 2015/03/05 10:35 \| 12 \| 60 \| 1.33 \| 6.10 \| 5.02 \| 9.0 \| ENE \| 2000 \| 100 Yellow Sea \| 2017/05/13 05:00 \| 10 \| 05 \| 1.93 \| 5.20 \| 4.02 \| 13.4 \| NW \| 2000 \| 100 La Jument \| 2017/12/15 14:20 \| 10 \| 30 \| 5.88 \| 10.00 \| 6.80 \| 13.3 \| W \| 2000 \| 100 La Jument \| 2018/01/03 09:39 \| 10 \| 30 \| 10.03 \| 12.80 \| 9.30 \| 17.9 \| W \| 2000 \| 100 La Jument \| 2018/01/04 11:43 \| 10 \| 30 \| 7.52 \| 11.10 \| - \| 14.7 \| W \| 2000 \| 100 \| \| \| \| \| \| \| \| Total: \| 19000 \| 1300 The last step of the method consisted of grouping the active wave breaking events in space and time (note that at this stage bright pixels are only grouped in space). Time-discrete wave breaking events had their bounding ellipse region filled in pixel space ($i,j$) and were stacked in time ($t$) resulting in a point-cloud-like three-dimensional ($i,j,t$) structure. The DBSCAN[34] algorithm was then used to cluster these data and obtain uniquely labelled clusters. The two parameters for the DBSCAN algorithm, that is, the minimum number of samples in each cluster ($n_{min}$) and the minimum distance allowed between two samples ($eps$), were set to be equals to the minimum number of points inside a fitted ellipse among all ellipses ($n_{min}$) and equals to the sampling frequency in Hertz ($eps$). These values for $n_{min}$ and $eps$ were constant among the analyzed datasets. Note that this final step can be replaced by any other density-based clustering algorithm or other more sophisticated algorithms such as SORT[52] (which is also available on the project’s code repository but not was used here). Note that up to the clustering step, all calculations were done in pixel domain and "time" was obtained from sequential frame numbers. To convert between pixel and metric coordinates, the grids constructed using the stereo- video dataset available from Guimarães et al. [36] were used in the present study. If stereo video data are not available, such conversions could be done knowing the camera position in real-world (that is, metric) coordinates, which is usually done using surveyed ground control points[53, 54]. Conversion from sequential frame numbers to time (in seconds) can be done by knowing the sample rate (in frames per second, that is, Hertz) for the data. The total amount of computational time required to process 20 minutes of raw video recorded at 10Hz with an image size of 5 megapixels is approximately two hours on a modern six-core computer (assuming that a pre-trained neural network is available). Much shorter processing times are achievable for smaller images sizes and higher number of computation threads. ### Evaluation Metrics Due to the imbalanced characteristic of the dataset, the classification accuracy score in isolation is not an appropriated metric to evaluated the present test dataset. For instance, a classifier that guesses that all labels are negative (that is, 0) would automatically obtain a high score ($\approx$ 90%). To properly assess the performance of the classifier, more robust metrics were defined. These metrics are defined as follows: * • True Positives ($TF$) and True Negatives ($TN$) are samples that were correctly classified. False Positives ($FP$) or Type I errors are false samples that were incorrectly classified as true, and False Negatives ($FN$) or Type II errors are true samples that were classified as false. * • Accuracy is the percentage of examples correctly classified considering both classes, that is, Accuracy=$\frac{TP+TN}{T+N}$, where $T$ and $N$ are the total number of positive and negative samples, respectively. * • Precision is the percentage of predicted positives that were correctly classified, that is, Precision=$\frac{TP}{TP+FP}$. * • Recall is the percentage of actual positives that were correctly classified, that is, Recall=$\frac{TP}{TP+FN}$ * • Area under the curve ($AUC$) is the area defined by plotting the FP rate against the TP rate (also referred to as receiver operating characteristic curve). This metric indicates the probability of a classifier ranking a random positive sample higher than a random negative sample. All metrics described above were monitored during the training process and were individually assessed to rank model performance. The training curves shown in Figure 2-a were used to assess when overfitting, that is, decreases in the loss function value for the training dataset that were not reflected in the validation dataset, started to occur. Training epochs after overfitting started were discarded. Here we favor the $AUC$ curves as shown in Figure 2-b to indicate better performing models because AUC is a more robust metric than the classification score and at the same time presents a smooth evolution with training epochs. Finally, a confusion matrix (or table of confusion) as shown in Figure 2-c was plotted for each model to assess Type I and Type II errors. ## Results ### Classifier performance From the analysis of all training curves, confusion matrices, and from Table 2, the best performing backbone architecture during training was ResNet50V2 by a considerable margin ($AUC$=0.989). These results however, did not translate to the validation dataset ($AUC$=0.873). Considering only the validation data, VGG16 was the best performing backbone with $AUC$=0.946. Considering only the test dataset, the best performing model was also VGG16 ($AUC$=0.855). Overall, VGG16 was selected as the best performing model and the results presented in the next sections will use this backbone. Other evaluation metrics such as the accuracy score, precision, and recall closely followed the evolution of AUC with training epochs (compare Figures 2-a and b, for example). In general, as the loss value decreased, the number of false positives decreased, which made the precision and recall to increase. This behavior was consistent for all models. Given that different architectures may perform better for other datasets and further optimization could change the model ranking presented here, all pre-trained models and their training metrics evolution are made available in the project’s code repository. Finally, it is worth mentioning that larger models (for example, VGG19, ResNET152 and EfficientNetB7) could achieve better results but this was not attempted there due to hardware limitations (that is, these models required more memory than what was available on the NVIDIA GTX 1080 GPU used in this study). Figure 2: Examples of training curves and confusion matrix for the best overall performing model (VGG16). a) Loss function value for training and validation data. b) $AUC$ value for training and validation data. In a) and b) the hatched area indicates the epochs after which the model started to overfit, the thick colored lines show smoothed loss or $AUC$ values (average at every 10 epochs), and the transparent lines show raw loss or $AUC$ values. c) confusion matrix calculated using test data only. Table 2: Evaluation metrics for all tested backbone architectures. Refer to the main manuscript text for definition of evaluation metrics. The best performing models are shown in boldface. Results are sorted by AUC. Model \| Accuracy \| TP \| FP \| TN \| FN \| Precision \| Recall \| AUC ---\|---\|---\|---\|---\|---\|---\|---\|--- Train ResNetV250 \| 0.97 \| 1414 \| 198 \| 13978 \| 280 \| 0.877 \| 0.835 \| 0.989 VGG16 \| 0.93 \| 855 \| 273 \| 13911 \| 831 \| 0.758 \| 0.507 \| 0.943 InceptionResnetV2 \| 0.927 \| 886 \| 359 \| 13823 \| 802 \| 0.712 \| 0.525 \| 0.932 EfficientNetB5 \| 0.772 \| 1403 \| 3346 \| 10920 \| 297 \| 0.295 \| 0.825 \| 0.874 MobileNet \| 0.904 \| 436 \| 268 \| 13916 \| 1250 \| 0.619 \| 0.259 \| 0.848 Validation VGG16 \| 0.932 \| 221 \| 65 \| 3478 \| 204 \| 0.773 \| 0.52 \| 0.946 InceptionResnetV2 \| 0.921 \| 190 \| 81 \| 3466 \| 231 \| 0.701 \| 0.451 \| 0.93 EfficientNetB5 \| 0.809 \| 353 \| 687 \| 2856 \| 72 \| 0.339 \| 0.831 \| 0.897 MobileNet \| 0.908 \| 123 \| 64 \| 3479 \| 302 \| 0.658 \| 0.289 \| 0.878 ResNetV250 \| 0.919 \| 197 \| 97 \| 3450 \| 224 \| 0.67 \| 0.468 \| 0.873 Test VGG16 \| 0.876 \| 106 \| 80 \| 945 \| 69 \| 0.57 \| 0.606 \| 0.855 ResNetV250 \| 0.881 \| 95 \| 63 \| 962 \| 80 \| 0.601 \| 0.543 \| 0.843 InceptionResnetV2 \| 0.882 \| 91 \| 57 \| 968 \| 84 \| 0.615 \| 0.52 \| 0.839 EfficientNetB5 \| 0.873 \| 88 \| 65 \| 960 \| 87 \| 0.575 \| 0.503 \| 0.827 MobileNet \| 0.875 \| 30 \| 5 \| 1020 \| 145 \| 0.857 \| 0.171 \| 0.768 ### Test Case with Real-world Data Figure 3 shows the results of the application of the best performing model architecture (VGG16) on La Jument (2018/01/03 09:39) and Black Sea (2011/10/04 09:38) data. Visual inspection of these results confirmed the ability of the neural network to correctly classify the naïvely identified wave breaking candidates and only keep instances that represented active wave breaking. Moreover, the same neural network was able to correctly classify active wave breaking events for the rogue waves seen at La Jument[29] and for the small wind-generated breakers seen in the Black Sea data. This result highlights the ability of the neural network to generalize well on the dataset, which is a difficult result to achieve. From the analysis of the training curves, the averaged classification error (accounting for the imbalance in the data) should be of the order of $\approx$10%, which to the authors’ knowledge it was not assessed by other wave breaking detection methods. We strongly recommend that future research should report active wave breaking detection errors when used to assess or further develop models that depend on wave breaking data. Figure 3: Example of the application of the method. a) Results of the naïve wave breaking detection (thresholding + DBSCAN) for La Jument data (03/01/2018 09:39). Note the great amount of passive foam being detected as active wave breaking. b) Results of active wave breaking detection using VGG16 as backbone for La Jument data (03/01/2018 09:39). Note the significant reduction in the amount of passive foam being detected. In both plots, number of clusters refers to the number of clusters identified by DBSCAN. The red dashed rectangle indicates the region of interest which, in these examples, was the same as the stereo-video reconstruction area. c) Results of the naïve wave breaking detection (thresholding + DBSCAN) for Black sea data (04/10/2011 09:38). d) Results of active wave breaking detection using VGG16 as backbone for Black Sea data (04/10/2011 09:38). Animations of these results are available at https://github.com/caiostringari/deepwaves. In this particular example, the image was subdivided into blocks of 256x256 pixels for processing. Note that identical results were seen using other architectures other than VGG16 to classify these data. ### Comparison with Mironov & Dulov (2008) To highlight the improvements obtained by the present method, this section presents a comparison between Mironov & Dulov’s (2008)[30] automatic active wave breaking detection method and the present method. To perform this task, data from the 2013 Black Sea experiments in Table 1 that had previously been classified and investigated in detail by Guimarães[55] were used. All the active breaking events (that is, considering all 900s of data) detected using Mironov & Dulov’s (2008)[30] were manually classified as true or false and compared to the labels predicted by our method. To the authors’ knowledge, these data are the only currently available data that has been classified by both methods as well as manually. Figure 4 shows the result of the compassion between models. On average, our method has relatively $\approx$50% less error then Mironov & Dulov’s (2008) method with an averaged absolute reduction in error of $\approx$15%. The results in Figure 4 are also consistent with the results seen in Table 2 which showed that our model had errors in the order of $\approx$15% when considering the validation and test datasets. Note that all tested neural network architectures performed very similarly with only a slight advantage for InceptionResnetV2. It is also worth mentioning that Mironov & Dulov’s (2008) method was designed and optimized to work specifically with data from the Black Sea and it is not guaranteed that it will generalize to other datasets. Our method, on the contrary, has been shown to generalize well for very distinct datasets (see Figure 3, for example) and with further optimization could achieve even better performance. Figure 4: Comparison between averaged classification error for active wave breaking detection between Mironov & Dulov’s (2008)[30] method and all the neural network architectures implemented here. The error bars represent one standard deviation from the mean. The data used for this comparison are from the 2013 Black Sea experiments described in Table 1. ### Wave Breaking Statistics In this section we briefly present examples of wave breaking statistics that can be directly derived from the classified wave breaking data. For brevity, the analysis is limited to data from the Black Sea because our classifier performed best at this location (classification errors < 10% considering both 2011 and 2013 experiments). Five quantities will be analysed: the wave breaking duration ($T_{br}$), the wave breaking area ($A_{br}$), the major ($a$) and minor ($b$) axis of the fitted ellipses (representative of the wave breaking lengths at their maximum during the active part of the wave breaking), and Phillips’ distribution $\Lambda(c)dc$. These quantities were obtained directly from space-time clustered wave breaking events with the exception of the cumulative wave breaking area ($A_{br}$) which was calculated from the projections of pixels clustered in the first step of the method to metric coordinates. The results of this analyses are shown in Figure 5. The wave breaking duration ($T_{br}$ normalized by wave peak period ($T_{p}$), Figure 5-a) roughly followed a shifted Gamma probability density function (PDF) and had a mean value of 0.12 and most frequent value (mode) of 0.13. This result shows that the active part of the wave breaking process happens very quickly. The wave breaking area ($A_{br}$, Figure 5-b) closely followed a Pareto distribution which indicates that large wave breaking events are relatively rare in the data. The ratio between the major and minor axis of the fitted ellipses ($a/b$, Figure 5-c) followed a Beta PDF and had a mean of 2.5 and mode of 1.9, which indicates that the ellipses’ major axis is approximately double the size of the minor axis. Assuming a negligible wave front angle, the wave breaking area scaling relation from Duncan [56] ($A_{br}/b^{2}$, Figure 5-d) also followed a Beta PDF and had mean of 0.1 and mode of 0.8, which is consistent with the previously reported value[56] (0.1$\pm$0.01). The wave breaking area (Figure 5-e) showed a quadratic increase with wave breaking event duration, which is trivial but seems not to have been previously directly shown before. Finally, Figure 5-f shows the $\Lambda(c)dc$ distributions obtained using our method and considering that the ellipse major axis is representative of the wave breaking length. The observed distributions greatly deviated from the theoretical $c^{-6}$ relation[12]. Note that all PDF fits shown here were statistically significant at the 95% confidence level using the two-tailed Kolmogorov–Smirnov test. See Table 3 for the description of the parameters of the PDFs presented in this section and the Discussion section for the contextualization of these results and the possible implications that they have for future research. Figure 5: Examples of statistical properties of breaking waves that can be directly obtained from the proposed method. a) Probability distribution of the wave breaking duration ($T_{br}$) normalized by wave peak period ($T_{p}$). The blue line shows the Gamma fit to the data, the purple dashed line shows the mean value (0.13) and the orange dashed line shows the mode value (0.12). b) Probability distribution of the wave breaking area ($A_{br}$) normalized by wavelength ($\frac{g}{2\pi}T_{p}^{2}$). The blue line shows the Pareto fit to the data. c) Probability distribution of the ratio between major and minor axis ($a/b$). The blue line shows the Beta fit to the data, the purple dashed line shows the mean value (2.4) and the orange dashed line shows the mode value (1.8). d) Probability distribution of the wave breaking area scaling parameter ($A_{br}/b^{2}$). The blue line shows the Beta fit to the data, the purple dashed line shows the mean value (0.1), the orange dashed line shows the mode value (0.08), and the red line shows Duncan’s constant value (0.11). e) Evolution of the wave breaking area over time. The black markers show the observed values, the error bars show values binned at 0.15s intervals, the blue line shows the quadratic regression to the data, the blue swath shows the 95% confidence interval for the regression, and $r^{2}$ is the correlation coefficient. f) Phillips[12] $\Lambda(c)dc$ distributions assuming that the wave breaking speed is the same as the phase speed of the carrier wave (that is, $c=c_{br}$) grouped by wind speed. The black dashed line shows the $c^{-6}$ theoretical decay, the colored markers show the average crest length binned at 0.1$ms^{-1}$ wave speed intervals, and the colored lines show a running average with a window size of 10 bins. Table 3: Fitted parameters for the distributions seen in Figure 5 . All fits were done using Scipy[57] and, consequently, the values reported here follow Scipy’s conventions. \| Distribution \| Parameter 1 \| Parameter 2 \| Location \| Scale ---\|---\|---\|---\|---\|--- Figure 5-a \| Gamma \| 3.001 \| - \| 0.053 \| 0.0260 Figure 5-b \| Pareto \| 4.339 \| - \| -0.007 \| 0.007 Figure 5-c \| Beta \| 6.117 \| 73.948 \| 0.207 \| 30.440 Figure 5-d \| Beta \| 6.389 \| 40.470 \| -0.760 \| 24.438 ## Discussion We have presented a new method to detect active wave breaking in video imagery data that is robust and easily transferable to future studies. Overall, VGG16 was the best performing architecture which is a surprising result given that VGG is a considerably older architecture that has been superseded by more recent architectures such as ResNets and EfficientNets [43]. Also surprisingly, EfficientNet was one of the worst performers considering the test dataset despite the state-of-the-art results that this architecture achieved in recent years[43]. One explanation could be that given VGG16 is the model with the highest number of parameters (despite being the shallowest network), it adapted better to the characteristics of the present dataset. Another explanation could be that EfficientNet is currently overfit on the ImageNet[58] dataset that was used to train this network hence its high reported classification scores. Another aspect to take into consideration is the speed in which the models can predict on new data (inference). While VGG16 was the best performing model, its size slows down inference time. In this regard, MobileNetV2 offers real-time inference speed at 10Hz for small image sizes (for example $512\times 512$px). Consequently, a model based on MobileNetV2 could be used to detect active wave breaking in an operational fashion in, for example, offshore floating wind turbines that are susceptible to wave breaking impacts and used to adjust anchor and cable settings in real- time. From the analysis of the application of the method on real-world data, it was visually observed that small active wave breaking events were not always detected, particularly for the Black Sea data. There are two possible explanations for this error. The simplest explanation could be that this type of events is under-represented in the training dataset. The solution for this issue consists of adding more data representative of these instances to the training dataset. The second possibility is that the image size becomes too small in the deeper layers of the network which makes it impossible for the network to learn these events (see below for further discussion). A solution for this issue could be to increase the input image size but this was not attempted here due to hardware constraints (that is, memory limitation, as discussed above). Neural networks have been historically seen as black boxes in which only the final classification outputs are relevant. There has been, however, an increase in interest to understanding how neural networks are learning. One technique that is of particular interest for the present paper is the concept of Gradient-weighted Class Activation Mapping (Grad-CAM)[59]. Briefly, this technique shows which regions of a convolutional layer are more important for the results obtained by the classifier. Figure 6 shows the results of Grad-CAM applied to examples for all unique locations from Table 1 considering VGG16’s last convolutional layer. When considering only actively breaking waves (Figure 6-a to d) it is evident that VGG16 closely mimicked how a human would classify these data, that is, it directly searched for the regions of the image that corresponded to active wave breaking. In the case of passive foam (Figure 6-e to h), VGG16 seemed to use larger portions of the image but at the same time focused on the flocculent foam seen in the images as a human classifier would do. In general, these results show that our model truly learned how to classify the images and is not merely guessing the labels. Figure 6: Results of Grad-CAM [59] for all unique experiment locations described in Table 1 applied to actively breaking waves (top row) and to passive foam (bottom row). a) Actively breaking wave example recorded at Adriatic Sea (2015/03/05 10:35). b) Actively breaking wave example recorded at the Black Sea (2011/10/04 11:07). c) Actively breaking wave example recorded at La Jument (2018/01/03 09:39). d) Actively breaking wave example recorded at the Yellow Sea (2017/05/13 05:00). e) Passive foam example recorded at Adriatic Sea (2015/03/05 10:35). f) Passive foam example recorded at the Black Sea (2011/10/04 11:07). g) Passive foam example recorded at La Jument (2018/01/03 09:39). h) Passive foam example recorded at the (2017/05/13 05:00). In all panels, the color scale indicates the weights of the class activation map with brighter colors showing regions of the image which the neural network used to classify each particular sample. The promising results presented here indicate that the current method should be extended to an object detection framework. Such a framework would eliminate the need for the image thresholding and DBSCAN steps. This implementation could be done by using a strongly-supervised architecture such as UNet [60], or by using a weakly-supervised method derived from Grad-CAM, for example. As a final recommendation regarding the machine learning aspect of this paper, we strongly encourage future researchers to add samples to the training dataset that matches their specific research needs instead of blindly applying the provided pre-trained models. We have also presented examples of wave breaking statistics that can be obtained using the proposed method. In general, the patterns observed here agreed with previously reported scaling factors that support the idea of wave breaking self-similarity. For example, Figure 5-f directly showed that the scaling parameter $A_{br}/b^{2}$ approaches the constant 0.1 value from Duncan’s [56] laboratory experiments. Another variable that showed signs of a self-similar behavior was the wave breaking area ($A_{br}$) which was very well described by a Pareto distribution and presented a steady quadratic increase with wave breaking duration ($T_{br}$). Extensive research [11, 15] has been grounded on the assumption that wave breaking is self-similar, but inconsistencies with this approach have been reported before [14]. Contrarily to other studies [11, 31], however, the $\Lambda(c)$ distributions obtained here did not closely match the theoretical $c^{-6}$ for a sea-state in equilibrium. As reported before [13], these differences may be due to the fact that here we only considered actively breaking waves for our analysis whereas other studies seem to track the speeds of both actively breaking waves and passive foam [32, 31]. Another possibility is that other phenomena not accounted by Phillips’[12] theory (for example, wave modulation[61]) play important role on wave breaking. A future publication that investigates the mechanisms related to the wave breaking kinematics using the method and data obtained here will soon follow. ## Conclusion We described a novel method to detect and classify actively breaking waves in video imagery data. Our method achieved promising results when assessed using several different metrics. Further, by analyzing the deeper layers of our neural network, we showed that the model mimicked how a human classifier would perform a similar classification task. 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# Kähler geometry of quiver varieties and machine learning George Jeffreys Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston MA 02215, USA<EMAIL_ADDRESS>and Siu-Cheong Lau Department of Mathematics and Statistics, Boston University, 111 Cummington Mall, Boston MA 02215, USA<EMAIL_ADDRESS> ###### Abstract. We develop an algebro-geometric formulation for neural networks in machine learning using the moduli space of framed quiver representations. We find natural Hermitian metrics on the universal bundles over the moduli which are compatible with the GIT quotient construction by the general linear group, and show that their Ricci curvatures give a Kähler metric on the moduli. Moreover, we use toric moment maps to construct activation functions, and prove the universal approximation theorem for the multi-variable activation function constructed from the complex projective space. ## 1\. Introduction Machine learning by artificial neural networks has made exciting developments and has been applied to many branches of science in recent years. Mathematically, stochastic gradient flow over a matrix space (or called the weight space) is the central tool. The non-convex nature of the cost function has made the problem very interesting. Current research has focused on different types of stochastic gradient flows and finding new types of networks, which have brought great improvements of computational efficiency. In geometry and physics, the applications of gradient flow and Morse theory have a long history and have brought numerous fundamental breakthroughs. For instance, the gradient flow of the Yang-Mills functional is used to find Hermitian Yang-Mills connections, whose existence in a stable holomorphic vector bundle is proved by Donaldson [Don85] and Uhlenbeck-Yau [UY86]. The celebrated Ricci flow found by Hamilton [Ham82], which is a crucial tool to solve the three-dimensional Poincaré conjecture, is essentially a gradient flow [Per02, Per03]. Its Kähler analog has been an important tool in finding Kähler-Einstein metrics on Fano manifolds [Yau96, Tia97, Don12, CSW18, CDS15a, CDS15b, CDS15c]. In these works, GIT quotients and finite-dimensional models have provided important motivations and guidelines [Don99]. Hamiltonian Floer theory [Flo89], which is essentially Morse theory on the loop space, was invented to solve the Arnold conjecture [FHS95, Ono95, FO99]. Various versions of Floer theory have been crucial ingredients in the study of mirror symmetry. In this paper, we would like to develop a foundational _algebro-geometric formulation_ for neural networks in machine learning. The theory of _quiver representations_ , which is a well-developed branch of mathematics motivated from Lie theory and has been an important tool in mathematical physics, will be well suited for this purpose. A quiver representation assigns to a directed graph $Q$ a bunch of vector spaces for the vertices and a bunch of linear maps for the arrows. Such a construction is in common with neural networks. However, in order to use quiver theory to formulate machine-learning neural networks, there are two main differences between these two subjects that needs to be addressed. 1. (1) Compactness of moduli space. A moduli space of quiver representations [Kin94] is defined by identifying _isomorphic_ quiver representations using GIT quotients. As a result, the moduli space is _compact_ when the quiver has no oriented cycle. On the other hand, the matrix space used in neural networks is non-compact. In machine learning, isomorphic quiver representations may correspond to physically different input or output information and in general cannot be identified. 2. (2) Non-linearity. _Activation functions_ , which are non-linear maps on the vector spaces over the vertices, serve as a crucial ingredient to achieve machine learning of non-linear functions. Such non-linearity jumps out of the category of quiver representations. This is also related to the first point above. Namely, such non-linear maps are not necessarily equivariant under the group of automorphisms of quiver representations. For the first point, we shall use framed quiver representations, which were first found by Nakajima [Nak94] in the study of affine Lie algebras. A framed representation assigns to each vertex a vector space together with a choice of ‘framing’ (for instance it is a basis in the simplest situation). In the applications considered here, such a decoration makes sure that isomorphic framed quiver representations correspond to the same physical state. Note that framed quiver moduli $\mathcal{M}$ are also compact when $Q$ has no oriented cycle. _Compactness_ is one of the main advantages of our algebro-geometric formulation, which makes sure the convergence of a gradient flow. In this formulation, the weight matrices are encoded as morphisms between the _universal vector bundles_ (over the framed quiver moduli) associated to the vertices. The data flow is encoded by sections of the universal bundles, which are sent from one to another bundles by the morphisms associated to the arrows of $Q$. The cost function, and hence its gradient flow, is defined on the framed quiver moduli $\mathcal{M}$. In particular, the _critical points and the gradient flow are controlled by the topology of $\mathcal{M}$_ (for instance, the Morse inequalities). The topology of a framed quiver moduli is well-understood by the work of Reineke [Rei08] when $Q$ has no oriented cycle. $\mathcal{M}$ is an iterated Grassmann bundle, and its Poincaré polynomial is a product of that of the Grassmannians. For the purpose of gradient flow, one needs to choose a _Kähler metric_ on $\mathcal{M}$, and also Hermitian metrics on the universal vector bundles. As a result, we have found metrics that are defined by explicit beautiful formulae. These metrics are not just $U_{\vec{d}}$-equivariant so that they descend to symplectic quotients, but are also $\mathrm{GL}_{\vec{d}}$-equivariant and hence compatible with the GIT construction of $\mathcal{M}$. Moreover, they are compatible with the iterated Grassmann structure found by Reineke. In application, such metrics would simplify the actual computational algorithm over the quiver moduli. They are summarized as follows. ###### Theorem 1.1 (Combining Theorem 3.7, 3.15,3.18). Let $Q$ be an arbitrary quiver. Fix a vertex $i\in Q_{0}$. Let $\rho$ be the row vector whose entries are $V_{\gamma}e^{\left(t(\gamma)\right)}$, where $\gamma$ is any path whose head $h(\gamma)$ is $i$ (including the trivial path), $t(\gamma)$ denotes its tail, and $V_{\gamma}\in\mathrm{Hom}(\mathbb{C}^{d_{t(\gamma)}},\mathbb{C}^{d_{h(\gamma)}})$ is the representing matrix of $\gamma$. Then $(\rho_{i}\rho_{i}^{})^{-1}=\left(\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)^{-1}$ is $\mathrm{GL}_{\vec{d}}$-equivariant, and it descends to a metric on the universal bundle $\mathcal{V}_{i}$ over a certain domain of convergence $\mathcal{M}^{\circ}$. When $Q$ has no oriented cycle, $\mathcal{M}^{\circ}=\mathcal{M}$. Moreover, the Ricci curvature of the induced metric on $\bigotimes_{i\in Q_{0}}\mathcal{V}_{i}$ gives a Kähler metric on $\mathcal{M}$. The precise definition of $M^{\circ}$ is given in Section 3.3. In this paper, we focus on the framed quiver moduli defined over complex numbers. In actual applications, we can also restrict to real coefficients. Then the above formula defines a bundle metric over $\mathcal{M}_{\mathbb{R}}$, and the Ricci curvature gives a Riemannian metric on $\mathcal{M}_{\mathbb{R}}$. Now let us address the second point. Namely, we need to introduce _non- linearity_ in addition to the usual theory of quiver representations. By definition, morphisms between universal vector bundles over $\mathcal{M}$ are linear along fibers. They correspond to weight matrices in neural networks. To introduce non-linearity, we shall treat the universal bundles as fiber bundles and construct suitable _fiber-bundle maps_ that play the role of activation functions. One of the commonly-used activation functions is $\frac{e^{2x}}{1+e^{2x}}:\mathbb{R}\to(0,1).$ We observe that this function also appears in the base of the symplectic trivialization of the open dense toric orbit of $\mathbb{P}^{1}$ as a toric variety: $(\mathbb{C}^{\times},\omega_{\mathbb{C}\mathbb{P}^{1}})\cong((0,1)\times\mathbb{S}^{1},\omega_{\mathrm{std}}),$ or lifted to the universal cover: $(\mathbb{C},\pi^{}\omega_{\mathbb{C}\mathbb{P}^{1}})\cong((0,1)\times\mathbb{R},\omega_{\mathrm{std}}).$ Here, $\omega_{\mathbb{C}\mathbb{P}^{1}}$ denotes the Fubini-Study metric of $\mathbb{C}\mathbb{P}^{1}$, that is, the standard area form of the unit sphere; $\omega_{\mathrm{std}}=dx\wedge dy$ is the standard symplectic form. Similarly, another activation function $\frac{z}{\sqrt{1+\|z\|^{2}}}:\mathbb{C}\to\mathbb{C}$ also arises as a symplectic trivialization: $(\mathbb{C},\omega_{\mathbb{C}\mathbb{P}^{1}})\cong(\\{w\in\mathbb{C}:\|w\|<1\\},\omega_{\mathbb{C}}).$ Motivated from these observations, we consider $\sigma(x)=\left(\frac{e^{2x_{i}}}{1+\sum_{j=1}^{n}e^{2x_{j}}}\right)_{i=1}^{n}:\mathbb{R}^{n}\to\Delta$ and $\psi(\vec{z})=\left(\frac{z_{i}}{\sqrt{1+\\|\vec{z}\\|^{2}}}\right)_{i=1}^{n}:\mathbb{C}^{n}\to\\{\vec{w}\in\mathbb{C}^{n}:\\|\vec{w}\\|<1\\}$ as multi-variable activation functions, where $\Delta$ denotes the standard simplex with vertices $0$ and $\epsilon_{1},\ldots,\epsilon_{n}$, the standard basis of $\mathbb{R}^{n}$. They arise from symplectomorphisms $(\mathbb{C}^{n},\omega_{\mathbb{C}\mathbb{P}^{n}})\cong(\\{\vec{w}\in\mathbb{C}^{n}:\\|\vec{w}\\|<1\,\omega_{\mathbb{C}^{n}}\\})$. More generally, these come from moment maps of _toric varieties_ [Gui94, Abr98]. We note that $\psi$ has an advantage of being $U(n)$-equivariant. The _universal approximation theorem_ (see for instance [Cyb89, Pet99, MM92, Pin99]) provides a theoretical foundation for the success of neural networks. In existing literature, the theorem was proved for single-variable activation functions. In this paper, we prove the universal approximation theorem for the above multivariable function $\sigma=\sigma_{\mathbb{R}^{n}}$. Note that $\sigma$ is the softmax function $\left(\frac{e^{2x_{i}}}{\sum_{j=0}^{n}e^{2x_{j}}}\right)_{i=0}^{n}$ restricted to the hyperplane $x_{0}=0$ and composed with the projection along $x_{0}$-direction. We shall restrict to real coefficients in this theorem. ###### Theorem 1.2 (same as Theorem 5.2). Let $K$ be a compact set of $\mathbb{R}^{d_{1}}$, and $f:K\to\mathbb{R}^{d_{3}}$ a continuous function. For any $\epsilon>0$, there exists $d_{2}>0$ and $W_{1}\in\operatorname{Mat}(d_{2},d_{1})$, $W_{2}\in\operatorname{Mat}(d_{3},d_{2})$, $b\in\mathbb{R}^{d_{2}}$ such that $\\|f^{U}_{W_{1},W_{2},b}-f\\|_{L^{2}(K)}<\epsilon$. Here, $f^{U}_{W_{1},W_{2},b}(x)=W_{2}\cdot\sigma_{\mathbb{R}^{d_{1}}}(W_{1}\cdot x+b)$ is the function coming from the $A_{3}$ quiver. The $A_{3}$ quiver corresponds to the feed-forward network with one input layer, one middle layer and one output layer. See Figure 1. ${a}$${b}$${c}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$ Figure 1. The $A_{3}$ quiver. The above theorem is proved by using the tropical limit of the toric manifold $\mathbb{P}^{n}$, and a geometric object that we call a centered polyhedral web, which is an analog of a tropical variety in an integral affine manifold. Since we do not have integral structure in the context here, we need to invent this new notion. In above, we have focus on explaining non-linearity for a single vector space. We shall _globalize them as non-linear fiber-bundle morphisms_ for the universal bundles over $\mathcal{M}$. This can be achieved with the help of Hermitian metrics on the universal bundles, so that the Fubini-Study metric on $\mathbb{P}^{n}$ can be globalized as a fiberwise symplectic structure on projective bundles over $\mathcal{M}$. Actually, the globalization from a single framing vector space $V$ to the universal fiber bundle over $\mathcal{M}$ _works for any continuous function_ $V\to V$ (and in particular for a symplectomorphism from $V$ to its image). Combining the ingredients explained above together, we can construct a gradient flow over the framed quiver moduli to achieve machine learning. The detail is given in Section 4. Such an algebro-geometric formulation has several _advantages_. First, the gradient flow under consideration runs in a _compact_ manifold. This ensures the existence of absolute extrema, convergence of the flow, and upper bound for the norm of the gradient vector field. Second, because of compactness, the flow is constrained by _topology_ of the manifold due to Morse theory. See Section 4.4. Finally, the moduli space has extra _symmetry_ coming from framing. If we use activation functions that respect this symmetry (for instance $\psi$ above enjoys $U(n)$-equivariance), we can perform dimension reduction which improves the effectiveness of the network. (See Proposition 4.16 and 4.17.) In summary, from this point of view, the success of neural network is resulted from the interplay between algebraic morphisms and (transcendental) symplectomorphisms. Interestingly, such an interplay is also an important feature that occurs in the study of complete integrable systems and mirror symmetry for toric manifolds and flag varieties, see for instance [Gui94, FLTZ12, Abo06, CLL12, NNU10, HKL18]. ### Some related works The relation between neural network and quiver representation was investigated in the recent paper [AJ20]. Their work considered the quotient space by $(\mathbb{C}^{\times})^{N}$ of pairs $(W,f)$, where $W$ is a quiver representation of $Q$ with the dimension vector $\vec{1}$, and $f$ associates each vertex a function $\mathbb{C}\to\mathbb{C}$ (playing the role of an activation function). Moreover, in dimension $\vec{1}$ (which is a typical case for machine learning), they invented an interesting way of encoding the data flow as a quiver representation. (In our work, the data flow is given as sections of universal bundles over the quiver moduli.) The approach and the goal of this paper is rather different. We aim at formulating machine learning as a gradient flow over a compact quiver moduli. In [AJ20], quiver representations were used in encoding the data in the network; however, the machine learning process was independent of the quiver moduli. Moreover, ‘double-framing’ was used, and the corresponding moduli space is non-compact. The map $(W,f)\mapsto W$ gives an infinite-dimensional fiber bundle over the quiver moduli $\mathcal{M}$, whose fibers are the spaces of choices of activation functions. In a typical program of machine learning, the activation functions are fixed during the optimization process. In order to formulate the program as a gradient flow over the compact moduli $\mathcal{M}$, we found a non-trivial way by equipping intermediate vertices with additional framings and metrics, so that we can lift $f$ to be a well-defined fiber-bundle map over $\mathcal{M}$. Note that $f$ is not equivariant under the group action of $(\mathbb{C}^{\times})^{N}$ ($\mathrm{GL}_{\vec{d}}$ in the higher rank case). Such a lifting is an important non-trivial step. Furthermore, we have dealt with representations of general rank $\vec{d}$, and a class of activation functions coming from toric symplectomorphisms. Different functions (on the same domain and target) are obtained if we deform the toric Kähler metric. To also optimize the activation functions during the learning process (see also [GGL19]), we may consider a gradient flow on $\mathcal{M}\times\mathcal{K}$ where $\mathcal{K}$ denotes the moduli of toric Kähler metrics in the same class. By the celebrated works of [Don99, Sem92], $\mathcal{K}$ is an infinite-dimensional negatively curved symmetric space. Recently, there is a rising interest of applying geometric techniques to the study of neural networks. For instance, in the works [GBH18, CYRL19], hyperbolic spaces are applied to machine learning in graphs and achieved great performance. Moreover, the applications of symmetry and group equivariance in neural networks were studied and developed in [CW16, CGW19, CGKW18, CWKW19, CAW+19, dHCW20]. Overall, these works aim at capturing symmetry of the input data and designing networks that are adapted to such symmetry. Moreover, homogeneous spaces (in place of vector spaces) have been employed in layers of convolutional neural networks. In comparison, our paper aims at revealing the geometric nature of neural networks and build a connection with algebraic geometry. The resulting framed quiver moduli, which has interesting topology and metrics, is the main geometric object of interest. Furthermore, we study activation functions that respects the ‘intrinsic symmetry’ over the quiver moduli, which can provide a more effective algorithm by dimension reduction. In the reverse direction, there are interesting applications of machine learning in frontier geometry and physics. For instance, [HY20] used machine learning to solve problems in computing graph Laplacians, such as recognizing graph Ricci-flatness and predicting the spectral gap. In physics, [HSTT18a, HSTT18b] used deep learning to study AdS/CFT correspondence by discretizing the equation of motion. Since we have formulated machine learning using quiver representations, it will be interesting to find direct relations between quiver gauge theory and these problems that can be attacked via machine learning. ### Organization of this paper In Section 2, we will take a quick review on quiver representations and their moduli spaces. In Section 3, we will construct nice Hermitian metrics on universal bundles over the moduli. For readers who are mainly interested in machine learning, Section 3 can be skipped for the first reading. Then we give an algebro-geometric formulation of neural network using quiver representations in Section 4. In Section 5, we prove the universal approximation theorem for the multivariable activation function $\sigma$. ## Acknowledgment We are grateful to Marco Antonio Armenta for informing us about the work [AJ20] and the further useful discussions. We express our gratitude to Shing- Tung Yau for his generous encouragement. The work of S.C. Lau in this paper is partially supported by the Simons collaboration grant. ## 2\. Review of framed quiver moduli Let $Q$ be a directed graph. Denote by $Q_{0},Q_{1}$ the set of vertices and arrows respectively. A quiver representation $V$ with dimension vector $\vec{d}\in\mathbb{Z}_{\geq 0}^{Q_{0}}$ associates each arrow $a$ with a matrix $V(a)$ of size $d_{h(a)}\times d_{t(a)}$ (where $h(a),t(a)$ denote the head and tail vertices of $a$ respectively). The set of complex quiver representations with dimension $\vec{d}$ form a vector space denoted by $R_{\vec{d}}(Q)$. The moduli space of quiver representations is a GIT quotient of $R_{\vec{d}}(Q)$ by the group of isomorphisms $\mathrm{GL}(\vec{d})=\prod_{i\in Q_{0}}\mathrm{GL}(d_{i},\mathbb{C})$ [Kin94], where $\mathrm{GL}(\vec{d})$ acts on $R_{\vec{d}}(Q)$ via $g\cdot(V(a):a\in Q_{1})=(g_{h(a)}\cdot V(a)\cdot g_{t(a)}^{-1}:a\in Q_{1}).$ (2.1) In the applications we consider in this paper, since the vector space over the input and output vertices are equipped with fixed basis with physical meanings, we need to use framed quiver representations [Nak94, Nak01, CB03, Rei08]. Let $\vec{d},\vec{n}\in\mathbb{Z}_{\geq 0}^{Q_{0}}$. $\vec{n}$ will be the dimension vector for the framing, which is a linear map $e^{(i)}:\mathbb{C}^{n_{i}}\to V_{i}$ at each $i\in Q_{0}$ (where $V_{i}=\mathbb{C}^{d_{i}}$). Since we will take a quotient by $\mathrm{GL}(\vec{d})$, we shall think of $V_{i}$ as a vector space without a preferred basis, while $\mathbb{C}^{n_{i}}$ is equipped with the standard basis. ###### Definition 2.1. The vector space of framed representations is given by $R_{\vec{n},\vec{d}}=R_{\vec{d}}\times\bigoplus_{i\in Q_{0}}\mathrm{Hom}(\mathbb{C}^{n_{i}},\mathbb{C}^{d_{i}}).$ It carries a natural action of $\mathrm{GL}(\vec{d})$ given by $g\cdot(V,e)=(g\cdot V,(ge^{(i)}:i\in Q_{0}))$, where $g\cdot V$ is given by Equation (2.1). We need to remove unstable framed representations from $R_{\vec{n},\vec{d}}$ in order to get a nice quotient by $\mathrm{GL}(\vec{d})$. ###### Theorem 2.2 ([Nak96]). $(V,e)\in R_{\vec{n},\vec{d}}$ is called stable if there is no proper subrepresentation $U$ of $V$ which contains $\mathrm{Im}\,e$. The set of all stable points of $R_{\vec{n},\vec{d}}$ is denoted by $R_{\vec{n},\vec{d}}^{s}$. Then the quotient $\mathcal{M}_{\vec{n},\vec{d}}:=R_{\vec{n},\vec{d}}^{s}/\mathrm{GL}(\vec{d})$ is a smooth variety, which is called to be a framed quiver moduli. Actually $\mathcal{M}_{\vec{n},\vec{d}}$ can be formulated as a GIT quotient [CB03, Rei08]. Namely, by adding an extra vertex labeled as $\infty$ to the quiver and $n_{i}$ arrows from the vertex $\infty$ to the vertex $i$, $(V,e)$ can be identified as a usual representation of this bigger quiver with the dimension vector $(\vec{d},1)$. The above stability condition can be rewritten as slope stability, and hence it is a GIT quotient [Kin94]. Since $(d,1)$ is a primitive vector, $\mathcal{M}_{\vec{n},\vec{d}}$ is a smooth fine moduli. There are universal vector bundles $\mathcal{V}_{i}$ over $\mathcal{M}_{\vec{n},\vec{d}}$ corresponding to each vertex $i$, with fibers $\mathcal{V}_{i}\|_{[V,e]}=V_{i}$. ###### Example 2.3. For the quiver with a single vertex and no arrow, and $n>d$, $\mathcal{M}_{n,d}=\operatorname{Gr}(n,d)=\\{e\in\mathrm{Hom}(\mathbb{C}^{n},\mathbb{C}^{d}):e\textrm{ is surjective}\\}/\mathrm{GL}_{\vec{d}}$ is the (dual) Grassmannian. We have the tautological bundle $\mathcal{V}$ over $\operatorname{Gr}(n,d)$. (Note that this tautological bundle is dual to the one on $\operatorname{Gr}(d,n)\cong\operatorname{Gr}(n,d)$.) The topology of $\mathcal{M}_{\vec{n},\vec{d}}$ is well-understood. Let’s make an ordering of the vertices. Namely the vertices are labeled by $\\{1,\ldots,N\\}$, such that $i<j$ implies there is no arrow going from $j$ to $i$. Such a labeling exists if $Q$ has no oriented cycle. ###### Theorem 2.4 (Reineke [Rei08]). Assume $Q$ has no oriented cycle. Consider the chain of iterated Grassmann bundles $M^{(N)}\stackrel{{\scriptstyle p_{N}}}{{\to}}M^{(N-1)}\stackrel{{\scriptstyle p_{N-1}}}{{\to}}\ldots\stackrel{{\scriptstyle p_{2}}}{{\to}}M^{(1)}\stackrel{{\scriptstyle p_{1}}}{{\to}}\mathrm{pt}$ (where $\mathrm{pt}$ denotes a singleton) defined by induction: $M^{(i)}=\operatorname{Gr}_{M^{(i-1)}}\left(\underline{\mathbb{C}^{n_{i}}}\oplus\bigoplus_{j\to i}p_{i-1}^{}\dots p_{j+1}^{}(S_{j}),d_{i}\right)\to M^{(i-1)},$ where $S_{i}$ denotes the tautological bundle on $M_{i}$ (as a Grassmann bundle over $M_{i-1}$). (The direct sum is over each arrow $j\to i$.) Then $\mathcal{M}_{\vec{n},\vec{d}}\cong M^{(N)}$, with universal bundles $\mathcal{V}_{i}\cong p_{N}^{}\dots p_{i+1}^{}S_{i}$ for all $i\in Q_{0}$. ###### Corollary 2.5 (Reineke [Rei08]). The Poincare polynomial of nonempty $\mathcal{M}_{\vec{n},\vec{d}}$ is given by $\prod_{i\in Q_{0}}\binom{n_{i}+\sum_{j\to i}d_{j}}{d_{i}}_{q^{2}}$ where $\binom{n}{d}_{q}=\prod_{k=1}^{d}\frac{q^{n-d+k}-1}{q^{k}-1}.$ ###### Remark 2.6. In [Rei08], the framing $e$ goes in the other direction (from $\mathbb{C}^{d_{i}}$ to $\mathbb{C}^{n_{i}}$). The above theorem is stated in the dual way, which is the convention we take for the rest of this paper. ###### Example 2.7. Consider the $A_{3}$-quiver which has three vertices $i=1,2,3$ and two arrows $a_{1}:1\to 2,a_{2}:2\to 3$. Suppose $n_{1}=d_{1}$, $n_{2}=d_{2}+1$ and $n_{3}=d_{3}$. Then the iterated Grassmann bundle is $M^{(3)}\to M^{(2)}\to M^{(1)}$, where $M^{(1)}=\operatorname{Gr}(d_{1},d_{1})=\mathrm{pt}$ (and its tautological bundle is the vector space $\mathbb{C}^{d_{1}}$); $M^{(2)}=\operatorname{Gr}(d_{2}+1+d_{1},d_{2})$ is equipped with the tautological bundle $S_{2}$ of rank $d_{2}$; $M^{(3)}=\operatorname{Gr}_{\operatorname{Gr}(d_{2}+1+d_{1},d_{2})}(\underline{\mathbb{C}}^{d_{3}}\oplus S_{2},d_{3})$ is a Grassmannian bundle over $\operatorname{Gr}(d_{2}+1+d_{1},d_{2})$ with fibers $\operatorname{Gr}(d_{2}+d_{3},d_{3})$. The corresponding Poincare Polynomial will be $\left(\prod_{k=1}^{d_{3}}\frac{q^{2(d_{2}+k)}-1}{q^{2k}-1}\right)\left(\prod_{k=1}^{d_{2}}\frac{q^{2(d_{1}+1+k)}-1}{q^{2k}-1}\right).$ See Figure 2. ${1}$${Gr(d_{1},d_{1})=\mathrm{pt}}$${2}$${Gr(d_{2}+1+d_{1},d_{2})}$${3}$${\operatorname{Gr}_{\operatorname{Gr}(d_{2}+1+d_{1},d_{2})}(\underline{\mathbb{C}}^{d_{3}}\oplus S_{2},d_{3})}$$\scriptstyle{a_{1}}$$\scriptstyle{a_{2}}$ Figure 2. The iterated Grassmann bundles associated to the $A_{3}$ quiver. ## 3\. Hermitian Metric over framed quiver moduli In constructing fiber-bundle endomorphisms, it will be crucial to consider Kähler metrics on universal bundles. In this section, we find a beautiful formula for the canonical metric on the universal bundle $\mathcal{V}_{i}$ over $\mathcal{M}_{n,d}$ written in homogeneous coordinates. Using this formula, we then show that the sum of Ricci curvatures over the vertices $i$ give a Kähler metric on $\mathcal{M}_{\vec{n},\vec{d}}$. First, let us begin by recalling the typical example $\operatorname{Gr}(n,k)$. ### 3.1. The Grassmannian Consider $\operatorname{Gr}\left(n,k\right)=\operatorname{Mat}^{\mathbb{C}}_{n,k}\sslash_{\chi=1}U(k)=\left\\{e\in\operatorname{Mat}^{\mathbb{C}}_{n,k}:ee^{}=I_{k}\right\\}\big{/}U\left(k\right)$ for $n\geq k$. Here we have used the dual description which better matches the frame convention used in this paper. Namely, $\operatorname{Gr}\left(n,k\right)$ parametrizes $k$-dimensional quotient vector spaces of a fixed $n$-dimensional vector space as opposed to $k$-dimensional subspaces. The moment map for the standard $U(k)$-action on $\operatorname{Mat}^{\mathbb{C}}_{n,k}$ is $ee^{}:\operatorname{Mat}^{\mathbb{C}}_{n,k}\to\mathbf{i}\mathfrak{u}_{k}$. We have taken the moment map level $\chi=1$ in the above symplectic reduction. Note that $U(k)$ is acting on the left, although in the above expression $U(k)$ appears on the right. Writing $e=\left(b,p\right)$ where $b\in\operatorname{Mat}^{\mathbb{C}}_{k,k}$ and $p\in\operatorname{Mat}^{\mathbb{C}}_{n-k,k}$, the moment-map equation $ee^{}=I_{k}$ becomes $bb^{}+pp^{}=I_{k}.$ We shall consider the chart defined by $U=\\{[b,p]\in\operatorname{Gr}(n,k):\det b\not=0\\}\cong\operatorname{Mat}^{\mathbb{C}}_{n-k,k}$ where the identification is given by the holomorphic coordinates $\zeta^{h}=b^{-1}p\in\operatorname{Mat}^{\mathbb{C}}_{n-k,k}.$ We also have the symplectic coordinates $\zeta^{u}=\left(b^{}b\right)^{\frac{1}{2}}b^{-1}p\in\operatorname{Mat}^{\mathbb{C}}_{n-k,k}.$ The entries of $\zeta^{u}$ are not meromorphic functions. On the other hand, $\zeta^{u}$ has the advantage that it satisfies the moment-map equation $b^{}b+\zeta^{u}(\zeta^{u})^{}=I_{k}.$ (3.1) (Note that the first term is $b^{}b$ instead of $bb^{}$.) The construction of $\zeta^{u}$ uses the polar decomposition $b=\left(b\left(b^{}b\right)^{-\frac{1}{2}}\right)\left(b^{}b\right)^{\frac{1}{2}}$ where $\left(b\left(b^{}b\right)^{-\frac{1}{2}}\right)\in U(k)$ and $\left(b^{}b\right)^{\frac{1}{2}}\in\mathbf{i}\mathfrak{u}(k)$ is positive definite. We obtain the coordinates $\zeta^{u}$ by observing $[b,p]=\left[\left(b^{}b\right)^{\frac{1}{2}},\zeta^{u}\right]$ using the left-$U(k)$-action, such that the first component $\left(b^{}b\right)^{\frac{1}{2}}$ is Hermitian, and is determined $\zeta^{u}$ due to the moment-map equation (3.1). The two coordinate systems are related by $\zeta^{u}=\left(b^{}b\right)^{\frac{1}{2}}\cdot\zeta^{h}.$ (3.2) Let $S$ be the tautological vector bundle whose fibers are the quotient vector spaces. (This is dual to the tautological bundle of $\operatorname{Gr}(k,n)\cong\operatorname{Gr}(n,k)$.) It can be written as the quotient of the trivial bundle: $S=\left(\left\\{ee^{}=I_{k}\right\\}\times\mathbb{C}^{k}\right)\big{/}U\left(k\right)$ where the left action of $U(k)$ on $\mathbb{C}^{k}$ is the standard one. We now take the standard metric on $\mathbb{C}^{k}$, which is preserved by $U(k)$ and hence descends to a metric $H$ of $S$. Denote the standard basis of $\mathbb{C}^{k}$ by $\epsilon_{j}$ for $j=1,\ldots,k$. Under this metric, we have the lifting of a local Hermitian frame over the chart $U=\\{\det b\not=0\\}$ being $u_{i}=b\left(b^{}b\right)^{-\frac{1}{2}}\cdot\epsilon_{i}$ since $\left[b,p,b\left(b^{}b\right)^{-\frac{1}{2}}\epsilon_{i}\right]\sim\left[\left(b^{}b\right)^{\frac{1}{2}},\zeta^{u},\epsilon_{i}\right]$. We also have the lifting of a local holomorphic frame $h_{i}=b\epsilon_{i}=b\left(b^{}b\right)^{\frac{1}{2}}b^{-1}u_{i}$ since $\left[b,p,b\epsilon_{i}\right]\sim\left(I_{k},\zeta^{h},\epsilon_{i}\right)$. The two frames are related as follows. ###### Lemma 3.1. $h_{i}=u_{j}a_{i}^{j}$ where $\left(a_{i}^{j}\right)=\left(b^{}b\right)^{\frac{1}{2}}$, $i$ is indexing the colomns and $j$ is indexing the rows. ###### Proof. Consider $\left(h_{1}\ldots h_{k}\right)=b\left(b^{}b\right)^{\frac{1}{2}}b^{-1}\left(u_{1}\ldots u_{k}\right)=\left(u_{1}\ldots u_{k}\right)\left(a_{i}^{j}\right).$ Thus $\left(a_{i}^{j}\right)=\left(u_{1}\ldots u_{k}\right)^{-1}b\left(b^{}b\right)^{\frac{1}{2}}b^{-1}\left(u_{1}\ldots u_{k}\right)$. $\left(u_{1}\ldots u_{k}\right)=b\left(b^{}b\right)^{-\frac{1}{2}}$ since $u_{i}=b\left(b^{}b\right)^{-\frac{1}{2}}\epsilon_{i}$. Result follows. ∎ ###### Proposition 3.2. The metric $H$ defined above on the tautological bundle $S$ is represented by the matrix $\left(I_{k}+\zeta^{h}\left(\zeta^{h}\right)^{}\right)^{-1}$ in the local holomorphic frame $h_{i}$ and the local coordinates $\zeta^{h}$. ###### Proof. Using Lemma 3.1, $\left(H\left(h_{i},h_{p}\right)\right)=\sum_{j,l}\left(H\left(a_{i}^{j}u_{j},a_{p}^{l}u_{l}\right)\right)=\sum_{j,l}\left(\overline{a_{i}^{j}}a_{p}^{l}H\left(u_{j},u_{l}\right)\right)=\sum_{j}\left(\overline{a_{i}^{j}}a_{p}^{j}\right)=b^{}b$ where $i$ is indexing the rows and $p$ is indexing the columns. By the moment map equation (3.1) and the relation (3.2), $b^{}b+\left(b^{}b\right)^{\frac{1}{2}}\cdot\zeta^{h}\left(\zeta^{h}\right)^{}\left(b^{}b\right)^{\frac{1}{2}}=I_{k}.$ Then $I_{k}+\zeta^{h}\left(\zeta^{h}\right)^{}=\left(b^{}b\right)^{-1}.$ Hence $H=\left(I_{k}+\zeta^{h}\left(\zeta^{h}\right)^{}\right)^{-1}.$ ∎ ###### Example 3.3. Let’s consider the simplest example: $\mathbb{P}^{1}=\operatorname{Gr}(2,1)=(\mathbb{C}^{2}-\\{0\\})/\mathbb{C}^{\times}=\mathbb{S}^{3}/\mathrm{U}\left(1\right).$ The tautological bundle for $\operatorname{Gr}(2,1)$ is $S=\left(\mathbb{S}^{3}\times\mathbb{C}\right)/U\left(1\right)$ where $U(1)$ acts on $\mathbb{C}$ in the standard way, and it acts on both factors on the left. (Note that this is dual to the usual notion of the tautological bundle of $\operatorname{Gr}(1,2)=\mathbb{P}^{1}$, since we are now considering the family of quotient lines of $\mathbb{C}^{2}$, which are dual to subspaces of $\mathbb{C}^{2}$.) Let’s take the standard metric on $\mathbb{C}$. We have the local Hermitian frame (over $z_{1}\neq 0$) $u$ given by $\left(z_{1},z_{2},z_{1}/\left\|z_{1}\right\|\right)\stackrel{{\scriptstyle U(1)}}{{\sim}}\left(\left\|z_{1}\right\|,\zeta^{u},1\right)$ where $\zeta^{u}$ is the coordinate of $\mathbb{P}^{1}$ which belongs to the open unit disc, and $\|z_{1}\|$ is determined by the moment-map equation $\left\|z_{1}\right\|^{2}+\left\|z_{2}\right\|^{2}=\left\|z_{1}\right\|^{2}+\left\|\zeta^{u}\right\|^{2}=1.$ We also have the local holomorphic frame $h$ defined by $\left(z_{1},z_{2},z_{1}\right)\stackrel{{\scriptstyle\mathbb{C}^{\times}}}{{\sim}}\left(1,\zeta^{h},1\right).$ The unitary and holomorphic coordinates are related by $\zeta^{u}=\left\|z_{1}\right\|\cdot\zeta^{h}=(1-\|\zeta^{u}\|^{2})\cdot\zeta^{h}$ for $\zeta^{h}\in\mathbb{C}$. The frames are related by $h=\left\|z_{1}\right\|\cdot u.$ The Hermitian frame $u$ always have length one. Writing the metric in the holomorphic frame $h$: $\displaystyle\left\|h\right\|^{2}$ $\displaystyle=\left\|z_{1}\right\|^{2}=1-\left\|\zeta^{u}\right\|^{2}=1-\left\|z_{1}\right\|^{2}\left\|\zeta^{h}\right\|^{2}$ $\displaystyle=1-\left(1-\left\|z_{1}\right\|^{2}\left\|\zeta^{h}\right\|^{2}\right)\left\|\zeta^{h}\right\|^{2}=\ldots$ $\displaystyle=\frac{1}{1+\left\|\zeta^{h}\right\|^{2}}.$ This is the standard metric on $\mathcal{O}_{\mathbb{P}^{1}}(1)$, whose curvature gives the Fubini-Study metric on $\mathbb{P}^{1}$. ### 3.2. Metric on framed quiver moduli We have seen that the standard metric on the trivial bundle over $\operatorname{Mat}^{\mathbb{C}}_{n,k}$ descends to give the standard metric on $\operatorname{Gr}(n,k)$. However, it turns out that for the framed quiver moduli, the standard metric on the trivial bundle over $R_{\vec{n},\vec{d}}$ is not good from the GIT quotient point of view, namely it is not equivariant under $\mathrm{GL}_{\vec{d}}$. In this section, we find a nice metric over $R_{\vec{n},\vec{d}}$ which is equivariant under $\mathrm{GL}_{\vec{d}}$. Recall from the last section that $\mathcal{M}_{\vec{n},\vec{d}}=R_{\vec{n},\vec{d}}^{s}/\mathrm{GL}_{\vec{d}}$. The universal bundle over the vertex $i$ is given by $\mathcal{V}_{i}=\left(R_{\vec{n},\vec{d}}^{s}\times\mathbb{C}^{d_{i}}\right)\big{/}\mathrm{GL}_{\vec{d}}$ where $\mathrm{GL}_{\vec{d}}$ acts diagonally on the left, the factor $\mathrm{GL}(d_{i},\mathbb{C})$ of $\mathrm{GL}_{\vec{d}}$ acts on $\mathbb{C}^{d_{i}}$ in the standard way, and other factors of $\mathrm{GL}_{\vec{d}}$ act trivially on $\mathbb{C}^{d_{i}}$. There is an equivalent description of $\mathcal{M}=\mathcal{M}_{\vec{n},\vec{d}}$ and the universal bundle $\mathcal{V}_{i}$ in terms of symplectic quotient. Namely, let $\mu:R_{\vec{n},\vec{d}}\to\mathbf{i}\mathfrak{u}_{\vec{d}}$ be the moment map. Explicitly, $\mu=(\mu_{i})_{i\in Q_{0}}$ where $\mu_{i}=e^{(i)}(e^{(i)})^{}-\sum_{t(a)=i}V_{a}^{}V_{a}+\sum_{h(a^{\prime})=i}V_{a^{\prime}}V_{a^{\prime}}^{}.$ Then define $\mathcal{M}_{\vec{n},\vec{d}}=\mu^{-1}\left\\{-c\right\\}\big{/}U_{\vec{d}}$ for the following level $c$. ###### Lemma 3.4. The slope stability condition $\left(1,\vec{0}\right)\in\mathbb{C}^{\hat{Q}_{0}}$ corresponds to the moment- map level $c=\left(-I_{d_{i}}\right)_{i\in Q_{0}}\in\mathbf{i}\mathfrak{u}_{\vec{d}}$, where $I_{k}$ denotes the identity matrix of rank $k$. ###### Proof. The character taken in King’s stability [Kin94] corresponding to $\left(1,\vec{0}\right)$ is $\left(1+\Sigma\vec{d}\right)\left(\left(1,\vec{0}\right)-\frac{1}{1+\Sigma\vec{d}}\left(1,\vec{1}\right)\right)=\left(\Sigma\vec{d},-1,\ldots,-1\right)$ where $\Sigma\vec{d}=\sum_{i\in Q_{0}}d_{i}$, and the first entry is over the root vertex. (Note that there is no group action over the root vertex.) Thus we should take $c$ to be $-I_{d_{i}}$ over each vertex $i$. ∎ The universal bundle over the vertex $i$ is then given by $\mathcal{V}_{i}=\left(\mu^{-1}\left\\{I_{\vec{d}}\right\\}\times\mathbb{C}^{d_{i}}\right)\big{/}U_{\vec{d}}.$ Let’s review some very basic definitions about group actions. ###### Definition 3.5. Suppose a Lie group $G$ acts on a vector bundle $V\stackrel{{\scriptstyle\pi}}{{\to}}M$ equivariantly, namely, $g\circ\pi=\pi\circ g$ for all $g\in G$, and the action is fiberwise linear. A metric $H$ on $V$ is said to be $G$-equivariant if $H_{x}\left(v,w\right)=H_{g\cdot x}\left(g\cdot v,g\cdot w\right).$ Writing in matrix form when $G=\mathrm{GL}(n,\mathbb{C})$, the above equation is $v^{}\cdot H_{x}\cdot w=v^{}\cdot\left(g^{}\cdot H_{g\cdot x}\cdot g\right)\cdot w,$ that is, $(g^{})^{-1}\cdot H_{x}\cdot g^{-1}=H_{g\cdot x}.$ (3.3) The following easily follows from the definition. ###### Lemma 3.6. Suppose a Lie group $G$ acts on a vector bundle $V\stackrel{{\scriptstyle\pi}}{{\to}}M$ equivariantly and fiberwise linearly, and the action of $G$ on $M$ is free and proper. A Hermitian form $H$ on $V$ descends to the corresponding bundle over the quotient $M/G$ if and only if $H$ is $G$-equivariant. For framed quiver varieties, we have the framing map $e^{(j)}\colon R_{\vec{n},\vec{d}}^{s}\rightarrow\mathrm{Hom}\left(\mathbb{C}^{n_{j}},\mathbb{C}^{d_{j}}\right)$ for each vertex $j$. Using this, we cook up a $\mathrm{GL}_{\vec{d}}$-invariant Hermitian form on the trivial bundle $\underline{\mathbb{C}^{d_{i}}}\to R_{\vec{n},\vec{d}}$, which descends to a metric on $\mathcal{V}_{i}\to\mathcal{M}_{\vec{n},\vec{d}}$. ###### Theorem 3.7. Suppose $Q$ has no oriented cycle. Fix $i\in Q_{0}$. Let $\rho$ be the row vector whose entries are $V_{\gamma}e^{\left(t(\gamma)\right)}$, where $\gamma$ is any path whose head $h(\gamma)$ is $i$ (including the trivial path), $t(\gamma)$ denotes its tail, and $V_{\gamma}\in\mathrm{Hom}(\mathbb{C}^{d_{t(\gamma)}},\mathbb{C}^{d_{h(\gamma)}})$ is the representing matrix of $\gamma$. This defines a map $V_{\gamma}e^{\left(t(\gamma)\right)}:R_{\vec{n},\vec{d}}\to\mathrm{Hom}(\mathbb{C}^{n_{t(\gamma)}},\mathbb{C}^{d_{i}}).$ Take $\rho\rho^{}=\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}:R_{\vec{n},\vec{d}}\to\mathrm{End}(\mathbb{C}^{d_{i}}).$ Then $H=(\rho\rho^{})^{-1}$ is $\mathrm{GL}_{\vec{d}}$-equivariant, and it descends to a metric on $\mathcal{V}_{i}$ over $\mathcal{M}_{\vec{n},\vec{d}}$. ###### Proof. $(\rho\rho^{})^{-1}$ is $\mathrm{GL}_{\vec{d}}$-equivariant: $\displaystyle H_{\left(g_{h(a)}V_{a}g^{-1}_{t(a)},g_{j}e^{(j)}\right)_{a\in Q_{1},j\in Q_{0}}}$ $\displaystyle=\left(\sum_{\gamma}\left(g_{h(\gamma)}V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(g_{h(\gamma)}V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)^{-1}$ $\displaystyle=(g^{}_{i})^{-1}\left(\sum_{\gamma}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)^{-1}g_{i}^{-1}$ for all $g\in\mathrm{GL}_{\vec{d}}$. By Lemma 3.6, it descends to the bundle $\mathcal{V}_{i}$ of the quotient. Then we prove that the matrix-valued function $H_{(V_{a},e^{(j)})_{a\in Q_{1},j\in Q_{0}}}=(\rho\rho^{})^{-1}$ defines a positive-definite metric on the trivial bundle $\underline{\mathbb{C}^{d_{i}}}$ _over the moment map level $\mu^{-1}(I_{\vec{d}})$_ (rather than the whole $R_{\vec{n},\vec{d}}$). We prove by induction on the vertices that $\rho\rho^{}=I_{d_{i}}+B$ where $B$ is a semi-positive-definite Hermitian matrix, and hence $\rho\rho^{}$ is positive definite (and so does $(\rho\rho^{})^{-1}$). Since the quiver does not have oriented cycle, $Q_{0}$ can be ordered such that $i<j$ whenever there is an arrow $i\to j$. Let $i_{0}$ be the minimal vertex. At $i_{0}$, there is no incoming arrow (other than the framing), and the moment-map equation reads $e^{(i_{0})}(e^{(i_{0})})^{}=I_{d_{i_{0}}}+\sum_{t(a)=i_{0}}V_{a}^{}V_{a}.$ $\sum_{t(a)=i_{0}}V_{a}^{}V_{a}$ is semi-positive definite: $v^{}\cdot V_{a}^{}V_{a}\cdot v=\\|V_{a}\cdot v\\|^{2}\geq 0$ for any column vector $v$. Thus the statement is true for $\rho\rho^{}=e_{i_{0}}e_{i_{0}}^{}$. Suppose the statement is true for all vertices less than $i\in Q_{0}$. At $i$, the moment-map equation is $e^{(i)}(e^{(i)})^{}=I_{d_{i}}+\sum_{t(a)=i}V_{a}^{}V_{a}-\sum_{h(a^{\prime})=i}V_{a^{\prime}}V_{a^{\prime}}^{}.$ Then $\displaystyle\rho\rho^{}$ $\displaystyle=e^{(i)}(e^{(i)})^{}+\sum_{h(a)=i}V_{a}\rho_{(t(a))}\rho_{(t(a))}^{}V_{a}^{}$ $\displaystyle=I_{d_{i}}+\sum_{t(a)=i}V_{a}^{}V_{a}-\sum_{h(a^{\prime})=i}V_{a^{\prime}}V_{a^{\prime}}^{}+\sum_{h(a^{\prime})=i}V_{a^{\prime}}(I_{d_{t(a^{\prime})}}+B_{t(a^{\prime})})V_{a^{\prime}}^{}$ $\displaystyle=I_{d_{i}}+\sum_{t(a)=i}V_{a}^{}V_{a}+\sum_{h(a^{\prime})=i}V_{a^{\prime}}B_{t(a^{\prime})}V_{a^{\prime}}^{}$ where $\rho_{(t(a))}=\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)_{h(\gamma)=t(a)}$, which by inductive assumption can be written as $I_{d_{t(a^{\prime})}}+B_{t(a^{\prime})}$ where $B_{t(a^{\prime})}$ is semi- positive definite. The matrices $V_{a}^{}V_{a}$ and $V_{a^{\prime}}B_{t(a^{\prime})}V_{a^{\prime}}^{}$ are semi-positive definite: $v^{}V_{a^{\prime}}B_{t(a^{\prime})}V_{a^{\prime}}^{}v=(V_{a^{\prime}}^{}v)^{}B_{t(a^{\prime})}(V_{a^{\prime}}^{}v)\geq 0$ for all $v$. This proves the statement for the vertex $i$. ∎ The expression $(\rho\rho^{})^{-1}$ can be understood as follows. $\rho^{}$ embeds the dual $V_{i}^{}$ into the dual frame which is a trivial bundle equipped with the standard metric. This gives an induced metric on $V_{i}^{}$, which is $\rho\rho^{}$ written in matrix form. Taking the dual, we get the metric $H_{i}=(\rho\rho^{})^{-1}$ on $V_{i}$. By construction, the metrics on the dual $\mathcal{V}_{i}^{}$ (still denoted as $H_{i}$) have the following nice property. Inductively, it gives nice expressions of $H_{i}$ in terms of _holomorphic coordinates_. ###### Proposition 3.8. Suppose $Q$ has no oriented cycle. For $v,w\in(\mathcal{V}_{i})^{}$, $H_{i}(v,w)=H_{0}((e^{(i)})^{}(v),(e^{(i)})^{}(w))+\sum_{h(a)=i}H_{t(a)}(a^{}(v),a^{}(w))$ where $H_{0}$ denotes the trivial metric on the trivial bundle, and $e^{(i)},a$ are denoting the holomorphic bundle maps corresponding to the framing and arrow maps respectively. ###### Proof. The metric on $\mathcal{V}_{i}^{}$ is given by the matrix $\rho\rho^{}$. Then the above equation follows from $\rho\rho^{}=e^{(i)}(e^{(i)})^{}+\sum_{h(a)=i}V_{a}\rho_{(t(a))}\rho_{(t(a))}^{}V_{a}^{}.$ ∎ ###### Remark 3.9. As we have seen, the Grassmannian $\operatorname{Gr}(n,k)$ can be understood as the framed moduli for the quiver which has one vertex and no arrow. The matrix $e\in\mathrm{Hom}(\mathbb{C}^{n},\mathbb{C}^{k})$ is the framing map. Then the moment map equation implies $\rho\rho^{}=ee^{}=I_{k}$ in the above proposition. This is the standard metric on the trivial bundle $\underline{\mathbb{C}^{k}}$ that we have used in the last subsection. In particular $\rho\rho^{}=(\rho\rho^{})^{-1}$ in this case. But this is not true for other quivers. ###### Remark 3.10. Note that the above becomes an infinite sum if the quiver has oriented cycles. The $\mathrm{GL}_{\vec{d}}$-equivariance still holds. We should restrict to the open subset of $R^{s}_{n,d}$ that $(\rho\rho^{})^{-1}$ is convergent. In the next subsection, we will prove that the same expression defines a metric for any given quiver. The GIT description will be important to the proof of Theorem 3.14. There is a residual symmetry $U_{\vec{n}}=\prod_{i\in Q_{0}}U({n_{i}})$ acting on $\mathcal{M}_{\vec{n},\vec{d}}$. Actually, there is a bigger symmetry by the non-compact group $\mathrm{GL}(\hat{W})$ [Rei08]. $U_{\vec{n}}$ is considered here since this is the symmetry of the metric $H$ on $\mathcal{V}_{i}$ as we shall see. ###### Definition 3.11. The right residual action of $U_{\vec{n}}$ on $\mathcal{M}$ is defined as $\left[(V_{a},e^{(j)})_{a\in Q_{1},j\in Q_{0}}\right]\cdot g=\left[(V_{a},e^{(j)}\circ g_{j})_{a\in Q_{1},j\in Q_{0}}\right]$ for $g=(g_{j}\in U(d_{i}))_{j\in Q_{0}}\in U_{\vec{n}}$. Since the above commutes with the left action of $\mathrm{GL}_{\vec{d}}$, the action is well-defined on $\mathcal{M}$. ###### Lemma 3.12. There is a canonical lift of the action of $U_{\vec{n}}$ on $\mathcal{M}_{\vec{n},\vec{d}}$ to the universal bundle $\mathcal{V}_{i}$, so that the bundle map $\mathcal{V}_{i}\rightarrow\mathcal{M}_{\vec{n},\vec{d}}$ is equivariant. ###### Proof. $\mathcal{V}_{i}$ is the $\mathrm{GL}_{\vec{d}}$-quotient of the trivial bundle $R^{s}_{n,d}\times V_{i}$. $U_{\vec{n}}$ acts on this by acting on the component $V_{i}$ trivially. This action commutes with the left action of $\mathrm{GL}_{\vec{d}}$ on $R^{s}_{n,d}\times V_{i}$, and hence descends to act on $\mathcal{V}_{i}$. ∎ ###### Lemma 3.13. The metric defined in Theorem 3.7 are $U_{\vec{n}}$-invariant. ###### Proof. For any $g\in U_{\vec{n}}$, since $g_{j}g_{j}^{}=I_{n_{j}}$ for any $j\in Q_{0}$, $\displaystyle H_{\left(V_{a},e^{(j)}\right)_{a\in Q_{1},j\in Q_{0}}\cdot g}$ $\displaystyle=\left(\sum_{\gamma}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\cdot g_{t(\gamma)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\cdot g_{t(\gamma)}\right)^{}\right)^{-1}$ $\displaystyle=\left(\sum_{\gamma}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)^{-1}=H_{\left(V_{a},e^{(j)}\right)_{a\in Q_{1},j\in Q_{0}}}.$ ∎ Recall from Theorem 2.4 that $\mathcal{M}_{\vec{n},\vec{d}}$ is the total space of an iterated Grassmann bundle $M^{(N)}\stackrel{{\scriptstyle p_{N}}}{{\to}}M^{(N-1)}\stackrel{{\scriptstyle p_{N-1}}}{{\to}}\ldots\stackrel{{\scriptstyle p_{2}}}{{\to}}M^{(1)}\stackrel{{\scriptstyle p_{1}}}{{\to}}\mathrm{pt}$. Moreover, $\mathcal{V}_{i}$ is the pull-back of the tautological bundle $S_{i}$ of the Grassmann bundle $M^{(i)}=\operatorname{Gr}_{M^{(i)}}(\underline{\mathbb{C}^{n_{i}}}\oplus\bigoplus_{j\to i}p_{i}^{}\ldots p_{j+1}^{}(S_{j}),d_{i})\to M^{(i-1)}$. The tautological bundle of the Grassmannian is equipped with a standard metric as illustrated in Section 3.1. Inductively, $\mathcal{V}_{i}$ is also equipped with a pull- back metric. We show that this equals to the metric $H$ we defined by an explicit formula. ###### Theorem 3.14. For all $i\in Q_{0}$, the metric $H=\left(\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)^{-1}$ equals to the metric on $\mathcal{V}_{i}$ constructed from the iterated Grassmann bundle. ###### Proof. As the proof of Theorem 3.7, we do induction on the vertices, which are totally ordered such that $i<j$ whenever there is an arrow $i\to j$. First, we have the GIT fiber-bundle map $\mathcal{M}\to M^{(j)}$, where $M^{(j)}$ is the framed moduli of the quiver $Q^{(j)}$ (which is obtained by removing all vertices $k>j$ and the corresponding arrows from $Q$). This map $(V,e)\mapsto(V^{\prime},e^{\prime})$ is simply forgetting all the irrelevant arrow maps and frame maps that are not supported on the subquiver $Q^{(j)}$. Note that the stability condition is preserved: any subrepresentation $R^{\prime}\subset V^{\prime}$ can be extended to a subrepresentation $R\subset V$ by assigning the whole $V_{k}$ to the additional vertices $k>j$ (and the arrow maps just come from restriction). If $\mathrm{Im}(e^{\prime})\subset R^{\prime}$, then $\mathrm{Im}(e)\subset R$. Note that here we use the GIT description instead of symplectic reduction since $(V^{\prime},e^{\prime})$ no longer satisfies the moment-map equation in defining $M^{(j)}$ (even when $(V,e)$ satisfies the moment-map equation for $\mathcal{M}$). We start with the minimal vertex $i_{0}$. The quiver $Q^{(i_{0})}$ is simply a single vertex, and the corresponding framed moduli is $M^{(i_{0})}=\operatorname{Gr}(n_{i_{0}},d_{i_{0}}).$ The universal bundle $\mathcal{V}_{i_{0}}^{Q^{(i_{0})}}$ is the tautological bundle of $M^{(i_{0})}=\operatorname{Gr}(n_{i_{0}},d_{i_{0}})$. From the last subsection, the standard metric of $\mathcal{V}_{i_{0}}^{Q^{(i_{0})}}$ is descended from $I_{d_{i_{0}}}$, which is exactly $(e^{Q^{(i_{0})}}(e^{Q^{(i_{0})}})^{})^{-1}$ by the moment map equation for $Q^{(i_{0})}$. The statement is trivial in this case. Now consider the vertex $i$. Denote the vertex right before $i$ by $i-1$. For the quiver $Q^{(i-1)}$, assume that the two metrics on the universal bundle $\mathcal{V}^{Q^{(i-1)}}_{j}$ agree for every $j\in Q^{(i-1)}_{0}$. We have the bundle map $\pi:M^{(i)}\to M^{(i-1)}$, and $\mathcal{V}^{Q^{(i)}}_{j}=\pi^{}\mathcal{V}^{Q^{(i-1)}}_{j}$ for all $j<i$. Moreover, the metric on $\mathcal{V}^{Q^{(i)}}_{j}$ is pull-back from $\mathcal{V}^{Q^{(i-1)}}_{j}$, which equals to $\left(\sum_{h(\gamma)=j}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)^{-1}$ by inductive assumption. The pull-back map does not change the arrow and framing maps of $Q^{(i-1)}$. This proves the statement for $S^{Q^{(i)}}_{j}$ for $j<i$. Consider $\mathcal{V}^{Q^{(i)}}_{i}$, which is the tautological bundle associated to the Grassmannian bundle over $M^{(i-1)}$ parametrizing quotients of $(e^{(i)},(V_{a})_{h(a)=i}):\left.\left(\underline{\mathbb{C}^{d_{i}}}\oplus\left(\bigoplus_{h(a)=i}\mathcal{V}^{Q^{(i-1)}}_{t(a)}\right)\right)\right\|_{x\in M^{(i-1)}}\to\mathbb{C}^{d_{i}}.$ The metric on $(\mathcal{V}^{Q^{(i)}}_{i})^{}$ is induced by the embedding $(e^{(i)},(V_{a})_{h(a)=i})^{}$ to $\left(\underline{\mathbb{C}^{d_{i}}}\oplus\left(\bigoplus_{h(a)=i}\mathcal{V}_{t(a)}\right)\right)^{}$, whose metric is given by $I_{d_{i}}\oplus\bigoplus_{h(a)=i}\left(\sum_{h(\gamma)=t(a)}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)$ by inductive assumption. Thus the induced metric on $(\mathcal{V}^{Q^{(i)}}_{i})^{}$ is $\displaystyle(e^{(i)},(V_{a})_{h(a)=i})\cdot\left(I_{d_{i}}\oplus\bigoplus_{h(a)=i}\left(\sum_{h(\gamma)=t(a)}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)\right)\cdot(e^{(i)},(V_{a})_{h(a)=i})^{}$ $\displaystyle=e^{(i)}(e^{(i)})^{}+\sum_{\begin{subarray}{c}h(a)=i\\\ h(\gamma)=t(a)\end{subarray}}V_{a}\cdot\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\cdot V_{a}^{}$ $\displaystyle=\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}.$ Taking reciprocal gives the metric on $\mathcal{V}^{Q^{(i)}}_{i}$. This proves the metric has the given expression. ∎ ###### Theorem 3.15. The Ricci curvature of the metric on $\bigotimes_{i\in Q_{0}}\mathcal{V}_{i}$ given in Theorem 3.7 defines a Kähler metric on $\mathcal{M}$. ###### Proof. As in Theorem 3.7, denote $\rho=\rho^{\left(i\right)}=\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)_{\gamma:h(\gamma)=i}$ which is a matrix-valued function on the vector space $R_{\vec{n},\vec{d}}$. At each point of $R_{\vec{n},\vec{d}}$, $\rho$ is a linear map from $\widehat{W}_{i}:=\bigoplus_{\gamma:h(\gamma)=i}\mathbb{C}^{n_{t(\gamma)}}$ (3.4) to $V_{i}$. The Ricci curvature of the metric $(\rho\rho^{})^{-1}$ is given by $i\partial\overline{\partial}\log\det\rho\rho^{\mathrm{}}$. We have $\displaystyle\partial\overline{\partial}\log\det\rho\rho^{\mathrm{}}$ $\displaystyle=\partial\left(\mathrm{tr~{}}\left(\left(\rho\rho^{\mathrm{}}\right)^{-1}\overline{\partial}\left(\rho\rho^{\mathrm{}}\right)\right)\right)$ $\displaystyle=\mathrm{tr}\left(\partial\left(\left(\rho\rho^{\mathrm{}}\right)^{-1}\rho\left(\partial\rho\right)^{\mathrm{}}\right)\right)$ $\displaystyle=\mathrm{tr~{}}\left(\left(\rho\rho^{\mathrm{}}\right)^{-1}\partial\rho\left(\partial\rho\right)^{\mathrm{}}+\left(\partial\left(\rho\rho^{\mathrm{}}\right)^{-1}\right)\rho\left(\partial\rho\right)^{\mathrm{}}\right)$ $\displaystyle=\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\partial\rho\right)-\mathrm{tr~{}}\left(\left(\rho\rho^{\mathrm{}}\right)^{-1}\left(\partial\left(\rho\rho^{\mathrm{}}\right)\right)\left(\rho\rho^{\mathrm{}}\right)^{-1}\rho\left(\partial\rho\right)^{\mathrm{}}\right)$ $\displaystyle=\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\partial\rho\right)-\mathrm{tr~{}}\left(\left(\rho\rho^{\mathrm{}}\right)^{-1}\rho\left(\partial\rho\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\left(\left(\partial\rho\right)\rho^{\mathrm{}}\right)\right)$ $\displaystyle=\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\partial\rho\right)-\mathrm{tr~{}}\left(\left(\partial\rho\cdot\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\left(\partial\rho\cdot\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)\right)\right)$ where $\overline{\partial}\rho=0$ since the matrix $\rho$ has polynomial entries in holomorphic coordinates. We can take the singular value decomposition $\rho=U\cdot\left(\text{diag}\left(\lambda_{1},\ldots,\lambda_{d_{i}}\right)\,\,\,0\right)\cdot V^{\mathrm{}}$ where $U\in U\left(d_{i}\right),~{}V\in U\left(\dim\widehat{W}_{i}\right)$, and $\lambda_{i}>0$. ($\lambda_{i}\neq 0$ since $\rho$ is surjective.) Then $\displaystyle\rho\rho^{\mathrm{}}$ $\displaystyle=U\left(\text{diag}\left(\lambda_{1}^{2},\ldots,\lambda_{d_{i}}^{2}\right)\right)U^{\mathrm{}}.$ $\displaystyle\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}$ $\displaystyle=V\left(\begin{array}[]{c}\text{diag}\left(\lambda_{1},\ldots,\lambda_{\alpha\left(i\right)}\right)\\\ 0\end{array}\right)\left(\text{diag}\left(\lambda_{1}^{-1},\ldots,\lambda_{\alpha\left(i\right)}^{-1}\right)\right)U^{\mathrm{}}=V\left(\begin{array}[]{c}I_{\alpha\left(i\right)}\\\ 0\end{array}\right)U^{\mathrm{}}.$ In other words, $\rho^{\mathrm{}}=\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)\left(\rho\rho^{\mathrm{}}\right)^{\frac{1}{2}}$ is decomposed into the rescaling $\rho\rho^{\mathrm{}}$ and the orthogonal embedding $\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)$ to $\mathrm{Im}\rho^{\mathrm{}}\subset\widehat{W}_{i}$. Now take a vector $v\in T^{1,0}R_{\alpha,d}\cong TR_{\alpha,d}$, and evaluate the above two-form by $(v,\bar{v})$. The first term $\mathrm{tr~{}}\left(\left(\partial_{v}\rho\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\partial_{v}\rho\right)$ is the square norm of the linear map $\partial_{v}\rho\colon\left(\widehat{W}_{i},h_{\mathrm{std}}\right)\rightarrow\left(V_{i},~{}h_{{\left(\rho\rho^{\mathrm{}}\right)^{-1}}}\right).$ Namely we take the standard basis in $\widehat{W}_{i}$ (which is orthonormal under the standard metric $h_{\mathrm{std}}$), map it to $V_{i}$ by $\partial_{v}\rho$, and take the sum of their square norms with respect to the metric $h_{{\left(\rho\rho^{\mathrm{}}\right)^{-1}}}$. The second term $\mathrm{tr~{}}\left(\left(\partial_{v}\rho\cdot\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\left(\partial_{v}\rho\cdot\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)\right)\right)$ is the square norm of the following component $\left(\partial_{v}\rho\right)_{1}$ of $\partial_{v}\rho$. Namely, we decompose $\widehat{W}_{i}=\left(\mathrm{Im}~{}\rho^{\mathrm{}}\right)\oplus\left(\mathrm{Im}~{}\rho^{\mathrm{}}\right)^{\bot}$ and write $\partial_{v}\rho=\left(\left(\partial_{v}\rho\right)_{1},\left(\partial_{v}\rho\right)_{2}\right)$ where $\left(\partial_{v}\rho\right)_{1}\colon\mathrm{Im}~{}\rho^{\mathrm{}}\rightarrow V_{i}$ and $\left(\partial_{v}\rho\right)_{2}\colon\left(\mathrm{Im}~{}\rho^{\mathrm{}}\right)^{\bot}\rightarrow V_{i}$. We have $\partial_{v}\overline{\partial_{v}}\log\det\rho\rho^{\mathrm{}}=\left\\|\partial_{v}\rho\right\\|_{H}^{2}-\left\\|\left(\partial_{v}\rho\right)_{1}\right\\|_{H}^{2}=\left\\|\left(\partial_{v}\rho\right)_{2}\right\\|_{H}^{2}\geq 0$ (3.5) for all $v\in T^{1,0}$. ($H$ stands for the metric $(\rho\rho^{\mathrm{}})^{-1}$.) This proves that the Ricci curvature of the metric for each $i$ is semi-positive definite. Now consider $H_{T}=\sum_{i\in Q_{0}}\partial\overline{\partial}\log\det\rho^{(i)}\left(\rho^{(i)}\right)^{\mathrm{}}.$ Suppose it is zero when evaluated at $(v,\bar{v})$. Then each individual term equals to zero. This forces $\left(\partial_{v}\rho^{(i)}\right)_{2}=0$ for all $i$, that is, image of $\left(\partial_{v}\rho^{(i)}\right)^{\mathrm{}}=\partial_{v}(\rho^{(i)})^{}$ sits in the image of $(\rho^{(i)})^{\mathrm{}}$. This exactly means $v$ descends to the zero tangent vector in the quotient $\mathcal{M}$: $v$ does not alter the subspaces given by $(\rho^{(i)})^{}:V_{i}\to\hat{W}_{i}$ for all $i$. By the identification of $\mathcal{M}$ as a quiver Grassmannian [Rei08], it means $v$ does not change the position of the point $((\rho^{(i)})^{}:i\in Q_{0})$ in the quiver Grassmannian, and hence must be the zero tangent vector. This proves the above expression is positive definite. ∎ By Equation (3.5), the metric on $T\mathcal{M}$ produced from the Ricci curvature of $\mathcal{V}_{i}$ is $H_{T}(v,v)=\sum_{i\in Q_{0}}\\|(\partial_{v}\rho^{(i)})_{2}\\|_{H_{i}}^{2}.$ We have the tautological exact sequence of vector bundles over $\mathcal{M}$: $0\to\bigoplus_{i\in Q_{0}}\mathrm{End}(\mathcal{V}_{i})\to\bigoplus_{a\in Q_{1}}\mathrm{Hom}(\mathcal{V}_{t_{a}},\mathcal{V}_{h_{a}})\oplus\bigoplus_{i\in Q_{0}}\mathcal{V}_{i}^{n_{i}}\to T\mathcal{M}\to 0$ where the second arrow is given by sending $X\|_{[\phi,e]\in\mathcal{M}}\in\bigoplus_{i\in Q_{0}}\mathrm{End}(\mathcal{V}_{i})$ to $\left((X_{h_{a}}\phi_{a}-\phi_{a}X_{t_{a}})_{a\in Q_{1}},(X_{i}e^{(i)})_{i\in Q_{0}}\right)$ (which is the derivative of the action of $\mathrm{GL}_{\vec{d}}$), and $T\mathcal{M}$ is obtained as the quotient bundle (of the middle one by the first one). $H(v,v)$ can be defined for $v\in\bigoplus_{a\in Q_{1}}\mathrm{Hom}(\mathcal{V}_{t_{a}},\mathcal{V}_{h_{a}})\oplus\bigoplus_{i\in Q_{0}}\mathcal{V}_{i}^{n_{i}}$. $H_{TM}$ is zero on $\bigoplus_{i\in Q_{0}}\mathrm{End}(\mathcal{V}_{i})$: the action of $\mathrm{GL}(d_{j},\mathbb{C})$ does not change $\rho^{(i)}$ for $j\not=i$; for $X\in\mathrm{End}(\mathcal{V}_{i})$, $(X\cdot\rho^{(i)}\cdot w)^{}v=w^{}(\rho^{(i)})^{}(X^{}\cdot v)=0$ for all $v\in\mathcal{V}_{i},w\in(\operatorname{Im}(\rho^{(i)})^{})^{\perp}$, and so $(\partial_{X}\rho^{(i)})_{2}=0$. ### 3.3. Quiver with oriented cycles When the quiver $Q$ has an oriented cycle, the framed moduli $\mathcal{M}_{\vec{n},\vec{d}}$ is no longer projective. Examples of such quivers were studied algebraically by [Fed13, ER09] along the line of Reineke. On the other hand, the metric given in Theorem 3.7 still makes sense for quiver with oriented cycles, as long as we stay in the domain of convergence and prove that it is positive-definite. Below we will prove this for any given quiver. Denote the moment-map level by $R^{\mu=I}_{n,d}=\\{\mu=1\\}\subset R_{\vec{n},\vec{d}}$. Let $\\|A\\|=\sup_{\\|v\\|=1}\\|A\cdot v\\|$ be the operator norm of a matrix $A$. We take the following open subset of $R^{\mu=I}_{n,d}$. ###### Definition 3.16. Define $R_{\vec{n},\vec{d}}^{\mu=I,\circ}:=\\{(V,e)\in R^{\mu=I}_{n,d}:\\|V(\gamma)\\|<1\textrm{ for every oriented cycle }\gamma\\}$ where $V(\gamma)=V(a_{k})\ldots V(a_{1})$ for $\gamma=a_{k}\ldots a_{1}$. $R_{\vec{n},\vec{d}}^{\circ}:=\mathrm{GL}_{\vec{d}}\cdot R_{\vec{n},\vec{d}}^{\mu=I,\circ}.$ and $\mathcal{M}_{\vec{n},\vec{d}}^{\circ}:=R_{\vec{n},\vec{d}}^{\mu=I,\circ}/U_{\vec{d}}=R_{\vec{n},\vec{d}}^{\circ}/\mathrm{GL}_{\vec{d}}.$ The above definition of $\mathcal{M}_{\vec{n},\vec{d}}^{\circ}$ makes sense because of the following. ###### Lemma 3.17. $R_{\vec{n},\vec{d}}^{\mu=I,\circ}$ is invariant under $U_{\vec{d}}$. ###### Proof. For every oriented cycle $\gamma$ at $i\in Q_{0}$, $(g\cdot V)(\gamma)=g_{i}V(\gamma)g_{i}^{-1}$, and hence the condition $\\|V(\gamma)\\|<1$ is respected for $g\in U_{\vec{d}}$. ∎ The main theorem in this section is the following. ###### Theorem 3.18. Let $Q$ be an arbitrary quiver. As in Theorem 3.7, for each $i\in Q_{0}$, set $H_{i}=(\rho_{i}\rho_{i}^{})^{-1}=\left(\sum_{h(\gamma)=i}\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right)^{-1}$ which is an infinite sum, whose terms are ordered by the length of the path $\gamma$. (There are just finitely many paths under each fixed length.) This gives a convergent function $H_{i}:R_{\vec{n},\vec{d}}^{\circ}\to\mathrm{End}(\mathbb{C}^{d_{i}})$. $H_{i}$ is $\mathrm{GL}_{\vec{d}}$-equivariant, and it descends to a metric on $\mathcal{V}_{i}$ over $\mathcal{M}_{\vec{n},\vec{d}}^{\circ}$. We break into several steps to prove the above theorem. First, consider the convergence. ###### Lemma 3.19. $\rho\rho^{}$ is absolutely convergent over $R_{\vec{n},\vec{d}}^{\mu=I,\circ}$. Hence $(\rho\rho^{})^{-1}$ is well- defined and $\mathrm{GL}_{\vec{d}}$-equivariant on $R_{\vec{n},\vec{d}}^{\circ}$. ###### Proof. For $(V,e)\in R_{\vec{n},\vec{d}}^{\mu=I,\circ}$, we consider the expression $\sum_{h(\gamma)=i}\left\\|V_{\gamma}e^{\left(t(\gamma)\right)}\right\\|\left\\|\left(V_{\gamma}e^{\left(t(\gamma)\right)}\right)^{}\right\\|\leq\sum_{h(\gamma)=i}\left\\|V_{\gamma}e^{\left(t(\gamma)\right)}\right\\|^{2}.$ There are only finitely many paths $\gamma_{1},\ldots,\gamma_{k}$ with $h(\gamma_{l})=i$ which do not contain any oriented cycle. Any other path (with $h(\gamma)=i$) can be written as concatenation of one of these $\gamma_{l}$ and some oriented cycles at some vertices. Thus $\sum_{h(\gamma)=i}\left\\|V_{\gamma}e^{\left(t(\gamma)\right)}\right\\|^{2}\leq\sum_{l=1}^{k}\\|\gamma_{l}e^{(t(\gamma_{l}))}\\|^{2}\sum_{p=0}^{\infty}(1-\epsilon)^{p}=\sum_{l=1}^{k}\\|\gamma_{l}e^{(t(\gamma_{l}))}\\|^{2}\sum_{p=0}^{\infty}(1-\epsilon)^{p}=\sum_{l=1}^{k}\frac{\\|\gamma_{l}e^{(t(\gamma_{l}))}\\|^{2}}{\epsilon}<\infty$ where given $V$, there is a fixed $\epsilon\in(0,1)$ such that $\\|V_{\gamma}\\|^{2}<1-\epsilon$ for all oriented cycles $\gamma$. Hence $\rho\rho^{}$ is absolutely convergent for every $(V,e)\in R_{\vec{n},\vec{d}}^{\mu=I,\circ}$. Every element in $R_{\vec{n},\vec{d}}^{\circ}$ can be written as $g\cdot(V,e)$ for $g\in\mathrm{GL}_{\vec{d}}$ and $(V,e)\in R_{\vec{n},\vec{d}}^{\mu=I,\circ}$. $(\rho\rho^{})^{-1}\|_{g\cdot(V,e)}=(g_{i}^{})^{-1}(\rho\rho^{})^{-1}\|_{(V,e)}g_{i}^{-1}$ where $(\rho\rho^{})^{-1}\|_{(V,e)}$ is convergent. ∎ It remains to prove positive definiteness of $H_{i}$. First we consider the following specific quiver which is simply a single oriented cycle. ###### Lemma 3.20. If $Q$ is a single oriented cycle with $N$ vertices, then for each vertex $i$, $\rho_{i}\rho_{i}^{}$ is positive definite. (In particular, when $N=1$, $Q$ consists of one vertex and a self loop.) ###### Proof. By symmetry, we just need to prove for $i=N$. Let $l=a_{N}\ldots a_{1}$, where $a_{k}$ is the arrow $(k-1)\to k$. $\rho_{N}\rho_{N}^{}=\sum_{k=1}^{N}\left(a_{N}\ldots a_{k+1}e_{k}e_{k}^{}a_{k+1}^{}\ldots a_{N}^{}+\sum_{p>0}l^{p}a_{N}\ldots a_{k+1}e_{k}e_{k}^{}a_{k+1}^{}\ldots a_{N}^{}(l^{p})^{}\right).$ The moment map equation at the vertex $k$ is $e_{k}e_{k}^{}=I-a_{k}a_{k}^{}+a_{k+1}^{}a_{k+1}$. Then the first term gives $\displaystyle\sum_{k=1}^{N}a_{N}\ldots a_{k+1}(I-a_{k}a_{k}^{}+a_{k+1}^{}a_{k+1})a_{k+1}^{}\ldots a_{N}^{}$ $\displaystyle=$ $\displaystyle\sum_{k=1}^{N}a_{N}\ldots a_{k+1}a_{k+1}^{}\ldots a_{N}^{}-\sum_{k=1}^{N}a_{N}\ldots a_{k+1}a_{k}a_{k}^{}a_{k+1}^{}\ldots a_{N}^{}+\sum_{k=1}^{N}a_{N}\ldots a_{k+1}a_{k+1}^{}a_{k+1}a_{k+1}^{}\ldots a_{N}^{}$ $\displaystyle=$ $\displaystyle I-ll^{}+\sum_{k=1}^{N}a_{N}\ldots a_{k+1}a_{k+1}^{}a_{k+1}a_{k+1}^{}\ldots a_{N}^{}.$ Similarly, the second term gives $\displaystyle\sum_{k=1}^{N}\sum_{p>0}l^{p}a_{N}\ldots a_{k+1}(I-a_{k}a_{k}^{}+a_{k+1}^{}a_{k+1})a_{k+1}^{}\ldots a_{N}^{}(l^{p})^{}$ $\displaystyle=$ $\displaystyle\sum_{p>0}l^{p}(l^{p})^{}-\sum_{p>0}l^{p}ll^{}(l^{p})^{}+\sum_{p>0}\sum_{k=1}^{N}l^{p}a_{N}\ldots a_{k+1}a_{k+1}^{}a_{k+1}a_{k+1}^{}\ldots a_{N}^{}(l^{p})^{}$ $\displaystyle=$ $\displaystyle ll^{}+\sum_{p>0}\sum_{k=1}^{N}l^{p}a_{N}\ldots a_{k+1}a_{k+1}^{}a_{k+1}a_{k+1}^{}\ldots a_{N}^{}(l^{p})^{}.$ Combining the two terms, $\rho_{N}\rho_{N}^{}=I+\sum_{p\geq 0}\sum_{k=1}^{N}l^{p}a_{N}\ldots a_{k+1}a_{k+1}^{}a_{k+1}a_{k+1}^{}\ldots a_{N}^{}(l^{p})^{}$ and the second term is semi-positive-definite. Hence $\rho_{N}\rho_{N}^{}$ is positive definite. ∎ The following is the key lemma to prove positive-definiteness for a general quiver. ###### Lemma 3.21. Suppose $Q$ has the property that for every $i\in Q_{0}$, restricted to the intersection of the moment map locus and $R^{Q,\circ}_{n,d}$, $\rho^{Q}_{i}(\rho^{Q}_{i})^{}=I+B_{i}$ for some semi-positive definite matrix $B_{i}$. Let $Q^{\prime}$ be obtained by concatenating to $Q$ a chain $x_{1}\stackrel{{\scriptstyle a_{1}^{\prime}}}{{\to}}\ldots\stackrel{{\scriptstyle a_{k-1}^{\prime}}}{{\to}}x_{k}$ where $x_{1}$ and $x_{k}$ are certain vertices in $Q$. ($x_{1}$ can be equal to $x_{k}$, meaning what we have added is an oriented cycle. When $k=1$, we have added a loop; $a^{\prime}_{0}=a^{\prime}_{1}$. When $k=2$, there is no intermediate vertex in the chain.) Then $Q^{\prime}$ has the same property. Namely, for every $i\in Q^{\prime}_{0}$, restricted to the intersection of the moment map locus and $R^{Q^{\prime},\circ}_{n,d}$, $\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{}=I+B_{i}^{\prime}$ for some semi-positive-definite matrix $B_{i}^{\prime}$. ###### Proof. First, consider the case that the given vertex $i\in Q_{0}^{\prime}$ belongs to $Q_{0}$. For the original quiver $Q$, $\rho^{Q}_{i}(\rho^{Q}_{i})^{}=I+B_{i}$ where $B_{i}$ is semi-positive definite. After concatenating the chain, the terms in $\rho^{Q}_{i}$, $\gamma^{x_{1}\to i}_{Q}e^{(1)}(e^{(1)})^{}(\gamma^{x_{1}\to i}_{Q})^{}\textrm{ and }\gamma^{x_{k}\to i}_{Q}e^{(k)}(e^{(k)})^{}(\gamma^{x_{k}\to i}_{Q})^{}$ where $\gamma^{x_{1}\to i}_{Q}$ ($\gamma^{x_{k}\to i}_{Q}$ resp.) is a path from $x_{1}$ (from $x_{k}$ resp.) to $i$, get affected. Namely, the moment map equation for $e^{(1)}(e^{(1)})^{}$ (or $e^{(k)}(e^{(k)})^{}$) gets an extra term $(a_{1}^{\prime})^{}a_{1}^{\prime}$ ($-a_{k-1}^{\prime}(a_{k-1}^{\prime})^{}$ resp.). (If $k=1$, $x_{1}=x_{k}$ and $a^{\prime}_{1}=a^{\prime}_{0}$, and the moment map equation for $e^{(1)}(e^{(1)})^{}$ gets both the extra terms $(a_{0}^{\prime})^{}a_{0}^{\prime}$ and $-a_{0}^{\prime}(a_{0}^{\prime})^{}$.) As a result, we have an extra negative term $-\gamma^{x_{k}\to i}_{Q}a_{k-1}^{\prime}(a_{k-1}^{\prime})^{}(\gamma^{x_{k}\to i}_{Q})^{}$ (3.6) for each path $\gamma^{x_{k}\to i}_{Q}$. We shall show that these negative terms can be canceled. We also have additional paths in $Q^{\prime}$ heading to $i$, which can be divided into the following types: 1. (1) $\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime}e^{(x_{l})}$ for $l=2,\ldots,k-1$. (This is an empty case when $k=1,2$.) 2. (2) $\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}}e_{j}$ where $\gamma_{\mathrm{new}}:=a_{k-1}^{\prime}\ldots a_{1}^{\prime}$ and $\gamma_{Q}^{j\to x_{1}}$ is any path in $Q$ from $j\in Q_{0}$ to $x_{1}$. ($\gamma_{Q}^{x_{1}\to x_{1}}$ can be the trivial path. $\gamma_{\mathrm{new}}=a_{0}^{\prime}$ when $k=1$.) 3. (3) $\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}e^{(x_{l})}$ for some $p>0$ and $l=2,\ldots,k-1$. 4. (4) $\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}}e_{j}$ for some $p>0$ and $j\in Q_{0}$. For (1), by the moment-map equation $e^{(x_{l})}(e^{(x_{l})})^{}=I+(a^{\prime}_{l})^{}a^{\prime}_{l}-a^{\prime}_{l-1}(a^{\prime}_{l-1})^{}$, we have $\displaystyle\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime}\cdot e^{(x_{l})}(e^{(x_{l})})^{}\cdot(\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime})^{}$ $\displaystyle=$ $\displaystyle\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime}(\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime})^{}+\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime}(a^{\prime}_{l})^{}a^{\prime}_{l}(\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime})^{}$ $\displaystyle-\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime}a^{\prime}_{l-1}(a^{\prime}_{l-1})^{}(\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a_{l}^{\prime})^{}.$ The first term above for $l=k-1$ cancel with the extra negative term (3.6) for $\rho_{i}^{Q}$. The third term (which is negative) for $l\in\\{3,\ldots,k-1\\}$ cancel with the first term of $l-1$. As a result, after combining (1) with the modified $\rho_{i}^{Q}$, the remaining negative terms are $-\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a^{\prime}_{1}(\gamma_{Q}^{x_{k}\to i}a_{k-1}^{\prime}\ldots a^{\prime}_{1})^{}=-\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}})^{}.$ (3.7) (For the case $k=1$, this trivially holds since $\gamma_{\mathrm{new}}=a^{\prime}_{0}$, and the above equals to (3.6).) Now consider (2): $\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}}e_{j}e_{j}^{}(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}})^{}$. The moment map equation for $e_{j}e_{j}^{}$ when $j\not=x_{1},x_{k}$ are the same for $Q$ and $Q^{\prime}$. Summing $\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}}e_{j}e_{j}^{}(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{j\to x_{1}})^{}$ over arbitrary $\gamma_{Q}^{j\to x_{1}}$ and $j\in Q_{0}$, we obtain $(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}})\cdot\rho^{Q}_{x_{1}}\cdot(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}})^{}=(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}})\cdot(I+B_{1})\cdot(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}})^{}$ plus $\sum_{\gamma_{Q}^{x_{1}\to x_{1}}}\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{1}\to x_{1}}(a_{1}^{\prime})^{}a_{1}^{\prime}(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{1}\to x_{1}})^{}-\sum_{\gamma_{Q}^{x_{k}\to x_{1}}}\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}}a_{k-1}^{\prime}(a_{k-1}^{\prime})^{}(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}})^{}$ which is due to the additional terms in the moment map equations for $e^{(x_{1})}(e^{(x_{1})})^{}$ and $e^{(x_{k})}(e^{(x_{k})})^{}$. (When $k=1$, $\gamma_{Q}^{x_{0}\to x_{0}}$ being the trivial path at $x_{0}$ is one of the possibilities.) In above, $B_{1}$ is semi-positive-definite by the assumption on $Q$. Then the negative terms (3.7) cancel with the first term. After combining the paths in $Q$ and (1) and (2), the remaining negative terms are $-\sum_{\gamma_{Q}^{x_{k}\to x_{1}}}\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}}a_{k-1}^{\prime}(a_{k-1}^{\prime})^{}(\gamma_{Q}^{x_{k}\to i}\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}})^{}.$ (3.8) (If there is no path $\gamma_{Q}^{x_{k}\to x_{1}}$ in $Q$ from $x_{k}$ to $x_{1}$, then this is zero, and we do not have (3) nor (4). We stop here and get that $\rho_{i}^{Q^{\prime}}(\rho_{i}^{Q^{\prime}})^{}=I+B_{i}^{\prime}$ for a semi-positive-definite matrix $B_{i}^{\prime}$.) The terms in (3) for $p=1$ and $l=k-1$ cancel with the above (3.8): $\displaystyle\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\cdot e^{(x_{l})}(e^{(x_{l})})^{}\cdot\left(\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\right)^{}$ $\displaystyle=$ $\displaystyle\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\left(\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\right)^{}$ $\displaystyle+\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}(a^{\prime}_{l})^{}a^{\prime}_{l}\left(\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\right)^{}$ $\displaystyle-\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}a^{\prime}_{l-1}(a^{\prime}_{l-1})^{}\left(\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)a_{k-1}^{\prime}\ldots a_{l}^{\prime}\right)^{}.$ Like in (1), for each $p$, the third term (which is negative) for $l\in\\{3,\ldots,k-1\\}$ cancel with the first term for $l-1$. (When $p=0$ and $l=k-1$, the third term is exactly (3.8).) Then the remaining negative terms are (3.7) modified by inserting the loops $\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)$ for $p>0$, that is, $-\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)\gamma_{\mathrm{new}}(\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)\gamma_{\mathrm{new}})^{}$. Then like in (2), these negative terms cancel with terms in (4). Summing up to finite $p$, the only negative terms left are (3.8) modified by inserting the loops $\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)$: $-\sum_{\gamma_{Q}^{x_{k}\to x_{1}}}\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}}a_{k-1}^{\prime}(a_{k-1}^{\prime})^{}\left(\gamma_{Q}^{x_{k}\to i}\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)\gamma_{\mathrm{new}}\gamma_{Q}^{x_{k}\to x_{1}}\right)^{}$ which cancel with terms in (3) for $(p+1)$. As $p\to\infty$, $\\|\left(\prod_{r=1}^{p}\gamma_{\mathrm{new}}\gamma_{Q,r}^{x_{k}\to x_{1}}\right)\\|\to 0$. This finishes the proof that $\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{}=I+B_{i}$, for $i\in Q_{0}$. For the case that $i=x_{j}$ for $j=2,\ldots,k-1$, the proof is similar. (We do not need to consider this case when $k=1,2$.) The paths in $Q^{\prime}$ heading to $x_{j}$ are divided into the following types: 1. (1) $a_{j-1}^{\prime}\ldots a_{l}^{\prime}e^{(x_{l})}$ for $l=2,\ldots,j$. 2. (2) $a_{j-1}^{\prime}\ldots a_{1}^{\prime}\left(\prod_{r=1}^{p}\gamma_{Q,r}^{x_{k}\to x_{1}}\gamma_{\mathrm{new}}\right)\gamma_{Q}^{j\to x_{1}}e_{j}$ for some $p\geq 0$, $j\in Q_{0}$. 3. (3) $a_{j-1}^{\prime}\ldots a_{1}^{\prime}\left(\prod_{r=1}^{p}\gamma_{Q,r}^{x_{k}\to x_{1}}\gamma_{\mathrm{new}}\right)\gamma_{Q}^{x_{k}\to x_{1}}a_{k-1}^{\prime}\ldots a_{l}^{\prime}e^{(x_{l})}$ for some $p\geq 0$, $l=2,\ldots,k-1$. The cancellation is similar and we do not repeat here. ∎ Similarly, adding a chain at a single vertex of $Q$ preserves the positive- definiteness property. ###### Lemma 3.22. Suppose $Q$ as in Lemma 3.21. Let $Q^{\prime}$ be obtained by concatenating to $Q$ a chain $x_{1}\stackrel{{\scriptstyle a_{1}^{\prime}}}{{\to}}\ldots\stackrel{{\scriptstyle a_{k-1}^{\prime}}}{{\to}}x_{k}$ at either $x_{1}$ or $x_{k}$ in $Q_{0}$. Then for every $i\in Q^{\prime}_{0}$, $\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{}=I+B_{i}^{\prime}$ for some semi-positive-definite matrix $B_{i}^{\prime}$. ###### Proof. The proof in this case is simpler than that of Lemma 3.21, since there is no new oriented cycle. Consider the case that $x_{k}\in Q_{0}$. If $i$ belongs to the chain, then the only paths that head to $i$ are contained in the chain. Since no oriented cycle is involved in all such paths, Theorem 3.7 already gives the result. If $i\in Q_{0}$, then $\displaystyle\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{}=$ $\displaystyle\sum_{j\in Q_{0}-\\{x_{k}\\}}\sum_{\gamma_{Q}^{j\to i}}(\gamma_{Q}^{j\to i}e^{(j)})(\gamma_{Q}^{j\to i}e^{(j)})^{}+\sum_{\gamma_{Q}^{x_{k}\to i}}\gamma_{Q}^{x_{k}\to i}\cdot e^{(x_{k})}(e^{(x_{k})})^{}\cdot(\gamma_{Q}^{x_{k}\to i})^{}$ $\displaystyle+\sum_{\gamma_{Q}^{x_{k}\to i}}\sum_{r=1}^{k-1}(\gamma_{Q}^{x_{k}\to i}a_{k-1}\ldots a_{r})e_{x_{r}}e_{x_{r}}^{}\cdot(\gamma_{Q}^{x_{k}\to i}a^{\prime}_{k-1}\ldots a^{\prime}_{r})^{}$ $\displaystyle=$ $\displaystyle\rho_{i}^{Q}(\rho_{i}^{Q})^{}-\sum_{\gamma_{Q}^{x_{k}\to i}}\gamma_{Q}^{x_{k}\to i}(a^{\prime}_{k-1}(a^{\prime}_{k-1})^{})(\gamma_{Q}^{x_{k}\to i})^{}$ $\displaystyle+\sum_{\gamma_{Q}^{x_{k}\to i}}\sum_{r=1}^{k-1}(\gamma_{Q}^{x_{k}\to i}a^{\prime}_{k-1}\ldots a^{\prime}_{r})(I-a^{\prime}_{r-1}(a^{\prime}_{r-1})^{}+(a^{\prime}_{r})^{}a^{\prime}_{r})\cdot(\gamma_{Q}^{x_{k}\to i}a^{\prime}_{k-1}\ldots a^{\prime}_{r})^{}$ $\displaystyle=$ $\displaystyle\rho_{i}^{Q}(\rho_{i}^{Q})^{}+\sum_{\gamma_{Q}^{x_{k}\to i}}\sum_{r=1}^{k-1}(\gamma_{Q}^{x_{k}\to i}a^{\prime}_{k-1}\ldots a^{\prime}_{r})(a^{\prime}_{r})^{}a^{\prime}_{r}\cdot(\gamma_{Q}^{x_{k}\to i}a^{\prime}_{k-1}\ldots a^{\prime}_{r})^{}.$ Since $\rho_{i}^{Q}(\rho_{i}^{Q})^{}=I+B_{i}$ for some semi-positive-definite matrix $B_{i}$, and the second term is semi-positive-definite, $\rho^{Q^{\prime}}_{i}(\rho^{Q^{\prime}}_{i})^{}$ satisfies the requirement. The case that $x_{1}\in Q_{0}$ is similar and the proof is omitted. ∎ ###### Proof of Theorem 3.18. Without loss of generality, suppose $Q$ is connected. (Otherwise $\mathcal{M}_{\vec{n},\vec{d}}$ and $\mathcal{V}_{i}$ decompose into products coming from the connected components, and we just need to study each component.) The case without oriented cycle is given in Theorem 3.7. Suppose $Q$ has at least one oriented cycle. By Lemma 3.20, the statement is true for this oriented cycle as a quiver. There must be additional arrows if this single oriented cycle is not yet the whole $Q$. Then we can either add a chain as in Lemma 3.21 or 3.22, and the statement still holds. (Both the cases of loop at a vertex or multiple edge are covered by Lemma 3.21.) Inductively the statement holds for $Q$. ∎ ###### Example 3.23. For the $A_{2}$-quiver, $\rho_{i}\rho_{i}^{}$ gives a metric for all $i$. By Lemma 3.21, this is still true if we add an oriented cycle with arrows $l_{1},\ldots,l_{p}$. See Figure 3. ${1}$${4}$${2}$${5}$${3}$$\scriptstyle{e^{(1)}}$$\scriptstyle{a_{1}}$$\scriptstyle{e_{2,1}}$$\scriptstyle{e_{2}}$$\scriptstyle{a_{2}}$$\scriptstyle{\ell_{1}}$$\scriptstyle{e_{3}}$$\scriptstyle{\ell_{p}}$$\scriptstyle{e_{2,p-1}}$ Figure 3. $A_{2}$ modified by adding an oriented cycle. Note that the following equality still holds over $R^{\circ}_{n,d}$: $\rho\rho^{}=e^{(i)}(e^{(i)})^{}+\sum_{h(a)=i}V_{a}\rho_{(t(a))}\rho_{(t(a))}^{}V_{a}^{}.$ Thus Proposition 3.8 still holds for quivers with oriented cycles. ###### Proposition 3.24. For any quiver $Q$ and every $v,w\in(\mathcal{V}_{i})^{}$, $H_{i}(v,w)=H_{0}((e^{(i)})^{}(v),(e^{(i)})^{}(w))+\sum_{h(a)=i}H_{t(a)}(a^{}(v),a^{}(w)).$ Now we consider a version of Theorem 3.15 in this case. We define $\hat{W}_{i}$ by Equation (3.4). But this time, it is an infinite direct sum of Hilbert spaces (meaning that it consists of infinite sequence $w=(w_{\gamma}:h(\gamma)=i)$ with $\\|w\\|^{2}=\sum_{\gamma}\\|w_{\gamma}\\|^{2}<\infty$). ###### Lemma 3.25. For each $(V,e)\in R^{\circ}_{n,d}$, $\rho_{i}(V,e)$ defines a bounded linear map $\hat{W}_{i}\to V_{i}$. Its adjoint $\rho_{i}(V,e)^{}:V_{i}\to\hat{W}_{i}$ has a singular-value decomposition. ###### Proof. For $w=(w_{\gamma}:h(\gamma)=i)\in\hat{W}_{i}$, $\rho_{i}(V,e)$ maps it to $\sum_{\gamma:h(\gamma)=i}V_{\gamma}e^{(t(\gamma))}w_{\gamma}.$ Like in the proof of Lemma 3.19, consider $\sum_{\gamma:h(\gamma)=i}\\|V_{\gamma}e^{(t(\gamma))}\\|\\|w_{\gamma}\\|\leq\\|w\\|\sum_{\gamma:h(\gamma)=i}\\|V_{\gamma}e^{(t(\gamma))}\\|<+\infty.$ This also shows that if $\\|w\\|=1$, then the image of $w$ is also bounded. $\rho_{i}(V,e)^{}$ has image being finite-dimensional (since $V_{i}$ is finite-dimensional), and hence is a compact operator. Thus it has a singular- value decomposition. ∎ ###### Proposition 3.26. The Ricci curvature of the metric given by $(\rho_{i}\rho_{i}^{})^{-1}$ is semi-positive definite on $\mathcal{M}^{\circ}$. ###### Proof. By the previous lemma, the proof of Theorem 3.15 on semi-positive definiteness still works. Namely, $\partial\overline{\partial}\log\det\rho\rho^{\mathrm{}}=\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\partial\rho\right)-\mathrm{tr~{}}\left(\left(\partial\rho\cdot\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\left(\partial\rho\cdot\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)\right)\right).$ Note that the two terms on the RHS are finite: $\mathrm{tr~{}}\left(\left(\partial\rho\right)^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\partial\rho\right)=\mathrm{tr~{}}\left(\rho\rho^{\mathrm{}}\right)^{-1}\partial\rho\left(\left(\partial\rho\right)^{\mathrm{}}\right)=\sum_{j=1}^{d_{i}}\langle\partial\rho\left(\partial\rho\right)^{\mathrm{}}\epsilon_{j},\left(\rho\rho^{\mathrm{}}\right)^{-1}\epsilon_{j}\rangle_{V_{i}}$ which is a finite sum, and similar for the second term. $\rho^{\mathrm{}}=\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)\left(\rho\rho^{\mathrm{}}\right)^{\frac{1}{2}}$ is decomposed into the rescaling $(\rho\rho^{\mathrm{}})^{\frac{1}{2}}$ and the orthogonal embedding $\left(\rho^{\mathrm{}}\left(\rho\rho^{\mathrm{}}\right)^{-\frac{1}{2}}\right)$ to $\mathrm{Im}\,\rho^{\mathrm{}}\subset\widehat{W}_{i}$. Then the above equals to $\\|(\partial\rho)_{2}\\|_{H}^{2}\geq 0$ as in Theorem 3.15. ∎ ## 4\. Fiberwise Nonlinearity In the mathematical study of quivers, we mostly focused on linear representations. In particular, the morphisms between universal vector bundles are linear along fibers. On the other hand, nonlinear ‘activation functions’ play a key role in machine learning. In this section, we construct some natural non-linear fiber-bundle endomorphisms of the universal bundles $\mathcal{V}_{i}$ over $\mathcal{M}_{\vec{n},\vec{d}}$ by using fiberwise symplectomorphisms. For simplicity, we shall take $n_{i}=d_{i}+1$ for all $i\in Q_{0}$ in this section. ### 4.1. Activation functions arising from toric moment maps and symplectomorphisms In this section, we make the observation that several activation functions commonly used in machine learning actually belong to a much bigger class, namely the $T$-equivariant symplectomorphisms on open subsets of a symplectic toric variety. First, let’s recall the basic setup for toric varieties. Let’s equip $\mathbb{C}^{m}$ with the standard Kähler structure. We obtain a symplectic toric variety $(X,\omega_{X})$ as a symplectic quotient by the real torus $T^{m-d}$. We assume $X$ is smooth. The $T^{m-d}$-action can be specified by an injective homomorphism $\mathbb{Z}^{m-d}\to\mathbb{Z}^{m}$, which induces a map $T^{m-d}\to T^{m}$, and $T^{m}$ acts on $\mathbb{C}^{m}$ by coordinate- wise multiplication. We assume that the quotient of $\mathbb{Z}^{m}$ by the image of $\mathbb{Z}^{m-d}$ is again a lattice, which we identify as $\mathbb{Z}^{d}$. We denote by $v_{i}\in\mathbb{Z}^{d}$ the images of the standard basic vectors of $\mathbb{Z}^{m}$ under the quotient map $\mathbb{Z}^{m}\to\mathbb{Z}^{d}$. The residual action of $T^{d}$ on $X$ gives a moment-map fibration over a polytope $P$, which is given by the intersection of $m$ half-spaces in $\mathbb{R}^{d}$: $\\{x\in\mathbb{R}^{d}:\ell_{j}(x):=v_{j}\cdot x-c_{j}\geq 0\\}$ where the constants $c_{j}\in\mathbb{R}^{d}$ are determined by the level taken in the symplectic quotient. We assume that the level is chosen such that for all $j$, $\\{\ell_{j}(x)=0\\}\cap P$ is a (non-empty) codimension-one boundary of the polytope $P$. Consider the open toric orbit of $X$, which can be identified as $(\mathbb{C}^{\times})^{d}$ by fixing a basis of $\mathbb{Z}^{d}$. Denote by $\omega_{X}$ the Kähler form induced on the symplectic quotient. Let $\omega_{\mathrm{std}}=\sum_{i=1}^{d}dx_{i}\wedge d\theta_{i}$ be the standard symplectic form on $\mathbb{R}^{d}\times T^{d}$ (where $T^{d}$ denotes the real $d$-torus). The symplectic form $\omega_{X}\|_{(\mathbb{C}^{\times})^{d}}$ has an explicit description by the following beautiful formula. ###### Theorem 4.1 ([Gui94, Abr98]). $\left(\frac{1}{2}\left(\sum_{i=1}^{m}v_{i}\log\ell_{i}(x)\right),\mathrm{Id}\right):(P^{\circ}\times T^{d},\omega_{\mathrm{std}}\|_{P^{\circ}\times T^{d}})\to\mathbb{R}^{d}\times T^{d}\stackrel{{\scriptstyle\exp}}{{\cong}}((\mathbb{C}^{\times})^{d},\omega_{X}\|_{(\mathbb{C}^{\times})^{d}})$ is a symplectomorphism. Taking the universal cover $\mathbb{R}^{d}\to T^{d}$ and lifting the above, one obtains the following. ###### Corollary 4.2. The inverse of $\left(\frac{1}{2}\left(\sum_{i=1}^{m}v_{i}\log\ell_{i}(x)\right),\,\mathrm{Id}\right)$ gives a symplectomorphism $\sigma_{\mathbb{C}}=(\sigma(\mathrm{Re}(\vec{z})),\mathrm{Im}(\vec{z})):(\mathbb{C}^{d},\exp^{}\omega_{X})\to(P^{\circ}\times\mathbb{R}^{d},\omega_{\mathrm{std}})$ where $\exp:\mathbb{C}^{d}\to(\mathbb{C}^{\times})^{d}\subset X$, and $\sigma:\mathbb{R}^{d}\to P^{\circ}$ is the inverse of $\left(\frac{1}{2}\left(\sum_{i=1}^{m}v_{i}\log\ell_{i}(x)\right),\,\mathrm{Id}\right)$. ###### Example 4.3. For $\mathbb{C}^{d}$, the moment polytope $P$ is $\mathbb{R}_{\geq 0}^{d}=\\{x_{i}\geq 0:i=1,\ldots,d\\}$. $v=(1,\ldots,1)$. The above map is simply $\left(\frac{\log x_{i}}{2}\right)_{i=1}^{d}:\mathbb{R}_{>0}^{d}\to\mathbb{R}^{d}$. The symplectomorphism $((\mathbb{C}^{\times})^{d},\omega_{\mathbb{C}^{n}}\|_{\mathbb{C}^{\times})^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}(\mathbb{R}_{>0}^{d}\times T^{d},\omega_{\mathrm{std}}\|_{P^{\circ}\times T^{d}})$ is $\left(\|z_{i}\|^{2},\frac{\log z_{i}-\log\bar{z_{i}}}{2i}\right)_{i=1}^{d}$. ###### Example 4.4. For the complex projective space $\mathbb{P}^{d}$, the corresponding moment polytope $P$ (for a chosen level) is the $d$-simplex given by $\ell_{i}\geq 0$ where $\ell_{i}(x)=x_{i}$ for $i=1,\ldots,d$, and $\ell_{d+1}(x)=1-x_{1}-x_{2}-\dots-x_{d}$. The generators are $v_{i}=\epsilon_{i}$ for $i=1,\ldots,d$ (the standard basis) and $v_{d+1}=-\sum_{i=1}^{d}v_{i}$. We have $d\left(\frac{1}{2}\sum_{i=1}^{d+1}\ell_{i}(x)\log\ell_{i}(x)\right)=\frac{1}{2}\sum_{i=1}^{d}\log\left(\frac{x_{i}}{1-\sum_{j=1}^{d}x_{j}}\right)dx_{i}$ as a map $P^{\circ}\to\mathbb{R}^{d}$. By direct computation, the inverse of this map equals to $\sigma(\vec{r})=\left(\frac{e^{2r_{i}}}{1+\sum_{j=1}^{d}e^{2r_{j}}}\right)_{i=1}^{d}:\mathbb{R}^{d}\to P^{\circ}.$ (4.1) Written in terms of the complex coordinates $\vec{z}\in(\mathbb{C}^{\times})^{d}$, the symplectomorphism $((\mathbb{C}^{\times})^{d},\omega_{\mathbb{P}^{n}}\|_{(\mathbb{C}^{\times})^{d}})\to(P^{\circ}\times T^{d},\omega_{\mathrm{std}}\|_{P^{\circ}\times T^{d}})$ is given by $\left(\frac{\|z_{i}\|^{2}}{1+\sum_{j=1}^{d}\|z_{j}\|^{2}},\frac{\log z_{i}-\log\bar{z_{i}}}{2i}\right)_{i=1}^{d}.$ Pulling back by $\mathbb{C}^{d}\to(\mathbb{C}^{\times})^{d}$, we have the symplectomorphism $\sigma_{\mathbb{C}}=\left(\left(\frac{e^{2r_{i}}}{1+\sum_{j=1}^{d}e^{2r_{j}}}\right)_{i=1}^{d},\mathrm{Id}\right):(\mathbb{C}^{d},\exp^{}\omega_{X})\to(P^{\circ}\times\mathbb{R}^{d},\omega_{\mathrm{std}})$. When $d=1$, $\frac{e^{2r}}{1+e^{2r}}$ is a commonly-used activation function. See Figure 4. By taking a direct product, $\left(\frac{e^{2r_{i}}}{1+e^{2r_{i}}}\right)_{i=1}^{d}:\mathbb{R}^{d}\to[0,1]^{d}$ corresponds to $(\mathbb{P}^{1})^{d}$. $/2\pi i\mathbb{Z}$$\cong$$\left(\frac{e^{2x}}{1+e^{2x}},y\right)$ Figure 4. $\mathbb{C}$ as a Covering Space Mapped to an Open Strip ###### Remark 4.5. In above, we have taken the quotient Kähler structure from $\mathbb{C}^{m}$. For a general toric Kähler structure, the symplectomorphism in Theorem 4.1 is given by $\left(d\left(\frac{1}{2}\left(\sum_{i=1}^{m}\ell_{i}\log\ell_{i}(x)\right)+h\right),\mathrm{Id}\right)$ where $h$ is a smooth function on the closed polytope $P$ such that the Hessian of $\left(\frac{1}{2}\left(\sum_{i=1}^{m}\ell_{i}\log\ell_{i}(x)\right)+h\right)$ is positive definite in $P^{\circ}$ [Abr98]. In particular, for a general projective toric variety $X$, we can take an embedding of $X$ to $\mathbb{P}^{N}$ by toric holomorphic sections of a very ample line bundle $L$, and use the induced toric Kähler structure from $\mathbb{P}^{N}$. Then the symplectomorphism $\sigma_{\mathbb{C}}$ is given by $(\sigma,\mathrm{Id})$ where $\sigma(\vec{r})=\frac{\sum_{i=1}^{d}e^{2(\vec{u_{i}},\vec{r})}\vec{u_{i}}}{\sum_{j=1}^{d}e^{2(\vec{u_{j}},\vec{r})}}:\mathbb{R}^{d}\to P^{\circ},$ $\vec{u_{i}}$ are points such that their convex hull equal to $P$, and $(\vec{u},\vec{r})$ is the standard dot product on $\mathbb{R}^{d}$. See [Ful93, Section 4.2]. Now we have the symplectomorphisms $\phi_{X}:(P^{\circ}\times T^{d},\omega_{\mathrm{std}}\|_{P^{\circ}\times T^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}((\mathbb{C}^{\times})^{d},\omega_{X}\|_{(\mathbb{C}^{\times})^{d}})$ and $\phi_{\mathbb{C}^{d}}:\mathbb{R}_{>0}^{d}\times T^{d}\stackrel{{\scriptstyle\cong}}{{\to}}((\mathbb{C}^{\times})^{d},\omega_{\mathbb{C}^{d}}\|_{(\mathbb{C}^{\times})^{d}})$ (Example 4.3). For the toric structure of $X$, let’s arrange the order of the indices such that the first $d$ vectors $v_{i}$ for $i=1,\ldots,d$ form a basis of $\mathbb{Z}^{d}$. (We assume $m\geq d$.) Moreover, we take the first $d$ constants $c_{j}=0$ for $j=1,\ldots,d$. Then $P^{\circ}\subset\mathbb{R}_{>0}^{d}$. Consider the composition $\phi_{\mathbb{C}^{d}}\circ\phi_{X}^{-1}:((\mathbb{C}^{\times})^{d},\omega_{X}\|_{(\mathbb{C}^{\times})^{d}})\to((\mathbb{C}^{\times})^{d},\omega_{\mathbb{C}^{d}}\|_{(\mathbb{C}^{\times})^{d}})$. It is a symplectomorphism onto the image $\phi_{\mathbb{C}^{d}}(P^{\circ}\times T^{d})$. ###### Proposition 4.6. $\phi_{\mathbb{C}^{d}}\circ\phi_{X}^{-1}$ extends to a $T$-equivariant symplectomorphism $\psi:(\mathbb{C}^{d},\omega_{X}\|_{\mathbb{C}^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}(\pi_{\mathbb{C}^{d}}^{-1}(P-B),\omega_{\mathbb{C}^{d}}\|_{\pi_{\mathbb{C}^{d}}^{-1}(P-B)})$ where $B=\bigcup_{i=d+1}^{m}\\{\ell_{i}(x)=0\\}$, and $\pi_{\mathbb{C}^{d}}=(\|z_{i}\|^{2})_{i=1}^{d}:\mathbb{C}^{d}\to\mathbb{R}_{\geq 0}^{d}$ is the moment map for $\mathbb{C}^{d}$. ###### Proof. $\phi_{X}$ is given by $2r^{X}_{i}=\log x_{i}+\sum_{j=d+1}^{m}(v_{j}^{(i)}\log\ell_{j}(x))$, where $v_{j}=(v_{j}^{(1)},\ldots,v_{j}^{(d)})$. $\phi_{\mathbb{C}^{d}}$ is given by $2r^{\mathbb{C}^{d}}_{i}=\log x_{i}$. Hence $e^{2r^{X}_{i}}=e^{2r^{\mathbb{C}^{d}}_{i}}\prod_{j=d+1}^{m}\ell_{j}^{v_{j}^{(i)}}(e^{2r^{\mathbb{C}^{d}}_{1}},\ldots,e^{2r^{\mathbb{C}^{d}}_{d}})$. In terms of the complex coordinates, this gives $z^{X}_{i}=z^{\mathbb{C}^{d}}_{i}\left(\prod_{j=d+1}^{m}\ell_{j}^{v_{j}^{(i)}}\left(\|z^{\mathbb{C}^{d}}_{1}\|^{2},\ldots,\|z^{\mathbb{C}^{d}}_{d}\|^{2}\right)\right)^{1/2}.$ It is obviously well-defined over $\pi_{\mathbb{C}^{d}}^{-1}(P-B)$. We need to show that it has inverse, which gives the required extension of $\phi_{\mathbb{C}^{n}}\circ\phi_{X}^{-1}$. Since $(\phi_{\mathbb{C}^{n}}\circ\phi_{X}^{-1})^{}(\omega_{\mathbb{C}^{n}})=\omega_{X}$ on $(\mathbb{C}^{\times})^{d}$, this still holds over $\mathbb{C}^{d}$ as the equality is a closed condition. Consider the Jacobian of $\vec{z}^{X}(\vec{z}^{\mathbb{C}^{d}},\overline{\vec{z}^{\mathbb{C}^{d}}})$. We shall show it is positive definite, and hence invertible. To simplify, we write $z=z^{\mathbb{C}^{d}}$. Denote $G=\frac{1}{2}\left(\sum_{i=1}^{m}\ell_{i}(x)\log\ell_{i}(x)-v\cdot x\right)$. For any non-zero vector $(a_{1},\ldots,a_{d})$, $\displaystyle\sum_{i,j}\overline{a_{j}}a_{i}\partial_{z_{i}}z_{j}^{X}$ $\displaystyle=$ $\displaystyle\sum_{i,j}\overline{a_{j}}a_{i}\partial_{z_{i}}\left(\exp(\partial_{x_{j}}\|_{x_{p}=z_{p}\bar{z_{p}}}G)\cdot\frac{z_{j}}{\|z_{j}\|}\right)$ $\displaystyle=$ $\displaystyle\sum_{i,j}\overline{a_{j}}a_{i}\exp(\partial_{x_{j}}\|_{x_{p}=z_{p}\bar{z_{p}}}G)\cdot\partial_{z_{i}}\left(\frac{z_{j}}{\|z_{j}\|}\right)+\sum_{i,j}\overline{a_{j}}a_{i}\frac{z_{j}}{\|z_{j}\|}\cdot\exp(\partial_{x_{j}}\|_{x_{p}=z_{p}\bar{z_{p}}}G)\cdot\frac{\partial(z_{i}\bar{z_{i}})}{\partial z_{i}}\cdot\frac{\partial^{2}G}{\partial x_{i}\partial x_{j}}$ $\displaystyle=$ $\displaystyle\sum_{i}\|a_{i}\|^{2}\|z_{i}^{X}\|\cdot\left(\frac{1}{2\|z_{i}\|}\right)+\sum_{i,j}\|z_{j}\|^{-1}\cdot\|z_{j}^{X}\|\cdot\overline{a_{j}}z_{j}\cdot a_{i}\bar{z_{i}}\cdot\frac{\partial^{2}G}{\partial x_{i}\partial x_{j}}.$ Note that $\|z_{j}\|^{-1}\|z_{j}^{X}\|=\left(\prod_{k=d+1}^{m}\ell_{k}^{v_{k}^{(j)}}\left(\|z^{\mathbb{C}^{d}}_{1}\|^{2},\ldots,\|z^{\mathbb{C}^{d}}_{d}\|^{2}\right)\right)^{1/2}$ which is positive. Let $c=\min\\{\|z_{j}\|^{-1}\|z_{j}^{X}\|:j=1,\ldots,d\\}$. Then the second term is no less than $ca_{i}\bar{z_{i}}\cdot\frac{\partial^{2}G}{\partial x_{i}\partial x_{j}}$. Since $\frac{\partial^{2}G}{\partial x_{i}\partial x_{j}}$ is positive definite on $P^{\circ}$, it is semi-positive definite on $P-B$. Thus this term is non-negative. The first term is positive. Thus $\sum_{i,j}\overline{a_{j}}a_{i}\partial_{z_{i}}z_{j}^{X}>0$. Similarly $\sum_{i,j}a_{j}\overline{a_{i}}\overline{\partial_{z_{i}}}z_{j}^{X}$. Hence the Jacobian is positive-definite and hence invertible. ∎ ###### Example 4.7. We continue to consider $\mathbb{P}^{d}$. From Example 4.4, $x_{i}=\frac{\|z^{\mathbb{P}^{d}}_{i}\|^{2}}{1+\sum_{j=1}^{d}\|z^{\mathbb{P}^{d}}_{j}\|^{2}}$. $\pi^{-1}_{\mathbb{C}^{d}}(P-B)=\\{\\|\vec{z}^{\mathbb{C}^{d}}\\|<1\\}$. From Example 4.3, $x_{i}=\|z^{\mathbb{C}^{d}}_{i}\|^{2}$. Hence the symplectomorphism $(\mathbb{C}^{d},\omega_{\mathbb{P}^{d}}\|_{\mathbb{C}^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}(\\{\\|\vec{z}^{\mathbb{C}^{d}}\\|<1\\},\omega_{\mathbb{C}^{n}}\|_{\pi_{\mathbb{C}^{d}}^{-1}(P-B)})$ is $z^{\mathbb{C}^{d}}_{i}=\frac{z^{\mathbb{P}^{d}}_{i}}{\sqrt{1+\\|\vec{z}^{\mathbb{P}^{d}}\\|^{2}}}.$ (4.2) When $d=1$, this gives $\frac{z}{\sqrt{1+\|z\|^{2}}}$ which is another activation function used in machine learning. ($z$ is restricted in $\mathbb{R}$ in most algorithms.) The symplectomorphism can be easily checked in this case: ($z=z^{\mathbb{P}^{1}}$ for simplicity) $\displaystyle dz^{\mathbb{C}}\wedge\overline{dz^{\mathbb{C}}}=$ $\displaystyle d\frac{z}{\sqrt{1+\|z\|^{2}}}\wedge d\frac{\overline{z}}{\sqrt{1+\|z\|^{2}}}$ $\displaystyle=$ $\displaystyle\frac{(1+z\bar{z})dz-(\|z\|^{2}dz+z^{2}\overline{dz})/2}{(1+z\bar{z})^{3/2}}\wedge\frac{(1+z\bar{z})\overline{dz}-(\|z\|^{2}\overline{dz}+\bar{z}^{2}dz)/2}{(1+z\bar{z})^{3/2}}=\frac{dz\wedge\overline{dz}}{(1+z\bar{z})^{2}}$ giving the Fubini-Study metric. See Figure 5. By taking the direct product, the symplectomorphism for the case $X=(\mathbb{P}^{1})^{d}$ is $\left(\frac{z_{j}}{\sqrt{1+\|z_{j}\|^{2}}}\right)_{i=1}^{d}$. $\frac{z}{\sqrt{1+\|z\|^{2}}}$ Figure 5. $\mathbb{C}$ as a Chart in $\mathbb{P}^{1}$ Mapped to an Open Disk Due to the nice fact that the Fubini-Study metric on $\mathbb{P}^{d}$ is $U(d)$-invariant (and so does the standard metric on $\mathbb{C}^{d}$), we have the following (which is not true for $(\mathbb{P}^{1})^{d}$ nor general toric manifolds). ###### Lemma 4.8. For $X=\mathbb{P}^{d}$, the symplectomorphism $\psi:(\mathbb{C}^{d},\omega_{\mathbb{P}^{d}}\|_{\mathbb{C}^{d}})\stackrel{{\scriptstyle\cong}}{{\to}}\left(\pi_{\mathbb{C}^{d}}^{-1}(P-B),\omega_{\mathbb{C}^{n}}\|_{\pi_{\mathbb{C}^{d}}^{-1}(P-B)}\right)$ in Proposition 4.6 is $U(d)$-equivariant. ###### Proof. Using the explicit expression (4.2), $\psi(U\cdot\vec{z}^{\mathbb{P}^{d}})=\frac{U\cdot\vec{z}^{\mathbb{P}^{d}}}{\sqrt{1+\\|U\cdot\vec{z}^{\mathbb{P}^{d}}\\|^{2}}}=\frac{U\cdot\vec{z}^{\mathbb{P}^{d}}}{\sqrt{1+\\|\vec{z}^{\mathbb{P}^{d}}\\|^{2}}}=U\cdot\psi(\vec{z}^{\mathbb{P}^{d}})$ for all $U\in U(d)$. ∎ As explained in Remark 4.5, we can also equip $X$ with another $T$-invariant Kähler form (that do not come from the standard Kähler structure on $\mathbb{C}^{m}$). The symplectomorphism is given by $\left(\frac{1}{2}d\left(\sum_{i=1}^{m}\ell_{i}(x)\log\ell_{i}(x)+h(x)\right),\mathrm{Id}\right):(P^{\circ}\times T^{d},\omega_{\mathrm{std}}\|_{P^{\circ}\times T^{d}})\to\mathbb{R}^{d}\times T^{d}\stackrel{{\scriptstyle\exp}}{{\cong}}((\mathbb{C}^{\times})^{d},\omega^{h}\|_{(\mathbb{C}^{\times})^{d}}).$ Thus the toric construction is rather flexible. ###### Example 4.9. The ‘softplus’ function $x=\log(1+e^{2y}):\mathbb{R}\to\mathbb{R}_{>0}$ gives an example of such a Kähler structure on $\mathbb{C}$ (by identifying it with $\mathbb{R}_{>0}\times\mathbb{R}$ with the standard symplectic structure). The inverse is $y=\frac{1}{2}\log(e^{x}-1)$, whose difference with $\frac{1}{2}\log x$ is $h^{\prime}=\frac{1}{2}\log\frac{e^{x}-1}{x}=\frac{1}{2}\log\left(1+\sum_{k=1}^{\infty}\frac{x^{k}}{(k+1)!}\right)$ which is indeed a smooth function on $\mathbb{R}_{\geq 0}$. Moreover, $y^{\prime}=\frac{e^{x}}{2(e^{x}-1)}>0$ on $\mathbb{R}_{>0}$. ### 4.2. Symplectomorphisms of fiber bundles over the moduli In the last subsection, we have exhibited various symplectic embeddings for $\mathbb{C}^{n}$. Now we want to make a family version of these maps over the framed quiver moduli $\mathcal{M}_{\vec{n},\vec{d}}$. The last subsection can be understood as constructing self-maps on a fiber of a vector bundle over $\mathcal{M}_{\vec{n},\vec{d}}$. To globalize (4.2), we consider the universal bundle $\mathcal{V}_{i}$ equipped with a Hermitian metric $H_{i}$. (We have constructed a nice Hermitian metric on $\mathcal{V}_{i}$ in Section 3.1.) We have a fiberwise symplectic structure $\omega_{\mathcal{V}_{i}}$ induced from the Hermitian metric. Moreover, we have the projective bundle $\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})$ which is a fiberwise compactification of $\mathcal{V}_{i}$. Then the fiber bundle $\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})$ is equipped with a fiberwise Kähler metric $\omega_{\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})}$ induced from $H_{i}$ (namely, $\frac{i}{2}\partial\bar{\partial}\log(H_{i}\oplus H_{0})$ where $H_{0}$ is the trivial metric on $\mathcal{O}_{M}$). ###### Proposition 4.10. There is a fiberwise symplectomorphism $\psi_{\mathcal{V}_{i}}:(\mathcal{V}_{i},\omega_{\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})}\|_{\mathcal{V}_{i}})\stackrel{{\scriptstyle\cong}}{{\to}}\left(\\{v\in\mathcal{V}_{i}:H_{i}(v,v)<1\\},\omega_{\mathcal{V}_{i}}\|_{\\{H_{i}(v,v)<1\\}}\right).$ ###### Proof. For each $p\in\mathcal{M}$, we have computed the symplectomorphism $(\mathcal{V}_{i}\|_{p},\omega_{\mathbb{P}(\mathcal{V}_{i}\oplus\mathcal{O}_{\mathcal{M}})}\|_{\mathcal{V}_{i}\|_{p}})\stackrel{{\scriptstyle\cong}}{{\to}}\left(\\{v\in\mathcal{V}_{i}\|_{p}:H_{i}(v,v)<1\\},\omega_{\mathcal{V}_{i}}\|_{\\{H_{i}(v,v)<1\\}}\right)$ in (4.2), with the metric given by $H_{i}\|_{p}$ here. Thus $\psi_{\mathcal{V}_{i}}(v)=\frac{v}{\sqrt{1+H_{i}(v,v)}}$ (4.3) gives a fiberwise symplectomorphism whose image is ${\\{H_{i}(v,v)<1\\}}$. ∎ Recall that the universal bundle $\mathcal{V}_{i}\to\mathcal{M}$ admits an action of $U_{\vec{n}}$ coming from framing (Definition 3.11). One advantage of $\psi_{\mathcal{V}_{i}}$ is that it is equivariant under this action. ###### Lemma 4.11. For $g\in U_{\vec{n}}$, $\psi_{\mathcal{V}_{i}}\circ g=g\circ\psi_{\mathcal{V}_{i}}$ if we used the metric $H_{i}$ given in Theorem 3.7. ###### Proof. This follows from (4.3) and $H_{i}(g\cdot v,g\cdot v)=H_{i}(v,v)$ by Lemma 3.13. ∎ ###### Remark 4.12. Equation (4.3) has an alternative derivation using the framing. Namely, we have the surjective morphism $\rho:\underline{\hat{W}_{i}}\to\mathcal{V}_{i}$ (see Equation (3.4)), whose dual give a fiberwise-linear embedding $\rho^{}:\mathcal{V}_{i}\stackrel{{\scriptstyle H_{i}}}{{\cong}}\mathcal{V}_{i}^{}\to\underline{\hat{W}_{i}}^{}$. (The underline means the trivial bundle over $\mathcal{M}$ associated with the vector space.) Then $\mathbb{P}(\underline{\hat{W}_{i}}^{}\oplus\mathcal{O})$ (with the standard metric) induces a fiberwise Kähler form on $\mathcal{V}_{i}^{}$, and we have a fiberwise symplectic embedding $(\mathcal{V}_{i},\omega_{\mathbb{P}(\underline{\hat{W}_{i}}^{}\oplus\mathcal{O})})\hookrightarrow(\mathcal{V}_{i},\omega_{\mathcal{V}_{i}})$. This gives $\frac{\rho\rho^{}\cdot H_{i}\cdot v}{\sqrt{1+H_{0}(\rho^{}\cdot H_{i}\cdot v,\rho^{}\cdot H_{i}\cdot v)}}=\frac{\rho\rho^{}\cdot H_{i}\cdot v}{\sqrt{1+v^{}\cdot H_{i}^{}\cdot\rho\rho^{}\cdot H_{i}\cdot v}}.$ Now if we use the metric $H_{i}=(\rho\rho^{})^{-1}$ given by Theorem 3.7, then the above equals to the expression in (4.3). We also have a more flexible construction using the framing, which globalize _any given non-linear continuous map_ $\sigma_{\mathbb{C}}:\mathbb{C}^{n_{i}}\to\mathbb{C}^{n_{i}}$. Namely, $\sigma_{\mathbb{C}}$ can be regarded as a fiberwise non-linear self-map on the trivial bundle $\underline{\mathbb{C}^{n_{i}}}\to\underline{\mathbb{C}^{n_{i}}}$ (still denoted by $\sigma_{\mathbb{C}}$). Then we take the composition $\mathcal{V}_{i}\stackrel{{\scriptstyle H_{i}}}{{\cong}}\mathcal{V}_{i}^{}\stackrel{{\scriptstyle(e^{(i)})^{}}}{{\to}}\underline{\mathbb{C}^{n_{i}}}\stackrel{{\scriptstyle\sigma_{\mathbb{C}}}}{{\to}}\underline{\mathbb{C}^{n_{i}}}\stackrel{{\scriptstyle e^{(i)}}}{{\to}}\mathcal{V}_{i}$ and denote it by $\sigma_{\mathcal{V}_{i}}$. See Figure 6. ${\underline{\mathbb{C}^{n}}}$${\underline{\mathbb{C}^{n}}^{}}$${\mathcal{V}_{i}}$${\mathcal{V}_{i}^{}}$$\scriptstyle{e^{(i)}}$$\scriptstyle{\sigma_{\mathbb{C}}}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$$\scriptstyle{(e^{(i)})^{}}$ Figure 6. Using the framing to globalize an activation function $\sigma_{\mathbb{C}}:\mathbb{C}^{n}\to\mathbb{C}^{n}$ over $\mathcal{M}$. It is easy to get the following explicit expression in terms of $\sigma_{\mathbb{C}}$. ###### Lemma 4.13. The above fiber-bundle map $\sigma_{\mathcal{V}_{i}}:\mathcal{V}_{i}\to\mathcal{V}_{i}$ equals to $\sigma_{\mathcal{V}_{i}}(v)=\sum_{k=1}^{n_{i}}(\sigma_{\mathbb{C}})_{k}\left(H_{i}(e^{(i)}_{1},v),\ldots,H_{i}(e^{(i)}_{n_{i}},v)\right)\cdot e^{(i)}_{k}$ (4.4) where we write $\sigma_{\mathbb{C}}=((\sigma_{\mathbb{C}})_{1},\ldots,(\sigma_{\mathbb{C}})_{n_{i}})$ and $e^{(i)}=(e^{(i)}_{1}\ldots e^{(i)}_{n_{i}})$. For instance, we can take $\sigma$ to be the one coming from the symplectomorphism in Corollary 4.2. We shall prove the universal approximation theorem for such $\sigma$ in Section 5. ### 4.3. A machine learning program using the framed quiver moduli Let $Q$ be a digraph and denote by $\mathbb{C}\cdot Q$ its path algebra over $\mathbb{C}$. Let $\vec{d}\in\mathbb{Z}_{\geq 0}^{Q_{0}}$ be a dimension vector. We take the framing dimension vector to be $\vec{n}=\vec{d}+\vec{1}$, where $\vec{1}_{i}:=1$ for all $i\in Q_{0}$. The one additional framing vector is used for translation (called a ‘bias’ vector). We fix a collection of input vertices and a collection of output vertices $I_{\mathrm{in}},I_{\mathrm{out}}\subset Q_{0}$, and $\gamma\in I_{\mathrm{out}}\cdot(\mathbb{C}\cdot Q)\cdot I_{\mathrm{in}}=\bigoplus_{i\in I_{\mathrm{in}},j\in I_{\mathrm{out}}}j\cdot(\mathbb{C}\cdot Q)\cdot i.$ (The trivial path at a vertex $i$ is again denoted by $i$.) #### 4.3.1. Machine learning using a flat space. Let’s first formulate a typical machine learning program in the quiver setup. The following flat space $U\cong\prod_{a\in Q_{1}}\mathrm{Hom}(V_{t(a)},V_{h(a)})\times\prod_{i\in Q_{0}}V_{i}$ is used frequently in the subject. ###### Lemma 4.14. The open subset $U=\\{[V,e]\in\mathcal{M}_{\vec{d}+\vec{1},\vec{d}}:(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})=I_{d_{i}}\textrm{ for all }i\in Q_{0}\\}\subset\mathcal{M}$ gives a coordinate chart of $\mathcal{M}$. ($I_{d_{i}}$ denotes the identity matrix of rank $d_{i}$.) ###### Proof. First, such $(V,e)$ are stable: $\mathrm{Im}(e)$ is the whole $V$. Second, for distinct $(V,e),(V^{\prime},e^{\prime})$ satisfying the above condition, $[V,e]\not=[V^{\prime},e^{\prime}]$: since they are stable, their orbits are closed. Suppose $g\cdot(V,e)=(V^{\prime},e^{\prime})$ for some $g\in\mathrm{GL}_{\vec{d}}$. Then $g_{i}\cdot e^{(i)}=(e^{\prime})^{(i)}$. But since $(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})=I_{d_{i}}=((e^{\prime})^{(i)}_{1},\ldots,(e^{\prime})^{(i)}_{d_{i}})$, this forces $g=\mathrm{Id}$ and so $(V,e)=(V^{\prime},e^{\prime})$, contradicting that they are distinct. Then we have the chart map $U\stackrel{{\scriptstyle\cong}}{{\to}}\prod_{a\in Q_{1}}\mathrm{Hom}(V_{t(a)},V_{h(a)})\times\prod_{i\in Q_{0}}V_{i}$ defined by $W_{a}:=(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})^{-1}V(a),\,\,b_{i}:=(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})^{-1}e^{(i)}_{d_{i}+1}.$ ∎ Now we fix a path $\gamma$ from the input vertices $I_{\mathrm{in}}$ to the output vertices $I_{\mathrm{out}}$. For each element $[V,e]\in U$, by composing the affine linear maps $V_{a}(\cdot)+e^{(t(a))}_{d_{t(a)}+1}$ attached to arrows $a$ in the path $\gamma$, together with some non-linear functions $\sigma_{i}:V_{i}\to V_{i}$ that are called ‘activation functions’, one obtains a non-linear function $f^{U}_{[V,e]}:\bigoplus_{i\in I_{\mathrm{in}}}\mathbb{C}^{d_{i}}\stackrel{{\scriptstyle(e^{(i)}_{1},\ldots,e^{(i)}_{d_{i}})}}{{\cong}}\bigoplus_{i\in I_{\mathrm{in}}}V_{i}\to\bigoplus_{j\in I_{\mathrm{out}}}V_{j}\stackrel{{\scriptstyle(e^{(j)}_{1},\ldots,e^{(j)}_{d_{i}})}}{{\cong}}\bigoplus_{j\in I_{\mathrm{out}}}\mathbb{C}^{d_{j}}$ which is used to approximate a non-explicitly given function $f$. A stochastic gradient flow of the error function on $U$ is employed to find the optimal point in $U$. Let’s write down the symmetry in Lemma 4.11 in the chart $U$. ###### Lemma 4.15. The chart $U$ is invariant under the right action by $\prod_{i\in Q_{0}}U(d_{i})\subset\prod_{i\in Q_{0}}U(d_{i}+1)=U_{\vec{n}}$ where $U(d_{i})\subset U(d_{i}+1)$ is embedded as $\left(\begin{array}[]{cc}U(d_{i})&0\\\ 0&1\end{array}\right)$. Consider the trivialization $\mathcal{V}_{i}\|_{U}\cong\left(\prod_{a\in Q_{1}}\mathrm{Hom}(V_{t(a)},V_{h(a)})\times\prod_{j\in Q_{0}}V_{j}\right)\times V_{i}$. The right action of $g\in U(d_{i})$ on $\mathcal{V}_{i}\|_{U}$ is given by $(W_{a},b_{j},v)_{\begin{subarray}{c}a\in Q_{1}\\\ j\in Q_{0},b_{j}\in V_{j}\\\ v\in V_{i}\end{subarray}}\cdot g=(g^{-1}\cdot W_{a},g^{-1}\cdot b_{j},g^{-1}v)_{\begin{subarray}{c}a\in Q_{1}\\\ j\in Q_{0},b_{j}\in V_{j}\\\ v\in V_{i}\end{subarray}}$ where $g^{-1}\cdot W_{a}$ equals to $g^{-1}W_{a}$ if $h(a)=i$, $W_{a}g$ if $t(a)=i$, and $W_{a}$ otherwise; $g^{-1}\cdot b_{j}$ equals to $g^{-1}b_{j}$ if $j=i$, and $b_{j}$ if $j\not=i$. ###### Proof. $U$ consists of points $[V,e]$ where $(e^{(j)}_{1},\ldots,e^{(j)}_{d_{i}})$ are invertible for all $j\in Q_{0}$. This property is invariant under the action of $\prod_{i\in Q_{0}}U(d_{i})$. Hence $U$ is an invariant subset. $(W_{a},b_{j},v)_{\begin{subarray}{c}a\in Q_{1}\\\ j\in Q_{0},b_{j}\in V_{j}\\\ v\in V_{i}\end{subarray}}$ corresponds to the point $[W,e,v]\in\mathcal{V}_{i}=(R^{s}_{n,d}\times V_{i})/\mathrm{GL}_{\vec{d}}$. where $e^{(j)}=(I_{d_{j}}\,\,b_{j})$. $\displaystyle[W,e,v]\cdot g=[W,e\cdot g,v]=[W,(g\,\,b_{j})_{j\in Q_{1}},v]$ $\displaystyle=$ $\displaystyle[g^{-1}\cdot W,(I_{d_{j}}\,\,g^{-1}\cdot b_{j})_{j\in Q_{1}},g^{-1}\cdot v]$ where the left action by $g^{-1}$ (as an element in $\mathrm{GL}_{\vec{d}}$) is as specified by definition. ∎ Now we prove an important symmetric property that the activation function (4.2) enjoys, which can be used to reduce the dimensions. ###### Proposition 4.16. Suppose the activation functions $\sigma_{i}:V_{i}\to V_{i}$ are taken to be the one given in Equation (4.2). Then $f^{U}$ is invariant under $\prod_{j\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(d_{j})\subset U_{\vec{n}}$. (See the embedding in Lemma 4.15.) ###### Proof. The terms of $f^{U}$ are of the form $\sigma_{h(a_{k})}\left(W_{a_{k}}\ldots\left(\sigma_{h(a_{1})}\left(W_{a_{1}}(v)+b_{h(a_{1})}\right)\ldots\right)+b_{h(a_{k})}\right)$. By the above lemma, for $g\in U(d_{h(a_{i})})$ where $h(a_{i})\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}$, the action of $g$ results in $\sigma_{h(a_{i})}\mapsto g\cdot\sigma_{h(a_{i})}g^{-1}$ in the above expression and does not affect any other part. By Lemma 4.8, $\sigma_{i}$ is $U(d_{i})$-equivariant, and hence $f^{U}$ remains invariant. ∎ By the above proposition, we can descend $f^{U}$ to the orbit space $U/\prod_{j\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(d_{j})$ to reduce the dimensions. However, the quotient space will be highly singular. Instead, we can realize the dimension reduction by restricting $f^{U}$ to a submanifold $U^{\prime}$ of $U$ whose orbit occupies the whole $U$, and do stochastic gradient flow on $U^{\prime}$ instead of $U$. The following gives one simple possibility. ###### Proposition 4.17. Consider a subset $\\{a_{1},\ldots,a_{p}\\}$ of arrows whose heads and tails do not belong to $I_{\mathrm{in}}\cup I_{\mathrm{out}}$, and for any two distinct arrows $a_{1},a_{2}$ in the subset, $t(a_{1})\not=h(a_{2})$. Then the vector subspace $U^{\prime}:=\\{(W_{a},b_{j})_{\begin{subarray}{c}a\in Q_{1}\\\ j\in Q_{0},b_{j}\in V_{j}\end{subarray}}\in U:W_{a_{i}}\textrm{ is of the form }\left(D\,\,W_{a_{i}}^{\prime}\right)\,\,\forall i=1,\ldots,p\\}$ has its orbit being the whole $U$, that is, $\left(\prod_{j\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(d_{j})\right)\cdot U^{\prime}=U$. In above, $D$ is a diagonal matrix (of the maximum possible size) and $W_{a_{i}}^{\prime}$ is any matrix occupying the rest. ###### Proof. This follows from the singular-value decomposition of a matrix $W$ as $A\cdot\left(D\,\,W^{\prime}\right)\cdot B$ where $A$ and $B$ are unitary matrices of appropriate sizes. ∎ #### 4.3.2. Machine learning using the quiver moduli. The quiver moduli $\mathcal{M}=\mathcal{M}_{\vec{n},\vec{d}}$ gives a compactification of $U$. Compactness is important for the formulation of Morse theory and convergence of a gradient flow. We would like to use the whole $\mathcal{M}$ in application of machine learning. Non-trivial metrics over the moduli will play a crucial role. First, consider the situation before adding in activation functions. Each arrow $a\in Q_{1}$ is associated with a vector-bundle morphism $a_{\mathcal{M}}:\mathcal{V}_{t(a)}\to\mathcal{V}_{h(a)}$. For each path $a_{k}\ldots a_{1}$, we take the map $(a_{k})_{\mathcal{M}}\left(\ldots\left((a_{2})_{\mathcal{M}}\left((a_{1})_{\mathcal{M}}(v)+e^{(h(a_{1}))}_{d_{h(a_{1})}+1}\right)+e^{(h(a_{2}))}_{d_{h(a_{2})}+1}\right)\ldots\right)+e^{(h(a_{k}))}_{d_{h(a_{k})}+1}$ (4.5) which is fiberwise affine linear. Thus a path $\gamma$ gives an affine bundle morphism $\gamma_{\mathcal{M}}:\mathcal{V}_{I_{\mathrm{in}}}:=\bigoplus_{i\in I_{\mathrm{in}}}\mathcal{V}_{i}\to\mathcal{V}_{I_{\mathrm{out}}}:=\bigoplus_{j\in I_{\mathrm{out}}}\mathcal{V}_{j}.$ Then we have $L_{\gamma}\left(\left(s^{(i)}_{k}\right)_{\begin{subarray}{c}i\in I_{\mathrm{in}}\\\ k\in\\{1,\ldots,d_{i}\\}\end{subarray}}\right)=\left(H_{j}\left(e^{(j)}_{p},\sum_{i\in I_{\mathrm{in}}}\gamma_{\mathcal{M}}\cdot\sum_{k=1}^{d_{i}}s^{(i)}_{k}e^{(i)}_{k}\right)\right)_{\begin{subarray}{c}j\in I_{\mathrm{out}},\\\ p\in\\{1,\ldots,d_{j}\\}\end{subarray}}:\bigoplus_{i\in I_{\mathrm{in}}}\underline{\mathbb{C}^{d_{i}}}\to\bigoplus_{j\in I_{\mathrm{out}}}\underline{\mathbb{C}^{d_{j}}}.$ (4.6) Suppose a continuous function $f=K\to\bigoplus_{j\in I_{\mathrm{out}}}\mathbb{C}^{d_{j}}$ is given, where $K$ is a compact subset of $\bigoplus_{i\in I_{\mathrm{in}}}\mathbb{C}^{d_{i}}$. Then the fiberwise integral $\mathcal{E}:=\int_{K}\left\\|f-L_{\gamma}\right\\|^{2}_{\mathcal{V}_{I_{\mathrm{out}}}}d\mu_{K}$ (4.7) gives a smooth function on $\mathcal{M}$. ###### Remark 4.18. Alternatively, we can define the fiber-bundle morphism $f_{\mathcal{M}}:\sum_{j\in I_{\mathrm{out}}}\sum_{l=1}^{d_{j}}f^{(j)}_{l}\cdot e^{(j)}_{l}:\underline{K}\to\mathcal{V}_{I_{\mathrm{out}}}$ and take $\int_{K}\left\\|f_{\mathcal{M}}\left(\left(s^{(i)}_{k}\right)_{\begin{subarray}{c}i\in I_{\mathrm{in}}\\\ k\in\\{1,\ldots,d_{i}\\}\end{subarray}}\right)-\sum_{i\in I_{\mathrm{in}}}\gamma_{\mathcal{M}}\cdot\sum_{k=1}^{d_{i}}s^{(i)}_{k}e^{(i)}_{k}\right\\|^{2}_{\mathcal{V}_{I_{\mathrm{out}}}}d\mu_{K}.$ However, with such a definition, we need to worry that $(e^{(j)}_{1},\ldots,e^{(j)}_{d_{j}})$ for some output $j$ degenerates (as a frame), in which case approximating $f$ and approximating $f_{\mathcal{M}}$ are different. We have a gradient flow $r:\mathbb{R}\to\mathcal{M}$ which can be used to minimize $\mathcal{E}$: $\frac{dr}{dt}=-(\nabla\mathcal{E})(r(t))$ where $\nabla\mathcal{E}=(d\mathcal{E})^{\\#_{g}}$ where $(\cdot)^{\\#}_{g}:T^{}\mathcal{M}\stackrel{{\scriptstyle g}}{{\cong}}T\mathcal{M}$ is the identification by a metric $g$ on $\mathcal{M}$. ($(\nabla\mathcal{E})^{p}=g^{pq}\partial_{q}\mathcal{E}$ in local coordinates.) $g$ can be taken to be the induced metric from the trivial metric on the vector space $R_{\vec{n},\vec{d}}$ via symplectic reduction. Alternatively, $g$ can be taken to be the metric given by the Ricci curvature in Theorem 3.15 (when $Q$ has no oriented cycle), which has a better expression in homogeneous coordinates. Note that $(L_{\gamma})\|_{[V,e]}$ is affine linear on $\bigoplus_{i\in I_{\mathrm{in}}}\underline{\mathbb{C}^{d_{i}}}$ for every $[V,e]\in\mathcal{M}$, which is not good enough for the purpose of approximating $f$. We introduce fiberwise non-linearity below. ###### Definition 4.19. Let $A$ be a finite set whose every element is associated with two vertices (head $h$ and tail $t$) in $Q_{0}$. Elements in $A$ are called activation arrows. (These are not arrows in $Q_{1}$.) The semiring generated by $Q_{1}$ and $A$, denoted by $\Gamma(Q,A)$, has the underlying vector space spanned by the independent set $\coprod_{p=0}^{\infty}S_{p}$ where $S_{p}$ is defined inductively as follows. 1. (1) $S_{0}$ consists of all paths of $Q$. 2. (2) Suppose $S_{p}$ has been defined, and each element in $S_{p}$ has a head $h$ and a tail $t$. $S_{p+1}$ consists of $\gamma\cdot\alpha\cdot\tilde{\gamma}$, where $\alpha\in A$, $\gamma$ is any path of $Q$ with $h(\alpha)=t(\gamma)$, and $\tilde{\gamma}\in t(\alpha)\cdot(\mathbb{C}\cdot S_{p})$. ($\mathbb{C}\cdot S_{p}$ denotes the vector space generated by $S_{p}$; $t(\alpha)\cdot(\mathbb{C}\cdot S_{p})$ is the subspace generated by elements of $S_{p}$ with head being $t(\alpha)$.) The above element has the head $h(\gamma)$ and the tail $t(\tilde{\gamma})$. The above vector space $\Gamma(Q,A)$ has an obvious product by concatenation. ($s_{1}\cdot s_{2}=0$ if $h(s_{2})\not=t(s_{1})$.) Note that for $\alpha\in A$, $(s_{1}+cs_{2})\cdot\alpha=s_{1}\cdot\alpha+cs_{2}\cdot\alpha$, but $\alpha\cdot(s_{1}+cs_{2})\not=\alpha\cdot s_{1}+c\alpha\cdot s_{2}$. Now, suppose each element $\alpha\in A$ is associated with a fiber-bundle morphism $\alpha_{\mathcal{M}}:\mathcal{V}_{t(\alpha)}\to\mathcal{V}_{h(\alpha)}.$ Then by composing the corresponding affine linear morphisms (as in Equation (4.5)) and $\alpha_{\mathcal{M}}$, an element $\tilde{\gamma}\in\Gamma(Q,A)$ induces a fiber-bundle morphism $\tilde{\gamma}_{\mathcal{M}}:\mathcal{V}_{t(\tilde{\gamma})}\to\mathcal{V}_{h(\tilde{\gamma})}.$ In particular, if we fix $\tilde{\gamma}\in I_{\mathrm{out}}\cdot\Gamma(Q,A)\cdot I_{\mathrm{in}}$, then we define $f_{\tilde{\gamma}}:\bigoplus_{i\in I_{\mathrm{in}}}\underline{\mathbb{C}^{d_{i}}}\to\bigoplus_{j\in I_{\mathrm{out}}}\underline{\mathbb{C}^{d_{j}}}$ like in the definition of $L_{\gamma}$ in (4.6) by replacing $\gamma_{\mathcal{M}}$ by $\tilde{\gamma}_{\mathcal{M}}$. Then a stochastic gradient flow for the corresponding error function (by replacing $L_{\gamma}$ by $f_{\tilde{\gamma}}$ in (4.7)) can be carried out. For now, we set $A$ to be $Q_{0}$ as a set, and each element $\mathfrak{o}_{i}$ has the head and tail being $i\in Q_{0}$. We can associate $\mathfrak{o}_{i}$ with the fiber-bundle morphism $\psi_{\mathcal{V}_{i}}$ given in (4.3), or $\sigma_{\mathcal{V}_{i}}$ in (4.4). Then we obtain $f_{\tilde{\gamma}}$ above which is non-linear along fibers for the purpose of machine learning. If we associate $\mathfrak{o}_{i}$ with $\psi_{\mathcal{V}_{i}}$ given in (4.3), then the symmetry of $\prod_{j\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(d_{j})\subset U_{\vec{n}}$ is respected, by Lemma 4.11. The proof is similar to that for Proposition 4.16 and is omitted here. ###### Proposition 4.20. Suppose $\mathfrak{o}_{i}$ is assigned as $\psi_{\mathcal{V}_{i}}$ given in (4.3), and $H_{i}$ is taken to be the metric in Theorem 3.7. Then $f_{\tilde{\gamma}}$ is invariant under $\prod_{j\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(d_{j})\subset U_{\vec{n}}$. (See the embedding in Lemma 4.15.) ###### Remark 4.21. Since we are taking affine linear morphisms $(a_{i})_{\mathcal{M}}(v)+e^{(h(a_{i}))}_{d_{h(a_{i})}+1}$ for the arrows which involves the term $e^{(h(a_{i}))}_{d_{h(a_{i})}+1}$, only the symmetry $U(d_{j})$ rather than $U(d_{j}+1)=U(n_{j})$ is respected. If the bias vector $e^{(h(a_{i}))}_{d_{h(a_{i})}+1}$ is not used in the program, then $f_{\tilde{\gamma}}$ will be invariant under the bigger group $\prod_{j\not\in I_{\mathrm{in}}\cup I_{\mathrm{out}}}U(n_{j})$. #### 4.3.3. A simple example Recall the quiver in Example 2.7. Let’s take $n_{1}=d_{1},n_{2}=d_{2}+1,n_{3}=d_{3}$ instead of $n_{i}=d_{i}+1\,\forall i$, since we do not need to use bias vectors at the input and output vertices. The path $\gamma$ is simply $a_{2}a_{1}$, and $\tilde{\gamma}=a_{2}\cdot\mathfrak{o}_{1}\cdot a_{1}$. We have the universal bundles $\mathcal{V}_{1}\cong\underline{\mathbb{C}}^{d_{1}}$, $\mathcal{V}_{2}$ and $\mathcal{V}_{3}$. We have $\rho^{(1)}=e^{(1)}$, $\rho^{(2)}=(e^{(2)}\,\,a_{1}e^{(1)})$, $\rho^{(3)}=(e^{(3)}\,\,a_{2}e^{(2)}\,\,a_{2}a_{1}e^{(1)})$. In terms of homogeneous coordinates (namely the coordinates $(e^{(1)},e^{(2)},e^{(3)},a_{1},a_{2})$ on the vector space $R_{\vec{n},\vec{d}}$, where each entry is a matrix of suitable size), the metrics given in Theorem 3.7 are $\displaystyle H_{1}$ $\displaystyle=\left(e^{(1)}(e^{(1)})^{}\right)^{-1},\,H_{2}=\left(e^{(2)}(e^{(2)})^{}+a_{1}e^{(1)}(e^{(1)})^{}a_{1}^{}\right)^{-1},\,$ $\displaystyle H_{3}$ $\displaystyle=\left(e^{(3)}(e^{(3)})^{}+a_{2}e^{(2)}(e^{(2)})^{}a_{2}^{}+a_{2}a_{1}e^{(1)}(e^{(1)})^{}a_{1}^{}a_{2}^{}\right)^{-1}.$ on $\mathcal{V}_{i},i=1,2,3$ respectively. The activation functions we constructed in the previous subsection are $\psi_{i}(v)=\frac{v}{\sqrt{1+v^{}H_{i}v}}\textrm{ and }\sigma_{i}(v)=\sum_{k=1}^{d_{i}+1}(\sigma_{\mathbb{C}})_{k}\left((e^{(i)}_{1})^{}H_{i}v,\ldots,(e^{(i)}_{d_{i}+1})^{}H_{i}v\right)\cdot e^{(i)}_{k},$ where we can take $(\sigma_{\mathbb{C}})_{k}(\vec{z})=\frac{e^{2\mathrm{Re}(z_{k})}}{1+\sum_{j=1}^{d_{i}+1}e^{2\mathrm{Re}(z_{j})}}+\mathbf{i}\,\mathrm{Im}(z_{k})$ for instance. Both have the $\mathrm{GL}$-equivariance property $\psi_{i}(g\cdot v)=g\cdot\psi_{i}(v)$ and $\sigma_{i}(g\cdot v)=g\cdot\sigma_{i}(v)$. The function (over $\mathcal{M}$) cooked up from this quiver is $f_{\tilde{\gamma}}:\mathbb{C}^{d_{1}}\to\mathbb{C}^{d_{3}}$, $f_{\tilde{\gamma}}(s_{1},\ldots,s_{d_{1}})=\left(H_{3}\left(e^{(3)}_{p},a_{2}\cdot\sigma_{2}\left(a_{1}\cdot\sum_{k=1}^{d_{1}}s_{k}e^{(1)}_{k}+e^{(2)}_{d_{2}+1}\right)\right)\right)_{p=1}^{d_{3}}$ if we use $\sigma_{2}$ as the activation function, or the same expression with $\sigma_{2}$ replaced by $\psi_{2}$. Then we run a stochastic gradient flow on $\mathcal{M}$ (or on the vector space $R_{\vec{n},\vec{d}}$ upstairs) to minimize the distance of $f_{\tilde{\gamma}}$ and a function $f$ coming from reality. To run the gradient flow, we need to take a metric on $\mathcal{M}$. Recall that we have the metric on the tangent bundle of $\mathcal{M}$ coming from the Ricci curvatures of $H_{i}$: $H_{T}=\sum_{i=1}^{3}\mathrm{tr~{}}\left(\left(\partial_{v}\rho^{(i)}\right)^{\mathrm{}}\cdot H_{i}\cdot\partial_{w}\rho^{(i)}\right)-\sum_{i=1}^{3}\mathrm{tr~{}}\left(\left(\partial_{v}\rho^{(i)}\cdot(\rho^{(i)})^{\mathrm{}}\cdot H_{i}^{\frac{1}{2}}\right)^{\mathrm{}}\cdot H_{i}\cdot\left(\partial_{w}\rho^{(i)}\cdot(\rho^{(i)})^{\mathrm{}}\cdot H_{i}^{\frac{1}{2}}\right)\right).$ (4.8) Consider the open subset of the vector space $R_{\vec{n},\vec{d}}$ in which $(e_{1}^{(i)},\ldots,e_{d_{i}}^{(i)})$ is invertible. (This is the preimage of the chart $U\subset\mathcal{M}$.) A tangent vector $v$ of $\mathcal{M}$ is lifted as $(\delta a_{1},\delta a_{2},\delta e_{d_{2}+1}^{(2)})$ (and all other components are set to be zero). Since $\rho^{(1)}=e^{(1)}$, $\rho^{(2)}=(e^{(2)},a_{1}e^{(1)})$, $\rho^{(3)}=(e^{(3)},a_{2}e^{(2)},a_{2}a_{1}e^{(1)})$, we have $\partial_{v}\rho^{(1)}=0$, $\partial_{v}\rho^{(2)}=\left((0\,\,\delta e_{d_{2}+1}^{(2)}),\,\,(\delta a_{1})e^{(1)}\right)$, and $\partial_{v}\rho^{(3)}=\left(0,\,\,(\delta a_{2})e^{(2)}+(0\,\,a_{2}\delta e_{d_{2}+1}^{(2)}),\,\,\,(\delta a_{2})a_{1}e^{(1)}+a_{2}(\delta a_{1})e^{(1)}\right).$ Then the above metric $H_{T}$ can be computed explicitly in terms of the homogeneous coordinates. ###### Remark 4.22. In above, if we use trivial metrics over the vector space $R_{\vec{n},\vec{d}}$ instead of $H_{i}$ and $H_{T}$, the expressions will get simpler; however they will only be $U_{\vec{d}}$-equivariant rather than $\mathrm{GL}_{\vec{d}}$-equivariant. Then we need to restrict to the moment- map level $\mu^{-1}(I)\subset R_{\vec{n},\vec{d}}$ and its tangent bundle, in order to stay in the same moduli $\mathcal{M}$ downstairs. This would increase the computational complexity. We can also write in inhomogeneous coordinates $(W_{1},W_{2},b)$ in the chart $U\subset\mathcal{M}$, where $(e_{1}^{(i)},\ldots,e_{d_{i}}^{(i)})=I_{d_{i}}$ for all $i=1,2,3$. $W_{1},W_{2}$ are the matrices of the arrows $a_{1},a_{2}$ and $e_{d_{2}+1}^{(2)}=b$ is the bias vector. Then $H_{1}=I$, $H_{2}=(I+bb^{}+W_{1}W_{1}^{})^{-1}$, and $H_{3}=(I+W_{2}W_{2}^{}+W_{2}bb^{}W_{2}^{}+W_{2}W_{1}W_{1}^{}W_{2}^{})^{-1}$. Then we can run the gradient flow in $U\subset\mathcal{M}$ (which has lower dimensions than $R_{\vec{n},\vec{d}}$). Note that when $W_{1},W_{2},b$ are close to zero, $H_{i}$ are close to the identity matrix, and $\sigma_{i}$ is close to $\sigma_{\mathbb{C}}$. Moreover, the second term of (4.8) is close to zero, and the first term is close to the standard metric. Thus when $W_{1},W_{2},b$ are small, the function $f_{\tilde{\gamma}}$ is close to the commonly used one $f^{U}_{W_{1},W_{2},b}(v)=W_{2}\cdot(\sigma_{\mathbb{C}}(W_{1}\cdot v+b)),$ (4.9) and the gradient flow is close to the usual one on the flat space $U$ (see Section 4.3.1). The additional terms can be understood as modifications to ensure the flow converges in $\mathcal{M}$. ### 4.4. A discussion on Morse inequalities By the work of Reineke [Rei08], the framed quiver moduli $\mathcal{M}$ is a tower of Grassmannians (Theorem 2.4), and hence its Poincaré polynomial is a product of that of Grassmannians (Corollary 2.5). Such topological invariants give important information about a gradient flow on $\mathcal{M}$. In particular the Morse inequalities for a Morse function $\mathcal{E}$ on a compact manifold $\mathcal{M}$ state as follows. Let $c^{j}(\mathcal{E})$ be the number of critical points of index $j$ for $\mathcal{E}$. Then for every $j$, $c^{j}(\mathcal{E})\geq h^{j}(\mathcal{M})$ where $h^{j}(\mathcal{M})$ denotes the cohomological numbers (which are coefficients of the Poincaré polynomial). Given a gradient flow, which is a path $\gamma:\mathbb{R}\to\mathcal{M}$ satisfying the gradient flow equation, $\lim_{t\to\pm\infty}\gamma(t)$ are critical points. Moreover, critical points carry important effect to the rate of the gradient flow. Namely, when the flow $\gamma$ gets close to a critical point with index being $1,\ldots,\dim\mathcal{M}-1$, $\\|\gamma^{\prime}(t)\\|=\\|\mathrm{grad}\,\mathcal{E}(\gamma(t))\\|$ becomes small. In other words the flow slows down when it passes through a neighborhood of a critical point. Such a slowing-down effect of saddle points was studied in machine learning in [PDGB14, DPG+14]. The cohomological numbers $h^{j}(\mathcal{M})$ give the minimum number of critical points and hence are important invariants of a neural network (which simply means a directed graph $Q$ together with a dimension vector $\vec{d}$ here). Over $\mathbb{C}$, $h^{j}(\mathcal{M})=0$ when $j$ is odd. Thus the Euler characteristic $\chi_{Q,\vec{d}}$ equals to $\sum_{j}h^{j}(\mathcal{M})$, which is the minimal total number of critical points. It is computed by simply setting $q=1$ in Corollary 2.5. Another important invariant is $\dim\mathcal{M}$ (that is, the number of training parameters of the network), which is simply $\mathcal{D}_{Q,\vec{d}}=\sum_{i\in Q_{0}}d_{i}\left(n_{i}+\sum_{\begin{subarray}{c}j\to i\\\ j\not=i\end{subarray}}d_{j}\right)$ using the notation of Corollary 2.5. (We take $n_{i}=d_{i}+1$ in this section.) These are illustrated in the two practical examples below. ###### Example 4.23. Consider $Q$ being the $A_{k+2}$ quiver, which has $k+2$ vertices labeled by $0,\ldots,k+1$, and there is exactly one arrow from $i$ to $i+1$ for $i=0,\ldots,k$, and no arrow otherwise. Set $d_{-1}:=0$. Then the minimal total number of critical points is $\chi_{Q,\vec{d}}=\prod_{i=-1}^{k}\binom{d_{i}+d_{i+1}+1}{d_{i+1}}$ and $\mathcal{D}_{Q,\vec{d}}=\sum_{i=-1}^{k}d_{i+1}(d_{i}+1).$ ###### Example 4.24. Now consider the following quiver $A_{k+2}^{\prime}$, which has vertices labeled by $0,\ldots,k+1$, and there is one arrow from vertex $i$ to vertex $j$ for every $i<j$. Set $d_{-1}:=0$. Then $\chi_{Q,\vec{d}}=\prod_{i=-1}^{k}\binom{\sum_{j=0}^{i+1}d_{j}+1}{d_{i+1}}$ and $\mathcal{D}_{Q,\vec{d}}=\sum_{i=-1}^{k}d_{i+1}\left(\sum_{j=0}^{i}d_{j}+1\right).$ Figure 7 shows the graph of $\log\chi_{Q,\vec{d}}$ versus $\mathcal{D}_{Q,\vec{d}}$ for the two examples, where we set $k=3$, $d_{1}=600,d_{5}=10$, and $d_{2}=d_{3}=d_{4}$. Note that the quiver denoted by $A_{k+2}^{\prime}$ in Example 4.24 is a simple analog of the network known as ResNet, which adds arrows to the $A_{k+2}$-quiver that skip the middle vertices to get around with the ‘gradient-vanishing problem’. Namely, in the $A_{k+2}$ case, the derivatives of $\mathcal{E}$ with respect to matrix entries for arrows in the early stage are typically very small by chain rule, which is not good for the flow rate. Arrows that skip the middle vertices are added, so that there are short paths which involve the early arrows. From Figure 7, we see that in the same dimensions, the minimal number of critical points in $\mathcal{M}$ is smaller for $A_{5}^{\prime}$ than that for $A_{5}$. (We have numerically verified this for general $k$.) This gives a supporting evidence that $\chi_{Q,\vec{d}}$ is an important invariant in applications to machine learning. Figure 7. A plot of $\log\chi_{Q,\vec{d}}$ (y-axis) versus $\mathcal{D}_{Q,\vec{d}}$ (x-axis) for $A_{5}$ and $A_{5}^{\prime}$. ### 4.5. A remark on Abelianization In many basic neural networks, each vertex of $Q$ is associated with a vector space of only dimension one. When $\vec{d}=\vec{1}$, that is, all entries of the dimension vector equal to one, $\mathcal{M}_{\vec{n},\vec{1}}$ is a quotient by the Abelian group $(\mathbb{C}^{\times})^{\Sigma\vec{d}}$, and hence a toric variety. Indeed, by Theorem 2.4, $\mathcal{M}_{\vec{n},\vec{1}}$ is a tower of projective spaces $\mathbb{P}^{k}$ for a sequence of $k$. Given $Q$ and $\vec{d}$, we can always construct a bigger quiver $Q^{\textrm{Ab},\vec{d}}$as follows. For each vertex $i\in Q_{0}$, we make $d_{i}$ copies indexed by $(i,p)$ for $p=1,\ldots,d_{i}$. For each arrow of $Q$ from $i$ to $j$, we make a corresponding arrow for $Q^{\textrm{Ab},\vec{d}}$ from $(i,p)$ to $(j,q)$ for every $p=1,\ldots,d_{i}$ and $q=1,\ldots,d_{j}$. See Figure 8 for an example. Figure 8. An example of Abelianization. The LHS shows a quiver $Q$ together with the dimension vector $\vec{d}$. The RHS shows $Q^{\mathrm{Ab},\vec{d}}$. Given a dimension vector $\vec{n}\in Q_{0}$, define $\vec{n}^{\textrm{Ab}}\in Q^{\textrm{Ab},\vec{d}}_{0}$ by $\vec{n}^{\textrm{Ab}}_{(i,p)}=\vec{n}_{i}$ for all $p$. The relation between $\mathcal{M}^{Q}_{\vec{n},\vec{d}}$ and the toric variety $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$ is known as Abelianization and is well-studied in [Mar00]. The basic example is $\operatorname{Gr}(n,d)$ (which is the framed moduli for the quiver with a single vertex), whose Abelianization is $(\mathbb{P}^{n})^{d}$ (the disconnected quiver with $d$ vertices and no arrow). Namely, the moduli spaces $\mathcal{M}^{Q}_{\vec{n},\vec{d}}$ and $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$ are GIT quotients of the same vector space $R^{Q}_{\vec{n},\vec{d}}=R^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$ by $\mathrm{GL}_{\vec{d}}$ and $(\mathbb{C}^{\times})^{\Sigma\vec{d}}$ respectively. More precisely, we have the fiber bundle $\mu_{U_{\vec{d}}}^{-1}(\\{I\\})/T\to\mathcal{M}^{Q}_{\vec{n},\vec{d}}$ with fibers being a product of complete flags $U_{\vec{d}}/T^{\Sigma\vec{d}}$, and the inclusion $\mu_{U_{\vec{d}}}^{-1}(\\{I\\})/T\subset\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$. The universal bundles $\mathcal{V}_{i}$ over $\mathcal{M}^{Q}_{\vec{n},\vec{d}}$ is descended from the direct sum of universal line bundles $\bigoplus_{p=1}^{d_{i}}\mathcal{V}_{(i,p)}$ over $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$ (restricted to the above subset). The cohomology of $\mathcal{M}^{Q}_{\vec{n},\vec{d}}$ is generated by the Chern classes $c_{k}(\mathcal{V}_{i})$, which can be written as the $k$-th elementary symmetric polynomials in $c_{1}(\mathcal{V}_{(i,p)})$ for $p=1,\ldots,d_{i}$. On the other side, the cohomology of $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$ is generated by $c_{1}(\mathcal{V}_{(i,p)})$. Note that the functions $f^{Q}_{\tilde{\gamma}}$ and $\mathcal{E}^{Q}$ over $\mathcal{M}^{Q}_{\vec{n},\vec{d}}$ constructed in Section 4.3.2 _cannot_ be lifted to $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$. The reason is that, $f^{Q}_{\tilde{\gamma}}$ and $\mathcal{E}^{Q}$ are $\mathrm{GL}_{\vec{d}}$-equivariant functions on $R^{Q,s}_{n,d}$, the subset of stable representations, rather than the whole $R^{Q}_{n,d}$. The definition of $f^{Q}_{\tilde{\gamma}}$ involves the metrics on the universal bundles $\mathcal{V}_{i}$, which take the expression $(\rho_{i}\rho_{i}^{})^{-1}$, and it is only defined over $R^{Q,s}_{n,d}$ where $\rho_{i}$ is surjective. Rather, we have the functions $f^{Q^{\textrm{Ab},\vec{d}}}_{\tilde{\gamma}}$ and $\mathcal{E}^{Q^{\textrm{Ab},\vec{d}}}$ on $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$, which uses the metrics $(\rho_{i,p}\rho_{i,p}^{})^{-1}$ on the line bundles $\mathcal{V}_{i,p}$. Previously we have taken $\vec{n}=\vec{d}+\vec{1}$ for $\mathcal{M}^{Q}_{n,d}$. After Abelianization, the dimension vectors $\vec{n}^{\mathrm{Ab}}$ and $\vec{1}$ for $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$ no longer satisfy such equality. This is actually not a problem, since the function $f^{Q^{\textrm{Ab},\vec{d}}}_{\tilde{\gamma}}$ defined on $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{2},\vec{1}}$ can be lifted to $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$. Alternatively, we can set the $k$-th framing vectors to be zero for all $k=2,\ldots,d_{i}+1$ and for all vertices $(i,p)\in Q^{\textrm{Ab},\vec{d}}_{0}$. This gives a subvariety of $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{n}^{\textrm{Ab}},\vec{1}}$ which is isomorphic to $\mathcal{M}^{Q^{\textrm{Ab},\vec{d}}}_{\vec{2},\vec{1}}$. ## 5\. Universal Approximation Theorem In Section 4.1, we have introduced the multi-variable functions $\sigma$ coming from moment maps of toric varieties. For instance, $\sigma_{k}(x)=\frac{e^{2x_{k}}}{1+\sum_{j=1}^{d}e^{2x_{j}}}$ for $X=\mathbb{P}^{d}$. In this section, we will give a theoretical basis for using this as an activation function, by proving the universal approximation theorem for this function. The universal approximation theorem ensures that in theory, any given function on a compact set can be approximated (as close as you want) by the functions produced from directed graphs (denoted by $f^{U}_{[V,e]}$ in Section 4.3.1). There are several different versions of this theorem [Cyb89, LLPS93, LJ18, LPW+17]. To the authors’ knowledge, the past works have focused on proving the theorem for single-variable activation functions. In the work of Cybenko in proving the theorem below, rescaling on the domain of the activation function plays a key role. The rescaling technique will also be very useful in our situation. ###### Theorem 5.1 ([Cyb89]). Let $\phi:\mathbb{R}\to\mathbb{R}$ be any continuous function with $\lim_{x\to\infty}\phi(x)=1$ and $\lim_{x\to-\infty}\phi(x)=0$. Let $K$ be a compact set in $\mathbb{R}^{d}$. Then the collection of functions $G:K\to\mathbb{R}$ of the form $G(x)=\sum_{j=1}^{N}\alpha_{j}\phi(y_{j}^{T}x+\theta_{j})$ where $N\in\mathbb{Z}_{>0}$,$y_{j}\in\mathbb{R}^{d}$, and $\theta_{j},\alpha_{j}\in\mathbb{R}$, are dense in the space of continuous functions $C(K)$. The above function $G$ can be understood as $f^{U}_{W_{1},W_{2},b}$ (4.9) produced from the graph $A_{3}$, when the dimension at the output vertex is $d_{3}=1$, and $\sigma:\mathbb{R}^{d_{2}}\to\mathbb{R}^{d_{2}}$ is taken to be $\sigma(\vec{x})=(\phi(x_{1}),\ldots,\phi(x_{d_{2}}))$. (Take $d_{2}=N$, $W_{1}=(y_{j}^{T})_{j=1,\ldots,N}$, $b=(\theta_{j})_{j=1,\ldots,N}$, and $W_{2}=(\alpha_{j})_{j=1,\ldots,N}$.) For general dimension $d_{3}$, we simply have $(G_{1},\ldots,G_{d_{3}})$, where $G_{i}$ are of the same form as above (with different $\alpha_{j,i}$) which can be used to approximate any given continuous function $K\to\mathbb{R}^{d_{3}}$. We will prove the following theorem. Consider the quiver with three vertices as in Section 4.3.3, and the function $f^{U}_{W_{1},W_{2},b}(v)=W_{2}(\sigma(W_{1}\cdot v+b))$ in (4.9), where $\sigma$ is the multi-variable activation function on $\mathbb{R}^{d_{2}}$ made from $\mathbb{P}^{d_{2}}$. ###### Theorem 5.2. Let $K$ be a compact set of $\mathbb{R}^{d_{1}}$, and $f:K\to\mathbb{R}^{d_{3}}$ a continuous function. For any $\epsilon>0$, there exists $d_{2}>0$ and $W_{1}\in\operatorname{Mat}(d_{2},d_{1})$, $W_{2}\in\operatorname{Mat}(d_{3},d_{2})$, $b\in\mathbb{R}^{d_{2}}$ such that $\\|f^{U}_{W_{1},W_{2},b}-f\\|_{L^{2}(K)}<\epsilon$. The compact set is given as a subset in $\mathbb{R}^{n}$. Thus from now on we restrict to the real field, which will suffice for the theorem. This means we take real-valued matrices and the real part $\sigma$ of $\sigma_{\mathbb{C}}$. ### 5.1. Tropical limit A crucial idea in the work of [Cyb89] is to compose $\phi$ with a rescaling, so that it tends to a step function in the limit. We can apply such a rescaling to the multi-variable function $\sigma:\mathbb{R}^{d}\to P^{\circ}$. This is well-known in toric geometry and is called the tropical limit. Let $\Sigma$ be the dual fan of the moment polytope $P$. $\Sigma$ is the collection of cones that are dual to the boundary strata of the polytope $P$. In particular, maximal cones of $\Sigma$ are one-to-one corresponding to corners of $P$. We assume that $\|\Sigma\|=\mathbb{R}^{d}$. $\mathbb{R}^{d}$ is stratified into the relative interiors of cones in $\Sigma$. We recall the following interesting fact from toric geometry. It plays an important role in the study of holomorphic discs and Lagrangian Floer theory for toric varieties. ###### Lemma 5.3. Let $X_{\Sigma}$ be a toric variety (equipped with any toric Kähler form). Let $C$ be a cone in $\Sigma$. For any $x\not=0$ which lies in the relative interior of $C$, $p_{C}=\lim_{t\to-\infty}\sigma(tx)$ exists and equal to a point (which is independent of $x$) in the boundary stratum of $P$ that is dual to $C$. In other words, the family of functions $\sigma_{t}(x)=\sigma(tx)$ converges (as $t\to+\infty$) to the discontinuous function $\sigma_{\infty}$, where $\sigma_{\infty}(x)=p_{C}$ if $x$ belongs to the relative interior of $C$. ###### Proof. The cone $C$ corresponds to a complex torus orbit of the toric variety $X$. To be more explicit, consider a maximal cone $C^{\mathrm{max}}$ that contains $C$. Without loss of generality, let $C^{\mathrm{max}}=\mathbb{R}_{\geq 0}\cdot\\{v_{1},\ldots,v_{d}\\}$, and $C=\mathbb{R}_{\geq 0}\cdot\\{v_{1},\ldots,v_{k}\\}$. $C^{\mathrm{max}}$ gives a local chart $\mathbb{C}^{d}$ of the toric variety, and the complex torus orbit corresponding to $C$ is given by $z_{1}=\ldots=z_{k}=0$. We have a special point given by $z_{1}=\ldots=z_{k}=0,\,z_{k+1}=\ldots=z_{d}=1$ in the orbit. We assert that $p_{C}=\lim_{t\to-\infty}\sigma(tx)\in P$ (for any $x$ in the relative interior of $C$) is the moment-map image of this point. To see this, we write $x=\sum_{i=1}^{k}x_{i}v_{i}$ where $x_{i}\not=0$ for all $i=1,\ldots,k$. Consider the lifting of $tx=(tx_{1},\ldots,tx_{k},0,\ldots,0)$: $(e^{tx_{1}},\ldots,e^{tx_{k}},1,\ldots,1)$ in the chart $\mathbb{C}^{d}\subset X$. Then $\sigma(tx)$ is the moment-map image of $(e^{tx_{1}},\ldots,e^{tx_{k}},1,\ldots,1)$. Taking $t\to-\infty$, $(e^{tx_{1}},\ldots,e^{tx_{k}},1,\ldots,1)\to(z_{1}=\ldots=z_{k}=0,z_{k+1}=\ldots=z_{d}=1)$. Thus $\sigma(tx)$ converges to the above special point $p_{C}$. ∎ In terms of solving equations, $p_{C}$ is the solution of the simultaneous equations $x_{1}=\ldots=x_{k}=0$ and $x_{i}\prod_{j=d+1}^{m}\ell_{j}^{v_{j,i}}\left(0,\ldots,0,x_{k+1},\ldots,x_{d}\right)=1$ for $i=k+1,\ldots,m$. (We have used the dual basis of $\\{v_{1},\ldots,v_{d}\\}$ to write the coordinates of $P$, and $v_{j}=\sum_{i=1}^{d}v_{j,i}v_{i}$.) By above, the solution exists and is unique. ###### Example 5.4. Consider $X=\mathbb{P}^{d}$. Denote the coordinates of $\mathbb{R}^{d}$ by $(x_{1},\ldots,x_{d})$, and set $x_{0}\equiv 0$. The $(d+1-l)$-cones $C$ of $\Sigma$ are given by $\left\\{x_{i_{1}}=\ldots=x_{i_{l}}>x_{k}\textrm{ for all }k\in\\{0,\ldots,d\\}-\\{i_{1},\ldots,i_{l}\\}\right\\}$, where $i_{1},\ldots,i_{l}\in\\{0,\ldots,d\\}$ are fixed, $l=1,\ldots,d+1$. For $\sigma=\left(\frac{e^{2x_{p}}}{1+\sum_{j=1}^{d}e^{2x_{j}}}\right)_{p=1}^{d}$, the point $p_{C}=\lim_{t\to-\infty}\sigma(tx)$ has coordinates $(p_{C})_{i_{r}}=1/l$ for $r=1,\ldots,l$ and $i_{r}\not=0$, and $(p_{C})_{j}=0$ for all other $j\not=i_{1},\ldots,i_{l},0$. In particular, for the maximal cones $S_{i}=\\{x_{i}>x_{k}\textrm{ for all }k\in\\{0,\ldots,d\\}-\\{i\\}\\}$, $p_{S_{i}}=\epsilon_{i}$ for $i=0,\ldots,d$ where $\epsilon_{0}:=0$ and $\\{\epsilon_{1},\ldots,\epsilon_{d}\\}$ denotes the standard basis. Figure 9. The left shows the fan picture of $\mathbb{P}^{2}$, and the right shows the moment-map polytope. The dots show the limit points $p_{C}$ for each cone $C$ of the fan. ###### Corollary 5.5. Let $K$ be any compact set in $\mathbb{R}^{d}$. For any $\epsilon>0$ and an open neighborhood of the union $U$ of codimension-one strata of $\Sigma$, there exists $t\gg 0$ such that $\|\sigma_{t}(x)-\sigma_{\infty}(x)\|<\epsilon$ for all $x\in K-U$. ($\sigma_{\infty}$ is defined in Lemma 5.3.) ###### Proof. Any $x\in K-U$ belongs to one of the maximal cones $C$. By Lemma 5.3, $\sigma_{t}(x)$ converges to $p_{C}$. Moreover, both $\sigma_{t}$ and $\sigma_{\infty}$ are continuous on $K-U$. Then the result follows from the compactness of $K-U$. ∎ In order to prove Theorem 5.2, we consider a particular type of polyhedral decompositions of $\mathbb{R}^{n}$, which we call to be a centered simplicial web. ### 5.2. Centered polyhedral web ###### Definition 5.6. A centered simplicial web with $N$ ordered compact chambers in $\mathbb{R}^{n}$ is a polyhedral decomposition of $\mathbb{R}^{n}$ whose vertices are all trivalent, defined inductively on the number of compact chambers as follows. A centered simplicial web with zero compact chamber is the polyhedral decomposition given by the fan of $\mathbb{P}^{n}$, up to an affine linear isomorphism in $\mathrm{GL}(n,\mathbb{R})\ltimes\mathbb{R}^{n}$. Now suppose the notion of a centered simplicial web with $N$ ordered compact chambers has been defined, which has exactly $(n+1)$ non-compact rays (which we call the outer rays), whose corresponding infinite lines intersect at exactly one point called the $N$-th center that lies in the union of the $N$ compact chambers. Moreover, the web is required to have $(n+1)$ non-compact chambers; each non-compact chamber is adjacent to $n$ outer rays and opposite to the remaining one outer ray. (‘Opposite’ here means that the non-compact chamber is disjoint from the corresponding outer ray, whose infinite line intersects the chamber at a half-line.) The outer rays are one-to-one corresponding to their opposite non-compact chambers. A centered simplicial web with $(N+1)$ ordered compact chambers is defined as follows. First, take a centered simplicial web with $N$ ordered compact chambers. Second, we choose a non-compact chamber, and denotes the direction of its opposite ray by a non-zero vector $v$. Third, we take an affine hyperplane which intersects all the relative interior of the $n$ adjacent rays of the non-compact chamber. This bounds a new compact chamber and the intersection points $V_{i}$ are the new vertices. Finally, we choose the $(N+1)$-th center to be $c_{N+1}=c_{N}-tv$, where $c_{N}$ is the $N$-th center, and $t\in\mathbb{R}_{>0}$ is taken such that $c_{N+1}$ lies in the union of the compact chambers (including the new one). Then we have $n$ new rays emanated from the vertices $V_{i}$ whose infinite lines pass through $c_{N+1}$. This gives a new web with $(N+1)$ ordered compact chambers, and it still has $(n+1)$ non-compact chambers, each of which is adjacent to $n$ outer rays and opposite to one outer ray. See Figure 10 for some examples of centered simplicial webs in $\mathbb{R}^{2}$. Figure 10. Examples of centered simplicial web in $\mathbb{R}^{2}$. They have zero,one, and two compact chambers respectively. From the above definition, there is a one-to-one correspondence between the centers and compact chambers. Moreover, the $(k+1)$-th compact chamber $C_{k+1}$ (for $k=1,\ldots,N-1$) is associated with a one-strata of the web, which is a subset of the opposite ray of the non-compact chamber containing $C_{k+1}$ in the $(k+1)$-th inductive step. Furthermore, both the $k$-th and $(k+1)$-th centers lie in the infinite line of the associated 1-strata of $C_{k+1}$. ###### Remark 5.7. The above notion is closely related to tropical subvarieties. In the tropical context, there is an integral structure on the ambient space and the balancing condition (whose definition requires the integral structure) is imposed on each vertex of a tropical variety. However, we do not have an integral structure here, since $\sigma_{\mathbb{C}}$ is defined on the universal cover $\mathbb{C}^{d}$ rather than $(\mathbb{C}^{\times})^{d}$ (see Corollary 4.2 and Example 4.4). It means affine linear maps are taken over $\mathbb{R}$ rather than over $\mathbb{Z}$. Instead of the balancing condition, we impose the notion of centers in the above definition. ###### Theorem 5.8. Given a centered simplicial web $A$ with $N$ ordered compact chambers in $\mathbb{R}^{n}$, there exists an affine-linear embedding $L:\mathbb{R}^{n}\to\mathbb{R}^{d}$, where $d=n+N$, such that the $L$-preimage of the fan of $\mathbb{P}^{d}$ in $\mathbb{R}^{d}$ equals to the web $A$. ###### Proof. We shall prove the following statement: given a centered simplicial web $A$ with $N\geq 1$ compact chambers in $\mathbb{R}^{n}$, there exists a centered simplicial web $B$ with $N-1$ compact chambers in $\mathbb{R}^{n+1}$ such that the intersection of $\mathbb{R}^{n}\cong\mathbb{R}^{n}\times\\{0\\}$ with $B$ equals to $A$. Then by applying this statement $N$ times, we obtain a web $B^{(N)}$ with zero compact chamber in $\mathbb{R}^{n+N}$ whose intersection with $\mathbb{R}^{n}\times\\{0\\}$ gives $A$. By an affine linear isomorphism on $\mathbb{R}^{n+N}$, $B^{(N)}$ is identified with the fan of $\mathbb{P}^{n+N}$. The required map $L$ is given by the composition of the inclusion $\mathbb{R}^{n}\times\\{0\\}\subset\mathbb{R}^{n+N}$ with this linear isomorphism. The above statement is proved by induction on $N$. First consider the case $N=1$. We take a point $V$ away from $\mathbb{R}^{n}\times\\{0\\}\subset\mathbb{R}^{n+1}$. Then we take a cone at $V$ over the compact simplicial chamber of $A$. Moreover, the line joining $V$ with the given center of $A$ intersects with the complement of the cone and produces a ray emanated from $V$. This gives a simplicial web in $\mathbb{R}^{n+1}$ with no compact chamber, whose intersection with $\mathbb{R}^{n}\times\\{0\\}$ is exactly $A$. (See Figure 11.) Now suppose it is true for $N$. Consider a centered simplicial web $A$ in $\mathbb{R}^{n}\cong\mathbb{R}^{n}\times\\{0\\}$ with $N+1$ ordered compact chambers. We can take away the $(N+1)$-th compact chamber $C=C_{N+1}$ (and forget the corresponding center $c_{N+1}$) and obtain a centered simplicial web $A^{\prime}$ with $N$ compact chambers. By inductive hypothesis, there exists a centered simplicial web $B^{\prime}$ in $\mathbb{R}^{n+1}$ with $N-1$ compact chambers whose intersection with $\mathbb{R}^{n}\times\\{0\\}$ gives $A^{\prime}$. The compact chamber $C$ of $A$ is contained in a non-compact chamber $C^{\prime}$ of $A^{\prime}$, which is the intersection of $\mathbb{R}^{n}\times\\{0\\}$ with a non-compact chamber $D^{\prime}$ of $B^{\prime}$. $C^{\prime}$ is opposite to an outer ray $R$ of $A^{\prime}$, which is the intersection of $\mathbb{R}^{n}\times\\{0\\}$ with a non-compact 2-plane $P$ of $B^{\prime}$. Note that the last two centers $c^{A}_{N}$ and $c^{A}_{N+1}$ of $A$ are contained in the line of $R$, and hence contained in the infinite 2-plane of $P$. Consider the two rays of $B$ that are adjacent to $P$. Denote the one which is opposite to the chamber $D^{\prime}$ by $L$. The other one is denoted by $L^{\prime}$, which must be adjacent to $D^{\prime}$. Now we construct a web $B$ in $\mathbb{R}^{n+1}$ whose intersection with $\mathbb{R}^{n}\times\\{0\\}$ gives $A$. A point $V$ in the relative interior of the ray $L^{\prime}$ is taken to be a new vertex. Consider the line passing through $V$ and the last center $c^{A}_{N+1}$. This line lies in the infinite 2-plane of $P$. Thus for a generic choice of $V$, it must intersect with the infinite line of $L$ at a point, which we shall define as the new center $c^{B}_{N}$ for $B$. $V$ is taken far away enough in the ray $L^{\prime}$ so that the intersection point $c^{B}_{N}$ equals to $c^{B}_{N-1}-v$ for some vector $v$ in the direction of $L$. Consider the $n$ outer rays of $A^{\prime}$ that are adjacent to the chamber $C^{\prime}$. They are the intersections of $\mathbb{R}^{n}\times\\{0\\}$ with the corresponding $n$ outer-2-planes of $B^{\prime}$ that are adjacent to $L^{\prime}$. The last chamber $C$ of $A$ is formed by the hyperplane through the vertices taken in the relative interior of the $n$ outer rays of $A^{\prime}$. The lines joining $V$ to these vertices in $A$ lie in the outer-2-planes of $B^{\prime}$, and hence intersect with the corresponding outer rays of $B^{\prime}$ at certain points, which we take to be new vertices of $B$. The hyperplane through $V$ and these new vertices bounds a new chamber. The lines joining $c^{B}_{N}$ with the new vertices produce the outer rays of the new web $B$. This gives $B$ whose intersection with $\mathbb{R}^{n}\times\\{0\\}$ equals to $A$. ∎ Figure 11. Construction of a tropical web with zero compact chamber whose intersection with a hyperplane equals to a given tropical web with one compact chamber. The inductive step in the above proof is illustrated by Figure 12. Figure 12. Construction of a tropical web with $3$ compact chambers whose intersection with a hyperplane equals to a given tropical web with $2$ compact chambers. Next, we consider polytopes rather than simplices and the corresponding webs formed from polytopes. Motivated from the well-known fact below, we define a centered polyhedral web to be the intersection of a centered simplicial web with an affine subspace. ###### Proposition 5.9. For a polytope $P$ with $m$ facets in $\mathbb{R}^{n}$ where $m>n+1$, there exists a simplex $S$ in $\mathbb{R}^{m-1}$ such that $S\cap(\mathbb{R}^{n}\times\\{0\\})=P$ (where $\mathbb{R}^{n}$ is identified with $\mathbb{R}^{n}\times\\{0\\}$). The simplex $S$ in Proposition 5.9 can be constructed as follows. Without loss of generality suppose $0\in P$. Consider the dual polytope $P^{\vee}=\\{\nu\in(\mathbb{R}^{n})^{}:(\nu,v)\leq 1\textrm{ for all }v\in P\\}$, which is the convex hull of its vertices $\nu_{i}$ for $i=1,\ldots,m$. Then we have a surjective map from the standard simplex $S^{\vee}=\\{\sum_{i}a_{i}\epsilon_{i}^{}\in(\mathbb{R}_{\geq 0}^{m})^{}:\sum_{i}a_{i}=1\\}$ (where $\\{\epsilon_{i}^{}:i=1,\ldots,m\\}$ is the standard basis) to $P^{\vee}$ by sending $\epsilon_{i}^{}$ to $\nu_{i}$. $S^{\vee}$ in the affine subspace $\\{\sum_{i}a_{i}=1\\}$ can be identified as a simplex in $(\mathbb{R}_{\geq 0}^{m-1})^{}$ by the projection along the direction $-\sum_{i}\epsilon_{i}$. Then the dual linear map gives the desired linear injection $\mathbb{R}^{n}\to\mathbb{R}^{m-1}$ which sends $P$ into $S$. By composing with a linear isomorphism, the image of $\mathbb{R}^{n}$ can be made to be $\mathbb{R}^{n}\times\\{0\\}$. As a result, a centered polygonal web with one compact chamber (which is constructed by taking a polygon with a chosen center $c$ and outer rays at vertices whose lines pass through $c$) can be obtained as an intersection with $\mathbb{R}^{n}\times\\{0\\}$ of a centered simplicial web with one compact chamber in $\mathbb{R}^{m-1}$ (where $m$ is the number of non-compact chambers). See the left of Figure 13 for an example. Figure 13. The left shows an example of a polyhedral web with one compact chamber, which is given as an intersection of a simplicial web with a subspace. The right two figures show a concentric simplicial and a polygonal web. The following degenerate configuration will be helpful. In Definition 5.6, suppose we take all the centers to be the same. Moreover, suppose the new hyperplane introduced to bound a new chamber is allowed to intersect the outer rays at the original vertices (rather than their relative interior). Then we can construct the following configuration. ###### Definition 5.10. Let’s take $(n+1)$ rays emanated from $0\in\mathbb{R}^{n}$, such that any $n$ of them are linearly independent. For each ray, we take a sequence of distinct points $V_{i,k}$ (where $i=1,\ldots,n+1$ is indexing the ray) such that $V_{i,k+1}-V_{i,k}$ is pointing in the ray direction. Then for each $k>0$, we take a simplex with vertices at $V_{i,k}$ for $i=1,\ldots,n+1$. This gives a polyhedral decomposition of $\mathbb{R}^{n}$. This is called a concentric simplicial web. By taking an intersection of the above degenerate simplicial configuration with a subspace (that passes through the center $0$), we get a configuration made from a sequence of polytopes whose vertices lie in a fixed collection of $p$ rays, where $p$ is the number of vertices of each polytope. We call this a concentric polyhedral web. See the right of Figure 13. ### 5.3. Proof of the approximation theorem We are now ready to prove Theorem 5.2. ###### Proof of Theorem 5.2. For any $\delta>0$, we can take a concentric polyhedral web $B$ in $\mathbb{R}^{d_{1}}$, such that for every chamber $C$ of $B$, $C\cap K$ is contained in a $\delta$-ball. $B$ is constructed as follows. Without loss of generality, let $K\ni 0$. First, we take a polytope $P$ that lies in a $\delta$-ball centered at $0\in\mathbb{R}^{d_{1}}$. $P$ induces a subdivision on the unit sphere $\mathbb{S}^{d_{1}-1}$ by projecting its boundary strata onto $\mathbb{S}^{d_{1}-1}$. $P$ is taken with sufficiently many vertices such that the induced subdivision on the unit sphere $\mathbb{S}^{d_{1}-1}$ lies in a $\delta^{\prime}$-ball for a chosen $\delta^{\prime}$. By Proposition 5.9, $P=S\cap(\mathbb{R}^{d_{1}}\times\\{0\\})$ for some simplex $S$ in $\mathbb{R}^{m-1}$ where $m$ is the number of facets of $P$. We take $0\in S$ to be the center. Then we take rays from $0$ through the vertices of $S$ and construct a concentric simplicial web. Since $K$ is compact, by taking $\delta^{\prime}$ sufficiently small, and the sequences of vertices in the rays sufficiently close to each other, the resulting concentric polyhedral web $B$ can be made such that every chamber intersects $K$ in a $\delta$-ball. Next, we take a centered simplicial web $A$ in $\mathbb{R}^{m-1}$ whose centers are chosen sufficiently close to each other, and the vertices in the inductive steps are taken such that $A\cap K$ is sufficiently close $B\cap K$. Namely, for every chamber $C$ of $A$, $C\cap K$ lies in the $\delta$-neighborhood of $C^{\prime}\cap K$ for the corresponding chamber $C^{\prime}$ of $B$. In particular, $C\cap K$ is contained in a $2\delta$-ball. By Theorem 5.8, there exists $L^{\prime}:\mathbb{R}^{m-1}\to\mathbb{R}^{d_{2}}$ such that $A$ is the $L^{\prime}$-preimage of the fan $\Sigma_{\mathbb{P}^{d_{2}}}$. By composing $L^{\prime}$ with $\mathbb{R}^{d_{1}}\times\\{0\\}\subset\mathbb{R}^{m-1}$, we obtain $L:\mathbb{R}^{d_{1}}\to\mathbb{R}^{d_{2}}$ such that for every maximal cone $S_{i}$ of $\Sigma_{\mathbb{P}^{d_{2}}}$, $L^{-1}(S_{i})\cap K$ lies in a $(2\delta)$-ball. Since $f$ is uniformly continuous in $K$, for every $\epsilon>0$, $\delta$ can be taken such that $\|f(x)-f(y)\|<\epsilon$ for every $x,y\in K$ lying in a $2\delta$-ball. In particular, we have a step function $s=\sum_{C}r_{C}\delta_{C}$ supported over $A$ (where $\delta_{C}(x)=1$ for $x\in C$ and $0$ otherwise, and $C$ are chambers of $A$) such that $\\|f-s\\|_{L^{2}(K)}<\epsilon\sqrt{\mathrm{Vol}(K)}$. We have the step function $\sigma_{\infty}$ which sends the interior of the maximal cones $S_{i}$ of $\Sigma_{\mathbb{P}^{d_{2}}}$ to $e_{i}$ for $i=0,\ldots,d_{2}$, where $e_{0}=0$. (See Example 5.4.) The cone $S_{i}$ corresponds to chambers $L^{-1}(S_{i})$ of $A$ under $L$. Since $\\{e_{i}-e_{0}:i=1,\ldots,d_{2}\\}$ forms a basis, there exists a unique affine linear map $W_{2}:\mathbb{R}^{d_{2}}\to\mathbb{R}^{d_{3}}$ which sends $e_{i}$ to $r_{L^{-1}(S_{i})}$ for all $i=0,\ldots,d_{2}$. Thus $s=W_{2}\circ\sigma_{\infty}\circ L$. Finally, by Corollary 5.5, there exists $t\gg 0$ such that $\|W_{2}\circ\sigma_{t}-W_{2}\circ\sigma_{\infty}\|<\epsilon/2$ on $L(K)-U$, where $U$ is an arbitrary open neighborhood of the codimension-one strata of $\Sigma_{\mathbb{P}^{d}_{2}}$. Moreover, $\|W_{2}\circ\sigma_{t}-W_{2}\circ\sigma_{\infty}\|$ is bounded. Hence by taking $\mathrm{Vol}(L^{-1}(U)\cap K)$ sufficiently small, we have $\\|W_{2}\circ\sigma_{t}\circ L-W_{2}\circ\sigma_{\infty}\circ L\\|_{L^{2}(K)}<\epsilon$. In conclusion, $\\|W_{2}\circ\sigma\circ(tL)-f\\|_{L^{2}(K)}\leq\\|W_{2}\circ\sigma\circ(tL)-W_{2}\circ\sigma_{\infty}\circ L\\|_{L^{2}(K)}+\\|W_{2}\circ\sigma_{\infty}\circ L-f\\|_{L^{2}(K)}$ can be made arbitrarily small. ∎ ## References * [Abo06] M. 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# Forward Beam Monitor for the KATRIN experiment A. Beglarian E. Ellinger111Corresponding author. N. Haußmann K. Helbing S. Hickford U. Naumann H.-W. Ortjohann M. Steidl J. Wolf and S. Wüstling ###### Abstract The KArlsruhe TRItium Neutrino (KATRIN) experiment aims to measure the neutrino mass with a sensitivity of $0.2\text{\,}\mathrm{eV}$ ($90\text{\,}\mathrm{\char 37\relax}$ CL). This will be achieved by a precision measurement of the endpoint region of the $\upbeta$-electron spectrum of tritium decay. The $\upbeta$-electrons are produced in the Windowless Gaseous Tritium Source (WGTS) and guided magnetically through the beamline. In order to accurately extract the neutrino mass the source activity is required to be stable and known to a high precision. The WGTS therefore undergoes constant extensive monitoring from several measurement systems. The Forward Beam Monitor (FBM) is one such monitoring system. The FBM system comprises a complex mechanical setup capable of inserting a detector board into the KATRIN beamline with a positioning precision of better than $0.3\text{\,}\mathrm{mm}$. The electron flux density at that position is on the order of ${10}^{6}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{mm}}^{-2}$. The detector board contains two silicon detector chips of p-i-n diode type which can measure the $\upbeta$-electron flux from the source with a precision of $0.1\text{\,}\mathrm{\char 37\relax}$ within $60\text{\,}\mathrm{s}$ with an energy resolution of FWHM $=$ $2\text{\,}\mathrm{keV}$. The unique challenge in developing the FBM arise from its designated operating environment inside the Cryogenic Pumping Section which is a potentially tritium contaminated ultra-high vacuum chamber at cryogenic temperatures in the presence of a $1\text{\,}\mathrm{T}$ strong magnetic field. Each of theses parameters do strongly limit the choice of possible materials which e.g. caused difficulties in detector noise reduction, heat dissipation and lubrication. In order to completely remove the FBM from the beam tube a $2\text{\,}\mathrm{m}$ long traveling distance into the beamline is needed demanding a robust as well as highly precise moving mechanism. ## 1 Introduction The KATRIN experiment will improve the sensitivity of neutrino mass measurements to $m_{\nu}=$ $0.2\text{\,}\mathrm{eV}$ ($90\text{\,}\mathrm{\char 37\relax}$ C.L.) corresponding to a $5\text{\,}\sigma$ discovery potential for a mass signal of $m_{\nu}=$ $0.35\text{\,}\mathrm{eV}$ [1, 2] in the most sensitive direct neutrino mass experiment to date. The neutrino mass will be derived from a precise measurement of the shape of the tritium $\upbeta$-decay spectrum near its endpoint at $E_{0}=$ $18\,573.7\pm 0.1\text{\,}\mathrm{eV}$ [3]. The source of $\upbeta$-electrons is a Windowless Gaseous Tritium Source (WGTS) which has an activity of ${10}^{11}\text{\,}\mathrm{Bq}$. The layout of the KATRIN beamline [4] is shown in figure 1. The Source and Transport Section (STS) consists of the WGTS, the Differential Pumping Section (DPS), the Cryogenic Pumping Section (CPS), and several source monitoring and calibration systems [5]. Along the beamline superconducting solenoids generate a magnetic field of several Tesla strength which adiabatically guides the $\upbeta$-electrons towards the spectrometers while excess tritium is pumped out of the system. The Spectrometer and Detector Section (SDS) consists of the pre-spectrometer, the main-spectrometer, the monitor-spectrometer, and the Focal Plane Detector (FPD). All spectrometers are of MAC-E-Filter type which transmit electrons with energies above a chosen retarding energy [6], and reject those with lower energies. The main-spectrometer can perform an energy analysis of the $\upbeta$-electrons with an energy resolution of $0.93\text{\,}\mathrm{eV}$ at $18.6\text{\,}\mathrm{keV}$. Figure 1: The KATRIN beamline. The FBM is located at the end of the CPS and represents the final source monitoring system before the $\upbeta$-electrons enter the spectrometer and detector section. The source-related parameters associated with the main systematic uncertainties in the determination of the neutrino mass are activity fluctuations of the WGTS, energy loss corrections (of $\upbeta$-electron scattering in the WGTS), the final state distribution, the source magnetic field, and the source plasma condition. In order to analyse the tritium $\upbeta$-spectrum and determine the neutrino mass the WGTS needs to be extremely stable, particularly in its activity and isotopic composition. Therefore, the WGTS properties need to be known with high precision, and are continuously monitored for short and long term fluctuations. There are several monitoring and calibration subsystems associated with the WGTS [5]. Results from the various subsystems are combined over long time periods during extended measurement time. This paper focuses on one such activity monitoring system, the Forward Beam Monitor (FBM). The FBM is the final monitoring subsystem for $\upbeta$-electrons from the source before they enter into the spectrometer and detector section. It has been commissioned prior to the KATRIN krypton measurement campaign in June 2017 [7]. Initial data was then obtained during the krypton measurement campaign and during the KATRIN first tritium measurement campaign in May 2018 [8]. The FBM is capable of continuously monitoring variations of the electron flux and changes in the measured shape of the $\upbeta$-decay spectrum during the KATRIN neutrino mass measurement phases. This paper is organised as follows. In section 2 the WGTS and its operating parameters are introduced and in section 3 the FBM measurement principle for the monitoring of the relevant WGTS parameters is explained. Section 4 contains a technical description of the FBM. In section 5 the FBM commissioning and results from the krypton and first tritium measurement phases are presented, and section 6 contains the summary. ## 2 Tritium source The WGTS is the origin of $\upbeta$-electrons whose observed spectrum will ultimately lead to the measurement of the neutrino mass [9]. The general setup of the WGTS is shown in figure 2. It is a column of tritium gas inside a cylinder with a diameter of $90\text{\,}\mathrm{mm}$ and a length of $10\text{\,}\mathrm{m}$. The latter is situated in a homogeneous magnetic field of $3.6\text{\,}\mathrm{T}$ generated by superconducting solenoid magnets. The tritium gas is injected in the middle of the beam tube with an adjustable pressure $p_{\text{in}}$ = ${10}^{-3}\text{\,}\mathrm{mbar}$, and is pumped out at both ends with a constant outlet pressure of $p_{\text{out}}=$ $0.05$ $p_{\text{in}}$. Figure 2: Setup of the WGTS. Tritium is injected into the centre of the cylinder and pumped out at both ends. The flux tube is surrounded by superconducting magnets to guide the $\upbeta$-electrons. The longitudinal density profile of the tritium molecules along the column is shown above. ### 2.1 Column density The column density is defined as tritium molecule density integrated along the central axis of the source, i.e., the number of tritium molecules per source cross section area. The neutrino mass measurement depends on the accurate description of inelastic scattering of electrons by the gas molecules inside the source. There are several key parameters of the WGTS that need to be kept stable with high precision in order to achieve a high sensitivity in the neutrino mass measurement. These include * • Beam tube temperature The molecular tritium gas must be at cryogenic temperatures of $<80\text{\,}\mathrm{K}$ to minimise corrections to the electrons energy due to thermal movement of the decaying mother atoms. The cooling concept is based on a two-phase liquid neon thermosiphon [10, 11]. * • Pressure The amount of tritium inside the source scales with the inlet pressure. Stabilisation is achieved using a pressurised control vessel from which tritium flows via a capillary to the beam tube. * • Tritium purity A high isotopic purity of molecular tritium gas ($>95\text{\,}\mathrm{\char 37\relax}$) is required. The tritium purity $\mathbf{\epsilon_{T}}$ is given by the ratio of the number of tritium atoms to the total sum of atoms in the WGTS. In addition to T2 other isotopolouges include DT, HT, D2, HD, and H2. The tritiated hydrogen isotopolouges differ in their mass, recoil energies, and the rotational and vibrational final state distributions of their daughter molecules following tritium decay. The gas composition is measured via LAser RAman spectroscopy (LARA) [12, 13]. These key parameters have an effect on the rate and/or energy of the electrons emitted from the source. There are several control and monitoring systems in the KATRIN experiment with the purpose of meeting the precision and stability requirements of the key source parameters. The column density, $\mathcal{N}$, can be obtained by combining an in-situ measurement of the tritium purity with an activity (decay rate) measurement. The count rate $S$ of $\upbeta$-electrons from the source as measured by dedicated particle detectors (activity monitors, see section 2.3) scales as $S=C\cdot\epsilon_{T}\cdot\mathcal{N}$ (2.1) where $C$ is a proportionality constant encompassing experimental properties such as detector efficiency and acceptance, and the half-life of tritium. Small fluctuations of the source parameters lead to changes in the observed shape of the differential $\upbeta$-electron spectrum. Fluctuations in the column density are expected to be in the ${10}^{-3}$ regime. Given the targeted sensitivity for the neutrino mass measurement, column density and tritium purity must not give rise to an uncertainty beyond $\delta m^{2}_{\nu}=$ $7.5\text{\times}{10}^{-3}\text{\,}{\mathrm{eV}}^{2}$ to the neutrino mass analysis. ### 2.2 Electron transport The $\upbeta$-electrons resulting from the decay of the tritium are adiabatically guided towards the spectrometer and detector section. The transport section is also used to eliminate the tritium flow towards the spectrometers which must be free of tritium in order to meet the necessary background requirements for neutrino mass measurements. The transport section consists of a Differential Pumping Section (DPS) and a Cryogenic Pumping Section (CPS). The DPS consists of five beam tube segments within superconducting solenoids with turbomolecular pumps between each [14]. The CPS traps all remaining traces of tritium by cryo-sorption on argon frost at $4\text{\,}\mathrm{K}$ condensed on the gold plated surfaces of the beam tube [15, 16]. Both the DPS and CPS have $20\text{\,}\mathrm{\SIUnitSymbolDegree}$ chicanes to block the line of sight for the diffusing tritium gas and to increase the probability that the tritium molecules get pumped away or hit the walls of the beam tube. At the end of the transport section the tritium flow is suppressed by $14$ orders of magnitude compared to the center of the WGTS. The electron flow is unaffected and all electrons are guided adiabatically towards the spectrometer and detector section. ### 2.3 Activity monitors The activity of the tritium source is monitored by two systems. These activity monitors 1. 1. provide information about fluctuations of the WGTS activity on a timescale of minutes and 2. 2. are used (together with the measured tritium purity) to monitor the column density with $0.1\text{\,}\mathrm{\char 37\relax}$ precision, via equation (2.1). One of these activity monitors is located at the rear wall upstream of the WGTS. This detector measures the X-rays created when the $\upbeta$-electrons impact on the rear wall [5]. The second activity monitor is called the Forward Beam Monitor (FBM). It is located downstream towards the main-spectrometer in the transport section, mounted between the last two superconducting solenoids of the CPS. Here the tritium flow has been suppressed by a factor of $10^{$14$}$, to approximately ${10}^{-14}\text{\,}\mathrm{mbar}\text{\,}\mathrm{l}\text{\,}{\mathrm{s}}^{-1}$, which minimises background effects and contamination from tritium. The magnetic field in this position is axially symmetric with a magnitude of $0.84\text{\,}\mathrm{T}$ so the spatial homogeneity of the source profile can be studied. The FBM is the final measurement component before the spectrometer and detector section. ## 3 Measurement principle The FBM measures $\upbeta$-electrons from the tritium source as they are guided to the spectrometer and detector section. Hence, the $\upbeta$-electrons are following the beamline when they are detected by the FBM. Such a detector must not shadow any part of the electron flux tube that will be used for the measurement of the neutrino mass. Therefore, the FBM configuration is such that the detector is located in the outer rim of the electron flux during neutrino mass measurements. The active radius of the flux tube used for measurement is approximately $71\text{\,}\mathrm{mm}$ and the outer rim in which the detector is situated is up to $7\text{\,}\mathrm{mm}$ wide. The p-i-n diode detectors have an energy threshold of approximately $5\text{\,}\mathrm{keV}$, dependent on the background noise and the type of diode used. This lower energy value is determined during calibration of each diode. For an accurate rate measurement the lower energy threshold needs to be stable. Figure 3: Cross section of the FBM setup with the electron flux tube in the KATRIN beamline. During nominal monitoring operation the FBM is situated in the outer rim of the flux tube, up to approximately $7\text{\,}\mathrm{mm}$ in thickness. It is assumed that the activity measurement in the outer rim of the flux tube is representative of the activity across the entire beamline cross section. Variations of the column density in the radial direction are expected to be on the ${10}^{-4}$ level [17]. The assumption that the outer rim is representative of the entire flux tube is verified during repeated calibration runs when the FBM is moved across the beamline. These two operation modes of the FBM are standard “monitoring mode” and calibration “scanning mode” and are described in the following sections. ### 3.1 Monitoring mode Monitoring mode is the standard mode of operation for the FBM. It is intended for permanent and continuous monitoring of the source activity and the main observable is the electron count rate. Together with the measurement of the tritium purity, the FBM monitoring mode provides continuously information on the column density of the source. ### 3.2 Scanning mode Flux tube scans are performed during calibration of the KATRIN experiment. The purpose of scanning is to 1. 1. confirm that the activity in the beamline outer rim is representative of the entire flux tube, 2. 2. map any irregularities in the cross section of the flux tube, and 3. 3. investigate the area of the flux tube entering the spectrometer and detector section (i.e. measure possible shadow effects by STS instrumentation). During the KATRIN experiment calibration runs are performed between neutrino mass measurement runs once every $\sim 60\text{\,}\mathrm{d}\mathrm{a}\mathrm{y}\mathrm{s}$. During commissioning and initial measurement campaigns the scanning mode was used more frequently. ## 4 Technical description In the following sections a technical description of the FBM is given. A more detailed description can be found in ref. [18]. Further information on the basic concept and the early development of the FBM can be found in ref. [19] and [20]. ### 4.1 Vacuum manipulator The measurement of the electron flux is performed under ultra high vacuum (UHV) conditions in a potentially tritium contaminated environment. The main mechanical requirements for the vacuum manipulator are: 1. 1. to situate the FBM detector board in the outer rim of the flux tube without shadowing the main detector and additionally to move it throughout the cross section of the flux tube, 2. 2. to be capable of removing all FBM components out of the CPS allowing full metal sealed vacuum gate valves to separate the FBM volume from the CPS volume, and 3. 3. to provide a safe enclosure for tritium, complying with all radiation safety regulations of the tritium laboratory. An overview of the complete FBM setup is shown in figure 4 and figure 5. Figure 4: The FBM hardware setup. The FBM is attached to the CPS perpendicular to the beamline. The CPS and FBM gate valves separate the FBM from the CPS if the detector is in parking position within the FBM six-way cross. With the help of the $2\text{\,}\mathrm{m}$ long bellow the detector arm can be driven into the flux tube within the CPS. The vacuum components of the FBM setup are separated from the CPS by a gate valve. Behind this valve the FBM detector board is completely removed from the KATRIN beamline. Attached to this volume are a turbomolecular pump and pressure gauges. Behind the main FBM vacuum volume are bellows, support structures, stepper motors, rotary encoders, and electrical feedthroughs. These components provide the movement of the FBM detector board and the readout of the measured data. The movement of the detector board is realised by combining two linear drive mechanisms. A long stainless steel support tube with an outer diameter of $54\text{\,}\mathrm{mm}$ can be moved over a distance of $1.8\text{\,}\mathrm{m}$ along its symmetry axis. At its forward end the detector holder (hereafter known as the “front end”, see figure 6) is attached. The support tube provides space for electrical feeding and a driving rod which can be moved coaxially along the tube by approximately $10\text{\,}\mathrm{cm}$. The latter linear movement is converted by the front end into a rotary movement with a rack and pinion drive such that the combination of these two movements enable the positioning of the detector board in a two-dimensional plane. Two edge-welded bellows are used to realise the linear movements in the vacuum. The large bellow has an unusually long extended length of $2223\text{\,}\mathrm{mm}$ with a working-stroke of $1800\text{\,}\mathrm{mm}$. The back end provides electrical feedthroughs as well as the mechanics for the rotary movement. The system is moved with a $2\text{\,}\mathrm{m}$ long spindle drive featuring low play and two carriages for more stability. To prevent the long bellows, the driving rod, and the support tube from sagging and hanging down, several supports are added to the setup. These include 3D printed trolleys outside the vacuum which can move freely over the slider and are automatically pulled along from the motion of the bellows, and structures with full ceramic ball bearings for supporting the long tube and driving rod inside the vacuum chamber. The front end which contains the FBM detector board is the mechanical and electrical connection between the detector board and the manipulator. It converts the linear movement of the driving rod into a rotary movement with a low play rack-and-pinion drive to allow the movement in the vertical direction. It is shown in figure 6. To reduce magnetic force acting on the system, as well as to reduce influences onto the electron guiding magnetic field, the front end, similar to all other vacuum parts of the FBM, is made of low permeability $\mu_{\text{p}}$ materials (such as stainless steel $1.4429$ with $\mu_{\text{p}}$ $<1.005$). To prevent cold welding of moveable parts the materials of the pinion (stainless steel), rack (titanium), and the front end’s cylindrically shaped main body (stainless steel), are alternated. A precise groove in the main cylinder allows leading the rack with low play. To reduce friction, an ultra low friction and UHV compatible dry lubrication is added, which mostly consists of a coating with tungsten disulfide. Figure 5: CAD drawing of the FBM as it is inserted into the CPS. In contrast to figure 4 parts of the vacuum equipment, of the CPS and half of the FBM’s long bellows are invisible for a better illustration of the mechanics. To facilitate an easy slipping onto the second support flange the cylinder has a chamber at its forward end. Two cut-outs extend the movement limits in $y$-direction and provide space for the electrical feeding. The axis of the detector holder is made of steel $1.4429$ like the pinion and is mounted via dry full-ceramic ball bearings. The lever arm is also made of steel $1.4429$, but the detector board holder (back plate) of aluminum, to reduce weight. To shield the detector board from radio frequency and, even more importantly, from the electron beam, a steel $1.4429$ cover was designed featuring two small holes which allow electrons to reach the p-i-n diode chips. The full lever arm length from the axis to the tip of the FBM (including the cover, compare figure 6 ) is $130\text{\,}\mathrm{mm}$ and the maximum height of the detector equals the height of the cover which is $50\text{\,}\mathrm{mm}$. The electrical connector is shielded from the electron beam by a thin steel plate. The turbomolecular pump is located vertically above the main FBM vacuum volume and is capable of pumping speeds up to $260\text{\,}\mathrm{l}\text{\,}{\mathrm{s}}^{-1}$ (nitrogen). Two pressure gauges are mounted below the FBM vacuum volume which cover the range from $1.3\text{\times}{10}^{-10}\text{\,}\mathrm{mbar}$ to $1.3\text{\times}{10}^{-2}\text{\,}\mathrm{mbar}$. In order to reach the required vacuum level the setup is baked out periodically after being exposed to atmosphere. Figure 6: The FBM manipulator front end. The detector board is fixed on the end of a lever arm which is rotated by a rack and pinion drive. ### 4.2 Motion control The two stepper motors mentioned in subsection 4.1 ($12.1\text{\,}\mathrm{N}\text{\,}\mathrm{m}$ and $2.7\text{\,}\mathrm{N}\text{\,}\mathrm{m}$ holding torque, $1.8\text{\,}\mathrm{\SIUnitSymbolDegree}$ resolution) are not directly acting on the spindle axes but with one stage transmissions using toothed wheels. Since the FBM is not equipped with motor breaks the $x$-transmission is chosen such that the torque at the motor is sufficiently small to withstand the vacuum forces even if it is not powered anymore. Since it is possible that the stepper motors miss steps without being noticed, absolute rotary encoders are used to determine the position of the FBM because they retain the full information of the position even during a power cut. These optical encoders work with up to $16$-bit single turn and $14$-bit multi turn resolution, i.e. $2^{16}$ steps per revolution can be counted. This sums up to an overall resolution of $2^{30}$ steps. To minimise mechanical play both encoders are connected directly to their corresponding spindle axes. The main spindle has a slope of $2.5\text{\,}\mathrm{mm}$, hence a theoretical precision of ${10}^{-5}\text{\,}\mathrm{\SIUnitSymbolMicro m}$ can be reached. However, due to mechanical tolerances the actual precision is significantly lower as will be described in subsection 5.1. To fulfill stringent safety requirements the motion control of the FBM is implemented on a Field-Programmable Gate Array (FPGA) which continues to run during power cuts with the help of three backup accumulators. It directly monitors and controls the motor, encoders, and sensors and also includes a fast full safety retraction of the FBM which allows closure of the safety gate valves to separate the FBM volume from the CPS. The FPGA communicates with two KATRIN internal database systems: the ZEntrale datenerfassung Und Steuerung (ZEUS) server and the Advanced Data Extraction Infrastructure (ADEI) server [21]. All data obtained by the FPGA is automatically transferred and available on both servers. Safety-critical systems, such as vacuum pumps, valves, pressure gauges, and end switches, are integrated within the KATRIN PCS7 safety system. ### 4.3 Detector The main tasks of the FBM are to monitor the electron flux within the electron beam and to obtain the beta spectrum of tritium. Detector chips with a thin entrance window (dead layer) are used to allow the detection of electrons with energies below $10\text{\,}\mathrm{keV}$. In addition this also allows detection of low energy ($<60\text{\,}\mathrm{keV}$) photons which is important for calibrating the detector. The FBM features a UHV compatible two channel detector board, including detector chips of silicon type and additional sensors, as described below. Figure 7: Left: The FBM detector board is made of polyimide and equipped with SMD parts. The two p-i-n diodes are glued to the tip of the board and their signals are amplified by two separate transimpedance amplifiers. Close to the p-i-n diodes a PT-1000 temperature sensor and a Hall sensor are located. Center: Collection of p-i-n diodes in TO-18 casing as they are delivered by the producer [28]. Right: Picture of the Hamamatsu S9055 p-i-n diode with the lid removed. The silicon diode itself is mounted on a ceramic carrier which can be taken out of the casing. #### 4.3.1 Detector board and back plate The detector board (PCB) is made of polyimide which meets the vacuum and material requirements because of its relatively low outgassing rate compared to most other polymers and excellent thermal resistance. To enhance the thermal through-plane conductivity of the board for dissipating the heat produced by the electrical components, the PCB is a flexible, thin ($0.2\text{\,}\mathrm{mm}$) multilayer board which consists of alternating polyimide and copper layers. Here the back plate acts as a passive heat sink which is permanently cooled in the cold environment of the CPS. The board contains two detector channels, each consisting of a detector chip and a preamplifier. A Hall sensor on the detector board determines the local magnetic field. In this region of the CPS the magnetic field is approximately $0.84\text{\,}\mathrm{T}$ in the centre of the flux tube and is axially symmetric. The magnetic field is measured in only one axis and the electron flux should follow this magnetic field exactly with the exception of upstream blockages. The measurement of the magnetic field is therefore also useful for additional positioning and alignment measurements. Temperature stabilisation is important as the p-i-n diode leakage current rises exponentially with detector temperature. Therefore, the energy resolution and stability of the energy threshold are dependent on the detector temperature and effect the spectra obtained. To record the temperature a PT-1000 sensor is placed on the detector board near the p-i-n diodes and the Hall sensor. The PCB is mounted on a $5\text{\,}\mathrm{mm}$-thick aluminum back plate attached to the moving components. It is glued to the back plate with a UHV compatible two-component adhesive to ease the mounting of the electrical parts and to improve the thermal contact. The electronics are covered by a stainless steel metal shield to protect them from electrons and ions in the beamline as well as from radio frequency interference. To allow electrons to reach the p-i-n diodes the cover features two holes. The detector board has “cut out” corners in order to reduce the area of the flux tube that is covered. The electronics and detectors on the FBM detector board are connected via a custom-made PEEK connector with cabling running through the FBM manipulator to the vacuum feedthroughs. #### 4.3.2 Preamplifiers and p-i-n diodes The preamplifiers of the two p-i-n diode detector channels are DC coupled charge sensitive amplifiers which operate in a continuous reset mode. Each preamplifier consists of a low-noise JFET front end in common-source configuration and an operational amplifier (op-amp) connected in a non- inverting scheme. The feedback loop stretching across both stages consists of a $R=$ $1\text{\,}\mathrm{G\SIUnitSymbolOhm}$ resistor in parallel with a $C=$ $0.5\text{\,}\mathrm{pF}$ capacitor, forming a time constant of $\tau=C\cdot R=$ $0.5\text{\,}\mathrm{ms}$. Thanks to the DC coupled circuitry, not only individual charge-generating events can be read out with a $F_{AC}=U/Q=1/C=$ $2\text{\,}\mathrm{V}\text{\,}{\mathrm{pC}}^{-1}$ translation factor, but also a current readout can be performed by looking at the DC voltage offset at the output of the preamplifier with $F_{DC}=$ $1\text{\,}\mathrm{V}\text{\,}{\mathrm{nA}}^{-1}$. The fundamental components of the FBM are the p-i-n diode detector chips. There are two silicon p-i-n diodes mounted on the detector board which detect the $\upbeta$-electrons from the tritium source. These two p-i-n diodes can have different active sensitive areas. The silicon p-i-n diodes are manufactured by Hamamatsu Photonics and can be type S5971, S5972, S5973, or S9055-01 which have sensitive areas of different sizes (see table 1). One advantage of these detectors is that their casing and properties are all identical, the only difference is their respective sensitive area. This means the electronic design of the detector board can remain the same and the board with the p-i-n diodes that most suits the measurement purposes can be mounted and inserted into the flux tube. Furthermore the dead layer does not exceed $1\text{\,}\mathrm{\SIUnitSymbolMicro m}$. Diode \| $\boldsymbol{A_{s}}$ [${\mathrm{mm}}^{2}$] \| $\boldsymbol{C}$ [$\mathrm{pF}$] \| $\boldsymbol{I_{\text{dark}}}$ [$\mathrm{pA}$] \| $\boldsymbol{d_{\text{dead}}}$ [$\mathrm{nm}$] \| $\boldsymbol{T_{s}}$ [$\mathrm{s}$] \| $\boldsymbol{T_{m}}$ [$\mathrm{s}$] ---\|---\|---\|---\|---\|---\|--- \| data sheet \| (at $10\text{\,}\mathrm{V}$) \| (at $10\text{\,}\mathrm{V}$) \| data sheet \| measured \| \| S9055-01 \| 0.008 \| 0.5 \| 2.0 \| <1000 \| 300–500 \| 498 \| 252 S9055 \| 0.031 \| 0.8 \| 2.0 \| 129 \| 64.6 S5973 \| 0.126 \| 1.6 \| 1.9 \| 31.7 \| 15.9 S5972 \| 0.503 \| 3.0 \| 10 \| 8.0 \| 4.0 S5971 \| 1.131 \| 3.0 \| 70 \| 3.5 \| 1.8 Table 1: Parameters of the FBM p-i-n diodes. Capacitance $C$ and dark current $I_{\text{dark}}$ are taken from the data sheets. The thickness of the dead layer $d_{\text{dead}}$ was determined with an electron gun and Monte Carlo simulations. The two right columns show the time to build a sufficiently detailed spectrum in monitoring mode ($T_{m}$) and scanning mode ($T_{s}$). The casing of these diodes is metal and includes a large glass window. Since the windows of these TO-18 casings would prevent the detection of any electrons the diodes are removed from the housing and directly mounted (using two-component adhesive) onto the FBM detector board. The Hamamatsu S5971 p-i-n diode detector chip is shown in figure 7. The choice of the p-i-n diode size is based on the expected rate from the tritium source within each measurement phase (larger diodes are used for commissioning measurements where the amount of tritium is lower). The statistical error of the measurement is dominated by the number of electrons that are counted by the detector and is given by $\frac{\Delta N}{N}=\frac{1}{\sqrt{N}}=\frac{1}{\sqrt{A\phi\epsilon t}}$ (4.1) where $A$ is the sensitive area of the p-i-n diode, $\phi$ is the electron flux density, $\epsilon$ is the detector efficiency, and $t$ is the measurement time. The detector efficiency includes losses due to back reflected electrons and pile-up effects. To reach the required precision of $\Delta N/N=$ $0.1\text{\,}\mathrm{\char 37\relax}$ the measurement time is $t=\frac{1}{0.001^{2}A\phi\epsilon}$ (4.2) Assuming an energy threshold of $7\text{\,}\mathrm{keV}$ approximately $\frac{1}{3}$ of the tritium spectrum is measured. Using this reduction factor, an electron flux density of ${10}^{6}\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{mm}}^{-2}$ and a detector efficiency of $\epsilon=$ $65\text{\,}\mathrm{\char 37\relax}$ the measurement time needed to reach the required $0.1\text{\,}\mathrm{\char 37\relax}$ precision for each of these p-i-n diodes is calculated and listed in table 1. The one unknown property of these p-i-n diodes is their individual dead layer. During manufacturing the thickness of the dead layer is not measured and therefore not available a priori, but limited to $1000\text{\,}\mathrm{nm}$. The thickness of the dead layer is indicated by the minimum energy that can be detected. The measurement of the dead layer is done by analysing the shape of the peak from monoenergetic electrons originating from an electron gun (see section 5.2). Figure 10 illustrates such an analysis. Measurements of the dead layer are performed for each p-i-n diode before they are mounted on the FBM detector board. It is assumed that the dead layer remains constant over time, even after bakeout cycles of the vacuum setup. This is because the dead layer is silicon oxide which is not affected by heat and requires approximately ${10}^{13}$ electrons (on the order of several years in the FBM location) to suffer from radiation damage. ### 4.4 Data acquisition For the two p-i-n diode detector channels an Amptek [27] PX5 and an Amptek DP5 are used for the data readout. These are digital pulse processors with build- in amplifiers used to amplify the signal by up to a factor of $100$. These Amptek devices are connected to an Apple (Mac mini) computer running the Object-orientated Real-time Control and Acquisition (ORCA) software [22]. An ORCA readout module was specifically designed for the FBM Amptek devices. The raw ORCA data is converted into ROOT [29] files for analysis. The preamplifier outputs of the two p-i-n diode detector channels can also be connected to separate low-pass filters to measure the DC offset occurring from the event rate on the respective p-i-n diode chip. The pulse processing parameters of each detector channel can be optimised to obtain either the count rate or a spectrum of the $\upbeta$-electrons from the source. The peaking time is set to * • Fast channel: $1.0\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to measure the count rate (larger p-i-n diode with higher count rate) * • Slow channel: $3.2\text{\,}\mathrm{\SIUnitSymbolMicro s}$ to measure the spectrum (smaller p-i-n diode with lower count rate) During scanning the required measurement time at each point is reduced due to the increased electron flux towards the centre of the beam tube. The analysis of the FBM data is based on the established analysis systems of the KATRIN experiment. Therefore, all data, slow control, and run files are available on the ADEI server and KATRIN databases. ## 5 Measurements This section presents selected results [18] of the measurements performed with the FBM during its commissioning phases as well as during the first KATRIN measurement campaigns. These results serve as an evaluation tool for the positioning accuracy of the vacuum manipulator and the performance of the detector. In some cases the data is compared to the results of numerical simulations of the detector response. ### 5.1 Alignment and positioning precision Figure 8: Left: Data and fit result of the calibrated and temperature corrected $z$-component of the magnetic field in the CPS. Right: The residuals of the simulated magnetic field which shows a good agreement with the data. Positioning reproducibility is the ability of the FBM to find a position relative to a former position. This is different to the absolute positioning accuracy which includes external reference points with respect to the KATRIN coordinate system. The reproducibility is validated by using a laser setup as well as a portable Coordinate Measuring Machine (CMM). It was determined to be better than $0.1\text{\,}\mathrm{mm}$. However, the overall alignment uncertainties (also CMM) dominate the absolute positioning accuracy as shown in table 2. \| $\boldsymbol{\sigma_{\text{alignment}}}$ [$\mathrm{mm}$] \| $\boldsymbol{\tilde{\sigma}_{\text{max}}}$ [$\mathrm{mm}$] \| $\boldsymbol{\sigma_{\text{full}}}$ [$\mathrm{mm}$] \| Offset [$\mathrm{mm}$] ---\|---\|---\|---\|--- $x$ \| $0.28$ \| $0.042$ \| $0.28$ \| $-1.2\pm 0.2$ $y$ \| $0.1$ \| $0.07$ \| $0.13$ \| $4.2\pm 0.2$ Table 2: The overall positioning accuracy $\sigma_{\text{full}}$ results from the combination of the uncertainties of the alignment $\sigma_{\text{alignment}}$ and the positioning reproducibility $\tilde{\sigma}_{\text{max}}$. There is a misalignment between the FBM and the flux tube expressed by a constant offset which was determined from flux tube scans. To calibrate the movement system, as well as to find the center of the flux tube, the magnetic field in the CPS can be used (see left panel in figure 8). The shape of the magnetic flux can be described by a two-dimensional Gaussian. The required calibration values, namely the encoder value for the horizontal lever arm and the offset of the magnetic flux center to the FBM system (listed in the last row in table 2), are given by the free parameters in a fit of data taken during a flux tube scan. To demonstrate the excellent positioning accuracy of the manipulator a thin ($0.14\text{\,}\mathrm{mm}$ diameter) electron beam was scanned with the FBM by moving the detector (type S5971 with $1.2\text{\,}\mathrm{mm}$ diameter) through the fixed beam in a grid pattern with $0.1\text{\,}\mathrm{mm}$ spacing [18]. Since the beam is far smaller than the p-i-n diode, it is rather the diode being scanned by the beam than vice versa. The plot in figure 9 shows the measured intensities as a function of detector position. The background is removed by a noise cut at $100\text{\,}\mathrm{c}\mathrm{p}\mathrm{s}$ and as a result only positions where electrons were detected are shown. The large circular contours represent the entrance window of the diode (small, $1.2\text{\,}\mathrm{mm}$ diameter) as it is stated in the data sheets and the visual surface (large, $1.3\text{\,}\mathrm{mm}$) of the diode as it was measured. The position of the contours is adjusted such that the number of events within the contours is maximised. The center represents the actual position of the beam at $x_{\text{FBM}}=$ $-1.2\text{\,}\mathrm{mm}$ and $y_{\text{FBM}}=$ $7.6\text{\,}\mathrm{mm}$. One spot close to the center has low statistics, which is associated with a small dirt particle which was found to be located at the equivalent position of the p-i-n diode chips’s active surface. ### 5.2 Detector response and dead layer For calibration KATRIN is equipped with an electron gun which is situated in the rear section and can provide a mono-energetic electron beam with energies up to $20\text{\,}\mathrm{keV}$. In the left panel of figure 10 the measured detector response to $18.6\text{\,}\mathrm{keV}$ electrons is shown. Due to partial charge collection the peak is shifted to lower energies, widens and develops a long low energy tail descending into an almost flat plateau. To understand the related effects and to reach the required precision for the FBM, numerical simulations [18] were performed using Geant4 [23]. The model of the detector comprises a detection volume and a fully insensitive dead layer (also known as slab-model). Within the detector the electrons undergo elastic and inelastic scattering losing their energy subsequently and are finally either completely stopped in the detector or reflected from it. For the interaction model the Penelope physics package was used. The low energy region A (see figure 10) is predominantly produced by reflected electrons and region B mainly by the dead layer losses which also cause the shift of the main peak in region C by approximately $0.6\text{\,}\mathrm{keV}$ (mean energy loss in the dead layer). The measured electron spectra were compared to a library of simulated electron spectra for different dead layers, with a $\chi^{2}$ test used to find the most accurate match. The best result was obtained with a dead layer thickness of $340\text{\,}\mathrm{nm}$. The simulations always overestimated the data in the low energy tail at approximately $5\text{\,}\mathrm{keV}$ (region A). This is caused by an incomplete model which does not include the magnetic field configuration in the CPS for the purpose of simplicity. Due to magnetic mirroring in the CPS electrons which were initially reflected from the detector are guided back within the peaking time of the DAQ contributing to regions B and C instead of A. It was possible to determine the dead layers of the FBM p-i-n diodes which range from $300\text{\,}\mathrm{nm}500\text{\,}\mathrm{nm}$ causing an energy dependent shift of the measured peak of $0.5\text{\,}\mathrm{keV}2\text{\,}\mathrm{keV}$ for electron energies up to $20\text{\,}\mathrm{keV}$. With these simulations the detection efficiency for electrons as a function of kinetic energy could also be determined as shown in the right plot in figure 10. Figure 9: Scan of stationary electron gun beam. The FBM detector board is moved through the beam in a grid with $0.1\text{\,}\mathrm{mm}$ step length. Each colored dot represents the measured rate at this detector position. The background is removed by a noise cut at $100\text{\,}\mathrm{c}\mathrm{p}\mathrm{s}$. Note that the size of the data points is arbitrary and does not represent the size of the beam spot. The inner circle represents the active surface (here $1.13\text{\,}{\mathrm{mm}}^{2}$) as stated by the manufacturer and the outer circle the visual surface of the p-i-n diode. The detector chip profiles are positioned such that they comprise the highest rate. Figure 10: Left: Measured and simulated electron gun peaks obtained during the first tritium campaign. The simulation includes a detector energy resolution with FWHM $=$ $2.35\text{\,}\mathrm{keV}$. The best match was obtained with a dead layer thickness of $340\text{\,}\mathrm{nm}$. Right: The simulated efficiencies for electrons not to get reflected from the detector ($\epsilon_{r}$), not to get stopped in the dead layer ($\epsilon_{d}$), and for exceeding the energy threshold of $5\text{\,}\mathrm{keV}$ ($\epsilon_{\text{th}}$ ($5\text{\,}\mathrm{keV}$)). The intrinsic efficiency of the detector is then given by $\varepsilon_{i}$($5\text{\,}\mathrm{keV}$) $=\varepsilon_{r}\cdot\varepsilon_{d}\cdot\varepsilon_{\text{th}}$($5\text{\,}\mathrm{keV}$). ### 5.3 First tritium measurement campaign Figure 11: Data and fit of the ${}^{241}\text{Am}$ spectrum. The spectrum is recorded with the FBM with an energy resolution of $\sigma_{\text{FWHM}}\approx$ $2\text{\,}\mathrm{keV}$. The positions of all ${}^{241}\text{Am}$ lines are identified and labeled. The full fit function consists of a combination of Gaussians and error functions for each (strong) ${}^{241}\text{Am}$ line in the peak. The global fit comprises $33$ free fit parameters. Only the Gaussian parts representing the strongest lines are used for the calibration of the FBM. Before the actual tritium measurement an alternative front end, equipped with a Faraday cup, was installed to the FBM in order to check ion blocking, measure the radial ion distribution in the beamline, and check the simulated source gas models by measuring secondary electrons [24]. The measurements with the p-i-n diode detector started with the “first tritium measurement campaign“ [25] which took place from the 5 to the 20 of May 2018 with a gas mixture of $0.5\text{\,}\mathrm{\char 37\relax}$ tritium in deuterium. In the following sections the results of this first data-taking period with tritium are presented. #### 5.3.1 Configuration With a fraction of only $0.5\text{\,}\mathrm{\char 37\relax}$ of tritium in the source gas an electron flux of approximately $5000\text{\,}{\mathrm{s}}^{-1}\text{\,}{\mathrm{mm}}^{-2}$ was expected at the FBM measuring plane. Therefore, the largest p-i-n diodes have been chosen ($1.1\text{\,}{\mathrm{mm}}^{2}$) to optimise counting statistics. The peaking time of the DAQ for both channels was $6.4\text{\,}\mathrm{\SIUnitSymbolMicro s}$, resulting in a pile-up rate of about $3\text{\,}\mathrm{\char 37\relax}$ which can be neglected for stability analyses (see section 5.3.4). Acceptance tests were performed prior to the campaign to extract calibration parameters, energy resolutions, and noise thresholds of the detectors. These measurements were performed with an 241Am source in the vented system with the FBM in parking position. The source was placed at a close distance between the two p-i-n diodes. The desired diode could then be irradiated using the movement mechanics and be adjusted to find the maximum count rate. In figure 11 one of the 241Am spectra extracted from these measurements is shown. The calibration parameters are obtained by a global fit to the whole spectrum. #### 5.3.2 Spectrum The spectrum shown in figure 12 is the first tritium spectrum recorded with the FBM. Between $6\text{\,}\mathrm{keV}20\text{\,}\mathrm{keV}$ the spectrum agrees with the expectation, however below $6\text{\,}\mathrm{keV}$ the slope is unexpectedly increasing. This is probably due to background counts from noise and edge effects from the diodes. This may also explain why the spectra of the two channels do not match perfectly for lower energies. Other likely sources for this mismatch, which is also the reason for about $2\text{\,}\mathrm{\char 37\relax}$ lower rate in channel $1$ than in channel $2$ during the whole campaign, are * • uncertainties in the energy calibrations which cause the deviations among the channels for lower energies, * • small differences in the active area, or * • small differences in the dead layer thickness of the two p-i-n diodes. Figure 12: Tritium $\upbeta$-spectrum measured with both detector channels of the FBM during the first tritium campaign. The rate deviation of approximately $2\text{\,}\mathrm{\char 37\relax}$ between the two channels is probably caused by the uncertainties in the calibrations or differences in the active surface or dead layer thickness of the p-i-n diodes. #### 5.3.3 Flux tube scans Several scans of the $\upbeta$-electron flux cross section were performed recording the tritium count rate, the magnetic field, and the temperature. During a scan, the temperature usually drops by about $1\text{\,}\mathrm{\SIUnitSymbolCelsius}$. This occurs when the detector is moved further into the cold CPS where the detector directly faces the $4\text{\,}\mathrm{K}$ cold beam tube of the CPS in which the argon frost layer is prepared. To check the calibration of the magnetic field sensor as well as to verify our understanding of the magnetic field configuration in the FBM measuring plane the magnetic field data is compared with simulations. The residual analysis displayed in figure 8 (right) shows a good agreement between simulation and data. Figure 13: Radial dependence of the count rate derived from a cross scan during the first tritium campaign with channel $1$ of the FBM. Top: 1D Gaussians are fit to the data for each horizontal ($\text{X}_{1,2}$) and vertical ($\text{Y}_{1,2}$) scan. The Gaussian means are compatible with the results from magnetic field measurements. One can clearly see that for identical positions slightly different rates are measured, for example the rate increased during the $x$-scans such that the mean of the $\text{X}_{2}$ fit is lower than for $\text{X}_{1}$. The Gaussian widths are approximately $\sigma=$ $165\text{\,}\mathrm{mm}$. Bottom: 2D scatter plot of the same data. The scans for $y$ are not perfectly on a vertical line due to the chosen scan pattern which explains the larger uncertainties in the fits. In figure 13 (top and bottom) the results of a cross scans over the cross section of the flux tube for both detector channels are shown. The electron flux shows the expected Gaussian shape where the rate drops from the center to the outer rim by approximately $10\text{\,}\mathrm{\char 37\relax}$ as predicted by simulations [26]. It can be seen that the event rate for identical positions changes during the scans which affects the extracted mean of the fits. Nevertheless, the means are compatible to the results from the alignment measurements in section 5.1 which use the magnetic field data. This is expected as the electron flux scales with the magnetic flux. #### 5.3.4 Rate stability Figure 14: $\upbeta$-electron rate trend summary of the first tritium campaign. The full available data from the stability measurements at the monitoring position for both channels is plotted. The count rates for channel 1 are approximately $0.7\text{\,}\mathrm{\char 37\relax}$ smaller compared to channel 2 using the same energy threshold (here $5.3\text{\,}\mathrm{keV}$). Apart from that, the channels follow the same trend. The full linear fit reveals a mean relative increase of about $0.02\text{\,}\mathrm{\char 37\relax}$ per hour while for the single regions this value is smaller than $0.01\text{\,}\mathrm{\char 37\relax}$ per hour. During the two weeks of the first tritium campaign the FBM was mainly monitoring the flux in the CPS at position $x_{\text{FBM}}=$ $65\text{\,}\mathrm{mm}$ (outer rim of the flux tube, see figure 3 and 13). From time to time background measurements were taken slightly out of the beam at $x_{\text{FBM}}=$ $80\text{\,}\mathrm{mm}$. The full rate trend graphs are shown in figure 14 for both detector channels including linear fits to the data. The entire monitoring time is separated into six time regions. There is a long term drift of approximately $0.02\text{\,}\mathrm{\char 37\relax}\mathrm{/}\mathrm{h}$ determined from all regions, while for single regions the drift is generally smaller, especially for the longer regions $2$, $3$, $5$, and $6$, hence the reason for the larger long term drift must mainly originate from incidences which occur between the regions. Several investigations have been performed to find the source of this long-term drift, and there are hints that the detector response changes over time due to an increase in the noise level and degrading effects of the detector chip. Hence this drift is probably caused by the FBM and not by a change of the incoming electron flux. The latter assumption is supported by the results of the other monitoring systems which do not observe such a drift. However, this long-term drift is sufficiently small as the FBM is designed to monitor relative source fluctuations over short time intervals, such as seconds, minutes and at maximum a few hours. Within these time ranges the drift is within the required sensitivity of $0.1\text{\,}\mathrm{\char 37\relax}$. Therefore, despite the observed long term drift, the FBM shows a stability fulfilling its design goal. In the first tritium campaign of KATRIN the FBM was utilised to reduce systematic uncertainties in the tritium concentration $\epsilon_{\text{T}}$ measurement performed by the LARA system. In this campaign the tritium amount was limited to about $1\text{\,}\mathrm{\char 37\relax}$ in deuterium, and consequentially statistical fluctuations in the determination of the concentration of the tritiated hydrogen isotopolouges were much stronger than it is the case for standard operation ($\epsilon_{\text{T}}>$ $0.95$). An average tritium concentration was determined over a long duration ($\approx 3\text{\,}\mathrm{h}$) with LARA, and this average tritium concentration was fluctuated according to the higher statistics FBM data for short durations. Note that these are time-scales over which the drift of the FBM is negligible. This way the uncertainty of the short-term fluctuations measurement of the tritium concentration were reduced from about $2\text{\,}\mathrm{\char 37\relax}$ down to about $0.5\text{\,}\mathrm{\char 37\relax}$. The cooperation of these two monitoring systems was crucial to reduce the tritium concentration systematic input for an upcoming $\mathrm{keV}$-scale sterile neutrino analysis of the first tritium data. ## 6 Summary The KATRIN experiment aims for a precise measurement of the electron antineutrino mass with a sensitivity of $0.2\text{\,}\mathrm{eV}$ ($90\text{\,}\mathrm{\char 37\relax}$ CL). One of the systematic uncertainties in this measurement arises from fluctuations of the column density of high luminosity tritium source. In order to reach the design goal of KATRIN, the latter must be measured on the per-mille level over time scales of a few minutes. Therefore the source is continuously monitored by several monitoring systems, one of which is the Forward Beam Monitor (FBM). The FBM has the advantage of being capable of continuously monitoring variations of the electron flux and changes in the observed shape of the $\upbeta$-decay spectrum with high accuracy on short time scales. A UHV compatible vacuum manipulator was commissioned. It is able to place a detector board directly into the beta-electron flux originating from the tritium source. Although the mounting position of the apparatus demands a movement mechanism with a working stroke of $1.8\text{\,}\mathrm{m}$ the FBM is able to reach any position within the electron flux cross-section with a precision of better than $0.3\text{\,}\mathrm{mm}$ which can be determined with magnetic field measurements. The detector board at the tip of the FBM manipulator measures the electron flux with two silicon p-i-n diodes. The FBM reaches an energy resolution of about $\sigma_{\text{FWHM}}=$ $2\text{\,}\mathrm{keV}$ at an energy threshold of $5\text{\,}\mathrm{keV}$. The readout electronics are optimised to register electron events at a rate of $\mathcal{O}$(${10}^{4}\text{\,}\mathrm{c}\mathrm{p}\mathrm{s}$) and thus to measure relative changes in the electron flux with $0.1\text{\,}\mathrm{\char 37\relax}$ precision in about $100\text{\,}\mathrm{s}$. The entrance window (dead layer) of the p-i-n diodes has a large impact on the detector response when measuring electrons. It was found that the dead layer thickness of the p-i-n diodes used for the FBM range from $300\text{\,}\mathrm{nm}500\text{\,}\mathrm{nm}$. After commissioning, the FBM was employed for several KATRIN measurement campaigns. The capabilities of the FBM detector were confirmed as well as the positioning accuracy of the manipulator. A small long term (days to weeks) drift of the rate was observed which correlates to a drift of the noise level of the electronics. On short time scales (hours) the FBM is stable to the per- mille level. With this the FBM is a monitoring device which reaches all its design goals. With its good performance the FBM data already played a key role in reducing the systematic uncertainties of the tritium concentration $\epsilon_{\text{T}}$ fluctuations during the first tritium campaign. 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# Some cluster tilting modules for weighted surface algebras Karin Erdmann Mathematical Institute, Oxford University, ROQ, Oxford OX2 6GG, United Kingdom<EMAIL_ADDRESS> ###### Abstract. Non-singular weighted surface algebras satisfy the necessary condition found in [6] for existence of cluster tilting modules. We show that any such algebra whose Gabriel quiver is bipartite, has a module satisfying the necessary ext vanishing condition. We show that it is 3-cluster tilting precisely for non- singular triangular or spherical algebras, but not for any other weighted surface algebra with bipartite Gabriel quiver. Keywords: Symmetric algebra, Surface algebra, cluster tilting modules. 2020 MSC: 16D50, 16E30, 16G20, 16G60 ###### 2020 Mathematics Subject Classification: 16D50, 16E30, 16G20, 16G60 ## 1\. Introduction A module $M$ of a finite-dimensional algebra $A$ is an $n$-cluster tilting module (or maximal $(n-1)$-orthogonal) provided $\displaystyle{\rm add}(M)=$ $\displaystyle\\{N\mid{\rm Ext}^{i}(M,N)=0\mbox{ for }1\leq i\leq n-1\\}$ $\displaystyle=$ $\displaystyle\\{N\mid{\rm Ext}^{i}(N,M)=0\mbox{ for }1\leq i\leq n-1\\},$ (see [12], [13]). We would like to know whether non-singular weighted surface algebras have cluster tilting modules. Weighted surface algebras are a class of tame symmetric algebras, periodic as bimodules, of period $4$ (see [7] and [8], [9]). This means that they satisfy the necessary condition found in [6], requiring that all non-projective modules should have bounded periodic resolutions. As observed in [6] if such an algebra has an $n$-cluster tilting module then the only option is $n=3$. Here we study weighted surface algebras which have a bipartite Gabriel quiver, which means that in the presentation as in [8] (see also [9]) it has many virtual arrows. We introduce a module $M$, defined in the same way for each of the algebras, which satisfies ${\rm Ext}^{1}(M,M)=0$ and ${\rm Ext}^{2}(M,M)=0$. We show that it is 3-cluster tilting when $\Lambda$ is either a triangle algebra $T(\lambda)$, or a spherical algebra $S(\lambda)$ (see §3 for the definition) with $\lambda\in K$ and $\lambda\neq 0,1$. We also show that for any other weighted surface algebra whose Gabriel quiver is bipartite, $M$ cannot be a direct summand of a 3-cluster tilting module. The algebra $T(\lambda)$ occurs in various places in the literature. It is an algebra with $k=1$ in the family $Q(3\mathcal{A})_{1}$ of algebras of quaternion type, in [5]. Furthermore, it occurs with the name $B_{1,1}(\lambda)$ in [3]. As well, it occurs in [1] with the name $A_{1}(\lambda)$. In [8] it is called the triangle algebra $T(\lambda)$ (in Example 3.4). Similarly the spherical algebra $S(\lambda)$ was introduced in [8] (Example 3.6). Spherical algebras are a special case of the family of algebras which come from the triangulation $T(n)$ of the sphere as defined in Example 7.5 of [10]. We call these algebras $n$-spherical; when $n=2$ they are the same as the spherical algebras. One may also observe that the spherical algebra with $n=1$ with the multiplicities $2,2,1$ is the same as the triangle algebra (see [8], Example 3.3). Algebras whose Gabriel quiver is the same as that of $T(\lambda)$, allowing a multiplicity $k>1$, coincide (up to a scalar parameter) with the algebras $Q(3\mathcal{A})_{2}^{k}$ in the labelling of [11]. When the characteristic of $K$ is $2$ these occur as the basic algebras of blocks of finite groups. Very recently B. Böhmler and R. Marcinczik proved using computer calculations that for $k=2$, it has a 3-cluster tilting module (see [2]). Much of this note was written five years ago, when talking to Idun Reiten about [3], and it was extended first when spherical algebras had been discovered, and then again, inspired by an email from R. Marcinczik (for which I am grateful). ## 2\. Preliminaries Throughout $K$ is an algebraically closed field, of arbitrary characteristic. Assume $\Lambda$ is a finite-dimensional symmetric $K$-algebra. We recall some identities for the stable category $\underline{\rm mod}\Lambda$. (1) $D{\rm Ext}^{1}(M,N)\cong\underline{\rm Hom}(\tau^{-1}N,M)$, and in this case $\tau\cong\Omega^{2}$. (2) ${\rm Ext}^{i}(U,V)\cong\underline{\rm Hom}(\Omega^{i}U,V)\cong\underline{\rm Hom}(U,\Omega^{-i}V)$. This implies that $\dim{\rm Ext}^{i}(M,N)=\dim{\rm Ext}^{1}(N,\Omega^{i+2}M)$. The algebras we consider have the property that all non-projective indecomposable right $\Lambda$-modules are $\Omega$ periodic of periods dividing $4$. This gives us the following, we refer to this as ext symmetry. ###### Corollary 2.1. Assume $\Lambda$ is symmetric and all modules have $\Omega$-period dividing $4$. Then for all $M,N$ we have $\dim{\rm Ext}^{2}(M,N)=\dim{\rm Ext}^{1}(N,M)$ as vector spaces. This simplifies the search for 3-cluster tilting modules. If we know that ${\rm Ext}^{1}(N,X)=0$ and ${\rm Ext}^{1}(X,N)=0$ then automatically ${\rm Ext}^{2}(X,N)=0$ and ${\rm Ext}^{2}(N,X)=0$. ## 3\. The algebras ### 3.1. Weighted surface algebras We review the definition from [9], for details see [7], [8], [9]. Assume $Q$ is a finite quiver. Denote by $KQ$ the path algebra of $Q$ over $K$. We will consider algebras of the form $A=KQ/I$ where $I$ is an ideal of $KQ$ which contains all paths of length $\geq m$ for some $m>>0$, so that the algebra is finite-dimensional and basic. The Gabriel quiver $Q_{A}$ of $A$ is then the full subquiver of $Q$ obtained from $Q$ by removing all arrows $\alpha$ with $\alpha+I\in R_{Q}^{2}+I$. A quiver $Q$ is _$2$ -regular_ if for each vertex $i\in Q_{0}$ there are precisely two arrows starting at $i$ and two arrows ending at $i$. Such a quiver has an involution on the arrows, $\alpha\mapsto\bar{\alpha}$, such that for each arrow $\alpha$, the arrow $\bar{\alpha}$ is the arrow $\neq\alpha$ such that $s(\alpha)=s(\bar{\alpha})$. A _triangulation quiver_ is a pair $(Q,f)$ where $Q$ is a (finite) connected 2-regular quiver, with at least two vertices, and where $f$ is a fixed permutation of the arrows such that $t(\alpha)=s(f(\alpha))$ for each arrow $\alpha$, and such that $f^{3}$ is the identity. The permutation $f$ uniquely determines a permutation $g$ of the arrows, defined by $g(\alpha):=\overline{f(\alpha)}$ for any arrow $\alpha$. We assume throughout that $(Q,f)$ is a triangulation quiver. To give the presentations of the algebras in question, we use the following notation. For each arrow $\alpha$, we fix $\displaystyle m_{\alpha}\in\mathbb{N}^{}$ a weight, constant on $g$-cycles, and $\displaystyle c_{\alpha}\in K^{}$ a parameter, constant on $g$-cycles, and define $\displaystyle n_{\alpha}:=$ the length of the $g$-cycle of $\alpha$, $\displaystyle B_{\alpha}:=\alpha g(\alpha)\ldots g^{m_{\alpha}n_{\alpha}-1}(\alpha)$ $\displaystyle\mbox{ the path along the $g$-cycle of $\alpha$ of length $m_{\alpha}n_{\alpha}$},$ $\displaystyle A_{\alpha}:=\alpha g(\alpha)\ldots g^{m_{\alpha}n_{\alpha}-2}(\alpha)$ the path along the $g$-cycle of $\alpha$ of length $m_{\alpha}n_{\alpha}-1$. ###### Definition 3.1. We say that an arrow $\alpha$ of $Q$ is virtual if $m_{\alpha}n_{\alpha}=2$, that is $A_{\alpha}$ has length $1$. Note that this condition is preserved under the permutation $g$, and that virtual arrows form $g$-orbits of sizes 1 or 2. We assume that the following conditions hold. (1) $m_{\alpha}n_{\alpha}\geq 2$ for all arrows $\alpha$, and (2) $m_{\alpha}n_{\alpha}\geq 3$ for all arrows $\alpha$ such that $\bar{\alpha}$ is virtual and $\bar{\alpha}$ is not a loop, and $m_{\alpha}n_{\alpha}\geq 4$ for all arrows $\alpha$ such that $\bar{\alpha}$ is virtual and $\bar{\alpha}$ is a loop. Condition (1) is a general assumption, and (2) is needed to eliminate two small algebras (see [8]). We also assume that $Q$ has at least three vertices. With this, the definition of a weighted surface algebra (as revised in [9]) is as follows. ###### Definition 3.2. The algebra $\Lambda=\Lambda(Q,f,m_{\bullet},c_{\bullet})=KQ/I$ is a weighted surface algebra if $(Q,f)$ is a triangulation quiver, with $\|Q_{0}\|\geq 2$, and $I=I(Q,f,m_{\bullet},c_{\bullet})$ is the ideal of $KQ$ generated by: 1. (1) $\alpha f(\alpha)-c_{\bar{\alpha}}A_{\bar{\alpha}}$ for all arrows $\alpha$ of $Q$, 2. (2) $\alpha f(\alpha)g(f(\alpha))$ for all arrows $\alpha$ of $Q$ unless $f^{2}(\alpha)$ is virtual, or unless $f(\bar{\alpha})$ is virtual and $m_{\bar{\alpha}}=1,\ n_{\bar{\alpha}}=3$. 3. (3) $\alpha g(\alpha)f(g(\alpha))$ for all arrows $\alpha$ of $Q$ unless $f(\alpha)$ is virtual, or unless $f^{2}(\alpha)$ is virtual and $m_{f(\alpha)}=1,\ n_{f(\alpha)}=3$. The Gabriel quiver $Q_{\Lambda}$ is the subquiver of $Q$ obtained by removing all virtual arrows. We recall a few properties. (1) Any such algebra is symmetric and tame. (2) The dimension of $e_{i}\Lambda$ is equal to $m_{\alpha}n_{\alpha}+m_{\bar{\alpha}}n_{\bar{\alpha}}$ where $\alpha,\bar{\alpha}$ are the arrows starting at $i$. The relations also imply that $c_{\alpha}B_{\alpha}=c_{\bar{\alpha}}B_{\bar{\alpha}}$ in $\Lambda$. One can show that this spans the socle of $e_{i}\Lambda$. We wish to define a module $M$ such that ${\rm Ext}^{1}(M,M)=0$ and ${\rm Ext}^{2}(M,M)=0$, as a candidate to be 3-cluster tilting. This can be done for a weighted surface algebra whose quiver is bipartite; this requires that each triangle of $f$ must contain a virtual arrow. Such a quiver can be thought of made up of three building blocks, first a quiver of the form $a_{1}$$b_{1}$$d_{1}$$a_{2}$$b_{2}$$d_{2}$$a_{3}$$\cdots$$a_{n}$$b_{n}$$d_{n}$$a_{n+1}\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!$$\xi_{1}$$\delta_{1}$$\gamma_{1}$$\eta_{1}$$\sigma_{1}$$\varrho_{1}$$\xi_{2}$$\delta_{2}$$\gamma_{2}$$\eta_{2}$$\sigma_{2}$$\varrho_{2}$$\xi_{n}$$\delta_{n}$$\gamma_{n}$$\eta_{n}$$\sigma_{n}$$\varrho_{n}$ , where the shaded triangles define the $f$-orbits. Next, quivers of the form $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\alpha}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$ or $\textstyle{2^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\gamma}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\varepsilon^{\prime}}$ We describe the quivers and algebras we consider. We always take the multiplicities at $2$-cycles of $g$ equal to $1$, and at loops we take multiplicity $2$. That is, all arrows in 2-cycles and loops are virtual and not part of the Gabriel quiver. ### 3.2. Algebras with Gabriel quiver $3\mathcal{A}$ We take the quiver $Q$ obtained by glueing the second and the third type above, identifying vertex $2$ with vertex $2^{\prime}$. $\textstyle{1\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varepsilon}$$\scriptstyle{\alpha}$$\textstyle{2\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\scriptstyle{\gamma}$$\textstyle{3\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\delta}$$\scriptstyle{\varepsilon^{\prime}}$ The permutation $g$ is of the form $(\alpha\ \gamma\ \delta\ \beta)(\varepsilon)(\varepsilon^{\prime})$. Let $m_{\alpha}=k\geq 2$. The case $k=1$ is special, this gives the triangular algebra, called $T(\lambda)$ in [8], here $\lambda\neq 1$. With suitable choice of $c_{\bullet}$, the presentation of the weighted surface algebra induces the (Gabriel) presentation of $T(\lambda)$ $\displaystyle\alpha\beta\alpha$ $\displaystyle=\alpha\gamma\delta,$ $\displaystyle\delta\beta\alpha$ $\displaystyle=\lambda(\delta\gamma\delta),$ $\displaystyle\beta\alpha\beta$ $\displaystyle=\gamma\delta\beta,$ $\displaystyle\beta\alpha\gamma$ $\displaystyle=\lambda(\gamma\delta\gamma),$ $\displaystyle\alpha\beta\alpha\gamma$ $\displaystyle=0,$ $\displaystyle\beta\alpha\beta\alpha\beta$ $\displaystyle=0,$ $\displaystyle\delta\gamma\delta\beta$ $\displaystyle=0,$ $\displaystyle\gamma\delta\gamma\delta\gamma$ $\displaystyle=0,$ $\displaystyle\alpha\beta\alpha\beta\alpha$ $\displaystyle=0,$ $\displaystyle\delta\gamma\delta\gamma\delta$ $\displaystyle=0,$ $\displaystyle\delta\beta\alpha\beta$ $\displaystyle=0.$ One can show that $T(\lambda)$ and $T(\mu)$ are not isomorphic for $\lambda\neq\mu$. The weighted surface algebras with the same quiver and $\varepsilon,\varepsilon^{\prime}$ virtual loops have Gabriel quiver denoted by $Q(3\mathcal{A})_{2}^{k}$ in [11] (which is the algebra with parameter $k$ in the the family $Q(3\mathcal{A})_{2}$ of [5]). ### 3.3. Spherical algebras We have the algebras whose quiver is given by the first building block where we identify $a_{1}=a_{n+1}$, for $n\geq 2$. The case $n=2$ gives the algebra $S(\lambda)$, called spherical algebra, introduced in [8], Example 3.6, as follows. 123456$\alpha$$\xi$$\delta$$\eta$$\beta$$\nu$$\varrho$$\varepsilon$$\sigma$$\mu$$\omega$$\gamma$ where the four shaded triangles denote the $f$-orbits. We take all multiplicities equal to $1$, the presentation induced by the weighted surface algebra presentation is, with suitable choice of $c_{\bullet}$, $\displaystyle\alpha\beta\nu$ $\displaystyle=\varrho\omega\nu,$ $\displaystyle\beta\nu\delta$ $\displaystyle=\lambda\beta\gamma\sigma,$ $\displaystyle\nu\delta\alpha$ $\displaystyle=\lambda\gamma\sigma\alpha,$ $\displaystyle\delta\alpha\beta$ $\displaystyle=\delta\varrho\omega,$ $\displaystyle\gamma\sigma\varrho$ $\displaystyle=\nu\delta\varrho,$ $\displaystyle\sigma\varrho\omega$ $\displaystyle=\lambda\sigma\alpha\beta,$ $\displaystyle\varrho\omega\gamma$ $\displaystyle=\lambda\alpha\beta\gamma,$ $\displaystyle\omega\gamma\sigma$ $\displaystyle=\omega\nu\delta,$ $\displaystyle\alpha\beta\nu\delta\alpha$ $\displaystyle=0,$ $\displaystyle\beta\nu\delta\varrho$ $\displaystyle=0,$ $\displaystyle\nu\delta\alpha\beta\nu$ $\displaystyle=0,$ $\displaystyle\delta\alpha\beta\gamma$ $\displaystyle=0,$ $\displaystyle\gamma\sigma\varrho\omega\gamma$ $\displaystyle=0,$ $\displaystyle\sigma\varrho\omega\nu$ $\displaystyle=0,$ $\displaystyle\varrho\omega\gamma\sigma\varrho$ $\displaystyle=0,$ $\displaystyle\omega\gamma\sigma\alpha$ $\displaystyle=0,$ $\displaystyle\beta\gamma\sigma\varrho$ $\displaystyle=0,$ $\displaystyle\sigma\alpha\beta\nu$ $\displaystyle=0,$ $\displaystyle\delta\varrho\omega\gamma$ $\displaystyle=0,$ $\displaystyle\omega\nu\delta\alpha$ $\displaystyle=0,$ $\displaystyle\beta\nu\delta\alpha\beta$ $\displaystyle=0,$ $\displaystyle\delta\alpha\beta\nu\delta$ $\displaystyle=0,$ $\displaystyle\sigma\varrho\omega\gamma\sigma$ $\displaystyle=0,$ $\displaystyle\omega\gamma\sigma\varrho\omega$ $\displaystyle=0.$ ### 3.4. The $n$-spherical algebra When $n\geq 2$, the permutation $g$ is of the form $\prod_{i=1}^{n}(\xi_{i}\,\eta_{i})\cdot(\gamma_{1}\,\sigma_{1}\,\gamma_{2}\,\sigma_{2}\,\dots\,\gamma_{n}\,\sigma_{n})\cdot(\varrho_{n}\,\delta_{n}\,\varrho_{n-1}\,\delta_{n-1}\,\dots\,\varrho_{1}\,\delta_{1}).$ We take the multiplicities for the $2n$-cycles to be $m,m^{\prime}\geq 1$, and write $c,c^{\prime}$ for the parameters at these cycles. ### 3.5. A mixed algebra We can glue together the three building blocks by identifying $2=a_{1}$, and $2^{\prime}=a_{n+1}$. In this case, the permutation $g$ is the product of one large cycle with $n$ cycles of length $2$, and two loops: $\prod_{i=1}^{n}(\xi_{i}\ \eta_{i})\cdot(\gamma_{1}\,\sigma_{1}\,\gamma_{2}\,\sigma_{2}\,\dots\,\gamma_{n}\,\sigma_{n}\ \gamma\ \delta\ \varrho_{n}\,\delta_{n}\,\varrho_{n-1}\,\delta_{n-1}\,\dots\,\varrho_{1}\,\delta_{1}\ \beta\ \alpha)(\varepsilon)(\varepsilon^{\prime}).$ We take again the multiplicities equal to $1$ on 2-cycles of $g$, or $m_{\gamma_{1}}=m$, and the parameter function with value $1$ on each virtual arrow. These algebras were not studied in previous papers but they fit into the same scheme. ## 4\. Construction of the module $M$ Let $\Lambda$ be one of the algebras as described above. Let $\Gamma\subset Q_{0}$ be the set of vertices which are not adjacent to a virtual arrow. ###### Definition 4.1. Let $M$ be the (right) $\Lambda$-module $M:=\Lambda\oplus[\bigoplus_{i\in\Gamma}S_{i}]\oplus[\bigoplus_{\nu\not\in\Gamma}\Omega^{2}(S_{\nu})]$ In the following we write down the details for the case of the $n$-spherical algebra, for the other algebras they are essentially the same. In this case $\Gamma=\\{a_{i}\mid 1\leq i\leq n\\}$. ### 4.1. The $\Omega$-translates of the simple modules For the algebra in question, the dimensions of the indecomposable projectives are: $\dim P_{a_{i}}=2n(m+m^{\prime}),\ \ \dim P_{b_{i}}=2nm+2,\ \ \dim P_{d_{i}}=2nm^{\prime}+2.$ Let $a_{i}\in\Gamma$. The structure of $\Omega^{\pm 1}(S_{a_{i}})$ can be seen from the presentation of the algebra. The module $\Omega^{2}(S_{a_{i}})$ has dimension 5, the Loewy structure is $\begin{matrix}b_{i-1}&&d_{i}\cr&a_{i}&\cr b_{i}&&d_{i-1}\end{matrix}$ That is, the module has a ’simple waist’. Now let $\nu\in Q\setminus\Gamma$ we set $U_{\nu}:=\Omega^{2}(S_{\nu})$. Then $\Omega(U_{\nu})=\Omega^{-1}(S_{\nu})$ and $\Omega^{-1}(U_{\nu})=\Omega(S_{\nu})$, their structure can also be seen from the presentation. We describe $U_{\nu}$. ###### Lemma 4.2. The module $U_{b_{i}}$ is uniserial of length $2nm^{\prime}-1$, with composition series $U_{b_{i}}=\mathcal{U}(a_{i},d_{i-1},a_{i-1},d_{i-2},a_{i-2},\ldots,a_{i+1})$ The module $U_{d_{i}}$ is uniserial of length $2nm-1$, with composition series $U_{d_{i}}=\mathcal{U}(a_{i+1},b_{i+1},a_{i+2},b_{i+2},\ldots,a_{i})$ (taking indices modulo $n$ and writing $a_{i}$, $d_{i}$, $b_{i}$ meaning the corresponding simple module). Proof We compute $U_{b_{1}}$, that is $\Omega^{2}(S_{b_{1}})=\Omega(S_{b_{1}})=\\{x\in P_{a_{2}}\mid\sigma_{1}x=0\\}\subset P_{a_{2}}$. From the relations for the algebra, we have $\sigma_{1}\varrho_{1}\delta_{1}=cA_{\sigma_{1}}=c\sigma_{1}A_{\sigma_{1}}^{\prime}$ Hence $\sigma_{1}\psi=0$ if we set $\psi=\psi_{\varrho_{1}\delta_{1}}:=\varrho_{1}\delta_{1}-cA_{\sigma_{1}}^{\prime}.$ One exhibits a basis for $\psi\Lambda$, showing that it has the same dimension as $\Omega^{2}(S_{b_{1}})$, hence we have equality. The submodule structure follows directly. The case $U_{d_{i}}$ is similar. $\Box$ ###### Proposition 4.3. We have ${\rm Ext}^{1}(M,M)=0$ and ${\rm Ext}^{2}(M,M)=0$. Proof By ext symmetry, it suffices to show that for any non-projective indecomposable summand $X$ of $M$ we have ${\rm Ext}^{1}(M,X)=0$ and ${\rm Ext}^{1}(X,M)=0$. For this, we use the following short exact sequences: Let $a_{i}\in\Gamma$, $None$ $0\to\Omega(S_{a_{i}})\to P_{a_{i}}\to S_{a_{i}}\to 0,\ \ 0\to\Omega^{2}(S_{a_{i}})\ \to P_{c_{i}}\oplus P_{d_{i}}\to\Omega(S_{a_{i}})\to 0$ Consider a vertex $\nu$ not in $\Gamma$, let $\nu=c_{i}$ $None$ $0\to\Omega^{-1}S_{b_{i}}\to P_{a_{i}}\to U_{b_{i}}\to 0,\ \ 0\to S_{b_{i}}\to P_{b_{i}}\to\Omega^{-1}(S_{b_{i}})\to 0$ Let $\nu=d_{i}$, then $None$ $0\to\Omega^{-1}(S_{d_{i}})\to P_{a_{i}+1}\to U_{d_{i}}\to 0,\ \ 0\to S_{d_{i}}\to P_{d_{i}}\to\Omega^{-1}(S_{d_{i}})\to 0$ We apply the functor ${\rm Hom}_{A}(-,X)$ to the above exact sequences. (I) Assume $X=S_{a_{j}}$ for some $j$. We know from the quiver that ${\rm Ext}^{1}(S_{a_{i}},S_{a_{j}})=0$ already. To show that ${\rm Ext}^{2}(S_{a_{i}},S_{a_{j}})=0$ we apply the functor to the second sequence in (1). From the structure of $\Omega^{2}(S_{a_{i}})$ we see directly that ${\rm Hom}(\Omega^{2}(S_{a_{i}}),S_{a_{j}})=0$ and hence ${\rm Ext}^{2}(S_{a_{i}},S_{a_{j}})=0$. We have ${\rm Ext}^{1}(U_{b_{i}},X)=0$ since ${\rm Hom}(\Omega^{-1}(S_{b_{i}}),S_{a_{j}})=0$ Furthermore ${\rm Ext}^{2}(U_{b_{i}},X)=0$ since ${\rm Hom}(S_{b_{i}},S_{a_{j}})=0$. Similarly one shows ${\rm Ext}^{1}(U_{d_{i}},X)=0$ and ${\rm Ext}^{2}(U_{d_{i}},X)=0$. (II) Now assume $X=U_{b_{j}}$ for some $j$. First, by dimension shift ${\rm Ext}^{1}(U_{\nu},U_{b_{j}})\cong{\rm Ext}^{1}(S_{\nu},S_{b_{j}})=0$ for any $\nu$ of valency $1$, from the quiver. Next, consider ${\rm Ext}^{2}(U_{\nu},X)$, by applying the functor ${\rm Hom}(-,X)$ to the second exact sequence in (2). We have ${\rm Hom}(S_{\mu},U_{b_{j}})=0$ (the socle of $U_{b_{i}}$ is always some $S_{a}$), and hence ${\rm Ext}^{2}(U_{\nu},X)=0$. Now consider ${\rm Ext}^{t}(S_{a_{i}},X)$ for $t=1,2$. By the ext symmetry, it is isomorphic to ${\rm Ext}^{t}(X,S_{a_{i}})$ for $t=2,1$. By part (I) we know that it is zero. The proof for $X=U_{d_{i}}$ is analogous. $\Box$ ###### Remark 4.4. For possible later use, we write down sequences which may be used to show ${\rm Ext}^{1}(X,M)=0$ and ${\rm Ext}^{2}(X,M)=0$: Let $a_{i}\in\Gamma$, $None$ $0\to S_{a_{i}}\to P_{a_{i}}\to\Omega^{-1}(S_{a_{i}})\to 0,\ \ 0\to\Omega^{-1}(S_{a_{i}})\to P_{b_{i}}\oplus P_{d_{i}}\to\Omega^{-2}(S_{a_{i}})\to 0$ Consider a vertex $\nu$ not in $\Gamma$, let $\nu=c_{i}$ $None$ $0\to U_{b_{i}}\to P_{a_{i+1}}\to\Omega(S_{b_{i}})\to 0,\ \ 0\to\Omega(S_{b_{i}})\to P_{b_{i}}\to S_{b_{i}}\to 0$ Let $\nu=d_{i}$, then $None$ $0\to U_{d_{i}}\to P_{a_{i}}\to\Omega(S_{d_{i}})\to 0,\ \ 0\to\Omega(S_{d_{i}})\to P_{d_{i}}\to S_{d_{i}}\to 0.$ ## 5\. Ext vanishing and 3-cluster tilting We would like to determine when $M$ is 3-cluster tilting. Hence take $X$ indecomposable and not projective, and assume ${\rm Ext}^{1}(M,X)=0={\rm Ext}^{2}(M,X).$ By ext symmetry, we get for free that ${\rm Ext}^{1}(X,M)=0={\rm Ext}^{2}(X,M).$ The aim is to show that $X$ is in add$(M)$, or if not, to identify $X$. ###### Lemma 5.1. The socle and the top of $X$ belong to ${\rm add}(\oplus_{i}S_{a_{i}})$. Proof Let $\nu$ be a vertex $\neq a_{i}$ for any $i$. Apply the functor ${\rm Hom}(-,X)$ to the second sequence of (2), this gives the exact sequence $0\to{\rm Hom}(\Omega^{-1}(S_{b_{i}}),X)\to{\rm Hom}(P_{b_{i}},X)\to{\rm Hom}(S_{b_{i}},X)\to 0$ Any homomorphsim $P_{b_{i}}\to X$ must map the socle to zero, otherwise it would be split. Hence it lies in ${\rm Hom}(\Omega^{-1}(S_{b_{i}}),X)$ and therefore the first two terms are isomorphic. Hence the last term is zero, as required. To show that also ${\rm Hom}(X,S_{\nu})=0$ we use a sequence from $(2^{})$. $\Box$ ###### Lemma 5.2. We have ${\rm Hom}(\Omega(X),S_{a_{i}})=0$ and ${\rm Hom}(S_{a_{i}},\Omega^{-1}(X))=0$. Proof Since ${\rm Ext}^{1}(X,S_{a_{i}})=0$, from a minimal projective cover of $X$ we obtain the exact sequence $0\to{\rm Hom}(X,S_{a_{i}})\to{\rm Hom}(P_{X},S_{a_{i}})\to{\rm Hom}(\Omega(X),S_{a_{i}})\to 0$ The first two terms are isomorphic since we start with a projective cover. Hence the last term is zero. Similarly by using an injective hull we get ${\rm Hom}(\Omega^{-1}(X),S_{a_{i}})=0$. $\Box$ Let $\mathcal{X}$ be the category of $A$-modules which have socle and top in ${\rm add}(S_{a_{i}})$. This category is equivalent to ${\rm mod}-e\Lambda e$. where $e$ is the idempotent $e:=\sum_{i}e_{a_{i}}$. An equivalence is given by the functor $V\mapsto Ve$, with inverse the composite of $(-)\otimes_{e\Lambda e}(e\Lambda)$ follows by factoring out the largest $A$-submodule $V^{\prime}$ with $V^{\prime}e=0$ (see for example [4]). We may write down quiver and presentation of the algebra $e\Lambda e$. The arrows are $x_{i}:=\gamma_{i}\sigma_{i}$ and $y_{i}:=\varrho_{i}\delta_{i}$, for $1\leq i\leq n$ where $x_{i}:a_{i}\mapsto a_{i+1}$ and $y_{i}:a_{i+1}\to a_{i}$. From the relations for $\Lambda$ we see We claim that $x_{i}y_{i}=0$ and $y_{i}x_{i-1}=0$. That is, $e\Lambda e$ is special biserial. Moreover, for any $i$, the longest non-zero monomial $x_{i}x_{i+1}\ldots$ is up to a scalar equal to the longest non-zero monomial $y_{i-1}y_{i-2}\ldots$, and this gives the socle relations. ###### Lemma 5.3. The module $X$ has simple socle and top. Proof The module $X$, and as well, all projectives (injectives) $P_{a_{i}}$ belong to the category $\mathcal{X}$, and hence we may fix an injective hull, or projective cover, of $X$ by identifying with the image of a suitable injective hull, or projective cover, of $Xe$, in ${\rm mod}e\Lambda e$. The indecomposable $e\Lambda e$-modules are ’strings’ or ’bands’, and their injective hulls or projective covers may be written down explicitly. Assume the socle of $X$ is not simple, then consider the injective hull $I_{X}$, it has at least two indecomposable summands, say it is $\oplus_{i\in R}P_{a_{i}}$. We may assume, with the above convention, and taking $X\to I_{X}$ as inclusion, that $X$ has a generator $\omega=(\omega_{1},\omega_{2},0,\ldots)$ such that $\omega A$ has socle of length two, and moreover, that $\omega x_{j}=(\omega_{1}x_{j},0,\ldots)$ and $\omega y_{j-1}=(0,\omega_{2}y_{j-1},0,\ldots)$ and $\omega x_{r}=0$, $\omega y_{s}=0$ for all other generators $x_{r},y_{s}$ of $e\Lambda e$. This implies then that $\omega J\subseteq X$ where $J$ is the radical of $\Lambda$. Now consider $\pi:I_{X}\to\Omega(X)$. The element $\pi(\omega_{1},0,\ldots)$ is non-zero (since $\omega$ is a generator for $X$). Furthermore, $[\pi(\omega_{1},0,\ldots)]J=\pi[(\omega_{1},0)J]=0$ since $(\omega_{1},0,\ldots)J$ is contained in $X$. Now $\pi(\omega_{1},0)=\pi(\omega_{1},0)e$, (since ${\rm top}X$ is in add$(\oplus S_{a_{i}})$. Hence for some $i$ we have ${\rm Hom}(S_{a_{i}},\Omega^{-1}(X))\neq 0$. This contradicts the previous Lemma. Similarly by exploiting a projective cover, one shows that the top of $X$ must be simple. $\Box$ ###### Proposition 5.4. The module $X$ is uniserial. Proof If $X$ is not uniserial then $Xe$ is not uniserial (using the structure of the projectives in this case). Then $Xe$ is a ’band module’. This means that $X$ contains a submodule isomorphic to the second socle of some $P_{a_{j}}$. That is ${\rm Hom}(\Omega^{2}(S_{a_{j}}),X)\neq 0$. Applying ${\rm Hom}(-,X)$ to the exact sequence $0\to\Omega^{2}(S_{a_{j}})\stackrel{{\scriptstyle\iota}}{{\to}}P:=P_{b_{j}}\oplus P_{d_{j-1}}\longrightarrow\Omega(S_{a_{j}})\to 0$ gives an excact sequence, that is a non-zero homomorphism $\theta:\Omega^{2}(S_{a_{j}})\to X$ factors through $\iota$, say $\theta=\psi\circ\iota$. The kernel of $\theta$ is the socle of $\Omega^{2}(S_{a_{j}})$ which also is the socle of $P$. We factor out these socles, then for the induced maps we have $\bar{\theta}=\bar{\psi}\circ\bar{\iota}.$ Now, the map $\bar{\psi}$ on the socle of $\bar{P}$ is non-zero on each component. It follows that the image of $\bar{\psi}$ has Loewy length equal to the Loewy length of $P/{\rm soc}P$. Note that all modules $P_{a_{i}}$ have the same Loewy length $\ell$ say. As well $P_{b_{j}}\oplus P_{d_{j-1}}$ has Loewy length $\ell$. Hence the Loewy length of $P/{\rm soc}P$ is $\ell-1$. The image of $\bar{\psi}$ is contained in the radical of $X$, which is the unique maximal submodule. It follows that the Loewy length of $X$ is $\ell$. But this means that $X$ must be projective, a contradiction. This shows that $Xe$ is uniserial, and then from the structure of the projectives, also $X$ is uniserial. $\Box$ We summarize. We have shown that if $X$ is indecomposable and not projective such that ${\rm Ext}^{1}(M,X)=0={\rm Ext}^{2}(M,X)$ then $()$ $X$ is uniserial, and ${\rm soc}X$ and ${\rm top}X$ are in add$(\oplus_{i}S_{a_{i}})$. That is, $X$ is a subquotient of some $U_{b_{i}}$ or $U_{d_{j}}$. We show now that if $X$ is any module satifying $()$ then ${\rm Ext}^{1}(M,X)=0$ and ${\rm Ext}^{2}(M,X)=0$. ###### Lemma 5.5. Let $X=\mathcal{U}(a_{j},b_{j},a_{j+1},b_{j+1},\ldots a_{l})$, a subquotient of some $U_{\nu}$. Then ${\rm Ext}^{1}(M,X)=0$ and ${\rm Ext}^{2}(M,X)=0$. Proof We use the sequences in the proof of Proposition 4.3. We apply the functor $(-,X):={\rm Hom}(-,X)$ to the exact sequences in (1). We start with the second, this gives $0\to(\Omega(S_{a_{i}}),X)\longrightarrow(P_{b_{i}}\oplus P_{d_{i-1}},X)\longrightarrow(\Omega^{2}(S_{a_{i}}),X)\to{\rm Ext}^{1}(\Omega(S_{a_{i}}),X)\to 0$ We see that ${\rm Hom}(\Omega^{2}(S_{a_{i}}),X)=0$ ($X$ is uniserial). Hence the ext space is zero. Moreover, it follows that the first two terms are isomorphic, which we can use for the first sequence: $0\to(S_{a_{i}},X)\longrightarrow Xe_{a_{i}}\longrightarrow Xe_{b_{i}}\oplus Xe_{d_{i}}\to{\rm Ext}^{1}(S_{a_{i}},X)\to 0$ In our case, $Xe_{d_{i}}=0$. Note that in the composition series we have length two subquotients $a_{r},b_{r}$, except that for $l=r$ we have an extra copy of $a_{l}$. Hence if $i=l$ then the first term is $K$, and $\|Xe_{a_{l}}\|=1+\|Xe_{b_{l}}\|$ and ext is zero. Suppose $i\neq\ell$, then the first term is zero and the second and third are isomorphic. Again ext is zero. Next, we apply $(-,X)$ to the sequences in (3). Since $S_{d_{i}}$ does not occur in $X$, the functor takes the second sequence to zero. From the first sequence we get $0\to{\rm Hom}(U_{d_{i}},X)\to Xe_{a_{i}+1}\to 0\to{\rm Ext}^{1}(U_{d_{i}},X)\to 0$ and the ext space is zero. Now consider $(-,X)$ applied to sequences in (2). The second sequence gives ${\rm Hom}(\Omega^{-1}(S_{b_{i}}),X)\cong Xe_{b_{i}}$ and ${\rm Ext}^{1}(\Omega^{-1}(S_{b_{i}}),X)=0$. Consider the first sequence, this gives $0\to{\rm Hom}(U_{b_{i}},X)\to Xe_{a_{i}}\to Xe_{b_{i}}\to{\rm Ext}^{1}(U_{b_{i}},X)\to 0$ If the top (ie $S_{a_{i}}$) of $U_{b_{i}}$ is not the same as the socle of $X$ then the hom space is zero and the second and third term are isomorphic, and ext is zero. Supoose $i=l$, then the first term is $K$, and $\|Xe_{a_{i}}\|=1+\|Xe_{b_{i}}\|$ and again the ext space is zero. $\Box$ ###### Corollary 5.6. Assume $\Lambda$ is the triangle algebra, or the spherical algebra. Then $M$ is 3-cluster tilting. Proof For these algebras, all indecomposables satisfying $()$ are in add$(M)$. $\Box$ Consider an $n$-spherical algebra for $n\geq 3$ and $m=m^{\prime}=1$. Then the (finite) set of modules $X$ satifying $()$ contains all modules of the form $\mathcal{U}(a_{i},b_{i},a_{i+1}),\ \ \mathcal{U}(a_{i},d_{i-1},a_{i-1}).$ To have a 3-cluster tilting module with $M$ as a summand, we would need to take $\widetilde{M}=M\oplus\mathcal{V}$ where $\mathcal{V}$ is the direct sum of all modules satisfying $()$. However, $\widetilde{M}$ has self-extensions. For example there is a non-split exact sequence $0\to\mathcal{U}(a_{2},b_{2},a_{3})\to S_{a_{2}}\oplus\mathcal{U}(a_{1},b_{1},a_{2},b_{2},a_{3})\to\mathcal{U}(a_{1},b_{1},a_{2})\to 0$ Hence $M$ cannot be extended to a 3-cluster tilting module for the $n$-spherical algebra when $n\geq 3$. We also consider the algebra with triangular quiver and $k\geq 2$. In this case the list of uniserial modules $X$ which are subquotients of $U_{1}$ and $U_{3}$ contains the modules $\mathcal{U}(2,3,2),\ \mathcal{U}(2,1,2)$ Let $\widetilde{M}=M\oplus\mathcal{V}$ where $\mathcal{V}$ is the direct sum of all indecomposable modules satisfying $()$. This is not a 3-cluster tilting module since it has self-extensions: we have the non-split exact sequence $0\to\mathcal{U}(2,3,2)\to S_{2}\oplus\mathcal{U}(2,1,2,3,2)\to\mathcal{U}(2,1,2)\to 0$ ## References [1] J. Bialkowski, A. Skowroński, On tame weakly symmetric algebras having only periodic modules. Arch. Math. 81(2003), 142-154. * [2] B. Böhmler, R. Marczinzik, A cluster tilting module for a representation-infinite block of a group algebra. arxiv:2101.10217 * [3] I. Burban, O. Iyama, B. Keller, I. Reiten, Cluster tilting for one-dimensional hypersurface singularities, Adv. Math. 217 (2008) 2443–2484. * [4] J. Brundan, R. Dipper, A. Kleshchev, Quantum linear groups and representations of $GL_{n}(\mathbb{F}_{q})$, AMS Memoirs vol. 149(2001) no. 706. * [5] K. Erdmann, Blocks of Tame Representation Type and Related Algebras, in: Lecture Notes in Math., vol. 1428, Springer-Verlag, Berlin-Heidelberg, 1990. * [6] K. Erdmann, T. Holm Maximal $n$-orthogonal modules for selfinjective algebras, Proc. Amer. Math. Soc. 136(2008), no. 9, 3069-3078. * [7] K. Erdmann, A. Skowroński, Weighted surface algebras, J. Algebra 505 (2018) 490–558. * [8] K. Erdmann, A. Skowroński, Weighted surface algebras: general version, J. Algebra 544 (2020), 170-227. * [9] K. Erdmann, A. Skowroński, Weighted surface algebras: general version, Corrigendum. J. Algebra 569(2021), 875-889. * [10] K. Erdmann, A. Skowroński, Algebras of generalized dihedral type. Nagoya Math. J. 240(2020), 181-236. * [11] T. Holm, Derived equivalence classification of algebras of dihedral, semidihedral, and quaternion type. J. Algebra 211(1999), 159-205. * [12] O. Iyama, Higher-dimensional Auslander-Reiten theory on maximal orthogonal subcategories. Adv. Math. 210 (2007), 22-50. * [13] O. Iyama, Auslander correspondence, Adv. Math. 210(2007), 51-82.
# A Deterministic Algorithm for the Discrete Logarithm Problem in a Semigroup Simran Tinani Institute of Mathematics, University of Zurich, Switzerland Joachim Rosenthal Institute of Mathematics, University of Zurich, Switzerland ###### Abstract The discrete logarithm problem in a finite group is the basis for many protocols in cryptography. The best general algorithms which solve this problem have time complexity of $\mathcal{O}(\sqrt{N}\log N)$, and a space complexity of $\mathcal{O}(\sqrt{N})$ where $N$ is the order of the group. (If $N$ is unknown, a simple modification would achieve a time complexity of $\mathcal{O}(\sqrt{N}(\log N)^{2})$.) These algorithms require the inversion of some group elements or rely on finding collisions and the existence of inverses, and thus do not adapt to work in the general semigroup setting. For semigroups, probabilistic algorithms with similar time complexity have been proposed. The main result of this paper is a deterministic algorithm for solving the discrete logarithm problem in a semigroup. Specifically, let $x$ be an element in a semigroup having finite order $N_{x}$. The paper provides an algorithm, which, given any element $y\in\langle x\rangle$, provides all natural numbers $m$ with $x^{m}=y$, and has time complexity $O(\sqrt{N_{x}}(\log N_{x})^{2})$ steps. The paper also gives an analysis of the success rates of the existing probabilistic algorithms, which were so far only conjectured or stated loosely. ## 1 Introduction Let $G$ be a group and assume $x,y\in G$ are two elements of the group. We refer to $x$ as the base element. The discrete logarithm problem (referred to henceforth as DLP) asks for the computation of an integer $m\in\mathbb{Z}$ (assuming such integers exist) such that $x^{m}=y$. The DLP plays an important role in a multitude of algebraic and number theoretic cryptographic systems. Its use was introduced in the Diffie-Hellman protocol for public key exchange [7] and has since seen a tremendous amount of development, generalisations and extensions [12]. Many modern-day systems for public key exchange use the discrete logarithm problem in a suitable group. The most commonly used groups have been the multiplicative group of finite fields and the group of points on an elliptic curve. The DLP in Jacobians of hyperelliptic curves and more general abelian varieties has also been studied extensively [5]. In this paper, we compute complexities using group multiplications as one fundamental step. Thus, an exponentiation $x^{e}$ is performed in $\mathcal{O}(\log e)$ steps. We will use the fact that for two lists of length $n$ in which a match exists, a match can be found in $\mathcal{O}(n\log n)$ steps using standard sorting and searching algorithms (for details, the interested reader may refer to [6]). For a general finite group of order $N$, there exist algorithms that solve the DLP in $\mathcal{O}(\sqrt{N}\log N)$ steps. Such algorithms are said to produce a square root attack. The most well-known examples are Shank’s Baby Step-Giant Step algorithm [18] and the Pollard-Rho algorithm [16]. Note that Shank’s algorithm is a deterministic algorithm having time complexity $\mathcal{O}(\sqrt{N}\log N)$ space complexity $\mathcal{O}(\sqrt{N})$. In contrast, Pollard’s algorithm is a probabilistic algorithm having time complexity $\mathcal{O}(\sqrt{N}\log N)$ group multiplications and space complexity $\mathcal{O}(1)$. If $N$ is unknown, a simple modification of these algorithms would achieve a time complexity of $\mathcal{O}(\sqrt{N}(\log N)^{2})$. Elliptic curve groups have been widely implemented in practice since for a carefully selected elliptic curve group the best known classical algorithm for solving DLP has running time $\mathcal{O}(\sqrt{N}\log N)$, where $N$ is the group order. This is in contrast to many other finite groups such as the multiplicative group of a finite field and the group of invertible matrices over a finite field where algorithms with subexponential running time are known [1]. In cryptography the Diffie-Hellman protocol using a finite group has been generalized to situations where the underlying problem is a discrete logarithm problem in a semigroup or even to situations where a semigroup acts on a set [10, 11]. The interested reader will find more material in a recent survey by Goel et al. [8]. It is naturally interesting to ask whether the DLP also has a square root attack in more generalized structures such as semigroups. Here, we define a semigroup as any set of elements with an associative binary operation. Since the best algorithms for the DLP all make use of the existence of inverses, it is unclear whether they can be generalized to a semigroup. However, when a special type of semigroup element, called a torsion element, is used as the base, it turns out that the DLP is reducible in polynomial time to the DLP in a finite group. A torsion element is one whose powers eventually repeat to form a cycle, and will be defined more precisely in Section 2. This section also elaborates more on why the standard collision-based algorithms are not directly adaptable to the semigroup case. A semigroup in which every element is torsion is called a torsion semigroup. The DLP in semigroups with a torsion base element, in a classical setting, was first discussed by Chris Monico [14] in 2002, and later in a paper by Banin and Tsaban [3] in 2016. While the discussion in the present paper is entirely on classical algorithms, it is also worth mentioning the paper [4], where the authors independently provide a quantum algorithm that solves the DLP in a torsion semigroup. Both the algorithm of Monico and the one of Banin and Tsaban are probabilistic and might fail with low probability. Further, some of their methods are heuristic, dependent on an oracle or some additional assumption, and their success rates and expected number of steps are either conjectured or stated loosely. It is therefore of interest to come up with an algorithm which deterministically computes the discrete logarithm in a semigroup. In this regard we like to make some analogy to the problem of determining if an integer is a prime number, a problem of great importance in cryptography. Nowadays in practice the algorithm of Miller and Rabin [13, 17] has been implemented for many years. Still it was a great result when Agrawal, Kayal and Saxena [2] came up with a deterministic polynomial time algorithm to achieve this goal. A key step in finding the discrete logarithm in a semigroup is computing the cycle length of an element. Both the algorithms of [3] and [14] rely on computing some multiple of the cycle length, and then removing “extra” factors by taking gcd’s until the cycle length is obtained. Once the cycle length value is obtained, the discrete logarithm may easily be computed with a few more simple steps. While Monico does not provide further elaboration on how this is done, the paper by Banin and Tsaban bridges this knowledge gap by showing how the problem is reduced to a DLP in a group once the cycle length and start values are known. Denote by $N_{x}$ the order of $x$ (formally defined in Definition 4). The complexity of the algorithm in [3] is $\mathcal{O}(\sqrt{N_{x}}(\log N_{x})^{2}\log\log N_{x})$, and the of the one in [14] is $\mathcal{O}(\sqrt{N_{x}}(\log N_{x})^{2})$. While both of the existing methods seem to succeed with high probability for practical values, we show that the process of taking successive gcd’s/factors is unnecessary, and that one can deterministically find the cycle length. The main contribution of this paper will be a deterministic algorithm for computing the discrete logarithm of an element $y$ in some semigroup $S$ with respect to some torsion base element $x\in S$. The time complexity of our algorithm is $\mathcal{O}(\sqrt{N_{x}}(\log N_{x})^{2})$. The paper is structured as follows: After providing preliminaries and basic definitions in Section 2, we will analyse in Section 3 the success rates and expected number of steps involved in the probabilistic algorithms for cycle length by Banin and Tsaban (Algorithm 1) and Monico (Algorithm 3). In Section 4, which is the main section of this paper, we provide a deterministic algorithm to calculate the cycle length $L_{x}$ of a torsion element $x$ of a semigroup and thus to also solve the DLP, without the use of an oracle. This algorithm has complexity $\mathcal{O}\left(\sqrt{N_{x}}\cdot(\log N_{x})^{2}\ \right).$ For completeness, we will also demonstrate the use of Pohlig–Hellman algorithm [15] for a semigroup. ## 2 Preliminaries A semigroup $S$ is a set together with an associative binary operation. Like in group theory where a torsion group consists of elements of finite order only we define: ###### Definition 1 (Torsion Element). Let $S$ be a semigroup. An element $x\in S$ is called a torsion element if the sub-semigroup $\langle x\rangle:=\\{x^{k}\mid k\in\mathbb{N}\\}$ generated by $x$, is finite. $S$ is called a torsion semigroup if every $x\in S$ is a torsion element. Throughout the paper the following definitions will be assumed: ###### Definition 2 (Cycle Start). Let $x\in S$. The cycle start $s_{x}$ of $x$ is defined as the smallest positive integer such that $x^{s_{x}}=x^{b}$ for some $b\in\mathbb{N}$, $b>s_{x}$. ###### Definition 3 (Cycle Length). Let $x\in S$. The cycle length $L_{x}$ of $x$ is defined as the smallest positive integer such that $x^{s_{x}+L_{x}}=x^{s_{x}}$. ###### Definition 4 (Element order). Let $x\in S$. With notation as above, we define the order $N_{x}$ of $x$ as the cardinality of the sub-semigroup $\langle x\rangle$. Note that $N_{x}=s_{x}+L_{x}-1.$ ###### Definition 5 (Semigroup DLP). Let $S$ be a semigroup and $x\in S$. The semigroup DLP is defined as follows. Given $y\in\langle x\rangle:=\\{x^{k}\mid k\in\mathbb{N}\\}$, find all $m\in N$ such that $x^{m}=y$. We state below a key result first proved in [3]. ###### Lemma 1 ([3]). Let $S$ be a semigroup and $x\in S$ be an element with cycle start $s_{x}$. The set of powers $G_{x}=\\{x^{s_{x}+k}\mid k\geq 0\\}$ of $x$ forms a finite cyclic group. The identity element of $G_{x}$ is given by $x^{tL_{x}}$, where $t$ is the minimum positive integer such that $x^{tL_{x}}\in G_{x}$. The following result is stated in [14] in a slightly different formulation. We provide an equivalent proof based on the group structure of $G_{x}$. ###### Lemma 2 ([14]). Let $x\in S$ have cycle start $s_{x}$ and cycle length $L_{x}$. For all integers $n,m\geq s_{x}$, we have $x^{m}=x^{n}\iff n\equiv m\mod L_{x}$. ###### Proof. We can assume without loss of generality that $n\geq m$, and so we can write $n=m+kL_{x}+u$, with $k\geq 0$ and $0\leq u<L_{x}$. First suppose that $n\equiv m\mod L_{x}$, i.e. $u=0$. Since $m,n\geq s_{x}$, we have $x^{n}=x^{m+kL_{x}}=x^{m}$. Conversely, if $x^{n}=x^{m}$, write $n_{1}=n-s_{x}\geq 0$, and $m_{1}=m-s_{x}\geq 0$. We have $x^{s_{x}+m_{1}}=x^{s_{x}+n_{1}}=x^{s_{x}+m_{1}+kL_{x}+u}=x^{s_{x}+m_{1}+u}.$ Now, without loss of generality, $m_{1}\geq s_{x}$, because if not, one can always increment $m_{1}$ and $n_{1}$ by multiples of $L_{x}$ until this happens. So, we can assume that $x^{m_{1}}$ lies in $G_{x}$ and is thus invertible. We multiply by the inverse on both sides to finally get $x^{s_{x}}=x^{s_{x}+u}.$ Thus, we must have $u=0$ or $n\equiv m\mod L_{x}$, as required. ∎ ###### Remark 1. It becomes clear from the above discussion that the standard collision-based algorithms for order and discrete log computations in a group do not adapt directly to a general semigroup. Collision-based algorithms for the computation of the order $N$ of a group element $x$ (for instance, see [19]) are based on the principle that whenever $N$ can be expressed in the form $N=A-B$ for non-negative integers $A$ and $B$, the collision $x^{A}=x^{B}$ always occurs. However, this principle does not work in a semigroup, where there are two independent components of the order. More specifically, for a semigroup element $x$ with cycle length $L_{x}$ and cycle start $s_{x}$, whenever $L_{x}$ may be expressed in the form $A-B$ for non-negative integers $A$ and $B$, the equality $x^{A}=x^{B}$ holds if and only if $A,B\geq s_{x}$. As an example, consider a semigroup element $x$ with cycle length $L_{x}=12$ and cycle start $s_{x}=5$. Then, $L_{x}=15-3$, but $x^{15}\neq x^{3}$. Thus without prior knowledge of the cycle start, the semigroup order $N_{x}$ or cycle length $L_{x}$ cannot directly be found using the same collision-based algorithms for groups. Similarly, collision-based algorithms fail for discrete log computations in a semigroup. As an example, consider a semigroup element $x$ with cycle length $L_{x}=15$ and cycle start $s_{x}=10$, and suppose that the discrete log of $y=x^{5}$ is to be found. Then $y\cdot x^{6}=x^{11}=x^{26}$ is obtained as a collision. However, unlike in the group case, the conclusion $y=x^{26-6}=x^{20}$ is wrong since $x^{5}\neq x^{20}$. This happens because even though $x$ is torsion and forms a cycle of powers, it is not invertible. This concludes the prerequisite knowledge on torsion elements in semigroups. In the next section, we study the existing probabalistic algorithms for cycle lengths, and analyse their assumptions, working and complexities. ## 3 Existing Probabalistic Agorithms ### 3.1 Banin and Tsaban’s Algorithm In this section, we study the probabalistic algorithm described in [3] for computing the cycle length of a torsion element in a semigroup. While the authors of the original paper describe their theory only for torsion semigroups, it will become clear that the same discussion holds true for any semigroup when the base element chosen is torsion. Let $S$ be a semigroup and $x$ be a torsion element of $S$. Let $s_{x}$ denote the cycle start of $x$ and $L_{x}$ its cycle length. Then, recall from Lemma 1 that $G_{x}:=\\{x^{s_{x}},x^{s_{x}+1},\ldots,x^{s_{x}+L_{x}-1}\\}$ is a cyclic group, and that it has order $L_{x}$. The authors of [3] assume the availability of a ‘Discrete Logarithm Oracle’ for the group $G_{x}$, which returns values $\log_{x}h$ for $h\in G_{x}$. They state that these values need not be smaller than the group order but are polynomial in the size of $G_{x}$ and the element $x$. The representation of the identity in $G_{x}$ is unknown, and a method to compute inverses is not available. The authors claim that the well-known algorithms for discrete logarithm computations in groups do not explicitly require inverses, or can easily be modified to work without the use of inverses. While it is true that these algorithms make use of mainly the existence of inverses rather than their explicit computation, we believe that the fact that easy modification is possible is not immediate without some justification. In fact, it will become clear in the later sections that the modified Baby-Step-Giant-Step algorithm devised by Monico [14] (and also the deterministic algorithm presented in Section 4) is a crucial and non-trivial part of any such modification. We make the following observation from the proof of Lemma 1 found in [3]. For any $k\geq 0$, denote by $v_{k}$ the smallest positive integer such that $v_{k}L_{x}\geq 2s_{x}+k.$ We then have $x^{v_{k}L_{x}-s_{x}-k}\in G_{x}$ and $x^{s_{x}+k}x^{v_{k}L_{x}-s_{x}-k}=x^{v_{k}L_{x}}=x^{tL_{x}},$ (1) so the inverse of the element $x^{s_{x}+k}$ of $G_{x}$ is given by $x^{v_{k}(L_{x})-s_{x}-k}$. In particular, the computation of inverses requires prior knowledge of the cycle start. As will be explained below, the cycle start may be computed only once the value of the cycle length is known, using a binary search. This explains why the authors insist that their Discrete Logarithm Oracle does not need to use the computation of inverses. Below, we describe Algorithm 1, which is the algorithm suggested in [3] to compute the order of the group $G_{x}$, i.e. the cycle length $L_{x}$ of $x$. Input A semigroup $S$ and a torsion element $x\in S$; a DLP oracle for groups Output The cycle length $L_{x}$ of $x$ 1: Initialize $i,j,g,L_{x}\leftarrow 1$, $N>>s_{x}+L_{x}$. Fix bounds $r>1,s>1$. 2: while $j<s$ 1. 1. Fix a random $z\in\\{\lfloor M/2\rfloor,\ldots,M\\}$ and set $h=x^{z}$. 2. 2. while $i<r$ 1. (a) Choose a random number $k_{i}>0$. 2. (b) Use the DLP oracle to compute $k_{i}^{\prime}=\log_{h}(h^{k_{i}})$. 3. (c) Set $g\leftarrow\gcd\limits_{j\leq i}(k_{j}-k_{j}^{\prime})=\gcd\left(\gcd\limits_{j<i}(k_{j}-k_{j}^{\prime}),\ k_{i}-{k_{i}}^{\prime}\right)$. 4. (d) Set $i\leftarrow i+1$. 3. 3. end while 4. 4. Set $L_{x}\leftarrow lcm(L_{x},g)$, $j\leftarrow j+1$. 6: Return $L_{x}$. Algorithm 1 Banin-Tsaban Algorithm for Cycle Length We first note that the authors state complexities in terms of $L_{x}$, which are valid when the bound $N$ for $N_{x}$ is known. If the algorithm fails for a value of $N$, the authors suggest to double $N$ and try again. In this case, which we will assume from now on, we assert that the complexities need to be taken in terms of $N_{x}$ instead of $L_{x}$. The oracle may be assumed to have the standard complexity of $O(\sqrt{N_{x}}\log N_{x})$ steps for discrete logarithm calculations. Step 2.2(c) takes $\mathcal{O}(\log(\max_{j\leq i}(k_{j}-k_{i}))=\mathcal{O}(\log N_{x})$ integer operations by the assumption on the oracle, which does not contribute to the total complexity. Thus, the total complexity of step (2.2) comes from the oracle alone, and is $\mathcal{O}(\sqrt{N_{x}}\log N_{x})$. Now, the authors of [3] remark that $r$ and $s$ can be taken to satisfy $r=\mathcal{O}(1)$ and $s=\mathcal{O}(\log\log N_{x})$. Thus, the total complexity is $\mathcal{O}(\log N_{x})$ times the complexity of Algorithm 1, and thus $\mathcal{O}(\log\log(N_{x})\log N_{x})$ times the complexity of step (2.2). Therefore, we get the total complexity of $\mathcal{O}(\log\log N_{x}(\log N_{x})^{2}\sqrt{N_{x}})$. Finally, in Algorithm 2, we present the application of the binary search method to find the cycle start once $L_{x}$ is known. This algorithm is formulated as below for this purpose in [3], though the idea to use a binary search is also originally mentioned in [14]. Input A semigroup element $x$ with cycle length $L_{x}$ Output Cycle start $s_{x}$ of $x$ 1: Initialize $s_{x}\leftarrow 1$ 2: while $x^{s_{x}+L_{x}}\neq x^{s_{x}}$ do $s_{x}\leftarrow 2s_{x}$ 3: end while 4: Set $a\leftarrow s_{x}/2$ 5: while $\|a-s_{x}\|\geq 2$ $c\leftarrow(a+s_{x})/2$ if $x^{c+L_{x}}\neq x^{c}$ then $a\leftarrow c$ else $s_{x}\leftarrow c$ 6: end while Algorithm 2 Calculating Cycle Start (Binary Search) ###### Lemma 3. Let $N_{x}$ be the order of the element $x$. Then Algorithm 2 requires $\mathcal{O}\left((\log N_{x})^{2}\right).$ steps. ###### Proof. Each of Steps (2) and (5) involves $\mathcal{O}(\log N_{x})$ rounds, each of which computes requires $\mathcal{O}(\log N_{x})$ semigroup multiplications and one comparison. The total complexity is thus $\mathcal{O}\left((\log N_{x})^{2}\right)$. ∎ ### 3.2 Monico’s Algorithm In his PhD thesis [14], Chris Monico provides a probabilistic algorithm (described below as Algorithm 3) that calculates the cycle length of an element in a finite ring of order $N$. This algorithm makes use of the multiplicative semigroup structure of the finite ring, and of the availability of the explicit bound $N$ for every cycle length, and is in fact applicable to any semigroup where such a bound $N$ is available. In this subsection, we analyse this algorithm, provide a more concrete bound on its success rate, and compute its complexity in terms of $N$. We will discuss this algorithm in terms of torsion semigroups, as opposed to finite rings. Input A finite semigroup $S$ with $\lvert S\rvert=N$ and an element $x\in S$ Output The cycle length $L_{x}$ of $x$ 1: Set $m=\lceil\sqrt{N}\rceil$. Choose a prime $q>N$. 2: For $0\leq i\leq m$, compute and store in a table the pairs $(i;x^{q+im})$. Sort the table by the second component. 3: Find the least positive integer $b_{1}$ such that $x^{q+b_{1}}$ is in the table: $x^{q+b_{1}}=x^{q+a_{1}m}$. (Note: $0<b_{1}<m$). 4: Find the least positive integer $b_{2}$ such that $x^{2q+b_{2}}$ is in the table: $x^{2q+b_{2}}=x^{q+a_{2}m}$. (Again, $0<b_{2}<m$). 5: Compute $g=\gcd(a_{1}m-b_{1},a_{2}m-b_{2}-q)$. 6: For each divisor $d$ of $g$ below some bound $B$, do the following. If $x^{N+g/d}=x^{N}$: set $g\leftarrow g/d$; 7: Output $L_{x}=g$ and stop. Algorithm 3 Monico’s Baby-Step Giant-Step for Cycle Length We first note that if $L_{x}>m$ and the table in Step (2) has repeated entries $x^{q+i_{1}m}=x^{q+i_{2}m}$, then numbers $b_{1}$ and $b_{2}$ may not exist below $m$. In this case the algorithm needs to be modified to take $g\leftarrow(i_{1}-i_{2})m$. However, whenever this case does not arise, it can be shown that steps 3 and 4 are always successful in finding a collision. We further remark that in step 6, the list of divisors of $g$ is kept fixed, while $g$ is updated to $g/d$ whenever the condition is satisfied. In the subsequent steps, non-divisors of $g/d$ can be immediately discarded. However, the end result depends on the order in which divisors are tested, which the algorithm does not mention explicitly. However, we note that it is, in fact, possible to restrict the testing to only the prime power divisors of $g$ below $B$, and with this setting, the optimal performance is obtained by taking divisors in decreasing order. We will assume this set-up for the rest of the analysis. Step (2) involves $\mathcal{O}(\log N)$ steps compute $x^{q}$ and $x^{m}$ and another $\mathcal{O}(\sqrt{N})$ multiplications to compute $x^{q},x^{q}\cdot x^{m},x^{q}\cdot x^{2}m,\ldots,x^{q}\cdot x^{m^{2}}$. Step 3 involves at most $m$ multiplications $x^{q+1}=x^{q}\cdot x,x^{q+1}\cdot x,\ldots,x^{q+m-1}$, with complexity $\mathcal{O}(\sqrt{N})$, and match-finding with the first list, with complexity $\mathcal{O}(\sqrt{N}\log N)$ with standard sorting and search algorithms. The same is true for step (4). Step (5) has complexity $\mathcal{O}(\log\max(a_{1}m-b_{1},a_{2}m-b_{2}-q))=\mathcal{O}(\log N)$ and so does not contribute to the overall complexity. Step 6 involves $B$ iterations of a multiplication and an exponentiation $x^{g/d}$, and thus has a time complexity of $\mathcal{O}(B(\log g+1))=\mathcal{O}(B\log N)$ multiplications. In the original work, Monico states that the bound $B$ of Algorithm 3 can always be chosen so that $B<\sqrt{a_{1}m-b_{1}}$. We remark that this claim does not hold in the current setting of the algorithm. For example, with a cycle length value of 4, and $a_{1}m-b_{1}=104$, $a_{2}m-b_{2}-q=52$, we get $g=52$. If $B<\sqrt{a_{1}m-b_{1}}=\sqrt{104}<11$, then we would only test divisors $d$ below 11, and would never factor out 13 to obtain the true cycle length. For such a bound to work, one needs to modify the algorithm to test both divisors $g$ and $g/d$ in step 6. However, we will show in Lemma 4 that it is almost always sufficient to take $B$ to be a reasonably large fixed constant, thus the complexity of step 6 can be counted as $\mathcal{O}(\log N)$, and does not contribute to the overall complexity. Thus, the overall time complexity is $\mathcal{O}(\sqrt{N}\log N)$. If $N$ is unavailable, the algorithm can also be modified to update the value of $N$ step-by-step until a large enough value is found. In this case, Algorithm 3 has a total complexity of $\mathcal{O}(\sqrt{N_{x}}(\log N_{x})^{2})$. Further, Monico suggests a modification to the above algorithm, viz. to find several such $a_{i}$ and $b_{i}$ and compute all the gcd’s. It is clear that this suggestion is exactly the method used in Banin and Tsaban’s algorithm as discussed in Section 3.1. We now analyze the probability of success. The algorithm first looks for a collisions of the form $x^{q+{a_{1}}m}=x^{q+b_{1}}$. The working principle is that in this case, the cycle length $L_{x}$ divides $a_{1}m-b_{1}$. Similarly, if also $x^{q+{a_{2}}m}=x^{2q+b_{2}}$ then $g=\gcd(a_{1}m-b_{1},a_{2}m-b_{2}-q)$ is a multiple of $L_{x}$. So far, the process is essentially the same in both Algorithms 1 and 3: while the former uses a discrete logarithm oracle to obtain multiples of the cycle length, the latter directly finds these multiples by finding collisions. However, in Algorithm 3, we do not proceed with computing multiple factors of $L_{x}$, but work with the fixed multiple $g$ of $L_{x}$, whereas in Algorithm 1 this multiple shrinks several times. Algorithm 3 then proceeds by fixing a bound $B$ and iterating over every number $d$ below $B$ to check if $d\mid g$. If yes, it executes the next part, i.e. checks if $x^{N+D/d}=x^{N}$, and if this holds, it sets $D\leftarrow D/d$. Note that if the factorization of the number $g$ is known (or if $g$ can be factored in time negligible compared to $O(\sqrt{N})$, then we do not need this fixed bound $B$, and can instead iterate over every prime factor $d$ of $g$. It is well-known that the number of prime factors of $g$ counted with multiplicity is $\mathcal{O}(\log g)$, so Step (5) of the algorithm can find $L_{x}$ in $\mathcal{O}(\log N)$ steps. However, in general, factoring $g$ may be difficult, so we assume from here on that the algorithm proceeds by fixing a bound $B$ for the divisors of $g$. Below we analyse the probability of the algorithm succeeding in terms of $B$ and $g$. ###### Lemma 4. The probability that Algorithm 3 succeeds is bounded below by $\left(1-\frac{1}{B}\right)^{\log g}$. ###### Proof. We write $g=L_{x}\cdot F$ for some number $F$ and suppose that the algorithm fails. This means that there is a divisor, and hence also a prime power divisor of $F$, which the algorithm fails to factor out. Let $p$ be a prime dividing $F$, $\alpha_{p}$ denote its largest power dividing $F$, and $\beta_{p}$ be its largest power below the fixed bound $B$. So, we have $p^{\alpha_{p}}\mid F$, $p^{\alpha_{p}+1}\nmid F$, $p^{\beta_{p}}<B$, $p^{\beta_{p}+1}>B$. Since the number of times the algorithm divides $g$ by $p$ is $\sum\limits_{i=1}^{\beta_{p}}i=\beta_{p}\cdot(\beta_{p}+1)/2,$ we must have $\beta_{p}\cdot(\beta_{p}+1)/2<\alpha_{p}$ if the algorithm fails. So, the algorithm succeeds as long as $\beta_{p}\cdot(\beta_{p}+1)/2\geq\alpha_{p}$ for every prime divisor $p$ of $F$. Thus, the probability of success for the algorithm can be bounded below by $\displaystyle\prod_{p\mid g}\mathrm{Prob}\left(\frac{\beta_{p}\cdot(\beta_{p}+1)}{2}\geq\alpha_{p}\right).$ Write $v_{p}=\frac{\beta_{p}(\beta_{p}+1)}{2}$ for simplicity. We may assume that $g$ is a random multiple of $L_{x}$ below the bound $B$, so $F$ is a random number in $\\{1,\ldots,\frac{B}{L_{x}}\\}$. We have, $\displaystyle\mathrm{Prob}(\alpha_{p}\leq v_{p})=$ $\displaystyle 1-\mathrm{Prob}(p^{v_{p}+1}\mid F)$ $\displaystyle=$ $\displaystyle 1-\left(\frac{B/L_{x}}{p^{v_{p}+1}(B/L_{x})}\right)$ $\displaystyle=$ $\displaystyle 1-1/p^{v_{p}+1}=1-\dfrac{1}{p^{\frac{\beta_{p}(\beta_{p}+1)}{2}+1}}.$ Hence, a lower bound for the probability of the algorithm’s success is $\prod\limits_{p\mid F}\left(1-\dfrac{1}{p^{\frac{\beta_{p}\cdot(\beta_{p}+1)}{2}+1}}\right).$ Now, we have, $\displaystyle p^{\beta_{p}+1}>B\;\iff\dfrac{1}{p^{\beta_{p}+1}}<\dfrac{1}{B}$ $\displaystyle\implies$ $\displaystyle 1-\dfrac{1}{p^{\frac{\beta_{p}(\beta_{p}+1)}{2}+1}}>1-\dfrac{1}{B^{\frac{\beta_{p}}{2}+1}}>1-\dfrac{1}{B}.$ We further make the following observation. Let $\omega(n)$ denote the number of distinct prime divisors of integer $n$ (note, however, that the same statement also holds if counted with multiplicity). Then clearly, $2^{\omega(n)}\leq n,$ and so, taking logarithms, $\omega(n)\leq\log_{2}n.$ Collecting all the above results, we conclude that the probability of success Prob (success) of Algorithm 3 is bounded below as follows. Prob (success) $\displaystyle\geq\prod_{p\mid F}\left(1-\frac{1}{B}\right)$ $\displaystyle=\left(1-\frac{1}{B}\right)^{\omega(F)}\geq\left(1-\frac{1}{B}\right)^{\log F}$ $\displaystyle\geq\left(1-\frac{1}{B}\right)^{\log g}.$ ∎ Note that this bound shows that Algorithm 3 is indeed successful with overwhelming probability, as conjectured by the author. For example, with $B=10^{6}$, even when $g$ is several orders of magnitude larger than $B$, say $g=2^{4000}$, the probability of success is greater than 99.6 percent, by the bound derived in Lemma 4. ## 4 Deterministic Solution of the DLP The solution of the DLP in a semigroup involves two parts: the calculation of the cycle length and start of the base element $x$, and the use of this value to find the discrete log. ### 4.1 Deterministic Algorithm for Cycle Length Computation We now present our deterministic algorithm for the computation of the cycle length. It works by finding a suitable collision, and also guarantees finding the actual cycle length rather than just a multiple of it, in a fixed number of steps. Input A semigroup $S$ and a torsion element $x\in S$. Assume $N_{x}$ is the order of $x$. Output Cycle length $L_{x}$ of $x$ 1: Initialize $N\leftarrow 1$. 2: Set $q\leftarrow\lceil\sqrt{N}\rceil$. 3: Compute, one by one, $x^{N},x^{N+1},\ldots,x^{N+q}$ and check for the equality $x^{N}=x^{N+j}$ at each step $j\geq 1$. Store these values in a table as pairs $(N+j,x^{N+j})$, $0\leq j<q$. If $x^{N}=x^{N+j}$ for any $j<q$, then set $L_{x}\leftarrow j$ and end the process. If not, proceed to the next step. 4: For $0\leq i\leq q$, compute, one by one, the values $x^{N+q},x^{N+2q},\ldots,x^{N+iq}$ and at each step $i$, look for a match in the table of values calculated in Step (3). 5: Suppose that a match $x^{N+iq}=x^{N+j}$ is found, and $i$ is the smallest integer such that this happens. Set $L_{x}\leftarrow iq-j$ and end the process. 6: If no match is found in steps 3 or 5, set $N\leftarrow 2\cdot N$ and go back to Step (2). Algorithm 4 Deterministic Algorithm for Cycle Length ###### Theorem 1. Let $S$ be a semigroup and $x\in S$ a torsion element with order $N_{x}$. If an upper bound on $N_{x}$ is known, Algorithm 4 returns the correct value of the cycle length $L_{x}$ with $\mathcal{O}\left(\sqrt{N_{x}}\cdot(\log N_{x})^{2}\ \right)$ steps. The total space complexity is $\mathcal{O}\left(\sqrt{N_{x}}\right)$ semigroup elements. ###### Proof. We first assume $N\geq\max(L_{x},s_{x})$ and show that steps 1 to 5 succeed in finding $L_{x}$. We have $q=\lceil\sqrt{N}\rceil$. If $L_{x}<q$, then the equality $x^{N}=x^{N+L_{x}}$ is found in the first step and the statement of the theorem follows. Else if $L_{x}\geq q$, we can write uniquely $L_{x}=iq-j,$ for some positive integers $i>0$, $0\leq j<q$. Now, we must have $i\leq q$, because otherwise if $i\geq q+1$, we would have $L_{x}\geq(q+1)q-j>q^{2}+q-q=q^{2}\geq N,$ a contradiction. We have $\displaystyle L_{x}=iq-j,\;0<i\leq q,0\leq j<q$ $\displaystyle\implies$ $\displaystyle N+j+L_{x}=N+iq$ $\displaystyle\implies$ $\displaystyle x^{N+j}=x^{N+j+L_{x}}=x^{N+iq},$ where the last step follows because $N>s_{x}$ by assumption. So, such a collision always occurs between elements of the two lists in the algorithm. We now claim that for the smallest such integer $i$ computed in Step (5) of Algorithm 4, $L_{x}=iq-j$. To see this, let $i$ be the smallest positive integer such that $x^{N+j}=x^{N+iq}.$ Also let $L_{x}=i^{\prime}q-j^{\prime}$, $0<i^{\prime}\leq q$, $0\leq j^{\prime}<q$. We have already shown above that such integers $i^{\prime}$ and $j^{\prime}$ exist for our choice of $N$. By the definition of $L_{x}$, we must have $L_{x}\mid iq-j$. Now suppose that $i^{\prime}>i$. Then, $\displaystyle i^{\prime}q-j^{\prime}\geq$ $\displaystyle(i+1)q-j^{\prime}$ $\displaystyle=$ $\displaystyle iq+(q-j^{\prime})>iq$ $\displaystyle\geq$ $\displaystyle iq-j.$ But, $L_{x}=i^{\prime}q-j^{\prime}\mid iq-j$, so we must have $iq-j=i^{\prime}q-j^{\prime}$. Since $i^{\prime}>i$, this means that $q\leq(i^{\prime}-i)q=(j^{\prime}-j)<j^{\prime},$ which is a contradiction because $0\leq j^{\prime}<q$. So, we must have $i^{\prime}=i$, $j^{\prime}=j$. This proves the claim. We have shown above that the algorithm finds the correct cycle length when $N>\max(s_{x},L_{x})$. Since the algorithm doubles the value of $N$ until a match is found, it always terminates and outputs the correct cycle length. We now look at the time complexity. For a given $N$, step (2) involves one exponentiation, or $\mathcal{O}(\log N)$ multiplications to find $x^{N}$ and then at most another $q=\mathcal{O}(\sqrt{N})$ multiplications and equality checks for $x^{N}\cdot x,x^{N}\cdot x^{2},\ldots,x^{N}\cdot x^{q}$. This step also needs a storage space of at most $q=\mathcal{O}(\sqrt{N})$ elements. Step 5 needs one exponentiation or $\mathcal{O}(\log N)$ multiplications to find $x^{q}$, and then another $q=\mathcal{O}(\sqrt{N})$ multiplications to find $x^{N+q}\cdot x^{q},x^{N+q}\cdot x^{2q}\ldots,x^{N+q^{2}}$. Finding matches in steps 3 and 5 can be done in $\mathcal{O}(q\log q)=\mathcal{O}(\sqrt{N}\log\sqrt{N})$ comparisons with the use of sorting and efficient look-up methods. Thus, clearly, steps 1 to 5 in algorithm 4 have a total complexity of $\mathcal{O}(\sqrt{N}\log N)$. Moreover, the algorithm starts at $N=1$ and doubles $N$ until the cycle length is found, i.e. until $N>\max(s_{x},L_{x})$. Thus, the number of times steps 2 to 5 are performed is $\displaystyle\left\lceil\log\left(\max\left(L_{x},s_{x}\right)\right)\right\rceil=\mathcal{O}\left(\max\left(\log\left(L_{x}\right),\log(s_{x})\right)\right)=\mathcal{O}(\log N_{x})$ Thus, the total number of steps involved is $\mathcal{O}\left(\sqrt{N_{x}}\cdot(\log N_{x})^{2}\right).$ Clearly, Step (3) involves the storage of $q=\lceil\sqrt{N}\rceil=\mathcal{O}\left(\sqrt{\max(s_{x},L_{x})}\right)=\mathcal{O}\left(\sqrt{N_{x}}\right)$ elements, so this value gives the total space complexity. This completes the proof. ∎ ###### Remark 2. If a bound $N$ on the order $N_{x}$ is known a priori, then Algorithm 4 can clearly be completed in a single round, with time complexity $\mathcal{O}\left(\sqrt{N}\cdot(\log N)\right)$. ###### Remark 3. For the case of a group, there exist better algorithms for the computation of the order of an element even when the total group order is unbounded. For instance, Algorithm 3.3 in [19] uses a growth function $d(t)$, which generalizes the square root function used above, to compute the order $N$ of a group element $x$, and achieves time and space complexities of $\mathcal{O}\left(\sqrt{N}\right)$, thus eliminating the additional $\log N$ multiplier introduced by the method in Algorithm 4. However, this method fails when used for a general semigroup due to the presence of two independent unknown components of the order. To see this, note that the algorithm would need to be modified for a semigroup as follows. At stage $t$, one has $g(t-1)\leq N_{x}<g(t)$. On the completion of the baby steps, one has a table with the powers $x^{g(t)},x^{g(t)+1},\ldots,x^{g(t)+b(t)}$ (the addition of $g(t)$ is necessary in the semigroup case to ensure that the loop is entered). The giant steps compute $x^{g(t)+g(t-1)+\cdot b(t)},x^{g(t)+g(t-1)+2\cdot b(t)},\ldots,x^{g(t)+g(t-1)+d(t)\cdot b(t)}=x^{2g(t)}$. Now, while $N_{x}$ is guaranteed to have a unique expression as $g(t-1)+ib(t)-j$ with $0<i\leq d(t)$ and $0\leq j\leq b(t)$, this does not necessarily lead to a collision. In fact, if $b(t)<L_{x}<g(t-1)$ and $2L_{x}>g(t)=g(t-1)+d(t)\cdot b(t)$, then neither the baby steps nor the giant steps leads to a collision, and the cycle length is never found (note that this can happen only if $L_{x}>s_{x}$). Moreover, if a collision $x^{g(t)+g(t-1)+ib(t)}=x^{g(t)+j}$ is obtained in the giant step phase, the only conclusion that can be drawn is that $L_{x}\mid g(t-1)+ib(t)-j$. If instead we forced the condition $g(t-1)\leq N_{x}<g(t)$, a collision again may never occur because there is no control on the cycle start (For instance, in matrix semigroups over finite simple semirings, the cycle start is often found to be much larger than the cycle length. In such cases, adapting group-based algorithms would fail). See Remark 1 for further details. #### 4.1.1 Experimental Results for Cycle Length Computations We used Algorithm 4 to compute cycle length values in several common semigroups, such as matrix semigroups over finite fields, matrix semigroups over the finite simple semiring $S_{20}$ (see [20] for a construction and [11] for the addition and multiplication tables), and the symmetric and alternating groups (where the cycle length is precisely the order of the element). We further used the obtained cycle lengths to compute the cycle start values using Algorithm 2. The working code may be found at https://github.com/simran-tinani/semigroup-cycle-length. ### 4.2 Solving the DLP once the Cycle Length is known In this section, we demonstrate the solution of the DLP for a torsion element $x$ in the semigroup $S$ once the cycle length is known. As before let $N_{x}$ be the order of the sub-semigroup $\langle x\rangle$, let $L_{x}$ be the cycle length of the torsion element $x$ (which we assume is already computed) and let $y\in\langle x\rangle$ be an element. In [3], the authors demonstrate the next steps in solving for $\log_{x}(y)$, via a reduction to a DLP in the group $G_{x}$, once $L_{x}$ and $s_{x}$ are known. The procedure is described in Algorithm 5 below, which has been adapted from the original formulation in [3]. Input A semigroup $S$, a torsion element $x\in S$, with cycle length $L_{x}$ and cycle start $s_{x}$, and $y\in S$ with $y=x^{m}$ Output The discrete logarithm $m$ of $y$ with base $x$ 1: Compute $t=\left\lceil\frac{s_{x}}{L_{x}}\right\rceil$ and define $x^{\prime}=x^{tL_{x}+1}\in G_{x}$. 2: Find the minimum number $0\leq b\leq t$ such that $y^{\prime}=y\cdot x^{bL_{x}}\in G_{x}$ using binary search. 3: Use Shank’s Baby-Step Giant-Step algorithm for the group $\langle x^{\prime}\rangle\subseteq G_{x}$ to compute $m^{\prime}\in\\{0,1,\ldots,L_{x}-1\\}$ such that ${(x^{\prime})}^{m^{\prime}}={y}^{\prime}$. 4: Find the maximum number $c\geq 0$ such that $x^{(tL_{x}+1)m^{\prime}-cL_{x}}\in G_{x}$ using binary search. 5: Return $m=m^{\prime}(tL_{x}+1)-(b+c)L_{x}$. Algorithm 5 Algorithm for Discrete Logarithm Since authors of [3] do not give an explicit proof of correctness Step 5 in Algorithm 5, we provide it in Theorem 2. Before this, we will need the following technical result. ###### Lemma 5. Let $L_{x}$ be the cycle length of $x\in S$, and $n$, $a$, and $a^{\prime}$ be fixed positive integers. Suppose that $x^{bL_{x}+n}=x^{a}\in G_{x}$, where $b$ is the minimum number such that $x^{bL_{x}+n}\in G_{x}$, and $x^{n-cL_{x}}=x^{a^{\prime}}\in G_{x}$, where $c$ the maximum number such that $x^{n-cL_{x}}\in G_{x}$. Then $bL_{x}+n\leq a,\ \text{and}\ n-cL_{x}\leq a^{\prime}.$ ###### Proof. First let $x^{bL_{x}+n}=x^{a}$ with $b$ minimal such that $x^{bL_{x}+n}\in G_{x}$. Suppose, to the contrary, that $bL_{x}+n>a$. We must have, by the minimality of $b$, $x^{(b-1)L_{x}+n}\not\in G_{x}$, so $(b-1)L_{x}+n<a$. $\displaystyle\text{But},\;\;\;$ $\displaystyle x^{bL_{x}+n}=x^{a}\in G_{x}$ $\displaystyle\implies$ $\displaystyle bL_{x}+n-a=kL_{x},\ k\geq 1$ $\displaystyle\implies$ $\displaystyle(b-k)L_{x}+n=a$ $\displaystyle\implies$ $\displaystyle x^{(b-k)L_{x}+n}=x^{a}\in G_{x},\ k\geq 1.$ This is a contradiction to the minimality of $b$. So, $bL_{x}+n\leq a$. Now suppose that $x^{x-cL_{x}}=x^{a}\in G_{x}$, with $c$ maximal, and suppose that $n-cL_{x}>a^{\prime}$. We argue as above: $\displaystyle L_{x}\mid n-cL_{x}-a^{\prime}$ $\displaystyle\implies$ $\displaystyle n-(k+c)L_{x}=a^{\prime},\ \text{for some}\ k\geq 1$ $\displaystyle\implies$ $\displaystyle x^{n-(k+c)L_{x}}=x^{a^{\prime}}\in G_{x},$ which is a contradiction to the maximality of $c$. Thus $n-cL_{x}\leq a^{\prime}$. ∎ ###### Theorem 2. Let $S$ be a semigroup, $x\in S$ a torsion element and $y\in\langle x\rangle$ any element. Assume the cycle length $L_{x}$ and cycle start $s_{x}$ of $x$ are known. Then Algorithm 5 returns the correct values of the discrete logarithm $m=\log_{x}(y)$ in $\mathcal{O}\left(\sqrt{L_{x}}+(\log N_{x})^{2}\right)$ semigroup multiplications, with a required storage of $\mathcal{O}\left(\sqrt{L_{x}}\right)$ semigroup elements. ###### Proof. We use the notations of Algorithm 5, and also write $n=\log_{x}y$. We will show that the output $m$ is equal to the correct discrete logarithm value $n$. Recall that we have a group $G_{x}$, generated by ${x}^{\prime}:=x^{tL_{x}+1}$, and with identity $x^{tL_{x}}$. The parameter $t$ is given by the formula $t=\left\lceil\frac{s_{x}}{L_{x}}\right\rceil$. Inverses in $G_{x}$ can be computed in polynomial time using the formula (1). Note that membership in $G_{x}$ can be tested with one equality check: $y\in G_{x}\iff y\cdot x^{L_{x}}=y$. There are now two cases: 1. 1. When $y\in G_{x}$, we have $b=0$. Here, it is possible to use Shank’s Baby Step-Giant Step algorithm [18] which is a deterministic algorithm and which requires $\mathcal{O}\left(\sqrt{L_{x}}\right)$ semigroup multiplications and storage space $\mathcal{O}\left(\sqrt{L_{x}}\right)$, in order to compute $\log_{{x}^{\prime}}(y)$. This is done in Step (3). From this value, $n=\log_{x}(y)$ is readily computed, as shown below. Note that in this case, $\log_{x}(y)$ is determined modulo $L_{x}$. 2. 2. When $y\not\in G_{x}$, Algorithm 5 first computes, using binary search, the smallest power $b$ of $x^{L_{x}}$ such that the product $y\cdot x^{bL_{x}}$ lies in the group $G_{x}$, and then proceeds as in case 1 via the Baby Step- Giant Step algorithm to find the discrete logarithm $m^{\prime}$ of $y\cdot x^{bL_{x}}$ with base $x^{\prime}$ (i.e. ${(x^{\prime})}^{m^{\prime}}=y\cdot x^{bL_{x}}$). Note that in this case, the value of $\log_{x}(y)$ is less than $s_{x}$, and is thus determined uniquely in $\mathbb{N}$. Again, the time and space complexity are both $\mathcal{O}\left(\sqrt{L_{x}}\right)$. In both cases above, we have the maximal value $c$ such that $x^{m^{\prime}(tL_{x}+1)-cL_{x}}\in G_{x}$, and so $c\leq L_{x}+s_{x}+1=N_{x}+1$, since $m^{\prime}\leq L_{x}$ and $tL_{x}\leq L_{x}+s_{x}$. We also clearly have $b\leq t\leq N_{x}$. Since the computations of both $b$ and $c$ are done via binary searches, they contribute $\mathcal{O}((\log N_{x})^{2})$ steps to the overall time complexity. Now, $x^{m^{\prime}(tL_{x}+1)-cL_{x}}=x^{m^{\prime}(tL_{x}+1)}=(x^{\prime})^{m^{\prime}}=x^{bL_{x}+n}.$ Applying Lemma 5 to the above equation, we must have $m^{\prime}(tL_{x}+1)-cL_{x}\leq bL_{x}+n,\ \text{and}\ bL_{x}+n\leq m^{\prime}(tL_{x}+1)-cL_{x}.$ Therefore, $bL_{x}+n=m^{\prime}(tL_{x}+1)-cL_{x}$, or $n=m^{\prime}(tL_{x}+1)-(b+c)L_{x},$ which is precisely equal to $m$, the value returned by the Algorithm 5. Thus, $m=n$. This completes the proof. ∎ Combining Theorem 1, Lemma 3 and Theorem 2 we arrive at the main proposition of the paper: ###### Proposition 1. Let $S$ be a semigroup, $x\in S$ a torsion element and $y\in\langle x\rangle$ any element. The discrete logarithm $m=\log_{x}(y)$ can be computed deterministically in $\mathcal{O}\left(\sqrt{N_{x}}\cdot(\log N_{x})^{2}\ \right)$ steps, with a required storage of $\mathcal{O}\left(\sqrt{N_{x}}\right)$ semigroup elements. ###### Proof. For the solution, one begins by finding $L_{x}$. This can be done using Algorithm 4 and according to Theorem 1 this requires $\mathcal{O}\left(\sqrt{N_{x}}\cdot(\log N_{x})^{2}\right)$ steps and the storage of $\mathcal{O}\left(\sqrt{N_{x}}\right)$ elements. By Lemma 3 the computation of the cycle start $s_{x}$ is achieved in $\mathcal{O}((\log N_{x})^{2})$ semigroup multiplications, which does not contribute to the overall cost of the algorithm. By Theorem 2, the discrete logarithm $m$ can then be retrieved using Algorithm 5, in $\mathcal{O}\left((\log N_{x})^{2}+\sqrt{L_{x}}\right)$ steps, with a required storage of $\mathcal{O}\left(\sqrt{L_{x}}\right)$ semigroup elements. As $L_{x}\leq N_{x}$, the overall complexity is dominated by the computation of the cycle length, and the proof of the result is now clear. ∎ ### 4.3 Solving the DLP once the Factorization of the Cycle Length is known We mentioned in the introduction that for a general group of order $N$ the best general known algorithms for solving the discrete logarithm problem have complexity $\mathcal{O}(\sqrt{N})$ operations. In case the order $N$ has a prime factorization into small primes there is the famous Pohlig–Hellman algorithm [15] for solving the DLP whose complexity is dominated by the largest prime factor in the integer factorization of $N$. In case that we have available the integer factorization of the cycle length $L_{x}$ we can adapt the Pohlig–Hellman algorithm for groups to a Pohlig–Hellman algorithm for solving the DLP in a semigroup. Algorithm 6 represents this adapted Pohlig–Hellman algorithm. Input A semigroup $S$, a torsion element $x\in S$, with cycle length $L_{x}=\prod_{i=1}^{r}p_{i}^{e_{i}}$ and cycle start $s_{x}$, and $y\in S$ with $y=x^{m}$ Output The discrete logarithm $m$ of $y$ with base $x$ 1: Compute $t=\left\lceil\frac{s_{x}}{L_{x}}\right\rceil$ and define $x^{\prime}=x^{tL_{x}+1}\in G_{x}$. 2: Find the minimum number $0\leq b\leq t$ such that $y^{\prime}=y\cdot x^{bL_{x}}\in G_{x}$ using binary search. 3: for $i\in\\{1,\ldots,r\\}$ 1. 1. Compute the values $x_{i}^{\prime}=(x^{\prime})^{L_{x}/p_{i}^{e_{i}}}$, $y_{i}^{\prime}=(y^{\prime})^{L_{x}/p_{i}^{e_{i}}}$, and $\gamma_{i}:=(x_{i}^{\prime})^{p^{e_{i}-1}}$. 2. 2. Calculate the inverse $z_{i}$ of ${x_{i}}^{\prime}$ in $G_{x}$ using (1). 3. 3. Set $k\leftarrow 0$ and $n_{0}\leftarrow 0$. 4. 4. while $k<e_{i}$ do 1. (a) Compute ${y_{k}^{\prime}}=(y_{i}^{\prime}z_{i}^{n_{k}})^{p^{e_{i}-1-k}}\in\langle\gamma_{i}\rangle$. 2. (b) Use Shank’s Baby-Step Giant-Step algorithm for the group $\langle\gamma_{i}\rangle\subseteq G_{x}$ to compute $d_{k}\in\\{0,1,\ldots,p_{i}-1\\}$ such that ${\gamma_{i}}^{d_{k}}={y_{k}}^{\prime}$. 3. (c) Set $n_{k+1}\leftarrow n_{k}+p_{i}^{k}d_{k}$, and $k\leftarrow k+1$. 5. 5. end while 6. 6. Set $m_{i}:=n_{e_{i}}$. 4: end for 5: Use the Chinese Remainder Theorem to solve the congruence equations ${\displaystyle m^{\prime}\equiv m_{i}{\pmod{p_{i}^{e_{i}}},}\quad\forall\ i\in\\{1,\dots,r\\}}$ uniquely for $m^{\prime}\mod L_{x}$. This gives the discrete logarithm of $y^{\prime}$ with respect to the base $x^{\prime}$ in the group $G_{x}$. 6: Find the maximum number $c\geq 0$ such that $x^{(tL_{x}+1)m^{\prime}-cL_{x}}\in G_{x}$ using binary search. 7: Return $m=m^{\prime}(tL_{x}+1)-(b+c)L_{x}$. Algorithm 6 Pohlig–Hellman Algorithm for solving the Discrete Logarithm Problem in a Semigroup ###### Theorem 3. Let $S$ be a semigroup, $x\in S$ a torsion element and $y\in\langle x\rangle$ any element. Assume the cycle start $s_{x}$ of $x$ is known and assume the integer factorization of the cycle length $L_{x}$ is known to be $L_{x}=\prod_{i=1}^{r}p_{i}^{e_{i}}$. Then Algorithm 6 computes the discrete logarithm $\log_{x}y$ requiring $\mathcal{O}\left(\sum\limits_{i=1}^{r}e_{i}\left(\log L_{x}+\sqrt{p_{i}}\right)+\left(\log N_{x}\right)^{2}\right)$ steps. The space complexity of the algorithm consists in $\mathcal{O}\left(\sum\limits_{i=1}^{r}e_{i}\sqrt{p_{i}}\right)$ semigroup elements. ###### Proof. Steps 1 and 2 are in analogy to the corresponding steps of Algorithm 5. Steps 3 to 5 represent the Pohlig–Hellman algorithm for groups with the implied complexity dominated by the largest prime factor $p_{i}$ of the integer factorization of $L_{x}$ (for a reference on Pohlig–Hellman in groups, see in [9, Theorem 2.32]). It follows that the running time of the algorithm is $\mathcal{O}\left(\sum\limits_{i=1}^{r}e_{i}\left(\log L_{x}+\sqrt{p_{i}}\right)\right)$ steps. The computation of $b$ and $c$ require in addition $\left(\log N_{x}\right)^{2}$ steps. The total space complexity is $\mathcal{O}\left(\sum\limits_{i=1}^{r}e_{i}\sqrt{p_{i}}\right)$ semigroup elements and that completes the proof. ∎ ## 5 Conclusion The DLP in a finite group has noteworthy significance for cryptography, and so an extension of existing solutions to other algebraic structures like semigroups, where inverses may not be available, is of natural interest. In particular, the DLP in a semigroup has been discussed before in two places, namely [3] and [14]. Both these authors provide probabilistic generalizations of existing collision-based methods for the case of a semigroup. The time complexity of the algorithm in [3] is $\mathcal{O}(\sqrt{N_{x}}(\log N_{x})^{2}\log\log N_{x})$, and the one in [14] is $\mathcal{O}(\sqrt{N_{x}}(\log N_{x})^{2})$. Both these methods rely on computing a multiple of the cycle length and then taking gcd’s or factors, and could fail with a small probability that depends on the parameters chosen. In this paper, we provided a deterministic solution of the semigroup DLP, which computes the cycle length directly and does not rely on finding a factor of it. The time complexity of our algorithm is $\mathcal{O}(\sqrt{N_{x}}(\log N_{x})^{2})$. We further demonstrated the application of the Pohlig-Hellman algorithm to semigroups. A direct consequence of our findings is that for cryptographic purposes, generalizing the type of algebraic structure for the DLP offers no additional advantage, at least in the torsion case, both in a classical and a quantum setting. ## 6 Acknowledgement This research is supported by armasuisse Science and Technology. The second author is also supported by Swiss National Science Foundation grant n. 188430. ## References * [1] L. M. Adleman and J. DeMarrais. A subexponential algorithm for discrete logarithms over all finite fields. Math. Comp., 61(203):1–15, 1993. * [2] M. Agrawal, N. Kayal, and N. Saxena. PRIMES is in P. Ann. of Math. (2), 160(2):781–793, 2004. * [3] M. Banin and B. Tsaban. A reduction of semigroup DLP to classic DLP. Des. Codes Cryptography, 81(1):75–82, October 2016. * [4] A.M. Childs and G. Ivanyos. Quantum computation of discrete logarithms in semigroups. Journal of Mathematical Cryptology, 8(4):405 – 416, 01 Dec. 2014\. * [5] H. Cohen, G. Frey, R. Avanzi, C. 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# Compact groups with a set of positive Haar measure satisfying a nilpotent law Alireza Abdollahi Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, Iran. <EMAIL_ADDRESS>and Meisam Soleimani Malekan Department of Mathematics, University of Isfahan, Isfahan 81746-73441, Iran; Institute for Research in Fundamental Sciences, School of Mathematics, Tehran, Iran. <EMAIL_ADDRESS> ###### Abstract. The following question is proposed in [4, Question 1.20]. Let $G$ be a compact group, and suppose that ${\mathcal{N}}_{k}(G)=\\{(x_{1},\dots,x_{k+1})\in G^{k+1}\;\|\;[x_{1},\dots,x_{k+1}]=1\\}$ has positive Haar measure in $G^{k+1}$. Does $G$ have an open $k$-step nilpotent subgroup? The case $k=1$ is already known. We positively answer it for $k=2$. ###### Key words and phrases: Compact groups, subsets with positive Haar measure, $2$-step nilpotent groups ###### 2010 Mathematics Subject Classification: 20E18; 20P05 ## 1\. Introduction and Results Let $G$ be a (Hausdorff) compact group. Then $G$ has a unique normalized Haar measure denoted by ${\mathbf{m}}_{G}$. The following question is proposed in [4]. ###### Question 1.1. [4, Question 1.20] Let $G$ be a compact group, and suppose that ${\mathcal{N}}_{k}(G)=\\{(x_{1},\dots,x_{k+1})\in G^{k+1}\;\|\;[x_{1},\dots,x_{k+1}]=1\\}$ has positive Haar measure in $G^{k+1}$. Does $G$ have an open $k$-step nilpotent subgroup? Positive answer to Question 1.1 is known for $k=1$ (see [3, Theorem 1.2]). It follows from [4, Theorem 1.19] that Question 1.1 has positive answer for arbitrary $k$ whenever we further assume that $G$ is totally disconnected i.e. $G$ is a profinite group. Here we answer positively Question 1.1 for $k=2$ (see Theorem 3.2 below). ## 2\. A preliminary lemma We need the following lemma in the proof of our main result. ###### Lemma 2.1. Suppose that $G$ is a group and $x_{1},x_{2},x_{3},g_{1},g_{2},g_{3}\in G$ are such that $\displaystyle 1$ $\displaystyle\overset{1}{=}[x_{1},x_{2},x_{3}]\overset{2}{=}[x_{1}g_{1},x_{2}g_{2},x_{3}g_{3}]\overset{3}{=}[x_{1}g_{1},x_{2}g_{2},x_{3}]\overset{4}{=}[x_{1}g_{1},x_{2},x_{3}g_{2}]$ $\displaystyle\overset{5}{=}[x_{1}g_{1},x_{2},x_{3}]\overset{6}{=}[x_{1}g_{1},x_{2},x_{3}g_{3}]\overset{7}{=}[x_{1},x_{2},x_{3}g_{1}]\overset{8}{=}[x_{1},x_{2}g_{2},x_{3}g_{1}]$ $\displaystyle\overset{9}{=}[x_{1},x_{2}g_{2},x_{3}]\overset{10}{=}[x_{1},x_{2},x_{3}g_{2}]\overset{11}{=}[x_{1},x_{2}g_{2},x_{3}g_{3}]\overset{12}{=}[x_{1},x_{2},x_{3}g_{3}].$ Then $[g_{1},g_{2},g_{3}]=1$. ###### Proof. We will throughout using famous commutator calculus identities. $\displaystyle 1$ $\displaystyle=[x_{1}g_{1},x_{2}g_{2},g_{3}]=[[x_{1}g_{1},g_{2}][x_{1}g_{1},x_{2}]^{g_{2}},g_{3}]\;\;\text{\rm by (2) and (3)}$ $\displaystyle=[[x_{1}g_{1},g_{2}][x_{1}g_{1},x_{2}],g_{3}]\;\;\textbf{\rm by (4) and (5)}$ $\displaystyle=[x_{1}g_{1},g_{2},g_{3}]=[[x_{1},g_{2}]^{g_{1}}[g_{1},g_{2}],g_{3}]\;\;\textbf{\rm by (5) and (6). \;\; (I)}$ On the other hand, $\displaystyle 1$ $\displaystyle=[x_{1},x_{2}g_{2},g_{1}]\;\;\text{\rm by (8) and (9)}$ $\displaystyle=[[x_{1},g_{2}][x_{1},x_{2}]^{g_{2}},g_{1}]=[[x_{1},g_{2}][x_{1},x_{2}],g_{1}]\;\;\text{by (1) and (10)}$ $\displaystyle=[x_{1},g_{2},g_{1}]\;\;\text{by (1) and (7). \;\; (II)}$ Also, $\displaystyle 1$ $\displaystyle=[x_{1},x_{2}g_{2},g_{3}]\;\;\text{\rm by (9) and (11)}$ $\displaystyle=[[x_{1},g_{2}][x_{1},x_{2}]^{g_{2}},g_{3}]=[[x_{1},g_{2}][x_{1},x_{2}],g_{3}]\;\;\text{by (1) and (10)}$ $\displaystyle=[x_{1},g_{2},g_{3}]\;\;\text{by (1) and (12). \;\; (III)}$ Now it follows from (I), (II) and (III) that $[g_{1},g_{2},g_{3}]=1$. ∎ ###### Remark 2.2. The “left version” ($g_{i}x_{j}$ instead of $x_{j}g_{i}$) of Lemma 3.2 is not clear to hold. The validity of a similar result to Lemma 3.2 for commutators with length more than $3$ is also under question. ## 3\. Compact groups with many elements satisfying the $2$-step nilpotent law We need the “right version” of [5, Theorem 2.3] as follows. ###### Theorem 3.1. If $A$ is a measurable subset with positive Haar measure in a compact group $G$, then for any positive integer $k$ there exists an open subset $U$ of $G$ containing $1$ such that $\mathbf{m}_{G}(A\cap Au_{1}\cap\cdots\cap Au_{k})>0$ for all $u_{1},\dots,u_{k}\in U$. ###### Proof. Since $\mathbf{m}_{G}(A)=\mathbf{m}_{G}(A^{-1})$, it follows from Theorem 2.3 of [5] that there exists an open subset $V$ of $G$ containing $1$ such that $\mathbf{m}_{G}(A^{-1}\cap v_{1}A^{-1}\cap\cdots\cap v_{k}A^{-1})>0$ for all $v_{1},\dots,v_{k}\in V$. By [2, Theorem 4.5], there exists an open subset $U\subseteq V$ such that $1\in U$ and $U=U^{-1}$. Thus for all $u_{1},\dots,u_{k}\in U$ $\displaystyle 0<$ $\displaystyle\mathbf{m}_{G}(A^{-1}\cap u_{1}^{-1}A^{-1}\cap\cdots\cap u_{k}^{-1}A^{-1})$ $\displaystyle=$ $\displaystyle\mathbf{m}_{G}((A\cap Au_{1}\cap\cdots\cap Au_{k})^{-1})$ $\displaystyle=$ $\displaystyle\mathbf{m}_{G}(A\cap Au_{1}\cap\cdots\cap Au_{k})$ This completes the proof. ∎ Now we can prove our main result. ###### Theorem 3.2. Let $G$ be a compact group, and suppose that ${\mathcal{N}}_{2}(G)=\\{(x_{1},x_{2},x_{3})\in G\times G\times G\;\|\;[x_{1},x_{2},x_{3}]=1\\}$ has positive Haar measure in $G\times G\times G$. Then $G$ has an open $2$-step nilpotent subgroup. ###### Proof. Let $X:={\mathcal{N}}_{2}(G)$. It follows from Theorem 3.1 and [2, Theorem 4.5] that there exists an open subset $U=U^{-1}$ of $G$ containing $1$ such that $X\cap X\bar{u}_{1}\cap\cdots\cap X\bar{u}_{11}\neq\varnothing\;\;\;()$ for all $\bar{u}_{1},\dots,\bar{u}_{11}\in U\times U\times U$. Now take arbitrary elements $g_{1},g_{2},g_{3}\in U$ and consider $\displaystyle\bar{u}_{1}$ $\displaystyle=(g_{1}^{-1},g_{2}^{-1},g_{3}^{-1}),\bar{u}_{2}=(g_{1}^{-1},g_{2}^{-1},1),\bar{u}_{3}=(g_{1}^{-1},1,g_{2}^{-1})$ $\displaystyle\bar{u}_{4}$ $\displaystyle=(g_{1}^{-1},1,1),\bar{u}_{5}=(g_{1}^{-1},1,g_{3}^{-1}),\bar{u}_{6}=(1,1,g_{1}^{-1}),\bar{u}_{7}=(1,g_{2}^{-1},g_{1}^{-1})$ $\displaystyle\bar{u}_{8}$ $\displaystyle=(1,g_{2}^{-1},1),\bar{u}_{9}=(1,1,g_{2}^{-1}),\bar{u}_{10}=(1,g_{2}^{-1},g_{3}^{-1}),\bar{u}_{11}=(1,1,g_{3}^{-1}).$ By $()$, there exists $(x_{1},x_{2},x_{3})\in X$ such that all the following 3-tuples are in $X$. $\displaystyle(x_{1}g_{1},x_{2}g_{2},x_{3}g_{3}),(x_{1}g_{1},x_{2}g_{2},x_{3}),(x_{1}g_{1},x_{2},x_{3}g_{2})$ $\displaystyle(x_{1}g_{1},x_{2},x_{3}),(x_{1}g_{1},x_{2},x_{3}g_{3}),(x_{1},x_{2},x_{3}g_{1}),(x_{1},x_{2}g_{2},x_{3}g_{1})$ $\displaystyle(x_{1},x_{2}g_{2},x_{3}),(x_{1},x_{2},x_{3}g_{2}),(x_{1},x_{2}g_{2},x_{3}g_{3}),(x_{1},x_{2},x_{3}g_{3}).$ Now Lemma 2.1 implies that $\langle g_{1},g_{2},g_{3}\rangle$ is nilpotent of class at most $2$. Therefore the subgroup $H:=\langle U\rangle$ generated by $U$ is $2$-step nilpotent. Since $H=\bigcup_{n\in{\mathbb{N}}}U^{n}$, $H$ is open in $G$. This completes the proof. ∎ ## References * [1] G. B. Folland, A Course in Abstract Harmonic Analysis, Stud. Adv. Math., Taylor & Francis, London, 1994. * [2] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis: Structure and Analysis for Compact Groups Analysis on Locally Compact Abelian Groups, Grundlehren Math. Wiss., Springer, Berlin, 2013. * [3] K. H. Hofmann and F. G. Russo, The probability that $x$ and $y$ commute in a compact group, Math. Proc. Cambridge Philos. Soc. 153 (2012), no. 3, 557-571. * [4] A. Martino, M. C. H. Tointon, M. Valiunas and E. Ventura, Probabilistic Nilpotence in infinite groups, to appear in Israel J. Math. * [5] M. Soleimani Malekan, A. Abdollahi and M. Ebrahimi, Compact groups with many elements of bounded order, J. Group Theory, 23 (2020) no. 6 991–998.
††thanks: This work has been partially funded by the Ministry of Science, Innovation and Universities of Spain through the grant PGC2018-095998-B-I00 and by the Agency for Management of University and Research Grants of Catalonia through the grant 2017SGR1725. # Geometry of certain foliations on the complex projective plane Samir Bedrouni Faculté de Mathématiques, USTHB, BP $32$, El-Alia, $16111$ Bab- Ezzouar, Alger, Algérie<EMAIL_ADDRESS><EMAIL_ADDRESS>David Marín Departament de Matemàtiques, Edifici Cc, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès (Barcelona), Spain. Centre de Recerca Matemàtica, Edifici Cc, Campus de Bellaterra, 08193 Cerdanyola del Vallès (Barcelona), Spain<EMAIL_ADDRESS> (August 27, 2024) ###### Abstract Let $d\geq 2$ be an integer. The set $\mathbf{F}(d)$ of foliations of degree $d$ on the complex projective plane can be identified with a Zariski’s open set of a projective space of dimension $d^{2}+4d+2$ on which $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$ acts. We show that there are exactly two orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ of minimal dimension $6$, necessarily closed in $\mathbf{F}(d)$. This generalizes known results in degrees $2$ and $3.$ We deduce that an orbit $\mathcal{O}(\mathcal{F})$ of an element $\mathcal{F}\in\mathbf{F}(d)$ of dimension $7$ is closed in $\mathbf{F}(d)$ if and only if $\mathcal{F}_{i}^{d}\not\in\overline{\mathcal{O}(\mathcal{F})}$ for $i=1,2.$ This allows us to show that in any degree $d\geq 3$ there are closed orbits in $\mathbf{F}(d)$ other than the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d}),$ unlike the situation in degree $2.$ On the other hand, we introduce the notion of the basin of attraction $\mathbf{B}(\mathcal{F})$ of a foliation $\mathcal{F}\in\mathbf{F}(d)$ as the set of $\mathcal{G}\in\mathbf{F}(d)$ such that $\mathcal{F}\in\overline{\mathcal{O}(\mathcal{G})}.$ We show that the basin of attraction $\mathbf{B}(\mathcal{F}_{1}^{d})$, resp. $\mathbf{B}(\mathcal{F}_{2}^{d})$, contains a quasi-projective subvariety of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-1)$, resp. $\dim\mathbf{F}(d)-(d-3)$. In particular, we obtain that the basin $\mathbf{B}(\mathcal{F}_{2}^{3})$ contains a non-empty Zariski open subset of $\mathbf{F}(3)$. This is an analog in degree $3$ of a result on foliations of degree $2$ due to Cerveau, Déserti, Garba Belko and Meziani. 2010 Mathematics Subject Classification. — 37F75, 32S65, 32M25, 32M05. ###### keywords: foliation, singularity, inflection point, orbit, isotropy group ## Introduction The set $\mathbf{F}(d)$ of holomorphic foliations of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is identified with a Zariski open subset of the projective space $\mathbb{P}_{\mathbb{C}}^{\hskip 0.56905ptd^{2}+4d+2}$. We are interested here in the action of the group $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})=\mathrm{PGL}_{3}(\mathbb{C})$ on $\mathbf{F}(d).$ We generalize to arbitrary degree some results known in small degrees [9, 1, 5] on this action. For $\mathcal{F}\in\mathbf{F}(d)$, we will respectively denote by $\mathcal{O}(\mathcal{F})$ and $\mathrm{Iso}(\mathcal{F})$ the orbit and the isotropy group of $\mathcal{F}$ under the action of $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}}),$ i.e. $\displaystyle\mathcal{O}(\mathcal{F}):=\\{\varphi^{}\mathcal{F}\in\mathbf{F}(d)\hskip 2.84526pt\|\hskip 2.84526pt\varphi\in\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})\\}$ and $\displaystyle\mathrm{Iso}(\mathcal{F}):=\\{\varphi\in\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})\hskip 2.84526pt\|\hskip 2.84526pt\varphi^{}\mathcal{F}=\mathcal{F}\\}.$ $\mathcal{O}(\mathcal{F})$ is a Zariski irreducible subset of $\mathbf{F}(d)$ and $\mathrm{Iso}(\mathcal{F})$ is an algebraic subgroup of $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}}).$ Following [15] we will say that a foliation of $\mathbf{F}(d)$ is convex if its leaves other than straight lines have no inflection points. We will denote by $\mathbf{FC}(d)$ the subset of $\mathbf{F}(d)$ consisting of convex foliations, which is a Zariski closed subset of $\mathbf{F}(d).$ According to [7, Proposition 2.2] every foliation of degree $0$ or $1$ is convex. For $d\geq 2$, $\mathbf{FC}(d)$ is a proper closed subset of $\mathbf{F}(d)$ and it contains the foliation $\mathcal{F}_{1}^{d}$ defined in the affine chart $(x,y)$ by the $1$-form (see [3, page 75]) $\omega_{1}^{d}=y^{d}\mathrm{d}x+x^{d}(x\mathrm{d}y-y\mathrm{d}x).$ We know from [9, Proposition 2.3] that if $\mathcal{F}$ is an element of $\mathbf{F}(d)$ with $d\geq 2,$ then the dimension of $\mathcal{O}(\mathcal{F})$ is at least $6,$ or equivalently, the dimension of $\mathrm{Iso}(\mathcal{F})$ is at most $2$. In addition these bounds are attained by the convex foliation $\mathcal{F}_{1}^{d}$ and the non convex foliation $\mathcal{F}_{2}^{d}$ defined by the $1$-form (see [3]) $\omega_{2}^{d}=x^{d}\mathrm{d}x+y^{d}(x\mathrm{d}y-y\mathrm{d}x).$ The main result of this paper is the following. ###### Theorem A. Let $d$ be an integer greater than or equal to $2$ and let $\mathcal{F}$ be an element of $\mathbf{F}(d).$ Assume that the isotropy group $\mathrm{Iso}(\mathcal{F})$ of $\mathcal{F}$ has dimension $2$. Then $\mathcal{F}$ is linearly conjugated to one of the two foliations $\mathcal{F}_{1}^{d}$ and $\mathcal{F}_{2}^{d}$ defined respectively by the $1$-forms * 1\. $\omega_{1}^{d}=y^{d}\mathrm{d}x+x^{d}(x\mathrm{d}y-y\mathrm{d}x);$ * 2\. $\omega_{2}^{d}=x^{d}\mathrm{d}x+y^{d}(x\mathrm{d}y-y\mathrm{d}x).$ In other words, $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ are the only orbits of dimension $6$. They are closed in $\mathbf{F}(d).$ Moreover we have $\displaystyle\mathrm{Iso}(\mathcal{F}_{1}^{d})=\left\\{\left(\frac{\alpha^{d-1}x}{1+\beta x},\frac{\alpha^{d}y}{1+\beta x}\right)\hskip 2.84526pt\Big{\|}\hskip 2.84526pt\alpha\in\mathbb{C}^{},\hskip 2.84526pt\beta\in\mathbb{C}\right\\},$ $\displaystyle\mathrm{Iso}(\mathcal{F}_{2}^{d})=\left\\{\left(\frac{\alpha^{d+1}x}{1+\beta x},\frac{\alpha^{d}y}{1+\beta x}\right)\hskip 2.84526pt\Big{\|}\hskip 2.84526pt\alpha\in\mathbb{C}^{},\hskip 2.84526pt\beta\in\mathbb{C}\right\\};$ these two groups are not conjugated. This theorem is a generalization in arbitrary degree of previous results on foliations of degrees $d=2$ ([9, Proposition 2.7]) and $d=3$ ([1, Theorem 10], [5, Corollary B]). We also obtain the following corollary, which generalizes [5, Corollary 3.9]: ###### Corollary B. Let $d$ be an integer greater than or equal to $2$ and let $\mathcal{F}$ be an element of $\mathbf{F}(d).$ If $\dim\mathcal{O}(\mathcal{F})\leq\leavevmode\nobreak\ 7$, then $\overline{\mathcal{O}(\mathcal{F})}\subset\mathcal{O}(\mathcal{F})\cup\mathcal{O}(\mathcal{F}_{1}^{d})\cup\mathcal{O}(\mathcal{F}_{2}^{d}).$ In particular, when $\dim\mathcal{O}(\mathcal{F})=7,$ the orbit $\mathcal{O}(\mathcal{F})$ of $\mathcal{F}$ is closed in $\mathbf{F}(d)$ if and only if $\mathcal{F}_{i}^{d}\not\in\overline{\mathcal{O}(\mathcal{F})}$ for $i=1,2.$ In the spirit of Corollary B we can ask under what condition the closure in $\mathbf{F}(d)$ of the orbit $\mathcal{O}(\mathcal{F})$ of an element $\mathcal{F}$ of $\mathbf{F}(d)$ contains the foliations $\mathcal{F}_{1}^{d}$ and $\mathcal{F}_{2}^{d}$, a question that we have already asked and studied in degree $3$ in [5, Section 3]. In Section §3, we extend (Propositions 3.4 and 3.11) in arbitrary degree $d$ our previous results in [5, Propositions 3.10, 3.12, 3.15, 3.17] concerning this question. For $\mathcal{F}\in\mathbf{F}(d),$ we call basin of attraction of $\mathcal{F}$ the subset $\mathbf{B}(\mathcal{F})$ of $\mathbf{F}(d)$ defined by $\displaystyle\mathbf{B}(\mathcal{F}):=\\{\mathcal{G}\in\mathbf{F}(d)\hskip 2.84526pt\|\hskip 2.84526pt\mathcal{F}\in\overline{\mathcal{O}(\mathcal{G})}\\}.$ It follows from [9, Theorem 2.15] that in degree $2$ the basin $\mathbf{B}(\mathcal{F}_{1}^{2})$ contains a quasi-projective subvariety of $\mathbf{F}(2)$ of dimension greater than or equal to $\dim\mathbf{F}(2)-1.$ In Section §3, we establish an analogous result in any degree greater than $2.$ ###### Theorem C (Theorem 3.10). For any integer $d\geq 2$, the basin of attraction $\mathbf{B}(\mathcal{F}_{1}^{d})$ of $\mathcal{F}_{1}^{d}$ contains a quasi- projective subvariety of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-1).$ Notice that the non-convexity of $\mathcal{F}_{2}^{d}$ and the fact that $\mathbf{FC}(d)$ is closed in $\mathbf{F}(d)$ imply that (0.1) $\displaystyle\mathbf{B}(\mathcal{F}_{2}^{d})\subset\mathbf{F}(d)\setminus\mathbf{FC}(d).$ In degree $2$, according to [9, Theorem 3], inclusion (0.1) is an equality: (0.2) $\displaystyle\mathbf{B}(\mathcal{F}_{2}^{2})=\mathbf{F}(2)\setminus\mathbf{FC}(2).$ It follows in particular from equality (0.2) that the basin $\mathbf{B}(\mathcal{F}_{2}^{2})$ is a Zariski open subset of $\mathbf{F}(2).$ For $d\geq 3$ we show the following result. ###### Theorem D (Theorem 3.18). In any degree $d\geq 3$, the basin of attraction $\mathbf{B}(\mathcal{F}_{2}^{d})$ of $\mathcal{F}_{2}^{d}$ contains a quasi- projective subvariety of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-3)$. In particular, the basin $\mathbf{B}(\mathcal{F}_{2}^{3})$ contains a non-empty Zariski open subset of $\mathbf{F}(3).$ Along the same order of ideas, we prove the following result. ###### Theorem E (Theorem 3.21). For any integer $d\geq 2$, the intersection $\mathbf{B}(\mathcal{F}_{1}^{d})\cap\mathbf{B}(\mathcal{F}_{2}^{d})$ is non- empty and it contains a quasi-projective subvariety of $\mathbf{F}(d)$ of dimension equal to $\dim\mathbf{F}(d)-3d.$ By combining equality (0.2) with the classification of C. Favre and J. V. Pereira of convex foliations of degree two (_cf._ [10, Proposition 7.4] or [6, Theorem A]), we see that the only closed orbits in $\mathbf{F}(2)$ under the action of $\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})$ are those of $\mathcal{F}_{1}^{2}$ and $\mathcal{F}_{2}^{2}.$ We show in Section §4 that in any degree $d\geq 3$ there are closed orbits in $\mathbf{F}(d)$ other than the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d}),$ unlike the situation in degree $2.$ More precisely, we will consider a family of elements of $\mathbf{F}(d)$ which has been already studied in degree $d=2$ in [9, page 189], namely the family $(\mathcal{F}_{0}^{d}(\lambda))_{\lambda\in\mathbb{C}^{}}$ of foliations of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ defined by the $1$-form $\omega_{0}^{d}(\lambda)=x\mathrm{d}y-\lambda y\mathrm{d}x+y^{d}\mathrm{d}y.$ We will see that, for $\lambda=1,$ $\mathcal{F}_{0}^{d}(1)$ is linearly conjugated to the foliation $\mathcal{F}_{1}^{d}$ and that, for any $\lambda\neq 1,$ $\dim\mathcal{O}\big{(}\mathcal{F}_{0}^{d}(\lambda)\big{)}=7.$ Moreover, we will show (Proposition 4.2) that the orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{d}(\lambda)\big{)}$ is closed for any $d\geq 3$ and $\lambda=-\frac{1}{d-1}$, resp. for any $d\in\\{3,4,5\\}$ and any $\lambda\in\mathbb{C}^{},$ and we conjecture that it is so for any $d\geq 6$ and any $\lambda\in\mathbb{C}^{}$ (see Conjectures 1 and 2). ## 1 Some definitions and notations ### 1.1 Singularities and local invariants A degree $d$ holomorphic foliation $\mathcal{F}$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is defined in homogeneous coordinates $[x:y:z]$ by a $1$-form $\omega=a(x,y,z)\mathrm{d}x+b(x,y,z)\mathrm{d}y+c(x,y,z)\mathrm{d}z,$ where $a,$ $b$ and $c$ are homogeneous polynomials of degree $d+1$ without common factor and satisfying the Euler condition $i_{\mathrm{R}}\omega=0$, where $\mathrm{R}=x\frac{\partial{}}{\partial{x}}+y\frac{\partial{}}{\partial{y}}+z\frac{\partial{}}{\partial{z}}$ denotes the radial vector field and $i_{\mathrm{R}}$ is the interior product by $\mathrm{R}$. Dually the foliation $\mathcal{F}$ can also be defined by a homogeneous vector field $\hskip 14.22636pt\mathrm{Z}=U(x,y,z)\frac{\partial{}}{\partial{x}}+V(x,y,z)\frac{\partial{}}{\partial{y}}+W(x,y,z)\frac{\partial{}}{\partial{z}},$ the coefficients $U,V$ and $W$ are homogeneous polynomials of degree $d$ without common factor. The relation between $\mathrm{Z}$ and $\omega$ is given by $\omega=i_{\mathrm{R}}i_{\mathrm{Z}}(\mathrm{d}x\wedge\mathrm{d}y\wedge\mathrm{d}z).$ The singular locus $\mathrm{Sing}\mathcal{F}$ of $\mathcal{F}$ is the projectivization of the singular locus of $\omega$ $\mathrm{Sing}\,\omega=\\{(x,y,z)\in\mathbb{C}^{3}\,\|\,a(x,y,z)=b(x,y,z)=c(x,y,z)=0\\}.$ Let $\mathcal{C}\subset\mathbb{P}^{2}_{\mathbb{C}}$ be an algebraic curve with homogeneous equation $F(x,y,z)=0.$ We say that $\mathcal{C}$ is an invariant curve by $\mathcal{F}$ if $\mathcal{C}\smallsetminus\mathrm{Sing}\mathcal{F}$ is a union of (ordinary) leaves of the regular foliation $\mathcal{F}\|_{\mathbb{P}^{2}_{\mathbb{C}}\smallsetminus\mathrm{Sing}\mathcal{F}}$. In algebraic terms, this is equivalent to require that the $2$-form $\omega\wedge\mathrm{d}F$ is divisible by $F$, i.e. it vanishes along each irreducible component of $\mathcal{C}.$ Let $p$ be an arbitrary point of $\mathcal{C}.$ When each irreducible component of $\mathcal{C}$ passing through $p$ is not $\mathcal{F}$-invariant, we define the tangency order $\mathrm{Tang}(\mathcal{F},\mathcal{C},p)$ of $\mathcal{F}$ with $\mathcal{C}$ at $p$ as follows. We fix a local chart $(\mathrm{u},\mathrm{v})$ such that $p=(0,0)$; let $f(\mathrm{u},\mathrm{v})=0$ be a reduced local equation of $\mathcal{C}$ in a neighborhood of $p$ and let $\mathrm{X}$ be a vector field defining the germ of $\mathcal{F}$ at $p.$ We denote by $\mathrm{X}(f)$ the Lie derivative of $f$ along $\mathrm{X}$ and by $\langle f,\mathrm{X}(f)\rangle$ the ideal of $\mathbb{C}\\{\mathrm{u},\mathrm{v}\\}$ generated by $f$ and $\mathrm{X}(f)$. Then $\mathrm{Tang}(\mathcal{F},\mathcal{C},p)=\dim_{\mathbb{C}}\frac{\mathbb{C}\\{\mathrm{u},\mathrm{v}\\}}{\langle f,\mathrm{X}(f)\rangle}.$ Notice that $\mathrm{Tang}(\mathcal{F},\mathcal{C},p)$ coincides with the intersection multiplicity $(\mathcal{C}.\mathcal{C}^{\prime})_{p}$ at $p$ of the two algebraic curves $\mathcal{C}=\\{F=0\\}$ and $\mathcal{C}^{\prime}=\\{\mathrm{Z}(F)=0\\}.$ Moreover, $\mathrm{Tang}(\mathcal{F},\mathcal{C},p)<+\infty$ by the non-invariance of the irreducible components of $\mathcal{C}$ passing through $p.$ By convention, we put $\mathrm{Tang}(\mathcal{F},\mathcal{C},p)=+\infty$ if there is at least one irreducible component of $\mathcal{C}$ invariant by $\mathcal{F}$ and passing through $p.$ Let us recall some local notions attached to the pair $(\mathcal{F},s)$, where $s\in\mathrm{Sing}\mathcal{F}$. The germ of $\mathcal{F}$ at $s$ is defined, up to multiplication by a unity in the local ring $\mathcal{O}_{s}$ at $s$, by a vector field $\mathrm{X}=A(\mathrm{u},\mathrm{v})\frac{\partial{}}{\partial{\mathrm{u}}}+B(\mathrm{u},\mathrm{v})\frac{\partial{}}{\partial{\mathrm{v}}}$. The algebraic multiplicity $\nu(\mathcal{F},s)$ of $\mathcal{F}$ at $s$ is given by $\nu(\mathcal{F},s)=\min\\{\nu(A,s),\nu(B,s)\\},$ where $\nu(g,s)$ denotes the algebraic multiplicity of the function $g$ at $s.$ Let us denote by $\mathfrak{L}_{s}(\mathcal{F})$ the family of straight lines through $s$ which are not invariant by $\mathcal{F}.$ For any line $\ell$ of $\mathfrak{L}_{s}(\mathcal{F}),$ we have the inequalities $1\leq\mathrm{Tang}(\mathcal{F},\ell_{s},s)\leq d$. This allows us to associate to the pair $(\mathcal{F},s)$ the following (invariant) integers $\displaystyle\hskip 22.76228pt\tau(\mathcal{F},s)=\min\\{\mathrm{Tang}(\mathcal{F},\ell,s)\hskip 2.84526pt\|\hskip 2.84526pt\ell\in\mathfrak{L}_{s}(\mathcal{F})\\},$ $\displaystyle\hskip 14.22636pt\kappa(\mathcal{F},s)=\max\\{\mathrm{Tang}(\mathcal{F},\ell,s)\hskip 2.84526pt\|\hskip 2.84526pt\ell\in\mathfrak{L}_{s}(\mathcal{F})\\}.$ The invariant $\tau(\mathcal{F},s)$ represents the tangency order of $\mathcal{F}$ with a generic line passing through $s.$ It is easy to see that $\tau(\mathcal{F},s)=\min\\{k\geq 1\hskip 2.84526pt\|\hskip 2.84526pt\det(J^{k}_{s}\,\mathrm{X},\mathrm{R}_{s})\not\equiv 0\\}\geq\nu(\mathcal{F},s),$ where $J^{k}_{s}\,\mathrm{X}$ denotes the $k$-jet of $\mathrm{X}$ at $s$ and $\mathrm{R}_{s}$ is the radial vector field centered at $s.$ The Milnor number of $\mathcal{F}$ at $s$ is the integer $\mu(\mathcal{F},s)=\dim_{\mathbb{C}}\mathcal{O}_{s}/\langle A,B\rangle,$ where $\langle A,B\rangle$ denotes the ideal of $\mathcal{O}_{s}$ generated by $A$ and $B$. The singularity $s$ is called radial of order $n-1$ if $\nu(\mathcal{F},s)=1$ and $\tau(\mathcal{F},s)=n.$ The singularity $s$ is called non-degenerate if $\mu(\mathcal{F},s)=1,$ or equivalently if the Jacobian matrix of $\mathrm{X}$ at $s$, denoted by $\mathrm{Jac}\hskip 0.28453pt\mathrm{X}(s),$ possesses two nonzero eigenvalues $\lambda,\mu.$ In this case, the quantity $\displaystyle\mathrm{BB}(\mathcal{F},s)=\frac{\mathrm{tr}^{2}(\mathrm{Jac}\hskip 0.28453pt\mathrm{X}(s))}{\det(\mathrm{Jac}\hskip 0.28453pt\mathrm{X}(s))}=\frac{\lambda}{\mu}+\frac{\mu}{\lambda}+2$ is called the Baum-Bott index of $\mathcal{F}$ at $s,$ see [2]. We will say that the singularity $s$ is quasi-radial of order $n-1$ if $\mu(\mathcal{F},s)=1,$ $\mathrm{BB}(\mathcal{F},s)=4$ and $\kappa(\mathcal{F},s)=n.$ In the sequel we will denote by $\mathrm{QRad}(\mathcal{F},n-1)$ the set of quasi-radial singularities of $\mathcal{F}$ of order $n-1$ and by $\widehat{\mathrm{QRad}}(\mathcal{F},n-1)$ the subset of $\mathrm{Sing}(\mathcal{F})\times\mathfrak{L}_{s}(\mathcal{F})$ defined by $\displaystyle\widehat{\mathrm{QRad}}(\mathcal{F},n-1):=\Big{\\{}(s,\ell)\in\mathrm{Sing}(\mathcal{F})\times\mathfrak{L}_{s}(\mathcal{F})\hskip 2.84526pt\|\hskip 2.84526pt\mu(\mathcal{F},s)=1,\hskip 2.84526pt\mathrm{BB}(\mathcal{F},s)=4,\hskip 2.84526pt\mathrm{Tang}(\mathcal{F},\ell,s)=n\Big{\\}}.$ ###### Remark 1.1. Every radial singularity of order $n-1$ of a foliation $\mathcal{F}$ of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}$ is quasi-radial of order $n-1.$ The converse is false: for instance, for the foliation defined in the affine chart $z=1$ by the $1$-form $(x+y)\mathrm{d}y-y\mathrm{d}x+(x^{n}+y^{d})\mathrm{d}x,$ with $n\in\\{2,3,\ldots,d\\},$ the point $[0:0:1]$ is a quasi-radial singularity of order $n-1,$ but it is not radial. ### 1.2 Inflection points Let us consider a foliation $\mathcal{F}$ of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ and let $p$ be a regular point of $\mathcal{F}$. Let us denote by $\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F}$ the tangent line to the leaf of $\mathcal{F}$ passing through $p$; it is the straight line of $\mathbb{P}^{2}_{\mathbb{C}}$ passing through $p$ with direction $\mathrm{T}_{p}\mathcal{F}.$ If $k\in\\{2,\ldots,d\\},$ we will say that $p$ is a (transverse) inflection point of order $k-1$ of $\mathcal{F}$ if $\mathrm{Tang}(\mathcal{F},\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F},p)=k,$ in which case the line $\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F}$ is not invariant by $\mathcal{F}.$ When $\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F}$ is $\mathcal{F}$-invariant, the point $p$ will be called a trivial inflection point of $\mathcal{F}$. If we denote by $\mathrm{Inv}(\mathcal{F})$ the set of invariant lines of $\mathcal{F},$ then the set of trivial inflection points of $\mathcal{F}$ is precisely $\mathrm{Inv}(\mathcal{F})\setminus\mathrm{Sing}(\mathcal{F}).$ In the sequel, we will denote by $\mathrm{Flex}(\mathcal{F})$ the set of inflection points of $\mathcal{F}$ and by $\mathrm{Flex}(\mathcal{F},k-1)$ the subset of $\mathrm{Flex}(\mathcal{F})$ consisting of transverse inflection points of $\mathcal{F}$ of order $k-1,$ i.e. $\displaystyle\mathrm{Flex}(\mathcal{F},k-1):=\Big{\\{}p\in\mathbb{P}^{2}_{\mathbb{C}}\hskip 2.84526pt\|\hskip 2.84526ptp\not\in\mathrm{Sing}(\mathcal{F}),\hskip 2.84526pt\mathrm{Tang}(\mathcal{F},\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F},p)=k\Big{\\}}.$ Let us recall the notion of inflection divisor of $\mathcal{F}$, introduced by Pereira [16], which allows to determine the set $\mathrm{Flex}(\mathcal{F}).$ Let $\mathrm{Z}$ be a homogeneous vector field of degree $d$ on $\mathbb{C}^{3}$ defining $\mathcal{F}.$ The inflection divisor of $\mathcal{F}$, denoted by $\mathrm{I}_{\mathcal{F}}$, is the divisor of $\mathbb{P}^{2}_{\mathbb{C}}$ defined by the homogeneous equation (1.1) $\left\|\begin{array}[]{ccc}x&\mathrm{Z}(x)&\mathrm{Z}^{2}(x)\\\ y&\mathrm{Z}(y)&\mathrm{Z}^{2}(y)\\\ z&\mathrm{Z}(z)&\mathrm{Z}^{2}(z)\end{array}\right\|=0.$ According to [16], $\mathrm{I}_{\mathcal{F}}$ satisfies the following properties: 1. 1. The support of $\mathrm{I}_{\mathcal{F}}$ is exactly the closure of the set $\mathrm{Flex}(\mathcal{F})$ of inflection points of $\mathcal{F}.$ More precisely, $\mathrm{I}_{\mathcal{F}}$ can be decomposed as $\mathrm{I}_{\mathcal{F}}=\mathrm{I}_{\mathcal{F}}^{\mathrm{inv}}+\mathrm{I}_{\mathcal{F}}^{\hskip 0.56905pt\mathrm{tr}},$ where the support of $\mathrm{I}_{\mathcal{F}}^{\mathrm{inv}}$ is the set $\mathrm{Inv}(\mathcal{F})$ of $\mathcal{F}$-invariant lines and the support of $\mathrm{I}_{\mathcal{F}}^{\hskip 0.56905pt\mathrm{tr}}$ is the closure of the set of transverse inflection points of $\mathcal{F}.$ 2. 2. If $\mathcal{C}$ is an algebraic curve invariant by $\mathcal{F},$ then $\mathcal{C}\subset\mathrm{I}_{\mathcal{F}}$ if and only if $\mathcal{C}\subset\mathrm{Inv}(\mathcal{F}).$ 3. 3. The degree of the divisor $\mathrm{I}_{\mathcal{F}}$ is $3d.$ The foliation $\mathcal{F}$ will be called convex if its inflection divisor $\mathrm{I}_{\mathcal{F}}$ is totally invariant by $\mathcal{F}$, i.e. if $\mathrm{I}_{\mathcal{F}}$ is a product of invariant lines. ## 2 Description of the foliations $\mathcal{F}$ of degree greater than or equal to $2$ such that $\dim\mathcal{O}(\mathcal{F})=6$ Recall that the foliations $\mathcal{F}_{1}^{d}$ and $\mathcal{F}_{2}^{d}$ are respectively defined in the affine chart $z=1$ by the $1$-forms $\displaystyle\omega_{1}^{d}=y^{d}\mathrm{d}x+x^{d}(x\mathrm{d}y-y\mathrm{d}x)$ and $\displaystyle\omega_{2}^{d}=x^{d}\mathrm{d}x+y^{d}(x\mathrm{d}y-y\mathrm{d}x).$ The foliation $\mathcal{F}_{1}^{d}$ is convex with inflection divisor $\mathrm{I}_{\mathcal{F}_{1}^{d}}=\mathrm{I}_{\mathcal{F}_{1}^{d}}^{\hskip 0.56905pt\mathrm{inv}}=x^{d+1}y^{2d-1}$ and it has two singular points $s_{1}=[0:0:1]$ and $s_{2}=[0:1:0]$; the singularity $s_{1}$ has maximal algebraic multiplicity $d$ and $s_{2}$ is radial of maximal order $d-1.$ The foliation $\mathcal{F}_{2}^{d}$ is not convex with invariant inflection divisor $\mathrm{I}_{\mathcal{F}_{2}^{d}}^{\hskip 0.56905pt\mathrm{inv}}=x^{2d+1}$ and transverse inflection divisor $\mathrm{I}_{\mathcal{F}_{2}^{d}}^{\hskip 0.56905pt\mathrm{tr}}=y^{d-1}.$ The singular locus $\mathrm{Sing}(\mathcal{F}_{2}^{d})$ is reduced to the point $s_{1}=[0:0:1]$; moreover $\nu(\mathcal{F}_{2}^{d},s_{1})=d.$ We note that the $1$-forms $\dfrac{\omega_{1}^{d}}{x^{2}y^{d}}$ and $\dfrac{\omega_{2}^{d}}{x^{d+2}}$ are closed and they respectively admit as first integrals $\displaystyle\frac{1}{d-1}\left(\frac{x}{y}\right)^{d-1}+\frac{1}{x}$ and $\displaystyle\frac{1}{d+1}\left(\frac{y}{x}\right)^{d+1}-\frac{1}{x};$ this allows to check that $\displaystyle\mathrm{Iso}(\mathcal{F}_{1}^{d})=\left\\{\left(\frac{\alpha^{d-1}x}{1+\beta x},\frac{\alpha^{d}y}{1+\beta x}\right)\hskip 2.84526pt\Big{\|}\hskip 2.84526pt\alpha\in\mathbb{C}^{},\hskip 2.84526pt\beta\in\mathbb{C}\right\\}$ and $\displaystyle\mathrm{Iso}(\mathcal{F}_{2}^{d})=\left\\{\left(\frac{\alpha^{d+1}x}{1+\beta x},\frac{\alpha^{d}y}{1+\beta x}\right)\hskip 2.84526pt\Big{\|}\hskip 2.84526pt\alpha\in\mathbb{C}^{},\hskip 2.84526pt\beta\in\mathbb{C}\right\\}.$ In particular, $\dim\mathrm{Iso}(\mathcal{F}_{i}^{d})=2$ for $i=1,2.$ Thus the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ are both of dimension $6,$ which is the minimal dimension possible in any degree $d\geq 2$ ([9, Proposition 2.3]). Theorem A announced in the Introduction shows that the orbits $\mathcal{O}(\mathcal{F}_{1}^{d})$ and $\mathcal{O}(\mathcal{F}_{2}^{d})$ are the only orbits having minimal dimension $6.$ The goal of this section is to prove this theorem. Let us denote by $\chi(\mathbb{P}^{2}_{\mathbb{C}})$ the Lie algebra of holomorphic vector fields on $\mathbb{P}^{2}_{\mathbb{C}}$: $\chi(\mathbb{P}^{2}_{\mathbb{C}})$ is of course the Lie algebra of the automorphism group of $\mathbb{P}^{2}_{\mathbb{C}}.$ Let $\mathcal{F}$ be a foliation on $\mathbb{P}^{2}_{\mathbb{C}}$ and let $\mathrm{X}$ be an element of $\chi(\mathbb{P}^{2}_{\mathbb{C}}).$ Following [9] we will say that $\mathrm{X}$ is a symmetry of the foliation $\mathcal{F}$ if the flow $\exp(t\mathrm{X})$ is, for each $t,$ in the isotropy group $\mathrm{Iso}(\mathcal{F})$ of $\mathcal{F}.$ If $\omega$ defines $\mathcal{F}$ in an affine chart, $\mathrm{X}$ is a symmetry of $\mathcal{F}$ if and only if $\mathrm{L}_{\mathrm{X}}\omega\wedge\omega=0,$ where $\mathrm{L}_{\mathrm{X}}\omega$ denotes the Lie derivative of $\omega$ along $\mathrm{X}.$ ###### Lemma 2.1. Let $\mathcal{F}$ be a foliation of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ and let $\mathrm{X}$ be a symmetry of $\mathcal{F}.$ Assume that there is an affine chart $\mathbb{C}^{2}\subset\mathbb{P}^{2}_{\mathbb{C}}$ such that the vector field $\mathrm{X}$ is affine (i.e. $\deg X\leq 1$) and let $\omega$ be a $1$-form defining $\mathcal{F}$ in this chart. Then there is a constant $\lambda\in\mathbb{C}$ such that $\mathrm{L}_{\mathrm{X}}\omega=\lambda\omega.$ ###### Proof. We will use an argument similar to one in [9, Proposition 2.5]. Since $\mathrm{L}_{\mathrm{X}}\omega\wedge\omega=0$ and $\omega$ has isolated singularities, the De Rham-Saito division theorem (_cf._ [17] or [8, Proposition 1.14]) ensures the existence of a holomorphic function $g$ on $\mathbb{C}^{2}$ such that $\mathrm{L}_{\mathrm{X}}\omega=g\omega.$ The $1$-form $\omega$ and the vector field $\mathrm{X}$ being polynomials, $\mathrm{L}_{\mathrm{X}}\omega$ is also polynomial; therefore $g$ is rational and holomorphic on $\mathbb{C}^{2}$ hence polynomial. The vector field $\mathrm{X}$ being affine we have $\deg\mathrm{L}_{\mathrm{X}}\omega\leq\deg\omega$; the equality $\mathrm{L}_{\mathrm{X}}\omega=g\omega$ implies that $g$ is constant. ∎ If $\mathcal{F}$ is a foliation on $\mathbb{P}^{2}_{\mathbb{C}}$, we will denote by $\mathfrak{iso}(\mathcal{F})$ the Lie algebra of the algebraic group $\mathrm{Iso}(\mathcal{F})$; $\mathfrak{iso}(\mathcal{F})$ is a Lie subalgebra of $\chi(\mathbb{P}^{2}_{\mathbb{C}})$ and it consists of symmetries of $\mathcal{F}.$ We know from [9, Proposition 2.5] that if $\dim\mathfrak{iso}(\mathcal{F})=2$ then $\mathfrak{iso}(\mathcal{F})$ is affine, i.e. generated by two vector fields $\mathrm{X}$ and $\mathrm{Y}$ such that $[\mathrm{X},\mathrm{Y}]=\mathrm{Y}.$ The following lemma classifies the affine Lie subalgebras of $\chi(\mathbb{P}^{2}_{\mathbb{C}})$ and it will be used to prove Theorem A. ###### Lemma 2.2. Every affine Lie subalgebra of $\chi(\mathbb{P}^{2}_{\mathbb{C}})$ is linearly conjugated to one of the following models ($\mathfrak{a}$) $\big{\langle}\gamma\,x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y},\frac{\partial}{\partial y}\big{\rangle}$ with $\gamma\in\mathbb{C}^{};$ ($\mathfrak{b}$) $\big{\langle}y\frac{\partial}{\partial y},\frac{\partial}{\partial y}\big{\rangle};$ * ($\mathfrak{c}$) $\big{\langle}\frac{\partial}{\partial x}+y\frac{\partial}{\partial y},\frac{\partial}{\partial y}\big{\rangle};$ * ($\mathfrak{d}$) $\big{\langle}x\frac{\partial}{\partial x}+(x+y)\frac{\partial}{\partial y},\frac{\partial}{\partial y}\big{\rangle};$ * ($\mathfrak{e}$) $\big{\langle}x\frac{\partial}{\partial x}+2y\frac{\partial}{\partial y},\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}\big{\rangle}.$ ###### Proof. Let $\mathfrak{g}$ be an affine Lie subalgebra of $\chi(\mathbb{P}^{2}_{\mathbb{C}}).$ Then there exist $\mathrm{X}$ and $\mathrm{Y}$ in $\chi(\mathbb{P}^{2}_{\mathbb{C}})$ such that $\mathfrak{g}=\langle\mathrm{X},\mathrm{Y}\rangle$ and $[\mathrm{X},\mathrm{Y}]=\mathrm{Y}.$ Fixing homogeneous coordinates $[x:y:z]$ in $\mathbb{P}^{2}_{\mathbb{C}}$ we have an isomorphism of Lie algebras $\tau\hskip 2.84544pt\colon\mathfrak{sl}_{3}(\mathbb{C})\rightarrow\chi(\mathbb{P}^{2}_{\mathbb{C}})$ defined, for $A\in\mathfrak{sl}_{3}(\mathbb{C}),$ by $\tau(A)=(x\hskip 5.69054pty\hskip 5.69054ptz)A\left(\begin{array}[]{c}\vspace{0.5mm}\frac{\partial}{\partial x}\\\ \vspace{0.5mm}\frac{\partial}{\partial y}\\\ \vspace{0.5mm}\frac{\partial}{\partial z}\end{array}\right).$ Notice that if $A=\left(\begin{array}[]{ccc}a_{11}&a_{12}&a_{13}\\\ a_{21}&a_{22}&a_{23}\\\ a_{31}&a_{32}&a_{33}\end{array}\right)\in\mathfrak{sl}_{3}(\mathbb{C}),$ then in the affine chart $z=1$ the vector field $\tau(A)\in\chi(\mathbb{P}^{2}_{\mathbb{C}})$ writes as $\displaystyle\left(a_{31}+(a_{11}-a_{33})x+a_{21}y-a_{13}x^{2}-a_{23}xy\right)\frac{\partial}{\partial x}+\left(a_{32}+a_{12}x+(a_{22}-a_{33})y-a_{13}xy- a_{23}y^{2}\right)\frac{\partial}{\partial y}.$ Let $M$ and $N$ be the matrices of $\mathfrak{sl}_{3}(\mathbb{C})$ associated to the vector fields $\mathrm{X}$ and $\mathrm{Y}$ respectively, i.e. $M=\tau^{-1}(\mathrm{X})$ and $N=\tau^{-1}(\mathrm{Y})$. Then the Lie bracket $[\mathrm{X},\mathrm{Y}]$ corresponds to $[M,N]:=MN-NM$ and therefore $[M,N]=N.$ Let us write $M=\left(\begin{array}[]{ccc}-m_{22}-m_{33}&m_{12}&m_{13}\\\ m_{21}&m_{22}&m_{23}\\\ m_{31}&m_{32}&m_{33}\end{array}\right).$ Taking into account the possible Jordan forms of a matrix of $\mathfrak{sl}_{3}(\mathbb{C}),$ it suffices us to treat the following possibilities $\displaystyle N=\left(\begin{array}[]{ccc}-a-b&0&0\\\ 0&a&0\\\ 0&0&b\end{array}\right),$ $\displaystyle N=\left(\begin{array}[]{ccc}-2c&0&0\\\ 0&c&0\\\ 0&1&c\end{array}\right),$ $\displaystyle N=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&1&0\end{array}\right),$ $\displaystyle N=\left(\begin{array}[]{ccc}0&0&0\\\ 1&0&0\\\ 0&1&0\end{array}\right).$ where $a,b\in\mathbb{C},c\in\mathbb{C}^{},$ with $(a,b)\neq(0,0).$ 1. If $N=\left(\begin{array}[]{ccc}-a-b&0&0\\\ 0&a&0\\\ 0&0&b\end{array}\right)$ then the equality $[M,N]=N$ implies that $a=b=0$: contradiction. 2. If $N=\left(\begin{array}[]{ccc}-2c&0&0\\\ 0&c&0\\\ 0&1&c\end{array}\right)$ then the $(1,1)$ coefficient of the matrix $[M,N]-N$ is equal to $2c$ and is therefore nonzero: contradiction. 3. Assume that $N=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&0\\\ 0&1&0\end{array}\right)$; the equality $[M,N]=N$ then leads to $M=\left(\begin{array}[]{ccc}1-2m_{33}&m_{12}&0\\\ 0&m_{33}-1&0\\\ m_{31}&m_{32}&m_{33}\end{array}\right)$. Up to replacing $M$ by $M-m_{32}N$ we can assume that $m_{32}=0.$ Now we will distinguish several eventualities: 3.1. When $(3m_{33}-1)(3m_{33}-2)\neq 0$ the matrix $P=\left(\begin{array}[]{ccc}3m_{33}-1&m_{12}&0\\\ 0&3m_{33}-2&0\\\ -m_{31}&-m_{31}m_{12}&3m_{33}-2\end{array}\right)$ commutes with $N$ and $P^{-1}MP=\left(\begin{array}[]{ccc}1-2m_{33}&0&0\\\ 0&m_{33}-1&0\\\ 0&0&m_{33}\end{array}\right)$. Thus $\mathfrak{g}$ is linearly conjugated to $\displaystyle\big{\langle}\tau(P^{-1}MP),\tau(N)\big{\rangle}=\big{\langle}(1-3m_{33})x\tfrac{\partial}{\partial x}-y\tfrac{\partial}{\partial y},\tfrac{\partial}{\partial y}\big{\rangle}=\big{\langle}\gamma\,x\tfrac{\partial}{\partial x}+y\tfrac{\partial}{\partial y},\tfrac{\partial}{\partial y}\big{\rangle},\qquad\text{where }\gamma=3m_{33}-1\in\mathbb{C}^{}.$ 3.2. Assume that $m_{33}=\frac{1}{3}.$ If $\delta\in\mathbb{C}^{}$ then the matrix $P=\left(\begin{array}[]{ccc}\frac{1}{\delta}&-m_{12}&0\\\ 0&1&0\\\ 0&m_{12}m_{31}&1\end{array}\right)$ commutes with $N$ and $P^{-1}MP=\left(\begin{array}[]{ccc}\frac{1}{3}&0&0\\\ 0&-\frac{2}{3}&0\\\ \frac{m_{31}}{\delta}&0&\frac{1}{3}\end{array}\right)$. As a result $\mathfrak{g}$ is linearly conjugated to $\displaystyle\big{\langle}\tau(P^{-1}MP),\tau(N)\big{\rangle}=\big{\langle}\tfrac{m_{31}}{\delta}\tfrac{\partial}{\partial x}-y\tfrac{\partial}{\partial y},\tfrac{\partial}{\partial y}\big{\rangle}=\big{\langle}-\tfrac{m_{31}}{\delta}\tfrac{\partial}{\partial x}+y\tfrac{\partial}{\partial y},\tfrac{\partial}{\partial y}\big{\rangle}.$ The case where $m_{31}=0$ leads to the model $(\mathfrak{b}).$ If $m_{31}\neq 0$ then by taking $\delta=-m_{31}$ we get the model $(\mathfrak{c}).$ 3.3. Assume that $m_{33}=\frac{2}{3}.$ If $\delta\in\mathbb{C}^{}$ then the matrix $P=\left(\begin{array}[]{ccc}\delta&0&0\\\ 0&1&0\\\ -\delta m_{31}&-m_{12}m_{31}&1\end{array}\right)$ commutes with $N$ and $P^{-1}MP=\left(\begin{array}[]{ccc}-\frac{1}{3}&\frac{m_{12}}{\delta}&0\\\ 0&-\frac{1}{3}&0\\\ 0&0&\frac{2}{3}\end{array}\right)$. As a consequence $\mathfrak{g}$ is linearly conjugated to $\displaystyle\big{\langle}\tau(P^{-1}MP),\tau(N)\big{\rangle}=\big{\langle}-x\tfrac{\partial}{\partial x}+(\tfrac{m_{12}}{\delta}x-y)\tfrac{\partial}{\partial y},\tfrac{\partial}{\partial y}\big{\rangle}=\big{\langle}x\tfrac{\partial}{\partial x}+(y-\tfrac{m_{12}}{\delta}x)\tfrac{\partial}{\partial y},\tfrac{\partial}{\partial y}\big{\rangle}.$ The case $m_{12}=0$ leads to the model $(\mathfrak{a})$ with $\gamma=1.$ If $m_{12}\neq 0$ then by taking $\delta=-m_{12}$ we obtain the model $(\mathfrak{d}).$ 4. Assume that $N=\left(\begin{array}[]{ccc}0&0&0\\\ 1&0&0\\\ 0&1&0\end{array}\right)$; then the equality $[M,N]=N$ implies that $M=\left(\begin{array}[]{ccc}-1&0&0\\\ m_{32}&0&0\\\ m_{31}&m_{32}&1\end{array}\right)$. Up to replacing $M$ by $M-m_{32}N$ we can assume that $m_{32}=0.$ The matrix $P=\left(\begin{array}[]{ccc}1&0&0\\\ 0&1&0\\\ -\frac{m_{31}}{2}&0&1\end{array}\right)$ commutes with $N$ and $P^{-1}MP=\left(\begin{array}[]{ccc}-1&0&0\\\ 0&0&0\\\ 0&0&1\end{array}\right)$. Therefore $\mathfrak{g}$ is linearly conjugated to $\displaystyle\big{\langle}\tau(P^{-1}MP),\tau(N)\big{\rangle}=\big{\langle}-2x\tfrac{\partial}{\partial x}-y\tfrac{\partial}{\partial y},y\tfrac{\partial}{\partial x}+\tfrac{\partial}{\partial y}\big{\rangle}=\big{\langle}2x\tfrac{\partial}{\partial x}+y\tfrac{\partial}{\partial y},y\tfrac{\partial}{\partial x}+\tfrac{\partial}{\partial y}\big{\rangle}.$ By permuting the coordinates $x$ and $y$ we obtain the model $(\mathfrak{e}).$ ∎ ###### Proof of Theorem A. Since $\dim\mathfrak{iso}(\mathcal{F})=\dim\mathrm{Iso}(\mathcal{F})=2,$ [9, Proposition 2.5] implies that $\mathfrak{iso}(\mathcal{F})$ is affine. Therefore, up to linear conjugation, $\mathfrak{iso}(\mathcal{F})$ is one of the models ($\mathfrak{a}$)–($\mathfrak{e}$) of Lemma 2.2. Let $\omega$ be a $1$-form defining $\mathcal{F}$ in the affine chart $z=1$ $\hskip 28.45274pt\omega=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,\quad A,B\in\mathbb{C}[x,y],\hskip 5.69054pt\gcd(A,B)=1.$ We will study the five possible models ($\mathfrak{a}$)–($\mathfrak{e}$) of the Lie algebra $\mathfrak{iso}(\mathcal{F})$ and show that $\omega$ is linearly conjugated to one of the two $1$-forms $\omega_{1}^{d}$ or $\omega_{2}^{d}.$ 1. Assume that $\mathfrak{iso}(\mathcal{F})$ is of one of the types ($\mathfrak{a}$)–($\mathfrak{d}$), i.e. that $\mathfrak{iso}(\mathcal{F})=\big{\langle}\mathrm{X},\mathrm{Y}\big{\rangle}$ where $\mathrm{X}\in\\{\gamma\,x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y},\varepsilon\frac{\partial}{\partial x}+y\frac{\partial}{\partial y},x\frac{\partial}{\partial x}+(x+y)\frac{\partial}{\partial y}\\},\mathrm{Y}=\frac{\partial}{\partial y}$ with $\varepsilon\in\\{0,1\\}$ and $\gamma\in\mathbb{C}^{}.$ By Lemma 2.1 there exist $\lambda,\mu\in\mathbb{C}$ such that $\mathrm{L}_{\mathrm{X}}\omega=\lambda\omega$ and $\mathrm{L}_{\mathrm{Y}}\omega=\mu\omega.$ Since $\mathrm{L}_{\mathrm{Y}}\mathrm{d}x=\mathrm{d}\mathrm{L}_{\mathrm{Y}}x=0$ and $\mathrm{L}_{\mathrm{Y}}\mathrm{d}y=\mathrm{d}\mathrm{L}_{\mathrm{Y}}y=0,$ we have $\mathrm{L}_{\mathrm{Y}}\omega=\mathrm{Y}(A)\mathrm{d}x+\mathrm{Y}(B)\mathrm{d}y=\frac{\partial A}{\partial y}\mathrm{d}x+\frac{\partial B}{\partial y}\mathrm{d}y.$ Therefore $\mathrm{L}_{\mathrm{Y}}\omega=\mu\omega$ if and only if $\frac{\partial A}{\partial y}=\mu A$ and $\frac{\partial B}{\partial y}=\mu B.$ Since $A,B\in\mathbb{C}[x,y]$ and $\mu\in\mathbb{C},$ it follows that $\mu=0$, $A(x,y)=A(x)$ and $B(x,y)=B(x).$ Thus $\hskip 28.45274pt\omega=A(x)\mathrm{d}x+B(x)\mathrm{d}y,\quad A,B\in\mathbb{C}[x],\hskip 5.69054pt\gcd(A,B)=1.$ 1.1. Let us consider the case where $\mathrm{X}=\gamma\,x\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ with $\gamma\in\mathbb{C}^{}.$ We have $\displaystyle\mathrm{L}_{\mathrm{X}}\omega=\mathrm{X}(A)\mathrm{d}x+A\mathrm{d}\mathrm{X}(x)+\mathrm{X}(B)\mathrm{d}y+B\mathrm{d}\mathrm{X}(y)=\big{(}\gamma\,xA^{\prime}+\gamma\,A\big{)}\mathrm{d}x+\big{(}\gamma\,xB^{\prime}+B\big{)}\mathrm{d}y,$ so that $\mathrm{L}_{\mathrm{X}}\omega=\lambda\omega$ if and only if $\gamma\,xA^{\prime}=(\lambda-\gamma)A$ and $\gamma\,xB^{\prime}=(\lambda-1)B.$ By putting $\kappa=\frac{\lambda-\gamma}{\gamma}$ and $\nu=\frac{\lambda-1}{\gamma},$ the last two equations can be rewritten as $xA^{\prime}=\kappa\,A$ and $xB^{\prime}=\nu B$ and can be immediately integrated to give $A(x)=\alpha\,x^{\kappa}$ and $B(x)=\beta\,x^{\nu},$ where $\alpha,\beta\in\mathbb{C}.$ Since $A,B\in\mathbb{C}[x]$ and $\gcd(A,B)=1,$ we deduce that $\alpha,\beta\in\mathbb{C}^{},$ $\kappa,\nu\in\mathbb{N}$ and $\kappa\,\nu=0.$ The equality $\deg\mathcal{F}=d$ then implies that • either $\kappa=0$ and $\nu=d,$ in which case $\omega=\alpha\mathrm{d}x+\beta\,x^{d}\mathrm{d}y;$ * • or $\nu=0$ and $\kappa=d,$ in which case $\omega=\alpha\,x^{d}\mathrm{d}x+\beta\mathrm{d}y.$ If $\omega=\alpha\mathrm{d}x+\beta\,x^{d}\mathrm{d}y,$ resp. $\omega=\alpha\,x^{d}\mathrm{d}x+\beta\mathrm{d}y,$ by making the change of coordinates $\left(x,y\right)\mapsto\left(\frac{y}{x},-\frac{\alpha}{\beta\,x}\right)$, we reduce ourselves to $\omega=\omega_{1}^{d}=y^{d}\mathrm{d}x+x^{d}(x\mathrm{d}y-y\mathrm{d}x),$ resp. $\omega=\omega_{2}^{d}=x^{d}\mathrm{d}x+y^{d}(x\mathrm{d}y-y\mathrm{d}x).$ 1.2. Let us examine the case where $X=\varepsilon\frac{\partial}{\partial x}+y\frac{\partial}{\partial y}$ with $\varepsilon\in\\{0,1\\}.$ Since $\mathrm{L}_{\mathrm{X}}\mathrm{d}x=\mathrm{d}\mathrm{L}_{\mathrm{X}}x=0$ and $\mathrm{L}_{\mathrm{X}}\mathrm{d}y=\mathrm{d}\mathrm{L}_{\mathrm{X}}y=\mathrm{d}y,$ we have $\mathrm{L}_{\mathrm{X}}\omega=\mathrm{X}(A)\mathrm{d}x+\mathrm{X}(B)\mathrm{d}y+B\mathrm{d}y=\varepsilon\,A^{\prime}\mathrm{d}x+(\varepsilon\hskip 0.56905ptB^{\prime}+B)\mathrm{d}y.$ Therefore $\mathrm{L}_{\mathrm{X}}\omega=\lambda\omega$ if and only if $\varepsilon\,A^{\prime}=\lambda\,A$ and $\varepsilon\hskip 0.56905ptB^{\prime}=(\lambda-1)B.$ Since $A,B\in\mathbb{C}[x]$ and $\lambda\in\mathbb{C},$ it follows that $AB=0$: contradiction with $\gcd(A,B)=1.$ 1.3. Let us study the eventuality: $\mathrm{X}=x\frac{\partial}{\partial x}+(x+y)\frac{\partial}{\partial y}.$ We have $\mathrm{d}\mathrm{X}(x)=\mathrm{d}x$ and $\mathrm{d}\mathrm{X}(y)=\mathrm{d}x+\mathrm{d}y,$ so that $\mathrm{L}_{\mathrm{X}}\omega=\mathrm{X}(A)\mathrm{d}x+A\mathrm{d}x+\mathrm{X}(B)\mathrm{d}y+B(\mathrm{d}x+\mathrm{d}y)=(xA^{\prime}+A+B)\mathrm{d}x+(xB^{\prime}+B)\mathrm{d}y.$ The condition $\mathrm{L}_{\mathrm{X}}\omega=\lambda\omega$ is then equivalent to the system of differential equations $xA^{\prime}+B=(\lambda-1)A$ and $xB^{\prime}=(\lambda-1)B,$ which can be easily integrated to yield $A(x)=(a-b\ln x)x^{\lambda-1}$ and $B(x)=bx^{\lambda-1},$ where $a,b\in\mathbb{C}.$ Since $A\in\mathbb{C}[x]$, we deduce that $b=0$ and therefore $B\equiv 0$: contradiction with $\gcd(A,B)=1.$ 2. Assume that $\mathfrak{iso}(\mathcal{F})$ is of type ($\mathfrak{e}$), i.e. $\mathfrak{iso}(\mathcal{F})=\big{\langle}\mathrm{X},\mathrm{Y}\big{\rangle}$ where $\mathrm{X}=x\frac{\partial}{\partial x}+2y\frac{\partial}{\partial y},\mathrm{Y}=\frac{\partial}{\partial x}+x\frac{\partial}{\partial y}.$ As before by writing explicitly that $\mathrm{L}_{\mathrm{X}}\omega=\lambda\omega$ and $\mathrm{L}_{\mathrm{Y}}\omega=\mu\omega,$ with $\lambda,\mu\in\mathbb{C}$ (Lemma 2.1), we obtain the system of partial differential equations $\displaystyle x\tfrac{\partial A}{\partial x}+2y\tfrac{\partial A}{\partial y}=(\lambda-1)A,$ $\displaystyle x\tfrac{\partial B}{\partial x}+2y\tfrac{\partial B}{\partial y}=(\lambda-2)B,$ $\displaystyle\tfrac{\partial A}{\partial x}+x\tfrac{\partial A}{\partial y}=\mu\hskip 1.13809ptA-B,$ $\displaystyle\tfrac{\partial B}{\partial x}+x\tfrac{\partial B}{\partial y}=\mu\hskip 0.56905ptB.$ It follows in particular that $\displaystyle(x^{2}-2y)\tfrac{\partial B}{\partial y}=(\mu\,x+2-\lambda)B$ and $\displaystyle(x^{2}-2y)\tfrac{\partial A}{\partial y}=(\mu\,x+1-\lambda)A-xB.$ Elementary integrations then lead to $\displaystyle B(x,y)=b(x)(x^{2}-2y)^{\frac{\lambda-2-\mu\,x}{2}}$ and $\displaystyle A(x,y)=\left(a(x)\sqrt{x^{2}-2y}-xb(x)\right)(x^{2}-2y)^{\frac{\lambda-2-\mu\,x}{2}},$ for some functions $a$ and $b$ of the coordinate $x.$ Since $A,B\in\mathbb{C}[x,y]$ and $\gcd(A,B)=1,$ we deduce that $\lambda-2-\mu\,x=0$ and $a(x)=0$ for any $x\in\mathbb{C}$, hence $\lambda=2,\mu=0$ and $a\equiv 0.$ Therefore $B(x,y)=b(x)$ and $A(x,y)=-xb(x)=-xB(x,y)$: contradiction with $\gcd(A,B)=1.$ ∎ ## 3 Foliations of $\mathbf{F}(d)$ degenerating onto $\mathcal{F}_{1}^{d}$ and $\mathcal{F}_{2}^{d}$ In this section we will study the problem of knowing whether the closure of the orbit of a foliation of $\mathbf{F}(d)$ contains the foliations $\mathcal{F}_{1}^{d}$ and $\mathcal{F}_{2}^{d}.$ The following definition will be useful. ###### Definition 3.1 ([9]). Let $\mathcal{F}$ and $\mathcal{F}^{\prime}$ be two foliations of $\mathbf{F}(d).$ We say that $\mathcal{F}$ degenerates onto $\mathcal{F}^{\prime}$ if the closure $\overline{\mathcal{O}(\mathcal{F})}$ (in $\mathbf{F}(d)$) of the orbit $\mathcal{O}(\mathcal{F})$ contains $\mathcal{O}(\mathcal{F}^{\prime})$ and $\mathcal{O}(\mathcal{F})\not=\mathcal{O}(\mathcal{F}^{\prime}).$ ###### Remarks 3.2. Let $\mathcal{F}$ and $\mathcal{F}^{\prime}$ be two foliations such that $\mathcal{F}$ degenerates onto $\mathcal{F}^{\prime}$. Then: * (i) $\dim\mathcal{O}(\mathcal{F}^{\prime})<\dim\mathcal{O}(\mathcal{F})$; * (ii) $\deg\mathrm{I}_{\mathcal{F}}^{\mathrm{inv}}\leq\deg\mathrm{I}_{\mathcal{F}^{\prime}}^{\mathrm{inv}},$ which is equivalent to $\deg\mathrm{I}_{\mathcal{F}}^{\hskip 0.56905pt\mathrm{tr}}\geq\deg\mathrm{I}_{\mathcal{F}^{\prime}}^{\hskip 0.56905pt\mathrm{tr}}$. In particular, if $\mathcal{F}$ is convex, $\mathcal{F}^{\prime}$ is also convex. Corollary B is an immediate consequence of Theorem A and Remark 3.2 (i). ###### Remark 3.3. Corollary B applies particularly to any foliation of $\mathbf{F}(d)$ which is homogeneous, i.e. which is given, for a suitable choice of affine coordinates $(x,y),$ by a homogeneous $1$-form $\omega=A(x,y)\mathrm{d}x+B(x,y)\mathrm{d}y,$ where $A,B\in\mathbb{C}[x,y]_{d}$ and $\gcd(A,B)=1.$ Indeed, for such a foliation $\mathcal{H}\in\mathbf{F}(d)$, we have (_cf._ [4]) $\mathrm{Iso}(\mathcal{H})=\\{(\alpha\,x,\alpha\,y)\hskip 2.84526pt\|\hskip 2.84526pt\alpha\in\mathbb{C}^{}\\};$ in particular, $\dim\mathcal{O}(\mathcal{H})=7$ and consequently $\hskip 15.649pt\overline{\mathcal{O}(\mathcal{H})}\subset\mathcal{O}(\mathcal{H})\cup\mathcal{O}(\mathcal{F}_{1}^{d})\cup\mathcal{O}(\mathcal{F}_{2}^{d}).$ Assertion 1. (resp. 2.) of the following proposition gives a necessary (resp. sufficient) condition for a foliation of $\mathbf{F}(d)$ to degenerate onto the foliation $\mathcal{F}_{1}^{d}.$ ###### Proposition 3.4. Let $\mathcal{F}$ be an element of $\mathbf{F}(d)$ such that $\mathcal{F}_{1}^{d}\not\in\mathcal{O}(\mathcal{F}).$ The following assertions hold: 1. If $\mathcal{F}$ degenerates onto $\mathcal{F}_{1}^{d}$, then $\mathcal{F}$ possesses a non-degenerate singularity $m$ satisfying $\mathrm{BB}(\mathcal{F},m)=4.$ 2. If $\mathcal{F}$ possesses a quasi-radial singularity of maximal order $d-1$, i.e. if $\mathrm{QRad}(\mathcal{F},d-1)\neq\emptyset$, then $\mathcal{F}$ degenerates onto $\mathcal{F}_{1}^{d}.$ ###### Proof. 1. Assume that $\mathcal{F}$ degenerates onto $\mathcal{F}_{1}^{d}$. Then there is an analytic family of foliations $(\mathcal{F}_{\varepsilon})$ defined by a family of $1$-forms $(\omega_{\varepsilon})$ such that $\mathcal{F}_{\varepsilon}$ belongs to $\mathcal{O}(\mathcal{F})$ for $\varepsilon\neq 0$ and $\mathcal{F}_{\varepsilon=0}=\mathcal{F}_{1}^{d}.$ The non-degenerate singular point of $\mathcal{F}_{1}^{d}$, denoted by $m_{0},$ is $\scriptscriptstyle\langle\\!\langle$ stable $\\!\scriptscriptstyle\,\rangle\\!\rangle$ in the sense that there is an analytic family $(m_{\varepsilon})$ of non-degenerate singular points of $\mathcal{F}_{\varepsilon}$ such that $m_{\varepsilon=0}=m_{0}.$ The $\mathcal{F}_{\varepsilon}$’s being conjugated to $\mathcal{F}$ for $\varepsilon\neq 0$, the foliation $\mathcal{F}$ admits a non-degenerate singular point $m$ such that $\forall\hskip 2.84526pt\varepsilon\in\mathbb{C}^{},\hskip 2.84526pt\mathrm{BB}(\mathcal{F}_{\varepsilon},m_{\varepsilon})=\mathrm{BB}(\mathcal{F},m).$ Since $\mu(\mathcal{F}_{\varepsilon},m_{\varepsilon})=1$ for any $\varepsilon\in\mathbb{C}$, the function $\varepsilon\mapsto\mathrm{BB}(\mathcal{F}_{\varepsilon},m_{\varepsilon})$ is continuous, hence constant on $\mathbb{C}$. As a consequence $\displaystyle\mathrm{BB}(\mathcal{F},m)=\mathrm{BB}(\mathcal{F}_{\varepsilon=0},m_{\varepsilon=0})=\mathrm{BB}(\mathcal{F}_{1}^{d},m_{0})=4.$ 2. Assume that $\mathcal{F}$ has a quasi-radial singularity $m$ of order $d-1$. Then $\mu(\mathcal{F},m)=1,\mathrm{BB}(\mathcal{F},m)=4$ and $\kappa(\mathcal{F},m)=d.$ This last equality ensures the existence of a line $\ell_{m}$ passing through $m$, not invariant by $\mathcal{F}$ and such that $\mathrm{Tang}(\mathcal{F},\ell_{m},m)=d.$ Let us choose an affine coordinate system $(x,y)$ such that $m=(0,0)$ and $\ell_{m}=\\{x=0\\}.$ The foliation $\mathcal{F}$ is defined in these coordinates by a $1$-form $\omega$ of type $\displaystyle\omega=C_{d}(x,y)(x\mathrm{d}y-y\mathrm{d}x)+\sum_{i=1}^{d}\big{(}A_{i}(x,y)\mathrm{d}x+B_{i}(x,y)\mathrm{d}y\big{)},$ $\displaystyle\text{where }A_{i},\,B_{i}\in\mathbb{C}[x,y]_{i},\hskip 2.84526ptC_{d}\in\mathbb{C}[x,y]_{d}.$ We have $\displaystyle\omega\wedge\mathrm{d}x\Big{\|}_{x=0}=\sum_{i=1}^{d}B_{i}(0,y)\mathrm{d}y\wedge\mathrm{d}x=\sum_{i=1}^{d}B_{i}(0,1)y^{i}\mathrm{d}y\wedge\mathrm{d}x.$ Then the equality $\mathrm{Tang}(\mathcal{F},\ell_{m},m)=d$ translates into $B_{i}(0,1)=0$ for $i\in\\{1,2,\ldots,d-1\\}$ and $B_{d}(0,1)\neq 0.$ This allows to write $\displaystyle B_{1}(x,y)=\alpha\hskip 0.56905ptx,$ $\displaystyle B_{d}(x,y)=x\widehat{B}_{d-1}(x,y)+\gamma\,y^{d},$ $\displaystyle B_{i}(x,y)=x\widetilde{B}_{i-1}(x,y)\hskip 2.84526pt\text{for }i\in\\{2,3,\ldots,d-1\\},$ where $\widetilde{B}_{i-1}\in\mathbb{C}[x,y]_{i-1},\hskip 2.84526pt\widehat{B}_{d-1}\in\mathbb{C}[x,y]_{d-1},\hskip 2.84526pt\gamma\in\mathbb{C}^{},\hskip 2.84526pt\alpha\in\mathbb{C}.$ The equalities $\mu(\mathcal{F},m)=1$ and $\mathrm{BB}(\mathcal{F},m)=4$ imply that $\alpha\neq 0$ and $A_{1}(x,y)=\delta\hskip 0.56905ptx-\alpha y$ for some $\delta\in\mathbb{C}.$ Thus $\omega$ is of type $\displaystyle\omega=\delta\hskip 0.56905ptx\mathrm{d}x+\big{(}x\widehat{B}_{d-1}(x,y)+\gamma\,y^{d}\big{)}\mathrm{d}y+\big{(}C_{d}(x,y)+\alpha\big{)}\big{(}x\mathrm{d}y-y\mathrm{d}x\big{)}+\sum_{i=2}^{d}A_{i}(x,y)\mathrm{d}x+x\sum_{i=2}^{d-1}\widetilde{B}_{i-1}(x,y)\mathrm{d}y,$ where $A_{i}\in\mathbb{C}[x,y]_{i},\hskip 2.84526pt\widetilde{B}_{i-1}\in\mathbb{C}[x,y]_{i-1},\hskip 2.84526pt\widehat{B}_{d-1}\in\mathbb{C}[x,y]_{d-1},\hskip 2.84526pt\delta\in\mathbb{C},\hskip 2.84526pt\alpha,\gamma\in\mathbb{C}^{}.$ By putting $\varphi=\left(\varepsilon^{d}\hskip 0.28453ptx,\varepsilon\hskip 0.28453pty\right)$ and $\theta=\alpha(x\mathrm{d}y-y\mathrm{d}x)+\gamma\,y^{d}\mathrm{d}y$, we obtain $\displaystyle\frac{1}{\varepsilon^{d+1}}\varphi^{}\omega=\theta+\varepsilon^{d-1}\left(\delta\hskip 0.56905ptx\mathrm{d}x+x\widehat{B}_{d-1}(\varepsilon^{d-1}x,y)\mathrm{d}y\right)+\varepsilon^{d}C_{d}(\varepsilon^{d-1}x,y)\left(x\mathrm{d}y-y\mathrm{d}x\right)+\sum_{i=2}^{d}\varepsilon^{i-1}A_{i}(\varepsilon^{d-1}x,y)\mathrm{d}x+x\sum_{i=2}^{d-1}\varepsilon^{i-1}\widetilde{B}_{i-1}(\varepsilon^{d-1}x,y)\mathrm{d}y$ which tends to $\theta$ as $\varepsilon$ tends to $0.$ By making the change of coordinates $\left(x,y\right)\mapsto\left(\frac{x}{y}-\frac{\gamma}{\alpha y},\frac{x}{y}\right)$, we reduce ourselves to $\theta=\omega_{1}^{d}=y^{d}\mathrm{d}x+x^{d}(x\mathrm{d}y-y\mathrm{d}x)$. As a result $\mathcal{F}$ degenerates onto $\mathcal{F}_{1}^{d}.$ ∎ ###### Example 3.5. Let us consider the homogeneous foliation $\mathcal{H}_{1}^{d}$ defined in the affine chart $z=1$ by the $1$-form ${\mspace{2.0mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2.0mu}}_{1}^{d}=y^{d}\mathrm{d}x-x^{d}\mathrm{d}y.$ We know from [4, Proposition 4.1] that $\mathcal{H}_{1}^{d}$ is convex and admits the points $[1:0:0]$ and $[0:1:0]$ as radial singularities of maximal order $d-1.$ Therefore $\mathcal{H}_{1}^{d}$ degenerates onto $\mathcal{F}_{1}^{d}$ (Proposition 3.4) and it does not degenerate onto $\mathcal{F}_{2}^{d}$, because $\mathcal{F}_{2}^{d}$ is not convex. Thus, according to Remark 3.3, we have $\hskip 42.67912pt\overline{\mathcal{O}(\mathcal{H}_{1}^{d})}=\mathcal{O}(\mathcal{H}_{1}^{d})\cup\mathcal{O}(\mathcal{F}_{1}^{d}).$ ###### Example 3.6. Let us consider the family $(\mathcal{G}^{d}(\gamma))_{\gamma\in\mathbb{C}}$ of foliations of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ defined in the affine chart $z=1$ by $\eta^{d}(\gamma)=(x-\gamma\,y)\mathrm{d}y-y\mathrm{d}x+x^{d}\mathrm{d}x-y^{d}\mathrm{d}y.$ We remark that the point $m=[0:0:1]$ is a non-degenerate singularity of $\mathcal{G}^{d}(\gamma)$ with Baum-Bott index $4.$ Moreover, along the line $\ell=\\{y=0\\}$ we have $\eta^{d}(\gamma)\wedge\mathrm{d}y\Big{\|}_{y=0}=x^{d}\mathrm{d}x\wedge\mathrm{d}y,$ so that $\mathrm{Tang}(\mathcal{G}^{d}(\gamma),\ell,m)=d$. It follows that the singularity $m$ of $\mathcal{G}^{d}(\gamma)$ is quasi-radial of maximal order $d-1.$ As a consequence $\mathcal{G}^{d}(\gamma)$ degenerates onto $\mathcal{F}_{1}^{d}$ (Proposition 3.4). The converse of assertion 2. of Proposition 3.4 is false as the following example shows. ###### Example 3.7. Let $\mathcal{F}$ be the foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}$ defined in the affine chart $z=1$ by $\displaystyle\omega=x\mathrm{d}y-y\mathrm{d}x+P(y)\mathrm{d}y,$ where $P$ is a polynomial of $\mathbb{C}[y]$ of degree $d$ admitting $0$ as a root of multiplicity $\leq d-1,$ i.e. $P$ is of the form $\displaystyle P(y)=y^{\nu}(a_{0}+a_{1}y+\cdots+a_{d-\nu}y^{d-\nu}),$ $\displaystyle\text{where}\hskip 2.84526pt\nu\in\\{1,2,\ldots,d-1\\},\hskip 2.84526pta_{i}\in\mathbb{C},\hskip 2.84526pta_{0}a_{d-\nu}\neq 0.$ The singular locus of $\mathcal{F}$ consists of the two points $m=[0:0:1]$ and $m^{\prime}=[1:0:0]$; moreover $\displaystyle\mu(\mathcal{F},m)=1,$ $\displaystyle\mathrm{BB}(\mathcal{F},m)=4,$ $\displaystyle\kappa(\mathcal{F},m)=\nu<d,$ $\displaystyle\mu(\mathcal{F},m^{\prime})>1.$ It follows that $\mathcal{F}$ has no quasi-radial singularity of maximal order $d-1,$ i.e. $\mathrm{QRad}(\mathcal{F},d-1)=\emptyset.$ However, $\mathcal{F}$ degenerates onto $\mathcal{F}_{1}^{d}$. Indeed, by putting $\varphi=$$\left(\dfrac{a_{d-\nu}}{\varepsilon^{d}}x,\dfrac{1}{\varepsilon}y\right)$, we see that $\displaystyle\lim_{\varepsilon\to 0}\frac{\varepsilon^{d+1}}{a_{d-\nu}}\varphi^{}\omega=x\mathrm{d}y-y\mathrm{d}x+y^{d}\mathrm{d}y.$ ###### Question 1. Let $\mathcal{F}$ be a foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ Is it true that if $\mathcal{F}$ degenerates onto $\mathcal{F}_{1}^{d}$ then * – either $\mathcal{F}$ admits a quasi-radial singularity of maximal order $d-1$, * – or $\mathcal{F}$ is conjugated to Example 3.7, i.e. up to linear conjugation $\mathcal{F}$ is given by a $1$-form of type $x\mathrm{d}y-y\mathrm{d}x+P(y)\mathrm{d}y$ with $P\in\mathbb{C}[y],$ $\deg P=d$ and $P(0)=0$? ###### Proposition 3.8. Let $d$ be an integer greater than or equal to $2.$ Let us denote by $U_{1}(d)$ the subset of $\mathbf{F}(d)$ defined by $\displaystyle U_{1}(d):=\Big{\\{}\mathcal{F}\in\mathbf{F}(d)\hskip 2.84526pt\|\hskip 2.84526pt\forall\hskip 0.56905pts\in\mathrm{Sing}(\mathcal{F}),\hskip 2.84526pt\mu(\mathcal{F},s)=1,\tau(\mathcal{F},s)=1\Big{\\}}.$ Then: * (i) $U_{1}(d)$ is a non-empty Zariski open subset of $\mathbf{F}(d);$ in particular, for any $\gamma\in\mathbb{C},$ $\mathcal{G}^{d}(\gamma)\in U_{1}(d)$ if and only if $\gamma\left(\gamma^{d+1}+\dfrac{(d+1)^{d+1}}{d^{d}}\right)\neq 0.$ * (ii) Let $\mathcal{F}$ be an element of $U_{1}(d).$ For any singular point $s\in\mathrm{Sing}(\mathcal{F}),$ the set $\Lambda(\mathcal{F},s):=\Big{\\{}\ell_{s}\in\mathfrak{L}_{s}(\mathcal{F})\hskip 2.84526pt\|\hskip 2.84526pt\mathrm{Tang}(\mathcal{F},\ell_{s},s)>1\Big{\\}}$ has at most $2$ elements. In particular, the set $\bigcup\limits_{n=2}^{d}\widehat{\mathrm{QRad}}(\mathcal{F},n-1)$ is finite. To prove this proposition, we need the following lemma. ###### Lemma 3.9. Let $\mathcal{F}$ be a foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}},$ $s$ a singular point of $\mathcal{F},$ $\ell_{s}$ a line passing through $s$ and not invariant by $\mathcal{F}$ and $\mathrm{X}=A(x,y)\frac{\partial{}}{\partial{x}}+B(x,y)\frac{\partial{}}{\partial{y}}$ a polynomial vector field defining $\mathcal{F}$ in an affine chart $(x,y)$ containing $s.$ Let us denote by $(x_{0},y_{0})$ the coordinates of $s$ and let $a(x-x_{0})+b(y-y_{0})=0$ be an equation of the line $\ell_{s}.$ Then, for any $n\in\\{2,3,\ldots,d\\},$ $\mathrm{Tang}(\mathcal{F},\ell_{s},s)\geq n$ if and only if $\displaystyle\frac{\mathrm{d}^{j}}{\mathrm{d}t^{j}}\Big{(}a\hskip 0.56905ptA(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)+bB(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)\Big{)}\Big{\|}_{t=0}=0,\quad\forall\hskip 0.56905ptj\in\\{1,2,\ldots,n-1\\}.$ In particular, the set $\Lambda(\mathcal{F},s):=\Big{\\{}\ell_{s}\in\mathfrak{L}_{s}(\mathcal{F})\hskip 2.84526pt\|\hskip 2.84526pt\mathrm{Tang}(\mathcal{F},\ell_{s},s)>\tau(\mathcal{F},s)\Big{\\}}$ is finite and its cardinality is at most $\tau(\mathcal{F},s)+1.$ ###### Proof. The $1$-form $\omega=A(x,y)\mathrm{d}y-B(x,y)\mathrm{\mathrm{d}}x$ also defines the foliation $\mathcal{F}$ because $i_{\mathrm{X}}\omega=0.$ We have $\displaystyle\omega\wedge\mathrm{d}\big{(}a(x-x_{0})+b(y-y_{0})\big{)}\Big{\|}_{(x,y)=(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)}=P(t)\mathrm{d}y\wedge\mathrm{d}x,$ where $P(t)=a\hskip 0.56905ptA(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)+bB(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)$. Thus $\mathrm{Tang}(\mathcal{F},\ell_{s},s)=\nu(P(t),0).$ Notice that $P(0)=\leavevmode\nobreak\ 0$ because the point $s$ being singular for $\mathcal{F},$ we have $A(x_{0},y_{0})=B(x_{0},y_{0})=0.$ Then $\mathrm{Tang}(\mathcal{F},\ell_{s},s)\geq n$ if and only if the root $t=0$ of the polynomial $P$ has multiplicity at least $n$, that is if and only if $P^{\prime}(0)=P^{\prime\prime}(0)=\cdots=P^{(n-1)}(0)=0$, hence the announced equivalence holds. By conjugating $\omega$ by the translation $(x+x_{0},y+y_{0})$, we can assume that $s=(0,0)$. Let us denote $\tau(\mathcal{F},s)$ simply by $\tau$. Then the vector field $\mathrm{X}$ decomposes in the form $\displaystyle\mathrm{X}=C_{\tau-2}(x,y)\mathrm{R}+\sum_{i=\tau}^{d+1}\mathrm{X}_{i},$ where $\mathrm{R}=x\frac{\partial{}}{\partial{x}}+y\frac{\partial{}}{\partial{y}}$, $C_{\tau-2}$ is a polynomial of degree $\leq\tau-2,$ $\mathrm{X}_{i}=A_{i}(x,y)\frac{\partial{}}{\partial{x}}+B_{i}(x,y)\frac{\partial{}}{\partial{y}}$ is a homogeneous vector field of degree $i$, with $\det(\mathrm{X}_{\tau},\mathrm{R})\not\equiv 0.$ Thus, we have $\displaystyle a\hskip 0.56905ptA(b\hskip 0.56905ptt,-a\hskip 0.56905ptt)+bB(b\hskip 0.56905ptt,-a\hskip 0.56905ptt)$ $\displaystyle=\bigg{(}a\Big{(}xC_{\tau-2}(x,y)+\sum_{i=\tau}^{d+1}A_{i}(x,y)\Big{)}+b\Big{(}yC_{\tau-2}(x,y)+\sum_{i=\tau}^{d+1}B_{i}(x,y)\Big{)}\bigg{)}\bigg{\|}_{(x,y)=(b\hskip 0.56905ptt,-a\hskip 0.56905ptt)}$ $\displaystyle=\sum_{i=\tau}^{d+1}\Big{(}a\hskip 0.56905ptA_{i}(b\hskip 0.56905ptt,-a\hskip 0.56905ptt)+bB_{i}(b\hskip 0.56905ptt,-a\hskip 0.56905ptt)\Big{)}$ $\displaystyle=\sum_{i=\tau}^{d+1}t^{i}Q_{i+1}(a,b),$ where $Q_{i+1}(a,b):=a\hskip 0.56905ptA_{i}(b,-a)+bB_{i}(b,-a)$ is a homogeneous polynomial of degree $i+1$ in $(a,b).$ From this, we deduce that $\mathrm{Tang}(\mathcal{F},\ell_{s},s)>\tau$ if and only if $Q_{\tau+1}(a,b)=0.$ As a result $\displaystyle\Lambda(\mathcal{F},s)=\Big{\\{}\ell_{s}=\\{ax+by=0\\}\in\mathfrak{L}_{s}(\mathcal{F})\hskip 2.84526pt\|\hskip 2.84526ptQ_{\tau+1}(a,b)=0\Big{\\}}.$ Now, the polynomial $Q_{\tau+1}$ is not identically zero because $Q_{\tau+1}(a,b)=-\det(\mathrm{X}_{\tau},\mathrm{R})\big{\|}_{(x,y)=(b,-a)}\not\equiv 0.$ It follows that $\Lambda(\mathcal{F},s)$ has cardinality at most $\tau+1.$ ∎ ###### Proof of Proposition 3.8. We have $\displaystyle U_{1}(d)=\Big{\\{}\mathcal{F}\in\mathbf{F}(d)\hskip 2.84526pt\|\hskip 2.84526pt\forall\hskip 0.56905pts\in\mathrm{Sing}(\mathcal{F}),\hskip 2.84526pt\det(\mathrm{Jac}\hskip 0.28453pt\mathrm{X}(s))\neq 0,\hskip 2.84526pt\det(J^{1}_{s}\mathrm{X},\mathrm{R}_{s})\not\equiv 0\Big{\\}},$ where $\mathrm{X}$ denotes a polynomial vector field defining $\mathcal{F}$ in an affine chart containing $s$ and $\mathrm{R}_{s}$ is the radial vector field centered at $s.$ It follows that $U_{1}(d)$ is a Zariski open subset of $\mathbf{F}(d).$ To establish assertion (i), it remains to show that for any $\gamma\in\mathbb{C},$ $\mathcal{G}^{d}(\gamma)\in U_{1}(d)$ if and only if $\gamma\left(\gamma^{d+1}+\frac{(d+1)^{d+1}}{d^{d}}\right)\neq 0.$ In homogeneous coordinates, the foliation $\mathcal{G}^{d}(\gamma)$ is defined by the $1$-form $\displaystyle\hskip 28.45274pt\Omega^{d}(\gamma)=z\big{(}x^{d}-yz^{d-1}\big{)}\mathrm{d}x-z\big{(}y^{d}+\gamma\,yz^{d-1}-xz^{d-1}\big{)}\mathrm{d}y+\big{(}y^{d+1}-x^{d+1}+\gamma\,y^{2}z^{d-1}\big{)}\mathrm{d}z\hskip 1.42262pt.$ The singular locus $\mathrm{Sing}\big{(}\mathcal{G}^{d}(\gamma)\big{)}$ consists of the points $\displaystyle s_{0}=[0:0:1],$ $\displaystyle s_{k}=[x_{k}:x_{k}^{d}:1],$ $\displaystyle s^{\prime}_{l}=[1:\xi^{l}:0],$ $\displaystyle k\in\\{1,2,\ldots,d^{2}-1\\},\,l\in\\{0,1,\ldots,d\\},$ where $\xi=\exp\left(\frac{2\mathrm{i}\pi}{d+1}\right)$ and the $x_{k}$’s are the roots of the polynomial $P(x)=x^{d^{2}-1}+\gamma\,x^{d-1}-1.$ In the affine chart $z=1,$ resp. $x=1,$ $\mathcal{G}^{d}(\gamma)$ is given by the vector field $\displaystyle\mathrm{Y}=\big{(}y^{d}+\gamma\,y-x\big{)}\frac{\partial{}}{\partial{x}}+\big{(}x^{d}-y\big{)}\frac{\partial{}}{\partial{y}},$ $\displaystyle\text{resp. }\mathrm{Z}=\big{(}y^{d+1}+\gamma\,y^{2}z^{d-1}-1\big{)}\frac{\partial{}}{\partial{y}}+z\big{(}y^{d}+\gamma\,yz^{d-1}-z^{d-1}\big{)}\frac{\partial{}}{\partial{z}}\hskip 1.42262pt.$ A direct computation show that $\det(\mathrm{Jac}\hskip 0.28453pt\mathrm{Y}(s_{0}))=1\neq 0$, $\det(J^{1}_{s_{0}}\mathrm{Y},\mathrm{R}_{s_{0}})=\gamma\,y^{2}$ and $\displaystyle\det(\mathrm{Jac}\hskip 0.28453pt\mathrm{Z}(s^{\prime}_{l}))=(d+1)\xi^{-2l}\neq 0,$ $\displaystyle\det(\mathrm{Jac}\hskip 0.28453pt\mathrm{Y}(s_{k}))=1-d\gamma\,x_{k}^{d-1}-d^{2}x_{k}^{d^{2}-1}=(d-1)\big{(}d\gamma\,x_{k}^{d-1}-d-1\big{)},\hskip 2.84526pt\text{\normalsize because}\hskip 2.84526ptP(x_{k})=0,$ $\displaystyle\det(J^{1}_{s^{\prime}_{l}}\mathrm{Z},\mathrm{R}_{s^{\prime}_{l}})=d\xi^{-l}\big{(}y-\xi^{l}\big{)}z\not\equiv 0,$ $\displaystyle\det(J^{1}_{s_{k}}\mathrm{Y},\mathrm{R}_{s_{k}})=\big{(}dx_{k}^{d^{2}-d}+\gamma\big{)}\big{(}y-x_{k}^{d}\big{)}^{2}-dx_{k}^{d-1}\big{(}x-x_{k}\big{)}^{2}\not\equiv 0,\hskip 2.84526pt\text{\normalsize because}\hskip 2.84526ptx_{k}\neq 0.$ From these we deduce that $\mathcal{G}^{d}(\gamma)\in U_{1}(d)$ if and only if $\gamma\neq 0$ and $d\gamma\,x_{k}^{d-1}-d-1\neq 0$, i.e. if and only if $\gamma\neq 0$ and $x_{k}^{d-1}\neq\frac{d+1}{d\gamma}.$ Now, by putting $Q(t)=t^{d+1}+\gamma\,t-1,$ we have $P(x)=Q(x^{d-1})$ so that $t_{0}\in\mathbb{C}$ is a root of the polynomial $Q(t)$ if and only if there exists $k\in\\{1,2,\ldots,d^{2}-1\\}$ such that $t_{0}=x_{k}^{d-1}.$ It follows that $\displaystyle\mathcal{G}^{d}(\gamma)\in U_{1}(d)\Longleftrightarrow\gamma\,Q\left(\frac{d+1}{d\gamma}\right)\neq 0\Longleftrightarrow\gamma\left(\gamma^{d+1}+\frac{(d+1)^{d+1}}{d^{d}}\right)\neq 0.$ Assertion (ii) is an immediate consequence of Lemma 3.9. ∎ ###### Theorem 3.10. Let $d$ be an integer greater than or equal to $2.$ Let us denote by $\Sigma_{1}(d)$ the subset of $\mathbf{F}(d)$ defined by $\displaystyle\Sigma_{1}(d):=\Big{\\{}\mathcal{F}\in\mathbf{F}(d)\hskip 2.84526pt\|\hskip 2.84526pt\mathrm{QRad}(\mathcal{F},d-1)\neq\emptyset\Big{\\}}.$ Then * (a) $\emptyset\neq\Sigma_{1}(d)\varsubsetneq\mathbf{B}(\mathcal{F}_{1}^{d});$ * (b) $\Sigma_{1}(d)$ is a constructible subset of $\mathbf{F}(d)$ of dimension greater than or equal to $\dim\mathbf{F}(d)-(d-1).$ ###### Proof. (a) $\Sigma_{1}(d)$ contains the foliations $\mathcal{H}_{1}^{d}$ and $\mathcal{G}^{d}(\gamma),\gamma\in\mathbb{C}$ (Examples 3.5 and 3.6) and is therefore non-empty. Assertion 2. of Proposition 3.4 ensures that $\Sigma_{1}(d)\subset\mathbf{B}(\mathcal{F}_{1}^{d})$; this inclusion is strict as Example 3.7 shows. (b) Let us denote by $\mathbb{\check{P}}^{2}_{\mathbb{C}}$ the dual projective plane of $\mathbb{P}^{2}_{\mathbb{C}}.$ Let $\pi\hskip 2.84526pt\colon\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}\times\mathbb{\check{P}}^{2}_{\mathbb{C}}\to\mathbf{F}(d)$ be the projection onto the first factor; we have $\Sigma_{1}(d)=\pi(W_{1}(d)),$ where $\displaystyle W_{1}(d):$ $\displaystyle=\bigcup_{\mathcal{F}\in\Sigma_{1}(d)}\\{\mathcal{F}\\}\times\widehat{\mathrm{QRad}}(\mathcal{F},d-1)$ $\displaystyle=\Big{\\{}(\mathcal{F},s,\ell)\in\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}\times\mathbb{\check{P}}^{2}_{\mathbb{C}}\hskip 2.84526pt\|\hskip 2.84526pts\in\mathrm{Sing}(\mathcal{F}),\hskip 2.84526pt\ell\in\mathfrak{L}_{s}(\mathcal{F}),\hskip 2.84526pt\mu(\mathcal{F},s)=1,\hskip 2.84526pt\mathrm{BB}(\mathcal{F},s)=4,\hskip 2.84526pt\mathrm{Tang}(\mathcal{F},\ell,s)=d\Big{\\}}.$ According to Lemma 3.9, $W_{1}(d)$ can be rewritten as (3.5) $\displaystyle W_{1}(d)=\left\\{(\mathcal{F},s,\ell)\in\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}\times\mathbb{\check{P}}^{2}_{\mathbb{C}}\hskip 1.42262pt\left\|\begin{array}[c]{l}\vspace{2mm}s=(x_{0},y_{0})\in\ell=\\{a(x-x_{0})+b(y-y_{0})=0\\}\\\ \vspace{2mm}A(x_{0},y_{0})=0,\hskip 2.84526ptB(x_{0},y_{0})=0,\hskip 2.84526pt\det(\mathrm{Jac}\hskip 0.28453pt\mathrm{X}(s))\neq 0,\hskip 2.84526pt\dfrac{\mathrm{tr}^{2}(\mathrm{Jac}\hskip 0.28453pt\mathrm{X}(s))}{\det(\mathrm{Jac}\hskip 0.28453pt\mathrm{X}(s))}=4\\\ \vspace{2mm}a\hskip 0.56905ptA(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)+bB(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)\not\equiv 0\\\ \dfrac{\mathrm{d}^{j}}{\mathrm{d}t^{j}}\Big{(}a\hskip 0.56905ptA(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)+bB(x_{0}+b\hskip 0.56905ptt,y_{0}-a\hskip 0.56905ptt)\Big{)}\Big{\|}_{t=0}=0,j=1,\ldots,d-1\end{array}\right.\right\\},$ where $\mathrm{X}=A(x,y)\frac{\partial{}}{\partial{x}}+B(x,y)\frac{\partial{}}{\partial{y}}$ is a polynomial vector field defining $\mathcal{F}$ in an affine chart $(x,y)$ containing $s.$ It follows that $W_{1}(d)$ is a quasi-projective subvariety of $\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}\times\mathbb{\check{P}}^{2}_{\mathbb{C}}.$ Thus, by Chevalley’s Theorem [11, Exercise II.3.19], the set $\Sigma_{1}(d)=\pi(W_{1}(d))$ is constructible. According to the above discussion and Proposition 3.8 (i), the intersection $U_{1}(d)\cap\Sigma_{1}(d)$ contains the foliations $\mathcal{G}^{d}(\gamma)$, with $\gamma\left(\gamma^{d+1}+\frac{(d+1)^{d+1}}{d^{d}}\right)\neq 0,$ and is therefore non-empty ($U_{1}(d)$ being the set of $\mathcal{F}\in\mathbf{F}(d)$ such that for any $s\in\mathrm{Sing}\mathcal{F},$ $\mu(\mathcal{F},s)=1$ and $\tau(\mathcal{F},s)=1$). Then there exists an irreducible component $\Sigma_{1}^{0}(d)$ of $\Sigma_{1}(d)$ such that $U_{1}(d)\cap\Sigma_{1}^{0}(d)\neq\emptyset.$ Let $W_{1}(d)=\bigcup\limits_{i=1}^{k}W_{1}^{i}(d)$ be the decomposition of $W_{1}(d)$ into its irreducible components. Let us denote by $\pi_{0}\hskip 2.84526pt\colon W_{1}(d)\to\mathbf{F}(d)$ the restriction of $\pi$ to $W_{1}(d).$ Then, there is $n\in\\{1,\ldots,k\\}$ such that $\overline{\pi_{0}(W_{1}^{n}(d))}=\overline{\Sigma_{1}^{0}(d)}.$ Indeed, since $\Sigma_{1}(d)=\pi_{0}(W_{1}(d)),$ we have $\overline{\Sigma_{1}^{0}(d)}\subset\overline{\Sigma_{1}(d)}=\bigcup\limits_{i=1}^{k}\overline{\pi_{0}(W_{1}^{i}(d))}.$ The irreducibility of $\Sigma_{1}^{0}(d)$ therefore ensures the existence of $n\in\\{1,\ldots,k\\}$ such that $\overline{\Sigma_{1}^{0}(d)}\subset\overline{\pi_{0}(W_{1}^{n}(d))}\subset\overline{\Sigma_{1}(d)}.$ Since $\overline{\Sigma_{1}^{0}(d)}$ is an irreducible component of $\overline{\Sigma_{1}(d)}$ and since $\overline{\pi_{0}(W_{1}^{n}(d))}$ is irreducible by continuity of $\pi_{0},$ we deduce that $\overline{\pi_{0}(W_{1}^{n}(d))}=\overline{\Sigma_{1}^{0}(d)}.$ Thus, since $U_{1}(d)$ is a Zariski open subset of $\mathbf{F}(d)$ (Proposition 3.8 (i)), the morphism $\pi_{0}$ induces by restriction a dominant morphism of quasi-projective varieties $\pi_{0}^{n}\hskip 2.84526pt\colon W_{1}^{n}(d)\cap\pi_{0}^{-1}(U_{1}(d))\to\overline{\Sigma_{1}^{0}(d)}\cap U_{1}(d).$ Notice that all the fibers of $\pi_{0}$ over the elements of $U_{1}(d)\cap\Sigma_{1}(d)$ are finite and non-empty. Indeed, if $\mathcal{F}\in U_{1}(d)\cap\Sigma_{1}(d)$ then, by Proposition 3.8 (ii), the set $\widehat{\mathrm{QRad}}(\mathcal{F},d-1)$ is finite and non-empty; therefore so is $\pi_{0}^{-1}(\mathcal{F})=\\{\mathcal{F}\\}\times\widehat{\mathrm{QRad}}(\mathcal{F},d-1).$ Since $\pi_{0}(W_{1}^{n}(d)\cap\pi_{0}^{-1}(U_{1}(d)))\subset U_{1}(d)\cap\Sigma_{1}(d),$ it follows that all the non-empty fibers of $\pi_{0}^{n}$ are finite and therefore zero-dimensional. The fiber dimension theorem (_cf._ [14, Theorem 3, page 49]) then implies that $\dim(W_{1}^{n}(d)\cap\pi_{0}^{-1}(U_{1}(d)))=\dim(\overline{\Sigma_{1}^{0}(d)}\cap U_{1}(d));$ since $W_{1}^{n}(d)\cap\pi_{0}^{-1}(U_{1}(d))$ and $\overline{\Sigma_{1}^{0}(d)}\cap U_{1}(d)$ are non-empty open subsets of the irreducible varieties $W_{1}^{n}(d)$ and $\overline{\Sigma_{1}^{0}(d)}$ respectively, we have $\displaystyle\hskip 65.44142pt\dim\overline{\Sigma_{1}^{0}(d)}=\dim(\overline{\Sigma_{1}^{0}(d)}\cap U_{1}(d))=\dim(W_{1}^{n}(d)\cap\pi_{0}^{-1}(U_{1}(d)))=\dim W_{1}^{n}(d).$ Now, from (3.5) we deduce that each irreducible component $W_{1}^{i}(d)$ of $W_{1}(d)$ has dimension $\hskip 41.25641pt\dim W_{1}^{i}(d)\geq\dim(\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}\times\mathbb{\check{P}}^{2}_{\mathbb{C}})-4-(d-1)=\dim\mathbf{F}(d)-(d-1),$ hence $\hskip 44.67102pt\dim\Sigma_{1}(d)=\dim\overline{\Sigma_{1}(d)}\geq\dim\overline{\Sigma_{1}^{0}(d)}=\dim W_{1}^{n}(d)\geq\dim\mathbf{F}(d)-(d-1).$ ∎ Assertion 1. (resp. 2.) of the following proposition gives a necessary (resp. sufficient) condition for a foliation of $\mathbf{F}(d)$ to degenerate onto the foliation $\mathcal{F}_{2}^{d}.$ ###### Proposition 3.11. Let $\mathcal{F}$ be an element of $\mathbf{F}(d)$ such that $\mathcal{F}_{2}^{d}\not\in\mathcal{O}(\mathcal{F}).$ The following assertions hold: 1. If $\mathcal{F}$ degenerates onto $\mathcal{F}_{2}^{d}$, then $\deg\mathrm{I}_{\mathcal{F}}^{\hskip 0.56905pt\mathrm{tr}}\geq d-1.$ 2. If $\mathcal{F}$ admits an inflection point of maximal order $d-1$, i.e. if $\mathrm{Flex}(\mathcal{F},d-1)\neq\emptyset,$ then $\mathcal{F}$ degenerates onto $\mathcal{F}_{2}^{d}.$ ###### Proof. 1. If $\mathcal{F}$ degenerates onto $\mathcal{F}_{2}^{d},$ then $\deg\mathrm{I}_{\mathcal{F}}^{\hskip 0.56905pt\mathrm{tr}}\geq\deg\mathrm{I}_{\mathcal{F}_{2}^{d}}^{\hskip 0.56905pt\mathrm{tr}}.$ An immediate computation shows that $\mathrm{I}_{\mathcal{F}_{2}^{d}}^{\hskip 0.56905pt\mathrm{tr}}=\leavevmode\nobreak\ y^{d-1}$ so that $\deg\mathrm{I}_{\mathcal{F}_{2}^{d}}^{\hskip 0.56905pt\mathrm{tr}}=d-1$, hence the announced inequality holds. 2. Assume that $\mathcal{F}$ possesses such a point. We choose an affine coordinate system $(x,y)$ such that $p=(0,0)$ is an inflection point of order $d-1$ of $\mathcal{F}$ and $x=0$ is the tangent line to the leaf of $\mathcal{F}$ passing through $p.$ Let $\omega$ be a $1$-form defining $\mathcal{F}$ in these coordinates. Since $\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F}=\\{x=0\\}$, $\omega$ is of type $\displaystyle\omega=C_{d}(x,y)(x\mathrm{d}y-y\mathrm{d}x)+\alpha\mathrm{d}x+\sum_{i=1}^{d}\big{(}A_{i}(x,y)\mathrm{d}x+B_{i}(x,y)\mathrm{d}y\big{)},$ $\displaystyle\text{where }A_{i},\,B_{i}\in\mathbb{C}[x,y]_{i},\hskip 2.84526ptC_{d}\in\mathbb{C}[x,y]_{d},\hskip 2.84526pt\alpha\in\mathbb{C}^{}.$ We have $\displaystyle\omega\wedge\mathrm{d}x\Big{\|}_{x=0}=\sum_{i=1}^{d}B_{i}(0,y)\mathrm{d}y\wedge\mathrm{d}x=\sum_{i=1}^{d}B_{i}(0,1)y^{i}\mathrm{d}y\wedge\mathrm{d}x.$ Therefore the hypothesis that $(0,0)$ is an inflection point of order $d-1$ of $\mathcal{F}$ translates into $B_{i}(0,1)=0$ for $i\in\\{1,2,\ldots,d-1\\}$ and $B_{d}(0,1)\neq 0.$ Then we can write $\displaystyle B_{d}(x,y)=x\widehat{B}_{d-1}(x,y)+\beta y^{d},$ $\displaystyle B_{i}(x,y)=x\widetilde{B}_{i-1}(x,y)\hskip 2.84526pt\text{for }i\in\\{1,2,\ldots,d-1\\},$ where $\widetilde{B}_{i-1}\in\mathbb{C}[x,y]_{i-1},\hskip 2.84526pt\widehat{B}_{d-1}\in\mathbb{C}[x,y]_{d-1},\hskip 2.84526pt\beta\in\mathbb{C}^{}.$ Thus $\omega$ is of type $\displaystyle\omega=\alpha\mathrm{d}x+\big{(}x\widehat{B}_{d-1}(x,y)+\beta y^{d}\big{)}\mathrm{d}y+C_{d}(x,y)\big{(}x\mathrm{d}y-y\mathrm{d}x\big{)}+\sum_{i=1}^{d}A_{i}(x,y)\mathrm{d}x+x\sum_{i=1}^{d-1}\widetilde{B}_{i-1}(x,y)\mathrm{d}y,$ where $A_{i}\in\mathbb{C}[x,y]_{i},\hskip 2.84526pt\widetilde{B}_{i-1}\in\mathbb{C}[x,y]_{i-1},\hskip 2.84526pt\widehat{B}_{d-1}\in\mathbb{C}[x,y]_{d-1},\hskip 2.84526pt\alpha,\beta\in\mathbb{C}^{}.$ Let us consider the family of automorphisms $\varphi=\varphi_{\varepsilon}=(\varepsilon^{d+1}x,\varepsilon y).$ We have $\displaystyle\frac{1}{\varepsilon^{d+1}}\varphi^{}\omega=\alpha\mathrm{d}x+\Big{(}\varepsilon^{d}x\widehat{B}_{d-1}(\varepsilon^{d}x,y)+\beta y^{d}\Big{)}\mathrm{d}y+\varepsilon^{d+1}C_{d}(\varepsilon^{d}x,y)\left(x\mathrm{d}y-y\mathrm{d}x\right)+\sum_{i=1}^{d}\varepsilon^{i}A_{i}(\varepsilon^{d}x,y)\mathrm{d}x+x\sum_{i=1}^{d-1}\varepsilon^{i}\widetilde{B}_{i-1}(\varepsilon^{d}x,y)\mathrm{d}y$ which tends to $\alpha\mathrm{d}x+\beta y^{d}\mathrm{d}y$ as $\varepsilon$ tends to $0.$ Clearly $\alpha\mathrm{d}x+\beta y^{d}\mathrm{d}y$ defines a foliation conjugated to $\mathcal{F}_{2}^{d}$; as a result $\mathcal{F}$ degenerates onto $\mathcal{F}_{2}^{d}.$ ∎ ###### Example 3.12. Let us consider the homogeneous foliation $\mathcal{H}_{2}^{d}$ defined in the affine chart $z=1$ by the $1$-form ${\mspace{2.0mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2.0mu}}_{2}^{d}=x^{d}\mathrm{d}x-y^{d}\mathrm{d}y.$ We know from [4, Proposition 4.1] that $\mathcal{H}_{2}^{d}$ has no non- degenerate singularity with Baum-Bott index $4$ and that $\hskip 123.48485pt\mathrm{Flex}(\mathcal{H}_{2}^{d},d-1)=\\{xy=0\\}\setminus\\{[0:0:1]\\}\neq\emptyset.$ Thus $\mathcal{H}_{2}^{d}$ degenerates onto $\mathcal{F}_{2}^{d}$ (Proposition 3.11) and it does not degenerate onto $\mathcal{F}_{1}^{d}$ (Proposition 3.4). Consequently, according to Remark 3.3, we have $\hskip 42.67912pt\overline{\mathcal{O}(\mathcal{H}_{2}^{d})}=\mathcal{O}(\mathcal{H}_{2}^{d})\cup\mathcal{O}(\mathcal{F}_{2}^{d}).$ ###### Example 3.13 (Jouanolou’s foliation). Let us consider the foliation $\mathcal{F}_{J}^{d}$ of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}$ defined, in the affine chart $z=1,$ by $\omega_{J}^{d}=(x^{d}y-1)\mathrm{d}x+(y^{d}-x^{d+1})\mathrm{d}y.$ This example is due to Jouanolou and is historically the first explicit example of foliation without invariant algebraic curve ([12]). The point $p=(0,0)$ is an inflection point of maximal order $d-1$ of $\mathcal{F}_{J}^{d}$ because $\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F}_{J}^{d}=\\{x=0\\}\hskip 5.69054pt\text{and}\hskip 5.69054pt\omega_{J}^{d}\wedge\mathrm{d}x\Big{\|}_{x=0}=y^{d}\mathrm{d}y\wedge\mathrm{d}x.$ As a result $\mathcal{F}_{J}^{d}$ degenerates onto $\mathcal{F}_{2}^{d}$ (Proposition 3.11). However, we know from [13, Section 3] that every singularity $s$ of $\mathcal{F}_{J}^{d}$ is non-degenerate with Baum-Bott index $\mathrm{BB}(\mathcal{F}_{J}^{d},s)=\frac{(d+2)^{2}}{d^{2}+d+1}\neq 4,$ so that $\mathcal{F}_{J}^{d}$ does not degenerate onto $\mathcal{F}_{1}^{d}$ (Proposition 3.4). The converse of assertion 2. of Proposition 3.11 is false as the following example shows. ###### Example 3.14. Let $\mathcal{F}$ be the foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}}$ defined in the affine chart $z=1$ by $\displaystyle\omega=\mathrm{d}x+P(y)\mathrm{d}y,$ $\displaystyle\text{where }P\in\mathbb{C}[y],\hskip 2.84526pt\deg P=d.$ It is easy to check that $\mathrm{Sing}(\mathcal{F})=\big{\\{}[1:0:0]\big{\\}}$ and $\mathrm{I}_{\mathcal{F}}^{\hskip 0.56905pt\mathrm{tr}}=P^{\prime}(y).$ If the derivative $P^{\prime}$ has a single root, i.e if $P$ is of the form $P(y)=a(y-\alpha)^{d}+b,$ where $\alpha,a,b\in\mathbb{C},a\neq 0,$ then $\mathcal{F}$ is conjugated to $\mathcal{F}_{2}^{d}$; indeed, we have $\displaystyle\frac{1}{a}\varphi^{}\omega=\mathrm{d}x+y^{d}\mathrm{d}y,\quad\text{where}\hskip 2.84526pt\varphi=(ax-by,y+\alpha).$ We assume that the derivative $P^{\prime}$ has at least two distinct roots; this implies that $d\geq 3.$ A straightforward computation shows that $\mathcal{F}$ has no inflection point of maximal order $d-1$, i.e. $\mathrm{Flex}(\mathcal{F},d-1)=\emptyset.$ However, $\mathcal{F}$ degenerates onto $\mathcal{F}_{2}^{d}$. Indeed, by writing $P(y)=a_{0}+a_{1}y+\cdots+a_{d}y^{d}$, $a_{i}\in\mathbb{C},\,a_{d}\neq 0,$ and by putting $\psi=$$\left(\dfrac{a_{d}}{\varepsilon^{d+1}}x,\dfrac{1}{\varepsilon}y\right)$, we obtain that $\displaystyle\lim_{\varepsilon\to 0}\frac{\varepsilon^{d+1}}{a_{d}}\psi^{}\omega=\mathrm{d}x+y^{d}\mathrm{d}y.$ ###### Question 2. Let $\mathcal{F}$ be a foliation of degree $d\geq 3$ on $\mathbb{P}^{2}_{\mathbb{C}}.$ Is it true that if $\mathcal{F}$ degenerates onto $\mathcal{F}_{2}^{d}$ then * – either $\mathcal{F}$ possesses an inflection point of maximal order $d-1$, * – or $\mathcal{F}$ is conjugated to Example 3.14, i.e. up to linear conjugation $\mathcal{F}$ is given by a $1$-form of type $\mathrm{d}x+P(y)\mathrm{d}y$ with $P\in\mathbb{C}[y],$ $\deg P=d$? ###### Proposition 3.15. Let $d$ be an integer greater than or equal to $2.$ Let us denote by $U_{2}(d)$ the set of foliations $\mathcal{F}\in\mathbf{F}(d)$ whose inflection divisor $\mathrm{I}_{\mathcal{F}}$ is transverse (i.e. $\mathrm{I}_{\mathcal{F}}=\mathrm{I}_{\mathcal{F}}^{\hskip 0.56905pt\mathrm{tr}}$) and reduced. Then * (i) $U_{2}(d)$ contains the Jouanolou’s foliation $\mathcal{F}_{J}^{d}$ and it is a (non-empty) Zariski open subset of $\mathbf{F}(d);$ * (ii) for any $d\geq 3,$ every foliation $\mathcal{F}\in U_{2}(d)$ has a finite number (possibly zero) of transverse inflection points of order greater than or equal to $2$; in other words, the set $\bigcup\limits_{k=3}^{d}\mathrm{Flex}(\mathcal{F},k-1)$ is finite. To establish this proposition, let us first prove the following lemma. ###### Lemma 3.16. Let $\mathcal{F}$ be a foliation of degree $d\geq 2$ on $\mathbb{P}^{2}_{\mathbb{C}},$ $p$ a regular point of $\mathcal{F}$ and $\mathrm{X}$ a polynomial vector field defining $\mathcal{F}$ in an affine chart $(x,y)$ containing $p.$ Then, for any $k\in\\{2,3,\ldots,d\\},$ $\mathrm{Tang}(\mathcal{F},\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F},p)\geq k$ if and only if the matrix $\left(\begin{array}[]{cccc}\mathrm{X}(x)&\mathrm{X}^{2}(x)&\cdots&\mathrm{X}^{k}(x)\\\ \mathrm{X}(y)&\mathrm{X}^{2}(y)&\cdots&\mathrm{X}^{k}(y)\end{array}\right)\Big{\|}_{p}$ has rank $1.$ ###### Remark 3.17. If $\mathrm{X}=\sum\limits_{i=1}^{n}\mathrm{X}_{i}(z_{1},\ldots,z_{n})\dfrac{\raisebox{-1.42262pt}{$\partial$}}{\partial z_{i}}$ is a holomorphic vector field on $\mathbb{C}^{n}$ and if $t\mapsto\alpha(t)$ is an integral curve of $\mathrm{X},$ then we have the following formula which can be easily proved by induction on $j:$ (3.6) $\displaystyle\frac{\raisebox{-2.27621pt}{$\mathrm{d}^{j}$}}{\mathrm{d}\hskip 0.56905ptt^{j}}\alpha(t)=(\mathrm{X}^{j}(z_{1}),\ldots,\mathrm{X}^{j}(z_{n}))\circ\alpha(t).$ ###### Proof. Let $t\mapsto\alpha(t)$ be the integral curve of $\mathrm{X}$ passing through $p$ at $t=0.$ The point $p$ being regular for $\mathcal{F},$ we have $\mathrm{T}_{p}\mathcal{F}\ni\alpha^{\prime}(0)=\mathrm{X}(p)\neq 0.$ Up to linear conjugation, we can assume that $p=(0,0)$ and $\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F}=\\{y=0\\}.$ We can then write $\alpha(t)=\left(\sum\limits_{i\geq 1}x_{i}\frac{t^{i}}{i!},\sum\limits_{i\geq 1}y_{i}\frac{t^{i}}{i!}\right)$ with $y_{1}=0$ and $x_{1}\neq 0.$ Thus, $\mathrm{Tang}(\mathcal{F},\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F},p)=\nu(g(t),0),$ where $g(t)=\sum\limits_{i\geq 2}y_{i}\frac{t^{i}}{i!}.$ As a result, $\mathrm{Tang}(\mathcal{F},\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F},p)\geq k$ if and only if $y_{2}=y_{3}=\cdots=y_{k}=0$, or equivalently if and only if the matrix $\left(\begin{array}[]{cccc}x_{1}&x_{2}&\cdots&x_{k}\\\ 0&y_{2}&\cdots&y_{k}\end{array}\right)$ has rank $1.$ Now, by using formula (3.6), we see that $\left(\begin{array}[]{cccc}x_{1}&x_{2}&\cdots&x_{k}\\\ 0&y_{2}&\cdots&y_{k}\end{array}\right)=\left(\begin{array}[]{cccc}\mathrm{X}(x)&\mathrm{X}^{2}(x)&\cdots&\mathrm{X}^{k}(x)\\\ \mathrm{X}(y)&\mathrm{X}^{2}(y)&\cdots&\mathrm{X}^{k}(y)\end{array}\right)\Big{\|}_{(x,y)=(0,0)},$ hence the lemma follows. ∎ ###### Proof of Proposition 3.15. (i) For $\mathcal{F}\in\mathbf{F}(d),$ to say that $\mathrm{I}_{\mathcal{F}}$ is transverse and reduced means that $\mathcal{F}$ has no invariant line and that $\mathrm{I}_{\mathcal{F}}$ has no multiple component, which shows that $U_{2}(d)$ is a Zariski open subset of $\mathbf{F}(d).$ As we have already mentioned in Example 3.13, the Jouanolou’s foliation $\mathcal{F}_{J}^{d}$ has no invariant algebraic curve [12]; in particular, it has no invariant line and consequently $\mathrm{I}_{\mathcal{F}_{J}^{d}}=\mathrm{I}_{\mathcal{F}_{J}^{d}}^{\hskip 0.56905pt\mathrm{tr}}.$ To establish the first announced assertion, it remains to prove that $\mathrm{I}_{\mathcal{F}_{J}^{d}}$ is reduced. In homogeneous coordinates, the foliation $\mathcal{F}_{J}^{d}$ is defined by the vector field $y^{d}\frac{\partial}{\partial x}+z^{d}\frac{\partial}{\partial y}+x^{d}\frac{\partial}{\partial z}$; an immediate computation, using formula (1.1), shows that $\mathrm{I}_{\mathcal{F}_{J}^{d}}$ has equation $F(x,y,z)=0,$ where $F(x,y,z)=x^{2d+1}z^{d-1}+y^{2d+1}x^{d-1}+z^{2d+1}y^{d-1}-3x^{d}y^{d}z^{d}.$ We must show that $F$ has no multiple factor in $\mathbb{C}[x,y,z]$. Since $F\in\mathbb{Z}[x,y,z],$ it suffices to show that $F$ has no multiple factor in $\mathbb{F}_{2}[x,y,z].$ Indeed, if $F$ had a multiple factor in $\mathbb{C}[x,y,z],$ then one of the resultants $\mathrm{Res}_{x}(F,\frac{\partial F}{\partial x})\in\mathbb{Z}[y,z]\hskip 1.13809pt$ or $\hskip 1.13809pt\mathrm{Res}_{y}(F,\frac{\partial F}{\partial y})\in\mathbb{Z}[x,z]\hskip 1.13809pt$ or $\hskip 1.13809pt\mathrm{Res}_{z}(F,\frac{\partial F}{\partial z})\in\mathbb{Z}[x,y]$ would be identically zero and therefore so would be its reduction modulo $2$; so that $F$ would also have a multiple factor in $\mathbb{F}_{\mathrm{2}}[x,y,z].$ We have to show that $\mathrm{gcd}(F,\frac{\partial F}{\partial x},\frac{\partial F}{\partial y},\frac{\partial F}{\partial z})=1$ in $\mathbb{F}_{2}[x,y,z],$ or equivalently that $\displaystyle\mathrm{gcd}(F,\tfrac{\partial F}{\partial x})=1\hskip 2.84526pt\text{in}\hskip 2.84526pt\mathbb{F}_{2}(y,z)[x],$ $\displaystyle\mathrm{gcd}(F,\tfrac{\partial F}{\partial y})=1\hskip 2.84526pt\text{in}\hskip 2.84526pt\mathbb{F}_{2}(x,z)[y],$ $\displaystyle\mathrm{gcd}(F,\tfrac{\partial F}{\partial z})=1\hskip 2.84526pt\text{in}\hskip 2.84526pt\mathbb{F}_{2}(x,y)[z].$ The coordinates $x,y,z$ playing a symmetric role, it suffices again to show that $\mathrm{gcd}(F,\tfrac{\partial F}{\partial x})=1$ in $\mathbb{F}_{2}(y,z)[x].$ In $\mathbb{F}_{2}[x,y,z]$ we have $\displaystyle F=x^{2d+1}z^{d-1}+y^{2d+1}x^{d-1}+z^{2d+1}y^{d-1}+x^{d}y^{d}z^{d}$ and $\displaystyle\tfrac{\partial F}{\partial x}=x^{d-2}\left(x^{d+2}z^{d-1}+dxy^{d}z^{d}+(d+1)y^{2d+1}\right).$ Then $x=0$ is not a root of $F\in\mathbb{F}_{2}(y,z)[x]$ and consequently $\displaystyle\mathbb{F}_{2}(y,z)[x]\ni\gcd(F,\tfrac{\partial F}{\partial x})=\gcd(F,\varphi),\quad\text{where}\quad\varphi=x^{d+2}+dxzy^{d}+(d+1)\frac{y^{2d+1}}{z^{d-1}}.$ Moreover, a straightforward computation shows that $\displaystyle x^{3}F=\left(x^{d+2}z^{d-1}-(d-1)xy^{d}z^{d}-dy^{2d+1}\right)\varphi+y^{d-1}z^{2d+1}\left(x+\frac{y^{d+1}}{z^{d}}\right)\left(x^{2}+(d^{2}-d-1)\frac{y^{d+1}}{z^{d}}x+d(d+1)\frac{y^{2d+2}}{z^{2d}}\right),$ so that $\displaystyle\mathbb{F}_{2}(y,z)[x]\ni\gcd(F,\varphi)$ $\displaystyle=\gcd\left(\Big{(}x+\frac{y^{d+1}}{z^{d}}\Big{)}\Big{(}x-\frac{y^{d+1}}{z^{d}}\Big{)},\,\varphi\right),\hskip 2.84526pt\text{because}\hskip 2.84526ptd^{2}-d\equiv d(d+1)\equiv 0\mod 2$ $\displaystyle=\gcd\left(x-\frac{y^{d+1}}{z^{d}},\,x^{d+2}+dxzy^{d}+(d+1)\frac{y^{2d+1}}{z^{d-1}}\right)$ $\displaystyle=\gcd\left(x-\frac{y^{d+1}}{z^{d}},\,x^{d+2}-\frac{y^{2d+1}}{z^{d-1}}\right)$ $\displaystyle=1,$ because $\left(\dfrac{y^{d+1}}{z^{d}}\right)^{d+2}\neq\dfrac{y^{2d+1}}{z^{d-1}}$ in the field $\mathbb{F}_{2}(y,z).$ As a result $\mathbb{F}_{2}(y,z)[x]\ni\gcd(F,\tfrac{\partial F}{\partial x})=1.$ (ii) Let $\mathcal{F}$ be a foliation of degree $d\geq 3$ on $\mathbb{P}^{2}_{\mathbb{C}}$ with reduced and transverse inflection divisor $\mathrm{I}_{\mathcal{F}}$, i.e. $\mathcal{F}\in U_{2}(d).$ We want to show that the set $\Gamma(\mathcal{F}):=\bigcup\limits_{k=3}^{d}\mathrm{Flex}(\mathcal{F},k-1)$ is finite. By definition of $\Gamma(\mathcal{F})$ we have (3.7) $\displaystyle\Gamma(\mathcal{F})\subset\Big{\\{}p\in\mathbb{P}^{2}_{\mathbb{C}}\hskip 2.84526pt\|\hskip 2.84526ptp\not\in\mathrm{Sing}(\mathcal{F}),\hskip 2.84526pt\mathrm{Tang}(\mathcal{F},\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F},p)\geq 3\Big{\\}}.$ Let $\mathrm{X}$ be a vector field defining $\mathcal{F}$ in an affine chart $\mathbb{C}^{2}=\\{(x,y)\\}\subset\mathbb{P}^{2}_{\mathbb{C}}.$ Lemma 3.16 and inclusion (3.7) imply that $\Gamma(\mathcal{F})\cap\mathbb{C}^{2}$ is contained in the set of points $p\in\mathbb{C}^{2}$ such that $\displaystyle\left(\begin{array}[]{c}\mathrm{X}(x)\\\ \mathrm{X}(y)\end{array}\right)(p)\neq\left(\begin{array}[]{c}0\\\ 0\end{array}\right),$ $\displaystyle\mathrm{I_{X}}(p):=\left\|\begin{array}[]{cc}\mathrm{X}(x)&\mathrm{X}^{2}(x)\\\ \mathrm{X}(y)&\mathrm{X}^{2}(y)\end{array}\right\|(p)=0,$ $\displaystyle\mathrm{X}(\mathrm{I_{X}})(p)=\left\|\begin{array}[]{cc}\mathrm{X}(x)&\mathrm{X}^{3}(x)\\\ \mathrm{X}(y)&\mathrm{X}^{3}(y)\end{array}\right\|(p)=0.$ Now, the affine chart $\mathbb{C}^{2}=\\{(x,y)\\}\subset\mathbb{P}^{2}_{\mathbb{C}}$ being arbitrary, $\Gamma(\mathcal{F})$ is finite if and only if $\Gamma(\mathcal{F})\cap\mathbb{C}^{2}$ is finite. It suffices therefore to show that the algebraic curves $\mathrm{I}_{\mathcal{F}}\cap\mathbb{C}^{2}=\\{\mathrm{I_{X}}(x,y)=0\\}$ and $\mathcal{C}:=\\{\mathrm{X}(\mathrm{I}_{\mathrm{X}})(x,y)=0\\}$ intersect at a finite number of points, i.e. that they have no common component. Let us argue by contradiction and assume that there exist $K,L,L^{\prime}\in\mathbb{C}[x,y],$ with $\deg K>0,$ such that $\mathrm{I_{X}}=KL$ and $\mathrm{X}(\mathrm{I}_{\mathrm{X}})=KL^{\prime}.$ Then $KL^{\prime}=\mathrm{X}(KL)=\mathrm{X}(K)L+K\mathrm{X}(L)$ and therefore $\mathrm{X}(K)L=K(L^{\prime}-\mathrm{X}(L)).$ Moreover, the hypothesis that $\mathrm{I}_{\mathcal{F}}$ is reduced implies that $\mathrm{gcd}(K,L)=1.$ It follows that there is $L^{\prime\prime}\in\mathbb{C}[x,y]$ such that $\mathrm{X}(K)=KL^{\prime\prime}$, which means that the algebraic curve $\mathcal{C}^{\prime}:=\\{K(x,y)=0\\},$ contained in $\mathrm{I}_{\mathcal{F}}$, is invariant by $\mathcal{F},$ contradicting the hypothesis that $\mathrm{I}_{\mathcal{F}}$ is transverse. ∎ ###### Theorem 3.18. Let $d$ be an integer greater than or equal to $2.$ Let us denote by $\Sigma_{2}(d)$ the subset of $\mathbf{F}(d)$ defined by $\displaystyle\Sigma_{2}(d):=\Big{\\{}\mathcal{F}\in\mathbf{F}(d)\hskip 2.84526pt\|\hskip 2.84526pt\mathrm{Flex}(\mathcal{F},d-1)\neq\emptyset\Big{\\}}.$ Then * (a) $\mathbf{B}(\mathcal{F}_{2}^{2})=\mathbf{F}(2)\setminus\mathbf{FC}(2)=\Sigma_{2}(2)$ and, for any $d\geq 3$, we have $\emptyset\neq\Sigma_{2}(d)\varsubsetneq\mathbf{B}(\mathcal{F}_{2}^{d});$ * (b) $\Sigma_{2}(d)$ is a constructible subset of $\mathbf{F}(d)$; * (c) for any $d\geq 3$, we have $\dim\Sigma_{2}(d)\geq\dim\mathbf{F}(d)-(d-3).$ In particular, the set $\Sigma_{2}(3)$, and therefore $\mathbf{B}(\mathcal{F}_{2}^{3}),$ contains a non-empty Zariski open subset of $\mathbf{F}(3).$ ###### Proof. (a) As we have already said in Introduction, the first equality $\mathbf{B}(\mathcal{F}_{2}^{2})=\mathbf{F}(2)\setminus\mathbf{FC}(2)$ follows from [9, Theorem 3]. The second equality $\mathbf{F}(2)\setminus\mathbf{FC}(2)=\Sigma_{2}(2)$ is a consequence of the following obvious remark: if $\mathcal{F}\in\mathbf{F}(2)\setminus\mathbf{FC}(2)$ then every transverse inflection point of $\mathcal{F}$ is of order $1$. The set $\Sigma_{2}(d)$ contains the foliations $\mathcal{H}_{2}^{d}$ and $\mathcal{F}_{J}^{d}$ (Examples 3.12 and 3.13) and is therefore non-empty. According to assertion 2. of Proposition 3.11, we have $\Sigma_{2}(d)\subset\mathbf{B}(\mathcal{F}_{2}^{d})$; this inclusion is strict for any $d\geq 3$ as Example 3.14 shows. (b) Let $\pi\hskip 2.84526pt\colon\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}\to\mathbf{F}(d)$ be the projection onto the first factor; notice that $\Sigma_{2}(d)=\pi(W_{2}(d)),$ where $\displaystyle W_{2}(d):$ $\displaystyle=\bigcup_{\mathcal{F}\in\Sigma_{2}(d)}\\{\mathcal{F}\\}\times\mathrm{Flex}(\mathcal{F},d-1)$ $\displaystyle=\Big{\\{}(\mathcal{F},p)\in\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}\hskip 2.84526pt\|\hskip 2.84526ptp\not\in\mathrm{Sing}(\mathcal{F}),\hskip 2.84526pt\mathrm{Tang}(\mathcal{F},\mathrm{T}^{\mathbb{P}}_{p}\mathcal{F},p)=d\Big{\\}}.$ By Lemma 3.16, $W_{2}(d)$ can be rewritten as (3.14) $\displaystyle W_{2}(d)=\Big{\\{}(\mathcal{F},p)\in\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}\hskip 2.84526pt\|\hskip 2.84526pt\left(\begin{array}[]{c}\mathrm{X}(x)\\\ \mathrm{X}(y)\end{array}\right)(p)\neq\left(\begin{array}[]{c}0\\\ 0\end{array}\right),\hskip 2.84526pt\left\|\begin{array}[]{cc}\mathrm{X}(x)&\mathrm{X}^{j}(x)\\\ \mathrm{X}(y)&\mathrm{X}^{j}(y)\end{array}\right\|(p)=0,j=2,\ldots,d\Big{\\}},$ where $\mathrm{X}$ denotes a polynomial vector field defining $\mathcal{F}$ in an affine chart $(x,y)$ containing $p.$ It follows that $W_{2}(d)$ is a quasi- projective subvariety of $\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}}.$ Therefore, by Chevalley’s theorem [11, Exercise II.3.19], the set $\Sigma_{2}(d)=\pi(W_{2}(d))$ is constructible. (c) From the above discussion and Proposition 3.15 (i), we have $\mathcal{F}_{J}^{d}\in U_{2}(d)\cap\Sigma_{2}(d)\neq\emptyset$ ($U_{2}(d)$ being the set of foliations of $\mathbf{F}(d)$ with reduced and transverse inflection divisor). Therefore there exists an irreducible component $\Sigma_{2}^{0}(d)$ of $\Sigma_{2}(d)$ such that $U_{2}(d)\cap\Sigma_{2}^{0}(d)\neq\emptyset.$ We denote by $\pi_{0}\hskip 2.84526pt\colon W_{2}(d)\to\mathbf{F}(d)$ the restriction of $\pi$ to $W_{2}(d).$ Let $W_{2}(d)=\bigcup\limits_{i=1}^{n}W_{2}^{i}(d)$ be the decomposition of $W_{2}(d)$ into its irreducible components. Then, by arguing as in the proof of Theorem 3.10, we see that there is $k\in\\{1,\ldots,n\\}$ such that $\overline{\pi_{0}(W_{2}^{k}(d))}=\overline{\Sigma_{2}^{0}(d)}.$ Since $U_{2}(d)$ is a Zariski open subset of $\mathbf{F}(d)$ (Proposition 3.15 (i)), the morphism $\pi_{0}$ therefore induces by restriction a dominant morphism of quasi-projective varieties $\pi_{0}^{k}\hskip 2.84526pt\colon W_{2}^{k}(d)\cap\pi_{0}^{-1}(U_{2}(d))\to\overline{\Sigma_{2}^{0}(d)}\cap U_{2}(d).$ Notice that, for any $\mathcal{F}\in U_{2}(d)\cap\Sigma_{2}(d),$ the fiber $\pi_{0}^{-1}(\mathcal{F})$ is finite and non-empty, because $\pi_{0}^{-1}(\mathcal{F})=\\{\mathcal{F}\\}\times\mathrm{Flex}(\mathcal{F},d-1)$ and $\mathrm{Flex}(\mathcal{F},d-1)$ is finite and non-empty by assertion (ii) of Proposition 3.15. Since $\pi_{0}(W_{2}^{k}(d)\cap\pi_{0}^{-1}(U_{2}(d)))\subset U_{2}(d)\cap\Sigma_{2}(d),$ we deduce that all the non-empty fibers of $\pi_{0}^{k}$ are finite and therefore zero-dimensional. The fiber dimension theorem (_cf._ [14, Theorem 3, page 49]) then ensures that $\dim(W_{2}^{k}(d)\cap\pi_{0}^{-1}(U_{2}(d)))=\dim(\overline{\Sigma_{2}^{0}(d)}\cap U_{2}(d))$; since $W_{2}^{k}(d)\cap\pi_{0}^{-1}(U_{2}(d))$ and $\overline{\Sigma_{2}^{0}(d)}\cap U_{2}(d)$ are non-empty open subsets of the irreducible varieties $W_{2}^{k}(d)$ and $\overline{\Sigma_{2}^{0}(d)}$ respectively, we have $\displaystyle\hskip 65.44142pt\dim\overline{\Sigma_{2}^{0}(d)}=\dim(\overline{\Sigma_{2}^{0}(d)}\cap U_{2}(d))=\dim(W_{2}^{k}(d)\cap\pi_{0}^{-1}(U_{2}(d)))=\dim W_{2}^{k}(d).$ Now, it follows from (3.14) that each irreducible component $W_{2}^{i}(d)$ of $W_{2}(d)$ has dimension $\dim W_{2}^{i}(d)\geq\dim(\mathbf{F}(d)\times\mathbb{P}^{2}_{\mathbb{C}})-(d-1)=\dim\mathbf{F}(d)-(d-3),$ hence $\hskip 44.10185pt\dim\Sigma_{2}(d)=\dim\overline{\Sigma_{2}(d)}\geq\dim\overline{\Sigma_{2}^{0}(d)}=\dim W_{2}^{k}(d)\geq\dim\mathbf{F}(d)-(d-3).$ The subset $\Sigma_{2}(d)\subset\mathbf{F}(d)$ being constructible, it contains a dense open subset of its closure $\overline{\Sigma_{2}(d)}$. In degree $d=3$ we have $\dim\overline{\Sigma_{2}(3)}\geq\dim\mathbf{F}(3)$ and therefore $\dim\overline{\Sigma_{2}(3)}=\dim\mathbf{F}(3),$ so that $\overline{\Sigma_{2}(3)}=\mathbf{F}(3)$ because $\mathbf{F}(3)$ is irreducible. It follows that $\Sigma_{2}(3)$ contains a dense open subset of $\mathbf{F}(3)$. This ends the proof of the theorem. ∎ ###### Remark 3.19. The set $\mathbf{F}(d)$ contains elements which degenerate onto both $\mathcal{F}_{1}^{d}$ and $\mathcal{F}_{2}^{d},$ e.g. the family of foliations $\mathcal{G}^{d}(\gamma)$, $\gamma\in\mathbb{C}.$ Indeed, on the one hand, we have seen (Example 3.6) that $\mathcal{G}^{d}(\gamma)$ degenerates onto $\mathcal{F}_{1}^{d}.$ On the other hand, by putting $\varphi=(\frac{x}{\varepsilon},\frac{y}{\varepsilon})$ we obtain that $\lim\limits_{\varepsilon\to 0}\varepsilon^{d+1}\varphi^{}\eta^{d}(\gamma)={\mspace{2.0mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2.0mu}}_{2}^{d},$ which shows that $\mathcal{G}^{d}(\gamma)$ degenerates onto the homogeneous foliation $\mathcal{H}_{2}^{d}$ (Example 3.12) and therefore, by transitivity, onto $\mathcal{F}_{2}^{d}.$ ###### Example 3.20. Let us consider the homogeneous foliation $\mathcal{H}_{1,2}^{d}$ defined in the affine chart $z=1$ by the $1$-form ${\mspace{2.0mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2.0mu}}_{1,2}^{d}=(x^{d}+y^{d})\mathrm{d}x+x^{d}\mathrm{d}y.$ This foliation degenerates onto both $\mathcal{F}_{1}^{d}$ and $\mathcal{F}_{2}^{d}.$ Indeed, on the one hand, $\mathcal{H}_{1,2}^{d}$ is given in the affine chart $y=1$ by $\hskip 29.87547pt{\mspace{2.0mu}\overline{\mspace{-1.4mu}\theta\mspace{-1.4mu}}\mspace{2.0mu}}_{1,2}^{d}=x\mathrm{d}z-z\mathrm{d}x+x^{d}\mathrm{d}z+x^{d}(x\mathrm{d}z-z\mathrm{d}x);$ we see that the point $[0:1:0]$ is a radial singularity of maximal order $d-1$ of $\mathcal{H}_{1,2}^{d}.$ Thus, by Proposition 3.4, $\mathcal{H}_{1,2}^{d}$ degenerates onto $\mathcal{F}_{1}^{d}.$ On the other hand, a straightforward computation shows that $\hskip 56.9055pt\mathrm{Flex}(\mathcal{H}_{1,2}^{d},d-1)=\\{y=0\\}\setminus\\{[0:0:1]\\}\neq\emptyset;$ consequently, $\mathcal{H}_{1,2}^{d}$ also degenerates onto $\mathcal{F}_{2}^{d}$ (Proposition 3.11). Since $\overline{\mathcal{O}(\mathcal{H}_{1,2}^{d})}\subset\mathcal{O}(\mathcal{H}_{1,2}^{d})\cup\mathcal{O}(\mathcal{F}_{1}^{d})\cup\mathcal{O}(\mathcal{F}_{2}^{d})$ (Remark 3.3), we deduce that in fact $\hskip 19.91684pt\overline{\mathcal{O}(\mathcal{H}_{1,2}^{d})}=\mathcal{O}(\mathcal{H}_{1,2}^{d})\cup\mathcal{O}(\mathcal{F}_{1}^{d})\cup\mathcal{O}(\mathcal{F}_{2}^{d}).$ ###### Theorem 3.21. Let $d$ be an integer greater than or equal to $2.$ Then (a) $\emptyset\neq\Sigma_{1}(d)\cap\Sigma_{2}(d)\subset\mathbf{B}(\mathcal{F}_{1}^{d})\cap\mathbf{B}(\mathcal{F}_{2}^{d})\supset\mathbf{B}(\mathcal{H}_{1,2}^{d});$ * (b) $\mathbf{B}(\mathcal{H}_{1,2}^{d})$ contains a quasi-projective subvariety of $\mathbf{F}(d)$ of dimension equal to $\dim\mathbf{F}(d)-3d.$ ###### Proof. (a) The intersection $\Sigma_{1}(d)\cap\Sigma_{2}(d)$ contains the homogeneous foliation $\mathcal{H}_{1,2}^{d}$ (Example 3.20) and is therefore non-empty. The inclusion $\Sigma_{1}(d)\cap\Sigma_{2}(d)\subset\mathbf{B}(\mathcal{F}_{1}^{d})\cap\mathbf{B}(\mathcal{F}_{2}^{d})$ follows from Theorems 3.10 and 3.18. Let us show the inclusion $\mathbf{B}(\mathcal{H}_{1,2}^{d})\subset\mathbf{B}(\mathcal{F}_{1}^{d})\cap\mathbf{B}(\mathcal{F}_{2}^{d}).$ Let $\mathcal{F}\in\mathbf{B}(\mathcal{H}_{1,2}^{d})$, i.e. $\mathcal{F}\in\mathbf{F}(d)$ such that $\mathcal{H}_{1,2}^{d}\in\overline{\mathcal{O}(\mathcal{F})}.$ Since $\mathcal{H}_{1,2}^{d}$ degenerates onto $\mathcal{F}_{i}^{d},i=1,2,$ it follows that $\mathcal{F}_{i}^{d}\in\overline{\mathcal{O}(\mathcal{H}_{1,2}^{d})}\subset\overline{\mathcal{O}(\mathcal{F})}$, hence $\mathcal{F}\in\mathbf{B}(\mathcal{F}_{1}^{d})\cap\mathbf{B}(\mathcal{F}_{2}^{d}).$ (b) Let us denote by $\Sigma(\mathcal{H}_{1,2}^{d})$ the subset of $\mathbf{F}(d)$ defined as follows: an element $\mathcal{F}$ of $\mathbf{F}(d)$ belongs to $\Sigma(\mathcal{H}_{1,2}^{d})$ if and only if * (1) $\mathcal{F}$ admits an invariant line $\ell$; * (2) there is a system of homogeneous coordinates $[x:y:z]\in\mathbb{P}^{2}_{\mathbb{C}}$ in which $\ell=\\{z=0\\}$ and $\mathcal{F}$ is defined in the affine chart $z=1$ by a $1$-form $\omega$ of type $\displaystyle\omega=\sum\limits_{i=0}^{d-1}\omega_{i}+\lambda{\mspace{2.0mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2.0mu}}_{1,2}^{d}=\sum\limits_{i=0}^{d-1}\omega_{i}+\lambda\left((x^{d}+y^{d})\mathrm{d}x+x^{d}\mathrm{d}y\right),$ where $\lambda\in\mathbb{C}^{}$ and the $\omega_{i}$’s are homogeneous $1$-forms of degree $i.$ Notice that $\Sigma(\mathcal{H}_{1,2}^{d})\subset\mathbf{B}(\mathcal{H}_{1,2}^{d}).$ Indeed, by putting $\varphi=(\frac{x}{\varepsilon},\frac{y}{\varepsilon})$ and by writing $\omega_{i}=P_{i}(x,y)\mathrm{d}x+Q_{i}(x,y)\mathrm{d}y,$ where $P_{i},Q_{i}\in\mathbb{C}[x,y]_{i},$ we obtain $\varepsilon^{d+1}\varphi^{}\omega=\sum_{i=0}^{d-1}(\varepsilon^{d-i}P_{i}(x,y)\mathrm{d}x+\varepsilon^{d-i}Q_{i}(x,y)\mathrm{d}y)+\lambda{\mspace{2.0mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2.0mu}}_{1,2}^{d}$ which tends to $\lambda{\mspace{2.0mu}\overline{\mspace{-1.4mu}\omega\mspace{-1.4mu}}\mspace{2.0mu}}_{1,2}^{d}$ as $\varepsilon$ tends to $0$. It follows that $\mathcal{H}_{1,2}^{d}\in\overline{\mathcal{O}(\mathcal{F})}$ for any $\mathcal{F}\in\Sigma(\mathcal{H}_{1,2}^{d}),$ hence the inclusion $\Sigma(\mathcal{H}_{1,2}^{d})\subset\mathbf{B}(\mathcal{H}_{1,2}^{d})$ holds. Moreover, every foliation $\mathcal{F}\in\mathbf{F}(d)$ is given in the affine chart $z=1$ by a $1$-form of type $\displaystyle\sum\limits_{i=0}^{d}\big{(}A_{i}(x,y)\mathrm{d}x+B_{i}(x,y)\mathrm{d}y\big{)}+C_{d}(x,y)(x\mathrm{d}y-y\mathrm{d}x),$ where $A_{i},B_{i}\in\mathbb{C}[x,y]_{i},C_{d}\in\mathbb{C}[x,y]_{d}$ with $\mathrm{gcd}\big{(}yC_{d}-\sum\limits_{i=0}^{d}A_{i},xC_{d}+\sum\limits_{i=0}^{d}B_{i}\big{)}=1$. Condition (2) is then equivalent to taking $C_{d}\equiv 0,\,A_{d}(x,y)=\lambda(x^{d}+y^{d}),\,B_{d}(x,y)=\lambda\,x^{d}.$ Since the set of foliations of $\mathbf{F}(d)$ admitting an invariant line is a Zariski closed subset of $\mathbf{F}(d),$ we deduce that $\Sigma(\mathcal{H}_{1,2}^{d})$ is a quasi-projective subvariety of $\mathbf{F}(d).$ Since $\omega$ and $\mu\omega$ define the same foliation if $\mu\neq 0,$ and the choice of a line $\ell\subset\mathbb{P}^{2}_{\mathbb{C}}$ is equivalent to the choice of a point in $\mathbb{\check{P}}^{2}_{\mathbb{C}},$ conditions (1) and (2) imply that $\dim\Sigma(\mathcal{H}_{1,2}^{d})=2+2\sum_{i=0}^{d-1}(i+1)=d^{2}+d+2=\dim\mathbf{F}(d)-3d.$ ∎ ## 4 A family of foliations of $\mathbf{F}(d)$ with orbits of dimension less than or equal to $7$ In this section we will establish some properties of the family $(\mathcal{F}_{0}^{d}(\lambda))_{\lambda\in\mathbb{C}^{}}$ of foliations of degree $d$ on $\mathbb{P}^{2}_{\mathbb{C}}$ defined in the affine chart $z=1$ by $\omega_{0}^{d}(\lambda)=x\mathrm{d}y-\lambda y\mathrm{d}x+y^{d}\mathrm{d}y.$ In homogeneous coordinates, $\mathcal{F}_{0}^{d}(\lambda)$ is given by $\Omega_{0}^{d}(\lambda)=-\lambda yz^{d}\mathrm{d}x+z\left(xz^{d-1}+y^{d}\right)\mathrm{d}y+y\left((\lambda-1)xz^{d-1}-y^{d}\right)\mathrm{d}z.$ Thus, the singular locus of $\mathcal{F}_{0}^{d}(\lambda)$ consists of the two points $s_{1}=[0:0:1]$ and $s_{2}=[1:0:0]$. The singularity $s_{1}$ is non- degenerate with Baum-Bott index $\mathrm{BB}(\mathcal{F}_{0}^{d}(\lambda),s_{1})=2+\lambda+\frac{1}{\lambda}$ and the singular point $s_{2}$ has maximal algebraic multiplicity $d.$ We see that for $\lambda=1$ the $1$-form $\Omega_{0}^{d}(1)$ writes in the affine chart $x=1$ as $z^{d}\mathrm{d}y+y^{d}(z\mathrm{d}y-y\mathrm{\mathrm{d}}z);$ we deduce that $\mathcal{F}_{0}^{d}(1)$ is conjugated to the foliation $\mathcal{F}_{1}^{d}$ and is therefore convex. In the sequel we assume that $\lambda\in\mathbb{C}\setminus\\{0,1\\}$. A direct computation, using formula (1.1), leads to (4.1) $\displaystyle\mathrm{I}_{\mathcal{F}_{0}^{d}(\lambda)}^{\hskip 0.56905pt\mathrm{inv}}=yz^{2d-1}$ and $\displaystyle\mathrm{I}_{\mathcal{F}_{0}^{d}(\lambda)}^{\hskip 0.56905pt\mathrm{tr}}=(\lambda-1)x-\big{(}(d-1)\lambda+1\big{)}y^{d};$ it follows that, for any $\lambda\in\mathbb{C}\setminus\\{0,1\\}$, $\mathcal{F}_{0}^{d}(\lambda)$ is not convex. A straightforward computation shows that the algebraic curve $(1-\lambda\hskip 0.56905ptd)x+y^{d}=0$ is invariant by $\mathcal{F}_{0}^{d}(\lambda).$ What is more, the rational $1$-form $\eta_{0}^{d}(\lambda)=\dfrac{\omega_{0}^{d}(\lambda)}{y\left(\left(1-\lambda\hskip 0.56905ptd\right)x+y^{d}\right)}$ is closed. For $\lambda=\frac{1}{d}$ we note that $\eta_{0}^{d}(\frac{1}{d})=\dfrac{\omega_{0}^{d}(\lambda)}{y^{d+1}}$ has as first integral $\dfrac{x}{dy^{d}}-\ln y;$ this allows to see that $\mathrm{Iso}(\mathcal{F}_{0}^{d}(\frac{1}{d}))$ is the group $\\{(\alpha^{d}x,\alpha y)\hskip 2.84526pt\|\hskip 2.84526pt\alpha\in\mathbb{C}^{}\\}.$ When $\lambda\in\mathbb{C}\setminus\\{0,1,\frac{1}{d}\\}$ a straightforward computation shows that $\eta_{0}^{d}(\lambda)$ integrates into $\lambda\ln\left(\left(1-\lambda\hskip 0.56905ptd\right)x+y^{d}\right)-\ln y,$ which allows to verify that the isotropy group is here again $\mathrm{Iso}(\mathcal{F}_{0}^{d}(\lambda))=\\{(\alpha^{d}x,\alpha y)\hskip 2.84526pt\|\hskip 2.84526pt\alpha\in\mathbb{C}^{}\\}.$ It follows in particular that, for any $\lambda\in\mathbb{C}\setminus\\{0,1\\}$, $\mathcal{O}(\mathcal{F}_{0}^{d}(\lambda))$ has dimension $7.$ Notice that two foliations $\mathcal{F}_{0}^{d}(\lambda)$ and $\mathcal{F}_{0}^{d}(\lambda^{\prime})$ are conjugated if and only if $\lambda=\lambda^{\prime}.$ ###### Proposition 4.1. Let $\lambda$ be a nonzero complex number. Let $\mathcal{F}$ be an element of $\mathbf{F}(d)$ such that $\mathcal{F}_{0}^{d}(\lambda)\not\in\mathcal{O}(\mathcal{F}).$ 1. If $\mathcal{F}$ degenerates onto $\mathcal{F}_{0}^{d}(\lambda)$, then $\mathcal{F}$ admits a non-degenerate singular point $m$ satisfying $\mathrm{BB}(\mathcal{F},m)=2+\lambda+\frac{1}{\lambda}$. 2. If $\mathcal{F}$ possesses a non-degenerate singular point $m$ such that $\mathrm{BB}(\mathcal{F},m)=2+\lambda+\frac{1}{\lambda}\qquad\text{and}\qquad\kappa(\mathcal{F},m)=d,$ then $\mathcal{F}$ degenerates onto $\mathcal{F}_{0}^{d}(\lambda).$ ###### Proof. It suffices to argue as in the proof of Proposition 3.4, replacing the foliation $\mathcal{F}_{1}^{d}$ by $\mathcal{F}_{0}^{d}(\lambda)$ and the equality $\mathrm{BB}(\mathcal{F},m)=4$ by $\mathrm{BB}(\mathcal{F},m)=2+\lambda+\frac{1}{\lambda}.$ ∎ ###### Proposition 4.2. The orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{d}(\lambda)\big{)}$ is closed in $\mathbf{F}(d)$ in the following two cases: (i) $d\geq 3$ and $\lambda=-\dfrac{1}{d-1};$ * (ii) $d\in\\{3,4,5\\}$ and $\lambda\in\mathbb{C}^{}.$ The proof of this proposition uses the following lemma. ###### Lemma 4.3. Let $\lambda$ be a nonzero complex number. Then, the orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{d}(\lambda)\big{)}$ is closed in $\mathbf{F}(d)$ if and only if $\mathcal{F}_{0}^{d}(\lambda)$ does not degenerate onto $\mathcal{F}_{2}^{d}.$ ###### Proof. The direct implication is obvious. Let us prove the converse. From the above discussion, $\mathcal{F}_{0}^{d}(1)$ is conjugated to the convex foliation $\mathcal{F}_{1}^{d}$; therefore its orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{d}(1))$ is closed in $\mathbf{F}(d).$ For any $\lambda\in\mathbb{C}\setminus\\{0,1\\}$, the unique non-degenerate singular point $s_{1}=[0:0:1]$ of $\mathcal{F}_{0}^{d}(\lambda)$ has Baum-Bott index $\mathrm{BB}(\mathcal{F}_{0}^{d}(\lambda),s_{1})=2+\lambda+\frac{1}{\lambda}\neq 4$; this implies, according to assertion 1. of Proposition 3.4, that $\mathcal{F}_{0}^{d}(\lambda)$ does not degenerate onto $\mathcal{F}_{1}^{d}.$ Moreover, for any $\lambda\in\mathbb{C}\setminus\\{0,1\\}$, $\mathcal{O}(\mathcal{F}_{0}^{d}(\lambda))$ has dimension $7.$ The converse implication then follows immediately from Corollary B. ∎ ###### Proof of Proposition 4.2. (i) Let us put $\lambda_{0}=-\frac{1}{d-1}$; according to (4.1) we have $\mathrm{I}_{\mathcal{F}_{0}^{d}(\lambda_{0})}^{\hskip 0.56905pt\mathrm{tr}}=(\lambda_{0}-1)x,$ hence $\deg\mathrm{I}_{\mathcal{F}_{0}^{d}(\lambda_{0})}^{\hskip 0.56905pt\mathrm{tr}}=1<d-1$ for any $d\geq 3.$ According to the first assertion of Proposition 3.11, it follows that, for any $d\geq 3,$ the foliation $\mathcal{F}_{0}^{d}(\lambda_{0})$ does not degenerate onto $\mathcal{F}_{2}^{d},$ so that its orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{d}(\lambda_{0})\big{)}$ is closed in $\mathbf{F}(d)$ (Lemma 4.3). (ii) Let $[x:y:z]$ be homogeneous coordinates in $\mathbb{P}^{2}_{\mathbb{C}}.$ For $n\in\mathbb{N}$, let us denote by $\Lambda^{1}_{n}$ the $\mathbb{C}$-vector space of $1$-forms in the variables $x,y,z,$ whose coefficients are homogeneous polynomials of degree $n.$ Let us put $\alpha=y\mathrm{d}z-z\mathrm{d}y,$ $\beta=z\mathrm{d}x-x\mathrm{d}z$ and $\gamma=x\mathrm{d}y-y\mathrm{d}x.$ We have the identification $\displaystyle\mathbf{F}(d)$ $\displaystyle=$ $\displaystyle\left\\{[\Omega]\in\mathbb{P}(\Lambda^{1}_{d+1})\hskip 2.84526pt\|\hskip 2.84526pt\Omega=p\mathrm{d}x+q\mathrm{d}y+r\mathrm{d}z,\hskip 2.84526ptp,q,r\in\mathbb{C}[x,y,z]_{d+1},\hskip 2.84526ptxp+yq+zr=0,\gcd(p,q,r)=1\right\\}$ $\displaystyle=$ $\displaystyle\left\\{[\Omega]\in\mathbb{P}(\Lambda^{1}_{d+1})\hskip 2.84526pt\|\hskip 2.84526pt\Omega=A\alpha+B\beta+C\gamma,\hskip 2.84526ptA,B\in\mathbb{C}[x,y,z]_{d},\hskip 2.84526ptC\in\mathbb{C}[x,y]_{d},\hskip 2.84526pt\gcd\big{(}yA-xB,zB-yC,xC- zA\big{)}=1\right\\}.$ By writting $\displaystyle A=\xi_{1}\hskip 0.28453ptx^{d}+\xi_{3}\hskip 0.28453ptx^{d-1}y+\cdots+\xi_{2d+1}\hskip 0.28453pty^{d}+\Big{(}\xi_{2d+3}\hskip 0.28453ptx^{d-1}+\xi_{2d+5}\hskip 0.28453ptx^{d-2}y+\cdots+\xi_{4d+1}\hskip 0.28453pty^{d-1}\Big{)}z+\Big{(}\xi_{4d+3}\hskip 0.28453ptx^{d-2}+\xi_{4d+5}\hskip 0.28453ptx^{d-3}y+\cdots+\xi_{6d-1}\hskip 0.28453pty^{d-2}\Big{)}z^{2}+\cdots+\xi_{d^{2}+3d+1}\hskip 0.28453ptz^{d},$ $\displaystyle B=\xi_{2}\hskip 0.28453ptx^{d}+\xi_{4}\hskip 0.28453ptx^{d-1}y+\cdots+\xi_{2d+2}\hskip 0.28453pty^{d}+\Big{(}\xi_{2d+4}\hskip 0.28453ptx^{d-1}+\xi_{2d+6}\hskip 0.28453ptx^{d-2}y+\cdots+\xi_{4d+2}\hskip 0.28453pty^{d-1}\Big{)}z+\Big{(}\xi_{4d+4}\hskip 0.28453ptx^{d-2}+\xi_{4d+6}\hskip 0.28453ptx^{d-3}y+\cdots+\xi_{6d}\hskip 0.28453pty^{d-2}\Big{)}z^{2}+\cdots+\xi_{d^{2}+3d+2}\hskip 0.28453ptz^{d},$ $\displaystyle C=\xi_{d^{2}+3d+3}\hskip 0.28453ptx^{d}+\xi_{d^{2}+3d+4}\hskip 0.28453ptx^{d-1}y+\xi_{d^{2}+3d+5}\hskip 0.28453ptx^{d-2}y^{2}+\cdots+\xi_{d^{2}+4d+2}\hskip 0.28453ptxy^{d-1}+\xi_{d^{2}+4d+3}\hskip 0.28453pty^{d},$ we can identify the class $[\Omega]$ of $\Omega=A\alpha+B\beta+C\gamma$ to the element $[\xi_{1}:\xi_{2}:\cdots:\xi_{d^{2}+4d+3}]\in\mathbb{P}^{\hskip 0.56905ptd^{2}+4d+2}_{\mathbb{C}}.$ Thus, we can identify $\mathbf{F}(d)$ with the Zariski open set: $\displaystyle\left\\{[\xi_{1}:\xi_{2}:\cdots:\xi_{d^{2}+4d+3}]\in\mathbb{P}^{\hskip 0.56905ptd^{2}+4d+2}_{\mathbb{C}}\hskip 1.42262pt\left\|\begin{array}[c]{l}\vspace{2mm}A=\xi_{1}\hskip 0.28453ptx^{d}+\xi_{3}\hskip 0.28453ptx^{d-1}y+\cdots+\xi_{2d+1}\hskip 0.28453pty^{d}+\big{(}\xi_{2d+3}\hskip 0.28453ptx^{d-1}+\xi_{2d+5}\hskip 0.28453ptx^{d-2}y+\cdots+\xi_{4d+1}\hskip 0.28453pty^{d-1}\big{)}z+\cdots+\xi_{d^{2}+3d+1}\hskip 0.28453ptz^{d}\\\ \vspace{2mm}B=\xi_{2}\hskip 0.28453ptx^{d}+\xi_{4}\hskip 0.28453ptx^{d-1}y+\cdots+\xi_{2d+2}\hskip 0.28453pty^{d}+\big{(}\xi_{2d+4}\hskip 0.28453ptx^{d-1}+\xi_{2d+6}\hskip 0.28453ptx^{d-2}y+\cdots+\xi_{4d+2}\hskip 0.28453pty^{d-1}\big{)}z+\cdots+\xi_{d^{2}+3d+2}\hskip 0.28453ptz^{d}\\\ \vspace{2mm}C=\xi_{d^{2}+3d+3}\hskip 0.28453ptx^{d}+\xi_{d^{2}+3d+4}\hskip 0.28453ptx^{d-1}y+\xi_{d^{2}+3d+5}\hskip 0.28453ptx^{d-2}y^{2}+\cdots+\xi_{d^{2}+4d+2}\hskip 0.28453ptxy^{d-1}+\xi_{d^{2}+4d+3}\hskip 0.28453pty^{d}\\\ \gcd\big{(}yA- xB,zB-yC,xC-zA\big{)}=1\end{array}\right.\right\\}.$ Then, via this identification, we have $\displaystyle\mathcal{F}_{2}^{d}=\big{[}\Omega_{2}^{d}\big{]}=\big{[}x^{d}\beta+y^{d}\gamma]=[0:1:0:0:\cdots:0:0:1\big{]}$ and $\displaystyle\mathcal{F}_{0}^{d}(\lambda)=\big{[}\Omega_{0}^{d}(\lambda)\big{]}=\big{[}(y^{d}+xz^{d-1})\alpha+\lambda\,yz^{d-1}\beta\big{]}=\big{[}\underbrace{0:0:\cdots:0}_{2d}:1:\underbrace{0:0:\cdots:0}_{d^{2}+d-5}:1:0:0:\lambda:\underbrace{0:0:\cdots:0}_{d+3}\big{]}.$ In addition, the orbit of a foliation $\mathcal{F}=[\Omega]\in\mathbf{F}(d)$ is $\displaystyle\mathcal{O}(\mathcal{F})=\left\\{[\varphi^{}\Omega]\hskip 2.84526pt\Big{\|}\hskip 2.84526pt\varphi=[a_{1}x+a_{2}y+a_{3}z:a_{4}x+a_{5}y+a_{6}z:a_{7}x+a_{8}y+a_{9}z]\in\mathrm{Aut}(\mathbb{P}^{2}_{\mathbb{C}})\right\\}.$ Let $[x_{1}:x_{2}:\cdots:x_{d^{2}+4d+3}]$ be a system of homogeneous coordinates in $\mathbb{P}^{\hskip 0.56905ptd^{2}+4d+2}_{\mathbb{C}}.$ For $d=3,$ let us consider the following homogeneous polynomial in $x_{1},x_{2},\ldots,x_{24}$ of degree $5$: $\displaystyle\hskip 8.5359ptP_{3}=-90x_{2}\Big{(}x_{1}\left(294x_{1}-269x_{4}\right)+10x_{2}\left(29x_{3}+4x_{6}\right)+86x_{4}^{2}\Big{)}x_{22}x_{24}-1125x_{2}^{2}\left(21x_{1}-23x_{4}\right)x_{23}x_{24}$ $\displaystyle\hskip 24.75375pt+45x_{2}\Big{(}2x_{3}\left(294x_{1}+13x_{4}\right)-x_{6}\left(552x_{1}-271x_{4}\right)+1125x_{2}x_{5}\Big{)}x_{21}x_{24}+28125x_{2}x_{10}x_{21}x_{23}x_{24}$ $\displaystyle\hskip 24.75375pt+25\Big{(}108\left(x_{9}-2x_{12}\right)\left(3x_{1}-4x_{4}\right)+9x_{10}\left(112x_{3}-93x_{6}\right)+675x_{2}x_{11}\Big{)}x_{21}^{2}x_{24}-6000x_{2}x_{10}x_{22}^{2}x_{24}$ $\displaystyle\hskip 24.75375pt-5625x_{5}x_{11}x_{21}^{3}+20\Big{(}\left(2x_{1}-x_{4}\right)\left(41x_{9}-7x_{12}\right)+30x_{10}\left(2x_{3}-3x_{6}\right)+50x_{2}x_{11}\Big{)}x_{22}^{3}-50625x_{2}^{3}x_{24}^{2}$ $\displaystyle\hskip 24.75375pt-5\Big{(}2x_{9}\left(207x_{1}-116x_{4}\right)-x_{12}\left(153x_{1}-314x_{4}\right)+5x_{10}\left(356x_{3}-359x_{6}\right)+1350x_{2}x_{11}\Big{)}x_{21}x_{22}x_{23}$ $\displaystyle\hskip 24.75375pt+1875\Big{(}x_{11}\left(2x_{3}-x_{6}\right)+x_{5}\left(2x_{9}-x_{12}\right)\Big{)}x_{21}^{2}x_{22}-375x_{2}\Big{(}2x_{1}\left(3x_{1}-7x_{4}\right)-x_{2}\left(3x_{3}-2x_{6}\right)+8x_{4}^{2}\Big{)}x_{23}^{2}$ $\displaystyle\hskip 24.75375pt+50\Big{(}5x_{10}\left(39x_{1}-38x_{4}\right)-3x_{2}\left(x_{9}-32x_{12}\right)\Big{)}x_{21}x_{23}^{2}-50\Big{(}x_{10}\left(14x_{1}-37x_{4}\right)-3x_{2}\left(7x_{9}+x_{12}\right)\Big{)}x_{22}^{2}x_{23}$ $\displaystyle\hskip 24.75375pt+15\Big{(}5x_{11}\left(21x_{1}+22x_{4}\right)-8x_{3}\left(14x_{9}-43x_{12}\right)+6x_{6}\left(13x_{9}-56x_{12}\right)-350x_{5}x_{10}\Big{)}x_{21}^{2}x_{23}+R\,x_{21}^{2}$ $\displaystyle\hskip 24.75375pt-5\Big{(}20x_{11}\left(24x_{1}-7x_{4}\right)+4x_{9}\left(97x_{3}-43x_{6}\right)+x_{12}\left(94x_{3}-211x_{6}\right)-600x_{5}x_{10}\Big{)}x_{21}x_{22}^{2}+S\,x_{21}x_{22}$ $\displaystyle\hskip 24.75375pt-75\Big{(}2x_{10}\left(78x_{1}-29x_{4}\right)-15x_{2}\left(2x_{9}-19x_{12}\right)\Big{)}x_{21}x_{22}x_{24}+125x_{2}x_{10}x_{22}x_{23}^{2}+Tx_{22}^{2}+Ux_{21}x_{23}$ $\displaystyle\hskip 24.75375pt+Vx_{22}x_{23},$ where $\displaystyle\hskip 8.5359ptR=5568x_{6}x_{5}\left(3x_{1}-4x_{4}\right)-18x_{3}x_{5}\left(1612x_{1}-1941x_{4}\right)+6x_{3}^{2}\left(1952x_{3}-4389x_{6}\right)+3x_{6}^{2}\left(7057x_{3}-2136x_{6}\right)-11250x_{2}x_{5}^{2}$ $\displaystyle\hskip 23.33147pt+2700x_{7}\left(3x_{1}-4x_{4}\right)^{2}+54x_{8}\left(3x_{1}-4x_{4}\right)\left(106x_{3}-89x_{6}\right),$ $\displaystyle\hskip 8.5359ptS=27000x_{2}x_{7}\left(3x_{1}-4x_{4}\right)-24x_{3}^{2}\left(658x_{1}-249x_{4}\right)+1512x_{4}x_{8}\left(11x_{1}-4x_{4}\right)+252x_{1}^{2}\left(83x_{5}-36x_{8}\right)-90x_{2}x_{3}\left(329x_{5}-318x_{8}\right)$ $\displaystyle\hskip 23.33147pt-2x_{4}x_{5}\left(17073x_{1}-6047x_{4}\right)+3x_{1}x_{6}\left(8712x_{3}-3599x_{6}\right)-x_{4}x_{6}\left(11658x_{3}-6041x_{6}\right)+90x_{2}x_{6}\left(226x_{5}-267x_{8}\right),$ $\displaystyle\hskip 8.5359ptT=20x_{1}x_{3}\left(294x_{1}-253x_{4}\right)-40x_{1}x_{6}\left(159x_{1}-152x_{4}\right)+1900x_{2}x_{3}\left(x_{3}-x_{6}\right)+20x_{4}^{2}\left(68x_{3}-95x_{6}\right)-25x_{2}x_{6}\left(40x_{3}-33x_{6}\right)$ $\displaystyle\hskip 23.33147pt+60x_{1}x_{2}\left(361x_{5}-252x_{8}\right)-10x_{2}x_{4}\left(983x_{5}-756x_{8}\right)+67500x_{2}^{2}x_{7},$ $\displaystyle\hskip 8.5359ptU=90x_{1}x_{3}\left(98x_{1}-117x_{4}\right)-30x_{1}x_{6}\left(171x_{1}-284x_{4}\right)-150x_{2}x_{6}\left(68x_{3}-35x_{6}\right)-30x_{2}x_{4}\left(167x_{5}+396x_{8}\right)+7050x_{2}x_{3}^{2}$ $\displaystyle\hskip 23.33147pt+20x_{4}^{2}\left(73x_{3}-157x_{6}\right)+270x_{1}x_{2}\left(41x_{5}+33x_{8}\right),$ $\displaystyle\hskip 8.5359ptV=5x_{2}x_{4}\left(1604x_{3}-611x_{6}\right)-30x_{1}^{2}\left(294x_{1}-563x_{4}\right)-30x_{4}^{2}\left(355x_{1}-86x_{4}\right)-30x_{1}x_{2}\left(463x_{3}-242x_{6}\right)-75x_{2}^{2}\left(109x_{5}-198x_{8}\right).$ A computation carried out with Maple shows that evaluating $P_{3}$ at an arbitrary element $[\xi_{1}:\xi_{2}:\cdots:\xi_{24}]$ of $\mathcal{O}\big{(}\mathcal{F}_{0}^{3}(\lambda)\big{)},$ we find $P_{3}\big{(}[\xi_{1}:\xi_{2}:\cdots:\xi_{24}]\big{)}=0,$ i.e. $\mathcal{O}\big{(}\mathcal{F}_{0}^{3}(\lambda)\big{)}$ is contained in the zero locus of $P_{3}$ $\displaystyle\mathrm{\text{Zeros}}(P_{3}):=\left\\{[x_{1}:x_{2}:\cdots:x_{24}]\in\mathbb{P}^{23}_{\mathbb{C}}\hskip 2.84526pt\|\hskip 2.84526ptP_{3}\big{(}[x_{1}:x_{2}:\cdots:x_{24}]\big{)}=0\right\\},$ which is a Zariski closed subset of $\mathbb{P}^{23}_{\mathbb{C}}.$ Therefore we have $\overline{\mathcal{O}\big{(}\mathcal{F}_{0}^{3}(\lambda)\big{)}}\subset\mathrm{\text{Zeros}}(P_{3})$ for any $\lambda\in\mathbb{C}^{}.$ Moreover, we have $\displaystyle P_{3}\left(0,1,0,0,\cdots,0,0,1\right)=-50625\neq 0,$ hence $\mathcal{F}_{2}^{3}\not\in\mathrm{\text{Zeros}}(P_{3}).$ It follows that, for any $\lambda\in\mathbb{C}^{},$ we have $\mathcal{F}_{2}^{3}\not\in\overline{\mathcal{O}\big{(}\mathcal{F}_{0}^{3}(\lambda)\big{)}}$, so that $\mathcal{F}_{0}^{3}(\lambda)$ does not degenerate onto $\mathcal{F}_{2}^{3}.$ Consequently, according to Lemma 4.3, the orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{3}(\lambda)\big{)}$ is closed in $\mathbf{F}(3)$. To show that the orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{4}(\lambda)\big{)},$ resp. $\mathcal{O}\big{(}\mathcal{F}_{0}^{5}(\lambda)\big{)},$ is closed in $\mathbf{F}(4),$ resp. $\mathbf{F}(5),$ it suffices to argue as in degree $d=3$, replacing the polynomial $P_{3}$ by the following polynomial $P_{4},$ resp. $P_{5}$: $\displaystyle P_{4}=\Big{(}3x_{3}\left(129x_{3}-212x_{6}\right)+3x_{4}\left(178x_{5}+15x_{8}\right)+12x_{1}\left(22x_{5}-3x_{8}\right)+5184x_{2}x_{7}-20x_{6}^{2}\Big{)}x_{31}+1728x_{15}x_{31}^{2}$ $\displaystyle\hskip 16.21828pt-432\left(2x_{13}-x_{16}\right)x_{31}x_{32}+48\left(42x_{11}-31x_{14}\right)x_{31}x_{33}-18\left(24x_{11}-19x_{14}\right)x_{32}^{2}-162x_{2}\left(4x_{1}-15x_{4}\right)x_{34}$ $\displaystyle\hskip 16.21828pt-18\Big{(}2x_{1}\left(27x_{3}-20x_{6}\right)-x_{4}\left(15x_{3}-x_{6}\right)+x_{2}\left(170x_{5}-69x_{8}\right)\Big{)}x_{32}+4212x_{12}x_{31}x_{34}-486x_{12}x_{32}x_{33}$ $\displaystyle\hskip 16.21828pt+36\Big{(}3\left(x_{1}-x_{4}\right)\left(12x_{1}-x_{4}\right)+22x_{2}\left(3x_{3}-2x_{6}\right)\Big{)}x_{33}-10368x_{2}^{2}x_{35},$ resp. $\displaystyle P_{5}=\Big{(}50x_{7}\left(4906x_{1}-4749x_{4}\right)-27040x_{10}\left(5x_{1}-6x_{4}\right)-5x_{5}\left(10596x_{3}-13469x_{6}\right)+20x_{8}\left(1019x_{3}-2028x_{6}\right)$ $\displaystyle\hskip 16.21828pt+569100x_{2}x_{9}\Big{)}x_{43}+142275x_{19}x_{43}^{2}-11690x_{17}x_{43}x_{44}+98140x_{14}x_{43}x_{47}-140x_{2}\left(2180x_{1}-1691x_{4}\right)x_{47}$ $\displaystyle\hskip 16.21828pt+35\left(1564x_{13}-1645x_{16}\right)x_{43}x_{46}+\Big{(}8620x_{8}\left(2x_{1}-x_{4}\right)-50x_{5}\left(141x_{1}-11x_{4}\right)+10x_{3}\left(513x_{3}-1580x_{6}\right)$ $\displaystyle\hskip 16.21828pt+70x_{2}\left(2779x_{7}-2704x_{10}\right)+9875x_{6}^{2}\Big{)}x_{44}-35\Big{(}\left(x_{1}-x_{4}\right)\left(295x_{1}+683x_{4}\right)-x_{2}\left(3776x_{3}-4427x_{6}\right)\Big{)}x_{46}$ $\displaystyle\hskip 16.21828pt+70\left(323x_{18}-253x_{15}\right)x_{43}x_{45}+7\left(686x_{13}-293x_{16}\right)x_{44}x_{45}-2975x_{15}x_{44}^{2}-15946x_{14}x_{45}^{2}-1422750x_{2}^{2}x_{48}$ $\displaystyle\hskip 16.21828pt+\Big{(}14x_{3}\left(15x_{1}+1124x_{4}\right)-14x_{6}\left(10x_{1}+1129x_{4}\right)-595x_{2}\left(221x_{5}-250x_{8}\right)\Big{)}x_{45}+49210x_{14}x_{44}x_{46}.$ ∎ For $d\geq 6,$ we propose: ###### Conjecture 1. Let $d$ be an integer greater than or equal to $6$ and $\lambda$ a nonzero complex number. A homogeneous coordinate system $[x_{1}:x_{2}:\cdots:x_{d^{2}+4d+3}]$ being fixed in $\mathbb{P}^{\hskip 0.56905ptd^{2}+4d+2}_{\mathbb{C}},$ there exists a homogeneous polynomial $Q_{d}\in\mathbb{C}[x_{1},x_{2},\cdots,x_{d^{2}+4d+3}]$ of degree $3$, not depending on $\lambda,$ which vanishes on the orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{d}(\lambda)\big{)}$ and does not vanish at the point $\mathcal{F}_{2}^{d}=[0:1:0:0:\cdots:0:0:1\big{]}.$ Computations made with Maple by the first author show the validity of this conjecture for $d$ small ($d\leq 30$) by taking the polynomial $Q_{d}$ in the following form: $\displaystyle Q_{d}=x_{d^{2}+3d+3}\left(\sum\limits_{i=1}^{d-1}\alpha_{i}\,x_{2d+2i+1}\,x_{d^{2}+4d+2-i}+\sum\limits_{i=0}^{4}\beta_{i}\,x_{2d+2i+4}\,x_{d^{2}+4d+2-i}\right)+(x_{1}\hskip 5.69054ptx_{2}\hskip 5.69054pt\cdots\hskip 5.69054ptx_{d+1})M\left(\begin{array}[]{c}x_{d^{2}+4d+3}\\\ x_{d^{2}+4d+2}\\\ \vdots\\\ x_{d^{2}+3d+3}\end{array}\right)$ $\displaystyle\hskip 20.48601pt+x_{d^{2}+3d+4}\left(\delta_{0}\,x_{2d+4}\,x_{d^{2}+4d+1}+\delta_{1}\,x_{2d+6}\,x_{d^{2}+4d}+\sum\limits_{i=1}^{d-3}\gamma_{i}\,x_{2d+2i+1}\,x_{d^{2}+4d+1-i}\right),$ where $M=\left(\begin{array}[]{c}L_{1}\\\ L_{2}\\\ \vdots\\\ L_{d+1}\end{array}\right)$ is a square matrix of order $d+1$ whose lines are of the form: $\displaystyle L_{1}=\left[0\hskip 5.69054pt0\hskip 5.69054pta_{1,3}x_{1}+b_{1,3}x_{4}\hskip 5.69054pta_{1,4}x_{3}+b_{1,4}x_{6}\hskip 5.69054pta_{1,5}x_{5}+b_{1,5}x_{8}\hskip 5.69054pt\cdots\hskip 5.69054pta_{1,\,d+1}x_{2d-3}+b_{1,\,d+1}x_{2d}\right]$ $\displaystyle L_{2}=\left[b_{2,1}x_{2}\hskip 5.69054pta_{2,2}x_{1}+b_{2,2}x_{4}\hskip 5.69054pta_{2,3}x_{3}+b_{2,3}x_{6}\hskip 5.69054pta_{2,4}x_{5}+b_{2,4}x_{8}\hskip 5.69054pta_{2,5}x_{7}+b_{2,5}x_{10}\hskip 5.69054pt\cdots\hskip 5.69054pta_{2,\,d+1}x_{2d-1}+b_{2,\,d+1}x_{2d+2}\right]$ $\displaystyle\vdots$ $\displaystyle L_{2k-1}=\left[\underbrace{0\hskip 5.69054pt0\hskip 5.69054pt\cdots\hskip 5.69054pt0}_{\min(2k,\,d+1)}\hskip 5.69054pta_{2k-1,\,2k+1}x_{2k-1}+b_{2k-1,\,2k+1}x_{2k+2}\hskip 5.69054pta_{2k-1,\,2k+2}x_{2k+1}+b_{2k-1,\,2k+2}x_{2k+4}\hskip 5.69054pt\cdots\hskip 5.69054pta_{2k-1,\,d+1}x_{2d-2k-1}+b_{2k-1,\,d+1}x_{2d-2k+2}\right]$ $\displaystyle L_{2k}=\left[\underbrace{0\hskip 5.69054pt0\hskip 5.69054pt\cdots\hskip 5.69054pt0}_{2k-2}\hskip 5.69054ptb_{2k,\,2k-1}x_{2k}\hskip 5.69054pta_{2k,\,2k}x_{2k-1}+b_{2k,\,2k}x_{2k+2}\hskip 5.69054pta_{2k,\,2k+1}x_{2k+1}+b_{2k,\,2k+1}x_{2k+4}\hskip 5.69054pt\cdots\hskip 5.69054pta_{2k,\,d+1}x_{2d-2k+1}+b_{2k,\,d+1}x_{2d-2k+4}\right],$ where $\alpha_{i},\beta_{i},\gamma_{i},\delta_{i},a_{i,j},b_{i,j}\in\mathbb{C}$ with $b_{2,1}\neq 0.$ It is clear that Conjecture 1 and Lemma 4.3 imply the following conjecture. ###### Conjecture 2. For any integer $d\geq 6$ and any $\lambda\in\mathbb{C}^{},$ the orbit $\mathcal{O}\big{(}\mathcal{F}_{0}^{d}(\lambda)\big{)}$ is closed in $\mathbf{F}(d).$ ## References [1] C. R. Alcántara and R. Ronzón-Lavie. Classification of foliations on $\mathbb{CP}^{2}$ of degree $3$ with degenerate singularities. J. Singul. 14:52–73, 2016. * [2] P. Baum and R. Bott. Singularities of holomorphic foliations. J. Differential Geometry, 7:279–342, 1972. * [3] S. Bedrouni. Feuilletages de degré trois du plan projectif complexe ayant une transformée de Legendre plate. PhD thesis, University of Sciences and Technology Houari Boumediene, 2017. Available on https://arxiv.org/abs/1712.03895. * [4] S. Bedrouni and D. Marín. Tissus plats et feuilletages homogènes sur le plan projectif complexe. Bull. Soc. Math. France, 146(3):479–516, 2018. * [5] S. Bedrouni and D. Marín. Classification of foliations of degree three on $\mathbb{P}^{2}_{\mathbb{C}}$ with a flat Legendre transform. To appear in Ann. Inst. Fourier (Grenoble), 2019. * [6] S. Bedrouni and D. Marín. Une nouvelle démonstration de la classification des feuilletages convexes de degré deux sur $\mathbb{P}^{2}_{\mathbb{C}}$. Bull. Soc. Math. France, 148(4):613–622, 2020. * [7] M. Brunella. Birational geometry of foliations, volume 1 of IMPA Monographs. Springer, Cham, 2015. * [8] F. Cano, D. Cerveau, and J. Déserti. Théorie élémentaire des feuilletages holomorphes singuliers. Echelles. Belin, 2013. * [9] D. Cerveau, J. Déserti, D. Garba Belko, and R. Meziani. Géométrie classique de certains feuilletages de degré deux. Bull. Braz. Math. Soc. (N.S.), 41(2):161–198, 2010. * [10] C. Favre and J. V. Pereira. Webs invariant by rational maps on surfaces. Rend. Circ. Mat. Palermo (2), 64(3):403–431, 2015. * [11] R. Hartshorne. Algebraic geometry. Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. * [12] J. P. Jouanolou. Équations de Pfaff algébriques, volume 708 of Lecture Notes in Mathematics. 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# State estimation with limited sensors – A deep learning based approach Yash Kumar Department of Mechanical Engineering Delhi Technological University Shahbad Daulatpur, Main Bawana Road, Delhi-110042, India <EMAIL_ADDRESS> &Pranav Bahl Department of Mechanical Engineering Delhi Technological University Shahbad Daulatpur, Main Bawana Road, Delhi-110042, India <EMAIL_ADDRESS> &Souvik Chakraborty Department of Applied Mechanics Indian Institute of Technology Delhi Hauz Khas - 110042, New Delhi, India <EMAIL_ADDRESS> ###### Abstract The importance of state estimation in fluid mechanics is well-established; it is required for accomplishing several tasks, including design/optimization, active control, and future state prediction. A common tactic in this regard is to rely on reduced-order models. Such approaches, in general, use measurement data of a one-time instance. However, often data available from sensors is sequential, and ignoring it results in information loss. In this paper, we propose a novel deep learning-based state estimation framework that learns from sequential data. The proposed model structure consists of the recurrent cell to pass information from different time steps, enabling this information to recover the full state. We illustrate that utilizing sequential data allows for state recovery from minimal and noisy sensor measurements. For efficient recovery of the state, the proposed approach is coupled with an auto-encoder based reduced-order model. We illustrate the performance of the proposed approach using three examples, and it is found to outperform other alternatives existing in the literature. _K_ eywords state estimation $\cdot$ autoencoder $\cdot$ recurrent neural network $\cdot$ limited sensors ## 1 Introduction The integration of deep learning has benefited modern algorithms in modeling, data processing, prediction, and control of various engineering systems. In fluid mechanics, work on machine learning implementation started last decade and has grown since then. [1] used a neural network to reconstruct turbulence flow fields and the flow in the near-wall region of a channel flow using wall information. [2] and [3], in two separate works, used deep learning algorithms to improve a Reynolds-averaged Navier Stokes turbulence model. Recently, [4] proposed a multi-fidelity deep learning framework for turbulent flow. [5] used a shallow network for estimating 2D state from measurements over cylinder surface. In this paper, we are particularly interested in applying deep learning for state estimation in fluid mechanics, and hence, the discussion hereafter is focused on the same. State estimation is the ability to recover flow based on a few noisy measurements. It is an inverse problem and arises in many engineering applications such as remote sensing, medical imaging, ocean dynamics, reservoir modeling, and blood flow modeling. Controlling the flow and optimizing machine design in these applications depend upon the ability to predict the state with given sensors. The challenge associated with state estimation is two-fold. Firstly, for almost all practical cases, state- estimation is an ill-posed problem, and hence, a unique solution to the problem does not exist [6]. Secondly, for practical problems, the number of sensors available is often limited. As a consequence, one has to deal with a sparse data set [7]. Attempts for state estimation dates back to 1960 when the Kalman filter based approaches were used for state estimation [8]. This method assumes the system’s dynamics to produce full state and updates it based on new measurements to reduce estimation error forming a closed feedback loop. However, the classical Kalman filter based approaches are only applicable for linear dynamical systems [9]. Improvements to Kalman filter algorithms, such as Extended Kalman filter [10] and Unscented Kalman filter [11] algorithms can also be found in the literature. [12] attempts to generalize Kalman filters by using the Gaussian process. Approaches based on observer dynamical system uses a reduced-order model to predict the future based on the past while simultaneously corrected by receiving measurements. [13] applies dynamic mode decomposition as a reduced-order model to Kalman smoother estimate to identify coherent structures. [14] used a nonlinear observer-based on Galerkin projection of Navier-Stokes equation to estimate POD coefficients. The use of Bayes filters, such as the Kalman and particle filters, in conjugation with POD based ROM on various flow problems, can also be found in the literature [15, 16, 17]. Another major category of approaches includes library-based approaches and stochastic approaches. Library-based approaches use offline data, and the library consists of generic modes such as Fourier, wavelet, discrete cosine transform basis or data specific POD or DMD modes, or training data. Library based approaches using sparse representation assumes state can be expressed as the combination of library elements. Sparse coefficients are obtained by solving pursuit problem [18, 19]. [20] used sparse representation and training data as the library with localized reconstruction for reconstructing complex fluid flows. Gappy POD [21] estimates POD coefficients in a least-square sense and uses a library of POD modes. However, it is prone to ill-conditioning and is dealt with using the best sensor placements [22] to improve the condition number[23]. The most explored approach for state estimation is perhaps the one based on stochastic estimation. The idea was first proposed by [24] for a turbulence study where the conditional mean was approximated using a power series. In a linear setting, coefficients are computed using a two-point, second-order correlation tensor. Other variants like quadratic stochastic estimation [25] and spectral linear stochastic estimation [26] can also be found in the literature. [27] proposed to include time delayed measurements to further improve accuracy. [28] extended stochastic approach to estimate POD coefficients. A linear mapping between sensors and coefficients was assumed. Recently, [29] used a neural network to learn a non-linear mapping between sensor measurements and POD coefficients. These approaches allow more flexibility in sensor placements and have been applied for flow control over airfoil [30] and analyzing isotropic turbulence [31, 32] and turbulent boundary layers [27, 25]. One limitation associated with all the approaches mentioned above resides in the fact that spatial information of a single sample is used to recover the full state, but often, data is sequential. Ignoring the sequence of the data during state information invariably results in information loss. To address this apparent shortcoming, we propose a novel deep learning-based non- intrusive framework for state estimation that learns from sequential data. The proposed framework couples recurrent neural network (RNN) with auto-encoder (AE). While AE is used to learn the nonlinear manifold, RNN is employed to take advantage of the time series data. We illustrate that by utilizing sequential data, the proposed framework is able to estimate the state in a more accurate fashion. Perhaps, more importantly, the number of sensors required is significantly less. For showcasing the performance of the proposed framework, two benchmark problems involving periodic vortex shedding and transient flow past a cylinder are considered. Results obtained are compared with those obtained using other state-of-the-art techniques. The remainder of the paper is organized as follows. In Section 2, details on the problem statement is provided. A brief review of RNN and AE are furnished in Section 3. Details on the proposed approach are provided in Section 4. Section 5 presents two numerical examples to illustrate the performance of the proposed approach. Finally, Section 6 provides the concluding remarks. ## 2 Problem statement We consider a dynamical system obtained by partial discretization of the governing differential equations: $\dot{\bm{w}}=\bm{f}\left(\bm{w},t;\bm{\theta}\right),\;\;\;\bm{w}(t_{n},\bm{\theta})=\bm{w}^{n}\left(\bm{\theta}\right),$ (1) where $\bm{w}\in\mathbb{R}^{N_{w}}$ represents the high-dimensional state vector that depends on parameters $\bm{\theta}\in\mathbb{R}^{N_{d}}$ and time $t\in\left[0,t_{max}\right]$. $\bm{f}$ in Eq. (1) is a nonlinear function that governs the dynamical evolution of the state vector $\bm{w}$. Note that for brevity of representation, we have not shown the dependence of the state $\bm{w}$ on $t$ and $\bm{\theta}$, and the dependence of $\bm{f}$ on $\bm{w},t$ and $\bm{\theta}$. We note that the state vector $\bm{w}$ is high-dimensional in nature and it is extremely difficult to directly work with $\bm{w}$. A commonly used strategy in this regards is to approximate the high-dimensional state vector $\bm{w}$ on a low-dimensional manifold, $\bm{w}\left(t;\bm{\theta}\right)\approx\bm{w}_{r}\left(t;\bm{\theta}\right)=\bm{\Phi}\left(\bm{a}\left(t;\bm{\theta}\right)\right),$ (2) where $\bm{a}\left(t;\bm{\theta}\right)\in\mathbb{R}^{N_{a}}$ represents the reduced space and $\bm{\Phi}\left(\cdot\right):\bm{a}\left(t;\bm{\theta}\right)\mapsto\bm{w}\left(t;\bm{\theta}\right)$ is the manifold. $N_{a}$ in Eq. (2) represent the dimension of reduced space such that $N_{a}\ll N_{w}$. Substituting Eq. (2) into Eq. (1), we obtain $\dot{\bm{a}}=\bm{\Psi}\left(\bm{f}\left(\bm{\Phi}\left(a\right),t;\bm{\theta}\right)\right),\;\bm{a}(t_{n};\bm{\theta})=\bm{a}^{n}(\bm{\theta}),$ (3) where $\bm{\Psi}(\cdot)=\left(\bm{\Lambda}\bm{\Gamma}^{-1}\right)\circ\bm{\Lambda}\left(\cdot\right)$ and $\bm{\Lambda}:\mathbb{R}^{N_{w}}\mapsto\mathbb{R}^{N_{a}}$. In Eq. (3), we have assumed that $\bm{\Phi}(\bm{a})$ is continuously differentiable such that $\bm{\Gamma}(\dot{\bm{a}})=\dot{\bm{\Phi}}(\bm{a})$; $\bm{\Gamma}:\mathbb{R}^{N_{a}}\mapsto\mathbb{R}^{N_{w}}$. $a^{n}(\bm{\theta})$ in Eq. (3) represents the initial condition. With this representation, the objective in state-estimation reduces to estimating the reduced order state variable $\bm{a}^{n}\left(\bm{\theta}\right)$. Generally, this is achieved by determining a mapping $\mathcal{M}:\mathbb{R}^{N_{s}}\mapsto\mathbb{R}^{N_{a}}$ between the sensor measurements and reduced state, $\bm{a}^{n}=\mathcal{M}(\bm{s}^{n}(\bm{\theta})),$ (4) where $\bm{s}^{n}\in\mathbb{R}^{N_{s}}$ represents the sensor measurements at time-step $n$ and $N_{s}$ indicates the number of sensors present. A schematic representation of the same is shown in Fig. 1. Figure 1: Schematic representation of state estimation framework. It consists of two components, (a) $\mathcal{M}(\bm{s})$ \- it maps the sensor data to a reduced state and (b) a projection operator - it maps the predicted reduced state to the original state. Performance of a state estimation framework is dependent on performance of this two components and the quality and quantity of data. The state estimation framework discussed above has two major limitations. * • We note that the state estimation framework only relies on sensor responses at the current step for predicting the state variables. In other words, the sequential nature of the sensor measurements is ignored. This invariably results in loss of information, and hence, the accuracy of the state estimation is compromised. * • Secondly, as shown in Eq. (2), the use of reduced-order model results in information loss. While completely avoiding this information loss is unavoidable, it is necessary to ensure that this information loss is minimized. This paper aims to develop a deep learning-based framework for state estimation that addresses the two limitations discussed above. ## 3 A brief review of AE and RNN This section briefly reviews two poplar deep learning approaches, namely auto- encoders (AE) and recurrent neural networks (RNN). It is to be noted that AE and RNN form the backbone of the proposed approach. ### 3.1 Auto-encoders AE is a class of unsupervised deep learning techniques trained to copy the inputs to the output. It consists of a latent space/hidden layer $\bm{h}\in\mathbb{R}^{h}$ that represents a compressed representation of the input; this is often referred to as the bottleneck layer. The network architecture for an AE can be viewed as having two parts, an encoder that maps the input to the latent space $\bm{h}$ and a decoder that reconstructs the inputs from the latent space. Mathematically, this is represented as $\bm{h}=\bm{\omega}\left(\bm{\xi}\right),\;\hat{\bm{\xi}}=\bm{g}\left(\bm{h}\right),$ (5) where $\bm{\omega}$ represents the encoder network and $\bm{g}$ represents the decoder network. $\bm{\xi}\in\mathbb{R}^{N}$ in Eq. (5) represents the input variables with $N$ being the number of variables. A schematic representation of AE is shown in Fig. 2. Figure 2: Schematic representation of an auto-encoder (AE). It consists of encoder ($\omega$) and ($g$). The red nodes represents the bottleneck layer. AE is integral to the neural network landscape and was initially developed for model reduction and feature extraction. As far as training an AE is concerned, an adaptive learning rate optimization algorithm(ADAM) is a popular choice. The learning in AE is generally expressed as $\bm{\alpha}=\arg\min_{\bm{\alpha}}\mathcal{L}(\bm{\xi}_{t},\bm{g}\left(\bm{\omega}(\bm{\xi})\right);\bm{\alpha}),$ (6) where $\mathcal{L}(\bm{\xi}_{t},\bm{g}\left(\bm{\omega}(\bm{\xi})\right);\bm{\alpha})$ represents the loss-function and $\bm{\alpha}=\left[\bm{\alpha}_{e},\bm{\alpha}_{d}\right]$ are the hyperparameters (weights and biases) of the neural network. $\bm{\alpha}_{e}$ corresponds to the hyperparameters of the encoder while $\bm{\alpha}_{d}$ corresponds to the hyperparameters of the decoder. Some important remarks on AE are furnished below Remark 1: A situation where $\bm{g}\left(\bm{\omega}(\bm{\xi})\right)=\bm{\xi}$ everywhere needs to be avoided [33]. In other words, the training algorithm is designed in such a way to restrict direct copying of the input. Remark 2: In AE, the dimensionality of the latent space $\bm{h}\in\mathbb{R}^{h}$ is generally much smaller than the dimensionality of the input variable $\bm{\xi}\in\mathbb{R}_{N}$, $h\ll N$. Therefore, $\bm{h}$ can be thought of as a reduced-order representation of $\bm{\xi}$. Remark 3: When the decoder $\bm{g}$ is linear, and $\mathcal{L}$ is a mean- squared error, an AE learns to span the same subspace as principal component analysis. Remark 4: AE with nonlinear encoder $\bm{\omega}$ and nonlinear decoder $\bm{g}$ learns a nonlinear reduced-order manifold; however, an AE with too much expressive capacity learns to copy the inputs (Remark 1). It is to be noted that researchers are still working on developing AE for various types of tasks. Some of the popular AE available in the literature includes variational AE [34], sparse AE [35], stochastic AE [36] and capsule AE [37] among other. For further details on different types of AE, interested readers may refer [33]. ### 3.2 Recurrent neural networks Many of the learning tasks involved in artificial intelligence necessitate handling data that are sequential in nature. Examples of sequential data include image captioning, time-series forecasting, and speech synthesis, among others. A recurrent neural network (RNN) is a type of neural network that is particularly suitable for sequential data. RNN captures the time dynamics by using cycles in the graph. Consider $\bm{\xi}$ to be inputs and $\bm{y}_{t}$ to be the output at time $t$. The output $\bm{y}_{t}$ in RNN is expressed as a function of $\bm{\xi}$ and $\bm{h}_{t}$; however, owing to the cyclic graph in RNN, the hidden state $\bm{h}_{t}$ is continuously updated as the sequence is processed. A schematic representation of a simple RNN is shown in Fig. 3. Note that different variants to the classical RNN can be found in the literature. In this work, we have used a Long Short-Term Memory [38], and hence, the discussion hereafter is focused on the same. Readers interested in other types of RNN may refer to [39]. Figure 3: Schematic representation of a typical RNN. $\xi$ (blue node) represents the input, $h_{i},\forall i$ (red node) represents the hidden layer and $y_{i}$ (green node) is the output sequence. LSTM, first proposed by [38], is a type of gated RNN cell that overcomes the well-know issue of vanishing gradient with the help of the gates that control the flow of information, i.e., differentiates between the information to be updated and that to be deleted [40, 41, 42]. LSTM cell comprises a forget gate, input gate, output gate, and a cell state. Each of these has its significance. The cell state refers to the information that has to be transferred in the sequence, and the respective gates determine the information that has to updated or deleted from the cell state [43]. The output of the current cell, also referred to as the hidden state, helps retain the short-term memory, and the cell state, on the other hand, is used to retain the Long-term Memory. Cell state in LSTM is multiplied with the forget gate in each cell along with the addition from the input gate, and this provides the opportunity for forgetting gate to eradicate the unimportant information and input gate to enhance state with useful information [43]. A schematic representation of LSTM cell is shown in Fig. 4. Figure 4: Schematic of the LSTM cell Mathematical, the operations being carried out inside a LSTM cell is represented using the following equations. $C_{t}=\mathbf{F}_{t}\odot\mathbf{C}_{t-1}+\mathbf{I}_{t}\odot\mathbf{\tilde{C}}_{t}$ (7a) $\mathbf{\tilde{C}}_{t}=\tanh(\mathbf{\xi}_{t}\mathbf{W}_{xc}+\mathbf{h}_{t-1}\mathbf{W}_{hc}+\mathbf{b}_{c}$ (7b) $\mathbf{I}_{t}=\sigma(\mathbf{\xi}_{t}\mathbf{\xi}_{xi}+\mathbf{h}_{t-1}\mathbf{W}_{hi}+\mathbf{b}_{i}$ (7c) $\mathbf{F}_{t}=\sigma(\mathbf{\xi}_{t}\mathbf{\xi}_{xf}+\mathbf{h}_{t-1}\mathbf{W}_{hf}+\mathbf{b}),$ (7d) where $\mathbf{I}_{t}$, $\mathbf{F}_{t}$ and $\tilde{\mathbf{C}}_{t}$ are respectively the input gate, forget gate, and a candidate cell state. Note that the update is carried out is additive in nature; this allows long-term information to pass through and avoids the gradient from vanishing. The short term state in LSTM is calculated as $\mathbf{O}_{t}=\sigma\left(\mathbf{\xi}_{t}\mathbf{\xi}_{xo}+\mathbf{h}_{t-1}\mathbf{W}_{ho}+\mathbf{b}_{o}\right)$ (8a) $\mathbf{h}_{t}=\mathbf{O}_{t}\odot\tanh(\mathbf{C}_{t}),$ (8b) where $\mathbf{O}_{t}$ represents the output gate. It is to be noted that $\mathbf{h}_{t}$ is used as the output of the cell as well as the hidden state for the next time-step; this is responsible for the short term memory of LSTM. $\mathbf{C}_{t}$ on the other hand is responsible for long term memory. The use of RNN for complex dynamical systems has attracted significant interest from the research community; this is primarily because of its capability in capturing temporal dependencies [40]. Multiple architectures have been proposed for using RNN for accomplishing the task future state prediction of a dynamical system. Recently [44] and [45] used LSTM and transformers as time integrator to predict flow evolution state. [46] used simple RNN for IMU modeling in deep Kalman filter. [47] used autoencoder with linear recurrence to learns important dynamical features. In this work also, we use LSTM for extracting useful information from sequential sensor data. The next section provides more details on the same. ## 4 Proposed approach In this section, we propose a novel deep learning based framework for state estimation. The proposed approach integrates the two deep learning approaches discussed in Section 3, namely AE and RNN. Within the proposed framework, AE learns the reduced nonlinear subspace. It helps in reducing the information loss due to the compressed representation. RNN, on the other hand, extends the capability of the proposed approach and allows it to reuse sensor data collected at previous time-steps. AE (by reducing the state variable) and RNN (by incorporating information from the past) also helps address the ill- posedness associated with solving a state estimation problem. Consider $\bm{s}=\left[\bm{s}_{1},\bm{s}_{2},\ldots,\bm{s}_{n}\right]\in\mathbb{R}^{N_{t}\times N_{s}}$ represents the measurement data obtained from $N_{s}$ sensors over $N_{t}$ time-steps. Also consider $\bm{w}\in\mathbb{R}_{N_{w}}$ to be state variables. We can express the state variable $\bm{w}^{t}$ at time $t$ as $\bm{w}^{t}=g\left(\mathcal{M}\left(\bm{s}^{t-k:t};\bm{\theta}_{M}\right);\bm{\theta}_{g}\right)$ (9) where $\mathcal{M}\left(\cdot;\bm{\theta}_{M}\right)$ represents a mapping between the sensor data and the reduced state variable and $g\left(\cdot;\bm{\theta}_{g}\right)$ projects the reduced state variable back to the original space. $\bm{\theta}_{M}$ and $\bm{\theta}_{g}$ represent parameters associated with $\mathcal{M}\left(\cdot;\bm{\theta}_{M}\right)$ and $g\left(\cdot;\bm{\theta}_{g}\right)$, respectively. Unlike existing methods, sensor data corresponding to current and previous time-steps have been used for predicting $\bm{w}^{t}$ in Eq. (9). A schematic representation of the same is shown in Fig. 5. Figure 5: Schematic representation of the proposed approach. The difference with Fig. 1 resides in the fact that information from previous time-steps are also utilized within the proposed framework. We propose to model $\mathcal{M}\left(\cdot;\bm{\theta}_{M}\right)$ using RNN and $g\left(\cdot;\bm{\theta}_{g}\right)$ by using AE. We also note that the sensor data is sequential and hypothesize that modeling this sequential nature of the sensor data will improve the predictive capability of model $\mathcal{M}\left(\cdot;\bm{\theta}_{M}\right)$. Therefore, we propose to model $\mathcal{M}\left(\cdot;\bm{\theta}_{M}\right)$ by using RNN. As stated in Section 3.2, RNN is suited explicitly for modeling such sequential data. Another aspect in Eq. (9) is associated with the projection operator $g\left(\cdot;\bm{\theta}_{g}\right)$. We reiterate that accuracy and efficiency of Eq. (9) is significantly dependent on $g\left(\cdot;\bm{\theta}_{g}\right)$. One popular choice among researchers is to use proper orthogonal decomposition for computing the projection operator $g$. However, proper orthogonal decomposition being a linear projection scheme has limited expressive capability. In this work, we propose to use AE as $g\left(\cdot;\bm{\theta}_{g}\right)$. Owing to the fact that AE is a nonlinear reduced order model, we expect the accuracy to enhance. However, one must note that training AE demands more computational effort as compared to proper orthogonal decomposition. Hereafter, we refer to the proposed framework as Auto-encoder and Recurrent neural network based state Estimation (ARE) framework. Next, details on network architecture and training algorithm for ARE are furnished. ### 4.1 Network architecture and training The ARE architecture proposed in this paper involves an AE and an RNN. For state estimation, the trained AE is split into the encoder and the decoder parts. The encoder part is used during training the RNN within the ARE framework. The decoder part is used while estimating the state variable. The AE and RNN networks are trained separately, with the latter following the former (see Fig. 6). The AE architecture considered in this paper consists of 5 hidden layers, with the 3rd layer being the bottleneck layer. The flow-field is vectorized before providing it as an input to the AE. Rectified linear unit (ReLU) activation function has been used for all but the last layer. For the last layer, a linear activation function is used. For training the network, ADAM [48] optimizer with a learning rate of $0.0007$ is used. We denote the trained parameters of the AE as $\bm{\theta}_{AE}=\left[\bm{\theta}_{w},\bm{\theta}_{g}\right]$, where $\bm{\theta}_{w}$ and $\bm{\theta}_{g}$ corresponds to the network parameters for encoder and decoder part respectively. For making the model robust to noisy data, two batch-norm layers and one dropout layer is added to AE. Batch normalization [49] is used to normalizes the activation distribution. It reduces the model’s sensitivity to learning rate [50], reduces training time, and increases stability. Additionally, this also acts as a regularizer. Dropout [51] is also an effective regularization technique that works by dropping connections between neurons during training with a specified probability $p$. Using the dropout layer just before the bottleneck layer proved to be most efficient in increasing the robustness of the network to noise as it simulates the noise in latent vector from the RNN network. A dropout probability of 0.35 is used in the network. Considering $Li$ to be the $i$-th layer of AE, BN to be the batch normalization and DR to be the dropbout, the architecture used in this paper is as follows: $L1\rightarrow BN\rightarrow L2\rightarrow BN\rightarrow DR\rightarrow L3\rightarrow L4\rightarrow L5\rightarrow L5\rightarrow Output$ Once the AE trains, we proceed to train the RNN part. The objective here is to learn a mapping between the sequential sensor data and the reduced state variable obtained by using the encoder part of the trained AE. First, the sensor data passes through RNN. It helps in capturing information from the sequential data. After that, the RNN outputs are mapped to the reduced states by using a feed-forward neural network. Reduced states of the training outputs are obtained by using the trained AE (encoder part). The parameters $\bm{\theta}_{M}$ (see Eq. (9)) corresponds to the RNN and the feedforward neural network and are obtained by solving the following optimization problem $\bm{\theta}_{M}^{}=\arg\min_{\bm{\theta}}\sum_{i=1}^{N_{\text{samp}}}\left\\|h^{t}-\mathcal{N}\left(\bm{s}^{t-k:t};\bm{\theta}_{M}\right)\right\\|+\lambda\left\\|\bm{\theta}_{M}\right\|_{2}^{2},$ (10) Where $\mathcal{N}$ represents the combined RNN and feed-forward neural network mapping. The second term in Eq. (10) represents $L^{2}$ regularization and is adopted to avoid overfitting. $\lambda$ is a tuning parameter and needs to be tuned manually. Similar to AE, the optimization problem is solved by using ADAM optimizer [48]. We have used weight decay of $10^{-5}$ and a learning rate of $0.0008$. Early stopping is used, which also acts as a regularizer [52]. RNN training is schematically shown in Fig. 6(b). The steps involved in training the proposed (ARE) are shown in Algorithm 1. Figure 6: Schematic representation of ARE (training phase) - (a) Training AE for reconstructing the flow-field. (b) Training the RNN for mapping the sensor data to the reduced state. 1 Library data generation: Perform experiment or run CFD simulations to generate a library of data, $\mathcal{D}=\left[\bm{s},\bm{w}\right]$ where, $\bm{s}=[\bm{s}^{1}_{1:N_{s}},\ldots,\bm{s}^{N_{t}}_{1:N_{s}},\;\;\bm{w}=\left[\bm{w}^{1},\ldots,\bm{w}^{N_{t}}\right],$ where $\bm{w}^{i}\in\mathbb{R}^{N_{w}}$. 2 Initialize: Initialize $\bm{\theta}_{g}$ and $\bm{\theta}_{M}$. Provide sequence length $k+1$ for RNN, learning rate parameters, network architectures and number of training epochs. 3 Train AE for $\bm{w}=\left[\bm{w}^{1},\ldots,\bm{w}^{N_{t}}\right]$. $\bm{a}^{t}\leftarrow\omega(\bm{w}^{t};\bm{\theta}_{w})$; $\triangleright$ Reducing dimensionality using AE Train RNN for sequence length $k+1$; $\triangleright$ Eq. (10) Algorithm 1 Training ARE ### 4.2 State estimation using the proposed approach Once the proposed ARE is trained by following the procedure detailed in Algorithm 1, one can use it for estimating the state. A trained ARE performs state estimation in two simple steps. In the first step, the trained RNN is used for estimating the reduced state based on the sensor measurement. Once the reduced state has been estimated, the decoder part of the AE is used to project the reduced state onto the original state. For clarity of readers, the steps for predicting state using ARE are shown in Algorithm 2. A schematic representation of the same is also shown in Fig. 7. 1 Pre-requisite: Trained ARE model, $\bm{\theta}=\left[\bm{\theta}_{M},\bm{\theta}_{g}\right]$ and sensor data $\bm{s}_{}^{t-k:t}$. $\bm{a}^{t}_{}\leftarrow\mathcal{N}(\bm{s}_{}^{t-k:t};\bm{\theta}_{M})$; $\triangleright$ Reduced state prediction using RNN $\bm{w}^{t}_{}\leftarrow=g(\bm{a}^{t}_{};\bm{\theta}_{g})$; $\triangleright$ Full-state estimation using decoder part of AE Algorithm 2 State estimation using ARE Figure 7: Schematic representation of the proposed ARE for state estimation ## 5 Numerical experiments In this section, two examples are presented to illustrate the performance of the proposed approach. The examples selected are well-known benchmark problems in the fluid mechanic’s community. For both examples, we have considered that the sensor measures vorticity, and the objective is to reconstruct the vorticity field. We present case studies by varying the number of sensors and the number of sequences available. To illustrate the excellent performance of our approach, a comparison with another state-of-the-art method has been provided. Comparison among results is carried out based on a qualitative and quantitative metric. To be specific, visual inspection is used as a qualitative metric, and the relative error is used as a quantitative metric, $\epsilon=\frac{\left\\|\bm{\omega}^{n}(\bm{\theta}^{})-\bm{\omega}^{n}_{r}(\bm{\theta}^{})\right\\|_{2}}{\left\\|\bm{\omega}^{n}(\bm{\theta}^{})\right\\|_{2}}\times 100.$ (11) where $\epsilon$ represents the error, $\bm{\omega}^{n}(\bm{\theta}^{})$ is the true state and $\bm{\omega}^{n}_{r}(\bm{\theta}^{})$ is the state vector predicted using the proposed approach. $\left\\|\cdot\right\\|_{2}$ represents the $l_{2}$ norm. The dataset for solving the state estimation problems is generated using OpenFoam [53]. The proposed approach has been implemented using PyTorch [54]. The software associated with the proposed approach, along with the implementation of both the examples, will be made available on acceptance of the paper. ### 5.1 Periodic Vortex shedding past a cylinder As the first example, we consider two-dimensional flow past a circular cylinder at Reynolds’ number $Re=190$. It is a well known canonical problem and is characterized by periodic laminar flow vortex shedding. A schematic representation of the computational domain is shown in Fig. 8. The circular cylinder is considered to have a diameter of $1$ unit. The center of the cylinder is located at a distance of $8$ units from the inlet. The outlet is located at a distance of $25$ units from the center of the cylinder. The sidewalls are at $4$ units distance from the center of the cylinder. At the inlet boundary, a uniform velocity of $1$ unit along the $X$-direction is applied. Pressure boundary condition with $P=0$ is considered at the outlet. A no-slip boundary at the cylinder surface is considered. Figure 8: (a) Schematic representation of the computational domain with boundary conditions at the inlet and the outlet. The cylinder is having a diameter of 1 unit. A no-slip boundary is considered at the cylinder wall. Zero pressure gradient at the inlet and zero velocity gradient at the outlet are considered. (b) Schematic of the problem domain with snapshot cutout of $a\times b$. For periodic vorticity problem, $a=6$ units and $b=4$ units. For transient flow problem, $a=12$ units and $b=6$ units. The schematics are not to scale. The dataset necessary for training the proposed model is generated by using Unsteady Reynolds-averaged Navier Stokes (URANS) simulation in OpenFoam [53]. The selection of URANS for data generation is based on the fact that results obtained using URANS are highly accurate in $Re\in[30,300]$ [55]. Nonetheless, the method proposed is not dependent on the fluid simulator used and can be seamlessly used in conjunction with more accurate simulator like DNS. The overall problem domain is discretized into 63420 elements with finer mesh near the cylinder. Time step $\delta t=0.02$ units is considered. For training the model, a library of 180 snapshots is generated by running OpenFoam. Additional 120 snapshots, 60 for validation and 60 for testing, have also been generated. Two consecutive snapshots are separated by 10$\delta t$. Coordinate of the snapshot cutout stretches from $[0,-2]\times[6,2]$ which is discretized into $168\times 251$ points in $x$ and $y$ directions (see Fig. 8(b)). The objective here is to recover the complete vorticity-field in the cutout by using the sensor measurements. Details on the network architecture are provided in Table 1. Table 1: Network architecture of proposed ARE and SD for periodic vortex shedding problem. BN indicates batch normalization and DR represents dropout. HS is the number of features in the hidden state of lstm. Networks \| Architecture ---\|--- ARE(Auto-Encoder) \| $251\times 168\rightarrow BN^{}\rightarrow 1024\rightarrow BN^{}\rightarrow DR(0.35)^{}\rightarrow 256\rightarrow 25\rightarrow 256\rightarrow 1024\rightarrow 168\times 251$ ARE(RNN) \| LSTM(HS=50) $\rightarrow 50\rightarrow 50\rightarrow 25$ SD \| $N_{s}\rightarrow 35\rightarrow BN\rightarrow DR(0.1)^{}\rightarrow 40\rightarrow BN\rightarrow 168\times 251$ Components with ∗ are not used in case of noise-free data. To illustrate the superiority of the proposed approach, the results obtained using the proposed approach are compared with those obtained using proper orthogonal decomposition-based deep state estimation (PDS) proposed by [29] and shallow decoder (SD) proposed by [5]. In PDS, the first 25 modes are used, which is the same as the number of neurons in the bottleneck layer of the proposed approach. The feed-forward neural network used in PDS for mapping the sensor measurements to the latent state is the same as the feed-forward network used in ARE. Note that comparison with other popular approaches such as gappy-POD and linear stochastic estimation is not shown as it is already established in [29] that PDS outperforms both the approaches. Brief details on PDS, gappy-POD, and linear stochastic estimation are provided in A. Fig. 9 shows the results obtained using different approaches. It is an idealized case where we have considered the sensor data to be noise free. The sequence length of four is used for training ARE. We have considered the extreme case where data from only one sensor is available. We observe that the proposed ARE, with only one senor, is able to recover the full state accurately. PDS and SD, on the other hand, yields less accurate results. (a) PDS (b) ARE (c) SD (d) Ground Truth Figure 9: Figure depicts results of periodic vortex shedding for one sensor. Orange dots in the images represents the location of sensor. SEQ-LEN used for ARE model is 4. (a) PDS (b) ARE (c) SD (d) Ground Truth Figure 10: Figure depicts results of periodic vortex shedding for one sensor. The data is corrupted with white Gaussian noise having SNR = 20. Orange dots in the images represents the location of sensor. SEQ-LEN used for ARE model is 4. Fig. 10 shows results corresponding to the case where the sensor is corrupted by white Gaussian noise. This is a more realistic case. Again, a sequence length of 4 is considered and it is assumed that data from only one sensor is available. In this case also, we observe that ARE yields highly accurate results and outperforms PDS and SD. The effect of noise on the proposed ARE is shown in Fig. 11. We observe that both ARE (with one and two sensors) and SD (with two sensors) yield the best results. The fact that the proposed approach is able to correctly predict the state from only one sensor data (noisy) is really impressive. SD with one sensor and PDS predicted results are significantly less accurate as compared to the proposed approach. Figure 11: (a) Performance of the proposed approach ARE, SD and PDS for periodic vortex shedding problem at different noise level. For both ARE and PDS, results corresponding to one and two sensors are presented. (b) Performance of proposed ARE, SD and PDS with increase in number of sensors. Data used is noise free. Next, we investigate the effect of varying the number of sensors. Fig. 11 shows the performance of different methods with the increase in the number of sensors. The data considered is idealistic with no noise. Cases corresponding to one, two, five, and ten sensors are presented. For all the cases, the proposed ARE is found to yield the best result. Results obtained using SD and PDS follows a similar trend. We also carried out an additional case study where we considered that sensor data from both past and future is available. A bi-directional RNN (B-RNN) based ARE developed for the same. However, due to the paucity of space, the same is not presented here. Those interested can refer to B for details on the same. ### 5.2 Transient Flow past a cylinder As the second example, we consider the problem involving transient flow past a cylinder. Because of the transient nature of the flow, this is much more challenging than the periodic vortex shedding problem in Section 5.1. The problem domain, meshing, and solution strategy for this problem are considered the same as the periodic vortex shedding problem. However, unlike the previous problem, we have considered variation in the Reynolds’ number. This exponentially increases the complexity of the problem. The training library for this problem was created by running OpenFoam. Total 1200 snapshots consisting of $400$ snapshots at $Re=[180,190,200]$ were generated. Given the fact that $Re$ considered for this problem, resides in $[30,300]$, we have used URANS for generating the data [55]. Validation and test set consists of $400$ sequential snapshots at $Re=[185,195]$. Similar to the previous problem, the snapshots were separated by a time interval of $10\delta t$. Co-ordinate of snapshot cutout stretches from $\left[0,-3\right]\times\left[12.0,3.0\right]$ (see Fig. 8(b)). The cutout is discretized into $252$ and $502$ points in the $x$ and $y$ direction, respectively. Similar to the previous example, the objective here is to recover the vorticity field based on sensor measurements. Similar to the previous example, results obtained have been compared with PDS and SD. Details on the network architecture used for this problem are provided in Table 2. Table 2: Network architecture of proposed ARE for transient flow problem. HS is the number of features in the hidden state of lstm. Networks \| Architecture ---\|--- ARE(Auto-Encoder) \| $252\times 502\rightarrow BN^{}\rightarrow 2000\rightarrow BN^{}\rightarrow DR(0.35)^{}\rightarrow 300\rightarrow 25\rightarrow 300\rightarrow 2000\rightarrow 252\times 502$ ARE(RNN) \| LSTM(HS=50) $\rightarrow 200\rightarrow 200\rightarrow 25$ SD \| $N_{s}\rightarrow 350\rightarrow BN\rightarrow DR(0.1)^{}\rightarrow 400\rightarrow BN\rightarrow 252\times 502$ Components with ∗ are not used in case of noise-free data. (a) PDS (b) ARE (c) SD (d) Ground truth Figure 12: Figure depicts results of Transient Flow for one sensor. Orange dots in the images represents the location of sensor. SEQ-LEN used for ARE model is 4 Fig. 12 shows the reconstructed vorticity field using ARE, SD and PDS. Ground truth has also been reported. It is an idealized case where we have considered the sensor data to be noise-free. A sequence length of four is used for ARE. We have considered the extreme case where data from only one sensor is available. For this problem also, the proposed is able to recover the full vorticity field accurately. PDS and SD, on the other hand, fails to accurately recover the vorticity field. Next, we consider a more realistic scenario where the sensor data is corrupted by noise. Fig. 13 shows the results corresponding to noisy sensor measurement. In this case, we have considered that measurements from two sensors are available. For this case also, the results obtained using the proposed ARE is found to be superior as compared to those in the literature. (a) PDS (b) ARE (c) SD (d) Ground truth Figure 13: Figure depicts results of Transient Flow for two sensors. The data is corrupted with white Gaussian noise with SNR = 28. Orange dots in the images represents the location of sensor. SEQ-LEN used for ARE model is 4 The effect of noise on the performance is shown Fig. 14. Overall, results obtained using the proposed ARE are found to be more accurate as compared to those obtained using PDS and SD. The improved accuracy of ARE can be attribute to the fact that it learns from a sequence of data. Surprisingly, we observe that PDS with one sensor yields superior result as compared to PDS with two sensors. This is because the POD modes for two sensor cases is susceptible to the noise in the data. As noise in the data reduces, the POD modes are identified correctly and performance of PDS with two sensors is found to be superior as expected. The proposed ARE and SD are immune to such overfitting phenomenon. We also investigate the effect of a number of sensors and sequence length considered in the proposed approach. Fig. 15 shows the performance of different methods with an increase in the number of sensors. Similar to previous example, cases corresponding to one, two, five, and ten sensors are presented. As expected, increase in number of sensors results in a improved accuracy. As for accuracy, for all the four cases, the proposed approach yields the best result outperforming both PDS and SD. Fig. 15 illustrates the performance of the proposed approach corresponding to the different sequence length. As expected, initially, the results are found to improve with an increase in the sequence length. ARE reaches a saturation point at sequence length three, and no significant improvement in results is observed on further increasing the sequence length. Figure 14: Performance of the proposed approach ARE, SD and PDS for the transient flow problem at different noise level. ARE1 corresponds to results with one sensor and ARE2 corresponds to results with 2 sensors. Figure 15: (a) Performance of PDS, SD, and proposed ARE with increase in number of sensors. (b) Performance of proposed ARE with increase in sequence length. Data used in both is without noise. Lastly, similar to previous example, we have also considered the case where both past and future data are available. B-RNN based ARE is used for solving the same. For details on the same, interested readers may refer B. ### 5.3 NOAA Optimum Interpolation (OI) SST V2 Dataset As the last example, we test our approach on the well-known Sear Surface Temperature (SST) dataset. SST Data is publicly available and uploaded by the Physical Sciences Division at National Oceanic and Atmospheric Administration (NOAA). The data is made available by the respective organization after the analysis of in-situ and satellite observation of the SST. The resolution of the data available corresponding to the variable “time” is Weekly, Monthly and monthly long-term mean. The resolution corresponding to the spatial domain is of 1 deg latitude and 1 deg longitude, which translates to 180 x 360 grid points. In this study we make use of weekly mean data from the year 1990 on- wards. The data is said to be centred around Wednesday for this period. Data for initial 400 weeks was chosen as the training data-set. Following 100 weeks data was chosen as validation data-set and following 200 weeks data for the testing. The problem description here is to reconstruct the SST data on the spatial domain with the help of limited sensor data measurements. Details on the network architecture used for this problem are provided in Table 3. Table 3: Network architecture of proposed ARE and SD for SST dataset. BN indicates batch normalization and DR represents dropout. HS is the number of features in the hidden state of lstm. Networks \| Architecture ---\|--- ARE(Auto-Encoder) \| $44219\rightarrow BN^{}\rightarrow 512\rightarrow BN^{}\rightarrow DR(0.35)^{}\rightarrow 256\rightarrow 25\rightarrow 256\rightarrow 512\rightarrow 44219$ ARE(RNN) \| LSTM(HS=50) $\rightarrow 100\rightarrow 100\rightarrow 25$ SD \| $N_{s}\rightarrow 350\rightarrow BN\rightarrow DR(0.1)^{}\rightarrow 400\rightarrow BN\rightarrow 44219$ Components with ∗ are not used in case of noise-free data. Fig. 16 shows the comparison of results between three different architectures we chose for our study i.e., PDS, SD, and ARE. To show the applicability of the techniques in real world scenarios, we make use of corrupted or noisy data at different signal to noise ratios and at three different sensor measurements i.e. 2, 4 and 8. The proposed problem reconstructs the SST data over the globe accurately for different sensor data input. We observe, with decreasing noise in the data, either of the approaches perform well; however, when noise in the data is significant, the proposed ARE is found to be significantly more accurate as compared to PDS and SD. Even with higher number of sensors, the proposed approach is found to be superior (at higher noise level) as compared to PDS and SD. For better visualization, the contour plot of reconstructed sea surface temperature is shown in Fig. 17. We have considered SNR to be 20 and assumed data to be available from four sensors. Sequence length of four is used for the proposed ARE. It is clearly visible that the reconstructed field using the proposed ARE is more accurate compared to the PDS and SD reconstructed fields. (a) sensors 2 (b) sensors 4 (c) sensors 8 Figure 16: Change in performance of different approaches is presented with increasing noise using three cases (a) PDS (b) ARE (c) SD (d) Ground truth Figure 17: Figure depicts results of SST with 4 sensors and at noise of SNR 20. Blue dots in the images represents the location of sensor. SEQ-LEN used for ARE model is 4 (a) SNR 10 (b) SNR 20 (c) SNR 30 Figure 18: Performance of the three model is compared with increasing no. of sensors at SNR 10, 20, 30 (a) sensor 2 (b) sensor 4 (c) sensor 8 Figure 19: Performance of ARE and PDS is compared at different number of neurons in bottleneck layer with noise of SNR 20 In Fig. 18, we compare the results of the approaches corresponding to the increasing number of sensors at three different SNR ratios. The investigation further strengthens our claim for the robustness of our proposed approach, ARE. As the number of sensors increases, it is evident that all the approaches tend to present better results. For the sake of comparison, we considered corrupted data at three different SNR, 10, 20 and 30, since the approaches mentioned perform more or less the same without noise. At SNR 10, the proposed approach tends to outperform its counterparts by a larger margin compared to SNR 30. As the corruption in data increases, ARE tends to perform much better than its counterparts. SD performs almost similar to ARE at lower noise level and better than PDS in all cases. Lastly, we investigate the sensitivity of the proposed approach to the number of neurons in the bottleneck layer. For PDS, this corresponds to the number of POD modes. The concept of bottleneck layer doesn’t exist for SD and hence, the same is not reported. Results obtained are shown in Fig. 19. The results are presented corresponding to cases with 2, 4 and 8 sensors. The SNR is kept fixed at 20. The number of neurons in the bottleneck layer is varied in $[5,50]$. We observe that the proposed approach is less sensitive to the number of neurons in the bottleneck layer, although slight improvement can be observed as the layer size is increased. This observation presents an insightful understanding of our approach which could be translated to a computationally less expensive approach, with decreased layer size of our bottleneck layer. As for PDS, we observe that the accuracy reduces with increase in the number of POD modes. This seems counter-intuitive as increase in POD modes is expected to reduce the error. However, on a closer examination, we observed that with more number of POD modes, the number of parameters in neural network increases, resulting in overfitting. This can be addressed by increasing the number of sensors. Overall, the results presented in this section illustrate the accuracy and robustness of the proposed approach in state estimation from limited sensor. ## 6 Conclusions In this work, we introduced a novel deep learning based approach for state estimation. The proposed approach uses an auto-encoder as a reduced-order model and recurrent neural network to map the sensor measurements to the reduced state. The proposed framework is superior to existing state estimation frameworks in two aspects. First, auto-encoder, being a nonlinear manifold learning framework, is superior to the usually used proper orthogonal decomposition. Secondly, unlike existing state estimation frameworks, the proposed approach utilizes present and past sensor measurements using the recurrent neural network. It results in improved accuracy, specifically for cases where limited sensors are deployed. Experiments performed on simulation of flow past a cylinder and sea-surface data illustrated the capability of the proposed approach in learning from sequential data. Comparison carried out with respect to proper orthogonal decomposition-based deep state estimation and shallow-decoder showed the superior accuracy of the proposed approach. Moreover, the proposed approach was also found to be robust to the noise in the sensor measurements. Utilizing sequential information can prove beneficial in many other state estimation tasks. Future work can be aimed at exploiting effects of other models used on sequential information such as transformers, comparing between other auto-encoder network variations, the effect of varying time step between measurements, developing models capable of transfer learning, i.e., trained on smaller time step but can use larger time step. 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Exploiting the past and the future in protein secondary structure prediction. Bioinformatics (Oxford, England), 15:937–46, 12 1999. ## Appendix A Proper orthogonal decomposition based deep state estimation (PDS) In this section, we briefly provide the theoretical background of PDS. We note that development of PDS is motivated from gappy-POD and linear stochastic estimation and hence, the discussion in this section also starts from gappy- POD and then proceeds to PDS via linear stochastic estimation. Let training set consist of $N_{t}$ flattened data images, $\mathbf{W}=(\bm{w}_{1},\bm{w}_{2},\ldots,\bm{w}_{N_{t}})$, where $\bm{w}_{t}\in\mathbb{R}^{N_{w}}$ and corresponding sensor measurements $\bm{s}=(\bm{s}_{1},\bm{s}_{2},\dots,\bm{s}_{N_{t}})$. As already state in Section 2, the aim in state estimation is to recover full state $\bm{w}$ via sensor measurements $\bm{s}$. ### A.1 Gappy-POD Consider, $\mathbf{H}\in\mathbb{R}^{N_{s}\times N_{w}}$ is measurement operator that maps full state to measurements and have ones as sensor location and zeros otherwise. Mathematically, $\bm{s}=\mathbf{H}\bm{w},$ (12) where $\bm{w}$ can be approximated by linear combination of $k$ modes. Modes used here are $k$ most dominant modes, $\bm{\Phi}$, and are obtained by proper orthogonal decomposition of training data $\mathbf{W}$ and selecting first $k$ singular vectors, $\bm{w}\approx\sum_{i=1}^{k}\phi_{i}a_{i}=\bm{\Phi}\bm{A},$ (13) where $\mathbf{W}\approx\bm{\Phi}DB^{},truncated\hskip 5.69054ptk-rank,$ (14) and $\bm{s}\approx\mathbf{H}\bm{\Phi}a.$ (15) In Eq. (14), $\bm{\Phi}\in\mathbb{R}^{N_{w}\times k}$. During testing $\bm{a}$ is obtained by solving the following minimization problem. $\bm{a}\in\underset{\tilde{\bm{a}}}{\arg\min}\left\\|\bm{s}-\mathbf{H}\bm{\Phi}\tilde{\bm{a}}\right\\|\|_{2}^{2}.$ (16) Solution to this problem is obtained by taking Moore-Penrose pseudo inverse $\bm{a}=(\mathbf{H}\bm{\Phi})^{+}\bm{s}$ (17) This approach requires previous knowledge of operator $\mathbf{H}$. This operator is only available for simple systems but is often unknown for systems of practical interest. ### A.2 Linear stochastic estimation Linear Stochastic Estimation overcomes this issue by defining another operator which will map latent state to sensor. Operator $P:\mathbb{R}^{k}\mapsto\mathbb{R}^{N_{s}}$ is learned from training data via the following minimization problem. $P\in\underset{\tilde{P}}{\arg\min}\|\|\bm{s}-\tilde{P}\bm{A}\|\|^{2}_{2},$ (18) where $\mathbf{S}\in\mathbb{R}^{N_{s}\times N_{t}}$ and $A\in\mathbb{R}^{k\times N_{t}}$. This approach completely skips over the operator $\mathbf{H}$; instead, $P$ is considered to be the empirical estimate of the linear operator $\mathbf{H}\Phi$. Subsequently, $\bm{a}$ is obtained as $\bm{a}\in\underset{\tilde{\bm{a}}}{\arg\min}\|\|\bm{s}-P\tilde{\bm{a}}\|\|^{2}_{2}$ (19) ### A.3 Deep state estimation Deep state estimation [29] approache replaces the linear mapping between latent state and sensors with nonlinear mapping to further generalize the approach. It uses a neural network $G:\mathbb{R}^{N_{s}}\mapsto\mathbb{R}^{bn}$ parametrized by $\theta$ for nonlinear mapping of sensor measurements to approximate embeddings. $\bm{a}=G(\bm{s},\bm{\theta})$ (20) The neural network is trained as $\bm{\theta}\in\underset{\tilde{\bm{\theta}}}{\arg\min}\sum_{i=1}^{N_{t}}\|\|\bm{a}^{(i)}-G(\bm{s}^{(i)},\tilde{\bm{\theta}})\|\|_{2}^{2},$ (21) where $\bm{a}\in\underset{\tilde{\bm{a}}}{\arg\min}\|\|\bm{w}-\Phi\tilde{\bm{a}}\|\|^{2}_{2}.$ (22) $\bm{\Phi}$ as before are the POD-modes. During testing trained neural network $G$ approximate embeddings for sensor measurements which is used to recover full state. We refer to this method as POD based deep state estimation (PDS). ## Appendix B Bi-directional RNN The basic idea of bidirectional recurrent neural nets (B-RNN) [56, 57] is to present each training sequence to two separate recurrent nets namely forward and backward, both of which are connected to the same output layer. (In some cases a third network is used in place of the output layer, but here we have used the simpler model). This means that for every point in a given sequence, the BRNN has complete, sequential information about all points before and after it. In this work, we have used B-RNN for solving a special state estimation problem where sensor measurements from both past and future states are available. A Schematic representation of B-RNN used is shown in the Fig. 20. It is modified to stop propagating information after the middle time step in forward and backward network. This lowers computational time and reduce number of training weights. Thus architecture proposed is capable of using information from both ahead and behind in time. Figure 20: Schematic representation of bi-directional ARE. This is useful when both past and future sensor measurements are available. (a) Periodic vortex shedding (b) Transient flow Figure 21: Comparison between B-RNN based ARE and RNN based ARE To feed sensor measurements $s^{t-k,\ldots,t+k}$ into network, data is splitted into $s^{t-k,\ldots,t}$ and $s^{t,\ldots,t+k}$ as shown in the Fig. 20. Note that although more accurate, this network can use only be used when delay in time of reconstruction is acceptable also it uses odd sequence length of sensor measurements. Output of forward and backward network is concatenated and passed to feedforward network which yields the final estimated latent representation of middle sensor measurement $s^{t}$. Thus training of $\mathcal{N}$ can be formulated as follows $\bm{\theta}_{M}^{}=\arg\min_{\bm{\theta}}\sum_{i=1}^{N_{\text{samp}}}\left\\|h^{t}-\mathcal{N}\left(\bm{s}^{t-k:t+k};\bm{\theta}_{M}\right)\right\\|+\lambda\left\\|\bm{\theta}_{M}\right\|_{2}^{2},$ (23) where $\theta_{M}$ are parameters of combined RNN and feed-forward network $\mathcal{N}:\mathbb{R}^{N_{s}*(2k+1)}\mapsto\mathbb{R}^{h}$. ### B.1 Results Fig. 21 shows a comparative assessment between B-RNN based ARE, and RNN based ARE. Results corresponding to one, two, five, and ten sensors are presented. For one sensor, results obtained using B-RNN based ARE significantly outperforms those obtained using RNN based ARE. However, as the number of sensors increases, the results obtained using the two approaches becomes identical. One counter-intuitive result is obtained for the periodic vortex shedding problem where the result obtained using B-RNN based ARE found to be worse than that obtained using RNN based ARE. It is probably because the neural network parameters for B-RNN has converged to a local minimum.
# Investigating Bi-Level Optimization for Learning and Vision from a Unified Perspective: A Survey and Beyond Risheng Liu, Jiaxin Gao, Jin Zhang, Deyu Meng, and Zhouchen Lin, R. Liu and J. Gao are with the DUT-RU International School of Information Science & Engineering, Dalian University of Technology, and also with the Key Laboratory for Ubiquitous Network and Service Software of Liaoning Province, Dalian 116024, China. E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>R. Liu is the corresponding author. J. Zhang is with the Department of Mathematics, Southern University of Science and Technology, and National Center for Applied Mathematics Shenzhen, China, E-mail<EMAIL_ADDRESS>Meng is with School of Mathematics and Statistics and Ministry of Education Key Lab of Intelligent Networks and Network Security, Xi’an Jiaotong University, Xi’an, Shaanxi, China. E-mail<EMAIL_ADDRESS>Lin is with the Key Laboratory of Machine Perception (Ministry of Education), School of Electronics Engineering and Computer Science, Peking University, Beijing 100871, China, and also with the Cooperative Medianet Innovation Center, Shanghai Jiao Tong University, Shanghai 200240, China. E-mail: <EMAIL_ADDRESS>received April 19, 2005; revised August 26, 2015. ###### Abstract Bi-Level Optimization (BLO) is originated from the area of economic game theory and then introduced into the optimization community. BLO is able to handle problems with a hierarchical structure, involving two levels of optimization tasks, where one task is nested inside the other. In machine learning and computer vision fields, despite the different motivations and mechanisms, a lot of complex problems, such as hyper-parameter optimization, multi-task and meta learning, neural architecture search, adversarial learning and deep reinforcement learning, actually all contain a series of closely related subproblms. In this paper, we first uniformly express these complex learning and vision problems from the perspective of BLO. Then we construct a best-response-based single-level reformulation and establish a unified algorithmic framework to understand and formulate mainstream gradient-based BLO methodologies, covering aspects ranging from fundamental automatic differentiation schemes to various accelerations, simplifications, extensions and their convergence and complexity properties. Last but not least, we discuss the potentials of our unified BLO framework for designing new algorithms and point out some promising directions for future research. ###### Index Terms: Bi-level optimization, Learning and vision applications, Value-function-based reformulation, Best-response mapping, Explicit and implicit gradients. ## 1 Introduction Bi-Level Optimization (BLO) is the hierarchical mathematical program where the feasible region of one optimization task is restricted by the solution set mapping of another optimization task (i.e., the second task is embedded within the first one) [1]. The outer optimization task is commonly referred to as the Upper-Level (UL) problem, and the inner optimization task is commonly referred to as the Lower-Level (LL) problem. BLOs involve two kinds of variables, referred to as the UL and LL variables, accordingly. The origin of BLOs can be traced to the domain of game theory and is known as Stackelberg competition [2]. Subsequently, it has been investigated in view of many important applications in various fields of science and engineering, particularly in economics, management, chemistry, optimal control, and resource allocation problems [3, 4, 5, 6]. Especially, in recent years, a great amount of modern applications in the fields of machine learning and computer vision (e.g., hyper-parameter optimization [7, 8, 9, 10], multi-task and meta learning [11, 12, 13], neural architecture search [14, 15, 16], generative adversarial learning [17, 18, 19], deep reinforcement learning [20, 21, 22] and image processing and analysis [23, 24, 25], just name a few) have arisen that fit the BLO framework. In general, most of the earlier BLOs are highly complicated and computationally challenging to solve due to their nonconvexity and non- differentiability [26, 27]. Despite their apparent simplicity, BLOs are nonconvex problems with an implicitly determined feasible region even if the UL and LL subproblems are convex [28, 29]. Indeed, it has been proved that even strictly checking the local optimality of the simplest BLO model (e.g., linear BLO) is still a NP-hard problem [30, 31]. In addition, the existence of multiple optima for the LL subproblem can result in an inadequate formulation of BLOs, which could aggravate the difficulty of theoretical analysis [32]. Despite the challenges, a lot of research topics consisting of methods and applications of BLOs have followed in this field, see [27, 33]. Early studies focused on numerical methods, including extreme-point methods [34], branch- and-bound methods [3, 35], descent methods [36, 37], penalty function methods [38, 39], trust-region methods [40, 41], and so on. The most often used procedure is to replace the LL subproblem with its Karush–Kuhn–Tucker (KKT) conditions, and if assumptions are made (such as smoothness, convexity, among others) the BLOs can be transformed into single-level optimization problems [42, 43, 44]. However, due to the high complexity of bi-level models, solving BLOs for large-scale and high-dimensional practical applications in learning and vision fields is still challenging [45]. The classical idea (e.g., the first-order approach in economics literature) to reformulate BLO is to replace the LL subproblem Eq. (1) by its KKT conditions and minimize over the original variables $\mathbf{x}$ and $\mathbf{y}$ as well as the multipliers. The resulting problem is a so-called Mathematical Program with Equilibrium Constraints (MPEC) [46, 47]. Unfortunately, MPECs are still a challenging class of problems because of the presence of the complementarity constraint [48]. Solution methods for MPECs can be categorized into two types of approaches. The first one, namely, the nonlinear programming approach rewrites the complementarity constraint into nonlinear inequalities, and then allows to leverage powerful numerical nonlinear programming solvers. The other one, namely, the combinatorial approach tackles the combinatorial nature of the disjunctive constraint. Despite the difficulties, MPEC has been studied intensively in the last three decades [49]. Recently, some progress on the MPEC approach in dealing with BLOs have been witnessed by the community of mathematical programming, in the context of selecting optimal hyper-parameters in regression and classification problems. There are two issues caused by the multipliers in the MPEC approach. First, in theory, if there exist more than one multipliers for the LL subproblem, MPEC will not be equivalent to the original BLO (in the local optimality scenario) [50]. Second, the introduced auxiliary multiplier variables can limit the numerical efficiency when solving the BLO problem. In recent years, a variety of machine learning and computer vision tasks, including but not limited to, hyper-parameter optimization [51, 52, 13, 53]), multi-task and meta learning [54, 55, 56, 57], neural architecture search [14, 16, 58, 59], adversarial learning [18, 17, 60, 21], and deep reinforcement learning [20, 61, 21, 62], have been investigated in application scenarios. Despite the different motivations and mechanisms, all these problems contain a series of closely related subproblems and have a natural hierarchical optimization structure. However, although received increasing attentions in both academic and industrial communities, there still lack a unified perspective to understand and formulate these different categories of hierarchical learning and vision problems. We notice that most previous surveys on BLOs (e.g., [1, 63, 64, 65, 66, 67, 68]) are purely from the viewpoint of mathematical programming and mainly focus on the formulations, properties, optimality conditions and these classical solution algorithms, such as evolutionary methods [5]. In contrast, the aim of this paper is to utilize BLO to express a variety of complex learning and vision problems, which explicitly or implicitly contain closely related subproblems. Furthermore, we present a unified perspective to comprehensively survey different categories of gradient-based BLO methodologies in specific learning and vision applications. In particular, we first provide a literature review on various complex learning and vision problems, including hyper-parameter optimization, multi-task and meta learning, neural architecture search, adversarial learning, deep reinforcement learning and so on. We demonstrate that all these tasks can be modeled as a general BLO formulation. Following this perspective, we then establish a best- response-based single-level reformulation to express these existing BLO models. By further introducing a unified algorithmic framework on the single- level reformulation, we can uniformly understand and formulate these existing gradient-based BLOs and analyze their accelerations, simplifications, extensions, and convergence and complexity proprieties. Finally, we demonstrate the potentials of our framework for designing new algorithms and point out some promising research directions for BLO in learning and vision fields. Compared with existing surveys on BLOs, our major contributions can be summarized as follows: 1. 1. To the best of our knowledge, this is the first survey paper to focus on uniformly understanding and (re)formulating different categories of complex machine learning and computer vision tasks and their solution methods (especially in the context of deep learning) from the perspective of BLO. 2. 2. By introducing a best-response-based single-level reformulation and constructing a best-response-based algorithmic framework, we obtain a general and flexible platform that can successfully unify different existing gradient- based BLO methodologies and uniformly analyze these accelerations, simplifications, and extensions in literature. 3. 3. The convergence behaviors of gradient-based BLOs are comprehensively analyzed. Especially, we establish a general convergence analysis template to investigate the iteration behaviors of a series of gradient-based BLOs from a unified perspective. The time and space complexity of various mainstream schemes is also systematically analyzed. 4. 4. Our gradient-based BLO platform not only comprehensively covers mainstream gradient-based BLO methods, but also has potentials for designing new BLO algorithms to deal with more challenging tasks. We also point out some promising directions for future research. We summarize our mathematical notations in Table I. The remainder of this paper is organized as follows. We first introduce some necessary fundamentals of BLOs in Section 2. Then, Section 3 provides a comprehensive survey of various learning and vision applications that all can be modeled as BLOs. In Section 4, we establish an algorithmic framework in a unified manner for existing gradient-based BLO schemes. Within this framework, we further understand and formulate two different categories of BLOs (i.e., explicit and implicit gradients for best-response) in Section 5 and Section 6, respectively. We also discuss the so-called lower-level singleton issue of BLOs in Section 7. The convergence and complexity properties of these gradient-based BLOs are discussed in Section 8. Section 9 puts forward potentials of our framework for designing new algorithms to deal with more challenging pessimistic BLOs. Finally, Section 10 points out some promising directions for future research. TABLE I: Summary of mathematical notations. Notation \| Description \| Notation \| Description ---\|---\|---\|--- $\mathcal{D}_{\mathtt{tr}}$/$\mathcal{D}_{\mathtt{val}}$ \| Training/Validation data \| $o^{ij}/\mathbf{x}_{o}^{ij}$ \| Operations/Operation weights $\pi/r$ \| Policy/Reward \| $s/a$ \| State/Action $Q^{\pi}$ \| Q-function under $\pi$ \| $Q^{\pi}(s,a)$ \| State-action value-function $G/D$ \| Generator/Discriminator \| $\mathbf{u}/\mathbf{v}$ \| Real-world image/Random noise $\rho_{t}$ \| Aggregation parameters \| $\mathbf{x}\in\mathbb{R}^{m}$/$\mathbf{y}\in\mathbb{R}^{n}$ \| UL/LL variable $F/f$ \| UL/LL objective \| $\mathcal{S}(\mathbf{x})$ \| Solution set of the LL subproblem (given $\mathbf{x}$) $\mathbf{y}^{}(\mathbf{x})$ \| BR mapping \| $\widetilde{\mathcal{S}}(\mathbf{x})$ \| Solution set of the ISB subproblem (given $\mathbf{x}$) $\mathtt{dist}(\cdot)$ \| Point-to-set distance \| $\circ$ \| Compound operation $\Psi(\mathbf{x})$ \| $\Psi=\Psi_{T}\circ\cdots\circ\Psi_{1}\circ\Psi_{0}$ \| $\Psi_{\bm{\theta}}(\mathbf{x})$ \| Hyper-network with parameters $\bm{\theta}$ $\Psi_{t}$ \| Dynamical system at $t$-th stage \| $\frac{\partial\varphi(\mathbf{x})}{\partial\mathbf{x}}$ \| Gradient of $\mathbf{x}$ $\frac{\partial F(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{x}}$ \| Direct gradient of $\mathbf{x}$ \| $\mathbf{G}(\mathbf{x})$ \| Indirect gradient of $\mathbf{x}$ $\frac{\partial\mathbf{y}^{}(\mathbf{x})}{\partial\mathbf{x}}$ \| BR Jacobian \| $\frac{\partial\varphi(\mathbf{x}^{k})}{\partial\mathbf{x}^{k}}$ \| Numerical BR Jacobian w.r.t. $\mathbf{x}^{k}$ $\varphi(\mathbf{x})$ \| UL value-function \| $\psi(\mathbf{x})$ \| LL value-function $\mathbf{d}(\mathbf{y}_{t-1};\mathbf{x})$ \| Optimistic aggregated gradient \| $\widetilde{\mathbf{d}}(\mathbf{y}_{t-1};\mathbf{x})$ \| Pessimistic aggregated gradient $\left(\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}}\right)^{-1}$ \| Inverse Hessian matrix \| $\left(\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}}\right)^{-1}\frac{\partial F}{\partial\mathbf{y}}$ \| Inverse Hessian-vector product $t$/$k$ \| Index of the LL/UL iteration \| $T/K$ \| Maximum LL/UL iteration number $(\cdot)_{t}$ \| $t$-th LL iteration \| $(\cdot)^{k}$ \| $k$-th UL iteration $(\cdot)^{{}^{\prime}}$ \| Transposition operation \| $(\mathbf{x}^{},\mathbf{y}^{})$ \| The optimal UL and LL solutions $\mathbf{Z}_{T}$ \| $\sum_{t=1}^{T}\left(\prod_{i=t+1}^{T}\mathbf{A}_{i}\right)\mathbf{B}_{t}$ \| $\mathbf{Z}_{T-M}$ \| $\sum_{t=T-M+1}^{T}\left(\prod_{i=t+1}^{T}\mathbf{A}_{i}\right)\mathbf{B}_{t}$ $\mathbf{P}(\mathbf{y}_{t-1},\bm{\omega})$ \| Layer-wise transformation \| $\sum\limits_{j=0}^{\infty}\left(\mathbf{I}-\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}}\right)^{j}$ \| Neumann series $\mathbf{A}_{t}$ \| $\frac{\partial\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})}{\partial\mathbf{y}_{t-1}}$ \| $\mathbf{B}_{t}$ \| $\frac{\partial\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})}{\partial\mathbf{x}}$ $\inf_{\mathbf{y}\in\mathcal{S}(\mathbf{x})}F(\mathbf{x},\mathbf{y})$ \| Optimistic objective \| $\psi_{\mu}(\mathbf{x})$ \| parameterized LL value-function (with $\mu$) $\sup_{\mathbf{y}\in\mathcal{S}(\mathbf{x})}F(\mathbf{x},\mathbf{y})$ \| Pessimistic objective \| $\varphi_{\mu,\theta,\tau}\left(\mathbf{x}\right)$ \| parameterized UL value-function (with $\mu$, $\theta$ and $\tau$) ## 2 Fundamentals of Bi-Level Optimization Bi-Level Optimization (BLO) contains two levels of optimization tasks, where one is nested within the other as a constraint. The inner (or nested) and outer optimization tasks are often respectively referred to as the Lower-Level (LL) and Upper-Level (UL) subproblems [1]. Correspondingly, there are two types of variables, namely, the LL ($\mathbf{y}\in\mathbb{R}^{n}$) and UL ($\mathbf{x}\in\mathbb{R}^{m}$) variables. Specifically, the LL subproblem can be formulated as the following parametric optimization task $\min_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y}),\ (\mbox{parameterized by $\mathbf{x}$}),$ (1) where we consider a continuous function $f:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}$ as the LL objective and $\mathcal{Y}\subseteq\mathbb{R}^{n}$ is a nonempty set. By denoting the value- function as $\psi(\mathbf{x}):=\min_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y})$, we can define the solution set of the LL subproblem with given $\mathbf{x}$ as $\mathcal{S}(\mathbf{x}):=\left\\{\mathbf{y}\in\mathcal{Y}~{}\|~{}f(\mathbf{x},\mathbf{y})\leq\psi(\mathbf{x})\right\\}.$ Then the standard BLO problem can be formally expressed as $\min\limits_{\mathbf{x}\in\mathcal{X}}F(\mathbf{x},\mathbf{y}),\ s.t.\ \mathbf{y}\in\mathcal{S}(\mathbf{x}),$ (2) where the UL objective $F:\mathbb{R}^{m}\times\mathbb{R}^{n}\to\mathbb{R}$ is also a continuous function and the feasible set $\mathcal{X}\subseteq\mathbb{R}^{m}$. In fact, a feasible solution to BLO in Eq. (2) should be a vector of UL and LL variables, such that it satisfies all the constraints in Eq. (2), and the LL variables are optimal to the LL subproblem in Eq. (1) for the given UL variables as parameters. In Fig. 1, we provide a simple visual illustration for BLOs stated in Eq. (2). Figure 1: Illustrating the problem of BLO. (a) first shows a standard BLO problem with the situation of multiple solutions of $f$. Green curves denote LL objectives denoted by $f$, and their corresponding minimizers given by $\mathcal{S}(\mathbf{x})$ were shown as green dots. The red curve represents the UL objective $F$, whose minimizer is shown as the red dot. (b) further illustrates that, in general, not all points (green dots) in $\mathcal{S}(\mathbf{x})$ could minimize the UL objective denoted by $F$. The above BLO problem has a natural interpretation as a non-cooperative game between two players (i.e., Stackelberg game [1]). Correspondingly, we may also call the UL and LL subproblems as the leader and follower, respectively. Then the “leader” chooses the decision $\mathbf{x}$ first, and afterwards the “follower” observes $\mathbf{x}$ so as to respond with a decision $\mathbf{y}$. Therefore, the follower may depend on the leader’s decision. Likewise, the leader has to satisfy a constraint that depends on the follower’s decision. It is worthwhile noting that the LL subproblem may have multiple solutions for every (or some) fixed value of the UL decision making variable $\mathbf{x}$. When the solution of the LL subproblem is not unique, it is difficult for the leader to predict which point in $\mathcal{S}(\mathbf{x})$ the follower will choose (see Fig. 1 (b) for example). ## 3 Understanding and Modeling Practical Problems by BLOs In this section, we demonstrate that even with different motivations and mechanisms, a variety of modern complex learning and vision tasks (e.g, hyper- parameter optimization, multi-task and meta learning, neural architecture search, adversarial learning, deep reinforcement learning and so on) actually share close relationships from the BLO perspective. Moreover, we provide a uniform BLO expression to (re)formulate all these problems. Table II provides a summary of learning and vision applications, which can be understood and modeled by BLO. TABLE II: Summary of related learning and vision applications that can be (re)formulated as BLOs. The abbreviations are listed as follows: Hyper-parameter Optimization (HO), Meta-Feature Learning (MFL), Meta-Initialization Learning (MIL), Neural Architecture Search (NAS), Adversarial Learning (AL), and Deep Reinforcement Learning (DRL). Task \| Important work \| Other work ---\|---\|--- HO \| [51] (ICML, 2017), [13] (AISTATS, 2019), [32] (ICML, 2020) \| [69] (SIAM, 2014), [70] (EURO, 2020), [71] (ICML, 2015), [72] (AISTATS, 2020), [73] (ICML, 2018), [74] (ICML, 2016), [75] (arXiv, 2019), [9] (ICLR, 2019), [53] (ICML, 2021), [76] (ICML, 2017), [77] (NIPS, 2020), [12] (ICML, 2018) MFL \| [12] (ICML, 2018), [56] (ICML, 2017) \| [11] (arXiv, 2017), [79] (CVPR, 2018), [80] (CVPR, 2018), [81] (ICLR, 2018), [82] (ICLR, 2017), [83] (ICLR, 2018), [84] (ICML, 2019) MIL \| [85] (NIPS, 2019), [57] (ICLR, 2019), [86] (ICLR, 2019), [87] (NIPS, 2019) \| [88] (ICLR, 2017), [55] (ICML, 2017), [89] (ICML, 2017), [90] (arXiv, 2018), [91] (arXiv, 2019), [92] (CVPR, 2020),[93] (AAAI, 2020), [94] (arXiv, 2018), [95] (NIPS, 2019), [96] (ICLR, 2020), [97] (ICML, 2019),[98] (ICML, 2018), [99] (CVPR, 2020), [100] (AAAI, 2020), [101] (ICASSP, 2020) NAS \| [14] (ICLR, 2019), [102] (NIPS, 2018), [58] (TGRS, 2020), [103] (CVPR, 2020), [59] (ICLR, 2019) \| [104] (ICLR, 2019), [105] (CVPR, 2018), [16] (ICLR, 2019), [106] (ICCV, 2019), [107] (AISTATS, 2020), [108] (CVPR, 2020), [109] (AAAI, 2020), [110] (CVPR, 2020), [111] (CVPR, 2019), [112] (NIPS, 2019), [113] (ICCV, 2019), [114] (NIPS, 2019), [115] (CVPR, 2020), [116] (arXiv, 2019), [117] (SIGKDD, 2020), [118] (CVPR, 2020), [119] (CVPR, 2020), [120] (arXiv, 2020) AL \| [121] (arXiv, 2016), [21] (arXiv, 2016) \| [122] (AAAI, 2020), [123] (CVPR, 2020), [18] (arXiv, 2018), [17] (PR, 2019), [124] (ICML, 2020), [60] (CVPR, 2020), [19] (CVPR, 2020), [125] (ICML, 2018) DRL \| [20] (AAAI, 2020), [21] (arXiv, 2016), [61] (ICML, 2020) \| [126] (CIRED, 2019), [62] (arXiv, 2020), [127] (NIPS, 2019), [128] (ICML, 2020), [129] (AAMAS, 2020), [130] (arXiv, 2019), [131] (TSG, 2019), [132] (NeurIPS, 2016), [133] (NeurIPS, 2017), [134] (arXiv, 2018), [61] (ICML, 2020), [135] (ICML, 2019) Others \| [136] (SIAM, 2013), [23] (SSVM, 2015), [137] (TIP, 2020), [138] (TNNLS, 2020), [139] (TIP, 2016) \| [140] (ICML, 2016),[141] (ICLR, 2019), [25] (IJCAI, 2020), [142] (arXiv, 2019), [143] (UAI, 2020), [144] (arXiv, 2021), [145] (arXiv, 2020), [146] (TIP, 2020), [147] (arXiv, 2019), [148] (arXiv, 2020), [149] (T-RO, 2020), [150] (WACV, 2020), [151] (arXiv, 2020), [152] (NIPS, 2020), [153] (arXiv, 2020), [154] (arXiv, 2020), [155] (CVPR, 2020), [156] (ICLR, 2018) ### 3.1 Hyper-parameter Optimization Hyper-parameter Optimization (HO) refers to the problem of identifying the optimal set of hyper-parameters that can’t be learned using the training data alone. Early in learning and vision areas, designing regularized models or support vector machines are generally the recommended approaches of selecting hyper-parameters [157]. Based on the representation of the hierarchical structure, these approaches are first expressed as a BLO problem and then transformed into the single-level optimization problem by replacing the LL subproblem with its optimality condition [158]. Due to the high computational cost, especially in high-dimensional hyper-parameter space, these original methods even could not guarantee a local optimal solution [159]. In recent years, gradient-based HO methods with deep neural networks have received extensive attention, which are generally divided into two categories: iterative differentiation (i.e., [7, 160, 23, 71, 10, 11, 12, 75, 13, 73]) and implicit differentiation (i.e., [161, 162, 136, 163, 9, 8, 74, 164, 72]), depending on how the gradient (w.r.t. hyper-parameters) can be computed. The former approximates the best-response function by performing several steps of gradient descent on the loss function, while the latter derives the hyper- gradients through the implicit function theory. One particular type of gradient-based HO is the data hyper-cleaning problem [51, 13], which generally trains a linear classifier with a cross-entropy function (w.r.t. parameters $\mathbf{y}$) and learns to optimize the hyper-parameters $\mathbf{x}$ with a $\ell_{2}$ regularization function. --- Figure 2: Schematic diagram of HO. The UL subproblem involves optimization of hyper-parameters $\mathbf{x}$ based on $\left(\mathcal{D}_{\mathtt{tr}},\mathcal{D}_{\mathtt{val}}\right)$, while the LL subproblem involves optimization of weight parameters $\mathbf{y}$, aiming to find the learning algorithm $g_{\mathbf{y}}(\cdot)$ based on $\mathcal{D}_{\mathtt{tr}}$. Indeed, HO can be understood as the most straightforward application of BLO in learning and vision fields [157]. Specifically, the UL objective $F(\mathbf{x},\mathbf{y};\mathcal{D}_{\mathtt{val}})$ aims to minimize the validation set loss with respect to the hyper-parameters (e.g. weight decay), and the LL objective $f(\mathbf{x},\mathbf{y};\mathcal{D}_{\mathtt{tr}})$ needs to output a learning algorithm by minimizing the training loss with respect to the model parameters (e.g. weights and biases). As illustrated in Fig. 2, the full dataset $\mathcal{D}$ is divided into the training and validation datasets (i.e., $\mathcal{D}_{\mathtt{tr}}\cup\mathcal{D}_{\mathtt{val}}$) and we instantiate how to model the HO task from the perspective of BLO. Inspired by this nested optimization, most HO applications can be characterized by the bi-level structure and formulated as the BLO problems. The UL subproblem involves the optimization of hyper-parameters $\mathbf{x}$ and the LL subproblem (w.r.t. weight parameters $\mathbf{y}$) aims to find the learning algorithm $g_{\mathbf{y}}(\cdot)$ by minimizing the training loss. ### 3.2 Multi-task and Meta Learning The goal of meta learning (a.k.a., learning to learn) is to design models that can learn new skills or adapt to new environments rapidly with a few training examples (see Fig. 3 for a schematic diagram). As a variant of meta learning, multi-task learning just intends to jointly perform all the given tasks [165, 166]. One of the most well-known instances of meta learning is few-shot classification (i.e., $N$-way $M$-shot). Each task is a $N$-way classification designed to learn the meta-parameter with $M$ training samples selected from each of the class. Specially, the full meta training data set $\mathcal{D}=\\{\mathcal{D}^{j}\\}$ ($j=1,\cdots,N$) can be segmented into $\mathcal{D}^{j}=\mathcal{D}_{\mathtt{tr}}^{j}\cup\mathcal{D}_{\mathtt{val}}^{j}$, where $\mathcal{D}^{j}$ is linked to the $j$-th task. --- Figure 3: Illustrating the training process of meta learning. The whole process is visualized to learn new tasks quickly by drawing upon related tasks on corresponding data sets. It can be decomposed into two parts: the “base- learner” trained for operating a given task and the “meta-learner” trained to learn how to optimize the base-learner. According to the dependency between the meta-parameters and the network parameters, current meta learning based methods can be roughly categorized as two groups, i.e., meta-feature learning and meta-initialization learning, as can be seen in Fig. 4. Specifically, meta-initialization learning aims to investigate the meta information of multiple tasks by the network initialization, which can also be understood as the promotion of fine-tuning [85, 87]. From the BLO perspective, we actually formulate the network parameters and their initialization (based on multi-task information) by the LL and UL subproblems, respectively. In contrast, meta-feature learning methods first separate the network architecture as the meta feature extraction part and the task-specific part. Then they formulate a hierarchical learning process [11, 12, 56]. So in such tasks, we use the UL and LL subproblems to model the meta-feature part and the task-specific part, respectively. #### 3.2.1 Meta-feature Learning Meta-Feature Learning (MFL) aims to learn a sharing meta feature representation of all tasks. Recently, series of meta learning based approaches show that multi-task with hard parameter sharing and meta-feature representation are essentially similar [167, 168]. The optimization of meta- learner with respect to meta-parameters based on the UL subproblem is similar to HO [11, 12, 73]. The cross-entropy function $\ell(\mathbf{x},\mathbf{y}^{j};\mathcal{D}_{\mathtt{tr}}^{j})$ is actually considered as the task-specific loss for the $j$-th task on the meta training data set to define the LL objective. As illustrated in the subfigure (a) of Fig. 4, following the bi-level framework, the network architecture in this category can be subdivided into two groups. The first is the cross-task intermediate representation layer parameterized by $\mathbf{x}$ (illustrated by the blue block), outputting the meta features. The second is the logistic regression layer parameterized by $\mathbf{y}^{j}$ (illustrated by the green block), as the ground classifier for the $j$-th task. As can be seen, the feature layers are shared across all episodes, while the softmax regression layer is episode (task) specific. We can also observe that the process of network forward propagation corresponds to the process of passing from the feature extraction part to the softmax part. #### 3.2.2 Meta-initialization Learning Meta-Initialization Learning (MIL) aims to learn a meta initialization for all tasks. MAML [89], known for its simplicity, estimates initialization parameters with the cross-entropy and mean-squared error for supervised classification and regression tasks purely by the gradient-based search. Except for initial parameters, recent approaches have focused on learning other meta variables, such as updating strategies (e.g., descent direction and learning rate [169, 88, 91]) and an extra preconditioning matrix (i.e., [78, 83, 98]). Moreover, implicit gradient methods have a rapid development in the context of few-shot meta learning. There exist a large variety of algorithms replacing the gradient process of the optimization of base-learner through calculation of implicit meta gradient [85, 95, 170, 86]. Due to the large amount of computation required to calculate the Hessian vector product in the training process, various Hessian-free algorithms have been proposed to alleviate the costly computation of second-order derivatives, including but not limited to [94, 54, 96, 57, 55, 56]. In particular, various first-order approximation BLO algorithms have been proposed to avoid the time-consuming calculation of second-order derivatives in [90]. For instance, a modularized optimization library was proposed in [53] to unify several meta learning algorithms into a common BLO framework111The code for this library is available at https://github.com/dut-media-lab/BOML.. As can be shown in subfigure (b) of Fig. 4, $\mathbf{x}$ denoted by blue blocks corresponds to network initialization parameters, and $\mathbf{y}$ denoted by green blocks corresponds to model parameters and is treated as the updated variable satisfying the condition $\mathbf{y}_{0}^{j}=\mathbf{x}$. Compared to MFL, there is no deeply intertwined and entangled relationship between two variables $(\mathbf{x},\mathbf{y}^{j})$, and $\mathbf{x}$ is only explicitly related to $\mathbf{y}$ in the initial state. As a bi-level coupled nested loop strategy, the LL subproblem based on base-learner is trained for operating a given task, and the UL subproblem based on meta-learner aims to learn how to optimize the base-learner. Among the well-known approaches in this direction, most recent approaches (i.e., [171, 90]) have claimed that the LL objective is denoted by the task-specific loss on the training data set, i.e., $f(\mathbf{x},\\{\mathbf{y}^{j}\\})=\ell(\mathbf{x},\mathbf{y}^{j};\mathcal{D}_{\mathtt{tr}}^{j})$. By utilizing cross-entropy function, the UL objective is given by $F(\mathbf{x},\\{\mathbf{y}^{j}\\})=\sum_{j}\ell(\mathbf{x},\mathbf{y}^{j};\mathcal{D}_{\mathtt{val}}^{j}).$ (a) MFL (b) MIL Figure 4: Illustration of two architectures that are generally applied to multi-task and meta learning: MFL and MIL. Both of them can be separated into two parts: meta-parameters denoted by $\mathbf{x}$ (blue blocks) and parameters denoted by $\mathbf{y}^{j}$ (green blocks). (a) shows meta- parameters for features shared across tasks and parameters of the logistic regression layer. (b) shows meta (initial) parameters shared across tasks and parameters of the task specific layer. Both MFL and MIL are essential solution strategies of one optimizer based on another optimizer, thus conforming to the construction of the BLO scheme. As a task-specific loss associated with the $j$-th task, the LL objective can be defined as $\mathbf{y}^{j}\in\arg\min_{\mathbf{y}^{j}\in\mathcal{Y}}f\left(\mathbf{x},\mathbf{y}^{j};\mathcal{D}_{\mathtt{tr}}^{j}\right)$, $j=1,\cdots,N$. Also, based on $\\{\mathcal{D}_{\mathtt{val}}^{j}\\}$, the UL objective can be given by $\min_{\mathbf{x}\in\mathcal{X}}F\left(\mathbf{x},\\{\mathbf{y}^{j}\\};\\{\mathcal{D}_{\mathtt{val}}^{j}\\}\right)$. To summarize, the UL meta-learner performs gradient descent operations and updates the meta-parameter with feedback from base-learners to extract generalized meta knowledge. Subsequently, the better meta knowledge is fed into the base-learner (i.e., the LL subproblem) as part of its model for optimizing $\mathbf{y}$, thereby forming an optimization cycle. ### 3.3 Neural Architecture Search Neural Architecture Search (NAS) seeks to automate the process of choosing the optimal neural network architecture [172]. Recently, there has aroused a great deal of interest in gradient-based differentiable NAS methods [14, 173, 174]. Specifically, these gradient-based differentiable NAS methods mainly contain three main concepts: search space, search strategy and performance estimation strategy. As shown in Fig. 5, by designing an architecture search space, they generally use a certain search strategy to find the optimal network architecture. Such a process can be regarded as the system of optimizing the operation and connection of each node. --- Figure 5: Schematic diagram of NAS. Derived from a predefined search space $\mathcal{A}$, NAS first selects an architecture $A$ to transport into the performance estimation strategy, then returns the estimated performance of $A$ to the search strategy. DARTS [14], the most well-known instance, relaxed the search space to be continuous and conducted searching for architectures in a differentiable way to simultaneously optimize the architectures and weights. Actually, each operation corresponds to a coefficient in DARTS. By denoting $\mathbf{x}=\\{\mathbf{x}^{ij}\\}$ as the architecture parameters and $\mathbf{x}^{ij}$ as the form of connection between two nodes, the expression formula of mixed operations $\bar{o}^{ij}(\cdot)$ based on the softmax function can be written as $\bar{o}^{ij}(\cdot)=\sum\limits_{o\in\mathcal{O}}\frac{\exp(\mathbf{x}_{o}^{ij})}{\sum\limits_{o^{\prime}\in\mathcal{O}}\exp(\mathbf{x}_{o^{\prime}}^{ij})}o(\cdot),$ where $o$ and $o^{\prime}$ are operations and $\mathcal{O}$ is the set of all candidate operations. Then, $o^{ij}=\arg\max_{o\in\mathcal{O}}\mathbf{x}_{o}^{ij}$ is further evaluated and performed in order to obtain the optimal architecture. However, due to the sharp deterioration in performance caused by the large number of skip connections, a great deal of improved approaches have emerged, such as ENAS [105], PC-DARTS [59], P-DARTS [106], just to name a few. Currently, a series of gradient-based differentiable NAS methods combined with meta learning have been proposed, see [175, 16, 176, 108]. Based on the bi- level coupling mechanism, these gradient-based differentiable NAS methods have achieved promising results in the numerous visual and learning applications, such as image classification [58], semantic segmentation [115, 177, 111], object detection [112, 113, 118, 103, 117], medical image analysis [177, 115], video classification [140], recommendation system [120], graph network [116, 130] and representation learning [130], etc. Given the proper search space, it is helpful for these gradient-based differentiable NAS methods to derive the optimal architecture for different vision and learning tasks. From the BLO’s point of view, the UL objective w.r.t. the architecture weights (e.g. block/cell) can be parameterized by $\mathbf{x}$. And the LL objective w.r.t. the model weights can be parameterized by $\mathbf{y}$. Therefore, the full searching process can virtually be formulated as a BLO paradigm, where the UL objective is defined by $F(\mathbf{x},\mathbf{y};\mathcal{D}_{\mathtt{val}})$ based on the validation data set $\mathcal{D}_{\mathtt{val}}$, and the LL objective is given by $f(\mathbf{x},\mathbf{y};\mathcal{D}_{\mathtt{tr}})$ based on the training data set $\mathcal{D}_{\mathtt{tr}}$. ### 3.4 Adversarial Learning Adversarial Learning (AL) is currently deemed as one of the most important learning tasks. It has been applied in a large variety of application areas, i.e., image generation [60, 18, 123], adversarial attacks [178] and face verification [17]. For example, the work proposed in [60] introduced an adaptive BLO model for image generation, which guided the generator to reasonably modify the parameters in a complementary and promoting way. Moreover, a new adversarial training strategy has been proposed by learning a parametric optimizer with neural networks to study the adversarial attack [18]. As the current influential model, Generative Adversarial Network (GAN) can be deemed as deep generative models [179]. Recently, targeting at finding pure Nash equilibrium of generator and discriminator, the author proposed to exploit a fully differentiable search framework by formalizing as solving a bi-level mini-max optimization problem [19]. --- Figure 6: Illustrating the architecture of GAN. The generator $G$ is represented as a deterministic feed forward neural network (red blocks), through which a fixed random noise $\mathbf{v}$ is passed to output $G(\mathbf{v})$. The discriminator $D$ is another neural network (green blocks) which maps the sampled real-world image $\mathbf{u}\sim p_{data}$ and $G(\mathbf{v})$ to a binary classification probability. Most of the AL approaches can formulate the unsupervised learning problem as a bi-level game between two opponents: a generator which samples from a distribution, and a discriminator which classifies the samples as real or false, as shown in Fig. 6. The goal of GAN is to minimize the duality gap denoted by $\mathcal{V}(D,G)$: $\begin{split}\min\limits_{G}\max\limits_{D}\mathcal{V}(D,G)&=\mathbb{E}_{\mathbf{u}\sim p_{data}(\mathbf{u})}\log D(\mathbf{u})\\\ &+\mathbb{E}_{\mathbf{v}\sim\mathcal{N}_{(0,1)}}\log(1-D(G(\mathbf{v}))),\end{split}$ where the fixed random noise source $\mathbf{v}$ obtained from $\mathbf{v}\sim\mathcal{N}_{(0,1)}$ is input into the generator $G$, which, together with the sampled real-world image $\mathbf{u}\sim p_{data}$, is then authenticated by the discriminator $D$. Notice that $\mathbb{E}$ denotes the expectation which implies that the average value of some functions under a probability distribution. Indeed, AL problems generally correspond to the mini-max BLO problems, where the UL discriminator denoted by $F$ targets on learning a robust classifier, and the LL generator denoted by $f$ tries to generate the adversarial samples. Specifically, the UL and LL objectives can be respectively formulated as $\begin{split}F(\mathbf{x},\mathbf{y})&=-\mathbb{E}_{\mathbf{u}\sim p_{data}(\mathbf{u})}\log D(\mathbf{u})\\\ &-\mathbb{E}_{\mathbf{v}\sim\mathcal{N}_{(0,1)}}\log(1-D(G(\mathbf{v}))),\end{split}$ $f(\mathbf{x},\mathbf{y})=-\mathbb{E}_{\mathbf{v}\sim\mathcal{N}_{(0,1)}}\log(D(G(\mathbf{v}))),$ where $G$ and $D$ are parameterized with variables $\mathbf{y}$ and $\mathbf{x}$, respectively. In other words, the UL subproblem aims to reduce the duality gap $\mathcal{V}(D,G)$ and the LL subproblem interactively optimizes the discriminator parameters denoted by $\mathbf{x}$ to obtain the optimal solution. ### 3.5 Deep Reinforcement Learning Deep Reinforcement Learning (DRL) aims to make optimal decisions by interacting with the environment and learning from the experiences. Indeed, a variety of DRL tasks, including Single-Agent Reinforcement Learning (SARL) [21, 22, 62], Multi-Agent Reinforcement Learning (MARL) [126, 129, 20, 180], Meta Reinforcement Learning (MRL) [61, 135, 181, 182], and Imitation Learning (IL) [132, 134, 133], which all can be modeled and tackled by BLO techniques. As for SARL problems, Actor-Critic (AC) type methods have been widely studied and viewed as a bi-level or two-time-scale optimization problems [62, 22], as illustrated in Fig. 7. Indeed, AC type DRL methods often aim to simultaneously learn a state-action value-function $Q^{\pi}$ that predicts to expect the discounted cumulative reward and a policy which is optimal for that value function: $\leavevmode\resizebox{446.2658pt}{}{$Q^{\pi}(s,a)=\mathbb{E}_{s_{i+j}\sim\mathcal{P},r_{i+j}\sim\mathcal{R},a_{i+j}\sim\pi}\left(\sum\limits_{k=0}^{\infty}\gamma^{j}r_{i+j}\|s_{i}=s,a_{i}=a\right)$},$ where $\mathcal{P}$ and $\mathcal{R}$ denote dynamics of the environment and reward function, $s$ and $a$ are the state and action, $i$ and $j$ represent the i-th and j-th steps, and $\mathbb{E}$ is the expectation which implies that the average value of some function under a probability distribution. The policy maximizes the expected discounted cumulative reward for that state- action value-function, i.e., $\pi^{}=\arg\max_{\pi}\mathbb{E}_{s_{0}\sim p_{0},a_{0}\sim\pi}\left(Q^{\pi}(s_{0},a_{0})\right),$ where $s_{0}$, $a_{0}$ and $p_{0}$ correspond to the initial state, initial action and the initial state distribution, respectively. Under the BLO paradigm, the actor and critic correspond to the UL and LL variables, respectively. Let $\mathbf{x}$ denote the parameters of the state-action value-function and $\mathbf{y}$ denote the parameters of the policy $\pi$. The UL and LL objectives respectively take the form $\displaystyle F(\mathbf{x},\mathbf{y})$ $\displaystyle=\mathbb{E}{s_{i},a_{i}\sim\pi}(\mathtt{div}(\mathbb{E}_{s_{i+1},a_{i+1},r_{i+1}}$ $\displaystyle\left(r_{i+1}+\gamma Q(s_{i+1},a_{i+1})\right)\parallel Q(s_{i},a_{i}))),$ $f(\mathbf{x},\mathbf{y})=-\mathbb{E}_{s_{0}\sim p_{0},a_{0}\sim\pi}Q^{\pi}(s_{0},a_{0}),$ where $\mathtt{div}(\cdot\|\|\cdot)$ represents any divergence. MARL studies how multiple agents can collectively learn, collaborate, and interact with each other in an environment. In the classical MARL system, agents are treated equally and the goal is to solve the Markov game to an arbitrary Nash equilibrium when multiple equilibria exist, thus lacking a solution for selection. To address this issue, the work in [20] formulates MARL as the multi-state model-free Stackelberg equilibrium learning problem. Thus, under Markov games, they construct a BLO formulation to find Stackelberg equilibrium to address the MARL task. Similarly, a multi-agent bi-level cooperative reinforcement learning algorithm was proposed in [126] to solve the stochastic decision-making problem. In recent years, MRL approaches (a.k.a., meta learning on reinforcement learning tasks), which aim to learn a policy that adapts fast to new tasks and/or environments, have achieved remarkable success [183, 181]. For example, the work in [182] learns a policy that can quickly adapt to other related models only with one policy gradient step. By adding control variables into gradient estimation, the work in [135] can obtain low variance estimates for policy gradients. While the work in [61] characterizes the optimality gap of the stationary points attained by MAML for both reinforcement learning and supervised learning. Since all these works are based on the meta- initialization platform, it is also nature to formulate these meta reinforcement learning methods from the perspective of BLOs. Generally, IL techniques are very useful when it is easier for an expert to demonstrate the desired behavior rather than to specify a reward function which would generate the same behavior or to directly learn the policy in DRL tasks [184]. In recent years, by connecting imitation learning with generative adversarial learning, a series of Generative Adversarial Imitation Learning (GAIL) techniques [132, 134, 133] have been investigated to imitate an expert in a model-free DRL scenario. Since GAIL type methods have a natural connection to the mechanism of GANs, we can definitely formulate these models using BLOs. --- Figure 7: Illustrating the schematic diagram of AC learning. First the actor $\pi$ interacts with the environment to learn the state-action value-function $Q^{\pi}(s,a)$, and then the actor $\pi$ is again obtained based on $Q^{\pi}(s,a)$. ### 3.6 Other Related Applications The rapid development of deep learning has claimed its domination in the area of image processing and analysis. In addition to the above mentioned tasks, there exist a significant amount of other related learning and vision tasks that can be re(formulated) as BLO problems, such as image enhancement [136, 24, 139, 185, 142], image registration [25], image-to-image translation [186], image recognition [187], image compression [188] and other related works [143, 152, 141]. For example, the earlier work presented in [136] considered the problem of parameter learning for image denoising models and incorporated $p$-norm–based analysis priors. Under a BLO formulation, the LL subproblem was given by a variational model which consisted of the data fidelity and regularization term, and the UL subproblem was expressed by the loss function. Furthermore, the work proposed in [139] formulated the discriminant dictionary learning method for image recognition tasks as a BLO. From this point of view, the UL subproblem can directly minimize the classification error, while the LL subproblem can use the sparsity term and the Laplacian term to characterize the intrinsic data structure. By addressing a unified BLO problem, the LL subproblem is usually expressed as fundamental models that conform to the laws or principles of physics, while the UL subproblem usually considers the further constraints on variables [23, 140]. ## 4 Gradient-based BLOs In past years, gradient-based techniques have became the most popular BLO solution strategies in learning and vision fields. In fact, one of the first gradient-based BLO methodology is [30]. Currently, a variety of explicit gradient-based methods have been investigated to solve BLOs [71, 73, 189]. Specifically, the works in [12, 51] first calculate gradient flow of the LL objective and then perform either reverse or forward gradient computations for the UL subproblem. Similar ideas have also been considered in [190, 23, 75], but with different specific implementations. On the other hand, there also exist some implicit gradient based methods [72, 85, 191] to use the implicit function theorem to obtain the gradient. In this section, we first review three categories of mainstream BLO formulations, which have been considered in various application scenarios. We then demonstrate how to uniformly reformulate these different BLOs from a single-level optimization perspective and investigate the intrinsic structures of existing gradient-based BLO algorithms within a unified algorithmic platform. ### 4.1 Different Formulations of BLO It is worthwhile to notice that the original BLO model given in Eq. (2) is not clear in case of the multiple LL optimal solutions for some of the selections of the UL decision maker [1]. Therefore, it is necessary to define, which solution out of the multiple LL solutions in $\mathcal{S}(\mathbf{x})$ should be considered. Here we actually consider three categories of viewpoints, i.e., singleton, optimistic and pessimistic BLOs. The most straightforward idea in existing learning and vision literature is to assume that $\mathcal{S}(\mathbf{x})$ is a singleton. Formally, we call the BLO model is with the Lower-Level Singleton (LLS) condition if $\forall\mathbf{x}\in\mathcal{X}$, the solution set of the LL subproblem (i.e., $\mathcal{S}(\mathbf{x})$) is a singleton. In this case, we can simplify the original model as $\min\limits_{\mathbf{x}\in\mathcal{X}}F(\mathbf{x},\mathbf{y}),\ s.t.\ \mathbf{y}=\arg\min\limits_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y}).$ (3) Such singleton version of BLOs is well-defined and could cover a variety of learning and vision tasks (e.g., [7, 71, 74, 9], just name a few). Thus, in recent years, dozens of methods have been developed to address this nested optimization task in different application scenarios (see the following sections for more details). Furthermore, the situation becomes more intricate if the LL subproblem is not uniquely solvable for each $\mathbf{x}\in\mathcal{X}$. Essentially, if the follower can be motivated to select an optimal solution in $\mathcal{S}(\mathbf{x})$ that is also best for the leader (i.e., with respect to $F$), it yields the so-called optimistic (strong) formulation of BLO $\min\limits_{\mathbf{x}\in\mathcal{X}}\left\\{\min\limits_{\mathbf{y}\in\mathcal{Y}}F(\mathbf{x},\mathbf{y}),\ s.t.\ \mathbf{y}\in\arg\min\limits_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y})\right\\}.$ (4) The above stated optimistic viewpoint has drawn increasing attention in BLO literature [192, 193, 194] and recently also been investigated in learning and vision fields [32, 189, 195]. In Section 7, we will further explore how to solve such optimistic BLOs in detail. If the leader does not have the information whether the follower returns the best response $\mathbf{y}$ from $\mathcal{S}(\mathbf{x})$ in terms of the UL objective $F$, then we have to assume that the follower is not cooperate with the leader. This is known as the pessimistic (weak) formulation of BLO [196, 197] and can be given as: $\min\limits_{\mathbf{x}\in\mathcal{X}}\left\\{\max\limits_{\mathbf{y}\in\mathcal{Y}}F(\mathbf{x},\mathbf{y}),\ s.t.\ \mathbf{y}\in\arg\min\limits_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y})\right\\}.$ (5) It should be pointed out that till now we still lack efficient gradient-based algorithms to address the pessimistic BLO problems222In Section 9, we will demonstrate that we can also obtain some practical gradient-based iteration scheme within our general algorithmic platform for the pessimistic formulation of BLO.. ### 4.2 BR-based Single-Level Reformulation In this work, we consider the optimal solution of the LL subproblem with a given UL variable $\mathbf{x}$ as the Best-Response (BR) of the follower (denoted as $\mathbf{y}^{}(\mathbf{x})$). Then we can interpret BLO as a game process, in which the leader $\mathbf{x}$ considers what BR of the follower $\mathbf{y}$ is, i.e., how it will respond once it has observed the quantity of the leader [198, 1]. Based on the above understanding, we can reformulate the three different categories of BLOs as a unified single-level optimization problem. Specifically, given the UL variable $\mathbf{x}$, we denote the corresponding BR mapping as $\mathbf{y}^{}(\mathbf{x})$. In fact, if considering the singleton BLO, $\mathbf{y}^{}(\mathbf{x})$ can be directly obtained by the unique LL solution. While for the optimistic and pessimistic BLOs, we actually first define their Inner Simple Bi-level (ISB) subproblems (w.r.t., $\mathbf{y}$)333It is known that the simple bi-level optimization is just a specific BLO problem with only one variable [32, 199]. as $\small\mbox{Optimistic ISB:}\min\limits_{\mathbf{y}\in\mathcal{S}(\mathbf{x})}F(\mathbf{x},\mathbf{y})\ \mbox{and}\ \mbox{Pessimistic ISB:}\max\limits_{\mathbf{y}\in\mathcal{S}(\mathbf{x})}F(\mathbf{x},\mathbf{y}).$ (6) Then by defining the solution set of ISB as $\widetilde{\mathcal{S}}(\mathbf{x})$, we could consider any $\mathbf{y}^{}(\mathbf{x})\in\widetilde{\mathcal{S}}(\mathbf{x})$ as the BR mapping, because points in $\widetilde{\mathcal{S}}(\mathbf{x})$ all obtain the minimum/maximum of $F(\mathbf{x},\mathbf{y})$ in $\mathcal{S}(\mathbf{x})$. Therefore, we can formulate the general BR mapping for different categories of BLOs as follows: $\left\\{\begin{array}[]{ll}\mathbf{y}^{}(\mathbf{x}):=\arg\min\limits_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y}),\quad\mbox{Singleton},\\\ \mathbf{y}^{}(\mathbf{x})\in\widetilde{\mathcal{S}}(\mathbf{x}):=\left\\{\begin{array}[]{l}\arg\min\limits_{\mathbf{y}\in\mathcal{S}(\mathbf{x})}F(\mathbf{x},\mathbf{y}),\quad\mbox{Optimistic},\\\ \arg\max\limits_{\mathbf{y}\in\mathcal{S}(\mathbf{x})}F(\mathbf{x},\mathbf{y}),\quad\mbox{Pessimistic}.\end{array}\right.\end{array}\right.$ (7) Based on Eq. (7), we actually obtain the following value-function-based reformulation (a single-level optimization model) for BLOs stated in Eq. (2), i.e., $\min\limits_{\mathbf{x}\in\mathcal{X}}\varphi(\mathbf{x}):=F(\mathbf{x},\mathbf{y}^{}(\mathbf{x})),$ (8) in which $\varphi(\mathbf{x})$ actually can be used to uniformly represent the UL value-function of $F$ from the singleton, optimistic (i.e., $\inf_{\mathbf{y}\in\mathcal{S}(\mathbf{x})}F(\mathbf{x},\mathbf{y})$) and pessimistic (i.e., $\sup_{\mathbf{y}\in\mathcal{S}(\mathbf{x})}F(\mathbf{x},\mathbf{y})$) viewpoints. ### 4.3 A Unified Platform for Gradient-based BLOs Moving one step forward, the gradient of $\varphi$ w.r.t. the UL variable $\mathbf{x}$ can be written as444Please notice that we actually do not distinguish between the operation of the derivatives and partial derivatives to simplify our presentation. $\underbrace{\frac{\partial\varphi(\mathbf{x})}{\partial\mathbf{x}}}_{\text{grad. of $\mathbf{x}$}}=\underbrace{\frac{\partial F(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{x}}}_{\text{direct grad. of $\mathbf{x}$}}\quad+\underbrace{\mathbf{G}(\mathbf{x}),}_{\text{indirect grad. of $\mathbf{x}$}}$ (9) where the indirect gradient $G(\mathbf{x})$ can be further specified as the following two components: $\mathbf{G}(\mathbf{x})=\underbrace{\overbrace{\left(\frac{\partial\mathbf{y}^{}(\mathbf{x})}{\partial\mathbf{x}^{\prime}}\right)^{\prime}}^{\text{BR Jacobian}}\overbrace{\frac{\partial F(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{y}}.}^{\text{direct grad. of $\mathbf{y}$}}}_{\text{indirect grad. of $\mathbf{x}$}}$ (10) Here we use “grad.” as the abbreviation of gradient and denote the transpose operation as $(\cdot)^{\prime}$. Note that, $\mathbf{y}^{}(\mathbf{x})$ as a general mapping, can be given specific constraints and necessary assumptions to fit their particular requirements for these specific gradient-based BLO approaches in order to obtain different iteration formats and theoretical properties. For details, please refer to the following contents. In fact, by simple computation, the direct gradient is easy to obtain. However, the indirect gradient is intractable to obtain because we must compute the changing rate of the optimal LL solution with respect to the UL variable (i.e., the BR Jacobian $\frac{\partial\mathbf{y}^{}(\mathbf{x})}{\partial\mathbf{x}}$). Please notice that we will also call $\frac{\partial\varphi(\mathbf{x}^{k})}{\partial\mathbf{x}^{k}}$ as the practical BR Jacobian w.r.t. $\mathbf{x}^{k}$ in the following statements. The computation of the indirect gradient $\mathbf{G}(\mathbf{x})$ naturally motives formulating $\mathbf{y}^{}(\mathbf{x})$ and hence $\frac{\partial\mathbf{y}^{}(\mathbf{x})}{\partial\mathbf{x}}$. For this purpose, a series of techniques have recently been developed from either explicit or implicit perspectives, which obtain their optimal solutions by recurrent differentiation through dynamic system and based on implicit differentiation theory, respectively. Now we demonstrate how to formulate various existing gradient-based BLOs from a unified algorithmic platform. We first summarize a general BLO updating scheme in Alg. 1. It can be seen that the key component of this algorithm is to calculate the BR Jacobian. Then with $\frac{\partial\varphi(\mathbf{x}^{k})}{\partial\mathbf{x}^{k}}$, we can just perform standard (stochastic) gradient descent schemes to update $\mathbf{x}^{k}$. Based upon our general algorithmic platform, we can observe that the main differences of these existing BLO approaches are just their specific strategies for calculating Jacobian of the BR mapping under different conditions (i.e., w/ LLS and w/o LLS). Algorithm 1 A General Gradient-based BLO Scheme 0: The UL and LL initialization. 0: The optimal UL and LL solutions. 1: for $k=1,\cdots,K$ do 2: Calculate the BR Jacobian $\frac{\partial\varphi(\mathbf{x}^{k})}{\partial\mathbf{x}^{k}}$. % (Mainstream calculation strategies are summarized in Figs. 8-9 and thoroughly surveyed in the following sections) 3: Perform (stochastic) gradient descent to update $\mathbf{x}^{k}$. % (based on $\frac{\partial\varphi(\mathbf{x}^{k})}{\partial\mathbf{x}^{k}}$) 4: end for In Fig. 8, we summarize mainstream gradient-based BLOs and illustrate their intrinsic relationships within our general algorithmic platform. It can be observed that in the LLS scenario, from the BR-based perspective, existing gradient methods can be categorized as two groups: Explicit Gradient for Best- Response (EGBR, stated in Section 5) and Implicit Gradient for Best-Response (IGBR, stated in Section 6). As for EGBR, there are mainly three types of methods, namely, recurrence-based EGBR (e.g., [51, 71, 12, 13, 14]), initialization-based EGBR (e.g., [90, 94] ) and proxy-based EGBR methods (e.g., [82, 171, 87, 78]), differing from each other in the way of formulating the BR mapping. For IGBR, existing works consider two groups of techniques (e.g., linear system [74, 85] and Neumann series [72]) to alleviate the computational complexity issue for the BR Jacobian. We emphasize that the validity of above BLO methodologies must depend on the singleton of their LL solution set. When solving BLOs without the LLS assumption, recent works in [32, 189] have demonstrated that we need to first construct BR mapping based on both UL and LL subproblems, and then solve two optimization subproblems, namely, the single-level optimization subproblem (w.r.t. $\mathbf{x}$) and the ISB subproblem (w.r.t. $\mathbf{y}$). While the work in [195] has introduced a series of barrier functions and utilized interior point methods to obtain the BR mapping for each given $\mathbf{x}$. To end up this section, we plot Fig. 9 to illustrate the optimization processes of existing mainstream gradient-based BLO methods from the BR mapping perspective and within our unified algorithmic platform. In the following (i.e., Sections 5-7), we will thoroughly survey these different categories of gradient-based BLO algorithms (including their acceleration, simplification and extension techniques) and their theoretical properties (convergence behaviors and computational complexity), accordingly. --- Figure 8: Summary of the mainstream gradient-based BLOs. We categorize these existing approaches into two main groups, i.e., w/ and w/o LLS assumptions. When solving BLOs with LLS assumption, these methods can be further divided into two categories: EGBR and IGBR. As for EGBRs, they can be solved by different Automatic Differentiation (AD) techniques (as denoted by the dashed rectangle). Very recently, two algorithms have also been proposed to address BLOs without the LLS assumption. In particular, they actually introduce a bi- level gradient aggregation or a value-function-based interior-point method to calculate the indirect gradient. --- Figure 9: Illustrating the roadmap of different categories of gradient-based BLOs. In the left bottom region, the formulations in the solid rectangles (i.e., singleton and optimistic) have been widely studied. In contrast, since gradient-based methods for pessimistic BLOs have not been properly investigated in existing literature, we denote this category of formulation by a dashed rectangle. In Section 9, we demonstrate that we can also obtain a practical pessimistic BLO scheme within our general algorithmic platform. ## 5 Explicit Gradient for Best-Response With the LLS condition, we delve deep into the EGBR category of methods, which aims to perform automatic differentiation through the LL dynamic system [200, 201] to solve the BLO problem. Specifically, given an initialization $\mathbf{y}_{0}=\Psi_{0}(\mathbf{x})$ at $t=0$, the iteration process of EGBRs can be generally written as $\mathbf{y}_{t}=\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x}),\ t=1,\cdots,T,$ (11) where $\Psi_{t}$ denotes some given updating scheme (based on the LL subproblem) at $t$-th stage and $T$ denotes the overall LL iterations number. For example, we can formulate $\Psi_{t}$ based on the gradient descent rule, i.e., $\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})=\mathbf{y}_{t-1}-\eta_{t}\mathbf{d}_{f}(\mathbf{y}_{t-1},\mathbf{x}),$ (12) where $\mathbf{d}_{f}(\mathbf{y}_{t-1},\mathbf{x})$ is the descent mapping of $f$ at $t$-th stage (e.g., $\mathbf{d}_{f}(\mathbf{y}_{t-1},\mathbf{x})=\frac{\partial f(\mathbf{x},\mathbf{y}_{t-1})}{\partial\mathbf{y}_{t-1}}$) and $\eta_{t}$ denotes the corresponding step size . Then we can calculate $\frac{\partial\varphi(\mathbf{x}^{k})}{\partial\mathbf{x}^{k}}$ by substituting $\mathbf{y}_{T}:=\Psi(\mathbf{x})$ approximately for $\mathbf{y}^{}(\mathbf{x})$, and the full dynamical system can be defined as $\Psi(\mathbf{x}):=\Psi_{T}\circ\cdots\circ\Psi_{1}\circ\Psi_{0}(\mathbf{x}).$ (13) Here the notation $\circ$ represents the compound dynamical operation of the entire iteration. That is, we actually consider the following optimization model $\min\limits_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x}):=F(\mathbf{x},\mathbf{y}_{T}(\mathbf{x})),$ (14) and need to calculate $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}$ (instead of Eq. (9)) in the practical optimization scenario. Since it should be noted that $\Psi$ actually obtains an explicit gradient for best-response of the follower, we call this category of gradient-based BLOs as EGBR approaches hereafter. Starting from the Eq. (11), it is obvious to notice that $\mathbf{y}_{t}$ may be affected coupling with the variable $\mathbf{x}$ throughout the iteration. This coupling relationship will have a direct impact on the optimization process of UL variable in Eq. (9). In fact, existing EGBR algorithms can be summarized from three perspectives. The first is that, if $\mathbf{x}$ closely acts on $\mathbf{y}_{t}$ during the whole iteration process, the subsequent optimization of variable $\mathbf{x}$ will be carried out recursively. The second is that when $\mathbf{x}$ only acts in the initial step, the subsequent optimization of variable $\mathbf{x}$ will be simplified. The third class is to replace the whole iterative process with a hyper- network, so as to efficiently approximate the BR mapping. Ultimately, in such cases, we divide them into three categories in terms of the coupling dependence of the two variables and the solution procedures, namely recurrence-based EGBR (stated in Section 5.1), initialization-based EGBR (stated in Section 5.2) and proxy-based EGBR (stated in Section 5.3). ### 5.1 Recurrence-based EGBR It can be seen from Eq. (11) that all the LL iterative variables $\mathbf{y}_{0},\mathbf{y}_{1},\cdots,\mathbf{y}_{T}$ depend on $\mathbf{x}$, and $\mathbf{x}$ acts as a recurrent variable of the dynamical system. One of the most well-known approaches for calculating $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}$ (with the above recurrent structure) is Automatic Differentiation (AD) [160, 202], which is also called algorithmic differentiation or simply “AutoDiff”. There exist two diametrically opposite ways on computing gradients for recurrent neural networks, of which one corresponds to back-propagation through time in a reverse-mode way [203, 204], and the other corresponds to real-time recurrent learning in a forward-mode way [205, 206]. Quite a number of methods, closely related to this subject, have been proposed since then [71, 51, 12, 13]. Here we would like to review recurrence-based BR methods, covering forward-mode, reverse-mode AD, truncated and one-stage simplifications. Forward-mode AD (FAD): To compute $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}$, FAD appeals to the chain rule for the derivative of the dynamical system [51]. Specifically, recalling that $\mathbf{y}_{t}=\Psi_{t}(\mathbf{y}_{t-1},\mathbf{x})$, we have that the operation $\Psi_{t}$ indeed depends on $\mathbf{x}$ both directly by its expression and indirectly through $\mathbf{y}_{t-1}$. Hence, by drawing upon the chain rule, the formulation is given as555Please notice that here we actually require $\mathbf{y}_{t}(\mathbf{x})$ to be a continuously differentiable function (w.r.t. $\mathbf{x}$) for all $t=1,\cdots,T$. In existing EGBRs, they just introduce differentiable $\Psi_{t}$ to meet this requirement. $\frac{\partial\mathbf{y}_{t}}{\partial\mathbf{x}}=\frac{\partial\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})}{\partial\mathbf{y}_{t-1}}\frac{\partial\mathbf{y}_{t-1}}{\partial\mathbf{x}}+\frac{\partial\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})}{\partial\mathbf{x}}.$ (15) To simplify the notation, we denote $\mathbf{Z}_{t}=\frac{\partial\mathbf{y}_{t}}{\partial\mathbf{x}}$, $\mathbf{A}_{t}=\frac{\partial\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})}{\partial\mathbf{y}_{t-1}}$, $\mathbf{B}_{t}=\frac{\partial\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})}{\partial\mathbf{x}}$ for $t>0$ and $\mathbf{Z}_{0}=\mathbf{B}_{0}=\frac{\partial\Psi_{0}(\mathbf{x})}{\partial\mathbf{x}}$. Then we can rewrite Eq. (15) as $\mathbf{Z}_{t}=\mathbf{A}_{t}\mathbf{Z}_{t-1}+\mathbf{B}_{t}$ $(t=1,\cdots,T)$. In this way, we have the following formulation to approximate the BR Jacobian $\frac{\partial\mathbf{y}_{T}(\mathbf{x})}{\partial\mathbf{x}}=\mathbf{Z}_{T}=\sum\limits_{t=0}^{T}\left(\prod\limits_{i=t+1}^{T}\mathbf{A}_{i}\right)\mathbf{B}_{t}.$ (16) Based on the above derivation, it is apparent that $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}$ can be computed by an iterative algorithm summarized in Alg. 2. Actually, FAD allows the program to update parameters after each step, which may significantly speed up the dynamic iterator and take up less memory resources when the number of hyper-parameters is much smaller than the number of parameters. It can be time-prohibitive for many hyper-parameters with a more efficient and convenient way. Algorithm 2 Forward-mode AD (FAD) 0: The UL variable at the current stage $\mathbf{x}$ and the LL initialization $\mathbf{y}_{0}$. 0: The gradient of $\varphi_{T}$ with respect to $\mathbf{x}$, i.e., $\frac{\partial\varphi_{T}}{\partial\mathbf{x}}$. 1: $\mathbf{Z}_{0}=\frac{\partial\Psi_{0}(\mathbf{x})}{\partial\mathbf{x}}$. 2: for $t=1,\cdots,T$ do 3: $\mathbf{y}_{t}=\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})$. 4: $\mathbf{Z}_{t}=\mathbf{A}_{t}\mathbf{Z}_{t-1}+\mathbf{B}_{t}$. 5: end for 6: return $\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{x}}+\mathbf{Z}_{T}^{\prime}\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{y}_{T}}$. Reverse-mode AD (RAD): RAD is a generalization of the back-propagation algorithm and based on a Lagrangian formulation associated with the parameter optimization dynamics. By replacing $\mathbf{y}^{}(\mathbf{x})$ by $\mathbf{y}_{T}$ and incorporating Eq. (16) into Eq. (9), a series of RAD works (e.g., [71, 12, 51]) derived $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}=\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{x}}+\mathbf{Z}_{T}^{\prime}\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{y}_{T}}.$ (17) Rather than calculating $\mathbf{Z}_{T}$ by forward propagation as that in FAD (i.e., Alg. 2), the computation of Eq. (17) can also be implemented by back- propagation. That is, we first define $\mathbf{g}_{T}=\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{x}}$ and $\bm{\lambda}_{T}=\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{y}_{T}}$. Then we update $\mathbf{g}_{t-1}=\mathbf{g}_{t}+\mathbf{B}_{t}^{\prime}\bm{\lambda}_{t}$, and $\bm{\lambda}_{t-1}=\mathbf{A}_{t}^{\prime}\bm{\lambda}_{t}$, with $t=T,\cdots,0$. Finally, we have that $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}=\mathbf{g}_{-1}$. Indeed, the above RAD calculation is structurally identical to back- propagation through time [51]. Moreover, we can also derive it following the classical Lagrangian approach. That is, we reformulate Eq. (14) as the following constrained model $\min\limits_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x})\ \ s.t.\ \ \left\\{\begin{array}[]{l}\mathbf{y}_{0}=\Psi_{0}(\mathbf{x}),\\\ \mathbf{y}_{t}=\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x}),\ t=1,\cdots,T.\end{array}\right.$ (18) The corresponding Lagrangian function can be written as $\begin{array}[]{l}\mathcal{L}(\mathbf{x},\\{\mathbf{y}_{t}\\},\\{\bm{\lambda}_{t}\\})=\varphi_{T}(\mathbf{x})+\bm{\lambda}_{0}^{\prime}\left(\Psi_{0}(\mathbf{x})-\mathbf{y}_{0}\right)\\\ +\sum\limits_{t=1}^{T}\bm{\lambda}_{t}^{\prime}\left(\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})-\mathbf{y}_{t}\right),\end{array}$ (19) where $\bm{\lambda}_{t}$ denotes the Lagrange multiplier associated with the $t$-th stage of the dynamic system. The KKT optimality condition of Eq. (18) is obtained by setting all derivatives of $\mathcal{L}$ to zero, satisfying the condition that $\mathbf{y}_{t}(\mathbf{x})$ is a continuously differentiable function w.r.t. $\mathbf{x}$ for the case that $t=1,\cdots,T$. Then by some simple algebras, we have $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}=\frac{\partial\mathcal{L}}{\partial\mathbf{x}}$. Overall, we present the RAD algorithm in Alg. 3. Algorithm 3 Reverse-mode AD (RAD) 0: The UL variable at the current stage $\mathbf{x}$ and the LL initialization $\mathbf{y}_{0}$. 0: The gradient of $\varphi_{T}$ with respect to $\mathbf{x}$, i.e., $\frac{\partial\varphi_{T}}{\partial\mathbf{x}}$. 1: $\mathbf{y}_{0}=\Psi_{0}(\mathbf{x})$. 2: for $t=1,\cdots,T$ do 3: $\mathbf{y}_{t}=\Psi_{t}(\mathbf{y}_{t-1};\mathbf{x})$. 4: end for 5: $\mathbf{g}_{T}=\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{x}}$ and $\bm{\lambda}_{T}=\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{y}_{T}}$. 6: for $t=T,\cdots,0$ do 7: $\mathbf{g}_{t-1}=\mathbf{g}_{t}+\mathbf{B}_{t}^{\prime}\bm{\lambda}_{t}$ and $\bm{\lambda}_{t-1}=\mathbf{A}_{t}^{\prime}\bm{\lambda}_{t}$. 8: end for 9: return $\mathbf{g}_{-1}$. Truncated RAD (TRAD): The above two precise calculation methods in many practical applications are tedious and time-consuming with full back- propagation training. As aforementioned, due to the complicated long-term dependencies of the UL subproblem on $\mathbf{y}_{T}(\mathbf{x})$, calculating Eq. (17) in RAD is a challenging task. This difficulty is further aggravated when both $\mathbf{x}$ and $\mathbf{y}$ are high-dimensional vectors. More recently, the truncation idea has been revisited to address the above issue and shows competitive performance with significantly less computation time and memory [207, 208, 13]. Specifically, by ignoring the long-term dependencies and approximating Eq. (17) with partial sums (i.e., storing only the last $M$ iterations), we have $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}\approx\mathbf{g}_{T-M}:=\frac{\partial F(\mathbf{x},\mathbf{y}_{T})}{\partial\mathbf{x}}+\mathbf{Z}_{T-M}^{\prime}\frac{\partial F(\mathbf{x},\mathbf{y}_{T}(\mathbf{x}))}{\partial\mathbf{y}_{T}},$ (20) where $\mathbf{Z}_{T-M}=\sum_{t=T-M+1}^{T}\left(\prod_{i=t+1}^{T}\mathbf{A}_{i}\right)\mathbf{B}_{t}$. It can be seen that ignoring the long-term dependencies can greatly reduce the time and space complexity for computing the approximate gradients. Recently, the work in [13] has investigated the theoretical properties of the above truncated RAD scheme, and confirmed this fact that using few-step back- propagation could perform comparably to optimization with the exact gradient, while requiring far less memory and half computation time. One-stage RAD: Limited and expensive memory is often a bottleneck in modern massive-scale deep learning applications. For instance, multi-step iteration of the inner program will cause a lot of memory consumption [89]. Inspired by BLO, a variety of simplified and elegant techniques have been adopted to circumvent this issue. The work in [14] proposes another simplification of RAD, which considers a fixed initialization $\mathbf{y}_{0}$ and only performs one-step iteration in Eq. (11) to remove the recurrent structure for the gradient computation in Eq. (17), i.e., $\frac{\partial\varphi_{1}(\mathbf{x})}{\partial\mathbf{x}}=\frac{\partial F(\mathbf{x},\mathbf{y}_{1}(\mathbf{x}))}{\partial\mathbf{x}}+\left(\frac{\partial\mathbf{y}_{1}(\mathbf{x})}{\partial\mathbf{x}^{\prime}}\right)^{\prime}\frac{\partial F(\mathbf{x},\mathbf{y}_{1}(\mathbf{x}))}{\partial\mathbf{y}_{1}(\mathbf{x})}.$ (21) By formulating the dynamical system as that in Eq. (12), we then write $\frac{\partial\mathbf{y}_{1}(\mathbf{x})}{\partial\mathbf{x}}$ as $\frac{\partial\mathbf{y}_{1}}{\partial\mathbf{x}^{\prime}}=\frac{\partial\left(\mathbf{y}_{0}-\frac{\partial f(\mathbf{x},\mathbf{y}_{0})}{\partial\mathbf{y}_{0}}\right)}{\partial\mathbf{x}^{\prime}}=-\frac{\partial^{2}f(\mathbf{x},\mathbf{y}_{0})}{\partial\mathbf{y}_{0}\partial\mathbf{x}^{\prime}}.$ (22) Since calculating Hessian in Eq. (22) is still time consuming, to further simplify the calculation, we can adopt finite approximation [14] to cancel the calculation of the Hessian matrix (e.g., central difference approximation). The specific derivation can be formalized as follows: $\frac{\partial F(\mathbf{x},\mathbf{y}_{1})}{\partial\mathbf{y}_{1}}\frac{\partial^{2}f(\mathbf{x},\mathbf{y}_{0})}{\partial\mathbf{y}_{0}\partial\mathbf{x}^{\prime}}\approx\frac{\frac{\partial f(\mathbf{x},\mathbf{y}_{0}^{+})}{\partial\mathbf{x}}-\frac{\partial f(\mathbf{x},\mathbf{y}_{0}^{-})}{\partial\mathbf{x}}}{2\epsilon},$ (23) in which $\mathbf{y}_{0}^{\pm}=\mathbf{y}_{0}\pm\epsilon\frac{\partial F(\mathbf{x},\mathbf{y}_{1})}{\partial\mathbf{y}_{1}}$. Note that $\epsilon$ is set to be a small scalar equal to the learning rate [59]. ### 5.2 Initialization-based EGBR The research community has started moving towards the challenging goal of building general purpose initialization-based optimization systems whose ability to learn the initial parameters better. Regardless of the recurrent structure, we need to consider the special setting to analyze a family of algorithms for learning the initialization parameters, named initialization- based EGBR methods. In this series, MAML [89] is considered as the most representative and important work. By making more practical assumptions about the coupling dependence of two variables, these methods no longer use the full dynamical system to explicitly and accurately describe the dependency between $\mathbf{x}$ and $\mathbf{y}$ as discussed above in Eq. (18), but adopt a further simplified paradigm. Specifically, by treating the iterative dynamical system with only the first step that $\mathbf{y}$ is explicitly related to $\mathbf{x}$, this process can be formulated as $\min\limits_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x})\ \ s.t.\ \ \left\\{\begin{array}[]{l}\mathbf{y}_{0}=\Psi_{0}(\mathbf{x}),\\\ \mathbf{y}_{t}=\Psi_{t}(\mathbf{y}_{t-1}),\ t=1,\cdots,T,\end{array}\right.$ (24) where $\mathbf{x}$ represents the network initialization parameters, and $\mathbf{y}_{t}$ represents the network parameters after performing some sort of update. Given initial condition $\Psi_{0}(\mathbf{x})$, then we obtain the following simplified formula $\mathbf{y}_{T}=\Psi_{0}(\mathbf{x})-\sum\limits_{t=1}^{T}\mathbf{d}_{f}(\mathbf{y}_{t-1}),$ (25) where $\mathbf{d}_{f}(\mathbf{y}_{t-1})$ is the descent mapping of $f$ at the $t$-th stage (e.g., $\mathbf{d}_{f}(\mathbf{y}_{t-1})=\frac{\partial f(\mathbf{x},\mathbf{y}_{t-1})}{\partial\mathbf{y}_{t-1}}$). Finally, we have the Jacobian matrix as follows $\frac{\partial\mathbf{y}_{T}}{\partial\mathbf{x}}=\frac{\partial\left(\Psi_{0}(\mathbf{x})-\sum\limits_{t=1}^{T}\mathbf{d}_{f}(\mathbf{y}_{t-1})\right)}{\partial\mathbf{x}}.$ (26) Then we have to calculate the Hessian matrix term $\frac{\partial^{2}f}{\partial\mathbf{y}_{t-1}\partial\mathbf{x}^{\prime}}$, which is time consuming in real computation scenario. To reduce the computational load, we will introduce two remarkably simple algorithms via a series of approximate transformation operations below. Among various schemes to simplify the algorithm based on initialization-based EGBR approaches, first-order approximation (e.g., [90, 94]) and layer-wise transformation (e.g., [82, 171, 87, 78]) are among the more popular. Very recently, the works in [209, 210] also consider the initialization as an auxiliary variable to improve the performance of RAD. First-order Approximation: For example, the most representative algorithms (i.e., FOMAML [90] and Reptile [94]) adopted the operation by first-order approximation, a way to alleviate the problem of Hessian term computation while not sacrificing much performance. Specifically, this approximation ignores the second derivative term by removing the Hessian matrix $\frac{\partial^{2}f}{\partial\mathbf{y}_{t-1}\partial\mathbf{x}^{\prime}}$, and then simplifies substitution of $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}$ performed by $\frac{\partial\varphi_{T}(\mathbf{x})}{\partial\mathbf{x}}=\frac{\partial F(\mathbf{x},\mathbf{y}_{T}(\mathbf{x}))}{\partial\mathbf{x}}+\left(\frac{\partial\Psi_{0}(\mathbf{x})}{\partial\mathbf{x}^{\prime}}\right)^{\prime}\frac{\partial F(\mathbf{x},\mathbf{y}_{T}(\mathbf{x}))}{\partial\mathbf{y}_{T}(\mathbf{x})}.$ (27) In addition, there is another way of first-order extension to simplify Eq. (26) through the operation of difference approximation [94]. It no longer avoids the Hessian term but tries another soft way to approximate $\frac{\partial\mathbf{y}_{T}}{\partial\mathbf{x}}$ (i.e.,$\mathbf{y}_{T}-\mathbf{x}$ and $(\mathbf{y}_{T}-\mathbf{x})/\alpha$), in which $\alpha$ is the step size used in gradient decent operation. Unlike [90], this method proposed to use different linear combinations of all steps rather than using just the final step. But overall, the above algorithm could significantly reduce the computing costs while keeping roughly equivalent performance. Layer-wise Transformation: Indeed, there are also a series of learning-based BLOs related to layer-wise transformation, i.e., Meta-SGD [82], T-Net [171], Meta-Curvature [87] and WarpGrad [78]. In addition to initial parameters, this type of work focuses on learning some additional parameters (or transformation) at each layer of the network. From the above Eq. (25), it can be uniformly formulated as $\mathbf{y}_{T}=\Psi_{0}(\mathbf{x})-\sum\limits_{t=1}^{T}\mathbf{P}(\mathbf{y}_{t-1},\bm{\omega})\mathbf{d}_{f}(\mathbf{y}_{t-1}),$ (28) where $\mathbf{P}(\mathbf{y}_{t-1},\bm{\omega})$ defines the matrix transformation learned at each layer and $\bm{\omega}$ is an auxiliary vector (e.g., learning rate). For example, Meta-SGD [82] learns a vector $\bm{\omega}$ of learning rates and $\mathbf{P}$ corresponded to $\mathtt{diag}(\bm{\omega})$, and T-Net [171] aims to learn block-diagonal preconditioning linear projections. Similarly, an additional the block- diagonal preconditioning transformation is also performed by Meta-Curvature [87]. WarpGrad [78] is closely related to the concurrent work of Meta- Curvature [87], which defines the preconditions gradient from a geometrical point of view and replaces the linear projection with a non-linear preconditioning matrix as a warp layer. ### 5.3 Proxy-based EGBR Generally speaking, calculating the BR mapping (or BR Jacobian) is key to solve BLOs. Recently, several proxy-based EGBR methods (e.g., [9, 77, 164]) utilize the differentiable hyper-network (denoted as $\Psi_{\bm{\theta}}(\mathbf{x})$ with parameters $\bm{\theta}$) to substitute the dynamic system $\Psi(\mathbf{x})$ and then approximate the BR mapping666Note that, these methods assume $\mathbf{y}^{}(\mathbf{x})$ is a continuously differentiable function and $\mathcal{X}$ and $\mathcal{Y}$ denote the whole space [9, 77, 164]., i.e., $\Psi_{\bm{\theta}}(\mathbf{x})\to\Psi(\mathbf{x})\approx\mathbf{y}^{}(\mathbf{x}).$ (29) Specifically, they train a hyper-network that takes hyper-parameters $\mathbf{x}$ as input and outputs the approximate optimal set of weights as the optimal solution of the LL subproblem. In fact, both global and local proxy techniques have been considered to approximate the BR mapping. From the perspective of global approximation, first, if the distribution $p(\mathbf{x})\subseteq\mathcal{X}$ is fixed, they learn $\bm{\theta}$ by minimizing $\mathbb{E}_{\mathbf{x}\sim p(\mathbf{x})}f(\mathbf{x},\Psi_{\bm{\theta}}(\mathbf{x}))$, so that $\Psi_{\bm{\theta}}(\mathbf{x})$ can approximate the BR mapping in a neighborhood around the current $\mathbf{x}$, and second update $\mathbf{x}$ with $\Psi_{\bm{\theta}}$ as a proxy substituted into Eq. (14), i.e., $\mathbf{x}^{}\approx\arg\min_{\mathbf{x}\in\mathcal{X}}F(\mathbf{x},\Psi_{\bm{\theta}}(\mathbf{x})).$ (30) For local approximation, by introducing a small UL disturbing term, they first minimize the objective $\mathbb{E}_{\epsilon\sim p(\epsilon\|\delta)}f(\mathbf{x}+\epsilon,\Psi_{\bm{\theta}}(\mathbf{x}+\epsilon))$, where $\epsilon$ represents the perturbation noise added to $\mathbf{x}$, and $p(\epsilon\|\delta)$ is defined as a factorized Gaussian noise distribution with a fixed scale parameter $\delta$. After that, the UL variable $\mathbf{x}$ is updated by minimizing the proxy function, i.e., Eq. (30). In comparison to other type EGBRs, proxy-based EGBRs can easily replace existing modules in deep learning libraries with hyper-counterparts that accept an additional vector of UL variable as input and adapt online, thereby requiring less memory consumption to meet the performance requirements. ## 6 Implicit Gradient for Best-Response In contrast to the EGBR methods surveyed above, IGBR methods in essence can be interpreted as introducing Implicit Function Theory (IFT) to derive BR Jacobian [211]. In particular, IGBR type BLOs only rely on the solution to the LL optimization and can effectively decouple the UL gradient computation from the choice of LL optimizer. Indeed, the gradient-based BLO methodologies with implicit differentiation are radically different from EGBR methods, which have been extensively applied in a string of applications (e.g., [72, 85, 191]). As an example, a set of early IGBR approaches (e.g., [161, 162]) used implicit differentiation to select hyper-parameters of kernel-based models. Recently, IGBR type approaches have been applied in different application scenarios, such as learning hyper-parameter for neural networks [72] and variational models [136]. Now we demonstrate how to derive IGBRs to solve BLOs. Specifically, in the LLS optimization scenario, we first require that $f(\mathbf{x},\mathbf{y})$ satisfies the smooth condition (or at least twice continuously differentiable) w.t.r. both the UL and LL variables, and $\mathbf{y}^{}(\mathbf{x})$ is a continuously differentiable function w.r.t. $\mathbf{x}$. Then we can directly obtain the implicit gradient of $\mathbf{x}$ (i.e., $\mathbf{G}(\mathbf{x})$) based on the first-order optimality condition (i.e., $\frac{\partial f(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{y}^{}(\mathbf{x})}=0$). That is, by deriving the above equation w.r.t. $\mathbf{x}$, we have that $\frac{\partial\mathbf{y}^{}(\mathbf{x})}{\partial\mathbf{x}^{\prime}}+\left(\frac{\partial^{2}f(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{y}^{}(\mathbf{x})\partial\mathbf{y}^{}(\mathbf{x})^{\prime}}\right)^{-1}\frac{\partial^{2}f(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{y}^{}(\mathbf{x})\partial\mathbf{x}^{\prime}}=0.$ By further assuming that $\frac{\partial^{2}f(\mathbf{x},\mathbf{y}^{}(\mathbf{x})}{\partial\mathbf{y}^{}(\mathbf{x})\partial\mathbf{y}^{}(\mathbf{x})^{\prime}}$ is invertible, and drawing upon the chain rule, the indirect gradient $\mathbf{G}(\mathbf{x})$ can be obtained as follows: $\displaystyle\mathbf{G}(\mathbf{x})=-\left(\frac{\partial^{2}f(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{y}^{}(\mathbf{x})\partial\mathbf{x}^{\prime}}\right)^{\prime}\left(\frac{\partial^{2}f(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{y}^{}(\mathbf{x})\partial\mathbf{y}^{}(\mathbf{x})^{\prime}}\right)^{-1}$ (31) $\displaystyle\frac{\partial F(\mathbf{x},\mathbf{y}^{}(\mathbf{x}))}{\partial\mathbf{y}^{}(\mathbf{x})}$ . Intuitively, Eq. (31) has offered the exact indirect gradient formulation but is generally calculated based on numerical approximations in practice. From a computational point of view, due to involving a large number of repeated product operations of Hessian-vector and Jacobian-vector, EGBRs based on high- dimensional data are usually computationally expensive and time-consuming. Thus a few implicit techniques, such as IGBR based on linear system [74, 85] and Neumann series [72], have been proposed to address this computational issue. Based on Linear System: To calculate the Hessian matrix inverse more efficiently, it is generally assumed that solving linear systems is a common operation (e.g., HOAG [74], IMAML [85]). Specially, $(\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}})^{-1}\frac{\partial F}{\partial\mathbf{y}}$ can be computed as the solution to the linear system $(\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}})\mathbf{q}=\frac{\partial F}{\partial\mathbf{y}}$ for $\mathbf{q}$. Based on the above derivation, it is apparent that $\frac{\partial F}{\partial\mathbf{x}}$ can be directly computed by the algorithm summarized in Alg. 4. Algorithm 4 Implicit Gradient by Solving Linear System 0: The UL variable at the current state, i.e., $\mathbf{x}$ 0: The gradient of $F$ with respect to $\mathbf{x}$, i.e., $\frac{\partial F}{\partial\mathbf{x}}$ 1: Optimize the LL variable up to tolerance $\epsilon$. That is, find ${\mathbf{y}_{\varepsilon}}$ such that $\\|\mathbf{y}^{}(\mathbf{x})-{\mathbf{y}_{\varepsilon}}\\|\leq\epsilon.$ 2: Solve the linear system $\left(\frac{\partial^{2}f(\mathbf{x},\mathbf{y}_{\varepsilon})}{\partial\mathbf{y}_{\varepsilon}\partial\mathbf{y}_{\varepsilon}^{\prime}}\right)\mathbf{q}=\frac{\partial F(\mathbf{x},\mathbf{y}_{\varepsilon})}{\partial\mathbf{y}_{\varepsilon}},$ for $\mathbf{q}$ up to the tolerance $\epsilon$, i.e., $\left\\|\left(\frac{\partial^{2}f}{\partial\mathbf{y}_{\varepsilon}\partial\mathbf{y}_{\varepsilon}^{\prime}}\right)\mathbf{q}-\frac{\partial F}{\partial\mathbf{y}_{\varepsilon}}\right\\|\leq\epsilon.$ 3: Compute approximate gradient by $\mathbf{p}=\frac{\partial F(\mathbf{x},\mathbf{y}_{\varepsilon})}{\partial\mathbf{x}}-\left(\frac{\partial^{2}f(\mathbf{x},\mathbf{y}_{\varepsilon})}{\partial\mathbf{y}_{\varepsilon}\partial\mathbf{x}^{\prime}}\right)^{\prime}\mathbf{q}.$ 4: return $\mathbf{p}$. Based on Neumann Series: Instead of solving the linear system, another type of IGBM (i.e., Neumann IFT [72]) method aims to calculate the Neumann series to approximate the inverse of Hessian matrix. Specifically, rather than solving the linear system in the second step of Alg. 4, the inverse Hessian is expressed as the following Neumann series: $\left(\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}}\right)^{-1}=\lim_{i\to\infty}\sum\limits_{j=0}^{i}\left(\mathbf{I}-\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}}\right)^{j},$ where $\mathbf{I}$ denotes an identity matrix with proper size. If the operator $\mathbf{I}-\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}}$ is contractive, it leverages that unrolling differentiation for $i$ steps around locally optimal weights $\mathbf{y}^{}(\mathbf{x})$ is equivalent to approximating the inverse with the first $i$ terms in Neumann series. In this way, the entire computation can efficiently perform vector-Jacobian products, thus providing a cheap approximation to the inverse-Hessian-vector product. ## 7 BLO beyond Lower-Level Singleton As stated in the above Sections 4-6, different categories of gradient-based algorithms have been proposed to address BLOs. However, most of these approaches rely on the LLS assumption (i.e., the solution set of the LL subproblem is a singleton) stated in Section 4 to simplify their optimization process and theoretical analysis. That is to say, the sequence $\\{\mathbf{y}_{t}\\}_{t=0}^{T}$ generated by these mainstream methods could converge to the true optimal solution only if the LLS condition is satisfied. Unfortunately, it has been demonstrated that such LLS assumption is too restrictive to be satisfied in most real-world learning and vision applications. For example, the works in [32, 189] have designed a series of counter-examples to illustrate that these existing EGBRs cannot obtain the correct solution if the LLS assumption is not satisfied. In this section, we review some recent works [32, 189, 195], which can efficiently address the LLS issue in the optimistic BLO scenario. The key optimization process of these works is to obtain the solution set of the ISB (i.e., Eq. (6)). That is, these works actually adopted different techniques, such as the UL and LL gradient aggregation [32, 189] and value-function-based interior-point method [195] to solve Eq. (6) for BLOs without the LLS condition. UL and LL Gradient Aggregation: Differing from previous EGBR type methods which only rely on the gradient information of the LL subproblem to update $\mathbf{y}$, a more generalized EGBR type method, Bi-level Descent Aggregation (BDA) method [32], characterizes an aggregate computation of both the LL and the UL descent information. With a given UL variable $\mathbf{x}$, the aggregated descent direction w.r.t. the ISB subproblem (i.e., Eq. (6)) can be defined as $\mathbf{d}(\mathbf{y}_{t-1};\mathbf{x})=\rho_{t}\frac{\partial F(\mathbf{x},\mathbf{y}_{t-1})}{\partial\mathbf{y}_{t-1}}+(1-\rho_{t})\frac{\partial f(\mathbf{x},\mathbf{y}_{t-1})}{\partial\mathbf{y}_{t-1}},$ (32) where ${\rho_{t}\in(0,1]}$ is the aggregation parameter (tending to zero [212, 213]), and $\frac{\partial F(\mathbf{x},\mathbf{y}_{t-1})}{\partial\mathbf{y}_{t-1}}$ (or $\frac{\partial f(\mathbf{x},\mathbf{y}_{t-1})}{\partial\mathbf{y}_{t-1}}$) stands for the descent directions of the UL (or LL) objectives. Value-Function-based Interior-point Method: Different from EGBRs and IGBRs, a more recent Value-Function Best-Response (VFBR) type BLO methods reformulate BLO into a ISB optimization problem by the value function of the UL objective. After that, they further transform it into a single-level optimization problem with an inequality constraint through the value function of the LL objective. Recently, a typical VFBR work, named Bi-level Value-Function-based Interior- point Method (BVFIM) [195], has designed a log-barrier penalty-based single- level reformulation for Eq. (6) to address the LLS issue in the non-convex scenario. Specifically, BVFIM first reformulates the ISB subproblem in Eq. (6) as follows: $\min\limits_{\mathbf{y}\in\mathcal{Y}}F(\mathbf{x},\mathbf{y}),\ \mathrm{\ s.t.\ }\ f(\mathbf{x},\mathbf{y})\leq\psi_{\mu}(\mathbf{x}),$ (33) where $\psi_{\mu}(\mathbf{x})$ is a regularized value function of the LL subproblem, i.e., $\psi_{\mu}(\mathbf{x})=\min\limits_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y})+\frac{\mu_{1}}{2}\\|\mathbf{y}\\|^{2}+\mu_{2}.$ (34) Here $\mu_{1},\mu_{2}$ are two positive constants and we denote $\mu=(\mu_{1},\mu_{2})$. Then the relaxed inequality constraint $f(\mathbf{x},\mathbf{y})\leq\psi_{\mu}(\mathbf{x})$ is penalized to the objective by a log-barrier penalty and thus Eq. (33) can be approximated by $\small\varphi_{\mu,\theta,\tau}\left(\mathbf{x}\right)=\min_{\mathbf{y}\in\mathcal{Y}}F(\mathbf{x},\mathbf{y})+\frac{\theta}{2}\\|\mathbf{y}\\|^{2}-\tau\ln(\psi_{\mu}(\mathbf{x})-f(\mathbf{x},\mathbf{y})),$ (35) where $(\mu,\theta,\tau)>0$. Finally, BVFIM proves that indirect gradient $\mathbf{G}(\mathbf{x})$ in Eq. (10) can be obtained by solving a series of Eq. (35) with decreasing parameters $(\mu,\theta,\tau)$ (tending to zero). It should be noticed that BVFIM can successfully avoid these time-consuming Hessian-vector and Jacobian-vector products, which are necessary in previous gradient-based BLOs. So this method is more suitable for BLO tasks with complex LL subproblems. ## 8 Theoretical Investigations In addition to modeling various learning and vision applications from the perspective of BLO and establishing a general algorithmic framework to unify different categories of existing BLO algorithms, in this section, we further investigate some important theoretical issues of BLOs, including the convergence behaviors and computational complexity of gradient-based BLOs, which actually can provide us insights and guidance in practical application scenarios (e.g., adopt/design proper BLO methods). ### 8.1 Convergence Properties and Required Conditions In existing literature, two categories of convergence properties have been proved for gradient-based BLOs. The first type is “convergence towards stationarity”, which guarantees that the UL value-function can converge to a first-order stationary point satisfying $\lim_{K\rightarrow\infty}\\|\frac{\partial\varphi(\mathbf{x}_{T}^{K})}{\partial\mathbf{x}_{T}^{K}}\\|=0$. Here we actually consider the convergence property of the UL variable, i.e., the number of UL iteration $K$ tends to infinity (with fixed number of LL iteration $T$). The other convergence results actually characterize the following properties: $\mathbf{x}_{T}\xrightarrow[]{s}\mathbf{x}^{}$777Here we use “$\xrightarrow[]{s}$” to denote subsequential convergence. and $\inf_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x})\rightarrow\inf_{\mathbf{x}\in\mathcal{X}}\varphi(\mathbf{x})$) when $T\to\infty$. That is, they prove that for any limit point $\bar{\mathbf{x}}$ of the sequence $\\{\mathbf{x}_{T}\\}$, if $\mathbf{x}_{T}$ is a global (resp. local) minimum of $\varphi_{T}(\mathbf{x})$, then $\bar{\mathbf{x}}$ is a global (resp. local) minimum of $\varphi(\mathbf{x})$. For convenience, we call this type of property as “convergence towards global/local minimum”888We will provide more details on this convergence property in the following subsection (i.e., Theorem 1).. In Table III, we analyze the convergence properties and conditions required by the UL and LL subproblems for different categories of gradient-based BLOs, including EGBRs (e.g., RHG [12], TRAD [13], HF-MAML [214], STN [9] and BDA [32]), IGBRs (e.g., HOAG [74] and IMAML [85]) and VFBR (e.g., BVFIM [195]). To guarantee the convergence to stationary solutions, some EGBRs (e.g., TRAD [13], HF-MAML [214] and STN [9]) required the first-order Lipshitz assumption for the UL and LL objectives (i.e., “${L}_{F}$” and “${L}_{f}$” for short) and the twice continuously differentiable property for the LL objective. In addition, there are also some EGBRs that require additional strong assumptions to obtain the first-order stationary points. For instance, HF-MAML [214] relies on second-order Lipshitz assumption (denoted as “Lipschitz-Hessian”) for the LL objective, while STN [9] needs the nonsingular Hessian assumption for the LL objective. As for IGBRs (e.g., HOAG [74] and IMAML [85]), they generally require that the gradient (w.r.t. $\mathbf{y}$) of both the UL objective and the LL objective are Lipschitz continuous. As another mainstream EGBR, the work [12] requires that the LL dynamic system $\\{\mathbf{y}_{T}(\mathbf{x})\\}$ is uniformly bounded on $\mathcal{X}$ and $\mathbf{y}_{T}(\mathbf{x})$ uniformly converges to $\mathbf{y}^{}(\mathbf{x})$ when $T\rightarrow\infty$. Then we can obtain the convergence towards the global/local minimum. As for IGBRs, both the Lipshitz Hessian and nonsingular Hessian are key properties to guarantee their stationarity convergence [74, 214, 85]. TABLE III: Summarizing the convergence results of mainstream gradient-based methods for BLOs within our framework. Category \| Method \| LLS \| UL \| LL \| Main convergence results ---\|---\|---\|---\|---\|--- ${L}_{F}$ \| SC \| ${L}_{f}$ \| $\mathcal{C}^{2}$ \| SC \| Lip-Hess \| NS-Hess EGBR \| TRAD [13] \| ✓ \| ✗ \| ✗ \| ✓ \| ✓ \| ✓ \| ✗ \| ✓ \| Stationarity: $\frac{\partial\varphi(\mathbf{x}_{T}^{K})}{\partial\mathbf{x}_{T}^{K}}\rightarrow 0$. HF-MAML [214] \| ✓ \| ✓ \| ✗ \| ✓ \| ✓ \| ✗ \| ✓ \| ✗ STN [9] \| ✓ \| ✗ \| ✗ \| ✓ \| ✓ \| ✓ \| ✗ \| ✓ RHG [12] \| ✓ \| ✗ \| ✗ \| ✗ \| ✗ \| ✗ \| ✗ \| ✗ \| Global/local minimum: $\mathbf{x}_{T}\xrightarrow[]{s}\mathbf{x}^{},$ $\inf_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x})\rightarrow\inf_{\mathbf{x}\in\mathcal{X}}\varphi(\mathbf{x})$. BDA [32] \| ✓ \| ✓ \| ✗ \| ✓ \| ✗ \| ✗ \| ✗ \| ✗ ✗ \| ✓ \| ✓ \| ✓ \| ✗ \| ✗ \| ✗ \| ✗ BDA [189] \| ✗ \| ✓ \| ✗ \| ✓ \| ✗ \| ✗ \| ✗ \| ✗ VFBR \| BVFIM [195] \| ✗ \| ✗ \| ✗ \| ✗ \| ✗ \| ✗ \| ✗ \| ✗ IGBR \| HOAG [74] \| ✓ \| ✓ \| ✗ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| Stationarity: $\frac{\partial\varphi(\mathbf{x}_{T}^{K})}{\partial\mathbf{x}_{T}^{K}}\rightarrow 0$. IMAML [85] \| ✓ \| ✓ \| ✗ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ 1 Notice that $F(\mathbf{x},\mathbf{y})$ is continuously differentiable on $\mathcal{X}\times\mathcal{Y}$ ($\mathcal{X}$ is a compact set) and $f(\mathbf{x},\mathbf{y})$ is continuously differentiable on $\mathbb{R}^{m}\times\mathcal{Y}$. The feasible solution set $\mathcal{Y}$ represents the whole space $\mathbb{R}^{n}$. * 2 “${L}_{F}$ (resp. ${L}_{f}$)” means the gradient of $F(\mathbf{x},\cdot)$ (resp. $f(\mathbf{x},\cdot)$) is Lipschitz continuous with Lipschitz constant ${L}_{F}$ (resp. ${L}_{f}$). SC means strongly convex and $\mathcal{C}^{2}$ implies that $f(\mathbf{x},\cdot)$ is second-order continuously differentiable w.r.t. $\mathbf{y}$. “NS-Hess” and “Lip-Hess” represent the nonsingularity and Lipschitz properties of Hessian $\frac{\partial^{2}f}{\partial\mathbf{y}\partial\mathbf{y}^{\prime}}$, respectively. Please refer to [214, 85, 74] for more details on these variational analysis concepts. * 3 Here we respectively represent “required” and “not required” by “✓” and “✗” for these properties. * 4 We summarize two kinds of convergent properties, i.e., “stationarity” and “global/local minimum”. The former implies that the gradient descent on the UL value-function converges to first-order stationary points satisfying $\lim_{K\rightarrow\infty}\\|\frac{\partial\varphi(\mathbf{x}_{T}^{K})}{\partial\mathbf{x}_{T}^{K}}\\|=0$ (with fixed number of LL iterations $T$), while the latter characterizes the convergence towards global/local minimum satisfying $\mathbf{x}_{T}\xrightarrow[]{s}\mathbf{x}^{}$ and $\inf_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x})\rightarrow\inf_{\mathbf{x}\in\mathcal{X}}\varphi(\mathbf{x})$ as $T\to\infty$. ### 8.2 A General Proof Template for EGBRs In this subsection, we would like to further provide a general proof template to analyze the convergence behaviors (i.e., convergence towards global/local minimum) of EGBR methods in more detail. In particular, given the output of the LL dynamic system (i.e., $\mathbf{y}_{T}(\mathbf{x})$), we first introduce two elementary properties on it as follows: 1. (1) Uniform approximation quality to the LL solution: $\\{\mathbf{y}_{T}(\mathbf{x})\\}$ is uniformly bounded on $\mathcal{X}$, and for any $\epsilon>0$, there exists $t(\epsilon)>0$ such that whenever $T>t(\epsilon)$, we have $\sup_{\mathbf{x}\in\mathcal{X}}\left\\{f(\mathbf{x},\mathbf{y}_{T}(\mathbf{x}))-\psi(\mathbf{x})\right\\}\leq\epsilon,$ or $\sup_{\mathbf{x}\in\mathcal{X}}\\|\frac{\partial f(\mathbf{x},\mathbf{y}_{T}(\mathbf{x}))}{\partial\mathbf{y}_{T}(\mathbf{x})}\\|\leq\epsilon,$ where $\psi(\mathbf{x})$ denotes the LL value-function, i.e., $\psi(\mathbf{x}):=\min_{\mathbf{y}\in\mathcal{Y}}f(\mathbf{x},\mathbf{y})$. 2. (2) Point-wise approximation quality to the ISB solution: For each $\mathbf{x}\in\mathcal{X}$, we have $\lim_{T\rightarrow\infty}\mathtt{dist}(\mathbf{y}_{T}(\mathbf{x}),\widetilde{\mathcal{S}}(\mathbf{x}))=0,$ where $\widetilde{\mathcal{S}}(\mathbf{x})$ represents the solution set of the ISB subproblem in Eq. (6) and $\mathtt{dist}(\cdot,\cdot)$ denotes the point- to-set distance. Equipped with the above two properties on $\\{\mathbf{y}_{T}(\mathbf{x})\\}$, we can present general convergence results of Eqs. (11)-(14) in the following theorem999Here we actually provide a brief proof roadmap, which is summarized based on theoretical studies in existing works [51, 195, 32, 189]. . ###### Theorem 1. (Convergence towards global/local minimum) Suppose that the generated sequence $\left\\{\mathbf{y}_{t}(\mathbf{x})\right\\}$ satisfies the above two properties. Let $\mathbf{x}_{T}$ be a global (resp. local) minimum of $\varphi_{T}(\mathbf{x})$, i.e., $\mathbf{x}_{T}\in\arg\min_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x})$. Then we have (1) Any limit point $\bar{\mathbf{x}}$ of the sequence $\\{\mathbf{x}_{T}\\}$ is a global (resp. local) minimum of $\varphi(\mathbf{x})$, i.e., $\bar{\mathbf{x}}\in\arg\min_{\mathbf{x}\in\mathcal{X}}\varphi(\mathbf{x})$. * (2) $\inf_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x})\rightarrow\inf_{\mathbf{x}\in\mathcal{X}}\varphi(\mathbf{x})$ as $T\rightarrow\infty$. ###### Proof. In the following, we first state the key steps for proving convergence properties in the global scenario and then demonstrate how to obtain the local convergence properties accordingly. Step 1. We should first verify that for $\bar{\mathbf{x}}\in\mathcal{X}$, $\psi$ satisfies $\limsup_{\mathbf{x}\rightarrow\bar{\mathbf{x}}}\psi(\mathbf{x})=\psi(\bar{\mathbf{x}}).$ Step 2. Then for any limit point $\bar{\mathbf{x}}$ of the sequence $\\{\mathbf{x}_{T}\\}$, there exist $\mathbf{y}_{m}(\mathbf{x}_{m})\rightarrow\bar{\mathbf{y}}$ for a subsequence $\\{\mathbf{x}_{m}\\}$ and some $\bar{\mathbf{y}}$. Thus we can obtain $\bar{\mathbf{y}}\in\mathcal{S}(\bar{\mathbf{x}})$. Step 3. Next, we verify the convergence property of the UL objective as follows: $\lim_{T\rightarrow\infty}\varphi_{T}(\mathbf{x})=\varphi(\mathbf{x}).$ Step 4. For any $\epsilon>0$, we verify the following inequality: $\varphi(\bar{\mathbf{x}})\leq F(\mathbf{x}_{m},\mathbf{y}_{m}(\mathbf{x}_{m}))+\epsilon\leq\lim_{m\rightarrow\infty}\varphi_{m}(\mathbf{x})+\epsilon,\ \forall\mathbf{x}\in\mathcal{X}.$ Step 5. Finally, we verify the following inequality $\limsup_{T\rightarrow\infty}\left\\{\inf_{\mathbf{x}\in\mathcal{X}}\varphi_{T}(\mathbf{x})\right\\}\leq\inf_{\mathbf{x}\in\mathcal{X}}\varphi(\mathbf{x}).$ Thus we can obtain convergence results stated in Theorem 1. For convergence to the local minimum, we actually consider $\mathbf{x}_{T}$ as a local minimum of $\varphi_{T}(\mathbf{x})$ with uniform neighborhood modulus $\delta>0$. Then any limit point $\bar{\mathbf{x}}$ of the sequence $\\{\mathbf{x}_{T}\\}$ is a local minimum of $\varphi(\mathbf{x})$, i.e., there exists $\tilde{\delta}>0$ such that $\varphi(\bar{\mathbf{x}})\leq\varphi(\mathbf{x}),\forall\mathbf{x}\in\mathbb{B}_{\delta}(\bar{\mathbf{x}})\cap\mathcal{X}$. According to the neighborhood property to spread out the analysis, the result of convergence towards local minimum can also be proved by the same steps. ∎ The above theoretical results actually provide us a general recipe to analyze the iteration behaviors and convergence properties of gradient-based BLOs, especially for EGBRs. In other words, we can understand that these existing numerical schemes and their required assumptions on the UL and LL subproblems are just to meet the above elementary iteration properties. It can be observed that classical EGBRs (e.g., [12, 13]) require to first enforce the LLS assumption on the BLO problem. The work in [12] assumes that the UL and LL objectives are continuously differentiable and also enforces the restrictive (local) strong convexity assumption on the LL objective. In fact, such properties can ensure the uniform convergence of $\\{\mathbf{y}_{T}(\mathbf{x})\\}$ towards $\mathbf{y}^{}(\mathbf{x})$, thus lead to the two elementary properties. In fact, the LLS assumption considered in [12] is more strict than that required in the proof template. The works in [32, 189] also consider that the UL and LL objectives are continuously differentiable, but make a weaker assumption on the LL objective, i.e., $f(\mathbf{x},\mathbf{y})$ is level-bounded in $\mathbf{y}$ and locally uniform in $\mathbf{x}\in\mathcal{X}$ (or the gradient of $f(\mathbf{x},\cdot)$ is Lipschitz continuous). Indeed, it can be verified that the conditions in [32, 189] can also ensure two elementary properties required by our proof template. Therefore, we have that the above two elementary convergence properties hold and we can obtain the convergence results stated in Theorem 1. It has been verified in [32, 189] that these classical EGBRs [12, 13] may lead to incorrect solutions if the LLS assumption is not satisfied. As stated in the above Section 7, BDA [32, 189] has been proposed to extend the EGBR method to address this issue. Theoretically, the work in [32] actually introduces the LL solution set property and the UL objective convergence property. Theoretical investigations in [189] further demonstrate that the iterative gradient-aggregation dynamics can solve the ISB subproblem without the LL singleton assumption and the UL strong convexity. Again, in order to remove the restrictive singleton and convex assumptions on the LL objective, BVFIM [195] further proves the same convergence results by introducing the strong constraints on a series of positive decreasing parameters $(\mu,\theta,\tau)$ when $f(\mathbf{x},\mathbf{y})$ and $F(\mathbf{x},\mathbf{y})$ are level- bounded in $\mathbf{y}$ and locally uniformly in $\mathbf{x}\in\mathcal{X}$. ### 8.3 Time and Space Complexity In this subsection, we analyze the complexity of time and space for these mainstream gradient-based BLO methods (i.e., EGBRs [51, 13, 32], IGBRs [74, 85, 72] and VFBR [195]), as summarized in Table IV. Please notice that here we just follow most BLO literature (e.g., [51, 13, 14]) to only estimate the complexity of computing the gradient of $\varphi$ w.r.t. $\mathbf{x}$ (defined in Eq. (9)) with a fixed (e.g., $T$-step) LL iteration. EGBR: As discussed in Section 5, EGBRs generally construct the BR mapping $\mathbf{y}^{}(\mathbf{x})$ or the indirect gradient $\mathbf{G}(\mathbf{x})$ with the implementation of an unrolled dynamic system (see Eq. (13)). In [51], the dynamic system can be implemented in either a forward automatic differential mode (i.e., FAD) or a reverse automatic differential mode (i.e., RAD). Especially, BDA implements a reverse aggregated gradient flow from the UL and LL subproblems to approximate the BR mapping. More specifically, taking into account the fact that the Hessian-matrix product is repeatedly calculated (i.e., $\sum_{t=0}^{T}\left(\prod_{i=t+1}^{T}\mathbf{A}_{i}\right)\mathbf{B}_{t}$) in the forward propagation, FAD requires the space complexity $O(mn)$ and the time complexity $O(m^{2}nT)$. RAD in the backward pass needs to evaluate Hessian- and Jacobian-vector products, and stores all the intermediate variables $\\{\mathbf{y}_{t}\in\mathbb{R}^{n}\\}_{t=1}^{T}$ in memory. So we have that the time and space costs are $O(n(m+n)T)$ and $O(m+nT)$, respectively. By ignoring the long-term dependencies, TRAD uses the truncated back-propagation trajectory with a smaller number of steps (i.e., $M<T$). As for BDA, with the similar backward propagation manner, we have that the complexity of time and space is the same as that for RAD. IGBR: As for IGBRs, we have that they require to derive the indirect gradient based on the implicit function theorem, which results in the overloaded computation with respect to the inverse of Hessian (see Eq. (31)). To mitigate this problem, IGBRs generally solve a linear system by Conjugate Gradient (CG) [74, 85] or Neumann series [72], as stated in Section 6. Without loss of generality, we uniformly assume that these methods perform $J$-step iterations to solve the linear system. Each step contains a hessian-vector product computation requiring the time cost $O(m+n^{2}J)$. Then with a $T$-step gradient descent on the LL subproblem, we have that the overall time and space complexities can be written as $O(m+nT+n^{2}J)$ and $O(m+n)$, respectively. It should be noted that the iteration step $J$ generally relies on the properties of Hessian-matrix, thus it should be set much larger than $T$. VFBR: It has been stated in Section 7 that VFBR type method (i.e., BVFIM) does not require to solve the unrolled dynamic system or approximate the inverse of Hessian, thus can obtain lower time and space complexity than EGBRs and IGBRs, especially on BLOs with high-dimensional LL subproblems. Specifically, we use $Q_{1}$ and $Q_{2}$ to represent the number of gradient iterations for solving the regularized subproblems in Eqs. (34) and (35), respectively. Then it can be checked that the time costs of calculating each gradient descent for the LL and UL value-functions are $O(nQ_{1})$ and $O(nQ_{2})$, respectively. Moreover, we require additional $O(m)$ time to perform the UL gradient updating. Thus the overall time cost of BVFIM is $O(m+n(Q_{1}+Q_{2}))$. As for the space complexity, it is easy to check that BVFIM requires $O(m+n)$ space cost and is the same as that in IGBRs. TABLE IV: Comparison of the time and space complexity for several gradient-based mainstream BLOs. Category \| Method \| Time \| Space ---\|---\|---\|--- EGBR \| FAD [51] \| $O(m^{2}nT)$ \| $O(mn)$ RAD [51] \| $O(n(m+n)T)$ \| $O(m+nT)$ BDA [32] \| $O(n(m+n)T)$ \| $O(m+nT)$ TRAD [13], \| $O(n(m+n)M)$ \| $O(m+nM)$ IGBR \| CG [74, 85] \| $O(m+nT+n^{2}J)$ \| $O(m+n)$ Neumann [72] \| $O(m+nT+n^{2}J)$ \| $O(m+n)$ VFBR \| BVFIM [195] \| $O(m+n(Q_{1}+Q_{2}))$ \| $O(m+n)$ It can be seen in Table IV that the reverse propagation methods (i.e., RAD, TRAD and BDA) have benefited from the lightweight matrix-vector multiplication (rather than the overweight Hessian-matrix), thus can obtain less computational complexity in comparison to the forward propagation approach (e.g., FAD). Especially for TRAD, the time and space complexity can be further reduced by the truncated back-propagation strategy. Compared with EGBRs, IGBRs maintain higher computational complexity due to the overloaded computation in terms of the inverse of Hessian. In contrast, VFBR can obtain lower time consuming than both EGBRs and IGBRs. It actually also outperforms EGBRs in costing less memory, especially when solving the LL subproblem on high- dimensional tasks (e.g., with extremely large $n$). ## 9 Potentials for New Algorithms Design As the last but not least part of the survey, this section aims to demonstrate the potentials of our general algorithmic framework for designing new gradient schemes for challenging BLO formulations, such as pessimistic BLOs (stated in Eq. (5)). In fact, pessimistic BLO formulation can be naturally interpreted as a non- cooperative game between two players and has been utilized to formulate problems in the area of mathematical programming [215, 216, 217] and other application fields, such as economics [218, 219] and biology [220]. However, from the pessimistic viewpoint, the UL player (i.e., leader) cannot anticipate the LL player (i.e., follower)’s decision, the constraint must be satisfied for any rational decision of the follower, thus pessimistic BLO is perceived to be very difficult to solve, especially in high-dimensional application scenarios [196]. Now we demonstrate how to develop a practical algorithm within our BR mapping based BLO algorithmic framework for pessimistic BLO formulations101010We emphasize that we just present an example to demonstrate the potentials of our framework for new algorithm design. Strict theoretical analysis and evaluations are definitely out of the scope in this paper and will be considered as the future work.. Concretely, based on Eq. (5) and pessimistic BR mapping (defined in Eq. (7)), we can follow the similar idea in Eq. (32) to aggregate the UL and LL gradients $\widetilde{\mathbf{d}}(\mathbf{y}_{t-1};\mathbf{x})=-\rho_{t}\frac{\partial F(\mathbf{x},\mathbf{y}_{t-1})}{\partial\mathbf{y}_{t-1}}+(1-\rho_{t})\frac{\partial f(\mathbf{x},\mathbf{y}_{t-1})}{\partial\mathbf{y}_{t-1}}.$ With the above procedure, it can be seen that the only difference between $\mathbf{d}$ and $\widetilde{\mathbf{d}}$ is just the sign of the UL gradient. Thus we can adopt the same calculation scheme as that in [32, 53] to solve Eq. (5). The corresponding roadmap is also illustrated in Fig. 9. ## 10 Conclusions and Future Prospects Bi-Level Optimization (BLO) is an important mathematical tool for modeling and solving machine learning and computer vision problems that have hierarchical optimization structures, such as hyper-parameter optimization, multi-task and meta learning, neural architecture search, adversarial learning and deep reinforcement learning, etc. In the above sections, we first demonstrated how to formulate different learning and vision tasks from a uniform BLO perspective. We then established a value-function-based single-level reformulation for different categories of BLO models and proposed a best- response-based optimization platform to uniformly understand and formulate a variety of existing gradient-based BLO methods. The convergence behaviors and complexity properties of these BLO algorithms have also been discussed. We also demonstrated potentials of our BLO platform for designing new algorithms to solve the more challenging pessimistic BLOs tasks. The future research of BLOs may focus but is not limited to the following aspects: * • Theoretical breakthrough: The convergence behaviors of gradient-based algorithms on various challenging BLOs, such as pessimistic BLOs [215, 221, 222], BLOs with complex constraints [223, 224], nonconvex objectives [225] and multiple followers [226], should be investigated. * • Computational improvement: It is also urgent to design efficient acceleration techniques (e.g., momentum and its variations) to speed up gradient-based BLOs in high-dimensional optimization scenario [227, 228, 229]. * • Wider applications: Recent deep learning tasks (e.g., knowledge distillation [230], self-supervised learning [231], and transformer [232]) are more and more sophisticated. 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From 2010 to 2012, he was doing research as joint Ph.D. in robotics institute at Carnegie Mellon University. From 2016 to 2018, He was doing research as Hong Kong Scholar at the Hong Kong Polytechnic University. He is currently a full professor with the Digital Media Department at International School of Information Science & Engineering, Dalian University of Technology. He was awarded the “Outstanding Youth Science Foundation” of the National Natural Science Foundation of China. His research interests include optimization, computer vision and multimedia. ---\|--- \| Jiaxin Gao received the B.S. degree in Applied Mathematics from Dalian University of Technology, China, in 2018. She is currently pursuing the PhD degree in software engineering at Dalian University of Technology, Dalian, China. She is with the Key Laboratory for Ubiquitous Network and Service Software of Liaoning Province, Dalian University of Technology, Dalian, China. Her research interests include computer vision, machine learning and optimization. ---\|--- \| Jin Zhang received the B.A. degree in Journalism from the Dalian University of Technology in 2007. He pursued a degree in mathematics and received the M.S. degree in Operational Research and Cybernetics from the Dalian University of Technology, China, in 2010, and the PhD degree in Applied Mathematics from University of Victoria, Canada, in 2015. After working in Hong Kong Baptist University for 3 years, he joined Southern University of Science and Technology as a tenure-track assistant professor in the Department of Mathematics. His broad research area is comprised of optimization, variational analysis and their applications in economics, engineering and data science. ---\|--- \| Deyu Meng (Member, IEEE) received the B.Sc. degree in information science, the M.Sc. degree in applied mathematics, and the Ph.D. degree in computer science from Xi’an Jiaotong University, Xi’an, China, in 2001, 2004, and 2008, respectively. He is currently a Professor with the School of Mathematics and Statistics, Xi’an Jiaotong University, and adjunct Professor with the Faculty of Information Technology, The Macau University of Science and Technology, Macao. From 2012 to 2014, he took his two-year sabbatical leave at Carnegie Mellon University, Pittsburgh, PA, USA. His current research interests include model-based deep learning, variational networks, and meta learning. ---\|--- \| Zhouchen Lin (M’00-SM’08-F’18) received the PhD degree from Peking University in 2000. He is currently a professor with the Key Laboratory of Machine Perception, School of EECS, Peking University. His research interests include computer vision, image processing, machine learning, pattern recognition, and numerical optimization. He has been an area chair of CVPR, ICCV, NIPS/NeurIPS, AAAI, IJCAI, ICLR and ICML many times, and is a Program Co-Chair of ICPR 2022. He was an associate editor of the IEEE Transactions on Pattern Analysis and Machine Intelligence and currently is an associate editor of the International Journal of Computer Vision. He is a Fellow of IAPR and IEEE. ---\|---
# DESY 20–214 ISSN 0418-9833 Dezember 2020 Angular analysis of bottom-flavored hadron production in semileptonic decays of polarized top quarks Bernd A. Kniehl111E-mail<EMAIL_ADDRESS> II. Institut für Theoretische Physik, Universität Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany S. Mohammad Moosavi Nejad222E-mail<EMAIL_ADDRESS> Faculty of Physics, Yazd University, P.O. Box 89195–741, Yazd, Iran ###### Abstract We study the inclusive production of bottom-flavored hadrons from semileptonic decays of polarized top quarks at next-to-leading order in QCD using fragmentation functions recently determined from a global fit to $e^{+}e^{-}$ data. We provide the relevant differential decay widths at parton level in analytic form. These results fill an important gap in the theoretical interpretation of recent measurements of the top-quark polarization and the $t\bar{t}$ spin correlations using dilepton final states in proton-proton collisions at the CERN Large Hadron Collider. As an illustration, we study the distributions in the scaled bottom-hadron energy of the polarized-top-quark decay widths for different $W$-boson helicities. PACS numbers: 12.38.Bx, 13.85.Ni, 14.40.Nd, 14.65.Ha ## 1 Introduction The top quark $t$ of the standard model (SM) is the heaviest known elementary particle. Due to its high mass, it plays a crucial role in testing the electroweak symmetry breaking mechanism and in searching for new physics beyond the SM. The precise determination of its properties, including its mass $m_{t}$ and total decay width $\Gamma_{t}$, and its process-dependent features, like its polarization or the correlation of its spin with that of a co-produced antitop quark, is of prime importance. The latter quantities are particularly sensitive probes of deviations from the SM and allow us to constrain, e.g., the anomalous chromoelectric and chromomagnetic dipole moments of the top quark. The top-quark lifetime $\tau_{t}=\hbar/\Gamma_{t}\approx 5\times 10^{-25}$ s [1] is shorter than the typical time scale of quantum chromodynamics (QCD) $\hbar/\Lambda_{\mathrm{QCD}}\approx 10^{-24}$ s and much shorter than the spin correlation time scale $\hbar m_{t}/\Lambda_{\mathrm{QCD}}^{2}\approx 10^{-21}$ s, where $\Lambda_{\mathrm{QCD}}$ is the asymptotic scale parameter of QCD. Therefore, the top quark decays before it can hadronize, and its full spin information is preserved during its decay process and fully encoded in the angular distribution of its decay products. The top-quark polarization and the $t\bar{t}$ spin correlations have recently been measured using dilepton final states in Run 2 at the CERN Large Hadron Collider (LHC) with center-of-mass energy $\sqrt{s}=13$ TeV, by the ATLAS [2] and CMS [3] Collaborations. Intriguingly, ATLAS found a deviation of 3.8 standard deviations ($\sigma$) from the SM prediction of the asymmetry $A_{\|\Delta\phi_{\ell\ell}\|}$ of the distribution in the azimuthal angular difference $\Delta\phi_{\ell\ell}$ of the decay leptons. This deviation was confirmed by CMS, albeit with a smaller significance of about $2\sigma$. These analyses are relying on the factorization of the squared matrix element of the full process, $\|\mathcal{M}(q\bar{q}/gg\to t\bar{t}\to b\ell^{+}\nu\bar{b}\ell^{-}\bar{\nu})\|^{2}\propto\rho\times R\times\bar{\rho}$, into the spin density matrices $R$, $\rho$, and $\bar{\rho}$ for on-shell $t\bar{t}$ hadroproduction and semileptonic $t$ and $\bar{t}$ decays, respectively, via the narrow-width approximation. While $R$ is treated in Refs. [2, 3] at next-to-leading order (NLO) in QCD [4, 5], $\rho$ and $\bar{\rho}$ are only modeled at leading order (LO) using the program package madspin [6]. Moreover, the formation of bottom-flavored hadrons is not taken into account within the rigorous framework of the QCD parton model with fragmentation functions (FFs) whose universality is guaranteed by the factorization theorem [7]. It is an urgent matter to clarify in how far the observed deviations may be related to a lack of precision in the theoretical treatment of the semileptonic $t$ and $\bar{t}$ decays. It is the purpose of the present paper to provide theoretical input needed to fill this gap. Specifically, we calculate the partial width of the inclusive decay$t(\uparrow)\to bW^{+}(\uparrow)\to Bl^{+}\nu_{l}+X$, where $B$ generically denotes a bottom-flavored hadron, at NLO in QCD allowing for top- quark polarization and definite $W$-boson helicity and properly accounting for parton-to-hadron fragmentation and finite-hadron-mass effects. By doing so, we generalize our previous work [8], where the top-quark spin was averaged over. The general-mass variable-flavor-number scheme (GM-VFNS), which has been elaborated for inclusive heavy-flavored-hadron production in $e^{+}e^{-}$ annihilation [9], two-photon collisions [10], photoproduction [11], and hadroproduction [12, 13, 14, 15], provides an ideal theoretical framework also here. However, owing to the large mass hierarchy $m_{b}\ll m_{t}$, finite-$m_{b}$ corrections are expected to be negligible in the case at hand. This expectation was actually confirmed in Ref. [8], by a comparative analysis of the partial width of the decay $t\to B+W^{+}$ in the GM-VFNS and the zero- mass variable-flavor-number scheme (ZM-VFNS), where bottom is included among the massless quark flavors. In fact, the finite-$m_{b}$ corrections were found to be much smaller than the contribution from gluon fragmentation. Therefore, we will adopt the ZM-VFNS in the following. However, we will include finite-$m_{B}$ effects, which modify the relations between partonic and hadronic variables and reduce the available phase space, as explained in Sec. 2 of Ref. [8]. In Ref. [8], we adopted the $B$ FFs from Ref. [16], which were determined at NLO in the ZM-VFNS through a joint fit to $e^{+}e^{-}$ annihilation data taken by ALEPH [17] and OPAL [18] at CERN LEP1 and by SLD [19] at SLAC SLC. Specifically, the power ansatz $D_{b}(z,\mu_{F}^{\text{ini}})=Nz^{\alpha}(1-z)^{\beta}$ was used as the initial condition for the $b\to B$ FF at factorization scale $\mu_{F}^{\text{ini}}=m_{b}=4.5$ GeV, while the gluon and light-quark FFs were generated via the Dokshitzer-Gribov-Lipatov-Altatelli-Parisi (DGLAP) [20, 21, 22] evolution. In Ref. [23], the analysis of Ref. [16] was updated by including the data taken by DELPHI [24] at CERN LEP1, which were published after Ref. [16], working both at NLO and next-to-next-to-leading order (NNLO) with the same theoretical assumptions. In our numerical analysis, we employ the new $B$ FFs from Ref. [23]. A similar analysis, albeit without fragmentation and finite-$m_{B}$ effects, was reported in Refs. [25, 26]. Our analysis provides an independent check of analytic results presented therein. A related NLO analysis with a different treatment of the final state, for bottom jets instead of bottom hadrons, was performed in Ref. [27], leading to results that, unfortunately, cannot be compared with ours in any straightforward way. Recently, the analysis of Refs. [25, 26] has been extended to NNLO in QCD using the optical theorem [28]. Due to the totally inclusive treatment of the hadronic part of the final state, this result does not allow for the implementation of FFs. This paper is organized as follows. In Sec. 2, we list our parton-level results in analytic form, relegating lengthy formulas to the Appendix. In Sec. 3, we present our numerical analysis. In Sec. 4, we summarize our conclusions. ## 2 Analytic results We work at NLO in the ZM-VFNS, implemented in the modified minimal-subtraction ($\overline{\mathrm{MS}}$) scheme, and consider the decay process $t(p_{t},s)\to b(p_{b})+W^{+}(p_{W},\lambda)({}+g(p_{g}))\to B(p_{B})+\ell^{+}(p_{\ell})+\nu_{\ell}(p_{\nu})+X,$ (1) where $X$ collectively denotes the unobserved final-state hadrons and the four-momentum, spin, and helicity assignments are indicated in parentheses. We have $s=\pm 1/2$ and $\lambda=0,\pm 1$. The gluon in Eq. (1) contributes to the real radiation at NLO. Both the $b$ quark and the gluon may hadronize into the $B$ hadron. For simplicity, we employ the narrow-width approximation, where $p_{W}^{2}=m_{W}^{2}$ and small terms of order $\mathcal{O}(\Gamma_{W}^{2}/m_{W}^{2})$ are neglected. As mentioned in Sec. 1, we put $m_{b}=0$, but keep $m_{B}$ finite. In the top-quark rest frame, the $b$ quark, gluon, and $B$ hadron have energies $E_{i}=p_{t}\cdot p_{i}/m_{t}$ ($i=b,g,B$), which nominally range from $E_{b}^{\text{min}}=E_{g}^{\text{min}}=0$ and $E_{B}^{\text{min}}=m_{B}$ to $E_{b}^{\text{max}}=E_{g}^{\text{max}}=(m_{t}^{2}-m_{W}^{2})/(2m_{t})$ and $E_{B}^{\text{max}}=(m_{t}^{2}+m_{B}^{2}-m_{W}^{2})/(2m_{t})$, respectively. As in Ref. [8], we choose the scaling variable $z$ by setting $E_{B}=zE_{a}$, with $a=b,g$, in the range $0\leq z\leq 1$. Introducing the scaled energies $x_{i}=E_{i}/E_{b}^{\text{max}}$ ($i=b,g,B$), we then have $m_{B}/E_{b}^{\text{max}}\leq x_{B}\leq x_{a}\leq 1$. An alternative definition of the scaling variable, in terms of light-cone variables, is discussed in Sec. 3 of Ref. [8], to where we refer the interested reader. We implement the polarization of the top quark by writing its (average) spin four-vector in its rest frame as $s_{t}^{\mu}=P(0,\sin\theta_{P}\cos\phi_{P},\sin\theta_{P}\sin\phi_{P},\cos\theta_{P})$, where $P$ is the magnitude of the top-quark polarization, taking values in the range $0\leq P\leq 1$. Here, it is understood that the $z$ axis is chosen to point along the $W$-boson flight direction, so that $s_{t}\cdot p_{W}=-P\|\vec{p}_{W}\|\cos\theta_{P}$. To describe the leptonic decay of the $W$ boson, we boost into the rest frame of the latter, which leaves the $z$ axis invariant, and define the charged-lepton four-momentum to be $p_{\ell}^{\mu}=E_{\ell}(1,\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$. The polar angles $\theta_{P}$ and $\theta$, which appear in our final results, are also illustrated in Fig. 1. Figure 1: Definitions of the polar angle $\theta_{P}$ of the top-quark polarization three-vector in the top-quark rest frame and of the polar angle $\theta$ of the charged-lepton three-momentum in the $W$-boson rest frame. In both cases, the $z$ axis is chosen to point along the $W$-boson three-momentum in the top-quark rest frame. We wish to calculate the triply differential partial decay width $d^{3}\Gamma/(dx_{B}\,d\cos\theta\,d\cos\theta_{P})$ of process (1). Analogously to Eq. (3) in Ref. [8], we have $\frac{d^{3}\Gamma}{dx_{B}\,d\cos\theta\,d\cos\theta_{P}}=\sum_{a=b,g}\int_{x_{B}}^{1}\frac{dx_{a}}{x_{a}}\,\frac{d^{3}\hat{\Gamma}_{a}}{dx_{a}\,d\cos\theta\,d\cos\theta_{P}}(\mu_{R},\mu_{F})D_{a}\left(\frac{x_{B}}{x_{a}},\mu_{F}\right),$ (2) where $\mu_{R}$ is the renormalization scale, $\mu_{R}$ is the factorization scale, $D_{a}(z,\mu_{F})$ is the $a\to B$ FF, and $\frac{d^{3}\hat{\Gamma}_{a}}{dx_{a}\,d\cos\theta\,d\cos\theta_{P}}=\frac{1}{2}\left(\frac{d^{2}\hat{\Gamma}_{a}^{\text{unpol}}}{dx_{a}\,d\cos\theta}+P\frac{d^{2}\hat{\Gamma}_{a}^{\text{pol}}}{dx_{a}\,d\cos\theta}\cos\theta_{P}\right).$ (3) The factor $1/2$ on the right-hand side of Eq. (3) ensures that $\int_{-1}^{1}d\cos\theta_{P}\,\frac{d^{3}\hat{\Gamma}}{dx_{a}\,d\cos\theta\,d\cos\theta_{P}}=\frac{d^{2}\hat{\Gamma}_{a}^{\text{unpol}}}{dx_{a}\,d\cos\theta}.$ (4) Each $W$-boson helicity is featured by a characteristic $\theta$ dependence, which is encoded in the structure $\frac{d^{2}\hat{\Gamma}_{a}}{dx_{a}\,d\cos\theta}=\frac{3}{8}(1+\cos\theta)^{2}\frac{d\hat{\Gamma}_{a}^{+}}{dx_{a}}+\frac{3}{8}(1-\cos\theta)^{2}\frac{d\hat{\Gamma}_{a}^{-}}{dx_{a}}+\frac{3}{4}\sin^{2}\theta\frac{d\hat{\Gamma}_{a}^{0}}{dx_{a}}.$ (5) This holds for both the unpolarized and polarized terms in Eq. (3). Notice that the $\theta$-dependent coefficients on the right-hand side of Eq. (5) are normalized so that, upon integration over $\cos\theta$, we have $\frac{d\hat{\Gamma}_{a}}{dx_{a}}=\sum_{\lambda=-1}^{1}\frac{d\hat{\Gamma}_{a}^{\lambda}}{dx_{a}}.$ (6) At LO, we only have $a=b$, and $x_{b}=1$ is fixed, i.e. the $x_{b}$ dependence comes as a delta-function peak. Specifically, we have $\displaystyle\hat{\Gamma}_{b,\text{LO}}^{0,\text{unpol}}$ $\displaystyle=$ $\displaystyle\hat{\Gamma}_{b,\text{LO}}^{0,\text{pol}}=F(1-\omega)^{2},$ $\displaystyle\hat{\Gamma}_{b,\text{LO}}^{-,\text{unpol}}$ $\displaystyle=$ $\displaystyle-\hat{\Gamma}_{b,\text{LO}}^{-,\text{pol}}=F(2\omega)(1-\omega)^{2},$ $\displaystyle\hat{\Gamma}_{b,\text{LO}}^{+,\text{unpol}}$ $\displaystyle=$ $\displaystyle\hat{\Gamma}_{b,\text{LO}}^{+,\text{pol}}=0,$ (7) where $\omega=m_{W}^{2}/m_{t}^{2}$ and $F=G_{F}m_{t}^{3}\|V_{tb}\|^{2}B(W^{+}\to\ell^{+}\nu_{\ell})/(8\pi\sqrt{2})$. Here, $G_{F}$ is Fermi’s constant, $V_{ij}$ is the $ij$ element of the Cabibbo-Kobayashi-Matrix quark mixing matrix [29, 30], and $B(W^{+}\to\ell^{+}\nu_{\ell})$ is the branching ratio of the leptonic $W$-boson decay mode considered. Neglecting the masses of the charged leptons and the first five quark flavors, we have $\displaystyle B(W^{+}\to\ell^{+}\nu_{\ell})$ $\displaystyle=$ $\displaystyle\frac{1}{3+N_{c}\sum_{\begin{subarray}{c}i=u,c\\\ j=d,s,b\end{subarray}}\|V_{ij}\|^{2}\left[1+3C_{F}\alpha_{s}(\mu_{R})/(4\pi)\right]}$ (8) $\displaystyle\approx$ $\displaystyle\frac{1}{9}\left[1-\frac{2}{3}\,\frac{\alpha_{s}(\mu_{R})}{\pi}\right],$ where we have included the NLO QCD correction, with color factors $N_{c}=3$ and $C_{F}=(N_{c}^{2}-1)/(2N_{c})=4/3$. In the last equality in Eq. (8), we have approximated $V_{ij}\approx\delta_{ij}$. At LO, the top-quark spin is passed on to the bottom quark for $\lambda=0$, while it is flipped for $\lambda=-1$; the case $\lambda=+1$ is forbidden by angular-momentum conservation in the limit $m_{b}\to 0$, as is reflected in Eq. (7). Using Eq. (6), we have $\displaystyle\hat{\Gamma}_{b,\text{LO}}^{\text{unpol}}$ $\displaystyle=$ $\displaystyle F(1+2\omega)(1-\omega)^{2},$ $\displaystyle\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}$ $\displaystyle=$ $\displaystyle F(1-2\omega)(1-\omega)^{2}.$ (9) Inserting Eq. (7) in Eq. (2), we obtain a very simple formula for the final LO result: $\displaystyle\frac{d^{3}\Gamma_{\text{LO}}}{dx_{B}\,d\cos\theta\,d\cos\theta_{P}}$ $\displaystyle=$ $\displaystyle\frac{3}{8}F(1-\omega)^{2}D_{b}(x_{B},\mu_{F})[\sin\theta(1+P\cos\theta_{P})$ (10) $\displaystyle{}+\omega(1-\cos\theta)^{2}(1-P\cos\theta_{P})].$ Figure 2: Feynman diagrams contributing to the partial decay width of the process in Eq. (1) at NLO: (a) initial-state radiation; (b) final-state radiation; (c) vertex correction; and (d) combination of wave function renormalizations and vertex counterterm. The leptonic $W$-boson decay is not shown. The Feynman diagrams contributing to the partial decay width of the process in Eq. (1) at NLO are depicted in Fig. 2. The NLO coefficient functions of the unpolarized case may be found in Appendix A of Ref. [8] and those of the polarized case are presented in the Appendix of this paper. At NLO, the case $\lambda=+1$ is enabled by the presence of the additional spin-one gluon even for $m_{b}=0$. At this point, we compare our new analytic results with the literature. In Refs. [25, 26], process (1) was also considered at NLO using the narrow-width approximation and putting $m_{b}=0$, but treating the final state in a less differential fashion, which does not allow for the convolution with $a\to B$ FFs on the basis of the ZM-VFNS. We can compare our results for $\hat{\Gamma}_{b}^{\lambda,\text{pol}}$ with Refs. [25, 26] upon integration over $x_{b}$ in the range $0\leq x_{b}\leq 1$. Specifically, the quantities $\hat{\Gamma}_{U}^{P}$, $\hat{\Gamma}_{L}^{P}$, and $\hat{\Gamma}_{F}^{P}$ listed in Eqs. (18)–(20) of Ref. [25] (see also Eqs. (42)–(44) in Ref. [26]) are related to $\hat{\Gamma}_{b}^{\lambda,\text{pol}}$ as $\hat{\Gamma}_{b}^{0,\text{pol}}/\hat{\Gamma}_{b,\text{LO}}^{\text{unpol}}=\hat{\Gamma}_{L}^{P}$ and $\hat{\Gamma}_{b}^{\pm,\text{pol}}/\hat{\Gamma}_{b,\text{LO}}^{\text{unpol}}=(\hat{\Gamma}_{U}^{P}\pm\hat{\Gamma}_{F}^{P})/2$. With these identifications, we fully agree with Refs. [25, 26]. ## 3 Numerical results We are now in a position to explore the phenomenological consequences of our results by performing a numerical analysis. We adopt from Ref. [1] the input parameter values $G_{F}=1.1663787\times 10^{-5}$ GeV-2, $m_{W}=80.379$ GeV, $m_{t}=172.4$ GeV, $m_{B}=5.279$ GeV, $\|V_{tb}\|=1$, and $B(W^{+}\to\ell^{+}\nu_{\ell})=10.86\%$. We evaluate $\alpha_{s}^{(n_{f})}(\mu_{R})$ at NLO (NNLO) in the $\overline{\text{MS}}$ scheme using Eq. (4) of Ref. [31], retaining only the first two (three) terms on the right-hand side, with $n_{f}=5$ active quark flavors and asymptotic scale parameter $\Lambda_{\overline{\text{MS}}}^{(5)}=225$ MeV (207 MeV) adjusted such that $\alpha_{s}^{(5)}(m_{Z})=0.1179$ for $m_{Z}=91.1876$ GeV [1]. This yields $\alpha_{s}^{(5)}(m_{t})=0.1076$ (0.1076). As already mentioned in Sec. 1, we use the up-to-date $B$ FFs from Ref. [23], both at NLO and NNLO. For definiteness, we identify $\mu_{R}=\mu_{F}=\xi m_{t}$ and vary $\xi$ from 1/2 to 2 about the default value 1 to estimate the theoretical uncertainty due to the lack of knowledge of higher-order corrections. The angular dependencies at the parton level, in Eqs. (3) and (5), are passed on to the hadron level via Eq. (2). The hadron level counterparts, $d\Gamma^{\lambda}/dx_{B}$, of the coefficient functions $d\hat{\Gamma}_{a}^{\lambda}/dx_{a}$ in Eq. (5) may be projected out from the measured $\theta$ distribution $d\Gamma/(dx_{B}\,d\cos\theta)$ as explained in Sec. 4 of Ref. [8] (see Eqs. (21)–(23) therein) and thus represent physical observables by themselves. In Ref. [8], the top quarks were taken to be unpolarized, i.e., according to our present notation, $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ were considered. In the following, we complement the study of Ref. [8] by presenting predictions for $d\Gamma^{\lambda,\text{pol}}/dx_{B}$. For the sake of a coherent treatment, we also provide the analogous predictions for $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$, thus updating the analysis of Ref. [8]. Our central predictions are of NLO. To assess their significance, we compare them with the respective LO results. Our LO predictions are slightly inconsistent because they are evaluated with NLO FFs. Unfortunately, Ref. [23] does not provide a LO FF set. By the way, the same is true for Ref. [16], to which we could have resorted otherwise. On the other hand, Ref. [23] also supplies a NNLO set. While consistent NNLO predictions are unfeasible in the absence of NNLO parton-level results, this still offers us the opportunity to get a first impression of the typical magnitude of the NNLO effects. In our pseudo-NNLO analysis, besides using NNLO FFs, we also evaluate $\alpha_{s}(\mu_{R})$ at NNLO as explained above. Figure 3: LO and NLO results for (a) $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and (b) $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ with $\lambda=0,\pm 1$ as functions of $x_{B}$. $d\Gamma_{\text{NLO}}^{+,\text{unpol}}/dx_{B}$ and $d\Gamma_{\text{NLO}}^{+,\text{pol}}/dx_{B}$ are rescaled by a factor of 10 for better visibility. The theoretical uncertainties are indicated by the shaded bands. Figure 4: NLO results for (a) $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and (b) $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ with $\lambda=0,-1$, normalized to the respective default LO results, as functions of $x_{B}$. The theoretical uncertainties of the NLO results are indicated by the shaded bands. In Fig. 3, we study the $x_{B}$ dependencies of $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ for $\lambda=0,\pm 1$ at NLO and compare them with the respective LO results for $\lambda=0,-1$. As explained in Sec. 2, $d\Gamma^{+,\text{unpol}}/dx_{B}$ and $d\Gamma^{+,\text{pol}}/dx_{B}$ vanish at LO in our approximation. The theoretical uncertainties are indicated by the shaded bands. The slight inconsistency in our LO analysis mentioned above, not only affects the default predictions, but also the estimation of the theoretical uncertainty, which is expected to be slightly larger if the $\mu_{F}$ dependence is subject to LO DGLAP evolution. Comparing Figs. 3(a) and (b), we observe that $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ are very similar as for normalization and line shape. Each $x_{B}$ distribution exhibits a maximum close to $x_{B}=0.8$. Longitudinal $W$-boson helicity is favored, with $d\Gamma_{\text{LO}/\text{NLO}}^{0,\text{unpol}/\text{pol}}/dx_{B}$ being more than twice as large as $d\Gamma_{\text{LO}/\text{NLO}}^{-,\text{unpol}/\text{pol}}/dx_{B}$ for negative $W$-boson helicity $\lambda=-1$. On the other hand, positive $W$-boson helicity $\lambda=+1$ is perturbatively suppressed; rescaling $d\Gamma_{\text{NLO}}^{+,\text{unpol}/\text{pol}}/dx_{B}$ by the inverse couplant, $2\pi/\alpha_{s}^{(5)}(m_{t})$, brings it up to the level of $d\Gamma_{\text{NLO}}^{-,\text{unpol}/\text{pol}}/dx_{B}$. The NLO corrections have a significant effect on the $x_{B}$ distributions for $\lambda=0,-1$, by raising their peaks and lowering their small-$x_{B}$ tails. To render these features more visible, we present, in Fig. 4, the QCD correction ($K$) factors for $\lambda=0,-1$, which we evaluate by normalizing the NLO predictions including their theoretical-uncertainty bands relative to the default LO predictions. From Fig. 4(a) and (b), we observe that, both for unpolarized and polarized top quarks, the $K$ factors steadily increase by one order of magnitude, typically from 0.2 to 2, as $x_{B}$ runs across its range of values. We conclude from Figs. 3 and 4 that the NLO corrections are quite significant and should be taken into account in theoretical interpretations of future top-quark polarization measurements. Figure 5: NLO and pseudo-NNLO results for (a) $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and (b) $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ with $\lambda=0,\pm 1$ as functions of $x_{B}$. $d\Gamma_{\text{NLO}/\text{NNLO}}^{+,\text{unpol}}/dx_{B}$ and $d\Gamma_{\text{NLO}/\text{NNLO}}^{+,\text{pol}}/dx_{B}$ are rescaled by a factor of 10 for better visibility. The theoretical uncertainties are indicated by the shaded bands. Figure 6: Pseudo-NNLO results for (a) $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and (b) $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ with $\lambda=0,\pm 1$, normalized to the respective default NLO results, as functions of $x_{B}$. The theoretical uncertainties of the pseudo-NNLO results are indicated by the shaded bands. In Fig. 5, compare our pseudo-NNLO evaluations of $d\Gamma^{\lambda,\text{unpol}}/dx_{B}$ and $d\Gamma^{\lambda,\text{pol}}/dx_{B}$ for $\lambda=0,\pm 1$ with the respective NLO results already presented in Fig. 3. For better visibility, we also plot, in Fig. 6, the ratios of the pseudo-NNLO results including their theoretical-uncertainty bands and the default NLO predictions. We conclude from Figs. 5 and 6 that the NNLO effects are likely to be relatively modest, of the order of 10% or less. ## 4 Summary and Conclusions We studied the inclusive production of bottom-flavored hadrons from semileptonic decays of polarized top quarks at NLO in the ZM-FVNS using the narrow-width approximation for the intermediate $W$ bosons, whose helicities we distinguished. Specifically, we considered the partial decay width differential in the scaled bottom-hadron energy $x_{B}$, the azimuthal angle $\theta$ of the charged lepton in the $W$-boson rest frame, and the azimuthal angle $\theta_{P}$ of the top-quark polarization in the top-quark rest frame. In our numerical analysis, we employed up-to-date $B$ FFs, recently determined from a global fit to all available $e^{+}e^{-}$ data [23]. We thus extended our previous study in Ref. [8], which was restricted to unpolarized top quarks. In Ref. [8], we had convinced ourselves, by comparing evaluations in the GM-VFNS and ZM-VFNS, that finite-$m_{b}$ corrections may be safely neglected. On the other hand, we retained finite-$m_{B}$ corrections, which reduce the available phase space and lead to visible effects at small values of $x_{B}$. We provided full analytic results, ready to be used by the interested reader. We found the NLO QCD corrections to be quite significant, inducing a reduction in the lower $x_{B}$ range and an enhancement in the upper $x_{B}$ range, with $K$ factors ranging from 0.2 to 2. On the other hand, including partial information from NNLO, contained in the evaluation of the $B$ FFs and $\alpha_{s}(\mu_{R})$, turned out to yield only mild modifications, with the due caveat that such results suffer from a violation of renormalization group invariance already in the considered order. Our combined results, from Ref. [8] and this paper, help us to fill an important gap in the theoretical interpretation of recent measurements of the top-quark polarization and the $t\bar{t}$ spin correlations in dilepton final states at the LHC [2, 3]. In particular, it will be interesting to see if these theoretical improvements will contribute to reconciling the ATLAS [2] and CMS [3] measurements of $A_{\|\Delta\phi_{\ell\ell}\|}$ with the SM expectations. ## Acknowledgments We thank G. Kramer for useful discussions at the initial stage of this work and C. Schwanenberger for detailed information on Ref. [3]. The work of B.A.K. was supported in part by the German Federal Ministry for Education and Research BMBF through Grant No. 05H18GUCC1. The work of S.M.M.N. was supported in part by the Iran National Science Foundation INSF through Grant No. 97005414. ## Appendix In this appendix, we list the coefficient functions $d\hat{\Gamma}_{a}^{\lambda,\text{pol}}/dx_{a}$ appearing in Eq. (5) at NLO in the ZM-VFNS with $m_{b}=0$. They possess the following structure: $\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{a}^{0,\text{pol}}}{dx_{a}}$ $\displaystyle=$ $\displaystyle\frac{1}{1-2\omega}\left[\delta_{ab}\delta(1-x_{a})+\frac{\alpha_{s}(\mu_{R})}{2\pi}\left(P_{ab}(x_{a})\ln\frac{m_{t}^{2}}{\mu_{F}^{2}}+C_{F}C_{a}^{0,\text{pol}}(x_{a})\right)\right],$ $\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{a}^{-,\text{pol}}}{dx_{a}}$ $\displaystyle=$ $\displaystyle\frac{-2\omega}{1-2\omega}\left[\delta_{ab}\delta(1-x_{a})+\frac{\alpha_{s}(\mu_{R})}{2\pi}\left(P_{ab}(x_{a})\ln\frac{m_{t}^{2}}{\mu_{F}^{2}}+C_{F}C_{a}^{-,\text{pol}}(x_{a})\right)\right],$ $\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{a}^{+,\text{pol}}}{dx_{a}}$ $\displaystyle=$ $\displaystyle\frac{4\omega}{(1-2\omega)(1-\omega)^{2}}\frac{\alpha_{s}(\mu_{R})}{2\pi}C_{F}C_{a}^{+,\text{pol}}(x_{a}),$ (11) where $\displaystyle P_{qq}(x)$ $\displaystyle=$ $\displaystyle C_{F}\left(\frac{1+x^{2}}{1-x}\right)_{+},$ $\displaystyle P_{gq}(x)$ $\displaystyle=$ $\displaystyle C_{F}\frac{1+(1-x)^{2}}{x}$ (12) are the timelike $q\to q$ and $q\to g$ splitting functions at LO. For $a=b$, we have $\displaystyle C_{b}^{0,\text{pol}}(x)$ $\displaystyle=$ $\displaystyle-\delta(1-x)\big{[}2\ln\omega\ln(1-\omega)+4\mathop{\mathrm{Li}}\nolimits_{2}(\omega)+\frac{2\omega}{1-\omega}\ln\omega+6\big{]}+2(1+x^{2})\bigg{(}\frac{\ln(1-x)}{1-x}\bigg{)}_{+}$ (13) $\displaystyle{}+2\frac{1+x^{2}}{(1-x)_{+}}\ln(x(1-\omega))+\frac{8(x-2)(1-x)^{2}}{x^{2}(Sx^{2}-2x+2)}+2\omega(1-x)-\frac{12}{x}+\frac{8}{x^{2}}+\frac{6}{1-x}-\frac{(1+x)^{2}}{1-x}-\frac{1+x^{2}}{1-x}R_{1}$ $\displaystyle{}+\bigg{(}1-\|1-2x+2Sx^{2}\|\bigg{)}\bigg{(}6\frac{1-x}{x^{2}}-\frac{1}{(1-x)(1-\omega)}+6\frac{(1-x)^{2}(x-2)^{2}}{x^{2}(Sx^{2}-2x+2)^{2}}+\frac{1}{2x^{2}(Sx^{2}-2x+2)}\big{[}9x^{3}$ $\displaystyle{}-57x^{2}+97x-\frac{1}{1-x}-47\big{]}\bigg{)}+\frac{R_{2}}{S^{3/2}(Sx^{2}-2x+2)^{5/2}}\bigg{(}x^{3}S^{4}(-x^{3}+7x^{2}-22x+10+\frac{2}{1-x})+xS^{3}(x^{4}$ $\displaystyle{}+4x^{3}+25x^{2}-32x+12)-2(1-x)^{2}+S^{2}(1-x)(5x^{3}+21x^{2}-12x+4)-S(1-x)(9x^{2}-4x-4)\bigg{)},$ $\displaystyle C_{b}^{-,\text{pol}}(x)$ $\displaystyle=$ $\displaystyle-\delta(1-x)\big{[}2\ln\omega\ln(1-\omega)+4\mathop{\mathrm{Li}}\nolimits_{2}(\omega)+\frac{2\omega}{1-\omega}\ln\omega+6+\frac{1-\omega}{\omega}\ln(1-\omega)\big{]}+2(1+x^{2})\bigg{(}\frac{\ln(1-x)}{1-x}\bigg{)}_{+}$ (14) $\displaystyle{}+2\frac{1+x^{2}}{(1-x)_{+}}\ln(x(1-\omega))-\frac{1+x^{2}}{2(1-x)}R_{1}+\frac{(1-S)^{2}}{\omega S}+\frac{\omega+Sx}{\omega}-\frac{\omega}{2S(1-x)}+\frac{(1-S-Sx)}{S}\ln(1-2Sx)$ $\displaystyle{}-\frac{3S(2x-1)+6(1-x)}{2S(Sx^{2}-2x+2)^{2}}+\frac{S^{2}(1+3x)+4x\omega-3S}{2\omega S(Sx^{2}-2x+2)}+B_{3}\|1-2x+2Sx^{2}\|+B_{1}\ln\|2Sx^{2}-2x+1\|$ $\displaystyle{}+\frac{R_{3}}{S^{2}\sqrt{\omega}}B_{2}-\frac{R_{2}}{2\sqrt{Sx^{2}-2x+2}}\bigg{(}x^{2}\sqrt{S}+\frac{(S-1)x}{\sqrt{S}}-\frac{2\sqrt{S}}{1-x}+\frac{2S^{2}-5S+7}{S^{\frac{3}{2}}}$ $\displaystyle{}+\frac{Sx(4S^{2}-23S+16)-S(28-19S)+6}{S^{\frac{5}{2}}(Sx^{2}-2x+2)}-\frac{3Sx(12-9S+S^{2})+15S(S-2)+12(1-x)}{S^{\frac{5}{2}}(Sx^{2}-2x+2)^{2}}\bigg{)},$ $\displaystyle C_{b}^{+,\text{pol}}(x)$ $\displaystyle=$ $\displaystyle\frac{(1+x^{2})R_{1}}{1-x}S^{2}-\frac{2(1+x)S^{3}}{\omega}+2S\big{[}1+Sx+(1-S-Sx)\ln(1-2Sx)\big{]}$ (15) $\displaystyle{}-\frac{(17-21x)S^{3}+(20x-27)S^{2}+4(2-x)S}{\omega(Sx^{2}-2x+2)}+\frac{6(1-x)S-3(1-2x)S^{2}}{(Sx^{2}-2x+2)^{2}}+\frac{\omega S}{1-x}+\frac{2R_{3}}{\sqrt{\omega}}B_{2}$ $\displaystyle{}-2S^{2}B_{3}\|1-2x+2Sx^{2}\|+2S^{2}B_{1}\ln\|2Sx^{2}-2x+1\|-\frac{R_{2}(1-x)\sqrt{S}}{x^{3}\sqrt{Sx^{2}-2x+2}}\bigg{(}11+7x^{2}-16x$ $\displaystyle{}+\frac{6(2-x)(1-x)^{3}}{(2-2x+Sx^{2})^{2}}+\frac{16x^{3}-58x^{2}+69x-28}{2-2x+Sx^{2}}+\frac{Sx^{2}(x^{2}+5x-4)}{1-x}+\frac{S^{2}x^{4}(1+x^{2})}{(1-x)^{2}}\bigg{)},$ where $S=(1-\omega)/2$, $\displaystyle R_{1}$ $\displaystyle=$ $\displaystyle\ln\bigg{[}(1-S)x^{2}-x+\frac{1}{2}+\frac{1}{2}\|2Sx^{2}-2x+1\|\bigg{]},$ $\displaystyle R_{2}$ $\displaystyle=$ $\displaystyle\ln\bigg{[}(1-x)(1-3Sx)+Sx^{2}(1-2Sx)+\sqrt{S(Sx^{2}-2x+2)}\|2Sx^{2}-2x+1\|\bigg{]}-$ $\displaystyle\ln\bigg{[}1+(S-1)x+\sqrt{S(Sx^{2}-2x+2)}\bigg{]},$ $\displaystyle R_{3}$ $\displaystyle=$ $\displaystyle\ln\big{[}1-x(1-\sqrt{\omega})\big{]}-\ln\big{\|}1-x(1+\sqrt{\omega})\|,$ $\displaystyle B_{1}$ $\displaystyle=$ $\displaystyle x-\frac{1-S}{S}-\frac{1-x}{2S(2-2x+Sx^{2})^{2}}\big{[}5S^{2}x^{2}-2(1-x)-2S(x^{2}+3x-3)\big{]},$ $\displaystyle B_{2}$ $\displaystyle=$ $\displaystyle-\omega S+xS^{2}+\frac{Sx(-4S^{2}+7S-2)+S(8-10S)-2}{2(Sx^{2}-2x+2)}-\frac{\omega(S^{2}x-4Sx+3S+2x-2)}{(Sx^{2}-2x+2)^{2}},$ $\displaystyle B_{3}$ $\displaystyle=$ $\displaystyle\frac{1}{2S(1-x)}+\frac{1}{1-2Sx}+\frac{6(1-x)-3S(1-2x)}{2S(Sx^{2}-2x+2)^{2}}+\frac{S^{2}(3x-5)+S(11-2x)-4}{2\omega S(Sx^{2}-2x+2)}.$ (16) For $a=g$, we have $\displaystyle C_{g}^{0,\text{pol}}(x)$ $\displaystyle=$ $\displaystyle\frac{1+(1-x)^{2}}{x}\bigg{(}-R_{1}+2\ln[x(1-\omega)(1-x)]\bigg{)}+\bigg{(}8S-x-26+\frac{13}{S}+\frac{26S^{2}-38S+13}{2S^{3}x^{2}}$ (17) $\displaystyle{}+\frac{-44S^{2}+88S-35}{2xS^{2}}\bigg{)}R_{4}+\bigg{(}\frac{1-6S}{2(2Sx-1)}+9\frac{1-S}{2Sx}+\frac{-4S^{2}+25S-13}{2S^{2}x^{2}}+\frac{\omega}{2(1-2Sx)^{2}}\bigg{)}\|2Sx^{2}-2x+1\|$ $\displaystyle{}+2(7-6S)+(1-4S)x+7\frac{7S-5}{2Sx}+\frac{4S^{2}-25S+13}{2S^{2}x^{2}},$ $\displaystyle C_{g}^{-,\text{pol}}(x)$ $\displaystyle=$ $\displaystyle\frac{1+(1-x)^{2}}{2x}\bigg{(}4\ln[x(1-\omega)(1-x)]-R_{1}\bigg{)}+\frac{1-6S}{4\omega}x-\frac{\omega^{2}}{32S^{3}x^{2}(1-2Sx)^{2}}-\frac{(10S-7)\omega}{32S^{3}x^{2}(1-2Sx)}$ (18) $\displaystyle{}+\frac{1+5S-8S^{2}}{2S\omega}-A_{3}R_{4}+\frac{\|2Sx^{2}-2x+1\|}{8\omega S^{2}x^{2}}A_{5}+\frac{A_{2}}{2S^{3}x^{2}}\ln(1-2Sx)+\frac{R_{3}A_{4}}{2\sqrt{\omega}S^{3}x^{2}}-A_{1}\frac{\ln\|2Sx^{2}-2x+1\|}{2x^{2}S^{3}}$ $\displaystyle{}+\frac{1}{16\omega xS^{2}}(48S^{3}+72S^{2}-50S-5)+\frac{1}{16\omega S^{3}x^{2}}(32S^{3}-88S^{2}+44S-3),$ $\displaystyle C_{g}^{+,\text{pol}}(x)$ $\displaystyle=$ $\displaystyle\frac{1}{2Sx^{2}}\Big{[}\frac{2R_{3}}{\sqrt{\omega}}A_{4}+2x(1+(1-x)^{2})S^{3}R_{1}+4S^{3}x^{2}A_{3}R_{4}+2A_{2}\ln(1-2Sx)-2A_{1}\ln\|2Sx^{2}-2x+1\|$ (19) $\displaystyle{}-\frac{SA_{5}}{2\omega}\|2Sx^{2}-2x+1\|+\frac{S}{4\omega}\bigg{(}4S^{2}(1+2S)x^{3}+8S(4S^{2}-7S+1)x^{2}+\frac{8\omega^{2}x}{S(1-4S)}-(96S^{2}-70S+5)x$ $\displaystyle{}+\frac{32S^{2}-12S-3}{S}-\frac{40S^{3}-68S^{2}+38S-7}{2S(1-2Sx)}+\frac{8S^{3}-12S^{2}+6S-1}{2S(1-2Sx)^{2}}\bigg{)}\Big{]},$ where $\displaystyle R_{4}$ $\displaystyle=$ $\displaystyle\ln\bigg{[}1-S(-2Sx^{2}+2x+1-\|2Sx^{2}-2x+1\|)\bigg{]}-\ln[1-2Sx],$ $\displaystyle A_{1}$ $\displaystyle=$ $\displaystyle 2x^{2}(x-1)S^{3}+(-2x^{2}+5x+1)S^{2}-2(2+x)S+2,$ $\displaystyle A_{2}$ $\displaystyle=$ $\displaystyle x(x^{2}-2)S^{3}+(-2x^{2}+5x+1)S^{2}-2(2+x)S+2,$ $\displaystyle A_{3}$ $\displaystyle=$ $\displaystyle\frac{2+x^{2}}{2x}-\frac{5x^{2}+5x+1}{2Sx^{2}}-\frac{7-S(8+15x)}{4x^{2}S^{3}},$ $\displaystyle A_{4}$ $\displaystyle=$ $\displaystyle 2x(2-x^{2})S^{3}+(2x^{2}-7x-4)S^{2}+2(x+3)S-2,$ $\displaystyle A_{5}$ $\displaystyle=$ $\displaystyle 20S-11-Sx(10S-7)-\frac{\omega(7-10S)}{2(1-2Sx)}+\frac{\omega^{2}}{2(1-2Sx)^{2}}.$ (20) Adding Eqs. (13)–(15) for $a=b$ and Eqs. (17)–(19) for $a=g$ according to Eq. (6), we obtain the respective coefficient functions pertaining to the case where $\theta$ is integrated over. Specifically, we have $\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{b}^{\text{pol}}}{dx_{b}}$ $\displaystyle=$ $\displaystyle\delta(1-x_{b})+\frac{\alpha_{s}(\mu_{R})}{2\pi}\left(P_{qq}(x_{b})\ln\frac{m_{t}^{2}}{\mu_{F}^{2}}+C_{F}C_{b}^{\text{pol}}(x_{b})\right),$ $\displaystyle\frac{1}{\hat{\Gamma}_{b,\text{LO}}^{\text{pol}}}\,\frac{d\hat{\Gamma}_{g}^{\text{pol}}}{dx_{g}}$ $\displaystyle=$ $\displaystyle\frac{\alpha_{s}(\mu_{R})}{2\pi}\left(P_{gq}(x_{g})\ln\frac{m_{t}^{2}}{\mu_{F}^{2}}+C_{F}C_{g}^{\text{pol}}(x_{g})\right),$ (21) where $\displaystyle C_{b}^{\text{pol}}(x)$ $\displaystyle=$ $\displaystyle\delta(1-x)\big{[}-2\ln\omega\ln(1-\omega)+\frac{2(1-\omega)}{1-2\omega}\ln(1-\omega)-4\mathop{\mathrm{Li}}\nolimits_{2}(\omega)-\frac{2\omega}{1-\omega}\ln\omega-6\big{]}$ $\displaystyle{}+2(1+x^{2})\bigg{(}\frac{\ln(1-x)}{1-x}\bigg{)}_{+}+2\frac{1+x^{2}}{(1-x)_{+}}\ln(x(1-\omega))+\frac{1-S}{S(Sx^{2}-2x+2)}-\frac{2}{1-4S}-\frac{\omega}{S(1-x)}-1-x$ $\displaystyle{}-\frac{1+x^{2}}{1-x}R_{1}+\bigg{(}\frac{1}{S(1-x)}+\frac{4\omega}{(1-4S)(1-2Sx)}-\frac{1-S}{S(Sx^{2}-2x+2)}\bigg{)}\|2Sx^{2}-2x+1\|$ $\displaystyle{}-\frac{R_{2}}{\sqrt{S(Sx^{2}-2x+2)}}\bigg{(}\frac{1}{S}+\frac{2}{1-4S}+\frac{(13S-5-4S^{2})x}{1-4S}-\frac{Sx(x^{2}-x+2)}{1-x}-\frac{(1-S)[1-(1-S)x]}{S(Sx^{2}-2x+2)}\bigg{)},$ $\displaystyle C_{g}^{\text{pol}}(x)$ $\displaystyle=$ $\displaystyle\frac{1+(1-x)^{2}}{x}\bigg{(}-R_{1}+2\ln[x(1-\omega)(1-x)]\bigg{)}-\bigg{(}x-\frac{1+\omega^{2}}{4S^{3}x^{2}}-\frac{8S^{2}-6S+3}{S(4S-1)}-\frac{2-\omega^{2}(2\omega-5)}{2S^{2}(1-4S)x}\bigg{)}R_{4}$ (22) $\displaystyle{}+\bigg{(}\frac{S-1}{2S^{2}x^{2}}+\frac{\omega}{2(1-2Sx)^{2}}-\frac{12S^{2}-15S+7}{2S(1-4S)x}-\frac{24S^{2}-26S+9}{2(1-4S)(1-2Sx)}\bigg{)}\|2Sx^{2}-2x+1\|+x+\frac{1-S}{2S^{2}x^{2}}$ $\displaystyle{}-\frac{2}{1-4S}+\frac{12S^{2}-7S+5}{2S(1-4S)x}.$ ## References * [1] P. 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# Practical Utility PV Multilevel Inverter Solutions John Buczek and Viktor Ivankevych <EMAIL_ADDRESS> <EMAIL_ADDRESS> _Abstract–_ Multilevel inverters are used to improve power quality and reduce component stresses. This paper describes and compares two multilevel cascaded three phase inverter implementations with two different modulation techniques: Phase Shifted Pulse Width Modulation, and Nearest Level Control. Further analysis will show required number of inverter levels with respect to modulation techniques to provide desired power and power quality to resistive load or grid. Cascaded inverter will be designed and simulated to draw power from PV cells. ## I Introduction A multi-level inverter is a power electronic system that synthesizes a desired voltage output from several levels of DC voltages as inputs [9]. Today, there are many different topologies of multilevel converters including, but not limited to, Diode-Clamped, Flying Capacitor, and Cascade H-bridge (CHB). While the topologies may be different, they all offer similar beneficial features. For sinusoidal outputs, multilevel converters improve their output voltage in quality as the number of levels of the converter increase, thus decreasing the Total Harmonic Distortion (THD) [6]. For this reason and others, multilevel converters have been used for high power photovoltaic (PV) inversion, electric motor drivers in electric vehicles, and other research and commercial applications [6, 3, 2, 3, 4]. Although, technological problems such as reliability, efficiency, the increase of the control complexity, and the design of simple modulation methods have slowed down the application of multilevel converters [6]. Figure 1: 5 Level Cascade H-bridge Converter [2] Figure 1 shows a 5 level CHB converter. As can be seen, CHB converters consist of multiple MOSFET (or equivalent) H-bridges that are connected in series. Each H-bridge having its own isolated DC voltage source. For the shown 5 level case, it requires two H-bridges that can be configured to output the 5 levels: +V, +V/2, 0, -V/2, and -V. While the theory of adding and subtracting isolated voltage sources is simple, such is harder to do in practice. Common methods include using isolated DC-DC converters such as flyback and forward converters that have transformers with multiple secondary windings [6]. Others use individual, or multiple isolated sets of PVs that power individual H-bridges [9, 2]. This paper uses two of the common modulation techniques for CHB converters, Phase Shifted PWM (PSPWM) and Nearest Level Control (NLS), to propose designs for a utility 3 phase PV inverter. Our designs will display many of the common advantages and disadvantages of the different modulation techniques for CHB. Our design requirements were to develop a 3 phase utility PV CHB inverter to supply 125kW at $480V_{L-L}^{RMS}$ with a THD below 5%. ## II Nearest Level Switching The Nearest-Level Switching control for a multilevel converter compares a control sine wave to DC voltage levels. For a Cascading Multi-level Converter consisting of _N_ H-bridges, each trigger when the voltage control sinusoidal is greater than their respective threshold given by the equation: $\displaystyle V_{thresh}(i)=V_{pk}\frac{2i-1}{2N}$ (1) Where _N_ is the number of H-bridges equal to $N=\frac{L-1}{2}$ for an _L_ level Cascade H-bridge (CHB), and _i_ is the switch number which ranges from 0 to _N-1_. Figure 2: Nearest Level Switching Waveform Synthesis [6] Figure 2 shows the waveform for the Nearest Level Switching. It should be noted that for every DC voltage source, the threshold voltage is at half of the DC value. Additionally, the peak output voltage is equal to $NV_{DC}$. Figure 3: 9-Level NLS CHB PWM Waveforms Figure 3 shows the PWM waveforms for the top H-bridge of a 9-level NLS CHB. The PWM for switches 1 and 3 are not the same, as is the case for switches 2 and 4. For a single H-bridge inverter, the diagonal switches could have the same PWM input, but that is not possible for multilevel converters. In order to achieve all of the voltage steps, either voltages need to be added and subtracted, or the outputs of H-bridges need to be shorted [4]. Our approach was to use the shorting technique. This essentially adds an equivalent of a zero state in Space Vector Modulation [6]. This output shorting was implemented by making the PWM for switch 3 the inverse of PWM 2 and PWM 4 the inverse of PWM 1. This shorted the outputs through switches 4 and 3. In practice, both zero states of switches 3,4 as well as switches 1,2 should be used in order to have more even wear on individual switches. ## III Nearest Level Switching Resistive Load RMS and THD Calculations For a cascading H-Bridge Multilevel Inverter with $L$ levels, using the Nearest-Level-Switching technique, the switching point $\alpha_{i}$ for level $i=0$ to $i=\frac{L-1}{2}$, are given by the equation: $\displaystyle\alpha_{i}=sin^{-1}\left(\frac{2i+1}{L-1}\right)$ (2) ### III-A Root Mean Squared From [8], the equation for calculating the RMS ($X^{RMS}$) of a function $x(t)$ is given as: $\displaystyle X^{RMS}=\sqrt{\frac{1}{T}\int_{0}^{T}(x(t))^{2}dt}$ (3) #### III-A1 3-Level For the simple 3-level Inverter with NLS, the RMS voltage of a resistive load can be calculated to be: $\displaystyle V_{3-L}^{RMS}$ $\displaystyle=\sqrt{\frac{1}{\pi}\int_{\alpha_{0}}^{\pi-\alpha_{0}}(V_{m})^{2}dt}$ $\displaystyle=V_{m}\sqrt{\frac{1}{\pi}\left[t\right]_{\alpha_{0}}^{\pi-\alpha_{0}}}$ $\displaystyle=V_{m}\sqrt{\frac{1}{\pi}\left(\pi-\alpha_{0}-\alpha_{0}\right)}$ $\displaystyle=V_{m}\sqrt{1-\frac{2}{\pi}\alpha_{0}}$ For $\alpha_{0}=sin^{-1}\left(\frac{1}{2}\right)=\frac{\pi}{6}$ $\displaystyle V_{3-L}^{RMS}$ $\displaystyle=V_{m}\sqrt{\frac{2}{3}}$ ### III-B 5-Level For the simple 5-level Inverter with NLS, the RMS voltage of a resistive load can be calculated to be: $\displaystyle V_{5-L}^{RMS}$ $\displaystyle=\biggl{\\{}\frac{1}{\pi}\biggl{(}\int_{\alpha_{0}}^{\alpha_{1}}\biggl{(}\frac{V_{m}}{2}\biggl{)}^{2}dt+\int_{\alpha_{1}}^{\pi-\alpha_{1}}\biggl{(}V_{m}\biggl{)}^{2}dt$ $\displaystyle+\int_{\pi-\alpha_{1}}^{\pi-\alpha_{0}}\biggl{(}\frac{V_{m}}{2}\biggl{)}^{2}dt\biggl{)}\biggr{\\}}^{1/2}$ $\displaystyle=V_{m}\sqrt{\frac{1}{\pi}\left(\left[\frac{t}{4}\right]_{\alpha_{0}}^{\alpha_{1}}+\left[t\right]_{\alpha_{1}}^{\pi-\alpha_{1}}+\left[\frac{t}{4}\right]_{\pi-\alpha_{1}}^{\pi-\alpha_{0}}\right)}$ $\displaystyle=V_{m}\sqrt{\frac{1}{\pi}\left(\pi-\frac{1}{2}\alpha_{0}-\frac{3}{2}\alpha_{1}\right)}$ $\displaystyle=V_{m}\sqrt{1-\frac{1}{2\pi}\alpha_{0}+\frac{3}{2\pi}\alpha_{1}}$ For $\alpha_{0}=sin^{-1}\left(\frac{1}{4}\right)$ and $\alpha_{1}=sin^{-1}\left(\frac{3}{4}\right)$ $\displaystyle V_{5-L}^{RMS}$ $\displaystyle\approx V_{m}0.7449$ ### III-C 7-Level For the simple 7-level Inverter with NLS, the RMS voltage of a resistive load can be calculated to be: $\displaystyle V_{7-L}^{RMS}$ $\displaystyle=\biggl{\\{}\frac{1}{\pi}\biggl{(}\int_{\alpha_{0}}^{\alpha_{1}}\left(\frac{V_{m}}{3}\right)^{2}dt+\int_{\alpha_{1}}^{\alpha_{2}}\left(\frac{2V_{m}}{3}\right)^{2}dt$ $\displaystyle+\int_{\alpha_{2}}^{\pi-\alpha_{2}}\left(V_{m}\right)^{2}dt+\int_{\pi-\alpha_{2}}^{\pi-\alpha_{1}}\left(\frac{2V_{m}}{3}\right)^{2}dt$ $\displaystyle+\int_{\pi-\alpha_{1}}^{\pi-\alpha_{0}}\left(\frac{V_{m}}{3}\right)^{2}dt\biggl{)}\biggl{\\}}^{1/2}$ $\displaystyle=V_{m}\biggl{\\{}\frac{1}{\pi}\biggl{(}\left[\frac{t}{3}\right]_{\alpha_{0}}^{\alpha_{1}}+\left[\frac{2t}{3}\right]_{\alpha_{1}}^{\alpha_{2}}$ $\displaystyle+\left[t\right]_{\pi-\alpha_{2}}^{\alpha_{2}}+\left[\frac{2t}{3}\right]_{\pi-\alpha_{1}}^{\pi-\alpha_{2}}+\left[\frac{t}{3}\right]_{\pi-\alpha_{1}}^{\pi-\alpha_{0}}\biggl{)}\biggl{\\}}^{1/2}$ $\displaystyle=V_{m}\sqrt{\frac{1}{\pi}\left(\pi-\frac{2}{9}\alpha_{0}-\frac{6}{9}\alpha_{1}-\frac{10}{9}\alpha_{2}\right)}$ $\displaystyle=V_{m}\sqrt{1-\frac{2}{9\pi}\alpha_{0}-\frac{6}{9\pi}\alpha_{1}-\frac{10}{9\pi}\alpha_{2}}$ For $\alpha_{0}=sin^{-1}\left(\frac{1}{6}\right)$, $\alpha_{1}=sin^{-1}\left(\frac{3}{6}\right)$, and $\alpha_{2}=sin^{-1}\left(\frac{5}{6}\right)$ $\displaystyle V_{7-L}^{RMS}$ $\displaystyle\approx V_{m}0.7217$ ### III-D L-Level From the previous derivations and Equation 2, a pattern for the RMS voltage can be seen. For a multilevel cascade H-Bridge Inverter with L levels, the equation for the RMS voltage of a resistive load can be given by: $\displaystyle V_{L}^{RMS}$ $\displaystyle=V_{m}\sqrt{1-\sum^{N-1}_{i=0}\left(\frac{2(2i+1)}{\piN^{2}}\alpha_{i}\right)}$ $\displaystyle=V_{m}\sqrt{1-\frac{2}{\pi N^{2}}\sum^{N-1}_{i=0}\left(sin^{-1}\left(\frac{2i+1}{L-1}\right)\left(2i+1\right)\right)}$ (4) ### III-E Fourier Series Expansion The Fourier Series Expansion was also performed on the ideal cascade H-bridge Inverter. In the ideal case with a purely resistive load, the output waveform has odd symmetry and quarter-wavelength symmetry inherently. From [8], the Fourier Series Expansion Coefficients for such symmetry are given as: $\displaystyle a_{h}$ $\displaystyle=0$ $\displaystyle b_{h}$ $\displaystyle=\frac{4}{\pi}\int_{0}^{\frac{\pi}{2}}\left(x(t)sin(h\omega t)\right)d\omega t$ #### III-E1 3-Level For the simple 3-level Inverter with NLS, the Fourier Series Expansion Coefficients of a resistive load can be expressed as: $\displaystyle b_{h}(3)$ $\displaystyle=\frac{4}{\pi}\int_{\alpha_{0}}^{\frac{\pi}{2}}\left(V_{m}sin(h\omega t)\right)d\omega t$ $\displaystyle=\frac{4V_{m}}{\pi h}\left[-cos(h\omega t)\right]^{\frac{\pi}{2}}_{\alpha_{0}}$ $\displaystyle=\frac{4V_{m}}{\pi h}cos(h\alpha_{0})$ ### III-F 5-Level For the simple 3-level Inverter with NLS, the Fourier Series Expansion Coefficients of a resistive load can be expressed as: $\displaystyle b_{h}(5)$ $\displaystyle=\frac{4}{\pi}\biggl{(}\int_{\alpha_{0}}^{\frac{\pi}{2}}\left(\frac{V_{m}}{2}sin(h\omega t)\right)d\omega t$ $\displaystyle+\int_{\alpha_{1}}^{\frac{\pi}{2}}\left(\frac{V_{m}}{2}sin(h\omega t)\right)d\omega t\biggl{)}$ $\displaystyle=\frac{2V_{m}}{\pi h}\left(\left[-cos(h\omega t)\right]_{\alpha_{0}}^{\frac{\pi}{2}}+\left[-cos(h\omega t)\right]_{\alpha_{1}}^{\frac{\pi}{2}}\right)$ $\displaystyle=\frac{2V_{m}}{\pi h}\left(cos(h\alpha_{0})+cos(h\alpha_{1})\right)$ $\displaystyle=\frac{2V_{m}}{\pi h}\left(cos(h\alpha_{0})+cos(h\alpha_{1})\right)$ ### III-G L-Level For a multilevel cascade H-Bridge Inverter with L levels, the Fourier Series Expansion Coefficients of a resistive load can be expressed as: $\displaystyle b_{h}(L)$ $\displaystyle=\frac{4}{\pi}\left(\sum^{N-1}_{i=0}\int_{\alpha_{i}}^{\frac{\pi}{2}}\left(\frac{V_{m}}{N}sin(h\omega t)\right)d\omega t\right)$ $\displaystyle=\frac{4V_{m}}{\pi hN}\sum^{N-1}_{i=0}\left(cos\left(h\alpha_{i}\right)\right)$ This can be simplified using the trigonometry identity: $\displaystyle cos(sin^{-1}(x))=\sqrt{1-x^{2}}$ From Equation 2, the Fourier Series first Coefficient can be simplified to be: $\displaystyle b_{1}(L)$ $\displaystyle=\frac{4V_{m}}{\pi N}\sum^{N-1}_{i=0}\left(\sqrt{1-\left(\frac{(2i+1)}{L-1}\right)^{2}}\right)$ (5) ### III-H Total Harmonic Distortion From [8], the equation for the Total Harmonic Distortion (THD) is given by: $\displaystyle THD=\frac{\sqrt{(X^{RMS})^{2}-(X^{RMS}_{1})^{2}}}{X^{RMS}_{1}}$ (6) From Equation 5, the equation for the RMS magnitude of the first harmonic can be calculated as: $\displaystyle V^{RMS}_{1}(L)=\frac{4V_{m}}{\pi N\sqrt{2}}\sum^{N-1}_{i=0}\left(\sqrt{1-\left(\frac{(2i+1)}{L-1}\right)^{2}}\right)$ (7) Equations 7, 4, and 6 were combined and plotted using python (Appendix A) to calculate the THD of an L-level CHB inverter with a resistive load. At the same time, PSIM simulations for the same number of levels were run with resistive loads and compared against one another. Table I: PSIM Simulated and Theoretically Calculated THD Levels (L) \| PSIM THD (%) \| Calculated THD(%) ---\|---\|--- 3 \| 31.0512 \| 31.08419 5 \| 17.5799 \| 17.6012 7 \| 12.2126 \| 12.2272 9 \| 9.35322 \| 9.363669 11 \| 7.58321 \| 7.587252 13 \| 6.3712 \| 6.378124 15 \| 5.49467 \| 5.502021 17 \| 4.83621 \| 4.837995 19 \| 4.31314 \| 4.317328 21 \| 3.89612 \| 3.89809 23 \| 3.55342 \| 3.553263 25 \| 3.26193 \| 3.264629 27 \| 3.017 \| 3.01947 Table I shows the THD results from both the Theoretical Calculated THD and the PSIM simulated THD for 3 levels to 27 levels. The two data sets are consistent with each other. From the Table, as the number of levels increases, the THD decreases. This quantitatively confirms that the output sinusoidal quality improves with more and more additional levels of the CHB. ## IV Nearest Level Switching Simulation Design As previously mentioned, the design goals were to produce a 60 Hz 3 phase inverter at $480V_{L-L}^{RMS}$ with a real power output of 125 kW. The THD of our inverter also needed to be below 5 %. These specifications were consistent with other commercially available PV inverters [5]. Figure 4: Nearest Level Switching Simulated H-bridge and Buck Converter Figure 4 shows the PSIM simulation for the NLS-CHB inverter individual H-bridge and buck converter. From the previous section, it was found that for a resistive load, the theoretical THD of a 27 level NLS-CHB was approximately 3%, so a 27 level inverter was chosen. From the number of levels and the peak grid voltage of $480V_{L-L}^{RMS}$, the individual H-bridge voltage was calculated to be $V_{dc}=480\sqrt{2/3}/13\approx 30.15V$. Our simulations did not include PV maximum power point tracking, so our simulations used a constant voltage source equal to the maximum voltage output of a single PV from the datasheet in [10]. From [7], for a buck converter, we know: $\displaystyle V_{o}$ $\displaystyle=DVs$ (8) $\displaystyle\Delta i_{L}$ $\displaystyle\approx\frac{V_{l}\Delta t}{L}$ $\displaystyle\approx\frac{DV_{s}(1-D)T}{L}$ $\displaystyle L$ $\displaystyle\approx\frac{DV_{s}(1-D)T}{\Delta i_{L}}$ (9) $\displaystyle\Delta V_{c}$ $\displaystyle\approx\frac{T\Delta i_{L}}{8C}$ (10) We expect $I_{Lpk}\approx 30A$. Let $f_{s}=200kHz$, $V_{s}=48.9V$, $V_{o}=480\sqrt{2/3}/13\approx 30.15V$,$\Delta V_{c}=4V$ , and $\Delta i_{L}=5\%=6A$. Then: $\displaystyle D$ $\displaystyle=0.6165$ $\displaystyle L$ $\displaystyle=9.633\mu H\approx 10\mu H$ $\displaystyle C$ $\displaystyle=937nF\approx 1\mu F$ The capacitor and inductor were intentionally kept small, but above critical values such to reduce their equivalent impedance when placed in series. Once the buck converters were designed, a single phase was simulated against a grid ac voltage source with a phase shift of -2.5 degrees in order to determine the desired cutoff frequency for the output filter. From our simulations, a cutoff frequency of approximately 700Hz was selected based on the output current harmonics. A simple LC low pass filter was chosen, with a cutoff frequency given by [7] as: $\displaystyle f_{c}=\frac{1}{2\pi\sqrt{LC}}$ (11) An inductor $L_{f}=1mH$ and a capacitor $C_{f}=50\mu F$ were selected. ## V Nearest Level Switching Simulation Simulation ### V-A NLS Results The NLS-CHB was first simulated with a single phase using ideal switches and a phase angle of -2.5 degrees from the grid voltage source. The simulation was able to reach a power level of 8.5 kW per phase at the desired line to neutral voltage level. The output current RMS was 30.66A and the current THD was 3.02%. At this current level, using the PV arrays from [10], we would need two in parallel going connected to each buck converter. The NLS-CHB was then simulated with a single phase using the PSIM default lossy switching models for NMOS MOSFETs. In addition, all reactive components were given series resistance values of 50m $\Omega$. The circuit was simulated for 0.5s and the output THD increased to 4.486% at a power factor of 99.56%. Figure 5: Nearest Level Switching PSIM 3-Phases (Y Connection) Figure 6: Nearest Level Switching 3 Phase Voltage and Current Waveforms Figure 7: Nearest Level Switching 3 Phase Voltage and Current Characteristics The NLS-CHB was finally simulated with all three phases, each with a respective phase shift of -2.5 degrees. Figure 5 shows the PSIM file of the NLS-CHB three phase simulation. Figures 6 and 7 show the output waveforms and characteristics respectively. The desired output voltage of $480V_{L-L}^{RMS}$ was achieved. The output real power was approximately 25kW with an output current THD of approximately 3.12%. From our simulation, in order to achieve our design requirements, 5 of such three phase inverters would need to be placed in parallel. Figure 8: Nearest Level Switching H-bridge Switch Stresses Figure 8 displays the individual switch stresses for the H-Bridge MOSFETs. Each MOSFET in each H-bridge needed to conduct a peak of 50A and block a peak of 40V. Base on design experience, voltage and current ratings are desired to be increased by 150 % to maintain safe operation at extreme performance cases. For a $480V_{L-L}^{RMS}$ system, these are comparatively small values [4]. While in this topology there are more switches, each switch experiences less stress. This leads to the possibility of using more efficient switches that have lower voltage and current ratings. Our simulation did kept each H-bridge at a constant duty ratio. The duty cycle for individual H-Bridges would need to be different, or a more complex controller would need to be used in order to evenly apply duty cycles to all the H-bridges. ### V-B NLS Comments The benefits that the Nearest Level Switching brings include greater efficiency and the ability to use additional active elements (additional H-bridges) in order to improve output waveform quality [3, 4]. From our analysis it has been shown that by increasing the number of H-bridges and levels, the THD of the output waveform decreases, thus making the output closer to a pure sinusoid. Due to the slow switching nature of the NLS technique, all switches turn on and off only once for the fundamental period of the output waveform, reducing the commutation losses of the switches but increasing the conduction losses. Also, as the number of H-bridges increases, the voltages across each of the switches in the H-bridges decreases (Equation 1), thus reducing individual switch losses and stresses. At the same time, as the number of H-bridges increases, the number of total switches will increase, thus increasing the total losses as well as the overall complexity of the control for all of the switches [4]. Thus the optimized number of switches depends on the specifications of the switches used as well as the output voltage and power. As a result, NLS CHB circuits are better suited for some applications more than others [3, 6]. ## VI Phase Shifted PWM The Phase Shifted PWM control for a multilevel converter applies a triangular waveform in comparison to a control sinusoidal function in order to obtain the desired PWM for each H-bridge. Each H-bridge’s triangle waveform has a phase shift depending on the number of levels: $\displaystyle\theta_{shift}=\frac{360^{\circ}}{L-1}=\frac{360^{\circ}}{2N}$ (12) Where N is the number of H-bridges in the cascade. The DC voltage for each H-bridge level is defined as: $\displaystyle V_{DC}=\frac{V_{DC,0}}{N}$ (13) Where $V_{DC,0}$ is the DC voltage required to generate desired AC output voltage in case of a single level inverter. ## VII Phase Shifted PWM Inverter Design For this project, it has been decided to choose cascade consisting of 6 level H-bridge inverters with PSPWM control. The carrier waveforms for all of the 6 levels were triangle waves of 100kHz. The overall 3 phase circuit with load, and level circuitry are shown in Figures 9 and 10. Note that the inverter is connected to the grid and the grid has an associated inductance is 1 mH. The grid voltage was given a -2.5° phase shift with respect to inverter output voltage to facilitate current flow from inverter to grid. From our project requirements, the required output voltage was $480V_{L-L}^{RMS}$ , or $\approx 277V_{L-N}^{RMS}$. From [8] and Equation 13, the total required DC voltage $V_{DC,0}$ and the individual H-bridge DC voltages $V_{DC,level}$ were found as: $\displaystyle V_{DC,0}=\frac{V_{rms,LL}}{m_{a}\|_{m_{a}=0.8}}$ $\displaystyle\sqrt{\frac{2}{3}}=\sqrt{\frac{2}{3}}\frac{480}{0.8}\approx 490V$ $\displaystyle V_{DC,level}$ $\displaystyle=\frac{490}{6}\approx 81.67V$ Similarly, from equation 12, the individual carrier wave phase shift can be found as: $\displaystyle\theta_{shift}=\frac{360^{\circ}}{26}=30^{\circ}$ Figure 9: 3 phase 6 level cascaded H-bridge inverter with grid as a load Figure 10: PSPWM carriers (leg A, leg B) for 6 level cascaded H-bridge inverter. Phase shift is $30^{\circ}$ Figure 10 shows the carrier signals for a 6 level cascaded H-bridge inverter (positive leg control signals are shown in top part, and negative leg control signals are shown in bottom part) Figure 11: Buck converter required to facilitate 81.67 VDC for inverter level Figure 11 displays the PS-PWM buck converter used. The inductor and capacitor values were calculated using Equations 9 and 10 (Final inductor and capacitor values were chosen to be $100\mu H$ and $100\mu F$).This circuit will facilitate power flow from the PV network to the cascaded inverter. Based on the single phase inverter, current draw from the circuit was $\approx 13.7ADC$. That leads to conclude that the inverter’s levels will behave as resistive load of $6\Omega$ for the buck converter. This figure has been used to design the appropriate buck converter (Equation 11). Based on the data for PV cells in [10], it was estimated that two sets of three series PV cells in parallel connected to each buck converter would be needed in order to provide sufficient voltage and current for our inverter. Hence, the input voltage and available current for buck converter are: 120.6 VDC @ 19.42 ADC. This defines duty cycle of buck converter to be 0.677 from Equation 8. Figure 12: Filtered and unfiltered voltage and current output waveforms Based on the unfiltered voltage and current simulation data, the voltage high frequency harmonics at frequencies above 2 kHz. Therefore, in order to have a low filtered THD with reasonably high L and C values for filter, it was decided to set cut-off frequency of LC filter at $\approx 1453Hz$. In addition to LC filter, the equivalent grid line inductance of 1 mH also acts as a filter. From Equation 11 the calculated filter values were: $\displaystyle L_{f}=200\mu H$ $\displaystyle C_{f}=60\mu F$ ## VIII PS-PWM Simulation Results ### VIII-A $3\phi$ Simulation Data Analysis First, the PSIM simulations for a single phase using the default PSIM lossy MOSFET models and lossy reactive elements ($R_{series}=50m\Omega$) were examined. Figure 13: Phase A filtered/unfiltered voltage and current waveforms Figure 14: Phase A filtered/unfiltered voltage/current waveform characteristic Figures Figures 13 and 14 display the output voltage and current filtered and unfiltered waveforms as well as the wave characteristics respectively for the single phase lossy simulation. As can be seen, the output voltage met the required line-line voltage of $480V_{L-L}^{RMS}$. In addition, all of the THD values were below 5%. The single phase output power was approximately 6.8kW. ### VIII-B $3\phi$ Simulation Data Analysis Next, the PSIM simulations for a three phase circuit using the default PSIM lossy MOSFET models and lossy reactive elements ($R_{series}=50m\Omega$) were examined. Figure 15: Three phase filtered/unfiltered voltage and current waveforms Figure 16: Three phase filtered/unfiltered voltage/current waveform characteristic Figures Figures 15 and 16 display the output voltage and current filtered and unfiltered waveforms as well as wave their characteristics for the single phase lossy simulation. As can be seen, the output voltage and output current THD values are below the required 5%. The total output power from the three phase simulation was calculated to be approximately 20.3kW. Thus in order to meet the design requirement of 125kW, at least seven three phase H-bridge inverters of this topology would need to be used. Figure 17: 6 level cascaded H-bridge inverter. Switch voltage/current stresses Figure 17 shows the voltages and currents experienced by the H-Bridges of the PS-PWM inverter. Based on the simulation data, the voltage stress on MOSFETs was approximately 85 VDC, and current stress was approximately 40 ADC. Base on design experience, voltage and current ratings are desired to be increased by 150 % to maintain safe operation at extreme performance cases. That implies that actual switch ratings should be 150 VDC / 60 ADC. ## IX Conclusions Multilevel converters inherently provide desired characteristics for high powered applications; but with them come inherent issues such as more complex structure and operation. The CHB in particular has a structure that allows for very high power applications due to their series connections of isolated power supplies. The drawback of this structure is that the if a single power source is used to supply each of the levels, then the isolation transformer would require currently non-standard transformers with large numbers of secondary windings[4]. Our solutions, as well as [9, 2], solve this problem by having isolated PV cells. This method assumes that all of the PV’s output the same current, but in practice would require more complex control in order to achieve the desired output voltage and power from PV cells that are not providing equal power. In addition, the gate control of each H-bridge would require to be isolated. This paper proposes two solutions for creating CHB inverters capable of outputting 125kW at $480V_{L-L}^{RMS}$. Our simulation results show that such is possible while maintaining a current THD below 5% as required for IEEE-519 [1]. Multilevel converters such as the CHB have unique features for power quality and modularity. Although they are not commonly used in industry now, they have great potential for the future. ## References * [1] IEEE Recommended Practice and Requirements for Harmonic Control in Electric Power Systems. IEEE Std 519-2014 (Revision of IEEE Std 519-1992), pages 1–29, 2014\. * [2] A. R. Beig, U. R. Y. Kumar, and V. T. Ranganathan. A novel fifteen level inverter for photovoltaic power supply system. In Conference Record of the 2004 IEEE Industry Applications Conference, 2004. 39th IAS Annual Meeting., volume 2, pages 1165–1171 vol.2, 2004. * [3] Z. Du, L. M. Tolbert, J. N. Chiasson, B. Ozpineci, H. Li, and A. Q. Huang. Hybrid cascaded H-bridges multilevel motor drive control for electric vehicles. In 2006 37th IEEE Power Electronics Specialists Conference, pages 1–6, 2006. * [4] L. G. Franquelo, J. Rodriguez, J. I. Leon, S. Kouro, R. Portillo, and M. A. M. Prats. The age of multilevel converters arrives. IEEE Industrial Electronics Magazine, 2(2):28–39, 2008. * [5] Kaco. blueplanet 87.0 – 150 TL3 Transformerless, three-phase string inverters. 125TL3. * [6] S. Kouro, R. Bernal, H. Miranda, C. A. Silva, and J. Rodriguez. High-Performance Torque and Flux Control for Multilevel Inverter Fed Induction Motors. IEEE Transactions on Power Electronics, 22(6):2116–2123, 2007. * [7] Philip T. Krein. Elements of Power Electronics. Oxford University Press, 1998. * [8] Ned Mohan, Tore M. Undeland, and Willian P. Robbins. Power Electronics: Converters, Applications, and Designs. John Wiley & Sons, Inc., 3rd edition, 2003. * [9] R. L. Naik and K. R. Y. Udaya. A novel technique for control of cascaded multilevel inverter for photovoltaic power supplies. In 2005 European Conference on Power Electronics and Applications, pages 9 pp.–P.9, 2005. * [10] Trina Solar. Bifacial Dual Glass 144 Cell Multi Busbar Module, 2019. TSM-DEG15MC.20(II). \| John Buczek currently pursuing a BSMS in Electrical Engineering with concentration in Power Systems at Northeastern University, Boston, MA. His research interests include power electronics, UAVs, and the Wireless Internet of Things. ---\|--- \| Viktor Ivankevych received B.S (2016) and M.S (2017) degrees in Electrical Engineering from New York University, Brooklyn, NY. He is currently pursuing the Ph.D. degree in Electrical Engineering at Northeastern University, Boston, MA. His research interests include power electronics and power converters. ---\|--- ### Appendix A: THD Python3 Code from math import pi, asin, sqrt def rms_value(L): ’’’ Function to return the rms value for the ’L’ level cascade H-bridge inverter :param L: int for the number of levels (needs to be odd) :retutn: float of the magnitude of rms value ’’’ N = (L-1)/2 # Number of H-bridges sum_steps = 0 # summation variable for ii in range(0, int(N), 1): # loop from 0 to N-1 sum_steps = sum_steps + asin((2ii+1)/(L-1))(2ii + 1) return sqrt(1 - (2/(piNN))sum_steps) def first_harmonic_rms(L): ’’’ Funciton to return the rms value of the first harmonic for the ’L’ level cascade H-bridge inverter :param L: int for the number of levels (needs to be odd) :return: float of the magnitude of the rms of the first harmonic ’’’ N = (L-1)/2 # Number of H-bridges sum_steps = 0 # summation variable for ii in range(0, int(N), 1): # loop from 0 to N-1 sum_steps = sum_steps + sqrt(1-((2ii+1)/(L-1))2) return (8/(pi1.0(L-1)))sum_steps/sqrt(2) def main(): ’’’ Main Method ’’’ L_array = range(3, 29, 2) # L from 3 to 27 odd integers for L in L_array: rms = rms_value(L) # get rms f_harm = first_harmonic_rms(L) # get first harmonic rms thd = sqrt(rms2 - f_harm2) / f_harm # calc THD print("Levels: ",L, "Vrms: " ,rms, "V,1,rms: ",f_harm , "THD: ", thd) if __name__ == "__main__": main()
TCID_50 PFU infectious doses SIN # Time to revisit the endpoint dilution assay and to replace and as measures of a virus sample’s infection concentration Daniel Cresta Department of Physics, Ryerson University, Toronto, ON, M5B 2K3, Canada Donald C. Warren Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) program, RIKEN, Wako-shi, Saitama, 351-0198, Japan Christian Quirouette Department of Physics, Ryerson University, Toronto, ON, M5B 2K3, Canada Amanda P. Smith Department of Pediatrics, University of Tennessee Health Science Center, Memphis, TN, 38163, USA Lindey C. Lane Department of Pediatrics, University of Tennessee Health Science Center, Memphis, TN, 38163, USA Amber M. Smith Department of Pediatrics, University of Tennessee Health Science Center, Memphis, TN, 38163, USA Catherine A. A. Beauchemin Department of Physics, Ryerson University, Toronto, ON, M5B 2K3, Canada Interdisciplinary Theoretical and Mathematical Sciences (iTHEMS) program, RIKEN, Wako-shi, Saitama, 351-0198, Japan ###### Abstract The infectivity of a virus sample is measured by the infections it causes, via a plaque or focus forming assay ( or FFU) or an endpoint dilution (ED) assay (, CCID50, EID50, etc., hereafter collectively $\text{ID}_{50}$). The counting of plaques or foci at a given dilution intuitively and directly provides the concentration of infectious doses in the undiluted sample. However, it has many technical and experimental limitations. For example, it is subjective as it relies on one’s judgement in distinguishing between two merged plaques and a larger one, or between small plaques and staining artifacts. In this regard, ED assays are more robust because one need only determine whether or not infection occurred. The output of the ED assay, the 50% infectious dose ($\text{ID}_{50}$), is calculated using either the Spearman-Kärber (1908\|1931) or Reed-Muench (1938) mathematical approximations. However, these are often miscalculated and their approximation of the $\text{ID}_{50}$ cannot be reliably related to the infectious dose. Herein, we propose that the plaque and focus forming assays be abandoned, and that the measured output of the ED assay, the $\text{ID}_{50}$, be replaced by a more useful measure we coined _specific infections_ (). We introduce a free, open-source web-application, midSIN, that computes the concentration in a virus sample from a standard ED assay, requiring no changes to current experimental protocols. We use midSIN to analyze sets of influenza and respiratory syncytial virus samples, and demonstrate that the /mL of a sample reliably corresponds to the number of infections a sample will cause per unit volume. The /mL concentration of a virus sample estimated by midSIN, unlike the $\text{ID}_{50}$/mL, can be used directly to achieve the desired multiplicity of infection. Estimates obtained with midSIN are shown to be more accurate and robust than those obtained using the Reed-Muench and Spearman-Kärber approximations. The impact of ED plate design choices (dilution factor, replicates per dilution) on measurement accuracy is also explored. The simplicity of as a measure and the greater accuracy provided by midSIN make them an easy and superior replacement for the , FFU, and other ID50 measures. We hope to see their universal adoption to measure the infectivity of virus samples. ## 1 Introduction The progression of a virus infection _in vivo_ or _in vitro_ , or the effectiveness of therapeutic interventions in reducing viral loads, are monitored over time through sample collections to measure changes (increases or decreases) in virus concentrations. As such, accurate measurement of the concentration in a sample is critical to study and manage virus infections. The most direct method is to count individual virions as observed under an electron microscope. However, this technique is costly, time consuming, and largely destructive of the samples, and is thus almost never used. Viral RNA can be counted via quantitative polymerase chain reaction (qPCR), a method that amplifies a specific virus genome segment (RNA or DNA) within the sample over multiple cycles. The growth curve resulting from successive amplification cycles, compared against the standard curve for a sample of known concentration, provides an estimate of the number of viral segments in the sample. The major limitation of this method is that it measures not only viral RNA from intact virions, only some of which are infection-competent, but also debris from apoptotic or lysed cells, and antibody- or antiviral-neutralized virions, which misrepresents the effective virion concentration. For this reason, a count of infectious particles rather than, or in addition to, total viral genome segments is preferred. Infectious virions do not systematically differ in any observable way from replication-defective virions, nor do they differ in a physical way that would allow for their mechanical or chemical separation. For this reason, methods to count infectious virions are based on counting the infections they cause, rather than the particles themselves. In practice, however, not all infection- competent virions contained in a sample will go on to successfully cause infection. Certain experimental conditions, such as temperature or acidity of the medium, can hasten the rate at which virions that were infection-competent in the sample lose infectivity before they can cause infection. This is why, hereafter, we will refer to the quantity measured by infectivity assays as the _infection concentration_ or the number of infections the sample will cause per unit volume, rather than its concentration of infectious virions, which is not a measurable quantity. Two main types of assays are used to quantify the infection concentration within a virus sample: (1) the plaque forming and focus forming assays; and (2) assays we will collectively refer to as endpoint dilution (ED) assays111Technically, the plaque and focus forming assays are also endpoint dilution assays because they rely on the counting of plaques or foci (the endpoint) as a function of dilutions. However, herein, we will refer to them as plaque or foci forming assays rather than endpoint dilution assays., which include the 50% tissue culture infectious dose () or cell culture infectious dose (CCID50) or egg infectious dose (EID50) assays, etc. The plaque forming assay was introduced by Renato Dulbecco in 1952 [3], as an improvement over the ED assay. The plaque forming assay and the focus forming assay, which rely on the same principles, suffer from a number of critical issues that cannot be overcome. For example, the liquid accumulation (meniscus) that forms around well edges means some infectious doses will not get quantified correctly or at all. It can be hard to distinguish two merged plaques from a single large plaque, or to decide how small a plaque one should consider when counting. Some of the difficulties in establishing a robust, unambiguous plaque or focus count for a given well are illustrated in Figure 1. For these reasons, different researchers will count a different number of plaques or foci when observing the same well. This subjectivity in the count means there is opportunity to (sub)consciously count a few more plaques or foci, for example, when expecting a virus strain to be more severe than another or in the absence of an antiviral compound. Ideally, there would be no discretion involved in the counting process of a quantification assay. Indeed, the decision process should be made by a physical, automated measurement, without the possibility of post-facto adjustments of any kind, for any reason. Figure 1: Examples of challenges in establishing a robust count of infection plaques or foci. MDCK cells were infected with a sample containing influenza A (H3N2) virions, and cell infection was visualized via staining by antibodies against the matrix (M) viral protein. The uneven liquid distribution along the well’s edges means some infectivity is lost or miscounted. It can be hard to distinguish between two merged foci and a single larger uneven focus. It is difficult to determine how small a focus should be counted, and doing so to decide on a focus size threshold to be used consistently for all wells and all samples within a particular experiment. As a result of these difficulties, different individuals will commonly count a different number of foci in the same well. Stained well image graciously provided by Frederick Koster (Lovelace Research Institute, NM, USA). In contrast, the ED assay offers a more decisive and robust binary determination as to whether or not infection has taken place in each well (or egg, animal, etc.). This determination is insensitive to small, spatially localized irregularities and is typically unanimously agreed upon by all observers. Therefore, it is less subject to (sub)conscious bias. In fact, this feature of the ED assay makes it ideal for systematic, machine-based determination of positive wells (or eggs or other culture types), eliminating subjectivity. Furthermore, infection of wells in the ED assay can be carried out in exactly the same way as planned infection experiments where they will make up the inoculum, e.g., in the same cell type, reproducing whether the inoculum is rinsed or not post-inoculation, and the duration of incubation with the inoculum. In contrast, plaque and focus forming assays can require the use of a semi-solid cellular overlay (e.g., agarose) to restrict the spread of virus beyond cells neighbouring those initially infected by the inoculum. The need to rinse or remove the inoculum to add the semi-solid overlay imposes strict constraints on the timing of this rinse. Because a longer incubation provides more opportunities for infectious virus to cause infection, the number of infectious doses counted via a plaque or focus assay can underestimate the true number of infections that will result when the quantified sample is later used to infect cells under longer incubation periods. The plaque assay can also require the use of different cells than those used in the infection experiments whenever the latter fail to die or detach (form clearly visible plaques) post-infection, making it difficult to predict the number of infections that will result when the quantified sample is later used to infect different cells. For all its many advantages, the ED assay currently has one key, remediable weakness: its output quantity, the (or CCID50 or EID50), does not directly correspond, or trivially relate, to one infectious dose. The simplistic calculations, introduced by Spearman-Kärber (SK) [9, 5] and Reed-Muench (RM) [8] nearly a century ago, remain the primary methods to quantify a virus sample’s infectivity using the ED assay. Many research groups rely on spreadsheet calculators that are passed down through generations of trainees or found on the internet, and can contain errors222For example, versions 2 and 3 of the Excel spreadsheet calculator provided by the Lindenbach Lab at Yale University (http://lindenbachlab.org/resources.html), which have since been removed.. While, theoretically, a dose of $1\text{\,}\tcid$ is expected to cause $-1/\ln(50\%)=$1.44\text{\,}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$$ [2], the approximation used by the SK and RM methods introduces an often overlooked bias where $$1\text{\,}\tcid$\approx$1.781\text{\,}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$$ where $1.781=\mathrm{e}^{\gamma}$ and $\gamma=0.5772$ is the Euler-Mascheroni constant [10, 4]. This makes it problematic to experimentally achieve the desired multiplicity of infection when inoculating from a sample quantified via the SK or RM methods. Many have proposed replacements for the RM and SK calculations with some based on logit or probit transforms of the data [4, 6, 2] and others on statistical analysis of the ED assay output [6, 7]. Sadly, none of these improvements were widely adopted, possibly due to a lack of visibility of these publications, or the lack of widespread awareness of the limitations of the RM and SK methods. Thus, the one issue with the ED assay is not with the assay itself but with the calculation of the . We submit that for all the reasons outlined above, the ED assay is experimentally more robust and reliable than the plaque and focus forming assays, and should be preferred over the latter. We propose to: 1. 1. Continue the use, or encourage the adoption, of the ED assay (e.g., assay), but to replace its output, the TCID50/mL (or CCID50/mL, EID50/mL, etc.), with a new quantity in units of Specific INfections or /mL corresponding to the number of infections the sample will cause per mL. The word _specific_ highlights the fact that the infectivity of a sample is specific to the particulars of the experimental conditions (temperature, medium, cell type, incubation time, etc.). 2. 2. Replace the Reed-Muench and Spearman-Kärber approximations with a computer software, midSIN (measure of infectious dose in SIN), that relies on Bayesian inference to measure the /mL of a virus sample. To avoid calculation errors and make the new method widely accessible, midSIN is maintained and distributed as free, open-source software on GitHub (https://github.com/cbeauc/midSIN) for user installation, but also via a free- to-use website application (https://midsin.physics.ryerson.ca) with an intuitive user interface. Here, we present examples of midSIN being used to analyze influenza and respiratory syncytial virus samples. We demonstrate that midSIN’s output, /mL, is an accurate estimate of the number of infections the sample will cause per unit volume. We show how the accuracy of the concentration estimate is affected by experimental choice of plate layout, including the dilution factor, and the number of replicates per dilution. We compare midSIN’s performance to that of the RM and SK methods, and demonstrate how the latter estimators are inaccurate under various circumstances, underlining the need to adopt midSIN to quantify virus samples via the ED assay. ## 2 Results ### 2.1 Key features of midSIN’s output Let us consider a fictitious ED experiment, with 11 dilutions and 8 replicate wells per dilutions, in which the minimum sample dilution, $\mathcal{D}_{1}=1/100=10^{-2}$, is serially diluted by a factor of $10^{-0.5}\approx 0.32$ ($\mathcal{D}_{2}=10^{-2.5}$, $\mathcal{D}_{3}=10^{-3}$, …, $\mathcal{D}_{11}=10^{-7}$), and the total volume of inoculum (diluted virus sample + dilutant) placed in each well is $V_{\text{inoc}}=$0.1\text{\,}\mathrm{mL}$$. Now, consider that a virus sample is measured using this ED experiment and one observes (8,8,8,8,8,7,7,5,2,0,0) infected wells out of 8 replicates at each of the 11 dilutions, as illustrated in Figure 2A. Figure 2: Visual representation of midSIN’s output for the example ED plate. (A) Illustration of the example ED plate where $\mathcal{D}_{i}$ are the chosen serial dilutions of the sample. For the example described in the text, $\mathcal{D}_{1}=10^{-2}$, $\mathcal{D}_{2}=10^{-2.5}$, …, $\mathcal{D}_{11}=10^{-7}$, with 8 replicates per dilution. The number of infected wells (# inf) is indicated at the bottom of each dilution column. (B) The midSIN-estimated likelihood distribution of the $\log_{10}$ infection concentration, $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$, for the example ED experiment. The vertical lines correspond to $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$, based on the most likely value (mode) of midSIN’s likelihood distribution (solid blue), or computed from the RM (solid orange) and SK (dashed green) approximations of the $\log_{10}(\tcid)$ (see Methods). The $x$-value of the white and light grey region on either sides of the mode indicate the edges of the 68% and 95% credible interval (CI), respectively. The midSIN-estimated $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$ $\text{mode}\pm 68\%\left[\pm 95\%\right]$ CI are indicated numerically above the graph. (C) The number of infected wells (black circles) out of the 8 replicates, as a function of the 11 serial dilutions of the example ED plate, from the least (leftmost) to the most (rightmost) diluted. For example, $x=3.0$ corresponds to a sample dilution of $10^{-3}$ or 1/1,000. The average (expected) number of infected wells, as a function of sample dilution, is shown for the most likely value of $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$ (blue curve) or its 68% and 95% CI (inner and outer edge of the grey bands, respectively). The sample dilution ($x$-value) at which the blue curve crosses the horizontal dotted line (50% infected wells) corresponds to a concentration of $1\text{\,}\tcid$ per ED well volume. The vertical lines indicate the sample dilution that yields a concentration of $1\text{\,}\tcid$ according to the RM and SK approximations. midSIN provides a graphical output of its results, shown in Figure 2B,C for this example. Note how the likelihood distribution for $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$ (Fig. 2B) is approximately a normal distribution. This is why $\log_{10}$ of the infection concentration should be used and reported, rather than the concentration itself. midSIN also graphically compares the number of infected wells observed experimentally (Fig. 2C, black dots) against the theoretically expected values (blue curve and grey CI bands). This graphical representation makes it easy to identify issues with the data entered or with the experiment itself. Importantly, midSIN provides a more useful quantity to the user than the : an estimate of the concentration of infections the sample will cause, /mL. For this example, the concentration is $10^{6.2\pm 0.1}\text{\,}\idnew\mathrm{/}\mathrm{m}\mathrm{L}$, where 6.2 is the mode (most likely value) of $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$, and $\pm 0.1$ is its 68% credible interval (CI). The /mL corresponds to the number of infections that will be caused per m L of the sample, which can be directly used to determine the sample dilution required to obtain a desired multiplicity of infection (MOI). Figure 3: Quantification of RSV sampled from _in vitro_ infections. Each row corresponds to a different experiment (mock-yield [my] or single-cycle [sc]) and sampling time point (e.g., $8\text{\,}\mathrm{h}$, $36\text{\,}\mathrm{h}$), and each sample was measured in duplicate (rep1, rep2). These data were collected from in vitro infections with the RSV A Long strain, and were previously reported in [1]. The ED measurement experiment were conducted using a plate layout of 11 dilutions, with 8 replicates per dilution, an inoculum volume of $V_{\text{inoc}}=$0.1\text{\,}\mathrm{mL}$$, serial dilutions from $\mathcal{D}_{1}=10^{-1}$ to $\mathcal{D}_{11}=10^{-6}$, separated by a dilution factor of $10^{-0.5}$. In a laboratory setting, ED experiments can be performed in batches, such as to quantify the infectious concentration in samples collected at several time points over the course of a cell culture infection. For such applications, midSIN provides a comma separated value (csv) template file readily editable in a spreadsheet program, to collect and submit the results for batch processing. Details on the format of the template file are available on midSIN’s website (https://midsin.physics.ryerson.ca). Figure 3 illustrates the output for a subset of measurements for _in vitro_ infection with the respiratory syncytial virus (RSV). Each sample was measured twice, and midSIN’s estimates are in good agreement with one another (within 95% CI). The $y$-axis in the left graph panels of midSIN’s graphical output is the non- normalized scale of the likelihood distribution for $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$, which ranges between $10^{-7}$ and $10^{-2}$. The scale loosely relates to the likelihood of observing a particular ED experimental outcome (see Methods). Unlikely ED outcomes appear as large departures of the observed number of infected wells (right panels, black dots) from what is theoretically expected (right panels, curve). It is interesting that the uncertainty (CI) of midSIN’s estimated $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$ appears to be independent of how much the ED outcome deviates from theoretical expectations. That is, the accuracy of midSIN is not strongly affected even when it is provided more unlikely, noisy experimental data. This robustness is explored further below. ### 2.2 Comparing to and virus sample concentrations The midSIN calculator provides an estimate of the number of infections that will be caused per mL of a virus sample (/mL). In principle, a plaque assay also measures the number of infections a sample will cause, with each infection expected to develop into a plaque. If a plaque assay is performed under experimental conditions and protocols as similar as possible to those of the ED assay (i.e., using the same cells, medium, period of incubation, rinsing method, etc.), midSIN’s /mL estimate is expected to be comparable, in theory, to the number of /mL observed in the plaque assay. In practice, however, the plaque assay likely provides a biased estimate of the concentration of infections in a sample due to its many experimental issues, discussed in the Introduction. To evaluate midSIN’s performance compared to existing methods, the infection concentration in two influenza A (H1N1) virus strain samples were measured via both plaque and ED assays, and their concentration in units of , , and were compared (Fig. 4). Details regarding the samples, and how the plaque and ED assays were performed are provided in Methods. Figure 4: Comparing to and for influenza A virus samples. (A,B) The infection concentration in two influenza A (H1N1) virus strain samples was measured via both an ED assay and a plaque assay (x, PFU). The ED assay was quantified in $\log_{10}(\tcid)$ using the RM (square) or SK (triangle) methods, or in $\log_{10}(\idnew)$ using midSIN (circle with 68%,95% CI). Each of the 2 strain samples was measured over 2 separate experiments (Exp. #1, #2), performed each time by 2 different researchers (Researcher A or B), with 5 biological replicates each. The grey bars indicate the range of $\log_{10}(\idnew)$ values across the 5 replicates. The RM, SK, and SIN measures were estimated for each replicate based on the same ED plate. The experimental details are provided in Methods. (C,D) The $\log_{10}$ of the ratio between either the via the RM or SK method or the , over the via midSIN. The ratios were computed for each replicate ($5\times 5$ replicates), per experiment, per researcher (25 replicates $\times$ 2 researchers $\times$ 2 experiments = 100 ratios) shown as individual symbols (dots) for each method (RM, SK, ). The mean and 68% CI of the 100 ratios are indicated numerically and as black circles with error bars. The concentrations estimated via the RM and SK methods are $\sim$1.5–1.7 times larger (Fig. 4C,D) than the concentration, and the set of ratios are statistically inconsistent with the assumption of equality ($p$-value: 0.01–0.03). Theoretically, $1\text{\,}\tcid$ is expected to cause $1.44\text{\,}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$ ($=1/\ln(2)$) [2]. However, the RM or SK approximations are known to introduce a bias such that $1\text{\,}\tcid$ estimated by these methods is expected to cause $1.781\text{\,}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$ ($=\mathrm{e}^{\gamma}$ where $\gamma=0.5772$ is the Euler-Mascheroni constant) [10, 4]. Using the RM, SK, and measurements presented in Figure 4A,B, we confirmed333The mean $\log_{10}(\mathrm{ratio})$ was re-computed for $\mathrm{ratio}=(\mathrm{RM}/1.781)/\idnew$ and (SK/1.781)/, and found to be 0.85–0.93. This is statistically consistent ($p$-value: 0.1–0.3) with the assumption of equality, i.e., $\mathrm{ratio}=10^{0}=1$. that $$1.781\text{\,}\idnew$\approx$1\text{\,}\tcid$$ when the latter is estimated via the RM or SK approximations, as expected theoretically if is indeed measuring the infection concentration in a sample. Similarly, the ratio of the concentration determined via the plaque assay and the concentrations estimated by midSIN is $\sim$0.89–0.93, which is statistically consistent with the assumption of equality ($p$-value: 0.2–0.5). These results confirm the theoretical expectation that $$1\text{\,}\pfu$\approx$1\text{\,}\idnew$$ when the plaque and ED assays are performed in the same manner, as was the case here. This provides further support, via two independent assays, that the concentration estimated by midSIN from the ED assay is a robust measure of the infection concentration of a virus sample. ### 2.3 Comparing midSIN’s performance to that of the RM and SK methods Figure 5: midSIN’s estimate of a sample’s infection concentration based on a single dilution. This is a simulated example of an ED plate with an inoculation volume of $V_{\text{inoc}}=$0.1\text{\,}\mathrm{mL}$$. Instead of serial dilutions, a single dilution ($\mathcal{D}_{1}=0.01$) is used, and either 1, 2 or 3 well(s) out of the 4 replicate wells are infected. As the fraction of infected wells increases, the uncertainty on the estimate (68% and 95% CIs) decreases, and the likelihood distribution becomes more symmetric (Normal-like). The RM and SK methods rely on the number of infected wells decreasing as dilution increases. Their estimates are affected when the number of infected wells remains unchanged or even increases as dilution increases, which statistics tell us can reasonably occur experimentally. The RM and SK methods also mostly require that at the lowest and highest sample dilutions, all wells be infected and uninfected, respectively. In contrast, midSIN is robust to these issues. Figure 5 demonstrates how midSIN can provide an estimate for the $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$ in a sample using the number of infected wells at a single dilution, as long as at least one well is uninfected if all others are infected or vice-versa. This is because midSIN relies on Bayesian inference, i.e., when more than one column is available, it uses information from each column successively to revise and improve its estimate. This allows midSIN to correct for even large deviations from theoretical expectations, and thus improves its accuracy. Figure 6: Comparing known input to estimated output concentrations. For each input concentration between $10^{2.2}$ and $10^{9.4}$, one million random ED experiment outcomes (# of positive wells in each dilution column) were generated. For each ED outcome, either (A) midSIN was used to determine the most likely $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$; or the (B) RM or (C) SK method was used to estimate the $\log_{10}(\tcid/\mathrm{m}\mathrm{L})$. Vertically stacked grey bands at each input concentration are sideways histograms, proportional to the number of ED outcomes that yield a given $y$-axis value. The black curves join the median (thick), 68${}^{\text{th}}$ (thin) and 95${}^{\text{th}}$ (dashed) percentile of the histograms, determined at (but not between) each input concentration. A plate layout of 11 dilutions, with 8 replicates per dilution, an inoculum volume of $V_{\text{inoc}}=$0.1\text{\,}\mathrm{mL}$$, serial dilutions from $\mathcal{D}_{1}=10^{-2}$ to $\mathcal{D}_{11}=10^{-8}$, separated by a dilution factor of $10^{-0.6}\approx 1/4$ were used in the simulated ED experiments. Figure 6 illustrates how well the midSIN, RM, and SK methods recover a known input sample concentration in simulated ED experiments, based on a plate layout consisting of 11 dilutions ($\mathcal{D}_{1}=10^{-2}$ to $\mathcal{D}_{11}=10^{-8}$), a dilution factor of $1/4$, and 8 replicates per dilutions. The infection concentration estimated by midSIN is in excellent agreement with the input concentration. For the RM and SK methods, which estimate the $\log_{10}(\tcid/\mathrm{m}\mathrm{L})$ rather than the $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$, the agreement is generally poor due to the bias they introduce. Furthermore, the RM and SK predictions are more variable (wavy pattern), and lose accuracy dramatically as the sample concentration approaches the limits of detection (the 2 ends) which, for the example plate layout simulated here, is around $10^{3}\text{\,}\idnew\mathrm{/}\mathrm{m}\mathrm{L}$ and $10^{9}\text{\,}\idnew\mathrm{/}\mathrm{m}\mathrm{L}$. Interestingly, the basic calculations behind the RM and SK methods constrain the set of values they can return (sparsely populated grey histograms), compared to the more continuous range returned by midSIN, which contributes to its increased accuracy. ### 2.4 Estimate accuracy as a function of plate layout In Figure 3, we observed that even for large discrepancies between the expected (right panels, blue curve) and observed (right panels, black dots) ED assay outcome, the uncertainty (CI) of midSIN’s estimate remains relatively unchanged. This apparent robustness is because the uncertainty is primarily determined by the experimental design, namely the change in dilution between columns (dilution factor) and the number of replicate wells per dilution. Figure 7 explores the impact of varying either only the dilution factor, or only the number of replicates at each dilution, or varying one at the expense of the other by using a fixed number of wells (96 wells). When using midSIN, smaller changes in dilution (e.g., going from a dilution factor of 2.2/100 to 61/100) or more replicates per dilution (4 to 24) each improves the measure’s accuracy (narrower CIs) by comparable amounts, but only when the total number of wells is allowed to increase to accommodate the change. When the total number of wells used is fixed, changing one at the expense of the other leaves the accuracy (CI) unchanged. This is somewhat also true for the $\log_{10}(\tcid)$ output concentration estimated by the RM and SK methods. However, at the smallest dilution factors (10/100 and 2.2/100), the bias introduced by the RM and SK methods becomes even larger and more unpredictable. For the input concentration considered in Figure 7 ($10^{5}\text{\,}\idnew\mathrm{/}\mathrm{m}\mathrm{L}$), the dilution at which 50% of wells are infected is near the middle dilution. For sample concentrations such that 50% infected wells occur near or at the lowest or highest dilution chosen, the effect is even more significant. Figure 7: Comparing the effect of the dilution factor and number of replicates per dilution. The effect of either (A,D,G) decreasing the change in dilution (from a dilution factor of $2.2/100$ to $61/100$) while keeping 8 replicates per dilution; or (B,E,H) increasing the number of replicates per dilutions (4 to 24) while keeping a fixed dilution factor ($\approx 35/100$); or (C,F,I) increasing the dilution factor while decreasing the number of replicates, keeping a fixed number of 96 wells used in total to titer one virus sample. Different rows represent the ratio of the estimated output concentration using (A–C) midSIN in /mL, (D–F) RM or (G–I) SK in /mL, and the input concentration. In all cases (A–I), the input concentration was $10^{5}\text{\,}\idnew\mathrm{/}\mathrm{m}\mathrm{L}$, and as the dilution factor was varied, the highest and lowest dilutions in the simulated ED plate were held fixed to $\mathcal{D}_{1}=10^{-2}$ and $\mathcal{D}_{\text{last}}=10^{-7}$, respectively, by changing the total # of dilutions performed (simulated). Everything else is generated or computed as described in the caption of Figure 6. Figure 7 also demonstrates that varying the dilution by smaller increments (e.g., a dilution factor of 61/100 rather than 10/100) provides greater granularity (uniqueness) of ED plate outcomes, and thus, greater accuracy of the $\log_{10}$ infection concentration estimates. Here, a distinct plate outcome means a distinct number of infected wells at each dilution, with no distinction as to exactly which of the replicate wells (e.g., the second versus the fourth) is infected at each dilution. An ED plate with serial dilutions ranging over 6 orders of magnitude (e.g., $10^{-2}$ to $10^{-7}$), with 4 different dilutions and 24 replicates/dilution (i.e., dilution factor of 2.2/100) provides $\sim 10^{6}$ ($[24+1]^{4}$) possible, distinct ED plate outcomes. In contrast, a plate with the same serial dilution range, but with 24 different dilutions and 4 replicates/dilution (i.e., dilution factor of 61/100) yields $\sim 10^{17}$ ($[4+1]^{24}$) distinct outcomes. More generally, $[\text{reps}+1]^{\text{dils}}$ is the number of distinct plate outcomes for a chosen number of dilutions (dils) and replicates (reps). Having fewer possible plate outcomes means that a larger range of concentrations would share the same most-likely ED plate outcome, yet each plate outcome only maps to one (the most likely) concentration estimate. This means that with fewer dilutions, the concentration estimate is forced to take on the nearest possible value it can take (Fig. 7, the next grey bar), and the accuracy of the concentration estimate is therefore reduced. So although having a greater number of dilutions is more labour intensive, it should be preferred over having a greater number of replicates per dilution. ## 3 Discussion We have introduced a new calculator tool called midSIN to replace the Reed- Muench (RM) and Spearman-Kärber (SK) calculations to quantify the infectivity of a virus sample based on a endpoint dilution (hereafter ED) assay. Rather than estimating the of a virus sample, midSIN calculates the number of infections the sample will cause, reported in units of specific infections (). It does so without requiring any changes to current ED assay protocols, and can be accessed for free via an open-source web-application (https://midsin.physics.ryerson.ca). Importantly, since the of a virus sample corresponds to the number of infections it will cause, it can be used directly to determine what dilution of the sample will achieve the desired multiplicity of infection (MOI). We showed that midSIN provides more accurate and robust estimates than the biased RM and SK approximations. We confirmed that the RM and SK approximations overestimate the by 23.5%, such that $1\text{\,}\tcid$ estimated by these methods will cause 1.781 rather than $1.44\text{\,}\mathrm{i}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}$ [10, 4]. While in theory one can obtain the intended MOI by multiplying the by 0.7 (or rather $\ln(2)=0.693$), one should instead multiply by 0.561 to account for the overestimation by RM and SK. Even when accounting for the overestimation, we showed that these methods perform particularly poorly when too few replicate wells per dilutions are used or when the change in dilution is large between successive serial dilutions. The two methods perform especially poorly when quantifying samples whose infection concentration approaches, but is still well within, the detection limit of the ED assay. In such cases, the bias introduced by these methods becomes even larger and more significant. For example, if the minimum and maximum dilutions of an ED plate are $10^{-2}$ and $10^{-8}$, virus samples with a concentration less than $10^{2.2}\text{\,}\idnew$ or greater than $10^{7.6}\text{\,}\idnew$ per inoculated well volume (typically $0.1\text{\,}\mathrm{m}\mathrm{L}$), will see their concentration estimated with an even larger bias by the RM and SK methods. Using midSIN, rather than RM or SK, to measure the infectivity of a virus sample based on an ED assay does not require any change to ED experimental protocols and methods currently in use in one’s laboratory (e.g., dilution factor, replicate per dilution, minimum dilution). Indeed, we demonstrated that midSIN can estimate a virus sample’s concentration based on even just a single dilution, as long as only a fraction of the replicate wells are infected at that dilution. For a given number of ED wells used to titrate the sample and fixed minimum and maximum dilutions (ED detection range), we showed that having smaller changes between dilutions (a larger number of serial dilutions) is better than having more replicates per dilution. So those wishing to improve the accuracy in estimating the infectivity of their virus samples should consider using more wells in titrating each virus sample, and favouring smaller dilution changes over more replicates. For example, using 11 dilutions, with a 4-fold dilution factor between dilutions and 8 replicate wells per dilution uses up 88 wells, leaving 8 wells of a 96-well plate for controls. This ED plate design, analyzed using midSIN, accurately measures virus sample concentrations ranging over $\sim$6 orders of magnitude (e.g., [$10^{1}$–$10^{7}$] /mL, or [$10^{6}$–$10^{12}$] /mL, etc.) with an accuracy of $\sim$1.6-fold ($\times 10^{\pm 0.2}$, 95% CI). In comparison, using 7 dilutions, with a 10-fold dilution factor, and 4 replicates (which uses 28 rather than 88 wells) would also span 6 orders of magnitude, but with an accuracy of $\sim$3.2-fold ($\times 10^{\pm 0.5}$, 95% CI). To put these 2 accuracies in perspective: $1\text{\,}\mathrm{mL}$ of a sample measured to contain $10\text{\,}\idnew\mathrm{/}\mathrm{m}\mathrm{L}$, is expected to yield either 6–16 or 3–31 infections 95% of the time, given an accuracy of either $\times 10^{\pm 0.2}$ or $\times 10^{\pm 0.5}\text{\,}\idnew\mathrm{/}\mathrm{m}\mathrm{L}$, respectively. Such an important decrease in accuracy means a reduced ability to detect experimental changes as statistically signficant, with the $\times 10^{\pm 0.5}$ accuracy requiring a $>$10-fold change for statistical significance. Failing to identify a change as statistically significant as part of a study is far more costly than using a few more wells for each sample to increase measurement accuracy, and thus the statistical power of the study. The midSIN-estimated obtained from an ED assay was also compared to the from a plaque assay for a set of influenza A virus samples. When the plaque and ED assays are performed as identically as possible (cell type, incubation time, etc.), as was the case here, $$1\text{\,}\idnew$\approx$1\text{\,}\pfu$$. This demonstrates that indeed midSIN’s is a measure of the number of infections a virus sample will cause. However, as mentioned, the plaque and focus forming assays often impose experimental requirements (e.g., an early rinse of the inoculum to add agarose, use of cells with pronounced CPE). Such constraints on the plaque or focus assay inoculation protocol make it nearly impossible to relate the number of plaques or foci observed to the number of infections the virus sample will cause under the intended, experimental infection conditions (e.g., late or no inoculum rinse, no agarose, to infect cells exhibiting no significant CPE). Adding to this the subjectivity of counting plaques or foci, it is clear the ED assay combined with midSIN to estimate the concentration of a virus sample is more accessible, accurate, and predictive. Beyond the work presented herein, the development of midSIN will continue online, as we implement new features and inputs for integration with various colorimetric and fluorescence instruments. The ease of use of midSIN and the greater usefulness and relevance of as a measure of a virus sample’s infectivity make them far superior to all currently available alternatives, including the PFU, FFU, , and other $\text{ID}_{50}$ measures. We hope to see them adopted widely. ### Acknowledgements The authors wish to thank Frederick Koster (Lovelace Respiratory Research Institute, NM, USA) for providing the antibody stained well image, and Evan Williams (UTHSC, UT, USA) for technical assistance. ### Funding This work was supported in part by Discovery Grant 355837-2013 (CAAB) from the Natural Sciences and Engineering Research Council of Canada (www.nserc- crsng.gc.ca), Early Researcher Award ER13-09-040 (CAAB) from the Ministry of Research and Innovation of the Government of Ontario (www.ontario.ca/page/early-researcher-awards), by the Interdisciplinary Theoretical and Mathematical Sciences programme (iTHEMS, ithems.riken.jp) at RIKEN (CAAB), and by R01 AI139088 (AMS, APS, LCL) from the NIH NIAID (www.niaid.nih.gov). The funders had no role in study design, data collection and analysis, or decision to publish. ### Authors contribution CAAB was responsible for study conceptualization, and project administration, DC, DCW, CQ, CAAB all contributed to the development of the methodology, data analysis, software, visualization. AMS, APS, LCL carried out experimentation, CAAB, AMS were responsible for funding acquisition, experimental planning, and supervision. DC and CAAB contributed to the original manuscript draft, and all authors contributed to its review and editing. ## 4 Methods ### 4.1 The mathematics of the dose-response assay #### 4.1.1 Considering a single well Consider a virus sample of volume $V_{\text{sample}}$ which contains an unknown concentration of infectious virions, $C_{\mathrm{inf}}$, which we aim to determine. Drawing a small volume, $V_{\text{inoc}}<V_{\text{sample}}$, from the sample of volume $V_{\text{sample}}$, is analogous to drawing balls out of a bag containing green and yellow balls, and considering green balls a success, and yellow ones a failure. It is a series of Bernoulli trials where $n=V_{\text{inoc}}/V_{\text{vir}}$ is the number of draws, i.e., the number of virion-size volumes ($V_{\text{vir}}$) drawn from the sample to form the inoculum volume ($V_{\text{inoc}}$), analogous to the number of balls drawn. $k$ is the number of successes, i.e., the number of infectious virions drawn from the sample to form the inoculum, analogous to the number of green balls drawn. $p$ is the probability of success, i.e., the fraction of virion-size volumes in the sample that are occupied by infectious virions, analogous to the probability of drawing a green ball. The probability of success, $p$, is related to the concentration of infectious virus in the sample, $C_{\mathrm{inf}}$, as $p=\frac{\text{Number of virions in sample}}{\text{Number of virion-size volumes in the sample}}=\frac{C_{\mathrm{inf}}V_{\text{sample}}}{V_{\text{sample}}/V_{\text{vir}}}=C_{\mathrm{inf}}V_{\text{vir}}\ ,$ where $C_{\mathrm{inf}}$ is the quantity we aim to estimate. Unlike the ball analogy where it is easy to count how many green balls $k$ were drawn, after having drawn $n$ virion-size volumes from the sample into our inoculum, we cannot count how many infectious virions were drawn into the inoculum. However, if this inoculum is deposited onto a susceptible cell culture, we can observe whether or not infection occurs, and this would indicate that the inoculum contained at least one or more infectious virions. Note that, as explained in the Introduction, even a productively infectious virion, i.e., one capable of completing the full virus replication from attachment to progeny release, might not result in a productive infection. As such, from hereon, $C_{\mathrm{inf}}$ is used to designate the concentration of specific infections in the sample, which is smaller or equal to the concentration of infectious virions, i.e., measures a subset of the infectious virions. Having deposited the inoculum into one well of the 96-well plate of our ED experiment, the likelihood that the well will _not_ become infected corresponds to the likelihood of having drawn $k=0$ infectious virions (or rather, specific infections) out of the $n$ virion volumes that make up our inoculum, namely $\displaystyle q_{\text{noinf}}$ $\displaystyle=\text{Binomial}(k=0\|n=V_{\text{inoc}}/V_{\text{vir}},p=C_{\mathrm{inf}}V_{\text{vir}})$ (1) $\displaystyle=\frac{n!}{0!(n-0)!}p^{0}(1-p)^{n-0}=(1-p)^{n}$ $\displaystyle q_{\text{noinf}}$ $\displaystyle=\left(1-C_{\mathrm{inf}}\,V_{\text{vir}}\right)^{V_{\text{inoc}}/V_{\text{vir}}}$ where $q_{\text{noinf}}$ can be simplified by realizing that $\displaystyle\ln(1-x)$ $\displaystyle\stackrel{{\scriptstyle\|x\|<1}}{{=}}-x-\frac{x^{2}}{2}-\frac{x^{3}}{3}-...\stackrel{{\scriptstyle\|x\|\ll 1}}{{\approx}}-x$ $\displaystyle\ln(q_{\text{noinf}})$ $\displaystyle=\frac{V_{\text{inoc}}}{V_{\text{vir}}}\ \ln(1-C_{\mathrm{inf}}\,V_{\text{vir}})\approx\frac{V_{\text{inoc}}}{V_{\text{vir}}}\ (-C_{\mathrm{inf}}\,V_{\text{vir}})=-C_{\mathrm{inf}}\,V_{\text{inoc}}\ .$ As such, $q_{\text{noinf}}=(1-C_{\mathrm{inf}}\,V_{\text{vir}})^{V_{\text{inoc}}/V_{\text{vir}}}\approx\exp\left[-C_{\mathrm{inf}}\,V_{\text{inoc}}\right]$ (2) where $q_{\text{noinf}}$ and $(C_{\mathrm{inf}}V_{\text{vir}})\in[0,1]$ because $C_{\mathrm{inf}}=N_{\text{vir}}/V_{\text{sample}}$ and the number of specific infections in the sample, $N_{\text{vir}}$, is at a minimum zero, and at most the maximum number of virion-size volumes that can physically fit in the sample volume, namely $V_{\text{sample}}/V_{\text{vir}}$. As such, the maximum possible infection concentration, given a sample of volume $V_{\text{sample}}$, is $C_{\mathrm{inf}}=(V_{\text{sample}}/V_{\text{vir}})/V_{\text{sample}}=1/V_{\text{vir}}$, and $C_{\mathrm{inf}}\in[0,1/V_{\text{vir}}]$. #### 4.1.2 Considering replicate wells at a given dilution The ED assay is based on serial dilutions of the sample, with each dilution separated by a fixed dilution factor. We define the dilution factor $\in(0,1)$ as the fraction of the inoculum volume drawn from the previous dilution. For example, if the inoculum for a well, $V_{\text{inoc}}=$100\text{\,}\mathrm{\SIUnitSymbolMicro L}$$, comprises $10\text{\,}\mathrm{\SIUnitSymbolMicro L}$ drawn from the previous dilution and $90\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of dilution media, the dilution factor is $10/100=0.1$. If the serial dilution begins with a dilution of $\mathcal{D}_{1}=0.2$, then the following dilution will be $\mathcal{D}_{2}=0.02$. In Eqn. (1), the dilution under consideration, $\mathcal{D}_{i}$, will affect $n$, the number of virion-sized volumes drawn from the sample and deposited into the wells of the $i^{\mathrm{th}}$ dilution, such that $n=\mathcal{D}_{i}V_{\text{inoc}}/V_{\text{vir}}$. Therefore, the probability that a well at the $i^{\mathrm{th}}$ dilution will _not_ become infected is given by $q_{i}\equiv q_{\text{noinf}}^{\mathcal{D}_{i}}=(1-C_{\mathrm{inf}}V_{\text{vir}})^{\mathcal{D}_{i}V_{\text{inoc}}/V_{\text{vir}}}\approx\exp\left[-C_{\mathrm{inf}}V_{\text{inoc}}\mathcal{D}_{i}\right]$ (3) where $1-q_{i}$ is the probability of infection for a well at the $i^{\mathrm{th}}$ dilution, where $\mathcal{D}_{i}\in[0,1]$. When conducting an ED assay, each dilution in the assay contains a number of independent infection wells (replicates), all inoculated with the same dilution, $\mathcal{D}_{i}$. This is analogous again to drawing balls out of a bag, but this time there are $n_{i}$ draws (replicate wells), and the probability of success (i.e., that a well becomes infected) is simply one minus the probability of failure (i.e., that a well does not become infected, $q_{i}$). The probability that $k_{i}$ out of the $n_{i}$ wells become infected at dilution $\mathcal{D}_{i}$, is described by the Binomial distribution $\text{Binomial}(k=k_{i}\|n=n_{i},p=1-q_{i})=\frac{n_{i}!}{k_{i}!(n_{i}-k_{i})!}\,(1-q_{i})^{k_{i}}\,q_{i}^{n_{i}-k_{i}}\propto(1-q_{\text{noinf}}^{\mathcal{D}_{i}})^{k_{i}}\,q_{\text{noinf}}^{\mathcal{D}_{i}(n_{i}-k_{i})}$ where $n_{i}$ is the number of replicate wells at each dilution, but could be less if any well at dilution $\mathcal{D}_{i}$ are spoiled or contaminated. However, our interest is not in determining $k_{1}$ given $q_{\text{noinf}}$, but rather in determining $q_{\text{noinf}}$ given that we observed $k_{1}$ infected wells out of $n_{1}$ wells in the first column. To this aim, we can make use of Bayes’ theorem which, in our context, can be expressed as $\mathcal{P}(p\|\text{data})=\frac{\mathcal{P}(\text{data}\|p)\ \mathcal{P}(p)}{\int_{0}^{1}\mathcal{P}(\text{data}\|p)\,\mathcal{P}(p)\,\mathrm{d}p}$ or rather $\displaystyle\mathcal{P}_{\text{post,1}}(q_{\text{noinf}}\|k_{1})$ $\displaystyle=\frac{\mathcal{P}(k_{1}\|q_{\text{noinf}})\ \mathcal{P}_{\text{prior}}(q_{\text{noinf}})}{\int_{0}^{1}\mathcal{P}(k_{1}\|q_{\text{noinf}})\,\mathcal{P}_{\text{prior}}(q_{\text{noinf}})\,\mathrm{d}q_{\text{noinf}}}$ $\displaystyle=\frac{\left[(1-q_{\text{noinf}}^{\mathcal{D}_{1}})^{k_{1}}\,q_{\text{noinf}}^{\mathcal{D}_{1}(n_{1}-k_{1})}\right]\,\mathcal{P}_{\text{prior}}(q_{\text{noinf}})}{\int_{0}^{1}\mathcal{P}(k_{1}\|q_{\text{noinf}})\,\mathcal{P}(q_{\text{noinf}})\,\mathrm{d}q_{\text{noinf}}}$ $\displaystyle\mathcal{P}_{\text{post,1}}(q_{\text{noinf}}\|k_{1})$ $\displaystyle\propto\left[(1-q_{\text{noinf}}^{\mathcal{D}_{1}})^{k_{1}}\,q_{\text{noinf}}^{\mathcal{D}_{1}(n_{1}-k_{1})}\right]\,\mathcal{P}_{\text{prior}}(q_{\text{noinf}})$ where $\mathcal{P}_{\text{post,1}}(q_{\text{noinf}}\|k_{1})$ is our updated, posterior belief about $q_{\text{noinf}}$ after having observed $k_{1}$ successes out of $n_{1}$ trials in the first column ($i=1$), and given our prior belief, $\mathcal{P}_{\text{prior}}(q_{\text{noinf}})$, about $q_{\text{noinf}}$ before making this observation. #### 4.1.3 Considering all dilutions of the ED assay As mentioned above, in the 96-well ED assay, each dilution contains a number of independent infection wells (replicates) inoculated with the same sample concentration. This process is then repeated over a series of dilutions, each separated from the previous by a fixed dilution factor. Having observed the fraction of wells infected at the first dilution considered, $\mathcal{D}_{1}$, we have updated our posterior belief about $q_{\text{noinf}}$. We will now use this updated belief as our new prior as we observe our second dilution ($\mathcal{D}_{2}$), such that $\displaystyle\mathcal{P}_{\text{post,2}}(q_{\text{noinf}}\|\vec{k}_{2})$ $\displaystyle\propto\mathcal{P}(k_{2}\|q_{\text{noinf}})\ \mathcal{P}_{\text{post,1}}(q_{\text{noinf}}\|k_{1})$ $\displaystyle\mathcal{P}_{\text{post,2}}(q_{\text{noinf}}\|\vec{k}_{2})$ $\displaystyle\propto\left[(1-q_{\text{noinf}}^{D_{2}})^{k_{2}}\,q_{\text{noinf}}^{D_{2}(n_{2}-k_{2})}\right]\,\left[(1-q_{\text{noinf}}^{\mathcal{D}_{1}})^{k_{1}}\,q_{\text{noinf}}^{\mathcal{D}_{1}(n_{1}-k_{1})}\right]\mathcal{P}_{\text{prior}}(q_{\text{noinf}})$ $\displaystyle\mathcal{P}_{\text{post,2}}(q_{\text{noinf}}\|\vec{k}_{2})$ $\displaystyle\propto\mathcal{Q}(\vec{k}_{2}\|q_{\text{noinf}})\,\mathcal{P}_{\text{prior}}(q_{\text{noinf}})\ ,$ where we introduce $\vec{k}_{2}=\\{k_{1},k_{2}\\}$ and $\mathcal{Q}(\vec{k}_{2}\|q_{\text{noinf}})=\left[(1-q_{\text{noinf}}^{D_{2}})^{k_{2}}\,q_{\text{noinf}}^{D_{2}(n_{2}-k_{2})}\right]\,\left[(1-q_{\text{noinf}}^{\mathcal{D}_{1}})^{k_{1}}\,q_{\text{noinf}}^{\mathcal{D}_{1}(n_{1}-k_{1})}\right]$ as short-hands for convenience. From this, it is easy to extrapolate the posterior likelihood distribution (pPLD) after having observed all $J$ dilutions ($\mathcal{D}_{1}$, $\mathcal{D}_{2}$, …, $\mathcal{D}_{J}$) of the ED assay, namely $\mathcal{P}_{\text{post,J}}(q_{\text{noinf}}\|\vec{k}_{J})\propto\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}})\,\mathcal{P}_{\text{prior}}(q_{\text{noinf}})$ (4) where $\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}})=\left[\prod_{j=1}^{J}(1-q_{\text{noinf}}^{\mathcal{D}_{j}})^{k_{j}}\right]q_{\text{noinf}}^{\sum_{j=1}^{J}\mathcal{D}_{j}(n_{j}-k_{j})}\ .$ (5) Note that this expression is largely equivalent to that obtained by Mistry et al. [7]. #### 4.1.4 Considering the choice of prior In Eqn. (4), we obtained a pPLD for $q_{\text{noinf}}$. Our objective, however, is to estimate the pPLD of $C_{\mathrm{inf}}$, the specific infection concentration in our sample, rather than $q_{\text{noinf}}$. In fact, because both the plaque and ED assays provide an accuracy that is normally distributed in $\log_{10}(C_{\mathrm{inf}})$ rather than $C_{\mathrm{inf}}$, it follows that $\log_{10}(C_{\mathrm{inf}})$ (hereafter $\ell_{\mathrm{Cinf}}$) rather than $C_{\mathrm{inf}}$ is the quantity of interest. We note that $\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}})$ in Eqn. (4) is a probability density function in $\vec{k}_{J}$ rather than in $q_{\text{noinf}}$. As such, a change of variables, say from $q_{\text{noinf}}$ to $\ell_{\mathrm{Cinf}}(q_{\text{noinf}})$, would affect only the prior because $\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}})=\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}}(\ell_{\mathrm{Cinf}}))=\mathcal{Q}(\vec{k}_{J}\|\ell_{\mathrm{Cinf}})$. Thus, the pPLD for $\ell_{\mathrm{Cinf}}$ is given by $\mathcal{P}_{\text{post},J}(\ell_{\mathrm{Cinf}}\|\vec{k}_{J})\propto\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}}(\ell_{\mathrm{Cinf}}))\ \mathcal{P}_{\text{prior}}(\ell_{\mathrm{Cinf}})\ ,$ (6) where $\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}})=\mathcal{Q}(\vec{k}_{J}\|C_{\mathrm{inf}})=\mathcal{Q}(\vec{k}_{J}\|\ell_{\mathrm{Cinf}})$ because $\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}}(...))$ can be written in terms of $q_{\text{noinf}}$, $C_{\mathrm{inf}}$, or $\ell_{\mathrm{Cinf}}$, because it is a probability density function in $\vec{k}_{J}=\\{k_{1},k_{2},...,k_{J}\\}$ rather than in $q_{\text{noinf}}$. To complete this expression, we need to choose a physically and biologically appropriate prior belief regarding $\ell_{\mathrm{Cinf}}$. Prior to conducting the ED assay, we know at least that $C_{\mathrm{inf}}\in[1/V_{\text{Earth}},1/V_{\text{vir}}]$, where $1/V_{\text{vir}}$ is the maximum possible concentration, namely that if the entire volume of the sample is constituted solely of infectious virions, and $1/V_{\text{Earth}}$ is the minimum possible concentration, namely that if there was only one infectious virion left on Earth. As we explain below, these limits are not important; only the fact that they are convincingly physically bounded both from above and below, i.e., $\in(0,\infty)$, is relevant. If we choose our prior to be uniform in $C_{\mathrm{inf}}\in[1/V_{\text{Earth}},1/V_{\text{vir}}]$, namely $\mathcal{P}_{\text{prior}}(C_{\mathrm{inf}})=1/(1/V_{\text{vir}}-1/V_{\text{Earth}})\approx V_{\text{vir}}$, and using the fact that $\mathcal{P}_{\text{prior}}(C_{\mathrm{inf}})\,\mathrm{d}C_{\mathrm{inf}}=\mathcal{P}_{\text{prior}}(\ell_{\mathrm{Cinf}})\,\mathrm{d}\ell_{\mathrm{Cinf}}$, we can write $\mathcal{P}_{\text{prior}}(\ell_{\mathrm{Cinf}})=\mathcal{P}_{\text{prior}}(C_{\mathrm{inf}})\frac{\mathrm{d}C_{\mathrm{inf}}}{\mathrm{d}\ell_{\mathrm{Cinf}}}=V_{\text{vir}}\frac{\mathrm{d}\left[10^{\ell_{\mathrm{Cinf}}}\right]}{\mathrm{d}\ell_{\mathrm{Cinf}}}=V_{\text{vir}}\ln(10)10^{\ell_{\mathrm{Cinf}}}\propto 10^{\ell_{\mathrm{Cinf}}}$ which yields $\mathcal{P}_{\text{post},J}(\ell_{\mathrm{Cinf}}\|\vec{k}_{J})\propto\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}}(\ell_{\mathrm{Cinf}}))\ 10^{\ell_{\mathrm{Cinf}}}\ .$ (7) We see here that the range chosen for the uniform prior in $C_{\mathrm{inf}}$ is not important because it only contributes a constant to our proportionality Eqn. (6). Alternatively, because the ED assay estimates $\ell_{\mathrm{Cinf}}$ rather than $C_{\mathrm{inf}}$, our prior belief about the virus concentration is more appropriately expressed in $\ell_{\mathrm{Cinf}}$ rather than $C_{\mathrm{inf}}$. Again, the bounds of the uniform distribution in $\ell_{\mathrm{Cinf}}$ is unimportant, provided that it is finite in extent such that $\ell_{\mathrm{Cinf}}\in[{\ell_{\mathrm{Cinf}}}_{\text{min}},\log_{10}(1/V_{\text{vir}})]$ where ${\ell_{\mathrm{Cinf}}}_{\text{min}}>-\infty$, such that we can write $\mathcal{P}_{\text{post},J}(\ell_{\mathrm{Cinf}}\|\vec{k}_{J})\propto\mathcal{Q}(\vec{k}_{J}\|q_{\text{noinf}}(\ell_{\mathrm{Cinf}}))\ .$ (8) Figure 8 illustrates the two distinct priors assumed to arrive at Eqns. (7) and (8) and their impact on the posterior $\mathcal{P}_{\text{post},J}(\ell_{\mathrm{Cinf}}\|\vec{k}_{J})$ for the example ED experiment described in Section 2.1. Figure 8A illustrates the consequence of choosing a prior uniform in $C_{\mathrm{inf}}$, i.e., a bias towards higher virus concentrations. This is because a uniform prior in $C_{\mathrm{inf}}$ corresponds to a belief that one is as likely to measure a set of virus concentrations in the range $[0.001,\,0.002]$ as in the range $[1,000,000.001,\,1,000,000.002]$. When plotted on a log-scale, there are $100\times$ more intervals of width 0.001 in $[10^{4},10^{5}]$ than in $[10^{2},10^{3}]$. Thus, this prior corresponds to a belief that the likelihood of measuring a certain virus concentration increases exponentially as $\ell_{\mathrm{Cinf}}$ increases linearly. In contrast, a prior uniform in $\ell_{\mathrm{Cinf}}$ corresponds to a belief that one is as likely to measure a set of virus concentrations in the range $[0.001,\,0.002]$ than in the range $[1,000,000,\,2,000,000]$, or rather in the range $[1,2]\times 10^{-3}$ than in the range $[1,2]\times 10^{6}$. As such, a uniform distribution in $\ell_{\mathrm{Cinf}}$ is more physically and biologically sensible and therefore was chosen for our estimation method. Figure 8: Impact of the choice of prior on the posterior distribution for $\ell_{\mathrm{Cinf}}$. (A) Non-normalized priors for $\log_{10}$(specific infections, /mL)$=\ell_{\mathrm{Cinf}}$ that are uniform in either $C_{\mathrm{inf}}$ or $\ell_{\mathrm{Cinf}}$ are shown. A prior uniform in $C_{\mathrm{inf}}$ is biased towards larger values of $\ell_{\mathrm{Cinf}}$. (B) Updated posterior belief about $\ell_{\mathrm{Cinf}}$ for each of the two prior beliefs shown in (A), as per Eqns. (7) and (8), after having observed the ED assay example provided in Section 2.1. While the prior uniform in $C_{\mathrm{inf}}$ yields a pPLD with a mode of $\ell_{\mathrm{Cinf}}=6.21$, that for a prior uniform in $\ell_{\mathrm{Cinf}}$ yields a mode of $\ell_{\mathrm{Cinf}}=6.18$. ### 4.2 Calculation of midSIN’s outputs One of the graphical outputs of midSIN is the non-normalized PLD of $\ell_{\mathrm{Cinf}}$ given the number of wells that were infected at each dilution, $\vec{k}_{J}$, like that shown in Figure 2(left panel), computed as $\displaystyle\mathcal{U}_{\text{post}}(\,\ell_{\mathrm{Cinf}}\,\|\,\vec{k}_{J}\,)$ $\displaystyle=\prod_{j=1}^{J}\ \frac{n_{j}!}{k_{j}!\ (n_{j}-k_{j})!}\ \cdot\ p_{j}^{k_{j}}\ \cdot\ (1-p_{j})^{n_{j}-k_{j}}$ (9) $\displaystyle\text{where}\ \ p_{j}$ $\displaystyle=1-\exp\left[\ -10^{\ell_{\mathrm{Cinf}}}\cdot V_{\text{inoc}}\cdot\mathcal{D}_{j}\ \right]\ .$ (10) While $\mathcal{U}_{\text{post}}$ is not the normalized likelihood of $\ell_{\mathrm{Cinf}}$, its maximum value at its mode (${\ell_{\mathrm{Cinf}}}_{\text{,mode}}$) is the normalized probability of observing this particular ED plate outcome ($\vec{k}_{J}$) out of all other possible plate outcomes, assuming the true, specific infection concentration in the sample is ${\ell_{\mathrm{Cinf}}}_{\text{,mode}}$. Another visual output of midSIN is a graphical representation of the theoretical number of wells that would be infected given the most likely $\ell_{\mathrm{Cinf}}$, like that shown in Figure 2(right panel). It is computed following $N_{\text{wells infected}}(x)=N_{\text{wells total}}\left[1-\exp\left(-10^{{\ell_{\mathrm{Cinf}}}_{\text{,mode}}}\,V_{\text{inoc}}\,10^{-x}\right)\right]\ ,$ (11) where $x$ is the $\log_{10}$ of the dilution such that $\mathcal{D}=10^{-x}$ is the dilution. It corresponds to the continuous equivalent of this quantity which is discrete in the ED assay, namely $\mathcal{D}_{i}=10^{-x_{i}}$ which is the $i^{\mathrm{th}}$ dilution of the sample. As such, $\mathcal{D}_{i}=$ (minimum dilution) $\cdot$ $\text{(dilution factor between columns)}^{i-1}$ where $i\in{[1,J]}$. For example, if the dilution of the least diluted column is $0.1=10^{-1}$ and the dilution factor between dilutions in the ED assay is such that it halves the concentration between each dilution, i.e., $1/2=2^{-1}=10^{-\log_{10}(2)}\approx 10^{-0.301}$, then $\mathcal{D}_{i}=10^{-1}\cdot 10^{-0.301\cdot(i-1)}$ such that $\mathcal{D}_{1}=10^{-1}$, $\mathcal{D}_{2}=10^{-1.301}$, $\mathcal{D}_{3}=10^{-1.602}$, and so on, such that $x_{1}=1$, $x_{2}=1.301$, $x_{3}=1.602$, and so on. In the graphical representation of the ED assay, the edges of the grey bands flanking the theoretical blue curve correspond to Eqn. (11) wherein ${\ell_{\mathrm{Cinf}}}_{\text{,mode}}$ has been replace by the 68% and 95% CI values for $\ell_{\mathrm{Cinf}}$. These CI bands _do not_ correspond to the 68% and 95% CI of the expected number of infected wells at each dilution given ${\ell_{\mathrm{Cinf}}}_{\text{,mode}}$. The sample dilution corresponding to $1\text{\,}\tcid$ estimated based on the biased RM and SK approximations (right panels) are converted to (left panels) based on $$1\text{\,}\tcid$=$\mathrm{e}^{\gamma=0.5772}\text{\,}\idnew$=$1.781\text{\,}\idnew$$ [10, 4]. In contrast, the $\log_{10}(\idnew/\mathrm{m}\mathrm{L})$ computed by midSIN can be converted to a true (unbiased) estimate of $\log_{10}(\tcid)$ using $$1\text{\,}\tcid$=$1/\ln(2)\text{\,}\idnew$=$1.44\text{\,}\idnew$$ [2]. ### 4.3 Infection concentration measures of influenza A virus samples #### 4.3.1 Cell culture Madin-Darby canine kidney cells (MDCKs) were cultured in growth media (complete MEM media with 5% heat-inactivated FBS), in tissue culture treated T75 flasks, at 37 °C with 5% CO2 and 95% relative humidity. Cells were split 1/10 every 3–4 days or upon reaching approximately 95% confluency. One passage of cells was expanded for use by both researchers in one experiment to quantify the 50% tissue culture infectious dose () and plaque forming units () of one viral strain. #### 4.3.2 Viral stocks Stocks of influenza A/Puerto Rico/8/34 (H1N1) (PR8) and influenza A/California/4/09 (Cali/09) were stored at -80 °C and thawed on ice immediately before use. The and of stock viruses was known to both researchers prior to this study. Serial dilutions were made in MDCK infection media (complete MEM media with 4.25% BSA) and dilutions were made by each researcher independently for titering. ‘Researcher A’ and ‘Researcher B’ independently performed the and assays of one viral strain for one experiment on the same day using the same viral stock, reagents, and passage of cells. Each experiment was performed on a separate day (Fig. 4). #### 4.3.3 Plaque assay MDCKs were seeded in six-well plates ($5.5\times 10^{5}\text{\,}\mathrm{c}\mathrm{e}\mathrm{l}\mathrm{l}\mathrm{s}\mathrm{/}\mathrm{m}\mathrm{L}$, $2\text{\,}\mathrm{m}\mathrm{L}\mathrm{/}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}$) and grown to 90% confluency overnight (37 °C, 5% CO2, 95% relative humidity). Each six-well plate contained 10-fold serial dilutions plated in singlet as well as a negative control and five 6-well plates were carried out per experiment. Cells were washed twice with PBS w/ Ca2+Mg2+ before the addition of $500\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of viral dilutions per well. After $1\text{\,}\mathrm{h}$ at room temperature on a rocker, the inoculum was aspirated, cells were washed with PBS containing Ca2+Mg2+ (PBS w/ Ca2+Mg2+) (Gibco), and gently covered with $2\text{\,}\mathrm{mL}$ of agarose overlay (complete media, 4.25% BSA, 0.9% agarose, $1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{L}$ TPCK-Trypsin). After drying the overlay at room temperature, plates were inverted and incubated (37 °C, 5% CO2, 95% relative humidity) for $3\text{\,}\mathrm{d}$ (PR8) or $4\text{\,}\mathrm{d}$ (Cali/09). Plaques were visualized by staining cells with 0.1% crystal violet solution in 37% formaldehyde for $30\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$ and counted by ‘Researcher A’ or ‘Researcher B’ on their respective experiments (Fig. 4). #### 4.3.4 assay MDCKs were seeded in 96-well flat bottom plates ($5\times 10^{4}$ cells/$100\text{\,}\mathrm{\SIUnitSymbolMicro L}$, $100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{L}\mathrm{/}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}$) and grown to 80% confluency overnight (37 °C, 5% CO2, 95% relative humidity). For each experiment, 4 replicate wells, at each of 7 different dilutions separated by a 10-fold dilution, were infected, and the dilution series was performed 5 times. Cells were washed with PBS w/ Ca2+Mg2+ before the addition of $100\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of viral dilutions per well. After $1\text{\,}\mathrm{h}$ at room temperature on a rocker, the inoculum was aspirated and replaced with $100\text{\,}\mathrm{\SIUnitSymbolMicro L}$ of infection media containing $1\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{g}\mathrm{/}\mathrm{m}\mathrm{L}$ TPCK-Trypsin. Cells were incubated (37 °C, 5% CO2, 95% relative humidity) for $3\text{\,}\mathrm{d}$ (PR8) or $4\text{\,}\mathrm{d}$ (Cali/09). Supernatants were used to do a hemagglutination (HA) assay with chicken red blood cells. HA assays were performed and read by ‘Researcher A’ or ‘Researcher B’ on their respective experiments. #### 4.3.5 Statistical analysis The data points reported in Figure 4C,D were computed by taking each of the 5 replicates measured with either the PFU, RM, or SK and the 5 replicates measured via (5 replicates $\times$ 5 replicates = 25 pairs) for each of the 2 experiments by each of the 2 researchers, yielding 100 pairs. For each pair, the $\log_{10}$ of ratio of either PFU, RM or SK over SIN was computed. The mean and standard deviation of the resulting 100 $\log_{10}(\mathrm{ratio})$ were computed and are reported in Figure 4C,D. The statistical significance ($p$-value) of the differences between (PFU,RM,SK) and () was computed using the Mann-Whitney U test (scipy.stats.mannwhitneyu). ## References * [1] C. A. A. Beauchemin, Y.-I. Kim, Q. Yu, G. Ciaramella, and J. P. DeVincenzo. Uncovering critical properties of the human respiratory syncytial virus by combining in vitro assays and in silico analyses. PLOS ONE, 14(4):e0214708, 15 April 2019. doi:10.1371/journal.pone.0214708. * [2] W. R. Bryan. Interpretation of host response in quantitative studies on animal viruses. Ann. N. Y. Acad. Sci., 69(4):698–728, 16 December 1957. doi:10.1111/j.1749-6632.1957.tb49710.x. * [3] R. Dulbecco. Production of plaques in monolayer tissue cultures by single particles of an animal virus. Proc. Natl. Acad. Sci. U.S.A., 38(8):747–752, August 1952. doi:10.1073/pnas.38.8.747. * [4] Z. Govindarajulu. Statistical techniques in bioassay, chapter 4. The Logit Approach, pages 35–90. Karger, Basel; New York, 2nd edition, 2001. doi:10.1159/isbn.978-3-318-00617-9. * [5] G. Kärber. Beitrag zur kollecktiven behandlung pharmakologischer reihenversuche. Archiv f. Experiment. Pathol. u. Pharmakol., 162(4):480–483, July 1931. doi:10.1007/BF01863914. * [6] D. D. LaBarre and R. J. Lowy. Improvements in methods for calculating virus titer estimates from TCID50 and plaque assays. J. Virol. Methods, 96(2):107–126, August 2001. doi:10.1016/S0166-0934(01)00316-0. * [7] B. A. Mistry, M. R. D’Orsogna, and T. Chou. The effects of statistical multiplicity of infection on virus quantification and infectivity assays. Biophys. J., 114(12):2974–2985, 19 June 2018. doi:10.1016/j.bpj.2018.05.005. * [8] L. J. Reed and H. Muench. A simple method of estimating fifty per cent endpoints. Am. J. 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# SYNTACTICALLY GUIDED GENERATIVE EMBEDDINGS FOR ZERO-SHOT SKELETON ACTION RECOGNITION ###### Abstract We introduce SynSE, a novel syntactically guided generative approach for Zero- Shot Learning (ZSL). Our end-to-end approach learns progressively refined generative embedding spaces constrained within and across the involved modalities (visual, language). The inter-modal constraints are defined between action sequence embedding and embeddings of Parts of Speech (PoS) tagged words in the corresponding action description. We deploy SynSE for the task of skeleton-based action sequence recognition. Our design choices enable SynSE to generalize compositionally, i.e., recognize sequences whose action descriptions contain words not encountered during training. We also extend our approach to the more challenging Generalized Zero-Shot Learning (GZSL) problem via a confidence-based gating mechanism. We are the first to present zero-shot skeleton action recognition results on the large-scale NTU-60 and NTU-120 skeleton action datasets with multiple splits. Our results demonstrate SynSE’s state of the art performance in both ZSL and GZSL settings compared to strong baselines on the NTU-60 and NTU-120 datasets. Index Terms— ZSL, skeleton action recognition, VAE, deep learning, language and vision ## 1 Introduction Advances in human action recognition have been predominantly driven by the abundance of online RGB videos. However, with the advent of accurate depth sensing technologies (e.g. Microsoft Kinect, Intel Real Sense), action recognition from 3D human skeleton data has also gained traction. Skeleton representations can be advantageous since they are compact, robustly separate the action subject (human) from background and enable privacy-preserving action capture. The introduction of large scale skeleton action datasets such as NTU-60 [1] and NTU-120 [2], have allowed researchers to develop high-performance approaches for skeleton action recognition [3, 4, 5, 6]. However, these approaches are resource intensive, prone to overfitting and fail to generalize on classes outside the training set. Therefore, there is a strong motivation for Zero-Shot Learning (ZSL) approaches in an attempt to readily generalize across actions outside the training set. In ZSL, visual representations and corresponding labels for seen classes are assumed to be available. During test time, the model is evaluated using data from unseen classes which are not present during training. Typically, side information (e.g. class attributes) is leveraged to transfer knowledge from the seen to unseen classes. As a popular approach, ZSL approaches employ a shared embedding strategy wherein the visual (image or video) features and semantic attribute features of the corresponding class labels are projected into a common embedding space [7, 8, 9, 10, 11, 12, 13]. Generative ZSL approaches present an alternative strategy wherein unseen samples [14] or features from unseen samples [11] are generated using Generative Adversarial Networks(GANs). Owing to the instability in training GANs, Variational Auto- Encoders(VAEs) [15, 16, 17] have also been used for feature generation. ZSL has been previously explored for skeleton action recognition. In the only available work [18] (arXiv), embedding based methods [8, 19] are used to align visual feature embedding of skeleton action sequence with the text embedding of the descriptive action phrase (e.g ‘take off jacket’, ‘put on glasses’). The visual features are represented by the final layer features of a skeleton action recognition model and the action phrase embedding is typically obtained by pooling the individual embeddings of words comprising the phrase. However, this approach does not enable alignment of visual embedding with respect to the individual contributors of phrase semantics - the verb (‘action’) and the noun(s) (‘participating entities’). This inability is a major shortcoming since it does not enable generalization, i.e., being able to map the test action sequences to a description containing novel combinations of verbs and nouns, some of which might be from training action descriptions themselves. To address these shortcomings, we propose an approach wherein the visual embedding is aligned based on the Parts of Speech (PoS) tags (verb, noun) of the phrasal words. Instead of directly mapping the visual and PoS-wise embeddings in a discriminative setting [20], we use group (per-PoS, visual) specific generative models with cross-group latent objective [17] for improved ZSL performance (Section 2). We also extend our approach to the Generalized Zero-Shot Learning (GZSL) problem, a more challenging and realistic variant of ZSL wherein good performance is required from seen and unseen classes. We do so by incorporating a confidence-based gating mechanism. (Section 2.5). Our approach enables state-of-the-art performance for ZSL and GZSL compared to strong baselines on the NTU-60 and the much larger NTU-120 dataset. (Section 4). The source code and pre-trained models can be accessed at https://github.com/skelemoa/synse-zsl. Fig. 1: Architectural diagram for our approach (SynSE). (left) The dotted path represents the process flow for the GZSL setting while the solid arrows represent the flow for ZSL. The Generative Multimodal Alignment Module is detailed on right side. It contains modality VAEs, where Part-of-Speech (PoS) specific latent generative embeddings $z_{v}$ (verb), $z_{n}$ (noun) are jointly aligned with segments ($z_{s,v},z_{s,n}$) of latent generative skeleton embedding $z_{s}$ via cross-modal alignment - refer Section 2 for more details. Note that the RGB images have been included for reference. Only the skeleton sequence is provided as input to the network. ## 2 SynSE ### 2.1 Problem definition Let $D_{tr}=\\{(x_{s}^{tr},y_{s}^{tr})\\}$ denote the set of $N_{tr}$ training samples where $x_{s}^{tr}$ denotes visual feature embedding of a skeleton action sequence, $y_{s}^{tr}\in Y_{s}$ is the corresponding member from the label set of seen classes. On similar lines, $D_{u}=\\{(x_{s}^{u},y_{s}^{u})\\}$ denotes the set of test samples with the subscript $u$ standing for unseen. Suppose $\hat{y}$ represents the test time class prediction. For ZSL, we have $\hat{y}\in Y_{u},Y_{s}\cap Y_{u}=\emptyset$ while for GZSL, we have $\hat{y}\in Y_{u}\cup Y_{s},Y_{s}\cap Y_{u}=\emptyset$. For simplicity, we drop the subscript for seen, unseen and refer to the class names as $y$ and the visual feature embedding as $x_{s}$. ### 2.2 Learning modality-wise latent generative spaces A crucial requirement for a ZSL approach is the ability to correctly map novel inputs. For this, we employ a Variational Auto Encoder (VAE) [21] as the base architecture to learn the generative space of latent representations. A VAE is trained by maximizing the Evidence Lower Bound (ELBO): $\mathcal{L}=\mathbb{E}_{q_{\phi}(z\|x)}[\log p_{\theta}(x\|z)]-\beta D_{KL}(q_{\phi}(z\|x)\|\|p_{\theta}(z))$ (1) Here, the first term on the right hand side is the reconstruction error and the second term is the Kullback-Leibler divergence between likelihood $p_{\theta}(z)$ and the prior $q_{\phi}(z\|x)$. $\beta$ is a hyperparameter which acts as a trade-off between the two error terms. A popular choice for the prior is the multivariate Gaussian distribution, $q_{\phi}(z\|x)=\mathcal{N}(\mu,\Sigma)$. The VAE maps the input $x$ initially to representations for $\mu,\Sigma$ and eventually to the randomized latent representation $z$ via the reparameterization trick [21]. The first stage in our approach involves learning individual latent generative latent spaces for visual and linguistic representations. This is achieved by using a VAE for each space. To enable semantically aware compositional generalization, the text description for class label $y$ is tokenized into constituent Part-of-Speech (PoS) specific sets - $y_{v}$ for verb and $y_{n}$ for noun. The tokens are encoded using a natural language encoder module to obtain the corresponding PoS-wise embeddings $e_{v}$ and $e_{n}$ (see Figure 1). Since our approach employs independent VAEs for skeleton ($s$) and linguistic ($v,n$) representations, the overall cost function for a single sample can be written as: $\displaystyle\mathcal{L}_{VAE}=\sum_{m\in\\{s,v,n\\}}\mathbb{E}_{q_{\phi}(z_{m}\|x_{m})}[\log p_{\theta}(x_{m}\|z_{m})]-$ (2) $\displaystyle\beta D_{KL}(q_{\phi}(z_{m}\|x_{m})\|\|p_{\theta}(z_{m}))$ ### 2.3 Cross-modal alignment The VAEs optimize latent representations for individual modalities. To achieve alignment between the skeleton sequence and linguistic latent representations, a cross-modal reconstruction objective is formulated [17]. First, the latent embeddings from the PoS embeddings ($z_{v}$, $z_{n}$) are concatenated (see $z_{l}$ in Figure 1) and the result $x_{l}$ is used to reconstruct the visual representation via the skeleton representation VAE’s posterior decoder $D_{s}$. Next, the skeleton sequence latent embedding ($z_{s}$) is uniformly mapped to as many embeddings ($z_{s,v}$, $z_{s,n}$) as the number of PoS tags. Complementary to the processing of $z_{l}$, each of the split embedding is used to reconstruct the corresponding PoS token embedding ($e_{n}$, $e_{v}$) via the corresponding PoS token embedding’s decoder ($D_{v}$, $D_{n}$). Overall, the cross-modal reconstruction objective for a training sample is formulated as: $\mathcal{L}_{CMR}=\|x_{s}-D_{s}(z_{l})\|_{2}+\sum_{m\in\\{v,n\\}}\|e_{m}-D_{i}(z_{s,m})\|_{2}$ (3) Finally, the VAE loss and the cross-modal reconstruction loss are optimized together as: $\mathcal{L}=\mathcal{L}_{VAE}+\alpha\mathcal{L}_{CMR}$ (4) where $\alpha$ is a trade-off weight factor. ### 2.4 Zero-shot Classification using Latent Embedding The PoS tag embeddings of each unseen class are respectively transformed by the PoS encoders ($E_{v},E_{n}$) and used to obtain samples from the latent generative space ($z_{l}$ \- see Figure 1). A softmax classifier $f_{u}:z_{l}\rightarrow Y_{u}$ is trained to classify these latent samples into the unseen classes. The cross-modal VAE setup described earlier aims to align the visual features with the language features in the common latent generative space. In other words, $z_{s}$ and $z_{l}$ are optimized to be interchangeable. Taking advantage of this, during inference, the unseen class skeleton sequence representation $x_{s}$ is first obtained. Supplying $x_{s}$ to the visual VAE encoder ($E_{s}$) enables us to obtain the mean visual latent embedding ($\mu_{s}$) of the sequence111Note that $z_{s}=\mu_{s}+\Sigma_{s}\odot\mathcal{E}$, where $\mathcal{E}=\mathcal{N}(0,I)$ by the VAE reparameterization trick.. The corresponding class prediction is obtained using $\mu_{s}$ and the classifier $f_{u}$ mentioned previously. ### 2.5 Gating Module for GZSL For a given skeleton sequence representation $x_{s}$, the probability distribution $c_{s}$ over seen classes is obtained from the skeleton action recognition model $f_{{s}}:x_{s}\rightarrow Y_{s}$ from which the action sequence embedding has been sourced all along. The unseen class classifier $f_{{u}}:E_{s}(x_{s})\rightarrow Y_{u}$, is a part of our ZSL approach described in the previous section which provides the unseen class probabilities $c_{u}$. The probability distribution over all the classes can be written as: $\displaystyle p(y\|x)=c_{s}p^{gate}(s;c_{s},c_{u})+c_{u}p^{gate}(u;c_{s},c_{u})$ (5) Further, we use a gating model (due to its superior performance for GZSL in other domains) to first decide whether the sample belongs to a seen class or an unseen class [22]. For this, the seen and unseen class probabilities are used as features to train a probabilistic binary classifier $p^{gate}(s;c_{s},c_{u})$ [22]. The resulting outputs are used to determine the probability distribution over all ($Y_{s}\cup Y_{u}$) classes (Equation 5). ### 2.6 Implementation Details Method \| NTU-60 \| NTU-120 ---\|---\|--- $55/5$ split \| $48/12$ split \| $110/10$ split \| $96/24$ split Jasani [18] (preprint) \| $65.53$ \| - \| - \| - ReViSE [12] \| $53.91$ \| $17.49$ \| $55.04$ \| $32.38$ JPoSE [20] \| $64.82$ \| $28.75$ \| $51.93$ \| $32.44$ CADA-VAE [17] \| $\mathbf{76.84}$ \| $28.96$ \| $59.53$ \| $35.77$ SynSE (ours) \| $75.81$ \| $\mathbf{33.30}$ \| $\mathbf{62.69}$ \| $\mathbf{38.70}$ Table 1: ZSL accuracy (%) on the NTU-60 and NTU-120 datasets. \| NTU-60 \| NTU-120 ---\|---\|--- Method \| ($55/5$) random split \| ($48/12$) random split \| ($110/10$) random split \| ($96/24$) random split \| s \| u \| h \| s \| u \| h \| s \| u \| h \| s \| u \| h ReViSE [12] \| $74.23$ \| $34.73$ \| $29.22$ \| $62.36$ \| $20.77$ \| $31.16$ \| $48.69$ \| $44.84$ \| $46.68$ \| $49.66$ \| $25.06$ \| $33.31$ JPoSE [20] \| $64.44$ \| $50.29$ \| $56.49$ \| $60.49$ \| $20.62$ \| $30.75$ \| $47.66$ \| $46.40$ \| $47.05$ \| $38.62$ \| $22.79$ \| $28.67$ CADA-VAE [17] \| $69.38$ \| $61.79$ \| $\mathbf{65.37}$ \| $51.32$ \| $27.03$ \| $35.41$ \| $47.16$ \| $49.78$ \| $48.44$ \| $41.11$ \| $34.14$ \| $37.31$ SynSE \| $61.27$ \| $56.93$ \| $59.02$ \| $52.21$ \| $27.85$ \| $\mathbf{36.33}$ \| $52.51$ \| $57.60$ \| $\mathbf{54.94}$ \| $56.39$ \| $32.25$ \| $\mathbf{41.04}$ SynSE (+ softgating) \| $65.17$ \| $59.51$ \| $62.21$ \| $69.23$ \| $21.74$ \| $33.09$ \| $74.76$ \| $37.68$ \| $50.10$ \| $72.54$ \| $21.09$ \| $32.67$ SynSE (- temp. scaling) \| $74.45$ \| $37.46$ \| $49.84$ \| $45.74$ \| $25.87$ \| $33.05$ \| $67.87$ \| $38.05$ \| $48.77$ \| $66.97$ \| $25.55$ \| $36.99$ SynSE (+ CADA-VAE’s GZSL) \| $82.70$ \| $0$ \| $0$ \| $87.63$ \| $0$ \| $0$ \| $80.43$ \| $0$ \| $0$ \| $82.46$ \| $0$ \| $0$ Table 2: GZSL Accuracy (%) for seen (s) classes, unseen (u) classes and their harmonic mean (h) on NTU-60 and NTU-120 datasets Visual and Textual features: The visual features $x_{s}$, are realised using the $256$ dimensional penultimate layer feature from 4s-ShiftGCN [3], a state- of-the-art deep network for skeleton action recognition. To maintain the zero- shot assumption, we train 4s-ShiftGCN only on the seen classes. We use the Sentence BERT model [23] to obtain 1024-dimensional PoS-wise word embeddings. Before splitting into verbs and nouns, the class names are modified to fill the missing PoS tag, e.g. ‘reading’ is changed to ‘reading book’, ‘drop’ to ‘drop object’, ‘headache’ to ‘have headache’. For actions where adding the missing tag (usually a noun) would be unreasonable (e.g. ‘jump up’, ‘stand up’), the average of all noun embeddings is used as a placeholder. Architectural Details: We have a single dense layer as the encoder ($E_{s}$) and decoder ($D_{s}$), which map the input features ($x_{s},e_{v},e_{n}$) to the latent space ($z_{s},z_{v},z_{n}$) and vice versa. $x_{s}$ is $256$-dimensional and $e_{v},e_{n}$ are $1024$-dimensional. The size of the latent dimension is based on the number of unseen classes. For small number ($5$) of unseen classes, the skeleton latent dimension is set as $100$ and the latent dimension for the PoS tags is $50$. For larger number of unseen classes, the latent dimensions are doubled to $200$ and $100$ for the skeleton latent dimension and PoS tags respectively. The ZSL classifier has a single dense layer which takes latent features as input and returns the softmax probabilities for unseen classes. Training Details: The VAEs within the Generative Multimodal Alignment Module are optimized using the Adam optimizer with a learning rate of $1e^{-4}$ and a batch size of $64$. The VAEs are trained using a cyclic annealing schedule [24] for multiple cycles to mitigate the vanishing KL divergence problem. The $\beta$ hyperparameter for the KL divergence is turned on after $1000$ epochs, starting with $0$ and is increased with a rate of $0.0021$ per epoch in each cycle. Similarly, the $\alpha$ parameter for the cross modal reconstruction is turned on after $1400$ epochs for experiments on NTU-60 and $1500$ epochs on NTU-120 and is kept constant with a value of $1$. One cycle is completed in $1700$ epochs for NTU-60 and $1900$ epochs for NTU-120 dataset. The zero-shot classifier (Section 2.4) is also optimized using Adam with a learning rate for $1e^{-3}$. $500$ features per unseen class are generated and the classifier is trained for $300$ epochs. The input to the gating model (Section 2.5) is the concatenation of the top $k$ softmax probabilities from the outputs of the seen and unseen classifiers. We set $k$ equal to the number of unseen classes and we temperature scale [25] the seen classifier probabilities as well. The gating model is implemented as a binary logistic regression classifier and optimized using LBFGS solver from the scikit-learn library with the default aggressiveness hyperparameter ($C=1$). For training the gating model, we set aside a few samples from the training set and refer to them as the gating train set. Similarly, we set aside a few samples from validation set (gating validation set). We train the gating model using the gating train set and determine the hyperparameters (temperature coefficient, threshold) using the gating validation set. The gating module is configured for use in ‘hard’ gating mode wherein $p^{gate}(s;c_{s},c_{u})$ and $p^{gate}(u;c_{u},c_{u})$ in Equation 5 take binary values [22]. ## 3 Experiments ### 3.1 Datasets NTU-60 [1]: This is a large-scale dataset curated for 3D human action analysis. It contains $56{,}880$ samples belonging to $60$ action classes, with $40$ different subjects captured from $80$ distinct camera viewpoints. The action sequences of skeleton representations are in the form of 3D coordinates for $25$ human body joints. We create two splits for ZSL evaluation - a $55/5$ split with $55$ seen classes, $5$ randomly chosen unseen classes and a more challenging $48/12$ split. NTU-120 [2]: NTU-120 builds upon NTU-60 and contains $60$ additional fine- grained action classes. It contains a total of $114{,}480$ samples spread across $120$ actions performed by $106$ different subjects captured from $155$ different camera viewpoints. Analogous to the NTU-60 ZSL evaluation setup, we create two splits - $110$ (seen)/$10$ (unseen) and $96$ (seen)/$24$ (unseen). ### 3.2 Experimental Details We perform ZSL and GZSL experiments on the NTU-60 and NTU-120 datasets on the described splits. Since no previous works for skeleton ZSL exist, we modify representative state-of-the-art approaches from other problem domains and implement from scratch. CADA-VAE [17] learns a generative latent space under a cross aligned and distribution aligned objective. Since we found the distribution alignment objective to induce instability in training, we omit it during optimization. ReViSE [12] aims to align the latent embeddings realised via autoencoders using a Maximum Mean Discrepancy criterion. JPose [20] attempts to learn PoS aware embeddings of word2vec representations for video retrieval tasks. It learns a series of progressively refined embeddings under inter/intra modal constraints in a discriminative setting. For fair comparison, the visual features and PoS embeddings are the same as ones used in our approach (Section 2.6). Component \| Default in SynSE \| Ablation \| Accuracy ---\|---\|---\|--- Language Embedding \| Sentence-BERT [23] \| Word2Vec [26] \| $60.76$ Visual Features \| 4s-ShiftGCN [3] \| MS-G3D [4] \| $68.80$ Latent Dimension \| $100$ \| $50$ \| $73.83$ Latent Dimension \| $100$ \| $200$ \| $74.67$ Latent features \| $500$ \| $250$ \| $73.89$ Latent features \| $500$ \| $1000$ \| $73.82$ \| original \| \| $\mathbf{75.81}$ Table 3: SynSE ZSL accuracy (%) on the NTU-60 dataset for various ablations (55/5 split). ## 4 Results ### 4.1 ZSL results Table 1 shows the ZSL results of the various approaches on the NTU-60 and NTU-120 datasets. For the $55/5$ split of NTU-60, the VAE-based generative approaches significantly outperform the discriminative embedding based approaches. SynSE’s performance is comparable to that of CADA-VAE. Predictably, results on the more challenging $48/12$ split show that having a larger number of unseen classes impacts performance across the board. However, SynSE offers significant improvement over other baseline approaches, including CADA-VAE. On the larger NTU-120 dataset, SynSE outperforms other methods on both the splits. ### 4.2 GZSL results Since we use a gating-based strategy for GZSL in SynSE (Section 2.5), we compare against other baselines by incorporating the same strategy. Specifically, the seen class classifier is kept the same while the specific baseline approach provides the corresponding unseen class probabilities. Following standard convention for GZSL, we report the average seen class accuracy (s), the average unseen class accuracy (u) and their harmonic mean (h). Table 2 shows the results for datasets and the associated pre-defined splits. Similar to the trend in ZSL for the $55/5$ split of NTU-60, SynSE performs poorer compared to CADA-VAE on the harmonic scale for the 55/5 NTU-60 split. However, it outperforms other approaches on the harmonic scale for other splits of NTU-60 and NTU-120. We also compare our hard gating strategy [27] with the soft gating based strategy [22]. The results in Table 2 show that soft gating is biased towards seen classes, resulting in poor harmonic accuracy. Additionally, Table 2 also shows the significant performance hit when temperature scaling (Sec. 2.6) is removed [25]. To further demonstrate the effectiveness of our GZSL strategy (i.e. gating model), we explored an alternative based on the approach used for CADA-VAE [17], which does not involve gating. As Table 2 shows, the resulting setup ends up too heavily skewed for seen classes and is unable to classify the unseen classes. ### 4.3 Ablations We perform ablation experiments on the 55/5 split of the NTU-60 dataset to analyse the importance of the building blocks of our alignment module and design choices affecting its ZSL performance. As the results show (Table 3), Sentence-BERT is a superior choice to Word2Vec [26] for embedding PoS-tagged words. Similarly, 4s-ShiftGCN provides better visual embeddings compared to another state-of-the-art skeleton action recognition model MS-G3D [4]. We further ablate on the architectural choices by varying the size of the latent dimension. As shown in Table 3, we see that both an increase and decrease in the size of the latent embedding causes reduction in ZSL performance. In order to validate our choice of $500$ latent features per class, we experiment with varying number of latent features with results as shown in Table 3. ## 5 Conclusion In this work, we have presented SynSE, a compositional approach for infusing latent visual representations of skeleton-based human actions with syntactic information derived from corresponding textual descriptions. We present the first set of zero-shot skeleton action recognition results on the large-scale NTU-60 and NTU-120 datasets. Our experiments show that SynSE outperforms strong baselines for ZSL and the more challenging GZSL setup. 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[0]RY is partially funded by NSF DMS 1916037. HM is partially funded by JSPS KAKENHI 19H04987, 19H05024 and 19K06957. KM is partially funded by JSPS KAKENHI 18K11485 url]polytopes.net 1]organization=Department of Operations Research, Naval Postgraduate School, city=Monterey, postcode=93943, state=CA, country=USA 2]organization=Graduate School of Science and Technology, Kwansei Gakuin University, city=Sanda, postcode=669-1337, state=Hyogo, country=Japan 3]organization=Graduate School of Medicine, Osaka City University, postcode=558-8585, state=Osaka, country=Japan [orcid=0000-0002-9258-6541] [1] [1]Corresponding author # Tropical Support Vector Machines: Evaluations and Extension to Function Spaces Ruriko Yoshida<EMAIL_ADDRESS>[ Misaki Takamori [ Hideyuki Matsumoto [ Keiji Miura<EMAIL_ADDRESS> ###### Abstract Support Vector Machines (SVMs) are one of the most popular supervised learning models to classify using a hyperplane in an Euclidean space. Similar to SVMs, tropical SVMs classify data points using a tropical hyperplane under the tropical metric with the max-plus algebra. In this paper, first we show generalization error bounds of tropical SVMs over the tropical projective torus. While the generalization error bounds attained via Vapnik-Chervonenkis (VC) dimensions in a distribution-free manner still depend on the dimension, we also show numerically and theoretically by extreme value statistics that the tropical SVMs for classifying data points from two Gaussian distributions as well as empirical data sets of different neuron types are fairly robust against the curse of dimensionality. Extreme value statistics also underlie the anomalous scaling behaviors of the tropical distance between random vectors with additional noise dimensions. Finally, we define tropical SVMs over a function space with the tropical metric. ###### keywords: Extreme Value Statistics Function Spaces Max-plus Algebra Supervised Learning Tropical Geometry We obtained generalization error bounds of tropical Support Vector Machines (SVMs) via the Vapnik-Chervonenkis dimensions of tropical hyperplanes using Tropical Radon Lemma by Jaggi et al. (2008). We demonstrated theoretically by extreme value statistics that the tropical SVMs are robust against the curse of dimensionality for classifying data points from two Gaussian distributions as well as experimentally recorded activities from different neuron types. We defined tropical SVMs over a function space to enable the classification of curves, which is a task we encounter frequently in practice, such as classifications on neuronal tuning curves. ## 1 Introduction In data science, one of the well-known challenges we face is to classify data points with a large number of predictors. For example, in image processing, in order to discriminate one object, such as missiles or face recognition, from others in images, these images are described as a vector in pixels which are typically in a very high dimensional vector space (Hung-Chih Chiang et al. (2000)). In bioinformatics, researchers try to classify particular diseases using high dimensional data sets such as micorarrays or SNPs (Fan and Ren (2006)). Fan and Ren (2006) summarize challenges and difficulties with high dimensionality in classification. Support Vector Machines (SVMs) are one of well-known supervised learning models to classify data points using a hyperplane and introduced by Boser et al. (1992) and Cortes and Vapnik (1995). The classical SVMs introduced by Boser et al. (1992) can be written as the $L_{2}$ norm SVMs, that is a hinge loss plus the $L_{2}$ norm penalty formulation. Zhang and Zhou (2010) showed that including many redundant features can cause difficulties in the performance of $L_{2}$ norm SVM. Bradley and Mangasarian (1998) showed that the SVM with the $L_{1}$ norm penalty instead of the $L_{2}$ norm penalty works well with variable selection as well as classification at the same time. This is called $L_{1}$ norm SVMs. While there are several frameworks to analyze the generalization performance, such as covering numbers (Mohri et al., 2018), real log canonical thresholds (Hayashi and Watanabe, 2017), mean- field regime (Nitanda et al., 2021), and Langevin dynamics regime (Suzuki, 2020), the SVMs especially allow us to derive the upper bounds for the generalization error via Vapnik-Chervonenkis (VC) dimensions (Vapnik, 2000). Although the bounds via VC dimensions may not necessarily be tight, it is still significant to have the first bound for a new variant of SVMs. Peng et al. (2016) showed the generalization bound on the probability of errors of the $L_{1}$ norm SVMs and it still depends on the number of predictors, i.e., the dimension of the normal vector of the hyperplane. Because of advances in computational algebraic geometry (Ren (2015)), tropical geometry finds applications in data science. For example, it can be applied to principal component analysis (Page et al. (2020); Yoshida et al. (2019)) and Bayesian Networks (Améndola et al. (2020)). With these statistical methodologies, tropical geometry is applied to phylogenomics analyses on Apicomplexa, African coelacanth whole genome data sets, and 1089 full length sequences of hemagglutinin (HA) for influenza A H3N2 from 1993 to 2017 in the state of New York obtained from the GI-SAID EpiFlu (Page et al. (2020); Yoshida et al. (2019)). Gärtner and Jaggi (2006) applied tropical geometry to SVMs. Instead of using a hyperplane defined by the $L_{2}$ metric, Gärtner and Jaggi (2006) used a tropical hyperplane with the tropical metric with the min- plus algebra to classify the data points. Same as the $L_{2}$ norm SVMs, tropical SVMs are the tropicalization of the $L_{2}$ norm hard margin SVMs which maximizes the margin, the distance from the tropical hyperplane to the closest data points in terms of the min-plus algebra. Gärtner and Jaggi (2006) also showed that a hard margin tropical SVM can be formulated as a linear programming problem. Then, Tang et al. (2020) showed necessary and sufficient conditions of the optimal solutions of the linear programming problem for finding a hard margin tropical SVM if it is feasible in terms of the max-plus algebra. They also introduced soft margin tropical SVMs and showed that finding a soft margin tropical SVM can be formulated as a linear programming problem. While there are some developments in computational sides of tropical SVMs, there has not been much in its theoretical evaluation and statistical analysis. For example, it is unclear how the tropical metric with the max-plus algebra and tropical SVMs handle the curse of dimensionality. Whether tropical SVMs are robust against the curse of dimensionality is an important question to answer, because we can extend tropical SVMs to various types of data including a function space. In fact, there are many classification problems with functions as predictors, such as neuronal tuning curves (Dayan and Abbott (2001)), instead of vectors. In this paper we focus on hard margin tropical SVMs with the max-plus algebra. Like the $L_{2}$ norm and the $L_{1}$ norm SVMs, we assume that there exists the optimal tropical hyperplane defined by the normal vector $\omega\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ such that the probability of the loss function being positive equals to zero. In addition, in order for the tropical metric to be well defined, we consider the _tropical projective torus_ , that is, the projective torus $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$, where ${\bf 1}:=(1,1,\ldots,1)$ is defined as the all-one vector. Note that any vector $(v_{1},\ldots,v_{d})\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ is equal to $(v_{1}+c,\ldots,v_{d}+c)$ with any scalar $c\in\mathbb{R}$ and $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ is isometric to $\mathbb{R}^{d-1}$. In applications, the tropical projective torus is useful for subtracting the baseline $\mathbb{R}{\bf 1}$ components from feature vectors, as we will see. Our main contribution of this paper primarily consists of two items: (1) evaluation of generalization errors for tropical SVMs using VC dimensions for tropical hyperplanes over $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$; and (2) extension to tropical SVMs on a function space, which consists of the tropical metric and a set of functions from a multi-dimensional vector space to a real number. While generalization error bounds of tropical SVMs over $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ still depends on the dimension, our simulations with a set of Gaussian distribution functions show that errors rates of tropical SVMs on $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ grow much slower than ones with $L_{2}$ norm SVMs when we increase the number of predictors while we fix the sample sizes. Then we extend a notion of tropical SVMs to a function space with the tropical metric. In fact, we show that a set of all functions from a multi-dimensional vector space to a real number with the tropical metric is a normed vector space. This manuscript is organized as follows: In Section 2, we set up basics in tropical geometry under the max-plus algebra. In Section 3, as an application, we show anomalous scaling behaviors of tropical distances. In Section 4, we set up tropical SVMs with the tropical metric over the tropical projective torus as linear programming problems and show the generalization error bounds via the VC dimension of a tropical hyperplane. In Section 5, as an application, we show theoretically by the extreme value statistics the robustness of the tropical SVMs against the curse of dimensionality in a case of two different multivariate Gaussian distributions and empirical data set of neuron types. In Section 6, we first define the tropical distance over a function space. Then we show that the tropical distance between functions is metric and we also show that a function space with the tropical metric forms a normed vector space. In addition, we define tropical SVMs over a function space with the tropical metric. ## 2 Definitions of Tropical Distances and Hyperplanes In this section, we remind readers some basics in tropical arithmetic and algebra using the max-plus algebra. Through this paper, we consider the tropical projective torus $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ which is isometric to $\mathbb{R}^{d-1}$. For more details, see Joswig (2022); Maclagan and Sturmfels (2015); Tang et al. (2020). ###### Definition 2.1 (Tropical Arithmetic Operations). In this tropical semiring $(\,\mathbb{R}\cup\\{-\infty\\},\boxplus,\odot)\,$, the basic tropical arithmetic operations of addition and multiplication are defined as: $a\boxplus b:=\max\\{a,b\\},~{}~{}~{}~{}a\odot b:=a+b~{}~{}~{}~{}\mbox{ where }a,b\in\mathbb{R}\cup\\{-\infty\\}.$ Over this semiring, $-\infty$ is the identity element under addition and 0 is the identity element under multiplication. ###### Definition 2.2 (Tropical Scalar Multiplication and Vector Addition). For any scalars $a,b\in\mathbb{R}\cup\\{-\infty\\}$ and for any vectors $v=(v_{1},\ldots,v_{d}),w=(w_{1},\ldots,w_{d})\in(\mathbb{R}\cup-\\{\infty\\})^{d}$, we define tropical scalar multiplication and tropical vector addition as follows: $a\odot v\boxplus b\odot w:=(\max\\{a+v_{1},b+w_{1}\\},\ldots,\max\\{a+v_{d},b+w_{d}\\}).$ ###### Definition 2.3. Suppose $S\subset\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$. If $a\odot v\boxplus b\odot w\in S$ for any $a,b\in\mathbb{R}$ and for any $v,w\in S$, then $S$ is called tropically convex. ###### Definition 2.4 (Tropical Convex Hull). Suppose we have a finite subset $V=\\{v^{1},\ldots,v^{s}\\}\subset\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$. The tropical convex hull or tropical polytope of $V$ is the smallest tropically- convex subset containing $V$. It can be written as the set of all tropical linear combinations of $V$ such that: $\mathrm{tconv}(V)=\\{a_{1}\odot v^{1}\boxplus a_{2}\odot v^{2}\boxplus\cdots\boxplus a_{s}\odot v^{s}\mid a_{1},\ldots,a_{s}\in\mathbb{R}\\}.$ A tropical polytope of a set of two points $\\{v^{1},\,v^{2}\\}\subset\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ is called a tropical line segment between two points $v^{1},\,v^{2}$. Note that over the tropical projective torus $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$, any point $x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ can be also written as $(x_{1},\ldots,x_{d})=(x_{1}+c,\ldots,x_{d}+c)$ for any $c\in\mathbb{R}$ by the definition of taking mod by ${\bf 1}:=(1,\ldots,1)$. Therefore, we can assume that $(x_{1},\ldots,x_{d})=(x_{1}-x_{d},\ldots,x_{d-1}-x_{d},0)$ for any point $x:=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$. Therefore over this manuscript we assume that the last coordinate of any point in the tropical projective torus $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ is equal to $0$ by taking this normalization. Figure 1: The tropical line segments between the origin and the three example points in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$ are represented by the blue, green, and red lines. The tropical distances from the origin to the three example points are $3$ (blue), $2$ (green), and $3$ (red). ###### Example 2.5 (Examples of Tropical Line Segments). Suppose we have $(2,3,0),\,(2,1,0),\,(2,-1,0)\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$. Then, the tropical line segments between the origin $(0,0,0)\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ and these three points in $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ are drawn by blue, green, and red lines in Figure 1. ###### Definition 2.6 (Generalized Hilbert Projective Metric). For any $v,\,w\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ such that $v=(v_{1},\ldots,v_{d})$ and $w=(w_{1},\ldots,w_{d})$, the tropical distance $d_{\rm tr}$ between them is defined such that: $d_{\rm tr}(v,w):=\max_{i}\bigl{\\{}v_{i}-w_{i}\bigr{\\}}-\min_{i}\bigl{\\{}v_{i}-w_{i}\bigr{\\}}.$ ###### Remark 2.7. The tropical metric $d_{\rm tr}$ is a metric over $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$. ###### Example 2.8. The tropical distances from the origin $(0,0,0)$ to the three example points in Figure1 in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$ are $d_{\rm tr}((2,3,0),(0,0,0))=3-0=3,$ $d_{\rm tr}((2,1,0),(0,0,0))=2-{\color[rgb]{0,0,0}0=2},$ $d_{\rm tr}((2,-1,0),(0,0,0))=2-(-1)=3.$ As one of many geodesics (shortest paths) toward the blue (green, red) point, the tropical line segment is denoted by the blue (green, red) line. Intuitively, a tropical distance is a shortest path length as far as you are allowed to go only parallel to the dotted lines. ###### Definition 2.9 (Tropical Hyperplane, Joswig (2005)). For any $\omega:=(\omega_{1},\ldots,\omega_{d})\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$, the tropical hyperplane defined by $\omega$ is the set of points $x\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ such that $\begin{array}[]{l}H_{\omega}:=\Big{\\{}(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}\|\exists i,j\in\\{1,\ldots,d\\}\mbox{ such that }\\\ \omega_{i}+x_{i}=\omega_{j}+x_{j}=\max\\{\omega_{1}+x_{1},\ldots\omega_{d}+x_{d}\\}\Big{\\}}.\\\ \end{array}$ In addition, we call $\omega$ the normal vector of the tropical hyperplane $H_{\omega}$. ###### Remark 2.10. In terms of tropical geometry, $H_{\omega}$ is the solutions of the tropical linear function with unknown $x_{1},\ldots,x_{d}$ such that for a fixed $\omega=(\omega_{1},\ldots,\omega_{d})\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$, $\displaystyle\omega_{1}\odot x_{1}\boxplus\ldots\boxplus\omega_{d}\odot x_{d}\mbox{ for }x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1},$ which is a special case of tropical polynomials. For more details on finding solutions of a tropical polynomial, see Maclagan and Sturmfels (2015). ###### Example 2.11 (Tropical Hyperplane in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$). Suppose we have $\omega=(1,2,0)\in\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$. Then $\begin{array}[]{rcl}H_{\omega}:&=&\Big{\\{}(x_{1},x_{2},x_{3})\in\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}\|1+x_{1}=2+x_{2}\geq x_{3}\mbox{ or}\\\ &&1+x_{1}=x_{3}\geq 2+x_{2}\mbox{ or }2+x_{2}=x_{3}\geq 1+x_{1}\Big{\\}}.\\\ \end{array}$ This means that we have four cases: 1. 1. Case 1: The first term and the second term are equal and max. $1+x_{1}=2+x_{2}>x_{3}(=0)$. 2. 2. Case 2: The first term and the third term are equal and max. $2+x_{2}<1+x_{1}=x_{3}(=0)$. 3. 3. Case 3: The second term and the third term are equal and max. $1+x_{1}<2+x_{2}=x_{3}(=0)$. 4. 4. Case 4: All terms are equal and max. $1+x_{1}=2+x_{2}=x_{3}(=0)$. We can set $x_{3}=0$ without loss of generality, since $(x_{1},x_{2},x_{3})$ and $(x_{1}-x_{3},x_{2}-x_{3},0)$ represent the same point in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$. Thus $H_{(1,2,0)}$ is the set of $(x_{1},x_{2},0)$ on the three half lines in Figure 2. Figure 2: Tropical hyperplane defined by $\omega=(1,2,0)$ over $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$. ###### Definition 2.12 (Sectors of Tropical Hyperplane, Joswig (2005)). Each tropical hyperplane $H_{\omega}$ divides the tropical projective torus $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ into $n$ connected components. These connected components are open sectors defined as: $S_{\omega}^{i}~{}:=~{}\\{x\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}\;\|\;\omega_{i}+x_{i}>\omega_{j}+x_{j},\;\forall j\neq i\;\\},\;\;\\!i\\!=\\!1,\\!\ldots\\!,d.$ ###### Definition 2.13 (Tropical Distance to a Tropical Hyperplane). The tropical distance $d_{\rm tr}$ from a point $x\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ to a tropical hyperplane $H_{\omega}$ is: $d_{\rm tr}(x,H_{\omega})\;:=\;\min\\{d_{\rm tr}(x,y)\;\|\;y\in H_{\omega}\\}.$ ###### Proposition 2.14 (Lemma 2.1 in Gärtner and Jaggi (2006)). Let $H_{\bf 0}$ be the tropical hyperplane defined by the zero vector ${\bf 0}=(0,0,\ldots,0)\in{\mathbb{R}}^{d}\\!/\mathbb{R}{\bf 1}$. For any $x=(x_{1},\ldots,x_{d})\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$, $d_{\rm tr}(x,H_{\bf 0})={\rm max}(x)-{\rm second\;max}(x).$ ###### Corollary 2.15 (Corollary 2.3 in Gärtner and Jaggi (2006)). For any $\omega\in{\mathbb{R}}^{d}\\!/\mathbb{R}{\bf 1}$ and for any $x\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$, $d_{\rm tr}(x,H_{\omega})=d_{\rm tr}(\omega+x,H_{\bf 0}).$ ###### Example 2.16. We use the sample $H_{\omega}$ from Example 2.11. Let $d=3$ and suppose $\omega=(1,2,0)\in\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$. In addition, suppose we have a point $x=(1,1,0)\in\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$. By Corollary 2.15, $d_{\rm tr}(x,H_{\bf\omega})=d_{\rm tr}(x+\omega,H_{\bf 0})=d_{\rm tr}((2,3,0),H_{\bf 0})=3-2=1.$ Figure 3: Tropical and Euclidean distances between two bell-shaped tuning curves which are horizontally shifted (left column) or vertically lifted (right column). The tropical distance is insensitive to the vertical lift of tuning curves. ## 3 Application 1: Valuable Scaling Behaviors of Tropical Distances ### 3.1 Bell-shaped Tuning Curves To characterize the properties of the tropical distances $d_{\rm tr}(v,w)$ proposed in this paper, we computed the tropical distances for the mathematical models of neuronal tuning curves as benchmarks. The tuning curves are the vectors consisting of neural responses to different stimuli. For example, the responses of neurons in the primary visual cortex to oriented bars on the screens as visual stimuli are known to show famous bell-shaped curves as a function of the orientation. We modeled neuronal tuning curves basically as standard Gaussian distribution functions and examined how the distances scale when one of two curves is horizontally shifted or vertically lifted. Both the tropical and Euclidean distances between two bell-shaped tuning curves increased with the increasing horizontal shift between them (Figure 3, left). Meanwhile, the tropical distances between the two bell-shaped tuning curves was insensitive to the vertical lift of tuning curves, while the Euclidean distance was not (Figure 3, right). The insensitivity to the vertical shift originates from the definition of the tropical distances $d_{\rm tr}(v,w)$ where tuning curves are considered in $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ and $(1,1,\ldots,1)$-direction is collapsed and ignored in Def.2.6. That is, $d_{\rm tr}(v+c{\bf 1},w+d{\bf 1})=d_{\rm tr}(v,w)$ for any $c$ and $d$ in $\mathbb{R}$. Figure 4: Tropical and Euclidean distances between random and flat tuning curves. The tropical distance scales with $\log n$ as a result of extreme value statistics. The circles denote simulation results and the lines denote theoretical results. Note that the entire lifting of tuning curves are often observed as a result of drifting background neural activities (Okun et al. (2015); Stringer et al. (2019); Miura et al. (2005, 2006, 2007); Miura (2013); Takahashi et al. (2019)). Thus, practically, the tropical distance can be useful for the sake of ignoring the temporal drift of tuning curves. ### 3.2 Random Tuning Curves Next we considered a random tuning curve consisting of noisy responses. When we computed the distances between a random tuning curve $v=(5,-x_{2},-x_{3},\ldots,-x_{n}),$ (1) where $x_{i}\sim\textrm{Exp(1)}$ and a flat tuning curve, $w=(0,0,0,\ldots,0),$ (2) the tropical distances was relatively insensitive to the additional noise dimensions. That is, the tropical distance scaled with $\log n$ while the Euclidean distance scaled with $\sqrt{n}$. The anomalous scaling for the tropical distance can be explained by the extreme value statistics (Gumbel (2004)) as, $\displaystyle d_{\rm tr}(v,w)$ $\displaystyle:=$ $\displaystyle\max_{i}\bigl{\\{}v_{i}-w_{i}\bigr{\\}}-\min_{i}\bigl{\\{}v_{i}-w_{i}\bigr{\\}}$ (3) $\displaystyle\approx$ $\displaystyle 5+\max_{i}\bigl{\\{}x_{i}\bigr{\\}}.$ Then, because $\max_{i}\bigl{\\{}x_{i}\bigr{\\}}-\log n\sim\textrm{Gumbel(0,1)}$ (Gumbel (2004)), the expectation is given as $\overline{d_{\rm tr}(v,w)}=5+\gamma+\log n.$ (4) This theoretical result explains the simulation result in Figure 4. In this way, the extreme value statistics plays the crucial role in the computation of the tropical distance between random vectors. ## 4 Evaluation of Tropical Support Vector Machines (SVMs) ### 4.1 $L_{2}$-norm SVMs In this section we provide an overview of the $L_{2}$ norm SVMs. Let $\mathcal{D}_{2}$ be the distribution on the joint random variable $(X,Y)$ for $X\in\mathbb{R}^{d}$ and $Y\in\\{-1,1\\}$ and let $\mathcal{S}_{2}$ be the sample $\mathcal{S}_{2}:=\\{(X^{1},Y^{1}),\ldots,(X^{n},Y^{n})\\}$. Then the standard linear SVM is a statistical model by solving the following regularization problem: $\min_{\omega}\left(\underbrace{\lambda\|\|\omega\|\|^{2}}_{\text{regularizer}}+\underbrace{n^{-1}\sum_{i=1}^{n}(1-Y^{i}((X^{i})^{T}\omega+\omega_{0}))_{+}}_{\text{error}}\right),$ where $\|\|x\|\|$ is the $L_{2}$ norm of a vector $x\in\mathbb{R}^{d}$, $(1-u)_{+}=\max\\{1-u,0\\}$ is the hinge loss function, $\lambda$ is a tuning parameter, $\omega_{0}$ is a constant term of the hyperplane, and $\omega$ is the normal vector of the hyperplane to separate the data points. Vapnik (2000) discussed the generalization bounds on the $L_{2}$ norm SVMs using Theorem 4.11 and the VC dimension of the $L_{2}$ norm SVMs. In addition, using the Rademacher complexity of $L_{2}$ norm of unit vectors (Bartlett and Mendelson (2003)), one can show the following generalization bound of the error rate: ###### Theorem 4.1. Let $w^{s}$ be an output from the classical hard margin SVM from a sample $\mathcal{S}$. Here $Y\in\\{-1,1\\}$ with the distribution $\pi_{+}=P(Y=1)$ and $\pi_{-}=P(Y=-1)$. Let $X\in\mathbb{R}^{d}$ be a random variable and $\mathcal{D}_{2}$ be the distribution for a random variable $(X,Y)$. Let $\|\|x\|\|$ be the $l_{2}$ norm of a vector $x\in\mathbb{R}^{d}$. Assuming that $\|\|X\|\|\leq R$ with probability 1 and there exists $\omega^{}$ with $P_{\mathcal{D}_{2}}(Y^{i}((X^{i})^{T}\omega+\omega_{0})\geq 1)=1$. Then, for any $\eta>0$ with the probability greater than or equal to $1-\eta$ we have $P_{\mathcal{D}_{2}}\left(Y\not=sign(X\cdot\omega^{s})\right)\leq\frac{4R\|\|\omega^{s}\|\|}{\sqrt{n}}+(1+2R\|\|\omega^{s}\|\|)\sqrt{\frac{2\log(4\|\|\omega^{s}\|\|/\eta)}{n}}.$ (5) There has been much work done in tighter generalization bounds on the $L_{2}$ norm SVMs using the VC dimension. As it is not this paper’s focus to discuss the details on the generalization bounds for the $L_{2}$ norm SVMs, for more details, see Burges (1998); Guermeur (2007) and the references therein. ###### Remark 4.2. Recall that the $L_{p}$ norm SVMs, for $0\leq p\leq\infty$, can be written as the following optimization problem: $\min_{\omega}\left(\underbrace{\lambda\|\|\omega\|\|_{p}^{2}}_{\text{regularizer}}+\underbrace{n^{-1}\sum_{i=1}^{n}(1-Y^{i}((X^{i})^{T}\omega+\omega_{0}))_{+}}_{\text{error}}\right),$ where $\|\|x\|\|_{p}$ is the $L_{1}$ norm of a vector $x\in R^{d}$, $(1-u)_{+}=\max\\{1-u,0\\}$ is the hinge loss function, $\lambda$ is a tuning parameter, and $\omega$ is the normal vector of the hyperplane to separate the data points. ### 4.2 Tropical SVMs In this section we provide an overview the hard margin tropical SVMs originally developed by Gärtner and Jaggi (2006) with the random variable $X\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ given the response variable $Y\in\\{0,1\\}^{d}$ for one-hot encoding. Note that tropical SVMs can perform multiclass classification naturally. Before we formally define the hard margin tropical SVMs, we need to define some notation. Let $S(x)$ be a set of indices of nonzero elements in a vector $x\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$, i.e., $S(x)\subset\\{1,\ldots,d\\}$ where $x_{i}\not=0$. Let $I_{\omega}(x)\in\\{0,1\\}^{d}$ be a vector of indicator functions of index set $\\{1,\ldots,d\\}$ of a vector $x$ with a vector $\omega\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ where $I_{\omega}(x)_{i}=\begin{cases}1&\text{if }x_{i}+\omega_{i}=\max(x+\omega)\\\ 0&\text{otherwise.}\end{cases}$ Let $J_{\omega}{\color[rgb]{0,0,0}(x)}\in\\{0,1\\}^{d}$ be also a vector of indicator functions of index set $\\{1,\ldots,d\\}$ of a vector $x$ such that $J_{\omega}(x)_{i}=\begin{cases}1&\text{if }x_{i}+\omega_{i}=\text{second max}(x+\omega)\\\ 0&\text{otherwise.}\end{cases}$ Suppose we have a categorical response variable $Y^{\prime}\in\\{g_{1},\ldots,g_{q}\\}$ with $q\leq d$ many levels. Then we set a vector of indicator functions $Y=(Y_{1},\ldots,Y_{d})$ such that $Y_{i}=\begin{cases}1&\mbox{if }Y^{\prime}=g_{i}\\\ 0&\mbox{else.}\end{cases}$ ###### Example 4.3. Suppose that we have a binary categorical response variable $Y^{\prime}\in\\{\mbox{"yes" , "no"}\\}$. Then we can set as a vector of indicator functions $Y=(Y_{1},\ldots,Y_{d})$ such that $Y_{1}=\begin{cases}1&\mbox{if }Y^{\prime}=\mbox{yes}\\\ 0&\mbox{else,}\end{cases}$ $Y_{2}=\begin{cases}1&\mbox{if }Y^{\prime}=\mbox{no}\\\ 0&\mbox{else,}\end{cases}$ and $Y_{3}=Y_{4}=\ldots=Y_{d}=0.$ In order to simplify the problem, here we consider a case that there are only two classes $Y^{A}$ and $Y^{B}$ in the response variable. More precisely, we have a random variable $Y^{A}:=(Y^{A}_{1},\ldots,Y^{A}_{d}),Y^{B}:=(Y^{B}_{1},\ldots,Y^{B}_{d})\in\\{0,1\\}^{d}$ such that for fixed $i,k\in\\{1,\ldots,d\\}$ with $i\neq k$, $Y^{A}_{j}=\begin{cases}1&\mbox{if }j=i\\\ 0&\mbox{else,}\end{cases}$ and $Y^{B}_{l}=\begin{cases}1&\mbox{if }l=k\\\ 0&\mbox{else}\end{cases}$ with a discrete probability $\pi_{A}:=P(Y=Y^{A})$ and $\pi_{B}:=P(Y=Y^{B})$ . Then, suppose we have a multivariate random variable $X\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ given $Y$ with the probability density function $f_{A}$ if $Y=Y^{A}$ and the probability density function $f_{B}$ if $Y=Y^{B}$ such that there exists a tropical hyperplane $H_{\omega^{}}$ with a normal vector $\omega^{}\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ with the following properties: (i) there exists an index $i\in\\{1,\ldots,d\\}$ such that $\text{for any}\;j\in\\{1,\ldots,d\\}\backslash\\{i\\},\;\;\omega^{}_{i}+X_{i}\;>\;\omega^{}_{j}+X_{j},\;\;\;\text{and}$ * (ii) $\max\left\\{I_{\omega^{}}(X)-Y\right\\}=0,$ with probability $1$. Let $\mathcal{D}$ be the distribution on the joint random variable $(X,Y)$ and let $\mathcal{S}$ be the sample $\mathcal{S}:=\\{(X^{1},Y^{1}),\ldots,(X^{n},Y^{n})\\}$. Then we formulate an optimization problem for solving the normal vector $\omega$ of an optimal tropical separating hyperplane $H_{\omega}$ for random variables $X$ given $Y$: For some cost $C\in\mathbb{R}$ $\begin{matrix}\displaystyle\left[\max\limits_{\omega\in\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}}\;\min\limits_{X\in\mathcal{S},i\in S(I_{\omega}(X)),j\in S(J_{\omega}(X))}\left\\{\underbrace{\left(X_{i}+\omega_{i}-X_{j}-\omega_{j}\right)}_{\text{margin}}+\underbrace{\frac{C}{n}\sum_{k=1}^{n}\min\left\\{Y^{k}-I_{\omega}(X^{k})\right\\}}_{\text{error}}\right\\}\right].\\\ \end{matrix}$ (6) Here, the expectation of the random variable $\max\left\\{I_{\omega}(X)-Y\right\\}$ is the $0-1$ loss function. Also note that $d_{\rm tr}(X,H_{\omega})\;=\;X_{i}+\omega_{i}-X_{j}-\omega_{j},$ where $i\in S(I_{\omega}(X)),j\in S(J_{\omega}(X))$. Thus, this optimization problem can be explicitly written as a linear programming problem (7)–(10) below, where the optimal solution $z$ means the margin of the tropical SVM: For some cost $C\in\mathbb{R}$ $\displaystyle\max\limits_{(z,\omega)\in\mathbb{R}\times\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}}\;\left(z+\frac{C}{n}\sum_{k=1}^{n}\min\left\\{Y^{k}-I_{\omega}(X^{k})\right\\}\right)$ (7) $\displaystyle\textrm{s.t.}\;\;\forall X$ $\displaystyle\in\mathcal{S},\forall i\in S(I_{\omega}(X)),\forall j\in S(J_{\omega}(X)),\;\;z+{\color[rgb]{0,0,0}X_{j}}+\omega_{j}{\color[rgb]{0,0,0}-X_{i}}-\omega_{i}\leq 0,$ (8) $\displaystyle\forall X$ $\displaystyle\in\mathcal{S},\forall i\in S(I_{\omega}(X)),\forall j\in S(J_{\omega}(X)),\;\;\omega_{j}-\omega_{i}\leq X_{i}-X_{j},$ (9) $\displaystyle\forall X$ $\displaystyle\in\mathcal{S},\forall l\not\in S(I_{\omega}(X))\cup S(J_{\omega}(X)),j\in S(J_{\omega}(X))\;\;\omega_{l}-\omega_{j}\leq X_{j}-X_{l}.$ (10) ###### Remark 4.4. With the regularization with the tropical metric to the tropical hyperplane, it minimizes the coefficients of the tropical hyperplane $\omega_{i}$ for $i=1,\ldots d$ and tries to minimize the difference between the coordinate of the maximum and second maximum of $X+\omega$. This means that this regularization behaves similar to the $L_{1}$ or $L_{\infty}$ norm regularization, that is, to select features which contribute to discriminate $X\|Y^{A}$ and $X\|Y^{B}$ . ### 4.3 Bound on Generalization Errors via VC Dimensions In this section we show the generalization bound for the error rate of the hard margin tropical SVM. In order to compute them, we define the loss function in terms of the distribution of $X\|Y$ defined in the previous section and the sample loss function. In order to prove the generalization bound, we use the VC dimension (Vapnik, 1995; Mohri et al., 2018) Let $\mathcal{D}$ be the distribution of $X\|Y$ defined in Section 4. In addition, let $L_{\mathcal{D}}(\omega)=\mathbb{E}_{\mathcal{D}}\left(\max\left\\{I_{\omega}(X)-Y\right\\}\right)$ be the loss function with respect to $\mathcal{D}$ and let $L_{\mathcal{S}}(\omega)=\mathbb{E}_{\mathcal{S}}\left(\max\left\\{I_{\omega}(X)-Y\right\\}\right)$ be the loss function with respect to the sample $\mathcal{S}$. Let $(Z_{s},\omega^{s})$ be an output of the tropical SVM. Then we want to find the upper bound for $P\left(\max\left\\{I_{\omega^{s}}(X)-Y\right\\}>0\right)=L_{\mathcal{D}}(\omega^{s}).$ (11) In order to prove our bound, we have to define VC dimension. ###### Definition 4.5. Suppose $M$ is a set and a family of a subset $R\subset 2^{M}$. A set $T\subset M$ is called shattered by $R$ if $\|\left\\{r\cap T\|r\in R\right\\}\|=2^{\|T\|}.$ The pair $(M,R)$ is called range space. The VC dimension of a range space, $VC-dim(M,R)$ is the maximal cardinality of a subset of $M$ which can be shattered by $R$. ###### Example 4.6 (Linear Classifiers). In $(d-1)$-dimensional Euclidean space, a linear classifier can shatter or successfully classify $d$ points in general positions with arbitrary labels. Thus, the VC dimension of the linear classifier is $d$. ###### Lemma 4.7 (Tropical Radon Lemma in Jaggi et al. (2008)). For $d+1$ points in $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$, there exists a partition into two sets with intersecting convex hulls, which are therefore not separable by any tropical hyperplane $H_{\omega}$. ###### Example 4.8. Any tropical hyperplane in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$ cannot shatter the four points shown in Figure 5. The fact that the two line segments intersect to each other suggests that it is impossible to classify them by tropical hyperplanes. In fact, ANY four points cannot be shattered by a tropical hyperplane in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$. This is because, given four or more points, there always exist ways of partitioning so that the tropical convex hull of the points in each class intersects. Figure 5: Example positions where four points cannot be shattered by tropical hyperplanes in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$. The gray lines represent the tropical convex hulls of the two points in the same class. The fact that the line segments intersect suggests that it is impossible to classify them by tropical hyperplanes. ###### Lemma 4.9 (Lemma 27 in Jaggi et al. (2008)). There exists a set of $d$ points in $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$ that can be shattered by single sectors defined by a tropical hyperplane $H_{\omega}$. While Tropical Radon Lemma 4.7 shows that the VC dimension of a tropical hyperplane is smaller than $d+1$, (trivial) Lemma 4.9 shows that it is larger than or equal to $d$ (as single sectors should have less ability than hyperplanes). ###### Theorem 4.10 (Jaggi et al. (2008)). Let $\mathbf{H}$ be a family of tropical hyperplanes in $\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1}$. Then the VC dimension of a range space $(\mathbb{R}^{d}\\!/\mathbb{R}{\bf 1},\mathbf{H})$ is $d$. ###### Theorem 4.11 (Vapnik (2000)). Let $\mathcal{H}$ be a family of classification models whose VC dimension is $h$. Let $n$ be the sample size of the sample $\mathcal{S}$. Then, for any $\eta>0$ and for a sufficiently large sample size $n$ with the probability greater than or equal to $1-\eta$, we have $\ell_{\mathcal{D}^{\prime}}(\mathcal{H})-\ell_{\mathcal{S}}(\mathcal{H})\leq\sqrt{\frac{h(\log(2n/h)+1)-\log({\color[rgb]{0,0,0}\eta/4})}{n}},$ where $\ell_{\mathcal{D}^{\prime}}(\mathcal{H})$ is the $0-1$ loss for the distribution $\mathcal{D}^{\prime}$ with $\mathcal{H}$ and $\ell_{\mathcal{S}}(\mathcal{H})$ is the $0-1$ loss for the sample $\mathcal{S}$ with $\mathcal{H}$. The generalization error bound for tropical SVMs is obtained by plugging in the VC dimension of tropical hyperplanes in Theorem 4.10 for $h$ in Theorem 4.11. ###### Theorem 4.12. Let $n$ be the sample size of the sample $\mathcal{S}$. Then, for any $\eta>0$ and for a sufficiently large sample size $n$ with the probability greater than or equal to $1-\eta$, we have $P_{\mathcal{D}}\left(\max\left\\{I_{\omega^{s}}(X)-Y\right\\}>0\right)\leq\sqrt{\frac{d(\log(2n/d)+1)-\log({\color[rgb]{0,0,0}\eta/4})}{n}}.$ (12) ###### Proof. Let $\mathcal{S}$ be a sample for a test set. Since there is a tropical hard margin and $(z_{s},\omega^{s})$ is a feasible solution on the test set $\mathcal{S}$, $L_{\mathcal{S}}(\omega^{s})=0$. With Theorem 4.10 and Theorem 4.11, we have $L_{\mathcal{D}}(\omega^{s})-L_{\mathcal{S}}(\omega^{s})\leq\sqrt{\frac{d(\log(2n/d)+1)-\log({\color[rgb]{0,0,0}\eta/4})}{n}}.$ Since $L_{\mathcal{S}}(\omega^{s})=0$ by the assumption and $L_{\mathcal{D}}(\omega^{s})=P_{\mathcal{D}}\left(\max\left\\{I_{\omega^{s}}(X)-Y\right\\}>0\right),$ we have the result. ∎ ## 5 Application 2: Robustness of Tropical SVMs against Curse of Dimensionality In the previous section, we derive the generalization error bound for tropical SVMs. The bound is stated in a distribution-free manner as a general feature applicable to all data. Specifically, the bound is derived purely combinatorially only based on the shapes of the hyperplanes. Algorithmic details of tropical SVMs are not yet taken into account there. Therefore it is possible that tropical SVMs outperform in specific situations, for example, when only a few features are prominently informative among many. Thus the evaluation of the performance of tropical SVMs by computational experiments can give complementary information. Here we applied tropical SVMs to specific problems of computational experiments and real data analyses to evaluate its performance originating from the max-plus algebra. ### 5.1 Implementation of Tropical SVMs Soft margin tropical SVMs are defined in Tang et al. (2020) which allow some data points in the wrong open sectors with the hinge loss function as similar to the $L_{2}$ norm SVMs. Since the loss function deals with $L_{0}$ norm of $I_{\omega}(X)-Y$, we have to run exponentially many linear programming problems in order to find the optimal tropical hyperplane to fit the data set. Therefore, Tang et al. introduced several heuristic algorithms to estimate the optimal tropical hyperplane to fit the data: Algorithms 1 – 4. In the computational experiments in this paper, we applied Algorithms 3, the simplest one, from Tang et al. (2020) to fit each data to the model, as it always performed the best for our simple examples. We obtained the R code from https://github.com/HoujieWang/Tropical-SVM. ### 5.2 Computational Experiments As we observe in Section 3.2, the tropical distance can ignore the nuisance dimensions that are purely noises. Therefore it is expected that tropical SVMs can show a good performance in the curse-of-dimensionality regime where extra dimensions are all uninformative. Here we performed a simple computational experiment of the two-class classification using the uncorrelated unit-variance Gaussian distributions whose means are ${\color[rgb]{0,0,0}m_{1}=}$ $(s,-s,0,0,\ldots,0)$ for class $1$ and ${\color[rgb]{0,0,0}m_{2}=}$ $(-s,s,0,0,\ldots,0)$ for class $2$, where $s=\sqrt{2}$ or $5$. To be specific, the unit covariance matrices are common for the two distributions as $\Sigma_{1}=\Sigma_{2}=\begin{pmatrix}1&0&\ldots&0\\\ 0&1&\ldots&0\\\ \vdots&\vdots&\vdots&\vdots\\\ 0&0&\ldots&1\\\ \end{pmatrix}$. Note that the signal or the difference of the means, $m_{2}-m_{1}=(2s,-2s,0,0,\ldots,0)$, is orthogonal to the irrelevant direction $(1,1,1,1,\ldots,1)$. As a measure of separation, the d-prime or the Euclidean distance between $m_{1}$ and $m_{2}$ divided by the S.D. (=1) is given by $d^{\prime}=2s\sqrt{2}$. Before we look at the curse-of-dimensionality regime or small samples, we first consider the case when the numbers of training and test samples $N$ are fairly large to compare the numerically obtained generalization errors with the theoretical bounds in Figure 6. We draw the lower bound of the classification success rates by subtracting the penalty term in the right hand side of the inequality in Theorem 4.11 from the hit rate for the training set. The gaps between the numerical computations and the theoretical bounds suggest that the bounds are not so tight. In fact, the lower bounds are mostly smaller than the chance level $50\%$ and thus not informative. Although we solely use $\eta=0.1$, the bounds are rather insensitive to the value of $\eta$. Even if we have as large as $N=1000$ or $10000$ observations, the gap is still quite noticeable as expected from the penalty term in the right hand side of the inequality in Theorem 4.11. Figure 6: Numerically obtained classification success rates (thick lines) and theoretical lower bounds (dotted lines) for the classification of two multivariate Gaussian distributions whose means are separated by $d^{\prime}=4$ and covariances are the unit matrices. The results for tropical SVMs are red colored while those for classical SVMs are black colored. The numbers of training and test samples for each class are $100$ (left column) and $200$ (right column). The cross-validated performance averaged over $1000$ realizations of data were plotted. Note that the lower bounds of the hit rates (=upper bounds of error rates) are not tight. Next, we consider small samples as a curse-of-dimensionality regime. Figure 7 shows the classification success rates of two multivariate Gaussian distributions separated by ${\color[rgb]{0,0,0}d^{\prime}=}4$ (top) or ${\color[rgb]{0,0,0}d^{\prime}=}10\sqrt{2}$ (bottom). The numbers of training and test samples for each class are 5 (left) and 10 (right). In all four cases, the hit rate decreases significantly for classical SVMs, while it decreases only slightly for tropical SVMs. In fact, it almost stays constant for the ${\color[rgb]{0,0,0}d^{\prime}=}10\sqrt{2}$ cases. Thus the tropical SVM is robust against the curse of dimensionality in the case for separating two Gaussian distributions. Figure 7: Classification success rates of two multivariate Gaussian distributions whose means are separated by $d^{\prime}=4$ (top row) or $10\sqrt{2}$ (bottom row) and covariances are the unit matrices. The numbers of training and test samples for each class are $5$ (left column) and $10$ (right column). The cross-validated performance averaged over $1000$ realizations of data were plotted. The errorbars denote the standard deviations for the $1000$ realizations. Note that the tropical SVM is robust against the curse of dimensionality. As our setting of classifying two Gaussians was so simple, it is expected that the results can be quite general. The only assumption here is that the difference of the two means is not parallel to ${\bf 1}=(1,1,\ldots,1)$. We believe that in real data the difference of the means rarely becomes by chance parallel to ${\bf 1}$. We will examine the robustness against the curse of dimensionality for the real data in what follows. But before doing that, let us explain theoretically why tropical SVMs are robust against the curse of dimensionality. ### 5.3 Understanding Robustness by Numerical Examples First, as a basic example, let us consider the simplest case with two points in three dimensions: $x_{1}=(5,-5,0)$ for class $1$ and $x_{2}=(-5,5,0)$ for class $2$. As our problem is to find $\omega=(\omega_{1},\omega_{2},\omega_{3})$ in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$, we can limit to $\omega=(\omega_{1},\omega_{2},0)$ without loss of generality. By maximizing the following margin over $\omega$ in $\mathbb{R}^{3}\\!/\mathbb{R}{\bf 1}$ under the condition with no classification error, $M(\omega):=\min\left(d_{\rm tr}(x_{1},H_{\omega}),d_{\rm tr}(x_{2},H_{\omega})\right),$ (13) we obtain $\omega^{}=(5+c,5+c,0)$ for $c>0$ as in Figure 8. Figure 8: (left) Margin $M(\omega)$ for $\omega=(\omega_{1},\omega_{2},0)$. For the regions A, it is $10-\|\omega_{1}-\omega_{2}\|$. For the region $\times$ containing $(0,0)$, it is $5$+min$(\omega_{1},\omega_{2})=\frac{\omega_{1}+\omega_{2}+10}{2}-\frac{\|\omega_{1}-\omega_{2}\|}{2}$. For the regions B, it is $\frac{\|\omega_{1}-\omega_{2}\|}{2}-\frac{\|\omega_{1}+\omega_{2}+10\|}{2}$. For the regions C, it is $5-\|\max(\omega_{1},\omega_{2})\|$. The other non-marked regions have non-zero classification errors and are ineligible. Note that on the verge of misclassification, it is always $0$. The maximum is attained at $\omega^{}=(5+c,5+c,0)$ for $c>0$. (right) Configuration in optimal solutions where margin is $10$. The red crosses denote the data points to be classified when $\omega^{}=(5,5,0)$. The orange crosses denote the data points to be classified when $\omega^{}=(25,25,0)$. Note that both configurations attain the maximum margin. In fact, $x_{1}+\omega^{}=(10+c,c,0)$ and $x_{2}+\omega^{}=(c,10+c,0)$ lead to $d_{\rm tr}(x_{1},H_{\omega^{}})=d_{\rm tr}(x_{2},H_{\omega^{}})=10$. In Figure 8 (left) for the margin $M(\omega)$, error-free regions for $\omega$ are only A, B and C. The other regions have classification errors. The entire shape of the error-free region reflect the Y-shape of the hyperplane $H_{0}$. For example, for $\omega=(-10,-10,0)$, $x_{1}+\omega$ and $x_{2}+\omega$ belong to the same bottom left sector of $H_{0}$, meaning misclassification. Note that as the margin function $M(\omega)$ is convex unless $x_{1}+\omega$ or $x_{2}+\omega$ crosses the sector borders, the linear programming is needed only once for each possible combination of labels and sectors (Tang et al. (2020)). Here we can impose the regularization term, $d_{\rm tr}(\omega,0)$, so that we choose the solution with the minimum norm even if the evaluation function outputs ties. In this tradition, the unique solution is $\omega^{}=(5,5,0)$. Second, let us consider the general case with $N$ points for each class in $d$ dimensions, whose coordinates are perturbed from $x_{1}$ or $x_{2}$ by standard Gaussian noises $\xi$ or $\eta$ (Figure 9) as in the previous computational experiments. We again need to maximize the margin: $M(\omega):=\min_{1\leq i\leq N,~{}1\leq l\leq 2}d_{\rm tr}(x_{Y=l}^{i},H_{\omega}),$ (14) with $\displaystyle x_{Y=1}^{i}=(5+\xi_{1}^{i},-5+\xi_{2}^{i},\xi_{3}^{i},\xi_{4}^{i},\ldots,\xi_{d}^{i}),$ $\displaystyle x_{Y=2}^{j}=(-5+\eta_{1}^{j},5+\eta_{2}^{j},\eta_{3}^{j},\eta_{4}^{j},\ldots,\eta_{d}^{j}),$ (15) where $\xi_{p}^{i}\sim N(0,1)$ and $\eta_{p}^{i}\sim N(0,1)$ for $1\leq p\leq d$ and $1\leq i\leq 2$. Let us try to find the solution near $\omega^{}=(5+c,5+c,0,0,\ldots,0)$ and $d\rightarrow\infty$. In the previous computational experiments, we observed that for all cases of $1000$ data realizations the best classifiers used the first and the second sectors as in Figure 9. Furthermore, among $x_{p}+\omega_{p}$ for $1\leq p\leq d$, the maximum and the second maximum were attained by $x_{1}+\omega_{1}$ and $x_{2}+\omega_{2}$ when the training data were classified. Therefore we focus on this situation, i.e. classification by using only sector $1$ and sector $2$, in what follows. Figure 9: A solution of tropical SVM that uses first and second sectors. $\omega$ is adjusted so that the tropical hyperplane is located right in the middle of the two support vectors. Under the assumption of $\omega_{2}\gg 5$, which will be justified later, the tropical distance to $H_{\omega}$ is $\displaystyle d_{\rm tr}(x_{Y=1}^{i},H_{\omega})=d_{\rm tr}(x_{Y=1}^{i}+\omega,H_{0})$ $\displaystyle={\rm\\{max-2nd~{}max~{}of\\}}~{}(5+\xi_{1}^{i}+\omega_{1},-5+\xi_{2}^{i}+\omega_{2},\xi_{3}^{i}+\omega_{3},$ $\displaystyle\ldots,\xi_{d}^{i}+\omega_{d})$ $\displaystyle=10+\xi_{1}^{i}+\omega_{1}-\xi_{2}^{i}-\omega_{2}$ (16) The minimum over $N$ points with class $1$ is $\displaystyle\min_{1\leq i\leq N}d_{\rm tr}(x_{Y=1}^{i},H_{\omega})$ $\displaystyle=$ $\displaystyle 10+\omega_{1}-\omega_{2}+\min_{1\leq i\leq N}\left(\xi_{1}^{i}-\xi_{2}^{i}\right)$ (17) $\displaystyle=$ $\displaystyle 10+\omega_{1}-\omega_{2}+\xi_{1}^{i_{}}-\xi_{2}^{i_{}}.$ where $i_{}(\omega)$ denotes the minimizer. Similarly for the class $2$, $\displaystyle\min_{1\leq j\leq N}d_{\rm tr}(x_{Y=2}^{j},H_{\omega})$ $\displaystyle=$ $\displaystyle 10+\omega_{2}-\omega_{1}+\min_{1\leq j\leq N}\left(\eta_{2}^{j}-\eta_{1}^{j}\right)$ (18) $\displaystyle=$ $\displaystyle 10+\omega_{2}-\omega_{1}+\eta_{2}^{j_{}}-\eta_{1}^{j_{}},$ where $j_{}(\omega)$ denotes the minimizer. By equating the minimum distances for the two classes, $10+\omega_{1}-\omega_{2}+\xi_{1}^{i_{}}-\xi_{2}^{i_{}}=10+\omega_{2}-\omega_{1}+\eta_{2}^{j_{}}-\eta_{1}^{j_{}},$ (19) we get $\omega_{2}=\omega_{1}+\frac{\xi_{1}^{i_{}}-\xi_{2}^{i_{}}+\eta_{1}^{j_{}}-\eta_{2}^{j_{}}}{2}.$ (20) This theoretical prediction perfectly matched with the computational experiments. This can be rewritten as, $\displaystyle\omega_{1}$ $\displaystyle=$ $\displaystyle 5+c$ $\displaystyle\omega_{2}$ $\displaystyle=$ $\displaystyle 5+\frac{\xi_{1}^{i_{}}-\xi_{2}^{i_{}}+\eta_{1}^{j_{}}-\eta_{2}^{j_{}}}{2}+c,$ (21) for $c>0$, which give the (evenly scored) maximizers of the margin under $\omega_{2}\gg 5$. Note that $\omega$ is perturbed from $(5+c,5+c)$ only slightly of order $O(1)$. This equation illustrates that the hyperplane just listens to the positions of the support vectors. Adding $c$ might look strange at first glance, but it is interpreted as the hyperplane’s freedom to shift along the irrelevant coordinates of the support vectors. In fact, the hyperplane for classical SVMs actually has the same freedom but you just cannot distinguish the shifted planes. Now we can explain why tropical SVMs are robust against the curse of dimensionality. In the computational experiments, the other $\omega_{p}(p\geq 3)$ were smaller by around five or more. Therefore the misclassification never happened as the maximum of $(x+\omega)$ always occurred at the first or second coordinate appropriately. Figure 10: Classification success rates of dopaminergic and nondopaminergic neurons by using various numbers of features for tropical and classical SVMs. All features are the firing rates, i.e. the number of spikes per second, in $300$ms bins. The first $7$ features we preferentially used are the responses to $7$ different reward stimuli, which are informative. The following $7$ features we appended are spontaneous activities after reward stimuli, which are less informative. The other features are spontaneous activities before reward stimuli, which are least informative. Note that the classification success rates decreased slower for the tropical SVM than the classical SVM, suggesting that the tropical SVM is relatively robust against the curse of dimensionality. ### 5.4 Real Data Analysis We now turn to empirical data from brain science. To summarize the data acquisition in this paragraph, we recorded the spiking activity of $493$ total neurons in the ventral tegmental area (VTA) and the substantia nigra (SN) of DAT (dopamine transporter)-cre mice using tetrodes while these mice performed various classical conditioning tasks (Matsumoto et al. (2016)). We optogenetically identified $179$ dopaminergic neurons (Cohen et al. (2012); Matsumoto et al. (2016)), which enabled us to assess the classification success rates based on the neuronal tuning curves. In each trial of the task, we delivered to a mouse one of seven stimuli, which were water rewards, air puff, etc. The conventional tuning curve for these neurons consists of the neural responses to the seven stimuli measured as the spike count within a $300$ms time bin. The tropical SVM for classifying dopaminergic and nondopaminergic neurons based on their activities as features was also robust against the curse of dimensionality (Figure 10). In order to use the same set of features, we limited to $41$ neurons including $17$ dopaminergic and $24$ nondopaminergic neurons. That is, the sample size is $41$. Both the training and test data consist of, randomly selected, $5$ dopaminergic and $5$ nondopaminergic neurons. The classification performance for the test data was used to select the model or which two sectors to use in Algorithm 3 from Tang et al. (2020). The cross validated classification success rates were computed for the remaining $21$ neurons. The cross validated classification success rates averaged over the $1000$ ways of selections of training and test data were plotted in Figure 10. Here we did not necessarily use the entire features but rather changed the numbers of features used for the classification. As the order of adding features can matter, we ordered and added the putatively informative features first. After the classification success rates showed the peaks at the beginning, they decreased slower for the tropical SVM than the classical SVM (Figure 10). In general, we believe that although the classification success rate is enhanced by adding some most informative features at the beginning, soon it will be deteriorated by less informative features, which can be a majority in real world big data. The tropical SVM may be superior in the latter regime of curse of dimensionality. ## 6 Extension of Tropical SVMs to Function Spaces In this section, we define tropical SVMs over a function space to enable the classification of curves, which we encounter frequently in practice. For example, we consider neuronal tuning curves in Figure 3 as a feature vector. That is, we can treat a tuning curve as a point in a function space. For example, it is valuable for identifying dopaminergic neurons in clinical researches (Ishikawa et al., 2018, 2019) to measure the distances between neuronal tuning curves and classify neurons using a tropical hyperplane boundary, which itself can be another curve in tropical SVMs. ### 6.1 Tropical Distances and Hyperplanes on Function Spaces Suppose we have a set of functions mapping from $\mathbb{R}^{d}$ to $\mathbb{R}$ denoted as $\mathcal{F}$ such that: $\begin{array}[]{c}\mathcal{F}:=\\{f:\mathbb{R}^{s}\to\mathbb{R}:\|f(x)\|\mbox{ is bounded,}\,f(x)=f(x)+c,\\\ \forall x\in\mathbb{R}^{s},\,\forall c\in\mathbb{R}\\}.\end{array}$ Let $d_{\rm tr}$ be a distance between functions in $\mathcal{F}$ such that $d_{\rm tr}(f,g):=\max\\{f(x)-g(x):\forall x\in\mathbb{R}^{s}\\}-\min\\{f(x)-g(x):\forall x\in\mathbb{R}^{s}\\}.$ ###### Example 6.1 (Functional Tropical Distances, Figure 11). Let $\mathcal{F}$ be a set of univariate Gaussian distribution functions with $\mu$ and $\sigma$, and mixtures of these Gaussian distributions with real valued coefficients. Suppose $F_{1}$, $F_{2}$, $F_{3}$ and $F_{4}$ are univariate Gaussian distribution functions with $\mu_{1}=\mu_{2}=-2$, $\mu_{3}=\mu_{4}=2$, $\sigma_{1}=\sigma_{3}=1$, and $\sigma_{2}=\sigma_{4}=1/2$. Then, $d_{\rm tr}(F_{1},F_{2})=d_{\rm tr}(F_{3},F_{4})=\frac{11}{8\sqrt{2\pi}}=0.549$, $d_{\rm tr}(F_{1},F_{3})=0.798$, and $d_{\rm tr}(F_{1},F_{4})=d_{\rm tr}(F_{2},F_{3})=1.197$. Figure 11: (left) Four example functions, $F_{1}$, $F_{2}$, $F_{3}$ and $F_{4}\in\mathcal{F}$, which are Gaussian distribution functions with $\mu=-2$ or $2$ and $\sigma=1$ or $0.5$. (middle) $f(x)-F_{1}(x)$ for the four functions in the left figure as $f(x)$. $F_{1}(x)$ denotes the reference black function with $\mu=-2$ and $\sigma=1$. (right) $d_{\rm tr}(f(x),F_{1}(x))$ is gray-color coded for various $\mu$ and $\sigma$ of $f(x)$. Note that the distance is $0$ for $(\mu,\sigma)=(-2,1)$ and increases with increasing $\mu$ or decreasing $\sigma$: $d_{\rm tr}(F_{2},F_{1})=0.549$, $d_{\rm tr}(F_{3},F_{1})=0.798$, and $d_{\rm tr}(F_{4},F_{1})=1.197$. ###### Lemma 6.2. $d_{\rm tr}(f(x),g(x))=d_{\rm tr}(-f(x),-g(x))=d_{\rm tr}(g(x),f(x))$ ###### Proof. $\begin{array}[]{cl}&d_{\rm tr}(f(x),g(x))\\\ =&-\min\\{f(x)-g(x):\forall x\in\mathbb{R}^{s}\\}+\max\\{f(x)-g(x):\forall x\in\mathbb{R}^{s}\\}\\\ =&\max\\{-f(x)+g(x):\forall x\in\mathbb{R}^{s}\\}-\min\\{-f(x)+g(x):\forall x\in\mathbb{R}^{s}\\}\\\ =&d_{\rm tr}(-f(x),-g(x))\\\ =&d_{\rm tr}(g(x)-f(x),0)\\\ =&d_{\rm tr}(g(x),f(x)).\end{array}$ ∎ ###### Lemma 6.3. $d_{\rm tr}(f(x),0)$ is a norm on $\mathcal{F}$. ###### Proof. For the identity, $d_{\rm tr}(f,0)=0$ if and only if $\max\\{f(x):\forall x\in\mathbb{R}^{s}\\}=\min\\{f(x):\forall x\in\mathbb{R}^{s}\\},$ meaning that $f(x)$ is constant. Next, by Lemma 6.2 with $g(x)=0$ $d_{\rm tr}(af,0)=d_{\rm tr}(\|a\|f,0)=\|a\|d_{\rm tr}(f,0)$ for $a\in\mathbb{R}$. For triangle inequality, $\begin{array}[]{cl}&d_{\rm tr}(f(x)+g(x),0)\\\ =&\max\\{f(x)+g(x):\forall x\in\mathbb{R}^{s}\\}-\min\\{f(x)+g(x):\forall x\in\mathbb{R}^{s}\\}\\\ =&\max\\{f(x)+g(x):\forall x\in\mathbb{R}^{s}\\}+\max\\{-f(x)-g(x):\forall x\in\mathbb{R}^{s}\\}\\\ \leq&\max\\{f(x):\forall x\in\mathbb{R}^{s}\\}+\max\\{g(x):\forall x\in\mathbb{R}^{s}\\}\\\ &+\max\\{-f(x):\forall x\in\mathbb{R}^{s}\\}+\max\\{-g(x):\forall x\in\mathbb{R}^{s}\\}\\\ =&\max\\{f(x):\forall x\in\mathbb{R}^{s}\\}+\max\\{g(x):\forall x\in\mathbb{R}^{s}\\}\\\ &-\min\\{f(x):\forall x\in\mathbb{R}^{s}\\}-\min\\{g(x):\forall x\in\mathbb{R}^{s}\\}\\\ =&d_{\rm tr}(f(x),0)+d_{\rm tr}(g(x),0).\\\ \end{array}$ ∎ ###### Lemma 6.4. $d_{\rm tr}$ is a metric on $\mathcal{F}$. ###### Proof. $d_{\rm tr}$ is a metric because it is induced by the norm of the difference, $d_{\rm tr}(f(x),g(x))=d_{\rm tr}(f(x)-g(x),0).$ ∎ By the proof of Lemma 6.4, we have the following theorem: ###### Theorem 6.5. $(\mathcal{F},d_{\rm tr})$ is a normed vector space. ###### Definition 6.6 (Tropical Hyperplane in $\mathcal{F}$). Let $B_{\epsilon}(x)$ be an open ball around $x\in\mathbb{R}^{s}$ with its radius $\epsilon>0$. For any $\omega\in\mathcal{F}$ and for $\epsilon>0$, the tropical hyperplane defined by $\omega$ and $\epsilon$, denoted by $H_{\omega,\epsilon}$, is the set of points $f\in\mathcal{F}$ such that $\|\\{\omega(y)+f(y)=\omega(x^{})+f(x^{}):y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}\|\geq 1,$ where $x^{}\in\operatorname{arg\,max}_{x\in\mathbb{R}^{s}}\left(f(x)+\omega(x)\right).$ We call $\omega\in\mathcal{F}$ the normal vector of $H_{\omega,\epsilon}$. ###### Definition 6.7 (Tropical Distance to a Tropical Hyperplane in $\mathcal{F}$). The tropical distance from a point $f\in\mathcal{F}$ to a tropical hyperplane $H_{\omega,\epsilon}$ is defined as $d_{\rm tr}(f,H_{\omega,\epsilon})\;:=\;\min\\{d_{\rm tr}(f,g)\;\|\;g\in H_{\omega,\epsilon}\\}.$ ###### Lemma 6.8. $d_{\rm tr}(f,H_{\omega,\epsilon})\;=\;d_{\rm tr}(f+\omega,H_{0,\epsilon}).$ ###### Proof. $H_{\omega,\epsilon}$ is defined as: $\begin{array}[]{c}H_{\omega,\epsilon}=\\{g\in\mathcal{F}:\|\\{\omega(y)+g(y)=\omega(x^{})+g(x^{}):\\\ y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}\|\geq 1\\},\end{array}$ where $x^{}\in\operatorname{arg\,max}_{x\in\mathbb{R}^{s}}(\omega(x)+g(x)).$ Thus $\begin{array}[]{rl}&d_{\rm tr}(f,H_{\omega})\\\ =&\Bigl{\\{}d_{\rm tr}(f,g):\|\\{\omega(y)+g(y)=\omega(x^{})+g(x^{}):\\\ &y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}\|\geq 1\Bigr{\\}}\\\ =&\Bigl{\\{}\max\\{f(x)-g(x)\\}-\min\\{f(x)-g(x)\\}:\\\ &\|\\{\omega(y)+g(y)=\omega(x^{})+g(x^{}):y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}\|\geq 1\Bigr{\\}}\\\ =&\Bigl{\\{}\max\\{f(x)-(h(x)-\omega(x))\\}-\min\\{f(x)-(h(x)-\omega(x))\\}:\\\ &\|\\{h(y)=h(x^{}):y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}\|\geq 1\Bigr{\\}}\\\ =&\Bigl{\\{}\max\\{(f(x)+\omega(x))-h(x)\\}-\min\\{(f(x)+\omega(x))-h(x)\\}:\\\ &\|\\{h(y)=h(x^{}):y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}\|\geq 1\Bigr{\\}}\\\ =&\left\\{d_{\rm tr}(f+w,g):\|\\{g(y)=g(x^{}):y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}\|\geq 1\right\\}\\\ =&d_{\rm tr}(f+\omega,H_{0,\epsilon}).\end{array}$ ∎ ###### Lemma 6.9. Let $x^{}\in\operatorname{arg\,max}_{x\in\mathbb{R}^{d}}f(x)$. Then $d_{\rm tr}(f,H_{0,\epsilon})=\max\\{f(x):x\in\mathbb{R}^{d}\\}-\max\\{f(y):y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}$. ###### Proof. Let $\alpha=\max\\{f(x):x\in\mathbb{R}^{d}\\}-\max\\{f(y):y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}$. Then, $g(x)=f(x)+\alpha\mathcal{I}(y)$, where $y\in(\mathbb{R}^{s}-B_{\epsilon}(x^{}))\\}$ and $\mathcal{I}(y)$ is the indicator function. Notice that $g(x)$ is in $H_{0,\epsilon}$. ∎ ###### Example 6.10 (Tropical Hyperplanes in $\mathcal{F}$). We will use the same set up as in Example 6.1 and Figure 11. Let $\epsilon=1$. Then by Lemma 6.9, we have $\begin{array}[]{rl}&d_{\rm tr}(F_{1},H_{0,\epsilon})\\\ =&\max\\{F_{1}(x):x\in\mathbb{R}\\}-\max\\{F_{1}(y):y\in(\mathbb{R}-B_{\epsilon}(x^{})),\\\ &x^{}=\operatorname{arg\,max}(F_{1}(x):x\in\mathbb{R})\\}\\\ =&F_{1}(-2)-F_{1}(-1)\\\ =&0.157.\end{array}$ Similarly, $\begin{array}[]{rl}d_{\rm tr}(F_{3},H_{0,\epsilon})=&d_{\rm tr}(F_{1},H_{0,\epsilon})=F_{1}(-2)-F_{1}(-1)=0.157,\\\ d_{\rm tr}(F_{4},H_{0,\epsilon})=&d_{\rm tr}(F_{2},H_{0,\epsilon})=F_{2}(-2)-F_{2}(-1)=0.690.\end{array}$ Suppose we have $\omega=F_{3}$. Then, by Lemma 6.8, we have $\begin{array}[]{rl}&d_{\rm tr}(F_{1},H_{\omega,\epsilon})\\\ =&d_{\rm tr}(F_{1}+\omega,H_{0,\epsilon})\\\ =&d_{\rm tr}(F_{1}+F_{3},H_{0,\epsilon})\\\ =&(F_{1}(-2)+F_{3}(-2))-(F_{1}(2)+F_{3}(2))\\\ =&0.399-0.399\\\ =&0.\end{array}$ Thus, $F_{1}\in H_{F_{3},\epsilon}$. Similarly, $F_{3}\in H_{F_{1},\epsilon}$, $F_{2}\in H_{F_{4},\epsilon}$ and $F_{4}\in H_{F_{2},\epsilon}$. ###### Definition 6.11 (Sectors of Tropical Hyperplane in $\mathcal{F}$). Each tropical hyperplane $H_{\omega,\epsilon}$ divides $\mathcal{F}$ into components, which are open sectors, $\begin{array}[]{l}S_{\omega,\epsilon}^{x}:=\\{\;f\in\mathcal{F}\|\\\ \omega(x_{0})+f(x_{0})>\omega(y)+f(y),\forall x_{0}\in B_{\epsilon}(x),\forall y\not\in B_{\epsilon}(x)\\},x\in\mathbb{R}^{s}\\}.\end{array}$ ###### Example 6.12. We will use the same set up as in Example 6.1. Again, let $\mathcal{F}$ be a set of univariate Gaussian distribution functions with $\mu$ and $\sigma$, and mixtures of these Gaussian distributions with real valued coefficients. Suppose $\omega\equiv 0$ which is a Gaussian distribution function with $\mu=0$ and $\sigma\to\infty$, $\epsilon=1$, and $x=0$. Then $\begin{array}[]{l}S_{\omega,1}^{0}:=\\{\mbox{Mixtures of univariate Gaussian}\\\ \mbox{distributions with real coefficients whose argmax in }[-1,1]]\\}.\end{array}$ Also note that a set of $\begin{array}[]{l}\\{\mbox{Gaussian distribution functions with }\mu\in[-1,1]\\\ \mbox{ and }\sigma>L\mbox{ for some }L>0\\}\end{array}$ is in this open sector. ### 6.2 Tropical SVMs on Function Spaces For a given $\epsilon>0$, suppose $F\in\mathcal{F}$ and $Y^{1},\,Y^{2}\in\\{0,1\\}$ are random variables such that there exist $\omega^{}\in\mathcal{F}$ with $\Bigl{(}\bigcup_{x^{}\in\operatorname{arg\,max}_{x\in\mathbb{R}^{s}}(\omega^{}(x)+F(x)\|Y^{1})}B_{\epsilon}(x^{})\Bigr{)}\bigcap\Bigl{(}\bigcup_{x^{}\in\operatorname{arg\,max}_{x\in\mathbb{R}^{s}}(\omega^{}(x)+F(x)\|Y^{2})}B_{\epsilon}(x^{})\Bigr{)}=\emptyset.$ (22) Now we set up a tropical SVM over $\mathcal{F}$, whose solution $\omega$ satisfies Equation 22. Let $\mathcal{D}_{\mathcal{F}}$ be the distribution on the joint random variable $(F,Y)$ for $F\in\mathcal{F}$ and $Y\in\\{0,1\\}$, and let $\mathcal{S}_{\mathcal{F}}$ be the sample $\mathcal{S}_{\mathcal{F}}:=\\{(F^{1},Y^{1}),\ldots,(F^{n},Y^{n})\\}$. For a given $\epsilon>0$, we formulate an optimization problem for solving the normal vector $\omega\in\mathcal{F}$ of an optimal tropical separating hyperplane $H_{\omega,\epsilon}$ for random variables $X\in\mathcal{F}$ given $Y\in\\{0,1\\}$: For some cost $C\in\mathbb{R}$, $\begin{matrix}\displaystyle\max\limits_{\omega\in\mathcal{F}}\min\limits_{F(x)\in\mathcal{S}_{\mathcal{F}}}\left(\underbrace{\left(\max_{x\in\mathbb{R}^{s}}(F(x)+\omega(x))-\max_{z\in\mathcal{Z}}(F(z)+\omega(z))\right)}_{\text{margin}}+\underbrace{\frac{C}{n}\sum_{k=1}^{n}\max\left\\{\mathcal{I}_{B_{\epsilon}(X^{})}(x)-Y^{k}\right\\}}_{\text{error}}\right),\end{matrix}$ (23) where $X^{}\in\operatorname{arg\,max}_{x\in\mathbb{R}^{s}}(\omega(x)+F(x))$ and $\mathcal{Z}:=\operatorname{arg\,max}_{x\in(\mathbb{R}^{s}-B_{\epsilon}(X^{}))}(\omega(x)+F(x))$. In practice, we approximate each function $F^{i}$, for $i=1,\ldots n$, by its empirical function $\hat{F}^{i}$ by taking some point $x\in\mathbb{R}^{s}$ to evaluate $F^{i}(x)$. In this paper we propose Algorithm 1 to heuristically conduct a tropical SVM using empirical functions $\hat{F}^{i}$, for $i=1,\ldots n$, using finite set of points $\\{x^{1},\ldots,x^{k}\\}\subset\mathbb{R}^{s}$ with $x^{j}\not\in B_{\epsilon}(x^{i})$ for $j\not=i$ and for $i=1,\ldots,k$. Let $\hat{F}^{i}=(F^{i}(x^{1}),\ldots,F^{i}(x^{k}))$, for $i=1,\ldots n$. Input: A train set $\mathcal{S}=\\{(\hat{F}^{1},Y^{1}),\ldots,(\hat{F}^{n},Y^{n})\\}$ Output: Estimated normal vector $\hat{\omega}$ of a tropical hyperplane Apply $\\{(\hat{F}^{1},Y^{1}),\ldots,(\hat{F}^{n},Y^{n})\\}$ to a tropical SVM over $\mathbb{R}^{k}/\mathbb{R}{\bf 1}$ return _The output from the tropical SVM over $\mathbb{R}^{k}/\mathbb{R}{\bf 1}$_ Algorithm 1 Heuristic tropical SVM over $\mathcal{F}$ ## 7 Discussion We show the generalization error bounds for tropical SVMs over the tropical projection space $\mathbb{R}^{d}/\mathbb{R}{\bf 1}$ which is isometric to $\mathbb{R}^{d-1}$. These bounds still depend on the dimension $d$ and if we fix the sample size $n$, these bounds do not make sense and we cannot extend these bounds for tropical SVMs over a function space $\mathcal{F}$ with tropical metric $d_{\rm tr}$. For future work it is interesting to obtain generalization error bounds for tropical SVMs over a function space $\mathcal{F}$ with tropical metric $d_{\rm tr}$. In addition, computational experiments show that tropical SVMs over $\mathbb{R}^{d}/\mathbb{R}{\bf 1}$ have much lower error rates than ones of $L_{2}$ norm SVMs over $\mathbb{R}^{d}$ when we fix the sample size and grow the dimension $d$. It seems these error rates are bounded by some constant. We are interested in tighter generalization error bounds for tropical SVMs over the tropical projection space $\mathbb{R}^{d}/\mathbb{R}{\bf 1}$. The generalization error bound for tropical SVMs was derived without any assumptions on data distributions. Specifically, the bound was derived purely combinatorially only based on the shapes of the hyperplanes. Algorithmic details of tropical SVMs are not yet taken into account. Thus it is not surprising that the tropical SVMs under the max-plus algebra outperform in the case where a few axes are much more informative than the others. Thus the evaluation of the tropical SVM in computational experiments gave complementary information. The max-plus algebra also leads to the anomalous $\log n$ scaling behaviors in the tropical distance $d_{\rm tr}$ between random vectors of length $n$ and its scaling behavior is the direct consequence of the extreme value statistics. As it is also essential for the theoretical explanation of the robustness of tropical SVMs against the curse of dimensionality, extreme value statistics play the key role in the computation with the tropical metric. For computational experiments, we used software developed by Tang et al. (2020). Tang et al. (2020) developed heuristic methods for hard margin and soft margin tropical SVMs over $\mathbb{R}^{d}/\mathbb{R}{\bf 1}$ since it is infeasible to compute them in practice. We conjecture that computing optimal separating tropical hyperplanes for tropical SVMs is NP-hard. However, we still do not know the exact computational time complexity in terms of $n$ and $d$. ## References Améndola et al. 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# A Characterization for Optimal Bundling of Products with Non-Additive Values Soheil Ghili Yale University E-mail<EMAIL_ADDRESS>Click here for the most current version. I thank Dirk Bergemann, Nima Haghpanah, Johannes Horner, Peter Klibanoff, Barry Nalebuff, Larry Samuelson, Kai Hao Yang, Jidong Zhou, and seminar participants for helpful comments. All errors are mine. ###### Abstract This paper studies optimal bundling of products with non-additive values. Under monotonic preferences and single-peaked profits, I show a monopolist finds pure bundling optimal if and only if the optimal sales volume for the grand bundle is larger than the optimal sales volume for any smaller bundle. I then (i) detail how my analysis relates to “ratio monotonicity” results on bundling; and (ii) describe the implications for non-linear pricing. ## 1 Introduction This paper studies optimal bundling decisions by a multi-product monopolist, and carries out an analysis with two main features. First, I allow for non- additive values: a consumer’s valuation for a given product can depend on whether s/he has purchased other products as well. Second, I seek to obtain a full characterization of when pure bundling (i.e., the act of selling only the package of all available products together as one bundle) is optimal. Under monotonic preferences and single-peaked profits, I prove that optimal bundling admits a simple characterization: Pure bundling is optimal if the optimal sales volume for the grand bundle (if sold alone) is strictly larger than that for any other bundle. Conversely, if there is at least one bundle whose optimal sales volume (if sold alone) is strictly larger than that of the grand bundle, then pure bundling is sub-optimal.111Note that this is slightly short of a full characterization given it does not determine the optimal bundling strategy when the optimal sales volume for the grand bundle is larger than those for smaller bundles but not always strictly. This small gap can be closed but with stronger assumptions that I decided not to make. See section 3 for more details. In simpler terms, bundling is optimal if and only if it helps sell more. The rest of the paper is organized as follows. Section 2 reviews the related literature. Section 3 sets up the model and formally presents the assumptions and the main result. Section 4 provides the proof. Section 5 discusses the assumptions and delves into the implications and interpretations of the main result. It also discusses additional results which are provided in the appendix. Section 6 concludes. ## 2 Related Literature The study of bundling dates at least as far back as Stigler (1963). Most papers in this literature focus on the case of “additive values,” meaning the valuation by each consumer of any given product $i$ is not impacted by whether she also possesses product $i^{\prime}\neq i$. Pioneering in this area was Adams and Yellen (1976), pointing out that bundling can be more profitable than unbundling when there is negative correlation among consumers in how they value individual products. Other studies such as McAfee et al. (1989); Menicucci et al. (2015); Pavlov (2011); Schmalensee (1984); Fang and Norman (2006); Manelli and Vincent (2007); Palfrey (1983); Bergemann et al. (2021) further develop results on optimal bundling (or optimal upgrade pricing) under additive values. Many of these studies focus on a setting with two products only. Also, with few exceptions– e.g., Daskalakis et al. (2017) who provide necessary and sufficient conditions–most of these studies concentrate on sufficient conditions for bundling. Although most of this literature examines a monopolist seller (which is also the focus of this paper), some studies have analyzed multiple sellers (McAfee et al. (1989); Zhou (2017, 2019)). The literature on non-additive values, to which this paper belongs, is considerably smaller. Part of this literature focuses directly on bundling (e.g., Haghpanah and Hartline (2021); Armstrong (2013, 2016); Long (1984)) whereas some study price discrimination settings which have implications for bundling (e.g., Anderson and Dana Jr (2009); Deneckere and Preston McAfee (1996)). This paper complements the literature on non-additive values in that it imposes a different set of assumptions (stronger only than those imposed by Haghpanah and Hartline (2021)) and delivers simple-to-interpret but necessary and sufficient conditions for bundling based on optimal quantities sold. I also connect the interpretation of my results to those based on price elasticities (such as Long (1984); Armstrong (2013)) as well as those based on ratio monotonicity (such as Haghpanah and Hartline (2021); Anderson and Dana Jr (2009); Deneckere and Preston McAfee (1996); Salant (1989)). Finally, I describe implications of my bundling results for screening models and nonlinear tariff design (e.g., Mussa and Rosen (1978); Maskin and Riley (1984)). ## 3 Main Result ### 3.1 Setup and Notations A monopolist has $n$ products to sell, indexed 1 through $n$. Possible bundles of these products are denoted $b\subseteq\\{1,...,n\\}$. Set $\mathcal{B}=\\{b\|b\subseteq\\{1,...,n\\}\\}$ represents the set of all possible bundles.222My notation, in part, follows Haghpanah and Hartline (2021). By $\bar{b}$ denote the grand bundle $\\{1,...,n\\}$. For any bundle $b$, denote $b^{C}=\bar{b}\setminus b$. There is a unit mass of customers whose types are represented by $t\in T\subset\mathbb{R}^{m}$ where $T$ is compact. As will be shown later, one of the model assumptions will imply that types are one-dimensional (i.e., $m=1$.)333The reason why I start with general $m$ and later show $m=1$ is that this exposition clarifies that $m=1$ is a an implication of the rest of the model assumptions, rather than a separate assumption itself. Probability distribution over types $f(\cdot)>0$ has no atoms.444The main result should hold without these assumptions on $f$. But I expect the proof to be less clean. The valuation by type $t$ for bundle $b$ is denoted $v(b,t)$. Assume $v(\emptyset,t)=0$. Also, for all $b$, suppose that $v(b,t)$ is continuous in $t$ except, possibly, for finitely many points. The per-unit cost of production for each product $i$ is $c_{i}\geq 0$. The problem the monopolist solves has two components. First, she makes a bundling decision. She chooses the optimal set $B^{}$ of bundles $b$ among subsets $B$ of $\mathcal{B}$ that satisfy $\emptyset\notin B$. Note that there are $2^{2^{n}-1}$ possible bundling strategies. Thus, characterizing the conditions under which the monopolist can simply choose $B^{}=\\{\bar{b}\\}$ should indeed be of value. The second decision by the firm is choosing prices $p(\cdot):B\rightarrow\mathbb{R}$ for the bundles offered.555Note that, in principle, one could model the bundling decision through pricing; because not offering a product would be equivalent to pricing it so high that no customer would purchase it. As such, separating the bundling and pricing decisions in the model is redundant. Nevertheless, I decided to carry out this separation because it makes the notation easier. Denote by $\mathcal{P}_{B}$ the set of all possible such pricing functions. Once the firm decides on set $B$ and prices $p(\cdot)$, customers decide which bundles to purchase (note that this model only considers deterministic selling procedures.) Each customer $t$’s decision $\beta(t\|B,p)\subseteq B$ is determined by: $\beta(t\|B,p)=\arg\max_{\hat{\beta}\subseteq B}v(\cup_{b\in\hat{\beta}}b,t)-\Sigma_{b\in\hat{\beta}}p(b)$ (1) Throughout, I assume customers break ties in favor of the seller. Also, note that equation 1 implies that customers want at most one unit of each product $i$ and find additional units redundant. Demand for bundle $b$ is given by the measure of customers $t$ who would choose to purchase bundle $b$: $D(b\|B,p)=\int_{t}\mathds{1}_{b\in\beta(t\|B,p)}f(t)dt$ (2) Firm profit under strategy $(B,p)$ is: $\pi(B,p)=\int_{t}\Sigma_{b\in\beta(t\|B,p)}\bigg{(}p(b)-\Sigma_{i\in b}c_{i}\bigg{)}f(t)dt$ (3) The monopolist chooses $(B^{},p^{})$ to maximize profit: $(B^{},p^{})=\arg\max_{B\in\mathcal{B},p\in\mathcal{P}_{B}}\pi(B,p)$ (4) With the setup of the firm problem laid out, I now introduce a few more definitions and notations. For disjoint bundles $b$ and $b^{\prime}$, denote by $v(b,t\|b^{\prime})$ the valuation by type $t$ for $b$ conditional on possessing $b^{\prime}$. Formally: $v(b,t\|b^{\prime})\equiv v(b\cup b^{\prime},t)-v(b^{\prime},t)$ In a similar manner, denote $\beta(t\|B,p,b^{\prime})=\arg\max_{\hat{\beta}\subseteq(B\setminus\\{b^{\prime}\\})}v(\cup_{b\in\hat{\beta}}b,t\|b^{\prime})-\Sigma_{b\in\hat{\beta}}p(b)$. Also denote $D(b\|B,p,b^{\prime})=\int_{t}1_{b\in\beta(t\|B,p,b^{\prime})}f(t)dt$. Moreover, denote $\pi(B,p\|b^{\prime})=\int_{t}\Sigma_{b\in\beta(t\|B,p,b^{\prime})}\bigg{(}p(b)-\Sigma_{i\in b}c_{i}\bigg{)}f(t)dt.$ Finally, for any set $\beta$ of bundles, denote $\beta^{\cup}=\cup_{b\in\beta}b$. Similarly, I write $\beta^{\cup}(t\|B,p,b^{\prime})$ instead of $\cup_{b\in\beta(t\|B,p,b^{\prime})}b$. I next turn to the assumptions and the main result. ### 3.2 Assumptions and Characterization The main results of the paper is about how optimal bundling decisions are informed by the comparison among optimal sales volumes for different bundles. I start with some necessary assumptions and definitions. ###### Assumption 1. Monotonicity: For all $b$, $v(b,t)$ is increasing in $v(\bar{b},t)$, and strictly so whenever $v(b,t)>0$. The same applies to $v(b,t\|b^{C})$ for all $b$. ###### Assumption 2. Quasi-concavity: For any $b\in\mathcal{B}$, profit functions $\pi(b,p)$ and $\pi(b,p\|b^{C})$ are strictly quasiconcave in $p(b)$ for all values of $p(b)$ that yield strictly positive demand for $b$.666This simply means the profit peaks only once as we vary each price. ###### Definition 1. By $D^{}(b)$ denote the “optimal quantity sold” of bundle $b$ if no other bundle were offered by the firm. Formally, $D^{}(b)$ is defined as $D(b\|\\{b\\},p_{b}^{})$ where $p_{b}^{}$ is the optimal price for bundle $b$ when $B=\\{b\\}$. ###### Definition 2. A given firm strategy $(B,p)$ involves pure bundling if: $\forall t:\beta^{\cup}(t\|B,p)\in\\{\emptyset,\bar{b}\\}$ We are now ready to state the main result. ###### Theorem 1. Under assumptions 1 and 2, the optimal strategy $(B^{},p^{})$ involves pure bundling if: $D^{}(\bar{b})>\max_{b\in\mathcal{B}\setminus\\{\bar{b}\\}}D^{}(b)$ Conversely, the optimal strategy does not involve pure bundling if: $D^{}(\bar{b})<\max_{b\in\mathcal{B}\setminus\\{\bar{b}\\}}D^{}(b)$ In words, this result says that the firm should pure bundle if and only if it helps “sell more.”777Note that this result is slightly short of a full characterization because it does not specify whether pure bundling is optimal when $D^{}(\bar{b})=\max_{b\in\mathcal{B}\setminus\\{\bar{b}\\}}D^{}(b)$. One can show that under this last possibility, pure bundling is optimal if instead of assuming profits are strictly quasi-concave in each price, we assume they are strictly concave and differntiable at peak. Even though this would yield a full characterization, I decided that the ability to speak to the “measure-zero” case of $D^{}(\bar{b})=\max_{b\in\mathcal{B}\setminus\\{\bar{b}\\}}D^{}(b)$ is too small a return to justify such a restrictive assumption as strict concavity. As such, I maintain the quasi-concavity assumption. ## 4 Proof of Theorem 1 I start by some remarks, definitions, and lemmas. ###### Lemma 1. There is a mapping $\tau$ from the set $T$ of types $t$ on to the interval $[0,1]$ such that: 1. 1. $\forall t,t^{\prime}\in T:v(\bar{b},t)>v(\bar{b},t^{\prime})\Leftrightarrow\tau(t)>\tau(t^{\prime})$. 2. 2. $\tau$ is a sufficient statistic: Once $\tau(t)$ is known, one can fully pin down all $v(b,t)$ without having to know $t$. Proof of Lemma 1. Set $\tau(t)\triangleq\frac{v(\bar{b},t)-\min_{t^{\prime}\in T}v(\bar{b},t^{\prime})}{\max_{t^{\prime}\in T}v(\bar{b},t^{\prime})-\min_{t^{\prime}\in T}v(\bar{b},t^{\prime})}$. By construction, it satisfies (1). To see why it satisfies (2), first note that by monotonicity, for any $t,t^{\prime}$ such that $v(\bar{b},t)=v(\bar{b},t^{\prime})$, we have $v(b,t)=v(b,t^{\prime})$ for any other $b$. As a result, $v(\bar{b},t)$ is sufficient information for determining $v(b,t)$ for all $b$. Next, observe that one can recover $v(\bar{b},t)$ from $\tau(t)$: $v(\bar{b},t)=\min_{t^{\prime}\in T}v(\bar{b},t^{\prime})+\tau(t)\times(\max_{t^{\prime}\in T}v(\bar{b},t^{\prime})-\min_{t^{\prime}\in T}v(\bar{b},t^{\prime}))$. As a result, once one knows $\tau(t)$, one would also know $v(b,t)$ for all $b$. Q.E.D. Based on this lemma, it is without loss to think of $t$ as $\tau(t)$ and, hence, the set of all possible $t$ as $[0,1]$. Therefore, we can use expressions such as $t\geq t^{\prime}$. Going forward, I assume $t\in[0,1]$. ###### Remark 1. Suppose functions $f_{1}(x)$, $f_{2}(x)$ and $f_{1}(x)+f_{2}(x)$ are all strictly quasi-concave over the interval $[a,b]$. Then either (i) $\arg\max f_{1}\leq\arg\max f_{2}$ or (ii) $\arg\max f_{1}\leq\arg\max(f_{1}+f_{2})$ will imply: $\arg\max f_{1}\leq\arg\max(f_{1}+f_{2})\leq\arg\max f_{2}.$ The proof is left to the reader. ###### Definition 3. For disjoint bundles $b$ and $b^{\prime}$, denote by $D^{}(b\|b^{\prime})$ the “optimal quantity sold” of bundle $b$ if all customers are already endowed with $b^{\prime}$ and only $b$ is offered by the firm at optimal price. Formally, $D^{}(b\|b^{\prime})\equiv D(b\|\\{b\\},p_{b\|b^{\prime}}^{},b^{\prime})$ where $p_{b\|b^{\prime}}^{}:\\{b\\}\rightarrow\mathbb{R}$ is effectively one real number, and it is chosen among other possible $p$ so that $\pi(\\{b\\},p\|b^{\prime})$ is maximized. Next, I show that the problem of finding the optimal price for a bundle is equivalent to the problem of finding the right type $t^{}$ and sell to types $t\geq t^{}$. ###### Definition 4. Define by $t^{}(b\|b^{\prime})$ the largest $t$ such that $1-F(t)\geq D^{}(b\|b^{\prime})$. Also, for simplicity, denote $t^{}(b\|\emptyset)$ by $t^{}(b)$. ###### Lemma 2. Consider bundles $b$ and $b^{C}=\bar{b}\setminus b$. Suppose that all types are endowed with bundle $b^{C}$, and that the firm is selling only bundle $b$, optimally choosing $p^{}_{b\|b^{C}}$. The set of types who will buy the product at this price is the interval $[t^{}(b\|b^{C}),1]$. Proof of Lemma 2. Follows directly from monotonicity. Monotonicity implies that the optimal sales volume $D^{}(b\|b^{C})$ would be purchased by the highest types $t$ with $t$ weakly above some cutoff $\hat{t}$. Definition 4 says that for the demand volume to equal $D^{}(b\|b^{C})$, the cutoff $\hat{t}$ has to equal $t^{}(b\|b^{C})$. Q.E.D. Lemma 2 is important in that it shows the problem of choosing $p^{}_{b\|b^{C}}$ can equivalently be thought of as the problem of choosing $t^{}_{b\|b^{C}}$. This allows us to set up the firm’s problem based on $t$. Next definition introduces a necessary notation for this purpose. ###### Definition 5. Consider disjoint bundles $b$ and $b^{\prime}$. Suppose that all types have already been endowed with $b^{\prime}$, and that the firm is to sell only bundle $b$. By $\pi_{b}(t\|b^{\prime})$ denote the profit to the firm if it chose a price for bundle $b$ such that all types $t^{\prime}\geq t$ would purchase bundle $b$: $\pi_{b}(t\|b^{\prime})=(1-F(t))\times\big{(}(v(t,b\|b^{\prime})-\Sigma_{i\in b}c_{i}\big{)}$ $=\pi(\\{b\\},v(b,t\|b^{\prime})\|b^{\prime})$ ###### Lemma 3. $\pi_{b}(t\|b^{C})$ is strictly quasi-concave in $t$. Proof of Lemma 3. Suppose $\pi_{b}(t\|b^{C})$ is not quasi-concave in $t$. This means there are $t_{1}<t_{2}<t_{3}$ such that $\pi_{b}(t_{2}\|b^{C})\leq\min(\pi_{b}(t_{1}\|b^{C}),\pi_{b}(t_{3}\|b^{C}))$. Then construct $p_{1},p_{2}$ and $p_{3}$ from $t_{1},t_{2}$ and $t_{3}$ according to the procedure in definition 5. That is, set $p_{i}=v(b,t\|b^{C})$ for each $i$. Monotonicity puts $p_{2}$ strictly between $p_{1}$ and $p_{3}$. Note that for these prices, we have: $\pi(\\{b\\},p_{2}\|b^{C})\leq\min(\pi(\\{b\\},p_{1}\|b^{C}),\pi(\\{b\\},p_{3}\|b^{C}))$ which violates the quasi-concavity assumption in $p$. Q.E.D. With the above definitions and lemmas in hand, we are ready to prove the main theorem. I start by the necessity condition (i.e., the condition that $D^{}(\bar{b})\geq D^{}(b)$ for all $b$ is necessary for pure bundling to optimal). Proof of necessity. We want to show that if there is some $b$ such that $D^{}(b)>D^{}(\bar{b})$, then pure bundling is sub-optimal. Specifically, I show that offering bundles $b$ and $\bar{b}$ would be strictly more profitable to the firm compared to offering $\bar{b}$ alone. The argument follows. ###### Lemma 4. $D^{}(b)>D^{}(\bar{b})$ implies $D^{}(b)>D^{}(b^{C}\|b)$. Proof of Lemma 4. Suppose, on the contrary, that $D^{}(b)\leq D^{}(b^{C}\|b)$. This means $t^{}(b)\geq t^{}(b^{C}\|b)$. We know: $t^{}(b)=\arg\max_{t}\pi_{b}(t)$ and $t^{}(b^{C}\|b)=\arg\max_{t}\pi_{b^{C}}(t\|b).$ Also, given definition 5, it is straightforward to verify that: $\pi_{\bar{b}}(t)\equiv\pi_{b^{C}}(t\|b)+\pi_{b}(t)$ By strict quasi-concavity of all profits in $t$ and by remark 1, it has to be that the argmax of $\pi_{\bar{b}}(t)$ falls in between the argmax values $t^{}(b^{C}\|b)$ and $t^{}(b)$. Therefore, we get: $t^{}(\bar{b})\leq t^{}(b)$, which implies $D^{}(b)\leq D^{}(\bar{b})$, contradicting a premise of the lemma. Q.E.D. ###### Lemma 5. Selling $D^{}(b^{C}\|b)$ units of the grand bundle $\bar{b}$ along with $D^{}(b)-D^{}(b^{C}\|b)$ units of bundle $b$ would be strictly more profitable to the monopolist compared to selling $D^{}(\bar{b})$ units of the grand bundle alone. Proof of Lemma 5. Note that given monotonicity and given Lemma 4, selling $D^{}(b^{C}\|b)$ units of the grand bundle $\bar{b}$ along with $D^{}(b)-D^{}(b^{C}\|b)$ units of bundle $b$ would simply mean selling $b$ to types $[t^{}(b),t^{}(b^{C}\|b))$ and selling $\bar{b}$ to types $[t^{}(b^{C}\|b),1]$. This can be implemented by offering bundles $b$ and $\bar{b}$ and pricing them at $p^{}_{b}$ and $p^{}_{b}+p^{}_{b^{C}\|b}$ respectively.888Recall that I use the notation $p^{}_{b}$ for the optimal price of $b$ when only $b$ is offered. Similarly, $p^{}_{b^{C}\|b}$ is the optimal price of $b^{C}$ when everyone is endowed by $b$ and the monopolist is only selling $b^{C}$. This would deliver the following profit: $\pi_{1}=\pi_{b^{C}}(t^{}(b^{C}\|b)\|b)+\pi_{b}(t^{}(b))$ Again, by monotonicity, selling $D^{}(\bar{b})$ units of the grand bundle can be thought of as selling $\bar{b}$ to types $t^{}(\bar{b})$ and above. This would deliver a profit of $\pi_{2}=\pi_{\bar{b}}(t^{}(\bar{b}))$, which can be expanded and written as: $\pi_{2}=\pi_{b^{C}}(t^{}(\bar{b})\|b)+\pi_{b}(t^{}(\bar{b}))$ Note that each term in $\pi_{2}$ is weakly less than its corresponding term in $\pi_{1}$ (due to the optimality of the terms in $\pi_{1}$). Also by the fact that $t^{}(b^{C}\|b)>t^{}(b)$, then either $t^{}(b^{C}\|b)\neq t^{}(\bar{b})$ or $t^{}(b)\neq t^{}(\bar{b})$. Thus, by strict quasi- concavity, at least one of the two inequalities between corresponding terms in $\pi_{1}$ and $\pi_{2}$ has to be strict, yielding $\pi_{1}>\pi_{2}$. Q.E.D. Given this lemma, the proof of the if side of the theorem is now complete. Q.E.D. Note that the proof of the first side did not use the constant marginal costs assumption. Next, I turn to the proof of the sufficiency conditions (i.e., that $D^{}(\bar{b})>\max_{b\in\mathcal{B}\setminus\\{\bar{b}\\}}D^{}(b)$ implies that pure bundling is optimal). Proof of sufficiency. I start with some lemmas. ###### Lemma 6. Under assumptions 1 and 2, there is a firm optimal strategy $(B^{},p^{})$ such that non-measure-zero set of customers $t$ we have $\beta^{\cup}(t\|B^{},p^{})=\bar{b}$. Proof of Lemma 6. I start by assuming that no optimal strategy $(B^{},p^{})$ involves selling $\bar{b}$ to a non-measure-zero set of consumers. Then I reach a contradiction by constructing a weakly profitable deviation from an assumed optimal $(B^{},p^{})$ such that the deviation sells $\bar{b}$ to a non-measure-zero set of consumers. By $D^{}(\bar{b})>D^{}(b)$ for all $b\neq\bar{b}$, we get: $D^{}(\bar{b})>0$, which in turn yields $t^{}(\bar{b})<1$. Next, note that the number of possibilities for $\beta(t\|B^{},p^{})$ is finite. By this finiteness and by piece-wise continuity of value functions, there is some $\tilde{t}\geq t^{}(\bar{b})$ such that all consumers with types higher than $\tilde{t}$ have the same purchase behavior.999Perhaps with the exception of type $t=1$; but that does not matter given its zero measure. Formally: $\forall t,t^{\prime}\in(\tilde{t},1):\beta^{\cup}(t\|B^{},p^{})=\beta^{\cup}(t^{\prime}\|B^{},p^{})$ Denote this commonly purchased bundle $\tilde{b}$. Given that our contrapositive assumption is that no non-measure-zero set of types purchases the grand bundle $\bar{b}$, it has to be that $\tilde{b}\neq\bar{b}$. Next, I construct a profitable deviation for the monopolist from $(B^{},p^{})$. First, note that given that currently no consumer purchases $\bar{b}$ or constructs it from other bundles, it has to be that either $\bar{b}$ is not part of $B^{}$ or it is expensive enough for no customer to prefer to obtain it. Next, assume that the monopolist deviates from $(B^{},p^{})$ by adding $\bar{b}$ to the set of bundles and pricing it at $p^{}(\tilde{b})+\Sigma_{i\in\tilde{b}^{C}}c_{i}+\epsilon$ where $\epsilon$ is chosen so that (i) type $\tilde{t}$ would weakly prefer $\tilde{b}$ over $\bar{b}$ but (ii) type 1 would weakly prefer $\bar{b}$ over $\tilde{b}$. Next, I proceed to show two things. First: finding such an $\epsilon$ is feasible. Second: with that $\epsilon$, the monopolist will see a weak profit increase relative to $(B^{},p^{})$ and sell $\bar{b}$ to a positive-measure set of types. For type $\tilde{t}$ to weakly prefer $\tilde{b}$ over $\bar{b}$, it has to be that $v(\bar{b},\tilde{t})-p^{}(\tilde{b})-\Sigma_{i\in\tilde{b}^{C}}c_{i}-\epsilon\leq v(\tilde{b},\tilde{t})-p^{}(\tilde{b})$ $\Leftrightarrow\epsilon\geq v(\tilde{b}^{C},\tilde{t}\|\tilde{b})-\Sigma_{i\in\tilde{b}^{C}}c_{i}$ Similarly, for type to 1 weakly to prefer to purchase $\bar{b}$, one can show that $\epsilon$ must satisfy: $\epsilon\leq v(\tilde{b}^{C},1\|\tilde{b})-\Sigma_{i\in\tilde{b}^{C}}c_{i}$ But by monotonicity, we have $v(\tilde{b}^{C},\tilde{t}\|\tilde{b})\leq v(\tilde{b}^{C},1\|\tilde{b})$. Therefore, $\epsilon$ may be chosen within the following interval: $[v(\tilde{b}^{C},\tilde{t}\|\tilde{b})-\Sigma_{i\in\tilde{b}^{C}}c_{i},v(\tilde{b}^{C},1\|\tilde{b})-\Sigma_{i\in\tilde{b}^{C}}c_{i}]$ If the interval is not a singleton, choose $\epsilon$ in the interior. Next, I show that once such $\epsilon$ is chosen, the new bundling and pricing strategy by the firm will weakly increase the profit to the insurer while selling $\bar{b}$ to a non-measure zero set of consumers. To see the latter, denote by $\tilde{t}^{\prime}$ the set of types who weakly prefer $\bar{b}$ over $\tilde{b}$ under this new strategy. By the choice of $\epsilon$ and by monotonicity, we have $\tilde{t}\leq\tilde{t}^{\prime}<1$. Therefore the new strategy will change the purchase behavior by types $t\geq\tilde{t}^{\prime}$ (and only those types.) Next note that this behavior-change is weakly profitable to the monopolist. Prior to this change, the profit to the monopolist from these types was: $\pi_{1}=(1-\tilde{t}^{\prime})\times(p^{}(\tilde{b})-\Sigma_{i\in\tilde{b}}c_{i})$ Under the new strategy (i.e., with the introduction of $\bar{b}$ at the price of $p^{}(\tilde{b})+\Sigma_{i\in\tilde{b}^{C}}c_{i}+\epsilon$,) the new profit level from these types is: $\pi_{2}=(1-\tilde{t}^{\prime})\times\bigg{(}(p^{}(\tilde{b})+\Sigma_{i\in\tilde{b}^{C}}c_{i}+\epsilon)-\Sigma_{i\in\bar{b}}c_{i}\bigg{)}$ $=\pi_{1}+(1-\tilde{t}^{\prime})\times\epsilon$ Thus, it remains to show $\epsilon\geq 0$. To this end, note that by $D^{}(\bar{b})>D^{}(\tilde{b})$ we have $t^{}(\bar{b})\leq t^{}(\tilde{b})$. This, by monotonicity, quasi-concavity, and remark 1, implies $t^{}(\tilde{b}^{C}\|\tilde{b})\leq t^{}(\bar{b})$ which in turn yields $t^{}(\tilde{b}^{C}\|\tilde{b})\leq\tilde{t}$. That is, all types weakly above $\tilde{t}$ would purchase $\tilde{b}^{C}$ if (i) they were offered it at the optimal price for the monopolist and (ii) they were pre- endowed with $\tilde{b}$. This implies that $\forall t\geq\tilde{t}:v(\tilde{b}^{C},t\|\tilde{b})\geq p^{}_{\tilde{b}^{C}\|\tilde{b}}\geq\Sigma_{i\in\tilde{b}^{C}}c_{i}$. But this means $\epsilon=v(\tilde{b}^{C},\tilde{t}\|\tilde{b})-\Sigma_{i\in\tilde{b}^{C}}c_{i}\geq 0$. As a result, we get $\pi_{2}\geq\pi_{1}$, which completes the proof of the lemma.Q.E.D. Next, it is useful to observe that the monotonicity assumption imposes a vertical relationship not only on consumers’ preferences, but also on their purchase behaviors. ###### Lemma 7. Consider bundling strategy $B$ and pricing strategy $p$. Consider types $t$ and $t^{\prime}$ such that $\beta^{\cup}(t\|B,p)\neq\beta^{\cup}(t^{\prime}\|B,p)$. Then the following statements hold: 1. 1. If $\beta^{\cup}(t^{\prime}\|B,p)=\emptyset$, we have $t^{\prime}<t$. 2. 2. If $\beta^{\cup}(t^{\prime}\|B,p)=\bar{b}$, we have $t^{\prime}>t$. This lemma says that if there are types who buy $\bar{b}$, they are the highest types. Also if there are types who buy nothing, they are the lowest types. Proof of Lemma 7. I prove the second statement in the lemma. The first statement would be proved in a similar way. Suppose that $\beta^{\cup}(t\|B,p)\neq\beta^{\cup}(t^{\prime}\|B,p)=\bar{b}.$ For simplicity, denote $\beta^{\cup}(t\|B,p)=\tilde{b}$. Now suppose, contrary to the statement of the lemma, that $t^{\prime}\leq t$. Given $\beta(t\|B,p)^{\cup}\neq\beta^{\cup}(t^{\prime}\|B,p)$, we know $t\neq t^{\prime}$ which implies $t^{\prime}<t$. Next, observe the following two inequalities: First, note that under $(B,p)$, type $t$ prefers purchasing $\beta(t\|B,p)$ and forming $\tilde{b}$ over purchasing $\beta(t^{\prime}\|B,p)$ and forming $\bar{b}$. Formally: $v(\tilde{b},t)-\Sigma_{b\in\beta(t\|B,p)}p(b)\geq v(\bar{b},t)-\Sigma_{b\in\beta(t^{\prime}\|B,p)}p(b)$ (5) Similarly, type $t^{\prime}$ prefers to purchase $\beta(t^{\prime}\|B,p)$ and forming $\bar{b}$ over purchasing $\beta(t\|B,p)$ and forming $\tilde{b}$. Formally: $v(\bar{b},t^{\prime})-\Sigma_{b\in\beta(t^{\prime}\|B,p)}p(b)\geq v(\tilde{b},t^{\prime})-\Sigma_{b\in\beta(t\|B,p)}p(b)$ (6) At least one of the two inequalities above has to be strict (because if both types were indifferent between $\bar{b}$ and $\tilde{b}$, they would break this tie the same way.) Adding these two inequalities together, we get: $v(\bar{b},t^{\prime})+v(\tilde{b},t)>v(\bar{b},t)+v(\tilde{b},t^{\prime})$ $\Leftrightarrow v(\bar{b},t^{\prime})-v(\tilde{b},t^{\prime})>v(\bar{b},t)-v(\tilde{b},t)$ $v(\tilde{b}^{C},t^{\prime}\|\tilde{b})>v(\tilde{b}^{C},t\|\tilde{b})$ This latter statement, combined with $t^{\prime}<t$, contradicts monotonicity. Q.E.D. In light of lemma 6, the following two corollaries of lemma 7 are useful. ###### Corollary 1. Under $(B^{},p^{})$, the set of types to for which $\beta^{\cup}(t\|B^{},p^{})=\bar{b}$ takes the form of $[t_{1},1]$ for some $t_{1}<1$. ###### Corollary 2. Under $(B^{},p^{})$, the set of types to for which $\beta^{\cup}(t\|B^{},p^{})=\emptyset$ takes the form of $[0,t_{2})$ for some $t_{2}<1$. With these lemmas in hand, I next turn to the proof of the sufficiency conditions. The strategy is, again, contrapositive. Assume on the contrary that we have, at the same time: (i) $\forall b\neq\bar{b}:D^{}(b)<D^{}(\bar{b})$ and (ii) the firm’s optimal strategy does not involve pure bundling. This latter statement implies that the set of all distinct bundles chosen by customers under $(B^{},t^{})$ includes members other than $\emptyset$ or $\bar{b}$. Formally, if we denote $\beta^{}=\\{b\|\exists t:\beta^{\cup}(t\|B^{},p^{})=b\\}$ then $\beta^{}\setminus\\{\emptyset,\bar{b}\\}\neq\emptyset$. In other words, our contrapositive assumption implies that $t_{1}$ in corollary 1 is strictly larger than $t_{2}$ in corollary 2. Then, note that by corollary 1 and the piece-wise continuity of values in $t$, there is some bundle $b_{1}\in\beta^{}\setminus\\{\emptyset,\bar{b}\\}$ such that for $t_{1}^{\prime}$ close enough to but smaller than $t_{1}$, we have: $\forall t\in[t_{1}^{\prime},t_{1}):\beta^{\cup}(t\|B^{},p^{})=b_{1}$ (7) Also, by corollary 2 and by piece-wise continuity of values in $t$, there is some bundle $b_{2}\in\beta^{}\setminus\\{\emptyset,\bar{b}\\}$ (which may or may not be the same as $b_{1}$) such that for $t_{2}^{\prime}$ close enough to but larger than $t_{2}$, we have: $\forall t\in[t_{2},t_{2}^{\prime}]:\beta^{\cup}(t\|B^{},p^{})=b_{2}$ (8) The rest of the proof of the sufficiency conditions of the theorem is organized as follows. I first make a series of claims (without proving them). Next I use the claims to prove the sufficiency conditions of the theorem. Finally, I will return to the proofs of the claims. ###### Claim 1. $t^{}(b_{1}^{C}\|b_{1})=t_{1}$. In words, claim 1 says that the set of customers who purchase the grand bundle $\beta(t\|B^{},p^{})=\bar{b}$ under the firm optimal strategy $(B^{},p^{})$ is the same as those who purchase $b_{1}^{C}$ and construct the grand bundle if (i) everyone is endowed with $b_{1}$ and (ii) the firm offers only $b_{1}^{C}$, pricing it optimally. ###### Claim 2. $t^{}(b_{2})=t_{2}$. Claim 2 says that the set of customers who purchase $\emptyset$ under the firm optimal strategy $(B^{},p^{})$ is the same as those who purchase $\emptyset$ if the firm offers only $b_{2}$ and prices it optimally. Next, note that the assumption $D^{}(\bar{b})>D^{}(b_{2})$, combined with monotonicity and claim 2, implies $t^{}(\bar{b})\leq t_{2}$. By $t_{1}>t_{2}$, we get $t^{}(\bar{b})<t_{1}=t^{}(b_{1}^{C}\|b_{1})$. Also note that: $\forall t:\pi_{\bar{b}}(t)=\pi_{b_{1}^{C}}(t\|b_{1})+\pi_{b_{1}}(t)$ As such, by strict quasi-concavity of profits, by $t^{}(\bar{b})<t^{}(b_{1}^{C}\|b_{1})$, and by remark 1, the peak of $\pi_{\bar{b}}(t)$ should happen in between those of $\pi_{b_{1}^{C}}(t\|b_{1})$ and $\pi_{b_{1}}(t)$. Therefore, we should have: $t^{}(b_{1})\leq t^{}(\bar{b})\leq t^{}(b_{1}^{C}\|b_{1})$. But $t^{}(b_{1})\leq t^{}(\bar{b})$ implies: $D^{}(b_{1})\geq D^{}(\bar{b})$ which is a contradiction. Therefore, the sufficiency part of the theorem is true provided that claims 1 and 2 are true. I now turn to the proofs of these claims. Proof of Claim 1. Suppose on the contrary that $t^{}(b_{1}^{C}\|b_{1})\neq t_{1}$. In that case, it can be shown that the firm can strictly improve its profit upon the optimal strategy $(B^{},p^{})$. The proof of this claim constructs such improvement. To this end, the following remark is useful to state. ###### Remark 2. Construct the bundling strategy $(\hat{B},\hat{p})$ from $(B^{},p^{})$ in the following way: * • $\hat{B}=\\{\beta^{\cup}(t\|B^{},p^{})\forall t\\}$ * • For each $t$, or in other words for each $\hat{b}=\beta^{\cup}(t\|B^{},p^{})\in\hat{B}$, set $\hat{p}(\hat{b})=\Sigma_{b\in\hat{b}}p^{}(b)$. For such $(\hat{B},\hat{p})$, we have: 1. 1. $\forall t:\beta(t\|\hat{B},\hat{p})=\beta(t\|B^{},p^{})$ 2. 2. $\pi(\hat{B},\hat{p})=\pi(B^{},p^{})$ This remark simply states that there is an optimal strategy by the seller under which each buyer type only purchases a single bundle instead of combining different bundles to construct her/his desired one. I skip the proof of this remark. Also, in order to save on notation, I assume from now on that it is $(B^{},p^{})$ itself that has the feature of every $\beta(t\|B^{},p^{})$ being a singleton. I now return to the proof of claim 1 and construct a strict improvement upon the profit of $(B^{},p^{})$. I do so by slightly adjusting the price of $\bar{b}$. That is, I show that there is a pricing strategy $p$ with $p(b)=p^{}(b)$ for all $b\neq\bar{b}$, but with $p(\bar{b})\neq p^{}(\bar{b})$, such that $\pi(B^{},p)>\pi(B^{},p^{})$. To see why this is the case, construct bundling strategy $B^{\prime}$ in the following way: $B^{\prime}=\\{b_{1},\bar{b}\\}$ (9) Also construct pricing strategy $p^{\prime}$ by fixing $p^{\prime}(b_{1})=\min_{t}v(b_{1})$ and setting $p^{\prime}(\bar{b})=p^{\prime}(b_{1})+\rho$ where $\rho$ is a parameter that we will vary. More specifically, I show that as long as $\rho\in[p^{}(\bar{b})-p^{}(b_{1})-\epsilon,p^{}(\bar{b})-p^{}(b_{1})+\epsilon]$ for a small enough $\epsilon$, then $\pi(B^{},p)$ and $\pi(B^{\prime},p^{\prime})$ move in parallel if we set $p(\bar{b})=p^{}(b_{1})+\rho$ and $p^{\prime}(\bar{b})=p^{\prime}(b_{1})+\rho$, and move $\rho$ (that is, as we change $\rho$, the difference $\pi(B^{},p)-\pi(B^{\prime},p^{\prime})$ remains constant). The range parameter $\epsilon$ should be chosen so that for any pricing strategy $p$ constructed with a $\rho$ in this interval we have: $D(\bar{b}\|B^{},p)<1-F(t_{1}^{\prime})$ where $t_{1}^{\prime}$ was constructed in equation 7. In other words, $\epsilon$ should be small enough (or, alternatively, $p(\bar{b})$ should be close enough to $p^{}(\bar{b})$ ) so that as we change $\rho$, the only types who are affected are those around $t_{1}$; and, hence, the only purchase decisions that are affected are choices between $b_{1}$ and $\bar{b}$. With this setup, note that if we set $\epsilon=0$, then demand for grand bundle $\bar{b}$ under both strategies will be equal to demand for the grand bundle under the optimal strategy: $\epsilon=0\Rightarrow D(\bar{b}\|B^{},p)=D(\bar{b}\|B^{\prime},p^{\prime})=D(\bar{b}\|B^{},p^{})$ Next, note that for small $\epsilon\neq 0$, we will still have $D(\bar{b}\|B^{},p)=D(\bar{b}\|B^{\prime},p^{\prime})$ because such changes in $\rho$ will lead the exact same set of types to switch their purchase decisions between $\bar{b}$ and $b_{1}$, under both strategies $(B^{},p)$ and $(B^{\prime},p^{\prime})$. This leads to the exact same revenue change between the two strategies as a result of the change in $\epsilon$ (or, equivalently, in $\rho$). Also the change in total costs are the same given the constant- marginal-costs assumption. As a result, $\pi(B^{},p)$ and $\pi(B^{\prime},p^{\prime})$ change in the same way as a result of small changes in $\rho$.101010One can show we do not need the constant marginal cost assumption if $n=2$. or if the monotonicity condition is strengthened. Now note that if claim 1 does not hold, then $\pi(B^{\prime},p^{\prime})$ under $\epsilon=0$ is not the global maximum for $\pi(B^{\prime},p^{\prime})$. By strict quasi-concavity, it is not a local maximum either. As a result, there is a small change in $\rho$ that would strictly increase $\pi(B^{\prime},p^{\prime})$. Given that $\pi(B^{\prime},p^{\prime})$ and $\pi(B^{},p)$ move in parallel if we slightly change $\rho$, then $\pi(B^{},p)$ should also strictly increase relative to $\pi(B^{},p^{})$, a contradiction.Q.E.D. Proof of Claim 2. The proof of this claim is similar to that of the previous claim. We start by assuming, on the contrary, that $t^{}(b_{2})\neq t_{2}$ and reach a contradiction. Construct $(B^{\prime},p^{\prime})$ by assuming $B^{\prime}=\\{b_{2}\\}$, which makes $p^{\prime}$ just one number (for the price of $b_{2}$). Similar to the previous claim, one can show that for prices $\rho$ for $b_{2}$ sufficiently close to $p^{}(b_{2})$ the two profit functions $\pi(B^{},p)$ and $\pi(B^{\prime},p^{\prime})$ move in parallel as we move $\rho$. Again, similarly to the previous claim, this implies that $(B^{},p^{})$ can be improved upon if $t_{2}\neq t^{}(b_{2})$. Q.E.D. The completion of the proofs for claims 1 and 2 finishes the proof of the sufficiency side of the theorem, and hence the theorem itself. Q.E.D. ## 5 Discussion ### 5.1 Discussion of the assumptions Monotonicity: The most restrictive assumption in this model is monotonicity, which, as Lemma 1 shows, makes the type space uni-dimensional. It is worth noting that monotonicity is quite common in the literature. A prevalent version of it in the screening literature is the single crossing condition imposed by Maskin and Riley (1984).111111Single crossing in Maskin and Riley (1984) is in fact stronger than my monotonicity condition. In order for monotonicity to be as strong as the single crossing condition in Maskin and Riley (1984), it has to be that $v(\bar{b},t^{\prime})\geq(>)v(\bar{b},t)\Rightarrow\forall b\cap b^{\prime}=\emptyset:v(b,t^{\prime}\|b^{\prime})\geq(>)v(b,t\|b^{\prime})$. Within the bundling literature most of the papers that focus on products with non-additive values focus on versions of the product-line pricing problem–which can be thought of as a special case of the bundling problem–and each impose a form of monotonicity (e.g., Anderson and Dana Jr (2009); Deneckere and Preston McAfee (1996); Long (1984)). This usually comes in the form of assumed increasing difference of values in the (unidimensional) type and the product quality level. Also the seminal paper by Mussa and Rosen (1978) on product line pricing assumes the valuation by each type of each quality level is proportional to both type and quality, which implies monotonicity.121212It is worth noting that unlike other papers on product line pricing such as Anderson and Dana Jr (2009), the Mussa and Rosen (1978) study does not focus on whether and when bundling is optimal. Mussa and Rosen (1978) make a series of assumptions that imply pure bundling (i.e., offering only the highest quality version) is always sub-optimal. One can indeed show that an appropriate translation of the Mussa and Rosen (1978) problem into the setting of this paper will satisfy the optimal sales volume conditions for sub- optimality of pure bundling. See appendix for more details. To my knowledge, the only studies of bundling of products with non-additive values that do not impose a version of monotonicity are Haghpanah and Hartline (2021) and Armstrong (2013). That said, the ratio-monotonicity conditions in Haghpanah and Hartline (2021) –at least in one direction– have a similar implication to my monotonicity condition.131313To be clear, there are many papers in the bundling literature that do not impose a version of monotonicity as a model assumption or as part of the conditions in their theorems. To the best of my knowledge, however, all such papers study environments with additive values, meaning they impose $\forall b,t:v(b,t)=\Sigma_{i\in b}v(\\{i\\},t)$. Finally, note that the monotonicity assumption does not force consumers to rank the products the same way. That is, it does not rule out $v(b,t)>v(b^{\prime},t)$ co-existing with $v(b,t^{\prime})<v(b^{\prime},t^{\prime})$. It, rather, rules out $v(b,t)>v(b,t^{\prime})$ co-existing with $v(b^{\prime},t)<v(b^{\prime},t^{\prime})$. Quasi-concavity: Quasi-concavity simply requires that each relevant profit function be single-peaked. This assumption has been made in the literature before (e.g., see assumption 5 in Maskin and Riley (1984)). Without this assumption, one can still prove a version of Theorem 1; but that version would be weaker and less straightforward to state. Finally, it is worth specifying what this assumption would look like if expressed based on the model primitives rather than profit functions. This assumption would require that $\frac{\partial\log(v(b,t)-\Sigma_{i\in b}c_{i})}{\partial t}-\frac{f(t)}{1-F(t)}$ cross zero only once from above for all $b\neq\emptyset$. Other notes: Most papers on optimal bundling of products with non-additive values assume there are only two products whereas this paper examines arbitrarily many. Additionally, one direction of the results in Theorem 1 does not use the assumption that marginal costs are constant. Furthermore, to my knowledge, with the exception of this paper and Haghpanah and Hartline (2021) which do not make assumptions on substitution patterns, other papers on bundling with non-additive values impose relatively strong complementarity assumptions.141414These assumptions are mostly implicit in the form of product-line pricing which may be thought of as a “base product” plus complementary add-ons. As an exception, Armstrong (2013) allows for complementarity or substitution separately, but does not allow the same products to be complementary for some types and substitutes for others. ### 5.2 Interpretation of the Result and Relation to Literature In the literature on bundling of products with non-additive values, a commonly mentioned condition for (sub-)optimality of pure bundling is ratio monotonicity. According to ratio-monotonicity, pure bundling is optimal if $\frac{v(b,t)}{v(\bar{b},t)}$ is everywhere weakly increasing in $v(\bar{b},t)$ for all $b$. Also pure-bundling is sub-optimal if this fraction is everywhere strictly increasing for at least one $b$. Versions of it have been mentioned in Anderson and Dana Jr (2009); Salant (1989); Deneckere and Preston McAfee (1996), and, most generally, Haghpanah and Hartline (2021).151515Haghpanah and Hartline (2021) prove their results under weaker underlying assumptions, use a stochastic version of ratio-monotonicity (which is weaker than the deterministic version and allows for multi-dimensional types,) and are to my knowledge the only paper that states both sides of the condition. Theorem 1 shows that, at the cost of having to make assumptions, 1 and 2, optimality of pure bundling could be linked to local–as opposed to global–ratio monotonicity. ###### Proposition 1. Suppose that assumptions 1 and 2 hold and that production costs are zero. Also suppose each $\pi(b,p)$ is differentiable in $p$ and that the derivative is only zero at the peak. Then for any two bundles $b$ and $b^{\prime}$ we have $D^{}(b^{\prime})>D^{}(b)$ if and only if for the unique $\tilde{t}$ with $\frac{\partial\log(v(b^{\prime},\tilde{t}))}{\partial t}=\frac{f(\tilde{t})}{1-F(\tilde{t})}$, we have: $\frac{\partial\log(v(b,\tilde{t}))}{\partial t}>\frac{f(\tilde{t})}{1-F(\tilde{t})}$. The proof follows directly from the assumptions. To see why this is related to local ratio monotonicity, note that according to Proposition 1, for $D^{}(\bar{b})>D^{}(b)$ to hold, we need the $\tilde{t}$ that satisfies $\frac{\partial\log(v(\bar{b},\tilde{t}))}{\partial t}=\frac{f(\tilde{t})}{1-F(\tilde{t})}$ to also satisfy $\frac{\partial\log(v(b,\tilde{t}))}{\partial t}>\frac{f(\tilde{t})}{1-F(\tilde{t})}$. In other words: $\frac{\partial\log(v(\bar{b},\tilde{t}))}{\partial t}<\frac{\partial\log(v(b,\tilde{t}))}{\partial t}$. This means that for $t$ slightly larger than $\tilde{t}$, we get: $\frac{v(\bar{b},t)}{v(\bar{b},\tilde{t})}<\frac{v(b,t)}{v(b,\tilde{t})}$. Rearranging, we get $\frac{v(b,\tilde{t})}{v(\bar{b},\tilde{t})}<\frac{v(b,t)}{v(\bar{b},t)}$. This is exactly ratio monotonicity.161616Note that this also to some extent resembles elasticity-comparison results such as those in Armstrong (2013) and Long (1984). Note that, according to this result, in order to check whether pure bundling is optimal, one does not need to calculate $D^{}(b)$ for all $b$. It would suffice to calculate $D^{}(\bar{b})$ and then check ratio-monotonicity at that local point. Parametric Examples: One can construct simple examples in which optimal sales volumes/local ratio monotonicity can pin down optimal bundling strategy whereas global ratio monotonicity cannot. Suppose $n=2$, and $t$ is uniformly distributed between 0 and 1. Assume the firm can produce these products at no cost. By $b$ denote the bundle $\\{1\\}$. For simplicity, assume the complementary bundle $b^{c}=\\{2\\}$ is not valued by any type: $\forall t:v(\\{2\\},t)=0$. A common example of this is when $\\{2\\}$ is an “add on,” which is not of value by itself but can add value once the “base product” is present (e.g., additional memory for a smart phone). Suppose $\forall t:v(b,t)=t+k_{1}$ where $k_{1}$ is a fixed real number. Finally, assume $\forall t:v(\bar{b},t)=t+k_{1}+t^{k_{2}}$ where $k_{2}$ is a positive real number. That is, each type $t$’s valuation of the add-on on top of the original product is $t^{k_{2}}$. One can verify that this setup satisfies monotonicity and quasi-concavity. Now fix $k_{1}=0.2$ and allow $k_{2}$ to vary. One can verify that for $k_{2}\leq\frac{2}{3}$, pure bundling is optimal while for $k_{2}>\frac{2}{3}$ it is optimal to mixed-bundle. Also, once $k_{2}$ passes $\frac{2}{3}$, the optimal sales volume for the grand bundle passes $0.6$ (which is the optimal sales volume for partial bundle $b$) from above. Additionally, $\frac{\partial\log\big{(}v(\bar{b},t)\big{)}}{\partial t}-\frac{f(t)}{1-F(t)}$ evaluated at $t=t^{}(b)=0.4$ passes 0 from below once $k_{2}$ passes $\frac{2}{3}$. Finally, one can verify that for an interval around $k_{2}=\frac{2}{3}$, (say $k_{2}\in[0.4,0.8]$) the ratio $\frac{v(b,t)}{v(\bar{b}),t}$ first decreases and then increases as $v(\bar{b})$ grows. To sum up, the optimal bundling decision is informed by optimal sales volumes/local ratio monotonicity but not by global ratio monotonicity. The appendix provides a visualization for this parametric class of problems. I finish this section by noting that the relation to ratio-monotonicity implies that the intuition provided by Haghpanah and Hartline (2021) is relevant to my framework as well: the correlation between WTP and perceived complementarity/substitution levels has implications for bundling decisions. Pure bundling is optimal if higher WTP customers see a lower degree of complementarity (or higher degree of substitution) across products, compared to lower WTP customers. ### 5.3 Implications for Non-linear Pricing The traditional screening/non-linear pricing problem (a la Maskin and Riley (1984) or Mussa and Rosen (1978)) can be cast as a form of bundling problem in which each product $i$ represents the $i$-th quality/quantity unit. The appendix shows that a version of the sales volumes result in Theorem 1 can be used to fully characterize what the optimal tariff should look like in a non- linear pricing problem (observe that this goes beyond the result in Theorem 1 which, instead of fully characterizing the bundling strategy, only specifies when pure bundling is optimal.) Leaving the details to the appendix, here I only discuss one important implication of it through an example: Example. Suppose types $t$ are uniformly distributed within $[0,1]$. The monopolist sells a single product with continuous quality levels $q\in[0,2]$. The monopolist optimally prices each quality level at $p^{}(q)$ where $p^{}(0)$ is fixed at 0. Production costs are zero. Valuations are given by $v(q,t)=q\sqrt[q]{t}$,171717This function simply captures the idea that the higher $q$, the more concave the value is across types $t$. For instance, $v(0,t)\equiv 0$, $v(\frac{1}{2},t)\equiv\frac{t^{2}}{2}$, $v(1,t)\equiv t$, and $v(2,t)\equiv 2\sqrt{t}$. and each type $t$ decides which quality level to purchase. In this setting, one can show that it is optimal to charge a flat fee of $\frac{2}{\sqrt{3}}=1.15$ for any strictly positive level of quality. Under this tariff, types $t<\frac{1}{3}$ purchase nothing while the rest of the types purchase the highest quality version of the product $q=1$. To see why this is true, one can apply our main result and check local ratio monotonicity at $t=\frac{1}{3}$: we have $\frac{\partial\log\big{(}v(q,t)\big{)}}{\partial t}=\frac{f(t)}{1-F(t)}$ when evaluated at $t=\frac{1}{3}$ and $q=2$. But for $t=\frac{1}{3}$ and any $q<2$, we have $\frac{\partial\log\big{(}v(q,t)\big{)}}{\partial t}=\frac{3}{q}>\frac{3}{2}=\frac{f(t)}{1-F(t)}$.181818Of course for our main result, which was developed in the bundling domain, to apply in this non- linear pricing domain, additional propositions are needed. The appendix provides those. This example shows a case where the optimal non-linear tariff leads to a “jump” across types in the quality purchased (from $q=0$ to $q=1$ at $t=\frac{1}{3}$.) The literature tends to assume this away. Mussa and Rosen (1978) state that “[t]he economic rationale for the conclusion that jumps in $q(\theta)$ are not optimal is that the monopolist would not be making full use of his power to discriminate among different types of buyers.”191919They denote type by $\theta$ instead of $t$ in this paper; and they denote optimal quality purchased by type $\theta$ under optimal contract by $q(\theta)$. By applying bundling results to non-linear pricing, this paper offers a different perspective: Changing the tariff from one that induces a continuous range of purchased quantities/qualities in an interval $[\underline{q},\bar{q}]$ to one that induces a jump from $\underline{q}$ to $\bar{q}$ may indeed be optimal, as long as this discontinuity leads sufficiently many consumers to “upgrade to $\bar{q}$” as opposed to “downgrade to $\underline{q}$.” In other words, inducing a jump in purchase behavior in tariff design may be optimal for similar economic reasons to those that make pure bundling of products optimal. ## 6 Conclusion This paper studied when pure bundling is optimal for a monopolist who sells products with non-additive values (i.e., with no restriction on complementarity/substitution patterns.) Under monotonicity and quasi-concavity assumptions, I showed that pure bundling is optimal if and only if the grand bundle, once sold on its own and optimally priced, would “sell more” than any smaller bundle. The appendix provides additional results on the relation to non-linear tariff design, and the implications of the model for the case of additive values. This paper provides the first if-and-only-if characterization for optimality of pure bundling under non-additive values; but at the expense of a monotonicity assumption that makes the type space single-dimensional. A full characterization result under non-additive values and multi-dimensional types would, in my view, be the most important way to extend the results in this paper. ## References * (1) * Adams and Yellen (1976) William James Adams and Janet L Yellen. 1976. 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Finally, Appendix C studies the implications of our results (as well as those of the ratio-monotonicity results from the literature) for environments where values are additive. ## Appendix A Visualizations of Examples from the Main Text The two panels of Figure 1 visualize the example from Section 5.2 of the paper. As a reminder, the example was about a parametric class of value functions in which $v(b,t)=t+k_{1}$ and $v(\bar{b},t)=t+k_{1}+t^{k_{2}}$. Each panel of the figure has a fixed value for $k_{1}$ and examines a range of values for $k_{2}$. For both cases of both panels, the ratio $\frac{v(b,t)}{v(\bar{b},t)}$ is non-monotonic in $v(\bar{b},t)$, making ratio monotonicity results less useful in determining the optimal bundling strategy. However, optimal sales volume results are in line with the optimal bundling decisions. In panel (a), pure bundling is sub-optimal in case 1 and optimal in case 2. In panel (b), pure bundling is optimal in case 1 and sub-optimal in case 2. (a) $k_{1}=0.2$, Case 1: $k_{2}=0.4$, Case 2: $k_{2}=0.8$ (b) $k_{1}=-0.2$, Case 1: $k_{2}=2$, Case 2: $k_{2}=1.2$ Figure 1: Family of examples in which $v(b,t)=t+k_{1}$ and $v(\bar{b},t)=t+k_{1}+t^{k_{2}}$. Also, Figure 2 visualizes the value function $v(q,t)=q\sqrt[q]{t}$ from Section 5.3 on the implications of our results for nonlinear tariff design. It plots $v(q,t)$ as a function of $t$ for different values of $q$. As can be seen and also algebraically verified, this function is increasing in both arguments. The key feature of this value function is that it becomes more concave in $t$ as $q$ increases (which implies the optimal sales volume increases in $q$.) Other value functions that share this feature should also yield a similar result, yielding a jump in consumer purchase behavior under the optimal tariff. Figure 2: Value function $v(q,t)=q\sqrt[q]{t}$ as a function of $t$ for different values of $q$ ## Appendix B Implications for Non-Linear Pricing The main focus of this paper is the problem of optimal bundling. Nevertheless, the main result also has implications for the non-linear pricing problem studied, among others, by Maskin and Riley (1984) and Mussa and Rosen (1978). Translated to the context of non-linear pricing, the analysis in this paper will take the following form: A monopolist selling a single product faces the problem of determining the optimal price schedule $T^{}(q)$ for different quantities $q\in\\{0,1,...,n\\}$. The monopolist’s objective is to maximize total profit $\pi(T^{})$. Valuations are denoted $v(q,t)$ with $v(0,t)=0$ for all $t$. Conditional on a prior endowment of $q^{\prime}$, we denote the valuations by $v(q,t\|q^{\prime})$. Similarly, we use notation $\pi_{q}(p)$ to denote the firm’s profit when it sells only $q$-size batches of the product, pricing each batch at $p\in\mathbb{R}$. Also $\pi_{q}(p\|q^{\prime})$ denotes the same thing under the condition that all consumers have been pre-endowed with $q^{\prime}\leq n-q$ unites of the product. $D^{}(q)$, and $D^{}(q\|q^{\prime})$ are defined in the expected way. Proposition 2 shows that in the above setting, one can fully characterize the optimal nonlinear tariff. ###### Proposition 2. Consider the non-linear pricing setup described above. Assume the following: (i) $v(q,t)$ is increasing in both arguments (and strictly so whenever positive), and $\forall t^{\prime}>t,q>0$ we have $v(q,t^{\prime})-v(q-1,t^{\prime})>v(q,t)-v(q-1,t)$. (ii) For all $q+q^{\prime}\leq n$, the profit function $\pi_{q}(p\|q^{\prime})$ is strictly quasi-concave in $p$ over the range of $p$ that generates strictly positive demand. (iii) $v(q,t)$ is continuous in $t$ except possibly for finitely many points. (iv) For any $q^{\prime}<n$, the set $\arg\max_{q\in\\{1,...,n-q^{\prime}\\}}D^{}(q\|q^{\prime})$ is a singleton denoted $q^{}(q^{\prime})$.202020That is, $q^{}({q^{\prime}})$ is the batch- size that, if sold and optimally priced in a market where all consumers have already been endowed with ${q^{\prime}}$ unites, would generate the highest demand. Similar to the main result, if instead of quasi-concavity we have concavity, one need not impose the assumption that the argmax is unique. Then the optimal price schedule will involve $m\leq n$ distinct quantities $(q_{1}^{},...,q_{m}^{})$ such that $q_{1}^{}=q^{}(0)$ and $\forall i\in\\{2,...,m\\}:q_{i}^{}=q_{i-1}^{}+q^{}(q_{i-1}^{})$. Precisely: $\forall q\in\\{q_{i-1}^{}+1,...,q_{i}^{}\\}:T^{}(q)=p^{}(q_{i}^{}-q_{i-1}^{}\|q_{i-1}^{})$ This proposition fully characterizes the optimal tariff $T^{}$. It starts by stating that the smallest quantity that any consumer can buy is $q_{1}^{}=q^{}(0)\equiv\arg\max_{q\in\\{1,...,n\\}}D^{}(q)$, i.e., the batch size that would sell the most if sold alone and priced optimally. In other words, at least $q_{1}^{}$ units of the products are always bundled together. The price of this $q^{}_{1}$-bundle is the optimal price that the monopolist would set if it were to only sell this bundle. From this point on, the proposition takes a recursive structure and states that the second distinct quantity sold of the product is $q_{2}^{}=q_{1}^{}+\arg\max_{q\in\\{1,...,n-q_{1}^{}\\}}D^{}(q\|q_{1}^{})$ and so on. Though Proposition 2 in some ways resembles the demand profile approach of Wilson (1993), it has fundamental differences. Before turning to the proof of this results, I make a few notes. First, as mentioned shortly before, Proposition 2 takes a step beyond the main result in Theorem 1: it fully characterizes what the optimal tariff looks like as opposed to only characterizing the conditions under which the optimal tariff will involve a flat fee for selling all $n$ units of the product.212121That condition would be $n=\arg\max_{q\in\\{1,...,n\\}}D^{}(q)$, mirroring the condition in Theorem 1. This property of Proposition 2 should naturally raise the question that whether Theorem 1 can also be strengthened to fully characterize the optimal bundling strategy as opposed to only characterizing when pure bundling is optimal. Unfortunately the answer turns out to be no; a bundling strategy constructed in a similar way to how the optimal tariff in Proposition 2 is constructed may or may not be optimal.222222Counterexamples are available upon request. Also, it is worth noting that the Maskin-Riley type monotonicity condition used in Proposition 2 is stronger than that used in Theorem 1. Nevertheless, this is not the reason why the optimal bundling strategy cannot be fully characterized. Second, there is a difference between Proposition 2 and the usual form in which the nonlinear pricing problem is studied in the literature. The origin of Proposition 2 being in bundling makes the domain of quantities $q$ by construction discrete and bounded, whereas the literature (e.g., Maskin and Riley (1984)) examines a continuous and possibly unbounded environment. That said, I do not see this feature of Proposition 2 as too restrictive, given that one an always examine the limit case as $n\rightarrow\infty$. Fourth, $q$ in this proposition need not be interpreted as quantity. One can also think of $q$ as quality in a similar fashion to Mussa and Rosen (1978). In that case, the result holds if the cost function is non-linear in quality, as long as it is still linear in the quantity of the consumers that purchase each quality level. Next, I turn to proving the proposition. Proof of Proposition 2. First, let us introduce some notations. In a similar spirit to the consumer choice notation $\beta(t\|B,p)$ in the bundling problem, define the following consumer choice function: $\beta(t\|T)\equiv\arg\max_{q}v(t,q)-T(q)$ (10) I use the same notation $\beta$ as I did in the formulation of the bundling problem in order for the parallels between the two settings to be clearer. However, there are some differences. Most notably, the output of the $\beta$ function here is just an integer number, not a bundle. Suppose consumers break ties in favor of higher quantities. I now proceed to state the following lemma. ###### Lemma 8. For any price schedule $T$ and any two types $t,t^{\prime}$ with $t^{\prime}>t$ we have: $\beta(t^{\prime}\|T)\geq\beta(t\|T)$. This lemma, whose proof is rather straightforward and is left to the reader, simply says that the consumption quantity is weakly monotonic in type. Also, in this proof, I will assume that the optimal schedule $T^{}(q)$ is weakly monotonic in $q$. This assumption is without loss, given that one can show that any non-monotonic $T$ can be modified in a way that (i) makes it weakly monotonic and (ii) delivers the same amount of profit to the monopolist. Given this weak monotonicity, $T^{}$ has to take the following form for a strictly increasing sequence $q_{0}=0,q_{1},...,q_{m}=n$ and a weakly increasing sequence $T_{1},...,T_{m}$: $\forall q\in\\{q_{i-1}+1,...,q_{i}\\}:T^{}(q)=T_{i}$ (11) The following lemma will be useful for the proof: ###### Lemma 9. $T_{1}=p^{}(q_{1}\|0)$. That is, the lowest type $t$ purchasing $q_{1}$ under the optimal contract will satisfy $1-F(t)=D^{}(q_{1})$. Proof of Lemma 9. Suppose this lemma’s claim is not true: $T_{1}\neq p^{}(q_{1}\|0)$. Then consider a scenario in which only $q_{1}$ is being sold by the seller at the price of $T_{1}$. Given that $T_{1}$ is not the optimal price, there is a small but nonzero $\epsilon$ such that if the seller prices $q_{1}$ at $T_{1}+\epsilon$ instead of $T_{1}$, the seller will strictly improve its profit when selling only $q_{1}$. Next, I move from the scenario of selling only a batch of $q_{1}$ units back to the full tariff design problem. I use the above deviation to construct a similar deviation from the full schedule $T^{}$ and show that the seller can strictly improve its profit. Construct price schedule $T\equiv T^{}+\epsilon$ for any positive $q$. I now claim that $T$ is strictly more profitable to the seller than is $T^{}$. To see why this claim is true, note that for any $q\in\\{q_{2},...,q_{m}\\}$ and any type $t$ such that $\beta(t\|T^{})=q$, we have $\beta(t\|T)=q$. This is because (i) for those types it is the IC constraint (and not the IR) that is binding; and (ii) the construction of $T$ from $T^{}$ preserves all the IC constraints. Therefore a move from $T^{}$ to $T$ will lead to the exact same revenue change that a move from $T_{1}$ to $T_{1}+\epsilon$ does in the sceniario of selling only $q_{1}$: the exact same new types are added to (or removed from) the set that purchases $q_{1}$, and the same change (i.e., $\epsilon$) has been made to the amount made off of each type that buys. In addition to the change in the revenue, a move from $T^{}$ to $T$ also leads to the exact same change in total costs as would a change from $T_{1}$ to $T_{1}+\epsilon$. This is because in both cases, the only change made in the production is the number of $q_{1}$-size batches (or, in the Mussa and Rosen (1978) interpretation, the number of $q_{1}$-quality products). Given that we have assumed the cost function to be linear in this change, the changes in total cost is the same between the two scenarios. As a result, a move from $T^{}$ to $T$ will lead to the exact same change in the total profit as would a move from $T_{1}$ to $T_{1}+\epsilon$. Therefore, a change from $T^{}$ to the new schedule $T$ strictly profitable, finishing the proof of the lemma. Q.E.D. Next, I introduce a lemma which will be the building block of the proof of this proposition. ###### Lemma 10. In the presentation of $T^{}$ in equation 11, it has to be that $q_{1}=q_{1}^{}$ where $q_{1}^{}=q^{}(0)$ as defined in the statement of Proposition 2. As a reminder, $q^{}(0)=\arg\max_{q=1,...,n}D^{}(q)$. Thus, the lemma simply says that the smallest quantity that the optimal schedule $T^{}$ offers to consumers is the quantity that, if sold alone, would sell the highest volume. Proof of Lemma 10. Suppose $q_{1}\neq q^{}_{1}$. Then one can construct a deviation from $T^{}$ that would strictly improve the seller’s profit. This will suffice to finish the proof of the lemma. First, for convenience, assume that even though $q_{1}\neq q^{}_{1}$, there is some $k>1$ such that $q_{k}=q^{}_{1}$. In other words, even though $q^{}_{1}$ is not the smallest package on the schedule, it is nonetheless somewhere on the schedule. Later I will show this assumption is not necessary. But for now, it will make the steps of the proof more straightforward. For each $i\in\\{1,...,m\\}$ denote by $t_{i}$ the lowest type that buys $q_{i}$ units under $T^{}$. That is: $t_{i}=\min\\{t:\,\beta(t\|T^{})=q_{i}\\}$. From Lemma 8 we know larger types buy weakly more units of the product. That is: $t_{1}<t_{2}<...<t_{m}$. From Lemma 9 we know that $1-F(t_{1})=D^{}(q_{1})$. Given $t_{k-1}\geq t_{1}$, we get $1-F(t_{k-1})\leq D^{}(q_{1})$. But we also know, by assumption, that $D^{}(q^{}_{1})>D^{}(q_{1})$. Therefore: $1-F(t_{k-1})<D^{}(q^{}_{1})\equiv D^{}(q_{k})$ (12) In addition, again by the definition of $q_{k}=q_{1}^{}$, we know that $D^{}(q_{k-1})<D^{}(q_{k})$. This inequality, along with a similar quasi- concavity argument to that used in the proof of the main result yields: $D^{}(q_{k})\leq D^{}(q_{k}-q_{k-1}\|q_{k-1})$ (13) Together, inequalities 12 and 13 yield: $1-F(t_{k-1})<D^{}(q_{k}-q_{k-1}\|q_{k-1})$ (14) Obviously, by $t_{k}>t_{k-1}$, we also know: $1-F(t_{k})<D^{}(q_{k}-q_{k-1}\|q_{k-1})$ (15) Note that equation 15 implies that if all consumers have already been endowed with $q_{k-1}$ units of the product and the monopolist is selling only batches of size $q_{k}-q_{k-1}$ and pricing them so that types $t_{k}$ and above purchase, then the monopolist, by quasi-concavity, will strictly profit from a small price reduction $\rho$. Next, I move from the scenario of selling only $q_{k}-q_{k-1}$ packages under a pre-endowment of $q_{k-1}$ to the main scenario of designing the full schedule $T^{}$. I argue that the monopolist will enjoy the same profit increase as the one described in the previous paragraph if it modifies $T^{}$ by reducing all $T_{j}$ for $j\geq k$ by $\rho$. The argument is similar to that in the proof of Lemma 9. This modification does not alter the behavior of any type $t$ that purchases $q_{k+1}$ or more units due to the fact that it preserves all of the binding IC constraints for those types. As a result, the impact of this change on the firm profit is (i) a cut in margin by $\rho$ across all consumers, combined by the change in the behavior of those who used to purchase less than $q_{k}$ units under $T^{}$ but will now switch to $q_{k}$.232323None of these types would switch to buying more than $q_{k}$ units, due to the single-crossing condition. Note that if the price reduction $\rho$ is small enough, these types will only consist of those who under $T^{}$ purchase $q_{k-1}$. From equation 14, we know there is a non-zero mass of such types. Therefore, the effect of this price change parallels that of the price change described in the previous paragraph, making it strictly profitable. But the above argument contradicts the optimality of $T^{}$. Therefore, the contrapositive assumption must have been incorrect. That is: it has to be that $q_{1}=q^{}_{1}$. With the above argument, the proof is complete for the case where there is some $k>1$ with $q_{k}=q^{}_{1}$. That is, when $q_{1}^{}$ is “on the price schedule.” Thus, it remains to show that the proof also works when $\nexists k:q_{k}=q^{}_{1}$. Suppose this is the case, and take $k$ to be the smallest index with $q_{k}\geq q^{}_{1}$. Like before, denote by $t_{i}$ the smallest type that purchases $q_{i}$ under $T^{}$ I now construct schedule $T^{}$ in the following way: • For any $q\leq q_{k-1}$ or $q>q^{}_{1}$, set $T^{}(q)=T^{}(q)$ * • For all $q_{k-1}<q\leq q^{}_{1}$, set $T^{}(q)=T^{}(q_{k})+v(q_{1}^{},t_{k})-v(q_{k},t_{k})$ Next, I take two steps. First, I show that $T^{}$ delivers the same profit to the monopolist as does $T^{}$. Then, I will construct a deviation from $T^{}$ that yields a strict profit improvement. As for the first step, note that by construction, $T^{}(q^{}_{1})$ is designed to make the type $t_{k}$ consumer indifferent between purchasing $q_{k}$ units and $q^{}_{1}$ units. But we know, by construction of $t_{k}$, that this type is also indifferent between buying $q_{k}$ units and buying $q_{k-1}$ units. This makes this type indifferent among all three quantities $q_{k-1}<q^{}_{1}<q_{k}$ under the tariff $T^{}$. But by the monotonicity condition (i.e., single crossing,) any type $t<t_{k}$ will strictly prefer $q_{k-1}$ to $q^{}_{1}$ and any type $t>t_{k}$ will strictly prefer $q_{k}$ over $q^{}_{1}$. In other words, this “addition of $q^{}_{1}$ to the schedule” will not change any consumer’s purchase behavior: $\forall t:\beta(t\|T^{})=\beta(t\|T^{})$. Thus the two tariffs deliver the same profit to the monopolist. But now the structure of $T^{}$ allows us to construct the same profit enhancing modification that we applied to $T^{}$ when we assumed it did have $q^{}_{1}$ on the schedule. All of the steps are the same. This finishes the proof of the lemma. Q.E.D. The rest of the proof is straightforward and involves recursive use of Lemma 10. First note that by quasi-concavity, and by $\forall q\in\\{1,...,n-q^{}_{1}\\}:D^{}(q+q_{1}^{})<D^{}(q_{1}^{})$, we have: $\forall q\in\\{1,...,n-q^{}_{1}\\}:D^{}(q\|q_{1}^{})<D^{}(q_{1}^{})$ Thus, all of the optimal strategies will sell only to types $t_{1}$ and above. This means that in order to construct “the rest of the optimal schedule” conditional on having set $T^{}(q)$ for all $q\leq q^{}_{1}$ equal to $p^{}(q\|0)$, one can just focus attention on designing the optimal schedule for selling $n-q^{}_{1}$ units when all consumers have been endowed with $q^{}_{1}$ units already. This means we are facing a version of the same problem. It is straightforward to check that all of the conditions of the main problem are satisfied for this “sub-problem.” Thus, one can apply Lemma 10 again and find that $q_{2}$ in the optimal schedule should be equal to $q^{}_{2}$ which was defined as $q^{}_{1}+q^{}(q^{}_{1})$. Repeating this procedure will fully characterize the optimal tariff and the outcome matches what the statement of the proposition predicted. Q.E.D. Note that Proposition 2 can now be used to characterize the optimal contract in the non-linear pricing example given in the main text (i.e., the one in Section 5.3 with $v(q,t)\equiv q\sqrt[q]{t}$.) That example could be cast as a limit case of Proposition 2 as $n\rightarrow\infty$. It can be seen that the quality level that would generate the highest sales volume would be $q=2$. To check this, it would be sufficient to verify that ratio monotonicity holds at $t=\frac{1}{3}$ which is the lowest type that would purchase if only quality $q=2$ is offered and optimally priced. ## Appendix C A Brief Analysis of Environments with Additive Values In this section, I explore the implications of Theorem 1 for environments with additive values (i.e., environments in which $\forall b,t:v(b,t)=\Sigma_{i\in b}v(\\{i\\},t)$), and compare that to implications of ratio monotonicity conditions that are used in the literature. Unfortunately, both ratio monotonicity conditions and conditions of Theorem 1 are of limited use when reduced to environments with additive values. This point is especially pronounced about the former. The results below, further clarify this matter. ###### Proposition 3. Suppose values are additive. Do not impose assumptions 1-2 but instead assume ratio monotonicity: $\forall t,t^{\prime}:v(\bar{b},t^{\prime})\geq v(\bar{b},t)\Rightarrow\frac{v(b,t^{\prime})}{v(\bar{b},t^{\prime})}\geq(\leq)\frac{v(b,t)}{v(\bar{b},t)}$ Under both of these weak forms of ratio-monotonicity (i.e., $\geq$ or $\leq$,) all value functions are proportional. That is, for all $b,t,t^{\prime}$ we have: $\frac{v(b,t)}{v(\bar{b},t)}=\frac{v(b,t^{\prime})}{v(\bar{b},t^{\prime})}$. As a corollary, strict ratio monotonicity under additive values is infeasible. Proof. Define $\alpha(i,t)=\frac{v(\\{i\\},t)}{v(\bar{b},t)}$. By ratio monotonicity, we have $\forall i:\alpha(i,t^{\prime})\geq\alpha(i,t)$. But we also know by additivity of values that $\Sigma_{i=1,...n}\alpha(i,t^{\prime})=\Sigma_{i=1,...n}\alpha(i,t)=1$. As a result, it has to be that $\forall i:\alpha(i,t^{\prime})=\alpha(i,t)$. In other words: $\forall i:\frac{v(\\{i\\},t^{\prime})}{v(\bar{b},t^{\prime})}=\frac{v(\\{i\\},t)}{v(\bar{b},t)}$ By additivity of values, we can say the same for all $b$: $\forall b:\frac{v(b,t^{\prime})}{v(\bar{b},t^{\prime})}=\frac{v(b,t)}{v(\bar{b},t)}$ which finishes the proof. The proof for the other direction of ratio monotonicity is similar. Q.E.D. In other words, all products and bundles will have proportional demand curves and the exact same optimal quantity sold. As a result, the monopolist is indifferent among all possible bundling strategies.242424Note that the above results assumed a “deterministic” version of ratio monotonicity. I add (without proving) that if we consider a “stochastic” version of ratio- monotonicity (a la Haghpanah and Hartline (2021),) we can show that the two random variables $v(\bar{b},t)$ and $\frac{v(b,t)}{v(\bar{b},t)}$ would have to be independent of each other under weak ratio-monotonicity in either direction. Similarly, a strict form of ratio monotonicity would be infeasible. This implies that pure bundling is always optimal. Next, I turn to the implications of this paper’s framework for additive values. ###### Proposition 4. Suppose values are additive and assumptions 1-2 hold. Then pure bundling is optimal if and only if $\forall b:D^{}(b)=D^{}(b^{C})$. Proof. If $\forall b:D^{}(b)=D^{}(b^{C})$, then the monopolist will be indifferent among all possible bundling strategies. If $\exists b:D^{}(b)>D^{}(b^{C})$, then one can use the fact that linearity implies $D^{}(b^{C}\|b)=D^{}(b^{C})$, in order to show that $D^{}(\bar{b})<D^{}(b)$. Thus, pure bundling is not optimal. Q.E.D. The fact that under assumptions 1-2 pure bundling is only optimal in such “measure-zero” cases should not be surprising. Additive values, once combined with monotonicity, will closely resemble s “positive-correlation” case in the Adams and Yellen (1976) context. We know that under positive correlation mixed bundling is preferable to pure bundling. Overall, the tools originally developed to analyze bundling under non-additive values (including this paper) seem to be less powerful when restricted to environments with additive values.
# A rotational and vibrational investigation of phenylpropiolonitrile (C6H5C3N) Zachary Buchanan Kin Long Kelvin Lee Olivia Chitarra Michael C. McCarthy Olivier Pirali Marie-Aline Martin-Drumel Université Paris-Saclay, CNRS, Institut des Sciences Moléculaires d’Orsay, 91405 Orsay, France Department of Chemistry, The University of California Davis, Davis, CA, USA Center for Astrophysics $\|$ Harvard & Smithsonian, Cambridge, Massachusetts 02138, United States SOLEIL Synchrotron, AILES beamline, l’Orme des Merisiers, Saint-Aubin, 91190 Gif-sur-Yvette, France ###### Abstract The evidence for benzonitrile (C6H5CN) in the starless cloud core TMC–1 makes high-resolution studies of other aromatic nitriles and their ring-chain derivatives especially timely. One such species is phenylpropiolonitrile (3-phenyl-2-propynenitrile, C6H5C3N), whose spectroscopic characterization is reported here for the first time. The low resolution (0.5 cm-1) vibrational spectrum of C6H5C3N has been recorded at far- and mid-infrared wavelengths (50–3500 cm-1) using a Fourier Transform interferometer, allowing for the assignment of band centers of 14 fundamental vibrational bands. The pure rotational spectrum of the species has been investigated using a chirped-pulse Fourier transform microwave (FTMW) spectrometer (6–18 GHz), a cavity enhanced FTMW instrument (6–20 GHz), and a millimeter-wave one (75–100 GHz, 140–214 GHz). Through the assignment of more than 6200 lines, accurate ground state spectroscopic constants (rotational, centrifugal distortion up to octics, and nuclear quadrupole hyperfine constants) have been derived from our measurements, with a plausible prediction of the weaker bands through calculations. Interstellar searches for this highly polar species can now be undertaken with confidence since the astronomically most interesting radio lines have either been measured or can be calculated to very high accuracy below 300 GHz. ###### keywords: pure rotation , vibration , astrophysical species , PAH derivative , phenylpropiolonitrile ††journal: Journal of Molecular Spectroscopy ## 1 Introduction The ubiquity of aromatic molecules is closely-correlated to their stability and lack of reactivity, with functionalized aromatics serving as a common motif in biological chemistry. Polycyclic aromatic hydrocarbons are a prominent class of aromatics; they are well-known constituents in the outflows of certain evolved carbon stars and common byproducts in incomplete combustion processes [1, 2, 3]. From a purely spectroscopic viewpoint, a number of simple derivatives of benzene —the prototypical aromatic ring C6H6— have either not been characterized at all or at insufficient resolution to undertake an astronomical search in the coldest most quiescent molecular clouds in space. The recent discovery of benzonitrile —the simplest aromatic nitrile (C6H5CN or PhCN hereafter)— using radio observations towards Taurus Molecular Cloud (TMC-1) [4] has reignited the interest in nitrogen-containing aromatics generally and CN-functionalized aromatics specifically [5]. The subsequent identification of cyanocyclopentadiene, C5H5CN, in the same cloud [6] has only intensified this interest. Nitriles are also known to be important constituents in the chemistry of Titan’s atmosphere [7], and they are very prominent in the interstellar medium (ISM), accounting for roughly 20% of the 220 or so molecules (47) detected in the ISM to date, including cyanopolyynes as long as HC11N [8]. The presence of a nitrile group often imparts a molecule with a large permanent dipole moment and an intense rotational spectrum. In the case of PhCN, replacing a single H atom in benzene with a CN group transforms an otherwise highly symmetric ring into a highly polar species ($\mu_{a}=4.5$ D; [9]) thereby greatly aiding its detection both in the laboratory and in space. Whilst the rotational spectrum of PhCN has been known for more than half-a- century [10], its interstellar detection was greatly aided by measurements at very high accuracy ($i.e.$, at a resolving power $f/\Delta f>10^{6}$) at centimeter wavelengths [4, 9]. In light of this finding, new —or in many cases improved— high-resolution studies of molecules closely related in structure and composition are worth pursuing. In this work, we report a combined pure rotational and vibrational investigation of a derivative of PhCN, phenylpropiolonitrile (3-phenyl-2-propynenitrile, C6H5C3N, abbreviated as PhC3N in the following) where the nitrile group is replaced by a longer chain variant (C3N) (Fig. 1). This species has previously been identified as a possible product from the reaction between the cyano radical and phenylacetylene (C6H5CCH) [11]. To the best of our knowledge, however, spectroscopic investigations have been limited to experimental and theoretical vibronic spectroscopy [12, 13] while high- resolution, rotationally-resolved studies are apparently lacking. In light of the large permanent electric dipole moment calculated here (5.9 D), laboratory measurement of rotational frequencies would allow astronomical searches for this ring-chain to be undertaken with little or no ambiguity. If found in space, the abundance of PhC3N would provide a key test for models of aromatic chemistry which are poorly constrained at present. The infrared spectrum of PhC3N is also of interest as a point of comparison with other benzene derivatives whose vibrational spectra are often plagued by a myriad of perturbations and resonances, notably Fermi and Darling-Dennison in the C-H and C#C stretching regions [14, 15, 16], in addition to Fermi and Coriolis interactions for low frequency (${\sim}150$ cm-1), large amplitude modes that are prominent at room temperature [17, 18, 19]—even under astronomical conditions. Figure 1: Molecular structure of PhC3N; $a$ and $b$ principal inertial axes are indicated (in red and green, respectively); the $c$-axis extends out of the molecular plane and is not shown. ## 2 Experimental and computational methods ### 2.1 Quantum chemical calculations Calculations were performed using the Gaussian’16 suite of electronic structure programs [20]. The goal of these calculations was to provide accurate estimates of spectroscopic parameters, including rotational constants, dipole moment, rotation-vibration corrections, and fundamental vibrational frequencies using second-order vibrational perturbation theory (VPT2). Geometry optimizations of PhC3N and four of its isomers—namely the isocyanide analog PhCCNC and three variants where CN is substituted for a H atom on the ring of phenyl acetylene to yield ortho-, meta-, and para- cyanoethynylbenzene (CEB)—were carried out at the $\omega$B97X-D/cc-pVQZ level on an ultrafine integration grid, in which optimized structures are those in which convergence to better than $10^{-5}$ of the root-mean-squared (RMS) value of the gradient has been achieved. For PhC3N, we also performed (an)harmonic frequency analysis, obtaining both the harmonic and anharmonic vibrational frequencies and intensities at the same level of theory; with the latter, the rotation-vibration interaction constants $\alpha$ were computed. Cartesian coordinates of the optimized equilibrium structures can be found in Tables S1-S5 in the Supporting Information. In addition to spectroscopic parameters, we have also performed rudimentary thermochemical calculations on the relative energies of PhC3N and its isomers using the G3//B3LYP composite method [21], which has been shown to provide near-chemical accuracy at excellent computational cost [22]. Given that semi- empirical methods typically perform best on closed-shell molecules with relatively simpler electronic structure, as the ones studied here, we believe these calculations provide a quantitative determination of relative stabilities, accurate to $\pm 4$ kJ/mol. ### 2.2 Fourier-transform infrared measurements The gas-phase vibrational spectrum of PhC3N was recorded in the far-infrared (far-IR) and mid-infrared (mid-IR) using the Bruker IFS 125 FT interferometer located at the AILES beamline of the SOLEIL synchrotron facility (no synchrotron radiation was used in the present study) [23]. For the far-IR measurements, the spectrometer was equipped with a $6$ µm mylar-silicon composite beamsplitter and a liquid helium-cooled silicon bolometer. A KBr beamsplitter and a sensitive HgCdTe detector, equipped with a cryogenically cooled entrance iris and optical filters [24], were used in the mid-IR region. Vapor of PhC3N was injected in a White-type multipass cell aligned to yield a 150 m optical path length [25] and isolated from the interferometer by 50 µm- thick polypropylene windows in the far-IR and wedged ZnSe windows in the mid- IR range. In both spectral regions, the interferometer was continuously evacuated to $10^{-5}$ mbar to minimize absorption from residual water. Spectra were recorded at a resolution of 0.5 cm-1 using a globar light source and an entrance iris of 4 mm, and consist of 100 and 500 co-added interferograms for the far-IR and mid-IR regions, respectively. Both spectra were recorded at a sample pressure of 5 µbar. It is worth noting that the rotational structure within the vibrational bands could not be resolved even at the highest resolution of the spectrometer (0.001 cm-1). ### 2.3 Chirped-pulse Fourier-transform microwave measurements Microwave measurements were performed using a chirped-pulse Fourier transform microwave (CP-FTMW) spectrometer located at the Center for Astrophysics [26] which operates between 8 and 18 GHz. About 0.3 g of solid PhC3N was introduced into a reservoir located behind the pulsed nozzle and Ne carrier gas (at a flow of $\sim$20 sccm at standard temperature and pressure) passed through the reservoir; the resulting gas mixture was then injected into the vacuum chamber by operating the pulsed valve at a very low repetition rate (5 Hz). Because the vapor pressure of PhC3N is relatively high at room temperature (several mbar), it was not necessary to heat the reservoir to observe rotational lines in the CP-FTMW spectrum with good signal-to-noise ratios. Approximately 17,000 nozzle pulses, each probed by 10 microwave chirps, were acquired. Additional details on the experimental set-up are provided in Ref. [26]. The resulting spectrum, with electronics artifacts and well-known contaminant lines ($e.g.$, acetone) removed, is displayed in Fig. 2. Transitions with $5\leq J^{\prime\prime}\leq 15$ and $K_{a}^{\prime\prime}\leq 4$ are visible. Figure 2: Experimental CP-FTMW spectrum (in black) of PhC3N in comparison with a simulation at $T_{\mathrm{rot}}=1$ K (in purple) using the calculated set of rotational constants; the simulation is plotted here with negative intensity to more easily compare the two spectra. Left panel: the full spectrum, noting that a few lines originating from impurities are visible around 10 GHz; Right panel: expanded view around the qR(7) transitions, as indicated by the gray rectangle on the left panel. The simulation was performed using PGOPHER [27]. ### 2.4 Cavity-based Fourier-transform microwave measurements In parallel to the present investigation, pure rotational transitions of PhC3N were also identified by several of the co-authors of this paper while analyzing the discharge products of a benzene/nitrogen (N2) mixture [28] . Using a cavity-enhanced FTMW (CE-FTMW) spectrometer operating between 6 and 40 GHz, transitions of PhC3N were measured at roughly ten times higher spectral resolution than can be achieved with the CP-FTMW instrument. In the following, we only briefly summarize the experimental conditions relevant to this work; further details about the CE-FTMW instrument and the discharge mixture experiments are presented in Refs. [26] and [28], respectively. PhC3N was synthesized by subjecting a mixture of C6H6 and N2 diluted heavily in Ne to a high-voltage discharge (800 V); typical flow rates were 14, 12, and 20 sccm respectively. Compared to the CP-FTMW measurements where PhC3N was the precursor, the discharge experiment results in a slightly higher rotational temperature, $i.e.$, $\sim 10$ K and produces rotational lines with demonstratively lower signal-to-noise ratio. Nevertheless, at the highest resolving power ($\sim 10^{7}$) the nitrogen nuclear-quadrupole structure for several low-$K_{a}$ transitions ($K_{a}^{\prime\prime}=0-2$) in the 6–20 GHz region was partially resolved. ### 2.5 Absorption millimeter-wave measurements The room temperature spectrum of PhC3N was recorded at ISMO with an absorption spectrometer in which a frequency multiplication scheme is used to generate millimeter-wave radiation [29]. By combining the output of a radiofrequency synthesizer (Rohde & Schwarz) operating in the 2–20 GHz region with one of two amplifier / multiplier chains, it is possible to produce broadly tunable radiation with modest power (of a few mW) throughout the millimeter band; a Radiometer Physics GmbH (RPG) SMZ110 for 75–97 GHz, and a Virginia Diodes Inc. (VDI) for 141–214 GHz. The millimeter-wave radiation was collimated using a 10 mm focal length Teflon lens into a 2 m long Pyrex absorption cell and further focused onto a Schottky diode detector from VDI, using a second identical lens. The input radiation was frequency modulated at a frequency of 48.2 kHz and a commercial lock-in amplifier (Ametek) demodulated the signal at the second harmonic. The spectrum was recorded using 50 kHz frequency steps, a time constant of 50/100 ms, and an FM deviation of 200/250 kHz (where the two values refer the lower/higher frequency measurements, respectively). A flow of PhC3N, not exceeding a pressure of 2 µbar, was pumped through the cell and evacuated by a turbomolecular pump. Above this pressure, significant pressure broadening of the rotational lines was observed. ## 3 Results and discussion ### 3.1 Spectroscopic considerations PhC3N belongs to the $C_{2\mathrm{v}}$ symmetry group and possesses a $\tilde{\mathrm{X}}\,^{1}\mathrm{A}_{1}$ electronic ground state with 39 modes of vibration following the irreducible representation $\Gamma=14\hskip 2.84526pt\mathrm{A}_{1}\oplus 3\hskip 2.84526pt\mathrm{A}_{2}\oplus 9\hskip 2.84526pt\mathrm{B}_{1}\oplus 13\hskip 2.84526pt\mathrm{B}_{2}$. All vibrational modes except those of $\mathrm{A}_{2}$ symmetry are IR active; $\mathrm{A}_{1}$ and $\mathrm{B}_{2}$ modes correspond to in-plane vibrations (respectively $a$\- and $b$-type bands) while $\mathrm{B}_{1}$ modes are out- of-plane vibrations ($c$-type bands). PhC3N is a prolate asymmetric top molecule with a permanent dipole moment of 5.9 Debye along the $a$ axis (axis of the C3N bonds, Fig. 1), according to our calculations. PhC3N also has nitrogen nuclear quadrupole hyperfine structure, but this splitting is only partially resolved at low frequency in the CE-FTMW measurements. Given that the molecule is $C_{2\mathrm{v}}$ symmetry, we also include statistical weights for equivalent exchangeable nuclei for correct transition intensities. Identical to PhCN, there are two sets of equivalent hydrogen atoms ($I_{1}=I_{2}=\frac{1}{2}$) that give rise to Fermi-Dirac statistics for symmetric (even $K$) and antisymmetric (odd $K$) rotational states with a ratio of $10:6$ [4]. ### 3.2 Vibrational spectroscopy Figure 3: Experimental (top traces, in black) and simulated (bottom traces, at 300 K, where different colors correspond to different symmetries) vibrational spectrum of PhC3N. The three bottom panels are expanded portions of the full spectrum, as indicated by gray rectangles. Simulation performed using the PGOPHER software [27] and the results of the anharmonic calculations (band centers, intensities, and rotational constants) and normalized to the strongest IR active mode ($\nu_{4}$). The simulations are inverted relative to the observed spectrum solely for comparison purposes. Secure band assignments are indicated by dash lines; additional labelled bands are those for which an assignment is proposed; simulated bands without any labels remain unassigned. The infrared spectrum of PhC3N is presented in Fig. 3 together with a simulation of the vibrational fundamentals predicted by our anharmonic quantum chemical calculations. At our experimental resolution, the rotational contour of most bands is evident in the spectrum. Because the simulation is in very good agreement with the experiment, numerous assignments can be made with confidence. For ambiguous assignments, two criteria can be invoked: (i) $c$-type bands (B1 symmetry) exhibit a sharp Q-branch; and (ii) $a$-type bands (A1) are usually narrower than the others (as can be seen on the simulated spectrum). Out of the 36 active infrared modes of PhC3N, 14 can be assigned with little or no ambiguity while 9 other bands have tentative assignments at this juncture; the remaining bands are either weak or are predicted in crowded regions of the spectrum (see Table 1 for a detailed list of the proposed assignments). Experimental band centers are taken as the frequency of the $Q$-branch when one exists ($c$-type bands), equidistant between the $P$\- and $R$-branches, or at the maximum of the envelope when no clear contour is visible. Considering $c$-type $Q$-branches are several wavenumbers wide, an accuracy of $\pm 2$ cm-1 can be expected for the band centers with B1 symmetry. For the others, a conservative value of $\pm 5$ cm-1 is proposed. Table 1: Fundamental vibrational bands (position and intensity) of PhC3N from the quantum chemical calculations performed in this work at the harmonic and anharmonic levels of theory, and proposition of assignments. Modes are numbered following the anharmonic calculations frequency order. Modes energy are in wavenumbers, intensities are in km/mol, $\delta$ values are in %. The experimental assignments are split in two categories, the relatively secure ones (column “Assign.") and the tentative ones (column “Prop."); $\delta$ values of the latter are reported in italics. Mode \| \| Harm. Calc. \| \| Anharm. Calc. \| \| Exp. ---\|---\|---\|---\|---\|---\|--- $\nu$ \| Sym. \| \| Energy \| Int. \| \| Energy \| Int. \| \| Assign. \| Prop. \| $\delta\,^{a}$ 1 \| A1 \| \| 3214 \| 6.7 \| \| 3055 \| 13.5 \| \| 3078 \| \| 0.8 2 \| A1 \| \| 3226 \| 5.6 \| \| 3026 \| 0.0 \| \| \| 3047 \| 0.7 3 \| A1 \| \| 3197 \| 0.0 \| \| 2999 \| 1.7 \| \| \| 3047 \| 1.6 4 \| A1 \| \| 2431 \| 242.3 \| \| 2399 \| 220.1 \| \| 2284 \| \| -4.8 5 \| A1 \| \| 2292 \| 10.7 \| \| 2265 \| 7.5 \| \| \| 2154/2198 \| -4.9/-3.0 6 \| A1 \| \| 1675 \| 1.0 \| \| 1633 \| 0.5 \| \| \| \| 7 \| A1 \| \| 1545 \| 7.7 \| \| 1510 \| 4.9 \| \| 1495 \| \| -1.0 8 \| A1 \| \| 1306 \| 1.6 \| \| 1284 \| 0.8 \| \| \| \| 9 \| A1 \| \| 1214 \| 2.1 \| \| 1193 \| 2.0 \| \| 1182 \| \| -0.9 10 \| A1 \| \| 1064 \| 4.6 \| \| 1045 \| 4.5 \| \| 1029 \| \| -1.5 11 \| A1 \| \| 1027 \| 1.3 \| \| 1012 \| 1.0 \| \| \| 1004/981 \| -0.8/-3.1 12 \| A1 \| \| 981 \| 0.1 \| \| 960 \| 0.0 \| \| \| \| 13 \| A1 \| \| 706 \| 3.4 \| \| 693 \| 3.5 \| \| \| \| 14 \| A1 \| \| 371 \| 0.8 \| \| 362 \| 0.5 \| \| \| \| 15 \| A2 \| \| 1019 \| 0.0 \| \| 1000 \| 0.0 \| \| \| \| 16 \| A2 \| \| 876 \| 0.0 \| \| 862 \| 0.0 \| \| \| \| 17 \| A2 \| \| 413 \| 0.0 \| \| 412 \| 0.0 \| \| \| \| 18 \| B1 \| \| 1042 \| 0.0 \| \| 1023 \| 0.0 \| \| \| \| 19 \| B1 \| \| 967 \| 2.8 \| \| 953 \| 2.4 \| \| 920 \| \| -3.5 20 \| B1 \| \| 792 \| 38.7 \| \| 784 \| 32.7 \| \| 757 \| \| -3.4 21 \| B1 \| \| 716 \| 36.7 \| \| 706 \| 42.7 \| \| 687 \| \| -2.6 22 \| B1 \| \| 576 \| 13.2 \| \| 568 \| 12.4 \| \| 532 \| \| -6.3 23 \| B1 \| \| 523 \| 0.2 \| \| 515 \| 0.3 \| \| \| \| 24 \| B1 \| \| 384 \| 2.8 \| \| 377 \| 2.8 \| \| 359 \| \| -4.9 25 \| B1 \| \| 195 \| 2.6 \| \| 192 \| 2.5 \| \| \| 185 \| -3.4 26 \| B1 \| \| 74 \| 0.6 \| \| 71 \| 0.7 \| \| 69 \| \| -3.3 27 \| B2 \| \| 3206 \| 1.9 \| \| 3030 \| 9.5 \| \| \| 3095 \| 2.1 28 \| B2 \| \| 3222 \| 8.4 \| \| 3012 \| 2.6 \| \| \| \| 29 \| B2 \| \| 1646 \| 0.7 \| \| 1608 \| 0.5 \| \| \| \| 30 \| B2 \| \| 1493 \| 8.6 \| \| 1475 \| 4.5 \| \| 1448 \| \| -1.8 31 \| B2 \| \| 1364 \| 1.2 \| \| 1331 \| 0.9 \| \| \| 1336 \| 0.4 32 \| B2 \| \| 1320 \| 1.3 \| \| 1292 \| 1.0 \| \| \| 1280 \| -0.9 33 \| B2 \| \| 1193 \| 0.4 \| \| 1183 \| 0.5 \| \| \| \| 34 \| B2 \| \| 1117 \| 4.4 \| \| 1094 \| 3.8 \| \| 1069 \| \| -2.3 35 \| B2 \| \| 644 \| 0.0 \| \| 635 \| 0.0 \| \| \| \| 36 \| B2 \| \| 584 \| 0.5 \| \| 574 \| 0.4 \| \| \| \| 37 \| B2 \| \| 511 \| 5.3 \| \| 504 \| 5.6 \| \| 482 \| \| -4.3 38 \| B2 \| \| 226 \| 2.3 \| \| 221 \| 2.4 \| \| \| 209 \| -5.6 39 \| B2 \| \| 71 \| 1.1 \| \| 71 \| 1.2 \| \| \| 69 \| -2.7 ${}^{a}\ \delta=(\mathrm{Exp.}-\mathrm{Anharm.Calc.})/\mathrm{Anharm.Calc.}\times 100$ A few of the assignments proposed in Fig. 3 and Table 1 warrant further discussion. While the assignment of the $\nu_{26}$ band (out-of-plane backbone motion) is relatively secure thanks to the presence of a sharp $Q$-branch, the band appears stronger than expected from our calculations. A possible explanation is that $\nu_{39}$ (C#C-C#N rocking), predicted roughly at the same frequency as $\nu_{26}$, contributes to the experimental band profile, thus enhancing absorption. In absence of any specific feature arising from $\nu_{39}$ on the experimental spectrum, we have chosen to assign this band to the same band center as $\nu_{26}$, at 69 cm-1. Slightly higher in frequency are $\nu_{25}$ and $\nu_{38}$, which involve similar nuclear motion as $\nu_{26}$ and $\nu_{39}$ and are also predicted to lie close in energy (at 192 and 221 cm-1, respectively), which in combination likely yield the feature observed around 200 cm-1. It is not straightforward, however, to unambiguously assign each band center, in part because $\nu_{25}$ is the only band of $B_{1}$ symmetry with a sizeable transition moment that does not exhibit a sharp Q-branch. Tentative assignments are proposed in Table 1, but their accuracy should be taken with caution as the actual bands centers could lie in a 10 to 20 cm-1 window from the proposed assignments. Concerning $\nu_{11}$ (ring symmetric breathing) and $\nu_{5}$ (C#C-C#N stretch)—predicted at 1012 and 2265 cm-1, respectively—two bands lie systematically close to the expected energy, with reasonable band profiles, thus in each case both assignments are reported in Table 1. In the case of $\nu_{5}$, the lowest frequency assignment, $i.e.$, 2154 cm-1 appears most likely as the relatively strong intensity of the observed band would indicate a fundamental that could otherwise not be predicted. However the shape of the band lying at 2198 cm-1 is closer to that expected for an $a$-type band, assuming that our aforementioned criterion remains valid at these frequencies (i.e. hot bands and combination bands could significantly affect the simplistic fundamentals- only picture). The most difficult analysis lies in the ${\sim}3000-3200$ cm-1 region, where C-H stretching motions are observed. From previous studies on similar molecules like phenylacetylene [14] and naphthalene [16], it is well-known that this frequency range is plagued by anharmonic resonances which can significantly complicate assignment. For PhC3N, three modes of $\mathrm{A}_{1}$ ($\nu_{1}$, $\nu_{2}$, $\nu_{3}$) and two of $\mathrm{B_{2}}$ ($\nu_{27}$, $\nu_{28}$) symmetries (Table 1) very close in energy (of order of ${\sim}$10 cm-1 for the harmonic frequencies) and can mix strongly, thus qualitatively shifting the fundamental energies and altering band intensities. In such cases, an approximate deperturbation analysis was performed using the generalized VPT2 calculations to identify and treat anharmonic resonances, e.g. those arising from Fermi (so-called “1–2” resonances) and Darling- Dennison (“1–1” interactions in the present case). These resonances are identified based on small differences in the energies of states and a model variational calculation [30]; the former is a zeroth order estimate for resonances, while the latter tests for the magnitude of the coupling [31]. The VPT2 routines in Gaussian treat the problem of state-to-state coupling as effective $2\times 2$ Hamiltonians, where the diagonal elements correspond to the state energies, and the off-diagonal elements represent coupling between the two states. These values are shown in Table 2, and give rise to significant deviation from the harmonic frequencies and intensities shown in Table 1. Of particular note is the unintuitive complete loss of intensity in $\nu_{2}$: $\nu_{3}$ is “dark” in the harmonic approximation, and it gains intensity primarily through borrowing intensity from the stronger $\nu_{1}$ and $\nu_{2}$ bands. In a general resonance picture, this interaction shares intensity, and to render a mode completely inactive is extremely rare if not unheard of. Given the VPT2 treatment here is only approximate, our effective deperturbation analysis is likely inadequate to properly treat these bands, and a fully coupled model involving fundamentals, combination bands, and overtones, is required instead. In light of this preliminary analysis, however, as well as the fact that the ${\sim}3000$ cm-1 region is heavily congested, we assume all three $\mathrm{A}_{1}$ modes are IR active. Measurement under cold conditions—either in a supersonic jet or buffer gas cell—will help future analysis of this molecule by eliminating the possibility of combination bands and overtones, in addition to minimizing lineshape blending from rotational contours. Similarly, selective deuteration might clarify the assignment of some features [14]. Table 2: Strong anharmonic resonances and their corresponding off-diagonal matrix elements for bands in the ${\sim}3100-3200$ cm-1 region. Darling-Dennison (DDR) and Fermi (FR) resonances are indicated; in this table, the former corresponds to $1-1$ type DDR, referring to the number of quanta for states involved. State 1 \| State 2 \| Type \| Coupling ---\|---\|---\|--- $v_{1}=1$ \| $v_{3}=1$ \| DD \| -6.6 $v_{2}=1$ \| $v_{3}=1$ \| DD \| 14.5 $v_{3}=1$ \| $v_{6}=1+v_{7}=1$ \| FR \| -23.2 $v_{3}=1$ \| $v_{29}=1+v_{30}=1$ \| FR \| -29.3 $v_{27}=1$ \| $v_{28}=1$ \| DD \| 9.4 ### 3.3 Rotational spectroscopy Using the ground state rotational and quartic centrifugal distortion constants from the anharmonic calculation, 65 strong lines of the CP-FTMW spectrum were assigned in a straightforward fashion using the PGOPHER software [27] (Fig. 2). The derived values are extremely close to those predicted by the calculation (the weighted frequency difference $\delta$ is less than 1 % for the rotational constants, see Table 3). This initial set of constants was then used to assign the millimeter-wave data. Loomis-Wood diagrams were produced by means of the LWWa software from Lodyga et al. [32] to aid in the assignment of the high-$J$ transitions. In total, 6151 $a$-type transitions (3780 different frequencies as a result of unresolved asymmetric splitting) of PhC3N in its ground vibrational state were assigned in the millimeter-wave spectrum, with values of $J^{\prime\prime}$ up to 199 and $K_{a}^{\prime\prime}$ up to 42. The SPFIT/SPCAT suite of programs [33] using the Watson $S$-reduced Hamiltonian in the $I^{r}$ representation was used to determine best-fit spectroscopic constants. All transitions were weighted accorded to their expected experimental accuracy, $i.e.$, 2 kHz and 25 kHz for the CE-FTMW and CP-FTMW transitions, respectively, and 50 kHz for the millimeter-wave transitions. To reproduce the data to their experimental accuracy, inclusion of several sextic and octic centrifugal distortion constants was required. Finally, the CE-FTMW transitions—the only ones for which the nuclear quadrupole splitting was resolved—were added to the fit to determine the $\chi$(N) terms. All 57 hyperfine components were reproduced to the measurement uncertainty by adjusting only $\chi_{aa}$(N) and $\chi_{bb}$(N). Both parameters are close to those expected from calculation (to within about 15 %, Table 3). When a transition was observed by both CP-FTMW and CE-FTMW spectroscopy, only the latter was retained in the fit, owing to the higher frequency accuracy of the cavity instrument. The rotational constants derived from a fit to all of the assigned rotational transitions are reported in Table 3. The final 49 kHz RMS value of the fit, corresponding to a reduced standard deviation $\sigma=1.00$, indicates that our present model adequately reproduces the ground state rotational spectrum of PhC3N. Table 3: Spectroscopic constants (rotational, centrifugal distorsion, and nuclear quadrupole constants) of PhC3N in its vibrational ground state (in MHz) and relevant fit parameters. Numbers in parenthesis are $1\sigma$ uncertainties expressed in the unit of the last digit. Parameters in brackets were kept fixed to the calculated values. Constant \| Calc.a \| Exp. \| $\delta\,^{b}$ ---\|---\|---\|--- $A_{0}$ \| $5656.9$ \| $5659.722\,(15)$ \| 0.05 $B_{0}$ \| $567.0$ \| $569.582206\,(39)$ \| 0.46 $C_{0}$ \| $515.4$ \| $517.404488\,(37)$ \| 0.39 $D_{J}\times 10^{6}$ \| $3.6$ \| $3.85110\,(77)$ \| 7.0 $D_{JK}\times 10^{3}$ \| $0.78$ \| $0.827177\,(85)$ \| 6.1 $D_{K}\times 10^{3}$ \| $0.42$ \| $0.400\,(68)$ \| -4.8 $d_{1}\times 10^{6}$ \| $-0.50$ \| $-0.55191\,(24)$ \| 10 $d_{2}\times 10^{6}$ \| $-0.52$ \| $-0.57101\,(79)$ \| 9.8 $H_{J}\times 10^{12}$ \| \| $-0.957\,(11)$ \| $H_{JK}\times 10^{9}$ \| \| $0.7916\,(18)$ \| $H_{KJ}\times 10^{6}$ \| \| $-0.03051\,(10)$ \| $h_{2}\times 10^{12}$ \| \| $0.575\,(15)$ \| $h_{3}\times 10^{12}$ \| \| $0.10329\,(55)$ \| $L_{JJK}\times 10^{15}$ \| \| $-1.295\,(20)$ \| $L_{JK}\times 10^{12}$ \| \| $0.06954\,(86)$ \| $L_{KKJ}\times 10^{12}$ \| \| $-3.512\,(36)$ \| $\chi_{aa}$ \| $-4.96$ \| $-4.219\,(77)$ \| -15 $\chi_{bb}$ \| $2.39$ \| $2.114\,(68)$ \| -12 $\chi_{cc}$ \| $2.57$ \| $2.143\,^{c}$ \| -17 $N\,^{d}$ \| \| 6256/3877/57 \| $J^{\prime\prime}_{\mathrm{max}}$, $K^{\prime\prime}_{a\,\mathrm{max}}$ \| \| 199, 42 \| rms /kHz \| \| 0.049 \| $\sigma\,^{e}$ \| \| 1.00 \| a $\omega$B97XD/cc-pVQZ level of theory, Bayesian corrected for $A$, $B$, and $C$, and anharmonic values for the centrifugal distortion constants, and equilibrium values for the hyperfine constants ${}^{b}\delta=(B_{\mathrm{exp.}}-B_{\mathrm{calc.}})/B_{\mathrm{calc.}}\times 100$ (in %) c Derived value d Total number of lines in the fit / Number of different frequencies / Number of lines with resolved nuclear quadrupole structure e Reduced standard deviation, unitless Figure 4: Portions of the millimeter-wave spectrum of PhC3N in comparison with a simulation of the pure rotational transitions in the ground vibrational state using the best-fit set of spectroscopic constants (Table 3). The simulation has been performed using the PGOPHER software and the resulting trace was then post-processed with a second derivative to allow a more straightforward comparison with the experimental spectrum. The line density in the experimental trace is far greater than our simulation, very likely because of lines from vibrational satellites. Figure 4 shows a portion of the millimeter-wave spectrum in comparison with a simulation of PhC3N in its vibrational ground state using the experimentally determined best-fit parameters from Table 3. As illustrated in this figure, many lines remain unassigned, but most of these likely arise from vibrational satellites, for which no attempt at assigning was made in the present study. Although longer integration times would allow a more in-depth analysis of these satellites, the spectrum is already fairly dense, implying that we may be close to the confusion limit. Indeed, the 2 µbar pressure used in this study, although quite low, was actually a compromise between reasonable signal-to-noise ratio and pressure broadening. Even at this pressure many lines are broader then expected from the effects of pressure broadening alone, and consequently may in fact be a spectral superposition of several transitions. Regarding hyperfine splitting, its magnitude rapidly collapses with increasing $J$, as expected from this type of interaction. No splitting is observed in the CP-FTMW nor in millimeter-wave measurements, and in the cavity experiments it is marginally resolved above $J^{\prime\prime}\approx 11$ (${\sim}12$ GHz). Even at the lowest-$J$ transitions measured here ($J^{\prime\prime}=5$, ${\sim}6$ GHz), the splitting due to hyperfine and Doppler effects are comparable (Fig. 5). The experimentally derived value of $\chi_{aa}$(N) [$-4.219$ (77) MHz; Table 3] is very similar to that reported for PhCN [-4.23738(36) MHz [9]], and the relative magnitudes are in agreement with the theoretical values of $\chi_{aa}$(N) calculated at the $\omega$B97X-D/cc-pVQZ level of theory for PhC3N (-4.959 MHz) and PhCN (-4.962 MHz). The small changes in $\chi$(N) are an indication that the local electronic structure of the nitrogen atom is relatively insensitive to the distance from the aromatic ring. Equivalently, this finding implies that electron delocalization through ring conjugation is very poorly coupled to the chain regardless of length. Similar behavior is seen for cyanopolyyne chains, where the value of $eQq$ equivalent to $\chi_{aa}$(N) (around $-4.1$ MHz) is relatively invariant with the length of the chain as well [34]. Figure 5: Example of a transition showing resolved hyperfine structure on the CE-FTMW spectrum, and comparison with 10 K simulations using the final set of spectroscopic parameters, with and without taking into account the Doppler splitting (simulations performed using the PGOPHER software, assuming a Lorentzian lineshape). As before, the simulation with Doppler splitting is inverted relative to the observed spectrum solely for comparison purposes. ### 3.4 Astronomical considerations Detection of molecules in space by radio astronomy is heavily dependent on the magnitude of their permanent dipole moment. In comparison with PhCN, our theoretical predictions suggest PhC3N is substantially more polar (5.9 D vs. of 4.5 D, where the former value has statistical uncertainty of $\pm 0.25$ D based on our prior benchmarking at the $\omega$B97X-D/cc-pVQZ level of theory [35]). Currently, PhCN is theorized to form in cold, dark clouds via a barrierless reaction between C6H6 and CN radical [4, 36]. We thus speculate that PhC3N could be formed via C2 insertion to PhCN or through an analogous C#N addition reaction between CN radical and phenylacetylene (PhC2H), or C3N and C6H6. If the latter mode is operative, then the abundance ratio of PhCN/PhC3N will be dependent on CN/C3N, assuming similar reaction cross- sections. The CN + PhC2H process has been studied in crossed-beam experiments by Bennett et al. [11], where the authors identify PhC3N as a potential reaction product, albeit not definitively so due to the lack of isomer specificity and at collision energies well in excess of interstellar cloud conditions (${\sim}30$ kJ/mol). This finding suggests other isomers, namely ortho, meta, para-CEB, might plausibly be formed from this reaction. While we have not attempted to experimentally characterized these species, estimates of their spectroscopic parameters are provided here. By scaling the experimental parameters for PhC3N (Table 3) to correct for vibrational and electronic effects (Table S6 in the Supporting Information), in conjunction to Bayesian uncertainties obtained from benchmarking [35], reliable constraints of these constants should aid future experimental searches for these species. In terms of astronomical detection, although the hyperfine structure does not take a large part the present rotational analysis, this splitting is partially resolved with cavity measurements up to 16 GHz. In cold, dark clouds such as TMC-1 where the linewidths are comparable to those measured with our CE-FTMW spectrometer, it is thus necessary to consider hyperfine splitting, as recent work on PhCN [4] demonstrates. This is particularly true for a relatively heavy molecule like PhC3N whose strongest lines should lie at centimeter- wavelengths at low temperatures (Fig. 6): at 10 K, the strongest features correspond to $J^{\prime\prime}=21$ around 23 GHz, while at 6 K —a typical temperature for molecules in TMC-1— the strongest features are near 15 GHz ($J^{\prime\prime}=14$). Thus, the $\mathrm{X/K_{u}}$ (8–12/12–18 GHz) bands appear to be the most promising to detect PhC3N in TMC–1. In sources with somewhat warmer temperatures, the intensity of individual transitions is significantly decreased due to the larger partition function, and the peak intensity, although relatively flat, falls in the W (75–110 GHz) and N (100–200 GHz) bands. Figure 6: Calculated rotational spectrum of PhC3N at 10 K (purple), 100 K (blue), and 300 K (orange). For the purposes of display, these relative intensities for the 100 K plot have increased by a factor of 5, while for the 300 K plot, the increase is 10. Detection of multiple isomers is also a powerful tool for constraining physical and chemical conditions in astrophysical environments. To assist in this process, we have calculated the relative energy of the aforementioned ortho, meta, and para-CEB isomers, along with the isocyanide isomer, PhCCNC. As shown in Fig. 7, apart from the isocyanide isomer, placement of the acetylenic unit on different parts of the ring produces isomers with comparable stability to PhC3N, i.e., they are effectively degenerate at the level of uncertainty afforded by G3//B3LYP ($\pm 4$ kJ/mol). As such, a determination of their relative abundances would provide a sensitive test of thermodynamics vs. kinetics in molecule formation. To aid further laboratory and hopefully astronomical efforts, Table S6, in addition to providing estimates of rotational constants, also reports calculated dipole moments at the $\omega$B97X-D/cc-pVQZ level of theory using two methods of empirical scaling to correct for vibrational effects and deficiencies in the electronic structure method. We note that the Bayesian scaling factors obtained in Ref. [35] applied to PhC3N —where we now have accurately determined parameters— exceed the performance of the purely theoretical VPT2 corrections, and bring the theoretical predictions within a few MHz of the experimentally measured ones (see also Fig. S1). We expect a similar degree of precision and uncertainty for other isomers. Figure 7: 0 K energetics of isomers of interest calculated at the G3//B3LYP level of theory, relative to PhC3N. Energies to the nearest kJ/mol are annotated above each bar. ## 4 Conclusion Using a combination of gas-phase measurements and quantum chemical calculations, the fundamental vibrational frequencies of most of the strongest IR-active fundamentals of PhC3N and over 6000 pure rotational transitions in its ground state between 6.5 and 220 GHz have been measured. The assignment of spectra in different regions was guided by theoretical predictions: in the infrared, anharmonic calculations helped to assign most bands, with the exception of the most congested and perturbed region around $3000$ cm-1; in the radio domain, estimates of centrifugal distortion and hyperfine terms proved useful. Comparisons with PhCN suggest that the local electronic structure of the terminal nitrogen —as probed through its quadrupole moment $\chi_{aa}$(N)— is relatively unaffected by chain lengthening, indicating a similar electric field gradient for nitrogen in both PhCN and PhC3N. In addition, we provide accurate and reliable predictions of the thermochemistry and spectroscopic parameters for several isomers of PhC3N, which should prove useful in guiding future laboratory experiments. These isomers include the isocyanide isomer of PhC3N and the three cyanoethynylbenzenes alluded to in previous work [11]; their discovery along with PhC3N in the ISM would provide sensitive measurements of local chemical and physical interstellar environments. With highly precise measurements of the rest frequencies and corresponding spectroscopic constants, a search for PhC3N can now be undertaken with considerable confidence in the ISM. The large permanent dipole moment (predicted to be 5.9 D), in addition to its chemical and structural similarity to astronomical PhCN makes PhC3N an excellent candidate for detection towards cold, dark molecular clouds such as TMC-1. Hyperfine-resolved measurements are expected to be highly relevant in a potential discovery, given that at low temperatures the strongest transitions lie in the $\mathrm{X/K_{u}}$ bands where the splitting is comparable to the source linewidth. ## Acknowledgements O.C., O.P., and M.-A.M.-D. acknowledge funding support from the Région Ile-de- France through DIM-ACAV+, from the Agence Nationale de la Recherche (ANR-19-CE30-0017-01), from the “Investissements d’Avenir” LabEx PALM (ANR-10-LABX-0039-PALM), and from the Programme National “Physique et Chimie du Milieu Interstellaire” (PCMI) of CNRS/INSU with INC/INP co-funded by CEA and CNES. K.L.K.L. and M.C.M. acknowledge funding support from NSF grant AST-1908576 and NASA grant 80NSSC18K0396. Z.S.B. acknowledges support from the Chateaubriand Fellowship of the Office for Science & Technology of the Embassy of France in the United States. 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# Constructing new APN functions through relative trace functions Lijing Zheng, Haibin Kan, Yanjun Li, Jie Peng, Deng Tang L. Zheng is with the School of Mathematics and Physics, University of South China, Hengyang, Hunan, 421001, China, (E-mail: zhenglijing817@163.com). H. Kan is with the School of Computer Sciences, Fudan University, Shanghai, 200433, China, (E-mail: hbkan@fudan.edu.cn). Y. Li is with the Mathematics and Science College of Shanghai Normal University, Shanghai, 200234, China, (yanjlmath90@163.com). J. Peng is with the Mathematics and Science College of Shanghai Normal University, Shanghai, 200234, China, (jpeng@shnu.edu.cn). D. Tang is with the School of Electronic Information and Electrical Engineering, Shanghai Jiao Tong University, Shanghai, 200240, China, (dtang@foxmail.com). Abstract: In 2020, Budaghyan, Helleseth and Kaleyski [IEEE TIT 66(11): 7081-7087, 2020] considered an infinite family of quadrinomials over $\mathbb{F}_{2^{n}}$ of the form $x^{3}+a(x^{2^{s}+1})^{2^{k}}+bx^{3\cdot 2^{m}}+c(x^{2^{s+m}+2^{m}})^{2^{k}}$, where $n=2m$ with $m$ odd. They proved that such kind of quadrinomials can provide new almost perfect nonlinear (APN) functions when $\gcd(3,m)=1$, $k=0$, and $(s,a,b,c)=(m-2,\omega,\omega^{2},1)$ or $((m-2)^{-1}~{}{\rm mod}~{}n,\omega,\omega^{2},1)$ in which $\omega\in\mathbb{F}_{4}\setminus\mathbb{F}_{2}$. By taking $a=\omega$ and $b=c=\omega^{2}$, we observe that such kind of quadrinomials can be rewritten as $a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(cx^{2^{s}+1})$, where $q=2^{m}$ and ${\rm Tr}^{n}_{m}(x)=x+x^{2^{m}}$ for $n=2m$. Inspired by the quadrinomials and our observation, in this paper we study a class of functions with the form $f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^{q}{\rm Tr}^{n}_{m}(G(x))$ and determine the APN-ness of this new kind of functions, where $a\in\mathbb{F}_{2^{n}}$ such that $a+a^{q}\neq 0$, and both $F$ and $G$ are quadratic functions over $\mathbb{F}_{2^{n}}$. We first obtain a characterization of the conditions for $f(x)$ such that $f(x)$ is an APN function. With the help of this characterization, we obtain an infinite family of APN functions for $n=2m$ with $m$ being an odd positive integer: $f(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(b^{3}x^{9})$, where $a\in\mathbb{F}_{2^{n}}$ such that $a+a^{q}\neq 0$ and $b$ is a non-cube in $\mathbb{F}_{2^{n}}$. We verify that the aforementioned APN quadrinomials are CCZ-inequivalent to any other known APN functions over $\mathbb{F}_{2^{10}}$. We also obtain two infinite families of APN functions: $a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(gx^{5}+ex^{4q+1})$, where $b,~{}g,~{}e$ satisfy: $i)$ $b$ not a cube, $g=1$, $e=\frac{1}{b^{2q-2}}$; or $ii)$ $b$ not a cube, and $g=e=b$. We can also find (at least) two new sporadic instances of APN functions over $\mathbb{F}_{2^{10}}$ up to CCZ- equivalence. Keywords: APN functions; relative trace functions; quadratic functions; CCZ- equivalence ## 1 Introduction Throughout this paper, we often identify the finite field $\mathbb{F}_{2^{n}}$ with $\mathbb{F}^{n}_{2}$ which is the $n$-dimensional vector space over $\mathbb{F}_{2}$. Any function $F:\mathbb{F}_{2^{n}}\rightarrow\mathbb{F}_{2^{m}}$ is called an $(n,m)$-function or vectorial Boolean functions if the values $n$ and $m$ are omitted. Vectorial Boolean functions are of critical importance in the field of symmetric cryptography, and the security of encryption algorithms heavily depends on the cryptographic properties of the vectorial Boolean functions. Researchers have proposed various properties to measure the resistance of a vectorial Boolean function to different kinds of cryptanalysis, including differential uniformity, nonlinearity, boomerang uniformity, algebraic degree, and so on. The lower the differential uniformity of a vectorial Boolean function, the better its security against differential cryptanalysis. In this paper, we mainly focus on the $(n,n)$-functions. The differential uniformity of any such functions is at least 2, and the functions achieving this bound are called almost perfect nonlinear (APN). It is difficult to find new infinite families of APN functions up to CCZ- equivalence. Up to now, only 6 infinite families of APN monomials and 14 infinite families of APN polynomials are known, since the early 90’s. On the other hand, in contrast to these facts, there are a lot of APN functions even over “small” field: for example, thousands of CCZ-inequivalent APN functions have been found over $\mathbb{F}_{2^{8}}$ [25]. Constructing new instances of infinite families is an area of deep heading research. We present Tables I and II including all currently known infinite families of APN functions. To Table II, we add the new function found with Theorem 3.3 in Section 3 below. We refer the readers to a recent nice work of Budaghyan et al. for more details on the classification of the known families of APN functions [7]. TABLE I: Known infinite families of APN power functions over $\mathbb{F}_{2^{n}}$ Family \| Exponent \| Conditions \| Algebraic degree \| Source ---\|---\|---\|---\|--- Gold \| $2^{i}+1$ \| ${\rm gcd}(i,n)=1$ \| 2 \| [18] Kasami \| $2^{2i}-2^{i}+1$ \| ${\rm gcd}(i,n)=1$ \| $i+1$ \| [19] Welch \| $2^{t}+3$ \| $n=2t+1$ \| $3$ \| [14] Niho \| \| $2^{t}+2^{t/2}-1$, $t$ even --- $2^{t}+2^{(3t+1)/2}-1$, $t$ odd $n=2t+1$ \| \| $t/2+1$ --- $t+1$ [15] Inverse \| $2^{2t}-1$ \| $n=2t+1$ \| $n-1$ \| [1, 22] Dobbertin \| $2^{4i}+2^{3i}+2^{2i}+2^{i}-1$ \| $n=5i$ \| $i+3$ \| [16] Throughout this paper, let $\omega\in\mathbb{F}_{4}\backslash\\{0,1\\}.$ Very recently, Budaghyan, Helleseth, and Kaleyski introduced an infinite family of quadrinomials over $\mathbb{F}_{2^{n}}$ of the following form: $g_{s}(x)=x^{3}+a(x^{2^{s}+1})^{2^{k}}+bx^{3\cdot 2^{m}}+c(x^{2^{s+m}+2^{m}})^{2^{k}},$ where $n=2m$. They showed that this family can provide new infinite families of APN functions [12]. More precisely, they showed that $g_{s}(x)$ is a new APN function if $k=0$, $(s,a,b,c)=(m-2,\omega,\omega^{2},1)$, or $((m-2)^{-1}~{}{\rm mod}~{}n,\omega,\omega^{2},1)$, if $m$ is odd with ${\rm gcd}(3,m)=1$. They also pointed out that when $k\geq 1$, $g_{s}(x)$ can also be APN, however, CCZ-equivalent to some known ones. Let $n=2m$ and $q=2^{m}$. In this paper, our motivation is to find new infinite families of APN functions over $\mathbb{F}_{2^{n}}$. We revisit the above-mentioned two infinite families of APN quadrionomials obtained in [12]. Observing that for any odd positive integer $s$, $\omega^{2^{s}}=\omega^{2}$, the APN functions for $s=m-2$, or $(m-2)^{-1}{\rm mod}~{}n$ can be rewritten as $g_{s}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(cx^{2^{s}+1})$, $a=\omega$, $b=c=\omega^{2}$. Here ${\rm Tr}^{n}_{m}(x):=x+x^{2^{m}}$ for $n=2m$. Inspired by the quadrinomials and our observation, let $a\in\mathbb{F}_{2^{n}}$, we study a class of functions with the following form: $f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^{q}{\rm Tr}^{n}_{m}(G(x)),~{}a+a^{q}\neq 0,$ (1) where $F$ and $G$ are quadratic functions with $F(0)=G(0)=0$. Based on the framework (1), we carefully choose quadratic functions $F$ and $G$ for finding APN functions. We mainly consider two kinds of functions in (1) by setting $F$ and $G$ as follows. $i)$ $F(x)=bx^{3}$, $G(x)=cx^{2^{s}+1}$; $ii)$ $F(x)=bx^{2^{i}+1}+cx^{2^{i+m}+1}$, $G(x)=gx^{2^{s}+1}+ex^{2^{s+m}+1}$, where $b,c,g,e\in\mathbb{F}_{2^{n}}$, and $i,s$ are positive integers. Let $n=2m$ with $m$ odd. Let $a\in\mathbb{F}_{2^{n}}$, and $f_{s}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(cx^{2^{s}+1}),~{}a+a^{q}\neq 0.$ We can find two more exponents $s=3$, or $m+2$, and the corresponding conditions on the coefficients such that $f_{s}(x)$ is an APN function over $\mathbb{F}_{2^{n}}$. Code isomorphism tests (see Sec. 2 below) indicate that for the exponent $s=3$, the APN function found with Theorem 3.3: $f_{3}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(b^{3}x^{9}),$ where $b$ is a non-cube, is new up to CCZ-equivalence over $\mathbb{F}_{2^{10}}$. We can also discover more coefficients for these two exponents $s=m-2$, and $(m-2)^{-1}{\rm mod}~{}n$ discovered by Budaghyan et al. such that $f_{s}(x)$ is APN without the assumption that ${\rm gcd}(3,m)=1$. In this way, some new instances of APN functions over $\mathbb{F}_{2^{10}}$ and $\mathbb{F}_{2^{14}}$ of the form $f_{s}(x)$ can also be found. Let $n=2m$, $q=2^{m}$, $a\in\mathbb{F}_{2^{n}}$, and $\displaystyle h_{i,s,b,c,g,e}(x)=a{\rm Tr}^{n}_{m}(bx^{2^{i}+1}+cx^{2^{i+m}+1})+a^{q}{\rm Tr}^{n}_{m}(gx^{2^{s}+1}+ex^{2^{s+m}+1}),~{}a+a^{q}\neq 0.$ We can find two infinite families of APN functions as follows, by letting $i=1$, $s=2$, $c=0$. $\displaystyle h_{1,2,b,0,g,e}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(gx^{5}+ex^{4q+1}),$ where $a\in\mathbb{F}_{2^{n}}$ such that $a+a^{q}\neq 0$, $m$ is odd, and $b,~{}g,~{}e$ satisfy: $i)$ $b$ not cube, $g=1$, $e=\frac{1}{b^{2q-2}}$; or $ii)$ $b$ not cube in $\mathbb{F}^{\ast}_{2^{n}}$, and $g=e=b$. By means of the code isomorphism test, we find that these two classes of APN functions are CCZ-inequivalent to each other, however, CCZ-equivalent to some functions in family F12 of Taniguchi over $\mathbb{F}_{2^{10}}$. The critical technique needed in the proof is to forge links between the cube-ness of some certain elements and the number of solutions to the equation of the following form: $\displaystyle Ax^{3}+Bx^{2}+B^{q}x+A^{q}=0.$ The rest of the paper is organized as follows. Some basic definitions are given in Section 2. We characterize the condition for $f(x)$ with the form (1) such that $f(x)$ is an APN function over $\mathbb{F}_{2^{n}}$, $n=2m$. In Section 3, we investigate the APN property of the functions with the form (1) by letting $F$, $G$ are both Gold functions or both quadratic binomials. We can find a new infinite family of APN quadrinomials, and generalize the two infinite families of APN functions found by Budaghyan et al. in [12]. We can find two infinite families of APN hexanomials, which computationally proved that they belong to family F12 over $\mathbb{F}_{2^{10}}$. We can also find (at least) two new APN instances over $\mathbb{F}_{2^{10}}$. A few concluding remarks are given in Section 4. ## 2 Preliminaries Let $\mathbb{F}_{2^{n}}$ be the finite field consisting of $2^{n}$ elements, then the group of units of $\mathbb{F}_{2^{n}}$, denoted by $\mathbb{F}^{\ast}_{2^{n}}$, is a cyclic group of order $2^{n}-1$. Let $\alpha\in\mathbb{F}_{2^{n}}.$ It is called a cube in $\mathbb{F}_{2^{n}}$, if $\alpha=\beta^{3}$ for some $\beta\in\mathbb{F}_{2^{n}}$; otherwise, it is called a non-cube. Let $m$ and $n$ be two positive integers satisfying $m~{}\|~{}n$, we use ${\rm Tr}^{n}_{m}(\cdot)$ to denote the trace function form $\mathbb{F}_{2^{n}}$ to $\mathbb{F}_{2^{m}}$, i.e., ${\rm Tr}^{n}_{m}(x)=x+x^{2^{m}}+x^{2^{2m}}+\cdots+x^{2^{(n/m-1)m}}.$ Let $f(x)$ be a function over $\mathbb{F}_{2^{n}}$. Then it can be uniquely represented as $f(x)=\sum^{2^{n}-1}_{i=0}a_{i}x^{i}$. This is the univariate representation of $f$. Let $0\leq i\leq 2^{n}-1$. The binary weight of $i$ is $w_{2}(i)=\sum^{n-1}_{s=0}i_{s}$, where $i=\sum^{n-1}_{s=0}i_{s}2^{s}$, $i_{s}\in\\{0,1\\}$. The algebraic degree of $f$, denoted by ${\rm deg}(f)$, is the largest binary weight of an exponent $i$ with $a_{i}\neq 0$ in the univariate representation of $f$. Functions of algebraic degree one, and two are called affine, quadratic, respectively. Given an $(n,n)$-function $F$, we denote by $\Delta_{F}(a,b)$ the number of solutions to the equation $D_{a}F(x)=b$, where $D_{a}F(x)=F(x)+F(x+a)$ is the _derivative_ of $F$ in direction $a\in\mathbb{F}_{2^{n}}$. $F$ is called _differentially $\delta$-uniform_ if the largest value of $\Delta_{F}(a,b)$ equals to $\delta$, for every nonzero $a$ and every $b$. If $F$ is differentially 2-uniform, we say that $F$ is _almost perfect nonlinear_ (APN). Two $(n,m)$-functions $F$ and $G$ are called extended affine equivalent (EA- equivalent) if there exist some affine permutation $L_{1}$ over $\mathbb{F}_{2^{n}}$ and some affine permutation $L_{2}$ over $\mathbb{F}_{2^{m}}$, and some affine function $A$ such that $F=L_{2}\circ G\circ L_{1}+A$. They are called Carlet-Charpin-Zinoviev equivalent (CCZ- equivalent) if there exists some affine automorphism $L=(L_{1},L_{2})$ of $\mathbb{F}_{2^{n}}\times\mathbb{F}_{2^{m}}$, where $L_{1}:\mathbb{F}_{2^{n}}\times\mathbb{F}_{2^{m}}\rightarrow\mathbb{F}_{2^{n}}$ and $L_{2}:\mathbb{F}_{2^{n}}\times\mathbb{F}_{2^{m}}\rightarrow\mathbb{F}_{2^{m}}$ are affine functions, such that $y=G(x)$ if and only if $L_{2}(x,y)=F\circ L_{1}(x,y)$. It is well known that EA-equivalence is a special kind of CCZ- equivalence, and that CCZ-equivalence preserves the differential uniformity [13]. Proving CCZ-inequivalence of functions can be very difficult in general, and this is resolved through code isomorphism. Let $\alpha$ be the primitive element in $\mathbb{F}_{2^{n}}$. Then two $(n,n)$-functions functions $F$ and $G$ are CCZ-equivalent if and only if $\mathcal{C}_{F}$, $\mathcal{C}_{G}$ are isomorphic [3], where $\mathcal{C}_{F}$ is the linear code corresponding to $F$ with the generating matrix as follows. $\mathcal{C}_{F}=\left(\begin{array}[]{cccc}1&1&\cdots&1\\\ 0&\alpha&\cdots&\alpha^{2^{n}-1}\\\ F(0)&F(\alpha)&\cdots&F(\alpha^{2^{n}-1})\\\ \end{array}\right)$ Let $f$ be a quadratic function over $\mathbb{F}_{2^{n}}$ with $f(0)=0$. Denote $\Delta_{d,f}(x):=f(dx)+f(dx+d)+f(d).$ Then it is well known that $f$ is APN if and only if for every $d\neq 0$, $\Delta_{d,f}(x)=0$ only has trivial solutions in $x$, i.e., only $x\in\mathbb{F}_{2}$ can be a solution to $\Delta_{d,f}(x)=0$. In the following, we determine the APN-ness of the functions with the form (1). ###### Lemma 2.1. Let $n=2m$, and $q=2^{m}$. Let $F$, $G$ be quadratic functions over $\mathbb{F}_{2^{n}}$ satisfying that $F(0)=0$, and $G(0)=0$. Let $f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^{q}{\rm Tr}^{n}_{m}(G(x)),$ where $a\in\mathbb{F}_{2^{n}}$ such that $a+a^{q}\neq 0$. Then $f(x)$ is APN over $\mathbb{F}_{2^{n}}$, if and only if the following system $\displaystyle\begin{cases}\Delta_{d,F}(x)\in\mathbb{F}_{2^{m}}&\\\ \Delta_{d,G}(x)\in\mathbb{F}_{2^{m}}&\end{cases}$ (2) only has $x=0,1$ as its solutions for any $d\neq 0\in\mathbb{F}_{2^{n}}$. ###### Proof. Since $f(x)$ is quadratic with $f(0)=0$, it is equivalent to showing that the following equation only has $x=0,1$ as its solutions for any $d\neq 0$ $\Delta_{d,f}(x)=f(dx)+f(dx+d)+f(d)=0.$ (3) We have $\Delta_{d,f}(x)=a{\rm Tr}^{n}_{m}(\Delta_{d,F}(x))+a^{q}{\rm Tr}^{n}_{m}(\Delta_{d,G}(x))=0.$ (4) In the following, we shall show that (4) holds if and only if ${\rm Tr}^{n}_{m}(\Delta_{d,F}(x))={\rm Tr}^{n}_{m}(\Delta_{d,G}(x))=0.$ The sufficiency is clear. Let us show the necessity. Raising (4) to its $q$-th power, we have $a^{q}{\rm Tr}^{n}_{m}(\Delta_{d,F}(x))+a{\rm Tr}^{n}_{m}(\Delta_{d,G}(x))=0.$ (5) Adding (4) and (5), $(a+a^{q}){\rm Tr}^{n}_{m}(\Delta_{d,F}(x))+(a+a^{q}){\rm Tr}^{n}_{m}(\Delta_{d,G}(x))=0,$ which infers, since $a+a^{q}\neq 0$, that ${\rm Tr}^{n}_{m}(\Delta_{d,F}(x))={\rm Tr}^{n}_{m}(\Delta_{d,G}(x)).$ (6) Substituting (6) into (4), we can obtain ${\rm Tr}^{n}_{m}(\Delta_{d,F}(x))={\rm Tr}^{n}_{m}(\Delta_{d,G}(x))=0,$ which is exactly the system (2). Therefore, $f(x)$ is APN, if and only if the system (2) only has trivial solutions $x=0,1$, for any $d\neq 0$. ∎ TABLE II: Known infinite families of quadratic APN polynomials over $\mathbb{F}_{2^{n}}$ ID \| Functions \| Conditions \| Source ---\|---\|---\|--- F1-F2 \| $x^{2^{s}+1}+u^{2^{k}-1}x^{2^{ik}+2^{mk+s}}$ \| $n=pk$, ${\rm gcd}(k,p)={\rm gcd}(s,pk)=1$, $p\in\\{3,4\\}$, $i=sk~{}{\rm mod}~{}p$, $m=p-i$, $n\geq 12$, $u$ primitive in $\mathbb{F}^{\ast}_{2^{n}}$ \| [9] F3 \| $sx^{q+1}+x^{2^{i}+1}+x^{q(2^{i}+1)}+dx^{2^{i}q+1}+d^{q}x^{2^{i}+q}$ \| $n=2m$, $q=2^{m}$, ${\rm gcd}(i,m)=1$, $d\in\mathbb{F}_{2^{n}}$, $s\in\mathbb{F}_{2^{n}}\backslash\mathbb{F}_{2^{m}}$, $X^{2^{i}+1}+dX^{2^{i}}+d^{q}X+1$ has no solution $x$ s.t. $x^{q+1}=1$ \| [8, 7] F4 \| $x^{3}+a^{-1}{\rm Tr}^{n}_{1}(a^{3}x^{9})$ \| $a\neq 0$ \| [10] F5 \| $x^{3}+a^{-1}{\rm Tr}^{n}_{3}(a^{3}x^{9}+a^{6}x^{18})$ \| $3~{}\|~{}n$, $a\neq 0$ \| [11] F6 \| $x^{3}+a^{-1}{\rm Tr}^{n}_{3}(a^{6}x^{18}+a^{12}x^{36})$ \| $3~{}\|~{}n$, $a\neq 0$ \| [11] F7-F9 \| $ux^{2^{s}+1}+u^{2^{k}}x^{2^{-k}+2^{k+s}}+vx^{2^{-k}+1}+\omega u^{2^{k}+1}x^{2^{s}+2^{k+s}}$ \| $n=3k$, ${\rm gcd}(k,3)={\rm gcd}(s,3k)=1$, $v$, $\omega\in\mathbb{F}_{2^{k}}$, $v\omega\neq 1$, $3~{}\|~{}(k+s)$, $u$ primitive in $\mathbb{F}^{\ast}_{2^{n}}$ \| [3, 4] F10 \| $cx^{q+1}+dx^{2^{i}+1}+d^{q}x^{q(2^{i}+1)}+\sum^{m-1}_{s=1}\gamma_{s}x^{2^{s}(q+1)}$ \| $n=2m$, $q=2^{m}$, ${\rm gcd}(i,m)=1$, $i$, $m$ odd, $\gamma_{s}\in\mathbb{F}_{q}$, $c\notin\mathbb{F}_{q}$, $d$ not a cube \| [3] F11 \| $(x+x^{q})^{2^{k}+1}+u^{\prime}(ux+u^{q}x^{q})^{(2^{k}+1)2^{i}}+u(x+x^{q})(ux+u^{q}x^{q})$ \| $n=2m$, $m\geq 2$ even, ${\rm gcd}(k,m)=1$, $q=2^{m}$, and $i\geq 2$ even, $u$ primitive in $\mathbb{F}^{\ast}_{2^{n}}$, $u^{\prime}\in\mathbb{F}_{2^{m}}$ not a cube \| [26] F12 \| $u(u^{q}x+ux^{q})(x+x^{q})+(u^{q}x+ux^{q})^{2^{2i}+2^{3i}}+\alpha(u^{q}x+ux^{q})^{2^{2i}}(x+x^{q})^{2^{i}}+\beta(x+x^{q})^{2^{i}+1}$ \| $n=2m$, $q=2^{m}$, ${\rm gcd}(i,m)=1$, $u$ primitive in $\mathbb{F}^{\ast}_{2^{n}}$, $\alpha$, $\beta\in\mathbb{F}_{2^{m}}$, and $X^{2^{i}+1}+\alpha X+\beta$ has no solution in $\mathbb{F}_{2^{m}}$ \| [23] F13 \| $L(x)^{2^{i}}x+L(x)x^{2^{i}}$ \| $n=km$, $m\geq 2$, ${\rm gcd}(n,i)=1$, $L(x)=\sum^{k-1}_{j=0}a_{j}x^{2^{jm}}$ satisfies the conditions in Theorem 6.3 of [6] \| [6] F14 \| $x^{3}+\omega x^{2^{s}+1}+\omega^{2}x^{3q}+x^{(2^{s}+1)q}$ \| $n=2m$, $q=2^{m}$, $m$ odd, $3\nmid m$, $\omega$ primitive in $\mathbb{F}^{\ast}_{2^{2}}$, $s=m-2$, $(m-2)^{-1}~{}{\rm mod}~{}n$ \| [12] F15 \| $a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(b^{3}x^{9})$ \| $n=2m$, $m$ odd, $q=2^{m}$, $a\notin\mathbb{F}_{q}$, $b$ not a cube \| new ## 3 Three infinite families of APN functions We want to find new APN functions of the form (1). In the following two subsections, the functions $F$ and $G$ were chosen very carefully to satisfy the conditions characterized in Lemma 2.1. This will yield a new infinite family of APN quadrinomails, two infinite families of APN hexanomials, and (at least) two sporadic APN functions CCZ-inequivalent to any other known APN functions over $\mathbb{F}_{2^{10}}$. ### 3-A F, G are both of Gold type We need the following two lemmas, which will be used in the proof of Theorem 3.3. ###### Lemma 3.1. Let $n=2m$ for $m$ odd, $q=2^{m}$. Suppose that for some $c\in\mathbb{F}_{2^{n}}$ we have $c^{3}(c+c^{2}+c^{4})^{q}\in\mathbb{F}_{2^{m}}.$ Then $c$ is a cube in $\mathbb{F}_{2^{n}}$. ###### Proof. Since ${\rm gcd}(3,2^{m}-1)=1$, any element of $\mathbb{F}_{2^{m}}$ is a cube. In the following, we assume that $c\notin\mathbb{F}_{2^{m}}$. Noting that $c^{3}(c+c^{2}+c^{4})^{q}=c^{(q+1)+2}+c^{2(q+1)+1}+c^{3(q+1)+q}$, we have $c^{q+1}(c+c^{q})^{2}+c^{2(q+1)}(c+c^{q})+c^{3(q+1)}(c+c^{q})=0$ by the assumption that $c^{3}(c+c^{2}+c^{4})^{q}\in\mathbb{F}_{2^{m}}$. Since $c+c^{q}\neq 0$, we have $c^{q+1}(c+c^{q})+c^{2(q+1)}+c^{3(q+1)}=0$, and hence $c+c^{q}=c^{q+1}+c^{2(q+1)}$. Note that any nonzero element $c$ of $\mathbb{F}_{2^{n}}$ has a unique polar decomposition of the form $c=vk$, where $k^{q+1}=1$, and $v^{q-1}=1$. Substituting $c=vk$ into $c+c^{q}=c^{q+1}+c^{2(q+1)}$, we have $k+k^{-1}=v+v^{3}$. By assumption that $c\notin\mathbb{F}_{2^{m}}$, we have $k\neq 1$. Then according to [21, Theorem 7], we have that $k$ is a cube in $U:=\\{x\in\mathbb{F}_{2^{n}}~{}\|~{}x^{q+1}=1\\}$. Therefore, $c=vk$ is a cube in $\mathbb{F}_{2^{n}}$. ∎ Let $s$ be a positive integer with ${\rm gcd}(s,n)=1$. Let $x\in\mathbb{F}_{2^{n}}$. It is clear that $x+x^{2^{s}}\neq 0$, if and only if $x\neq 0,1$. We have the following lemma. ###### Lemma 3.2. Let $n=2m$ for $m$ odd with ${\rm gcd}(3,m)=1$. Let s be a positive integer such that $3s\equiv 1~{}{\rm mod}~{}n$. Suppose that for some $x\in\mathbb{F}_{2^{n}}\backslash\\{0,1\\}$, we have $\frac{x+x^{2}}{(x+x^{2^{s}})^{2^{2s}-2^{s}+1}}\in\mathbb{F}_{2^{m}}.$ Then $x+x^{2^{s}}$ is a cube. ###### Proof. Let $d=x+x^{2^{s}}$. Then $d\neq 0$, since $x\neq 0,1$, and ${\rm gcd}(s,n)=1$. We can express $x+x^{2}=d+d^{2^{s}}+d^{2^{2s}}$. Then $\frac{x+x^{2}}{(x+x^{2^{s}})^{2^{2s}-2^{s}+1}}=\frac{d+d^{2^{s}}+d^{2^{2s}}}{d^{2^{2s}-2^{s}+1}}=d^{-2^{s}(2^{s}-1)}+d^{-(2^{s}-1)^{2}}+d^{2^{s}-1}=A^{-2^{s}}+A^{-2^{s}+1}+A,$ where $A=d^{2^{s}-1}$. Then the condition of this lemma is equivalent to that $A^{-2^{s}}+A^{-2^{s}+1}+A+1\in\mathbb{F}_{2^{m}},$ which is exaclty $\frac{(A+1)^{2^{s}+1}}{A^{2^{s}}}\in\mathbb{F}_{2^{m}}.$ If $A=1$, i.e., $d^{2^{s}-1}=1$, then $d=1$, and hence $x+x^{2^{s}}=1$ is a cube. In fact, since ${\rm gcd}(2^{s}-1,2^{n}-1)=1$, $g(x)=x^{2^{s}-1}$ is a permutation of $\mathbb{F}_{2^{n}}$. Then by $g(d)=g(1)=1$, we have $d=1$. If $A\neq 1$, then there exists some $\alpha\in\mathbb{F}^{\ast}_{2^{m}}$ such that $A^{2^{s}}=(A+1)^{2^{s}+1}\alpha$. Since $s$ is odd, $3~{}\|~{}2^{s}+1$, we have $A^{2^{s}+1}\alpha$ is a cube, and hence $A^{2^{s}}$ is a cube, that is, $A$ is a cube. However, note that ${\rm gcd}(3,2^{s}-1)=1$, we have that $d$ is a cube, when $A=d^{2^{s}-1}$ is. ∎ In the following theorem, we investigate the APN property of the functions with the form (1) by letting $F(x)=bx^{3}$, and $G(x)=cx^{2^{s}+1}$. This allows us to find a new infinite family of APN quadrinomials $f(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(b^{3}x^{9})$, where $b$ is a non- cube in $\mathbb{F}_{2^{n}}$. ###### Theorem 3.3. Let $n=2m$ with $m\geq 1$ odd, and $q=2^{m}$. Let $a\in\mathbb{F}_{2^{n}}$, and $f_{s}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(cx^{2^{s}+1})$ with $a\notin\mathbb{F}_{q}$, $bc\neq 0$, $s$ odd. Then $f_{s}(x)$ is APN over $\mathbb{F}_{2^{n}}$, if $s,b,c$ satisfy the following i) $s=m-2$, $b$ not a cube, $\frac{c^{4}}{b}\in\mathbb{F}_{2^{m}}$; or ii) $s=(m-2)^{-1}~{}{\rm mod}~{}n$, $b$ not a cube, $\frac{c^{2^{s}-1}}{b^{2^{2s}}}\in\mathbb{F}_{2^{m}}$; or iii) $s=3$, $b$ not a cube, $\frac{c}{b^{3}}\in\mathbb{F}_{2^{m}}$; or iv) ${\rm gcd}(3,m)=1$, $3s\equiv 1{\rm~{}mod~{}}n$, $b$ not a cube, $\frac{c}{b^{2^{2s}-2^{s}+1}}\in\mathbb{F}_{2^{m}}$; or v) $s=m$, $b$ not a cube, $c\notin\mathbb{F}_{2^{m}};$ or vi) $s=m+2$, $b$ not a cube, $bc\in\mathbb{F}_{2^{m}}$; or vii) $s=n-1$, $\frac{c^{2}}{b}\notin\mathbb{F}_{2^{m}}$. ###### Proof. Let $F(x)=bx^{3}$, $G(x)=cx^{2^{s}+1}$. Then $\Delta_{d,F}(x)=d^{3}b(x^{2}+x),~{}{\rm and}~{}\Delta_{d,G}(x)=d^{2^{s}+1}c(x^{2^{s}}+x).$ According to Lemma 2.1, proving $f_{s}(x)$ is an APN function over $\mathbb{F}_{2^{n}}$ is equivalent to showing that the system: $\Delta_{d,F}(x)\in\mathbb{F}_{2^{m}}$, and $\Delta_{d,G}(x)\in\mathbb{F}_{2^{m}}$ can only has trivial solutions $x=0,1$ for any $d\neq 0$. Assume, to the contrary, that $f_{s}(x)$ is not an APN function, when $s,b,c$ satisfy the conditions of one item in this theorem. Then the following system $\displaystyle\begin{cases}d^{3}b(x^{2}+x)=\alpha,&\\\ d^{2^{s}+1}c(x^{2^{s}}+x)=\beta.&\end{cases}$ (7) has a non-trivial solution $x\notin\mathbb{F}_{2}$ for some $d\neq 0$, where $\alpha,~{}\beta\in\mathbb{F}_{2^{m}}$ with $\alpha\neq 0$. Since $m$ is odd, ${\rm gcd}(3,2^{m}-1)=1$, we have that $\alpha=e^{3}$ for some $e\in\mathbb{F}^{\ast}_{2^{n}}.$ Dividing both sides of the first equation in (7) by $e^{3}$, we obtain that $(d/e)^{3}b(x^{2}+x)=1$. Dividing both sides of the second equation in (7) by $e^{2^{s}+1}$, we have $(d/e)^{2^{s}+1}c(x^{2^{s}}+x)=\beta e^{-(2^{s}+1)}$. Since $s$ is odd, we have $3~{}\|~{}2^{s}+1$, and $e^{2^{s}+1}\in\mathbb{F}_{2^{m}}.$ Therefore, the system (7) has a non-trivial solution $x\notin\\{0,1\\}$ if and only if the system $\displaystyle\begin{cases}d^{3}b(x^{2}+x)=1,&\\\ d^{2^{s}+1}c(x^{2^{s}}+x)=\beta.&\end{cases}$ (8) has a solution for some $d\in\mathbb{F}^{\ast}_{2^{n}}$ and $\beta\in\mathbb{F}_{2^{m}}.$ $i)$ $s=m-2$, $b$ is a non-cube in $\mathbb{F}_{2^{n}}$ and $\frac{c^{4}}{b}\in\mathbb{F}^{\ast}_{2^{m}}$. Raising the second equation in (8) to its fourth power, we have $d^{q+4}c^{4}(x^{q}+x^{4})=\beta^{4}$. From the first equation, we have $d^{3}=\frac{1}{b(x^{2}+x)}$. Substituting this relation into the previous equation, we have $d^{q+1}\frac{c^{4}}{b}\frac{x^{q}+x^{4}}{x^{2}+x}\in\mathbb{F}_{2^{m}}$. Since $d^{q+1}\in\mathbb{F}^{\ast}_{2^{m}}$, and $\frac{c^{4}}{b}\in\mathbb{F}^{\ast}_{2^{m}}$ by assumption, we have $\frac{x^{q}+x^{4}}{x+x^{2}}\in\mathbb{F}_{2^{m}}$. By [12, Lemma 1], we have $x+x^{2}$ is a cube in $\mathbb{F}_{2^{n}}$, and hence $b$ is a cube by $d^{3}b(x^{2}+x)=1$, a contradiction to the assumption that $b$ is a non-cube. $ii)$ $s=(m-2)^{-1}~{}{\rm mod}~{}n$, $b$ is a non-cube in $\mathbb{F}^{\ast}_{2^{n}}$ and $\frac{c^{2^{s}-1}}{b^{2^{2s}}}\in\mathbb{F}^{\ast}_{2^{m}}$. It can be seen from the proof of Theorem 2 in [12] that the critical conditions ensuring the APN-ness of this $f_{s}(x)$ are exactly that $b$ is a non-cube in $\mathbb{F}_{2^{n}}$ and $\frac{c^{2^{s}-1}}{b^{2^{2s}}}\in\mathbb{F}^{\ast}_{2^{m}}$. We invite the readers to check it, and we omit the arguments here. $iii)$ $s=3$, $b$ is a non-cube in $\mathbb{F}_{2^{n}}$ and $\frac{c}{b^{3}}\in\mathbb{F}^{\ast}_{2^{m}}.$ It can be seen that in this case (8) becomes $\displaystyle\begin{cases}d^{3}b(x^{2}+x)=1,&\\\ d^{9}c(x^{8}+x)=\beta.&\end{cases}$ Substituting $d^{3}=\frac{1}{b(x+x^{2})}$ into the second equation of the above system, we have $\displaystyle\frac{c}{b^{3}}\cdot\frac{x+x^{8}}{(x+x^{2})^{3}}=\beta,$ which infers that $\frac{x+x^{8}}{(x+x^{2})^{3}}\in\mathbb{F}_{2^{m}}$, since $\frac{c}{b^{3}}\in\mathbb{F}^{\ast}_{2^{m}}$ by assumption. It implies that $(x+x^{2})^{3}(x+x^{8})^{q}\in\mathbb{F}_{2^{m}}$. Denoting $e=x+x^{2}$, we have $x+x^{8}=e+e^{2}+e^{4}$, and hence $e^{3}(e+e^{2}+e^{4})^{q}\in\mathbb{F}_{2^{m}}$. Now, according to Lemma 3.1, $e=x+x^{2}$ is a cube. Then $b$ is a cube by $d^{3}b(x+x^{2})=1$, which contradicts to the assumption that $b$ is a non-cube. $iv)$ ${\rm gcd}(3,m)=1$, $3s\equiv 1{\rm~{}mod~{}}n$, $b$ is a non-cube in $\mathbb{F}_{2^{n}}$ and $\frac{c^{2^{2s}-2^{s}+1}}{b}\in\mathbb{F}^{\ast}_{2^{m}}$. Since ${\rm gcd}(2^{s}-1,2^{n}-1)=2^{{\rm gcd}(s,n)}-1=1$, we have that $x+x^{2^{s}}\neq 0$, when $x\neq 0,1$. Then (8) becomes $\displaystyle\begin{cases}d^{2^{3s}+1}b(x+x^{2})=1,&\\\ d^{2^{s}+1}c(x+x^{2^{s}})=\beta,&\end{cases}$ where $\beta\in\mathbb{F}_{2^{m}}$ with $\beta\neq 0$, since $x+x^{2^{s}}\neq 0$. By the second equation, we have $d^{2^{s}+1}=\frac{\beta}{c(x+x^{2^{s}})}$. Substituting this relation into the first equation, noting that $2^{3s}+1=(2^{s}+1)(2^{2s}-2^{s}+1)$, we have $\displaystyle\frac{b}{c^{2^{2s}-2^{s}+1}}\cdot\frac{x+x^{2}}{(x+x^{2^{s}})^{2^{2s}-2^{s}+1}}\in\mathbb{F}_{2^{m}},$ which infers, since $\frac{b}{c^{2^{2s}-2^{s}+1}}\in\mathbb{F}^{\ast}_{2^{m}}$ by assumption, that $\displaystyle\frac{x+x^{2}}{(x+x^{2^{s}})^{2^{2s}-2^{s}+1}}\in\mathbb{F}^{\ast}_{2^{m}}.$ (9) Now, by the assumption that $b$ is a non-cube in $\mathbb{F}_{2^{n}}$ and $\frac{c^{2^{2s}-2^{s}+1}}{b}\in\mathbb{F}^{\ast}_{2^{m}}$, we have that $c$ is a non-cube. On the other hand, by (9) and Lemma 3.2, we have that $x+x^{2^{s}}$ is a cube, which infers that $c$ is a cube from the second equation $d^{2^{s}+1}c(x+x^{2^{s}})=\beta$ of the above system, a contradiction. $v)$ $s=m$, $b$ is a non-cube in $\mathbb{F}_{2^{n}}$, and $c\notin\mathbb{F}_{2^{m}}.$ It can be seen that (8) becomes $\displaystyle\begin{cases}d^{3}b(x+x^{2})=1,&\\\ d^{2^{m}+1}c(x+x^{2^{m}})=\beta,&\end{cases}$ where $\beta\in\mathbb{F}_{2^{m}}$. Since $c\notin\mathbb{F}_{2^{m}}$, and $d^{2^{m}+1}\in\mathbb{F}^{\ast}_{2^{m}},$ $x+x^{2^{m}}\in\mathbb{F}_{2^{m}}$ for any $d\neq 0$, $x\in\mathbb{F}_{2^{n}}$, by the second equation, we have $\beta$ must equal to zero, which infers that $x\in\mathbb{F}_{2^{m}}$. Then by the fact that any element of $\mathbb{F}_{2^{m}}$ is a cube, we have $d^{3}(x+x^{2})$ is a cube in $\mathbb{F}^{\ast}_{2^{n}}$, which implies that $b$ is a cube in $\mathbb{F}^{\ast}_{2^{n}}$, a contradiction to the assumption that $b$ is a non-cube. $vi)$ $s=m+2$, $b$ is a non-cube in $\mathbb{F}_{2^{n}}$ and $bc\in\mathbb{F}^{\ast}_{2^{m}}$. It can be seen (8) becomes $\displaystyle\begin{cases}d^{3}b(x+x^{2})=1,&\\\ d^{4(q+1)-3}c(x+x^{4q})=\beta,&\end{cases}$ where $\beta\in\mathbb{F}_{2^{m}}$ with $\beta\neq 0$ since $x+x^{4q}\neq 0$ when $x\neq 0,1$. Since $d^{3}b(x+x^{2})=1$, we have $d^{3}=\frac{1}{b(x+x^{2})}$. Substituting this relation into the second equation, we have $\displaystyle d^{4(q+1)}bc(x+x^{2})(x+x^{4q})=\beta.$ Then by the assumption that $bc\in\mathbb{F}^{\ast}_{2^{m}}$, we have $(x+x^{2})(x+x^{4q})\in\mathbb{F}_{2^{m}}$. According to [12, Lemma 1], we have $x+x^{2}\neq 0$ is a cube, which infers that $b$ is a cube by $d^{3}b(x+x^{2})=1$, a contradiction to the assumption that $b$ is a non-cube. $vii)$ $s=n-1$, $\frac{c^{2}}{b}\notin\mathbb{F}_{2^{m}}$. Since ${\rm gcd}(2^{s}-1,2^{n}-1)=2^{{\rm gcd}(s,n)}-1=1$, we have that $x+x^{2^{s}}\neq 0$, if $x\neq 0,1$. It can be seen that (8) becomes $\displaystyle\begin{cases}d^{3}b(x+x^{2})=1,&\\\ d^{2^{s}+1}c(x+x^{2^{s}})=\beta,&\end{cases}$ where $\beta\in\mathbb{F}_{2^{m}}$ with $\beta\neq 0$. Squaring the second equation, we have $d^{3}c^{2}(x+x^{2})=\beta^{2}$. Comparing with the first equation, we have $\frac{c^{2}}{b}=\beta^{2}\in\mathbb{F}_{2^{m}}$, which contradicts with the assumption that $\frac{c^{2}}{b}\notin\mathbb{F}_{2^{m}}.$ ∎ ###### Remark 3.4. Code isomorphism tests described in Section 2 suggest that all the polynomials from the same item of Theorem 3.3 are all CCZ-equivalent; the APN function $x^{3}+\omega x^{2^{s}+1}+\omega^{2}x^{3q}+x^{(2^{s}+1)q}$ discovered in [12] is CCZ-equivalent to all the functions in i), ii), respectively, for $s=m-2$, and $s=(m-2)^{-1}~{}{\rm mod}~{}n$, if ${\rm gcd}(3,m)=1$; the polynomials $f_{s}(x)$ for $s=m+2$ in vi) are equivalent to the ones for $s=m-2$ in i); the polynomials $f_{s}(x)$ for $s=m$ in v) are equivalent to some functions in family F10 from Table II, see also the arguments in Remark 3.7 below; the polynomial $f_{s}(x)$ for $s=n-1$ in vii) is CCZ-equivalent to $x^{3}$. The remaining value of $s=3$ in iii) yields APN quadrinomials $f_{3}(x)$, which are CCZ-inequivalent to any currently known APN function over $\mathbb{F}_{2^{10}}$. By the arguments above that all the polynomials in the same item are all CCZ-equivalent, we only take a representative of iii). We let $f_{3}(x)=\omega{\rm Tr}^{n}_{m}(bx^{3})+\omega^{2}{\rm Tr}^{n}_{m}(b^{3}x^{9})$, where $b$ is a non-cube, $\omega\in\mathbb{F}_{2^{2}}\backslash\mathbb{F}_{2}$. We use this $f_{3}(x)$ to compare against representatives from all the known infinite families including $f_{s}(x)$, $s=m-2$, $(m-2)^{-1}~{}{\rm mod}~{}n$ in i), ii) which are essentially due to Budaghyan, Helleseth, and Kaleyski ([12]). Note that, Budaghyan et al. had presented a table listing all the representatives, except family F12, of all the known CCZ-inequivalent APN functions over $\mathbb{F}_{2^{10}}$, see Table III of [12]. To complete the work of code isomorphism test, we have to find all the representatives of F12 over $\mathbb{F}_{2^{10}}$. Thanks to the nice work [20], we can obtain these representatives. In fact, let $\gamma$ be a primitive element in $\mathbb{F}^{\ast}_{2^{5}}$, according to [20, Theorem 4.5], there are exactly 6 of CCZ-inequivalent Taniguchi APN functions from F12: $i=1$, take $\alpha=1$, $\beta=1,~{}\gamma^{7},~{}\gamma^{11}$; $i=2$, take $\alpha=1$, $\beta=1,~{}\gamma^{3},~{}\gamma^{15}$. The notations $i,~{}\alpha,~{}\beta$ used here are the same as the ones used in family F12 of Table II. ###### Remark 3.5. Let $n=2m$ with $m$ odd, and ${\rm gcd}(m,3)=1$. Let $q=2^{m}$. Let $z$ be a primitive element in $\mathbb{F}^{\ast}_{2^{n}}$, and $\omega=z^{\frac{2^{n}-1}{3}}$. Then $\omega$ is a primitive element in $\mathbb{F}_{2^{2}}$. Let $s=m-2$ or $(m-2)^{-1}{\rm~{}mod}~{}n$. Then $g_{s}(x)=x^{3}+\omega x^{2^{s}+1}+\omega^{2}x^{3q}+x^{(2^{s}+1)q}$ is an APN function ([12]). It can be seen that $g_{s}(x)$ can be covered by our theorem. In fact, noting that $\omega^{2^{s}}=\omega^{2}$ for any odd $s$, $g_{s}(x)=\omega{\rm Tr}^{n}_{m}(\omega^{2}x^{3})+\omega^{2}{\rm Tr}^{n}_{m}(\omega^{2}x^{2^{s}+1})=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(cx^{2^{s}+1})$, where $a=\omega,b=c=\omega^{2}$. It is clear that $a+a^{q}=1\neq 0$, and $b=\omega$ is a non-cube since ${\rm gcd}(m,3)=1$, and $\frac{c^{4}}{b}=1=\frac{c^{2^{t}-1}}{b^{2^{2t}}}$, where $t=(m-2)^{-1}{\rm~{}mod}~{}n$. Then by $i)$, $ii)$ of the above theorem, we have that $g_{s}(x)$ is APN over $\mathbb{F}_{2^{n}}$, for $s=m-2$, and $(m-2)^{-1}{\rm~{}mod}~{}n$, respectively. ###### Remark 3.6. Let $n=2m$ with $m$ odd. Let us investigate the APN property of $f_{m-2}(x)$ further. A pair ($b,c$) is said to satisfy property $\mathbf{P}_{m-2}$, if $b$ is a cube in $\mathbb{F}^{\ast}_{2^{n}}$, and $c\in\mathbb{F}^{\ast}_{2^{n}}$ such that the following assertion holds: For any $x\in\mathbb{F}_{2^{n}}$ with $x\neq 0,1$, $x+x^{2}$ is a non-cube in $\mathbb{F}_{2^{n}}$, if $\frac{c^{4}}{b}\cdot\frac{x^{q}+x^{4}}{x+x^{2}}\in\mathbb{F}_{2^{m}}$. Then $f_{m-2}(x)$ is APN over $\mathbb{F}_{2^{n}}$ for these $b$, $c$. In fact, this assertion can be seen from the proof of $i)$ in the above theorem. With the help of computer, we find that when $m=5$, $7$, there exist a lot of pairs ($b,c$) satisfying $\mathbf{P}_{m-2}$. More precisely, let $m=5$ or $7$, $z$ be a primitive element in $\mathbb{F}^{\ast}_{2^{2m}}$, $j=\frac{(2^{m}+1)}{3}$, and $U=\\{(z^{j})^{i}~{}\|~{}{\rm gcd}(3,i)=1,~{}1\leq i\leq 2^{n}-1\\}$. Then any pair ($b,c$) with $b\neq 0$ a cube, and $\frac{c^{4}}{b}\in U$ satisfies $\mathbf{P}_{m-2}$. However, when $m=9$, $11$, there does not exist such ($b,c$). We therefore propose the following: Open Problem 1. Does there exist infinite odd integer $m\geq 1$ such that $\mathbf{P}_{m-2}$ holds? ###### Remark 3.7. Let $n=2m$ with $m$ odd, and $q=2^{m}$. Let us revisit the function $f_{m}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(cx^{2^{m}+1})$ investigated in $v)$. Replacing $bx^{3}$ by $bx^{2^{i}+1}$, we let $f(x)=a{\rm Tr}^{n}_{m}(bx^{2^{i}+1})+a^{q}{\rm Tr}^{n}_{m}(cx^{2^{m}+1})$, where $i$ is an odd positive integer with ${\rm gcd}(i,m)=1$. With similar arguments, by $3~{}\|~{}2^{i}+1$ and ${\rm gcd}(i,m)=1$, we can obtain that $f(x)$ is APN, if $b$ is a non-cube in $\mathbb{F}_{2^{n}}$, and $c\notin\mathbb{F}_{2^{m}}$. Note that $\frac{1}{a}f(x)=dx^{2^{m}+1}+{\rm Tr}^{n}_{m}(bx^{2^{i}+1})$, where $d=a^{q-1}(c+c^{q})$ can be chosen as any element in $\mathbb{F}_{2^{n}}\backslash\mathbb{F}_{2^{m}}$, since $a,~{}c\notin\mathbb{F}_{q}$, we have that $f(x)$ in fact are exactly the functions in family F10 up to EA-equivalence. This observation suggests that it is worthy to finding APN functions with the following form: $\displaystyle f_{i,s}(x)=a{\rm Tr}^{n}_{m}(bx^{2^{i}+1})+a^{q}{\rm Tr}^{n}_{m}(cx^{2^{s}+1}),~{}\text{where~{}}a\in\mathbb{F}_{2^{n}}~{}such~{}that~{}a+a^{q}\neq 0,~{}n=2m~{}\text{is a positive integer}.$ (10) ###### Remark 3.8. It is noted that there does not exist elements satisfying the conditions in $iv)$. However, we decide to preserve this item, because we feel that the technique used in the proof may provide some insights for the constructions of APN functions. ### 3-B F, G are both quadratic binomials Let us consider more general case. Let $n=2m$ with $m$ a positive integer. Let $\displaystyle h_{i,s,b,c,g,e}(x)=a{\rm Tr}^{n}_{m}(bx^{2^{i}+1}+cx^{2^{i+m}+1})+a^{q}{\rm Tr}^{n}_{m}(gx^{2^{s}+1}+ex^{2^{s+m}+1}),$ (11) where $a\in\mathbb{F}_{2^{n}}$ such that $a+a^{q}\neq 0$, $b,c,g,e\in\mathbb{F}_{2^{n}}$. In this subsection, we want to find APN functions of the form (11). We remark first that the APN polynomials considered in family F3 can be covered by $h_{i,s,b,c,g,e}(x)$. In fact, let $i=m$, $b\notin\mathbb{F}_{2^{m}}$, $c=0$, $g=1$, then (11) becomes $a^{q-1}(b+b^{q})x^{q+1}+x^{2^{s}+1}+x^{(2^{s}+1)q}+ex^{2^{s}q+1}+e^{q}x^{2^{s}+q}$, which are exactly the functions in F3, since $a^{q-1}(b+b^{q})$ can be choosen as any elements in $\mathbb{F}_{2^{n}}\backslash\mathbb{F}_{2^{m}}$. We can find two infinite families of APN functions with the above form (11), and computationally prove that they are CCZ-inequivalent to any APN power functions over $\mathbb{F}_{2^{10}}$, and we can find a new sporadic instance of APN functions over $\mathbb{F}_{2^{10}}$. ###### Theorem 3.9. [24] Let $n=2m$, and $a\in\mathbb{F}^{\ast}_{2^{n}}$. Let $t_{1}$ be one solution in $\mathbb{F}_{2^{n}}$ of $t^{2}+at+1=0$ (if ${\rm Tr}^{n}_{1}\Big{(}\frac{1}{a^{2}}\Big{)}=0$). Let $f(x)=x^{3}+x+a$, then $\bullet$ $f$ has no zeros in $\mathbb{F}_{2^{n}}$ if and only if ${\rm Tr}^{n}_{1}\Big{(}\frac{1}{a^{2}}\Big{)}=0$, and $t_{1}$ is not a cube in $\mathbb{F}_{2^{n}}$. $\bullet$ $f$ has three zeros in $\mathbb{F}_{2^{n}}$ if and only if ${\rm Tr}^{n}_{1}\Big{(}\frac{1}{a^{2}}\Big{)}=0$, and $t_{1}$ is a cube in $\mathbb{F}_{2^{n}}$. We need the following theorem, which will be used for generating APN functions (see Corollary 1). Let $n=2m$ with $m$ being an odd positive integer, and $q=2^{m}$. Let $x\in\mathbb{F}_{2^{n}}$ with $x\neq 0,1$. Then fix the following notations for this given element $x$. $\displaystyle r:=x^{q+1};~{}h:=x+x^{q};~{}c:=x+x^{2};$ $\displaystyle D:=A(A^{q+1}+B^{q+1});~{}H:=A^{2}(A^{q}B^{3}+AB^{3q}+B^{2+2q}),$ where $A,B$ are some elements determined by $x$. By a routine work, we have that $\displaystyle h+h^{2}=c+c^{q}.$ The following result can not only give rise to APN functions of the form (11) but can also yield Budaghyan-Carlet APN hexanomials (family F3), and hence it has its own importance and we state it as a theorem. The proof can be seen in the appendix. ###### Theorem 3.10. Let $n=2m$ with $m$ being an odd positive integer. Let $x$ be any given element in $\mathbb{F}_{2^{n}}\backslash\\{0,1\\}$. Use the notations given as above. Let $\displaystyle f(y)=Ay^{3}+By^{2}+B^{q}y+A^{q}=0.$ (12) Then equation (12) has no solutions in $\mathbb{F}_{2^{n}}$, if A, B, c satisfy 1) $A=c^{2-2q}(h+c+c^{2})$, $B=c+c^{2}$, and $c=x+x^{2}$ is a non-cube in $\mathbb{F}_{2^{n}}$; or 2) $A=\frac{h+c+c^{2}}{c^{q}}$, $B=1+c$, and $c=x+x^{2}$ is a non-cube in $\mathbb{F}_{2^{n}}$. ###### Remark 3.11. Let $n=2m$, and $q=2^{m}$. Recall first that the condition needed in family F3 is that $\displaystyle y^{2^{i}+1}+dy^{2^{i}}+d^{q}y+1=0$ (13) has no solutions in $U=\\{x\in\mathbb{F}_{2^{n}}~{}\|~{}x^{q+1}=1\\}$. Here $i$ is a positive integer with ${\rm gcd}(i,m)=1$. When $i=1$, this condition is exactly that $y^{3}+dy^{2}+d^{q}y+1=0$ has no solutions in $U$. With the same notations as in Theorem 3.10. Let $A$ be the elements given in 1) or 2). Let $\Gamma=\\{A\in\mathbb{F}^{\ast}_{2^{m}}~{}\|~{}x\in\mathbb{F}_{2^{n}}\backslash\mathbb{F}_{2^{m}},~{}c=x+x^{2}~{}\text{not cube}\\}$. Numerical experiments suggest that $\Gamma$ is always nonempty for any odd $m$. This can yield Budaghyan-Carlet APN functions in family F3. In fact, let $A\in\Gamma$, then (12) becomes $\displaystyle y^{3}+dy^{2}+d^{q}y+1=0,~{}d=\frac{B}{A}.$ According to Theorem 3.10, the above equation has no solutions in $\mathbb{F}_{2^{n}}$. Therefore, this theorem can be used to yield APN functions in family F3. It is noted that the existence of the coefficients $d$ such that the equation (13) has no solutions in $U$ (or $\mathbb{F}_{2^{n}}$) for a given positive integer $i$ had also been studied in [2, 5].We expect that $\Gamma$ does indeed empty for any odd positive integer $m$, and hence propose the following: Open problem 2. Let $n=2m$ with $m$ odd. Show that $\Gamma$ is always nonempty. It is also interesting and important to consider the following question. Open problem 3. Let $n=2m$ with $m$ a positive integer, $q=2^{m}$. Let $i$ be a positive with ${\rm gcd}(m,i)=1$. Find more exponents $i$, and elements $A,B$ such that the following equation has no solutions in $\mathbb{F}_{2^{n}}$. $\displaystyle Ay^{2^{i}+1}+By^{2^{i}}+B^{q}y+A^{q}=0.$ In the following, we investigate the APN property of the functions with the form (11) by letting $i=1,c=0$. We does indeed find two infinite families of APN functions. But, astonishingly enough, the function obtained happened to be CCZ-equivalent to some functions in family F12 with a completely different from that of Taniguchi. ###### Corollary 1. Let $n=2m$ be a positive integer with $m$ odd, and $q=2^{m}$. Let $h_{s}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(gx^{2^{s}+1}+ex^{2^{s+m}+1})$ with $a\notin\mathbb{F}_{q}$, $bge\neq 0$. Then $h_{s}(x)$ is APN over $\mathbb{F}_{2^{n}}$, if $s,b,g,e$ satisfy $\displaystyle 1)$ $\displaystyle~{}~{}~{}s=2,~{}b{\rm~{}is~{}not~{}a~{}cube},~{}g=1,~{}e=\frac{1}{b^{2q-2}};{~{}~{}\rm or}$ $\displaystyle 2)$ $\displaystyle~{}~{}~{}s=2,~{}b{\rm~{}is~{}not~{}a~{}cube},~{}g=e=b.$ ###### Proof. 1) $s=2$, $b$ is not a cube, $g=1$, $e=\frac{1}{b^{2q-2}}$. Let $F(x)=bx^{3}$, $G(x)=x^{2^{s}+1}+ex^{2^{s+m}+1}$. Then we have $\displaystyle\Delta_{d,F}=d^{3}b(x+x^{2}),~{}\Delta_{d,G}=d^{2^{s}+1}(x+x^{2^{s}})+d^{2^{s+m}+1}e(x+x^{2^{s+m}}).$ According to Lemma 2.1, we have that $h_{s}(x)$ is APN if the following system $\displaystyle\begin{cases}d^{3}b(x+x^{2})=\alpha&\\\ d^{2^{s}+1}(x+x^{2^{s}})+d^{2^{s+m}+1}e(x+x^{2^{s+m}})=\beta&\end{cases}$ only has $x=0,1$ as its solutions for any $d\neq 0$, where $\alpha$, $\beta\in\mathbb{F}_{2^{m}}.$ Assume, to the contrary, that there exists some $d\neq 0$, $x\neq 0,1$ such that the above system holds. Now let $s=2$, $b$ is a non-cube, $e=\frac{1}{b^{2q-2}}$. Then $\alpha\neq 0$, $b=\frac{\alpha}{d^{3}(x+x^{2})}$, $e=b^{-(2q-2)}=d^{6q-6}(x+x^{2})^{2q-2}$ (note that $\alpha^{2q-2}=1$). Substituting it into the second equation of the above system, we have $\displaystyle d^{5}(x+x^{4})+d^{10q-5}(x+x^{2})^{2q-2}(x+x^{4q})=\beta,$ which is equivalent to $\displaystyle d^{5}(x+x^{4})+d^{10q-5}(x+x^{2})^{2q-2}(x+x^{4q})+\Big{(}d^{5}(x+x^{4})+d^{10q-5}(x+x^{2})^{2q-2}(x+x^{4q})\Big{)}^{q}=0.$ (14) Let $u=d^{5}$. Then the above equation becomes $\displaystyle u(x+x^{4})+u^{2q-1}(x+x^{2})^{2q-2}(x+x^{4q})+\Big{(}u(x+x^{4})+u^{2q-1}(x+x^{2})^{2q-2}(x+x^{4q})\Big{)}^{q}=0.$ (15) Note that any nonzero element $u$ of $\mathbb{F}_{2^{n}}$ has a unique polar decomposition of the form $u=vk$, where $v^{q+1}=1$, and $k^{q-1}=1$. Substituting $u=vk$ into (15), then (15) can be reduced as $\displaystyle v(x+x^{4})+v^{2q-1}(x+x^{2})^{2q-2}(x+x^{4q})+\Big{(}v(x+x^{4})+v^{2q-1}(x+x^{2})^{2q-2}(x+x^{4q})\Big{)}^{q}=0.$ Multiplying both sides by $v^{3}$ of the above equation, by the fact that $v^{q}=v^{-1}$, we have $\displaystyle Ay^{3}+By^{2}+B^{q}y+A^{q}=0,$ where $y=v^{2}\in\mathbb{F}_{2^{n}}$, and $A$, $B$ are given in 1) of Theorem 3.10. Now, according to 1) of Theorem 3.10, we obtian that the element $x+x^{2}$ is a cube, and hence $b$ is a cube from the first equation $d^{3}b(x+x^{2})=\alpha$ of the system, since $\alpha\in\mathbb{F}^{\ast}_{2^{m}}$ is a cube. This derives a contradiction to the assumption that $b$ is a non-cube. 2) $s=2$, $b$ is not a cube, $g=e=b$. Let $F(x)=bx^{3}$ and $G(x)=bx^{5}+bx^{4q+1}$. We have $\displaystyle\Delta_{d,F}(x)=d^{3}b(x+x^{2})\hskip 5.69046pt{\rm and}\hskip 5.69046pt\Delta_{d,G}(x)=d^{5}b(x+x^{4})+d^{4q+1}b(x+x^{4q}).$ By Lemma 2.1, $h_{s}(x)$ is APN if and only if the following system $\displaystyle\begin{cases}d^{3}b(x+x^{2})=\alpha\\\ d^{5}b(x+x^{4})+d^{4q+1}b(x+x^{4q})=\beta\end{cases}$ only has trivial solutions $x\in\mathbb{F}_{2}$ for any $d\in\mathbb{F}_{2^{n}}^{}$ and $\alpha,\beta\in\mathbb{F}_{2^{m}}$. Assume now that there exist some $d\in\mathbb{F}_{2^{n}}^{}$, $\alpha\in\mathbb{F}_{2^{m}}$, $\beta\in\mathbb{F}_{2^{m}}$ such that the system has non-trivial solutions $x\in\mathbb{F}_{2^{n}}\backslash\mathbb{F}_{2}$. Then $\alpha\neq 0$. By the first equation, we have $b=\frac{\alpha}{d^{3}(x+x^{2})}$. Substituting this relation into the second equation, we have $\displaystyle\frac{d^{2}(x+x^{4})}{x+x^{2}}+\frac{d^{4q-2}(x+x^{4q})}{x+x^{2}}=\frac{\beta}{\alpha},$ which implies that $\displaystyle\frac{d^{2}(x+x^{4})}{x+x^{2}}+\frac{d^{4q-2}(x+x^{4q})}{x+x^{2}}+\bigg{(}\frac{d^{2}(x+x^{4})}{x+x^{2}}+\frac{d^{4q-2}(x+x^{4q})}{x+x^{2}}\bigg{)}^{q}=0,$ since $\alpha,~{}\beta\in\mathbb{F}_{2^{m}}$. Let $\mu=d^{2}$. We have $\displaystyle\frac{\mu(x+x^{4})}{x+x^{2}}+\frac{\mu^{2q-1}(x+x^{4q})}{x+x^{2}}+\bigg{(}\frac{\mu(x+x^{4})}{x+x^{2}}+\frac{\mu^{2q-1}(x+x^{4q})}{x+x^{2}}\bigg{)}^{q}=0.$ (16) To complete the proof, it suffices to show that $x+x^{2}$ is a cube of $\mathbb{F}_{2^{n}}$, which will derive that $b$ is a cube from the first equation of the above system and this will yield a contradiction to the assumption that $b$ is a non-cube. Let $\mu=\nu k$, where $\nu^{q+1}=1$ and $k\in\mathbb{F}_{2^{m}}^{}$, and substitute $\mu=\nu k$ into (16), we have $\displaystyle\frac{\nu(x+x^{4})}{x+x^{2}}+\frac{\nu^{2q-1}(x+x^{4q})}{x+x^{2}}+\bigg{(}\frac{\nu(x+x^{4})}{x+x^{2}}+\frac{\nu^{2q-1}(x+x^{4q})}{x+x^{2}}\bigg{)}^{q}=0.$ Multiplying both sides of the above equation by $\nu^{3}$, we have $\displaystyle Ay^{3}+By^{2}+B^{q}y+A^{q}=0,$ where $y=\nu^{2}$, $A=\Big{(}\frac{x+x^{4q}}{x+x^{2}}\Big{)}^{q}$ and $B=\frac{x+x^{4}}{x+x^{2}}=1+x+x^{2}$. According to 2) of Theorem 3.10, $x+x^{2}$ is a cube in $\mathbb{F}_{2^{n}}$, otherwise, the above equation has no solutions in $\mathbb{F}_{2^{n}}$. ∎ Example 1. Besides the two infinite classes of APN functions presented in Corollary 1, we can also find a new instance of APN functions over $\mathbb{F}_{2^{10}}$ CCZ-inequivalent to any other known APN functions. Let $z$ be a primitive element in $\mathbb{F}^{\ast}_{2^{10}}$. Then $\displaystyle h_{s}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(gx^{5}+ex^{4q+1})$ is an APN function over $\mathbb{F}_{2^{10}}$, where $b=1$, $g=z$, $e=z^{369}$. TABLE III: All Known CCZ-inequivalent APN functions over $\mathbb{F}_{2^{10}}$, $q=2^{5}$ Function \| Conditions \| Family ---\|---\|--- $x^{2^{i}+1}$ \| $i=1,3$ \| Gold $x^{57}$ \| $-$ \| Kasami $x^{339}$ \| $-$ \| Dobbertin $x^{6}+x^{33}+\alpha^{31}x^{192}$ \| $\alpha$ primitive in $\mathbb{F}^{\ast}_{2^{10}}$ \| F3 $x^{33}+x^{72}+\alpha^{31}x^{258}$ \| $\alpha$ primitive in $\mathbb{F}^{\ast}_{2^{10}}$ \| F3 $x^{3}+{\rm Tr}^{10}_{1}(x^{9})$ \| $-$ \| F4 $x^{3}+\alpha^{-1}{\rm Tr}^{10}_{1}(\alpha^{3}x^{9})$ \| $\alpha$ primitive in $\mathbb{F}^{\ast}_{2^{10}}$ \| F4 \| $u(u^{q}x+ux^{q})(x+x^{q})+$ --- $(u^{q}x+ux^{q})^{2^{2i}+2^{3i}}+$ $\alpha(u^{q}x+ux^{q})^{2^{2i}}(x+x^{q})^{2^{i}}+$ $\beta(x+x^{q})^{2^{i}+1}$ \| $u$ primitive in $\mathbb{F}^{\ast}_{2^{10}}$, --- $z$ primitive in $\mathbb{F}^{\ast}_{2^{5}}$, $i=1$, $\alpha=1$, $\beta=1,z^{7},z^{11}$; $i=2$, $\alpha=1$, $\beta=1,z^{3},z^{15}$ F12 $B(x)=x^{3}+\alpha^{341}x^{36}$ \| $-$ \| sporadic, see [17] \| $x^{3}+\omega x^{2^{s}+1}+$$\omega^{2}x^{3q}+x^{(2^{s}+1)q}$ --- \| $s=3,7,$ $\omega$ primitive in $\mathbb{F}^{\ast}_{2^{2}}$ --- F14 \| $\alpha{\rm Tr}^{n}_{m}(\alpha x^{3})+\alpha^{q}{\rm Tr}^{n}_{m}(\alpha^{3}x^{9})$ --- \| $\alpha$ primitive in $\mathbb{F}^{\ast}_{2^{10}}$ --- F15 \| $\alpha{\rm Tr}^{n}_{m}(x^{3})+\alpha^{q}{\rm Tr}^{n}_{m}(\alpha^{11}x^{9})$ --- \| $\alpha$ primitive in $\mathbb{F}^{\ast}_{2^{10}}$ --- \| sporadic, see --- Remark 3.6 \| $\alpha{\rm Tr}^{n}_{m}(x^{3})+\alpha^{q}{\rm Tr}^{n}_{m}(\alpha x^{5}+\alpha^{369}x^{4q+1})$ --- \| $\alpha$ primitive in $\mathbb{F}^{\ast}_{2^{10}}$ --- \| sporadic, see --- Example 1 ## 4 Conclusions Let $n=2m$, and $q=2^{m}$. We studied a class of quadratic functions with the form $f(x)=a{\rm Tr}^{n}_{m}(F(x))+a^{q}{\rm Tr}^{n}_{m}(G(x))$, where $F$, $G$ are quadratic functions. We found a new infinite family of APN quadrinomials over $\mathbb{F}_{2^{n}}$, $a\in\mathbb{F}_{2^{n}}$, $n=2m$ with $m$ odd as follows. $\displaystyle f_{1}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(b^{3}x^{9}),~{}b\text{~{}not~{}a~{}cube},~{}a\notin\mathbb{F}_{q}.$ We generalized the two infinite families of APN functions obtained in [12] to a broader condition on $m$, that is, the assumption that ${\rm gcd}(3,m)=1$ needed in [12] can be removed, up to CCZ-equivalence. We also found two infinite families of APN functions over $\mathbb{F}_{2^{2m}}$ for odd $m$, which turned out to be in family F12, that is, the the Taniguchi APN functions when $m=5$, as follows. $\displaystyle f_{2}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(x^{5}+\frac{1}{b^{2q-2}}x^{4q+1}),~{}b\text{~{}not~{}a~{}cube},~{}a\in\mathbb{F}_{2^{n}}\backslash\mathbb{F}_{2^{m}},$ and $\displaystyle f_{3}(x)=a{\rm Tr}^{n}_{m}(bx^{3})+a^{q}{\rm Tr}^{n}_{m}(bx^{5}+bx^{4q+1}),~{}b\text{~{}not~{}a~{}cube},~{}a\in\mathbb{F}_{2^{n}}\backslash\mathbb{F}_{2^{m}}.$ Code isomorphism tests showed that $f_{2}$ and $f_{3}$ are CCZ-inequivalent to each other over $\mathbb{F}_{2^{10}}$. We found two new instances of APN functions over $\mathbb{F}_{2^{10}}$. We also proposed three open problems, and we cordially invite the readers to attack these open problems. ## References [1] T. Beth., C. 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In the following, we assume that $c$ is a non-cube in $\mathbb{F}_{2^{n}}$. Note that $A\neq 0$. In fact, if $A=0$, then $h+c+c^{2}=x^{4}+x^{q}=0$, which implies that $x\in\mathbb{F}_{2^{n}}\cap\mathbb{F}_{2^{m-2}}=\mathbb{F}_{2}$, since $m$ is odd, and ${\rm gcd}(n,m-2)=1$, a contradiction to the assumption that $x\neq 0,1$. Let $y:=y+\frac{B}{A}$. Then equation $(\ref{key-1})$ becomes $\displaystyle y^{3}+\frac{AB^{q}+B^{2}}{A^{2}}y+\frac{A^{q+1}+B^{q+1}}{A^{2}}=0.$ Let $y=Ez$, where $E$ satisfies that $E^{2}=\frac{AB^{q}+B^{2}}{A^{2}}$. Note that $E\neq 0$. In fact, this would imply that $AB^{q}=B^{2}$, and hence $A^{q}B=B^{2q}$, $A^{q+1}B^{q+1}=B^{2(q+1)}$. However, by the fact that $B\neq 0$ (if $B=0$, then $c=0,1$, a contradiction to the assumption that $c$ is a non-cube), we have $A^{q+1}+B^{q+1}=0$, which implies that $(x+x^{q})^{5}=0$, i.e., $x\in\mathbb{F}_{q}$, and then $c=x+x^{2}\in\mathbb{F}_{2^{m}}$ is a cube in $\mathbb{F}_{2^{n}}$, since every element in $\mathbb{F}_{2^{m}}$ is a cube by the fact that ${\rm gcd}(3,2^{m}-1)=1$ (since $m$ is odd), a contradiction. Then the above equation becomes $\displaystyle z^{3}+z+a=0,$ (17) where $a\neq 0$ satisfies that $\displaystyle a=\frac{A^{q+1}+B^{q+1}}{A^{2}E^{3}}.$ It can be checked that $\displaystyle\frac{1}{a^{2}}=\frac{(AB^{q}+B^{2})^{3}}{A^{2}(A^{q+1}+B^{q+1})^{2}}.$ (18) It is clear that equation (12) has no solutions in $\mathbb{F}_{2^{n}}$ if and only if (17) has no solutions. To complete the proof, according to Theorem 3.9, we have to show that ${\rm Tr}^{n}_{1}\Big{(}\frac{1}{a^{2}}\Big{)}=0$, and $t_{1}$ is a non-cube in $\mathbb{F}_{2^{n}}$, where $t_{1}$ is one solution in $\mathbb{F}_{2^{n}}$ of $t^{2}+at+1=0$. Claim 1. ${\rm Tr}^{n}_{1}\Big{(}\frac{1}{a^{2}}\Big{)}=0$. In fact, we have $\displaystyle\frac{1}{a^{2}}=\frac{(AB^{q}+B^{2})^{3}}{A^{2}(A^{q+1}+B^{q+1})^{2}}=\frac{B^{3}+M}{A(A^{q+1}+B^{q+1})}+\Bigg{(}\frac{B^{3}+M}{A(A^{q+1}+B^{q+1})}\Bigg{)}^{2},$ (19) where $M$ is one solution of the following equation $\displaystyle M^{2}+DM+H=0.$ (20) Recall the notations that $D=A(A^{q+1}+B^{q+1})$, $H=A^{2}(A^{q}B^{3}+AB^{3q}+B^{2+2q})$, $A=c^{2-2q}(h+c+c^{2})$, $B=c+c^{2}$, $c=x+x^{2}$. We need only to show that the above equation in $M$ has solutions in $\mathbb{F}_{2^{n}}$, i.e., ${\rm Tr}^{n}_{1}\Big{(}{\frac{H}{D^{2}}}\Big{)}=0$. This can be seen from the following fact. $\displaystyle\frac{H}{D^{2}}=\frac{A^{q}B^{3}+AB^{3q}+B^{2+2q}}{(A^{q+1}+B^{q+1})^{2}}$ is an element in $\mathbb{F}_{2^{m}}$, since $A^{q}B^{3}+AB^{3q}={\rm Tr}^{n}_{m}(A^{q}B^{3})$, $A^{q+1}$, $B^{q+1}\in\mathbb{F}_{2^{m}}$. Next, we need to find one solution $t_{1}$ in $\mathbb{F}_{2^{n}}$ of $t^{2}+at+1=0$, and show that $t_{1}$ is a non-cube. It is clear that $t_{1}$ can be represented as $av$, where $v=\frac{B^{3}+M}{A(A^{q+1}+B^{q+1})}$, since $\frac{1}{a^{2}}=v+v^{2}$ according to (19). Note that $t_{1}=av$ satisfies that $\displaystyle t^{2}_{1}=a^{2}v^{2}=\frac{(B^{3}+M)^{2}}{(AB^{q}+B^{2})^{3}}.$ Therefore, to show $t_{1}$ is a non-cube in $\mathbb{F}_{2^{n}}$, we have to show that $B^{3}+M$ is a non-cube. Claim 2. $B^{3}+M$ is a non-cube in $\mathbb{F}_{2^{n}}$. Our strategy is to find the explicit expression of $M$, and then show that $B^{3}+M$ is a non-cube. To this end, we have to revisit equation (20), and explore more information on the element $\frac{H}{D^{2}}$ (it is in $\mathbb{F}_{2^{m}}$). Very fortunately, we find that ${\rm Tr}^{m}_{1}\Big{(}\frac{H}{D^{2}}\Big{)}=0.$ In fact, recall the notations that $h=x+x^{q}$, and $r=x^{q+1}$, we find (with computer assistance) that (a surprise) $\displaystyle\frac{H}{D^{2}}=u+u^{2},$ (21) where $\displaystyle u=\frac{h^{2}(r+r^{2})+r+r^{4}+hr^{2}}{h^{5}}.$ Then $M$ can be chosen as $Du$ (this is because it suffices to find one solution of $M^{2}+DM+H=0$). We find that $\displaystyle M=Du=A(A^{q+1}+B^{q+1})u$ $\displaystyle=$ $\displaystyle c^{2-2q}(h+c+c^{2})h^{5}\cdot\frac{h^{2}(r+r^{2})+r+r^{4}+hr^{2}}{h^{5}}$ $\displaystyle=$ $\displaystyle\frac{c^{2}(h+c+c^{2})({h^{2}(r+r^{2})+r+r^{4}+hr^{2})}}{c^{2q}}.$ Then, recall the notation that $B=c+c^{2}$, we can obtain the expression of $B^{3}+M$ as follows. $\displaystyle\begin{aligned} B^{3}+M&=&\frac{h(c^{2q+4}+c^{q+5}+c^{q+4})+c^{2q+4}+c^{q+5}}{c^{2q}}\\\ &=&\frac{c^{2}\Big{(}h(c^{2q+4}+c^{q+5}+c^{q+4})+c^{2q+4}+c^{q+5}\Big{)}}{c^{2+2q}}.\end{aligned}$ (22) The above expression can be deduced from $\displaystyle h+h^{2}=c+c^{q},~{}c^{q+1}=r+r^{2}+hr.$ Note that $h,~{}c^{2+2q}\in\mathbb{F}^{\ast}_{2^{m}}$ is a cube, it suffices to show that $\displaystyle hc^{2}\Big{(}h(c^{2q+4}+c^{q+5}+c^{q+4})+c^{2q+4}+c^{q+5}\Big{)}$ is a non-cube. By the fact that $h+h^{2}=c+c^{q}$, we have $\displaystyle hc^{2}\Big{(}h(c^{2q+4}+c^{q+5}+c^{q+4})+c^{2q+4}+c^{q+5}\Big{)}=c^{5}c^{q+1}((c+c^{q})^{2}+h^{2}).$ Since $c^{q+1}$, $c+c^{q}$, $h\in\mathbb{F}^{\ast}_{2^{m}}$ are all cubes in $\mathbb{F}_{2^{n}}$, we have that the above element is a non-cube, when $c$ is a non-cube. ∎ ### 5-B Proof of 2) in Theorem 3.10 ###### Proof. The proof is similar to that of 1) in Theorem 3.10. Recall the following notations: $r=x^{q+1};h=x+x^{q};c=x+x^{2};$$A=\frac{h+c+c^{2}}{c^{q}};B=1+c,$ from which we can obtain that $h+h^{2}=c+c^{q}$ and $A^{q+1}+B^{q+1}=\frac{(x+x^{q})^{5}}{(x+x^{2})^{q+1}}=\frac{h^{5}}{c^{q+1}}$. Note that $A\neq 0$, otherwise, we have $x+x^{4q}=0$ that means that $x\in\mathbb{F}_{2^{n}}\cap\mathbb{F}_{4q}=\mathbb{F}_{2}$, since $\gcd(m+2,n)=1$. Then setting $y:=y+\frac{B}{A}$, this can transform (12) into $\displaystyle y^{3}+\frac{AB^{q}+B^{2}}{A^{2}}y+\frac{A^{q+1}+B^{q+1}}{A^{2}}=0.$ (23) Observe that $B\neq 0$ (otherwise $c=1$ is a cube) and $AB^{q}+B^{2}\neq 0$, otherwise, we have $A^{q+1}+B^{q+1}=0$, that is, $h=0$, which implies that $c\in\mathbb{F}_{q}$ contracting to the assumption that $c$ is a non-cube, since $\gcd(3,2^{m}-1)=1$ for any odd $m$. Thus we can transform the equation (23) into $\displaystyle z^{3}+z+a=0$ (24) by setting $y=Ez$, where $a,E\in\mathbb{F}_{2^{n}}^{*}$ such that $\displaystyle E^{2}=\frac{AB^{q}+B^{2}}{A^{2}}\hskip 5.69046pt{\rm and}\hskip 5.69046pta^{2}=\frac{A^{2}(A^{q+1}+B^{q+1})^{2}}{(AB^{q}+B^{2})^{3}}.$ We need now to prove that equation (24) has no solutions in $\mathbb{F}_{2^{n}}$. According to Theorem 3.9, we have to show that ${\rm Tr}_{1}^{n}\Big{(}\frac{1}{a^{2}}\Big{)}=0$ and the solutions in $\mathbb{F}_{2^{n}}$ of equation $t^{2}+at+1=0$ are not cubes of $\mathbb{F}_{2^{n}}$. Firstly, we prove that ${\rm Tr}_{1}^{n}\Big{(}\frac{1}{a^{2}}\Big{)}=0$. Note that $\frac{1}{a^{2}}$ can be written as $\displaystyle\frac{1}{a^{2}}=\frac{B^{3}+M}{A(A^{q+1}+B^{q+1})}+\bigg{(}\frac{B^{3}+M}{A(A^{q+1}+B^{q+1})}\bigg{)}^{2},$ (25) where $M$ is a solution of $\displaystyle M^{2}+DM+H=0,$ (26) where $D=A(A^{q+1}+B^{q+1})$ and $H=A^{2}(AB^{3q}+A^{q}B^{3}+B^{2(q+1)})$. Then we transform the problem into showing that equation (26) has solutions in $\mathbb{F}_{2^{n}}$, which is equivalent to ${\rm Tr}_{1}^{n}\Big{(}\frac{H}{D^{2}}\Big{)}=0$. Indeed, it can be seen that $\displaystyle\frac{H}{D^{2}}=\frac{AB^{3q}+A^{q}B^{3}+B^{2(q+1)}}{(A^{q+1}+B^{q+1})^{2}}=\frac{{\rm Tr}_{m}^{n}(AB^{3q})+B^{2(q+1)}}{(A^{q+1}+B^{q+1})^{2}},$ which is clearly in $\mathbb{F}_{q}$. Thus, ${\rm Tr}_{1}^{n}\Big{(}\frac{H}{D^{2}}\Big{)}=0$. Then, we show that the solutions of $t^{2}+at+1=0$ are not cubes in $\mathbb{F}_{2^{n}}$. Assume that $t_{1}$ is a solution of $t^{2}+at+1=0$. Then by (25), it can be represented by $t_{1}=a\nu$, where $\nu=\frac{B^{3}+M}{A(A^{q+1}+B^{q+1})}$, and thus $\displaystyle t_{1}^{2}=a^{2}\nu^{2}=\frac{(B^{3}+M)^{2}}{(AB^{q}+B^{2})^{3}}.$ Therefore, to show $t_{1}$ is not a cube, it suffices to show $(B^{3}+M)^{2}$ and thus $B^{3}+M$ is not a cube of $\mathbb{F}_{2^{n}}$. In the following, we show this fact by giving the explicit expression of $M$ by revisiting (26) again. By the above discussion, we have obtained that $\frac{H}{D^{2}}\in\mathbb{F}_{q}$. We further want to show that ${\rm Tr}_{1}^{m}\Big{(}\frac{H}{D^{2}}\Big{)}=0$, which is equivalent to showing $\displaystyle\frac{H}{D^{2}}=\mu+\mu^{2}$ (27) for some $\mu\in\mathbb{F}_{2^{m}}$. Recall that $A=\frac{h+c+c^{2}}{c^{q}}$, $B=1+c$ and $A^{q+1}+B^{q+1}=\frac{h^{5}}{c^{q+1}}$, we have $\displaystyle A^{q}B^{3}+AB^{3q}=$ $\displaystyle\frac{(h+c^{q}+c^{2q})B^{3}}{c}+\frac{(h+c+c^{2})B^{3q}}{c^{q}}$ $\displaystyle=$ $\displaystyle\frac{c^{q}(h+c^{q}+c^{2q})B^{3}+c(h+c+c^{2})B^{3q}}{c^{q+1}}$ $\displaystyle=$ $\displaystyle\frac{h(c^{q}B^{3}+cB^{3q})+c^{q}B^{3}(c^{q}+c^{2q})+cB^{3q}(c+c^{2})}{c^{q+1}}.$ While $\displaystyle h(c^{q}B^{3}+cB^{3q})=$ $\displaystyle h\big{(}c^{q}(1+c+c^{2}+c^{3})+c(1+c^{q}+c^{2q}+c^{3q})\big{)}$ $\displaystyle=$ $\displaystyle h(c+c^{q}+c^{q+1}(c+c^{q})+c^{q+1}(c+c^{q})^{2})$ and $\displaystyle c^{q}B^{3}(c^{q}+c^{2q})+cB^{3q}(c+c^{2})=$ $\displaystyle c^{q}(1+c+c^{2}+c^{3})(c^{q}+c^{2q})+(c^{q}(1+c+c^{2}+c^{3})(c^{q}+c^{2q}))^{q}$ $\displaystyle=$ $\displaystyle c^{2q}+c^{3q}+c^{2q+1}+c^{3q+1}+c^{2q+2}+c^{3q+2}+c^{2q+3}+c^{3q+3}+$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}(c^{2q}+c^{3q}+c^{2q+1}+c^{3q+1}+c^{2q+2}+c^{3q+2}+c^{2q+3}+c^{3q+3})^{q}$ $\displaystyle=$ $\displaystyle(c+c^{q})^{2}+c^{3}+c^{3q}+c^{q+1}(c+c^{q})+c^{q+1}(c+c^{q})^{2}.$ We have $\displaystyle c+c^{q}=x+x^{q}+(x+x^{q})^{2},c^{q+1}=x^{q+1}+x^{2(q+1)}+x^{q+1}(x+x^{q}),$ from which we can obtain that $\displaystyle h(c^{q}B^{3}+cB^{3q})=$ $\displaystyle(x+x^{q})^{2}+(x+x^{q})^{3}+x^{q+1}(x+x^{q})^{2}+x^{q+1}(x+x^{q})^{3}+x^{q+1}(x+x^{q})^{5}$ $\displaystyle+x^{q+1}(x+x^{q})^{6}+x^{(2q+1)}(x+x^{q})^{2}+x^{(2q+1)}(x+x^{q})^{5}$ and $\displaystyle c^{q}B^{3}(c^{q}+c^{2q})+cB^{3q}(c+c^{2})=$ $\displaystyle(x+x^{q})^{2}+(x+x^{q})^{3}+(x+x^{q})^{5}+(x+x^{q})^{6}+x^{q+1}(x+x^{q})^{2}$ $\displaystyle+x^{q+1}(x+x^{q})^{3}+x^{q+1}(x+x^{q})^{4}+x^{q+1}(x+x^{q})^{5}$ $\displaystyle+x^{2(q+1)}(x+x^{q})^{2}+x^{2(q+1)}(x+x^{q})^{4}.$ Thus we have $\displaystyle c^{q+1}(A^{q}B^{3}+AB^{3q})=$ $\displaystyle(x+x^{q})^{5}+(x+x^{q})^{6}+x^{q+1}(x+x^{q})^{4}+x^{q+1}(x+x^{q})^{6}$ $\displaystyle+x^{2(q+1)}(x+x^{q})^{4}+x^{2(q+1)}(x+x^{q})^{5}$ and $\displaystyle c^{2(q+1)}(A^{q}B^{3}+AB^{3q})=$ $\displaystyle x^{q+1}(x+x^{q})^{5}+x^{q+1}(x+x^{q})^{7}+x^{2(q+1)}(x+x^{q})^{4}+x^{2(q+1)}(x+x^{q})^{7}$ $\displaystyle+x^{4(q+1)}(x+x^{q})^{4}+x^{4(q+1)}(x+x^{q})^{5}.$ We further have $\displaystyle c^{2(q+1)}B^{2(q+1)}=$ $\displaystyle c^{2(q+1)}(1+c)^{2(q+2)}$ $\displaystyle=$ $\displaystyle c^{2(q+1)}+c^{2(q+1)}(c+c^{q})^{2}+c^{4(q+1)}$ $\displaystyle=$ $\displaystyle x^{2(q+1)}+x^{2(q+1)}(x+x^{q})^{6}+x^{4(q+1)}(x+x^{q})^{2}+x^{8(q+1)}.$ Recall that $h=x+x^{q}$, $r=x^{q+1}$. Thus, we have $\displaystyle c^{2(q+1)}(A^{q}B^{3}+AB^{3q}+B^{2(q+1)})=$ $\displaystyle rh^{5}+rh^{7}+r^{2}+r^{2}h^{4}+r^{2}h^{6}+r^{2}h^{7}+r^{4}h^{2}+r^{4}h^{4}+r^{4}h^{5}+r^{8},$ and $\displaystyle\frac{H}{D^{2}}=\frac{r}{h^{5}}+\frac{r}{h^{3}}+\frac{r^{2}}{h^{10}}+\frac{r^{2}}{h^{6}}+\frac{r^{2}}{h^{4}}+\frac{r^{2}}{h^{3}}+\frac{r^{4}}{h^{8}}+\frac{r^{4}}{h^{6}}+\frac{r^{4}}{h^{5}}+\frac{r^{8}}{h^{10}}=\mu+\mu^{2}$ where $\mu=\frac{r+r^{4}+r^{2}h+(r+r^{2})h^{2}}{h^{5}}$. The rest of this proof is similar to that of Theorem 3.10, so we omit it here.∎
# Tractable higher-order under-approximating AE extensions for non-linear systems Eric Goubault Sylvie Putot LIX, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris, France (email<EMAIL_ADDRESS> ###### Abstract We consider the problem of under and over-approximating the image of general vector-valued functions over bounded sets, and apply the proposed solution to the estimation of reachable sets of uncertain non-linear discrete-time dynamical systems. Such a combination of under and over-approximations is very valuable for the verification of properties of embedded and cyber-physical controlled systems. Over-approximations prove properties correct, while under- approximations can be used for falsification. Coupled, they provide a measure of the conservatism of the analysis. This work introduces a general framework relying on computations of robust ranges of vector-valued functions. This framework allows us to extend for under-approximation many precision refinements that are classically used for over-approximations, such as affine approximations, Taylor models, quadrature formulae and preconditioning methods. We end by evaluating the efficiency and precision of our approach, focusing on the application to the analysis of discrete-time dynamical systems with inputs and disturbances, on different examples from the literature. ###### keywords: Uncertain systems, Computer-aided control design ## 1 Introduction Guaranteed state estimation and reachability analysis are central to many problems in control, such as robust and optimal control of dynamical systems, set invariance, safety verification, or control synthesis. This ultimately relies on computing ranges of functions over a domain, that we have to approximate since this is an intractable problem. Much of the existing work focuses on over-approximations of images of functions, or of reachable sets, generally based on convex set representations (in particular intervals, ellipsoids, polyhedra). We are interested here in the much less studied problem of computing under-approximations, that is, sets of states guaranteed to be reached. Combining over and under approximations is fundamental for the validation of control systems.When the over-approximation is not sufficient to prove a property, an under-approximation is helpful to state the quality of the over-approximation. Additionally, when an under- approximation of the reachable set intersects the set of error states, it provides a proof of falsification of the property. For general controlled systems, the reachability properties will depend on the initial conditions of the system, but also on the sensitivity of the system to some control inputs and external disturbances, as reflected by the notions of minimal and maximal reachability Mitchell (2007). We generalize these notions here to robust reachability, when both control inputs and adversarial disturbances are present. The robust image will be the intersection, for all possible disturbances, of the images of a function or reachable sets of a system. #### Contents and contributions The computation of the reachable set of a dynamical system can be reduced to a series of images of sets by some vector-valued function. In previous work Goubault and Putot (2020), we introduced mean-value extensions allowing us to compute under approximations (also called inner-approximations) of such images in a very efficient way. In this article, we generalize this approach to higher-order extensions, and develop quadrature formulas for more precise under-approximations. We also address many questions that are to be solved for an accurate and efficient implementation: * • Section 2 recaps the necessary background from previous work. Section 3 generalizes the mean-value extension of Goubault and Putot (2020) for the under and over-approximation of the robust range of sufficiently smooth real- valued functions $f:\mathbb{R}^{m}\rightarrow\mathbb{R}$; this generalization allows us to propose higher-order Taylor extensions for under-approximating robust ranges of functions. These new extensions, just as the mean-value extensions, is the basis for under-approximations of elementary vector-valued functions from $\mathbb{R}^{m}$ to $\mathbb{R}^{n}$, as detailed in Section 2.2, which is instrumental in the reachability analysis of dicrete-time nonlinear systems proposed later; * • Section 4 proposes a novel approach to subdivisions for mean-value and Taylor extensions, based on the idea of quadratures (in numerical calculations of integrals): we show that this improves precision of the computation of under and over-approximations, while still scaling with the dimension of the system; * • Section 5 applies this work to the approximation of robust reachable sets of discrete-time dynamical systems; we present results on representative systems from the literature, demonstrating the tractability and precision of our approach. #### Related work Our approach is related to and partially relies on work on modal intervals and mean-value extensions, which applications include the computation of under- approximations of function images Goldsztejn (2012a, b). It is also related to over-approximations of nonlinear functions and dynamical systems, on which we rely to compute under-approximations. Many methods for over-approximating reachable sets for non-linear systems have been developed, among which Taylor methods Makino and Berz (2003) or polytopes-based methods Guernic and Girard (2009); Dreossi et al. (2016). There exist less methods for the harder problem of under-approximating images of functions or sets of reachable states. Some approaches have been proposed for linear discrete-time systems Kurzhanski and Varaiya (2000); Girard et al. (2006); Raković and Fiacchini (2008). Interval-based methods, relying on space discretization, have been used for under-approximating the image of nonlinear functions Goldsztejn and Jaulin (2010). They were also used to over and under approximate solutions of differential systems with uncertain initial conditions Mézo et al. (2018). Tight approximations for reachable sets of nonlinear continuous systems can be found via expensive Eulerian methods: the zero sub-level set of the Lipschitz viscosity solution to a Hamilton-Jacobi (HJB) partial differential equation gives the (backward) reachable set Chen et al. (2016). Other approaches, using SoS methods and LMI relaxations have been proposed for inner approximations, see e.g. Korda et al. (2013). In Xue et al. (2020), under-approximations for polynomial systems are obtained by solving semi-definite programs. Taylor models are used on the inverse flow map to derive under-approximations Chen et al. (2014), but using topological conditions that are checked with interval constraints solving, which have difficulties to scale up with dimension. In Xue et al. (2016), the computation of the under-approximated reachable set is based on a costly analysis of the boundary of the reachable sets and polytopic approximations. In Kochdumper and Althoff (2020), some non-convex under-approximations are computed with polynomial zonotopes, relying on a computation of the outer-approximation of the reachable set, of an enclosure of the boundary of the reachable set, and a reduction of the outer-approximation until it is fully included in the region delimited by the boundary. #### Notations For a continuously differentiable vector-valued function $f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$, we note $f_{i}$ its $i$-th component and $\nabla f=(\nabla_{j}f_{i})_{ij}=(\frac{\partial f_{i}}{\partial x_{j}})_{1\leq i\leq n,1\leq j\leq m}$ its Jacobian matrix. We note $\langle x,y\rangle$ the scalar product of vectors $x$ and $y$, and $\lvert x\rvert$ the absolute value extended componentwise. Intervals are used in many situations to rigorously compute with interval domains instead of reals, usually leading to over-approximations of function ranges over boxes. Set valued quantities, whether scalar or vector-valued, will be noted with bold letters, e.g $x$. We denote ${\mathbb{I}\mathbb{R}}=\\{{\makebox{\boldmath$x$}}=[\underline{x},\overline{x}],\>\underline{x}\in\mathbb{R},\>\overline{x}\in\mathbb{R}\\}$ the set of intervals with real-valued bounds. If $\overline{x}<\underline{x}$, the interval represents the empty set. For a (possibly vector-valued) interval ${\makebox{\boldmath$x$}}\in{\mathbb{I}\mathbb{R}}^{m}$, we note $c({{\makebox{\boldmath$x$}}})=(\underline{x}+\overline{x})/2$ its center and $r({{\makebox{\boldmath$x$}}})=(\overline{x}-\underline{x})/2$ its radius. The operators over real numbers are lifted in intervals using the same notation. ## 2 Background: AE extensions for computing function images We recall in this section the notations and results of Goubault and Putot (2020): we state mean-value over and under-approximating extensions for scalar and vector-valued functions. An _over-approximating extension_ , also called _outer-approximating extension_ , of a function $f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ is a function ${\makebox{\boldmath$f$}}:\mathcal{P}(\mathbb{R}^{m})\rightarrow\mathcal{P}(\mathbb{R}^{n})$, such that for all $x$ in $\mathcal{P}(\mathbb{R}^{m})$, $\mbox{range}(f,{\makebox{\boldmath$x$}})=\\{f(x),x\in{\makebox{\boldmath$x$}}\\}\subseteq{\makebox{\boldmath$f$}}({\makebox{\boldmath$x$}})$. Dually, under-approximations determine a set of values proved to belong to the range of the function over some input set. An _under-approximating extension_ , also called _inner-approximating extension_ , of $f$ is a function ${\makebox{\boldmath$f$}}:\mathcal{P}(\mathbb{R}^{m})\rightarrow\mathcal{P}(\mathbb{R}^{n})$, such that for all $x$ in $\mathcal{P}(\mathbb{R}^{m})$, ${\makebox{\boldmath$f$}}({\makebox{\boldmath$x$}})\subseteq\mbox{range}(f,{\makebox{\boldmath$x$}})$. Under- and over-approximations can be interpreted as quantified propositions: $\operatorname{range}(f,{\makebox{\boldmath$x$}})\subseteq{\makebox{\boldmath$z$}}$ can be written $\forall x\in{\makebox{\boldmath$x$}},\,\exists z\in{\makebox{\boldmath$z$}},\,f(x)=z,$ while ${\makebox{\boldmath$z$}}\,\subseteq\operatorname{range}(f,{\makebox{\boldmath$x$}})$ can be written $\forall z\in\,{\makebox{\boldmath$z$}},\,\exists x\in{\makebox{\boldmath$x$}},\,f(x)=z.$ Both these propositions are what we will call _AE propositions_ , for quantified propositions where universal quantifiers (A) precede existential quantifiers (E). ### 2.1 Mean-value AE extensions for scalar-valued functions We consider a function $f:\mathbb{R}^{m}\rightarrow\mathbb{R}$. The natural interval extension consists in replacing real operations by their interval counterparts in the expression of the function. A generally more accurate extension relies on a linearization by the mean-value theorem. #### 2.1.1 Mean-value AE extensions Suppose $f$ is differentiable over the box $x$. The mean-value theorem implies that $\forall x^{0}\in{\makebox{\boldmath$x$}},\,\forall x\in{\makebox{\boldmath$x$}},\exists\xi\in{\makebox{\boldmath$x$}},\,f(x)=f(x^{0})+\langle\nabla f(\xi),x-x^{0}\rangle.$ If we can bound the range of the gradient of $f$ over $x$, by ${\makebox{\boldmath$\nabla f$}}({\makebox{\boldmath$x$}})$, then we can derive an interval enclosure, called the mean-value extension. Let us choose $x^{0}$ to be the center $c({{\makebox{\boldmath$x$}}})$ of $x$ and recall we note $r({{\makebox{\boldmath$x$}}})=(\overline{x}-\underline{x})/2$ its radius. ###### Theorem 1 (Thm. 1, Goubault and Putot (2020)) Let $f$ be a continuously differentiable function from $\mathbb{R}^{m}$ to $\mathbb{R}$ and ${\makebox{\boldmath$x$}}\in{\mathbb{I}\mathbb{R}}^{m}$. Let ${\makebox{\boldmath$f$}}^{0}=[\underline{f^{0}},\overline{f^{0}}]$ include $f(c({{\makebox{\boldmath$x$}}}))$ and $\nabla$ a vector of intervals ${\makebox{\boldmath$\nabla_{i}$}}=[\underline{\nabla}_{i},\overline{\nabla}_{i}]$ for $i\in\\{1,\ldots,m\\}$ such that $\left\\{\left\lvert\nabla_{i}f(c({\makebox{\boldmath$x$}}_{1}),\ldots,c({\makebox{\boldmath$x$}}_{i-1}),x_{i},\ldots,x_{m})\right\rvert,x\in{\makebox{\boldmath$x$}}\right\\}\subseteq{\makebox{\boldmath$\nabla$}}_{i}.$ We have the over- and under-approximating extensions $\displaystyle\mbox{range}(f,{\makebox{\boldmath$x$}})\subseteq[\underline{f^{0}},\overline{f^{0}}]+\langle\overline{\nabla},r({{\makebox{\boldmath$x$}}})\rangle[-1,1]$ (1) $\displaystyle[\overline{f^{0}}-\langle\underline{\nabla},r({\makebox{\boldmath$x$}})\rangle,\underline{f^{0}}+\langle\underline{\nabla},r({\makebox{\boldmath$x$}})\rangle]\subseteq\mbox{range}(f,{\makebox{\boldmath$x$}})$ (2) ###### Example 2.1 Let us consider the range of $f$ defined by $f(x)=x^{2}-x$ over ${\makebox{\boldmath$x$}}=[2,3]$. We can compute $f(2.5)=3.75$ and $\lvert\nabla f([2,3])\rvert\subseteq[3,5]$. Then (1) and (2) yield $3.75+1.5[-1,1]\subseteq\mbox{range}(f,[2,3])\subseteq 3.75+2.5[-1,1],$ from which we deduce $[2.25,5.25]\subseteq\mbox{range}(f,[2,3])\subseteq[1.25,6.25]$. We refer to extensions (1) and (2) as _AE extensions_ , as they can be interpreted as _AE propositions_. Note that the wider, lesser quality, are the over-approximations of $f$ and its derivatives, the tighter, less quality, are the under-approximations. The under-approximation can even become empty if the width $\overline{f_{0}}-\underline{f_{0}}$ of the approximation of $f(c({{\makebox{\boldmath$x$}}}))$ exceeds $2\langle\overline{\nabla},r({{\makebox{\boldmath$x$}}})\rangle$: in this case the lower bound of the resulting interval is larger than the upper bound, which by convention we identify with the empty interval. Note also that when $0\in{\makebox{\boldmath$\nabla_{i}{\makebox{\boldmath$f$}}$}}$, then $\underline{\nabla_{i}}=0$ and if this is the case for all $i$, the under- approximation is empty or reduced to a point. A special attention to the practical evaluation of these extensions over the region $x$ of interest is thus crucial, this is the object of Section 4. #### 2.1.2 Mean-value AE extensions of the robust range Mean-value AE extensions can be generalized to compute ranges that are robust to disturbances, identified as some input components. Let us partition the indices of the input space in two subsets $I_{\mathcal{A}}$ and $I_{\mathcal{E}}$, where $I_{\mathcal{A}}$ defines the indices of the inputs that correspond to disturbances, and $I_{\mathcal{E}}$ the remaining dimensions. We decompose the input box $x$ accordingly by ${\makebox{\boldmath$x$}}={\makebox{\boldmath$x$}}_{\mathcal{A}}\times{\makebox{\boldmath$x$}}_{\mathcal{E}}$. We define the robust range of function $f$ on $x$, robust on ${\makebox{\boldmath$x$}}_{\mathcal{E}}$ with respect to disturbances ${\makebox{\boldmath$x$}}_{\mathcal{A}}$, as $\mbox{range}(f,{\makebox{\boldmath$x$}},I_{\mathcal{A}},I_{\mathcal{E}})=\\{z\,\|\,\forall w\in{\makebox{\boldmath$x$}}_{\mathcal{A}},\,\exists u\in{\makebox{\boldmath$x$}}_{\mathcal{E}},\,z=f(w,u)\\}$. Intuitively, $u$ will be control components, $w$ disturbances to which the output range should be robust. ###### Theorem 2 (Thm. 2, Goubault and Putot (2020)) Let $f$ be continuously differentiable function from $\mathbb{R}^{m}$ to $\mathbb{R}$ and ${\makebox{\boldmath$x$}}={\makebox{\boldmath$x$}}_{\mathcal{A}}\times{\makebox{\boldmath$x$}}_{\mathcal{E}}\in{\mathbb{I}\mathbb{R}}^{m}$. Let ${\makebox{\boldmath$f$}}^{0}$, ${\makebox{\boldmath$\nabla$}}_{w}$ and ${\makebox{\boldmath$\nabla$}}_{u}$ be vectors of intervals such that $f(c({\makebox{\boldmath$x$}}))\subseteq{\makebox{\boldmath$f$}}^{0}$, $\\{\left\|\nabla_{w}f(w,c({\makebox{\boldmath$x$}}_{\mathcal{E}}))\right\|\,,\,w\in{\makebox{\boldmath$x$}}_{\mathcal{A}}\\}\subseteq{\makebox{\boldmath$\nabla$}}_{w}$ and $\\{\left\|\nabla_{u}f(w,u)\right\|\,,w\in{\makebox{\boldmath$x$}}_{\mathcal{A}},\,u\in{\makebox{\boldmath$x$}}_{\mathcal{E}}\\}\subseteq{\makebox{\boldmath$\nabla$}}_{u}$. We have: $\mbox{range}(f,{\makebox{\boldmath$x$}},I_{\mathcal{A}},I_{\mathcal{E}})\subseteq[\underline{f^{0}}-\langle\overline{\nabla}_{u},r({{\makebox{\boldmath$x$}}_{\mathcal{E}}})\rangle+\langle\underline{\nabla}_{w},r({{\makebox{\boldmath$x$}}_{\mathcal{A}}})\rangle,\\\ \overline{f^{0}}+\langle\overline{\nabla}_{u},r({{\makebox{\boldmath$x$}}_{\mathcal{E}}})\rangle-\langle\underline{\nabla}_{w},r({{\makebox{\boldmath$x$}}_{\mathcal{A}}})\rangle]$ (3) $[\overline{f^{0}}-\langle\underline{\nabla}_{u},r({{\makebox{\boldmath$x$}}_{\mathcal{E}}})\rangle+\langle\overline{\nabla}_{w},r({{\makebox{\boldmath$x$}}_{\mathcal{A}}})\rangle,\underline{f^{0}}+\\\ \langle\underline{\nabla}_{u},r({{\makebox{\boldmath$x$}}_{\mathcal{E}}})\rangle-\langle\overline{\nabla}_{w},r({{\makebox{\boldmath$x$}}_{\mathcal{A}}})\rangle]\subseteq\mbox{range}(f,{\makebox{\boldmath$x$}},I_{\mathcal{A}},I_{\mathcal{E}})$ (4) We refer to Example 2 of Goubault and Putot (2020) for a sample computation. ### 2.2 AE extensions for vector-valued functions Following Goubault and Putot (2020), we now detail how full n-dimensional boxes can be included in the image of vector-valued functions $f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$, for $m\geq n$, using AE extensions of robust ranges. Theorem 3 and Definition 2.2 will be instrumental in Algorithm 1 for discrete-time reachability of Section 5. The mean-value extensions of Theorem 1 or the generalization of Theorem 5 give us under and over-approximations of projections of the image of the function. The Cartesian product of the over-approximations of each component provides an over-approximation of a vector-valued function $f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$. This is however not the case for under-approximation. Suppose for example that we have $\forall z_{1}\in{\makebox{\boldmath$z$}}_{1},\exists x_{1}\in{\makebox{\boldmath$x$}}_{1},\,\exists x_{2}\in{\makebox{\boldmath$x$}}_{2},\>z_{1}=f_{1}(x)$ and $\forall z_{2}\in{\makebox{\boldmath$z$}}_{2},\exists x_{1}\in{\makebox{\boldmath$x$}}_{1},\,\exists x_{2}\in{\makebox{\boldmath$x$}}_{2},\>z_{2}=f_{2}(x)$. We cannot deduce directly that for all $\forall z_{1}\in{\makebox{\boldmath$z$}}_{1}$ and $\forall z_{2}\in{\makebox{\boldmath$z$}}_{2}$ there exists $x_{1}$ and $x_{2}$ such that $z=f(x)$. Suppose now that we have: $\forall z_{1}\in{\makebox{\boldmath$z$}}_{1},\forall x_{1}\in{\makebox{\boldmath$x$}}_{1},\,\exists x_{2}\in{\makebox{\boldmath$x$}}_{2},\>z_{1}=f_{1}(x)$ and $\forall z_{2}\in{\makebox{\boldmath$z$}}_{2},\forall x_{2}\in{\makebox{\boldmath$x$}}_{2},\,\exists x_{1}\in{\makebox{\boldmath$x$}}_{1},\>z_{2}=f_{2}(x)$ with continuous selections $x_{2}$ and $x_{1}$. Then there exists functions $g_{2}(z_{1},x_{1}):{\makebox{\boldmath$z$}}_{1}\times{\makebox{\boldmath$x$}}_{1}\rightarrow{\makebox{\boldmath$x$}}_{2}$ and $g_{1}(z_{2},x_{2}):{\makebox{\boldmath$z$}}_{2}\times{\makebox{\boldmath$x$}}_{2}\rightarrow{\makebox{\boldmath$x$}}_{1}$ that are continuous in $x_{1}$ (resp. $x_{2}$), and such that $\forall(z_{1},z_{2})\in{\makebox{\boldmath$z$}}$, $\forall(x_{1},x_{2})\in{\makebox{\boldmath$x$}}$, $z_{1}=f_{1}(x_{1},g_{2}(z_{1},x_{1}))$ and $z_{2}=f_{2}(g_{1}(z_{2},x_{2}),x_{2})$. Using the Brouwer fixed point theorem on the continuous map $g~{}:(x_{1},x_{2})\rightarrow(g_{1}(z_{2},x_{2}),g_{2}(z_{1},x_{1}))$ on the compact set ${\makebox{\boldmath$x$}}_{1}\times{\makebox{\boldmath$x$}}_{2}$, then $\forall(z_{1},z_{2})\in{\makebox{\boldmath$z$}}$, $\exists(x^{z}_{1},x^{z}_{2})\in{\makebox{\boldmath$x$}}$ such that $(z_{1},z_{2})=f(x^{z}_{1},x^{z}_{2})$. This result can be generalized to functions $f~{}:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ for any $n$, as stated in Theorem 3. ###### Theorem 3 (Theorem 3 in Goubault and Putot (2020)) Let $f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ be an elementary function and $\pi:[1\ldots m]\rightarrow[1\ldots n]$ Let us note, for all $i\in[1\ldots n]$, $J_{E}^{(z_{i})}=\\{j\in[1\ldots m],\;\pi(j)=i\\}$ and $J_{A}^{(z_{i})}=\\{j\in[1\ldots m]\\}\setminus J_{E}^{(z_{i})}$. Consider the $n$ AE-extensions $i\in[1\ldots n]$ built from Theorems 2, 4 or 5 and such that $\forall z_{i}\in{\makebox{\boldmath$z$}}_{i},\>(\forall x_{j}\in{\makebox{\boldmath$x$}}_{j})_{j\in J_{A}^{(z_{i})}},\>(\exists x_{j}\in{\makebox{\boldmath$x$}}_{j})_{j\in J_{E}^{(z_{i})}},\>z_{i}=f_{i}(x)$ (5) Then ${\makebox{\boldmath$z$}}={\makebox{\boldmath$z$}}_{1}\times{\makebox{\boldmath$z$}}_{2}\times\ldots\times{\makebox{\boldmath$z$}}_{n}$, if non-empty, is an under-approximation of the image of $f$: $\forall z\in{\makebox{\boldmath$z$}},\,\exists x\in{\makebox{\boldmath$x$}},\>z=f(x).$ Theorem 3 gives us directly a computation of an under-approximation of $\mbox{range}(f,{\makebox{\boldmath$x$}})$ for $f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$. It can also be used to compute an under-approximation of the robust range $\mbox{range}(f,{\makebox{\boldmath$x$}},I_{\mathcal{A}},I_{\mathcal{E}})$. For this, we need to choose $\pi:[1\ldots m]\rightarrow([1\ldots n]\setminus I_{\mathcal{A}})$, which corresponds to the fact that the disturbance part of the input components will always be quantified universally. We define below the result of this process, which will be later used in reachability algorithms for discrete-time dynamical systems. ###### Definition 2.2 Let $f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ and $\pi:[1\ldots m]\rightarrow([1\ldots n]\setminus I_{\mathcal{A}})$. We define ${\mathcal{I}}(f,{\makebox{\boldmath$x$}},I_{\mathcal{A}},I_{\mathcal{E}},\pi)$ an under-approximation of $\mbox{range}(f,{\makebox{\boldmath$x$}},I_{\mathcal{A}},I_{\mathcal{E}})$ obtained using Theorem 3 with function $\pi$, in which the under-approximation of each component is obtained using Theorem 2 (or Corollary 3.2). We define ${\mathcal{O}}(f,{\makebox{\boldmath$x$}},I_{\mathcal{A}},I_{\mathcal{E}},\pi)$ the over-approximation of the robust range obtained using Theorem 2 component- wise. ## 3 Generalization to new AE extensions We now introduce new robust AE extensions for a function $f:\mathbb{R}^{m}\rightarrow\mathbb{R}$, which are no longer necessarily based on the mean-value theorem. ###### Theorem 4 Suppose we have an approximation function $g$ for $f$, which is an elementary111Elementary functions are compositions of +, -, $\times$, /, sine, cosine, log, exp functions in particular. function in the sense of Goldsztejn (2012a), satisfying $\forall w\in{\bf x}_{\mathcal{A}},\ \forall u\in{\bf x}_{\mathcal{E}},\ \exists\xi\in{\bf x},\ f(w,u)=g(w,u,\xi)$. Then any under- approximation (resp. over-approximation) of the robust range of $g$ with respect to $x_{\mathcal{A}}$ and $\xi$, ${\mathcal{I}}_{g}\subseteq\mbox{range}(g,{\bf x}\times{\bf x},I_{\mathcal{A}}\cup\\{m+1,\ldots,2m\\},I_{\mathcal{E}})$ is an under- approximation (resp. over-approximation) of the robust range of $f$ with respect to $x_{\mathcal{A}}$, i.e. ${\mathcal{I}}_{g}\subseteq\mbox{range}(f,{\bf x},I_{\mathcal{A}},I_{\mathcal{E}})$. For instance, for a continuously $(n+1)$-differentiable $f$, the following $g$, obtained by a Taylor-Lagrange expansion and noting $x=(w,u)$, is an approximation function for $f$ $g(x,\xi)=f(x^{0})+\sum\limits_{i=1}^{n}\frac{(x-x^{0})^{i}}{i!}D^{i}f(x^{0})\\\ +D^{n+1}f(\xi)\frac{(x-x^{0})^{n+1}}{(n+1)!}$ (6) where $D^{\alpha}f$ denotes the higher order partial derivative of $f$. For $n=0$, $g$ is the mean-value approximation. We focus on the under-approximation. As $g$ is elementary, by Proposition 10.1 of Goldsztejn (2012a), we have a continuous Skolem function $(w,u)\rightarrow\xi(w,u)$, i.e. a function such that for all $x=(w,u)\in\mathbb{R}^{m}$, $f(w,u)=g(w,u,\xi(x))$. Consider an under- approximation ${\mathcal{I}}_{g}$ of the robust range of $g$ with respect to $\xi$ and $x_{\mathcal{A}}$. It satisfies $\forall z\in{\mathcal{I}}_{g},\ \forall w\in{\bf x}_{\mathcal{A}},\ \forall\xi\in{\bf x},\ \exists u\in{\bf x}_{\mathcal{E}},\ z=g(w,u,\xi)$. Let $z\in{\mathcal{I}}_{g}$ and $w\in{\bf x}_{\mathcal{A}}$ be fixed. As $g$ is elementary, we have a corresponding continuous Skolem function $\xi\rightarrow u(\xi)$, i.e. a function such that for all $\xi\in{\bf x}$, $z=g(w,u(\xi),\xi)$. For this $z\in{\mathcal{I}}_{g}$ and $w\in{\bf x}_{\mathcal{A}}$, the continuous map $u\rightarrow u(\xi(w,u))$ defined from ${\bf x}_{\mathcal{E}}$ over itself, has a fixed point $u^{\infty}$, by Brouwer’s theorem. It is such that $z=f(w,u^{\infty})=g(w,u^{\infty},\xi(w,u^{\infty}))$. Hence $z$ is in the robust range of $f$ with respect to $x_{\mathcal{A}}$. ###### Example 3.1 Consider function $f(x)=x^{3}+x^{2}+x+1$ on $[-\frac{1}{4},\frac{1}{4}]$. The exact range is $[0.796875,1.328125]$. Let us approximate $f$ by a quadratic function, using an order 2 Taylor-Lagrange expansion. We compute $f^{(1)}(x)=3x^{2}+2x+1$ and $f^{(2)}(x)=6x+2$. By Theorem 4, the range of $f$ over $[-\frac{1}{4},\frac{1}{4}]$ is under (resp. over) approximated by any under (resp. over) approximation of the robust range with respect to $\xi$ of $\begin{array}[]{rcl}g(x,\xi)&=&f(x^{0})+(x-x^{0})f^{(1)}(x^{0})+f^{(2)}(\xi)\frac{(x-x^{0})^{2}}{2}\\\ &=&1+x+x^{2}(3\xi+1)\end{array}$ In the general case, it may still be difficult to compute the under- approximated robust range of $g$. However, Theorem 5 gives a simple way which is well suited in particular for quadratic Taylor-based approximations. ###### Theorem 5 Let $g$ be an elementary function $g(w,u,\xi)=\alpha(w,u)+\beta(w,u,\xi)$ over $x=(w,u)\in{\bf x}\subseteq{\mathbb{I}\mathbb{R}}^{m}$ and $\xi\in{\bf x}$. Let ${\mathcal{I}}_{\alpha}$ be an under-approximation of the robust range of $\alpha$ with respect to $w$, i.e. $\mbox{range}(\alpha,{\bf x},I_{\mathcal{A}},I_{\mathcal{E}})$, and ${\mathcal{O}}_{\beta}$ an over- approximation of the range of $\beta$, i.e. $\mbox{range}(\beta,{\bf x}\times{\bf x},\emptyset,\\{1,\ldots,2m\\})$. The robust range of $g$ with respect to $w\in{\bf x}_{\mathcal{A}}$ and $\xi\in{\bf x}$, i.e. $\mbox{range}(g,{\bf x}\times{\bf x},I_{\mathcal{A}}\cup\\{m+1,\ldots,2m\\},I_{\mathcal{E}})$, is under- approximated by ${\mathcal{I}}_{g}=[\underline{\mathcal{I}}_{\alpha}+\overline{\mathcal{O}}_{\beta},\overline{\mathcal{I}}_{\alpha}+\underline{\mathcal{O}}_{\beta}].$ This is the case of Taylor expansions (6), where $\alpha$ is the degree $n$ polynomial, and $\beta$ the degree $n+1$ remainder. ${\mathcal{I}}_{\alpha}$ under-approximating $\mbox{range}(\alpha,{\bf x},I_{\mathcal{A}},I_{\mathcal{E}})$ means $\forall a\in{\mathcal{I}}_{\alpha},\ \forall w\in{\bf x}_{\mathcal{A}},\ \exists u\in{\bf x}_{\mathcal{E}},\ a=\alpha(w,u)$. As $\alpha$ is elementary, we have a continuous Skolem function $(w,a)\rightarrow u(w,a)$. Moreover, for all $z$ in ${\mathcal{I}}_{g}$, for all $b\in{\mathcal{O}}_{\beta}$, we have $z-b\in[\underline{\mathcal{I}}_{\alpha}+\overline{\mathcal{O}}_{\beta}-\overline{\mathcal{O}}_{\beta},\overline{\mathcal{I}}_{\alpha}+\underline{\mathcal{O}}_{\beta}-\underline{\mathcal{O}}_{\beta}]={\mathcal{I}}_{\alpha}$. For given $z\in{\mathcal{I}}_{g}$, $w\in{\bf x}_{\mathcal{A}}$ and $\xi\in{\bf x}$, consider the continuous function $b\rightarrow\beta(w,u(w,z-b),\xi)$ over ${\mathcal{O}}_{\beta}$. By Brouwer fixed point theorem we have $b^{\infty}=\beta(w,u(w,z-b^{\infty}),\xi)$. Therefore, for any $z\in{\mathcal{I}}_{g}$, $\xi\in{\bf x}$, there exist $a=z-b^{\infty}\in{\mathcal{I}}_{\alpha}$, $b=b^{\infty}\in{\mathcal{O}}_{\beta}$ and $x=(w,u(w,a))\in{\bf x}$ such that $z=a+b$. This implies that ${\mathcal{I}}_{g}$ is a robust under-approximation of $g(x,\xi)=\alpha(x)+\beta(x,\xi)$ with respect to $w$ and $\xi$. A direct consequence is a simple order 2 under-approximating Taylor method: ###### Corollary 3.2 Consider $f:\mathbb{R}^{m}\rightarrow\mathbb{R}$ a function in $C^{2}$. Let ${\makebox{\boldmath$f$}}^{0}$, ${\makebox{\boldmath$\nabla$}}_{w}^{0}$ and ${\makebox{\boldmath$\nabla$}}_{u}^{0}$ be such that $f(x^{0})\subseteq{\makebox{\boldmath$f$}}^{0}$, $\left\|\nabla_{w}f(x^{0})\right\|\subseteq{\makebox{\boldmath$\nabla$}}_{w}^{0}$ and $\left\|\nabla_{u}f(x^{0})\right\|\subseteq{\makebox{\boldmath$\nabla$}}_{u}^{0}$ with $x^{0}=c({\makebox{\boldmath$x$}})$. Then $\mbox{range}(f,{\makebox{\boldmath$x$}},I_{\mathcal{A}},I_{\mathcal{E}})$ is under-approximated by $[\underline{\mathcal{I}}_{\alpha}+\overline{\mathcal{O}}_{\beta},\overline{\mathcal{I}}_{\alpha}+\underline{\mathcal{O}}_{\beta}]$ where ${\mathcal{I}}_{\alpha}=[\overline{f^{0}}-\langle\underline{\nabla}_{u}^{0},r({{\makebox{\boldmath$x$}}_{\mathcal{E}}})\rangle+\langle\overline{\nabla}_{w}^{0},r({{\makebox{\boldmath$x$}}_{\mathcal{A}}})\rangle,$ $\underline{f^{0}}+\langle\underline{\nabla}_{u}^{0},r({{\makebox{\boldmath$x$}}_{\mathcal{E}}})\rangle-\langle\overline{\nabla}_{w}^{0},r({{\makebox{\boldmath$x$}}_{\mathcal{A}}})\rangle]$ and ${\mathcal{O}}_{\beta}$ is any over-approximation of $\\{\frac{1}{2}D^{2}f(x)(r({\makebox{\boldmath$x$}}))^{2},x\in{\makebox{\boldmath$x$}}\\}$. This is a direct application of Theorems 4 and 5 where $g$ is the 2nd order Taylor approximant, i.e. given by Equation (6) for $n=1$, combined with Theorem 2 to compute the robust under-approximation of the order 1 approximation. We use a particular case of Theorem 2: the under-approximation of an order 1 polynomial is almost trivial, we can compute it exactly if the computation is performed in real numbers with exact evaluation of $f$ and its gradient at a point. The expression we give here accounts for computation errors. ###### Example 3.3 We carry on with Example 3.1. The computation of the under approximation of the range of $1+x$ over $[-\frac{1}{4},\frac{1}{4}]$ yields $[\frac{3}{4},\frac{5}{4}]$. We also need an over approximation of the range of $x^{2}(3\xi+1)$ for $x$ and $\xi$ in $[-\frac{1}{4},\frac{1}{4}]$. Standard interval computation yields $[0,\frac{1}{16}][\frac{1}{4},\frac{7}{4}]=[0,\frac{7}{64}]$. Overall, we deduce $[0.859375,1.25]\subseteq\mbox{range}(f,{\makebox{\boldmath$x$}})$. In comparison, the mean-value AE extension of Theorem 2 would have given us the less precise under-approximation $[0.875,1.125]$. ## 4 Refinements: preconditioning and quadrature formulae The n-dimensional inner boxes that we compute with the techniques of Section 2.2 can sometimes be small or empty, even when the projected inner- approximations on each component are tight. There are different reasons, for which we propose solutions in this section. ### 4.1 Preconditioning for computing inner skewed boxes The first difficulty is when the image of the vector-valued function cannot be precisely approximated by a centered box. ###### Example 4.1 We consider $f(x)=(2x_{1}^{2}-x_{1}x_{2}-1,x_{1}^{2}+x_{2}^{2}-2)^{\intercal}$ with ${\makebox{\boldmath$x$}}=[0.9,1.1]^{2}$. The under-approximated projections on the two components (respectively $[-0.38,0.38]$ and $[-0.38,0.38]$) are close to the over-approximated ranges ($[-0.42,0.42]^{2}$), but we only find empty inner boxes. This problem can be partly solved, as already described in Goubault and Putot (2020) by computing a skewed box as under-approximation, that is the image of a box by a linear map, instead of a box. This can be achieved by combining preconditioning to the mean-value theorem. Let $C\in\mathbb{R}^{n\times n}$ be a non-singular matrix. If $z$ is an interval vector such that ${\makebox{\boldmath$z$}}\subseteq\mbox{range}(Cf,{\makebox{\boldmath$x$}})$ , we can deduce a skewed box to be in the range of $f$, that is $\\{C^{-1}z\|z\in{\makebox{\boldmath$z$}}\\}\subseteq\mbox{range}(f,{\makebox{\boldmath$x$}})$. A natural choice for $C$ is the inverse of the center of the interval Jacobian matrix $C=(c({\makebox{\boldmath$\nabla$}}))^{-1}$. ###### Example 4.2 On Example 4.1, using this preconditioning and $pi:(1\rightarrow 1,2\rightarrow 2)$, we obtain for $f_{2}$ as a function of $f_{1}$ the yellow under-approximating parallelotope of Figure 1(a). We estimate the image $\mbox{range}(f,{\makebox{\boldmath$x$}})$ by sampling points in the input domain. This sampling-based estimation is represented as the dark dots-filled region. The green parallelotope and box are the over-approximations with and without preconditioning. (a) Example 4.2: under- (dotted lines) and over-approximation (plain lines) $x^{0}$$x^{1}$${\makebox{\boldmath$x$}}^{k}={\makebox{\boldmath$x$}}\setminus{\makebox{\boldmath$x$}}^{k-1},\nabla^{k}=[\lvert\nabla f\rvert]({\makebox{\boldmath$x$}}^{k})$$x_{1}^{-k}$$x_{1}^{-1}$$x_{1}^{0}$$x_{1}^{1}$$x_{1}^{k}$$x_{2}^{-k}$$x_{2}^{0}$$x_{2}^{k}$${\makebox{\boldmath$x$}}^{1},\nabla^{1}$${\makebox{\boldmath$x$}}^{2},\nabla^{2}$ (b) Partitioning the input domain (c) Example 4.3: approximations for quadrature and order 2 extensions Figure 1: Illustrations for Sections 2.2 and 4 ### 4.2 Quadrature formulae for the mean-value extension The mean-value interval extension can yield a rough approximation. This is especially the case when the variation of the gradient is important over the input range, the extreme case for under-approximation being when this variation contains zero: the under-approximation is empty or reduced to a point. Using simple quadrature formulae partially solves this problem. Let $f:\mathbb{R}^{m}\rightarrow\mathbb{R}$. We partition each dimension $j=[1\ldots m]$ of the $m$-dimensional input box ${\makebox{\boldmath$x$}}={\makebox{\boldmath$x$}}_{1}\times\ldots\times{\makebox{\boldmath$x$}}_{m}$ in $2k$ sub-intervals and define, for all $j=[1\ldots m]$, $x_{j}^{-k}\leq x_{j}^{-(k-1)}\leq\ldots\leq x_{j}^{0}\leq\ldots\leq x_{j}^{k}$, with $x_{j}^{-k}=\underline{{\makebox{\boldmath$x$}}_{j}}$, $x_{j}^{0}=\mbox{mid}({{\makebox{\boldmath$x$}}_{j}})$, $x_{j}^{k}=\overline{{\makebox{\boldmath$x$}}_{j}}$. We note $dx^{i}=x^{i}-x^{i-1}$ the vector-valued deviation. Let us refine the mean-value AE extensions using such a partition. The first natural idea is to compute an under-approximation for each sub-box obtained as product of sub-intervals in each dimension. But in general, the under- approximating boxes will be non-contiguous, and their convex union is in general not an under-approximation of $\mbox{range}(f,{\makebox{\boldmath$x$}})$. Moreover, this approach would not scale well. We now propose a scheme that avoids these unions, and remains linear in $k$ with respect to the non-partitioned case. We note ${\makebox{\boldmath$x$}}^{1}=[x_{1}^{-1},x_{1}^{1}]\times[x_{2}^{-1},x_{2}^{1}]\times\ldots\times[x_{m}^{-1},x_{m}^{1}]$, and for all $i$ between $2$ and $k$, ${\makebox{\boldmath$x$}}^{i}=[x_{1}^{-i},x_{1}^{i}]\times\ldots\times[x_{m}^{-i},x_{m}^{i}]\setminus\mathring{{\makebox{\boldmath$x$}}}^{i-1}$, where $\setminus$ denotes the set difference and $\mathring{{\makebox{\boldmath$x$}}}$ the interior of $x$. This partition is represented in Figure 1(b) for a two-dimensional input space. In practice, each ”square ring” ${\makebox{\boldmath$x$}}^{i}$ will be decomposed in $2n$ sub-boxes for the Jacobian evaluation. By the mean-value theorem, $\forall x\in[x^{-1},x^{1}]$, $\exists\xi^{1}\in[x^{-1},x^{1}],$ $f(x)=f(x^{0})+\langle\nabla f(\xi^{1}),x-x^{0}\rangle$. Suppose we can compute ${\makebox{\boldmath$f$}}^{0}\supseteq f(x^{0})$ and ${\makebox{\boldmath$\nabla$}}^{i}$ for $i$ in $[1,k]$ such that $\\{\lvert\nabla f(x)\rvert,x\in{\makebox{\boldmath$x$}}^{i}\\}\subseteq{\makebox{\boldmath$\nabla$}}^{i}$. We have $\mbox{range}(f,{\makebox{\boldmath$x$}}^{1})\subseteq{\makebox{\boldmath$f$}}^{0}+\langle\overline{\nabla}^{1},dx^{1}\rangle[-1,1]$ and $[\overline{f^{0}}-\langle\underline{\nabla}^{1},dx^{1}\rangle,\underline{f^{0}}+\langle\underline{\nabla^{1}},dx^{1}\rangle]\subseteq\mbox{range}(f,[x^{-1},x^{1}])$. Let us now take $x\in{\makebox{\boldmath$x$}}^{2}$. We can iterate the mean- value theorem on the adjacent input subdivision and write that for all $x\in{\makebox{\boldmath$x$}}^{2}$, there exists $x^{1}\in{\makebox{\boldmath$x$}}^{1}\cap{\makebox{\boldmath$x$}}^{2}$ (that is on the border between ${\makebox{\boldmath$x$}}^{1}$ and ${\makebox{\boldmath$x$}}^{2}$), there exists $\xi^{2}\in{\makebox{\boldmath$x$}}^{2}$ such that $f(x)=f(x^{1})+\langle\nabla f(\xi^{2}),x-x^{1}\rangle$ and $\lvert x_{1}-x^{1}_{1}\rvert\leq dx_{1}^{2}$ and $\lvert x_{2}-x^{1}_{2}\rvert\leq dx_{2}^{2}$. (take for example for $x^{1}$ the intersection of the line from $x^{0}$ to $x$ with the border between ${\makebox{\boldmath$x$}}^{1}$ and ${\makebox{\boldmath$x$}}^{2}$). We have $\mbox{range}(f,{\makebox{\boldmath$x$}}^{1}\cup{\makebox{\boldmath$x$}}^{2})\subseteq{\makebox{\boldmath$f$}}^{0}+\langle\overline{\nabla}^{1},dx^{1}\rangle[-1,1]+\langle\overline{\nabla}^{2},dx^{2}\rangle[-1,1].$ There also exists $(x,x^{1})\in{\makebox{\boldmath$x$}}^{2}\times{\makebox{\boldmath$x$}}^{1}$ such that $\lvert x_{1}-x^{1}_{1}\rvert=dx^{2}_{1}$ and $\lvert x_{2}-x^{1}_{2}\rvert=dx^{2}_{2}$ (take the corners of the boxes ${\makebox{\boldmath$x$}}^{1}$ and ${\makebox{\boldmath$x$}}^{2}$), so that we also have $[\overline{f^{0}}-\langle\underline{\nabla}^{1},dx^{1}\rangle-\langle\underline{\nabla}^{2},dx^{2}\rangle,\underline{f^{0}}+\langle\underline{\nabla}^{1},dx^{1}\rangle+\langle\underline{\nabla}^{2},dx^{2}\rangle]\subseteq\mbox{range}(f,{\makebox{\boldmath$x$}}^{1}\cup{\makebox{\boldmath$x$}}^{2}).$ This generalizes to the $k$ subdivisions: $\displaystyle\mbox{range}(f,{\makebox{\boldmath$x$}})\subseteq{\makebox{\boldmath$f$}}^{0}+\sum_{i=1}^{k}\langle\overline{\nabla}^{i},dx^{i}\rangle[-1,1]$ (7) $\displaystyle[\overline{f^{0}}-\sum_{i=1}^{k}\langle\underline{\nabla}^{i},dx\rangle,\underline{f^{0}}+\sum_{i=1}^{k}\langle\underline{\nabla}^{i},dx\rangle]\subseteq\mbox{range}(f,{\makebox{\boldmath$x$}})$ (8) The same idea applies to the estimation of robust ranges. Naturally, other schemes can be proposed, relying on the idea that this technique can be seen as using a quadrature formula for integrating the Jacobian of a function. ###### Example 4.3 We consider $f(x)=(2x_{1}^{2}+2x_{2}^{2}-2x_{1}x_{2}-2,x_{1}^{3}-x_{2}^{3}+4x_{1}x_{2}-3)^{\intercal}$ with ${\makebox{\boldmath$x$}}=[0.9,1.1]^{2}$. The results are represented in Figure 1(c). The sampling-based estimation of the image is the dark dots- filled region. We choose $pi:(1\rightarrow 1,2\rightarrow 2)$. Using the preconditioned mean-value extension without partitioning, the over- approximation is the largest green parallelotope and the under-approximation for the joint range is empty. The quadrature formula for the mean-value extension with $k=10$ partitions on one hand, and the order 2 extension of Corollary 3.2 on the other hand, yield two very similar under-approximating yellow parallelotopes. They also yield two very similar green over- approximating parallelotopes. Both approaches actually have comparable precision on the different examples tested. The light green box is the order 2 over-approximation without preconditioning. ###### Remark 4.4 One could be tempted to use more partitions to improve the quality of the approximation. However, the computations yield approximations that are centered at $f(x^{0})$. We can observe that the under-approximating skewed box is actually already very close to being the largest skewed box entirely included in the image, given a fixed skewing and a center at $f(x^{0})$. The order 2 estimation allows for slight decentering, but it would also be possible to use less basic quadrature formulae for that purpose. Quadrature can also be combined with order 2 extensions. Quadrature can also be combined with some classical partition of the inputs, or with different center choices. However, a possibly disjoint union of approximations (corresponding to a classical partition of inputs) is inconvenient, so we would recommend this use only as a property-driven refinement. Finally, as a sound under-approximation is still obtained by considering a sub-region of the input set, refinements can be obtained by detecting and removing sub-regions where Jacobian coefficients are either very sensitive to the inputs or close to zero. ### 4.3 Bounding the Jacobian matrix The approach relies on being able to compute over-approximations of $\nabla f(x)$ over some sub-sets of input box $x$, namely ${\makebox{\boldmath$\nabla$}}^{i}$ for $i$ in $[1,k]$ such that $\\{\lvert\nabla f(x)\rvert,x\in{\makebox{\boldmath$x$}}^{i}\\}\subseteq{\makebox{\boldmath$\nabla$}}^{i}$. Automatic differentiation allows to compute the derivatives, but need to be combined with set-membership methods to handle uncertainties. We have found the combination of automatic differentiation with an evaluation in affine arithmetic to provide a good trade-off between efficiency and precision. All the experiments presented in this work were performed with an implementation relying on this combination. Affine forms provide an interesting combination of parameterization and set-based estimation: a parametric approximate form for $\nabla f(x)$ valid on all box $x$ is computed, that can be instantiated on ${\makebox{\boldmath$x$}}^{i}$ to yield tight over-approximations $\nabla^{i}$, without the need for performing several evaluations of the differentiation. The full description is out of scope, we give below a flavor of the use of affine arithmetic for Jacobian estimation on a simple example. ###### Example 4.5 Consider in Example 4.3 the derivative of $f_{1}(x_{1},x_{2})=2x_{1}^{2}+2x_{2}^{2}-2x_{1}x_{2}-2$ with respect to $x_{1}$, with $(x_{1},x_{2})\in[0.9,1.1]^{2}$. This derivative is $\nabla_{1}f_{1}(x_{1},x_{2})=4x_{1}-2x_{2}$. Evaluation with affine arithmetic first consists in creating a centered form with a fresh noise symbol for each input: $\hat{x}_{1}=1.+0.1\varepsilon_{1}$ and $\hat{x}_{2}=1.+0.1\varepsilon_{2}$, with $(\varepsilon_{1},\varepsilon_{2})\in[-1,1]^{2}$. The gradient evaluated on these affine forms is $\hat{\nabla}_{1}\hat{f}_{1}(x)=2+0.4\varepsilon_{1}-0.2\varepsilon_{2}.$ Here the abstraction is exact. In the general case, affine arithmetic will compute an approximate affine form and bound the approximation error in a new noise term. Let us now use this affine form to over-approximate $\nabla_{1}f_{1}(x)$ over all $x$. This amounts to computing the interval bounds when $\varepsilon_{1}$ and $\varepsilon_{2}$ range in $[-1,1]$, for which we get ${\makebox{\boldmath$\nabla$}}_{1,1}=[1.4,2.6]$. Now we use the same affine form to over-approximate $\nabla_{1}f_{1}(x)$ over ${\makebox{\boldmath$x$}}^{1}$. This amounts to take $\varepsilon_{1}$ and $\varepsilon_{2}$ both ranging in $[-\frac{1}{k},\frac{1}{k}]$. Let us consider $k=10$ subdivisions, we obtain ${\makebox{\boldmath$\nabla$}}^{1}_{1,1}=2+0.4[-0.1,0.1]-0.2[-0.1,0.1]=[1.94,2.06]$. The process can be iterated to compute ${\makebox{\boldmath$\nabla$}}^{i}_{1,1}$ for $i\in[2\ldots k]$, expressing the membership to a subset ${\makebox{\boldmath$x$}}^{i}$ of $x$ as constraints on $\varepsilon_{1}$ and $\varepsilon_{2}$, used to instantiate to ${\makebox{\boldmath$x$}}^{i}$ the estimation of $\nabla_{1}f_{1}(x)$. For instance, in order to compute ${\makebox{\boldmath$\nabla$}}^{2}_{1,1}$, we decompose the computation on the 4 rectangles that define ${\makebox{\boldmath$x$}}_{2}$ (see Figure 1(b)): $(\varepsilon_{1},\varepsilon_{2})\in([\frac{1}{k},\frac{2}{k}]\times[-\frac{2}{k},\frac{2}{k}])\cup([-\frac{2}{k},-\frac{1}{k}]\times[-\frac{2}{k},\frac{2}{k}])\cup([-\frac{1}{k},\frac{1}{k}]\times[-\frac{2}{k},-\frac{1}{k}])\cup([-\frac{1}{k},\frac{1}{k}]\times[\frac{1}{k},\frac{2}{k}])$. ## 5 Application to the reachability of discrete-time systems We consider discrete-time non-linear dynamical systems with inputs of the form $\begin{cases}z^{k+1}=f(z^{k},u^{k})\\\ z^{0}\in{\makebox{\boldmath$z$}}^{0}\end{cases}$ (9) where $f:\mathbb{R}^{m}\rightarrow\mathbb{R}^{n}$ is a vector-valued non- linear function with $m\geq n$, $z\in\mathbb{R}^{n}$ the vector of state variables, $u\in{{\makebox{\boldmath$u$}}}\subseteq\mathbb{R}^{m-n}$ the input signal, and ${\makebox{\boldmath$z$}}^{0}$ the initial set. Given an initial set ${\makebox{\boldmath$z$}}^{0}$, we want to compute the bounded time reachable set of the dynamical system, i.e, the set of states visited by the dynamical system up to a fixed time horizon $K\in\mathbb{N}$. The reachable set can be obtained as the solution of the recursion ${\makebox{\boldmath$z$}}^{k+1}=\\{f(z^{k},u^{k})\|z^{k}\in{\makebox{\boldmath$z$}}^{k},u^{k}\in{{\makebox{\boldmath$u$}}}\\}$, for $k\in[0,K]$. The computation of the reachable set can be seen as a series of images of sets by vector-valued function $f$. We thus can use the results of Sections 2 to 4 to compute approximations of these reachable sets. For conciseness, we consider systems without disturbances and compute maximal (or classical) reachable sets. The algorithms can be straightforwardly extended to robust reach set of systems with disturbances, basically replacing ranges by robust ranges. This allows us to use the lighter notations ${\mathcal{I}}(f,{\makebox{\boldmath$x$}},\pi)$ and ${\mathcal{O}}(f,{\makebox{\boldmath$x$}},\pi)$ to note the under and over- approximating sets introduced in Definition 2.2. #### 5.0.1 Method 1 the first method consists in iteratively using function image, independently for under and over-approximation, taking as input the previously computed approximation of the image. We compute under and over-approximations $I^{k}$ and $O^{k}$ of the reachable set ${\makebox{\boldmath$z$}}^{k}$ by $\begin{cases}I^{0}={\makebox{\boldmath$z$}}^{0},\;O^{0}=z^{0}\\\ I^{k+1}={\mathcal{I}}(f,I^{k},\pi),\;O^{k+1}={\mathcal{O}}(f,O^{k},\pi)\end{cases}$ (10) Indeed, at each step $k$, we have $I^{k+1}\subseteq\mbox{range}(f,I^{k})\subseteq\mbox{range}(f,{\makebox{\boldmath$z$}}^{k})={\makebox{\boldmath$z$}}^{k+1}\subseteq\mbox{range}(f,O^{k})\subseteq O^{k+1}$. With this approach, at each step $k$, under and over-approximations $I^{k}$ and $O^{k}$ of the joint range are used as input for the next step. It is thus particularly important to compute tight under and over-approximations of this joint range. In particular, using the preconditioning of Section 4.1 will often be crucial, both for under and over approximation. This yields Algorithm 1. Algorithm 1 Iterated discrete-time reachability 0: $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$, ${\makebox{\boldmath$z$}}^{0}\subseteq{\mathbb{I}\mathbb{R}}^{n}$ initial state, $K\in\mathbb{N}^{+}$, an over-approximating extension $[\nabla f]$ (see Section 4) 0: $I^{k}$ and $O^{k}$: under and over-approximations of the reachable set $\mbox{range}(f^{k},{\makebox{\boldmath$z$}}^{0})$ for $k\in[1,K]$ $I^{0}\coloneqq{\makebox{\boldmath$z$}}^{0},O^{0}\coloneqq{\makebox{\boldmath$z$}}^{0}$; choose $\pi:[1\ldots n]\mapsto[1\ldots n]$ for $k$ from 0 to $K-1$ do ${\makebox{\boldmath$\nabla$}}_{I}^{k}\coloneqq\|[\nabla f](I^{k})\|$, ${\makebox{\boldmath$\nabla$}}_{O}^{k}\coloneqq\|[\nabla f](O^{k})\|$ $A_{I}^{k}\coloneqq c({\makebox{\boldmath$\nabla$}}_{I}^{k})$, $A_{O}^{k}\coloneqq c({\makebox{\boldmath$\nabla$}}_{O}^{k})$ (supposed non- singular, otherwise taken to identity matrix) $C_{I}^{k}\coloneqq(A_{I}^{k})^{-1}$, $C_{O}^{k}\coloneqq(A_{O}^{k})^{-1}$ ${\makebox{\boldmath$z$}}_{I}^{k+1}\coloneqq{\mathcal{I}}(C_{I}^{k}f,I^{k},\pi)$, ${\makebox{\boldmath$z$}}_{O}^{k+1}\coloneqq{\mathcal{O}}(C_{O}^{k}f,O^{k},\pi)$ if ${\makebox{\boldmath$z$}}_{I}^{k}=\emptyset$ then return end $I^{k+1}\coloneqq A_{I}^{k}{\makebox{\boldmath$z$}}_{I}^{k+1}$, $O^{k+1}\coloneqq A_{O}^{k}{\makebox{\boldmath$z$}}_{O}^{k+1}$ end for At each step $k=0\ldots K-1$, ${\makebox{\boldmath$z$}}_{I}^{k+1}$ is an interval vector such that, if it is non empty, $I^{k+1}=A_{I}^{k}{\makebox{\boldmath$z$}}_{I}^{k+1}\subseteq\mbox{range}(f,I^{k})\subseteq\mbox{range}(f^{k+1},{\makebox{\boldmath$z$}}^{0})$. The over-approximation is computed similarly, and is fully decoupled. #### 5.0.2 Method 2 the second method consists in computing the sensitivity to initial state by approximating the gradient of the iterated function. At each step $k$, we compute the under and over-approximation of $\mbox{range}(f^{k},{\makebox{\boldmath$z$}}^{0})$, i.e. the loop body $f$ iterated $k$ times, starting from the initial state ${\makebox{\boldmath$z$}}^{0}$. This yields the schematic Algorithm 2, with same inputs and hypotheses as in Algorithm 1. Algorithm 2 Discrete-time reachability computed on $f^{k}$ for $k$ from 0 to $K-1$ do $I^{k+1}\coloneqq{\mathcal{I}}(f^{k+1},{\makebox{\boldmath$z$}}^{0},\pi)$, $O^{k+1}\coloneqq{\mathcal{O}}(f^{k+1},{\makebox{\boldmath$z$}}^{O},\pi)$ end for Here, at each step $k$, the under- and over-approximation are both obtained from an over-approximation of $f^{k+1}$ evaluated at the center of ${\makebox{\boldmath$z$}}^{0}$ and an over-approximation of the gradient of $f^{k+1}$ over ${\makebox{\boldmath$z$}}^{0}$: at each step, the gradient can be obtained by differentiating the gradient from the previous step. Of course, this can also be combined with preconditioning. #### 5.0.3 Discussion While relying on the same techniques for range estimation, Algorithm 1 and Algorithm 2 are different in spirit: Algorithm 2 relies only on the propagation of over-approximations to deduce under-approximations. In particular, the under-approximation may be empty at some step $k$, and become non-empty again at further steps (a similar remark was made in the context of continuous systems in Goubault and Putot (2017)). In comparison, Algorithm 1 needs at each step an under-approximating box or skew box that is non-empty for all components. On the other hand, Algorithm 2 is more costly as it requires a differentiation of the iterated function. ## 6 Implementation and examples We now present results on small systems from the literature. The approach is implemented as part of the RINO C++ prototype, available from https://github.com/cosynus-lix/RINO. The prototype allows to experiment the function range estimation, but its actual target is discrete and continuous- time reachability, combining for continuous-time the techniques presented here with Taylor model methods Goubault and Putot (2019, 2020). The prototype uses the fadbad++ (http://www.fadbad.com/fadbad.html) automatic differentiation library and the aaflib (http://aaflib.sourceforge.net/) affine arithmetic library. The timings are given on a Macbook Pro 2.6GHz Intel Core i7 and 32Gb of RAM. ##### Test Model We consider the test model Dreossi et al. (2016): $\begin{split}x_{1}^{k+1}&=x_{1}^{k}+(0.5(x_{1}^{k})^{2}-0.5(x_{2}^{k})^{2})\Delta\\\ x_{2}^{k+1}&=x_{2}^{k}+2x_{1}^{k}x_{2}^{k}\Delta\end{split}$ with as initial set a box $x_{1}\in[0.05,0.1]$ and $x_{2}\in[0.99,1.00]$, and $\Delta=0.01$. Figure 2 shows the under and over-approximated reachable sets (respectively the filled yellow region and green parallelotope) over time up to 25 steps with Algorithm 1. They are obtained in 0.02 seconds. We can observe that the under and over-approximations are very close one to another, confirming the accuracy of the results. Figure 2: Skewed box under and over-approximations for 25 steps (Algorithm 1) for the test model ##### SIR Epidemic Model We now consider the SIR epidemic model with the parameters of Dreossi et al. (2016). $\begin{split}x_{1}^{k+1}&=x_{1}^{k}-\beta x_{1}^{k}x_{2}^{k}\Delta\\\ x_{2}^{k+1}&=x_{2}^{k}+(\beta x_{1}^{k}x_{2}^{k}-\gamma x_{2}^{k})\Delta\\\ x_{3}^{k+1}&=x_{3}^{k}+\gamma x_{2}^{k}\Delta\end{split}$ We compute the reachable set up to 60 steps from the initial box $(x_{1},x_{2},x_{3})\in[0.79,0.80]\times[0.19,0.20]\times[0,0.1]$. The parameter values are $\beta=0.34$, $\gamma=0.05$, and $\Delta=0.5$. The reachable sets computed in 0.05 seconds with Algorithm 1 up to 60 steps are represented for $(x_{1},x_{2})$ in Figure 3. Figure 3: Skewed box under and over-approximations for 60 steps (Algorithm 1) of the SIR epidemic model We can note in particular from the zoomed reachable set in the Figure, which corresponds to last step (60), that the under-approximation (in yellow) is still of good quality (the purple dots correspond to sample executions). However, only Algorithm 2 is able to compute non-empty approximations when taking as initial condition $x_{3}=0$ which is of empty interior, instead of $x_{3}\in[0,0.1]$. We obtain in 0.05 seconds the very tight approximations of the projections of the components represented Figure4(a). (a) SIR epidemic model, 60 steps (b) Bees model, 1500 steps Figure 4: Projected under and over-approximations, Algorithm 2 ##### Honeybees Site Choice Model We consider the reachable sets up to 1500 steps of the model studied in Dreossi et al. (2016): $\begin{split}x_{1}^{k+1}&=x_{1}^{k}-(\beta_{1}x_{1}^{k}x_{2}^{k}+\beta_{2}x_{1}^{k}x_{3}^{k})\Delta\\\ x_{2}^{k+1}&=x_{2}^{k}+(\beta_{1}x_{1}^{k}x_{2}^{k}-\gamma x_{2}^{k}+\delta\beta_{1}x_{2}^{k}x_{4}^{k}+\alpha\beta_{1}x_{2}^{k}x_{5}^{k})\Delta\\\ x_{3}^{k+1}&=x_{3}^{k}+(\beta_{2}x_{1}^{k}x_{3}^{k}-\gamma x_{3}^{k}+\delta\beta_{2}x_{3}^{k}x_{5}^{k}+\alpha\beta_{2}x_{3}^{k}x_{4}^{k})\Delta\\\ x_{4}^{k+1}&=x_{4}^{k}+(\gamma x_{2}^{k}-\delta\beta_{1}x_{2}^{k}x_{4}^{k}-\alpha\beta_{2}x_{3}^{k}x_{4}^{k})\Delta\\\ x_{5}^{k+1}&=x_{5}^{k}+(\gamma x_{3}^{k}-\delta\beta_{2}x_{3}^{k}x_{5}^{k}-\alpha\beta_{1}x_{2}^{k}x_{5}^{k})\Delta\\\ \end{split}$ with as initial set the box $x_{1}=500$, $x_{2}\in[390,400]$, $x_{3}\in[90,100]$, $x_{4}=x_{5}=0$ and the parameter values $\beta_{1}=\beta_{2}=0.001$, $\gamma=0.3$, $\delta=0.5$, $\alpha=0.7$, and $\Delta=0.01$. Algorithm 1 runs very fast, taking only 1.7 seconds for the 1500 steps. But the under-approximation is very soon empty and the over-approximation tends to strongly widen after 800 steps. In comparison, Algorithm 2 takes 57 seconds to complete the 1500 reachability steps. But the projected under-approximations are very tight, close to the over-approximations, as can be seen on Figure 4(b) which represents under and over-approximations for all components as functions of steps. These results should also be compared to the much wider over-approximation of Dreossi et al. (2016) (Figure 7, where time is number of steps divided by 100), obtained in 81 seconds. ## 7 Conclusion and future work We focused on new AE under-approximating extensions and their accurate practical evaluation for non-linear vector-valued functions, and exemplified their interest for the reachability of discrete-time systems. 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# Multi-agent simulation of voter’s behaviour Albin Soutif Carole Adam Sylvain Bouveret Univ. Grenoble-Alpes, Grenoble Informatics Laboratory (_This is an internship report originally written in June 2018 by A. Soutif, ENSIMAG intern_ _under the supervision of C. Adam and S. Bouveret_ ) ###### Abstract The goal of this paper is to simulate the voters’ behaviour given a voting method. Our approach uses a multi-agent simulation in order to model a voting process through many iterations, so that the voters can vote by taking into account the results of polls. Here we only tried basic rules and a single voting method, but further attempts could explore new features. Keywords: Computational social choice, Iterative voting, multi-agent simulation ## 1 INTRODUCTION A voting process involves the participation of many people that interact together in order to reach a common decision. In this paper, we focus on voting processes in which a single person is elected. A voting method is defined as the set of rules that determine the winner of the election, given an input from each voter, for example their preferred candidate or an order relation between all candidates. Social Choice Theory is the field that studies the aggregation of individual preferences towards a collective choice, like for example electing a candidate or choosing a movie. Computational social choice is a recent field which aim is to apply computer science to social choice problems [3]. Here we aim to design a multi-agent simulation in order to study the collective decision process involved in an election. A multi-agent simulation is a simulation in which we make several autonomous agent interact with each other in a defined environment. The goal of this kind of simulation is to study the global behaviour emerging from the local actions of the agents [4, 6, 5]. There are only a few attempts to mix the field of multi-agent simulation and computational social choice to model a voting process, and that is the kind of approach we use in this paper. The goal of this approach is to study the dynamics of voting methods through many iterations. Here, we will focus on one particular voting method which is plurality voting. In this method, voters vote for only one candidate and the winner of the election is the candidate with the most votes. Our approach was supported by the results of a study on the French presidential election of 2017 [7]. In this study, people were asked to vote with different voting methods, and to tell for whom they officially voted. The goal of this study was to better understand the properties of several voting methods. Among these voting methods, one in particular helped to build this model, because it gave a precise insight of voters’ voting intentions (not just their favourite candidate), it is the continuous evaluation method. In this method, a voter gives a mark between 0 and 100 to each of the 11 candidates. Thanks to this study, we had access to over 30 000 of these continuous evaluations, which helped us initialise our model. A voting process can be modelled intuitively as a multi-agent system. Indeed, voters and candidates can be modelled as autonomous agents, the real world, or the internet, is the environment in which these agents evolve, and the political debates, polls, conversations … Are the interactions which can influence their final decision. In a multi-agent simulation, it is reasonable to limit ourselves to a simpler model, in which the agents evolve in a limited environment and have much simpler interactions with each other. Otherwise it would be too difficult to model, to interpret, and too computationally expensive. First we will review the state of the art in this field (Section 2), then we will present the way we built our model (Section 3), then our results along with our interpretation (Section 4), and eventually we will conclude and discuss some possible extensions of the model (Section 5). ## 2 STATE OF THE ART Some interesting attempts have been made at building voting process models. Here we will review the ones that inspired us the most in building our model. ### 2.1 Static models A static simulation has been made by Ka-Ping Yee [10] in which the voters are modelled by 2-dimensional points. In this simulation a normal distribution of voters is drawn around every point, and each point is coloured according to the candidate that won the election with that distribution of voters. The conclusion of these simulations highlighted the different behaviours of some voting rules. ### 2.2 Iterative models Airiau et al. [1] attempted to model the outcome of an election where voters could react strategically by learning from the others and trying to react in order to maximise an utility function. The authors found out that for a given set of rule and a precise methodology, the outcome was convergent and had good mathematical properties. The drawback is that for the convergence to be certain, the voters had to vote one by one and know at each step which candidate was winning, which is not a realistic procedure. Another attempt has been made for educational purpose by a network of researchers in multi-agent simulation (MAPS) [2]. They made a multi-agent simulation called “the Iznogoud model”, with simple interactions between 2-dimensional agents. The interactions used were local (each voter directly influenced its surroundings), whereas in our model there are no local interactions because we only focused on the global effect of polls, and not on the effect of local interactions such as political conversations between individuals. Also in the Iznogoud model, the voters were initialised randomly, making it impossible to compare the results to any real case scenario. In both the approaches from Ka-Ping Yee [10] and the Iznogoud simulation [2], the voters are placed in a 2-dimensional space. In Ka-Ping Yee simulation the model is not dynamic, so such a model would not help us study the dynamics of voting methods. In the Iznogoud simulation, the interactions between voters are local and very simple, and the model is initialised with a random population of voters. In our model, we will try to integrate interactions that are more realistic, and to initialise the population of agents with the help of the results of the ”voter autrement” experimentation [7]. ## 3 OUR MODEL We choose to model each voter as a point in a 11 dimensional space, and we initialise the model with the study’s continuous evaluation results, giving as coordinates to a voter the marks he has given to the different candidates. Because we also know the official vote of each voter, we integrate the candidates into that space by taking the centroids of their voter’s continuous evaluation. The model is dynamic and at each iteration, the voters move and then a plurality vote is computed. To compute this vote, we make each voter vote for the closest candidate from him and the candidate with the most votes is the winner. In order to make the agents interact between each other, we use polls that give information to everyone on the current plurality vote outcome. The voter compute their move taking into account the results of the last poll. That’s the way the agents interact between each other. The model ran with different set of utility functions and agent behaviour. Basically, we make each agent move with a velocity vector computed at each iteration, and several methods were used to compute this one. We will discuss those methods and their expected properties in this section. ### 3.1 Formal notations Formally, we note as $V_{i}$ the set of voters at iteration $i$, and $C$ the (static) set of candidates. Both of these sets contain a finite number of 11-dimension vectors. Thus $v_{k,i}$ is the position of voter $v_{k}$ at iteration $i$. $s_{i}$ is the last poll available at iteration $i$, also represented as an 11-dimension vector, containing the number of votes for each candidate in the poll’s sample. $V(v_{k,i},s_{i})$ is the velocity vector of voter $v$, knowing its current position and the result of the last poll. At each iteration, the next position vector of each voter $v_{k,i+1}\in V_{i+1}$ is given by the equation (1), as the sum of its previous position and its velocity vector computed from this position and the last poll. $v_{k,i+1}=v_{k,i}+V(v_{k,i},s_{i})$ (1) ### 3.2 Utility functions Before introducing the movement rules of our model, we introduce the concept of utility function that we will later use in these rules. A utility function is defined for each voter, as the utility a candidate would have for this voter if he was elected. We define a utility function $u:\mathbb{R}^{11}\times\mathbb{R}^{11}\rightarrow\mathbb{R}$, giving for the first point (a voter) the utility that the second point (a candidate) would have if winning the election. In practice, we will use $u(v_{k,i},c)$ for $v_{k,i}\in V_{i}$ and $c\in C$. In our model, we want that utility function to be in line with the fact that the voters vote for the closest candidate from them. So intuitively, the closer a voter is to a candidate, the more he likes him. The main property that we want from these functions is that they are decreasing functions of the distance between the voter and the candidates, and that they are not linear (to model that some voters are strongly attached to a candidate). Noting $dist(v,c)$ the Euclidean distance between voter $v$ and candidate $c$, we designed 2 different utility functions representing different reasoning. The first utility function given by equation (2) below simply expresses utility as inversely proportional to the distance with the candidate. $u:v,c\rightarrow\frac{1}{dist(v,c)}$ (2) The second utility function that we designed also models the fact that a voter can be strongly opposed to a candidate when he is too far away from him (further than a parameter $\alpha$). In this case this utility function becomes negative, ensuring that the voter will not get any closer to this candidate (repulsion). This second function is given by equation (3) below. The drawback is that some voters just flee away from the whole set of candidates, which we model as abstention in the vote. $u:v,c\rightarrow\frac{\alpha-dist(v,c)}{(1+dist(v,c))^{2}}$ (3) ### 3.3 Velocity computation In this section we describe three techniques that can be used to compute the velocity vector of the voters. #### First approach: 3-pragmatist rule A first approach is to use a rule described in [1] which is the three pragmatist rule. This rule makes the voter vote for his favourite candidate amongst the first three candidates of the election, which represent a strategic vote for a candidate who has a chance to win. We integrate that in our model by making each voter move towards this candidate, once given the results of the last poll. The 3-pragmatist velocity vector towards candidate $c$, the closest to voter $v_{k,i}$ among the first 3 candidates in results of poll $s_{i}$, is given by the following equation: $V(v_{k,i},s_{i})=e_{v_{k,i},c}$ (4) where $e_{a,b}$ is the unit vector from point $a$ to $b$. #### Second approach: maximum expected utility A second approach is to use a utility function and to move toward the candidate that maximises the expected utility. In order to approximate this expectation, we use the poll results to compute a probability that each candidate wins, given by the following equation as the proportion of expressed votes in favour of this candidate. $p(wins(c)\|s_{i})=\frac{numberofvotesforcins_{i}}{totalnumberofvotesins_{i}}$ (5) Once we have the probability that each candidate wins, computed from the polls, the expected utility of their victory is the product of this probability by the utility of them winning. This approximation gives a chance even to candidates who have a small number of votes, provided that the utility of them winning is high enough to compensate. On the contrary, other approaches would just dismiss these candidates altogether, considering that voting for them is useless because they have no chance to win in the end. Thus we can compute the candidate selected by the voter as the one that maximises this expected utility, as expressed by the following equation: $c=argmax_{c\in C}[u(v_{k,i},c)p(wins(c)\|s_{i})]$ (6) In the end the velocity vector is computed as the unit vector from voter $v_{k,i}$ to that candidate $c$: $V(v_{k,i},s_{i})=e_{v_{k,i},c}$ (7) #### Third approach: personalised opinion centre The third approach is different in the sense that rather than moving towards a single candidate, it makes the voter move towards a personalised opinion centre. In this approach we compute velocity as a linear combination of the expected utilities of the different candidates, as expressed by the following equation. $V(v_{k,i},s_{i})=\sum_{c\in C}e_{v_{k,i},c}u(v_{k,i},c)p(wins(c)\|s_{i})$ (8) Since movement towards each candidate is weighed by the expected utility of that candidate, we can expect that this vector will somehow be pointing in a direction that will make the voter closer to the candidate that maximises its expected utility. ## 4 RESULTS ### 4.1 Visualization In order to visualise the multi-agent model behaviour, we choose to use dimensionality reduction techniques. Two of them were tested, PCA (Principal Component Analysis, [9]) and T-SNE (t-distributed stochastic neighbor embedding, [8]). We choose PCA because it is less computationally expensive and is not stochastic. The PCA results on initialisation of the model are shown on Figure 1. Figure 1: Principal Component Analysis to initialise the model ### 4.2 Parameters Before presenting the results, we will review the set of parameters used and the circumstances of the simulations. The simulations were ran on a sample of 1000 voters which are drawn among the 30000 voters who took part in the “Voter Autrement” experiment. We will use the same seed for the random generator for this sample to allow reproducibility of the experiments. We have defined above two utility functions and three velocity formulas that can be tested. We are not going to present nor simulate all the combinations, but we will rather focus on the most interesting ones, and try to identify the impact of the parameters on the simulation. Below are the official votes given by the sample’s voters. Compared to the actual results of the 2017 presidential election, it is clearly not representative, but it will still be interesting to compare those results to the outputs of our model. • Jean-Luc Melenchon 403 * • Jean Lassalle 7 * • François Asselineau 5 * • Nicolas Dupont-Aignan 11 * • François Fillon 34 * • Benoit Hamon 144 * • Philippe Poutou 12 * • Emmanuel Macron 200 * • Marine Le Pen 18 * • Nathalie Arthaud 3 * • Jacques Cheminade 2 ### 4.3 Three-pragmatist rule We tried the three-pragmatist velocity rule with different poll results. We can see that this rule is very sensitive to the order of candidates given by the poll (because it only draws electors towards the first three candidates in the results of the poll). Indeed, with a poll giving Nathalie Arthaud above Emmanuel Macron (Figure 2, the voters closer to Macron (who is not in the top 3) move toward their closest candidate among the three winners, in this case Benoit Hamon. This leads to Hamon winning the election against Jean-Luc Melenchon. Figure 2: With a poll where Nathalie Arthaud is above Emmanuel Macron On the contrary, if the order in the poll results is the actual order (Macron above Arthaud), the real order is just confirmed by the next iterations (Figure 3). Figure 3: With a poll where Emmanuel Macron is above Nathalie Arthaud This rule gives accurate results regarding the actual official votes of our sample, but only for the first three candidates of the poll. It also gives totally wrong results if the poll order is not the same as the actual order. In fact, some people behave in such a way that they will not change their vote, even if their candidate has no chance to win at all. Explanations can be a desire to show contestation, or a strong attachment to this candidate and their ideas. ### 4.4 Maximum expected utility In our next experiment, we used our second expected utility function and one poll, and computed velocity with the second technique, of maximising expected utility. Results are shown on Figure 4. Figure 4: with utility function (2) and one poll In this experiment, Emmanuel Macron has nearly as many votes as expected in the sample. However, the utility function still needs to be tuned, because all the voters who initially preferred Benoit Hamon, reported their votes (after the poll) towards Jean-Luc Melenchon, which is not the case in the real sample results. One explanation is that our expected utility model is imperfect; indeed, people have a non-objective way of thinking in this situation, and this is hard to model in a homogeneous way. ### 4.5 Utility speed (c) In the next scenario, we used the third velocity formula, that moves voters towards a personalised opinion centre. We compared the results obtained with our 2 utility functions, and also with a single poll or regular polls along the campaign. Figure 5 illustrates the results with utility directly inversely proportional to distance (first formula), and one single poll. In this figure, the number of votes for Jean-Luc Melenchon is accurate, but Emmanuel Macron’s votes are too low. Figure 5: With utility function (1) and one poll With that same utility function, regular polls do not greatly change the results (Figure 6). We can observe nearly the same behaviour as in the previous figure, even though we updated the information available to the agents every 20 iterations. Figure 6: with utility function (1) and one poll every 20 iteration Finally we also tested the second utility function introducing repulsion towards candidates that are too far, still with regular update polls. In Figure 7 the results are much more tied, but they are also less accurate with respect to the actual official votes of the sample. It is also hard to conclude on a final order in this case, given that the rule does not converge. Figure 7: With utility function (2) and one poll every 20 iteration ### 4.6 Periodicity of polls The simulations run show that periodical polls do not fundamentally change the result of the election. Indeed, quite often the results of the polls (here made on 10% of the total population) are consistent with the order of the real results, so the movement tendency and the order of candidates are just confirmed a bit more by each new poll. And whenever a poll gives a different order, it is often counter-balanced by the next poll. However, in a real situation, other events can have an effect on the voters preferences, such as media scandals (for instance the Penelope gate scandal involving François Fillon during the French 2017 presidential elections, which probably made him lose that election). In this case, even if some people are not directly deterred from voting for their favourite candidate by the scandal, they may be indirectly led to do so by strategy, if the subsequent polls show that this candidate’s chance to win has dropped too much. ### 4.7 Notes about properties The movement rules that we presented above were mainly built to make the voters in the simulation behave somehow close to how the actual voters behave. Because we focused on those properties, and did not make any mathematical analysis of the rules, they do not necessarily have good mathematical properties, in particular convergence. It is hard to interpret the results if the simulations are not converging towards at least a fixed order of the candidates, if not a fixed number of votes. Indeed, we have no way to link the number of iterations in the simulation to a real duration (in days), so any interpretation would remain a strong supposition. The results presented here have been obtained after a large number of iterations, when the electors are ”caught in a pit” and mainly move towards one candidate, so that the order of the candidates does converge, but the convergence of the number of votes is not guaranteed. ## 5 CONCLUSION AND OPENING The models presented above have shown different interesting behaviours. The one which gave the most accurate results is the simplest one, using the three- pragmatist rule, even though it does not allow more than 3 candidates to receive votes. To extend this approach it would be interesting to experiment a k-pragmatist rule for different values of k, and play around it (maybe compute a velocity vector using different k-pragmatist results). The maximum expected utility rule should have really good properties, but the utility functions were chosen quite arbitrarily. We struggled to accurately compute an expected utility, because people do not really conduce complex probabilistic calculus, and it is hard to model what really happens in someones’ mind when making a voting decision. Many factors might play a role such as emotions or cognitive biases. Also, the second utility function uses a customisable $\alpha$ parameter to represent the tolerance before a voter is repulsed by a candidate. Here we used the same $\alpha$ for each voter, but we could also compute an individual value for each voter, as a function of his starting position. Finally, in this paper we only gave information to the voters through polls, unique or recurrent, but more complex events could be modelled, as for example media coverage of scandals, the use of social networks to communicate, or the influence of other voters on each voter’s opinions. ## References * [1] Stéphane Airiau, Umberto Grandi, and Filipo Studzinski Perotto. Learning agents for iterative voting. In Rothe J., editor, International Conference on Algorithmic Decision Theory, volume 10576 of Lecture Notes in Computer Science, pages 139–152, Cham, 2017. Springer. * [2] Frédéric Amblard, Thomas Louail, Romain Reulier, Paul Salze, and Patrick Taillandier. Modèle iznogoud : fiche pédagogique. https://maps.hypotheses.org/production-pedagogique-de-maps/modeles-mapsiens/modele-iznogoud, 2014\. * [3] Felix Brandt, Vincent Conitzer, Ulle Endriss, Jérôme Lang, and Ariel D Procaccia. Handbook of computational social choice. Cambridge University Press, New York, NY, USA, 1st edition, 2016. * [4] Joshua M Epstein and Robert Axtell. Growing artificial societies: social science from the bottom up. Brookings Institution Press, 1996. * [5] Charles M. Macal. Everything you need to know about agent-based modelling and simulation. Journal of Simulation, 10(2):144–156, 2016. * [6] Charles M. Macal and Michael J. North. Tutorial on agent-based modelling and simulation. Journal of Simulation, 4(3):151–162, 2010. * [7] CoCoRICo-CoDec project. Experimentation voter autrement. https://vote.imag.fr/, 2017. * [8] Laurens Van der Maaten and Geoffrey Hinton. Visualizing data using t-sne. Journal of machine learning research, 9(11), 2008. * [9] Svante Wold, Kim Esbensen, and Paul Geladi. Principal component analysis. Chemometrics and intelligent laboratory systems, 2(1-3):37–52, 1987\. * [10] Ka-Ping Yee. Voting simulation visualizations. http://zesty.ca/voting/sim/, 2005.
# Autoencoder-based Condition Monitoring and Anomaly Detection Method for Rotating Machines Sabtain Ahmad, Kevin Styp-Rekowski, Sasho Nedelkoski, Odej Kao Distributed and Operating Systems TU Berlin Berlin, Germany <EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract Rotating machines like engines, pumps, or turbines are ubiquitous in modern day societies. Their mechanical parts such as electrical engines, rotors, or bearings are the major components and any failure in them may result in their total shutdown. Anomaly detection in such critical systems is very important to monitor the system’s health. As the requirement to obtain a dataset from rotating machines where all possible faults are explicitly labeled is difficult to satisfy, we propose a method that focuses on the normal behavior of the machine instead. We propose an autoencoder model-based method for condition monitoring of rotating machines by using an anomaly detection approach. The method learns the characteristics of a rotating machine using the normal vibration signals to model the healthy state of the machine. A threshold-based approach is then applied to the reconstruction error of unseen data, thus enabling the detection of unseen anomalies. The proposed method can directly extract the salient features from raw vibration signals and eliminate the need for manually engineered features. We demonstrate the effectiveness of the proposed method by employing two rotating machine datasets and the quality of the automatically learned features is compared with a set of handcrafted features by training an Isolation Forest model on either of these two sets. Experimental results on two real-world datasets indicate that our proposed solution gives promising results, achieving an average F1-score of 99.6%. ###### Index Terms: LSTM autoencoder, feature extraction, anomaly detection, condition monitoring, rotating machines ## I Introduction In the modern day industry, machine systems are becoming more complex and fulfill critical tasks. To enhance their reliability, the condition of the system should be monitored. Any rotating machine, e.g. a pump, compressor, or steam turbine, will eventually reach a point of poor health. One effective strategy for enhancing their reliability and cost-effective maintenance is to utilize Condition Monitoring (CM) and Prognostics and Health Management (PHM). The aim is to identify unexpected anomalies, faults, and failures [1]. Prognostics systems built on a data-driven approach acquire data in-situ using a network of sensors that monitor the system [3]. The dataset evolving from the measurements of the sensors usually has a high dimensionality and may also contain unwanted interference and noise. This dimensionality of the data has a direct impact on the training time as well as the accuracy of neural network- based models. A workaround for dealing with such a problem is to reduce the dimensionality of the input signal by extracting features that carry the health information of the system. Several methods have been proposed for achieving this, for instance, extracting temporal features by computing the root mean square (RMS), Skewness, Kurtosis, or Peak to Peak distance [2]. The main attributes of an ideal anomaly detection and condition monitoring system include the ability to collect useful features and the utilization of these features to identify the deteriorating condition of the machine by observing the deviation from the normal (healthy) behavior. Manual feature engineering methods require domain knowledge combined with “trial and error” strategies. However, the advent of deep learning and recent progress in autoencoder based models has provided an alternative way for feature extraction and dimensionality reduction. By stacking up layers to form deep autoencoders and by reducing the number of units in the hidden layers, it is expected that hidden units will extract features that will represent the data. The best features for the task at hand are learned directly from the data, thus avoiding ad-hoc trial and error strategies. Since the autoencoder creates a reduced representation of the data, it seems intuitive to use this representation for anomaly detection. The assumption is that the autoencoder only learns to map normal data points or inliers and does not include anomalies in the trained representation. Hence, trying to reconstruct anomalous data points will fail and incur huge losses. Finally, this reconstruction error can be used as a basis for calculating anomaly scores and labeling of unseen data points. For rotating machine (RM) anomaly detection in general, it is much easier to collect large amounts of data than to accurately obtain their corresponding labels. Particularly in cases where faults or degradation evolve naturally over time. Correctly assigning labels is susceptible to the data ambiguity issue, especially at a pivotal stage when the machines exhibit early signs of failure but are far from obvious when equated with the fully developed faults. Moreover, anomalies are rare events, having prior knowledge about all possible anomalies is almost impossible, so the attention is shifted from anomalous to normal states for which data is available in large quantities. The motivation is that by modeling the normal behavior of the machine the system will also be able to detect anomalies that have not been observed previously. We propose a framework for RM condition monitoring and anomaly detection based on Long short-term memory (LSTM) autoencoder networks. We demonstrate its applicability in both anomaly detection and condition monitoring by evaluating it on two real-world datasets. The advantages of the autoencoder based approach include the ability to work without any preprocessing, without any predetermined transformations such as FFT, without any manual feature engineering, without any feature selection, and the fact that it does not limit itself to the preidentified anomalies; it has the potential to detect new anomalies which have never been seen before. We also show that the time- domain features learned in the process can be used to enhance the performance of a simpler detection model such as Isolation Forest and thus can obviate the need for manual feature extraction. The flowchart of the proposed method is displayed in Fig. 1. The measured vibration signals will be preprocessed and afterward, the training of the model is completed that is able to extract features and reconstruct the signal for the final anomaly detection. Figure 1: Framework flowchart of the proposed method. The remainder of this paper is organized as follows. We will discuss related works in Section II. Section III discusses our proposed LSTM-based autoencoder method for robust feature extraction and anomaly detection. Sections IV and V discuss the experimental results, while the conclusions and future works are presented in Section VI. ## II Related Work A plethora of studies has been conducted in the field of RM fault/anomaly detection and time-series anomaly detection in general [4]. Features and their generation are important concepts in data analysis and anomaly detection. The use and selection of features are crucial for measuring differences in data thus detecting anomalies. Traditionally, features extracted from time or frequency domain have been used for monitoring the condition or computing the remaining useful life (RUL) of the machine [5, 6]. Many extracted features are influenced by operating conditions and are insensitive to anomalies. Researchers have tried to increase the performance of anomaly detection methods for RMs by performing multivariate analysis while using multiple features together, [7] computed 21 bearing features using signal processing techniques. Others have tried to achieve this by constructing features for specific failure modes and performing multivariate analysis based on them as did the authors in [8] by constructing 5 features each of which was sensitive to a different failure mode. However, such manually extracted features are not generalizable and fail to provide useful machine health information in especially unknown cases. More recently, [9] tried to classify various bearing classes by using a set of time-frequency domain features and artificial neural networks. [10] replaced the manual feature extraction step by applying 1-D convolutional neural networks (CNN) to raw motor signals. The evaluation performed on bearing fault detection demonstrated the superiority of their approach compared to conventional feature extraction methods. [11] trained an autoencoder for feature extraction and used these features to train a supervised fault detection classification model. [12] studied several one-class classifiers such as nearest neighbors and k-means for detecting faulty rotor bars in an induction motor. They concluded that the k-nearest neighbor method stood out among all the tested methods. Deep neural network-based architectures in particular autoencoders are successfully employed for supervised classification of faults into different fault categories by using time or frequency domain features extracted using prior knowledge [13]. A probabilistic framework for anomaly detection in natural gas consumption time series is introduced in [14]. However, the prediction method predicts the consumption levels using other independent variables and does not incorporate the temporal information that the data had to offer. [15] employed an vanilla autoencoder to detect anomalies in the electric power system by embedding the temporal information using sliding windows. Reconstruction errors obtained on sliding windows were used to compute anomaly scores. The Inclusion of temporal information using a sliding window works well in some cases but is not scalable usually. The current RM anomaly detection methods, as discussed above, face one or several of the following limitations: labor-intensive manual feature extraction, the requirement of accurately labeled datasets, or failure to incorporate temporal information. Time series data such as sensor data is best modeled as a sequence where the data point at each timestep is dependent on the previous data points. LSTM-based autoencoders are capable of dealing with the time-series sequences and can take variable length input. To overcome issues described above, in this study, we train an LSTM-based autoencoder over the vibration signals to autonomously monitor the condition of RMs and extract time-domain features to provide an alternative for manual feature extraction. To the best of our knowledge, this is the first time that an unsupervised method based on LSTM-autoencoders is used for identifying fault/anomalies in accelerometer vibration signals. ## III Rotating Machine Anomaly Detection We address the problem of anomaly detection in abnormal vibratory phenomena captured through the accelerometer sensors mounted on RMs which should indicate a deterioration of the system. As opposed to traditional health prognostic systems that usually encapsulate feature extraction and anomaly detection as distinct blocks, the proposed system takes directly raw time- series vibration signals as input and it can efficiently learn optimal features and based on these features determine the system’s health. In the case of multi-dimensional input, the set of M sensors $\\{m_{1},...,m_{M}\\}$ (also called generators) are used to capture the behavior of a RM. This measured data is fed as input to an LSTM-based autoencoder (LSTM-AE) that is trained over the vibration signals via batch gradient descent to minimize a reconstruction error term between an original signal and its reconstruction. Specifically, the encoder maps an input vector x to a lower-dimensional hidden representation h by an affine mapping following a nonlinearity and the decoder correspondingly generates an estimation x’ of the input vector x. AEs belong to the unsupervised representation learning class which try to model the data distribution through the discovery of a set of latent representations, also called embeddings, whose variations capture most of the structure of the original data distribution [16]. These hidden-layer units or low-dimensional embeddings force the AE model to learn the key representations from the original vibration signal. The encoder generates a rich non-linear set of features from the sensory data and the decoder learns to reconstruct the original signal using these features. The motivation to use autoencoders is their ability to detect anomalies based on the fact that anomalies are quite rare and deviate greatly from the general pattern in normal healthy data. The model is trained with the aim to learn the normal behavior of a RM, thus not recognizing anomalies. Figure 2: Condition monitoring of RMs through diagnosis of anomalies using LSTM-AE. The encoder and decoder are two main components of the network and both are based on Long short-term memory network (LSTM) units. LSTMs are a type of recurrent neural network (RNN) that can integrate the temporal information into the network and maintain a hidden state vector which acts as a memory for the past information [17]. ##### Encoder We observe the input data sequence denoted by ${\mathbf{X=(x^{(1)},x^{(2)},...,x^{(N)})}}$, where ${\mathbf{x^{N}=(x_{1}^{(N)},x_{2}^{(N)},...,x_{T}^{(N)})}}$ ${\in{R^{Txd}}}$, meaning that for each index N there is a d-dimensional time-series sequence with T timesteps each. We use the RNN to process the variable input sequence and to extract the sequential information from the time-series data. For each sequence this is given by; $C_{t}^{\prime}=tanh(W_{C}.x_{t}+R_{C}h_{t-1}+b_{C})$ (1) where ${C_{t}}^{\prime}$, $x_{t}$ and $h_{t-1}$ are memory state, input and output vectors from last step and $W_{C}$, $R_{C}$ and $b_{C}$ are input weights, recurrent weights and the bias. Tanh is used as a non-linear activation function, whose output range lies in the interval $[-1,1]$. The input sequences are passed through the encoder part of the LSTM network which encodes an input sequence or batch of sequences using LSTM units and updates its hidden state. The output of the encoder is given by; $h_{t}=\sigma^{e}_{\phi}(x_{t},h_{t-1})$ (2) where $h_{t}$ is the output of the $i^{th}$ LSTM-encoder, $\phi$ represents the parameter set of the encoder and to avoid the vanishing and exploding gradients issue, $\sigma$ for both encoder and decoder is chosen as ReLU activation function. ##### Decoder: The representations obtained from mapping the vibration signal to lower- dimensional embeddings through the encoder are used by the decoder to reproduce the original signal. The Output from the last encoder of the network becomes the input to the LSTM-decoder network: $h_{t}{{}^{\prime}}=\sigma^{d}_{\varphi}(h_{t},h^{\prime}_{t-1});\;\;\;x^{\prime}_{t}=\sigma(h^{\prime}_{t})$ (3) where the set of parameters of the decoder is represented by $\varphi$ and $x^{\prime}$ is the reconstructed input which is used to compute the reconstruction error (RE) = ${\|\|x_{t}-x^{\prime}_{t}\|\|^{2}}$, also known as the Mean Squared Error. The error is required to update the network’s encoder and decoder parameters and later, compute the anomaly scores. The AE model should be sensitive enough to reproduce the original signal but insensitive enough to the training data and noise, such that the model learns a generalizable representation of the data. During training, the focus of the model lies in learning the normal behavior of the machine, hence anomalies are not included in the training data. Consequently, during prediction, a slight deviation from the normal behavior would increase the reconstruction error (RE). So, monitoring the increase in reconstruction error gives the possibility to detect anomalies but also anticipate the fault in advance by identifying the degradation point, the timestamp after which the RE starts to increase. ### III-A Model Training The raw vibration data is normalized first to have zero mean and unit variance. We divide the RM’s vibration data into four sets: a training set ($T_{N}$), two validation sets ($V_{N}$ & $V_{A}$), and a test set ($T_{A}$). The distribution of data among these four sets is as follows: $T_{N}$ 70%, $V_{N}$ & $V_{A}$ 5% (each) and $T_{A}$ 20%. $T_{N}$ consists only of normal sequences and is used for training the LSTM-AE model. NASA bearing dataset (dataset-1, details in section IV ) does not provide explicit labels for each sequence, so we assume initial 70% of the data to be normal as machines have a low probability of being in a faulty state, meaning that we assume anomalies to occur rarely compared to the normal data. According to the ground truth in the provided datasets, failures occurred only at the end of each run-to- failure experiment and thus the first 70% is used for the training of the model. The architecture of the LSTM-AE is designed as such to allow the model to have enough capability for feature learning and secondly, the number of units of the next hidden layer is set smaller than that of the previous layer so that feature learning can be viewed as a signal compression process, as shown in Fig. 2 (a). Here, the training process with the train set $T_{n}$ is depicted, together with the compression process of the raw signal within the autoencoder. The weights of the LSTM-AE are updated via stochastic gradient descent using mini-batches and the Adam [24] optimizer is utilized to speed up the training process. Batch normalization allows faster and stable training of deep neural networks. To restrict the model from overfitting, dropout is used and to avoid exploding gradients, gradient clipping is applied. Model hyperparameters are learned using the Bayesian optimization method, including the size of mini-batches, learning rate, and weight decay. ### III-B Anomaly Monitoring Process The detection process of anomalies consists of converting reconstruction errors into anomaly scores(AS) for each input sequence and using these scores to obtain a threshold, characterizing the normal behavior of the machine. After the model has been trained, sequences in $V_{N}$ are passed through the model to get the reconstruction errors which in turn are used to estimate the parameters of a Normal distribution ($\mu$ $\Sigma$) using maximum likelihood estimation (MLE), similar to [18]. The probability $p_{i}$ of obtaining the reconstruction error $e_{i}$ is given by the value of the Normal distribution at $e_{i}$. Using the $\mu$ & $\Sigma$, the anomaly score for a datapoint ${\mathbf{x_{t}^{(N)}}}$ is computed as follows: $a_{i}=(e_{i}-\mu)^{T}\Sigma^{-1}(e_{i}-\mu)$ (4) where $a_{i}$ is the desired anomaly score, $e_{i}$ is RE obtained for a sequence ${x_{t}^{(i)}}$ and $\mu$ $\Sigma$ are the mean and variance of a multivariate Gaussian distribution. During the initialization phase, an anomaly score threshold $\tau$ is also learned using a validation set $V_{A}$ that may contain examples from anomalous sequences alongside with the normal sequences. Unseen sequences within $T_{A}$ are classified as follows: if a sequence has an anomaly score $>$ $\tau$ it will be labeled as an anomaly, otherwise as normal. This is also depicted in Fig. 2 (b) where the raw measured signal is input to the trained autoencoder model that generates reconstruction errors and anomaly scores. These are then used within the described decision process to determine whether the signal was normal or anomalous. The assumption here is, that as the monitored equipment degrades or faces a failure, this disrupts the normal working of the machine and affects the interaction between different variables which can be measured by accelerometers, especially in form of vibrations. As the sensor values start deviating from the normal working condition of the machine, it is expected to see an increased error in the reconstruction of the input. By monitoring the reconstruction error and the anomaly score, an indication of the health of the monitored machine can be derived. ### III-C Feature Extraction Feature learning is a critical step in improving the performance of anomaly detection models due to the multidimensionality of data that is input into the model. In general, machine vibration signals comprise a stationary vibration part, a random vibration part, and noise [19]. We study two methods for the inspection of vibration signals: automated feature extraction based on LSTM-AE and manual feature extraction based on classic signal processing methods. Isolation Forest is used to evaluate the effectiveness of these two feature extraction methods. Isolation Forest is a random forest-based anomaly detection algorithm that utilizes isolation to determine anomalies in data [20]. The Isolation Forest (IF) algorithm is based on Decision Forests, an ensemble method that uses the averages of outputs from many different trees. #### III-C1 LSTM-AE based Feature Extraction One of the main characteristics of an autoencoder that is more powerful for finding intrinsic data structures by reducing data dimensionality through non- linear transformations than Principal Component Analysis (PCA) [21]. The encoder from the trained LSTM-AE is used to perform feature extraction. The process is done by reducing the number of units in the hidden layer, it is expected that the hidden units in the encoder network will extract features that will represent the data. To learn more abstract features, multiple AEs are stacked together to form a stacked AE, in which the output of each hidden layer is connected to the input of successive hidden layers. A stacked AE applies dimensionality reduction in a hierarchical manner, obtaining more abstract features in higher hidden layers which lead to a better reconstruction of the data [22]. #### III-C2 Manual Feature Extraction Following time-domain statistical features are generally used to detect incipient machine faults/anomalies: mean ($\mu_{x}$), root mean square (RMS), percentiles (25th, 50th & 75th ), max absolute value, standard deviation ($\sigma_{x}$),peek-to-peek, skewness, kurtosis, entropy, and AR-coefficients. The last five features are described through the following equations: $Peak{-}peak(p{-}p)=abs(Max(x))+abs(Min(x))$ (5) $Skewness=\frac{\sum_{i=1}^{N}(x_{i}-\mu_{x})^{3}}{N\sigma_{x}^{3}}$ (6) $Kurtosis=\frac{\sum_{i=1}^{N}(x_{i}-\mu_{x})^{4}}{N\sigma_{x}^{4}}$ (7) $Entropy=\sum_{i}p_{i}\log p_{i}$ (8) $AR{-}coefficients=\sum_{k=1}^{p}a_{k}x[n-k]+e[n]$ (9) where in equation 9; p is the degree of the AR model, x[n] is a signal composed of b data points, $a_{k}$ is real-values AR coefficient and e[n] is white noise. ## IV Evaluation There are two datasets considered for the evaluation of the proposed method. We evaluate and test the applicability of the proposed method in Condition Monitoring using the IMS bearing dataset [23](Dataset-1). In addition, the effectiveness to find anomalies in unsupervised settings is evaluated using a private industry dataset from vibration data of RMs. Evaluations performed on these two datasets demonstrate the ability of the method to detect anomalies within the industry dataset and perform condition monitoring on the bearing dataset, as the IMS bearing dataset doesn’t provide explicit labels but instead contains a degrading health state scenario of the bearing under experiment. Below, we introduce the datasets and illustrate the performance of the proposed method on these datasets. ### IV-A Dataset-1 This dataset was gathered from a run-to-failure experimental setting, involving four bearings and is subdivided into three datasets, each of which consists of the vibration signals from these four bearings [23]. These sets were collected from three test-to-failure experiments which were performed independently, and failures occurred at the end of the test. Table I represents the properties of the collected data from these experiments. TABLE I: Dataset-1: NASA Bearing dataset description Set # \| Batches \| Batch Size \| Anomaly ---\|---\|---\|--- Set1 \| 2156 \| 4 x 20480 \| B3 and B4 Set2 \| 984 \| 4 x 20480 \| B1 Set3 \| 6324 \| 4 x 20480 \| B3 TABLE II: Dataset-2: Industry dataset description Machine \| Batches \| Batch Size \| Anomalies ---\|---\|---\|--- RM-1 \| 1176 \| 6144 \| 53 RM-2 \| 1463 \| 2401 \| 4 RM-3 \| 2204 \| 2401 \| 4 RM-4 \| 1452 \| 2401 \| 3 RM-5 \| 1452 \| 2401 \| 4 ### IV-B Dataset-2 This dataset consists of data from five different RMs (RM-1 to RM-5) which are of the same kind but build-wise unique. Data from each RM contains 3-dimensional vibration signals measured over time, captured using the accelerometer sensors attached to the housing of the RM. The signal recordings were taken in batches and the batch sizes vary, depending upon the RM. Considering the condition of the machine at the time of recording, labels are assigned to each batch, 0 representing normal and 1 anomalous, respectively. Table II presents the summarized statistics of the industry dataset. The difference between normal and anomalous signals is shown in Fig. 3, vibrations in all three directions (x, y, z) differ significantly for anomalous signals compared to normal vibration signals. These differences can have a variety of characteristics as the amplitude of different frequencies differs between the two signals. Figure 3: Example of a normal (Left) and anomalous (Right) signal from RM-1, rows corresponding to the x, y and z-axis of the accelerometer. ## V Experiments and Results We demonstrate our method’s applicability in condition monitoring & anomaly detection with three general settings where in the first one, the autoencoder is trained with 70% of the data, 10% (split between two validation sets) is used to calculate the threshold, and remaining 20% of it is kept for evaluation to determine the prediction ability of the proposed approach for an anomaly detection task. In the second experimental setting, each run-to- failure experiment is simulated by training the LSTM-AE model with the available data from the non-failing RMs to monitor the condition of the machine in which failure occurred during the run-to-failure experiment. The trained model is used to monitor the status of the faulty machines to demonstrate that once the model is trained, it is capable of being applied to different RMs of a similar kind, enabling the transfer of rare anomaly knowledge from RMs to other RMs of a similar kind. The third setting consists of a method for performing automated feature extraction from vibration signals. Features extracted using this setting are compared against a set of handcrafted features by training Isolation Forest on these two features sets separately. ### V-A Setting 1: Online Prognostic In this setting, as discussed, the autoencoder learns to reconstruct the normal behavior of an RM using only 70% of the available healthy/normal data in order to evaluate the prediction performance of the model in an online monitoring phase. The threshold $\tau$ is obtained using a validation set of size 5% to classify the samples as normal/healthy or anomalous. Anomaly scores for each data sample are calculated using the reconstruction errors and data points with anomaly scores larger than $\tau$ are labeled as anomalies. The point at which the anomaly score crosses the threshold and starts to increase gradually is considered as the degradation point. ##### Result: Dataset-1 Here we study 4 cases of failing bearings (B1 to B4) of the different sub- datasets (S1 to S3), S1-B3, S1-B4, S2-B1, and S3-B3, for early fault detection. The model trained on individual bearings is able to predict the degradation point and capture the propagation of fault in the simulated run- to-failure experiment as is depicted in Fig. 4. As the fault in every failing bearing occurs at the end, Fig. 4 displays the anomaly scores of the last 620, 450, and 780 batch samples for S1-B3 & S1-B4, S2-B1, and S3-B3, respectively. For every failing bearing in each dataset, the proposed method generates a trend corresponding to the health status of the bearing based on the vibration sensory data. As the bearing health starts to degrade, the anomaly score tends to increase and once it passes the degradation point (marked with the filled circles) it increases gradually. By following this trend and abrupt changes in the scores one can feasibly detect the failure. The degradation point for each bearing was calculated by identifying the abrupt change and continuous increasing trend in anomaly scores. Sample no. 2039, 1704, 667, and 5241 were identified as degradation points for S1-B3, S1-B4, S2-B1, and S3-B3, respectively. The circle on the score line indicates the degradation starting point. Figure 4: Deteriorating condition of the failing bearings captured by the model. High anomaly scores indicate that system is more likely to be in faulty state. Circles mark the start of the degradation for each bearing. Figure 5: Anomaly scores for RM-1 from the second dataset: Red straight line indicates the threshold, data points above this red line are labeled as anomalous, otherwise normal. Black circles indicate false positives. TABLE III: Setting 1 results: individual RM anomaly detection scores. Machine \| Precision \| Recall \| TPR \| FPR \| F1-score ---\|---\|---\|---\|---\|--- RM-1 \| 0.993 \| 0.993 \| 0.981 \| 0.010 \| 0.993 RM-2 \| 1.0 \| 1.0 \| 1.0 \| 0.0 \| 1.0 RM-3 \| 1.0 \| 1.0 \| 1.0 \| 0.0 \| 1.0 RM-4 \| 0.994 \| 0.991 \| 1.0 \| 0.007 \| 0.993 RM-5 \| 0.965 \| 0.972 \| 0.25 \| 0.007 \| 0.967 ##### Result: Dataset-2 The second dataset consists of 5 different RMs with anomalies, for each machine there is a different number of anomalies (see Table II). To detect anomalies from each individual machine, we conduct five different experiments (one for each machine) using only the non-anomalous (normal) data to train the autoencoder network. The anomaly detection results are evaluated using, Precision, Recall, True positive rate (TPR), False positive rate (FPR), and F1-score. These evaluation metrics are weighted per class. As described, model performance is evaluated over the test set consisting of 20% of the data. For instance, the test set for RM-1 consisted of 145 (92 normal, 53 anomalous) sequences in total. Each sample from the test set was reconstructed using the trained autoencoder network and was labeled as anomalous if its anomaly score was larger than the threshold, normal otherwise. The trained model was able to detect 52/53 anomalies without raising many false alarms (just one false positive). Fig. 5 displays the anomaly scores for each data point; sequences below the threshold (marked by the red line) are classified as normal and data points with anomaly scores larger than the threshold are labeled as anomalies. Evaluation scores (precision, recall, TPR, FPR, F1-score) for individual RMs from the second dataset, averaged over 10 repetitions of the experiments are presented in Table III in which the model performs very well with F1-scores close to 1 and is able to detect anomalies without generating many false positives which is indicated by the high TPR and low FPR. Based on the results, the LSTM-AE can effectively extract discriminative features directly from the raw vibration data and achieve a competitive anomaly detection rate. For the test data, an overall detection F1-score of 99% and TPR over 93% for each machine is obtained. ### V-B Setting 2: Condition Monitoring In dataset-1, during each run-to-failure experiment, only one out of 4 bearings faced failure while the other 3 remained in healthy condition except for experiment-1 in which a fault occurred in two bearings at the end. The question of interest is whether the knowledge extracted from the healthy bearings can be applied to detect the deteriorating condition of a faulty bearing and raise alarm well before the total failure actually happens. To validate this, in this setting only the data from the healthy bearings is used to train the model and this trained model is then used to evaluate the condition of the previously unseen faulty bearings. Three simulations, one per sub-dataset are performed as follows: The Set1 model is trained using only training data from the non-failing bearings S1-B1 & S1-B2 with the aim to learn the functioning of healthy bearing and apply this learned knowledge to monitor and detect the deteriorating condition of the failing bearings S1-B3 & S1-B4. Similarly, for Set2, the training data comprised of S2-B2, S2-B3, and S2-B4 and while the test set consisted of S2-B1 and for Set3, training data consists of S3-B1, S3-B2, S3-B4 and test set of S3-B3. Fig. 6 displays the output anomaly scores of the four failing bearings from these experiments. The trend it generates for each bearing is in accordance with the ground truth, from the beginning until near the end the bearings remain in healthy condition, thus low anomaly scores are correctly calculated. The fault starts to appear only at the very end, which is captured by the model by a continuous increase in the anomaly scores. A low anomaly score corresponds to healthy behavior while an upward trend (high anomaly score) highlights the abnormal behavior of the machine. Arrival and propagation of fault of for every failing bearing is shown in Fig. 6. It is clearly visible that the proposed approach is able to identify the initiation of the faulty trend as well as the increasing effect of the deterioration for all four bearings. Fig. 7 visualizes the anomaly scores and visually indicates the health status of the four faulty bearings S1-B3, S1-B4, and S2-B1 and S3-B3 in parts B3, B4, B1, and B3, respectively. The color bar represents the anomaly score, from 0 (blue) to 1 (red). TABLE IV: Model trained on combined healthy data from four RMs is able to detect anomalies without raising any false alarms across all four RMs. Machine \| Precision \| Recall \| TPR \| FPR \| F1-score ---\|---\|---\|---\|---\|--- RM-2 \| 1.0 \| 1.0 \| 1.0 \| 0.0 \| 1.0 RM-3 \| 0.997 \| 0.997 \| 0.75 \| 0.0 \| 0.997 RM-4 \| 1.0 \| 1.0 \| 1.0 \| 0.0 \| 1.0 RM-5 \| 1.0 \| 1.0 \| 1.0 \| 0.0 \| 1.0 Figure 6: Anomaly scores for the condition monitoring of the faulty bearings by modeling the normal working behavior of the remaining bearings using the other non-failing bearings from the run-to-failure experiments. Figure 7: Visualization of anomaly scores from the run-to-failure experiments, where blue color indicates that the system is less likely to be in a faulty state while an abnormal behavior is indicated by the red color bars. The same setting was applied to our second dataset where the results were obtained by combining the normal data points from four RMs (RM-2 to RM-5). We divided each RM’s data into three sets: training ${\mathit{Train_{Pi}}}$ (70%), validation ${\mathit{V_{Pi}}}$ (10%) and test ${\mathit{Test_{Pi}}}$ (20%). Train set T was created by combining the four ${{Train_{Pi}}}$ and was used to train the model. The validation sets ${\mathit{V_{Pi}}}$ were used to calculate the threshold ${\tau}$ using the anomaly scores for each RM ${\mathit{P{i}}}$. The final anomaly detection model trained on T was then evaluated on each ${\mathit{Test_{Pi}}}$ set and the obtained results are presented in Table IV, in which the model performs very well in detecting anomalies across four different machines without raising any false alarms. It is worthwhile to notice that combining the training data from multiple RMs increases the performance of the model compared to the performance of the model on individual RMs. The trained LSTM-AE is able to model the healthy working behavior of the RMs and is sensitive to anomalies, as it can be seen in Fig. 8 which shows that the identified anomalies match well with the ground truth. The model can reconstruct the normal signals very well, indicated by a low anomaly score, while it fails to do so whenever it encounters anomalies. Figure 8: Anomaly detection results obtained from mixing the normal signals from four machines to train the LSTM-AE model and then using the validation data from each machine to calculate the threshold to detect the anomalous samples. All identified anomalies except for one false negative in case of RM-3 were identified correctly by the model. ### V-C Setting 3: Feature Extraction We employ the same LSTM-based autoencoder model to automatically extract time- domain features from vibration signals. The automatically extracted features are compared with a set of manually crafted features, by training an Isolation Forest on both of these two sets and monitoring the performance of the model. Results obtained from the IMS bearing dataset using the proposed method are compared with the IF-based anomaly detection trained only on manually extracted features. The comparison of results between two Isolation Forest based models is shown in Table V, which shows the total number of anomalies detected by the models in each bearing. It’s evident that the S1-B1, S1-B4, S2-B1, and S3-B3 are having comparatively more anomalous samples than the rest of the bearings as these four bearings result in failure according to the ground truth provided. The comparison illustrates that the latent features extracted by the LSTM-AE perform better than the manually engineered features demonstrating the effectiveness of the proposed approach in providing a robust set of features that boost the performance of the anomaly detection model. This can be confirmed by looking at the number of anomalies detected by each model; for each failing bearing, the IF model based on automatically extracted features detects a greater number of anomalies than the model using the handcrafted set of features. One interpretation of this could be that LSTM-AE provides a set of features that are more sensitive and thus can detect the subtle changes in the vibrations when the fault appears initially and starts developing gradually until the failure finally happens and bearing stops functioning. While on the other hand manually handcrafted features only start labeling signals as anomalous when the fault has fully developed already, and vibrations deviate significantly from the normal signals. TABLE V: Number of anomalies detected by Isolation Forest using automatically extracted features (A_FE) and manually extracted features (M_FE): Failing bearings are marked in red and results of the best performing model on these bearings is highlighted with bold entries. Bearing \| No. of Anomalies (A_FE) \| No. of Anomalies (M_FE) ---\|---\|--- S1-B1 \| 04 \| 06 S1-B2 \| 07 \| 05 S1-B3 \| 39 \| 26 S1-B4 \| 441 \| 213 S2-B1 \| 269 \| 133 S2-B2 \| 09 \| 14 S2-B3 \| 07 \| 16 S2-B4 \| 13 \| 11 S3-B1 \| 24 \| 57 S3-B2 \| 41 \| 35 S3-B3 \| 1016 \| 693 S3-B4 \| 39 \| 51 Multi-step anomaly detection results, where the first step consists of automated features extraction using LSTM-AE and in the second step these features are utilized to train an anomaly detection model (Isolation Forest), for the second dataset are summarized in Table VI. It is important to mention that each RM except for RM-1 in dataset-2 contains very few anomalies (see Table II). IF based on features provided by LSTM-AE (AFE$\\_$IF) outperforms IF trained using the manually engineered features (MFE$\\_$IF) for all RMs in terms of precision. Although, for RM-3 and RM-5 MFE$\\_$IF performed better in terms of TPR. However, it is worthwhile to notice that the model based on automatically engineered features results in less false positives in comparison to the model trained on handcrafted features which had significantly more false positives. As previously stated, the studies that have used handcrafted features obviously are not able to carry the complete system health representation and may not represent the characteristics of the underlying vibration signal under all circumstances. As seen from the results, fault/anomaly detection performance of the conventional methods such as Isolation Forest depends highly on the carefully crafted features. Consequently, this limits the general applicability of these methods. TABLE VI: Comparison of Isolation Forest trained on automatically extracted features (AFE_If) and the same model trained using the handcrafted features (MFE_IF) using the evaluation metrics, row-wise for each RM in Dataset-2. \| AFE_IF \| MFE_IF ---\|---\|--- \| P \| TPR \| FPR \| P \| TPR \| FPR RM-1 \| 0.972 \| 0.943 \| 0.01 \| 0.91 \| 1.0 \| 0.19 RM-2 \| 1.0 \| 1.0 \| 0.0 \| 0.976 \| 1.0 \| 0.23 RM-3 \| 0.991 \| 0.75 \| 0.004 \| 0.984 \| 1.0 \| 0.15 RM-4 \| 1.0 \| 1.0 \| 0.0 \| 0.983 \| 1.0 \| 0.09 RM-5 \| 0.979 \| 0.25 \| 0.0 \| 0.977 \| 1.0 \| 0.14 The experimental results obtained in all three settings are able to support the earlier claims that LSTM-AE trained only on the normal/healthy signals is not just able to monitor the health condition of an individual RM but knowledge learned can also be effectively used to detect anomalies in similar yet different RMs. Additionally, it is also shown that latent representations obtained from LSTM-AE can be used as an alternative to manual feature extraction, which requires prior domain and signal processing knowledge. ## VI Conclusion and Future Work In conclusion, we have demonstrated an unsupervised method for automated feature extraction from raw vibration signals that can be used for detecting faults and anomalies in rotating machines. A typical condition monitoring system requires feature extraction and decision about the health of the system. The feature extraction component of such systems involves the implementation of signal processing methods for preprocessing the data before using it for anomaly detection. The proposed method fuses the features extraction and anomaly detection modules within one condition monitoring system. Encoding the time-dependent as well as inherent features of the measured vibration time series into a hidden state of an LSTM-based autoencoder enabled the usage of the resulting reconstruction error to be used in an anomaly detection setup. The experimental results on two real-world datasets illustrate the effectiveness of the proposed model and show that LSTM-based autoencoders can extract salient features from vibration signals and achieve high accuracy in fault diagnosis. The model was able to detect the deteriorating condition of all four failing bearings for IMS dataset and achieved an overall F1 score of 99.6% for the second dataset. As demonstrated in the experiments, the latent representations obtained from the LSMT-AE carry system health information and achieve a higher F1 score than the manually extracted features (97% and 93% respectively), indicating a better representation of the characteristics of the vibration signal for the proposed method. For future work, other alternatives for choosing a threshold value shall be investigated. Also more advanced types of autoencoders such as variational autoencoder can be tried out, which alternatively try to model the data distribution instead of learning to re-create the normal data points, which means datasets with mixed examples (anomalous $\&$ non-anomalous) could also be used for training the model. 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# On Erdős’s Method for Bounding the Partition Function Asaf Cohen Antonir School of Mathematical Sciences, Tel Aviv University, Tel Aviv, 6997801, Israel. Email<EMAIL_ADDRESS>Asaf Shapira School of Mathematics, Tel Aviv University, Tel Aviv 69978, Israel. Email: <EMAIL_ADDRESS>Supported in part by ISF Grant 1028/16, ERC Starting Grant 633509 and NSF-BSF Grant 2019679. ###### Abstract For fixed $m$ and $R\subseteq\\{0,1,\ldots,m-1\\}$, take $A$ to be the set of positive integers congruent modulo $m$ to one of the elements of $R$, and let $p_{A}(n)$ be the number of ways to write $n$ as a sum of elements of $A$. Nathanson proved that $\log p_{A}(n)\leq(1+o(1))\pi\sqrt{2n\|R\|/3m}$ using a variant of a remarkably simple method devised by Erdős in order to bound the partition function. In this short note we describe a simpler and shorter proof of Nathanson’s bound. ## 1 Introduction. A partition of an integer $n$ is a sequence of positive integers $a_{1}\leq a_{2}\leq\cdots$ whose sum is $n$. Let $p(n)$ denote the classical partition function of $n$, namely, the number of ways to write $n$ as a sum of positive integers. The celebrated Hardy–Ramanujan formula [2] (discovered independently by Uspensky [6]) states that $p(n)\sim\frac{1}{4n\sqrt{3}}\exp(\pi\sqrt{2n/3})$. Erdős [1] later devised a remarkably simple proof of the slightly weaker upper bound $\log p(n)\leq\pi\sqrt{2n/3}\;.$ (1) Let $\mathbb{N}$ denote the set of positive integers, and suppose $S\subseteq\mathbb{N}$. We define $p_{S}(n)$ to be the number of partitions of $n$ with all summands in $S$. For a fixed positive integer $m$ and $R\subseteq\\{0,1,\ldots,m-1\\}$, we take $A=A(m,R)$ to be the set of all positive integers $a$ with $a\;(\mathrm{mod}\;m)\in R$. Nathanson [4] used Erdős’s method for proving (1) to obtain111Nathanson [4] also proves that $\log p_{A}(n)\geq(1-o(1))\pi\sqrt{2n\|R\|/3m}\;$. $\log p_{A}(n)\leq(1+o(1))\pi\sqrt{2n\|R\|/3m}\;.$ (2) The argument in [4] was more complicated than Erdős’s due to the need to control various error parameters (but was still simpler than the original proof of this result [3]); see the remark at the end of the proof. Our goal in this short note is to give a proof of (2) which is as simple as Erdős’s proof of (1). The main trick is that, instead of directly bounding $p_{A}(n)$, we will instead bound $p_{A^{+}}(n)$, where given $m$ and $R$ as above, we take $A^{+}=A\setminus R$, that is, the set of all integers $a\geq m$ with $a\;(\mathrm{mod}\;m)\in R$. Our main result here is the following generalization222Indeed, when $m=1$ and $R=\\{0\\}$, we have $p_{A^{+}}(n)=p(n)\;$. of (1). ###### Theorem 1. For every $A^{+}$ as above, $\log p_{A^{+}}(n)\leq\pi\sqrt{2n\|R\|/3m}$ . It is easy to obtain (2) from the upper bound given by Theorem 1. Indeed, we first note that for every $n^{\prime}$ we have $p_{R^{+}}(n^{\prime})\leq(n^{\prime}+1)^{\|R\|}$, where $R^{+}=R\setminus\\{0\\}$. This follows immediately from the fact that in every partition of $n^{\prime}$, each of the integers of $R^{+}$ is used at most $n^{\prime}$ times. We thus infer that $p_{A}(n)=\sum_{0\leq n^{\prime}\leq n}p_{R^{+}}(n^{\prime})\cdot p_{A^{+}}(n-n^{\prime})\leq(n+1)^{\|R\|}\sum_{0\leq n^{\prime}\leq n}e^{c\sqrt{n-n^{\prime}}}\leq(n+1)^{\|R\|+1}e^{c\sqrt{n}}\;,$ where $c=\pi\sqrt{2\|R\|/3m}$. Taking logs from both sides, we obtain (2). The proof of Theorem 1 appears in the next section. At the end of that section we briefly explain why our proof is simpler than that of [4]. ## 2 Proof of Theorem 1. For a given fixed integer $m\geq 1$ and $R\subseteq\\{0,1,\ldots,m-1\\}$, let $A^{+}$ denote the set of all integers $a\geq m$ with $a\;(\mathrm{mod}\;m)\in R$. We start with a few observations that extend those used in [1]. We first note that, for every $0<t<1$, we have $\sum_{a\in A^{+}}at^{a}=\sum_{r\in R}\frac{(r+m)t^{r+m}-rt^{2m+r}}{(1-t^{m})^{2}}\;.$ (3) Indeed, $\sum_{a\in A^{+}}at^{a}=\sum_{r\in R}\sum_{a\in A_{r}^{+}}at^{a}$ where $A_{r}^{+}$ is the set of all integers $a\geq m$ with $a=r\;(\mathrm{mod}\;m)$ (i.e., $A_{r}^{+}=\\{r+m,r+2m,r+3m,\ldots\\}$). Hence, without loss of generality we may assume $\|R\|=1$. Letting $r\in R$, we have $\displaystyle\sum_{a\in A_{r}^{+}}at^{a}$ $\displaystyle=t\sum_{a\in A_{r}^{+}}\frac{d}{dt}t^{a}=t\cdot\frac{d}{dt}\sum_{a\in A_{r}^{+}}t^{a}=t\cdot\frac{d}{dt}\frac{t^{r+m}}{1-t^{m}}=\frac{(r+m)t^{r+m}-rt^{2m+r}}{(1-t^{m})^{2}}\;.$ This proves (3). We next claim that, if $0\leq r\leq m-1$ is an integer, then for all $x>0$, we have $\frac{(r+m)e^{-(r+m)x}-re^{-(2m+r)x}}{(1-e^{-mx})^{2}}\leq\frac{1}{mx^{2}}\;.$ (4) Indeed, since $x>0$, the power series expansion of $e^{x}$ gives $e^{x/2}-e^{-x/2}=2\sum_{k=0}^{\infty}\frac{1}{(2k+1)!}\left(\frac{x}{2}\right)^{2k+1}=x+x^{3}\sum_{k=1}^{\infty}\frac{x^{2k-2}}{(2k+1)!\cdot 2^{2k}}>x\;,$ implying that $\frac{e^{-x}}{(1-e^{-x})^{2}}=\frac{1}{(e^{x/2}-e^{-x/2})^{2}}<1/x^{2}\;.$ We can thus infer that $\displaystyle\frac{(r+m)e^{-(r+m)x}-re^{-(2m+r)x}}{(1-e^{-mx})^{2}}$ $\displaystyle={((r+m)e^{-rx}-re^{-(m+r)x}})\frac{e^{-mx}}{(1-e^{-mx})^{2}}$ $\displaystyle\leq({(r+m)e^{-rx}-re^{-(m+r)x})}\frac{1}{m^{2}x^{2}}\;.$ It remains to check that the expression in parentheses is bounded by $m$. Since the derivative of ${(r+m)e^{-rx}-re^{-(m+r)x}}$ (which is $r(r+m)(e^{-(m+r)x}-e^{-rx})$) is always nonpositive for $x\geq 0$, it is enough to check its value at $x=0$ where it attains the value $m$. This proves (4). We now note that (3) and (4) imply that, for every $x>0$, $\sum_{a\in A^{+}}ae^{-ax}\leq\frac{\|R\|}{mx^{2}}\;.$ (5) The final observation we will need is the well-known fact that, for every set of positive integers $S$, we have $n\cdot p_{S}(n)=\sum_{s\in S\cap[n]}s\sum_{1\leq k\leq n/s}p_{S}(n-sk)\;,$ (6) where we use $[n]$ for the integers $\\{1,\ldots,n\\}$. To see this, let $p_{S}(n,s,t)$ and $p_{S}^{\prime}(n,s,t)$ be the number of partitions of $n$ with summands in $S$ where $s$ appears exactly $t$ times, and at least $t$ times, respectively. Then by double counting,333The two sides of the first equality count the sum of all integers that appear in all partitions of $n$ using integers from $S$ (there are $p_{S}(n)$ such partitions). As to the third equality, it follows by observing that each partition of $n$ with exactly $t$ occurrences of $s$ contributes $1$ to $t$ of the summands $p_{S}^{\prime}(n,s,t)$, namely $p_{S}^{\prime}(n,s,1),p_{S}^{\prime}(n,s,2),\ldots,p_{S}^{\prime}(n,s,t)$. See Theorem 15.1 in [5] for a full detailed proof. we have $\displaystyle n\cdot p_{S}(n)$ $\displaystyle=\sum_{s\in S,t\in\mathbb{N}}s\cdot t\cdot p_{S}(n,s,t)=\sum_{s\in S\cap[n]}s\sum_{t\in\mathbb{N}}t\cdot p_{S}(n,s,t)$ $\displaystyle=\sum_{s\in S\cap[n]}s\sum_{t\in\mathbb{N}}p_{S}^{\prime}(n,s,t)=\sum_{s\in S\cap[n]}s\sum_{1\leq k\leq n/s}p_{S}(n-sk)\;.$ This proves (6). We are now ready to complete the proof of Theorem 1. We use induction on $n$, with the base case trivially holding. We have $\displaystyle n\cdot p_{A^{+}}(n)$ $\displaystyle=\sum_{a\in A^{+}\cap[n]}a\sum_{1\leq k\leq n/a}p_{A^{+}}(n-ak)\leq\sum_{a\in A^{+}\cap[n]}a\sum_{1\leq k\leq n/a}e^{c\sqrt{n-ak}}$ $\displaystyle\leq e^{c\sqrt{n}}\sum_{a\in A^{+}\cap[n]}a\sum_{1\leq k\leq n/a}e^{-\frac{cak}{2\sqrt{n}}}\leq e^{c\sqrt{n}}\sum_{k=1}^{\infty}\sum_{a\in A^{+}}ae^{-\frac{cak}{2\sqrt{n}}}$ $\displaystyle\leq e^{c\sqrt{n}}\sum_{k=1}^{\infty}\frac{4\|R\|n}{mc^{2}k^{2}}=ne^{c\sqrt{n}}\frac{4\|R\|}{mc^{2}}\sum_{k=1}^{\infty}\frac{1}{k^{2}}=n\cdot e^{c\sqrt{n}}\;,$ where the first equality is (6), the first inequality is by the induction hypothesis, the second inequality uses the elementary fact $\sqrt{n-rk}\leq\sqrt{n}-\frac{rk}{2\sqrt{n}}$, and in the last inequality we applied (5) with $x=\frac{ck}{2\sqrt{n}}$. Dividing both sides by $n$ we obtain the theorem. ### Bounding $p_{A^{+}}(n)$ vs. bounding $p_{A}(n)$. The reader might be wondering why bounding $p_{A^{+}}(n)$ is so much easier than bounding $p_{A}(n)$. The answer is that the former gives us inequality (4) from which we obtain the clean inequality (5). To illustrate the complication that arises when working with $p_{A}(n)$, let us take $A$ to be the set of odd integers. Then, running the same argument, instead of (4), one would have liked to use the inequality $\frac{e^{-x}+e^{-3x}}{(1-e^{-2x})^{2}}\leq\frac{1}{2x^{2}}$, which is false. To overcome this, one then needs to use the fact that this inequality is approximately correct for small $x$, which significantly complicates the proof. Acknowledgement: We would like to thank the referees for their detailed and helpful comments. ## References * [1] Erdős, P. (1942). On an elementary proof of some asymptotic formulas in the theory of partitions. Ann. Math. 43(3): 437–450. * [2] Hardy, G. H., Ramanujan S. (1917). Asymptotic formulae for the distribution of integers of various types. Proc. London Math. Soc. 16(2): 112–132. * [3] Meinardus, G. (1954). Asymptotische Aussagen über Partitionen. Math. Z. 61: 388–398. * [4] Nathanson, M. B. (2002). On Erdős’s elementary method in the asymptotic theory of partitions. In: Halász, G., Lovász, L., Simonovits, M., Sós, V. T., eds. Paul Erdős and his Mathematics, I. Bolyai Soc. Math. Stud., volume 11. Budapest: János Bolyai Mathematical Society, pp. 515–531. * [5] Nathanson, M. B. (2000). Elementary Methods in Number Theory. Graduate Texts in Mathematics, vol. 195. Berlin: Springer. * [6] Uspensky, J. V. (1920). Asymptotic expressions of numerical functions occurring in problems concerning the partition of numbers into summands. Bull. Acad. Sci. de Russie. 14(6): 199–218.
∎ ††thanks: A.R.Usha Devi and Sudha acknowledge financial support from the Department of Science and Technology, India through Project No. DST/ICPS/QuST/Theme-2/2019/Project#107 11institutetext: Seeta Vasudevrao 22institutetext: Department of Physics, Bangalore University, Bangalore. 22email<EMAIL_ADDRESS>33institutetext: I. Reena 44institutetext: Department of Physics, Bangalore University, Bangalore. 44email<EMAIL_ADDRESS>55institutetext: Sudha 66institutetext: Department of Physics, Kuvempu University, Shankaraghatta. Inspire Institute Inc., Alexandria, Virginia, 22303, USA. 66email: <EMAIL_ADDRESS>77institutetext: A. R. Usha Devi 88institutetext: Department of Physics, Bangalore University, Bangalore Inspire Institute Inc., Alexandria, Virginia, 22303, USA. 88email: <EMAIL_ADDRESS>99institutetext: A. K. Rajagopal 1010institutetext: Inspire Institute Inc., Alexandria, Virginia, 22303, USA. 1010email: <EMAIL_ADDRESS> # Sum Uncertainty Relations: Uncertainty Regions for Qubits and Qutrits Seeta Vasudevrao I. Reena Sudha A. R. Usha Devi A. K. Rajagopal ###### Abstract We investigate the notion of uncertainty region using the variance based sum uncertainty relation for qubits and qutrits. We compare uncertainty region of the qubit (a 2-level system) with that of the qutrit (3-level system) by considering sum uncertainty relation for two non-commuting Pauli-like observables, acting on the two dimensional qubit Hilbert space. We identify that physically valid uncertainty region of a qubit is smaller than that of a qutrit. This implies that an enhanced precision can be achieved in the measurement of incompatible Pauli-like observables acting on the 2-dimensional subspace of a qutrit Hilbert space. We discuss the implication of the reduced uncertainties in the steady states of $\Lambda$, V, $\Xi$ types of 3-level atomic systems. Furthermore, we construct a two-qubit permutation symmetric state, corresponding to a 3-level system and show that the reduction in the sum uncertainty value - or equivalently, increased uncertainty region of a qutrit system – is a consequence of quantum entanglement in the two-qubit system. Our results suggest that uncertainty region can be used as a dimensional witness. ###### Keywords: Sum Uncertainty Relation Uncertainty Region Entanglement 3-Level System Steady-state Population ††journal: International Journal of Theoretical Physics ## 1 Introduction Quantum theory prevents assignment of precise values for two or more incompatible observables simultaneously. Heisenberg’s heuristic argument Heisenberg highlighted this uncertainty associated with the non-commuting position ($Q$) and momentum ($P$) observables, in terms of the constraints placed on the product of their standard deviations. If position of a particle is measured, prediction of its momentum gets inaccurate and vice versa. A mathematically formal version of the position-momentum uncertainty relation $(\Delta Q)(\Delta P)\geq\frac{\hbar}{2}$ was subsequently formulated by Kennard ken . Furthermore, Robertson rob (motivated by Weyl’s arguments Weyl ) extended the uncertainty relations to any arbitrary pairs of non-commuting observables $A_{1}$, $A_{2}$. Different forms of uncertainty relations have been formulated over the years AK ; ozawa ; Hall ; BLW ; Hoffman ; sw1 ; arun ; kraus ; MU ; sw ; Berta ; Coles ; Werner ; Li ; Abbot , capturing the trade-off between two or more non- commuting observables. It has been shown that uncertainty relations play a crucial role in quantum information processing tasks like quantum key distribution Berta ; Coles ; Cerf1 ; Cerf2 ; Koshi . Non-trivial state- dependent uncertainty relations which are experimentally verifiable are found to be of importance in device-independent cryptography sw1 . A broader perspective on uncertainty relations, based on the concept of _uncertainty regions_ , is recently being explored Werner ; Li ; Abbot ; Busch and provides a geometric visualization of the uncertainty relation. For any uncertainty relation, the corresponding uncertainty region is the _legitimate domain_ of standard deviation (or any other measure of uncertainties) of a pair (or triple) of observables, in the entire range of their possible values Busch . Points $(\Delta A_{1},\,\Delta A_{2})$ inside the uncertainty region specify the uncertainty in the simultaneous measurement of a pair of observables $A_{1}$, $A_{2}$. Different types of uncertainty relations can be chosen for analysing the uncertainty regions Werner ; Li ; Abbot ; Busch . In this work we have chosen variance based sum uncertainty relation, a state- independent uncertainty relation, proposed by Hofmann and Takeuchi Hoffman , to analyze the uncertainty regions/minimum of the variance based sum uncertainty relation for two non-commuting Pauli-like observables. The structure of the paper is as follows: In Section 2, we outline the geometry of uncertainty region corresponding to variance based sum-uncertainty relation for a pair of incompatible Pauli-like observables, when they are measured in quantum states of qubits (2-level systems) and qutrits (3-level systems). We show that uncertainty region of qutrits is larger, containing points with enhanced measurement precision for incompatible Pauli-like observables, in comparison with that of qubits. In Section 3, we express the minimum of the sum of variances of two noncommuting Pauli-like observables $A^{(ij)}_{1}=\vec{\sigma}^{(ij)}\cdot\hat{a}$, $A^{(ij)}_{2}=\vec{\sigma}^{(ij)}\cdot\hat{b},\ \ \hat{a}\cdot\hat{b}=0$ of a 3-level system, given that both the observables are restricted to the 2-dimensional (qubit) subspace (labelled by the pair of indices $(ij),\,i<j=1,2,3$) of a 3-level (qutrit) system, in terms of the populations $\rho_{ii}$, $\rho_{jj}$ in the $i^{\rm th}$ and $j^{\rm th}$ levels. We discuss implication of the reduced uncertainties in the steady states of $\Lambda$, V and $\Xi$ types of 3-level atomic systems. Section 4 details the construction of permutation symmetric two-qubit system corresponding to a qutrit state and explicit evaluation of equivalent sum uncertainty relation for Pauli-like observables. In Section 5, we establish that separable two- qubit states can never achieve utmost precision in the measurement of incompatible Pauli-like observables. We also show that maximum precision in measurement of the non-commuting Pauli observables is possible using entangled symmetric two-qubit states. Section 6 provides concluding remarks. ## 2 Uncertainty regions for qubits and qutrits: The well-known generalized uncertainty relation rob for observables $A_{1}$, $A_{2}$ is given by $(\Delta A_{1})(\Delta A_{2})\geq\frac{1}{2}\left\|\langle[A_{1},\,A_{2}]\rangle\right\|$ (1) where $[A_{1},\,A_{2}]=A_{1}A_{2}-A_{2}A_{1}$ is the commutator and $\Delta A_{1}$, $\Delta A_{2}$ defined by $\displaystyle\Delta A_{1}$ $\displaystyle=$ $\displaystyle\sqrt{\Delta^{2}\,A_{1}},\ \ \ \ \ \Delta^{2}A_{1}=\langle A^{2}_{1}\rangle-\langle A_{1}\rangle^{2}$ $\displaystyle\Delta B$ $\displaystyle=$ $\displaystyle\sqrt{\Delta^{2}\,A_{2}},\ \ \ \ \ \Delta^{2}A_{2}=\langle A^{2}_{2}\rangle-\langle A_{2}\rangle^{2}.$ (2) are the standard deviations of $A_{1}$, $A_{2}$ in any quantum state $\rho$. Here, $\langle\cdots\rangle=\mbox{Tr}\,(\rho\cdots)$ is the expectation value of any observable in the state $\rho$. Hofmann and Takeuchi Hoffman reformulated the uncertainty relation (1) in the form of sum of variances. Given a set of non-commuting operators $A_{i}$, $i=1,\,2,\,\ldots,\ n$, they have shown that $\sum_{i=1}^{n}\,\Delta^{2}\,A_{i}\geq\,k_{A},\ \ k_{A}\ \mbox{being a non- negative real number.}$ (3) It is a _state-independent_ uncertainty relation Hoffman with non-trivial bound for incompatible observables. We now set up the sum-uncertainty relation in (3) for a pair of observables $A_{1}$, $A_{2}$ acting on the most general state of a qubit: $\rho_{\rm qubit}=\frac{1}{2}\left[I_{2}+r_{1}\sigma_{1}+r_{2}\sigma_{2}+r_{3}\sigma_{3}\right],\ \ \ r_{1}^{2}+r_{2}^{2}+r_{3}^{2}\leq 1$ (4) where $\sigma_{i}$, $i=1,\,2,\,3$ are Pauli spin operators, $I_{2}$ is the two-dimensional identity operator and $\vec{r}=(r_{1},\,r_{2},\,r_{3})$, ($\|\vec{r}\|\leq 1$) is a real three dimensional vector, the mean spin vector of the state $\rho_{\rm qubit}$. On choosing $A_{1}=\sigma_{1}$ and $A_{2}=\sigma_{2}$, we get $\langle A_{1}^{2}\rangle=\langle A_{2}^{2}\rangle=1$, $\langle A_{1}\rangle=r_{1}$, $\langle A_{2}\rangle=r_{2}$. With $\Delta^{2}\,A_{1}=1-r_{1}^{2}$, $\Delta^{2}\,A_{2}=1-r_{2}^{2}$, the sum-uncertainty relation becomes $\displaystyle\Delta^{2}\,A_{1}+\Delta^{2}\,A_{2}=2-(r_{1}^{2}+r_{2}^{2})\geq 1,$ (5) $\displaystyle 0\leq\Delta A_{1}\leq 1,\ \ \ \ \ \ 0\leq\Delta A_{2}\leq 1.$ In general, we consider the _orthogonal_ Pauli-observables $A_{1}=\vec{\sigma}\cdot\hat{a},\ \ \ A_{2}=\vec{\sigma}\cdot\hat{b},\ \ \vec{\sigma}=\left(\sigma_{1},\,\sigma_{2},\,\sigma_{3}\right),\ \ \ \hat{a}\cdot\hat{b}=0$ (6) and we readily have $\langle A_{1}\rangle=\hat{a}\cdot\vec{r}$, $\langle A_{2}\rangle=\hat{b}\cdot\vec{r}$, $\langle A_{1}^{2}\rangle=\hat{a}\cdot\hat{a}=1$, $\langle A_{2}^{2}\rangle=\hat{b}\cdot\hat{b}=1$ leading to $\displaystyle\Delta^{2}\,A_{1}=1-\left(\hat{a}\cdot\vec{r}\right)^{2},\ \ \ \ \Delta^{2}\,A_{2}=1-\left(\hat{b}\cdot\vec{r}\right)^{2}$ $\displaystyle\Delta^{2}\,A_{1}+\Delta^{2}\,A_{2}=2-\left(\hat{a}\cdot\vec{r}\right)^{2}-\left(\hat{b}\cdot\vec{r}\right)^{2}.$ (7) As $\|\vec{r}\|\leq 1$, $\|\hat{a}\|=\|\hat{b}\|=1$, we have $\hat{a}\cdot\vec{r}\leq 1,\ \ \hat{b}\cdot\vec{r}\leq 1$ and from Eq. (2) we obtain the following sum-uncertainty relation for orthogonal Pauli-observables on a qubit: $\Delta^{2}\,A_{1}+\Delta^{2}\,A_{2}\geq 1\ \ \ \mbox{with}\ \ \ 0\leq\Delta A_{1}\leq 1,\ \ \ 0\leq\Delta A_{2}\leq 1.$ (8) The sum uncertainty relation (8) implies that the points $(\Delta A_{1},\,\Delta A_{2})$ lying outside the circular quadrant form the uncertainty region for Pauli-observables $A_{1}$, $A_{2}$ measured on a qubit, as can be seen in Fig. 1. Figure 1: The uncertainty region (light-blue shaded) of a qubit for orthogonal Pauli observables $A_{1}$, $A_{2}$ in (6): Notice that the points $(\Delta A_{1},\,\Delta A_{2})$ close to the origin correspond to measurements that result in better accuracy. But as the uncertainty region does not contain points below the circular arc (See Fig. 1), precise joint measurements of Pauli-observables on a qubit are impossible and Fig. 1 provides clear visualization of this fact. We now consider a 3-level system, $\rho_{\rm qutrit}=\omega\|\psi\rangle\langle\psi\|\oplus(1-\omega),\ \ 0\leq\omega\leq 1,$ (9) obtained by appending an ancillary level to a single-qubit pure state $\|\psi\rangle$. Here (9) corresponds to the state of a qutrit whose explicit form is given by $\rho_{\rm qutrit}=\left(\begin{array}[]{ccc}\frac{\omega(1+r_{3})}{2}&\frac{\omega(r_{1}-ir_{2})}{2}&0\\\ \frac{\omega(r_{1}+ir_{2})}{2}&\frac{\omega(1-r_{3})}{2}&0\\\ 0&0&1-\omega\end{array}\right),\ \ \ r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=1$ (10) Here, $r_{1}$, $r_{2}$, $r_{3}$ are the components of the _unit mean spin vector_ $\hat{r}$, corresponding to the _pure state_ $\|\psi\rangle$ (See (9)) and $\omega$ is a real parameter. The uncertainty region of the qutrit in (9) has been examined in Ref. Busch , for orthogonal Pauli observables $A_{1}=\vec{\sigma}\cdot\hat{a}\oplus 0,\ \ \ \ A_{2}=\vec{\sigma}\cdot\hat{b}\oplus 0,\ \ \hat{a}\cdot\hat{b}=0.$ (11) It can be seen that Busch $\langle A_{1}\rangle=\omega(\hat{a}\cdot\hat{r}),\ \ \ \langle A_{2}\rangle=\omega(\hat{b}\cdot\hat{r}),\ \ \langle A_{1}^{2}\rangle=\langle A_{2}^{2}\rangle=\omega,$ (12) leading to $\Delta^{2}A_{1}=\omega-\omega^{2}(\hat{a}\cdot\hat{r})^{2},\ \ \ \Delta^{2}A_{2}=\omega-\omega^{2}(\hat{b}\cdot\hat{r})^{2}.$ (13) On fixing $\Delta^{2}A_{1}$ and minimizing $\Delta^{2}A_{2}$, one obtains Busch $\left(\Delta A_{2}\right)_{\rm min}=\Delta A_{1}\sqrt{(1-\Delta^{2}A_{1})}.$ (14) Similarly, fixing $\Delta^{2}A_{2}$ and minimizing $\Delta^{2}A_{1}$ results in Busch $\left(\Delta A_{1}\right)_{\rm min}=\Delta A_{2}\sqrt{(1-\Delta^{2}A_{2})}.$ (15) From (14), (15), we see that when $\Delta A_{1}=0$, $\Delta A_{2}$ can also become zero and vice versa. This means, origin $(0,\,0)$ of the co-ordinate system, a point corresponding to utmost precision in simultaneous measurement, is physically realizable for measurement of orthogonal Pauli observables on the qutrit state $\rho_{\rm qutrit}$ Busch . The uncertainty region of the qutrit is larger in comparison with that of a qubit, containing points near the origin and origin itself, as can be readily seen in Fig. 2. Figure 2: The uncertainty region of a qutrit (light blue): The origin, with $\Delta A_{1}=\Delta A_{2}=0$ and points closer to it lie in the admissible region. It is evident from the above observations that simultaneous measurements with utmost precision are impossible when non-commuting Pauli measurements are performed on a 2-dimensional Hilbert space (qubit) whereas a 3-dimensional Hilbert space (qutrit) admits enhanced precision. In the following, we show that the uncertainty sum for two Pauli-like observables in an arbitrary 3-level system reduces below that for a 2-level system. ## 3 Sum uncertainty relation for 3-level atomic systems In this section we explore the sum uncertainty relation for two Pauli-like observables (i.e., atomic operators acting on any 2-level subspace of a 3-level atomic system) $\displaystyle A^{(ij)}_{1}$ $\displaystyle=$ $\displaystyle\vec{\sigma}^{(ij)}\cdot\hat{a},\ \ \vec{\sigma}^{(ij)}=\left(\sigma^{(ij)}_{1},\,\sigma^{(ij)}_{2},\,\sigma^{(ij)}_{3}\right)$ $\displaystyle A^{(ij)}_{2}$ $\displaystyle=$ $\displaystyle\vec{\sigma}^{(ij)}\cdot\hat{b},\ \ \ \ i<j=1,\,2,\,3,$ (16) where $\hat{a}\cdot\hat{b}=0$, $\hat{a}\cdot\hat{a}=1=\hat{b}\cdot\hat{b}$ and $\displaystyle\sigma^{(12)}_{1}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}0&1&0\\\ 1&0&0\\\ 0&0&0\end{array}\right),\ \ \ \ \sigma^{(13)}_{1}=\left(\begin{array}[]{ccc}0&0&1\\\ 0&0&0\\\ 1&0&0\end{array}\right),\ \ \ \sigma^{(23)}_{1}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&1\\\ 0&1&0\end{array}\right)$ $\displaystyle\sigma^{(12)}_{2}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}0&-i&0\\\ i&0&0\\\ 0&0&0\end{array}\right),\ \ \sigma^{(13)}_{2}=\left(\begin{array}[]{ccc}0&0&-i\\\ 0&0&0\\\ i&0&0\end{array}\right),\ \ \sigma^{(23)}_{2}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&0&-i\\\ 0&i&0\end{array}\right)$ $\displaystyle\sigma^{(12)}_{3}$ $\displaystyle=$ $\displaystyle\left(\begin{array}[]{ccc}1&0&0\\\ 0&-1&0\\\ 0&0&0\end{array}\right),\ \ \sigma^{(13)}_{3}=\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\ 0&0&-1\end{array}\right),\ \ \sigma^{(23)}_{3}=\left(\begin{array}[]{ccc}0&0&0\\\ 0&1&0\\\ 0&0&-1\end{array}\right).$ In an arbitrary 3-level atomic system, characterized by the density matrix, $\varrho_{{\rm qutrit}}=\left(\begin{array}[]{ccc}\varrho_{11}&\varrho_{12}&\varrho_{13}\\\ \varrho^{}_{12}&\varrho_{22}&\varrho_{23}\\\ \varrho^{}_{13}&\varrho^{}_{23}&\varrho_{33}\\\ \end{array}\right),$ (20) we obtain $\displaystyle\left\langle\left(A_{1}^{(ij)}\right)^{2}\right\rangle$ $\displaystyle=$ $\displaystyle{\rm Tr}\,\left[\varrho_{\rm qutrit}\,\left(\vec{\sigma}^{(ij)}\cdot\hat{a}\right)^{2}\right]=\varrho_{ii}+\varrho_{jj},$ (21) $\displaystyle\left\langle\left(A_{2}^{(ij)}\right)^{2}\right\rangle$ $\displaystyle=$ $\displaystyle{\rm Tr}\,\left[\varrho_{\rm qutrit}\,\left(\vec{\sigma}^{(ij)}\cdot\hat{b}\right)^{2}\right]=\varrho_{ii}+\varrho_{jj},$ (22) and $\displaystyle\left\langle A_{1}^{(ij)}\right\rangle$ $\displaystyle=$ $\displaystyle{\rm Tr}\,\left[\varrho_{\rm qutrit}\,\left(\vec{\sigma}^{(ij)}\cdot\hat{a}\right)\right]=\vec{n}^{(ij)}\cdot\hat{a},$ (23) $\displaystyle\left\langle A_{1}^{(ij)}\right\rangle$ $\displaystyle=$ $\displaystyle{\rm Tr}\,\left[\varrho_{\rm qutrit}\,\left(\vec{\sigma}^{(ij)}\cdot\hat{b}\right)\right]=\vec{n}^{(ij)}\cdot\hat{b},$ (24) where $\displaystyle\vec{n}^{(ij)}={\rm Tr}\,[\varrho_{\rm qutrit}\,\vec{\sigma}^{(ij)}]=\left(2\,{\rm Re}\,\varrho_{ij},\,2\,{\rm Im}\,\varrho_{ij},\,\varrho_{ii}-\varrho_{jj}\right).$ (25) Choosing $\hat{a}=\hat{n}^{(ij)}=\vec{n}^{(ij)}/\|\vec{n}^{(ij)}\|$, $\hat{b}=\hat{n}^{(ij)}_{\perp}$ and simplifying the sum of variances $\Delta^{2}A_{1}^{(ij)}+\Delta^{2}A_{2}^{(ij)}$ in the 3-level system (20) (with the help of (21), (22), (23), (24), (25))), we obtain $\displaystyle\left[\Delta^{2}A_{1}^{(ij)}+\Delta^{2}A_{2}^{(ij)}\right]$ $\displaystyle=$ $\displaystyle 2\,(\varrho_{ii}+\varrho_{jj})-\|\vec{n}^{(ij)}\|^{2}$ (26) $\displaystyle=$ $\displaystyle 2\,(\varrho_{ii}+\varrho_{jj})-\left[4\,\|\varrho_{ij}\|^{2}+(\varrho_{ii}-\varrho_{jj})^{2}\right].$ Positive semidefiniteness of the $(ij)^{\rm th}$ $2\times 2$ block of $\varrho_{\rm qutrit}$ imposes the condition $\|\vec{n}^{(ij)}\|\leq\varrho_{ii}+\varrho_{jj}$ (27) leading to the following minimum value for the uncertainty sum: $\displaystyle\left[\Delta^{2}A_{1}^{(ij)}+\Delta^{2}A_{2}^{(ij)}\right]_{\rm min}$ $\displaystyle=$ $\displaystyle 2\,\left(\varrho_{ii}+\varrho_{jj}\right)-\left(\varrho_{ii}+\varrho_{jj}\right)^{2}.$ (28) It may be noted that if we restrict ourselves to the 2-level system i.e., $i,j=1,2$, we obtain $\left[\Delta^{2}A_{1}^{(12)}+\Delta^{2}A_{2}^{(12)}\right]_{\rm min}=1$, as $(\varrho_{11}+\varrho_{22})={\rm Tr}[\varrho]=1$. In other words, the uncertainty sum is always greater than 1 in a 2-level system indicating that joint measurement of the non-commuting Pauli observables $\vec{\sigma}\cdot\hat{a},\ \vec{\sigma}\cdot\hat{b}$ is limited by the sum uncertainty relation (8). On the other hand, when an additional level is included, the populations $\varrho_{ii}$, $\varrho_{jj}$ of the $i^{\rm th}$ and $j^{\rm th}$ levels ($i>j=1,2,3$) play a crucial role in enhancing the measurement precision of the atomic observables $\vec{\sigma}^{(ij)}\cdot\hat{a}$, $\vec{\sigma}^{(ij)}\cdot\hat{b}$. In Fig. 3 we have plotted the minimum value of the uncertainty sum $\left[\Delta^{2}A_{1}^{(ij)}+\Delta^{2}A_{2}^{(ij)}\right]_{\rm min}$ as a function of the populations $\varrho_{ii}$, $\varrho_{jj}$ (see (28)). It is clearly seen that the minimum value of the uncertainty sum (28) can take values smaller than 1. This establishes the advantage of the additional level for improving precision in the measurement of atomic observables. Figure 3: Minimum value of the uncertainty sum $\left[\Delta^{2}A_{1}^{(ij)}+\Delta^{2}A_{2}^{(ij)}\right]_{\rm min}$ as a function of the populations $\rho_{ii}$, $\rho_{jj}$ (see (28)). Enhanced measurement precision of incompatible Pauli-like atomic observables is ensured whenever $\left[\Delta^{2}A_{1}^{(ij)}+\Delta^{2}A_{2}^{(ij)}\right]_{\rm min}<1$. Considerable research interest has been evinced in exploring the response of $\Lambda$, $\Xi$, V types of 3-level atomic systems to lasing radiation 3levelReview . In $\Lambda$-type system (see Fig. 4(a)) transition between the two lower levels $\|3\rangle$, $\|2\rangle$ is forbidden and the upper level $\|1\rangle$ is commonly shared in atomic transitions with levels $\|3\rangle$ and $\|2\rangle$; in V-type system (Fig. 4(b)), transitions from the lower level $\|3\rangle$ with the two upper levels $\|1\rangle$ and $\|2\rangle$ are allowed, but the transition $\|1\rangle\leftrightarrow\|2\rangle$ between the upper levels is forbidden. While atomic transitions $\|1\rangle\leftrightarrow\|2\rangle$ and $\|2\rangle\leftrightarrow\|3\rangle$ are allowed in $\Xi$ type system (Fig. 4(c)), the transition $\|1\rangle\leftrightarrow\|3\rangle$ is forbidden. It is of interest to consider any two levels of the atomic 3-level system, between which transitions are allowed, as a qubit, and explore if there is any enhanced precision in the measurement of two non-commuting qubit operators. To this end, we consider the coherent population trapping state in a $\Lambda$ type atomic system radmore82 ; pra95 ; pro_opt96 ; pra97 : $\displaystyle\varrho^{\Lambda}_{33}=\frac{1}{2}=\varrho^{\Lambda}_{22},\ \ \varrho^{\Lambda}_{11}=0.$ (29) It is clearly seen that the uncertainty sum (28) with $i=1,\,j=2$ and $i=1,\,j=3$ is given by $\displaystyle\left[\Delta^{2}A_{1}^{(12)}+\Delta^{2}A_{2}^{(12)}\right]^{\Lambda}_{\rm min}=2\,(\varrho^{\Lambda}_{11}+\varrho^{\Lambda}_{22})-(\varrho^{\Lambda}_{11}+\varrho^{\Lambda}_{22})^{2}=0.75$ $\displaystyle\left[\Delta^{2}A_{1}^{(13)}+\Delta^{2}A_{2}^{(13)}\right]^{\Lambda}_{\rm min}=2\,(\varrho^{\Lambda}_{11}+\varrho^{\Lambda}_{33})-(\varrho^{\Lambda}_{11}+\varrho^{\Lambda}_{33})^{2}=0.75$ revealing improved precision in the measurements of the qubit operator pairs $\left\\{A_{1}^{(12)},\,A_{2}^{(12)}\right\\}$ and $\left\\{A_{1}^{(13)},\,A_{2}^{(13)}\right\\}$. In the case of V-type 3-level atom, with the transition $\|3\rangle\leftrightarrow\|2\rangle$ driven by a strong-coupling laser field and $\|3\rangle\leftrightarrow\|1\rangle$ transition driven by an incoherent pump field the steady state populations are given by pra96 $\displaystyle\varrho^{V}_{11}\approx 0.2,\ \varrho^{V}_{22}\approx\varrho^{V}_{33}\approx 0.4.$ (30) The uncertainty sum (28) of Pauli-like atomic observables associated with $i=1,\,j=3$ and $i=2,\,j=3$ are given by $\displaystyle\left[\Delta^{2}A_{1}^{(13)}+\Delta^{2}A_{2}^{(13)}\right]^{V}_{\rm min}=2\,(\varrho^{V}_{11}+\varrho^{V}_{33})-(\varrho^{V}_{11}+\varrho^{V}_{33})^{2}=0.84$ $\displaystyle\left[\Delta^{2}A_{1}^{(23)}+\Delta^{2}A_{2}^{(23)}\right]^{V}_{\rm min}=2\,(\varrho^{V}_{22}+\varrho^{V}_{33})-(\varrho^{V}_{22}+\varrho^{V}_{33})^{2}=0.96.$ Thus the steady state of V-type atomic qutrit (see (30)) offers advantage over 2-level atomic system in reducing the uncertainty sum of non-commuting Pauli- like atomic observables. Figure 4: Schematic diagrams of (a) $\Lambda$-type (b) V-type and (c) $\Xi$-type 3-level atomic systems. Arrows between the energy levels indicate allowed transitions. Populations in the steady state of a 3-level $\Xi$ atomic system pra97 ; pra98 satisfy the condition $\displaystyle\varrho^{\Xi}_{11}=\varrho^{\Xi}_{22}\leq\frac{1}{3},\ \ \ \frac{1}{3}\leq\varrho^{\Xi}_{33}\leq\frac{1}{2}.$ (31) We thus obtain the limiting value of the uncertainty sum as, $\displaystyle\left[\Delta^{2}A_{1}^{(12)}+\Delta^{2}A_{2}^{(12)}\right]^{\Xi}_{\rm min}$ $\displaystyle=$ $\displaystyle 2\,\left(1-\varrho^{\Xi}_{33}\right)-\,\left(1-\varrho^{\Xi}_{33}\right)^{2};\ \ \ \frac{1}{3}\leq\varrho^{\Xi}_{33}\leq\frac{1}{2},$ $\displaystyle\implies$ $\displaystyle\hskip 14.45377pt\frac{3}{4}\leq\left[\Delta^{2}A_{1}^{(12)}+\Delta^{2}A_{2}^{(12)}\right]^{\Xi}_{\rm min}\leq\frac{8}{9},$ $\displaystyle\left[\Delta^{2}A_{1}^{(23)}+\Delta^{2}A_{2}^{(23)}\right]^{\Xi}_{\rm min}$ $\displaystyle=$ $\displaystyle 2\,\left(1-\varrho^{\Xi}_{11}\right)-\,\left(1-\varrho^{\Xi}_{11}\right)^{2};\ \ \ 0\leq\varrho^{\Xi}_{11}\leq\frac{1}{3},$ $\displaystyle\implies$ $\displaystyle\hskip 14.45377pt\frac{8}{9}\leq\left[\Delta^{2}A_{1}^{(12)}+\Delta^{2}A_{2}^{(12)}\right]^{\Xi}_{\rm min}\leq 1,$ for the atomic Pauli-like observables associated with $i=1,\,j=2$ and $i=2,j=3$ atomic levels of the $\Xi$-type atomic system. More detailed investigations on improving measurement precision of non-commuting Pauli-like atomic observables of a driven 3-level system, beyond what can be achieved in a 2-level system, will be reported separately. ## 4 Transformation of qutrit state into a two-qubit symmetric state We now analyze the enhanced accuracy of orthogonal Pauli measurements on qutrit states from a perspective based on the separability/non-separability of two-qubit symmetric states corresponding to qutrit states. In the following, we detail the construction of two-qubit _symmetric states_ from qutrit states, in particular the state in (9). We also carry out an analysis of the role, if any, of two-qubit entanglement in the measurement precision possible in the qutrit state (9). A two-qubit symmetric state belongs to the 3-dimensional maximum multiplicity space of the collective angular momentum $j=j_{1}+j_{2}$, $j_{1}=j_{2}=\frac{1}{2}$. With the dimensions of the qutrit space and that of the two-qubit symmetric state (expressed in angular momentum basis) being equal, they have a one-one correspondence. Here, we outline this correspondence and accomplish the construction of two-qubit symmetric states corresponding to qutrit states. The qutrit state in (9), expressed as a $3\times 3$ matrix in (10), can be written equivalently as $\rho_{\rm qutrit}\equiv\left(\begin{array}[]{cccc}\frac{\omega(1+r_{3})}{2}&\frac{\omega(r_{1}-ir_{2})}{2}&0&0\\\ \frac{\omega(r_{1}+ir_{2})}{2}&\frac{\omega(1-r_{3})}{2}&0&0\\\ 0&0&1-\omega&0\\\ 0&0&0&0\end{array}\right),$ (32) The density matrix $\rho_{AB}$ of a two-qubit symmetric state is given by uma $\rho_{AB}=\frac{1}{4}\left[I_{2}\otimes I_{2}+\sum_{i=1}^{3}\,s_{i}\left(\sigma_{i}\otimes I_{2}+I_{2}\otimes\sigma_{i}\right)+\sum_{i,j=1}^{3}\,t_{ij}\,(\sigma_{i}\otimes\sigma_{j})\right],$ (33) where $t_{ij}=t_{ji}$. The elements $\rho_{ij}$, $i,\,j=1,\,2,\,3,\,4$ of the $4\times 4$ matrix $\rho_{AB}$ are explicitly given by $\displaystyle\rho_{11}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\left(1+2s_{3}+t_{33}\right),\ \ \ \rho_{22}=\frac{1}{4}\left(1-t_{33}\right)=\rho_{33}$ $\displaystyle\rho_{12}$ $\displaystyle=$ $\displaystyle\rho_{13}=\frac{1}{4}\left(s_{1}-is_{2}+t_{13}-it_{23}\right)=\rho^{\ast}_{21}=\rho^{\ast}_{31}$ $\displaystyle\rho_{14}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\left(t_{11}-t_{22}-2it_{12}\right)=\rho^{\ast}_{14}$ $\displaystyle\rho_{24}$ $\displaystyle=$ $\displaystyle\rho_{34}=\frac{1}{4}\left(s_{1}-is_{2}-t_{13}+it_{23}\right)=\rho^{\ast}_{42}=\rho^{\ast}_{43}$ $\displaystyle\rho_{23}$ $\displaystyle=$ $\displaystyle\frac{1}{4}\left(t_{11}+t_{22}\right)=\rho_{32},\ \ \ \rho_{44}=\frac{1}{4}\left(1-2s_{3}+t_{33}\right)$ (34) The standard basis for a two-qubit state is given by the direct product basis (uncoupled basis) consisting of orthonormal vectors $\|1/2;1/2\rangle,\ \ \|1/2;-1/2\rangle,\ \ \|-1/2;1/2\rangle,\ \ \|-1/2;-1/2\rangle$ (35) where $\|m_{1};m_{2}\rangle=\|m_{1}\rangle\otimes m_{2}\rangle$, $m_{1},\,m_{2}=1/2,\,-1/2$, One can readily express the direct product basis $\\{\|m_{1};m_{2}\rangle\\}$ in terms of the collective angular momentum basis (coupled basis) $\\{\|jm\rangle\\}$, ($j=1,\,0$, $-j\leq m\leq j$ for each $j$) and vice versa, through $\displaystyle\|11\rangle$ $\displaystyle=$ $\displaystyle\|1/2;1/2\rangle,\ \ \ \|10\rangle=\frac{1}{\sqrt{2}}\left(\|1/2;-1/2\rangle+\|-1/2;1/2\rangle\right)$ (36) $\displaystyle\|1-1\rangle$ $\displaystyle=$ $\displaystyle\|-1/2;-1/2\rangle,\ \ \ \|00\rangle=\frac{1}{\sqrt{2}}\left(\|1/2;-1/2\rangle-\|-1/2;1/2\rangle\right).$ From (36), it follows that the unitary matrix $U=\left(\begin{array}[]{cccc}1&0&0&0\\\ 0&0&0&1\\\ 0&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\ 0&\frac{1}{\sqrt{2}}&\frac{-1}{\sqrt{2}}&0\end{array}\right),\ \ U^{\dagger}\,U=U\,U^{\dagger}=I_{4}$ (37) where † denotes hermitian conjugate, corresponds to the transformation from coupled basis to uncoupled basis. The similarity transformation $U^{\dagger}\rho_{\rm qutrit}U$ effects the transformation of the state $\rho_{\rm qutrit}$ (See (9), (32)) to its equivalent two-qubit symmetric state $\rho_{AB}$. Explicitly, we have $\rho_{AB}=U^{\dagger}\rho_{\rm qutrit}U=\frac{1}{2}\left(\begin{array}[]{cccc}(1+r_{3})\omega&0&0&(r_{1}-ir_{2})\omega\\\ 0&1-\omega&1-\omega&0\\\ 0&1-\omega&1-\omega&0\\\ (r_{1}+ir_{2})\omega&0&0&(1-r_{3})\omega\end{array}\right).$ (38) On comparing the elements (See (4)) of $\rho_{AB}$ with the corresponding elements in Eq. (38), we obtain the following relation between the parameters $\omega$, $r_{1}$, $r_{2}$, $r_{3}$, ($r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=1$) of the qutrit state in (9) and the parameters $s_{i}$, $t_{ij}$, $i,\,j=1,\,2,\,3$ of the symmetric two-qubit state $\rho_{AB}$ in (33). That is, the non-zero parameters $s_{i}$, $t_{ij}$, $i,\,j=1,\,2,\,3$ of $\rho_{AB}$ (See (33)) are seen to be $\displaystyle s_{3}$ $\displaystyle=$ $\displaystyle\omega r_{3},\ \ \ t_{12}=t_{21}=\omega r_{2}$ (39) $\displaystyle t_{11}$ $\displaystyle=$ $\displaystyle(1-\omega)+\omega r_{1},\ \ \ \ t_{22}=(1-\omega)-\omega r_{1},\ \ t_{33}=2\omega-1.$ Thus, a two-qubit symmetric state $\rho_{AB}$ in (33)) with its elements given in (39) corresponds to the qutrit state $\rho_{\rm qutrit}$ in (9). ## 5 Sum uncertainty relation for two-qubit state $\rho_{AB}$ Here, we set up the sum uncertainty relation for the two-qubit state $\rho_{AB}$ (See (38)) in order to establish the equivalence of its uncertainty region with that of the qutrit state $\rho_{\rm qutrit}$ (See (9)). In order to do this, we need to recognize the two-qubit observables ${\mathcal{A}}_{1}$, ${\mathcal{A}}_{2}$, which are equivalent to $A_{1}$, $A_{2}$ (See (11)). For simplicity, and without loss of generality, we choose the orthonormal vectors in (11) to be $\hat{a}=(1,\,0,\,0)$, $\hat{b}=(0,\,1,\,0)$ so that $A_{1}=\sigma_{1}\oplus 0$, $A_{2}=\sigma_{2}\oplus 0$. It is not difficult to see that we can express $A_{1}$, $A_{2}$ as $A_{1}=\sigma_{1}\oplus{\bf 0}_{2},\ \ A_{2}=\sigma_{2}\oplus{\bf 0}_{2},\ \ {\bf 0}_{2}=\left(\begin{array}[]{cc}0&0\\\ 0&0\end{array}\right),$ (40) to facilitate their action on the qutrit state $\rho_{\rm qutrit}$ expressed as a $4\times 4$ matrix in (32). Corresponding to the basis transformation $\rho_{\rm qutrit}\longrightarrow\rho_{AB}$ in (38), the observables $A_{1}$, $A_{2}$ (See (40)) undergo the similarity transformation $\displaystyle{\mathcal{A}}_{1}$ $\displaystyle=$ $\displaystyle U^{\dagger}\,A_{1}U=\frac{1}{2}\left[\sigma_{1}\otimes\sigma_{1}-\sigma_{2}\otimes\sigma_{2}\right],$ $\displaystyle{\mathcal{A}}_{2}$ $\displaystyle=$ $\displaystyle U^{\dagger}\,A_{2}U=\frac{1}{2}\left[\sigma_{1}\otimes\sigma_{2}+\sigma_{2}\otimes\sigma_{1}\right].$ (41) Here $U$ is the basis transformation matrix (See (37)) that takes $\rho_{\rm qutrit}$ to $\rho_{AB}$ (See (38)). On explicit evaluation, we get $\displaystyle\langle{\mathcal{A}}_{1}\rangle=\mbox{Tr}\,({\mathcal{A}}_{1}\rho_{AB})=\omega\,r_{1},\ \ \ \langle{\mathcal{A}}_{1}^{2}\rangle=\mbox{Tr}\,({\mathcal{A}}_{1}^{2}\rho_{AB})=\omega$ $\displaystyle\langle{\mathcal{A}}_{2}\rangle=\mbox{Tr}\,({\mathcal{A}}_{2}\rho_{AB})=\omega\,r_{2},\ \ \ \langle{\mathcal{A}}_{2}^{2}\rangle=\mbox{Tr}\,({\mathcal{A}}_{2}^{2}\rho_{AB})=\omega$ $\displaystyle\Delta^{2}{\mathcal{A}}_{1}=\omega-\omega^{2}\,r_{1}^{2},\ \ \ \ \ \ \ \ \ \ \ \Delta^{2}{\mathcal{A}}_{2}=\omega-\omega^{2}\,r_{2}^{2}.$ (42) The expressions for $\Delta^{2}{\mathcal{A}}_{1}$, $\Delta^{2}{\mathcal{A}}_{2}$ in (5) are the same as that obtained in (13) for the qutrit state $\rho_{\rm qutrit}$ (See (9)). The uncertainty region of the two-qubit state $\rho_{AB}$ is thus the same as that of the qutrit state $\rho_{\rm qutrit}$, with the origin $(\Delta{\mathcal{A}}_{1},\,\Delta{\mathcal{A}}_{2})=(0,\,0)$ being a physically realizable point (See Fig. 2). As $r_{1}^{2}+r_{2}^{2}+r_{3}^{2}=1$, the sum uncertainty relation of the two- qubit state $\rho_{AB}$ can be simplified to (See (5)) $\Delta^{2}{\mathcal{A}}_{1}+\Delta^{2}{\mathcal{A}}_{2}=2\omega-\omega^{2}\,\kappa^{2},\ \ \ \kappa=\sqrt{r_{1}^{2}+r_{2}^{2}}=\sqrt{1-r_{3}^{2}}.$ (43) Fig. 4 shows the variation of the uncertainty sum $\Delta^{2}{\mathcal{A}}_{1}+\Delta^{2}{\mathcal{A}}_{2}$ as a function of the parameters $0\leq\omega,\kappa\leq 1$. Figure 5: The contour plot showing the variation of the uncertainty sum $\Delta^{2}{\mathcal{A}}_{1}+\Delta^{2}{\mathcal{A}}_{2}$ with respect to the parameters $\omega$ and $\kappa$. The dotted line corresponds to $\Delta^{2}{\mathcal{A}}_{1}+\Delta^{2}{\mathcal{A}}_{2}=\frac{3}{4}$. ### 5.1 Sum uncertainty relation for symmetric two-qubit separable states Our intention in obtaining the two-qubit counterpart $\rho_{AB}$ of the qutrit state $\rho_{\rm qutrit}$ lies in utilizing its _composite nature_ and examine whether separability/non-separability of $\rho_{AB}$ has any role in the better precision observed in joint measurement of Pauli observables on the _single_ party state $\rho_{\rm qutrit}$. In order to carry out this task, we consider the most general bipartite, symmetric separable state $\rho_{\rm sep}=\sum_{i}\,p_{i}\left(\rho_{i}\otimes\rho_{i}\right),\ \ i=1,\,2,\,3\ldots$ (44) with $0\leq p_{i}\leq 1$, $\sum_{i}\,p_{i}=1$ being the probabilities. It has been shown in Ref. pfun that the single qubit density operators $\rho_{i}$, $i=1,\,2,\,3\ldots$ constituting any symmetric separable state $\rho_{\rm sep}$ are necessarily _pure_. Thus, $\rho_{i}=\frac{1}{2}\,\left(I_{2}+\vec{\sigma}\cdot\hat{s}_{i}\right),\ \ \hat{s}_{i}=\left(s_{1i},\,s_{2i},\,s_{3i}\right),\ \ s_{1i}^{2}+s_{2i}^{2}+s_{3i}^{2}=1.$ (45) We evaluate the expectation values of ${\mathcal{A}}_{\alpha},$ and ${\mathcal{A}}^{2}_{\alpha}$, $\alpha=1,2$ (See (5)) in a product state $\rho_{i}\otimes\rho_{i}$ (where $\rho_{i}$ is given by (45)): $\displaystyle\langle{\mathcal{A}}_{1}\rangle_{i}$ $\displaystyle=$ $\displaystyle\mbox{Tr}\,\left[\left(\rho_{i}\otimes\rho_{i}\right){\mathcal{A}}_{1}\right]=\frac{1}{2}\,\left(s_{1i}^{2}-s_{2i}^{2}\right),$ $\displaystyle\langle{\mathcal{A}}_{1}^{2}\rangle_{i}$ $\displaystyle=$ $\displaystyle\mbox{Tr}\,\left[\left(\rho_{i}\otimes\rho_{i}\right){\mathcal{A}}_{1}^{2}\right]=\frac{1}{2}\,\left(1+s_{3i}^{2}\right),$ $\displaystyle\langle{\mathcal{A}}_{2}\rangle_{i}$ $\displaystyle=$ $\displaystyle\mbox{Tr}\,\left[\left(\rho_{i}\otimes\rho_{i}\right){\mathcal{A}}_{2}\right]=s_{1i}s_{2i},$ $\displaystyle\langle{\mathcal{A}}_{2}^{2}\rangle_{i}$ $\displaystyle=$ $\displaystyle\mbox{Tr}\,\left[\left(\rho_{i}\otimes\rho_{i}\right){\mathcal{A}}_{2}^{2}\right]=\frac{1}{2}\,\left(1+s_{3i}^{2}\right),$ leading to $\displaystyle\left(\Delta^{2}\,{\mathcal{A}}_{1}\right)_{i}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[1+s_{3i}^{2}-\frac{1}{2}(s_{1i}^{2}-s_{2i}^{2})^{2}\right]$ $\displaystyle\left(\Delta^{2}\,{\mathcal{A}}_{2}\right)_{i}$ $\displaystyle=$ $\displaystyle\frac{1}{2}\left[1+s_{3i}^{2}-2s_{1i}^{2}s_{2i}^{2}\right]$ (46) Using $s_{1i}^{2}+s_{2i}^{2}+s_{3i}^{2}=1$ and on simplification, we get $\left(\Delta^{2}\,{\mathcal{A}}_{1}\right)_{i}+\left(\Delta^{2}\,{\mathcal{A}}_{2}\right)_{i}=\frac{3}{4}+\frac{3}{2}s_{3i}^{2}-\frac{1}{4}s_{3i}^{4}$ (47) From the structure of $\rho_{\rm{sep}}$ (See (44)), we readily have $\Delta^{2}{\mathcal{A}}_{1}=\sum_{i}\,p_{i}\,\left(\Delta^{2}\,{\mathcal{A}}_{1}\right)_{i}$, $\Delta^{2}{\mathcal{A}}_{2}=\sum_{i}\,p_{i}\,\left(\Delta^{2}\,{\mathcal{A}}_{2}\right)_{i}$ and hence we get (See (47)) $\Delta^{2}\,{\mathcal{A}}_{1}+\Delta^{2}\,{\mathcal{A}}_{2}=\frac{3}{4}+\frac{3}{2}\sum_{i}\,p_{i}\,s_{3i}^{2}-\frac{1}{4}\sum_{i}\,p_{i}\,s_{3i}^{4}$ (48) As $0\leq s_{3i}\leq 1$, it readily follows that $\left(\Delta^{2}\,{\mathcal{A}}_{1}+\Delta^{2}\,{\mathcal{A}}_{2}\right)_{\rm min}=\frac{3}{4}$ (49) which happens when $s_{3i}=0$ for all $i=1,\,2,\,3,\cdots$. In other words, $\Delta^{2}\,{\mathcal{A}}_{1}+\Delta^{2}\,{\mathcal{A}}_{2}\geq\frac{3}{4}$ (50) is the sum-uncertainty relation for symmetric separable two-qubit states, with its lowest bound being $3/4$. Thus the uncertainty sum $\rho_{\rm sep}$ (See (44)), set up for the two-qubit observables ${\mathcal{A}}_{1}$, ${\mathcal{A}}_{2}$ in (5) cannot even go close to zero. This implies that symmetric separable states (See (44)) can never achieve maximum accuracy in joint measurements by ${\mathcal{A}}_{1}$, ${\mathcal{A}}_{2}$ in (5). We now wish to check whether entanglement in the two-qubit state $\rho_{AB}$ contributes to enhanced precision in the measurements of the observables ${\mathcal{A}}_{1}$, ${\mathcal{A}}_{2}$. To this end, we evaluate the concurrence Wootters ; hill , a measure of two-qubit entanglement, of the state $\rho_{AB}$. Concurrence of any arbitrary two-qubit state $\rho$ is defined as Wootters ; hill $C=\mbox{max}\,\left(0,\,\sqrt{\lambda_{1}}-\sqrt{\lambda_{2}}-\sqrt{\lambda_{3}}-\sqrt{\lambda_{4}}\right)$ (51) where $\lambda_{k}$, $k=1,\,2,\,3,\,4$ are the eigenvalues of the matrix $\rho(\sigma_{y}\otimes\sigma_{y})\rho^{\ast}(\sigma_{y}\otimes\sigma_{y})$, arranged in the descending order (i.e., $\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq\lambda_{4}$). The structure of $\rho_{AB}$ in (38) allows us to make use of the simplified expression for concurrence given in Ref. wang , and leads to $C_{AB}=\bigg{\\{}\begin{array}[]{cc}\omega(1+\kappa)-1\ \ \ \mbox{for}\ \ \ \omega(1+\kappa)\geq 1\\\ 1-\omega(1+\kappa)\ \ \ \mbox{for}\ \ \ \ \omega(1+\kappa)\leq 1\end{array}$ (52) where $\kappa=\sqrt{1-r_{3}^{2}}$ and $0\leq r_{3}\leq 1$. In other words, we have $C_{AB}=\left\|\omega(1+\kappa)-1\right\|,\ \ \kappa=\sqrt{1-r_{3}^{2}},\ \ 0\leq r_{3}\leq 1.$ (53) A contour plot of $C_{AB}$ as a function of the parameters $\omega$ and $\kappa$ is shown in Fig. 6. Figure 6: The contour plot showing the variation of concurrence $C_{AB}$ with respect to the parameters $\omega$ and $\kappa$ Based on Figs. 5 and 6, we reach the following conclusions: 1. 1. The uncertainty sum $\Delta^{2}\,{\mathcal{A}}_{1}+\Delta^{2}\,{\mathcal{A}}_{2}$ can be reduced below the value $3/4$ only in an entangled state $\rho_{AB}$ (i.e., when $C_{AB}\neq 0$). In particular, $\Delta^{2}\,{\mathcal{A}}_{1}+\Delta^{2}\,{\mathcal{A}}_{2}\longrightarrow 0$, in maximally entangled two-qubit symmetric states (i.e., when $C_{AB}\rightarrow 1$). 2. 2. While no separable state can reduce the uncertainty sum $\Delta^{2}\,{\mathcal{A}}_{1}+\Delta^{2}\,{\mathcal{A}}_{2}$ below the value $3/4$, there indeed exist entangled states with $\Delta^{2}\,{\mathcal{A}}_{1}+\Delta^{2}\,{\mathcal{A}}_{2}\geq 3/4$. This implies that, while qutrit states $\rho_{\rm qutrit}$ that permit accurate simultaneous measurements of the observables $A_{1},\ A_{2}$ are necessarily associated with entangled two-qubit states, the converse is not always true. It can thus be concluded that entanglement in a two-qubit state constructed from a qutrit (a qubit appended with an additional level) plays a significant role in the precise joint measurements by a pair of orthogonal Pauli observables. It would be of interest to examine whether two-qubit states constructed from a qudit (a qubit with $d-1$ ancillary levels) exhibit a similar feature. A study of uncertainty region of such two-qubit states, dimensional dependence of uncertainty sum and accuracy of simultaneous measurements by _any_ incompatible pair of observables in $d$-dimensional spaces form topics of further interest. ## 6 Conclusion This work is a contribution to the ongoing study on uncertainty regions Werner ; Li ; Abbot ; Busch providing a different perspective in accounting for better measurement precision seen in qutrits. For any arbitrary 3-level atomic systems we have obtained an expression for minimum value of uncertainty sum for Pauli-like observables in terms of atomic populations. This is useful to study if enhanced measurement precision (reduction in the uncertainty sum) can be realized in $\Lambda$, V and $\Xi$ types of 3-level atomic systems, which are characterised by different schemes of allowed/forbidden atomic transitions between any two levels. 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# Modelling the Impact of Scandals: the case of the 2017 French Presidential Election Yassine Bouachrine and Carole Adam (This is an ENSIMAG internship report originally written in June 2019 by intern Yassine Bouachrine under the supervision of Carole Adam) ###### Abstract This paper proposes an agent-based simulation of a presidential election, inspired by the French 2017 presidential election. The simulation is based on data extracted from polls, media coverage, and Twitter. The main contribution is to consider the impact of scandals and media bashing on the result of the election. In particular, it is shown that scandals can lead to higher abstention at the election, as voters have no relevant candidate left to vote for. The simulation is implemented in Unity 3D and is available to play online. Keywords: agent-based simulation, computational social choice, voting models ## 1 Introduction During the 2017 French presidential election, the media had a very impactful role in the shift of the opinion away from the election’s favorite François Fillon. The seriousness of the accusations against the candidate led to Fillon plummeting in the polls. We will try to model the impact of both conventional and social media through scandal diffusion, in order to better understand the dynamics underlying the voting process. There are a variety of existing models for the voting process. However, most of the models see voting as a discrete event, while it is actually a sample at a given instant of a system in continuous evolution. This paper tries to shed some light on improvements which could take into account the inherent dynamism that comes with the interactions of the voters. Another issue is that most computer voting models are adapted to the American context. In France, there are multiple candidates participating in the first round of the elections but in reality, few of them are actually considered by the voters, despite being aligned with their ideals. This is due to the nature of the voting process in France which takes place in two rounds. This leads the voters to cast their vote strategically towards candidates who have a chance of making it out of the first round. The Voter Autrement111Voter Autrement : https://vote.imag.fr/ experiment explored the effects of this strategic voting in 2017 by testing various voting systems during the presidential election. The results showed that the alternative voting methods yield vastly different results (in terms of who is elected), especially for candidates such as J-L. Mélenchon and H. Hamon (systematic improvement) or F. Fillon and M. Le Pen (systematic decline). Other voting methods such as candidate ranking make the strategic vote useless, since every candidate gets a chance to make it to the second turn. Finally, the context for the 2017 presidential election is even more unique in regard to the state of the French political scene. It is the first presidential election since the split of the centre-right party (UMP), and it is taking place amid the overall dissatisfaction of the population with the French Socialist Party (PS, left wing). This will be further elaborated on in the Voting models section as it is relevant to studying the impact of such a context on the existing models. Our goal here is to model the impact of scandals over the course of the election and the change in the opinion of the voters. Last year, A. Soutif [13] tried to model the election process using an agent-based model by feeding the agents the results of the polls reported by the media. His goal in that study was to model the impact of the polls on the votes and on the strategic vote in particular, as polls give information on which candidates have chances of making it out of the first round. Our approach aims to complement this work by showing the additional impact of media through the diffusion of scandals during the campaign. The first part of the paper (Section 2) focuses on the data analysis upon which the model is built. The second part of the paper (Section 3) addresses voting models and our implementation of the suggested model. ## 2 Data analysis Building a model requires the availability of sufficient and relevant data about the phenomenon we are trying to model. ### 2.1 Comparing poll results and media trends In our case, we took the aggregated poll results from various organisms [14] shown in Figure 1, and interpolated linearly when we had missing data (mostly B. Hamon in the early polls). We then compared the result of these polls with the evolution of the presence of the candidates across traditional media and Twitter. Figure 1: Evolution of the polls during the elections (Source: Wikipedia [15]) Google Trends222https://trends.google.fr/trends/ is a Google website that offers statistics on Google Search queries. We used it to evaluate the searched queries associated with the the top five candidates in the News section, the results are shown on Figure 2. An interesting point can be made around how F. Fillon dominates the media presence in the early months of the election, alongside B. Hamon which we can ignore since it is mostly due to the French Socialist Party presidential primary. Figure 2: Evolution of the search queries for the candidates We can see that media coverage of this candidate spikes at multiple points in time (Figure 3), which is likely due to the “Penelope gate”, which is a scandal associated with the alleged fictitious employment of members of Fillon’s family. Further, the media frenzy surrounding the “Penelope gate” looks correlated with the evolution of the polls, as shown in Figure 3. Figure 3: Evolution of the polls and search queries for Fillon The takeaway from this comparison is that further media bashing does not seem to impact polls as much. This can be attributed to the nature of the scandal being a punctual event, and that people willing to cast away their vote already did so with the initial media outlets. The biggest beneficiary of this is definitely the closest candidate in the political spectrum: E. Macron. Another interesting candidate is M. Le Pen who dropped significantly in the polls (Figure 4) once the media picked up that she could actually finish first at the first round. Figure 4: Evolution of the polls and search queries for Le Pen An hypothesis could be that there is actually a bidirectional relationship between polls and media coverage, with each one affecting the other. For example, in F. Fillon’s case it is the disclosure of the scandal by Le Canard Enchaîné that led to him dropping significantly, whereas in M. Le Pen’s case, it is the poll results that led to an increase in coverage, and then her poll results dropped as a result of this increased coverage. ### 2.2 Processing Twitter data The limits of the information we can leverage from Google Trends is that it does not tell us about the nature or content of the coverage. We cannot know for sure if the increasing number of articles are rather positive or negative ones. We can deduce that a posteriori by observing the impact on the polls, but it is what we are actually trying to model. Therefore, we used Twitter data, and analyze the tweets during the primary round to try and visualize opinion trends during the election. #### 2.2.1 Data sets Two datasets were used in our work: * • The first one is from Kaggle 333https://www.kaggle.com/jeanmidev/french- presidential-election: Kaggle is an online exchange platform for datascientist, users can publish datasets among other things. This Kaggle dataset contains tweets sampled during the elections. It is very rich but the data collection rate varies and some tweets appear to be truncated. More details about this dataset can be found at the source. * • The other dataset is a courtesy of E. Duble, research engineer at LIG. It contains an anonymized collection, sampling only geotagged tweets, that are mentioning the top hashtags during the election. The importance of hashtags has been shown for instance by the Politoscope project [10] #### 2.2.2 Clustering Building a vectorial representation of the tweets can be done in various ways. M. Campr and K. Jezek [3] provided a performance evaluation of various methods for paragraph vectorization. At first, we opted for Tweet2Vec [6], which relies on character-based representations (as opposed to word representations for Doc2Vec [8]) that perform better for content such as tweets. However, Tweet2vec uses hashtag prediction to train the model, which is limited for our use-case since we already have a restricted number of hashtags. It also takes longer to train compared to word-based models. We used Facebook fastText 444Facebook FastText: https://fasttext.cc/ to generate embeddings, and enhanced them with their term frequency-inverse document frequency (TF-IDF, [12]) in the corpus. We then computed the tweet embeddings as the average of the word embeddings. Performing Principal Component Analysis (PCA, [16]) on the 100-dimensional tweet embeddings did not yield very good results as the embeddings are already built to minimize colinearity. ## 3 Voting model In this section, we provide an overview of existing models and their limitations before presenting the model we built through the observation of the data. ### 3.1 Existing models and their limitations Doing a taxonomy of existing models is outside the scope of this paper, there are good resources available for that [1, 9, 7]. We will focus on some limitations of the existing models, starting with the psychological model. As hinted to in the introduction, the context of the election makes partisan identification hard to rely on: the schism of the centre-right party reshaped the political scene entirely. Also, the overall dissatisfaction with F. Hollande hurt the socialist party. Towards the end of his mandate, his popularity rating was lower than Macron’s was during the Gilet Jaunes protests [11]. On top of that, the appearance of new actors on the scene such as En Marche further shook the scene. En Marche made retrospective voting irrelevant as the party had never held responsibilities. The novelty Macron brings to the table, and his ambition of uniting the political parties, gave him a considerable advantage. All of these circumstances made the elections very volatile. Even more sophisticated models such as the funnel of causality have to be rethought. Figure 5: Funnel of causality, source [4] Media has to have a bigger role in this funnel, especially social media as it has been shown to be a good indicator of standings [2], almost as good as traditional polls. And that is, despite it being sensitive to social engineering (cf. Cambridge Analytica’s impact on the American presidential election). ### 3.2 Proposed model Our model is an enhancement of proximity models, in order to take into account the diffusion of scandals and the movement of neighboring agents. #### 3.2.1 Simulation environment and initialization The environment is a 100 X 100 units 2D plan. There are two types of active agents: candidates and voters. For the simulation, we define: * • The appeasement delta $\Delta\alpha\in[0,1]$, rate at which the repulsion of the candidate diminishes * • The falloff rate for the potential of the scandals $\Delta\rho\in[0,1]$ * • The maximum openness for the voters $\sigma_{max}\in[0,100]$, defines how far a voter considers his surroundings * • The maximum tolerance for the voters $\theta_{max}\in[0,+\infty[$ #### 3.2.2 Agents and their attributes For candidates $C$, we define: * • Position at time $t$ as $\psi_{t}\in[0,1]^{2}$ initialized manually * • Repulsion at time $t$ as $\gamma_{t}\in[0,1]$ with $\gamma_{0}=0$ * • A list S of scandals with $S_{i}$ being the i-th one. For voters $V$, we define: * • Position at time $t$ as $\psi_{t}\in[0,1]^{2}$ with $\psi_{0}\sim\mathcal{U}^{2}(0,1)$ (distance to a candidate inversely proportional to agreement with this candidate) * • Openness as $\sigma\in[0,\sigma_{max}]$ with $\sigma\sim clamp_{[0,1]}(\mathcal{N}(0.5,0.2^{2}))\sigma_{max}$ (the radius is which a voter considers agents around him) * • Charisma as $\kappa\in[0,1]$ with $\kappa\sim clamp_{[0,1]}(\mathcal{N}(0.5,0.2^{2}))$ (the chance to influence others around) * • Tolerance as $\theta\in[0,1]$ with $\theta\sim clamp_{[0,1]}(\mathcal{N}(0.5,0.2^{2}))\theta_{max}$ (the threshold for repulsion before dismissing a candidate completely) * • Conformity as $\eta\in[0,1]$ with $\eta\sim clamp_{[0,1]}(\mathcal{N}(0.5,0.2^{2}))$ For scandals, we define: * • Potential at time $t$ as $\rho_{t}\in[0,1]$ with $\rho_{0}$ initialized manually by the user (parameter in the simulation) #### 3.2.3 Simulation update At each time-step of the simulation, we update the entities. To simplify the equations, we assume that the values are clamped to their domain. For scandals, the potential decreases with time, at the falloff rate: $\rho_{t+1}(x)=\rho_{t}(x)-\Delta\rho$ (1) For candidates, the position is static, and the repulsion increases with each scandal, and decreases with time at the pace set by the appeasement delta: $\gamma_{t+1}(x)=\gamma_{t}(x)-\Delta\alpha+\sum_{y\in S(x)}\rho_{t+1}(y)$ (2) For voters, the position evolves as their opinion about the different candidates evolves based on their surroundings: $\begin{split}\psi_{t+1}(x)&=\psi_{t}(x)\\\ &+\operatorname{argmin}_{y}\\{\|\|\psi_{t}(x)-\psi_{t}(y)\|\|_{2}\mid y\in C\land\gamma_{t+1}(y)<\theta(x)\\}\frac{1}{1+\gamma_{t+1}(y)}\\\ &+\eta\sum_{y\in V\land\|\|\psi_{t}(x)-\psi_{t}(y)\|\|_{2}<\sigma(x)}\kappa(y)(\psi_{t}(x)-\psi_{t}(y))\end{split}$ (3) When the simulation stops, each voter votes for the closest candidate still considered in the openness radius around him. If there are none, the voter withholds his vote (chooses abstention). #### 3.2.4 Implementation details The simulation is built in Unity 2018.3.5f1 555Unity 3D: https://unity.com/. It allowed for faster prototyping and also supports a wide range of platforms to run the simulation on. The simulation is available to play online at http://lig-tdcge.imag.fr/votsim/ or to download as a WebGL export666WebGL export of the simulator: https://ensiwiki.ensimag.fr/index.php?title=IRL_- _Modélisation_de_la_dynamique_des_opinions_des_électeurs. A first screen lets the user select the values of the global parameters of the simulation (Figure 6): the number of voters and candidates, the appeasement delta (rate at which the scandals decrease, which determines the duration of their effect on opinions), and the maximum values for tolerance and openness (individual values of all agents are then set randomly under this boundary). Figure 6: Parameter selection before starting the simulation On the next screen, the user can modify the simulation speed. Voters are moving in the environment towards or away from the candidates. The user can also trigger a scandal and choose its intensity and target candidate, by using a button at the bottom of the window, in order to observe the influence on the movements of the voters (see Figure 7). The intensity of the scandal then decreases with time. Figure 7: Screenshot of the simulation after triggering a scandal #### 3.2.5 Discussion and results Regarding the motivations behind the model, we wanted to enhance the existing models with the observations made from our data analysis. First, regarding the initialization, the justification behind the uniform distribution for the voters’ positions is the unique context behind the 2017 election that we detailed earlier, with many voters not knowing which parties to consider. A fine-tuned Gaussian mixture model could also be explored. Most of the reasoning behind the model is based on the reactions to the Penelopegate, with some voters completely turning their backs on F. Fillon (which we model by a scandal being above their tolerance threshold) and some only showing hesitation (tolerance threshold not reached). A temporary dip followed by a partial recovery in the polls supports the model: regardless of the further media bashing around the event, voters have a threshold over which further coverage has no effect. Over the simulated scenarios, one of the most interesting observations is that scandals tend to be tied with an increased abstention rate. In our model this is represented by the voter moving too far from all candidates (due to repulsion generated by scandals about their favourite candidates, or to diverging opinions with the others), so that no valid candidate is still present in the openness radius when the election comes; in that case the voter prefers to choose abstention. Our model can therefore reproduce and explain the 2017 presidential election’s high abstention rate in the first round, at 22.23% [5]. ## 4 Conclusion We have seen that media coverage of the campaign scandals can have a big impact on the election results. The simulation showed that scandals can totally shape the result of the election and that scandals profit to the closer candidates on the political spectrum. The more interesting finding was how scandals impact the abstention rate, which is in agreement with the observations made in the context of the French 2017 presidential election and the high abstention rate recorded. There is still much to do to reach a unified model, a first step in that direction would be enhancing the simulation with the results from A. Soutif’s experiments regarding the impact of the polls [13]. We could then initialize the model to match the French political scene at the beginning of the first round and test if it corresponds to the observed election results. If the model is validated, we could explore alternative scenarios for the election: how different scandals could have led to different results and particularly what would have happened if there were no scandals involving the pre-campaign favorite F. Fillon. ## References [1] Rui Antunes. Theoretical models of voting behaviour. Exedra, 4(1):145–70, 2010. * [2] David Anuta, Josh Churchin, and Jiebo Luo. Election bias: Comparing polls and twitter in the 2016 us election. arXiv preprint arXiv:1701.06232, 2017. * [3] Michal Campr and Karel Ježek. Comparing semantic models for evaluating automatic document summarization. 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A Study On Some Geometric and Physical Properties of Hyper-Generalised Quasi-Einstein Spacetime Kaushik Chattopadhyay1111This is the corresponding author., Arindam Bhattacharyya2 and Dipankar Debnath3 1Department of Mathematics, Jadavpur University, Kolkata-700032, India <EMAIL_ADDRESS> 2Department of Mathematics, Jadavpur University, Kolkata-700032, India <EMAIL_ADDRESS> 3Department of Mathematics, Bamanpukur High School(H.S), Nabadwip, India <EMAIL_ADDRESS> ###### Abstract In the present paper we discuss about a set of geometric and physical properties of hyper-generalised quasi-Einstein spacetime. At the beginning we discuss about pseudosymmetry over a hyper-generalised quasi-Einstein spacetime. Here we discuss about $W_{2}$-Ricci pseudosymmetry, $Z$-Ricci pseudosymmetry, Ricci pseudosymmetry and projective pseudosymmetry over a hyper-generalised quasi-Einstein spacetime. Later on we take over Ricci symmetric hyper-generalised quasi-Einstein spacetime and derive a set of important geometric and physical theorems over it. Moving further we consider some physical applications of the hyper-generalised quasi-Einstein spacetime. Lastly we prove the existence of a hyper-generalised quasi-Einstein spacetime by constructing a non-trivial example. M.S.C.2010: 53C15, 53C25, 53C35. Keywords: $W_{2}$-curvature tensor, $Z$ tensor, projective curvature tensor, Riemannian curvature tensor, hyper generalised quasi-Einstein spacetime, Einstein equation, heat flux, stress tensor. ## 1 Introduction The General theory of Relativity is unarguably the most beautiful theory the World of Physics has ever produced. It is the most powerful result of the human intellect. This is an extremely important theory to study the nature of this universe, cosmology and gravity. Three most important things that Modern Scientists/ Mathematical Physicists can learn from special to general relativity are as follows: (i) The laws of Physics should be the same in every inertial reference frame, i. e., the abandonment of the privileged states of inertial frame of reference. (ii) The acceptance of the dynamical role of the metric $g$, i.e., the study of non-linear behaviour of nature. and (iii) The spacetime has to be considered as a class of semi-Riemannian geometry. The semi-Riemannian geometry has become more and more relevant and significant in dealing with the nature of this universe with every passing day. The theory of general relativity is mainly studied on a semi-Riemannian manifold which sometimes is not an Einstein spacetime. Thus it was always necessary to expand the concept of Einstein manifold to quasi-Einstein, then generalised quasi-Einstein, mixed generalised quasi-Einstein and lastly to hyper-generalised quasi-Einstein manifold. We demonstrate the introduction to this procedure as follows: An Einstein manifold is a Riemannian or pseudo-Riemannian manifold whose Ricci tensor S of type $(0,2)$ is non-zero and proportional to the metric tensor. Einstein manifolds form a natural subclass of various classes of Riemannian or semi-Riemannian manifolds by a curvature condition imposed on their Ricci tensor [4]. Also in Riemannian geometry as well as in general relativity theory, the Einstein manifold plays a very important role. M. C. Chaki and R. K. Maity had given the notion of quasi Einstein manifold [18] in $2000$. A non flat $n$-dimensional Riemannian manifold $(M^{n},g)$, $n(>2)$ is said to be a quasi-Einstein manifold if its nonzero Ricci tensor $S$ of type $(0,2)$ satisfies the following condition $S(X,Y)=\alpha g(X,Y)+\beta A(X)A(Y),$ (1.1) where for all vector fields $X$, $g(X,\xi_{1})=A(X),\leavevmode\nobreak\ g(\xi_{1},\xi_{1})=1.$ (1.2) That is, $A$ being the associated $1$-form, $\xi_{1}$ is generally known as the generator of the manifold. $\alpha$ and $\beta$ are associated nonzero scalar functions. This manifold is denoted by $(QE)_{n}$. Clearly, for $\beta=0$, this manifold reduces an Einstein manifold. We can note that Robertson-Walker spacetimes are quasi-Einstein spacetimes. In the recent papers [2], [19], the application of quasi-Einstein spacetime and generalised quasi-Einstein spacetime in general relativity have been studied. Many more works have been done in the spacetime of general relativity [3], [13], [14], [21], [22], [23], [24], [25], [26]. Then M. C. Chaki initiated the notion of generalized quasi-Einstein manifold [17] in 2001. A Riemannian manifold of dimension $n(>2)$ is said to be generalized quasi Einstein manifold if its Ricci tensor $S$ of type $(0,2)$ is not identically zero and satisfies the following condition $S(X,Y)=\alpha g(X,Y)+\beta A(X)A(Y)+\gamma[A(X)B(Y)+A(Y)B(X)],$ (1.3) where $\alpha$, $\beta$ and $\gamma$ are real valued, nonzero scalar functions on $(M^{n},g)$, $A$ and $B$ are called two non zero $1$-forms such that $g(X,\xi_{1})=A(X),g(X,\xi_{2})=B(X),g(\xi_{1},\xi_{2})=0,g(\xi_{1},\xi_{1})=1,g(\xi_{2},\xi_{2})=1.$ (1.4) Here $\xi_{1}$ and $\xi_{2}$ are two unit vector fields which are orthogonal to each other. $\alpha$, $\beta$ and $\gamma$ are called associated scalars, $A$ and $B$ are called associated $1$-forms. $\xi_{1}$ and $\xi_{2}$ are two generators of the manifold. This manifold is denoted by $(GQE)_{n}$. Clearly, for $\gamma=0$, then it takes the form of a quasi-Einstein manifold and for $\beta=\gamma=0$, it takes the form of an Einstein manifold. The notion of hyper-generalized quasi-Einstein manifold has been introduced by A. A. Shaikh, C. Özgür and A. Patra [1] in 2011. According to them, a Riemannian manifold $(M^{n},g)$, $(n>2)$ is said to be a hyper-generalized quasi-Einstein manifold if its Ricci tensor S of type $(0,2)$ is non-zero and satisfies the following condition $\displaystyle S(X,Y)=\alpha g(X,Y)+\beta A(X)A(Y)+\gamma[A(X)B(Y)+A(Y)B(X)]$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ +\delta[A(X)D(Y)+A(Y)D(X)],$ (1.5) for all $X,Y,Z\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(M)$. Here $\alpha$, $\beta$, $\gamma$, $\delta$ are non-zero scalar functions on $(M^{n},g)$. $A$, $B$, $D$ are non-zero $1$-forms such that $g(X,\xi_{1})=A(X),\leavevmode\nobreak\ g(X,\xi_{2})=B(X),\leavevmode\nobreak\ g(X,\xi_{3})=D(X),$ (1.6) where $\xi_{1}$, $\xi_{2}$, $\xi_{3}$ are mutually orthogonal unit vector fields. i. e., $g(\xi_{1},\xi_{2})=g(\xi_{2},\xi_{3})=g(\xi_{1},\xi_{3})=0;\leavevmode\nobreak\ g(\xi_{1},\xi_{1})=g(\xi_{2},\xi_{2})=g(\xi_{3},\xi_{3})=1.$ (1.7) An $n$-dimensional hyper-generalized quasi-Einstein manifold is generally denoted as $(HGQE)_{n}$. Shaikh, Özgür and Patra in [1] studied on hyper- generalized quasi-Einstein manifolds with some geometric properties of it. Güler and Demirbaǧ [20] dealt with some Ricci conditions on hyper-generalized quasi-Einstein manifolds, D. Debnath [9] proved few theorems about the properties of the hyper-generalized quasi-Einstein manifolds. The concept of perfect fluid spacetime arose while discussing the structure of this universe. Perfect fluids are often used in the general relativity to model the idealised distribution of matter, such as the interior of a star or isotropic pressure. In general relativity the matter content of the spacetime is described by the energy-momentum tensor. The matter content is assumed to be a fluid having density and pressure and possessing dynamical and kinematical quantities like velocity, acceleration, vorticity, shear and expansion. The energy-momentum tensor $T$ of a perfect fluid spacetime is given by the following equation $\cite[cite]{[\@@bibref{}{jmh12}{}{}]},\cite[cite]{[\@@bibref{}{jmh13}{}{}]}$ $T(X,Y)=(\sigma+p)A(X)A(Y)+pg(X,Y).$ (1.8) Here $g(X,\xi_{1})=A(X),A(\xi_{1})=-1$, for any $X,Y$. $p$ and $\sigma$ are called the isotropic pressure and the energy density respectively. $\xi_{1}$ being the unit timelike velocity vector field. The Einstein field equation [6] is given by $S(X,Y)-\frac{r}{2}g(X,Y)+\lambda g(X,Y)=kT(X,Y);\leavevmode\nobreak\ \forall X,Y\in TM,$ (1.9) here $r$ being the scalar curvature, $S$ being the Ricci tensor of type $(0,2)$. $k$ and $\lambda$ are the gravitational constant and cosmological constant respectively. From Einstein’s field equation it follows that energy momentum tensor is a symmetric $(0,2)$ type tensor of divergence zero. In the year of 2012, Mantica and Molinari[7] defined a new generalized symmetric $(0,2)$ tensor called Z tensor. According to them it is given as, $Z(X,Y)=S(X,Y)+\phi g(X,Y),$ (1.10) where $\phi$ is an arbitrary scalar function. A set of properties of $Z$ tensor have been studied in the papers [7] and [8]. Like the $Z$ curvature tensor projective curvature tensor also plays a very significant role in studying different properties of semi-Riemannian geometry. Let $M^{n}(n\geq 3)$ be a semi-Riemannian manifold. The projective curvature tensor[15] is defined by, $P(X,Y)Z=R(X,Y)Z-\frac{1}{n-1}\\{S(Y,Z)X-S(X,Z)Y\\}.$ (1.11) Hyper-generalized quasi-Einstein manifolds is considered as the base space of general relativistic viscous fluid spacetime, which inspired us to take a look on some geometric properties of the $(HGQE)_{4}$ spacetime under certain conditions which we study on sections 2, 3, 4 and 5. Then we discuss about Ricci symmetric hyper-generalized quasi-Einstein spacetimes in section 6. In section 7 we derive a result about the energy-momentum tensor on hyper- generalized quasi-Einstein spacetime of constant curvature with cyclic parallel Ricci tensor. Also the spacetime has wide applications in general relativistic viscous fluid spacetime admitting heat flux and stress, which motivated us to discuss about some physical applications of an $(HGQE)_{4}$ spacetime in section 8. Finally in section 9 we construct a non-trivial example of an $(HGQE)_{4}$ spacetime to prove the existence of such spacetime. ## 2 $W_{2}$-Ricci pseudosymmetric $(HGQE)_{4}$ spacetime The $W_{2}$-curvature tensor was introduced by G. P. Pokhariyal and R. S. Mishra [12] in $1970$ and they studied some properties of it. A $W_{2}$-curvature tensor on a manifold $(M^{n},g)$, $n(>3)$ is defined by $W_{2}(X,Y)Z=R(X,Y)Z-\frac{1}{n-1}[g(Y,Z)QX-g(X,Z)QY].$ (2.1) Here $r$ being the curvature tensor and $Q$ is the Ricci operator defined by $g(QX,Y)=S(X,Y),\forall\leavevmode\nobreak\ X,Y$. For an $(HGQE)_{4}$ quasi-Einstein spacetime the $(0,4)$ $W_{2}$ curvature tensor takes the following form, $W_{2}(X,Y,Z,W)=R(X,Y,Z,W)-\frac{1}{3}[g(Y,Z)S(X,W)-g(X,Z)S(Y,W)].$ (2.2) Firstly, we take a hyper generalized quasi-Einstein spacetime satisfying the condition $W_{2}.S=F_{S}Q(g,S)$. Here $F_{S}$ being a certain function on the set $U_{S}=\\{x\in M:S\neq\frac{r}{n}g$ at $x\\}$ and $Q(g,S)$ being the Tachibana tensor working on the metric tensor and the Ricci tensor. This spacetime is called $W_{2}$-Ricci pseudosymmetric $(HGQE)_{4}$. Now for all $X,Y,Z\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(M^{4})$; $\displaystyle S(W_{2}(X,Y)Z,W)+S(Z,W_{2}(X,Y)W)$ $\displaystyle=F_{S}[g(Y,Z)S(X,W)-g(X,Z)S(Y,W)$ $\displaystyle+g(Y,W)S(Z,X)-g(X,W)S(Y,Z)].$ (2.3) From the equation (1) from the equation (2) we get, $\displaystyle\alpha g(W_{2}(X,Y)Z,W)+\beta A(W_{2}(X,Y)Z)A(W)$ $\displaystyle+\gamma[A(W_{2}(X,Y)Z)B(W)+B(W_{2}(X,Y)Z)A(W)]$ $\displaystyle+\delta[A(W_{2}(X,Y)Z)D(W)+D(W_{2}(X,Y)Z)A(W)]$ $\displaystyle+\alpha g(W_{2}(X,Y)W,Z)+\beta A(W_{2}(X,Y)W)A(Z)$ $\displaystyle+\gamma[A(W_{2}(X,Y)W)B(Z)+B(W_{2}(X,Y)W)A(Z)]$ $\displaystyle+\delta[A(W_{2}(X,Y)W)D(Z)+D(W_{2}(X,Y)W)A(Z)]$ $\displaystyle=F_{S}[g(Y,Z)S(X,W)-g(X,Z)S(Y,W)$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ +g(Y,W)S(Z,X)-g(X,W)S(Y,Z)].$ (2.4) Contracting the equation over $X$ and $W$ and putting $Z=\xi_{1}$ we get, $\displaystyle\alpha[\frac{4}{3}\\{(\alpha-\beta)A(Y)-\gamma B(Y)-\delta D(Y)\\}-\frac{r}{3}A(Y)]-\frac{1}{3}[-\gamma B(Y)-\delta D(Y)]$ (2.5) $\displaystyle+\frac{\gamma^{2}}{3}A(Y)+\gamma R(\xi_{1},Y,\xi_{1},\xi_{2})-\frac{\alpha\gamma}{3}B(Y)+\frac{\delta^{2}}{3}A(Y)$ $\displaystyle+\delta R(\xi_{1},Y,\xi_{1},\xi_{3})-\frac{\delta\alpha}{3}D(Y)$ $\displaystyle=F_{S}[rA(Y)-4\\{(\alpha-\beta)A(Y)-\gamma B(Y)-\delta D(Y)\\}].$ Putting $X=\xi_{1},Z=\xi_{2},W=\xi_{3}$ in the equation (2) we get, $\displaystyle\delta R(\xi_{1},Y,\xi_{2},\xi_{1})+\gamma R(\xi_{1},Y,\xi_{3},\xi_{1})-\frac{\alpha}{3}B(Y)(-\delta)$ (2.6) $\displaystyle-\frac{\alpha}{3}D(Y)(-\gamma)+\frac{\delta}{3}B(Y)(\alpha-\beta)+\frac{\gamma}{3}D(Y)(\alpha-\beta)$ $\displaystyle=F_{S}[B(Y)(-\delta)+D(Y)(-\gamma)].$ combining the equations (2.5) and (2.6) we get, $\displaystyle\alpha[\frac{4}{3}\\{(\alpha-\beta)A(Y)-\gamma B(Y)-\delta D(Y)\\}-\frac{r}{3}A(Y)]-\frac{1}{3}[-\gamma B(Y)-\delta D(Y)]$ (2.7) $\displaystyle+\frac{\gamma^{2}}{3}A(Y)-\frac{\alpha\gamma}{3}B(Y)+\frac{\delta^{2}}{3}A(Y)$ $\displaystyle-\frac{\delta\alpha}{3}D(Y)-F_{S}[B(Y)(-\delta)+D(Y)(-\gamma)]-\frac{\alpha}{3}B(Y)(-\delta)$ $\displaystyle-\frac{\alpha}{3}D(Y)(-\gamma)+\frac{\delta}{3}B(Y)(\alpha-\beta)+\frac{\gamma}{3}D(Y)(\alpha-\beta)$ $\displaystyle=F_{S}[rA(Y)-4\\{(\alpha-\beta)A(Y)-\gamma B(Y)-\delta D(Y)\\}].$ setting $\gamma=\delta$ yields, $\displaystyle\alpha[\frac{4}{3}\\{(\alpha-\beta)A(Y)-\gamma B(Y)-\gamma D(Y)\\}-\frac{r}{3}A(Y)]-\frac{1}{3}[-\gamma B(Y)-\gamma D(Y)]$ (2.8) $\displaystyle+\frac{\gamma^{2}}{3}A(Y)-\frac{\alpha\gamma}{3}B(Y)+\frac{\gamma^{2}}{3}A(Y)$ $\displaystyle-\frac{\gamma\alpha}{3}D(Y)-F_{S}[B(Y)(-\gamma)+D(Y)(-\gamma)]-\frac{\alpha}{3}B(Y)(-\gamma)$ $\displaystyle-\frac{\alpha}{3}D(Y)(-\gamma)+\frac{\gamma}{3}B(Y)(\alpha-\beta)+\frac{\gamma}{3}D(Y)(\alpha-\beta)$ $\displaystyle=F_{S}[rA(Y)-4\\{(\alpha-\beta)A(Y)-\gamma B(Y)-\gamma D(Y)\\}].$ Putting $Y=\xi_{1}$ in the above equation we get, $-3\beta F_{S}=\alpha\beta-\frac{2\gamma^{2}}{3}.$ (2.9) Again putting $Y=\xi_{2}$ in the equation (2.8) we get, $\gamma(\alpha+3F_{S})=0,$ (2.10) which gives either $\gamma=0$ or $F_{S}=\frac{-\alpha}{3}$. Now, if $\gamma=0$ then since $\gamma=\delta$ thus $\gamma=\delta=0$. Hence the spacetime becomes a quasi-Einstein spacetime. On the other hand if $F_{S}=\frac{-\alpha}{3}$ then from the equation (2.9) we get $\gamma=0$. Again since $\gamma=\delta$ thus $\gamma=\delta=0$. Thus in both the cases the manifold is reduced to a quasi-Einstein spacetime. Hence we conclude the following theorem as: Theorem 2.1: A $W_{2}$-Ricci pseudosymmetric $(HGQE)_{4}$ spacetime is a $(QE)_{4}$ spacetime if $\gamma=\delta$. ## 3 Z-Ricci pseudosymmetric $(HGQE)_{4}$ spacetime A semi-Riemannian manifold $(M^{n},g),n\geq 3$, is called Z-Ricci pseudosymmetric iff the following relation holds: $Z\cdot K=F_{K}P(g,K),$ (3.1) on the set $U_{K}=\\{x\in M:P(g,K)\neq 0\leavevmode\nobreak\ at\leavevmode\nobreak\ x\\}$, where $K$ is the Ricci operator defined by $S(X,Y)=g(KX,Y)$ and $F_{K}$ is a smooth function on $U_{K}$. The operator $P$ is defined by the following way: $P(g,K)(W;X,Y)=K((X\wedge_{g}Y)W)$ (3.2) for all vector fields $X,Y,W$. Now if the spacetime is a Z-Ricci pseudosymmetric then from the equation (3.1) we get, $(Z(X,Y)\cdot K)W=F_{K}P(g,K)(W;X,Y),$ (3.3) which takes the following form, $\displaystyle Z(Y,KW)X-Z(X,KW)Y-Z(Y,W)KX-Z(X,W)KY$ $\displaystyle=F_{K}\\{g(Y,W)KX-g(X,W)KY\\}.$ (3.4) With the help of the equation $(\ref{e1.5})$ we demonstrate the Ricci operator by the following equation: $KX=\alpha X+\beta A(X)\xi_{1}+\gamma[A(X)\xi_{2}+B(X)\xi_{1}]+\delta[A(X)\xi_{3}+D(X)\xi_{1}].$ (3.5) Applying the equation (3.5) in equation (3) we get, $\displaystyle Z(Y,\alpha W)X+\beta A(W)Z(Y,\xi_{1})X+\gamma A(W)Z(Y,\xi_{2})X$ (3.6) $\displaystyle+\gamma B(W)Z(Y,\xi_{1})X+\delta A(W)Z(Y,\xi_{3})X+\delta D(W)Z(Y,\xi_{1})X$ $\displaystyle-[Z(X,\alpha W)Y+\beta A(W)Z(X,\xi_{1})Y+\gamma A(W)Z(X,\xi_{2})Y$ $\displaystyle+\gamma B(W)Z(X,\xi_{1})Y+\delta A(W)Z(X,\xi_{3})Y+\delta D(W)Z(X,\xi_{1})Y]$ $\displaystyle=\\{F_{K}g(Y,W)+Z(Y,W)\\}KX-\\{F_{K}g(X,W)-Z(X,W)\\}KY.$ With the help of the equation (1.10) this further implies, $\displaystyle\alpha Z(Y,W)X+\beta A(W)\\{(\alpha-\beta+\phi)A(Y)-\gamma B(Y)-\delta D(Y)\\}X$ (3.7) $\displaystyle+\gamma A(W)\\{(\alpha+\phi)B(Y)+\gamma A(Y)\\}X$ $\displaystyle+\gamma B(W)\\{(\alpha-\beta+\phi)A(Y)-\gamma B(Y)-\delta D(Y)\\}X$ $\displaystyle+\delta A(W)\\{(\alpha+\phi)D(Y)+\gamma A(Y)\\}X$ $\displaystyle+\delta D(W)\\{(\alpha-\beta+\phi)A(Y)-\gamma B(Y)-\delta D(Y)\\}X$ $\displaystyle-\alpha Z(X,W)Y-\beta A(W)\\{(\alpha-\beta+\phi)A(X)-\gamma B(X)-\delta D(X)\\}Y$ $\displaystyle-\gamma A(W)\\{(\alpha+\phi)B(X)+\gamma A(X)\\}Y$ $\displaystyle-\gamma B(W)\\{(\alpha-\beta+\phi)A(X)-\gamma B(X)-\delta D(X)\\}Y$ $\displaystyle-\delta A(W)\\{(\alpha+\phi)D(X)+\gamma A(X)\\}Y$ $\displaystyle-\delta D(W)\\{(\alpha-\beta+\phi)A(X)-\gamma B(X)-\delta D(X)\\}Y$ $\displaystyle=\\{F_{K}g(Y,W)+S(Y,W)+\phi g(Y,W)\\}KX$ $\displaystyle-\\{F_{K}g(X,W)-S(X,W)-\phi g(X,W)\\}KY.$ Putting $X=\xi_{1},Y=\xi_{2}$ equation (3.7) yields, $\displaystyle\alpha\\{(\alpha+\phi)B(W)+\gamma A(W)\\}\xi_{1}+\beta A(W)\\{-\gamma\\}\xi_{1}+\gamma A(W)\\{\alpha+\phi\\}\xi_{1}$ (3.8) $\displaystyle+\gamma B(W)\\{-\gamma\\}\xi_{1}+\delta D(W)\\{-\gamma\\}\xi_{1}-\alpha\\{(\alpha-\beta+\phi)A(W)-\gamma B(W)-\delta D(W)\\}\xi_{2}$ $\displaystyle+\beta A(W)\\{\alpha-\beta+\phi)\\}\xi_{2}+\gamma^{2}A(W)\xi_{2}$ $\displaystyle+\gamma B(W)\\{\alpha-\beta+\phi)\\}\xi_{2}+\delta^{2}A(W)\xi{2}+\delta D(W)\\{\alpha-\beta+\phi)\\}\xi_{2}$ $\displaystyle=\\{F_{K}B(W)+\phi B(W)+\alpha B(W)+\gamma A(W)\\}K\xi_{1}$ $\displaystyle-\\{F_{K}A(W)-\phi A(W)-(\alpha-\beta)A(W)+\gamma B(W)+\delta D(W)\\}K\xi_{2}.$ Taking inner product with $\xi_{1}$ to both the sides of the equation (3.8) we get, $\displaystyle A(W)\\{\alpha\gamma+\phi\gamma\\}+B(W)\\{-\gamma^{2}-F_{K}(\alpha-\beta)+\phi\beta+\alpha\beta\\}-D(W)\\{\delta\gamma\\}=0.$ (3.9) Putting $W=\xi_{3}$ in equation (3.9) we get, $-\gamma\delta=0.$ (3.10) That means at least one of $\gamma$ or $\delta$ must be zero. Which means the manifold is reduced to a generalized quasi-Einstein spacetime. This allows us to derive the next theorem as: Theorem 3.1: A Z-Ricci pseudosymmetric $(HGQE)_{4}$ spacetime is a $(GQE)_{4}$ spacetime. ## 4 Ricci pseudosymmetric $(HGQE)_{4}$ spacetime A semi-Riemannian manifold $M^{n}(n\geq 3)$ is called Ricci-pseudosymmetric if the following relation $(R(X,Y)\cdot S)(Z,W)=F_{S}Q(g,S)$ (4.1) holds on $U_{S}=\\{x\in M:S\neq\frac{r}{n}\leavevmode\nobreak\ at\leavevmode\nobreak\ x\\}$ and $L_{S}$ is a function on $U_{S}$. From the equation (4.1) we get, $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ S(R(X,Y)Z,W)+S(Z,R(X,Y)W)$ $\displaystyle=F_{S}[g(Y,Z)S(X,W)-g(X,Z)S(Y,W)$ $\displaystyle+g(Y,W)S(Z,X)-g(X,W)S(Y,Z)].$ (4.2) Using the equation (1) we demonstrate the equation (4) as follows: $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \beta[A(R(X,Y)Z)A(W)+A(Z)A(R(X,Y)W)]$ $\displaystyle+\gamma[A(R(X,Y)Z)B(W)+A(W)B(R(X,Y)Z)+A(Z)B(R(X,Y)W)$ $\displaystyle+A(R(X,Y)W)B(Z)]+\delta[A(R(X,Y)Z)D(W)+A(W)D(R(X,Y)Z)$ $\displaystyle+A(Z)D(R(X,Y)W)+A(R(X,Y)W)D(Z)]$ $\displaystyle=F_{S}[\beta\\{g(Y,Z)A(X)A(W)-g(X,Z)A(Y)A(W)+g(Y,Z)A(Z)A(X)$ $\displaystyle-g(X,W)A(Y)A(Z)\\}+\gamma\\{g(Y,Z)[A(X)B(W)+A(W)B(X)]$ $\displaystyle-g(X,Z)[A(Y)B(W)+A(W)B(Y)]+g(Y,W)[A(X)B(Z)+A(Z)B(X)]$ $\displaystyle-g(X,W)[A(Y)B(Z)+A(Z)B(Y)]\\}+\delta\\{g(Y,Z)[A(X)D(W)+A(W)D(X)]$ $\displaystyle-g(X,Z)[A(Y)D(W)+A(W)D(Y)]+g(Y,Z)[A(X)D(Z)+A(Z)D(X)]$ $\displaystyle-g(X,W)[A(Y)D(Z)+A(Z)D(Y)]\\}].$ (4.3) Contracting equation (4) over X and W we get, $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \beta[A(R(\xi_{1},Y)Z)-A(Z)S(Y,\xi_{1})]+\gamma[A(R(\xi_{2},Y)Z)$ $\displaystyle+B(R(\xi_{1},Y)Z)-A(Z)S(Y,\xi_{2})-B(Z)S(Y,\xi_{1})]$ $\displaystyle+\delta[A(R(X,Y)Z)D(W)+D(R(\xi_{1},Y)Z)$ $\displaystyle-A(Z)S(Y,\xi_{3})-S(Y,\xi_{1})D(Z)]$ $\displaystyle=F_{S}\\{\beta[-g(Y,Z)-4A(Y)A(Z)]-4\gamma[A(Y)B(Z)$ $\displaystyle A(Z)B(Y)]-4\delta[A(Y)D(Z)+A(Z)D(Y)]\\}.$ (4.4) Setting $Z=\xi_{1}$ in equation (4) we get, $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \beta(Y,\xi_{1})+\gamma[R(\xi_{1},Y,\xi_{1},\xi_{2})+S(Y,\xi_{2})]$ $\displaystyle+\delta[R(\xi_{1},Y,\xi_{1},\xi_{3})+S(Y,\xi_{3})]$ $\displaystyle=F_{S}[3\beta A(Y)+4\gamma B(Y)+4\delta D(Y)].$ (4.5) Now, putting $Z=\xi_{2},W=\xi_{3}$ in the equation (4) we have, $\displaystyle-\gamma R(\xi_{1},Y,\xi_{1},\xi_{3})-\delta R(\xi_{1},Y,\xi_{1},\xi_{2})$ (4.6) $\displaystyle=F_{S}\\{\gamma[D(Y)A(X)-D(X)A(Y)]+\delta[A(X)B(Y)-A(Y)B(X)]\\}.$ If $\gamma=\delta$ then from the equations (4) and (4.6) we conclude, $\displaystyle[\beta S(Y,\xi_{1})+\gamma S(Y,\xi_{2})+\gamma S(Y,\xi_{3})]$ $\displaystyle-F_{S}[3\beta A(Y)+4\gamma B(Y)+4\gamma D(Y)]$ $\displaystyle=F_{S}\\{\gamma[D(Y)A(X)-D(X)A(Y)]+\gamma[A(X)B(Y)-A(Y)B(X)]\\}.$ (4.7) Putting $X=\xi_{1}$ we get, $\displaystyle A(Y)[\alpha\beta-\beta^{2}+2\gamma^{2}]+B(Y)[-\beta\gamma+\gamma\alpha]+D(Y)[-\beta\gamma+\gamma\alpha]$ $\displaystyle-F_{S}[3\beta A(Y)+4\gamma B(Y)+4\gamma D(Y)]=-F_{S}\gamma[D(Y)+B(Y)].$ (4.8) Putting $Y=\xi_{2}$ in equation (4) we get, $\gamma[(\alpha-\beta)-3F_{S}]=0.$ (4.9) Again putting $Y=\xi_{1}$ in equation (4) we get, $F_{S}=\frac{\alpha\beta-\beta^{2}+2\gamma^{2}}{3\beta}.$ (4.10) From the equation (4.9) we have either $\gamma=0$ or $F_{S}=\frac{\alpha-\beta}{3}$. Now, if $\gamma=0$ then since $\gamma=\delta$, thus $\gamma=\delta=0$, implying the manifold reduces to a quasi-Einstein spacetime. Again if $F_{S}=\frac{\alpha-\beta}{3}$ then from the equation (4.10) we get $\gamma=0$ and hence $\gamma=\delta=0$, which again implies the manifold is reduced to a quasi-Einstein spacetime. This allows us to deduce the following theorem: Theorem 4.1: A Ricci pseudosymmetric $(HGQE)_{4}$ spacetime is a $(QE)_{4}$ spacetime if $\gamma=\delta$. ## 5 Projectively pseudosymmetric $(HGQE)_{4}$ spacetime From the equation $(\ref{e1.11})$ we see that for an $(HGQE)_{4}$ quasi- Einstein spacetime the $(0,4)$ projective curvature tensor takes the following form, $P(X,Y,Z,W)=R(X,Y,Z,W)-\frac{1}{3}[S(Y,Z)g(X,W)-S(X,Z)g(Y,W)].$ (5.1) A semi-Riemannian manifold $M^{n}(n\geq 3)$ is called projectively pseudosymmetric if the following relation $(P(X,Y)\cdot S)(Z,W)=F_{S}Q(g,S)$ (5.2) holds on $U_{S}=\\{x\in M:S\neq\frac{r}{n}\leavevmode\nobreak\ at\leavevmode\nobreak\ x\\}$ and $L_{S}$ is a function on $U_{S}$. From the equation (5.2) we get, $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ S(P(X,Y)Z,W)+S(Z,P(X,Y)W)$ $\displaystyle=F_{S}[g(Y,Z)S(X,W)-g(X,Z)S(Y,W)$ $\displaystyle+g(Y,W)S(Z,X)-g(X,W)S(Y,Z)].$ (5.3) From the equation (1) from the equation (5) we get, $\displaystyle\alpha g(P(X,Y)Z,W)+\beta A(P(X,Y)Z)A(W)$ $\displaystyle+\gamma[A(P(X,Y)Z)B(W)+B(P(X,Y)Z)A(W)]$ $\displaystyle+\delta[A(P(X,Y)Z)D(W)+D(P(X,Y)Z)A(W)]$ $\displaystyle+\alpha g(P(X,Y)W,Z)+\beta A(P(X,Y)W)A(Z)$ $\displaystyle+\gamma[A(P(X,Y)W)B(Z)+B(P(X,Y)W)A(Z)]$ $\displaystyle+\delta[A(P(X,Y)W)D(Z)+D(P(X,Y)W)A(Z)]$ $\displaystyle=F_{S}[g(Y,Z)S(X,W)-g(X,Z)S(Y,W)$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ +g(Y,W)S(Z,X)-g(X,W)S(Y,Z)].$ (5.4) Putting $X=\xi_{1},Z=\xi_{2},W=\xi_{3}$ in the equation (5) we get, $\displaystyle-\delta R(\xi_{1},Y,\xi_{2},\xi_{1})-\gamma R(\xi_{1},Y,\xi_{3},\xi_{1})$ $\displaystyle=-\frac{\alpha\gamma}{3}D(Y)-\frac{\alpha\delta}{3}B(Y)+\frac{\gamma}{3}(\alpha D(Y)+\delta A(Y))$ $\displaystyle+\frac{\delta}{3}(\alpha B(Y)+\gamma A(Y))-\frac{\gamma^{2}}{3}A(Y)-\frac{\delta^{2}}{3}A(Y)$ $\displaystyle+F_{S}\\{\gamma D(Y)+\delta B(Y)\\}.$ (5.5) Contracting equation (5) over $X,W$ and putting $Z=\xi_{1}$ we get, $\displaystyle-\frac{\beta\gamma}{3}B(Y)-\frac{\beta\delta}{3}D(Y)-\frac{\gamma^{2}}{3}A(Y)$ $\displaystyle+\frac{4\delta}{3}[\alpha D(Y)+\delta A(Y)]+\gamma(\beta-\alpha)B(Y)-\frac{\delta^{2}}{3}A(Y)$ $\displaystyle-\frac{\delta r}{3}D(Y)+\delta(\beta-\alpha)D(Y)$ $\displaystyle-\frac{4\alpha}{3}[(\alpha-\beta)A(Y)-\gamma B(Y)-\delta D(Y)]+\frac{\alpha r}{3}A(Y)$ $\displaystyle+\frac{4\beta}{3}[(\alpha-\beta)A(Y)-\gamma B(Y)-\delta D(Y)]-\frac{\beta r}{3}A(Y)$ $\displaystyle+\frac{4\gamma}{3}[\alpha B(Y)+\gamma A(Y)]-\frac{\gamma r}{3}B(Y)$ $\displaystyle- F_{S}\\{rA(Y)-4[(\alpha-\beta)A(Y)-\gamma B(Y)-\delta D(Y)]\\}$ $\displaystyle=-\gamma R(\xi_{1},Y,\xi_{1},\xi_{2})-\delta R(\xi_{1},Y,\xi_{1},\xi_{3}).$ (5.6) If $\gamma=\delta$ then rom the equations $(\ref{e5.5}),(\ref{e5.6})$, by using the property of $R$ we get, $\displaystyle-\frac{\beta\gamma}{3}B(Y)-\frac{\beta\gamma}{3}D(Y)-\frac{\gamma^{2}}{3}A(Y)$ $\displaystyle+\frac{4\gamma}{3}[\alpha D(Y)+\gamma A(Y)]+\gamma(\beta-\alpha)B(Y)-\frac{\gamma^{2}}{3}A(Y)$ $\displaystyle-\frac{\gamma r}{3}D(Y)+\gamma(\beta-\alpha)D(Y)$ $\displaystyle-\frac{4\alpha}{3}[(\alpha-\beta)A(Y)-\gamma B(Y)-\gamma D(Y)]+\frac{\alpha r}{3}A(Y)$ $\displaystyle+\frac{4\beta}{3}[(\alpha-\beta)A(Y)-\gamma B(Y)-\gamma D(Y)]-\frac{\beta r}{3}A(Y)$ $\displaystyle+\frac{4\gamma}{3}[\alpha B(Y)+\gamma A(Y)]-\frac{\gamma r}{3}B(Y)$ $\displaystyle- F_{S}\\{rA(Y)-4[(\alpha-\beta)A(Y)-\gamma B(Y)-\gamma D(Y)]\\}$ $\displaystyle=-\\{-\frac{\alpha\gamma}{3}D(Y)-\frac{\alpha\gamma}{3}B(Y)+\frac{\gamma}{3}(\alpha D(Y)+\gamma A(Y))$ $\displaystyle+\frac{\gamma}{3}(\alpha B(Y)+\gamma A(Y))-\frac{\gamma^{2}}{3}A(Y)-\frac{\gamma^{2}}{3}A(Y)$ $\displaystyle+F_{S}\\{\gamma D(Y)+\gamma B(Y)\\}\\}.$ (5.7) Putting $Y=\xi_{2}$ in equation (5) we get, $\frac{\gamma}{3}(-\beta+\alpha+3F_{S})=0.$ (5.8) Again putting $Y=\xi_{1}$ in equation (5) we get, $3\beta F_{S}=\frac{2\gamma^{2}}{3}+\alpha\beta-\beta^{2}.$ (5.9) From the equation (5.8) we have either $\gamma=0$ or $F_{S}=\frac{\alpha-\beta}{3}$. Now, if $\gamma=0$ then since $\gamma=\delta$, thus $\gamma=\delta=0$, implying the manifold reduces to a quasi-Einstein spacetime. Again if $F_{S}=\frac{\alpha-\beta}{3}$ then from the equation (5.9) we get $\gamma=0$ and hence $\gamma=\delta=0$, which again implies the manifold is reduced to a quasi-Einstein spacetime. This allows us to deduce the following theorem: Theorem 5.1: A projectively pseudosymmetric $(HGQE)_{4}$ spacetime is a $(QE)_{4}$ spacetime if $\gamma=\delta$. ## 6 Ricci symmetric $(HGQE)_{4}$ spacetime A semi-Riemannian manifold $M^{n}(n\geq 3)$ is called Ricci-symmetric if $\nabla S=0$, where $S$ is the Ricci tensor of the manifold. Considering the spacetime as a $(HGQE)_{4}$ spacetime we observe from equation $(\ref{e1.5})$ that the manifold becomes Ricci symmetric if it satisfies the following relation: $\displaystyle\nabla_{Z}S(X,Y)=d\alpha(Z)g(X,Y)+d\beta(Z)A(X)A(Y)$ $\displaystyle+\beta[(\nabla_{Z}A)(X)A(Y)+A(X)(\nabla_{Z}A)(Y)]$ $\displaystyle+d\gamma(Z)[A(X)B(Y)+A(Y)B(X)]$ $\displaystyle+\gamma[(\nabla_{Z}A)(X)B(Y)+A(X)(\nabla_{Z}B)(Y)]$ $\displaystyle+(\nabla_{Z}A)(Y)B(X)+A(Y)(\nabla_{Z}B)(X)]$ $\displaystyle+d\delta(Z)[A(X)D(Y)+A(Y)D(X)]$ $\displaystyle+\delta[(\nabla_{Z}A)(X)D(Y)+A(X)(\nabla_{Z}D)(Y)]$ $\displaystyle+(\nabla_{Z}A)(Y)D(X)+A(Y)(\nabla_{Z}D)(X)]=0.$ (6.1) Putting $X=Y=\xi_{1}$ in $(\ref{e6.1})$ we get, $-d\alpha(Z)+d\beta(Z)-2\gamma(\nabla_{Z}B)(\xi_{1})-2\delta(\nabla_{Z}D)(\xi_{1})=0.$ (6.2) Putting $X=Y=\xi_{2}$ in $(\ref{e6.1})$ we get, $d\alpha(Z)+2\gamma(\nabla_{Z}A)(\xi_{2})=0.$ (6.3) Again putting $X=Y=\xi_{3}$ in $(\ref{e6.1})$ we get, $d\alpha(Z)+2\delta(\nabla_{Z}A)(\xi_{3})=0.$ (6.4) Since the vector fields $\xi_{1},\xi_{2},\xi_{3}$ are mutually orthogonal then $g(\xi_{1},\xi_{2})=g(\xi_{1},\xi_{3})=0$, this implies that $Z(g(\xi_{1},\xi_{2}))=Z(g(\xi_{1},\xi_{3}))=0$. Which further implies, $(\nabla_{Z}B)(\xi_{1})=-(\nabla_{Z}A)(\xi_{2})$ (6.5) and $(\nabla_{Z}D)(\xi_{1})=-(\nabla_{Z}A)(\xi_{3}).$ (6.6) Subtracting equations (6.3),(6.4) from (6.2) and using the relations (6.5) and (6.6) we get, $d(\beta-3\alpha)(Z)=0,\leavevmode\nobreak\ \forall\leavevmode\nobreak\ Z\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(M).$ (6.7) That implies $\beta-3\alpha$ is a constant. Now contracting the equation $(\ref{e6.1})$ over $X,W$ and using $(\ref{6.5}),(\ref{6.6})$ we get, $d(\beta-4\alpha)(Z)=0,\leavevmode\nobreak\ \forall\leavevmode\nobreak\ Z\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(M).$ (6.8) From the equations (6.7) and (6.8) it is clear that $\alpha,\beta$ are constants. So, from the equations (6.3) and (6.4) we get $\gamma(\nabla_{Z})(\xi_{1})=0$ (6.9) and $\delta(\nabla_{Z})(\xi_{2})=0.$ (6.10) The equation (6.9) shows $\gamma=0$ or $\gamma(\nabla_{Z})(\xi_{1})=0$. If $\gamma=\delta$ then from (6) we get, $\beta[(\nabla_{Z}A)(X)A(Y)+A(X)(\nabla_{Z}A)(Y)=0.$ (6.11) Putting $x=\xi_{1}$ in (6.11) we get, $\beta(\nabla_{Z}A)(Y)=0.$ (6.12) If $\beta=0$ then since $\gamma=\delta=0$ the manifold reduces to an Einstein manifold which is a contradiction. So, $\beta\neq 0$. This implies that, $(\nabla_{Z}A)(Y)=0.$ (6.13) Again if $\gamma\neq 0$ then from (6.9) we get, $\beta(\nabla_{Z}A)(\xi_{2})=0.$ (6.14) Using (6.14) and putting $X=\xi_{1},Y=\xi_{2}$ in the equation (6) we get, $d\gamma(Z)=0.$ (6.15) Which imply $\gamma$ is also a constant. Then, putting $X=\xi_{2}$ in (6) and using (6.15) we get, $\gamma(\nabla_{Z}A)(Y)=0.$ (6.16) Since $\gamma\neq 0$ thus we get the equation (6.13) once again. Hence, we always obtain $(\nabla_{Z}A)(Y)=0\leavevmode\nobreak\ \forall\leavevmode\nobreak\ Z,Y\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(M).$ Which can be written as $g(\nabla_{Z}\xi_{1},Y)=0\leavevmode\nobreak\ \forall\leavevmode\nobreak\ Z,Y\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(M).$ (6.17) Thus we obtain $\nabla_{Z}\xi_{1}=0.$ (6.18) Which implies that the generator vector field $\xi_{1}$ is always parallel. Hence we obtain the following theorem as: Theorem 6.1: In a Ricci symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ the generator vector field $\xi_{1}$ is always parallel. Again putting $Z=\xi_{1}$ in (6.18) we get, $\nabla_{\xi_{1}}\xi_{1}=0.$ (6.19) Which implies that the integral curves of $\xi{1}$ are geodesics. This leads us to the next theorem as: Theorem 6.2: In a Ricci symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ the integral curves of the generator vector field $\xi_{1}$ are geodesics. With the help of the theorems (6.1) and (6.2) we arrive at the following condition $R(X,Y)\xi_{1}=\nabla_{X}\nabla_{Y}\xi_{1}-\nabla_{Y}\nabla_{X}\xi_{1}-\nabla_{[X,Y]}\xi_{1}=0.$ (6.20) Hence we obtain the following theorem as: Theorem 6.3: In a Ricci symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ the Riemannian curvature tensor vanishes at the generator vector field. Now contracting (6.20) we get $S(X,\xi_{1})=0.$ (6.21) Thus from (1) we get, $(\alpha-\beta)A(X)-\gamma[B(X)-D(X)]=0.$ (6.22) Since $\gamma=\delta$ thus putting $X=\xi_{1}$ in (6.22) we get, $\alpha=\beta.$ (6.23) Again putting $X=\xi_{2}$ in (6.22) we get, $\gamma=0.$ (6.24) Thus $\gamma=\delta=0$, hence from (1) we get, $S(X,Y)=\alpha[g(X,Y)+A(X)A(Y)].$ (6.25) This allows us to arrive at the next theorem as: Theorem 6.4: Every Ricci symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ is a $(QE)_{4}$ spacetime with the scalar functions are constants and equal. Using the equations (6.25) from the equation (1.9) we get, $T(X,Y)=\frac{2\lambda-\alpha}{2k}g(X,Y)+\frac{\alpha}{k}A(X)A(Y).$ (6.26) Since $\alpha,\lambda,k$ all are constants thus taking derivative to both the sides the equation (6.26) we get, $(\nabla_{Z}T)(X,Y)=0.$ (6.27) Therefore we see in this case the energy-momentum tensor is covariantly constant. This leads us to the following theorem: Theorem 6.5: In a Ricci-symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ satisfying Einstein field equation with cosmological constant the energy- momentum tensor is covariantly constant. Again from (6.26) we observe that since the velocity vector field $\xi_{1}$ is parallel and $\alpha$ is a constant thus the energy-momentum tensor is of Codazzi type. Hence we derive the next theorem as: Theorem 6.6: In a Ricci-symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ satisfying Einstein field equations with cosmological constant the energy- momentum tensor is of Codazzi type. Now from the equations (1.8), (1.9) and theorem (6.4) we conclude the values of $\sigma$ and $p$ as, $\sigma=\frac{3\alpha-2\lambda}{2k},\leavevmode\nobreak\ p=\frac{2\lambda-\alpha}{2k}.$ (6.28) Since, $\alpha,\lambda,k$ all are constants thus we get $\sigma,p$ are also constants. This leads us to the following theorem: Theorem 6.7: In a Ricci-symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ satisfying Einstein field equation with cosmological constant the energy density and the isotropic pressure are constants. It is proved [2] that in a perfect fluid spacetime if the energy-momentum tensor is of Codazzi type then the vorticity and shear of the spacetime vanish. Hence we derive the next theorem: Theorem 6.8: In a Ricci-symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ satisfying Einstein field equations with cosmological constant the vorticity and the shear tensor vanish. Here we see that the velocity vector $\xi_{1}$ is constant over the spacelike hypersurface orthogonal to $\xi_{1}$. But it is described in [5] that perfect fluid spacetime that is vorticity free and shear free is of petrov type $I,D$ or $O$. Thus we state the next theorem as: Theorem 6.9: The local cosmological structure of a Ricci-symmetric $(HGQE)_{4}$ spacetime with $\gamma=\delta$ satisfying Einstein field equation with cosmological constant can be identified as petrov type $I,D$ or $O$. ## 7 $(HGQE)_{4}$ spacetime with cyclic parallel Ricci tensor Consider an $(HGQE)_{4}$ with cyclic parallel Ricci tensor. Then we get the following equation, $(\nabla_{X}S)(Y,Z)+(\nabla_{Y}S)(X,Z)+(\nabla_{Z}S)(X,Y)=0.$ (7.1) From the equation $(\ref{e1.9})$ we have $(\nabla_{X}S)(Y,Z)=\frac{1}{2}dr(Z)g(X,Y)+k(\nabla_{Z}T)(X,Y).$ (7.2) Now, if in an $(HGQE)_{4}$ with cyclic parallel Ricci tensor the scalar curvature of the spacetime is constant then, $dr(X)=0,$ (7.3) for all $X\in{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}(M)$. Using the equation $(\ref{e7.3})$ in the equation $(\ref{e8.2})$ we get, $(\nabla_{X}S)(Y,Z)=k(\nabla_{Z}T)(X,Y).$ (7.4) Now, if the $(HGQE)_{4}$ spacetime is cyclic parallel then, $\displaystyle k\\{(\nabla_{X}T)(Y,Z)+(\nabla_{Y}T)(X,Z)+(\nabla_{Z}T)(X,Y)\\}$ $\displaystyle=(\nabla_{X}S)(Y,Z)+(\nabla_{Y}S)(X,Z)+(\nabla_{Z}S)(X,Y)=0.$ (7.5) Since $k$, being the gravitational constant is always nonzero, from the equation (LABEL:e8.5) we have $(\nabla_{X}T)(Y,Z)+(\nabla_{Y}T)(X,Z)+(\nabla_{Z}T)(X,Y)=0.$ (7.6) This allows us to obtain the following theorem as: Theorem 7.1: In a hyper-generalised quasi-Einstein spacetime with cyclic parallel Ricci tensor if the scalar curvature is constant then the energy- momentum tensor is also cyclic parallel. ## 8 On the physical applications of an $(HGQE)_{4}$ spacetime Here we study some physical applications of the $(HGQE)_{4}$ spacetime. In [10], [11] G. F. R. Ellis has given the energy momentum tensor of a fluid matter distribution as follows: $\displaystyle T(X,Y)=(\sigma+p)A(X)A(Y)+pg(X,Y)+A(X)B(Y)+A(Y)B(X)$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ +A(X)D(Y)+A(Y)D(X),$ (8.1) where, $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ g(X,\xi_{1})=A(X),\leavevmode\nobreak\ g(X,\xi_{2})=B(X),\leavevmode\nobreak\ g(X,\xi_{3})=D(X),$ $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ A(\xi_{1})=-1,\leavevmode\nobreak\ B(\xi_{2})=1,\leavevmode\nobreak\ D(\xi_{3})=1,$ $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ g(\xi_{1},\leavevmode\nobreak\ \xi_{2})=0,\leavevmode\nobreak\ g(\xi_{2},\leavevmode\nobreak\ \xi_{3})=0,\leavevmode\nobreak\ g(\xi_{3},\leavevmode\nobreak\ \xi_{1})=0,$ and $\sigma$ is the matter density, $p$ is the isotropic pressure, $\xi_{1}$ is the timelike velocity vector field, $\xi_{2}$ is the heat conduction vector field and $\xi_{3}$ is the stress vector field. Combining equation (8) with equation (1.9) we get, $\displaystyle S(X,Y)=(kp+\frac{r}{2}-\lambda)g(X,Y)+k(\sigma+p)A(X)A(Y)$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ +k[A(X)B(Y)+A(Y)B(X)]$ $\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ +k[A(X)D(Y)+A(Y)D(X)].$ (8.2) Comparing equation (8) with equation (1.9) we get that this spacetime is clearly an $(HGQE)_{4}$ spacetime with the constants $\gamma=\delta=k.$ Hence all the results that we derived in the earlier sections are absolutely effective in this spacetime, and thus we derive the following set of theorems: From theorem (2.1) we get, Theorem 8.1: A $W_{2}$-Ricci pseudosymmetric viscous fluid $(HGQE)_{4}$ spacetime is a $(QE)_{4}$ spacetime. From theorem (3.1) we get, Theorem 8.2: A $Z$-Ricci pseudosymmetric viscous fluid $(HGQE)_{4}$ spacetime is a $(GQE)_{4}$ spacetime. From theorem (4.1) we get, Theorem 8.3: A Ricci pseudosymmetric viscous fluid $(HGQE)_{4}$ spacetime is a $(QE)_{4}$ spacetime. From theorem (5.1) we get, Theorem 8.4: A projectively pseudosymmetric viscous fluid $(HGQE)_{4}$ spacetime is a $(QE)_{4}$ spacetime. From theorem (6.1) we get, Theorem 8.5: In a Ricci symmetric $(HGQE)_{4}$ viscous fluid spacetime the generator vector field $\xi_{1}$ is always parallel. From theorem (6.2) we get, Theorem 8.6: In a Ricci symmetric $(HGQE)_{4}$ viscous fluid spacetime the integral curves of the generator vector field $\xi_{1}$ are geodesics. From theorem (6.3) we get, Theorem 8.7: In a Ricci symmetric $(HGQE)_{4}$ viscous fluid spacetime the Riemannian curvature tensor vanishes at the generator vector field. From theorem (6.4) we get, Theorem 8.8: Every Ricci symmetric $(HGQE)_{4}$ viscous fluid spacetime is a $(QE)_{4}$ spacetime with the scalar functions are constants and equal. From theorem (6.5) we get, Theorem 8.9: In a Ricci symmetric $(HGQE)_{4}$ viscous fluid spacetime satisfying Einstein field equation with cosmological constant the energy- momentum tensor is covariantly constant. From theorem (6.6) we get, Theorem 8.10: In a Ricci symmetric $(HGQE)_{4}$ viscous fluid spacetime satisfying Einstein field equation with cosmological constant the energy- momentum tensor is of Codazzi type. From theorem (6.7) we get, Theorem 8.11: In a Ricci symmetric $(HGQE)_{4}$ viscous fluid spacetime satisfying Einstein field equation with cosmological constant the energy density and the isotropic pressure are constants. From theorem (6.8) we get, Theorem 8.12: In a Ricci symmetric $(HGQE)_{4}$ viscous fluid spacetime satisfying Einstein field equation with cosmological constant the vorticity and the shear tensor vanish. From theorem (6.9) we get, Theorem 8.13: The local cosmological structure of a Ricci-symmetric $(HGQE)_{4}$ viscous fluid spacetime satisfying Einstein field equation with cosmological constant can be identified as petrov type $I,D$ or $O$. ## 9 Example of $(HGQE)_{4}$ spacetime Finally we give a non-trivial example to establish the existence of $(HGQE)_{4}$ spacetime non-trivially. For this we consider a metric known as Lorentzian metric $g$ on $M^{4}$ given by $ds^{2}=g_{ij}dx^{i}dx^{j}=-\frac{k}{r}(dt)^{2}+\frac{1}{\frac{c}{r}-4}(dr)^{2}+r^{2}(d\theta)^{2}+(r\sin\theta)^{2}(d\phi)^{2},$ where $i,j=1,2,3,4$ and $k,c$ are constants. Thus we obtain the nonzero components of Christofell symbols, curvature tensors and Ricci tensors as follows: $\displaystyle\Gamma_{33}^{2}=4r-c,\Gamma_{12}^{1}=-\frac{1}{2r},\Gamma_{22}^{2}=\frac{c}{2r(c-4r)},\Gamma_{32}^{3}=\Gamma_{42}^{4}=\frac{1}{r},$ $\displaystyle\Gamma_{43}^{4}=\cot\theta,\Gamma_{44}^{2}=(4r-c)(\sin\theta)^{2},\Gamma_{44}^{3}=-\frac{\sin(2\theta)}{2}$ (9.1) $\displaystyle R_{1221}=-\frac{k(c-3r)}{r^{3}(c-4r)},R_{1331}=\frac{k(c-4r)}{2r^{2}},R_{1441}=\frac{k(c-4r)(\sin\theta)^{2}}{2r^{2}}$ $\displaystyle R_{2332}=\frac{c}{2(4r-c)},R_{2442}=\frac{c(\sin\theta)^{2}}{2(4r-c)},R_{3443}=r(c-5r)(\sin\theta)^{2}$ $\displaystyle R_{11}=-\frac{k}{r^{3}},R_{22}=-\frac{3}{r(c-4r)},R_{33}=-3,R_{44}=-3(\sin\theta)^{2}$ (9.2) From $(\ref{9.1})$ and $(\ref{9.2})$ it follows that $M^{4}$ is a Lorentzian manifold of nonzero scalar curvature ($=-\frac{8}{r^{2}}$). Now we will prove that this is an $(HGQE)_{4}$ manifold. We consider $\alpha,\beta,\gamma$ and $\delta$ as the associated scalars and we consider them as follows: $\alpha=-\frac{5}{r^{2}},\beta=-\frac{12}{r^{2}},\gamma=\frac{3}{r^{2}},\delta=-\frac{4}{r^{2}}$ (9.3) and the associated $1$-forms are as follows : $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ A_{i}(x)=\left\\{\begin{array}[]{cccl}\sqrt{\frac{k}{2r}}&\mbox{for}&i=1\\\ \sqrt{\frac{r}{6(c-4r)}}&\mbox{for}&i=2\\\ 0&\mbox{for}&i=3,4\\\ \end{array}\right.$ ; $B_{i}(x)=\left\\{\begin{array}[]{ccl}\sqrt{\frac{k}{r}}&\mbox{for}&i=1\\\ 0&\mbox{for}&i=2,3,4\end{array}\right.$ $\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ D_{i}(x)=\left\\{\begin{array}[]{ccl}\sqrt{\frac{k}{r}}&\mbox{for}&i=1\\\ 0&\mbox{for}&i=2,3,4\end{array}\right.$ Hence we gain, $(i)R_{11}=\alpha g_{11}+\beta A_{1}A_{1}+\gamma[A_{1}B_{1}+B_{1}A_{1}]+\delta[A_{1}D_{1}+D_{1}A_{1}]$ $(ii)R_{22}=\alpha g_{22}+\beta A_{2}A_{2}+\gamma[A_{2}B_{2}+B_{2}A_{2}]+\delta[A_{2}D_{2}+D_{2}A_{2}]$ $(iii)R_{33}=\alpha g_{33}+\beta A_{3}A_{3}+\gamma[A_{3}B_{3}+B_{3}A_{3}]+\delta[A_{3}D_{3}+D_{3}A_{3}]$ $(iv)R_{44}=\alpha g_{44}+\beta A_{4}A_{4}+\gamma[A_{4}B_{4}+B_{4}A_{4}]+\delta[A_{4}D_{4}+D_{4}A_{4}]$ As every Ricci tensor other than $R_{11},R_{22},R_{33}$ and $R_{44}$ are zero, so we obtain $R_{ij}=\alpha g_{ij}+\beta A_{i}A_{j}+\gamma[A_{i}B_{j}+B_{i}A_{j}]+\delta[A_{i}D_{j}+D_{i}A_{j}],i,j=1,2,3,4.$ Consequently, scalar curvature $=4\alpha-\beta=-\frac{8}{r^{2}}$. Hence, $(M^{4},g)$ is a hyper-generalized quasi Einstein manifold. Conclusion: The general theory of relativity is the most prominent flagship of modern physics. It deals with the curvature of spacetime. As hyper-generalized quasi-Einstein spacetime is considered as the base space of the fluid matter distribution. Thus it has been very necessary to study about the geometric and physical applications of hyper-generalized quasi-Einstein spacetime. It deals with the relativistic viscous fluid spacetime admitting heat flux and stress. The general theory of relativity describes gravity as a geometric property of spacetime. The curvature of spacetime is directly related to the energy- momentum tensor. Also we know the cosmological constant to be of homogeneous energy density which causes the expansion of the universe to accelerate. 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# Powering COVID-19 community Q&A with Curated Side Information Manisha Verma, Kapil Thadani, and Shaunak Mishra Yahoo! Research, VerizonMedia, New York<EMAIL_ADDRESS> (2019) ###### Abstract. Community question answering and discussion platforms such as Reddit, Yahoo! answers or Quora provide users the flexibility of asking open ended questions to a large audience, and replies to such questions maybe useful both to the user and the community on certain topics such as health, sports or finance. Given the recent events around COVID-19, some of these platforms have attracted 2000+ questions from users about several aspects associated with the disease. Given the impact of this disease on general public, in this work we investigate ways to improve the ranking of user generated answers on COVID-19. We specifically explore the utility of external technical sources of side information (such as CDC guidelines or WHO FAQs) in improving answer ranking on such platforms. We found that ranking user answers based on question-answer similarity is not sufficient, and existing models cannot effectively exploit external (side) information. In this work, we demonstrate the effectiveness of different attention based neural models that can directly exploit side information available in technical documents or verified forums (e.g., research publications on COVID-19 or WHO website). Augmented with a temperature mechanism, the attention based neural models can selectively determine the relevance of side information for a given user question, while ranking answers. ††journalyear: 2019 ## 1\. Introduction Question answering systems are key to finding relevant and timely information about several issues. Community question answering (cQ&A) platforms such as Reddit, Yahoo! answers or Quora have been used to ask questions about wide ranging topics. Most of these platforms let users ask, answer, vote or comment on questions present on the platform. However, question answering platforms are useful not only for getting public opinions or votes about areas such as entertainment or sports but can also serve as information hot-spots for more sensitive topics such as health, injuries or legal topics. Thus, it is imperative that when the user visits _sensitive_ topics content, answer ranking also takes into account curated side information from reliable (external) sources. Most prior work on cQ&A has focused on incorporating question-answer similarity (LTR_2008, ; semantic_sim2015, ), user reputation (yang2016beyond, ; hong2009classification, ; user_interaction_2013, ), integration of multi-modal content (medical_qa2019, ), community interaction features (comm_interaction2018, ) associated with answers or just the question answering network (comm_net2019, ) on the platform. However, there is very limited work on incorporating _curated content_ from external sources. Existing work only exploits knowledge bases (medical_qa2019, ) that consist of different entities and relationships between these entities to score answers. However, there are some limitations of knowledge bases that would make it difficult to use them for community Q&A for rapidly evolving topics such as disease outbreaks (e.g. ebola, COVID-19), wild-fires or earthquakes. Figure 1. Illustrative example of COVID-19 community answer ranking powered by side information in the form of research papers, and information from verified sources (such as CDC, WHO, and NHS). Firstly, knowledge bases contain information about established entities, and do not rapidly evolve to incorporate new information which makes them unreliable for novel disease outbreaks such as COVID-19 where information rapidly changes and its verification is time sensitive. Secondly, it may be hard to determine what even _constitutes an entity_ as new information arrives about the topic. To overcome these limitation in this work, we posit that external _curated_ free-text or semi-structured informational sources can also be used effectively for cQ&A tasks. In this work, we demonstrate that free text or semi structured external information sources such as CDC111https://www.cdc.gov/, WHO222https://www.who.int/ or NHS333https://www.nhs.uk/ can be very useful for ranking answers on community Q&A platforms since they contain frequently updated information about several topics such as ongoing disease outbreaks, vaccines or resources about other topics such as surgeries, birth control or historical numerical data about diseases across the world. We argue that for sensitive topics such as COVID-19, it is useful to use publicly available _vetted_ information for improving our ranking systems. In this work, we explore the utility of publicly available information for ranking answers for questions associated with COVID-19. We specifically focus on ranking answers for questions in two publicly available _primary_ Q&A datasets: a) Yahoo! Answers444https://answers.yahoo.com/ and b) recently released annotated Q&A dataset (emnlp_data2020, ) in presence of two _external_ semi-structured curated sources: a) TREC-COVID (voorhees2020treccovid, ) and b) WHO questions and answers 555https://www.who.int/emergencies/diseases/novel-coronavirus-2019/question- and-answers-hub on COVID-19. We explore the utility of deep learning models with attention in this work to improve upon existing state-of-the-art systems. More specifically, we propose a temperature regulated attention mechanism to rank answers in presence of external (side) information. Our experiments on 10K+ questions from both _source_ datasets on COVID-19 show that our models can improve ranking quality by a significant margin over question-answer matching baselines in presence of external information. Figure 1 demonstrates the overall design of our system. We specifically use attention based neural architecture with temperature to automatically determine which components in the external information are useful for ranking user answers with respect to a question. Ranking performance, when evaluated with three metrics shows that precision and recall for correct answer retrieval improves by $\sim$17% and $\sim$9% for both source datasets respectively over several other cQ&A models. ## 2\. Related work Community Question and Answering (cQ&A) systems is a well researched sub-field both in information retrieval and NLP communities. Several systems have been proposed to rank user submitted answers to questions on community platforms such as Yahoo! answers, Reddit and Quora. Ranking user submitted answers on community question-answering platforms has been addressed with several approaches. Primary method is to determine the relevance of the answer given an input question. Text based matching is one of the most common approaches to rank answers. Researchers have used several methods to compute _similarity_ between a question and user generated answers to determine relevance. For instance, feature based question-answer matching is used in (LTR_2008, ) with 17 features extracted from unigrams, bigrams and web correlation features using unstructured user search logs to rank answers. It is worth noting that user features and community features when incorporated may still yield further improvements in the performance of these models but this is not the focus of our work. The authors in (LTR_2008, ) used questions extracted from Yahoo! answers for their experiments. Researchers have used different approaches such representation learning, for instance, in (LSTMans2017IWCS, ; cohen_2016, ) authors use LSTM to represent questions and answers respectively. Convolutional networks have also been used in (yang2016beyond, ; zhou2015answer, ) to rank answers. Other approaches such as doc2vec (nie2017data, ), tree-kernels (severyn2012structural, ), adversarial learning (yang2019adversarial, ), attention (attention_question2018, ; medical_qa2019, ; LSTMans2017IWCS, ; attentive2017AAAI, ) or deep belief networks (wang2010modeling, ) have been used to score question and answer pairs. There have also been studies exploring community, user interaction or question based features (yang2016beyond, ; hong2009classification, ; user_interaction_2013, ; comm_interaction2018, ) to rank answers. While these approaches are relevant, it is not always evident how one can incorporate external information when it is either in free-text or semi-structured format into these systems. We explore some question-answer based matching approaches as baselines in this work and show that for rapidly evolving topics such as COVID-19, inclusion of external _curated_ information can boost model performance. The line of work most closely related to ours is incorporation of knowledge bases in Q&A systems. Existing work (rankQA_2019, ; kb_rank2018Sig, ; medical_qa2019, ), however, approaches different tasks. For instance, authors in (kb_rank2018Sig, ; rankQA_2019, ) focus on finding factual answers to questions using a knowledge base. This does not extend easily to cQ&A where neither the questions nor the answers may request or refer to any facts. Most recent work is (medical_qa2019, ) on incorporating medical KB for ranking answers on medical Q&A platforms. They propose to learn path based representation of _entities (from KB)_ present in question and answers posted by users. This approach relies on reliable detection of entities first, which may be absent for emerging topics such as COVID-19 pandemic. Another limitation of this work is that external knowledge may not always be present in a _structured_ format. For example, CDC guidelines are usually simple question-answer pairs posted on the website. This makes it difficult to apply their approach to our problem. The proposed approach in this work incorporates semi-structured information directly with help of temperature regulated attention. Finally, with the rise of COVID-19, researchers across disciplines are actively publishing information and datasets to share understanding of the virus and its impact on people. Researchers routinely organize dedicated challenges such as SemEval (nakov2019semeval, ) with tasks such as ranking answers on QA forums. One such initiative is TREC-COVID track (voorhees2020treccovid, ) which released queries, documents and manual relevance judgements to power search for COVID related information. Authors in (cairecovid, ) also released COVID-19 related QA dataset with 100+ questions and answers pairs extracted from TREC COVID 666https://ir.nist.gov/covidSubmit/data.html initiative. These questions/answer pairs are not user generated content, hence, do not reflect real user questions. We also rely on recently released Q&A dataset from (emnlp_data2020, ) for our task. We also compile a dataset of 2000+ COVID-19 questions with 10K+ answers all submitted by users on Yahoo! answers for this work. ## 3\. Method ### 3.1. Problem formulation In this work, we focus on ranking answers for $n$ questions $q_{1},\ldots,q_{n}$ related to an emerging topics such as COVID. Each $q_{i}$ is associated with a set of two or more answers $A_{i}=\\{a_{ij}:j\geq 2\\}$ and corresponding labels $Y_{i}=\\{y_{ij}:j\geq 2\\}$ representing answer relevance. We use binary indicator for relevance where relevant judgments (e.g., favorite, upvoted) are provided by the user, i.e., $y_{ij}\in\\{0,1\\}$ respectively. We attempt to model the relevance of each answer $a_{ij}$ to its corresponding question using an external source which may contain free text or semi- structured information. For example, the _external_ source could consist of information-seeking queries or questions $eq_{1},\ldots,eq_{m}$ related to a topic, with each $eq_{k}$ linked to a set of relevant scientific articles or answers $ED_{k}$, where each answer/document $ed_{1},\ldots,ed_{p}$ may be judged for relevance by human judges (voorhees2020treccovid, ) or some experts. We hypothesize that this semi-structured or free-text information may be valuable in identifying user answer quality for certain kinds of questions, although not all. We investigate this with our model to recover the true labels $y_{ij}$ for each user answer $a_{ij}\in A_{i}$ given its question $q_{i}$, category information, and information from the _external_ source $\langle eq_{k},ED_{k}\rangle_{k=1}^{m}$. ### 3.2. Proposed Model Figure 2. External source augmentation model In this work, we explore token-level matching mechanism to determine the relevance of information in the external source that may inform the label prediction task. Our model (_$\tau$ -att_) aims to match a given user question with all the submitted answers in the presence of external information about the same domain. First, the question $q_{i}$, an answer $a_{ij}$ and additional metadata can be encoded into a $d$-dimensional vector $x_{i}$ using a text encoder $f_{\text{input}}$. We use LSTM based encoder for both question and answer in the _primary_ source which can handle input sequences of variable length. ##### Question Encoding: Each word $w_{i}^{q}$ in a question is represented as a $K$ dimensional vector with pre-trained word embeddings. LSTM takes each token embedding as input and updates hidden state $h_{i}^{q}$ based on previous state $h_{i-1}^{q}$. Finally, the hidden state is input to a feed forward layer with smaller dimension $F<K$ to compress question encoding as follows: (1) $h_{i}^{q}=LSTM(h_{i-1}^{q},w_{i}^{q}),\>f_{i}^{q}=RELU(h_{i}^{q}W_{q}+b_{q})$ ##### Answer Encoding: Each word $w_{j}^{a}$ in the answer is also represented as a $K$ dimensional vector with pre-trained word embeddings. LSTM takes each token embedding as input and updates hidden state $h_{j}^{a}$. We also reduce the dimension of answer encoding with a feed forward layer with dimension $F<K$ as follows: (2) $h_{j}^{a}=LSTM(h_{j-1}^{a},w_{j}^{a}),\>f_{j}^{a}=RELU(h_{j}^{a}W_{a}+b_{a})$ We concatenate the question and answer representations for further processing. (3) $f_{ij}=[f_{i}^{q},f_{j}^{a}]$ ##### External source encoding: External sources of information can vary from task-to-task. We encode each segment of data individually. For instance, if there are two segments in the source (e.g. question/answer or query/document), our system encodes both segments individually. We use the same encoding architecture used for primary source question/answer encoding above. Encoding example for two segment _external_ source is given below. (4) $\begin{split}h_{t}^{eq}=LSTM(h_{t-1}^{eq},w_{t}^{eq}),\>f_{t}^{eq}=RELU(h_{t}^{eq}W_{eq}+b_{eq})\\\ h_{t}^{ed}=LSTM(h_{t-1}^{ed},w_{t}^{ed}),\>f_{t}^{ed}=RELU(h_{t}^{ed}W_{ed}+b_{ed})\\\ \end{split}$ Source \| Question \| Rel answer \| Non-rel answer ---\|---\|---\|--- Yahoo! Ans \| 3cmI am really scared to go places for St. Patrick’s day because of the coronavirus. what do I do? \| 7cmUnfortunately, there’s not enough people that care and will still go out and party despite the coronavirus epidemic. I’m proud of you in that you’re taking extra precautions … Good for you! \| 5cmStop being scared of viruses. What’s the problem? Infobot \| 3cmCan corona live on cardboard? \| 7cmA recent study shows that the virus can live in the air … On cardboard, it can live up to 24 hours (1 day) \| 5cmThe risk is quite low for one to become infected with COVID19 through mail/packages - especially because…(over a period of a few days/weeks). Table 1. Sample rel/non-rel answers from both sources We incorporate external source encoding with a temperature $(\tau)$ based variant of scaled dot-product attention, which provides a straightforward conditioning approach over a set of query-document pairs. Question encoding vector $f_{ij}$ serves as a query over keys $f_{t}^{eq}$. If two segments are present in the external source such as query/document, the model uses the attention weights over first segment (e.g. query) to determine the importance of the second segment (e.g. document) respectively. It is easy to extend this framework to _external_ sources with multiple segments. The two segment attention is described below. (5) $z_{it}=\frac{f_{ij}^{\top}f_{t}^{eq}}{\sqrt{d}}\\\ \alpha_{it}=\frac{e^{z_{it}/\tau}}{\sum_{l}e^{z_{l}/\tau}}\\\ s_{itd}^{\prime}=\sum_{d}\alpha_{it}f_{t}^{ed}$ To summarize, temperature $(\tau)$ based attention helps determine the relevance of each $f_{t}^{ed}$ corresponding $f_{t}^{eq}$ with respect to the question encoding. Temperature $(\tau)$ parameter helps us control the uniformity of attention weights $\alpha_{it}$. Finally, labels are predicted using a multi-layer perceptron over the input vector $f_{ij}$ and the learned weighted average of side information $s_{itd}^{\prime}$. We use binary cross entropy loss to train the proposed model. (6) $\hat{y}_{ij}=F_{\text{output}}([f_{ij};s_{itd}^{\prime}])$ where $F_{\text{output}}$ uses sigmoid activation function. Since community questions may often be entirely unrelated to external sources, a key aspect of this approach is determining whether the _external source_ is useful, not merely attending to its entries that are most relevant. Temperature based attention mechanism is useful in controlling which external source entries are useful for user questions. It is worth noting that one will have to experiment and tune the value of temperature $\tau$ such that ranking performance improves. ## 4\. Experimental Setup Given the model architecture, in this section, we provide a detailed overview of different datasets, metrics and baselines used in our experiments. (a) Yahoo! ques length (b) Yahoo! ans length (c) Infobot ques length (d) Infobot ans length Figure 3. Token distribution in different sources ### 4.1. Data We compiled two question answering datasets. The first was collected from Yahoo! answers and the second was recently released in (emnlp_data2020, ) where both datasets have questions raised by real users. In this work we focus specifically on questions associated with COVID-19. Different statistics about the train and test split of both q&a datasets are given in Table 2 respectively. A pair of relevant and non-relevant answers for a question in both datasets is also shown in Table 1 for reference. More details about them is given below. Stat \| Yahoo! Ans \| Infobot ---\|---\|--- Train Q-A \| 9341 \| 6354 Train ans/q \| 6.25$\pm$2.9 \| 4.40$\pm$0.77 Train #qwords \| 12.71$\pm$5.8 \| 6.55$\pm$3.93 Train #awords \| 36.31$\pm$93.59 \| 92.17$\pm$59.27 Test Q-A \| 2232 \| 1592 Test ans/q \| 5.96$\pm$2.87 \| 4.41$\pm$0.76 Test #qwords \| 13.07$\pm$5.89 \| 6.21$\pm$2.94 Test #awords \| 35.64$\pm$80.31 \| 92.39$\pm$59.47 Table 2. Train and test data from primary sources ##### Yahoo! Dataset : We crawled COVID-19 related questions from Yahoo! answers 777https://answers.search.yahoo.com/search?p=coronavirus using several keywords such as ‘coronavirus’, ‘covid-19’, ‘covid’, ‘sars-cov2’ and ‘corona virus’ between the period of Jan 2020 to July 2020 to ensure we gather all possible questions for our experiments. We keep only those questions have two or more answers. In total, we obtained 1880 questions with 11500 answers. We used favorite answers as positive labels (similar to previous work (LTR_2008, )), assuming that users, over time rate answers (with upvotes/downvotes) that are most relevant to the submitted question. We normalized the question and answer text by removing a small list of stop words, numbers, links or any symbols. Figure 3(a) and 3(b) show the distribution of question and answer lengths respectively. Questions contain $12.7\pm 5.8$ _(qwords)_ words and answers consist of $36.3\pm 93.5$ (mean$\pm$std) words _(awords)_ respectively which indicates that user submitted answers can vary widely on Yahoo! answers. On average, a question has about 6 answers _(ans/q)_ in Yahoo! ans dataset. We spilt the data into three sets: train (64%, 1196 questions, 7435 answers), validation (16%, 298 questions, 1858 answers) and test (20%, 374 questions, 2310 answers) set where questions for each set were uniformly sampled. ##### Infobot Dataset (emnlp_data2020, ) : Researchers at JHU (emnlp_data2020, ) have recently compiled a list of user submitted questions on different platforms and manually labeled 22K+ question- answer pairs. We cleaned this set by removing questions with less than two answers or no relevant answers. In total, our dataset contains 8000+ question answer pairs where each question may have _multiple_ relevant answers which is not the same as Yahoo! answers dataset. Figure 3(c) and 3(d) show the distribution of question and answer lengths respectively. #### 4.1.1. External sources We use two external datasets to rank answers. Details of each dataset are given below: ##### TREC COVID (voorhees2020treccovid, ): We use recently released TREC COVID-19 track data with 50 queries which also contain manually drafted query descriptions and narratives. Expert judges have labeled over 5000 scientific documents for these 50 queries from the CORD-19 dataset 888https://www.semanticscholar.org/cord19. These documents contain coronavirus related research. Given the documents are scientific literature, we initialize document embeddings using SPECTER (cohan2020specter, ). ##### WHO: We use data released on question and answer hub of WHO website999https://www.who.int/emergencies/diseases/novel- coronavirus-2019/question-and-answers-hub to create a list of question-answer pairs. There are 147 question and answer pairs in this dataset where questions contain 13.28$\pm$5.36 words and answers contain 133.2$\pm$100.9 words respectively. ### 4.2. Baselines We evaluated our model against embedding similarity baseline. We computed four baselines as follows: ##### Random: An answer is chosen at random as relevant for a user question. This is expected to provide a lower bound on retrieval performance. ##### Linear Attention (_att_) : When $\tau=1.0$, our model defaults to simple linear attention over all the information present in the external sources. This gives an indication of how well the model performs when its forced to look at all the information in the external source. ##### Linear combination (_$\lambda$ -sim_) : We linearly combine similarities between Yahoo! question-answer and Trec query-answer as shown below: (7) $\emph{$\lambda$-sim}=\lambda\;cos(ya,yq)+(1-\lambda)\;\max_{tq}(cos(ya,tq))$ where $ya$, $yq$ and $tq$ are Yahoo! answer, question and concatenated trec query, narrative and description embeddings respectively. This is a more crude version of temperature attention where $\lambda$ controls the contribution of each component directly. We vary $\lambda$ to determine the optimal combination. Question-Answer similarity (_qasim_) is similarity between question and answer embedding i.e. $\lambda=1$. Both question and answer embeddings are obtained by averaging over their individual token embeddings. ##### BERT Q&A (_bert_) : Large scale pre-trained transformers (devlin2019bert, ) are widely popular for NLP tasks. BERT like models have shown effectiveness on Q&A datasets such as SQUAD 101010https://rajpurkar.github.io/SQuAD-explorer/. We fine-tune BERT base model with two different answer lengths a) 128 _(bert-sl128)_ and b) 256 tokens _(bert-sl256)_ respectively. The intuition is that large scale pre- trained models are adept at language understanding and can be fine-tuned for new tasks with small number of samples. We finetune BERT for both datasets Yahoo! ans and Infobot respectively. It is non-trivial to include external information in BERT and we leave this for future work. \| Yahoo! Ans \| Infobot ---\|---\|--- Model \| P$@$1 \| R$@$3 \| MRR \| P$@$1 \| R$@$3 \| MRR $\tau$-att \| 0.393 \| 0.644 \| 0.598 \| 0.673 \| 0.868 \| 0.802 $\lambda$-sim \| 0.3743 \| 0.633 \| 0.578 \| 0.551 \| 0.817 \| 0.7207 bert-sl256 \| 0.406 \| 0.657 \| 0.615 \| 0.581 \| 0.803 \| 0.744 bert-sl128 \| 0.363 \| 0.604 \| 0.589 \| 0.557 \| 0.799 \| 0.731 att \| 0.377 \| 0.645 \| 0.589 \| 0.567 \| 0.821 \| 0.739 qasim \| 0.318 \| 0.608 \| 0.546 \| 0.551 \| 0.817 \| 0.720 random \| 0.21 \| - \| - \| 0.239 \| - \| - Table 3. Evaluation with WHO external data ### 4.3. Evaluation Metrics We evaluate the performance of our model using three popular ranking metrics, mainly Precision (P$@$1), Mean Reciprocal Rank (MRR), and Recall (R$@$3). Each metric is described below: * • Precision (P$@$k): Precision at position $k$ evaluates the fraction of relevant answers retrieved until position k. For, both datasets Yahoo! ans and Infobot (emnlp_data2020, ), we evaluate whether the top answer i.e. $(k=1)$ in the ranked list is indeed correct. It is defined as follows: (8) $Prec@k=\frac{1}{\|Q\|}\sum_{i=1}^{\|Q\|}\frac{\sum_{j=1}^{k}\mathbb{I}\\{rel_{ij}=1\\}}{k}$ where $\mathbb{I}\\{rel_{ij}=1\\}$ indicates whether the answer at position $j$ is relevant to the $i^{th}$ question. * • Recall (R$@$k): Recall at position $k$ evaluates the fraction of relevant answers retrieved from all the answers marked relevant for a question. We report recall averaged for all the queries in test set. For recall, we take a cutoff as $(k=3)$, which evaluates whether the model is able to retrieve the correct answers in top 3 positions. It is defined as follows: (9) $Recall@k=\frac{1}{\|Q\|}\sum_{i=1}^{\|Q\|}\frac{\sum_{j=1}^{k}\mathbb{I}\\{rel_{ij}=1\\}}{\|rel_{i}\|}$ where $\|rel_{i}\|$ is the number of relevant answers for the $i$th question. * • MRR (MRR): evaluates the average of the reciprocal ranks corresponding to the most relevant answer for the questions in test set, which is given by: (10) $MRR=\frac{1}{\|Q\|}\sum_{i=1}^{\|Q\|}\frac{1}{rank_{i}}$ where $\|Q\|$ indicates the number of queries in the test set and $rank_{i}$ is the rank of the _first_ relevant answer for the $i^{th}$ query. ### 4.4. Parameter Settings Both primary datasets, Yahoo! ans and Infobot, were divied into three parts: train ($\sim$60%), validation and test (20%) respectively. The baseline models $\lambda$-sim and $att$ are initialized with glove embeddings 111111https://nlp.stanford.edu/projects/glove/ of 100 dimensions. We performed a parameter sweep over $\lambda$ and $\tau$ for $\lambda$-sim and $\tau$-att models with step size of 0.1 between $\\{0,1.0\\}$ respectively. We used base uncased model for $bert$ implementation. We fine-tuned the model between 1-10 epochs and found that 3 epochs gave the best result on validation set. We used LSTM with 64 hidden units to represent question, answer and all the information in external datasets. We experimented with higher embedding size and hidden units, but the performance degraded significantly as the model tends to overfit on training data. Lastly we used batch size of 64 and trained the model for 30 epochs with early stopping. \| Yahoo! Ans \| Infobot ---\|---\|--- Model \| P$@$1 \| R$@$3 \| MRR \| P$@$1 \| R$@$3 \| MRR $\tau$-att \| 0.532 \| 0.778 \| 0.715 \| 0.606 \| 0.842 \| 0.766 $\lambda$-sim \| 0.326 \| 0.616 \| 0.555 \| 0.556 \| 0.813 \| 0.722 bert-sl256 \| 0.406 \| 0.657 \| 0.615 \| 0.581 \| 0.803 \| 0.744 bert-sl128 \| 0.363 \| 0.604 \| 0.589 \| 0.557 \| 0.799 \| 0.731 att \| 0.291 \| 0.495 \| 0.494 \| 0.601 \| 0.833 \| 0.762 qasim \| 0.318 \| 0.608 \| 0.546 \| 0.551 \| 0.817 \| 0.720 random \| 0.21 \| - \| - \| 0.239 \| - \| - Table 4. Evaluation with TREC-COVID external data ## 5\. Results In this work, our focus is to evaluate the utility of external information in improving answer ranking for cQ&A task. Thus, we performed experiments to answer three main research questions listed below. RQ1: Does external information improve answer ranking? RQ2: How does temperature ($\tau$) compare with $\lambda$ parameter? RQ3: What kind of queries/questions does the model attend to when ranking relevant/non-relevant answers? ##### RQ1: Does external information improve answer ranking? We evaluated different models for ranking answers in Yahoo! ans and Infobot dataset in presence of TREC and WHO datasets respectively. We found that temperature regulated attention models that incorporate external sources indeed outperform the baselines as shown in Table 4 and Table 3 respectively. Category \| $\tau$-att \| $\lambda$-sim \| qasim ---\|---\|---\|--- Entertainment (47) \| 0.829 \| 0.702 \| 0.59 Health (62) \| 0.693 \| 0.69 \| 0.645 Politics (143) \| 0.727 \| 0.629 \| 0.587 Society (38) \| 0.578 \| 0.473 \| 0.42 Family (20) \| 0.85 \| 0.750 \| 0.65 Table 5. Recall$@$3 of models across categories (_$\tau$ -att_) model beats _bert_ models by $\sim$30% in precision, $\sim$18% in recall and $\sim$16% in MRR respectively on TREC data. However, (_$\tau$ -att_) does only marginally better than _att_ model in precision and MRR on Infobot data. We suspect that is due to the large set of query-document pairs in TREC-COVID data compared to fewer number of question-answer pairs in Infobot dataset. Our results also clearly suggest that embedding based matching of question-answer pair (_qasim_) would not yield a good ranker, though it is better than choosing an answer at random (_random_). When WHO is used as an external dataset, we find that (_$\tau$ -att_) model is slightly worse than _bert_. This suggests that not all sources would equally benefit cQ&A task. Since attention is dependent on the input query and key embedding lengths, it would be interesting to scale the computation in our model to incorporate several open external datasets to overcome this limitation in the future. Yahoo! ans questions are also assigned categories by users. Category based breakdown of performance on test set is given in Table 6 and Table 5 respectively, where categories with largest number of questions in test set are listed. In all the categories, our model outperforms best $\lambda$-sim and _qasim_ model respectively. The largest improvement happens for questions in Family category where our model achieves an improvement of 71% over the $\lambda$-sim model. It seems that ranking answers for questions from society and politics are harder than other categories. All the models, however, are able to rank the top answer in first three positions effectively as Recall$@$3 is high for all the categories. Category \| $\tau$-att \| $\lambda$-sim \| qasim ---\|---\|---\|--- Entertainment (47) \| 0.446 \| 0.382 \| 0.297 Health (62) \| 0.483 \| 0.419 \| 0.354 Politics (143) \| 0.45 \| 0.300 \| 0.272 Society (38) \| 0.28 \| 0.157 \| 0.236 Family (20) \| 0.6 \| 0.350 \| 0.40 Table 6. Precision$@$1 of models across categories \| Temperature ($\tau$) ¿ 1.0 ---\|--- \| 10 \| 100 \| 1000 \| 10 \| 100 \| 1000 Src+ Ext \| Prec$@$1 \| Recall$@$3 Yahoo! + TREC \| 0.46 \| 0.38 \| 0.38 \| 0.73 \| 0.644 \| 0.64 Yahoo! + WHO \| 0.37 \| 0.38 \| 0.36 \| 0.64 \| 0.65 \| 0.64 Infobot + TREC \| 0.44 \| 0.59 \| 0.39 \| 0.72 \| 0.81 \| 0.75 Infobot + WHO \| 0.65 \| 0.41 \| 0.44 \| 0.85 \| 0.76 \| 0.79 Table 7. Variation in P$@$1 and R$@$3 across different temperature values. (a) Yahoo!+TREC (b) Yahoo!+WHO (c) Infobot + TREC (d) Infobot + WHO Figure 4. Temperature and $\lambda$ variation impact on Prec$@$1 ##### RQ2: How does temperature ($\tau$) compare with $\lambda$ parameter? We argued that linearly combining similarities between question-answer in primary dataset and between question-external source may not be sufficient to boost performance. We observe that in our results too i.e. $\lambda$-sim models do not perform better than (_$\tau$ -att_) models. This clearly indicates that more sophisticated models can learn to combine this information directly from training data. However, our experiments indicate that optimal value of (_$\tau$_) varies across primary datasets and external sources of information. For instance, (_$\tau$ -att_) model performed best when _$\tau=0.4$_ and _$\tau=0.9$_ for Yahoo! ans and Infobot dataset respectively when TREC was used as external source. It performed best when _$\tau=0.1$_ and _$\tau=0.5$_ for Yahoo! ans and Infobot dataset respectively when WHO was used as external source. We also tried to vary _$\tau$_ beyond 1.0 to determine whether it yielded a trend as shown in Table 7. Higher values of temperature seem to degrade model performance. We found that optimal temperature range is between $[0.1-1]$. Existing research in model distillation (hinton2015distilling, ) has also empirically found that lower values of temperature yield better performance. We also compared model performance in terms of precision when $\lambda$ and $\tau$ are varied for $\lambda$-sim models and temperature based models respectively as shown in Figure 4. Temperature based models peak at one value but do not have a clear trend indicating that one needs to explore different $\tau$ values at the time of training for better performance. On the other hand, we observe that adding external information also helps the $\lambda$-sim models until a certain threshold. Overall, both sets of models show that free- text external information can be incorporated to improve answer ranking performance. ##### RQ3: What kind of queries/questions does the model attend to when ranking relevant/non-relevant answers? Attention based models have a very unique feature: they can aid explaining the internal workings of neural network models. We inspect what kind of queries/questions in external datasets does our model pay attention to while ranking relevant or non-relevant answers. Figure 5 shows one such example of Yahoo! question and incorporation of TREC data. At the time of scoring relevant answer, the model gives higher weight to some queries compared to others. In the example, for instance, it assigns more weight to queries associated with masks or COVID virus response to weather changes. We observe higher attention weights for questions when relevant answers are ranked than when non-relevant answers are scored. An example question, a relevant and non- relevant answer along with model attention weights on TREC queries are shown from the Infobot data in Figure 6 respectively. It shows a similar trend where attention weights are high for external queries that are closely associated with the question answer text. Overall, our experiments show that curated external information is useful for improving community question answering task. Our experiments also indicate that this external knowledge need not always be structured text. However, it is worth noting that curated and reliable external sources may not always be available for all domains. We addressed a very niche task in this work, and further research is required to extend it to incorporate multiple external sources. We posit that with scalable attention mechanisms, this work can be easily made tractable for large external sources containing thousands or millions of entries in the future. Figure 5. Y! ques, its rel and non-rel ans and questions with $\tau$-att model’s attention values for TREC queries. Figure 6. Infobot ques, its rel and non-rel ans and questions with $\tau$-att model’s attention values for TREC queries. ## 6\. Conclusion Question answering platforms provide users with effective and easy access to information. These platforms also provide content on rapidly evolving _sensitive_ topics such as disease outbreaks (such as COVID-19) where it is also useful to use external _vetted_ information for ranking answers. Existing work only exploits knowledge bases which have some limitations that makes it difficult to use them for community Q&A for rapidly evolving topics such as wild-fires or earthquakes. In this work, we tried to evaluate the effectiveness of external (free text or semi-structured) information in improving answer ranking models. We argue that simple question-answer text matching may be insufficient and in presence of external knowledge, but temperature regulated attention models can distill information better which in turn yields higher performance. Our proposed model with temperature regulated attention, when evaluated on two public datasets showed significant improvements by augmenting information from two _external_ curated sources of information. In future, we aim to expand these experiments to other categories such as disaster relief and scale the attention mechanism to include multiple external sources in one model. ## References * [1] A. Cohan, S. Feldman, I. Beltagy, D. 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# Gain distance matrices for complex unit gain graphs Aniruddha Samanta Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India. Email<EMAIL_ADDRESS>M. Rajesh Kannan Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India. Email<EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract A complex unit gain graph ($\mathbb{T}$-gain graph), $\Phi=(G,\varphi)$ is a graph where the function $\varphi$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation. In this article, we propose gain distance matrices for $\mathbb{T}$-gain graphs. These notions generalize the corresponding known concepts of distance matrices and signed distance matrices. Shahul K. Hameed et al. introduced signed distance matrices and developed their properties. Motivated by their work, we establish several spectral properties, including some equivalences between balanced $\mathbb{T}$-gain graphs and gain distance matrices. Furthermore, we introduce the notion of positively weighted $\mathbb{T}$-gain graphs and study some of their properties. Using these properties, Acharya’s and Stanić’s spectral criteria for balance are deduced. Moreover, the notions of order independence and distance compatibility are studied. Besides, we obtain some characterizations for distance compatibility. Mathematics Subject Classification(2010): 05C22(primary); 05C50, 05C35(secondary). Keywords. Complex unit gain graph, Signed distance matrix, Distance matrix, Adjacency matrix, Hadamard product of matrices. ## 1 Introduction Let $\Phi=(G,\varphi)$ be a connected complex unit gain graph ($\mathbb{T}$-gain graph) on a simple graph $G$ with $n$ vertices. Let $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$ and $E(G)$ be the vertex set and the edge set of $G$, respectively. If two vertices $v_{i}$ and $v_{j}$ are connected by an edge, then we write $v_{i}\sim v_{j}$. If $v_{i}\sim v_{j}$, then the edge between them is denoted by $e_{i,j}$. The _adjacency matrix_ $A(G)$ of a graph $G$ is a symmetric matrix whose $(i,j)th$ entry is $1$ if $v_{i}\sim v_{j}$ and zero otherwise. A _path_ $P$ in $G$ between the vertices $s$ and $t$ is denoted by $sPt$. The _distance_ between two vertices $s$ and $t$ in $G$ is the length of the shortest path between $s$ and $t$, and is denoted by $d_{G}(s,t)$ (or simply $d(s,t)$). The _distance matrix_ of an undirected graph $G$, denoted by $D(G)$, is the symmetric $n\times n$ matrix whose $(i,j)$th entry is $d(v_{i},v_{j})$. The distance matrix of an undirected graph has been widely studied in the literature, see [2, 3, 4, 5] and the references therein. Let $G$ be a simple undirected graph. An oriented edge from the vertex $v_{s}$ to the vertex $v_{t}$ is denoted by $\overrightarrow{e}_{s,t}$. For each undirected edge $e_{s,t}\in E(G)$, there is a pair of oriented edges $\overrightarrow{e}_{s,t}$ and $\overrightarrow{e}_{t,s}$. The collection $\overrightarrow{E}(G):=\\{\overrightarrow{e}_{s,t},\overrightarrow{e}_{t,s}:e_{s,t}\in E(G)\\}$ is the _oriented edge set associated with $G$_. Let $\mathbb{T}=\\{z\in\mathbb{C}:\|z\|=1\\}$. A _complex unit gain graph (or $\mathbb{T}$-gain graph)_ on a simple graph $G$ is an ordered pair $(G,\varphi)$, where the gain function $\varphi:\overrightarrow{E}(G)\rightarrow\mathbb{T}$ is a mapping such that $\varphi(\overrightarrow{e}_{s,t})=\varphi(\overrightarrow{e}_{t,s})^{-1}$, for every $e_{s,t}\in E(G)$. A $\mathbb{T}$-gain graph $(G,\varphi)$ is denoted by $\Phi$. The _adjacency matrix_ of a $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$ is a Hermitian matrix, denoted by $A(\Phi)$ and its $(s,t)th$ entry is defined as follows: $a_{st}=\begin{cases}\varphi(\overrightarrow{e}_{s,t})&\text{if }\mbox{$v_{s}\sim v_{t}$},\\\ 0&\text{otherwise.}\end{cases}$ The spectrum and the spectral radius of $\Phi$ are the spectrum and the spectral radius of $A(\Phi)$ and denoted by $\operatorname{spec}(\Phi)$ and $\rho(\Phi)$, respectively. A _signed graph_ is a graph $G$ together with a signature function $\psi:E(G)\rightarrow\\{\pm 1\\}$, and is denoted by $\Psi=(G,\psi)$. The adjacency matrix of $\Psi$, denoted by $A(\Psi)$, is an $n\times n$ matrix whose $(i,j)$th entry is $\psi(e_{i,j})$. Therefore, a signed graph can be considered as a $\mathbb{T}$-gain graph $\Psi=(G,\psi)$, where $\psi$ is a signature function. The notion of adjacency matrix of $\mathbb{T}$-gain graphs generalize the notion of adjacency matrix of undirected graphs, adjacency matrix of signed graphs and the Hermitian adjacency matrix of a mixed graph. For more information about the properties of gain graphs and $\mathbb{T}$-gain graphs, we refer to [11, 12, 16, 17]. Let $\Psi=(G,\psi)$ be a signed graph. The sign of a path in $\Psi$ is the product of sign of all edges of the path [15]. Recently, in [6] the authors introduced the notion of signed distance matrices $D^{\max}(\Psi)$ and $D^{\min}(\Psi)$ for a signed graph $\Psi$. ###### Definition 1.1 ([6, Definition 1.1]). Let $\Psi=(G,\psi)$ be a signed graph with vertex set $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$. Then two auxiliary signs are defined as follows: * (a) $\psi_{\max}(v_{i},v_{j})=-1$ if all shortest $v_{i}v_{j}$-paths are negative, $+1$ otherwise, * (b) $\psi_{\min}(v_{i},v_{j})=+1$ if all shortest $v_{i}v_{j}$-paths are positive, $-1$ otherwise. The two signed distance matrices are defined as follows: * (a) $D^{\max}(\Psi)=(d_{\max}(v_{i},v_{j}))_{n\times n}$, * (b) $D^{\min}(\Psi)=(d_{\min}(v_{i},v_{j}))_{n\times n}$, where $d_{\max}(v_{i},v_{j})=\psi_{\max}(v_{i},v_{j})d(v_{i},v_{j})$ and $d_{\min}(v_{i},v_{j})=\psi_{\min}(v_{i},v_{j})d(v_{i},v_{j})$. A signed graph $\Psi$ is distance compatible if and only if $D^{\max}(\Psi)=D^{\min}(\Psi)$. A characterization of balanced signed graph in terms of signed distance matrices is obtained in [6]. For more about signed distance matrices, see [6, 13]. In this article, we introduce the notion of gain distance matrices $D^{\max}_{<}(\Phi)$ and $D^{\min}_{<}(\Phi)$ for a $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$ associated with an ordered vertex set $(V(G),<)$. These concepts generalize the notions of signed distance matrices of signed graphs and distance matrices of undirected graphs. We define positively weighted $\mathbb{T}$-gain graphs and establish two new characterizations for balance of gain graphs. Acharya’s Spectral criterion and Stanić’s spectral criterion are particular cases of these characterizations. Besides, we introduce two properties of a $\mathbb{T}$-gain graph, ordered-independence, and distance compatibility to gain distance matrices. Thereupon we establish two characterizations for the balance of $\mathbb{T}$-gain graphs in terms of gain distance matrices and distance compatibility properties. Subsequently, we present some results on the characterization of distance compatibility. This paper is organized as follows: In section 2, we collect needed known definitions and results. In section 3, we define the notion of gain distance matrices, order-independent and distance compatibility, and discuss their properties. In section 4, we discuss the positively weighted $\mathbb{T}$-gain graphs and establish two spectral characterizations for the balance(Theorem 4.2, Theorem 4.3). In section 5, we derive two characterizations for balance $\mathbb{T}$-gain graph in terms of the gain distance matrices (Theorem 5.2, Theorem 5.3). In section 6, we obtain a couple of characterizations for distance compatible $\mathbb{T}$-gain graphs (Theorem 6.1, Theorem 6.2, Theorem 6.3). ## 2 Definitions, notation and preliminary results Let $G=(V(G),E(G))$ be a connected undirected graph with no loops and multiple edges, where $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$ is the vertex set and $E(G)$ is the edge set of $G$. A graph $G$ is _geodetic_ , if there exists a unique shortest path between any two vertices of $G$. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph on $G$. For $s,t\in V(G)$, $sPt$ denotes a path starts at $s$ and ends at $t$ in $G$. In case of gain graph, $sPt$ denotes the oriented path from the vertex $s$ to the vertex $t$. The gain of the path $sPt$ is $\varphi(sPt)=\prod\limits_{j=1}^{k}\varphi(\overrightarrow{e_{j}})$, where $\overrightarrow{e_{1}},\overrightarrow{e_{2}},\dots,\overrightarrow{e_{k}}$ are the consecutive oriented edges in $sPt$. Therefore, $\varphi(tPs)=\overline{\varphi(sPt)}$. The gain of an oriented cycle $\overrightarrow{C_{n}}$ with edges $\overrightarrow{e_{1}},\overrightarrow{e_{2}},\dots,\overrightarrow{e_{n}}$ is $\varphi(\overrightarrow{C_{n}})=\prod\limits_{j=1}^{n}\varphi(\overrightarrow{e_{j}})$. A cycle $C$ is _neutral_ in $\Phi$ if $\varphi(\overrightarrow{C})=1$. A gain graph $\Phi$ is _balanced_ , if all cycles in $\Phi$ are neutral. A $\mathbb{T}$-gain graph $\Phi$ is _anti-balanced_ if $-\Phi$ is balanced. Let $\operatorname{Re}(x)$ and $\operatorname{Im}(x)$ denote the real and imaginary part of a complex number $x$, respectively. A function $\zeta:V(G)\rightarrow\mathbb{T}$ is a _switching function_. Let $\Phi_{1}=(G,\varphi_{1})$ and $\Phi_{2}=(G,\varphi_{2})$ be two $\mathbb{T}$-gain graphs. Then $\Phi_{1}$ and $\Phi_{2}$ are _switching equivalent_ , denoted by $\Phi_{1}\sim\Phi_{2}$, if there exists a switching function $\zeta$ such that $\varphi_{1}(\overrightarrow{e}_{i,j})=\zeta(v_{i})^{-1}\varphi_{2}(\overrightarrow{e}_{i,j})\zeta(v_{j})$, for all $e_{i,j}\in E(G)$. If $\Phi_{1}\sim\Phi_{2}$, then $A(\Phi_{1})$ and $A(\Phi_{2})$ are diagonally similar and hence have the same spectra. ###### Lemma 2.1 ([12, Corollary 3.2]). Let $\Phi_{1}$ and $\Phi_{2}$ be two $\mathbb{T}$-gain graphs on a connected graph $G$ with a normal spanning tree $T$. Then $\Phi_{1}\sim\Phi_{2}$ if and only if $\varphi_{1}(\overrightarrow{C_{j}})=\varphi_{2}(\overrightarrow{C_{j}})$, for all fundamental cycles $C_{j}$ with respect to $T$. A signed graph is a $\mathbb{T}$-gain graph $\Psi=(G,\psi)$, where $\psi(\overrightarrow{e}_{i,j})=1$ or $-1$ for $e_{i,j}\in E(G)$. The _sign_ of a path in $\Psi$ is the product of the signs (the gains) of the edges in the path. ###### Theorem 2.1 (Harary’s path criterion [7] ). Let $\Psi$ be a signed graph on an underlying graph $G$. Then $\Psi$ is balanced if and only if any pair of vertices $s,t$, every $st$-path have the same signature. Let $\mathbb{C}^{m\times n}$ denote the set of all $m\times n$ matrices with complex entries. For $A=(a_{ij})\in\mathbb{C}^{n\times n}$, define $\|A\|=(\|a_{ij}\|)$. For two matrices $A=(a_{ij})$ and $B=(b_{ij})$, we write $A\leq B$ if $a_{ij}\leq b_{ij}$ for all $i,j$. A matrix is _non-negative_ , if all entries of a matrix are non-negative. The spectral radius of a matrix $A$ is denoted by $\rho(A)$. ###### Theorem 2.2 ([9, Theorem 8.4.5]). Let $A,B\in\mathbb{C}^{n\times n}$. Suppose $A$ is irreducible and non- negative and $A\geq\|B\|$. Let $\mu=e^{i\theta}\rho(B)$ be a given maximum modulus eigenvalue of $B$. If $\rho(A)=\rho(B)$, then there is a unitary diagonal matrix $D$ such that $B=e^{i\theta}DAD^{-1}$. Let $A=(a_{ij}),B=(b_{ij})\in\mathbb{C}^{m\times n}$. The _Hadamard product_ of $A$ and $B$, denoted by $A\circ B$, is defined as $A\circ B=(a_{ij}b_{ij})_{m\times n}$. For any three matrices $A,B,C$ of same order, $(A\circ B)\circ C=A\circ(B\circ C)$. Let us recall the following property of Hadamard product of matrices. ###### Proposition 2.1 ([8, Lemma 5.1.2]). Let $A,B,C$ be three $n\times n$ matrices and $D,E$ be two $n\times n$ diagonal matrices. Then $D(A\circ B)E=(DAE)\circ B=(DA)\circ(BE)=(AE)\circ(DB)=A\circ(DBE).$ ## 3 Gain distance matrices This section introduces the notion of gain distance matrices of $\mathbb{T}$-gain graphs, which generalize the notion of distance matrices of undirected graphs and signed distance matrices of signed graphs. Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph on $G$. For $s,t\in V(G)$, $sPt$ denotes the oriented path from the vertex $s$ to the vertex $t$. Define three sets of paths $\mathcal{P}(s,t),\mathcal{P}^{\max}(s,t)$ and $\mathcal{P}^{\min}(s,t)$ as follows: $\mathcal{P}(s,t)=\left\\{sPt:sPt\text{ is a shortest path}\right\\},$ $\mathcal{P}^{\max}(s,t)=\left\\{sPt\in\mathcal{P}(s,t):\operatorname{Re}(\varphi(sPt))=\max\limits_{s\tilde{P}t\in\mathcal{P}(s,t)}\operatorname{Re}(\varphi(s\tilde{P}t))\right\\}$ and $\mathcal{P}^{\min}(s,t)=\left\\{sPt\in\mathcal{P}(s,t):\operatorname{Re}(\varphi(sPt))=\min\limits_{s\tilde{P}t\in\mathcal{P}(s,t)}\operatorname{Re}(\varphi(s\tilde{P}t))\right\\}.$ Note that $\mathcal{P}^{\max}(s,t)=\mathcal{P}^{\max}(t,s)$ and $\mathcal{P}^{\min}(s,t)=\mathcal{P}^{\min}(t,s)$. Let $G$ be a simple graph with vertex set $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$. We denote $\left(V(G),<\right)$ as an ordered vertex set, where $`<`$ is a total ordering of the vertices of $G$. An ordering $`<_{r}`$ is the _reverse ordering_ of $`<`$ if $v_{i}<_{r}v_{j}$ if and only if $v_{j}<v_{i}$, for any $i,j$. An ordering $`<`$ is the _standard vertex ordering_ if $v_{1}<v_{2}<\dots<v_{n}$. ###### Definition 3.1 (Auxiliary gains). Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with an ordered vertex set $(V(G),<)$. We define two auxiliary gains with respect to $<$ as follows. 1. (1) The function $\varphi^{<}_{\max}:V(G)\times V(G)\rightarrow\mathbb{T}$ is the maximum auxiliary gain with respect to the vertex order $<$ such that $\varphi^{<}_{\max}(s,t)=\overline{\varphi^{<}_{\max}(t,s)}$ for each $(s,t)\in V(G)\times V(G)$ and $\varphi^{<}_{\max}$ is defined by $\varphi^{<}_{\max}(s,t)=\varphi(sPt)$ where $s<t$, and $sPt\in\mathcal{P}^{\max}(s,t)$ and $\operatorname{Im}(\varphi(sPt))=\max\limits_{s\tilde{P}t\in\mathcal{P}^{\max}(s,t)}\operatorname{Im}(\varphi(s\tilde{P}t)).$ 2. (2) The function $\varphi^{<}_{\min}:V(G)\times V(G)\rightarrow\mathbb{T}$ is the minimum auxiliary gain with respect to the vertex order $<$ such that $\varphi^{<}_{\min}(s,t)=\overline{\varphi^{<}_{\min}(t,s)}$ for each $(s,t)\in V(G)\times V(G)$ and $\varphi^{<}_{\min}$ is defined by $\varphi^{<}_{\min}(s,t)=\varphi(sPt)$ where $s<t$, and $sPt\in\mathcal{P}^{\min}(s,t)$ and $\operatorname{Im}(\varphi(sPt))=\min\limits_{s\tilde{P}t\in\mathcal{P}^{\min}(s,t)}\operatorname{Im}(\varphi(s\tilde{P}t)).$ Note that, for $s<t$, $\varphi^{<}_{\max}(s,t)(\mbox{~{}resp.,~{}}\varphi^{<}_{\min}(s,t))$ is the maximum (resp., minimum) gain, with respect to the lexicographic order, over all the shortest paths between the vertices $s$ and $t$. ###### Definition 3.2 (Gain distances). Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with an ordered vertex set $(V(G),<)$. For any two vertices $s,t\in V(G)$, there are two gain distances from the vertex $s$ to the vertex $t$ which are defined as follows: 1. (1) $d^{<}_{\max}(s,t)=\varphi^{<}_{\max}(s,t)d(s,t),$ 2. (2) $d^{<}_{\min}(s,t)=\varphi^{<}_{\min}(s,t)d(s,t).$ Next we define the gain distance matrices for $\mathbb{T}$-gain graphs. ###### Definition 3.3 (Gain distance matrices). Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with an order $<$ on the vertex set $V(G)$, where $V(G)=\\{v_{1},v_{2},\cdots,v_{n}\\}$. The gain distance matrices $D_{<}^{\max}(\Phi)$ and $D_{<}^{\min}(\Phi)$ associated with $<$ are defined as follows: 1. (1) $D_{<}^{\max}(\Phi)=\left(d^{<}_{\max}(v_{i},v_{j})\right)$, 2. (2) $D_{<}^{\min}(\Phi)=\left(d^{<}_{\min}(v_{i},v_{j})\right)$. Here $d^{<}_{\max}(v_{i},v_{j})$ is the $(i,j)$th entry of $D_{<}^{\max}(\Phi).$ The gain distance matrices are the generalization of the distance matrix of an undirected graph and signed distance matrices of a signed graph. It is easy to see that the gain distance matrices $D^{\max}_{<}(\Phi)$ and $D^{\min}_{<}(\Phi)$ are Hermitian, and hence have real eigenvalues. For any pair of vertices $(v_{s},v_{t})$, there are two different maximum gain distances $d^{<}_{\max}(v_{s},v_{t})$ and $d^{<}_{\max}(v_{t},v_{s})$ which have same absolute value but they are the complex conjugate to each other. The distance $d^{<}_{\max}(v_{s},v_{t})$ is the maximum gain distance from the vertex $v_{s}$ to the vertex $v_{t}$ in $\Phi$ with respect to the vertex ordering $(V(G),<)$. Likewise, the minimum gain distance is defined. Now we illustrate the definitions with the following example. Figure 1: $\mathbb{T}$-gain graphs $\Phi=(G,\varphi)$ ###### Example 3.1. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph shown in Figure 1. Let us consider the standard order $<$ on vertex set $V(G)$, where $V(G)=\\{v_{1},v_{2},v_{3},v_{4},v_{5}\\}$. Then $D^{\max}_{<}(\Phi)=\left[\begin{array}[]{ccccc}0&1&2e^{\frac{i\pi}{6}}&1&e^{\frac{i\pi}{3}}\\\ 1&0&e^{\frac{i\pi}{6}}&1&2e^{\frac{i\pi}{3}}\\\ 2e^{-\frac{i\pi}{6}}&e^{-\frac{i\pi}{6}}&0&e^{\frac{i\pi}{6}}&3e^{\frac{i\pi}{6}}\\\ 1&1&e^{-\frac{i\pi}{6}}&0&2e^{\frac{i\pi}{3}}\\\ e^{-\frac{i\pi}{3}}&2e^{-\frac{i\pi}{3}}&3e^{-\frac{i\pi}{6}}&2e^{-\frac{i\pi}{3}}&0\end{array}\right].$ Now, consider the reverse ordering $`<_{r}`$ of the standard order $`<`$. Then $D^{\max}_{<_{r}}(\Phi)=\left[\begin{array}[]{ccccc}0&1&2e^{-\frac{i\pi}{6}}&1&e^{\frac{i\pi}{3}}\\\ 1&0&e^{\frac{i\pi}{6}}&1&2e^{\frac{i\pi}{3}}\\\ 2e^{\frac{i\pi}{6}}&e^{-\frac{i\pi}{6}}&0&e^{\frac{i\pi}{6}}&3e^{-\frac{i\pi}{6}}\\\ 1&1&e^{-\frac{i\pi}{6}}&0&2e^{\frac{i\pi}{3}}\\\ e^{-\frac{i\pi}{3}}&2e^{-\frac{i\pi}{3}}&3e^{\frac{i\pi}{6}}&2e^{-\frac{i\pi}{3}}&0\end{array}\right].$ Here $D^{\max}_{<}(\Phi)\neq D^{\max}_{<_{r}}(\Phi)$. In fact, $\operatorname{spec}(D^{\max}_{<}(\Phi))\neq\operatorname{spec}(D^{\max}_{<_{r}}(\Phi))$. Similarly,$D^{\min}_{<}(\Phi)\neq D^{\min}_{<_{r}}(\Phi)$. ###### Definition 3.4 (Vertex order independent). A $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$ is vertex order-independent (simply, order independent), if $D^{\max}_{<}(\Phi)=D^{\max}_{<_{r}}(\Phi)$ and $D^{\min}_{<}(\Phi)=D^{\min}_{<_{r}}(\Phi)$, where $<$ is the standard vertex order on $V(G)$. In this case, we define$D^{\max}(\Phi)=D^{\max}_{<}(\Phi)=D^{\max}_{<_{r}}(\Phi)$ and $D^{\min}(\Phi)=D^{\min}_{<}(\Phi)=D^{\min}_{<_{r}}(\Phi)$. Now we present a characterization for order independent $\mathbb{T}$-gain graph. ###### Theorem 3.1. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph. Then $\Phi$ is not order independent if and only if at least one of the following holds. 1. (i) There exists $v_{s},v_{t}\in V(G)$ with at least two shortest paths from $v_{s}$ to $v_{t}$ in $\mathcal{P}^{\max}(v_{s},v_{t})$ have different gains. 2. (ii) There exists $v_{s},v_{t}\in V(G)$ with at least two shortest paths from $v_{s}$ to $v_{t}$ in $\mathcal{P}^{\min}(v_{s},v_{t})$ have different gains. ###### Proof. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with vertex set $V(G)=\\{v_{1},v_{2},\cdots,v_{n}\\}$. Let $<$ be the standard vertex order on $V(G)$. Suppose $\Phi$ is not order independent, then either $D^{\max}_{<}(\Phi)\neq D^{\max}_{<_{r}}(\Phi)$ or $D^{\min}_{<}(\Phi)\neq D^{\min}_{<_{r}}(\Phi)$ hold. Suppose $D^{\max}_{<}(\Phi)\neq D^{\max}_{<_{r}}(\Phi)$. Then there exists $v_{s},v_{t}\in V(G)$ such that $d_{\max}^{<}(v_{s},v_{t})\neq d_{\max}^{<_{r}}(v_{s},v_{t})$. Then $\varphi_{\max}^{<}(v_{s},v_{t})\neq\varphi_{\max}^{<_{r}}(v_{s},v_{t})$. Let $v_{s}<v_{t}$ and $\varphi_{\max}^{<}(v_{s},v_{t})=\varphi(v_{s}P_{1}v_{t})=x+iy\in\mathbb{T}$, for some $v_{s}P_{1}v_{t}\in\mathcal{P}^{\max}(v_{s},v_{t})$. It is clear that $y\neq 0$. Now $v_{t}<_{r}v_{s}$ and $\varphi_{\max}^{<_{r}}(v_{t},v_{s})=\varphi(v_{t}P_{2}v_{s})$, for some $v_{t}P_{2}v_{s}\in\mathcal{P}^{\max}(v_{s},v_{t})$. Then $\varphi_{\max}^{<_{r}}(v_{s},v_{t})=\overline{\varphi_{\max}^{<_{r}}(v_{t},v_{s})}=\overline{\varphi(v_{t}P_{2}v_{s})}=\varphi(v_{s}P_{2}v_{t})$. Since $v_{t}P_{2}v_{s}\in\mathcal{P}^{\max}(v_{s},v_{t})$, so $\varphi(v_{s}P_{2}v_{t})=x-iy_{1}\in\mathbb{T}$, where either $y_{1}=y$ or $y_{1}=-y$. Also, $\varphi_{\max}^{<}(v_{s},v_{t})\neq\varphi_{\max}^{<_{r}}(v_{s},v_{t})$, so $\varphi(v_{s}P_{1}v_{t})\neq\varphi(v_{s}P_{2}v_{t})$. Thus $\varphi(v_{s}P_{2}v_{t})=x-iy$ and $y>0$. Hence $(i)$ holds. Similarly, $D^{\min}_{<}(\Phi)\neq D^{\min}_{<_{r}}(\Phi)$ implies $(ii)$. Conversely, suppose statement $(i)$ holds. Then there exist two shortest $\overrightarrow{v_{s}v_{t}}$-paths $v_{s}P_{1}v_{t}$ and $v_{s}P_{2}v_{t}$ in $\mathcal{P}^{\max}(v_{s},v_{t})$ with different gains. If $\varphi(v_{s}P_{1}v_{t})=x+iy\in\mathbb{T}$, then $\varphi(v_{s}P_{2}v_{t})=x-iy$, $y\neq 0$. Without loss of generality, assume that $y>0$. If $v_{s}<v_{t}$, then $\varphi_{\max}^{<}(v_{s},v_{t})=x+iy$ and $\varphi_{\max}^{<_{r}}(v_{s},v_{t})=\overline{\varphi_{\max}^{<_{r}}(v_{t},v_{s})}=\overline{x+iy}=x-iy$. Thus $\varphi_{\max}^{<}(v_{s},v_{t})\neq\varphi_{\max}^{<_{r}}(v_{s},v_{t})$. Therefore $D^{\max}_{<}(\Phi)\neq D^{\max}_{<_{r}}(\Phi)$ and hence $\Phi$ is not order independent. Similarly if the statement $(ii)$ holds, then $\Phi$ is not order independent. ∎ ###### Proposition 3.1. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph and $`<`$ be the standard vertex order. Then $D_{<}^{\max}(\Phi)=D_{<}^{\min}(\Phi)$ if and only if $D^{\max}(\Phi)$ and $D^{\min}(\Phi)$ are well defined and $D^{\max}(\Phi)=D^{\min}(\Phi)$. ###### Proof. Suppose $D_{<}^{\max}(\Phi)=D_{<}^{\min}(\Phi)$. Let $s,t\in V(G)$ such that $s<t$. Then $d^{<}_{\max}(s,t)=d^{<}_{\min}(s,t)$. Therefore $\varphi^{\max}_{<}(s,t)=\varphi^{\min}_{<}(s,t)$. Thus all the shortest paths from $s$ to $t$ have the same gain. Therefore, $\varphi^{\max}_{<}(s,t)=\varphi^{\max}_{<_{r}}(s,t)$ and $\varphi^{\min}_{<}(s,t)=\varphi^{\min}_{<_{r}}(s,t)$. Thus $D^{\max}(\Phi)$ and $D^{\min}(\Phi)$ are well defined. Since $d^{<}_{\max}(s,t)=d^{<}_{\min}(s,t)$, so $D^{\max}(\Phi)=D^{\min}(\Phi)$. The converse is easy to verify. ∎ ###### Theorem 3.2. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph and $`<`$ be the standard vertex order. Let $<_{a}$ be any vertex order on $V(G)$. Then $D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$ if and only if $D^{\max}_{<_{a}}(\Phi)=D^{\min}_{<_{a}}(\Phi)$. ###### Proof. Let $<$ be the standard vertex order, and $D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$. Let $v_{i},v_{j}\in V(G)$. Then $d^{<}_{\max}(v_{i},v_{j})=d^{<}_{\min}(v_{i},v_{j})$ and $\varphi^{<}_{\max}(v_{i},v_{j})=\varphi^{<}_{\min}(v_{i},v_{j})$. Thus all the shortest paths from $v_{i}$ to $v_{j}$ have the same gain. Therefore, for any arbitrary vertex ordering $<_{a}$, we have $\varphi^{<_{a}}_{\max}(v_{i},v_{j})=\varphi^{<_{a}}_{\min}(v_{i},v_{j})$. Hence $d^{<_{a}}_{\max}(v_{i},v_{j})=d^{<_{a}}_{\min}(v_{i},v_{j})$. Since $v_{i}$ and $v_{j}$ are arbitrary, so $D^{\max}_{<_{a}}(\Phi)=D^{\min}_{<_{a}}(\Phi)$. Proof of the converse is similar to that of the previous part. ∎ ###### Definition 3.5 (Distance compatible). A $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$ is called _gain distance compatible (simply, distance compatible)_ if $D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$, where $<$ is the standard order. In this case, we define $D(\Phi)=D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$. The proof of the following theorem is easy to verify. ###### Theorem 3.3. For a $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$, the following are equivalent: 1. (1) $\Phi$ is distance compatible. 2. (2) $D(\Phi)=D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)=D^{\max}(\Phi)=D^{\min}(\Phi)$. 3. (3) $D(\Phi)$ is well defined. ###### Proof. $(1)\implies(2):$ Let $\Phi$ be distance compatible. Then, by the definition, $D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$ for standard vertex order $<$. Also $D(\Phi)=D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$. Now by Proposition 3.1, $D^{\max}(\Phi)=D^{\min}(\Phi)$. Hence $D^{\max}(\Phi)=D^{\min}_{<}(\Phi)$ and $D^{\min}(\Phi)=D^{\min}_{<}(\Phi)$. $(2)\implies(3):$ By the definition of distance-compatible $\mathbb{T}$-gain graph, $D(\Phi)$ exists. $(3)\implies(1):$ If $D(\Phi)$ is well defined, then $D(\Phi)=D^{\max}(\Phi)=D^{\min}(\Phi)$. Hence $\Phi$ is distance-compatible. ∎ ###### Proposition 3.2. Let $\Phi=(G,\varphi)$ be any distance compatible $\mathbb{T}$-gain graph. If $\Phi\sim\Psi$, then $\Psi$ is distance compatible and $\operatorname{spec}(D(\Phi))=\operatorname{spec}(D(\Psi))$. ###### Proof. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with the standard vertex order $<$. Let $s,t\in V(G)$. Since $\Phi$ is distance compatible, all oriented shortest paths $sPt$ from $s$ to $t$ have the same gain. As $\Phi\sim\Psi$, so there exists a switching function $\zeta$ such that $\psi(sPt)=\zeta(s)^{-1}\varphi(sPt)\zeta(t)$, for any shortest path $sPt$. For any shortest path $sPt$, $\varphi(sPt)$ is unique, so $\psi(sPt)$ is unique. Thus $\psi_{\max}^{<}(s,t)=\psi_{\min}^{<}(s,t)$ and hence $D^{\max}_{<}(\Psi)=D^{\min}_{<}(\Psi)$. That is, $\Psi$ is distance compatible and $D(\Psi)$ is well defined. Let $d_{\psi}(s,t)$ and $d_{\varphi}(s,t)$ be the unique gain distance from $s$ to $t$ in $\Psi$ and $\Phi$, respectively. Then $d_{\varphi}(s,t)=\zeta(s)^{-1}d_{\psi}(s,t)\zeta(t)$. Thus $D(\Phi)$ and $D(\Psi)$ are similar and hence $\operatorname{spec}(D(\Phi))=\operatorname{spec}(D(\Psi))$. ∎ Converse of the above statement holds for balanced $\mathbb{T}$ gain graph, see Corollary 5.1. ###### Proposition 3.3. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph. Then 1. (1) If $\Phi$ is a signed graph, then $\Phi$ is order-independent. 2. (2) If $\Phi$ is balanced or anti-balanced, then $\Phi$ is order-independent. 3. (3) If $\Phi$ is distance compatible, then $\Phi$ is order-independent. 4. (4) If $\Phi$ is geodetic, then $\Phi$ is order-independent. However, converse of the above statements need not be true in general, see Example 3.2. The following result is an extension of the Harary’s path criterion for $\mathbb{T}$-gain graphs. ###### Lemma 3.1. Let $\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph. Then $\Phi$ is balanced if and only if every directed $(s,t)$-path have the same gain in $\Phi$, for any two vertices $s,t$. ###### Proof. Let $\Phi=(G,\varphi)$ be balanced. Suppose that the two oriented paths $sP_{1}t$ and $sP_{2}t$ have different gains. That is $\varphi(sP_{1}t)\neq\varphi(sP_{2}t)$. Then $\varphi(sP_{1}t)\varphi(tP_{2}s)=\sum\limits_{j=1}^{k}\varphi(\overrightarrow{C_{j}})\neq 1$, where $C_{1},C_{2},\dots,C_{k}$ are the cycles formed by these two paths. Therefore, there exist at least one cycle, say $C_{j}$ such that $\varphi(\overrightarrow{C_{j}})\neq 1$ . Thus $\Phi$ is not balanced, a contradiction. Converse is easy to verify. ∎ Let $\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph. Then $\Phi$ is either order-independent or order-dependent. If $\Phi$ is ordered-independent, then $\Phi$ may or may not be balanced, anti-balanced, geodetic. If $\Phi$ is ordered-dependent, then, by Proposition 3.3, $\Phi$ is neither balanced nor anti-balanced nor geodetic. Therefore, any $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$ belongs to one of the following classes: 1. (A) $\Phi$ is balanced or anti-balanced or geodetic and $D^{\max}(\Phi)=D^{\min}(\Phi)$. 2. (B) $\Phi$ is neither balanced nor anti-balanced nor geodetic and $D^{\max}(\Phi)=D^{\min}(\Phi)$. 3. (C) $\Phi$ is neither balanced nor anti-balanced nor geodetic and $D^{\max}(\Phi)\neq D^{\min}(\Phi)$. 4. (D) $\Phi$ is neither balanced nor anti-balanced nor geodetic and at least one of $D^{\max}(\Phi)$ and $D^{\min}(\Phi)$ are not well defined. Next we give some examples. Examples of Type $(A)$ can be constructed easily. Example 3.1 is of Type $(D)$. Examples of Type $(B)$ and Type $(C)$ are given below. Figure 2: $\mathbb{T}$-gain graphs $\Phi_{1}$ and $\Phi_{2}$ ###### Example 3.2. Let us consider the $\mathbb{T}$-gain graph $\Phi_{1}=(G,\varphi_{1})$ (in Figure 2) with the standard vertex order. The graph $\Phi_{1}$ is neither balanced nor anti-balanced nor geodetic. Also $\Phi$ is order-independent. Now $D^{\max}(\Phi_{1})=\left[\begin{array}[]{ccccc}0&1&1&2e^{\frac{i\pi}{6}}&e^{\frac{i\pi}{6}}\\\ 1&0&1&2&1\\\ 1&1&0&e^{\frac{i\pi}{3}}&1\\\ 2e^{-\frac{i\pi}{6}}&2&e^{-\frac{i\pi}{3}}&0&1\\\ e^{-\frac{i\pi}{6}}&1&1&1&0\end{array}\right],D^{\min}(\Phi_{1})=\left[\begin{array}[]{ccccc}0&1&1&2e^{\frac{i\pi}{3}}&e^{\frac{i\pi}{6}}\\\ 1&0&1&2e^{\frac{i\pi}{3}}&1\\\ 1&1&0&e^{\frac{i\pi}{3}}&1\\\ 2e^{-\frac{i\pi}{3}}&2e^{-\frac{i\pi}{3}}&e^{-\frac{i\pi}{3}}&0&1\\\ e^{-\frac{i\pi}{6}}&1&1&1&0\end{array}\right].$ Thus $D^{\max}(\Phi_{1})\neq D^{\min}(\Phi_{1})$. Therefore, by Theorem 3.3, $\Phi_{1}$ is distance incompatible. ###### Example 3.3. The $\mathbb{T}$-gain graph $\Phi_{2}$ (in Figure 2) with the standard vertex ordering is neither balanced nor anti-balanced nor geodetic. Here $\Phi_{2}$ is order-independent. However, it is distance compatible and $D^{\max}(\Phi_{2})=D^{\min}(\Phi_{2})=\left[\begin{array}[]{ccccc}0&1&e^{\frac{i\pi}{4}}&2e^{\frac{i\pi}{2}}&e^{\frac{i\pi}{4}}\\\ 1&0&1&2e^{\frac{i\pi}{4}}&1\\\ e^{-\frac{i\pi}{4}}&1&0&e^{\frac{i\pi}{4}}&1\\\ 2e^{-\frac{i\pi}{2}}&2e^{-\frac{i\pi}{4}}&e^{-\frac{i\pi}{4}}&0&e^{-\frac{i\pi}{4}}\\\ e^{-\frac{i\pi}{4}}&1&1&e^{\frac{i\pi}{4}}&0\end{array}\right].$ ## 4 Positively weighted $\mathbb{T}$-gain graph In this section, we introduce the notion of a positively weighted $\mathbb{T}$-gain graph. The adjacency matrices of positively weighted $\mathbb{T}$-gain graphs generalize the following notions: $\mathbb{T}$-gain adjacency matrices, Hermitian adjacency matrices of mixed graphs, adjacency matrices of signed graphs, adjacency matrices of undirected graphs. ###### Definition 4.1 (Positively weighted $\mathbb{T}$-gain graph). Let $\Phi=(G,\varphi)$ be any $\mathbb{T}$-gain graph with $E(G)$ be the undirected edge set of $G$. Let $w:E(G)\rightarrow\mathbb{R}^{+}$ be a weight function on the edges of $G$. The positively weighted $\mathbb{T}$-gain graph associated with $\Phi$ and $w$ is the graph $G$ together with the weighted gain function $\varphi_{w}$ defined as follows: $\varphi_{w}(\overrightarrow{e}_{i,j})=\varphi(\overrightarrow{e}_{i,j})w(e_{i,j}).$ A positively weighted $\mathbb{T}$-gain graph on $(G,\varphi)$ is denoted by $(G,\varphi,w)$(or simply $\Phi_{w}$). The adjacency matrix associated with $\Phi_{w}$, denoted by $A(\Phi_{w})$, is an $n\times n$ Hermitian matrix whose $(i,j)th$ entry is $\varphi_{w}(\overrightarrow{e}_{i,j})$ if $e_{i,j}\in E(G)$, and zero otherwise. Since $A(\Phi_{w})$ is Hermitian, so all its eigenvalues are real. The spectrum of $A(\Phi_{w})$ is the spectrum of $\Phi_{w}$. If $\varphi=1$, then the corresponding positively weighted $\mathbb{T}$-gain graph is the weighted graph $(G,w)$, and is denoted by $G_{w}$. The adjacency matrix of $G_{w}$, denoted by $A(G_{w})$, is an $n\times n$ symmetric matrix with the $(i,j)$th entry $w(e_{i,j})$. Then $A(\Phi_{w})=A(\Phi)\circ A(G_{w})$, where ${}^{\prime}\circ^{\prime}$ is the Hadamard product. We establish an expression for the characteristic polynomial of $A(\Phi_{w})$, which is a generalization of the weighted Sachs formula. Let $\Phi_{w}=(G,\varphi,w)$ be a positively weighted $\mathbb{T}$-gain graph. The weight of a cycle $C$ in $G_{w}$ is defined as $w(C)=\prod\limits_{e\in E(C)}w(e)$. Now $\overrightarrow{C}$ is an oriented cycle. Then the weighted $\mathbb{T}$-gain of $\overrightarrow{C}$ is $\varphi_{w}(\overrightarrow{C})=\prod\limits_{\overrightarrow{e}\in\overrightarrow{E(C)}}\varphi_{w}(\overrightarrow{e})=\prod\limits_{\overrightarrow{e}\in\overrightarrow{E(C)}}\varphi(\overrightarrow{e})w(e)=w(C)\varphi(\overrightarrow{C})$. An elementary subgraph $H$ of $G$ is a subgraph of $G$ such that each component of $H$ is either a cycle or an edge of $G$. For an elementary subgraph $H$, $H_{e}$ denotes the set of isolated edges in $H$. The collection of all elementary subgraphs with $k$ vertices is denoted by $\mathcal{H}_{k}$. Next, we state the weighted gain Sachs formula. As the proof is similar to that of the weighted case, so we skip it. ###### Theorem 4.1 (Weighted gain Sachs formula). Let $\Phi_{w}=(G,\varphi,w)$ be a positively weighted $\mathbb{T}$-gain graph with characteristic polynomial $\chi(\Phi_{w};x)=x^{n}+a_{1}x^{n-1}+\dots+a_{n}$. Then $a_{i}=\sum\limits_{H\in\mathcal{H}_{i}}(-1)^{p(H)}2^{c(H)}w(H_{e})w(H)\prod\limits_{C\in H}\operatorname{Re}(\varphi(C)),$ (1) where $c(H),p(H)$ and $C$ denote the number of cycles, the number of components and cycle in $H$ , respectively. If we choose $\mathbb{T}$-gain graph $\Phi$ to be the underlying graph $G$, then above formula become the known Weighted Sachs formula. ###### Corollary 4.1 (Weighted Sachs formula, [6]). Let $(G,w)$ be a weighted graph with characteristic polynomial $\chi(G,w;x)=x^{n}+a_{1}x^{n-1}+\dots+a_{n}$. Then the coefficients can be expressed as $a_{i}=\sum\limits_{H\in\mathcal{H}_{i}}(-1)^{p(H)}2^{c(H)}w(H_{e})w(H),$ (2) where $c(H),p(H)$ denote the number of cycles and the number of components in $H$, respectively. Now we are ready to state two interesting results which generalize the corresponding known result for $\mathbb{T}$-gain graph and signed graph. ###### Theorem 4.2. Let $\Phi_{w}=(G,\varphi,w)$ be a positively weighted $\mathbb{T}$-gain graph. Then $\Phi_{w}$ and $G_{w}$ are cospectral if and only if $\Phi$ is balanced. ###### Proof. If $\Phi=(G,\varphi)$ is balanced, then there exists a diagonal unitary matrix $U$ such that $A(\Phi)=UA(G)U^{}$. Now, by Proposition 2.1, $\displaystyle A(\Phi_{w})=A(\Phi)\circ A(G_{w})$ $\displaystyle=UA(G)U^{}\circ A(G_{w})=U(A(G)\circ A(G_{w}))U^{}=UA(G_{w})U^{}.$ Thus $\Phi_{w}$ and $G_{w}$ are cospectral Conversely, suppose $\Phi_{w}$ and $G_{w}$ are cospectral. Let $\chi(\Phi_{w};x)=\sum\limits_{i=0}^{n}a_{i}x^{n-i}$ and $\chi(G,w;x)=\sum\limits_{i=0}^{n}b_{i}x^{n-i}$ be the characteristic polynomials of $\Phi_{w}$ and $G_{w}$, respectively. Suppose that $\Phi$ is not balanced. Then there exists a cycle of smallest length, say $k$, which is not balanced. Let $\mathcal{C}_{k}$ be the collection of all unbalanced $k$ cycles. Then, by Theorem 4.1, $b_{k}-a_{k}=2\sum\limits_{C\in\mathcal{C}_{k}}w(C).\\{1-\operatorname{Re}(\varphi(C))\\}>0,$ a contradiction. Thus $\Phi$ is balanced. ∎ The well known Acharya’s spectral criterion for the balance of signed graphs follows from Theorem 4.2. ###### Corollary 4.2 ([1, Corollary 1.1]). Let $\Psi=(G,\psi)$ be a signed graph. Then spectra of $\Psi$ and $G$ coincide if and only if $\Psi$ is balanced. ###### Proof. By taking $\varphi=\pm 1$ and $w=1$ in Theorem 4.2, we get the result. ∎ Another consequence is the following recent result about the signed graph. ###### Corollary 4.3 ([6, Theorem 2.4]). Let $\Psi=(G,\psi)$ be a signed graph and $w$ be a positively weighted function, where $\psi=\pm 1$. Then $\Psi_{w}$ and $G_{w}$ are cospectral if and only if $\Psi$ is balanced. Next, we prove one of the main results of this article. ###### Theorem 4.3. Let $\Phi_{w}=(G,\varphi,w)$ be a connected positively weighted $\mathbb{T}$-gain graph. Then the spectral radius of $\Phi_{w}$ and $G_{w}$ are equal if and only if either $\Phi$ or $-\Phi$ is balanced. ###### Proof. Suppose either $\Phi$ or $-\Phi$ is balanced. Then, it is easy to see that, the spectral radius $\Phi_{w}$ and $G_{w}$ are equal. Conversely, suppose $\rho(\Phi_{w})=\rho(G_{w})$. Let $\mu_{n}\leq\mu_{n-1}\leq\dots\leq\mu_{1}$ be the eigenvalues of $\Phi_{w}$. Then either $\rho(\Phi_{w})=\mu_{1}$ or $\rho(\Phi_{w})=-\mu_{n}$. Case 1: If $\rho(\Phi_{w})=\mu_{1}$, then, by Theorem 2.2, there exists a diagonal unitary matrix $D$ such that $A(\Phi_{w})=DA(G_{w})D^{}$. Now $A(\Phi)\circ A(G_{w})=D(A(G)\circ A(G_{w}))D^{}$. Then, by Proposition 2.1, $A(\Phi)\circ A(G_{w})=(DA(G)D^{})\circ A(G_{w})$. Define $B=(b_{ij})$ as follows: $b_{ij}$ is the inverse of the nonzero $(i,j)th$-entry of $A(G_{w})$, otherwise zero. Then $(A(\Phi)\circ A(G_{w}))\circ B=((DA(G)D^{})\circ A(G_{w}))\circ B$. Thus, by Proposition 2.1, we have $A(\Phi)=DA(G)D^{}$. Thus $\Phi$ is balanced. Case 2: If $\rho(\Phi_{w})=-\mu_{n}$, then $\mu_{n}=e^{i\pi}\rho(\Phi_{w})$. By Theorem 2.2, there exists a diagonal unitary matrix $D$, such that $A(\Phi_{w})=e^{i\pi}DA(G_{w})D^{}$. That is, $-A(\Phi_{w})=DA(G_{w})D^{}$. Then $A(-\Phi)\circ A(G_{w})=D(A(G)\circ A(G_{w}))D^{}$. By Proposition 2.1, we have $A(-\Phi)=DA(G)D^{}$. Thus $-\Phi$ is balanced. ∎ Now we present the following consequences of the above results. ###### Corollary 4.4. Let $\Phi_{w}=(G,\varphi,w)$ be a connected positively weighted $\mathbb{T}$-gain graph. Then the largest eigenvalue of $\Phi_{w}$ and $G_{w}$ are equal if and only if $\Phi$ is balanced. ###### Corollary 4.5. Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph. Then the largest eigenvalue of $\Phi$ and $G$ are equal if and only if $\Phi$ is balanced. ###### Proof. The proof follows from Corollary 4.4 by assuming $w=1$. ∎ Also Theorem 4.3 unifies the following recent results. ###### Corollary 4.6 ([6, Corollary 2.7]). Let $\Psi_{w}=(G,\psi,w)$ be a connected positively weighted signed graph. Then $\Psi$ is balanced if and only if the largest eigenvalue of $\Psi_{w}$ and $G_{w}$ coincide. ###### Proof. By taking $\varphi=\pm 1$, the result follows from Corollary 4.4. ∎ ###### Corollary 4.7 ([10, Theorem 4.4]). Let $\Phi=(G,\varphi)$ be a connected $\mathbb{T}$-gain graph. Then spectral radius of $\Phi$ and $G$ coincide if and only if either $\Phi$ is balanced or $-\Phi$ is balanced. ###### Proof. Take $w=1$ in Theorem 4.3. ∎ ###### Corollary 4.8 ((Stanić’s spectral criterion [14, Lemma 2.1])). Let $\Psi=(G,\psi)$ be a connected signed graph. Then the largest eigenvalue of $\Psi$ and $G$ coincide if and only if $\Psi$ is balanced. ###### Proof. Result follows from Corollary 4.4 by choosing $\varphi=\pm 1$ and $w=1$. ∎ ## 5 Characterizations of balanced $\mathbb{T}$-gain graphs in terms of gain distance matrices In this section, we establish two characterizations for balanced $\mathbb{T}$-gain graphs using the gain distance matrices. Let us define two complete $\mathbb{T}$-gain graphs which are obtained from gain distance matrices $D^{\max}_{<}(\Phi)$ and $D^{\min}_{<}(\Phi)$. ###### Definition 5.1. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph and $`<`$ be an order on $V(G)$. The complete $\mathbb{T}$-gain graph with respect to $D^{\max}_{<}(\Phi)$, denoted by $K^{D^{\max}_{<}}(\Phi)$, is defined as follows: keep the edges of $\Phi$ unchanged, and join non adjacent vertices $v_{i}$ and $v_{j}$ with gain $\varphi(\overrightarrow{e}_{i,j})=\varphi^{<}_{\max}(v_{i},v_{j})$ for all $v_{i},v_{j}$. Similarly $K^{D^{\min}_{<}}(\Phi)$ is defined using $D^{\min}_{<}(\Phi)$. For a $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$ with order $<$, if $D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$, then the associated complete $\mathbb{T}$-gain graphs $K^{D^{\max}}(\Phi)$ and $K^{D^{\max}}(\Phi)$ are the same, and it is denoted by $K^{D}(\Phi)$. Then the proof of the following theorem is easy to verify. ###### Theorem 5.1. For a $\mathbb{T}$-gain graph $\Phi=(G,\varphi)$, the following are equivalent: 1. (1) $K^{D}(\Phi)$ is well defined. 2. (2) $K^{D^{\max}}(\Phi)=K^{D^{\max}}(\Phi)=K^{D^{\max}_{<}}(\Phi)=K^{D^{\min}_{<}}(\Phi)=K^{D}(\Phi)$. 3. (3) $D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$, for some ordering $<$. ###### Theorem 5.2. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with vertex order $<$. Then the following statements are equivalent. 1. (i) $\Phi$ is balanced. 2. (ii) $K^{D^{\max}}(\Phi)$ is balanced. 3. (iii) $K^{D^{\min}}(\Phi)$ is balanced. 4. (iv) $D^{\max}(\Phi)=D^{\min}(\Phi)$ and associated complete $\mathbb{T}$-gain graph $K^{D}(\Phi)$ is balanced. ###### Proof. $(i)\implies(iv)$ Let $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$. Suppose $\Phi$ is balanced. Let $v_{i},v_{j}\in V(G)$. Then, by Lemma 3.1, all shortest oriented paths $v_{i}Pv_{j}$ have the same gain. Thus $\varphi_{\max}^{<}(v_{i},v_{j})=\varphi_{\min}^{<}(v_{i},v_{j})$. Therefore, $D^{\max}_{<}(\Phi)=D^{\min}_{<}(\Phi)$. By Proposition 3.1 and Corollary 3.2, $D^{\max}(\Phi)=D^{\min}(\Phi)$. Hence $K^{D}(\Phi)$ is well defined. Claim: $K^{D}(\Phi)$ is balanced. Let $v_{i}\nsim v_{j}$ in $G$ and $e_{i,j}$ be the edge joining $v_{i}$ and $v_{j}$ in $K^{D}(\Phi)$. For every oriented path $v_{i}Pv_{j}$ in $\Phi$ have the same gain. So every cycle passing through the edge $e_{i,j}$ has gain $1$. Let $T$ be a normal spanning tree of $G$. Suppose $v_{i}\nsim v_{j}$ in $G$. In $T$, joining the edge $e_{i,j}$ creates a fundamental cycle of $K^{D}(\Phi)$, say $C_{T}$. Now by previous observation, $\varphi(C_{T})=1$. Thus all the fundamental cycles in $K^{D}(\Phi)$ are neutral, and hence, by Lemma 2.1, $K^{D}(\Phi)$ is balanced. If $K^{D}(\Phi)$ is balanced, then, as $\Phi$ is a subgraph of $K^{D}(\Phi)$, so $\Phi$ is balanced. Therefore, $(iv)\implies(i),(iii)\implies(i)$ and $(ii)\implies(i)$ follow. The proofs $(iv)\implies(iii)$ and $(iv)\implies(ii)$ are easy to see. ∎ ###### Theorem 5.3. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph with vertex order $<$.Then the following statements are equivalent: 1. (i) $\Phi$ is balanced. 2. (ii) $D^{\max}(\Phi)$ is cospectral with $D(G)$. 3. (iii) $D^{\min}(\Phi)$ is cospectral with $D(G)$. 4. (iv) The largest eigenvalue of $D^{\max}(\Phi)$ and $D(G)$ are equal. 5. (v) The largest eigenvalue of $D^{\min}(\Phi)$ and $D(G)$ are equal. ###### Proof. Let $V(G)=\\{v_{1},v_{2},\dots,v_{n}\\}$, and $\Phi$ be balanced. Then by Theorem 5.2, $K^{D^{\max}}(\Phi)$ is balanced. Note that $K^{D^{\max}}(\Phi)=(K_{n},\psi)$ with $\psi(\overrightarrow{e}_{i,j})=\varphi_{\max}(v_{i},v_{j})=\varphi(v_{i}Pv_{j})$, where $v_{i}Pv_{j}$ is a shortest path in $\Phi$. Consider ${D^{\max}}(\Phi)$ as the adjacency matrix of a positively weighted $\mathbb{T}$-gain graph $(K_{n},\psi,w)$ with weight function $w:E(K_{n})\rightarrow\mathbb{R}^{+}$ is defined as $w(e_{i,j})=d(v_{i},v_{j})$, where $d(v_{i},v_{j})$ is the distance between $v_{i}$ and $v_{j}$ in $G$. Then the adjacency matrix of $(K_{n},w)$ is same as $D(G)$. By Theorem 4.2 and Theorem 5.2, $\Phi$ is balanced if and only if $D^{\max}(\Phi)$ is cospectral with $D(G)$. Thus $(i)\Leftrightarrow(ii)$. Now, by Corollary 4.4, $\Phi$ is balanced if and only if the largest eigenvalue of $D^{\max}(\Phi)$ and $D(G)$ coincide. This proves $(i)\Leftrightarrow(iv)$. The proofs of $(i)\Leftrightarrow(iii)$ and $(i)\Leftrightarrow(v)$ are similar . ∎ Both of the above characterizations extend the corresponding known characterizations [6, Theorem 3.1] and [6, Theorem 3.5] for signed graph. ###### Corollary 5.1. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph. Then $\Phi$ is balanced if and only if $D(\Phi)$ exist and it is cospectral with $D(G)$. ## 6 Distance compatible gain graphs In this final section, we establish a couple of characterizations for distance compatible $\mathbb{T}$-gain graphs. These results extend the corresponding known results for signed graph [6]. ###### Theorem 6.1. Let $\Phi=(G,\varphi)$ be any bipartite $\mathbb{T}$-gain graph. Then $\Phi$ is distance compatible if and only if $\Phi$ is balanced. ###### Proof. If $\Phi$ is balanced, by Theorem 5.3, $\Phi$ is distance compatible. Conversely, suppose $\Phi$ is distance compatible. Then, by Proposition 3.3, $\Phi$ is order-independent and $D^{\max}(\Phi)=D^{\min}(\Phi)$. Suppose that $\Phi$ is unbalanced. Since $\Phi$ is bipartite, there exists an unbalanced even cycle $C$. Let $v_{i}$ and $v_{j}$ be two diametrical vertices of $C$. Then $C$ contains two disjoint paths $v_{i}P_{1}v_{j}$ and $v_{i}P_{2}v_{j}$ of same length. Since $\varphi(C)\neq 1$ and $\varphi(\overrightarrow{C})=\varphi(v_{i}P_{1}v_{j})\varphi(v_{j}P_{2}v_{i})\neq 1$, so $\varphi(v_{i}P_{1}v_{j})\neq\varphi(v_{i}P_{2}v_{j})$. Claim: $v_{i}P_{1}v_{j}$ and $v_{i}P_{2}v_{j}$ are shortest paths between $v_{i}$ and $v_{j}$. Suppose $v_{i}P_{1}v_{j}$ and $v_{i}P_{2}v_{j}$ are not shortest paths. Let $v_{i}Pv_{j}$ be a shortest path. Then at least one of the even cycle formed by $v_{i}P_{1}v_{j}$, $v_{i}Pv_{j}$ and $v_{i}P_{2}v_{j}$, $v_{i}Pv_{j}$ is unbalanced, and has length strictly smaller than that of $C$, which is a contradiction. Since $\varphi(v_{i}P_{1}v_{j})\neq\varphi(v_{i}P_{2}v_{j})$, for any ordered vertex set $(V(G),<)$, $\varphi_{\max}^{<}(v_{i},v_{j})\neq\varphi_{\min}^{<}(v_{i},v_{j})$, a contradiction. Thus $\Phi$ is balanced. ∎ A _cut vertex_ in a graph $G$ is a vertex whose removal creates more components than the number of components of $G$. A _block_ of a graph $G$ is a maximum connected subgraph of $G$ that has no cut vertex. ###### Theorem 6.2. Let $\Phi=(G,\varphi)$ be a $\mathbb{T}$-gain graph. Then, $\Phi$ is distance compatible if and only if every block of $\Phi$ is distance compatible. ###### Proof. Let $B_{1},B_{2},\dots,B_{k}$ be the blocks of $\Phi$. Suppose every block is distance compatible. Let $s,t\in V(G)$. If $s$ and $t$ are in the same block then they are distance compatible. Suppose $s$ and $t$ are in different blocks. Without loss of generality, suppose $s$ is in $B_{1}$ and $t$ is in $B_{2}$. Then any path $sPt$ in $G$ must passes through the cut vertices $v_{i}$ and $v_{j}$ where $v_{i}$ and $v_{j}$ are in $B_{1}$ and $B_{2}$, respectively ($v_{i}$ may be same as $v_{j}$). Any shortest path $sPt$ can be decompose into $sPv_{i}\cup v_{i}Pv_{j}\cup v_{j}Pt$. Since $B_{1}$ is distance compatible, so any shortest path $sPv_{i}$ has unique gain. As the vertices $v_{i}$ and $v_{j}$ are connected by a unique path, so $\varphi(v_{i}Pv_{j})$ is unique. Proofs of the other cases are similar. Therefore, any shortest path from $s$ to $t$ has same gain. Thus $s$ and $t$ are distance compatible. Hence $\Phi$ is distance compatible. Converse is easy to verify. ∎ Let $\Phi=(G,\varphi)$ be $\mathbb{T}$-gain graph with the standard order $<$ on the vertex set. Let $s,t\in V(G)$. If $\varphi_{\max}^{<}(s,t)=\varphi_{\min}^{<}(s,t)$, then $s$ and $t$ are called distance compatible. Note that $\varphi_{\max}^{<}(s,t)=\varphi_{\min}^{<}(s,t)$ if and only if $\varphi_{\max}^{<_{a}}(s,t)=\varphi_{\min}^{<_{a}}(s,t)$, for any other vertex order $<_{a}$. Therefore, the vertices $s$ and $t$ are called _distance-incompatible_ if for some order $<_{a}$, $\varphi_{\max}^{<_{a}}(s,t)\neq\varphi_{\min}^{<_{a}}(s,t)$ holds. ###### Lemma 6.1. Let $\Phi=(G,\varphi)$ be a $2$-connected non-geodetic $\mathbb{T}$-gain graph. If $s$ and $t$ are two incompatible vertices of least distance in $G$ then there exists at least two internally disjoint shortest paths between $s$ and $t$ which have different gains. ###### Proof. Since $s,t$ are distance-incompatible and $G$ is non-geodetic, so there exist at least two shortest paths say $sP_{1}t$ and $sP_{2}t$ such that $\varphi(sP_{1}t)\neq\varphi(sP_{2}t)$. If $sP_{1}t$ and $sP_{2}t$ are internally disjoint, then we are done. Suppose $sP_{1}t$ and $sP_{2}t$ are not internally disjoint. Let $v_{1},v_{2},\dots,v_{p}$ be the common internal vertices of the paths $sP_{1}t$ and $sP_{2}t$ . Let $C_{1},C_{2},\dots,C_{r}$ be the only cycles formed by $sP_{1}t$ and $sP_{2}t$. Thus $\varphi(sP_{1}t)\varphi(tP_{2}s)=\sum\limits_{i=1}^{r}\varphi(\overrightarrow{C_{i}})\neq 1$. Then there exist a cycle $C_{j}$ which is not balanced. Let $C_{j}$ be formed by $v_{j}P_{1}v_{j+1}$ and $v_{j}P_{2}v_{j+1}$. Since $sP_{1}t$ and $sP_{2}t$ are shortest paths, so $v_{j}P_{1}v_{j+1}$ and $v_{j}P_{2}v_{j+1}$ must be shortest paths in between $v_{j}$ and $v_{j+1}$ and of same lengths. Also $\varphi(\overrightarrow{C_{j}})=\varphi(v_{j}P_{1}v_{j+1})\varphi(v_{j+1}P_{2}v_{j})\neq 1$. Thus $\varphi(v_{j}P_{1}v_{j+1})\neq\varphi(v_{j}P_{2}v_{j+1})$. Hence $v_{j}$ and $v_{j+1}$ are distance-incompatible, and distance between them in $G$ is smaller than the distance between the vertices $s$ and $t$ in $G$, a contradiction. ∎ ###### Theorem 6.3. Let $\Phi=(G,\varphi)$ be any $2$-connected non-geodetic $\mathbb{T}$-gain graph. Then $\Phi$ is distance-incompatible if and only if there is an unbalanced even cycle such that there exist two diametrically opposite vertices $s$ and $t$ which have no other smaller length path. ###### Proof. If $\Phi$ is distance-incompatible, then there exist vertices $s,t$ which are distance-incompatible and of least distance. Then by Lemma 6.1, there exists a pair of shortest disjoint paths in between $s$ and $t$ such that they have different gains. Let $C_{2l}$ be the cycle formed by the two disjoin paths. Therefore, $\varphi(\overrightarrow{C_{2l}})\neq 1$ and $s,t$ do not have any other shorter length path. 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# A Neighborhood-preserving Graph Summarization Abd Errahmane KIOUCHE Univ Lyon, Université Lyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France. E-mail<EMAIL_ADDRESS>LCSI, Ecole nationale Supérieure d’Informatique (ESI),Algeria. Julien BASTE Univ. Lille, CNRS, Centrale Lille, UMR 9189 - CRIStAL - Centre de Recherche en Informatique Signal et Automatique de Lille, F-59000 Lille, France. E-mail: <EMAIL_ADDRESS>Mohammed HADDAD Univ Lyon, Université Lyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France. E-mail<EMAIL_ADDRESS>Hamida SEBA Univ Lyon, Université Lyon 1, LIRIS UMR CNRS 5205, F-69621, Lyon, France. E-mail<EMAIL_ADDRESS> (date) ###### Abstract We introduce in this paper a new summarization method for large graphs. Our summarization approach retains only a user-specified proportion of the neighbors of each node in the graph. Our main aim is to simplify large graphs so that they can be analyzed and processed effectively while preserving as many of the node neighborhood properties as possible. Since many graph algorithms are based on the neighborhood information available for each node, the idea is to produce a smaller graph which can be used to allow these algorithms to handle large graphs and run faster while providing good approximations. Moreover, our compression allows users to control the size of the compressed graph by adjusting the amount of information loss that can be tolerated. The experiments conducted on various real and synthetic graphs show that our compression reduces considerably the size of the graphs. Moreover, we conducted several experiments on the obtained summaries using various graph algorithms and applications, such as node embedding, graph classification and shortest path approximations. The obtained results show interesting trade-offs between the algorithms runtime speed-up and the precision loss. _K_ eywords Graph compression $\cdot$ Graph summarization $\cdot$ Algorithm speed-up $\cdot$ Node embedding $\cdot$ Graph embedding ## 1 Introduction Graphs are widely used in data modeling because of their ability to represent, in a simple and intuitive way, complex relations and interactions between objects: social interactions, protein-protein interactions, chemical molecule bonds, transport networks, etc. We recall that a graph $G=(V,E)$ is a data modeling tool consisting of a set $V$ of vertices, also called nodes, and a set $E$ of edges that connect vertices. Vertices represent objects, while edges represent relationships between them. Edges can be directed, and both vertices and edges can have labels. As we are witnessing an explosion in the number of data generated and processed by our applications, it becomes important to deal efficiently with large graphs the processing of which remains a challenging issue. In fact, the amount of generated data is continuously increasing. Our basic and simple daily activities such as sending emails, surfing websites, purchasing online, and interacting via social networks, generate, on their own, a huge amount of data each day. For example, in 2019 Facebook social network had more than 2.4 Billion monthly active users with an average of 155 friendship links for each user111https://www.omnicoreagency.com/facebook-statistics/ visited July.2020. This large volume of data makes graph querying and analysis a very challenging task. However, a viable solution seems to emerge from the possibilities offered by graph summarizing. Graph summarization, also known by graph compression or simplification, is a solution that tackles scalability and performance issues when dealing with massive graph data. Beyond the reduction of the volume of data which is the main aim of compression, graph summarization looks for significant summaries that can be used, in graph analysis, without decompression. In fact, using graph summaries helps to speed-up graph algorithms so that they can efficiently run on large graphs. Compression algorithms produce smaller graphs or simpler graph representations, which can be maintained in main memory and queried and analyzed in reasonable time. Many graph algorithms, such as node embedding, node classification, recommendations, shortest path approximation and graph comparison, are based on the neighborhood information available for each node. Finding this information may be difficult in practice as dealing with all the neighbors for each node requires all the edges (links) of the graph to be processed, which is time and space consuming. This motivated us to introduce a graph compression that controls the size of the preserved neighborhood of vertices in the computed summary. So, in this paper, we propose a new graph compression which retains only a user-specified proportion of the neighbors of each node to reduce the size of the graph while preserving neighborhood queries. The main idea is to sparsify the graph by removing edges, while ensuring that a predefined proportion of the neighbors of each node is included in the set of $t$-hops neighbors of the node in the compressed graph ($t\geq 1$). The main advantages of our neighborhood-preserving compression are: * • Reduction in storage space of the graph: our compression can decrease drastically the number of edges in the graph, thus allowing the compressed graph to be loaded into main memory. The size of the compressed graph can be controlled by adjusting the proportion of the preserved neighborhood’s information. * • Fast Approximation of graph algorithms: Many graph algorithms, such as community detection, shortest path lengths and graph comparison, are mainly based on the neighborhood information of the graph nodes. These algorithms cannot efficiently run on large graphs since they require all the edges of the original graph to be loaded in main memory. Since our compression produces a smaller graph that maintains the principal neighborhood’s information, it can be used to allow these algorithms to handle large graphs and run faster while providing good approximations of the original results. * • User-controlled trade-off between compression ratio and information loss: with our compression, the user can control the size of the compressed graph by adjusting the amount of information loss that can be tolerated. This is a very useful property, since the amount of tolerated information loss differs significantly from one application to another, and the desired size of the compressed graph depends mainly on the available memory. The remainder of this paper is organized as follows: Section 2 reviews related works on graph compression methods and their applications. Section 3 formally defines the problem of neighborhood-preserving graph compression and studies its complexity. Then, Section 4 provides a description of the algorithms used to implement this compression. Section 5 presents the results obtained through the extensive experiments we undertook to evaluate the compression approach, as well as the usefulness of the obtained summaries. Finally, Section 6 concludes the paper and points out some research perspectives. ## 2 Related Work Graph compression is attracting increasing interest in various domains and applications [1]. The aim of graph compression, considered here, is to compute a graph summary that retains all or part of the original graph properties, thus allowing use of the summary instead of the original graph in certain applications. The obtained summary can be either a graph that is simpler or smaller than the original graph, or any other data structure that is more compact or is simpler to use than the original graph. Compression algorithms can be classified in three main categories according to how they simplify the input graph: (1) sampling, (2) sparsification, and (3) regularity encoding. Sampling and sparsification based methods generate lossy graph compression and their results are generally a graph. Regularity encoding based methods allow having lossy, as well as loss-less, graph summaries, either as graphs or other data structures: 1. 1. Graph sampling [2, 3] consists in using a fraction of the graph to make inferences about the whole dataset. It is generally used for dynamic graphs for which a sample at time $t$ is a likely representation of the graph. It is also used with very large graphs, such as protein to protein interactions, where dealing with the whole graph is too slow. Several graph sampling methods are proposed in the literature. They generally start with a set of initial vertices (and/or edges) which can be empty and expand the sample based on a specific algorithm such as graph exploration and traversal algorithms. As examples, Breadth-First sampling is used for social network analysis [4] and graph mining [5]. In [6], the authors apply a traversal based sampling that utilizes only the local information of nodes, combined with estimated values of a set of properties, to guide the sampling process and extract tiny samples that preserve the properties of the graph and closely approximate their distributions in the original graph. Random walks based methods are also largely applied in large-scale graph analysis [7, 8, 9]. Frontier sampling, an edge sampling method using multidimensional random walkers, is used to estimate the degree distributions and the global clustering coefficient in [10]. In [11], the authors approximate betweenness centrality based on a sampled set of shortest paths. 2. 2. Graph sparsification stands for the methods that compute a sparse subgraph of the input graph, which preserves some of its properties such as cuts or shortest paths [12]. Graph sparsification methods can also rely on sampling as a tool to achieve sparsification. Given a social graph and a log of past propagations, the authors of [13] prune the network to a prefixed extent, while maximizing the likelihood of generating the propagation traces in the log. A similar work is described in [14]. It tackles the problem of simplifying a graph, while maintaining the connectivity recorded in a given set of observed activity traces represented by a set of DAGs (or trees) with specified roots. The problem consists in selecting a subset of arcs in the graph so as to maximize the number of nodes reachable in all DAGs by the corresponding DAG roots. This is a cover-maximization problem that the authors bring to a problem of minimizing a submodular function under size constraints and using an algorithm introduced in [15] to solve it. 3. 3. Regularity encoding based methods search for regularities within the graph structure, i.e., particular patterns or just repetitive patterns, and then encode these regularities so as to obtain a compact representation of the graph. Several approaches are proposed in the literature and differ by both the kind of considered regularities and how these regularities are encoded within the computed summary. Some methods of this class consist in merging or combining similar nodes, or subgraphs into super-nodes, and similar edges into super-edges. Others work directly on the adjacency matrix of the graph using for example $k2$-trees [16]. In [17], the authors propose a summarizing approach that iteratively aggregates similar nodes, i.e., those that have the greatest number of common neighbors. This aggregation is controlled by an objective function that represents the cost of the compressed output graph and is defined according to the principle of Minimum Description Length (MDL) [18]. The graph is encoded with a summary and a set of correcting edges. These corrections, applied to the summary, enable the initial graph to be reconstructed. Identifying vertices with a similar neighborhood is a well- studied topic known as modular decomposition of graphs [19, 20], which aims to highlight groups of vertices that have the same neighbors outside the group. These subsets of vertices are called modules. Modular decomposition is used in [21] to compress a graph and compute its exact list of triangles using solely the computed summary. The compression consists in considering each module as a super-node. Several works such as [22] and [23] take advantage of the regularities of the web graph structure, such as locality and similarity properties, to compress its adjacency lists and reduce the number of bits needed to encode a link. In [24, 25], the authors compress graphs using MDL on a predefined vocabulary of substructures. In [26], graphs are compressed by recursively detecting repeated substructures and representing them through grammar rules. In [27], the authors use a clustering algorithm to partition the original set of vertices into a number of clusters, which will be super- nodes connected by super-edges to form a complete weighted graph. The super- edge weights are the edge densities between vertices in the corresponding super-nodes. The goal is to produce a summary that minimizes the reconstruction error of the original graph. In [28], the authors merge into super-nodes graph vertices that have common neighbors so that the obtained compression ensures that the efficiency of a given task does not drop below a user-specified threshold. In [29], the authors accelerate node grouping using a divide and conquer approach that allows parallel node merging. In [30], the authors use tensor decomposition to group nodes of an evolving graph according to their connectivity patterns into super-nodes. In [31], the authors address the problem of preserving node attributes while summarizing diffusion networks. They propose a sub-quadratic parallelizable algorithm that finds the best set of candidate nodes and merges them to construct a smaller network of super-nodes that ensures similar diffusion properties to the original graph. In [32], the authors use MDL to measure motif relevance based on motif capacity to compress a graph. In [33], the authors incrementally compute a summary of an evolving graph using frequent patterns and MDL principle, combined with a set of operations (merge, split, etc.) on the patterns, in order to provide changes that have occurred in the data since the previous state. In [34], the authors hybrid regularity encoding with sparsification by using both node grouping and edge sparsification. This allows optimizing the size of the obtained summary and the graph reconstruction error. It is interesting to note that few works explore the usefulness of the computed summaries beyond simple neighborhood or reachability queries. Summaries obtained by graph sampling are used to estimate graph parameters and are rarely used as input for graph applications. Most regularity-encoding based methods do not investigate this issue at all. Graph sparsification methods are generally designed for specific applications because it is difficult to have lossy summaries that can be used in several kind of graph applications. By targeting neighborhood information in our compression and allowing to control the amount of information loss in the computed summary, we aim to be able to use our summaries in a variety of graph applications. In fact, several graph applications, such as node embedding, node classification, recommendations, etc. are based on the availability of node neighborhood information. In the remainder of the paper, we show that controlling the amount of this information in the computed summary allows to reach good trade- off between algorithm speed-up and precision loss when using the summary as input instead of the original graph. ## 3 A Neighborhood-preserving graph compression In this section, we explore a new graph sparsification method that aims to control the amount of neighborhood information available for each node in the graph. Our goal is to compute a graph summary that can be used instead of the original graph in several applications. ### 3.1 Problem Statement Let $t\geq 1$ be a positive integer. The main idea of neighborhood-preserving graph compression is to sparsify the input graph by removing edges, while ensuring that, for all $1\leq i\leq t$, a proportion $p(i)$ of the neighbors of each node $v$ is included in the set of the $i$-hops neighbors of $v$ in the compressed graph. We denote such compression by $(p,t)$-compression where: * • $p:\mathbb{N}^{}\to[0,1]$ is a monotonically increasing function, which represents the proportion of each node’s original neighbors that must be retrieved in its $i$-hops neighborhood in the compressed graph. • $t$ : is the minimum value for which $p$ reaches its maximal value i.e., $p(x)=p(t),\forall x\geq t$. More formally, given an undirected graph $G=(V,E)$, where $V$ is the set of vertices and $E$ is the set of edges, a $(p,t)$-compression of $G$ is defined as follows: ###### Definition 1 Let $t$ be an integer and $p:\mathbb{N}\to[0,1]$ be a monotonically increasing function satisfying $p(x)=p(t)$, $\forall x>t$. A _$(p,t)$ -compression_ of a graph $G=(V,E)$ consists in finding a subgraph $G_{c}=(V_{c},E_{c})$ of $G$ such that $V_{c}=V$, $E_{c}\subseteq E$, and, for each $0<x\leq t$ and each $v\in V$, $\left\|N_{G}^{1}(v)\cap N_{G_{c}}^{x}(v)\right\|\geq\left\|N_{G}^{1}(v)\right\|p(x)$, where $N_{G}^{x}(v)$ is the set of all $x$-hop neighbors of $v$ in $G$. In other words, the compressed graph $G_{c}$ contains less edges than $G$ and preserves a given amount (equal to $p(t)$) of the original neighbors for each node. In fact, it is required that, for each $0<x\leq t$, a proportion $p(x)$ of the original neighbors must be accessible within a maximum of $x$ hops in $G_{c}$ using a simple BFS traversal with depth $x$. Figure 1 illustrates an example of our compression in which $50\%$ of the original neighbors of each vertex are preserved in the compressed graph and are reachable within maximum $2$ hops. The resulting compressed graph is $30\%$ smaller than the original one. (a) (a) Original graph (b) (b) Compressed graph Figure 1: $(p,2)$-compression of Zachary’s karate club network [35] where $p(1)=0.5$ and $p(2)=1$ for each node. With $(p,t)$-compression, function $p$ aims to control the loss of neighborhood information at each neighborhood depth. It is obvious that the smaller the preserved proportions, the better the compression ratio, and vice- versa. As regards parameter t, the higher it is, the bigger the stretch factor of the compressed graph, and vice-versa. The following corollary gives a lower bound of the size of the compressed graph. ###### property 1 For any $(p,t)$-compression, the number of edges $\|E_{c}\|$ of the compressed graph satisfies the following inequality: $\|E\|p(1)\leq\|E_{c}\|$ ###### proof 1 According to the handshaking theorem, we have $\sum_{v\in V}deg(v)=2\|E\|$. Since at least a proportion equal to $p(1)$ of the original neighbors of each node must be kept in the compressed graph, we have $\sum_{v^{{}^{\prime}}\in V_{c}}deg(v^{{}^{\prime}})\geq\sum_{v\in V}deg(v)p(1)=2\|E\|p(1)$, thus $\|E_{c}\|\geq p(1)\|E\|$. It is interesting to note that spanners [36] are special cases of $(p,t)$-compression. Given a graph $G$, possibly edge-weighted, a _graph spanner_ (or _spanner_ for short) is a subgraph $G^{\prime}$ which preserves lengths of shortest paths in $G$ up to a multiplicative and/or additive error. A $t$-spanner is a subgraph $G^{\prime}$ such that the distance between two vertices in $G^{\prime}$ is at most $t$ times the distance between the same two vertices in $G$. Thus, a $t$-spanner is a particular $(p,t)$-compression that could be defined such that $p(i)=0$ for every positive integer $i<t$ and $p(i)=1$ for $i\geq t$. In other words, a $t$-spanner is a $(p,t)$-compression whose proportion function $p(x)$ is the Shifted Unit Step Function $u(x-t)$. From a structural point of view, Figure 2 gives an illustration of a 2-spanner of the Diamond graph (Figure 2.(b)) a 2-spanner of the Diamond graph which is also a $(p,2)$-compression where $p(1)=\frac{1}{2}$ and $p(2)=1$ (Figure 2.(c)) and a $(p,2)$-compression where $p(1)=\frac{1}{3}$ and $p(2)=\frac{2}{3}$ which is not a 2-spanner (Figure 2.(d)). Figure 2: a Diamond graph and some of its compressions. We can see in this figure that with a $2$-spanner, the left-most vertex is only connected to $\frac{1}{3}$ of its original neighbors in the Diamond graph (see Figure 2 (b)), while all the vertices keep at least half of their original neighbors with a $(p,t)$-compression (see Figure 2 (c)). ### 3.2 NP-hardness and hardness of approximation Peleg and Schäffer [36] showed that, given an unweighted graph $G$, and integers $t\geq 2$, $m\geq 1$, determining if $G$ has a $t$-spanner containing $m$ or fewer edges is NP-complete, even when $t$ is fixed to be $2$. The reduction is from the edge dominating set problem on bipartite graphs. Since a $t$-spanner is a particular $(p,t)$-compression of the graph $G$, we can deduce the following result: ###### theorem 1 Finding the optimal (smallest) compressed graph satisfying the $(p,t)$-compression constraints for $t\geq 2$ is an NP-Hard problem. Another work giving us an alternative proof is Cai’s [37]. He showed that for any fixed $t\geq 2$, the minimum $t$-spanner problem is NP-hard, and for $t\geq 3$, the problem is NP-hard even when restricted to bipartite graphs. The reduction is from the 3-SAT problem. Dinitz et al. [38] show that for $t\leq 3$, and for all $\epsilon>0$, the $t$-spanner problem cannot be approximated with a ratio better than $2^{(log^{1-\epsilon}n)/k}$ unless $NP\subseteq BPTIME(2^{polylog(n)})$. This implies the same inapproximability result for $(p,t)$-compression. Concerning the best known approximation ratio, Elkin and Peleg [39] propose approximation algorithms with a sublinear approximation ratio, and study certain classes of graphs for which logarithmic approximation is feasible. It is also shown in [40] that for $t=2$, the $t$-spanner problem admits an $O(log(n))$ approximation. All these results strongly indicate that finding better or even equivalent approximations for the $(p,t)$-compression problem will be a hard task. In particular, finding a better result should begin with finding better approximation than $O(log(n))$ for the $t$-spanner problem. ### 3.3 Integer linear programming formulation In the following, we provide an integer linear programming formulation of our problem. Given an input graph $G=(V,E)$, we denote by $\mathcal{W}$ the set of all paths in $G$. Given $e=\\{u,v\\}\in E$, we denote by $\mathcal{W}_{uv}$ the set of all paths in $\mathcal{W}$ from $u$ to $v$. Note that the graph is undirected and so $\mathcal{W}_{uv}$ also corresponds to the paths from $v$ to $u$. Moreover, given $i\in\mathbb{N}$ we denote by $\mathcal{W}_{uv}^{i}$ the set of all paths of $\mathcal{W}_{uv}$ of size at most $i$. We then define the used variables: * • $x_{e}$, for each $e\in E$, denotes whether $e\in E$ is selected ($x_{e}=1$) or deleted ($x_{e}=0$). * • $f_{w}$, for each $w\in\mathcal{W}$, is such that $f_{w}=0$ if at least one edge $e$ of $w$ is such that $x_{e}=0$. Note that we can have a path $w\in\mathcal{W}$ such that every edge $e$ of the path is such that $x_{e}=1$ but still have $f_{w}=0$. We can now write the integer linear programming equation: $\displaystyle\min\quad$ $\displaystyle\sum_{e\in E}x_{e}$ (1) s.t. $\displaystyle f_{w}\leq x_{e}$ $\displaystyle\forall w\in\mathcal{W},e\in E:~{}e\in w$ (2) $\displaystyle\sum_{w\in\mathcal{W}_{uv}}f_{w}\leq 1$ $\displaystyle\forall uv\in E,i\in\mathbb{N}$ (3) $\displaystyle\sum_{v\in N(u)}\sum_{w\in\mathcal{W}_{uv}^{i}}f_{w}\geq p(i)\|N(u)\|$ $\displaystyle\forall u\in V,i\in\mathbb{N}$ (4) $\displaystyle x_{e}\in\\{0,1\\}$ $\displaystyle e\in E$ (5) $\displaystyle f_{w}\in\\{0,1\\}$ $\displaystyle w\in\mathcal{W}$ (6) The variable $f_{w}$ can be seen as a flow from the source to the sink. (2) ensures that if a path $w\in\mathcal{W}$ uses a removed edge $e$ (i.e., $e$ is such that $x_{e}=0$), then the flow $f_{w}=0$. Using (3), we know that for each $uv\in E$ there exists at most one path $w\in\mathcal{W}_{uv}$ such that $f_{w}=1$. By intuition, we assume that we took the path $w\in\mathcal{W}_{uv}$ of shortest length that is still available in the remaining graph after removing the edges $e\in E$ such that $x_{e}=0$. Then condition (4) ensures that the number of neighbors of a vertex $u$, which are now at distance at most $i$ in the new graph, is at least the proportion given by $p(i)$. ## 4 Algorithms and approximations In this section, we present four algorithms for finding the $(p,t)$-compression of an input graph $G$. Since finding the optimal $(p,t)$-compression is NP-Hard and cannot be resolved in polynomial time, we propose polynomial time approximations (sub-optimal algorithms). Algorithm 1 gives the basic implementation of our compression. It has the advantage of simplicity and speed. Algorithm 1 takes as input a simple graph to compress $G=(V,E)$, the compression parameters $p$ and $t$ and an order $E_{o}$ for processing the edges of the input graph. $E_{o}$ is by default a random ordering of the vertices. The idea is to process the edges of the initial graph in the order $E_{o}$. The algorithm processes the edges of $G$ incrementally as follows: If an edge $e$ can be removed from $G$ without violating the neighborhood preservation constraints, the algorithm does not keep this edge in the summary. Otherwise, the algorithm keeps the edge in the summary. Assume that the average branching factor ( Out/In degree) of the graph is equal to $b$, then average time complexity of Algorithm 1 is $O(\|E\|b^{t})$. Data: $G=(V,E)$ a simple Graph, $t$ an integer, $p$ a monotonically increasing function $p:\mathbb{N}\to[0,1]$, $E_{o}$ a possible order of the graph edges Result: $G_{c}=(V_{c},E_{c})$ a compressed graph 1 $//$ Initialization Step ; 2 $G_{c}=(V_{c},E_{c})\leftarrow(V,\emptyset)$; 3 $G^{\prime}=(V^{\prime},E^{\prime})\leftarrow(V,\emptyset)$; 4 for _$e=(u,v)\in E_{o}$_ do 5 $E^{\prime}\leftarrow E^{\prime}\cup\\{(u,v)\\}$; 6 $N_{G^{\prime}}^{1}(u)\leftarrow$ direct neighbors of node $u$ in $G^{\prime}$; 7 $N_{G^{\prime}}^{1}(v)\leftarrow$ direct neighbors of node $v$ in $G^{\prime}$; 8 insert $\leftarrow False$; 9 for _$i=1$ to $t$_ do 10 $N_{Gc}^{i}(u)\leftarrow$ neighbors of node $u$ in $G_{c}$ within at most $i$-hops; 11 $N_{Gc}^{i}(v)\leftarrow$ neighbors of node $v$ in graph $G_{c}$ within at most $i$-hops; 12 if _$\|N_{Gc}^{i}(u)\cap N_{G^{\prime}}{1}(u)\| <p(i)\|N_{G^{\prime}}^{1}(u)\|$ or $\|N_{Gc}^{i}(v)\cap N_{G^{\prime}}^{1}(v)\|<p(i)\|N_{G^{\prime}}^{1}(v)\|$_ then 13 insert $\leftarrow True$; 14 Break; 15 16 end if 17 18 end for 19 if _insert_ then 20 $E_{c}\leftarrow E_{c}\cup\\{(u,v)\\}$; 21 22 end if 23 24 end for Algorithm 1 Basic Algorithm We note that different edge orderings lead to different compression performances. Therefore, in order to improve the compression performance of our algorithm, we propose in the three following subsections, 3 sub-optimal algorithms which are based on the basic algorithm and try to find a near optimal edge processing order. ### 4.1 Linear programming We provide in Section 3.3, an optimal integer linear programming formulation of $(p,t)$-compression. However, such a resolution is NP-hard to solve, so we use the standard tricks consisting in relaxing the problem. For this, we keep the same formulation but allow the values $x_{e}$, $e\in E$, and $f_{w}$, $w\in\mathcal{W}$, to be any real values between $0$ and $1$. As we only have to consider paths of a length at most $t$, we have a polynomial number of variables (the degree of which depends on the fixed value $t$). The average number of variables is of the order $O(\|E\|+\|V\|b^{t})$, where $b$ is the average branching factor of the graph. This resolution provides a value for each $x_{e}$, $e\in E$. The interpretation we give to this resolution is that the higher the value of $x_{e}$, the more likely we want to keep $e$ in our solution. In reverse, the lower the value of $x_{e}$, the more likely we want to remove the edge $e$. Thus, we can use the values of $x_{e}$, $e\in E$, in order to obtain an ordering for the edges and give this ordering to the basic compression algorithm (see Algorithm 2). Since this linear problem is solvable in polynomial time, the time complexity of Algorithm 2 is $O(poly(\|E\|+\|V\|b^{t}))$. Data: $G=(V,E)$ a simple Graph, $t$ an integer, $p$ a monotonically increasing function $p:\mathbb{N}\to[0,1]$ Result: $G_{c}=(V_{c},E_{c})$ a compressed graph 1 $//$ Computing the greedy edge order $E_{go}$ ; 2 3Solve the LP Relaxed problem to compute the edge scores $x_{e}$; 4 5$E_{go}\leftarrow$ sort edges E in descending order according to their score $x_{e}$; 6 7$G_{c}\leftarrow$ Basic Algorithm ($G$,$t$,$p$,$E_{go}$); Algorithm 2 LP Algorithm ### 4.2 Greedy order based on edge connectivity Computing the LP-based order is time-consuming for large graphs according to its time complexity. So, we propose in this subsection another edge ordering that can be computed much faster than the LP order. The idea is to first process the edges with a high centrality value. The centrality we consider here is a relaxation of local edge betweenness defined in [41]. An edge with a high edge betweenness centrality represents a bridge-like connector between two parts of a network, the removal of which may affect the shortest paths between them. The local edge betweenness of an edge $e$ is the number of shortest paths running along $e$, the length of which is less than or equal to some constant $t$. In our relaxation, we consider all simple paths of a length at most $t$, i.e., not necessarily shortest paths. Thus, we compute for every edge $e$ a centrality score $s(e)$ according to Equation 7. In Equation 7, $\sigma_{t}(u,v\|e)$ is the number of simple paths from $u$ to $v$ of length $\leq t$ that pass through the edge $e$. Once all scores are computed, we sort the edges in descending order according to their score $s(e)$ and pass the obtained order as input to the basic algorithm (See Algorithm 3). The average time complexity of Algorithm 3 is $O((\|E\|+\|V\|b^{t})log(\|E\|+\|V\|b^{t}))$. $s(e)=\sum_{(u,v)\in E}\sigma_{t}(u,v\|e)\ \forall(u,v)\in E$ (7) Data: $G=(V,E)$ a simple Graph, $t$ an integer, $p$ a monotonically increasing function $p:\mathbb{N}\to[0,1]$ 1 Result: $G_{c}=(V_{c},E_{c})$ a compressed graph 2 $//$ Computing the greedy edge order $E_{go}$ ; 3 4for _$e\in E$_ do 5 compute the score s(e) using Equation 7; 6 7 end for 8 9$E_{go}\leftarrow$ sort the edges of $G$ in descending order according to their score $s(e)$; 10 11$G_{c}\leftarrow$ Basic Algorithm ($G$,$t$,$p$,$E_{go}$); Algorithm 3 Greedy Algorithm based on edge connectivity (EC) ### 4.3 Sub-optimal order based on Simulated Annealing In the previous two subsections, we have proposed two greedy edge orderings to improve compression performance. However, the drawback of these two solutions is that they are more time-consumingthan Algorithm 1 with a random edge ordering, as we will reveal in the next section with the experimental evaluation. Moreover, the computation time cannot be controlled by the user since the computation of both orderings, i.e., the LP ordering and the greedy ordering based on edge connectivity, cannot be suspended, and we need to go to the end of the calculation. To overcome this problem, we propose a third algorithm based on Simulated Annealing (SA) [42]. The advantage of this solution is that the computation time can be controlled by the user by adjusting the number of SA iterations. SA is an optimization scheme that allows efficient search space exploration by accepting, with a given probability, worst solutions to avoid a premature convergence [42]. SA for $(p,t)$-compression is illustrated in Algorithm 4. The initial state of the algorithm is a random order of edges. Then, in each iteration, the algorithm makes a slight modification to edge order by performing two permutations of two elements in the vector representing the order and recomputes the cost of the new solution. If the new order is better, the algorithm keeps the order. Otherwise, the algorithm keeps it with a probability which increases over the iterations (see line 19 of Algorithm 4). Data: $G=(V,E)$ a simple Graph, $t$ an integer, $p$ a monotonically increasing function $p:\mathbb{N}\to[0,1]$, $N$ an integer (Number of iterations), $T_{0}$ a double ( Initial temperature), $\alpha$ a double ( decreasing factor) Result: $G_{c}=(V_{c},E_{c})$ a compressed graph 1 2$S\leftarrow$ Random order of $E$; 3 $T\leftarrow T_{0}$; 4 $G_{t}(V_{t},E_{t})\leftarrow$ Basic Algorithm( $G$,$t$,$p$,$S$); 5 $C_{best}\leftarrow\|E_{t}\|$; 6 $C_{S}\leftarrow\|E_{t}\|$; 7 for _$i=1$ to $N$_ do 8 $S_{2}\leftarrow$ Perturbing $S$ by swapping the order of two random edges; 9 $G_{t}(V_{t},E_{t})\leftarrow$ Basic Algorithm($G$,$t$,$p$,$S_{2}$); 10 11 if _$\|E_{t}\| <C_{best}$_ then 12 $E_{best}\leftarrow S$; 13 $C_{best}\leftarrow\|E_{t}\|$; 14 15 end if 16 if _$\|E_{t}\| <COST_{S}$_ then 17 $S\leftarrow S_{2}$; 18 $C_{S}\leftarrow\|E_{t}\|$; 19 20 end if 21 else 22 $r\leftarrow$ random number between $0$ and $1$; 23 if _$\exp(\frac{C_{S}-\|E_{t}\|}{T}) >r$_ then 24 $S\leftarrow S_{2}$; 25 $COST_{S}\leftarrow\|E_{t}\|$; 26 27 end if 28 29 end if 30 $T\leftarrow\alphaT$; 31 end for 32$G_{c}\leftarrow$ Basic Algorithm ($G$,$t$,$p$,$E_{best}$); Algorithm 4 $(p,t)$ Compression based on simulated annealing ## 5 Experimental Analysis In this section, we present an experimental analysis of our compression. First, we evaluate the approximation algorithms provided to compute the compression. Then, we provide an analysis of the sensitivity of the compression to parameters $p$ and $t$. Finally, we evaluate its effectiveness on several tasks such as graph properties estimation, node embedding and whole graph embedding. All the experiments are carried out on an Intel core $i7$ processor with $64$ Gigabytes of memory. ### 5.1 Evaluation of approximation algorithms In this subsection, we present a comparative experimental study of the four proposed approximations of $(p,t)$-compression. For this, we launched the 4 approximations, i.e., the basic algorithm with random edge ordering (Algorithm 1), the LP approximation (Algorithm 2), the EC approximation (Algorithm 3), and the SA approximation (Algorithm 4), on 3 families of synthetic graphs the properties of which are given in Table 1. Table 1: Characteristics of the synthetic graph families Name \| number of graphs \| $\|V\|$ \| $\|E\|$ ---\|---\|---\|--- SYNTHETIC 1 \| $30$ \| $20$ \| $60$ SYNTHETIC 2 \| $30$ \| $50$ \| $350$ SYNTHETIC 3 \| $30$ \| $100$ \| $1.4K$ We use the following compression parameters $t=2$ , $p(1)=0.0$ and $p(2)=0.5$. For a reliable and accurate comparison, we carried out around thirty tests on each family of graphs for each algorithm. The results of the comparison are depicted in Table 2. Note that the user configuration of the SA is $T_{0}=10$, $N=1000$ and $\alpha=0.99$. We notice that the two greedy algorithms LP and EC, and the SA algorithm outperform the basic algorithm with a random ordering of edges in terms of compression performance. The results clearly show that the greedy (EC) and the SA algorithms are the best algorithms. The greedy (EC) algorithm seems really interesting and offers the best trade-off between compression performance and runtime. However, all the approximations are still much slower than the basic algorithm with a random order of edges. Therefore, we recommend using the basic algorithm for large graphs. Table 2: Evaluation of the approximation algorithms Dataset \| Basic \| Greedy ( LP) \| Greedy ( EC) \| SA ---\|---\|---\|---\|--- Avg $\|E_{c}\|$ \| time \| Avg $\|E_{c}\|$ \| time \| Avg $\|E_{c}\|$ \| time \| Avg $\|E_{c}\|$ \| time SYNTHETIC 1 \| 28 \| 0.001 \| 25.24 \| 0.02 \| 23.55 \| 0.01 \| 21.56 \| 0.5 SYNTHETIC 2 \| 121.63 \| 0.008 \| 113.26 \| 2.5 \| 105.66 \| 0.02 \| 105.9 \| 5.2 SYNTHETIC 3 \| 367.03 \| 0.05 \| 354.36 \| 212 \| 323.23 \| 0.09 \| 340.4 \| 40 ### 5.2 Impact of the compression parameters In this series of experiments, we study the effect of parameters $p$ and $t$ on compression performance. To this end, we evaluate our compression using two metrics: the compression runtime measured in seconds and the compression ratio that represents the ratio of the number of deleted edges over the total number of edges (see Equation 8). $compression\;ratio=\frac{\|E\|-\|E^{{}^{\prime}}\|}{\|E\|}$ (8) Note that higher is the compression ratio better is the storage space gain ensured by the compression. For these experiments and all the following ones we use real graph datasets. Table 3 gives the main characteristics of these datasets. Table 3: Characteristics of the real datasets used in our experiments Name \| number of graphs \| $\|V\|$ \| $\|E\|$ ---\|---\|---\|--- BLOG-CATALOG \| $1$ \| $10.31K$ \| $333.98K$ CA-ASTROPH \| $1$ \| $18.77K$ \| $198.11K$ CA-HEPTH \| $1$ \| $9.8K$ \| $25.9K$ COLLAB \| $5000$ \| $372.5K$ \| $49.1M$ ENZYMES \| $600$ \| $19.5K$ \| $74.6K$ FLICKR \| $1$ \| $80.51K$ \| $5.89M$ PROTEINS \| $1113$ \| $43.5K$ \| $162.1K$ Table 4 gives the compression ratio obtained by our compression on the CA- AstroPh dataset, while varying the neighborhood preservation proportion $p$. As expected, the compression ratio decreases as the preserved proportion of neighborhood increases and vice-versa. Most of the values of the compression ratio obtained with the various combinations of parameters are satisfactory. In addition, we remark that the compression ratio range is wide (from $7\%$ to $75\%$) which confirms the possibility of controlling effectively the trade- off information loss/compression ratio using parameters $p$ and $t$. Furthermore, we set $p(t)=1$ in all experiments, which means that the whole initial neighborhood of each node can be retrieved in a neighborhood of radius $r=t$ at maximum. This ensures that reachability queries are fully preserved for all vertices. The choice of the best combination of parameters depends essentially on the nature of the graph to be compressed and the user needs. Particularly, for this example, the combinations $(0.5,1)$ and $(0.7,1)$ seem really interesting. Table 4: Compression ratio of the Ca-AstroPh dataset with different combinations of parameters $p$ and $t$ $t$ \| $p(1)$ \| $p(2)$ \| $p(3)$ \| compression ratio ---\|---\|---\|---\|--- $2$ \| $0.2$ \| $1.0$ \| - \| $58.13\%$ $0.5$ \| $1.0$ \| - \| $45.82\%$ $0.7$ \| $1.0$ \| - \| $26.39\%$ $0.9$ \| $1.0$ \| - \| $7.43\%$ $3$ \| $0.0$ \| $0.2$ \| $1.0$ \| $75.00\%$ $0.2$ \| $0.5$ \| $1.0$ \| $71.50\%$ $0.5$ \| $0.7$ \| $1.0$ \| $46.73\%$ $0.7$ \| $0.9$ \| $1.0$ \| $26.43\%$ The curves depicted in Figure 3 show the runtime and the compression ratio as a function of the value of $t$ where $p(0<x<t)=0$ and $p(t)=1$. Note that this combination of parameters gives a particular type of subgraphs called $t$-spanners [36]. We notice that the compression ratio grows slowly and starts to level off from $t=5$. However, the execution time increases exponentially and rapidly. This is due to the complexity of the compression, which is of the order $O(\|E\|b^{t})$ in the average case. Although spanners give good compression ratios on this dataset ranging from $58\%$ up to $85\%$, they do not allow good control of the trade-off between information loss and compression ratio. Indeed, for this dataset, spanners give a control margin of $(85\%-58\%=27\%)$ for a maximal stretch factor $t=6$, which represents a significant loss of neighborhood information. However, with $(p,t)$-compression we get a larger control margin of $(75\%-8\%=67\%)$ with a maximal stretch factor $t=3$. This confirms once again the efficiency and usefulness of our compression and its parameter $p$ when compared to spanners. Figure 3: Compression performance of t-spanners on the Ca-AstroPh dataset ### 5.3 Applications of $(p,t)$compression Several graph algorithms are based on the availability of neighborhood information of nodes. Our first motivation is to be able to use these kinds of algorithms directly on the compressed graphs. So, the purpose of these experiments is to show the effectiveness of our compression in terms of speeding-up for such graph algorithms, while handling large graphs and providing good approximations of the original results. For this, and for all the following experiments, we use the datasets presented in Table 3 and compute two new metrics in addition to the compression ratio namely: • Speed-up factor: the ratio between the algorithm run-time on the original graph and its run-time on the compressed graph. he higher the speed-up factor, the faster the graph algorithm on the compressed graph. * • Performance loss: the difference between the performance metric value on the original graph and the performance metric value on the compressed graph. For example, for a classification task the performance loss is the difference between the accuracy on the original graph and the accuracy on the compressed graph.The smaller the performance loss, the better the approximation of the graph properties on the compressed graph. #### 5.3.1 Shortest paths approximation The most suitable application for our compression is the approximation of the shortest paths between all nodes, since every $(p,t)$-compression where $p(t)=1.0$ preserves all the connectivity properties between the nodes of the graph, by stretching all the connecting paths by a factor equal to $t$ in the worst case. In this experimental phase, we compressed three unweighted undirected graphs with the following combination of parameters $t=2$, $p(1)=0.5$, and $p(2)=1.0$. Then, we applied the BFS (Breadth First Search) algorithm to compute all shortest paths between all nodes. Table 5 summarizes the obtained compression ratio and the shortest path speed-up obtained on three datasets: CA-ASTROPH, CA-HEPTH and BLOG-CATALOG. We notice that our compression saves a considerable amount of storage space (compression ratio $>31\%$) while approximating faster (speed-up ranges from $1.06$ to $1.49$) the shortest path lengths for the 3 chosen datasets. Indeed, reducing the number of edges of the graphs reduces the run-time of the shortest paths computed by the BFS algorithm, which is of complexity $O(\|V\|(\|E\|+\|V\|))$. This speed-up is more noticeable for denser graphs. Table 5: Speeding-up all shortest paths computation Dataset \| Space gain \| Speed-up ---\|---\|--- CA-ASTROPH \| $45.82\%$ \| $1.496$ BLOG-CATALOG \| $46.52\%$ \| $1.323$ CA-HEPTH \| $31.08\%$ \| $1.065$ Figure 4 shows the distribution of the shortest path lengths in the original and compressed graphs for the three datasets. We note that the two curves have almost the same pace. This shows that our compression preserves the distribution of the lengths of the shortest paths in the 3 datasets. However, the curves of the compressed graphs are slightly stretched and shifted from the original curves. This is due to the stretching of the paths as a result of compression. This stretch is not really considerable because of the preservation of $50\%$ of the direct neighbors of each vertex in the graph. In addition, unlike $t$-spanners, $(p,t)$-compression compression controls the shift between the two curves by adjusting the value of parameter $p$. (a) (a) Ca-AstroPh dataset (b) (b) Blog-Catalog dataset (c) (c) Ca-HepTh dataset Figure 4: Distribution of shortest path lengths of the original and compressed graphs. #### 5.3.2 Whole graph embedding speed-up Many whole graph embedding methods are based on the local neighborhood information of the nodes. These methods learn graph representations by exploring the node neighborhood and extracting some features such as walks, shortest paths, and local substructures. Since our compression preserves the local neighborhood within radius $t$, it is thus worth running these algorithms on compressed graphs to see if our compression speeds up these algorithms and to evaluate the performance loss. For this, we compressed three different datasets COLLAB, ENZYMES, and PROTEINS where $t=3$, $p(x<t)=0.0$ and $p(t)=1.0$ and we run three graph embedding algorithms on the compressed graphs: shortest path delta kernel [43], graphlet kernel [44] and Graph2vec [45]. We evaluated the performance of these algorithms on both the original and the compressed graphs in graph classification tasks as follows: we train an SVM classifier with $90\%$ of the graphs chosen randomly and then compute classification accuracy on the test set composed of the remaining $10\%$ graphs. For Graph2vec, we used the best configuration of parameters given in the original paper [45]. Table 6: Graph kernel performance on the compressed graphs cr: compression ratio. pl: performance loss Dataset \| Kernel \| cr \| Speed-up \| Original accuracy \| Accuracy \| pl ---\|---\|---\|---\|---\|---\|--- COLLAB \| SP delta \| $79\%$ \| $4.52$ \| $65.78\%$ \| $64.78\%$ \| $1.00\%$ 3-Graphlet \| $9.39$ \| $64.62\%$ \| $53.80\%$ \| $10.82\%$ 4-Graphlet \| $\textgreater 10$ \| (out of time) \| $53.68\%$ \| - PROTEINS \| SP delta \| $40\%$ \| $1.123$ \| $71.91\%$ \| $72.18\%$ \| $0.0\%$ 3-Graphlet \| $1.49$ \| $71.60\%$ \| $71.18\%$ \| $0.42\%$ 4-Graphlet \| $2.02$ \| $71.58\%$ \| $71.35\%$ \| $0.23\%$ ENZYMES \| SP delta \| $39\%$ \| $1.06$ \| $29.31\%$ \| $24.71\%$ \| $4.60\%$ 3-Graphlet \| $1.6$ \| $24.58\%$ \| $19.58\%$ \| $5.00\%$ 4-Graphlet \| $1.87$ \| $30.03\%$ \| $19.70\%$ \| $10.33\%$ Table 6 shows the performance of the graph kernels on the compressed graphs. We notice that the three kernels run faster on compressed graphs in all experiments. This Kernel computation speed-up is more noticeable on denser datasets, especially for the graphlet kernels. Indeed, we notice that the 4-Graphlet kernel exceeded the time limit (10 hours) on COLLAB’s original graphs, while it takes less than one hour on the compressed dataset. Regarding performance loss, we notice a small loss in values for the shortest path kernels ranging from $0.0$ to $4.6\%$. However, the loss is more noticeable for the graphlet kernel but it remains acceptable ($<11.0\%$ in all experiments). This is due to the fact that our $(p,t)$-compression does not preserve all graphlets, for example only $6$ graphlets are preserved among the $11$ graphlets of size=4. Despite this, kernel computation speed-up is very satisfactory. Table 7: Graph2vec performance on compressed datasets Dataset \| Compression ratio \| Speed-up \| Performance Loss ---\|---\|---\|--- COLLAB \| $79\%$ \| $2.54$ \| $0.00\%$ PROTEINS \| $40\%$ \| $1.004$ \| $2.93\%$ ENZYMES \| $39\%$ \| $1.27$ \| $5.99\%$ Table 7 shows the performance of Graph2vec on the compressed datasets. Globally, the algorithm runs faster on the compressed graphs, and the speed-up factor is bigger on denser datasets. Performance loss is acceptable on the ENZYMES dataset ($<6\%$) and very satisfactory on the first two datasets ($<3.0\%$). This is due to the fact that Graph2vec considers graphs as sets of Weisfeiler-Lehman relabeled subgraphs [46] that encompass higher order neighborhoods of graph nodes. These subgraphs are highly preserved by our compression. Figure 5 depicts the distribution of the classification accuracy obtained using Graph2vec on the 3 datasets (compressed and original) by running 10 experiments on each dataset and for each type of graph (compressed or original). We notice that the performances on compressed graphs and original graphs are nearly equivalent. The performance on original graphs is slightly better for the ENZYMES and PROTEINS datasets. Moreover, the loss of performance due to the compression is not great. Figure 5: Graph2vec performance boxplot #### 5.3.3 Node embedding speed-up In this series of experiments, we use two main algorithms, i.e., Node2vec [47] and DeepWalk [48], to learn node representations for both compressed and original graphs. We use BLOG-CATALOG and FLICKR datsets. The compressed graphs are obtained using $(p,t)$-compression where $t=2$, $p(1)=0.5$ and $p(2)=1.0$. The compression ratios are $\geq 45\%$ for the two datasets. Node2vec and DeepWalk are run using the best parameter combinations given in the original papers. To evaluate their performance on both compressed and original graphs, we run multiple multilabel classification tasks on the obtained representations. To this end, a sample $P_{tr}=\\{0.1,...,0.9\\}$ of the labeled nodes is used as training data. The rest of the nodes are used for testing. This process is repeated $10$ times, and we use Macro-F1 and Micro-F1 as performance metrics. Table 8 shows the performance of Node2vec and Deepwalk on the compressed graphs where $P_{tr}=0.5$. We notice that Deepwalk runs at the same speed on both the original and compressed graphs. This is because the time complexity of Deepwalk depends only on the number of nodes in the graph, which remains the same after our compression. However, Node2vec runs much faster on the compressed graphs than on the original graphs. This is justified by the fact that Node2vec’s time complexity depends on the square of the branching factor $b$ of the graph [49], which is implicitly related to the number of edges in the graph. The Micro-F1 and Macro-F1 scores obtained on the compressed graphs are nearly equivalent to the original scores. The performance loss rates are low ($\leq 4\%$) for both methods, and insignificant on the FLICKR dataset ($\leq 1.2\%$). For more fine-grained results, we also compared the performance of the two algorithms on compressed and original graphs while varying the size of the training sample $P_{tr}$ from $0.1$ to $0.9$. We summarize the results graphically for the Micro-F1 and Macro-F1 scores for both methods, i.e., Node2vec and Deepwalk, in Figures 6 and 7 respectively. Here we make the same observations: the performances of the two algorithms on compressed graphs are almost similar to their performances on the original graphs for all training rates. With the BLOG-CATALOG dataset, the performance drops by $4\%$ on compressed graphs in the worst case. However the performance curves on original and compressed graphs are almost identical in the FLICKR dataset. Table 8: Performance of node embedding algorithms on compressed datasets Dataset \| Method \| Space gain \| Speed-up \| Loss ( Micro F1) \| Loss ( Macro F1) ---\|---\|---\|---\|---\|--- BlogCatalog \| DeepWalk \| $46.52\%$ \| $0.99$ \| $3.6\%$ \| $3.4\%$ Node2vec \| $2.87$ \| $3.4\%$ \| $2.5\%$ Flickr \| DeepWalk \| $45.59\%$ \| $0.99$ \| $0.9\%$ \| $1.2\%$ Node2vec \| $5.32$ \| $0.3\%$ \| $0.0\%$ Figure 6: Performance of Node2vec on compressed graphs Figure 7: Performance of DeepWalk on compressed graphs ## 6 Conclusion and future work In this paper, we presented a graph summarization approach designed to control the amount of neighborhood information preserved in the computed summary. This approach relies on two parameters: a function $p$ that gives the proportion of each node’s original neighbors to be preserved in its $i$-hops neighborhood in the compressed graph, and a threshold $t$ for which $p$ reaches its maximal value. We presented algorithms to compute this compression with the minimum cost, and showed their effectiveness in compressing input graphs through experimental evaluation on multiple real life as well as synthetic graph datasets. We also showed that the summaries computed by the proposed approach can be used without any decompression as input to multiple graph applications, such as node embedding, graph classification, and shortest path approximations. The results show interesting trade-offs between algorithm runtime speed-up and precision loss. As for future work, we consider a more thorough analysis of $(p,t)$-compression impact on walks based graph learning algorithms such as Node2vec and DeepWalk. In fact, we observed some situations where learning accuracy increased when the graph was compressed. This was a quite unexpected observation. While we guess that walks are biased in the right direction by removing edges, characterizing such edges remains an open question. Another important open question is to find an efficient method to order graph edges. This would allow us to significantly improve the time complexity of the approach. In addition, we aim to design an incremental version of our compression to deal with dynamic graphs or graph streams. We note also that our approach can be used on both directed and undirected graph. However, our compression do not consider the labels of the edges. To compress edge-labelled graphs, a new model need to be defined so as to take into account these labels for example when defining the edge ordering. ##### Acknowledgement: This work is funded by ANR under grant ANR-20-CE23-0002. ## References * [1] Y. Liu, T. Safavi, A. Dighe, and D. 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# Detecting Deepfake Videos Using Euler Video Magnification Rashmiranjan Das 1, Gaurav Negi 1 and Alan F. Smeaton 1,2 1School of Computing and 2Insight Centre for Data Analytics Dublin City University, Glasnevin, Dublin, 9, Ireland. Email<EMAIL_ADDRESS> ###### Abstract Recent advances in artificial intelligence make it progressively hard to distinguish between genuine and counterfeit media, especially images and videos. One recent development is the rise of deepfake videos, based on manipulating videos using advanced machine learning techniques. This involves replacing the face of an individual from a source video with the face of a second person, in the destination video. This idea is becoming progressively refined as deepfakes are getting progressively seamless and simpler to compute. Combined with the outreach and speed of social media, deepfakes could easily fool individuals when depicting someone saying things that never happened and thus could persuade people in believing fictional scenarios, creating distress, and spreading fake news. In this paper, we examine a technique for possible identification of deepfake videos. We use Euler video magnification which applies spatial decomposition and temporal filtering on video data to highlight and magnify hidden features like skin pulsation and subtle motions. Our approach uses features extracted from the Euler technique to train three models to classify counterfeit and unaltered videos and compare the results with existing techniques. ## Introduction Deepfakes describes a technique for artificially manipulating video, initially applied to swap celebrities faces into video recordings which were shared on sites like Reddit [13]. They operate by replacing the face of one person in an original video, with a second person, inserted so that head movement, facial expressions, lighting and lip syncing when talking, are all exactly as in the original video. While many thousands of images of the second person to be superimposed into the deepfake, are usually required, recent work has shown that good deepfakes can be generated with a reduced number [21]. This means fake videos can be realistic and can be generated with small amounts of training data. Deepfakes can be recognised as both an opportunity and a threat because they permit users with relatively little computing experience in machine learning or computer programming to generate almost seamless fake videos. The availability of state of the art deep learning libraries such as TensorFlow [1] and Keras [7], with enough accessible training data of facial images, allows generation of fake video recordings whose quality is so good they can be very persuading [11]. The initial implementations of deepfakes relied on convolutional autoencoders [22]. Images of both subjects are reduced to lower dimensions using an encoder and reconstructed using a decoder. This training is performed for both source and destination facial expressions. In order to perform a face swap, a trained encoder of the source is mapped with a decoder trained on the target subject’s face. An upgrade to this technique is by adding a generative adversarial network (GAN) in the decoder [10, 2]. GANs consist of two modules, a generator and a discriminator. The task of the generator is to develop images resembling the source while the discriminator determines if the image is counterfeited. It is an iterative process, which makes deepfakes realistic as they are constantly learning. The availability of such sophisticated techniques for deepfake generation in the hands of ordinary researchers and their possible exploitation by other persons have escalated concerns about their possible misuse. Applications such as Deepfacelab [20], FakeApp and OpenFaceSwap are GUI based tools made accessible to relatively untrained researchers to create deepfake videos. With these tools, it becomes progressively possible for video evidence to be altered for political tension, false video evidence and fake news. Hence, this poses a challenge for society as well as an opportunity for creating novel entertainment, but it demands an effective technique for the detection of such counterfeit video. ## Related Work on Deepfake Detection One approach to deepfake detection focuses on psychological signals in the video [16], proposing a detection method by observing eye blinking in videos, a psychological signal not well presented in synthesised videos. This is based on a novel deep learning model combining a convolutional neural network (CNN) with a recursive neural network (RNN) that captures phenomenological and temporal regularities in the eye blinking process. Since the training images used to generate deep fakes do not usually include images of the subject with eyes closed, this is a clever approach, though it can be circumvented by intentionally integrating images into the training data with eyes closed. The work in [15] exploits colour disparity between GAN-generated images and real images in the non-RGB colour space to classify them. Again in the work reported in [18] analysed the colour difference between GAN images and real images. However, it is not clear if this approach can be applied to inspecting local areas in the image, as would be needed in the deepFake case. The paper which is most similar to our work is by Fernandes et al. [9], where they investigate the heart rates of people in deepfake videos and real videos using Neural ODE. The objective of this paper is to generate a heart rate from deepfake videos, which are assumed to have no heartbeat. However, this approach may not perform very well in case of detecting deepfakes as deepfakes are usually not stable and are not under perfect lighting conditions, so it can be difficult to obtain a stable heartbeat. ## Euler Video Magnification Eulerian Video Magnification (EVM) [25] is a technique which can uncover fleeting and hidden details in videos that will, in general, be hard to see with the human eye. EVM magnifies and visualises temporal variations in spatial and/or colour aspects in videos. The method emphasises subtle changes which occur naturally and are encoded within the video but not seen when viewed. For instance, one can enhance the slight colour changes in videos which include exposed human skin such as around the face where the capillaries in the skin show pulsation due to the bloodstream and blood flow changes caused, in turn from heart rate. This is similar to Differential Imaging Forensics introduced in [4] which also reveal latent visual features in videos that are not perceivable by human observers. The basis for EVM is that these difficult-to-see changes happen at specific frequencies that we can expand using a static window in the frequency space. For instance, in a plucked guitar, each guitar string resonates at a different frequency, so to amplify a string’s vibration EVM looks at pixel variations in the respective frequencies of the note being played. Similarly to amplifying human pulse visualisation in exposed skin around the face, one can consider pixel changes in frequencies somewhere in the range 1.0 to 3Hz and process this as it resonates to between 60 and 180 beats per minute. Figure 1: Euler magnification architecture taken from [25] The Eulerian amplification process consists of steps shown in Figure 1. A video is decomposed into images broken down using a Laplacian pyramid into various frequency ranges. The temporal changes concerning pixels in all frequency ranges of the Laplacian pyramid are bandpass filtered to select important and relevant frequency bands which are amplified by a magnification factor and this outcome is added to the respective signal. Amplified signals which belong to different frequency bands in the Laplacian pyramid are flattened to generate the last yield. The key attribute is the temporal frequency band which can be specified by adjusting the high and low cutoff frequencies for the filter. One way to consider this is as continuously stacking small variants of the picture on top of each other. This procedure makes a pyramid shape with the base as the first picture and gradually compressing as the pyramid rises. When we extract the frequency band of interest we amplify the signal and add it back to the source data. Adjusting the amplification factor of the bandpass signal results in a larger boost to the temporal bandpass. Changing these parameters can make variations in the scene more apparent but large amplification can add artefacts to the result. ## Video Data For Deepfake Detection During 2020, AWS, Facebook, Microsoft, and others joined together to build and run a Deepfake Detection Challenge, offering a prize fund of US$1M to researchers taking part. The Deepfake Detection Challenge Dataset (DFDC) is described in [8]. The challenge was hosted on the Kaggle website and 2,265 teams took part in the activity, making more than 3,000 submissions. The full DFDC dataset consists of 124,000 videos, some of which are real, some are deepfakes. Subjects in the videos are from varied ethnicities and have different skin tones, genders, lighting conditions and head poses, and activities. DFDC deepfakes were generated using the whole range of manipulation techniques, such tampering with the intent of representing the real adversarial space of facial manipulation, though no further details of methods used were provided to participants in the challenge. All video clips in the DFDC training set were left at their original resolution and quality, so deriving appropriate augmentations of the training set was left as an exercise to researchers. When the DFDC was complete and results processed, the best system achieved 82.56% accuracy, based on an ensemble of techniques. This is important for two reasons. First, because there are many ways to generate deepfakes, these need to be counteracted by using many ways to detect deepfakes. Secondly, because an individual generation technique may require different detection techniques, depending on the video that is generated, thus an ensemble of detection techniques, is appropriate. Along with the DFDC data set, we generated our own set of deepfake videos, created using the Deepfacelab application. Participants’ consent was given and participants signed an agreement approved by the DCU School of Computing Research Ethics Committee. A group of 30 participants each submitted a video of 10 seconds recorded in a controlled environment with suitable lighting at 1080p resolution and the participants remained quite still unlike some of the DFDC videos. Several face swaps were performed using the H64, H128 and SAE techniques [17]. These are autoencoder techniques that reduce the data to smaller dimensions, for example, the H64 model compresses the data into 64x64 pixels. Each face swap video was trained for 30,000 epochs with a mean loss of 0.0630. All these videos are of the same dimension as in DFDC dataset with 30 frames per second. The duration of the videos is 10 secs. A labelled data set was then created by merging the above data sets and this was also used in our experiments. All processing was done on an IdeaPad L340 with an NVidia graphic card GTX 1650 and 8 cores, Cuda enabled version 10.1. ## Methodology This paper explores the use of Euler Video Magnification as a way to pre- process video and use the result as an indicator of whether a video is a deepfake or not. We perform both EVM-based colour and movement amplification on videos to explore if the resulting differences can distinguish between original and deepfake videos. A related technique to our work on colour-based EVM is photoplethysmography (PPG) [19]. This is a process to identify fluctuations in blood volume by shining light of a given wavelength onto the skin and measuring changes in light assimilation. The pumping of heart drives blood to the skin surface in an oscillation cycle and it is the differences in the colour of oxygenated blood that causes changes in light assimilation which is thus a measure of heart rate. Photoplethysmography can also be used to recognise human activities [6] but in this paper we are interested in using colour-based EVM on exposed skin areas such as the face, to see if a pulse can be detected. Such a process is robust to different skin tones and small motion of the subject and our interest is to see if such EVM-based pulses are present in deepfakes as well as in original videos. Along with measuring minute colour changes, EVM can also magnify tiny motions by a subject in a video. One of the characteristics of people is natural tremor. This is a naturally occurring oscillatory motion which is frequent but not observable to the naked eye due to its very small amplitude. Their recurrence is within the scope of 8 to 12 Hz. These periodic motions have been observed to stay with age and are caused by compressions that are caused in muscles of the limbs. To illustrate this, Figure 2 taken from [24] displays vertical and horizontal displacements that occur naturally to a participant while taking a series of burst shot images with a smartphone camera, those displacements being caused by natural human tremors. The graph was based on 86 burst shots and the circle formed with red dots marks one standard deviation of movement in any direction. The observation we make from the graph is that human tremors are symmetrically distributed across all directions. Such small motions due to natural tremors could be magnified by Euler magnification so that they may be detectable as a differentiator between real and deepfake videos. Figure 2: Horizontal and vertical displacement of camera shots taken in burst mode, caused by natural tremors, taken from [24]. We processed videos with EVM, both deepfakes and originals, and then subjected them to three different techniques to extract features which we used for video classification. ### Technique 1: SSIM The first of the techniques we use is SSIM which is a comparison technique used to compare two frames, evaluate their likeness and calculate a similarity index for the video based on visual structures. The Structural Similarity Index (SSIM) is a perceptual metric that quantifies image quality degradation caused by processing such as data compression or by losses in data transmission, and for deepfake videos, a quality degradation in frames are due to a less well-trained neural net. It is a full reference metric that requires two images from similar image capture. SSIM is calculated based on luminance, contrast and structure. Comparing the time series of SSIM of an original video and its EVM equivalent in Figure 3 it can be seen that Euler magnification enhances inconsistencies among adjacent frames of deepfake videos. Such inconsistencies in frames are due to pixelated faces and such irregularities will have been intensified and magnified as a result of Euler magnification where the spatial amplification factor of EVM magnifies the irregularity and hence there are drops in the similarity index. SSIM is defined in Figure 4 [23]. Figure 3: Inter-frame dissimilarity for original, deepfake, Euler magnified original and Euler magnified deepfake videos. Figure 4: Structural similarity index of two windows x and y of common size NxN. $\mu$ is the average value of (x and y). $\sigma^{2}$ is the variance. $\sigma$ is the covariance of x and y. c1 and c2 two variables to stabilise the division with weak denominator. ### Technique 2: LSTM The idea of using a long short term memory (LSTM) network within a neural network architecture is to help the model learn long term dependencies across the data series. LSTM networks were first presented by Hochreiter & Schmidhuber in 1997 [12] and their original idea has been upgraded numerous times. The LSTM model’s primary objective is to recollect information over an extensive stretch of time. Unlike an RNN with its single tanh layer, LSTMs have four strategically arranged modules. On this basis, LSTM has been used by us for our classification task. The second technique we use builds on the amount of success achieved by CNN models in video analysis, by adding a Long Short Term Memory (LSTM) network into a neural network architecture. This can be used to learn any long term dependencies in a data sequence. The LSTM is coupled with the inception module to learn discriminative features from video frames [14]. Inception V3 includes an Inception module where there are changes in the spatial convolutions to depth-wise separable convolutions. Our model is build using CNN network layers for feature extraction followed by an LSTM layer for temporal analysis of Euler magnified videos. Our network has fully connected layers and a dropout layer to make sure that there is no over-fitting. The total number of trainable parameters used is 5,500,898 and these are used as input for the LSTM network and 2 node network working as a detector for deepfake videos from original videos. To obtain ground truth, the neural network was trained on videos which were not modified by the Euler magnified method. The hidden layer had a ‘relu’ activation function, while the last layer had ‘softmax’ as the activation function. We calculated the loss and accuracy of the technique on both the training and test sets. We performed the same sequential steps on the same sets of the video but on the Euler magnified form of the video for comparison on how the technique compares to the standard classification techniques. We used an LSTM network with 512 widths and dropout of 0.5 to randomly set values of outgoing edges of hidden layers to zero. The last layer is constructed using a softmax activation layer to predict video class. ### Technique 3: Heart Rate Estimation When computing Euler magnification on a deepfake video, we observed that deepfake videos also exhibit pulsation as seen in Figure 5. Our third technique is to estimate the heart rates of subjects appearing in videos by focusing on an area of facial skin, and analysing if any differences in heart rate calculations due to skin pulsation could be observed between them. Figure 5: EVM on deepfake video The main hypothesis behind the work in this paper is to Euler magnify both source and counterfeit videos in order to extract features which could highlight skin pulsation or minuscule movement by a subject in the videos. Through pulsation, we can visualise and thus be able to extract the heart rate of the subject in the video by calculating the number of colour change peaks and counting each one as a heartbeat. EVM can amplify spatial as well as the temporal aspects of a video. Spatial magnifies the motion while temporal magnifies colour changes on skin tone. We use the temporal aspect to visualise the pulse on exposed facial skin. The features of this that are customisable are filter type, magnification factor and range of frequency. Videos in our test set were subjected to a range of EVMs with a frequency range of between 1Hz (60 BPM) and 1.33 Hz (80 BPM). The amplification factor was set to 50. To fetch a heartbeat, a Fast Fourier transform algorithm was used. The temporal signal was transformed into a frequency domain to fetch the signal measured in hertz. Our data set consists of 19.25% of REAL videos which have not been altered, With the deepfakes accounting for 80.75% of the samples. There is a huge imbalance between the categories thus models might be biased towards categorise videos as deepfake hence the data needs to be up-scaled to balance it. We used OpenCV [5] to detect the locations of faces in videos using the face recognition package. We observe that in some cases, when the subject is not looking frontally at the camera or when the luminosity is low, the algorithm for face detection does not detect the face or eyes correctly. ## Results The results of our experiments were produced using Python 3 on a computer with 8 GB RAM and a 4 core Ryzen 5 AMD processor. We used 400 sets of videos from the DFDC Kaggle dataset [8] and 30 assembled directly by us. We generated 5 video datasets by changing parameters of Euler magnification, notably amplification factors. The complete sequence can be divided into the following steps 1. 1. Create metadata of the videos extracted from multiple sources; 2. 2. Detect faces and crop video to leave only the face in the video; 3. 3. Euler magnify the video with a specific frequency range and amplification factor; 4. 4. Train and evaluate a model on Euler magnified video. ### Results for Technique 1: SSIM As observed anecdotally from multiple graphs of SSIM scores for videos, there were more similarity score drops in counterfeit deepfake videos when compared to their real video counterparts when magnified by the Euler magnification process. After magnifying the data using EVM, we calculated SSIM scores for all videos in the dataset. Deepfake video detection is particularly difficult to train as the manipulation can be observed only on a few frames and it is restricted to certain areas of the face. When there is much movement in the video, there can be inconsistencies and important areas in the frame appear only briefly. Below are the results we obtained when we used a range of standard machine learning models to classify videos: Technique \| Logistic \| Decision \| NNet \| NNet + ---\|---\|---\|---\|--- Regression \| Tree \| LSTM Original Videos \| 68.7% \| 65% \| 77% \| 77% EVM Videos \| 53.7% \| 62% \| 70% \| 62% These results are below the 82.56% accuracy achieved by the best-performing submission to the DFDC but that was based on an ensemble while our results are one single technique. The results show consistently that SSIM-derived features are more discriminative of real vs. deepfake videos, before Euler Video Magnification was applied, suggesting that our default settings for EVM could be tweaked to improve performance. In the table below, we processed our test videos with Euler magnification with multiple amplification factors (15, 20, 30, 40 and 50). The frequency band was restricted to between 0.8hz and 1.0 Hz. As we can see from the results the lower amplification factor performed better than the higher amplification factor. In these videos, a higher amplification factor led to additional noise which blocked some of the features in the videos. This indicates that Euler magnification is introducing noise as the amplification factor increases. Amplification \| Accuracy \| Loss ---\|---\|--- Factor 10 \| 70.24% \| 0.6036 20 \| 68.36% \| 0.6043 30 \| 65.77% \| 0.6051 40 \| 63.54% \| 0.6243 50 \| 60.49% \| 0.6189 ### Results for Technique 2: LSTM We ran the Inception v3 inspired LSTM model for 100 epochs to give the following results on our test video set. For these experiments we found that classification on the original unprocessed videos was more accurate and outperformed classification accuracy when EVM had been applied to the videos on an amplification factor of 30. Video set \| Accuracy \| Loss ---\|---\|--- Original Videos \| 77.24% \| 0.88 EVM Videos \| 61.79% \| 2.52 This result is another apparent setback to the idea of using EVM for deepfake detection. The difference in accuracy between original and EVM videos is even more pronounced than in Model 1. ### Results for Technique 3: Heart Rate Estimation A comparison of heart rate based on pulsation observed in the Euler magnified video of an original and a deepfake video revealed that beats per minute (BPM) for both the videos were very similar, with slight changes only in the first decimal position. As an example in Figure 6, the heart rate of both the original and deepfake videos came to 65.78 beats per minute (1.096 Hz). Thus the temporal variance calculated through Euler video magnification is insufficient to differentiate deepfake from original videos [3]. Subjects appearing in both deepfake videos and original videos have an estimated heartbeat which is indistinguishable. This reveals that the GANs used for creating the deepfakes does not simply superimpose a new image of the subject on top of the real image which could have concealed facial pulsation colour changes, but the GAN manages to faithfully model the true data distribution of the real data at the pixel level and in this way it keeps the colour-temporal changes from the genuine video, intact. Figure 6: Heartbeat calculation using Fast Fourier transform on EVM of Original and Deepfake video ## Conclusions In this paper, we tested the effect and impact of Euler Video Magnification as a technique for video pre-processing leading to possible detection of deepfake videos. Both the colour and the spatial aspects of EVM were tested as possibilities for a number of classification models we built for discriminating between real and deepfake. We used accuracy as a metric, even though accuracy is known not to be a great metric for evaluation when using imbalanced datasets like ours, which is why we include accuracy figures for both original and EVM processed videos, so we can compare. What the accuracy performance figures do not show is that using EVM vs. not using EVM as a video pre-process actually detects different videos, so it is not the case that EVM pre-processing simply eliminates some videos from being accurately classified as real or deepfake. Thus we should include a range of metrics including Precision, Recall, F1 and others so as to more fully understand what EVM is doing, but as this paper is preliminary work, that forms part of future work. The results of the best performing systems at the Deepfake Detection Challenge [8] achieved an accuracy of 82.56% across a much larger test dataset than we have used here, so we cannot compare our results directly against the data in the full DFDC results. However because these best results were based on ensembles of techniques, it follows that the more variety among the systems used in the ensemble, the better will be will be the final, overall result. We believe that our results help us to understand how deepfake video generation incorporate pulsation information and subject tremor motion into their generated videos. In future work, we would like to explore the most difficult facial objects to alter like lips and eyes, for fake detection as well as a deeper exploration into the results by using different evaluation metrics. We would also investigate different feature sets and limit the parameters of EVM to focus on its impact. We would also like to explore how compression artefacts in video storage interact with the EVM process as it is possible it is masking the contribution of EVM. ### Acknowledgements This work was part-funded by Science Foundation Ireland through the Insight Centre for Data Analytics (SFI/12/RC/2289_P2), co-funded by the European Regional Development Fund. ## Author Biography Rashmiranjan Das works in the field of Artificial Intelligence at Deciphex Limited. He received his Masters in Data Analytics from Dublin City University in 2020 and holds a Bachelors in Computer Engineering from Mumbai University. After his undergraduate education, he joined Vertisystem limited as a software developer and then as an Analyst at Ernst & Young. Gaurav Negi completed his Masters in Data Analytics from Dublin City University in 2020 and is now working at Xcelerator Machine Translations Ltd. Gaurav holds a bachelor’s degree in Computer Science from Amity University, India (2016) and worked for 3 years as a Data Analyst for Genpact India. Alan Smeaton is Professor of Computing at Dublin City University. He received his PhD from University College Dublin (1987). He is an elected member of the Royal Irish Academy and a winner of the Academy’s Gold Medal in Engineering Sciences, an award given to individuals who have made a demonstrable and internationally recognised outstanding scholarly contribution in their fields. Alan is Chair of ACM SIGMM. ## References * [1] Martín Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, et al. Tensorflow: A system for large-scale machine learning. In 12th USENIX Symposium on Operating Systems Design and Implementation, pages 265–283, 2016. * [2] Martin Arjovsky, Soumith Chintala, and Léon Bottou. 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Further author information: Send correspondence to C. P.; E-mail: <EMAIL_ADDRESS> # Performance of the Dark Energy Spectroscopic Instrument (DESI) Fiber System Claire Poppett Space Sciences Laboratory, University of California, Berkeley, Berkeley, CA Patrick Jelinsky Space Sciences Laboratory, University of California, Berkeley, Berkeley, CA Julien Guy Lawrence Berkeley National Laboratory, Berkeley, CA Jerry Edelstein Space Sciences Laboratory, University of California, Berkeley, Berkeley, CA Sharon Jelinsky Space Sciences Laboratory, University of California, Berkeley, Berkeley, CA Jessica Aguilar Lawrence Berkeley National Laboratory, Berkeley, CA Ray Sharples Centre for Advanced Instrumentation, Durham University, Durham, UK Jurgen Schmoll Centre for Advanced Instrumentation, Durham University, Durham, UK David Bramall Centre for Advanced Instrumentation, Durham University, Durham, UK Luke Tyas Centre for Advanced Instrumentation, Durham University, Durham, UK Paul Martini The Ohio State University Kevin Fanning The Ohio State University Michael Levi Lawrence Berkeley National Laboratory, Berkeley, CA David Brooks University College London Peter Doel University College London Duan Yutong Boston University Gregory Tarle University of Michigan Erique Gazta$\tilde{\text{n}}$aga University of Barcelona Francisco Prada Instituto de Astrofisica de Andalucia CSIC the DESI Collaboration ###### Abstract The recently commissioned Dark Energy Spectroscopic Instrument (DESI) will measure the expansion history of the universe using the Baryon Acoustic Oscillation technique. The spectra of 35 million galaxies and quasars over 14000 sq deg will be measured during the life of the experiment. A new prime focus corrector for the KPNO Mayall telescope delivers light to 5000 fiber optic positioners. The fibers in turn feed ten broad-band spectrographs. We describe key aspects and lessons learned from the development, delivery and installation of the fiber system at the Mayall telescope. ###### keywords: DESI, fiber system, focal plane, robotic positioners ## 1 INTRODUCTION The Dark Energy Spectroscopic Instrument (DESI) is a fiber-fed spectroscopic instrument installed on the 4-meter Mayall telescope at Kitt Peak National Observatory (KPNO). During its 5 year survey, DESI will measure the spectroscopic redshift of up to 35 million galaxies and quasars, enabling the completion of the largest 3D map of the universe out to a redshift of $\sim$3.5. This unprecedented dataset will be used to measure Baryon Acoustic Oscillations and Redshift Space Distortions leading to a more accurate measurement of the expansion history of the universe and ultimately furthering our understanding of dark energy [1]. The instrument delivers light from the 4-m telescope through a new corrector [2], which provides a 3 degree field of view to the focal plane [3]. The focal plane contains 5,000 optical fibers that are reconfigurable using robotic positioners [4]. These fibers carry the light from the focal plane to ten broad-band spectrographs, sensitive from 360-980nm with a resolution of 2,000-5,000 [5]. The DESI fiber system consists of 5020 custom built positioner fiber assemblies (PFAs) that are installed into 5020 robotic fiber positioners. 5000 of these fibers are used for science and 20 are routed to a sky camera that is used to estimate the exposure times. The 107$\mu$m core fibers are precision cleaved and then bonded into fused silica ferrules that allow them to be bonded into the positioner. A polyimide tube is bonded into the back of the ferrule to provide strain relief from the positioner as it moves between targets. Finally, an anti-reflection (AR) coating is applied to the front face [6]. The PFA is installed into a positioner and the ferrule is bonded into the ferrule arm after being aligned in focus. Ten 47.5 m cables run from the focal plane enclosure to spectrograph slits. Each slit consists of 500 fibers in 20 blocks of 25 fibers, where the blocks follow the slit curvature. The cable and slit assembles are connected to the focal plane via fusion splicing. The fusion splice not only facilitates an effective fabrication, integration and testing flow but also allows the system to retain maximum throughput and minimize the focal ratio degradation (FRD). A schematic overview of the DESI fiber system is shown in figure 1. --- Figure 1: Schematic of the DESI fiber system. The 5000 fibers are divided into ten identical bundles of 500 fibers, each of which is integrated with one focal plane petal. An international collaboration of institutes worked together with industry to design, produce and test the fiber system. The fiber system was installed at the 4m-Mayall telescope with $\geq 99\%$ of fibers intact, $\geq 90\%$ fibers with collimated FRD $\leq 1.8^{\circ}$, and $\geq 90\%$ throughput delivered from the prime focus corrector to the spectrograph. This high performance is due to many innovations such as precision cleaved front end fibers [6], and fusion splices between the focal plane and spectrograph slit assemblies [7]. In this paper we describe key aspects and lessons learned from the successful development, delivery and installation of the fiber system at the Mayall telescope. This paper will focus on several new topics. In section 2 we discuss the requirements and specifications that drove the testing performed prior to shipping and the performance that was achieved. Section 3 focuses on the installation of the fiber system at the Mayall Telescope as the cables were routed from the focal plane to the spectrograph shack. The final section (4), shows the preliminary performance of the fiber system using on-sky data taken during early commissioning. ## 2 Fiber system Requirements, Specifications, and Throughput Measurements As light propagates through an optical fiber, it suffers from fabrication- and stress-induced increases in entropy that manifest as beam attenuation and diffusion of the input entrance angles (modal diffusion). Mechanical fiber stresses can be induced by the fiber termination method, optomechanical bonding method, and other mechanical forces. Broadly speaking, optimizing fiber performance is therefore motivated by controlling the entropy gain in the system by, e.g., minimizing mechanical stresses and/or intentionally scrambling the light path. The first-order concern when optimizing fiber performance is throughput loss. For an end-to-end system, throughput losses due to the fiber cable can be broken into (1) unavoidable attenuation as the light propagates through the cable (e.g., cladding-mode losses) and (2) losses that occur when attempting to capture the fiber output beam. Fiber manufacturers continue to improve the former with new materials and fabrication techniques. A critical aspect of the latter is the increase in angles from the input to the output beam. This non conservation of étendue is referred to as focal-ratio degradation (FRD) [8]. There are two standard experiments used to characterize fiber FRD: (1) Cone (or solid angle) tests measure the angular diffusion of a fiber by illuminating it with a filled cone of light with constant surface brightness over a specified angle.[9] To better simulate the telescope illumination pattern, scaled obscurations, such as the secondary mirror and support structure, are added to mimic the telescope. Cone tests therefore directly measure how much energy will be enclosed within some $f$-ratio in a fiber, which is the ratio of the focal length to the diameter of the entrance pupil ($f/\\#$), for a known input illumination geometry. (2) Ring (or collimated) tests measure FRD by injecting a collimated beam (e.g., from a low-power laser) into the fiber.[10] The fiber azimuthally scrambles the beam to form a ring illumination pattern, recorded by an imaging detector. The diameter of the ring gives a direct measurement of the incidence angle and FRD is measured by the thickness of the ring in the radial direction. Although less direct than the cone test, the ring test is very simple to perform and interpret since there are very few sources of measurement error. The requirement on the focal ratio degradation (FRD) for the fiber system states that the FRD-induced throughput of the fiber system should be $\geq 90~{}\%$ enclosed energy (EE) averaged over all fibers. This requirement must be met within an f/3.57 output beam, illuminated by a uniform f/3.9 (7.3 degree half angle) input beam at a wavelength of 625 nm that includes a scaled Mayall-secondary obstruction (3.46 degree half angle) with a chief ray that is less than 26 arc minutes from the fiber’s optical axis. This requirement is verified through full cone FRD testing as described above. However, since it was impractical to perform a full cone FRD test for every science fiber, it was necessary to find a relationship between full cone FRD and collimated FRD. It was found through testing that at f/in=3.9, a $0.2^{\circ}$ increase in the ring width of a collimated FRD test would result in a 2$\%$ loss in throughput to the spectrograph as described in Poppett (2018) [11]. Through multiple tests of splicing fibers with differed FRD performances together we established through that a model that fits the spliced FRD fairly well is given in the following equation: $\text{FRD}_{SP}=\sqrt{\text{FRD}^{2}_{PFA}+\text{FRD}^{2}_{slit}}$ (1) where $\text{FRD}_{SP}$ is the collimated FRD of the resultant splice, $\text{FRD}_{PFA}$ is the collimated FRD of the positioner fiber assembly and $\text{FRD}_{slit}$ is the FRD of the slit and cable. ### 2.1 FRD Budget Collimated FRD testing of first 1000 PFAs is shown in figure 2 and allowed us to establish a basline for PFA performance. Any PFA with an FRD greater than 1.5∘ should be rejected. As is shown by this figure, the best fit normal distribution has an average of 0.88∘ and a sigma of 0.18∘, however, the distribution is not normal and has an extended wing with increasing FRD. In order to establish the FRD requirements for the fiber cable (fiber in cable with terminated slit) we assumed that the FRD distribution would be a Gaussian. We then ran a Monte Carlo simulation of this Gaussian distribution convolved with the PFA FRD distribution using equation 1 and throughput measurements, obtained from full cone testing during R&D, in order to estimate the distribution of throughputs for the fibers. The FRD mean was varied from 1.2∘ to 2.2∘ and the $\sigma$ from 0.2∘ to 0.8∘. The results of a few throughput distributions are plotted in Figure 2. From this analysis, and since the distribution is not likely to be Gaussian and will have a tail for larger FRDs, it was determined that the fiber cable+slit FRD should have a mean $\leq 1.8^{\circ}$ with a $\sigma$ $\leq 0.6^{\circ}$ in order to meet the FRD budget for the final spliced fiber. --- Figure 2: Left: Collimated FRD budget of the first 1000 PFAs manufactured and Right: the modelled throughput distribution for various Cable FRD Gaussian distributions. The mean and the $\sigma$ of the distribution was varied. The blue curve has a $\sigma$ of 0.6, and the red curve has a $\sigma$ of 0.8. ### 2.2 As Delivered Throughput In order to measure the absolute throughput of the full fiber system we combined collimated FRD results with the measured throughput of a calibrated fiber. By comparing the total counts in the FRD image of the spliced fiber with the calibrated fiber it was possible to determine the absolute throughput of the end-to end system. Figure 3 shows the results from the measurements of one petal (1 wedge of the focal plane containing 500 fibers) as an example. All petals met the throughput requirements and the fiber system was delivered with $\geq 99\%$ of fibers intact and $\geq 90\%$ throughput from the prime focus corrector to the spectrograph. --- Figure 3: Collimated FRD performance, Absolute throughput, and throughput normalised for FRD performance for 1 section of the focal plane prior to shipment to the telescope. ## 3 Fiber system Installation The installation of the fiber system at the Mayall telescope was performed in 2019 in three main stages [12, 13]. The first stage was to install the focal plane system onto the back of the prime focus corrector assembly. In June and July, 2019, the telescope was parked in the southeast annex location in the Mayall telescope dome and the petals were installed using a custom installation system, referred to as the sled, from a platform that enabled the petals to be guided into position. This process is shown in the first two panels of figure 4. The final panel in this figure shows the cable management system, which was designed to route the cables without violating their minimum bend radius of 200 mm whilst retaining the ability to remove any petal without others being affected. --- Figure 4: Ten optical fiber cables from the focal plane petals are strain relieved to a frame that will later be covered within the FPE. The routing of each cable was carefully engineered to ensure the bend radius of the cables was everywhere greater than 200 mm. The panel on the left shows the first petal loaded onto the sled prior to insertion. The middle panel shows the back of the focal plane when 8/10 were loaded. The panel on the right shows the focal plane cable management system. During August and September, 2019, the thermally insulated focal plane enclosure (FPE, from Berkeley Lab) was installed and the fiber cables were routed down the telescope to where the spectrographs are located. The cables and slits were stored on their shipping carts during the focal plane installation and handled by an expert team of NOIRlab engineers (see figure 5). --- Figure 5: The fiber cables and slits were stored on their shipping carts during the focal plane installation and handled by an expert team of NOIRlab engineers during cable routing. The fiber cables emerge from the FPE and cross to the upper ring, along both the southeast and northeast spider vanes. They are stacked parallel to the optical axis when they cross the spider vanes in order to to minimize obscuration. From the upper ring, the two groups of five cables are constrained by custom hardware designed, built, and installed by NOIRLab. The two sets of cables converge into a single group of ten cables about halfway down the serrurier truss. The fiber cables then follow an arc-shaped path through the declination and hour axes. They were installed in articulated cable carriers with reversible bend directions to ensure that they do not twist or otherwise violate their minimum bend radius. The bundle of ten cables coming down the telescope truss were loaded into the declination wrap carrier on a bench near the east declination bearing, which was then craned into its custom guides in the gap between the telescope center section and the hour angle “horseshoe” [13]. The bundle of ten cables was strain relieved to the oval tube of the horseshoe, and then the bundle passed into the hour angle cable wrap in one layer, ten cables wide. The fixed end of the hour angle cable wrap was attached to the fixed telescope mount, and from there the cables passed straight through penetrations in the wall of the large coudé room on the main floor to the east of the telescope mount. The cables continue from that wall through penetrations in the DESI spectrograph clean room shack. The penetrations through the walls of the large coudé room and the shack were sized to allow the passage of the slitheads while they were in their protective shipping boxes. --- Figure 6: Routing of the cables through the Dec and Polar Axis before they penetrate the wall of the spectrograph shack. ## 4 On Sky Fiber system Performance On sky fiber system performance was evaluated by measuring the stability of both the bulk throughput and the PSF stability. If the PSF changes from the calibration to the science exposures, this will result in an extraction bias. It was a requirement that the spectroscopic PSF should be characterized for all fibers in each science exposure over the full wavelength range such that the PSF bias did not exceed 1%. ### 4.1 Fiber Throughput Stability Throughput variations in the fiber system were measured under a variety of conditions. The first test measured throughput variations as the robotic fiber positioners were moved. The measurements were made by moving the telescope from Zenith and pointing instead to the calibration screen in the dome and measuring PSF flux variations in the spectrographs as the positioners moved around their patrol zone. Figure 7 shows that most fibers were found to have a rms flux variation below 1%. Only 3% of fibers have a flux variation that exceeds 1%. --- Figure 7: Example of the aperture flux variations around isolated arc lines when moving the positioners. Most fibers were found to have a rms flux variation below 1 % The second test measured the PSF stability under different telescope pointings. This was necessary in order to prove that different bends in the cable did not result in different fiber output distributions. Again, the telescope was pointed to the white spot and then both the dome and telescope were moved through 4 different dome azimuth angles (0, 107, 180, 253 deg). Figure 8 show that the variation in throughput is within $\pm$1%, and there is no evidence for an effect of the dome azimuth / telescope pointing on the PSF stability. --- Figure 8: Example of fiber throughput stability as measured in Spectrograph-0 NIR camera. The variation in throughput is within $\pm$1%, and there is no evidence for an effect of the dome azimuth / telescope pointing on the PSF stability. The fibers in the slit are organised into 20 blocks of 25 fibers and this is clearly seen in the data. Finally, an analysis of sky residuals was performed on multiple night time exposures taken during early commissioning. When reducing science data, the sky spectrum recorded during each integration is used as the normalization, when considering the residual sky-subtraction error in any given data set [14]. If the PSF is not stable, this error will increase. The results show that the sky residuals RMS on emission lines is $\leq$ CCD noise + 1 % of sky lines. These results are shown in figure 9 --- Figure 9: Sky residuals RMS on emission lines is $\leq$ noise + 1 % of sky. Results shown are from 1 sky tile in the red and blue cameras observed during instrument commissioning ## 5 Conclusions The Dark Energy Spectroscopic Instrument (DESI) is a fiber-fed spectroscopic instrument installed on the 4-meter Mayall telescope at Kitt Peak National Observatory (KPNO). DESI fiber system starts with 5000 custom built positioner fiber assemblies (PFAs) that are installed into 5000 robotic fiber positioners. Ten 47.5 m cables run from the focal plane enclosure to spectrograph slits. The DESI fiber system was delivered to the 4m-Mayall telescope with $\geq 99\%$ of fibers intact and more than $\geq 90\%$ throughput delivered from the prime focus corrector to the spectrograph. It was installed on the Mayall telescope at KPNO in 2019. The instrument was commissioned over a 6 month period in 2019/2020 and finished earlier than planned due to the global pandemic of COVID-19. Preliminary analysis from this commissioning data has shown that the PSF stability is $\leq$1 % for most fibers. This requirement is met when moving or not the positioners, and as a function of dome azimuth. Finally it has been proven that the PSF stability is sufficient to obtain a sky subtraction precision better than 1%. ###### Acknowledgements. This research is supported by the Director, Office of Science, Office of High Energy Physics of the U.S. Department of Energy under Contract No. DE–AC02–05CH1123, and by the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility under the same contract; additional support for DESI is provided by the U.S. National Science Foundation, Division of Astronomical Sciences under Contract No. AST-0950945 to the NSF’s National Optical-Infrared Astronomy Research Laboratory; the Science and Technologies Facilities Council of the United Kingdom; the Gordon and Betty Moore Foundation; the Heising-Simons Foundation; the French Alternative Energies and Atomic Energy Commission (CEA); the National Council of Science and Technology of Mexico; the Ministry of Economy of Spain, and by the DESI Member Institutions. The authors are honored to be permitted to conduct astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with particular significance to the Tohono O’odham Nation. ## References * [1] DESI Collaboration, “The DESI Experiment Part I: Science,Targeting, and Survey Design,” arXiv e-prints , arXiv:1611.00036 (Oct. 2016). * [2] Doel, P., Besuner, R., Brooks, D., Flaugher, B., Gallo, G., Gutierrez, G., Kent, S., Lampton, M., Levi, M., Liang, M., Miller, T., and Sprayberry, D., “The prime focus corrector for dark energy spectroscopic instrument,” in [Ground-based and Airborne Instrumentation for Astronomy VI ], Evans, C. J., Simard, L., and Takami, H., eds., Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series 9908, 99088D (Aug. 2016). * [3] Lambert, A. R., Besuner, R. W., Claybaugh, T. M., and Silber, J. H., “DESI focal plate mechanical integration and cooling,” in [Ground-based and Airborne Instrumentation for Astronomy VI ], Evans, C. 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∎ 11institutetext: The authors are with the Department of Informatics, Bioengineering, Robotics, and Systems Engineering, University of Genoa, Via All’Opera Pia 13, 16145 Genoa, Italy. 11email<EMAIL_ADDRESS> # An Integrated Localisation, Motion Planning and Obstacle Avoidance Algorithm in Belief Space Antony Thomas Fulvio Mastrogiovanni Marco Baglietto (Received: date / Accepted: date) ###### Abstract As robots are being increasingly used in close proximity to humans and objects, it is imperative that robots operate safely and efficiently under real-world conditions. Yet, the environment is seldom known perfectly. Noisy sensors and actuation errors compound to the errors introduced while estimating features of the environment. We present a novel approach (1) to incorporate these uncertainties for robot state estimation and (2) to compute the probability of collision pertaining to the estimated robot configurations. The expression for collision probability is obtained as an infinite series and we prove its convergence. An upper bound for the truncation error is also derived and the number of terms required is demonstrated by analyzing the convergence for different robot and obstacle configurations. We evaluate our approach using two simulation domains which use a roadmap-based strategy to synthesize trajectories that satisfy collision probability bounds. ###### Keywords: Motion Planning Belief Space Planning Collision Probability ## 1 Introduction Planning and decision making under uncertainty are fundamental requirements for autonomous robots. Uncertainties often arise due to insufficient knowledge about the environment, imperfect sensing and inexact robot motion. In these conditions, the robot poses or other variables of interest can only be dealt with in terms of probabilities. Planning is therefore performed in the belief space, which corresponds to the set of all probability distributions over possible robot states platt2010RSS . At a given time instant, we consider the belief or the belief state of the robot which corresponds to a probability distribution of the robot state (or other variables of interest) given the measurements and controls thus far van_den_berg2012IJRR . Consequently, for efficient planning and decision making, it is required to reason about future belief distributions due to candidate actions and the corresponding expected observations. Such a problem falls under the category of Partially Observable Markov Decision Processes (POMDPs) kaelbling1998AI . Robots are becoming ubiquitous in our day-to-day lives and are being increasingly used in close proximity to humans and other objects in service- oriented scenarios such as factories, living spaces, or elderly care facilities. It is therefore of vital importance that robots operate efficiently and safely in real-world conditions. Localization is a key aspect for safe and efficient robot motion as it is a precursor to solving the problems “where to move to” and “how to reach there”. A robot perceives the environment through its sensors and distinct objects known as landmarks aid the robot in localizing. However, most approaches assume that these landmarks are known with high certainty. For example, given the map of the environment, while planning for future actions the standard Markov localization111The application of Bayes filter to the localization problem is called Markov localization thrun2005book . does not take into account the map uncertainty, that is, the landmark location uncertainties are ignored and the locations are assumed to be perfectly known. This means that given the map and the sensor range222Note that the concepts discussed here are applicable to any sensor used for robot localization. In particular, in this work (Section 5) we use a laser range finder and beacons that give signal measurements in terms of the distance to the beacons., for any landmark, there exists a set of viewpoints from which an observation may be obtained. Let us consider for example a robot equipped with a laser range finder and observing a landmark. Whenever the robot location is such that the landmark falls within the sensing range, a measurement is obtained. Thus, there exists a set of robot locations or viewpoints from which a measurement of the landmark may be obtained. Therefore, when the landmark locations are assumed to be perfect, this set of viewpoints can be easily determined since it depends on the environment map and the sensing capabilities of the sensor employed. Yet, this might not be true in practice. For example, consider the map of an environment obtained from a Simultaneous Localization and Mapping (SLAM) session. Due to the dynamic nature of the environment, the objects of interests could be occluded when viewed from the set of viewpoints which would have otherwise produced a full observation. Moreover, an erroneous localization, for example due to wrong data association, could lead to wrongly estimated object poses. Thus, in such cases it is more fitting to consider the uncertainty in the landmark locations. This landmark uncertainty directly translates to the fact that the viewpoints whence the object can be observed is uncertain. This is visualized in Fig. 1. As seen on the left hand side of the figure, when the object location is known perfectly, there exists a region (green) from which the object can be observed. Note that as discussed before this region is determined from the environment map and the sensor capabilities. Some of the viewpoints inside this region are shown in black. On the right hand side of the figure, we consider the uncertainty in landmark location the red shaded region denotes the uncertainty in landmark location. Since the object location is not known precisely (object can be anywhere within the uncertainty region), given a viewpoint, it cannot be said with certainty that the object will be observed. This is so because given a viewpoint, a landmark is observed if it falls within the sensing range. However, since the landmark location is not fixed and is uncertain, the landmark may or may not be within the sensing range. For example, if the landmark location is Gaussian distributed, then the landmark, in practice can be anywhere within the (say) 3-$\sigma$ uncertainty region. Thus we cannot define a precise region from which the landmark can be observed. Therefore, one can only reason in terms of the probability of observing the object from the considered viewpoint. This results in a probability distribution function for the viewpoints. Consequently, not considering this uncertainty can wrongly localize the robot, leading to inefficient plans causing catastrophes. From now on, we will use the term object uncertainty to refer to the notion of uncertainty in landmark location. Figure 1: The red blob denotes an object in the environment. The green region corresponds to the set of viewpoints from which the object can be observed; some of these viewpoints are shown as black dots. On the right hand side, the red shaded region denotes the uncertainty in object location, with the red blob denoting its mean position. The corresponding viewpoint region is visualized as the intersection between different viewpoint regions that correspond to the object being at different locations (left hand side shows one instance of this). In order to ensure safe robots’ motion, it is also essential to consider collision avoidance strategies. As robots are being increasingly used in service-oriented scenarios with both static and dynamic obstacles, deterministic approaches do not fare well. Moreover, in the case of dynamic obstacles, their future states have to be predicted. Yet this is an added difficulty due to the lack of perfect knowledge of their motions. As a result, providing safety guarantees is difficult. ### 1.1 Notations and Problem Definition Throughout this paper vectors will be assumed to be column vectors and will be denoted by lower case letters, that is x. The transpose of x will be denoted by $\textbf{x}^{T}$ and its Euclidean norm by $\mathinner{\\!\left\lVert\textbf{x}\right\rVert}=\sqrt{\textbf{x}^{T}\textbf{x}}$. A multivariate Gaussian distribution of x with mean $\bm{\mu}$ and covariance $\Sigma$ will be denoted using the notation $\textbf{x}\sim\mathcal{N}(\bm{\mu},\Sigma)$. Matrices will be denoted by capital letters. The trace of a square matrix $M$ will be denoted by $tr(M)$. The identity matrix will be denoted by $I$ or $I_{n}$ when the dimension needs to be stressed. A diagonal matrix with diagonal elements $\lambda_{1},\ldots,\lambda_{n}$ will be denoted by $diag(\lambda_{1},\ldots,\lambda_{n})$. Sets will be denoted using calligraphic capital letters like $\mathcal{S}$ or $\mathcal{R}$. Unless otherwise mentioned, subscripts on vectors/matrices will be used to denote time indexes and (whenever necessary) superscripts will be used to indicate the robot or the object that it refers to. For example, $\textbf{x}_{k}^{i}$ represents the state of robot $i$ at time instant $k$. The notation $P(\cdot)$ will be used to denote the probability of an event and the probability density function (pdf) will be denoted by $p(\cdot)$. While deriving the Belief Space Planning (BSP) framework to incorporate object uncertainties we will mainly follow the notations and formalisms in thrun2005book . We now formally define the problem that we tackle in this paper. Consider a robot operating in a partially-observable environment. The map of the environment is either known a priori or is built using a standard SLAM algorithm. At any time $k$, we denote the robot pose (or configuration) by $\textbf{x}_{k}\doteq(x_{k},y_{k},\theta_{k})$, the acquired measurement from objects is denoted by $\textbf{z}_{k}$ and the applied control action is denoted as $\textbf{u}_{k}$. Note that by objects we refer to both the landmarks and the obstacles in the environment. We consider a standard motion model with Gaussian noise $\textbf{x}_{k+1}=f(\textbf{x}_{k},\textbf{u}_{k})+\textbf{w}_{k}\ ,\ \textbf{w}_{k}\sim\mathcal{N}(0,R_{k})$ (1) where $\textbf{w}_{k}$ is the random unobservable noise, modeled as a zero mean Gaussian. We note that modeling the random unobservable noise variables as Gaussians with zero mean is a common practice in robotics thrun2005book . The objects are detected through the robot’s sensors and, assuming data association is known, the observation model can be written as $\textbf{z}_{k}=h(\textbf{x}_{k},O_{k}^{i})+\textbf{v}_{k}\ ,\ \textbf{v}_{k}\sim\mathcal{N}(0,Q_{k})$ (2) where $O_{k}^{i}$ is the detected $i$-th object and $\textbf{v}_{k}$ is the zero mean Gaussian noise. The function $h(\mathbf{x}_{k},O_{k}^{i})$ denotes the fact that at time $k$, the measurement $\mathbf{z}_{k}$ is obtained by observing the $i-th$ object $O_{k}^{i}$ from viewpoint (robot location) $\textbf{x}_{k}$. In the case of a laser-range finder the function $h$ could be defined as the distance between $\mathbf{x}_{k}$ and the location of the object (or any particular point on the object) $O_{k}^{i}$. If we consider the case of a camera, $h$ may be defined as a pinhole projection operator, projecting the object $O_{k}^{i}$ onto the image plane. Given the models in (1) and (2), in this paper we focus on two aspects. First, we consider the object uncertainties while localizing the robot. Second, we compute the exact probability of collision under obstacle uncertainty, which is modeled as a Gaussian distribution. Finally, we evaluate our approach in two simulation domains: a 2D mobile robot domain and a 2D manipulator domain. It is to be noted that for the manipulator domain we will be concerned with the collision avoidance of the manipulator’s end-effector. ### 1.2 Related Work BSP has been researched extensively in the past with applications spanning a variety of areas including autonomous navigation, multi-modal planning, and active SLAM van_den_berg2012IJRR ; Kurniawati2016ISRR ; agha_mohammadi2014IJRR ; prentice2009IJRR ; thomas2019ISRR ; thomas2020STAIRS ; kaelbling2013IJRR ; pathak2018IJRR .kaelbling2013IJRR consider object uncertainty since they are planning in an unknown environment and require several measurements to obtain confidence estimates of object locations. Thus they perform active perception, that is, to look for robot actions that enhances information to reduce the object uncertainty. This context is different from ours since we consider a known environment with object uncertainty and focus on active localization incorporating these uncertainties. In pathak2018IJRR , the concept of object uncertainty is commented upon (they call it scene uncertainty); however they do not show how it affects the state estimation. Dynamic environments are considered in Kurniawati2016ISRR ; agha_mohammadi2014IJRR however the landmark/beacon locations are assumed to be known perfectly; thomas2019ISRR ; thomas2020STAIRS also consider perfect landmark locations in the context of task and motion planning. Thus most active and passive localization-based approaches focus on robot state uncertainty and assume perfect knowledge about the location of the objects in the environment. However, in practice, the environment is seldom known with high certainty and hence providing formal guarantees for safe navigation is imperative. Patil et al. patil2012ICRA estimate the probability of collision under robot state uncertainty by truncating the state distributions. In bry2011ICRA , future state distributions are predicted and the uncertainties are used to compute bounded collision probabilities. Lee et al. lee2013IROS use sigma hulls333 Sigma hulls are convex hulls of the geometry of individual robot links transformed according to the sigma points in joint space lee2013IROS . to formulate collision avoidance constraints in terms of the signed distance to the obstacles. Du Toit and Burdick dutoit2011IEEE , Park et al. park2018IEEE compute the collision probability by marginalizing the joint distribution between the robot and obstacle location. The distributions are assumed to be Gaussian and the marginalization is computed with an indicator function that is true under the collision condition. However, since there is no closed-form solution to this formulation, an approximation is assumed. Furthermore, Park et al. compute an upper bound for the collision probability. An approximation is computed using Monte Carlo Integration in lambert2008ICCARV , albeit computationally intensive. Another impressive work that uses Monte Carlo approach is Monte Carlo Motion Planning (MCMP) janson2018ISRR . This approach first solves a deterministic motion planning problem with inflated obstacles and then adjusts the inflation to compute a path that is exactly as safe as desired. Linear chance constraints are used to compute bounded collision-free trajectories with dynamic obstacles in zhu2019RAL . Axelrod et al. axelrod2018IJRR focus exclusively on obstacle uncertainty. They formalize a notion of “shadows”, which are the geometric equivalent of confidence intervals for uncertain obstacles. The shadows fundamentally give rise to loose bounds but the computational complexity of bounding the collision probability is greatly reduced. Uncertain obstacles are modelled as polytopes with Gaussian-distributed faces in shimanuki2018WAFR . Planning a collision- free path in the presence of “risk zones” is considered in salzman2017ICAPS by penalizing the time spent in these risk zones. Risk contour maps which give the risk information (uncertainties in location, size and geometry of obstacles) in uncertain environments are used in jasour2019RSS to obtain safe paths with bounded risks. A related approach for randomly moving obstacles is presented in hakobyan2019RAL . Formal verification methods have also been used to construct safe plans ding2013ICRA ; sadigh2016RSS . Most approaches discussed above compute the collision probability along a path by summing or multiplying the probabilities along different waypoints in the path. Boole’s inequality is used to decouple the total probability in terms of individual waypoint probabilities. Such approaches tend to be overly conservative and rather than computing bounded collision probabilities along a path, the bound should be checked for each configuration along a path. Moreover, in most approaches, the collision probability computed along each waypoint is an approximation of the true value. On the one hand, such approximations can overly penalize paths and could gauge all plans to be infeasible. On the other hand some approximations can be lower444For example, the approach in dutoit2011IEEE computes a value lower than the actual when the robot state covariance is small. than the true collision probability values and can lead to synthesizing unsafe plans. ### 1.3 Contributions In this paper two main theoretical contributions are presented. First, we incorporate object uncertainties in the BSP planning framework and derive the resulting Bayes filter in terms of the prediction and measurement updates of the Extended Kalman Filter (EKF). The second is the computation of the probability of collision under environment uncertainty. We formulate the collision avoidance constraint as a quadratic form in random variables. This provides an exact expression for the collision probability in terms of a converging infinite series. A notion of safety is also formalized to compute configurations that satisfy the required collision probability bounds. We make the following assumptions: (1) the uncertainties are modelled using Gaussian distributions; (2) while formulating the collision constraint, we assume that the robot and obstacles have circular geometries. However, this is by no means a limitation and the approach can be extended to objects with different geometries by considering the configuration spaces. ## 2 Object Uncertainty In this Section, we focus on a BSP formulation that incorporates object uncertainties, that is, the viewpoints whence the objects can be observed are not precisely known. We define the object space $\mathcal{O}=\\{O^{i}\|\text{$O^{i}$ is an object, and}\ 1\leq i\leq\|\mathcal{O}\|\\}$ to be the set of all objects in the environment. The motion (1) and observation (2) models can be written in a probabilistic framework as $p(\textbf{x}_{k+1}\|\textbf{x}_{k},\textbf{u}_{k})$ and $p(\textbf{z}_{k}\|\textbf{x}_{k},O_{k}^{i})$, respectively. Let us consider that at time $k$ the robot received a measurement $\textbf{z}_{k}$ which was originated by observing object $O^{i}_{k}$. Given an initial distribution $p(\textbf{x}_{0})$, and the motion and observation models $p(\textbf{x}_{k+1}\|\textbf{x}_{k},\textbf{u}_{k})$ and $p(\textbf{z}_{k}\|\textbf{x}_{k},O_{k}^{i})$, the posterior probability distribution at time $k$ is the belief $b[\textbf{x}_{k}]$ and can be written as $p(\textbf{x}_{k}\|\textbf{z}_{k},O_{k}^{i},\textbf{z}_{0\mathrel{\mathop{\mathchar 58\relax}}k-1},\textbf{u}_{0\mathrel{\mathop{\mathchar 58\relax}}k-1})$, where $O^{i}_{k}$ is the object observed at time $k$, $\textbf{z}_{0\mathrel{\mathop{\mathchar 58\relax}}k-1}\doteq\\{\textbf{z}_{0},...,\textbf{z}_{k-1}\\}$ is the sequence of measurements up to $k-1$ and $\textbf{u}_{0\mathrel{\mathop{\mathchar 58\relax}}k-1}\doteq\\{\textbf{u}_{0},...,\textbf{u}_{k-1}\\}$ is the sequence of controls up to $k-1$. Using Bayes rule and theorem of total probability, $b[\textbf{x}_{k}]$ can be expanded as $p(\textbf{x}_{k}\|\textbf{z}_{k},O_{k}^{i},\textbf{z}_{0\mathrel{\mathop{\mathchar 58\relax}}k-1},\textbf{u}_{0\mathrel{\mathop{\mathchar 58\relax}}k-1})\\\ =\eta_{k}p(\textbf{z}_{k}\|\textbf{x}_{k},O_{k}^{i})p(O_{k}^{i}\|\textbf{x}_{k})\int_{\textbf{x}_{k-1}}p(\textbf{x}_{k}\|\textbf{x}_{k-1},\textbf{u}_{k-1})b[\textbf{x}_{k-1}]$ (3) where $\eta_{k}=1/p(\textbf{z}_{k}\|\textbf{z}_{0\mathrel{\mathop{\mathchar 58\relax}}k-1},\textbf{u}_{0\mathrel{\mathop{\mathchar 58\relax}}k-1})$ is the normalization constant and $b[\textbf{x}_{k-1}]\sim\mathcal{N}(\bm{\mu_{k-1}},\Sigma_{k-1})$ is the belief at time $k-1$. The term $p(O_{k}^{i}\|\textbf{x}_{k})$ denotes the probability of observing the object $O_{k}^{i}$ from the pose $\textbf{x}_{k}$ and models the object uncertainty. Similarly, given an action $\textbf{u}_{k}$, the propagated belief can be written as $b[\bar{\textbf{x}_{k+1}}]=\int_{\textbf{x}_{k}}p(\textbf{x}_{k+1}\|\textbf{x}_{k},\textbf{u}_{k})b[\textbf{x}_{k}]$ (4) Given the current belief $b[\textbf{x}_{k}]$ and the control $\textbf{u}_{k}$, the propagated belief parameters, that is, mean and covariance, can be computed using the standard EKF prediction as $\begin{split}\bar{\bm{\mu}}_{k+1}&=f(\bm{\mu}_{k},\bm{u}_{k})\\\ \bar{\Sigma}_{k+1}&=F_{k}\Sigma_{k}F_{k}^{T}+R_{k}\end{split}$ (5) where $F_{k}$ is the Jacobian of $f(\cdot)$ with respect to $\textbf{x}_{k}$. To compute the posterior belief using EKF update equations, we first need to model the term $p(O_{k}^{i}\|\textbf{x}_{k})$. In this work we model the object distribution as a Gaussian distribution given by $p(O_{k}^{i}\|x_{k})\sim\mathcal{N}(\bm{\mu}_{O_{k}^{i}},\Sigma_{O_{k}^{i}})$ (6) where $\bm{\mu}_{O_{k}^{i}}$ is the mean viewpoint/pose that corresponds to the maximum probability of observing $O_{k}^{i}$ and $\Sigma_{O_{k}^{i}}$ is the associated covariance. For convenience we state the probability density function (pdf) of multivariate Gaussian distributions. For $\textbf{x}\sim\mathcal{N}(\bm{\mu},\Sigma)$ the pdf is of the form $p(\textbf{x})=det\left(2\pi\Sigma\right)^{-\frac{1}{2}}\textrm{exp}\left(-\frac{1}{2}(\textbf{x}-\bm{\mu})^{T}\Sigma^{-1}(\textbf{x}-\bm{\mu})\right)$ (7) where $det(\cdot)$ denotes the determinant. Expanding the right hand side of (3), we have $b[\textbf{x}_{k+1}]=\eta^{\prime}_{k}\int\text{exp}(-\mathcal{J}_{k+1})$, where $\eta^{\prime}_{k}$ contains the non-exponential terms and $\mathcal{J}_{k+1}$ is given by $\mathcal{J}_{k+1}=\frac{1}{2}\left(\textbf{z}_{k+1}-h\left(\bar{\bm{\mu}}_{k+1}\right)-H_{k+1}\left(\textbf{x}_{k+1}-\bar{\bm{\mu}}_{k+1}\right)\right)^{T}\\\ Q_{k+1}^{-1}\left(\textbf{z}_{k+1}-h\left(\bar{\bm{\mu}}_{k+1}\right)-H_{k+1}\left(\textbf{x}_{k+1}-\bar{\bm{\mu}}_{k+1}\right)\right)\\\ +\frac{1}{2}(\bm{x}_{k+1}-\bm{\mu}_{O_{k+1}^{i}})^{T}\Sigma_{O_{k+1}^{i}}^{-1}(\textbf{x}_{k+1}-\bm{\mu}_{O_{k+1}^{i}})\\\ +\frac{1}{2}(\textbf{x}_{k+1}-\bar{\bm{\mu}}_{k+1})^{T}\bar{\Sigma}_{k+1}^{-1}(\textbf{x}_{k+1}-\bar{\bm{\mu}}_{k+1})$ (8) where $H_{k+1}$ is the Jacobian of $h(\cdot)$ with respect to $\textbf{x}_{k+1}$. We note that when object uncertainty is not considered, the second term in (8) disappears and the results that we derive below reduce to that of the standard EKF update case. The parameters of this Gaussian can be obtained by taking the first and second derivatives of $\mathcal{J}_{k+1}$ with respect to $\textbf{x}_{k+1}$, $\frac{\partial\mathcal{J}_{k+1}}{\partial\textbf{x}_{k+1}}=-H_{k+1}^{T}Q_{k+1}^{-1}\left(\textbf{z}_{k+1}-h(\bar{\bm{\mu}}_{k+1})-\right.\\\ \left.H_{k+1}(\textbf{x}_{k+1}-\bar{\bm{\mu}}_{k+1})\right)+\Sigma_{O_{k+1}^{i}}^{-1}\left(\textbf{x}_{k+1}-\bm{\mu}_{O_{k+1}^{i}}\right)+\\\ \bar{\Sigma}_{k+1}^{-1}\left(\textbf{x}_{k+1}-\bar{\bm{\mu}}_{k+1}\right)$ (9) $\frac{\partial^{2}\mathcal{J}_{k+1}}{\partial\textbf{x}_{k+1}^{2}}=H_{k+1}^{T}Q_{k+1}^{-1}H_{k+1}+\Sigma_{O_{k+1}^{i}}^{-1}+\bar{\Sigma}_{k+1}^{-1}$ (10) The term (10) is the inverse of the covariance of $b[\textbf{x}_{k+1}]$ thrun2005book , that is, $\Sigma_{k+1}=\left(H_{k+1}^{T}Q_{k+1}^{-1}H_{k+1}+\Sigma_{O_{k+1}^{i}}^{-1}+\bar{\Sigma}_{k+1}^{-1}\right)^{-1}$ (11) Since the mean of $b[\textbf{x}_{k+1}]$ is the value that minimizes $\mathcal{J}_{k+1}$, it is obtained by equating (9) to zero $H_{k+1}^{T}Q_{k+1}^{-1}\left(\textbf{z}_{k+1}-h\left(\bar{\bm{\mu}}_{k+1}\right)-H_{k+1}\left(\textbf{x}_{k+1}-\bar{\mu}_{k+1}\right)\right)\\\ =\Sigma_{k+1}^{-1}\left(\bm{\mu}_{k+1}-\bar{\bm{\mu}}_{k+1}\right)-\Sigma_{O_{k+1}^{i}}^{-1}\left(\bm{\mu}_{O_{k+1}^{i}}-\bar{\bm{\mu}}_{k+1}\right)\\\ \implies\bm{\mu}_{k+1}=\bar{\bm{\mu}}_{k+1}+K_{k+1}\left(\textbf{z}_{k+1}-h\left(\bar{\bm{\mu}}_{k+1}\right)\right)\\\ +\Sigma_{k+1}\Sigma_{O_{k+1}^{i}}^{-1}\left(\bm{\mu}_{O_{k+1}^{i}}-\bar{\bm{\mu}}_{k+1}\right)$ (12) where $K_{k+1}=\Sigma_{k+1}H_{k+1}^{T}Q_{k+1}^{-1}$ is the Kalman gain. As in the case of standard EKF, the gain $K_{k+1}$ can be transformed to an expression that does not depend on $\Sigma_{k+1}$, by post-multiplying with an identity matrix $I=AA^{-1}$, where $A=\\\ \left(H_{k+1}\bar{\Sigma}_{k+1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\Sigma_{O_{k+1}^{i}}H_{k+1}^{T}+Q_{k+1}\right)$ (13) This gives $K_{k+1}=\Sigma_{k+1}\left(H_{k+1}^{T}Q_{k+1}^{-1}H_{k+1}\bar{\Sigma}_{k+1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\right.\\\ \left.\Sigma_{O_{k+1}^{i}}H_{k+1}^{T}+H_{k+1}^{T}\right)A^{-1}$ (14) In order to simplify the above expression for $K_{k+1}$, we first compute the inverse of the term $\bar{\Sigma}_{k+1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\Sigma_{O_{k+1}^{i}}$ (15) The inverse is computed as $\left(\bar{\Sigma}_{k+1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\Sigma_{O_{k+1}^{i}}\right)^{-1}\\\ =\Sigma_{O_{k+1}^{i}}^{-1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)\bar{\Sigma}_{k+1}^{-1}\\\ =\Sigma_{O_{k+1}^{i}}^{-1}\bar{\Sigma}_{k+1}\bar{\Sigma}_{k+1}^{-1}+\Sigma_{O_{k+1}^{i}}^{-1}\Sigma_{O_{k+1}^{i}}\bar{\Sigma}_{k+1}^{-1}\\\ =\Sigma_{O_{k+1}^{i}}^{-1}+\bar{\Sigma}_{k+1}^{-1}$ (16) Using (16) and (11), the expression in (14) simplifies to $K_{k+1}=\Sigma_{k+1}\left(H_{k+1}^{T}Q_{k+1}^{-1}H_{k+1}+\Sigma_{O_{k+1}^{i}}^{-1}+\bar{\Sigma}_{k+1}^{-1}\right)\bar{\Sigma}_{k+1}\\\ \left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\Sigma_{O_{k+1}^{i}}H_{k+1}^{T}\\\ \left(H_{k+1}\bar{\Sigma}_{k+1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\Sigma_{O_{k+1}^{i}}H_{k+1}^{T}+Q_{k+1}\right)^{-1}\\\ =\bar{\Sigma}_{k+1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\Sigma_{O_{k+1}^{i}}H_{k+1}^{T}\\\ \left(H_{k+1}\bar{\Sigma}_{k+1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\Sigma_{O_{k+1}^{i}}H_{k+1}^{T}+Q_{k+1}\right)^{-1}\\\ \vspace{-0.2cm}$ (17) By treating the sum $\Sigma_{O_{k+1}^{i}}^{-1}+\bar{\Sigma}_{k+1}^{-1}$ in (11) as a single term and applying the matrix inversion lemma on the right hand side of (11) and further simplifying using the expression for the inverse computed in (16), it can be shown that $\Sigma_{k+1}=\left(I-K_{k+1}H_{k+1}\right)\bar{\Sigma}_{k+1}\left(\bar{\Sigma}_{k+1}+\Sigma_{O_{k+1}^{i}}\right)^{-1}\Sigma_{O_{k+1}^{i}}$ (18) We note that when no object uncertainty is considered the update step of the standard EKF gives $\bm{\mu}_{k+1}=\bar{\bm{\mu}}_{k+1}+K_{k+1}\left(\textbf{z}_{k+1}-h\left(\bar{\bm{\mu}}_{k+1}\right)\right)$ and $\Sigma_{k+1}=\left(I-K_{k+1}H_{k+1}\right)\bar{\Sigma}_{k+1}$. The additional term in (12) rightly adjusts the mean $\bm{\mu}_{k+1}$ accounting for the fact that the object location is uncertain. Similarly, the extra terms in (18) account for the object uncertainty and scale the posterior covariance accordingly. ## 3 Collision Probability Let $\mathcal{R}$ represent the set of all points occupied by a rigid-body robot at any given time. Thus, $\mathcal{R}$ represents the collection of points that form the rigid-body robot. Similarly, let $\mathcal{S}$ represent the set of all points occupied by a rigid-body obstacle. A collision occurs if $\mathcal{R}\cap\mathcal{S}\neq\\{\phi\\}$ and we denote the probability of collision as $P\left(\mathcal{R}\cap\mathcal{S}\neq\\{\phi\\}\right)$. In this work we assume circular geometries for $\mathcal{R}$ and $\mathcal{S}$ with radii $r_{1}$ and $s_{1}$, receptively and we denote the center of mass of the robot and the obstacle by $\textbf{x}_{k}$ and s, receptively. By abuse of notation we will use $\textbf{x}_{k}$ and s equivalently to $\mathcal{R}$ and $\mathcal{S}$. The collision condition will be written in terms of the center of mass as $\mathcal{C}_{\textbf{x}_{k},\textbf{s}}\mathrel{\mathop{\mathchar 58\relax}}\mathcal{R}\cap\mathcal{S}\neq\\{\phi\\}$. It is noteworthy that both $\textbf{x}_{k}$ and s are not known precisely but can only be estimated probabilistically, as seen in the previous section. At this point we would like to stress the fact that the concepts and the derivations herein are valid for any 2D rigid-body robot. A mobile robot may be represented by a minimum area enclosing circle. In the case of a 2D manipulator robot each link can be approximated by bounding circles that tightly enclose the link. For such robots, the collision with an obstacle has to be checked for each bounding circle. For example, consider a manipulator robot with $l$ bounding circles. Then the collision condition for the $i-$th circle ($1\leq i\leq l$) is given by $\mathcal{C}_{\textbf{x}_{k}^{i},\textbf{s}}$, where $\textbf{x}_{k}^{i}$ is the center of the $i-$th circle. Let us now consider an obstacle at any given time instant, distributed according to the Gaussian $\textbf{s}\sim\mathcal{N}\left(\bar{\textbf{s}},\Sigma_{s}\right)$, where $\bar{s}$ represents the mean and $\Sigma_{s}$ the uncertainty in the estimation of the object. Given the belief at time $k$, that is, $b[\textbf{x}_{k}]$, the probability of collision is given by $P\left(\mathcal{C}_{\textbf{x}_{k},\textbf{s}}\right)=\int_{\textbf{x}_{k}}\int_{\textbf{s}}I_{c}(\textbf{x}_{k},\textbf{s})p(\textbf{x}_{k},\textbf{s})$ (19) where $\mathcal{C}_{\textbf{x}_{k},\textbf{s}}$ as defined above represents the fact that robot configuration $\textbf{x}_{k}$ and its collision with obstacle at location s is considered, and $I_{c}$ is an indicator function defined as $I_{c}(\textbf{x}_{k},\textbf{s})=\begin{cases}1\ &\text{if}\ \mathcal{R}\cap\mathcal{S}\neq\\{\phi\\}\\\ 0\ &\text{otherwise}.\end{cases}$ (20) Du Toit and Burdick dutoit2011IEEE , Park et al. park2018IEEE approximate the integral in (19) as $Vp(\textbf{x}_{k},\textbf{s})$, where $V$ is the 2D footprint (area) occupied by the robot. For this approximation, in dutoit2011IEEE it is assumed that the robot radius $\varepsilon$ is negligible and a point obstacle is considered for this derivation. To do away with this approximation, we formulate the above problem by considering an alternative approach. Since the robot and obstacle are assumed to be spherical objects, the collision constraint can be written as $\mathinner{\\!\left\lVert\textbf{x}_{k}-\textbf{s}\right\rVert}^{2}\leq(r_{1}+s_{1})^{2}$ (21) where $\textbf{x}_{k}$ and s are the random vectors that denote the robot and obstacle pose respectively. Here, $\textbf{x}_{k}$ and s corresponds to the body-fixed frames in the global frame. As noted before, the two random vectors in (19) are distributed according to $\textbf{s}\sim\mathcal{N}\left(\mathbf{\bar{\textbf{s}}},\Sigma_{s}\right)$ and $\textbf{x}_{k}\sim\mathcal{N}\left(\bm{\mu}_{k},\Sigma_{k}\right)$. Let us denote by $\textbf{w}=\textbf{x}_{k}-\textbf{s}$, the difference between the two random variables. Then we know that w is also a Gaussian, distributed as $\textbf{w}\sim\mathcal{N}\left(\bm{\mu}_{k}-\bm{\bar{s}},\Sigma_{k}+\Sigma_{s}\right)$. The collision constraint can now be written as $\textbf{v}=\mathinner{\\!\left\lVert\textbf{w}\right\rVert}^{2}=\textbf{w}^{T}\textbf{w}\leq(r_{1}+s_{1})^{2}$ (22) where v is a random vector distributed according to the squared $L_{2}$-norm of w. Now, given the probability density function (pdf) of v, the collision constraint in (21)reduces to solving the integral $P\left(\mathcal{C}_{\textbf{x}_{k},\textbf{s}}\right)=\int_{0}^{(r_{1}+s_{1})^{2}}p(v)$ (23) where $p(v)=P_{\textbf{v}}(\textbf{v}=v)$ is the pdf of v. It is noteworthy that the above expression is the cumulative distribution function (cdf) of v, which is defined as $F_{\textbf{v}}\left((r_{1}+s_{1})^{2}\right)=P\left(\textbf{v}\leq(r_{1}+s_{1})^{2}\right)$. ### 3.1 Quadratic Form in Random Variables A quadratic form in random variables is defined as provost1992book , ###### Definition 1 Let $\textbf{x}=\left(x_{1},\ldots,x_{n}\right)^{T}$ denote a random vector with mean $\bm{\mu}=\left(\mu_{1},\ldots,\mu_{n}\right)^{T}$ and covariance matrix $\Sigma$. Then the quadratic form in the random variables $x_{1},\ldots,x_{n}$ associated with an $n\times n$ symmetric matrix $A=(a_{ij})$ is $Q(\textbf{x})=Q(x_{1},\ldots,x_{n})=\textbf{x}^{T}A\textbf{x}=\sum_{i=1}^{n}\sum_{j=1}^{n}a_{ij}X_{i}X_{j}$ (24) Let us define $\textbf{y}=\Sigma^{-\frac{1}{2}}\textbf{x}$ and define a random vector $\textbf{z}=\left(\textbf{y}-\Sigma^{-\frac{1}{2}}\bm{\mu}\right)$. The resulting distribution of z is thus zero mean with covariance being the identity matrix. Thus the quadratic form becomes $Q(\textbf{x})=\left(\textbf{z}+\Sigma^{-\frac{1}{2}}\bm{\mu}\right)^{T}\Sigma^{\frac{1}{2}}A\Sigma^{\frac{1}{2}}\left(\textbf{z}+\Sigma^{-\frac{1}{2}}\bm{\mu}\right)$ (25) Suppose there exists an orthogonal matrix $P$, that is, $PP^{T}=I$ which diagonalizes $\Sigma^{\frac{1}{2}}A\Sigma^{\frac{1}{2}}$, then $P^{T}\Sigma^{\frac{1}{2}}A\Sigma^{\frac{1}{2}}P=\textrm{diag}\left(\lambda_{1},\ldots,\lambda_{n}\right)$, where $\lambda_{1},\ldots,\lambda_{n}$ are the eigenvalues of $\Sigma^{\frac{1}{2}}A\Sigma^{\frac{1}{2}}$. The quadratic form can now be written as $\begin{split}Q(\textbf{x})&=\left(\textbf{z}+\Sigma^{-\frac{1}{2}}\bm{\mu}\right)^{T}\Sigma^{\frac{1}{2}}A\Sigma^{\frac{1}{2}}\left(\textbf{Z}+\Sigma^{-\frac{1}{2}}\bm{\mu}\right)\\\ &=\left(\textbf{u}+\textbf{b}\right)^{T}\textrm{diag}\left(\lambda_{1},\ldots,\lambda_{n}\right)\left(\textbf{u}+\textbf{b}\right)\end{split}$ (26) where $\textbf{u}=P^{T}\textbf{z}=(u_{1},\ldots,u_{n})^{T}$ and $\textbf{b}=P^{T}\Sigma^{-\frac{1}{2}}\bm{\mu}=(b_{1},\ldots,b_{n})^{T}$. The expression in (26) can be written concisely, $Q(\textbf{x})=\textbf{x}^{T}A\textbf{x}=\sum_{i=1}^{n}\lambda_{i}(u_{i}+b_{i})^{2}$ (27) ###### Theorem 3.1 The cdf of $Q(\textnormal{{x}})=\textnormal{{y}}=\textnormal{{x}}^{T}A\textnormal{{x}}$ with $A=A^{T}>0,\textnormal{{x}}\sim\mathcal{N}(\bm{\mu},\Sigma),\Sigma>0$ is $F_{\textnormal{{y}}}(y)=P(\textnormal{{y}}\leq y)=\sum_{k=0}^{\infty}(-1)^{k}c_{k}\frac{y^{\frac{n}{2}+k}}{\Gamma\left(\frac{n}{2}+k+1\right)}$ (28) and its pdf is given by $p_{\textnormal{{y}}}(y)=P(\textnormal{{y}}=y)=\sum_{k=0}^{\infty}(-1)^{k}c_{k}\frac{y^{\frac{n}{2}+k-1}}{\Gamma\left(\frac{n}{2}+k\right)}$ (29) where $\Gamma$ denotes the gamma function and $\begin{split}&c_{0}=\textrm{exp}(-\frac{1}{2}\sum\limits_{i=1}^{n}b_{i}^{2})\prod_{i=1}^{n}\left(2\lambda_{i}\right)^{-\frac{1}{2}}\\\ &c_{k}=\frac{1}{k}\sum\limits_{i=0}^{k-1}d_{k-i}c_{i}\\\ &d_{k}=\frac{1}{2}\sum\limits_{i=1}^{n}\left(1-kb_{i}^{2}\right)\left(2\lambda_{i}\right)^{-k}\end{split}$ The proof of the above theorem is beyond the scope of this paper and we refer the interested readers to provost1992book . It is easily seen that the left hand side of (22), is in the quadratic form $Q(\textbf{y})$ with $A=I$, the identity matrix. Thus the collision probability can be computed from (28) as $P\left(\mathcal{C}_{\textbf{x}_{k},\textbf{s}}\right)=F_{\textbf{y}}\left((r_{1}+s_{1})^{2}\right)$ (30) ### 3.2 Convergence and Truncation Error In this section we will prove the convergence the infinite series in (28) and (29). Note that the series expansion of the pdf in Theorem 29 is of the form $p_{\textbf{y}}(y)=\sum\limits_{k=0}^{\infty}c_{k}h_{k}(y)$ (31) From kotz1967AMS we have the following lemma. ###### Lemma 1 Let $\\{h_{k}\\}_{0}^{\infty}$ be a sequence of measurable complex valued functions on $[0,\infty]$ and $\\{c_{k}\\}_{0}^{\infty}$ be a sequence of complex numbers such that $\sum_{k=0}^{\infty}\|c_{k}\|\|h_{k}(y)\|\leq\alpha e^{(\beta y)}\ \textrm{for}\ y\in[0,\infty]$ (32) where $\alpha$, $\beta$ are real constants. Then $L\left(h_{k}(y)\right)$ and $L(p_{\textbf{y}}(y))$ exist for $Re(s)>\beta$, and $L(p_{\textbf{y}}(y))=\sum_{k=0}^{\infty}c_{k}L(h_{k}(s))$ (33) where $L(\cdot)$ denotes the Laplace transform. Let us now define the term $M(\theta)$ such that $M(\theta)=\sum_{k=0}^{\infty}c_{k}\theta^{k}$ (34) where the infinite series is a uniformly convergent series for $\theta$ in some region with $M(\theta)>0$. Let the Laplace transform of $h_{k}(y)$ be the form $L(h_{k}(y))=\xi(s)\eta^{k}(s)$, where, for $Re(s)>\beta$ with $\beta$ being a real constant, $\xi(s)$ is a non-vanishing analytic function and $\eta(s)$ is an analytic function with an inverse function $\eta(\zeta(\theta))=\theta$. For $h_{k}(y)$ in (29), we have, $\xi(s)=(2s)^{-n/2}$, $\eta(s)=-(2s)^{-1}$ and $\zeta(\theta)=-(2\theta)^{-1}$. Now let us define, $\begin{split}M(\theta)&=\left(L(p_{\textbf{y}})\circ\zeta/\xi\circ\zeta\right)(\theta)=\sum_{k=0}^{\infty}c_{k}\theta^{k}\end{split}$ (35) where $\circ$ denotes function composition. Using Cauchy’s inequality, we get $\|c_{k}\|\leq\frac{m(\rho)}{\rho^{k}},\ \ m(\rho)=\textrm{max}_{\|\theta\|=\rho}\|M(\theta)\|$ (36) Since $h_{k}(y)$ is bounded and, using (36), the condition (32) in Lemma 33 is satisfied and the series $p_{\textbf{y}}(y)$ converges uniformly in every bounded interval of $y>0$. As a result, integrating $p_{\textbf{y}}(y)$ term- by-term, the obtained series $F_{\textbf{y}}(y)$ is uniformly convergent in every bounded interval of $y>0$. If the series in (29) is truncated after $N$ terms, the truncation error is $e(N)=\sum_{k=N+1}^{\infty}\|c_{k}h_{k}(y)\|=\left\|\sum_{k=N+1}^{\infty}c_{k}\frac{y^{\frac{n}{2}+k-1}}{\Gamma\left(\frac{n}{2}+k\right)}\right\|$ (37) Using (36), an upper bound for the truncation error can hence be obtained as $e(N)\leq\frac{m(\rho)}{\rho^{k}}\left\|\sum_{k=N+1}^{\infty}\frac{y^{\frac{n}{2}+k-1}}{\Gamma\left(\frac{n}{2}+k\right)}\right\|$ (38) where the summation term can be further simplified using the gamma function identity, $\forall\varsigma>0,\ \Gamma(\varsigma+1)=\varsigma\Gamma(\varsigma)$, giving $e(N)\leq m(\rho)\left(\Gamma\left(\frac{n}{2}\right)N!\right)^{-1}(\frac{y}{2})^{\frac{n}{2}-1}(\frac{y}{2\rho})^{N+1}\text{exp}(\frac{y}{2\rho})$ (39) The truncation error for (28) is obtained in a similar manner, $\displaystyle E(N)\leq m(\rho)\left(\Gamma\left(\frac{n}{2}\right)(N+1)!\right)^{-1}(\frac{y}{2})^{\frac{n}{2}}(\frac{y}{2\rho})^{N+1}\text{exp}(\frac{y}{2\rho})$ (40) The expression for $m(\rho)$ is obtained from kotz1967AMS2 , $m(\rho)=\prod_{j=1}^{n}\lambda_{j}^{-\frac{1}{2}}\text{exp}\left(-\frac{1}{2}\sum_{j=1}^{n}\frac{b_{j}^{2}\lambda_{j}}{\lambda_{j}+\rho}\right)\prod_{j=1}^{n}(1-\frac{\rho}{\lambda_{j}})^{-\frac{1}{2}}$ (41) The expression in (41) is valid only if $\rho<\lambda_{j}$ kotz1967AMS and hence $\rho<\textrm{min}\ \lambda_{j}$. Thus we have $m(\rho)\rightarrow 0$ with $\sum_{j=1}^{n}b_{j}^{2}\rightarrow\infty$. The larger the distance from the obstacles and the higher the certainty in the robot and obstacle positions, the greater is the $b_{j}$ (see 26) value. In such scenarios, convergence is often attained within the first few terms of the series. For a given robot configuration and obstacle parameters, we see that the only varying term in (40) is $(y/2\rho)^{N+1}/(N+1)!$ which depends on $\lambda_{j}$’s, that is the eigenvalues of $\Sigma_{k}+\Sigma_{s}$. Clearly, at time instant $k$, the parameter that influences the convergence is the degree of uncertainty in both the robot and obstacle location, that is, $\Sigma_{k}+\Sigma_{s}$. The convergence is visualized for different configurations in Fig. 2. The blue and green circles represent a robot and an obstacle, respectively. The red ellipses corresponds to the 3$\sigma$ uncertainties for different covariances $diag(0.04,0.04),\ diag(0.08,0.08),\ \ldots,\ diag(0.74,0.74)$. In Fig. 2(a) the robot and the obstacle are touching each other. For each of these covariances, the number of terms for convergence is shown in Fig. 2(b). The worst case corresponds to the covariance of $diag(0.04,0.04)$, requiring 16 terms for convergence (dashed blue line with spikes in Fig. 2(b)). In Fig. 2(c) the distance between the robot and the obstacle is increased by 0.2$m$ and the covariance $diag(0.04,0.04)$ needed 12 terms for convergence. The distances are further increased by 0.4$m$ and 0.8$m$ in Fig. 2(e), (g) and their worst case convergences are 9 and 5 respectively as seen in Fig.2(f), (h). The number of terms for worst case convergence that corresponds to covariance $diag(0.04,0.04)$ and the respective time for collision probability computation are shown in Table 1. Configuration \| Terms for convergence \| Computation time (s) ---\|---\|--- A \| 16 \| 0.0412 $\pm$ 0.0086 B \| 12 \| 0.0044 $\pm$ 0.0041 C \| 9 \| 0.0008 $\pm$ 0.0003 D \| 5 \| 0.0004 $\pm$ 0.0002 Table 1: The maximum number of terms required for convergence and the corresponding collision probability computation time. The values correspond to the covariance $diag(0.04,0.04)$ for each of the configurations. (a) Configuration A (b) Collision probability evolution (c) Configuration B (d) Collision probability evolution (e) Configuration C (f) Collision probability evolution (g) Configuration D (h) Collision probability evolution Figure 2: Different configurations for a robot of radius 0.3$m$ and obstacle of radius 0.5$m$. For each configuration the evolution of probability of collision is plotted for different covariances. In each of the 4 configurations, maximum terms for convergence is for the minimum covariance of $diag(0.04,0.04)$. ### 3.3 Safe Configuration In the presence of perception and motion uncertainty, providing safety guarantees for robot motion is imperative. Let us assume that the obstacle position is known with high certainty as a result of perfect sensing. However, since the true state of the robot is not known and only a distribution of these states can be estimated, collision checking has to be performed for this distribution of states. Moreover, in practice, the observations are noisy and this renders the estimated obstacle location (and shape) uncertain. Hence, this uncertainty should be taken into account while considering collision avoidance. Given a robot configuration $\textbf{x}_{k}$, we define the following notion of $\epsilon-$safe configuration. ###### Definition 2 A robot configuration $\textbf{x}_{k}$ is an $\epsilon-$safe configuration with respect to an obstacle location s, if the probability of collision is such that $P\left(\mathcal{C}_{\textbf{x}_{k},\textbf{s}}\right)\leq 1-\epsilon$. For example, a $0.99-$safe configuration implies that the probability of this configuration colliding with the obstacle is at most $0.01$. We use the sampling based Probabilistic Roadmap (PRM) kavraki1996IEEE to compute motion plans. As a result we can only guarantee probabilistic completeness for returning $\epsilon-$safe configurations since the PRM motion planner is probabilistically complete karaman2011IJRR , that is the probability of failure decays to zero exponentially with the number of samples used in the construction of the roadmap. The failure to find an $\epsilon-$safe configuration might be because such a configuration indeed does not exist or simply because there were not enough samples. ### 3.4 Complexity Analysis It is known that for $m$ nodes, the computational complexity of PRM is $O(m\log m)$ karaman2011IJRR . First let us consider the case of belief space planning over the PRM graph, without computing the collision probabilities. Finding a trajectory to the goal requires performing Bayesian (EKF) update operations. This basically involves performing matrix operations— matrix multiplication and inversion of matrices. For a state of dimension $n$, the covariance matrix is of dimension $O(n^{2})$. Therefore, each step of the Bayesian update has a complexity of $O(n^{3})$. If $T$ denotes the number of time steps in the trajectory, then the overall computational complexity is $O(n^{3}T)$. Let us now analyze the complexity of collision probability computation. From (40) we see that for each iteration, the truncation error varies with $(y/2\rho)$. Therefore, to achieve $E(N)\leq\delta$, for an $\epsilon-$safe configuration, $k=O\left(\log\frac{\delta\rho}{y(1-\epsilon)}\right)$ iterations are required. We note that for each obstacle, the runtime is increased by this factor. ## 4 Cost Function At each time instant the robot is required to minimize its control usage and proceed towards the goal $\textbf{x}^{g}$, while minimizing its state uncertainty. We quantify the state uncertainty by computing the trace of the marginal covariance of the robot position. As a result, we have the following cost function $c\doteq\mathinner{\\!\left\lVert\xi(\textbf{u}_{k})\right\rVert}^{2}_{M_{u}}+\mathinner{\\!\left\lVert\textbf{x}_{k}-\textbf{x}^{g}\right\rVert}^{2}_{M_{g}}+tr\left(\mathinner{\\!\left\lVert M_{\Sigma}\right\rVert}^{2}_{\Sigma_{k}}\right)+M_{C}P(\mathcal{C})$ (42) where $\mathinner{\\!\left\lVert x\right\rVert}_{S}=\sqrt{x^{T}Sx}$ is the Mahalanobis norm, $M_{u},M_{g},M_{C}$ are weight matrices and $\xi(\textbf{u}_{k})$ is a function that quantifies control usage. The choice of weight matrices and the control function vary with application. The term $tr\left(\mathinner{\\!\left\lVert M_{\Sigma}\right\rVert}^{2}_{\Sigma_{k}}\right)=tr\left(M_{\Sigma}^{T}\Sigma_{k}M_{\Sigma}\right)$, returns the marginal covariance of the robot location. Therefore, $M_{\Sigma}=\tau\bar{M}_{\Sigma}$, where $\tau$ is a positive scalar and $\bar{M}_{\Sigma}$ is a matrix filled with zero or identity entries. $P(\mathcal{C})$ represents the probability of collision and $M_{C}$ penalizes the belief states with higher collision probabilities. The failure to find an $\epsilon-$safe configuration might be because such a configuration indeed does not exist or simply because there was not enough samples in the roadmap. In such scenario the roadmap has to be extended. Different strategies could be implemented to efficiently extend the roadmap but is not the main focus of the current paper. Therefore we follow a straightforward approach to add more samples when an $\epsilon-$safe configuration cannot be found. Given a node from which no $\epsilon-$safe configuration can be found, a circle of certain radius (half the maximum distance allowed between two edges) is drawn. Samples are then added to the roadmap and the PRM graph is updated until an $\epsilon-$safe configuration is found or until time-out. ## 5 Simulation Results In this section we first provide a comparison of our approach with park2018IEEE and dutoit2011IEEE . We then explore the capabilities of our approach in two simulation domains. Performance are evaluated on an Intel® Core i7-6500U<EMAIL_ADDRESS>with 8GB RAM under Ubuntu 16.04 LTS. ### 5.1 Comparison to Other Approaches (a) (b) (c) Figure 3: Comparison of our approach to other methods. (a) The robot state is known perfectly, however the obstacle location is uncertain. (b) Robot state uncertainty is considered (contours in blue). The collision probability value computed with park2018IEEE gave a much higher value. (c) A point-like robot and obstacle are considered. The values computed with park2018IEEE ; dutoit2011IEEE are much lower than expected. Case \| Algorithm \| Collision probability \| Computation time (s) \| Feasible ---\|---\|---\|---\|--- (a) \| Numerical integral \| $4.62\%$ \| 0.8896 $\pm$ 0.0356 \| Yes Du Toit and Burdick dutoit2011IEEE \| $5.84\%$ \| 0.0026 $\pm$ 0.0003 \| Yes Park et al. park2018IEEE \| $33.26\%$ \| 0.2367 $\pm$ 0.2081 \| No Our approach \| $4.61\%$ \| 0.0232 $\pm$ 0.0024 \| Yes (b) \| Numerical integral \| $8.25\%$ \| 1.2309 $\pm$ 0.0298 \| Yes Du Toit and Burdick dutoit2011IEEE \| $14.20\%$ \| 0.0021$\pm$ 0.0001 \| No Park et al. park2018IEEE \| $36.31\%$ \| 0.2108 $\pm$ 0.3067 \| No Our approach \| $8.22\%$ \| 0.0208 $\pm$ 0.0021 \| Yes (c) \| Numerical integral \| $14.82\%$ \| 1.2450 $\pm$ 0.0301 \| No Du Toit and Burdick dutoit2011IEEE \| $0.46\%$ \| 0.0019 $\pm$ 0.0004 \| Yes Park et al. park2018IEEE \| $0.61\%$ \| 0.3145 $\pm$ 0.4610 \| Yes Our approach \| $14.83\%$ \| 0.0271 $\pm$ 0.0087 \| No Table 2: Comparison of collision probability methods. Park et al. park2018IEEE approximate the integral in (19) as $Vp(\textbf{x}_{k},\textbf{s})$, where $V$ is the 2D footprint or area occupied by the robot. For computing $p(\textbf{x}_{k},\textbf{s})$, they first assume a distribution centered around the obstacle with the covariance being the sum of the robot and obstacle location uncertainties. The collision probability is then computed by finding the $\textbf{x}_{k}$ that maximizes $p(\textbf{x}_{k},\textbf{s})$ and formulate the problem as an optimization problem with a Lagrange multiplier. In dutoit2011IEEE the density of the center of the robot is used. For comparing with these approaches, we formulate the problem as given in each of these works555For the comparison, the approaches in dutoit2011IEEE ; park2018IEEE have been reproduced to the best our understanding and the reproduced codes (including numerical integration and our approach) can be found here— https://bitbucket.org/1729antony/comparison_cp_methods/src/master/. In order to validate the values computed using our approach, we perform numerical integration of the expression in (19), which gives the exact collision probability value. Three different cases are considered as shown in Fig. 3. The solid green circle denotes an obstacle of radius 0.5m and its corresponding uncertainty contours are shown as green circles. The solid blue circle denotes a robot of radius 0.3m with the blue circles showing the Gaussian contours. We define a collision probability threshold of $0.1$, that is, a $0.9-$safe configuration. The collision probability values and the computation times are provided in Table 2. In Fig. 3(a), the robot position known with high certainty and our approach computes collision probability as $4.61\%$ and hence the given configuration is a $0.9-$safe configuration. The numerical integral provides the actual value and as seen in Table 2, it is computed to be $4.62\%$, thus proving the exactness of our method. However, the collision probability computed as given in park2018IEEE is $33.26\%$ (almost seven times our value), predicting the configuration to be unsafe. The approach in dutoit2011IEEE gave the value of $5.84\%$, a much tighter upper bound. In Fig. 3(b), there is robot uncertainty along the horizontal axis and the collision probability computed using our approach is $8.22\%$. The actual value is computed to be $8.25\%$. As compared to the previous case, the probability has almost doubled. This is quite intuitive as seen from the robot uncertainty spread and hence there is greater chance for intersection between the robot and the obstacle. The value computed using the approach in park2018IEEE is $36.31\%$ ($4.5$ times our value). The approach in dutoit2011IEEE also gave a higher value of $14.20\%$. Unlike the approaches in dutoit2011IEEE ; park2018IEEE our approach rightly predicts the configuration to be a $0.9-$safe configuration. The higher values obtained using dutoit2011IEEE ; park2018IEEE are due to the overly conservative nature of the estimates. The approach of Park et al. park2018IEEE and dutoit2011IEEE assumes that the robot radius is very small. We also compute the collision probabilities for a robot and an obstacle with radius $0.05$m each, where the robot and the obstacle are touching each other (Fig. 3(c)). The obstacle location is also much more certain, with the uncertainty reduced by $97\%$ as compared to cases in Fig. 3(a),(b). Actual value obtained using numerical integral is $14.82\%$. The probability of collision computed using our approach is $14.83\%$, whereas, using the approach in park2018IEEE the computed value is $0.61\%$ and the approach in dutoit2011IEEE computes it to be $0.46\%$. Thus our approach predicts the configuration to be unsafe. To get a sense of the actual value, we compute the area of the covariance matrix, which is $6.28\times 10^{-4}m^{2}$. This clearly indicates that $0.61\%$ is too small a value and the configuration is not $0.9-$safe configuration. Using the approaches in dutoit2011IEEE ; park2018IEEE would lead to collision as it predicts the configuration to be safe. Our approach computes the exact probability of collision and outperforms the approaches in dutoit2011IEEE ; park2018IEEE . ### 5.2 2D Environment Domain (a) (b) Figure 4: Simulation environment. (a) Scaled-down ($\times\frac{1}{4}$) top view of the environment with the sampled roadmap and start and goal locations of the robot. (b) Pioneer robot at the starting node of the roadmap. (a) (b) (c) (d) (e) (f) Figure 5: Trajectory and the covariance evolution for single planning instantiations are shown. Different cases with obstacle uncertainty for a point robot and a robot of radius 0.3$m$ are shown in (a), (b), (c) and (d). (e) The planned trajectory when there is uncertainty in beacon locations. (f) True beacon locations are shown in yellow. Approach \| Robot radius \| Obstacle uncertainty \| Beacon (object) uncertainty \| Planned trajectory ---\|---\|---\|---\|--- Our \| Point \| No \| No \| Fig. 5(a) Our \| Point \| Yes \| No \| Fig. 5(b) Our \| 0.3 m \| No \| No \| Fig. 5(a) dutoit2011IEEE \| 0.3 m \| No \| No \| Fig. 5(a) park2018IEEE \| 0.3 m \| No \| No \| Fig. 5(d) Our \| 0.3 m \| Yes \| No \| Fig. 5(c) Our \| 0.3 m \| No \| Yes \| Fig. 5(e) Our \| 0.3 m \| No \| No (true beacon location) \| Fig. 5(f) Our \| 0.3 m \| No \| No (mean beacon location) \| Fig. 5(a) Table 3: Different configurations used for the 2D environment domain. We consider the case of a environment where a mobile robot moving in an environment of $30m\times 20m$. A scaled-down top view is seen in Fig. 4(a). The underlying PRM graph, the start (S in the figure) and goal (G in the figure) locations can also be seen. The gray circles denote the obstacles in the environment. Fig. 4(b) shows a Pioneer P3DX robot at the start location. For the robot motion model, we consider the following non-linear dynamics thrun2005book $\begin{split}x_{k+1}&=x_{k}+\delta_{trans}\cos(\theta_{k}+\delta_{rot1})\\\ y_{k+1}&=y_{k}+\delta_{trans}\sin(\theta_{k}+\delta_{rot1})\\\ \theta_{k+1}&=\theta_{k}+\delta_{rot1}+\delta_{rot2}\end{split}$ (43) where $\textbf{x}_{k}\doteq(x,y,\theta)$ is the robot pose at time $k$ and $\textbf{u}_{k}\doteq(\delta_{rot1},\delta_{trans},\delta_{rot2})$ is the applied control. The model assumes that the robot ideally implements the following commands in order: rotation by an angle of $\delta_{rot1}$, translation of $\delta_{trans}$ and a final rotation of $\delta_{rot2}$ orienting the robot in the required direction. The robot accrue translational and rotational errors while executing $\textbf{u}_{k}$ and localizes itself by estimating its position using signal measurements from beacons $\bar{b}_{1},\ldots,\bar{b}_{7}$, which are located at $(x_{\bar{b}_{1}},y_{\bar{b}_{1}}),\ldots,(x_{\bar{b}_{7}},y_{\bar{b}_{7}})$. The signal strength decays quadratically with the distance to the beacon, giving the following observation model with sensor noise $v_{k}$, $\textbf{z}_{k}=\begin{bmatrix}1/\left((x_{k}-x_{\bar{b}_{1}})^{2}+(y_{k}-y_{\bar{b}_{1}})^{2}+1\right)\\\ \vdots\\\ 1/\left((x_{k}-x_{\bar{b}_{7}})^{2}+(y_{k}-y_{\bar{b}_{7}})^{2}+1\right)\end{bmatrix}+v_{k}$ (44) We validate our approach in the above discussed environment by varying different parameters, a summary of which is provided in Table 3. Below we detail each of cases considered in Table 3. We first consider the motion planning approach for a point-like robot. The cost function is of the form in (42) with $M_{u}=0.3$, $M_{g}=diag(0.8,0.8)$, $M_{\Sigma}=diag(1,1)$ and $M_{C}=10$. The underlying PRM graph with 65 nodes is shown in Fig. 5, with the green dots denoting the sampled nodes. The robot, starting from its initial belief state (mean pose denoted by S in the figure) has to reach the node $\textbf{x}_{g}$ (G in the figure), while reducing its uncertainty. The blue triangles denote the beacons that aid in localization. The solid black circles with radius 0.5m, represent obstacles in the environment and the red ellipses denote the 3$\sigma$ covariances (only the ($x$,$y$) portion is shown). Unless otherwise mentioned, in all the experiments, $0.99-$safe configurations are solicited and the total planning time is the average time for 25 different runs. We first consider a case with a point robot and no uncertainty in obstacle location. The planned trajectory in this case is seen in cyan in Fig. 5(a) with total planning time of $0.0051s(\pm 0.0008s)$. Please note that the total planning time also includes the collision probability computation time. Next, we consider uncertainty in one of the obstacle location, whose covariance ellipse is shown in gray. The planned trajectory is seen in cyan in Fig. 5(b) and the planning was completed under $0.0279s(\pm 0.0043s)$. Due to the uncertainty in the obstacle location, the robot takes a longer route to avoid collision. A robot of radius 0.3$m$ and certain (negligible uncertainty) obstacles gave the same trajectory as in Fig. 5(a) with a planning time of $0.0055s(\pm 0.0009s)$. However, when the obstacle location is uncertain the resulting trajectory is as shown in Fig. 5(c). A change in the trajectory is observed, as compared to the case of a point robot in Fig. 5(b). The planning time in this case is $0.0294s(\pm 0.0047s)$. It is also worth mentioning that in Fig. 5(b) and (c), the roadmap was updated by adding a node since a $0.99-$safe configuration could not be found. The added node is seen in brown, with its coordinates being approximately $(9,11)$. We also run the case with no obstacle uncertainty and a robot of radius 0.3$m$ using the approach of Park et al. park2018IEEE . In this case the planned trajectory is as given in Fig. 5(d). Note that using our approach, the same scenario gives a shorter trajectory (Fig. 5(a)). The longer trajectory computed using the approach in park2018IEEE is due to the fact that a loose upper bound is computed for the collision probability. As a result a longer trajectory is obtained. Contrary to this, we compute the exact collision probability and hence a shorter trajectory is synthesized. The same scenario is also run with the approach in dutoit2011IEEE and produced a trajectory similar to ours. However, since the uncertainties are significantly lower, the approximate collision probability values computed using dutoit2011IEEE are much smaller than the actual values. Next, we consider the case with uncertainty in the location of the beacons. The considered robot radius is 0.3$m$ with the bottom obstacle being uncertain with covariance $diag(0.49,0.49)$. Taking object uncertainty into account, the planned trajectory with covariance evolution is as shown in Fig .5(e). Fig. 5(f), shows the trajectory planned with true beacon locations. The beacons are shown in yellow to denote the true location. Considering only the mean position of the beacons and neglecting the position uncertainty, the planned trajectory is as shown in Fig. 5(a). Actual execution of this would lead to collision with the bottom obstacle. However, executing the planned trajectory obtained by considering the uncertainty in beacon locations does not violate the $\epsilon-$safety criterion and all the configurations are $0.99-$safe. It is noteworthy that though we have discussed a 2D environment, the approach directly extends to a mobile robot navigating in a 3D environment. In such domains, the mobile robot may be represented by a minimum volume enclosing sphere. Similarly, the obstacles can also be approximated by their corresponding minimum volume enclosing spheres. Hence the collision condition is the same as given in (21) and therefore the approach discussed in this paper remains valid. ### 5.3 Laser-grasp Domain (a) (b) Figure 6: Trajectory of the end-effector; green dots denote its mean and the red ellipses denote the covariance matrix. The puck is shown in black and the end-effector is shown to its right. (a) Trajectory and covariance evolution when object uncertainty is not considered and (b) when object uncertainty is considered. (a) (b) Figure 7: Green dots denote the mean of the state trajectory and the red ellipses denote the covariance matrix. Mean position of the obstacle at each time instant is visualized in blue. (a) State trajectory and covariance evolution during offline collision avoidance planning. (b) More information is acquired during online planning, reducing the uncertainty of the obstacle and thereby leading to a change in the planned trajectory. We consider two modified versions of the laser-grasp domain as suggested in platt2010RSS . In this domain, a planar robot manipulator must locate and proceed towards a round puck. The state space is the position of the manipulator’s end-effector relative to a grasping point defined directly in front of the puck. Though the end-effector position is assumed to be known completely, the state is not directly observed since the puck position is unknown. Its position can be determined using the laser range finder that points out as a horizontal line from the end-effector. The underlying system dynamics is $f(\textbf{x}_{t},\textbf{u}_{t})=\textbf{x}_{t}\ +\ \textbf{u}_{t}$ (45) where $\textbf{x}\in\mathbb{R}^{2}$ denotes the state space and $\textbf{u}\in\mathbb{R}^{2}$ is the end-effector velocity. The cost function is of the form in (42) with $M_{u}=diag(10,10)$, $M_{g}=diag(100,100)$, $M_{\Sigma}=diag(10000,10000)$ and $M_{C}=10$. First, we consider a scenario wherein an additional object is placed that aids in localization. In this scenario, the state is the end-effector position which is not known precisely due to actuation errors. The goal is to place the end-effector directly in front of the puck so as to be able to grasp it. Both the object and the puck can be detected by the horizontal laser. However, the object location is not known exactly and the $3\sigma$ uncertainty ellipse is shown in light blue in Fig. 6(a) and (b). The mean position is visualized by the blue blob and the yellow blob denotes the actual object location. The red ellipses represent the state covariance at different points along the trajectory. Fig. 6(a) shows the case in which object uncertainty is not considered and the object is assumed to be its mean position. The manipulator moves towards the object first, localizing the end-effector position and then proceeds further to place the end-effector at the grasping point. However, as seen in Fig. 6(a), while executing this plan produced offline, not considering the object uncertainty leads to the collision of the end-effector with the true object (in yellow). When the object uncertainty is considered, the execution of the plan do not lead to collision, as it can be seen in Fig. 6(b). This illustrates the fact that not considering object uncertainty can wrongly localize the robot, leading to catastrophes. Next, we consider a scenario wherein the state space is the position of the manipulator’s end-effector relative to a grasping point defined directly in front of the puck. The state is not directly observed since the puck position is unknown. However, as soon as the manipulator starts to move, a ball starts to roll in between the manipulator and the puck. The ball follows a Gaussian velocity distribution, and therefore at each time instant, the mean position of the ball and the corresponding uncertainty can be estimated. The mean position of the ball at each time instant is shown in blue in Fig. 7(a) and (b). The green dots denote the mean of the state trajectory. As seen in Fig. 7(a), the manipulator initially moves downwards. However, as the ball comes closer, the manipulator retraces its path and move upwards towards its starting position to avoid collision. This is so because the safety constraint for $\epsilon=0.99$ is violated. As the ball keeps moving upwards, after a while, it is seen that the manipulator takes a downward action just before reaching its starting position since the configuration is a $0.99-$safe configuration. The scenario in Fig. 7(b) is similar to that of Fig. 7(a). However, it is seen that once the manipulator retraces its path backward towards the starting position, it takes a downward action much earlier. This is because more information is acquired during online planning and the uncertainty bound on the obstacle changes with time. The 2D manipulator domain studied here directly extends to 3D manipulator scenarios for both static and mobile manipulators. In the case of static manipulators, the end-effector is approximated as a sphere. Each link is approximated as a set of spheres kept side by side. However, in heavily cluttered environments such an approximation can be computationally intensive since each sphere has to be checked for collision with obstacles. An alternative and effective approach is to consider the minimum-volume enclosing ellipsoid for each link rimon1997JINT . It is known that for every convex polyhedron, there exits a unique ellipsoid of minimal volume that contains the polyhedron and is called the Löwner-John ellipsoid of the polyhedron grotschel1988geometric . Thus each link can be represented by their corresponding Löwner-John ellipsoids. The distance between two ellipsoids is used to modify the collision condition in (21). For mobile manipulators, the collision condition should also checked for the base as discussed in the 2D robot section. ## 6 Conclusion In this paper, we have addressed a novel approach to compute the probability of collision under robot and obstacle pose uncertainties. The collision probability is computed as an infinite series whose convergence is proved. An upper bound for the truncation error is also derived. As shown in Fig. 2, convergence analysis is performed for different configurations and it is seen that our approach is of the order of milliseconds and therefore can be used in online planning. We also provide a comparison with the approaches in park2018IEEE ; dutoit2011IEEE . In addition, we incorporate landmark uncertainties in belief space planning and derive the resulting Bayes filter in terms of the prediction and measurement updates of the EKF. Finally, experimental evaluation for a mobile robot scenario and a 2D manipulator is performed to illustrate our approach. We have considered static obstacles in this paper and the immediate future work is to realize the approach in simulated and real-world environments with dynamic obstacles. ## References * [1] Ali-Akbar Agha-Mohammadi, Suman Chakravorty, and Nancy M Amato. 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# Closed-form Quadrangulation of $n$-Sided Patches Marco Tarini Università degli Studi di Milano<EMAIL_ADDRESS> ###### Abstract We analyze the problem of quadrangulating a $n$-sided patch, each side at its boundary subdivided into a given number of edges, using a single irregular vertex (or none, when $n=4$) that breaks the otherwise fully regular lattice. We derive, in an analytical closed-form, (1) the necessary and sufficient conditions that a patch must meet to admit this quadrangulation, and (2) a full description of the resulting tessellation(s). ## 1 Introduction Consider a polygonal-shaped, planar region patch $P$, delimited by $n>1$ sides, each subdivided in a number of edges. Let $e_{i}\in\mathbb{N}$ be the number of edges found on side $i$, $i\in[0..n-1]$. We are interested in determining whether or not $P$ can be quad-tessellated using only one irregular vertex, of valency $n$, somewhere in the interior (even this vertex is regular when $n=4$). This tessellation, when it exists, can also be described as the one obtained by applying one step of Catmull-Clark (CC) subdivision [3] to the polygon $P$, which creates $n$ quadrilateral regions, followed by a conforming, fully regular tessellation of each of these regions, each at some appropriate grid resolution. For this reason, we refer to this property of $P$ as it being “CC- able”. For example, the construction in Fig. 1 shows that a pentagonal-shaped patch delimited by $e_{i}=(6,4,3,5,4)$ edges happens to be CC-able. We show a closed-form formulation to determine the existence (and the uniqueness) of this quadrangulation, for an arbitrary $P$, and to construct it, when it exists. Figure 1: An example: a pentagonal patch with $e_{0..4}=(6,4,3,5,4)$ is CC- able, as graphically shown here. ### 1.1 Motivations and context An intensively studied recent problem in Geometry Processing is Coarse layout decomposition (see [2] for a survey), where a given surface must be partitioned into a layout of patches homologous to a disk. This is a precondition for numerous tasks (such as surface parametrization, shape segmentation, shape matching, application of machine learning on 3D shapes), each coming with its own set of useful final applications (ranging from texture mapping, digital fabrication, shape recognition, shape modelling, etc). Clearly, each such scenario dictates different requirements on the layout. Among others, one central application is surface semi-regular quad- remeshing (see [1] for a survey, and specifically Section 3.2 there). For other potential contexts, see the Conclusions. The layout is typically required to be comprised of rectangular regions, as they admit a “natural” tessellation consisting of a fully-regular lattice – but only if opposite sides of the region are subdivided in an equal number of edges. The present work can be understood as a way to _extend the concept of a “natural tessellation” to non-rectangular patches_. Figure 2: Schema for parameters $e_{i}$ and unknowns $s_{i}$ for patches with $n$ = 2, 3, 4, 5 and 6 sides. The same schema generalises to larger $n$. ## 2 Related work The literature addresses a problem similar to ours, but where multiple internal irregular vertices are allowed. In this case, solutions are now known to always exists [10] (as long as the “parity” condition holds, see below); note that layouts presenting more internal irregular vertices are typically considered less desirable in many contexts. This version of the problem was solved in [10] for $n\leqslant 6$, and an algorithm is offered that always constructs one valid solution. The problem was previously tackled for $n=3$ or $4$, with similar results [9]. These algorithms are combinatorial in nature; this is in contrast with our closed- form characterization of the instances admitting a solution with a single- irregular vertex (in addition to solving also for $n>6$). The algorithm in [10] returns the one solution considered the “best” among valid alternatives, according to some definition of desirability (although not all possibilities are always considered). In a similar spirit, Machine- Learning [6] and Data-Driven [5] heuristics have been leveraged to pick one “best” solution among the ones admitted by a given instance (again, according to some targeted criteria). In older proposals, users, such as digital modellers, are allowed to interactively navigate inside the space of admissible solutions, in search of the “preferred” one for a given mesh [8]. In most cases (although not necessarily all), the “best” or “preferred” solution is exactly the one featuring a single irregular vertex inside the patch, when such a solution exists; to our knowledge, our is the first criterion to determine _a priori_ whether that is the case or not. ## 3 Solution One well-known precondition for $P$ to be tasselable with only quads is that $\left(\sum_{i}{e_{i}}\right)\text{mod}\,2=0.$ (1) We refer to this as the _parity condition_ , and we assume it always holds. We are looking for a tessellation with a single, internal irregular vertex. In the only possible construction for a solution is depicted in Figure 2, where each side around $P$ is split into two sub-sides. Due to the constraint on regularity, a pair of sub-sides at the left and the right of another edge $i$ must share the same number of edges $s_{i}$. Thus, we have that $\forall i<n,\;\;e_{i}=s_{i-1}+s_{i+1}$ (2) (all indices, in the above and in all following equations, are considered modulo $n$). Because each side of the polygon must be split in two sub-sides, we also need to assume $e_{i}>1$. For $P$ to be CC-able, equations 2 must be fulfilled for some choice of unknown positive integer values $s_{i}$. This general property translates into different sets of conditions for each value of $n$, as we analyze in the following sections. ### 3.1 Two-sided shapes ($n=2$) For $n=2$, the problem is trivial. Equation (2) becomes simply $e_{0}=2\;s_{1}\;\;\;\;\text{ , }\;\;\;\;e_{1}=2\;s_{0}$ (3) (as $i-1$ and $i+1$ denote the same index modulo 2). Note that it is not necessarily the case that valency-2 vertices (sometimes called _doublets_ , [4]) are considered valid configurations, but they can be [11]. See also Fig. 3. Figure 3: A CC-able 2-sided patch, shown with its the corresponding internal quadrangulation. In conclusion, _a two-sided patch is CC-able (in only one way) iff both $e_{0}$ and $e_{1}$ are even numbers_ (a stronger condition than the Parity condition). ### 3.2 Triangular shapes ($n=3$) For $n=3$, we rewrite Equation (2) in matrix form: $\left(\begin{matrix}0&1&1\\\ 1&0&1\\\ 1&1&0\end{matrix}\right)\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\end{matrix}\right)=\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\end{matrix}\right)$ (4) which implies (by matrix inversion): $\frac{1}{2}\left(\begin{matrix}-1&+1&+1\\\ +1&-1&+1\\\ +1&+1&-1\end{matrix}\right)\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\end{matrix}\right)=\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\end{matrix}\right)$ (5) In other terms, $\forall i,\;\;s_{i}=\frac{1}{2}\Big{(}e_{i-1}-e_{i}+e_{i+1}\Big{)}.$ (6) For $s_{i}$ to be integer, the expression in parenthesis must be even, which is already guaranteed by Eq. (1). For $s_{i}$ to be positive, we also need the above expression to be positive, therefore: $\forall i,\;\;e_{i-1}+e_{i+1}>e_{i}.$ (7) In conclusion, _a triangular patch is CC-able iff each side has fewer edges than the other two sides combined._ This condition is a discrete version of the familiar triangle inequality. ### 3.3 Quadrilateral shapes ($n=4$) For a quadrilateral shape, to be CC-able amounts to be regularly griddable. The conditions for this to be the case are well-known and obvious, and are re- derived here (consistently with the other cases) only for completeness. With $n=4$, Eq. (2) can be written as $\left(\begin{matrix}0&1&0&1\\\ 1&0&1&0\\\ 0&1&0&1\\\ 1&0&1&0\end{matrix}\right)\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\\\ s_{3}\end{matrix}\right)=\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\ e_{3}\end{matrix}\right)$ (8) The matrix is non-singular, as the first two rows match the second two; therefore, the system has either multiple solutions for $s_{i}$, when $e_{0}=e_{2}$ and $e_{1}=e_{3}$, or no solution otherwise. This condition also implies the parity condition, as $\sum e_{i}=2(e_{0}+e_{1})$. Because $n=4$, the internal “irregular” vertex is, actually, regular like the others, and every valid choice of $s_{i}$ produces the same tessellation. In conclusion, _a quadrilateral patch is CC-able (in only one way) when the opposite sides have a matching number of edges._ ### 3.4 Pentagonal shapes ($n=5$) For $n=5$, we rewrite Eq. (2) as $\left(\begin{matrix}0&1&0&0&1\\\ 1&0&1&0&0\\\ 0&1&0&1&0\\\ 0&0&1&0&1\\\ 1&0&0&1&0\end{matrix}\right)\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\\\ s_{3}\\\ s_{4}\end{matrix}\right)=\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\end{matrix}\right)$ (9) which implies (by matrix inversion) $\frac{1}{2}\left(\begin{matrix}+1&+1&-1&-1&+1\\\ +1&+1&+1&-1&-1\\\ -1&+1&+1&+1&-1\\\ -1&-1&+1&+1&+1\\\ +1&-1&-1&+1&+1\end{matrix}\right)\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\end{matrix}\right)=\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\\\ s_{3}\\\ s_{4}\end{matrix}\right)$ (10) In other terms, $\forall i,\;\;s_{i}=\frac{1}{2}\Big{(}\left(e_{i-1}+e_{i}+e_{i+1}\right)-\left(e_{i-2}+e_{i+2}\right)\Big{)}.$ (11) For $s_{i}$ to be integer, the expressions above must be even before halving, which is already guaranteed by Eq. (1). The sign constraint ($s_{i}>0$) produces $\forall i,\;\;e_{i-1}+e_{i}+e_{i+1}>e_{i-2}+e_{i+2}$ (12) In conclusion, _a pentagonal patch is CC-able (in only one way) iff the total number of edges in any three consecutive edges is larger than the number of edges in the other two._ ### 3.5 Hexagonal shapes ($n=6$) For $n=6$, we rewrite Eq. (2) as $\left(\begin{matrix}0&1&0&0&0&1\\\ 1&0&1&0&0&0\\\ 0&1&0&1&0&0\\\ 0&0&1&0&1&0\\\ 0&0&0&1&0&1\\\ 1&0&0&0&1&0\end{matrix}\right)\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\\\ s_{3}\\\ s_{4}\\\ s_{5}\end{matrix}\right)=\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\\\ e_{5}\end{matrix}\right)$ (13) which implies (by matrix inversion) $\frac{1}{2}\left(\begin{matrix}[r]0&+1&0&-1&0&+1\\\ +1&0&+1&0&-1&0\\\ 0&+1&0&+1&0&-1\\\ -1&0&+1&0&+1&0\\\ 0&-1&0&+1&0&+1\\\ +1&0&-1&0&+1&0\\\ \end{matrix}\right)\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\\\ e_{5}\end{matrix}\right)=\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\\\ s_{3}\\\ s_{4}\\\ s_{5}\end{matrix}\right)$ (14) In other terms, $\forall i,\;\;s_{i}=\frac{1}{2}\Big{(}e_{i-1}+e_{i+1}-e_{i+3}\Big{)}.$ (15) The integrity constraint ($s_{i}\in\mathbb{Z}$) translates in the requirement for the sum of $e_{0,2,4}$, and the sum of $e_{1,3,5}$, to be even numbers. The sign constraint ($s_{i}>0$) gives, by $j=(i+3)$ modulo 6: $\forall j,\;\;e_{j-2}+e_{j+2}>e_{j}.$ (16) In conclusion, _a hexagonal patch is CC-able (in only one way) iff (1) each even side has fewer edges than the other two even sides combined (and likewise, for odd sides), and (2), both the even sides, and the odd sides, have an even total number of edges._ ### 3.6 Heptagonal shapes ($n=7$) For $n=7$, we rewrite Eq. (2) as $\left(\begin{matrix}0&1&0&0&0&0&1\\\ 1&0&1&0&0&0&0\\\ 0&1&0&1&0&0&0\\\ 0&0&1&0&1&0&0\\\ 0&0&0&1&0&1&0\\\ 0&0&0&0&1&0&1\\\ 1&0&0&0&0&1&0\end{matrix}\right)\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\\\ s_{3}\\\ s_{4}\\\ s_{5}\\\ s_{6}\end{matrix}\right)=\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\\\ e_{5}\\\ e_{6}\end{matrix}\right)$ (17) which implies (by matrix inversion) $\frac{1}{2}\left(\begin{matrix}-1\\!&\\!+1\\!&\\!+1\\!&\\!-1\\!&\\!-1\\!&\\!+1\\!&\\!+1\\\ +1\\!&\\!-1\\!&\\!+1\\!&\\!+1\\!&\\!-1\\!&\\!-1\\!&\\!+1\\\ +1\\!&\\!+1\\!&\\!-1\\!&\\!+1\\!&\\!+1\\!&\\!-1\\!&\\!-1\\\ -1\\!&\\!+1\\!&\\!+1\\!&\\!-1\\!&\\!+1\\!&\\!+1\\!&\\!-1\\\ -1\\!&\\!-1\\!&\\!+1\\!&\\!+1\\!&\\!-1\\!&\\!+1\\!&\\!+1\\\ +1\\!&\\!-1\\!&\\!-1\\!&\\!+1\\!&\\!+1\\!&\\!-1\\!&\\!+1\\\ +1\\!&\\!+1\\!&\\!-1\\!&\\!-1\\!&\\!+1\\!&\\!+1\\!&\\!-1\end{matrix}\right)\left(\begin{matrix}e_{0}\\\ e_{1}\\\ e_{2}\\\ e_{3}\\\ e_{4}\\\ e_{5}\\\ e_{6}\end{matrix}\right)=\left(\begin{matrix}s_{0}\\\ s_{1}\\\ s_{2}\\\ s_{3}\\\ s_{4}\\\ s_{5}\\\ s_{6}\end{matrix}\right)$ (18) In other terms, $\displaystyle\begin{split}\forall i,\;\;s_{i}=\frac{1}{2}\Big{(}e_{i-2}+e_{i-1}+e_{i+1}+e_{i+2}&\\\ -\left(e_{i}+e_{i+3}+e_{i-3}\right)\;\;\;\;&\Big{)}.\end{split}$ (19) For $s_{i}$ to be integer, the expression in parenthesis must be even, which is already guaranteed by Eq. (1). The sign constraint ($s_{i}>0$) becomes $\forall i,\;\;e_{i-2}+e_{i-1}+e_{i+1}+e_{i+2}>e_{i}+e_{i+3}+e_{i-3}.$ (20) In conclusion, _a heptagonal patch is CC-able (in only one way) iff each side plus its two opposite sides have fewer edges than the remaining four sides combined._ ### 3.7 Octagonal shapes ($n=8$) For $n=8$, we can rewrite Eq. (2) as two separate linear systems: $\begin{split}\left(\begin{matrix}1&0&0&1\\\ 1&1&0&0\\\ 0&1&1&0\\\ 0&0&1&1\\\ \end{matrix}\right)\left(\begin{matrix}s_{1}\\\ s_{3}\\\ s_{5}\\\ s_{7}\end{matrix}\right)&=\left(\begin{matrix}e_{0}\\\ e_{2}\\\ e_{4}\\\ e_{6}\end{matrix}\right)\\\ \left(\begin{matrix}1&1&0&0\\\ 0&1&1&0\\\ 0&0&1&1\\\ 1&0&0&1\\\ \end{matrix}\right)\left(\begin{matrix}s_{0}\\\ s_{2}\\\ s_{4}\\\ s_{6}\end{matrix}\right)&=\left(\begin{matrix}e_{1}\\\ e_{3}\\\ e_{5}\\\ e_{7}\end{matrix}\right)\end{split}$ (21) Neither matrix is invertible, being deficient by one rank: specifically, their two even rows and two odd rows sum up to the same row-vector. Therefore, the system can have solutions only when $\begin{split}e_{0}+e_{4}&=e_{2}+e_{6},\\\ e_{1}+e_{5}&=e_{3}+e_{7}.\end{split}$ (22) In the following, we show that this condition is also sufficient. The parity condition is already implied by Eq. (22), because $\sum{e_{i}}=2(e_{0}+e_{4}+e_{1}+e_{5})$. The only remaining condition is that $s_{i}>0$. Let $k_{0},k_{1}$ be the choices for $s_{0},s_{1}$ (consented by the rank deficits). Using Eq. (2), we get the values of all other $s_{i}$: $\displaystyle s_{0}$ $\displaystyle=k_{0},$ $\displaystyle s_{1}$ $\displaystyle=k_{1},$ $\displaystyle s_{2}$ $\displaystyle=e_{1}-s_{0}$ $\displaystyle s_{3}$ $\displaystyle=e_{2}-s_{1}$ $\displaystyle=e_{1}-k_{0},$ $\displaystyle=e_{2}-k_{1},$ $\displaystyle s_{4}$ $\displaystyle=e_{3}-s_{2}$ $\displaystyle s_{5}$ $\displaystyle=e_{4}-s_{3}$ (23) $\displaystyle=e_{3}-e_{1}+k_{0},$ $\displaystyle=e_{4}-e_{2}+k_{1},$ $\displaystyle s_{6}$ $\displaystyle=e_{5}-s_{4}$ $\displaystyle s_{7}$ $\displaystyle=e_{6}-s_{5}$ $\displaystyle=e_{5}-e_{3}+e_{1}-k_{0}$ $\displaystyle=e_{6}-e_{4}+e_{2}-k_{1}$ $\displaystyle=e_{7}-k_{0},$ $\displaystyle=e_{0}-k_{1}.$ (the bottom line uses Eq. (22)). Therefore, to have $s_{i}>0$, the choices $k_{0}$ and $k_{1}$ must be, each, subject to two upper-bounds and two lower- bounds: $\displaystyle 0$ $\displaystyle<k_{0}<e_{1},$ $\displaystyle 0$ $\displaystyle<k_{1}<e_{2},\;\;\;\;\;\;$ $\displaystyle e_{1}-e_{3}$ $\displaystyle<k_{0}<e_{7},$ $\displaystyle e_{2}-e_{4}$ $\displaystyle<k_{1}<e_{0}.\;\;\;\;\;\;$ The above set of constraints is always feasible. The existence of a valid integer solution for $k_{0}$ (and likewise, for $k_{1}$) is guaranteed because each of the two lower bounds is smaller, by at least two units, than each of the two upper bounds. _Proof:_ it is immediate to verify this for $0<e_{1,7}$ and $e_{1}-e_{3}<e_{1}$ (as $e_{1,3,7}>1$); finally, it also holds for $e_{1}-e_{3}<e_{7}$ because $e_{1}<e_{7}+e_{3}=e_{1}+e_{5}$ (and $e_{5}>1$; the last equality is Eq. 22). $\square\;$ In conclusion, _an octagonal patch is CC-able (in general, in multiple ways) iff one pair of even opposite sides have the same combined number of edges as the other pair of even opposite sides, and same for odd sides._ Figure 4: Examples of (strict or non-strict) CC-ability being met or not met for triangular (left) and pentagonal (right) patches. The solitary singular vertex is the red dot; the inequalities (respectively, equations 7,12) which are not met, or met in a non-strict sense, are annotated in red. See Section 4. ### 3.8 Generalization to $n>8$ Our CC-ability analysis for each $n\leqslant 8$ covers (we think) the cases used by most practical scenarios. The case for any other $n>8$ can be constructed similarly, presenting only more instances of the situations encountered for $n\leqslant 8$. We only briefly surmise a generalization here. For odd values of $n$, Eq. (2) can always be expressed with an invertible linear system, which can be inverted to extract the values of $s_{i}$ as a linear expression of the form $s_{i}=\frac{1}{2}\sum_{j}\pm e_{j}>0$ (25) (as exemplified for $n=3,5,7$). Because $\sum\pm e_{j}$ has always the same parity as $\sum e_{j}$, the condition that $s_{i}\in\mathbb{Z}$ is already guaranteed by the Parity condition, Eq. (1). The condition for CC-ability is thus entirely determined by the signs of $s_{i}$, which translates into a set of $n$ linear inequalities, each involving all $e_{i}$. For a given patch, there is either one or no solution. For $n=2h$ ($h\in\mathbb{N}$), Eq. (2) can be expressed as two separate linear systems, partitioning the patch sides in two alternating subsets of $h$ elements each. When $h$ is odd (i.e., $n$ is not a multiple of 4), both systems are invertible, and CC-ability is then determined by a total of $n$ inequalities, each involving $h$ sides, plus the condition that the total number of edges on each subset must be even; the solution is always unique, if it exists. When $h$ is even (i.e., $n$ is a multiple of 4) neither system is invertible, each being deficient by one rank; the CC-ability is determined by the equality conditions on $e_{i}$ required by either system to admit solutions. Solutions can be multiple (but equivalent when $n=4$, as noted), and their space is spanned by two degrees of freedom. ## 4 Non-strict CC-ability The definition of CC-ability can be relaxed by accepting that the one singularity is found on the boundary of the patch, rather than in its interior (see Figure 4 for examples). We term the relaxed definition “non-strict CC- ability”. For a patch to be non-strictly CC-able, we need $s_{i}$ to be just non- negative, rather than strictly positive, thus allowing for all the inequalities in the conditions for CC-ability to be fulfilled in the non- strict sense, in Equations (7,12,16,20,LABEL:eq:octaKi) for the cases $n=$ 3,5,6,7,8, respectively. The assumption that $e_{i}>1$ can also be dropped, allowing for $e_{i}=1$ in the input patch. When _one_ inequality is fulfilled as equality, the irregular vertex will be on the boundary, i.e. as a boundary vertex with edge-valency $\neq 3$. When _two_ inequalities are fulfilled as equalities, the irregular vertex will be found on a corner, i.e. as a corner vertex with edge-valency $\neq 2$ (as shown in bottom-right of Figure 4). It is never possible for more than two inequalities to be fulfilled as equalities (without infringing at least another inequality). Figure 5: An octagonal patch (top) fulfills the conditions in Eq. (22) and is therefore CC-able. After Eq. (LABEL:eq:octaKi), we have two choices $k_{0},k_{1}$, ranging in the intervals $0<k_{0}<3$ and $0<k_{1}<4$; all resulting quadrangulations are shown. ## 5 Resulting quadrangulation In addition to stating the conditions that must be met for a patch to be CC- able (either strictly or non-strictly), our construction also provides a closed-form description of the resulting quadrangulation. The set of values $s_{i}$ can be understood as a compact and complete way to describe the internal tessellation of the patch. Specifically, they describe the grid-sizes of the regular rectangular areas constructed on the corners of the original patch; or, equivalently, the number of edges of the sub-sides splitting each side, and, thus, the topological position of the singularity inside $P$ (see Figure 2). The Equations (3,6,11,15,19,23) define the values of $s_{i}$ as a closed function of $e_{i}$ for the cases of $n=2,3,5,6,7,8$ respectively. In the last case, the equations use two arbitrary values $k_{i}$, which must be chosen inside the intervals defined in closed-form by Eq. (LABEL:eq:octaKi), spanning all and only the possible solutions. ## 6 Discussion In this work, we identified a problem statement for a self-contained simple task, useful in the context of Geometry Processing, and derived a complete answer. Specifically, we expressed in closed form the conditions for CC- ability, the description of the resulting quadrangulation, and the set of available choices (if multiple solutions are possible) #### About the uniqueness of the solution As we have shown, the sought single-irregular-vertex quadrangulation can be non-unique for $n=8$. This contradicts the commonly held notion that, in a quadrangulation, no irregular vertex can be “moved alone” (e.g. [8]), meaning that its topological position inside a region cannot be modified without also affecting either the tessellation of the region boundary or the topological position of another irregular vertex in the same region. As it turns out, valency-8 vertices are exceptional in that they _can_ be “moved alone”, as exemplified in Figure 5. As we have shown, this is the lowest valency for which this happens (but it also occurs for valencies 12, 16, 20, and so on). #### Usability The property of being CC-able can be easily embedded in optimization systems, because it is expressed in closed form as a set of linear inequality, equality, or parity constraints, which can be enforced for example in convex numerical solvers. #### Potential applications While the main motivation for our analysis stems from Coarse-layout construction (see discussion in Section 1.1), another potential context is that of shape modelling, where digital modellers often define 3D shape high- level description, leaving an automatic system in the background to deal with the minutiae of the surface tessellation (for example [10]), such as in the software suits 3D Coat (PilgWay), Z-Brush (Pixologic) and others. Due to the mentioned direct relationship with Catmull-Clark subdivision, our formulation can be a useful tool in the context of “reverse subdivision” (for example, [7]), where the objective is to seek a subdivision surface approximating a given shape. More broadly speaking, the generality of the analyzed problem statement leads us to believe that our closed-form formulation can find numerous applications. ## References * [1] Bommes, D., Lévy, B., Pietroni, N., Puppo, E., Silva, C. T., Tarini, M., and Zorin, D. Quad-mesh generation and processing: A survey. Comput. Graph. Forum 32, 6 (2013), 51–76. * [2] Campen, M. Partitioning surfaces into quadrilateral patches: A survey. Comput. Graph. Forum 36, 8 (2017), 567–588. * [3] Catmull, E., and Clark, J. Recursively generated b-spline surfaces on arbitrary topological meshes. Computer-Aided Design 10, 6 (1978), 350 – 355. * [4] Daniels, J., Silva, C. T., Shepherd, J., and Cohen, E. Quadrilateral mesh simplification. ACM Trans. Graph. 27, 5 (Dec. 2008). * [5] Marcias, G., Takayama, K., Pietroni, N., Panozzo, D., Sorkine-Hornung, O., Puppo, E., and Cignoni, P. Data-driven interactive quadrangulation. ACM Trans. Graph. 34, 4 (2015), 65:1–65:10. * [6] Matveev, A., Artemov, A., Rakhimov, R., Bobrovskikh, G., Panozzo, D., Zorin, D., and Burnaev, E. Def: Deep estimation of sharp geometric features in 3d shapes, 2020. * [7] Panozzo, D., Puppo, E., Tarini, M., Pietroni, N., and Cignoni, P. Automatic construction of quad-based subdivision surfaces using fitmaps. IEEE Transactions on Visualization and Computer Graphics 17, 10 (2011), 1510–1520. * [8] Peng, C.-H., Zhang, E., Kobayashi, Y., and Wonka, P. Connectivity editing for quadrilateral meshes. ACM Trans. Graph. 30, 6 (Dec. 2011), 1–12. * [9] Takayama, K., Panozzo, D., Sorkine-Hornung, A., and Sorkine-Hornung, O. Robust and controllable quadrangulation of triangular and rectangular regions. Technical Report/ETH Zurich, Department of Computer Science 784 (2013). * [10] Takayama, K., Panozzo, D., and Sorkine-Hornung, O. Pattern-based quadrangulation for _N_ -sided patches. Comput. Graph. Forum 33, 5 (2014), 177–184. * [11] Tarini, M., Pietroni, N., Cignoni, P., Panozzo, D., and Puppo, E. Practical quad mesh simplification. Computer Graphics Forum (2010).
# Iterated and mixed discriminants Alicia Dickenstein, Sandra di Rocco, Ralph Morrison Department of Mathematics, FCEN, University of Buenos Aires and IMAS (UBA-CONICET), Ciudad Universitaria, Pab. I, C1428EGA Buenos Aires, Argentina<EMAIL_ADDRESS>KTH, Royal Institute of Technology, 10044, Stockholm, Sweden<EMAIL_ADDRESS>Department of Mathematics, Williams College, Williamstown, MA 01267, USA <EMAIL_ADDRESS> ###### Abstract. We consider systems of Laurent polynomials with support on a fixed point configuration. In the non-defective case, the closure of the locus of coefficients giving a non-degenerate multiple root of the system is defined by a polynomial called the mixed discriminant. We define a related polynomial called the multivariate iterated discriminant, generalizing the classical Schäfli method for hyperdeterminants. This iterated discriminant is easier to compute and we prove that it is always divisible by the mixed discriminant. We show that tangent intersections can be computed via iteration if and only if the singular locus of a corresponding dual variety has sufficiently high codimension. We also study when point configurations corresponding to Segre- Veronese varieties and to the lattice points of planar smooth polygons, have their iterated discriminant equal to their mixed discriminant. The first author acknowledges the support of ANPCyT PICT 2016-0398, UBACYT 20020170100048BA, CONICET PIP 11220150100473 and SOURCES, Argentina. The second author acknowledges supported by KTH and Williams College. The third author acknowledges support by ICERM and VR grants NT:2014-4763, NT:2018-03688. All three authors acknowledge support by the Knut and Alice Wallenberg foundation. ## 1\. Introduction Let $K$ be an algebraically closed field of characteristic zero and $A\subset\mathbb{Z}^{n}$ a finite lattice subset. A (Laurent) polynomial with support on the point configuration $A$, $p=\sum_{a\in A}c_{a}x^{a}\in K[x_{1},\ldots,x_{n}],$ is called an _$A$ -polynomial_. Consider a system of $(r+1)$ polynomials equations supported on $A_{0},\ldots,A_{r}\subset\mathbb{Z}^{n}$ respectively: (1.1) $p_{0}=p_{1}=\ldots=p_{r}=0,\quad p_{i}=\sum_{a\in A_{i}}c_{i,a}x^{a}.$ When $r+1=n$, the Bernstein-Kouchnirenko Theorem says that for a generic choice of the coefficients $c_{i,a}$ there is a finite number of non- degenerate solutions to (1.1) in $(K^{})^{n},$ equal to the mixed volume $MV(A_{0},\ldots,A_{n-1})$ of the support subsets. These intersections are transverse except for solutions on which the Jacobian associated to the system vanishes. When $r+1<n,$ vanishing points of the Jacobian correspond to singularities of the variety cut out by the hypersurfaces $p_{i}=0$. Classical work by Salmon [Sal82] and Bromwich [Bro71] classified singular intersections of two quadric surfaces, corresponding to the case $r=1$ and $A$ equal to the lattice points in the dilated simplex $2\Delta_{3}$ in $\mathbb{R}^{3}$. The basic idea of these results was already pursued by Cayley in connection with tangent intersections of conics in $\mathbb{C}^{2}.$ More recently, the problem has been revisited with similar tools in [FNO89], in the context of geometric modeling with focus on the real case; and in [ZJT+19], where these techniques are used to classify singular Darboux cyclides. These are surfaces in 3-space that are the projection of the intersection of two quadrics in dimension four. A generalization to the case of two higher dimensional quadric hypersurfaces is given in [Ott13]. Consider two space quadrics, given in matrix form by (1.2) $p_{i}=\begin{bmatrix}1&x_{1}&x_{2}&x_{3}\end{bmatrix}\,M_{i}\,\begin{bmatrix}1\\\ x_{1}\\\ x_{2}\\\ x_{3}\end{bmatrix},\quad i=0,1.$ For generic matrices, the intersection $(p_{1}=p_{2}=0)$ describes a non- sigular curve of degree $4$. The non-generic intersections are described in [Sch53, GKZ94] in the following way. Consider the pencil of quadrics given by $p_{0}+tp_{1}$. Using the Schäfli decomposition method, the existence of a tangential intersection can be studied by considering the zero locus of the the following polynomial in the entries of $M_{0},M_{1}$: (1.3) $D_{4\Delta_{1}}(\det(M_{0}+tM_{1})),$ where $D_{4,\Delta_{1}}$ is the univariate discriminant of the degree $4$ polynomial $\det(M_{0}+tM_{1})$, considered as a polynomial in $t.$ For generic matrices this is a polynomial of degree $6$ in its entries (that is, in the coefficients of $p_{0},p_{1}$), and it vanishes whenever $\det(M_{0}+tM_{1})$ does not have four simple roots. To classify the different singular intersections, they then studied the Segre characteristics arising from the Jordan normal form of $M_{0}+tM_{1}$. Figure 1. Transverse (left) and non-transverse (right) hyperbolas In this paper, we propose and study a generalization of this approach for any support $A$. We consider Equation (1.3) to be an iterated process, as we are computing the discriminant of a discriminant. Factorizations of iterated discriminants and resultants for polynomials of three variables where studied in [BM09]. Our aim is to define and study an iterated discriminant generalizing Schäfli’s method for hyperdeterminants, and to show when tangent intersections can be computed via iteration. The theory of $A$-discriminants was introduced in [GKZ94] and has been extensively studied both from a geometric and a computational viewpoint [DFS07, DRRS07, Est10, GHRS16]. Denote by $X_{A}\subset{\mathbb{P}^{\|A\|-1}}$ the projective variety defined as the closed image of the monomial embedding given by the $A$-monomials. The dual variety $X_{A}^{\nu}\subset{\mathbb{P}^{\|A\|-1}}^{\nu}$ is the closure of the coefficient vectors of the $A$-polynomials $p$ whose zero-locus $(p=0)$ has a singular point $x\in(K^{})^{n}$ with nonzero coordinates. Equivalently, the dual variety is the closure of the hyperplane sections of $X_{A}$ which are singular at a point with nonzero coordinates. The expected codimension of $X_{A}^{\nu}$ is one and when this is the case we say that $A$ is _non- defective_. When $A$ is non-defective the irreducible polynomial $D_{A}\in{\mathbb{Z}}[(c_{a})_{a\in A}]$ defining (up to sign) the dual variety: $X_{A}^{\nu}=(D_{A}=0),$ is called the _$A$ -discriminant_ [GKZ94]. We will use the notation $D_{A}((c_{a})_{a\in A})=D_{A}(p)$. If $n=1$ and $A=\\{0,1,2\\}$, then $p=c_{2}x^{2}+c_{1}x+c_{0}$ and $D_{A}(p)=c_{1}^{2}-4c_{0}c_{2}$ is the classical discriminant of a degree two polynomial. More generally, $D_{0,1,\ldots,\delta}$ coincides with the classical discriminant of univariate polynomials of degree $\delta$. This $D_{0,1,\ldots,\delta}$ is a polynomial of degree $2(\delta-1)$ in the coefficients, which we denote by $D_{\delta\Delta_{1}}$. The case of multi- linear polynomials (i.e. tensors) corresponds to the case in which the convex hull of $A$ equals the product $\Delta_{n_{1}}\times\ldots\times\Delta_{n_{l}}$ where $\Delta_{s}$ denotes the unit simplex of dimension $s.$ This multivariate $A$-discriminant is also referred to as the hyperdeterminant of size $(n_{1}+1)\times\ldots\times(n_{l}+1)$ [GKZ94, Chapter 14]. This is a classical object defined originally by Cayley [Cay45]. Note that for a quadratic polynomial $p$ with associated matrix $M$ as in (1.2), that is for $A$ consisting of the lattice points in $2\Delta_{3}$, the existence of a singular point in $(p=0)$ implies that the linear forms given by its partial derivatives vanish and so $\det(M)=0$. Indeed, $D_{A}(p)=\det(M)$. This suggests that an iterated discriminant should be connected to the notion of discriminant for a system of polynomials. This notion is called the _mixed discriminant_ [GKZ94, CCD+13, DEK14], which is a natural generalization of the classical $A$-discriminant. Given $r+1$ finite configurations $A_{0},\ldots,A_{r}\subset\mathbb{Z}^{n}$, we call an isolated solution $x\in(K^{})^{n}$ a _non-degenerate multiple root_ for the system (1.1) if the $r+1$ gradient vectors $\nabla_{x}p_{i}(x),i=0,\ldots,r$ are linearly dependent but any subset of $r$ of them is linearly independent. The associated _mixed discriminantal variety_ is the closure of the locus of coefficients for which the system has a non- degenerate multiple root. If this variety is a hypersurface, it is defined by a single irreducible polynomial which we call the mixed discriminant, denoted $MD_{A_{0},\ldots A_{r}}$. If it is not a hypersurface, we call the system _defective_ and set $MD_{A_{0},\ldots A_{r}}=1$. Observe that when $r=0$, $MD_{0,A_{0}}=D_{A_{0}}$ equals the $A_{0}$ discriminant. In fact, in the non-defective case, mixed discriminants are special cases of discriminants of a single polynomial. This was settled in [GKZ94], but without the hypothesis of non-degeneracy of the common multiple root and in [CCD+13] for the case $r+1=n$. Given $A_{0},\ldots A_{r}$, the associated Cayley configuration $C=C(A_{0},\dots,A_{r})\subset\mathbb{Z}^{n+r}$ is the union of the lifted configurations $e_{i}\times A_{i}\in\mathbb{Z}^{n+r}$ for $i=0,\ldots,r$, where $e_{0}=0$ and $e_{i}$ is the standard $i^{th}$ basis vector in $\mathbb{Z}^{r}$ for $i\geq 1.$ As sparse discriminants are affine invariants of lattice configurations [GKZ94], we could equivalently consider $C\subset\mathbb{Z}^{n+r+1}$, where now $e_{0},\dots,e_{r}$ denote the canonical basis in $\mathbb{Z}^{r+1}$. In Proposition 3.3 we prove that: ###### Proposition. If $C$ is non-defective, then $MD(A_{0},\ldots,A_{r})$ equals $D_{C}$. This characterization leads to the following definition of multivariate iterated discriminant of order $r.$ We introduce $(r+1)$ new variables $\lambda_{0},\ldots,\lambda_{r}$ and encode the initial system by one auxiliary $C$-polynomial: $P_{\lambda}=\lambda_{0}p_{0}+\ldots+\lambda_{r}p_{r}\in K[\lambda_{0},\dots,\lambda_{r},x_{1},\dots,x_{n}].$ We will denote both this polynomial and its tuple of coefficients by $P_{\lambda}$ where $\lambda=(\lambda_{0},\ldots,\lambda_{r}).$ In the present paper we consider the case when $A_{0}=\ldots=A_{r}=A$ and use the notation $MD_{r,A}:=MD_{A,\ldots,A}.$ Notice that $D_{A}(P_{\lambda})$ is a homogeneous polynomial of degree $\deg(D_{A})$ in $\lambda_{0},\ldots,\lambda_{r}.$ ###### Definition 1.1. Given $A\subset\mathbb{Z}^{n}$ non-defective, denote by $d$ the codimension of the singular locus of the dual variety $X_{A}^{\nu}$. Given $r\geq 0$, the _multivariate iterated discriminant of order $r$_ is the polynomial $ID_{r,A}$ on the coefficients of $(r+1)$ $A$-polynomials $p_{0},\dots,p_{r}$ defined by $\begin{cases}ID_{r,A}(p_{0},\dots,p_{r}):=D_{\delta_{A}\Delta_{r}}(D_{A}(P_{\lambda})),\,\text{ if }d\geq r+1,\\\ ID_{r,A}(p_{0},\dots,p_{r}):=0,\,\text{ otherwise. }\end{cases}$ It is worth noting that in the classical case of $r=0,$ all these polynomials coincide by definition: $MD_{0,A}=ID_{0,A}=D_{A},\;\;\text{ and }D_{\delta\Delta_{0}}(D_{A}(\lambda p_{A}))=D_{A}.$ The latter equality is a consequence of the fact that the discriminant (in the variable $\lambda$) of the monomial $D_{A}\lambda^{\delta}$ is the coefficient $D_{A}$ [Jou91]. Moreover, when $A$ consists of the vertices of a simplex, $ID_{r,A}$ coincides with the hyperdeterminant Schäfli decomposition [GKZ94, Ch. 14]. Our main results give a precise relation between $MD_{r,A}$ and $ID_{r,A}$. The advantage of relating $MD_{r,A}$ with $ID_{r,A}$ is that the latter polynomial is much easier to compute. We show that in the non-defective case $MD_{r,A}$ is always an irreducible factor of $ID_{r,A}$, as a consequence of biduality (see Section 4). Therefore, if $ID_{r,A}(p_{0},\dots,p_{r})\neq 0$, we get a certificate that the intersection $(p_{0}=\dots=p_{r})$ is smooth. When $A$ is non-defective, we denote by $\textrm{sing}(X_{A}^{\nu})$ the subscheme of $X_{A}^{\nu}$ defined by the ideal generated by the partial derivatives of $D_{A}.$ We show that $ID_{r,A}$ can have other irreducible factors given by the Chow forms $Ch_{Y_{k}}$ of the higher dimensional irreducible components of the schematic singular locus of the dual variety $X_{A}^{\nu}$. We recall the notion of Chow forms at the beginning of Section 4. Theorem 4.4 and Proposition 4.6 imply the following Theorem. ###### Theorem. Assume $A\subset\mathbb{Z}^{n}$ is non-defective and let $r\in\mathbb{Z}$ with $0\leq r\leq\dim(X_{A}).$ Then, the mixed discriminant $MD_{r,A}$ always divides the iterated discriminant $ID_{r,A}$. Moreover: 1. (1) If ${\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))>r$, then $ID_{r,A}=MD_{r,A}$. 2. (2) If ${\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))=r$, then $ID_{r,A}=MD_{r,A}\prod_{i=k}^{\ell}Ch_{Y_{k}}^{\mu_{k}},$ where $Y_{1},\dots,Y_{\ell}$ are the irreducible components of $sing(X_{A}^{\nu})$ of maximal dimension $r$, with respective multiplicities $\mu_{k}\geq 1$. 3. (3) If ${\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))<r$, then $ID_{r,A}=0.$ As a corollary of our results, we show in Proposition 5.2 that an iterated method to characterize singular complete intersections is unfortunately only possible for $r+1$ hypersurfaces of the same degree $d$ in $\mathbb{P}^{n}$ and only if $r=1$ and $d=2$ (the case already found in [Ott13, Theorem 8.2]): ###### Proposition. Let $1<d$ and $1\leq r\leq n$. Then $MD_{r,d\Delta_{n}}=ID_{r,d\Delta_{n}}$ if and only if $r=1$ and $d=2$. The paper is organized as follows. In Section 2 we present some examples that motivate the theory of iterated discriminants. In Section 3 we present material on mixed discriminants and Cayley configurations, and in Section 4 we develop the theory of iterated discriminants and prove our main results. In Section 5 we ask more broadly when mixed and iterated discriminants are equal, for products of scaled simplices, that is, when $X_{A}$ is a Segre- Veronese variety. The case of Segre varieties was solved in [WZ96], via a careful study of the singularities of hyperdeterminant varieties. Our Conjecture 5.3 is the following, notation as in Section 5: ###### Conjecture. The equality $\deg(ID_{r,A_{\ell,d,k}})=\deg(MD_{r,A_{\ell,d,k}})$ holds if and only if $\mathbb{P}^{r}(1)\times\mathbb{P}^{k_{1}}(d_{1})\times\cdots\times\mathbb{P}^{k_{\ell}}(d_{\ell})$ is of one of the following cases: 1. (1) $\mathbb{P}^{r}\times\mathbb{P}^{m}\times\mathbb{P}^{m},\,m\geq 1,\,r=1,2$, 2. (2) $(\mathbb{P}^{1})^{4}$, 3. (3) $\mathbb{P}^{1}\times\mathbb{P}^{n}(2)$. A partial answer is given in Theorem 5.6 and Proposition 5.2. Finally, in Section 6 we analyze the case of plane curves. Theorem 6.3 shows that for planar configurations $A$ consisting of the lattice points of a smooth polygon, the only case where $MD_{1,A}$ equals $ID_{1,A}$ are the known cases in which the polygon is the unit square (the bilinear case) or $2\Delta_{2}$, the standard triangle of size $2$. This implies that in all other cases, the singularities of the discriminant locus have codimension one; that is, there are “many” different types of singular hypersurfaces defined by $A$-polynomials. A factorization of the iterated discriminants gives all components of the singular locus of codimension one. ### Acknowledgements We thank Carlos D’Andrea, Frédéric Bihan, Laurent Busé, Bernard Mourrain, and Giorgio Ottaviani for helpful discussions and references to previous work in this direction. ## 2\. Motivating examples In this section we present some motivating examples that we abstract in the paper. The first two correspond to two classical cases in which the iterated discriminant actually computes the mixed discriminant. The last two are the simplest cases which already show the occurrence of other factors of the iterated discriminant. ###### Example 2.1. Let $A=\\{(0,0),(1,0),(0,1),(1,1)\\}$ be the vertices of the unit cube and let $f=c_{00}+c_{10}x_{1}+c_{01}x_{2}+c_{11}x_{1}x_{2}$ be an $A$-polynomial. In this case, $D_{A}(f)=c_{00}c_{11}-c_{10}c_{01}$ is a polynomial of degree $2$, which equals the determinant of the matrix $\left(\begin{array}[]{cc}c_{00}&c_{01}\\\ c_{10}&c_{11}\end{array}\right).$ In case $r+1=2$, the mixed discriminant associated with two $A$-polynomials $p_{0}=c^{1}_{00}+c^{1}_{10}x_{1}+c^{1}_{01}x_{2}+c^{1}_{11}x_{1}x_{2}$ and $p_{1}=c^{2}_{00}+c^{2}_{10}x_{1}+c^{2}_{01}x_{2}+c^{2}_{11}x_{1}x_{2}$, is the following degree four irreducible polynomial, which is the hyperdeterminant of format $2\times 2\times 2$ (see [GKZ94], pp. 475–479): $MD_{1,A}(p_{0},p_{1})={c^{2}_{00}}^{2}{c^{1}_{11}}^{2}-2c^{2}_{00}c^{2}_{01}c^{1}_{10}c^{1}_{11}-2c^{2}_{00}c^{2}_{10}c^{1}_{01}c^{1}_{11}-2c^{2}_{00}c^{2}_{11}c^{1}_{00}c^{1}_{11}+$ $4c^{2}_{00}c^{2}_{11}c^{1}_{01}c^{1}_{10}+2c^{2}_{00}c^{1}_{00}{c^{1}_{11}}^{2}-4c^{2}_{00}c^{1}_{01}c^{1}_{10}c^{1}_{11}+{c^{2}_{01}}^{2}{c^{1}_{10}}^{2}+4c^{2}_{01}c^{2}_{10}c^{1}_{00}c^{1}_{11}-$ $2c^{2}_{01}c^{2}_{10}c^{1}_{01}c^{1}_{10}-2c^{2}_{01}c^{2}_{11}c^{1}_{00}c^{1}_{10}+2c^{2}_{01}c^{1}_{00}c^{1}_{10}c_{11}+{c^{2}_{10}}^{2}{c^{1}_{01}}^{2}-$ $2c^{2}_{10}c^{2}_{11}c^{1}_{00}c^{1}_{01}+2c^{2}_{10}c^{1}_{00}c^{1}_{01}c^{1}_{11}+{c^{2}_{11}}^{2}{c^{1}_{00}}^{2}-2c^{2}_{11}{c^{1}_{00}}^{2}c^{1}_{11}+{c^{1}_{00}}^{2}{c^{1}_{11}}^{2},$ It vanishes at $(p_{0},p_{1})$ with respective coefficient vectors $(1,1,-2,-1)$ and $(1,1,-3,-2)$, corresponding to the tangent hyperbolas in Figure 1. One form of computing $D_{A}$ is as the iterated discriminant $ID_{1,A}$. Write $\det\left(\begin{array}[]{cc}c^{1}_{00}+\lambda c^{2}_{00}&c^{1}_{01}+\lambda c^{2}_{01}\\\ c^{1}_{10}+\lambda c^{2}_{10}&c^{1}_{11}+\lambda c^{2}_{11}\end{array}\right)\,=\,\Delta_{0}+\Delta_{1}\lambda+\Delta_{2}\lambda^{2},$ and then compute $MD_{1,A}(c^{1},c^{2})=\Delta_{1}^{2}-4\Delta_{0}\Delta_{2},$ as the univariate resultant of the degree $2$ polynomial $\Delta_{0}+\Delta_{1}\lambda+\Delta_{2}\lambda^{2}$ in $\lambda$ with coefficients in $\mathbb{Z}[c^{1},c^{2}]$. This compact formula is the simplest case of Schäfli’s formula to compute the mixed discriminant $MD_{1,A}$. ###### Example 2.2. Let us consider again the case discussed in the Introduction corresponding to the singular intersections of two quadric surfaces $p_{0},p_{1}$ in three- space. We display their common support $A$ as the columns of the following $3\times 10$ matrix: $\begin{bmatrix}0&1&0&0&2&1&1&0&0&0\\\ 0&0&1&0&0&1&0&2&1&0\\\ 0&0&0&1&0&0&1&0&1&2\end{bmatrix}.$ We also display the corresponding Cayley configuration $C=\Delta_{1}\times A$ as the columns of the following $5\times 20$-matrix: $\small\setcounter{MaxMatrixCols}{20}\begin{bmatrix}0&1&0&0&2&1&1&0&0&0&0&1&0&0&2&1&1&0&0&0\\\ 0&0&1&0&0&1&0&2&1&0&0&0&1&0&0&1&0&2&1&0\\\ 0&0&0&1&0&0&1&0&1&2&0&0&0&1&0&0&1&0&1&2\\\ 1&1&1&1&1&1&1&1&1&1&0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0&1&1&1&1&1&1&1&1&1&1\end{bmatrix}.$ In this case, we know that $(X_{C})^{\nu}$ is a hypersurface by [DR14]. Thus we have that the polynomial $MD_{1,A}(p_{0},p_{1})$ cuts out the closure of the locus of coefficients for which the two quadrics lie tangent to one another at a point and it can be computed via the discriminant $D_{C}$ by Proposition 3.3. It can be also computed as the iterated discriminant in (1.3). This polynomial can be studied through tropical discriminants as in [DFS07]. Moreover, one can compute the univariate discriminant $D_{4\Delta_{1}}$ of a degree $4$ polynomial as the discriminant of its cubic resolvent from Galois theory. Let $\Delta_{i}$ denote the coefficient $\lambda^{i}$ in $\det(M_{0}+tM_{1})$. Then $MD_{1,A}(p_{0},p_{1})=\frac{4p^{3}-q^{2}}{27},$ where $p=12\Delta_{4}\Delta_{0}-3\Delta_{3}\Delta_{1}+\Delta_{2}^{2}$, and $q=72\Delta_{4}\Delta_{2}\Delta_{0}+9\Delta_{3}\Delta_{2}\Delta_{1}-27\Delta_{4}\Delta_{1}^{2}-27\Delta_{0}\Delta_{3}^{2}-2\Delta_{2}^{3}$. This gives a compact and feasible way of computing the mixed discriminant $MD_{1,A}$ we are interested in. In fact, expanding this expression in terms of the coefficients of $p_{0},p_{1}$ is beyond the capabilities of the excellent Computer Algebra System Macaulay2 [GS] in a standard computer, because it is a polynomial of degree $24$ which has degree $12$ in both the coefficients of $p_{0}$ and $p_{1}$. Note that a general polynomial of bidegree $(12,12)$ in two groups of $10$ variables has more than $4\cdot 10^{11}$ monomials! The general case is hinted in the following simple examples. ###### Example 2.3. Consider the two dimensional configuration $A=\\{(0,0),(1,0),(2,0),(0,1),(1,1)\\}$ corresponding to the first Hirzebruch surface ${\mathbb{F}}_{1}$. Given a generic polynomial $f$ with support $A$: $f(x,y)=a_{0}+a_{1}x+a_{2}x^{2}+y(b_{0}+b_{1}x),$ the $A$-discriminant coincides with the resultant of the two univariate polynomials $a_{0}+a_{1}x+a_{2}x^{2}$ and $b_{0}+b_{1}x$ and thus is equal to the degree $3$ polynomial $D_{A}(f)\,=a_{0}b_{1}^{2}-a_{1}b_{0}b_{1}+a_{2}b_{0}^{2}.$ The mixed discriminant $MD_{1,A}$ has degree $8$, while the iterated discriminant $ID_{2,A}$ has degree $2\cdot 3\cdot(3-1)^{1}=12$ by (4.9) . There is another irreducible factor that we explain in Theorem 4.4 and compute in Example 4.7. ###### Example 2.4. We now consider the case of a univariate polynomial of degree $3$ with $A=\\{0,1,2,3\\}$ and $r=1$. Given two cubic polynomials $p_{0},p_{1}$ depending on a variable $x$, their mixed discriminant equals the discriminant of the Cayley configuration $C=\\{(0,0),(0,1),(0,2),(0,3),(1,0),(1,1),(1,2),(1,3)\\}$ at the polynomial $p_{0}+tp_{1}$ in one more variable $t$. In fact, $D_{C}(p_{0}+tp_{1})$ equals the univariate resultant ${\rm Res}_{3,3}(p_{0},p_{1})$. This resultant can be computed as the determinant of the associated Sylvester matrix and therefore has degree $6$ in the vectors of coefficients of $p_{0},p_{1}$. Since the discriminant $D_{A}$ of a cubic univariate polynomial has degree $4$, the iterated discriminant $ID_{1,A}=D_{4\Delta 1}(D_{A}(p_{0}+tp_{1}))$ instead has degree $2\cdot 4\cdot 3=24$ according to (4.9). It has another irreducible factor of degree $6$ raised to the third power, which corresponds to the Chow form of the singular locus of $D_{A}=0$ corresponding to degree $3$ polynomials with a triple root (a degenerate multiple root), predicted by Theorem 4.4. ## 3\. The Mixed Discriminant and the discriminant of the Cayley configuration In this section we show in Proposition 3.3 that in the non-defective case the mixed discriminant $MD(A_{0},\ldots,A_{r})$ coincides with the discriminant of the associated Cayley configuration $D_{C}$, thus generalizing Theorem 2.1 in [CCD+13]. Note that when $C$ is defective these varieties need not coincide, as shown in Example 2.2 in [CCD+13]. We also characterize, in Proposition 3.4, non-defectivity of $C$ when all $A_{i}$ are equal. The latter result relies on a classical criterion by Katz, stated as Lemma 3.1 below, and is a simple consequence of [WZ94, Th. 0.1]. Recall that for a projective variety $X$, the _dual defect_ of $X$ is defined to be (3.1) $\text{def}(X):=\text{codim}(X^{\nu})-1,$ where $X^{\nu}$ is the dual variety consisting of singular hyperplane sections to $X$. In particular, if the dual variety is a hypersurface as expected, then the dual defect is equal to $0$ and $X$ is said to be non-defective. When $X=X_{A}$ for some finite lattice configuration $A$, we also say that $A$ is non-defective. In this context, we have the following lemma due to Katz. ###### Lemma 3.1. [Kat73] Let $A\subset\mathbb{Z}^{n}$ be a lattice configuration with $\|A\|=N+1.$ Let $H_{p}(f)$ denote the Hessian matrix of an $A$-polynomial $f$. Then ${\rm codim}{X_{A}^{\nu}}=1+{\rm min}_{f}\left({{\rm corank}(H_{u}(f))}\right)$ where $u$ is a general point and $f$ varies among the polynomials with support in $A$ vanishing at $u.$ In particular, ${\rm codim}{X_{A}^{\nu}}=1$ implies that polynomials vanishing at a general point $u$ together with their partial derivatives have Hessian of maximal rank. Observe that Lemma 3.1 is equivalent to saying that in the non-defective case, the closure of the singular $A$-polynomials coincides with the closure of the nodal $A$-polynomials, that is, polynomials only admitting non-degenerate multiple roots (see Introduction). ###### Corollary 3.2. If $A$ is a non-defective finite lattice configuration, then $X_{A}^{\nu}=\overline{\left\\{f\in{\mathbb{P}^{N}}^{\nu}\,:\,f(u)=0,\frac{\partial f}{\partial x_{i}}(u)=0\text{ and }\det(H(f))(u)\neq 0\text{ for some }u\in(K^{})^{n}\right\\}}.$ ###### Proof. The inclusion “$\subseteq$” follows by definition and the inclusion “$\supseteq$” follows by Lemma 3.1. ∎ Let us now consider the Cayley configuration $C$ associated to $r+1$ finite lattice configurations $A_{0},\ldots,A_{r}\in\mathbb{Z}^{n}.$ We remark that the $r$ in [GKZ94] corresponds to our $r-1.$ We use the following notation: $(y,x)=(y_{0},\ldots,y_{r},x_{1},\dots,x_{n})$. A polynomial $f$ with support on $C$ has the form $f\,=\,\sum_{0}^{r}y_{i}p_{i},$ where $p_{i}$ are $A_{i}$-polynomials in the variables $x$. Consider the Jacobian matrix of $p_{0},\dots,p_{r}$ at $u$ $\begin{bmatrix}\nabla_{x}(p_{0})(u)&\ldots&\nabla_{x}(p_{r})(u)\end{bmatrix}.$ Notice that $f\in X_{C}^{\nu}$ if there exists $(\lambda,u)\in(\mathbb{C}^{\nu})^{r+1}\times(\mathbb{C}^{\nu})^{n}\text{ s.t. }p_{0}(u)=\dots=p_{r}(u)=0\text{ and }\lambda\in\ker\begin{bmatrix}\nabla_{x}(p_{0})(u)&\ldots&\nabla_{x}(p_{r})(u)\end{bmatrix}^{T};$ or equivalently, if $\sum_{i=0}^{r}\lambda_{i}\nabla_{x}(p_{i})(u)=0$ and thus the gradients are linearly dependent. In particular, ${\rm rank}(\begin{bmatrix}\nabla_{x}(p_{0})(u)&\ldots&\nabla_{x}(p_{r})(u)\end{bmatrix})\leq r.$ We will now prove that the locus where the rank is exactly $r$ characterizes the dual variety $X_{C}^{\nu},$ assuming it is a hypersurface. ###### Proposition 3.3. Let $A_{0},\dots,A_{r}$ and $C$ as above and assume that $C$ is non-defective. Then, $MD(A_{0},\ldots,A_{r})(p_{0},\dots,p_{r})=D_{C}(\sum_{i=0}^{r}y_{i}p_{i}),$ where $p_{i}$ are $A_{i}$-polynomials for $i=0,\dots,r$ and $(y_{0},\dots,y_{r})$ are variables. ###### Proof. Let $\phi_{f}$ be the tuple of coefficients of $f$. Corollary 3.2 implies that $X_{C}^{\nu}=\overline{\\{\phi_{f}\;:\;f(\lambda,u)=0,p_{A_{i}}(u)=0,i=0,\dots,r,\,\sum_{0}^{r}\lambda_{j}\frac{\partial p_{A_{j}}}{\partial x_{i}}(u)=0\text{ and }\det(H(f))(\lambda,u)\neq 0\\}},$ where $(\lambda,u)\in(\mathbb{C}^{\nu})^{r+1}\times(\mathbb{C}^{\nu})^{n}.$ Here, $H(f)(\lambda,u)$ means the following: as $f$ is homogeneous in the variables $y$ and $\lambda\in(\mathbb{C}^{\nu})^{r+1}$, we assume that $\lambda_{0}=1$ and that $(\lambda_{1},\dots,\lambda_{r})$ are its affine coordinates. Thus, $H(f)(\lambda,u)$ is the Hessian of $f$ with respect to the variables $(\lambda_{1},\dots,\lambda_{r},x_{1},\dots,x_{n})$ . This Hessian matrix is of the form $H_{(u,\lambda)}(\phi)=\begin{bmatrix}\lambda_{0}H(p_{A_{0}})(u)+\sum_{1}^{r}\lambda_{i}H(p_{A_{i}})(u)&\begin{bmatrix}\nabla(p_{A_{1}})(u)\\\ \ldots\\\ \nabla(p_{A_{r}})(u)\end{bmatrix}^{T}\\\ \begin{bmatrix}\nabla(p_{A_{1}})(u)\\\ \ldots\\\ \nabla(p_{A_{r}})(u)\end{bmatrix}&0\end{bmatrix}.$ It follows that if $\phi_{f}\in X_{C}^{\nu}$ and $\det H(f)(\lambda,u)\neq 0,$ which happens for generic points in $X_{C}^{\nu}$ by Corollary 3.2, then $\textrm{rank}\begin{bmatrix}\nabla(p_{A_{1}})(u)\\\ \ldots\\\ \nabla(p_{A_{r}})(u)\end{bmatrix}=r.$ As $\lambda_{j}\neq 0$ for any $j=0,\dots,r$, we can choose instead the chart setting $\lambda_{j}=1$. We would then get a Hessian matrix whose determinant will vanish if and only if the one we are computing vanishes, and we would deduce that the gradients of the polynomials $p_{A_{i}}(u)$ for $i\neq j$ form a matrix of rank $r$, that is, they are linearly independent. Moreover, this is exactly the condition implying that $\phi_{f}$ belongs to the mixed-discriminantal variety $MD(A_{0},\ldots,A_{r})=0$ which we denote by $X_{MD}$. It follows that $X_{C}^{\nu}\subseteq X_{MD}$ and that $X_{MD}$ is also a hypersurface, i.e. $MD(A_{0},\ldots,A_{r})\neq 1.$ The reverse inclusion follows essentially from the definition. In fact if $\phi_{f}\in X_{MD}$ is generic, then there is a common zero $u\in(C^{\nu})^{n}$ of $p_{A_{0}},\dots,p_{A_{r}}$ and a linear dependency $\sum\lambda_{i}\nabla(p_{A_{i}})(u)=0$ with all $\lambda_{i}\neq 0,$ because all the maximal minors in the matrix $\begin{bmatrix}\nabla(p_{A_{0}})(u)\\\ \ldots\\\ \nabla(p_{A_{r}})(u)\end{bmatrix}$ are assumed to be nonzero. It follows that $\phi_{f}\in X_{C}^{\nu}.$ ∎ Notice that if $A_{0}=A_{1}=\ldots=A_{r}=A$ then $C=\\{e_{0},\dots,e_{r}\\}\times A$, which is usually written as $C=\Delta_{r}\times A$. Following [WZ94], we define the following quantity associated to a projective variety $X$: (3.2) $\mu(X)\,=\,\dim(X)+{\rm def}(X),$ where the defect of $X$ has been defined in (3.1). We end this section with the following result about non-defectivity. ###### Proposition 3.4. Let $A$ be a non-defective finite lattice configuration. Then, the associated Cayley configuration $C=\Delta_{r}\times A$ is non-defective if and only if $r\leq\dim(X_{A})$. ###### Proof. Note that $X_{C}=\mathbb{P}^{r}\times X_{A}$. We can then use Theorem 0.1 in [WZ96], which says that $\mu(\mathbb{P}^{r}\times X_{A})={\rm max}(r+\dim(X_{A}),r+{\rm def}(\mathbb{P}^{r}),\dim(X_{A})+{\rm def}(X_{A})).$ According to (3.2), we have that $\mu(X_{C})=r+\dim(X_{A})+{\rm def}(X_{C})$. Since ${\rm def}(\mathbb{P}^{r})=r$, and by hypothesis ${\rm def}(X_{A})=0$, we get that $\mu(X_{C})={\rm max}(r+\dim(X_{A}),2r,\dim(X_{A})).$ When $r\leq\dim(X_{A})$, we get that $\mu(X_{C})=r+\dim(X_{A})$ which implies that ${\rm def}(X_{C})=0$. On the other side, when $r>\dim(X_{A})$, we have that $\mu(X_{C})=2r$ and so ${\rm def}(X_{C})=r-\dim(X_{A})>0$. ∎ ## 4\. The multivariate iterated discriminant In the remainder of the paper we will consider the case $A_{i}=A$ for $i=0,\ldots,r.$ In order to establish an iterated process for the mixed discriminant it is convenient to consider the geometric iterated discriminant $JD_{r,A}$ introduced in Definition 4.2 below. In Proposition 4.6 we prove that this polynomial coincides with the iterated discriminant $ID_{r,A}$ from Definition 1.1. It implies that Theorem 4.4, which can be considered the main result of this paper, also holds for $ID_{r,A}$, as stated in the Introduction. Recall that given an irreducible and reduced projective variety $Y\subset\mathbb{P}^{N}$ of codimension $s$, its Chow form $Ch_{Y}$ is defined as follows. Consider linear subspaces of dimension $\ell$ in $\mathbb{P}^{N}$, $L\in Gr(\ell+1,N+1).$ If $s\geq\ell+1$, any generic $L$ will not intersect $Y.$ The irreducible subvariety $\\{L\in Gr(\ell+1,N+1)\,:\,L\cap Y\neq\emptyset\\}$ parametrizing the exceptional intersection locus, has codimension $(s-\ell)$ in $Gr(\ell+1,N+1).$ In case $\ell=s-1$ the defining polynomial is denoted by $Ch_{Y}$ and it is called the Chow form of $Y$ [GKZ94, page 99]. We also need to recall two classical facts which will be used in the proof of our main Theorem 4.4. ###### Remark 4.1. Given a finite lattice configuration $A$ and a generic singular hyperplane section of $X_{A}$, we can recover the intersection point by means of the gradient of the discriminant $D_{A}$. Precisely, 1. (1) As we are assuming that ${\rm char}(K)=0$, if a regular point $H$ in the dual variety $X_{A}^{\nu}$ is tangent to $X_{A}$ at a regular point $y_{H}$, then this projective point is unique and $y_{H}=\nabla D_{A}(H)$ [GKZ94, Th.1.1, Ch. 1]. This is referred to as biduality. 2. (2) When $X_{A}^{\nu}\subset(\mathbb{P}^{N})^{\nu}$ is a hypersurface, biduality implies that the Gauss map $\gamma:X_{A}^{\nu}\dashrightarrow\mathbb{P}^{n}$ is defined by $H\mapsto\nabla D_{A}(H)=y_{H}$ and the closure of its image equals $X_{A}$. Let $A=\\{a_{0},\ldots,a_{N}\\}\subset\mathbb{Z}^{n}$ be a lattice configuration. We will assume henceforth that $A$ is non-defective and that it is a homogeneous polynomial of degree $\delta>0.$ Given $A$-polynomials $p_{i}=\sum_{j=0}^{N}c_{ij}x^{a_{j}},\quad\quad i=0,\ldots r,$ we also denote by $(p_{0},\ldots,p_{r})\in(\mathbb{P}^{(r+1)(N+1)-1})^{\nu}$ the vector of their coefficients. For any $\lambda=(\lambda_{0},\ldots,\lambda_{r})\in\mathbb{P}^{r}$ we write $P_{\lambda}:=\lambda_{0}p_{0}+\ldots+\lambda_{r}p_{r}\in(\mathbb{P}^{N})^{\nu}.$ ###### Definition 4.2. Consider the incidence variety (4.1) $\Sigma=\left\\{((p_{0},p_{1},\ldots,p_{r}),\lambda)\in(\mathbb{P}^{(r+1)(N+1)-1})^{\nu}\times\mathbb{P}^{r}\,:\,\sum_{j}c_{ij}\frac{\partial D_{A}}{\partial c_{j}}(P_{\lambda})=0,\,i=0,\ldots,r\right\\}.$ Let $\pi:\Sigma\to(\mathbb{P}^{(r+1)(N+1)-1})^{\nu}$ be the linear projection onto the first factor. The $r$-multivariate iterated dual scheme $\pi(\Sigma)$ is defined by the projective elimination ideal $\pi I=(I:m^{\infty})\cap\mathbb{C}[c],$ where $\mathbb{C}[c]$ is the ring of polynomials in the variables $c_{ij}$ and $I$ is the ideal $I=\left\langle\sum_{j}c_{ij}\frac{\partial D_{A}}{\partial c_{j}}(P_{\lambda}),i=0,\ldots,r\right\rangle.$ When $\pi(\Sigma)$ has codimension one, we denote by $JD_{r,A}\in\mathbb{Z}[c]$ a generator (unique up to multiplication by a nonzero constant) of the union of the codimension one components of $\pi I$ and we call it the _geometric iterated discriminant_. Notice that the projection is in general not irreducible; see for instance Example 2.3. We will see in Proposition 4.6 below that the geometric iterated discriminant $JD_{r,A}$ coincides with the more naive definition of the iterated discriminant $ID_{r,A}$ from Definition 1.1. Let $(p,\lambda)=((p_{0},\ldots,p_{r}),(\lambda_{0},\ldots,\lambda_{r}))\in\Sigma$. In order to understand the projection $\pi$ we consider two auxiliary maps, $\phi:\Sigma\to X_{A}^{\nu}$ and $T:(\mathbb{P}^{(r+1)(N+1)-1})^{\nu}\dashedrightarrow Gr(r+1,N+1):$ ${\Sigma}$${X_{A}^{\nu}}$${\pi(\Sigma)}$${Gr(r+1,N+1)}$$\scriptstyle{\phi}$$\scriptstyle{T}$ defined by $\phi(p,\lambda)=P_{\lambda}$ and $T(p_{0},\ldots,p_{r})=T_{p},$ where we denote by $T_{p}$ the projective linear span of $p_{0},\ldots,p_{r}.$ ###### Lemma 4.3. Let $p=(p_{0},\ldots,p_{r})\in(\mathbb{P}^{(r+1)(N+1)-1})^{\nu},$ then $JD_{r,A}(p)=0$ if and only if $T_{p}$ is tangent to $X_{A}^{\nu}$ at some point $\xi.$ ###### Proof. If $JD_{r,A}(p_{0},\ldots,p_{r})=0$ then there exists $\lambda$ such that $(p,\lambda)\in\Sigma$; let $\xi=P_{\lambda}.$ Consider the equalities (4.2) $\sum_{j}c_{ij}\frac{\partial D_{A}}{\partial c_{j}}(P_{\lambda})=0,\quad i=0,\ldots,r.$ The Euler relation implies $\deg(D_{A})\cdot D_{A}(P_{\lambda})=0,$ so $D_{A}(P_{\lambda})=0$ and thus $P_{\lambda}\in X_{A}^{\nu}.$ Moreover (4.2) implies that each $p_{i}$ lies in $T_{X^{\nu},\xi}$, which is equivalent to $T_{p}\subset T_{X^{\nu},\xi}.$ The converse follows the same line of proof. ∎ Recall that we denote by $\textrm{sing}(X_{A}^{\nu})$ the subscheme of $X_{A}^{\nu}$ defined by the ideal generated by the partial derivatives of $D_{A}.$ ###### Theorem 4.4. Assume $A\subset\mathbb{Z}^{n}$ is non-defective and let $r\in\mathbb{Z}$ with $0\leq r\leq\dim(X_{A}).$ Then, the mixed discriminant $MD_{r,A}$ always divides the geometric iterated discriminant $JD_{r,A}$. Moreover: 1. (1) If ${\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))>r$, then $JD_{r,A}=MD_{r,A}$. 2. (2) If ${\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))=r$, then $JD_{r,A}=MD_{r,A}\prod_{i=k}^{\ell}Ch_{Y_{k}}^{\mu_{k}},$ where $Y_{1},\dots,Y_{\ell}$ are the irreducible components of $sing(X_{A}^{\nu})$ of maximal dimension $r$, with respective multiplicities $\mu_{k}\geq 1$. 3. (3) If ${\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))<r$, then $\pi(\Sigma)=(\mathbb{P}^{(r+1)(N+1)-1})^{\nu},$ and $JD_{r,A}=0.$ ###### Proof. As already observed, in the classical case of $r=0$ we have $ID_{0,A}=MD_{0,A}=D_{A}.$ Note also that by Propositions 3.4 and 3.3, $\deg(MD_{r,A})>0$ and it is irreducible. Observe that the map $\phi$ is surjective since for any $F\in X_{A}^{\nu},$ $F=\phi(F,\ldots,F,\frac{1}{(r+1)},\ldots,\frac{1}{(r+1)})$ and we have $(F,\ldots,F,\frac{1}{(r+1)},\ldots,\frac{1}{(r+1)})\in\Sigma.$ The rational map $T$ is defined over the open dense subset $U_{T}=\\{p\;:\,T_{p}\simeq\mathbb{P}^{r}\\}$ of all $p$ with linear span of projective dimension $r$. Notice also that $T$ is surjective and that for each $H\in Gr(r+1,N+1)$, the fiber $T^{-1}(H)$ has dimension $(r+1)^{2}-1.$ Let $\Sigma^{\circ}=\phi^{-1}((X_{A}^{\nu})_{reg})$ and $\Sigma^{\prime}=\phi^{-1}(sing(X_{A}^{\nu}))$; that is, let $\Sigma^{\circ}=\\{(p,\lambda)\in\Sigma\,:\,P_{\lambda}\in(X_{A}^{\nu})_{reg}\\},\quad\Sigma^{\prime}=\\{(p,\lambda)\in\Sigma\,:\,P_{\lambda}\in sing(X_{A}^{\nu})\\}.$ It follows that $\pi(\Sigma)=\pi(\Sigma^{\circ})\cup\pi(\Sigma^{\prime}).$ We claim that $\overline{\pi(\Sigma^{\circ})}\subseteq V(MD_{r,A}).$ In fact, take a generic point $(p,\lambda)\in\pi(\Sigma^{\circ})$. We can then assume that not only $P_{\lambda}\in(X_{A}^{\nu})_{reg}$, but also there is a unique regular point $y=(x^{m_{0}}:\ldots:x^{m_{N}})\in X_{A}$ with $x\in(K^{})^{n}$ such that $P_{\lambda}(x)=0$ and $\frac{\partial P_{\lambda}}{\partial x_{i}}(x)=0,\,\,i=1,\ldots,n.$ By Remark 4.1, $y=(\frac{\partial D_{A}}{\partial c_{0}}(P_{\lambda}):\ldots:\frac{\partial D_{A}}{\partial c_{n}}(P_{\lambda})).$ The equations $\frac{\partial P_{\lambda}}{\partial x_{i}}(x)=0$ for $i=1,\ldots,n$ mean that $\sum\lambda_{i}\nabla(p_{i})(x)=0.$ Moreover $p_{i}(x)=0$ for all $i$ because (4.3) $p_{i}(x)=\sum_{j}c_{ij}y_{j}=k\sum_{j}c_{ij}\frac{\partial D_{A}}{\partial c_{j}}(P_{\lambda})=0,\quad i=0,\ldots r$ for some $k\in K^{}$ such that $y=k\,\nabla D_{A}(P_{\lambda}).$ This implies that $MD_{r,A}(p_{0},\ldots,p_{r})=0.$ We now show that $V(MD_{r,A})\subseteq\pi(\Sigma).$ Let $(p_{0},\ldots,p_{r})$ be a generic element in the zero locus of $MD_{r,A}.$ Then there exists $(u,\lambda)\in(K^{})^{n+r+1}$ such that $p_{0}(u)=\cdots=p_{r}(u)=0\text{ and }\begin{bmatrix}\nabla p_{0}(u)&\cdots&\nabla p_{r}(u)\end{bmatrix}\begin{bmatrix}\lambda_{0}\\\ \vdots\\\ \lambda_{r}\end{bmatrix}=0.$ We claim that $(p,\lambda)\in\Sigma.$ If $P_{\lambda}\in sing(X_{A}^{\nu})$ is generic then $\frac{\partial D_{A}}{\partial c_{j}}(P_{\lambda})=0$ and thus $(p_{0},\ldots,p_{r})\in\pi(\Sigma).$ If instead $P_{\lambda}\in(X_{A}^{\nu})_{reg}$ is generic, then biduality gives $\nabla D_{A}(P_{\lambda})=y$ with $y=(u^{m_{0}}:\ldots:u^{m_{N}})$ and thus $\sum_{j}c_{ij}\frac{\partial D_{A}}{\partial c_{j}}(P_{\lambda})=p_{i}(u)=0$ as in (4.3), implying again that $(p_{0},\ldots,p_{r})\in\pi(\Sigma).$ We have then proved that (4.4) $\overline{\pi(\Sigma^{\circ})}\subseteq V(MD_{r,A})\subseteq\pi(\Sigma).$ Consider the non-embedded primary components of the ideal $\langle\frac{\partial D_{A}}{\partial c_{j}},j=0,\ldots,N\rangle$ defining the singular locus of $X_{A}^{\nu}.$ Correspondingly, we consider the decomposition into irreducible components $sing(X_{A}^{\nu})=\bigcup Y_{k}.$ Define (4.5) $V_{k}=\\{H\in Gr(r+1,N+1)\,:H\cap Y_{k}\neq\emptyset\\}\text{ and }\Sigma_{k}=\phi^{-1}(Y_{k}).$ Recall that $\text{\rm codim}_{Gr(r+1,N+1)}(V_{k})=\max\\{0,\text{\rm codim}_{\mathbb{P}^{N}}(Y_{k})-r\\}.$ Assume that $\text{\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))>r$. Then $\text{\rm codim}_{Gr(r+1,N+1)}(V_{k})\geq 2$ for all $k.$ It follows that $\text{\rm codim}_{{\mathbb{P}^{(r+1)(N+1)-1}}^{\nu}}(\pi(\Sigma_{i}))=\text{\rm codim}_{{\mathbb{P}^{(r+1)(N+1)-1}}^{\nu}}(\overline{\pi(\Sigma_{i})\cap U_{T}})\geq\text{\rm codim}_{{\mathbb{P}^{(r+1)(N+1)-1}}^{\nu}}(\overline{T^{-1}(V_{i})})\geq 2$ for all $i$. The containment in Equation (4.4) then implies that $\pi(\Sigma)$ is of codimension one and set-theoretically coincides with $V(MD_{r,A})$. As the mixed discriminant $MD_{r,A}$ is irreducible, it remains to show that the multiplicity of $MD_{r,A}$ in $ID_{r,A}$ is one. For that, it is enough to show that there exists $(p_{0}^{\nu},\ldots,p_{r}^{\nu})\in V(MD_{r,A})$ and $\lambda^{\nu}$ such that $(p^{\nu},\lambda^{\nu})\in\Sigma$ and $d\pi((p^{\nu},\lambda^{\nu}))$ has maximal rank. We start by choosing a point $\xi\in reg(X_{A}^{\nu})$ such that $\textrm{rank}(H)=n,$ where $H=Hess(D_{A})(\xi).$ Notice that $H={\rm Jac}(\gamma)(\xi),$ where $\gamma:X_{A}^{\nu}\dasharrow X_{A}$ is the Gauss map defined as $\gamma(y)=\nabla D_{A}(y)$ which has generic rank equal to $n=\dim(X_{A}).$ Up to a change of coordinates, $H$ can be assumed to be of the form (4.6) $H=\begin{bmatrix}I_{n}&0\\\ 0&0\end{bmatrix}.$ Consider $Z=\\{M\in Gr(r+1,N+1)\,:\,\xi\in M\subset T_{X_{A}^{\nu},\xi}\\}$ and recall that we assume that $r\leq n.$ Note that for every $p\in T^{-1}(Z),$ we have $\xi\in T_{p}\subset T_{X_{A}^{\nu},\xi}$ and $\dim(T_{p})=r$ as $p\in U_{T}$. It follows that there exists a unique $\lambda$ such that $(p,\lambda)\in\Sigma$ and $\phi(p,\lambda)=\xi.$ Consider the $(r+1)\times(N+1)$ matrix $M_{p}$ whose $i$-th row corresponds to the coefficients of $p_{i}$. Without loss of generality, the matrix $M_{p}$ can be assumed to be of the form $[I_{r+1},C].$ Let $\chi(p)=\det\begin{bmatrix}I_{r+1}&C\\\ -C&I_{r+1}\end{bmatrix}.$ Notice that if all the entries of $C$ are $0$ then $\chi(p)=1$, and thus $\chi$ is not identically zero. Choose now $p\in T^{-1}(Z)$ such that $\chi(p)\neq 0$ and let $\lambda^{\nu}$ be the unique solution to $\xi=\sum\lambda^{\nu}_{i}p_{i}.$ Note that $d\pi(p,\lambda^{\nu})=M_{p}HM_{p}^{t}.$ If this matrix is not of maximal rank then there is $\alpha\in\mathbb{P}^{r}$ such that $M_{p}HM_{p}^{t}\alpha=0.$ Since $H$ is as in (4.6) and $M_{p}^{t}\alpha\neq 0$ then $HM_{p}^{t}\alpha\in\ker(M_{p}),$ which is a contradiction to $\chi(p)\neq 0.$ Assume that $\text{\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))<r$. Then $\text{\rm codim}_{\mathbb{P}^{N}}(Y_{k})<r+1$ for all $k.$ The assumption also implies that any element of the Grassmannian belongs to $V_{k}$ for all $k$ (defined in (4.5)) and that $\pi(\Sigma^{\prime})\cap U_{T}=T^{-1}(Gr(r+1,N+1))=U_{T}.$ It follows that $\mathbb{P}^{(r+1)(N+1)-1}=\overline{U_{T}}=\overline{\pi(\Sigma_{i})\cap U_{T}}=\pi(\Sigma_{i})\subset\pi(\Sigma)$ and thus $\pi(\Sigma)=(\mathbb{P}^{(r+1)(N+1)-1})^{\nu}$. Assume now that $\text{\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))=r.$ Then $\text{\rm codim}_{Gr(r+1,N+1)}(V_{k})=1$ and thus by definition $\pi(\Sigma_{k})=V(Ch_{Y_{k}}^{\mu_{k}})$. ∎ ###### Remark 4.5. = We have not identified the exponents $\mu_{k}$ occurring in the factorization of the iterated discriminant. Based on the evidence we have collected through examples (see for instance Examples 2.3 and Example 4.7) and the results in [BM09], we see some evidence of the following. We conjecture that $\mu_{k}=2$ if $Y_{k}$ is a component corresponding to the closure of the locus of those $p$ for which there are two different non-degenerate multiple roots (the double point locus), while $\mu_{k}=3$ when $Y_{k}$ is a component corresponding to the locus of those $p$ for which there is a degenerate multiple root (the cusp locus). Weyman and Zelevinsky showed in [WZ96] that for some hyperdeterminants there can be more than one irreducible component of each type. The following Proposition 4.6 explains the name geometric iterated discriminant: we show that under the hypotheses of Theorem 4.4, $JD_{r,A}$ equals $ID_{r,A}$ and thus when it is nonzero it can be computed as a discriminant of a discriminant. Recall that, given a natural number $d$, we denote by $d\Delta_{r}$ in $\mathbb{R}^{r+1}$ the lattice configuration given by the integer points in the dilated unit simplex $d$ times, and by $D_{d\Delta_{r}}$ the associated discriminant. For any homogeneous polynomial $H=H(\lambda_{0},\dots,\lambda_{r})$ of degree $d$, the discriminant of $H$ equals, up to constant, the resultant of its partial derivatives: (4.7) $D_{d\Delta_{r}}(H)\,=\,{\rm Res}_{d-1}\left(\frac{\partial H}{\partial\lambda_{0}},\dots,\frac{\partial H}{\partial\lambda_{r}}\right),$ where ${\rm Res}_{d-1}$ denotes the homogeneous resultant associated to $r+1$ homogeneous polynomials of degree $d-1$ [GKZ94, Jou91]. Moreover, the following universal property is proved in [Jou91]. Let $G_{0},\dots,G_{r}\in\mathbb{Z}[u][\lambda_{0},\dots,\lambda_{r}]$ have degree $d-1$ with generic coefficients $u$: $G_{i}(u,\lambda)=\sum_{\|\alpha\|=d_{i}}\,u_{i,\alpha}\,\lambda^{\alpha}.$ Denote by $\langle G_{0},\dots,G_{r}\rangle\subset\mathbb{Z}[u][\lambda_{0},\dots,\lambda_{r}]$, the ideal generated by $G_{0},\dots,G_{r}$ in this ring. Then for any variable $\lambda_{i}$ and any $N>\sum_{i}d_{i}-n$, it holds that (4.8) $\lambda_{i}^{N}\,{\rm Res}_{d-1}(G_{0},\dots,G_{r})\in\langle G_{0},\dots,G_{r}\rangle.$ Thus, such an equality holds for any specialization of the coefficients $u$ in a ring. Let $A$ be a non-defective configuration with $\text{\rm codim}(sing(X_{A}^{\nu}))\geq r$. Call $\delta=\deg(D_{A})$ and for a choice of $A$-polynomials $p_{0},\dots,p_{r}$ consider the evaluation $D_{A}(P_{\lambda})=D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i})$, which is either zero or a homogeneous polynomial in $\lambda=(\lambda_{0},\dots,\lambda_{r})$ of degree $\delta$. ###### Proposition 4.6. Under the hypotheses of Theorem 4.4, the following equality holds: $JD_{r,A}=ID_{r,A}.$ Moreover, when ${\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))\geq r$, the degree of the iterated discriminant equals (4.9) $\deg(ID_{r,A})=(r+1)\delta(\delta-1)^{r}.$ ###### Proof. By Theorem 4.4 and Definition 1.1, we can assume that ${\rm codim}_{X_{A}^{\nu}}(sing(X_{A}^{\nu}))\geq r$. Let $(p_{0}^{0},\dots,p_{r}^{0},\lambda^{0})$ be a point in the incidence variety $\Sigma$ defined in (4.1). Note that for any $i=0,\dots,r$, we have that $0\,=\,\sum_{j}c_{ij}\frac{\partial D_{A}}{\partial c_{j}}(\sum_{i=0}^{r}\lambda_{i}^{0}p_{i}^{0})\,=\frac{\partial}{\partial\lambda_{i}}D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i}^{0})(\lambda^{0}).$ Then, $D_{\delta\Delta_{r}}(D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i}^{0}))=0.$ Conversely, as we pointed out in (4.7), this homogeneous discriminant equals the resultant ${\rm Res}_{\delta-1}\left(\frac{\partial}{\partial\lambda_{0}}D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i}^{0}),\dots,\frac{\partial}{\partial\lambda_{r}}D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i}^{0})\right).$ It then follows from [Jou91] that if $D_{\delta\Delta_{r}}(D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i}^{0}))=0$, then there exists $\lambda^{0}\in\mathbb{P}^{r}$ which is a common zero of all these partial derivatives. Moreover, we deduce from Equation (4.8) that for any ring $R$ containing the coefficients of $D_{A}(P_{\lambda})$ and for any $i=0,\dots,r$, the iterated discriminant $D_{\delta\Delta_{r}}(P_{\lambda})$ lies in the ideal generated by the partial derivatives $\frac{\partial}{\partial\lambda_{j}}D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i}^{0}),j=0,\dots,r,$ in the localization $R[\lambda_{0},\dots,\lambda_{r}]_{\lambda_{i}}$. It follows that $D_{\delta\Delta_{r}}(D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i}^{0}))=0$ defines the schematic image of $\Sigma$ and thus it coincides with $ID_{r,A}(p_{0},\dots,p_{r})$. To see that Equation (4.9) holds, recall that $\deg(D_{A})=\delta$ and so the degree of $D_{A}(\sum_{i=0}^{r}\lambda_{i}p_{i})$ in the coefficients of $p_{0},\dots,p_{r}$ as well as in the $\lambda$ variables is equal to $\delta$. On the other side, the degree of $D_{\delta\Delta_{r}}$ is equal to $(r+1)\delta^{r}$. ∎ We conclude this section with two examples that illustrate Theorem 4.4 with $r+1=2.$ In the first one, the singular locus has codimension $r+1=2$, which implies a factor (with multiplicity $2$) of the iterated discriminant. In the second one, the singular locus has codimension bigger than $2$, which implies equality between $MD_{1,A}$ and $ID_{2,A}.$ ###### Example 4.7. [Example 2.3, continued.] Consider again the two dimensional configuration corresponding to the first Hirzebruch surface ${\mathbb{F}}_{1}$: $A=\\{(0,0),(1,0),(2,0),(0,1),(1,1)\\}.$ Given a generic $A$-polynomial $f(x,y)=a_{0}+a_{1}x+a_{2}x^{2}+y(b_{0}+b_{1}x)$, we saw that $D_{A}(f)\,=a_{0}b_{1}^{2}-a_{1}b_{0}b_{1}+a_{2}b_{0}^{2}$. The ideal defining the singular locus $S$ of $X_{A}^{\nu}$ is generated by $b_{0}^{2},b_{0}b_{1},b_{1}^{2},-a_{1}b_{1}+2a_{2}b_{0},2a_{0}b_{1}-a_{1},b_{0}.$ This ideal has multiplicity $2$ and its radical is generated by $b_{0},b_{1}$. In this case, $ID_{2,A}$ has another irreducible factor $Ch_{S}$ of degree $2$ coming from the Chow form of $S$, to the second power: $ID_{2,A}=MD_{1,A}\,\cdot\,Ch_{S}^{2},$ where $Ch_{S}((a_{0},a_{1},a_{2},b_{0},b_{1}),(A_{0},A_{1},A_{2},B_{0},B_{1}))=B_{0}b_{1}-B_{1}b_{0}$. ###### Example 4.8. Let $X_{A}$ be the Segre embedding of $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$, so $D_{A}$ is the hyperdeterminant of format $(2,2,2)$ of degree $4$, whose singular locus has codimension greater than $2$ by [WZ96]. Take $r=2$, so that $MD_{1,A}$ equals the discriminant of the hyperdeterminant of format $(2,2,2,2)$ (corresponding to the Segre embedding of $\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}\times\mathbb{P}^{1}$.) In this case, $MD_{1,A}$ equals the iterated discriminant $ID_{2,A}$ and thus has degree $2\cdot 4\cdot(4-1)^{1}=24$. This is the only known case of polynomials of degree bigger than $2$ for which the iterated and the mixed discriminants coincide. ## 5\. Comparing mixed and iterated discriminants In this section we consider the case when $A$ equals the lattice points in a cartesian product of dilates of standard simplices: $d_{1}\Delta_{k_{1}}\times\cdots\times d_{\ell}\Delta_{k_{l}}$, for some $\ell\geq 1$. In other words we investigate Segre-Veronese varieties $X_{A}=\mathbb{P}^{k_{1}}(d_{1})\times\cdots\times\mathbb{P}^{k_{\ell}}(d_{\ell}).$ The symbol $\mathbb{P}^{k}(d)$ denotes the Veronese embedding of degree $d$ in dimension $k$, i.e. the variety $\mathbb{P}^{k}$ embedded in $\mathbb{P}^{{k+d\choose d}-1}$ by the global sections of the line bundle ${\mathcal{O}}_{{\mathbb{P}}^{k}}(d).$ We occasionally denote $\mathbb{P}^{k}(1)$ by $\mathbb{P}^{k}.$ The symbol $\mathbb{P}^{k_{1}}(d_{1})\times\cdots\times\mathbb{P}^{k_{\ell}}(d_{\ell})$ denotes the Segre embedding of the above defined Veronese embeddings, more precisely the variety $\mathbb{P}^{k_{1}}\times\cdots\times\mathbb{P}^{k_{\ell}}$ embedded via the global sections of the line bundle $\pi^{}_{1}{\mathcal{O}}_{{\mathbb{P}}^{k_{1}}}(d_{1})\otimes\ldots\otimes\pi_{\ell}^{}{\mathcal{O}}_{{\mathbb{P}}^{k_{\ell}}}(d_{\ell}),$ where $\pi_{i}$ denotes the $i^{th}$ projection $\pi_{i}:\mathbb{P}^{k_{1}}\times\cdots\times\mathbb{P}^{k_{\ell}}\to\mathbb{P}^{k_{i}}.$ These are toric embeddings corresponding to the configurations of lattice points of the polytopes $d_{1}\Delta_{k_{1}}\times\ldots\times d_{l}\Delta_{k_{\ell}}.$ When $d_{i}=1$ we recover the case of hyperdeterminants, which has been completely solved in [WZ96]. In Proposition 5.2 we show that that when $\ell=1$ there is equality if and only if $r=1$ and $d_{1}=2$. We then conjecture that these are all the possible cases (see Conjecture 5.3), that is, in all other cases the singularities of the discriminantal locus have codimension one in the dual variety. We conclude with Theorem 5.6, which covers the case in which all $d_{i}>1$. To determine when the iterated and mixed discriminants of Segre-Veronese varieties are equal, we start with the following lemma, which allows us to compute the degree of $MD_{r,d\Delta_{n}}$, that is, the case in which we consider $(r+1)$ polynomials of degree $d$ in $n$ variables. Recall that when $r\leq n$, we know by Proposition 3.3 that the mixed discriminant equals the discriminant of the Cayley configuration given by the lattice points in the product of simplices $\Delta_{r}\times d\Delta_{n}$. ###### Lemma 5.1. If $r\leq n$ then $\deg(MD_{r,d\Delta_{n}})=(n+1)\binom{n}{r}d^{r}(d-1)^{n-r}$. ###### Proof. We will use [GKZ94, Ch. 13, Theorem 2.4], which tells us that this degree is equal to the coefficient of the monomial $x^{r}y^{n}$ in the expansion of $S(x,y)=\frac{1}{\left((1+x)(1+y)-x(1+y)-dy(1+x)\right)^{2}}=\frac{1}{(1-(d-1)y-dxy)^{2}}.$ We may write $S(x,y)=\left(\frac{1}{1-q}\right)^{2}=\sum_{n\geq 0}(n+1)q^{n},$ where $q=(d-1)y+dxy=y((d-1)+dx)$. Since $q^{n}=\left(\sum_{j=0}^{n}\binom{n}{j}(d-1)^{n-j}d^{j}x^{j}\right)y^{n},$ we have (5.1) $S(x,y)\,=\,\sum_{n\geq 0}\sum_{j=0}^{n}(n+1)\binom{n}{j}(d-1)^{n-j}d^{j}x^{j}y^{n}.$ From this expansion, we see that the coefficient of $x^{r}y^{n}$ is equal to $(n+1)\binom{n}{r}(d-1)^{n-r}d^{r}$ when $r\leq n$, and is equal to $0$ if $r>n$. This completes the proof. ∎ ###### Proposition 5.2. Let $1<d$ and $1\leq r\leq n$. Then $MD_{r,d\Delta_{n}}=ID_{r,d\Delta_{n}}$ if and only if $r=1$ and $d=2$. The fact that this equality holds in the case of $r=1$ and $d=2$ was shown in [Ott13, Theorem 8.2]. Although we include this in our proof for completeness, the main contribution of this result is that equality does not hold in any other case. ###### Proof. For any $d>1$, the configuration of lattice points in $d\Delta_{n}$ is non- defective [BJ14] and as $r\leq n$, it is enough to check that $\deg(MD_{r,d\Delta_{n}})=\deg(ID_{r,d\Delta_{n}})$ by Propositions 3.3 and 3.4. From Lemma 5.1 we know that $\deg(MD_{r,d\Delta_{n})})=(n+1)\binom{n}{r}d^{r}(d-1)^{n-r}.$ By Proposition 4.6 we also know that $\deg(ID_{r,d\Delta_{n}})=(n+1)(d-1)^{n}(r+1)((n+1)(d-1)^{n}-1)^{r}.$ To determine when these are equal, we will consider the ratio of the two degrees, both of which are nonzero for $d>1$. We have $\frac{\deg(ID_{r,d\Delta_{n}})}{\deg(MD_{r,d\Delta_{n}})}=\frac{(n+1)(d-1)^{n}(r+1)((n+1)(d-1)^{n}-1)^{r}}{(n+1)\binom{n}{r}d^{r}(d-1)^{n-r}}=\frac{(d-1)^{r}(r+1)((n+1)(d-1)^{n}-1)^{r})}{\binom{n}{r}d^{r}}.$ For any $d>1$ we have $(n+1)(d-1)^{n}-1=n(d-1)^{n}+(d-1)^{n}-1\geq n(d-1)^{n},$ with equality if and only if $d=2$. Thus the numerator satisfies $(d-1)^{r}(r+1)((n+1)(d-1)^{n}-1)^{r}\geq(d-1)^{r}(r+1)(n(d-1)^{n})^{r}=(d-1)^{r(n+1)}(r+1)n^{r}.$ Since $\binom{n}{r}\leq\frac{n^{r}}{r!}$, we have $\frac{\deg(ID_{r,d\Delta_{n}})}{\deg(MD_{r,d\Delta_{n}})}\geq\frac{(d-1)^{r(n+1)}(r+1)n^{r}}{\frac{n^{r}}{r!}d^{r}}=\frac{(d-1)^{r(n+1)}(r+1)!}{d^{r}}=\left(\frac{(d-1)^{n+1}}{d}\right)^{r}\cdot(r+1)!$ If $d=2$, then this ratio is $\frac{(r+1)!}{d^{r}}=\frac{(r+1)!}{2^{r}}\geq 1$, with equality if and only if $r=1$. If $d>2$, then $(d-1)^{n+1}\geq(d-1)^{2}\geq 2(d-1)=2d-2>d$. Thus $\frac{(d-1)^{n+1}}{d}>1$, and so $\frac{\deg(ID_{r,d\Delta_{n}})}{\deg(MD_{r,d\Delta_{n}})}>(r+1)!.$ Thus except possibly in the case of $d=2$ and $r=1$, we have $\deg(ID_{r,d\Delta_{n}})>\deg(MD_{r,d\Delta_{n}})$. To see that $d=2$ and $r=1$ gives $\deg(MD_{r,d\Delta_{n})})=\deg(ID_{r,d\Delta_{n}})$, note that in this case the ratio of the degrees is $\frac{(2-1)^{1}(1+1)((n+1)(2-1)^{n}-1)^{1})}{\binom{n}{1}2^{1}}=\frac{2(n+1-1)}{2n}=1.$ ∎ Geometrically, Proposition 5.2 shows that for $\mathbb{P}^{r}(1)\times\mathbb{P}^{n}(d)$ the associated mixed discriminant is equal to the iterated discriminant only when $r=1$ and $d=2$. Note that we don’t consider the case $d=1$ because this case is defective. It is natural to consider the same question for any product-of-simplices : $\mathbb{P}^{r}(1)\times\mathbb{P}^{k_{1}}(d_{1})\times\cdots\times\mathbb{P}^{k_{\ell}}(d_{\ell}).$ Setting $d=(d_{1},\ldots,d_{\ell})$ and $k=(k_{1},\ldots,k_{\ell})$, let $A_{\ell,d,k}$ denote the configuration corresponding to $\mathbb{P}^{k_{1}}(d_{1})\times\cdots\times\mathbb{P}^{k_{\ell}}(d_{\ell})$. We conjecture the following: ###### Conjecture 5.3. We have $ID_{r,A_{\ell,d,k}}=MD_{r,A_{\ell,d,k}}$ if and only if $\mathbb{P}^{r}(1)\times\mathbb{P}^{k_{1}}(d_{1})\times\cdots\times\mathbb{P}^{k_{\ell}}(d_{\ell})$ is of one of the following forms: 1. (1) $\mathbb{P}^{r}\times\mathbb{P}^{m}\times\mathbb{P}^{m},m\geq 1,r=1,2$, 2. (2) $(\mathbb{P}^{1})^{4}$, 3. (3) $\mathbb{P}^{1}\times\mathbb{P}^{n}(2)$. This conjecture was inspired by the question posed in [GKZ94, Chapter 14, pg 479], which coincides with the above conjecture when $d_{i}=1$ for all $i$. Their conjecture (and thus our conjecture in this special case) was proved in [WZ96]. Note that Proposition 5.2 implies that Conjecture 5.3 is true when $\ell=1$, which puts us into case (3). To study our conjecture in general, the following theorem giving the degree of the mixed discriminant $MD_{r,A_{\ell,d,k}}$ will be useful. Let $B=B_{\ell}$ be the set of all non-empty subsets $\Omega\subset\\{0,1,\ldots,\ell\\}$. For each $\Omega\in B$, let $d_{\Omega}=\sum_{j\in\Omega}d_{j}.$ Let $\delta(\Omega)\in\mathbb{Z}^{\ell+1}_{+}$ be the characteristic vector of $\Omega$. For every $\kappa=(r,k_{1},\ldots,k_{\ell})\in\mathbb{Z}_{+}^{r+1}$, let $\mathcal{P}(\kappa)$ denote the set of all partitions of $\kappa$ into a sum of vectors $\delta(\Omega)$; in other words, $\mathcal{P}(\kappa)$ is the set of all non-negative integral vectors $(m_{\Omega})_{\Omega\in B}$ such that $\sum_{\Omega\in B}m_{\Omega}\delta(\Omega)=\kappa$. ###### Theorem 5.4 (Theorem 13.2.5, [GKZ94]). The degree of $MD_{r,A_{\ell,d,k}}$ is given by $\sum_{(m_{\Omega})\in\mathcal{P}(\kappa)}\left(1+\sum_{\Omega\in B}m_{\Omega}\right)!\prod_{\Omega\in B}\frac{(d_{\Omega}-1)^{m_{\Omega}}}{m_{\Omega}!}.$ Note that any partition using the vector $\delta(\\{0\\})=(1,0,\ldots,0)$ will not contribute to this sum, since $d_{\\{0\\}}-1=0$. Letting $C$ be the set of all nonempty subsets of $\\{1,\ldots,\ell\\}$ and letting $k=(k_{1},\ldots,k_{\ell})$ as before, we have that $\deg\left(ID_{r,A_{\ell,d,k}}\right)=(r+1)\delta(\delta-1)^{r},$ where $\delta=\sum_{(m_{\Omega})\in\mathcal{P}(k)}\left(1+\sum_{\Omega\in C}m_{\Omega}\right)!\prod_{\Omega\in C}\frac{(d_{\Omega}-1)^{m_{\Omega}}}{m_{\Omega}!}.$ When it is clear from context, we will abbreviate $\deg(MD_{r,A_{\ell,d,k}})$ as $\deg(MD)$ and $\deg(ID_{r,A_{\ell,d,k}})$ as $\deg(ID)$. ###### Example 5.5. Let us compare the degrees of the mixed and the iterated discriminant when $r=1$, $\ell=2$, and $k_{1}=k_{2}=1$. To compute the degree of the mixed discriminant, we consider all partitions of $\kappa=(1,1,1)$. We may discount any partition with the vector $(1,0,0)$, as this partition would contribute a term of $0$ to $\deg(MD)$. Thus, the only relevant partitions are • $(1,1,1)$, * • $(1,1,0)+(0,0,1)$, and * • $(1,0,1)+(0,1,0)$. The contributions from these terms to $\deg(MD)$ are * • $2!\cdot\frac{(d_{1}+d_{2})^{1}}{1!}=2(d_{1}+d_{2})$ , * • $3!\cdot\frac{d_{1}^{1}\cdot(d_{2}-1)^{1}}{1!\cdot 1!}=6d_{1}(d_{2}-1)$, and * • $3!\cdot\frac{d_{2}^{1}\cdot(d_{1}-1)^{1}}{1!\cdot 1!}=6d_{2}(d_{1}-1)$, respectively. (Note that some of these contributions will be zero if one or both of $d_{1}$ and $d_{2}$ are equal to $1$.) Adding these gives $\deg(MD)=2(d_{1}+d_{2})+6d_{1}(d_{2}-1)+6d_{2}(d_{1}-1)=12d_{1}d_{2}-4d_{1}-4d_{2}=4(3d_{1}d_{2}-d_{1}-d_{2}).$ To compute $\deg(ID)$, we must consider the partitions of $k=(1,1)$. There are only two: $(1,1)$ and $(1,0)+(0,1)$. The contributions of these to $\delta$ are * • $2!\cdot\frac{(d_{1}+d_{2}-1)^{1}}{1!}=2(d_{1}+d_{2}-1)$ and * • $3!\cdot\frac{(d_{1}-1)^{1}(d_{2}-1)^{1}}{1!\cdot 1!}=6(d_{1}-1)(d_{2}-1),$ respectively. Thus $\delta=2(d_{1}+d_{2}-1)+6(d_{1}-1)(d_{2}-1)=6d_{1}d_{2}-4d_{1}-4d_{2}+4.$ It follows that $\deg(ID)=2\delta(\delta-1)=2(6d_{1}d_{2}-4d_{1}-4d_{2}+4)(6d_{1}d_{2}-4d_{1}-4d_{2}+3).$ We will now argue that $\deg(ID)>\deg(MD)$, unless $d_{1}=d_{2}=1$. First we perform the change of variables $d_{1}=d_{1}^{\prime}+1$ and $d_{2}=d_{2}^{\prime}+1$, to remove some of the negatives. This gives $\deg(MD)=4(3d_{1}^{\prime}d_{2}^{\prime}+2d_{1}^{\prime}+2d_{2}^{\prime}+1)$ and $\deg(ID)=2(6d_{1}^{\prime}d_{2}^{\prime}+2d_{1}^{\prime}+2d_{2}^{\prime}+2)(6d_{1}^{\prime}d_{2}^{\prime}+2d_{1}^{\prime}+2d_{2}^{\prime}+1).$ Now, if either $d_{1}$ or $d_{2}$ is greater than $1$, then $(6d_{1}^{\prime}d_{2}^{\prime}+2d_{1}^{\prime}+2d_{2}^{\prime}+1)$ is at least $3$, meaning that $\deg(ID)\geq 6(6d_{1}^{\prime}d_{2}^{\prime}+2d_{1}^{\prime}+2d_{2}^{\prime}+2)=36d_{1}^{\prime}d_{2}^{\prime}+12d_{1}^{\prime}+12d_{2}^{\prime}+12.$ This is certainly greater than $\deg(MD)=12d_{1}^{\prime}d_{2}^{\prime}+8d_{1}^{\prime}+8d_{2}^{\prime}+4,$ since $d_{1}^{\prime}$ and $d_{2}^{\prime}$ are nonnegative. So, in this case $\deg(ID)>\deg(MD)$. If we do have $d_{1}=d_{2}=1$, then $\deg(MD)=4=\deg(ID)$. This equality was predicted by case (1) of Conjecture 5.3. We will now prove that Conjecture 5.3 holds in the case that $r=1$ and $d_{i}>1$ for all $i$. ###### Theorem 5.6. Suppose $d_{i}>1$ for all $i$. Then the only case where $MD_{1,A_{\ell,d,k}}=ID_{1,A_{\ell,d,k}}$ is when $\ell=1$ and $d_{1}=2$. ###### Proof. This proposition holds when $\ell=1$ by Proposition 5.2, and when $\ell=2$ and $k_{1}=k_{2}=1$ by Example 5.5. Thus it suffices to prove that for $\ell=2$ with $(k_{1},k_{2})\neq(1,1)$, and for $\ell\geq 3$, we have $\deg\left(MD_{1,A_{\ell,d,k}}\right)<\deg\left(ID_{1,A_{\ell,d,k}}\right)$. First we consider how partitions of $\kappa=(1,k_{1},\ldots,k_{\ell})$ relate to partitions of $k=(k_{1},\ldots,k_{\ell})$. Each partition of $\kappa$ gives rise to a partition of $k$ simply by deleting the first coordinate and grouping together vectors that are now identical. Note that no partition $(m_{\Omega})$ contributing to $\deg(MD)$ uses the vector $(1,0,\ldots,0)$. Also, exactly one vector in each partition of $\kappa$ is of the form $(1,,\ldots,)$. Call the support of this vector $\Psi((m_{\Omega}))$, or simply $\Psi$ when the context is clear. Note that $m_{\Psi}=1$. Let $\Xi$ denote $\Psi\setminus\\{0\\}$. Isolating $\Psi$ and $\Xi$, we may write $\displaystyle\deg\left(MD\right)=$ $\displaystyle\sum_{(m_{\Omega})\in\mathcal{P}(\kappa)}\left(1+\sum_{\Omega\in B}m_{\Omega}\right)!\prod_{\Omega\in B}\frac{(d_{\Omega}-1)^{m_{\Omega}}}{m_{\Omega}!}$ $\displaystyle=$ $\displaystyle\sum_{(m_{\Omega})\in\mathcal{P}(\kappa)}\left(1+\sum_{\Omega\in B}m_{\Omega}\right)!\cdot\frac{(d_{\Psi}-1)^{m_{\Psi}}}{m_{\Psi}!}\cdot\frac{(d_{\Xi}-1)^{m_{\Xi}}}{m_{\Xi}!}\prod_{\Omega\in B,\Omega\neq\Psi,\Xi}\frac{(d_{\Omega}-1)^{m_{\Omega}}}{m_{\Omega}!}$ $\displaystyle=$ $\displaystyle\sum_{(m_{\Omega})\in\mathcal{P}(\kappa)}\left(1+\sum_{\Omega\in B}m_{\Omega}\right)!\cdot(d_{\Psi}-1)\cdot\frac{(d_{\Xi}-1)^{m_{\Xi}}}{m_{\Xi}!}\prod_{\Omega\in B,\Omega\neq\Psi,\Xi}\frac{(d_{\Omega}-1)^{m_{\Omega}}}{m_{\Omega}!}.$ Given $(m_{\Omega})$ a partition of $\kappa$, let $(n_{\Omega})$ be the corresponding partition of $k$. So, if the term in $\deg(MD)$ coming from $(m_{\Omega})$ is $\left(1+\sum_{\Omega\in B}m_{\Omega}\right)!\cdot(d_{\Psi}-1)\cdot\frac{(d_{\Xi}-1)^{m_{\Xi}}}{m_{\Xi}!}\prod_{\Omega\in B,\Omega\neq\Psi,\Xi}\frac{(d_{\Omega}-1)^{m_{\Omega}}}{m_{\Omega}!},$ then the term in $\delta$ coming from $(n_{\Omega})$ is $\left(1+\sum_{\Omega\in B}m_{\Omega}\right)!\cdot\frac{(d_{\Xi}-1)^{m_{\Xi}+1}}{(m_{\Xi}+1)!}\prod_{\Omega\in B,\Omega\neq\Psi,\Xi}\frac{(d_{\Omega}-1)^{m_{\Omega}}}{m_{\Omega}!}.$ Note that $d_{\Xi}=d_{\Psi}-1$. We know that $d_{\Xi}\neq 1$ by our assumption that $d_{i}>1$ for all $i$, so the change in factor between these two contributions is $\frac{d_{\Xi}-1}{d_{\Xi}(m_{\Xi}+1)}.$ Since $d_{\Xi}>1$, this is at least $\frac{1}{2(m_{\Xi}+1)}$. In general, $m_{\Omega}\leq\max\\{k_{i}\\}$ for any $\Omega$; since the vector $\delta(\Psi)$ also appears in the partition of $\kappa$, we in fact have $m_{\Xi}\leq\max\\{k_{i}\\}-1$. So, $m_{\Xi}+1\leq\max\\{k_{i}\\}$. It follows that $\frac{1}{2(m_{\Xi}+1)}$ is greater than or equal to $\frac{1}{2\max\\{k_{i}\\}}$. So, passing from a partition of $\kappa$ to a partition of $k$, the corresponding term in $\delta$ is at least $\frac{1}{2\max\\{k_{i}\\}}$ times the corresponding term in $\deg(MD)$. Now we consider how many partitions of $\kappa$ give rise to the same partition of $k$. Given a partition $(n_{\Omega})$ of $k$, all relevant partitions of $\kappa$ that map to it can be constructed by choosing a single vector used in $(n_{\Omega})$, and appending a $1$ to the $0^{th}$ coordinate. Thus, the number of partitions of $\kappa$ mapping to $(n_{\Omega})$ is equal to the number of distinct vectors used in $(n_{\Omega})$. The number of distinct vectors in this partition can be bounded by $k_{1}+\ldots+k_{\ell}$, since this is the total sum of all the entries of all the vectors used. Thus, we have that $\delta\geq\frac{1}{2\max\\{k_{i}\\}(k_{1}+\cdots+k_{\ell})}\deg(MD).$ It follows that $\displaystyle\deg(ID)=2\delta(\delta-1)\geq$ $\displaystyle 2\frac{1}{2\max\\{k_{i}\\}(k_{1}+\cdots+k_{\ell})}\deg(MD)\cdot(\delta-1)$ $\displaystyle=$ $\displaystyle\frac{\delta-1}{\max\\{k_{i}\\}(k_{1}+\cdots+k_{\ell})}\cdot\deg(MD).$ To show that $\deg(ID)>\deg(MD)$, it remains to show that $\delta-1>\max\\{k_{i}\\}(k_{1}+\cdots+k_{\ell})$. First, rewrite $\displaystyle\delta=$ $\displaystyle\sum_{(m_{\Omega})\in\mathcal{P}(k)}\left(1+\sum_{\Omega\in C}m_{\Omega}\right)!\prod_{\Omega\in C}\frac{(d_{\Omega}-1)^{m_{\Omega}}}{m_{\Omega}!}$ $\displaystyle=$ $\displaystyle\sum_{(m_{\Omega})\in\mathcal{P}(k)}\left(1+\sum_{\Omega\in C}m_{\Omega}\right)\cdot\frac{\left(\sum_{\Omega\in C}m_{\Omega}\right)!}{\prod_{\Omega\in C}m_{\Omega}!}\prod_{\Omega\in C}(d_{\Omega}-1)^{m_{\Omega}}$ $\displaystyle=$ $\displaystyle\sum_{(m_{\Omega})\in\mathcal{P}(k)}\left(1+\sum_{\Omega\in C}m_{\Omega}\right)\cdot{\sum_{\Omega\in C}m_{\Omega}\choose m_{\Omega_{1}},\ldots,m_{\Omega_{t}}}\prod_{\Omega\in C}(d_{\Omega}-1)^{m_{\Omega}}.$ For any partition of $k$, we have that $\sum_{\Omega\in C}m_{\Omega}\geq\max\\{k_{i}\\}$, since $k_{i}$ vectors (counted with multiplicity) must have nonzero $i^{th}$ coordinate. Moreover, a multinomial coefficient ${a_{1}+a_{2}+\cdots+a_{s}\choose a_{1},a_{2},\cdots,a_{s}}$ can be rewritten as the product ${a_{1}\choose a_{1}}{a_{1}+a_{2}\choose a_{2}}\cdots{a_{1}+a_{2}+\cdots+a_{s}\choose a_{s}}$, so it is at least as large as ${a_{1}+a_{2}+\cdots+a_{s}\choose a_{s}}$. Of course, we may reorder the $a_{i}$’s in any way we desire. So, as long as some $a_{i}$ satisfies $0<a_{i}<a_{1}+\cdots+a_{s}$, we have ${a_{1}+a_{2}+\cdots+a_{s}\choose a_{1},a_{2},\cdots,a_{s}}\geq{a_{1}+a_{2}+\cdots+a_{s}\choose a_{i}}\geq{a_{1}+a_{2}+\cdots+a_{s}\choose 1}=(a_{1}+\cdots+a_{s})$. This means that if $(m_{\Omega})$ is a partition of $k$ that uses at least two different vectors, we have ${\sum_{\Omega\in C}m_{\Omega}\choose m_{\Omega_{1}},\ldots,m_{\Omega_{t}}}\geq\sum_{\Omega\in C}m_{\Omega}\geq\max\\{k_{i}\\}$. Finally, the product $\prod_{\Omega\in C}(d_{\Omega}-1)^{m_{\Omega}}$ is greater than or equal to $1$. Thus, every partition of $k$ that uses at least distinct two vectors contributes at least $(1+\max\\{k_{i}\\})\cdot\max\\{k_{i}\\}$ to $\delta$. We will now argue that there are at least $\ell$ such partitions of $k$. To do this, we split into two cases: where $\ell=2$, and where $\ell\geq 3$. If $\ell=2$ and $(k_{1},k_{2})\neq(1,1)$, then there are indeed at least two such partitions of $k=(k_{1},k_{2})$. For instance, we could use $(1,1)+(k_{1}-1)(1,0)+(k_{2}-1)(0,1)$ and $k_{1}(1,0)+k_{2}(0,1)$. Both do indeed use at least two distinct vectors since at least one of $k_{1}-1$ and $k_{2}-1$ is nonzero. Assume now $\ell\geq 3$. We can construct a partition of $k$ that uses at least two vectors by choosing any $0-1$ vector with support size at least $2$ and at most $\ell-1$, and then completing the partition by using standard basis vectors. The condition on the support size guarantees that at least one other vector will be used, and that this new standard basis vector has not already been used. There are $2^{\ell}-2-\ell$ such initial vectors, which is greater than or equal to $\ell$ since $\ell\geq 3$. Thus, at least $\ell$ partitions of $k$ contribute at least $(1+\max\\{k_{i}\\})\cdot\max\\{k_{i}\\}$ to $\delta$. Note that $\ell\max\\{k_{1}\\}\geq k_{1}+\cdots+k_{\ell}$. It follows that $\displaystyle\delta\geq$ $\displaystyle\ell(1+\max\\{k_{i}\\})\cdot\max\\{k_{i}\\}$ $\displaystyle\geq$ $\displaystyle\ell\max\\{k_{i}\\}\cdot\max\\{k_{i}\\}+\ell$ $\displaystyle>$ $\displaystyle\ell\max\\{k_{i}\\}\cdot\max\\{k_{i}\\}+1$ $\displaystyle\geq$ $\displaystyle\max\\{k_{i}\\}(k_{1}+\cdots+k_{\ell})+1$ Equivalently, $\delta-1>\max\\{k_{i}\\}(k_{1}+\cdots+k_{\ell})$. This implies that $\deg(ID)>\deg(MD)$, as desired. ∎ ## 6\. Curves in the plane In this section we will determine when the mixed and iterated discriminants associated to a planar configuration are equal. Let $A=P\cap\mathbb{Z}^{2}$, where $P$ is a smooth lattice polygon of dimension $2$. Let $v_{A}$, $p_{A}$, and $V_{A}$ denote the normalized area ${\rm area}_{\mathbb{Z}}(P)$ (that is, twice its Euclidean area), the lattice perimeter (that is, the number of points in $A$ on the edges of $P$), and the number of vertices of $P$, respectively. It is well known [GKZ94] that in this smooth case the degree $\delta_{A}$ of $D_{A}$ equals $\delta_{A}=3v_{A}-2p_{A}+V_{A}.$ The degree of the mixed discriminant can be computed from Corollary 3.15 in [CCD+13] as $\deg(MD(A,A))=2({\rm area}_{\mathbb{Z}}(2P)-{\rm area}_{\mathbb{Z}}(P)-p_{A})=2(4v_{A}-v_{A}-p_{A})=6v_{A}-2p_{A}.$ We can reformulate these equations in terms of the number of interior lattice points of $P$. Let $i_{A}$ denote the number of interior lattice points of $P$. Then we know by Pick’s Theorem that $v_{A}=2i_{A}+p_{A}-2,$ which can be rewritten as $v_{A}-p_{A}=2i_{A}-2$. This allows us to write $\deg(MD(A,A))=6v_{A}-2p_{A}=4v_{A}+2(v_{A}-p_{A})=4v_{A}+4i_{A}-4=4(v_{A}+i_{A}-1).$ and $\delta_{A}=3v_{A}-2p_{A}+V_{A}=v_{A}+2(v_{A}-p_{A})+V_{A}=v_{A}+4(i_{A}-1)+V_{A}$ ###### Example 6.1. Let $A=((0,0),(2,0),(0,2))$. Let us verify that $\deg(MD(A,A))=\deg(ID_{1,A})$, as implied by Proposition 5.2. We have $v_{A}=4$, $i_{A}=0$, and $V_{A}=3$. This gives us $\deg(MD(A,A))=4(v_{A}+i_{A}-1)=4(4-0-1)=12$ and $\delta_{A}=v_{A}+4(i_{A}-1)+V_{A}=4+4(0-1)+3=3.$ This means that $\deg(ID_{1,A})=2\delta_{A}(\delta_{A}-1)=2\cdot 3\cdot 2=12=\deg(MD(A,A))$. ###### Example 6.2. Assume $A=\text{conv}((0,0),(1,0),(0,1),(1,1))$. Let us verify that $\deg(MD(A,A))=\deg(ID_{1,A})$, as implied by Example 2.1. We have $v_{A}=2$, $i_{A}=0$, and $V_{A}=4$. This gives us $\deg(MD(A,A))=4(v_{A}+i_{A}-1)=4(2-0-1)=4$ and $\delta_{A}=v_{A}+4(i_{A}-1)+V_{A}=2+4(0-1)+4=2.$ This means that $\deg(ID_{1,A})=2\delta_{A}(\delta_{A}-1)=2\cdot 2\cdot 1=4=\deg(MD(A,A))$. It turns out that these two examples are the only smooth polygons $P$ where the iterated and the mixed discriminants associated to the configuration of lattice points in $P$ coincide. ###### Theorem 6.3. The only smooth polygons $P$ with an associated discriminant without singularities in codimension bigger than $1$ are the known cases of the triangle $2\Delta_{2}$ and the unit square. ###### Proof. Assume $P$ is such a polygon, $A=P\cap\mathbb{Z}^{2}$ and $\delta_{A}=\deg(D_{A})$. Using our formulas for $MD(A,A)$ and $\delta_{A}$, we have that $4(v_{A}+i_{A}-1)=2(v_{A}+4(i_{A}-1)+V_{A})(v_{A}+4(i_{A}-1)+V_{A}-1),$ which is equivalent to $2(v_{A}+i_{A}-1)=(v_{A}+4(i_{A}-1)+V_{A})(v_{A}+4(i_{A}-1)+V_{A}-1).$ Suppose for the sake of contradiction that $i_{A}>0$. Then $v_{A}+i_{A}-1\leq v_{A}+4(i_{A}-1)<v_{A}+4(i_{A}-1)+V_{A}$. Now, if $a,b,c,d$ are positive real numbers with $ab=cd$, then $b<c$ implies $a>d$. This means that $2>v_{A}+4(i_{A}-1)+V_{A}-1\geq v_{A}+V_{A}-1\geq v_{A}+2$. In other words, $v_{A}<0$, a contradiction. Thus we know that $i_{A}=0$. Setting $i_{A}=0$ reduces our equation to $2(v_{A}-1)=(v_{A}+V_{A}-4)(v_{A}+V_{A}-5).$ By a classification result due to [Koe91] and presented again in [Cas12], all convex lattice polygons with no interior lattice points are equivalent to either the triangle $2\Delta_{2}=\text{conv}((0,0),(2,0),(0,2))$, or to a polygon of the form $\text{conv}((0,0),(0,1),(a,0)),(b,1)$, where $a\geq b\geq 0$ and $a\geq 1$. These polygons are illustrated in Figure 2. All of these polygons have either three or four vertices. So, we must have $V_{A}=3$ or $V_{A}=4$. Figure 2. All lattice polygons with no interior lattice points If $V_{A}=3$, our equation becomes $2(v_{A}-1)=(v_{A}-1)(v_{A}-2).$ This means that either $v_{A}=1$, or $2=v_{A}-2$; that is, $v_{A}=1$ or $v_{A}=4$. If $v_{A}=1$, the only possibility for $P$ is the primitive lattice triangle of normalized $1$; but this gives a degenerate system, and so is removed from our consideration. If $v_{A}=4$, the only possibilities of $P$ are $\text{conv}((0,0),(2,0),(0,2))$ and $\text{conv}((0,0),(4,0),(0,1))$. The second polygon is not smooth, so the only possible triangle is $\text{conv}((0,0),(2,0),(0,2))$. If $V_{A}=4$, our equation becomes $2(v_{A}-1)=v_{A}(v_{A}-1).$ This means that either $v_{A}=1$ (which is impossible impossible with $V_{A}=4$), or that $v_{A}=2$. The only polygon with $4$ vertices and area $2$ is the square $\text{conv}((0,0),(1,0),(0,1),(1,1))$ Thus we have shown that $\text{conv}((0,0),(2,0),(0,2))$ and $\text{conv}((0,0),(1,0),(0,1),(1,1))$ are the only possibilities for $P$. 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11institutetext: Auckland University of Technology, Auckland, New Zealand, 22institutetext: University of Tasmania, Tasmania, Australia 33institutetext: Nagoya Institute of Technology, Japan 33email<EMAIL_ADDRESS>weihua.li<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS> # Privacy Information Classification: A Hybrid Approach Jiaqi Wu 11 Weihua Li 11 Quan Bai 22 Takayuki Ito 33 Ahmed Moustafa 33 ###### Abstract A large amount of information has been published to online social networks every day. Individual privacy-related information is also possibly disclosed unconsciously by the end users. Identifying privacy-related data and protecting the online social network users from privacy leakage turn out to be significant. Under such a motivation, this study aims to propose and develop a hybrid privacy classification approach to detect and classify privacy information from OSNs. The proposed hybrid approach employs both deep learning models and ontology-based models for privacy-related information extraction. Extensive experiments are conducted to validate the proposed hybrid approach, and the empirical results demonstrate its superiority in assisting online social network users against privacy leakage. ###### Keywords: Privacy detection Online Social Networks Deep Learning Privacy Information. ## 1 Introduction With the proliferation and popularisation of the World Wide Web, Online Social Networks (OSNs) become of one of the essential channels for social interactions and communications [1, 2]. OSNs provide great convenience to the users, but these online social platforms also raise potential risks, such as privacy leakage. A vast amount of private information can be accessed publicly through OSNs, such as preferences, email address, marital status, hometown, activities attended, etc., which may lead to severe security issues. Therefore, it is significant to explore a useful and practical approach to protect OSN users from privacy information disclosure. Users should be reminded before posting any privacy-related messages to the public. In the context of OSNs, “user privacy” refers to a sequence of words, stating or implying any individual’ s personal information, preferences, events that he or she involved; privacy leakage describes a situation when an individual shares stories including private information with their contacts or even those they are not familiar with. Thus, it is necessary to develop a tool to detect and identify all the possible privacy-related information contained in any posting messages [20, 7]. More importantly, the justifications of privacy- related information classification would be helpful for OSN users to get rid of posting similar messages again. In our previous research work [11], we conducted preliminary studies and developed a generic framework of privacy leakage detection for OSN users based on deep learning models. The proposed framework is capable of capturing privacy-related entities after giving sufficient training. However, two significant limitations are to be covered. Firstly, it can only remind users regarding the possibility of privacy leakage. As a result, detailed leaking information in terms of what kind of leakage is missing. Secondly, the privacy model is not conceptually modelled or presented. Ontology models conceptually reflect the domain-specific knowledge in the form of terms and demonstrate two apparent advantages, i.e., shareability and reusability [22]. Therefore, ontology-based privacy models can be easily extended and applied to various OSNs [16]. Moreover, as ontology organises the concepts in the form of taxonomy or hierarchy based on a pre-defined natural relationship, it is suitable to introduce ontology into the research of privacy-related information extraction and classification. As an extension of our previous work [11], in this paper, we leverage a hybrid privacy classification approach, incorporating both deep learning and ontology models, for individual users of OSNs. The extended framework is capable of addressing the privacy leakage problem for individuals by effectively identifying privacy information and classifying into a detailed category. More specifically, the proposed hybrid approach is composed of two major components, i.e., a deep learning based approach to detect the privacy leakage on online social data and an ontology privacy model classifying the privacy- related information into fine-grained privacy categories. Deep learning models are utilised to conduct the Name Entity Recognition (NER) and detect the pre- defined privacy-related entities. An ontology model is developed based on the analysis of massive data collected from real-world OSNs. Given the predictive results carried out by the deep learning model, the privacy ontology model further classifies the recognised privacy-related entities into sub-classes. The rest of the paper is organised as follows. Section 2 reviews the existing literature of privacy information classification approaches on OSNs. Section 3 introduces the entire privacy information detection and fine-grained classification approach. In Section 4, experiments are conducted to evaluate the ontology-based approach on the dataset crawled from twitter. Section 5 concludes the findings of this paper and points out the limitations and future direction. ## 2 Related Work Most contemporary privacy information classification approaches aim to detect some specific categories of privacy information on OSNs or perform a binary classification, i.e., sensitive or non-sensitive, rather than identifying and classifying privacy information for the end users. [8] indicates that it is easy for users to leak the matters or activities that one involves anytime anywhere unconsciously. For example, a tweet saying that user’s family will go out for a holiday implies no one stays at home, which may cause robbery accidents. Therefore, the privacy information of OSNs users should be prompted before sharing with the public. [6] proposes a machine learning classification approach by adopting Named Entity Recognition technology, which can classify privacy information on OSNs into different categories, e.g., electronic devices and brands. Similarly, [13] present a privacy classifier for three kinds of sensitive tweets, i.e., drunk, holiday and disease, and classifies tweets into binary results, i.e., sensitive tweets or nonsensitive tweets. From the research above, we can see even there are several papers about classifying privacy revealing information on OSNs, most of them pay attention to specific categories by machine learning approaches. Very few studies classify them according to a domain privacy ontology and protect individual OSNs users from online privacy leaks. Consequently, an ontology-based classification approach is comparatively a new research field of privacy information classification on OSNs. Ontology-based classification approaches show outstanding performance in many areas, e.g., online job offers [3] and trust requirements on semantic web services [4]. Whereas, it is necessary to build an ontology into the OSNs [10]. [5] survey the privacy ontology and point out that although some security ontologies for fulfilling security requirements have been presented, these studies focus on security rather than privacy [19]. They present a novel privacy ontology to identify the key concepts and relations to satisfy the privacy requirement. However, it aims to deal with the privacy requirements for software engineers. Actually, there is little research about privacy information ontology which can apply for OSNs. Therefore, in this paper, we aim to build an ontology about privacy information on online social media. Because almost all the privacy-related entities are words or phrases, the semantic similarity degree calculating plays a significant role in the proposed ontology-based privacy approach. ## 3 Automated Hybrid Privacy Detection Framework The proposed automated hybrid privacy information detection and classification approach are demonstrated in Fig 1. There are two key parts in the proposed framework, i.e., the privacy-related entities recognition and further classification based on ontology models. In the former, a deep learning model is trained to detect four types of entities that potentially cause privacy leakage. While, in the latter, a privacy ontology model is developed based on the analysis of messages posted by the OSN users. Figure 1: Automated Hybrid Privacy Detection Framework The rationale of utilising a hybrid model of both deep learning and ontology- based classification is clarified as follows: ontology-only approach classifies privacy information merely based on the domain-specific vocabulary of terms or concepts, which have to be systematically defined [17]. Deep learning based NER does not necessarily require a lexicon or domain-specific words, but it is difficult to control the classification of specific terms as deep learning models turn out to be a “black-box”. By considering the factors mentioned above, a combination of deep learning and ontology model is adopted. The details of these two key modules are introduced in the following subsections. ### 3.1 Deep-learning Based Privacy Information Detection Users’ data collection turns out to be the preliminary of deep-learning based privacy information detection. OSN users’ public data can be collected from OSNs through web crawlers or available APIs. The collected raw data are supposed to be filtered and pre-processed, e.g., removing messages which are advertisements or spams, removing meaningless words and characters and parsing word sequences to tokens. Next, the processed data are enriched by running through the pre-annotation process if a privacy-detection NER model is already available. Given the pre- annotated dataset, privacy-related entities are required to be annotated manually, and the data size of the training and testing data for the deep learning model are nearly 20k. As mentioned previously, in the context of OSNs, “privacy” is associated with an individual’ s personal information, preferences, events that he or she involved. Thus, messages containing four types of entities, i.e., “PERSON”, “TRAIT”, “PREF”, and “EVENT”, potentially cause privacy leakage. After annotation, the annotated dataset is then fed into the deep learning model for training. ### 3.2 Ontology-based Privacy Information Classification As the second part of our hybrid privacy detection approach, an ontology-based classification approach can remind individual users regarding what is to be disclosed instead of simply giving general information. #### 3.2.1 The Domain and Scope of The Privacy Domain Ontology The defining domain and scope of “privacy” is the first step to build a private-related information ontology model [17]. Naturally, the privacy information concepts describing different subclasses (class corresponds to the entity in this thesis) of the four privacy-related entities will be fed into our ontology. Specifically, the privacy domain ontology on OSNs includes: 1. 1. A hierarchical classification of privacy concepts from general classes to specific subclasses. 2. 2. A set of relations between privacy classes to link concepts in a more complicated way that implied by an underlying hierarchy. #### 3.2.2 Privacy-related Keywords Extraction Initially, it is significant to obtain a comprehensive list of privacy-related terms and concepts in order to form the hierarchy of privacy ontology. To construct an ontology, we extracted all the values of privacy-related entities recognized by the deep learning based model. Next, based on the word frequencies, representative keywords are selected as the major indicators of the subcategories of the entities. For example, we want to find some keywords regarding private events under the “Event” main class. Some verbs representative of private events, e.g., eating, shopping, etc., as well as some nouns, e.g., concert, meeting, journey, etc., are frequently mentioned in event-related entities. Additionally, some words are significant indicators of privacy-related entities, which can imply the user is leaking his/her ’TRAIT’ sensitive information. Because the creation process of an ontology is an interactive process [12], we searched the selected keywords in the dataset to find out the occurrence of these words and how important of them according to the term frequency [17]. Through the interactive procedure, the terms and concepts of the privacy information ontology are finally determined. For example, because the keyword of “interview” frequently appears in the extracted entities, it turns out to be a keyword, representing the subclass of “Corporate Event”. Table 1 shows the representative keywords extracted from the collected data, which are used for building the privacy domain ontology. Table 1: Corresponding Keywords with Classes and Subclasses Class \| Subclass \| Keywords ---\|---\|--- Person \| Individual \| I \| Third Party \| you, we, they, he, she, classmate, uncle Preference \| Item \| book, chocolate, keyboard, tea \| Hobby \| cosplay, paint, fishing, dancing, reading \| Specific Person \| girlfriend, teacher Event \| Private Event \| eat, shopping, concert, movie, exercise, spa \| Corporate Event \| wedding, interview, meeting, conference, festival, party, parade, salon \| Journey \| fly, holiday, travel, island, hotel, airport Trait \| Individual Identity \| years-old, Auckland \| Linked Information \| lawyer, female, gay, Christian, married, white, disable #### 3.2.3 Privacy Ontology Among the possible approaches in developing a class hierarchy, a top-down process has been selected by considering the relationships among the privacy concepts in this paper [17]. The ontology hierarchy presents a tree structure, having most general classes on top and specific associative classes connected with the general ones. For example, given “PERSON” as a superclass, “Individual” and “Third Party” can be the subclasses based on the recognised entity-value pairs. Three other superclasses, i.e., “TRAIT”, “EVENT”, and “PREFERENCE”, are also included. TRAIT describes personally identifiable information(PII) or sensitive personal information, which is defined and classified into two types from the usage of the PII in United Stated legal fields, i.e., distinguish identity and relating information [14]. Similarly, TRAITS can be classified into two subclasses as below: 1. 1. Any information can be used to distinguish an individuals identity, such as birth date and hometown. 2. 2. Any information can be linked to an individual, such as medical, educational, marital status and employment information. Through this kind of classification approach, two subclasses of Trait are identified: Individual identity and Linked information. For example, the date of birth can be recognised as an individuals identity. Whereas, race, gender, sexual orientation, marital status, religion, belief, and education background are categorized as linked information. Similarly, EVENT can be classified as a private event, corporate event, and journey in terms of the event is social or non-social. Among the subclasses of “Event”, tweets about the journey is individually classified because we think users who reveal their journey plans will make them very vulnerable to theft crimes. Preference can be classified as a specific person, item and hobby according to the characteristic of the leaking hobby information. Therefore, the privacy domain ontology for OSNs is presented in Fig. 2: Figure 2: Privacy Ontology #### 3.2.4 Semantic Phrase Similarity Degree Based On GloVe To classify the privacy-related entities into fine-grained subclasses, it is essential to find an approach in finding the subclass where the entities have the highest belonging degree. Moreover, the highest belonging degree is decided by the highest similarity degree between the extracted entities and the representative terms. Comparing the similarity of word embedding of each word can decide their similarity degree [9]. Word embedding is defined as semantic vector space models that use vectors to represent each word. Consequently, in this subsection, we will propose a semantic phrase similarity degree approach based on Glove. It is divided into two steps. Firstly, a word semantics vector space model is decided. Secondly, the construction of the classification model will be described by taking “Event” entity as an example. $\bullet$ Word Semantics Vector Space Model In the first step, the main step is to choose a word semantics vector space model. A pre-trained statistical model (called “en_core_web_lg” in spacy) is used in this paper, which is trained on blogs, news, and comments with GloVe. Global Vectors for Word Representation (GloVe) is a state-of-art tool using word embedding techniques. Most word embedding approaches like Word2Vec exist a disadvantage, which is the lack of co-occurrence between words. Luckily, the GloVe approach trains global word-word co-occurrence counts which fills this gap, outperforming than other current word embedding approaches in common word similarity tasks[18] [15]. That is the reason we choose this word semantics vector space model for us to use in order to compare the similarity degree between words. $\bullet$ The Construction of The Classification Model To explain the construction of the classification model, we make use of “Event” entities as an example. In Fig. 2, the entity consists of three subclasses, “Private Event”, “Corporate Event” and “Journey”. Moreover, in Table 1, there are 20 representative terms representing the subclasses associated with them. Each representative term can be represented as $term_{i}$, where $i$ belongs to {0,1,2,…,19}. Among them, ’Private Event’:{0,1,2,…,5}, ’Corporate Event’:{6,7…,13}, and ’Journey’:{14,15…,19}. Similarly, each word in the extracted entities can be represented as $entity_{j}$. Consequently, $S_{ij}$ can represent the semantic similarity degree between each word in the extracted entities and each representative term: $S_{ij}=similarity(entity_{j},term_{i})$ (1) So the similarity of event-related entities and the event representative terms in the ontology can be shown as follows: $S_{i(sum)}=\sum_{j=1}^{n}S_{ij}$ (2) After the calculation of all the similarity degree, then the subclass of the extracted event entities can be decided according to the maximum degree of $S_{i(sum)}$, which means the subclass which obtains the maximum $S_{i(sum)}$ is the corresponding subclass of the event-related entity: $\lambda=\max\left(S_{0}{(sum)},S_{1}{(sum)},...,S_{19}{(sum)}\right)$ (3) Then the $i$ which obtains the maximum degree of $S_{i(sum)}$ can be decided and the corresponding subclass can be decided as below: $subclass=\left\\{\begin{array}[]{ll}PrivateEvent&\textrm{$i\in{0,1,...,5}$}\\\ CorporateEvent&\textrm{$i\in{6,7,...,13}$}\\\ Journey&\textrm{$i\in{14,15,...,19}$}\end{array}\right.$ Hence, by using the semantic information which the word embedding technique capture [21], we can classify privacy-related information into the corresponding subclass in the privacy ontology like the “Event” procedure we list. ## 4 Experiments Experiments have been conducted to evaluate the proposed hybrid privacy detection approach. The experiment uses a real-world testing dataset that aims to classify fine-grained privacy-related entities. Moreover, we also focus on one interesting problem around private tweets: What type of personal private information is leaked most on OSNs? ### 4.1 Data Description Twitter 111https://twitter.com/ is one of the largest Online Social Media platforms as a micro-blogging service, where a large amount of information is broadcast publicly by individual users. In Twitter, the information posted by end users is named as tweets. Twitter provides APIs, allowing developers to search and store tweets based on certain criteria. Therefore, we search and collect 18k tweets from Twitter through API. Most tweets are selected by searching for keywords related to sensitive activities and plans. Consequently, we use Twitter API to search for some terms which contain sensitive keywords, e.g., sensitive activities and plans, which may result in privacy information leakage and cause negative consequences. Then we collect around 18k tweets as a testing dataset in this experiment. As we demonstrated before, the tweets will be conducted with the deep learning based NER approach and the ontology-based classification approach in this chapter, then the corresponding subclasses will be decided. ### 4.2 Evaluation To evaluate the performance of the hybrid privacy detection approach, we manually annotate the subclasses of privacy leaking information. Moreover, the ”ground truth” is prepared by allowing the users themselves to provide opinions on whether they leak the privacy and what types of private information they are leaking on tweets. For example, a tweet “I watch a movie.”, has “I” annotated as “Individual” and “watch a movie” annotated as “Private Event”. Four traditional measures to utilised to evaluate the performance of the proposed hybrid approach: 1. 1. Accuracy: the fraction of correct classification in the testing data set; 2. 2. Precision: the fraction of correct classification among all results are classified in this subclass in the testing data set; 3. 3. Recall: the fraction of correct classification among all actual results belong to this subclass in the testing data set; 4. 4. F1-value: the harmonic value of precision and recall, which is a balance measurement. ### 4.3 Experimental Results Our hybrid privacy classification approach utilizes a deep learning based NER approach and an ontology-based approach to perform classification of specific privacy information. After the NER, we use the ontology vocabulary for performing semantic phrase similarity degree calculation with the extracted privacy-related entities. We evaluate our hybrid approach on the testing dataset and get a considerable performance, as shown in Table 2, with high accuracy in each type. We believe this accuracy is high enough to demonstrate the effectiveness of our automated detection and classification approach. However, we observe the accuracy of categories under “TRAIT” entity is much lower than other types of entities. The extraction result of “TRAIT” entity is lower than other types of entity and a trait-related privacy entity will be classified to categories under other types of entities. That is why the accuracy value of classification of “TRAIT” entity is lower than other entities. Table 2: Performance of Hybrid Privacy Information Classification Class \| Subclass \| Accuracy ---\|---\|--- Person \| Individual \| 0.94 \| Third Party \| 0.85 Preference \| Item \| 0.82 \| Hobby \| 0.81 \| Specific Person \| 0.76 Event \| Private Event \| 0.74 \| Corporate Event \| 0.76 \| Journey \| 0.77 Trait \| Individual Identity \| 0.62 \| Linked Information \| 0.64 ### 4.4 Privacy Leaking Information Categorization After the whole automated hybrid privacy information detection approach, all the types of private information leaks on the 18k testing dataset can be extracted. Then we plot the distribution of the results of the privacy information types of the testing dataset in Fig. 3, which includes all the subclasses under the “TRAIT”, “PREF”, and “EVENT” entities. According to the privacy rule in this detection approach, each tweet with privacy-related entities which is identified as the private tweet contains the “Person” entity, so the distribution of the “Person” entity is not necessary to be analyzed its categorization. Figure 3: Distribution of Different Types of Privacy Information Leaking In this section, we also explore the question of what type of sensitive information users leak most. We counted the percentage of the eight types of privacy information in the 18k testing dataset. The results are shown in Fig. 3. From Fig. 3, we can see the most leaking information is the “Event” entity, where “Private Event” counts the most in it. Additionally, “Journey” under the “Event” and “Specific person” under the “Preference” are also leaked a lot in tweets, which means the information about all subclasses under the entity of “Event” both is leaked a lot on OSNs. From the results, we suggest that OSNs users should exercise a little more restraint about posting relevant tweets about these types. On the other hand, people are more conservative about “Individual Identity” and “Hobby” information because the privacy leaks (as the percentage) of the two types is much smaller than other types. ## 5 Conclusions and Future Works In this paper, we propose a hybrid approach for classifying private information generated by OSN users. Through characterising the nature of privacy information leaking on OSNs, a deep learning model has been employed for privacy-related entities recognition, and an ontology-based classification approach is conducted to automatically classify fine-grained privacy information, i.e., nine subtypes of private leaking. The ontology-based approach calculated the semantic similarity between entities extracted from the deep learning model and the representative terms. We evaluated the result with the accuracy value, which demonstrates it gains a considerable performance. Moreover, what specific types of personal private information users are leaking on OSNs can be understood. This research can be extended by investigating the following directions. Firstly, we can recognise privacy-related entities on tweets other than on all tweets by one user. In the future, all privacy information revealing by a user can be collected by the model and protect the user. Secondly, different tweets are associated with different degrees of privacy leakage. In this paper, we demonstrate what types of privacy-related entities OSNs users reveal. However, we do not analyse the detailed privacy leakage degree of users. In the future, we plan to evaluate the privacy leakage degree and explore the insights based on predictive results. ## References * [1] Batra, A., Sidhu, K., Sharma, S.: Characteristics of women whatsapp users and use pattern. 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# Mining Large-Scale Low-Resource Pronunciation Data From Wikipedia Tania Chakraborty, Manasa Prasad, Theresa Breiner, Sandy Ritchie, Daan van Esch Google Research {taniarini, pbmanasa, tbreiner, sandyritchie<EMAIL_ADDRESS> (June 2020) ###### Abstract Pronunciation modeling is a key task for building speech technology in new languages, and while solid grapheme-to-phoneme (G2P) mapping systems exist, language coverage can stand to be improved. The information needed to build G2P models for many more languages can easily be found on Wikipedia, but unfortunately, it is stored in disparate formats. We report on a system we built to mine a pronunciation data set in 819 languages from loosely structured tables within Wikipedia. The data includes phoneme inventories, and for 63 low-resource languages, also includes the grapheme-to-phoneme (G2P) mapping. 54 of these languages do not have easily findable G2P mappings online otherwise. We turned the information from Wikipedia into a structured, machine-readable TSV format, and make the resulting data set publicly available so it can be improved further and used in a variety of applications involving low-resource languages. ## 1 Introduction There are thousands of languages spoken around the world, and many efforts to learn about them and document them. However, information about low-resource languages is not always easy to find, or leveraged to its full potential. Wikipedia is a useful resource for language-specific information, with a community of native speakers and linguistic experts who continue to improve coverage and quality111See the LingWiki effort: https://en.wikipedia.org/wiki/Wikipedia:GLAM/SOAS/Lingwiki. While Wikipedia is often used to build monolingual or parallel text corpora [Prasad et al., 2018, Rahma et al., 2018], its articles on specific languages contain useful information that is not frequently extracted. In this paper we describe how we extracted pronunciation information from these Wikipedia pages for hundreds of languages, some of which are not represented in other sources, which can be used in technologies for low- resource languages. The data was mined from tables containing information such as phonemes, associated graphemes, and sample words in each language. While the tables are understandable to the human eye, it is nontrivial to automatically parse and generate standardized TSV files from them. We publish our mined data and encourage others to leverage and improve this source. ## 2 Uses for Pronunciation Data While large-scale pronunciation data can be useful in many linguistic pursuits, grapheme-to-phoneme (G2P) mappings are also a building block for automatic language processing systems. Tools to generate pronunciations from input words [Mortensen et al., 2018, Novak et al., 2016] typically support a specific set of languages but can often be extended using G2P rules for new languages. G2P data can also be used in cross-lingual transfer learning, e.g. in named entity recognition [Bharadwaj et al., 2016]. G2P mappings are also required in typical automatic speech recognition (ASR) and text-to-speech (TTS) systems. While these systems usually train on paired audio and transcription data, several recent areas of research leverage G2P mappings to overcome this requirement. If there is a G2P mapping and some text available in a low-resource language, an ASR system can be built by repurposing an acoustic model from a similar higher-resource language [Prasad et al., 2019]. G2P mappings can even be bootstrapped for new languages using only the language’s phoneme inventory combined with higher-resource language text written in the same script [Bleyan et al., 2019], making even simple phoneme data a more useful tool for scaling technologies to new languages. ## 3 Similar Resources One similar resource in this space is PHOIBLE [Moran et al., 2014], a database listing the phonemes of over 2,100 languages, their possible allophones, and how common each phoneme is in the language, but not any G2P information. Wikimedia’s Wiktionary and Incubator222https://meta.wikimedia.org/wiki/Wiktionary#List_of_Wiktionaries are open-content dictionaries written in about 500 languages, and can be used to automatically extract phoneme and pronunciation information [Schlippe et al., 2010, Lee et al., 2020]. Deri and Knight [Deri and Knight, 2016] as well as Peters et al. [Peters et al., 2017] have been able to adapt this data even to build G2P models for new languages that do not appear in the mined data. Deri and Knight also published the G2P mappings that they mined from Wikipedia IPA Help tables333https://en.wikipedia.org/wiki/Category:International_Phonetic_Alphabet_help for 98 languages. Our data set, which includes phoneme inventories for over 800 languages, offers G2P data in 63 low-resource languages, 54 of which are not covered by Deri and Knight. There is also more potential G2P information stored for an additional 224 languages, as we detail in section 4.3. We hope that this additional data set will be a useful supplement to the existing resources. ## 4 Mining Pronunciation Data from Wikipedia The English Wikipedia contains hundreds of articles about specific languages444https://en.wikipedia.org/wiki/Index_of_language_articles, often including details on the phonology and writing systems, and covers languages that are not represented in other data sources. We wanted to aggregate this data into an easily comparable and processable data set to make it more widely usable for the research that we described in section 2. ### 4.1 Extracting the Wikipedia Pages Wikipedia’s category structure enabled us to target relevant pages inter- linked from the category pages for Languages by Country555http://en.wikipedia.org/wiki/Category:Languages_by_country and Language Phonologies666http://en.wikipedia.org/wiki/Category:Language_phonologies. We also checked for other potential language pages by leveraging the Wikipedia search query for ISO639 codes777https://iso639-3.sil.org/; for example, the page on Amarasi (ISO 639 code ‘aaz’) can be found at https://en.wikipedia.org/wiki/ISO_639:aaz. Using tooling similar to wikitable2csv888https://github.com/gambolputty/wikitable2csv, we extracted the table data embedded in each targeted page. ### 4.2 Parsing the Tables From the tables on each language’s page, we wanted to parse: a) the phoneme inventory of the language b) the pronunciation features of the phonemes, including voicing, place/manner of articulation, and other phonetic features, useful for clarity and in case of nonstandard transcriptions c) grapheme to phoneme mappings d) words in the native script and e) the phonemic transcriptions of the words. While some or all of this data is present in most of the language pages, different pages format their tables in different ways, which makes data extraction and processing tricky. We aim to cover as many edge cases as possible and successfully parse a majority of the table formats. Some especially odd cases and ones that we left unaddressed can be found in section 4.3. We mainly saw two different ways that data in a table are presented, which we will call Type A and Type B. Type A has one set of headers, where the contents are related to each other horizontally (or vertically if headers are on the side). For example, Figure 1(a)999Screenshots of pages accessed on June 24, 2020: https://en.wikipedia.org/wiki/Amarasi_language and https://en.wikipedia.org/wiki/Alekano_language is a Type A table where the data in the column titled “Amarasi Alphabet” are graphemes, and the data in the column titled IPA are the corresponding phonemes. Type B has two sets of headers, in which case the data between columns is not related, but rather classified in a similar way. For example, in Figure 1(b), the phonemes in the columns are not related to each other; the data in each cell is classified by two or more pronunciation features, namely place of articulation (column headers) and manner of articulation/voicing (row headers). (a) Amarasi (iso:aaz) grapheme-phoneme table (b) Alekano (iso:gah) consonant phoneme table Figure 1: Example Wikipedia tables with relatively simple formats. Since there is no obvious indicator of a table’s type from its representation, we have to automatically determine how to parse the data by observing the structure of the table; i.e., how many headers the table has, if the table has both column and row headers, etc. We determine that a given row or column is a header if it has consecutive ‘header’ type cells, which are annotated with a ‘!’ symbol101010According to the Wikipedia Table Help page: https://en.wikipedia.org/wiki/Help:Table. We treat tables with headers only on the top or the side as Type A, while tables with both are Type B. Category \| Targeted Keywords ---\|--- Grapheme \| letter, grapheme, alphabet, written Phoneme \| IPA, pronunciation Pronunciation Feature \| description, vowel/consonant [found in table caption rather than header] Example word \| example, word Transcription \| transcription Unclassified \| [any unmatchable data] Table 1: Keywords in table headers that helped us determine the category of data in the table. The data extracted from the tables is classified into one of the categories listed in Table 1. The correct category for each data item is based on the type of the table and the text in the header corresponding to the item’s column. For Type A tables, a regular expression is used to determine the category, based on keywords as you can see in Table 1. If the header text contains any variation of the keywords we are searching for, we can classify that data into the matching category. For Type B tables, the data is classified as a pronunciation feature, with the cell value representing the associated phoneme and the actual pronunciation features extracted from the row and column headers. ### 4.3 Challenges As we described in Section 4, the primary challenge in mining the data was the variety of the table structures. We tried to account for the most common types of tables, including the simpler Type A and Type B tables described above, as well as tables with repeated headers, as seen in Figure 2111111Screenshot of page accessed on June 24, 2020: https://en.wikipedia.org/wiki/Wakhi_language. If repetition is not accounted for, the contents of an entire column would be incorrectly associated with each other. Instead, we need to associate only two adjacent rows with each other as we parse the table. The same principle applies to tables that have a repetitions every N rows or every N columns for any value of N. Some more table examples can be found in Appendix A. Figure 2: Wakhi (iso:wbl) data table with repeated headers across rows Another challenge is that the keywords in Table 1 may be insufficient to classify data correctly for all cases. For example, if a table uses the name of the language as the header for the graphemes in a G2P table instead of a more generic keyword, we will not parse the data correctly. As we developed the parser, we added in new keywords to cover more cases, but do not cover all edge cases. When we find a table whose header is not a keyword, we store the data item and its associated header together in the Unclassified category rather than discarding it. Some examples can be found in Appendix A. If a keyword is used in a way we do not expect, or if there is a very intricate table format, there can be other problems. Some tables list multiple data within a cell, and it is not clear what the relationship is in order to split the data or classify it correctly. Some examples of these tables can be found in Appendix B. Our dataset may include some of this noise, although we hope that in most cases, tables that confused our system would store the data in the Unclassified category to at least be human readable. ## 5 Our Data Set We found phoneme data for 819 languages, and G2P mappings for 63 languages. 54 of these do not appear in Deri and Knight’s G2P data for 98 languages mined from the Wikipedia IPA Help Tables 121212https://drive.google.com/drive/u/1/folders/0B7R_gATfZJ2aWkpSWHpXUklWUmM [Deri and Knight, 2016]. An additional 224 languages had some data we could only parse as Unclassified, but further analysis hints that 72 of them may contain G2P information. Since the Wikipedia pages for higher-resource languages do not often contain G2P tables (but tend to focus on grammar or history), our data set does not include G2P for these languages, which are more likely covered by Deri and Knight, and instead represents more low- resource languages. We publish our data 131313https://github.com/google/language- resources/tree/master/mined-wiki-phoneme-tables in TSV format, where the columns correspond to: grapheme, phoneme, pronunciation features, example word, IPA transcription of the example word, and Unclassified data. Phonemes can appear in more than one row if, for example, they are mapped to more than one grapheme or example word. If a field is empty, it is filled in with ‘(n/a)’. See Figure 3 for an example excerpt, with more examples available in the appendix. We also include a txt with all Wikipedia links that were used. Figure 3: Excerpt of Amarasi (iso:aaz) TSV data file, containing g2p mappings ## 6 Conclusion We mined a data set covering 819 languages’ phonemes, including 63 languages’ grapheme-to-phoneme mappings, 54 of which are low-resource languages that do not have easily findable G2P mappings in other available sources. This information is available on Wikipedia, but inconsistencies with how it is formatted make it quite challenging to use in extending language technology. While our extraction system might be improved to handle more edge cases in future work, it may in fact be even more fruitful to collaborate with relevant Wikipedia editors to standardize their contributions to make the data more machine-readable. At any rate, we hope this work helps the research community continue to scale research and technologies to more and more languages. ## Appendix A Additional Examples of Table Formats We Handle This appendix gives some further examples of table formats that are nontrivial to parse in order to associate and classify the data correctly. Figure 4: Ocaina (iso:oca) data table with repeated headers across columns In Figure 4 141414Screenshot of page accessed on June 24, 2020: https://en.wikipedia.org/wiki/Ocaina_language, we see that tables may have not only repeated row headers (as seen in Figure 2) but may have repeated column headers. We handle this case. Figure 5: Albanian (iso:sq) data table with multiple layers of headers In Figure 5 151515Screenshot of page accessed on June 24, 2020: https://en.wikipedia.org/wiki/Albanian_language every element must be connected with multiple columns and rows of headers to ensure that we do not miss any information. We handle this case. (a) Adyghe (iso:ady) data table with multiple orthographies (b) Adyghe (iso:ady) extracted data example Figure 6: Adyghe Language extraction example Figure 6(a) 161616Screenshot of page accessed on June 24, 2020: https://en.wikipedia.org/wiki/Adyghe_language combines the G2P mappings for two scripts used by the language, with headers that are too specific compared to our keywords for the correct category (grapheme). We handle this case by storing the data and its associated header in the Unclassified category, which is labeled as “Best Guess” in Figure 6(b), and allows the user to see some more information for the language that wasn’t straightforward to parse automatically. ## Appendix B Additional Examples of Table Formats We Don’t Handle Well (a) E (iso:eee) data table where keyword is used unexpectedly (b) Madurese (iso:mad) data table with multiple items per cell In Figure 7(a) 171717Screenshot of page accessed on June 25, 2020: https://en.wikipedia.org/wiki/E_language, a table has a header containing one of the keywords, “letter”, but not in the way that we generally expect, which would be in a table that shows G2P. We do not handle this case and this data is parsed into the data set mistakenly. In Figure 7(b) 181818Screenshot of page accessed on June 24, 2020: https://en.wikipedia.org/wiki/Madurese_language, there are multiple data items within each cell, and there is no clear way to understand automatically how to split up or classify these items. We do not handle this case. ## Appendix C Additional Example Excerpts from TSV Data Files Figure 8 is an snippet from the oca.tsv file, and shows how G2P mappings could be captured in the Unclassified column. Figure 9 is another example of the type of data that could be captured by the Unclassified column. This language did not have corresponding phonemes in the table, instead it had words from the language and their English meaning. Figure 8: Ocaina (iso:oca) tsv format Figure 9: Krio (iso:kri) tsv format ## References * [Bharadwaj et al., 2016] Akash Bharadwaj, David Mortensen, Chris Dyer, and Jaime Carbonell. 2016\. Phonologically aware neural model for named entity recognition in low resource transfer settings. 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capbtabboxtable[][] Modeling of Nonlinear Interference Power for Dual-Polarization 4D Formats Gabriele Liga1, Bin Chen1,2, Astrid Barreiro1, and Alex Alvarado1 1Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands 2School of Computer Science and Information Engineering, Hefei University of Technology, Hefei, China <EMAIL_ADDRESS> ###### Abstract We assess the accuracy of a recently introduced nonlinear interference model for general dual-polarization 4D formats. Unlike previous models for polarization-multiplexed 2D formats, an average gap from split-step Fourier simulations within 0.1 dB is demonstrated. ## 1 Introduction Nonlinear interference (NLI) modeling in optical fiber transmission is a key tool to analyze and optimize the performance of optical communication systems. In recent years, remarkable progress has been made in this direction, and as new model’s applications emerge, further work is required to meet new accuracy and computational requirements. Constellation shaping is today a popular area of application for NLI models for two reasons: i) the performance of a given constellation significantly depends on the amount of NLI power induced during propagation; ii) NLI models can provide easy-to-compute as well as accurate cost functions for the performance optimization. However, as research focus moves towards multidimensional constellation shaping, where the different dimensions are mapped onto the degrees of freedom of the fiber channel (quadratures, polarization, wavelengths, etc), extensions of the established NLI models for two-dimensional (2D) formats are needed. NLI power models such as the enhanced Gaussian noise (EGN) model [1] introduced analitical expressions explicitly linking the properties of the transmitted constellation to the resulting NLI power after propagation. Nonetheless, such expressions have been developed only for so-called polarization-multiplexed 2D (PM-2D) formats, i.e. when a single 2D constellation is used to independently map information over 2 orthogonal polarization modes of the optical field. The resulting dual polarization four- dimensional (DP-4D) constellation is, thus, given by the Cartesian product of the component 2D constellation by itself. However, the class of PM-2D formats represents only a limited subset of all possible DP-4D constellations. A vast literature on assessing the performance of DP-4D formats that do not fall within the PM-2D class is available, (see, e.g., [2]), and, lately, DP-4D constellations have shown improved shaping gains compared to other conventional PM-2D formats [3, 4, 5]. However, no NLI model has been so far available to support the design of nonlinearity-tolerant DP-4D formats. Recently, in [6], we extended the model in [1] to account for the entire DP-4D class of modulation formats. In this contribution, we present a first numerical validation of our 4D model. Moreover, we show that heuristic extensions of the EGN model to non PM-2D formats may lead to inaccuracies in the prediction of the NLI power coefficient beyond 1 dB, even for fairly regular 4D modulation formats. Our 4D model [6] is instead proven to be accurate on average within 0.1 dB for all 4D modulation formats studied in this work. ## 2 Analytical Formulation of the NLI coefficient for DP-4D formats The model we aim to validate in this work consists of an analytical formula for the computation of the vector $(\sigma^{2}_{x},\sigma^{2}_{y})$, where $\sigma^{2}_{x}$ and $\sigma^{2}_{y}$ represent the NLI power within the received signal bandwidth over the $x$\- and $y$ polarizations, respectively. The corresponding NLI power coefficient vector $(\eta_{x},\eta_{y})\triangleq(\sigma^{2}_{x},\sigma^{2}_{y})/P^{3}$, where $P$ denotes the transmitted signal power. The model’s expression was derived using a first-order perturbational approach under the hypothesis of single- channel transmission with quasi-rectangular pulse spectrum. A comprehensive discussion on the assumptions of the model as well as its mathematical derivation can be found in [6]. Based on [6, eqs. (42)-(43)], the NLI power coefficient for the $x$ polarization can be found as $\displaystyle\begin{split}\eta_{x}&=\left(\frac{8}{9}\right)^{2}\frac{\gamma^{2}}{P^{3}}\left[R_{s}^{3}\left(\Phi_{1}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{1}+\Phi_{2}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{2}+\Phi_{3}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{3}\right)+R_{s}^{2}\left(\Psi_{1}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{4}+2\operatorname{Re}\\{\Psi_{2}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{5}+\Psi_{3}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{5}^{}\\}+\Psi_{4}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{6}\right.\right.\\\ &\left.\left.+2\operatorname{Re}\\{\Lambda_{1}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{7}+\Lambda_{2}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{7}^{}(f)\\}+\Lambda_{3}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{8}+2\operatorname{Re}\\{\Lambda_{4}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{9}+\Lambda_{5}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{9}^{}\\}+\Lambda_{6}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{10}\right)+R_{s}\Xi_{1}\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{11}\right],\end{split}$ (1) where $\gamma$ is the fiber nonlinearity coefficient, $R_{s}$ is the symbol rate and the coefficients $\Phi_{1}$, $\Phi_{2}$, $\Phi_{3}$, $\Psi_{1}$, $\Psi_{2}$, …, $\Psi_{4}$, $\Lambda_{1}$, $\Lambda_{2}$, …, $\Lambda_{6}$, $\Xi_{1}$ are functions of several different intra- and cross- polarization moments of the DP-4D transmitted modulation format, and are given in [6, Table 8]. The $\overline{{\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}}_{i}$ coefficients are obtained integrating over the channel bandwidth (see [6, eq. (42)-(43)]) the frequency-dependent integrals ${\mathchoice{\raisebox{0.0pt}{$\displaystyle\chi$}}{\raisebox{0.0pt}{$\textstyle\chi$}}{\raisebox{0.0pt}{$\scriptstyle\chi$}}{\raisebox{0.0pt}{$\scriptscriptstyle\chi$}}}_{i}(f)$, $i=1,2,\dots,11$ in [6, Table 8]. The expression for $\eta_{y}$ can be found from (1) by simply applying the transformation $x\rightarrow y$ and $y\rightarrow x$. As discussed in [6, Sec. 8], (1) reduces to the EGN formula for conventional PM-2D formats. 16641281284096$30$$30.5$$31$$31.5$$32$$32.5$c4_164D-64PRSl4_1284D_128OSPM-64QAM0.8 dB$M$$\eta_{x}$ [dB 1/W2]4D model (1)SSFM4D-EGN model (a) 16641281284096$30$$30.5$$31$$31.5$$32$$32.5$c4_164D-64PRSl4_1284D_128OSPM-64QAM0.9 dB$M$$\eta_{y}$ [dB 1/W2]4D model (1)SSFM4D-EGN model (b) Fig. 1: NLI power coefficients $\eta_{x}$ (a), and $\eta_{y}$ (b), for DP-4D constellations with different cardinality $M$. ## 3 Methodology Parameter \| Value ---\|--- TX parameters Symbol rate ($R_{s}$) \| 32 Gbaud No. of channels \| 1 RRC rolloff \| 0.01 % TX power ($P$) \| $-20$ dBm Fiber parameters Attenuation coeff. ($\alpha$) \| 0.2 (dB/km) Dispersion par. ($D$) \| 17 ps/nm/km Nonlinear coeff. ($\gamma$) \| 1.3 (W$\cdot\text{km})^{-1}$) Link parameters Span length \| 100 km No. of spans \| 10 SSFM parameters Step distribution \| Adaptive step $\phi_{NLmax}$ \| $10^{-3}$ rad Sim. bandwidth \| 96 GHz Table 1: System parameters. The numerical validation of the model in this work is performed via the estimation of the signal-to-noise ratio in split-step Fourier method (SSFM) simulations where optical nonlinearity is kept as the sole source of noise. Indeed, in such a scenario, $\mathbf{E}\approx(P_{x}(\text{SNR}_{x}\cdot P^{3})^{-1},P_{y}(\text{SNR}_{y}\cdot P^{3})^{-1})$, where $P_{x}$, SNRx, $P_{y}$, and SNRy are the transmitted powers and signal-to-noise ratios over $x$ and $y$ polarization, resp. The previous formula becomes increasingly accurate as higher-order NLI terms vanish compared to the first-order one, i.e. for small values of $P$. The simulated single-channel, multi-span optical system is described in Table 1. At the transmitter, 4D symbols are jointly modulated using a root-raised cosine (RRC) pulse shape. At the receiver, chromatic dispersion compensation and matched filtering followed by sampling are performed. $\text{SNR}_{x}$ and $\text{SNR}_{y}$ are then computed via a data-aided approach. Finally, an adaptive step size SSFM with a maximum nonlinear phase rotation $\phi_{NLmax}$ per step is used to simulate the fiber propagation. The EGN model is also used as a reference to show the increased accuracy of (1). However, in its standard formulation, the EGN model does not provide a suitable expression to account for general DP-4D formats. An intuitive way to extend the EGN model to general DP-4D formats is to consider the EGN expressions as separately applicable to the two 2D constellations obtained by the projection of the transmitted DP-4D format over the $x$ and $y$ polarization plane. In the following, we adopt this approach, labelled 4D-EGN model, to compute EGN-based estimates of E. ## 4 Results The numerical results in this section are based on a set of DP-4D formats which combines the full list of 4D constellations in [7] (4D sphere packings), and formats used in optical communications such as the 4D 64-ary polarization ring switched (4D-64PRS) format [4], the 4D orthant-symmetric 128 (4D-OS128) format [5], and the family of 4D 2-amplitudes 8 phase-shift keying (4D-2A8PSK) formats [3]. In Fig. 1, $\eta_{x}$ (Fig. 1(a)) and $\eta_{y}$ (Fig. 1(b)) are shown for some of the above mentioned constellations with different constellation cardinalities $M$, using: i) (1) (blue bars); ii) the 4D-EGN model (yellow bars); iii) the SSFM (red bars). For all constellations shown, our 4D model is within 0.1 dB from the SSFM estimates for both $\eta_{x}$ and $\eta_{y}$. The 4D-EGN model leads to inaccuracies of up to 0.9 dB, for $\eta_{y}$ in “l4_128”, or 0.8 dB for $\eta_{x}$ in “c4_16”. Neither “l4_128” nor “c4_16” are symmetric constellations with respect to the $y=x$ plane, and this is reflected by uneven values of $\eta_{x}$ and $\eta_{y}$. However, even for symmetric constellations such as 4D-64PRS and 4D-128OS, the 4D-EGN model deviates from SSFM estimates by approx. 0.6 dB in both $\eta_{x}$ and $\eta_{y}$. As expected, for a PM-2D quadrature amplitude modulation (QAM) format such as PM-64QAM, all three estimation methods are in perfect agreement. To further validate (1), the average deviation $\overline{\Delta\eta}\triangleq(\Delta\eta_{x}+\Delta\eta_{y})/2$ is computed, where $\Delta\eta_{x}$ and $\Delta\eta_{y}$ are the deviations (in dB and in absolute value) for $\eta_{x}$ and $\eta_{y}$, respectively, between a given model and the SSFM estimates. In Fig. 3, $\overline{\Delta\eta}$ is illustrated for (1) and the 4D-EGN model as a function of the constellation cardinality $M$. The solid lines show the average $\overline{\Delta\eta}$ across all considered constellations, whereas dashed lines indicate the maximum and minimum deviation. The results show an average $\overline{\Delta\eta}$ within 0.1 dB for (1) across all cardinalities, with maximum $\overline{\Delta\eta}$ of 0.25 dB (for $M=16$). On the contrary, the 4D-EGN model shows deviations in excess of 1 dB for low cardinality formats ($M\leq$64), with worst-case scenario $\overline{\Delta\eta}$ of 1.75 dB for $M=16$. At higher cardinalities ($M\geq$64), the 4D-EGN model accuracy improves as the average $\overline{\Delta{\eta}}$ lies between 0.3 dB and 0.5 dB. Const. label \| $M$ \| $(\eta_{x},\eta_{y})$ [dB 1/W2] ---\|---\|--- dicyclic4_16 [7] \| 16 \| (30.2, 30.2) 4D-2A-8PSK5b [3] \| 32 \| (30.3, 30.3) 4D-2A-8PSK6b [3] \| 64 \| (30.3, 30.3) 4D-2A-8PSK7b [3] \| 128 \| (30.3, 30.3) w4_256 [7] \| 256 \| (30.7, 30.7) sphere4_512 [7] \| 512 \| (30.7, 30.7) 120cell4_600 [7] \| 600 \| (30.3, 30.3) a4_2048 [7] \| 2048 \| (30.7, 30.8) a4_4096 [7] \| 4096 \| (30.8, 30.7) Table 2: $\overline{\eta}$-optimal formats in Fig. 3. Finally, in Fig. 3, the minimum values of the average NLI power coefficient $\overline{\eta}\triangleq(\eta_{x}+\eta_{y})/2$ are shown for each $M$ within the set of constellations analyzed in this work. The corresponding $\overline{\eta}$-optimal constellations are listed in Table 2 for some values of $M$. The 4D-EGN model (yellow markers) almost always overestimates $\overline{\eta}$, with deviations up to 0.75 dB ($M=81$). Conversely, (1) (blue markers) is consistently within 0.1 dB from the SSFM $\overline{\eta}$ (red markers). Among these optimal constellations, we highlight: i) “w4_256” and “a4_4096”, already studied in the context of coded modulation (see refs. in [7]), which outperform PM-16QAM and PM-64QAM, respectively; ii) “120cell600” (see $x$ and $y$ projections in the inset in Fig. 3), which is a constant-modulus constellation in 4D, and has the lowest $\overline{\eta}$ among all formats with $M\geq$128. [] 163264128256512102420484096$0.1$$0.5$$1$$1.5$$1.8$$2$Max. $\overline{\Delta\eta}$Min. $\overline{\Delta\eta}$$M$$\overline{\Delta\eta}\text{ [dB]}$4D-EGN model4D model (1) Fig. 2: Average NLI power coeff. gap across the two polarizations ($\overline{\Delta\eta}$) as a function of the cardinality $M$. [] 16326412825660020484096$30.5$$31$$31.5$$32$$32.5$$\approx$ 0.75 dB“120cell4_600”$x$ pol.$y$ pol.$M$$\overline{\eta}$ [dB 1/W2]4D-EGN model4D model (1)SSFM Fig. 3: Minimum values of $\overline{\eta}$ vs. cardinality $M$ for the 4D constellations investigated in this work. ## 5 Conclusions We tested a novel model predicting the NLI power for general DP-4D constellations. Based on this preliminary study, the assessed model shows a superior accuracy compared to a heuristic 4D extension of the EGN model. In particular, a 0.1 dB average deviation from SSFM simulations is demonstrated for the NLI power coefficient over a wide variety of regular 4D formats. Although further numerical validation is required, we foresee the model in this work as a powerful analytical tool for optimizing constellations in the DP-4D space of the optical field. Acknowledgements: The work of G. Liga is funded by the EuroTechPostdoc programme under the European Union’s Horizon 2020 research and innovation programme (Marie Skłodowska-Curie grant agreement No. 754462). This work has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 757791). ## References [1] A. Carena, G. Bosco, V. Curri, Y. Jiang, P. Poggiolini, and F. Forghieri, “EGN model of non-linear fiber propagation,” Opt.Express 22, 16335–16362 (2014). * [2] E. Agrell and M. Karlsson, “Power-efficient modulation formats in coherent transmission systems,” JLT 27, 5115–5126 (2009). * [3] K. Kojima, T. Yoshida, T. Koike-Akino, D. S. Millar, K. Parsons, M. Pajovic, and V. Arlunno, “Nonlinearity-tolerant four-dimensional 2A8PSK family for 5–7 bits/symbol spectral efficiency,” JLT 35, 1383–1391 (2017). * [4] B. Chen, C. Okonkwo, H. Hafermann, and A. Alvarado, “Polarization-ring-switching for nonlinearity-tolerant geometrically shaped four-dimensional formats maximizing generalized mutual information,” JLT 37, 3579–3591 (2019). * [5] B. Chen, A. Alvarado, S. van der Heide, M. v. d. Hout, H. Hafermann, and C. Okonkwo, “Analysis and experimental demonstration of orthant-symmetric four-dimensional 7 bit/4D-sym modulation for optical fiber communication,” arXiv:2003.12712 (2020). * [6] G. Liga, A. Barreiro, H. Rabbani, and A. Alvarado, “Extending fibre nonlinear interference power modelling to account for general dual-polarisation 4D modulation formats,” Entropy 22, 1324 (2020). * [7] “Sphere packings of dimension 4,” https://codes.se/packings/4.htm.
# A unified approach to Local Quantum Uncertainty and Interferometric Power by Metric Adjusted Skew Information Paolo Gibilisco Department of Economics and Finance, University of Rome “Tor Vergata”, Via Columbia 2, Rome 00133, Italy<EMAIL_ADDRESS>Davide Girolami<EMAIL_ADDRESS>Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 129, Italy Frank Hansen Department of Mathematical Sciences, University of Copenaghen, Universitetsparken 5 DK-2100 Copenhagen, Denmark<EMAIL_ADDRESS> ###### Abstract Local quantum uncertainty and interferometric power have been introduced by Girolami et al. in GTA:2013 ; GSGTFSSOA:2014 as geometric quantifiers of quantum correlations. The aim of the present paper is to discuss their properties in a unified manner by means of the the metric adjusted skew information defined by Hansen in Hansen:2006b . ## I Introduction One of the key traits of many-body quantum systems is that the full knowledge of their global configurations does not imply full knowledge of their constituents. The impossibility to reconstruct the local wave functions $\|\psi_{1}\rangle,\,\|\psi_{2}\rangle$ (pure states) of two interacting quantum particles from the wave function of the whole system, $\|\psi_{12}\rangle\neq\|\psi_{1}\rangle\otimes\|\psi_{2}\rangle$, is due to the existence of entanglement ent . Investigating open quantum systems, whose (mixed) states are described by density matrices $\rho_{12}=\sum_{i}p_{i}\|\psi_{i}\rangle_{12}\langle\psi_{i}\|$, revealed that the boundary between the classical and quantum worlds is more blurred than we thought. There exists a genuinely quantum kind of correlation, quantum discord, which manifests even in absence of entanglement, i.e. in separable density matrices $\rho_{12}=\sum_{i}p_{i}\rho_{1,i}\otimes\rho_{2,i}$ oz ; hv . The discovery triggered theoretical and experimental studies to understand the physical meaning of quantum discord, and the potential use of it as a resource for quantum technologies rev . Relying on the known interplay between geometrical and physical properties of mixed states Uhlmann:1992 ; book , a stream of works employed information geometry techniques to construct quantifiers of quantum discord AyGibiliscoMatus: 2018 ; BogaertGirolami:2017 ; CFTA:2018 ; GII:2009 ; FPA:2017 ; GibiliscoIsola:2011 . In particular, two of the most popular ones are the Local Quantum Uncertainty (LQU) and the Interferometric Power (IP) GTA:2013 ; GSGTFSSOA:2014 . A merit of these two measures is that they admit an analytical form for $N$ qubit states across the $1\,vs\,N-1$ qubit partition. Also, they have a clear-cut physical interpretation. The lack of certainty about quantum measurement outcomes is due to the fact that density matrices are changed by quantum operations. The LQU evaluates the minimum uncertainty about the outcome of a local quantum measurement, when performed on a bipartite system. It is proven that two- particle density matrices display quantum discord if and only if they are not “classical-quantum” states. That is, they are not (mixture of) eigenvalues of local observables, $\rho_{12}\neq\sum_{i}p_{i}\|i\rangle_{1}\langle i\|\otimes\rho_{2,i}$, or $\rho_{12}\neq\sum_{i}p_{i}\rho_{1,i}\otimes\|i\rangle_{2}\langle i\|$, in which $\\{\|i\rangle\\}$ is an orthonormal basis. Indeed, this is the only case in which one can identify a local measurement that does not change a bipartite quantum state, whose spectral decomposition reads $A_{1}=\sum_{i}\lambda_{i}\|i\rangle_{1}\langle i\|$, or $A_{2}=\sum_{i}\lambda_{i}\|i\rangle_{2}\langle i\|$. The LQU was built as the minimum of the Wigner-Yanase skew information, a well-known information geometry measure WignerYanase:1963 , between a density matrix and a finite- dimensional observable (Hermitian operator). It quantifies how much a density matrix $\rho_{12}$ is different from being a zero-discord state. The IP was concocted by following a similar line of thinking. Quantum discord implies a non-classical sensitivity to local perturbations. This feature of quantum particles, while apparently a limitation, translates into an advantage in the context of quantum metrology metro . It was theoretically proven and experimentally demonstrated that quantum systems sharing quantum discord are more sensitive probes for interferometric phase estimation. The figure of merit of such measurement protocols is the quantum Fisher information of the state under scrutiny with respect to a local Hamiltonian (in Information Geometry the QFI is known as the SLD or Bures-Uhlmann metric). The latter generates a unitary evolution that imprints information about a physical parameter on the quantum probe. The IP is the minimum quantum Fisher information over all the possible local Hamiltonians, being zero if and only if the probe states are classically correlated. Here, we polish and extend the mathematical formalization of information- geometric quantum correlation measures. We build a class of parent quantities of the LQU (and consequently of the IP) in terms of the the metric adjusted skew informations Hansen:2006b . In Sections 2,3, we review definition and main properties of operator means. In Sections 4-6, we discuss information- geometric quantities that capture complementarity between quantum states and observables. In particular, we focus on the quantum $f$-covariances and the quantum Fisher information. They quantify the inherent uncertainty about quantum measurement outcomes. After having recalled the definition of metric adjusted skew information (Section 7), we build a new quantum discord measure, the metric adjusted local quantum uncertainty ($f$-LQU), in Section 8. Finally we are able to show that LQU and IP are just two particular members of this family allowing a unified treatment of their fundamental properties. ## II Means for positive numbers We use the notation ${\mathbb{R}}_{+}=(0,+\infty)$. ###### Definition 1. A bivariate mean PetzTemesi:2005 is a function $m\colon{\mathbb{R}}_{+}\times{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ such that 1. 1. $m(x,x)=x.$ 2. 2. $m(x,y)=m(y,x).$ 3. 3. $x<y$ $\,\Rightarrow\,$ $x<m(x,y)<y.$ 4. 4. $x<x^{\prime}$ and $y<y^{\prime}$ $\,\Rightarrow\,$ $m(x,y)<m(x^{\prime},y^{\prime}).$ 5. 5. $m$ is continuous. 6. 6. $m$ is positively homogeneous; that is $m(tx,ty)=t\cdot m(x,y)$ for $t>0.$ We use the notation $\mathcal{M}_{num}$ for the set of bivariate means described above. ###### Definition 2. Let $\mathcal{F}_{num}$ denote the class of functions $f\colon\mathbb{R}_{+}\to\mathbb{R}_{+}$ such that 1. 1. $f$ is continuous. 2. 2. $f$ is monotone increasing. 3. 3. $f(1)=1.$ 4. 4. $tf(t^{-1})=f(t)$. The following result is straightforward. ###### Proposition 1. There is a bijection $f\mapsto m_{f}$ betwen ${\mathcal{F}}_{nu}$ and ${\mathcal{M}}_{nu}$ given by $m_{f}(x,y)=yf(y^{-1}x)\qquad\text{and in reverse}\qquad f(t)=m(1,t)$ for positive numbers $x,y$ and $t.$ In Table 1 we have some examples of means. Table 1: $\begin{array}[]{\|c\|c\|c\|}\hline\cr{\rm Name}&{f}&{m_{f}}\\\ \hline\cr{\rm arithmetic}&\displaystyle\frac{1+x}{2}&\displaystyle\frac{x+y}{2}\\\ \hline\cr{\rm WYD},\beta\in(0,1)&\displaystyle\frac{x^{\beta}+x^{1-\beta}}{2}&\displaystyle\frac{x^{\beta}y^{1-\beta}+x^{1-\beta}y^{\beta}}{2}\\\ \hline\cr{\rm geometric}&\sqrt{x}&\sqrt{xy}\\\ \hline\cr{\rm harmonic}&\displaystyle\frac{2x}{x+1}&\displaystyle\frac{2}{x^{-1}+y^{-1}}\\\ \hline\cr{\rm logarithmic}&\displaystyle\frac{x-1}{\log x}&\displaystyle\frac{x-y}{\log x-\log y}\\\ \hline\cr\end{array}$ ## III Means for positive operators in the sense of Kubo-Ando The celebrated Kubo-Ando theory of operator means KuboAndo79/80 ; PetzTemesi:2005 ; GibiliscoHansenIsola:2009 may be viewed as the operator version of the results of Section II. ###### Definition 3. A bivariate mean $m$ for pairs of positive operators is a function $(A,B)\to m(A,B),$ defined in and with values in positive definite operators on a Hilbert space, that satisfies, mutatis mutandis, conditions $(1)$ to $(5)$ in Definition 1. In addition, the transformer inequality $Cm(A,B)C^{}\leq m(CAC^{},CBC^{}),$ should also hold for positive definite $A,B$ and arbitrary $C.$ Note that the transformer inequality replaces condition $(6)$ in Definition 1. We denote by $\displaystyle{\mathcal{M}}_{op}$ the set of matrix means. ###### Example 1. The arithmetic, geometric and harmonic operator means are defined, respectively, by setting $\begin{array}[]{rcl}A\nabla B&=&\frac{1}{2}(A+B)\\\\[6.45831pt] A\\#B&=&A^{1/2}\bigl{(}A^{-1/2}BA^{-1/2}\bigr{)}^{1/2}A^{1/2}\\\\[8.61108pt] A{\rm!}B&=&2(A^{-1}+B^{-1})^{-1}.\end{array}$ We recall that a function $f\colon(0,\infty)\to\mathbb{R}$ is said to be operator monotone (increasing) if $A\leq B\quad\Rightarrow\quad f(A)\leq f(B)$ for positive definite matrices of arbitrary order. It then follows that the inequality also holds for positive operators on an arbitrary Hilbert space. An operator monotone function $f$ is said to be symmetric if $f(t)=tf(t^{-1})$ for $t>0$ and normalized if $f(1)=1.$ ###### Definition 4. ${\mathcal{F}}_{op}$ is the class of functions $f:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ such that 1. 1. $f$ is operator monotone increasing, 2. 2. $tf(t^{-1})=f(t)\qquad t>0,$ 3. 3. $f(1)=1.$ The fundamental result, due to Kubo and Ando, is the following. ###### Theorem. There is a bijection $f\mapsto m_{f}$ between ${\mathcal{M}}_{op}$ and ${\mathcal{F}}_{op}$ given by the formula $m_{f}(A,B)=A^{1/2}f(A^{-1/2}BA^{-1/2})A^{1/2}.$ ###### Remark 1. The function in ${\mathcal{F}}_{op}$ are (operator) concave which makes the operator case quite different from the numerical (commutative) case. For example, there exist convex functions in $\mathcal{F}_{num},$ see GH:2017 . If $\rho$ is a density matrix (a quantum state) and $A$ is a self-adjoint matrix (a quantum observable), then the expectation of $A$ in the state $\rho$ is defined by setting ${\rm E}_{\rho}(A)={\rm Tr\hskip-1.99997pt}~{}(\rho A).$ ## IV The correspondence between Fisher information and metric adjusted skew information We introduce now a technical tool which is useful to establish some fundamental relations between quantum covariance, quantum Fisher information and the metric adjusted skew information. ###### Definition 5. For $f\in{\mathcal{F}}_{op}$ we define $f(0)=\lim_{x\to 0}f(x).$ We say that a function $f\in{\mathcal{F}}_{op}$ is regular if $f(0)\not=0,$ and non-regular if $f(0)=0,$ cf. PetzSudar:1996 ; Hansen:2006b . ###### Definition 6. A quantum Fisher information is extendable if its radial limit exists and it is a Riemannian metric on the real projective space generated by the pure states. For the definition of the radial limit see PetzSudar:1996 where the following fundamental result is proved. ###### Theorem. An operator monotone function $f\in{\mathcal{F}}_{op}$ is regular, if and only if $\langle\cdot,\cdot\rangle_{\rho,f}$ is extendable. ###### Remark 2. The reader should be aware that there is no negative connotation associated with the qualification “non-regular”. For example, a very important quantum Fisher information in quantum physics (see FickSauermann:1990 ), namely the Kubo-Mori metric related to the function $f(x)=(x-1)/\log x,$ is non-regular. We introduce the sets of regular and non-regular functions ${\mathcal{F}}_{op}^{\,r}:=\\{f\in{\mathcal{F}}_{op}\mid f(0)\not=0\\},\quad{\mathcal{F}}_{op}^{\,n}:=\\{f\in{\mathcal{F}}_{op}\mid f(0)=0\\}$ and notice that trivially ${\mathcal{F}}_{op}={\mathcal{F}}_{op}^{\,r}\dot{\cup}{\mathcal{F}}_{op}^{\,n}$ . ###### Definition 7. We introduce to $f\in{\mathcal{F}}_{op}^{\,r}$ the transform $\tilde{f}$ given by $\tilde{f}(x)=\frac{1}{2}\left[(x+1)-(x-1)^{2}\frac{f(0)}{f(x)}\right]$ for $x>0.$ We may also write ${\tilde{f}}={\mathcal{G}}(f),$ cf. GibiliscoImparatoIsola:2007 ; GibiliscoHansenIsola:2009 . The following result is taken from (GibiliscoHansenIsola:2009, , Theorem 5.1). ###### Theorem. The correspondence $f\to\tilde{f}$ is a bijection between ${\mathcal{F}}_{op}^{\,r}$ and ${\mathcal{F}}_{op}^{\,n}\,.$ In Table 2 we have some examples (where $0<\beta<1$). Table 2: $\displaystyle\begin{array}[]{\|c\|c\|}\hline\cr f&\tilde{f}\\\ \hline\cr\displaystyle\frac{1+x}{2}&\displaystyle\frac{2x}{x+1}\\\ \hline\cr\displaystyle\frac{(\sqrt{x}+1)^{2}}{4}&\sqrt{x}\\\ \hline\cr\displaystyle\beta(1-\beta)\frac{(x-1)^{2}}{(x^{\beta}-1)(x^{1-\beta}-1)}&\displaystyle\frac{x^{\beta}+x^{1-\beta}}{2}\\\ \hline\cr\end{array}$ ## V Quantum f-Covariance The notion of quantum $f$-covariance has been introduced by Petz, see Petz:2003 ; GibiliscoHiaiPetz:2009 . Any Kubo-Ando function $m_{f}(x,y)=yf(y^{-1}x)$ for $x,y>0$ has a continuous extension to $[0,+\infty)\times[0,+\infty)$ given by $m_{f}(0,y)=f(0)y,\quad m_{f}(x,0)=f(0)x,\quad m_{f}(0,0)=0,\quad x,y>0.$ The operator $m_{f}(L_{\rho},R_{\rho})$ is well-defined by the spectral theorem for any state, see (GibiliscoImparatoIsola:2007, , Proposition 11.1 page 11). To self-adjoint $A$ we set $A_{0}=A-({\rm Tr\hskip-1.99997pt}~{}\rho A)I,$ where $I$ is the identity operator. Note that ${\rm Tr\hskip-1.99997pt}~{}\rho A_{0}={\rm Tr\hskip-1.99997pt}~{}\rho A-({\rm Tr\hskip-1.99997pt}~{}\rho A){\rm Tr\hskip-1.99997pt}~{}\rho=0,$ if $\rho$ is a state. ###### Definition 8. Given a state $\rho,$ a function $f\in\mathcal{F}_{op}$ and self-adjoint $A,B$ we define the quantum $f$-covariance by setting ${\rm Cov}_{\rho}^{f}(A,B)={\rm Tr\hskip-1.99997pt}~{}B_{0}\,m_{f}(L_{\rho},R_{\rho})A_{0}$ and the corresponding quantum $f$-variance by ${\rm Var}_{\rho}^{f}(A)={\rm Cov}_{\rho}^{f}(A,A).$ The $f$-variance is a positive semi-definite sesquilinear form and $f\leq g\quad\Rightarrow\quad{\rm Var}_{\rho}^{f}(A)\leq{\rm Var}_{\rho}^{g}(A).$ (1) Note that for the standard covariance we have ${\rm Cov}_{\rho}(A,B)={\rm Cov}_{\rho}^{SLD}(A,B),$ where the SLD or Bures-Uhlmann metric is the one associated with the function $(1+x)/2$. ###### Proposition 2. If $\rho$ is a pure state then ${\rm Var}_{\rho}^{f}(A)=2\,m_{f}(1,0)\cdot{\rm Var}_{\rho}(A),$ cf. TothPetz:2013 . ###### Corollary. If $\rho$ is a pure state and $f$ is non-regular, then ${\rm Var}_{\rho}^{f}(A)=0.$ ###### Proof. If $f$ is non regular $m_{f}(1,0)=0$ QED ## VI Quantum Fisher Information The theory of quantum Fisher information is due to Petz and we recall here the basic results. If ${\mathcal{N}}$ is a differentiable manifold we denote by $T_{\rho}\mathcal{N}$ the tangent space to $\mathcal{N}$ at the point $\rho\in{\mathcal{N}}$. Recall that there exists a natural identification of $T_{\rho}{\mathcal{D}}^{1}_{n}$ with the space of self-adjoint traceless matrices; namely, for any $\rho\in{\mathcal{D}}^{1}_{n}$ $T_{\rho}{\mathcal{D}}^{1}_{n}=\\{A\in M_{n}\mid A=A^{}\,,\,\hbox{Tr}\,A=0\\}.$ A stochastic map is a completely positive and trace preserving operator $T:M_{n}\to M_{m}$. A monotone metric is a family of Riemannian metrics $g=\\{g^{n}\\}$ on $\\{{\mathcal{D}}^{1}_{n}\\}$, $n\in\mathbb{N}$, such that $g^{m}_{T(\rho)}(TX,TX)\leq g^{n}_{\rho}(X,X)$ holds for every stochastic map $T:M_{n}\to M_{m}$, every faithful state $\rho\in{\mathcal{D}}^{1}_{n},$ and every $X\in T_{\rho}{\mathcal{D}}^{1}_{n}$. Usually monotone metrics are normalized in such a way that $[A,\rho]=0$ implies $g_{\rho}(A,A)={\rm Tr}({\rho}^{-1}A^{2})$. A monotone metric is also called (an example of) quantum Fisher information (QFI). This notation is inspired by Chentsov’s uniqueness theorem for commutative monotone metrics Chentsov:1982 . Define $L_{\rho}(A)=\rho A$ and $R_{\rho}(A)=A\rho$, and observe that $L_{\rho}$ and $R_{\rho}$ are commuting positive superoperators on $M_{n}.$ For any $f\in{\mathcal{F}}_{op}$ one may also define the positive (non-linear) superoperator $m_{f}(L_{\rho},R_{\rho})$. The fundamental theorem of monotone metrics may be stated in the following way: ###### Theorem. (See Petz:1996 ). There exists a bijective correspondence between monotone metrics (quantum Fisher information(s)) on ${\mathcal{D}}^{1}_{n}$ and functions $f\in{\mathcal{F}}_{op}$. The correspondence is given by the formula $\langle A,B\rangle_{\rho,f}={\rm Tr}(A\cdot m_{f}(L_{\rho},R_{\rho})^{-1}(B))$ for positive matrices $A$ and $B.$ ## VII Metric adjusted skew information By using the general form of the quantum Fisher information it is possible to greatly generalize the Wigner-Yanase information measure. To $f\in\mathcal{F}_{op}$ the so-called Morosova function $c_{f}(x,y)$ is defined by setting $c_{f}(x,y)=\frac{1}{yf(xy^{-1})}=m_{f}(x,y)^{-1}\qquad x,y>0.$ (2) The corresponding monotone symmetric metric $K_{\rho}$ is given by $K_{\rho}^{f}(A,B)={\rm Tr\hskip-1.99997pt}~{}A^{}c_{f}\bigl{(}L_{\rho},R_{\rho}\bigr{)}B,$ (3) where $L_{\rho}$ and $R_{\rho}$ denote left and right multiplication with $\rho.$ Note that $K^{f}_{\rho}(A)$ is increasing in $c_{f}$ and thus decreasing in $f.$ If furthermore $f$ is regular, the notion of metric adjusted skew information (Hansen:2006b, , Definition 1.2) is defined by setting $I^{f}_{\rho}(A)=I^{f}(\rho,A)=\frac{f(0)}{2}K^{f}_{\rho}\bigl{(}i[\rho,A^{}],i[\rho,A]\bigr{)},$ (4) where $\rho>0.$ We use the second notation, $I^{f}(\rho,A),$ when the expression of the state takes up too much space. We also tacitly extended the metric adjusted skew information to arbitrary (non-self-adjoint) operators $A.$ It is convex (Hansen:2006b, , Theorem 3.7) in the state variable $\rho$ and $0\leq I^{f}_{\rho}(A)\leq{\rm Var}_{\rho}(A)$ (5) with equality if $\rho$ is pure (Hansen:2006b, , Theorem 3.8), see also the summery with interpretations in (CaiHansen:2010, , Theorem 1.2). Furthermore, the notion of unbounded metric adjusted skew information for non-regular functions in $\mathcal{F}_{op}$ is introduced in (CaiHansen:2010, , Theorem 5.1). For regular $f\in\mathcal{F}_{op}$ the metric adjusted skew information may be written as $I_{\rho}^{f}(A)={\rm Tr\hskip-1.99997pt}~{}\rho A^{2}-{\rm Tr\hskip-1.99997pt}~{}A\,m_{\tilde{f}}(L_{\rho},R_{\rho})A,$ se (AudenaertCaiHansen:2008, , equation (7)). We thus obtain that the metric adjusted skew information is decreasing in the transform $\tilde{f}$ for arbitrary self-adjoint $A,$ that is $\tilde{f}\leq\tilde{g}\quad\Rightarrow\quad I_{\rho}^{f}(A)\geq I_{\rho}^{g}(A)\qquad\text{for}\quad f,g\in{\mathcal{F}}_{op}^{\,r}\,.$ (6) We may also write $\check{f}=\frac{f(0)}{f(t)}\qquad\text{and}\qquad\check{c}(x,y)=y^{-1}\check{f}(xy^{-1})$ and obtain $I_{\rho}^{f}(A)=\frac{1}{2}{\rm Tr\hskip-1.99997pt}~{}i[\rho,A^{}]\check{c}\bigl{(}L_{\rho},R_{\rho}\bigr{)}i[\rho,A],$ cf. (AudenaertCaiHansen:2008, , equation (10)). It follows that the metric adjusted skew information is increasing in $\check{f}$ for arbitrary $A.$ It may be derived from (GibiliscoImparatoIsola:2007, , Proposition 6.3, page 11), that the metric adjusted skew information can be expressed as the difference $I^{f}_{\rho}(A)={\rm Var}_{\rho}(A)-{\rm Var}^{\tilde{f}}_{\rho}(A)$ with extension to the sesquilinear form $I^{f}_{\rho}(A,B)={\rm Cov}_{\rho}(A,B)-{\rm Cov}^{\tilde{f}}_{\rho}(A,B).$ ### VII.1 Information inequalities A function $f\colon\mathbb{R_{+}}\to\mathbb{R_{+}}$ is in $\mathcal{F}_{op}$ if and only if it allows a representation of the form $f(t)=\frac{1+t}{2}\exp\int_{0}^{1}\frac{(\lambda^{2}-1)(1-t)^{2}}{(\lambda+t)(1+\lambda t)(1+\lambda)^{2}}\,h_{f}(\lambda)\,d\lambda,$ (7) where the weight function $h_{f}\colon[0,1]\to[0,1]$ is measurable. The equivalence class containing $h_{f}$ is uniquely determined by $f,$ cf. (AudenaertCaiHansen:2008, , Theorem 2.1). This representation gives rise to an order relation in $\mathcal{F}_{op}.$ ###### Definition 9. Let $f,g\in\mathcal{F}_{op}.$ We say that $f$ is majorized by $g$ and write $f\preceq g,$ if the function $\varphi(t)=\frac{t+1}{2}\,\frac{f(t)}{g(t)}\qquad t>0$ is in $\mathcal{F}_{op}\,.$ The partial order relation $\preceq$ is stronger that the usual order relation $\leq,$ and it renders $(\mathcal{F}_{op}\,,\preceq)$ into a lattice with $f_{\text{min}}(t)=\frac{2t}{t+1}\qquad\text{and}\qquad f_{\text{max}}(t)=\frac{t+1}{2}\qquad$ (8) as respectively minimal element and maximal element. Furthermore, $f\preceq g\quad\text{if and only if}\quad h_{f}\geq h_{g}\qquad\text{almost everywhere},$ (9) cf. (AudenaertCaiHansen:2008, , Theorem 2.4). The restriction of $\preceq$ to the regular part of $\mathcal{F}_{op}$ induces a partial order relation $\preceq$ on the set of metric adjusted skew informations. ###### Proposition 3. The restriction of the order relation $\preceq$ renders the regular part of $\mathcal{F}_{op}$ into a lattice. In addition, if one of two functions $f,g\in\mathcal{F}_{op}$ is non-regular, then the minorant $f\wedge g$ is also non-regular. ###### Proof. Take $f\in\mathcal{F}_{op}$ with representative function $h_{f}$ as given in (7). Then it follows that $f$ is regular if and only if the integral $\int_{0}^{1}\frac{h_{f}(\lambda)}{\lambda}\,\hskip-1.99997pt\operatorname{\mathit{d}}\hskip-2.29996pt{}\lambda<\infty.$ (10) Take now regular functions $f,g\in\mathcal{F}_{op\,.}$ We know that $\bigl{(}\mathcal{F}_{op}\,,\preceq\bigr{)}$ is a lattice (AudenaertCaiHansen:2008, , bottom of page 141), and that the representative function in (7) for the minorant $f\wedge g$ is given by $h_{f\wedge g}=\max\\{h_{f},h_{g}\\}\leq h_{f}+h_{g}$ showing that also $h_{f\wedge g}$ satisfies the integrability condition (10) implying that $f\wedge g$ is regular. Since $h_{f\vee g}=\min\\{h_{f},h_{g}\\}\leq h_{f}$ it also follows that the majorant is regular. We now take functions $f,g\in\mathcal{F}_{op}$ with representative functions $h_{f}$ and $h_{g}$ and assume that $f$ is non-regular. Since $h_{f\wedge g}=\max\\{h_{f},h_{g}\\}\qquad\text{and thus}\qquad h_{f}\leq h_{f\wedge g}$ we obtain that also the minorant $f\wedge g$ is non-regular. QED ### VII.2 The Wigner-Yanase-Dyson skew informations The Wigner-Yanase-Dyson skew information (with parameter $p)$ is defined by setting $I_{p}(\rho,A)=-\frac{1}{2}{\rm Tr\hskip-1.99997pt}~{}[\rho^{p},A[[\rho^{1-p},A],\qquad 0<p<1.$ It is an example of a metric adjusted skew information and reduces to the Wigner-Yanase skew information for $p=1/2\,.$ The representing function $f_{p}$ of $I_{p}(\rho,A)$ is given by $f_{p}(t)=p(1-p)\cdot\frac{(t-1)^{2}}{(t^{p}-1)(t^{1-p}-1)}\qquad 0<p<1,$ that is $I_{p}(\rho,A)=I^{f_{p}}_{\rho}(A).$ The weight-functions $h_{p}(\lambda)$ in equation (7) corresponding to the representing functions $f_{p}$ are given by $h_{p}(\lambda)=\frac{1}{\pi}\arctan\frac{(\lambda^{p}+\lambda^{1-p})\sin p\pi}{1-\lambda-(\lambda^{p}-\lambda^{1-p})\cos p\pi}\qquad 0<\lambda<1.$ It is a non-trivial result that the Wigner-Yanase-Dyson skew informations $I_{p}(\rho,A)$ are increasing in the parameter $p$ for $0<p\leq 1/2$ and decreasing in $p$ for $1/2\leq p<1$ with respect to the order relation $\preceq,$ cf. (AudenaertCaiHansen:2008, , Theorem 2.8). The Wigner-Yanase skew information is thus the maximal element among the Wigner-Yanase-Dyson skew informations with respect to the order relation $\preceq.$ ### VII.3 The monotonous bridge The family of metrics with representing functions $f_{\alpha}(t)=t^{\alpha}\left(\frac{1+t}{2}\right)^{1-2\alpha}\qquad t>0,$ decrease monotonously (with respect to $\preceq)$ from the largest monotone symmetric metric down to the Bures metric for $\alpha$ increasing from $0$ to $1.$ They correspond the the constant weight functions $h_{\alpha}(\lambda)=\alpha$ in equation (7). However, the only regular metric in this bridge is the Bures metric $(\alpha=1).$ It is however possible to construct a variant bridge by choosing the weight functions $h_{p}(\lambda)=\left\\{\begin{array}[]{lrl}0,&\lambda&<1-p\\\\[4.30554pt] p,&\lambda&\geq 1-p\end{array}\right.\qquad 0\leq p\leq 1$ in equation (7) instead of the constant weight functions. It is non-trivial that these weight functions provide a monotonously decreasing bridge (with respect to $\preceq)$ of monotone symmetric metrics between the smallest and the largest (monotone symmetric) metric. The benefit of this variant bridge is that all the constituent metrics are regular except for $p=1.$ ## VIII Metric adjusted local quantum uncertainty We consider a bipartite system $\mathcal{H}=\mathcal{H}_{1}\otimes\mathcal{H}_{2}$ of two finite dimensional Hilbert spaces. ###### Definition 10. Let $f\in\mathcal{F}_{\text{op}}$ be regular and take a vector $\Lambda\in\mathbf{R}^{d}.$ We define the Metric Adjusted Local Quantum Uncertainty (or $f$-LQU) by setting $\mathcal{U}_{1}^{\Lambda,f}(\rho)=\inf\\{I^{f}_{\rho}(K_{1}\otimes 1_{2})\mid K_{1}\text{ has spectrum $\Lambda$}\\},$ (11) where $\rho_{12}$ is a bipartite state, and $K_{1}$ is the partial trace of an observable $K$ on $\mathcal{H}.$ The minimum in the above definition is thus taken over local observables $K_{1}\otimes 1_{2}\in B(\mathcal{H}_{1}\otimes H_{2})$ such that $K_{1}$ is unitarily equivalent with the diagonal matrix $\text{diag}(\Lambda).$ ###### Remark 3. The metric adjusted LQU has been studied in the literature for specific choices of $f.$ • If $f(x)=f_{WY}(x)=\Bigl{(}\frac{1+\sqrt{x}}{2}\Bigr{)}^{2}$ then ${\mathcal{U}}_{1}^{\Lambda,f}$ coincide with the LQU introduced in (GTA:2013, , equation 2). * • If $f(x)=f_{SLD}(x)=\frac{1+x}{2}$ then $\mathcal{U}_{1}^{\Lambda,f}$ coincides with the Interferometric Power introduced in GSGTFSSOA:2014 . ###### Proposition 4. For $f,g\in\mathcal{F}_{op}^{\,r}$ with $\tilde{g}\leq\tilde{f}$ we have the inequality ${\mathcal{U}}^{\Lambda,f}_{1}(\rho_{12})\leq{\mathcal{U}}^{\Lambda,g}_{1}(\rho_{12}).$ In particular the LQU is smaller than the IP. ###### Proof. Let $\tilde{K}_{1}$ be the local observable with spectrum $\Lambda$ minimizing the metric adjusted skew information. Then ${\mathcal{U}}^{\Lambda,f}_{1}(\rho_{12})=I^{f}_{\rho_{12}}\bigl{(}\tilde{K}_{1}\otimes 1_{2}\bigr{)}\geq I^{g}_{\rho_{12}}\bigl{(}\tilde{K}_{1}\otimes 1_{2}\bigr{)}\geq{\mathcal{U}}^{\Lambda,g}_{1}(\rho_{12}),$ where we used the inequality in (6). QED ###### Corollary. Let $g_{1}$ and $g_{2}$ be regular functions in $\mathcal{F}_{op}$ and set $f=\tilde{g}_{1}\wedge\tilde{g}_{2}$ with respect to the lattice structure in $\mathcal{F}_{op}\,.$ Then there is a regular function $g$ in $\mathcal{F}_{op}$ such that $\tilde{g}=f=\tilde{g}_{1}\wedge\tilde{g}_{2}$ and $\max\bigl{\\{}{\mathcal{U}}^{\Lambda,g_{1}}_{1}(\rho_{12}),\,{\mathcal{U}}^{\Lambda,g_{2}}_{1}(\rho_{12})\bigr{\\}}\leq{\mathcal{U}}^{\Lambda,g}_{1}(\rho_{12})$ for arbitrary $\rho_{12}\,.$ ###### Proof. The functions $\tilde{g}_{1}$ and $\tilde{g}_{2}$ are non-regular by Theorem Theorem. By Proposition 3 we thus obtain that also the minorant $f$ is non- regular. Therefore there exists, by the correspondence in Theorem Theorem, a (unique) regular function $g$ in $\mathcal{F}_{op}$ such that $\tilde{g}=f.$ The assertion then follows by Proposition 4. QED Following BogaertGirolami:2017 we prove that the metric adjusted LQU is a measure of non-classical correlations, i.e. it meets the criteria which identify discord-like quantifiers, see rev . ###### Theorem. If the state $\rho$ is classical-quantum in the sense of Piani , then the metric adjusted LQU vanishes, that is $\mathcal{U}_{1}^{\Lambda,f}(\rho)=0.$ Conversely, if the coordinates of $\Lambda$ are mutually different (thus rendering the operator $K_{1}$ non-degenerate) and $\mathcal{U}_{1}^{\Lambda}(\rho)=0,$ then $\rho$ is classical-quantum. ###### Proof. We note that the metric adjusted skew information $I_{\rho}^{f}(A)$ for a faithful state $\rho$ is vanishing if and only if $\rho$ and $A$ commute. If $\rho$ is classical-quantum, then $P_{1}(\rho)=\sum_{i}(P_{1,i}\otimes 1_{2})\rho(P_{1,i}\otimes 1_{2})=\rho$ for some von Neumann measurement $P$ given by a resolution $(P_{i})$ of the identity $1_{1}$ in terms of one-dimensional projections. We may choose $K_{1}$ diagonal with respect to this resolution, so $K_{1}\otimes 1_{2}$ and $\rho$ commute and thus $\mathcal{U}_{1}^{\Lambda,f}(\rho)=0.$ If on the other hand the $f$-LQU $\mathcal{U}_{1}^{\Lambda,f}(\rho)=0,$ then there exist a local observable $K_{1}\otimes 1_{2}$ such that $[\rho,K_{1}\otimes 1_{2}]=0.$ By the spectral theorem we write $K_{1}=\sum_{i}\lambda_{i}P_{1,i}=\sum_{i}\lambda_{i}\|i\rangle_{1}\langle i\|$ and since $\rho(K_{1}\otimes 1_{2})=(K_{1}\otimes 1_{2})\rho$ we obtain by multiplying with $P_{1,i}\otimes 1_{2}$ from the left and $P_{1,j}\otimes 1_{2}$ from the right the identity $\lambda_{j}(P_{1,i}\otimes 1_{2})\rho(P_{1,j}\otimes 1_{2})=\lambda_{i}(P_{1,i}\otimes 1_{2})\rho(P_{1,j}\otimes 1_{2}).$ If $K_{1}$ is non-degenerate, it thus follows that $(P_{1,i}\otimes 1_{2})\rho(P_{1,j}\otimes 1_{2})=0\qquad\text{for}\quad i\neq j.$ By summing over all $j$ different from $i$ we obtain $(P_{1,i}\otimes 1_{2})\rho((1_{1}-P_{1,i})\otimes 1_{2})=0,$ thus $(P_{1,i}\otimes 1_{2})\rho=(P_{1,i}\otimes 1_{2})\rho(P_{1,i}\otimes 1_{2}),$ so $P_{1,i}\otimes 1_{2}$ and $\rho$ commute. It follows that $P_{1}(\rho)=\sum_{i}(P_{1,i}\otimes 1_{2})\rho(P_{1,i}\otimes 1_{2})=\rho,$ so $\rho$ is left invariant under the von Neumann measurement $P$ given by $(P_{i}).$ Therefore, $\rho$ is classical-quantum. QED Recall that Luo and Zhang LuoZang:2008 proved that a state $\rho$ is classical-quantum if and only if there exists a resolution $(P_{i})$ of the identity $1_{1}$ such that $\rho=\sum_{i}p_{i}P_{1,i}\otimes\rho_{i},$ where each $\rho_{i}$ is a state on $\mathcal{H}_{2}$ and $p_{i}\geq 0,$ and the sum $\sum_{i}p_{i}=1.$ By (CaiHansen:2010, , Lemma 3.1) the inequality $I_{\rho}^{f}(K_{1}\otimes 1_{2})\geq I_{\rho_{1}}^{f}(K_{1})$ is valid, where $\rho_{1}={\rm Tr\hskip-1.99997pt}~{}_{2}\,\rho_{12}\,.$ Consequently, we obtain that $\mathcal{U}_{1}^{\Lambda,f}(\rho)\geq\inf_{K_{1}}\,I^{f}_{\rho_{1}}(K_{1})=\inf_{\sigma_{1}}\,I^{f}_{\sigma_{1}}\bigl{(}K_{1}\bigr{)},$ (12) where the minimum is taken over states $\sigma_{1}$ on $\mathcal{H}_{1}$ unitarily equivalent with $\rho_{1}.$ ###### Theorem. The metric adjusted LQU is invariant under local unitary transformations. ###### Proof. For the metric adjusted skew information and local unitary transformations we have $\begin{array}[]{l}\mathcal{U}_{1}^{\Lambda,f}\bigl{(}(U_{1}\otimes U_{2})\rho_{12}(U_{1}\otimes U_{2})^{\dagger}\bigr{)}\\\\[6.45831pt] =\min_{K_{1}}I^{f}\bigl{(}(U_{1}\otimes U_{2})\rho_{12}(U_{1}\otimes U_{2})^{\dagger},K_{1}\otimes 1_{2}\bigr{)}\\\\[6.45831pt] =\min_{K_{1}}I^{f}\bigl{(}\rho_{12},(U_{1}\otimes U_{2})^{\dagger}(K_{1}\otimes 1_{2})(U_{1}\otimes U_{2})\bigr{)}\\\\[6.45831pt] =\min_{K_{1}}I^{f}\bigl{(}\rho_{12},(U_{1}^{\dagger}K_{1}U_{1}\otimes 1_{2}\bigr{)}=\mathcal{U}^{\Delta,f}_{1}(\rho_{12}),\end{array}$ where we used the definition in (11). QED ###### Theorem. The metric adjusted LQU is contractive under completely positive trace- preserving maps on the non-measured subsystem. ###### Proof. Let $\tilde{K}_{1}$ be the local observable minimizing the metric adjusted skew information. A completely positive trace preserving map $\Phi_{2}$ on system 2 is obtained as an amplification followed by a partial trace (Stinespring dilation): ${\rm Tr\hskip-1.99997pt}~{}_{3}(U_{23}\rho_{23}U_{23}^{\dagger})=\Phi_{2}\,\rho_{2}$. The metric adjusted LQU is invariant under local unitaries. Also, the metric adjusted skew information is contractive under partial trace. Calling $d_{3}$ the dimension of the Hilbert space of the ancillary system 3, one has $\displaystyle\mathcal{U}_{1}^{\Lambda,f}(\rho_{12})=$ $\displaystyle\,I^{f}\left(\rho_{12},\tilde{K}_{1}\otimes 1_{2}\right)=I^{f}\left(\rho_{12}\otimes\frac{1}{d_{3}}1_{3},\tilde{K}_{1}\otimes 1_{23}\right)$ $\displaystyle=$ $\displaystyle\,\ I^{f}\left((1_{1}\otimes U_{23})\left(\rho_{12}\otimes\frac{1}{d_{3}}1_{3}\right)(1_{1}\otimes U_{23}^{\dagger}),\tilde{K}_{1}\otimes 1_{23}\right)$ $\displaystyle\geq$ $\displaystyle\,I^{f}\left({\rm Tr\hskip-1.99997pt}~{}_{3}\left\\{(1_{1}\otimes U_{23})\left(\rho_{12}\otimes\frac{1}{d_{3}}1_{3}\right)(1_{1}\otimes U_{23}^{\dagger})\right\\},\tilde{K}_{1}\otimes 1_{2}\right)$ $\displaystyle=$ $\displaystyle\,I^{f}\left((1_{1}\otimes\Phi_{2})\rho_{12},\tilde{K}_{1}\otimes 1_{2}\right)$ $\displaystyle\geq$ $\displaystyle\,\mathcal{U}_{1}^{\Lambda,f}\bigl{(}(1_{1}\otimes\Phi_{2})\rho_{12}\bigr{)},$ as desired. QED ###### Theorem. The metric adjusted LQU reduces to an entanglement monotone for pure states. ###### Proof. The metric adjusted $f$-LQU coincides with the standard variance on pure states, that is $I_{\rho}^{f}(A)={\rm Var}_{\rho}(A)={\rm Tr\hskip-1.99997pt}~{}\rho A^{2}-({\rm Tr\hskip-1.99997pt}~{}\rho A)^{2}$ whenever $\rho$ is pure (Hansen:2006b, , Theorem 3.8). But in GTA:2013 it has been proved that the minimum local variance is an entanglement monotone for pure states. QED ## IX Conclusion In this work, we have built a unifying information-geometric framework to quantify quantum correlations in terms of metric adjusted skew informations. We extended the physically meaningful definition of LQU to a more general class of information measures. Crucially, metric adjusted quantum correlation quantifiers enjoy, by construction, a set of desirable properties which make them robust information measures. 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# Computational strategies and estimation performance with Bayesian semiparametric Item Response Theory models Sally Paganin<EMAIL_ADDRESS>University of California, Berkeley Christopher J. Paciorek University of California, Berkeley Claudia Wehrhahn University of California, Santa Cruz Abel Rodríguez University of Washington, Seattle Sophia Rabe-Hesketh University of California, Berkeley Perry de Valpine University of California, Berkeley ###### Abstract Item response theory (IRT) models typically rely on a normality assumption for subject-specific latent traits, which is often unrealistic in practice. Semiparametric extensions based on Dirichlet process mixtures offer a more flexible representation of the unknown distribution of the latent trait. However, the use of such models in the IRT literature has been extremely limited, in good part because of the lack of comprehensive studies and accessible software tools. This paper provides guidance for practitioners on semiparametric IRT models and their implementation. In particular, we rely on NIMBLE, a flexible software system for hierarchical models that enables the use of Dirichlet process mixtures. We highlight efficient sampling strategies for model estimation and compare inferential results under parametric and semiparametric models. Keywords: binary IRT models, Dirichlet process mixture, MCMC strategies, NIMBLE. ## 1 Introduction Traditional approaches in item response theory (IRT) modeling rely on the assumption that subject-specific latent traits follow a normal distribution. This assumption is often considered for computational convenience, but there are many situations in which it may be unrealistic (Samejima, 1997). For example, Micceri (1989) gives a comprehensive review of many psychometric datasets where the distribution of latent individual trait does not respect the normality assumption and presents instead asymmetries, heavy-tails or multimodality. In addition, estimation of IRT parameters in the presence of non-normal latent traits has been shown to produce biased estimates of the parameters (see, for example Seong, 1990; Kirisci ., 2001; Schmitt ., 2006; Finch Edwards, 2016). Different proposals have been made in the general IRT literature for relaxing this normality assumption, using either Markov chain Monte Carlo (MCMC) or Marginal Maximum Likelihood (MML) estimation methods. One option is to rely on more general parametric assumptions. For example, Azevedo . (2011) considered a skew-normal distribution (Azzalini, 1985), while others have suggested finite mixtures of normal distributions (Bolt ., 2001; Bambirra Gonçalves ., 2018). Alternatively, one can refrain from making distributional assumption on the latent abilities by using nonparametric maximum likelihood estimation (Laird, 1978; Mislevy, 1984), B-splines (Woods Thissen, 2006; Johnson, 2007) or empirical histograms (Woods, 2007). This paper considers a Bayesian nonparametric approach that uses a Dirichlet process mixture (Ferguson, 1973; Lo, 1984; Escobar West, 1995) as a nonparametric prior on the distribution of the subject-specific latent trait. Dirichlet process mixtures are often used as flexible models to describe the unknown distribution of an heterogeneous population of interest. These models are sometimes interpreted as mixture models with an infinite number of components. In practice these model treat the number of groups as an unknown parameter and estimate it from the data, so that the model can easily account for multi-modality, asymmetries or outliers in the latent trait distribution. We focus in particular on semiparametric extensions of logistic IRT models for binary responses. Such models are semiparametric because they retain other, parametric, assumptions of binomial mixed models, such as the functional form of the link function. Even though some Bayesian nonparametric extensions of binary IRT models have been presented in the literature, they have been given only limited consideration. Within this approach, the semiparametric 1PL model has been the focus of more effort as well as software (Jara ., 2011, DPpackage, no longer actively maintained). San Martín . (2011) investigated a semiparametric generalization of the 1PL model from a theoretical perspective, while Finch Edwards (2016) provided results from simulation studies. An example using the semiparametric 2PL model is given in Duncan MacEachern (2008). However, such semiparametric models have not received much attention in applied IRT modeling, in good part because of the lack of comprehensive studies and accessible software tools. The goal of this paper is to provide a practical guide to semiparametric IRT models for both (i) applied researchers interested in using Dirichlet process mixtures, and (ii) those familiar with Bayesian nonparametrics concepts who are interested in IRT models. To achieve these goals, we fill three major gaps that hinder widespread application of semiparametric Bayesian IRT models. First, we implement the semiparametric 1PL, 2PL and 3PL models in NIMBLE (de Valpine ., 2017) (R package nimble, de Valpine ., 2020), a flexible R-based system for hierarchical modeling. In particular, NIMBLE provides functionality for fitting hierarchical models that involve Dirichlet process priors either via a Chinese Restaurant Process (CRP) (Aldous, 1985; Pitman, 1996; Blackwell MacQueen, 1973) or a truncated stick-breaking (SB) (Sethuraman, 1994) representation of the prior. Hence, NIMBLE supports a much wider class of models than those that are implemented in standard software packages. Code is provided for all examples in a publicly accessible GitHub repository (https://github.com/salleuska/IRT_nimble_code). Second, focusing on the 2PL model, we study the efficiency of several MCMC sampling strategies in both simulated and real-data scenarios. We define sampling strategies as the combination of model parameterization, identifiability constraints and sampling algorithms, focusing on general MCMC algorithms available in easy-to-access software tools for Bayesian hierarchical models. We find that some choices of parameterization and identifiability constraints can yield order-of-magnitude differences in sampling efficiency compared to others. This approach also allows us to compare various random walk Metropolis-Hastings MCMC strategies to the Hamiltonian Monte Carlo (HMC) strategy implemented in the widely used Stan package (Stan Development Team, 2018). Although there is Stan support for many parametric IRT models (Bürkner, 2021; Furr, 2017), HMC algorithms are not readily available for Dirichlet process prior models, since HMC cannot sample discrete parameters (the component indicators), which cannot be easily integrated out in infinite mixture models. Finally, we present a comparison of inferential results for item and subject parameters under parametric and semiparametric specifications. To make these comparisons fair, we carefully elicit prior distributions for the models by matching the prior predictive distribution of the data to a common distribution (Berger Pericchi, 1996). We also illustrate how to estimate the entire distribution of latent traits and its functionals under the two specifications. As expected, we find that the semiparametric model improves recovery of item and individual latent trait parameters in the case of non- normal latent traits. More suprisingly, there seems to be little inferential penalty in using a semiparametric model when a parametric model would be correct, supporting the benefit of greater robustness to mis-specification. These conclusions are based on analyses carried out on simulated data as well as two real datasets related to education and medical assessments: the 2007 Trends in International Mathematics and Science Study (TIMSS) and the 1996 Health Survey for England. For both the real data examples, the semiparametric model performs better than the parametric counterpart, with the semiparametric model identifying distinct modes in the distribution of the latent trait missed by the parametric model. We note that other authors have considered nonparametric IRT models that rely on a general monotonic function in place of the logistic/probit link function. These models are sometimes referred as NIRT models. Some work using the Dirichlet process falls in this class of models (Qin, 1998; Miyazaki Hoshino, 2009; Karabatsos, 2017). While we do not pursue this direction in this paper, focusing instead on nonparametric modeling of the latent trait distribution, such an extension is relatively straightforward. The remainder of the paper is organized as follows. In Section 2 we present the standard IRT model and the Bayesian semiparametric extension along with considerations for identifiability. We then present different potential sampling strategies (Section 3) and discuss the goals of our experiments. To fairly compare the different strategies, we give guidance on selecting prior distributions in Section 4. We introduce simulated and real-world data in Section 5. Comparison of the results in terms of MCMC efficiency and statistical inference is presented in Sections 6 and 7. In Section 8, we conclude that having access to semiparametric models can be broadly useful, as it allows inference on the entire underlying latent trait distribution and its functionals, with NIMBLE being a flexible framework for estimation of such models. ## 2 IRT models and background IRT models are widely used in various social science disciplines to scale binary responses into continuous constructs. For conciseness, in this section we introduce model notation in the context of educational assessment, where typically data are answers to exam questions from a set of individuals and the latent trait is interpreted as an individual’s ability. In particular, let $y_{ij}$ denote the answer of individual $j$ to item $i$ for $j=1,\ldots,N$ and $i=1,\ldots,I$, with $y_{ij}=1$ when the answer is correct and $0$ otherwise. Responses from different individuals are assumed to be independent, while responses from the same individual are assumed independent conditional on the latent trait (this is sometimes called the _local independence assumption_ in the psychometric literature). ### 2.1 Binary logistic IRT models Let $\pi_{ij}$ denote the probability that individual $j$ answers item $i$ correctly, given the model parameters $\eta_{j},\lambda_{i},\beta_{i}$; i.e., $\pi_{ij}=\Pr(y_{ij}=1\mid\eta_{j},\lambda_{i},\beta_{i})$ for $i=1,\ldots,I$ and $j=1,\ldots,N$. The parameter $\eta_{j}$ represents the latent ability of the $j$-th individual, while $\beta_{i}$ and $\lambda_{i}$ encode the item characteristics for the $i$-th item. In the two-parameter logistic (2PL) model, the probability $\pi_{ij}$ is determined using the logistic function as $\text{logit}(\pi_{ij})=\lambda_{i}(\eta_{j}-\beta_{i}),\quad i=1,\ldots,I,\quad j=1,\ldots,N.$ (1) A further assumption for the latent abilities is that they are independently and identically distributed according to some distribution $G$, $\eta_{j}\stackrel{{\scriptstyle iid}}{{\sim}}G,\quad j=1,\ldots,N,$ (2) with $G$ traditionally a standard normal distribution. The parameter $\lambda_{i}>0$ is often referred to as _discrimination_ , since items with a large $\lambda_{i}$ are better at discriminating between subjects with similar abilities, while $\beta_{i}$ is called _difficulty_ because for any fixed $\eta_{j}$ the probability of a correct response to item $i$ is decreasing in $\beta_{i}$. Often, the log-odds in (1) are reparameterized as $\lambda_{i}\eta_{j}+\gamma_{i},$ with $\gamma_{i}=-\lambda_{i}\beta_{i}$. The two parameterizations are sometimes referred to as _IRT parameterization_ and _slope-intercept (SI) parameterization_ , respectively. While the slope-intercept parameterization is often considered for computational convenience, the IRT parameterization is the most traditional in terms of interpretation. In exploring different strategies for Bayesian estimation, we will consider both alternatives and investigate potential differences in terms of computational performance. Alternative models can be obtained by considering a different number of item parameters. When $\lambda_{i}=1$ for all $i=1,\ldots,I$, the model in (1) reduces to the one-parameter logistic (1PL) model, also known as Rasch model (Rasch, 1990). In some settings one may wish to account for the probability of answering correctly by chance, by introducing a third set of item parameters, $\upsilon_{i},i=1,\ldots,I$, referred to as _guessing_ parameters, so that $\Pr(y_{ij}=1\mid\eta_{j},\lambda_{i},\beta_{i},\upsilon_{i})=\upsilon_{i}+(1-\upsilon_{i})\text{expit}\\{\lambda_{i}(\eta_{j}-\beta_{i})\\},$ where $\text{expit}\\{\cdot\\}$ denotes the inverse of the logistic function. This model is typically referred to as the three-parameter logistic model (3PL), and is often relevant in educational assessments. ### 2.2 Semiparametric IRT models The classical formulation of IRT models assumes that the latent abilities in (2) follow a normal distribution. This assumption can be relaxed, modeling the distribution of ability as a mixture of normal distributions, where the number of mixture components does not need to be specified in advance but rather is learned from the data. This can be achieved using a Dirichlet process mixture (DPM) model for the distribution of ability. In particular, the distribution of ability $G$ in (2) can be specified as a convolution involving a Dirichlet process (DP) prior, i.e. $\displaystyle G=\int\mathcal{K}(\eta_{j}\mid\theta)F(d\theta),\quad F\sim\mbox{DP}(\alpha,G_{0}),$ (3) where $\mathcal{K}(\cdot\mid\theta)$ is a suitable probability kernel indexed by the parameter $\theta$, while $\alpha$ and $G_{0}$ are, respectively, the concentration parameter and the base distribution of the Dirichlet process. In the context of binary IRT models, it seems natural to choose a normal kernel for $\mathcal{K}(\cdot\mid\theta)$, indexed by parameters $\theta=\\{\mu,\sigma^{2}\\}$. This means the distribution of ability is a mixture of normal distributions, where the number of mixture components and their means and variances are unknown. Furthermore, under this choice, taking $\alpha\to 0$ leads to the original parametric model discussed in Section 2.1, in this case a single normal. Parameters characterizing each mixture component are drawn from the base distribution, $G_{0}$. For computational convenience the base distribution is typically the product of conjugate distributions, e.g., a normal distribution for $\mu$ and an inverse-gamma distribution for $\sigma^{2}$. We proceed now to discuss the Dirichlet process prior in more detail. There are two main representations of the Dirichlet process, each leading to a different MCMC posterior sampling strategy, namely the stick-breaking representation (SB) (Sethuraman, 1994) and the Chinese Restaurant Process (CRP) (Aldous, 1985; Pitman, 1996; Blackwell MacQueen, 1973). In this work we use the CRP representation. The CRP representation is derived from (3) integrating out the random measure $F$. More specifically, let $\theta_{1},\ldots,\theta_{N}$ be an independent sample from $F$, with some values possibly repeated. Integrating over $F$ one can obtain the joint prior distribution on $(\theta_{1},\ldots,\theta_{N})$, which can be written as the product of a sequence of conditional distributions, where $(\theta_{j}\mid\theta_{j-1},\ldots,\theta_{1})\sim\frac{\alpha}{\alpha+j-1}G_{0}+\sum_{l=1}^{j-1}\frac{1}{\alpha+j-1}\delta_{\theta_{l}},$ (4) for $j=1,\ldots,N$, where $\delta_{a}$ is the Dirac probability measure concentrated at $a$. The second term in (4) represents the probability that a new observation is equal to one of the previous ones, while the first term captures the possibility that we observe a new value, which would be drawn from the base measure $G_{0}$. The CRP name comes from an analogy often used to describe the process in (4). Consider a Chinese restaurant with an infinite number of tables, each table serving one dish shared by all customers sitting at that table. In this metaphor, each table represents a possible mixture component, while each dish represents the parameter indexing the distribution associated with the mixture component. Customers entering the restaurant can seat themselves at a previously occupied table and share the same dish (with probability proportional to the number of customers already sitting at the table), or go to a new table and order another dish (with probability proportional to $\alpha$). The dishes are selected according to the centering distribution $G_{0}$. One way to make the Chinese restaurant analogy clearer is by reparameterizing the model. Denote by $\theta_{k}^{}$ the dish served in table $k$ (which is a draw from $G_{0}$) and let $z_{j}$ be the variable denoting the table chosen by the $j$th customer. Then $\displaystyle p(z_{j}=k\mid z_{j-1},\ldots,z_{2},z_{1},\alpha)=\begin{cases}\frac{n_{k}^{j-1}}{\alpha+j-1},\quad k=1,\ldots,K^{j-1},\\\ \frac{\alpha}{\alpha+j-1},\quad k=K^{j-1}+1,\end{cases}$ (5) where $K^{j-1}$ is the total number of occupied tables by the first $j-1$ customers, and $n^{j-1}_{k}$ is the number of customers at table $k$ among the first $j-1$. This new parameterization can be related to the old one by noting that $\theta_{i}=\theta_{z_{i}}^{}$. The concentration parameter $\alpha$ controls the distribution of the number of tables (components), with larger values favoring more tables. Using the indicators $\mathbf{z}=\\{z_{j},j=1,\ldots,N\\}$, we can denote by $\mathbf{z}\mid\alpha\sim\mbox{CRP}(\alpha)$ the joint distribution induced by (5), and rewrite the DPM model for the distribution of ability in (3) using $\displaystyle\eta_{j}\mid z_{j},\theta^{}_{1},\theta^{}_{2},\ldots$ $\displaystyle\stackrel{{\scriptstyle ind}}{{\sim}}\mathcal{K}(\cdot\mid\theta^{}_{z_{j}}),\quad j=1,\ldots,N,$ $\displaystyle\mathbf{z}\mid\alpha$ $\displaystyle\sim\mbox{CRP}(\alpha),$ $\displaystyle\theta^{}_{k}$ $\displaystyle\stackrel{{\scriptstyle iid}}{{\sim}}G_{0},\quad k=1,2,\ldots.$ Together, the probability kernel, base measure, and CRP form a Dirichlet process mixture. Alternatively, the distribution $F$ can be written using the stick-breaking representation: $F(\cdot)=\sum_{k=1}^{\infty}w_{k}\delta_{\tilde{\theta}_{k}},$ where $\tilde{\theta}_{1},\tilde{\theta}_{2},\ldots$ is a sequence of independent draws from $G_{0}$ and the weights are constructed by letting $w_{k}=v_{k}\prod_{l=1}^{k-1}(1-v_{l})$, with $v_{1},v_{2},\ldots$ being a sequence of independent draws from a $\mbox{Beta}(1,\alpha)$ distribution. This construction makes it clear that, as long as the kernel $\mathcal{K}(\cdot\mid\theta)$ is continuous, the distribution of ability $G$ is also continuous, but $F$ is almost surely discrete, naturally inducing clustering via repeats in the parameter indexing the distribution of ability. Note that an alternative to the formulation described above is to model the distribution of the ability $G$ directly using a DP, e.g., centered around a normal distribution. Such a model also comprises the standard parametric model as a limiting case (now, when $\alpha\to\infty$) and leads to slightly simpler computational algorithms. However, we believe that such an approach has some serious drawbacks in the context of most IRT applications. By definition, realizations from a Dirichlet process are almost surely discrete. This property has made the Dirichlet process a useful tool in clustering applications. However, in our context, it implies that we believe that two (or more) individuals potentially have exactly the same ability. Not only is this assumption not realistic, but it potentially prevents us from distinguishing individuals based on their abilities, which is one common goal in IRT modeling. The use of a Dirichlet process mixture with a continuous kernel (Gaussian, in this case) sidesteps this issue. ### 2.3 Identifiability and constraints Without additional constraints, the parameters of the models presented in Section 2 are not identifiable (e.g., see Geweke Singleton, 1981; Bafumi ., 2005, as well as Section A in the Supplementary Materials). For example in the 2PL and 3PL models, increasing all $\eta_{j}$ and $\beta_{i}$ values by the same amount yields the same probabilities in (1) for all $i$ and $j$. More generally, the ability parameters are known up to a linear transformation, and constraints are needed to identify them. To address this problem, traditional work on parametric IRT models assumes that latent abilities in (2) come from a standard normal distribution, i.e., $G\equiv\mathcal{N}(0,1)$, and constrains the discrimination parameters $\lambda_{i}$ for $i=1,\ldots,I$ to be positive. Alternative constraints can also establish identifiability and could yield different computational performance for MCMC sampling. A common alternative considers sum-to-zero constraints for the item parameters (Fox, 2010) $\sum_{i=1}^{I}\beta_{i}=0,\quad\left(\text{or }\sum_{i=1}^{I}\gamma_{i}=0\right),\quad\textbf{}\sum_{i=1}^{I}\log(\lambda_{i})=0.$ (6) Centering the difficulty parameters addresses the invariance to translations, while centering the log of the discrimination parameters (setting their product to one) accounts for the invariance to rescalings of the latent space. Another potential set of constraints, popular in political science applications, involves fixing the value of the latent traits for two individuals (e.g., see Clinton ., 2004). Whatever the set of constraints, it is worthwhile to note that they can be either directly incorporated in the model as part of the prior (and, therefore in the structure of the sampling algorithms), or they can be applied as a postprocessing step (after running an unconstrained MCMC). This last approach is typical of parameter-expanded algorithms, which embed the target model in a larger specification. Parameter expansion has been proposed in the literature to accelerate EM (C. Liu ., 1998) and Gibbs sampler (JS. Liu Wu, 1999) convergence, as well as to induce new classes of priors (Gelman, 2004). Although targeting the same posterior, constrained priors and parameter expansion can lead to very different results in terms of convergence and mixing of the MCMC algorithms. Similar arguments apply for the semiparametric extensions using the Dirichlet process mixture. In that setting, one identifiability strategy may be to constrain the base distribution $G_{0}$, e.g., by letting $G_{0}\sim\mathcal{N}(0,1)$ (for example, see Duncan MacEachern, 2008). However, even if the prior expectation and variance of $G_{0}$ are zero and one, the corresponding posterior quantities can deviate substantially from these values, leading to biased inference (Yang Dunson, 2010). More general centering approaches have been proposed in the literature when a DP distribution is used to model random effects or latent variables in a hierarchical model (Yang ., 2010; Yang Dunson, 2010; Li ., 2011). These approaches rely on parameter expansion by sampling from the unconstrained DP model and then applying a post-processing procedure to the posterior samples. This post-processing procedure requires the analytical evaluation of the posterior mean and variance of the DP random measure, with Li . (2011) providing results under the CRP representation and Yang . (2010) under the stick-breaking one. Although such strategies are useful for general hierarchical models to avoid identifiability issues, for the semiparametric 1PL, 2PL, and 3PL models it is simpler to use the sum-to-zero constraints on the item parameters in (6), and that is the approach we adopt in this work. As in the parametric case, we can either include these constraints in the prior or use the parameter expansion approach for sampling and then center and rescale the posterior samples as appropriate. ## 3 Sampling strategies for logistic IRT models In this work we explore different _sampling strategies_ for Bayesian estimation of the logistic parametric and semiparametric IRT models. We define a _sampling strategy_ to include the combination of model parameterization, identifiability constraints and sampling algorithms. We focus on the case of the 2PL model, as it contains the 1PL as a special case and presents the same identifiability challenges as the 3PL model. The strategies considered are summarized in Table 1. Identifiability constraints Parametric Semi-parametric Slope-intercept IRT Slope-intercept IRT Constrained abilities MH/conjugate MH/conjugate Centered HMC (Stan) Constrained item parameters MH/conjugate MH/conjugate∗ MH/conjugate MH/conjugate∗ Unconstrained MH/conjugate MH/conjugate MH/conjugate MH/conjugate Centered Centered Table 1: Summary of the 14 sampling strategies considered for the parametric and semiparametric 2PL model. Each of the 14 entries is a different strategy with “MH/conjugate”, “Centered”, and “HMC (Stan)” referring to three different sampling algorithms discussed below. The asterisk symbol denotes the sampling strategies that lead directly to samples parameterized as model (3). Others need post-processing to correspond to model (3). We explore both parameterizations of the 2PL model mentioned in Section 2.1: the IRT and the slope-intercept parameterization. To compare estimates obtained from different parameterizations on a common scale, we post-process posterior samples (using transformations described in the Supplementary Materials, Section A) to respect the following base parameterization $\displaystyle\text{logit}(\pi_{ij})$ $\displaystyle=\lambda_{i}(\eta_{j}-\beta_{i}),\quad i=1,\ldots,I,$ $\displaystyle\sum_{i=1}^{I}\log(\lambda_{i})$ $\displaystyle=0\quad\sum_{i=1}^{I}\beta_{i}=0,$ $\displaystyle\eta_{j}$ $\displaystyle\sim G,\quad j=1,\ldots,N,$ (7) where $G$ denotes a general distribution for the latent abilities, either parametric or nonparametric. The model in (3) follows the IRT parameterization with sum-to-zero identifiability constraints, which is typically the target one for inference for interpretability reasons. As discussed in Section 2.3, our target inferential model in (3) can be estimated directly, accounting for identifiability constraints. This can be achieved by introducing in the model formulation a set of auxiliary item parameters, $\\{\lambda_{i}^{\prime},\beta^{\prime}_{i}\\}$ for each $i=1,\ldots,I$ and defining $\\{\lambda_{i},\beta_{i}\\}$ as $\displaystyle\log(\lambda_{i})$ $\displaystyle=\log(\lambda^{\prime}_{i})-\frac{1}{I}\sum_{i=1}^{I}\log(\lambda^{\prime}_{i})\quad\beta_{i}=\beta^{\prime}_{i}-\frac{1}{I}\sum_{i=1}^{I}\beta^{\prime}_{i},\quad i=1,\ldots,I.$ (8) In Table 1 we label this model as the _constrained item parameters model_. Unconstrained priors are then placed on the auxiliary parameters $\\{\lambda_{i}^{},\beta^{}_{i}\\}$. The same formulation applies under the slope-intercept parameterization, where the sum-to-zero constraints are placed on the pairs $\\{\log(\lambda_{i}),\gamma_{i}\\}$ for $i=1,\ldots,I$. Alternatively, we can consider _unconstrained_ versions of the 2PL model, treating the unconstrained model as a parameter-expanded version of the target inferential model in (3), where the redundant parameters are the means of the difficulty and log discrimination parameters. Hence we conduct MCMC sampling with known model unidentifiability, and before using the results for inference, we transform samples to follow the model in (3), as described in the Supplementary Materials, Section A. Finally, for the parametric case only, we consider the traditional version of the 2PL model that assumes the ability parameters $\eta_{j}$ for $j=1,\ldots,N$ follow a standard normal distribution (_constrained abilities model_). ### 3.1 Sampling algorithms We focus on general MCMC sampling algorithms available in easy-to-access software tools for Bayesian hierarchical models, such as NIMBLE and Stan, that can flexibly accommodate different choices of prior distributions and link functions. In this section we give an overall description of the algorithms considered for the different sampling strategies. We refer to the Supplementary Materials, Section B, for a detailed summary of the samplers used for each parameter. For both the parametric and semiparametric models we consider NIMBLE’s default sampling configuration (_MH/conjugate algorithm_). NIMBLE’s MCMC uses an overall one-at-a-time sampling strategy, cycling over individual parameters, or parameter blocks for parameters with a multivariate prior. By default, specific sampler types are assigned to the parameters or parameter blocks, but the user can choose to change sampler types, control blocking strategies, and modify details of sampling algorithm behavior. NIMBLE’s default MCMC configuration assigns a conjugate (sometimes called “Gibbs”) sampler where possible, sampling from the corresponding full conditional posterior distribution. For non-conjugate continuous-valued parameters, NIMBLE’s default sampler assignment is an adaptive random walk Metropolis-Hastings. For the parametric versions of the 2PL model, the strategies using the default NIMBLE assignments (_MH/conjugate algorithm_) correspond to these conjugate and adaptive random walk Metropolis-Hastings samplers, with the latter also used for most parametric components of the semiparametric 2PL. Specialized samplers are assigned when Bayesian nonparametric priors are considered in the semiparametric 2PL. In the case of the slope-intercept parameterization, we take advantage of NIMBLE’s flexibility to include user-programmed custom samplers (_centered sampler_). The proposed centered sampler uses an adaptive random walk Metropolis-Hastings sampler with a joint proposal for each pair of item parameters $\\{\lambda_{i},\gamma_{i}\\}$ for $i=1,\ldots,I$, thereby accounting for their posterior correlation. The proposal is made under a reparameterization of the model that centers the abilities to have mean zero. Implementation details are provided in the Supplementary Materials, Section B. Finally, in the parametric setting only, we consider a Hamiltonian Monte-Carlo (HMC) algorithm, as implemented in the Stan software (Carpenter ., 2017). Stan implements an adaptive HMC sampler (Betancourt ., 2017) based on the No-U-Turn sampler (NUTS) of Hoffman Gelman (2014). HMC algorithms are known to produce samples that are much less autocorrelated than those of other samplers but at more computational cost given the need to calculate the gradient of the log- posterior. In this work, we limit the comparison to the IRT parameterization with constraints on the abilities distribution, as that is the model provided in the edstan R package (Furr, 2017). ### 3.2 Aims In the remainder of the paper, we study the efficiency of the MCMC sampling strategies in Table 1 to fit binary logistic IRT models, and we compare inferential results under parametric and semiparametric specifications. Using both simulated and real-world data, we aim to answer the following questions: 1. Q.1 For the parametric binary logistic IRT model, which of the Metropolis- Hastings-based MCMC sampling strategies in Table 1 are most efficient? Do different strategies work better in different scenarios for the distribution of ability? 2. Q.2 For the parametric binary logistic IRT model, how does efficiency of random walk Metropolis-Hastings sampling compare to Hamiltonian Monte Carlo (HMC), as implemented in the popular Stan package? This question is of interest because HMC is not readily available for semiparametric models using Dirichlet process priors. 3. Q.3 How does MCMC efficiency of a semiparametric model compare to that of a parametric one? Does this comparison differ when the parametric model is correctly vs. incorrectly specified? 4. Q.4 To what degree does the use of a parametric model when its assumptions are violated yield bad inference? Does use of a semiparametric model change inference even when a parametric model would be valid? 5. Q.5 How much do results differ between the semiparametric and parametric models for the real data examples? In Section 6 we discuss the choice of efficiency metrics to address Q.1-Q.3 and present the results obtained using simulated and real-world data. In Section 7, we discuss differences in the inferential results to investigate Q.4-Q.5. ## 4 Choice of prior distributions Past research on Bayesian IRT models has warned about the use of either vague priors or highly informative priors when there is little information about the parameters (Sheng, 2010; Natesan ., 2016). In particular Natesan . (2016) investigated the use of different prior choices in 1PL and 2PL models using MCMC and variational Bayes algorithms and found that the use of vague priors tends to produce biased inference or convergence issues. Similarly, it is well known that highly informative prior distributions on parameters can strongly affect model comparison procedures. To ensure a fair comparison between results from different strategies, we chose the parameters of the priors in such a way that the induced prior predictive distribution of the data is similar across all the different model parameterizations. This “predictive matching approach” has been widely used to guide prior elicitation in model comparison settings (Berger Pericchi, 1996; Bedrick ., 1996; Ibrahim, 1997). In the context of binary logistic IRT models, we aim to match the prior marginal predictive distribution of a response $y_{ij}$, which in turn can be achieved by matching the induced prior distribution on the marginal prior probability of a correct response, $\pi_{ij}=\mbox{expit}\\{\lambda_{i}(\eta_{j}-\beta_{i})\\}$. Note that all the priors discussed in this paper are separately exchangeable, which means that this prior marginal will be the same for any values of $i$ and $j$. In particular, we attempt to match a $\mbox{Beta}(0.5,0.5)$ distribution, which is both the reference and the Jeffreys prior for the Bernoulli likelihood in the fully exchangeable case (Bernardo, 1979; Berger ., 2009). A similar approach to prior elicitation in the context of latent space models for networks can be found in Guhaniyogi Rodriguez (2020) and Sosa Rodrìguez (2021). Because there are no analytical expressions available for the prior distribution of $\pi_{ij}$, we use simulations to estimate the shape of the prior distribution and obtain an approximate match. This is facilitated by our implementation in NIMBLE. Indeed, one of the advantages of the NIMBLE system is that it provides a seamless way to simulate from the model of interest. Histograms of samples from the resulting induced priors can be seen in Figure 1 for a set of parametric and semiparametric models. Further details are presented in the following subsections. Figure 1: Histogram of samples from the induced prior on $\pi_{ij}$ under each of the considered models. Dashed line indicates the density function of a $\mbox{Beta}(0.5,0.5)$ distribution. Samples for the semiparametric models use a prior distribution $\mbox{Gamma}(2,4)$ for the DP concentration parameter $\alpha$, but similar results are obtained under the other settings presented Section 4. ### 4.1 Priors for the item parameters In Bayesian IRT modeling, normal distributions are typically chosen as priors for the item parameters. This is true under both parameterizations. In addition, the discrimination parameters, $\\{\lambda_{i}\\}_{i=1}^{I}$, are typically assumed positive, so we consider a normal distribution on the log- scale. To summarize, priors on the item parameters are: $\log{\lambda_{i}}\sim\mathcal{N}(\mu_{\lambda},\sigma^{2}_{\lambda}),\quad\beta_{i}\sim\mathcal{N}(0,\sigma^{2}_{\beta}),\quad\gamma_{i}\sim\mathcal{N}(0,\sigma^{2}_{\gamma})\quad i=1,\ldots,I.$ By default, we center on the difficulty parameters $\beta_{i}$ (or the reparameterized version $\gamma_{i}$) on 0 for $i=1,\ldots,I$, while we set $\sigma^{2}_{\beta}=\sigma^{2}_{\gamma}=3$. For the discrimination parameters, we set $\mu_{\lambda}=\sigma^{2}_{\lambda}=0.5$ such that the prior probability mass on the original scale is mostly in the range $(0.5,2.5)$. ### 4.2 Priors for the distribution of ability In choosing priors for the abilities, we distinguish between the parametric and semiparametric cases. In the parametric case, excluding the strategies in which the distribution is a standard normal, we assume $G\equiv\mathcal{N}(\mu_{\eta},\sigma^{2}_{\eta})$. We specify hyperpriors for the unknown mean and variance, using a normal distribution for the mean $\mu_{\eta}\sim\mathcal{N}(0,3)$, and an inverse-gamma distribution for the variance, $\sigma^{2}_{\eta}\sim\mbox{InvGamma}(2.01,1.01)$ as in Paulon . (2018), with hyperparameter values implying an a priori marginal expected value of $1$ and an a priori variance equal to $100$. In the semiparametric case, we need to specify the base distribution $G_{0}$ of the DP mixture prior along with the hyperparameters. We choose $G_{0}\equiv\mathcal{N}(0,\sigma^{2}_{0})\times\mbox{InvGamma}(\nu_{1},\nu_{2})$ where $\mbox{InvGamma}(\nu_{1},\nu_{2})$ denotes an inverse-gamma distribution with shape parameter $\nu_{1}$ and mean $\nu_{2}/(\nu_{1}-1)$. In choosing values for the hyperparameters $\\{\sigma^{2}_{0},\nu_{1},\nu_{2}\\}$, we first considered the concentration parameter $\alpha$ as fixed and evaluated the induced prior distribution on $\boldsymbol{\pi}$ for values of $\alpha\in\\{0.01,0.05,0.5,1,1.5,2\\}$. Recall that $\alpha$ controls the prior expectation and variance of the number of clusters induced by the DP, which are both of the order $\alpha\log(N)$. We discuss prior choice for the $\alpha$ in Section 4.3. As in the parametric case, we center the normal distribution for the mixture component means on $0$ with $\sigma_{0}^{2}=3$ and set $\nu_{1}=2.01$ and $\nu_{2}=1.01$ for the inverse-gamma distribution. Given these settings, we found that choosing $\alpha\in\\{0.01,0.05,0.5,1,1.5,2\\}$ does not have much effect on the marginal prior distribution of the $\pi_{ij}$s. ### 4.3 Prior on the DP concentration parameter One may be interested in placing a prior distribution on the concentration parameter $\alpha$ of the Dirichlet process. A typical choice for the DP concentration parameter is a $\mbox{Gamma}(a,b)$, with shape $a>0$ and scale $b>0$, due to its computational convenience (Escobar West, 1995). As previously stated, the concentration parameter controls the prior distribution of the number of clusters (Escobar West, 1995; SJ. Liu, 1996). In choosing values $a$ and $b$, we considered the implied prior mean and variance of the number of clusters. Let $K_{N}$ denote the number of clusters for a sample of size $N$. Results from Antoniak (1974) and SJ. Liu (1996) show that the expected value and variance of $K_{N}$ given $\alpha$ is $\mathbb{E}(K_{N}\mid\alpha)=\sum_{i=1}^{N}\frac{\alpha}{\alpha+N-i},\quad\mathbb{V}ar(K_{N}\mid\alpha)=\sum_{i=1}^{N}\frac{\alpha(i-1)}{(\alpha+N-i)^{2}}.$ (9) We exploit these results to choose values $a$ and $b$ that lead to reasonable a priori values for the moments of the number of clusters for each of our applications. For a given N and for different values of $a$ and $b$, we evaluated the marginal expectation and variance of the quantities in (9) via Monte Carlo approximation. We sample $\alpha_{r}$ for $r=1,\ldots,R$ from its prior and compute $\widehat{\mathbb{E}}(K_{N})=\frac{1}{R}\sum_{r=1}^{R}\mathbb{E}[K_{N}\mid\alpha_{r}],\quad\widehat{\mathbb{V}ar}(K_{N})=\frac{1}{R}\sum_{r=1}^{R}\mathbb{V}ar(K_{N}\mid\alpha_{r})+\widehat{\mathbb{V}ar}\left(\mathbb{E}[K_{N}\mid\alpha]\right),$ where $\widehat{\mathbb{V}ar}\left(\mathbb{E}[K_{N}\mid\alpha]\right)=R^{-1}\sum_{r=1}^{R}\left[\mathbb{E}[K_{N}\mid\alpha_{r}]-\widehat{\mathbb{E}}[K_{N}]\right]^{2}$. We explored a few prior choices and tabulate approximated moments in Table 2, for the values of $N$ in our datasets. We consider the popular choice of $a=2$, $b=4$ for the hyperparameters as in Escobar West (1995) along with values favoring a small number of clusters ($a=1,b=3$) and values leading to a more vague prior ($a=1,b=1$). For our applications we decided to favor a relatively small number of clusters, choosing $a=2,b=4$ as hyperparameters for the simulated data, and $a=1,b=3$ for the real-world data. $\alpha\sim Gamma(a,b)$ \| $\mathbb{E}[\alpha]$ \| $\mathbb{V}ar(\alpha)$ \| $\widehat{\mathbb{E}}(K_{2,000})$ \| $\widehat{\mathbb{V}ar}(K_{2,000})$ \| $\widehat{\mathbb{E}}(K_{14,525})$ \| $\widehat{\mathbb{V}ar}(K_{14,525})$ \| $\widehat{\mathbb{E}}(K_{7,377})$ \| $\widehat{\mathbb{V}ar}(K_{7,377})$ ---\|---\|---\|---\|---\|---\|---\|---\|--- $a=2,b=4$ \| $0.5$ \| $0.12$ \| $4.7$ \| $9.3$ \| $5.6$ \| $14.12$ \| $5.3$ \| $12.2$ $a=1,b=3$ \| $0.3$ \| $0.11$ \| $3.5$ \| $7.6$ \| $4.2$ \| $11.76$ \| $3.9$ \| $10.2$ $a=1,b=1$ \| $1$ \| $1$ \| $7.8$ \| $43.7$ \| $9.8$ \| $73.35$ \| $9.2$ \| $64.5$ Table 2: Approximate expectation and variance of the a priori number of clusters, $K_{N}$, under different choices of the concentration parameter distribution, for $N=\mbox{2,000, 14,525, 7,377}$ as in our data example presented in Section 5. ## 5 Data examples ### 5.1 Synthetic data We specify three different simulation scenarios for the distribution of ability: _unimodal_ , _bimodal_ and _multimodal_ distributions. For all the scenarios, we simulate responses from $N=2,000$ individuals on $I=15$ binary items. Values for the discrimination parameters $\\{\lambda_{i}\\}_{i=1}^{15}$ are sampled from a $\mbox{Uniform}(0.5,1.5)$ distribution, while values for difficulty parameters $\\{\beta_{i}\\}_{i=1}^{15}$ are taken to be equally spaced in $(-3,3)$. We center the log of the discrimination parameters on zero, while the difficulty parameters are already centered based on how they are generated. We consider three different underlying distributions for the latent abilities, $\eta_{j}$ for $j=1,\ldots,2,000$. In the _unimodal_ scenario, latent abilities are generated from a normal distribution with mean $0$ and variance $(1.25)^{2}$. In the _bimodal_ scenario, we use a equal- weights mixture of two normal distributions with means $\\{-2,2\\}$ and common variance $(1.25)^{2}$. Finally, for the _multimodal_ scenario, latent abilities are generated from the following mixture $\frac{1}{5}\mathcal{N}(-2,1)+\frac{2}{5}\mathcal{N}(0,(0.5)^{2})+\frac{2}{5}\mathcal{SN}(3,1,-3),$ where $\mathcal{SN}(\xi,\omega,\zeta)$ indicates a skew-normal distribution (Azzalini, 1985) with location parameter $\xi$, scale parameter $\omega>0$ and parameter $\zeta$ that controls the asymmetry of the distribution. As a sensitivity analysis, we considered other values of $N$ and $I$, simulating data for each scenario following a factorial design with $I\in\\{10,30\\}$ and $N\in\\{1,000,5,000\\}$. We discuss efficiency results of the different sampling strategies in Section 6 and report results in the Supplementary Materials, Section D. ### 5.2 Real world data The first example is a subset of data from the 1996 Health Survey for England (Joint Health Surveys Unit of Social and Community Planning Research and University College London, 2017), a survey conducted yearly to collect information concerning health and behavior of households in England. In particular, we have data for 10 items measuring Physical Functioning (PF-10), which is a sub-scale of the SF-36 Health Survey (Ware, 2003) administered to people aged $16$ and above. In this case the latent trait quantifies the physical status of a given individual (McHorney ., 1997; Hays ., 2000). Participants in the survey were asked whether they perceived limitations in a variety of physical activities (e.g., running, walking, lifting heavy objects) and if so the degree of limitation. We list the original questions in the the Supplementary Materials, Section C. Answers to items comprised three possible responses (“yes, a lot”, “yes, limited a little”, “no, not limited at all”); however, in our analysis we consider the dichotomous indicator for not being limited at all. The left panel of Figure 2 shows the distribution of raw scores, i.e. the total of correct answers. We consider the 2PL model for this data, as it reasonable to assume that some of the questions are more informative in defining individuals with high physical impairment (see Supplementary Materials, Section C). For simplicity, we analyzed complete case data from $14,525$ individuals out of $15,592$ respondents, although the model can easily accommodate missing data. The second example uses data from the TIMSS (Trends in International Mathematics and Science Study) survey, which is an international comparative educational survey dedicated to improving teaching and learning in mathematics and science for students around the world (http://timssandpirls.bc.edu/TIMSS2007/about.html). We used data from the 2007 eighth-grade mathematics assessment for the United States (N = $7,377$), publicly available at https://timssandpirls.bc.edu/TIMSS2007/idb_ug.html. The dataset comprises $214$ items, with $192$ of them dichotomous, while the remaining $22$ have three category responses (“incorrect”, “partially correct”, “correct”). We dichotomized these latter questions, considering partially correct answers as incorrect ones. Like other large-scale assessments, participants in TIMSS only received a subset of the items according to a booklet design, resulting in $28$-$32$ item responses per student. Distribution of the raw scores for the data are shown in the right panel of Figure 2. As for the previous example, it is reasonable to assume that some items discriminate differently between students with high and low ability. However, in the context of educational testing, the 3PL model is often considered because it accounts for the probability of answering correctly by chance. Hence we consider both the 2PL and the 3PL models when evaluating the different strategies. Figure 2: Distribution of the raw scores (total of correct answers) for the real data examples: health data (left panel) and TIMSS data (right panel). Finally, we note that participants in the survey were sampled according to a complex two-stage clustered sampling design that we did not consider in our application. In other contexts the design is typically taken into account using sampling weights for model estimation, as discussed for example by Rutkowski . (2010). ## 6 Comparing results in terms of efficiency MCMC performance is often evaluated in terms of mixing, often by calculating the effective sample size (ESS), which is the equivalent number of independent samples that would contain the same statistical information as the actual non- independent samples. However, comparison between different MCMC algorithms based solely on mixing can be misleading, as different samplers can vary greatly in terms of computational cost (Nguyen ., 2020). Hence, it is appropriate to consider ESS per computation time (in seconds), the rate at which effectively independent samples are generated. A second issue is how to combine ESS results for multiple parameters. For this purpose, we used the _multivariate ESS_ (mESS) recently introduced by Vats . (2019), which accounts for cross-correlations among parameters. Computation time is typically measured for the actual MCMC run, not accounting for steps to prepare for a run, thereby focusing on the algorithms of interest rather than unrelated aspects of the software. Comparison between HMC and MCMC algorithms raises the question of how to fairly account for computation times, given that these two classes of algorithms use different tuning phases. Since there is not an established way to compare these two algorithms in the literature, we decided to consider different timings when using the two algorithms: (i) _sampling time_ , which accounts only for the time to draw the posterior samples, hence discarding the time needed for the burn-in and warm- up phases of the two algorithms and (ii) _total time_ comprising also the burn-in and warm-up phases. Although one can use alternative metrics for the comparison, this choice can provide interesting and useful insights. When computing efficiency based on the sampling time, we can assess pure efficiency of sampling from the posterior. Using total time accounts for potentially different times needed for warm-up/burn-in by the different algorithms but introduces the difficulty of determining the optimal burn-in/warm-up time, which we avoided here in favor of using basic defaults. Estimate of the mESS is based on the multivariate batch means estimator as described in Vats . (2019) and implemented in the mcmcse (Flegal ., 2021) package. Since we used different specifications for the distribution of ability across the different sampling strategies, we calculate the mESS considering only the common parameters (i.e., the item parameters and sampled abilities) after transforming samples to the parameterization of our target inferential model (3). The mESS provides a single scalar measure of joint mixing for all the parameters of interest in a model, but it does not necessarily reflect univariate ESS values of each parameter. For example, mESS can be larger than all the univariate ESS values (see Supplementary Materials, Section D, Figures 16-18 for some insights). Given this, it can be useful as a simple overall performance metric but does not replace ESS for specific parameters of interest. Given the large number of examples and sampling strategies, we performed a preliminary experiment to choose the number of iterations and number of burn- in or warm-up samples. In particular, for a portion of the simulations we used multiple runs to determine the number of iterations and samples needed to obtain reliable estimates of the mESS. For all MCMCs using NIMBLE, we decided to use a total of $50,000$ iterations, with a $10\%$ burn-in of $5,000$ for all examples. When running the HMC algorithm via Stan, we used a total of $15,000$ iterations, with the first $5,000$ iterations as warm-up steps. However, we observed highly variable values of the mESS estimates when using the HMC algorithm. We also found that values of mESS are correlated with those of the HMC tuning parameters (see Supplementary Materials, Section D). Given this variability in mixing performance, we decided to limit comparisons with HMC to the simulation scenarios, reporting results from the run with the median ESS across multiple replications. All the models were estimated using a Linux cluster with 4 nodes having 24 cores and 128 GB RAM per node (Intel(R) Xeon(R) CPU E5-2643 v2 @ 3.50GHz). Across simulations, the running times ranged between 15-94 minutes for parametric models and between 37-124 minutes for semiparametric ones. For the data applications, running times ranged between 85-384 minutes for the parametric models, and 410-610 minutes for semiparametric ones. ### 6.1 Efficiency results for simulated data For the three simulation scenarios we estimated the 2PL parametric model using the different sampling strategies summarized in Table 1. Figure 3 compares efficiency for all these strategies using the multivariate ESS per second, computed with respect to both the total and sampling time. Figure 3: Multivariate ESS per second for various sampling strategies used to estimate the 2PL parametric model for the unimodal (left column), bimodal (middle column), and multimodal (right column) scenarios. Results are computed using total time (top row) and sampling time (bottom row). Note that SI stands for “slope-intercept”. Using different time baselines when computing efficiency changes the ranking of the MCMC algorithms for the unimodal and multimodal simulation, highlighting the trade-off for the HMC algorithm between sampling efficiency and computational cost. While the HMC is highly efficient in producing samples with low correlation, warm-up steps are computationally expensive. Recall that efficiency values presented for the HMC strategy are relative to a median performance across multiple runs. Amongst the non-HMC strategies, unconstrained scenarios generally mix well, as do scenarios with constraints on the abilities. However, imposing constraints on the item parameters directly in the sampling performs poorly because obtaining each sample is time-consuming. This is because the constraints in (8) require calculation of all the likelihood terms for each parameter update, whereas for other strategies only the likelihood terms for individuals’ responses on the item under consideration need to be calculated. While incorporating constraints on the item parameters is time-consuming, such a strategy could be useful in more complicated hierarchical models, in particular when it is unclear how to rescale posterior samples. The centered strategy for the slope-intercept parameterization seems to have little impact across the scenarios. Moving to the semiparametric models, recall that we did not consider identifiability constraints on the abilities. We either included identifiability constraints on the item parameters in the sampling or sampled from the unconstrained model and rescaled the posterior samples. Figure 4: Multivariate ESS per second (computed using the total time) for semiparametric 2PL models (bottom row) in comparison with their parametric version (top row) under the two simulation scenarios. Note that SI stands for “slope-intercept”. As expected when using a more complicated model, we observed some reduction in efficiency in the semiparametric model compared to the parametric model, but not a drastic one (Figure 4). Results for the semiparametric case are similar in relative terms, but not in absolute magnitudes, when comparing the different parameterizations and constraints. For the non-HMC strategies, we also looked at how different combinations of numbers of items and individuals affect efficiency of the different sampling strategies (as discussed in Section 5.1), with results shown in the Supplementary Materials, Section D. We found that the ranking across strategies is generally stable across the different scenarios; strategies using constraints on items are the worst overall, and the benefit of using other strategies is most evident when the number of individuals $N$ is low. ### 6.2 Efficiency results for real-world data We did similar comparisons using the real-data examples (Figure 5), noting that we excluded the constrained items strategy, given its poor performance on the simulated datasets, and the HMC strategy, because of the high variability in mixing performance. Figure 5: Multivariate ESS per second (computed using the total time) for the health data (left column) parametric and semiparametric 2PL models, and TIMSS data for the 2PL (middle column) and 3PL (right column) models. Note the scales are different for the different rows and that SI stands for “slope- intercept”. The efficiency is lower than for the simulated datasets because the real data have many more individuals or items, and therefore more parameters. The same consideration applies when comparing efficiency of the 2PL and 3PL models for the TIMSS data. We also noted that for the health data, when considering the posteriors for the abilities in the semiparametric model, we saw evidence for multimodality and some difficulty moving between modes for individuals with high raw scores. The multimodality is likely related to it being difficult for the semiparametric model to identify the exact magnitude of the ability for such individuals. The use of more informative priors, with careful elicitation of the prior distribution, may be important in such cases. ## 7 Comparing results in terms of statistical inference In this section we compare results for the parametric and semiparametric models in terms of statistical inference, regardless of the sampling strategy used to obtain posterior samples. All posterior samples follow the parameterization in (3), and we use samples from the most efficient sampling strategy for each dataset. We use posterior means as point estimates for the item and ability parameters. For the simulated datasets, we measure how well the models recover the (known) true value of the parameters using absolute error, e.g., $\|\hat{\beta_{i}}-\beta_{i}\|$, and squared error, e.g., $(\hat{\beta_{i}}-\beta_{i})^{2}$. A crucial point of this paper is to make inference on the distribution of latent abilities. An estimate of this distribution is sometimes based on the posterior means of the abilities (Duncan MacEachern, 2008; Bambirra Gonçalves ., 2018), and histograms or kernel density plots are reported. Such an estimate ignores uncertainty in the estimates of individual abilities. Instead, one should directly obtain the point estimate of the posterior distribution of the latent abilities $p(\eta\mid\mathbf{Y})$ (for any value of $\eta$) using the posterior samples. In the parametric case, this reduces to: $\widehat{p(\eta\mid\mathbf{Y})}=\frac{1}{T}\sum_{t=1}^{T}\mathcal{N}(\eta;\mu^{(t)},\sigma^{2(t)}),$ (10) with $\mathcal{N}(\cdot;\mu,\sigma^{2})$ indicating the probability density function of a normal distribution with mean $\mu$ and variance $\sigma^{2}$, and $t=1,\ldots,T$ denoting an MCMC iteration. In the semiparametric case, a point estimate of $p(\eta\mid\mathbf{Y})$ is the posterior mean of the mixing measure $G$ of the Dirichlet process. This can be obtained using posterior samples, averaging over the DP conditional distribution in (4) computed for each iteration $t=1,\ldots,T$, $p\widehat{(\eta\mid\mathbf{Y})}=\frac{1}{T}\sum_{t=1}^{T}\left\\{\left[\sum_{k=1}^{K^{(t)}}\frac{n_{k}^{(t)}}{\alpha^{(t)}+N}\mathcal{N}(\eta;\mu_{k}^{(t)},\sigma_{k}^{2^{(t)}})\right]+\frac{\alpha^{(t)}}{\alpha^{(t)}+N}\mathcal{N}(\eta;\mu_{K^{(t)}+1},\sigma_{K^{(t)}+1}^{2})\right\\},$ (11) with $n_{k}^{(t)}$ the number of observations in cluster $k$ at iteration $t$, $K^{(t)}$ the total number of clusters at iteration $t$, and $\mu_{K^{(t)}+1}$ and $\sigma_{K^{(t)}+1}^{2}$ sampled from $G_{0}$ (conditional on the data). We graphically compare estimates for the distribution of ability resulting from (10)–(11) with the estimates obtained using the posterior means. It is possible to make full inference on $p(\eta\mid\mathbf{Y})$ in the semiparametric setting; this requires sampling from the posterior of the mixing distribution $F$. A computational approach to obtain the entire posterior distribution has been presented in Gelfand Kottas (2002), a version of whose algorithm is implemented in NIMBLE in the function getSamplesDPMeasure. This function provides samples of a truncated version of the infinite mixture to a level $L$. The value of $L$ varies at each iteration of the MCMC’s output when $\alpha$ is random, while it is the same at each iteration when $\alpha$ is fixed. In our case, for every MCMC iteration, we can obtain samples of the vector of mixture weights $\\{w_{1}^{(t)},\ldots,w_{L^{(t)}}^{(t)}\\}$ and parameters of the mixture components. We can use these samples to make inference on functionals of the distribution, such as the percentile for an individual, $100\times p_{j}$, where $p_{j}=\int_{-\infty}^{\eta_{j}}p(\eta\mid\mathbf{Y})d\eta$, typically paired with test scores when giving results for educational assessments. For an individual $j$ for $j=1,\ldots,N$ we estimate $p_{j}$ at each MCMC iteration as $p_{j}^{(t)}=\sum_{l=1}^{L^{(t)}}w_{l}^{(t)}F_{\mathcal{N}}(\eta_{j}^{(t)};\mu_{l}^{(t)},\sigma_{l}^{2(t)}),$ (12) where $F_{\mathcal{N}}$ denotes the distribution function of the normal distribution. For comparison, we define the parametric counterpart as $p_{j}^{(t)}=F_{\mathcal{N}}(\eta_{j}^{(t)};\mu^{(t)},\sigma^{2(t)})$. \| Unimodal Simulation \| Bimodal Simulation \| Multimodal Simulation ---\|---\|---\|--- \| Parametric \| Semi-parametric \| Parametric \| Semi-parametric \| Parametric \| Semi-parametric \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE \| MAE \| MSE Difficulty parameters \| 0.0996 \| 0.0185 \| 0.0988 \| 0.0183 \| 0.0895 \| 0.0137 \| 0.0673 \| 0.0083 \| 0.0647 \| 0.0080 \| 0.0737 \| 0.0103 Discrimination parameters \| 0.0734 \| 0.0069 \| 0.0731 \| 0.0070 \| 0.0832 \| 0.0105 \| 0.0397 \| 0.0020 \| 0.0721 \| 0.0082 \| 0.0677 \| 0.0062 Ability parameters \| 0.4836 \| 0.3719 \| 0.4836 \| 0.3720 \| 0.5944 \| 0.5571 \| 0.5501 \| 0.4775 \| 0.5477 \| 0.4753 \| 0.5212 \| 0.4444 Table 3: MAE and MSE for the item and ability parameters estimates, under the three simulation scenarios, using samples from most efficient MCMC-based sampling strategies. ### 7.1 Inferential results for the simulated datasets We report results using posterior samples from the unconstrained sampling strategy under the IRT parameterization. Using the absolute error and the squared error for each parameter, we report in Table 3 the mean absolute error (MAE) and the mean squared error (MSE) across item and ability parameters. Figure 6: Unimodal simulated data. Comparison of the posterior mean estimates (with $95\%$ credible interval) of item parameters (difficulties $\boldsymbol{\beta}$ and discriminations $\boldsymbol{\lambda}$) for parametric and semiparametric 2PL models and true simulated values. Note that some estimates from the parametric model overlap almost exactly with semiparametric ones. In the unimodal scenario, we observe similar performance for the parametric and semiparametric 2PL in estimating item parameters (Figure 6). As expected, we observe some differences when considering the bimodal and multimodal scenario (Figure 7). The use of a model with a more flexible distribution improves recovery of both item and ability parameters in the bimodal scenarios (Table 3). This is especially evident when comparing estimates of the discrimination parameters, in particular for larger values. Instead, in the multimodal case the inference for the item parameters seems relatively insensitive to the specification. Figure 7: Bimodal Simulation (top row) and multimodal simulation (bottom row). Comparison of the posterior mean estimates (with $95\%$ credible interval) of item parameters (difficulties $\boldsymbol{\beta}$ and discriminations $\boldsymbol{\lambda}$) for parametric and semiparametric 2PL models and true simulated values. Results for the ability parameters are similar when estimating abilities using the posterior means of the individual abilities (Figure 8). However, results are very different when one looks at estimates of the distribution of ability (Figure 9). Figure 8: Histogram and density estimate of individual posterior mean abilities, under unimodal (left column), bimodal (middle column), and multimodal (right column) scenarios compared with the true density (dotted line). Figure 9: Distribution of ability estimated under the unimodal (left column), bimodal (middle column), and multimodal (right column) scenarios, compared with the true density (dotted line). Dashed lines indicate $95\%$ credible intervals for the estimated distributions. The normality assumption of the parametric model leads to unimodal density estimates, inconsistent with the true distribution, whereas the semiparametric model can recover it. The posterior means of individual abilities in Figure 8 are a compromise between the inferred distribution of ability and the information in the data, so with sufficient observations, one can obtain estimates of the distribution that are reasonable even with severe model mis- specification. In other words, for mis-specified parametric models, the in- sample predictions for observed individuals can be reasonable, while the out- of-sample predictions based on (10) for new individuals are poor. Note that when using the parametric model, inspection of the posterior means of individual abilities can be used to assess model mis-specification relative to the assumed parametric distribution. Mis-specification of the distribution of ability has limited effect when estimating individual percentiles. In Figure 10 we compare the posterior mean estimates of individual percentiles with the percentiles calculated using the true distribution assumed for the simulation, for a subset of $50$ individuals. Overall, the parametric and semiparametric estimates produce similar results even when the estimated density is largely different. This is because estimation of percentile is basically ranking the individuals; it makes sense that ranking is relatively insensitive to the estimation of the distribution of ability. Figure 10: Estimates of individual percentiles (with $95\%$ credible interval) for a subset of $50$ individuals with varying (true) ability levels under the unimodal (left column), bimodal (middle column), and multimodal (right column). Black dots correspond to true percentiles. ### 7.2 Inferential results for real-world data For the real data examples we graphically inspect results from the parametric and semiparametric models. To compare the overall model fit in the parametric and semiparametric cases, we computed the Widely Applicable Information Criterion (WAIC) (Watanabe Opper, 2010). We refer to the NIMBLE user manual for details about WAIC calculation (de Valpine ., 2022, see Section 7.7). We found that for the health data the semiparametric model performs better than the parametric one (WAIC of 68,527 versus 70,414), while for the TIMSS data the best model is the semiparametric 3PL (WAIC of 228,871; parametric 3PL: 229,123; semiparametric 2PL: 229,529; parametric 2PL: 229,575). Hence in this Section, we show inferential results using samples from the IRT unconstrained model for both the health and TIMSS data, using the 3PL model for TIMMS. We also report inferential results for the 2PL model in the Supplementary Materials (Section E). In Figure 11 we compare item parameter estimates from the two models for the health data application, while Figure 12 shows estimates for the distribution of abilities. Recall that, in this case, we interpret the latent ability as physical ability, with high values characterizing healthy individuals. As with the bimodal simulation, estimates from the parametric model of the distribution of physical ability are quite different than the distribution of individual posterior mean abilities. It is clear that the parametric model is badly mis-specified and would produce bad out-of-sample predictions. In contrast, the semiparametric model seems to nicely characterize multi-modality in the latent distribution. We observe in Figure 12 large credible intervals for high values of this distribution that can be explained by the presence of many individuals with high raw scores, (i.e., 9 or 10 out of 10, Figure 2) for whom the model can clearly determine that their physical abilities are high, but with the exact magnitudes being difficult to identify. The two modeling assumptions yield different estimates of the item parameters (Figure 11), with this difference being higher for extreme values. However, the relative ranking of the items is roughly the same in both cases, with for example item 1 (_Vigorous activities_) being the most difficult item and the one with lowest value of the discrimination parameter. According to the parametric model, discrimination parameters for item $3$ (_Lift/carry_) and item $10$ (_Bathing/dressing_) should have similar values, while the semiparametric model separates them. Figure 11: Health data. Comparison of item parameter estimates from the parametric and semiparametric models. In each panel items are ordered by increasing values of the parameter estimate under the semiparametric model. Figure 12: Health Data. Histogram and density estimate of the posterior means of the latent abilities (left panel) and estimate of the posterior distribution for the latent abilities (right panel). Dashed lines indicate $95\%$ credible intervals for the estimated distributions. For the TIMSS data we only compare point estimates of the item parameters (Figure 13), due to the large number of parameters. Estimates for the difficulty parameters and discrimination parameters have the largest difference for low/high values of the parameters, while estimates of the guessing parameters are quite different. Figure 14 shows estimates of the distribution of ability. In this case the semiparametric model estimate shows departure from the normal parametric assumption, with multimodality in the estimated distribution. Differences in the distribution of ability between the semiparametric and parametric model may explain the large discrepancies between the guessing parameter estimates; however, further investigation of this matter is outside the scope of the paper. We found also that the estimate of the ability distribution is quite different under the semiparametric 3PL and semiparametric 2PL model with the distribution under the 2PL model being unimodal and right-skewed instead of multimodal (see Supplementary Materials, Section F). Figure 13: TIMSS data. Comparison of posterior estimates of the item parameters between the parametric and semiparametric 3PL model both using the SI unconstrained centered sampling strategy. Figure 14: TIMSS Data. Histogram and density estimate of the posterior means of the latent abilities (left panel), and estimate of the posterior distribution for the latent abilities (right panel). Dashed lines indicate $95\%$ credible intervals for the estimated distributions. Figure 15: Estimates of individual percentiles (with $95\%$ credible interval) for a subset of $50$ individuals, for the health data (left panel) and TIMSS data (right panel). Figure 15 compares estimates of the percentiles for both the health and TIMSS data for a sample of $50$ individuals sorted according to the point estimates of the abilities from the semiparametric model. There are moderate differences in percentile values and individual ordering between the parametric and semiparametric models; in particular some estimates are associated with larger intervals than in the semiparametric case. ## 8 Discussion In this paper, we consider a semiparametric extension for binary logistic IRT models, using Dirichlet process mixtures as a nonparametric prior to flexibly characterize the distribution of ability. We provide an overview of these models and study how different sets of constraints can address identifiability issue and lead to different MCMC estimation strategies. Focusing on the 2PL and 3PL models, we compare efficiency and inferential results under different sampling strategies based on model parametrization, constraints and sampling algorithms. We find that MCMC performance across strategies can vary in relation to underlying shape of the latent distribution and the total number of parameters. When moving to semiparametric modeling, the computational cost can be high for large datasets, given that sampling from the Dirichlet process requires iteration through all individuals. However we find computational costs to be reasonable in our applications in light of the better inferential results. In particular under model mis-specification, inference for item parameters worsens noticeably in the parametric model compared to the semiparametric model. With sufficient data, inference for the abilities of observed individuals can be decent even under mis-specification of the distribution of ability, but inference for the unknown latent distribution (i.e., the predictive distribution for new individuals) as a whole can be quite bad. Although parametric IRT models can work well in applications in educational assessment, having access to semiparametric models can be broadly useful as it allows inference on the entire underlying distribution of ability and its functionals. This is particularly relevant in contexts where the distribution of the individual latent trait is more complicated, for example, when measuring health (Smits ., 2020) or psychological outcomes (Reise Rodriguez, 2016). Results of this work potentially can be generalized to versions of binary IRT models using different prior distributions or link functions (e.g. probit), since we considered general MCMC sampling algorithms as opposed to algorithms tailored to specific choices of such model components. As a general recommendation for IRT models, we found that sampling strategies using parameter expansion are more efficient than those embedding the constraints. In this work we extensively use the NIMBLE software for hierarchical modeling, with code reproducing results in the paper available at https://github.com/salleuska/IRT_nimble_code. Although there are other software solutions enabling Bayesian nonparametric modeling, these are often limited in the type of algorithms or in the class of models available. NIMBLE offers a high degree of flexibility in that the models considered in this paper could be easily embedded in more complicated ones. Sampler assignment can be highly customized by the user, including user-defined sampling algorithms. This customizability makes NIMBLE a powerful platform for comparing different sampling strategies. At the same time, NIMBLE allows easy sharing of the most successful strategies as block-box implementations for end users. ## Supplementary materials ## A. Identifiability The 2PL model is not identifiable based on the likelihood. Here we demonstrate the non-identifiability for the two parameterizations we consider, showing how different linear transformations lead to the same probabilities. Note that these transformations are defined for every parameter associated with each item $i=1,\ldots,I$ and individual $j=1,\ldots,N$. Under the IRT parameterization: 1. 1. $\eta_{j}^{\prime}=\eta_{j}/s$ and $\lambda_{i}^{\prime}=s\lambda_{i}$ $\lambda_{i}^{\prime}(\eta_{j}^{\prime}-\beta_{i})=s\lambda_{i}(\eta_{j}/s-\beta_{i})=\lambda_{i}\eta_{j}-\lambda_{i}\beta_{i}=\lambda_{i}(\eta_{j}-\beta_{i}),$ 2. 2. $\eta_{j}^{\prime}=\eta_{j}+c$ and $\beta_{i}^{\prime}=\beta_{i}+c$, $\lambda_{i}(\eta_{j}^{\prime}-\beta_{i}^{\prime})=\lambda_{i}(\eta_{j}+c-(\beta_{i}+c))=\lambda_{i}(\eta_{j}-\beta_{i}).$ Under the slope-intercept parameterization: 1. 1. $\eta_{j}^{\prime}=\eta_{j}/s$ and $\lambda_{i}^{\prime}=s\lambda_{i}$, $\lambda_{i}^{\prime}\eta_{j}^{\prime}+\gamma_{i}=s\lambda_{i}\eta_{j}/s+\gamma_{i}=\lambda_{i}\eta_{j}+\gamma_{i},$ 2. 2. $(\lambda_{i}\eta_{j})^{\prime}=\lambda_{i}\eta_{j}+c$ and $\gamma_{i}^{\prime}=\gamma_{i}-c$, or $\eta_{j}^{\prime}=\eta_{j}+c$ and $\gamma_{i}^{\prime}=\gamma_{i}-\lambda_{i}c$ $\lambda_{i}\eta_{j}^{\prime}+\gamma_{i}^{\prime}=\lambda_{i}(\eta_{j}+c)+\gamma_{i}-\lambda_{i}c=\lambda_{i}\eta_{j}+\gamma_{i}.$ ### Post-processing to satisfy identifiability constraints This section reports the transformations we apply to item and ability parameters in order to satisfy the identifiability constraints in our base parameterization (3). These transformations are applied to each posterior sample. Under the IRT parameterization, the set of transformations for each posterior sample of $\\{\lambda_{i},\beta_{i},\eta_{j}\\}$ for $i=1,\ldots,I,j=1,\ldots,N$ takes these forms: $\displaystyle\lambda^{}_{i}=s\lambda_{i},\quad\beta^{}_{i}=\frac{\beta_{i}-b}{s},\quad\eta^{}_{j}=\frac{\eta_{j}-b}{s},$ subject to $\prod_{i=1}^{I}\lambda^{}_{i}=1$, $\sum_{i=1}^{I}\beta^{}_{i}=0$. By solving the system of equations given by the transformations and the set of identifiability constraints, we obtain $\displaystyle s=\exp\left\\{\sum_{i=1}^{I}\log(\lambda_{i})/I\right\\},\quad b=\frac{\sum_{i=1}^{I}\beta_{i}}{I}.$ (A1) Under the slope-intercept parameterization, the set of transformations for each posterior sample of $\\{\lambda_{i},\gamma_{i},\eta_{j}\\}$ for $i=1,\ldots,I,j=1,\ldots,N$ takes these forms: $\displaystyle\tilde{\lambda}_{i}=s\lambda_{i},\quad\tilde{\gamma}_{i}=\gamma_{i}-\lambda_{i}c,\quad\tilde{\eta}_{j}=\frac{\eta_{j}+c}{s},$ subject to $\prod_{i=1}^{I}\tilde{\lambda}_{i}=1$, $\sum_{i=1}^{I}\tilde{\gamma}_{i}=0$. Similarly, by solving the system of equations given by the transformations and the set of identifiability constraints, we obtain $\displaystyle s=\exp\left\\{\sum_{i=1}^{I}\log(\lambda_{i})/I\right\\},\quad c=\frac{\sum_{i=1}^{I}\gamma_{i}}{\sum_{i=1}^{I}\lambda_{i}}.$ (A2) Finally, to get from the slope-intercept parameterization to the IRT parameterization, we define $\tilde{\beta}_{i}\vcentcolon=-\tilde{\gamma}_{i}/\tilde{\lambda}_{i}$ and then calculate $\beta_{i}^{}=\tilde{\beta}_{i}-\sum_{i}\tilde{\beta}_{i}/I$. ### Rescaling the DP density We can obtained posterior samples from the mixing distribution $F$ via NIMBLE’s getSamplesDPmeasure function, allowing us to estimate the density for the latent ability distribution. However, when comparing these estimated densities between models, for some of the sampling strategies, we need to transform the estimated density to account for the transformations of the abilities from the scale on which sampling is done to the scale in (3). As an example, consider the IRT parameterization without constraints. From the MCMC output we can obtain $p(\tilde{\eta})$ evaluated for different values of $\tilde{\eta}$, but we want $p(\tilde{\eta}^{})$ with $\tilde{\eta}^{}=(\tilde{\eta}-b)/s$. To do so we need the Jacobian of the transformation, which is simply $s$. Then, we obtain $p(\tilde{\eta}^{})$ $p(\tilde{\eta}^{})=p_{\tilde{\eta}}(s\tilde{\eta}^{}+b)\left\|\frac{\partial(s\tilde{\eta}^{}+b)}{\partial\tilde{\eta}^{}}\right\|=p_{\tilde{\eta}}(s\tilde{\eta}^{}+b)s.$ (A3) ## B. Details about the sampling algorithms Tables 4-5 summarize the sampling algorithms used for each sampling strategy in Table 1. Table 4: Summary of the sampling algorithms used for each parameter under the sampling strategies considered for the parametric 2PL model. Model constraints IRT parameterization SI parameterization MH/conjugate HMC (Stan) MH/conjugate Centered Constrained abilities Adaptive MH $\\{\log(\lambda_{i}),\beta_{i},\eta_{j}\\}$ HMC $\left\\{\\{\log(\lambda_{i})\\},\\{\beta_{i}\\},\\{\eta_{j}\\}\right\\}$ Adaptive MH $\\{\log(\lambda_{i}),\gamma_{i},\eta_{j}\\}$ Centered sampler for pairs $\\{\log(\lambda_{i}),\gamma_{i}\\}$ Adaptive MH $\\{\eta_{j}\\}$ Constrained item Adaptive MH $\\{\log(\lambda^{}_{i}),\beta^{}_{i},\eta_{j}\\}$ - Adaptive MH $\\{\log(\lambda^{}_{i}),\gamma^{}_{i},\eta_{j}\\}$ - Unconstrained Adaptive MH $\\{\log(\lambda_{i}),\beta_{i},\eta_{j}\\}$ Conjugate $\\{\mu,\sigma^{2}\\}$ - Adaptive MH $\\{\log(\lambda_{i}),\gamma_{i},\eta_{j}\\}$ Conjugate $\\{\mu,\sigma^{2}\\}$ Centered sampler for pairs $\\{\log(\lambda_{i}),\gamma_{i}\\}$ Adaptive MH $\\{\eta_{j}\\}$ Conjugate $\\{\mu,\sigma^{2}\\}$ Table 5: Summary of the sampling algorithms used for each parameter under the sampling strategies considered for the semiparametric 2PL model. Model constraints IRT parameterization SI parameterization MH/conjugate MH/conjugate Centered Constrained item Adaptive MH $\\{\log(\lambda^{}_{i}),\beta^{}_{i},\eta_{j}\\}$ CRP sampler $\\{\alpha,\\{z_{j}\\}\\}$ Conjugate $\\{\\{\mu^{}_{k}\\},\\{\sigma^{2}_{k}\\}\\}$ Adaptive MH $\\{\log(\lambda^{}_{i}),\gamma^{}_{i},\eta_{j}\\}$ CRP sampler $\\{\alpha,\\{z_{j}\\}\\}$ Conjugate $\\{\\{\mu^{}_{k}\\},\\{\sigma^{2}_{k}\\}\\}$ - Unconstrained Adaptive MH $\\{\log(\lambda_{i}),\beta_{i},\eta_{j}\\}$ CRP sampler $\\{\alpha,\\{z_{j}\\}\\}$ Conjugate $\\{\\{\mu^{}_{k}\\},\\{\sigma^{2}_{k}\\}\\}$ Adaptive MH $\\{\log(\lambda_{i}),\gamma_{i},\eta_{j}\\}$ CRP sampler $\\{\alpha,\\{z_{j}\\}\\}$ Conjugate $\\{\\{\mu^{}_{k}\\},\\{\sigma^{2}_{k}\\}\\}$ Centered sampler for pairs $\\{\log(\lambda_{i}),\gamma_{i}\\}$ Adaptive MH $\\{\eta_{j}\\}$ CRP sampler $\\{\alpha,\\{z_{j}\\}\\}$ Conjugate $\\{\\{\mu^{}_{k}\\},\\{\sigma^{2}_{k}\\}\\}$ #### Description of the sampling algorithms * • Conjugate sampler: for the models used in the paper, we exploit conjugancy results for the Normal distribution with Normal Inverse-gamma priors for the mean and variance. * • Adaptive MH (Metropolis-Hastings): uses a normal proposal distribution, with initial proposal variance equal to $1$ and and adaptation interval of $200$ iterations. The adaptation routine is implemented as given in Shaby Wells (2010). * • Centered sampler: this is a custom defined sampler implemented by the authors. Details are given in the below. * • CRP sampler: under the CRP specification, the random measure $G$ is integrated out from the model and NIMBLE assigns a collapsed sampler. * – clustering indicators $\mathbf{z}$ are updated as in described in Neal (2000); * – the DP concentration parameter $\alpha$ is sampled as described in Escobar West (1995) (Section 6) when a Gamma prior is used, as in the models considered in the paper. If another prior is considered, NIMBLE uses a random walk Metropolis-Hastings. ### Centered sampler We consider a custom sampler for the 2PL model under the slope-intercept parameterization. Intuition for this sampling strategy comes from the resemblance to a linear model. In order to sample $\\{\lambda_{i},\gamma_{i}\\}$ efficiently, we propose centering the implied covariate, $\eta_{j}$, to have mean zero. This is analogous to centering covariates in a linear model, but in this case the ”covariate” values are not fixed, so the centering needs to be done in each iteration. For a given item $i$ for $i=1,\ldots,I$ we can rewrite $\displaystyle\lambda_{i}\eta_{j}+\gamma_{i}$ $\displaystyle=\lambda_{i}(\eta_{j}-\bar{\eta})+\lambda_{i}\bar{\eta}+\gamma_{i},$ $\displaystyle=\lambda_{i}\eta_{j}^{c}+\gamma_{i}^{c},$ such that the quantity $\eta_{j}^{c}=\eta_{j}-\bar{\eta}$ is centered. The idea is to propose a new value $\lambda_{i}^{}$ in this new parameterization at each MCMC iteration, using a random walk on the log scale. Translating to the original parameterization, we have: $\displaystyle\lambda_{i}^{}\eta_{j}^{c}+\gamma_{i}^{c}$ $\displaystyle=\lambda_{i}^{}(\eta_{j}-\bar{\eta})+\lambda_{i}\bar{\eta}+\gamma_{i},$ $\displaystyle=\lambda_{i}^{}\eta_{j}-\lambda_{i}^{}\bar{\eta}+\lambda_{i}\bar{\eta}+\gamma_{i}.$ This means that we are proposing $\gamma_{i}^{}=\gamma_{i}+\bar{\eta}(\lambda_{i}-\lambda_{i}^{})$. Thus we have a joint proposal $(\lambda_{i}^{},\gamma_{i}^{})$ that accounts for the usual correlation in a regression between intercept and slope. Apart from accounting for sampling $\lambda_{i}$ on the log scale, the proposal is symmetric, so no Hastings correction is needed. The original sampler for $\gamma_{i}$ can stay the same. This is because in the reparameterization with $\gamma_{i}^{c}$ above, shifting $\gamma_{i}$ by a certain amount is equivalent to shifting $\gamma_{i}^{c}$. ## C. Health data questions The following items are about activities you might do during a typical day. Does your health now limit you in these activities? If so, how much? 1. 1. Vigorous activities: Vigorous activities, such as running, lifting heavy objects, participating in strenuous sports. 2. 2. Moderate activities: Moderate activities, such as moving a table, pushing a vacuum cleaner, bowling or playing golf. 3. 3. Lift/Carry: Lifting or carrying groceries. 4. 4. Several stairs: Climbing several flights of stairs. 5. 5. One flight stairs: Climbing one flight of stairs. 6. 6. Bend/Kneel/Stoop: Bending, kneeling, or stooping. 7. 7. Walk more mile: Walking more than a mile. 8. 8. Walk several blocks: Walking several blocks. 9. 9. Walk one block: Walking one block. 10. 10. Bathing/Dressing: Bathing or dressing yourself. ## D. A note on efficiency comparisons ### Comments on the multivariate ESS In this section we compare univariate and multivariate efficiency metrics. In particular, we compare efficiency values based on the mESS (multivariate efficiency), with the distribution of efficiencies calculated for each parameter (univariate efficiency) using the total time. We report these metrics for each simulation scenario, selecting three representative strategies under the IRT parameterization for the parametric 2PL model. Figures 16-18 show the distribution of univariate efficiency for difficulty, discrimination and ability parameters, along with a table reporting information for the multivariate efficiency. There are some differences between the multivariate and univariate efficiency results. This is expected because the mESS provides a single scalar measure of mixing performance that accounts for cross-correlation among the parameters and does not necessarily reflect the distribution of univariate ESSs. In fact, values of the mESS can be larger than all the univariate ESSs. For example, the IRT HMC strategy has larger univariate ESS values compared to other sampling strategies, but lower mESS. Strategy \| mESS \| total time (second) \| efficiency (mESS/second) ---\|---\|---\|--- IRT HMC (Stan)* \| 31802 \| 1405 \| 23 IRT unconstrained \| 27540 \| 934 \| 29 IRT constrained item \| 27904 \| 7318 \| 4 Figure 16: Unimodal simulation ($N=2000,I=15)$. Distribution of univariate efficiencies (univariate ESS/seconds) for each group of parameters used to compute the multivariate efficiency (mESS/second) using the total time. The symbol denotes median results across 11 runs. Strategy \| mESS \| total time (second) \| efficiency (mESS/second) ---\|---\|---\|--- IRT HMC (Stan) \| 30473 \| 2191 \| 14 IRT unconstrained \| 27167 \| 937 \| 29 IRT constrained item \| 27463 \| 4508 \| 6 Figure 17: Bimodal simulation ($N=2000,I=15)$. Distribution of univariate efficiencies (univariate ESS/seconds) for each group of parameters used to compute the multivariate efficiency (mESS/second) using the total time. The symbol denotes median results across 11 runs. Strategy \| mESS \| total time (second) \| efficiency (mESS/second) ---\|---\|---\|--- IRT HMC (Stan) \| 22698 \| 1176 \| 19 IRT unconstrained \| 27442 \| 1407 \| 19 IRT constrained item \| 27756 \| 5248 \| 5 Figure 18: Multimodal simulation ($N=2000,I=15)$. Distribution of univariate efficiencies (univariate ESS/seconds) for each group of parameters used to compute the multivariate efficiency (mESS/second) using the total time. The symbol ** denotes median results across 11 runs. ### Variability of mESS when using HMC from the Stan software When deciding on the number of posterior , burn-in and warm-up samples, we tried to obtain a reliable estimate of the multivariate ESS. To ensure the chains were long enough, we used multiple runs for some of the experiments. We found that mESS estimates based on the chosen settings of the MCMC algorithm (i.e., number of iterations, number of burn-in or warm-up samples) have negligible variability across multiple runs for all strategies with the exception of using HMC as implemented in Stan. As an example, Figure 19 shows the distribution of multivariate ESS for the IRT constrained abilities approach. Figure 19: Comparison of mESS estimates across multiple runs for the IRT constrained abilities approach using the unimodal simulation scenario. We also found that lower and higher values of ESS for the strategies using the HMC are highly correlated with values of the tuning parameters of the HMC (i.e., leapfrog and step-size parameters), which are typically estimated during the warm-up phase (see Figure 20). Figure 20: Estimates of the mESS versus average values of the HMC tuning parameters (step-size and leapfrog) across post warm-up iterations. ## E. Results from additional simulations We investigated how different combinations of numbers of items and individuals affect efficiency of the different sampling strategies. In particular, we simulated data under the three scenarios presented in Section 5.1 following a factorial design with $I\in\\{10,30\\}$ and individuals $N\in\\{1,000,5,000\\}$. We omit results for the strategy using HMC due to the high variability in estimating the mESS. Figure 21: Multivariate ESS per second of the parametric sampling strategies (excluding HMC) across different combinations of numbers of items and individuals. Figure 22: Multivariate ESS per second of the semiparametric sampling strategies (excluding HMC) across different combinations of numbers of items and individuals. ## F. Inferential results for the TIMSS data using the 2PL model Figure 23: TIMSS data, 2PL model. 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# The number of prime factors of integers with dense divisors Andreas Weingartner Department of Mathematics, 351 West University Boulevard, Southern Utah University, Cedar City, Utah 84720, USA<EMAIL_ADDRESS> (Date: November 11, 2021) ###### Abstract. We show that for integers $n$, whose ratios of consecutive divisors are bounded above by an arbitrary constant, the normal order of the number of prime factors is $C\log\log n$, where $C=(1-e^{-\gamma})^{-1}=2.280...$ and $\gamma$ is Euler’s constant. We explore several applications and resolve a conjecture of Margenstern about practical numbers. ###### 2010 Mathematics Subject Classification: 11N25, 11N37 ## 1\. Introduction We say that a positive integer $n$ is $t$-dense if the ratios of consecutive divisors of $n$ do not exceed $t$. Let $\mathcal{D}(x,t)$ denote the set of $t$-dense integers $n\leq x$ and write $D(x,t)=\|\mathcal{D}(x,t)\|$. Let $\omega(n)$ (resp. $\Omega(n)$) be the number of prime factors of $n$, counted without (resp. with) multiplicity. Theorem 1 gives the average and normal order of $\omega(n)$ and $\Omega(n)$ for the $t$-dense integers. We write $\log_{2}x$ for $\log\log x$ and define $E(x,t)$, the approximate expected value of $\omega(n)$ on $\mathcal{D}(x,t)$, by $E(x,t):=C\log_{2}x-(C-1)\log_{2}t,\quad C:=(1-e^{-\gamma})^{-1}=2.280291...$ ###### Theorem 1. Let $\xi(x)\to\infty$. Uniformly for $x\geq t\geq 2$, $\frac{\sum_{n\in\mathcal{D}(x,t)}\omega(n)}{D(x,t)}=E(x,t)+O\left(1\right)$ (1) and $\Bigl{\|}\left\\{n\in\mathcal{D}(x,t):\|\omega(n)-E(x,t)\|>\xi(x)\sqrt{\log_{2}x}\right\\}\Bigr{\|}\ll\frac{D(x,t)}{\xi(x)^{2}}.$ (2) These results also hold with $\Omega$ in place of $\omega$. Note that $\log_{2}x\leq E(x,t)\leq C\log_{2}x+O(1)$, as $x\geq t\geq 2$. If $t=x$, then $E(x,t)=\log_{2}x$ and $\mathcal{D}(x,t)=[1,x]\cap\mathbb{N},$ so Theorem 1 contains the well-known results about the average and normal order of $\omega(n)$ on $\mathbb{N}$. If $t\geq 2$ is constant, then $E(x,t)=C\log_{2}x+O(1)$, so that the average and normal order of $\omega(n)$ for $t$-dense integers $n$ is $C\log_{2}n$. The $t$-dense integers are a special case of a family of integer sequences that arise as follows. Let $\theta$ be an arithmetic function. Let $\mathcal{B}=\mathcal{B}_{\theta}$ be the set of positive integers containing $n=1$ and all those $n\geq 2$ with prime factorization $n=p_{1}^{\alpha_{1}}\cdots p_{k}^{\alpha_{k}}$, $p_{1}<p_{2}<\ldots<p_{k}$, which satisfy $p_{i}\leq\theta\big{(}p_{1}^{\alpha_{1}}\cdots p_{i-1}^{\alpha_{i-1}}\big{)}\qquad(1\leq i\leq k).$ (3) We write $\mathcal{B}(x)=\mathcal{B}\cap[1,x]$ and $B(x)=\|\mathcal{B}(x)\|$. When $\theta(n)=nt$, then $\mathcal{B}$ is the set of $t$-dense integers [6, 10, 14]. If $\theta(n)=\sigma(n)+1$, where $\sigma(n)$ is the sum of the positive divisors of $n$, then $\mathcal{B}$ is the set of practical numbers [2, 6, 7, 8, 9, 10, 14], i.e. integers $n$ such that every $m\leq n$ can be expressed as a sum of distinct positive divisors of $n$. We will derive Corollaries 1 through 3 from Theorem 1 in Section 2. ###### Corollary 1. Assume $\theta$ satisfies $\max(2,n)\leq\theta(n)\leq nf(n)$, where $f$ is non-decreasing. Let $\xi(x)\to\infty$. We have $\frac{\sum_{n\in\mathcal{B}(x)}\omega(n)}{B(x)}=C\log_{2}x\left\\{1+O\left(\left(\frac{\log f(x)}{\log_{2}x}\right)^{1/3}\right)\right\\}$ (4) and $\Bigl{\|}\bigl{\\{}n\in\mathcal{B}(x):\|\omega(n)-C\log_{2}x\|>\xi(x)\sqrt{\log_{2}x}\bigr{\\}}\Bigr{\|}\ll B(x)\frac{\log f(x)}{\xi(x)^{2}}.$ (5) These results also hold with $\Omega$ in place of $\omega$. Conjecture 5 of Margenstern [2] proposes that, for practical numbers, $\sum_{n\in\mathcal{B}(x)}\omega(n)\sim\mu x/(\log x)^{\eta}$, for some constants $\mu>0$ and $1/2<\eta<1$. The estimate (4) disproves this conjecture, since $B(x)\sim c_{\theta}x/\log x$ by [14, Thm. 1.2], if $\max(2,n)\leq\theta(n)\ll n(\log 2n)/(\log_{2}3n)^{1+\varepsilon}$ for $n\geq 1$. Corollary 2 shows that almost all large practical numbers $n$ have about $C\log_{2}n$ prime factors. ###### Corollary 2. If $\theta$ satisfies $\max(2,n)\leq\theta(n)\ll n(\log 2n)^{o(1)}$, then the average and normal order of $\omega(n)$ on $\mathcal{B}$ is $C\log_{2}n$. That is, as $x\to\infty$, $\sum_{n\in\mathcal{B}(x)}\omega(n)\sim\sum_{n\in\mathcal{B}(x)}C\log_{2}n$ (6) and all but $o(B(x))$ integers $n\in\mathcal{B}(x)$ satisfy $\omega(n)=(1+o(1))C\log_{2}n.$ (7) These results also hold with $\Omega$ in place of $\omega$. Let $\tau(n)$ be the number of positive divisors of $n$. ###### Corollary 3. If $\theta$ satisfies $\max(2,n)\leq\theta(n)\ll n(\log 2n)^{o(1)}$, then $(\log x)^{C\log 2-o(1)}<\frac{\sum_{n\in\mathcal{B}(x)}\tau(n)}{B(x)}\ll(\log x)^{e\log 2},$ (8) as $x\to\infty$, and all but $o(B(x))$ integers $n\in\mathcal{B}(x)$ satisfy $\tau(n)=(\log n)^{C\log 2+o(1)}=(\log n)^{1.580577...+o(1)}.$ (9) Conjecture 4 of Margenstern [2] says that, in the case of practical numbers, $\sum_{n\in\mathcal{B}(x)}\tau(n)\sim\nu x(\log x)^{\delta}$, for constants $1/2<\nu,\,\delta<1$. The estimate (8) implies $\delta\in[C\log 2-1,e\log 2-1]=[0.580...,0.884...]$, since $B(x)\asymp x/\log x$. In [17], we prove this conjecture with $\delta=0.713...$ and some constant $\nu>0$. The next two corollaries are improvements to the lower bounds of Theorems 1 and 3 of [5]. The proofs of both of these theorems rely on the fact that almost all $n\in\mathcal{B}$ satisfy $\Omega(n)<(e+\varepsilon)\log_{2}n$. With Corollary 2, this can be improved to $\Omega(n)<(C+\varepsilon)\log_{2}n$, under the assumption $\theta(n)\ll n(\log n)^{o(1)}$. In the lower bound of [5, Thm. 1] for the count of practical numbers that are also shifted primes, which has an exponent of $(e+1)\log(e+1)-e\log(e)+1+\varepsilon=3.16470...$, we can replace $e$ by $C$ to get $3.01711...$. ###### Corollary 4. Fix a nonzero integer $h$ and assume $\theta$ satisfies $\max(2,n)\leq\theta(n)\ll n(\log 2n)^{o(1)},\quad\theta(mn)\ll m^{O(1)}\theta(n)\quad(n,m\in\mathbb{N}).$ (10) We have $\frac{x}{(\log x)^{3.01712}}\ll_{h}\bigl{\|}\\{p\leq x:p\mbox{ prime},\ p-h\in\mathcal{B}\\}\bigr{\|}\ll_{h}\frac{x}{(\log x)^{2}},$ where $h$ is not divisible by $\prod_{p\leq\theta(1)}p$ in the lower bound. Similarly, in the lower bound of [5, Thm. 3] for the count of twin practical numbers, which has an exponent of $2+4e\log 2+\varepsilon=9.53667...$, we can replace $e$ by $C$ to get $8.32230...$ ###### Corollary 5. Fix a nonzero integer $h$ and assume $\theta$ satisfies (10). We have $\frac{x}{(\log x)^{8.32231}}\ll_{h}\bigl{\|}\\{n\leq x:n\in\mathcal{B},n+h\in\mathcal{B}\\}\bigr{\|}\ll_{h}\frac{x}{(\log x)^{2}}.$ For the lower bound, assume that (i) $n\in\mathcal{B}$ and $m\leq 3n/\|h\|$ imply $mn\in\mathcal{B}$, and (ii) if $\theta(1)<3$, then $h\in 2\mathbb{Z}$ if $\theta(2)\geq 3$, and $h\in 4\mathbb{Z}$ if $\theta(2)<3$. Conditions (i) and (ii) in Corollary 5 are satisfied by the practical numbers and by the $2$-dense integers for any nonzero even integer $h$, and by the $t$-dense integers for any nonzero integer $h$, provided $t\geq 3$. The $\varphi$-practical numbers [4, 12] are integers $n$ such that $x^{n}-1$ has a divisor in $\mathbb{Z}[x]$ of every degree up to $n$. Although not an example of a set $\mathcal{B}_{\theta}$, they are a superset of $\mathcal{B}_{\theta_{1}}$ with $\theta_{1}(n)=n+1$, and a subset of $\mathcal{B}_{\theta_{2}}$ with $\theta_{2}(n)=n+2$. Therefore, Corollaries 1 through 5 also apply to the $\varphi$-practical numbers, provided $h$ is odd in the lower bound of Corollary 4, while $h$ is even in the lower bound of Corollary 5. Theorem 1 is a consequence of Theorems 2 and 3. Theorem 2 gives an estimate for $D_{q}(x)=D_{q}(x,t):=\|\\{n\in\mathcal{D}(x,t):q\|n\\}\|,$ when $q$ has a bounded number of prime factors. As in the case $q=1$ (see [14, Thm. 1.3]), the main term contains the function $d(v)$, which is defined by $d(v)=0$ for $v<0$ and $d(v)=1-\int_{0}^{\frac{v-1}{2}}\frac{d(u)}{u+1}\ w\left(\frac{v-u}{u+1}\right)du\qquad(v\geq 0),$ (11) where $w(u)$ denotes Buchstab’s function. ###### Theorem 2. Let $k\in\mathbb{N}\cup\\{0\\}$ be fixed. Uniformly for $x\geq 1$, $t\geq 2$, $q\in\mathbb{N}$ with $\Omega(q)=k$, $v=\log x/\log t$, we have $\displaystyle D_{q}(x,t)$ $\displaystyle=xd(v)\eta_{q,t}\left\\{1+O_{k}\left(\frac{1}{\log xt}+\frac{\log 2q\log qt}{\log^{2}xt}\right)\right\\}+O(1),$ (12) $\displaystyle D_{q}(x,t)$ $\displaystyle=xd(v)\eta_{q,t}\left\\{1+O_{k}\left(\frac{\log 2q}{\log 2x}\right)\right\\},$ (13) where $q^{-1}\ll\eta_{q,t}\ll_{k}q^{-1}.$ (14) ###### Corollary 6. Let $k\in\mathbb{N}\cup\\{0\\}$ be fixed. Uniformly for $x\geq 1$, $t\geq 2$, $q\in\mathbb{N}$ with $\Omega(q)=k$, we have $D_{q}(x,t)=\frac{c_{q}x}{\log xt}\left\\{1+O_{k}\left(\frac{1}{\log xt}+\frac{\log^{2}qt}{\log^{2}xt}\right)\right\\}+O(1),$ (15) where $c_{q}=c_{q,t}=C\eta_{q,t}\log t,\quad q^{-1}\log t\ll c_{q}\ll_{k}q^{-1}\log t,$ (16) and $D_{q}(x,t)\ll_{k}\frac{x\log t}{q\log xt}.$ (17) The estimates (15) and (16) follow from (12) and (14), since $d(v)=C(v+1)^{-1}\left\\{1+O\left((v+1)^{-2}\right)\right\\}$ by [13, Thm. 1]. The upper bound (17) follows from (13), because $D_{q}(x,t)=0$ if $q>x$. Theorem 3 gives estimates for $c_{q}$ when $q$ is a prime or a product of two primes. These estimates are needed to derive Theorem 1 from Theorem 2. ###### Theorem 3. Let $p\leq q$ be primes. The constant factor in (15) satisfies $c_{\theta}:=c_{1}=C(\log t-\gamma)+O\left(e^{-\sqrt{\log t}}\right),$ (18) $qc_{q}=Cc_{\theta}\left\\{1+O\left(\frac{1}{\log q}+\frac{\log^{2}t}{\log^{2}q}\right)\right\\}$ (19) $qc_{q}=c_{\theta}+C\log q+O\left(\exp\left(-\sqrt{\log t}\right)\right)\quad(q\leq t),$ (20) $pqc_{pq}=C^{2}c_{\theta}\left\\{1+O\left(\frac{1}{\log p}+\frac{\log^{2}t}{\log^{2}p}+\frac{\log^{2}p}{\log^{2}q}\right)\right\\},$ (21) $pqc_{pq}=\left(Cc_{\theta}+C^{2}\log p\right)\left\\{1+O\left(\frac{1}{\log q}+\frac{\log^{2}t}{\log^{2}q}+e^{-\sqrt{\log t}}\right)\right\\}\quad(p\leq t),$ (22) $pqc_{pq}=c_{\theta}+C\log pq+O\left(e^{-\sqrt{\log t}}\right)\quad(p\leq q\leq t).$ (23) In Section 2 we derive Corollaries 1, 2 and 3 from Theorem 1. Section 3 contains several lemmas, used in the proofs of Theorems 2 and 3, about members of $\mathcal{B}$ that are multiples of a natural number $q$. The proof of Theorem 2 is given in Section 4. In Section 5 we establish Theorem 3 with the help of Corollary 6, which is a consequence of Theorem 2. Finally, in Section 6 we apply Theorems 2 and 3 to prove Theorem 1. ## 2\. Proof of Corollaries 1, 2 and 3 ###### Lemma 1. We have $D(x,t)\ll\frac{x\log t}{\log xt}\qquad(x>1/t,t\geq 2),$ $D(x,t)\gg\frac{x\log t}{\log xt}\qquad(x\geq 1,t\geq 2),$ $D(x/q,t)-D(q/t,t)\leq D_{q}(x,t)\leq D(x/q,qt)\qquad(x\geq 0,t\geq 2,q\geq 1).$ ###### Proof. The first two estimates follow from [6, Thm. 1]. If $m\in\mathcal{D}(x/q,t)$ and $m>q/t$, then $mq\in\mathcal{D}_{q}(x,t)$. This shows that $D(x/q,t)-D(q/t,t)\leq D_{q}(x,t)$. If $n\in\mathcal{D}(x,t)$ and $q\|n$, we write $n=qm$ and observe that $m\in\mathcal{D}(x/q,qt)$. Thus, $D_{q}(x,t)\leq D(x/q,qt)$. ∎ ###### Proof of Corollary 1. We first show (5). For $1\leq n\leq x$, we have $\max(2,n)\leq\theta(n)\leq nf(n)\leq nf(x)$. Thus, $\mathcal{B}(x)\subset\mathcal{D}(x,f(x))$. If $f(x)\leq x$, (2) yields $\left\|\left\\{n\in\mathcal{B}(x):\|\omega(n)-E(x,f(x))\|>\frac{\xi(x)}{2}\sqrt{\log_{2}x}\right\\}\right\|\ll\frac{D(x,f(x))}{\xi(x)^{2}}.$ The assumption $\theta(n)\geq\max(2,n)$ implies $B(x)\gg x/\log x$, by [14, Thm. 1.2]. By Lemma 1, $\frac{D(x,f(x))}{\xi(x)^{2}}\ll\frac{x\log f(x)}{\xi(x)^{2}\log x}\ll B(x)\frac{\log f(x)}{\xi(x)^{2}}.$ The result being trivial if $\log f(x)>\xi(x)^{2}$, we may assume $\log f(x)\leq\xi(x)^{2}$, so that $\|E(x,f(x))-C\log_{2}x\|=\|(C-1)\log_{2}f(x)\|\leq\frac{\xi(x)}{2}\sqrt{\log_{2}x},$ for $x\geq x_{0}$. Thus, (5) holds if $f(x)\leq x$. If $f(x)>x$, then $\xi(x)^{2}\geq\log f(x)>\log x$, so that $\|\omega(n)-C\log_{2}x\|>\xi(x)\sqrt{\log_{2}x}$ implies $\omega(n)>\xi(x)>\sqrt{\log x}$. The result now follows from Nicolas’ Theorem [3], an asymptotic estimate for the quantity $\|\\{n\leq x:\Omega(n)=k\\}\|$, which easily implies $\|\\{n\leq x:\Omega(n)\geq y\log_{2}x\\}\|\ll\frac{x}{(\log x)^{y\log 2-1}},$ (24) uniformly for $x\geq 2$ and $y\geq 2+\delta$, for any fixed $\delta>0$. Next, we show that (5) implies (4). From (24) we have $\|\\{n\leq x:\omega(n)\geq 6\log_{2}x\\}\|\leq\|\\{n\leq x:\Omega(n)\geq 6\log_{2}x\\}\|\ll\frac{B(x)}{\log^{2}x}.$ (25) Since $\omega(n)\leq\Omega(n)\ll\log n$, the contribution to $\sum_{n\in\mathcal{B}(x)}\omega(n)$ from $n$ with $\omega(n)>6\log_{2}x$ is $\ll B(x)/\log x$, while the contribution from $n$ with $\omega(n)\leq 6\log_{2}x$ and $\|\omega(n)-C\log_{2}x\|>\xi(x)\sqrt{\log_{2}x}$ is $\ll(6\log_{2}x)B(x)\frac{\log f(x)}{\xi(x)^{2}},$ by (5). The contribution to $\sum_{n\in\mathcal{B}(x)}\omega(n)$ from $n$ with $\omega(n)\leq 6\log_{2}x$ and $\|\omega(n)-C\log_{2}x\|\leq\xi(x)\sqrt{\log_{2}x}$ is $B(x)\left(1+O\left(\frac{\log f(x)}{\xi(x)^{2}}+\frac{1}{\log^{2}x}\right)\right)C\log_{2}x\left(1+O\left(\frac{\xi(x)}{\sqrt{\log_{2}x}}\right)\right).$ If $\log f(x)\leq\log_{2}x$, (4) now follows with $\xi(x)=(\log f(x))^{1/3}(\log_{2}x)^{1/6}$. If $\log f(x)>\log_{2}x$, (4) follows directly from (25). The argument works the same with $\Omega(n)$ in place of $\omega(n)$. ∎ ###### Proof of Corollary 2. Assume $\max(2,n)\leq\theta(n)\ll n(\log 2n)^{o(1)}$. Define $f(x)=\max_{n\leq x}\theta(n)/n$, so that $f$ is non-decreasing and $f(x)=(\log x)^{o(1)}$ as $x\to\infty$, that is $\log f(x)=o(\log_{2}x)$. The relation (6) follows from (4). Choosing $\xi(x)=(\log f(x)\log_{2}x)^{1/4}$ in (5) yields (7). ∎ ###### Lemma 2. Let $\varepsilon>0$. For $2\leq\alpha\leq 4-\varepsilon$ we have $\sum_{n\leq x\atop\Omega(n)\geq\alpha\log_{2}x}\tau(n)\ll x(\log x)^{\alpha(\log 2-\log\alpha+1)-1}.$ ###### Proof. This is a variation of Exercise 05 in [1]. Write $y^{\Omega(n)}=\sum_{d\|n}f(d)$, so that $f(n)$ is multiplicative and $f(p^{k})=y^{k}(1-1/y)$ for $k\geq 1$, by Möbius inversion. For $0\leq y\leq 2-\varepsilon$, $\begin{split}\sum_{n\leq x}\tau(n)y^{\Omega(n)}&=\sum_{n\leq x}\tau(n)\sum_{d\|n}f(d)\leq\sum_{d\leq x}f(d)\tau(d)\sum_{m\leq x/d}\tau(m)\\\ &\leq x\log x\sum_{d\leq x}f(d)\tau(d)/d\leq x\log x\sum_{P^{+}(d)\leq x}f(d)\tau(d)/d\\\ &=x\log x\prod_{p\leq x}\sum_{k\geq 0}f(p^{k})\tau(p^{k})/p^{k}\ll x(\log x)^{2y-1}.\end{split}$ If $1\leq y\leq 2-\varepsilon$, we get $\sum_{n\leq x\atop\Omega(n)\geq\alpha\log_{2}x}\tau(n)y^{\alpha\log_{2}x}\ll x(\log x)^{2y-1}.$ The result now follows with $y=\alpha/2$. ∎ ###### Proof of Corollary 3. Since $2^{\omega(n)}\leq\tau(n)\leq 2^{\Omega(n)}$ for all $n\geq 1$, the estimate (9) and the lower bound in (8) follow at once from (7). For the upper bound in (8), we write $\sum_{n\in\mathcal{B}(x)\atop\Omega(n)\leq e\log_{2}x}\tau(n)\leq B(x)2^{e\log_{2}x}=B(x)(\log x)^{e\log 2}$ and $\sum_{n\in\mathcal{B}(x)\atop\Omega(n)\geq e\log_{2}x}\tau(n)\leq\sum_{n\leq x\atop\Omega(n)\geq e\log_{2}x}\tau(n)\ll x(\log x)^{e\log 2-1}\ll B(x)(\log x)^{e\log 2},$ by Lemma 2 with $\alpha=e$. ∎ ## 3\. Multiples of $q$ in $\mathcal{B}$ In this section we develop some general identities for sets $\mathcal{B}$, defined by (3), with $\theta:\mathbb{N}\to\mathbb{R}\cup\\{\infty\\},\quad\theta(1)\geq 2,\quad\theta(n)\geq P^{+}(n)\quad(n\geq 2),$ (26) where $P^{+}(n)$ denotes the largest prime factor of $n$. Let $\Phi(x,y)=1_{x\geq 1}+\|\\{2\leq n\leq x:P^{-}(n)>y\\}\|,$ where $P^{-}(n)$ denotes the smallest prime factor of $n$. Let $\psi(n):=\begin{cases}1&\mbox{if }n\in\mathcal{B}\\\ 0&\mbox{else.}\end{cases}$ and define $\lambda_{n}(s):=\frac{\psi(n)}{n^{s}}\prod_{p\leq\theta(n)}\left(1-\frac{1}{p^{s}}\right),\quad\lambda_{n}:=\lambda_{n}(1),$ $\mu_{n}(s):=\sum_{p\leq\theta(n)}\frac{\log p}{p^{s}-1}-\log n,\quad\mu_{n}:=\mu_{n}(1).$ ###### Lemma 3. Let $\theta$ satisfy (26) and let $q_{1}\leq q_{2}\leq\ldots\leq q_{k}$ be primes. For $x\geq 0$, $\sum_{n\geq 1}\psi(n)\Phi\left(\frac{x}{n},\theta(n)\right)=\lfloor x\rfloor$ (27) and $\sum_{n\geq 1\atop q_{1}\cdots q_{k}\|n}\psi(n)\Phi\left(\frac{x}{n},\theta(n)\right)=\sum_{\theta(n)\geq q_{k}\atop q_{1}\cdots q_{k-1}\|n}\psi(n)\Phi\left(\frac{x}{nq_{k}},\theta(n)\right).$ (28) ###### Proof. The relation (27) is [15, Lemma 3]. We will show (28). Every $m\in q_{1}\cdots q_{k}\mathbb{N}$ factors uniquely as $m=nr$ where $n\in\mathcal{B}$ and $P^{-}(r)>\theta(n)$ if $r>1$. If $q_{1}\nmid n$ then $\theta(n)<q_{1}$. If $q_{1}\|n$, let $j$ be the largest index such that $q_{1}\cdots q_{j}\|n$, so that $q_{j+1}>\theta(n)$ if $j<k$. We count all integer multiples of $q_{1}\cdots q_{k}$ up to $x$ according to $j$ and $n$: $\left\lfloor\frac{x}{q_{1}\cdots q_{k}}\right\rfloor=\sum_{\theta(n)<q_{1}}\psi(n)\Phi\left(\frac{x}{nq_{1}\cdots q_{k}},\theta(n)\right)\\\ +\sum_{j=1}^{k-1}\sum_{\theta(n)<q_{j+1}\atop q_{1}\cdots q_{j}\|n}\psi(n)\Phi\left(\frac{x}{nq_{j+1}\cdots q_{k}},\theta(n)\right)+\sum_{q_{1}\cdots q_{k}\|n}\psi(n)\Phi\left(\frac{x}{n},\theta(n)\right).$ (29) We can now establish (28) by induction on $k$. When $k=1$, (29) and (27) yield (28). For the inductive step, we write the inner sum of (29) as $\sum_{\theta(n)<q_{j+1}\atop q_{1}\cdots q_{j}\|n}=\sum_{q_{1}\cdots q_{j}\|n}-\sum_{\theta(n)\geq q_{j+1}\atop q_{1}\cdots q_{j}\|n}$ and use the inductive hypothesis on the sum $\sum_{q_{1}\cdots q_{j}\|n}$ to get $\sum_{\theta(n)<q_{j+1}\atop q_{1}\cdots q_{j}\|n}\psi(n)\Phi\left(\frac{x}{nq_{j+1}\cdots q_{k}},\theta(n)\right)=\sum_{\theta(n)\geq q_{j}\atop q_{1}\cdots q_{j-1}\|n}\psi(n)\Phi\left(\frac{x}{nq_{j}\cdots q_{k}},\theta(n)\right)\\\ -\sum_{\theta(n)\geq q_{j+1}\atop q_{1}\cdots q_{j}\|n}\psi(n)\Phi\left(\frac{x}{nq_{j+1}\cdots q_{k}},\theta(n)\right).$ Thus, the sum over $j$ in (29) is telescoping and the result follows from (27). ∎ ###### Lemma 4. Let $\theta$ satisfy (26) and let $q_{1}\leq q_{2}\leq\ldots\leq q_{k}$ be primes. For $\operatorname{Re}(s)>1$ we have $\sum_{n\geq 1}\lambda_{n}(s)=1$ (30) and $\sum_{n\geq 1\atop q_{1}\cdots q_{k}\|n}\lambda_{n}(s)=\frac{1}{q_{k}^{s}}\sum_{\theta(n)\geq q_{k}\atop q_{1}\cdots q_{k-1}\|n}\lambda_{n}(s).$ (31) Both relations hold at $s=1$ if $B(x)=o(x)$. ###### Proof. The relation (30) is [16, Lemma 1] when $\operatorname{Re}(s)>1$ and [15, Theorem 1] when $s=1$. The proof of (31) mirrors that of (28). We first assume $\operatorname{Re}(s)>1$. Every $m\in q_{1}\cdots q_{k}\mathbb{N}$ factors uniquely as $m=nr$ where $n\in\mathcal{B}$ and $P^{-}(r)>\theta(n)$ if $r>1$. If $q_{1}\nmid n$ then $\theta(n)<q_{1}$. If $q_{1}\|n$, let $j$ be the largest index such that $q_{1}\cdots q_{j}\|n$, so that $q_{j+1}>\theta(n)$ if $j<k$. We rearrange the terms of the Dirichlet series $\sum_{q_{1}\cdots q_{k}\|m}m^{-s}$ according to $n$ and $j$. After dividing by $\zeta(s)$, this shows that, for $\operatorname{Re}(s)>1$, $\frac{1}{(q_{1}\cdots q_{k})^{s}}=\sum_{\theta(n)<q_{1}}\frac{\lambda_{n}(s)}{(q_{1}\cdots q_{k})^{s}}+\sum_{j=1}^{k-1}\sum_{\theta(n)<q_{j+1}\atop q_{1}\cdots q_{j}\|n}\frac{\lambda_{n}(s)}{(q_{j+1}\cdots q_{k})^{s}}+\sum_{q_{1}\cdots q_{k}\|n}\lambda_{n}(s).$ (32) We establish (31) by induction on $k$. When $k=1$, the result follows from applying (30) to the first sum of (32). For the inductive step, note that the inner sum in (32) is $\begin{split}\sum_{\theta(n)<q_{j+1}\atop q_{1}\cdots q_{j}\|n}\frac{\lambda_{n}(s)}{(q_{j+1}\cdots q_{k})^{s}}&=\sum_{q_{1}\cdots q_{j}\|n}\frac{\lambda_{n}(s)}{(q_{j+1}\cdots q_{k})^{s}}-\sum_{\theta(n)\geq q_{j+1}\atop q_{1}\cdots q_{j}\|n}\frac{\lambda_{n}(s)}{(q_{j+1}\cdots q_{k})^{s}}\\\ &=\sum_{\theta(n)\geq q_{j}\atop q_{1}\cdots q_{j-1}\|n}\frac{\lambda_{n}(s)}{(q_{j}\cdots q_{k})^{s}}-\sum_{\theta(n)\geq q_{j+1}\atop q_{1}\cdots q_{j}\|n}\frac{\lambda_{n}(s)}{(q_{j+1}\cdots q_{k})^{s}},\\\ \end{split}$ by the inductive hypothesis. Thus, the sum over $j$ in (32) is a telescoping sum and the result follows from (30). If $B(x)=o(x)$, the validity of (31) at $s=1$ follows from (28), in much the same way that the validity of (30) at $s=1$ follows from (27), which is demonstrated in the proof of [15, Thm. 1]. ∎ ###### Lemma 5. Let $\theta$ satisfy (26) and let $q_{1}\leq q_{2}\leq\ldots\leq q_{k}$ be primes. For $\operatorname{Re}(s)>1$ we have $\sum_{n\geq 1}\lambda_{n}(s)\mu_{n}(s)=0$ (33) and $\sum_{n\geq 1\atop q_{1}\cdots q_{k}\|n}\lambda_{n}(s)\mu_{n}(s)=\frac{1}{q_{k}^{s}}\sum_{\theta(n)\geq q_{k}\atop q_{1}\cdots q_{k-1}\|n}\lambda_{n}(s)\bigl{(}\mu_{n}(s)-\log q_{k}\bigr{)}.$ (34) ###### Proof. Differentiate (30) and (31) with respect to $s$. ∎ ## 4\. Proof of Theorem 2 The following estimate for $\Phi(x,y)$, which differs from the one we used in [14], simplifies the proof of Theorem 2. ###### Lemma 6. Uniformly, for $x\geq 0$, $y\geq 2$, we have $\begin{split}\Phi(x,y)&=1_{x\geq 1}+x\prod_{p\leq y}\left(1-\frac{1}{p}\right)+\frac{x}{\log y}\\!\left\\{w(u)-e^{-\gamma}\\!-\left.\frac{y}{x}\right\|_{x\geq y}\\!\\!+O\left(\frac{e^{-u/3}}{\log y}\right)\\!\right\\}\\\ &=1_{x\geq 1}+x\prod_{p\leq y}\left(1-\frac{1}{p}\right)+O\left(\frac{xe^{-u/3}}{\log y}\right),\end{split}$ where $u=\frac{\log\max(1,x)}{\log y}$ and $w(u)$ is Buchstab’s function. ###### Proof. The second estimate follows from the first, since $w(u)-e^{-\gamma}\ll e^{-u}$ and, if $x\geq y\geq 2$, then $y/x\ll e^{-u/2}$. When $x\geq y\geq 2$ and $\log y\geq(\log_{2}x)^{2}$, the first estimate follows from combining equations (49), (52), (59) and (60) of [11, Sec. III.6], with equation (6) of [11, Sec. III.5], where we estimate the integral in (52) as $\int_{0}^{\infty}\|w^{\prime}(u-v)\|y^{-v}dv\ll\int_{0}^{\infty}e^{-(u-v)/2}y^{-v}dv=\frac{e^{-u/2}}{\log y-1/2}\asymp\frac{e^{-u/2}}{\log y}.$ When $x\geq y\geq 2$ and $\log y<(\log_{2}x)^{2}$, then $u\gg\sqrt{\log x}$ and the result follows from [11, Thm. III.6.1 and Thm. III.5.1]. When $x<y$, then $\Phi(x,y)=1_{x\geq 1}$ and $w(u)=0$, so that the result follows from Mertens’ formula [11, Thm. I.1.11]. ∎ ###### Lemma 7. Assume $B(x)=B_{t}(x)$ is the counting function of a set $\mathcal{B}_{t}\subset\mathbb{N}$ that depends on the parameter $t$. Assume $B(x)=x\int_{1}^{\infty}\frac{B(y)}{y^{2}\log yt}\left(e^{-\gamma}-w\left(\frac{\log x/y}{\log yt}\right)\right)dy+R(x)\quad(x\geq 1),$ such that the integrals $\alpha_{t}:=e^{-\gamma}\int_{1}^{\infty}\frac{B(y)}{y^{2}\log yt}dy,\quad\beta_{t}:=\frac{-1}{\log t}\int_{1}^{\infty}R(y)\frac{dy}{y^{2}}$ converge. Then $B(x)=x\eta_{t}d(v)+O\Bigl{\\{}1+x\beta_{t}(v+1)^{-3.03}+R(x)+I(x)+J(x)\Bigr{\\}},$ where $v=\log x/\log t$, $d(v)$ is given by (11), $\eta_{t}=\alpha_{t}+\beta_{t}$, $I(x)=\frac{x}{\log xt}\int_{x}^{\infty}R(y)\frac{dy}{y^{2}},\quad J(x)=\frac{x}{(\log xt)^{3.03}}\int_{1}^{x}R(y)(\log yt)^{2.03}\frac{dy}{y^{2}}.$ ###### Proof. We follow the second half of the proof of [14, Thm. 1.3]. The only modification needed is the use of the improved estimates $(v+1)d(v)=C+O((v+1)^{-2.03}),\quad(v\geq 0),$ (35) and $(v+1)^{2}d^{\prime}(v)=-C+O((v+1)^{-2.03}),\quad(v\geq 0).$ (36) The estimate (35) is a consequence of [13, Cor. 6], while (36) follows from inserting (35) in the proof of [13, Cor. 5]. In [14], we used slightly weaker estimates for simplicity, with an exponent of $-2$ instead of $-2.03$ in the error terms. In the proof of Theorem 2, the improved exponent will save a factor of $\log_{2}x$ (when estimating the contribution from $R_{2}(x)$ to $J(x)$). ∎ For $n\in\mathbb{N}$ with prime factorization $n=p_{1}\cdots p_{k}$, where $p_{1}\leq\ldots\leq p_{k}$, define $F(n):=\max_{1\leq j\leq k}p_{j}^{2}p_{j+1}\cdots p_{k}.$ ###### Lemma 8. If $x<\max(m,F(m)/t)$, then $D_{m}(x,t)=0$. ###### Proof. Note that $n\in\mathcal{D}_{m}(x,t)$ if and only if $n\leq x$, $m\|n$ and $F(n)\leq nt$. Also, $m\|n$ implies $F(m)\leq F(n)$. Thus, if $D_{m}(x,t)\neq 0$ and $n\in\mathcal{D}_{m}(x,t)$, then $m\leq n\leq x$ and $F(m)\leq F(n)\leq nt\leq xt$, so $x\geq\max(m,F(m)/t)$. ∎ ###### Lemma 9. Let $n\geq 2$ with prime factorization $n=p_{1}\cdots p_{k}$, $p_{1}\leq\ldots\leq p_{k}$. If $F(n)\leq x$, then $p_{k-j+1}\cdots p_{k}\leq x^{1-2^{-j}}$ for $1\leq j\leq k$. ###### Proof. We use induction on $j$. When $j=1$, the claim is that $p_{k}\leq x^{1/2}$, which follows from $p_{k}^{2}\leq F(n)\leq x$, for all $k\geq 1$. Assume now that the claim is correct for some $j\in\mathbb{N}$ and all $k\geq j$. Let $k\geq j+1$. We have $p_{k-j}^{2}p_{k-j+1}\cdots p_{k}\leq F(n)\leq x$. Thus, if $p_{k-j}p_{k-j+1}\cdots p_{k}\geq x^{1-2^{-j-1}}$, then $p_{k-j}\leq x^{2^{-j-1}}$. By the inductive hypothesis, $p_{k-j}(p_{k-j+1}\cdots p_{k})\leq x^{2^{-j-1}}x^{1-2^{-j}}=x^{1-2^{-j-1}},$ for all $k\geq j+1$. ∎ ###### Lemma 10. Let $k\geq 0$ be fixed. For $m\geq 1$ with $\Omega(m)=k$, $t\geq 2$ and $x\geq\max(m,F(m)/t)$, we have $\frac{x\log t\log 2m}{m\log xt\log 2x}\gg_{k}1.$ ###### Proof. This is obvious if $m=1$ or if $t\geq x^{2^{-k}}$. If $m\geq 2$ and $t<x^{2^{-k}}$, then $F(m)\leq xt$ and Lemma 9 imply $m\leq(xt)^{1-2^{-k}}<x^{1-4^{-k}}\ll_{k}x/(\log xt)^{2}$, from which the claim follows. ∎ ###### Proof of Theorem 2. In the remainder of this paper, we write $D_{q}(x)$ for $D_{q}(x,t)$ and $D(x)$ for $D(x,t)$. We will show by induction on $k\geq 0$ that, for $k=\Omega(q)$, the estimates (12), (13) and (14) hold and that, for primes $r$ with $r\geq P^{+}(q)$, we have $\sum_{n\in\mathcal{D}_{q}\atop n\geq r/t}\Phi(x/rn,nt)-\frac{x}{r}\sum_{n\in\mathcal{D}_{q}\atop n\geq r/t}\lambda_{n}\ll 1+\frac{x\log t\log qr\log qrt}{qr\log^{3}xt}=:R_{2}(x,qr),$ (37) for $x\geq\max(qr,F(qr)/t)$. Note that (13) and (35) imply $D_{q}(x)\ll_{k}xd(v)\eta_{q,t}\asymp_{k}\frac{x\log t}{q\log xt},\quad(x\geq 1,q\geq 1,t\geq 2),$ (38) since $D_{q}(x)=0$ if $q>x$. When $k=0$, $q=1$, (12) is [14, Eq. (13)], (13) is [14, Thm. 1.3] and (14) is [14, Eq. (6)]. To show (37) for $k=0$ and $q=1$, assume that $r$ is prime and $x\geq\max(r,r^{2}/t)$. Equations (27) and (30) show that $\begin{split}&\sum_{n\in\mathcal{D}\atop n\geq r/t}\Phi(x/rn,nt)-\frac{x}{r}\sum_{n\in\mathcal{D}\atop n\geq r/t}\lambda_{n}\\\ &=-\\{x/r\\}-\sum_{n\in\mathcal{D}\atop n<r/t}\Phi(x/rn,nt)+\frac{x}{r}\sum_{n\in\mathcal{D}\atop n<r/t}\lambda_{n}\\\ &\ll 1+D(r/t)+\sum_{n\in\mathcal{D}\atop n<r/t}\frac{x}{rn\log nt}\exp\left(-\frac{\log xt/r}{3\log nt}\right),\end{split}$ (39) by the second estimate in Lemma 6. We have $D(r/t)\ll r\log t/(t\log r)$ by Lemma 1, so $D(r/t)\ll R_{2}(x,r)$ follows from $\frac{r}{\log^{2}r\log rt}\ll\frac{xt/r}{\log^{3}xt}\asymp\frac{xt/r}{\log^{3}(xt/r)},$ since $xt/r\geq\sqrt{xt}$. This holds because $r\leq xt/r$. Finally, the last sum in (39) is $\ll R_{2}(x,r)$ by Lemma 1. Thus, (37) holds for $k=0$. For the inductive step, assume that (13) (and hence (38)) and (14) hold for $q\in\mathbb{N}$ with $\Omega(q)=k$ for some $k\geq 0$. If $k\geq 1$, assume that (37) holds for $\Omega(q)=k-1$. Let $r$ be a prime with $r\geq P^{+}(q)$ and write $m=qr$. We note that in the remainder of this proof, all implied constants in the $\ll$ and big-O notation may depend on $k$. We estimate the first sum in (28) with Lemma 6 and apply (31) to get $\begin{split}D_{m}(x)&+\sum_{n\in\mathcal{D}_{m}}\frac{x}{n\log nt}\left\\{w\left(\frac{\log x/n}{\log nt}\right)-e^{-\gamma}-\left.\frac{n^{2}t}{x}\right\|_{n^{2}\leq\frac{x}{t}}\\!+O\left(\frac{e^{-\frac{\log x/n}{3\log nt}}}{\log nt}\right)\right\\}\\\ &=\sum_{n\in\mathcal{D}_{q}\atop n\geq r/t}\Phi(x/rn,nt)-\frac{x}{r}\sum_{n\in\mathcal{D}_{q}\atop n\geq r/t}\lambda_{n}.\end{split}$ (40) The contribution from the last two terms in the first sum in (40) is $\ll\tilde{R}_{1}(x):=\frac{x\log mt}{m(\log xt)^{2}},$ by Lemma 1. In the second application of this argument we will be able to replace $\tilde{R}_{1}(x)$ by the smaller $R_{1}(x):=\frac{x\log t}{m(\log xt)^{2}}.$ The error from applying Abel summation to the remaining terms of the first sum in (40) is also $\ll\tilde{R}_{1}(x)$, since $w(u)-e^{-\gamma}\ll e^{-u}$ and $w^{\prime}(u)\ll e^{-u}$. Thus, $\begin{split}D_{m}(x)=&x\int_{1}^{\infty}\frac{D_{m}(y)}{y^{2}\log yt}\left(e^{-\gamma}-w\left(\frac{\log x/y}{\log yt}\right)\right)dy+O(\tilde{R}_{1}(x))\\\ &+\sum_{n\in\mathcal{D}_{q}\atop n\geq r/t}\Phi(x/rn,nt)-\frac{x}{r}\sum_{n\in\mathcal{D}_{q}\atop n\geq r/t}\lambda_{n}.\end{split}$ (41) If $x/r<\max(q,F(q)/t,r/t)$, that is $x<\max(m,F(m)/t)$, then Lemma 8 shows that the first sum in (41) vanishes, while the second sum is $\ll\frac{x}{m}\min\left(1,\frac{\log t}{\log r}\right)\ll\frac{x}{m}\min\left(1,\frac{\log t}{\log m}\right),$ by the inductive hypothesis (38) and since $\log r\leq\log m\leq\log r^{k+1}\ll_{k}\log r$. Define $R_{2}(x):=\begin{cases}\frac{x}{m}\min\left(1,\frac{\log t}{\log m}\right)&\text{if }x<\max(m,F(m)/t)\\\ 1+\frac{x\log t\log m\log mt}{m\log^{3}xt}&\text{if }x\geq\max(m,F(m)/t).\end{cases}$ Thus, $D_{m}(x)=x\int_{1}^{\infty}\frac{D_{m}(y)}{y^{2}\log yt}\left(e^{-\gamma}-w\left(\frac{\log x/y}{\log yt}\right)\right)dy+R(x)$ (42) holds with $R(x)\ll\tilde{R}_{1}(x)+R_{2}(x)$ when $x<\max(m,F(m)/t)$. Assume now that $x\geq\max(m,F(m)/t)$. If $k=0$ and $q=1$, we have already shown that the last row of (41) is $\ll R_{2}(x)$. If $k\geq 1$, write $q=uv$ where $v=P^{+}(q)$. Equations (28) and (31) show that the last row of (41) equals $\begin{split}=&\sum_{n\in\mathcal{D}_{u}\atop n\geq v/t}\Phi(x/nrv,nt)-\frac{x}{rv}\sum_{n\in\mathcal{D}_{u}\atop n\geq v/t}\lambda_{n}-\sum_{n\in\mathcal{D}_{q}\atop n<r/t}\Phi(x/nr,nt)+\frac{x}{r}\sum_{n\in\mathcal{D}_{q}\atop n<r/t}\lambda_{n}\\\ =&S-T-(U-V),\end{split}$ say. Now $x\geq\max(m,F(m)/t)$ implies $x/r\geq\max(q,F(q)/t,r/t)$. The inductive hypothesis (37) yields $S-T\ll 1+\frac{(x/r)\log t\log q\log qt}{q\log^{3}(xt/r)}\ll 1+\frac{x\log t\log m\log mt}{m\log^{3}xt}=R_{2}(x),$ since $xt/r\geq\sqrt{xt}$. The second estimate in Lemma 6 and (38) show that $\begin{split}U-V&\ll D_{q}(r/t)+\sum_{n\in\mathcal{D}_{q}\atop n<r/t}\frac{x}{nr\log nt}\exp\left(-\frac{\log\frac{x}{nr}}{3\log nt}\right)\\\ &\ll\frac{r^{2}\log t}{mt\log r}+R_{2}(x)\ll R_{2}(x),\end{split}$ where the last assertion is implied by $\frac{r}{\log r\log m\log mt}\ll\frac{xt/r}{\log^{3}(xt/r)},$ which holds because $mt>m\geq r$ and $r\leq xt/r$. As the last row of (41) is $\ll R_{2}(x)$, we have established that (37) holds for $\Omega(q)\leq k$ and that (42) holds with $R(x)\ll\tilde{R}_{1}(x)+R_{2}(x)$. We need to estimate $I(x)$ and $J(x)$ from Lemma 7. For this purpose we may assume that $x\geq\max(m,F(m)/t)$. If not, Lemma 8 shows that $D_{m}(x)=0$, so that (12) holds since the main term in (12) is absorbed by the error terms. We find that $I(x)+J(x)\ll\tilde{R}_{1}(x)+R_{2}(x)$ and $\beta_{t}\ll\frac{\log m}{m\log t}$ so that $x\beta_{t}(v+1)^{-3}\ll R_{2}(x)$. The conclusion of Lemma 7 is that $D_{m}(x)=x\eta_{m,t}d(v)+O\bigl{(}\tilde{R}_{1}(x)+R_{2}(x)\bigr{)}.$ (43) The lower bound in Lemma 1 yields $\eta_{m,t}\gg m^{-1}$. Lemma 11 shows that $\eta_{m,t}\ll_{k}m^{-1}$. Together with (43), Lemmas 8 and 10, this implies $D_{m}(x)\ll_{k}\frac{x\log t}{m\log xt},\quad(x\geq 1,t\geq 2,\Omega(m)\leq k+1).$ (44) Running through this proof a second time with this upper bound replacing the one in Lemma 1, we can replace $\tilde{R}_{1}(x)$ by $R_{1}(x)$ to obtain $D_{m}(x)=x\eta_{m,t}d(v)+O\bigl{(}R_{1}(x)+R_{2}(x)\bigr{)},$ where $1/m\ll\eta_{m,t}\ll_{k}1/m$. This shows that (12) and (14) hold with $q$ replaced by $m=qr$. To see that (12) implies (13), note that the term $O(1)$ is acceptable by Lemma 10, provided $x\geq\max(m,F(m)/t)$. If $x<\max(m,F(m)/t)$, then $D_{m}(x)=0$ and $x<F(m)\leq m^{2}$, so (13) (with $q$ replaced by $m$) follows from $\log x\leq 2\log m$. This completes the proof of Theorem 2. ∎ ###### Lemma 11. Assume that (43) holds for $m$ with $\Omega(m)\leq k+1$, and that $\eta_{q,t}\ll_{k}q^{-1}$ for $\Omega(q)\leq k$. Then $\eta_{m,t}\ll_{k}m^{-1}$ for $\Omega(m)=k+1$. ###### Proof. Let $m=qr$ where $r$ is prime and $r\geq P^{+}(q)$. Equation (34) yields $\sum_{n\geq 1\atop qr\|n}\lambda_{n}(s)\mu_{n}(s)=\frac{1}{r^{s}}\sum_{n\geq r/t\atop q\|n}\lambda_{n}(s)\bigl{(}\mu_{n}(s)-\log r\bigr{)}.$ As in [16], we let $s=1+1/\log^{2}N$ and split the sum on the left according to $n\leq N$ and $n>N$. As $N\to\infty$, the contribution from $n\leq N$ converges to (see [16, Lemma 3]) $\sum_{qr\|n}\lambda_{n}\mu_{n}$, while the contribution from $n>N$ converges to (see [16, Lemma 4]) $-c_{qr}(1-e^{-\gamma}),$ by (43), where $\eta_{qr,t}=Cc_{qr}\log t$. Applying the same reasoning to the sum on the right-hand side, we get $\sum_{n\geq 1\atop qr\|n}\lambda_{n}\mu_{n}-c_{qr}(1-e^{-\gamma})=\frac{1}{r}\Bigl{(}\sum_{n\geq r/t\atop q\|n}\lambda_{n}\bigl{(}\mu_{n}-\log r\bigr{)}-c_{q}(1-e^{-\gamma})\Bigr{)}.$ Since $\mu_{n}\ll\log t$ , we obtain $rc_{qr}\ll S\log t+S\log r+c_{q},\quad S:=r\sum_{n\geq 1\atop qr\|n}\lambda_{n}=\sum_{n\geq r/t\atop q\|n}\lambda_{n},$ by (31). Now $\eta_{q,t}\ll q^{-1}$ and (44) holds with $q$ in place of $m$. Thus, $c_{q}\ll q^{-1}\log t$ and $S=\sum_{n\geq r/t\atop q\|n}\lambda_{n}\ll q^{-1}\min(1,\log t/\log r).$ This shows that $rc_{qr}\ll q^{-1}\log t$, which is the desired result. ∎ ## 5\. Proof of Theorem 3 ###### Proof of (18), (19) and (20). The estimate (18) is [16, Cor. 3]. Equation (34), with $k=1$ and $q_{1}=q$ prime, is $\sum_{n\geq 1\atop q\|n}\lambda_{n}(s)\mu_{n}(s)=\frac{1}{q^{s}}\sum_{n\geq q/t}\lambda_{n}(s)\bigl{(}\mu_{n}(s)-\log q\bigr{)}.$ (45) As in [16], we let $s=1+1/\log^{2}N$ and split the sum on the left according to $n\leq N$ and $n>N$. As $N\to\infty$, the contribution from $n\leq N$ converges to (see [16, Lemma 3]) $\sum_{n\geq 1\atop q\|n}\lambda_{n}\mu_{n}$, while the contribution from $n>N$ converges to (see [16, Lemma 4]) $-c_{q}(1-e^{-\gamma}),$ by Corollary 6. Applying the same reasoning to the right-hand side of (45), we get $\sum_{n\geq 1\atop q\|n}\lambda_{n}\mu_{n}-c_{q}(1-e^{-\gamma})=\frac{1}{q}\Bigl{(}\sum_{n\geq q/t}\lambda_{n}\bigl{(}\mu_{n}-\log q\bigr{)}-c_{\theta}(1-e^{-\gamma})\Bigr{)}.$ With the estimate [16, Lemma 13] $\mu_{n}=\log t-\gamma+O\left(e^{-\sqrt{\log nt}}\right),$ (46) we obtain $qc_{q}(1-e^{-\gamma})=S\log q+c_{\theta}(1-e^{-\gamma})+O\left(Se^{-\sqrt{\log\max(q,t)}}\right),$ (47) where $S:=\sum_{n\geq q/t}\lambda_{n}=q\sum_{n\geq 1\atop q\|n}\lambda_{n},$ by (31). If $q\leq t$, then $S=1$ by (30), so that (20) follows from (47). In general, we have $S=\frac{e^{-\gamma}c_{\theta}}{\log q}\left(1+O\left(\frac{1}{\log q}+\frac{\log^{2}t}{\log^{2}q}\right)\right),\quad(q\geq 2,\,t\geq 2).$ (48) If $q\leq t$, this holds because $S=1$ and $c_{\theta}\asymp\log t$. If $q\geq t$, (48) follows from applying Abel summation to $S=\sum_{n\geq q/t}\lambda_{n}$, together with Mertens’ formula and Corollary 6 (with $q=1$). Combining (48) with (47) proves (19). ∎ ###### Proof of (21), (22) and (23). Equation (34), with $k=2$ and $q_{1}=p\leq q_{2}=q$ is $\sum_{n\geq 1\atop pq\|n}\lambda_{n}(s)\mu_{n}(s)=\frac{1}{q^{s}}\sum_{n\geq q/t\atop p\|n}\lambda_{n}(s)\bigl{(}\mu_{n}(s)-\log q\bigr{)}.$ (49) As in [16], we let $s=1+1/\log^{2}N$ and split the sum on the left according to $n\leq N$ and $n>N$. As $N\to\infty$, the contribution from $n\leq N$ converges to (see [16, Lemma 3]) $\sum_{pq\|n}\lambda_{n}\mu_{n}$, while the contribution from $n>N$ converges to (see [16, Lemma 4]) $-c_{pq}(1-e^{-\gamma}),$ by Corollary 6. Applying the same reasoning to the right-hand side of (49), we get $\sum_{n\geq 1\atop pq\|n}\lambda_{n}\mu_{n}-c_{pq}(1-e^{-\gamma})=\frac{1}{q}\Bigl{(}\sum_{n\geq q/t\atop p\|n}\lambda_{n}\bigl{(}\mu_{n}-\log q\bigr{)}-c_{p}(1-e^{-\gamma})\Bigr{)}.$ We estimate $\mu_{n}$ with (46) to obtain $qc_{pq}(1-e^{-\gamma})=T\log q+c_{p}(1-e^{-\gamma})+O\left(T\exp\left(-\sqrt{\log\max(pt,q)}\right)\right),$ (50) where $T:=\sum_{n\geq q/t\atop p\|n}\lambda_{n}=q\sum_{n\geq 1\atop pq\|n}\lambda_{n},$ by (31). If $p\leq q\leq t$, then $T=\sum_{n\geq 1\atop p\|n}\lambda_{n}=\frac{1}{p}\sum_{n\geq p/t}\lambda_{n}=\frac{1}{p}\sum_{n\geq 1}\lambda_{n}=\frac{1}{p},$ by (31) and (30). The estimate (23) now follows from (50) with $T=1/p$ and $c_{p}$ estimated by (20). In general, we have $T=\frac{e^{-\gamma}c_{p}}{\log q}\left(1+O\left(\frac{1}{\log q}+\frac{\log^{2}pt}{\log^{2}q}\right)\right),\quad(q\geq p\geq 2,\,t\geq 2).$ (51) If $q<pt$, this is implied by $T\ll c_{p}/\log q$, which follows from (17) (with $q$ replaced by $p$). If $q\geq pt$, we estimate $\lambda_{n}$ with Mertens’ formula and use Abel summation and the estimate (15). The contribution from the first two error terms in (15) is clearly acceptable, while the term $O(1)$ contributes $\ll\int_{q/t}^{\infty}\frac{dy}{y^{2}\log yt}\leq\frac{t}{q\log q}\asymp\frac{c_{p}}{\log q}\cdot\frac{pt}{q\log t}\ll\frac{c_{p}}{\log q}\cdot\frac{\log^{2}pt}{\log^{2}q}.$ Now substitute (51) into (50) and estimate $c_{p}$ with (19) to get (21), and with (20) to get (22). ∎ ## 6\. Proof of Theorem 1 ###### Proof of (1).. We have $\sum_{n\in\mathcal{D}(x)}\omega(n)=\sum_{p\leq x}D_{p}(x)=xd(v)\sum_{p\leq x}\eta_{p,t}\left(1+O\left(\frac{\log p}{\log x}\right)\right),$ by (13). The contribution from the error term is $\ll xd(v)\asymp D(x)$, since $\eta_{p,t}\ll 1/p$ by (14). Now $\eta_{p,t}C\log t=c_{p}$ and $\eta_{1,t}C\log t=c_{\theta}$, by Corollary 6. With (18), (19) and (20), we find that $\sum_{p\leq x}\eta_{p,t}=\frac{1}{C\log t}\sum_{p\leq x}c_{p}=\frac{c_{\theta}}{C\log t}(E(x,t)+O(1))=\eta_{1,t}(E(x,t)+O(1)).$ The result now follows from (13) with $q=1$, that is $D(x)=xd(v)\eta_{1,t}(1+O(1/\log x))$. To see that (1) remains valid when $\omega$ is replaced by $\Omega$, note that $\sum_{n\in\mathcal{D}(x)}(\Omega(n)-\omega(n))=\sum_{n\in\mathcal{D}(x)}\sum_{k\geq 2}\sum_{p^{k}\|n}1=\sum_{p\leq x}\sum_{k\geq 2}D_{p^{k}}(x)\ll\frac{x\log t}{\log xt}\ll D(x),$ by Lemma 1. ∎ ###### Proof of (2).. We have $\sum_{n\in\mathcal{D}(x)}\omega(n)^{2}=\sum_{p,q\leq x}D_{pq}(x)+O\Bigl{(}\sum_{p\leq x}D_{p}(x)\Bigr{)},$ where $p,q$ run over primes. The last term is $\ll D(x)\log_{2}x$, by (1). Thus, (13) yields $\sum_{n\in\mathcal{D}(x)}\omega(n)^{2}=O(D(x)\log_{2}x)+xd(v)\sum_{p,q\leq x}\eta_{pq,t}\left(1+O\left(\frac{\log pq}{\log x}\right)\right).$ Since $\eta_{pq,t}\ll 1/pq$, the contribution from the error term is $\ll xd(v)\log_{2}x\ll D(x)\log_{2}x$. With (21), (22) and (23), we find that $\begin{split}\sum_{p,q\leq x}\eta_{pq,t}=\frac{1}{C\log t}\sum_{p,q\leq x}c_{pq}&=\frac{c_{\theta}}{C\log t}\bigl{(}E(x,t)^{2}+O(\log_{2}x)\bigr{)}\\\ &=\eta_{1,t}\bigl{(}E(x,t)^{2}+O(\log_{2}x)\bigr{)}.\end{split}$ Combining this with (1) and (13) (with $q=1$), we get $\sum_{n\in\mathcal{D}(x)}(\omega(n)-E(x,t))^{2}\ll D(x)\log_{2}x,$ which implies (2). This estimate remains valid if $\omega(n)$ is replaced by $\Omega(n)$, since $\sum_{n\in\mathcal{D}(x)}(\Omega(n)-\omega(n))^{2}\leq\sum_{p\neq q\leq x}\sum_{k,j\geq 2}D_{p^{k}q^{j}}(x)+\sum_{p\leq x}\sum_{k\geq 2}2kD_{p^{k}}(x)\ll D(x),$ by Lemma 1. ∎ ## Acknowledgments The author thanks Eric Saias and the anonymous referee for several very helpful suggestions. ## References * [1] R. R. Hall and G. Tenenbaum, Divisors, Cambridge University Press, 1988. * [2] M. Margenstern, Les nombres pratiques: théorie, observations et conjectures, J. Number Theory 37 (1991), 1–36. * [3] J.-L. Nicolas, Sur la distribution des nombres entiers ayant une quantité fixée de facteurs premiers, Acta Arith. 44 (1984), 191–200. * [4] C. Pomerance, L. Thompson, A. Weingartner, On integers $n$ for which $X^{n}-1$ has a divisor of every degree, Acta Arith. 175 (2016), no. 3, 225–243. * [5] C. Pomerance and A. Weingartner, On primes and practical numbers, arXiv: 2007.11062 * [6] E. 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# QCD EQUATION OF STATE AT FINITE CHEMICAL POTENTIALS FOR RELATIVISTIC NUCLEAR COLLISIONS AKIHIKO MONNAI Department of Mathematical and Physical Sciences, Japan Women’s University Bunkyo-ku, Tokyo 112-8681, Japan <EMAIL_ADDRESS>BJÖRN SCHENKE Physics Department, Brookhaven National Laboratory Upton, New York 11973, USA <EMAIL_ADDRESS>CHUN SHEN Department of Physics and Astronomy, Wayne State University Detroit, Michigan 48201, USA RIKEN BNL Research Center, Brookhaven National Laboratory Upton, New York 11973, USA <EMAIL_ADDRESS> (Day Month Year) ###### Abstract We review the equation of state of QCD matter at finite densities. We discuss the construction of the equation of state with net baryon number, electric charge, and strangeness using the results of lattice QCD simulations and hadron resonance gas models. Its application to the hydrodynamic analyses of relativistic nuclear collisions suggests that the interplay of multiple conserved charges is important in the quantitative understanding of the dense nuclear matter created at lower beam energies. Several different models of the QCD equation of state are discussed for comparison. ###### keywords: quantum chromodynamics; equation of state; nuclear collision. PACS numbers: ## 1 Introduction The collective properties of quantum chromodynamic (QCD) matter have been a topic of great interest in nuclear physics. A milestone has been the discovery of the quark-gluon plasma (QGP) [1, 2, 3, 4], a high-temperature phase of QCD, at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory (BNL) in the year 2000 [5, 6, 7, 8]. The QGP is speculated to have filled the universe about $10^{-5}$-$10^{-4}$ seconds after the Big Bang. The collider experiments have allowed the quantitative study of QCD matter through comparison of theoretical calculations and experimental data and consequently provided a glimpse of the early universe. The nearly-perfect fluidity and rapid thermalization of the QGP are major discoveries and have opened up a world of possibilities to study thermodynamics of strongly-interacting elementary particles in collider experiments. The high-energy frontier has been explored by the Large Hadron Collider (LHC) at the European Organization for Nuclear Research (CERN), which has been in operation since 2009 [9, 10, 11]. It has extended our experimental knowledge of the QCD phase diagram (Fig. 1) in the direction of temperature, getting closer to the beginning of the universe. The fluidity has been shown to persist at higher temperatures, though the fluid may become less perfect [12] as the system would be less strongly-coupled. Figure 1: A schematic illustration of the QCD phase diagram and the beam energy scan experiments. The phase structure in the dense regions are conjectured based on model estimations. The next frontier on the phase diagram is the high-density regime [13, 14], where first principles calculations are known to suffer from the fermion sign problem [15]. Estimations based on the chiral model indicate that the quark- hadron transition turns from a crossover to a first-order phase transition at a finite baryon chemical potential, suggesting the existence of a critical point [16]. Further theoretical model analyses indicate that the QCD phase structure can be quite nontrivial; possible scenarios include the color superconducting (CSC) phase at low temperature and high baryon density, where quarks form a condensate of Cooper pairs [17, 18, 19], the second critical point at the high-density end of the quark-hadron phase boundary implied by the QCD axial anomaly [20], the chiral and color superconducting phase transitions enhanced with vector interaction [21], and the quarkyonic phase, suggested by studies in the large $N_{c}$ limit [22]. Exploration of the dense quark matter is of particular importance since the experimental detection of gravitational waves, emerging from e.g. neutron star mergers, now give more stringent constraints on the properties of the compact stars themselves, including the equation of state [23, 24]. See 25, 26 for recent reviews. The collider experiments are a powerful tool to obtain bottom-up insight into the QCD matter at finite baryon chemical potential with high precision in controlled environments (Fig. 2). The Beam Energy Scan programs, being preformed at BNL RHIC and planned at various facilities including the GSI Facility for Antiproton and Ion Research (FAIR), JINR Nuclotron-based Ion Collider fAility (NICA), and JAEA/KEK Japan Proton Accelerator Research Complex (J-PARC). The heavy-ion programs at BNL Alternating Gradient Synchrotron (AGS), CERN Super Proton Synchrotron (SPS) and GSI Schwerionensynchrotron 18 (SIS 18), provide complimentary data for understanding the properties of dense quark matter. Figure 2: A schematic illustration of the nuclear collisions at high energies where the system has larger temperature and smaller baryon density (left) and at intermediate to low energies where the system has lower temperature and larger baryon density (right). One of the most successful models for the description of the dynamical evolution of QGP is the relativistic hydrodynamic model [27, 28]. The observed spectra of hadronic particles up to moderate transverse momenta ($p_{T}\lesssim 3\,{\rm GeV}$) are known to be in quantitative agreement with hydrodynamic model calculations. Azimuthal momentum anisotropies, characterized with flow harmonics $v_{n}$ [29, 30, 31, 32], are considered to be one of the most prominent pieces of evidence for the nearly-perfect fluidity of the produced QCD medium, because they are found to clearly reflect the geometrical anisotropy of the overlap region of colliding nuclei, implying that the system is strongly coupled. The physics of QCD enters the model through the equation of state – and the transport coefficients in off- equilibrium cases – along with details of initial conditions. Thus, once a realistic initial geometry is given, it is in principle possible to extract information on the QCD equation of state by comparing numerical results based on trial input with experimental data [33, 34, 35, 36, 37, 38]. The equation of state is a fundamental relation among thermodynamic variables. Earliest studies of the QCD equation of state date back to the MIT bag model [39, 40], where confinement is introduced phenomenologically. Hadrons are treated as quarks in bags within the QCD vacuum. Consequently, the model has a first-order phase transition between the hadron and QGP phases. Since then, our understanding of QCD thermodynamics has been deepened with the advent of model approaches such as the potential model [41] and the Nambu-Jona-Lasinio model [42, 43]. A breakthrough was brought when first principle calculations became possible with the advancement of the computational method of lattice QCD (at zero chemical potentials). SU(3) pure glue studies predict a first- order QCD phase transition while more realistic (2+1)-flavor calculations suggest a crossover transition, implying the importance of quark contributions in the phenomenon [44, 45, 46]. State-of-the-art lattice QCD simulations with a physical pion mass provide high-precision results of the QCD equation of state over a wide-range of temperatures [47, 48, 49]. The lattice QCD equation of state, when embedded in a hydrodynamic model, is known to reproduce the experimental data of nuclear collisions at top RHIC and LHC energies well. It was considered in the earlier days of QGP phenomenology that the fluidity appears only at and above energies around $\sqrt{s_{NN}}=\mathcal{O}(10^{2})$ GeV. The net baryon density in most cases was not considered important because it would be small and have negligible effects for such systems, except at forward rapidities. As the hydrodynamic model became more sophisticated, on the other hand, it was rediscovered that the hydrodynamic description can be valid for hadronic yields at lower energies, down to $\sqrt{s_{NN}}=\mathcal{O}(10)$ GeV, which is the typical energy scale covered by the beam energy scan programs[50, 51]. This may be partially owing to the fact that one has a better understanding of non-equilibrium processes – initial dynamics of local equilibration, viscosity and diffusion, and hadronic transport, which occur before, during, and after the hydrodynamic evolution, respectively – and can now show that hydrodynamics, which is based on the idea of local equilibrium, is compatible with experimental results. For a quantitative description of nuclear collisions in the beam energy scan programs, one needs the equation of state at finite densities [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69] as input for hydrodynamic simulations. First principle calculations are known to be challenging at finite densities, owing to the aforementioned sign problem. Several intriguing methods to circumvent the sign problem have been proposed, such as the Taylor expansion method [70, 71], the reweighting method [72, 71, 73], the imaginary chemical potential method [74, 75, 76], the complex Langevin method [77, 78, 79, 80], the Lefschetz thimble method [81, 82], and the path optimization method [83, 84], but so far no complete description is available at small temperatures and large chemical potentials. In this review, we will discuss the phenomenological construction of the QCD equation of state at finite chemical potentials for relativistic nuclear collisions. Oftentimes, only net baryon number is taken into account as the conserved charge in the equation of state, especially when hydrodynamic modeling is concerned. We review the neos model [85, 86] based on the lattice QCD equation of state and susceptibilities at vanishing densities from Refs. 48, 87, 88, 89, 90 (see also Refs. 91, 92, 93, 49, 94), and the hadron resonance gas equation of state, that include three conserved charges relevant in nuclear collisions: net baryon (B), electric charge (Q), and strangeness (S) as a successor to the version including only net baryon chemical potentials [95, 96, 97, 98, 99, 100, 101]. The selection of conserved charges is based on the assumption that only light quarks ($u$, $d$, and $s$) would thermalize in nuclear collisions. It has been employed in recent hydrodynamic model analyses [102, 103, 104]. A similar approach also has recently been proposed in Ref. 105, and the importance of multiple conserved charges has been discussed in various situations before our model realization [106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119]. We demonstrate by explicit calculations within a hydrodynamic model that the description of experimental data is improved by our comprehensive treatment of the conserved charges in the construction of the equation of state. Finally, we compare different models for the QCD equation of state, both at zero and finite densities, and present conclusions and summary. The natural units $c=\hbar=k_{B}=1$ and the mostly-minus Minkowski metric $g^{\mu\nu}=\mathrm{diag}(+,-,-,-)$ are used. ## 2 Status We review the status of the study of the QCD equation of state. QCD thermodynamics is a topic of interest to a broad range of studies from nuclear physics to particle physics to astrophysics. Here we focus on the phenomenological equations of state intended for use in hydrodynamic studies of relativistic nuclear collisions. ### 2.1 From bag model to lattice QCD Early hydrodynamic models often employed the equation of state inspired by the MIT bag model, such as EOS Q, which has a first-order phase transition at zero densities. The hadronic phase is described using a resonance gas and the QGP phase using a parton gas with a bag constant [120, 121, 122, 27]. A finite net baryon density was relatively easy to implement in such models, though it was neglected in many cases, because it would have small effects around mid- rapidity at top RHIC energies. These equations of state are, despite involving a first-order phase transition, able to reproduce the experimental data of hadronic spectra and elliptic flow reasonably well with an appropriate choice of initial conditions in inviscid models. Crossover-like equations of state have also been discussed in the literature. Early studies include a functional parametrization of pioneering lattice QCD results [123, 124, 125] by matching a parton gas with a pion gas equation of state and encoding the details of the transition using the choice of the connection width $\Delta T$ (defined later as in Eq. (8)). A more sophisticated equation of state was developed by connecting the results of the hadron resonance gas and an effective theory for finite temperature SU(3) gauge theory, [126, 127] and was used for viscous hydrodynamic analyses [128]. With the advent of first principle calculations, the connection of the results of lattice QCD simulations and hadron resonance gas became a topic of interest [129, 130, 131, 132, 133, 134, 135]. A variation of such approach includes the quasi-particle model fit to the lattice QCD data [53, 136]. Sometimes the lattice QCD equation of state is used directly at vanishing densities down into the hadronic phase, though caution is needed because the energy-momentum conservation at particlization is no longer automatically guaranteed if the hadron resonance gas description is not used [137], and the inconsistency may be hidden by the normalization of initial conditions. Additionally, the lattice QCD data in the continuum limit typically have uncertainty bands of a few percent, though they have been improved considerably in recent simulations. As mentioned earlier, the lattice-based QCD equation of state is considered to give an accurate description of the hot matter created and observed in the collider experiments at RHIC and LHC, where the conserved charges can be neglected [33, 34, 35, 36, 37, 38]. It is important to next elucidate the high density regions of the QCD phase diagram for fully utilizing the data from the ongoing and upcoming beam energy scan programs and for understanding microscopic properties of the QCD matter near equilibrium. ### 2.2 Equation of state at finite densities The finite-density version of a hybrid equation of state s95p-v1 [57] is one of the pioneering studies to use the coefficients of the Taylor expansion method for construction. A temperature shift was introduced to the susceptibilities estimated in lattice QCD calculations with larger than physical pion mass, which tend to produce a higher $T_{c}$ than those with physical pion mass, for smooth matching to the resonance gas results. As the lattice QCD calculations improved, one has begun to use the bare result of the baryon susceptibility in hydrodynamic simulations [138, 139]. The connection of the lattice QCD results using the physical pion mass to the hadron resonance gas results has been discussed at finite density of net baryons [96] and of net baryons, electric charge and strangeness [85, 105]. The implementation of a critical point in the equation of state is also a topic of importance [16, 140, 141]. The 3-dimensional Ising model is often used for this purpose because it is considered to be in the same universality class as QCD [52, 64]. The rescaled magnetic field and the reduced temperature in the latter model [142] are mapped onto the reduced temperature and (baryon) chemical potential in QCD, respectively. Experimental elucidation of the critical point is a long-standing goal to which no complete answer is available yet [143, 144]. Perturbative QCD calculations have also been improved to include higher order contributions [145, 146, 147, 148, 149, 150, 151], though the convergence of the weak-coupling expansion for the pressure is slow, even at the $g^{6}$ order, and the dependence on the renormalization scale is large [152]. In light of this situation, improved versions of the perturbation theory have been proposed, such as the two-particle irreducible (2PI) formalism [153, 154, 155] and hard thermal loop (HTL) perturbation theory [156, 157, 158]. There are quantitative studies to match perturbative results [159, 160, 161, 162, 163] to the hadron resonance gas ones with the help of lattice QCD data to approach the finite density regime [59]. The effective model approaches to the finite-density phase structure include the Nambu-Jona-Lasinio model with the Polyakov loop [164, 106, 165, 56], polyakov loop enhanced quark-meson models [66], and a more phenomenological quasi-particle model, [54] where the result of Ref. 56 has been employed in one of the first modern hydrodynamic simulations of the beam energy scan experiments [115]. An alternative approach to describe strongly-coupled matter is via holographic gauge-string duality [166, 167, 168]. The original anti-de Sitter/conformal field theory (AdS/CFT) correspondence is conjectured for the $\mathcal{N}=4$ super Yang-Mills theory, which is scale invariant and thus has no phase transition. The primary role of the conjecture in the phenomenology of nuclear collisions is perhaps the prediction of transport coefficients [169, 170, 171, 172], for which first principle calculations are difficult. Extensions of this method to non-conformal theories have been proposed in order to preserve consistency with the thermodynamic properties of QCD. Such examples include the Einstein-Maxwell-Dilation model [173, 174, 175] which can reproduce the known lattice QCD data. The QCD equation of state is also a topic of importance for compact stars. The typical chemical potential is larger and the temperature is smaller in such systems compared with those in nuclear collisions [176], though it may be possible to have occasional baryonic dense spots in the latter through event- by-event fluctuations. There have been extensive studies on the neutron star equation of state – see, e.g., Refs. 25, 26 for recent reviews. In addition to the intriguing observation regarding the Shapiro delay[177], the experimental discovery of gravity waves has brought a plethora of new data, which can constrain the nuclear equation of state in the cold and dense regime [24]. ## 3 NEOS – hybrid QCD equation of state We discuss the construction of the QCD equation of state at finite chemical potentials of net baryon number, electric charge, and strangeness. neos is an equation of state model, which takes one of the latest lattice QCD equations of state at vanishing chemical potentials as a baseline. Following the Taylor expansion method, the second- and fourth-order diagonal and off-diagonal susceptibilities are implemented. This expansion method has the advantage of being able to express the thermodynamic variables at finite density with those at zero density. On the other hand, one needs an additional prescription at low temperatures, because the Taylor expansion becomes less reliable when the chemical potential over temperature ratio is large [178]. Thus, the hadron resonance gas model, which is a framework to understand the low-temperature QCD system in terms of stable hadrons and meta-stable resonances, is used at lower temperatures, and its pressure is matched to the lattice-based pressure near the crossover. There are additional motivations for the connection procedure. First, all the thermodynamic variables and second- and fourth- order susceptibilities of the hadron resonance gas model are known to show excellent agreement with those of lattice QCD. The fact that the hadron resonance gas model shares basic thermodynamic properties with the first principle calculation motivates one to assume that the model captures essential physics. Second, the success of hydrodynamic modeling relies on the hadron resonance gas picture when the flow field is converted into hadronic particles.111It should be noted that the experimental data for particle spectra, chemical ratios, and chemical freeze- out indicate that the concept of temperature is valid near particlization in nuclear collisions, even though there are arguments regarding hydrodynamization without thermalization at the earliest stage of hydrodynamic evolution [179, 180, 181]. In anisotropic hydrodynamics thermodynamic variables can receive modifications from local momentum anisotropies [182, 183, 184, 185, 186]. The Cooper-Frye prescription [137] for particlization also requires that the equation of state in the hydrodynamic evolution is the same as that in the hadronic transport model to allow for energy-momentum and charge conservation. Direct use of the hadronic resonance gas equation of state at low temperatures is the most practical way to achieve this. ### 3.1 Lattice QCD equation of state in the Taylor expansion method The higher temperature side of the equation of state is constructed using the Taylor-expanded pressure $P_{\mathrm{lat}}$, $\displaystyle\frac{P_{\mathrm{lat}}}{T^{4}}$ $\displaystyle=$ $\displaystyle\frac{P_{0}}{T^{4}}+\sum_{l,m,n}\frac{\chi^{B,Q,S}_{l,m,n}}{l!m!n!}\bigg{(}\frac{\mu_{B}}{T}\bigg{)}^{l}\bigg{(}\frac{\mu_{Q}}{T}\bigg{)}^{m}\bigg{(}\frac{\mu_{S}}{T}\bigg{)}^{n},$ (1) where $P_{0}$ and $\chi^{B,Q,S}_{l,m,n}$ are the pressure and $(l+m+n)$-th order susceptibilities at zero chemical potentials, calculated in lattice QCD simulations. $T$ is the temperature and $\mu_{B}$, $\mu_{Q}$, $\mu_{S}$ are the chemical potentials for net baryon, electric charge, and strangeness, respectively. They are related to the quark chemical potentials as $\displaystyle\mu_{u}$ $\displaystyle=$ $\displaystyle\frac{1}{3}\mu_{B}+\frac{2}{3}\mu_{Q},$ (2) $\displaystyle\mu_{d}$ $\displaystyle=$ $\displaystyle\frac{1}{3}\mu_{B}-\frac{1}{3}\mu_{Q},$ (3) $\displaystyle\mu_{s}$ $\displaystyle=$ $\displaystyle\frac{1}{3}\mu_{B}-\frac{1}{3}\mu_{Q}-\mu_{S}.$ (4) The susceptibilities can be expressed as $\chi_{l,m,n}^{B,Q,S}=\frac{\partial^{l}\partial^{m}\partial^{n}P(T,\mu_{B},\mu_{Q},\mu_{S})/T^{4}}{\partial(\mu_{B}/T)^{l}\partial(\mu_{Q}/T)^{m}\partial(\mu_{S}/T)^{n}}\bigg{\|}_{\mu_{B,Q,S}=0}.$ (5) $l+m+n$ is constrained by the matter-antimatter symmetry to be even. One can alternatively consider isospin instead of electric charge. ### 3.2 Hadron resonance gas equation of state The hadron resonance gas picture is used for calculating the lower temperature side of the equation of state. Its pressure reads $\displaystyle P_{\mathrm{had}}$ $\displaystyle=$ $\displaystyle\pm T\sum_{i}\int\frac{g_{i}d^{3}p}{(2\pi)^{3}}\ln[1\pm e^{-(E_{i}-\mu_{i})/T}]$ (6) $\displaystyle=$ $\displaystyle\sum_{i}\sum_{k}(\mp 1)^{k+1}\frac{1}{k^{2}}\frac{g_{i}}{2\pi^{2}}m_{i}^{2}T^{2}e^{k\mu_{i}/T}K_{2}\bigg{(}\frac{km_{i}}{T}\bigg{)},$ where $E_{i}=\sqrt{p^{2}+m_{i}^{2}}$ is the energy, $m_{i}$ is the mass, $g_{i}$ is the degeneracy, and $\mu_{i}$ is the chemical potential of the $i$-th hadronic species. $\mu_{i}$ can be expressed as $\mu_{i}=B_{i}\mu_{B}+Q_{i}\mu_{Q}+S_{i}\mu_{S}$ using the quantum numbers $B_{i}$, $Q_{i}$, and $S_{i}$ for net baryon, electric charge, and strangeness. $K_{2}(x)$ is the modified Bessel function of the second kind. The expansion with $k$ takes account of the correction of quantum statistics. It is usually sufficient to consider the $k\leq 3$ terms for pions, the $k\leq 2$ terms for kaons, and the $k=1$ term for heavier particles. The Boltzmann limit corresponds to the $k=1$ case. We treat hadrons as on-shell particles and do not include spectral functions for the resonance states in our model [187]. ### 3.3 Hybrid equation of state The neos equation of state is obtained by connecting the ones from lattice QCD and the hadron resonance gas model. The pressure is given as $\displaystyle\frac{P}{T^{4}}$ $\displaystyle=$ $\displaystyle\frac{1}{2}[1-f(T,\mu_{B},\mu_{Q},\mu_{S})]\frac{P_{\mathrm{had}}(T,\mu_{B},\mu_{Q},\mu_{S})}{T^{4}}$ (7) $\displaystyle+$ $\displaystyle\frac{1}{2}[1+f(T,\mu_{B},\mu_{Q},\mu_{S})]\frac{P_{\mathrm{lat}}(T,\mu_{B},\mu_{Q},\mu_{S})}{T^{4}},$ where the connecting function $f$ should satisfy $f\to 1$ and $f\to-1$ in the high and low temperature limits, respectively. Here we choose a smooth hyperbolic function $f(T,\mu_{B},\mu_{Q},\mu_{S})=\tanh\bigg{[}\frac{T-T_{c}(\mu_{B})}{\Delta T_{c}}\bigg{]}.$ (8) $T_{c}(\mu_{B})$ is the connecting temperature for which we use $T_{c}(\mu_{B})=0.16\ \mathrm{GeV}-0.4\,(0.139\ \mathrm{GeV}^{-1}\mu_{B}^{2}+0.053\ \mathrm{GeV}^{-3}\mu_{B}^{4})$ motivated by the $\mu_{B}$ dependence of the chemical freeze-out line [188]. The connecting width is chosen to be $\Delta T_{c}=0.1T_{c}(0)$. The dependencies on electric charge and strangeness chemical potentials are assumed to be small and neglected here. The choices of possible parameter values and their effects are limited for the following reasons. First, the thermodynamic conditions $\displaystyle\frac{\partial^{2}P}{\partial T^{2}}$ $\displaystyle=$ $\displaystyle\frac{\partial s}{\partial T}>0,$ (9) $\displaystyle\frac{\partial^{2}P}{\partial\mu_{B,Q,S}^{2}}$ $\displaystyle=$ $\displaystyle\frac{\partial n_{B,Q,S}}{\partial\mu_{B,Q,S}}>0.$ (10) have to be imposed near the connection range because they would no longer be trivially satisfied when two different frameworks are being connected by another function. The procedure leaves a narrow window for the possible choice of parameters. Second, the fact that the lattice QCD and hadron resonance gas equations of state match over a finite temperature range implies that the overall thermodynamic properties of the system do not and should not depend on the detailed parameter choice. The above procedure gives a crossover equation of state by construction. One may argue that there could be a critical point in the accessible range of the QCD phase diagram. We consider the crossover-type equation of state here to allow for baseline hydrodynamic calculations without critical behavior. Future experimental observation of deviations from that baseline can then be analyzed to deduce the existence and location of the QCD critical point. If one introduces $f$ with a non-differentiable kink, an equation of state with a first-order phase transition is easily obtained [69]. It is useful here to introduce basic thermodynamic relations for estimating other macroscopic variables. The entropy density $s$, the net baryon, electric charge, and strangeness densities $n_{B,Q,S}$, the energy density $e$, and the sound velocity $c_{s}$ are obtained via $\displaystyle s$ $\displaystyle=$ $\displaystyle\left.\frac{\partial P}{\partial T}\right\|_{\mu_{B},\mu_{Q},\mu_{S}},\ \ n_{B}=\left.\frac{\partial P}{\partial\mu_{B}}\right\|_{T,\mu_{Q},\mu_{S}},$ (11) $\displaystyle n_{Q}$ $\displaystyle=$ $\displaystyle\left.\frac{\partial P}{\partial\mu_{Q}}\right\|_{T,\mu_{B},\mu_{S}},\ \ n_{S}=\left.\frac{\partial P}{\partial\mu_{S}}\right\|_{T,\mu_{B},\mu_{Q}},$ (12) $\displaystyle e$ $\displaystyle=$ $\displaystyle Ts-P+\mu_{B}n_{B}+\mu_{Q}n_{Q}+\mu_{S}n_{S},$ (13) $\displaystyle c_{s}^{2}$ $\displaystyle=$ $\displaystyle\left.\frac{\partial P}{\partial e}\right\|_{n_{B},n_{Q},n_{S}}+\frac{n_{B}}{e+P}\left.\frac{\partial P}{\partial n_{B}}\right\|_{e,n_{Q},n_{S}}$ (14) $\displaystyle+$ $\displaystyle\frac{n_{Q}}{e+P}\left.\frac{\partial P}{\partial n_{Q}}\right\|_{e,n_{B},n_{S}}+\frac{n_{S}}{e+P}\left.\frac{\partial P}{\partial n_{S}}\right\|_{e,n_{B},n_{Q}},$ for the system with multiple conserved charges. ### 3.4 Multiple charges in nuclear collisions The strangeness density in nuclear collisions on average is vanishing because the colliding nuclei are net strangeness free. This is called the strangeness neutrality condition. The condition leads to positive strangeness chemical potential in the presence of positive baryon chemical potential, because the number of strange quarks would exceed that of anti-quarks in the QGP phase if $\mu_{S}=0$ was assumed. An interpretation based on the parton picture is that $\mu_{S}\sim\mu_{B}/3$ follows from Eq. (4) when $\mu_{Q}\sim 0$. For the hadronic phase, the strangeness chemical potential can be suppressed because the lightest baryon with strangeness is $\Lambda$, the mass of which is already large compared with the temperature of the system. The electric charge density is related to the net baryon density via the proton-to-nucleon number ratio $Z/A$. $Z/A$ of the nuclei used or planned in the collider experiments at RHIC and LHC are listed in Table 3.4. The primarily-used heavy ions Au and Pb have $Z/A\approx 0.4$. For neutron-rich nuclei, the chemical potential of $d$ quarks is larger than that of $u$ quarks, i.e., $\mu_{d}=\mu_{B}/3-\mu_{Q}/3>\mu_{u}=\mu_{B}/3+2\mu_{Q}/3$, which implies that $\mu_{Q}<0$ when $\mu_{B}>0$ in the QGP phase. The trend remains in the hadronic phase because negative pions would be abundant compared with positive pions, which leads to $\mu_{\pi^{-}}=-\mu_{Q}>\mu_{\pi^{+}}=\mu_{Q}$. One would have the opposite situation $\mu_{Q}>0$ for proton-rich nuclei, which are relevant in smaller systems. Number ratios of protons to nucleons $Z/A$ for the nuclei used or planned at RHIC and LHC. Nucleus ${}^{1}_{1}$H ${}^{2}_{1}$H ${}^{3}_{2}$He ${}^{8}_{16}$O ${}^{27}_{13}$Al ${}^{63}_{29}$Cu $Z/A$ 1.000 0.500 0.667 0.500 0.481 0.460 Nucleus ${}^{96}_{40}$Zr ${}^{96}_{44}$Ru ${}^{127}_{54}$Xe ${}^{197}_{\ 79}$Au ${}^{208}_{\ 82}$Pb ${}^{238}_{\ 92}$U $Z/A$ 0.417 0.458 0.425 0.401 0.394 0.387 ### 3.5 Numerical construction Results of (2+1)-flavor lattice QCD simulations are used to evaluate the pressure [48] and the second- and fourth-order susceptibilities [87, 88, 89, 90] at vanishing densities in the numerical construction of the hybrid equation of state. In addition, $\chi_{6}^{B}$, $\chi_{5,1}^{B,Q}$, and $\chi_{5,1}^{B,S}$ of the sixth-order susceptibilities are phenomenologically introduced for a proper matching of the thermodynamic variables because the results of the Taylor expansion method of lattice QCD simulations cannot be naïvely used when they have large error bars, as small displacement of the crossover temperature can lead to unphysical gaps in thermodynamic quantities when $\mu_{B}/T$ is large. The Stefan-Boltzmann limits are used as anchors on the high temperature side so the basic thermodynamic properties are preserved when lattice QCD data points are scarce. Those treatments could be improved in the future when more data become available. The functional forms for the parametrization of all the susceptibilities used in the model are found in Ref. 85. The hadron resonance gas model includes all the hadrons and resonances which have $u$, $d$ and/or $s$ as constituent components and have masses smaller than 2 GeV in the Particle Data Group list [187]. The pressure and susceptibilities up to the fourth order are found to agree well with those of lattice QCD calculations. The following three situations are simulated: (i) the conventional situation $\mu_{S}=\mu_{Q}=0$ where only the net baryon number is considered as conserved charge, (ii) the situation with the strangeness neutrality condition $n_{S}=0$ and vanishing electric charge chemical potential $\mu_{Q}=0$, and (iii) the realistic situation in collisions of heavy nuclei where $n_{S}=0$ and $n_{Q}=0.4\,n_{B}$. They are labeled as neos B, neos BS, and neos BQS, respectively, in the article. Figure 3: (a) The dimensionless pressure $P/T^{4}$ and (b) the dimensionless strangeness density $-n_{S}/T^{3}$ of neos B, (c) the dimensionless pressure $P/T^{4}$ and (d) the strangeness chemical potential $\mu_{S}$ of neos BS, and (e) the dimensionless pressure $P/T^{4}$ and (f) the electric charge chemical potential $-\mu_{Q}$ of neos BQS as functions of $T$ and $\mu_{B}$ [85]. The solid, long-dashed, dash-dotted, and short-dashed lines indicate the constant $s/n_{B}$ trajectories at 420, 144, 51, and 30, respectively. The dimensionless pressure $P/T^{4}$ as a function of $T$ and $\mu_{B}$ is shown in Fig. 3 (a) where $\mu_{S}=\mu_{Q}=0$ (neos B). The trajectories of the constant entropy density to net baryon density ratio $s/n_{B}$ indicate the typical trajectory in the $T$-$\mu_{B}$ plane explored by collider experiments at each center-of-mass energy, because the net baryon density and – in the ideal hydrodynamic approximation – entropy density are conserved during the hydrodynamic evolution. $s/n_{B}=420,144,51$, and $30$ correspond to $\sqrt{s_{NN}}=200,62.4,19.6$, and $14.5$ GeV [76], respectively. It should be noted that there will be a range of $s/n_{B}$ for every collision since the medium is spatially inhomogeneous. Also, event-by-event fluctuations further smear the trajectories on the phase diagram. $\mu_{B}/T$ is fixed on those trajectories when $s\sim T^{3}$ and $n_{B}\sim\mu_{B}T^{2}$ in the QGP phase. Once the trajectories enter the hadronic phase, they are bent toward larger $\mu_{B}$ because protons, the lightest baryons, are considerably heavier than pions. While this situation leads to a thermodynamically consistent crossover equation of state, it does not reflect the situation in nuclear collisions because the strangeness neutrality condition is violated as shown in Fig. 3 (b). The negative strangeness density is consistent with the expectation that positive $\mu_{B}$ leads to a system with more $s$ quarks and fewer $\bar{s}$ quarks. It approaches zero on the low temperature side because the lightest hadrons with strangeness are kaons, whose mass is non-negligible in the hadronic phase. Once the strangeness neutrality condition $n_{S}=0$ is imposed, the pressure is meaningfully modified in the region where the $\mu_{B}/T$ is relatively large, as demonstrated in Fig. 3 (c). Figure 3 (d) shows that the strangeness neutrality condition leads to positive strangeness chemical potentials. The trajectories are shifted to the larger $\mu_{B}$ side by about 50% in the QGP phase, because only $u$ and $d$ quarks contribute to $n_{B}$ in neos BS instead of $u,d$, and $s$ quarks in neos B, because in strangeness neutral systems strange quarks and antiquarks do not contribute to the net baryon number. The larger values of $\mu_{B}$ can be important in the hydrodynamic model because baryon diffusion, which is primarily driven by the spatial gradient of $\mu_{B}/T$ [138, 189, 99], would be enhanced. The differences of the trajectories in neos B and BS are smaller in the hadronic phase because, as mentioned earlier, the lightest hadron with net baryon number and strangeness is the $\Lambda$ baryon, which is already heavy compared with the medium temperature. Figure 4: The thick solid, thin solid, and thick dotted lines are the sound velocity squared of neos B, BS, and BQS, respectively, as a function of temperature along the trajectories of (a) $s/n_{B}=420$ and (b) $s/n_{B}=30$. [85]. Finally, we study the situation of matter with fixed electric charge-to-baryon ratio $n_{Q}/n_{B}=0.4$ and strangeness neutrality. Shown in Fig. 3 (e) and (f) are the dimensionless pressure and electric chemical potential, respectively. The pressure does not change much going from neos BS to BQS and neither do the trajectories, because $\mu_{Q}=0$ implies $n_{Q}/n_{B}\sim 0.5$ which happens to be not too far from the more realistic situation. The negative electric chemical potential, nevertheless, is important in heavy-ion phenomenology, as it presents a quantitative explanation for the abundance of negative pions over positive pions observed in the experiments [190, 191, 192]. Figure 4 shows the sound velocity as a function of temperature for neos B, BS, and BQS. The sound velocity has a minimum because the pressure does not change significantly as a function of the energy density in the vicinity of the quark-hadron crossover (Eq. 14). Comparing the low baryon density ($s/n_{B}=420$) and high baryon density ($s/n_{B}=30$) results, the sound velocity in the hadronic phase is found to be suppressed and the minimum is shifted toward the lower temperature side for larger densities by a few MeV. The strangeness neutrality condition slightly increases the sound velocity, while the realistic electric charge-to-baryon ratio leads to negligible change. The quantity approaches the Stefan-Boltzmann limit $c_{s}^{2}=1/3$ at high temperatures, it reaches 94.8% at $T=0.6$ GeV and 97.2% at $T=0.8$ GeV of the limit for $s/n_{B}=420$. Figure 5: Isopressure planes in the chemical potential space in (a) the hadronic phase where $P/T^{4}=0.8$ and $T=0.14$ GeV and (b) the QGP phase where $P/T^{4}=2$ and $T=0.2$ GeV [85]. The isopressure surface at constant temperatures in the chemical potential space is investigated to illustrate the interplay of multiple conserved charges. The numerical result in the hadronic phase where $P/T^{4}=0.8$ and $T=0.14$ GeV is shown in Fig. 5 (a). The intercepts can be defined as $P(\mu_{B}^{\mathrm{int}},0,0)=P(0,\mu_{Q}^{\mathrm{int}},0)=P(0,0,\mu_{S}^{\mathrm{int}})$. They are ordered as $\mu_{B}^{\mathrm{int}}>\mu_{S}^{\mathrm{int}}>\mu_{Q}^{\mathrm{int}}$, reflecting the mass ordering of the lightest hadrons to carry the respective charges, $m_{p}>m_{K}>m_{\pi}$. The situation is different in the QGP phase as shown in Fig. 5 (b) where $P/T^{4}=2$ and $T=0.2$ GeV. The intercept ordering $\mu_{B}^{\mathrm{int}}>\mu_{Q}^{\mathrm{int}}>\mu_{S}^{\mathrm{int}}$ is consistent with a parton gas interpretation that $\mu_{B}^{\mathrm{int}}/3\sim 2\mu_{Q}^{\mathrm{int}}/3\sim\mu_{S}^{\mathrm{int}}$, though $\mu_{S}^{\mathrm{int}}$ is not as small owing to the fact that the strange quark mass is not negligible at the chosen temperature. Figure 6: The solid, long-dashed, dash-dotted, and short-dashed lines indicate the constant $s/n_{B}$ trajectories at 420, 144, 51, and 30, respectively, in the $\mu_{B}$-$\mu_{S}$-$\mu_{Q}$ space [85]. The constant $s/n_{B}$ trajectories of neos BQS are plotted in the chemical potential space to illustrate typical regions explored in nuclear collisions (Fig. 6). The end with larger values of $\mu_{S}$ and $\|\mu_{Q}\|$ corresponds to the high temperature region. One can see that the trajectories form a straight line in the QGP phase, because of the constraints $n_{S}=0$ and $n_{Q}=0.4n_{B}$ under the leading order approximation of the partonic results [85] $\displaystyle\begin{pmatrix}n_{B}\\\ n_{Q}\\\ n_{S}\end{pmatrix}=T^{2}\begin{pmatrix}\chi_{2}^{B}&\chi_{1,1}^{B,Q}&\chi_{1,1}^{B,S}\\\ \chi_{1,1}^{B,Q}&\chi_{2}^{Q}&\chi_{1,1}^{Q,S}\\\ \chi_{1,1}^{B,S}&\chi_{1,1}^{Q,S}&\chi_{2}^{S}\end{pmatrix}\begin{pmatrix}\mu_{B}\\\ \mu_{Q}\\\ \mu_{S}\end{pmatrix},$ (15) which leads to $\mu_{B}=4.6n_{B}/T^{2}$, $\mu_{Q}=-0.2n_{B}/T^{2}$, and $\mu_{S}=1.6n_{B}/T^{2}$. As mentioned earlier, they deviate from the straight line in the hadronic phase in the direction of larger baryon chemical potentials because of the mass difference between protons and pions. The second bend towards larger strangeness chemical potential near the low temperature end is induced by the mass difference between kaons and pions. It is important to note that one does not explore the $T$-$\mu_{B}$ plane but the $T$-$\mu_{B}$-$\mu_{Q}$-$\mu_{S}$ space in nuclear collider experiments. This should be taken into account when analyzing the experimental data to learn about the phase structure of QCD. ### 3.6 Applications to nuclear collisions The phenomenological consequences of the conditions of strangeness neutrality and electric charge-to-baryon ratio of heavy nuclei are studied by using the hydrodynamic model [193] of relativistic nuclear collisions. We consider Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV as conducted at the CERN SPS [191, 194, 195, 190, 196, 197]. The initial conditions for the hydrodynamic model are calculated using an event-by-event dynamical Glauber model [98]. The Glauber model is a framework that provides initial geometrical configurations in the transverse plane based on the Woods-Saxon potential and inelastic nucleon-nucleon cross section [198]. Its improved version, the Glauber-Lexus model [96], takes into account the exchange of longitudinal momentum [199]. The dynamical Glauber model is the four-dimensional version in the sense that the energy and net baryon number densities are introduced to the system as each sub-collision of target and projectile nucleons occurs over time. The numerical implementation music [200, 28, 201] is used to perform the three-dimensional hydrodynamic simulation. A simple choice of transport coefficients, namely a shear viscosity of $\eta/s=0.08$ and vanishing bulk viscosity and baryon diffusion, is employed to minimize ambiguities. Particlization is assumed to occur on a surface of constant energy density, defined by the switching energy density $e_{\textrm{sw}}$. Particles are then further evolved using the hadron cascade model Ultra-relativistic Quantum Molecular Dynamics (UrQMD) [202, 203]. Figure 7: (Left) The hadronic yields for the most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV calculated with neos B, BS, and BQS represented by circular, triangular and cross symbols, respectively. [85]. (Right) The particle-antiparticle ratios with the same conditions compared with the experimental data [204]. List of hadronic chemical potentials. Hadrons Chemical potentials $\pi^{+}$ $\mu_{Q}$ $K^{+}$ $\mu_{Q}+\mu_{S}$ $\phi$ 0 $p$ $\mu_{B}+\mu_{Q}$ $\Lambda$ $\mu_{B}-\mu_{S}$ $\Xi^{-}$ $\mu_{B}-\mu_{Q}-2\mu_{S}$ $\Omega$ $\mu_{B}-\mu_{Q}-3\mu_{S}$ The simulated yields of particles and antiparticles in most central events (using $e_{\textrm{sw}}=0.26$ GeV/fm3) are shown in Fig. 7 (left) and their ratios in Fig. 7 (right) for the three different versions of the equation of state. Comparison of neos B and BS results to the experimental data from SPS shows that the strangeness neutrality condition improves the description of the particle-antiparticle ratios of the hadrons with finite strangeness chemical potential, $K$, $\Lambda$, $\Xi$, and $\Omega$. The antiproton-proton ratio is also modified and moves closer to the data because of the aforementioned enhancement in the baryon chemical potential, when imposing strangeness neutrality. The differences between the neos BS and BQS results are rather small because the electric chemical potential is small for the collisions of heavy nuclei. As mentioned before, it is still phenomenologically important because it explains the experimental result that the anti-pion to pion ratio is greater than one. A list of particle species along with the chemical potentials that affect their respective yields is given in Table 3.6. In Fig. 8, the switching energy density dependence is studied for particle yields (left) and ratios (right) using neos BQS. Chemical equilibrium is assumed down to lower temperatures when a lower switching energy density is considered. The results mostly agree with the experimental data when $e_{\textrm{sw}}=0.16$-$0.36$ GeV/fm3. The yields of antibaryons are most affected as $e_{\textrm{sw}}$ decreases, possibly because of the interplay of the enhancement of the baryon chemical potential at particlization (see Fig. 3 (e)) and the suppression of heavier particle production in the thermal bath. The two effects tend to cancel for baryons while they are additive for antibaryons. Figure 8: (Left) The hadronic yields for the most central Pb+Pb collisions at $\sqrt{s_{NN}}=17.3$ GeV estimated with neos BQS at $e_{\textrm{sw}}=0.16,0.26$, and $0.36$ GeV/fm3 represented by circular, triangular and cross symbols, respectively. [85]. (Right) The particle- antiparticle ratios with the same conditions compared with experimental data [204]. ## 4 Comparison of equation of state models We numerically compare different models of the QCD equation of state used in hydrodynamic simulations of relativistic nuclear collisions. Then effects of the differences on hydrodynamic evolution are investigated. ### 4.1 Thermodynamic properties Shown in Fig. 9 (a) are the trace anomalies $(e-3P)/T^{4}$ from lattice QCD simulations and several equation of state models alongside neos, that we discussed in the previous section. neos and Duke [205] equations of state show agreement with the continuum limit result of the HotQCD Collaboration, on which their structure is based. Similarly, BEST [64] and University of Houston (denoted as UH) [105] equations of state agree with the results from the Wuppertal-Budapest (WB) Collaboration, utilized for their construction. s95p-v1 (s95p) [132] is one of the earliest works on the hybrid equation of state and the deviation from the rest of the models may be owing in part to the difference in the lattice QCD data used. Its parametrization and matching procedure are also different owing to the now-resolved discrepancy between the resonance gas and early lattice data with non-physical pion mass. An updated version, employing more recent lattice QCD results, s83s18, has recently been released [135]. It is noteworthy that Duke and s95p results are very similar in the hadronic phase. Finally, we point out that the two shown lattice QCD results for the trace anomaly in the continuum limit are consistent. The sound velocities are shown in Fig. 9 (b). The basic structure of having a minimum of $c_{s}^{2}$ near the crossover is found in all models and lattice simulations. The exact location of the minimum is sensitive to the details of the construction of each model, such as the connecting temperature and width. Again, by construction neos and Duke equations of state agree with the HotQCD result – and BEST and UH equations of state with the WB result – at higher temperatures. Figure 9: (a) Comparison of the trace anomalies from lattice QCD simulations and lattice QCD based equation of state models. (b) Comparison of the sound velocities extracted from the equation of state models and lattice QCD calculations. The comparison of the trajectories on the phase diagram for constant $s/n_{B}=94$, which approximately corresponds to the collision energy of $\sqrt{s_{NN}}=39$ GeV [206], is shown in Fig. 10 (a) to illustrate the properties of the equations of state at finite density. The phase trajectories for neos B and BEST as well as those for neos BQS and UH BQS behave similarly with small differences in the QGP phase, which may come from the difference in the lattice QCD data employed. The effect of additional charges to the baryon chemical potential is as discussed in the previous section. The difference between neos BQS and UH BQS results in the hadronic phase may come from the difference in the hadronic components used in the resonance gas [207] and the structural difference that the UH BQS equation of state is expanded up to the second and fourth order in $\mu_{B,Q,S}/T$ in the hadronic phase to perform matching to the lattice data in the susceptibilities, whereas neos uses the hadron resonance gas without truncation as the matching is done for the pressure. Figure 10 (b) shows the sound velocities at finite densities on the $s/n_{B}=94$ trajectories. One can see that they are sensitive to the details of the equation of state used. The results obtained with neos B and BQS are similar as previously observed in Fig. 4. It should be noted that the small wiggles in UH BQS and BEST equations of state at low temperature are artifacts caused by a cut-off at $\mu_{B}=0.45$ GeV. Figure 10: (a) Comparison of the phase trajectories for constant $s/n_{B}=94$ from BEST equation of state vs. neos B and UH BQS vs. neos BQS. (b) Comparison of the sound velocities extracted from the equation of state models. ### 4.2 Hydrodynamic evolution We now compare the hydrodynamic evolution in heavy ion collisions with different equations of state. Figure 11 (a) shows the time-evolution of the average time-like flow component $\langle u^{\tau}\rangle$ in one 30-40% Au+Au collision at $\sqrt{s_{NN}}=200$ GeV with the IP-Glasma initial condition [208, 209]. The quantity is closely related to radial flow, which affects the slope of the particles’ transverse momentum spectra ($u^{\tau}$ is closely related to the transverse flow velocity $u_{\perp}$ via the flow normalization condition $u\cdot u=1$, particularly when neglecting longitudinal flow $u^{\eta}$). $\langle u^{\tau}\rangle$ increases with time and exhibits similar behavior in all cases. neos BQS and Duke equations of state lead to similar results. The BEST and UH BQS equations of state lead to larger $\langle u^{\tau}\rangle$, while $\langle u^{\tau}\rangle$ of the s95p equation of state is smaller than that for neos BQS and Duke at later times, but is slightly larger at earlier times before around $\tau=2$ fm. The orderings are consistent with those of the sound velocity and trace anomaly, considering that the average medium temperature is larger ($\sim 0.4$ GeV) at earlier times and smaller ($\sim 0.2$ GeV) at later times. The time evolution of the system’s averaged momentum anisotropy $\varepsilon_{p}=\frac{\sqrt{\langle T^{xx}-T^{yy}\rangle^{2}+\langle 2T^{xy}\rangle^{2}}}{\langle T^{xx}+T^{yy}\rangle},$ (16) in the same hydrodynamic setup is shown in Fig. 11 (b). This quantity is closely related to the final elliptic momentum anisotropy of produced particles. The differences in the momentum anisotropy between the equation of state models are rather small. The s95p result rises and falls slightly earlier than the others. The UH BQS equation of state has the largest momentum anisotropy at later times, followed by the BEST equation of state. neos BQS and Duke equations of state have similar $\varepsilon_{p}$, though the former is slightly larger than the latter around $\tau=4$ fm. Figure 11: (a) The time evolution of averaged $u^{\tau}$ for an Au+Au collision in 30-40% centrality at 200 GeV with different equations of state. (b) Similar comparison for the time evolution of the momentum anisotropy $\varepsilon_{p}$. We make a similar comparison at finite net baryon density by simulating (3+1)D hydrodynamic evolution for 20-30% Au+Au collisions at 39 GeV with the event- averaged initial condition [102]. Figure 12 shows that the four equations of state produce a very similar evolution for the development of hydrodynamic radial flow and the momentum anisotropy. Similar to the zero density case, the larger speed of sound in the BEST equation of state leads to slightly stronger radial flow compared to the neos and UH equations of state in Fig. 12(a). The system’s momentum anisotropy at late time has the order BEST $>$ UH BQS $>$ neos. Figure 12: (a) The time evolution of averaged $u^{\tau}$ for an Au+Au collision in 20-30% centrality at 39 GeV with different equations of state. (b) Similar comparison for the time evolution of the momentum anisotropy $\varepsilon_{p}$. Figure 13: The averaged phase trajectories for a fireball at mid-rapidity in 20-30% Au+Au collisions at 39 GeV. Finally, we show the phase trajectories of mid-rapidity Au+Au collisions with the four equations of state in Figure 13. These trajectories are averaged over fluid cells from realistic (3+1)D hydrodynamic simulations. The difference among the four equations of state are in qualitative agreement with the difference in constant $s/n_{B}$ trajectories shown in Fig. 10. The strangeness neutrality condition moves the trajectories towards larger $\mu_{B}$ compared to those without this constraint. It indicates that, as mentioned earlier, having multiple conserved charges is phenomenologically important for the exploration of the QCD phase diagram, including the critical point search, as well as for the estimation of dissipative processes, such as baryon diffusion. The trajectory from the UH BQS has slightly larger $\mu_{B}$ values compared to the neos BQS in the QGP phase. ## 5 Conclusion and summary We reviewed QCD equations of state at finite chemical potentials. All current models for the equation of state generally agree at zero densities, because of the advances in lattice QCD simulations, which all agree now that the quark- hadron transition is a crossover at around $T=155$-$160$ MeV, and the information from lattice QCD is used as input to determine parameters in the various phenomenological models. On the other hand, the finite-density structure of the QCD phase diagram, such as the critical point and first-order phase transition, is less well understood. Going beyond zero density is not directly possible on the lattice due to the fermion sign problem. Besides lattice based methods such as Taylor expansion or the use of imaginary chemical potentials, various approaches to obtain a finite density QCD equation of state have been proposed, including the perturbative QCD method, the Polyakov loop-extended Nambu-Jona-Lasinio model, and holographic conjecture. We have introduced the neos model where the three conserved charges in the strongly-interacting medium – net baryon, electric charge and strangeness – are explicitly considered. The model is built from a state-of-the-art lattice QCD equation of state and the second- and fourth-order susceptibilities from the lattice, together with the hadron resonance gas result, which includes all known hadrons and resonances with masses below 2 GeV. Lattice and hadron gas equations of state are connected near the quark-hadron transition in a thermodynamically consistent way to obtain a crossover-type equation of state at finite temperatures and chemical potentials. We have considered the strangeness neutrality condition $n_{S}=0$ and the electric charge-to-baryon ratio $n_{Q}=0.4n_{B}$, that reflect the situation in collisions of heavy nuclei to elucidate the effects of multiple conserved charges. The multi-dimensional neos QCD equation of state has been included in the viscous hydrodynamic model of heavy-ion collisions music at intermediate relativistic energies. We showed in the comparison of the theoretical predictions with SPS experimental data of particle-antiparticle ratios that the model description is visibly improved when the strangeness neutrality condition is imposed for the hadrons with finite strangeness quantum numbers and – through the interplay of conserved charges – also for those with finite baryon number. The realistic electric charge-to-baryon ratio induces smaller effect because the electric charge chemical potential is small when heavy stable nuclei such as Au and Pb are used. Nevertheless, its inclusion leads to a correct description of the antipion-to-pion number ratio exceeding one in the observed data. Our results also clarify that in the beam energy scan one is really exploring the $T$-$\mu_{B}$-$\mu_{Q}$-$\mu_{S}$ phase diagram, instead of just one in the $T$-$\mu_{B}$ plane. This is important when extracting the information of the QCD medium properties and phase structures from experimental data. We have then compared several models of the QCD equation of state used in relativistic hydrodynamic studies of nuclear collisions. In the zero density limit, the continuum limit results for the trace anomaly and sound velocity of the Wuppertal-Budapest collaboration and the HotQCD collaboration agree within their error bands. neos and Duke equations of state exhibit good agreement with the latter results while BEST and UH equations of state with the former results, as expected from their construction. The widely-used s95p-v1 has a larger trace anomaly and smaller sound velocity in the crossover region, and newer versions, such as s83s18, should be used. The comparison of the constant $s/n_{B}$ trajectories in the phase diagram with different equations of state demonstrated that neos B and BEST equations of state are close to each other at finite density. Once the strangeness neutrality condition and the realistic charge-to-baryon ratio is taken into account, the trajectories are shifted to larger $\mu_{B}$. neos BQS and UH BQS equations of state behave similarly in the QGP phase with a slight difference coming likely from the choice of lattice data. The sound velocity differs between the two equations of state in the vicinity of the crossover, possibly because of the difference in the connection of the hadron resonance gas to the lattice QCD results. We performed several hydrodynamic simulations of heavy ion collisions using different equations of state and studied their effect on the time evolution of flow velocities and momentum anisotropies. At $\sqrt{s_{NN}}=200$ GeV, the time evolution of the averaged time-like flow component $\langle u^{\tau}\rangle$ is in the order of BEST, UH BQS, neos BQS, Duke, and s95p, from the fastest to the slowest buildup (and largest to smallest final values), which is consistent with the ordering of the sound velocity in the zero density case near the crossover temperature. Comparing the averaged momentum anisotropies, final values for UH BQS and BEST are larger than for neos BQS and Duke, which in turn are larger than those for s95p. The differences in $\langle u^{\tau}\rangle$ at finite density at $\sqrt{s_{NN}}=39$ GeV is rather small, but if closely observed, BEST produces the largest, followed by UH BQS and neos at later times because of the differences in the speed of sound. The average momentum anisotropy is also ordered similarly, but BEST and UH BQS are closest to each other. We also studied the trajectories of the average temperature and baryon chemical potential of the system, and found them to be qualitatively consistent with the constant $s/n_{B}$ trajectories of the corresponding equation of state models. Progress in determining the nuclear equation of state at finite densities as been significant in the last several years. Advances in lattice QCD have allowed to move towards realistic modeling with finite chemical potentials and even studies including potential critical points are possible. Experimental advances have also been tremendous, both on the front of heavy ion collision beam energy scans and gravitational wave observations of neutron star (and black hole) binary systems, which will allow for ever improving constraints on the nuclear equation of state over a wide range in the phase diagram. A public version of the neos tabulated results is available online [210] for the use in relativistic hydrodynamic models and other related studies. Other codes/data are also publicly available for BEST [211], UH BQS [212], s95p-v1 [213], s83s18 [214], and Duke [215] equations of state. ## Acknowledgments The authors thank Frithjof Karsch, Swagato Mukherjee, and Sayantan Sharma for useful discussions. AM is supported by JSPS KAKENHI Grant Number JP19K14722. BPS is supported under DOE Contract No. DE-SC0012704. CS is supported in part under DOE Contract No. DE-SC0013460 and in part by the National Science Foundation (NSF) under grant number PHY-2012922. This research used resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. 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11institutetext: Institute for Health and Society, Medical College of Wisconsin, 8701 Watertown Plank Rd, Wauwatosa, WI, 53226 22institutetext: Department of Statistics, University of Missouri, 600 S State St., Apt. 408 Bellingham, WA 98225 # Most Powerful Test Sequences with Early Stopping Options Sergey Tarima Nancy Flournoy (Received: date / Accepted: date) ###### Abstract Sequential likelihood ratio testing is found to be most powerful in sequential studies with early stopping rules when grouped data come from the one- parameter exponential family. First, to obtain this elusive result, the probability measure of a group sequential design is constructed with support for all possible outcome events, as is useful for designing an experiment prior to having data. This construction identifies impossible events that are not part of the support. The overall probability distribution is dissected into stage specific components. These components are sub-densities of interim test statistics first described by Armitage, McPherson and Rowe (1969) that are commonly used to create stopping boundaries given an $\alpha$-spending function and a set of interim analysis times. Likelihood expressions conditional on reaching a stage are given to connect pieces of the probability anatomy together. The reduction of the support caused by the adoption of an early stopping rule induces sequential truncation (not nesting) in the probability distributions of possible events. Multiple testing induces mixtures on the adapted support. Even asymptotic distributions of inferential statistics are mixtures of truncated distributions. In contrast to the classical result on local asymptotic normality (Le Cam 1960), statistics that are asymptotically normal without stopping options have asymptotic distributions that are mixtures of truncated normal distributions under local alternatives with stopping options; under fixed alternatives, asymptotic distributions of test statistics are degenerate. ###### Keywords: Adaptive designs adapted support group sequential designs local asymptotics interim hypothesis testing likelihood ratio tests ## 1 Introduction We define a _sequential experiment_ to be one in which the decision to stop collecting data is based on data collected previously in the study. Wetherill and Glazebrook (1986) emphasize that ” _two aspects of a sequential procedure must be clearly distinguished, the stopping rule, and the manner in which inferences are made once observations are stopped. $\ldots$ in the design problem, it is important to know how probable are various possible results.”_ We distinguish the probability framework underpinning these two activities, and also a third - the probability framework underlying interim hypothesis tests. Dodge and Romig (1929) proposed the first known sequential test procedure in which a decision to stop or continue collecting data was based on prior data, recognizing that decisions to stop or continue a trial made based on prior observations could substantially reduce the expected numbers of required subjects; it was a two-stage design. Bartky (1943) devised a multiple sequential testing procedure for binomial data based on Neyman and Pearson (1933)’s likelihood ratio test that Wald (1947) cites as a ”forerunner” to his more general _sequential probability ratio test_ (SPRT) procedure in which the probabilities of type I and II errors are controlled. Extensions with stopping decisions based on groups of subjects [_Group Sequential Designs (GSDs)_] are given in Jennison and Turnbull (1999). Neyman and Pearson (1933) show that likelihood ratio tests are most powerful for testing a simple null versus a simple alternative hypotheses. In Ferguson (2014), the Karlin-Rubin theorem is viewed as an extension of Neyman-Pearson approach to most powerful testing of composite hypotheses. The Karlin-Rubin theorem applies to the one-dimensional exponential family. With sequential stopping options, some elements of the sample space become impossible, which changes the distributions of statistics. Sufficient statistics become dependent on the random sample size; see Blackwell (1947). If statistics belong to the exponential family without early stopping options, then they belong to a curved exponential family when exposed to early stopping options (Efron (1975); Liu and Hall (1999); Liu et al. (2006)). Section 3.2 shows that despite being from a curved exponential family, the sequential tests based on likelihood ratios continue to be most powerful for any $\alpha$-spending function. ### 1.1 Notation and A Simple Example This example demonstrates a couple important repercussions of sequential stopping rules: * • The support is reduced, * • Bivariate normal random variables become non-observable; the observable bivariate random variable is a mixture of truncated normal random variables, Let $X_{1}$ and $X_{2}$ be $N(\theta,1)$ random variables with an unknown location parameter $\theta$. #### 1.1.1 Non-sequential experiments If $X_{1}=x_{1}$ alone is observed, the log-likelihood $l(\theta\|X_{1}=x_{1})=-2\log\sqrt{2\pi}-0.5(x_{1}-\theta)^{2}$ is maximized at $\widehat{\theta}_{1}=x_{1}$. If both $X_{1}=x_{1}$ and $X_{2}=x_{2}$ are observed independently, the random variable $(X_{1},X_{2})$ is defined on the probability space $\left(R^{2},{\cal{B}},P\right)$, where $R^{2}$ is the sample space [$R=\left(-\infty,\infty\right)$], ${\cal{B}}$ is the Borel $\sigma$-algebra on $R^{2}$ and $P$ is a bivariate normal distribution with mean vector $(\theta,\theta)$, units variances and zero correlation. The log-likelihood function is $l(\theta\|X_{1}=x_{1},X_{2}=x_{2})=-2\log\sqrt{2\pi}-0.5(x_{1}-\theta)^{2}-0.5(x_{2}-\theta)^{2}$ and the maximum likelihood estimator (MLE) of $\theta$ is $\widehat{\theta}_{2}=(x_{1}+x_{2})/2$. #### 1.1.2 Sequential experiments: likelihood, support and probability measures What happens in sequential settings when $X_{2}$ is only observed if $X_{1}<2.18$? Let $D$ denote the random stopping stage. Then $D=1+I(X_{1}<2.18)$, where $I(\cdot)$ is an indicator function. In this simple example, $D$ is also the random sample size. The joint distribution of (X,D) and the marginal distribution of X are the same: $\displaystyle f_{\textbf{X}}\left(\textbf{x}\|\theta\right)=f_{\textbf{X},D}\left(\textbf{x},d\|\theta\right)$ $\displaystyle=\phi(x_{1}-\theta)\left[\phi(x_{2}-\theta)\right]^{I\left(d=2\right)}$ (1) $\displaystyle=[\phi(x_{1}-\theta)]^{I(d=1)}\left[\phi(x_{1}-\theta)\phi(x_{2}-\theta)\right]^{I\left(d=2\right)},$ where $\phi(\cdot)$ is the standard normal density. The representation of $f_{\textbf{X}}\left(\textbf{x}\|\theta\right)$ in the first line of (1) partitions the density according to data collection stages, while the representation in the second line partitions the density according to stopping stages. In canonical form, $\displaystyle f_{\textbf{X},D}\left(\textbf{x},d\|\theta\right)$ $\displaystyle=$ $\displaystyle h(\mathbf{x})\exp\Big{(}\left[x_{1}+I(d=2)x_{2}\right]\theta-\left[1+I(d=2)\right]\frac{\theta^{2}}{2}\Big{)},$ (2) where $h(\mathbf{x})=\exp\Big{(}-\frac{1}{2}\left[x_{1}^{2}+I(d=2)x_{2}^{2}+I(d=2)\log\left(\sqrt{\pi}\right)+\log\left(\sqrt{\pi}\right)\right]\Big{)}$. Thus, the density (2) belongs to the curved exponential family with a sufficient statistic $\left(\sum_{k=1}^{d}x_{i},d\right)=\left[x_{1}+I(d=2)x_{2},\,1+I(d=2)\right].$ Curved exponential families were defined by Efron (1975) and the sufficient statistic with a random number of summands $\sum_{k=1}^{D}x_{k}$ was derived by Blackwell (1947). Probability distribution (2) is a special case of the exponential family derived in Liu et al. (2006) [see their formula (2.6)]. A more general probability distribution of $\mathbf{X}$ is presented in Section 2. The log-likelihood function $\displaystyle l(\theta\|X_{1}=x_{1},X_{2}=x_{2},D=d)$ $\displaystyle=\begin{cases}[\phi(x_{1}-\theta)]^{I(d=1)}&\textrm{ if }D=1,\\\ \left[\phi(x_{1}-\theta)\phi(x_{2}-\theta)\right]^{I\left(d=2\right)}&\textrm{ if }D=2\end{cases}$ is maximized at $\widehat{\theta}=I(d=1)x_{1}+I(d=2)(x_{1}+x_{2})/2.$ Consequently, the score function, the MLE and the observed information are the same as for the non-sequential experiment. What changes? The joint support of $X_{1}$ and $X_{2}$ changes because $X_{2}$ becomes impossible (not just missing) when $X_{1}\geq 2.18$. The random variable $(X_{1},X_{2})$ is non-observable when $X_{1}\geq 2.18$ and the joint distribution of $X_{1}$ and $X_{2}$ is therefore truncated and not normal. Formally, the support for joint density can be decomposed into support for the experiment stopping with $X_{1}$ and support for the experiment continuing to observe $X_{2}$: ${\cal{T}}=\\{x_{1}\geq 2.18\\}\ \cup\ \\{\\{x_{1}<2.18\\}\cap\\{x_{2}\in R\\}\\}\ \subset R^{2}.$ This ${\cal{T}}$ is a special case of the support formalized in Liu et al. (2006). If $A\in{\cal{T}}$, then $\displaystyle P_{(D)}=\text{Pr}((X_{1},X_{2})\in A)$ $\displaystyle=$ $\displaystyle\text{Pr}(X_{1}\geq 2.18)\text{Pr}(X_{1}\in A\|X_{1}\geq 2.18)$ (3) $\displaystyle+$ $\displaystyle\text{Pr}(X_{1}<2.18)\text{Pr}((X_{1},X_{2})\in A\|X_{1}<2.18)$ is a probability measure on the $\sigma$-algebra $\sigma({\cal{T}})$. Thus, the observable random variable $\displaystyle\mathbf{X}_{{\cal{T}}}=\begin{cases}X_{1}&\textrm{ if }$D=1$\\\ (X_{1},X_{2})&\textrm{ if }$D=2$.\end{cases}$ is defined on the probability space $\left({\cal{T}},\sigma({\cal{T}}),P_{(D)}\right).$ In contrast to $(X_{1},X_{2})$, $\mathbf{X}_{{\cal{T}}}$ is observable for this sequential experiment. The MLE, is a random variable defined on $\left({\cal{T}},\sigma({\cal{T}}),P_{(D)}\right)$ and its probability distribution is a mixture of the left truncated normal random variable $\\{X_{1}\|D=1\\}$ and an average of the right truncated normal random variable $\\{X_{1}\|D=2\\}$ and the normal random variable $X_{2}$. Thus, in this sequential experiment, MLEs are not normal random variables as illustrated in Figure 1. The first column shows the distribution of the MLE if the experiment stopped at stage 1, which is left truncated normal. The seconds of histograms shows the distribution of the MLE if the experiment proceeded to stage 2. This distribution is a mixture of right truncated data from stage 1 and untruncated normal data from stage 2. The final column shows unconditional distribution of the MLE. (a) $x_{1}:x_{1}\geq 2.18$, $\theta=0$ (b) $\frac{x_{1}+x_{2}}{2}:x_{1}<2.18$, $\theta=0$ (c) $\widehat{\theta}$, $\theta=0$ (d) $x_{1}:x_{1}\geq 2.18$, $\theta=2.18$ (e) $\frac{x_{1}+x_{2}}{2}:x_{1}<2.18$, $\theta=2.18$ (f) $\widehat{\theta}$, $\theta=2.18$ Figure 1: Distribution of MLEs from the experiment described in Section 1.1. $n_{1}=n_{2}=1$; $10^{6}$ Monte-Carlo simulations. ### 1.2 The Scope of this Paper We consider sequential experiments having a small finite number of interim decision points, that is, experimental set-ups for which Martingale central limit theorems and Brownian theory are not suitable. Our interest is in experiments that aim primarily on a hypothesis test of effect size. We focus on characterizing the effect of sequential stopping rules on probability distributions of test statistics. In this manuscript, Section 2 introduces notation for GSDs with stopping rules dependent on a parameter of interest [through the distributions of the test statistics]. This section also presents distributions of cumulative test statistics conditional on reaching a stage, conditional on stopping at a stage, and unconditional defined on the probability space with truncation- adapted support. All of these probability distributions are truncated or truncated-mixtures. Section 3 presents likelihood-based inference on the truncation-adapted probability space. In section 3.1 a local asymptotic distribution of the MLEs is found to be non-degenerate; it is a mixture of truncated normal distributions. Section 3.2 shows that the possibility of early stopping does not change the monotonicity of likelihood ratios in the one-parameter exponential family. Thus, stage-specific tests continue to be uniformly most powerful by the Karlin-Rubin theorem, which makes sequential tests based on monotone likelihood ratio uniformly most powerful. Throughout Sections 2 and 3 theoretical results are illustrated by a two stage example, Pocock (1977). Finally, Section 4 concludes this article with a summary and a discussion of impact within the contemporary research environment. ## 2 Probability Distributions with Early Stopping Let $X$ denote subjects’ outcome variable and assume that, when observed in isolation, it has a probability distribution function or a probability mass function $f_{X}=f_{X}\left(x\|\theta\right)$. To simplify the material, the term density is used to refer to probability measures without formally distinguishing between them. Let a sequence of $X$s be observed with the primary objective of testing the null hypothesis $H_{0}:\theta=0$ with overall $\alpha$-level type 1 error and $1-\beta$ power at an alternative $H_{1}:\theta=\theta_{1}$. It is convenient to group the random sample into _stages_ separated by the interim analysis times: $\mathbf{X}_{1},\mathbf{X}_{2},\ldots$, where $\mathbf{X}_{k}=(X_{n_{(k-1)}+1},\ldots,X_{n_{(k)}})$; $n_{(k)}=\sum_{j=1}^{k}n_{j}$; here for simplicity $n_{j}$ is a pre-specified number of observations in stage $j$, $1\leq k\leq K$; $n_{(0)}=0$ and $K-1$ is the maximum number of interim analyses permitted. Every stage is assumed to be “reachable”, that is, there is a positive probability of reaching each stage. Data collection at each stage is followed by a hypothesis test that results in a decision to stop the study or to enroll a new group of patients; except that if stage $K$ is reached, the experiment stops after $n_{(K)}$ observations regardless of the last $n_{K}$ observations’ values. ### 2.1 Stopping Decisions Let $T_{(k)}$ be a function of observations $\mathbf{X}_{(k)}=(X_{1},\ldots,X_{n_{(k)}})$ that is compared against a cutoff value $c_{k}$ to determine whether to stop at stage $k$ or continue through stage $k+1$, $1\leq k<K-1$. These decisions are defined by the events $\left\\{\cap_{j=1}^{k-1}\left\\{T_{(j)}\leq c_{j}\right\\}\right\\}\cap\left\\{T_{(k)}>c_{k}\right\\}$ and $\cap_{j=1}^{K-1}\left\\{T_{(j)}\leq c_{k}\right\\}$, respectively, and are conveniently summarized by a random variable denoting the stopping stage: $D=K\cdot I\left(\cap_{j=1}^{K-1}\left\\{T_{(j)}\leq c_{j}\right\\}\right)+\sum_{k=1}^{K-1}k\cdot I\left(\left(\cap_{j=1}^{k-1}\left\\{T_{(j)}\leq c_{j}\right\\}\right)\cap\left\\{T_{(k)}>c_{k}\right\\}\right);$ $D\in\\{1,\ldots,K\\}$ will appear as random index such as in $\mathbf{X}_{(D)}$ to emphasize that the stopping stage is unknown and is described probabilistically though the random variable $D$. $\mathbf{X}_{(d)}$ is the random variable $\mathbf{X}_{(D)}$ conditioned on stopping with stage $D=d,d=1,\ldots,K$. It is important to account for $D$ in probability statements about $T_{(d)}$ because $D$ determines the observations’ probability support as illustrated in Section 1.1. ### 2.2 After deciding to stop (a) Sketch of disjoint support regions for subdensities by (scaleless) stage- specific test statistics (b) Sketch of disjoint support regions for subdensities by (scaleless) cumulative test statistics Figure 2: Support associated with different stopping decisions when $K=3$; $c_{k}$ is the critical value for stopping at stage $k$. If study stops at stage $D=d<K$, $H_{0}$ is rejected. If $d=K$, the final hypothesis test determines acceptance or rejection of $H_{0}$. At the time of the final analysis, the density of observations conditional on stopping at stage $d$ (i.e., the density of $\mathbf{X}_{(d)}=\mathbf{X}_{(D)}\|\\{D=d\\}$) is $\displaystyle f_{\mathbf{X}_{(d)}}^{C}=f_{\mathbf{X}_{(D)}}\left(\textbf{{x}}_{(D)}\|D=d,\theta\right)=\frac{I\left(D=d\right)}{\text{Pr}_{\theta}\left(D=d\right)}f_{\mathbf{X}_{(d)}}\left(\textbf{{x}}_{(d)}\|\theta\right)=\frac{f_{\mathbf{X}_{(d)}}^{sub}}{\text{Pr}_{\theta}\left(D=d\right)},$ (4) where $f_{\mathbf{X}_{(d)}}^{sub}=[f_{\mathbf{X}_{(d)}}(\textbf{{x}}_{(d)}\|\theta)]^{I\left(D=d\right)}$ denotes the sub-density with support defined by $D=d$ (see the exemplary sketches in Figure 2). Similarly, for a statistic $T_{(d)}=T(\mathbf{X}_{(d)})$, the conditional on $D=d$ density is $f_{T_{(d)}}^{C}$. In contrast, if the stopping rule is not random and the experiment stops with $n_{(d)}$ observations, $\text{Pr}_{\theta}(D=d)\equiv 1$ and the observations have density $f_{\mathbf{X}_{(d)}}$. ### 2.3 At the Time of Experimental Design Prior to data collection, both $D$ and $X_{(D)}$ are unknown and the joint density of $X_{(D)}$ can be written in several ways: $\displaystyle f_{\mathbf{X}_{(D)}}\left(\textbf{{x}}_{(D)}\|\theta\right)$ $\displaystyle=$ $\displaystyle\sum_{d=1}^{K}[f_{\mathbf{X}_{(d)}}(\textbf{{x}}_{(d)}\|\theta)]^{I\left(D=d\right)}$ (5) $\displaystyle=$ $\displaystyle\sum_{d=1}^{K}f_{\mathbf{X}_{(d)}}^{sub}=\sum_{d=1}^{K}\text{Pr}_{\theta}\left(D=d\right)f_{\mathbf{X}_{(d)}}^{C}.$ The joint density is a mixture of densities corresponding to possible outcome vectors; i.e., these densities are defined on non-overlapping regions of the density’s support. When a test statistic $T_{d}$ summarizes $d$th stage data, the density of $T_{(D)}$ can be written analogous to Equation (5) as $\displaystyle f_{T_{(D)}}\left({\textit{t}}_{(D)}\|\theta\right)$ $\displaystyle=$ $\displaystyle\sum_{d=1}^{K}[f_{{T}_{(d)}}({\textit{t}}_{(d)}\|\theta)]^{I\left(D=d\right)}$ (6) $\displaystyle=$ $\displaystyle\sum_{d=1}^{K}f_{{T}_{(d)}}^{sub}=\sum_{d=1}^{K}\text{Pr}_{\theta}\left(D=d\right)f_{{T}_{(d)}}^{C}.$ Even if every stage-specific test statistic $T_{d}$ is normally distributed, the distribution of $T_{(D)}$ is not. ### 2.4 At Interim Hypothesis Testing Suppose at stage $d-1$, the decision was made to continue sampling, the support for the $d$th test statistic is characterized by $D\geq d$. The density of $T_{(D)}$ conditional on $D\geq d$ is $\displaystyle f_{T_{(D)}}\left(t_{(D)}\|D\geq d,\theta\right)$ $\displaystyle=$ $\displaystyle\frac{\sum_{k=d}^{K}f_{T_{(k)}}^{sub}}{\text{Pr}_{\theta}\left(D\geq d\right)}=\sum_{k=d}^{K}\frac{\text{Pr}_{\theta}\left(D=k\right)}{\text{Pr}_{\theta}\left(D\geq d\right)}f_{T_{(k)}}^{C}.$ (7) Again, even if each $T_{k}$ is normal, the distributions of $T_{(D)}\|D\geq d$ are not. #### 2.4.1 Connection with Armitage’s algorithm SAS’s popular SEQDESIGN procedure, R’s gsDesign package, Cytel’s EAST and others assess type I and power properties using a recursive sub-density formula (Armitage et al. (1969)) to evaluate the distribution of $T_{(D)}\|D\geq d$. Armitage’s subdensity is $f_{T_{(D)}\|D\geq d}^{sub}=\sum_{k=d}^{K}f_{T_{(k)}}^{sub}$ and $\displaystyle f_{T_{(D)}}\left(t_{(D)}\|D\geq d,\theta\right)$ $\displaystyle=$ $\displaystyle\frac{f_{T_{(D)}\|D\geq d}^{sub}}{\text{Pr}_{\theta}\left(D\geq d\right)}.$ (8) For example, at $K=2$, the sub-density of $T_{(D)}\|D\geq d$ is $f^{sub}_{T_{(D)}\|D\geq 2}(t\|\theta)=\int_{-\infty}^{c_{1}}f_{T_{(D\geq 2)}\|T_{(1)}}(t\|t_{1},\theta)f_{T_{(1)}}(t_{1}\|\theta)dt_{1}$ (9) and its density is $f_{T_{(D)}\|D\geq 2}(t\|\theta)=f^{sub}_{T_{(D)}\|D\geq d}(t\|\theta)\left(\int_{-\infty}^{c_{1}}f_{T_{(1)}}(t_{1}\|\theta)dt_{1}\right)^{-1}$. Recursively, the density conditional on reaching the $d$th interim analysis is $f_{T_{(D)}}(t\|D\geq d,\theta)=\frac{\int_{-\infty}^{c_{d-1}}f_{T_{(D\geq d)}\|T_{(d-1)}}(t\|t_{d-1},\theta)f_{T_{(d-1)}}(t_{d-1}\|\theta)dt_{d-1}}{\int_{-\infty}^{c_{d-1}}f_{T_{(d-1)}}(t_{d-1}\|\theta)dt_{d-1}}.$ (10) ### 2.5 $\sigma-$fields and support defined by a set of critical values At the time of $d$th hypothesis test, given $D\geq d$, the values $\\{x_{j}:j>n_{(d)}\\}$ are not observable and hence do not contribute to the density; indeed, they do not belong to the adaptation-rule driven sample space, and consequently, they do not belong to a $\sigma$-field of the random process being monitored: hence, they do not belong to the sequential experiment as a whole. In this paper, by analogy with structural zeroes in contingency tables, these values are excluded from the sample space. The $d$th stage-specific test statistic $T_{d}=T\left(\textbf{X}_{d}\right)$ is defined on a probability space $\left({\cal{T}}_{d},\sigma\left({\cal{T}}_{d}\right),P_{d}\right)$, where ${\cal{T}}_{d}$ is typically a real line (${\cal{R}}$), $\sigma\left({\cal{T}}_{d}\right)$ is Borel $\sigma-$field and $P_{d}$ is a probability measure on the measurable space $\left({\cal{T}}_{d},\sigma\left({\cal{T}}_{d}\right)\right)$. A sequence of nested $\sigma$-fields ${\cal{F}}_{(d)}:=\sigma\left({\cal{T}}_{1}\right)\times\cdots\times\sigma\left({\cal{T}}_{d}\right),$ $d=1,\ldots,K$, creates a filtration $\textbf{F}=\left({\cal{F}}_{(d)}\right)_{d\leq K}$ on the product probability space $\left({\cal{T}},{\cal{F}}_{(K)},P\right)$, where ${\cal{T}}={\cal{T}}_{1}\times\cdots\times{\cal{T}}_{K}$ and $P=P_{1}\times\cdots\times P_{K}$. But in the presence of possible stopping, not all combinations of $(T_{1},\ldots,T_{K})$ are possible. The cumulative test statistics $T_{(d)}=T_{(d)}(T_{1},\ldots,T_{d})$ are defined only on a subspace of the sample space ${\cal{T}}_{1}\times\cdots\times{\cal{T}}_{d}$. Thus, probability environment substantially changes. #### 2.5.1 Interim hypothesis testing The statistic $T_{(1)}=T_{1}$ conditional on reaching stage $1$ $(D\geq 1)$ is defined on the sample space ${\cal{T}}_{(1)}={\cal{T}}_{1}$; so $\sigma\left({\cal{T}}_{(1)}\right)={\cal{F}}_{(1)}$. The statistic $T_{(2)}$ conditional on $D\geq 2$ is defined on ${\cal{T}}_{(2)}=\left[(t_{1},t_{2}):\\{t_{1}\leq c_{1}\\}\right],$ and $\sigma\left({\cal{T}}_{(2)}\right)\subset{\cal{F}}_{(2)}$. Further, for $d\in\\{3,\ldots,K\\}$, $T_{(d)}$ conditional on $D\geq d$ is defined on ${\cal{T}}_{(d)}=\left((t_{1},\ldots,t_{d}):\left\\{t_{(j)}\leq c_{j}\right\\},j=1,\ldots,d-1\right).$ For all $d$, the support ${\cal{T}}_{(d)}$ and the $\sigma$-field $\sigma\left({\cal{T}}_{(d)}\right)$ is reduced by the possibility of early stopping: $\sigma\left({\cal{T}}_{(d)}\right)\subset{\cal{F}}_{(d)}$. This creates new measurable spaces $\left({\cal{T}}_{(d)},\sigma\left({\cal{T}}_{(d)}\right)\right)$ for interim tests at every possible stage $1,\ldots,K$. Armitage’s recursive sub-density formula [Armitage et al. (1969)] is defined on this measurable space; see Equation (9) for $K=2$. Re-scaling yields a density function [see Equation (10)] which defines a probability measure to complete the probability space $\left({\cal{T}}_{(d)},\sigma\left({\cal{T}}_{(d)}\right),P_{(d)}\right)$, where the probability measure $P_{(d)}$ is determined by density (10): $P_{(d)}(A)=\int_{A}f_{T_{(D)}}(t\|D\geq d,\theta)dt$, $A\in\sigma\left({\cal{T}}_{(d)}\right)$. #### 2.5.2 After the stop decision is made At each interim stage $d=1,\ldots,K-1$, the decision to reject or accept $H_{0}$ splits ${\cal{T}}_{(d)}$ into two non-overlapping regions denoted ${\cal{T}}_{(d)}^{stop}$ and ${\cal{T}}_{(d)}^{cont}$, respectively. Since ${\cal{T}}_{(d)}^{stop}\cap{\cal{T}}_{(d+1)}=\emptyset$ and ${\cal{T}}_{(d)}^{cont}\subset{\cal{T}}_{(d+1)}$, then the sets ${\cal{T}}_{(1)}^{stop},\ldots,{\cal{T}}_{(K-1)}^{stop},$ and ${\cal{T}}_{(K)}$ make a partition of the sample space of $T_{(D)}$: ${\cal{T}}_{(K)}+\sum_{k=1}^{K-1}{\cal{T}}_{(k)}^{stop}=\left(\cup_{d=1}^{K-1}{\cal{T}}_{(d)}^{stop}\right)\cup{\cal{T}}_{(K)}=\cup_{d=1}^{K}{\cal{T}}_{(d)}\subset{\cal{T}}.$ We define a probability space for the observable random variable ${T}_{(d)}$ using the probability measures $P_{(d)}$ defined by $f_{T_{(d)}}^{C}$ on the measurable space $\left({\cal{T}}_{(d)}^{stop},\sigma\left({\cal{T}}_{(d)}^{stop}\right)\right)$. At $D=K$, the probability measure $P_{(K)}$ defined by $f_{T_{(K)}}^{C}$ completes the probability space for a measurable space $\left({\cal{T}}_{(K)},\sigma\left({\cal{T}}_{(K)}\right)\right)$. Thus, at the end of the study, at $D=d$, the researcher operates with an _observable_ random variable $T_{(d)}$. #### 2.5.3 At the design stage The conditional random variables, $T_{(d)}$, defined on non-overlapping $\sigma$-fields are combined together into the unconditional random variable $T_{(D)}\sim f_{T_{(D)}}$ defined on the sample space $\left(\cup_{d=1}^{K-1}{\cal{T}}_{(d)}^{stop}\right)\cup{\cal{T}}_{(K)}$. The probability distribution defined on the $\sigma$-field on this sample space is $P_{(D)}=\sum_{d=1}^{K}\text{Pr}\left(D=d\right)P_{(d)}$. #### 2.5.4 Impossible events The set ${\cal{T}}_{0}={\cal{T}}\setminus\cup_{d=1}^{K}{\cal{T}}_{(d)}$ contains all impossible combinations of $(t_{1},\ldots,t_{K})$ under a chosen stopping rule. If $K=2$, for example, then ${\cal{T}}\setminus\cup_{d=1}^{2}{\cal{T}}_{(d)}=\left\\{(t_{1},t_{2}):t_{1}>c_{1}\right\\}$. Without the possibility of early stopping, all combinations of $(t_{1},\ldots,t_{K})\in{\cal{T}}$ would be possible. ### 2.6 Pocock’s Example: One-Sided Two-Group Sequential Z-test Pocock (1977) proposed a simple two-stage procedure for testing $H_{0}:\theta=0$ with a pre-determined power at $H_{1}:\theta=\theta_{1}$ on normal data. With $n_{1}=n_{2}$, $c_{1}=c_{2}=2.18$ is used to secure the overall type 1 error rate $\alpha=0.025$ with a one-sided $z$ test. Let $n_{0}=0$ and $Z_{k}=\frac{1}{\sqrt{n_{k}}}\sum_{i=n_{(k-1)}+1}^{n_{(k)}}X_{i}=\sqrt{n_{k}}\cdot\bar{X}_{k}\overset{d}{=}\mathcal{N}(\theta,1),\quad k=1,2.$ The study is stopped for efficacy at stage 1 if $Z_{1}\geq 2.18$ and proceeds to stage 2 when $Z_{1}<2.18$. If the study is stopped at stage 1, the support for $\bar{X}_{1}\|Z_{1}\geq 2.18$ starts at $2.18/\sqrt{n_{1}}$ and stretches to $+\infty$. If the study continues through stage 2, support for $\bar{X}_{1}\|Z_{1}<2.18$ ranges from $-\infty$ to $2.18/\sqrt{n_{1}}$ . The test statistic for the second stage, under $Z_{1}<2.18$, $Z_{(2)}=\frac{\sqrt{n_{1}}}{\sqrt{n_{1}+n_{2}}}Z_{1}+\frac{\sqrt{n_{2}}}{\sqrt{n_{1}+n_{2}}}Z_{2}=\frac{1}{\sqrt{n_{1}+n_{2}}}\sum_{i=1}^{n_{1}+n_{2}}X_{i}.$ The distribution of $Z_{(2)}$ is a mixture of a right-truncated normal $Z_{1}\|Z_{1}<2.18$ and the normal $Z_{2}$. Figure 3 shows histograms of $Z_{1}\|Z_{1}\geq 2.18$, $Z_{(2)}\|Z_{1}<2.18$, and $Z_{(D)}$ estimated from $100,000$ Monte-Carlo samples assuming $\theta=2.18$. These histograms based on $n_{1}=n_{2}=100$ are almost identical to the histograms in Figure 1 based on $n_{1}=n_{2}=1$. This highlights on the important message that non-normality continues to be present even asymptotically Tarima and Flournoy (2019a). (a) $Z_{1}\|Z_{1}\geq 2.18$, $\theta=2.18$ (b) $Z_{(2)}\|Z_{1}<2.18,$ $\theta=2.18$ (c) $Z_{(D)},\theta=2.18$ Figure 3: Histograms of test statistics from a two-stage Pocock’s experiment with critical value $c_{1}=2.18$; $f_{\mathbf{X}_{k}}=\mathcal{N}(2.18,1)$ and $n_{k}=100,k=1,2$. ## 3 Likelihood-based Inference with Early Stopping If the stopping rule is not random and the experiment stopped with $n_{(d)}$ observations, $\text{Pr}_{\theta}(D=d)\equiv 1$, the likelihood is $\displaystyle{\mathcal{L}}^{fix}\left(\theta\|k,\textbf{{x}}_{(d)}\right)=f_{\mathbf{X}_{(d)}}.$ (11) Considering the joint density (5) conditional on the data $\left(d,\textbf{{x}}_{(d)}\right)$ observed at the end of a sequential experiment, the likelihood is $\displaystyle{\mathcal{L}}\left(\theta\|d,\textbf{{x}}_{(d)}\right)=f_{\mathbf{X}_{(d)}}^{sub}.$ (12) The indicator in (4) emphasizes that support for the random variables is reduced by the conditioning; see this illustrated in Figure 2. Note also in Figure 2 that the support conditional on stopping at one stage is disjoint from the support conditional on stopping at another stage. ${\cal{L}}$ is a function of $\theta$ and the observed data $(d,\textbf{{x}}_{(d)})$, but ${\cal{L}}$ is a continuous function only of $\theta$, as discontinuities in $(d,\textbf{{x}}_{(d)})$ arise from the mixture distribution of $\mathbf{X}_{(D)}$. These discontinuities are inherited by MLEs and other statistics derived from them. Conditional on $(d,\textbf{{x}}_{(d)})$, MLEs maximizing ${\mathcal{L}}$ are $\displaystyle\widehat{\theta}$ $\displaystyle=\arg\max_{\theta}f_{\mathbf{X}_{(d)}}^{sub}.$ For every $\left(d,\textbf{{x}}_{(d)}\right)$, $f_{\mathbf{X}_{(d)}}^{sub}=f_{\mathbf{X}_{(d)}}$ for all $\theta$; consequently, $\widehat{\theta}^{fix}:=\arg\max_{\theta}f_{\mathbf{X}_{(d)}}=\arg\max_{\theta}f_{\mathbf{X}_{(d)}}^{sub}=\widehat{\theta},$ that is, as is well known, making stopping decisions does not alter maximum likelihood point estimates; they are the same whether obtained by maximizing ${\mathcal{L}}$ or ${\mathcal{L}}^{fix}$ and other observed statistics derived from the likelihood (e.g., the score function and the observed information) are unaffected as well. The following example extends the simple one in Section 1.1 with $n_{1}=n_{2}=1$ to arbitrary $n_{1}$ and $n_{2}$ to illustrate (as is proven later) that, although maximum likelihood point estimates and test statistics are unaffected by early stopping decisions, their probability distribution does not tend to normality even with larger sample sizes. ### 3.1 Large Sample Properties The $d$th stage-specific MLEs $\widehat{\theta}_{d}$ of $\theta$ and their statistical models are called _regular_ if, without the possibility of early stopping, $\xi_{d}=\sqrt{n_{d}}\left(\widehat{\theta}_{d}-\theta\right)\overset{d}{\to}{\cal{N}}(0,\sigma^{2}),$ (13) where $0<\sigma<\infty$. Assumption (13) was described in Tarima and Flournoy (2019a) to include the more specific assumptions for * • independent, identically distributed observations by Cramér [e.g., for example, Ferguson (1996)], * • independent not identically distributed observations [e.g., Philippou et al. (1973)], * • dependent observations [e.g., Crowder (1976)], * • and densities whose support depends on parameters [e.g., Wang et al. (2014)]. All these specific sets of assumptions include assumptions of the existence and consistency of the MLE. With large samples, the MLE at the stage $d$ analysis can be approximated recursively by $\displaystyle\widehat{\theta}_{(d)}$ $\displaystyle\approx\frac{n_{(d-1)}}{n_{(d)}}\widehat{\theta}_{(d-1)}+\frac{n_{d}}{n_{(d)}}\widehat{\theta}_{d}\approx\sum_{j=1}^{d}\frac{n_{j}}{n_{(d)}}\widehat{\theta}_{d},$ (14) where $\widehat{\theta}_{(d-1)}$ is an MLE based on cumulative data from stages $1$ to $d-1$ and $\widehat{\theta}_{d}$ is the $d$th stage-specific MLE. The standardized $d$th stage-specific MLE is so, $T_{(d)}=\sqrt{n_{(d)}}\sum_{j=1}^{d}\frac{n_{j}}{n_{(d)}}\left(\frac{\widehat{\theta}_{j}-\theta}{\sigma}\right)=\sum_{j=1}^{d}\sqrt{\frac{n_{j}}{n_{(d)}}}\ \xi_{j}\approx\sqrt{n_{(d)}}\left(\frac{\widehat{\theta}_{(d)}-\theta}{\sigma}\right),$ that is, $T_{(d)}$ is (approximately) the standardized $d$th stage-specific MLE. The asymptotic properties of $T_{(D)}$ depend on the existence and distribution of a limiting random variable $r_{(D)}$ defined by $\sum_{d=1}^{K}I(D=d)\frac{n_{(d)}}{n_{d}}\overset{d}{\to}\sum_{d=1}^{K}I(D=d)r_{(d)}=r_{(D)},$ (15) where $r_{(d)}=\lim_{n_{d}\to\infty}n_{(d)}/n_{d}$ is the asymptotic ratio of the $d$th cumulative-stage and stage-specific sample sizes [Theorem 1 in Tarima and Flournoy (2019a)]; $r_{(D)}$ is a multinomial random variable with support on $r_{(d)}$. Assume the limits $n_{(d)}/n_{j}\to r_{(d)j}\in(0,\infty)$, $j\leq d$ exist with $r_{(d)d}=r_{(d)}$. Then given $D=d$, $T_{(d)}\to\sum_{j=1}^{d}\frac{\xi_{j}}{\sqrt{r_{(d)j}}}.$ While $\xi_{j}\overset{d}{\to}{\cal{N}}(0,1)$, the distributions used for the final analysis, the interim analysis and experimental design, respectively, are mixtures of truncated distributions: $\displaystyle{\text{Pr}_{\theta}}\left(T_{(D)}<v\|D=d\right)$ $\displaystyle\to$ $\displaystyle\text{Pr}_{\theta}\left(\sum_{j=1}^{d}\frac{\xi_{j}}{\sqrt{r_{(d)j}}}<v\Big{\|}D=d\right),$ $\displaystyle{\text{Pr}_{\theta}}\left(T_{(D)}<v\|D\geq d\right)$ $\displaystyle\to$ $\displaystyle\sum_{k=d}^{K}{\text{Pr}_{\theta}}\left(D=k\right){\text{Pr}_{\theta}}\left(\sum_{j=1}^{k}\frac{\xi_{j}}{\sqrt{r_{(k)j}}}<v\Big{\|}D=k\right),$ $\displaystyle\text{Pr}_{\theta}\left(T_{(D)}<v\right)$ $\displaystyle\to$ $\displaystyle\sum_{k=1}^{K}{\text{Pr}_{\theta}}\left(D=k\right){\text{Pr}_{\theta}}\left(\sum_{j=1}^{k}\frac{\xi_{j}}{\sqrt{r_{(k)j}}}<v\Big{\|}D=k\right).$ ### Pocock’s Example: Large Sample Properties Under assumption (13) with large sample sizes, $\widehat{\theta}_{(D)}=I(D=1)\widehat{\theta}_{(1)}+I(D=2)\widehat{\theta}_{(2)}$ are standardized as $\displaystyle T_{(D)}$ $\displaystyle=\sqrt{n_{(D)}}\left(\widehat{\theta}_{(D)}-\theta\right)/\sigma$ $\displaystyle=I(D=1)\sqrt{n_{1}}\left(\widehat{\theta}_{(1)}-\theta\right)/\sigma+I(D=2)\sqrt{n_{1}+n_{2}}\left(\widehat{\theta}_{(2)}-\theta\right)/\sigma.$ (16) Assume the limit $I(D=1)+I(D=2)\frac{n_{1}+n_{2}}{n_{2}}\overset{d}{\to}r_{(D)}$ exists as $n_{1}\to\infty$, Then, adopting the local alternative hypothesis $\theta=h/\sqrt{n_{1}}$ yields a non-degenerate limiting distribution of $T_{(D)}$ that models both stages of the experiment: $\displaystyle\text{Pr}_{\theta}\left(T_{(D)}<v\right)$ $\displaystyle\to$ $\displaystyle p_{1}\Phi\left(\left.v\right\|D=1\right)$ (17) $\displaystyle+$ $\displaystyle(1-p_{1})\int_{-\infty}^{c_{1}}\Phi\left(\sqrt{r_{(D)}}v-\sqrt{r_{(D)}-1}y\right)\phi\left(y\|D=2\right)dy,$ where $p_{1}=\lim_{n_{1}\to\infty}{\text{Pr}_{\theta}}\left(D=1\right)$ is the limiting stage 1 stopping probability; $\phi$ and $\Phi$ denote the standard normal density and cumulative distribution function, respectively. With a fixed alternative $\theta$, $\text{Pr}_{\theta}\left(T_{(D)}<v\right)\to\Phi(v)$ because $p_{1}\to 1$ with probability 1, and modeling related to stage 2 data is lost. In Pocock’s example, $T_{(D)}=I(D=1)Z_{1}+I(D=2)\left(Z_{1}+Z_{2}\right)/\sqrt{2}$ and $\displaystyle\text{Pr}_{\theta}\left(T_{(D)}<v\right)$ $\displaystyle\to$ $\displaystyle\text{Pr}_{\theta}\left(Z_{1}>c_{1}\right)\Phi\left(\left.v\right\|Z_{1}>c_{1}\right)$ (18) $\displaystyle+$ $\displaystyle\text{Pr}_{\theta}\left(Z_{1}\leq c_{1}\right)\int_{-\infty}^{c_{1}}\Phi\left(\sqrt{2}v-y\right)\frac{\phi\left(y\right)}{\text{Pr}\left(y\leq c_{1}\right)}dy$ Note if $Z_{1}\leq c_{1}$, then $\text{Pr}_{\theta}\left(\frac{Z_{1}+Z_{2}}{\sqrt{2}}<v\|Z_{1}\leq c_{1}\right)=\int_{-\infty}^{c_{1}}\Phi\left(\sqrt{2}v-y\right)\frac{\phi\left(y\right)}{\text{Pr}\left(y\leq c_{1}\right)}dy,$ which is a continuous mixture of distributions. ### 3.2 Most Powerful Group Sequential Tests Let $X\sim f_{X}(\theta)$, where $f$ belongs to the one-parameter exponential family. Without a possibility of early stopping, the likelihood for a realization $\textbf{{x}}=\left(x_{1},\ldots,x_{n}\right)$ of a random sample $\textbf{X}=\left(X_{1},\ldots,X_{n}\right)$ is $\displaystyle{\cal{L}}\left(\theta\|\textbf{{x}}\right)=\prod_{i=1}^{n}f_{X}\left(\textbf{{x}}\right)$ $\displaystyle=h\left(\textbf{{x}}\right)g\left(T\left(\textbf{{x}}\right)\|\theta\right)=h\left(\textbf{{x}}\right)e^{\eta(\theta)T\left(\textbf{{x}}\right)+A\left(\theta\right)}$ where all relevant information about $\theta$ is absorbed by a sufficient statistic $T\left(\textbf{{x}}\right)$. Assume the test statistic $Z$ is a one-to-one transformation of $T$. If $LR(t)=g(t\|\theta)/g(t\|\theta_{0})$ has monotone likelihood ratio (MLR) in $t$, then the Karlin-Rubin theorem provides uniformly most powerful (UMP) tests. In sequential testing settings, when $d$th stage is reached $\left(D\geq d\right)$, the interim likelihood $\displaystyle{\cal{L}}\left(\theta\|D\geq d,\textbf{{x}}_{(d)}\right)$ $\displaystyle=$ $\displaystyle I\left(D\geq d\right)h\left(\textbf{{x}}_{(d)}\right)e^{\eta(\theta)T\left(\textbf{{x}}_{(d)}\right)+A_{d}\left(\theta\right)}$ (19) has the associated interim likelihood ratio $\displaystyle LR(t\|D\geq d)=\frac{{\cal{L}}\left(\theta\|D\geq d,\textbf{{x}}_{(d)}\right)}{{\cal{L}}\left(\theta_{0}\|D\geq d,\textbf{{x}}_{(d)}\right)}.$ For every $d$th stage hypothesis test, $D\geq d$ and $LR(t\|D\geq d)=\exp\left[\left(\eta(\theta_{1})-\eta(\theta_{0})\right)T_{(d)}+\left(A_{d}(\theta_{1})-A_{d}(\theta_{0})\right)\right]$ which means that the MLR property is preserved with early stopping. Definition: A sequence of $\alpha_{d}$-level interim tests $\\{T_{(1)}>c_{1}\\},\ldots,\\{T_{(D)}>c_{D}\\}$ will be called the sequential test. ###### Theorem 3.1 For any fixed $(\alpha_{1},\ldots,\alpha_{K})$ and $\left(n_{1},\ldots,n_{K}\right)$ $(1)$ the interim LR test $\\{T_{(d)}>c_{d}\\}$ is a UMP $\alpha_{d}$-level test and $(2)$ no sequential test is more powerful than the sequential test based interim LRs. Feature \| Mathematical Definition ---\|--- overall type $1$ error \| $\alpha=\sum_{k=1}^{K}\alpha_{k}\prod_{j=1}^{k-1}\text{Pr}_{0}\left(Z_{(j)}\leq u_{j}\right)$ stage-specific type 1 error \| $\alpha_{d}=\text{Pr}_{0}\left(Z_{(d)}>u_{d}\big{\|}\cap_{j=1}^{d-1}\left\\{Z_{(j)}\leq u_{j}\right\\}\right)$ $\alpha$-spending function \| $\alpha_{(d)}=\sum_{j=1}^{d}\alpha_{j}\prod_{i=1}^{j-1}\text{Pr}_{0}\left(Z_{(j)}\leq u_{j}\right)$ overall type 2 error \| $\beta(\theta)=\sum_{j=1}^{K}\beta_{j}(\theta)\prod_{i=1}^{j-1}\text{Pr}_{\theta}\left(Z_{(j)}\leq u_{j}\right)$ stage-specific type 2 error \| $\beta_{d}(\theta)=\text{Pr}_{\theta}\left(Z_{(d)}>u_{d}\big{\|}\cap_{j=1}^{d-1}\left\\{Z_{(j)}\leq u_{j}\right\\}\right)$ $\beta$-spending function \| $\beta_{(d)}\left(\theta\right)=\sum_{j=1}^{d}\beta_{j}(\theta)\prod_{i=1}^{j-1}\text{Pr}_{\theta}\left(Z_{(j)}\leq u_{j}\right)$ overall power \| $1-\beta(\theta)$ stage-specific power \| $1-\beta_{d}(\theta)$ cumulative power \| $1-\beta_{(d)}\left(\theta\right)$ Table 1: Definitions of Operational Characteristics for Sequential Tests $d=1,\ldots,K$; $\prod_{i=1}^{0}[\cdot]=1$. Proof. $(1)$ As shown above, conditional on reaching stage $d$, $(D\geq d)$, $T_{(d)}$ continues to be sufficient in exponential families and the MLR property is preserved. By the Karlin-Rubin theorem, the test based on $T_{(d)}$ is uniformly most powerful at $d^{th}$ stage. $(2)$ Using Table 1 notation, $1-\beta(\theta)=\sum_{k=1}^{K}\left(\prod_{j=1}^{k-1}\left[\beta_{j}(\theta)\right]\right)[1-\beta_{k}(\theta)]=1-\prod_{k=1}^{K}\beta_{k}(\theta).$ At $K=1$, $\alpha_{1}$ and $n_{1}$ uniquely define $c_{1}$ and $\left\\{T_{(1)}>c_{1}\right\\}$ is a UMP by part 1 with a stage-specific power curve $1-\beta_{1}(\theta)$. At an arbitrary stage $d$, $\alpha_{d}$ and $n_{d}$ uniquely define $c_{d}$ and, by part 1 of the theorem, the stage- specific power $1-\beta_{d}(\theta)$, is the highest. Thus, for any choice of $(\alpha_{1},\ldots,\alpha_{K})$ and $\left(n_{1},\ldots,n_{K}\right)$, and consequently $\\{T_{(d)}>c_{d}\\}$, the power of the sequential test based on LRs is $1-\beta(\theta)=1-\prod_{k=1}^{K}\beta_{k}(\theta)$. This power is the highest for any given $\theta$, because the stage specific type 2 errors $\beta_{d}(\theta)$ are the lowest for each $d$ at any $\theta$. Q.E.D. ### Pocock’s Example: Likelihood Ratio For $X_{i}\sim{\cal{N}}(\theta,1)$, $i=1,\ldots,n$, the likelihood ratio for testing $H_{0}:\theta=\theta_{0}$ vs $H_{A}:\theta=\theta_{1}>\theta_{0}$ is $\log LR(t)=-n\left(\log\theta_{0}-\log\theta_{1}\right)-t/\theta_{1},$ which has rejection region $\left\\{t>c\right\\}$, where $t=\sum_{i=1}^{n}X_{i}$ is a sufficient statistic. Using $Z=t/\sqrt{n}\sim f_{Z}={\cal{N}}(\theta_{1},1)$, the likelihood ratio test is $\left\\{Z>c/\sqrt{n}\right\\}$. With Pocock’s example, under $D\geq 1$, $Z_{1}$ is a normal random variable used for stage 1 hypothesis testing, and $\log LR(Z_{1}\|D\geq 1)=-n_{1}\left(\log\theta_{0}-\log\theta_{1}\right)-\sqrt{n_{1}}Z_{1}/\theta_{1}.$ Then, $\log LR(Z_{(2)}\|D\geq 2)=-(n_{1}+n_{2})\left(\log\theta_{0}-\log\theta_{1}\right)-\sqrt{n_{1}+n_{2}}Z_{(2)}/\theta_{1}+C,$ where $C=\log\text{Pr}\left(Z_{1}<c_{1}\|\theta_{0}\right)-\log\text{Pr}\left(Z_{1}<c_{1}\|\theta_{1}\right)$ is independent of data given $D\geq 2$. By Theorem 3.1, $\\{Z_{(d)}\geq c_{d}\\}$ is the UMP test at stage $d$ given $\alpha_{d}$ and $n_{d}$, and no other sequential test is more powerful overall. ## 4 Impact and Summary To establish that a sequence of likelihood ratio tests are most powerful, we began by constructing the joint probability distribution over the set of possible events. The use of an early stopping criterion eliminates the possibility of some realizations of cumulative test statistics $\mathbf{Z}=\left(Z_{(1)},\ldots,Z_{(K)}\right)$. This makes an otherwise normal random process $\mathbf{Z}$ unobservable. On its true stopping rule adapted support, the distribution of $\mathbf{Z}$ is a mixture of truncated distributions at each stage. Thus, unadapted distributional assumptions should be modified to take into account the planned adaptation scheme. Liu and Hall (1999) and Liu et al. (2006) recognized change in support in the one-parameter exponential family and investigated bias estimation. Schou and Marschner (2013) recognized presence of truncation in the joint distribution of stage- specific test statistics, but they mostly focused on bias in meta-analytic studies. The adapted support is critical for derivation MLEs’ and test statistics’ distributions in Section 2. In Section 2, distinct probability measures are derived for design [unconditional], interim hypothesis testing [conditional on collecting data up to the time of interim testing] and when the study is completed [conditional on deciding to stop at a particular stage]. These probability distributions formalized a new probabilistic framework (Section 2) which is used in Section 3.2 to show no testing sequence is more powerful than sequential likelihood ratio tests. Likelihood ratio tests are most powerful for testing simple hypotheses and, under monotone likelihood ratio, for testing composite hypotheses; see Neyman and Pearson (1933) and Karlin-Rubin theorem in Ferguson (2014). The Karlin- Rubin theorem works with the one-parameter exponential family and Section 3.2 shows that most powerful testing continues to hold for sequential tests. Even though distributions of sample-size-dependent sufficient statistics (Blackwell (1947)) belong to a curved exponential family (Efron (1975); Liu and Hall (1999); Liu et al. (2006)), the distributions conditional on the sample size still belong to one-parameter exponential family. This fact is used in Section 3.2 to show that likelihood ratio sequential tests are most powerful tests with any pre-determined $\alpha$-spending function. This result is applicable to many common group sequential designs. Treatment-effect-dependent (non-ancillary) stopping rules are part of many common GSDs including Pocock Pocock (1977), O’Brien & Fleming O’Brien and Fleming (1979), and Haybittle-Peto designs Haybittle (1971); Peto et al. (1976). If the data follow a normal distribution with known variance (without possibility of early stopping), then these designs are most powerful for their $\alpha$-spending functions. Similarly, application of Simes test (see Simes (1986)) and its recently proposed modification for sequential testing (see Tamhane et al. (2020)) cannot have higher power than group sequential tests from Jennison and Turnbull (1999) with the same $\alpha$-spending function. Historically, many researchers have relied, and currently rely, on joint normality (e.g., Jennison and Turnbull (1999), Proschan et al. (2006), Kunz et al. (2020)). Critical adjustments ate often done with recursive sub-density estimation; see Armitage et al. (1969). Section 2.4.1 places Armitage’s sub- density formula in the broader probability framework of possible events and confirms, from the prospective of this $\sigma$-algebra, that the common current practice of using Armitage’s formula for calculating distributions of interim test statistics is appropriate. There is plenty of evidence that adaptive designs make statistics non-normally distributed. Demets and Lan (1994) point out that the distribution of stage- specific test statistics is not normal and should be estimated recursively. Jennison and Turnbull (1999) plot the density of a normal test statistic used in GSD settings [pages 174-177], where discontinuity points clearly show non- normality. Li et al. (2002) find the joint density of stage 1 and stage 2 standardized test statistics not to be bivariate normal. Local asymptotic non- normality was established following sample size recalculations (SSRs) that depend on an interim observed treatment effect (Tarima and Flournoy (2019a, b)); and a GSD with a single interim analysis can be viewed as a special case of an SSR. MLEs converging to random mixtures of normal variables have been found in other adaptive designs (Ivanova et al. (2000), Ivanova and Flournoy (2001), May and Flournoy (2009), Lane and Flournoy (2012), Flournoy et al. (2018)). Milanzi et al. (2015) developed a likelihood approach that applies when the early stopping rule does not depend on the parameter of interest. In this case, sample size adaptation is ancillary to the treatment effect and asymptotic normality of MLEs holds. Gnedenko and Korolev (1996); Bening et al. (2012); Christoph et al. (2020); Korolev and Zeifman (2019) assume that the distribution of the random sample size does not depend on previosly collected data; Molenberghs et al. (2012) assumes this asymptotically. Nevertheless, convergence of sample means to non-normal random variables was shown even for ancillary random sample sizes. In Gnedenko and Korolev (1996), Gnedenko and Korolev show convergence of standardized sums with random number of summands of infinitely divisible random variables to mixtures of stationary distributions. They give conditions for convergence to a mixture of normal distributions. Bening et al. Bening et al. (2012) and Christoph et al. Christoph et al. (2020) explore convergence to mixtures of normal distributions and to Student’s limit distribution. When convergence is _mixed_ (see, for example Häusler and Luschgy (2015)), Lin et al. (2020) shows how norming with the observed information can result in a normal limit. However, the requirement for mixed convergence appears strong, and it does not cover the limiting mixtures obtained in this paper. The impact of early stopping is pervasive. It affects the probabilistic characterisation of the tests (e.g., type I error and Fisher Information) as well as the distributions of MLEs and test statistics. Its effect on Fisher information, when stopped at different stages, is not widely recognized. During the design phase, before observations are taken, the full form of the joint density (5), accounting for all possible events, is appropriate. This contrasts with the current practice of using the density assuming the experiment will continue through stage $K$. More details on the differences resulting from these two design approaches will be the subject of another paper. However, normality and asymptotic normality assumptions continue to be directly used with non-normally distributed statistics. We identify two main reasons for this. 1. 1. Many researchers consider large sample properties against a fixed treatment effect independent of the sample size. From one point of view, a treatment effect is a population quantity which does not change with sample size. But if one develops an asymptotic approximation to the testing environment using a fixed treatment effect, the statistical experiment stops at the first interim analysis with probability one for any consistent test; test statistics degenerate to a point mass; see Section 7.4 in Fleming and Harrington (1991). Under a fixed treatment effect, the power converges to one and cannot be used to compare different testing procedures. This issue triggered development of various descriptions of asymptotic relative efficiency. The most popular approach is Pitman asymptotic relative efficiency Pitman (1948), where asymptotic power is evaluated under local alternatives; see Nikitin (1995). In addition, local alternatives clearly reflect actual practice for experiment planning. Experiments are never planned for a statistical power = 1. Small sample size studies (pre-clinical, animal studies) are planned to detect large effect sizes, moderate sample sizes (typical phase 3 studies) are used to detect moderate effect sizes, and large sample sizes (epidemiological studies, like vaccine studies) are used to detect small differences. Koopmeiners et al. (2012) explored MLEs conditional on stopping, but assumed asymptotic normality to evaluate their uncertainty. Martens and Logan (2018) relied on asymptotic normality for evaluating regression coefficients under the Fine–Gray model in GSD settings. Asendorf et al. (2019) evaluated asymptotic properties with SSR under a fixed alternative for negative binomial random variables. 2. 2. Some researchers investigate local asymptotic properties when early stopping is not possible or is ancillary to the treatment effect. This also leads to asymptotic normality. Scharfstein et al. (1997) show that without possibility of early stopping “time-sequential joint distributions of many statistics $\ldots$ are multivariate normal with an independent increments covariance structure” under local alternatives. These results are generally consistent with the classical results on local asymptotic normality of Le Cam (1960), where both mean and variance of the limiting normal distribution depend on the parameter of interest; see Chapter 7 of van der Vaart (1998). However, Section 3.1 shows that the possibility of early stopping destroys local asymptotic normality: the limiting distribution of standardized test statistics is a mixture of truncated normal distributions. Similar findings were previously proved for non-ancillary sample size recalculations; see Tarima and Flournoy (2019a). Gao et al. (2013) is a rare exception in not making a normality assumption; these authors mostly deal with set operations and probabilities and, using stage-wise ordering of events, they calculate P-values, confidence intervals, and a median unbiased estimate of the parameter of interest. It is recommended that the full form of the joint density (5), accounting for all possible events, be used for study design before observations are taken. 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# Dexterous Manipulation Primitives for the Real Robot Challenge Claire Chen &Krishnan Srinivasan 11footnotemark: 1 &Jeffrey Zhang 11footnotemark: 1 &Junwu Zhang 11footnotemark: 1 Stanford University, Palo Alto, CA ###### Abstract This report describes our approach for Phase 3 of the Real Robot Challenge. To solve cuboid manipulation tasks of varying difficulty, we decompose each task into the following primitives: moving the fingers to the cuboid to grasp it, turning it on the table to minimize orientation error, and re-positioning it to the goal position. We use model-based trajectory optimization and control to plan and execute these primitives. These grasping, turning, and re- positioning primitives are sequenced with a state-machine that determines which primitive to execute given the current object state and goal. Our method shows robust performance over multiple runs with randomized initial and goal positions. With this approach, our team placed second in the challenge, under the anonymous name “sombertortoise” on the leaderboard. Example runs of our method solving each of the four levels can be seen in this video. Advised by: Lin Shao11footnotemark: 1, Shenli Yuan11footnotemark: 1, Preston Culbertson11footnotemark: 1, Hongkai Dai **Toyota Research Institute, Los Altos, CA, Mac Schwager11footnotemark: 1, Jeannette Bohg 11footnotemark: 1 ## 1 Real Robot Challenge Overview Figure 1: The Phase 3 setup consists of the TriFinger robot and a cuboid object. The Real Robot Challenge [1] invited teams to design and implement methods for performing dexterous manipulation tasks using the TriFinger robotic platform [2]. The TriFinger robot has three fingers, each with three degrees of freedom. Phases 1 and 2 of the challenge involve manipulating a cube with a 6cm side length, first in simulation and then with the real robot. Phase 3 involves manipulating a 2cm x 2cm x 8cm cuboid object. Each phase is divided into the following 4 levels of increasing difficulty: • Level 1: Move object to randomly sampled goal position on the table. Orientation not considered. * • Level 2: Move object to a fixed goal position above the table. Orientation not considered. * • Level 3: Move object to randomly sampled goal position. Orientation not considered. * • Level 4: Move object to randomly sampled goal pose where both position and orientation are considered. In all phases, we have access to robot proprioception including joint angles and fingertip forces, as well as an estimate of the object pose obtained from three cameras. We are also given full information about the shape and mass of the object, as well as the mass properties of the robot. ## 2 Method Our approach rests on the idea that in-hand manipulation tasks can be broken down into manipulation primitives [3]. We decompose the task of moving the object to a goal pose into three primitives: * • grasp: Move fingertips from initial positions to pre-defined contact points on cuboid. Before executing the re-position and turn primitives, the state machine must choose the grasp primitive. * • re-position: With three fingers on the cuboid, translate and lift to goal position. * • turn: With three fingers on the cuboid, rotate it on the table and return fingers to initial positions above object. We use model-based trajectory optimization and control to plan and execute these primitives, the details of which are described in the following sub- sections. We design a state machine to sequence the primitives. At the start of a task, the state machine selects the grasp primitive to bring the fingers to desired contact points on the cuboid. Once the cuboid has been grasped, the state machine determines whether to re-position or turn the cuboid. For Levels 1-3, the state machine always chooses the re-position primitive, as orientation error is not considered in the score. For Level 4, the state machine chooses to turn the object to align its yaw rotation with that of the goal pose before choosing the re-position primitive to move it to the goal position. To grasp the object, we plan trajectories for each fingertip to reach desired contact points using trajectory optimization and track these trajectories using the simplified impedance controller from [2]. To re-position and turn the object, we first compute a trajectory for the object pose assuming that contact forces are applied at the pre-defined contact points from the grasping primitive, and then use this trajectory to compute the corresponding fingertip trajectories. In addition to assuming that the fingers remain fixed at the pre-defined contact points, we also assume that we have access to prior knowledge of robot kinematics and dynamics, physical parameters of the object, as well as accurate state feedback on object pose and finger joints. Given these assumptions, a model-based trajectory optimizer and simplified impedance controller to track fingertip trajectories are sufficient for executing all of the primitives. Figure 2: A diagram of the state machine. A primitive is selected based on the required level of difficulty and the difference between the object’s current and goal pose. The state machine begins with the grasp primitive to move the fingers to contact points on the object. For Levels 1-3, the state machine always chooses the re-position primitive, as orientation error is not considered in the score. For Level 4, the state machine chooses to turn the object to align the yaw rotation before choosing the re-position primitive to move it to the goal position. The re-position primitive is always assumed to be successful and is thus the terminal state. ### 2.1 Grasping To grasp the cuboid securely, we use a three-fingered grasp to pinch the cuboid across its short axis, as shown in Figure 2. This grasp is used for both the re-position and turn primitives. We pre-define the contact points that compose this grasp, with one finger placed on a long face of the cuboid and the other two fingers placed the opposite long face of the cuboid, equidistant from the center of the face. For each finger, we choose which face to contact based on the face that is closest to that finger. In our tests, we find that this simple face assignment strategy is enough to ensure that all fingers are able to reach the contact points without collision, since the cuboid always starts near the center of the arena. Since the object is very narrow, we perform a two-stage grasping motion by first lowering the fingers to the height of the object before then moving them towards the object to pinch it. We use trajectory optimization to first plan a trajectory to lower the fingers, and then again to pinch the cuboid. For the cube manipulation task in Phase 2, we included a collision penalty term in the trajectory optimization problem to prevent the fingertips from making unwanted contact with the cube. While this worked well for grasping the cube, we found it much more challenging to tune the collision penalty term for a slimmer object like the cuboid. Ultimately, the trajectory optimization we use for the cuboid tasks in Phase 3 does not include collision avoidance, as the two-stage grasping motion was more reliable for preventing unwanted collisions. We acknowledge that without the need for collision avoidance, an operational space controller could have been used instead of trajectory optimization for the simple task of moving the fingertips to desired goal positions; however, we already had the trajectory optimization method implemented. #### Fingertip position trajectory optimization: We formulate a direct collocation trajectory optimization problem to compute collision-free fingertip trajectories to the desired goal positions in Cartesian space. The optimization problem, shown below, considers only the kinematics of the fingers and assumes that the pose of the cuboid does not change. We find trajectories of length $T$ for joint angles $q$ and joint velocities $\dot{q}$ and use forward kinematics to compute the corresponding fingertip positions $x$ and velocities. $\displaystyle\underset{q,\dot{q},\alpha}{\text{minimize}}\;\;\;\sum_{t=0}^{T}(x_{\text{goal}}-x_{t})^{T}Q(x_{\text{goal}}-x_{t})+\sum_{t=0}^{T-1}\dot{q}^{T}R\dot{q}+\sum\alpha$ (1) subject to: $\displaystyle q_{k+1}-q_{k}$ $\displaystyle=\frac{1}{2}(t_{k+1}-t_{k})(\dot{q}_{k+1}+\dot{q}_{k})$ $\displaystyle\;\;\;\forall k\in(0,1,2,...,T-1)$ (2) $\displaystyle\|\|x_{k}\|\|_{2}$ $\displaystyle<r_{\text{arena}}$ $\displaystyle\;\;\;\forall k\in(0,1,2,...,T-1)$ (3) $\displaystyle(x_{\text{goal}}-x_{T})^{2}$ $\displaystyle>\alpha$ (4) $\displaystyle\alpha$ $\displaystyle>0$ (5) $\displaystyle q$ $\displaystyle\in[q_{\text{min}},q_{\text{max}}]$ (6) $\displaystyle\dot{q}$ $\displaystyle\in[\dot{q}_{\text{min}},\dot{q}_{\text{max}}]$ (7) The cost function includes the distance between the fingertip positions and goal positions at each time step weighted with weight matrix $Q$, joint velocities at each time step weighted with weight matrix $R$, and the sum of all the slack variables $\alpha$. The weight matrices were tuned to produce reasonable-looking solution trajectories for the fingers. Equation 2 enforces joint velocity constraints, Equation 3 ensures that the fingertips remain within the arena radius, and Equations 4 and 5 constrain the final fingertip positions to be equal to the goal fingertip positions, relaxed with slack variables $\alpha$. We use IPOPT [4] to solve all trajectory optimization problems. ### 2.2 Turning and re-positioning Given an initial and goal pose for the object’s center of mass (CoM), we use direct collocation to plan a trajectory for the object, as well as the associated forces that need to be applied at fixed contact points to move the object along this trajectory. The contact points are assumed to be fixed at the pre-defined positions specified in the grasp primitive. #### Object CoM trajectory optimization: The following optimization problem finds trajectories of length $T$ for object pose $o$, object velocity $\dot{o}$, and contact forces $\lambda^{cf}$ expressed in the corresponding local contact point reference frames $cf$. $\displaystyle\underset{o,\dot{o},\lambda^{cf},\alpha}{\text{minimize}}\;\;\;\sum_{t=0}^{T}(o_{\text{goal}}-o_{t})^{T}Q(o_{\text{goal}}-o_{t})+\sum_{t=0}^{T-1}(\lambda^{cf}_{\text{target}}-\lambda^{cf}_{t})^{T}R(\lambda^{cf}_{\text{target}}-\lambda^{cf}_{t})+\sum\alpha$ (8) subject to: $\displaystyle o_{k+1}-o_{k}$ $\displaystyle=\frac{1}{2}(t_{k+1}-t_{k})(\dot{o}_{k+1}+\dot{o}_{k})$ $\displaystyle\;\;\;\forall k\in(0,1,2,...,T-1)$ (9) $\displaystyle\dot{o}_{k+1}-\dot{o}_{k}$ $\displaystyle=\frac{1}{2}(t_{k+1}-t_{k})(\ddot{o}_{k+1}+\ddot{o}_{k})$ $\displaystyle\;\;\;\forall k\in(0,1,2,...,T-1)$ (10) where: $\displaystyle\ddot{o}=M_{\text{obj}}^{-1}(G\lambda^{cf}+g_{\text{obj}})$ (11) $\displaystyle(o_{\text{goal}}-o_{T})^{2}$ $\displaystyle>\alpha$ (12) $\displaystyle\alpha$ $\displaystyle>0$ (13) We use Equation 11 to define object dynamics, where $M_{\text{obj}}$ is the mass matrix of the object, $G$ is the grasp matrix, and $g_{\text{obj}}$ is the vector of gravitational forces on the object. The cost function includes the distance between the object positions and goal positions at each time step weighted with weight matrix $Q$, the difference between the contact forces and target contact forces $\lambda^{cf}_{\text{target}}$ at each time step weighted with weight matrix $R$, and the sum of all the slack variables $\alpha$. The weight matrices were tuned to produce trajectories where the object would move towards the goal pose gradually, rather than too quickly, as we found that it was most reliable to move the object slowly. The normal force components of $\lambda^{cf}_{\text{target}}$ are set to some desired normal force. We can also constrain the contact forces to lie within linear approximations of friction cones to prevent contact slippage, but in practice, we found that applying sufficient normal force was enough to prevent grasp slippage. Equations 9 \- 11 enforce the object’s dynamic constraints, and Equations 12 and 13 constrain the final object position to be equal to the goal object pose $o_{\text{goal}}$, relaxed with slack variables $\alpha$. Given the trajectories for object pose and contact forces expressed in local contact frames and given our assumption of fixed contact points, we compute the corresponding trajectories for fingertip positions $x$ and contact forces expressed in robot frame $\lambda^{rf}$. Compared to re-positioning the object, re-orienting the object as required in Level 4 is considerably less straight-forward. While re-positioning the cuboid can be achieved with a single grasp, rotating the cuboid could potentially require re-grasping. To enable this, we introduce additional logic into our state machine. Given the difficult nature of re-orienting the object in mid- air, we only attempt to turn the object while it rests on the table. To ensure that the contact points remain reachable by each finger while rotating the cuboid, we chose a simple heuristic: we turn the object in 45 degree increments, resetting and re-grasping the object between each rotation. For example, a 120 degree rotation would be broken down into two 45 degree rotations followed by a 30 degree rotation. ### 2.3 Impedance Controller We compute the joint torques necessary for tracking the desired fingertip trajectories in Cartesian space using the simplified impedance controller from [2] with additional gravity compensation for the fingers (time index omitted for clarity): $\tau=J^{T}(k_{p}(x_{\text{ref}}-x)+k_{v}(\dot{x}_{\text{ref}}-\dot{x})+\lambda^{rf})+g_{\text{hand}}$ (14) where $\tau\in\rm I\\!R^{9}$ is the vector of joint torques to be applied to each finger, $x_{\text{ref}}$ are the desired fingertip positions from the reference trajectory, $\lambda^{rf}$ is the vector of desired contact forces to be applied by each finger in robot frame, $J$ is the Jacobian of the 3 fingers, $g_{\text{hand}}$ is the gravity compensation vector, and $k_{p}$ and $k_{v}$ are hand-tuned controller gains. While we were able to grasp and move the object by just taking into account feedback on fingertip positions, we also tried incorporating object pose feedback to further reduce the steady state error between current and goal object pose. We follow the method presented in [2], which uses a PD law to compute the wrench that needs to be applied to the object to track a desired trajectory. The noisy object orientation estimates made it difficult for us to obtain stable performance with this additional PD law, so we chose to only track fingertip trajectories. ### 2.4 Reinforcement Learning To improve the score of the model-based method described above, we explored using several deep reinforcement learning algorithms for learning other primitives, such as pushing and turning the cuboid. We used Proximal Policy Optimization (PPO) to optimize a neural network policy $\pi_{\phi}$, parametrized by $\phi$, to push the object from an arbitrary starting position and orientation on the table to the center of the table with a desired goal orientation. The composite reward function integrates the objectives of minimizing position and orientation error, and is computed by the following equation: $R(o_{\text{pos}},o_{\text{ori}},g_{\text{pos}},g_{\text{ori}})=r_{\text{pos}}(\\|o_{\text{pos}}-g_{\text{pos}}\\|)+r_{\text{ori}}(\\|o_{\text{ori}}^{\text{yaw}}-g_{\text{ori}}^{\text{yaw}}\\|).$ (15) The position reward function, $r_{\text{pos}}$, is $r_{\text{pos}}(d)=\begin{cases}1,&\text{if $d<0.05$}\\\ \frac{1}{9(20d)^{2}+1}&\text{$0.05<d<0.075$}\\\ 0&\text{o.w.}\end{cases}$ and similarly, the orientation reward function $r_{\text{ori}}$, is computed by $r_{\text{ori}}(d)=\begin{cases}1,&\text{if $d<\frac{\pi}{8}$}\\\ \frac{1}{9(8^{-1}\pi d)^{2}+1}&\text{if $\frac{\pi}{8}<d<\frac{3\pi}{16}$}\\\ 0&\text{o.w.}\end{cases}$ The objective from [5] is then optimized to learn a policy: $L(s,a,\phi_{k},\phi)=\min\bigg{(}\frac{\pi_{\phi}(a\|s)}{\pi_{\phi_{k}}(a\|s)}A^{\pi_{\phi_{k}}}(s,a),\;\text{clip}\bigg{(}\frac{\pi_{\phi}(a\|s)}{\pi_{\phi_{k}}(a\|s)},1-\epsilon,1+\epsilon\bigg{)}A^{\pi_{\phi_{k}}}(s,a)\bigg{)},$ (16) where $s$ and $a$ are an arbitrary state and action, $A^{\pi_{\phi}}(s,a)$ is the advantage function, and $\phi_{k}$ is the policy parameters at the previous optimization step $k$. The resulting policy forms a new _Push-Turn_ primitive and pushes the object to a position that is a) easier for the grasp primitive to grasp the object from and b) avoids calling the _turn_ primitive, which requires more time to complete. While this policy was able to perform the rotation task in simulation, due to robustness issues when transferred to the real robot, we opted to use the optimal controller alone in the final submission for phase 3, as it showed less variance in the Level 4 task score. ## 3 Results and Discussion \| Level 1 \| Level 2 \| Level 3 \| Level 4 ---\|---\|---\|---\|--- Goal # \| 1 \| 1 \| 1 \| 2 \| 3 \| 1 \| 2 \| 3 Median score \| -3455 \| -7889 \| -12241 \| -8676 \| -11775 \| -49773 \| -26250 \| -39230 Table 1: The median scores of our method for different goals and levels. Per level and goal, we execute three runs each starting from a random initial position. For Levels 1 and 2, we show results for one randomly sampled goal position each. For Levels 3 and 4, we show results for three randomly sampled goal poses. Our current method is able to turn the cuboid on the table and re-position it to a goal pose. We evaluate our method on each level. Table 1 shows the median scores over three runs per goal. The score for an episode is computed by summing the negated error between current and goal object pose over each time step of a two minute episode, i.e. scores closer to zero are better. As expected, the scores depend heavily on the goal pose; goal poses closer to the center of the arena, where the cuboid starts, result in better scores, as it takes fewer time steps for the object to reach the goal. For example, goals #1 and #2 in Level 3 are approximately 15cm and 10cm away from the center of the arena, respectively. Although the median final error to the goal position is approximately 1cm for both goals, the final scores are quite different. On the real robot, the initial pose of the cuboid is always near the center of the arena, but with some randomness due to the robot initialization procedure. As a consequence, we observe that the varied initial poses of the cuboid can cause large variations in scores across runs for the same goal pose. This was particularly true for Level 4 goals, since the initial orientations varied greatly. Although our method is able to complete Levels 1, 2, and 3, as well as make good progress in Level 4, there are several improvements that could be made in the future. Firstly, our method assumes accurate knowledge of object pose, and is therefore not robust to the noisy object orientation measurements from the visual tracker. Secondly, our method still suffered from steady state errors when re-positioning the cuboid, which could be improved by taking into account object pose feedback with either another feedback law or a learned policy. For instance, a hybrid control policy combining RL with impedance control in a hierarchical or residual policy could be helpful in reducing some of these errors, and aid in planning a trajectory that requires reorienting the object. Finally, more sophisticated reasoning could be used to choose grasps and sequence rotations for re-orienting the object.well ## 4 Acknowledgements We thank the challenge organizers for providing us with the opportunity to use the TriFinger robot. We would especially like to thank Felix Widmaier for the very prompt responses to our many questions. We look forward to using the TriFinger platform for more exciting projects in the future! ## References * Rea [2020] Real robot challenge, 2020. URL https://real-robot-challenge.com/en. * Wüthrich et al. [2020] M. Wüthrich, F. Widmaier, F. Grimminger, J. Akpo, S. Joshi, V. Agrawal, B. Hammoud, M. Khadiv, M. Bogdanovic, V. Berenz, et al. Trifinger: An open-source robot for learning dexterity. _arXiv preprint arXiv:2008.03596_ , 2020. * Okamura et al. [2000] A. M. Okamura, N. Smaby, and M. R. Cutkosky. An overview of dexterous manipulation. In _Proceedings 2000 ICRA. Millennium Conference. IEEE International Conference on Robotics and Automation. Symposia Proceedings (Cat. No. 00CH37065)_ , volume 1, pages 255–262. IEEE, 2000. * Wächter and Biegler [2006] A. Wächter and L. T. Biegler. On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming. _Mathematical Programming_ , 2006. * Schulman et al. [2017] J. Schulman, F. Wolski, P. Dhariwal, A. Radford, and O. Klimov. Proximal policy optimization algorithms. _ArXiv_ , abs/1707.06347, 2017.
# The FRAM robotic telescope for atmospheric monitoring at the Pierre Auger Observatory The Pierre Auger Collaboration et al ###### Abstract FRAM (F/Photometric Robotic Atmospheric Monitor) is a robotic telescope operated at the Pierre Auger Observatory in Argentina for the purposes of atmospheric monitoring using stellar photometry. As a passive system which does not produce any light that could interfere with the observations of the fluorescence telescopes of the observatory, it complements the active monitoring systems that use lasers. We discuss the applications of stellar photometry for atmospheric monitoring at optical observatories in general and the particular modes of operation employed by the Auger FRAM. We describe in detail the technical aspects of FRAM, the hardware and software requirements for a successful operation of a robotic telescope for such a purpose and their implementation within the FRAM system. ## 1 Introduction The Pierre Auger Observatory [1], located near Malargüe, Argentina, is currently the largest ultra-high energy cosmic ray observatory in the world. It measures the properties of extensive air showers induced by cosmic rays in the atmosphere. The observatory uses a hybrid detection technique and consists in its basic configuration of an array of surface particle detectors and five stations with optical telescopes overlooking this array from each side. These telescopes detect the faint fluorescence light emitted in the atmosphere due to the passage of the secondary particles of the extensive air showers. While the surface detectors work continuously, the fluorescence telescopes can be operated only in dark conditions during nights with low Moon illumination (approximately 15% of the time). The measurement of the longitudinal profile of the fluorescence light by the fluorescence telescopes provides an estimate of the energy deposit of the showers applying the known air fluorescence yield and its dependence on local atmospheric conditions [2, 3] and the consideration of the so-called invisible energy [4]. The total energy of the primary cosmic ray is then determined as the sum of the calorimetric energy and the invisible energy. The longitudinal fluorescence profile also carries information about the character of the primary particle, and thus about the mass composition of the primary particle beam, and potentially also about the high-energy hadronic interactions that took place during the development of the shower. In order to fully exploit the fluorescence technique, many aspects of atmospheric conditions must be continually monitored. The production yield of the fluorescence light depends on vertical profiles of temperature, humidity, and pressure. The transmission of this light is affected by scattering by molecules and aerosols (the contribution of absorption is negligible in the wavelength range used by the fluorescence telescopes) over distances up to tens of kilometres to the fluorescence telescopes. While the Rayleigh scattering by molecules can be determined from the state variables of the atmosphere, the Mie scattering by aerosols requires frequent dedicated measurements [5, 6]. Using hourly vertical aerosol profiles, instead of an average profile, significantly improves the precision of the determination of both the energy of the primary particle and the depth of shower maximum [7]. The presence of any clouds can also affect the determination of the aforementioned observables as well as of any other properties inferred from the shape of the profile (such as those used for particle physics studies), as the presence of the clouds can significantly distort the apparent profile. A series of cloud cuts from different instruments is applied to select events for physics analysis. For these reasons, the operation of fluorescence telescopes requires a sophisticated atmospheric monitoring system. At the Pierre Auger Observatory, the primary devices to assess the transparency of the atmosphere are two laser installations near the center of the array, called the CLF (Central Laser Facility) and the XLF (eXtreme Laser Facility) [8], used as reference light sources. The scattered light is then observed with the fluorescence telescopes themselves to infer the vertical profiles of aerosol extinction [1]. Additionally, a Raman lidar is operated at site of the CLF, sharing the laser beam with the CLF [9]. Furthermore, elastic lidars at the sites of the fluorescence sites primarily provide information on cloud cover and height [10]; the cloud cover is also monitored using infra-red cloud cameras [11] and visible-light All-Sky Cameras that detect clouds by star counting [12]. The laser systems actively produce light which means that when used in the field of view of the fluorescence telescopes, the data taking is affected. The FRAM (F/Photometric Robotic Atmospheric Monitor) robotic telescope has been developed as an additional atmospheric monitoring device using stars as reference light sources. The first FRAM hardware was installed at the site of the first fluorescence telescope station (Los Leones) in 2005. Since then, the hardware, measurement methods, and applications of the system have gone through several major improvements. The layout of the observatory with the described devices is illustrated in Fig. 1. Figure 1: Schematic overview of selected atmospheric monitoring devices installed at the Pierre Auger Observatory. At each FD site, there is, among other devices, a lidar station, an infrared camera for cloud cover detection and a visible-light All-Sky Camera. Two laser facilities (CLF and XLF) are installed close to the center of the surface detector array. The FRAM telescope is located at the Los Leones FD site (with a second FRAM planned for the Cohuieco FD site in the future). Using stars as reference sources allows for determining the integral optical depth using a passive system. Indeed, the FRAM complements the laser-based methods where data are needed with high temporal and spacial resolution without interfering with the operation of the fluorescence telescopes. Currently the main application of FRAM at the Auger Observatory is the rapid monitoring of atmospheric conditions along the apparent path of showers that may have an anomalous longitudinal profile [13]. Knowing actual atmospheric conditions for those showers enables us to exclude the possibility that such anomaly was caused by the presence of any clouds along the shower track ("Shoot-the-Shower") and hadronic interactions models can be tested and further constrained [14]. The successful development of the Auger FRAM led to the proposal to include a similar instrument in the design of the Cherenkov Telescope Array (CTA), the future largest ground-based gamma-ray observatory in the world. The CTA FRAMs have a modified design tailored to the characteristics of the CTA observatory where they will monitor the changes in atmospheric transparency across the whole field of view of the Cherenkov telescopes simultaneously [15]. Based on data from the wide-field telescopes installed at the future sites of the Cherenkov Telescope Array, it has been be shown that wide-field photometry can be used to measure the integral atmospheric extinction with a precision better than 0.01 optical depths [16]. We plan to implement this method at the FRAM setup at the Pierre Auger Observatory in the near future. ## 2 Applications of stellar photometry for atmospheric monitoring The basic idea of using stars as reference sources for atmospheric monitoring is simple: the difference between the observed brightness of a star and the predicted value based on the star’s known properties depends on the transparency of the atmosphere between the detector and the upper edge of the atmosphere in the given direction. More specifically, for each star observed, a model brightness is calculated from the catalogue value (in magnitudes) $m_{\mathrm{cat}}$ as $m_{\mathrm{model}}=M\left(m_{\mathrm{cat}}+Z+f(C,x,y)+g(A,k,C)\right),$ (2.1) where $M$ accounts for possible non-linearity of the whole system (including the photometric method), $Z$ is the calibration constant of the system (so- called zeropoint), $f$ describes corrections due to the different response of the system to stars of different colors through some color index $C$ and in different parts of the field of view, and $g$ is a model of extinction depending on the airmass $A$, the color of the star and the extinction constant $k$, which is equal to the extinction measured at the zenith [17]. The airmass $A$ is a dimensionless quantity expressing the integral mass of atmosphere (or a component thereof) encountered by the light of a star at a given altitude above the horizon relative to the same quantity at the zenith. If stars at different values of airmass are observed at once or in a short time window, the parameters of the instrument and of the atmosphere can be determined simultaneously as the former do not depend on the airmass while the latter do and can thus be separated using a fitting procedure, akin to the well-known Langley method [18]. As some instrumental parameters change only slowly in time, they may be fitted over a larger set of observations globally. In reality, both the observed and the predicted brightness are affected by uncertainties which determine the applicability of the method for different purposes and the optimal detection setup. The availability of precise photometric data for stars varies from star to star as there are many catalogs with varying levels of sky coverage. Moreover, typically a brightness value measured in a bandpass that does not match the spectral response of a given setup and thus data from several bandpasses are required for a reliable prediction of the brightness of the star. In standard photometric fields, there are many stars in a few small fields of the sky for which precise data in many wavelength bands exist, but using only these would severely limit the possibility to do measurements in an arbitrary direction. Furthermore, these isolated fields are typically unsuitable for any method that uses the Langley calibration, because the only way to observe them at various values of airmass is to wait for them to move due to the rotation of the Earth – however during this time, the conditions may change. For the purpose described here, it is thus better to use a suitable all-sky catalog, such as Tycho2 [19]. Despite the precision of Tycho2 for individual stars being lower than that of dedicated photometric surveys, the homogeneity of its data over the entire sky is extremely valuable as it prevents the introduction of any biases when processing measurements of stars from large areas of the sky simultaneously. Using such a homogeneous catalog allows measurements of transparency in arbitrary directions, although the low precision for individual stars requires the observation of a large number of stars at once. The uncertainty of the catalog brightness in Tycho2 increases quickly above magnitudes of roughly 10. Since there are relatively few sufficiently bright stars in the sky, a wide-field setup (on the order of degrees) has to be preferred, although this property comes along with a relatively small aperture. The uncertainty in the measured brightness of the stars depends on the specific instrumental setup. For a long time, photoelectric photometers were considered the gold standard in astronomical photometry thanks to their stability and absolute calibration, but they have been mostly superseded by CCD cameras for most applications. For a CCD camera, the uncertainty in a photometric measurement depends on the noise level and stability of the camera electronics and other hardware effects and then chiefly on the amount of light registered from the star. That in turn depends on exposure length and the aperture of the optics; for a given aperture, the maximal field of view is limited by practical limits of optics and thus there is always a trade-off between the precision of measurement for individual stars and the number of stars observed. Based on these considerations and from practical experience, we have identified several basic operation modes in which a small robotic optical telescope can be employed in atmospheric monitoring for an optical observatory (not necessarily a set of fluorescence or Cherenkov telescopes). Figure 2: An example of an altitude scan taken by the Auger FRAM. Each individual point represents a single star and the y-value is equal to the observed extinction for the star: the difference between its catalog and measured brightness, corrected for various instrumental and atmospheric effects. The fit of the extinction as a function of altitude (shown by the dashed line) allows the determination of the vertical aerosol optical depth. This particular scan has been originally taken in Mode B, triggered by a cosmic ray shower, but the same data can be used for precise aerosol measurements. In this mode, besides the continuous coverage of the field of view of the fluorescence detector between 1.5 and 31.5 degrees of altitude, also an image of the arrival direction of the shower is taken, which is also included in the fit to improve the lever arm. The error bars include the statistical and systematic uncertainties of both the measurement and the catalog values. #### Mode A – Precise aerosol measurement using altitude scans. Taking a series of images at different altitudes above the horizon and thus at different values of airmass $A$ allows for the simultaneous determination of the atmospheric extinction and the instrument calibration constant $Z$. This is achieved by a fit to the altitude dependence of the difference between predicted and observed brightness of the stars when assuming horizontal stratification of aerosols in the atmosphere as described in detail in [16] – see Fig. 2. This method provides the most precise value of the integral vertical aerosol optical depth, which is obtained from the shape of the altitude dependence and thus does not require prior knowledge of the calibration constant $Z$. In fact it can be used to provide calibration for measurements in other modes. However in order for the fit to work, the extinction must follow the assumed altitude dependence, which is violated in particular when clouds are present. In the presence of some clouds on the sky, the probability for successful measurement can be improved by using external data to select a cloud-free path for the scan on the sky; this is implemented on all the current FRAMs using data from the All-Sky Cameras that are present both at the Auger Observatory and at the future CTA sites. Potentially it can be also problematic when aerosols are distributed in a non-stratified manner. Such instances are clearly visible from the data as deviations from the fitted shape and can be excluded, so the result is not an invalid value, but no value at all. Therefore, in order to produce results, the method requires some level of homogeneity over several kilometers of distance from the telescope towards the azimuth of the scan, which is the typical distance where most of the observed extinction happens (usually only stars more than 7 degrees above the horizon are taken into account in altitude scans). However this is still a much smaller scale than that of the whole Auger Observatory ($65\times 45$ km2). The method can also be used to investigate small inhomogeneities in the aerosol distribution by taking scans in different azimuth directions and this can also be repeated with high temporal resolution in order to quantify the changes in the aerosol conditions. At the Auger FRAM, the altitude scans are taken as a series of 30 s exposures taken with the photometric Johnson B filter (see Fig. 3 for an example) at an azimuth based on the available real- time cloud data from other instruments and the position of the Moon in the sky. The Johnson B filter has been chosen for its proximity in wavelength to the near-UV region in which the Auger fluorescence telescopes operate, as photometry directly in this region is complicated by the spectral properties of the used optics (cf. Fig. 8) and the lack of suitable all-sky stellar catalogs for comparison. The measured extinction can be translated to the near-UV region using Eq. 2.2. Figure 3: One of the images taken by the Auger FRAM during a Mode A altitude scan, in this case covering a well-known region of the Milky Way around the star $\eta$ Carinae and its surrounding nebula (seen left of the center). A single 30-second exposure in the Johnson B filter can be used to measure the brightness of hundreds of stars simultaneously in such a rich area. #### Mode B – Triggered operation in a large field of view. For a system such as the fluorescence detector of the Pierre Auger Observatory with a large field of view ($180^{\circ}\times 30^{\circ}$ for each of the four main FD telescope stations), it is not feasible to permanently monitor the whole field of view, but a robotic telescope can be used for timely determination of conditions in a selected part of the field of view. For triggering such a dedicated measurement, quasi-online reconstructed data from the fluorescence telescopes, indicating “interesting” events, can be used, based on a set of configurable cuts, as the rate of detection of showers by the fluorescence telescopes is too high to follow-up on all of them. When a shower is selected for observation by FRAM, the geometric parameters of its trajectory are passed to the FRAM system, which generates a set of fields to observe so that the trajectory is well covered, see Fig. 4. The investigated trajectory then also forms a scan in altitude and thus the method of mode A can be applied directly to obtain a calibrated measurement. However if the aim is to identify inhomogeneities in the investigated region (such as in the case of the Shoot-the-Shower program at Auger), it is sufficient to just search for deviations from the theoretical altitude dependence of extinction and thus the method is extremely sensitive to any such disturbances as those are completely independent of the calibration of the system. On the other hand a horizontally uniform layer is completely undetectable through stellar photometry, but may be important for the subject at hand as an extensive air shower for example may pass through the layer making its light affected by the extinction in the layer only for a part of its trajectory. In principle, a similar method could be used for any other purpose where a suitable trigger can be found to define an area of interest within a larger field of view. If only deviations from the theoretical altitude dependence are of interest, the area can be of any shape; if precise aerosol measurements are needed, the area either needs to span a large range of airmass, or separate altitude scans (mode A) shall be made for calibration. Figure 4: The apparent trajectory of an extensive air shower can be described by several geometric parameters (left) that are then passed to FRAM, where a series of observation fields (right) is generated in order to cover the trajectory across the whole FD field of view between 1.5 and 31.5 degrees of altitude as well as the direction of the arrival of the shower as a serendipity observation for the case that the cosmic ray was associated with a transient phenomenon. The observation depicted here is the same that produced the data shown in Fig. 2. #### Mode C – Continuous monitoring of a small field of view. If the area of interest on the sky is sufficiently small so that it can be contained entirely within a field of view of a robotic telescope (such as is the case with the field of view of the CTA telescopes), it can be monitored continuously (with temporal resolution given by the length of exposure and readout) for changes in transparency. As in the previous case (Mode B), it is much easier to detect changes in transparency than to measure the absolute value of it in a single field. However even the latter is readily possible using the calibration provided by taking altitude scans (Mode A) at regular intervals, even though the precision of the extinction measured is smaller than that of the value obtained from a full scan, as it combines the uncertainty of the measurement in the field and that of the calibration determined from the scans. This is the operation mode foreseen for the CTA FRAMs during their future operation. The field of view itself will be cut into smaller areas of comparable numbers of stars using adaptive Voronoi tessellation in order to provide a 2D view of possible changes in the transparency across the field of view; when large changes are detected, the observation of the Cherenkov telescopes will be interrupted and the vertical profile will be assessed with a lidar [20]. #### Mode D – Measurement of wavelength dependence of aerosol extinction. The aerosol extinction is usually assumed to be inversely proportional to a power of the wavelength with an Ångström exponent $\alpha$ between 0 and 2, which varies due to changes in the physical composition of the aerosols, so that for the optical lengths $\tau_{1}$ and $\tau_{2}$ measured at wavelengths $\lambda_{1}$ and $\lambda_{2}$, we find $\frac{\tau_{1}}{\tau_{2}}=\left(\frac{\lambda_{1}}{\lambda_{2}}\right)^{-\alpha}.$ (2.2) This can be determined in principle by taking altitude scans (Mode A) in several different wavebands (typically defined by photometric filters, such as the Johnson BVRI system used by both Auger and CTA FRAMs). Very precise values are needed to obtain a reasonable precision in $\alpha$ when the aerosol content of the atmosphere is small, as the absolute error of $\alpha$ is proportional to the relative errors of the individual measurements. For example when measuring at two mean wavelengths $\lambda_{1}$, $\lambda_{2}$, we receive $\Delta\alpha=\frac{\ln(1+\delta\tau_{1})+\ln(1+\delta\tau_{2})}{\ln\lambda_{1}-\ln\lambda_{2}}\approx\frac{\delta\tau_{1}+\delta\tau_{2}}{\ln\lambda_{1}-\ln\lambda_{2}}.$ (2.3) Note that the effective wavelength of the measured extinction depends on the spectrum of the individual stars and the Ångström exponent itself; the former effect is accounted for within Eq. (2.1), where the dependence of $k$ on the color of the star is explicitly modeled, the latter can be dealt with in an iterative manner – first processing the data while assuming e.g. $\alpha=1$ in Eq. (2.1), then using Eq. (2.2) to improve the estimate of $\alpha$ and processing the data again. For other modes of operation, the bandpass defined by the Johnson B filter (centered around 445 nm [21]) can be used, in which the molecular contribution to the extinction is dominated by Rayleigh scattering which is easily calculated from the overall column density of the atmosphere as provided by e.g. a global model, such as the GDAS [22]. For other Johnson filters (typically V, centered around 551 nm and R, centered around 658 nm), the contribution of molecular absorption mainly by water and ozone is both important and highly variable due to meteorological effects and thus must be carefully considered when subtracting the molecular component from the measured extinction in order to extract the aerosol optical depth. The Tycho2 catalog only has data in two filters, which are close to Johnson B and V which provide only a limited lever arm for the determination of $\alpha$ and thus other catalogs must be considered. For atmospheric monitoring at Auger and CTA, the effect of changing $\alpha$ is small as their primary calibration methods (lasers) operate at wavelengths close to those of the majority of the observed light – for example the contribution of the unknown wavelength dependence of the aerosol scattering to the uncertainty in shower energy is estimated to be only 0.5% [4]. Nevertheless the knowledge of physical properties of aerosols may be of general interest [23, 24, 25]. For the Auger FRAM, a measurement in this mode consists of a scan in altitude, during which a 30-second image is taken in each of the B, V and R filters for every field of the scan. Figure 5: A simplified schema of the major components of the FRAM setup at the Pierre Auger Observatory. Lines between components indicate either mechanical binding (dotted) or electronic communication (solid). ## 3 Hardware ### 3.1 General requirements Even though the Auger FRAM went through several major hardware changes, it has generally consisted of the same basic elements: a weatherproof enclosure, a German equatorial astronomical mount, two light detection systems (carried jointly by the same mount) – a small (20–30 cm) telescope first equipped with a photometer and later with a CCD camera (a “narrow-field”, NF, system) and a photographic lens with a CCD camera (a “wide-field", WF, system) – and a set of control and auxiliary devices. For a schematic overview of the system, see Fig. 5. The WF system allows both the measurement of a large number of stars for the purposes of precision aerosol measurement and the quick coverage of a large uninterrupted band of the sky for the purposes of the detection of clouds in the Shoot-the-Shower program and it is thus the primary tool used for atmospheric monitoring. The NF system currently provides the opportunity for additional astronomical observations (such as astrometry and/or photometry of asteroids, comets and variable stars, and gamma-ray burst follow-up [26]) within the spare time allowed by the atmospheric monitoring requirements, but it is also being developed for use in the atmospheric monitoring program. Such a possibility is particularly promising after the latest upgrade where the apparent field of view of the NF system has been considerably expanded, increasing the number of stars that can be measured at once. One of the benefits of using stellar photometry for atmospheric monitoring is the possibility to assemble a highly capable device from affordable off-the- shelf products. The unique operating conditions of FRAM impose specific requirements on those devices for several reasons: (i) The robotic nature of the telescope means that the observations are carried out continuously during every night, without breaks unless the weather conditions do not allow observations. Moreover, the exposures are typically short and in many observation modes, the target area on the sky is changed between each pair of exposures, resulting in many movements of the camera shutters and the mount. (ii) The remote locations where astroparticle experiments (such as the Auger Observatory) are located require that the instruments are as autonomous as possible and most issues are solved remotely without local intervention. Regular on-site maintenance also should be kept to a minimum (typically once per year in the case of the Auger FRAM). (iii) The environmental conditions at the sites are demanding, with considerable annual and diurnal changes in temperature, torrential rains, snow, periods of high humidity, but also of blowing dust. Additionally to the technical considerations, further requirements stem from the needs of the analysis of the atmospheric monitoring data and evolve with the progress of this analysis. The long experience with operating FRAM allows us to identify key issues, take appropriate steps to mitigate them, and select suitable products during hardware upgrades. We now describe these considerations for each part of the FRAM setup. Figure 6: The FRAM enclosure with one half open and one closed. Inside is an older version of the FRAM setup with the Meade SCT and Paramount ME mount. ### 3.2 Enclosure The enclosure protects the FRAM setup from the environment when it is not operating. As FRAM is expected to quickly react to triggers and to observe various parts of the sky in short sequence, a solution with a roof that fully opens to give access to the full sky at once is preferable to a more traditional rotating dome with a slit opening. In the FRAM case, the pyramidal roof consists of two independently operated halves (Fig. 6), each moved by a pair of hydraulic cylinders. Mounted in the corners of the dome, the cylinders automatically balance the load (they are connected in parallel), and can open each half to almost 180°. The hydraulic system is designed for operation with considerable wind-forces acting on the dome halves in transition. The mechanical linkages are also designed with sufficient safety margin as they must hold the dome closed against the most extreme wind forces that might be experienced at the site of the Auger Observatory. The hydraulic pressure is provided by a single pump driven by a single-phase AC motor. The whole system is now controlled by a Schneider Zelio smart-relay (PLC) module [27] using custom firmware. Since a custom-made board has proven to be not reliable enough, the industry-standard PLC solution has been implemented. The PLC has outputs that control the position of the hydraulic valves, the power to the pump motor, the source of this power (mains or backup) and the on/off state of the backup power inverter. Additionally, it controls the power to the mount, cameras and other devices to provide a reliable way of restarting them and it also provides a special output for the on/off control of the 10micron mount (see Sec. 3.3) For each half of the roof, mechanical end-switches are connected to the PLC inputs to indicate the position of a fully opened and fully closed roof. Further inputs provide the voltage of the (normally constantly charged) battery used to provide the 12 V DC voltage for the valves and the status of the mains AC power. The PLC communicates with the control PC via Modbus over TCP/IP. To achieve a fail-safe design to open the roof, the PC transmits a heartbeat signal (periodically changes a register value in the PLC). To close the roof, the PC simply stops the heartbeat. In case the communication with the PC is lost, the roof closes automatically, as this is simply equivalent to a command to close. On remote sites such as the Auger Observatory, a mains power cut is not uncommon. As the Auger Observatory does not provide global backup power to all its devices, and the Auger FRAM is fully dependent on this external power (unlike for example the solar-powered FRAMs at the southern CTA site), it has to be able to safely shut down in case of a power cut. Significant effort has been devoted to ensure the ability of the roof to close in such a situation. The pump motor creates a significant power spike on startup, which requires the use of a powerful inverter providing backup power from four lead-acid batteries. The inverter cannot be left powered during normal operation, because it causes electromagnetic interference in the CCD cameras, but it also must be given at least 30 seconds to stabilize after power on before the pump draws power from it – this is ensured through the logic in the PLC. During a power outage, the PLC runs from a UPS which also backs up the control PC, but the internal network is disconnected on purpose (the network switch is not backed up by the UPS) so that commands from the PC cannot interfere with the emergency closing of the roof. The hydraulic roof system is generally reliable, but regular maintenance (on the scale of once per year) is needed to check the status of the hydraulic connections and the oil level. The pump and all the hydraulic hoses had to be replaced after roughly 10 years of operation and a similar frequency of maintenance is needed in the future. The emergency closing system is a critical safety measure with several single points of failure and thus its integrity should be checked after any changes in the FRAM setup. ### 3.3 Mount The mount points the cameras to the target area in the sky and follows the movement of the target due to the rotation of the Earth. For a remote robotic observatory like FRAM, the mount must be able to resume observation after a power cut without human intervention, which requires at least that it is able to find a “home” position autonomously – the lack of this ability has proven the original Losmandy G11 mount [28] of FRAM as unsuitable. The replacement – Paramount ME [29] – had a simple optical sensor in each axis allowing it to return to its home position. All movements are evaluated relative to the home position from the movement of the stepper motors in each axis. After a power cut or mount restart, it must be ensured that the homing procedure is executed first before any attempt to reach a position on the sky, because commanding a move from a “desynchronized” state may cause movements of the mount beyond its physical boundaries. This does not harm the hardware, but requires human intervention, as reaching the physical stops prevents further autonomous movement of the mount in any direction. This is an intrinsic feature of the stepper motors used in Paramount ME, which need freedom of at least one step in both directions to properly initialize. A software daemon is installed at the control PC for ensuring the homing procedure. The Paramount ME required regular maintenance even beyond that prescribed by the vendor, due to the dusty environment of the Auger Observatory and the constant load of the robotic operation. Cleaning and greasing of the worm gear mechanisms had to be carried out at least once per year and the worms on both axes had to be replaced after five years of use. When the worms were in bad condition, the mount had a tendency to slip during movement and slew slightly off target – while this could be corrected using the astrometry of the star images, it occasionally led to the aforementioned problem of desynchronization and reaching the physical stops, requiring on-site intervention. The current equatorial mount – 10micron GM2000 HPS [30] – installed in September 2018, has absolute position sensors and its position is thus precisely known at any moment. Moreover, the manufacturer claims at least 10 years of maintenance-free operation. The GM2000 contains its own control computer, which needs to be booted and shut down properly. Normally this is done using a switch – for remote operation, the switch is driven by one of the outputs of the roof PLC. The switch is stateless and there is no output from the mount to easily allow the PLC to determine whether it is on or off. Thus the control PC checks the state of the mount (over TCP/IP) continually and stores this information in a register of the PLC so that the PLC can autonomously handle the safe shutdown of the mount in case of a power cut. Since commissioning, the GM2000 has operated reliably and without maintenance. The internal communication between parts of the mount seems to be quite sensitive to electromagnetic disturbance and requires careful grounding. ### 3.4 Light sensors We have tested a photoelectric photometer as the main light detector, but we have not found the technology practical for the purposes of atmospheric monitoring: the measurement of a large number of stars is slow and even for individual stars, the reproducibility of the measurement was poor in the context of a remote robotic telescope due to issues with centering the star image on the aperture of the photometer automatically and various other issues that do not occur when an observer is physically present on site. On the other hand, CCD photometry easily allows simultaneous measurements of a large number of stars, even though it brings its own issues, especially regarding the stability of absolute calibration due to a multitude of effects including temperature dependences of the behavior of the camera electronics and focusing of the optics. With the combination of a careful data analysis and a well designed operational mode, the CCD cameras still give much better results than when using the photometer. CCD cameras are thus currently used in both Auger and CTA FRAMs. In the near future, we plan to test the performance in this application of CMOS detectors which, although superficially similar (as a large matrix of micron-sized pixels), are technologically very different from CCD cameras. The possibility to use SiPMs as detectors has been also considered but never explored further. The Finger Lakes Instrumentation (FLI) [31] cameras used in the first iterations of FRAM were not suitable for such a heavy use – their three-bladed iris shutters suffered from the repeated short exposures and became quickly unusable despite maintenance. Since 2012, all cameras used at FRAM are supplied by Moravian Instruments (MI) [32] and use a rotating shutter made of a single butterfly-shaped piece of metal which does not show any degradation in function even after years of use. Since one of the cameras failed without an apparent reason, the vendor now provides a special version of the cameras with protective coating on all electronics and the problem has not appeared again. However, care must be taken to regularly (at least yearly) replace the desiccant in the cold chamber as one of the CTA FRAM cameras was considerably damaged most likely due to corrosion caused by internal humidity. The most problematic part of the cameras is the filter wheel, which typically holds at least BVR Johnson photometric filters. For most applications, it would be sufficient to include a stationary B filter, but for the measurements of the Ångström exponent and for various astronomical applications, the ability to change filters is highly desirable. The MI filter wheel contains a large circular filter holder with a rubber band at the edge which is rotated by a small driving wheel; the positions of the filters are marked by holes in the holder that are detected by an optical gate, with the first position indicated by two holes as a “home” location. This setup has the tendency to become worn out and slip, causing a de-synchronization which then causes the applied filters to be shifted by one position from the requested ones. The vendor has delivered several improvements to the rubber band as well as detailed procedures for the setting of the pressure between the driving wheel and the rubber band, but the problem still occasionally appears. We have developed a software routine that checks for possible problems and restarts the filter wheel if needed. Most of the filter changes occur during the Ångström measurements (Mode D), which is always done as a series of scans with different filters – the procedure is such that in case of a filter wheel malfunction, it usually still obtains a series of consistent scans, just in a different set of filters and a large fraction of affected data can be salvaged automatically because from a full scan, the filter used can be relatively easily inferred. Figure 7: The current FRAM setup with the narrow-field system on the top and the wide-field system on the bottom. ### 3.5 Wide-field system Originally using a Pentacon 200/2.8 lens with a small FLI camera and then a MI G2-1600, the wide-field system was significantly upgraded in 2013 to the current configuration of a Nikkor 300/2.8 lens and a large-format 36$\times$36 mm2 MI G4-16000 camera (Fig. 7). Even though the focal length of the Nikkor is longer, the large format camera covers an area of $7^{\circ}\times 7^{\circ}$, significantly larger than before the upgrade. The optical quality of the Nikkor lens degrades towards the corners of the image, as expected because the CCD chip is even larger than a traditional film frame for which the lens has been originally designed. The correction for this effect in Eq. (2.1) is acceptable in the whole field for the purposes of cloud detection in the Shoot-the-Shower program, but for precision aerosol measurements, only the inner circular part with a diameter of roughly 6.3 degrees, where this correction is less dependent on the momentary state of focus, is used; this still amounts to more than 60% of the area of the CCD chip. The fast f/2.8 lens has a narrow depth of focus. Moreover, it loses its focusing due to temperature changes and slippage due to the movement of the mount. Also due to residual color aberrations, the optimal focus position is slightly different for different filters, even though the filters themselves are confocal. Keeping proper focus is important in particular for operation in Mode C (monitoring of a single field) because changes in focusing have a strong effect on the calibration constant for stellar photometry as it affects the distribution of light around the center of the stellar image. This mode is not commonly employed at the Auger Observatory, but even for cloud detection or self-calibrated scans, it is necessary to have some means of controlling the focus as otherwise the lens eventually de-focuses to a useless state. The lens has only a manual focusing ring; to be able to focus it remotely, we thus use a focusing kit by Rigel Systems [33] which consists of a plastic cogwheel attached to the focus ring of the lens which is driven by a USB-controlled stepper motor. Ideally, the lens should be aligned so that its optical axis intersects the CCD chip perpendicularly in the center of the chip as then the correction on the position of the star on the chip would be strictly radial, reducing the number of free parameters. The alignment can be checked easily by measuring the star shapes across the imaged field. A custom-made holding system for the lens and camera was designed and then gradually improved. Eventually such a task proves near impossible to perform flawlessly as there are many combined effects causing optical misalignment and not all of them are due to the mutual alignment of the camera and the lens. However it turns out that if a rough alignment is achieved, the remaining variations in star shape across the field of view can be satisfactorily treated during data processing, even if they are not radial, without introducing a bias in the measured atmospheric properties. Thus, further improvement is not required. Figure 8: Spectral characteristics of the key components of the wide-field system – transmissivity of the lens and the filters and quantum efficiency of the CCD – as measured in an optical laboratory. Fig. 8 shows the spectral characteristics of the key components of the wide- field system. ### 3.6 Narrow-field system The narrow-field system is a standard small astronomical telescope. The original 20-centimeter Cassegrain telescope had significant problems with mechanical stability and has been replaced by a 30cm Meade [34] Schmidt- Cassegrain (on loan from Instituto de Astrofisica Andalusia) and later by a 30-centimeter Orion UK [35] Dall-Kirkham (Fig. 7). The change to the Dall- Kirkham system was accompanied by a change from a small camera with a 14$\times$10 mm2 chip to another G4-16000 with a 36$\times$36 mm2 chip, which the Dall-Kirkham fully covers, albeit with significant vignetting; the setup achieves a full square degree of field of view. To avoid contributing to the vignetting, the system is equipped with a 3-inch focuser from Astro Systeme Austria [36]. ## 4 Control, software and operation All components of the FRAM system are controlled from a single PC running Linux. Apart from reliable basic components, a key consideration is the exclusive use of SSDs as spinning disks universally succumb to the environmental conditions on site. The requirements for local storage are substantial, as the Pierre Auger Observatory (as is typical for astroparticle experiments) is located in a remote area and transferring large amounts of low-priority data, such as the raw images from the CCD cameras, is not always easy. The PC is equipped with an IPMI interface allowing remote management and power control using a dedicated second ethernet interface. The most problematic aspect of the PC control lies in the USB communication with the various devices. Even though the problems could be solved by applying custom- made kernel patches, recently problems in USB communication appeared again, presumably because of the specific situation of a telescope setup, where many USB devices are mounted together in such a manner that their enclosures are conductively connected, possibly causing multiple ground loops. One possible solution, which is currently under investigation, could be the installation of optical isolators on the USB cables to prevent the ground loops and other sources of interference. The whole FRAM operation is conducted within the RTS2 software framework, an open-source package for robotic observatories [37]. This modular framework was originally developed primarily for follow-ups of gamma-ray bursts (GRBs), but over time it has been extended for many purposes and is deployed on many observatories around the world. The Auger FRAM has been one of the test-beds for its development for years and some features of the package are tailored to the needs of the project [38]. RTS2 can run an astronomical telescope with all its accessories (including the enclosure) autonomously. It is based on the elementary concept of a “target”, which can be an astronomical object described by its coordinates, or a more complicated recipe for observation defined via a Python interface using a specific API. During the night, the targets can be chosen from a database based on priorities assigned to them or the telescope set to follow nightly queues of planned observations in a given order. Thanks to the GRB background, RTS2 also naturally supports reactive observations triggered by external sources of information. For the Shoot-the- Shower program, a dedicated module has been developed to receive and process near real-time data from the Central Data Acquisition System of the Auger Observatory and take a decision to follow up an observation of a cosmic ray shower based on configurable sets of parameters. Despite the autonomous capabilities of RTS2, the best results for FRAM in terms of uptime and data quality are still achieved with human supervision. Since 2012, we additionally rely on a dedicated observer, who checks the status of FRAM (and later also of the other FRAMs at CTA) every day before and sometimes during observations and resolves various issues or, if necessary, calls for on-site intervention. Over time, we have been able to identify the most common issues (such as race conditions among devices leading to data corruption, improper metadata recording, incorrect handling of the meridian flip of the mount etc.) and have either resolved them with improvements in hardware or by software workarounds, thus slowly eliminating the need for nightly human oversight. It is still not clear how to judge the quality of data taken automatically due to the variability of atmospheric conditions (how to determine whether any problems with data are due to unfavourable conditions or system issues) and thus regular checking of the outputs by an experienced observer is still desirable. Apart from a multitude of bug fixes, the most common among the software solutions to the operational issues is the use of watchdogs over the drivers for the necessary devices. Each of the drivers runs as a separate process and in case of a crash, it is restarted, sometimes including an automatic power- cycle of the associated device. An important improvement is the development of a routine for automatic focusing, that can reliably decide if it is actually detecting stars and if the results are meaningful. It is implemented by analyzing a set of stars that are automatically detected in a sequence of images acquired with different focus positions. The images are cross-matched in order to reject spurious detections, and then a common minimum is found in the measured sequence of stellar FWHMs. If not enough star sequences are detected or if no clear minima are seen in their FWHMs, the routine properly identifies the focusing failure and does not update an optimal focus estimation, thus avoiding artificial focus drifts in bad or unstable weather conditions. ## 5 Current FRAM setup and observation modes To summarize, the current FRAM setup at the Pierre Auger Observatory consists of a wide-field imager (MI G4-16000 CCD on a Nikkor 300/2.8 with a stepper motor external focuser) and a narrow-field imager (MI G4-16000 CCD on a 30-centimeter Orion UK Dall-Kirkham with 3” ASA focuser) jointly carried by a 10micron GM2000 HPS German equatorial mount, housed in a custom enclosure with a hydraulically opened roof controlled by a PLC and controlled from a PC running the RTS2 software with local modifications. The operation is mostly automated, but the status is checked almost nightly by a dedicated observer. The basis of the usual observation program is taking regular altitude scans (Mode A) taken as a series of 30 second exposures in the B filter in azimuths selected taking into account the data from the All-sky Camera and the position of the Moon (a too close passage to the Moon is avoided as such data are difficult to process properly); less frequently, scans are conducted in B, V and R filters for the Ångström coefficient measurements (Mode D). The scans are implemented using the Python API interface and a server-client configuration which allows multiple cameras (two in our case, NF and WF) to take data simultaneously and also allows clean resumption of scans in case of an interruption. In between these scans, some selected astronomical targets can be observed. The Shoot-the-Shower module is continually listening for triggers from the Central Data Acquisition System of the Auger Observatory and in case of an extensive air shower passing a set of predefined cuts, any ongoing observation is immediately stopped and the observation along the apparent path of the shower is commenced (Mode B). Regular runs of the auto- focusing routine are scheduled through the night to keep the focus consistent as the nightly decrease of outside temperatures causes contraction of the optical paths of the instruments. ## 6 Conclusions Stellar photometry can be used for atmospheric monitoring in various modes, several of which are employed by the Auger FRAM. The Shoot-the-Shower program is the most valuable for further applications in physics analyses using the data of the Pierre Auger Observatory. Currently, the FRAM data on triggered air showers are being integrated into an analysis of cloud-free anomalous air shower events for studies of aspects of the mass composition of the primary beam and particle physics. The highest quality dataset comes from the latest WF setup installed in 2013, thus comprising more than 7 years so far. The expected number of truly anomalous events for the limited field of view of the fluorescence telescopes is still very low (at the order of 1 ‰ of all detected events, according to the rough estimates of [14]) and thus another FRAM installation is planned at the Cohuieco FD station. At that station, the low- energy extension HEAT [39] is also located, further increasing the number of detected showers. This second Auger FRAM will consist of two WF setups with identical optics, but different light sensors – alongside the traditional CCD camera, a CMOS detector will be tested to assess its viability for atmospheric monitoring. The methods to process the aerosol data from Mode A scans and Ångström coefficients from Mode D observations from both the existing FRAM and the future one are being finalized. All images taken by the Auger FRAM are calibrated and stored in a database. As part of the Open Data project at FZU - Institute of Physics of the Czech Academy of Sciences, all images were made publicly available through a web interface111https://pc048b.fzu.cz/archive/, where they can be searched using various criteria for a range of astronomical applications, such as pre-discovery data on newly discovered Solar System objects or the study of specific variable stars. ## Acknowledgments The successful installation, commissioning, and operation of the Pierre Auger Observatory would not have been possible without the strong commitment and effort from the technical and administrative staff in Malargüe. We are very grateful to the following agencies and organizations for financial support: Argentina – Comisión Nacional de Energía Atómica; Agencia Nacional de Promoción Científica y Tecnológica (ANPCyT); Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET); Gobierno de la Provincia de Mendoza; Municipalidad de Malargüe; NDM Holdings and Valle Las Leñas; in gratitude for their continuing cooperation over land access; Australia – the Australian Research Council; Brazil – Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Financiadora de Estudos e Projetos (FINEP); Fundação de Amparo à Pesquisa do Estado de Rio de Janeiro (FAPERJ); São Paulo Research Foundation (FAPESP) Grants No. 2019/10151-2, No. 2010/07359-6 and No. 1999/05404-3; Ministério da Ciência, Tecnologia, Inovações e Comunicações (MCTIC); Czech Republic – Grant No. MSMT CR LTT18004, LM2015038, LM2018102, CZ.02.1.01/0.0/0.0/16_013/0001402, CZ.02.1.01/0.0/0.0/18_046/0016010 and CZ.02.1.01/0.0/0.0/17_049/0008422; France – Centre de Calcul IN2P3/CNRS; Centre National de la Recherche Scientifique (CNRS); Conseil Régional Ile-de- France; Département Physique Nucléaire et Corpusculaire (PNC-IN2P3/CNRS); Département Sciences de l’Univers (SDU-INSU/CNRS); Institut Lagrange de Paris (ILP) Grant No. LABEX ANR-10-LABX-63 within the Investissements d’Avenir Programme Grant No. ANR-11-IDEX-0004-02; Germany – Bundesministerium für Bildung und Forschung (BMBF); Deutsche Forschungsgemeinschaft (DFG); Finanzministerium Baden-Württemberg; Helmholtz Alliance for Astroparticle Physics (HAP); Helmholtz-Gemeinschaft Deutscher Forschungszentren (HGF); Ministerium für Innovation, Wissenschaft und Forschung des Landes Nordrhein- Westfalen; Ministerium für Wissenschaft, Forschung und Kunst des Landes Baden- Württemberg; Italy – Istituto Nazionale di Fisica Nucleare (INFN); Istituto Nazionale di Astrofisica (INAF); Ministero dell’Istruzione, dell’Universitá e della Ricerca (MIUR); CETEMPS Center of Excellence; Ministero degli Affari Esteri (MAE); México – Consejo Nacional de Ciencia y Tecnología (CONACYT) No. 167733; Universidad Nacional Autónoma de México (UNAM); PAPIIT DGAPA-UNAM; The Netherlands – Ministry of Education, Culture and Science; Netherlands Organisation for Scientific Research (NWO); Dutch national e-infrastructure with the support of SURF Cooperative; Poland -Ministry of Science and Higher Education, grant No. DIR/WK/2018/11; National Science Centre, Grants No. 2013/08/M/ST9/00322, No. 2016/23/B/ST9/01635 and No. HARMONIA 5–2013/10/M/ST9/00062, UMO-2016/22/M/ST9/00198; Portugal – Portuguese national funds and FEDER funds within Programa Operacional Factores de Competitividade through Fundação para a Ciência e a Tecnologia (COMPETE); Romania – Romanian Ministry of Education and Research, the Program Nucleu within MCI (PN19150201/16N/2019 and PN19060102) and project PN- III-P1-1.2-PCCDI-2017-0839/19PCCDI/2018 within PNCDI III; Slovenia – Slovenian Research Agency, grants P1-0031, P1-0385, I0-0033, N1-0111; Spain – Ministerio de Economía, Industria y Competitividad (FPA2017-85114-P and PID2019-104676GB-C32, Xunta de Galicia (ED431C 2017/07), Junta de Andalucía (SOMM17/6104/UGR, P18-FR-4314) Feder Funds, RENATA Red Nacional Temática de Astropartículas (FPA2015-68783-REDT) and María de Maeztu Unit of Excellence (MDM-2016-0692); USA – Department of Energy, Contracts No. DE-AC02-07CH11359, No. DE-FR02-04ER41300, No. DE-FG02-99ER41107 and No. DE-SC0011689; National Science Foundation, Grant No. 0450696; The Grainger Foundation; Marie Curie- IRSES/EPLANET; European Particle Physics Latin American Network; and UNESCO. ## References * [1] Pierre Auger collaboration, _The Pierre Auger Cosmic Ray Observatory_ , _Nucl. 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Huege39,14, J. Hulsman8,39, A. Insolia56,45, P.G. Isar74, P. Janecek30, J.A. Johnsen86, J. Jurysek30, A. Kääpä36, K.H. Kampert36, B. Keilhauer39, J. Kemp40, H.O. Klages39, M. Kleifges38, J. Kleinfeller9, M. Köpke37, N. Kunka38, B.L. Lago16, R.G. Lang18, N. Langner40, M.A. Leigui de Oliveira22, V. Lenok39, A. Letessier-Selvon33, I. Lhenry-Yvon32, D. Lo Presti56,45, L. Lopes72, R. López62, L. Lu94, Q. Luce37, A. Lucero8, J.P. Lundquist76, A. Machado Payeras20, G. Mancarella54,46, D. Mandat30, B.C. Manning12, J. Manshanden41, P. Mantscha, S. Marafico32, A.G. Mariazzi4, I.C. Mariş13, G. Marsella59,45, D. Martello54,46, H. Martinez18, O. Martínez Bravo62, M. Mastrodicasa55,44, H.J. Mathes39, J. Matthews88, G. Matthiae60,49, E. Mayotte36, P.O. Mazura, G. Medina-Tanco67, D. Melo8, A. Menshikov38, K.-D. Merenda86, S. Michal31, M.I. Micheletti6, L. Miramonti57,47, S. Mollerach1, F. Montanet34, C. Morello52,50, M. Mostafá91, A.L. Müller8, M.A. Muller20, K. Mulrey14, R. Mussa50, M. Muzio90, W.M. Namasaka36, A. Nasr-Esfahani36, L. Nellen67, M. Niculescu- Oglinzanu73, M. Niechciol42, D. Nitz89, D. Nosek29, V. Novotny29, L. Nožka31, A Nucita54,46, L.A. Núñez28, M. Palatka30, J. Pallotta2, P. Papenbreer36, G. Parente79, A. Parra62, M. Pech30, F. Pedreira79, J. Pȩkala69, R. Pelayo64, J. Peña-Rodriguez28, E.E. Pereira Martins37,8, J. Perez Armand19, C. Pérez Bertolli8,39, M. Perlin8,39, L. Perrone54,46, S. Petrera43,44, T. Pierog39, M. Pimenta72, V. Pirronello56,45, M. Platino8, B. Pont80, M. Pothast82,80, P. Privitera92, M. Prouza30, A. Puyleart89, S. Querchfeld36, J. Rautenberg36, D. Ravignani8, M. Reininghaus39,8, J. Ridky30, F. Riehn72, M. Risse42, V. Rizi55,44, W. Rodrigues de Carvalho19, J. Rodriguez Rojo10, M.J. Roncoroni8, M. Roth39, E. Roulet1, A.C. Rovero5, P. Ruehl42, S.J. Saffi12, A. Saftoiu73, F. Salamida55,44, H. Salazar62, G. Salina49, J.D. Sanabria Gomez28, F. Sánchez8, E.M. Santos19, E. Santos30, F. Sarazin86, R. Sarmento72, C. Sarmiento-Cano8, R. Sato10, P. Savina54,46,32, C.M. Schäfer39, V. Scherini46, H. Schieler39, M. Schimassek37,8, M. Schimp36, F. Schlüter39,8, D. Schmidt37, O. Scholten81,14, P. Schovánek30, F.G. Schröder93,39, S. Schröder36, J. Schulte40, S.J. Sciutto4, M. Scornavacche8,39, A. Segreto51,45, S. Sehgal36, R.C. Shellard15, G. Sigl41, G. Silli8,39, O. Sima73,f, R. Šmída92, P. Sommers91, J.F. Soriano87, J. Souchard34, R. Squartini9, M. Stadelmaier39,8, D. Stanca73, S. Stanič76, J. Stasielak69, P. Stassi34, A. Streich37,8, M. Suárez-Durán28, T. Sudholz12, T. Suomijärvi35, A.D. Supanitsky8, J. Šupík31, Z. Szadkowski71, A. Taboada37, A. Tapia27, C. Taricco61,50, C. Timmermans82,80, O. Tkachenko39, P. Tobiska30, C.J. Todero Peixoto17, B. Tomé72, A. Travaini9, P. Travnicek30, C. Trimarelli55,44, M. Trini76, M. Tueros4, R. Ulrich39, M. Unger39, L. Vaclavek31, M. Vacula31, J.F. Valdés Galicia67, L. Valore58,48, E. Varela62, V. Varma K.C.8,39, A. Vásquez- Ramírez28, D. Veberič39, C. Ventura25, I.D. Vergara Quispe4, V. Verzi49, J. Vicha30, J. Vink84, S. Vorobiov76, H. Wahlberg4, C. Watanabe24, A.A. Watsonc, M. Weber38, A. Weindl39, L. Wiencke86, H. Wilczyński69, T. Winchen14, M. Wirtz40, D. Wittkowski36, B. Wundheiler8, A. Yushkov30, O. Zapparrata13, E. Zas79, D. Zavrtanik76,77, M. Zavrtanik77,76, L. Zehrer76, A. Zepeda63 and R. Cunniffe30, J. Eliášek30, I. Ebrová,g M. Jelínek,h S. Karpov30, P. Kubánek,i M. Mašek30 1 Centro Atómico Bariloche and Instituto Balseiro (CNEA-UNCuyo-CONICET), San Carlos de Bariloche, Argentina 2 Centro de Investigaciones en Láseres y Aplicaciones, CITEDEF and CONICET, Villa Martelli, Argentina 3 Departamento de Física and Departamento de Ciencias de la Atmósfera y los Océanos, FCEyN, Universidad de Buenos Aires and CONICET, Buenos Aires, Argentina 4 IFLP, Universidad Nacional de La Plata and CONICET, La Plata, Argentina 5 Instituto de Astronomía y Física del Espacio (IAFE, CONICET-UBA), Buenos Aires, Argentina 6 Instituto de Física de Rosario (IFIR) – CONICET/U.N.R. and Facultad de Ciencias Bioquímicas y Farmacéuticas U.N.R., Rosario, Argentina 7 Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), and Universidad Tecnológica Nacional – Facultad Regional Mendoza (CONICET/CNEA), Mendoza, Argentina 8 Instituto de Tecnologías en Detección y Astropartículas (CNEA, CONICET, UNSAM), Buenos Aires, Argentina 9 Observatorio Pierre Auger, Malargüe, Argentina 10 Observatorio Pierre Auger and Comisión Nacional de Energía Atómica, Malargüe, Argentina 11 Universidad Tecnológica Nacional – Facultad Regional Buenos Aires, Buenos Aires, Argentina 12 University of Adelaide, Adelaide, S.A., Australia 13 Université Libre de Bruxelles (ULB), Brussels, Belgium 14 Vrije Universiteit Brussels, Brussels, Belgium 15 Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, RJ, Brazil 16 Centro Federal de Educação Tecnológica Celso Suckow da Fonseca, Nova Friburgo, Brazil 17 Universidade de São Paulo, Escola de Engenharia de Lorena, Lorena, SP, Brazil 18 Universidade de São Paulo, Instituto de Física de São Carlos, São Carlos, SP, Brazil 19 Universidade de São Paulo, Instituto de Física, São Paulo, SP, Brazil 20 Universidade Estadual de Campinas, IFGW, Campinas, SP, Brazil 21 Universidade Estadual de Feira de Santana, Feira de Santana, Brazil 22 Universidade Federal do ABC, Santo André, SP, Brazil 23 Universidade Federal do Paraná, Setor Palotina, Palotina, Brazil 24 Universidade Federal do Rio de Janeiro, Instituto de Física, Rio de Janeiro, RJ, Brazil 25 Universidade Federal do Rio de Janeiro (UFRJ), Observatório do Valongo, Rio de Janeiro, RJ, Brazil 26 Universidade Federal Fluminense, EEIMVR, Volta Redonda, RJ, Brazil 27 Universidad de Medellín, Medellín, Colombia 28 Universidad Industrial de Santander, Bucaramanga, Colombia 29 Charles University, Faculty of Mathematics and Physics, Institute of Particle and Nuclear Physics, Prague, Czech Republic 30 Institute of Physics of the Czech Academy of Sciences, Prague, Czech Republic 31 Palacky University, RCPTM, Olomouc, Czech Republic 32 CNRS/IN2P3, IJCLab, Université Paris-Saclay, Orsay, France 33 Laboratoire de Physique Nucléaire et de Hautes Energies (LPNHE), Sorbonne Université, Université de Paris, CNRS-IN2P3, Paris, France 34 Univ. Grenoble Alpes, CNRS, Grenoble Institute of Engineering Univ. Grenoble Alpes, LPSC-IN2P3, 38000 Grenoble, France 35 Université Paris-Saclay, CNRS/IN2P3, IJCLab, Orsay, France 36 Bergische Universität Wuppertal, Department of Physics, Wuppertal, Germany 37 Karlsruhe Institute of Technology (KIT), Institute for Experimental Particle Physics, Karlsruhe, Germany 38 Karlsruhe Institute of Technology (KIT), Institut für Prozessdatenverarbeitung und Elektronik, Karlsruhe, Germany 39 Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics, Karlsruhe, Germany 40 RWTH Aachen University, III. Physikalisches Institut A, Aachen, Germany 41 Universität Hamburg, II. Institut für Theoretische Physik, Hamburg, Germany 42 Universität Siegen, Department Physik – Experimentelle Teilchenphysik, Siegen, Germany 43 Gran Sasso Science Institute, L’Aquila, Italy 44 INFN Laboratori Nazionali del Gran Sasso, Assergi (L’Aquila), Italy 45 INFN, Sezione di Catania, Catania, Italy 46 INFN, Sezione di Lecce, Lecce, Italy 47 INFN, Sezione di Milano, Milano, Italy 48 INFN, Sezione di Napoli, Napoli, Italy 49 INFN, Sezione di Roma “Tor Vergata”, Roma, Italy 50 INFN, Sezione di Torino, Torino, Italy 51 Istituto di Astrofisica Spaziale e Fisica Cosmica di Palermo (INAF), Palermo, Italy 52 Osservatorio Astrofisico di Torino (INAF), Torino, Italy 53 Politecnico di Milano, Dipartimento di Scienze e Tecnologie Aerospaziali , Milano, Italy 54 Università del Salento, Dipartimento di Matematica e Fisica “E. De Giorgi”, Lecce, Italy 55 Università dell’Aquila, Dipartimento di Scienze Fisiche e Chimiche, L’Aquila, Italy 56 Università di Catania, Dipartimento di Fisica e Astronomia, Catania, Italy 57 Università di Milano, Dipartimento di Fisica, Milano, Italy 58 Università di Napoli “Federico II”, Dipartimento di Fisica “Ettore Pancini”, Napoli, Italy 59 Università di Palermo, Dipartimento di Fisica e Chimica ”E. Segrè”, Palermo, Italy 60 Università di Roma “Tor Vergata”, Dipartimento di Fisica, Roma, Italy 61 Università Torino, Dipartimento di Fisica, Torino, Italy 62 Benemérita Universidad Autónoma de Puebla, Puebla, México 63 Centro de Investigación y de Estudios Avanzados del IPN (CINVESTAV), México, D.F., México 64 Unidad Profesional Interdisciplinaria en Ingeniería y Tecnologías Avanzadas del Instituto Politécnico Nacional (UPIITA-IPN), México, D.F., México 65 Universidad Autónoma de Chiapas, Tuxtla Gutiérrez, Chiapas, México 66 Universidad Michoacana de San Nicolás de Hidalgo, Morelia, Michoacán, México 67 Universidad Nacional Autónoma de México, México, D.F., México 68 Universidad Nacional de San Agustin de Arequipa, Facultad de Ciencias Naturales y Formales, Arequipa, Peru 69 Institute of Nuclear Physics PAN, Krakow, Poland 70 University of Łódź, Faculty of Astrophysics, Łódź, Poland 71 University of Łódź, Faculty of High-Energy Astrophysics,Łódź, Poland 72 Laboratório de Instrumentação e Física Experimental de Partículas – LIP and Instituto Superior Técnico – IST, Universidade de Lisboa – UL, Lisboa, Portugal 73 “Horia Hulubei” National Institute for Physics and Nuclear Engineering, Bucharest-Magurele, Romania 74 Institute of Space Science, Bucharest-Magurele, Romania 75 University Politehnica of Bucharest, Bucharest, Romania 76 Center for Astrophysics and Cosmology (CAC), University of Nova Gorica, Nova Gorica, Slovenia 77 Experimental Particle Physics Department, J. Stefan Institute, Ljubljana, Slovenia 78 Universidad de Granada and C.A.F.P.E., Granada, Spain 79 Instituto Galego de Física de Altas Enerxías (IGFAE), Universidade de Santiago de Compostela, Santiago de Compostela, Spain 80 IMAPP, Radboud University Nijmegen, Nijmegen, The Netherlands 81 KVI – Center for Advanced Radiation Technology, University of Groningen, Groningen, The Netherlands 82 Nationaal Instituut voor Kernfysica en Hoge Energie Fysica (NIKHEF), Science Park, Amsterdam, The Netherlands 83 Stichting Astronomisch Onderzoek in Nederland (ASTRON), Dwingeloo, The Netherlands 84 Universiteit van Amsterdam, Faculty of Science, Amsterdam, The Netherlands 85 Case Western Reserve University, Cleveland, OH, USA 86 Colorado School of Mines, Golden, CO, USA 87 Department of Physics and Astronomy, Lehman College, City University of New York, Bronx, NY, USA 88 Louisiana State University, Baton Rouge, LA, USA 89 Michigan Technological University, Houghton, MI, USA 90 New York University, New York, NY, USA 91 Pennsylvania State University, University Park, PA, USA 92 University of Chicago, Enrico Fermi Institute, Chicago, IL, USA 93 University of Delaware, Department of Physics and Astronomy, Bartol Research Institute, Newark, DE, USA 94 University of Wisconsin-Madison, Department of Physics and WIPAC, Madison, WI, USA —– a Fermi National Accelerator Laboratory, Fermilab, Batavia, IL, USA b Max-Planck-Institut für Radioastronomie, Bonn, Germany c School of Physics and Astronomy, University of Leeds, Leeds, United Kingdom d Colorado State University, Fort Collins, CO, USA e now at Hakubi Center for Advanced Research and Graduate School of Science, Kyoto University, Kyoto, Japan f also at University of Bucharest, Physics Department, Bucharest, Romania g Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Warsaw, Poland h Astronomical Institute of the Czech Academy of Sciences, Ondřejov, Czech Republic i Vera C. Rubin Observatory, La Serena, Chile
# Sojourn times of Gaussian related random fields Krzysztof Dȩbicki Krzysztof Dȩbicki, Mathematical Institute, University of Wrocław, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland <EMAIL_ADDRESS>, Enkelejd Hashorva Enkelejd Hashorva, Department of Actuarial Science, University of Lausanne, UNIL-Dorigny, 1015 Lausanne, Switzerland<EMAIL_ADDRESS>, Peng Liu Peng Liu, Department of Mathematical Sciences, University of Essex, Colchester, UK <EMAIL_ADDRESS>and Zbigniew Michna Zbigniew Michna, Department of Logistics, Wrocław University of Economics and Business, Poland <EMAIL_ADDRESS> Abstract: This paper is concerned with the asymptotic analysis of sojourn times of random fields with continuous sample paths. Under a very general framework we show that there is an interesting relationship between tail asymptotics of sojourn times and that of supremum. Moreover, we establish the uniform double-sum method to derive the tail asymptotics of sojourn times. In the literature, based on the pioneering research of S. Berman the sojourn times have been utilised to derive the tail asymptotics of supremum of Gaussian processes. In this paper we show that the opposite direction is even more fruitful, namely knowing the asymptotics of supremum o f random processes and fields (in particular Gaussian) it is possible to establish the asymptotics of their sojourn times. We illustrate our findings considering i) two dimensional Gaussian random fields, ii) chi-process generated by stationary Gaussian processes and iii) stationary Gaussian queueing processes. Key Words: sojourn/occupation times; exact asymptotics; generalized Berman- type constants; Gaussian random fields; queueing process; chi-process. AMS Classification: Primary 60G15; secondary 60G70 ## 1\. Introduction & First Result Let $X(t),t\in E$ be a random field with compact parameter set $E\subset\mathbb{R}^{d},d\geq 1$ and almost surely continuous sample paths. For a given level $u\in\mathbb{R}$ define the excursion set of $X$ above the level $u$ by $A_{u}(X)\coloneqq\\{t\in E:X(t)>u\\}.$ The probability that $A_{u}$ is not empty $\mathbb{P}\left\\{A_{u}(X)\not=\emptyset\right\\}=\mathbb{P}\left\\{\exists t\in E:X(t)>u\right\\}=\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\}=:p_{u}$ is widely studied in the literature under the asymptotic regime $u\to\infty$, and the assumption that $X$ has marginals with infinite upper endpoint; see, e.g., [26, 1] for $X$ being Gaussian processes and related random fields. Define the Lebesgue volume of $A_{u}(X)$ by $Vol(A_{u}(X))=\int_{E}\mathbb{I}(X(t)>u)dt.$ For specific cases, commonly $d=1$ and $X$ is stationary, asymptotic results as $u\to\infty$ are also known for the probability that the volume of the excursion set (occupation time or sojourn time) exceeds $v(u)z$, i.e., approximations of $r_{u}(z)\coloneqq\mathbb{P}\left\\{Vol(A_{u}(X))>v(u)z\right\\},\quad u\to\infty$ for some specific positive scale function $v$ and $z\geq 0$ are available, see the seminal contribution [4]. The non-stationary case has been considered in [5, 6]. See also [7] for the comprehensive introduction of extremes of sojourns for Gaussian processes. In this contribution we are mainly interested in the formalisation of the uniform double-sum method for sojourns of random processes and fields focusing on the multidimensional case $d\geq 2$, for which no asymptotic results for $r_{u}(z)$ are available in the literature. The first question of our study is whether we can determine a positive scaling functions $v(u),u>0$ and some survival function $\bar{F}$ such that (1) $\lim_{u\to\infty}\mathbb{P}\left\\{Vol(A_{u}(X))>v(u)z\Bigl{\lvert}Vol(A_{u}(X))>0\right\\}=\lim_{u\to\infty}\mathbb{P}\left\\{Vol(A_{u}(X))>v(u)z\Bigl{\lvert}\sup_{t\in E}X(t)>u\right\\}=\bar{F}(z)$ is valid for all $z\geq 0$. If (1) holds for some $z$ positive such that $\bar{F}(z)>0$ the asymptotics of $r_{u}(z)$ is proportional to that of $p_{u}$, i.e., $r_{u}(z)\sim\bar{F}(z)p_{u},\quad u\to\infty.$ Here $a(t)\sim b(t)$ means asymptotic equivalence of two real-valued functions $a(t)$ and $b(t)$ when the argument $t$ tends to infinity or zero. For a given index set $K$ we write $\sharp K$ for the cardinality of $K$. The following theorem states tractable conditions that imply (1) for $X$ as above and $E=E_{u}$. In order to avoid repetition, all Gaussian processes hereafter are assume to have almost surely continuous sample paths. ###### Theorem 1.1. Let $E_{u},u>0$ be compact set of $\mathbb{R}^{d}$ such that $\lim_{u\to\infty}\mathbb{P}\left\\{\sup_{t\in E_{u}}X(t)>u\right\\}=0$. Suppose that there exist collections of Lebesgue measurable disjoint compact sets $I_{k}(u,n),k\in K_{u,n}$ with $K_{u,n}$ non-empty countable index sets such that $E(u,n)\coloneqq\bigcup_{k\in K_{u,n}}I_{k}(u,n)\subset{\color[rgb]{0,0,0}E_{u}},$ then (1) holds with $E=E_{u}$ if the following three conditions are satisfied: A1) (Reduction to relevant sets) $\lim_{n\rightarrow\infty}\limsup_{u\rightarrow\infty}\frac{\mathbb{P}\left\\{\sup_{t\in{\color[rgb]{0,0,0}E_{u}}\setminus E(u,n)}X(t)>u\right\\}}{\mathbb{P}\left\\{\sup_{t\in{\color[rgb]{0,0,0}E(u,n)}}X(t)>u\right\\}}=0.$ A2) (Uniform single-sum approximation) There exists $v(u)>0$ and $\bar{F}_{n},n\geq 1$ such that (2) $\lim_{u\to\infty}\sup_{k\in K_{u,n}}\biggl{\lvert}\frac{\mathbb{P}\left\\{Vol(\\{t\in I_{k}(u,n):X(t)>u\\})>v(u)x\right\\}}{\mathbb{P}\left\\{\sup_{t\in I_{k}(u,n)}X(t)>u\right\\}}-\bar{F}_{n}(x)\biggr{\rvert}=0,\quad x\geq 0,\ {n\geq 1}$ and for all $x\geq 0$ (3) $\displaystyle\bar{F}(x)\coloneqq\lim_{n\rightarrow\infty}\bar{F}_{n}(x)\in(0,1].$ A3) (Double-sum negligibility) For all large $n$ and large $u$, $\sharp K_{u,n}\geq 2$ and $\lim_{n\rightarrow\infty}\limsup_{u\rightarrow\infty}\frac{\sum_{i\neq j,i,j\in K_{u,n}}\mathbb{P}\left\\{\sup_{t\in I_{i}(u,n)}X(t)>u,\sup_{t\in I_{j}(u,n)}X(t)>u\right\\}}{\sum_{k\in K_{u,n}}\mathbb{P}\left\\{\sup_{t\in I_{k}(u,n)}X(t)>u\right\\}}=0.$ For $X(t),t\in\mathbb{R}$ being a Gaussian process, [10] shows that conditions A1)-A3) are satisfied under very general assumptions on $X$. From [10], we can formulate some general conditions on $X$ that imply (4) $\lim_{u\to\infty}\sup_{k\in K_{u,n}}\biggl{\lvert}\frac{\mathbb{P}\left\\{\sup_{t\in I_{k}(u,n)}X(t)>u\right\\}}{\Xi_{k}(u)}-C_{n}\biggr{\rvert}=0$ for some known deterministic functions $\Xi_{k}(u)$, $k\in K_{u,n}$ and $C_{n}$ positive constants such that $\lim_{n\to\infty}C_{n}=C\in(0,\infty)$. In order to prove (2) if (4) holds, we shall prove that (5) $\lim_{u\to\infty}\sup_{k\in K_{u,n}}\biggl{\lvert}\frac{\mathbb{P}\left\\{Vol(\\{t\in I_{k}(u,n):X(t)>u\\})>v(u)x\right\\}}{\Xi_{k}(u)}-D_{n}(x)\biggr{\rvert}=0,\quad$ where $D_{n},\ n\geq 1$ are deterministic functions such that $\lim_{n\to\infty}D_{n}(x)=D(x)>0$, $x\geq 0$. This then in turn implies that (3) holds with $\bar{F}(x)=\frac{D(x)}{C}.$ Note that in case that $D$ is continuous at $x=0$ we also expect that $C=D(0)$ for all $z\geq 0$. In the literature various results are known for supremum of functions of Gaussian vector processes, for instance for chi-square processes, chaos of Gaussian processes, order statistics of Gaussian processes, (see, e.g., [25, 26, 20, 2]) or reflected Gaussian processes modelling a queueing process with Gaussian input (see, e.g., [24, 21, 17, 27, 15, 22, 19, 23, 12, 13]). In Section 3 we illustrate the applicability of Theorem 1.1 by the analysis of three diverse families of stochastic processes: 1) Gaussian random fields (GRF’s), 2) chi-processes and 3) reflected fractional Brownian motions. For all this families of stochastic processes the available results in the literature show that both A1) and A3) hold under quite general conditions; see Section 2. Hence, in view of Theorem 1.1, in order to get (1) it suffices to determine $\bar{F}$ in A2). Except the above examples, our findings can also be applied to many other GRF’s. For instance, multi-dimensional GRF’s with $d\geq 3$, non-stationary chi-process or chi-square process, Gaussian chaos process, non-stationary Gaussian fluid queues and so on. However, we shall not analyze these random processes or fields in this paper. Brief organisation of the rest of the paper. In Section 2 we introduce some notation and Berman-type constants that play the core role in the description of $\bar{F}$. In Section 3, we provide examples that illustrate the derived in Theorem 1.1 technique for getting (1). Some technical lemmas are given in Section 4; their proofs are deferred to Section 6. The proofs of the main contributions of this paper are presented in Section 5. ## 2\. Berman-type constants We begin with the introduction of the Berman-type constants for given independent fBm’s $B_{\alpha_{i}}(s),s\in\mathbb{R}$ with Hurst index $\alpha_{i}/2\in(0,1]$, $i=1,2$. For given continuous functions $h_{1},h_{2}$ set $W_{\alpha_{1},\alpha_{2},h_{1},h_{2}}(t)\coloneqq\sum_{i=1}^{2}(W_{\alpha_{i}}(t_{i})-h_{i}(t_{i})),\quad t=(t_{1},t_{2})\in\mathbb{R}^{2},\quad W_{\alpha_{i}}(t_{i})=\sqrt{2}B_{\alpha_{i}}(t_{i})-\left\|t_{i}\right\|^{\alpha_{i}}\,.$ For simplicity, let $B_{0}(s)\equiv 0,s\in\mathbb{R}$. For $\alpha_{i}\in[0,2],i=1,2$, $x\geq 0$ and $E\subset\mathbb{R}^{2}$ a compact set, let $\mathcal{B}_{\alpha_{1},\alpha_{2}}^{h_{1},h_{2}}(x,E)=\int_{\mathbb{R}}\mathbb{P}\left\\{\int_{E}\mathbb{I}(W_{\alpha_{1},\alpha_{2},h_{1},h_{2}}(t)>z)dt>x\right\\}e^{z}dz$ and if the limit exists, define $\mathcal{B}_{\alpha_{1},\alpha_{2}}^{b_{1}\|t_{1}\|^{\beta_{1}},b_{2}\|t_{2}\|^{\beta_{2}}}(x)\coloneqq\lim_{S\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{b_{1}\|t_{1}\|^{\beta_{1}},b_{2}\|t_{2}\|^{\beta_{2}}}(x,G(S,\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}))}{S^{\mathbb{I}(\alpha_{1}<\beta_{1})+\mathbb{I}(\alpha_{2}<\beta_{2})}},$ where $G(S,\alpha_{1},\beta_{1},\alpha_{2},\beta_{2})=\left\\{\begin{array}[]{cc}[0,S]^{2},&\alpha_{1}<\beta_{1},\alpha_{2}<\beta_{2},\\\ {[-S,S]\times[0,S]},&\alpha_{1}\geq\beta_{1},\alpha_{2}<\beta_{2},\\\ {[0,S]\times[-S,S]},&\alpha_{1}<\beta_{1},\alpha_{2}\geq\beta_{2},\\\ {[-S,S]^{2}}&\alpha_{1}\geq\beta_{1},\alpha_{2}\geq\beta_{2}.\end{array}\right.$ We omit superscripts $h_{i}$’s if $h_{1}(s)=h_{2}(s)=0,s\in\mathbb{R}$ and then we put in our notation $\beta_{1}=\beta_{2}=\infty$ (this implies that $\alpha_{1}<\beta_{1}$ and $\alpha_{2}<\beta_{2}$). Notice that for $x=0$, $\mathcal{B}_{\alpha_{1},\alpha_{2}}^{h_{1},h_{2}}(x)$ reduces to the classical Pickands or Piterbarg constants, see e.g., [26]. The one-dimensional Berman type constant is given by $\mathcal{B}_{\alpha}(x,[a,b])=\int_{\mathbb{R}}\mathbb{P}\left\\{\int_{[a,b]}\mathbb{I}(W_{\alpha}(s)>z)ds>x\right\\}e^{z}dz$ for $\alpha\in(0,2],a<b,a,b\in\mathbb{R}$, and $\mathcal{B}_{\alpha}(x)=\lim_{S\to\infty}\frac{\mathcal{B}_{\alpha}(x,[0,S])}{S}.$ One can refer to [16] and [14] for the existence and properties of one- dimensional Berman constants. For $x=0$, $\mathcal{H}_{\alpha}\coloneqq\mathcal{B}_{\alpha}(0)$ reduces to the classical Pickands constant; see, e.g., [26]. The next lemma deals with properties of $\displaystyle\widehat{\mathcal{B}}_{\alpha_{1},\dots,\alpha_{m}}\left(x,\prod_{i=1}^{m}[0,n_{i}]\right)\coloneqq\int_{\mathbb{R}}\mathbb{P}\left\\{\int_{[0,n_{1}]}\mathbb{I}\left\\{\sup_{t_{i}\in[0,n_{i}],i=2,\dots,m}\sum_{i=1}^{m}W_{\alpha_{i}}(t_{i})>s\right\\}dt_{1}>x\right\\}e^{s}ds$ for $\alpha_{i}\in(0,2],i=1,\dots,m$ and $m\geq 1$. ###### Lemma 2.1. For any $x\geq 0$, and $n_{1}>0$ (6) $\displaystyle\widehat{\mathcal{B}}_{\alpha_{1},\dots,\alpha_{m}}(x,n_{1})$ $\displaystyle\coloneqq$ $\displaystyle\lim_{n_{i}\rightarrow\infty,i=2,\dots,m}\frac{\widehat{\mathcal{B}}_{\alpha_{1},\dots,\alpha_{m}}\left(x,\prod_{i=1}^{m}[0,n_{i}]\right)}{\prod_{i=2}^{m}n_{i}}$ $\displaystyle=$ $\displaystyle\prod_{i=2}^{m}\mathcal{H}_{\alpha_{i}}\int_{\mathbb{R}}\mathbb{P}\left\\{\int_{[0,n_{1}]}\mathbb{I}\left\\{W_{\alpha_{1}}(t)>s\right\\}dt>x\right\\}e^{s}ds\in(0,\infty)$ and (7) $\displaystyle\widehat{\mathcal{B}}_{\alpha_{1},\dots,\alpha_{m}}(x)\coloneqq\lim_{n\rightarrow\infty}\frac{\widehat{\mathcal{B}}_{\alpha_{1},\dots,\alpha_{m}}(x,n)}{n}=\mathcal{B}_{\alpha_{1}}(x)\prod_{i=2}^{m}\mathcal{H}_{\alpha_{i}}\in(0,\infty).$ ###### Remark 2.2. The limits in (6) are finite and positive and $\widehat{\mathcal{B}}_{\alpha_{1},\dots,\alpha_{m}}(x,n_{1})$ is a continuous function of $x$ over $[0,n_{1})$ which follows from the combination of Lemma 2.1 and Lemma 4.1 in [14]. The claim of Lemma 2.1 still holds if we replace $B_{\alpha_{i}}$ by $X_{i}$ being independent centered Gaussian processes with stationary increments and variance function satisfying some regular conditions as e.g. in [17]. ## 3\. Illustrating examples In this section we shall apply Theorem 1.1 to three classes of processes: i) GRF’s , ii) chi-process generated by a stationary Gaussian process and iii) stationary reflected fractional Brownian motions with drift. ### 3.1. Sojourns of GRF’s Although numerous results for the tail asymptotics of supremum of GRF’s are available for both stationary and non-stationary cases (see e.g., [26, 28]), sojourns have not been treated so far in the literature. It follows from the available results in the literature, that A1) holds under quite general conditions, for instance when the variance function has a unique point of maximum and $X$ satisfies a global Hölder continuity condition, see e.g., [26]. The main tool for proving A1) is the so-called Piterbarg inequality, see [26][Thm 8.1] and the recent contribution [9]. Under some further weak assumptions on the variance/covariance function of $X$, also A3) has been shown to hold for a wide collection of cases of interest, see [26, 8]. Thus, in light of Theorem 1.1, in order to prove (1) for GRF’s the main task is the explicit calculation of $\bar{F}$. #### 3.1.1. GRF’s with constant variance First we consider $X$ being a centred GRF with $Var(X(t))=1,t\in E\subset\mathbb{R}^{2}$ and the correlation function $r(t,s)$, $t,s\in\mathbb{R}^{2}$ satisfying (8) $\displaystyle 1-r(t_{1},t_{2},s_{1},s_{2})\sim a_{1}\|t_{1}-s_{1}\|^{\alpha_{1}}+a_{2}\|t_{2}-s_{2}\|^{\alpha_{2}},\quad(t_{1},t_{2}),(s_{1},s_{2})\in E,\|t_{i}-s_{i}\|\rightarrow 0,i=1,2,$ with $a_{i}>0$ and $\alpha_{i}\in(0,2]$, $i=1,2$. Moreover, (9) $\displaystyle r(t_{1},t_{2},s_{1},s_{2})<1,\quad(t_{1},t_{2}),(s_{1},s_{2})\in E,(t_{1},t_{2})\neq(s_{1},s_{2}).$ For notational simplicity we shall consider $E=[0,T_{1}]\times[0,T_{2}]$, the results for general hypercubes in $\mathbb{R}^{d}$ follows with similar calculations. The case that $T_{i}=T_{i,u},i=1,2$ depend on $u$ needs some extra care. $T_{i,u}$’s should not be too small, i.e., $\lim_{u\to\infty}T_{i,u}u^{2/\alpha_{i}}=\infty,i=1,2.$ On the other side $T_{i,u}$’s cannot be too large too. If the GRF is stationary, then for some $\beta\in(0,1)$ we should require that $\lim_{u\to\infty}T_{1,u}T_{2,u}e^{-\beta u^{2}/2}=0.$ In the more complex situation that we are looking at below the existence of $\beta$ is not clear. We suppress the discussion for long intervals in order to avoid further complications. ###### Proposition 3.1. Let $X(t),t\in E=[0,T_{1}]\times[0,T_{2}]$ be a centred GRF which satisfies (8) and (9) and assume that $v(u)=a_{1}^{-1/\alpha_{1}}a_{2}^{-1/\alpha_{2}}u^{-2/\alpha_{1}-2/\alpha_{2}}$. Then for all $x\geq 0$ $\displaystyle\lim_{u\rightarrow\infty}\mathbb{P}\left\\{\int_{E}\mathbb{I}(X(t)>u)dt>v(u)x\Bigl{\lvert}\sup_{t\in E}X(t)>u\right\\}=\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x)}{\mathcal{B}_{\alpha_{1},\alpha_{2}}(0)}.$ #### 3.1.2. GRF’s with non-constant variance Denote by $\sigma(t)=\sqrt{Var(X(t))}$ and assume that $t^{}={(t^{}_{1},t^{}_{2})}\in E=[-T_{1},T_{1}]\times[-T_{2},T_{2}]$ is the inner point of $E$, which is the unique point such that $\sigma(t^{})=\sup_{t\in E}\sigma(t)=1$ satisfying (10) $\displaystyle 1-\sigma(t)\sim b_{1}\|t_{1}-t_{1}^{}\|^{\beta_{1}}+b_{2}\|t_{2}-t_{2}^{}\|^{\beta_{2}},\quad t=(t_{1},t_{2})\in E,\,{\lVert t-t^{}\rVert\rightarrow 0},$ with $b_{i}>0,\beta_{i}>0$, $i=1,2$. Here $\lVert\cdot\rVert$ denotes the Euclidean norm. Moreover, let (11) $\displaystyle 1-r(t,s)\sim a_{1}\|t_{1}-s_{1}\|^{\alpha_{1}}+a_{2}\|t_{2}-s_{2}\|^{\alpha_{2}}$ as $t,s\in E,\lVert t-t^{}\rVert,\lVert s-t^{}\rVert\rightarrow 0$ with $a_{i}>0$ and $\alpha_{i}\in(0,2]$, $i=1,2$, $s=(s_{1},s_{2})$, where $r(t,s)$ is the correlation function of the random field $X$. In the notation below we interpret $\infty\cdot 0$ as $0$. ###### Proposition 3.2. If $X(t),t\in E$ is a centered GRF which satisfies (10) and (11) and $v(u)=\prod_{i=1}^{2}\left(a_{i}^{-1/\alpha_{i}^{}}u^{-2/\min(\alpha_{i},\beta_{i})}\right)$ with $\alpha_{i}^{}=\alpha_{i}\mathbb{I}(\alpha_{i}\leq\beta_{i})+\infty\mathbb{I}(\alpha_{i}>\beta_{i})$, then for all $x\geq 0$ $\lim_{u\rightarrow\infty}\mathbb{P}\left\\{\int_{E}\mathbb{I}(X(t)>u)dt>v(u)x\Bigl{\lvert}\sup_{t\in E}X(t)>u\right\\}=\frac{\mathcal{B}_{\hat{\alpha}_{1},\hat{\alpha}_{2}}^{\bar{a}_{1}b_{1}\left\|t_{1}\right\|^{\beta_{1}},\bar{a}_{2}b_{2}\left\|t_{2}\right\|^{\beta_{2}}}(x)}{\mathcal{B}_{\hat{\alpha}_{1},\hat{\alpha}_{2}}^{\bar{a}_{1}b_{1}\left\|t_{1}\right\|^{\beta_{1}},\bar{a}_{2}b_{2}\left\|t_{2}\right\|^{\beta_{2}}}(0)},$ where $\bar{a}_{i}=\left\\{\begin{array}[]{cc}0&\alpha_{i}<\beta_{i}\\\ \frac{1}{a_{i}}&\alpha_{i}=\beta_{i}\\\ 1&\alpha_{i}>\beta_{i}\end{array}\right.,\quad\hat{\alpha}_{i}=\left\\{\begin{array}[]{cc}\alpha_{i}&\alpha_{i}\leq\beta_{i}\\\ 0&\alpha_{i}>\beta_{i}\end{array}\right.,i=1,2.$ ### 3.2. Sojourns of chi-processes Let $X(t),t\in[0,T]$ be a centered stationary Gaussian process with unit variance and correlation function satisfying $\displaystyle 1-r(s,t)\sim a\|t-s\|^{\alpha},\quad\|s-t\|\rightarrow 0,$ where $\alpha\in(0,2]$ and for all $s\neq t$, $s,t\in[0,T]$ $\displaystyle r(s,t)<1.$ Define the chi-process of degree $m\geq 1$ by (12) $\displaystyle\chi(t)\coloneqq\sqrt{\sum_{i=1}^{m}X_{i}^{2}(t)},\quad t\in\mathbb{R},$ where $X_{i},1\leq i\leq m$ are iid copies of $X$. The exact asymptotics of $\mathbb{P}\left\\{\sup_{t\in[0,T]}\chi(t)>u\right\\}$ has been investigated in [25, 26, 20]. In the following theorem we consider the sojourn time of $\chi$. ###### Proposition 3.3. Let $\chi$ be defined as in (12). If $v(u)=a^{-1/\alpha}u^{-2/\alpha}$, then for all $x\geq 0$ $\lim_{u\rightarrow\infty}\mathbb{P}\left\\{\int_{[0,T]}\mathbb{I}(\chi(t)>u)dt>v(u)x\Bigl{\lvert}\sup_{t\in[0,T]}\chi(t)>u\right\\}=\frac{{\mathcal{B}}_{\alpha}(x)}{{\mathcal{B}}_{\alpha}(0)}.$ ### 3.3. Sojourns of stationary reflected fractional Brownian motion with drift Consider a stationary reflected fractional Brownian motion with drift $Q(t),t\geq 0$, i.e., $\displaystyle Q(t)\coloneqq\sup_{s\geq t}\left(B_{\alpha}(s)-B_{\alpha}(t)-c(s-t)\right),$ where $B_{\alpha}$ is an fBm with Hurst parameter $\alpha/2\in(0,1)$ and $c\in(0,\infty)$. Motivated by some applications of $Q(t)$ to queueing models, the seminal paper [21] studied the tail asymptotics of $Q(0)$. Later on, [27] considered the tail asymptotics of the supremum of $Q(t)$ over a time horizon. Recently, the findings of Piterbarg have been extended to Gaussian processes with stationary increments [12]. We consider next the case of fBm and note that a more general case of Gaussian processes with stationary increments can be also dealt with using results from [12]. In the following we consider $E_{u}=[0,T_{u}]$, where $T_{u}$ is a non-negative function of $u>0$. ###### Proposition 3.4. Let $v(u)=u^{\frac{2(\alpha-1)}{\alpha}}\left(\frac{\sqrt{2}(\tau^{})^{\alpha}}{1+c\tau^{}}\right)^{2/\alpha}$ with $\tau^{}=\frac{\alpha}{c(2-\alpha)}$ and $\alpha\in(0,2)$. i) If $\lim_{u\to\infty}\frac{T_{u}}{v(u)}=T\in(0,\infty)$, then for $T>x\geq 0$ $\lim_{u\rightarrow\infty}\mathbb{P}\left\\{\int_{[0,T_{u}]}\mathbb{I}(Q(t)>u)dt>v(u)x\Bigl{\lvert}\sup_{t\in[0,T_{u}]}Q(t)>u\right\\}=\frac{\mathcal{B}_{\alpha}(x,[0,T])}{\mathcal{B}_{\alpha}(0,[0,T])}.$ ii) If $\lim_{u\to\infty}\frac{T_{u}}{v(u)}=\infty$ and $T_{u}<e^{\beta u^{2-\alpha}}$ with $\beta\in\left(0,\left(\frac{1+c\tau^{}}{\sqrt{2}(\tau^{})^{\alpha/2}}\right)^{2}\right)$, then for all $x\geq 0$ $\lim_{u\rightarrow\infty}\mathbb{P}\left\\{\int_{[0,T_{u}]}\mathbb{I}(Q(t)>u)dt>v(u)x\Bigl{\lvert}\sup_{t\in[0,T_{u}]}Q(t)>u\right\\}=\frac{{\mathcal{B}}_{\alpha}(x)}{{\mathcal{B}}_{\alpha}(0)}.$ ###### Remark 3.5. 1) Note that $\lim_{u\rightarrow\infty}v(u)=\infty$ for $\alpha>1$, and $\lim_{u\rightarrow\infty}v(u)=0$ for $\alpha<1$. 2) Conclusion in i) of Proposition 3.4 still holds for $x>T$ since both sides in the equality of i) are $0$. However, it becomes tricky for the case $T=x$. We consider two special cases for $T=x$. If $T=x$ and $T_{u}\leq xv(u)$ for $u$ sufficiently large, then both sides in the equality of i) are $0$. If $T=x$ and $T_{u}>xv(u)$ for sufficiently large $u$, we get, as $u\to\infty$ $\mathbb{P}\left\\{\int_{[0,T_{u}]}\mathbb{I}(Q(t)>u)dt>v(u)x\Bigl{\lvert}\sup_{t\in[0,T_{u}]}Q(t)>u\right\\}\sim\frac{\mathbb{P}\left\\{\inf_{t\in[0,T_{u}]}Q(t)>u\right\\}}{\mathbb{P}\left\\{\sup_{t\in[0,T_{u}]}Q(t)>u\right\\}}.$ Combining the above two cases for $T=x$, we conclude that the limit for $T=x$ generally does not exist. ## 4\. Auxiliary lemmas In this section we collect some lemmas that play important, although mostly technical role in the proofs of results given in Sections 1-3. Their proofs are deferred to Section 6. We begin with a lemma which is an extension of Theorem 2.1 from [10]. Suppose that for a compact $d-$dimensional hyperrectangle $K\subset\mathbb{R}^{d}$ we have $I_{k}(u,n)=\\{t_{u,n,k}+(v_{1}(u)t_{{1}},\dots v_{d}(u)t_{d}):\,t\in K\\},$ where $v_{i}(u)>0$, $i=1,\ldots,d$ and $t=(t_{1},\ldots,t_{d})\in\mathbb{R}^{d}$. Then, by transforming time, we have $\displaystyle\mathbb{P}\left(Vol(\\{t\in I_{k}(u,n):X(t)>u\\})>v(u)z\right)$ $\displaystyle=$ $\displaystyle\mathbb{P}\left(\int_{I_{k}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)z\right)$ $\displaystyle=$ $\displaystyle\mathbb{P}\left(\int_{K}\mathbb{I}(X(t_{u,n,k}+(v_{1}(u)t_{{1}},\dots v_{d}(u)t_{d}))>u)dt>z\right),$ where $v(u)=\prod_{i=1}^{d}v_{i}(u)$. Motivated by these calculations, we consider next $\xi_{u,j}(t),t\in E_{1},\ j\in S_{u},\ {u\geq 0}$ a family of centered GRF’s with continuous sample paths and variance function $\sigma_{u,j}^{2}$. Suppose in the following that $S_{u}$ is a countable set for all $u$ large. For simplicity in the following we assume that ${0}\in E_{1}$. For a random variable $Z$, we set $\overline{Z}=\frac{Z}{\sqrt{Var(Z)}}$ if $Var(Z)>0$. We introduce next three assumptions: * C0: $\\{g_{u,j},j\in S_{u}\\}$ is a sequence of deterministic functions of $u$ satisfying $\displaystyle\lim_{u\to\infty}\inf_{j\in S_{u}}g_{u,j}=\infty.$ * C1: $Var(\xi_{u,j}({0}))=1$ for all large $u$ and any $j\in S_{u}$ and there exists some bounded continuous function $h$ on $E_{1}$ such that $\displaystyle\lim_{u\to\infty}\sup_{{s}\in E_{1},j\in S_{u}}\left\|g_{u,j}^{2}\left(1-\sigma_{u,j}({s})\right)-h({s})\right\|=0.$ * C2: There exists a centered GRF $\zeta({s}),{s}\in\mathbb{R}^{d}$ with a.s. continuous sample paths such that (13) $\displaystyle\lim_{u\to\infty}\sup_{s,s^{\prime}\in E_{1},j\in S_{u}}\left\|g_{u,j}^{2}\big{(}Var(\overline{\xi}_{u,j}({s})-\overline{\xi}_{u,j}({s}^{\prime}))\big{)}-2Var(\zeta(s)-\zeta(s^{\prime}))\right\|=0.$ * C3: There exist positive constants $C,\nu,u_{0}$ such that $\displaystyle\sup_{j\in S_{u}}g_{u,j}^{2}Var(\overline{\xi}_{u,j}({s})-\overline{\xi}_{u,j}({s}^{\prime}))\leq C\lVert{s}-{s}^{\prime}\rVert^{\nu}$ holds for all ${s},{s}^{\prime}\in E_{1},u\geq u_{0}$. Denote by $C(E_{i}),i=1,2$ the Banach space of all continuous functions $f:E_{i}\mapsto\mathbb{R}$, with $E_{i}\subset\mathbb{R}^{d_{i}},d_{i}\geq 1,i=1,2$ being compact rectangles equipped with the sup-norm. Let $\Gamma:C(E_{1})\rightarrow C(E_{2})$ be a continuous functional satisfying F1: For any $f\in C(E_{1})$, and $a>0,b\in\mathbb{R}$, $\Gamma(af+b)=a\Gamma(f)+b$; F2: There exists $c>0$ such that $\sup_{t\in E_{2}}\Gamma(f)(t)\leq c\sup_{s\in E_{1}}f(s),\ \ \forall f\in C(E_{1}).$ Hereafter, $Q_{i},i\in\mathbb{N}$ are some positive constants which might be different from line to line and $f(u,n)\sim g(u),u\to\infty,n\to\infty$ means that $\lim_{n\to\infty}\lim_{u\to\infty}\frac{f(u,n)}{g(u)}=1.$ ###### Lemma 4.1. Let $\\{\xi_{u,j}({s}),{s}\in E_{1},j\in S_{u},{u\geq 0}\\}$ be a family of centered GRF’s defined as above satisfying C0-C3 and let $\Gamma$ satisfy F1-F2. Let $\eta$ be a positive $\sigma$-finite measure on $E_{2}$ being equivalent with the Lebesgues measure on $E_{2}$. If for all large $u$ and all $j\in S_{u}$ $\mathbb{P}\left\\{\sup_{t\in E_{2}}\Gamma(\xi_{u,j})(t)>g_{u,j}\right\\}>0,$ then for all $x\in[0,\eta(E_{2}))$ (14) $\displaystyle\lim_{u\to\infty}\sup_{j\in S_{u}}\biggl{\lvert}\frac{\mathbb{P}\left\\{\int_{E_{2}}\mathbb{I}\left(\Gamma(\xi_{u,j})({t})>g_{u,j}\right)\eta(dt)>x\right\\}}{\Psi(g_{u,j})}-\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})\biggr{\rvert}=0,$ where $\Psi$ is the tail of the standard normal distribution and $\displaystyle\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})\coloneqq\int_{\mathbb{R}}\mathbb{P}\left\\{\int_{E_{2}}\mathbb{I}\big{(}\Gamma(\sqrt{2}\zeta- Var(\zeta)-h)(t)+y>0\big{)}\eta(dt)>x\right\\}e^{-y}dy$ and the constant $\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})$ is continuous at $x\in{(}0,\eta(E_{2}))$. ###### Lemma 4.2. Let $x\geq 0$. Then (i) $\mathcal{B}_{\alpha_{1},\alpha_{2}}(x)=\lim_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}\in(0,\infty),$ (ii) $\lim_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{n}\in(0,\infty),$ (iii) $\lim_{n\rightarrow\infty}\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},a_{2}^{-1}b_{2}\|t_{2}\|^{\alpha_{2}}}(x,[-n,n]^{2})\in(0,\infty).$ ## 5\. Proofs ### 5.1. Proof of Theorem 1.1 Let next $A_{u}(X)\coloneqq\\{t\in E_{u}:X(t)>u\\}.$ For all $x\geq 0$ and all $u$ positive, since $v(u)$ is non-negative we have $\displaystyle\pi(u)$ $\displaystyle\coloneqq$ $\displaystyle\mathbb{P}\left\\{Vol(A_{u}(X))>v(u)x\Bigl{\lvert}Vol(A_{u}(X))>0\right\\}$ $\displaystyle=$ $\displaystyle\mathbb{P}\left\\{Vol(A_{u}(X))>v(u)x\Bigl{\lvert}\sup_{t\in E_{u}}X(t)>u\right\\}$ $\displaystyle=$ $\displaystyle\frac{\mathbb{P}\left\\{\int_{E}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}}{\mathbb{P}\left\\{\sup_{t\in{\color[rgb]{0,0,0}E_{u}}}X(t)>u\right\\}}$ and further for all $n\geq 1$ $\displaystyle\pi(u)$ $\displaystyle\geq$ $\displaystyle\frac{\mathbb{P}\left\\{\int_{E(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}}{\mathbb{P}\left\\{\sup_{t\in E(u,n)}X(t)>u\right\\}+\mathbb{P}\left\\{\sup_{t\in{\color[rgb]{0,0,0}E_{u}}\setminus E(u,n)}X(t)>u\right\\}},$ $\displaystyle\pi(u)$ $\displaystyle\leq$ $\displaystyle\frac{\mathbb{P}\left\\{\int_{E(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}}{\mathbb{P}\left\\{\sup_{t\in E(u,n)}X(t)>u\right\\}}+\frac{\mathbb{P}\left\\{\sup_{t\in{\color[rgb]{0,0,0}E_{u}}\setminus E(u,n)}X(t)>u\right\\}}{\mathbb{P}\left\\{\sup_{t\in{\color[rgb]{0,0,0}E(u,n)}}X(t)>u\right\\}}.$ Applying A1, it follows that $\displaystyle\pi(u)\sim\frac{\mathbb{P}\left\\{\int_{E(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}}{\mathbb{P}\left\\{\sup_{t\in E(u,n)}X(t)>u\right\\}}=:\pi(u,n),\quad u\rightarrow\infty,n\rightarrow\infty.$ For the case that $\sharp K_{u,n}=1$ for $u$ and $n$ sufficiently large, the claim can be established straightforwardly by A2. Thus let us suppose that $\sharp K_{u,n}\geq 2$ for $n$ and $u$ sufficiently large. In order to proceed we shall apply the standard scheme utilising Bonferroni inequality. Set therefore $\Sigma_{u,n}\coloneqq\sum_{k\in K_{u,n}}\mathbb{P}\left\\{\sup_{t\in I_{k}(u,n)}X(t)>u\right\\},\quad\Sigma\Sigma_{u,n}\coloneqq\sum_{i\neq j,i,j\in K_{u,n}}\mathbb{P}\left\\{\sup_{t\in I_{i}(u,n)}X(t)>u,\sup_{t\in I_{j}(u,n))}X(t)>u\right\\}.$ By the Bonferroni inequality $\displaystyle\Sigma_{u,n}-\Sigma\Sigma_{u,n}\leq\mathbb{P}\left\\{\sup_{t\in E(u,n)}X(t)>u\right\\}\leq\Sigma_{u,n}.$ The asymptotic behaviour of the probability of interest in the above inequality can be derived if the following two-step procedure is successful (which will work in our settings here). First we determine the exact asymptotics of the upper bound and then in a second step we show that the correction in the lower bound is asymptotically negligible. Now we want to apply the same idea for the sojourn functional, here the analysis is however more involved. Observe first that for any $u>0$ $\displaystyle\mathbb{P}\left\\{\int_{E(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}$ $\displaystyle\quad\leq\mathbb{P}\left\\{\sum_{k\in K_{u,n}}\int_{I_{k}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}$ $\displaystyle\quad\leq\mathbb{P}\left\\{\exists k\in K_{u,n},\int_{I_{k}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}$ $\displaystyle\quad\quad+\mathbb{P}\left\\{\exists i,j\in K_{u,n},i\neq j,\int_{I_{i}(u,n)}\mathbb{I}(X(t)>u)dt>0,\int_{I_{j}(u,n)}\mathbb{I}(X(t)>u)dt>0\right\\}$ $\displaystyle\quad\leq\hat{\pi}(u,n)+\Sigma\Sigma_{u,n},$ where $\hat{\pi}(u,n)=\sum_{k\in K_{u,n}}\mathbb{P}\left\\{\int_{I_{k}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}.$ Using Bonferroni inequality again we have $\displaystyle\mathbb{P}\left\\{\int_{E(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}$ $\displaystyle\geq$ $\displaystyle\mathbb{P}\left\\{\exists k\in K_{u,n},\int_{I_{k}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}$ $\displaystyle\geq$ $\displaystyle\hat{\pi}(u,n)-\Sigma\Sigma_{u,n}.$ The sojourn integral can then be approximated by $\hat{\pi}(u,n)$ if we show the correction in the lower bound is negligible. We have $\displaystyle\limsup_{u\rightarrow\infty}\pi(u,n)$ $\displaystyle\leq$ $\displaystyle\limsup_{u\rightarrow\infty}\frac{\hat{\pi}(u,n)+\Sigma\Sigma_{u,n}}{\Sigma_{u,n}-\Sigma\Sigma_{u,n}}=\limsup_{u\rightarrow\infty}\frac{\hat{\pi}(u,n)}{\Sigma_{u,n}}\times\frac{1+\limsup_{u\rightarrow\infty}\frac{\Sigma\Sigma_{u,n}}{\hat{\pi}(u,n)}}{1-\limsup_{u\rightarrow\infty}\frac{\Sigma\Sigma_{u,n}}{\Sigma_{u,n}}},$ $\displaystyle\liminf_{u\rightarrow\infty}\pi(u,n)$ $\displaystyle\geq$ $\displaystyle\liminf_{u\rightarrow\infty}\frac{\hat{\pi}(u,n)-\Sigma\Sigma_{u,n}}{\Sigma_{u,n}}=\liminf_{u\rightarrow\infty}\frac{\hat{\pi}(u,n)}{\Sigma_{u,n}}-\limsup_{u\rightarrow\infty}\frac{\Sigma\Sigma_{u,n}}{\Sigma_{u,n}}.$ By (2) in A2 for any $n\geq 1$ and $x\geq 0$ $\displaystyle\limsup_{u\rightarrow\infty}\frac{\hat{\pi}(u,n)}{\Sigma_{u,n}}=\liminf_{u\rightarrow\infty}\frac{\hat{\pi}(u,n)}{\Sigma_{u,n}}=\bar{F}_{n}(x)$ implying (15) $\displaystyle\bar{F}_{n}(x)-\limsup_{u\rightarrow\infty}\frac{\Sigma\Sigma_{u,n}}{\Sigma_{u,n}}\leq\liminf_{u\rightarrow\infty}\pi(u,n)\leq\limsup_{u\rightarrow\infty}\pi(u,n)\leq\bar{F}_{n}(x)\times\frac{1+\limsup_{u\rightarrow\infty}\frac{\Sigma\Sigma_{u,n}}{\bar{F}_{n}(x)\Sigma_{u,n}}}{1-\limsup_{u\rightarrow\infty}\frac{\Sigma\Sigma_{u,n}}{\Sigma_{u,n}}}.$ In view of A3, letting $n\rightarrow\infty$ in the above inequalities we have that for $x\geq 0$ $\lim_{n\rightarrow\infty}\lim_{u\rightarrow\infty}\pi(u,n)=\bar{F}(x)\in{(0,1]}.$ This completes the proof. $\Box$ ### 5.2. Proof of Lemma 2.1 By the independence of $W_{\alpha_{i}}$’s for any positive $n_{1},\ldots,n_{m}$ $\displaystyle\widehat{\mathcal{B}}_{\alpha_{1},\dots,\alpha_{m}}\left(x,\prod_{i=1}^{m}[0,n_{i}]\right)$ $\displaystyle=$ $\displaystyle\mathbb{E}\left\\{\int_{\mathbb{R}}\mathbb{I}(\int_{[0,n_{1}]}\mathbb{I}\left\\{\sup_{t_{i}\in[0,n_{i}],i=2,\dots,m}\sum_{i=1}^{m}W_{\alpha_{i}}(t_{i})>s\right\\}dt_{1}>x)e^{s}ds\right\\}$ $\displaystyle=$ $\displaystyle\mathbb{E}\left\\{e^{\sum_{i=2}^{m}\sup_{t_{i}\in[0,n_{i}]}W_{\alpha_{i}}(t_{i})}\int_{\mathbb{R}}\mathbb{I}(\int_{[0,n_{1}]}\mathbb{I}\left\\{W_{\alpha_{1}}(t_{1})>s\right\\}dt_{1}>x)e^{s}ds\right\\}$ $\displaystyle=$ $\displaystyle\prod_{i=2}^{m}\mathbb{E}\left\\{\sup_{t_{i}\in[0,n_{i}]}e^{W_{\alpha_{i}}(t_{i})}\right\\}\int_{\mathbb{R}}\mathbb{P}\left\\{\int_{[0,n_{1}]}\mathbb{I}\left\\{W_{\alpha_{1}}(t_{1})>s\right\\}dt_{1}>x\right\\}e^{s}ds.$ Hence the claim follows by the definition of Pickands and Berman constants. $\Box$ ### 5.3. Proof of Proposition 3.1 The proof will be established by checking that A1-A3 in Theorem 1.1 are satisfied. We begin with the introduction of partition $I_{k_{1},k_{2}}(u,n)=\prod_{i=1}^{2}[a_{i}^{-1/\alpha_{i}}u^{-2/\alpha_{i}}k_{i}n,a_{i}^{-1/\alpha_{i}}u^{-2/\alpha_{i}}(k_{i}+1)n],$ for $0\leq k_{i}\leq[T_{i}a_{i}^{1/\alpha_{i}}u^{2/\alpha_{i}}n^{-1}]-1=:N_{i}(u,n),\ i=1,2.$ Let $K_{u,n}=\\{(k_{1},k_{2}):0\leq k_{1}\leq N_{1}(u,n),0\leq k_{2}\leq N_{2}(u,n)\\}$ and $E(u,n)=\bigcup_{(k_{1},k_{2})\in K_{u,n}}I_{k_{1},k_{2}}(u,n)$. Then $E(u,n)\subset E$. Condition A1. It follows straightforwardly from Lemma 7.1 in [26] that (16) $\displaystyle\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\}\sim\sum_{0\leq k_{i}\leq N_{i}(u,n),i=1,2}\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,{n})}X(t)>u\right\\},\quad u\rightarrow\infty,n\to\infty,$ which implies that condition A1 holds. Condition A2. Let for $t=(t_{1},t_{2})$ $\xi_{u,n,k_{1},k_{2}}(t)=X(a_{1}^{-1/\alpha_{1}}u^{-2/\alpha_{1}}(k_{1}n+t_{1}),a_{2}^{-1/\alpha_{2}}u^{-2/\alpha_{2}}(k_{2}n+t_{2})),\quad v(u)=a_{1}^{-1/\alpha_{1}}a_{2}^{-1/\alpha_{2}}u^{-2/\alpha_{1}-2/\alpha_{2}}.$ We derive the uniform asymptotics, as $u\rightarrow\infty$, of $\mathbb{P}\left\\{Vol(\\{t\in I_{k_{1},k_{2}}(u,n):X(t)>u\\})>v(u)x\right\\}=\mathbb{P}\left\\{\int_{[0,n]^{2}}\mathbb{I}(\xi_{u,n,k_{1},k_{2}}(t)>u)dt>x\right\\},$ with $x\geq 0$. For this, we check conditions C0-C3 of Lemma 4.1 with $\Gamma(f)=f,f\in C([0,n]^{2})$. First note that C0-C1 follow trivially with $h=0$ and $g_{u,j}=u$. Moreover, by (8), we have $\displaystyle\lim_{u\rightarrow\infty}\sup_{0\leq k_{i}\leq N_{i}(u,n),i=1,2}\sup_{s,t\in[0,n]^{2}}\left\|u^{2}Var(\xi_{u,n,k_{1},k_{2}}(t)-\xi_{u,n,k_{1},k_{2}}(s))-2Var\left(\sum_{i=1}^{2}B_{\alpha_{i}}(t_{i})-\sum_{i=1}^{2}B_{\alpha_{i}}(s_{i})\right)\right\|=0,$ with $B_{\alpha_{i}},i=1,2$ being two independent fBms’ with indices $\alpha_{i}/2$, respectively. This implies that C2 is satisfied with $\zeta(t)=\sum_{i=1}^{2}B_{\alpha_{i}}(t_{i}).$ Additionally, in light of (8), we have that $\sup_{0\leq k_{i}\leq N_{i}(u,n)+1,i=1,2}u^{2}Var(\xi_{u,n,k_{1},k_{2}}(t)-\xi_{u,n,k_{1},k_{2}}(s))\leq C\lVert{t}-{s}\rVert^{\min(\alpha_{1},\alpha_{2})},\quad{s},{t}\in[0,n]^{2}.$ This means that C3 holds. Thus, by Lemma 4.1, (17) $\displaystyle\lim_{u\rightarrow\infty}\sup_{0\leq k_{i}\leq N_{i}(u,n),i=1,2}\left\|\frac{\mathbb{P}\left\\{Vol(\\{t\in I_{k_{1},k_{2}}(u,n):X(t)>u\\})>v(u)x\right\\}}{\Psi(u)}-\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})\right\|=0.$ Therefore, by Lemma 6.1 in [26] we obtain $\lim_{u\rightarrow\infty}\sup_{0\leq k_{i}\leq N_{i}(u,n),i=1,2}\left\|\frac{\mathbb{P}\left\\{Vol(\\{t\in I_{k_{1},k_{2}}(u,n):X(t)>u\\})>v(u)x\right\\}}{\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,n)}X(t)>u\right\\}}-\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{\mathcal{B}_{\alpha_{1},\alpha_{2}}(0,[0,n]^{2})}\right\|=0.$ Since, by (i) of Lemma 4.2, for any $x\geq 0$ we have (18) $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}(x)=\lim_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}\in(0,\infty),$ then (19) $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x)}{\mathcal{B}_{\alpha_{1},\alpha_{2}}(0)}=\lim_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{\mathcal{B}_{\alpha_{1},\alpha_{2}}(0,[0,n]^{2})}\in(0,{1]},\quad x\geq 0,$ which confirms that A2 holds with $\bar{F}(x)=\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x)}{\mathcal{B}_{\alpha_{1},\alpha_{2}}(0)}$. Condition A3. By (7.4) in the proof of Lemma 7.1 in [26], for all large $u$ and $n$ $\displaystyle\sum_{0\leq k_{i},k_{i}^{\prime}\leq N_{i}(u,n),i=1,2,(k_{1},k_{2})\neq(k_{1}^{\prime},k_{2}^{\prime})}\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,n)}X(t)>u,\sup_{t\in I_{k_{1}^{\prime},k_{2}^{\prime}}(u,n)}X(t)>u\right\\}\leq\left(\frac{\mathbb{C}_{2}}{\sqrt{n}}+e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\right)\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\},$ where $\mathbb{C},\mathbb{C}_{1}$ and $\mathbb{C}_{2}$ are some positive constants, which gives that A3 is satisfied. This completes the proof. $\Box$ ### 5.4. Proof of Proposition 3.2 Without loss of generality, we assume that $t^{}=(0,0)$. The proof relies on verification that A1-A3 in Theorem 1.1 are satisfied. We begin by introducing some notation. Let (20) $\displaystyle I_{k_{1},k_{2}}(u,n)=\prod_{i=1}^{2}[k_{i}v_{i}(u)n,(k_{i}+1)v_{i}(u)n],\quad v_{i}(u)=a_{i}^{-1/\alpha_{i}^{}}u^{-2/\min(\alpha_{i},\beta_{i})},i=1,2,\quad v(u)=v_{1}(u)v_{2}(u),$ where $\alpha_{i}^{}=\alpha_{i}\mathbb{I}(\alpha_{i}\leq\beta_{i})+\infty\mathbb{I}(\alpha_{i}>\beta_{i})$. Additionally, let $e(t)=\frac{1-{\sigma}(t)}{\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}}-1,\|t\|\neq 0,~{}e_{u}=\sup_{0<\|t_{i}\|<\left(\frac{\ln u}{u}\right)^{2/\beta_{i}}}\|e(t)\|,$ and set $N_{i}^{\prime}(u,n)=\left[\frac{(e_{u}^{-1/4}\wedge\ln u)^{2/\beta_{i}}}{u^{2/\beta_{i}}v_{i}(u)n}\right],\ i=1,2.$ We distinguish different scenarios according to the values of $\alpha_{i},\beta_{i},i=1,2$. Case $\alpha_{i}<\beta_{i},i=1,2$. In this scenario $v_{i}(u)=a_{i}^{-1/\alpha_{i}}u^{-2/\alpha_{i}},i=1,2,\ \ K_{u,n}=\\{(k_{1},k_{2}):0\leq\|k_{i}\|\leq N_{i}^{\prime}(u,n),i=1,2\\}$ and $E(u,n)=\bigcup_{(k_{1},k_{2})\in K_{u,n}}I_{k_{1},k_{2}}(u,n)$. Conditions A1 and A3. Following the same reasoning as in the proof of Proposition 3.1, the validity of conditions A1 and A3 follows straightforwardly from (34), (40) and (41) in [18]. Condition A2. Let $\xi_{u,n,k_{1},k_{2}}(t)=\overline{X}(v_{1}(u)(k_{1}n+t_{1}),v_{2}(u)(k_{2}n+t_{2})),$ (21) $\displaystyle u_{n,k_{1},k_{2}}^{-}=u\inf_{t\in I_{k_{1},k_{2}}(u,n)}\frac{1}{{\sigma(t)}},\quad u_{n,k_{1},k_{2}}^{+}=u\sup_{t\in I_{k_{1},k_{2}}(u,n)}\frac{1}{{\sigma(t)}}.$ Then $\displaystyle\mathbb{P}\left\\{Vol(\\{t\in I_{k_{1},k_{2}}(u,n):X(t)>u\\})\geq v(u)x\right\\}\leq\mathbb{P}\left\\{\int_{[0,n]^{2}}\mathbb{I}(\xi_{u,n,k_{1},k_{2}}(t)>u_{n,k_{1},k_{2}}^{-})dt>x\right\\},$ $\displaystyle\mathbb{P}\left\\{Vol(\\{t\in I_{k_{1},k_{2}}(u,n):X(t)>u\\})\geq v(u)x\right\\}\geq\mathbb{P}\left\\{\int_{[0,n]^{2}}\mathbb{I}(\xi_{u,n,k_{1},k_{2}}(t)>u_{n,k_{1},k_{2}}^{+})dt>x\right\\}.$ In order to derive the uniform asymptotics of the above terms we check conditions C0-C3 of Lemma 4.1 with $\Gamma(f)=f,$ $f\in C([0,n]^{2})$ for $\xi_{u,n,k_{1},k_{2}}(t),\ (k_{1},k_{2})\in K_{u,n}.$ Note that C0-C1 holds with $h=0$ and $g_{u,j}=u_{n,k_{1},k_{2}}^{\pm}$. By (10) and (11), we have $\displaystyle\lim_{u\rightarrow\infty}\sup_{s,t\in[0,n]^{2},(k_{1},k_{2})\in K_{u,n}}\left\|(u_{n,k_{1},k_{2}}^{\pm})^{2}(Var(\xi_{u,n,k_{1},k_{2}}(t)-\xi_{u,n,k_{1},k_{2}}(s)))-2Var\left(\sum_{i=1}^{2}B_{\alpha_{i}}(t_{i})-\sum_{i=1}^{2}B_{\alpha_{i}}(s_{i})\right)\right\|=0,$ where $B_{\alpha_{i}},i=1,2$ are two independent fBm’s with indices $\alpha_{i},i=1,2$ respectively. This confirms that C2 holds with $\zeta(t_{1},t_{2})=B_{\alpha_{1}}(t_{1})+B_{\alpha_{2}}(t_{2})$. By (11), we have $\sup_{(k_{1},k_{2})\in K_{u,n}}(u_{n,k_{1},k_{2}}^{\pm})^{2}(Var(\xi_{u,n,k_{1},k_{2}}(t)-\xi_{u,n,k_{1},k_{2}}(s)))\leq Q\|\|s-t\|\|^{\min(\alpha_{1},\alpha_{2})},\quad s,t\in[0,n]^{2}.$ Thus C3 is satisfied. Therefore, by Lemma 4.1, we have that for $0\leq x<n^{2}$, (22) $\displaystyle\lim_{u\rightarrow\infty}\sup_{(k_{1},k_{2})\in K_{u,n}}\left\|\frac{\mathbb{P}\left\\{\int_{[0,n]^{2}}\mathbb{I}(\xi_{u,n,k_{1},k_{2}}(t)>u_{n,k_{1},k_{2}}^{\pm})dt>x\right\\}}{\Psi(u_{n,k_{1},k_{2}}^{\pm})}-\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})\right\|=0.$ Since (23) $\displaystyle\lim_{u\rightarrow\infty}\sup_{(k_{1},k_{2})\in K_{u,n}}\left\|\frac{\Psi(u_{n,k_{1},k_{2}}^{-})}{\Psi(u_{n,k_{1},k_{2}}^{+})}-1\right\|=0$ (see Section 6 for the validation of (23)), by (22) we obtain for $0\leq x<n^{2}$ $\lim_{u\rightarrow\infty}\sup_{(k_{1},k_{2})\in K_{u,n}}\left\|\frac{\mathbb{P}\left\\{Vol(\\{t\in I_{k_{1},k_{2}}(u,n):X(t)>u\\})\geq v(u)x\right\\}}{\Psi(u_{n,k_{1},k_{2}}^{-})}-\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})\right\|=0.$ Therefore, (2) holds with $\bar{F}_{n}(x)=\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{\mathcal{B}_{\alpha_{1},\alpha_{2}}(0,[0,n]^{2})},\quad x\geq 0.$ Finally, by (19), we have that A2 holds. Thus the claim is established with $\bar{F}(x)=\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x)}{\mathcal{B}_{\alpha_{1},\alpha_{2}}(0)}.$ Case $\alpha_{1}=\beta_{1},\alpha_{2}<\beta_{2}$. In this case $v_{i}(u)=a_{i}^{-1/\alpha_{i}}u^{-2/\alpha_{i}},i=1,2.$ Let (24) $\displaystyle\hat{I}_{k_{2}}(u,n)=I_{-1,k_{2}}(u,n)\cup I_{0,k_{2}}(u,n),\quad E_{1}(u,n)=\bigcup_{k_{2}\in K_{u,n}}\hat{I}_{k_{2}}(u,n),$ where $K_{u,n}\coloneqq\\{k_{2}\in\mathbb{Z}:\|k_{2}\|\leq N_{2}^{\prime}(u,n)\\}$. Conditions A1 and A3. Analogously to the previous case, conditions A1 and A3 hold with $E(u,n)\coloneqq E_{1}(u,n)$ and ${I}_{k}(u,n)\coloneqq\hat{I}_{k_{2}}(u,n)$, by (34), (46), (48) and (49) of [18]. Condition A2. Rewrite (10) as $\frac{1}{\sigma(t)}=\left(1+(1+e_{1}(t_{1}))b_{1}\|t_{1}\|^{\beta_{1}}\right)\left(1+(1+e_{2}(t_{2}))b_{2}\|t_{2}\|^{\beta_{2}}\right),$ for some functions $e_{1}(t_{1})$ and $e_{2}(t_{2})$ which satisfy $\lim_{u\rightarrow\infty}\sup_{t\in E_{1}(u,n)}\|e_{i}(t_{i})\|=0,\quad i=1,2.$ Let $\xi_{u,n,k_{2}}(t)=\frac{\overline{X}(v_{1}(u)t_{1},v_{2}(u)(k_{2}n+t_{2}))}{1+b_{1}\|v_{1}(u)t_{1}\|^{\beta_{1}}(1+e_{1}(v_{1}(u)t_{1}))},\quad v(u)=a_{1}^{-1/\alpha_{1}}a_{2}^{-1/\alpha_{2}}u^{-2/\alpha_{1}-2/\alpha_{2}},$ $u_{k_{2},n}^{-}=u\inf_{t\in\hat{I}_{k_{2}}(u,n)}(1+b_{2}\|t_{2}\|^{\beta_{2}}(1+e_{2}(t_{2}))),\quad u_{k_{2},n}^{+}=u\sup_{t\in\hat{I}_{k_{2}}(u,n)}(1+b_{2}\|t_{2}\|^{\beta_{2}}(1+e_{2}(t_{2}))).$ Then it follows that $\displaystyle\mathbb{P}\left\\{Vol(\\{t\in\hat{I}_{k_{2}}(u,n):X(t)>u\\})>v(u)x\right\\}\leq\mathbb{P}\left\\{\int_{[-n,n]\times[0,n]}\mathbb{I}(\xi_{u,n,k_{2}}(t)>u_{k_{2},n}^{-})dt>x\right\\},$ $\displaystyle\mathbb{P}\left\\{Vol(\\{t\in\hat{I}_{k_{2}}(u,n):X(t)>u\\})>v(u)x\right\\}\geq\mathbb{P}\left\\{\int_{[-n,n]\times[0,n]}\mathbb{I}(\xi_{u,n,k_{2}}(t)>u_{k_{2},n}^{+})dt>x\right\\}.$ Straightforward application of Lemma 4.1 with $\Gamma(f)=f,$ $f\in C([-n,n]\times[0,n])$ and $h(t)=a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}}$ in C1, gives that for $0\leq x<2n^{2}$, $\displaystyle\lim_{u\rightarrow\infty}\sup_{k_{2}\in K_{u,n}}\left\|\frac{\mathbb{P}\left\\{\int_{[-n,n]\times[0,n]}\mathbb{I}(\xi_{u,n,k_{2}}(t)>u_{k_{2},n}^{\pm})dt>x\right\\}}{\Psi(u_{k_{2},n}^{\pm})}-\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])\right\|=0.$ Similarly to (23), we have (25) $\displaystyle\lim_{u\rightarrow\infty}\sup_{k_{2}\in K_{u,n}}\left\|\frac{\Psi(u_{k_{2},n}^{-})}{\Psi(u_{k_{2},n}^{+})}-1\right\|=0.$ Consequently, for $0\leq x<2n^{2}$ $\displaystyle\lim_{u\rightarrow\infty}\sup_{k_{2}\in K_{u,n}}\left\|\frac{\mathbb{P}\left\\{Vol(\\{t\in\hat{I}_{k_{2}}(u,n):X(t)>u\\})>v(u)x\right\\}}{\Psi(u_{k_{2},n}^{-})}-\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])\right\|=0.$ Thus (2) holds with $\bar{F}_{n}(x)=\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(0,[-n,n]\times[0,n])}.$ By (ii) of Lemma 4.2 it follows that (27) $\lim_{n\to\infty}\bar{F}_{n}(x)=\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x)}{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(0)}\in(0,{1]},$ which confirms that A2 holds. Thus, applying Theorem 1.1, we establish the claim with $\bar{F}(x)=\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x)}{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(0)}.$ Case $\alpha_{1}=\beta_{1},\alpha_{2}=\beta_{2}$. In this case we have $v_{i}(u)=a_{i}^{-1/\alpha_{i}}u^{-2/\alpha_{i}},i=1,2.$ Let (28) $\displaystyle E(u,n)\coloneqq\hat{I}(u,n)\coloneqq\bigcup_{i,j=-1,0}I_{i,j}(u,n).$ Conditions A1 and A3. It follows from (34) and (52) in the proof of theorem 3.1 of [18] that A1 holds. Since we take only one interval $I_{1}(u,n)$, condition A3 is not applicable to this case. Condition A2. Let $\displaystyle\xi_{u,n}(t)=X(v_{1}(u)t_{1},v_{2}(u)t_{2})),\quad v(u)=a_{1}^{-1/\alpha_{1}}a_{2}^{-1/\alpha_{2}}u^{-2/\alpha_{1}}u^{-2/\alpha_{2}}.$ Then $\mathbb{P}\left\\{Vol(\\{t\in\hat{I}(u,n):X(t)>u\\})\geq v(u)x\right\\}=\mathbb{P}\left\\{\int_{[-n,n]^{2}}\mathbb{I}(\xi_{u,n}(t)>u)dt>x\right\\}.$ In order to derive the asymptotics of the above term, similarly to the previous cases, we observe that C1 in Lemma 4.1 holds with $h(t)=a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}}+a_{2}^{-1}b_{2}\|t_{2}\|^{\alpha_{2}}$ while C2 and C3 have been checked in the case of $\alpha_{i}<\beta_{i},i=1,2$. Hence we have $\displaystyle\lim_{u\rightarrow\infty}\left\|\frac{\mathbb{P}\left\\{\int_{[-n,n]^{2}}\mathbb{I}(\xi_{u,n}(t)>u)dt>x\right\\}}{\Psi(u)}-\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},a_{2}^{-1}b_{2}\|t_{2}\|^{\alpha_{2}}}(x,[-n,n]^{2})\right\|=0.$ Combining the above with the fact that, by (iii) of Lemma 4.2, $\displaystyle\lim_{n\rightarrow\infty}\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},a_{2}^{-1}b_{2}\|t_{2}\|^{\alpha_{2}}}(x,[-n,n]^{2})\in(0,\infty)$ we conclude that A2 holds with $\bar{F}(x)=\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},a_{2}^{-1}b_{2}\|t_{2}\|^{\alpha_{2}}}(x)}{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},a_{2}^{-1}b_{2}\|t_{2}\|^{\alpha_{2}}}(0)}\in(0,{1]}.$ Hence we establish the claim. For the cases $\alpha_{1}>\beta_{1},\alpha_{2}=\beta_{2}$, and $\alpha_{1}>\beta_{1},\alpha_{2}>\beta_{2}$, we can establish the claim similarly to the case of $\alpha_{1}=\beta_{1},\alpha_{2}=\beta_{2}$. For the case $\alpha_{1}>\beta_{1},\alpha_{2}<\beta_{2}$, the proof is similar to the case of $\alpha_{1}=\beta_{1}$, $\alpha_{2}<\beta_{2}$. This completes the proof. $\Box$ ### 5.5. Proof of Proposition 3.3 In order to apply Theorem 1.1, we introduce some useful notation. Let $\displaystyle I_{k}(u,n)=[kv(u)n,(k+1)v(u)n],\quad N(u,n)=\left[\frac{T}{v(u)n}\right]-1,$ and $E(u,n)=\bigcup_{k\in K_{u,n}}I_{k}(u,n),$ with $K_{u,n}=\\{k\in\mathbb{N}:0\leq k\leq N(u,n)\\}$ and $v(u)=a^{-1/\alpha}u^{-2/\alpha}.$ We denote by $Z(t,\theta)=\sum_{i=1}^{m}X_{i}(t)v_{i}(\theta),\quad A=[0,\pi]^{m-2}\times[0,2\pi),$ where $\theta=(\theta_{1},\ldots,\theta_{m-1})$ and $v_{1}(\theta)=\cos\theta_{1},v_{2}(\theta)=\sin\theta_{1}\cos\theta_{2},v_{3}(\theta)=\sin\theta_{1}\sin\theta_{2}\cos\theta_{3},\dots,v_{m-1}(\theta)=(\prod_{i=1}^{m-2}\sin\theta_{i})\cos\theta_{m-1},v_{m}(\theta)=\prod_{i=1}^{m-1}\sin\theta_{i}.$ In this proof, we will use that $\chi(t)=\sup_{\theta\in A}Z(t,\theta).$ We split the set $A$ into (setting $k=(k_{1},\dots,k_{m-1})$) $A=\bigcup_{k\in\Lambda}A_{k},\quad\Lambda=\\{(k_{1},\dots,k_{m-1}):1\leq k_{i}\leq L,1\leq i\leq m-2,1\leq k_{m-1}\leq 2L\\},$ where $A_{k}=\prod_{i=1}^{m-1}\left[\frac{(k_{i}-1)\pi}{L},\frac{k_{i}\pi}{L}\right],\quad k_{m-1}\leq 2L-1,$ $A_{k_{1},\dots,k_{m-2},2L}=\left(\prod_{i=1}^{m-2}\left[\frac{(k_{i}-1)\pi}{L},\frac{k_{i}\pi}{L}\right]\right)\times\left[2\pi-\frac{\pi}{L},2\pi\right),$ and $L$ is a positive integer. Moreover, let (29) $\displaystyle\pi_{1}(u)$ $\displaystyle\coloneqq$ $\displaystyle\sum_{k\neq k^{\prime},k,k^{\prime}\in\Lambda}\mathbb{P}\left\\{\sup_{t\in[0,v(u)n],\theta\in A_{k}}Z(t,\theta)>u,\sup_{t\in[0,v(u)n],\theta\in A_{k^{\prime}}}Z(t,\theta)>u\right\\},$ $\displaystyle\Sigma\Sigma_{u,n}$ $\displaystyle\coloneqq$ $\displaystyle\sum_{0\leq k_{1}<k_{2}\leq N(u,n)}\mathbb{P}\left\\{\sup_{t\in I_{k_{1}}(u,n)}\chi(t)>u,\sup_{t\in I_{k_{2}}(u,n)}\chi(t)>u\right\\}$ $\displaystyle=$ $\displaystyle\sum_{0\leq k_{1}<k_{2}\leq N(u,n)}\mathbb{P}\left\\{\sup_{(t,\theta)\in I_{k_{1}}(u,n)\times A}Z(t,\theta)>u,\sup_{(t,\theta)\in I_{k_{2}}(u,n)\times A}Z(t,\theta)>u\right\\}$ $\displaystyle\leq$ $\displaystyle\sum_{0\leq k_{1}<k_{2}\leq N(u,n),i,j\in\Lambda}\mathbb{P}\left\\{\sup_{(t,\theta)\in I_{k_{1}}(u,n)\times A_{i}}Z(t,\theta)>u,\sup_{(t,\theta)\in I_{k_{2}}(u,n)\times A_{j}}Z(t,\theta)>u\right\\}.$ Denote by (with $k=(k_{1},\dots,k_{m-1}),l=(l_{1},\dots,l_{m-1})$) $J_{k,l}(u)=\prod_{i=1}^{m-1}\left[\frac{(k_{i}-1)\pi}{L}+l_{i}u^{-1}n_{1},\frac{(k_{i}-1)\pi}{L}+(l_{i}+1)u^{-1}n_{1}\right],\quad\Lambda_{1}(u)=\left\\{l:0\leq l_{i}\leq\left[\frac{\pi u}{Ln_{1}}\right],1\leq i\leq m-1\right\\},$ and let (30) $\displaystyle p_{k}^{}(u)=\sum_{l,l^{\prime}\in\Lambda_{1}(u),l\neq l^{\prime}}\mathbb{P}\left\\{\sup_{t\in[0,v(u)n],\theta\in J_{k,l}(u)}Z(t,\theta)>u,\sup_{t\in[0,v(u)n],\theta\in J_{k,l^{\prime}}(u)}Z(t,\theta)>u\right\\}.$ Conditions A1 and A3. Condition A1 follows from Corollary 7.3 in [26] while A3 can be deduced from equations (7.4), (7.6) and (7.18) in the proofs of Lemma 7.1 and Theorem 7.1 in [26]. Condition A2. Let us put $\displaystyle\pi(n,u)\coloneqq\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}(\chi(t)>u)dt>v(u)x\right\\}=\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}\left(\sup_{\theta\in A}Z(t,\theta)>u\right)dt>v(u)x\right\\}.$ To verify A2, by stationarity we have to find the asymptotics of $\pi{(n,u)}$ as $u\to\infty$, which is given in the following lemma. ###### Lemma 5.1. For $n>x$ $\displaystyle\pi(n,u)\sim\frac{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,n)}{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(0,n)}\mathbb{P}\left\\{\sup_{[0,v(u)n]}\chi(t)>u\right\\},\quad u\rightarrow\infty.$ Proof of Lemma 5.1. Let $D_{k}=\\{t\in[0,v(u)n]:\sup_{\theta\in A_{k}}Z(t,\theta)>u\\}.$ Then we have $\displaystyle\int_{[0,v(u)n]}\mathbb{I}\left(\sup_{\theta\in A}Z(t,\theta)>u\right)dt$ $\displaystyle=$ $\displaystyle\int_{[0,v(u)n]}\mathbb{I}_{\bigcup_{k\in\Lambda}D_{k}}(t)dt$ $\displaystyle\leq$ $\displaystyle\sum_{k\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}}(t)dt$ and $\displaystyle\int_{[0,v(u)n]}\mathbb{I}\left(\sup_{\theta\in A}Z(t,\theta)>u\right)dt$ $\displaystyle\geq$ $\displaystyle\sum_{k\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}}(t)dt-\sum_{k\neq k^{\prime},k,k^{\prime}\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}\bigcap D_{k^{\prime}}}(t)dt.$ Note that $\displaystyle\pi(n,u)$ $\displaystyle\geq$ $\displaystyle\mathbb{P}\left(\sum_{k\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}}(t)dt-\sum_{k\neq k^{\prime},k,k^{\prime}\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}\bigcap D_{k^{\prime}}}(t)dt>v(u)x\right)$ $\displaystyle\geq$ $\displaystyle\mathbb{P}\left(\sum_{k\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}}(t)dt>v(u)(x+\epsilon),\sum_{k\neq k^{\prime},k,k^{\prime}\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}\bigcap D_{k^{\prime}}}(t)dt\leq v(u)\epsilon\right)$ $\displaystyle\geq$ $\displaystyle\mathbb{P}\left(\sum_{k\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}}(t)dt>v(u)(x+\epsilon)\right)-\mathbb{P}\left(\sum_{k\neq k^{\prime},k,k^{\prime}\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}\bigcap D_{k^{\prime}}}(t)dt>v(u)\epsilon\right)$ $\displaystyle\geq$ $\displaystyle\mathbb{P}\left(\sum_{k\in\Lambda}\int_{[0,v(u)n]}\mathbb{I}_{D_{k}}(t)dt>v(u)(x+\epsilon)\right)-\pi_{1}(u)$ $\displaystyle\geq$ $\displaystyle\sum_{k\in\Lambda^{}}p_{k}(x+\epsilon,u)-2\pi_{1}(u),$ where $\epsilon>0$ and $\pi_{1}(u)$ is given in (29) and $\displaystyle p_{k}(x,u)$ $\displaystyle=$ $\displaystyle\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}\left(\sup_{\theta\in A_{k}}Z(t,\theta)>u\right)dt>v(u)x\right\\},$ $\displaystyle\Lambda^{}$ $\displaystyle=$ $\displaystyle\\{k\in\Lambda,1<k_{i}<L,1\leq i\leq m-2,k_{m-1}\neq 1,L,2L\\}.$ Similarly we get $\pi(n,u)\leq\sum_{k\in\Lambda}p_{k}(x,u)+\pi_{1}(u).$ Hence (31) $\displaystyle\sum_{k\in\Lambda^{}}p_{k}(x+\epsilon,u)-2\pi_{1}(u)\leq\pi(n,u)\leq\sum_{k\in\Lambda}p_{k}(x,u)+\pi_{1}(u).$ $\diamond$ Upper bound for $p_{k}(x,u)$. A direct calculations show $\displaystyle Var(Z(t,\theta))$ $\displaystyle=$ $\displaystyle 1,$ $\displaystyle Corr(Z(t,\theta),Z(t^{\prime},\theta^{\prime}))$ $\displaystyle=$ $\displaystyle Corr(X(t),X(t^{\prime}))\left(\cos(\theta_{1}-\theta_{1}^{\prime})-\sin\theta_{1}\sin\theta_{1}^{\prime}(1-\cos(\theta_{2}-\theta_{2}^{\prime}))\right.$ $\displaystyle\quad\left.-\dots-\left(\prod_{i=1}^{m-2}\sin\theta_{i}\sin\theta_{i}^{\prime}\right)\left(1-\cos(\theta_{m-1}-\theta_{m-1}^{\prime})\right)\right).$ Hence (32) $\displaystyle 1-Corr(Z(t,\theta),Z(t^{\prime},\theta^{\prime}))$ $\displaystyle\sim$ $\displaystyle a\|t-t^{\prime}\|^{\alpha}+\frac{1}{2}(\theta_{1}-\theta_{1}^{\prime})^{2}+\frac{\sin^{2}\theta_{1}}{2}(\theta_{2}-\theta_{2}^{\prime})^{2}$ $\displaystyle\quad+\frac{1}{2}\left(\prod_{i=1}^{m-2}\sin^{2}\theta_{i}\right)(\theta_{m-1}-\theta_{m-1}^{\prime})^{2},\quad\|t-t^{\prime}\|\rightarrow 0,\|\|\theta-\theta^{\prime}\|\|\rightarrow 0.$ We have (33) $\displaystyle p_{k}(x,u)\leq\sum_{l\in\Lambda_{1}(u)}\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}\left(\sup_{\theta\in J_{k,l}(u)}Z(t,\theta)>u\right)dt>v(u)x\right\\}+p_{k}^{}(u),$ where $p_{k}^{}(u)$ is given in (30). Let $Z_{u,k,l}(t,\theta)=Z\left(v(u)t,\frac{(k_{1}-1)\pi}{L}+l_{1}u^{-1}n_{1}+u^{-1}c_{1}^{-1}(\theta_{k,l}(u))\theta_{1},\dots,\frac{(k_{m-1}-1)\pi}{L}+l_{m-1}u^{-1}n_{1}+u^{-1}c_{m-1}^{-1}(\theta_{k,l}(u))\theta_{m-1}\right),$ and $G_{l}=\prod_{i=1}^{m-1}[0,c_{i}(\theta_{k,l}(u))n_{1}],$ where $c_{k}(\theta)=2^{-1/2}\prod_{i=1}^{k-1}\|\sin\theta_{i}\|,2\leq k\leq m-1,\quad c_{1}(\theta)=2^{-1/2},\quad\theta_{k,l}(u)=\left(\frac{(k_{1}-1)\pi}{L}+l_{1}u^{-1}n_{1},\dots,\frac{(k_{m-1}-1)\pi}{L}+l_{m-1}u^{-1}n_{1}\right).$ Noting that $G_{l}=\prod_{i=1}^{m-1}[0,c_{i}(\theta_{l}(u))n_{1}]\subset\prod_{i=1}^{m-1}[0,c_{k,i}^{+}n_{1}]=:G_{k}^{+},\quad c_{k,i}^{+}=\sup_{\theta\in A_{k}}c_{i}(\theta),$ we have $\displaystyle\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}\left(\sup_{\theta\in J_{k,l}(u)}Z(t,\theta)>u\right)dt>v(u)x\right\\}$ $\displaystyle=$ $\displaystyle\mathbb{P}\left\\{\int_{[0,n]}\mathbb{I}\left(\sup_{\theta\in G_{l}}Z_{u,k,l}(t,\theta)>u\right)dt>v(u)x\right\\}$ $\displaystyle\leq$ $\displaystyle\mathbb{P}\left\\{\int_{[0,n]}\mathbb{I}\left(\sup_{\theta\in G_{k}^{+}}Z_{u,k,l}(t,\theta)>u\right)dt>v(u)x\right\\}.$ A straightforward application of Lemma 4.1 for $\Gamma:C([0,n]\times G_{k}^{+})\rightarrow C([0,n])$ defined by $\Gamma(f)=\sup_{\theta\in G_{k}^{+}}f(t,\theta),\quad f\in C([0,n]\times G_{k}^{+})$, where $h=0$ in C1 and $\zeta(t,\theta)=B_{\alpha}(t)+\sum_{i=1}^{m-1}N_{i}\theta_{i},$ with $N_{i},i=1,\dots,m-1$ being independent standard normal random variables independent of $B_{\alpha}$, implies that for all $x\geq 0$ we have (34) $\displaystyle\lim_{u\rightarrow\infty}\sup_{l\in\Lambda_{1}(u)}\left\|\frac{\mathbb{P}\left\\{\int_{[0,n]}\mathbb{I}\left(\sup_{\theta\in G_{l}^{+}}Z_{u,k,l}(t,\theta)>u\right)dt>v(u)x\right\\}}{\Psi(u)}-\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,[0,n]\times G_{l}^{+})\right\|=0.$ By (7.18) in the proof of Theorem 7.1 in [26], we have (35) $\displaystyle p_{k}^{}(u)=o\left(u^{m-1}\Psi(u)\right),\quad u\rightarrow\infty,n_{1}\rightarrow\infty.$ Hence, by (33)-(35) and using Lemma 2.1 we have $\displaystyle p_{k}(x,u)$ $\displaystyle\leq$ $\displaystyle\limsup_{n_{1}\rightarrow\infty}\frac{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,[0,n]\times G_{l}^{+})}{(n_{1})^{m-1}}\left(\frac{\pi}{L}\right)^{m-1}u^{m-1}\Psi(u)$ $\displaystyle\leq$ $\displaystyle\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,n)\prod_{i=1}^{m-1}c_{k,i}^{+}\left(\frac{\pi}{L}\right)^{m-1}u^{m-1}\Psi(u),\quad u\rightarrow\infty,n_{1}\rightarrow\infty.$ $\diamond$ Lower bound for $p_{k}(x,u)$. By (31), we have that for $\epsilon>0$ $\displaystyle p_{k}(x,u)$ $\displaystyle\geq$ $\displaystyle\sum_{l\in\Lambda_{2}(u)}\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}\left(\sup_{\theta\in J_{{k,l}}(u)}Z(t,\theta)>u\right)dt>v(u)(x+\epsilon)\right\\}-2p_{k}^{}(u)$ $\displaystyle\geq$ $\displaystyle\sum_{l\in\Lambda_{2}(u)}\mathbb{P}\left\\{\int_{[0,n]}\mathbb{I}\left(\sup_{\theta\in G_{k}^{-}}Z_{u,k,l}(t,\theta)>u\right)dt>v(u)(x+\epsilon)\right\\}-2p_{k}^{}(u),$ where $\Lambda_{2}(u)=\left\\{l:0\leq l_{i}\leq\left[\frac{\pi u}{Ln_{1}}\right]-1,1\leq i\leq m-1\right\\},$ $G_{l}=\prod_{i=1}^{m-1}[0,c_{i}(\theta_{l}(u))n_{1}]\supset\prod_{i=1}^{m-1}[0,c_{k,i}^{-}n_{1}]=:G_{k}^{-},\quad c_{k,i}^{-}=\min_{\theta\in A_{k}}c_{i}(\theta).$ By (34), (35), Lemma 2.1 and Remark 2.2 we have for $n>x$ $\displaystyle p_{k}(x,u)$ $\displaystyle\geq$ $\displaystyle\liminf_{n_{1}\rightarrow\infty}\frac{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x+\epsilon,[0,n]\times G_{l}^{-})}{(n_{1})^{m-1}}\left(\frac{\pi}{L}\right)^{m-1}u^{m-1}\Psi(u)$ $\displaystyle\geq$ $\displaystyle\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x+\epsilon,n)\prod_{i=1}^{m-1}c_{k,i}^{-}\left(\frac{\pi}{L}\right)^{m-1}u^{m-1}\Psi(u)$ $\displaystyle\geq$ $\displaystyle\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,n)\prod_{i=1}^{m-1}c_{k,i}^{-}\left(\frac{\pi}{L}\right)^{m-1}u^{m-1}\Psi(u),\quad u\rightarrow\infty,\epsilon\rightarrow 0.$ $\diamond$ Asymptotics for $\pi(u,n)$. By (7.6) in [26] $\pi_{1}(u)=o\left(u^{m-1}\Psi(u)\right),\quad u\rightarrow\infty,L\rightarrow\infty.$ Therefore, in view of (31), $\displaystyle\limsup_{u\to\infty}\frac{\pi(n,u)}{u^{m-1}\Psi(u)}$ $\displaystyle\leq$ $\displaystyle\limsup_{L\rightarrow\infty}\sum_{k\in\Lambda}\left(\prod_{i=1}^{m-1}c_{k,i}^{+}\right)\left(\frac{\pi}{L}\right)^{m-1}\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,n),$ $\displaystyle\liminf_{u\to\infty}\frac{\pi(n,u)}{u^{m-1}\Psi(u)}$ $\displaystyle\geq$ $\displaystyle\liminf_{L\rightarrow\infty}\sum_{k\in\Lambda^{}}\left(\prod_{i=1}^{m-1}c_{k,i}^{-}\right)\left(\frac{\pi}{L}\right)^{m-1}\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,n).$ Using the fact that $\limsup_{L\rightarrow\infty}\sum_{k\in\Lambda}\left(\prod_{i=1}^{m-1}c_{k,i}^{+}\right)\left(\frac{\pi}{L}\right)^{m-1}=\liminf_{L\rightarrow\infty}\sum_{k\in\Lambda^{}}\left(\prod_{i=1}^{m-1}c_{k,i}^{-}\right)\left(\frac{\pi}{L}\right)^{m-1}=Vol(S_{m-1}),$ it follows that $\displaystyle\pi(n,u)\sim\frac{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,n)}{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(0,n)}\mathbb{P}\left\\{\sup_{[0,v(u)n]}\chi(t)>u\right\\},\quad u\rightarrow\infty.$ This completes the proof of Lemma 5.1. $\Box$ Condition A2 continued. Lemma 2.1 yields that for $x\geq 0$ $\frac{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x)}{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(0)}=\lim_{n\rightarrow\infty}\frac{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x,[0,n])}{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(0,[0,n])}\in(0,{1]}.$ Hence A2 holds with $\bar{F}(x)=\frac{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(x)}{\widehat{\mathcal{B}}_{\alpha,2,\dots,2}(0)},\quad x\geq 0.$ Thus we establish the claim and hence the proof is complete. $\Box$ ### 5.6. Proof of Proposition 3.4 We first apply Theorem 1.1 to derive the asymptotics for case ii) of Proposition 3.4. Let $E(u,n)=\bigcup_{i=0}^{N(u,n)}I_{i}(u,n),\quad I_{i}(u,n)=[iv(u)n,(i+1)v(u)n],~{}N(u,n)=\left[\frac{T_{u}}{nv(u)}\right]-2,$ and $v(u)=u^{\frac{2(\alpha-1)}{\alpha}}\left(\frac{(\tau^{})^{\alpha/2}}{1+c\tau^{}}\right)^{2/\alpha},\quad\tau^{}=\frac{\alpha}{c(2-\alpha)}.$ Let $Z(s,t)=\frac{B_{\alpha}(s)-B_{\alpha}(t)}{1+c(s-t)},~{}I_{i}^{\prime}(u,n)=[iq(u)n,(i+1)q(u)n],~{}q(u)=u^{-1}v(u),$ and $\displaystyle\Sigma\Sigma(u,n)$ $\displaystyle\coloneqq\sum_{i\neq j,0\leq i,j\leq N(u,n)}\mathbb{P}\left\\{\sup_{t\in I_{i}(n,u)}Q(t)>u,\sup_{t\in I_{j}(n,u)}Q(t)>u\right\\}$ $\displaystyle{=}\sum_{i\neq j,0\leq i,j\leq N(u,n)}\mathbb{P}\left\\{\sup_{t\in I_{i}(n,u),s\geq t}(B_{\alpha}(s)-B_{\alpha}(t)-c(s-t))>u,\sup_{t\in I_{j}(n,u),s\geq t}(B_{\alpha}(s)-B_{\alpha}(t)-c(s-t))>u\right\\}$ $\displaystyle=\sum_{i\neq j,0\leq i,j\leq N(u,n)}\mathbb{P}\left\\{\sup_{t\in I_{i}^{\prime}(n,u),s\geq t}Z(s,t)>u^{1-\alpha/2},\sup_{t\in I_{j}^{\prime}(n,u),s\geq t}Z(s,t)>u^{1-\alpha/2}\right\\},$ where in the last equality we use the self-similarity of fBm. Moreover, let $L_{i}(u)=[\tau^{}+iq(u)n_{1},\tau^{}+(i+1)q(u)n_{1}],\quad M(u)=\left[\frac{u^{\alpha/2}\ln u}{v(u)n_{1}}\right],$ $G(u)=\\{s:\|s-\tau^{}\|<u^{\alpha/2-1}\ln u\\},\quad G^{c}(u)=[0,\infty)\setminus G(u),$ and (36) $\displaystyle\pi_{2}(u)=\sum_{-M(u)-1\leq i<j\leq M(u)+1}\mathbb{P}\left\\{\sup_{t\in[0,q(u)n],s\in L_{i}(u)}Z(s,t)>u^{1-\alpha/2},\sup_{t\in[0,q(u)n],s\in L_{j}(u)}Z(s,t)>u^{1-\alpha/2}\right\\}.$ Conditions A1 and A3. Condition A1 follows from Theorems 3.1-3.3 of [12] while A3 is due to Lemma 5.6 of [12] and the upper bounds of $\Sigma_{i}(u),i=1,2,3,4$ in the proof of Theorem 3.1 in [12]. Condition A2. Due to stationarity of the process $Q$, in order to show (2) it suffices to find the exact asymptotics of $\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}(Q(t)>u)dt>v(u)x\right\\}$ as $u\to\infty$. By the self-similarity of $B_{\alpha}$, we have $\displaystyle\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}(Q(t)>u)dt>v(u)x\right\\}$ $\displaystyle=$ $\displaystyle\mathbb{P}\left\\{\int_{[0,v(u)n]}\mathbb{I}\left(\sup_{s\geq t}(B_{\alpha}(s)-B_{\alpha}(t)-c(s-t))>u\right)dt>v(u)x\right\\}$ $\displaystyle=$ $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}.$ ###### Lemma 5.2. For $n>x$ (37) $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}\sim\widehat{\mathcal{B}}_{\alpha,\alpha}(x,n)\sqrt{\frac{2A}{B}}\frac{u}{m(u)v(u)}\Psi(u),\quad u\rightarrow\infty.$ Proof. Upper bound. Using the fact that $\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)\leq\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)+\mathbb{I}\left(\sup_{s\in G^{c}(u)}Z(s,t)>u^{1-\alpha/2}\right)$ we obtain $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}$ $\displaystyle\quad\leq\mathbb{P}\left\\{\int_{[0,q(u)n]}\left(\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)+\mathbb{I}\left(\sup_{s\in G^{c}(u)}Z(s,t)>u^{1-\alpha/2}\right)\right)dt>q(u)x\right\\}$ $\displaystyle\quad\leq\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}$ $\displaystyle\quad\quad+\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\in G^{c}(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}$ $\displaystyle\quad\quad+\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>0,\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\in G^{c}(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>0\right\\}$ $\displaystyle\quad\leq\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}+2\mathbb{P}\left\\{\sup_{t\in[0,q(u)n],s\in G^{c}(u)}Z(s,t)>u^{1-\alpha/2}\right\\}.$ Moreover, since $\displaystyle\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)\leq\sum_{\|i\|\leq M(u)+1}\mathbb{I}\left(\sup_{s\in L_{i}(u)}Z(s,t)>u^{1-\alpha/2}\right)$ we have $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}\leq\pi_{1}(u)+\pi_{2}(u),$ where $\pi_{2}(u)$ is given in (36) and (38) $\displaystyle\pi_{1}(u)=\sum_{\|i\|\leq M(u)+1}\mathbb{P}\left\\{\int_{t\in[0,q(u)n]}\mathbb{I}\left(\sup_{s\in L_{i}(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}.$ By Lemma 5.6 of [12] we obtain $\mathbb{P}\left\\{\sup_{t\in[0,q(u)n],s\in G^{c}(u)}Z(s,t)>u^{1-\alpha/2}\right\\}=o\left(\mathbb{P}\left\\{\sup_{t\in[0,v(u)n]}Q(t)>u\right\\}\right),\quad u\rightarrow\infty,$ and in light of the upper bounds of $\Lambda_{i}(u),i=1,2,3,4$ in the proof of Theorem 3.1 of [12] (39) $\displaystyle\pi_{2}(u)=o\left(\mathbb{P}\left\\{\sup_{t\in[0,v(u)n]}Q(t)>u\right\\}\right),\quad u\rightarrow\infty,n_{1}\rightarrow\infty.$ Next we focus on $\pi_{1}(u)$. We denote $m(u)=\frac{1+c\tau^{}}{(\tau^{})^{\alpha/2}}u^{1-\alpha/2},\quad\tau^{}=\frac{\alpha}{c(2-\alpha)},$ $A=\left(\frac{\alpha}{c(2-\alpha)}\right)^{-\alpha/2}\frac{2}{2-\alpha},\quad B=\left(\frac{\alpha}{c(2-\alpha)}\right)^{-\alpha/2-1}\frac{\alpha}{2}.$ Rewrite $\displaystyle\mathbb{P}\left\\{\int_{t\in[0,q(u)n]}\mathbb{I}\left(\sup_{s\in L_{i}(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}=\mathbb{P}\left\\{\int_{t\in[0,n]}\mathbb{I}\left(\sup_{s\in[0,n_{1}]}Z_{u,i}(s,t)>m(u)\right)dt>x\right\\},$ where $Z_{u,i}(s,t)=\frac{B_{\alpha}(\tau^{}+q(u)(in_{1}+s))-B_{\alpha}(q(u)t)}{1+c(\tau^{}+q(u)(in_{1}+s-t))}\cdot\frac{1+c\tau^{}}{(\tau^{})^{\alpha/2}}.$ Let for $0<\epsilon<1$ $m_{i}^{\pm}(u)=m(u)\left(1+\left(\frac{B}{2A}\pm\epsilon\right)q(u)(in_{1}\pm n)^{2}\right).$ A direct calculation shows (see also Lemmas 5.3-5.4 in [12]) that (40) $\displaystyle m_{i}^{-}(u)\leq m(u)(Var(Z_{u,i}(s,t)))^{-1/2}\leq m_{i}^{+}(u),\quad\|i\|\leq M(u)+1$ and (41) $\displaystyle\lim_{u\rightarrow\infty}\sup_{\|i\|\leq M(u)+1}\sup_{(s,t)\neq(s^{\prime},t^{\prime}),(s,t),(s^{\prime},t^{\prime})\in[0,n_{1}]\times[0,n]}\left\|(m_{i}^{\pm}(u))^{2}\frac{1-Corr(Z_{u,i}(s,t),Z_{u,i}(s^{\prime},t^{\prime}))}{\|t-t^{\prime}\|^{\alpha}+\|s-s^{\prime}\|^{\alpha}}-1\right\|=0.$ Hence $\displaystyle\mathbb{P}\left\\{\int_{t\in[0,n]}\mathbb{I}\left(\sup_{s\in[0,n_{1}]}Z_{u,i}(s,t)>m(u)\right)dt>x\right\\}\leq\mathbb{P}\left\\{\int_{t\in[0,n]}\mathbb{I}\left(\sup_{s\in[0,n_{1}]}\overline{Z}_{u,i}(s,t)>m_{i}^{-}(u)\right)dt>x\right\\}.$ Next, by Lemma 4.1 applied to $\Gamma:C([0,n]\times[0,n_{1}])\rightarrow C([0,n])$ defined by $\Gamma(f)=\sup_{t\in[0,n_{1}]}f(s,t),f\in C([0,n]\times[0,n_{1}])$, with $h=0$ in C0-C1 and C2 satisfied with $\zeta(s,t)=B_{\alpha}(s)+B_{\alpha}^{\prime}(t)$, we have (42) $\displaystyle\lim_{u\rightarrow\infty}\sup_{\|i\|\leq M(u)+1}\left\|\frac{\mathbb{P}\left\\{\int_{t\in[0,n]}\mathbb{I}\left(\sup_{s\in[0,n_{1}]}\overline{Z}_{u,i}(s,t)>m_{i}^{-}(u)\right)dt>x\right\\}}{\Psi(m_{i}^{-}(u))}-\widehat{\mathcal{B}}_{\alpha,\alpha}(x,[0,n]\times[0,n_{1}])\right\|=0,$ and in light of Lemma 2.1, we have (43) $\displaystyle\pi_{{1}}(u)$ $\displaystyle\leq$ $\displaystyle\widehat{\mathcal{B}}_{\alpha,\alpha}(x,[0,n]\times[0,n_{1}])\sum_{\|i\|\leq M(u)+1}\Psi(m_{i}^{-}(u))$ $\displaystyle\leq$ $\displaystyle\widehat{\mathcal{B}}_{\alpha,\alpha}(x,[0,n]\times[0,n_{1}])\Psi(u)\sum_{\|i\|\leq M(u)+1}e^{-m^{2}(u)\left(\frac{{\color[rgb]{0,0,0}B}}{2A}-\epsilon\right)\left(u^{-1}v(u)(in_{1})\right)^{2}}$ $\displaystyle\leq$ $\displaystyle\frac{\widehat{\mathcal{B}}_{\alpha,\alpha}(x,[0,n]\times[0,n_{1}])}{n_{1}}\sqrt{\frac{2A\pi}{B}}\frac{u}{m(u)v(u)}\Psi(u)$ $\displaystyle\sim$ $\displaystyle\widehat{\mathcal{B}}_{\alpha,\alpha}(x,n)\sqrt{\frac{2A\pi}{B}}\frac{u}{m(u)v(u)}\Psi(u),$ as $u\rightarrow\infty,n_{1}\rightarrow\infty,\epsilon\rightarrow 0$. Therefore, we conclude that $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}\leq\widehat{\mathcal{B}}_{\alpha,\alpha}(x,n)\sqrt{\frac{2A\pi}{B}}\frac{u}{m(u)v(u)}\Psi(u),\quad u\rightarrow\infty.$ Lower bound. Observe that for u sufficiently large, $s>t$ holds for all $s\in G(u),t\in[0,q(u)n]$. Therefore, $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}\geq\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}.$ By the fact that $\displaystyle\mathbb{I}\left(\sup_{s\in G(u)}Z(s,t)>u^{1-\alpha/2}\right)$ $\displaystyle\geq$ $\displaystyle\sum_{\|i\|\leq M(u)}\mathbb{I}\left(\sup_{s\in L_{i}(u)}Z(s,t)>u^{1-\alpha/2}\right)$ $\displaystyle-\sum_{-M(u)\leq i<j\leq M(u)}\mathbb{I}\left(\sup_{s\in L_{i}(u)}Z(s,t)>u^{1-\alpha/2},\sup_{s\in L_{j}(u)}Z(s,t)>u^{1-\alpha/2}\right)$ $\displaystyle=:$ $\displaystyle A_{1}(u,t)-A_{2}(u,t),$ it follows that for $\epsilon>0$ (recall $q(u)=u^{-1}v(u)$) $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}$ $\displaystyle\geq\mathbb{P}\left\\{\int_{[0,q(u)n]}\left(A_{1}(u,t)-A_{2}(u,t)\right)dt>q(u)x\right\\}$ $\displaystyle\geq\mathbb{P}\left\\{\int_{[0,q(u)n]}A_{1}(u,t)dt>q(u)(x+\epsilon),\int_{[0,q(u)n]}A_{2}(u,t)dt<q(u)\epsilon\right\\}$ $\displaystyle\geq\mathbb{P}\left\\{\int_{[0,q(u)n]}A_{1}(u,t)dt>q(u)(x+\epsilon)\right\\}-\mathbb{P}\left\\{\int_{[0,q(u)n]}A_{2}(u,t)dt\geq q(u)\epsilon\right\\}$ $\displaystyle\geq\mathbb{P}\left\\{\exists{\color[rgb]{0,0,0}i:}\,\|i\|\leq M(u),\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\in L_{i}(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)(x+\epsilon)\right\\}-\pi_{2}(u)$ (44) $\displaystyle\geq\sum_{\|i\|\leq M(u)}\mathbb{P}\left\\{\int_{t\in[0,q(u)n]}\mathbb{I}\left(\sup_{s\in L_{i}(u)}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)(x+\epsilon)\right\\}-2\pi_{2}(u),$ where $\pi_{2}(u)$ is defined in (36). Similarly as in (43) and in light of (39), we have $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}$ $\displaystyle\geq$ $\displaystyle\widehat{\mathcal{B}}_{\alpha,\alpha}(x+\epsilon,n)\sqrt{\frac{2A}{B}}\frac{u}{m(u)v(u)}\Psi(u)$ $\displaystyle\geq$ $\displaystyle\widehat{\mathcal{B}}_{\alpha,\alpha}(x,n)\sqrt{\frac{2A}{B}}\frac{u}{m(u)v(u)}\Psi(u),\quad u\rightarrow\infty,\epsilon\rightarrow 0.$ Consequently for $n>x$ (45) $\displaystyle\mathbb{P}\left\\{\int_{[0,q(u)n]}\mathbb{I}\left(\sup_{s\geq t}Z(s,t)>u^{1-\alpha/2}\right)dt>q(u)x\right\\}\sim\widehat{\mathcal{B}}_{\alpha,\alpha}(x,n)\sqrt{\frac{2A}{B}}\frac{u}{m(u)v(u)}\Psi(u),\quad u\rightarrow\infty.$ $\Box$ Moreover, by Lemma 2.1 $\displaystyle\frac{\mathcal{B}_{\alpha}(x)}{\mathcal{B}_{\alpha}(0)}=\frac{\widehat{\mathcal{B}}_{\alpha,\alpha}(x)}{\widehat{\mathcal{B}}_{\alpha,\alpha}(0)}=\lim_{n\rightarrow\infty}\frac{\widehat{\mathcal{B}}_{\alpha,\alpha}(x,n)}{\widehat{\mathcal{B}}_{\alpha,\alpha}(0,n)}\in(0,{1]}.$ Thus A2 holds with $\bar{F}(x)=\frac{\mathcal{B}_{\alpha}(x)}{\mathcal{B}_{\alpha}(0)},\quad x\geq 0.$ This completes the proof of case ii). For case i), note that if $x=0$, the claim clearly holds. Next we suppose that $0<x<T$. By (45) for any $0<\epsilon<\min(x/2,(T-x)/2)$, $\displaystyle\mathbb{P}\left\\{\int_{[0,T_{u}]}\mathbb{I}(Q(t)>u)dt>v(u)x\right\\}$ $\displaystyle\leq$ $\displaystyle\mathbb{P}\left\\{\int_{[0,v(u)(T+\epsilon)]}\mathbb{I}(Q(t)>u)dt>v(u)x\right\\}$ $\displaystyle\leq$ $\displaystyle\mathbb{P}\left\\{\int_{[0,v(u)T]}\mathbb{I}(Q(t)>u)dt>v(u)(x-\epsilon)\right\\}$ $\displaystyle\sim$ $\displaystyle\frac{\widehat{\mathcal{B}}_{\alpha,\alpha}(x-\epsilon,T)}{\widehat{\mathcal{B}}_{\alpha,\alpha}(0,T)}\mathbb{P}\left\\{\sup_{t\in[0,v(u)T]}Q(t)>u\right\\},\quad u\rightarrow\infty.$ Analogously, $\displaystyle\mathbb{P}\left\\{\int_{[0,T_{u}]}\mathbb{I}(Q(t)>u)dt>v(u)x\right\\}\geq\frac{\widehat{\mathcal{B}}_{\alpha,\alpha}(x+\epsilon,T)}{\widehat{\mathcal{B}}_{\alpha,\alpha}(0,T)}\mathbb{P}\left\\{\sup_{t\in[0,v(u)T]}Q(t)>u\right\\},\quad u\rightarrow\infty.$ In light of Remark 2.2 we establish the claim by letting $\epsilon\rightarrow 0$ in the above inequalities. This completes the proof. $\Box$ ## 6\. Appendix Proof of Lemma 4.1 For notational simplicity denote by $\rho_{u,j}$ the correlation function of the random field $\xi_{u,j}$. Further set $\chi_{u,j}(s)\coloneqq g_{u,j}(\overline{\xi}_{u,j}(s)-\rho_{u,j}(s,{0})\overline{\xi}_{u,j}({0})),\quad s\in E_{1}$ and $f_{u,j}(s,y)\coloneqq y\rho_{u,j}(s,{0})-g^{2}_{u,j}\left(1-\rho_{u,j}(s,{0})\right)-g^{2}_{u,j}\frac{1-\sigma_{u,j}(s)}{\sigma_{u,j}(s)},\ s\in E_{1},y\in\mathbb{R}.$ Conditioning on $\xi_{u,j}({0})$, by F1 and using that $\overline{\xi}_{u,j}({0})$ and $\overline{\xi}_{u,j}(s)-\rho_{u,j}(s,{0})\overline{\xi}_{u,j}({0})$ are mutually independent we obtain $\displaystyle\mathbb{P}\left\\{\int_{E_{2}}\mathbb{I}\left\\{\Gamma\left({g_{u,j}(\xi_{u,j}(s)-g_{u,j})}\right)(t)>0\right\\}\eta(dt)>x\right\\}$ $\displaystyle=\frac{e^{-g^{2}_{u,j}/2}}{\sqrt{2\pi}g_{u,j}}\int_{\mathbb{R}}\exp\left(-y-\frac{y^{2}}{2g^{2}_{u,j}}\right)\mathbb{P}\left\\{\int_{E_{2}}\mathbb{I}\left\\{\Gamma\left(g_{u,j}(\xi_{u,j}(s)-g_{u,j})\right)(t)>0\right\\}\eta(d{t})>x\|\xi_{u,j}({0})=g_{u,j}+yg_{u,j}^{-1}\right\\}dy$ $\displaystyle=\frac{e^{-g^{2}_{u,j}/2}}{\sqrt{2\pi}g_{u,j}}\int_{\mathbb{R}}\exp\left(-y-\frac{y^{2}}{2g^{2}_{u,j}}\right)\mathbb{P}\left\\{\int_{E_{2}}\mathbb{I}\left\\{\Gamma\left(\sigma_{u,j}(s)\left(\chi_{u,j}(s)+f_{u,j}(s,y)\right)\right)(t)>0\right\\}\eta(dt)>x\right\\}dy$ $\displaystyle=\frac{e^{-g^{2}_{u,j}/2}}{\sqrt{2\pi}g_{u,j}}\int_{\mathbb{R}}\exp\left(-y-\frac{y^{2}}{2g^{2}_{u,j}}\right)\mathcal{I}_{u,j}(y;x)dy,$ where $\mathcal{I}_{u,j}(y;x)\coloneqq\mathbb{P}\left\\{\int_{E_{2}}\mathbb{I}\left\\{\Gamma\left(\sigma_{u,j}(s)\left(\chi_{u,j}(s)+f_{u,j}(s,y)\right)\right)(t)>0\right\\}\eta(dt)>x\right\\}.$ Noting that $\lim_{u\rightarrow\infty}\sup_{j\in S_{u}}\left\|\frac{\frac{e^{-g^{2}_{u,j}/2}}{\sqrt{2\pi}g_{u,j}}}{\Psi(g_{u,j})}-1\right\|=0$ in order to show the claim it suffices to prove that (46) $\displaystyle\lim_{u\to\infty}\sup_{j\in S_{u}}\left\|\int_{\mathbb{R}}\exp\left(-y-\frac{y^{2}}{2g^{2}_{u,j}}\right)\mathcal{I}_{u,j}(y;x)dy-\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})\right\|=0$ for all $x\geq 0$. In view of C3 it follows that that for $u>u_{0}$ $Var(\chi_{u,j}({s})-\chi_{u,j}({s}^{\prime}))\leq g^{2}_{u,j}\mathbb{E}\left\\{\overline{\xi}_{u,j}({s})-\overline{\xi}_{u,j}({s^{\prime}})\right\\}^{2}\leq Q_{1}\lVert{s}-{s^{\prime}}\rVert^{\nu},\quad s,s^{\prime}\in E_{1},$ with $\nu>0$. Further, by C0-C2 for each $y\in\mathbb{R}$ (47) $\displaystyle\lim_{u\to\infty}\sup_{j\in S_{u},s\in E_{1}}\left\|f_{u,j}(s,y)-y+\sigma^{2}_{\zeta}(s)+h(s)\right\|=0.$ Hence, by F2 (48) $\displaystyle\sup_{j\in S_{u}}e^{-y}\mathcal{I}_{u,j}(y;x)$ $\displaystyle\leq$ $\displaystyle e^{-y}\sup_{j\in S_{u}}\mathbb{P}\left\\{\sup_{t\in E_{2}}\Gamma\left(\chi_{u,j}(s)+f_{u,j}(s,y)\right)(t)>0\right\\}$ $\displaystyle\leq$ $\displaystyle e^{-y}\sup_{j\in S_{u}}\mathbb{P}\left\\{\sup_{s\in E_{1}}\\{\chi_{u,j}(s)+f_{u,j}(s,y)\\}>0\right\\}$ $\displaystyle\leq$ $\displaystyle e^{-y}\sup_{j\in S_{u}}\mathbb{P}\left\\{\sup_{s\in E_{1}}\chi_{u,j}(s)>Q_{2}\left\|y\right\|-Q_{3}\right\\}$ $\displaystyle\leq$ $\displaystyle Q_{4}\left\|y\right\|^{2n/\nu-1}e^{-Q_{5}y^{2}-y},\quad y<-M,$ where in the last inequality we used Piterbarg inequality and $M>0$. Moreover, it follows trivially that for all $x\geq 0$ (49) $\displaystyle\sup_{j\in S_{u}}e^{-y}\mathcal{I}_{u,j}(y;x)\leq e^{-y},\quad y\in\mathbb{R}.$ Therefore by the dominated convergence theorem and assumption C0 $\displaystyle\sup_{j\in S_{u}}\left\|\int_{\mathbb{R}}\exp\left(-y-\frac{y^{2}}{2g^{2}_{u,j}}\right)\mathcal{I}_{u,j}(y;x)dy-\int_{\mathbb{R}}e^{-y}\mathcal{I}_{u,j}(y;x)dy\right\|$ $\displaystyle\leq\int_{\mathbb{R}}\sup_{j\in S_{u}}\left(e^{-y}\mathcal{I}_{u,j}(y;x)({1-e^{-y^{2}/(2g^{2}_{u,j})}})\right)dy\rightarrow 0,\quad u\to\infty.$ Hence in order to prove the convergence in (46) it suffices to show that (50) $\displaystyle\lim_{u\to\infty}\sup_{j\in S_{u}}\left\|\int_{\mathbb{R}}e^{-y}\mathcal{I}_{u,j}(y;x)dy-\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})\right\|=0$ for all $x\in[0,\eta(E_{2}))$. Weak convergence. The claim follows from the same arguments as in [11][Lem 4.3,4.7], where the precise meaning of uniform weak convergence is also given. Thus let $C(E_{1})$ denote the Banach space of all continuous functions on the compact set $E_{1}$ equipped with supremum norm. For any ${s},{s}^{\prime}\in E_{1}$, by C2 we have $Var(\chi_{u,j}(s)-\chi_{u,j}({s}^{\prime}))=g^{2}_{u,j}\left(\mathbb{E}\left\\{\overline{\xi}_{u,j}(s)-\overline{\xi}_{u,j}({s}^{\prime})\right\\}^{2}-\left(\rho_{u,j}(s,{0})-\rho_{u,j}({s}^{\prime},{0})\right)^{2}\right)\rightarrow 2Var(\zeta(s)-\zeta({s}^{\prime}))$ uniformly with respect to $j\in S_{u}$ as $u\to\infty$. Hence, the finite- dimensional distributions of $\chi_{u,j}(s),s\in E_{1}$ weakly converge to that of $\sqrt{2}\zeta(s),s\in E_{1}$ uniformly with respect to $j\in S_{u}$. In view of C3, we know that the measures on $C(E_{1})$ induced by $\\{\chi_{u,j}(s),s\in E_{1},j\in S_{u}\\}$ are uniformly tight for large $u$, and by C1, $\sigma_{u,j}(s)$ converges to $1$ uniformly for $s\in E_{1}$ and $j\in S_{u}$ as $u\to\infty$. Therefore, $\\{\sigma_{u,j}(s)\chi_{u,j}(s),s\in E_{1}\\}$ converge weakly to $\\{\sqrt{2}\zeta(s),s\in E_{1}\\}$ as $u\to\infty$ uniformly with respect to $j\in S_{u}$, which together with (47) implies that for each $y\in\mathbb{R}$, the probability measures on $C(E_{1})$ induced by $\\{\chi_{u,j}^{f}(s,y),s\in E_{1}\\}$ converges weakly as $u\to\infty$ to that induced by $\\{\zeta_{h}(s)+y,{t}\in E_{1}\\}$ uniformly with respect to $j\in S_{u}$, where $\chi_{u,j}^{f}({s},y)\coloneqq\sigma_{u,j}(s)\left(\chi_{u,j}(s)+f_{u,j}(s,y)\right)\quad\textrm{and}\quad\zeta_{h}(s)\coloneqq\sqrt{2}\zeta(s)-\sigma^{2}_{\zeta}(t)-h(s).$ Continuous mapping theorem implies that for each $y\in\mathbb{R}$, the push- forward probability measures $P_{u,y}$ on $C(E_{2})$ induced by $\\{\Gamma\left(\chi_{u,j}^{f}(\cdot,y)\right)(t),t\in E_{2}\\}$ converges weakly the push-forward probability measure $P_{y}$ induced by $\\{\Gamma\left(\zeta_{h}\right)(t)+y,t\in E_{2}\\}$ as $u\to\infty$ uniformly with respect to $j\in S_{u}$. The continuity of the sojourn functional is also discussed in [3][Lem 4.2]. A sequence of functions $f_{n}\in C(E_{2})$ converges to $f\in C(E_{2})$ as $n\to\infty$ with respect to uniform topology if $f_{n}\to f$ uniformly as $n\to\infty$. Since $\eta$ is absolutely continuous with respect to Lebesgue measure on $E_{2}$ we can define the set $A_{}=\left\\{f\in C(E_{2}):\int_{E_{2}}\mathbb{I}(f(t)=0)\eta(dt)>0\right\\},$ which is measurable in the completion $\mathcal{C}^{\mu}$ of $\mathcal{C}$ with respect to $\nu$, where $\mathcal{C}$ is the Borel $\sigma$-field of $C_{2}(E)$. Its complement belongs to ${\color[rgb]{0,0,0}\mathcal{C}^{\mu}}$, i.e., $A_{}^{c}=C(E_{2})\setminus A_{}\in{\color[rgb]{0,0,0}\mathcal{C}^{\mu}}.$ Any function $f\in A_{}^{c}$ is a continuity point of the sojourn functional $J:C(E_{2})\mapsto[0,\eta(E_{2})],$ where $J(f)=\int_{E_{2}}1(f(t)>0)\eta(dt),f\in C(E_{2}).$ This functional is measurable $\mathcal{C}/\mathcal{B}(\mathbb{R})$ by the assumption on $\eta$. We shall show that it is continuous at any $f\in A_{}^{c}$. Let such $f$ be given. By the definition of the integral such $f$ is not equal to zero on any compact interval of $\mathbb{R}$. Let $f_{n}\to f$ uniformly as $n\to\infty$. Then $1(f_{n}(t)>0)\to 1(f(t)>0)$ as $n\to\infty$ for almost all $t\in\mathbb{R}$ (with respect to Lebesgue measure). Hence by dominated convergence theorem we have $J(f_{n})\to J(f)$ as $n\to\infty$, which means that the functional is continuous for all $f\in A_{}^{c}$. Recall that $P_{y}$ is the push-forward (image measure) on $C(E_{2})$ with respect to $\Gamma(\xi_{h})+y$. We claim that $P_{y}(A_{})>0$ is possible only for $y$ in a countable set of $\mathbb{R}$. Indeed, any $f\in A_{}$ is such that it is constant equal to zero on a compact interval. Consequently, $P_{y}(A_{})>0$ means that the functions $f\in A_{}$ are constant equal to $-y$ on some interval of $\mathbb{R}$. If this is true for two different $y$’s, then the intervals where $f$ is constant equal $-y$ must be disjoint, therefore this can be true only for countable $y$’s. Alternatively, using the fact that $\mathbb{P}\left\\{\Gamma(\zeta_{h})(t)+y=0\right\\}=0$ a.e., $y\in\mathbb{R}$, by the $\sigma$-finiteness of $\eta$, Fubini-Tonelli theorem yields $\displaystyle\int_{\mathbb{R}}\mathbb{E}\left\\{\int_{E_{2}}\mathbb{I}(\Gamma(\zeta_{h})(t)+y=0)\eta(dt)\right\\}dy=\int_{E_{2}}\int_{\mathbb{R}}\mathbb{P}\left\\{\Gamma(\zeta_{h})(t)+y=0\right\\}dy\eta(dt)=0.$ Hence for almost all $y\in\mathbb{R}$ $\mathbb{E}\left\\{\int_{E_{2}}\mathbb{I}(\Gamma(\zeta_{h})(t)+y=0)\eta(dt)\right\\}=0,$ which means that, for almost all $y\in\mathbb{R}$ $P_{y}(A_{})=\mathbb{P}\left(\int_{E_{2}}\mathbb{I}(\Gamma(\zeta_{h})(t)+y=0)\eta(dt)>0\right)=0.$ Consequently, since $J(f)$ is continuous for $f\in A_{}^{c}$, by continuous mapping theorem, as $u\to\infty$ (51) $\displaystyle\int_{E_{2}}\mathbb{I}\left(\Gamma\left(\chi_{u,j}^{f}(\cdot,y)\right)(t)>0\right)\eta(dt)$ weakly converges to $\int_{E_{2}}\mathbb{I}\left(\Gamma\left(\zeta_{h}\right)(t)+y>0\right)\eta(dt)$ uniformly with respect to $j\in S_{u}$ for almost all $y\in\mathbb{R}$. Convergence on continuity points. Define $\mathcal{I}(y;x)\coloneqq\mathbb{P}\left\\{\int_{E_{2}}\mathbb{I}\left(\Gamma(\zeta_{h})(t)+y>0\right)\eta(d{t})>x\right\\}.$ We draw a similar argument as in Theorem 1.3.1 of [7] to verify (50) for all continuity points $x\in(0,\eta(E_{2}))$ of $\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})$. Let $x_{0}\in(0,\eta(E_{2}))$ be such a continuity point, that is $\lim_{\varepsilon\to 0}\int_{\mathbb{R}}\left(\mathcal{I}(y;x_{0}+\varepsilon)-\mathcal{I}(y;x_{0}-\varepsilon)\right)e^{-y}dy=0.$ Since for large $M$ and all $x\geq 0$ by F2 as in the derivation of (48) we have (52) $\displaystyle e^{-y}\mathcal{I}(y;x)\leq Q_{4}^{\prime}\left\|y\right\|^{2n/\nu-1}e^{-Q_{5}y^{2}-y},\quad y<-M$ it follows from the dominated convergence theorem that $\int_{\mathbb{R}}\left(\mathcal{I}(y;x_{0}+)-\mathcal{I}(y;x_{0}-)\right)e^{-y}dy=0$ and thus by the monotonicity of $\mathcal{I}(y;x)$ in $x$ for each fixed $y$, $x_{0}$ is a continuous point of $\mathcal{I}(y;x)$ for a.e. $y\in\mathbb{R}$. Thus by (51) for a.e. $y\in\mathbb{R}$ (53) $\displaystyle\lim_{u\to\infty}\sup_{j\in S_{u}}\left\|\mathcal{I}_{u,j}(y;x_{0})-\mathcal{I}(y;x_{0})\right\|=0.$ As shown in (48), (49) and (52) it follows from the dominated convergence theorem that $\displaystyle\sup_{j\in S_{u}}\left\|\int_{\mathbb{R}}e^{-y}\mathcal{I}_{u,j}(y;x_{0})dy-\int_{\mathbb{R}}e^{-y}\mathcal{I}(y;x_{0})dy\right\|$ (54) $\displaystyle\leq\int_{\mathbb{R}}\sup_{j\in S_{u}}\left\|\mathcal{I}_{u,j}(y;x_{0})-\mathcal{I}(y;x_{0})\right\|e^{-y}dy\rightarrow 0,\quad u\to\infty$ establishing the proof for all continuity points $x\in(0,\eta(E_{2}))$. Moreover, for the case that $x=0$, (6) also holds by replacing sojourn with supremum. This can be shown directly without any continuity requirement for $\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})$ at $x=0$. Continuity of $\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})$. Next we show that $\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})$ is continuous at any $x\in(0,\eta(E_{2}))$ using that $\eta$ is equivalent with Lebesgue measure on $E_{2}$. Note that $\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})$ is clearly right continuous at $0$. Next we show the continuity at $x\in(0,E_{2})$. The claimed continuity at $x$ follows if we show $\int_{\mathbb{R}}\mathbb{P}\left\\{A_{y}\right\\}e^{-y}dy=0,\quad A_{y}=\left\\{\int_{E_{2}}\mathbb{I}\big{(}\Gamma(\zeta_{h})(t)+y>0\big{)}\eta(d{t})=x\right\\},\quad y\in\mathbb{R}.$ If $\int_{E_{2}}\mathbb{I}\big{(}\Gamma(\zeta_{h})(t)+y>0\big{)}\eta(d{t})=x,$ with $0<x<\eta(E_{2})$, then using the fact that $\Gamma(\zeta_{h})(t)$ is continuous over $E_{2}$ and the Lebesgue measure is absolutely continuous with respect to $\eta$, we have that for any $y^{\prime}>y$ $\int_{E_{2}}\mathbb{I}\big{(}\Gamma(\zeta_{h})(t)+y^{\prime}>0\big{)}\eta(d{t})>x.$ This implies that $A_{y}\cap A_{y^{\prime}}=\emptyset,y\neq y^{\prime},y,y^{\prime}\in\mathbb{R}.$ Noting that the continuity of $\Gamma(\zeta_{h})$ guarantees the measurability of $A_{y}$, and $\\{y:y\in\mathbb{R}\quad\text{such that}\quad\mathbb{P}\left\\{A_{y}\right\\}>0\\}$ is a countable set because if it were not we would find countably many (disjoint) $A_{y}$ such that $\sum\mathbb{P}\left\\{A_{y}\right\\}=\infty$. Thus we get $\int_{\mathbb{R}}\mathbb{P}\left\\{A_{y}\right\\}e^{-y}dy=0,$ hence $\mathcal{B}^{\Gamma,h,\eta}_{\zeta}(x,E_{2})$ is continuous on $(0,\eta(E_{2}))$, establishing the claim. $\Box$ Before proceeding to the proof of Lemma 4.2, under notation introduced in the proof of Proposition 3.1, we denote and analyze (55) $\displaystyle\Sigma\Sigma_{1}(u,n)$ $\displaystyle\coloneqq$ $\displaystyle\sum_{0\leq k_{i},k_{i}^{\prime}\leq N_{i}(u,n),i=1,2,(k_{1},k_{2})\neq(k_{1}^{\prime},k_{2}^{\prime})}\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,n)}X(t)>u,\sup_{t\in I_{k_{1}^{\prime},k_{2}^{\prime}}(u,n)}X(t)>u\right\\},$ (56) $\displaystyle\Sigma\Sigma_{2}(u,n)$ $\displaystyle\coloneqq$ $\displaystyle\sum_{0\leq 2k_{i},2k_{i}^{\prime}\leq N_{i}(u,n),i=1,2,(k_{1},k_{2})\neq(k_{1}^{\prime},k_{2}^{\prime})}\mathbb{P}\left\\{\sup_{t\in I_{2k_{1},2k_{2}}(u,n)}X(t)>u,\sup_{t\in I_{2k^{\prime}_{1},2k^{\prime}_{2}}(u,n)}X(t)>u\right\\},$ (57) $\displaystyle\Theta(u)$ $\displaystyle\coloneqq$ $\displaystyle T_{1}T_{2}a_{1}^{1/\alpha_{1}}a_{2}^{1/\alpha_{2}}u^{2/\alpha_{1}+2/\alpha_{2}}\Psi(u).$ Moreover, following notation introduced in the proof of Proposition 3.2, let $\Sigma\Sigma_{3}^{\prime\prime}(u,n)\coloneqq\sum_{\|k_{i}\|,\|k_{i}^{\prime}\|\leq N_{i}^{\prime}(u,n),i=1,2,(k_{1},k_{2})\neq(k_{1}^{\prime},k_{2}^{\prime})}\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,n)}X(t)>u,\sup_{t\in I_{k_{1}^{\prime},k_{2}^{\prime}}(u,n)}X(t)>u\right\\}.$ (58) $\displaystyle\hat{I}_{k_{2}}(u,n)\coloneqq I_{-1,k_{2}}(u,n)\cup I_{0,k_{2}}(u,n),\quad E_{1}(u,n)\coloneqq\bigcup_{\|k_{2}\|\leq N_{2}^{\prime}(u,n)}\hat{I}_{k_{2}}(u,n),$ and (59) $\displaystyle\Sigma_{3}^{\prime}(u,n)$ $\displaystyle\coloneqq$ $\displaystyle\sum_{\|k_{i}\|\leq N_{i}^{\prime}(u,n)+1,i=1,2,~{}k_{1}\neq-1,0}\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,n)}X(t)>u\right\\},$ (60) $\displaystyle\Sigma\Sigma_{3}(u,n)$ $\displaystyle\coloneqq$ $\displaystyle\sum_{\|k_{2}\|,\|k_{2}^{\prime}\|\leq N_{2}^{\prime}(u,n),k_{2}\neq k_{2}^{\prime}}\mathbb{P}\left\\{\sup_{t\in\hat{I}_{k_{2}}(u,n)}X(t)>u,\sup_{t\in\hat{I}_{k_{2}^{\prime}}(u,n)}X(t)>u\right\\},$ (61) $\displaystyle\Sigma\Sigma_{4}(u,n)$ $\displaystyle\coloneqq$ $\displaystyle\sum_{\|2k_{2}\|,\|2k_{2}^{\prime}\|\leq N_{2}^{\prime}(u,n)-1,k_{2}\neq k_{2}^{\prime}}\mathbb{P}\left\\{\sup_{t\in\hat{I}_{2k_{2}}(u,n)}X(t)>u,\sup_{t\in\hat{I}_{2k_{2}^{\prime}}(u,n)}X(t)>u\right\\}.$ ###### Lemma 6.1. Under the assumptions of Proposition 3.1 (62) $\displaystyle\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\}\sim\sum_{0\leq k_{i}\leq N_{i}(u,n),i=1,2}\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,{n})}X(t)>u\right\\}\sim\mathbb{C}_{0}\Theta(u),\quad u\rightarrow\infty,n\to\infty,$ where $\mathbb{C}_{0}>0$. Moreover, for all large $u$ and $n$ $\displaystyle\Sigma\Sigma_{1}(u,n)$ $\displaystyle\leq$ $\displaystyle\left(\frac{\mathbb{C}_{2}}{\sqrt{n}}+e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\right)\Theta(u),\quad\Sigma\Sigma_{2}(u,n)\leq e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\Theta(u),$ where $\mathbb{C},\mathbb{C}_{1}$ and $\mathbb{C}_{2}$ are some positive constants. Proof of Lemma 6.1 Asymptotics (62) follow from Lemma 7.1 in [26], while the bounds can be deduced from equations (7.4) and (7.6) in the proof of Lemma 7.1 in [26]. $\Box$ ###### Lemma 6.2. Under the assumptions of Proposition 3.2, for $\alpha_{i}<\beta_{i},i=1,2$, $\mathbb{P}\left\\{\sup_{t\in E\setminus E(u,n)}X(t)>u\right\\}=o\left(\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\}\right)$ as $u\to\infty,n\to\infty$, and $\Sigma\Sigma_{3}^{\prime\prime}(u,n)=o\left(\sum_{0\leq k_{i}\leq N_{i}^{\prime}(u,n),i=1,2}\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,n)}X(t)>u\right\\}\right),$ as $u\to\infty,n\to\infty$. For $\alpha_{1}=\beta_{1},\alpha_{2}<\beta_{2}$ $\mathbb{P}\left\\{\sup_{t\in E\setminus E_{1}(u,n)}X(t)>u\right\\}=o\left(\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\}\right),$ as $u\to\infty$, $n\to\infty$, and for $u$ and $n$ sufficiently large $\displaystyle\Sigma\Sigma_{3}(u,n)$ $\displaystyle\leq\left(\frac{\mathbb{C}_{2}}{\sqrt{n}}+e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\right)\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\},$ $\displaystyle\Sigma_{3}^{\prime}(u,n)$ $\displaystyle\leq e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\},$ $\displaystyle\Sigma\Sigma_{4}(u,n)$ $\displaystyle\leq e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\}.$ For $\alpha_{1}=\beta_{1}$ and $\alpha_{2}=\beta_{2}$ $\mathbb{P}\left\\{\sup_{t\in E\setminus\bigcup_{i,j\in\\{-1,0\\}}I_{i,j}(u,n)}X(t)>u\right\\}=o\left(\mathbb{P}\left\\{\sup_{t\in E}X(t)>u\right\\}\right),$ as $u\to\infty,n\to\infty$. Proof of Lemma 6.2 The proof of Lemma 6.2 follows from [18]. Specifically, the first one follows from (34), the second one from (40) and (41), the third one from (34) and (46), the fourth one from (48) and (49), the fifth one from (46), the six one from (48), and the last one from (34) and (52) in the proof of Theorem 3.1 of [18]. $\Box$ Now we are in the position to prove Lemma 4.2. Proof of Lemma 4.2 Ad (i). We follow notation introduced in the proof of Proposition 3.1. For any $n,n_{1}>\sqrt{x}$, we have (63) $\displaystyle\Sigma_{1}^{-}(u,n_{1})-\Sigma\Sigma_{1}(u,n_{1})\leq\mathbb{P}\left\\{\int_{E(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}\leq\Sigma_{1}^{+}(u,n)+\Sigma\Sigma_{1}(u,n),$ where $\Sigma\Sigma_{1}(u,n)$ is given in (55) and $\displaystyle\Sigma_{1}^{\pm}(u,n)=\sum_{0\leq k_{i}\leq N_{i}(u,n)\pm 1,i=1,2}\mathbb{P}\left\\{\int_{I_{k_{1},k_{2}}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}.$ By (17), it follows that $\displaystyle\Sigma_{1}^{+}(u,n)$ $\displaystyle\leq$ $\displaystyle\sum_{0\leq k_{i}\leq N_{i}(u,n),i=1,2}\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})\Psi(u)$ $\displaystyle\leq$ $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}\Theta(u),\quad u\rightarrow\infty,$ where $\Theta(u)$ is defined in (57). Analogously, we obtain the lower bound $\displaystyle\Sigma_{1}^{-}(u,n)$ $\displaystyle\geq$ $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}\Theta(u),\quad u\rightarrow\infty.$ Lemma 6.1 shows that for $u$ and $n$ sufficiently large $\Sigma\Sigma_{1}(u,n)\leq\left(\frac{\mathbb{C}_{2}}{\sqrt{n}}+e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\right)\Theta(u).$ Dividing both sides of (63) by $\Theta(u)$ and letting $u\rightarrow\infty$, we have $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n_{1}]^{2})}{n_{1}^{2}}-\frac{\mathbb{C}_{2}}{\sqrt{n_{1}}}-e^{-\mathbb{C}_{1}n_{1}^{\mathbb{C}}}\leq\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}+\frac{\mathbb{C}_{2}}{\sqrt{n}}+e^{-\mathbb{C}_{1}n^{\mathbb{C}}}.$ The above implies that $\displaystyle\limsup_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}$ $\displaystyle=$ $\displaystyle\liminf_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}<\infty.$ Next we show that $\liminf_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}>0.$ Observe that (64) $\displaystyle\mathbb{P}\left\\{\int_{E}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}\geq\Sigma_{2}(u,n)-\Sigma\Sigma_{2}(u,n),$ where $\Sigma\Sigma_{2}(u)$ is given in (56) and $\displaystyle\Sigma_{2}(u,n)=\sum_{0\leq 2k_{i}\leq N_{i}^{\prime}(u,n),i=1,2}\mathbb{P}\left\\{\int_{I_{2k_{1},2k_{2}}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}.$ In light of (17), we have $\displaystyle\Sigma_{2}(u,n)$ $\displaystyle\geq$ $\displaystyle\sum_{0\leq 2k_{i}\leq N_{i}^{\prime}(u,n),i=1,2}\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})\Psi(u)$ $\displaystyle\geq$ $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{4n^{2}}\Theta(u),\quad u\rightarrow\infty.$ Moreover, by Lemma 6.1 we have, for $u$ and $n$ large enough $\Sigma\Sigma_{2}(u,n)\leq e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\Theta(u).$ Combination of upper bound in (63) and lower bound in (64) leads to (65) $\displaystyle\liminf_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}\geq\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n_{1}]^{2})}{4n_{1}^{2}}-e^{-\mathbb{C}_{1}n_{1}^{\mathbb{C}}}.$ For $n_{1}>\sqrt{x}$ $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n_{1}]^{2})$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\mathbb{P}\left\\{\int_{[0,n_{1}]^{2}}\mathbb{I}\left(\sum_{i=1}^{2}(\sqrt{2}B_{\alpha_{i}}(t_{i})-\left\|t_{i}\right\|^{\alpha_{i}})>s\right)dt>x\right\\}e^{s}ds$ $\displaystyle\geq$ $\displaystyle\int_{\mathbb{R}}\mathbb{P}\left\\{\inf_{t\in[0,n_{1}]^{2}}\sum_{i=1}^{2}\left(\sqrt{2}B_{\alpha_{i}}(t_{i})-\left\|t_{i}\right\|^{\alpha_{i}}\right)>s\right\\}e^{s}ds>0,$ which combined with the monotonicity of $\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n_{1}]^{2})$ in $n_{1}$ and (65) implies that for sufficiently large $n_{1}$ $\liminf_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})}{n^{2}}\geq\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n_{1}]^{2})-4n_{1}^{2}e^{-\mathbb{C}_{1}n_{1}^{\mathbb{C}}}}{4n_{1}^{2}}>0,$ establishing the proof of (i). Ad (ii). We follow notation introduced in the proof of Proposition 3.2 for the case $\alpha_{1}=\beta_{1}$ and $\alpha_{2}<\beta_{2}$. Let next for $u>0$ $E_{2}(u)\coloneqq\left[-\left(\frac{e_{u}^{-1/4}\wedge\ln u}{u}\right)^{2/\beta_{1}},\left(\frac{e_{u}^{-1/4}\wedge\ln u}{u}\right)^{2/\beta_{1}}\right]\times\left[-\left(\frac{e_{u}^{-1/4}\wedge\ln u}{u}\right)^{2/\beta_{2}},\left(\frac{e_{u}^{-1/4}\wedge\ln u}{u}\right)^{2/\beta_{2}}\right],$ $I_{k_{1},k_{2}}(u,n)\coloneqq[k_{1}v_{1}(u)n,(k_{1}+1)v_{1}(u)n]\times[k_{2}v_{2}(u)n,(k_{2}+1)v_{2}(u)n],$ $\Theta_{1}(u)\coloneqq 2\hat{\Gamma}(1/\beta_{2}+1)a_{2}^{1/\alpha_{2}}b_{2}^{-1/\beta_{2}}u^{2/\alpha_{2}-2/\beta_{2}}\Psi(u),$ where $\hat{\Gamma}(\cdot)$ is the gamma function and $e_{u}=\sup_{0<\|t_{i}\|<\left(\frac{\ln u}{u}\right)^{2/\beta_{i}},i=1,2}\|e(t)\|,\ \ ~{}e(t)=\frac{1-{\sigma}(t)}{\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}}-1,\|t\|\neq 0.$ Observe that $\displaystyle\mathbb{P}\left\\{\int_{E_{2}(u)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}$ $\displaystyle\geq$ $\displaystyle\mathbb{P}\left\\{\int_{E_{1}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\},$ $\displaystyle\mathbb{P}\left\\{\int_{E_{2}(u)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}$ $\displaystyle\leq$ $\displaystyle\mathbb{P}\left\\{\int_{\bigcup_{\|k_{2}\|\leq N_{2}^{\prime}(u,n)+1}\hat{I}_{k_{2}}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}$ $\displaystyle\quad+\mathbb{P}\left\\{\sup_{E_{2}(u)\setminus(\bigcup_{\|k_{2}\|\leq N_{2}^{\prime}(u,n)+1}\hat{I}_{k_{2}}(u,n))}X(t)>u\right\\}.$ Hence it follows that (66) $\displaystyle\Sigma_{3}^{-}(u,n_{1})-\Sigma\Sigma_{3}(u,n_{1})\leq\mathbb{P}\left\\{\int_{E_{2}(u)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}\leq\Sigma_{3}^{+}(u,n)+\Sigma_{3}^{\prime}(u,n),$ with $\displaystyle\Sigma_{3}^{\pm}(u,n)=\sum_{\|k_{2}\|\leq N_{2}^{\prime}(u,n)\pm 1}\mathbb{P}\left\\{\int_{\hat{I}_{k_{2}}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\},$ where $I_{k_{1},k_{2}}(u,n)$ is defined in (20) and $\Sigma_{3}^{\prime}$ and $\Sigma\Sigma_{3}$ are given in (59) and (60) respectively. Noting that (5.4) also holds for $\|k_{2}\|\leq N_{2}^{\prime}(u,n)+1$, we have for $x\geq 0$ $\displaystyle\Sigma_{3}^{\pm}(u,n)$ $\displaystyle\sim$ $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])\sum_{\|k_{2}\|\leq N_{2}^{\prime}(u,n)+1}\Psi(u_{k_{2},n}^{\pm})$ $\displaystyle\sim$ $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])\Psi(u)\sum_{\|k_{2}\|\leq N_{2}^{\prime}(u,n)+1}e^{-u^{2}b_{2}(\|k_{2}\|v_{2}(u)n)^{\beta_{2}}}$ $\displaystyle\sim$ $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{n}\Theta_{1}(u),\quad u\rightarrow\infty.$ In light of Lemma 6.2, we have that for $u$ and $n$ sufficiently large $\Sigma\Sigma_{3}(u,n)+\Sigma_{3}^{\prime}(u,n)\leq\left(\frac{\mathbb{C}_{2}}{\sqrt{n}}+e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\right)\Theta_{1}(u).$ Dividing both sides of (66) by $\Theta_{1}(u)$ respectively and letting $u\rightarrow\infty$, we have that $\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n_{1},n_{1}]\times[0,n_{1}])}{n_{1}}-\frac{\mathbb{C}_{2}}{\sqrt{n_{1}}}-e^{-\mathbb{C}_{1}n_{1}^{\mathbb{C}}}\leq\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{n}+\frac{\mathbb{C}_{2}}{\sqrt{n}}+e^{-\mathbb{C}_{1}n^{\mathbb{C}}},$ which gives that $\displaystyle\liminf_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{n}=\limsup_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{n}<\infty.$ Moreover, we have $\displaystyle\mathbb{P}\left\\{\int_{E_{2}(u)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}\geq\Sigma_{4}(u,n)-\Sigma\Sigma_{4}(u,n),$ where $\Sigma\Sigma_{4}(u,n)$ is defined in (61) and $\displaystyle\Sigma_{4}(u,n)=\sum_{\|2k_{2}\|\leq N_{2}^{\prime}(u,n)-1}\mathbb{P}\left\\{\int_{\hat{I}_{2k_{2}}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}.$ By (5.4), for $x\geq 0$ we have $\displaystyle\Sigma_{4}(u,n)$ $\displaystyle\sim$ $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])\sum_{\|2k_{2}\|\leq N_{2}^{\prime}(u,n)-1}\Psi(u_{k_{2},n}^{-})$ $\displaystyle\sim$ $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{2n}\Theta_{1}(u),\quad u\rightarrow\infty.$ By Lemma 6.2, for $u$ and $n$ sufficiently large, we have $\Sigma\Sigma_{4}(u,n)\leq e^{-\mathbb{C}_{1}n^{\mathbb{C}}}\Theta_{1}(u).$ In view of (66) for the upper bound, we have $\displaystyle\liminf_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{n}$ $\displaystyle\geq$ $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n_{1},n_{1}]\times[0,n_{1}])}{n_{1}}-e^{-\mathbb{C}_{1}n_{1}^{\mathbb{C}}}.$ Noting that for $n>\sqrt{x}$ $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])$ $\displaystyle=$ $\displaystyle\int_{\mathbb{R}}\mathbb{P}\left\\{\int_{[-n,n]\times[0,n]}\mathbb{I}\left(\sum_{i=1}^{2}(B_{\alpha_{i}}(t_{i})-\left\|t_{i}\right\|^{\alpha_{i}})-a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}}>s\right)dt>x\right\\}e^{s}ds$ $\displaystyle\geq$ $\displaystyle\int_{\mathbb{R}}\mathbb{P}\left\\{\inf_{t\in[-n,n]\times[0,n]}\left(\sum_{i=1}^{2}\left(B_{\alpha_{i}}(t_{i})-\left\|t_{i}\right\|^{\alpha_{i}}\right)-a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}}\right)>s\right\\}e^{s}ds>0,$ and by the monotonicity of $\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])$ with respect to $n$, we have, for $n_{1}$ sufficiently large, $\displaystyle\liminf_{n\rightarrow\infty}\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n,n]\times[0,n])}{n}$ $\displaystyle\geq$ $\displaystyle\frac{\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},0}(x,[-n_{1},n_{1}]\times[0,n_{1}])}{n_{1}}-e^{-\mathbb{C}_{1}n_{1}^{\mathbb{C}}}>0.$ This completes the proof of (ii). Ad (iii). We follow notation introduced in the proof of Proposition 3.2 for the case $\alpha_{i}=\beta_{i}$, $i=1,2$ Observe that (67) $\displaystyle\Sigma_{5}(u,n)\leq\mathbb{P}\left\\{\int_{E^{\prime}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\}\leq\Sigma_{5}(u,n)+\Sigma\Sigma_{5}(u,n),$ where $E^{\prime}(u,n)=\bigcup_{(k_{1},k_{2})\in K_{u,n}}I_{k_{1},k_{2}}(u,n)$ and $\Sigma_{5}(u,n)=\mathbb{P}\left\\{\int_{\hat{I}(u,n)}\mathbb{I}(X(t)>u)dt>v(u)x\right\\},$ $\Sigma\Sigma_{5}(u,n)=\sum_{\|k_{i}\|\leq N_{i}^{\prime}(u,n),k_{i}\neq-1,0,i=1,2}\mathbb{P}\left\\{\sup_{t\in I_{k_{1},k_{2}}(u,n)}\overline{X}(t)>u_{n,k_{1},k_{2}}^{-}\right\\},$ with $u_{n,k_{1},k_{2}}^{-}$ defined in (21) and $\hat{I}(u,n)$ in (28). In light of (22) and (10), we have that for $u$ sufficiently large $\displaystyle\Sigma\Sigma_{5}(u,n)$ $\displaystyle\leq$ $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})\sum_{\|k_{i}\|\leq N_{i}^{\prime}(u,n),k_{i}\neq-1,0,i=1,2}\Psi(u_{n,k_{1},k_{2}}^{-})$ $\displaystyle\leq$ $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})\Psi(u)\sum_{\|k_{i}\|\leq N_{i}^{\prime}(u,n),k_{i}\neq-1,0,i=1,2}e^{-a_{1}^{-1}b_{1}\|k_{1}^{}n\|^{\beta_{1}}-a_{2}^{-1}b_{2}\|k_{2}^{}n\|^{\beta_{2}}}$ $\displaystyle\leq$ $\displaystyle\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n]^{2})e^{-Q_{1}(n^{\beta_{1}}+n^{\beta_{2}})}\Psi(u),$ where $k_{i}^{}=k_{i}I_{\\{k_{i}>0\\}}+(\|k_{i}\|-1)I_{\\{k_{i}<0\\}},i=1,2.$ Hence dividing (67) by $\Psi(u)$ and letting $u\rightarrow\infty$, we have for any $n,n_{1}>\sqrt{x}$ $\displaystyle 0<\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},a_{2}^{-1}b_{2}\|t_{2}\|^{\alpha_{2}}}(x,[-n,n]^{2})\leq\mathcal{B}_{\alpha_{1},\alpha_{2}}^{a_{1}^{-1}b_{1}\|t_{1}\|^{\alpha_{1}},a_{2}^{-1}b_{2}\|t_{2}\|^{\alpha_{2}}}(x,[-n_{1},n_{1}]^{2})+\mathcal{B}_{\alpha_{1},\alpha_{2}}(x,[0,n_{1}]^{2})e^{-Q_{1}(n_{1}^{\beta_{1}}+n_{1}^{\beta_{2}})}.$ Letting $n\rightarrow\infty$ with $n_{1}$ fixed in the above inequality, we complete the proof. $\Box$ Proof of (23): Observe that $\displaystyle\frac{\Psi(u_{n,k_{1},k_{2}}^{-})}{\Psi(u_{n,k_{1},k_{2}}^{+})}\sim e^{\frac{\left(u_{n,k_{1},k_{2}}^{+}\right)^{2}-\left(u_{n,k_{1},k_{2}}^{-}\right)^{2}}{2}},\quad u\rightarrow\infty$ uniformly with respect to $0\leq\|k_{i}\|\leq N_{i}{{}^{\prime}}(u,n),i=1,2$. Furthermore, by (10), for $u$ sufficiently large $\displaystyle\left(u_{n,k_{1},k_{2}}^{+}\right)^{2}-\left(u_{n,k_{1},k_{2}}^{-}\right)^{2}$ $\displaystyle=$ $\displaystyle u^{2}\left(\sup_{t\in I_{k_{1},k_{2}}(u,n)}\frac{1}{{\sigma^{2}(t)}}-\inf_{t\in I_{k_{1},k_{2}}(u,n)}\frac{1}{{\sigma^{2}(t)}}\right)$ $\displaystyle=$ $\displaystyle u^{2}\sup_{s,t\in I_{k_{1},k_{2}}(u,n)}\frac{{\sigma^{2}(t)}-{\sigma^{2}(s)}}{{\sigma^{2}(t)}{\sigma^{2}(s)}}$ $\displaystyle\leq$ $\displaystyle 4u^{2}\sup_{s,t\in I_{k_{1},k_{2}}(u,n)}\|{\sigma(t)}-{\sigma(s)}\|$ $\displaystyle=$ $\displaystyle 4u^{2}\sup_{s,t\in I_{k_{1},k_{2}}(u,n)}\left\|(1+e(t))\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}-(1+e(s))\sum_{i=1}^{2}b_{i}\|s_{i}\|^{\beta_{i}}\right\|$ $\displaystyle\leq$ $\displaystyle 4u^{2}\sup_{s,t\in I_{k_{1},k_{2}}(u,n)}\left\|\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}-\sum_{i=1}^{2}b_{i}\|s_{i}\|^{\beta_{i}}\right\|+8u^{2}\sup_{t\in I_{k_{1},k_{2}}(u,n)}\|e(t)\|\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}$ $\displaystyle\leq$ $\displaystyle 4u^{2}\sum_{i=1}^{2}b_{i}\beta_{i}\|\theta_{i}\|^{\beta_{i}-1}v_{i}(u)n+8u^{2}\sup_{t\in I_{k_{1},k_{2}}(u,n)}\|e(t)\|\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}},$ where $e(t)=\frac{1-{\sigma(t)}}{\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}}-1,\|t\|\neq 0$ and $\theta_{i}\in(k_{i}v_{i}(u)n,(k_{i}+1)v_{i}(u)n)$. Using the fact that $N_{i}^{\prime}(u,n)=\left[\frac{(e_{u}^{-1/4}\wedge\ln u)^{2/\beta_{i}}}{u^{2/\beta_{i}}v_{i}(u)n}\right]~{}\text{and}~{}\lim_{u\rightarrow\infty}e_{u}=0,$ we have that $u^{2}\sup_{t\in I_{k_{1},k_{2}}(u,n)}\|e(t)\|\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}\leq 2e_{u}\sum_{i=1}^{2}b_{i}(e_{u}^{-1/4}\wedge\ln u)^{2}\rightarrow 0,$ as $u\rightarrow\infty$ uniformly with respect to $0\leq\|k_{i}\|\leq N_{i}{{}^{\prime}}(u,n),i=1,2$. For $\beta_{i}\geq 1,i=1,2,$ $\displaystyle u^{2}\sum_{i=1}^{2}b_{i}\beta_{i}\|\theta_{i}\|^{\beta_{i}-1}v_{i}(u)n$ $\displaystyle\leq$ $\displaystyle u^{2}\sum_{i=1}^{2}b_{i}\beta_{i}\left(\frac{\ln u}{u}\right)^{\frac{2(\beta_{i}-1)}{\beta_{i}}}v_{i}(u)n$ $\displaystyle\leq$ $\displaystyle\sum_{i=1}^{2}2a_{i}^{-1/\alpha_{i}}b_{i}\beta_{i}u^{2/\beta_{i}-2/\alpha_{i}}(\ln u)^{\frac{2(\beta_{i}-1)}{\beta_{i}}}n\rightarrow 0,\quad u\rightarrow\infty$ uniformly with respect to $0\leq\|k_{i}\|\leq N_{i}{{}^{\prime}}(u,n),i=1,2$, where $(\theta_{1},\theta_{2})\in I_{k_{1},k_{2}}(u,n)$. For $0<\beta_{i}<1,i=1,2,$ $\displaystyle u^{2}\sup_{s,t\in I_{k_{1},k_{2}}(u,n)}\left\|\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}-\sum_{i=1}^{2}b_{i}\|s_{i}\|^{\beta_{i}}\right\|$ $\displaystyle\leq$ $\displaystyle u^{2}\sum_{i=1}^{2}b_{i}\beta_{i}\|\theta_{i}\|^{\beta_{i}-1}v_{i}(u)n$ $\displaystyle\leq$ $\displaystyle u^{2}\sum_{i=1}^{2}b_{i}\beta_{i}\|v_{i}(u)n\|^{\beta_{i}}\rightarrow 0,\quad u\rightarrow\infty,$ holds uniformly for $0\leq\|k_{i}\|\leq N_{i}{{}^{\prime}}(u,n),k_{i}\neq-1,0,i=1,2.$ For $0<\beta_{i}<1,k_{i}=-1,0,i=1,2$ $\displaystyle u^{2}\sup_{s,t\in I_{k_{1},k_{2}}(u,n)}\left\|\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}-\sum_{i=1}^{2}b_{i}\|s_{i}\|^{\beta_{i}}\right\|$ $\displaystyle\leq$ $\displaystyle u^{2}\sup_{s,t\in I_{k_{1},k_{2}}(u,n)}\left(\sum_{i=1}^{2}b_{i}\|t_{i}\|^{\beta_{i}}+\sum_{i=1}^{2}b_{i}\|s_{i}\|^{\beta_{i}}\right)$ $\displaystyle\leq$ $\displaystyle 2u^{2}\sum_{i=1}^{2}b_{i}\|v_{i}(u)n\|^{\beta_{i}}$ $\displaystyle=$ $\displaystyle 2\sum_{i=1}^{2}a_{i}^{-\beta_{i}/\alpha_{i}}b_{i}n^{\beta_{i}}u^{2-2\beta_{i}/\alpha_{i}}\rightarrow 0,\quad u\rightarrow\infty.$ Therefore, we can conclude that $\displaystyle\left(u_{n,k_{1},k_{2}}^{+}\right)^{2}-\left(u_{n,k_{1},k_{2}}^{-}\right)^{2}\rightarrow 0$ as $u\rightarrow\infty$ uniformly with respect to $0\leq\|k_{i}\|\leq N_{i}^{\prime}(u,n),i=1,2$ establishing the proof. $\Box$ Acknowledgement: K. 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# Shape or Texture: Understanding Discriminative Features in CNNs Md Amirul Islam1,6, Matthew Kowal1, Patrick Esser3, Sen Jia2, Björn Ommer3, Konstantinos G. Derpanis1,5,6 & Neil Bruce4,6 1Department of Computer Science, Ryerson University, Canada 2University of Waterloo, Canada 3IWR, HCI, Heidelberg University, Germany 4School of Computer Science, University of Guelph, Canada 5Samsung AI Centre Toronto, Canada 6Vector Institute for AI, Canada <EMAIL_ADDRESS><EMAIL_ADDRESS> <EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract Contrasting the previous evidence that neurons in the later layers of a Convolutional Neural Network (CNN) respond to complex object shapes, recent studies have shown that CNNs actually exhibit a ‘texture bias’: given an image with both texture and shape cues (e.g., a stylized image), a CNN is biased towards predicting the category corresponding to the texture. However, these previous studies conduct experiments on the final classification output of the network, and fail to robustly evaluate the bias contained (i) in the latent representations, and (ii) on a per-pixel level. In this paper, we design a series of experiments that overcome these issues. We do this with the goal of better understanding what type of shape information contained in the network is discriminative, where shape information is encoded, as well as when the network learns about object shape during training. We show that a network learns the majority of overall shape information at the first few epochs of training and that this information is largely encoded in the last few layers of a CNN. Finally, we show that the encoding of shape does not imply the encoding of localized per-pixel semantic information. The experimental results and findings provide a more accurate understanding of the behaviour of current CNNs, thus helping to inform future design choices. ## 1 Introduction Convolutional neural networks (CNNs) have achieved unprecedented performance in various computer vision tasks, such as image classification (Krizhevsky et al., 2012; Simonyan & Zisserman, 2015; He et al., 2016), object detection (Ren et al., 2015; He et al., 2017) and semantic segmentation (Long et al., 2015; Chen et al., 2017; Islam et al., 2017). Despite their black box nature, various studies have shown that early layers in CNNs activate for low-level patterns, like edges and blobs, while deeper layers activate for more complex and high-level patterns (Zeiler & Fergus, 2014; Springenberg et al., 2014). The intuition is that this hierarchical learning of latent representations allows CNNs to recognize complex object shapes to correctly classify images (Kriegeskorte, 2015). In contrast, recent works (Brendel & Bethge, 2019; Hermann & Lampinen, 2020) have argued that CNNs trained on ImageNet (IN) (Deng et al., 2009) classify images mainly according to their texture, rather than object shape. These conflicting results have large implications for the field of computer vision as it suggests that CNNs trained for image classification might be making decisions based largely off spurious correlations rather than a full understanding of different object categories. One example of these spurious correlations is how the Inception CNN (Szegedy et al., 2015) recognizes the difference between ‘Wolf’ and ‘Husky’, based on whether there is snow in the background (Tulio Ribeiro et al., 2016). Recognizing object shapes is important for the generalization to out-of-domain examples (e.g., few-shot learning), as shape is more discriminative than texture when Figure 1: A shape biased model (trained on Stylized ImageNet) makes predictions based on the object’s shape, or does it? Extracting binary ($3^{\text{rd}}$ column) and semantic ($4^{\text{th}}$ col.) segmentation maps with a one convolutional layer read-out module shows that, while the model classifies the image level shape label correctly as a ‘bird’, it fails to encode the full object shape ($3^{\text{rd}}$ col.) as well as fails to categorically assign every object pixel to the ‘bird’ class ($4^{\text{th}}$ col.). texture-affecting phenomena arise, such as lighting, shading, weather, motion blur, or when switching between synthetic and real data. In addition to performance, identifying the discriminative features that CNNs use for decision making is critical for the transparency and further improvements of computer vision models. While the model may achieve good performance for a certain task, it cannot communicate to the user about the reasons it made certain predictions. In other words, successful models need to be good, and interpretable (Lipton, 2019). This is crucial for many domains where causal mechanisms should play a significant role in short or long-term decision making such as healthcare (e.g., what in the MRI indicates a patient has cancer?). Additionally, if researchers intend for their algorithms to be deployed, there must be a certain degree of trust in the decision making algorithm. One downside of the increasing abstraction capabilities of deep CNNs is the lack of interpretability of the latent representations due to hidden layer activations coding semantic concepts in a distributed fashion (Fong & Vedaldi, 2018). It has therefore been difficult to precisely quantify the type of information contained in the latent representations of CNNs. Some methods have looked at ways to analyze the latent representations of CNNs on a neuron-to- neuron level. For instance, (Bau et al., 2017) quantify the number of interpretable neurons for a CNN by evaluating the semantic segmentation performance of an individual neuron from an upsampled latent representation. Later work (Fong & Vedaldi, 2018) then removed the assumption that each neuron encodes a single semantic concept. These works successfully quantify the number of filters that recognize textures or specific objects in a CNN, but do not identify shape information within these representations. The most similar works to ours are those that aim to directly quantify the shape information in CNNs. For example, (Geirhos et al., 2018) analyzed the outputs of CNNs on images with conflicting shape and texture cues. By using image stylization (Huang & Belongie, 2017), they generated the Stylized ImageNet dataset (SIN), where each image has an associated shape and texture label. They then measured the ‘shape bias’ and ‘texture bias’ of a CNN by calculating the percentage of images a CNN predicts as either the shape or texture label, respectively. They conclude that CNNs are ‘texture biased’ and make predictions mainly from texture in an image. This metric has been used in subsequent work exploring shape and texture bias in CNNs (Hermann & Kornblith, 2019); however, the method only compares the output of a CNN, and fails to robustly quantify the amount of shape information contained in the latent representations (note that they refer to ‘shape’ as the entire 3D form of an object, including contours that are not part of the silhouette, while in our work, we define ‘shape’ as the 2D class-agnostic silhouette of an object). Thus, the method from (Hermann & Kornblith, 2019) cannot answer a question of focus in our paper: ‘What fraction of the object’s shape is actually encoded in the latent representation?’. Further, as their metric for shape relies solely on the semantic class label, it precludes them from evaluating the encoded shape and associated categorical information on a per-pixel level. For instance, we show in Fig. 1 that shape biased models (i.e., trained on stylized images) do not classify images based on the entire object shape: even though the CNN correctly classifies the image as a bird, only the partial binary mask (i.e., ‘shape’) can be extracted from the latent representations and it cannot attribute the correct class label to the entire object region (i.e., semantic segmentation mask). Contributions. To address these issues, we perform an empirical study on the ability of CNNs to encode shape information on a neuron-to-neuron and per- pixel level. To quantify these two aspects, we first approximate the mutual information of latent representations between pairs of semantically related images which allows us to estimate the number of dimensions in the feature space dedicated to encoding shape and texture. We then propose a simple strategy to evaluate the amount of shape information contained in the internal representations of a CNN, on a per-pixel level. The latter technique is utilized to distinguish the quality of different shape encodings, regardless of the number of neurons used in each encoding. After showing the efficacy of the two methods, we reveal a number of meaningful properties of CNNs with respect to their ability to encode shape information, including the following: (i) Biasing a CNN towards shape predominantly changes the number of shape encoding neurons in the last feature encoding stage. (ii) When a CNN is trained on ImageNet, the majority of shape information is learned during the first few epochs. (iii) A significant amount of shape is encoded in the early layers of CNNs, which can be utilized to extract additional shape information from the network, by combining with shape encodings from deeper layers. (iv) Encoding the shape and class of an object does not imply the useful encoding of localized per-pixel categorical information. All code will be released to reproduce data and results. ## 2 Do CNNs Spend More Learning Capacity on Shape or Texture? With the goal of revealing the characteristics of where, when, and how much shape information is encoded in CNNs, we first aim to quantify the number of dimensions which encode shape in a CNN’s latent representation. This analysis on the latent representations will allow us to determine where the network spends learning capacity on shape, while other methods that focus solely on the network outputs have difficulty measuring the difference in shape information between convolutional layers. Figure 2: Illustration of the techniques used to quantify shape in this paper. (A) Estimating the dimensionality of semantic concepts in latent representations: We stylize each image with five textures to generate image pairs which share the semantic concepts shape (right pair) and texture (left pair). We feed these image pairs (shown is shape) to an encoder, $E(\cdot)$, and calculate the mutual information between the two latent representations, $z^{a}$ and $z^{b}$, to estimate the dimensionality, $\|z_{\text{shape}}\|$. (B) We quantify the shape information encoded in a convolutional neural network by freezing the weights, and then training a small read-out module (i.e., three 3$\times$3 convolutional layers) on the latent representation to predict either a binary or semantic segmentation map. ### 2.1 Estimating shape and texture dimensionality Previous works (Bau et al., 2017; Esser et al., 2020) proposed various mechanisms to reveal the semantic concepts encoded in latent representations of CNNs. To quantify the amount of texture and shape information, we follow the approach of (Esser et al., 2020), where the number of neurons that represent a certain semantic concept is estimated. Given a pretrained CNN encoder, $E(I)=z$, where $z$ is a latent representation, we aim to estimate the dimensionality of the semantic concepts shape and texture within $z$. The main idea is that the mutual information between image pairs, $I^{a}$ and $I^{b}$, which are similar in a semantic concept, will be preserved in a neuron $z_{i}$ only if the neuron encodes that specific semantic concept. Hence, the mutual information between the corresponding neuron pairs, $z_{i}^{a}=E(I^{a})$ and $z_{i}^{b}=E(I^{b})$, can be used to quantify the degree to which a semantic concept is represented by the neuron. A simple and efficient estimate for their mutual information $\operatorname{MI}(z_{i}^{a},z_{i}^{b})$ can be obtained based on the correlation coefficient $\rho_{i}$. Indeed, under the assumption that the marginal distribution of the neuron $z_{i}$ is Gaussian, the correlation coefficient $\rho_{i}$ provides a lower bound on the true mutual information through the following relationship which becomes tight for jointly Gaussian $z_{i}^{a},z_{i}^{b}$ (Kraskov et al., 2004; Foster & Grassberger, 2011). $\operatorname{MI}(z_{i}^{a},z_{i}^{b})\geq-\frac{1}{2}\log(1-\rho_{i}^{2}),\quad\text{where }\rho_{i}=\frac{\text{Cov}\bigl{(}z_{i}^{a},z_{i}^{b}\bigr{)}}{\sqrt{\text{Var}(z_{i}^{a})\;\text{Var}(z_{i}^{b})}}.$ (1) To quantify how well a concept $k$ is represented in terms of the number of neurons $\|z_{k}\|$ that encode the concept, we compute a score for each concept and the relative number of neurons is determined with a softmax over these scores and a baseline score. The latter is given by the number of neurons $\|z\|$, and shape and texture scores are given by the sum of their respective correlation coefficients $\rho_{i}^{\text{shape}}$ and $\rho_{i}^{\text{texture}}$, which are computed according to Eq. 1 with statistics taken over image pairs that are similar in shape and texture, respectively. Note that $k\in\\{1,2\\}$ in our case, and the remaining dimensions not captured in any of the two semantic factors are allocated to the residual semantic factor, which by definition captures all other variability in the latent representation, $z$. Stylized PASCAL VOC 2012 Dataset. Our goal is to estimate the dimensionality of two semantic concepts: (i) shape and (ii) texture, and analyze pixel-wise shape information. Therefore we must generate a dataset that we can sample image pairs which share the semantic factors shape or texture, and have per- pixel object annotations. To accomplish this goal, we create the Stylized PASCAL VOC 2012 (SVOC) dataset. Similar to SIN, we use the AdaIN style transfer algorithm (Huang & Belongie, 2017) to generate stylized images from the PASCAL VOC 2012 dataset (Everingham et al., 2010) with the same settings and hyperparameters as in the original paper (Huang & Belongie, 2017). We choose five random textures from the Describable Textures Dataset (Cimpoi et al., 2014) as the styles and we stylize every PASCAL VOC image with all five of these textures. For a fair comparison with models trained on ImageNet variants, we take only the images from PASCAL VOC which contain a single object. With the SVOC dataset, we can now sample image pairs which are similar in texture, by using two images from different categories but stylized with the same texture (Fig. 2(A) left), or shape, by using the same image stylized with two different textures (Fig. 2(A) right). Table 1: Dimensionality estimation of semantic factors $\|z_{k}\|$ for the stage-5 latent representation. Note that the total dimension of the latent representation, $\|z\|$, is 2048 for all networks, and that the remaining dimensions are allocated to the ‘residual’ factor. (a) ResNet50 compared with BagNets. BagNets have more neurons which encode texture than shape due to their restricted receptive field. (b) Networks with varying levels of shape bias. The number of neurons which encode shape correlates with shape bias. (c) Deeper networks contain more shape encoding neurons. Model \| Factor $\|z_{k}\|$ ---\|--- Shape \| Texture ResNet50 \| 349 \| 692 BagNet33 \| 284 \| 825 BagNet17 \| 278 \| 839 BagNet9 \| 276 \| 841 Training Data \| Factor $\|z_{k}\|$ ---\|--- Shape \| Texture IN \| 349 \| 692 SIN \| 536 \| 477 (SIN+IN)$\rightarrow$IN \| 376 \| 640 Model \| Factor $\|z_{k}\|$ ---\|--- Shape \| Texture ResNet50 \| 349 \| 692 ResNet101 \| 365 \| 667 ResNet152 \| 371 \| 661 ### 2.2 Results We now evaluate the efficacy of the dimensionality estimation method by comparing two networks which differ significantly in their ability to encode shape information. The first is a standard ResNet50 architecture and the second is the recently proposed BagNet (Brendel & Bethge, 2019). BagNets are a modified version of ResNet50 that restrict the height and width of the effective receptive field of the CNN to be a fixed maximum, i.e., either 9, 17, or 33 pixels. This patch-based construction precludes BagNets from classifying images based on extended shape cues. The results of this comparison are presented in Table 1(a) where both the ResNet50 and the BagNet variants are trained on IN. Note that ‘Stage’ refers to a residual block in a ResNet (i.e., there are five stages in ResNet) and all experiments in Table 1 use the stage-5 features. As expected, BagNets have more neurons encoding texture than the ResNet50 and there is a clear correlation between the receptive field of the network and the amount of shape encoded. As the receptive field decreases, the number of neurons encoding texture increases even further, while the number of neurons encoding shape decreases. We now examine whether the ‘shape bias’ metric (Geirhos et al., 2018) correlates with the number of shape encoding neurons. Table 1(b) compares the estimated dimensionality of a ResNet50 trained on ImageNet against networks which are biased towards shape using two different training strategies: (i) training solely on SIN and (ii) training on SIN and IN simultaneously followed by fine-tuning on IN (denoted as (SIN+IN)$\rightarrow$IN, which achieves the best accuracy on ImageNet top-1% out of the three variations (Geirhos et al., 2018)). ResNet50 trained on IN has far more neurons dedicated to encoding texture than shape. There is a large difference when training a ResNet solely on SIN, where it has less neurons which encode texture than shape. When trained and fine-tuned on (SIN+IN) and IN, respectively, there is an increase in the number of neurons which encode shape compared to IN. We consider if there is any pattern in the number of neurons encoding shape or texture as the network depth increases. Table 2: Comparing dimensionality estimations for the texture factor between two methods. Both methods show that more texture neurons are found in representations with smaller receptive fields. Method \| Res152 \| Res101 \| Res50 \| Bag33 \| Bag17 \| Bag9 ---\|---\|---\|---\|---\|---\|--- Net. Diss. \| 433 \| 481 \| 499 \| 592 \| 623 \| 537 Dim. est. \| 661 \| 667 \| 692 \| 825 \| 839 \| 841 As can be seen in Table 1(c), networks have more shape and less texture neurons as depth increases. This may be due to the increase in learning capacity of the deeper networks, as more hierarchical representations allow for the network to learn increasingly complex shapes compared to the shallower networks. Further, deeper networks have stronger long range connections due to a larger effective receptive field potentially resulting in additional shape encoding neurons. The increase in shape understanding could be one of the reasons why deeper networks achieve better performance on various tasks, e.g., image classification. Finally, we assess the consistency between the dimensionality estimation technique and network dissection (Bau et al., 2017), another method which estimates the number of neurons representing different concepts (described in Sec. A.2). Since network dissection cannot estimate shape dimensionalities, the comparison is limited to the texture dimensions shown in Table 2. Except for the case of BagNet9 and a difference in the absolute numbers of neurons (discussed in Sec. A.2), both methods agree about the correlation between texture dimensionality and the receptive field, which provides further evidence that dimensionality estimates quantify the relevance of semantic concepts faithfully. Stage-Wise Analysis of Shape and Texture Dimensionality. We now explore where CNNs encode shape by applying the dimensionality estimation method with the latent representations from ResNet50 stages one to five with different amounts of shape bias. Due to the different dimensions Table 3: Percentage of neurons ($\|z_{k}\|/\|z\|$) encoding different semantic concepts, $k$, for different stages of ResNet50 trained for various levels of shape bias. While a moderate percentage of neurons encode shape in stages $f_{1}$, $f_{2}$, and $f_{3}$, the majority of shape neurons are found in stage $f_{5}$. Networks with shape bias learn additional shape information in stage $f_{5}$. Stage \| IN \| SIN \| (SIN+IN)$\rightarrow$IN ---\|---\|---\|--- Factor $\|z_{k}\|/\|z\|$ \| Factor $\|z_{k}\|/\|z\|$ \| Factor $\|z_{k}\|/\|z\|$ Shape \| Texture \| Shape \| Texture \| Shape \| Texture $f_{1}$ \| 12.5% \| 42.2% \| 12.5% \| 42.2% \| 12.5% \| 42.2% $f_{2}$ \| 14.1% \| 40.2% \| 14.1% \| 40.6% \| 14.1% \| 40.6% $f_{3}$ \| 14.6% \| 39.5% \| 14.8% \| 39.5% \| 14.8% \| 39.5% $f_{4}$ \| 15.3% \| 37.9% \| 17.7% \| 34.7% \| 17.7% \| 34.8% $f_{5}$ \| 17.0% \| 33.8% \| 26.2% \| 23.3% \| 18.4% \| 31.2% at each of these stages, we present the results as the percentage of dimensions encoding the particular semantic factor, $\|z_{k}\|/\|z\|$, where $\|z\|$ refers to length of the latent representation. Table 3 shows that all stages of the network encode shape with an increase in the last two stages. Further, biasing the models towards shape only changes the percentage of shape encoding in the final two stages. Beginning at the fourth stage, there is a significant jump in the number of shape dimensions for all three models with the shape biased models having a larger increase. At the final stage, latent representations encode even more shape, where SIN in particular has a large increase of 8.5%. This indicates that biasing a model towards shape mainly affects the last two stages of the network, suggesting that future work could focus on improving the shape bias of earlier layers. An increase in shape dimensions is inversely proportional to the amount of texture dimensions. Notably, from stage four to stage five, there is a large drop in the amount of texture dimensions for all networks. When Does Shape Become Relevant During Training? To answer the question ‘When do models learn to encode shape and texture during training?’, we capture the changes of shape and texture occurring over the course of training a classifier on ImageNet (IN) and Stylized ImageNet (SIN). We obtain 18 different instances of a ResNet50 model during training on IN and SIN, each representing a checkpoint between epochs 0 and 90 (equally distributed). For each checkpoint, we measure the dimensionality of shape and texture semantic factors and plot the results in Fig. 3. The shape factor in the stage-5 latent representations for both IN (Fig. 3 middle) and SIN (Fig. 3 right) models become increasingly more relevant during the course of training, however the percentage of dimensions grows much larger and faster in the case of the SIN trained model. The texture factor decreases as the training progresses in both cases as well. For the stage-4 representation in a model trained on IN (Fig. 3 left), note that the shape encoding neurons increase only marginally over the course of training. This further reveals that a large proportion of shape information is encoded at the deepest layer. Figure 3: Analyzing the number of dimensions in a ResNet50 which encode shape ($\|z_{shape}\|$) and texture ($\|z_{texture}\|$) over the course of ImageNet (IN, left two) and Stylized ImageNet (SIN, right) training. Dimensions are estimated using stage four, $\|z^{(4)}\|=1024$, and stage five, $\|z^{(5)}\|=2048$, latent representations. When training begins, $z$ is very sensitive to texture but over the course of training learns to focus on the shape instead (faster in SIN case). The vertical lines represent multiplying the learning rate by a factor of 0.1. Note that the estimated dimensions differ slightly from Table 1 as we trained the IN and SIN models used in this figure from scratch. ## 3 How Much Shape Information do CNNs Encode? The previous section measured the dimensionality of shape encodings for various CNNs and settings. We now aim to evaluate the quality of these encodings, and whether more shape encoding neurons implies that more robust shape information can be extracted from these latent representations. We also conduct a set of experiments by targeting the shape and texture-specific neurons (see Sec. 3.3 and Sec. A.1 for results and discussion), revealing an additional link between the two techniques used in our paper (i.e., dimensionality estimation and read-out module). Hermann & Kornblith (2019) measured the quality of shape encodings in a CNN’s latent representations by training a linear classifier on the CNN’s late-stage features to predict the shape label of SIN images. Quantifying shape information by using image level labels does not allow for the per-pixel evaluation of the encoded shape, and its relation to the associated categorical label, two key components for fully evaluating the characteristics of shape information contained in a particular encoding. ### 3.1 Quantifying Shape Information in CNN Latent Representations To overcome the aforementioned issues, we consider two tasks which require a detailed understanding of object shape: binary and semantic segmentation. A ‘shape encoding network’ (SEN), the network being analyzed, consists of a CNN with fixed weights. We then train a shallow read-out module that takes a latent representation from the SEN, to predict a segmentation map (i.e., binary or semantic). If the read-out module can accurately segment objects with a binary mask, we conclude the SEN encodes the precise shape of the objects of interest. Further, the read-out modules ability to perform semantic segmentation, measures how much of this encoded shape is successfully localized with per-pixel categorical information. We use ResNet networks of various depths (i.e., 34, 50, and 101) as SENs with a readout module containing either one or three convolution layers with 3$\times$3 kernels. Table 4: Left: We measure the amount of shape encoded in frozen CNN by training a read-out module on either binary (Bin) or semantic segmentation (Sem) under different training settings. ‘None’: random initialization, ‘End- to-End’: network is not frozen and trained with the read-out module, ‘IN’: pre-trained on ImageNet. Right: Shape information contained in various shape biased models. Training \| 1 Layer Readout \| 3 Layers Readout ---\|---\|--- ResNet34 \| ResNet50 \| ResNet101 \| ResNet34 \| ResNet50 \| ResNet101 Bin \| Sem \| Bin \| Sem \| Bin \| Sem \| Bin \| Sem \| Bin \| Sem \| Bin \| Sem None \| 46.5 \| 5.2 \| 48.0 \| 6.1 \| 44.9 \| 5.1 \| 58.0 \| 7.2 \| 58.0 \| 6.0 \| 55.0 \| 4.8 End-to-End \| 80.2 \| 63.4 \| 80.2 \| 62.7 \| 81.0 \| 65.8 \| 82.1 \| 67.7 \| 82.2 \| 68.1 \| 82.9 \| 71.5 IN \| 66.3 \| 48.1 \| 70.6 \| 50.9 \| 72.1 \| 51.9 \| 78.9 \| 59.1 \| 79.8 \| 61.6 \| 80.4 \| 63.4 Training \| ResNet50 ---\|--- Bin \| Sem IN \| 79.8 \| 61.6 SIN \| 76.4 \| 53.7 (SIN+IN)$\rightarrow$IN \| 77.8 \| 58.0 ### 3.2 Results We use the trainaug and val split of the VOC 2012 dataset to train and test the read-out module, respectively. The binary segmentation ground truth labels are generated by converting all semantic categories to a single ‘object’ class. Note that the binary segmentation and semantic segmentation experiments are done completely independently of one another. Table 4 presents the results in terms of mean-Intersection-over-Union (mIoU) under different initialization settings with; ‘IN’: a SEN trained for ImageNet classification, ‘None’: a SEN with random weight initialization and without any training, ‘End-to-End’: the SEN and readout module trained in an end-to- end manner on either the binary (Bin) or semantic (Sem) segmentation ground truth. The None and End-to-End networks represent lower and upper bounds for encoding shape, respectively. All read-out modules in this section are trained on the last layer’s latent representations. Interestingly, three convolutional layers can extract similar amounts of shape information from the IN-SEN as the End-to-End-SEN. For example, training the ResNet101 End-to-End-SEN for Bin improves the mIoU by merely 2.5% compared to the IN-SEN. ImageNet trained CNNs also contain shape encodings which successfully localize per-pixel categorical information as well, which can be seen when comparing the performance of the IN-SEN and End-to-End SEN, e.g., for ResNet50, the IN-SEN and End-to-End-SEN achieve 61.6% and 68.1%, respectively. This is an interesting result considering the difficulty of semantic segmentation and that none of the IN- SEN weights are trained for pixel-wise objectives. Shape information also increases relative to the depth of the network which supports the results presented in Table 1(c). As expected, the End-to-End-SEN and IN-SEN contain significantly more shape information in their latent representations than the baseline None-SEN. We now evaluate the shape information encoded in networks which have different levels of shape bias. We compare the Bin and Sem performance of the read-out module trained on the features of three different SENs trained on IN, SIN, and (SIN+IN)$\rightarrow$IN. As the validation is on non-stylized images, SIN-SEN has slightly lower performance for Bin, and significantly less performance on Sem. Such a large difference in performance implies that while the boundary of the object is known, it is difficult for the network to correctly assign per- pixel categorical information, a phenomenon further explored in Sec. 3.2.1. Interestingly, the (SIN+IN)$\rightarrow$IN-SEN also has slightly lower performance than the IN-SEN for Bin, but does not suffer in performance as much as the SIN-SEN in the case of Sem. Where is Shape Information Stored? Table 5: Shape encoding results for different stages of ResNet networks trained on ImageNet. Combining features from early stages increases shape encoding. $f_{1}$ \| $f_{2}$ \| $f_{3}$ \| $f_{4}$ \| $f_{5}$ \| ResNet50 \| ResNet101 ---\|---\|---\|---\|---\|---\|--- Bin \| Sem \| Bin \| Sem \| \| \| \| \| 44.7 \| 4.6 \| 42.3 \| 4.7 \| \| \| \| \| 52.7 \| 6.4 \| 53.0 \| 5.6 \| \| \| \| \| 59.6 \| 10.9 \| 57.7 \| 9.4 \| \| \| \| \| 70.8 \| 33.9 \| 73.2 \| 43.6 \| \| \| \| \| 70.6 \| 50.9 \| 72.1 \| 51.9 \| \| \| \| \| 66.0 \| 16.6 \| 63.8 \| 13.5 \| \| \| \| \| 74.5 \| 42.2 \| 76.8 \| 49.8 \| \| \| \| \| 77.3 \| 53.7 \| 78.2 \| 56.2 \| \| \| \| \| 77.3 \| 52.9 \| 78.2 \| 55.2 Figure 4: Stage-wise predictions of read-out module on binary (Bin) and semantic (Sem) segmentation. We now examine if the large amount of shape information contained in ImageNet pretrained models is equally distributed across different stages of the CNN. In this experiment, we train one layer read-out modules on features from different stages, ($f_{1},f_{2},f_{3},f_{4},f_{5}$), of the SEN to examine which stage of a CNN encodes shape information. As shown in Table 5, the read- out module trained on the last stage features, ($f_{4},f_{5}$), achieves higher performance compared to the earlier stage features, ($f_{1},f_{2},f_{3}$), for both Bin and Sem. This is to be expected, as feature maps from later stages have higher channel dimensions and larger effective receptive fields compared to the feature maps extracted from earlier layers. A surprising amount of shape information (i.e., Bin) can be extracted from stages $f_{1},f_{2}$ and $f_{3}$; however, these features lack high-level semantics to correlate with this shape information, which can be observed as the corresponding Sem performance is much lower. Figure 4 reveals this phenomenon; the horse and person are outlined even for the early stage binary masks, but are only labelled with correct per-pixel categorical assignments in the later stages. Considering the non-trivial amount of shape information contained in the early stages, we investigate if aggregating multi-stage features encodes more shape compared to the last stage feature, $f_{5}$. Table 5 (bottom) shows that training a readout module on multi-stage features significantly improves the Bin and Sem performance, suggesting that tasks requiring shape information may benefit from hypercolumn style architectures (Hariharan et al., 2015). This indicates that some shape information is encoded in earlier layers but not captured in the late stages, which agrees with the dimensionality estimation results in Table 3, as around 12.5% and 14% of neurons encode shape in the first stage and second stage, respectively. Figure 5: Quantifying the shape and semantic information encoded by a CNN over the course of ImageNet training. Vertical lines represent the learning rate decay. When do CNNs Encode Shape During ImageNet Training? Now we quantify the amount of shape encoded in the latent representations over the same ImageNet training snapshots as in Sec. 2.1. Fig. 5 shows the performance of the read-out module trained on the frozen SEN every five epochs, as well as the ResNet50’s validation accuracy on ImageNet classification. Note that we train a separate read-out module for every snapshot. Similar to the findings in Fig. 3, we see that the majority of both binary and semantic shape information is learned within the first 10 epochs i.e., $\frac{77.1\%}{79.4\%}=97.1\%$ of the final Bin mIoU and $\frac{52.9\%}{60.0\%}=88.2\%$ of the final Sem mIoU is obtained by the read-out module after only 10 epochs. Contrasting purely shape related information (i.e., Bin), a small but significant portion of per-pixel categorical information is learned after the initial 10 epochs, when the learning rate decay is employed. Table 6: Shape encoding results for shape biased ResNet50s on stylized VOC12 validation set. Training Data \| ResNet50 ---\|--- Bin. \| Sem. IN \| 45.7 \| 8.5 SIN \| 60.3 \| 26.4 (SIN+IN)$\rightarrow$ IN \| 44.5 \| 9.0 Figure 6: Binary and semantic segmentation masks extracted from CNNs trained on ImageNet (IN) and Stylized ImageNet (SIN). #### 3.2.1 Does Knowing an Object’s Shape Imply Knowing Its Semantic Class? We now explore whether a CNN encoding an object’s shape necessarily implies that it also encodes the correct semantic category on a per-pixel level. In other words, for a frozen CNN, can a read-out module (trained for binary segmentation) successfully extract the binary mask while another read-out module (trained for semantic segmentation) cannot successfully extract the semantic segmentation mask? Previous results (e.g., Table 4, Table 5) show that, for certain layers and networks, the binary segmentation performance of a read-out module is much higher relative to the semantic segmentation performance. This suggests that shape information (i.e., the binary mask) and semantic information can be encoded in a mutually exclusive manner, i.e., a CNN can encode the silhouette of the object without encoding the semantic category of each pixel of the silhouette belongs to. To this end, we validate various SENs and their read-out modules on stylized VOC12 val images as this ensures the networks must encode per-pixel semantic information based solely on the object’s shape (note that stylization removes all texture information, see Sec. 2.2). The difference in performance between the Bin mIoU and Sem mIoU can therefore approximate the amount of shape information that is not correlated to its corresponding semantic class. As shown in Table 6, the large difference in performance between Bin and Sem suggests that these SENs capture the shape (i.e., Bin mask) of the object but lack the ability to correctly assign per-pixel semantic labels to these objects. Qualitative results are presented in Fig. 6; note how the binary mask (presented as likelihood heatmaps) for the SIN trained model reasonably segments the objects, while the semantic masks fail to resemble realistic predictions, i.e., multiple object categories are placed spuriously over the object of interest. ### 3.3 Targeting Shape and Texture Neurons 7$10$$20$$50$$100$$40$$60$$80$Top $X$% of Neurons Kept (%)mean IoU (%)Binary Segmentation - IN$10$$20$$50$$100$$20$$40$$50$$60$Percentage of Neurons Kept (%)Semantic Segmentation - IN$10$$20$$50$$100$$40$$60$$80$Percentage of Neurons Kept (%)Binary Segmentation - SIN$10$$20$$50$$100$$20$$40$$50$Percentage of Neurons Kept (%)Semantic Segmentation - SIN Figure 7: Shape encoding results by means of training a read-out module on the latent representations of an ImageNet (i.e., texture-biased model) (left two) and stylized ImageNet (shape-biased model)(right two) trained ResNet-50 for binary and semantic segmentation when removing all but top $X\%$ shape or texture-specific neurons. In Sec. 2, we used a dimensionality estimation technique to estimate the number of dimensions which encode shape and texture in a CNNs latent representations. Given these neurons, we now validate if the most shape- specific, or texture-specific, neurons can influence the performance of a read-out module when keeping these specific neurons during training. We hypothesize that the network trained on Stylized ImageNet (i.e., a shape biased model) will be more reliant on the shape neurons than a network trained on ImageNet (i.e., a texture biased model) which are known to naturally exhibit a texture bias. See Appendix A.1 for additional experiments where we remove the targeted neurons instead of keeping them to further validate if the most shape or texture-specific neurons can influence the performance of a read-out module during inference. To asses this hypothesis, we conduct a series of read-out module experiment and the same settings as Sec. 3 are imposed. However, during this experiment we manipulate the latent representation as an image passes through the ResNet-50, before it is fed through the read-out module. More specifically, we rank the neurons by mutual information for both the shape and texture semantic factors, and then identify the top $X\%$ of neurons from either the shape or texture neurons. Then, we train read-out modules on the latent representations of two frozen ResNet-50s, one trained on ImageNet (IN) and another model trained on Stylized ImageNet (SIN). Before the latent representation is fed through the read-out module, we remove all other neurons except for the top $X\%$ of shape or texture-specific neurons. This forces the read-out modules to learn to perform binary segmentation and semantic segmentation solely from the top $X\%$ of neurons for either semantic factor, and we can identify which neurons are more heavily relied on for each network (i.e., the shape biased or texture biased model). Results. Figure 7 illustrates the binary and semantic segmentation results in terms of mIoU obtained from training read-out modules on IN (left two) and SIN ((right two)) trained ResNet50s, respectively. It is clear that for the model biased towards texture (IN pretrained), keeping texture neurons while removing all other neurons results in a better performance than keeping only the shape neurons. In contrast, Fig. 7 (right two) shows that for shape-biased model (SIN pretrained), keeping shape-specific neurons achieves better performance than keeping only texture-specific neurons. These results support the hypothesis that the network trained on Stylized ImageNet (i.e., a shape biased model) is not only biased towards making predictions based on object shape, but more reliant on shape-specific neurons than a network trained on IN. ## 4 Conclusion In this paper, we presented a systematic study of the capacity and quality of shape encoded in a CNNs latent representations. Approximating the mutual information between stylized PASCAL VOC images allowed us to estimate the dimensionality of the semantic concepts shape and texture (Sec. 2.1). We also designed a simple strategy for determining how much shape information is encoded in these latent representations, by training a read-out module on per- pixel binary segmentation ground truth labels. Additionally, we perform semantic segmentation to quantify how much of this shape encoding can be correctly attributed to per-pixel categorical information. We showed that a model pre-trained on ImageNet has weights that contain almost all the shape and categorical information needed to perform binary or semantic segmentation from the late stage features. We showed that CNNs encode a surprising amount of shape information at all stages of the network, but correctly assigning categorical labels to the corresponding shape only occurs at the last layers of the network, and that removing the image’s texture information severely hurts this correspondence. Finally, we showed how removing all but a certain number of targeted shape or texture-specific neurons affects performance differently depending on the reliance on these neurons. These findings reveal important mechanisms which characterize a network’s ability to encode shape information. We anticipate these findings will be valuable for designing more robust and trustworthy computer vision algorithms. ## Acknowledgements The authors gratefully acknowledge financial support from the Canadian NSERC Discovery Grants, Ontario Graduate Scholarship, and Vector Institute Post- graduate Affiliation award. 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In _CVPR_ , 2020. ## Appendix A Appendix ### A.1 Removing Shape and Texture Neurons In Sec. 3.3, we performed an experiment where we kept the top $X\%$ shape and texture-specific neurons to compare how much different CNNs relied on these neurons to encode shape. We now perform a similar experiment, but instead of keeping the top $X\%$ neurons, we first identify the top $N$ of neurons from either the shape or texture-specific neurons. We then remove these neurons before passing the latent representation to the read-out module by simply setting the features at other dimensions to zero. This allows us to identify which neurons in each network (i.e., the shape biased or texture biased model) are relied on more heavily to encode shape and semantic information. Note that for this experiment, no training occurs. The goal is to simply measure the difference in inference performance when removing $N$ shape-specific neurons, or $N$ texture-specific neurons. Therefore we simply take the trained models and read-out modules from Sec. 3 to perform inference while masking out the targeted neurons. Note that validation is done on the val split from (non- stylized) VOC 2012. Experimental Details. For this experiment, the dimensions sharing the most mutual information with respect to shape and texture are obtained from the same experiments from Sec. 2. We then rank the dimensions for each semantic factor by mutual information. Training and inference are done with the trainug and val split, respectively, from the (non-stylized) PASCAL VOC 2012 (Everingham et al., 2010) dataset. Table 7: Shape encoding results for ResNet50’s trained on ImageNet (IN) and stylized ImageNet (SIN) based read-out modules when the top $N$ shape or texture-specific neurons are removed from the latent representation during inference. Removing the top $N$ shape specific neurons from the SIN-read-out hurts the network’s shape-recognition abilities more compared to the IN-read- out model. $N$ \| Shape \| Texture \| Residual ---\|---\|---\|--- IN \| SIN \| IN \| SIN \| IN \| SIN Bin \| Sem \| Bin \| Sem \| Bin \| Sem \| Bin \| Sem \| Bin \| Sem \| Bin \| Sem 0 \| 70.6 \| 50.9 \| 76.4 \| 53.7 \| 70.6 \| 50.9 \| 76.4 \| 53.7 \| 70.6 \| 50.9 \| 76.4 \| 53.7 100 \| 68.8 \| 46.4 \| 64.6 \| 37.1 \| 69.3 \| 44.7 \| 65.7 \| 39.2 \| 68.3 \| 46.7 \| 64.7 \| 37.3 200 \| 67.9 \| 40.0 \| 57.3 \| 31.4 \| 67.9 \| 39.7 \| 62.5 \| 32.7 \| 66.1 \| 44.2 \| 64.9 \| 34.8 300 \| 61.6 \| 38.0 \| 58.6 \| 25.8 \| 66.3 \| 37.6 \| 55.0 \| 27.9 \| 62.9 \| 40.0 \| 62.3 \| 30.1 #### A.1.1 Results Table 7 presents the binary and semantic segmentation results in terms of mIoU. We report the results under three different settings; (i) top $N$ shape- specific neurons removed, (ii) top $N$ texture-specific neurons removed, and (iii) top $N$ residual neurons removed. Note that for this experiment, we do not train the read-out module. Instead, we first remove the specified neurons, and then run inference using the pretrained IN and SIN models as well as the already trained read-out modules. Interestingly, we find that gradually removing the shape-specific neurons from SIN pretrained model more significantly hurts performance than the IN pretrained model. For instance, removing 100 shape-specific neurons from SIN achieves 37.1% sem mIoU, while the performance dropped to 25.8% sem IoU when the top 300 shape-specific neurons are removed (i.e., an 11.3% drop). When comparing this to the performance drop of the IN trained model, we see that the difference is lower, from 46.4% to 38.0% (i.e., an 8.4% drop). This further supports the hypothesis that SIN trained models are more reliant on the individual shape encoding neurons than the texture encoding neurons. In addition, removing shape- specific neurons from SIN pretrained model hurts performance more than removing the texture neurons. For example, when removing 300 shape neurons for the SIN trained model, the performance drops to 25.8%, while removing 300 texture-specific neurons decreases the performance to only 27.9%. Finally, we observe that removing shape or texture specific neurons hurts performance more than removing the residual neurons. This suggests that shape and texture are the two most important semantic factors for a network to encode and that other semantic factors contained in the residual (e.g., color, lighting) are not as discriminative for the task of image classification. ### A.2 Consistency of Dimensionality Estimation Sec. 2.1 analyzed the consistency between the dimensionality estimate of (Esser et al., 2020) and that of (Bau et al., 2017). In order to quantify interpretability, the latter evaluates the alignment between neurons and semantic concepts using the Broden dataset, which consists of images with pixel-wise labelings of different semantic concepts. For each neuron $z_{i}$, it determines the top quantile level $T_{i}$ such that the activation value of $z_{i}$ exceeds $T_{i}$ only in $0.5\%$ of all observed cases over the dataset, i.e. $p(z_{i}>T_{i})=0.005$. Feature maps are then upsampled to the original image resolution and each neuron is thresholded according to its quantile $T_{i}$ to obtain a binary segmentation mask. A neuron is then termed a detector for a concept, if its segmentation mask has the highest intersection over union score (IoU) for this concept, and the IoU exceeds a threshold of $0.04$. Because the Broden dataset contains no shape concepts, the comparison in Table 2 is limited to estimates on the number of neurons encoding texture, which is determined by the number of detectors for concepts from the categories material, texture and color of the Broden validation dataset. We observe that both methods predict an inverse relationship between the estimated texture dimensionalities and the receptive field, except for the case of BagNet9, where a sudden drop in the number of texture detectors is observed for network dissection. Besides this qualitative agreement, the predicted absolute numbers differ. There are two main sources for the incompatibility in the absolute number of neurons. First, both approaches rely on hyperparameters,u i.e. the baseline score and the choice of the normalization function in the case of dimensionality estimation, and the chosen quantile and IoU threshold in the case of network dissection. Second, the semantic meaning of texture depends on the data, i.e. dimensionality estimation relies on the image pairs of SVOC, whereas network dissection relies on texture images of Broden. This might also explain the drop in detectors for BagNet9 if its receptive field is too small for some of the Broden textures. While absolute numbers depend on hyperparameters, results obtained with both methods are comparable across networks. The dimensionality estimate relies on an estimate of mutual information from samples. This remains a challenging problem, and even powerful variational bounds exhibit either high-bias or high-variance and suffer from sensitivity to batch sizes (Poole et al., 2019). Besides statistical limitations on the ability to accurately estimate mutual information (McAllester & Stratos, 2020), even estimates which give neither upper nor lower bounds or those which give loose bounds are still useful in practice. For example, (Tschannen et al., 2020) demonstrate that loose bounds can lead to better representations when they are learned by mutual information maximization. For dimensionality estimation, potential biases of estimates will cancel out when comparing them between shape and texture neurons, hence an estimate based on the correlation is a suitable and efficient choice for our purposes. ### A.3 Estimating Shape and Texture Dimensionality of Different Networks Trained on Stylized ImageNet We further estimate the dimensionality of shape and texture semantic concepts of different networks in Table 8 to test the consistency of the results reported in Table 1 on different architectures. We run the dimensionality estimation experiment (see Sec. 2.1) on AlexNet (Krizhevsky et al., 2012) and VGG-16 (Simonyan & Zisserman, 2015), trained on IN and SIN. Consistent with the findings for ResNet50, Table 8 shows that training on SIN increases the number of dimensions encoding shape and concurrently decreases the number of dimensions encoding texture: AlexNet-IN: [Shape=729, Texture=1299], AlexNet- SIN: [1119, 870]. VGG-16-IN: [710, 1321], VGG-16-SIN: [1090, 879]. The dimensionality estimation is done on the final representation before the last linear layer for all networks. Table 8: Comparison of shape bias and shape dimensionality for different networks. Network \| IN \| SIN ---\|---\|--- Factor $\|z_{k}\|$ \| Bias \| Factor $\|z_{k}\|$ \| Bias Shape \| Texture \| Shape \| Texture \| Shape \| Texture \| Shape \| Texture ResNet-50 \| 14.1% \| 40.2% \| 22.1% \| 77.9% \| 26.2% \| 23.3% \| 81.0% \| 19.0% AlexNet \| 18.0% \| 30.6% \| 42.9% \| 57.1% \| 26.0% \| 21.5% \| 75.5% \| 24.5% VGG-16 \| 15.3% \| 37.9% \| 17.2% \| 82.8% \| 26.6% \| 21.5% \| 77.4% \| 22.6% ### A.4 Estimating Shape and Texture Dimensionality of Self-Attention Networks We also experiment with the recently proposed Self-Attention Networks (Zhao et al., 2020), which replace convolutional layers with self-attention layers. Three different depths of Self-Attention Networks (SANs) were proposed. For fair comparison, we experiment with SAN-19, which the authors claim is the most similar size to ResNet50, in terms of the number of parameters in the network. Additional, SANs come with two types of layer operations, patch-based and pair-based. The patch-based SAN compares patches of pixels within the attention operations, while the pair-based SAN compares individual pixels and achieves lower performance on ImageNet. Due to the lower effective receptive field of the pair-based SAN compared to the patch-based SAN, we expect to see a larger number of neurons encoding shape in the patch-based SAN. The results are shown in Table 9. The patch-based SAN19 has the largest number of shape encoding dimensions and lowest number of texture encoding dimensions, when compared to the pair-based SAN19 and ResNet50. Table 9: Comparing the number of shape encoding neurons and texture encoding neurons for self-attention networks (Zhao et al., 2020). Model \| Factor $\|z_{k}\|$ \| Factor $\|z_{k}\|/\|z\|$ ---\|---\|--- Shape \| Texture \| Shape \| Texture ResNet50 \| 349 \| 692 \| 17.0% \| 33.8% SAN-19 (patch) \| 384 \| 610 \| 18.8% \| 29.8% SAN-19 (pair) \| 304 \| 764 \| 14.9% \| 37.3% ### A.5 Layer-Wise Dimensionality Estimation on AlexNet We now explore where another CNN encodes shape and texture at each layer of the network. More specifically, we apply the dimensionality estimation technique from Sec. 2.1 on AlexNet (Krizhevsky et al., 2012) on a number of different layers. Due to the different dimensions Table 10: Percentage of neurons ($\|z_{k}\|/\|z\|$) encoding different semantic concepts, $k$, for different stages of AlexNet (Krizhevsky et al., 2012) trained for various levels of shape bias. Stage \| IN \| SIN ---\|---\|--- Factor $\|z_{k}\|/\|z\|$ \| Factor $\|z_{k}\|/\|z\|$ Shape \| Texture \| Shape \| Texture conv1 \| 17.0% \| 35.0% \| 16.6% \| 35.3% pool1 \| 19.0% \| 31.7% \| 18.8% \| 31.8% conv2 \| 20.7% \| 27.2% \| 21.1% \| 27.7% pool2 \| 20.7% \| 26.1% \| 21.1% \| 25.9% conv3 \| 20.6% \| 27.0% \| 23.9% \| 23.2% conv4 \| 21.2% \| 25.8% \| 25.4% \| 21.7% conv5 \| 21.3% \| 24.5% \| 25.2% \| 21.0% pool3 \| 21.7% \| 23.8% \| 25.4% \| 20.9% fc6 \| 18.8% \| 28.7% \| 24.5% \| 21.9% fc7 \| 18.0% \| 30.6% \| 26.0% \| 21.5% at each of these stages, we present the results as the percentage of dimensions encoding the particular semantic factor, $\|z_{k}\|/\|z\|$, where $\|z\|$ refers to length of the latent representation. The results are presented in Table 10 and Fig. 8. Note that the output from the convolutional layers also include the ReLU activation function. 8 conv1pool1conv2pool2conv3conv4conv5pool3fc6fc7$15$$20$$25$$30$LayerDimensions (%)Shapeconv1pool1conv2pool2conv3conv4conv5pool3fc6fc7$20$$25$$30$$35$$40$LayerTexture Figure 8: Shape (left) and texture (right) encoding dimensions estimated on each layer of AlexNet (Krizhevsky et al., 2012). Shape biased AlexNet trained on Stylized ImageNet (Geirhos et al., 2018) encode more shape at the later layers of the network which is consistent with the findings for ResNets (He et al., 2016).
# Reduced Order and Surrogate Models for Gravitational Waves Manuel Tiglio111email<EMAIL_ADDRESS>Aarón Villanueva222email: <EMAIL_ADDRESS> (Facultad de Matemática, Astronomía, Física y Computación, Universidad Nacional de Córdoba, Córdoba (5000), Argentina ) ###### Abstract We present an introduction to some of the state of the art in reduced order and surrogate modeling in gravitational wave (GW) science. Approaches that we cover include Principal Component Analysis, Proper Orthogonal Decomposition, the Reduced Basis approach, the Empirical Interpolation Method, Reduced Order Quadratures, and Compressed Likelihood evaluations. We divide the review into three parts: representation/compression of known data, predictive models, and data analysis. The targeted audience is that one of practitioners in GW science, a field in which building predictive models and data analysis tools that are both accurate and fast to evaluate, especially when dealing with large amounts of data and intensive computations, are necessary yet can be challenging. As such, practical presentations and, sometimes, heuristic approaches are here preferred over rigor when the latter is not available. This review aims to be self-contained, within reasonable page limits, with little previous knowledge (at the undergraduate level) requirements in mathematics, scientific computing, and other disciplines. Emphasis is placed on optimality, as well as the curse of dimensionality and approaches that might have the promise of beating it. We also review most of the state of the art of GW surrogates. Some numerical algorithms, conditioning details, scalability, parallelization and other practical points are discussed. The approaches presented are to large extent non-intrusive and data-driven and can therefore be applicable to other disciplines. We close with open challenges in high dimension surrogates, which are not unique to GW science. ###### Contents 1. 1 Introduction 2. 2 Reduced Order Modeling and Gravitational Waves 3. 3 Mathematical Preliminaries 4. I Representation and Compression 1. 4 Principal Component Analysis 2. 5 Proper Orthogonal Decomposition 1. 5.1 PCA and POD 3. 6 Spectral expansions 1. 6.1 Spectral methods 1. 6.1.1 Fourier Expansions 2. 6.1.2 Jacobi Polynomials 4. 7 Parametrized problems and optimal approximations 5. 8 Reduced Basis 1. 8.1 Introduction 2. 8.2 The Training set 3. 8.3 Greedy algorithms 1. 8.3.1 Convergence rates and near-optimality 2. 8.3.2 Complexity, scaling, computational aspects 5. II Predictive Models 1. 9 Polynomial interpolation 1. 9.1 Representation versus prediction 2. 9.2 Prediction through polynomial interpolation 3. 9.3 Convergence rates, collocation points and Runge’s phenomenon 4. 9.4 Discrete expansions and interpolation 5. 9.5 Multiple dimensions 2. 10 The Empirical Interpolation Method 1. 10.1 From projection to interpolation 2. 10.2 EIM algorithm 3. 10.3 Accuracy and conditioning of the EIM 3. 11 Surrogate models 1. 11.1 Surrogate models for components 2. 11.2 Empirical interpolant based surrogates 4. 12 Surrogates of compact binaries 1. 12.1 Numerical relativity binary black holes 2. 12.2 Numerical relativity hybrid binary black holes 3. 12.3 Extreme mass ratio inspirals 1. 12.3.1 Eccentric inspirals 4. 12.4 Effective One Body 5. 12.5 Post-Newtonian (PN) 6. 12.6 Ringdown 6. III Data Analysis 1. 13 Reduced Order Quadratures 1. 13.1 ROQ, other quadrature methods, dependence on dimensionality 2. 14 Accelerating Parameter Estimation with ROQ 1. 14.1 Constructing the Reduced Order Quadrature 2. 14.2 Implementation in LALInference 3. 15 Challenges, open issues and comments 4. 16 Acknowledgments #### Notation * • $h_{\lambda}$: a function (for the purposes of this article, a GW waveform) associated with parameter value $\lambda$. It can be, for example, a time series $h_{\lambda}=h_{\lambda}(t)$, or a frequency domain function $h_{\lambda}=h_{\lambda}(f)$. * • $\lambda$: a (usually multi-dimensional) parameter for $h_{\lambda}$. * • $\tt dim$: number of parameters of $\lambda$. * • $L$: number of time/frequency samples. * • ${\cal F}$: an abstract space of functions of interest. Typically, ${\cal F}:=\\{h_{\lambda}\,\|\,\lambda\in\Phi\\}$, with $\lambda$ in some compact region (continuous or discrete) $\Phi$. We also refer to ${\cal F}$ as the fiducial or underlying model. * • $N$: size of the training set. Also, the number of points in standard quadratures, polynomial interpolation, etc. * • ${\cal T}$: training set of parameter points ${\cal T}:=\\{\lambda_{i}\\}_{i=1}^{N}$. It is assumed to be compact. * • ${\cal K}$: training set of functions in ${\cal F}$, ${\cal K}:=\\{f_{i}\\}_{i=1}^{N}=\\{f(\lambda_{i},\cdot)\\}_{i=1}^{N}\subset{\cal F}$. * • RB: Reduced Basis as a framework. * • $\Lambda_{i}$: selected (typically, by a greedy algorithm) parameter values. * • rb: a specific reduced basis. * • $t,f$: time, frequency domains. * • EIM: The Empirical Interpolation Method. * • $T,F$: selected (typically, by the EIM) time/frequency interpolation nodes. * • $n$: number of basis elements and of EIM nodes (they are equal by construction). The goal of ROM, if the problem is amenable to dimensional reduction, is to find an accurate basis such that $n\ll N$. * • $\Lambda_{n}$: the Lebesgue constant for an approximation with $n$ basis elements/interpolation points. This involves some ambiguity: we use the same symbol for greedy selected points, but it should be clear from context which one we refer to. * • $h_{\tt s}$: a surrogate function for $h$. * • Boldfaces are used for matrices, for example $\bf A$ has elements $A_{ij}$. * • $\langle a,b\rangle$: scalar product between two vectors or functions $a,b$. ### 1 Introduction Gravitational wave (GW) science has reached a level of maturity in which many tools from areas such as modern approximation theory, data science, machine learning, and artificial intelligence are being incorporated into the field. These attempt to address challenges such as dealing with complex modeling, analysis, and handling of big data. A common feature of these challenges is the computational cost involved, which in many cases can be prohibitive, to the point that it cannot be solely overcome with larger or faster (super)computers, specialized hardware such as GPUs, or software optimization. This is particularly the case for parametrized problems, where each query depends on multiple input parameters that might only be known at run time. This is exacerbated as the number of parameters grow, usually resulting in the curse of dimensionality. This refers to the complexity of the problem (here leaving the term complexity ambiguous on purpose) growing fast, sometimes exponentially, with the number of parameters. In the case of gravitational waves from binary systems, parameters can be intrinsic or extrinsic. The former relate to parameters such as the mass and spin of the binary components, the initial separation and eccentricity of the system, and equations of state if matter is present. Extrinsic parameters include distance of the source to Earth, sky position, orientation, time and phase of arrival. One of the first challenges of a parametrized problem is sampling it. With standard methods (for example, equally spaced, using the metric approach in GWs, or stochastic sampling) the accuracy of such catalogs for GW detection increases in many cases at best linearly with the number of samples, and their sizes typically increase exponentially with the number of parameter dimensions dim as ${\cal O}(N^{\tt dim})$. In fact, for the metric approach, the number of templates grows as $(1-\text{MM})^{-\text{dim}/2}$, with MM the minimal match. Producing such catalogs, as well as their storage and analysis, can become challenging and, again, even not tractable through raw computational power and storage. One approach to this problem is decreased fidelity: the modeling of the problem is simplified by approximations to Einstein’s General Relativity equations which are cheaper to solve for and analyze. Decreased fidelity is a delicate approach, though, since the accuracy of the approximations might not be known without access to the high fidelity models for an arbitrary query, which if not available could become an issue when attempting to assign error bars and a given precision needs to be guaranteed for statistical purposes. Decreased fidelity models can also lead to missed signals or biases in parameter estimation of detected GWs. Thus, another challenge is to simulate the GWs emitted by, for example, the coalescence of compact binary objects in real time, without any physical approximation. Here the gold standard is high accuracy numerical relativity (NR) solutions to the full Einstein field equations. It might appear unattainable to replace online (that is, not evaluated in advance), for an arbitrary query, a supercomputer simulation which might take per query hundreds to tens of thousands of hours of computing time with a substitute or surrogate model of equal accuracy but which can be evaluated in real time on a commodity laptop. As we show throughout this review, this can be – and is being – done, though there are still outstanding challenges left and the problem is not completely solved. Finally, another challenge of equal importance is to perform parameter estimation on the sources of any detected GW signal in quasi real time. Meaning, fast enough so as to process the large number of detections by modern GW laser interferometers. And, most important, in the case of sources with electromagnetic counterparts, fast enough to allow for rapid telescope followups. This requires both online fast evaluation of the GW waveforms, as well as rapid likelihood evaluations. In this review we discuss some state of the art approaches which attempt to (and in some, but not all, cases manage to) obtain accurate and fast-to- evaluate and analyze surrogate models of gravitational waves emitted by binary systems, accomplishing some of the aforementioned challenges. The common aspect underlying all these methods is reduced order modeling (ROM). The approaches here reviewed are Principal Component Analysis (PCA), Singular Value Decompositions (SVD), Reduced Basis (RB), the Empirical Interpolation Method (EIM), and derivatives, such as Reduced Order Quadratures (ROQ). The goal is intendedly not to be a definite survey, since it is a very active field. Instead, we attempt to provide an introduction to some of the approaches that are being used in practice and do deliver on some or several of the above challenges, along with some basic theory underlying each of them. The review is divided into three parts, each of which builds upon the previous one: 1. 1. Representation One example is the generation of compact catalogs or banks of gravitational waves for searches. Another one is to analyze a system and look for redundancies. 2. 2. Predictive Models The most ambitious goal here is to build surrogate models that can be evaluated in real time and are indistinguishable from numerical relativity supercomputer simulations of the Einstein equations, without any physical approximation. The target is the evaluation of a waveform in the order of milliseconds per angular mode; that is, a speed up of at least $\sim 10^{8}$ with respect to NR and without loss of accuracy. 3. 3. Data Analysis One of the main goals of ROM and other efforts in GW science is to achieve very fast parameter estimations, in particular so that real time alerts can be sent for searches of electromagnetic counterparts. From an astrophysical point of view, the target is from months using standard methods to around $10$ minutes using Focused Reduced Order Quadratures (discussed in Section 14), including millions to tens of millions waveform evaluations and likelihood evaluations. We encourage the reader to provide us with feedback, including topics to cover in future versions of this Living Review article. For briefness we have to skip many references, we apologize in advance for any and all omissions in such an active field. ### 2 Reduced Order Modeling and Gravitational Waves This article reviews approaches which attempt to solve many of the aforementioned problems in GW science through Reduced Order Modeling (ROM), also known as Dimensional or Complexity Reduction, and the related field of Surrogate Modeling. ROM as a field has been around for a long time, but over the last decade and a half there have been major theoretical advances followed by a rapid raise in the number of applications and pace at which they both take place, in many areas of science, engineering and technology. This is in part, again, due to powerful new approaches and results, from approximation theory to numerical analysis and scientific computing, but also due to the recognition of the power of dimensional reduction in many important problems, some of them long-standing. That is, problems which are only now being seen in the light of the latest developments of ROM. On top of that, fields such as data science (DS), Machine Learning (ML) and Artificial Intelligence (AI) are considerably benefiting from ROM to either eliminate redundancies in big data, making them more amenable to analysis, and/or identifying relevant features for further studies using techniques from these disciplines. In fact, depending on the definitions of these fields, some include ROM as a subdiscipline. There is a big difference, though. There are many books on established DS, ML, and AI practices and theory, while the literature on modern approaches to ROM for parametrized problems, with some important exceptions that we mention in this review, is largely composed of technical papers, which could be difficult to grasp for practitioners as introductory material. Also, these notable exceptions usually focus on time-independent partial differential equations (PDEs), and even as introductory material they might be hard to absorb by non- mathematicians. In contrast, this review covers approaches which are purely data-driven and attempts to be amenable as an introduction for GW science practitioners, keeping theory to the minimum necessary to build intuition on why a given approach works as it does, what are the challenges left, and possible approaches to them. One of the reasons why ROM for parametrized system has been largely devoted to time-independent PDEs is because in that case many rigorous results, such as a priori error bounds on the reduced model, can be proved; these are called certified approaches. Here we move beyond that particular arena to ROM for general parametrized systems. This includes problems which might not involve any differential equation, be time-dependent, to purely data-driven problems such as analysis and handling of large amounts of data. This broadening is at the expense, in many cases, of more heuristics and less rigor, such as a posteriori validation as opposed to rigorous a priori error bounds. The rationale for this is simply that many problems of interest in GW science are far too complex for existing detailed rigorous theorems. On the other hand, GW problems share many similarities with others for which approaches with proven properties have been developed and therefore certain algorithms can be adapted to cases of interest in GWs. In other instances it is quite the opposite: available rigorous results are quite generic, and can be abstracted from any previous application in which they were introduced. Half-rigorous, half-heuristic approaches are not as pessimistic as it might seem. Many widely used techniques in DS, ML and AI are somewhat heuristic in nature and require a posteriori validation. Similarly with half-rigorous, half-heuristic cutting-edge ROM approaches as those discussed here: they can (and should) always be a posteriori validated. Our presentation keeps technicalities and analyses to the minimum necessary for building intuition of why a technique works as it does, under which conditions it does so, and what can be expected given the properties of the problem of interest. At the same time, it is not a survey of all the work done in the field. There are two important books about Reduced Basis; though focused on partial differential equations there is a lot of valuable information (well beyond the scope of this short introduction): [70] and [111]. A website for ROM with several resources is [9], while [11] has the speakers slides from a workshop on ROM in General Relativity; even though the latter is several years old and there has been much progress since then, the topics covered are still highly relevant. A very recent and lengthier, year-long program at ICERM (Brown University) focused on ROM in GR can be found on the website [6], including many of the slides and talks of each workshop. ### 3 Mathematical Preliminaries This section serves as a brief recap of some basic mathematics and to introduce some notation used hereon. Since in this review we deal with model reduction for parametrized problems, we next consider an example amenable to it. Example 1. Consider the model $h(t;\lambda)=\exp(\lambda t)\,.$ (1) where $\lambda$ is a complex parameter and $\operatorname{Im}({\lambda})>0$. This represents oscillatory, exponentially damped functions; it can be seen as a toy model for the ringdown of a black hole (discussed in Section 12.6). In the notation that we use throughout this review, $h$ is a function of the physical variable $t$ parametrized by $\omega$ and for compactness we often use the following type of shortcuts: $h=h_{\lambda}=h(\lambda)=h(t;\lambda)\,.$ Suppose now that we have a set of $N$ samples of $\lambda$, leading to a training set ${\cal K}=\\{h_{i}\\}=\\{h_{\lambda_{i}}\\}=\\{h(\lambda_{i})\\}=\\{h(t;\lambda_{i})\\}\,,\quad i=1\ldots N.$ Since the functions of interest (1) are known in closed-form, building such a training set is straightforward, though in general this is not the case (as when solving the Einstein equations is required). One of the goals of ROM is to find a reduced basis, that is, a subset of ${\cal K}$ with number of elements $n\leq N$ – with, hopefully, $n\ll N$ –, such that its span (the span of a set of vectors is its set of linear combinations) represents ${\cal K}$ with arbitrarily high accuracy. The compression rate is then $C_{r}:=N/n\,.$ Furthermore, in practice one also needs to discretize time (or frequency). So let’s sample these training set elements at an arbitrary set of times $\\{t_{i}\\}_{i=1}^{L}$, where $L$ stands for Length. Using, for example, the Empirical Interpolation Method (EIM, discussed in Section 10), one can subsample these time samples to a subset $l\leq L$. In fact, in EIM, if the number of reduced basis is $n$, then by construction $l=n$. That is, the initial set being of size $N\times L$ can now be reduced to $n^{2}$, with a double compression rate (both in parameter and time domains), $C_{r}:=\frac{N\times L}{n^{2}}\,.$ #### Inner products and norms for functions or vectors Let $\Omega$ denote the physical domain. Throughout this review it represents time or frequency interval, $[t_{\rm min},t_{\rm max}]$ or $[f_{\rm min},f_{\rm max}]$ respectively, though in general it could be space, space- time, or some more abstract arena. For the sake of discussion we consider time intervals. For any two complex-valued functions, $h(t)$ and $g(t)$, we consider inner/scalar/dot products and their corresponding norms of the form $\langle h_{1},h_{2}\rangle:=\int_{\Omega}\bar{h}_{1}(t)h_{2}(t)\omega(x)dx,\quad\\|f\\|^{2}:=\langle f,f\rangle\,,$ (2) where the bar over $h_{1}$ denotes complex conjugation and $\omega(x)$ is a generic weight function. These are referred to as weighted $L_{2}$ scalar products. In data analysis/signal processing and usually in the frequency domain, $S:=\omega^{-1}$ characterizes the sensitivity of the detector and is referred to as the power spectral density (PSD). We consider also discrete inner products and norms of the form $\langle h_{1},h_{2}\rangle:=\sum_{i=1}^{L}\bar{h}_{1}(t_{i})h_{2}(t_{i})\omega_{i}\,\quad\\|h\\|^{2}:=\langle h,h\rangle\,.$ (3) This discrete version of an $L_{2}$ scalar product is usually denoted as $\ell_{2}$, with a lowercase to distinguish it from the continuum case. Whenever (3) is the discrete approximation of an integral (2) the values $t_{i}\in\Omega$ and $\omega_{i}$ are respectively referred to as quadrature nodes and weights. Together, $\\{t_{i},\omega_{i}\\}_{i=1}^{L}$ is called a quadrature rule. When $\omega_{i}=1$, (3) is referred to as the Euclidean scalar product. Throughout this review we also use the infinity, or max, norm $\\|h\\|_{\infty}:=\max_{1\leq i\leq L}\|h(t_{i})\|\,.$ of a vector $f\in\mathbb{C}^{L}$. This norm is not induced by an inner product. That is, it can be shown that there is no scalar product $\langle\cdot,\cdot\rangle$ for which $\\|h\\|_{\infty}=\langle h,h\rangle$. Formally speaking, (2) is defined for functions while (3) is for vectors. We use the same notation $\langle\cdot,\cdot\rangle$ for both cases; with the hope that the distinction is clear from the context. Some proofs are sometimes more convenient in a continuous setting while numerical computations are restricted to discrete values. Depending on the context, we sometimes switch between these two settings using the same notations for both of them. Example 2. Polynomials. Consider the space of degree $n$ polynomials defined on $\Omega=[-1,1]$. Any element $f(t)=\sum_{i=0}^{n}c_{i}t^{i}\,,$ (4) can be written as a sum of $(n+1)$ terms and such space is a linear one of dimension $(n+1)$. The first $(n+1)$ normalized Legendre polynomials $\\{P_{i}(t)\\}_{i=0}^{n}$ form an orthonormal basis with respect to the scalar product (2) with $\omega(t)\equiv 1$. #### Matrices Consider a complex $L\times N$ matrix $\textbf{H}\in\mathbb{C}^{L\times N}$, where $L$ is the number of rows (“length of the time series”) of H and $N$ is the number of columns (“number of parameter samples”), ${\bf H}:=\begin{pmatrix}h_{11}&h_{12}&\cdots&h_{1N}\\\ h_{21}&h_{22}&\cdots&h_{2N}\\\ \vdots&\vdots&\ddots&\vdots\\\ h_{L1}&h_{L2}&\cdots&h_{LN}\end{pmatrix}\,.$ (5) As one might imagine, for gravitational waves $h_{ij}=h_{j}(t_{i})$. Each column is referred to as a snapshot. A matrix ${\bf H}$ of shape $L\times N$ appears often throughout this review, where $N$ is the number of samples in the training set (or $n$, the dimensionality of the reduced basis) and $L$ is the length of each time series, so it is a very concrete example. A square matrix ${\bf H}$ is said to be non-singular if its inverse – denoted by ${\bf H}^{-1}$ – exists, i.e. ${\bf H}{\bf H}^{-1}={\bf H}^{-1}{\bf H}={\bf I}$, where ${\bf I}=\begin{pmatrix}1&0&\cdots&0\\\ 0&1&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\ 0&0&\cdots&1\end{pmatrix}=:diag(1,1,\ldots,1)\,.$ The (Hermitian) transpose of ${\bf H}$, denoted by $H^{\dagger}$ is defined as the matrix with elements $\left({\bf H}\right)^{\dagger}_{ij}=\bar{{\bf H}}_{ji}\,,$ the bar again indicates complex conjugation, and we say that ${\bf H}$ is symmetric (hermitian or self-adjoint, in the current context), if ${\bf H}={\bf H}^{\dagger}$. Eigenvectors and eigenvalues: A vector $x\neq 0$ is an eigenvector of ${\bf H}$ with eigenvalue $\alpha$ if ${\bf H}x=\alpha x\,.$ Eigenvectors are defined up to a normalization constant; that is, if $x$ is an eigenvector with eigenvalue $\alpha$ so is $ax$, for any non-zero $a$. ##### Matrix Norms The Frobenius norm. Imagine “unpacking” ${\bf H}$ into a long vector of size $L\times N$. Measuring this vector with the Euclidean norm defines the Frobenius norm $\\|{{\bf H}}\\|^{2}_{F}:=\sum_{i}^{L}\sum_{j}^{N}\,\left\|h_{ij}\right\|^{2}$ (6) of a matrix ${\bf H}$. Later we will see that the Frobenius norm plays a key role when discussing approximation by proper orthogonal decompositions. Induced norm. The matrix can be viewed as a linear operator mapping vectors from $\mathbb{C}^{N}$ to $\mathbb{C}^{L}$. Given norms for both $\mathbb{C}^{L}$ and $\mathbb{C}^{N}$, which need not be the same, the induced norm of a matrix is defined as $\\|{{\bf H}}\\|=\max_{\\|x\\|=1}\\|{{\bf H}}x\\|\,.$ (7) This norm characterizes the maximal possible “amplification” from the application of ${\bf H}$ to $x$. More precisely: the value of $\\|{{\bf H}}\\|$ is the smallest positive number $c$ such that $\\|{{\bf H}}x\\|\leq c\\|x\\|\,.$ (8) Notice that, following standard practice, we use the same notation – namely, $\|\|\cdot\|\|$ – for both the norm of a vector ($\\|x\\|$ and $\\|{\bf H}x\\|$ in the above definition) and a matrix ($\\|{\bf H}\\|$). It should be clear from the context, though, which one we are referring to. ##### Rank and Kernel The range of a complex $L\times N$ matrix ${\bf H}$ is $\mbox{range}({\bf H})=\\{y\in\mathbb{C}^{L}\|\,y={\bf H}x\;\;\mbox{for some }x\in\mathbb{C}^{N}\\}\,,$ its rank $\mbox{rank}({\bf H})=\dim\left(\mbox{range}({\bf H})\right)\,,$ its kernel $\mbox{ker}({\bf H})=\\{x\in\mathbb{C}^{L}\|\,{\bf H}x=0\\}\,,$ and its nullity $\text{nul}({\bf H})=\text{dim}(\text{ker}({\bf H}))\,.$ As a linear algebra exercise it can be shown that $\text{rank}({\bf H})+\text{nul}({\bf H})=N\,.$ For square $N\times N$ matrices, the following properties are equivalent: 1. 1. ${\bf A}$ is non-singular 2. 2. $\mbox{det}({\bf A})\neq 0$ 3. 3. $\mbox{ker}({\bf A})=\\{{\bf 0}\in\mathbb{C}^{N}\\}$ 4. 4. $\mbox{rank}({\bf A})=N$ 5. 5. ${\bf A}$ has linearly independent columns and vectors. #### Approximation by projection ##### The least squares problem It is rather easy to approximate one function by other functions, such as Fourier, wavelets, polynomial expansions, or somewhat physically inspired bases. In this article we focus on representing gravitational waves by themselves. Intuitively, this should be (and is) more efficient. In this subsection we briefly introduce projection-based approximations, with more details given in Section 6. To motivate the problem consider an $n$-dimensional vector space $W_{n}$ which is itself a subspace of a Hilbert’s one ${\cal H}$ (for the purposes of this article, this means that a scalar product is assumed to exist). A common approximation criteria is a least squares (LS) one. That is, one seeks to approximate $h\in{\cal H}$ by $h^{(n)}\in W_{n}$ which is the solution to $h^{(n)}={\tt argmin}_{\tilde{h}}\\|h-\tilde{h}\\|^{2}\,,$ (9) where $\tilde{h}\in W_{n}$. Equation (9) means finding the element of $W_{n}$ that minimizes the squared norm of the representation error. If $\\{e_{i}\\}_{i=1}^{n}$ is an orthonormal basis of $W_{n}$ and $\langle.,.\rangle$ the scalar product associated with ${\cal H}$, the following is the (unique) solution to the LS problem, $h^{(n)}=\sum_{i=1}^{n}c_{i}e_{i}\,,\quad\text{with}\;c_{i}=\langle e_{i},h\rangle.$ (10) The solution $h^{(n)}$ is the orthogonal projection of $h$ onto $W_{n}$, therefore denoted as ${\cal P}_{n}h:=h^{(n)}\,.$ What it means is that if $h\in W_{n}$ then ${\cal P}_{n}h=h\,,$ (11) and the residual $\delta h={\cal P}_{n}h-h$ satisfies $\langle{\cal P}_{n}h,\delta h\rangle=0\,.$ (12) The solution in 10 is basis-independent and $h^{(n)}$ is uniquely defined. The orthogonal projection onto any linear space is a geometric construction, so it is independent of the basis used to represent it: If $\\{e_{i}\\}_{i=1}^{n}$ and $\\{\tilde{e}_{i}\\}_{i=1}^{n}$ are any two orthonormal bases, and the projection coefficients $c_{i}$ and $\tilde{c}_{i}$ are computed according to (10) (replacing in the second case $e_{j}$ by $\tilde{e}_{j}$), then ${\cal P}_{n}h=\sum_{i=1}^{n}c_{i}e_{i}=\sum_{i=1}^{n}\tilde{c}_{i}\tilde{e}_{i}\,.$ $\sum_{i=1}^{n}\langle e_{i},v\rangle e_{i}=\sum_{i=1}^{n}\langle\tilde{e}_{i},v\rangle\tilde{e}_{i}\,,$ (13) leading to the same solution of the LS problem. One might ask why the emphasis on orthogonal or orthonormal bases; it is due to a conditioning issue which we discuss this in Section 6. For the time being the summary is that one should always use orthogonal or orthonormal bases. An exception is when building predictive models, as discussed in Section 11. ##### Representing a set of elements: collective error Suppose we seek to approximate not one function but a set of them: ${\cal F}=\\{h_{1},h_{2},\dots,h_{N}\|h_{i}\in{\cal H}\\}$ (14) From the above discussion in Section 3 we know that ${\cal P}_{n}h_{i}$ is the the best representation of any particular $h_{i}$. The question of how one measures the approximation error of the collective set naturally arises. Two reasonable and standard criteria are $\frac{1}{N}\sum_{i=1}^{N}\\|h_{i}-{\cal P}_{n}h_{i}\\|^{2}\,,$ (15) and $\max_{i}\\|h_{i}-{\cal P}_{n}h_{i}\\|^{2}\,,$ (16) which have the interpretations of the mean and maximum errors, respectively. Whether the notion of error ${\rm(1)}$ or ${\rm(2)}$ is chosen for an error minimization criteria will lead to the main two ROM approaches discussed in this review. Namely, Proper Orthogonal/Singular Value Decompositions, or the Reduced Basis-greedy approach. #### Further reading The material discussed can be found in standard linear algebra/numerical analysis books. There are many introductory good ones, we suggest [129] and [130]. ## Part I Representation and Compression ### 4 Principal Component Analysis Principal Component Analysis (PCA) is perhaps one of the most used tools when seeking for redundancy or a hierarchy of importance of variables in statistical analysis. It has a close relationship with Proper Orthogonal Decomposition (POD) as we discuss in Section 5.1. With this approach one seeks to determine the most statistically relevant variables and to potentially dimensionally discard from the problem the least relevant ones. In order to do this, we recall the definition of the covariance between two stochastic variables $X,Y$ as given by $\displaystyle{\rm Cov}(X,Y)$ $\displaystyle=\big{\langle}(X-\langle X\rangle)(Y-\langle Y\rangle)\big{\rangle}$ $\displaystyle=\langle XY\rangle-\langle X\rangle\langle Y\rangle\,,\,$ (17) where the brackets denote expectation values. The covariance between two variables provides a measure of the degree to which their fluctuations are correlated. A smaller (larger) covariance implies lower (higher) correlation. In particular, the covariance of a variable with itself is its variance (i.e., the standard deviation squared) and measures deviations from the mean value. When there are multiple stochastic variables $X_{i}$ ($i=1,\ldots,n$) one can construct their associated covariance matrix $\bf{C}$ with components $C_{ij}={\rm Cov}(X_{i},X_{j})$. This matrix is symmetric, non-negative definite and can therefore be diagonalized with an orthogonal transformation. Consider the $i^{\rm th}$ normalized eigenvector $\hat{{\bf V}}_{i}$. If we set ${\vec{\bf X}}:=(X_{1},\ldots,X_{n})$ then the principal components (PCs) are the associated eigenmodes, ${\cal E}_{i}={\vec{\bf X}}\cdot\hat{\bf V}_{i}\,.$ (18) The PCs are new variables representing directions in which the $X_{i}$ have different levels of variance. They are uncorrelated, a consequence of the orthogonality of the eigenvectors, and their associated eigenvalues $\lambda$ are their variances, ${\rm Cov}({\cal E}_{i},{\cal E}_{j})=\lambda_{i}\delta_{ij}\,.$ (19) The fact that, by construction, principal components are uncorrelated with each other is important since they provide independent pieces of statistical information. The smaller an eigenvalue $\lambda_{i}$, the more likely that the corresponding linear combination ${\cal E}_{i}$ will not deviate from its average value for a randomly set of variables. Therefore, if there exist small eigenvalues then the associated PCs are largely conserved in a statistical sense. Conversely, the larger an eigenvalue is then the more relevant the associated PC is in describing the dynamics and variations in the problem. There are two related but different senses in which for a parametrized time series a principal component with small variance can be semi-conserved. The first is being constant as a function of time for an arbitrary but fixed set of parameters. The second one is in the statistical sense that deviations of a principal component from the mean value are small for arbitrary but fixed initial and final times over a set of runs with the initial configurations. A small variance automatically implies approximate conservation in the second sense but not necessarily in the first one. The interest is not only in those PCs which have the smallest variances (and thus identify semi-conserved quantities in the second sense) but also in those with the largest variances, which encode the most information about the system dynamics. This should be clarified through the following example. Example 3. PCA for spin dynamics. One may wonder if given a uniform initial spin orientation distribution, after a while those orientations turn into some preferred orientation, such as being aligned into some preferred direction. This was studied in [57] through massive numerical simulations. We only briefly review some of the results of that reference. The interest at the time was motivated by the unexpectedly large kicks found in numerical relativity simulations and whether they were of a generic nature in a statistical sense; for a review on recent comprehensive studies on this topic see Section 12.1. In order to perform a large enough analysis the Post-Newtonian equations were used, up to 3.5PN order in the angular frequency and 2PN with the covariant spin supplementary condition. Next, in any PCA study one has to define which variables to analyze. In order for the quantities to be invariant under rotations of the system of reference, at least in a Newtonian sense, the first quantities chosen were scalar products between the normalized spins of each binary component and the orbital angular momentum of the system. Moreover, their differences between initial and final values: $\displaystyle\Delta({\bf\hat{S}}_{1}\cdot{\bf\hat{L}})$ $\displaystyle=$ $\displaystyle{\bf\hat{S}}_{1}\cdot{\bf\hat{L}}\|_{f}-{\bf\hat{S}}_{1}\cdot{\bf\hat{L}}\|_{i}=:{{\Delta}_{1\text{L}}}$ (20) $\displaystyle\Delta({\bf\hat{S}}_{2}\cdot{\bf\hat{L}})$ $\displaystyle=$ $\displaystyle{\bf\hat{S}}_{2}\cdot{\bf\hat{L}}\|_{f}-{\bf\hat{S}}_{2}\cdot{\bf\hat{L}}\|_{i}=:{{\Delta}_{2\text{L}}}$ (21) $\displaystyle\Delta({\bf\hat{S}}_{1}\cdot{\bf\hat{S}}_{2})$ $\displaystyle=$ $\displaystyle{\bf\hat{S}}_{1}\cdot{\bf\hat{S}}_{2}\|_{f}-{\bf\hat{S}}_{1}\cdot{\bf\hat{S}}_{2}\|_{i}=:{{\Delta}_{12}}\,,$ (22) where hats stand for unitary vectors. The choice of initial and final values is something to define and is discussed in [57], the summary is that the main results do not depend on these choices, which gives insight into the fact that the binary problem in GR is highly redundant. The orbital angular momentum and spin orientations naturally become correlated due to spin-orbit and spin-spin interactions as each of these binary black hole configurations evolve in time. However, at least within the PN approximation here considered, the orbital angular momentum and spin vectors remain perfectly uniformly distributed [68]. For example, a Kolmogorov-Smirnov test for a representative configuration returns a p-value of $\sim 10^{-5}$ when testing for lack of uniformness [68]. Higher PN expansions might introduce small biases [89] but if so they appear to be at a level in which approximating the mean of the above scalar products at any instant of time by zero is a very good approximation. Ref. [57] starts with a simple case and makes contact with previous conservation results. The authors start building towards the more general case by first doing a PCA using only the two spin-orbit (SO) variables in (20) and (21), ${{\Delta}_{1\text{L}}}=\Delta({\bf\hat{S}}_{1}\cdot{\bf\hat{L}})\,,\quad{{\Delta}_{2\text{L}}}\mathrel{\mathop{:}}=\Delta({\bf\hat{S}}_{2}\cdot{\bf\hat{L}})\,.$ (23) However, spin-spin interactions in both the numerical simulations and in the analytical calculations are included in the PN equations of motion used when solving for the evolution of each configuration. For mass and spin magnitudes ($m_{j},\chi_{j}$) of each black hole the covariance matrix for the variables (23) is ${\bf C}=\left(\begin{array}[]{cc}{\rm Cov}({{\Delta}_{1\text{L}}},{{\Delta}_{1\text{L}}})&{\rm Cov}({{\Delta}_{1\text{L}}},{{\Delta}_{2\text{L}}})\\\ {\rm Cov}({{\Delta}_{2\text{L}}},{{\Delta}_{1\text{L}}})&{\rm Cov}({{\Delta}_{2\text{L}}},{{\Delta}_{2\text{L}}})\end{array}\right)\,,$ (24) where the entries can come either from numerical simulations or from what the authors call the instantaneous approximation. The matrix $\bf C$ is then diagonalized to find the principal components. From numerical simulations the authors find that, sampling across many random initial spin orientations, each of the principal components has zero mean over time (to numerical accuracy), $\langle\Delta{\cal E}_{j}^{\text{SO}}\rangle=0$, a consequence of the spin orientation distributions remaining highly uniform during the inspiral. Furthermore, they find $\lambda_{2}$ to be in the range $\sim 10^{-9}-10^{-4}$ for the parameters sampled and that it grows with both spin magnitudes, which is expected from physical intuition, but also that it increases as the equal mass case is approached, which was unexpected. As an example with $(m_{1},m_{2},\chi_{1},\chi_{2})=(0.4,0.6,1.0,1.0)$, Figure 1 shows a graphical representation of the principal components overlaid on a scatter plot of the ${{\Delta}_{1\text{L}}}$ and ${{\Delta}_{2\text{L}}}$ data from $1,\\!000$ out of $100,\\!000$ numerical simulations using random initial spin orientations. Notice that the first PC, which points along the direction of the eigenvector ${\bf\hat{V}}_{1}$ with the largest eigenvalue $\lambda_{1}$, captures the largest variation in the data while the second PC, pointing along ${\bf\hat{V}}_{2}$, indicates that there is very little spread in the data in that direction, which is also implied by the smallness of $\lambda_{2}$ relative to $\lambda_{1}$. Therefore, for the time interval considered, the second PC is largely irrelevant. This figure is almost an ideal example of dimensionality reduction through PCA. For a detailed analysis see [57]. Figure 1: A graphical representation of the principal components for the spin- orbit variables and the numerical data of ${{\Delta}_{1\text{L}}}$ and ${{\Delta}_{2\text{L}}}$ for a binary black hole system with, as illustration, $m_{1}=0.4$ and maximal spin magnitudes. One can see that PC2 is largely irrelevant. For details see [57]. #### Further Reading Principal Component Analysis is a very well known tool to look for redundancies or categorize the relevance of different variables. One of its weaknesses is that it depends on which variables to look for. The case study that we chose is due to the fact that if looking in more detail at the original reference reviewed, there are strong indications that there are at least three (out of eight) redundant quantities in the problem, and therefore the problem is amenable to an unsupervised dimensional reduction approach, as presented throughout this review. Reference [76] is a very pedagogical book solely on PCA, with many case studies and modern developments. ### 5 Proper Orthogonal Decomposition Given the snapshots $H_{1},\ldots,H_{N}\in\mathbb{C}^{L}$ (the training samples), the snapshot matrix is ${\bf H}=[H_{1},\ldots H_{N}]\in\mathbb{C}^{L\times N}$ with, in general, $L\geq N$. The goal here is to find $n\leq N$ orthonormal vectors $\\{\phi_{i}\\}_{i=1}^{n}$ in $\mathbb{C}^{L}$ minimizing the average approximation error (15) $\frac{1}{N}\sum_{i=1}^{N}\\|H_{i}-{\cal P}_{n}H_{i}\\|^{2}\,,$ (25) where ${\cal P}_{n}$ is the orthogonal projector onto $W_{n}:=\text{span}\\{\phi_{i}\\}_{i=1}^{n}$ and $\\|\cdot\\|$ is the Euclidean norm. The minimizer subspace of (25) is called a Proper Orthogonal Decomposition (POD) of rank $n$, and its generating basis the POD one. Notice that in general the POD basis elements $\phi_{i}$ are not members of $\\{H_{i}\\}_{i=1}^{N}$. In the gravitational wave case, this means that the reduced basis is not a subset of waveforms, neither does the method provide a set of the “most representative” points in the space of waveforms for building a posteriori bases through numerical relativity simulations. Another disadvantage is that the problem, involving a potentially very large linear algebra system, is not easily parallelizable in distributed memory architectures. Still, the method is straightforward to implement and can, for example, quickly provide insight into whether a problem is amenable to dimensionality reduction: after performing a POD decomposition of the snapshot matrix, one can look at the decay rate of singular values as a function of $n$ in order to know if the problem can be codified in a space of lower dimensionality than the original one. This can be turned into an actual strategy: a POD for a small subset of the problem can provide a quick insight, and if the singular values decay fast enough with $n$, the larger problem of interest can be tackled with a greedy approach, as discussed in Section 8.3. The optimal solution to the minimization problem defined by Eq. (25) can be accomplished by means of a Singular Value Decomposition (SVD) procedure. In this framework the POD basis is given by the first $n$ left singular vectors of the snapshot matrix $\bf{H}$ (this can be proved by considering the first- order optimality conditions for the minimization problem). Consider then the following singular value decomposition of H: $\bf{H}=\bf{U\Sigma V}^{\dagger}\,,$ (26) where ${\bf U}=[u_{1},\ldots,u_{L}]\in\mathbb{C}^{L\times L}$, ${\bf V}=[v_{1},\ldots,v_{N}]\in\mathbb{C}^{N\times N}$ are orthogonal matrices and ${\bf\Sigma}=\begin{bmatrix}{\bf D}\\\ {\bf 0}\end{bmatrix}\in\mathbb{C}^{L\times N},{\bf D}=diag(\sigma_{1},\ldots,\sigma_{N})\in\mathbb{C}^{N\times N},\sigma_{1}\geq\ldots\geq\sigma_{N}\geq 0.$ (27) Then, the rank-$n$ POD basis is given by $\phi_{i}=u_{i},\,i=1,\ldots,n$. This basis provides a low-rank approximation to H and represents the minimizer basis for the minimization problem stated in (25). To see this, consider from (26) the relation $u_{i}^{\dagger}{\bf H}=\sigma_{i}v_{i}^{\dagger}\,,$ valid for all column vectors of ${\bf U}$. Next, multiply by $u_{i}$ on the left and sum $\sum_{i=1}^{n}u_{i}u_{i}^{\dagger}{\bf H}=\sum_{i=1}^{n}\sigma_{i}u_{i}v_{i}^{\dagger}\,.$ (28) The l.h.s. of (28) is exactly the orthogonal projector associated to the basis $\\{u_{i}\\}_{i=1}^{n}$ acting on $\bf H$ and the r.h.s. represents its rank-$n$ approximation ${\bf H}_{n}$. Therefore we can rewrite (28) as ${\cal P}_{n}{\bf H}={\bf H}_{n}\,.$ (29) Remarks on the SVD decomposition * • The numbers $\sigma_{i}$ are known as the singular values of the matrix H and correspond to the positive square roots of the eigenvalues of the associated matrix $\bf{K}=\bf{H}^{\dagger}\bf{H}$. They are usually chosen in descending order in practice to facilitate the recognition of the most relevant principal components. * • For the matrices U and V, only the first $\mathrm{rank}(\mathbf{H})$ columns are unique, whereas the remaining ones are arbitrarily extended such that orthogonality is maintained. Since $\sigma_{i}=0$ whenever $i>\mathrm{rank}(\mathbf{H})$ the factorization (26) does not depend on the choice of this extension, since such extension is annihilated by null entries of $\bf\Sigma$. * • The first $N$ columns of U and V are known as the left- and right-singular vectors of H respectively. The right-singular vectors, $v_{i}$, are the normalized eigenvectors of ${\bf H}^{\dagger}{\bf H}$ and the left-singular vectors, $u_{i}$, are the normalized eigenvectors of ${\bf H}{\bf H}^{\dagger}$. * • The number $K$ of non-zero elements in $\mathbf{\Sigma}$ is exactly $\text{rank}(\mathbf{H})$. Consequently, Eq. ((26)) is sometimes called a rank-revealing factorization. Error of a POD approximation The accuracy of the low rank-$n$ approximation ${\bf H}_{n}={\cal P}_{n}{\bf H}=[{\cal P}_{n}H_{1},\ldots,{\cal P}_{n}H_{N}]$ is given by the following lemma. Lemma 1. Define the approximation error by $\epsilon=\frac{1}{N}\sum_{i=1}^{N}\\|H_{i}-{\cal P}_{n}H_{i}\\|^{2}.$ (30) It can be shown that $\epsilon$ satisfies $\epsilon=\frac{1}{N}\sum_{i=n+1}^{N}\sigma_{i}^{2}.$ (31) In terms of the Frobenius norm, $\epsilon=\frac{1}{N}\\|{\bf H}-{\bf H}_{n}\\|^{2}_{F}.$ (32) Intuitively, the square of the Frobenius norm of the difference between a matrix and its low-rank approximation represents the total squared difference of the rows of ${\bf H}$ due to omitting the last $(N-n)$ singular values when forming ${\bf H}$. As discussed above, the projection ${\bf H}_{n}$ represents the best rank-$n$ approximation of H in the Frobenius norm. In summary, going back to section 3, one can see that the minimization problem (1) related to the average error of a set of functions ${\cal F}$ is optimally solved by the POD/SVD decomposition of the matrix associated to those functions. In the context of a gravitational-wave template bank the rows of the snapshot matrix are the waveforms evaluated at different time/frequency values, and the different rows correspond to different intrinsic parameters. The inner product of rows with themselves, in turn, correspond to the total power in the bank. Thus, the squared Frobenius norm of a template bank is the total power contained within all the templates in it. For normalized templates, this is simply the size of the bank. It can be seen how Eq. (32) directly corresponds to the total power in a template bank lost from a low-rank approximation to it. More precisely, the error measure is directly related to the average fractional SNR loss which, up to a constant, is the squared Frobenius norm of the difference between the full template bank and its low-rank approximation. Fixing the average fractional SNR loss thus determines the total number of non-zero singular values which must be retained to guarantee that the rank- reduced bank remains effective. This is discussed in practical terms through the following case study. Example 4. SVD for gravitational waves. In Ref. [35] the authors studied the application of SVD to gravitational wave templates to reduce the redundancies in the bank and build an orthogonal basis to represent the whole set. As a proof of concept, the authors applied a SVD approach to a set of CBC waveforms corresponding to a sliver of the BNS parameter space. Next, we summarize some of the results of this reference, closely following its notation, which might be different from the rest of this review. In order to detect a GW signal, the common choice of the minimal match between an arbitrary point in parameter space and its nearest point of the template bank is $97\%$. In order to compare and filter the data against the entire template bank, an approximation to the matched filtering $\rho_{\alpha}=\langle\bar{h}_{\alpha},s\rangle$ is sought for, where $h_{\alpha}$ is a complex waveform vector and $s$ is the data vector (the presumed signal). In this way the number of evaluation of inner products can be reduced as well as its computational cost. Let’s define the $N\times L$, where $N=2M$ and $M$ stands for the number of complex waveform (there is no obvious reason why to perform an SVD on real and imaginary parts of a waveform, given that the POD approach can handle complex snapshots) template matrix ${\bf H}=\\{h_{1}^{R},h_{1}^{I},\ldots,h_{M}^{R},h_{M}^{I}\\}\,,$ (33) where $h_{i}^{R,I}$ are the real and imaginary part of the $i$-template waveform, each one corresponding to the rows of $\bf H$. Applying an SVD decomposition to ${\bf H}$ and writing it in component form (here we follow the notation of [35]) ${\bf H}_{\mu\nu}=\sum_{\kappa=1}^{N}v_{\mu\kappa}\sigma_{\kappa}u_{\kappa\nu}\,,$ (34) one can define the truncated sum ${\bf H}^{\prime}_{\mu\nu}=\sum_{\kappa=1}^{N^{\prime}}v_{\nu\kappa}\sigma_{\kappa}u_{\kappa\nu}\,,$ (35) and approximate $\rho_{\alpha}$ by $\rho^{\prime}_{\alpha}=\langle H^{\prime}_{2\alpha-1}-iH^{\prime}_{2\alpha},s\rangle=\sum_{\nu=1}^{N^{\prime}}(v_{{2\alpha-1}\,\nu}\sigma_{\nu}-iv_{{2\alpha}\,\nu}\sigma_{\nu})\langle u_{\nu},s\rangle\,,$ (36) where the $H^{\prime}_{j}$ are the vector rows of ${\bf H}^{\prime}$. Figure 2 shows a representation of the matrix of waveforms H and its associated SVD-basis. One disadvantage of this kind of dimensional reduction is that the remaining basis barely resembles the structure of the original template. This can be fixed with the RB-greedy approach that is presented in Section 8. Figure 2: Left: Representation of template waveforms corresponding to $\bf H$ against time. Right: Representation of the first four SVD-basis. Notice that since the SVD basis functions do not correspond to actual waveforms, they display non-physical behavior which might be difficult to fit for when building a surrogate model as discussed in Section 11. Figures taken from [35]. Accuracy. An approximation to the data snapshot matrix is sought such that $\\|{\bf H}_{\mu}-{\bf H}^{\prime}_{\mu}\\|\sim 1\%$. Following this requirement, the fractional SNR $\Big{\langle}\frac{\delta\rho}{\rho}\Big{\rangle}:=\frac{1}{M}\sum_{\alpha=1}^{M}\frac{\delta\rho_{\alpha}}{\rho_{\alpha}}$ (37) can be approximated by: $\Big{\langle}\frac{\delta\rho}{\rho}\Big{\rangle}=\frac{1}{2N}\sum_{\mu=N^{\prime}+1}^{N}\sigma_{\mu}^{2}\,,$ (38) resulting from a Taylor expansion valid in the range $\langle\delta\rho/\rho\rangle<10\%$. Note that this approximation is proportional to the squared Frobenius norm of the truncation error of ${\bf H}$, $\Big{\langle}\frac{\delta\rho}{\rho}\Big{\rangle}=\frac{1}{2N}\\|{\bf H}-{\bf H}^{\prime}\\|_{F}^{2}\,.$ (39) As a case of application, [35] shows an SVD analysis to gravitational waves emitted by a CBC-BNS, with chirp masses $1.125M_{\odot}\leq M_{c}<1.240M_{\odot}$ (that is, a rather small sliver in parameter space) and component masses $1M_{\odot}\leq m_{1},m_{2}<3M_{\odot}$. In order to satisfy a minimal match of $96.8\%$, a number of templates $M=456$ ($N=912$) to cover the parameter space was found. In Fig. 3 (plot of $\langle\delta\rho/\rho\rangle$ vs. # of SVD-basis elements) it can be seen that, to obtain $\langle\delta\rho/\rho\rangle=10^{-3}$, the number of basis elements needed to reconstruct the whole template bank to that accuracy can be reduced from $N=912$ to $N^{\prime}=118$. Though POD/SVD is a good starting point for ROM, in following sections we will develop a more modern framework for modeling reduction. Figure 3: SNR (Eq. (37)) for a CBC-BNS gravitational waves as a function of the number of SVD-basis elements – valid for $SNR<10\%$ –. Figure taken from [35]. #### 5.1 PCA and POD Principal Component Analysis and Proper Orthogonal Decomposition are closely related from a mathematical point of view, though usually applied in different contexts. PCA is in general used in statistical analysis. It represents a solution to an optimization problem; namely, that one of finding the uncorrelated directions of maximum statistical variability in a stochastic data set. POD is usually applied in dimension reduction modeling. As in PCA, POD also represents a solution to an optimization problem: finding the best low-rank approximation to a data set matrix. This is achieved by minimizing the Frobenius norm of the error matrix (Eq. (32)). Suppose we have a data set matrix of size $L\times N$, $N$ being the number of variables and $L$, the number of samples corresponding to each variable. In PCA one is not interested in the “length” of each variable and their individual components, but rather on their correlations. Put differently, if there are $N$ stochastic variables, the covariance matrix, regardless of the number of components in each stochastic variable, is in all cases of size $N\times N$. Compression can be achieved by ignoring the directions in which the stochastic data have little variance. Mathematical relationship between PCA and POD Although conceptually different, both methods are mathematically related. To see this, we make a simple observation concerning the snapshot matrix and its covariance matrix. Let ${\bf X}:=[X_{1},\ldots,X_{N}]\in\mathbb{C}^{L\times N}$ be the snapshot matrix with each column $i$ storing $L$ independent observations of the random variable $X_{i}$. We assume that each $X_{i}$ is identically distributed. Subtracting to each column $X_{i}$ the corresponding mean value $\langle X_{i}\rangle$, $X_{s,i}\rightarrow X_{s,i}-\langle X_{i}\rangle\,,\,s=1,\ldots,L$, and replacing $\bf{X}$ by this “centered” matrix version, the associated covariance matrix $\bf{C}$ can be written as ${\bf C}=\frac{1}{L-1}{\bf X}^{\dagger}{\bf X}\,.$ (40) Without loss of generality, we will suppose $\bf X$ has full rank. Therefore, being an hermitian, non-negative definite matrix, $\bf C$ can be diagonalized as ${\bf C}={\bf V}{\bf\Lambda}{\bf V}^{\dagger}\,,$ where ${\bf\Lambda}=\text{diag}(\lambda_{1},...,\lambda_{N})$ is the matrix of non-zero descending-ordered real eigenvalues and $\bf V$ is the unitary matrix storing eigenvectors of $\bf C$. The columns of $\bf V$ are called principal axes or principal directions in the PCA framework. PCs correspond to the projection of the data onto these principal directions, the columns of the matrix product ${\bf X}{\bf V}$. Now, perform a POD/SVD decomposition of the snapshot matrix X ${\bf X}=\mathbf{U}\mathbf{\Sigma}\mathbf{\tilde{V}}^{\dagger}$ and rewrite Eq. (40) as ${\bf C}=\frac{1}{L-1}\,\,(\mathbf{U}\mathbf{\Sigma}\mathbf{\tilde{V}}^{\dagger})^{\dagger}\mathbf{U}\mathbf{\Sigma}\mathbf{\tilde{V}}^{\dagger}=\mathbf{\tilde{V}}\,\frac{\mathbf{\Sigma}^{\top}\mathbf{\Sigma}}{L-1}\,\mathbf{\tilde{V}}^{\dagger}\,.$ (41) The similarity relation between $\bf C$ and $\frac{\mathbf{\Sigma}^{\top}\mathbf{\Sigma}}{L-1}$ implies that both have the same eigenvalues: $\lambda_{i}=\frac{\sigma^{2}_{i}}{L-1}\,,\quad i=1,\ldots,N\,.$ Another benefit of this mathematical correspondence is that PC’s are straigthforward to compute as ${\bf X}{\bf V}={\bf U}\mathbf{\Sigma}$. #### Further reading The reader may consult [76, 140] for clear expositions about the classical matrix factorization methods presented in this section. QR decompositions form another class of matrix factorizations which can be used for low-rank approximations. A theoretical framework for rank revealing QR decompositions of a matrix was developed in the the late 1980’s and 1990’s [38, 73], motivated partly by the SVD’s high computational cost. QR decompositions form the basis for many modern fast algorithms [66, 64, 47, 42]. Actually, motivated by GW research an important resuIt has been shown: the RB-greedy algorithm is completely equivalent to a certain type of QR decomposition [16]. ### 6 Spectral expansions In this section we discuss fast converging classical linear approximations, since they naturally lead to Reduced Basis when considering parametrized problems. Consider a set of complex functions, ${\cal F}:=\\{h:\mathbb{C}\rightarrow\mathbb{C}\\}\,.$ A standard linear approximation consists of the following: a basis with $n$ (non necessary orthogonal) functions is somehow chosen, $\\{e_{i}\\}_{i=1}^{n}\,.$ Next, a function $h$ in ${\cal F}$ is approximated by a linear combination of the chosen basis elements, $h(z)\approx h^{(n)}(z):=\sum_{i=1}^{n}c_{i}e_{i}(z)\,,\quad c_{i}\in\mathbb{C}.$ (42) There are several criteria to choose the expansion coefficients $\\{c_{i}\\}$. We recap least-squares (LS), in Section 10 we will discuss interpolation. In the LS approach the coefficients $\\{c_{i}\\}$ are chosen such that the representation error is minimized with respect to some chosen norm, $\left\\|h-\sum_{i=1}^{n}c_{i}e_{i}\right\\|^{2}\,.$ (43) As was stated in Section 3, the solution to the LS problem (Eq. (9), Section 3) is the orthogonal projection onto the span of the basis, $h^{(n)}={\cal P}_{n}h\,.$ (44) That is, the approximation (42) minimizes the error (43) when the coefficients $\\{c_{i}\\}$ are chosen such that the residual $\delta^{(n)}h=h-h^{(n)}$ satisfies $\langle\delta h,h^{(n)}\rangle=0$, which implies $\left\langle e_{j}(\cdot),h(\cdot)-\sum_{i=1}^{n}c_{i}e_{i}(\cdot)\right\rangle=0\quad\quad\text{ for }j=1,\ldots,n\,.$ (45) The solution to (45) is $c_{i}=\sum_{j=1}^{n}({\bf G}^{-1})_{ij}\langle e_{j},h\rangle\,,$ (46) where ${\bf G}^{-1}$ is the inverse of the Gram matrix or Gramian ${\bf G}$, with entries ${\bf G}_{ij}:=\langle e_{i},e_{j}\rangle\,.$ If the basis is orthonormal, this matrix is the identity and one recovers the familiar expression ${\cal P}_{n}h=\sum_{i=1}^{n}\langle e_{i},h\rangle e_{i}\,.$ (47) The Gramian matrix can be very ill-conditioned in general, meaning that the calculation of its inverse, needed in (46), can have large numerical errors (see, for example [136]). Therefore, from a numerical conditioning point of view, it is convenient to work with an orthonormal (or orthogonal) basis since the Gram matrix is the identity. In iterative approximations one defines the convergence rate as the rate at which the representation error $\\|\delta^{(n)}h\\|=\left\\|h-{\cal P}_{n}h\right\\|$ (48) decreases as $n$ increases for any given $h$. In the context of RB, when one is dealing with parametrized systems, this error will depend also on the parameters of the system. #### 6.1 Spectral methods In terms of accuracy and optimal convergence rates for approximation of a space ${\cal F}$, we have not yet discussed two related aspects: 1. 1. The choice of a “good” basis or, more precisely, the approximation space $W_{n}$. 2. 2. The optimal choice of how many basis elements to use. One would think that the larger, the better. This is related to the regularity of the functions in ${\cal F}$. That is, how smooth or differentiable they are. The first comment might be puzzling, after all we have emphasized that given an approximation space $W_{n}$, the LS approximation is uniquely defined in a geometric way. The question really is what the approximation space $W_{n}$ should be to minimize the representation error. ##### 6.1.1 Fourier Expansions We start with the simplest and most familiar case: that one of periodic functions in $[0,2\pi]$, unit weight, $\omega(x)=1$ 333The choice of norm is crucial, see the next Section about Jacobi polynomials for more insights about this., and Fourier modes as (orthonormal) basis, $e_{j}(x)=\frac{1}{\sqrt{2\pi}}e^{ijx}\,,\quad j\in\mathbb{Z}$ Assuming for simplicity that $n$ is even, we have ${\cal P}_{n+1}h(x)=\sum_{-n/2}^{n/2}\hat{h}_{j}e_{j}(x)\,,$ where $\hat{h}_{j}$ are the Fourier coefficients of $h$. One can show that if $h$ has $s$ derivatives, then there exists a constant $C>0$ independent of $n$ such that $\\|h-{\cal P}_{n+1}h\\|\leq C(n+1)^{-s}\left\\|\frac{d^{s}h}{dx^{s}}\right\\|$ (49) for all $n\geq 0$. In particular, if $h$ is smooth ($h\in C^{\infty}$), then the representation error decays faster than any power law with $n\rightarrow\infty$, which is referred to as spectral convergence. Under further conditions, in particular the case in which $h$ is analytic, the error decay is actually exponential. The Fourier case gives a very intuitive way of how this happens just by using integration by parts, for more details see Chapter 9 of [120]. The summary here is that Fourier modes are not a good basis just because they are periodic functions, but they provide fast convergence as a representation space, as fast as the regularity of the function(s) being represented has. This is the core idea of spectral methods and, for parametrized systems, reduced basis. A brief summary of the non-periodic case follows. ##### 6.1.2 Jacobi Polynomials In the case of non periodic functions a similar result to that one discussed for periodic functions holds, whether the domain is bounded or infinite. Again, the choice of weight is crucial. Any interval can be mapped into $[-1,1]$ or $(-1,1)$, that is why we usually refer to those intervals. The open interval case is because in some cases (such as Chebyshev) the weights actually diverge at the $\pm 1$ end points – there is a reason for this but we shall skip it. Polynomials are a natural basis to use. Why? Just because after centuries we understand them well; going beyond that is one way of looking at Reduced Basis. Given a maximum degree, the span of polynomials is the same, so the question is only what kind of weights guarantee fast convergence upon regularity of the function to be represented. The following family of weights is a sufficient class, thought not a necessary one, $\omega(x)=(1-x)^{\alpha}(1+x)^{\beta}\,\quad x\in(-1,1)$ (50) with $\alpha,\beta>-1$. Under these assumptions, the following holds in a Jacobi Polynomial approximation: $\\|h-{\cal P}_{n+1}h\\|\leq C(n+1)^{-s}\left\\|(1-x^{2})^{s/2}\frac{d^{s}h}{dx^{s}}\right\\|$ (51) for all $n>(s-1)$ and $C$ independent of $n$. If $h$ is smooth there is asymptotic spectral convergence. Under additional conditions on the smoothness (or, but not necessarily, analyticity), the convergence is in fact exponential. In this review we will loosely associate smoothness with asymptotic exponential convergence of application-specific spectral expansions, without discussing these additional assumptions. Most of spectral methods literature is based on solving numerical problems, prominently differential equations. In this review we shift this focus to that one of a more fundamental representation problem, after which all the calculus for approximating quadratures, taking derivatives, and solving differential equations follows in a rather straightforward way. This is not just a matter of taste, but will make the introduction of Reduced Basis and all its associated calculus and applications very natural and almost trivial. #### Further reading The weights (50) lead to Jacobi polynomials, which are solutions to a singular Sturm-Liouville problem, the properties of which guarantee the above mentioned spectral convergence. Standard examples are Legendre and Chebyshev polynomials. Their span is the same, it is with respect to which scalar product they guarantee fast convergence and their discrete version and relation to interpolation, discussed in Section 10. They are also orthonormalized with respect to their own scalar products, which helps avoid the typical conditioning problem of inverting the associated Gramian matrix. For more details on and a quick glimpse at spectral methods, from a physicist or practitioner perspective, see Chapter 9 of Ref. [120]. Reference [139] is a surprisingly compact and efficient book to get started with spectral methods (despite its title, MATLAB is not necessary at all to digest the content). A long classic reference, especially for practitioners, is [29]. The book is legally available for free from the author’s webpage, though with some typos. A great book, in some sense targeted at ordinary differential equations is [56]. For a modern presentation and the latest results, from theory to current approaches to beat Gibb’s phenomena, see [70] and references therein. ### 7 Parametrized problems and optimal approximations We are interested in approximating some abstract space of parametrized functions ${\cal F}$, which in general is not linear (the sum of two waveforms does not need to be a waveform) but we assume that it can be embedded in a Hilbert one ${\cal H}$ (for example, the one of integrable functions in the $L_{2}$ sense). We denote the underlying space of parameters as $\Phi$, which we assume to be compact. For example, ${\cal F}$ can be the space of $\lambda$-parametrized solutions $u_{\lambda}(x)$ of a partial differential equation representing the dynamics of a physical system. In this review we place emphasis in gravitational waves, parametrized for example by the mass and spin of each black hole in a binary collision. The question discussed next is how well one can theoretically approximate all of ${\cal F}$ by a set of $n$ basis elements of ${\cal H}$ in a linear and most compact way. This leads to the Kolmogorov n-width 444There are other widths, see [107]. $d_{n}$ of ${\cal F}$ with respect to ${\cal H}$, $\displaystyle d_{n}=d_{n}({\cal F},{\cal H}):=\min_{\\{e_{i}\\}_{i=1}^{n}\in{\cal H}}\max_{\lambda\in\Phi}\min_{c_{i}\in\mathbb{C}}\bigg{\\|}h(\cdot;\lambda)-\sum_{i=1}^{n}c_{i}(\lambda)e_{i}(\cdot)\bigg{\\|}^{2}\,.$ (52) Comment 1. We explain the meaning of (52), from right to left, with respect to the $\min,\max,\min$ properties. * • The first $\min$ implies that the optimal representation with respect to the (so far arbitrary) basis $\\{e_{i}\\}_{i=1}^{n}$ is used. That is, given a basis, the best representation (in the induced norm $\\|.\\|$) is considered. We already discussed that this is the orthogonal projection ${\cal P}_{n}$ onto the span of the basis $\\{e_{i}\\}_{i=1}^{n}$, so we can replace (52) by $\displaystyle d_{n}:=\min_{\\{e_{i}\\}_{i=1}^{n}\in{\cal H}}\max_{\lambda\in\Phi}\bigg{\\|}h(\cdot;\lambda)-{\cal P}_{n}h(\cdot;\lambda)\bigg{\\|}^{2}\,.$ (53) * • Next, the largest error in parameter space for such a best approximation is picked. That is the “worst best” approximation given a choice of basis. * • Finally, a choice of basis (more precisely, the subspace $W_{n}$) which minimizes this worst best error is chosen and the associated error is by definition the n-width. The compactness of $\Phi$ is important to guarantee that these minimum and maximum exist in the search through the parameter space. In other words, the $n$-width is an upper bound of “what is the best that one can do” if one could optimally choose an optimal basis. This is mostly a theoretical problem, since solving for such a basis is impractical (or, actually, in most cases, intractable) from a computational point of view because it carries combinatorial complexity (all elements of the basis have to be simultaneously chosen). It is more of a theoretical upper bound against which to benchmark the quality of any computable approximation which seeks for an approximate solution to the $n$-width problem. Parameter regularity and fast convergence Exact expressions for the $n$-width in general can only be achieved in a few cases (see, for example, [91]). In general, the best one can do is to set up upper bounds under specific assumptions and infer the dependence of the $n$-width with respect to $n$. Indeed, Kolmogorov calculated this distance for a special class of functions [91] with its first $(r-1)$ derivatives with respect to parameter variation absolutely continuous and obtained a power law dependency for the $n$-width, $d_{n}\sim n^{-r}\,.$ If the set of functions considered were $C^{\infty}$ with respect to parameter variation – the functions themselves can be discontinuous – the rate of convergence of its optimal basis representation would be better than any power law. This is called spectral convergence and will be relevant in next chapters at the moment of quantifying the fast convergence of a Reduced Basis approach. This is exactly the case in many scenarios of GWs, since they depend smoothly on the parameters of their sources and one therefore expects an optimal approach to have very fast (in fact, exponential) convergence, as opposed to random sampling, for example, for which the convergence is sublinear. This is the main reason underlying the extreme high accuracy of very compact surrogate models based on reduced bases. That is, one expects in most GW scenarios $d_{n}\sim e^{-an^{b}}\,,$ and an optimal basis should be extremely accurate and compact; in fact in some cases super-exponential convergence ($b>1$) has been found in the GW context [53]. The question then turns into how to find an approximate basis which is not only computable but is also nearly optimal with respect to the n-width. It is “common knowledge” that parametrized problems with regularity with respect to parameter variation show spectral convergence of the n-width in practice. One might ask how this observation can be possible at all when we have mentioned that the n-width is not computable in practice: we will return to this point when we discuss the greedy algorithm. But in fact, up to our knowledge there is no rigorous proof of this expectation for general parametrized systems. But there is a very compelling argument. Asymptotic exponential convergence is also observed in practice in GWs and, being physicists, that is good enough for us. The argument is as follow (courtesy of Albert Cohen): The convergence rate of the n-width depends on the smoothness/regularity of the functions with respect to the parameters of the problem. We now argue that if there is regularity with respect to parameter variation, as in many cases of interest, then one can expect fast (in fact, up to exponential or even super-exponential) decay of the approximation error. The argument is a spectral standard one (spectral methods were discussed in the previous section) applied to parameter variation and is as follows: If the parametric map $\lambda\rightarrow h_{\lambda}(\cdot)$ is smooth enough with respect to $\lambda$, then it can be very well approximated in some appropriate basis in the $\lambda$ parameter variable, for instance by Fourier or Jacobi polynomials. This means that there is an expansion of the form $c_{1}(\lambda)h_{1}(\cdot)+\ldots+c_{n}(\lambda)h_{n}(\cdot)+\ldots\,,$ (54) with $h_{i}\in{\cal F}$, that converges fastly towards $h_{\lambda}(\cdot)$ in ${\cal F}$ uniformly in $\lambda$. Now, a partial sum of the form (54) is a member of the span of $\\{h_{1},\ldots,h_{n}\\}$, which means that this linear space approximates well all functions in ${\cal F}$. What “well” means depends on the approximation result used. If the number of parameters is finite and the dependence is analytic, then exponential rates of the form $exp(-n^{1/d})$ can easily be proved. If the dependence is only $C^{s}$, then a rate $n^{-s/d}$ can also easily be proved. This argument establishes, then, that there are bases in which expansions of the form (54) converge very fast, and as a consequence the n-width can only decay faster. To summarize, for problems with smooth parametric dependence, fast convergence in terms of greedy reduced bases can be expected and it is not surprising that such global methods outperform local ones. We mentioned that the functions themselves can be discontinuous, which might be confusing so it is worthwhile explaining it in more detail, even if qualitatively. Consider a problem of fluid dynamics, where the solutions might develop shocks at a finite time, and imagine that one is solving the partial differential equations for a parametrized family of initial data. This is completely fine in terms of the fast convergence of the n-width, since the location, shape, etc., of the shock depend smoothly on the initial data; the time series (in this case) itself does not need to have regularity with respect to the physical variable (time in this example) but with respect to parameter variation. ### 8 Reduced Basis In this section we present the RB-greedy framework for parametrized systems in order to address the resolution of the Kolmogorov problem in a quasi-optimal way. First mathematical conventions related to the RB-scheme are presented, followed by a discussion of its near-optimality with respect to different $n$-width behaviors. #### 8.1 Introduction In previous sections we have discussed linear approximation of functions by orthogonal projection onto the span of a basis. The latter was taken to be a generic, problem-independent one, such as Fourier modes or polynomials. The Reduced Basis (RB) approach is a framework for efficiently solving parametrized problems, representing the solutions in a compact way, and predicting new ones based on an offline-online decomposition. In parametrized problems one is interested in functions of the form $h_{\lambda}=h_{\lambda}(\cdot)=h(\cdot;\lambda)\,.$ where is (in general a multidimensional) parameter $\lambda$, the discretization of which will define the training space. In fields related to scientific computing and data science one is usually interested in multiple evaluations and analyses of functions in real time. The approach of RB is especially tailored to that one in which some numerical problem has to be solved in order to obtain each function, and obtaining such numerical solutions is very expensive. The approach is also very powerful when large existing data sets are known and a sparse representation is needed and/or multiple, fast operations on them. Then a minimal, nearly optimal set of representative such solutions (waveforms, for the purpose of this review) is sought for in order to construct a reduced basis (rb) for the whole solution set. These rb solutions constitute an application-specific basis. That is, a set $\\{h_{\Lambda_{i}}\\}_{i=1}^{n}$ (55) of functions in the space of interest for carefully chosen parameter values $\Lambda_{i}$ is used as a basis itself, as opposed to generic basis such as polynomials or Fourier modes. The scalar product in this parametrized space is taken at fixed parameter values; that is, with respect to the physical dimension(s), $\langle h_{i},h_{j}\rangle=\langle h_{\lambda_{i}},h_{\lambda_{j}}\rangle=\langle h({\cdot;\lambda_{i}}),h({\cdot;\lambda_{j}})\rangle\;.$ The same results of Section 6 follow through for the parametrized and application-specific case. Namely, the optimal solution to the least-squares approximation is $h(\cdot;\lambda)\approx{\cal P}_{n}h(\cdot;\lambda)=\sum_{i=1}^{n}c_{i}(\lambda)e_{i}(\cdot)\,,$ (56) with the coefficients $c_{i}$ given by the equivalent of Eq. (46), in the physical dimension(s) and the reduced basis 55 relabeled as $\\{e_{1},\ldots,e_{n}\\}$. Namely, $c_{i}(\lambda)=\sum_{j=1}^{n}({\bf G}^{-1})_{ij}\langle e_{j}(\cdot),h(\cdot;\lambda)\rangle\,,$ (57) where the entries of the Gramian matrix are ${\bf G}_{ij}:=\langle e_{i},e_{j}\rangle\,.$ Clearly, the Gramian coefficients depend on the special parameters $\Lambda_{i}$ associated to each basis element. So far, the span of the reduced basis uniquely determines the rb representation. This reduced space depends primarily on the choice of the selected parameter points in order to define a starting basis. As of a convenient basis itself, for the same reasons discussed in Section 6, from a conditioning numerical perspective, it is convenient to work with an orthonormal basis built out of the rb solutions through a simple orthonormalization procedure. This does not change the span of the rb and is numerically convenient. #### 8.2 The Training set A commonly used approach to construct a basis, described below in Section 8.3, is through a greedy algorithm. In its simplest version, the algorithm identifies a set of $n$ points in parameter space out of a representative enough set of functions of interest that are actually known. We call this set of functions the training set: ${{\cal K}}:=\\{h_{\lambda_{i}}\\}_{i=1}^{N}\,.$ (58) The subset $\\{h_{\Lambda_{i}}\\}_{i=1}^{n}$ constitutes a nearly optimal basis for application-specific spectral expansions of any function in ${\cal K}$ in a precise sense discussed in Section 8.3.1. If there is partial redundancy/similarity in the latter, then $n<N$ or even $n\ll N$. Note here that we have used the symbol ${\cal K}$ to represent a discretization of the space of interest ${\cal F}$ in order to perform actual computer calculations. The training set can be constructed by any means, including simple random or uniform sampling, more sophisticated stochastic methods [95], the metric approach [102], or those of Ref. [92], for example. For large problems, resampling it while constructing the basis can be critical; for an application in the case of GWs see for example [28]. Regardless of the method used to populate the training set, the RB-greedy formalism produces a compact and highly accurate representation of the training space catalog. The reduced basis is used to approximate other functions in the space of interest, whether they were in the training set catalog or not, through linear combinations that represent an orthogonal projection onto its span, $h_{\lambda}={\cal P}_{n}h_{\lambda}+\delta h_{\lambda}\,,$ (59) where the RB approximation is ${\cal P}_{n}h_{\lambda}$ and satisfies, by construction, $\langle{\cal P}_{n}h_{\lambda},\delta h_{\lambda}\rangle=0$. #### 8.3 Greedy algorithms We mentioned that a commonly used way to generate a reduced basis is through a greedy algorithm. In its simplest form, such as when the waveforms are inexpensive to compute or the data is already somehow known, the greedy algorithm, outlined in Algorithm 1, has as input a discretization of the parameter and solution space, ${\cal T}:=\\{\lambda_{i}\\}_{i=1}^{N}\quad{{\cal K}}=\\{h_{\lambda_{i}}\\}_{i=1}^{N}$ (60) with the elements of ${\cal T}$ usually called training points. To put emphasis on structure instead of form, all functions in the training set in this review are normalized. How to recover the norm, particularly for parameter estimation of any detected signal, is discussed in Section 14.1. The scheme needs an arbitrary seed $\Lambda_{1}\in{\cal T}$ to initialize it, and a threshold error $\epsilon$ for a target representation accuracy (or greedy error). Part of the output of the algorithm is a sequential selection of $n$ parameter greedy points $\\{\Lambda_{1},\Lambda_{2},\ldots,\Lambda_{n}\\}\subset{\cal T}$ and their associated waveforms $\\{h_{\Lambda_{1}},h_{\Lambda_{2}},\ldots,h_{\Lambda_{n}}\\}\subset\\{h_{\lambda_{i}}\\}_{i=1}^{N}\,.$ The set of waveforms $\\{h_{\Lambda_{i}}\\}_{i=1}^{n}$ constitutes the reduced basis. As was already discussed, for numerical conditioning purpose it is sometimes convenient (see Eq. (10)) to work with an orthonormalized set $\\{e_{i}\\}_{i=1}^{n}$ instead of directly the $\\{h_{\Lambda_{i}}\\}_{i=1}^{n}$. Another output of the algorithm is precisely the set of projection coefficients for functions in the training space catalog, whereas coefficients for any other known function can be computed through projection onto the basis. In Section 10 we discuss how to approximate these coefficients through interpolation in the physical dimension (time in the case of gravitational waveforms), and in Section 11 a predictive approach for accurate and fast evaluation of new (unknown) functions in ${\cal F}$ through surrogate models. These are predictive models: as opposed to a known function being projected into a compact basis, they predict new solutions (waveforms). Algorithm 1 Greedy algorithm for reduced basis 1:Input: $\\{\lambda_{i}\,,h(\cdot;\lambda_{i})\\}_{i=1}^{N}$, $\epsilon$ 2:Initialize $i=0$ and define $\sigma_{0}=1$ 3:Seed choice (arbitrary): $\Lambda_{1}\in{\cal T}$, $e_{1}=h(\cdot;\Lambda_{1})$ 4:rb = $\\{e_{1}\\}$ 5:while $\sigma_{i}\geq\epsilon$ do 6: $i=i+1$ 7: $\sigma_{i}=\max_{\lambda\in{\cal T}}\\|h(\cdot;\lambda)-{\cal P}_{i}h(\cdot;\lambda)\\|^{2}$ 8: $\Lambda_{i+1}=\text{argmax}_{\lambda\in{\cal T}}\\|h(\cdot;\lambda)-{\cal P}_{i}h(\cdot;\lambda)\\|^{2}$ 9: $e_{i}=h(\cdot;\Lambda_{i+1})-{\cal P}_{i}h(\cdot;\Lambda_{i+1})$ (Gram- Schmidt) 10: $e_{i+1}=e_{i+1}/\\|e_{i+1}\\|$ (normalization) 11: rb = rb $\cup\,e_{i+1}$ 12:end while 13:Output: rb $\\{e_{i}\\}_{i=1}^{n}$ and greedy points $\\{\Lambda_{i}\\}_{i=1}^{n}$ Comment 2. * • Greedy-type algorithms are global optimization procedures used in contexts outside reduced basis or even dimensional reduction. * • Being a global optimization algorithm, the choice of the seed $\Lambda_{1}$ is largely irrelevant. This was explicitly discussed as a sidenote in Ref. [36], see Fig. 4 below. We also refer to Section 12.6 for a discussion of Ref. [36] in the context of RB for multi-mode black hole ringdown. * • What the greedy algorithm does in Step 8 is to select the waveform for which its representation error onto the existing basis with $i-1$ elements is worst, and in Step 11 adding it to the enrichment of the rb representation. * • In steps 9 and 10 the rb waveforms are orthonormalized to avoid ill- conditioning of the computation of the projection (see the discussion in Section 6, before 6.1). * • Given an arbitrary user-defined tolerance error $\epsilon$, the algorithm stops when the approximation (56) meets the error tolerance – introduced as an input in the greedy algorithm –, $\\|h_{\lambda}-{\cal P}_{n}{h_{\lambda}}\\|^{2}\leq\epsilon\,\,\,\,\forall\,\,\lambda\in{\cal T}.$ * • The expected exponential convergence of the method for the problems of interest implies that $\epsilon$ can be made arbitrarily small with a relatively small number $n$ of basis element, with $n<N$ and, in many cases, $n\ll N$. Figure 4: Taken from [36]. This figure shows the representation error as a function of the number of reduced basis waveforms for a single QNM catalog. The authors iterate over all possible seed waveforms in the training set. The dark line represents the average, and the shaded area the maximum dispersion around it. This numerical experiment confirms that the seed choice becomes nearly irrelevant due to the global nature of the the greedy algorithm at each step, as intuitively expected being a global optimization. . In practice, one uses a better conditioned orthogonalization algorithm in step $9$ than the standard (i.e., “classical”) Gram-Schmidt one. The naive implementation of the classical Gram-Schmidt procedure is actually ill- conditioned. This is related to the fact that the Gramian matrix, which would have to be inverted, can become nearly singular [136]. To overcome this one can use an iterated Gram-Schmidt algorithm or a QR decomposition. A popular alternative to the classical Gram-Schmidt, largely used, and called the modified Gram-Schmidt, is also ill-conditioned, contrary to common perception since its ill conditioning only becomes apparent for large large data sets. See [117, 63, 131] for discussions about the conditioning and numerical stability of different orthonormalization procedures. Error definitions. We define the greedy error of the rb $\\{{e_{i}}\\}_{i=1}^{n}$ as $\displaystyle\sigma_{n}:=\max_{\lambda}\bigg{\\|}h_{\lambda}-{\cal P}_{n}h_{\lambda}\bigg{\\|}^{2}\leq\epsilon\,,$ (61) where $\epsilon$ is the user-defined tolerance error, which depends on the number of basis elements. The quantity $\sigma_{n}$ represents the largest error in the parameter space of the best approximation by the greedy-reduced basis. As discussed in [53], in the limit of sufficiently dense training spaces the greedy error is comparable to the minimal match (${\mathrm{MM}}$) through $\sigma_{n}\sim 1-{\mathrm{MM}}{\rm~{}~{}as~{}~{}}N\rightarrow\infty.$ (62) Since the RB framework allows us to compress the information presented in the training set, it is useful to introduce the quantity $C_{r}:=N/n\,,$ (63) called the compression ratio [119], to measure it. ##### 8.3.1 Convergence rates and near-optimality The greedy algorithm chooses, in a precise sense, a nearly-optimal basis for the function spaces ${\cal K}$ or ${\cal F}$, depending on whether one is discussing the discrete or continuum cases. In order to quantify this near- optimality, recall the definition of the $n$-width from Section 7, Eq. (53), but write it in a more geometric way: $\displaystyle d_{n}=d_{n}({\cal F},{\cal H})=\min_{W_{n}\in{\cal H}}\max_{\lambda}\bigg{\\|}h_{\lambda}-{\cal P}_{n}h_{\lambda}\bigg{\\|}^{2}\,.$ (64) The search space here is the whole Hilbert space and one looks for an optimal $n$-dimensional subspace $W_{n}$ for approximating the function manifold ${\cal F}$. As was stated in Section 7, $d_{n}$ is a theoretical upper bound to any practical algorithm to perform a linear approximation. In practice, solving such optimization problem becomes unfeasible due to its intrinsic combinatorial complexity. The nested nature of the greedy algorithm becomes crucial for reducing the complexity of the Kolmogorov problem. This means that each $H_{m}={\tt Span}\\{e_{i}\\}_{i=1}^{m}$ satisfies $H_{1}\subset H_{2}\subset\ldots\subset H_{n}$ and this feature dramatically reduces the search space in (64). One expects that this reduction of the search space is at the expense of losing completely – if not a modicum – the $n$-width optimality but, as we discuss in the next paragraph, this expectation is the opposite from being true. In which sense does the reduced basis-greedy procedure degrade the optimality in the Kolmogorov sense? To answer this, lets summarize two important results in relation with the greedy algorithm: if $d_{n}$ decays exponentially with $n$ then so does $\sigma_{n}$ [48], $d_{n}\leq D\mathrm{e}^{-an^{\alpha}}\implies\sigma_{n}\leq\sqrt{2D}\gamma^{-1}\mathrm{e}^{-a^{\prime}_{\alpha}n^{\alpha}}\,,$ (65) where $D$, $a$, $\alpha$ are positive constants. Similarly, if the n-width has polynomial decay then so does the greedy error, $d_{n}\leq Dn^{-\alpha}\implies\sigma_{n}\leq D^{\prime}_{\alpha}n^{-\alpha}\,,$ (66) where $D,\alpha>0$. More generally, for any decay rate of the n-width, $\sigma_{n}\leq 2\gamma^{-1}d_{n/2}\,.$ (67) The factor $\gamma$ in Eqs. (65) and (67) is a constant in $(0,1]$ [48]. In recent years there have been efforts to improve these bounds. For details see [148, 100]. In light of these results, we see that the reduced basis-greedy procedure inherites the optimality of the $n$-width. If the latter has exponential convergence, so it does the greedy error. In this precise sense the reduced basis-greedy approach is nearly-optimal: we cut down the complexity of the Kolmogorov problem at the very low expense of losing quality which, in most practical applications, becomes insignificant. The convergence rate of the $n$-width (and, in consequence, of the greedy error) depends on the parametric smoothness/regularity of the functions/waveforms. Indeed, if the functions are analytic and can be extended to a complex region, exponential decay for the greedy error can be proven (see [111], section 5.5, and citations therein). Therefore, the rb expansion is expected to have very fast convergence to the original waveform as the number of basis elements is increased. This is similar in spirit to the standard fast convergence of spectral methods, but here smoothness in the parameters of the problem is exploited, and the basis are elements themselves of the space of functions of interest. For this reason RB is sometimes referred to as an application-specific spectral expansion. To summarize, for problems with smooth parametric dependence, fast convergence in terms of greedy bases can be expected and it is not surprising that such global methods outperform local ones. Comment 3. Not only the choice of basis is important, but a also global expansion. To illustrate this consider again the example of the space of smooth periodic functions from section 6.1.1. If one is interested in a family of them, and is able to choose $n$ Fourier modes to represent them as best as possible, one could: * • build a discretization of the space of periodic functions and choose $n$ Fourier modes which when compared against the whole training set gives the best global approximation error; this approach is a global variation of the well known best m-term approximation problem, in which one is interested in approximating some function and wants the best set of $m$ elements chosen from a dictionary of functions to do it; * • follow the standard approach, in which the functions of interest are projected onto the span of the first $n$ Fourier modes. If the functions of interest are smooth, this approach gives exponentially decaying representation errors with the number of Fourier modes. This is, of course, not a predictive model, since the function to be projected has to be known. The first path would return a global and problem-dependent basis by construction. The second one, on the contrary, will return a standarized basis with no further feeling for the problem than the minimization of the maximum error due to the projection. ##### 8.3.2 Complexity, scaling, computational aspects Besides its near-optimality the greedy algorithm has several implementation advantages: 1. 1. It is simple to implement (see Algorithm 1). 2. 2. It follows a stricter aproximation criteria (see section 3) than POD/SVD, namely, the minimization of the worst error, as opposed to the average one, in parameter space due to the projection onto the rb basis. It provides a stricter error control. 3. 3. It is embarrassingly parallel. Each greedy sweep, defined by Step 7 of Alg. 1, can be carried out simultaneously and independently for different values of $\lambda\in{\cal T}$. 4. 4. As we remarked in Sec. 8.3.1, the greedy points and associated basis are hierarchical (nested). This is particularly important if the algorithm is to be used to guide which numerical relativity simulations should be carried out. Since they are so expensive, it is desirable that if further accuracy is needed, then it should be possible to achieve it by adding more points (as opposed to starting the process from scratch). 5. 5. Low complexity. That is, the cost of all steps 6-12 in Alg. 1 are independent of $i$, leading to a total cost linear in $n$, the total number of basis elements. This observation follows from the fact that ${\cal P}_{i+1}h_{\lambda}={\cal P}_{i}h_{\lambda}+\langle e_{i+1},h_{\lambda}\rangle e_{i+1}\,.$ (68) So, if the projection coefficients are stored while building the basis, at each sweep only the dot product between the elements of the training set and the last basis element has to be computed. #### Further reading A comprehensive review of reduced basis is [112]. For textbooks see [70] and [111]. ## Part II Predictive Models ### 9 Polynomial interpolation There are several reasons why we find it useful to discuss classical polynomial interpolation: * • It is used in ROM for gravitational wave physics and other fields; we discuss some advantages and disadvantages. * • There is a close relationship between interpolation, quadrature rules, spectral methods and Reduced Basis. It is easier to introduce the standard theory first, followed by the application specific version at the heart of ROM for parametrized systems. The emphasis of this section is on approximation quality quantified through convergence rates and error bounds. Except for some side comments pointing out implementations known to be poorly conditioned, we shall not discuss numerical algorithms or their implementation which can be found in standard references. #### 9.1 Representation versus prediction In standard approximation theory, discussed in Sections 5 and 6, one expands a function in some basis. Assuming that the approximation quality is measured by some weighted $L_{2}$ norm, the optimal approximation is the orthogonal projection onto the representation space. This projection is independent of the basis, in the sense that it only depends on its span. Assuming these are orthonormal for simplicity and conditioning purposes, the orthogonal projection is given by $h(x)\approx{\cal P}_{n}h(x)=\sum_{i=1}^{n}\langle e_{i},h\rangle e_{i}(x)\,.$ (69) Approximating $h$ by its projection as in (69), while perhaps useful by itself, is not a predictive tool: we need to know the values of $h(x)$ for all $x$ in order to calculate the projection coefficients in Eq. (69). Here, by prediction it is meant a new evaluation within the range used to build the approximating model. This means, for example, within the same parameter and physical (time/frequency for example) ranges in the case of GWs. We are not dealing with a much more difficult problem: namely, that one of prediction of time series, for example, based on previous history. Then, to get a predictive power we need to first turn to an interpolation scheme. Building an interpolant for $h(x)$ requires only the knowledge of $n$ values of $h(x)$ whereas the unknown values of the function are predicted from this interpolant. As previously mentioned, the applications are many, including numerical schemes to solve partial differential equations, the numerical computation of integrals and building predictive surrogate models. As we will see in Section 9.4 these apparently radically different approaches for approximation, such as interpolation and projection, are closely related, and in some cases both are equivalent within certain frameworks. #### 9.2 Prediction through polynomial interpolation Suppose we have a collection of $(n+1)$ different points $\\{x_{i}\\}_{i=0}^{n}$555Note that we count from $i=0$, only in this Section, in order to match our notation with the common definition of Lagrange polynomials., and a set of associated function values $\\{f_{i}:=f(x_{i})\\}_{i=0}^{n}$. We assume that $f$ is a real valued, one- dimensional function. We will discuss the more challenging multivariate case in Section 9.5. The interpolation problem consists of using the partial function sampling $\\{f_{i}\\}_{i=0}^{n}$ to approximate $f(x)$ by another function which agrees with $f$ at the interpolating nodes $\\{x_{i}\\}_{i=0}^{n}$. To be concrete, let’s focus on linear interpolation: we require the interpolant of $f$ to be expressible as a linear combination of the $(n+1)$ independent basis functions $\\{e_{i}(x)\\}_{i=0}^{n}$, ${\cal I}_{n}[f](x):=\sum_{i=0}^{n}C_{i}e_{i}(x)\,.$ (70) We already discussed in section 6 that the optimal least-squares-sense approximation of the above form is provided by orthogonal projection onto the basis $\\{e_{i}(x)\\}_{i=0}^{n}$. In the interpolation scheme one trades the least-squares criteria for one which requires less information about $f$. Let’s define this criteria by making the interpolant being equal to the function at the nodes, ${\cal I}_{n}[f](x_{j})=f(x_{j})\,,\quad j=0,\ldots,n\,.$ (71) When the basis are linearly independent polynomials of degree $\leq n$ we call this special case Polynomial interpolation. One can show by explicit construction that there is a unique polynomial which satisfies the interpolation problem (70,71). Before doing so we give an example. Example 5. Suppose we have two points, $\\{x_{0},x_{1}\\}$, sampling a function $f(x)$. Let the sample values be $f_{0}=f(x_{0})$ and $f_{1}=f(x_{1})$. The functions $\\{1,x\\}$ comprise a suitable basis for our $n=1$ polynomial interpolant. Then the interpolating polynomial is of degree $\leq 1$, i.e. a straight line joining the two points: ${\cal I}_{1}[f](x)=ax+b$ We explicitly solve for $(a,b)$ by writing the interpolation problem (71) $\displaystyle{\cal I}_{1}[f](x_{0})$ $\displaystyle=$ $\displaystyle ax_{0}+b=f_{0}$ (72) $\displaystyle{\cal I}_{1}[f](x_{1})$ $\displaystyle=$ $\displaystyle ax_{1}+b=f_{1}$ (73) which has as solutions $a=\frac{f_{0}-f_{1}}{x_{0}-x_{1}}\,\,,\,\,b=\frac{x_{0}f_{1}-x_{1}f_{0}}{x_{0}-x_{1}}\,.$ (74) Thus, ${\cal I}_{1}[f](x)=\left(\frac{f_{0}-f_{1}}{x_{0}-x_{1}}\right)x+\left(\frac{x_{0}f_{1}-x_{1}f_{0}}{x_{0}-x_{1}}\right)\,.$ (75) General solution: Lagrange polynomials Existence. The existence of a polynomial which solves the interpolation problem (71) can be shown by explicit construction. Suppose we have access to a basis $e_{i}(x)=\ell_{i}^{(n)}(x)$ which satisfies ${\cal I}_{n}[f](x)=\sum_{j=0}^{n}f_{j}l_{j}^{(n)}(x)$ (76) where, for each $j=0\ldots n$, $l_{j}^{(n)}(x)$ is a polynomial of degree $\leq n$ such that $l_{j}^{(n)}(x_{i})=\delta_{ij}\,\,\,\mbox{for }i=0\ldots n$ (77) and $\delta_{ij}$ is the Kronecker delta ($\delta_{ij}=1$ if $i=j$ and zero otherwise). Then ${\cal I}_{n}[f](x)$ as given by Eq. (76) would be a solution to the interpolation problem. The Lagrange polynomials are precisely those with the property (77). They are given by $l_{j}^{(n)}(x)=\left(\prod_{k=0,k\neq j}^{n}(x-x_{k})\right)/\left(\prod_{k=0,k\neq j}^{n}(x_{j}-x_{k})\right)\,.$ (78) Example 6. We look at the same case ($n=1$) of Example 9.2, but now we construct an interpolant through Lagrange polynomials. By Eq. (78) these are $\l_{0}^{(1)}(x)=\frac{x-x_{1}}{x_{0}-x_{1}}\;\;\;,\;\;\;\l_{1}^{(1)}(x)=\frac{x-x_{0}}{x_{1}-x_{0}}\,,$ (79) and using (76) the Lagrange form of the interpolant becomes $\displaystyle{\cal I}_{1}[f](x)$ $\displaystyle=$ $\displaystyle f_{0}l_{0}^{(1)}(x)+f_{1}l_{1}^{(1)}(x)$ (80) $\displaystyle=$ $\displaystyle\left(\frac{f_{0}-f_{1}}{x_{0}-x_{1}}\right)x+\left(\frac{x_{0}f_{1}-x_{1}f_{0}}{x_{0}-x_{1}}\right)$ The result (80) is the same as (75) due to the uniqueness of the polynomial interpolant, which we now prove. Uniqueness. Suppose there are two polynomials $P_{n}$ and $R_{n}$ of degree $\leq n$ satisfying the interpolation condition $P_{n}(x_{j})=R_{n}(x_{j})=f_{j},\,j=0,\ldots,n$. The polynomial $Q(x):=P_{n}(x)-R_{n}(x)$ then satisfies $Q(x_{j})=0,\,j=0,\ldots,n$. This means that $Q(x)$ has $(n+1)$ zeros, in contradiction with the hypothesis that it is at most of degree $n$. Ergo, $Q(x)=0$ and $P_{n}(x)=R_{n}(x)$. The Vandermonde form Yet another way of showing existence and uniqueness for the polynomial interpolant in a constructive way is the following. We write down the polynomial interpolant using monomial basis functions, ${\cal I}_{n}[f](x)=\sum_{j=0}^{n}a_{j}x^{j}\,.$ The interpolation conditions (71) become a set of linear equations for the coefficients $a_{j}$ ${\bf V}a=f\,,$ (81) where ${\bf V}_{ij}=x_{i}^{j-1}$, $a=[a_{0},\ldots,a_{n}]^{\top}$ and $f=[f(x_{0}),\ldots,f(x_{n})]^{\top}$. The matrix ${\bf V}$ is called the Vandermonde matrix and its determinant, also called the Vandermonde polynomial, takes the form $\det({\bf V})=\prod_{0\leq i<j\leq n}(x_{j}-x_{i})\,.$ In order to have one and only one solution to Eq. (81) this determinant has to be non-zero. In one spatial dimension this condition is automatically satisfied whenever the interpolation points are unique. There are ways of inverting ${\bf V}$ in order to solve Eq. (81) using Lagrange polynomials, which we will not pursue here. The main reason for briefly mentioning the Vandermonde matrix is because a generalization of it will play a prominent role in the Empirical Interpolation Method, described in Section 10. #### 9.3 Convergence rates, collocation points and Runge’s phenomenon Here we discuss the behavior of the error of the polynomial interpolant ${\cal I}_{n}[f](x)$ of some function $f(x)$ at $(n+1)$ nodes, to illustrate the difficulties in obtaining an accurate interpolation scheme on a unstructured set of nodes which, from the point of view of ROM, one wishes to be sparse. Leaving aside both sparsity and unstructured meshes, one might imagine that for “nice enough” functions (for example, infinite differentiable), this error decreases as the number of nodes and, therefore, degree of the polynomial basis $n$ increase. This is not the case, the classical counterexample is Runge’s one, here described, and the overall problem is referred to as Runge’s phenomenon. The distinction is between local approximations of fixed degree and global ones. Because these two concepts play a key role in reduced order modeling we discuss them in this context with some detail. Suppose that one has $\\{x_{i}\\}_{i=0}^{n}$ nodes at which to interpolate a function. Local interpolation We consider local interpolants of degree one because we have already explicitly discussed them in Example 75. One would partition the interval of interest, where on each subinterval $I_{j}:=[x_{j},x_{j+1}]\,,\quad j=0,\ldots,n-1$ (82) one builds a local interpolant of degree one using the end points as interpolating nodes, ${\cal I}[f](x)=\left(\frac{f_{j}-f_{j+1}}{x_{j}-x_{j+1}}\right)x+\left(\frac{x_{j}f_{j+1}-x_{j+1}f_{j}}{x_{j}-x_{j+1}}\right)\,,\quad\text{for }x\in I_{j}\,,\quad j=0,\ldots,n-1\,.$ (83) Such fixed order interpolants, where their degrees are independent of the number of nodes $n$, is – as explained later below – guaranteed to converge to the interpolated function as the number of nodes in a given fixed physical interval increases (the gridspacing decreases). Furthermore, such convergence is guaranteed regardless of the structure/location of the interpolating nodes, they can even be randomly located. Even more, to some degree, convergence is also guaranteed regardless of the smoothness of the data; this is why fixed order methods, especially very low order ones, are in general used for noisy data. This class of methods is usually very robust, but at the expense of accuracy. The latter is exacerbated in the absence of a dense amount of data. On the contrary, in Reduced Order Modeling one seeks very sparse representations of very high accuracy. This combination of, and even trade-off between, robustness (convergence) and accuracy is a delicate and challenging issue in ROM. Global interpolation Instead, one could use a global interpolant, where all $(n+1)$ nodes and function values are used to build a single interpolant, ${\cal I}_{n}[f](x)=\sum_{i=0}^{n}\ell_{i}^{(n)}(x)f_{i}\,.\quad\forall x\in[x_{0},x_{n}]\,.$ As the density of points and $n$ increases, one would keep using all of them, thereby increasing the degree of the interpolant. This might seem desirable and more accurate than a local approach of fixed degree, since more information is used. The classical counterexample is due to Runge [118], and it consists of interpolating the function $f(x)=\frac{1}{1+(5x)^{2}}\;\,,\quad x\in[-1,1]\,.$ (84) This function is $C^{\infty}$ (has infinite derivatives) yet the error of a global polynomial interpolant at equally spaced points diverges near the boundaries [52] at higher orders (see Fig. 5). For unstructured points, such as those usually appearing in reduced models, the lack of convergence of global approximations is in general worse. Figure 5: Runge’s phenomenon in polynomial interpolation. In this example the approximation uses 19 equispaced nodes over the interval $[-1,1]$ and the interpolant polynomial is of degree 18. The interpolation tends to diverge near the boundaries. This issue cannot be solved by increasing the resolution of the sampling but the opposite: divergencies get worse for equispaced nodes. As discussed below, this can be resolved and fast convergence achieved if the interpolation nodes can be appropriately chosen. However, standard choices for such special nodes are in general not suitable for reduced order modeling. In particular, they are neither necessarily sparse nor adapted to the problem of interest. They are generic and in general (and most important) not hierarchical, requiring recomputing from scratch the sought approximation if higher accuracy is required. Error analysis and convergence I In what follows we assume that $f$ is smooth enough as needed from the context. Denoting the interpolation error by $E_{n}(x):=\left\|f(x)-{\cal I}_{n}[f](x)\right\|$ the following result holds: $E_{n}(x)=\left\|\frac{1}{(n+1)!}f^{(n+1)}(\xi)\omega_{n+1}(x)\right\|$ (85) where $\omega_{n+1}(x):=\prod_{i=0}^{n}(x-x_{i})$ is called the nodal polynomial of order $(n+1)$, and $\xi$ is in the smallest interval $I_{x}$ containing $x_{0}\ldots x_{n}$ and $x$. Equation (85) can be proven as follows. Let’s define the function $F(t)=f(t)-{\cal I}_{n}[f](t)-\alpha(x)\omega_{n+1}(t)$ where $\alpha(x)=\frac{f(x)-{\cal I}_{n}[f](x)}{\omega_{n+1}(x)}$ This function is well defined since $x\neq x_{i}$, and it is as smooth as $f(t)$. Observe that $F(x)=0$ and, in particular, $F(x_{i})=0$ ($i=1,...,n$). This means that $F(t)$ has $(n+2)$ distinct zeros in the interval in which $f(t)$ is smooth. Now, the Mean Value Theorem tells us that, between any couple of zeros, there is one zero for the derivative $F^{\prime}(t)$. Then the function $F^{\prime}(t)$ has $(n+1)$ zeros. Repeating the previous argument, we conclude that $F^{(n+1)}(t)$ has exactly one zero. Let’s call it $\xi$. Thus $F^{(n+1)}(\xi)=f^{n+1}(\xi)-\alpha(\xi)(n+1)!=0$. The statement in (85) follows directly from this relation. Let’s make a comment here. The point $x$ at which the interpolant is evaluated when computing the error (85) does not need to be in the smallest interval containing $x_{0}\ldots x_{n}$. Assuming an ordering $x_{0}<\ldots<x_{n}$, sometimes the process of approximating $f(x)$ by ${\cal I}_{n}[f](x)$ is called interpolation only if $x\in[x_{0},x_{n}]$ and extrapolation otherwise. We get back to the error formula (85) and analyze its behavior as $n$ increases. In particular, we ask ourselves under what conditions does $E_{n}(x)\rightarrow 0\;\;\;\mbox{as}\;\;\;n\rightarrow\infty\,.$ The error (85) can be decomposed in two terms, one related to the behavior of the derivatives of $f$, $\left\|\frac{f^{(n+1)}(\xi)}{(n+1)!}\right\|$ and another one related to the distribution of nodes through $\omega_{n+1}(x)$. We first discuss why fixed order interpolants are guaranteed to converge, regardless of the distribution of nodes (assuming the interpolated function is smooth enough), since the argument is simple. Discussion 1. Local interpolants For definiteness consider the case of equally spaced points (the generalization to an arbitrary set is straightforward), $x_{j}=x_{0}+j\Delta x\,,\quad j=0,\ldots n\,.$ Then the error of the interpolant (83) on each subinterval (82) is bounded by $E\leq\frac{1}{(m+1)!}\max_{x\in I_{j}}\left\|f^{(m+1)}(x)\right\|(\Delta x)^{m}\,,$ (86) where $m$ is the (fixed) number of interpolation nodes and degree of the interpolants (clearly, $m\leq n$). Since $m$ is fixed, the convergence rate is determined by $\Delta x\rightarrow 0$, and is of order $m$. Discussion 2. Global interpolants In this case $m=n$. One can see from the interpolation error (85) that convergence is guaranteed if the derivatives of $f$ decay with $n$ sufficiently fast. In fact, the interpolant converges in the infinity norm if all derivatives $f^{(s)}$ are bounded by the same constant in the interval of interest. However, it is not obvious from (85) to find out in a straightforward way the allowed growth rate of the derivatives of $f$ such that convergence/divergence of the error takes place. In fact, it is not easy to see how such error formula explains Runge’s phenomenon without resorting to complex calculus [52]. However, we can ask ourselves the next related questions: (Q1) What can be said about the maximum of $\|\omega_{n+1}(x)\|$ in the interval of interest? (Q2) If the physical problem allows such freedom, how can one choose the nodes so as to minimize such maximum? For simplicity and without loss of generality we focus on the interval $x\in[-1,1]$. We can answer these questions as: Answer to (Q1): For all choices of nodes $\max_{[-1,1]}\|\omega_{n+1}(x)\|\geq 2^{-n}$ (87) Answer to (Q2): If $\omega_{n+1}(x)=2^{-n}T_{n+1}(x)\,,$ where $T_{j}$ denotes the Chebyshev polynomial of degree $n$, then $\max_{[-1,1]}\|\omega_{n+1}(x)\|=2^{-n}\,.$ In other words, using the roots of the Chebyshev polynomial as interpolating nodes minimizes the error associated with the location of such nodes. It is in this sense that Chebyshev nodes are many times referred to as optimal points for interpolation. They have the disadvantage, though, that they are note hierarchical/nested, or adapted to the problem of interest. The former failure means, from a ROM perspective, that any model relying on them, including perhaps expensive numerical simulations, have to be completely built from scratch if higher accuracy is required. This is solved by a nearly-optimal set of nested nodes for application-specific interpolants designed with ROM problems in mind, discussed in Section 10. Error analysis and convergence II In principle, direct approximation through projection requires the knowledge of the continuum waveform, while interpolation only needs information at a (specified by the user) set of nodes. We next address what is the relationship between these two strategies. In terms of representation accuracy, we focus on the $L_{2}$ norm, since it is directly related to the overlap error commonly used in data analysis. Since, as discussed, given a basis and linear approximation, projection is optimal in the least squares sense, one knows that $\\|h-{{\cal P}}_{n}h\\|^{2}\leq\\|h-{\cal I}_{n}[h]\\|^{2}\,.$ (88) The question is how “sub-optimal” in the above sense interpolation is. The answer obviously depends on the basis and interpolation nodes. A general framework to study this is through the definition of the Lebesgue constant. For any arbitrary basis and interpolation scheme one can derive the more precise bound related to (88): $\\|h-{\cal I}_{n}[h]\\|^{2}\leq\Lambda_{n}\\|h-{{\cal P}}_{n}h\\|^{2}\,,$ (89) where $\Lambda_{n}:=\\|{\cal I}_{n}\\|^{2}\,.$ (90) Comment 4. 1. 1. The $\Lambda_{n}$’s are referred as the Lebesgue constants and depend, given a vector norm, purely on the interpolation operator ${\cal I}_{n}$. 2. 2. In terms of accuracy, the strategy is to build an interpolant (by choosing both the basis and nodes) such that the Lebesgue constant grows as slow as possible with $n$. It is a computable quantity, so one can actually quantify how much is “lost” with respect to projection through Eq. (89). 3. 3. Strictly speaking, the Lebesgue constant is usually referred to in the context of the infinity norm, $\\|u\\|_{\infty}:=\max_{i}\|u_{i}\|\,,$ but throughout this review, unless otherwise stated, we refer to it in the $L_{2}$ norm. As an example, and going back to Runge’s phenomenon at equally spaced versus Chebyshev nodes for global polynomial interpolation, the following illustrates the advantages of the latter and explains why it is such a popular choice – when it is a feasible approach at all within the problem of interest, which is not always the case in practice. Example 7. Here the $\\|.\\|_{\infty}$ norm is used. For large $n$ one has the following behavior (see, for example, [71]). 1. 1. For equally spaced nodes, the Lebesgue constant grows exponentially, as $\Lambda_{n}\sim\frac{2^{n+1}}{n\log n}\,.$ (91) 2. 2. For Chebyshev nodes, it only grows logarithmically, $\Lambda_{n}\sim\log{(n)}\,.$ (92) In the context of polynomial interpolation, it can be proved that the Lebesgue constant grows at least logarithmically in the infinity norm for any selection of nodes. In particular, Chebyshev nodes belong to a family that induce a nearly-optimal behavior on the Lebesgue constant, since the growth (92) is only logarithmic. For thorough discussions around this topic see [71]. #### 9.4 Discrete expansions and interpolation In approximation through projection one in principle needs full knowledge of the function $f$ to be approximated through the projection coefficients $\langle e_{i},h\rangle$ in Eq. (47). In interpolation, instead, one only needs knowledge of the function at the interpolating nodes. In practice, the integrals to compute $\langle e_{i},h\rangle$ are replaced by a quadrature rule, i.e. a numerical approximation (we discuss quadrature rules and specifically those built using ROM in Section 13). These are called then discrete projection approximations. Here it suffices to say that, if the quadrature nodes can be chosen, one can find an optimal family of quadrature rules that maximizes the degree for which the quadrature is exact for polynomials. These are called Gaussian quadratures and the nodes Gaussian nodes; examples of the latter are Legendre and Chebyshev nodes. The relevant result here relating approximation by discrete projection and interpolation is the following: Result 1. Discrete projection and interpolation If the projection coefficients are approximated by Gaussian quadratures and are used Gaussian nodes for interpolation, then the resulting approximation ${\cal P}_{n}h$ exactly equals the polynomial interpolant ${\cal I}[h]$ [71]. Through this result one can guarantee fast convergence of interpolation using Gauss nodes. The problem is that in many parametrized problems of interest, one cannot choose Gaussian nodes in parameter space as representative solutions. Either because data is not available (for example, if taken from experiments), or because it is not efficient to do so from a modeling perspective. In Section 10 we discuss a generalization for ROM and parametrized systems that attempts to generalize Result 9.4. #### 9.5 Multiple dimensions Evaluation cost of multivariate polynomials Even when counting with enough data for multi-dimensional fits, enough for accurate local approximations which avoid Runge’s phenomenom, a perhaps not so well known fact is that the evaluation cost of multivariate polynomials in general grows exponentially with the dimensionality of the problem. Naive evaluation of a $1$-dimensional polynomial of degree $n$, that is, evaluating all the monomials in the standard format $p_{n}(x)=a_{0}+a_{1}x+\ldots+a_{n}x^{n}$ (93) carries an operation count of ${\cal O}(n^{2})$. Through a simple factorization, Horner’s rule (see [132]), the operation count can be reduced to $2n$. Suppose we want to evaluate $p_{n}(x)$. First note that $p_{n}(x)$ can be decomposed in a nested way $p_{n}(x)=a_{0}+x(a_{1}+x(a_{2}+\cdots x(a_{n-1}+a_{n}x)\cdots)\,.$ Horner’s rule consists in computing iteratively the recursion $\begin{split}b_{0}=&a_{n}\\\ b_{k}=&a_{n-k}+b_{k-1}x\quad k=1,\cdots,n\end{split}$ reaching $p_{n}(x)=b_{n}$ in $2n$ operations (multiplication and addition, $n$ times). It can be shown that this is the minimum number of operations to evaluate a one-dimensional problem unless some offline factorization is carried out for multiple online evaluations, in which case the computational cost can be slightly reduced. Next, consider multivariate polynomial evaluation. Suppose one attempts to evaluate the generalization of (93) to the $d$-dimensional case. For definiteness, the form in the ${\tt dim}=2$ case would take the form $p_{n}(x,y)=\sum_{i=0}^{n_{x}}\sum_{j=0}^{n_{y}}a_{ij}x^{i}y^{j}\,.$ The evaluation cost of such an approach in the $d$-dimensional case, assuming for simplicity the same degree $n$ in each dimension, is of order ${\cal O}(n^{2d})$. Ideally, one would wish a generalization of Horner’s rule with, say, evaluation cost of order ${\cal O}(d\times n)$. Unfortunately not such algorithm is known, neither a proof of which could be the optimal number of operations needed to perform a multivariate polynomial evaluation. In recent years there have been several proposals to face this problem, ranging from rigorous mathematical attempts (see e.g. [106, 98, 141, 81, 18, 18, 77, 142]) to the proposal of heuristic greedy algorithms [37] aiming to reduce the number of calculations. Multivariate polynomial evaluations scale exponentially with the dimensionality of the problem. This is a severe issue in most cases of interest which cannot be underestimated. In the context of ROM, the limitations of multivariate polynomial approximations appear in almost every possible context. This point is discussed in more detail and emphasized in Section 15. #### Further reading Reference [58] reviews multivariate polynomial interpolation, and [99] presents a proposal for polynomial interpolation on unstructured grids in multiple dimensions. There are schemes for fast multiple evaluations of multi-variate polynomials though. They are based on an offline factorization of the given polynomial (which can be expensive, but done only once), with a total cost of ${{\cal O}}(n^{{\tt dim}+1})$ for ${\cal O}(n^{\tt dim})$ evaluations, leading to an average of ${\cal O}(n)$ for each evaluation [84]. To our knowledge this problem of multi-dimensionality has not been yet tackled in ROM for gravitational waves and is still largely an open problem in approximation theory. Instead, in GW science local interpolants in the form of splines are being used, which results in degradation of the accuracy of the surrogate models being built. In terms of higher (than one) dimensions, sparse grids are being used. One of their problems is that they are not necessarily hierarchical, leading to a problem when building training sets from numerical relativity to later apply the RB framework. ### 10 The Empirical Interpolation Method The Empirical Interpolation Method (EIM) was proposed in 2004 [20] as a way of identifying a good set of interpolation nodes on multi-dimensional unstructured meshes and has since found numerous applications (for a very short list, see [1, 39, 40, 51, 90]). When the basis is application-specific, so is the EIM. Here, perhaps contrary to what might be suspected, interpolation is in the physical dimension(s), not on the parametric ones. For definiteness, in the case of gravitational waves this is either the time or frequency domain. Before getting into technical details, we highlight some of the properties of the EIM: * • It is hierarchical (nested). The “most relevant” points in the physical dimension(s) and associated interpolants are selected and built, respectively. This follows a process that is dual, in a precise sense, to the construction of a reduced basis using a greedy algorithm. * • It is designed to overcome the difficulties of stability, accuracy and evaluation cost on sparse and scattered grids, especially in multiple dimensions. * • It is highly accurate. By design the algorithm attempts to control the behavior of the Lebesgue constant (see Section 10.3) at each greedy EIM sweep. * • It provides an affine parametrization which has multiple consequences. In particular, it allows for the design of Reduced Order Quadratures, described in Section 13, which lead in particular to fast likelihood evaluations, as discussed in Section 14. The material below relies on standard interpolation ideas which we have summarized in Section 9. We will refer to the physical variable(s) as $x$, the nodes selected by the EIM as $X_{i}\,(i=1\ldots n)$ 666Except for the GW case, in which we will explicitly use $t$ and $T$ – or $f$ and $F$, in the case of frequency., the (in general multidimensional) parameter as $\lambda$ and its corresponding greedy points as $\Lambda_{i}\,(i=1\ldots n)$. It is not a coincidence that there is the same number of EIM nodes and greedy points, in fact they go hand by hand. #### EIM: algorithm and properties Within the EIM method one seeks to find an empirical (that is, problem- dependent) global interpolant, which also provides an affine parametrization of functions. That is, the EIM approximation of a function $h(x;\lambda)$ is of the form $h(x;\lambda)\approx{\cal I}_{n}[h](x;\lambda)=\sum_{i=1}^{n}B_{i}(x)h(X_{i};\lambda)\,.$ (94) The affine parametrization of ${\cal I}_{n}[h](x;\lambda)$ refers to the fact that the r.h.s. of (94) is a sum of terms which depend only on $x$ (the $B_{i}$ coefficients) multiplied by other ones which depend only on $\lambda$ (the $h(X_{i};\lambda)$); one might also want to refer to this as “separability”. Below in Section 10.1 we discuss how one arrives at (94), how to compute in the offline stage the $B_{i}$ coefficients, and the quality of this approximation. In addition, even though it is perhaps not usually viewed this way, or not emphasized enough compared to online evaluations to solutions of parametrized PDEs, the EIM also provides an application-specific down- and upsampling that, for practical purposes, beats Nyquist downsampling, with implications for signal or more generally data processing. See for example Section 11.2 and discussion around Figures 12 and 13. #### 10.1 From projection to interpolation In Section 3 we discussed that the optimal linear representation in any weighted $L_{2}$ norm, given a basis (assuming for simplicity and conditioning that it is orthonormal) of cardinality $n$ is through an expression of the form $h(x;\lambda)\approx{\cal P}_{n}h(x;\lambda):=\sum_{i=1}^{n}c_{i}(\lambda)e_{i}(x)\,,$ (95) where $c_{i}(\lambda)=\langle e_{i}(\cdot),h(\cdot;\lambda)\rangle\,.$ (96) This is the standard approximation by projection, the most common one being using polynomials or Fourier modes as basis. Now, approximation through projection requires knowledge of the function to be represented at sufficient values of $x$ so as to accurately compute the coefficients (96). In Section 9 we discussed how this is usually replaced by interpolation, and the strong relationship between projection and interpolation in standard spectral theory. Next we describe the EIM strategy, which mimics the spectral approach. The approximation through projection onto a reduced basis, Eq. (95), is replaced by interpolation (in the physical dimension(s), $x$) as follows. First a rb $\\{e_{i}(x)\\}$ is chosen, for example through a POD or greedy approach. The interpolant then is sought to have the form ${\cal I}_{n}[h](x;\lambda):=\sum_{i=1}^{n}C_{i}(\lambda)e_{i}(x)\,,$ (97) where on purpose the expansion, which are not projection, coefficients $C_{i}$ are denoted by capital letters to distinguish them from the projection ones (96). Instead, they are defined to be solutions of the interpolation problem; namely that the interpolant exactly agrees with the function at the EIM nodes (the construction of which we discuss below) $\displaystyle{\cal I}_{n}[h](X_{i};\lambda)=h(X_{i};\lambda),\qquad\forall\,i=1,\dots,n.$ (98) For the moment, we shall assume that the EIM nodes $X_{i}$ are known and proceed to describe how to use them to find the EIM interpolant. This can be somewhat misleading, since it is not the way it works in practice, which is: the first EIM node is found, its associated interpolant built, the second EIM node found, the interpolant enriched to take it into account, and so on. They are not disjoint processes, unlike (again) standard spectral methods, where all the Gaussian nodes are found and afterwards the interpolant built. This is not only a procedural difference, but highlights a big difference of the EIM: namely that the approach (nodes and interpolant) is hierarchical. Hopefully this gradual presentation is intuitive and pedagogical, later in this section we will present the full, coupled algorithm together with its numerical intricacies. Equation (98) is equivalent to solving the $n$-by-$n$ system $\sum_{i=1}^{n}{\bf V}_{ji}C_{i}(\lambda)=h(X_{j};\lambda)\,,\quad j=1,\ldots,n$ (99) for the coefficients $\\{C_{i}\\}_{i=1}^{n}$, where the interpolation matrix $\bf{V}:=\left(\begin{array}[]{cccc}e_{1}(X_{1})&e_{2}(X_{1})&\cdots&e_{n}(X_{1})\\\ e_{1}(X_{2})&e_{2}(X_{2})&\cdots&e_{n}(X_{2})\\\ e_{1}(X_{3})&e_{2}(X_{3})&\cdots&e_{n}(X_{3})\\\ \vdots&\vdots&\ddots&\vdots\\\ e_{1}(X_{n})&e_{2}(X_{n})&\cdots&e_{n}(X_{n})\\\ \end{array}\right)$ (100) is a generalization of the Vandermonde matrix – see Eq. (81) – when polynomial basis are used. We have already discussed that in that case the Vandermonde matrix can easily be very ill-conditioned if the nodes are chosen, for example, equally spaced, not to mention if they are scattered. So one can anticipate that the solution to the problem (97) is already non-trivial and one of the goals of the EIM is to make sure that it does not lead to an ill conditioned problem, as well as providing a high accuracy interpolant. The choice of empirical nodes given by the EIM together with the linear independence of the reduced basis ensure that $\bf{V}$ (Eq. (100)) is invertible so that $C_{i}=\sum_{j=1}^{n}\left({\bf V}^{-1}\right)_{ij}h(X_{j};\lambda)$ (101) is the unique solution to (98). It then follows upon substituting (101) into (97) that the empirical interpolant is ${\cal I}_{n}[h](x;\lambda)=\sum_{j=1}^{n}B_{j}(x)h(X_{j};\lambda)$ (102) where $B_{j}(x):=\sum_{i=1}^{n}e_{i}(x)\left({\bf V}^{-1}\right)_{ij}$ (103) is independent of $\lambda$. Note that (102) is a linear combination of the waveform itself evaluated at the empirical nodes. The coefficients $\\{B_{i}\\}_{i=1}^{n}$ satisfy $B_{i}(X_{j})=\delta_{ij}$ and are built directly from the reduced basis. They provide a clean offline/online separation. Because of this the $\\{B_{i}\\}_{i=1}^{n}$ functions can be pre-computed offline once the reduced basis is generated while the (fast) interpolation is evaluated during the online stage from (102) when the parameter $\lambda$ is specified by the user. Evaluations of the waveform at the EIM nodes are still needed at the arbitrarily chosen parameter $\lambda$ in order to construct the interpolant in (102). One can wonder how this can be of any use. In Section 11 we explain how to build surrogate predictive models for the evaluations of $h(X_{j};\lambda)$ for any $\lambda$ (that is, not present in the original training space). Next we discuss how to compute the EIM nodes in a rather qualitative way, before presenting the general algorithm later on. Example 8. Finding the EIM nodes and building the interpolant. Consider for definiteness a time series. The algorithm takes as input the basis set $\\{e_{i}\\}_{i=1}^{n}$ and an arbitrary number and choice of time samples $\\{t_{i}\\}_{i=1}^{L}$ from which the empirical interpolation nodes $\\{T_{i}\\}_{i=1}^{n}$ are to be selected. The EIM algorithm proceeds as follows 1. 1. The first time node is chosen to maximize the value of $\|e_{1}(t_{i})\|$; that is, $\left\|e_{1}(T_{1})\right\|\geq\left\|e_{1}(t_{i})\right\|$ for all time samples $t_{i}$. 2. 2. Next, an empirical interpolant for the second basis function is built using only the first basis function: From Eqs. (97,98) we have ${\cal I}_{1}[e_{2}](t)=C_{1}e_{1}(t)$ where $C_{1}=e_{2}(T_{1})/e_{1}(T_{1})$ has been found from Eq. (98) with $i=1$. 3. 3. The second empirical interpolation node is chosen to maximize the value of the pointwise interpolation error of ${\cal I}_{1}[e_{2}](t)-e_{2}(t)$; that is, $\left\|{\cal I}_{1}[e_{2}](T_{2})-e_{2}(T_{2})\right\|\geq\left\|{\cal I}_{1}[e_{2}](t_{i})-e_{2}(t_{i})\right\|$ for all data samples. 4. 4. Steps $2$ and $3$ are then repeated to select the remaining $(n-2)$ nodes. The full algorithm for generating the EIM nodes is shown in Algorithm 2. Comment 5. As described, the EIM follows a greedy approach, albeit somewhat different from that one usually used to build a reduced basis. While a greedy algorithm to build a RB selects the most relevant points in parameter space, the EIM selects the most relevant points in the physical dimension(s). In addition, the former uses a (possibly weighted) $L_{2}$ norm while the EIM uses the infinity one. Comment 6. Notice also that even though constructing the EIM is also done offline, it has a much smaller computational cost than constructing the basis, since it operates only on the latter, the training set is not involved anymore. The following example graphically shows the first three iterations of the EIM algorithm applied to Legendre polynomials. Example 9. Consider the set $\\{P_{i}(x)\\}_{i=0}^{n}$ of $(n+1)$ normalized Legendre polynomials defined on $[-1,1]$. These form an orthonormal basis for the space of degree $n$ polynomials wrt the weight function $\omega(x)\equiv 1$. Their ordering is important for approximations: expanding, by orthogonal projection, a smooth function using the first $n$ Legendre polynomials typically results in exponential convergence [71, 146]. Said another way, $P_{0}$ is the most important basis function, $P_{1}$ the next most important, and so on. Given a convergence rate for the aforementioned Legendre projection-based approximation we might wonder how much accuracy is lost by trading it for the interpolation (97) and how to optimally choose the nodes $\\{X_{i}\\}$. When the relevant error measurement is the maximum pointwise error, Chebyshev nodes are known to be well suited for interpolation, bringing an additional error which is bounded by $\log(n)$ [108, 113], as discussed in Section 9.3 (see Example 5). For this reason, Chebyshev nodes are specially tailored to benchmark the nodes selected by the EIM in this example. We select the EIM nodes according to the same preferential ordering as the Legendre polynomials themselves; the first six normalized polynomials are shown in Fig. 6. Figure 6: The first six normalized Legendre polynomials. For this example we chose the first 24 Legendre polynomials as the initial basis for feeding the EIM algorithm. Figure 7 shows the process of the EIM algorithm to choose the interpolation nodes for the Legendre basis. The first EIM node is defined by the location of max$(\|P_{0}\|)$. Since $P_{0}$ is a constant function there is no preference in the search, so we simply select the middle point $x=0$. To identify the second node we 1. 1. build the empirical interpolant ${\cal I}_{0}[P_{1}]$ of $P_{1}$ using $P_{0}$ as the basis and $x=0$ as interpolation node (the first EIM node), 2. 2. compute the pointwise error $r_{1}=P_{1}-{\cal I}_{0}[P_{1}]$, and 3. 3. select the second EIM node by the location of max$(\left\|r_{1}\right\|)$. In this case we find $r_{1}=P_{1}$ since ${\cal I}_{0}[P_{1}]$ is zero due to the fact that $P_{1}(0)=0$. The process continues until the number of EIM nodes equals the number of basis elements. Figure 7: Representation of the EIM algorithm acting on the first 24 normalized Legendre basis. We show the bases, residuals and EIM nodes involved in the iterations. The first three figures (upper-left, upper-right and bottom-left) illustrate the two steps of the algorithm that maximize residuals and select the EIM nodes. The last panel (bottom-right) shows the latest iteration of the algorithm, which corresponds to the $24$th basis. The vertical line corresponds to the residual’s maximum. In Fig. 8 we show the $24$ EIM nodes found by the algorithm and compare them to the first 24 Chebyshev nodes in the interval $[-1,1]$. Both node distributions are qualitatively similar. Note that the EIM distribution emulates the clustering of the Chebyshev nodes at the edges $x\pm 1$. Reference [90], which has inspired our numerical experiment with Legendre polynomials, also compares the Lebesgue constant(s) for the EIM nodes of this example and the and Chebyshev ones. Figure 8: A direct comparison between the distribution of the 24 EIM nodes for the Example 10.1 and the first 24 Chebyshev nodes. Example 10. This example, taken from Ref. [34], graphically shows the EIM algorithm’s application to basis functions for which a good set of interpolation nodes is unknown. This is a sine-Gaussian family of the form $h(t;\lambda):=Ae^{-(t-t_{c})^{2}/(2\alpha^{2})}\sin(2\pi f_{0}(t-t_{c}))\,,$ (104) which models gravitational waves from generic burst processes. Here $A$, $f_{0}$ and $\alpha$ are the amplitude, frequency and width of the waveform $h$ respectively, $t_{c}$ is the arrival time of the GW-burst signal, and $t\in[-\infty,\infty]$. The Fourier transform (FT) of this waveform is given by $\displaystyle{\tilde{h}}(f,t_{c};\lambda)=e^{i2\pi ft_{c}}{\tilde{h}}(f;\lambda)\,,$ (105) where ${\tilde{h}}(f;\lambda)$ is the FT of the GW-burst at $t_{c}=0$: $\displaystyle{\tilde{h}}(f;\lambda)=i2A\alpha\sqrt{2\pi}\sinh(4\pi^{2}\alpha^{2}f_{0}f)e^{-2\pi^{2}\alpha^{2}(f_{0}^{2}+f^{2})}.$ (106) This waveform family is, then, described by four free parameters $\lambda=(\alpha,f_{0},t_{c},A)$. We focus on the two parameters $(\alpha,f_{0})$, since the others are extrinsic and can be handled differently. The parameter space considered is $\alpha=[.02,2]\sec\quad\,,\quad f_{0}=[.01,1]{\rm Hz}\,,$ (107) Figure 9 provides a graphical illustration of the EIM first iterations. Figure 9: Iterations 1 (left) and 2 (right) of the EIM algorithm for Example 10.1. The first EIM node is defined by the location of max$(\|e_{1}\|)$. To identify the second node one: i) builds the empirical interpolant ${\cal I}_{1}[e_{2}]$ of $e_{2}$ using $e_{1}$ and the EIM node $F_{1}$, ii) computes the pointwise error ${\cal I}_{1}[e_{2}]-e_{2}$; iii) the second EIM node is then defined by the location of max$(\left\|{\cal I}_{1}[e_{2}]-e_{2}\right\|)$. The process continues until all $n$ empirical interpolation nodes are found. Figures taken from [34]. #### 10.2 EIM algorithm We present the EIM algorithm as used/introduced in Ref. [54]. Algorithm 2 The Empirical Interpolation Method 1:Input: rb = $\\{e_{i}\\}_{i=1}^{n}$ 2:$X_{1}=\text{argmax}_{x}\|e_{1}\|$ 3:for $j=2\to n$ do 4: Build ${\cal I}_{j-1}[e_{j}](x)$ 5: $r_{j}(x)=e_{j}(x)-{\cal I}_{j-1}[e_{j}](x)$ ($r$ stands for residual) 6: $X_{j}=\text{argmax}_{x}\|r_{j}\|$ 7:end for 8:Output: EIM nodes $\\{X_{i}\\}_{i=1}^{n}$ and interpolant ${\cal I}_{n}$ Comment 7. We make some remarks on the algorithm: * • It uses a greedy algorithm in the infinity norm. * • The iteration is performed only on the reduced basis, and there is no need of a training set as it was the case in the construction of the basis itself. * • At each greedy sweep it chooses the node that differs most from the interpolant built so far. * • The selection of EIM nodes is basis-dependent: different choices of the basis should lead to different nodes. However, once the nodes are computed, the interpolant becomes basis independent [40]. From a geometric viewpoint, one realizes that the interpolant is a projector (though not orthogonal) onto the span of the basis. Once the reduced space and the nodes are chosen, the interpolant becomes uniquely defined. #### 10.3 Accuracy and conditioning of the EIM In Section 9.3 we discussed how the quality of an interpolant in a weighted $L_{2}$ norm is related to the optimal projection representation. Interpolation in general loses accuracy but at the advantage of needing less information for computing the approximation. One can bound the interpolation error defining a Lebesgue constant (see Eq. (89)): $\\|h_{\lambda}-{\cal I}_{n}[h_{\lambda}]\\|^{2}\leq\Lambda_{n}\\|h_{\lambda}-{{\cal P}}_{n}h_{\lambda}\\|^{2}\,,\quad\Lambda_{n}:=\\|{\cal I}_{n}\\|^{2}\,.$ (108) Taking the maximum in both sides of the inequality, and recalling the definition of the greedy projection error in Eq. (61), we obtain $\max_{\lambda}\\|h{{}_{\lambda}}-{\cal I}_{n}[h_{\lambda}]\\|^{2}\leq\Lambda_{n}\sigma_{n}\,.$ (109) Comment 8. The maxima in (109) involve some ambiguities, as they might refer to those with respect to the training set or those of the underlying space of interest. It is problem and user-dependent to clarify which is the case, but the above results apply to both of them. Comment 9. As discussed below, the Lebesgue constant is an a posteriori computable number, so if one has an estimate of the projection error $\sigma_{n}$, also has an estimation of the interpolation one. Assuming there is enough training data, an estimate for $\sigma_{n}$ can be accomplished by standard validation tests. Otherwise, if the data is sparse, one may apply other standard methods from machine learning such as k-fold or leave-one-out cross-validations. In the context of the EIM, some obvious questions arise from these observations: how is the EIM related to the Lebesgue constant? Does it optimize for accuracy in some way? These questions were addressed in Ref. [138]. First notice that, at each step of the EIM, when building the interpolant, a Vandermonde matrix ${\bf V}_{n}$ (Eq. (100)) has to be inverted for computing the interpolation coefficients. The nature of the algorithm already ensures the invertibility of ${\bf V}_{n}$ at all steps. It turns out that the EIM in fact optimizes with respect to this invertibility. Theorem. Define $\textup{det}({\bf V}_{j}):=V_{j}(X_{1},\ldots,X_{j})$. Then, the residual $r_{j}(x)$ computed in the $(j-1)$-iteration of the EIM-loop satisfies $r_{j}(x)=\frac{V_{j}(X_{1},\ldots,X_{j-1},x)}{V_{j-1}(X_{1},\ldots,X_{j-1})}\quad j=2,3,\ldots n\,.$ (110) In consequence, once $X_{j}$ is chosen in Step 6, the residual at this node becomes $r_{j}(X_{j})=\frac{V_{j}(X_{1},\ldots,X_{j-1},X_{j})}{V_{j-1}(X_{1},\ldots,X_{j-1})}=\frac{\text{det}({\bf V}_{j})}{\text{det}({\bf V}_{j-1})}\,.$ (111) The proof can be found in [138]. This result says that, at each $j$-step, the EIM algorithm selects a new node $X_{j}$ in order to maximize the module of the determinant of ${\bf V}_{j}$, making the Vandermonde matrix as invertible as possible at each iteration. The following is a useful identity to compute Lebesgue constants in practice [17]: if the basis vectors are orthonormal in the 2–norm $\\|\cdot\\|_{2}$, then $\Lambda_{n}=\\|{\bf V}_{n}^{-1}\\|^{2}_{2}\,.$ (112) We use this relation and the formulation of the inverse of a matrix in terms of its adjoint to rewrite the Lebesgue constant as $\Lambda_{n}=\frac{\\|\text{adj}({\bf V}_{n})\\|^{2}_{2}}{\|\text{det}({\bf V}_{n})\|^{2}}\,.$ (113) In doing this, we see that when maximizing the determinant of the Vandermonde matrix at each step the EIM attempts to partially control the growth of $\Lambda_{n}$, in the sense of making the denominator of (113) as large as possible, but without controlling its numerator. So, the algorithm does not solve for the optimization problem of finding global (since it is a nested approach) nodes that maximize the determinant of the Vandermonde matrix, but neither the partial problem of minimizing the whole Lebesgue constant at each step, as is usually thought. Analogous observations can be made for the conditioning of ${\bf V}_{n}$, since both the Lebesgue constant and the condition number of ${\bf V}_{n}$ are directly related by $\kappa_{n}=\\|{\bf V}_{n}\\|_{2}\Lambda_{n}\,.$ For further discussions around these topics in the context of GWs, see [138]. Finally, we point out that, in the context of the EIM, a rather pessimistic bound for the Lebesgue constant can be derived [40]: $\Lambda_{n}\leq(1+\sqrt{2L})^{n-1}\\|e_{1}\\|_{\infty}^{-1}\,,$ where $L$ stands for the size of the sampling of the $x$ variable. As an a priori estimate, this bound is not useful in practice since it does not reflect the observed growth of $\Lambda_{n}$, which is, in most cases of interest, much slower. Sharper a priori estimates are difficult to prove. In practice, one computes the Lebesgue constant a posteriori. #### Further reading The original papers introducing the EIM are [20] and [90]. Reference [40] presents a very clearly explained discrete version with emphasis on solutions to non-linear time-dependent parametrized equations. In the latter case one is interested in approximating non-linear terms, in similarity with Galerkin versus collocation methods when solving partial differential equations. An hp- refinement of the standard EIM approach is presented in [51]; this in general should be of very practical importance for several scenarios, such as when there are discontinuities with respect to parameter variation, but it has so far not been used in GW science. The study of conditioning of Vandermonde matrices can be tracked to [59], where orthogonal polynomials are used to improve the condition number. See also [60, 61], and, for a short review, [72]. For different studies around Vandermonde-like matrices see [78, 46, 104, 115]. ### 11 Surrogate models The first kind of surrogate models for gravitational waves followed the lines of fitting for the projection coefficients of a reduced basis, be it obtained through a POD or greedy approach. That is, given a basis $\\{e_{i}\\}_{i=1}^{n}$ the RB approximation for a waveform $h$ is $h(t;\lambda):=\sum_{i=1}^{n}c(\lambda_{i})e_{i}(t)\,,$ (114) where the basis elements $\\{e_{i}\\}$ can be waveforms themselves as in the RB-greedy approach, or linear combinations of them as in the POD approach. This representation requires knowledge of the projection coefficients $\\{c_{i}\\}$, which are only known if the waveform itself is known. That is, as it stands, Eq. (114) is not a predictive tool but a representation one. A natural step to predict new waveforms is to build a surrogate by simply fitting in some way for new values of the parameter $\lambda$: $h_{\tt s}(t,\lambda):=\sum_{i=1}^{n}c_{{\tt s}\,,i}(\lambda)e_{i}(t)\,,$ (115) where $\\{c_{{\tt s}\,,i}\\}$ are $n$ approximations to the true projection coefficients $\\{c_{i}\\}$. There is a large variety of ways to fit for these coefficients, such as polynomial interpolation, splines, radial basis, or ML regression approaches. A number of them (without much success) were studied in Ref. [2]. One of the problems found when the basis elements are not gravitational waves themselves, as in the POD case, or if using the auxiliary orthonormal basis in the RB-greedy approach, is that they have non-physical complex structure in their parameter dependence that is difficult to represent, this is also discussed in Appendix F of [54]. One solution is to use as basis elements the waveforms themselves, something that is not possible within the POD approach, since it does not provide a set of the most representative waveforms to use. Another approach, if a dense training space is available, is to use local fits (such as splines) to find these unknown projection coefficients. The limitation of needing a dense training set is precisely being able to generate it. The main source of errors in these approaches is the fitting procedure, which in general has slow convergence. Therefore, fitting only at the EIM nodes, as described below, serves at least two goals: i) to minimize the source of fitting errors by doing so only at the EIM nodes, ii) to provide very fast to evaluate surrogate models. Impressive results have been achieved, which we discuss in Section 12, using this approach. Other approaches follow non- algorithmic schemes and the literature in this field is rather new and in constant evolution. To mention only a few, see for example [69, 147] in which neural networks are combined with reduced order modeling or [41] for the same approach in the context of GWs. We leave for future updates of this review a thorough discussion of these topics. #### 11.1 Surrogate models for components Even though we have emphasized data-driven approaches, some domain knowledge is still (very) useful. For example, the GWs emitted by two black holes have an apparent complexity, but can be decomposed into components with simple structure and easier to build models for. For example, in the non-precessing case the GWs are essentially oscillatory functions of time with an increasing amplitude, until around the time of merger, followed by the ringdown regime. Thus, it is advantageously to decompose them into phase and amplitude: $h(t;\lambda)=A(t,\lambda)e^{i\Phi(t,\lambda)}\,.$ (116) The structure of the waveforms themselves as well as phase and amplitude for the case of two black holes initially in quasi-circular orbit and without spin) are shown in Figures 10 and 11. One can notice the simplification in structure when decomposing into phase and amplitude. Figure 10: Representation of the $h_{+}$ polaritazion of a waveform corresponding to the mode $(2,2)$ for a BBH coalescence process initially in quasi-circular orbit without spin and mass ratio $q=m_{1}/m_{2}=1.7$. This picture corresponds to the surrogate model SpEC_q1_10_NoSpin [25], and was generated using the gwsurrogate Python package [5]. Figure 11: Representation of amplitude $A(t,\lambda=q)$ and phase $\phi(t,\lambda=q)$ surfaces for the waveform mode $(2,2)$ corresponding to a symbolic model [137] for the surrogate plotted in Fig. 10 above. Similarly, in the precessing case using a co-precessing frame simplifies the structure of the waveforms, and it is advantageous to include the coordinate transformations themselves as components to model for. A question arises of whether to build a reduced basis for the waveforms and later build surrogate models for the system components, or to start by building bases for the latter and then reconstruct the waveform from these surrogate components or pieces. In general, the latter approach should be more accurate, even if at the expense of higher offline cost. #### 11.2 Empirical interpolant based surrogates The surrogate approach here described was introduced in Ref. [54], the presentation of which we follow closely. It represents, with some conceptually relatively minor variations, the state-of-the art as of this writing. The approach has three offline stages, which are described below in decreasing order of computational cost, and a fourth, very fast to evaluate online surrogate model. Offline stages 1. 1. Select the most relevant $n$ points in parameter space (shown as red dots in Fig. 12) using, for example, a greedy algorithm as described in Section 8.3. The waveforms/functions associated with these selections (shown as red lines) provide a nearly optimal reduced basis for the space of interest ${\cal F}$. 2. 2. Identify $n$ time (or frequency) samples of the full time series using the EIM, to build an interpolant that accurately reconstructs any fiducial waveform for all times if it is known at the EIM nodes. These nodes are shown as blue dots on the vertical axis in Fig. 12. 3. 3. At each EIM node perform a fit with the method of choice in the parameter dimension for the amplitude and phase of the waveform using the greedy data from Step 1. The fits are indicated by blue horizontal lines in Fig. 12. Figure 12: A schematic of the method for building and evaluating the surrogate model, taken from [54]. The red dots show the greedy selection of parameter points for building the reduced basis (Step 1, offline), the blue dots (Step 2, offline) show the associated empirical nodes in time from which a waveform can be reconstructed by interpolation with high accuracy, and the blue lines (Step 3, offline) indicate a fit for the waveform’s parametric dependence at each EIM node. The yellow dot shows a generic parameter, which is predicted at the yellow diamonds and filled in between for arbitrary times using the EIM, represented as a dotted black line (Step 4, online). For further illustration, we show in Fig. 13 the distribution of EIM nodes for gravitational waves corresponding to an EOB model, taken from [54]. Notice the sparsity of the distribution of EIM subsamples (nodes): only 19 empirical nodes are needed to reconstruct the whole time series consisting of roughly $65-70$ waveform cycles for the entire parameter space. From a signal processing perspective, this may sound quite strange, since the EIM sampling does not seem to meet the minimal Nyquist sampling needed to reconstruct the time series. This apparent contradiction is resolved by realizing that the reduced basis already encodes the physical dependency of the model, so, there is indeed information in the background working for a faithful reconstruction of the waveforms. Figure 13: Example of a distribution of the EIM nodes corresponding to a fiducial model composed by EOB waveforms. Only $19$ nodes are needed to represent $65-70$ cycles within about machine accuracy. Figure taken from [54]. Step by step offline construction Put in a more specific way, the surrogate model is built as follows. Given $\\{\Lambda_{i}\\}_{i=1}^{n}$ greedy parameters and functions (waveforms) the empirical interpolant for an arbitrary parameter value $\lambda$ has the form (Section 10) ${\cal I}[h](t,\lambda)=\sum_{i=1}^{n}B_{i}(t)h(T_{i},\lambda)\approx h(t,\lambda)\,,$ (117) where the $\\{B_{i}\\}$ functions are computed offline from the basis, as well as the EIM nodes $\\{T_{i}\\}$. In the case of non-spinning binaries, it is convenient to decompose all waveforms $h$ into their phase and amplitudes as in Eq. (116) and, in particular at the EIM nodes, $h(T_{i},\lambda)=A(T_{i},\lambda)e^{i\Phi(T_{i},\lambda)}\,.$ (118) For each $i=1\ldots n$, build fits for amplitude and phase, $A(T_{i},\lambda)\approx A^{\tt fit}(T_{i},\lambda)\quad,\quad\Phi(T_{i},\lambda)\approx\Phi^{\tt fit}(T_{i},\lambda)\quad\forall\lambda\,.$ Replacing these approximants, which are so far only valid for the EIM nodes $\\{T_{i}\\}$, as opposed to arbitrary values of $t$, into (119) yields $h_{\tt s}(T_{i},\lambda):=A^{\tt fit}(T_{i},\lambda)e^{i\Phi^{\tt fit}(T_{i},\lambda)}\,.$ (119) Finally replacing $h$ by $h_{\tt s}$ in Eq. (117) leads to the final surrogate model for all values of $t$ and $\lambda$, $h_{\tt s}(t,\lambda):=\sum_{i=1}^{n}B_{i}(t)A^{\tt fit}(T_{i},\lambda)e^{i\Phi^{\tt fit}}(T_{i},\lambda)\quad\forall t,\lambda\,.$ (120) Expressions similar to Eq. (120) are the final expressions of the surrogates used so far within numerical relativity, while decomposing the waveforms into more components in the case of spin and precession. Importantly, once the training set has been processed to build the bases, the surrogate model only relies on the knowledge of the bases, the training set is no longer involved. Its accuracy (error estimates) and evaluation costs are discussed next. Online stage, evaluation cost We now discuss the cost, in terms of operation counts, to evaluate a surrogate model. For simplicity we count every arithmetic operation as a single one regardless of their actual complexity. Also for simplicity and definiteness, still restricting the discussion to the non-spinning case, the complete surrogate model is given by Eq. (120), where the $n$ coefficients $B_{i}(t)$ and the $2n$ fitting functions $\\{A_{i}(\lambda)\\}_{i=1}^{n}$ and $\\{\phi_{i}(\lambda)\\}_{i=1}^{n}$ are assembled offline as described above. In order to evaluate the surrogate model for some parameter $\lambda_{0}$ we only need to evaluate each of those $2n$ fitting functions at $\lambda_{0}$, recover the $n$ complex values $\\{A_{i}(\lambda_{0})e^{-i\phi_{i}(\lambda_{0})}\\}_{i=1}^{n}$, and finally perform the summation in (120). Each $B_{i}(t)$ is a complex-valued time series with $L$ samples, where $L$ is the desired number of time or frequency outputs. Therefore, the overall operation count to evaluate (120) at each $\lambda_{0}$ is $(2n-1)L$ plus the cost to evaluate the fitting functions. At least in one- dimension, the evaluation cost of polynomials of order $n_{\tt fit}$ using Horner’s method (which is known to be optimal in terms of operation counts [103]) is $2n_{\tt fit}$. We discussed the exponential cost of evaluating polynomials in higher (than one) dimensions in Section 9.5 and the final comments of that whole Section. We present some further discussions on this and related topics related to the fitting stage in Section 11. To summarize, the cost of evaluating the surrogate (120) is ${\tt cost_{s}}=\left[(2n-1)+c_{\tt fit}\right]L$ (121) where at least in $1$-dimension $c_{\tt fit}\leq 2n$ and therefore ${\tt cost_{s}}\leq\left(4n-1\right)L\sim{\cal O}(L)$ (122) Comment 10. * • The evaluation of the surrogate model (120) is embarrassingly parallel in the number of outputs $L$ (each output evaluation is completely independent of the other ones) and therefore the total computational time can be speed up as wished. In contrast, time dependent equations (ordinary or differential) are very difficult to parallelize in time. * • The cost of most ordinary and partial differential equations solvers is at best linear in $L$. The speedup of the described surrogate approach for evaluation, if parallelized, is actually independent of the number of desired outputs $L$. * • In Eq. (122), the linearity in $L$ is simply due to the number of independent output sample evaluations. The optimal cost would be ${\tt cost_{s}}=L$, corresponding to one arithmetic operation to evaluate the surrogate at any output $x$ (time or frequency, in the context of this review) of interest. If $n$ is small enough, as it happens in practice for GWs, the operation count $(\ref{eq:cost_surrogate})$ is in a precise sense, very efficient. Error estimates One of the errors of interest for the complete surrogate model is a discrete version of the $L_{2}$ normed difference between a fiducial waveform and its surrogate. For definiteness we consider equally spaced $L$ time output samples, $\\|h(\cdot,\lambda)-h_{\tt S}(\cdot,\lambda)\\|^{2}:=\Delta t\sum_{i=1}^{L}\left\|h(t_{i};\lambda)-h_{\rm S}(t_{i};\lambda)\right\|^{2}\,,$ (123) where $\Delta t=(t_{\mathrm{max}}-t_{\mathrm{min}})/(L-1)$. Other errors of interest are the pointwise ones for the phase and amplitude, $\left\|\frac{A(t;\lambda)-A_{\rm S}(t;\lambda)}{A(t;\lambda)}\right\|\,,\quad\|\phi(t;\lambda)-\phi_{\rm S}(t;\lambda)\|\,.$ (124) The following error bound for the discrete error (123) can be derived: $\\|h(\cdot,\lambda)-h_{\tt S}(\cdot,\lambda)\\|^{2}\leq\Lambda_{n}\sigma_{n}+\Lambda_{n}\Delta t\sum_{i=1}^{n}\left[h(T_{i},\lambda)-h_{\rm S}(T_{i},\lambda)\right]^{2}\,.$ (125) For a proof, see Ref. [54]. This bound identifies contributions from two sources. The first term in the r.h.s. of (125) describes how well the EIM interpolant (i.e., the basis and EIM nodes) represents $h(t;\lambda)$. The expected exponential decay of the greedy error $\sigma_{n}$ with $n$ along with a slowly growing Lebesgue constant $\Lambda_{n}$ results in this term being very small. The second term in the r.h.s of (125) is related to the quality of the fit. Currently, the fitting step is the dominant source of error in the surrogate models constructed, compared to the first two steps of generating the reduced basis and building the empirical interpolant. Improving on this source of error is still a remaining goal. #### Further reading Certified approaches refer to those which can a priori guarantee error bounds for the reduced model compared to the ground truth. Up to our knowledge these have been so far restricted to elliptic (coercive) partial differential equations. We again refer to [112], [70] and [111]. For an implementation of the Reduced Basis and Empirical Interpolation Methods in a concise and user- friendly API, see the Arby [3] Python package, a fully data-driven module for building surrogate models, reduced bases and empirical interpolants from training data. ### 12 Surrogates of compact binaries To date the most accurate binary black hole (BBH) numerical relativity (NR) surrogates have used a reduced bases-greedy approach, the EIM, and surrogate construction as discussed in this review, with some variations, and the SpEC code [13] for numerical relativity training simulations. However, other public catalogs of NR waveforms such as that one of the RIT group could be equally used [67, 12]. The surrogate catalog can be found in [4] and can be evaluated with the gwsurrogate [5] Python package. Another package for surrogate evaluations is surfinBH [14], built on top of gwsurrogate. What is found in all these surrogates is that when compared to numerical relativity, they are at least an order of magnitude more accurate than other existing models such as hybrid, phenomenological or effective ones. In addition to NR surrogates, we also discuss other ones using as fiducial models post-Newtonian, Effective One Body, and Ringdown approximations. #### 12.1 Numerical relativity binary black holes 1. 1. Non spinning (1 dim): The first BBH NR surrogate was presented in [25], and referred to as SpEC_q1_10_NoSpin. The black holes are initially non-spinning with initial orbital eccentricity smaller than $10^{-3}$, in the time range $[-2750,100]M$ where, as is common practice, the waveforms have been aligned so that $t=0$ stands for the peak of the amplitude (which is around the merger of the two black holes), corresponding to about $15$ orbits before merger. This is a one- dimensional parameter problem, with the mass ratio $q$ chosen in the range $q:=m_{1}/m_{2}\in[1,10]$ . The selected $17$ greedy values were taken from an EOB model, seeded with existing five parameters corresponding to numerical relativity simulations using the Einstein equations. Next, numerical relativity was used to solve for the remaining $17$ parameters and later the reduced basis built, of $22$ elements total. Impressively, the resulting surrogate model includes all spherical harmonic modes up to $\ell=8$. One could also ask for a symbolic model representing the numerical surrogate one. In this line, in [137] the authors constructed the first ab initio free- form symbolic model (that is, analytical expressions in terms of elementary functions) for gravitational waves using symbolic regression (SR) through genetic programming. The fiducial model corresponds to the principal $(2,2)$-mode of the surrogate SpEC_q1_10_NoSpin described in [25], which is taken as ground truth solution of the Einstein field equations, since it is practically indistinguishable (meaning, within the numerical relativity errors themselves) from supercomputer numerical simulations. The approach is ab initio, meaning no approximations to the Einstein equations are taken, such as stitching PN waveforms with EOB, NR and ringdown ones. The search for closed expressions is completely free, meaning that no prior hypothesis related to the type of functions is made. This is at the heart of genetic programming: successive models evolve under evolutionary pressure until reaching a tolerance error (or another stopping condition) without incurring in any human bias. To generate the training set of waveforms for SR, instead of performing naive samplings such as equally spaced grids (which showed prohibitive in terms of convergence times), the key step was to sample the time domain with the $22$ empirical interpolation nodes (EIM nodes) used in the assembly of the fiducial model SpEC_q1_10_NoSpin. This sampling represented the minimal set of pieces of information to represent the whole fiducial model without loss of structure, then accomplishing fast convergence in the regression instance and ending with closed-form expressions of maximum overlap error of 1% with respect to the NR surrogate model. One of the salient features of these closed expressions is that they are not divided into expressions for the inspiral regime, for the merger, and finally for the ringdown, but cover the whole inspiral-merger-ringdown regime. 2. 2. Spinning, motivated by GW150914 (4-5 dim): Continuing the work of Refs. [25], Ref. [26] presented the first spinning BBH surrogate model, with the parameter region motivated by the first GW detection, GW150914. Two surrogates were built: NRSur4d2s_TDROM_grid12 and NRSur4d2s_FDROM_grid12. The physical range is $q\leq 2$, dimensionless spin magnitudes $\chi_{1,2}\leq 0.8$, and the initial spin of the smaller black hole along the axis of the orbital angular momentum. The parameter region includes LIGO’s first detection, GW150914, though with less cycles (using for training NR simulations with $25-35$ cycles before merger, while GW150914 has around $55$ cycles). NRSur4d2s_TDROM_grid12 was built using $276$ NR simulations as training set and PN to find the greedy points, for which the PN surrogate reaches a floor error of $10^{-3}$. The parametric fits, used in the surrogate assembly, are fixed to a particular “order” – not necessarily polynomial order, since trigonometric functions were included as part of the dictionary of fitting functions . The authors attribute the reason for this floor to that fixing, the general problem of this issue is discussed in Section 15. Next, NR simulations were performed for those PN greedy points using a minimally rotating and co-precessing frame and, together with the coordinate transformations to an inertial frame, reduced bases for the different “pieces” or components of the waveform were built. The initial spin of the smaller black hole by construction lays along the axis of the orbital angular momentum, reducing the parameter dimensionality to $5$. Next, the azimuthal component of the spin of the larger black hole at a reference time $t_{0}=t_{\text{peak}}-4,500M$ is included through an analytical approximation, thus effectively reducing the parameter dimensionality to 4, while modeling $5$ dimensions. Thus, the model does include a physical approximation and is not purely based on NR simulations, while being able to include precession in the modeling. Model NRSur4d2s_FDROM_grid12 is built not from the 276 NR training set simulations but from the time domain surrogate NRSur4d2s_TDROM_grid12, which can be quickly evaluated at a much larger number of points to populate a new, more dense, training set. 3. 3. Spinning (7d): The first surrogate model, named NRSur7dq2, for the full 7-D parameter space of GWs emitted for a non-eccentric BBH coalescence was presented in [27]. It used $744$ NR simulations to construct the training set with parameter ranges $q\leq 2$, $\chi_{1},\chi_{2}\leq 0.8$, for about $20$ orbits prior to merger, and $\ell\leq 4$. The authors “recycled” the $276$ NR simulations used in the spinning 4-D case described above and complete the total of $744$ NR simulations with a metric-based population criteria to select the remaining parameter points. Then the authors extend the number of simulations by means of symmetry arguments to $886$. As in [26], but with certain improvements, they decompose waveforms in data pieces and proceed to construct the surrogate following as a guideline the 4-D case [26]. The in-sample errors computed for the $886$ NR waveforms show that the largest surrogate errors are comparable to the largest NR resolution errors ($\sim 10^{-2}$). For estimating the out-of-sample errors, the authors performed cross-validation over the training set by randomly dividing it in 20 sets of $44-45$ waveforms. They left out $1$ set at each step and built a trial surrogate for the remaining $19$ sets to compare it against the one that left out. This resulted in mistmatches similar to those of the in-sample case. The authors also compare the mismatches for a fully precessing EOB model (SEOBNRv3 [105]) and for a phenomenological waveform model which includes some effects of precession (IMRPhenomPv2 [65]). Mismatches more than an order of magnitude larger than the NRSur7dq2 surrogate model are found. In [143] the 7-D parameter space is covered, for about $20$ orbits before merger, mass ratios $q\leq 4$, arbitrary spin orientations with dimensionless magnitudes $\chi_{1},\chi_{2}\leq 0.8$, $\ell\leq 4$ multipole modes, and initial orbital eccentricity also less than $10^{-3}$. This model, NRSur7dq4, extends the previous 7-D one and is to date the most exhaustive and general surrogate model for BBHs from NR. The length of these simulations is sufficient to represent some but not all the BBH signals measured by LIGO and Virgo in the first two observation runs. It is proposed in the reference to hybridize the NR waveforms with PN approximations for higher values of mass ratio and spin magnitudes (see Sec. 12.2). In contrast to the above Ref. [27], here $1,528$ NR simulations were used for the training set. In addition, a surrogate model, named NRSur7dq4Remnant, is built for the mass, spin, and recoil kick velocity of the remnant black hole. To test their models, the authors perform a 20-fold cross-validation study on the training simulations. First they randomly divide the $1,528$ training simulations into 20 groups of $\sim 76$ simulations each. For each one, they built a trial surrogate using the $\sim 1,452$ remaining training simulations and test against these $\sim 76$ validation ones. They show that the mismatches for NRSur7dq4 against NR, computed with the Advanced LIGO design sensitivity noise curve, are always $\lesssim 8\times 10^{-3}$ at the 95 percentile level over the mass range $50-200M$. For NRSur7dq4Remnant the 95th percentile errors are $\sim 5\times 10^{-4}M$ for mass, $\sim 2\times 10^{-3}$ for spin magnitude, and $\sim 4\times 10^{-4}c$ for kick magnitude. Compared to the spinning EOB waveform model SEOBNRv3[105], they found in the two models an improvement of errors of at least one order of magnitude. Name \| mass ratio \| $\chi_{1,2}$ \| precession \| dim \| $\ell$ \| GW cyles ---\|---\|---\|---\|---\|---\|--- SpEC_q1_10_NoSpin \| $\leq 10$ \| 0 \| no \| 1 \| $\leq 8$ \| $\sim 25-31$ NRSur4d2s_TDROM_grid12 \| $\leq 2$ \| $\leq 0.8$ \| yes \| 4 \| $\leq 3$ \| $\sim 25-35$ NRSur7dq2 \| $\leq 2$ \| $\leq 0.8$ \| yes \| 7 \| $\leq 4$ \| $\sim 40$ NRSur7dq4 \| $\leq 4$ \| $\leq 0.8$ \| yes \| 7 \| $\leq 4$ \| $\sim 40$ Table 1: Characterization of the waveform surrogates described so far corresponding to NR binary black holes. ##### Kicks Before the release of accurate surrogates for GWs covering the full 7-D parameter space of BBH dynamics, black hole kicks were mostly modeled with fitting formulas based on PN theory with the subsequent calibration based on NR simulations. Reference [62] presented the first effort to determine remnant properties from BH mergers using a ROM-based surrogate model. More specifically, the authors used the NRSur7dq2 model [27] to generate a waveform template and analyze the linear momentum dissipation due to the emission of GWs. Their procedure provides the velocity accumulation profile ${\bf v}(t)$ and the final kick speed $v_{k}$ of the remnant black hole. The comparison for recoil speeds between NR simulations and NRSur7dq2 shows well agreement within an order of $\sim 10^{-4}c$, with some outliers in the order of $\sim 10^{-3}$c. The authors suggest that, even in the case where the surrogate accurately models post-merger strains, small errors might propagate to the phase of the center-of-mass-oscillation causing a relatively large error on the final kick velocity. More recently, in [145] the first surrogate models for remnant oscillations were constructed using Gaussian process regression (a type of machine learning regression method, see, for example, [49]) and NR simulations for training. These fits are able to provide remnant mass, spin vector and recoil kick vector with high accuracy for 1) precessing BHs with mass ratio $q\leq 2$ and spin magnitudes $\chi_{1},\chi_{2}\leq 0.8$; 2) non-precessing BHs with mass ratio $q\leq 8$ and anti-aligned spin magnitudes $\chi_{1},\chi_{2}\leq 0.8$. #### 12.2 Numerical relativity hybrid binary black holes The previous surrogate models do not cover the entire LIGO band. To remedy this, in [144] a hybridized non-precessing model named NRHybSur3dq8 was presented. In that work NR waveforms are “stitched” at early times with PN and EOB ones, thus being able to cover the entire band of advanced LIGO with a starting frequency of $20$Hz and for systems with mass as low as $2.25M_{\odot}$. This model is based on $104$ NR simulations for the 3-D parameter region $q\leq 8$, $\|\chi_{z1}\|,\|\chi_{z2}\|\leq 0.8$ for modes $\ell\leq 4$ and $(5,5)$, excluding $(4,1)$ and $(4,0)$. To populate the training space, the authors performed $91$ NR simulations and completed for a total of $104$ with $13$ waveforms added using BHs exchange symmetry (equal mass, unequal spin). The parameters for the $91$ NR simulations were selected by a greedy procedure, iteratively constructing a PN surrogate model, testing it with a dense validation set and selecting the next greedy-parameter for the largest model error. At the hybridization stage, the early-inspiral waveforms were stitched with NR ones minimizing a cost function by varying the time and frame shifts between waveforms in an appropiate matching region. The matching region was settled to start at $1,000M$ after the start of the NR waveform and end after $3$ orbits of the binary inspiral. At the end, the authors find that its hybridized surrogate model NRHybSur3dq8 performs well within NR truncation errors and outperforms the SEOBNRv4HM spinning-EOB model [43] by about two orders of magnitude. As an application case, NRHybSur3dq8 was used in [19] to generate GWs to study tidal effects by means of a PN tidal splicing method. The resulting model was named NRHybSur3dq8Tidal. It was also added to the last update of the SXS Collaboration catalog [30] of numerical simulations for BBH coalescences. #### 12.3 Extreme mass ratio inspirals In [116] the authors built a surrogate model, named EMRISur1dq1e4, for extreme mass ratio inspirals using non-spinning point particle black hole perturbation theory (ppBHPT) through numerical solutions of the Teukolsky equation with a point particle as source. The trajectory of the particle was determined by an adiabatic inspiral at early times, a late-stage geodesic plunge, and a transition region. The mass ratio was taken to be in the range $[3,10^{3}]$. After a mass rescaling the surrogate model agrees remarkably well with NR waveforms (solving the full Einstein equations), which are available for mass ratios $q\leq 10$. The mass rescaling was empirical, in the sense that it was chosen as a function of the mass ratio and numerically chosen to minimize the difference with NR waveforms for the $(2,2)$ mode. Even so, the degree of agreement after rescaling is surprisingly good and unexpected since a priori there is no reason why such a good agreement should be present at all. As the authors point out, however, their result seems to be in line with growing evidence that suggests perturbation theory with self-force corrections might be applicable to nearly equal mass systems. ##### 12.3.1 Eccentric inspirals In [21] the authors introduce a SVD-based ROM technique to model waveforms emitted by the coalescence of compact binaries with any residual orbital eccentricity. They apply this framework to eccentric-PN waveforms generated with the CBwaves open-source software [44] and build a reduced order model for a 3-D subset of waveforms of the full 8-D parameter space corresponding to total mass $M$, mass ratio $q$ and eccentricity $e_{0}$. The ranges covered by the template bank were $2.15M_{\odot}\leq M\leq 215M_{\odot}$, $0.01\leq q\leq 1$ and $0\leq e_{0}\leq 0.96$. The speedup in evaluating the surrogate model is $2-3$ orders of magnitude faster than generating the corresponding CBwaves waveforms, reaching a factor of several thousand around $10-50M_{\odot}$. #### 12.4 Effective One Body In [54] accurate surrogate models for EOB waveforms of non-spinning BBH coalescences were constructed using the Reduced Basis (RB) framework, corresponding to modes $(2,1),(2,2),(3,3),(4,4)$ and $(5,5)$ with mass ratios from $1$ to $10$. The authors benchmarked the surrogate model against a fiducial one generated with the EOB solver of the LAL software package. For a sampling rate of $2048$ Hz they found a speedup of $\approx 2,300$, about three orders of magnitude faster than the LAL waveform model. Reference [79] presented a surrogate model of a non-spinning EOB waveform model with $l=2,3,4$ tidal multipole moments that reproduces binary neutron star (BNS) numerical simulation waveforms up to merger. The authors find, within the RB framework, that $12$ amplitude and $7$ phase basis elements are sufficient to reconstruct any BNS waveform with a starting frequency of $10$ Hz. The surrogate has maximum errors of $3.8$ in amplitude ($0.04$ excluding the last 100M before merger) and $0.043$ radians in phase. Following a different trend, Ref. [80] implemented Gaussian process regression to build a frequency-domain surrogate version for an aligned-spin BNS waveform model using the EOB formalism. The resulting surrogate has a maximum mismatch of $4.5\times 10^{-4}$ and a speedup $O(10^{3})$ with respect to the original model. As an alternative to a greedy-based method for ROM, Singular Value Decomposition (SVD) can be used to generate a reduced basis for the GWs. Along this line, Refs. [109] and [110] presented two frequency-domain reduced order models for EOB models SEOBNRv1 [135] and SEOBNRv2 [134], respectively. These surrogates are built upon an SVD-based method to construct reduced basis and implement tensor product splines as interpolation method. The surrogate for SEOBNRv2 is a spin-aligned model for the GW dominant $(2,2)$ mode and extends the spin range of the first surrogate to almost the entire Kerr range. It also covers the entire parameter space in which the first one is defined: symmetric mass ratios $0.01\leq\eta\leq 0.25$ and spin magnitudes $-1\leq\chi_{i}\leq 0.99$. In general, the mistmatches are better than $\sim 0.1\%$ against SEOBNRv2 except in regions of parameter space in which the original model presents discontinuities, inducing mistmatches $\sim 1\%$ in the surrogate. #### 12.5 Post-Newtonian (PN) The first contact between the Reduced Basis method with gravitational waveform modeling occurred in [53]. The authors built a reduced basis for 2PN waveforms corresponding to a 2-D space of non-spinning BNS inspirals with mass components in the range $[1,3]M_{\odot}$. They found that, remarkably, to machine precision error, only $921$ basis elements are needed to represent the full template bank used as fiducial model. Also the greedy-approach here used was compared against a metric template placement method, finding exponential decay of the greedy error with the number of bases as opposed to the approximately linear convergence rate of the metric approach. Later on, in [55] a reduced basis was built for the non-precessing case of BBH inspirals (4-D parameter space: $2$ masses, $2$ aligned or anti-aligned spins) using the restricted TaylorF2 PN approximation with component masses in the range $[3,30]M_{\odot}$ and dimensionless spin magnitudes in the full range $[-1,1]$. The authors found that, for a tolerance error of $10^{-11}$, when increasing the dimensionality of the parameter space from 2-D to 4-D the number of basis elements needed to span the whole space of waveforms increased only in $6.6\%$ with respect to the $1,725$ bases needed for the 2-D case. Furthermore, going from the 3-D to the 4-D case implied adding only $15$ more basis elements. This opened the possibility that the curse of dimensionality could be beaten for the complete 8/7-D case, as discussed in Section 15. In [28] the problem of building a reduced basis for the full 7-D case, where there are no closed-form expressions and ordinary differential equations need to be solved, was tackled. The waveforms correspond to 3.5PN precessing inspirals (mass ratio $q\in[1,10]$ and dimensionless spin magnitudes $\|\|\chi_{i}\|\|\leq 0.9$). It used a modified version of the standard greedy algorithm to randomly resample the 7-D parameter space at each iteration with a fixed number $K$ of waveforms. This was crucial to overcome the limitations imposed by the curse of dimensionality against the construction of a densely populated training space. The sampling number $K$ was increased in each run until reaching $K=36,000$ for which the number of RB waveforms became independent of $K$. Choosing to work in the binary’s precessing frame, the authors exploited the fact that in this frame the waveforms have a weaker parametric dependence than they do in the inertial one. Another important ingredient was the choice of waveform parametrization in phase instead of time or frequency, taking advantage of the smooth dependence of waveforms on this variable. They find that with all these modifications, only $50$ waveforms are needed to represent the entire 7-D space with an error of $10^{-7}$. #### 12.6 Ringdown With the advent of GW astronomy since the first detection of the GW150914 event in 2015 a new era for testing general relativity and alternative theories of gravity in strong regimes [74, 24, 151, 114] has opened. In the case of compact binary coalescences different techniques were developed in recent years [133, 45, 101] to improve the extraction of information after merger. As it is well known, the account of post-merger properties is accomplished through the study of the quasinormal modes (QNMs) of the remnant Kerr black hole. Different models for ringdown waveforms have been constructed in rather recent years through several techniques. See for example [87, 88, 86, 85] for recent applications of greedy and regression methods in the construction of QNM models. In practice, single-mode ringdown searches can limit the number of measured events in advanced ground-based detectors such as LIGO. Moreover, parameter estimation errors can become large for such single-mode searches when the actual waveform contains a second mode. Besides this, there are further motivations on multi-mode searches, such as consistency tests of GR (e.g., the no-hair theorem [22, 50, 23]) and feature inference about the progenitors of the final black hole. Looking forward for future multi-mode ringdown searches, the Reduced Basis scheme was implemented in [36] to construct a compact representation of multi-mode QNM catalogs. A single QNM waveform has the form $h_{lmn}(t)={\cal A}_{lmn}(\Omega)\frac{M}{r}e^{-\pi\frac{f_{lmn}}{Q_{lmn}}t}\cos{(2\pi f_{lmn}t)}\,,$ (126) where the different symbols are defined as ${\cal A}_{lmn}(\Omega):=$ orientation-dependent dimensionless amplitude; $r:=$ distance to the source; $M:=$ black hole mass; $f_{lmn}:=$ central frequency; $Q_{lmn}:=$ quality factor, and geometric units $G=c=1$ are taken for granted. Starting from the one-mode case, a template bank of ringdown waveforms was constructed for the $(l,m,n)=(2,2,0)$ and $(3,3,0)$ QNMs. Asking for several minimal matches $(MM)$ between template and signal the parameter space is filled using a metric-based population criteria over the 2-D space $(f,Q)$. For example, a minimal match $MM=0.99$ corresponds to a catalog of $2,213$ waveforms with spin magnitude of the remnant black hole being in the range $[0,0.9947]$ and mass $M$ in the range $[2.9744,3025.7]M_{\odot}$. Table 14 shows the number of RB elements needed for each case. Figure 14: Taken from [36]. Number of reduced basis waveforms ($N_{RB}$) and metric-based templates $N_{metric}$ needed to represent one-mode QNM training spaces with $(l,m,n)=(2,2,0)$ and (3, 3, 0) for different minimal matches MM. The training space representation error is $\epsilon=10^{-12}$. For $\tt dim=2$, $N_{metric}$ scales with MM as $N_{metric}\propto(1-MM)^{-1}$. The RB representation was validated through a Monte Carlo simulation by randomly sampling the parameter space with $10^{7}$ waveforms. An average waveform representation error $\|\|h_{\lambda}-{\cal P}_{N}h_{\lambda}\|\|^{2}\approx 4.21\times 10^{-13}$ was found, one order of magnitude better than the maximum training space representation error $\epsilon^{2}=10^{-12}$. For the two-mode case, the authors linearly compose the previous single modes $(2,2,0)$ and $(3,3,0)$ and find similar results than in the one-mode case, since the algorithm takes advantage of the linearity of the QNMs. Following this observation, the authors finally propose a simple method to represent unconstrained multi-mode waveforms by approximating it by the sum of individual one-mode projections. ## Part III Data Analysis ### 13 Reduced Order Quadratures We have so far presented a method to produce a reduced model which can be used as a representation for an underlying set of functions. This combines the RB and EIM frameworks to produce accurate representations. Finally, one builds a surrogate predictive model through interpolation in parameter space at EIM nodes. Next we discuss how this framework can be used to compute numerical approximations of integrals (quadratures). Reduced order quadratures (ROQ) were introduced, at least in the context of GWs, in [17]. They use the EIM to build a set of nodes and weights to construct the integral, so the main difference with standard quadratures is that they are application-specific for parametrized problems. This results in fast online quadrature evaluations, which are at the core of data analysis when computing correlations, likelihoods, etc. Comment 11. Reduced Order Quadratures choose a nearly optimal subset of nodes from the training points in the physical domain (time, frequency, space) at which any given basis is known. These training points can be arbitrarily located: they can be equally spaced, randomly distributed, etc. This is an important aspect for many practical purposes, such as experimental data, where one might not be able to dictate when signals are measured, and it is in sharp contrast with fast converging Gaussian quadratures, which do dictate the time at which data should be measured. The latter might not only be impractical but also unfeasible in many experimental scenarios. Comment 12. Being based on the EIM, the method naturally applies to multiple dimensions, unstructured data and meshes of arbitrary shapes. Comment 13. By design, also due to being based on the EIM, very fast convergence with the number of ROQ nodes is observed, typically exponentially in the cases of interest. This would not be possible using standard methods which rely on smoothness of the functions to be integrated. More precisely, in the case of GWs, in order for Gaussian quadratures to converge fast the signal would have to be smooth as a function of time, which is never the case for GW signals due to the presence of measurement noise. In contrast, ROQ lift this requirement and in practice achieve exponential convergence even for noisy data. The idea of ROQ is remarkably simple for its impact: 1. 1. Step 1. Build a reduced basis for the problem of interest. The basis can be built, for example, using a POD or greedy approach. 2. 2. Step 2. Build an empirical interpolant, based on the previous basis, as discussed in Section 10. This approximates a parametrized function $f(x,\lambda)$ in the space of interest by an empirical interpolant of the form ${\cal I}_{n}[f](x,\lambda):=\sum_{i=1}^{n}B_{i}(x)f(X_{i},\lambda)\approx f(x,\lambda)$ (127) In GW science $f$ would typically be a waveform, and $x$ time or frequency but the method is generic. The affine parameterization in $\lambda$ and physical dimensions $x$ achieved by (127) is one of the critical ingredients that allows for ROQ. By affine parameterization one means a decomposition in terms of products of functions which depend only on parameter and physical variables, as in (127). 3. 3. Step 3. Reduced order quadratures follow what would otherwise be the standard procedure when building quadratures, but replacing standard polynomial interpolants with EIM based ones. Namely, $I_{n}[f](\lambda):=\int f(x,\lambda)dx\approx\int{\cal I}_{n}[f](x,\lambda)dx=\sum_{i=0}^{n}w_{i}f(X_{i},\lambda)\,,$ (128) where the ROQ weights $\omega_{i}:=\int B_{i}(x)dx$ (129) are computed offline, with any quadrature method of choice or availability for computing (approximating) the integrals in Eq. (129). Comment 14. If the number of available (say, time or frequency) samples is $L$, then one would typically use them to precompute the weights (129) in the offline stage. After this offline work, the online evaluation is decreased to $n$ ROQ nodes, with no practical loss of accuracy and usually $n\ll L$, as we discuss in Section 14, leading to dramatic speedups in likelihood computations, among other applications. #### 13.1 ROQ, other quadrature methods, dependence on dimensionality Gaussian quadratures (such as Chebyshev and Legendre) are considered some of the most efficient methods for integrating smooth generic functions. This is not just a perception: there is a whole theory of why this is generically the case. Compared to ROQ, though, they do suffer from several disadvantages, mentioned in the Discussion 5 of Section 9.3: 1. 1. The location of their nodes, at which the integrand has to be known, is dictated by the method, which is unrealistic for any experiment or application based on observations. 2. 2. They are not hierarchical. That is, more nodes cannot simply be added for higher accuracy but each new quadrature has to be built from scratch. 3. 3. Their fast convergence and near-optimality is subject to very specific conditions: smoothness of the integrand being the most important one after the imposition of the nodes location. This is an unrealistic assumption for signals taken from experiments. 4. 4. As with any polynomial-based approach, their efficiency in general (as when using grids based on tensor products of one-dimensional ones) decreases with the dimensionality of the problem, unlike ROQ for cases of interest in this review. As we discuss next, ROQ can beat these best known generic methods while working under more relaxed conditions. This is at the expense of offline work. Obstacles (1) and (2) are well known, so there is not much need to comment on them. In Refs. [33, 34] point (3) is discussed and how ROQ can lead to fast convergence even in the presence of noisy data. So next we discuss point (4), by considering two simple examples, in one and two dimensions (both physically and parametrically). They are taken from reference [17]. Example 11. The example function family of interest here is (there is nothing particular about this choice) $f(x,\lambda)=\left[\left(x-\lambda_{1}\right)^{2}+0.1^{2}\right]^{-1/2}\,,$ (130) where both $x$ and $\lambda$ are real and we want to compute an approximation to its integral, $\displaystyle I(\lambda)=\int_{\Omega}f(x,\lambda)dx\,,\quad$ (131) with, out of arbitrariness, $\Omega=[-1,1]$ and $\lambda_{1}\in[-0.1,0.1]$. We use two approaches to approximate (131): 1. 1. Gaussian quadratures: Numerical integration is carried out using up to $n=150$ nodes and a Gauss-Legendre rule. Recall that for each value of $\lambda$ a separate quadrature for each value of $n$ has to be performed since the method is not hierarchical. 2. 2. Reduced Order Quadratures: A reduced basis (in this case using a greedy approach) is first built for the family of functions (130), subsequently the empirical interpolant and nodes and, finally, the ROQ. Since ROQ are hierarchical the additional cost for showing a convergence test from $n$ to $(n+1)$ is independent of $n$, only an extra node needs to be added. This, again, is in contrast to regenerating each quadrature rule as in the Gaussian case. The results are shown in Figure (15), as black solid and dashed lines. Even for an error of at most $10^{-4}$, ROQ provide a factor of $\sim 4$ savings, with the savings dramatically increasing with higher accuracy. We recall once again that we are comparing ROQ with one of the best general purpose quadrature rules. Figure 15: Error curves for the 1-D and 2-D dimensional cases of Examples 13.1 and 15 from Ref. [17], respectively, using Gauss-Legendre and ROQ rules. Errors are computed by taking the maximum over the entire training set. The ROQ savings increase from $\sim 4$ to $\sim 12$ as the number of spatial dimensions is increased from one to two. Further savings are expected as the number of spatial dimensions increases. Note that the non-monotonicity of the ROQ curves is because the EIM does not optimize for accuracy at each step, this is discussed in Section 10.3. Approximating functions by products of functions (in particular of polynomials) in each physical dimension leads to evaluation costs of the approximants which scale exponentially with the dimentionality. As with higher accuracy, increasing the dimensionality of the problem leads to ROM providing larger savings when compared to generic methods. We next give an example. Example 12. The family of functions of interest is now $2$-dimensional, both in space and parameters, $f(x,\lambda)=\left[\left(x_{1}-\lambda_{1}\right)^{2}+\left(x_{2}-\lambda_{2}\right)^{2}+0.1^{2}\right]^{-1/2}\,,$ (132) where the physical domain is $x=(x_{1},x_{2})\in\Omega=[-1,1]\times[-1,1]$ and for the parametric one $\lambda\in[-0.1,0.1]\times[-0.1,0.1]$, and we are interested in multiple evaluations of the parametrized integral $I(\lambda)=\int_{\Omega}f(x,\lambda)dx\,.$ When considering Gaussian quadratures the curse of dimensionality already in two dimensions becomes apparent. In standard multidimensional quadratures the integrand is approximated by the product of one-dimensional polynomials. Therefore we consider up to $150^{2}$ Gauss nodes to integrate functions from the family (132). And, again, because Gaussian quadratures are not hierarchical, this requires a set of $150^{2}$ quadrature rules for each quadrature evaluation in a convergence test. Next, we build a reduced greedy basis, EIM, and ROQ. The convergence of both methods, Gaussian and ROQ, are displayed as solid and dashed blue lines in Fig. 15. Instead of a factor $\sim 4$ savings as in the 1-D case for a modest error of $10^{-4}$, for this same accuracy the savings of ROQ are $\sim 12$. As we will show in the next section, for realistic problems in GW physics, the savings are much larger than those of these test problems. ### 14 Accelerating Parameter Estimation with ROQ Once a detection of a gravitational wave is made one would like to infer the astrophysical properties of the source which emitted it. The goal here is to do a full Bayesian analysis to compute the posterior probability distribution function (PDF) of a set of astrophysical parameters ($\lambda$) which describe the source. Being able to do so in real time (or nearly so) is particularly important for rapid followups of electromagnetic counterparts enabling multi- messenger astronomy, among other motivations. For this reason we place emphasis on binary neutron stars, or mixed pairs. Assuming that the detector data $d$ contains the source’s signal $h(\lambda_{\tt true})$ and stationary Gaussian noise $n$ in an additive way 777Here the noise is denoted by $n$, as is the dimensionality of a reduced basis throughout this review. Hopefully, from context this should not cause confusion., $d=h(\lambda_{\tt true})+n\,,$ the likelihood of data $d$ corresponding to a parameter $\lambda$ is given by [75] $\mathcal{L}(d\|\lambda)=A\exp[-\frac{1}{2}(d-h(\lambda),d-h(\lambda)]\,.$ (133) The evaluations of likelihoods for parameter estimation (PE) is a high dimensional one. For binary systems, even excluding equations of state in the presence of matter, it is already a $15$-dimensional problem, with $8$ parameters being intrinsic (mass and spin vector of each binary component) and $7$ extrinsic ones such as luminosity distance, coalescence time, inclination of the binary plane, and sky localization. The bulk of the computational cost in (133) comes from two sources: 1. 1. Computing the gravitational wave candidates $h(\lambda)$. 2. 2. Evaluating the inner products in (133) . Surrogate models can decrease the computational cost of (1), and ROQ those of (2). In this section we briefly review (2); that is, the use of ROQ and ROM in general for accelerating parameter estimation (PE). Therefore, this section is intendently limited in scope, and it is not intended to be a general review of approaches for accelerating PE, which is a field by itself given its importance. Our presentation follows closely that one of Ref. [34]. In Eq. (133) $(a\|b)$ is a weighted inner product for discretely sampled data. More explicitly, and working in the frequency domain, of the form $(d\|h(\lambda))=4\mathbb{R}\ \Delta f\sum_{i=1}^{L}\frac{\bar{d}(f_{i})h(f_{i},\lambda)}{S_{n}(f_{i})}\,,$ (134) where $d(f_{i})$ and $h(f_{i},\lambda)$ are the (discrete) Fourier transforms at frequencies $\\{f_{i}\\}_{i=1}^{L}$, a bar denotes complex conjugation, and the power spectral density (PSD) $S_{n}(f_{i})$ characterizes the detector’s noise. For a given observation time $T=1/\Delta f$ and detection frequency window $(f_{\tt high}-f_{\tt low})$ there are $L={\tt int}\left(\left[f_{\tt high}-f_{\tt low}\right]T\right)$ (135) sampling points in the sum (134). Usually $L$ is large, which determines the second bulk of the computation when evaluating the likelihood (133). #### 14.1 Constructing the Reduced Order Quadrature The ROQ scheme for parameter estimation requires three steps, as described in Section 13. Here we slightly reformulate them in order to bring up some practical issues. These steps are: 1. 1. Construct a reduced basis, i.e. a set $n$ elements whose span reproduces the GW model with a desired precision. 2. 2. Using the EIM, construct the empirical interpolant and its nodes. 3. 3. The ROQ weights (138) are computed, and are used to replace, without loss of accuracy, inner product evaluations (134) by ROQ compressed ones. Comment 15. * • Step 1. The reduced basis only needs to be built over the space of intrinsic parameters for the waveform family. Furthermore, if the basis is generated using a PSD $\equiv 1$ the representation of the waveform family can be used with any PSD whenever the weights are built as in Eq. ((138)). In practice, for the purposes of this Section, basis generation proceeds in two stages. A reduced basis is first constructed, which can be evaluated for any value of. Next, this basis is evaluated at $L$ equally spaced frequency samples appropriate for the detector. * • Step 2. Given an accurate basis of dimension $n$, it is possible to uniquely and accurately reconstruct any waveform $h(\lambda)$ from only $n$ subsamples $\\{h(F_{k},\lambda)\\}_{k=1}^{n}$ using the EIM. The frequency nodes $\\{F_{k}\\}_{k=1}^{n}$, selected from the full set $\\{f_{i}\\}_{i=1}^{L}$, are also obtained by the EIM. This step provides a near-optimal compression strategy in frequency which is complimentary to the parameter one of Step (1). The model’s empirical interpolant, valid for all parameters, can be written as (cf. Eq. (19) of Ref. [54]) ${\cal I}[f](f_{i},\lambda)=\exp(-2\pi\mathrm{i}t_{c}f_{i})\sum_{j=1}^{n}B_{j}(f_{i})h(t_{c}=0,F_{j},\lambda)\approx h(f_{i},\lambda)\,,$ (136) where, for the sake of the discussion below, we have temporarily isolated the coalescence time $t_{c}$ from the other extrinsic parameters. * • Step 3. Except for $t_{c}$, no extrinsic parameter affects the construction of the ROQs. The coalescence time, however, requires special treatment. One can see this by substituting Eq. (136) into Eq. (134), $(d\|h(t_{c},\lambda))=\sum_{k=1}^{n}\omega_{k}(t_{c})h(t_{c}=0,F_{k},\lambda)\,,$ (137) with the ROQ weights given by $\omega_{k}(t_{c})=4\mathbb{R}\Delta f\sum_{i=1}^{L}\frac{\bar{d}(f_{i})B_{k}(f_{i})}{S_{n}(f_{i})}\exp(-2\pi\mathrm{i}t_{c}f_{i})\,.$ (138) In practice, the dependence of (138) on $t_{c}$ can be achieved through a simple domain decomposition, using that an estimate for the time window $W$ centered around the coalescence time $t_{\tt trigger}$ is given by the GW search pipeline. This suggests a prior interval $[t_{\tt trigger}-W,t_{\tt trigger}+W]$ to be used for $t_{c}$. This prior interval is then split into $n_{c}$ equal subintervals of size $\Delta t_{c}$. The number of subintervals is chosen so that the discretization error is below the measurement uncertainty on the coalescence time. Finally, on each subinterval a unique set of ROQ weights is constructed. Step (3) is currently implemented in the LALInference pipeline as summarized in Algorithm (3). The offline steps (1) and (2) are carried out independently. By construction, the approach guarantees that these offline steps need be to carried out only once for each waveform family model. Algorithm 3 Computing the ROQ weights for parameter estimation 1:Input: $d,S_{n},\\{B_{j}\\}_{j=1}^{N},\Delta f,t_{\tt trigger},W,\Delta t_{c}$. 2:Set $n_{c}={\tt int}\left(\left(2W\right)/\Delta t_{c}\right)+1$ 3:for $j=1\to n_{c}$ do 4: $T_{j}=t_{\tt trigger}-W+\left(j-1\right)\Delta t_{c}$ 5: for $k=1\to N$ do 6: Compute $\omega_{k}(T_{j})$ via Eq. (138) 7: end for 8:end for 9:Output: $\\{T_{j}\\}_{j=1}^{n_{c}},\\{\\{\omega_{k}(T_{j})\\}_{k=1}^{N}\\}_{j=1}^{n_{c}}$. ##### Compressed Norm Evaluations There are two more terms to consider for fast computations of inner products, which can be seen from Eq. (133). One of them is $(h(\lambda)\|h(\lambda))$. Unless the GW model is a closed-form expression, one needs to build a fast online evaluation for this norm for values of $\lambda$ that are only known at run time. This can be achieved by constructing a reduced basis for $(h,h)$, then its empirical interpolant, and finally its ROQ. The other term, $(d\|d)$, only needs to be computed once per parameter estimation or search analysis, so it does not require any special treatment to speed up its calculation. #### Total speedup Notice that even though the ROQ weights (Alg. 3) are computed in the online stage, they only depend on the detection-triggered data $d$ and not on any $h(\lambda)$. Therefore, they can be computed in what can be referred to as the startup stage, which requires $n$ full (of size $L$) inner product (134) evaluations for each $t_{c}$ interval. As discussed below, in practice this cost is negligible, while each likelihood is subsequently calculated millions of times, leading to significant speedups in parameter estimation studies, resulting in observed speedups in the whole PE study equal to $L/n$, which is the reduction in the number of terms needed to compute (137) instead of (134). This assumes that $n<L$ but, as we see below, this is indeed the case in problems of interest (furthermore, usually $n\ll L$). This speedup comes from operation counts, but it has also been observed in practice in actual implementations in the LIGO Algorithm Library (LAL) pipeline [7], as discussed below. For BNS, for the early advanced detectors’ configuration [83] ROQ showed to provide a factor of $\sim 30\times$ speedup in PE for low-frequency sensitivity of $40$Hz, and $\sim 70\times$ and $\sim 150\times$ as the sensitivity band is lowered to $20$Hz and $10$Hz, respectively; in all cases without practical loss of accuracy or systematic biases. Example 13. The material for this example is taken from Ref. [33]. The majority of a binary neutron star’s GW signal can be expected to be in the inspiral regime [121], which can be described by the closed-form TaylorF2 approximation [32]. While TaylorF2 does not incorporate spins or the merger- ringdown phases of the binary’s evolution, these might not be important for BNS parameter estimation and can therefore be neglected, at least in a first approximation [125]. For a thorough study on this point using the SpinTaylorF2 approximation, see [96]. Even for this simple to evaluate waveform family, inference on a single data set used to require significant computational wall-time with standard parameter estimation methods [15]. In Ref. [33] the authors first computed the observation time $T$ required to contain a typical BNS signal. Next, a reduced basis of dimensionality $n$ needed to represent this model for any pair of BNS masses was constructed. The upper frequency $f_{\tt high}$ was fixed to $1024$Hz while $f_{\tt low}$ varied between $10$Hz and $40$Hz. The time taken for a BNS system with an initial GW frequency of $f_{\tt low}$ to inspiral to $1024$Hz, $\displaystyle T_{\tt BNS}=\left[6.32+2.07\times\frac{10^{6}}{\left(f_{\tt low}/\text{Hz}\right)^{3}+5.86\left(f_{\tt low}/\text{Hz}\right)^{2}}\right]{\text{s}}\,,$ (139) was empirically found by generating a $\left(1+1\right)\,M_{\odot}$ waveform (directly given in the frequency domain) and Fourier transforming it to the time domain where the duration up to when the waveform’s evolution terminates is measured. Equation (139) and subsequent fits were found using a genetic algorithm-based symbolic regression software, Eureqa [10, 122, 123]. The length $L$, as implied by Eq. (135), is plotted in the top panel of Fig. 16. Figure 16: Top: Length $L$ (red dots) of a typical binary neutron star inspiral waveform, with the solid black curve connecting this data implied by the fit (139). Middle: Number of reduced basis waveforms (red crosses), with the solid black curve given by the fit (140). Bottom: Speedup implied by operation counts, as given by equation (141). Figure from [33], where what here is denoted as $n$ (the number of basis) in that reference is $N$ (here used for the size of the initial training set). As discussed, each basis only needs to be constructed over the space of intrinsic parameters – in this case the two-dimensional space of component masses, chosen to be in the range $\left[1,4\right]M_{\odot}$. This range is wider than expected for neutron stars, but ensures that the resulting PDFs do not have sharp cut-offs [93]. The number of reduced basis required to represent the TaylorF2 model within this range with a representation error around double precision ($\sim 10^{-14}$) can be fit by $\displaystyle n_{\tt BNS}=3.12\times 10^{5}\left(f_{\tt low}/\text{Hz}\right)^{-1.543}\,,$ (140) and is shown in the middle panel of Fig. 16. It was found that increasing the high-frequency cutoff to $4096$ Hz only adds a handful of basis elements, while $L$ changes by a factor of $4$, indicating that the speedup for an inspiral-merger-ringdown model might be higher, especially given that not many EIM nodes are needed for the merger and ringdown regimes [54]. This was indeed shown in [128], as discussed below. Recalling equation (135), the expected speedup from standard to ROQ-compressed likelihood evaluations is given by $\displaystyle\frac{L}{n}\approx\left(1024\text{Hz}-f_{\tt low}\right)\frac{T_{\tt BNS}}{n_{\tt BNS}}\,,$ (141) with $T_{\tt BNS}$ and $n_{\tt BNS}$ given by Eqs. (139) and (140), respectively. As reported in [128], this speedup is indeed observed using LALInference, and is shown in Fig. 16 (bottom), with a reduction in computational cost and time of $\sim 30$ for the initial detectors (with a cutoff of $f_{\tt low}=40\,$Hz) and $\sim 150$ once the advanced detectors reach $f_{\tt low}\sim 10\,$Hz. #### 14.2 Implementation in LALInference Reduced order quadratures for compressed likelihood evaluations and Algorithm (3) were originally implemented in the LAL parameter estimation pipeline, known as LALInference [7, 15], by Vivien Raymond and Rory Smith. The resulting variation is called LALInference$\\_$ROQ. This section provides unpublished details which were not included in [33], courtesy also of VR and RS. Below is a comparison between MCMC parameter estimation results using the standard version of LALInference and ROQ accelerated studies using LALInference$\\_$ROQ for the previous example, where TaylorF2 is the waveform model. Synthetic signals embedded in simulated Gaussian noise were injected into the LAL pipeline, for settings anticipating at the time the initial configuration of aLIGO, using the zero detuned high power PSD [82] and $f_{\tt low}=40$Hz. The time window was taken to be $W=0.1$s about the coalescence time $t_{c}$ of a binary neutron star signal [126, 15]. Following the discussed procedure, LALInference$\\_$ROQ discretizes this prior into $n_{c}=2,000$ sub-intervals, each of size $\Delta t_{c}=10^{-5}$s, for which it constructs a unique set of ROQ weights on each sub-interval. A width of $10^{-5}$s ensures that this discretization error is below the measurement uncertainty on the coalescence time, which is typically $\sim 10^{-3}$s [15]. As expected, the ROQ and standard likelihood approaches produce statistically indistinguishable results for posterior probability density functions over the full $9$-dimensional parameter space ($2$ intrinsic dimensiones and the full $7$ extrinsic ones). Figure 17 and Table 2 describe results for the nine parameters obtained in one particular MCMC simulation; other simulations were qualitatively similar. \| $\mathcal{M}_{c}\,(M_{\odot})$ \| $\eta$ \| $D\,$(Mpc) \| $t_{c}$(s) \| $\alpha$ (rad) \| $\delta$ (rad) \| $\phi_{c}$ (rad) \| $\psi$ (rad) \| $\iota$ (rad) \| SNR ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- injection \| 1.2188 \| 0.25 \| 172 \| 0 \| 4.91 \| -0.981 \| 1.46 \| 0 \| 1.89 \| 11.4 standard \| $1.2188^{1.2189}_{1.2184}$ \| $0.249^{0.250}_{0.243}$ \| $153^{107}_{334}$ \| $0^{0}_{0}$ \| $4.89_{1.79}^{4.98}$ \| $-0.978_{-1.01}^{1.20}$ \| $3.35_{0.309}^{6.07}$ \| $1.58_{1.54\times 10^{-2}}^{3.13}$ \| $1.82_{1.32}^{2.91}$ \| 12.9 ROQ \| $1.2188^{1.2189}_{1.2184}$ \| $0.249^{0.250}_{0.243}$ \| $149_{100}^{352}$ \| $0_{0}^{0}$ \| $4.89_{1.82}^{4.98}$ \| $-0.978_{-1.01}^{1.19}$ \| $3.01_{0.543}^{5.97}$ \| $1.58_{2.85\times 10^{-2}}^{3.11}$ \| $1.79_{1.39}^{2.82}$ \| 12.9 Table 2: Chirp mass $\mathcal{M}_{c}$, symmetric mass ratio $\eta$, source distance (from Earth) $D$, coalescence time $t_{c}$, right ascension $\alpha$, declination $\delta$, coalescence phase $\phi_{c}$, polarization phase $\psi$, inclination $\iota$ and Signal-to-Noise Ratio (SNR) of the analysis from Figure 17. Median value and 90% credible intervals are provided for both the standard likelihood (second line) and the ROQ compressed likelihood (third line). The SNR is empirically measured from $\mathrm{Likelihood_{max}}\approx\mathrm{SNR}^{2}/2$. The differences between the two methods are dominated by statistics from computing intervals with a finite number of samples. Credit: Vivien Raymond It is also useful to quantify the fractional difference in the 9-D likelihood function computed using ROQ and the standard approach. This fractional error has found to be $\Delta\log\,\mathcal{L}=1-\left(\frac{\log\,\mathcal{L}}{\log\,\mathcal{L}_{\tt{ROQ}}}\right)\lesssim 10^{-6}$ in all cases. That is, both approaches are indistinguishable for all practical purposes. In addition to providing indistinguishable results, ROQ accelerated inference is significantly faster: The ROQ-based MCMC study with the discussed settings takes $\sim 1$ hour, compared to $\sim 30$ hours using the standard likelihood approach, in remarkable agreement with the expected savings based on operation counts. The wall-time of the analysis is proportional to the total number of posterior samples of the MCMC simulation, which in this case was $\sim 10^{7}$. The startup stage required to build the ROQ weights has negligible cost and is completed in near real-time, $\sim 30$s, which is equivalent to $\sim 0.028\%$ of the total cost of a standard likelihood parameter estimation study. Figure 17: Probability density function for the chirp mass $\mathcal{M}_{c}$, symmetric mass ratio $\eta$, inclination $\iota$, coalescence phase $\phi_{c}$, right ascension $\alpha$, declination $\delta$, polarization phase $\psi$, source distance (from Earth) $D$ and coalescence time $t_{c}$ of a simulated event in LIGO/Virgo data. In green as obtained in $\sim 30$ hours by the standard likelihood, and in blue as obtained in $1$ hour with the ROQ. The injection values are in red, and are listed in Table 2. The overlap region of the sets of PDFs is the hatched region. Credit: Vivien Raymond. For a lower cutoff frequency of $f_{\tt low}\sim 20$Hz, the speedup reduction is from a couple of weeks to hours. For advanced detectors with $f_{\tt low}\sim 10$Hz, the longest BNS signals last around $2048$s in duration. Assuming a fiducial high frequency cut-off of $1024\,$Hz, which is approaching the upper limit of the sensitivity of aLIGO/AdV, datasets can be as large as $L\sim 1024\text{Hz}^{-1}\times 2048\text{s}\sim 10^{6}$. Assuming that the advanced detectors require at least $\sim 10^{7}$ posterior samples, this implies runtimes upwards of $\sim 100$ days and one Petabyte worth of model evaluations using the standard approach. On the other hand, ROQ reduces this to hours. Remarkably, this approach when applied to the advanced detectors operating at design sensitivity is faster than even the standard likelihood one used for the initial detectors. Additionally, with parallelization of the sum in each likelihood evaluation essentially real-time full Bayesian analysis might be achieved. More details can be found on the website [8]. #### Further reading Even though ROQs decrease the computational cost of likelihood evaluations for binary neutron stars for advanced detectors by about two orders of magnitude, this is still in the order of hours, which does not meet the need for real- time followup of GW detections through electromagnetic counterparts. Therefore, other ideas, beyond or on top of ROQ are needed. One approach, named Focused Reduced Order Quadratures (FROQ) [97], is to restrict the parameter estimation search based on trigger values from the detection pipeline, which further reduces the cost to around 10 minutes, which is a good target from an astrophysical point of view for searches of electromagnetic counterparts. Even though the study of [97] uses Post-Newtonian closed-form approximations, if NR surrogates are used for other scenarios, waveform evaluation cost should not be an issue for extending FROQ. In Reference [128] the authors built the first ROQ for precessing inspiral- merger-ringdown compact binaries, using the IMRPhenomPv2 model, which includes all $7$ intrinsic parameters. For low cutoff frequencies of $f_{\tt low}=20$Hz, the authors found speedups of up to between $\sim 4\times$ and $\sim 300\times$ (for short, BBH, and long, BNS systems, respectively), leading to an estimate of $6-12$ hours for a full PE study, as opposed to $\sim 1$day instead of $\sim 6$ months. The resulting code is also available from LALInference. This study was extended in [94] adding a parametrization for deviations from General Relativity using IMRPhenomPv2. We mentioned that one of the computational burdens of PE is the computation of the gravitational waveforms and that this can be solved using surrogate models. This speedup was analyzed in detail using SVD interpolated waveforms in [126]. Even though in this Section we have focused on the use of ROQ for parameter estimation, the procedure for GW searches is similar. As a proof of the generality of ROQ, it was used in Ref. [31] in the context of studies of laser light scattering with a speedup of $\sim 2,750$ with essentially no loss in precision. As emphasized in Section 15, an important topic that we have not touched in this review is that one of statistical machine learning approaches (as opposed to algorithmic ones as in this review) for GWs prediction and analysis. An exception in order here is that one of [41], since it bridges ROM with Artificial Neural Networks (ANN) and addresses an important problem. Namely, the challenge of interpolating the projection coefficients in a Reduced Basis approach. Instead of algorithmically interpolating them, the authors use ANN to map the relationship from parameters to projection coefficients. Though the focus of the reference is on parameter estimation of GWs, the fundamental idea is applicable to the predictive step itself; in fact it was proposed around the same time in a different context in [147]. ### 15 Challenges, open issues and comments To discuss one of the main challenges left in reduced order and surrogate models for gravitational waves we review the case of inspiral spinning non- precessing binary PN black hole waveforms, taken from Ref. [55]. The parameter space is 4-D and the sector covered corresponds to $m_{i}\in[3-30]M_{\odot}$, with $\chi_{i}\in[-1,1]$ ($i=1,2$). The authors study, in the context of Reduced Basis, the rb for parameter dimensions ${\tt dim}=2,3,4$, which correspond to the non-spinning, spinning-aligned, and spinning non-precessing cases, respectively. In Figure 18 there is a histogram corresponding to the 4-D case showing the greedy parameters count for mass ratio $m_{1}/m_{2}\in[0.1,10]$ for a representation error of $\sim 10^{-14}$ of any waveform in the considered parameter range. In Figure 19, for visual representation, we show the distribution of the greedy points in parameter space for the 3-D case where the spins are aligned or anti-aligned, with $\chi_{1}=\chi_{2}=\chi$. The lower cutoff frequency being considered at the time was $40$Hz. Figure 18: The greedy parameter choices for the mass ratio $m_{1}/m_{2}\in[0.1,10]$. Notice that the greedy points cluster in the near- equal mass sector. Figure taken from [55]. Figure 19: The greedy parameter choices for the mass ratio $m_{1}/m_{2}$ in the 4-D case. Right: the greedy parameter choices for the 3-D $(m_{1},m_{2},\chi)$ for the aligned spin case. Figure taken from [55]. Figure 20: The reduced basis representation error as a function of the dimensionality of the reduced bases for ${\tt dim}=2,3,4$. The number of reduced basis elements barely grows with the dimensionality of the problem $\tt dim$, suggesting that at least in the context of GWs from compact binaries, the curse of dimensionality might be beaten by the reduced basis approach. Figure taken from [55]. Comment 16. 1. 1. From Fig. 18 it can be seen that the largest number of selected greedy points is near the equal mass case. The reason for this is that since a fixed frequency range is considered, the longer waveforms correspond to nearly equal mass components, have a larger number of cycles and therefore more structure and representative information. 2. 2. In the non-spinning case around $1,700$ basis waveforms are needed for machine precision representation, see Fig. 20. This is a rather large number of data points to fit each waveform phase and amplitude at the EIM nodes for the considered mass range when building a surrogate as proposed in Section 11.2. Therefore, a low order local polynomial or spline interpolation would probably be accurate enough for practical purposes when the fiducial model is fast enough to evaluate – as it is in this example – while avoiding Runge’s phenomenon (discussed in Section 9). 3. 3. The reason why as increasing the dimensionality of the problem the reduced basis approach seems to beat the curse of dimensionality can be seen from the right panel of Fig. 19. Namely, the data becomes extremely sparse, and the number of basis elements remains approximately constant while the volume of the parameter space grows exponentially with the number of dimensions (Figure 20). This sparseness is a very much desired condition indeed for any ROM approach, but it introduces difficult challenges, as discussed below. In the case of projection and when the waveforms are known in closed form, or can be computed in a relatively inexpensive way, there are no major issues: in order to evaluate the representation of the waveform one simply computes the projection coefficients onto the basis. Similarly for predictive models, since in those cases there are an “arbitrary” number of local waveforms to use for fitting in Step 3 of an EIM-based interpolant (Section 11.2): a local fit of low order (be it in the form of interpolation, least squares, splines, etc.) can be used with high accuracy and while avoiding Runge’s phenomena. A very different situation is that one in which the training set is computationally expensive to build, as when it involves solving the full Einstein equations through numerical relativity and only a sparse training set can be constructed and there are not enough points nearby in parameter space to perform local fits with enough accuracy. One solution is to keep running NR simulations to enrich the training set. This approach might be affordable, with enough computational resources, for binary black holes. For binary neutron stars or mixed pairs, on the other hand, where the parameter dimensionality grows considerably, this might be impractical if not just impossible in reality. This challenge follows directly from the agnostic, data-driven, purposely design of the surrogate approach discussed in this review, in which no differential equations are invoked. The motivation for this design is that it is a major effort to build a production Einstein solver for realistic problems of interest, and an intrusive approach would require major algorithmic and software changes and as a consequence the NR community might be reluctant to it. Yet, that might be the only feasible approach in the long run. An intermediate solution might be doing further research in state of the art of global approaches to approximations in multiple dimensions which are fast to evaluate and do not suffer from Runge’s phenomena, be it in the form of deterministic algorithms or statistical machine learning/artificial intelligence type. In this review we have covered mostly algorithms of the first kind and tangentially mentioned some results of the second kind. But there is a lot of ongoing activity and interesting results on the ML/AI side, though it appears too premature to cover them here, so we will defer their discussion to a future, updated version of this Living Review. For a recent preliminary comparative study, though, on ML/AI surrogates, see [124]. In terms of parameter estimation and within reduced order modeling, the use of Focused Reduced Order Quadratures, discussed at the end of Section 14, might be a good enough solution for near real-time or fast-enough followups of electromagnetic counterparts. A related active field but not in GW science is that one of Uncertainty Quantification (UQ) [127] and, in particular, generalized polynomial chaos [149], which we have not discussed in this review but are closely related to Reduced Basis and can remove the usual assumption (and its consequences) of the noise being Gaussian and stationary noise, which is not the case in GW science. The application of these areas of research in GW science are unknown to us, so we also defer them to a further version of this review. However, it has to be highlighted that other techniques to deal with non-Gaussian noise have found new detections from public LIGO data [150], so the importance of the topic cannot be overemphasized. ### 16 Acknowledgments We thank, for valuable discussions about topics covered in this review through collaborations and/or interactions throughout the years: Harbir Antil, Peter Binev, Jonathan Blackman, Priscilla Canizares, Sarah Caudill, Wojciech Czaja, Albert Cohen, Scott Field, Jonathan Gair, Chad Galley, Chad Hanna, Frank Herrmann, Jan Hesthaven, Stephen Lau, Jason Kaye, David Knezevic, Tom Loredo, Akil Narayan, Ricardo Nochetto, Evan Ochsner, Dianne O’Leary, Michael Pürrer, Vivien Raymond, Gianluigi Rozza, Rory Smith, Benjamin Stamm, Béla Szilágyi, Eitan Tadmor, Michele Vallisneri, and Alan Weinstein. We apologize in advance for any omissions. We especially thank Scott Field, Chad Galley and Rory Smith for initial contributions to an initial draft of this review years ago, which we have attempted to write from scratch, given the results over the last years in the field, but some imprints from the original conception might still be left. Special credits are due to Frank Herrmann, who played a key role in the foundational efforts of reduced basis in gravitational wave science since 2009, bringing it from exploratory analyses and initial proofs of concept to a production-level stage. Special credits are also due to Jan S. Hesthaven, who introduced one of us (MT) to the idea of using the Reduced Basis framework for GW science back in 2009 at Brown University, and Ben Stamm for a short but intensive course at UMD that he gave us to get started. We also thank Jorge Pullin, the Horace Hearne Institute and the Center for Computation and Technology at LSU, for hospitality while part of this review was written. 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[type=editor, auid=000,bioid=1, prefix=, role=, orcid=] [role=, suffix=, ] # Steady-State Models of STATCOM and UPFC using Flexible Holomorphic Embedding Pradeep Singh<EMAIL_ADDRESS>Department of Electrical Engineering, Indian Institute of Technology, Delhi, 30332 INDIA Nilanjan Senroy <EMAIL_ADDRESS> ###### Abstract To investigate the effect and ability of FACTS devices using Fast and Flexible Holomorphic Embedding technique (FFHE), it is necessary to develop an embedded system for these devices. Therefore, this paper presents FFHE based embedded system for STATCOM and UPFC. The embedded system is also proposed for their controlling modes. The introduced embedded system for STATCOM and UPFC is flexible which allows to take any state as an initial guess instead of fixed state, which leads towards the reduced runtime and decrease the required number of terms, as compared to standard Holomorphic Embedded Load-Flow method (HELM). To demonstrate the effectiveness and practicability, the proposed models of STATCOM and UPFC have been tested for several cases. Further, the developed recursive formulas for power balance equations, devices’ physical constraints and their controlling modes are thoroughly investigated and examined. From several tests, it is found that the proposed embedded system requires less execution time and reduce the error at higher rate. ###### keywords: FACTS Holomorphic Embedding Power-Flow STATCOM UPFC Voltage Source Converter ## 1 Introduction FACTS controllers play vital roles in diversified domains of power system operation and control. These devices improve voltage profile, transient stability, voltage stability, available transfer capability etc., thereby improving the overall performance of the power system. But their optimal installation and performance analysis is indispensable to harness maximum benefit-cost ratio. Therefore, their accurate steady-state mathematical models are required to emulate their functionality and performance. These models are integrated with non-linear static load-flow equations of the power systems, which increases the complexity of load-flow problem [1, 2, 3, 4, 5]. The conventional steady-state models of FACTS controllers are based on numerical iterative techniques e.g. Gauss-Seidal (GS), Newton-Raphson (NR), Fast Decoupled (FD) etc. These iterative techniques are primarily constrained by the requirement of a good initial guess and they offer slow convergence or even divergence in some cases. Additionally, the solution provided by these methods is indecisive, because sometimes these techniques diverge due to a bad initial guess (although the solution exists) and sometimes converge to a spurious solution (even when the solution does not exists). These limitations were overcome by the Holomorphic Embedding Load-Flow Method (HELM) developed in [6]. The seminal work in development of HELM framework has been presented in [6, 7, 8, 9, 10, 11, 12]; and advanced applications of HELM in power system analysis can be found in [13, 14, 15, 16, 17, 18, 19, 20]. The main advantage of HELM are deterministic initial guess and assured convergence if the solution exists. However, it has been reported that HELM requires 10 to 20 times more execution time than NR method [21]. Therefore, Fast and Flexible Holomorphic Embedded (FFHE) method to reduce the execution time has been proposed in [22]. FFHE provides the flexibility to set any arbitrary state as an initial guess. This property circumvents the need to restart the load-flow program when bus- type switching takes place. Therefore, FFHE requires less number of terms to converge if initial guesses are promising. Hence, FFHE is fast and more suitable for load flow analysis than conventional iterative methods. HELM framework comprising of HE models of thyristor-based FACTS controllers have been proposed in [23]. Unified Power Flow Controller (UPFC) is considered as the superior controller among all FACTS controllers [2, 3, 4]. UPFC is a VSC-based FACTS controller which are more advanced and sophisticated as compared to Thyristor-based FACTS controller because they inject less harmonics, provide independent control, fast dynamic response and higher flexibility. In [24], a VSC-based FACTS controller, namely Static Synchronous Compensator (STATCOM), has been developed as a variable voltage source using standard HELM. However, the presented model of STATCOM requires larger runtime and also doesn’t offers the flexibility to use any state as the starting point. Therefore, the main aim of this work is to develop the FFHE based models of VSC-based FACTS controllers. In this research work, FFHE based models of two VSC-based FACTS controllers, namely Static Synchronous Compensator (STATCOM) and Unified Power Flow controller (UPFC) have been developed along-with their controlling modes. The remaining part of this paper is organised as follows: Section 2 deals with the methodology used to develop the FFHE based models of STATCOM and UPFC along-with their controlling modes. In Section 3, the key findings and numerical results of the introduced embedded system have been discussed. Section 4, presents the conclusion of this study. ## 2 Proposed Holomorphic Embedding Formulations This section addresses the procedure adopted to develop a flexible embedded system for STATCOM and UPFC. During development of steady-state model, it is assumed that the system and stated devices are 3-phase balanced and the harmonics generated by them are negligible. Firstly, the embedded system and recursive relationships for the STATCOM and UPFC have been developed and finally, numerical values of all unknown variables along-with their operation bounds are investigated. ### 2.1 STATCOM Modeling A STATCOM is a shunt connected FACTS device acts as a controllable voltage source. In steady-state, it can inject or absorb reactive power by regulating the injected voltage source magnitude and phase angle. Generally, it consists of a voltage source converter, a capacitor and a coupling transformer. The shunt connected FACTS devices are mainly used to regulate the bus voltage magnitude, although these devices can also be used to control other parameters of the system as discussed in [25]. Figure 1 shows the topological structure of a STATCOM. Figure 1: Equivalent circuit of a STATCOM In Figure 1, $y_{SH}$, $I_{SH}$ and $V_{SH}$ are the coupling transformer leakage impedance, complex current and voltage injected by the STATCOM at bus $i$ respectively. The required equations to represent the STATCOM are the power balance equation (PBE) at the connected bus and two equations from device’s physical constraint: real power injection criteria and control mode. Equations (1), (2), and (3e) provide the PBE at $i^{th}$ bus, real power injection criterion and various control modes for STATCOM. Equations (3a), (3b), (3c), (3d), and (3e) represent the control constraint equations for the bus voltage magnitude, injected voltage phasor magnitude, injected reactive power, reactive power-flow between buses and equivalent imaginary admittance respectively. $V^{}_{i}\sum_{k=1}^{N}Y_{ik}V_{k}+V^{}_{i}(V_{i}+V_{SH})y_{SH}=S^{}_{i}$ (1) $\Re\big{[}V_{SH}(V^{}_{i}+V^{}_{SH})y^{}_{SH}\big{]}=0$ (2) $V_{i}V^{}_{i}=\|V^{SP}_{i}\|^{2}$ (3a) $V_{SH}V^{}_{SH}=\|V^{SP}_{SH}\|^{2}$ (3b) $\Im\big{[}V_{SH}(V^{}_{i}+V^{}_{SH})y^{}_{SH}\big{]}=Q^{SP}_{SH}$ (3c) $\Im\big{[}V_{i}(V^{}_{i}-V^{}_{j})y^{}_{ij}\big{]}=Q^{SP}_{ij}$ (3d) $\Im\bigg{[}\frac{(V_{i}+V_{SH})}{V_{SH}}y_{SH}\bigg{]}=b^{SP}_{eq(SH)}$ (3e) where, $V_{i}$, $S_{i}$ and $Y_{ik}$ are the voltage phasor of bus $i$, injected complex power at bus $i$ and $(i,k)^{th}$ element of the bus admittance matrix respectively. The quantities with suffix $SP$ are the specified or targeted values. The symbols $N$, $N_{PV}$, and $N_{PQ}$ denotes the set of total, generator and load buses respectively. The operators $\Re(\bullet)$, $\Im(\bullet)$ and ${}^{\prime}^{\prime}$ represents real, imaginary and conjugation operation respectively. The injected voltage phasor $V_{SH}$ is treated as a free variable function of $\alpha$. The proposed holomorphic embedded system for (1), (2), and (3e) are as follows: $V^{}_{i}(\alpha^{})\sum_{k=1}^{N}Y_{ik}V_{k}(\alpha)+V^{}_{i}(\alpha^{})\big{\\{}V_{i}(\alpha)+V_{SH}(\alpha)\big{\\}}y_{SH}=\alpha\bigg{[}S^{}_{i}-C^{}_{i}\sum_{k=1}^{N}Y_{ik}C_{k}-C^{}_{i}(C_{i}+C_{SH})y_{SH}\bigg{]}+C^{}_{i}\sum_{k=1}^{N}Y_{ik}C_{k}\\\ +C^{}_{i}(C_{i}+C_{SH})y_{SH}$ (4) $\Re\big{[}V_{SH}(\alpha)\big{\\{}V^{}_{i}(\alpha^{})+V^{}_{SH}(\alpha^{})\big{\\}}y^{}_{SH}\big{]}=-\alpha\Re\big{[}C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}\big{]}+\Re[C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}]$ (5) $V_{i}(\alpha)V^{}_{i}(\alpha^{})=C_{i}C^{}_{i}+\alpha\big{[}\|V^{SP}_{i}\|^{2}-C_{i}C^{}_{i}\big{]}$ (6a) $V_{SH}(\alpha)V^{}_{SH}(\alpha^{})=C_{SH}C^{}_{SH}+\alpha\big{[}\|V^{SP}_{SH}\|^{2}-C_{SH}C^{}_{SH}\big{]}$ (6b) $\Im\big{[}V_{SH}(\alpha)\big{\\{}V^{}_{i}(\alpha^{})+V^{}_{SH}(\alpha^{})\big{\\}}y^{}_{SH}\big{]}=\Im\big{\\{}C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}\big{\\}}+\alpha\big{[}Q^{SP}_{SH}-\Im\big{\\{}C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}\big{\\}}\big{]}$ (6c) $\Im\big{[}V_{i}(\alpha)\big{\\{}V^{}_{i}(\alpha^{})-V^{}_{j}(\alpha^{})\big{\\}}y^{}_{ij}\big{]}=\Im\big{\\{}C_{i}(C^{}_{i}-C^{}_{j})y^{}_{ij}\big{\\}}+\alpha\big{[}Q^{SP}_{ij}-\Im\big{\\{}C_{i}(C^{}_{i}-C^{}_{j})y^{}_{ij}\big{\\}}\big{]}$ (6d) $\Im\bigg{[}\frac{\\{V_{i}(\alpha)+V_{SH}(\alpha)\\}}{V_{SH}(\alpha)}y_{SH}\bigg{]}=\Im\bigg{\\{}\frac{(C_{i}+C_{SH})y_{SH}}{C_{SH}}\bigg{\\}}+\alpha\bigg{[}b^{SP}_{eq(SH)}-\Im\bigg{\\{}\frac{(C_{i}+C_{SH})y_{SH}}{C_{SH}}\bigg{\\}}\bigg{]}$ (6e) Equations (4), (5) and (6e) correspond to (1), (2), and (3e) respectively and fulfil the requirement of embedding at the reference state $\alpha_{0}=0$ and the target state $\alpha_{1}=1$. The constants $C_{SH}\in\mathbb{C}\setminus\\{0\\}$ and $C_{k}\in\mathbb{C}\setminus\\{0\\}$ are adjustable and can be of any pre-specified values. The constant $C_{SH}$ is used to represent the initial value of injected voltage source $V_{SH}$, whereas constant $C_{k}$ represents the initial value of bus voltages. At $\alpha=0$, the solution of (4) and (5) gives $V_{SH}[0]=C_{SH}$ and $V_{k}[0]=C_{k}$. A STATCOM can’t inject active power into the system but it can inject reactive power. Therefore, $n_{ST}$ STATCOMs can independently control $2n_{ST}-1$ parameters. Therefore, this configuration can control only one quantity of the power system independently. $[A^{ST}]=\begin{bmatrix}\begin{array}[]{cccccccccc}1&0&0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0&0\\\ \mu_{\mathcal{GF}}&\xi_{\mathcal{GF}}&\mu_{\mathcal{GG}}&\xi_{\mathcal{GG}}&\mu_{\mathcal{GL}}&\xi_{\mathcal{GL}}&\mu_{\mathcal{G}i}&\xi_{\mathcal{G}i}&0&0\\\ 0&0&C_{\mathcal{G}re}&C_{\mathcal{G}im}&0&0&0&0&0&0\\\ \mu_{\mathcal{LF}}&\xi_{\mathcal{LF}}&\mu_{\mathcal{LG}}&\xi_{\mathcal{LG}}&\mu_{\mathcal{LL}}&\xi_{\mathcal{LL}}&\mu_{\mathcal{L}i}&\xi_{\mathcal{L}i}&0&0\\\ -\xi_{\mathcal{LF}}&\mu_{\mathcal{LF}}&-\xi_{\mathcal{LG}}&\mu_{\mathcal{LG}}&\mu^{\bigstar}_{\mathcal{LL}}&\xi^{\bigstar}_{\mathcal{LL}}&-\xi_{\mathcal{L}i}&\mu_{\mathcal{L}i}&0&0\\\ \mu_{i\mathcal{F}}&\xi_{i\mathcal{F}}&\mu_{i\mathcal{G}}&\xi_{i\mathcal{G}}&\mu_{i\mathcal{L}}&\xi_{i\mathcal{L}}&\mu_{ii}+2C_{ire}g_{SH}+C_{SHre}g_{SH}-C_{SHim}b_{SH}&\xi_{ii}+2C_{iim}g_{SH}+C_{SHim}g_{SH}+C_{SHre}b_{SH}&C_{ire}g_{SH}+C_{iim}b_{SH}&C_{iim}g_{SH}-C_{ire}b_{SH}\\\ -\xi_{i\mathcal{F}}&\mu_{i\mathcal{F}}&-\xi_{i\mathcal{G}}&\mu_{i\mathcal{G}}&-\xi_{i\mathcal{L}}&\mu_{i\mathcal{L}}&\mu^{\bigstar}_{ii}+2C_{ire}b_{SH}+C_{SHre}b_{SH}+C_{SHim}g_{SH}&\xi^{\bigstar}_{ii}+2C_{iim}b_{SH}+C_{SHim}b_{SH}-C_{SHre}g_{SH}&C_{ire}b_{SH}-C_{iim}g_{SH}&C_{iim}b_{SH}+C_{ire}g_{SH}\\\ 0&0&0&0&0&0&C_{SHre}g_{SH}+C_{SHim}b_{SH}&C_{SHim}g_{SH}-C_{SHre}b_{SH}&C_{ire}g_{SH}-C_{iim}b_{SH}+2C_{SHre}g_{SH}&C_{iim}g_{SH}+C_{ire}b_{SH}+2C_{SHim}g_{SH}\\\ \lx@intercol\hfil\cdots Select~{}row~{}vector~{}from~{}equation~{}(\ref{ST_Modes_Mat})~{}corresponding~{}to~{}desired~{}control~{}mode\cdots\hfil\lx@intercol\\\ \end{array}\end{bmatrix}$ (7) $\begin{bmatrix}\begin{array}[]{cccccccccc}0&0&0&0&0&0&C_{ire}&C_{iim}&0&0\\\ 0&0&0&0&0&0&0&0&C_{SHre}&C_{SHim}\\\ 0&0&0&0&0&0&C_{SEim}g_{SH}-C_{SHre}b_{SH}&-C_{SHre}g_{SH}-C_{SHim}b_{SH}&-C_{iim}g_{SH}-C_{ire}b_{SH}-2C_{SHre}b_{SH}&C_{ire}g_{SH}-C_{iim}b_{SH}-2C_{SHim}b_{SH}\\\ 0&0&0&0&C_{ire}b_{i\mathcal{L}}-C_{iim}g_{i\mathcal{L}}&C_{iim}b_{i\mathcal{L}}+C_{ire}g_{i\mathcal{L}}&C_{\mathcal{L}re}b_{i\mathcal{L}}+C_{\mathcal{L}im}g_{i\mathcal{L}}-2C_{ire}b_{i\mathcal{L}}&C_{\mathcal{L}im}b_{i\mathcal{L}}-C_{\mathcal{L}re}g_{i\mathcal{L}}-2C_{iim}b_{i\mathcal{L}}&0&0\\\ 0&0&0&0&0&0&\Im(\frac{y_{SH}}{C_{SH}})&\Re(\frac{y_{SH}}{C_{SH}})&-\Im(\frac{y_{SH}C_{i}}{C^{2}_{SH}})&-\Re(\frac{y_{SH}C_{i}}{C^{2}_{SH}})\\\ \end{array}\end{bmatrix}\begin{bmatrix}X^{ST}\end{bmatrix}=~{}\begin{bmatrix}\begin{array}[]{c}\Gamma_{M1}[n-1]\\\ \Gamma_{M2}[n-1]\\\ \Gamma_{M3}[n-1]\\\ \Gamma_{M4}[n-1]\\\ \Gamma_{M5}[n-1]\\\ \end{array}\end{bmatrix}$ (8) $[X^{ST}]=\begin{bmatrix}\begin{array}[]{cccccccccc}V_{\mathcal{F}re}[n]&V_{\mathcal{F}im}[n]&V_{\mathcal{G}re}[n]&V_{\mathcal{G}im}[n]&V_{\mathcal{L}re}[n]&V_{\mathcal{L}im}[n]&V_{ire}[n]&V_{iim}[n]&V_{SHre}[n]&V_{SHim}[n]\end{array}\end{bmatrix}^{{}^{\prime}}$ (9) $[B^{ST}]=\begin{bmatrix}\begin{array}[]{cccccccccc}\Re[\Gamma_{\mathcal{F}}[n-1]]&\Im[\Gamma_{\mathcal{F}}[n-1]]&\Gamma_{\mathcal{G}}[n-1]&\Gamma_{\mathcal{GV}}[n-1]&\Re[\Gamma_{\mathcal{L}}[n-1]]&\Im[\Gamma_{\mathcal{L}}[n-1]]&\Re[\Gamma^{ST}_{i}[n-1]]&\Im[\Gamma^{ST}_{i}[n-1]]&\Gamma^{ST}_{PBE}[n-1]&\Gamma^{ST}_{Mi}[n-1]\end{array}\end{bmatrix}^{{}^{\prime}}$ (10) The general recurrence relationships for $n\geq 1$ are obtained by comparing the coefficient of $\alpha^{n}$ and the system of linear equations for STATCOM $[A^{ST}]_{(\Upsilon\times\Upsilon)}[X^{ST}]_{(\Upsilon\times 1)}=[B^{ST}]_{(\Upsilon\times 1)}$ has been derived, where $\Upsilon=2(N+n_{ST})$. The derived coefficient matrix, unknown vector and known vector can be expressed as in (7), (9) and (10) respectively (shown at the top of this page). The general recurrence formula for stated control modes are also formulated and shown in (8). The some entries of known vector $[B^{ST}]$ can be expressed as follows: $\Gamma^{ST}_{i}[n-1]=\eta_{n1}\bigg{[}S^{}_{i}-C^{}_{i}\sum^{N}_{k=1}Y_{ik}C_{k}-C^{}_{i}(C_{i}+C_{SH})y_{SH}\bigg{]}-\sum^{n-1}_{d=1}\bigg{\\{}V^{}_{i}[d]V_{i}[n-d]-V^{}_{i}[d]V_{SH}[n-d]\bigg{\\}}y_{SH}\\\ -\sum^{N}_{k=1}\sum^{n-1}_{d=1}Y_{ik}V^{}_{i}[d]V_{k}[n-d]$ (11) $\Gamma^{ST}_{PBE}[n-1]=-\eta_{n1}\Re\big{[}C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}\big{]}-\Re\Bigg{[}\sum^{n-1}_{d=1}\bigg{\\{}V_{SH}[d]V^{}_{i}[n-d]+V_{SH}[d]V^{}_{SH}[n-d]\bigg{\\}}y^{}_{SH}\Bigg{]}$ (12) $\Gamma^{ST}_{M1}[n-1]=\frac{\eta_{n1}}{2}\big{[}(V^{SP}_{i})^{2}-C_{i}C^{}_{i}\big{]}-\frac{1}{2}\sum^{n-1}_{d=1}V_{i}[d]V^{}_{i}[n-d]$ (13) The embedding of (3a) and (3b) are similar to each other, therefore, the general recurrence relationship for $\Gamma^{ST}_{M2}[n-1]$ can be obtained by changing the subscript $i$ to $SH$ in (13). $\Gamma^{ST}_{M3}[n-1]=\eta_{n1}\big{[}Q^{SP}_{SH}-\Im\big{\\{}C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}\big{\\}}\big{]}-\Im\Bigg{[}\sum^{n-1}_{d=1}\bigg{\\{}V_{SH}[d]V^{}_{i}[n-d]+V_{SH}[d]V^{}_{SH}[n-d]\bigg{\\}}y^{}_{SH}\Bigg{]}$ (14) $\Gamma^{ST}_{M4}[n-1]=\eta_{n1}\big{[}Q^{SP}_{ij}-\Im\big{\\{}C_{i}(C^{}_{i}-C^{}_{j})y^{}_{ij}\big{\\}}\big{]}-\Im\Bigg{[}\sum^{n-1}_{d=1}\bigg{\\{}V_{i}[d]V^{}_{i}[n-d]-V_{i}[d]V^{}_{j}[n-d]\bigg{\\}}y^{}_{ij}\Bigg{]}$ (15) $\Gamma^{ST}_{M5}[n-1]=\eta_{n1}\Bigg{[}b^{SP}_{eq(SH)}-\Im\bigg{\\{}\frac{(C_{i}+C_{SH})y_{SH}}{C_{SH}}\bigg{\\}}\Bigg{]}-\Im\Bigg{[}\sum^{n-1}_{d=1}y_{SH}\bigg{\\{}-W_{SH}[d]V_{SH}[n-d]\frac{C_{i}}{C_{SH}}\\\ +V_{i}[d]W_{SH}[n-d]\bigg{\\}}\Bigg{]}$ (16) ### 2.2 UPFC Modeling Figure 2: Equivalent circuit of a UPFC A UPFC is a shunt-series connected FACTS device. Figure 2 shows the topological structure of a UPFC. It consists of two voltage source converters, two coupling transformers and a common capacitor. The shunt VSC is coupled to a local bus $i$ through a shunt connected transformer and the series VSC is coupled to a transmission line through a series connected transformer. The shunt converter can inject the reactive power into the system and it allows the exchange of the real power to the series converter through a common DC link to satisfy the device’s physical constraint. The UPFC with $n_{UP}$ branches (i.e. $2n_{UP}$ free variables) can independently control $2n_{UP}-1$ parameters because one variable is responsible for real power balance of the device. Therefore, the circuit configuration shown in Figure 2 can simultaneously control three quantities of the system independently. The real power can be exchanged amongst the shunt and series converters via a common DC link, but the sum of active power exchanged must be equal to zero. Due to incorporation of the UPFC in the system, two new complex variable $V_{SH}$ and $I_{SE}$ have been introduced, which further add four unknown variables. Therefore, to find out the unique solution of unknown variables, four additional equations are required. To model the UPFC into load-flow algorithms, one equation from device’s physical constraint and three equations from controlling modes of the device along with the modification of PBE at the local bus $i$ and receiving end bus $m$ are required. The mathematical formulation of the shunt and series converter controlling modes are similar to the controlling modes of STATCOM and SSSC [26] respectively. The PBE at bus $i$, $m$ and real power power exchange constraints can be expressed as follows: $V^{}_{i}\sum_{k=1}^{N}Y_{ik}V_{k}+V^{}_{i}I_{SE}+V^{}_{i}(V_{i}+V_{SH})y_{SH}=S^{}_{i}$ (17) $V^{}_{m}\sum_{k=1}^{N}Y_{mk}V_{k}-V^{}_{m}I_{SE}=S^{}_{m}$ (18) $\Re\big{[}(V_{m}-V_{i})I^{}_{SE}\big{]}+\Re\big{[}V_{SH}(V^{}_{i}+V^{}_{SH})y^{}_{SH}\big{]}=0$ (19) The following new embedded system is introduced for (17)-(19): $V^{}_{i}(\alpha^{})\sum_{k=1}^{N}Y_{ik}V_{k}(\alpha)+V^{}_{i}(\alpha^{})\big{\\{}(V_{i}(\alpha)+V_{SH}(\alpha))y_{SH}+I_{SE}(\alpha)\big{\\}}=C^{}_{i}\sum_{k=1}^{N}Y_{ik}C_{k}+C^{}_{i}\big{\\{}(C_{i}+C_{SH})y_{SH}+D_{SE}\big{\\}}\\\ +\alpha\Bigg{[}S^{}_{i}-C^{}_{i}\sum_{k=1}^{N}Y_{ik}C_{k}-C^{}_{i}\big{\\{}(C_{i}+C_{SH})y_{SH}+D_{SE}\big{\\}}\Bigg{]}$ (20) $V^{}_{m}(\alpha^{})\sum_{k=1}^{N}Y_{mk}V_{k}(\alpha)-V^{}_{m}(\alpha^{})I_{SE}(\alpha)=C^{}_{m}\sum_{k=1}^{N}Y_{mk}C_{k}-C^{}_{m}D_{SE}+\alpha\Bigg{[}S^{}_{m}-C^{}_{m}\sum_{k=1}^{N}Y_{mk}C_{k}+C^{}_{m}D_{SE}\Bigg{]}$ (21) $\Re\bigg{[}\big{\\{}V_{m}(\alpha)-V_{i}(\alpha)\big{\\}}I^{}_{SE}(\alpha^{})+V_{SH}(\alpha)y^{}_{SH}\big{\\{}V^{}_{i}(\alpha^{})+V^{}_{SH}(\alpha^{})\big{\\}}\bigg{]}=\Re\big{[}\\{C_{m}-C_{i}\\}D^{}_{SE}+C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}\big{]}\\\ -\alpha\Re\big{[}\\{C_{m}-C_{i}\\}D^{}_{SE}+C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}\big{]}$ (22) $[A^{UP}]=\begin{bmatrix}\begin{array}[]{cccccccccccccc}1&0&0&0&0&0&0&0&0&0&0&0&0&0\\\ 0&1&0&0&0&0&0&0&0&0&0&0&0&0\\\ \mu_{\mathcal{GF}}&\xi_{\mathcal{GF}}&\mu_{\mathcal{GG}}&\xi_{\mathcal{GG}}&\mu_{\mathcal{GL}}&\xi_{\mathcal{GL}}&\mu_{\mathcal{G}i}&\xi_{\mathcal{G}i}&\mu_{\mathcal{G}m}&\xi_{\mathcal{G}m}&0&0&0&0\\\ 0&0&C_{\mathcal{G}re}&C_{\mathcal{G}im}&0&0&0&0&0&0&0&0&0&0\\\ \mu_{\mathcal{LF}}&\xi_{\mathcal{LF}}&\mu_{\mathcal{LG}}&\xi_{\mathcal{LG}}&\mu_{\mathcal{LL}}&\xi_{\mathcal{LL}}&\mu_{\mathcal{L}i}&\xi_{\mathcal{L}i}&\mu_{\mathcal{L}m}&\xi_{\mathcal{L}m}&0&0&0&0\\\ -\xi_{\mathcal{LF}}&\mu_{\mathcal{LF}}&-\xi_{\mathcal{LG}}&\mu_{\mathcal{LG}}&\mu^{\bigstar}_{\mathcal{LL}}&\xi^{\bigstar}_{\mathcal{LL}}&-\xi_{\mathcal{L}i}&\mu_{\mathcal{L}i}&-\xi_{\mathcal{L}m}&\mu_{\mathcal{L}m}&0&0&0&0\\\ \mu_{i\mathcal{F}}&\xi_{i\mathcal{F}}&\mu_{i\mathcal{G}}&\xi_{i\mathcal{G}}&\mu_{i\mathcal{L}}&\xi_{i\mathcal{L}}&\mu_{ii}+D_{SE1re}+D_{SEre}+2C_{ire}g_{SH}+C_{SHre}g_{SH}-C_{SHim}b_{SH}&\xi_{ii}+D_{SEim}+2C_{iim}g_{SH}+C_{SHim}g_{SH}+C_{SHre}b_{SH}&\mu{im}&\xi_{im}&C_{ire}g_{SH}+C_{iim}b_{SH}&C_{iim}g_{SH}-C_{ire}b_{SH}&C_{ire}&C_{iim}\\\ -\xi_{i\mathcal{F}}&\mu_{i\mathcal{F}}&-\xi_{i\mathcal{G}}&\mu_{i\mathcal{G}}&-\xi_{i\mathcal{L}}&\mu_{i\mathcal{L}}&\mu^{\bigstar}_{ii}+D_{SEim}+2C_{ire}b_{SH}+C_{SHre}b_{SH}+C_{SHim}g_{SH}&\xi^{\bigstar}_{ii}-D_{SEre}+2C_{iim}b_{SH}+C_{SHim}b_{SH}-C_{SHre}g_{SH}&-\xi_{im}&\mu_{im}&C_{ire}b_{SH}-C_{iim}g_{SH}&C_{ire}g_{SH}+C_{iim}b_{SH}&-C_{iim}&C_{ire}\\\ \mu_{m\mathcal{F}}&\xi_{m\mathcal{F}}&\mu_{m\mathcal{G}}&\xi_{m\mathcal{G}}&\mu_{m\mathcal{L}}&\xi_{m\mathcal{L}}&\mu_{mi}&\xi_{mi}&\mu_{mm}-D_{SEre}&\xi_{mm}-D_{SEim}&0&0&-C_{mre}&-C_{mim}\\\ -\xi_{m\mathcal{F}}&\mu_{m\mathcal{F}}&-\xi_{m\mathcal{G}}&\mu_{m\mathcal{G}}&-\xi_{m\mathcal{L}}&\mu_{m\mathcal{L}}&-\xi_{mi}&\mu_{mi}&\mu^{\bigstar}_{mm}-D_{SEim}&\xi^{\bigstar}_{mm}+D_{SEim}&0&0&C_{mim}&-C_{mre}\\\ 0&0&0&0&0&0&-D_{SE1re}-D_{SE2re}&-D_{SE1im}-D_{SE2im}&D_{SE1re}&D_{SE2im}&C_{mre}-C_{ire}&C_{mim}-C_{iim}&C_{tre}-C_{ire}&C_{tim}-C_{iim}\\\ \lx@intercol\hfil\cdots Select~{}3~{}different~{}row~{}vector~{}from~{}equation~{}(21)~{}of~{}[26]~{}and~{}(\ref{ST_Modes_Mat})~{}corresponding~{}to~{}desired~{}control~{}mode\cdots\hfil\lx@intercol\\\ \end{array}\end{bmatrix}$ (23) $[X^{UP}]=\begin{bmatrix}\begin{array}[]{cccccccccccccc}V_{\mathcal{F}re}[n]&V_{\mathcal{F}im}[n]&V_{\mathcal{G}re}[n]&V_{\mathcal{G}im}[n]&V_{\mathcal{L}re}[n]&V_{\mathcal{L}im}[n]&V_{ire}[n]&V_{iim}[n]&V_{mre}[n]&V_{mim}[n]&V_{SHre}[n]&V_{SHim}[n]&I_{SEre}[n]&I_{SEim}[n]\end{array}\end{bmatrix}^{{}^{\prime}}$ (24) $[B^{UP}]=\begin{bmatrix}\begin{array}[]{cccccccccccccc}\Re[\Gamma_{\mathcal{F}}[n-1]]&\Im[\Gamma_{\mathcal{F}}[n-1]]&\Gamma_{\mathcal{G}}[n-1]&\Gamma_{\mathcal{GV}}[n-1]&\Re[\Gamma_{\mathcal{L}}[n-1]]&\Im[\Gamma_{\mathcal{L}}[n-1]]&\Re[\Gamma^{UP}_{i}[n-1]]&\Im[\Gamma^{UP}_{i}[n-1]]&\Re[\Gamma^{UP}_{m}[n-1]]&\Im[\Gamma^{UP}_{m}[n-1]]&\Gamma^{UP}_{PBE}[n-1]&\Gamma^{UP}_{Mi}[n-1]&\Gamma^{UP}_{Mi}[n-1]&\Gamma^{UP}_{Mi}[n-1]\end{array}\end{bmatrix}^{{}^{\prime}}$ (25) The similar procedure has been adopted to obtain the system of linear equations as discussed in Section 2.1. The system of linear equations for UPFC $[A^{UP}]_{(\Upsilon\times\Upsilon)}[X^{UP}]{(\Upsilon\times 1)}=[B^{UP}]{(\Upsilon\times 1)}$ is given in (23), (24) and (25) respectively (shown at the bottom of this page), where $\Upsilon=2(N+2n_{UP})$. For purpose of numerical studies, the different controlling modes for the shunt and series converters have been adopted from (8) and [26] respectively. Some entries of known vector $[B^{UP}]$ are directly taken from [26] (e.g. entries of slack, load and generator buses) and shown in Appendix A; and the remaining are as follows: $\Gamma^{UP}_{i}[n-1]=\eta_{n1}\bigg{[}S^{}_{i}-C^{}_{i}\sum^{N}_{k=1}Y_{ik}C_{k}-C^{}_{i}(C_{i}+C_{SH})y_{SH}-C^{}_{i}D_{SE}\bigg{]}-\sum^{n-1}_{d=1}\bigg{\\{}V^{}_{i}[d]V_{i}[n-d]+V^{}_{i}[d]V_{SH}[n-d]\bigg{\\}}y_{SH}\\\ -\sum^{N}_{k=1}\sum^{n-1}_{d=1}Y_{ik}V^{}_{i}[d]V_{k}[n-d]-\sum^{n-1}_{d=1}V^{}_{i}[d]I_{SE}[n-d]$ (26) $\Gamma^{UP}_{m}[n-1]=\eta_{n1}\bigg{[}S^{}_{m}-C^{}_{m}\sum^{N}_{k=1}Y_{mk}C_{k}+C^{}_{m}D_{SE}\bigg{]}-\sum^{N}_{k=1}\sum^{n-1}_{d=1}Y_{mk}V^{}_{m}[d]V_{k}[n-d]+\sum^{n-1}_{d=1}V^{}_{m}[d]I_{SE}[n-d]$ (27) $\Gamma^{UP}_{PBE}[n-1]=-\eta_{n1}\Re\big{[}C_{SH}(C^{}_{i}+C^{}_{SH})y^{}_{SH}+(C_{m}-C_{i})D^{}_{SE}\big{]}-\Re\Bigg{[}\sum^{n-1}_{d=1}\bigg{\\{}V_{m}[d]I^{}_{SE}[n-d]-V_{i}[d]I^{}_{SE}[n-d]\bigg{\\}}\Bigg{]}\\\ -\Re\Bigg{[}\sum^{n-1}_{d=1}\bigg{\\{}V_{SH}[d]V^{}_{i}[n-d]+V_{SH}[d]V^{}_{SH}[n-d]\bigg{\\}}y^{}_{SH}\Bigg{]}$ (28) ## 3 Results and Discussions The proposed models of STATCOM and UPFC; and their multi-control capabilities have been tested on IEEE 118-bus test systems [27]. But for demonstration purpose, only selected numerical results on IEEE 118-bus test system have been presented in this paper. To validate the proposed models, investigation has been done by incorporating the different devices at different locations and by changing the target values of quantities. All the quantities are in p.u. and the base power being is 100 MVA. The proposed FFHE based models have been tested and examined for all the control modes as discussed in Section 2. The proposed FFHE and NR based models of STATCOM and UPFC were modelled in MATLAB environment and simulated on Intel(R) Core(TM) i3-4150 CPU 3.50 GHz processor with 4-GB RAM. For all the investigations, the mismatch tolerance of $10^{-8}$ is considered for maximum bus power mismatch. The values of unknown variables obtained after 3 iterations of NR method have been used as initial guesses for the FFHE load-flow method throughout this paper. The numerical results of IEEE 118-bus test system in the absence of FACTS devices are presented in Table 1. The similar procedure as discussed in [26] has been adopted to calculate the percentage reduction in mismatch and runtime. Table 1: Numerical results of IEEE 118-bus test system without any FACTS device $V_{16}=0.9839\angle-17.80\degree$ \| $S_{16-17}=-0.1752-j0.0370$ ---\|--- $V_{20}=0.9581\angle-17.81\degree$ \| $S_{20-21}=-0.2869+j0.0496$ $V_{75}=0.9675\angle-7.07\degree$ \| $S_{75-74}=0.5236+j0.0604$ $V_{114}=0.9607\angle-15.27\degree$ \| $S_{114-115}=0.0134+j0.0088$ Table 2: Numerical results when STATCOM is incoporated at bus no. 16 in the IEEE 118-bus test system \| \| Case 1 \| Case 2 \| Case 3 \| Case 4 \| Case 5 ---\|---\|---\|---\|---\|---\|--- Specified Parameters \| $V^{SP}_{16}=1.1$ \| $V^{SP}_{SH}=1$ \| $Q^{SP}_{SH}=-0.5$ \| $Q^{SP}_{16-17}=0$ \| $b^{SP}_{eq(SH)}=-0.8$ Power-flow results \| $V_{16}$ \| $\textbf{1.1}\angle-20.28\degree$ \| $0.9977\angle-17.99\degree$ \| $0.9519\angle-17.45\degree$ \| $0.9867\angle-17.84\degree$ \| $1.0324\angle-18.61\degree$ $V_{SH}$ \| $1.1197\angle 158.68\degree$ \| $\textbf{1}\angle 161.87\degree$ \| $0.9466\angle 162.87\degree$ \| $0.9872\angle 162.13\degree$ \| $1.0407\angle 160.93\degree$ $S_{16-17}$ \| $-0.2121+j0.6672$ \| $-0.1769+j0.0578$ \| $-0.1736-j0.1739$ \| $-0.1754+\textbf{j0}$ \| $-0.1843+j0.2506$ $S_{SH}$ \| $0+j2.2210$ \| $0+j0.2352$ \| $0-\textbf{j0.5}$ \| $0+j0.0470$ \| $0+j0.8664$ $b_{eq(SH)}$ \| $-1.7717$ \| $-0.2352$ \| $0.5580$ \| $-0.0482$ \| -0.8 $\%\vartriangle E$ \| $25.91$ \| $24.24$ \| $27.92$ \| $28.95$ \| $33.87$ $\%T$ \| $11.63$ \| $9.25$ \| $8.10$ \| $10.97$ \| $9.23$ Table 3: Numerical results when STATCOM is incoporated at bus no. 114 in the IEEE 118-bus test system \| \| Case 1 \| Case 2 \| Case 3 \| Case 4 \| Case 5 ---\|---\|---\|---\|---\|---\|--- Specified Parameters \| $V^{SP}_{114}=0.9$ \| $V^{SP}_{SH}=1$ \| $Q^{SP}_{SH}=-0.1$ \| $Q^{SP}_{114-115}=0$ \| $b^{SP}_{eq(SH)}=0.2$ Power-flow results \| $V_{114}$ \| $\textbf{0.9}\angle-15.03\degree$ \| $0.9927\angle-15.76\degree$ \| $0.9569\angle-15.23\degree$ \| $0.9598\angle-15.26\degree$ \| $0.9539\angle-15.19\degree$ $V_{SH}$ \| $0.8843\angle 165.96\degree$ \| $\textbf{1}\angle 163.81\degree$ \| $0.9559\angle 164.83\degree$ \| $0.9595\angle 164.75\degree$ \| $0.9520\angle 164.92\degree$ $S_{114-115}$ \| $0.0095-j0.6308$ \| $0.0083+j0.3854$ \| $0.0135-j0.0319$ \| $0.0134-\textbf{j0}$ \| $0.0135-j0.0662$ $S_{SH}$ \| $0-j1.3773$ \| $0+j0.7374$ \| $0-\textbf{j0.1}$ \| $0-j0.0241$ \| $0-j0.1813$ $b_{eq(SH)}$ \| $1.7613$ \| $-0.7374$ \| $0.1094$ \| $0.0262$ \| 0.2 $\%\vartriangle E$ \| $22.35$ \| $23.16$ \| $28.48$ \| $24.23$ \| $29.18$ $\%T$ \| $12.43$ \| $9.47$ \| $7.69$ \| $9.78$ \| $10.89$ To validate the proposed embedding for STATCOM and its controlling modes, several test have been carried out on IEEE 118-bus test system. For demonstration purpose, only two location results have been presented in Table 2. For examining cases 1-5, STATCOM is assumed to be connected at bus 16. The specified reference values; and calculated values of STATCOM’s variables and selected system variables are given in row 2 and 3 respectively. In case 1, the voltage magnitude of STATCOM connected bus was chosen as a control variable. In this case, STATCOM is able to set the voltage of bus 16 to the specified reference 1.1 p.u. by providing the reactive power support to that bus. The examination of case 2 from this table proves that the proposed formulation also enables the model of STATCOM to control its injected voltage to a specified reference 1 p.u. Case 3 demonstrates the reactive power absorption capability of the device and it can be observed that the absorbed reactive power is equal to the desired value. As per literature, STATCOM also has the capability to control reactive power-flow through the line, case 4 verifies the same. In this case, STATCOM controls the reactive power-flow of line 16-17 from $-$0.0370 p.u. to target value 0 p.u. So, by forcing the reactive power-flow through the line to zero, the active power transferring capability of the line can be increased. This particular mode may be attractive in context of the electricity market environment. The proposed formulation also enables the model to operate like a controllable admittance and it can be verified from case 5 as shown in Table 2. Now, a STATCOM is installed at bus 114 and the desired values of control parameters, calculated values of system variables, STATCOM’s parameters for all cases are presented in Table 3. From this table, it can be observed that the all the desired values (shown in bold letters) have been achieved. From Tables 2 and 3, it can also be verified that the NR based model of STATCOM takes more runtime as compared to NR-assisted FFHE based model of STATCOM and the proposed model also reduces the error at faster rate. In the literature only STATCOM model based on standard HELM has been developed in [24], therefore, the performance of NR-assisted FFHE and basic HELM based model of STATCOM is also investigated. From the various test cases, it has been observed that the FFHE based model of STATCOM takes 80-90% less time and requires 40-70% less terms to converge as compared to basic HELM based model of STATCOM. Table 4: Numerical results when UPFC in the IEEE 118-bus test system (location: $75-74$) \| \| Case 1 \| Case 2 ---\|---\|---\|--- Specified Parameters \| $V^{SP}_{75}=1$ \| $V^{SP}_{SH}=1$ $P^{SP}_{75-74}=0.75$ \| $P^{SP}_{75-74}=0.2$ $Q^{SP}_{75-74}=0$ \| $Q^{SP}_{75-74}=0.1$ Power-flow results \| $V_{75}$ \| $\textbf{1}\angle-7.99\degree$ \| $0.9937\angle-6.80\degree$ $V_{SH}$ \| $1.0087\angle 171.52\degree$ \| $\textbf{1}\angle 172.83\degree$ $S_{SH}$ \| $0.0129+j0.8636$ \| $-0.002+j0.6315$ $b_{eq(SH)}$ \| $-0.8488$ \| $-0.6315$ $S_{75-74}$ \| 0.75+j0 \| 0.2+j0.1 $V_{SE}$ \| $0.0725\angle 95.78\degree$ \| $0.0855\angle-117.30\degree$ $I_{SE}$ \| $0.75\angle-7.99\degree$ \| $0.2250\angle-33.37\degree$ $S_{SE}$ \| $-0.0129+j0.0528$ \| $0.002-j0.0191$ $X_{eq(SE)}$ \| $0.0938$ \| $-0.3778$ $\%\vartriangle E$ \| $26.58$ \| $23.37$ $\%T$ \| $5.19$ \| $6.45$ Further, the proposed embedded system for UPFC has been also tested for two cases. Firstly, it is assumed that the UPFC is connected between the buses 75 and 74 at the location of bus 75. In case 1, the shunt control of UPFC is to control the voltage at bus 75 and the series controls of UPFC are to control active and reactive power flowing through the transmission line 75-74. Similarly, in case 2, the shunt control of UPFC is to control injected voltage and series control responsibilities of UPFC are the same except the desired values. The rows 2, 3, 4, 5 of Table 4 presents the specified parameters, calculated values of UPFC’s and system variables, percentage reduction in error and runtime respectively. The active power exchange between the shunt and series converters is also presented in this table. In Table 4, numerically it is shown that the all target values (shown in bold letters) have been met for both cases. The decrement in error ranges from 15% to 27% and reduction in runtime ranges from 3% to 7%. Table 5 presents the results for two cases when the UPFC is assumed to be connected between buses 20 and 21. The UPFC parameters i.e. magnitudes and phase angles of the UPFC’s injected voltages $V_{SE}$, $V_{SH}$ are also given in this table, which enforce the system operating point to set as per specified reference quantities. The calculated values of specified references are shown in bold letters and it can be verified that all the desired values have been achieved. From Tables 5, it can be observed that the error reduction is higher and runtime is lesser for FFHE based model of UPFC. Note that in all the cases, convergence has been achieved; and the proposed models exhibited very good convergence and also converge at faster rate as compared to models based on standard NR method. Table 5: Numerical results when UPFC in the IEEE 118-bus test system (location: $20-21$) \| \| Case 1 \| Case 2 ---\|---\|---\|--- Specified Parameters \| $V^{SP}_{20}=1$ \| $V^{SP}_{SH}=1$ $P^{SP}_{20-21}=0.6$ \| $P^{SP}_{20-21}=-0.4$ $Q^{SP}_{20-21}=0$ \| $Q^{SP}_{20-21}=0$ Power-flow results \| $V_{20}$ \| $\textbf{1}\angle-28.95\degree$ \| $0.9981\angle-17.48\degree$ $V_{SH}$ \| $1.0051\angle 150.77\degree$ \| $\textbf{1}\angle 162.38\degree$ $S_{SH}$ \| $0.0093+j0.5087$ \| $-0.0290+j0.2182$ $b_{eq(SH)}$ \| $-0.5035$ \| $-0.2182$ $S_{20-21}$ \| 0.6+j0 \| $-$0.4+j0 $V_{SE}$ \| $0.5545\angle 62.65\degree$ \| $0.0866\angle-164.27\degree$ $I_{SE}$ \| $0.6\angle-28.95\degree$ \| $0.4008\angle 162.53\degree$ $S_{SE}$ \| $-0.0093+j0.3326$ \| $0.0290+j0.0190$ $X_{eq(SE)}$ \| $0.9238$ \| $0.1183$ $\%\vartriangle E$ \| $24.31$ \| $25.20$ $\%T$ \| $7.54$ \| $6.01$ Table 6: Power-flow results when device limit constraints are imposed Device \| Specified Parameter (p.u.) \| Limit Violated \| Power-flow results ---\|---\|---\|--- STATCOM \| $V^{SP}_{16}=1.1$ \| Yes \| $V_{16}={\color[rgb]{0,0,1}1.0830}\angle-19.81\degree$ $V_{SH}=\textbf{1.1}\angle 159.28\degree$ $S_{16-17}=-0.2038+j0.5580$ $S_{SH}=0+j1.8806$ $b_{eq(SH)}=-1.5542$ $\%\vartriangle E=23.85$ $\%T=6.38$ UPFC \| \| Yes \| $V_{20}=\textbf{1}\angle-23.77\degree$ \| $V_{SE}=\textbf{0.3}\angle 72.46\degree$ \| $I_{SE}=0.1813\angle-23.77\degree$ \| $S_{20-121}={\color[rgb]{0,0,1}0.1813}+j\textbf{0}$ $V^{SP}_{20}=1$ \| $S_{SE}=-0.0059+j0.0541$ $P^{SP}_{20-21}=0.6$ \| $X_{eq(SE)}=1.6446$ $Q^{SP}_{20-21}=0$ \| $V_{SH}=1.0038\angle 156.63$ \| $S_{SH}=0.0059+j0.3724$ \| $b_{eq(SH)}=-0.3696$ \| $\%\vartriangle E=17.37$ \| $\%T=3.94$ The introduced embedded system has also been investigated with device limit constraints. To demonstrate the handling of constraints, one case from Table 2 and another case from Table 5 were chosen. The maximum value of injected voltage by SSSC and STATCOM were selected as 0.3 p.u. and 1.1 p.u. respectively. From Table 2, it can be observed that when the target voltage of bus 16 is chosen as 1.1 p.u. and no limits were set on the injected voltage, the STATCOM is able to control the voltage magnitude of bus 16 to the specified reference. But when the limits are imposed, STATCOM is not able to control the same because higher $V_{SH}$ is required to maintain the same. Therefore, whenever the limit constraints are violated, current mode is relaxed; and STATCOM will acts as a constant voltage source (i.e. injected voltage magnitude mode is activated) and reference value is set equal to the value of violated limit. From Table 6, it can be observed that the voltage magnitude of bus 16 is not equal to the reference value but it is equal to the 1.0830 p.u., while the injected voltage is maintained to 1.1 p.u. Table 6 also presents the results when the limit constraints for UPFC are imposed and similar results are achieved. Therefore, it can be concluded that the proposed FFHE based models of STATCOM and UPFC are also suitable when limit constraints of the devices are imposed. The NR and FFHE based models have been tested on same platform for comparison, although the derived inferences may not be general due to various differences in simulation structure, programming skills etc. It is observed that the percentage reduction in error and runtime are different for each case. Because there is no particular choice of the initial guess for the proposed models to ensure fast convergence in all cases. The selection of promising initial guess still remains an issue. Briefly, the runtime and rate of convergence of the proposed models will vary for different initial guess. Although, the performance of proposed FFHE based FACTS devices model are sensitive to the initial guess, but the proposed model perform better than basic HELM based models. Moreover, for most of cases, FFHE based models outperforms NR based models in context of error reduction. A common conclusion for all the stated devices and their controlling modes is that the proposed embedding for the devices are reliable and have good convergence characteristics. Further, the proposed formulation reduces the mismatch error at faster rate and therefore, converges slightly faster than the NR method based FACTS devices models. Therefore, the proposed FFHE based models offers a good alternative to the NR based models when promising initial guesses are available. ## 4 Conclusion The purpose of this research work is to develop flexible HELM models of STATCOM, and UPFC. To do the same, the power balance equations and devices’ controlling modes are embedded with a complex variable $\alpha$ in such a manner that the embedded equations satisfy the requirement of embedding. Afterwards, the unknown variables were represented by power series and recursive formulas have been obtained for $n\geq 1$. Lastly, at $\alpha=1$ numerical values of unknown variables are calculated using determinant method. The proposed embedded system of stated devices provides flexibility because any state can serve as an initial guess. From the numerical results, it is observed that the results of FFHE based models are similar to results obtained by NR based model of devices in perspective of the final calculated values of devices parameters, system variables and operational bounds. The comparison of runtime for FFHE based models with NR based models showed that later one took more time. Although the performance of FFHE based models is sensitive to the initial guess but the NR assisted FFHE based models outperform the NR based models in error reduction. The proposed model of STATCOM takes very less execution time as compared to the basic HELM model of STATCOM. The numerous results showed that the FFHE based models represents a step forward compared to the NR based models. So, the proposed models may be very useful at the planning stage of the power system. Therefore, the proposed FFHE based models offers a good alternative to the NR based models when promising initial guesses are available. In this paper, no sophisticated technique is used during implementation of FFHE based models, therefore, possibility of reducing runtime further is still large via parallel computing, code optimization etc. ## Appendix A Entries of Coefficient matrix and Known Vector The entries of coefficient matrix $[A^{ST}]$ and $[A^{UP}]$ are as follows: $\mu_{ik}=G_{ik}C_{ire}+B_{ik}C_{iim}~{}~{};~{}k\neq i$ (29) $\xi_{ik}=G_{ik}C_{iim}-B_{ik}C_{ire}~{}~{};~{}k\neq i$ (30) $\mu_{ii}=G_{ii}C_{ire}+B_{ii}C_{iim}+\sum^{N}_{k=1}\big{(}G_{ik}C_{kre}-B_{ik}C_{kim}\big{)}$ (31) $\xi_{ii}=G_{ii}C_{iim}-B_{ii}C_{ire}+\sum^{N}_{k=1}\big{(}G_{ik}C_{kim}+B_{ik}C_{kre}\big{)}$ (32) $\mu^{\bigstar}_{ii}=-G_{ii}C_{iim}+B_{ii}C_{ire}+\sum^{N}_{k=1}\big{(}G_{ik}C_{kim}+B_{ik}C_{kre}\big{)}$ (33) $\xi^{\bigstar}_{ii}=G_{ii}C_{ire}+B_{ii}C_{iim}+\sum^{N}_{k=1}\big{(}B_{ik}C_{kim}-G_{ik}C_{kre}\big{)}$ (34) The entries of known vector $[B^{ST}]$ and $[B^{UP}]$ are as follows: $\Gamma_{\mathcal{F}}=\eta_{n1}(V^{SP}_{\mathcal{F}}-C_{\mathcal{F}})$ (35) $\Gamma_{\mathcal{G}}=\frac{\eta_{n1}}{2}\Bigg{[}2P_{\mathcal{G}}-\Re\bigg{\\{}C_{\mathcal{G}}\sum^{N}_{k=1}Y^{}_{\mathcal{G}k}C^{}_{k}\bigg{\\}}\Bigg{]}-\Re\Bigg{[}\sum^{N}_{k=1}\sum^{n-1}_{d=1}Y^{}_{\mathcal{G}k}V^{}_{\mathcal{G}}[d]V_{k}[n-d]\Bigg{]}$ (36) $\Gamma_{\mathcal{GV}}=\frac{\eta_{n1}}{2}\big{[}(V^{SP}_{\mathcal{G}})^{2}-C_{\mathcal{G}}C^{}_{\mathcal{G}}\big{]}-\frac{1}{2}\sum^{n-1}_{d=1}V_{\mathcal{G}}[d]V^{}_{\mathcal{G}}[n-d]$ (37) $\Gamma_{\mathcal{L}}=\eta_{n1}\bigg{[}S^{}_{\mathcal{L}}-C^{}_{\mathcal{L}}\sum^{N}_{k=1}Y_{\mathcal{L}k}C_{k}\bigg{]}-\sum^{N}_{k=1}\sum^{n-1}_{d=1}Y_{\mathcal{L}k}V^{}_{\mathcal{L}}[d]V_{k}[n-d]$ (38) $\Gamma^{SC}_{M1}=\eta_{n1}\big{[}P^{SP}_{im}-\Re(C_{i}D^{}_{SE})\big{]}-\Re\Bigg{[}\sum^{n-1}_{d=1}V_{i}[d]I^{}_{SE}[n-d]\Bigg{]}$ (39) $\Gamma^{SC}_{M2}=\eta_{n1}\big{[}Q^{SP}_{im}-\Im(C_{i}D^{}_{SE})\big{]}-\Im\Bigg{[}\sum^{n-1}_{d=1}V_{i}[d]I^{}_{SE}[n-d]\Bigg{]}$ (40) As, $F_{SE}(\alpha)$ and $\|I_{SE}(\alpha)\|$ are the inverse and magnitude of $I_{SE}(\alpha)$, so, the general recurrence relationship between these can be expressed as given in (41) and (42). $F_{SE}[n]=\frac{-1}{D_{SE}}\sum^{n-1}_{d=0}F_{SE}[d]I_{SE}[n-d]$ (41) $\big{\|}I_{SE}[n]\big{\|}=\frac{1}{2\big{\|}D_{SE}\big{\|}}\bigg{\\{}\sum^{n}_{d=0}I_{SE}[d]I^{}_{SE}[n-d]-\sum^{n-1}_{d=1}\big{\|}I_{SE}[d]\big{\|}\big{\|}I_{SE}[n-d]\big{\|}\bigg{\\}}$ (42) ## References [1] X.-P. 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# Many-body effects in the excitations and dynamics of trapped Bose-Einstein condensates Ofir E. Alon<EMAIL_ADDRESS>Department of Mathematics, University of Haifa, Haifa 3498838, Israel Haifa Research Center for Theoretical Physics and Astrophysics, University of Haifa, Haifa 3498838, Israel Raphael Beinke <EMAIL_ADDRESS>Theoretische Chemie, Physikalisch- Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany Lorenz S. Cederbaum<EMAIL_ADDRESS>heidelberg.de Theoretische Chemie, Physikalisch-Chemisches Institut, Universität Heidelberg, Im Neuenheimer Feld 229, D-69120 Heidelberg, Germany ###### Abstract This review explores the dynamics and the low-energy excitation spectra of Bose-Einstein condensates (BECs) of interacting bosons in external potential traps putting particular emphasis on the emerging many-body effects beyond mean-field descriptions. To do so, methods have to be used that, in principle, can provide numerically exact results for both the dynamics and the excitation spectra in a systematic manner. Numerically exact results for the dynamics are presented employing the well-established multicongurational time-dependent Hartree for bosons (MCTDHB) method. The respective excitation spectra are calculated utilizing the more recently introduced linear-response theory atop it (LR-MCTDHB). The latter theory gives rise to an, in general, non-hermitian eigenvalue problem. The theory and its newly developed implementation are described in detail and benchmarked towards the exactly-solvable harmonic- interaction model. Several applications to BECs in one- and two-dimensional potential traps are discussed. With respect to dynamics, it is shown that both the out-of-equilibrium tunneling dynamics and the dynamics of trapped vortices are of many-body nature. Furthermore, many-body effects in the excitation spectra are presented for BECs in different trap geometries. It is demonstrated that even for essentially-condensed systems, the spectrum of the lowest-in-energy excitations computed at the many-body level can differ substantially from the standard mean-field description. In general, it is shown that bosons carrying angular momentum are more sensitive to many-body effects than bosons without. These effects are present in both the dynamics and the excitation spectrum. ###### Contents 1. 1 Introduction 2. 2 General concepts of many-boson physics 1. 2.1 Many-body Hamiltonian and Schrödinger equation 2. 2.2 Second quantization 1. 2.2.1 Fock states and commutation relations 2. 2.2.2 Bosonic field operator 3. 2.2.3 One- and two-body operators 4. 2.2.4 Hamiltonian in second quantization 3. 2.3 Dirac-Frenkel and least action variational principles 4. 2.4 Definition of Bose-Einstein condensation in traps 3. 3 Many-body theory 1. 3.1 Multiconfigurational time-dependent Hartree method for bosons (MCTDHB) 2. 3.2 Many-body linear response 1. 3.2.1 The connection between linear response and excitation spectra 2. 3.2.2 Linear-response MCTDHB (LR-MCTDHB) 4. 4 Numerical implementation of many-body linear response theory 1. 4.1 Code structure and challenges 2. 4.2 Construction of the LR matrix 3. 4.3 Calculation of eigenvalues 1. 4.3.1 The Arnoldi algorithm 2. 4.3.2 The QR-iteration 3. 4.3.3 Polynomial filtering 4. 4.3.4 The implicitly restarted Arnoldi method (IRAM) 5. 5 Comparison with an exactly-solvable model 1. 5.1 The harmonic-interaction model (HIM) 2. 5.2 Ground state and dynamics 3. 5.3 Excited states 6. 6 Applications to many-body dynamics and excitations of BECs 1. 6.1 Tunneling dynamics in traps 1. 6.1.1 Exact tunneling dynamics in a bosonic Josephson junction 2. 6.1.2 Many-body tunneling dynamics in a two-dimensional radial double well 2. 6.2 Dynamical fragmentation 1. 6.2.1 Tunneling to open space 2. 6.2.2 Phantom vortices 3. 6.3 Excitation spectra 1. 6.3.1 One-dimensional harmonic and double-well systems 2. 6.3.2 Triple wells and larger lattices 3. 6.3.3 Rotating Bose-Einstein condensates in an anharmonic trap 7. 7 Summary and Conclusions 8. Appendix A Mean-field theory 1. A.1 Single-orbital approaches 1. A.1.1 The GP equation 2. A.1.2 The BdG equations 2. A.2 Multi-orbital approaches 1. A.2.1 Time-dependent multi-orbital mean field (TDMF) 2. A.2.2 Linear-response best-mean-field (LR-BMF) 9. Appendix B LR-MCTDHB in block-diagonal form 10. Appendix C Further benchmarks of LR-MCTDHB 11. Appendix D Many-particle variance: A sensitive quantity to many-body effects ## 1 Introduction In 1924, Albert Einstein discovered that the spectral energy distribution in certain systems of massive particles is different from the Maxwell-Boltzmann statistics. He was inspired by a work of Satayendra N. Bose who derived the thermal energy distribution of black-body radiation solely based on assumptions from quantum theory [1], which is slightly different to the previous semi-classical approach of Max Planck. Einstein realized the far- reaching consequences of this, namely that a composite system of massive particles at sufficiently low temperatures could behave as a matter wave due to all particles sharing the same single-particle state. Thus, he applied Bose’s idea to a single-component ideal gas [2, 3]. This paved the way towards what is known as Bose-Einstein condensation. Consequently, all particles that obey the statistical distribution found by Bose and Einstein are termed bosons. It took approximately 70 years until the first gaseous Bose-Einstein condensate (BEC) was realized in an experiment. The group of Eric A. Cornell and Carl E. Wieman produced a repulsive BEC of rubidium atoms [4], shortly followed by the group of Wolfgang Ketterle using repulsive sodium atoms in a magneto-optical trap [5]. The latter three scientists obtained the Nobel prize in physics for this achievement in 2001. The first attractive BEC was realized by the group of Randall G. Hulet using lithium atoms [6]. In more recent experiments, condensation was observed in a large variety of systems, e.g., for exciton-polariton systems [7, 8], magnons at room temperature [9], or, rather astonishingly because of the lack of a rest mass and a vanishing chemical potential, even for photons in a cavity [10]. Over the past two decades, the experimental control of BECs has grown remarkably. This includes both the interaction of the constituent particles by employing Feshbach resonances [11, 12, 13] as well as the trap geometry and dimensionality [14]. This qualifies BECs for testing the fundamental principles of quantum mechanics, or very recently even to simulate certain processes in astrophysics [15, 16, 17]. Of particular interest is the understanding of the dynamics as well as the spectrum of excited states of trapped BECs made of interacting bosons. Theoretical descriptions for these were carried out many years before they were finally realized experimentally. The most prominent approach to obtain the ground state as well as the dynamical motion of condensates is the Gross- Pitaevskii (GP) equation [18, 19, 20, 21]. In this theory it is assumed that actually all bosons in the system occupy the same single-particle state, making it a mean-field theory. The GP equation has been successfully applied in many cases [22] and can be seen as the adequate starting point to solve many physical problems of BECs. Similarly, the linear-response (LR) theory atop the GP equation, resulting in the famous Bogoliubov-de Gennes (BdG) equations [23, 24], represents the commonly utilized approach to describe excitations of trapped ultracold bosons, e.g., for dark-bright soliton pairs [25], in rotating BECs with dipole-dipole interaction [26], or in a 2D hexagonal lattice with superimposed harmonic confinement [27], to name only a few recent applications. Furthermore, the BdG equations were extensively used for the description of Cooper pairs in superconductors [28] or for the understanding of the BCS-BEC crossover connecting superconductivity and superfluidity in correlated fermionic systems [29]. A crucial limitation of the GP mean-field theory is the fact that it ignores correlations between the bosons, and thus does not account for condensate depletion, i.e., the fraction of particles outside the condensed mode due to correlations and interactions, even at zero temperature. In a highly-cited review article from 1999, the condensate depletion is found to be “very small (less than $1\%$) in the presently available experimental conditions” [30], and thus the usage of mean-field methods like the BdG approach has been considered justified. However, the many-body methods developed in the following years were capable to demonstrate the remarkable impact of depletion and even fragmentation of a BEC onto its dynamics and its excitation spectrum. One of these many-body approaches is the multiconfigurational time-dependent Hartree method for bosons [MCTDHB, or MCTDHB($M$) with $M$ being the number of utilized single-particle states, called orbitals], which has been introduced [31, 32], benchmarked [33], and applied successfully to various systems in recent years. Being derived from a variational principle, its results can be improved systematically by increasing the number of orbitals $M$. Additionally, its LR theory, termed LR-MCTDHB($M$), has been introduced recently [34, 35]. The latter represents a full many-body description of excitations in trapped BECs. This review has two main objectives. At first, the newly developed numerical implementation of LR-MCTDHB, capable of treating large systems also in $D>1$ spatial dimensions which has not been realized before, is introduced and benchmarked against an exactly-solvable model. Secondly, various applications of both MCTDHB and LR-MCTDHB to trapped BECs are discussed in which the depletion has a strong impact on the dynamics and on the spectrum of excited states. With regard to dynamics, it is shown that fragmentation can develop over time although a system is initially almost entirely condensed. With regard to excitation spectra, it is demonstrated that substantial many-body effects appear even if the ground-state depletion is of the order of the above mentioned $1\%$ or lower. Even for the case of marginal depletion where one might expect mean-field theory to accurately describe the physics, it is instructive to compare the results obtained from the many-body approaches described above to the ones obtained from the GP and BdG theories. It is important to note that for only a single orbital, $M=1$, MCTDHB reduces exactly to the GP equation. Correspondingly, LR-MCTDHB($1$) yields the BdG equations. Hence, the commonly used mean-field approaches are included in the utilized many-body theories as their simplest limiting cases. To address these goals, the review is structured as follows. First, in Section 2, the main theoretical concepts of many-boson physics are presented. This includes the many-body Schrödinger equation, the framework of second quantization which is particularly useful for the many-body theory described in this work, the introduction of the Dirac-Frenkel variational principle from which the central equations of motion can be derived, as well as the definition of Bose-Einstein condensation. Second, in Section 3, a detailed derivation of MCTDHB is given. The essential ingredients like the ansatz for the many-boson wave function are discussed. Then, its LR theory is derived, which represents the central theory utilized in this work to calculate low- energy excitations of trapped bosonic systems on the many-body level. Third, in Section 4, the newly developed numerical implementation of LR-MCTDHB is described in detail. Therefore, general technical challenges and their solutions are explained. Furthermore, a detailed description of the numerical algorithms utilized is presented. Subsequently, in Section 5, the used implementations of MCTDHB and LR-MCTDHB are benchmarked against the exactly- solvable harmonic-interaction model, whose analytic solutions of the energies of the ground and excited states are presented beforehand. Applications of both implementations are then shown and their physics discussed in Section 6. Concerning the dynamics, the focus is twofold. First, applications dealing with the phenomenon of tunneling of BECs between wells inside potential traps in one and two spatial dimensions are discussed, demonstrating the many-body nature of quantum tunneling of ultracold BECs. Second, several cases of dynamical fragmentation of initially coherent bosonic systems are presented. For instance, the tunneling of a BEC to open space is analyzed. Furthermore, the connection between angular momentum, especially of quantized vortices, and the development of fragmentation is investigated. Concerning applications of LR-MCTDHB, the many-body effects in the low-energy spectra of BECs in one- dimensional multi-well traps and lattices are shown. Moreover, as a 2D application, the impact of angular momentum on the lowest-in-energy excitations of a rotating BEC is examined. Essential physical results and implications are provided. Finally, in Section 7, a summary and an outlook of possible constituent future research as well as of further enrichment of the implementation of LR-MCTDHB is provided. Additionally, the review contains four appendices which broaden its scope and provide additional insight. In Appendix A, the commonly-used mean-field theories, the GP theory and the constituent LR theory leading to the BdG equations, are described. A further mean-field approach for both the statics and dynamics of trapped BECs that has been developed prior to the many-body theories of Section 3 is presented, called the time-dependent multi-orbital mean field (TDMF) and the linear-response best-mean-field (LR-BMF). The latter can be seen as intermediate theories between the standard single-orbital mean- field approaches and the full many-body treatment and have their own merits. Appendix B demonstrates that LR-MCTDHB can be written in block-diagonal form, which, in some cases, can be beneficial. Advantages and possible weaknesses are discussed. In Appendix C, additional results of the LR-MCTDHB benchmark are provided. Finally, in Appendix D, the many-body variance, a quantity that turns out to be very sensitive to many-body effects, is introduced and derived for the example of the center-of-mass (c.m.) position of a BEC. Its relevance is discussed in several examples. ## 2 General concepts of many-boson physics In this section, the general concepts of describing trapped ultracold bosons are introduced. This includes the representation of the many-body Hamiltonian, the bosonic field as well as properties of the wave function for a system of identical bosons. Moreover, the definition of Bose-Einstein condensation and depletion which is used in the subsequent sections is presented. ### 2.1 Many-body Hamiltonian and Schrödinger equation The time-dependent Schrödinger equation for a generic system of $N$ identical spinless bosons in $D\leq 3$ dimensions is given by $\hat{H}\Psi(\mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{N};t)=i\frac{\partial}{\partial t}\Psi(\mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{N};t)$ (1) with the full many-body wave function $\Psi(\mathbf{r}_{1},\mathbf{r}_{2},...,\mathbf{r}_{N};t)$ depending on the spatial coordinates $\\{\mathbf{r}_{i}\\}$ of all $N$ bosons and the time $t$. The Hamiltonian $\hat{H}$ reads $\hat{H}=\sum_{i=1}^{N}\hat{h}(\mathbf{r}_{i})+\lambda_{0}\sum_{j>i}^{N}\hat{W}(\mathbf{r}_{i},\mathbf{r}_{j})$ (2) with the single-particle term $\displaystyle\hat{h}(\mathbf{r}_{i})$ $\displaystyle=\hat{T}(\mathbf{r}_{i})+V(\mathbf{r}_{i})$ (3) $\displaystyle\hat{T}(\mathbf{r}_{i})$ $\displaystyle=-\frac{1}{2}\Delta_{i}=-\frac{1}{2}\frac{\partial^{2}}{\partial\mathbf{r}_{i}^{\,2}}$ (4) comprising the contributions of the kinetic energy $\hat{T}$ and the external potential $V$ which is assumed to be real. For simplicity, the terms are given in dimensionless units with $\hbar=m=1$. The two-body operator $\hat{W}(\mathbf{r_{i}},\mathbf{r_{j}})$ represents the interaction potential of strength $\lambda_{0}$. In this work, interaction potentials which solely depend on the distance $r_{ij}=\|\mathbf{r_{i}}-\mathbf{r_{j}}\|$ between two bosons are considered. In general, both $V$ and $\hat{W}$ can be time- dependent. ### 2.2 Second quantization In this section, the formalism of second quantization is briefly explained, including the Fock state representation of an $N$-boson state, the symmetry properties of the wave function, and the Hamiltonian in second quantization. #### 2.2.1 Fock states and commutation relations As a consequence of the spin-statistics theorem [36, 37], the wave function $\Psi(\mathbf{r}_{1},...,\mathbf{r}_{N})$ of $N$ identical bosons is a fully symmetrized $N$-particle state, i.e., it is symmetric under the permutation of two bosons with coordinates $\mathbf{r}_{i}$ and $\mathbf{r}_{j}$, i.e., $\Psi(...,\mathbf{r}_{i},...,\mathbf{r}_{j},...)=+\Psi(...,\mathbf{r}_{j},...,\mathbf{r}_{i},...).$ (5) In second quantization, this is achieved by using a Fock state (or number state) basis of the $N$-particle Hilbert space $\mathcal{H}^{(N)}$ for the wave function $\Psi$. This basis incorporates the previously mentioned symmetrization requirements as well as the commutation relations of identical bosons shown below. For a complete set of orthonormalized single-particle states, also referred to as orbitals, $\\{\phi_{i}:i=1,...,\infty\\}$, a general Fock state reads $\|n_{1},n_{2},...\rangle=\prod_{i}\left[\frac{1}{\sqrt{n_{i}!}}\left(\hat{b}_{i}^{\dagger}\right)^{n_{i}}\right]\|\text{vac}\rangle$ (6) where the integers $\\{n_{i}\\}$ denote the occupation numbers of the individual orbitals and $\|\text{vac}\rangle$ the particle vacuum. In Eq. (6), the creation and annihilation operators $\hat{b}_{i}^{\dagger}$ and $\hat{b}_{i}$, which create or annihilate a particle in the state $\phi_{i}$, were introduced. The action of these operators on a Fock state is given by $\displaystyle\hat{b}_{i}\,\|n_{1},n_{2},...,n_{i},...\rangle=$ $\displaystyle\sqrt{n_{i}}\,\|n_{1},n_{2},...,n_{i}-1,...\rangle$ $\displaystyle\hat{b}_{i}^{\dagger}\,\|n_{1},n_{2},...,n_{i},...\rangle=$ $\displaystyle\sqrt{n_{i}+1}\,\|n_{1},n_{2},...,n_{i}+1,...\rangle$ and thus the number operator for any single-particle state $\phi_{i}$ reads $\hat{n}_{i}=\hat{b}_{i}^{\dagger}\hat{b}_{i}.$ (7) For a system with a fixed amount of bosons $N$, the occupation numbers of the orbitals sum up to the total number of particles $N$, i.e., $N=\sum_{i}\,n_{i}.$ (8) The creation and annihilation operators fulfill the conventional bosonic commutation relations $\left[\hat{b}_{i},\hat{b}_{k}\right]=0,\quad\quad\left[\hat{b}^{\dagger}_{i},\hat{b}^{\dagger}_{k}\right]=0,\quad\quad\left[\hat{b}_{i},\hat{b}_{k}^{\dagger}\right]=\delta_{ik}$ (9) which ensure the exchange symmetry of the Fock state basis in Eq. (6) and thus of the total wave function $\Psi$ as required in Eq. (5). #### 2.2.2 Bosonic field operator Utilizing the creation and annihilation operators of bosons for single- particle states from the previous section, one can define operators that create or annihilate a boson at position $\mathbf{r}$ at time $t$. These operators are the so-called field operators, $\displaystyle\hat{\Psi}^{\dagger}(\mathbf{r})=\sum_{i}\hat{b}^{\dagger}_{i}(t)\phi_{i}^{\ast}(\mathbf{r},t)$ (10) $\displaystyle\hat{\Psi}(\mathbf{r})=\sum_{i}\hat{b}_{i}(t)\phi_{i}(\mathbf{r},t),$ (11) where it is taken into account that the orbitals as well as the creation and annihilation operators can in general be time-dependent. Thus, $\hat{b}^{\dagger}(t)$ and $\hat{b}(t)$ can be expressed in terms of the field operators, $\hat{b}_{i}^{\dagger}(t)=\int\phi_{i}(\mathbf{r},t)\hat{\Psi}^{\dagger}(\mathbf{r})d\mathbf{r},\quad\quad\hat{b}_{i}(t)=\int\phi_{i}^{\ast}(\mathbf{r},t)\hat{\Psi}(\mathbf{r})d\mathbf{r}$ (12) where it is assumed that the set $\\{\phi_{i}(\mathbf{r},t)\\}$ consists of orthonormalized functions. In the following, the time argument in the above quantities is suppressed. The field operators in Eqs. (10) and (11) fulfill the bosonic commutation relations $\left[\hat{\Psi}(\mathbf{r}),\hat{\Psi}(\mathbf{r}^{\,\prime})\right]=0,\quad\quad\left[\hat{\Psi}^{\dagger}(\mathbf{r}),\hat{\Psi}^{\dagger}(\mathbf{r}^{\,\prime})\right]=0,\quad\quad\left[\hat{\Psi}(\mathbf{r}),\hat{\Psi}^{\dagger}(\mathbf{r}^{\,\prime})\right]=\delta\left(\mathbf{r}-\mathbf{r}^{\,\prime}\right).$ (13) The density operator is defined as $\hat{\rho}(\mathbf{r})=\hat{\Psi}^{\dagger}(\mathbf{r})\hat{\Psi}(\mathbf{r})$ (14) such that the operator of the total number of bosons in the system is given by $\hat{N}=\int\,d\mathbf{r}\,\hat{\rho}(\mathbf{r}).$ (15) #### 2.2.3 One- and two-body operators In second quantization, a general one-body operator $\hat{A}^{(1)}$, i.e., an operator acting on a single particle, reduces to the form $\hat{A}^{(1)}=\sum_{i,j}a_{ij}\,\hat{b}_{i}^{\dagger}\hat{b}_{j}$ (16) where the matrix elements $a_{ij}$ are given by $a_{ij}=\int\phi_{i}^{\ast}(\mathbf{r})\hat{A}^{(1)}\,\phi_{j}(\mathbf{r})\,d\mathbf{r}$ (17) or, equivalently in bra-ket notation, $a_{ij}=\langle i\|\hat{A}^{(1)}\|j\rangle$. Similarly, the general form of a two-body operator, i.e., an operator acting on two particles, is given by $\hat{A}^{(2)}=\frac{1}{2}\sum_{i,j,k,l}a_{ijkl}\,\hat{b}_{i}^{\dagger}\hat{b}^{\dagger}_{j}\hat{b}_{k}\hat{b}_{l}$ (18) with $a_{ijkl}=\iint\phi_{i}^{\ast}(\mathbf{r})\phi_{j}^{\ast}(\mathbf{r}^{\,\prime})\hat{A}^{(2)}\,\phi_{k}(\mathbf{r})\phi_{l}(\mathbf{r}^{\,\prime})\,d\mathbf{r}d\mathbf{r}^{\,\prime},$ (19) or $a_{ijkl}=\langle i,j\|\hat{A}^{(2)}\|k,l\rangle$ in short. The factor $1/2$ in Eq. (18) accounts for double counting of the individual terms. #### 2.2.4 Hamiltonian in second quantization The Hamiltonian in Eq. (2) can now be expressed within the framework of second quantization. Using the bosonic field operators in Eqs. (10) and (11) as well as the general expressions for one- and two-body operators, Eqs. (16) and (18), one obtains $\displaystyle\hat{H}$ $\displaystyle=\sum_{i=1}^{N}\hat{h}(\mathbf{r}_{i})+\lambda_{0}\sum_{j>i}^{N}\hat{W}(\mathbf{r}_{i},\mathbf{r}_{j})\Rightarrow$ $\displaystyle\hat{H}$ $\displaystyle=\int d\mathbf{r}\left(\hat{\Psi}^{\dagger}(\mathbf{r})\hat{h}(\mathbf{r})\hat{\Psi}(\mathbf{r})+\frac{\lambda_{0}}{2}\int d\mathbf{r}^{\,\prime}\hat{\Psi}^{\dagger}(\mathbf{r})\hat{\Psi}^{\dagger}(\mathbf{r}^{\,\prime})\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\hat{\Psi}(\mathbf{r})\hat{\Psi}(\mathbf{r}^{\,\prime})\right)$ $\displaystyle=\sum_{i,j}h_{ij}\,\hat{b}_{i}^{\dagger}\hat{b}_{j}+\frac{\lambda_{0}}{2}\sum_{i,j,k,l}W_{ijkl}\,\hat{b}_{i}^{\dagger}\hat{b}_{j}^{\dagger}\hat{b}_{k}\hat{b}_{l}$ (20) with the matrix elements $\displaystyle h_{ij}$ $\displaystyle=\int\phi_{i}^{\ast}(\mathbf{r})\,\hat{h}(\mathbf{r})\,\phi_{j}(\mathbf{r})\,d\mathbf{r}$ (21) $\displaystyle W_{ijkl}$ $\displaystyle=\iint\,\phi_{i}^{\ast}(\mathbf{r})\phi_{j}^{\ast}(\mathbf{r}^{\,\prime})\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\,\phi_{k}(\mathbf{r})\phi_{l}(\mathbf{r}^{\,\prime})\,d\mathbf{r}\,d\mathbf{r}^{\,\prime}.$ (22) To keep the notation simple, the time argument of the quantities in Eqs. (20)-(22) is suppressed. The above representation of $\hat{H}$ from Eq. (20) will be used in the many-body formalism presented in Section 3. ### 2.3 Dirac-Frenkel and least action variational principles The working equations of all utilized many-body methods in this work can be derived from the Dirac-Frenkel variational principle (DFVP) [38, 39, 40]. Like for any other variational method, it yields a condition for the dynamical evolution for any parametrized ansatz of the wave function $\Psi$. Within the DFVP, the equation that determines the dynamics is given by $\left\langle\delta\Psi(t)\left\|\hat{H}-i\hbar\frac{\partial}{\partial\,t}\right\|\Psi(t)\right\rangle=0$ (23) which basically means that any allowed variation $\|\delta\Psi(t)\rangle$ is orthogonal to the residual wave function $\left(\hat{H}-i\hbar\frac{\partial}{\partial\,t}\right)\|\Psi(t)\rangle$. It is important to note that under certain conditions, the DFVP is fully equivalent to two other commonly used time-dependent variational principles, which are the variational principle due to MacLachlan [41] and the least action principle. See in this context Ref. [42]. The least action principle is based on the assumption that the wave function minimizes the action $S=\int dt\,\left\langle\Psi\left\|\hat{H}-i\hbar\frac{\partial}{\partial t}\right\|\Psi\right\rangle,$ (24) meaning that its variation with respect to $\Psi$ should vanish, i.e., $\delta S=0.$ (25) Eqs. (24) and (25), in combination with normalization constraints, are used to derive the equations of motion for MCTDHB. The full derivation of the latter theory is carried out in detail in Section 3.1. Furthermore, the least action principle can also be employed to derive the standard mean-field equation for the ground state and dynamics of a trapped BEC, i.e., the GP equation. Its derivation is presented in Appendix A.1.1. ### 2.4 Definition of Bose-Einstein condensation in traps In 1956, Penrose and Onsager developed a theoretical criterion which is utilized in the definition of Bose-Einstein condensation and is based on standard quantities from statistical quantum mechanics [43]. It makes use of the one-body reduced density matrix (RDM) of an $N$-boson system which can be defined via the first order correlation function $\displaystyle\mathbf{\rho}^{(1)}(\mathbf{r}\|\mathbf{r}^{\,\prime})$ $\displaystyle=\left\langle\Psi\left\|\hat{\Psi}^{\dagger}(\mathbf{r}^{\,\prime})\hat{\Psi}(\mathbf{r})\right\|\Psi\right\rangle$ $\displaystyle=\sum_{i,j}\,\rho_{ij}\,\phi_{i}^{\ast}(\mathbf{r}^{\,\prime})\phi_{j}(\mathbf{r})$ (26) with $\rho_{ij}=\langle\Psi\|\hat{\rho}_{ij}\|\Psi\rangle,\quad\hat{\rho}_{ij}=\hat{b}_{i}^{\dagger}\hat{b}_{j}$ (27) where the time argument is suppressed in all quantities. More details on RDMs are given in Appendix D. For $\mathbf{r}=\mathbf{r}^{\,\prime}$, Eq. (2.4) denotes the one-particle density. Diagonalizing the one-body RDM yields $\mathbf{\rho}^{(1)}(\mathbf{r}\|\mathbf{r}^{\,\prime})=\sum_{i}n_{i}^{(1)}\,\alpha_{i}^{\ast}(\mathbf{r}^{\,\prime})\alpha_{i}(\mathbf{r})$ (28) where the eigenvalues $\left\\{n_{i}^{(1)}\|n_{1}^{(1)}\geq n_{2}^{(1)}\geq...\right\\}$ and the eigenfunctions $\\{\alpha_{i}(\mathbf{r})\\}$ are the natural occupation numbers and the natural orbitals, respectively. Whenever possible, the superscript ’$(1)$’ of the natural occupation numbers will be omitted throughout this work. According to Ref. [43], any $N$-boson system where only the largest eigenvalue is macroscopic, i.e., $n_{1}=\mathcal{O}(N)$, is said to be condensed. However, if more than one eigenvalue of the one-body RDM are macroscopic, the system is said to be depleted/fragmented [44, 45]. In this work, the degree of depletion (or fragmentation) is quantified by $f=\frac{1}{N}\sum_{i>1}n_{i}$ (29) which is the aggregated occupation of all but the first natural orbital. Another classification that uses the above relation distinguishes the terms depletion and fragmentation more clearly and was used in Ref. [46]. In Section 6, the dynamics and excited states of BECs in different trap geometries in one and two spatial dimensions are studied. There, the degree of depletion/fragmentation will be an essential quantity to distinguish the many- body results obtained from the methods described in the subsequent Section 3 from the mean-field results obtained by using the theories described in the Appendices A.1.1 and A.1.2. ## 3 Many-body theory In this section, the many-body approach utilized in this work to compute the ground state and the dynamics of trapped BECs as well as their low-energy excitation spectra is presented. First, an introduction to MCTDHB is made in Section 3.1, followed by the application of LR theory atop it, termed LR- MCTDHB, in Section 3.2.2. ### 3.1 Multiconfigurational time-dependent Hartree method for bosons (MCTDHB) In this section, a detailed derivation of the MCTDHB [or MCTDHB($M$)] theory which has been originally introduced in Refs. [31, 32], is presented. It originates from the closely related MCTDH method [47, 48, 49] which is a widely used technique to compute the dynamics and excited states of nuclei in molecules. Further extensions of MCTDHB to the case of type conversion of particles, to mixtures of bosons and fermions, to Hubbard-type Hamiltonians of BECs on lattices, as well as to BECs with internal degrees of freedom can be found in Refs. [50, 51, 52, 53]. Other many-body theories for the ground state and excitations of BECs were utilized as well. One example is the density matrix renormalization group (DMRG) theory [54]. It is particularly designed for applications in the field of one-dimensional condensed matter physics, but was extended beyond this to quantum chemistry and quantum information. An early review can be found in [55]. Furthermore, DMRG has been extended and applied to two-dimensional systems as well [56]. However, this review deals with MCTDHB and its LR theory. The ansatz for the many-body wave function to solve the many-boson Schrödinger equation, Eq. (1), is given by $\|\Psi(t)\rangle=\sum_{\mathbf{n}}C_{\mathbf{n}}(t)\,\|\mathbf{n};t\rangle,$ (30) which is a superposition of permanents $\\{\|\mathbf{n};t\rangle\\}$ comprised of $M$ single-particle orbitals $\\{\phi_{j}(\mathbf{r},t):1\leq j\leq M\\}$ and expansion coefficients $\\{C_{\mathbf{n}}(t)\\}$ where $\mathbf{n}=(n_{1},\ldots,n_{M})^{t}$ is a vector carrying the individual occupation numbers of the orbitals for a given permanent. The summation in Eq. (30) includes all $N_{\text{conf}}=\binom{N+M-1}{N}$ possibilities to distribute $N$ bosons onto $M$ single-particle orbitals. It is important to note that both the permanents and the coefficients are time-dependent, which is the main difference compared to the frequently used many-body approach of the full configuration interaction (FCI) method. In the latter theory, the amount of configurations for a given number $M$ of single-particle basis functions is the same as for MCTDHB. However, these basis states are fixed in shape, i.e., they remain unchanged and only the coefficients are time- dependent. For MCTDHB, the single-particle orbitals are time-adaptive and, as shown below, the corresponding set of equations of motion (EOMs) is coupled to the set of EOMs of the coefficients. In terms of numerical convergence, utilizing a time-adaptive instead of a shape-fixed basis dramatically improves the convergence towards exact results for both the ground state and the dynamics of trapped BECs, as discussed in Section 5. The working equations of MCTDHB are determined by the least action principle of Eqs. (24) and (25), which explicitly reads $0=\delta S=\delta\left(\int dt\,L(t)\right)$ (31) with $L(t)=\left\langle\Psi(t)\left\|\hat{H}-i\frac{\partial}{\partial t}\right\|\Psi(t)\right\rangle-\sum_{i,j}\mu_{ij}(t)[\langle\phi_{i}\|\phi_{j}\rangle-\delta_{ij}]$ (32) and $\hbar=1$. The time-dependent Langrange multipliers $\\{\mu_{ij}(t)\\}$ account for the orthonormalization of the orbitals in time. Whenever possible, the time-argument of all quantities is suppressed in the following. Plugging in the ansatz for the wave function, Eq. (30), and the expression for the Hamiltonian, Eq. (20), into Eq. (32) results in $\displaystyle L(t)$ $\displaystyle=\sum_{k,q}^{M}\rho_{kq}\left[h_{kq}-\left(i\frac{\partial}{\partial t}\right)_{kq}\right]+\frac{1}{2}\sum_{k,s,q,l}^{M}\rho_{ksql}W_{ksql}$ $\displaystyle-i\sum_{\mathbf{n}}C_{\mathbf{n}}^{\ast}(t)\frac{\partial C_{\mathbf{n}}(t)}{\partial t}-\sum_{k,q}\mu_{kq}[\langle\phi_{k}\|\phi_{q}\rangle-\delta_{kq}]$ (33) where the elements of the one- and two-body RDMs in this case read $\displaystyle\rho_{kq}$ $\displaystyle=\sum_{\mathbf{n},\mathbf{n}^{\,\prime}}C_{\mathbf{n}}^{\ast}(t)C_{\mathbf{n}^{\,\prime}}(t)\left\langle\mathbf{n};t\left\|\hat{b}_{k}^{\dagger}\hat{b}_{q}\right\|\mathbf{n}^{\,\prime};t\right\rangle$ (34) $\displaystyle\rho_{ksql}$ $\displaystyle=\sum_{\mathbf{n},\mathbf{n}^{\,\prime}}C_{\mathbf{n}}^{\ast}(t)C_{\mathbf{n}^{\,\prime}}(t)\left\langle\mathbf{n};t\left\|\hat{b}_{k}^{\dagger}\hat{b}_{s}^{\dagger}\hat{b}_{q}\hat{b}_{l}\right\|\mathbf{n}^{\,\prime};t\right\rangle.$ (35) The variation of $S$ with respect to $\phi_{k}^{\ast}(\mathbf{r},t)$ yields $\frac{\delta S}{\delta\phi_{k}^{\ast}(\mathbf{r},t)}=\sum_{q=1}^{M}\left[\rho_{kq}\left(\hat{h}-i\frac{\partial}{\partial t}\right)-\mu_{kq}+\sum_{s,l=1}^{M}\rho_{ksql}\hat{W}_{sl}\right]\|\phi_{q}\rangle$ (36) with $\hat{W}_{sl}(\mathbf{r},t)=\int\phi_{s}^{\ast}(\mathbf{r}^{\,\prime},t)\,\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\,\phi_{l}(\mathbf{r}^{\,\prime},t)\,d\mathbf{r}^{\,\prime}.$ (37) According to the DFVP, this variation should vanish, which determines the EOMs of the orbitals given by $\sum_{q=1}^{M}\left[\rho_{kq}\hat{h}-\mu_{kq}+\sum_{s,l=1}^{M}\rho_{ksql}\hat{W}_{sl}\right]\|\phi_{q}\rangle=i\sum_{q=1}^{M}\rho_{kq}\frac{\partial}{\partial t}\|\phi_{q}\rangle.$ (38) Taking the scalar product with $\langle\phi_{i}\|$ in Eq. (38) results in the expression for the Lagrange multipliers, $\mu_{ki}(t)=\sum_{q=1}^{M}\left(\rho_{kq}\left[h_{iq}-\left(i\frac{\partial}{\partial t}\right)_{iq}\right]+\sum_{s,l=1}^{M}\rho_{ksql}W_{isql}\right).$ (39) By reinserting them into Eq. (38), multiplying by the inverse of the one-body RDM $\\{\boldsymbol{\rho}^{-1}\\}_{ik}$ from the left-hand side (LHS), and summing over $k$, one arrives at $i\hat{\mathbf{P}}\frac{\partial}{\partial t}\|\phi_{i}\rangle=\hat{\mathbf{P}}\left[\hat{h}\|\phi_{i}\rangle+\sum_{k,s,l,q=1}^{M}\\{\boldsymbol{\rho}^{-1}\\}_{ik}\,\rho_{ksql}\hat{W}_{sl}\|\phi_{q}\rangle\right]$ (40) with $\hat{\mathbf{P}}=\mathbb{1}-\sum_{j^{\prime}=1}^{M}\|\phi_{j^{\prime}}\rangle\langle\phi_{j^{\prime}}\|$ (41) being a projection operator on the tangential space of $\text{span}\left(\phi_{1},...,\phi_{M}\right)$. Due to invariance properties of the ansatz for the wave function in Eq. (30), the orbitals can be chosen without loss of generality to be orthogonal to their corresponding time- derivatives [47, 48], i.e., $\langle\phi_{i}\|\dot{\phi_{j}}\rangle=0\quad\forall i,j=1,...,M.$ (42) The time evolution of the orbitals is therefore fully orthogonal and ensures the orthonormality of the orbitals in time. Hence, one can further simplify Eq. (40) to read $i\frac{\partial}{\partial t}\|\phi_{i}\rangle=\hat{\mathbf{P}}\left[\hat{h}\|\phi_{i}\rangle+\sum_{k,s,l,q=1}^{M}\\{\boldsymbol{\rho}^{-1}\\}_{ik}\,\rho_{ksql}\hat{W}_{sl}\|\phi_{q}\rangle\right],$ (43) and Eq. (43) is referred to as the orbitals’ EOM in the following. With respect to the coefficients, the least action principle $\displaystyle\frac{\delta S}{\delta C_{\mathbf{n}}^{\ast}(t)}=0$ (44) leads to $\sum_{\mathbf{n}^{\,\prime}}\left\langle\mathbf{n};t\left\|\hat{H}-i\frac{\partial}{\partial t}\right\|\mathbf{n}^{\,\prime};t\right\rangle C_{\mathbf{n}^{\,\prime}}(t)=i\frac{\partial}{\partial t}C_{\mathbf{n}}(t).$ (45) Thus, the EOM of the coefficients is given by $\mathbf{\mathcal{H}}(t)\mathbf{C}(t)=i\frac{\partial}{\partial t}\mathbf{C}(t)$ (46) with the matrix elements $\mathcal{H}_{\mathbf{n}\mathbf{n}^{\,\prime}}=\left\langle\mathbf{n};t\left\|\hat{H}-i\frac{\partial}{\partial t}\right\|\mathbf{n}^{\,\prime};t\right\rangle$ (47) and $\mathbf{C}(t)$ being a vector whose entries are the $N_{\text{conf}}$ coefficients $\\{C_{\mathbf{n}}(t)\\}$. It is worth noting that $\mathbf{\mathcal{H}}(t)$ is hermitian. Eq. (46) is referred to as the coefficients’ EOM in the following. One observes that the sets of working equations for the orbitals and coefficients are coupled to each other. With regard to the orbitals’ EOM, the coefficients appear in the expressions for the two-body RDM as well as in the inverse of the one-body RDM, see Eqs. (34), (35) and (43), respectively. With regard to the coefficients’ EOM, the implicit dependence on the orbitals is incorporated in the matrix elements of Eq. (47) since the permanents appearing therein are assembled by the orbitals. By employing imaginary time-propagation, i.e., by setting $it\rightarrow\tau$ in the working equations (43) and (46), the MCTDHB theory reduces to the multiconfigurational Hartree for bosons (MCHB) method which has been introduced in Ref. [57]. In that way, the ground state of a given system can be computed very efficiently, as will be shown in Section 5.2. It can be inferred from Eq. (46) that the evolution of $\mathbf{C}(t)$ is unitary and therefore its normalization is conserved in time. However, in comparison to the orbitals’ EOMs in Eq. (43), the coefficient vector $\mathbf{C}(t)$ does not automatically propagate in an orthogonal manner. The latter can be achieved by introducing a time-dependent phase of the form $\mathbf{C}(t)\rightarrow\mathbf{C}(t)e^{-i\int^{t}dt^{\prime}\mathbf{C^{\dagger}}(t^{\prime})\mathbf{\mathcal{H}}(t^{\prime})\mathbf{C}(t^{\prime})}$ (48) which, by inserting it into Eq. (46), yields $i\frac{\partial}{\partial t}\mathbf{C}(t)=\mathbf{\mathcal{P}_{c}}\mathbf{\mathcal{H}}(t)\mathbf{C}(t)$ (49) with the projector that maps onto the orthogonal space of the coefficients is given by $\mathbf{\mathcal{P}_{c}}=\mathbb{1}-\mathbf{C}(t)\mathbf{C^{\dagger}}(t).$ (50) Including the projector $\mathbf{\mathcal{P}_{c}}$ is however redundant in the sense that the evolution of the coefficient vector is unitary already without it, see Eq. (46). On the contrary, this is not the case for the orbitals, where the projector $\hat{\mathbf{P}}$ that stems from the Lagrange multipliers in Eq. (31) ensures that the orbitals stay normalized in time. Henceforth, only the latter projector is considered in the subsequent derivation of LR-MCTDHB (Section 3.2.2), whereas the coefficients’ projector $\mathbf{\mathcal{P}_{c}}$ is set to unity in the following without any loss of generality. It is important to note that for a single orbital, i.e., $M=1$, the MCTDHB theory reduces to the GP mean-field equation [18, 19, 20, 21], meaning that GP$\equiv$MCTDHB(1). The derivation of the latter theory using the variational principle is described in Appendix A.1.1. ### 3.2 Many-body linear response #### 3.2.1 The connection between linear response and excitation spectra Before carrying out the derivation of the many-body LR theory atop MCTDHB, a rather general perspective of applying LR to the time-dependent Schrödinger equation, Eq. (1), in order to calculate the excitation spectrum of a trapped BEC is discussed. The goal is to demonstrate that in general, a LR analysis atop an exact eigenstate of the unperturbed Hamiltonian $\hat{H}^{0}$, typically the ground state, yields the exact excitation spectrum of the many- particle system. The following derivation is closely related to the one in Ref. [35]. One considers the time-dependent Hamiltonian $\displaystyle\hat{H}(t)$ $\displaystyle=\hat{H}^{0}+\hat{H}_{\text{ext}}(t),$ (51) $\displaystyle\hat{H}_{\text{ext}}(t)$ $\displaystyle=\hat{f}^{+}(\mathbf{r})e^{-i\omega t}+\hat{f}^{-}(\mathbf{r})e^{i\omega t}$ (52) where a weak time-dependent external field with amplitudes $\hat{f}^{+}$ and $\hat{f}^{-}$ and oscillation frequency $\omega$ is added to the stationary hermitian Hamiltonian $\hat{H}^{0}$. The latter has eigenenergies $E_{k},\,k\in\mathbb{N}$, and the ground-state energy is denoted by $\varepsilon^{0}$. The projected time-dependent Schrödinger equation with $\hbar=1$, $\hat{\mathbf{P}}_{\Psi}\hat{H}(t)\Psi(t)=i\dot{\Psi}(t),\quad\hat{\mathbf{P}}_{\Psi}=\mathbb{1}-\|\Psi(t)\rangle\langle\Psi(t)\|,$ (53) which can be obtained by introducing the phase $\Psi(t)\rightarrow\Psi(t)e^{-i\int^{t}dt^{\prime}\langle\Psi(t^{\prime})\|\hat{H}(t^{\prime})\rangle\Psi(t^{\prime})}$ (54) to the system’s wave function $\Psi(t)$ and inserting it into Eq. (1), ensures that the time evolution is orthogonal in the sense that $\langle\Psi(t)\|\dot{\Psi}(t)\rangle=0$ (55) at any time $t$. To solve Eq. (53), the ansatz $\Psi(\mathbf{r},t)=e^{-\varepsilon^{0}t}\left(\Psi_{0}(\mathbf{r})+u(\mathbf{r})\,e^{-i\omega t}+v^{\ast}(\mathbf{r})\,e^{i\omega t}\right)$ (56) with is employed. The ground state $\Psi_{0}$ obtained by $\hat{H}^{0}\Psi_{0}=\varepsilon^{0}\Psi_{0}$ as well as the correction amplitudes $u$ and $v$ are assumed to be stationary. The attached time- dependent oscillations have the same frequency $\omega$ as the external perturbing field. Utilizing the orthogonality condition, Eq. (55), one obtains $\displaystyle\langle\Psi_{0}\|u\rangle=0\quad\text{and}\quad\langle\Psi_{0}\|v^{\ast}\rangle=0,$ (57) meaning that the correction amplitudes are orthogonal to the ground-state wave function and thus $\displaystyle\hat{\mathbf{P}}_{\Psi_{0}}\|u\rangle=\|u\rangle\quad\text{and}\quad\hat{\mathbf{P}}_{\Psi_{0}}\|v^{\ast}\rangle=\|v^{\ast}\rangle$ (58) for the operator projecting onto the tangential space of $\Psi_{0}$, i.e., $\hat{\mathbf{P}}_{\Psi_{0}}=\mathbb{1}-\|\Psi_{0}\rangle\langle\Psi_{0}\|$. Inserting the ansatz (56) into Eq. (53), one arrives at $\displaystyle\hat{\mathbf{P}}_{\Psi_{0}}\hat{H}^{0}(u\,e^{-i\omega t}+v^{\ast}\,e^{i\omega t})+\hat{\mathbf{P}}_{\Psi_{0}}(\hat{f}^{+}e^{-i\omega t}+\hat{f}^{-}e^{i\omega t})\Psi_{0}$ $\displaystyle=(\omega+\varepsilon^{0})(u\,e^{-i\omega t}-v^{\ast}\,e^{i\omega t})$ (59) where only terms linear to $u$ and $v$ are kept. This can be written in matrix form by ordering terms proportional to $e^{-i\omega t}$ and $e^{i\omega t}$, respectively. The result reads $\left[\begin{pmatrix}\hat{\mathbf{P}}_{\Psi_{0}}(\hat{H}^{0}-\varepsilon^{0})&0\\\ 0&-\hat{\mathbf{P}}^{\ast}_{\Psi_{0}}(\hat{H}^{0,\ast}-\varepsilon^{0})\end{pmatrix}-\omega\right]\begin{pmatrix}u\\\ v\end{pmatrix}=\begin{pmatrix}-\hat{\mathbf{P}}_{\Psi_{0}}\hat{f}^{+}\,\Psi_{0}\\\ \hat{\mathbf{P}}^{\ast}_{\Psi_{0}}\hat{f}^{-,\ast}\,\Psi_{0}\end{pmatrix}$ (60) where complex conjugation to the lower equation has been applied. Defining the LR matrix $\mathcal{L}$ as $\mathcal{L}=\begin{pmatrix}\hat{\mathbf{P}}_{\Psi_{0}}(\hat{H}^{0}-\varepsilon^{0})\hat{\mathbf{P}}_{\Psi_{0}}&0\\\ 0&-\hat{\mathbf{P}}^{\ast}_{\Psi_{0}}(\hat{H}^{0,\ast}-\varepsilon^{0})\hat{\mathbf{P}}^{\ast}_{\Psi_{0}}\end{pmatrix}$ (61) where redundantly the projectors $\hat{\mathbf{P}}_{\Psi_{0}}$ and $\hat{\mathbf{P}}^{\ast}_{\Psi_{0}}$ have been inserted from the right-hand side (RHS) to the upper and lower blocks, the homogeneous version of Eq. (60) is given by $\mathcal{L}\begin{pmatrix}u\\\ v\end{pmatrix}=\omega\begin{pmatrix}u\\\ v\end{pmatrix}.$ (62) For the upper block, one can immediately deduce that the spectrum of corresponding eigenvalues is given by $\omega_{k}=E_{k}-\varepsilon^{0}$, meaning that one obtains the eigenvalues of the unperturbed Hamiltonian $\hat{H}^{0}$. Moreover, one obtains $\omega_{k}=-(E_{k}-\varepsilon^{0})$ for the lower block. This is an important result because it means that the LR due to an external perturbation, as described in Eq. (52), yields, if applied to an exact eigenstate of $\hat{H}^{0}$ (not necessarily the ground state), the full spectrum of exact excitation energies. It is worth stressing the generality of the above derivation because no physical properties of the system were specified, i.e., whether $\hat{H}^{0}$ describes a single- or multi-particle system, bosons or fermions (or even mixtures), or identical or distinguishable particles. Thus, LR represents a generic method for obtaining the spectrum of excited states of a quantum system. Section 3.2.2, explicitly deals with the many-body linear response of systems consisting of identical bosons. This section is closed by referring to systems where no exact eigenstate of $\hat{H}^{0}$, e.g., the ground state, is known. This is naturally the most common case since the number of systems where analytic solutions are available is very small. Hence, in order to obtain accurate results for the excitation energies from a LR analysis, one is in need of a very accurate approximation of the system’s ground state. In other words, increasing the quality of the underlying ground-state approximation will increase the accuracy of the spectrum of excited states. From a numerical point of view, Section 5 shows that MCTDHB turns out to be a very powerful approach to calculate the ground state of a trapped BEC to very high accuracy, and that it is clearly superior to other widely used methods like the GP mean-field equation or the FCI method. As a result, LR-MCTDHB leads to highly accurate excitation spectra, for which numerical evidence is presented in Section 5.3. #### 3.2.2 Linear-response MCTDHB (LR-MCTDHB) In this section, LR-MCTDHB [or LR-MCTDHB($M$)] is derived. To this end, a trapped $N$-boson system is considered and its ground-state orbitals and coefficients, $\\{\phi_{k}^{0}:1\leq k\leq M\\}$ and $\mathbf{C}^{0}$, are calculated by using imaginary time-propagation of the MCTDHB working equations [57], Eqs. (43) and (46). Afterwards, the same time-dependent external perturbation as given in Eq. (52) is added to the Hamiltonian, such that the latter is finally given by Eqs. (51). The EOMs are linearized according to this. The subsequent derivation of the LR equations follows Refs. [34, 35]. The ansätze for the response orbitals and coefficients due to the external perturbation are $\displaystyle\phi_{k}(\mathbf{r},t)$ $\displaystyle\approx\phi_{k}^{0}(\mathbf{r})+\delta\phi_{k}(\mathbf{r},t)$ (63) $\displaystyle\delta\phi_{k}(\mathbf{r},t)$ $\displaystyle=u_{k}(\mathbf{r})e^{-i\omega t}+v_{k}^{\ast}(\mathbf{r})e^{i\omega t}$ (64) $\displaystyle\mathbf{C}(t)$ $\displaystyle\approx e^{-i\epsilon^{0}t}[\mathbf{C}^{0}+\delta\mathbf{C}(t)]$ (65) $\displaystyle\delta\mathbf{C}(t)$ $\displaystyle=\mathbf{C}_{u}e^{-i\omega t}+\mathbf{C}_{v}^{\ast}e^{i\omega t}$ (66) with $1\leq k\leq M$. The $k$-th response orbital is thus given by the zeroth- order orbital, typically the time-independent ground-state orbital $\phi_{k}^{0}(\mathbf{r})$, plus a weak, time-dependent perturbation $\delta\phi_{k}(\mathbf{r},t)$ with oscillation frequency $\omega$ and stationary response amplitudes $u_{k}(\mathbf{r})$ and $v_{k}(\mathbf{r})$. The response coefficients consist of similar parts and are additionally multiplied with a time-dependent phase $e^{-i\varepsilon^{0}t}$ to which it is referred below. The time-dependence is therefore essentially incorporated in the perturbation part. In the following, the linearization of the orbitals’ and coefficients’ EOMs are discussed separately. For linearizing the orbitals’ EOMs, it is instructive to utilize the version in Eq. (38) given by $i\sum_{q=1}^{M}\rho_{kq}\|\dot{\phi}_{q}(t)\rangle=\sum_{q=1}^{M}[\hat{Z}_{kq}-\mu_{kq}(t)]\|\phi_{q}(t)\rangle,$ (67) with the replacement $\hat{Z}_{kq}=\rho_{kq}[\hat{h}+\hat{H}_{\text{ext}}(t)]+\sum_{s,l=1}^{M}\rho_{ksql}\hat{W}_{sl}.$ (68) Inserting Eq. (63) into Eq. (67), one can group the resulting expression in terms of perturbation orders. In zeroth order, one obtains $0=\sum_{q}[\hat{Z}_{kq}^{0}-\mu^{0}_{kq}]\|\phi_{q}^{0}\rangle$ (69) with $\hat{Z}^{0}_{kq}=\rho^{0}_{kq}\hat{h}+\sum_{s,l=1}^{M}\rho^{0}_{ksql}\hat{W}^{0}_{sl}.$ (70) Here and in the following, quantities with the superscript ’0’ denote that they only contain zeroth-order orbitals and coefficients. Eq. (69) represents the orbital part of the MCHB working equations, see Ref. [57] in this context. For the first order, the LHS of Eq. (67) yields $i\sum_{q}(\delta\rho_{kq}\underbrace{\|\dot{\phi}_{q}^{0}\rangle}_{=0}+\rho_{kq}^{0}\|\dot{\delta\phi_{q}}(t)\rangle)=i\sum_{q}\rho_{kq}^{0}\|\dot{\delta\phi_{q}}(t)\rangle.$ (71) The RHS is given by $\sum_{q}[\hat{Z}_{kq}^{0}-\mu_{kq}^{0}]\,\|\delta\phi_{q}(t)\rangle+\sum_{q}[\delta\hat{Z}_{kq}-\delta\mu_{kq}(t)]\,\|\phi_{q}^{0}\rangle$ (72) with $\displaystyle\delta\hat{Z}_{kq}$ $\displaystyle=\delta\left(\rho_{kq}\hat{h}+\sum_{s,l}\rho_{ksql}\hat{W}_{sl}\right)+\rho_{kq}^{0}\hat{H}_{\text{ext}}(t)$ $\displaystyle=\delta\rho_{kq}\hat{h}+\sum_{s,l}\rho_{ksql}^{0}\delta\hat{W}_{sl}+\sum_{s,l}\delta\rho_{ksql}\hat{W}_{sl}^{0}+\rho_{kq}^{0}\hat{H}_{\text{ext}}(t).$ (73) To compute the RHS further, the variation of the chemical potential $\delta\mu_{kq}(t)$, given by $\displaystyle\delta\mu_{kq}(t)$ $\displaystyle=\left\langle\delta\phi_{q}\left\|\sum_{j}\left[\hat{Z}_{kj}^{0}-i\rho_{kj}\frac{\partial}{\partial t}\right]\,\right\|\phi_{j}^{0}\right\rangle+\left\langle\phi_{q}^{0}\left\|\sum_{j}\delta\left[\hat{Z}_{kj}^{0}-i\rho_{kj}\frac{\partial}{\partial t}\,\right\|\phi_{j}\right\rangle\right]$ $\displaystyle\overset{\text{Eq.}(\ref{LR- MCTDHB_zero})}{=}\sum_{j}\mu_{kj}^{0}\left\langle\delta\phi_{q}\|\phi_{j}^{0}\right\rangle+\left\langle\phi_{q}^{0}\left\|\sum_{j}\rho_{kj}^{0}\hat{H}_{\text{ext}}(t)\right\|\phi_{j}^{0}\right\rangle$ $\displaystyle+\left\langle\phi_{q}^{0}\left\|\left(\sum_{j}\delta\rho_{kj}\hat{h}+\sum_{j,s,l}(\rho_{ksjl}^{0}\delta\hat{W}_{sl}+\delta\rho_{ksjl}\hat{W}_{sl}^{0})\right)\right\|\phi_{j}^{0}\right\rangle$ $\displaystyle+\left\langle\phi_{q}^{0}\left\|\sum_{j}\left(\hat{Z}_{kj}^{0}-i\rho_{kj}^{0}\frac{\partial}{\partial t}\right)\right\|\delta\phi_{j}\right\rangle\,,$ (74) is employed. Thus, Eq. (72) can be written as $\displaystyle\sum_{q}[\hat{Z}_{kq}^{0}-\mu_{kq}^{0}]\,\|\delta\phi_{q}\rangle+\sum_{q}\left\\{\delta\rho_{kq}\hat{h}+\rho_{kq}^{0}\hat{H}_{\text{ext}}(t)+\sum_{s,l}(\rho_{ksql}\,\delta\hat{W}_{sl}+\delta\rho_{ksql}\,\hat{W}_{sl}^{0})\right\\}\|\phi_{q}^{0}\rangle$ $\displaystyle-\sum_{q}\left\\{\sum_{j}(-\mu_{kj}^{0})\left\langle\phi_{q}^{0}\|\delta\phi_{j}\right\rangle+\left\langle\phi_{q}^{0}\left\|\sum_{j}\rho_{kj}^{0}\hat{H}_{\text{ext}}(t)\right\|\phi_{j}^{0}\right\rangle\right.$ $\displaystyle+\left.\left\langle\phi_{q}^{0}\left\|\left(\sum_{j}\delta\rho_{kj}\hat{h}+\sum_{j,s,l}(\rho_{ksjl}^{0}\delta\hat{W}_{sl}+\delta\rho_{ksjl}\hat{W}_{sl}^{0})\right)\right\|\phi_{j}^{0}\right\rangle\right.$ $\displaystyle+\left.\left\langle\phi_{q}^{0}\left\|\sum_{j}\left(\hat{Z}_{kj}^{0}-i\rho_{kj}^{0}\frac{\partial}{\partial t}\right)\right\|\delta\phi_{j}\right\rangle\ \right\\}\|\phi_{q}^{0}\rangle$ (75) which, combined with the LHS, Eq. (71), leads to $\displaystyle\sum_{q}\hat{\mathbf{P}}\left\\{\left(\hat{Z}_{kq}^{0}-\mu_{kq}^{0}\right)\,\|\delta\phi_{q}\rangle+\left(\delta\rho_{kq}\hat{h}+\rho_{kq}^{0}\hat{H}_{\text{ext}}(t)\right.\right.$ $\displaystyle+\sum_{s,l}\delta\rho_{ksql}\,\hat{W}_{sl}^{0}+\sum_{s,l}\left.\left.\rho_{ksql}^{0}\,\delta\hat{W}_{sl}\right)\|\phi_{q}^{0}\rangle\right\\}$ $\displaystyle=i\hat{\mathbf{P}}\sum_{q}\rho_{kq}^{0}\|\delta\dot{\phi}_{q}\rangle$ (76) where the projection operator $\hat{\mathbf{P}}=\mathbb{1}-\sum_{j^{\prime}}\|\phi_{j^{\prime}}^{0}\rangle\langle\phi_{j^{\prime}}^{0}\|$ projects on the tangential space of the stationary orbitals. From the invariance condition, Eq. (42), one can deduce $\displaystyle 0$ $\displaystyle=\delta\langle\phi_{k}\|\dot{\phi}_{q}\rangle=\langle\delta\phi_{k}\underbrace{\|\dot{\phi}_{q}^{0}\rangle}_{=0}+\langle\phi_{k}^{0}\|\delta\dot{\phi}_{q}\rangle$ $\displaystyle=-i\omega\langle\phi_{k}^{0}\|u_{q}\rangle e^{-i\omega t}+i\omega\langle\phi_{k}^{0}\|v^{\ast}_{q}\rangle e^{i\omega t}$ $\displaystyle\Rightarrow\langle\phi_{k}^{0}\|u_{q}\rangle=\langle\phi_{k}^{0}\|v^{\ast}_{q}\rangle=0\quad\quad\forall\,1\leq k,q\leq M$ (77) such that $\hat{\mathbf{P}}\|\delta\dot{\phi}_{q}\rangle=\left(\mathbb{1}-\sum_{j^{\prime}=1}^{M}\|\phi^{0}_{j^{\prime}}\rangle\langle\phi^{0}_{j^{\prime}}\|\right)\|\delta\dot{\phi}_{q}\rangle=\|\delta\dot{\phi}_{q}\rangle$ (78) which makes the projection operator on the RHS of Eq. (3.2.2) redundant. One further utilizes $\displaystyle\delta\hat{W}_{sl}\left[u_{l},u_{s}^{\ast},v_{l}^{\ast},v_{s}\right]$ $\displaystyle=\int d\mathbf{r}^{\,\prime}\left\\{\delta\phi_{s}^{\ast}(\mathbf{r}^{\,\prime},t)\,\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\,\phi_{l}^{0}(\mathbf{r}^{\,\prime})\right.$ $\displaystyle+\left.\phi^{0,\ast}_{s}(\mathbf{r}^{\,\prime})\,\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\,\delta\phi_{l}(\mathbf{r}^{\,\prime},t)\right\\}$ (79) $\displaystyle\delta\rho_{kq}\left[\mathbf{C}_{u},\mathbf{C}_{v},\mathbf{C}_{u}^{\ast},\mathbf{C}_{v}^{\ast}\right]$ $\displaystyle=\langle\delta\mathbf{C}(t)\|\hat{b}^{\dagger}_{k}\hat{b}_{q}\|\mathbf{C}^{0}\rangle+\langle\mathbf{C}^{0}\|\hat{b}^{\dagger}_{k}\hat{b}_{q}\|\delta\mathbf{C}(t)\rangle$ (80) $\displaystyle\delta\rho_{ksql}\left[\mathbf{C}_{u},\mathbf{C}_{v},\mathbf{C}_{u}^{\ast},\mathbf{C}_{v}^{\ast}\right]$ $\displaystyle=\langle\delta\mathbf{C}(t)\|\hat{b}^{\dagger}_{k}\hat{b}^{\dagger}_{s}\hat{b}_{q}\hat{b}_{l}\|\mathbf{C}^{0}\rangle+\langle\mathbf{C}^{0}\|\hat{b}^{\dagger}_{k}\hat{b}^{\dagger}_{s}\hat{b}_{q}\hat{b}_{l}\|\delta\mathbf{C}(t)\rangle$ (81) where the action of, e.g., $\hat{b}^{\dagger}_{k}\hat{b}_{q}$ on $\|\mathbf{C}\rangle$ results in a new vector of coefficients for the same Fock states, i.e., $\hat{b}^{\dagger}_{k}\hat{b}_{q}\|\mathbf{C}\rangle=\sum_{\mathbf{n}}C_{\mathbf{n}}(t)\hat{b}^{\dagger}_{k}\hat{b}_{q}\|\mathbf{n};t\rangle=\sum_{\mathbf{n}^{\prime}}C_{\mathbf{n}^{\prime}}(t)\|\mathbf{n}^{\prime};t\rangle\equiv\|\mathbf{C}^{\prime}\rangle,$ (82) and inserts the above equations, together with Eqs. (52), (64), and (66), into Eq. (3.2.2). Collecting all terms proportional to $e^{-i\omega t}$ results in $\displaystyle\sum_{q}\hat{\mathbf{P}}\left\\{\left(\hat{Z}_{kq}^{0}-\mu_{kq}^{0}\right)\,\|u_{q}\rangle+\left(\delta\rho_{kq}\left[\mathbf{C}_{u},\mathbf{C}_{v}\right]\hat{h}+\rho_{kq}^{0}\,\hat{f}^{+}\right.\right.$ $\displaystyle+\sum_{s,l}\delta\rho_{ksql}\left[\mathbf{C}_{u},\mathbf{C}_{v}\right]\,\hat{W}_{sl}^{0}+\sum_{s,l}\left.\left.\rho_{ksql}^{0}\,\delta\hat{W}_{sl}[u_{l},v_{s}]\right)\|\phi_{q}^{0}\rangle\right\\}$ $\displaystyle=\omega\sum_{q}\rho_{kq}^{0}\|u_{q}\rangle.$ (83) Similarly, collecting all terms proportional to $e^{i\omega t}$ yields $\displaystyle\sum_{q}\hat{\mathbf{P}}\left\\{\left(\hat{Z}_{kq}^{0}-\mu_{kq}^{0}\right)\,\|v^{\ast}_{q}\rangle+\left(\delta\rho_{kq}\left[\mathbf{C}_{u}^{\ast},\mathbf{C}_{v}^{\ast}\right]\hat{h}+\rho_{kq}^{0}\,\hat{f}^{-}\right.\right.$ $\displaystyle+\sum_{s,l}\delta\rho_{ksql}\left[\mathbf{C}_{u}^{\ast},\mathbf{C}_{v}^{\ast}\right]\,\hat{W}_{sl}^{0}+\sum_{s,l}\left.\left.\rho_{ksql}^{0}\,\delta\hat{W}_{sl}[u_{s}^{\ast},v_{l}^{\ast}]\right)\|\phi_{q}^{0}\rangle\right\\}$ $\displaystyle=-\omega\sum_{q}\rho_{kq}^{0}\|v_{q}^{\ast}\rangle$ $\displaystyle\overset{\text{C.C.}}{\iff}$ $\displaystyle\sum_{q}\hat{\mathbf{P}}^{\ast}\left\\{\left(\hat{Z}_{kq}^{0,\ast}-\mu_{qk}^{0}\right)\,\|v_{q}\rangle+\left(\delta\rho_{qk}\left[\mathbf{C}_{u},\mathbf{C}_{v}\right]\hat{h}^{\ast}+\rho_{qk}^{0}\,\hat{f}^{-,\ast}\right.\right.$ $\displaystyle+\sum_{s,l}\delta\rho_{qlks}\left[\mathbf{C}_{u},\mathbf{C}_{v}\right]\,\hat{W}_{ls}^{0}+\sum_{s,l}\left.\left.\rho_{qlks}^{0}\,\delta\hat{W}_{ls}[u_{s},v_{l}]\right)\|\phi_{q}^{0,\ast}\rangle\right\\}$ $\displaystyle=-\omega\sum_{q}\rho_{qk}^{0}\|v_{q}\rangle$ (84) where ’C.C.’ denotes complex conjugation of the entire equation. Here, it is used that the matrix $\\{\mu_{ij}\\}$ of the Lagrange multipliers is hermitian. Furthermore, the two-body interaction potential is assumed to be real throughout this work. For linearizing the coefficients’ EOMs, one obtains in zeroth order the stationary equation $\boldsymbol{\mathcal{H}^{0}}\mathbf{C}^{0}=\varepsilon^{0}\,\mathbf{C}^{0}$ (85) with the matrix elements $\\{\mathcal{H}^{0}_{\mathbf{n}\mathbf{n}^{\,\prime}}\\}=\langle\mathbf{n}\|\hat{H}^{0}\|\mathbf{n}^{\,\prime}\rangle$. Eq. (85) is the working equation for the coefficients within the MCHB theory. In first order, one arrives at $i\frac{\partial}{\partial t}\delta\mathbf{C}(t)=\delta\boldsymbol{\mathcal{H}}\mathbf{C}^{0}+\left(\boldsymbol{\mathcal{H}^{0}}-\varepsilon^{0}\right)\delta\mathbf{C}(t)$ (86) with $\delta\boldsymbol{\mathcal{H}}=\sum_{k,q}\delta h_{kq}\,\hat{b}^{\dagger}_{k}\hat{b}_{q}+\frac{1}{2}\sum_{k,s,q,l}\delta W_{ksql}\,\hat{b}^{\dagger}_{k}\hat{b}^{\dagger}_{s}\hat{b}_{q}\hat{b}_{l}$ (87) and $\displaystyle\delta h_{kq}$ $\displaystyle=\int\,d\mathbf{r}\,\delta\phi^{\ast}_{k}(\mathbf{r},t)\hat{h}\phi_{q}^{0}(\mathbf{r})+\int\,d\mathbf{r}\,\phi^{0,\ast}_{k}(\mathbf{r})\hat{h}\delta\phi_{q}(\mathbf{r},t)$ $\displaystyle+\underbrace{\int\,d\mathbf{r}\,\phi^{0,\ast}_{k}(\mathbf{r})\hat{H}_{\text{ext}}(\mathbf{r},t)\phi_{q}^{0}(\mathbf{r})}_{\equiv(H_{\text{ext}})_{kq}}$ (88) $\displaystyle\delta W_{ksql}$ $\displaystyle=\iint\,d\mathbf{r}d\mathbf{r}^{\,\prime}\,\delta\phi^{\ast}_{k}(\mathbf{r},t)\phi_{s}^{0,\ast}(\mathbf{r}^{\,\prime})\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi_{q}^{0}(\mathbf{r})\phi_{l}^{0}(\mathbf{r}^{\,\prime})$ $\displaystyle+\iint\,d\mathbf{r}d\mathbf{r}^{\,\prime}\,\phi^{0,\ast}_{k}(\mathbf{r})\delta\phi_{s}^{\ast}(\mathbf{r}^{\,\prime},t)\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi_{q}^{0}(\mathbf{r})\phi_{l}^{0}(\mathbf{r}^{\,\prime})$ $\displaystyle+\iint\,d\mathbf{r}d\mathbf{r}^{\,\prime}\,\phi^{0,\ast}_{k}(\mathbf{r})\phi_{s}^{0,\ast}(\mathbf{r}^{\,\prime})\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\delta\phi_{q}(\mathbf{r},t)\phi_{l}^{0}(\mathbf{r}^{\,\prime})$ $\displaystyle+\iint\,d\mathbf{r}d\mathbf{r}^{\,\prime}\,\phi^{0,\ast}_{k}(\mathbf{r})\phi_{s}^{0,\ast}(\mathbf{r}^{\,\prime})\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi_{q}(\mathbf{r})\delta\phi_{l}(\mathbf{r}^{\,\prime},t).$ (89) Collecting all terms proportional to $e^{-i\omega t}$ yields $\displaystyle\omega\mathbf{C}_{u}$ $\displaystyle=(\boldsymbol{\mathcal{H}^{0}}-\varepsilon^{0})\mathbf{C}_{u}+\left[\sum_{k,q}\delta h_{k,q}[v_{k},u_{q},\hat{f}^{+}]\hat{b}^{\dagger}_{k}\hat{b}_{q}\right.$ $\displaystyle+\left.\frac{1}{2}\sum_{k,s,q,l}\delta W_{ksql}[v_{k},v_{s},u_{q},u_{l}]\hat{b}^{\dagger}_{k}\hat{b}^{\dagger}_{s}\hat{b}_{q}\hat{b}_{l}\right]\mathbf{C}^{0},$ (90) whereas collecting all terms proportional to $e^{+i\omega t}$ results in $\displaystyle-\omega\mathbf{C}_{v}^{\ast}$ $\displaystyle=(\boldsymbol{\mathcal{H}^{0}}-\varepsilon^{0})\mathbf{C}_{v}^{\ast}+\left[\sum_{k,q}\delta h_{k,q}[u^{\ast}_{k},v^{\ast}_{q},\hat{f}^{-}]\hat{b}^{\dagger}_{k}\hat{b}_{q}\right.$ $\displaystyle+\left.\frac{1}{2}\sum_{k,s,q,l}\delta W_{ksql}[u^{\ast}_{k},u^{\ast}_{s},v^{\ast}_{q},v^{\ast}_{l}]\hat{b}^{\dagger}_{k}\hat{b}^{\dagger}_{s}\hat{b}_{q}\hat{b}_{l}\right]\mathbf{C}^{0}$ $\displaystyle\overset{\text{C.C.}}{\iff}-\omega\mathbf{C}_{v}$ $\displaystyle=(\boldsymbol{\mathcal{H}^{0,\ast}}-\varepsilon^{0})\mathbf{C}_{v}+\left[\sum_{k,q}\delta h_{q,k}[u_{k},v_{q},\hat{f}^{-,\ast}]\left(\hat{b}^{\dagger}_{k}\hat{b}_{q}\right)^{\ast}\right.$ $\displaystyle+\left.\frac{1}{2}\sum_{q,l,k,s}\delta W_{qlks}[u_{k},u_{s},v_{q},v_{l}]\left(\hat{b}^{\dagger}_{k}\hat{b}^{\dagger}_{s}\hat{b}_{q}\hat{b}_{l}\right)^{\ast}\right]\mathbf{C}^{0,\ast}.$ (91) After the linearization of both the orbitals’ and coefficients’ EOMs, one can combine the sets of LR equations by casting Eqs. (3.2.2), (3.2.2), (3.2.2) and (3.2.2) into matrix form, leading to $\left(\mathbf{\mathcal{P}}\mathcal{L}-\mathcal{M}\omega\right)\begin{pmatrix}\mathbf{u}\\\ \mathbf{v}\\\ \mathbf{C}_{u}\\\ \mathbf{C}_{v}\end{pmatrix}=\mathbf{\mathcal{M}}\,\mathbf{\mathcal{P}}\,\mathbf{\mathcal{R}}[\hat{f}^{+},\hat{f}^{-}]$ (92) with $\mathbf{u}=(u_{1},...,u_{M})^{t}$, $\mathbf{v}=(v_{1},...,v_{M})^{t}$ and the LR matrix $\mathcal{L}=\begin{pmatrix}\mathbf{\mathcal{L}}_{oo}&\mathbf{\mathcal{L}}_{oc}\\\ \mathbf{\mathcal{L}}_{co}&\mathbf{\mathcal{L}}_{cc}\end{pmatrix}$ (93) where the submatrices refer to the couplings between the orbitals and the coefficients. In Eq. (92), $\mathbf{\mathcal{P}}$ is a projection matrix, $\mathbf{\mathcal{M}}$ is a metric which contains the one-body RDM, and the RHS $\mathbf{\mathcal{R}}$ contains the amplitudes of the external perturbation, $\hat{f}^{+}$ and $\hat{f}^{-}$. In the following, the submatrices in Eq. (93) are explicitly derived. The $(2M)$-dimensional square orbital matrix $\mathcal{L}_{oo}$ consists of four blocks and reads $\mathcal{L}_{oo}=\begin{pmatrix}\mathcal{L}^{u}_{oo}&\mathcal{L}^{v}_{oo}\\\ -\left(\mathcal{L}^{v}_{oo}\right)^{\ast}&-\left(\mathcal{L}^{u}_{oo}\right)^{\ast}\end{pmatrix}.$ (94) The upper part can be obtained from Eq. (3.2.2) by collecting all terms proportional to $\mathbf{u}$, yielding $\mathcal{L}^{u}_{oo}=\boldsymbol{\rho^{0}}\hat{h}-\boldsymbol{\mu^{0}}+\mathbf{\Omega^{0}}+\boldsymbol{\kappa}^{1}$ (95) with $\boldsymbol{\rho^{0}}=\\{\rho^{0}_{kq}\\}$, $\boldsymbol{\mu^{0}}=\\{\mu^{0}_{kq}\\}$, $\mathbf{\Omega^{0}}=\\{\Omega^{0}_{kq}\\}=\sum_{s,l}\rho^{0}_{ksql}\hat{W}^{0}_{sl}$ and $\boldsymbol{\kappa}^{1}=\\{\kappa^{1}_{kq}\\}=\sum_{s,l}\rho^{0}_{ksql}\hat{K}^{0}_{sl}$. Therein, the exchange interaction operator is defined as $\hat{K}_{sl}=\int d\mathbf{r}^{\,\prime}\phi_{s}^{\ast}(\mathbf{r}^{\,\prime})\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}\phi_{l}(\mathbf{r}^{\,\prime})$ (96) where $\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}$ permutes the coordinates $\mathbf{r}$ and $\mathbf{r}^{\,\prime}$, i.e., $\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}f(\mathbf{r})=f(\mathbf{r}^{\,\prime})$ (97) for any function $f(\mathbf{r})$. The action of $\hat{K}_{sl}$ on an arbitrary function $f(\mathbf{r})$ therefore reads $\hat{K}_{sl}\,f(\mathbf{r})=\hat{W}_{sf}\,\phi_{l}(\mathbf{r}).$ (98) Furthermore, collecting all terms proportional to $\mathbf{v}$ yields $\mathcal{L}^{v}_{oo}=\boldsymbol{\kappa}^{2},\quad\boldsymbol{\kappa}^{2}=\\{\kappa^{2}_{kq}\\}=\sum_{s,l}\rho_{kqsl}^{0}\hat{K}_{l^{\ast}s}$ (99) for the upper right block of $\mathcal{L}_{oo}$. The lower two submatrices can be obtained from Eq. (3.2.2) and turn out to be the negative complex conjugate of the upper submatrices. The upper right block of $\mathcal{L}$, i.e., the $2(M\times N_{\text{conf}})$-matrix $\mathcal{L}_{oc}$, has a similar structure than $\mathcal{L}_{oo}$, meaning it consists of four separate blocks given by $\mathcal{L}_{oc}=\begin{pmatrix}\mathcal{L}^{u}_{oc}&\mathcal{L}^{v}_{oc}\\\ -\left(\mathcal{L}^{v}_{oc}\right)^{\ast}&-\left(\mathcal{L}^{u}_{oc}\right)^{\ast}\end{pmatrix}.$ (100) The upper two submatrices can again be extracted from Eq. (3.2.2) and yield, collecting all terms proportional to $\mathbf{C_{u}}$, $\mathcal{L}^{u}_{oc}=\sum_{q}\left\\{\hat{h}\left(\hat{\rho}_{qk}\mathbf{C^{0}}\right)^{\dagger}+\sum_{s,l}\hat{W}_{sl}^{0}\left(\hat{\rho}_{qlks}\mathbf{C^{0}}\right)^{\dagger}\right\\}\phi_{q}^{0}\,,$ (101) whereas collecting all terms proportional to $\mathbf{C_{v}}$ results in $\mathcal{L}^{v}_{oc}=\sum_{q}\left\\{\hat{h}\left(\hat{\rho}_{kq}\mathbf{C^{0}}\right)^{t}+\sum_{s,l}\hat{W}_{sl}^{0}\left(\hat{\rho}_{ksql}\mathbf{C^{0}}\right)^{t}\right\\}\phi_{q}^{0}\,.$ (102) The lower two submatrices can be extracted from Eq. (3.2.2), see Ref. [35]. With respect to the $2(N_{\text{conf}}\times M)$-matrix $\mathcal{L}_{co}$, one finds $\mathcal{L}_{co}=\begin{pmatrix}\mathcal{L}^{u}_{co}&\mathcal{L}^{v}_{co}\\\ -\left(\mathcal{L}^{v}_{co}\right)^{\ast}&-\left(\mathcal{L}^{u}_{co}\right)^{\ast}\end{pmatrix}$ (103) with $\mathcal{L}_{co}^{u}=\sum_{k}\phi_{k}^{0,\ast}\left\\{\left(\hat{\rho}_{kq}\mathbf{C^{0}}\right)\hat{h}+\sum_{sl}\left(\hat{\rho}_{klqs}\mathbf{C^{0}}\right)\hat{W}_{ls}^{0}\right\\}$ (104) from collecting all terms proportional to $u_{q}$ in Eq. (3.2.2) and $\mathcal{L}_{co}^{v}=\sum_{k}\phi_{k}^{0}\left\\{\left(\hat{\rho}_{qk}\mathbf{C^{0}}\right)\hat{h}^{\ast}+\sum_{sl}\left(\hat{\rho}_{qskl}\mathbf{C^{0}}\right)\hat{W}_{sl}^{0}\right\\}$ (105) from collecting all terms proportional to $v_{q}$ in Eq. (3.2.2). It can thus be seen that $\left(\mathcal{L}_{oc}^{u}\right)^{\dagger}=\mathcal{L}_{co}^{u},\quad\left(\mathcal{L}_{oc}^{v}\right)^{t}=\mathcal{L}_{co}^{v}.$ (106) Furthermore, one can extract the $(2N_{\text{conf}})$-dimensional square matrix $\mathcal{L}_{cc}$ from Eqs. (3.2.2) and (3.2.2), which gives $\mathcal{L}_{cc}=\begin{pmatrix}\boldsymbol{\mathcal{H}^{0}}-\varepsilon^{0}&\mathbf{0_{cc}}\\\ \mathbf{0_{cc}}&-\left(\boldsymbol{\mathcal{H}^{0,\ast}}-\varepsilon^{0}\right)\end{pmatrix}$ (107) where $\mathbf{0_{cc}}$ is the ($N_{\text{conf}}$)-dimensional zero matrix. The projection operator $\mathbf{\mathcal{P}}$ is given by $\mathbf{\mathcal{P}}=\begin{pmatrix}\boldsymbol{\mathcal{P}}_{oo}&\mathbf{0_{oc}}\\\ \mathbf{0_{co}}&\boldsymbol{\mathbb{1}_{cc}}\end{pmatrix}$ (108) with $\boldsymbol{\mathcal{P}}_{oo}=\begin{pmatrix}\hat{\mathbf{P}}&0\\\ 0&\mathbf{\hat{P}^{\ast}}\end{pmatrix}$ (109) and $\mathbf{\mathbb{1}_{cc}}$ being the $(2N_{\text{conf}})$-dimensional unit matrix. The matrices $\mathbf{0_{oc}}$ and $\mathbf{0_{co}}$ denote the $2(M\times N_{\text{conf}})$\- and $2(N_{\text{conf}}\times M)$-dimensional zero matrices, respectively. The metric $\mathbf{\mathcal{M}}$ reads $\mathbf{\mathcal{M}}=\begin{pmatrix}\boldsymbol{\rho}_{oo}&\mathbf{0_{oc}}\\\ \mathbf{0_{co}}&\mathbf{\mathbb{1}_{cc}}\end{pmatrix}$ (110) where $\mathbf{\rho}_{oo}$ is given by $\boldsymbol{\rho}_{oo}=\begin{pmatrix}\boldsymbol{\rho^{0}}&\mathbf{0_{o}}\\\ \mathbf{0_{o}}&\boldsymbol{\rho^{0,\ast}}\end{pmatrix}$ (111) and $\mathbf{0_{o}}$ being the ($M\times M$)-dimensional zero matrix. In the following, the response amplitudes and coefficients are written in bra- ket notation for reasons that become clear below. In order to render Eq. (92) an eigenvalue equation, it is helpful to notice that only the term proportional to the external frequency $\omega$ has no projector, which implies $\mathbf{\mathcal{P}}\mathbf{\mathcal{M}}\begin{pmatrix}\|\mathbf{u}\rangle\\\ \|\mathbf{v}\rangle\\\ \|\mathbf{C}_{u}\rangle\\\ \|\mathbf{C}_{v}\rangle\end{pmatrix}=\mathbf{\mathcal{M}}\begin{pmatrix}\|\mathbf{u}\rangle\\\ \|\mathbf{v}\rangle\\\ \|\mathbf{C}_{u}\rangle\\\ \|\mathbf{C}_{v}\rangle\end{pmatrix}$ (112) and thus one may add a redundant projector $\mathbf{\mathcal{P}}\mathcal{L}\Rightarrow\mathbf{\mathcal{P}}\mathcal{L}\mathbf{\mathcal{P}}$ to the left term of the LHS of Eq. (92). Multiplying now with $\mathbf{\mathcal{M}^{-1/2}}$ from the left yields $\left(\mathbf{\mathcal{M}^{-1/2}}\mathbf{\mathcal{P}}\mathcal{L}\mathbf{\mathcal{P}}\mathbf{\mathcal{M}^{-1/2}}-\omega\right)\mathbf{\mathcal{M}^{+1/2}}\begin{pmatrix}\|\mathbf{u}\rangle\\\ \|\mathbf{v}\rangle\\\ \|\mathbf{C}_{u}\rangle\\\ \|\mathbf{C}_{v}\rangle\end{pmatrix}=\mathbf{\mathcal{M}^{+1/2}}\mathbf{\mathcal{P}}\mathbf{\mathcal{R}}[\hat{f}^{+},\hat{f}^{-}]$ (113) or, in a more compact notation, $\left(\bar{\mathcal{L}}-\omega\right)\begin{pmatrix}\|\mathbf{\overline{u}}\rangle\\\ \|\mathbf{\overline{v}}\rangle\\\ \|\overline{\mathbf{C}_{u}}\rangle\\\ \|\overline{\mathbf{C}_{v}}\rangle\end{pmatrix}=\mathbf{\overline{\mathcal{R}}}[\hat{f}^{+},\hat{f}^{-}]$ (114) where $\displaystyle\bar{\mathcal{L}}$ $\displaystyle=\mathbf{\mathcal{M}^{-1/2}}\mathbf{\mathcal{P}}\mathcal{L}\mathbf{\mathcal{P}}\mathbf{\mathcal{M}^{-1/2}}$ $\displaystyle=\begin{pmatrix}\boldsymbol{\rho_{oo}}^{-1/2}\,\boldsymbol{\mathcal{P}}_{oo}\,\mathcal{L}_{oo}\,\boldsymbol{\mathcal{P}}_{oo}\,\boldsymbol{\rho_{oo}}^{-1/2}&\quad\boldsymbol{\rho_{oo}}^{-1/2}\,\boldsymbol{\mathcal{P}}_{oo}\,\mathcal{L}_{oc}\\\ \mathcal{L}_{co}\,\boldsymbol{\mathcal{P}}_{oo}\,\boldsymbol{\rho_{oo}}^{-1/2}&\mathcal{L}_{cc}\end{pmatrix},$ (115) as well as $\left(\mathbf{\overline{u}},\mathbf{\overline{v}},\overline{\mathbf{C}_{u}},\overline{\mathbf{C}_{v}}\right)^{t}=\mathbf{\mathcal{M}^{+1/2}}\left(\mathbf{u},\mathbf{v},\mathbf{C}_{u},\mathbf{C}_{v}\right)^{t}$ and $\mathbf{\overline{\mathcal{R}}}=\mathbf{\mathcal{M}^{+1/2}}\mathbf{\mathcal{P}}\mathbf{\mathcal{R}}$ were introduced. To keep the notation simple, the bar over the individual quantities in Eq. (114) will be omitted in the following. For the homogeneous case, i.e., with $\mathbf{\mathcal{R}}=0$, one arrives at the central equation of this work, given by $(\mathcal{L}-\omega)\begin{pmatrix}\|\mathbf{u}\rangle\\\ \|\mathbf{v}\rangle\\\ \|\mathbf{C}_{u}\rangle\\\ \|\mathbf{C}_{v}\rangle\end{pmatrix}=0$ (116) which is a standard eigenvalue problem of the LR matrix $\mathcal{L}$. Its solutions, i.e., the set of eigenvalues $\\{\omega_{k}=E_{k}-\varepsilon^{0}\\}$ and eigenvectors $\\{(\|\mathbf{u}^{k}\rangle,\|\mathbf{v}^{k}\rangle,\|{\mathbf{C}_{u}}^{k}\rangle,\|{\mathbf{C}_{v}}^{k}\rangle)^{t}\\}$ where $k\in\mathbb{N}$ labels the excitations, can be used to solve the inhomogeneous problem of Eq. (114) by employing the ansätze $\begin{pmatrix}\|\mathbf{u}\rangle\\\ \|\mathbf{v}\rangle\\\ \|\mathbf{C}_{u}\rangle\\\ \|\mathbf{C}_{v}\rangle\end{pmatrix}=\sum_{k}c_{k}\begin{pmatrix}\|\mathbf{u}^{k}\rangle\\\ \|\mathbf{v}^{k}\rangle\\\ \|{\mathbf{C}_{u}}^{k}\rangle\\\ \|{\mathbf{C}_{v}}^{k}\rangle\end{pmatrix}$ (117) and $\mathbf{\mathcal{R}}=-\sum_{k}\gamma_{k}\begin{pmatrix}\|\mathbf{u}^{k}\rangle\\\ \|\mathbf{v}^{k}\rangle\\\ \|{\mathbf{C}_{u}}^{k}\rangle\\\ \|{\mathbf{C}_{v}}^{k}\rangle\end{pmatrix}$ (118) where the minus sign is chosen arbitrarily. One finally obtains $\sum_{k}c_{k}(\omega_{k}-\omega)\begin{pmatrix}\|\mathbf{u}^{k}\rangle\\\ \|\mathbf{v}^{k}\rangle\\\ \|{\mathbf{C}_{u}}^{k}\rangle\\\ \|{\mathbf{C}_{v}}^{k}\rangle\end{pmatrix}=-\sum_{k}\gamma_{k}\begin{pmatrix}\|\mathbf{u}^{k}\rangle\\\ \|\mathbf{v}^{k}\rangle\\\ \|{\mathbf{C}_{u}}^{k}\rangle\\\ \|{\mathbf{C}_{v}}^{k}\rangle\end{pmatrix}$ (119) which means that the expansion coefficients $c_{k}$ in Eq. (117) are given by $c_{k}=\frac{\gamma_{k}}{\omega-\omega_{k}}\,.$ (120) It is stressed that the response weights $\\{\gamma_{k}\\}$ are the only quantities that depend on the perturbing fields $\hat{f}^{+}$ and $\hat{f}^{-}$ and thus on the actual shape of the perturbation. All other quantities like the LR matrix $\mathcal{L}$, the eigenenergies $\\{\omega_{k}\\}$ and the eigenvectors $\\{(\|\mathbf{u}^{k}\rangle,\|\mathbf{v}^{k}\rangle,\|{\mathbf{C}_{u}}^{k}\rangle,\|{\mathbf{C}_{v}}^{k}\rangle)^{t}\\}$ are obtained from the homogeneous eigenvalue problem of Eq. (116) where the amplitudes $\hat{f}^{+}$ and $\hat{f}^{-}$ of the perturbing field do not appear. Multiplying Eq. (118) by $\left(\langle\mathbf{u}^{k}\|,-\langle\mathbf{v}^{k}\|,\langle{\mathbf{C}_{u}}^{k}\|,-\langle{\mathbf{C}_{v}}^{k}\|\right)$ from the left, taking into account the orthonormality conditions for the perturbed amplitudes and coefficients given by $\displaystyle\langle\mathbf{u}^{k}\|\mathbf{u}^{k^{\prime}}\rangle-\langle\mathbf{v}^{k}\|\mathbf{v}^{k^{\prime}}\rangle+\langle{\mathbf{C}_{u}}^{k}\|{\mathbf{C}_{u}}^{k^{\prime}}\rangle-\langle{\mathbf{C}_{v}}^{k}\|{\mathbf{C}_{v}}^{k^{\prime}}\rangle$ $\displaystyle=\delta_{kk^{\prime}}$ (121) $\displaystyle\langle\mathbf{v}^{k}\|\mathbf{u}^{k^{\prime},\ast}\rangle-\langle\mathbf{u}^{k}\|\mathbf{v}^{k^{\prime},\ast}\rangle+\langle{\mathbf{C}_{v}}^{k}\|{\mathbf{C}_{u}}^{k^{\prime},\ast}\rangle-\langle{\mathbf{C}_{u}}^{k}\|{\mathbf{C}_{v}}^{k^{\prime},\ast}\rangle$ $\displaystyle=0,$ (122) leads to the response weights $\displaystyle\gamma_{k}$ $\displaystyle=-\left(\langle\mathbf{u}^{k}\|,-\langle\mathbf{v}^{k}\|,\langle{\mathbf{C}_{u}}^{k}\|,-\langle{\mathbf{C}_{v}}^{k}\|\right)\,\mathbf{\mathcal{R}}$ $\displaystyle=-\left(\langle\mathbf{u}^{k}\|,-\langle\mathbf{v}^{k}\|,\langle{\mathbf{C}_{u}}^{k}\|,-\langle{\mathbf{C}_{v}}^{k}\|\right)\begin{pmatrix}-(\boldsymbol{\rho^{0}})^{1/2}\,\hat{\mathbf{P}}\hat{f}^{+}\|\boldsymbol{\phi^{0}}\rangle\\\ (\boldsymbol{\rho^{0,\ast}})^{1/2}\,\hat{\mathbf{P^{\ast}}}\hat{f}^{-,\ast}\|\boldsymbol{\phi^{0,\ast}}\rangle\\\ -\sum_{i,j}\langle\phi_{i}\|\hat{f}^{+}\|\phi_{j}\rangle\,\hat{b}_{i}^{\dagger}\hat{b}_{j}\,\|\mathbf{C^{0}}\rangle\\\ \sum_{i,j}\langle\phi_{j}\|\hat{f}^{-,\ast}\|\phi_{i}\rangle\,\left(\hat{b}_{i}^{\dagger}\hat{b}_{j}\right)^{\ast}\,\|\mathbf{C^{0,\ast}}\rangle\end{pmatrix}$ $\displaystyle=\langle\mathbf{u}^{k}\|\hat{f}^{+}(\boldsymbol{\rho^{0}})^{1/2}\|\boldsymbol{\phi^{0}}\rangle+\langle\mathbf{v}^{k}\|\hat{f}^{-,\ast}(\boldsymbol{\rho^{0,\ast}})^{1/2}\|\boldsymbol{\phi^{0,\ast}}\rangle$ $\displaystyle+\sum_{i,j}\langle\phi_{i}\|\hat{f}^{+}\|\phi_{j}\rangle\langle{\mathbf{C}_{u}}^{k}\|\hat{b}_{i}^{\dagger}\hat{b}_{j}\,\|\mathbf{C^{0}}\rangle+\sum_{i,j}\langle\phi_{j}\|\hat{f}^{-,\ast}\|\phi_{i}\rangle\left\langle{\mathbf{C}_{v}}^{k}\left\|\left(\hat{b}_{i}^{\dagger}\hat{b}_{j}\right)^{\ast}\right\|\mathbf{C^{0,\ast}}\right\rangle$ (123) where the vector $\boldsymbol{\|\phi^{0}\rangle}=\left(\|\phi_{1}^{0}\rangle,...,\|\phi_{M}^{0}\rangle\right)^{t}$ collects the stationary ground-state orbitals. Each response weight $\gamma_{k}$ quantifies how strong the $k$-th excitation contributes to the overall response of the system to an external perturbation with amplitudes $\hat{f}^{+}$ and $\hat{f}^{-}$. For instance, it may happen that for a chosen shape of the perturbing external fields, some excited states do not respond at all and thus appear to be transparent. The response density of the $k$-th excited state due to the perturbation can in general be expressed as $\Delta\rho^{k}(\mathbf{r})=\Delta\rho_{o}^{k}(\mathbf{r})+\Delta\rho_{c}^{k}(\mathbf{r})$ (124) where $\Delta\rho_{o}^{k}(\mathbf{r})$ and $\Delta\rho_{c}^{k}(\mathbf{r})$ denote the orbitals’ and coefficients’ contribution, respectively. Those are given by $\displaystyle\Delta\rho_{o}^{k}$ $\displaystyle=\langle\boldsymbol{\phi^{0}}\|(\boldsymbol{\rho^{0}})^{1/2}\left(\mathbf{\|u^{k}\rangle}+\mathbf{\|v^{k,\ast}\rangle}\right)$ $\displaystyle+\left(\mathbf{\|u^{k}\rangle}+\mathbf{\|v^{k,\ast}\rangle}\right)^{\dagger}(\boldsymbol{\rho^{0,\ast}})^{-1/2}\boldsymbol{\rho^{0}}\,\boldsymbol{\|\phi^{0}\rangle}$ (125) and $\Delta\rho_{c}^{k}=\sum_{i,j=1}^{M}\,\phi_{i}^{0,\ast}\phi_{j}^{0}\left(\langle\mathbf{C}^{0}\|\hat{b}_{i}^{\dagger}\hat{b}_{j}\|{\mathbf{C}_{u}}^{k}\rangle+\langle{\mathbf{C}_{v}}^{k,\ast}\|\hat{b}_{i}^{\dagger}\hat{b}_{j}\|\mathbf{C}^{0}\rangle\right).$ (126) To summarize, the many-body LR theory based on the MCTDHB working equations, termed LR-MCTDHB, was introduced and derived. For a given $N$-bosons system in a potential trap $V$ and a two-body interaction $\hat{W}$, the central equation in order to obtain the excitation spectrum is Eq. (116). Although this equation appears to be rather straightforward to solve, it in general represents a demanding task because firstly, $\mathcal{L}$ is not hermitian and secondly, it can become extremely large in size when the number of bosons $N$ and/or the number of orbitals $M$ grows. In Appendix B, it is shown that the LR matrix can be brought to block-diagonal form by using a complex transformation, which halves the dimensionality of the eigenvalue problem. However, it involves matrix-matrix products of the individual blocks of $\mathcal{L}$, which can be numerically very expensive. It hence remains a system-dependent question whether it is beneficial to perform this transformation or not. In the next section, Section 4, details on the numerical implementation of LR- MCTDHB are presented. First, general rather technical challenges of the eigenvalue problem in Eq. (116) are discussed before explaining the code structure and the numerical methods utilized. Finally, it is once again stressed that for $M=1$, LR-MCTDHB reduces to the (particle-conserving) BdG equations, i.e., BdG$\equiv$LR-GP$\equiv$LR-MCTDHB(1). The derivation of the BdG theory is described in Appendix A.1.2. Since the BdG is commonly used to calculate excitations of trapped BECs, its results for certain applications (see Section 6) are compared to the many-body results for $M>1$. ## 4 Numerical implementation of many-body linear response theory In this section, information about the LR-MCTDHB implementation are presented. Before dealing with the technical and numerical details in Sections 4.2 and 4.3, a brief overview about the general structure of the code is given in Section 4.1, together with a description of technical challenges that were to be solved. ### 4.1 Code structure and challenges As a starting point, the structure of the numerical implementation, consisting of several thousand lines of code, is explained. The entire task of calculating the energies of excited states and the corresponding correction amplitudes of a trapped interacting BEC, i.e., solving the eigenvalue problem of Eq. (116), is separated into two major parts. Those are (i) the construction of the LR matrix $\bar{\mathcal{L}}$ (simply denoted by $\mathcal{L}$ in the following) as described in Eq. (3.2.2) and (ii) finding a desired amount of eigenpairs, i.e., eigenvalues and the corresponding eigenvectors, of the low-energy spectrum. To this end, two running modes were implemented. Mode 1 is the construction mode. As an input, it requires a stationary state of a system of $N$ bosons in a trap $V$ with interaction $\hat{W}$, where the bosons can occupy $M$ single-particle orbitals. Typically, this stationary state is the ground state of the system, and it can therefore be obtained from a previous MCTDHB calculation using imaginary time propagation. The output of mode 1 is the LR matrix $\mathcal{L}$. Afterwards, the code can be restarted in mode 2, the diagonalization mode, which takes $\mathcal{L}$ as an input and calculates a desired amount of eigenpairs in the low-energy part of the excitation spectrum. Furthermore, the response weights and response densities, described in Section 3.2.2, are calculated. The main challenge of both the construction and the eigenvalue calculation is the memory consumption of $\mathcal{L}$, and, especially in mode 1, the size of intermediate matrices that appear during the construction, like the projector $\mathbf{\mathcal{P}}$ or the individual blocks $\mathcal{L}_{oo},\,\mathcal{L}_{oc},\,\mathcal{L}_{co}$ and $\mathcal{L}_{cc}$. Although the memory consumption might be a minor issue for systems in 1D with small values of $N$ and $M$, it can become a massive problem for larger systems. To give an example, one can consider a system with $N=100$ particles and $M=4$ orbitals. In such a case, already the coefficient matrix $\mathcal{L}_{cc}$ has a dimensionality of $353702$, which means it would need approximately $4$ terabytes of memory to store all matrix elements, assuming those are in general complex numbers of 32 bytes. However, the available working memory (RAM) of compute nodes on modern high performance computation clusters is typically much smaller, being of the order of $128$ or $256$ gigabytes. One possible solution of this problem is to store all (large) matrices that appear during the construction of $\mathcal{L}$, as well as $\mathcal{L}$ itself, in a sparse matrix storage format. There are multiple possibilities to do that, e.g., the compressed-sparse-row (CSR) or compressed- sparse-column (CSC) format (see, e.g., Ref. [58] for additional formats). To obtain eigenpairs of $\mathcal{L}$ by running the code in mode 2, one is in need of a numerical method that is capable of dealing with sparse and large non-hermitian matrices. The method of choice is the implicitly restarted Arnoldi method (IRAM), implemented in the ARPACK numerical library [59]. In the current implementation, the parallel version, termed PARPACK, is utilized [60]. The IRAM is presented in detail in Section 4.3. The PARPACK routines have a significant advantage over eigenvalue routines from other libraries like ScaLAPACK [61] because they do not require the matrix that one wishes to diagonalize as an input, not even in any sparse format. It is only required to hand over the result of matrix-vector multiplications (matvecs) as input, and the actual calculation of these matvecs takes place outside of any PARPACK routine, meaning that the implementation of the matvecs is completely independent. It can be carried out separately, utilizing routines from other numerical libraries, e.g., the Intel Math Kernel Library (MKL) [62], that calculate matvecs of large and sparse matrices in a very efficient manner. Moreover, calculating matvecs it is a process that can be easily parallelized which further improves the code performance. Due to the large amount of operations in both running modes, especially for large $N$ and $M$ in two- or three-dimensional space, sequential runs of the code would require very long runtimes. Therefore, a fully MPI-parallelized implementation of the code was carried out for both running modes. Whereas this was a comparatively straightforward task for mode 2 – since only the matvecs needed to be parallelized – it turned out to be a demanding challenge with respect to mode 1. A parallelization scheme that is a compromise between taking care of the least possible memory consumption on the one hand, and distributing the workload among all participating MPI processes (PEs) as equal as possible on the other hand, is employed. In Section 4.2, the overall construction process is explained in more detail. ### 4.2 Construction of the LR matrix In this section, details on the implementation of the LR matrix construction are presented. Figure 1 shows the structure of the code and its main steps. As input, the code requires the stationary state to which the LR analysis should be applied, usually the ground state of a trapped BEC. In particular, one needs to hand over the set of orbitals $\\{\phi_{i}^{0}:1\leq i\leq M\\}$ as well as the corresponding coefficients $\mathbf{C^{0}}$. Afterwards, intermediate matrices are calculated which are essential to built up the individual blocks of $\mathcal{L}$ as described in Eq. (93). This includes the kinetic energy operator $\hat{T}_{\text{kin}}$, the direct and exchange interaction operators $\hat{W}_{sl}$ and $\hat{K}_{sl}$, as well as the projection operator $\mathbf{\mathcal{P}}$. All of these calculations are done in parallel, and the final result is stored in a sparse storage format on the master process PE 0. Then, the matrices $\mathcal{L}_{oo}$ and $\mathcal{L}_{oc}$ are calculated. Whereas $\mathcal{L}_{oo}$ is a $2(M\cdot N_{\text{grid}})$ square matrix with $N_{\text{grid}}$ denoting the number of grid points, $\mathcal{L}_{oc}$ is a $2(M\cdot N_{\text{grid}}\times N_{\text{conf}})$ matrix. It is stressed that constructing these two matrices is usually the most time-consuming part of the entire calculation of $\mathcal{L}$ due to the large amount of operations involved. On the one hand, for problems where the grid is large (e.g., in 2D and 3D) and $M\geq 7$, the construction of $\mathcal{L}_{oo}$ represents the essential part of the calculations. On the other hand, if the number of particles is large such that $N_{\text{conf}}=\mathcal{O}(10^{5})$ or higher, the construction of $\mathcal{L}_{oc}$ becomes the most demanding task. Additionally, both matrices need to be multiplied by $\boldsymbol{\rho_{oo}}^{-1/2}\boldsymbol{\mathcal{P}}_{oo}$ from the left, which further increases the amount of operations significantly. Because of this, both the construction of $\mathcal{L}_{oo}$ and $\mathcal{L}_{oc}$ are MPI-parallelized, as well as the subsequent multiplication with the metric and projector matrices. As for the preliminary calculation described above, the final results of $(\boldsymbol{\rho_{oo}}^{-1/2}\boldsymbol{\mathcal{P}}_{oo}\mathcal{L}_{oo})$ and $(\boldsymbol{\rho_{oo}}^{-1/2}\boldsymbol{\mathcal{P}}_{oo}\mathcal{L}_{oc})$ are stored on PE 0 in a sparse storage format. Input: $\\{\phi_{i}^{0}\\},\,\mathbf{C^{0}}$ Preliminary calculations: $\hat{T}_{\text{kin}}$ $\hat{W}_{sl}$ $\hat{K}_{sl}$ $\mathbf{\mathcal{P}}$ • MPI-parallelized • On exit stored on PE 0 Calculate $\mathcal{L}_{oo}$ $\mathcal{L}_{oc}$ to do matrix multiplications $\boldsymbol{\rho_{oo}}^{-1/2}\boldsymbol{\mathcal{P}}_{oo}\mathcal{L}_{oo}$ $\boldsymbol{\rho_{oo}}^{-1/2}\boldsymbol{\mathcal{P}}_{oo}\mathcal{L}_{oc}$ • MPI-parallelized • On exit stored on PE 0 Row-by-row generation of upper blocks of $\mathcal{L}$ • Each PE generates different row • Written to file by PE 0 Row-by-row generation of lower blocks of $\mathcal{L}$ • For lower left block: Each PE generates different row • For $\mathcal{L}_{cc}$: All PEs share work of each row • Written to file by PE 0 Output: $\mathcal{L}$ Figure 1: Schematic diagram of the MPI-parallelized implementation for the construction of $\mathcal{L}$. The parallelization scheme shares the amount of calculations equally between all PEs. Before $\mathcal{L}$ is generated, preliminary calculations of important intermediate matrices are performed. To minimize the memory consumption, those are only stored on PE 0 in a sparse storage format. For the row-by-row construction of the upper and lower blocks of $\mathcal{L}$, PE 0 distributes all necessary information to the other PEs, collects the results afterwards and writes them to a file. See text for details. Next, the generation of $\mathcal{L}$ starts. Therefore, the problem is separated into the construction of the upper and lower part. With respect to the upper blocks, this means that only the upper left block needs to be multiplied by $\boldsymbol{\mathcal{P}}_{oo}\,\boldsymbol{\rho_{oo}}^{-1/2}$ from the right, whereas the upper right block is already fully calculated and can be written to the output file directly. To efficiently calculate the upper-left block, a row-by-row parallelization scheme is employed where each PE calculates a different row at a time. Therefore, PE 0 distributes all necessary matrix elements of $(\boldsymbol{\rho_{oo}}^{-1/2}\boldsymbol{\mathcal{P}}_{oo}\mathcal{L}_{oo})$ and $\mathbf{\mathcal{P}}$ to the other PEs. Finally, all PEs send back their calculated rows (i.e., the non-zero elements and their coordinates) to PE 0, which writes the results to the output file. Concerning the construction of the lower part, again a row-by-row parallelization scheme is utilized. For the lower-left block, each PE generates a different row at a time. However, in the case of $\mathcal{L}_{cc}$, all PEs work on the same row at a time. Upon finalization, the results are collected by PE 0 and written to the output file, again in a sparse storage format. This section is ended by briefly discussing the issue of small but non-zero matrix elements. As already stated, the matrices produced are typically sparse, i.e., most elements are zero. Moreover, many other elements can be very small but nonzero. Their impact on the subsequent eigenvalue calculation, which is described in the next sections, is very often negligible. However, for a problem where the dimensionality of $\mathcal{L}$ is very large, those elements can become challenging in terms of memory consumption. Thus, it is important to define a tolerance value $\tau$ that is used to minimize the amount of non-zero elements which need to be stored. In the current implementation, a matrix element is treated as zero and is therefore not stored if its magnitude is smaller than $\tau$. For very large matrices, the value of $\tau$ has to be defined such that on the one hand not too many matrix elements are set to zero in order to obtain numerically accurate results for the eigenvalues and eigenvectors, but, on the other hand, it has to be large enough because too many very small elements can significantly decrease the performance of the numerical method for calculating eigenpairs of $\mathcal{L}$. The latter numerical procedure will be explained in the next section. ### 4.3 Calculation of eigenvalues In order to calculate the lowest eigenvalues of a given LR matrix, the IRAM is employed. In the following, the main ideas of this approach are presented by first discussing the basic ingredients (the Arnoldi method, the QR-iteration, and polynomial filtering) before dealing with the IRAM explicitly. In short, it generates a Krylov subspace of a given matrix whose basis vectors are used to build a much smaller matrix. The eigenvalues of this matrix are also approximate eigenvalues of the original matrix. After each iteration of the IRAM, the convergence of a set of wanted eigenvalues is checked, and if it is still not converged, the entire process is restarted with an initial vector where the directions of unwanted eigenvalues are projected out. Thus, the convergence with respect to the wanted eigenvalues is gradually enhanced. Below, the individual steps of the IRAM are explained in more detail. #### 4.3.1 The Arnoldi algorithm The first step is to calculate an orthogonal projection of a sparse, complex, in general non-hermitian matrix $A\in\mathbb{C}^{n\times n}$. Therefore, the Arnoldi algorithm is used which generates an orthonormal basis (ONB) of the $k$-dimensional subspace given by $\mathcal{K}^{k}(A,v)=\text{span}(v,Av,...,A^{k-1}v)$ (127) where $v\in\mathbb{C}^{n}$ is an initial vector and $k\leq n$, typically even $k\ll n$. $\mathcal{K}^{k}$ is called the $k$-dimensional Krylov subspace of $A$. The entire Arnoldi algorithm is given by Algorithm 1. The ONB is obtained by performing a Gram-Schmidt-orthonormalization at each Arnoldi step, i.e., making the vector $v_{j}$ in the $j$-th step orthogonal to all previous basis vectors $v_{i},\,i\leq j,$ and normalizing it afterwards. The elements of the ONB are called the Arnoldi vectors. If the resulting Arnoldi vector $v_{j+1}$ of the $j$-th step is non-zero, its norm is stored in the matrix element $h_{j+1,j}$ of the matrix $\tilde{H}_{k+1}\in\mathbb{C}^{(k+1)\times k}$. If $j\leq k$, the next Arnoldi step is initialized. After the step for $j=k$, the iteration stops and returns the $k$-dimensional ONB of $\mathcal{K}^{k}$, plus an additional orthonormal vector $v_{k+1}\in\mathbb{C}^{n}$, together with the full matrix $\tilde{H}_{k+1}$. It is possible that for a certain step $j<k$, the norm of the Arnoldi vector $v_{j+1}$ is zero, meaning that one has found an ONB of an invariant Krylov subspace of $A$ with dimensionality $j$. In this case, the Arnoldi algorithm finishes immediately. 1: $v_{1}\leftarrow v,\,\|\|v\|\|_{2}=1$ 2: for $j=1,\,j\leq k$ do 3: $z\leftarrow Av_{j}$ 4: for $i=1,\,i\leq j$ do 5: $h_{ij}\leftarrow\langle v_{i},z\rangle$ 6: $z\leftarrow z-h_{ij}v_{i}$ 7: end for 8: $h_{j+1,j}\leftarrow\|\|z\|\|_{2}$ 9: if $h_{j+1,j}=0$ then 10: quit 11: else 12: $v_{j+1}\leftarrow z/h_{j+1,j}$ 13: end if 14: end for Algorithm 1 The Arnoldi algorithm As an essential result, one obtains the Arnoldi decomposition of $A$ given by $AV_{k}=V_{k}H_{k}+h_{k+1,k}\,v_{k+1}\,e_{k}^{t}$ (128) where $V_{k}\in\mathbb{C}^{n\times k}$ contains the Arnoldi basis as columns and $H_{k}\in\mathbb{C}^{k\times k}$ is an upper Hessenberg matrix, i.e., $\\{(H_{k})_{ij}\\}=0$ if $i>j+1$, built by the first $k$ rows of $\tilde{H}_{k+1}$. The second term in Eq. (128), which contains the matrix element $h_{k+1,k}$, the additional Arnoldi vector $v_{k+1}$ and the transpose of the $k$-th unit vector, i.e., $e_{k}=(0,0,...,0,1)^{t}\in\mathbb{C}^{k}$, takes care of the fact that the orthonormal projection of $A$, $A\approx V_{k}H_{k}V_{k}^{\dagger},$ (129) is in general not exact. However, for the case of having found an invariant Krylov subspace of $A$ of lower dimensionality $j\leq k$, it becomes exact. It is straightforward to realize that the eigenvalues of $H_{k}$ are approximate, or, in the case of an invariant subspace, exact eigenvalues of A. This is due to the cyclic permutation symmetry of the determinant that defines the characteristic polynomial of $A$, i.e., $\displaystyle 0$ $\displaystyle=\det(A-\lambda I)\overset{\text{Eq.\,}(\ref{Arnoldi_orth_proj})}{\approx}\det(V_{k}H_{k}V_{k}^{\dagger}-\lambda I)$ $\displaystyle=\det(V_{k}[H_{k}-\lambda I]V_{k}^{\dagger})$ $\displaystyle=\det(\underbrace{V_{k}^{\dagger}V_{k}}_{=I}[H_{k}-\lambda I])$ $\displaystyle=\underbrace{\det(I)}_{=1}\,\det(H_{k}-\lambda I)$ $\displaystyle=\det(H_{k}-\lambda I)$ (130) with $\lambda\in\mathbb{C}$ and $I$ denoting the identity matrix. If $u_{i}\in\mathbb{C}^{k}$ is an eigenvector of $H_{k}$ and $\theta_{i}\in\mathbb{C}$ the corresponding eigenvalue, the pair $(\theta_{i},V_{k}\,u_{i})$ is called a Ritz pair of $A$ with Ritz value $\theta_{i}$ and the associated Ritz vector $V_{k}\,u_{i}\in\mathbb{C}^{n}$. In order to estimate how accurate this Ritz pair approximates an exact eigenpair of $A$, one can calculate the residual norm $r$ defined as $\displaystyle r$ $\displaystyle\equiv\|\|A(V_{k}u_{i})-\theta_{i}(V_{k}u_{i})\|\|_{2}\overset{\text{Eq.\,}(\ref{Arnoldi_decomp})}{=}\|\|(V_{k}H_{k}+h_{k+1,k}\,v_{k+1}\,e_{k}^{t})u_{i}-\theta_{i}(V_{k}u_{i})\|\|_{2}$ $\displaystyle=\|\|\theta_{i}(V_{k}u_{i})+h_{k+1,k}\,v_{k+1}\,e_{k}^{t}\,u_{i}-\theta_{i}(V_{k}u_{i})\|\|_{2}$ $\displaystyle=\|h_{k+1,k}\|\cdot\underbrace{\|v_{k+1}\|}_{=1}\cdot\|e_{k}^{t}u_{i}\|$ $\displaystyle=\|h_{k+1,k}\|\cdot\|e_{k}^{t}u_{i}\|,$ (131) i.e., $r$ is the product of the magnitudes of the matrix element $h_{k+1,k}$ of $\tilde{H}_{k+1}$ and the $k$-th element of $u_{i}$. The smaller it is, the better does the Ritz pair $(\theta_{i},V_{k}u_{i})$ approximate the exact $i$-th eigenpair of $A$. To summarize, the eigenvalue problem of a typically large matrix $A$ can be mapped onto an eigenvalue problem of a much smaller matrix $H_{k}$ whose eigenvalues are believed to be good approximations of the eigenvalues of $A$. Due to the Hessenberg form of $H_{k}$, its eigenvalues can efficiently be calculated by utilizing the QR-iteration, which is explained in the next section. Additional details on the Arnoldi algorithm can be found in Ref. [63]. #### 4.3.2 The QR-iteration The QR-iteration is a technique to calculate the eigenvalues and eigenvectors of a given matrix $A\in\mathbb{C}^{n\times n}$. It is based on the so-called QR-decomposition which reads $A=QR,\quad Q,R\in\mathbb{C}^{n\times n}$ (132) where $Q$ is a unitary matrix, i.e., $QQ^{\dagger}=I$, and $R$ is upper triangular. This decomposition exists for any matrix [63]. Plugging the ansatz of Eq. (132) into the characteristic polynomial gives $\displaystyle 0$ $\displaystyle=\det(A-\lambda I)$ $\displaystyle=\det(QR-\lambda I)$ $\displaystyle=\det(QQ^{\dagger}(QR)QQ^{\dagger}-Q\lambda IQ^{\dagger})$ $\displaystyle=\det(Q(RQ-\lambda I)Q^{\dagger})$ $\displaystyle=\det(RQ-\lambda I)$ (133) which means that the eigenvalues of $A$ are the same as the eigenvalues of the matrix $RQ$. Furthermore, $A=QR\iff Q^{\dagger}AQ=RQ,$ (134) i.e., $RQ$ is just a unitary transformation of $A$. The QR-iteration is described in Algorithm 2. At each step $k$, the QR- decomposition of the current matrix $A_{k}$ is used to define the matrix of the next step as $A_{k+1}=R_{k}Q_{k}=Q_{k}^{\dagger}A_{k}Q_{k}$. It can be shown that this sequence converges to a triangular matrix, i.e., $\lim_{k\rightarrow\infty}A_{k}=\tilde{A}$ with $\\{\tilde{A}_{ij}\\}=0$ if $i>j$, and the eigenvalues of a triangular matrix are simply given by its diagonal elements. One can show that the eigenvalues of $A_{k}$ are the same as the ones of $A$ by considering again the characteristic polynomial $\displaystyle 0$ $\displaystyle=\det(A_{k}-\lambda I)$ $\displaystyle=\det(Q_{k-1}^{\dagger}A_{k-1}Q_{k-1}-\lambda I)$ $\displaystyle=\det(Q_{k-1}^{\dagger}Q_{k-2}^{\dagger}A_{k-2}Q_{k-2}Q_{k-1}-\lambda I)$ $\displaystyle=...=\det(Q_{k-1}^{\dagger}Q_{k-2}^{\dagger}...Q_{0}^{\dagger}\underbrace{A_{0}}_{=A}Q_{0}...Q_{k-2}Q_{k-1}-\lambda I)$ $\displaystyle=\det(A-\lambda I)$ (135) where the cyclic permutation symmetry of the determinant is used in the last step. Although the QR-iteration possibly requires a large amount of operations per step for a general matrix $A$, it turns out to be of very low computational cost with respect to upper Hessenberg matrices like the matrix $H_{k}$ obtained from a $k$-step Arnoldi process described in Algorithm 1. In such a case, one can apply $(k-1)$ Givens rotations $G_{i}$, $i\in\\{1,...,k-1\\}$, which are all unitary operations such that the overall QR-decomposition of $H_{k}$ at each QR-step $j$ is given by $(H_{k})_{j}=Q_{j}R_{j}=(G_{k-1})_{j}...(G_{1})_{j}R_{j}.$ (136) Further details on Gives rotations can be found in Ref. [64]. With respect to the IRAM, the QR-iteration turns out to be important not only because it is used to calculate the eigenvalues of $H_{k}$, but also in terms of restarting the Arnoldi iteration with an initial vector that incorporates shifts that filter out the direction of unwanted eigenvectors of $H_{k}$. Those shifts are obtained by utilizing QR-steps, which are explained in detail in Section 4.3.4. In Ref. [63], further details on the QR-iteration are presented. 1: $A_{0}\leftarrow A$ 2: $\text{Calculate the}\,\text{QR-decomposition}\,A_{0}=Q_{0}R_{0}$ 3: for $k=1\,\text{until convergence}$ do 4: $A_{k}\leftarrow R_{k-1}Q_{k-1}$ 5: if $A_{k}$ ”triangular enough” then 6: quit 7: else 8: $\text{Calculate the}\,\text{QR-decomposition}\,A_{k}=Q_{k}R_{k}$ 9: end if 10: end for Algorithm 2 The QR-iteration #### 4.3.3 Polynomial filtering Before dealing with the IRAM in particular, a brief discussion on polynomial filtering techniques of the initial vector $v$ of a Krylov subspace iteration, like the Arnoldi algorithm of Section 4.3.1, is realized. In general, the target of applying a filter to a vector $v$ is to make the contribution of unwanted directions very small, or even zero. To understand how polynomial filtering works in general, a square matrix $A\in\mathbb{C}^{n\times n}$ whose eigenpairs are given by the set $\\{(\lambda_{i},v_{i})\|1\leq i\leq n\\}$ with $\lambda_{i}\in\mathbb{C}$ and $v_{i}\in\mathbb{C}^{n}$ is considered. Suppose the eigenvectors form a basis. Each arbitrary vector $v\in\mathbb{C}^{n}$ can then be written as $v=\sum_{j=1}^{n}c_{j}v_{j}$ with complex expansion coefficients $\\{c_{j}\\}$. Assuming that one wants to filter out the contribution of the $k$-th eigenvector $v_{k}$, one can define $p(t)\equiv t-\lambda_{k}$ (137) and apply this very simple polynomial onto $v$: $\displaystyle p(A)v$ $\displaystyle=\sum_{j=1}^{n}c_{j}p(A)v_{j}=\sum_{j=1}^{n}c_{j}(A-\lambda_{k})v_{j}$ $\displaystyle=\sum_{j=1}^{n}c_{j}(\lambda_{j}-\lambda_{k})v_{j}$ $\displaystyle=\sum_{j\neq k}^{n}c_{j}(\lambda_{j}-\lambda_{k})v_{j}.$ (138) If $A$ is hermitian, the overlap with $v_{k}$ vanishes, i.e., $\langle v_{k}\|p(A)v\rangle=\sum_{j\neq k}^{n}c_{j}(\lambda_{j}-\lambda_{k})\underbrace{\langle v_{k}\|v_{j}\rangle}_{=\delta_{kj}}=0,$ (139) because in that case the eigenvectors of $A$ would be orthogonal. Thus, the vector $p(A)v$ is orthogonal to $v_{k}$, and any additional power iteration of this vector with respect to $A$ preserves this property, meaning that $\langle v_{k}\|A^{n}p(A)v\rangle=\sum_{j\neq k}^{n}c_{j}\lambda_{j}^{n}(\lambda_{j}-\lambda_{k})\underbrace{\langle v_{k}\|v_{j}\rangle}_{=\delta_{kj}}=0.$ (140) Therefore, the Arnoldi iteration, applied to an initial vector $p(A)v$, yields a Krylov basis which is orthogonal to $v_{k}$, and also the orthogonal projection $H_{k}$ would not contain $\lambda_{k}$ in its spectrum of eigenvalues. If $A$ is non-hermitian, the overlap of $p(A)v$ and $v_{k}$ is not strictly zero since the eigenvectors $\\{v_{i}\\}$ can be linearly dependent. However, within the IRAM, the overall contribution of $v_{k}$ gradually decreases upon increasing the number of restarts with properly shifted initial vectors. Nevertheless, this shows that the LR-matrix $\mathcal{L}$, which is in general non-hermitian, leads to additional numerical effort which would be absent in the case of a hermitian matrix. It is also possible to do multiple shifts at once by using a filter given by $p(t)=\prod_{i=1}^{N}\left(t-\lambda_{\pi(i)}\right)$ (141) where $N\leq n$ is the number of shifts and $\pi(i)\in\\{1,...,n\\}\,\forall\,i$. In this case, the directions of $N$ different eigenvectors of $A$ are filtered out. The above choices of $p(t)$ do only reflect the simplest possibilities of filter polynomials, but also more sophisticated ones like the Chebyshev polynomials can be chosen [64]. #### 4.3.4 The implicitly restarted Arnoldi method (IRAM) 1: Take an initial vector $v_{1}\leftarrow v$, $\|\|v\|\|_{2}=1$ 2: Perform a $k$-step Arnoldi iteration to get $\quad\quad AV_{k}=V_{k}H_{k}+f_{k}\,e_{k}^{t}$ with $f_{k}:=h_{k+1,k}\,v_{k+1}$ 3: Calculate the eigenvalues $\theta_{1},...,\theta_{k}$ of $H_{k}$ by using the QR-iteration 4: while $m$ wanted eigenvalues of $H_{k}$ not converged do 5: Select $p=k-m$ unwanted eigenvalues $\theta_{1},...,\theta_{p}$ 6: Perform $p$ QR-steps with unwanted eigenvalues as shifts: $\quad\quad[H^{(p)}_{k},Q]:=QR[H_{k},\theta_{1},...,\theta_{p}],\quad V^{(p)}_{k}:=V_{k}Q$ 7: Set $f_{m}:=h_{k+1,k}\,Q_{kp}\,v_{k+1}+h^{(p)}_{m+1,m}\,v^{(p)}_{m+1}$ 8: Define $H_{m}:=H^{(p)}_{k}(1:m,1:m)$, $V_{m}:=V^{(p)}_{k}(1:n,1:m)$ 9: Perform $p$ additional Arnoldi steps onto $V_{m},\,H_{m}$ and $f_{m}$ to obtain a $k$-step Arnoldi decomposition 10: Calculate the eigenvalues of the new $H_{k}$ by using the QR-iteration 11: end while Algorithm 3 The implicitly restarted Arnoldi method (IRAM) After the introduction of the three essential ingredients of the IRAM, namely the Arnoldi algorithm, the QR-iteration, and polynomial filtering, one can finally discuss the IRAM itself in detail. The full algorithm is given in Algorithm 3. At first, a $k$-step Arnoldi iteration is performed, yielding an Arnoldi decomposition of $A$ as described in Eq. (128). Then, the eigenvalues $(\theta_{1},...,\theta_{k})$ are calculated utilizing the QR-iteration. From this set, $p$ unwanted eigenvalues $(\theta_{1},...,\theta_{p})$ are selected. If one for example seeks the eigenvalues of $A$ that have the smallest magnitude, an appropriate selection would be the $p$ eigenvalues of $H_{k}$ that have the largest magnitude. Afterwards, a sequence of $p$ QR-shifts with the unwanted eigenvalues is performed. For the first shift, the Arnoldi decomposition reads $\displaystyle(A-\theta_{1}I)V_{k}$ $\displaystyle=V_{k}(H_{k}-\theta_{1}I)+h_{k+1,k}\,v_{k+1}\,e_{k}^{t}$ (142) $\displaystyle\iff(A-\theta_{1}I)V_{k}$ $\displaystyle=V_{k}Q_{1}R_{1}+h_{k+1,k}\,v_{k+1}\,e_{k}^{t}$ (143) $\displaystyle\iff AV_{k}Q_{1}$ $\displaystyle=V_{k}Q_{1}(R_{1}Q_{1}+\theta_{1}I)+h_{k+1,k}\,v_{k+1}\,e_{k}^{t}\,Q_{1}$ (144) where the QR-decomposition $H_{k}-\theta_{1}I=Q_{1}R_{1}$ is used in Eq. (143). Setting $\displaystyle H_{k}^{(1)}:=R_{1}Q_{1}+\theta_{1}I,\quad V_{k}^{(1)}:=V_{k}Q_{1},\quad\left(e_{k}^{(1)}\right)^{t}:=e_{k}^{t}Q_{1}$ (145) the next QR-shift can be applied: $\displaystyle(A-\theta_{2}I)V_{k}^{(1)}$ $\displaystyle=V_{k}^{(1)}(H^{(1)}_{k}-\theta_{2}I)+h_{k+1,k}\,v_{k+1}\,\left(e_{k}^{(1)}\right)^{t}$ (146) $\displaystyle\overset{H^{(1)}_{k}-\theta_{2}I=Q_{2}R_{2}}{\iff}AV_{k}^{(1)}Q_{2}$ $\displaystyle=V_{k}^{(1)}Q_{2}(R_{2}Q_{2}+\theta_{2}I)+h_{k+1,k}\,v_{k+1}\,\left(e_{k}^{(1)}\right)^{t}\,Q_{2}$ (147) where one can define, similar to Eq. (145), $\displaystyle H_{k}^{(2)}:=R_{2}Q_{2}+\theta_{2}I,\quad V_{k}^{(2)}:=V_{k}^{(1)}Q_{2},\quad\left(e_{k}^{(2)}\right)^{t}:=\left(e_{k}^{(1)}\right)^{t}Q_{2}.$ (148) This procedure is continued until all $p$ QR-shifts are applied. Multiplying Eq. (143) by $e_{1}$ yields $\displaystyle(A-\theta_{1}I)V_{k}e_{1}$ $\displaystyle=V_{k}Q_{1}R_{1}e_{1}+h_{k+1,k}\,v_{k+1}\,e_{k}^{t}e_{1}$ $\displaystyle\iff(A-\theta_{1}I)v_{1}$ $\displaystyle=V_{k}Q_{1}r_{11}e_{1}$ $\displaystyle\iff(A-\theta_{1}I)v_{1}$ $\displaystyle=v^{(1)}_{1}r_{11}$ (149) where $r_{11}$ is the upper left element of $R_{1}$ and $v_{1}^{(1)}$ is the first column of $V_{k}Q_{1}$. This in turn means that the first vector of the new basis, i.e., $v^{(1)}_{1}$, is proportional to $(A-\theta_{1}I)v_{1}$. As discussed in the previous section, this polynomial filter reduces the contribution of the unwanted eigenvector corresponding to $\theta_{1}$ in $v^{(1)}_{1}$, or, in the optimal case, makes its contribution even vanish completely. This holds for all successive vectors in $V_{k}^{(1)}$. After having applied the second QR-shift, one can show that $v_{1}^{(2)}\sim(A-\theta_{2}I)(A-\theta_{1}I)v_{1}$. Continuing in the same manner until the $p$-th QR-shift is applied yields $v_{1}^{(p)}\sim\prod_{i=1}^{p}(A-\theta_{i}I)v_{1}$. Then, by taking the first $m=k-p$ columns of $V_{k}^{(p)}$ as the new Arnoldi basis $V_{m}$, as well as taking the leading principal $(m\times m)$-matrix of $H_{k}^{(p)}$ as the new orthogonal projection $H_{m}$ of $A$, one obtains an $m$-step Arnoldi decomposition $AV_{m}=V_{m}H_{m}+f_{m}\,e_{m}^{t}$ with an error term given by $f_{m}=h_{k+1,k}\,Q_{kp}\,v_{k+1}+h^{(p)}_{m+1,m}\,v^{(p)}_{m+1}$ (150) where it is used that the unitary matrix $Q$ is of Hessenberg form. Based on this, one performs $p$ additional Arnoldi steps yielding again a $k$-step Arnoldi decomposition. The essential difference to the previous $k$-step Arnoldi decomposition is that now all basis vectors in $V_{k}$ are approximately or even fully orthogonal to the eigenvectors of the associated unwanted eigenvalues. The accuracy of the eigenvalues and eigenvectors of the newly computed matrix $H_{k}$ can be estimated with the residual norm $r$ introduced in Section 4.3.1, Eq. (4.3.1). If it is below a predefined tolerance for a desired amount of wanted eigenpairs, the algorithm finishes. Otherwise, the set of eigenvalues of $H_{k}$ is again separated into wanted and unwanted eigenvalues, and the procedure of QR-shifting and refining the given Arnoldi basis restarts. By employing the above described shift strategy without restarting the Arnoldi iteration from the very beginning, one obtains the same result as if an explicit restart of the entire $k$-step Arnoldi process with the initial vector $v_{1}^{(p)}$ had been performed. Because of this similarity, the overall algorithm is said to be implicitly restarted. The computational effort in terms of matvecs with $A$ is given by $(k+p)$ operations in both cases, i.e., for explicit and implicit restarting. However, implicit restarting turned out to be numerically more stable, and has additional advantages which are beyond the scope of this work. Figure 2 shows the general structure of the code for the matrix diagonalization utilizing the implementation of IRAM in the PARPACK library [60]. Essential for the performance is the efficient parallel implementation of the matvecs which are done outside of the PARPACK routine pznaupd. The latter carries out the IRAM iterations in parallel [60]. It is important to note that the IRAM, although it is particularly efficient for sparse matrices, would also work for dense matrices. Further details are presented in Refs. [64, 59, 63]. To summarize, the newly developed implementation of LR-MCTDHB was presented in this section and the code was explained in detail. Furthermore, the IRAM, a very powerful method for computing eigenpairs of very large and sparse matrices, was described. In the next section, the code is benchmarked against an exactly-solvable model. Input: $\mathcal{L}$ Distribution of $\mathcal{L}$ • Each PE gets the same amount of rows from the upper and lower blocks Define initial vector $v_{0}$ Arnoldi iteration • Call pznaupd (PARPACK routine that performs IRAM iterations) Eigenvalues converged? Matvec $z:=Av_{i}$ Calculate eigenvectors, response weights and response densities Outputnoyes Figure 2: Schematic diagram of the MPI-parallelized implementation for the diagonalization of $\mathcal{L}$. Different chunks of non-zero elements of $\mathcal{L}$ are distributed among the PEs. The Arnoldi iteration is carried out by the routine pznaupd of Parallel ARPACK (PARPACK) [60], which either finishes upon convergence of a desired amount of eigenvalues or expects the user to perform a matvec and return the result to it. Finally, the eigenvalues, eigenvectors, as well as response weights and response densities are calculated and written to output files. See text for details. ## 5 Comparison with an exactly-solvable model After the presentation of the theoretical and numerical frameworks utilized, along with the structure of the implemented code, the newly developed implementation of LR-MCTDHB is benchmarked in the following. Furthermore, since it builds the basis for the utilized LR theory, results from a benchmark of the MCTDHB implementation used in this work [65], which makes use of a sophisticated mapping of bosonic operators in Fock space [66], are presented as well. This implementation is now also available with a recently developed graphical user interface and works on all common operating systems [67]. Other codes became available during the last years, e.g., the MCTDH-X software package [68] which contains a separate graphical user interface. Furthermore, the multi-layer (ML-)MCTDHB approach [69, 70, 71], which is particularly suited for the description of Bose-Bose or Bose-Fermi mixtures, is now implemented. As a reference system, the exactly-solvable harmonic-interaction model (HIM) is used. In the latter model, both the single-particle and interaction potentials are of harmonic type. This model system has been used recently to benchmark the fermionic counterpart of MCTDHB, termed MCTDHF [221], as well as the time-dependent restricted-active-space self-consistent-field theory for bosons (TD-RASSCF-B) [222] and its variant for bosonic mixtures [223]. In the first part of this section, an introduction to the HIM and the analytic results of its energy spectrum are given (Section 5.1). Afterwards, an overview of the MCTDHB benchmark against the HIM, which has been carried out in great detail in Ref. [33], is presented (Section 5.2). This includes both the ground state and the out-of-equilibrium quantum dynamics of a bosonic many-particle system. Finally, a comparison of the numerical results of the low-energy spectrum obtained from LR-MCTDHB and the analytic values, both for the 1D and 2D cases, is made (Section 5.3). ### 5.1 The harmonic-interaction model (HIM) The HIM Hamiltonian contains the harmonic trap potential $V(\mathbf{r})=\frac{1}{2}\left(\Omega_{x}^{2}x^{2}+\Omega_{y}^{2}y^{2}+\Omega_{z}^{2}z^{2}\right),\quad\mathbf{r}=(x,y,z)^{t}$ (151) where the boson mass $m$ is set to unity. In $D>1$ dimensions, one distinguishes between the isotropic and anisotropic HIM, where the former refers to the case where all frequencies $\\{\Omega_{i}\\}$ are equal, whereas the latter refers to the case where at least one frequency differs from the others. The interaction between the $i$-th and $j$-th boson is described as $\hat{W}(\mathbf{r}_{i},\mathbf{r}_{j})=\lambda_{0}\|\mathbf{r}_{i}-\mathbf{r}_{j}\|^{2}$ (152) where a positive (negative) interaction strength $\lambda_{0}$ denotes attraction (repulsion) between the bosons. As described in Ref. [72], the following coordinate transformation yields the separation of the relative and center-of-mass (c.m.) coordinates, $\mathbf{Q}_{k}=\frac{1}{\sqrt{k(k+1)}}\sum_{i=1}^{k}(\mathbf{r}_{k+1}-\mathbf{r}_{i}),\quad 1\leq k\leq N-1$ (153) and $\mathbf{Q}_{N}=\frac{1}{\sqrt{N}}\sum_{i=1}^{k}\mathbf{r}_{i},$ (154) where $N$ is the number of bosons in the system. This leads to the Hamiltonian in the c.m. frame given by $\displaystyle\hat{H}$ $\displaystyle=\hat{H}_{rel}+\hat{H}_{c.m.}$ (155) $\displaystyle\hat{H}_{rel}$ $\displaystyle=\frac{1}{2}\sum_{j=x,y,z}\sum_{k=1}^{N-1}(p_{j,k}^{2}+\delta_{j}^{2}Q_{j,k}^{2})$ (156) $\displaystyle\hat{H}_{c.m.}$ $\displaystyle=\frac{1}{2}\sum_{j=x,y,z}(p_{j,N}^{2}+\Omega_{j}^{2}Q_{j,N}^{2})$ (157) with momenta $p_{j,N}=(1/i)\partial_{Q_{j,N}}$ of the c.m. coordinates, $p_{j,k}=(1/i)\partial_{Q_{j,k}}$ of the relative coordinates, and the parameters $\delta_{j}^{2}=\Omega_{j}^{2}\pm 2N\|\lambda_{0}\|$ where again the plus (minus) sign denotes attraction (repulsion) between the bosons. For simplicity $\hbar=1$ is assumed. It is instructive to interpret the Hamiltonian as a composite model system of $(N-1)$ identical non-interacting particles in a harmonic confinement with trap frequencies $\mathbf{\delta}=(\delta_{x},\delta_{y},\delta_{z})^{t}$ and the c.m. particle which moves in the harmonic confinement with the original trap frequencies $\\{\Omega_{i}\\}$. The ground-state wave function in the c.m. frame takes the form $\Psi_{0}(\mathbf{Q}_{1},...,\mathbf{Q}_{N})=\prod_{j=x,y,z}\left(\frac{\delta_{j}}{\pi}\right)^{\frac{N-1}{4}}\left(\frac{\Omega_{j}}{\pi}\right)^{\frac{1}{4}}\exp\left(-\frac{1}{2\delta_{j}}\sum_{k=1}^{N-1}Q_{j,k}^{2}-\frac{\Omega_{j}}{2}Q_{j,N}^{2}\right),$ (158) i.e., it is basically a product of $N$ harmonic oscillator (HO) ground-state wave functions in each spatial direction [72]. However, in the laboratory frame, the representation of the ground state is much more involved. Already for $N=2$ bosons in 1D, it is an infinite sum of Hartree products of HO eigenfunctions instead of a single product of two HO ground states in the c.m. frame [33]. Numerical convergence towards this state is therefore a very demanding challenge in the laboratory frame. The exact solution for the energy levels reads $\displaystyle E\left(\\{n_{j}\\},\\{m_{j}\\}\right)$ $\displaystyle=\sum_{j=x,y,z}\left\\{\left(n_{j}+\frac{N-1}{2}\right)\delta_{j}+\left(m_{j}+\frac{1}{2}\right)\Omega_{j}\right\\}$ (159) where the $\\{n_{j}\\}$ denote the quantum numbers of excitations of the relative coordinates and the $\\{m_{j}\\}$ denote the quantum numbers of c.m. excitations in the $x$-, $y$\- and $z$-directions. In particular, this means that the ground-state energy is given by $E_{0}\equiv\varepsilon^{0}=\sum_{j=x,y,z}\left\\{\frac{(N-1)}{2}\delta_{j}+\frac{1}{2}\Omega_{j}\right\\}$ (160) whereas the energy distance between any excited state and the ground state reads $\omega({\\{n_{j}\\},\\{m_{j}\\}})=\sum_{j=x,y,z}\left\\{n_{j}\delta_{j}+\Omega_{j}m_{j}\right\\}.$ (161) Eqs. (160) and (161) will be used for the comparison to the numerical results obtained from (LR-)MCTDHB. The degeneracies of excited states is discussed in [73]. It is worth noting that for any number of bosons $N$, there is no solution for an excitation corresponding to $\sum_{j=x,y,z}n_{j}=1$, i.e., a single-particle excitation with respect to the relative coordinates. This is due to the even permutation symmetry for identical bosons, Eq. (5), and it can be shown by mathematical induction as follows. For the sake of simplicity, the proof is restricted to the 1D case, i.e., $\mathbf{Q}_{i}=Q_{i}$, which does not limit the generality since in higher dimensions, the Hamiltonian decouples into the $x$-, $y$\- and $z$-components, see Eqs. (156) and (157). The wave function corresponding to a first order excitation of the relative coordinates, denoted in the following by $\Psi^{\prime}$, obeys $\Psi^{\prime}(Q_{1},...,Q_{N})\sim\frac{\partial}{\partial Q_{k}}\Psi_{0}(Q_{1},...,Q_{N}),\quad 1\leq k\leq N-1.$ (162) For $N=2$, this yields $\displaystyle\Psi^{\prime}(Q_{1},Q_{2})\sim Q_{1}\Psi_{0}\sim(x_{2}-x_{1})\Psi_{0},$ (163) see Eq. (158). Because the wave function needs to be fully symmetric under the exchange $x_{1}\leftrightarrow x_{2}$, one finds $\displaystyle\Psi^{\prime}\sim\left\\{(x_{2}-x_{1})+(x_{1}-x_{2})\right\\}\Psi_{0}=0,$ (164) meaning that there is no first order relative excited state for two bosons. Assuming that this is also true for $N$ bosons, i.e., $\Psi^{\prime}(Q_{1},...,Q_{N})\sim\frac{\partial}{\partial Q_{k}}\Psi_{0}(Q_{1},...,Q_{N})=0\quad\forall\,1\leq k\leq N-1,$ (165) one obtains for $(N+1)$ bosons and $K\leq N$ $\displaystyle\Psi^{\prime}(Q_{1},...,Q_{N+1})$ $\displaystyle\sim\frac{\partial}{\partial Q_{K}}\Psi_{0}(Q_{1},...,Q_{N+1})$ $\displaystyle\sim\sum_{i=1}^{K}(x_{K+1}-x_{i})\Psi_{0}$ $\displaystyle=\left(Kx_{K+1}-\sum_{i=1}^{K}x_{i}\right)\Psi_{0}$ $\displaystyle=\left(Kx_{K+1}-x_{K}-\sum_{i=1}^{K-1}x_{i}\right)\Psi_{0}$ $\displaystyle=\left(Kx_{K+1}-x_{K}-(K-1)x_{K}+\underbrace{(K-1)x_{K}-\sum_{i=1}^{K-1}x_{i}}_{\sim Q_{K-1}}\right)\Psi_{0}$ $\displaystyle\sim K\left(x_{K+1}-x_{K}\right)\Psi_{0}+Q_{K-1}\Psi_{0}$ (166) where the argument of $\Psi_{0}$ is suppressed from the second line onward. For the second term in Eq. (5.1), one can make use of the assumption from Eq. (165) that the contribution of $Q_{K-1}\Psi_{0}$ to $\Psi^{\prime}$ vanishes after symmetrization of the wave function. Moreover, the first term is proportional to the relative coordinate of only two bosons, for which it was already shown that its contribution to $\Psi^{\prime}$ vanishes as well after symmetrization. Thus, one infers that also for $(N+1)$ bosons $\Psi^{\prime}(Q_{1},...,Q_{N+1})=0$ (167) which ends the proof. ### 5.2 Ground state and dynamics At first, numerical results obtained from MCTDHB for the ground state of the 1D HIM for different particle numbers and the exact values from Eq. (160) are compared. The index ’$x$’ is suppressed in the following for simplicity. The trap frequency is set to $\Omega=1$ and the two-particle interaction strength $\lambda_{0}$ is adjusted such that the mean-field interaction parameter $\Lambda=\lambda_{0}(N-1)$ is kept fixed at $\Lambda=0.5$. Results are shown in Table 1, which originally appeared in Ref. [33] where it has been demonstrated that the MCTDHB method is much more efficient and accurate for the description of the ground state and dynamics of a many-boson system than the widely used FCI method. One observes that MCTDHB gives highly accurate results for the ground-state energy for all considered particle numbers. Furthermore, one needs less self-consistent orbitals $M$ in order to converge to the exact ground-state energy when $N$ increases. This is anticipated, since in the Hartree limit where $N\rightarrow\infty$ for fixed $\Lambda$, the energy per particle converges to the GP value. What is however surprising is the degree of accuracy. As an example, for $N=1000$ the relative error $(\varepsilon^{0}_{M=3}-\varepsilon^{0}_{\text{exact}})/\varepsilon^{0}_{\text{exact}}$ is lower than $10^{-10}\,\%$. The fact that MCTDHB can very accurately assemble the ground state of a large many-boson system with only very few orbitals has far-reaching consequences for the accuracy of excitation energies obtained by LR-MCTDHB. As discussed in Section 3.2.1, the latter will be closer to the exact values if the ground state to which LR is applied is very accurately described. Further examples for the ground-state energies in the 1D and 2D HIM will be given in Section 5.3 and in Appendix C, where in particular the convergence of excitation energies is discussed. With regard to the dynamics, comparisons of numerical and exact results were carried out in Ref. [33] for (i) the out-of-equilibrium quench dynamics and (ii) the dynamics due to time-dependent driving of the trapping frequency and the interaction strength. Concerning (i), Fig. 3 compares the exact, FCI and MCTDHB results for the oscillations of the density $\rho$ of $N=2$ bosons after a sudden quench of the interaction parameter from $\Lambda=0$ to $\Lambda=0.5$. The initial state $\Psi_{0}$ is therefore the ground state of the non-interacting system. In principle, the time-evolution is given by $\displaystyle\|\Psi(t)\rangle$ $\displaystyle=\sum_{n=0}^{\infty}c_{j}\,e^{-iE_{j}t}\|\alpha_{j}\rangle,$ (168) $\displaystyle c_{j}$ $\displaystyle=\langle\alpha_{j}\|\Psi(0)\rangle$ (169) where the $\\{\|\alpha_{j}\rangle\\}$ are an arbitrary basis of the two- particle Hilbert space and the $\\{E_{j}\\}$ the corresponding eigenenergies. One possible choice of this basis are the solutions of the time-independent problem. It remains to compute the overlap integrals in Eq. (169). In practice, the infinite sum in Eq. (168) has to be truncated, and for the case described in Fig. 3, numerical convergence at each time step of the expansion in Eq. (168) is ensured by including the $60$ lowest energy eigenstates of the time-independent problem. However, utilizing only $M=4$ time-adaptive orbitals, the MCTDHB calculations lead to a density oscillation that can no longer be distinguished from the exact result. Already for $M=3$ the results are highly accurate. If a fixed set of orbitals is used (FCI), even 8 orbitals do not lead to the same accuracy than MCTDHB(4). This remarkably shows the numerical benefit of using a time-adaptive, self-consistent basis set. $M$ \| $N=10$ \| $N=100$ \| $N=1000$ ---\|---\|---\|--- 1 \| 7.071067811865483 \| 70.71067811865483 \| 707.1067811865483 2 \| 7.038769026303168 \| 70.68016951747168 \| 707.0764334257315 3 \| 7.038350652406389 \| 70.68012541218675 \| 707.0764289871865 4 \| 7.038348424909910 \| 70.68012539174549 \| 5 \| 7.038348415349058 \| 70.68012539173762 \| 6 \| 7.038348415311494 \| \| 7 \| 7.038348415311018 \| \| $\varepsilon^{0}_{exact}$ \| 7.038348415311011 \| 70.68012539173752 \| 707.0764289869851 Table 1: Ground-state energies of $N=10,100,$ and $1000$ bosons in the 1D HIM with interaction parameter $\Lambda=0.5$ and trap frequency $\Omega=1.0$. Numerical results are shown for different numbers of orbitals $M$. Exact results are obtained from Eq. (160). The table is adapted from Ref. [33]. Figure 3: Out-of-equilibrium dynamics of the one-particle density $\rho(x=0)$ for $N=2$ bosons after an interaction quench from $\Lambda=0$ to $\Lambda=0.5$. Results are shown for different numbers of orbitals $M$. The trap frequency is $\Omega=1.0$. For the time scale shown, the MCTDHB(4) dynamics cannot be distinguished from the exact dynamics. The FCI method is less accurate even for $8$ orbitals. All quantities are dimensionless. See text for details. The figure is taken from Ref. [33]. Figure 4: Dynamics in the HIM due to a time-dependent driving $f_{2}(t)$ of the trapping frequency $\Omega(t)=\Omega_{0}\sqrt{1+f_{2}(t)}$ and the interaction strength $\lambda(t)=\lambda_{0}\left[1-\frac{\Omega_{0}^{2}}{2N\lambda_{0}}f_{2}(t)\right]$. The upper panel shows the complicated time-dependent profile of the driving frequency $f_{2}(t)$. The lower panel shows the evolution of the c.m. energy $\varepsilon_{2}(t)$ for different levels of MCTDHB($M$). For $N=10$ bosons, the curve of MCTDHB(7) can hardly be distinguished from the exact result for the shown time interval. The same holds true for $M=6$ and $N=50$ bosons. Even for less orbitals, MCTDHB yields an accurate description of the evolution of $\varepsilon_{2}(t)$ up to $t\approx 35$ for $N=10,\,M=6$ and $t\approx 25$ for $N=50,\,M=5$. On the contrary, the GP results do only follow the exact curves for very short times and clearly fail to describe the long-time behavior, even qualitatively. All quantities are dimensionless. See text for details. The figure is taken from Ref. [33]. Fig. 4 shows the time-evolution of the c.m. energy $\epsilon_{2}(t)$ when both the trapping frequency and the two-body interaction strength are time- dependent and given by $\Omega(t)=\Omega_{0}\sqrt{1+f_{2}(t)}$ with a constant $\Omega_{0}$ and $\lambda(t)=\lambda_{0}\left[1-\frac{\Omega_{0}^{2}}{2N\lambda_{0}}f_{2}(t)\right]$. The function $f_{2}(t)$ denotes a time-dependent driving frequency. In that way, the trapping frequency of the $(N-1)$ relative coordinates remains time- independent, i.e., $\delta=\sqrt{\Omega_{0}^{2}[1-f_{2}(t)]+2N\lambda_{0}\left[1+\frac{\Omega_{0}^{2}}{2N\lambda_{0}}f_{2}(t)\right]}=\sqrt{\Omega_{0}^{2}+2N\lambda_{0}}.$ (170) Thus, although in the laboratory frame the bosons move in a time-dependent trap and interact via a time-dependent potential, the same system in the c.m. frame appears to be much less complicated because only the trapping potential of the c.m. particle is time-dependent due to $f_{2}(t)$. Because the c.m. and relative motions separate, the time-dependent energy of the system is given by $E=\frac{N-1}{2}\delta+\varepsilon_{2}(t)$ where $\varepsilon_{2}(t)$ is determined by the energy expectation value $\varepsilon_{2}(t)=\langle\Psi_{N}(t)\|\hat{H}_{c.m.}(t)\|\Psi_{N}(t)\rangle$ (171) of the c.m. particle. The state $\|\Psi_{N}(t)\rangle$ is obtained from the effective single-particle Schrödinger equation $i\partial_{t}\|\Psi_{N}(t)\rangle=\hat{H}_{c.m.}(t)\|\Psi_{N}(t)\rangle$. From Fig. 4, one can deduce that MCTDHB is capable to account for the dynamical evolution of $\varepsilon_{2}(t)$ very accurately, even for longer times. Furthermore, one observes that for a higher number of particles $N$ a smaller number of time-adaptive orbitals $M$ is required to obtain the same level of numerical accuracy. In contrast to that, the GP evolution of $\varepsilon_{2}(t)$ only follows the exact curves of $N=10$ and $50$ bosons for a very short time up to $t\approx 1$, and becomes remarkably inaccurate afterwards. For the long-time behavior, it cannot even reproduce qualitatively the profile of $\varepsilon_{2}(t)$. To sum up, the previous examples clearly indicate that the MCTDHB theory qualifies for describing the ground state, the dynamics due to a sudden quench of system parameters as well as the dynamics due to time-dependent trapping and interaction potentials, and that it is clearly superior to the FCI many- body and GP mean-field methods. The full benchmark, together with additional valuable details, can be found in Ref. [33]. ### 5.3 Excited states In this section, the applicability of LR-MCTDHB to obtain highly accurate results for the energies of excited states of trapped multi-boson systems is discussed. As for the benchmark concerning the ground state and dynamics, the analytic results of the HIM from Eq. (161) are used to benchmark the numerical results. In the following, LR-MCTDHB is first benchmarked against the 1D HIM for repulsive bosons, i.e., with $\lambda_{0}<0$. Then, for the 2D case, the isotropic HIM with $\Omega_{x}=\Omega_{y}$ is considered. In Appendix C, further benchmarks against the 1D HIM with attractive bosons, against the anisotropic case in 2D, as well as against the isotropic HIM in the rotating frame of reference are presented. \| $M=1$ \| $M=3$ \| $M=4$ \| $(m_{x},n_{x})$ \| Exact ---\|---\|---\|---\|---\|--- $\varepsilon^{0}$ \| 4.527693 \| 4.524922 \| 4.524922 \| $(0,0)$ \| 4.524922 $\omega_{1}$ \| 1.000000 \| 1.000000 \| 1.000000 \| $(1,0)$ \| 1.000000 $\omega_{2}$ \| 1.811077 \| 1.788855 \| 1.788854 \| $(0,2)$ \| 1.788854 $\omega_{3}$ \| n/a \| 2.000005 \| 2.000000 \| $(2,0)$ \| 2.000000 $\omega_{4}$ \| 2.716616 \| 2.683282 \| 2.683282 \| $(0,3)$ \| 2.683282 $\omega_{5}$ \| n/a \| 2.788858 \| 2.788854 \| $(1,2)$ \| 2.788854 $\omega_{6}$ \| n/a \| 3.000009 \| 3.000000 \| $(3,0)$ \| 3.000000 $\omega_{7}$ \| 3.622154 \| 3.577961 \| 3.573164 \| $(0,4)$ \| 3.577709 $\omega_{8}$ \| n/a \| 3.580111 \| 3.577712 \| $(0,4)$ \| 3.577709 $\omega_{9}$ \| n/a \| 3.685038 \| 3.681347 \| $(1,3)$ \| 3.683282 $\omega_{10}$ \| n/a \| 3.790078 \| 3.788483 \| $(2,2)$ \| 3.788854 $\omega_{11}$ \| n/a \| 4.002097 \| 3.999997 \| $(4,0)$ \| 4.000000 Table 2: Benchmark of the LR-MCTDHB implementation to the repulsive 1D HIM with $N=10$ bosons. Shown are the ground-state energy $\varepsilon^{0}$ and the energies $\omega_{i}$ of the first few excitations for different values of $M$. The trapping frequency is $\Omega=1.0$ and the interaction strength is $\lambda_{0}=-0.01$, yielding a marginal ground-state depletion of $f\approx 0.03\%$ for $M=4$. Whereas the BdG ($M=1$) spectrum misses many states, the many-body spectra for $M=3$ and $4$ contain all low-lying excitations. Even the two-fold degenerate state $(0,4)$, as proposed in Ref. [73], is obtained (see $\omega_{7}$ and $\omega_{8}$). The assignment of the corresponding single GP state to $\omega_{7}$ is arbitrary, and one would need to analyse for instance the response density to identify whether it correlates with the 7th or 8th many-body excitation. The overall accuracy of excitation energies increases with the number of orbitals. Underlined digits denote deviations from the exact values from Eqs. (160) and (161). All quantities are dimensionless. See text for more details. For the benchmark against the 1D repulsive HIM, the system parameters are set to $\Omega=1$ for the trap frequency and $\lambda_{0}=-0.01$ for the interaction strength, which yields a slight depletion of $f\approx 0.03\%$ for $N=10$ bosons and $M=4$ orbitals. Calculations were carried out on a grid from $[-9,9)$ with 128 grid points. Results for the excitation energies relative to the ground-state energy, $\omega_{i}=E_{i}-\varepsilon^{0}$, are shown in Table 2. There are several comments to be made. At first, it can be observed that the mean-field BdG calculation ($M=1$) misses several excited states in the low-energy spectrum. This is due to the fact that the BdG theory by construction does not account for excitations were several particles at a time are excited from the ground-state manifold, so-called multi-particle excitations. For further details see Appendix A.1.1. The second observation is that the numerical accuracy is enhanced if more self-consistent orbitals are included into the ground-state description. Also the two-fold degenerate excitation $(0,4)$, as proposed in Ref. [73], is obtained. The accuracy of the BdG spectrum, apart from the c.m. excitation $(1,0)$, is clearly lower than for the many-body results of $M=3$ and $4$ orbitals. In addition, the convergence towards the exact ground-state energy is observed. Table 3 shows excitation energies obtained for the isotropic 2D HIM with trap frequencies $\Omega_{x}=\Omega_{y}=1$ and $N=100$ repulsive bosons with interaction parameter $\lambda_{0}=-0.001$. All computations were carried out on a $[-9,9)\times[-9,9)$ grid with $64\times 64$ grid points. At first, one sees that the BdG approach yields the energy of the lowest c.m. excited state, which is two-fold degenerate, to perfect accuracy. All other states obtained are pure relative excitations. Higher c.m. excited states, as well as combinations of those with relative excitations, are missing in the BdG spectrum. $\omega_{i}(\gamma)$ \| $M=1$ \| $M=2$ \| $M=3$ \| $(m_{x},m_{y},n_{x}$+$n_{y})$ \| Exact ---\|---\|---\|---\|---\|--- $\varepsilon^{0}(1)$ \| 89.554453 \| 89.551373 \| 89.548293 \| $(0,0,0)$ \| 89.548292 $\omega_{1}(2)$ \| 1.000000 \| 1.000000 \| 1.000000 \| $(1,0,0)$ \| 1.000000 \| 1.000000 \| 0.999792 \| 1.000000 \| $(0,1,0)$ \| 1.000000 $\omega_{2}(3)$ \| 1.791089 \| 1.788859 \| 1.788859 \| $(0,0,2)$ \| 1.788854 \| 1.791089 \| 1.791089 \| 1.788861 \| $(0,0,2)$ \| 1.788854 \| 1.791089 \| 1.791143 \| 1.788859 \| $(0,0,2)$ \| 1.788854 $\omega_{3}(3)$ \| n/a \| 2.000438 \| 2.000439 \| $(2,0,0)$ \| 2.000000 \| n/a \| n/a \| 2.000657 \| $(1,1,0)$ \| 2.000000 \| n/a \| n/a \| 2.000439 \| $(0,2,0)$ \| 2.000000 $\omega_{4}(4)$ \| 2.686634 \| 2.683292 \| 2.683226 \| $(0,0,3)$ \| 2.683282 \| 2.686634 \| 2.684452 \| 2.683226 \| $(0,0,3)$ \| 2.683282 \| 2.686634 \| 2.685519 \| 2.683312 \| $(0,0,3)$ \| 2.683282 \| 2.686634 \| 2.686634 \| 2.683312 \| $(0,0,3)$ \| 2.683282 $\omega_{5}(6)$ \| n/a \| 2.789191 \| 2.787088 \| $(1,0,2)$ \| 2.788854 \| n/a \| 2.791314 \| 2.787088 \| $(1,0,2)$ \| 2.788854 \| n/a \| 2.793711 \| 2.789270 \| $(1,0,2)$ \| 2.788854 \| n/a \| n/a \| 2.789270 \| $(0,1,2)$ \| 2.788854 \| n/a \| n/a \| 2.791536 \| $(0,1,2)$ \| 2.788854 \| n/a \| n/a \| 2.791535 \| $(0,1,2)$ \| 2.788854 Table 3: Benchmark of the LR-MCTDHB implementation to the isotropic HIM in 2D. Results are presented for $N=100$ bosons and different numbers of orbitals $M$. The trapping frequencies are $\Omega_{x}=\Omega_{y}=1.0$, and the interaction strength is $\lambda_{0}=-0.001$. The depletion is marginal with $f\approx 0.003\%$ for $M=2$ and $f\approx 0.006\%$ for $M=3$. Shown are the energies of the ground state $\varepsilon^{0}$ and the first $5$ excited states, $\omega_{i}=E_{i}-\varepsilon^{0}$, together with their multiplicities $\gamma$. In the BdG case, $M=1$, only the first pure c.m. excitation is found, together with all pure excitations of relative coordinates. For LR- MCTDHB(2), one obtains more c.m. excited states in the $x$-direction, together with combinations of excitations in the relative coordinates. For LR- MCTDHB(3), one obtains the missing c.m. excited states in the $y$-direction and their combinations with excitations of the relative coordinates. The overall accuracy increases with $M$, with few exceptions in the spectrum for $M=2$ which are discussed in the text. Underlined digits denote deviations from the exact values from Eqs. (160) and (161). All quantities are dimensionless. See text for more details. In contrast to the BdG approach, LR-MCTDHB(2) gives access to more excited states. Before discussing those, it is worth examining the two underlying ground-state orbitals. The first one is a Gaussian in the trap center, whereas the second one resembles a $p_{x}$-orbital. The degree of depletion is marginal with $f\approx 0.003\%$. In addition to the states obtained for $M=1$, the spectrum contains the second-order c.m. excitation in the $x$-direction, as well as three out of six states corresponding to $\omega_{5}$, which are excited states combining a first-order c.m. excitation in the $x$-direction and relative excitations. The accuracy of levels that were already accessible within the BdG approach has increased, e.g., for the first state of $\omega_{2}$ or the states of $\omega_{4}$. Nevertheless, some states have lost accuracy. The most remarkable example is the second first- order c.m. excitation of $\omega_{1}$, which is a c.m. excitation in the $y$-direction. Naturally the question arises why this is the case. The reason for this is that the second ground-state orbital, as discussed above, resembles a $p_{x}$-orbital. This means that the ground-state description for $M=2$ orbitals shows a preferred direction, which in this case is the direction along the $x$-axis. Apparently, the consequence of this is that excitations in the $y$-direction either lose in accuracy or remain inaccessible. A possible reason for the loss in accuracy is that, compared to the BdG case, a small amount of probability weight of the first orbital has been transferred to the second orbital that effectively only covers the $x$-direction. It is stressed that rotating the second orbital by an arbitrary angle, e.g., $\pi/4$ or $\pi/2$, leads to exactly the same spectrum, which means that it does not matter for the numerical results which direction is preferred. Including $M=3$ orbitals significantly enhances the obtained spectrum. One observes that both problems of the spectrum for $M=2$, i.e., the loss in accuracy for certain states as well as the absence of excitations in the low- energy spectrum, are solved. The third ground-state orbital resembles a $p_{y}$-orbital, i.e., it is perpendicular to the preferred direction of the $M=2$ ground state. One can deduce from this observations a fundamental difference between the computation of the ground state of a given $N$-boson system and its excited states. Whereas increasing the number of orbitals always leads to a lower ground state energy, it can happen that the spectrum for ($M+1$) orbitals contains states that are less accurate than for only $M$ orbitals. This is in particular likely to happen in 2D and 3D because preferred directions may occur. As a result, in order to obtain accurate excitation spectra, one has to pay much more attention to the symmetry of the problem in higher spatial dimensions. To summarize, it was demonstrated for the HIM that both the MCTDHB and LR- MCTDHB implementations utilized in this work are well-suited to obtain accurate results for the spectrum and dynamics of trapped interacting bosons. Furthermore, the many-body approaches clearly improve the GP and BdG mean- field methods. As mentioned above, further benchmarks are given in Appendix C. ## 6 Applications to many-body dynamics and excitations of BECs Since the advent of MCTDHB in 2007, the amount of scientific publications where it is utilized grew substantially. As one of the first applications, the splitting of 1D repulsive, fully-coherent BECs by a potential barrier has been investigated [74, 75], showing that it can lead to fragmented condensates. Similar observations were made by splitting a radially-symmetric BEC in 2D with a ring-shaped barrier [76]. Also the reverse is possible, meaning that two initially independent and thus fragmented BECs with no overlap in space can collide, interfere and build up coherence [77]. Also for attractive BECs, the splitting due to a Gaussian-shaped barrier results in the formation of a superposition of two distinct bosonic clouds, a so-called caton, which is not describable at the GP level [78]. Moreover, it has been shown for the first time that coherent attractive BECs in 1D can fragment even without a potential barrier, namely when the energy exceeds a threshold value [79]. Fragmentation persists and may even intensify when more orbitals are included in the calculation [80]. In the attractive case full convergence is difficult to achieve as it requires more orbitals, see also [81]. When an attractive BEC propagates towards a potential barrier and scatters from it, the system also evolves to be highly fragmented although it was initially condensed [82, 83]. The degree of fragmentation depends on the degree of transmission through the barrier. Even bright-soliton trains, i.e., stable multi-hump matter waves of attractive bosons, whose dynamics were believed to be fully describable at the mean-field level, are shown to lose their initial coherence [84]. The splitting of BECs has been further investigated with respect to the generation of number- or phase-squeezed states applying optimal control theory [85, 86, 87, 89, 88]. Optimal control has also been utilized in 1D bosonic Josephson junctions (BJJ), on the one hand to steer the system parameters such that an enhancement of the Shapiro effect was observed [90], and on the other hand to drive a BEC from an initial state to a target state at the quantum speed limit [91]. Moreover, it has been demonstrated that Mach-Zehnder interferometry with ultracold bosons is relatively robust against the nonlinear interaction between the particles [92]. A remarkable correspondence between the onset of wave chaos at the GP level and the onset of fragmentation at the many-body level has been found for BECs that scatter from either shallow periodic or disordered potential landscapes [93, 94], but the result holds also for other potentials. Thus, the development of wave chaos, i.e., the exponential separation of initially close states, can be seen as an indication that a many-body treatment is necessary. The distance between states is measured utilizing the $L^{2}$ norm between two Hilbert space vectors. Of potentially high relevance is the demonstration how the positions of individual particles can be constructed from the many-body wave function by simulating single shots [95]. The latter are connected to experiments on trapped BECs where typically an image of the density represents a histogram of a single shot. Over the last years, MCTDHB has been applied to a growing number of systems, e.g., to study the breathing dynamics in 1D harmonic traps due to a quantum quench [96] or the many-body effects in solitonic excitations of ultracold BECs [97, 98]. Other examples are the correlated dynamics of a single atom coupled to a trapped BEC by collisions [99] or the examination of the structure of mesoscopic molecular ions [100], which contributes to the currently very active research in this field. Recently, MCTDHB has also been applied to dipolar BECs in a double well [101] and in 1D lattices [102, 103]. In addition, the coupling of one- and two- component BECs in a cavity to a radiation field has been of interest [104, 105, 106]. One of the main results of the latter research was the finding of a fragmented superradiant phase when the pump power of the laser exceeds a critical threshold. This fragmented superradiance cannot be explained by the Dicke model which assumes the BEC to be a simple two-level system. In the remaining part of this section, the focus is laid on several applications where the tunneling dynamics in traps (Section 6.1) and the dynamical fragmentation of initially coherent BECs (Section 6.2) is investigated, both in 1D and 2D systems. Afterwards, all applications of LR- MCTDHB that exist up to now are presented in detail (Section 6.3). Special emphasis for the 2D systems under consideration is laid on the impact of angular momentum on the appearance of many-body effects in the dynamics and excitation spectra. ### 6.1 Tunneling dynamics in traps Subsequently, the tunneling dynamics of BECs held in double-well traps in both one and two spatial dimensions are discussed in this section. The main purpose is to apply MCTDHB to solve the many-boson Schrödinger equation and to discover novel phenomena that are not captured by the standard methods. #### 6.1.1 Exact tunneling dynamics in a bosonic Josephson junction The quantum dynamics of a tunneling BEC in a 1D BJJ have constituted a large research field in the past decades. Before the first experimental realizations [107, 108], theoretical predictions concerning Josephson oscillations and self-trapping, i.e., the suppression of tunneling between the wells, were obtained by a (multi-mode) GP description [109, 110, 111, 112, 113, 114], a path-integral formulation [115], a two-mode Bose-Hubbard (BH) approach [116] or a Fock-space WKB method [117]. A review of various theoretical and experimental results is given in [118]. A more recent study dealt with the tunneling dynamics in a controlled BJJ, where the spin state of an ionic impurity controls the tunneling between the wells [119]. The latter extends the findings on the dynamics of trapped ultracold BECs due to the presence of an ion [120, 121]. Furthermore, the orbital Josephson effect due to a time- dependent driving potential was investigated numerically by employing MCTDHB, not only in a double-well system [122]. The main objective of this section is however to study the out-of-equilibrium tunneling dynamics of a single-component BEC in a BJJ from a many-body perspective, and the method of choice is MCTDHB. As described below, the latter approach allows for the observation of fundamentally new physics dealing with, e.g., reduced self-trapping and dynamical fragmentation of initially fully-coherent bosonic clouds. A comparison to other popular theoretical tools like the GP approach and the BH model is made. The findings presented below were recently published in Refs. [123, 124, 125, 126], where further details can be found. Recently, results of a comparable study where both the BH model and MCTDHB were used to analyze the tunneling of few-boson systems in asymmetric double-wells, investigating the interaction blockade that isolates the motion of a single particle in the vicinity of others, became available [127]. The trapping confinement considered here reflects two harmonic wells of the form $V_{\pm}(x)=\frac{1}{2}(x\pm 2)^{2}$ (172) which are connected by a cubic spline in the interval $\|x\|\leq 0.5$ that generates a potential barrier between the wells. The plus sign refers to the left well, whereas the minus sign refers to the right well. The repulsion between the bosons is modeled by the contact interaction potential, $\lambda_{0}\hat{W}(x,x^{\prime})=\lambda_{0}\delta(x-x^{\prime})$. The repulsion strength is given by the mean-field parameter $\Lambda=\lambda_{0}(N-1)$. An important quantity with respect to the tunneling dynamics is the occupation probability of the left and right wells, where the former is given by $P_{L}(t)=\frac{1}{N}\int_{-\infty}^{0}\rho(x,t)dx$ (173) where $\rho(x,t)$ describes the time-dependent one-body density, see Eq. (2.4), and $N$ refers to the number of particles in the trap. Since the MCTDHB results are compared to the BH model below, it is instructive to define the left- and right-well orbitals, $\phi_{L}$ and $\phi_{R}$, with the ground and the first excited state of the trap, denoted by $\phi_{g}$ and $\phi_{u}$ due to their gerade and ungerade symmetry. These two states are energetically well below the barrier. The orbitals $\phi_{L}$ and $\phi_{R}$ are defined via $\phi_{L,R}(x)=\frac{\phi_{g}(x)\pm\phi_{u}(x)}{\sqrt{2}}.$ (174) The BH parameters for on-site interaction $U$ and hopping $J$ are derived from $\phi_{L}$ and $\phi_{R}$ by $U=\lambda_{0}\int\|\phi_{L}(x)\|^{4}\,dx,\quad J=-\int\phi_{L}^{\ast}(x)\hat{h}(x)\phi_{R}(x)\,dx$ (175) and utilized to define the BH interaction parameter $\Lambda_{\text{BH}}=UN/(2J)$. The two-site BH Hamiltonian then reads $\hat{H}_{\text{BH}}=-J(\hat{b}_{L}^{\dagger}\hat{b}_{R}+\hat{b}_{R}^{\dagger}\hat{b}_{L})+\frac{U}{2}(\hat{b}_{L}^{\dagger}\hat{b}_{L}^{\dagger}\hat{b}_{L}\hat{b}_{L}+\hat{b}_{R}^{\dagger}\hat{b}_{R}^{\dagger}\hat{b}_{R}\hat{b}_{R})$ (176) where the operators $\hat{b}_{L,R}^{(\dagger)}$ annihilate (create) a boson in the left and right well, respectively. Within the two-mode BH approach, the eigenvalues and eigenfunctions of the one-body RDM $\rho^{(1)}=\begin{pmatrix}\rho_{LL}&\rho_{LR}\\\ \rho_{RL}&\rho_{RR}\end{pmatrix}$ (177) with $\rho_{LL}=\langle N,0\|e^{+i\hat{H}_{\text{BH}}t}\,\hat{b}^{\dagger}_{L}\hat{b}_{L}\,e^{-i\hat{H}_{\text{BH}}t}\|N,0\rangle$ (with $\hbar=1$) and the other elements defined analogously, represent the natural occupations and natural orbitals of the BH method. Therefore, the population of the second natural orbital can be used as a measure for fragmentation. In the following, the tunneling dynamics when the BEC is initially trapped in the left well, i.e., $P_{L}(0)=1$, will be explored for different repulsion strengths. Most importantly, for a two-mode GP description of the system, self-trapping of the bosons is predicted for $\Lambda_{\text{BH}}>\Lambda_{c}=2$ when the magnitude of the initial imbalance between the occupations of the two wells, denoted by $z=\frac{N_{L}-N_{R}}{N}$, is unity [109, 110, 111]. Figure 5: Short-time dynamics of the occupation probability of the left well, $P_{L}(t)$, for (a) $N=20$, $\Lambda=0.152$, (b) $N=100$, $\Lambda=0.152$ ($\Lambda_{\text{BH}}<\Lambda_{c}$) (c) $N=20$, $\Lambda=0.245$, and (d) $N=100$, $\Lambda=0.245$ ($\Lambda_{\text{BH}}>\Lambda_{c}$). The GP [dotted black], BH [dashed magenta] and numerically exact results from MCTDHB($M$) [solid blue] are compared. For the weak interaction in panels (a) and (b), the GP theory predicts Rabi oscillations of the full density. At the many-body level, a collapse of these oscillations occurs after approximately three cycles. For stronger interaction in panels (c) and (d), GP still predicts that all bosons tunnel between the wells, whereas at the many-body level tunneling is clearly suppressed. However, MCTDHB and the BH model yield different dynamics in all cases. The deviations become more pronounced for stronger repulsion. The inset shows the numerical convergence with respect to $M$ for 2, 4, 6 and 8 orbitals [(a) and (c)] and 2 and 4 orbitals [(b) and (d)]. All quantities are dimensionless. See text for details. The figure is taken from Ref. [123]. Figure 6: Upper panel: Dynamics of $P_{L}(t)$ for very strong repulsion. The interaction parameter $\Lambda_{\text{BH}}=47.8$ ($43.4$) for $N=10$ [solid blue] ($N=100$, solid green) is clearly above the critical value of $\Lambda_{c}=2$. While the BH predicts full self-trapping of the bosons [dashed magenta], the exact results show tunneling between the wells. Lower panel: Evolution of the natural occupations $n_{i}^{(1)}(t)$. At the BH level, the system remains condensed throughout the propagation time, while it fragments at the exact many-body level where essentially four orbitals become macroscopically occupied. All quantities are dimensionless. See text for details. The figure is adapted from Ref. [123]. Figure 7: BH versus exact dynamics for both attractive and repulsive BECs with system parameters $N=20$, $\frac{\|U\|}{J}=0.226$ and $\|\lambda_{0}\|=0.0129$. Left panel: Occupation probability $P_{L}(t)$ of the left well. While the BH predicts the same evolution for attraction and repulsion, differences can be observed at the exact many-body level. Inset: Time-evolution of the ratio of $P_{L}(t)$ for attraction and repulsion. Right panel: The same for the first two natural occupation numbers $n_{i}^{(1)}(t)$. Again, the BH model predicts equivalent dynamics for attraction and repulsion, whereas the exact results show clear differences. All quantities are dimensionless. See text for details. The figures are taken from Ref. [124]. Fig. 5 shows a comparison between the short-time dynamics of $P_{L}(t)$ for BECs with $N=20$ ($100$) bosons obtained from MCTDHB (solid blue), BH (dashed magenta) and GP (dotted black) for two different interaction strengths where one is below and one is above the critical value $\Lambda_{c}$. One observes that for the case of $20$ bosons with weak interaction parameter $\Lambda_{\text{BH}}<\Lambda_{c}$ [panel (a)], the GP prediction is totally different than the BH and numerically exact MCTDHB predictions. According to GP, Rabi oscillations of the entire cloud between the wells occur, whereas the BH model and MCTDHB predict a density collapse after approximately three Rabi cycles, resulting in roughly $50\%$ of the bosons in each well. The time for one Rabi cycle is close to the theoretical prediction of $t_{\text{Rabi}}=\pi/J$. Although the BH curve is clearly superior to the GP curve, it also deviates from the numerically exact result after half a Rabi cycle. In the case of $100$ bosons [panel (b)], similar observations can be made. However, the density collapse occurs after a longer time. As mentioned above, for stronger repulsion with $\Lambda_{\text{BH}}>\Lambda_{c}$, tunneling should be entirely suppressed due to self-trapping according to a two-mode GP description. It can be deduced from the exact dynamics of $20$ bosons [panel (c)] that indeed less bosons tunnel between the wells, meaning that the oscillation amplitude is clearly smaller than for the weaker repulsion. As predicted by MCTDHB, at most $50$ bosons tunnel into the right well after half a Rabi cycle, and even less in the BH dynamics. Nevertheless, tunneling is not completely suppressed. This means that the effect of self-trapping, as predicted by the two-mode GP theory, is obviously reduced at the accurate many-body level. The deviations between the MCTDHB and BH results are more pronounced for stronger than for weaker repulsion. The dynamics of $N=100$ bosons yields similar observations [panel (d)]. In contrast to that, the GP theory predicts tunneling of the whole cloud. The reason why GP fails to describe the dynamics correctly is the development of fragmentation, already on the short time scales shown. For the case of weak repulsion in the upper panels of Fig. 5, the initially condensed BEC with $f=10^{-4}$ ($10^{-5}$) for $N=20$ ($100$) evolves to be fragmented by $33\%$ ($26\%$) after three Rabi cycles. Interestingly, for stronger repulsion, the degree of fragmentation of the again initially fully-coherent BEC is less after three Rabi cycles than for the case of weaker repulsion, yielding approximately $28\%$ and $18\%$ for $20$ and $100$ bosons, respectively. The predicted GP dynamics differ both quantitatively and qualitatively from the exact results. Most important, for all system parameters used in Fig. 5, GP predicts that essentially all bosons tunnel back and forth between the wells. This is clearly not the case at the many-body level, neither for BH nor for MCTDHB. It is stressed that although the chosen system parameters are within the expected regime of validity for both the GP and BH approaches, both theories fail to describe the short-time dynamics of the BECs. To further illustrate the effect of reduced self-trapping as well as the degree of fragmentation, Fig. 6 shows the evolution of $P_{L}(t)$ and of the occupation of the first few natural orbitals $n_{i}^{(1)}(t)$, both for MCTDHB and the BH model. The utilized repulsion parameter exceeds the critical value significantly with $\Lambda_{\text{BH}}=47.8\,(43.4)$ for $N=10\,(100)$. From the top panel, it can be inferred that the BH model predicts complete self- trapping for both values of $N$, meaning that all bosons remain in the left well throughout the time-evolution. On the contrary, the predictions of MCTDHB show that indeed bosons tunnel from left to right, and one can anticipate that $P_{L}(t)$ approaches the long-time average of $0.5$. With respect to the natural occupations, one observes that the BEC remains fully-condensed in time at the BH level, while the MCTDHB results show that fragmentation sets in already during the first Rabi cycle. Moreover, more than ten orbitals are necessary to accurately describe the dynamics. The different results are associated with a quick loss of coherence between the bosons that is only described by MCTDHB and not at the BH level. In this context, see Ref. [123] for more details. Another interesting aspect is the symmetry of the two-mode BH Hamiltonian in Eq. (176) under the unitary transformation $\hat{R}=\\{\hat{b}_{L}\rightarrow\hat{b}_{L},\hat{b}_{R}\rightarrow-\hat{b}_{R}\\}$, leading to $\hat{R}\hat{H}_{\text{BH}}(U)\hat{R}=-\hat{H}_{\text{BH}}(-U)$. A consequence of this is the equivalence of the occupation of the wells for attractive and repulsive interactions, i.e., $P_{L,R}(t;U)=P_{L,R}(t;-U).$ (178) Moreover, it can be shown that the eigenvalues of the BH one-body RDM in Eq. (177) do not depend on the sign of the interaction. Thus, $P_{L}(t)$ and the occupations $n_{i}^{(1)}(t)$ will be the same for attraction and repulsion of the same magnitude. Naturally, the full many-body Hamiltonian of the system does not possess such a symmetry, and one can therefore expect different dynamics for attraction and repulsion between the bosons. The left panel of Fig. 7 shows the short-time dynamics of $p_{L}(t)$ for a BEC with $N=20$ bosons with interaction parameter $\|U\|/J=0.226$, corresponding to $\|\lambda_{0}\|=0.0129$. As anticipated, the evolution at the BH level is identical for attraction and repulsion. However, the results obtained by MCTDHB show differences between the cases of positive and negative $\lambda_{0}$. On the time scale shown, the BH model mainly underestimates tunneling for the repulsive case, whereas it overestimates tunneling for the attractive case. The right panel of Fig. 7 shows the evolution of the first few natural occupation numbers for the same system as in the left panel, both at the BH and full many-body level. As expected, the BH model predicts the same evolution for $n_{1}^{(1)}(t)$ and $n_{2}^{(1)}(t)$, irrespective of the sign of the interaction parameter. In contrast to that, the results obtained from MCTDHB show that the evolution of the different occupations are distinct for the attractive and repulsive cases. Compared to the exact results, the BH model mainly overestimates the fragmentation of the attractive BEC, whereas it mainly underestimates it for the repulsive BEC. Figure 8: Universality of the degree of fragmentation. Upper panel: Evolution of $P_{L}(t)$ for different numbers of bosons ($N=100$ in blue, $N=1000$ in red and $N=10000$ in orange) where the interaction parameter $\Lambda=\lambda_{0}(N-1)=0.152$ is kept fixed. Intermediate particle numbers ($N=200,\,500,\,2000$ and $5000$) are shown in gray. The less particles are contained in the BEC, the faster the density collapses. Lower panel: Evolution of the first two natural occupations. Irrespective of the number of bosons, the degree of fragmentation upon the density collapse is the same. All quantities are dimensionless. See text for details. The figure is taken from [126]. The analysis of the dynamics in a 1D BJJ is closed by discussing the compelling feature of universal fragmentation. The latter means that all BECs with identical interaction parameter $\Lambda$ fragment to the same value after the density has collapsed. Fig. 8 shows exactly this behavior for $\Lambda=0.152$ and BECs with $N=100$, $1000$, and $10000$ particles where initially all bosons are kept in the left well, i.e., $P_{L}(0)=1$. From the upper panel, one sees that the density oscillations between the wells collapse after a few Rabi cycles, leading to the equal distribution of $50\%$ of the bosons in each well. In general, the less bosons are contained in the BEC, the faster the density collapses. Coming along with the damping of the oscillations, the BEC starts to fragment, reaching its maximal value after the collapse. Again, the less bosons are in the system, the faster the system fragments. Most importantly, irrespective of $N$, the final degree of fragmentation is the same. The fact that the degree of fragmentation appears to be universal is unexpected, since usually fragmentation strongly depends on the number of particles in the BEC. Moreover, the degree of fragmentation upon the density collapse, denoted by $f_{\text{col}}$, depends highly on the initial imbalance $z$ of bosons in the left and right wells. The larger $z$, the stronger does the system fragment in time. This holds true even in the limit of very weak interactions, $\Lambda\ll 1$, meaning that there is no weak-interaction regime where GP yields the correct dynamics. It is important to note that the universality is not an artifact of any approximation, since the result is obtained by solving the full many-boson Schrödinger equation numerically exact. Interestingly, the two-mode BH approach can be used to derive an analytic prediction of $f_{\text{col}}$ yielding results close to the exact ones obtained by MCTDHB. Additional details on the analytic expression can be found in Ref. [126]. To summarize, it was demonstrated that the out-of-equilibrium tunneling dynamics of BECs in a 1D BJJ show significant many-body features that are not accounted for at the GP mean-field level. This includes the appearance of a density collapse in the oscillations between the wells as well as dynamical fragmentation which the GP theory cannot capture by construction. The degree of fragmentation is universal for fixed $\Lambda=\lambda_{0}(N-1)$ for a wide range of initial states. Applying the commonly used BH model, it was shown that it is superior in certain aspects in comparison to the GP mean-field approach. However, it does also not predict the numerically exact tunneling dynamics as obtained by MCTDHB, e.g., on the short-time dynamics where the interaction parameter is in the vicinity of its critical value or for very strong repulsion where it predicts complete self-trapping of the bosonic cloud. However, the BH model can be utilized to derive an analytic expression for the universal degree of fragmentation that is in good agreement with the exact results. #### 6.1.2 Many-body tunneling dynamics in a two-dimensional radial double well After having shown the many-body nature of the tunneling dynamics of BECs in a 1D BJJ, the question of how a repulsive BEC behaves in a radially symmetric 2D system with a double-well structure is addressed. In particular, as for the previous section, the out-of-equilibrium tunneling dynamics are of interest, and it is studied whether many-body effects can be observed. The results presented in this section were published in Ref. [128], and additional details can be found therein. Figure 9: (a) Schematic plot of the radial double well. The IN region denotes the trap center, the OUT region denotes the outer annulus. Both are separated by the central ring-shaped barrier. (b) Ground-state energy per particle of a BEC with $N=100$ bosons and total angular momentum $L=0$ for different positions $R_{B}$ of the radial barrier. The results are calculated at the mean-field GP level ($M=1$). The quantities $E_{0}^{\text{IN}}$ (short-dashed red) and $E_{0}^{\text{OUT}}$ (solid green) denote the energies for the IN and OUT regions for interaction strength $\Lambda=0$, whereas $E_{2}^{\text{IN}}$ (dash-double-dotted blue) and $E_{2}^{\text{OUT}}$ (dash-dotted magenta) denote the energies for $\Lambda=2$. The crossing points, i.e., the radii at which the energies for IN and OUT are alike, are located at $R_{1}=3.271$ ($\Lambda=0$) and $R_{2}=3.412$ ($\Lambda=2$), see the vertical black lines. Larger repulsion between the bosons shifts the crossing point to larger radii. (c) Same analysis as in (b) but for total angular momentum $L=N$, meaning that each particle carries on average one unit of angular momentum. The location of the crossing points are $R_{3}=4.089$ ($\Lambda=0$) and $R_{4}=4.093$ ($\Lambda=0.2$). The angular momentum has a significant effect on the position of the crossing points. Increasing $L$ shifts them to larger radii, which is qualitatively similar as for increasing $\Lambda$. However, one would need to highly increase the interaction strength in order to shift the crossing points as much as adding one unit of angular momentum per particle does. All quantities are dimensionless. See text for details. Figs. (b) and (c) are adapted from Ref. [128]. To study tunneling phenomena in 2D geometries has been of high interest in recent years, especially for trapped vortices, i.e., macroscopic quantum states with definite angular momentum, characterized by a density node at its core and phase discontinuities around it. Research concerning tunneling of vortices has been carried out for example in 2D superfluids [129]. Others dealt with vortices tunneling through a Gaussian-shaped barrier [130], in pinning potentials [131, 132, 133], or between two Gaussian wells [134], to mention only a few. However, there has been no investigation of the tunneling dynamics of vortices in a radially-symmetric double well, in particular not at the many-body level. The latter geometry is very interesting to study many- body dynamics since the trap symmetry preserves the angular momentum of the bosonic cloud. It has been utilized recently to study many-body vortices which are separated in space due to the central barrier [135]. The following results, recently published in Ref. [128], especially deal with the impact of angular momentum on the tunneling dynamics. Accurate many-body dynamics are obtained by solving the full many-body Schrödinger equation utilizing MCTDHB and compared to the corresponding mean- field dynamics from the GP theory. For the dynamics, the radial double well in which a BEC with $N=100$ spinless bosons is confined reads $V(r)=\begin{cases}B\,e^{-2(r-R_{B})^{4}}+C\,e^{-0.5\,(r-R_{C})^{4}}\,\,&\mbox{if }r=(x^{2}+y^{2})^{1/2}\leq R_{C}\\\ C\,\,&\mbox{if }r>R_{C}\end{cases}$ (179) where $B$ is the height of the central ring-shaped barrier located at the radius $R_{B}$, and $C$ is the height of the crater wall at radius $R_{C}$. A schematic illustration of $V$ is shown in Fig. 9(a). The region enclosed by the barrier is denoted by IN, the external rim, i.e., the region between the barrier and the crater wall, is denoted by OUT. For the numerical calculations, $C=200.0$ and $B=1.0$ are set which ensures on the one hand that the energy per particle is sufficiently small such that the BEC stays trapped in the crater, and on the other hand allows for tunneling between IN and OUT, as described below. The radius of the crater is set to $R_{C}=9.0$. The short-range repulsion between the particles is modeled by a Gaussian that has been employed recently [76, 136], $\lambda_{0}\hat{W}(\mathbf{r}_{i},\mathbf{r}_{j})=\frac{\lambda_{0}}{2\pi\sigma^{2}}\,e^{-\|\mathbf{r}_{i}-\mathbf{r}_{j}\|^{2}/2\sigma^{2}},$ (180) which circumvents the regularization problems of the contact interaction [137]. The repulsion strength will again be expressed in terms of the mean- field parameter $\Lambda=\lambda_{0}(N-1)$ in the following. The effect of the range of the interaction on the degree of fragmentation has been investigated in detail for repulsive BECs in 1D and 2D single wells [138], essentially showing that it becomes increasingly dependent on the density the more long- ranged it is. The analysis starts with the ground-state energy of two distinct cases, namely for (i) having all particles in the IN region with closed external rim and (ii) having all particles in the OUT region with closed trap center. The notation $E^{\text{IN}}_{\Lambda}$ and $E^{\text{OUT}}_{\Lambda}$, where the superscript denotes whether the BEC is held in the IN or OUT region and the subscript denotes the repulsion strength $\Lambda$, is utilized. The ground- state energies for different radial positions of the barrier $R_{B}$ are calculated using the GP equation. The corresponding results are shown in Fig. 9(b) for total angular momentum $L=0$ and Fig. 9(c) for $L=N$, i.e., where each boson on average carries one unit of angular momentum. One observes that for all shown parameters ($\Lambda=0,\,2.0$ for $L=0$ and $\Lambda=0,\,0.2$ for $L=N$) the curves for $E^{\text{IN}}_{\Lambda}$ and $E^{\text{OUT}}_{\Lambda}$ intersect at certain radii which are referred to as the crossing points or simply the crossings in the following. These are defined as the positions where the ground-state energies of the BEC in the IN and OUT regions are identical. Putting the barrier onto the crossing point is the 2D analogue of the symmetric double well in 1D. However, the IN and OUT regions are only energetically equivalent, whereas their geometries are totally different. It is found that both the interaction strength $\Lambda$ and the angular momentum $L$ push the crossing point to larger radii once they are increased. Though, the effect of $L$ is significantly stronger since already for $\Lambda=0$, the transition from $L=0$ to $L/N=1$ leads to a shift of the crossing point from $R_{B}=3.271$ to $R_{B}=4.089$. In order to obtain a similar shift just by increasing $\Lambda$, the repulsion needs to be largely increased, and thus one would certainly leave the weakly-interacting regime. The depletion of the ground states with $M=4$ orbitals are $f\approx 10^{-3}$ for $L=0$ and $\Lambda=2.0$ and $f\approx 10^{-9}$ for $L=N$ and $\Lambda=0.2$, respectively. This means that in both cases, the systems’ ground states are essentially condensed. (a) (b) (c) (d) Figure 10: (a) Time evolution of the occupation probability of the external rim, $P_{\text{OUT}}(t)$, for $N=100$ bosons with $L=0$ and $\Lambda=2.0$. For $M=4$, initially almost all bosons tunnel. The oscillation amplitude is damped over time and effectively collapses after 12 Rabi cycles with approximately half of the bosons in each part (solid red). The system evolves from being almost fully condensed to two-fold fragmented (see solid green and dark blue curves for $n_{1}$ and $n_{2}$). The occupations of the third and fourth orbital are identical and negligible (solid light blue). On the contrary, the GP theory predicts undamped oscillations (dotted gray). (b) Same as in (a) but for $L=N$ and $\Lambda=0.2$. Again, the system evolves from being fully coherent to two-fold fragmented, and the particles finally occupy both the IN and OUT parts by roughly 50$\%$. Mind the different time scales ($\tau_{1}>\tau_{2}$). Inset: Detailed view between $150\tau_{2}\leq t\leq 155\tau_{2}$. The GP dynamics again do not show the density collapse. (c) Real, imaginary, and absolute value of the first two natural orbitals $\alpha_{1}$ and $\alpha_{2}$ at $t=18.4\,\tau_{1}$ where the density is already collapsed. The GP case is shown in the top panels. For $\alpha_{1}$ and $\alpha_{2}$, the density occupies both the IN and OUT regions. At the same time, the GP orbital only covers the external part. (d) Same as in (c) but for the dynamics described in (b). At the many-body level, the nodal structure of the vortex state is clearly visible in the trap center. All quantities are dimensionless. See text for details. All figures are adapted from Ref. [128]. To study the out-of-equilibrium dynamics for different values of $\Lambda$ and $L$, the focus is laid on the situation where the barrier is located on the corresponding crossing points. First, the case of zero angular momentum, i.e., $L=0$, is analyzed. As an initial state, the ground state of trapping the BEC in the OUT region with closed central part is chosen. To trigger the dynamics, the system is quenched by suddenly opening the trap center, such that the entire potential is given by Eq. (179). It was found numerically that there is no difference in starting from either the IN or OUT region when the barrier is located at the crossing point. Fig. 10(a) shows the dynamical evolution of the occupation probability of the external rim, $P_{\text{OUT}}(t)=1-P_{\text{IN}}(t)=\frac{1}{N}\int_{R_{B}<r\leq R_{C}}\rho(\mathbf{r};t)d\mathbf{r},$ (181) where $\rho(\mathbf{r},t)$ denotes the single-particle density. For the first two Rabi cycles of $\tau_{1}=110.23$, nearly the whole cloud tunnels between IN and OUT. Then, the occupation of the second natural orbital starts to grow, accompanied by a damping of the tunneling oscillations. After approximately 12 cycles, the density oscillations have collapsed and the particles are almost equally distributed between IN and OUT. The BEC is now highly (two-fold) fragmented, with occupations of 56.7% and 41.1% of the first two natural orbitals $\alpha_{1}$ and $\alpha_{2}$, respectively. In contrast to that, the GP dynamics do not show any damping of the oscillations, not even for the long-time behavior (not shown). Fig. 10(c) shows the real and imaginary parts as well as the density of the GP orbital (top panels) and of the first two natural orbitals in the many-body case (middle and lower panels) at an instant in time at which the density oscillations have already collapsed. Whereas there are no bosons in the trap center for the GP case, there is significant population of the IN region at the many-body level. Similar observations can be made for $L=N$ and $\Lambda=0.2$, presented in Fig. 10(b). The many-body dynamics show that the density collapses and that approximately 50 bosons are occupying the IN and OUT regions finally. The BEC is two-fold fragmented, the occupation of higher natural orbitals is negligible. The GP dynamics (see inset) do not account for the damped oscillations. With regard to the density of the GP and natural orbitals in Fig. 10(d), it can be seen that $\alpha_{1}$ and $\alpha_{2}$ clearly show the characteristic nodal vortex structure in the center, whereas at the GP level there are no bosons in the IN region at the chosen instant of time $t=183.3\,\tau_{2}$ with $\tau_{2}=66.79$. It was further found that many-body effects like the density collapse or the onset of fragmentation are more likely to occur when the particles carry angular momentum. For $L=N$ and $\Lambda=2.0$, the system quickly becomes four-fold fragmented, i.e., at least four orbitals are necessary to account for the many-body dynamics. For $L=0$, the dynamics are essentially governed by only two orbitals, as can be seen from Fig. 10(c). Thus, the weakly- interacting regime is apparently smaller for the case of $L>0$, meaning that vortices are more sensitive to many-body effects. In Section 6.3.3, this will be discussed again when excitations in rotating BECs are investigated. To summarize, the many-body nature of the tunneling dynamics of BECs in a 2D radial double well was demonstrated . The GP mean-field theory does not account for the collapse of the density oscillations between IN and OUT, and therefore fails to give an accurate description of the dynamics. Furthermore, the initially condensed BECs fragment. It was found that bosons carrying angular momentum are more sensitive to many-body effects than non-rotating ones because fragmentation sets in already for weaker interaction strengths. In agreement with these findings, a recent work demonstrates analytically that a many-body approach is necessary for describing the conservation of angular momentum in a 2D radially-symmetric system [139]. ### 6.2 Dynamical fragmentation #### 6.2.1 Tunneling to open space The obtained findings on the many-body tunneling process of initially trapped bosons to open space in 1D are summarized in this section. The tunneling process of a single particle was understood already at the beginning of the 20th century [140, 141, 142, 143] and is still of current interest, e.g., with respect to the exit time and momentum of an electron in the ionization process [144, 145]. Tunneling of a many-boson system, which has also been studied [146, 147, 148, 149], is naturally more complicated due to the interactions and correlations between the constituent particles. The following results, published in Refs. [150, 151, 152, 153, 154], explain the tunneling mechanism to open space from a many-body point-of-view utilizing MCTDHB. The main points that are addressed deal with the questions whether the particles tunnel one by one or if several particles tunnel simultaneously, and whether the entire process is indeed of many-body nature. To study the dynamics of the tunneling to open space, a system of $N$ bosons confined in a harmonic trap of the form $V(x,t=0)=\frac{1}{2}x^{2}$ is considered. For $t>0$, the trap is quenched such that it becomes open to one side. The region where the bosons are trapped initially and the open space are separated by a potential barrier. The full potential reads $V(x,t>0)=\theta(x_{c1}-x)\frac{1}{2}x^{2}+\theta(x-x_{c1})\theta(x_{c2}-x)P(x)+\theta(x-x_{c2})T$ (182) where $\theta(x)$ denotes the Heaviside step function, $P(x)$ denotes a third order polynomial that ensures a smooth connection between the harmonic trap, the potential barrier and the threshold potential $T$ whose role will be discussed below. The coordinates $x_{c1}$ and $x_{c2}$ refer to connection points, where the former connects the harmonic trap and the barrier and the latter connects the barrier and the threshold. The bosons repel each other via the contact interaction potential, and its strength is measured via the mean- field parameter $\Lambda=\lambda_{0}(N-1)$. Fig. 11 shows a schematic plot of the trapping potential for $t>0$ (solid black). The maximum height of the barrier separates the IN and OUT regions, denoted by the vertical red line. Moreover, the colored horizontal lines mark the total energies of states with different numbers of bosons $N$ in the trap. Depending on the threshold value $T$, certain states can become bound, meaning that no additional particles can escape from the trap via tunneling through the barrier. How the threshold can be utilized to control the tunneling-to- open-space dynamics will be discussed below. Figure 11: Schematic plot of the potential $V$ for $t>0$ (solid black). The initial Gaussian-shaped density $\rho(x,t=0)$ is shown in solid blue. The IN and OUT regions are separated at the position of the maximum of the potential barrier (see solid red lines and labels). Colored horizontal lines denote the energies of states with different numbers of bosons $N$ in the trap. The energy differences between those states are denoted by the chemical potentials $\mu_{i}$. The value $T$ denotes the height of the threshold potential in the OUT region. All quantities are dimensionless. See text for details. The figure is taken from Ref. [153]. Important quantities for the analysis are the density $\rho(x,t)$ [see Eq. (2.4)], as well as the density in momentum space $\rho(k,t)$, or simply the momentum distribution, which is obtained by performing a Fourier transformation (FT) of the one-particle reduced density in real space. Furthermore, the degree of fragmentation $f$, defined in Eq. (29), and especially its evolution in time, is of major interest. The above quantities are computed at the many-body level employing MCTDHB. The analysis starts by looking onto the shape of the momentum distribution in general. Fig. 12 shows $\rho(k,t)$ for a system of $N=101$ bosons with interaction parameter $\Lambda=0.3$ at four different instants in time $t_{1}<t_{2}<t_{3}<t_{4}$. It consists of two parts, a Gaussian background and a peak structure. The former refers to the particles that did not escape from the trap yet, since the initial real-space density is also a Gaussian. On the contrary, the peaks refer to the momenta of the already escaped particles. The heights of the peaks are time-dependent, which can be seen from the growth of the dominant peak. The position of the peaks can be understood from a static model introduced below. Figure 12: The momentum distribution $\rho(k,t)$ for $N=101$ bosons with repulsion strength $\Lambda=0.3$ at four instants in time with $t_{1}<t_{2}<t_{3}<t_{4}$. The Gaussian background refers to the bosons remaining in the trap, whereas the peaks refer to the momenta of the emitted bosons. The height of the dominant peak is clearly growing in time. All quantities are dimensionless. See text for details. The figure is adapted from Ref. [152]. The minimal total energy $E_{\text{TOT}}$ of the system is comprised of the energy of the bosons that remain in the IN region, denoted by $E_{\text{HO}}(\lambda_{0},N_{\text{IN}})$ indicating its dependence on the two-body repulsion strength $\lambda_{0}$ and the number of particles in the trap $N_{\text{IN}}$, as well as on the minimal energy of the $N_{\text{OUT}}$ bosons that tunneled through the barrier into open space. Their minimal energy is the height of the threshold $T$. One thus obtains $E_{\text{TOT}}=E_{\text{HO}}(\lambda_{0},N_{\text{IN}})+N_{\text{OUT}}\,T.$ (183) Assuming that the interaction among the emitted bosons is negligible, the kinetic energy of the $i$-th escaped boson is given by $E_{\text{kin}}(T,\mu_{i})=\mu_{i}-T$ (184) where the $\\{\mu_{i}\\}$ are chemical potentials that refer to the energy to bring an additional boson into the IN region with already $N-i$ bosons. The corresponding momenta read $k_{i}^{T}=\sqrt{2(\mu_{i}-T)}$ (185) where $m=\hbar=1$ is assumed. It is stressed that this model describes the tunneling of a single boson. Figure 13: The tunneling to open space of $N=2$ bosons. Left panel: Static consideration of the energy dependence of the possible final states $\|N_{\text{IN}},N_{\text{OUT}}\rangle$ on the threshold $T$. For $T\lesssim 0.5$, both particles are expected to tunnel, whereas for $0.5\lesssim T\lesssim 0.8$, only one boson is expected to tunnel. For even larger $T$, the bosons should remain trapped. Right panel: The obtained peaks in the momentum distribution $\rho(k,t=600,T)$ for different $T$. For $T\leq 0.4$, two peaks appear at $k_{1,2}^{T}$, for $T=0.5$ and $0.6$, only one peak appears. This is in agreement with the expected number of ejected bosons from the trap. For larger $T$, the peaks are shifted to smaller momenta since the bosons have to overcome a larger threshold. All quantities are dimensionless. See text for details. The figures are adapted from Ref. [153]. These model assumptions are tested against a system with $N=2$ bosons and two- body repulsion strength $\lambda_{0}=1$. The left panel of Fig. 13 shows the dependence of $E_{\text{TOT}}$ on the threshold value $T$ for the three possible final states $\|N_{\text{IN}},N_{\text{OUT}}\rangle$, given by $\|2,0\rangle$, $\|1,1\rangle$ and $\|0,2\rangle$. The utilized notation $\|N_{\text{IN}},N_{\text{OUT}}\rangle$ should not be confused with a Fock state and is just used to count the particles in the two regions IN and OUT. For $T\lesssim 0.5$, the lowest-in-energy state is $\|0,2\rangle$, i.e., the final state where both bosons have tunneled to open space. For $0.5\lesssim T\lesssim 0.8$, the state $\|1,1\rangle$ is the lowest, meaning that only one boson is expected to tunnel in the long-time dynamics. For even higher $T$, tunneling is suppressed completely because the state $\|2,0\rangle$ corresponds to the lowest energy. This is in full agreement with the results shown in the right panel of Fig. 13. There, the peaks in the momentum distribution $\rho(k,t=600)$, corresponding to the momenta $k_{1}^{T}$ and $k_{2}^{T}$ of the emitted particles, are presented for different values of the threshold $T$. The $k_{2}^{T}$-peaks are plotted upside-down for better visibility. The first observation is that increasing $T$ shifts the peaks to smaller momenta, which is due to the higher energy that is necessary to overcome the threshold. Secondly, two peaks appear up to $T=0.4$, whereas for $T=0.5$ and $0.6$ only one peak appears. This can be understood in terms of the static considerations from the left panel, where it was observed that for $T\lesssim 0.5$ both particles are expected to tunnel through the barrier, whereas only one boson is expected to tunnel for $0.5\lesssim T\lesssim 0.8$. Thus, the threshold can be utilized to control the number of tunneling bosons. The arrows in the bottom of the right panel mark the calculated positions of the peaks with respect to the model of Eq. (185). The good agreement with the positions of the peaks of the momentum distribution suggests that the tunneling of single particles can be well described by the above model. Figure 14: Evolution of the natural occupations of the first few natural orbitals. Left panel: The case of zero threshold, $T=0$, utilizing $M=4$ orbitals. The interaction strength is $\lambda_{0}=0.3$. For all particle numbers shown, the system evolves from being condensed to being multi-fold fragmented. Right panel: The occupation of the first two natural orbitals for $N=2$ particles and $\lambda_{0}=1$, but for different heights of the threshold $T$. In general, the higher the threshold the later the development of fragmentation sets in. Moreover, the system with larger $T$ fragment less than the ones with smaller $T$. All quantities are dimensionless. See text for details. The figures are taken from Refs. [152, 153]. It remains the question whether the bosons do only tunnel one by one, or if it is also possible that several bosons tunnel at once. To answer this question, one has to consider that the peaks for $T\lesssim 0.5$ in Fig. 13 (right panel) appear sequentially, i.e., first the dominant peak develops, and then the second, smaller peak appears. The fact that the height of the peaks are time-dependent suggests that the two processes, namely the separate ejection of the first and second boson, are not independent. If however a multi-boson ejection happened, peaks at higher momenta than the ones observed would appear. For instance, in the limit of large $N$, the chemical potential corresponding to a two-boson ejection would be $\mu^{2b}\approx 2\mu_{1}$, which, for the case of $T=0$, yields $k^{2b,0}\approx\sqrt{2}k_{1}^{0}$. This is simply not observed in the Fourier spectrum of $\rho(k,t)$. Thus, it can be deduced that the bosons tunnel to open space one at a time. Nevertheless, as mentioned above, the individual processes of the ejection of single bosons are not independent, meaning that they quantum-mechanically interfere. Finally, it is discussed whether the overall process of tunneling to open space is indeed a many-body process or whether it can also be described at the mean-field level. To this end, the time-evolution of the occupations of the first few natural orbitals for systems with different numbers of bosons and different heights of the threshold is studied in Fig 14. In the weakly- interacting regime considered here, all initial states of the bosonic clouds are essentially condensed. For the case of zero threshold (left panel), one observes that for all numbers of bosons considered the system becomes multi- fold fragmented. Especially for $N=4$ bosons, at least four orbitals become macroscopically occupied. With respect to the case of $N=2$ particles and non- zero thresholds, the behavior is similar, meaning that the initially coherent BEC becomes highly fragmented. In general, the onset of fragmentation happens at later times when $T$ is larger. Furthermore, the degree of fragmentation is largest in the absence of any threshold. To summarize, it has been found that the tunneling-to-open-space dynamics of repulsive bosons from a harmonic trap are of many-body nature. With time, the system evolves from being coherent to become highly fragmented, and thus a mean-field description is inapplicable even in the weakly-interacting regime. The bosons tunnel one by one, meaning that no multi-boson tunneling occurs where more than a single boson at the same time is emitted jointly from the trap. However, the individual tunneling processes are not independent, they quantum-mechanically interfere. The threshold in the open space can be utilized to control the number of emitted particles. In particular, it can be used to suppress tunneling completely. A detailed analysis on the coherence of the system, showing that the emitted bosons lose coherence with the bosons in the trap and among themselves, can be found in Refs. [150, 151, 152, 153, 154], as well as additional details. #### 6.2.2 Phantom vortices A recently found vortex type in a rotating and repulsive 2D single-component BEC, termed phantom vortex, is presented in this section. Different to the common understanding of a quantum vortex that, among other features, has a density node (or core) and a phase discontinuity surrounding it, a phantom vortex cannot be detected in the condensate density. Although they were not found in single-component condensates so far, coreless vortices have been predicted earlier for multi-component condensates, e.g., spinor condensates where the core of one species, i.e., bosons with a certain spin, is filled by another species [155, 156]. Since the experimental realization of gaseous BECs [4, 6, 5, 157, 158], the nucleation of vortices in rotating BECs was of interest and it was found that there is a critical rotation frequency below which no quantized vortices appear [161, 159, 162, 163, 164, 160]. In contrast to that, phantom vortices appear also below this critical rotation frequency. Instead of a node in the density, they rather manifest themselves in nodes of the densities of underlying natural orbitals. Thus, phantom vortices are a many-body object that cannot be described at the mean-field level. Moreover, their occurrence is accompanied by the onset of fragmentation. The following results were published in Ref. [165]. Earlier results on phantom vortices in a stirred BEC where, e.g., the fragmentation entropy is analyzed, can be found in Ref. [166]. The system under consideration is a BEC of $N=100$ spinless bosons. They interact via the same Gaussian repulsion as in Eq. (180) with interaction parameter $\Lambda=\lambda_{0}(N-1)=17.1$. They are confined in a time- dependent trap that is varied stepwise. It reads $V(\mathbf{r},t)=\frac{1}{2}\left(x(t)^{2}+y(t)^{2}\right)+\frac{1}{2}\eta(t)\left(x(t)^{2}-y(t)^{2}\right)$ (186) where $\eta$ is the parameter of the anisotropy given by the second term in Eq. (186). The $x-$ and $y-$coordinates are varied in time according to $\begin{pmatrix}x(t)\\\ y(t)\end{pmatrix}=\begin{pmatrix}\cos(\omega t)&\sin(\omega t)\\\ -\sin(\omega t)&\cos(\omega t)\end{pmatrix}$ (187) where the frequency $\omega$ is set to $0.78$. Initially, the trapping potential is isotropic and harmonic, meaning that $\eta(t=0)=0$, and the corresponding ground state is computed both at the mean-field level using the GP equation as well as at the many-body level using imaginary time-propagation of the MCTDHB equations. Afterwards, a slight anisotropy is added by increasing $\eta$ from zero to its maximal value of $0.1$ until $t=80$. The resulting elliptic trap is kept between $80<t\leq 300$, before $\eta$ is ramped down to zero again between $300<t\leq 380$. For the last interval until $t=500$, the isotropic harmonic trap is kept constant. The entire procedure is shown in the upper panel of Fig. 15, together with representative densities of the BEC for the individual intervals. These densities already suggest that at no point in time there is a vortex contained in it, as discussed in more detail below. Figure 15: Top panel: Time-evolution of the anisotropy parameter $\eta(t)$ (red line) for the 4 distinct intervals discussed in the text, together with representative densities (blue). The dynamic protocol is divided into a period of ’ramp up’, ’maximal anisotropy’, ’ramp down’ and ’isotropic’. Second panel: Time-evolution of the first 4 natural occupation numbers $\rho_{i}^{(\text{NO})}(t)$. Fragmentation sets in at the end of the ramp up of the anisotropy. Third panel: Time-evolution of the orbital angular momenta $(L_{z})_{ii}(t)$ and the total angular momentum per particle, $L_{z}(t)/N$. Fluctuations are maximal when fragmentation sets in. Bottom panel: Comparison between the time-evolution of the total energy $E$ relative to the initial energy $E_{0}$ and $L_{z}(t)/N$. The GP theory underestimates both quantities, especially the angular momentum is approximately three times less than at the many-body level. All quantities are dimensionless. See text for details. The figure is taken from Ref. [165]. Of particular interest for the analysis will be, apart from the density $\rho(\mathbf{r})$, the natural occupation numbers, denoted here by $\rho_{i}^{(\text{NO})}$ but with the same meaning as the $n_{i}$ in Eq. (28), and the densities $\|\phi_{i}\|^{2}$ of the natural orbitals (mind the different notation as compared to Section 3 where the $\phi_{i}$ denote the working orbitals of MCTDHB). Furthermore, the time-evolution of the expectation values of the orbital angular momenta, given by $(L_{z})_{ii}=\langle\phi_{i}\|\hat{L}_{z}\|\phi_{i}\rangle$, and of the total angular momentum per particle, $L_{z}/N=\frac{1}{N}\sum_{i,j=1}^{M}\rho_{ij}\,(L_{z})_{ij}$, are analyzed. The time-argument is suppressed in the above mentioned quantities. Computations are carried out for orbital numbers $M=4$ (many-body) and $M=1$ (GP). Additionally, signatures of phantom vortices can also be seen in the phase of the first-order correlation function $g^{(1)}(\mathbf{r}\|\mathbf{r}^{\prime};t)$ [see Eq. (D.6) in Appendix D] defined by $S_{g}(\mathbf{r}\|\mathbf{r}^{\prime};t)=\text{arg}[g^{(1)}(\mathbf{r}\|\mathbf{r}^{\prime};t)]$ (188) as well as in the phases of the natural orbitals given by $S_{i}(\mathbf{r},t)=\text{arg}[\phi_{i}(\mathbf{r};t)].$ (189) The discussion of the results is split into two parts. At first, the appearance of phantom vortices is analyzed at an instant in time when the trap is already isotropic again. Afterwards, the degree of fragmentation in the system is discussed. Figure 16: Top panels: Total density $\rho(\mathbf{r})$ (a) and densities of the natural orbitals (b)-(e) at $t=450$, i.e., when the trap is already isotropic again. Although no vortex is observed in the total density, phantom vortices in the orbitals $\phi_{1}$, $\phi_{3}$ and $\phi_{4}$ appear. Bottom panels: Phases of the first-order correlation function (f) and of the natural orbitals (g)-(j). Phase discontinuities are observed for all phantom vortices. All quantities are dimensionless. See text for details. The figure is taken from Ref. [165]. With respect to the first part, the total density and the densities of the natural orbitals at $t=450$, i.e., when the anisotropy is ramped down to zero again, are examined in Fig. 16. One observes that the total density [panel (a)] does not exhibit any vortex. On the contrary, several phantom vortices appear in the densities of the natural orbitals. The densities of the first and fourth natural orbitals contain a charge-1 and a charge-3 phantom vortex, respectively [panels (b) and (e)]. The density of the third orbital contains a pair of charge-1 phantom vortices [panel (d)]. The mechanism responsible for the nucleation of the phantom vortices in $\phi_{1}$ and $\phi_{4}$ is different to the one responsible for the pair in $\phi_{3}$. The former mechanism is called node mutation, and is explained for the charge-1 phantom vortex in $\phi_{1}$. This phantom vortex originates from the node of the initial second orbital $\phi_{2}$, which resembles a 2$p_{x}$ orbital that has an angular node. When angular momentum is added to the system due to the anisotropy, the lobes of the $2p_{x}$ orbital spread out and tend to close this angular node, leaving an elliptic density node in the center. After the trap is isotropic again, a phantom vortex is left at the center, which now appears in the density of $\phi_{1}$ since the first and second orbital have switched labels at $t\approx 220$ because of their changing occupation numbers [see second panel from the top in Fig. 15]. Thus, the remaining phantom vortex was nucleated from an initial node. A similar process happens for the charge-3 phantom vortex in the density of $\phi_{4}$. It originates from $\phi_{3}(t=0)$ which resembles a $2p_{y}$ orbital. There, the angular node mutated first to three single phantom vortices that finally merged into a larger charge-3 phantom vortex. The fact that it finally appears in the fourth natural orbital is due to the label switching of $\phi_{3}$ and $\phi_{4}$ during the ramp-up of the anisotropy. Figure 17: Time-evolution of the fragmentation for different numbers of particles $N$. The same dynamical protocol of the trapping potential as in Fig. 15 is used. For all considered boson numbers, the occupation $\rho_{1}^{(\text{NO})}$ decreases clearly. In general, fragmentation sets in later for higher particle numbers. Results are computed using $M=2$ orbitals. The interaction parameter is kept fixed at $\Lambda=17.1$. All quantities are dimensionless. See text for details. The figure is taken from the supplementary material of Ref. [165]. The mechanism responsible for the phantom-vortex pair in $\phi_{3}$ [panel (d)] is called slow orbital-orbital vorticity transfer because of the following observation. At $t\approx 220$, the pair of co-rotating phantom vortices appeared in the density of $\phi_{2}$. When time progressed, the two phantom vortices started to move towards the density edge and disappeared finally at $t\approx 275$. At the same time, two phantom vortices were build at the density edge of $\phi_{3}$, which afterwards continued to move towards the center. At $t\approx 340$, the separation between this pair is the same as before in the density of $\phi_{2}$. Thus, vorticity has been transferred between the two natural orbitals over a period much longer than the rotation period of $\tau=2\pi/\omega\approx 8$. This process between the second and third orbital can also be seen from the correlated orbital angular momenta $(L_{z})_{22}$ and $(L_{z})_{33}$ in Fig. 15 for $t\geq 250$. The position of the phase discontinuities shown in Figs. 16(g), (i) and (j) coincide with the corresponding positions of the phantom vortex cores. Especially in panel (j) one can see from the merging three discontinuity branches in the center that the corresponding phantom vortex in panel (e) is of charge 3. Comparing the density in panel (a) and the phase $S_{g}(\mathbf{r}\|0;450)$ in panel (f), one observes the occurrence of several so-called ghost vortices at the density edge. The latter are distinct from phantom vortices because they do not enter the bulk density of the condensate and do also not contribute much to the total angular momentum. The dynamical fragmentation that occurs during the time-evolution, as well as its consequences for the angular momenta and the energy of the system [see again Fig. 15] is now analyzed. The initial state is essentially condensed ($\rho_{1}^{(\text{NO})}\approx 99.7\%$), and it stays condensed for almost the entire period in which the anisotropy is ramped up to its maximal value. In this time frame, the total angular momentum and energy computed at the many-body and mean-field levels coincide as long as the BEC remains coherent. Once fragmentation sets in, the latter quantities start to deviate from each other. During the first part of the time interval in which the trap is kept maximally anisotropic ($80<t\leq 150$), $\rho_{1}^{(\text{NO})}$ decreases significantly to roughly $40\%$, and the second, third and fourth natural orbitals become macroscopically occupied. At the same time, the orbital angular momenta vary the most and several of them reach their maximum values. Then, the occupations and angular momenta of $\phi_{1}$ and $\phi_{4}$ remain more or less constant. For the orbitals $\phi_{2}$ and $\phi_{3}$, both quantities change slightly in a correlated manner, i.e., when $\rho_{2}^{(\text{NO})}$ increases $\rho_{3}^{(\text{NO})}$ decreases by the same amount. This again reflects the slow orbital-orbital vorticity transfer between $\phi_{2}$ and $\phi_{3}$ discussed above. With regard to the total angular momentum and energy, both quantities vary strongly during the periods of maximal anisotropy and ramp down, and saturate only afterwards. The corresponding GP values are substantially different since they underestimate the system’s gain of energy and angular momentum. In particular for the latter quantity, it stays below $0.5$, well below the threshold for the occurrence of vortices at the mean-field level at unit angular momentum per particle. Interestingly, $L_{z}/N$ for $M=4$ orbitals is greater than this threshold, but still no vortices are observed in the density $\rho(\mathbf{r})$. This can be rationalized because one can show analytically that a vortex in the density of a fragmented BEC, i.e., a fragmented vortex, requires more than unit angular momentum per particle [see the supplementary information of Ref. [165]]. The reason is that a fragmented vortex originates from coincident phantom vortices, meaning that all natural orbitals must have a phantom vortex at the same position, and all of these phantom vortices need to have different charges such that the natural orbitals are orthogonal. As seen from panels (b)-(d) in Fig. 16, there are no coincident phantom vortices, and thus the fragmented system does not show any vortices in the density. The onset of fragmentation was also observed for different numbers of bosons. Fig. 17 shows the evolution of $\rho_{1}^{(\text{NO})}$ for different $N$ between $10$ and $10^{4}$, utilizing $M=2$ time-adaptive orbitals. The interaction parameter $\Lambda=17.1$ is kept fixed. One generally observes that fragmentation sets in for all considered particle numbers, and that furthermore the onset of fragmentation occurs at later times for higher particle numbers. In conclusion, the nucleation of phantom vortices has been investigated for a rotating BEC that acquires angular momentum via a time-dependent modification of the trap. Two different mechanisms for the creation of phantom vortices were found, namely node mutation as well as slow orbital-orbital vorticity transfer. Phantom vortices, by definition many-body objects, cannot be described at the mean-field level. The initially coherent system highly fragments, also for different numbers of particles in the condensate. Further information can be found in Ref. [165], in particular in the supplementary material containing videos that show, e.g., the evolution of all quantities shown in Fig. 16. ### 6.3 Excitation spectra In this section, details on current applications of LR-MCTDHB are presented. The amount of scientific publications where it is utilized is still rather small because the theory itself has been developed only a few years ago [34, 35]. In addition to that, the efficient MPI-parallelized numerical implementation of LR-MCTDHB was a very demanding task, and all the technical challenges going along with it, extensively described in Section 4, were solved very recently. It is stressed again that without such an implementation, a vast majority of problems cannot be treated at all, especially in $D>1$ spatial dimensions. Before developing the sophisticated and complex parallel implementation, there was a first implementation available that could treat small 1D systems. The results of Refs. [34] and [167] were obtained with this code [see below and Section 6.3.1]. In more recent applications, discussed in Secs. 6.3.2 and 6.3.3, the new implementation was utilized since it was impossible to obtain the same results with the former code. (a) (b) Figure 18: (a) Numerical convergence of the first three center-of-mass excited states (labeled by 1, $2^{\prime}$ and $3^{\prime}$) with respect to the number of orbitals $M$ for $N=10$ bosons in a harmonic trap with trap frequency $\omega_{\text{ho}}=\sqrt{2}$. The strength of the contact repulsion is $\lambda_{0}N=1$. For $M=5$ orbitals, the energies obtained from LR-MCTDHB coincide with the analytic predictions. The numerical implementation used is the precursor of the implementation described and benchmarked in Sections 4 and 5. (b) Comparison between the excitation spectra obtained from BdG (bottom) and LR-MCTDHB($2$) (top) for $N=10$ bosons in a shallow double well. The interaction strength is the same as in panel (a). Excitations labeled with primes refer to states solely obtained at the many-body level. The intensity of the response to a linear perturbation ($f^{+}=f^{-}=x$) is denoted by red triangles, whereas the intensity of the response to a quadratic perturbation ($f^{+}=f^{-}=x^{2}$) is denoted by green squares. The insets show the trap potential and the ground-state densities. One observes that the many-body low- energy spectrum contains a lot more states than the mean-field spectrum. All quantities are dimensionless. See the main text and Ref. [34], from which both figures are taken, for further details, especially for the discussion of the excitation type shown at the top of panel (b). The main purpose of Ref. [34] is to introduce LR-MCTDHB by giving a derivation of its main equations, presented in Section 3.2.2 of this work. In particular, the equations for the submatrices as described in Eq. (93) are developed for the case of contact interaction between the bosons. As an application, excitations for BECs in harmonic and shallow double-well traps are analyzed. Fig. 18(a) shows the numerical convergence for the first three c.m. excited states (labeled by 1, $2^{\prime}$ and $3^{\prime}$) of $N=10$ bosons in a harmonic trap with trap frequency $\omega_{\text{ho}}=\sqrt{2}$ in terms of the number of orbitals $M$. The strength of the repulsion is given by $\lambda_{0}N=1$. One can observe that the c.m. excitations are accurately obtained for $M\geq 5$ orbitals. The BdG theory only yields the first c.m. excitation. In Fig. 18(b), the low-energy spectrum as well as the response intensities for linear and quadratic perturbations ($f^{+}=f^{-}=x,\,x^{2}$) are presented for $N=10$ bosons in a shallow double well of the form $V(x)=b/2\cos\left(\pi x/3\right)+\omega_{\text{ho}}^{2}x^{2}/2$ with barrier height $b=5$. The most important observation by comparing the number of obtained states for BdG and LR-MCTDHB($2$) is that the many-body spectrum contains a lot more states. This in turn means that although the degree of depletion is only about $0.2\%$, a many-body description for the lowest-in- energy excitation spectrum is unavoidable in order to obtain accurate results. In the subsequent Section 6.3.1, the excitation spectrum for the case of BECs trapped in harmonic and double-well potentials is elaborated in more detail, especially with respect to the question of how many-body excitations can be triggered in the out-of-equilibrium dynamics of the condensates. Before discussing further applications of LR-MCTDHB in the following, a remark on how mean-field and many-body excitations are defined in this work is made in order to avoid misconceptions of these important terms. Excitations calculated from Eq. (116), i.e., from LR-MCTDHB ($M>1$), are termed many-body excitations since they stem from a many-body theory. On the contrary, all excitations calculated with the BdG equations in Eq. (A.16) of Appendix A.1.2 are referred to as mean-field excitations since they stem from a mean-field theory. Moreover, excitations where only a single boson is excited from the condensed mode are called single-particle excitation, whereas excitations where more than a single boson is excited are called multi-particle excitations. These definitions are utilized consistently in the applications below. #### 6.3.1 One-dimensional harmonic and double-well systems As a first application of LR-MCTDHB, the many-body nature of excitations in a 1D harmonic trap as well as in shallow and deep symmetric double wells is analyzed. The low-energy spectra obtained from the BdG equation at the mean- field level and from LR-MCTDHB at the many-body level are compared. Furthermore, it is investigated which excitations are involved in the dynamics due to certain quench scenarios. The following results where published in Ref. [167]. Excited states are studied in a harmonic trapping potential with a Gaussian barrier given by $V(x)=ax^{2}+b\,\exp\left(-cx^{2}\right)$ (190) where $a$ denotes the harmonic trap frequency and $B$ denotes the barrier height. For simplicity, $\hbar=m=1$ is assumed. Furthermore, the trapping parameters are set to $a=1/2$ and $c=1$. The bosons interact via the zero- ranged contact interaction potential. The interaction strength is again measured by the mean-field parameter $\Lambda=\lambda_{0}(N-1)$ where $\lambda_{0}$ is chosen to be positive to account for repulsion. Fig. 19 shows $V(x)$ for different barrier heights, together with the corresponding ground- state densities. For the purely harmonic trap ($b=0$), the BEC is essentially condensed with $n_{1}=99.9\%$. On the contrary, for the shallow ($b=5$) and deep ($b=10$) double wells, the ground state is already fragmented by approximately $5\%$ and $40\%$, respectively. One can further observe that for the deep well, the overlap between the densities in the left and right wells is marginal, whereas there is still a clear overlap observable in the shallow case. Since the trap is reflection invariant, the excitations can be categorized in gerade ($g$) and ungerade ($u$) symmetry. Fig. 20 shows the low-energy spectrum of $N=10$ bosons for different barrier heights $0\leq b\leq 10$ and $\Lambda=1.0$. Excitation energies are given relative to the ground-state energy, i.e., $\Delta E=E-E_{0}$. One readily observes the formation of bands, which is even more pronounced for lower interaction strengths [see Fig. 2 in Ref. [167]]. The bands are labeled by integer numbers, where the zeroth band denotes the one that is lowest in energy. Comparing the mean-field and many- body approaches, one can see that the BdG spectrum only contains 5 different excitations, whereas the spectrum computed with LR-MCTDHB shows a much richer structure where a lot more excited states appear. In terms of numerical convergence, $M=2$ orbitals are sufficient to describe the spectrum up to $\Delta E\sim 2$ for not too low barrier heights, i.e., $b\geq 4$. For states higher in energy, at least four orbitals are necessary. Figure 19: Trap potentials (solid red) and ground-state densities (gray) for different barrier heights $b$. The ground states are computed with MCTDHB($2$). The BEC consists of $N=10$ bosons and the repulsion strength is $\Lambda=1.0$. The occupation number $n_{1}$ of the first natural orbital shows that increasing the barrier height leads to fragmentation. Vertical lines refer to the lowest-in-energy excited states computed with LR- MCTDHB($2$). The green potentials and the blue eigenvalues refer to the harmonic approximation of the right well. All quantities are dimensionless. See text for details. The figure is taken from Ref. [167]. Figure 20: Low- energy excitation spectrum of $N=10$ bosons with interaction parameter $\Lambda=1.0$ for different barrier heights $b$. Many-body results computed with LR-MCTDHB($2$) are denoted by solid lines (gerade excitations in green, ungerade excitations in red). Results for $M=4$ orbitals are given by dotted lines, and the corresponding BdG energies are given by open black circles. All states up to $\Delta E\sim 2$ are converged with two orbitals. The BdG approach misses many states in the shown energy range. All quantities are dimensionless. See text for details. The figure is taken from Ref. [167]. In the following, different shift and quench protocols are applied to trigger excitations from the low-energy spectrum. For gerade excitations, a quench of the harmonic confinement from $a=0.4\rightarrow 0.5$ is performed, whereas ungerade excitations are addressed by a shift of the trap position, meaning that $x\rightarrow x+x_{\text{shift}}$ with $x_{\text{shift}}=0.1$. The gerade excitations are referred to as breathing modes, and the ungerade excitations are referred to as dipole modes. The dynamics of the bosonic clouds for the two cases are shown in Fig. 21 on a short time scale. In the left panel, dipole oscillations for a BEC in the deep double well are depicted, which means that in each well, the bosons oscillate around the equilibrium position given by the minima of the wells. On the RHS, the breathing dynamics for $N=10$ bosons in the purely harmonic trap is shown. One can observe that the density spreads to the left and right in a symmetric manner, and returns to the minimum of the well. The Fourier spectra of the time-evolution of certain quantities is used to extract the spectrum of involved excitations in the dynamics. This comprises $(a)$ the expectation value $\langle x\rangle$, $(b)$ the variance per particle $\frac{1}{N}\Delta^{2}_{\hat{X}}$, and $(c)$ the density $\rho$ at the position $x=1$. The many-body variance is discussed in detail in Appendix D. In Ref. [167], the Fourier spectra for $b=0\,,5$ and $10$ are investigated. Here, the discussion is restricted to the latter. To remind the reader, the system is highly fragmented by approximately $40\%$ in this case. Figure 21: Many-body dynamics of the density of $N=10$ bosons with interaction parameter $\Lambda=1.0$ for a shift of the trap position in the deep double well (left panel) and a quench of the harmonic confinement (right panel). In the former scenario, one observes dipole oscillations of the bosonic clouds in the two wells, whereas one observes breathing oscillations for the dynamics due to the quench of the confinement. All quantities are dimensionless. See text for details. The figure is taken from Ref. [167]. Table 4 shows excitation and de-excitation energies where the obtained values from the BdG and LR-MCTDHB($2$) theories (columns 2 and 3) are compared to the obtained excitation energies of the FT spectra from the shift and quench scenarios (columns 4-7). Moreover, several de-excitations, i.e., transitions between a higher and a lower excited state, are presented (last 4 rows). Those refer to higher-order effects and are in principle not obtained directly by LR. However, since LR yields the exact excitation spectrum once it is applied to an exact eigenstate of the full many-body Hamiltonian, the de-excitations can be extracted from differences between the excitation energies of the LR spectrum. The excitation labels are shown in the first column. Many-body excitations, i.e., excitation that only appear in the many-body spectrum, are denoted by primed labels. \| $M=1$ \| $M=2$ \| Shift GP \| Shift MB \| Quench GP \| Quench MB ---\|---\|---\|---\|---\|---\|--- $0u$ \| 0.035 (0.46) \| 0.063 (0.19) \| 0.034 (0.02a) \| \| \| $0g^{\prime}$ \| \| 0.064 \| \| \| \| 0.066 (0.36b) $1g$ \| 2.085 (0.91) \| 2.092 (0.81) \| \| \| 2.085 (1.0b,0.93c) \| 2.092 (0.33b,0.89c) $1u$ \| 2.102 (0.46) \| 2.095 (0.57) \| 2.102 (0.94a,0.66c) \| 2.095 (0.93a,0.7c) \| \| $1u^{\prime}$ \| \| 2.158 (0.08) \| \| 2.163 (0.02a,0.02c) \| \| $1g^{\prime}$ \| \| 2.161 (0.09) \| \| \| \| 2.161 (0.12b) $2g$ \| 3.912 (0.04) \| 3.934 (0.03) \| 3.914 (0.04c) \| \| 3.914 (0.07c) \| 3.934 (0.07b,0.1c) $2u$ \| 3.997 (0.07) \| 3.943 (0.07) \| 3.997 (0.04a,0.23c) \| 3.943 (0.03a,0.15c) \| \| $2u^{\prime}$ \| \| 4.029 (0.05) \| \| 4.032 (0.02a,0.07c) \| \| $2g^{\prime}$ \| \| 4.039 \| \| \| \| $1g^{\prime}\leftrightarrow 1g$ \| \| 0.069 \| \| \| \| $2g\leftrightarrow 1g$ \| 1.827 \| 1.841 \| \| \| \| 1.893 (0.01b) $2u\leftrightarrow 1u$ \| 1.894 \| 1.848 \| 1.895 (0.01c) \| 1.849 (0.02c) \| \| $1g\leftrightarrow 0g^{\prime}$ \| \| 2.028 \| \| \| \| 2.032 (0.02b) Table 4: Table of excitation and de-excitation energies for $N=10$ bosons with $\Lambda=1.0$ in the deep double well. Listed are results from BdG and LR- MCTDHB($2$), as well as from the Fourier spectra of the expectation value of the position $\langle x\rangle$ (superscript $a$), the variance per particle $\frac{1}{N}\Delta^{2}_{\hat{X}}$ (superscript $b$) and the real-space density $\rho(x=1)$ (superscript $c$), for both the shift and quench scenarios. GP denotes the mean-field and MB the many-body case. The intensities are given in brackets. All quantities are dimensionless. See text for more details. The table is adapted from Ref. [167]. Up to $\Delta E\sim 4$, the many-body approach yields ten excited states, whereas on the contrary the BdG spectrum only consists of five excitations. The energies of states that appear in both spectra differ slightly from each other for $1g$, $1u$, $2g$ and $2u$, but significantly for the lowest excitation $0u$. That demonstrates that strong deviations between the mean- field and many-body excitation energies can already occur for the lowest excited states. Also the intensities of several states differ clearly from each other, especially for the lowest states $0u$, $1g$ and $1u$. The intensities from the LR calculations are identical to the response weights described in Eq. (3.2.2), with $f^{+}=f^{-}=x$ for the ungerade and $f^{+}=f^{-}=x^{2}$ for the gerade states. Figure 22: Fourier spectra of $\langle x\rangle$, $\frac{1}{N}\Delta^{2}_{\hat{X}}$, and $\rho(x=1)$ from the shift and quench dynamics in the deep double well. The GP spectra are denoted by solid black and the MCTDHB($2$) spectra by dashed red lines. The intensities are normalized by the sum of the intensities of all constituent peaks, and excitations with less than $1\%$ intensity are omitted. Especially the spectrum of the many-body variance consists of several peaks that are not observed in the corresponding GP spectrum. All quantities are dimensionless. See text for details. The figure is adapted from Ref. [167]. Columns 4 and 5 show the obtained peaks in the Fourier analysis of $\langle x\rangle$ and $\rho(x=1)$ for the shift scenario, both for the GP and many- body cases. The Fourier spectra are shown in the left panels of Fig. 22. Surprisingly, the shift also triggered a gerade excitation ($2g$) at the mean- field level, whereas at the many-body level only ungerade states are observed. Moreover, no peak is observed for $0u$ in the many-body dynamics, presumably because its intensity is too weak to be distinguished from the background noise. The obtained peak positions coincide with the predictions from BdG and LR-MCTDHB($2$), respectively. With regard to de-excitations, the mean-field and many-body approaches yield a very small peak for the $2u\leftrightarrow 1u$ transition in the density spectrum. The Fourier analysis of the quench dynamics of $\frac{1}{N}\Delta^{2}_{\hat{X}}$ and $\rho(x=1)$ are shown in column 6 (GP) and column 7 [MCTDHB($2$)]. Only peaks corresponding to gerade excitations are observed. Both approaches yield peaks for $1g$ and $2g$. At the many-body level, two additional peaks referring to $0g^{\prime}$ and $1g^{\prime}$ are obtained. The peak positions again accurately match the predictions from the static analysis. Most importantly, the intensities of these excitations in the variance spectrum are comparatively large, and therefore $0g^{\prime}$ and $1g^{\prime}$ are the most promising many-body excited states to be detected in an experiment. Furthermore, no de-excitation peaks are found in the GP spectra, while the variance spectrum shows small peaks for the $2g\leftrightarrow 1g$ and $1g\leftrightarrow 0g^{\prime}$ transitions at the many-body level, coinciding with the values from LR-MCTDHB($2$). In general, the sensitivity of the utilized operators to the detection of excited states can be seen as follows. The spectrum of the local operator $\rho(x=1)$ is in particular useful to detect ungerade excitations in the shift scenario. The intensities of the peaks of many-body excitations are however weak. To detect gerade excitation, nonlocal quantities, especially the variance, turn out to be more sensitive than the density. In addition, the variance spectrum yields intense peaks also for many-body excitations. To summarize, it has been demonstrated that from a methodological point of view, utilizing LR-MCTDHB in order to obtain the many-body excitation spectra of repulsive BECs in 1D harmonic and double-well trap geometries is a promising and highly accurate technique. Its predictions coincide with the Fourier spectra of the time-evolution of different quantities like the local density or the position variance. From a physical point of view, some of the obtained excited states corresponding to many-body states not included in standard mean-field approaches yield intense responses especially in the variance spectra and are therefore likely to be measured in experiments. More details on the presented results can be obtained from Ref. [167]. #### 6.3.2 Triple wells and larger lattices In this section, many-body effects in the excitation spectra of weakly- interacting BECs in 1D lattices are investigated. The main objective is to demonstrate with numerically converged results that a marginal depletion of the order of $1\%$ is sufficient in order to observe clear differences between the lowest-in-energy excitations computed at the mean-field and many-body levels. The results presented below were recently published in Ref. [168]. Studying dynamics and excitations of ultracold bosons in 1D optical lattices was of high interest in the past, and the literature is extensive in this research field. Examples are the dynamics in the superfluid phase [169], nonlinear dynamics in periodic potentials [170] or in fluctuation-driven binary condensates [171]. A review of earlier works can be found in Ref. [172]. Of particular relevance for the findings discussed in this section are studies were the BdG theory does not agree with numerical and experimental observations. It has been found that the oscillation frequency of a superfluid BEC in a large optical lattice with a superimposed harmonic confinement deviates clearly from the BdG prediction [173, 174]. In addition, the strong depletion of a gaseous BEC in optical lattices cannot be fully explained by the BdG theory, not only in 1D [175]. Optical lattices, in particular triple-well systems, have also been analyzed recently utilizing MCTDHB. This includes the out-of-equilibrium dynamics of rather small BECs consisting of a few particles only, induced by quenching the interaction and the lattice depth [176, 177, 178] and periodic driving of the system [179, 180]. Of particular interest was the analysis of cradle modes, dipole oscillations corresponding to the transport of bosons over the barrier between adjacent wells. Furthermore, the many-body dynamics of BECs initially prepared in Mott-insulating states have been of interest [181]. The different phases of the BEC, i.e., the superfluid, the Mott-insulating, and the fermionized gas phase, can be detected from the many-body entropy and the Glauber correlation functions [182]. It was even shown for a BEC in a tilted triple well with hard-wall boundary conditions how the correlations between wells can be controlled by the well depth, the interaction strength and the tilt [183]. Many-body effects in the ground state of a bosonic mixture in a more complex 2D lattice where studied using the multi-layer version of MCTDHB [184]. Many-body excitations in optical lattices were addressed as well, either by making a Fourier analysis of certain quantities [176] or by doing an exact diagonalization. There, the system’s wave function was expressed using a many- body ansatz based on a number state expansion including localized Wannier functions [179, 180, 177, 178]. The latter approach of exact diagonalization is possible for small systems with a few bosons only. (a) (b) (c) (d) Figure 23: Low-energy spectra of $N=10$ bosons in a triple well computed with LR-MCTDHB($7$). (a) Shallow case with lattice depth $V_{0}=1.01\,E_{R}$. For $\Lambda=0$, the excitations form levels of degenerate states. Degeneracies are lifted for $\Lambda>0$. The BdG results only contain the two doublets denoted by BdG(1) and BdG(2). (b) Same as in (a) but in the deep lattice with $V_{0}=10.13\,E_{R}$. The BEC for $\Lambda=2.0$ is fragmented by $8.4\%$. (c) Comparison between BdG(1) and its multiples (open circles) and the many-body results (colored closed symbols) for the system in (a) with $\Lambda=4.0$ where only the states $(+1)$ and $(-1)$ of the first single-particle band are occupied by $N^{(1)}$ bosons. The difference grows with $N^{(1)}$. Inset: Evolution of the relative error as defined in the main text. (d) Same as in (c), but for the system in (b) with $\Lambda=2.0$. All quantities are dimensionless. See text for details. The figures are adapted from Ref. [168]. Below, the lowest-in-energy excitations of BECs in 1D lattice potentials with periodic boundary conditions are investigated using LR-MCTDHB. This opens the possibility to study also larger systems with more wells and particles. The utilized trap potential reads $V(x)=V_{0}\,\cos^{2}\left(\frac{\pi}{l}\,x\right)$ (191) where $V_{0}$ denotes the lattice depth and $l$ denotes the distance between two neighboring lattice sites. The depth of the lattice is commonly expressed in terms of the recoil energy $E_{R}=\frac{\hbar^{2}k_{0}^{2}}{2m}$ with the lattice momentum $k_{0}=\frac{\pi}{l}$. For simplicity, $\hbar=m=l=1$ is assumed. The bosons repel each other via the contact interaction potential and its strength is again expressed in terms of the mean-field parameter $\Lambda=\lambda_{0}(N-1)$. (a) (b) Figure 24: (a) Same as in Fig. 23(a) but in a lattice with 10 sites. The depletion grows faster with $\Lambda$ than in the triple well, meaning that the system is even more sensitive for many-body effects. (b) Same as in Fig. 23(c) but for 10 sites and $\Lambda=1.0$. The relative error (inset) reaches similar values as for systems in the shallow triple well with comparable depletion. All quantities are dimensionless. See text for details. The figures are adapted from [168]. As a start, $N=10$ bosons confined in a triple-well potential with periodic boundary conditions are considered. The low-energy excitation spectra for both a shallow ($V_{0}=5.0\approx 1.01\,E_{R}$) and a deep well ($V_{0}=50.0\approx 10.13\,E_{R}$) are discussed. Results are presented in Figs. 23(a) and (b). In both wells, the spectra for the non-interacting BEC are structured in sequences of levels with different degeneracies. The states and levels can be associated with appropriate quantum numbers, envisioning that for $\Lambda=0$ eigenstates of a single particle in a periodic lattice are exact quasimomentum eigenstates. The eigenstates appear in a band structure that is more pronounced for deep lattices. For the shallow lattice in panel (a), the first level consisting of two degenerate states denotes the excitations of putting one boson in either a state with quasimomentum $(+1)$ or with quasimomentum $(-1)$. Similarly, the states of level 5 denote the excitations where one boson is put to either $(+1)$ or $(-1)$ from the second single-particle band. To this end, the notation $\left(n_{+1}^{(1)},n_{-1}^{(1)};n_{+1}^{(2)},n_{-1}^{(2)}\right)$ is introduced, where the first two entries in brackets denote the occupation of $(+1)$ and $(-1)$ from the first single-particle band and the last two entries denote the occupation of $(+1)$ and $(-1)$ from the second single-particle band. As another example, level 2 contains the states $(2,0;0,0)$, $(0,2;0,0)$ and $(1,1;0,0)$. It is stressed that the BdG spectrum only contains the excitations of levels 1 and 5, and misses all multi-particle excitations in the low-energy part. However, for $\Lambda=0$, the missing levels can be anticipated by taking sums and multiples of the energies of the first and fifth levels. As soon as there are correlations between the bosons, i.e., for $\Lambda>0$, this is no longer possible, as discussed below. In panel (b), only the first single-particle band appears in the shown energy range because, as shown in Fig. 25, the first and second band are strongly separated from each other. Figure 25: Proof of numerical convergence for the systems discussed in Figs. 23 and 24. LR-MCTDHB($7$) clearly suffices in all cases. For the BEC in the deep triple well [middle panel], all 65 single- and multi-particle excitations below the bottom of the second band [denoted by BdG(2)] are obtained. Already for $M=3$, the results are highly accurate which is due to the large separation of the first two single-particle bands. All quantities are dimensionless. See text for details. The figures are adapted from Ref. [168]. For non-zero repulsion in the shallow well in panel (a), the BECs are slightly depleted ($f=1.1\%$ for $\Lambda=4.0$), whereas for the deep well in panel (b) the BEC for $\Lambda=2.0$ is already fragmented by $8.4\%$. One can observe that all excited states grow in energy once the bosons interact, and that splittings of several states from the same level occur although they were degenerate for $\Lambda=0$. Thus, the clear level structure of the non- interacting case vanishes as $\Lambda$ increases, which makes it more difficult to identify the individual states with regard to the categorization in terms of quasimomentum states. The corresponding mean-field results contain only two doublets for the shallow and one doublet for the deep well, denoted by BdG(1) and BdG(2). All other states, representing multi-particle excitations where more than one boson at a time is excited from the ground state, are absent. Figs. 23(c) and (d) demonstrate in how far the BdG energies can be utilized to anticipate the positions of the missing multi-particle states. As mentioned above, this yields the exact energies for the missing states in the non- interacting system. In particular, multi-particle excitations of putting in total $N^{(1)}$ bosons into the states $(+1)$ and $(-1)$ of the first single- particle band are discussed. With regard to the BEC with $\Lambda=4.0$ in the shallow triple well [panel (c)], one observes that BdG overestimates the excitation energies for the states of $1\leq N^{(1)}\leq 5$, i.e., the accurate many-body values are lower. The discrepancy between mean-field and many-body results grows with $N^{(1)}$. Especially the relative error, defined as $E_{\text{rel}}=\left\|\frac{\omega_{\text{BdG}}\left(N^{(1)}\right)-\omega_{\text{MB}}\left(N^{(1)}\right)}{\omega_{\text{MB}}\left(N^{(1)}\right)}\right\|$ where $\omega_{\text{MB}}\left(N^{(1)}\right)$ denotes the many-body energy with the largest distance to $\omega_{\text{BdG}}\left(N^{(1)}\right)=N^{(1)}\cdot\text{BdG(1)}$, grows quickly to more than $10\%$. It is emphasized again that the BEC is only slightly depleted by approximately $1\%$. For the fragmented BEC with $\Lambda=2.0$ in the deep triple well [panel (d)], the differences become even more substantial because already for the single-particle excitation ($N^{(1)}=1$) the relative error is about $5\%$. This means that in this case BdG does not only miss all multi-particle excited states in the low-energy spectrum, but it is also very inaccurate for the single-particle excitations. Fig. 24 shows a similar analysis for $N=10$ bosons in a larger shallow lattice with 10 sites. In principle, the described many-body effects observed in the triple well also appear in this extended lattice. However, with respect to the degree of fragmentation, which is $f=0.84\%$ for $\Lambda=1.0$, one can deduce that these many-body effects set in even earlier than in the triple well because a comparable depletion is achieved already for weaker repulsion. Hence, an accurate many-body description is of even larger importance if the lattice size is increased. Figure 26: (a) Energies of the zero-quasimomentum modes (ZQMs) for the system in Fig. 23(a) with $\Lambda=4.0$, up to the top of the second single-particle band [denoted by BdG(3)]. All shown states are not obtained by the BdG mean- field approach. The ZQMs are categorized by the symmetry of their density responses, which is either gerade or ungerade on each lattice site. Mind the different index $n_{z}$, which enumerates only the ZQMs compared to the index $n$ of the upper figures which enumerates all excitations. (b) and (c): Real part of the density responses of the states $(3,0;0,0)$ and $(0,3;0,0)$ [panel (c)] as well as $(4,1;0,0)$ and $(1,4;0,0)$ [panel (d)]. The vertical lines separate the lattice sites from each other. Notice the different scales. All quantities are dimensionless. See text for more details. The figures are taken from Ref. [168]. The results shown in Fig. 25 demonstrate numerical convergence with respect to the number of self-consistent orbitals $M$ utilized. In all cases, $M=7$ orbitals clearly suffice for the low-energy excitations. In particular, for the middle panel describing a BEC in the deep triple well, one can see that firstly, $M=3$ orbitals already give very accurate values for the excitation energies, and, secondly, all multi-particle states build up by the states $(+1)$ and $(-1)$ from the first single-particle band are obtained (65 in total). To remind the reader, it is stressed again that the BEC is fragmented in this case, and nevertheless LR-MCTDHB is capable to give converged results. So far, it was shown that the BdG energies for a BEC in a 1D lattice can be largely inaccurate, even for weakly-depleted condensates with $f\approx 1\%$. Though it has been shown analytically that for a BEC with sufficiently weak and long-ranged repulsion in the Hartree limit, i.e., where $N\rightarrow\infty$ with fixed $\Lambda$, the BdG equation yields the exact energies of all single-particle excitations, and the energies of the multi- particle states are given by multiples and sums of the BdG energies [185] (extending the findings of a previous work on homogeneous systems [186]). Whether this result is applicable also for BECs with contact interaction remains unclear. However, naturally the question arises how fast, if at all, the excitation spectrum of a BEC with arbitrary shape of the repulsion approaches the BdG energies upon increasing the number of bosons. This question is discussed further in the subsequent section, Section 6.3.3, where excited states in rotating BECs are analyzed. Figure 27: Low-energy Fourier spectrum of the time evolution of the position variance per particle in $x$-direction, $\frac{1}{N}\Delta^{2}_{\hat{X}}$, due to a quantum quench of the lattice depth from $V_{0}=4.9$ to $V_{0}=5.0$. The system parameters are the same as in Fig. 23(a) with $\Lambda=4.0$. The many- body spectrum (dashed red), obtained from the evolution with $M=5$ orbitals, shows two sharp peaks referring to two-particle excitations and one less intense peak referring to a three-particle excitation. The first intense peak refers to the two-particle excitation $(1,1;0,0)$, whereas the second intense peak refers to $(1,0;0,1)$ [or $(0,1;1,0)$]. The less intense peak refers to $(3,0;0,0)$ [or $(0,3;0,0)$]. All obtained peaks are ZQMs with gerade symmetry. The GP spectrum (solid black) in the shown energy range is completely flat. The depletion of the BEC is $f\sim 1.1\%$ during the entire propagation time. All quantities are dimensionless. See text for more details. Finally, the zero-quasimomentum modes (ZQMs) are briefly discussed. The latter are defined in this work as the excitations where $\text{MOD}(P,L)=0$, with $P$ denoting the total quasimomentum up to the second single-particle band given by $P=n_{+1}^{(1)}+n_{+1}^{(2)}-n_{-1}^{(1)}-n_{-1}^{(2)},$ (192) and $L$ denoting the number of lattice sites. The spectrum of the ZQMs up to BdG(3), which marks the top of the second single-particle band, is shown in Fig. 26(a). The system parameters are the same as in Fig. 23(a) with $\Lambda=4.0$. It is worth noting that all of the shown ZQMs are multi- particle excitations, meaning that BdG does not account for a single ZQM in the low-energy spectrum although there exist more than 20. By analyzing the response densities of the ZQMs, it was found that they can be categorized by their symmetry with respect to the individual wells, which is either gerade or ungerade. Specific examples are given in Figs. 26(b) and (c). The ZQMs are most likely the easiest states to excite in an experiment where the BEC is subject to either a quench of the potential or the interaction strength. For example, a simple quench scenario to excite the gerade ZQMs would be a sudden change of the lattice depth. To this end, the ground state of a BEC with $N=10$ bosons in the triple well with interaction parameter $\Lambda=4.0$ and lattice depth $V_{0}=4.9$ is calculated. Then, the depth is suddenly quenched to $V_{0}=5.0$, and the subsequent time-evolution is investigated. Fig. 27 shows the Fourier spectrum of the dynamical evolution of the position variance per particle, $\frac{1}{N}\Delta^{2}_{\hat{X}}$, obtained from both MCTDHB($5$) and GP. Shown is the energy range $5.0\leq\omega\leq 15.5$. At the many-body level, 3 distinct peaks are observed, whose positions correspond to the first, second, and fourth gerade excitations of the spectrum in Fig. 26(a). They can thus be labeled according to the occupation of (+1) and (-1) from the first and second single-particle band. The peaks corresponding to the two-particle excitations at $\omega\sim 6.35$ and $\omega\sim 13.48$ are more intense than the peak corresponding to the three-particle excitation at $\omega\sim 9.02$. Excitations where higher particle numbers are involved are not intense enough to be distinguished from the background noise. At the GP level, the spectrum is flat, which means that none of the ZQMs obtained at the many-body level is included in the mean-field dynamics, which was anticipated due to the multi-particle nature of the ZQMs. To summarize, it was demonstrated that the excitation spectrum of a weakly- depleted BEC in a 1D optical lattice shows clear many-body effects which are not accounted for at the mean-field level. A depletion of $f\sim 1\%$ is sufficient to observe these effects. In larger lattices, the sensitivity to many-body features even increases. It was further shown that a special type of many-body excited states (ZQMs) are easily accessible by performing a simple quantum quench to the system. A promising quantity to detect these states is the position variance, as already seen in the previous section. #### 6.3.3 Rotating Bose-Einstein condensates in an anharmonic trap After the discussion of many-body excited states in 1D trapped BECs, an example in 2D, namely the case of rotating BECs in an anharmonic, radially- symmetric potential crater, is examined. Of particular interest is the weakly- interacting regime in which the ground-state depletion is marginal and one would naively expect mean-field approaches like the BdG theory to be valid and accurate. The results below were recently published in Ref. [187] and it is referred to this work for additional details. Rotating BECs were of high interest in the past. Research dealt with the occurrence of quantized vortices [161, 188, 162, 163, 164] or with the similarity to the fractional quantum Hall effect for charged particles in the vicinity of a magnetic field [189, 191, 190, 192]. Moreover, low-lying excitations in such systems were addressed, for example in vortex lattices [26, 193, 194, 195]. Furthermore, investigations were made on the decay of the counter-rotating quadrupole mode [196], on Tkachenko modes [197, 198], on the twiston spectrum [199], or on excitations in anharmonic traps [200, 201]. In the latter works, the mean-field excitations were calculated by utilizing the BdG theory. Computations employing a many-body description arerather rare. An examples is the analysis of the yrast spectra in a harmonic confinement obtained by exact diagonalization using the lowest Landau level approximation [203, 204, 202, 205]. As mentioned above, accurate many-body results for the low-energy spectrum of rotating BECs in an anharmonic, radially-symmetric trapping potential are presented. Especially, the effect of the angular momentum in the BECs’ ground states on the excitation energies is discussed. As for the examples in the earlier sections, the BdG and many-body results are compared to each other. Figure 28: (a)-(e) Ground-state densities of rotating BECs with $N=10$ bosons and interaction parameter $\Lambda=0.5$ for different vorticities $l$. Results are shown for MCTDHB($7$), the corresponding GP densities look alike (not shown). For non-zero vorticity, the ground state is a single vortex whose core size grows with its charge $l$. (f) Low-energy excitation spectra atop the ground states from the upper panels. Many-body results obtained by LR- MCTDHB($7$) are denoted by red and blue squares, the BdG results are denoted by black and blue lines. Once the ground state has non-zero vorticity, deviations between the mean-field and many-body excitation energies are observed, and they grow with $l$. Moreover, the density of (many-body) excited states grows strongly. All ground states are only slightly depleted, with a maximum of $f\approx 0.38\%$ for $l=4$. Symbols colored in blue refer to the (+1) excitation energies described in the main text. All quantities are dimensionless. See text for details. The figure is adapted from Ref. [187]. The crater in which the BEC is confined reads $V(r)=\begin{cases}C\,e^{-0.5\,(r-R_{C})^{4}}\,\,&\mbox{if }r\leq R_{C}\\\ C\,\,&\mbox{if }r>R_{C}\end{cases}$ (193) which is essentially the same as in Eq. (179) but without the central ring- shaped barrier. Here, the radius of the crater is set to $R_{C}=3.0$ and the height of the wall is kept at $C=200.0$. Furthermore, the same Gaussian repulsion as in Eq. (180) is employed. Excitations in this system are analyzed in the rotating frame of reference, where the Hamiltonian $\hat{H}$ in Eq. (2) contains an additional contribution accounting for the rotation around the $z$-axis. The whole Hamiltonian thus reads $\hat{H}_{\text{rot}}=\hat{H}-\Omega_{\text{rot}}\hat{L}_{z}$ (194) where $\Omega_{\text{rot}}$ denotes the angular velocity around the $z$-axis and $\hat{L}_{z}$ denotes the operator of the angular momentum in the $z$-direction. A LR analysis is applied atop the ground states of $N=10$ bosons with weak repulsion strength $\Lambda=0.5$ for different values of $\Omega_{\text{rot}}$, and thus of the angular momentum per particle $l$, which is referred to as the vorticity or the charge below. The degree of condensation, $N(1-f)$, ranges from $9.999$ ($l=0$) to $9.962$ ($l=4$) out of 10 particles, meaning that the BEC is highly condensed for all considered parameter values. The upper panels in Fig. 28 show the ground-state densities calculated with MCTDHB($7$) for vorticities $0\leq l\leq 4$. Whereas for $l=0$, the density resembles a Gaussian in the center of the crater, the shape of the densities for $l>0$ indicate that the ground state is a vortex with a density node at its core. One can further see that this core is growing with $l$. The corresponding GP densities look alike (not shown). The lower panel in Fig. 28 compares the low-energy excitation spectrum calculated atop these ground states, both for the mean-field BdG case (black and blue lines) and for LR-MCTDHB($7$) (red and blue squares). The excitations denoted by blue symbols are explained below. At first, the spectrum for $l=0$ is examined. Counting the number of excited states, one obtains 2 states from the BdG approach and 3 states from the many-body approach. Moreover, the energies of the two mean-field excitations coincide with the corresponding many-body results. Since the BdG theory only accounts for single-particle excitations, the third excited stated obtained from LR-MCTDHB is an excitation where more than one particle is excited from the ground state. In particular, it represents the state where two bosons are put into the orbital corresponding to the first excitation. The latter single-particle excitation is referred to as (+1) since it describes the case of putting a boson from the ground-state mode with angular momentum $l_{z}=0$ to an orbital with $l_{z}=1$. Examining the spectra for $l>0$, one observes two major changes. At first, the density of excitations in the low-energy spectrum clearly grows with $l$, which can be deduced from the large number of many-body excited states entering. The number of mean-field excitations also grows (from 2 states for $l=0$ to 4 states for $l=4$), but very slightly compared to the number of many-body states. Therefore, the amount of excited states that BdG misses is strongly growing with the vorticity of the BEC’s ground state. The second observation is that the values of the single-particle energies from BdG and LR-MCTDHB do not coincide any longer, as it was the case for $l=0$. In fact, the entire BdG spectrum for vorticity $l=4$ appears to be completely unrelated to the corresponding many-body spectrum. The increasing gap between the (+1) results of the two approaches is clearly visible. Fig. 29 addresses the evolution of the energy of (+1), denoted by $\omega_{(+1)}$, with respect to the angular velocity $\Omega_{\text{rot}}$. Shown are the results obtained from the BdG theory, as well as many-body results for different numbers of bosons ($N=10,\,100$ and $1000$). In the $l=0$ sector, all energies coincide, indicating that for this case the BdG theory is sufficient. Henceforth, this obviously changes. With growing vortex charge of the ground state, the mean-field and many-body lines for (+1) separate further from each other. Most importantly, increasing the number of bosons in the condensate only marginally changes this, meaning that the many- body and mean-field results for large $N$ approach each other very slowly (see inset). As mentioned in the previous section, it was proven analytically that for a trapped BEC with sufficiently weak and long-ranged repulsion, the excitation energies in the Hartree limit converge towards the BdG energies [185]. The same is believed to happen for a rotating BEC under certain conditions [206]. Figure 29: Excitation energies $\omega_{(+1)}$ for different angular velocities $\Omega_{\text{rot}}$. Compared are the BdG energies and the many- body results for different numbers of bosons $N$, obtained by LR-MCTDHB. The interaction strength is $\Lambda=0.5$. The transition from ground-state vorticity $l$ to $l+1$ between two adjacent analyzed values of $\Omega_{\text{rot}}$ are denoted by dotted vertical lines (mean-field in black and many-body in red). The higher the ground-state vorticity $l$, the larger is the separation between the BdG and many-body energies. Inset: Magnified view for $\Omega_{\text{rot}}=3.8$. All quantities are dimensionless. See text for details. The figure is adapted from Ref. [187]. Figure 30: Evolution of the energetic difference $\Delta_{N}$ between the mean-field and many-body results for $\omega_{(+1)}$ for different particle numbers $N$. The differences are shown relative to $\Delta_{10}$, i.e., the gap for $N=10$ bosons. Solid lines denote exponential fits. The gap becomes larger as $l$ grows. Even for $l=1$ and $N=1000$, it is approximately $97\%$ of $\Delta_{10}$, and the slopes of the fit curves do not indicate a sharp decent towards zero if $N$ is increased further. All quantities are dimensionless. See text for details. The figure is taken from Ref. [187]. In Fig. 30, the dependence of the difference $\Delta_{N}$ between the mean- field and many-body energies of (+1) on the number of bosons in the BEC is analyzed in more detail. The results are given relative to the energy for $N=10$ bosons, denoted by $\Delta_{10}$. For all shown vorticities, the gap becomes smaller with increasing $N$. However, it is obvious that even for BECs with an experimentally relevant number of bosons ($N=1000$) one is still far away from the predictions of the BdG theory. For $l=1$, one obtains that the gap size $\Delta_{1000}$ is still about 97$\%$ of $\Delta_{10}$, and correspondingly higher for $l>1$. Additionally, the shape of the exponential fit curves to the numerical data does not show any sign of a steep descend when $N$ is increased further. It can firstly be deduced that the excitation energies approach the BdG predictions for $l>0$ very slowly, and, secondly, one apparently needs a very high number of particles in order to enter the regime where BdG yields approximately the correct results. However, it is important to note that the convergence towards the BdG excitation energies in the Hartree limit is not proven for the system under consideration, since Ref. [206] only deals with the case of a rotating BEC with broken radial symmetry. The latter situation appears, e.g., for a BEC in a harmonic confinement when the rotation is fast enough to generate multiple quantized vortices in the ground state (see, e.g., Ref. [207]). Although the numerical results cannot give a definite answer for the Hartree limit, they at least suggest that the regime in which one is clearly in need of an accurate many-body theory for the excitation energies covers the experimentally relevant regime of rotating BECs, with $N$ being of the order of $10^{3}-10^{6}$ bosons. A further discussion on the applicability of the GP and BdG theories in the Hartree limit can be found in Appendix A. Figure 31: Low-energy Fourier spectrum of the time evolution of the position variance in $x$-direction, $\frac{1}{N}\Delta^{2}_{\hat{X}}$, due to a quantum quench of the radius $R_{C}$ from $3.01$ to $3.0$. The number of bosons is $N=10$. The many-body spectrum (red), obtained from the evolution with $M=7$ orbitals, shows two sharp peaks in the considered energy range, whereas the GP spectrum (black) is flat. The first peak refers to the two-particle excitation $(1_{+1},1_{-1})$, whereas the second peak indicates a de-excitation between two higher-lying states (see right inset). The left inset shows the evolution of $\frac{1}{N}\Delta^{2}_{\hat{X}}$ from $0\leq t\leq 30$. The depletion of the BEC stays below $0.4\%$ during the entire propagation. All quantities are dimensionless. See text for details. It is finally suggested, similar to the 1D systems discussed in Section 6.3.1, how a quench can be utilized in order to excite breathing modes. To this end, the ground state of a BEC with $N=10$ bosons and angular velocity $\Omega_{\text{rot}}=3.2$ in the same trap as described in Eq. (193), but with a slightly larger radius of $R_{C}=3.01$, is calculated. The obtained ground state has the same symmetry and vorticity $l=3$ as the one for $R_{C}=3.0$ and $\Omega_{\text{rot}}=3.2$. Then, the crater is suddenly shrinked back to $R_{C}=3.0$ and the dynamical evolution is computed with MCTDHB($7$), which is sufficient in order to obtain converged dynamics for the position variance. As could be seen in the previous sections, the latter quantity is very sensitive for many-body effects, and therefore, a spectral analysis of its dynamics by computing the Fourier spectrum of $\frac{1}{N}\Delta^{2}_{\hat{X}}(t)$ is carried out. Due to the radial symmetry, one could as well analyze the position variance in the $y$-direction. The result are shown in Fig. 31. The many-body spectrum shows two distinct peaks, where the first corresponds to the $(1_{+1},1_{-1})$ two-particle excitation, where one boson is excited to an orbital with $l=4$ and one is excited to an orbital with $l=2$, such that the total angular momentum is unchanged. The energy that LR-MCTDHB($7$) predicts for this state is $\omega=0.605$ which coincides with the peak position of the Fourier analysis. The second peak at $\omega\approx 1.12$ denotes a de-excitation between two states that are much higher in energy ($\omega\approx 10.6\rightarrow 9.47$, see right inset in Fig. 31). The occurrence of de-excitations is an indication that the quench described above already leads to second-order effects, meaning that one has left the LR regime. With regard to the mean-field spectrum from the GP dynamics, one observes that in the shown energy range, the Fourier spectrum is completely flat. The absence of the first peak is anticipated because the GP theory does not account for two-particle excitations by construction. The absence of the second peak can be understood from the right inset where the Fourier spectrum in the interval $9\leq\omega\leq 11$ is shown. One sees that in the GP case, there is only one peak at $\omega\approx 10.6$, but no second one at $\omega\approx 9.47$ as in the many-body spectrum. Thus, very different spectra from the dynamics of the variance at the mean-field and many-body levels are obtained, and it is shown that the suggested quench leads to an excitation of a many-body breathing mode. It is stressed that the depletion stays below $0.4\%$ for the entire propagation, which means that the BEC is essentially condensed. To summarize, the low-energy spectrum of a rotating BEC in an anharmonic, radially-symmetric crater shows significant many-body effects, even for weakly-interacting, highly-condensed systems. These effects can be observed already for the lowest-in-energy, single-particle excited states, and remain even for mesoscopic and large BECs. Furthermore, a simple quench triggers a many-body breathing mode, which can be observed, e.g., by analyzing the Fourier spectrum of the position variance. ## 7 Summary and Conclusions This review presents recently developed many-body approaches capable to describe and compute accurately the ground state and dynamics as well as the energy spectrum of trapped ultracold bosons. It is relevant to employ methods which are on the same footing. The dynamics is calculated by the well-known MCTDHB method, the ground state is obtained by imaginary-time propagation using the same method, and the energy spectrum of the system is then determined by applying the linear-response theory to the ground state computed via MCTDHB. The resulting theory is named LR-MCTDHB. The MCTDHB theory is briefly discussed and more attention is paid to the less spread LR-MCTDHB theory. As the implementation of the MCTDHB is well documented in the literature, only references are given here, while the newly developed LR-MCTDHB implementation which makes the theory widely applicable, is presented here in some detail for the first time. The latter posed a major challenge due to both the complexity of the underlying theory and the accompanying technical difficulties like the vast memory consumption and the necessity of parallelization. The code structure is explained in detail both with respect to the construction of the linear-response matrix and its subsequent partial diagonalization which provides the desired excitation energies. A comprehensive description of the procedure used to calculate the lowest eigenvalues is also provided. The corresponding benchmarks of the LR- MCTDHB code (see also Appendix C) with an analytically solvable many-boson non-trivial model demonstrate its correct functioning and show that the theory provides numerically exact results. The second main goal of this review is to present applications to many-body dynamics and excitations of BECs in interesting scenarios, and, of course, to discuss the underlying physics. Several systems of trapped BECs in one and two dimensions are addressed in detail and the main attention is drawn to the detection of many-body effects that are not describable by applying mean-field theory. Particularly emphasized are the tunneling dynamics in double-well systems and the development of fragmentation as a function of time in initially coherent BECs. The applications shown in this work have demonstrated that a full many-body treatment of the dynamics of trapped interacting BECs is inevitable. For instance, it is exhibited that the out-of-equilibrium tunneling dynamics of initially coherent BECs in 1D and 2D double wells are of many-body nature. After a few Rabi cycles, tunneling between the wells becomes suppressed at the many-body level, whereas unperturbed oscillations are predicted at the mean-field level. Surprisingly, the degree of fragmentation in 1D is universal for constant values of the interaction parameter. Going to 2D, the sensitivity of the tunneling dynamics to many-body effects is even enhanced as soon as the bosons carry angular momentum. This can also be deduced from the larger amount of orbitals necessary in the limit of weak repulsion as compared to the case of bosons with zero angular momentum. Further applications deal with the nucleation of phantom vortices, which are pure many-body objects not observable in the condensate density, as well as with the tunneling of bosons to open space. In both cases, the initially coherent systems become fragmented and are, therefore, in need of an accurate many-body description. With respect to excitations, it is observed that the ground-state depletion has a strong impact on the low-energy spectrum. Even for marginal depletion where one might expect mean-field approaches to give accurate results, substantial many-body effects like the shift of excitation energies and the appearance of complicated many-boson excitations occur. Applications in 1D explores the many-body effects in double- and triple-well systems as well as in larger lattices. In 2D, it is found that once the bosons carry angular momentum and the ground state becomes a vortex, the deviations between the mean-field and many-body excitation spectra grow strongly. Moreover, this discrepancy tends to persist when the number of particles in the condensate increases up to an experimentally relevant size. This has a potentially large impact on experiments on ultracold bosons since the dynamics are mostly determined by the lowest-in-energy excitations. Hopefully, these findings and the new insights into the physics of ultracold bosons will stimulate future experiments. Before closing, we would like to add several remarks concerning further general and technical developments of the methods which can further broaden their range of applicability. Until now the applications of the many-body linear-response theory were for excitations on top of the system’s ground state. We emphasize that the LR-MCTDHB can also be applied to any other state obtained via MCTDHB. In other words, it is possible to compute excitations on top of any other state. This will help to address excitations which are difficult to reach from the ground state. Another general possibility lies in the formulation of LR-MCTDHB in block-diagonal form, which is potentially advantageous in calculating the low-energy spectrum, as discussed in Appendix B. Apart from these, further general extensions and developments of both the LR theory itself as well as of its numerical implementation are worth of additional investigation. With regard to theoretical developments, an extension of LR-MCTDHB to BEC mixtures, i.e., compound systems of different types of bosons, is highly desirable as their popularity has substantially grown [208, 184, 209, 210, 211, 212, 213]. The underlying MCTDHB theory for bosonic mixtures, also including the possibility of internal degrees of freedom, has been formulated [53, 70, 71], but a linear-response theory is completely missing. It turns out that the so-called multi-layer MCTDHB, introduced and described in Ref. [70], is particularly advantageous numerically. To this end, an interesting goal would be to apply the linear- response theory to multi-layer MCTDHB. Moreover, mixtures of bosons and fermions might become even more relevant [71, 214, 218, 215, 216, 217, 219], meaning that also the equations of motion of the fermionic counterpart of MCTDHB, called MCTDHF (see, e.g., Ref. [220] and references therein, and the recently developed implementation described in Ref. [221]), should be utilized to develop the respective linear-response theory. Further technical developments which accelerate the computations are, of course, also welcome. One interesting possible extension lies in the application of linear-response theory to the time-dependent restricted-active- space self-consistent-field (TD-RASSCF) method, introduced in Ref. [222]. The latter theory is similar to MCTDHB, but is an approximation to it. It is, however, capable of reducing systematically the amount of necessary coefficients significantly. This is achieved by restricting the possible excitations in the active space of the single-particle orbitals according to certain conditions or symmetries of the physical problem at hand. Thus, the dimensionalities of the linear-response matrices which would result when constructed based on the linear-response theory on top of TD-RASSCF, could also be greatly reduced. Such an approach will substantially simplify the computation of the excitation spectrum and, moreover, enable the treatment of even larger systems, i.e., with more particles and if necessary higher numbers of orbitals. With regard to the technical development of the current implementation, the application of additional techniques to diagonalize the often huge linear- response matrix can be beneficial. A potentially very powerful method may be the FEAST algorithm [224], which is a contour integration and density matrix- based method. It is available in a free software package [225] and also allows for high-level parallelization. In particular, it enables the specification of an energy interval in which the desired eigenvalues should be found. Hence, it might become easier to explore not just the lowest but also intermediate eigenvalues. The above outlined avenues for future research demonstrate the large variety of perspectives, which this work will hopefully stimulate. ## Appendix A Mean-field theory In this Appendix, several mean-field approaches to describe the ground state and dynamics as well as the excitation spectra of trapped BECs are presented. At first, the GP single-orbital mean-field theory (Section A.1.1) and the LR theory atop it, yielding the famous BdG equations (Section A.1.2), are described. Those approaches are the most common ones in the literature, and are expected to accurately describe the physics of coherent BECs. They are also used in this work to compare mean-field and many-body results. Afterwards, a time-dependent multi-orbital mean-field approach, called TDMF (Section A.2.1), and its corresponding LR theory LR-BMF (Section. A.2.2), are discussed. The latter two theories are suitable for describing fragmented BECs at the mean-field level. They can be understood as bridging theories between the GP/BdG theories and the full many-body descriptions given by MCTDHB and LR-MCTDHB, respectively. ### A.1 Single-orbital approaches #### A.1.1 The GP equation The GP equation is commonly used to obtain the ground state and the time evolution of a coherent BEC, i.e., where all $N$ bosons occupy the same single-particle state $\phi(\mathbf{r},t)$. The many-boson wave function $\Psi$ is thus given by a single Hartree product, $\Psi(\mathbf{r}_{1},...,\mathbf{r}_{N};t)=\phi(\mathbf{r}_{1},t)\,...\,\phi(\mathbf{r}_{N},t)\equiv\|N;t\rangle.$ (A.1) Originally, the GP equation is derived assuming contact interaction between the bosons. However, to be more general, the two-body interaction potential is taken to be of general form $\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})$. The GP equation can be derived from the least action principle defined in Eq. (25). The expectation value that appears in the action functional, one obtains $\left\langle N;t\left\|\hat{H}-i\frac{\partial}{\partial\,t}\right\|N;t\right\rangle=Nh_{\text{\tiny{GP}}}+\frac{\lambda_{0}N(N-1)}{2}W_{\text{\tiny{GP}}}-iN\left(\frac{\partial}{\partial t}\right)_{\text{\tiny{GP}}}$ (A.2) where the quantities $h_{\text{\tiny{GP}}}$ and $W_{\text{\tiny{GP}}}$ are defined in Eqs. (21) and (22) utilizing $\phi(\mathbf{r},t)$. The time derivative is given by $\left(\frac{\partial}{\partial t}\right)_{\text{\tiny{GP}}}=\int\phi^{\ast}(\mathbf{r},t)\,\frac{\partial}{\partial t}\,\phi(\mathbf{r},t)\,d\mathbf{r}$. One arrives at $\displaystyle 0$ $\displaystyle=\frac{\delta S}{\delta\phi^{\ast}(\mathbf{r},t)}\left[Nh_{\text{\tiny{GP}}}+\frac{\lambda_{0}N(N-1)}{2}W_{\text{\tiny{GP}}}-iN\left(\frac{\partial}{\partial t}\right)_{\text{\tiny{GP}}}\right]$ $\displaystyle=N\left[\hat{h}+\lambda_{0}(N-1)\int\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\|\phi(\mathbf{r}^{\,\prime},t)\|^{2}d\mathbf{r}^{\,\prime}-i\frac{\partial}{\partial t}\right]\phi(\mathbf{r},t).$ (A.3) Thus, the time-dependent GP equation used in this work is thus given by $\left[\hat{h}+\Lambda\int\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\,\|\phi(\mathbf{r}^{\,\prime},t)\|^{2}\,d\mathbf{r}^{\,\prime}\right]\phi(\mathbf{r},t)=i\frac{\partial}{\partial t}\phi(\mathbf{r},t)$ (A.4) with the mean-field interaction parameter $\Lambda=\lambda_{0}(N-1)$. As mentioned above, for the contact interaction potential it takes the original form as derived in [18, 19] which can also be found in standard textbooks [20, 21]. The time-independent GP equation reads $\left[\hat{h}+\Lambda\int\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\,\|\phi^{0}(\mathbf{r}^{\,\prime})\|^{2}\,d\mathbf{r}^{\,\prime}\right]\phi^{0}(\mathbf{r})=\mu\phi^{0}(\mathbf{r})$ (A.5) where $\mu$ is the chemical potential. Whereas Eq. (A.4) describes the time- evolution of a fully-condensed BEC, Eq. (A.5) is used to determine the GP ground-state orbital $\phi^{0}(\mathbf{r})$. This section is closed with a brief discussion of the applicability of the GP theory in the Hartree limit of repulsive trapped BECs, meaning for $N\rightarrow\infty$ with fixed interaction parameter $\Lambda$. It has been proven analytically that in this limit, the GP theory provides the exact energy and density per particle of the system’s ground state [226]. This, however, does not necessarily mean that the corresponding GP wave function, Eq. (A.1), coincides with the exact wave function in this limit. To this end, it has been demonstrated recently that the overlap of the exact and GP wave function in the Hartree limit can be clearly less than unity or even be vanishingly small [227]. Moreover, a subsequent study has shown the many-body character of the exact wave function in the Hartree limit by applying many- body perturbation theory to the mean-field Hamiltonian [228], which demonstrates that the mean-field and exact wave functions can be very different from each other. Moreover, as a further interesting result, it was found that the condensate depletion in the Hartree limit becomes a constant. These findings potentially have severe consequences for the excitation spectrum of a BEC in or close to the Hartree limit, which is discussed at the end of Section A.1.2. #### A.1.2 The BdG equations The LR theory atop the time-dependent GP equation, Eq. (A.4), is derived below. Therefore, a weak time-dependent perturbation is added to the Hamiltonian, i.e., $\hat{H}(\mathbf{r})\rightarrow\hat{H}(\mathbf{r})+\hat{H}_{\text{ext}}(\mathbf{r},t)$. The perturbation $\hat{H}_{\text{ext}}(\mathbf{r},t)$ explicitly reads $\hat{H}_{\text{ext}}(\mathbf{r},t)=f^{+}(\mathbf{r})e^{-i\omega t}+f^{-}(\mathbf{r})e^{i\omega t}$ (A.6) with the real amplitudes $f^{+}(\mathbf{r})$ and $f^{-}(\mathbf{r})$ and the frequency $\omega$. The response of the system is calculated around the stationary GP ground-state orbital $\phi^{0}(\mathbf{r})$ obtained from Eq. (A.5). For the perturbed orbital $\phi(\mathbf{r},t)$, the ansatz $\sqrt{N}\phi(\mathbf{r},t)=e^{-i\mu t}\left(\sqrt{N}\phi^{0}(\mathbf{r})+u(\mathbf{r})e^{-i\omega t}+v^{\ast}(\mathbf{r})e^{i\omega t}\right)$ (A.7) is utilized. In Eq. (A.7), the quantities $u(\mathbf{r})$ and $v(\mathbf{r})$ denote small and time-independent response amplitudes. Plugging in Eqs. (A.6) and (A.7) into the time-dependent GP equation, Eq. (A.4), yields in zeroth order the time-independent GP equation, Eq. (A.5). In first order, collecting all terms proportional to $e^{-i\omega t}$ leads to $\displaystyle(\omega+\mu)u(\mathbf{r})$ $\displaystyle=\sqrt{N}f^{+}(\mathbf{r})\phi^{0}(\mathbf{r})+\hat{H}_{\text{\tiny{GP}}}u(\mathbf{r})+\Lambda\int\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r}^{\,\prime})v(\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r})d\mathbf{r}^{\,\prime}$ $\displaystyle+\Lambda\int\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi^{0,\ast}(\mathbf{r}^{\,\prime})u(\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r})d\mathbf{r}^{\,\prime}$ (A.8) where the GP Hamiltonian $\hat{H}{\text{\tiny{GP}}}$ is defined as $\hat{H}{\text{\tiny{GP}}}=\hat{h}+\Lambda\int\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\,\|\phi^{0}(\mathbf{r}^{\,\prime},t)\|^{2}\,d\mathbf{r}^{\,\prime}.$ (A.9) Collecting all terms proportional to $e^{i\omega t}$ results in $\displaystyle(\mu-\omega)v^{\ast}(\mathbf{r})$ $\displaystyle=\sqrt{N}f^{-}(\mathbf{r})\phi^{0}(\mathbf{r})+\hat{H}_{\text{\tiny{GP}}}v^{\ast}(\mathbf{r})+\Lambda\int\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r}^{\,\prime})u^{\ast}(\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r})\,d\mathbf{r}^{\,\prime}$ $\displaystyle+\Lambda\int\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi^{0,\ast}(\mathbf{r}^{\,\prime})v^{\ast}(\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r})\,d\mathbf{r}^{\,\prime}.$ (A.10) Complex conjugation and multiplication by $(-1)$ of Eq. (A.1.2) opens the possibility to cast Eqs. (A.1.2) and (A.1.2) in matrix form. The result reads $\left(\mathcal{L}_{\text{BdG}}-\omega\right)\begin{pmatrix}u(\mathbf{r})\\\ v(\mathbf{r})\end{pmatrix}=\begin{pmatrix}-\sqrt{N}f^{+}(\mathbf{r})\phi^{0}(\mathbf{r})\\\ \sqrt{N}f^{-,\ast}(\mathbf{r})\phi^{0,\ast}(\mathbf{r})\end{pmatrix}$ (A.11) with the BdG matrix $\mathcal{L}_{\text{BdG}}=\begin{pmatrix}A&B\\\ -B^{\ast}&-A^{\ast}\end{pmatrix}$ (A.12) where the quantities $A$ and $B$ are given by $A=\hat{H}_{\text{\tiny{GP}}}-\mu+\Lambda\int d\mathbf{r}^{\,\prime}\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi^{0,\ast}(\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r})\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}$ (A.13) and $B=\Lambda\int d\mathbf{r}^{\,\prime}\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r}^{\,\prime})\phi^{0}(\mathbf{r})\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}\,.$ (A.14) where the operator $\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}$ was introduced in Eq. (96). In the literature, Eq. (A.11) with contact interaction $\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})=\delta(\mathbf{r}-\mathbf{r}^{\,\prime})$ and zero RHS, i.e., $\left(\mathcal{L}_{\text{BdG}}-\omega\right)\begin{pmatrix}u(\mathbf{r})\\\ v(\mathbf{r})\end{pmatrix}=0,$ (A.15) is referred to as the BdG equations [23, 24]. In this work, however, the interaction potential is in general different from the contact interaction potential. Moreover, the particle-conserving BdG equations [229, 230, 231, 232], given by $\mathbf{\mathcal{P}}\mathcal{L}_{\text{BdG}}\mathbf{\mathcal{P}}\begin{pmatrix}u(\mathbf{r})\\\ v(\mathbf{r})\end{pmatrix}=\omega\begin{pmatrix}u(\mathbf{r})\\\ v(\mathbf{r})\end{pmatrix}$ (A.16) where the projector $\mathbf{\mathcal{P}}=\mathbb{1}-\|\phi^{0}\rangle\langle\phi^{0}\|$ removes any contribution of the ground-state orbital $\phi^{0}(\mathbf{r})$ from the response amplitudes $u(\mathbf{r})$ and $v(\mathbf{r})$, are used to calculate the low-energy excitation spectra at the mean-field level. From the above derivation one can readily see the close connection of the GP orbital and the BdG equations. As discussed in the previous section on the GP theory, the wave function of the system in the Hartree limit can be very different from the GP wave function, although the latter gives the exact energy and density per particle. Thus, it can be questioned whether the LR theory atop the GP wave function, resulting in the BdG equations, is in general reliable in the Hartree limit. Scientific works that prove the exactness of the BdG spectrum in this limit under certain conditions were discussed in Section 6.3.3. Nevertheless, the example of a rotating BEC in an anharmonic trap, which does not fulfill the assumptions made in Ref. [206], serves as an example where at least in the experimentally relevant regime of $N\approx 10^{3}-10^{6}$ bosons in the condensate the lowest-in-energy excitations substantially deviate from the BdG predictions. ### A.2 Multi-orbital approaches #### A.2.1 Time-dependent multi-orbital mean field (TDMF) The TDMF theory, introduced in Ref. [233], is briefly described. It can be seen as an intermediate step between the GP theory and MCTDHB because it allows for more than one single-particle orbital that the bosons can occupy, and is thus applicable to fragmented condensates. However, as a major difference to the full many-body description in Section 3, the occupation of these single-particle orbitals is fixed. The ansatz for the $N$-boson wave function is given by $\Psi(\mathbf{r}_{1},...,\mathbf{r}_{N};t)=\hat{\mathcal{S}}\phi_{1}(\mathbf{r}_{1},t)\phi_{2}(\mathbf{r}_{2},t)\,...\,\phi_{N}(\mathbf{r}_{N},t)$ (A.17) where the operator $\hat{\mathcal{S}}$ accounts for the symmetrization of the wave function. In general, not all single-particle orbitals $\phi_{i}$ need to be different. The only constraint is that the number of particles is fixed such that $N=\sum_{i=1}^{M}n_{i}$ where $M$ is the number of different orbitals in Eq. (A.17) and the $n_{i}$ are their fixed occupation numbers. Applying the least action principle to $S=\int dt\left(\langle\Psi\|\hat{H}-i\frac{\partial}{\partial t}\|\Psi\rangle-\sum_{i,j=1}^{M}\mu_{ij}(t)[\langle\phi_{i}\|\phi_{j}\rangle-\delta_{ij}]\right)$ (A.18) with the time-dependent Lagrange multipliers $\mu_{ij}(t)$ yields $\displaystyle 0=\frac{\delta S}{\delta\phi_{k}^{\ast}(\mathbf{r},t)}$ $\displaystyle=n_{k}\left(\hat{h}-i\frac{\partial}{\partial t}+\lambda_{0}(n_{k}-1)\hat{J}_{k}+\sum_{l\neq k}^{M}\lambda_{0}n_{l}(\hat{J}_{l}+\hat{K}_{l})\right)\|\phi_{k}\rangle$ $\displaystyle=\sum_{j=1}^{M}\mu_{kj}(t)\|\phi_{j}(t)\rangle\quad\forall\,k=1,...,M$ (A.19) where $\displaystyle\hat{J}_{l}(\mathbf{r},t)$ $\displaystyle=\int d\mathbf{r}^{\,\prime}\phi_{l}^{\ast}(\mathbf{r}^{\,\prime},t)\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi_{l}(\mathbf{r}^{\,\prime},t)$ (A.20) and $\displaystyle\hat{K}_{l}(\mathbf{r},t)$ $\displaystyle=\int d\mathbf{r}^{\,\prime}\phi_{l}^{\ast}(\mathbf{r}^{\,\prime},t)\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}\phi_{l}(\mathbf{r}^{\,\prime},t)$ (A.21) are the direct and exchange interaction potentials, respectively. Multiplying Eq. (A.19) by $\langle\phi_{l}\|$ gives $\mu_{kl}(t)=n_{k}\left(h_{lk}-\left(\frac{\partial}{\partial t}\right)_{lk}+\lambda_{0}(n_{k}-1)\hat{J}_{k}W_{lkkk}+\sum_{j\neq k}^{M}\lambda_{0}n_{j}(W_{ljjk}+W_{ljkj})\right)$ (A.22) for the Lagrange multipliers. Plugging this into Eq. (A.19) results in $\displaystyle i\mathbf{\hat{P}}\frac{\partial}{\partial t}\|\phi_{k}\rangle=\mathbf{\hat{P}}\left(\hat{h}+\lambda_{0}(n_{k}-1)\hat{J}_{k}+\sum_{l\neq k}^{M}\lambda_{0}n_{l}(\hat{J}_{l}+\hat{K}_{l})\right)\|\phi_{k}\rangle\quad\forall\,k=1,...,M$ (A.23) where the projector $\mathbf{\hat{P}}=\mathbb{1}-\sum_{j=1}^{M}\|\phi_{j}\rangle\langle\phi_{j}\|$ projects onto the tangential space of $\text{span}(\phi_{1},...,\phi_{M})$. To simplify the above EOMs for the orbitals, the condition $\langle\phi_{i}\|\dot{\phi}_{j}\rangle=0\quad\forall\,i,j=1,...,M$ (A.24) is invoked. This leads to the conservation of energy, and the projector on the LHS of Eq. (A.23) can be omitted, i.e., $i\frac{\partial}{\partial t}\|\phi_{k}\rangle=\mathbf{\hat{P}}\left(\hat{h}+\lambda_{0}(n_{k}-1)\hat{J}_{k}+\sum_{l\neq k}^{M}\lambda_{0}n_{l}(\hat{J}_{l}+\hat{K}_{l})\right)\|\phi_{k}\rangle\quad\forall\,k=1,...,M.$ (A.25) It is important to note that the condition in Eq. (A.24) is a manually-invoked restriction to the orbitals within the TDMF theory, whereas the same condition in MCTDHB appears naturally and rigorously. Furthermore, one observes that for $M=1$, i.e., when only a single orbital is available, Eq. (A.25) reduces to the (projected) time-dependent GP equation, Eq. (A.4). The TDMF theory has been extended to describe spinor condensates and Bose-Bose or Bose-Fermi mixtures as well, see in this context Ref. [234]. By imaginary-time propagation of the TDMF EOMs, one obtains the corresponding theory for the stationary equations, called the best mean field for condensates [235]. Applications can be found in, e.g., Refs. [236, 237, 238, 239]. #### A.2.2 Linear-response best-mean-field (LR-BMF) A possible LR approach based on a multi-orbital mean-field description is the LR-BMF theory which has been introduced in Ref. [240]. To derive it, the EOMs of the previously described TDMF theory, Eq. (A.19), are linearized for the case of adding a weak time-dependent external field to the Hamiltonian, i.e., $\hat{H}\rightarrow\hat{H}+\hat{H}_{\text{ext}}(t)$. The perturbation $\hat{H}_{\text{ext}}(t)$ is assumed to be of the same form as in Eq. (A.6). Plugging this into Eq. (A.19) yields $\displaystyle\left(\hat{Z}_{i}-i\frac{\partial}{\partial t}\right)\|\phi_{i}\rangle-\sum_{j=1}^{M}\frac{\mu_{ij}(t)}{n_{i}}\|\phi_{j}\rangle=-\hat{H}_{\text{ext}}(t)\|\phi_{i}\rangle$ (A.26) with the replacement $\hat{Z}_{i}=\hat{h}+\lambda_{0}(n_{i}-1)\hat{J}_{i}+\sum_{l\neq i}^{M}\lambda_{0}n_{l}(\hat{J}_{l}+\hat{K}_{l}).$ (A.27) The Lagrange multipliers, after redefining them as $\mu_{ij}(t)/n_{i}\rightarrow\mu_{ij}(t)$, can be expressed by $\mu_{ij}(t)=\langle\phi_{j}\|\hat{Z}_{i}+\hat{H}_{\text{ext}}(t)\|\phi_{i}\rangle$ (A.28) where the condition of Eq. (A.24) was used. The ansatz for the perturbed orbitals is given by $\phi_{i}(\mathbf{r},t)\approx\phi_{i}^{0}(\mathbf{r})+\delta\phi_{i}(\mathbf{r},t)$ (A.29) Plugging Eq. (A.29) into Eq. (A.26) leads to $\displaystyle-\hat{H}_{\text{ext}}(t)\|\phi_{i}^{0}\rangle$ $\displaystyle=\left(\hat{Z}_{i}^{0}-i\frac{\partial}{\partial t}\right)\|\delta\phi_{i}\rangle-\sum_{j=1}^{M}\mu_{ij}^{0}(t)\|\delta\phi_{j}\rangle$ $\displaystyle+\lambda_{0}(n_{i}-1)\delta\hat{J}_{i}\|\phi_{i}^{0}\rangle+\sum_{j\neq i}^{M}\lambda_{0}n_{j}(\delta\hat{J}_{j}+\delta\hat{K}_{j})\|\phi_{i}^{0}\rangle$ (A.30) $\displaystyle-\sum_{j=1}^{M}\delta\mu_{ij}(t)\|\phi_{j}^{0}\rangle$ with $\displaystyle\delta\hat{J}_{i}(\mathbf{r},t)$ $\displaystyle=\int d\mathbf{r}^{\,\prime}\delta\phi_{i}^{\ast}(\mathbf{r}^{\,\prime},t)\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\phi_{i}^{0}(\mathbf{r}^{\,\prime})+\int d\mathbf{r}^{\,\prime}\phi_{i}^{0,\ast}(\mathbf{r}^{\,\prime})\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\delta\phi_{i}^{0}(\mathbf{r}^{\,\prime},t)$ (A.31) and $\displaystyle\delta\hat{K}_{i}(\mathbf{r},t)$ $\displaystyle=\int d\mathbf{r}^{\,\prime}\delta\phi_{i}^{\ast}(\mathbf{r}^{\,\prime},t)\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}\phi_{i}^{0}(\mathbf{r}^{\,\prime})+\int d\mathbf{r}^{\,\prime}\phi_{i}^{0,\ast}(\mathbf{r}^{\,\prime})\hat{W}(\mathbf{r},\mathbf{r}^{\,\prime})\hat{\mathcal{P}}_{\mathbf{r}\mathbf{r}^{\,\prime}}\delta\phi_{i}^{0}(\mathbf{r}^{\,\prime},t).$ (A.32) The variation of the Lagrange multipliers can be expressed as $\displaystyle\delta\mu_{ij}(t)=-\sum_{l=1}^{M}\mu_{il}^{0}\langle\phi_{j}^{0}\|\delta\phi_{l}\rangle+\langle\phi_{j}^{0}\|\delta(\hat{Z}_{i}\|\phi_{i}\rangle)+\langle\phi_{j}^{0}\|\hat{H}_{\text{ext}}(t)\|\phi_{i}^{0}\rangle$ (A.33) where the identity $\hat{Z}_{i}^{0}\|\phi_{i}^{0}\rangle=\sum_{l=1}^{M}\mu_{il}^{0}\|\phi_{l}^{0}\rangle$ and integration by parts was used in the first term. One finally obtains $\displaystyle\mathbf{\hat{P}}\left[\hat{Z}_{i}^{0}\|\delta\phi_{i}\rangle-\sum_{j=1}^{M}\mu_{ij}^{0}\|\delta\phi_{j}\rangle+\left(\lambda_{0}(n_{i}-1)\delta\hat{J}_{i}+\sum_{j\neq i}^{M}\lambda_{0}n_{j}(\delta\hat{J}_{j}+\delta\hat{K}_{j})\right)\|\phi_{i}^{0}\rangle\right]$ $\displaystyle=-\mathbf{\hat{P}}\hat{H}_{\text{ext}}(t)\|\phi_{i}^{0}\rangle+i\frac{\partial}{\partial t}\|\delta\phi_{i}\rangle$ (A.34) with the projector $\mathbf{\hat{P}}=\mathbb{1}-\sum_{k}\|\phi_{k}^{0}\rangle\langle\phi_{k}^{0}\|$. By first making the ansatz $\delta\phi_{i}(\mathbf{r},t)=\frac{1}{\sqrt{n_{i}}}\left(u_{i}(\mathbf{r})e^{-i\omega t}+v_{i}^{\ast}(\mathbf{r})e^{i\omega t}\right)$ (A.35) with stationary response amplitudes $u_{i}(\mathbf{r})$ and $v_{i}(\mathbf{r})$ and plugging this into Eq. (A.2.2) afterwards, it is again possible, similar to derivation of the BdG equations in Section A.1.2, to group the terms of the resulting equations with respect to their linearity to either $e^{-i\omega t}$ or $e^{i\omega t}$. As a result, one arrives at an eigenvalue equation of the form $(\mathbf{\mathcal{P}}\mathbf{\mathcal{L}}-\omega)\begin{pmatrix}\|\mathbf{u}\rangle\\\ \|\mathbf{v}\rangle\end{pmatrix}=\mathbf{\mathcal{P}}\begin{pmatrix}-f^{+}(\mathbf{r})\|\boldsymbol{\phi_{n}^{0}}\rangle\\\ f^{-,\ast}(\mathbf{r})\|\boldsymbol{\phi_{n}^{0,\ast}}\rangle\end{pmatrix}$ (A.36) with the vector $\|\boldsymbol{\phi_{n}^{0}}\rangle=\|\sqrt{n_{1}}\phi_{1}^{0},...,\sqrt{n_{M}}\phi_{M}^{0}\rangle$, the $2(M\times M)$ square matrix $\mathbf{\mathcal{L}}=\begin{pmatrix}\mathbf{Z^{0}}-\boldsymbol{\mu^{0}}+\mathbf{A}&\mathbf{B}\\\ -\mathbf{B}^{\ast}&-(\mathbf{Z^{0}}-\boldsymbol{\mu^{0}}+\mathbf{A})^{\ast}\end{pmatrix},$ (A.37) and the projection matrix $\mathbf{\mathcal{P}}=\\{\mathcal{P}_{ij}\\}=\begin{cases}\mathbf{\hat{P}}&i=j\leq M\\\ \mathbf{\hat{P}}^{\ast}&i=j>M\\\ 0&i\neq j\end{cases}$ (A.38) of the same size. The matrix $\mathbf{Z^{0}}=\text{diag}(\hat{Z}_{1}^{0},...,\hat{Z}_{M}^{0})$ is a diagonal matrix, whereas $\boldsymbol{\mu^{0}}=\\{\mu_{ij}^{0}\\}$ contains the Lagrange multipliers. The matrices $\mathbf{A}$ and $\mathbf{B}$, for the case of the contact interaction potential, read $\mathbf{A}=\\{A_{ij}\\}=\begin{cases}\lambda_{0}(n_{i}-1)\|\phi_{i}^{0}\|^{2}&i=j\\\ 2\lambda_{0}\sqrt{n_{i}n_{j}}\phi_{i}^{0}\phi_{j}^{0,\ast}&i\neq j\end{cases}$ (A.39) and $\mathbf{B}=\\{B_{ij}\\}=\begin{cases}\lambda_{0}(n_{i}-1)(\phi_{i}^{0})^{2}&i=j\\\ 2\lambda_{0}\sqrt{n_{i}n_{j}}\phi_{i}^{0}\phi_{j}^{0}&i\neq j.\end{cases}$ (A.40) Since the term proportional to $\omega$ is the only one without a projector in Eq. (A.36), a redundant projector can be added to the RHS of $\mathbf{\mathcal{P}}\mathbf{\mathcal{L}}$, i.e., $\mathbf{\mathcal{P}}\mathbf{\mathcal{L}}\rightarrow\mathbf{\mathcal{P}}\mathbf{\mathcal{L}}\mathbf{\mathcal{P}}$. The resulting homogeneous eigenvalue equation $\mathbf{\mathcal{P}}\mathbf{\mathcal{L}}\mathbf{\mathcal{P}}\begin{pmatrix}\|\mathbf{u}\rangle\\\ \|\mathbf{v}\rangle\end{pmatrix}=\omega\begin{pmatrix}\|\mathbf{u}\rangle\\\ \|\mathbf{v}\rangle\end{pmatrix}$ (A.41) is the central equation in the LR-BMF theory. A comparison to the BdG matrix $\mathcal{L}_{\text{BdG}}$ in Eqs. (A.12)-(A.14) shows that for $M=1$, LR-BMF reduces to the BdG theory, i.e., LR-BMF($M=1$)$\equiv$BdG. If one however compares the LR matrix of Eq. (A.37) to the one of LR-MCTDHB in Section 3.2.2, it can be seen that the latter is clearly more complicated because it also includes the couplings between orbitals and coefficients. Again, in both mean- field approaches presented in this Appendix, there is only one possible configuration, i.e., the configuration $\|N;t\rangle$ for the BdG case and the configuration $\|n_{1},...,n_{M};t\rangle$ for the multi-orbital mean-field case. The occupations of the underlying single-particle orbitals are fixed. In fact, the LR-BMF matrix $\mathbf{\mathcal{P}}\mathbf{\mathcal{L}}\mathbf{\mathcal{P}}$ can be seen as the upper-left block $\mathcal{L}_{oo}$ of the LR matrix in Eq. (93). Further details on the LR-BMF theory, together with an application to a BEC in a double well, can be found in Ref. [240]. ## Appendix B LR-MCTDHB in block-diagonal form In this Appendix, a complex transformation that essentially halves the dimensionality of the homogeneous eigenvalue problem of Eq. (116) is introduced. However, as shown below, it involves matrix-matrix multiplications of individual blocks of the LR matrix $\mathcal{L}$. The latter products might be numerically expensive. It is therefore not in general beneficial to perform the transformation described below, which was first discussed in Ref. [241]. One considers the transformation matrix $Q=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\theta\\\ 1&-1\theta\end{pmatrix},\quad Q^{-1}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1\\\ 1\theta&-1\theta\end{pmatrix},\quad QQ^{-1}=Q^{-1}Q=\mathbb{1}$ (B.1) where the operation $\theta$ is defined via its action on any operator $\hat{O}$ given by $\theta\hat{O}=\hat{O}^{\ast}$. A matrix of the form $\mathcal{A}=\begin{pmatrix}A&B\\\ -B^{\ast}&-A^{\ast}\end{pmatrix}$ (B.2) behaves under the transformation in Eq. (B.1) as $\displaystyle Q\mathcal{A}Q^{-1}$ $\displaystyle=\frac{1}{2}\begin{pmatrix}1&1\\\ 1\theta&-1\theta\end{pmatrix}\begin{pmatrix}A&B\\\ -B^{\ast}&-A^{\ast}\end{pmatrix}\begin{pmatrix}1&1\theta\\\ 1&-1\theta\end{pmatrix}$ $\displaystyle=\begin{pmatrix}0&A-B\theta\\\ A+B\theta&0\end{pmatrix}.$ (B.3) Multiplying the result of Eq. (B) with itself yields the block-diagonal matrix $\left(Q\mathcal{A}Q^{-1}\right)\left(Q\mathcal{A}Q^{-1}\right)=\begin{pmatrix}(A-B\theta)(A+B\theta)&0\\\ 0&(A+B\theta)(A-B\theta)\end{pmatrix}.$ (B.4) This transforms the eigenvalue problem $\mathcal{A}\alpha=\lambda\alpha$ into $\displaystyle\lambda^{2}(Q\alpha)$ $\displaystyle=\left(Q\mathcal{A}Q^{-1}\right)\left(Q\mathcal{A}Q^{-1}\right)(Q\alpha)$ $\displaystyle\Leftrightarrow\,\lambda^{2}\frac{1}{\sqrt{2}}\begin{pmatrix}u+v^{\ast}\\\ u-v^{\ast}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}(A-B\theta)(A+B\theta)&0\\\ 0&(A+B\theta)(A-B\theta)\end{pmatrix}\frac{1}{\sqrt{2}}\begin{pmatrix}u+v^{\ast}\\\ u-v^{\ast}\end{pmatrix}$ (B.5) with $\alpha=\frac{1}{\sqrt{2}}\begin{pmatrix}u\\\ v\end{pmatrix}$. Thus, the original eigenvalue problem of $\mathcal{A}$ can be split up into two smaller eigenvalue problems of $\mathcal{L}^{(1)}\equiv(A-B\theta)(A+B\theta)$ and $\mathcal{L}^{(2)}\equiv(A+B\theta)(A-B\theta)$ with $\text{dim}(\mathcal{L}^{(1)})=\text{dim}(\mathcal{L}^{(2)})=\text{dim}(\mathcal{A})/2$. Moreover, both $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$ have the same eigenvalues, which are the squared eigenvalues of $\mathcal{A}$. Interchanging the second and third row as well as the second and third column of the LR matrix in Eq. (3.2.2) which explicitly reads $\displaystyle\mathcal{L}=\begin{pmatrix}\boldsymbol{\rho}^{-\frac{1}{2}}\hat{\mathbf{P}}\mathcal{L}_{oo}^{u}\hat{\mathbf{P}}\boldsymbol{\rho}^{-\frac{1}{2}}&\boldsymbol{\rho}^{-\frac{1}{2}}\hat{\mathbf{P}}\mathcal{L}_{oo}^{v}\hat{\mathbf{P}}^{\ast}(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}&\boldsymbol{\rho}^{-\frac{1}{2}}\hat{\mathbf{P}}\mathcal{L}_{oc}^{u}&\boldsymbol{\rho}^{-\frac{1}{2}}\hat{\mathbf{P}}\mathcal{L}_{oc}^{v}\\\ -(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}\hat{\mathbf{P}}^{\ast}\mathcal{L}_{oo}^{v,\ast}\hat{\mathbf{P}}\boldsymbol{\rho}^{-\frac{1}{2}}&-(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}\hat{\mathbf{P}}^{\ast}\mathcal{L}_{oo}^{u,\ast}\hat{\mathbf{P}}^{\ast}(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}&-(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}\hat{\mathbf{P}}^{\ast}\mathcal{L}_{oc}^{v,\ast}&-(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}\hat{\mathbf{P}}^{\ast}\mathcal{L}_{oc}^{u,\ast}\\\ \mathcal{L}_{co}^{u}\hat{\mathbf{P}}\boldsymbol{\rho}^{-\frac{1}{2}}&\mathcal{L}_{co}^{v}\hat{\mathbf{P}}^{\ast}(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}&\mathbf{\mathcal{H}}-\varepsilon^{0}&0\\\ -\mathcal{L}_{co}^{v,\ast}\hat{\mathbf{P}}\boldsymbol{\rho}^{-\frac{1}{2}}&-\mathcal{L}_{co}^{u,\ast}\hat{\mathbf{P}}^{\ast}(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}&0&-(\mathbf{\mathcal{H}}-\varepsilon^{0})^{\ast}\end{pmatrix}$ (B.6) yields a matrix of the form given in Eq. B.2. Its submatrices are $A=\begin{pmatrix}\boldsymbol{\rho}^{-\frac{1}{2}}\hat{\mathbf{P}}\mathcal{L}_{oo}^{u}\hat{\mathbf{P}}\boldsymbol{\rho}^{-\frac{1}{2}}&\boldsymbol{\rho}^{-\frac{1}{2}}\hat{\mathbf{P}}\mathcal{L}_{oc}^{u}\\\ \mathcal{L}_{co}^{u}\hat{\mathbf{P}}\boldsymbol{\rho}^{-\frac{1}{2}}&\mathbf{\mathcal{H}}-\varepsilon^{0}\end{pmatrix}$ (B.7) and $B=\begin{pmatrix}\boldsymbol{\rho}^{-\frac{1}{2}}\hat{\mathbf{P}}\mathcal{L}_{oo}^{v}\hat{\mathbf{P}}^{\ast}(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}&\boldsymbol{\rho}^{-\frac{1}{2}}\hat{\mathbf{P}}\mathcal{L}_{oc}^{v}\\\ \mathcal{L}_{co}^{v}\hat{\mathbf{P}}^{\ast}(\boldsymbol{\rho^{\ast}})^{-\frac{1}{2}}&0\end{pmatrix}.$ (B.8) Mind the suppressed superscript ’0’ for the density matrices as compared to, e.g., Eq. (111). It is hence possible to diagonalize either $\mathcal{L}^{(1)}$ or $\mathcal{L}^{(2)}$ with $A$ and $B$ defined in Eqs. (B.7) and (B.8) to obtain the squared eigenvalues of the original LR matrix $\mathcal{L}$. The advantage is, as mentioned above, that the dimensionality of $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$ is only half of the dimensionality of $\mathcal{L}$. Moreover, for two distinct eigenvalues $\omega_{1}$ and $\omega_{2}$ of $\mathcal{L}$ with $\omega_{2}\geq\omega_{1}>0.5$ and separation $\Delta\equiv\omega_{2}-\omega_{1}\geq 0$, the separation $\Delta^{(2)}$ in the spectrum of the squared eigenvalues is given by $\displaystyle\Delta^{(2)}$ $\displaystyle\equiv\omega_{2}^{2}-\omega_{1}^{2}=(\omega_{1}+\Delta)^{2}-\omega_{1}^{2}$ $\displaystyle=2\omega_{1}\Delta+\Delta^{2}=\Delta(2\omega_{1}+\Delta)$ $\displaystyle\geq 2\omega_{1}\Delta$ $\displaystyle>\Delta.$ (B.9) Thus, the separation grows in the spectra of $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$. Since in general, the IRAM converges in less iterations when the eigenvalues are more separated from each other, eigenvalues of larger magnitude than $0.5$ might be obtained with less iterations by diagonalizing either $\mathcal{L}^{(1)}$ or $\mathcal{L}^{(2)}$ instead of $\mathcal{L}$. On the contrary, for eigenvalues with $0\leq\omega_{1}\leq\omega_{2}\leq 0.5$, one finds that $\displaystyle\Delta^{(2)}$ $\displaystyle=\Delta(2\omega_{1}+\Delta)$ $\displaystyle=\Delta\underbrace{(\omega_{2}+\omega_{1})}_{\leq 1}$ $\displaystyle\leq\Delta,$ (B.10) which means that the separation of these eigenvalues is decreased in the spectra of $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$. Therefore, although each iteration of the IRAM is less expensive for the smaller matrices, more iterations might be necessary to obtain converged eigenvalues with smaller magnitude than 0.5. A possible solution to this problem is to shift $\mathcal{L}\rightarrow\mathcal{L}+0.5\cdot\mathbb{1}$, where $\mathbb{1}$ is the $2(M+N_{\text{conf}})$-dimensional unit matrix, before applying the transformation in Eq. (B.1). It therefore remains a system-dependent question whether it is beneficial to use the full LR matrix $\mathcal{L}$ or one of the smaller matrices $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$ to obtain the low-energy spectrum. One needs to take into account the computational cost of building the latter two matrices. Moreover, it is not in general true that, assuming the blocks $A$ and $B$ are sparse matrices, their product is also sparse. If one is in the unfortunate situation that the matrices $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$ are much denser than the original matrix $\mathcal{L}$, it is potentially also much more demanding to converge to their lowest eigenvalues, although they are much smaller. It is therefore a priori not possible to decide which approach is more efficient, and in principle, one would always need to first calculate the (shifted) matrices $\mathcal{L}^{(1)}$ and $\mathcal{L}^{(2)}$ and analyze how dense they are before making an appropriate decision. All applications of LR-MCTDHB discussed in this work, however, avoided this preliminary analysis, and used the full LR matrix $\mathcal{L}$ to obtain the low-energy spectra. For additional details, it is referred to Ref. [241]. ## Appendix C Further benchmarks of LR-MCTDHB Further results on the benchmark of the newly developed implementation of LR- MCTDHB against the HIM are presented in this Appendix. In particular, comparisons of the exact and numerical results for attractive bosons in 1D as well as for repulsive bosons in the anisotropic HIM in 2D are made. In addition, the numerical convergence in the rotating frame of reference, which is important for the validity of the results presented in Section 6.3.3, is demonstrated. \| $M=1$ \| $M=4$ \| $M=6$ \| $(m_{x},n_{x})$ \| Exact ---\|---\|---\|---\|---\|--- $\varepsilon^{0}$ \| 9.137833 \| 9.038151 \| 9.038150 \| $(0,0)$ \| 9.038150 $\omega_{1}$ \| 1.000000 \| 1.000000 \| 1.000000 \| $(1,0)$ \| 1.000000 $\omega_{2}$ \| n/a \| 2.000222 \| 2.000000 \| $(2,0)$ \| 2.000000 $\omega_{3}$ \| n/a \| 3.000432 \| 3.000000 \| $(3,0)$ \| 3.000000 $\omega_{4}$ \| 3.655133 \| 3.794752 \| 3.794733 \| $(0,2)$ \| 3.794733 $\omega_{5}$ \| n/a \| 4.011839 \| 4.000012 \| $(4,0)$ \| 4.000000 $\omega_{6}$ \| n/a \| 4.794870 \| 4.794733 \| $(1,2)$ \| 4.794733 $\omega_{7}$ \| n/a \| 5.022646 \| 5.000028 \| $(5,0)$ \| 5.000000 $\omega_{8}$ \| 5.482700 \| 5.692143 \| 5.692100 \| $(0,3)$ \| 5.692100 $\omega_{9}$ \| n/a \| 5.797912 \| 5.794737 \| $(2,2)$ \| 5.794733 $\omega_{10}$ \| n/a \| 6.142955 \| 6.000676 \| $(6,0)$ \| 6.000000 Table C.1: Benchmark of LR-MCTDHB to the attractive 1D HIM. Shown are the ground-state energy $\varepsilon^{0}$ and the energies $\omega_{i}$ of the first ten excitations for $N=10$ bosons and different numbers of orbitals $M$. The trapping frequency is $\Omega=1.0$ and the interaction strength is $\lambda_{0}=0.13$, yielding a ground-state depletion of $f=0.94\%$. This is comparable to the average depletion of the 1D lattice systems discussed in Section 6.3.2. Underlined digits denote deviations from the exact values from Eqs. (160) and (161). All quantities are dimensionless. See text for more details. For the case of attractive bosons in the 1D HIM, the system parameters are chosen as $\Omega=1.0$ for the trapping frequency and $\lambda_{0}=0.13$ for the two-body interaction strength. The number of bosons is $N=10$. The calculations were carried out on a grid with 128 grid points in the interval $[-9,9)$. Results for the excitation energies relative to the ground-state energy, $\omega_{i}=E_{i}-\varepsilon^{0}$, are shown in Table C.1. With respect to $\varepsilon^{0}$, one observes that $M=4$ and $M=6$ lead to highly accurate results. With respect to the excited states, it can be observed that the BdG approach, i.e., for $M=1$, misses several excitations in the low- energy spectrum. In contrast to that, the many-body calculations yield all states. The numerical accuracy clearly improves upon increasing the number of self-consistent orbitals. For $M=6$, even the higher c.m. excited states $\omega_{5},\,\omega_{7}$, and $\omega_{10}$ are obtained to very high accuracy. Together with the benchmark from Table 2, it can be deduced that the implementation of LR-MCTDHB is capable of accurately describing the low-energy spectrum of both repulsive and attractive bosons. \| $M=1$ \| $M=2$ \| $M=3$ \| $(m_{x},m_{y}$ $n_{x},n_{y})$ \| Exact ---\|---\|---\|---\|---\|--- $\varepsilon^{0}$ \| 110.007587 \| 110.004507 \| 110.003452 \| $(0,0,0,0)$ \| 110.003452 $\omega_{1}$ \| 1.000000 \| 0.999999 \| 1.000000 \| $(1,0,0,0)$ \| 1.000000 $\omega_{2}$ \| 1.378405 \| 1.378280 \| 1.378405 \| $(0,1,0,0)$ \| 1.378405 $\omega_{3}$ \| 1.791089 \| 1.788859 \| 1.788860 \| $(0,0,2,0)$ \| 1.788854 $\omega_{4}$ \| n/a \| 2.000438 \| 2.000477 \| $(2,0,0,0)$ \| 2.000000 $\omega_{5}$ \| 2.200152 \| 2.200520 \| 2.198271 \| $(0,0,1,1)$ \| 2.198268 $\omega_{6}$ \| n/a \| 1.940430 \| 2.378704 \| $(1,1,0,0)$ \| 2.378405 $\omega_{7}$ \| 2.609215 \| 2.609214 \| 2.607684 \| $(0,0,0,2)$ \| 2.607681 $\omega_{8}$ \| 2.686634 \| 2.683292 \| 2.683294 \| $(0,0,3,0)$ \| 2.683282 $\omega_{9}$ \| n/a \| n/a \| 2.757078 \| $(0,2,0,0)$ \| 2.756810 $\omega_{10}$ \| n/a \| 2.789191 \| 2.789227 \| $(1,0,2,0)$ \| 2.788854 $\omega_{11}$ \| n/a \| 3.000758 \| 3.000816 \| $(3,0,0,0)$ \| 3.000000 Table C.2: Benchmark of LR-MCTDHB to the anisotropic 2D HIM with $N=100$ bosons. The trapping frequencies are $\Omega_{x}=1.0$ and $\Omega_{y}=\sqrt{1.9}$ in $x$\- and $y$-directions, and the two-body interaction strength id $\lambda=-0.001$. All previously obtained degeneracies in the isotropic case (see Table 3) are lifted. The accuracy of the energies increases with the number of orbitals. Underlined digits denote deviations from the exact values from Eqs. (160) and (161). All quantities are dimensionless. See text for more details. Table C.2 shows the numerical results obtained for the anisotropic HIM in 2D with trap frequencies $\Omega_{x}=1.0$ and $\Omega_{y}=\sqrt{1.9}$. The remaining system parameters are the same as in the isotropic case from Table 3. One first observes that the anisotropy lifts all previous degeneracies of the isotropic case. Apart from that, the numerical results are qualitatively similar. In the BdG case, only the first c.m. excitation is obtained numerically exact, all other c.m. excited states are inaccessible. The pure relative excitations are also found to a good accuracy. For $M=2$, one gets access to higher c.m. excited states in the $x$-direction, together with combined excitations of that type with excitations of relative coordinates. As in the isotropic case, all excited states obtained in the $y$-coordinate deteriorate in accuracy compared to the BdG case, see, e.g., $\omega_{2}$ and $\omega_{5}$. The reason is the same as for the isotropic case, namely that the additional orbital resembles a $p_{x}$-orbital and leads to preferred direction in the description. Including a third orbital that resembles a $p_{y}$-orbital solves this problem. It additionally improves the overall accuracy of the low-energy excitations, and moreover ensures that all low- lying excited states are obtained. \| $M=1$ \| $M=3$ \| $(n,m,l_{z})$ \| Exact ---\|---\|---\|---\|--- $\varepsilon^{0}$ \| 89.554453 \| 89.548293 \| (0,0,0) \| 89.548292 $\omega_{1}$ \| 0.900000 \| 0.900000 \| (0,1,1) \| 0.900000 $\omega_{2}$ \| 1.100000 \| 1.100000 \| (0,1,-1) \| 1.100000 $\omega_{3}$ \| 1.591089 \| 1.588859 \| (2,0,2) \| 1.588854 $\omega_{4}$ \| 1.791089 \| 1.788862 \| (2,0,0) \| 1.788854 $\omega_{5}$ \| n/a \| 1.800439 \| (0,2,2) \| 1.800000 $\omega_{6}$ \| 1.991089 \| 1.988859 \| (2,0,-2) \| 1.988854 $\omega_{7}$ \| n/a \| 2.000657 \| (0,2,0) \| 2.000000 $\omega_{8}$ \| n/a \| 2.200438 \| (0,2,-2) \| 2.200000 $\omega_{9}$ \| 2.386633 \| 2.383226 \| (3,0,3) \| 2.383282 $\omega_{10}$ \| n/a \| 2.487088 \| (2,1,3) \| 2.488854 $\omega_{11}$ \| 2.586634 \| 2.583318 \| (3,0,1) \| 2.583282 $\omega_{12}$ \| n/a \| 2.689270 \| (2,1,1) \| 2.688854 $\omega_{13}$ \| n/a \| 2.691535 \| (2,1,1) \| 2.688854 $\omega_{14}$ \| n/a \| 2.700528 \| (0,3,3) \| 2.700000 $\omega_{15}$ \| 2.786634 \| 2.783318 \| (3,0,-1) \| 2.783282 Table C.3: Benchmark of LR-MCTDHB to the isotropic 2D HIM in the rotating frame. The angular velocity around the $z$-axis is $\Omega_{\text{rot}}=0.1$. Results are presented for $N=100$ bosons and different numbers of orbitals $M$. The trapping frequencies are $\Omega_{x}=\Omega_{y}=1.0$, whereas the strength of the repulsive interaction is $\lambda_{0}=-0.001$. Shown are the energies of the ground state, $\varepsilon^{0}$, and of the first $15$ excited states, $\omega_{i}=E_{i}-\varepsilon^{0}$. Underlined digits denote deviations from the exact values from Eqs. (160), (161) and (C.1). The quantum numbers $n$ and $m$ refer to the relative and c.m. coordinates, whereas $l_{z}$ refers to the angular momentum in the $z$-direction. All quantities are dimensionless. See text for more details. Finally, excitations in the rotating frame of reference are discussed. The energies of the ground and excited states, given by Eqs. (160) and (161), become shifted by the term $E_{\text{rot}}=-\Omega_{\text{rot}}\,l_{z}$ (C.1) where $\Omega_{\text{rot}}$ denotes the angular velocity around the $z$-axis and $l_{z}$ denotes the angular momentum of the exited state in the $z$-direction. In Table 3, all obtained excitations are labeled by corresponding quantum numbers for relative and c.m. excitations, $n$ and $m$, as well as by $l_{z}$. The increased numerical accuracy of LR-MCTDHB($3$) in comparison to the BdG results becomes clearly obvious. Moreover, all states are obtained for $M=3$, whereas the BdG spectrum misses several states. As a conclusion, the utilized implementation of LR-MCTDHB is also capable of accurately describing the low-energy excitation spectra in the rotating frame. ## Appendix D Many-particle variance: A sensitive quantity to many-body effects In this Appendix, a brief introduction of the many-body variance, which is employed in the main text to study many-body excitations involved in the dynamics of trapped BECs, is given. The many-body variance has been of interest in recent studies on correlations between bosons in the ground state [242] and for out-of-equilibrium BECs [243, 244], in particular in a 1D bosonic Josephson junction [245] or in anharmonic and anisotropic trapping potentials [246, 81]. It has also been found that the variance is a sensitive measure for the numerical convergence of MCTDHB [80]. The focus is laid on the position variance in the $x$-direction in the following. The position operator of $N$ bosons is given by $\hat{X}=\sum_{i=1}^{N}\hat{x}_{i}$ (D.1) where the operator $\hat{x}_{i}$ denotes the position operator of the $i$-th particle. To compute its variance per particle in the state $\Psi(\mathbf{r}_{1},...,\mathbf{r}_{N})$, which reads $\frac{1}{N}\Delta^{2}_{\hat{X}}=\frac{1}{N}\left(\langle\Psi\|\hat{X}^{2}\|\Psi\rangle-\langle\Psi\|\hat{X}\|\Psi\rangle^{2}\right),$ (D.2) one also needs to evaluate the expectation value of the square of the position operator. The latter is given by $\hat{X}^{2}=\sum_{i=1}^{N}\hat{x}_{i}^{2}+\sum_{i<k}2\,\hat{x}_{i}\hat{x}_{k},$ (D.3) i.e., it consists of a one-body and a two-body term. For the expectation value of $\hat{X}$ one obtains $\displaystyle\frac{1}{N}\langle\Psi\|\hat{X}\|\Psi\rangle$ $\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\int d\mathbf{r}_{1}...d\mathbf{r}_{N}\,\Psi^{\ast}(\mathbf{r}_{1},...,\mathbf{r}_{N})\,x_{i}\,\Psi(\mathbf{r}_{1},...,\mathbf{r}_{N})$ $\displaystyle=\frac{1}{N}\sum_{i=1}^{N}\int d\mathbf{r}_{i}\,\frac{\rho^{(1)}(\mathbf{r}_{i}\|\mathbf{r}_{i})}{N}x_{i}$ $\displaystyle=\int d\mathbf{r}\,\frac{\rho(\mathbf{r})}{N}x.$ (D.4) In the second step, the definition of the density $\rho(\mathbf{r})\equiv\rho^{(1)}(\mathbf{r}\|\mathbf{r})$, employing the diagonal of the one-body RDM, was used. In general, the time-dependent $p$-th order RDM of a given $N$-boson system in the state $\Psi(\mathbf{r}_{1},...,\mathbf{r}_{N};t)$ is defined as $\displaystyle\rho^{(p)}(\mathbf{r}_{1},...,\mathbf{r}_{p}\|\mathbf{r}_{1}^{\prime},...,\mathbf{r}_{p}^{\prime};t)$ $\displaystyle=\frac{N!}{(N-p)!}\int d\mathbf{r}_{p+1}...d\mathbf{r}_{N}\,\Psi(\mathbf{r}_{1},...,\mathbf{r}_{p},\mathbf{r}_{p+1},...,\mathbf{r}_{N};t)$ $\displaystyle\times\Psi^{\ast}(\mathbf{r}_{1}^{\prime},...,\mathbf{r}_{p}^{\prime},\mathbf{r}_{p+1},...,\mathbf{r}_{N};t).$ (D.5) From Eq. (D), one can define the $p$-th order correlation function $g^{(p)}$ as $\displaystyle g^{(p)}(\mathbf{r}_{1}^{\prime},...,\mathbf{r}_{p}^{\prime}\|\mathbf{r}_{1},...,\mathbf{r}_{p};t)=\frac{\rho^{(p)}(\mathbf{r}_{1}^{\prime},...,\mathbf{r}_{p}^{\prime}\|\mathbf{r}_{1},...,\mathbf{r}_{p};t)}{\sqrt{\Pi_{i=1}^{p}\,\rho^{(1)}(\mathbf{r}_{i}\|\mathbf{r}_{i};t)\,\rho^{(1)}(\mathbf{r}_{i}^{\prime}\|\mathbf{r}_{i}^{\prime};t)}}.$ (D.6) Whereas the wave function $\Psi$ is assumed to be normalized to unity, i.e., $\langle\Psi(t)\|\Psi(t)\rangle=1$, the $p$-th order RDM is not. One further obtains $\displaystyle\frac{1}{N}\langle\Psi\|\hat{X}^{2}\|\Psi\rangle$ $\displaystyle=\int d\mathbf{r}\,\frac{\rho(\mathbf{r})}{N}x^{2}$ $\displaystyle+\frac{2}{N}\sum_{i<k}\int d\mathbf{r}_{1}...d\mathbf{r}_{N}\,\Psi^{\ast}(\mathbf{r}_{1},...,\mathbf{r}_{N})\,x_{i}x_{k}\,\Psi(\mathbf{r}_{1},...,\mathbf{r}_{N})$ (D.7) where the second term can be simplified due to $\displaystyle\sum_{i<k}\int d\mathbf{r}_{1}...d\mathbf{r}_{N}\,\Psi^{\ast}(\mathbf{r}_{1},...,\mathbf{r}_{N})\,x_{i}x_{k}\,\Psi(\mathbf{r}_{1},...,\mathbf{r}_{N})$ $\displaystyle=\sum_{i<k}\int d\mathbf{r}_{i}d\mathbf{r}_{k}\,\frac{\rho^{(2)}(\mathbf{r}_{i},\mathbf{r}_{k},\mathbf{r}_{i},\mathbf{r}_{k})}{N(N-1)}x_{i}x_{k}$ $\displaystyle=\frac{N(N-1)}{2}\int d\mathbf{r}_{1}d\mathbf{r}_{2}\,\frac{\rho^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1},\mathbf{r}_{2})}{N(N-1)}x_{1}x_{2}$ (D.8) where the definition of the two-body RDM $\rho^{(2)}$ was utilized. The choice of the coordinates of the first and second particle in the last step is arbitrary. This yields $\displaystyle\frac{1}{N}\langle\Psi\|\hat{X}^{2}\|\Psi\rangle$ $\displaystyle=\int d\mathbf{r}\,\frac{\rho(\mathbf{r})}{N}x^{2}$ $\displaystyle+\int d\mathbf{r}_{1}d\mathbf{r}_{2}\,\frac{\rho^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1},\mathbf{r}_{2})}{N}x_{1}x_{2}$ (D.9) such that one obtains $\displaystyle\frac{1}{N}\Delta^{2}_{\hat{X}}$ $\displaystyle=\int d\mathbf{r}\,\frac{\rho(\mathbf{r})}{N}x^{2}-\frac{1}{N}\left[\int d\mathbf{r}\,\rho(\mathbf{r})\,x\right]^{2}$ $\displaystyle+\int d\mathbf{r}_{1}d\mathbf{r}_{2}\,\frac{\rho^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1},\mathbf{r}_{2})}{N}x_{1}x_{2}$ (D.10) for the variance of the position operator $\hat{X}$. To arrive at an expression in the Hartree limit, we first expand $\rho^{(2)}$ by the natural orbitals $\\{\alpha_{i}\\}$, which yields $\rho^{(2)}(\mathbf{r}_{1},\mathbf{r}_{2},\mathbf{r}_{1},\mathbf{r}_{2})=\sum_{i,j,k,l}\,\rho_{ijkl}\,\alpha^{\ast}_{i}(\mathbf{r}_{1})\alpha^{\ast}_{j}(\mathbf{r}_{2})\alpha_{k}(\mathbf{r}_{1})\alpha_{l}(\mathbf{r}_{2}).$ (D.11) Then, we can perform the limit which gives $\displaystyle\lim_{N\rightarrow\infty}\frac{1}{N}\Delta_{\hat{X}}^{2}$ $\displaystyle=\Delta_{\text{GP}}^{2}+\Delta_{\text{correlations}}^{2},$ (D.12) $\displaystyle\Delta_{\text{GP}}^{2}$ $\displaystyle=\int d\mathbf{r}\,\|\phi^{0}(\mathbf{r})\|^{2}\,x^{2}-\left[\int d\mathbf{r}\,\|\phi^{0}(\mathbf{r})\|^{2}\,x\right]^{2}$ (D.13) $\displaystyle\Delta_{\text{correlations}}^{2}$ $\displaystyle=\lim_{N\rightarrow\infty}\sum_{i,j,k,l\neq 1111}\int d\mathbf{r}_{1}d\mathbf{r}_{2}\,\frac{\rho_{ijkl}}{N}\alpha^{\ast}_{i}(\mathbf{r}_{1})\alpha^{\ast}_{j}(\mathbf{r}_{2})\,x_{1}x_{2}\,\alpha_{k}(\mathbf{r}_{1})\alpha_{l}(\mathbf{r}_{2}).$ (D.14) Eq. (D.13) expresses the fact that, in the Hartree limit, the GP orbital $\phi^{0}$, obtained from Eq. (A.5), exactly describes the ground-state density per particle, i.e., $\lim_{N\rightarrow\infty}\frac{\rho(\mathbf{r})}{N}=\|\phi^{0}(\mathbf{r})\|^{2}$. To remind the reader, only one natural orbital exists in the GP case. In obtaining Eq. (D.13) one makes use of the relation $\lim_{N\rightarrow\infty}\frac{\rho_{1111}}{N(N-1)}=1$ as well. This means that many-body contributions to the variance per particle, Eq. (D.14), can only occur due to the occupation of higher natural orbitals, as is expressed in the excluded summation $i,j,k,l\neq 1111$ in Eq. (D.14). It has been shown that even a small occupation of the higher natural orbitals, i.e., the number of depleted particles, can have a sizable effect at the infinite particle limit. For further details, also including the variance of other quantities like the momentum and angular momentum operators, see Refs. [242, 139]. Applications to exact many-body excitations from the dynamics of the variance were discussed in section 6.3. Here, we show an example that demonstrates the numerical convergence of the ground states of the rotating BECs of Fig. 28 with $M=7$ orbitals, the position variance in $x$-direction depending on different values of the vorticity $l$ is shown in Fig. 32. One readily observes that already for $M=5$ orbitals the ground state variances are essentially condensed, and the deviations are less than $\mathcal{O}(10^{-3})$ of the exact values (see inset). Figure 32: The position variance per particle in $x$-direction for the ground states of the rotating BECs treated in Fig. 28. For all vorticities $l$ shown, essentially $M=5$ self-consistent orbitals are sufficient to reach numerical convergence because the results lie atop the ones for higher values of $M$. Inset: Enlarged view for the case of $l=4$. All quantities are dimensionless. See text for details. The figure is taken from Ref. 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# Near-Field Millimeter Wave Vector Measurements - Experimental Design & Measurement Interpretation Laurent Chusseau1, Thibaut Auriac1, Jérémy Raoult1 1IES, Université de Montpellier, CNRS, Montpellier, France ###### Abstract Near-field imaging experiments exist both in optics and microwaves with often different methods and theoretical supports. For millimeter waves or THz waves, techniques from both fields can be merged to identify materials at the micron scale on the surface or in near-surface volumes. The principle of such near- field vector imaging at the frequency of 60 GHz is discussed in detail here. We develop techniques for extracting vector voltages and methods for extracting the normalized near-field vector reflection on the sample. In particular, the subharmonic IQ mixer imbalance, which produced corrupted outputs either due to amplitude or phase differences, must be taken into account and compensated for to avoid any systematic errors. We provide a method to fully characterize these imperfections and to isolate the only contribution of the near-field interaction between the probe and the sample. The effects of the mechanical modulation waveform and harmonic rank used for signal acquisition are also discussed. ###### Index Terms: Near-field measurement, mm-waves, microwave probes, vector local characterization, subharmonic IQ mixer. ## I Introduction Near-field imaging is now an essential technique for the characterization of devices and materials. Depending on the targeted application, reflection or transmission modes are used in the infrared [1], microwave [2] or THz [3], and for the last two take advantage of the transparency of many materials for non- destructive testing through the surface [4, 5, 6, 7]. If high spatial resolution is desired in respect to the measurement wavelength, a probe is required to concentrate the field and overcome the Abbe diffraction limit by means of evanescent waves. At GHz frequencies, the scanning microwave microscope (SMM) frequently uses coaxial resonant probes whose quality factor and resonance can be related to the interaction with the sample [8, 9, 10], or AFM tips [11, 12, 13] that can be modeled electromagnetically in 3D [14, 15, 16]. The latter technique is also used in the millimeter wave (mmW) region, which is the realm where it is possible to combine optical and microwave techniques, for example the joint use of propagating waves [17] and guided waves coupled to specific probes including small antennae capable of concentrating the electric or magnetic field with a very high resolution [18, 19, 20]. Although most of near-field experiments in the mmW have focused on intensity measurements [21, 22, 6, 23, 24, 17], local vector characterization is necessary to distinguish material changes from loss changes. Many such experiments have recently been proposed in scattering-type scanning near-field optical microscopy (s-SNOM) applied to the THz domain [25, 26, 27, 28], but only the latter seems to be easy to translate to mmW. We do the same transformation on our 60 GHz test bench [24] and chose to include a subharmonic IQ mixer to simultaneously detect the two quadratures of the reflected field, but we kept a complete waveguide configuration. This paper details the new experiment and elaborate the required processing of experimental signals in order to determine without systematic error and with reasonable accuracy the vector voltages and reflection coefficients related to the near-field probe-sample interaction. We show how the rapid phase rotation due to the path length in mmW can be used to achieve these values, and how neutralize the IQ imbalance of the mixer, either in amplitude or phase, by its theoretical account in data processing. In the end, this gives us an overall figure of merit for the subharmonic IQ mixer and the entire detection system. The paper is organized as follows. In §II we describe the experimental setup and its basic operation. §III details the processing to be applied to experimental data, namely the vector voltage determination and how mixer non- ideal behavior transforms the observed vector voltage. We then develop the method for tracing the true reflection coefficient, provided that a quality reference measurement is available. In §IV, we deal with the effect of the harmonic rank of the detection in relation to the modulation waveform applied to the mechanical modulation of the distance between the probe and the sample, which is a peculiarity of our experiment as compared to s-SNOM. Finally, in §V we qualify the frequency response of our experiment and the subharmonic mixer imbalance. ## II Measurement Setup The scheme of the new experiment is given in Fig. 1 with its central element for the vector output: the subharmonic IQ mixer which operates at zero intermediate frequency in the band 55-65 GHz. Compared to our previous intensity reflectometer [24], the whole mmW transmitting/detecting system has changed and now uses an HF synthesizer followed by multiplication chains for the detection and LO signal. In the measuring path, an active $\times 4$ multiplier delivering 10 dBm is used in the front of a 3 dB coupler which returns the reflected signal at the input of the mixer. In between, we take care of the mmW signal level entering the subharmonic IQ mixer by incorporating a variable attenuator to validate the -10 dBm requirement for linear operation. The LO path of the mixer is fed by an active doubler delivering 10 dBm. The $\times 4$ and $\times 2$ active multipliers are not really perfectly flat in frequency, unlike the reference synthesizer. As a result, the subharmonic IQ mixer’s conversion loss variations, which are typically 15 to 18 dB, may experience additional ripple due to these power variations of the mmW sources. The whole system was provided by LTEQ Microwave according to our specifications. As in [24], probes are homemade bow-ties attached to a WR15 open end. Metal triangles used for the bow-tie were produced using fs-laser cut on a 10 µm thick tungsten sheet. The Fig. 2 illustrates this realization and the probe positioning 10 µm above a surface. The lateral and vertical positioning over the sample is ensured by a motorized step-by-step system (Newport XPS-C8) and is optically monitored with two cameras at right angle as shown in the picture. When set at their highest magnification, cameras equipped with Navitar UltraZoom lenses can observe a displacement of 1 µm, which corresponds to one motor step. Figure 1: Scheme of the mmW near-field experimental setup. All mmW links are made using WR15 waveguides up to the subharmonic IQ mixer. The modulation of the probe-to-sample distance is carried out by the piezoelectric actuator (PI P-611.Z Precision Z-stage) driven by a signal generator whose frequency is set to 40 Hz due to the limited weight capacity of this actuator. Subharmonic IQ mixer outputs are thus modulated at this low- frequency that is also the reference of the two lock-in (AMETEK 7265 Dual Phase DSP). This allows the filtering of the near-field spatial component ideally split in its real and imaginary parts. In practice both lock-in outputs are complex and refer to the phase of the low-frequency modulation. Correct usage thus requires to set the same phase reference on the two lock- in. Figure 2: Photos of the experiment. Left: emission/detection board in place with the waveguide probe and the two cameras. Right up: tungsten bow-tie probe view from above with the microscope. Right down: Probe as observed by the camera 10 µm above the gold mirror, the reflection is used to set the minimum distance to $h=10$ µm. At first glance, the real part in channel I (respectively, the imaginary part in channel Q) of the complex near-field reflected voltage $\widetilde{v}$ is expected to have the module $m_{I}=\sqrt{X_{I}^{2}+Y_{I}^{2}}$ (respectively, $m_{Q}=\sqrt{X_{Q}^{2}+Y_{Q}^{2}}$). If these values are obtained straightforwardly, their respective sign must be determined to fully locate $\widetilde{v}$ in the entire complex plane. Our first test sample was chosen to obtain the highest possible near-field reflectivity. It is a gold optical mirror from Edmund Optics with an ultra- flat substrate ($\lambda/20$) and a thin (a few nm) dielectric protective layer. A far-field reflectivity of $\approx 97\%$ is certified in the mid- infrared, a value also expected in the far infrared and in the mmW range since gold is homogeneous and well known [29]. This sample is intended to be a reference for reflectivity, to which any other sample can be compared. When scanning the input frequency in a tiny range of 200 MHz, the four outputs of the lock-ins oscillate rapidly, as shown in Fig. 3. This is due to the total electrical length of the system here estimated at about $1.25$ m using the $240\pm 2$ MHz period deduced for the $X_{i}$ and the $Y_{i}$. Since the probe senses the reflection on a flat and homogeneous material, it is not anticipated that it will contribute to a rapidly changing $\widetilde{v}$ with frequency, but rather that it can be regarded as constant. Consequently, we assume in what follows that the almost sinusoidal evolution of the $X_{i}$ and $Y_{i}$ is solely attributed to the electrical length. Figure 3: Lock-in outputs as measured $h=10$ µm above a gold mirror with a sinusoidal vibration amplitude $\delta h=20$ µm while scanning the input frequency. In addition, a close inspection of Fig. 3 shows a deviation from the expected ideal subharmonic IQ mixer: at 60.765 GHz channel I cancels ($X_{I}=Y_{I}=0$) but channel Q is not maximum, which occurs at 60.785 GHz. Conversely, the same occurs at 60.720 GHz for channel Q and 60.705 GHz for channel I. This is quite unexpected for a perfect IQ mixer and shows the mixture of channel signals because of its imperfection. As a result, the measurement of near-field reflection at a single frequency would inevitably be flawed by a systematic error regarding the contribution of the material placed just under the probe. We therefore prefer to carry out frequency sweeps like the one shown in Fig. 3 and process the data to eliminate these errors by assuming that the near-field contribution is constant over the low measurement frequency band. ## III Measurement Data Processing ### III-A Vector Voltage Determination Proceeding from a measurement like the one shown in Fig. 3, the modulus of the reflected near-field voltage is directly obtained by $\left\lvert{\widetilde{v}}\right\rvert=\sqrt{m_{I}^{2}+m_{Q}^{2}}.$ (1) Accessing the phase is more tedious since the rough estimate $\angle\widetilde{v}=\arctan\frac{m_{Q}}{m_{I}}$ is inherently limited to $[0,\frac{\pi}{2}]$ because of the positive $\arctan$ argument. Going further is possible using the original lock-in outputs, which include the sign changes for channels I and Q when $m_{I}$ and $m_{Q}$ go through zero. Noting that these outputs always present opposite $X$ and $Y$ values, the sign are straightforwardly obtained from $\operatorname{sgn}(I)=\operatorname{sgn}(X_{I}-Y_{I})$ and $\operatorname{sgn}(Q)=\operatorname{sgn}(X_{Q}-Y_{Q})$. Keeping track of all these sign changes, we define the phase of the reflected voltage using the heuristic formula $\angle\widetilde{v}=\arctan\frac{\operatorname{sgn}(Q)m_{Q}}{\operatorname{sgn}(I)m_{I}}.$ (2) We have applied these formulas to the experimental case of the gold mirror in Fig. 3 and also for an unintentionally doped GaAs wafer (ITME, Warsaw). The same geometrical conditions were applied for the measurement, _i.e._ $h=10$ µm, $\delta h=20$ µm and a sinusoidal modulation. The results are shown in Fig. 4 with the modulus and phase of $\widetilde{v}$ and their locations in the complex plane. As expected, $\angle\widetilde{v}$ covers the whole range $[-\pi,\pi]$ which leads to elliptic representations in the complex plane. Figure 4: Evolutions of the complex reflected near-field voltages $\widetilde{v}$ for a gold mirror and a GaAs undoped wafer. Left up: modulus, left down: phase, right: complex representation. Dots are for gold and open triangles for GaAs. Although the two materials can be easily distinguished on these curves, especially by their average values $\left\lvert{\widetilde{v}}\right\rvert$ which are clearly different and by their shifted $\angle\widetilde{v}$ evolutions, it is impossible at this stage to characterize a material with a single complex value whereas this was hoped by assuming a constant influence of the material on the limited frequency range considered. In this framework, centered ellipses are distorted views of circles usually obtained in microwave by measurements using sliding loads. We attribute this distortion to the imbalance of the IQ mixer and model it in the sequel to refine the analysis. ### III-B Normalized Vector Reflection Let us consider the ideally reflected complex voltage $v$ in the near-field of the tested surface. As said it should be constant over the tiny frequency range considered because our materials are far from any resonance. The voltage $v$ thus undergoes only a phase shift $\exp j\phi$ due to the electrical length before being transformed by the subharmonic IQ mixer to produce $\widetilde{v}$. We assume that mixer imbalance is twofold: first, it affects the gain by a factor $\gamma$ possibly different from unity that we apply only to the imaginary part, and second, the imperfect quadrature is accounted for by an additional angle $\theta$ added to the $\pi/2$ phase shift of the Q channel. This is written $\widetilde{v}=\Re(v\exp j\phi)+j\gamma\Im(v\exp j(\phi+\theta)).$ (3) If we now divide the vector voltage measured on the sample $\widetilde{v}_{S}$, for instance the GaAs wafer of Fig. 4 by the vector voltage measured on a reference $\widetilde{v}_{R}$, namely the gold mirror in the same figure, which is highly conductive at mmW and supposed to produce the highest possible reflection, we obtain $\frac{\widetilde{v}_{S}}{\widetilde{v}_{R}}=\rho\frac{\cos(\phi+\varphi)+j\gamma\sin(\phi+\varphi+\theta)}{\cos\phi+j\gamma\sin(\phi+\theta)},$ (4) where $\rho=\left\lvert{v_{S}}\right\rvert/\left\lvert{v_{R}}\right\rvert$ and $\varphi=\angle v_{S}-\angle v_{R}$. As in reflectometry, the normalized vector reflection $\rho\exp j\varphi$ is thus intended to be the near-field mmW reflection coefficient $\Gamma$ of the tested material, provided that the reference is a perfect metal. Eq. 4 is easily identified as a circle in the complex plane parametrized by $\phi$. It is completely described if $\phi\in[0,\pi]$ and collapses to a single point with a perfect IQ mixer ($\gamma=1$ and $\theta=0$). Solving a set of 3 equations for 3 different points such as $\phi=0$, $\phi=\pi/2$ and $\phi=-\theta$ which must belong to the same circle $(X-X_{c})^{2}+(Y-Y_{c})^{2}-R^{2}=0$, provides the link with our parameters in Eq. (4). After some lengthly developments, we end up with the definition of the circle $\displaystyle X_{c}$ $\displaystyle=\rho\cos\varphi$ (5a) $\displaystyle Y_{c}$ $\displaystyle=\frac{\rho(1+\gamma^{2})\sin\varphi}{2\gamma\cos\theta}$ (5b) $\displaystyle R$ $\displaystyle=\zeta Y_{c}$ (5c) with $\zeta=\sqrt{1-\frac{4\gamma^{2}\cos^{2}\theta}{\left(1+\gamma^{2}\right)^{2}}}.$ (6) Interestingly the term $\zeta$ is completely independent of both the sample and reference. It is thus some kind of measure of the IQ mixer ideality within our experiment. The gain imbalance $\gamma$ and the quadrature error $\theta$ are mixed in $\zeta$ and cannot be solved separately because we are missing a fourth variable. Order of magnitude of $\zeta$-values are given in Fig. 5 as a function of $\gamma$ and $\theta$. For gain errors lower than 3 dB and phase errors lower than 30° it does not exceed 0.6 which is nonetheless significant compared to its limit value of 1 obtained when $\gamma$ or $1/\gamma\to\infty$ or $\theta\to\pm\pi/2$. We must keep in mind this limit value of 1 which cannot be exceeded under any circumstances in our experiments. It is a criterion of suitability of the measurements and their exploitation that will be implemented in the following. Values on the order of 0.4 and less, however, seem acceptable. Figure 5: Ideality factor $\zeta$ as a function of gain mismatch $\gamma$ and quadrature phase mismatch $\theta$. Go back to measurements to calculate and plot the ratio $\widetilde{v}_{S}/\widetilde{v}_{R}$ from data of Fig. 4. As shown in Fig. 6 we end up with a nearly perfect circle, just like predicted above. We have limited here the frequency range previously considered to $[60.690-60.815]$ GHz in order to describe the circle without overlap. Since the points are not evenly distributed on the circle, there is no simple criteria to immediately access the position of its center, so a least-squares fit was used to determine the triplet $(X_{c},Y_{c},R)$. A secondary result of the fit that minimizes the distance between the measured points and the circle is the error distance accumulated for measurement points to the circle. It is used in the sequel to estimate the uncertainties on each of the inferred values using Student’s t-test with 95% confidence level and classical propagation of errors at the first level on the equations. The error is only $\approx 6\,10^{-4}$ per point in this example, as could be expected given the excellent agreement between the best circle, represented by a solid line in the Fig. 6, with the measurement points. The position of the center is noted by a filled square. The module and phase of the near-field reflection are then straightforwardly calculated from Eqs. (5) by $\displaystyle\rho$ $\displaystyle=\sqrt{X_{c}^{2}+Y_{c}^{2}-R^{2}}$ (7a) $\displaystyle\varphi$ $\displaystyle=\arccos\frac{X_{c}}{\rho}.$ (7b) Figure 6: Ratio $\widetilde{v}_{S}/\widetilde{v}_{R}$ in the complex plane for the GaAs sample and the gold mirror reference of Fig. 4. Dots are measurements, the solid line is the corresponding best circle fit whose center is the filled square, and the filled diamond is the deduced $\Gamma=\rho\exp j\varphi$ normalized vector reflection. In the case of Fig. 6 this deduced point is represented by a diamond. It is obviously different from the center, even if according to Eq. (5a) their real parts are identical. In essence, we end up here with the normalized vectorial reflection $\Gamma=v_{S}/v_{R}=(0.482+j0.225)\pm 0.004$ with an ideality factor $\zeta=0.436\pm 0.005$, which illustrate the effectiveness of the entire measurement operation process, including error estimates. In the ideal case where our reference is perfect, it is expected that $\Gamma$ is the true reflection coefficient in near-field at mmW. ## IV Influence of the mechanical modulation In scattering-type scanning near-field optical microscopy (s-SNOM) the mechanical probe-to-sample distance modulation is always sinusoidal since it is provided by a quartz resonance [25, 12, 17, 28]. Unconventional modulation waveforms are possible in our experiment since the mechanical displacement is performed by a piezoelectric actuator controlled by a low-frequency generator. The actuator is optimized for 40 Hz but has a cut-off frequency of about 100 Hz, so we can use any arbitrary waveform validating these conditions. Measurements have been performed as test cases with our two samples: the GaAs wafer and the gold mirror. We try successively sinusoid, square and triangle modulations at constant 40 Hz fundamental frequency. In each case a measure is done on the reference and reproduce in exactly the same conditions for position, modulation and detection for the GaAs sample. Initial position of the probe is always the minimum distance $h=10$ µm above the surface before applying the modulation whose amplitude is intended to be $\delta h=20$ µm, as calibrated for sinusoid. Depending on the modulation waveform, the effective $h$ may be smaller, especially with square modulation where the transfer function of the piezoelectric actuator exhibits an overshoot. Although the displacement approximately follows the command, the rise and fall times are not instantaneous and the effective distance between the probe and the sample is distorted. This occurs also to a lesser extent with triangular modulation. TABLE I: Deduced normalized reflection coefficient $\Gamma$ of GaAs versus the modulation waveform and the harmonic rank. Rank \| Sinus Modulation \| Square Modulation \| Triangle Modulation ---\|---\|---\|--- $\Gamma$ \| $\zeta$ \| $\Gamma$ \| $\zeta$ \| $\Gamma$ \| $\zeta$ 1 \| $(0.482+j0.225)\pm 0.004$ \| $0.436$ \| $(0.489+j0.226)\pm 0.004$ \| $0.435$ \| $(0.487+j0.226)\pm 0.005$ \| $0.436$ 2 \| $(0.655+j0.280)\pm 0.006$ \| $0.408$ \| $(0.679+j0.270)\pm 0.009$ \| $0.440$ \| $(0.687+j0.283)\pm 0.006$ \| $0.410$ 3 \| — \| $5.6$ \| $(0.511+j0.231)\pm 0.005$ \| $0.436$ \| $(0.528+j0.234)\pm 0.004$ \| $0.431$ 4 \| — \| $1.00$ \| $(0.728+j0.289)\pm 0.005$ \| $0.414$ \| $(0.735+j0.283)\pm 0.021$ \| $0.418$ 5 \| \| \| — \| $2.9$ \| — \| $2.9$ 6 \| \| \| $(0.760+j0.263)\pm 0.017$ \| $0.418$ \| \| 7 \| \| \| $(0.411+j0.173)\pm 0.010$ \| $0.452$ \| \| 8 \| \| \| $(0.779+j0.276)\pm 0.026$ \| $0.428$ \| \| 9 \| \| \| $(0.355+j0.146)\pm 0.021$ \| $0.47$ \| \| The experimental procedure and the determination of the vectorial voltage described in Fig. 4 were reproduced for the three modulations with increasing harmonic ranks selected on the lock-ins. This was done as long as the detected signal emerged from the noise. As can be seen from the average $\left\lvert{\widetilde{v}}\right\rvert$ voltages plotted in Fig. 7, the detected signal decreases rapidly with harmonic rank, but it also depends a lot on the waveform of the modulation. Nevertheless, $\left\lvert{\widetilde{v}}\right\rvert$ is very similar for all waveforms at the first two harmonics. Beyond that, the decrease in intensity is huge for sinusoidal modulation, which generates those high harmonics only by the nonlinear probe-to-sample interaction caused by the rapid decrease of intensity in the near-field. This peculiarity of probe-to-sample interaction is often used to improve the spatial resolution [30, 31, 24]. For other modulation waveforms, higher harmonics are less attenuated, especially in the case of square modulation. Actually they tend to follow the intensities that could be computed by the Fourier transform of the initial waveforms, in particular the high attenuation of even harmonics. For such waveforms we can no more argue for an improvement in resolution by detecting high order harmonics. Figure 7: Average module of the voltage detected $\left\lvert{\widetilde{v}}\right\rvert$ for $f\in[60.690,60.815]$ GHz versus harmonic rank and modulation waveform: $\bullet$ are for sinus, $\blacksquare$ for square and $\blacklozenge$ for triangle modulations. The results were interpreted using Eqs. (7) applied to the best fitted circles on the experiments. We used $\zeta$ as a determinant: if its value exceeds 1, which according to the discussion in §III-B is impossible in the framework of our subharmonic IQ mixer model, then we rejected the measure as a whole. The practical reasons for such a mismatch can be multiple, such as non- reproducible positioning over our test surfaces, weak or noisy signals… The factor $\zeta$ is therefore an effective quality criterion for the control of our measurements. In the case of triangle modulation, the Fig. 8 illustrates the measurement points with the 4 circles obtained for the first 4 harmonics. Although the measurement points are all well located on circles, it is obvious that the even and odd harmonics are grouped together, but the two groups do not merge even if they overlap. As a result the extracted reflection coefficient with even harmonics shows real and imaginary parts which both have higher values. Figure 8: Normalized reflection measured for the four first harmonic ranks with a triangle modulation waveform. Dots are measurements and circles their corresponding best fits. The complete results for all harmonics of all waveforms are given in the Table I. Rejected cases occurring because $\zeta\geq 1$ are represented by a dash and this occurs especially for all harmonics 5. We believe that this is because this frequency of 200 Hz is the only one that is also a harmonic of the power supply frequency, which inevitably produces a large increase of noise in the detection circuit. As soon as $\left\lvert{\widetilde{v}}\right\rvert$ is high enough in all other cases, except for the higher harmonics of the sinusoidal modulation, we obtain $\zeta\approx 0.4$ and the results do not seem to be biased. The data of Table I are plotted in Fig. 9. The superposition of the results for all waveforms highlights the experimental independence in relation to this setting. On the contrary, the alternation of the values between even and odd harmonics is conspicuous. While the real parts have values that increase with the rank of even harmonics, the opposite occurs for odd harmonics, and the same trend is practically true for the imaginary part. Nevertheless the most notable thing is the dichotomy between even and odd harmonics. The most surprising fact of these results is that for a given harmonic rank the normalized reflection value determined is the same regardless of the modulation shape, and this is verified within the limit of the error bars for the first two harmonics and still very close for next two one. Note that increasing the harmonic rank rapidly decreases the signal and therefore mechanically increases the error. Figure 9: Real (left) and imaginary parts (right) of the normalized reflection $\Gamma$ versus harmonic rank and modulation waveform: $\bullet$ are for sinus, $\blacksquare$ for square and $\blacklozenge$ for triangle modulations. Obtaining different $\Gamma$ values from one rank to another is deeply counter-intuitive, as is the fact that even and odd harmonics seem to behave identically. Actually, if only the material tested in the near-field matters, the reflection should be expressed from the dielectric constant $\epsilon$ of the material as predicted by the Fresnel coefficients [32], $\Gamma=(\epsilon-1)/(\epsilon+1)$. For instance with sinusoidal modulation at rank 1, it leads to $\epsilon=2.863+j1.581$, _i.e._ a refractive index of $1.751+j0.451$, which is far from the tabulated value of $3.606+j0.0012$ at 300 GHz [29]. Therefore, the quantities determined here must not be regarded as reflection coefficients in the usual sense of microwaves and it is not realistic to extract any $\epsilon$ from them. Although we use the oscillation of the probe-to-sample distance, which concentrates the useful signal on the near- field, we had already shown by approach curves with these bow-tie probes that a longer distance interaction also exists [24], which certainly enters here in the final quantitative result. The probable origin of this interaction was identified and attributed to the mandatory openings between the two triangles constituting the bow-tie, and we had also noticed that such a long-distance contribution may be favored if a large dielectric support is used to support the two triangles. For the present probe all precautions had been taken to minimize these effects. Nonetheless, the quantities extracted are not reflection coefficients even if we used the usual $\Gamma$ notation of microwaves. They seem however characteristic of the materials placed in the near-field of our bow-tie probe and are obtained with a very good accuracy. ## V Measurement of subharmonic IQ mixer ideality Since the $\zeta$ factor is a measure of the ideality of our instrumentation, it is worth characterizing it over the entire available bandwidth. Under conditions of sinusoidal modulation with $h=10$ µm and $\delta h=20$ µm and detection on the fundamental frequency, we therefore measured both our gold mirror reference and our GaAs sample from 55 to 65 GHz with a 10 MHz stepsize. We then proceed to the extraction defined in §III to obtain $\zeta$ and $\Gamma$ using a 120 MHz sliding window which allows the application of Eqs. (5-7). Figure 10 shows the evolution of $\zeta$ as a function of frequency over the entire operating band of the subharmonic IQ mixer. Obviously, $\zeta$ varies a lot in the whole band with really high values above 61 GHz which exceed unity around 64GHz. This is in full agreement with the observation of a drastic drop of the measured $\widetilde{v}$ at such frequencies for both gold and GaAs. It is attributed to the conjunction of a power drop of the two multipliers, especially the one dedicated to the LO which provides 3dBm less than below, and an increase in the conversion losses of the subharmonic IQ mixer, leading to an almost $\widetilde{v}$ cancellation at lock-in outputs. Figure 10 shows that the best frequency range for our system is around 59 GHz, but if we assume a maximum value of $\approx 0.4$ it is still quite usable from about 56 GHz to 61 GHz. We set out this criterion here because although the frequency around 60.75 GHz used in §III and §IV is at the limit of this useful band, we saw that the results were quite satisfactory and accurate. Figure 10: Ideality factor $\zeta$ versus frequency. It should be noted here that due to a handling error, the spacing between the probe tips has been modified since the recordings in §III. If this spacing was previously 11 µm, it is now 15 µm due to a too abrupt touch that exceeded the tungsten elasticity limits. Nevertheless the probe has been repaired and kept its global shape with only this new spacing parameter changed. A fundamental consequence of this probe crash and repair is that $\zeta$ is independent of this spacing since we obtain here in Fig. 10 $\zeta=0.433\pm 0.024$ at 60.75 GHz whereas it was determined in §IV at $\zeta=0.436\pm 0.005$ (see Table I). The parameter $\zeta$ is therefore a characteristic that is essentially associated with the mixer imbalance but which nevertheless includes any possible variations in LO power that affect its operating point. The determination of $\zeta$ over the whole frequency band also yields $\Gamma$ thanks to Eqs. (7). It is plotted in Fig. 11 where frequencies leading to $\zeta\geq 1$, for instance around 64 GHz, have been rejected. If $\Gamma$ really represented the reflection coefficient only related to the interaction of a dipole with a homogeneous material, we should expect a value corresponding to the Fresnel reflection coefficient, $\Gamma=(\epsilon-1)/(\epsilon+1)$, which should be constant since both gold and GaAs are not dispersive at mmW. This is not the case here, even if in the whole range 56-60 GHz the values vary very little. Since, on the other hand, $\zeta$ encompasses all the imperfections of the experiment involving the subharmonic mixer and its LO, this variation can only be attributed here to the interaction of the probe and the sample which does not seem to follow a simple model. In truth this is confirmed by the fact that we have here at 60.75 GHz $\Gamma=(0.400+j0.152)\pm 0.014$ instead of $\Gamma=(0.482+j0.225)\pm 0.004$ found previously before the probe was modified. Figure 11: Real and imaginary parts of the normalized reflection $\Gamma$ versus frequency. We can therefore conclude that our near-field measurement remains dependent on the probe used, even if the uniqueness of $\zeta$ has shown that all other sources of variability related to the experiment have been compensated for. This is clearly a drawback as we cannot trace back to universal material characteristics but it is probably still possible to distinguish near-field materials after learning with each probe used. In future work, we will try to demonstrate and illustrate this behavior with other probes and materials. ## VI Conclusion We presented the realization of a new measurement bench for near-field reflectometry in mmW. A detailed analysis of the operation of this bench has been carried out, making it possible to take advantage of the electrical length that exists between the test and measurement paths. We illustrate its use to extract the near-field contribution that is independent of the mmW board from measurements distorted by systematic errors. A spin-off of this theoretical analysis of the operation of the experiment is the definition of a coefficient of ideality $\zeta$ to which the IQ mixer imbalance and the gain variations of the mmW sources mainly contribute. We applied this technique to the near-field measurement of a GaAs substrate and a gold mirror. At fixed frequency the study shows a result that is substantially independent of the waveform of the mechanical modulation but not of the order of the harmonic used for detection. Curiously the even and odd harmonics lead to two groups of results that are quite consistent with each other but remain unexplained. A study of our measurement system over the entire 55-65 GHz band has allowed us to qualify $\zeta$. For reasons related to our equipment, the latter one turns out to be rather bad in high frequency but allows very reliable measurements between 56 and 61 GHz. Moreover, $\zeta$ appears independent of the probe, which is not the case of the normalized vectorial voltage $\Gamma$ measured in the near-field. It is clear that our probe does not have a trivial interaction with the sample and is responsible for this unexpected variation. In the past, we had already observed similar effects with a long range detection effect from the lateral radiation of our probes [24]. However, here we have taken great care to minimize it by eliminating any protruding dielectric material used for mounting the probe. The values obtained for $\Gamma$ and its evolution with frequency cannot be explained with a simple model of the probe-sample interaction. Linking $\Gamma$ to the material parameters of the sample under test in the near-field will therefore require a further study, which can certainly only be carried out through full 3D electromagnetic modeling of the coupled probe-sample system. 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# Dimensional Reduction and (Anti) de Sitter Bounds ###### Abstract Dimensional reduction has proven to be a surprisingly powerful tool for delineating the boundary between the string landscape and the swampland. Bounds from the Weak Gravity Conjecture and the Repulsive Force Conjecture, for instance, are exactly preserved under dimensional reduction. Motivated by its success in these cases, we apply a similar dimensional reduction analysis to bounds on the gradient of the scalar field potential $V$ and the mass scale $m$ of a tower of light particles in terms of the cosmological constant $\Lambda$, which ideally may pin down ambiguous $O(1)$ constants appearing in the de Sitter Conjecture and the (Anti) de Sitter Distance Conjecture, respectively. We find that this analysis distinguishes the bounds $\|\nabla V\|/V\geq\sqrt{4/(d-2)}$, $m\lesssim\|\Lambda\|^{1/2}$, and $m\lesssim\|\Lambda\|^{1/d}$ in $d$-dimensional Planck units. The first of these bounds precludes accelerated expansion of the universe in Einstein-dilaton gravity and is almost certainly violated in our universe, though it may apply in asymptotic limits of scalar field space. The second bound cannot be satisfied in our universe, though it is saturated in supersymmetric AdS vacua with well-understood uplifts to 10d/11d supergravity. The third bound likely has a limited range of validity in quantum gravity as well, so it may or may not apply to our universe. However, if it does apply, it suggests a possible relation between the cosmological constant and the neutrino mass, which (by the see-saw mechanism) may further provide a relation between the cosmological constant problem and the hierarchy problem. We also work out the conditions for eternal inflation in general spacetime dimensions, and we comment on the behavior of these conditions under dimensional reduction. ###### Contents 1. 1 Introduction 2. 2 Review: Dimensional Reduction and Weak Gravity Conjectures 3. 3 Light Particles and the Cosmological Constant 4. 4 Interlude: Eternal Inflation in Higher Dimensions 5. 5 Derivatives of Scalar Field Potentials 6. 6 Conclusions ## 1 Introduction The search for universal features of quantum gravity–also known as the swampland program [1, 2]–has seen a resurgence in recent years. Strong evidence has been given in favor of certain conjectured properties of quantum gravities (so-called “swampland conjectures”), some longstanding conjectures have been discarded as counterexamples have emerged, and many seemingly- unrelated aspects of physics and mathematics have been connected through an ever-growing swampland web. But the swampland program also faces some serious difficulties. First and foremost is the infamous swampland tradeoff: conjectures that tend to have more evidence in their favor tend to have less interesting applications to cosmology and phenomenology, whereas conjectures that have significant implications for observable physics tend to be more speculative. Related to this is a lack of precision: many swampland statements are plagued by squiggles ($\sim$, $\lesssim$, $\gtrsim$) and $O(1)$ constants to be determined later. These squiggles make the conjectures difficult to test, and they suggest that we may not yet understand the underlying physics responsible for such conjectures. Without this, it is difficult to say when exactly the conjectures may be expected to hold, and when they may cease to be valid. The goal of this paper is to sharpen some of these more speculative swampland conjectures, primarily through a mechanism that has successfully sharpened swampland conjectures in the past: dimensional reduction. As we will see in Section 2 below, $O(1)$ factors in the Weak Gravity Conjecture (WGC) bound and the Repulsive Force Conjecture (RFC) bound can be fixed precisely by demanding that these bounds should be exactly preserved under dimensional reduction from $D$ to $d=D-1$ dimensions: the bound will be saturated in $d$ dimensions if and only if it is saturated in $D$ dimensions. In these cases, dimensional reduction succeeds in selecting bounds that are both physically meaningful and seemingly-universal: all known quantum gravities satisfy both the WGC and the RFC, and compelling evidence has been provided that these conjectures should hold more generally. Of course, this does not prove that _any_ bound that is preserved under dimensional reduction is necessarily a universal constraint on quantum gravity. But the success of dimensional reduction in the above cases suggests that it may serve as a useful diagnostic for distinguishing certain $O(1)$ values in conjectured bounds when previous arguments have left us with ambiguity. Relatedly, this paper will not address the ultimate question of whether certain proposed conjectures are universally true in quantum gravity. It seems very likely that many of these conjectures are true within a limited domain of validity. For instance, the relationship between the masses of a tower of light particles and the cosmological constant proposed in the AdS Distance Conjecture [3] is likely true in infinite families of quantum gravities, each member of which is distinguished by some discrete flux parameter. The lower bound on the gradient of a scalar field potential proposed in the de Sitter Conjecture [4] is likely true in asymptotic regions of scalar field space, and relatedly eternal inflation is likely forbidden in such regions [5]. The bounds we will obtain in this paper do not necessarily apply to all quantum gravities–in fact, some of them are violated even in our own universe–but they may point us to interesting, universal behavior in certain limits of quantum gravity. At the least, they give us a mark to aim for and to explore further. The main results and structure of our paper is as follows: in Section 2, we will review how dimensional reduction distinguishes the WGC and RFC bounds, fixing their $O(1)$ factors. In Section 3, we will apply a similar dimensional reduction analysis to bounds on the masses of light particles in terms of the cosmological constant of the form $m\lesssim\|\Lambda_{d}\|^{\alpha_{d}}M_{d}^{1-d\alpha_{d}}\,,$ (1.1) where $\Lambda_{d}$ and $M_{d}$ are the cosmological constant and the Planck mass in $d$ dimensions, respectively. In [3], it was conjectured that a tower of light particles should satisfy this bound in any AdS (and perhaps dS) vacuum. We will see that dimensional reduction picks out $\alpha_{d}=1/2$ and $\alpha_{d}=1/d$ as special values of $\alpha_{d}$, depending on whether or not we include the contribution from 1-loop Casimir effects. The former value is saturated in known supersymmetric compactifications [3] and was previously singled out in [6], whereas the latter is interesting from the perspective of the standard model, since neutrinos have a mass of roughly $m_{\nu}\sim\Lambda_{4}^{1/4}$. If this mass comes from the see-saw mechanism, $m_{\nu}\sim y^{2}v^{2}/\Lambda_{\text{UV}}$, where $y$ is the Yukawa coupling, $v$ is the Higgs vev, and $\Lambda_{\text{UV}}$ is some UV scale, this bound further becomes $v^{2}\lesssim\Lambda_{4}^{1/4}\Lambda_{\text{UV}}/y\,,$ (1.2) which bounds the Higgs vev in terms of the cosmological constant and therefore relates the hierarchy problem to the cosmological constant problem. In particular, if $\Lambda_{\text{UV}}\sim M_{\text{GUT}}$, $y\sim 0.1$, then (1.2) becomes $v\lesssim 100$ GeV, which is not far from the measured value $v=246$ GeV. (Note that we have swept a number of $O(1)$ factors here under the rug.) In Section 4, we will derive constraints for eternal inflation in $d\geq 4$ spacetime dimensions by solving the associated Fokker-Planck equation analytically, generalizing the $d=4$ result of [5]. We will find that eternal stochastic inflation occurs only if the following conditions are satisfied on the first and second derivatives of the potential: $\displaystyle\frac{\|\nabla V\|^{2}}{V^{(d+2)/2}}$ $\displaystyle<\frac{2(d-1)^{3}}{\pi\Omega_{d-2}}\left(\frac{2}{(d-1)(d-2)M_{d}^{d-2}}\right)^{(d+2)/2}\,,$ (1.3) $\displaystyle\frac{V^{\prime\prime}}{V}$ $\displaystyle>-\frac{2(d-1)}{d-2}\frac{1}{M_{d}^{d-2}}\,,$ (1.4) where $\Omega_{d-2}={2\pi^{(d-1)/2}/\Gamma(\frac{d-1}{2})}$. In multifield landscapes, $V^{\prime\prime}$ above should be replaced by a sum over the negative eigenvalues of the Hessian of the potential. In Section 5, we will consider bounds of the form $\frac{\|\nabla V_{d}\|}{V_{d}^{\gamma_{d}}}\geq c_{d}M_{d}^{d(1-\gamma_{d})-(d-2)/2}\,.$ (1.5) Preservation under dimensional reduction distinguishes $\gamma_{d}=1$ as a special value of $\gamma_{d}$, in agreement with the de Sitter Conjecture (dSC) bound of [4] and the Trans-Planckian Censorship Conjecture (TCC) bound. It further selects $c_{d}=\sqrt{4/(d-2)}$ as a special value of $c_{d}$, in which case (1.5) corresponds exactly to the condition required to avoid accelerated expansion of the universe. Since this particular value of $c_{d}$ is larger than the coefficient $c_{d}=2/\sqrt{(d-1)(d-2)}$ that appears in the TCC bound, we learn that a theory satisfying the TCC bound will still satisfy the TCC bound after dimensional reduction. Likewise, since our value $\gamma_{d}=1$ is smaller than the value $\gamma_{d}=(d+2)/4$ appearing in the eternal inflation bound (1.3), we see that the dimensionally reduced theory will not lead to eternal inflation in asymptotic regions of scalar field space; eternal inflation can occur only if the radion is stabilized. We end in Section 6 with conclusions and directions for future research. ## 2 Review: Dimensional Reduction and Weak Gravity Conjectures In this section, we review how the Weak Gravity Conjecture (WGC) [7] bound and the Repulsive Force Conjecture (RFC) [8] bounds are preserved under dimensional reduction. Our starting point is an Einstein-Maxwell-dilaton action for a $P$-form gauge field $A_{\mu_{1}\ldots\mu_{P}}$ in $D=d+1$ dimensions: $S=\frac{1}{2\kappa_{D}^{2}}\int\mathrm{d}^{D}x\sqrt{-g}\left({\cal R}_{D}-\frac{1}{2}(\nabla\phi)^{2}\right)-\frac{1}{2e_{P;D}^{2}}\int\mathrm{d}^{D}x\sqrt{-g}\mathrm{e}^{-\alpha_{P;D}\phi}F_{P+1}^{2}\,.$ (2.1) Here, $F_{P+1}=\mathrm{d}A_{P}$ is the field strength for $P$-form gauge field, and $F_{q}^{2}\mathrel{:=}\frac{1}{q!}F_{\mu_{1}\ldots\mu_{q}}F^{\mu_{1}\ldots\mu_{q}}\,.$ (2.2) Note that the action (2.1) is rather special and does not describe the low- energy theory of a generic quantum gravity, but it will suffice for our purposes to restrict our attention to this case. We define $\frac{1}{\kappa_{D}^{2}}\mathrel{:=}M_{D}^{D-2},$ (2.3) where $M_{D}$ is the reduced Planck mass in $D$ dimensions. With this convention, the WGC bound for a $(P-1)$-brane of quantized charge $q$ and tension $T_{P}$ is given by: $e_{P;D}^{2}q^{2}M_{D}^{D-2}\geq\left[\frac{\alpha_{P;D}^{2}}{2}+\frac{P(D-P-2)}{D-2}\right]T_{P}^{2}\mathrel{:=}\gamma^{-1}_{P;D}(\alpha)T_{P}^{2}\,.$ (2.4) The RFC bound in this theory is given by $e_{P;D}^{2}q^{2}M_{D}^{D-2}\geq 2(\partial_{\phi}T_{P})^{2}+\frac{P(D-P-2)}{D-2}T_{P}^{2}\,,$ (2.5) where $\partial_{\phi}T_{P}\mathrel{:=}\partial T_{P}/\partial\phi$ is the partial derivative of the brane tension with respect to the dilaton $\phi$, holding the Planck scale fixed. A $(P-1)$-brane that satisfies the WGC bound is said to be superextremal: its charge-to-mass ratio is greater than that of an extremal black brane. A $(P-1)$-brane that satisfies the RFC bound is self- repulsive: two such parallel branes separated by a parametrically large distance will feel a repulsive force between them. We now show how these bounds are preserved under dimensional reduction. We consider a dimensional reduction ansatz of the form, $\mathrm{d}s^{2}=\mathrm{e}^{\frac{\lambda(x)}{d-2}}\mathrm{d}{\hat{s}}^{2}(x)+\mathrm{e}^{-\lambda(x)}\mathrm{d}y^{2},$ (2.6) where $y\sim y+2\pi R$. This ansatz is chosen so that the dimensionally reduced action is in Einstein frame, i.e., there is no kinetic mixing between $\lambda$ and the $d$-dimensional metric. For simplicity, we do not include a Kaluza-Klein photon, but we do include a massless radion $\lambda(x)$, which controls the radius of the circle. Under such a dimensional reduction, the Einstein-Hilbert term reduces as $\frac{1}{2\kappa_{D}^{2}}\int\mathrm{d}^{D}x\sqrt{-g}{\cal R}_{D}\rightarrow\frac{1}{2\kappa_{d}^{2}}\int\mathrm{d}^{d}x\sqrt{-\hat{g}}{\cal R}_{d}-\frac{1}{2}\int\mathrm{d}^{d}x\sqrt{-\hat{g}}\,G_{\lambda\lambda}(\nabla\lambda)^{2}\,,$ (2.7) where $\displaystyle\frac{1}{\kappa_{d}^{2}}$ $\displaystyle=M_{d}^{d-2}=(2\pi R)M_{D}^{D-2}\,,$ (2.8) $\displaystyle G^{(d)}_{\lambda\lambda}$ $\displaystyle=\frac{(d-1)}{4\kappa_{d}^{2}(d-2)}=M_{d}^{d-2}\frac{d-1}{4(d-2)}\,.$ (2.9) The $P$-form in $D$ dimensions gives both a $P$-form and a $p\mathrel{:=}P-1$-form in $d$ dimensions. The former comes from taking all of the legs of the $P$-form to lie along noncompact directions, while the latter comes from taking one of the legs of the $P$-form to wrap the compact circle. Similarly, a $(P-1)$-brane charged under the $P$-form descends to both a $(P-1)$-brane and a $(p-1)$-brane, charged under the respective forms. For simplicity, we present only the former here: the latter computation is analogous. Consider then the $P$-form gauge field in $d$ dimensions. The associated gauge coupling is given by $e_{P;d}^{2}=\frac{1}{2\pi R}e_{P;D}^{2}\mathrm{e}^{\frac{P\lambda}{d-2}}\,.$ (2.10) The tension of a $(P-1)$-brane transverse to the compact circle is given by $T_{P}^{(d)}=\mathrm{e}^{\frac{P\lambda}{2(d-2)}}T_{P}^{(D)}\,.$ (2.11) Upon reduction, the radion $\lambda$ and dilaton $\phi$ each couple exponentially to the Maxwell term in the action, as in (2.1). We may then redefine the dilaton as the linear combination of $\lambda$ and $\phi$ that couples to the Maxwell term. This effectively shifts the coupling $\alpha$ appearing in the WGC bound to $\alpha_{P;d}^{2}=\alpha_{P;D}^{2}+\frac{2P^{2}}{(d-1)(d-2)}.$ (2.12) Plugging this into the WGC bound (2.4) with $D\rightarrow d$, we find $\displaystyle e_{P;d}^{2}q^{2}M_{d}^{d-2}\geq\left[\frac{\alpha_{P;d}^{2}}{2}+\frac{P(d-P-2)}{d-2}\right]T_{P}^{2}=\left[\frac{\alpha_{P;D}^{2}}{2}+\frac{P(D-P-2)}{D-2}\right]T_{P}^{2}.$ (2.13) The right-hand side of this bound matches that of (2.4). This shows that the WGC bound has been exactly preserved by the dimensional reduction process. An analogous result holds for the case of decreasing $P\rightarrow P-1$ and for Kaluza-Klein modes when $P=1$ provided we include a Kaluza-Klein photon in the dimensional reduction ansatz of (2.6) [9]: in all such cases, the WGC bound is exactly preserved under reduction. The RFC bound in $d$ dimensions is then given by $e_{P;d}^{2}q^{2}\geq\frac{1}{2\pi R}(\partial_{\phi}T_{P}^{(d)})^{2}+(G^{(d)})^{\lambda\lambda}(\partial_{\lambda}T_{P}^{(d)})^{2}-\frac{P(d-P-2)}{d-2}\frac{\big{(}T_{P}^{(d)}\big{)}^{2}}{M_{d}^{d-2}}.$ (2.14) Using (2.9) and (2.11), we have $(G^{(d)})^{\lambda\lambda}(\partial_{\lambda}T_{P}^{(d)})^{2}=\frac{4(d-2)}{M_{d}^{d-2}(d-1)}\left(\frac{P}{2(d-2)}\right)^{2}\big{(}T_{P}^{(d)}\big{)}^{2}=\frac{P^{2}}{(d-1)(d-2)}\frac{\big{(}T_{P}^{(d)}\big{)}^{2}}{M_{d}^{d-2}}\,.$ (2.15) This combines with the last term in (2.14) to give $\left[\frac{P^{2}}{(d-1)(d-2)}+\frac{P(d-P-2)}{d-2}\right]\frac{\big{(}T_{P}^{(d)}\big{)}^{2}}{M_{d}^{d-2}}=\frac{P(D-P-2)}{D-2}\frac{\big{(}T_{P}^{(d)}\big{)}^{2}}{M_{d}^{d-2}}\,.$ (2.16) With this, we see that the RFC bound in $d$ dimensions (2.14) is exactly the same as the RFC bound in $D$ dimensions (2.5). This result may be generalized to multiple scalar fields, multiple gauge fields, and general scalar field couplings. It holds also for the case of $P$ reduced to $p=P-1$ and for Kaluza-Klein modes when $P=1$ [10]: in all cases, the RFC bound is exactly preserved under dimensional reduction. Within the string landscape, there are no known counterexamples to either the WGC or the RFC. Many examples have been confirmed to satisfy these conjectures [7, 10, 11, 12, 13, 14, 15, 16, 17, 18], and a number of arguments suggest that they should hold more generally [19, 20, 21, 22, 23, 24]. The takeaway lesson here is that preservation under dimensional reduction can be a useful tool for identifying universal, physically meaningful behavior of quantum gravities. The value of $\gamma_{P;D}$ in (2.4) is distinguished in that it dictates the extremality bound for charged black branes in the theory of (2.1), and it is also distinguished in that it is preserved under dimensional reduction. Similarly, the bound (2.5) is distinguished by the fact that it dictates the self-repulsiveness of a brane at long distances, and simultaneously it is distinguished in that it is preserved under dimensional reduction. Both of these bounds are further distinguished by the fact that they seem to be satisfied in all quantum gravity theories. We see that, at least in these two cases, dimensional reduction picks out universal, physically meaningful constraints on quantum gravities. ## 3 Light Particles and the Cosmological Constant Suppose that a family of AdS or dS quantum gravities contain a tower of light particles whose masses satisfy $m_{n}\lesssim n\|\Lambda_{D}\|^{\alpha_{D}}M_{D}^{1-D\alpha_{D}}\,,~{}~{}~{}~{}~{}n\in\mathbb{Z}_{>0}.$ (3.1) Such towers of light particles occur, for instance, in all known AdS vacua of string theory, typically arising as Kaluza-Klein modes of some compactified dimensions. Such towers were discussed in detail in [3], where it was further conjectured that there exists a universal, $O(1)$ value of $\alpha_{D}$ that is obeyed in all (A)dS vacua of string theory. This conjecture was called the (A)dS Distance Conjecture ((A)dSDC). We now apply a dimensional reduction analysis to the bound (3.1). Our starting point is the $D$-dimensional action: $S=\int\mathrm{d}^{D}x\sqrt{-g}\left[\frac{M_{D}^{D-2}}{2}{\cal R}_{D}-\Lambda_{D}-\frac{1}{2}\sum_{n}\left((\nabla\phi_{n})^{2}+m_{n}^{2}\phi_{n}^{2}\right)\right]\,,$ (3.2) where $\phi_{n}$ represents the $n$th particle in the tower. We have written the action as if these particles are scalar fields, but this assumption is not necessary. We now perform a dimensional reduction to $d=D-1$ dimensions using the same ansatz as above: $\mathrm{d}s^{2}=\mathrm{e}^{\frac{\lambda(x)}{d-2}}\mathrm{d}{\hat{s}}^{2}(x)+\mathrm{e}^{-\lambda(x)}\mathrm{d}y^{2},$ (3.3) The resulting action in $d$ dimensions takes effectively the same form as (3.2): $S=\int\mathrm{d}^{d}x\sqrt{-\hat{g}}\left[\frac{M_{d}^{d-2}}{2}\left({\cal R}_{d}-\frac{d-1}{4(d-2)}(\nabla\lambda)^{2}\right)-\Lambda_{d}-\frac{1}{2}\sum_{n}\left((\nabla\phi_{n})^{2}+(m_{n}^{(d)})^{2}\phi_{n}^{2}\right)\right]\,,$ (3.4) At a classical level, the parameters are related by $M_{d}^{d-2}=(2\pi R)M_{D}^{D-2}\,,~{}~{}~{}~{}\Lambda_{d}=(2\pi R)\Lambda_{D}\mathrm{e}^{\frac{\lambda}{d-2}}\,,~{}~{}~{}~{}m_{n}^{(d)}=m_{n}^{(D)}\mathrm{e}^{\frac{\lambda}{2(d-2)}}\,.$ (3.5) There will also be Kaluza-Klein modes, which we have ignored. For simplicity, we assume here that $\langle\lambda\rangle=0$: we can ensure this by shifting $\lambda\rightarrow\lambda-\langle\lambda\rangle$, if necessary. From this, we see that if the masses $m_{n}^{(D)}$ satisfy (3.1) in $D$ dimensions, then the masses $m_{n}^{(d)}$ will satisfy $m_{n}\lesssim n\|\Lambda_{d}\|^{\alpha_{d}}M_{d}^{1-2\alpha_{d}}\,,~{}~{}~{}~{}~{}n\in\mathbb{Z}_{>0}\,,$ (3.6) provided $\|\Lambda_{d}\|^{\alpha_{d}}M_{d}^{1-d\alpha_{d}}\geq\|\Lambda_{D}\|^{\alpha_{D}}M_{D}^{1-D\alpha_{D}}=\|\Lambda_{d}\|^{\alpha_{D}}M_{d}^{\frac{d-2}{d-1}(1-D\alpha_{D})}(2\pi R)^{\frac{1}{D-2}(2\alpha_{D}-1)}\,.$ (3.7) From this, we see that the value $\alpha_{d}=1/2$ is distinguished by dimensional reduction: for this particular value, the $R$ dependence cancels, and the bound (3.1) is exactly preserved under dimensional reduction. As pointed out in [3], the value $\alpha_{d}=1/2$ is also distinguished physically in that it is saturated in all known string/M-theory backgrounds that can be understood as 10 or 11-dimensional solutions. The authors further conjecture that the bound (3.1) should be satisfied with $\alpha_{d}=1/2$ for all supersymmetric AdS vacua. This value of $\alpha_{d}$ was also singled out it [6], where it was connected to the dSC, the TCC, and the Swampland Distance Conjecture (SDC) [2]. Once again, dimensional reduction has distinguished a bound of physical interest. It is worth pointing out that this bound cannot be satisfied in our own universe, however. A tower of particles of this mass would run in loops, correct the graviton propagator, and lead gravity to become strongly coupled at the “species bound” scale $E_{\text{QG}}$ of order $E_{\text{QG}}\sim M_{4}/\sqrt{N(E_{\text{QG}})}\,,~{}~{}~{}N(E_{\text{QG}})\sim E_{\text{QG}}M_{4}/\Lambda_{4}^{1/2}\,.$ (3.8) This in turn implies $E_{\text{QG}}\sim\Lambda_{4}^{1/6}M_{4}^{1/3}\sim 0.02\text{ GeV}\,,$ (3.9) which conflicts with the experimental fact that gravity remains weakly coupled at energies accessible at the LHC. Our above analysis takes place entirely at a classical level. As shown in (3.5), the cosmological constant $\Lambda_{d}$ acquires an exponential dependence on the radion $\lambda$. If the radion is not stabilized by quantum effects, the dimensionally reduced theory does not have a vacuum. Suppose that we now insist that the theory should in fact have a vacuum, so the radion must be stabilized. This can be done most simply by including the 1-loop Casimir energy contributions to the radion potential from the light particles. Upon dimensional reduction to $d$ dimensions, the contribution from a particle of mass $m$ takes a very simple form in the massless limit ($mR\ll 1$) [25]: $V_{C}(\lambda)=\mp\frac{2}{(2\pi R)^{d}\Omega_{d}}\zeta(d+1)\mathrm{e}^{\frac{d(d-1)}{2(d-2)}\lambda}\,,~{}~{}~{}\Omega_{d}=\frac{2\pi^{(d+1)/2}}{\Gamma(\frac{d+1}{2})}\,.$ (3.10) Here, $\zeta(x)$ is the Riemann zeta function, $\Omega_{d}$ is the volume of the unit $d$-sphere, and the $+$ sign is for bosons or fermions with antiperiodic boundary conditions, while the $-$ sign is for fermions with periodic boundary conditions. The 1-loop Casimir energy for a particle much heavier than $1/R$ is exponentially suppressed as $\mathrm{e}^{-2\pi mR}$, so for our purposes, it suffices to consider contributions only from light particles with $m\lesssim 1/R$. For simplicity, we approximate all such particles to be massless. The full potential at 1-loop, including both the classical contribution and the 1-loop Casimir energy from light particles, is then given by $V(\lambda)=V_{\Lambda}(\lambda)+V_{C}(\lambda)=(2\pi R)\Lambda_{D}\mathrm{e}^{\frac{\lambda}{d-2}}-\sum_{n\|m_{n}<1/R}(-)^{F_{n}}\frac{2}{(2\pi R)^{d}\Omega_{d}}\zeta(d+1)\mathrm{e}^{\frac{d(d-1)}{2(d-2)}\lambda}\,.$ (3.11) We differentiate with respect to $\lambda$ and set this to 0 to find a critical point: $0=\partial_{\lambda}V(\lambda)=\frac{1}{d-2}(2\pi R)\Lambda_{D}\mathrm{e}^{\frac{\lambda}{d-2}}-\frac{d(d-1)}{2(d-2)}\sum_{n\|m_{n}<1/R}(-)^{F_{n}}\frac{2}{(2\pi R)^{d}\Omega_{d}}\zeta(d+1)\mathrm{e}^{\frac{d(d-1)}{2(d-2)}\lambda}\,.$ (3.12) The exponent of the second term is larger than that of the first term. This means that in order to find a minimum, the second term must be positive, while the first is negative. For $\Lambda_{D}<0$, we may therefore obtain a minimum in $d$ dimensions provided that fermions with periodic boundary conditions dominate the Casimir energy. For $\Lambda_{D}>0$, on the other hand, we may find a maximum in $d$ dimensions provided that bosons or fermions with antiperiodic boundary conditions dominate the Casimir energy. Creating a de Sitter minimum in this way would require a delicate interplay between bosons and fermions, and it would require us to go beyond the massless particle approximation we have employed here. Let us now suppose that we have a tower of particles of equal spin $F$, with masses $m_{n}=nm\,,~{}~{}~{}~{}n\in\mathbb{Z}_{>0}\,,$ (3.13) so that the sum in (3.12) becomes $\sum_{n\|m_{n}<1/R}(-)^{F_{n}}=\sum_{n=1}^{1/(mR)}(-)^{F}=(-)^{F}\frac{1}{mR}\,.$ (3.14) In order for the two terms in (3.12) to balance, we must therefore have (ignoring $O(1)$ factors): $\frac{1}{mR^{D}}\sim\|\Lambda_{d}\|\,.$ (3.15) We further impose that the bound (3.1) must be saturated both before and after reduction: $m\sim\|\Lambda_{D}\|^{\alpha_{D}}M_{D}^{1-2\alpha_{D}}\sim\|\Lambda_{d}\|^{\alpha_{d}}M_{d}^{1-2\alpha_{d}}\,.$ (3.16) Putting these together, we may eliminate $R$ and $m$ to find a relation between $\Lambda_{D}$ and $\Lambda_{d}$, which gives the following relation between $\alpha_{D}$ and $\alpha_{d}$: $\alpha_{D}=\alpha_{d}-\frac{2\alpha_{d}^{2}+\alpha_{d}-1}{2\alpha_{d}-d^{2}+3}\,.$ (3.17) This is a recursive relation between $\alpha_{d}$ and $\alpha_{d+1}$, which has a 1-parameter family of solutions. One such solution is the value $\alpha_{d}=1/2$, which we found above from a strictly classical analysis. Another solution is $\alpha_{d}=1/d$: this value is special in that it is the smallest possible value of $\alpha_{d}$ consistent with the recursion relation (3.17) that remains non-negative in the limit $d\rightarrow\infty$. Note that it is also the unique value of $\alpha_{d}$ for which the Planck scale $M_{d}$ drops out of (3.1), and in the reduction studied above it leads to $m\sim 1/R\sim\|\Lambda_{d}\|^{1/d}\sim\|\Lambda_{D}\|^{1/D}$, so the sum over $n$ in (3.14) runs over an $O(1)$ number of terms. The bound $m\lesssim\|\Lambda_{d}\|^{1/d}$ is rather tantalizing, as the current upper bound on the sum of the masses of the three standard model neutrinos is $0.3$ eV, coming from a combination of CMB measurements, galaxy surveys, and Lyman-alpha forest data. The lightest neutrino may in principle be massless, though there are lower bounds on the differences in the squared-masses of the neutrinos from observations of neutrino oscillations. This requires that at least one neutrino must have a mass above about $0.05$ eV. The cosmological constant in our universe is measured to be $\Lambda_{4}^{1/4}\simeq 10^{-30}M_{4}\simeq 0.002\text{ eV}\,,$ (3.18) which is quite close to the neutrino mass scale of roughly $0.01-0.1$ eV. As discussed in Section 1, the relationship between the neutrino mass and the cosmological constant implies a relationship between the Higgs vev and the cosmological constant, if we assume that the neutrino mass comes from the see- saw mechanism, thereby relating the cosmological constant problem to the hierarchy problem. This argument is reminiscent of the work of [26, 27, 28]. This application to our own universe comes with an important caveat, however: our 1-loop calculation produced minima of the radion potential only in the case of a negative cosmological constant $\Lambda_{d}<0$, whereas our universe has $\Lambda_{d}>0$, for which our simple analysis yields maxima of the potential. This could be remedied by moving beyond the massless limit and considering both fermions and bosons (or fermions with antiperiodic boundary conditions). Indeed, balancing the 1-loop Casimir energies of particles in the standard model leads to a vacuum in three dimensions upon dimensional reduction [25]. ## 4 Interlude: Eternal Inflation in Higher Dimensions We now take a short break from our dimensional reduction analysis to work out the conditions for eternal inflation in general spacetime dimension $d$ by solving the Fokker-Planck equation, generalizing the computations of [5] (see also [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39]).aaaWe are very thankful to Liam McAllister for discussions on the computations in this section. In Section 5 below, we will comment briefly on the behavior of the conditions for avoiding eternal inflation under dimensional reduction. We begin from the $d$-dimensional metric $ds^{2}=-dt^{2}+a(t)^{2}d\vec{x}^{2}.$ (4.1) and the action $S=\int d^{d}x\sqrt{-g}\left[\frac{M_{d}^{d-2}}{2}\mathcal{R}-\frac{1}{2}(\nabla\phi)^{2}-V(\phi)\right].$ (4.2) For simplicity, we take $\phi$ to be canonically normalized, and at 0th order we take it to be homogenous so that its spatial derivatives vanish. The equation of motion for the scalar field is thus $\ddot{\phi}+(d-1)H\dot{\phi}=-V^{\prime}(\phi)\,,$ (4.3) where $H=\dot{a}/a$. The stress-energy tensor is given by $\displaystyle T^{\mu}_{\nu}$ $\displaystyle=-g^{\mu\delta}\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\delta\nu}}=\text{diag}\left(-\frac{1}{2}\dot{\phi}^{2}-V(\phi),\frac{1}{2}\dot{\phi}^{2}-V(\phi),...,\frac{1}{2}\dot{\phi}^{2}-V(\phi)\right).$ (4.4) This is the stress-energy tensor of a perfect fluid of density $\rho$, pressure $p$, with $\rho=\frac{1}{2}\dot{\phi}^{2}+V(\phi)\,,~{}~{}~{}p=\frac{1}{2}\dot{\phi}^{2}-V(\phi)\,.$ (4.5) Einstein’s equations are given by $G_{\mu\nu}=\kappa_{d}^{2}T_{\mu\nu}=\frac{1}{M_{d}^{d-2}}T_{\mu\nu}\,.$ (4.6) Here, we have $\displaystyle G_{00}=\frac{1}{2}(d-1)(d-2)H^{2}=\frac{\rho}{M_{d}^{d-2}}\,,$ (4.7) $\displaystyle\frac{1}{a^{2}}G_{11}=...=\frac{1}{a^{2}}G_{d-1,d-1}=-(d-2){\ddot{a}\over a}-\frac{1}{2}(d-2)(d-3)H^{2}=\frac{p}{M_{d}^{d-2}}\,.$ (4.8) This gives the Friedmann equations: $H^{2}=\frac{2\rho}{(d-1)(d-2)M_{d}^{d-2}}\,,~{}~{}~{}\frac{\ddot{a}}{a}=-\frac{p}{(d-2)M_{d}^{d-2}}-\frac{(d-3)H^{2}}{2}\,.$ (4.9) Inflation takes place in the slow-roll regime, $\dot{\phi}^{2}\ll V(\phi)\,,~{}~{}~{}~{}~{}\|\ddot{\phi}\|\ll H\|\dot{\phi}\|,\|V^{\prime}(\phi)\|\,,$ (4.10) in which case the equation of motion for $\phi$ and the first Friedmann equation become $(d-1)H\dot{\phi}=-V^{\prime}(\phi)\,,~{}~{}~{}~{}H^{2}=\frac{2V(\phi)}{(d-1)(d-2)M_{d}^{d-2}}\,.$ (4.11) To study eternal inflation, we further need to incorporate backreaction from quantum fluctuations of the scalar field. The scalar 2-point function in dSd takes the form [40] $\langle\phi^{2}\rangle=\frac{H^{d-1}}{\pi\Omega_{d-2}}t+\ldots\,,~{}~{}~{}~{}~{}~{}\Omega_{d-2}=\frac{2\pi^{(d-1)/2}}{\Gamma(\frac{d-1}{2})}\,.$ (4.12) where $\ldots$ represents terms that are not linear in $t$. This linear term encodes the effect of Gaussian quantum fluctuations that exit the horizon, decohere, and backreact on the classical slow-roll equation of motion (4.11). This backreaction takes the form $(d-1)H\dot{\phi}+V^{\prime}(\phi)=N(t)\,,$ (4.13) where $N(t)$ is a Gaussian noise term, which induces a random walk of the field $\phi$ in the potential $V(\phi)$. In other words, in an infinitesimal time $\delta t$, $\phi$ will vary according to $\delta\phi=-\frac{1}{(d-1)H}V^{\prime}(\phi)\delta t+\delta\phi_{q}(\delta t)\,,~{}~{}~{}~{}~{}\delta\phi_{q}(\delta t)\sim\mathcal{N}(0,\frac{H^{d-1}}{\pi\Omega_{d-2}}\delta t)\,,$ (4.14) where the variance of the normal distribution comes from the coefficient of the linear term in (4.12). If we further approximate $H$ as a constant, which is well-justified in the slow-roll regime, the evolution of the probability distribution $P[\phi,t]$ of $\phi$ as a function of $t$ is then described by a Fokker-Planck equation [29, 30, 31]: $\dot{P}[\phi,t]=\frac{1}{2}\left(\frac{H^{d-1}}{\pi\Omega_{d-2}}\right)\partial_{\phi}^{2}P[\phi,t]+\frac{1}{(d-1)H}\partial_{\phi}\Big{(}(\partial_{\phi}V(\phi))P[\phi,t]\Big{)}\,.$ (4.15) This equation is difficult to solve for a general potential $V(\phi)$, but the solution takes a simple, analytic form when the potential is linear or quadratic in $\phi$. In both cases, the solution takes the form of a Gaussian distribution: $P[\phi,t]=\frac{1}{\sigma(t)\sqrt{2\pi}}\exp\left[-\frac{(\phi-\mu(t))^{2}}{2\sigma^{2}(t)}\right]\,.$ (4.16) For a linear potential $V=V_{0}-\alpha\phi$, the parameters $\mu(t)$, $\sigma^{2}(t)$ are given by $\mu(t)=\frac{\alpha}{(d-1)H}\,t\,,~{}~{}~{}~{}\sigma^{2}(t)=\frac{H^{d-1}}{\pi\Omega_{d-2}}\,t\,.$ (4.17) For a quadratic hilltop potential $V=V_{0}-\frac{1}{2}m^{2}\phi^{2}$, the parameters are instead given by $\mu(t)=0\,,~{}~{}~{}~{}\sigma^{2}(t)=\frac{(d-1)H^{d}}{2\pi\Omega_{d-2}m^{2}}\left[-1+\exp\left(\frac{2m^{2}}{(d-1)H}\,t\right)\right]\,.$ (4.18) For the linear potential, inflation occurs if $\phi<\phi_{c}$, where $\phi_{c}$ is some critical value whose precise value will be unimportant to us. The probability that $\phi<\phi_{c}$ at time $t$ is given by $\text{Pr}[\phi>\phi_{c},t]=\int_{-\infty}^{\phi_{c}}d\phi\,P[\phi,t],$ (4.19) where $P[\phi,t]$ is the probability density function for a Gaussian distribution, given in (4.16), with mean $\mu(t)$ and variance $\sigma^{2}(t)$ given by (4.17). The result is $\text{Pr}[\phi>\phi_{c},t]=\frac{1}{2}\text{erfc}\left[\frac{\mu(t)-\phi_{c}}{\sigma(t)\sqrt{2}}\right]=\frac{1}{2}\text{erfc}\left[\frac{\frac{\alpha}{(d-1)H}t-\phi_{c}}{\left(\frac{2H^{d-1}t}{\pi\Omega_{d-2}}\right)^{1/2}}\right],$ (4.20) with erfc the error function. For large $t$, this error function can be approximated to leading order as an exponential, $\text{Pr}[\phi>\phi_{c},t]\sim\exp\left[-\left(\frac{\frac{\alpha}{(d-1)H}t-\phi_{c}}{\left(\frac{2H^{d-1}t}{\pi\Omega_{d-2}}\right)^{1/2}}\right)^{2}\,\right]\sim\exp\left[-\frac{\pi\Omega_{d-2}\alpha^{2}}{2(d-1)^{2}H^{d+1}}t\right],$ (4.21) This means that the probability of inflation occurring at time $t$ for a fixed comoving observer decays exponentially with time. On the other hand, the volume of the inflating region grows exponential in time as $\exp((d-1)Ht)$. Eternal inflation occurs if this exponential growth beats the exponential decay, i.e., if $(d-1)H>\frac{\pi\Omega_{d-2}\alpha^{2}}{2(d-1)^{2}H^{d+1}}\,.$ (4.22) Substituting $V$ for $H$ using (4.11) and setting $\alpha=\nabla V(\phi)$, this becomes $\frac{\|\nabla V\|^{2}}{V^{(d+2)/2}}<\frac{2(d-1)^{3}}{\pi\Omega_{d-2}}\left(\frac{2}{(d-1)(d-2)M_{d}^{d-2}}\right)^{(d+2)/2}\,.$ (4.23) A similar analysis applies to the quadratic hilltop potential. Now, inflation occurs if $\|\phi\|<\phi_{c}$, and the probability of this occurring at time $t$ is given by $\text{Pr}[\|\phi\|<\phi_{c},t]=\int_{-\phi_{c}}^{\phi_{c}}d\phi\,P[\phi,t]\,,$ (4.24) The probability density function is a Gaussian with mean and variance given by (4.18). This gives $\displaystyle\text{Pr}[\|\phi\|<\phi_{c},t]=\text{erf}\left[\frac{\phi_{c}-\mu(t)}{\sigma(t)\sqrt{2}}\right]$ $\displaystyle=\text{erf}\left[\left(\frac{\pi\Omega_{d-2}m^{2}\phi_{c}^{2}}{(d-1)H^{d}\left(-1+\exp\left(\frac{2m^{2}}{(d-1)H}\,t\right)\right)}\right)^{1/2}\right]$ $\displaystyle\sim\exp\left[-\frac{m^{2}}{(d-1)H}\,t\right]\,,$ (4.25) where in the last line we have Taylor-expanded the error function at the origin. As before, eternal inflation occurs if the exponential expansion of the universe dominates the exponential decay of (4.25), which is equivalent to the condition $(d-1)H>\frac{m^{2}}{(d-1)H}\,.$ (4.26) Substituting $V$ for $H$ using (4.11) and setting $m^{2}=-V^{\prime\prime}$, this becomes $\frac{V^{\prime\prime}}{V}>-\frac{2(d-1)}{d-2}\frac{1}{M_{d}^{d-2}}\,.$ (4.27) As in [5], this analysis may be generalized straightforwardly to theories with multiple scalar fields: as long as the potential $V(\phi^{i})$ separates into a sum $\sum_{i}V_{i}(\phi^{i})$, where each $V_{i}$ is linear or quadratic and depends only on $\phi^{i}$, the solution to the Fokker-Planck equation will separate into a product of Gaussian wavepackets. The main upshot of this is that, at a critical point of a multifield potential, one should replace $V^{\prime\prime}$ in (4.27) with a sum over the negative eigenvalues of the Hessian. To conclude this section, let us compare the bound (4.27) to the analogous bound in the Refined de Sitter Conjecture (RdSC) [41, 42, 43]: $\frac{\text{min}(\nabla_{i}\nabla_{j}V)}{V}\leq-\frac{c_{d}^{\prime}}{M_{d}^{d-2}}\,,$ (4.28) where $c_{d}^{\prime}$ is some $O(1)$ constant, and $\text{min}(\nabla_{i}\nabla_{j}V)$ is the minimum eigenvalue of the Hessian. We see here that the RdSC is incompatible with eternal inflation provided $c_{d}^{\prime}>2(d-1)/(d-2)$. The RdSC bound with this value of $c_{d}^{\prime}$ seems to be violated in some 4d examples with 10d supergravity uplifts [44, 45], but these solutions do not lie in the classical regime of string theory, so they may be modified by stringy effects [46]. Furthermore, even if the RdSC bound with this value of $c_{d}^{\prime}$ is violated for the smallest eigenvalue of the Hessian, one might still violate the necessary conditions for eternal inflation upon summing over the other negative eigenvalues of the Hessian. ## 5 Derivatives of Scalar Field Potentials Next, we consider a bound on scalar field potentials of the form $\frac{\|\nabla V_{D}\|}{V_{D}^{\gamma_{D}}}\geq c_{D}M_{D}^{D(1-\gamma_{D})-(D-2)/2}\,,$ (5.1) for $V_{D}>0$, where $\gamma_{D}$ and $c_{D}$ are $O(1)$ constants. Here, $\|\nabla V_{D}\|^{2}=G^{ij}\partial_{i}V_{D}\partial_{j}V_{D}$, and the action takes the form $\displaystyle S=\int\mathrm{d}^{D}x\sqrt{-g}\left[\frac{M_{D}^{D-2}}{2}{\cal R}_{D}-\frac{1}{2}G_{ij}(\phi)\nabla\phi_{i}\nabla\phi_{j}-V_{D}(\phi)\right]\,.$ (5.2) We want to understand how this behaves under dimensional reduction. We take our usual dimensional reduction ansatz: $ds^{2}=e^{\lambda/(d-2)}d\hat{s}^{2}+e^{-\lambda}dy^{2}\,.$ (5.3) The resulting action in $d$ dimensions is given by $S=\int\mathrm{d}^{d}x\sqrt{-\hat{g}}\left[\frac{M_{d}^{d-2}}{2}\left({\cal R}_{d}-\frac{d-1}{4(d-2)}(\nabla\lambda)^{2}\right)-\frac{1}{2}G_{ij}^{(d)}(\phi)\nabla\phi_{i}\nabla\phi_{j}-V_{d}(\phi)\right]\,,$ (5.4) where we have $\displaystyle M_{d}^{d-2}=(2\pi R)M_{D}^{D-2}\,,~{}~{}~{}G_{ij}^{(d)}=(2\pi R)G_{ij}^{(D)}\,,~{}~{}~{}V_{d}=V_{D}(2\pi R)e^{\lambda/(d-2)}\,.$ (5.5) There is also a contribution to $V_{d}$ from the Casimir energy, but for large $R$, this will be parametrically subdominant to the classical term, and we will neglect it here. Thus we have $\displaystyle\|\nabla V_{d}\|^{2}$ $\displaystyle=(G^{(d)})^{ij}\partial_{i}V_{d}\partial_{j}V_{d}+\frac{4(d-2)}{d-1}\frac{1}{M_{d}^{d-2}}(\partial_{\lambda}V_{d})^{2}\,,$ $\displaystyle=(2\pi R)(G^{(D)})^{ij}\partial_{i}V_{D}\partial_{j}V_{D}+\frac{4}{(d-1)(d-2)}\frac{2\pi R}{M_{D}^{D-2}}V_{D}^{2}\,,$ (5.6) $\displaystyle=(2\pi R)\|\nabla V_{D}\|^{2}+\frac{4}{(d-1)(d-2)}\frac{2\pi R}{M_{D}^{D-2}}V_{D}^{2}\,,$ where we have set $\langle\lambda\rangle=0$ after taking the $\lambda$ derivative. This yields $\frac{\|\nabla V_{d}\|^{2}}{V_{d}^{2\gamma_{d}}}M_{d}^{d-2-2d(1-\gamma_{d})}=(2\pi R)^{\frac{4}{d-2}(\gamma_{d}-1)}M_{D}^{\frac{d-1}{d-2}(2d\gamma_{d}-d-2)}\frac{\|\nabla V_{D}\|^{2}+\frac{4}{(d-1)(d-2)}M_{D}^{2-D}V_{D}^{2}}{V_{D}^{2\gamma_{d}}}\,.$ (5.7) From this, we see that the value $\gamma_{D}=\gamma_{d}=1$ is distinguished by dimensional reduction: for this particular value, the $R$-dependence cancels, and if we ignore the $\lambda$ dependence of $V_{d}$, the bound (5.1) is exactly preserved under dimensional reduction. With $\gamma_{D}=1$, this bound matches the “de Sitter Conjecture” bound of [4]. Setting $\gamma_{D}=\gamma_{d}=1$, we may further fix the constant $c_{D}$ by including the $\lambda$ dependence of $V_{d}$. In particular, assuming that the bound (5.1) is exactly preserved under dimensional reduction, so that $V_{D}$ saturates the bound in $d$ dimensions precisely when $V_{d}$ saturates the bound in $d$ dimensions, we have $c_{d}^{2}=\frac{\|\nabla V_{d}\|^{2}}{V_{d}^{2}}M_{d}^{d-2}=\frac{c_{D}^{2}V_{D}^{2}+\frac{4}{(d-1)(d-2)}V_{D}^{2}}{V_{D}^{2}}\,,$ (5.8) where in the first equality we have assumed that the bound is saturated $d$ dimensions, and in the second we have used (5.6) and assumed that it is saturated in $D$ dimensions. This gives a recursive relation for $c_{D}$: $c_{d}^{2}=c_{D}^{2}+\frac{4}{(d-1)(d-2)}\,,$ (5.9) which is solved by $c_{d}^{2}=\beta+\frac{4}{d-2}\,,$ (5.10) where $\beta$ is a free parameter. This leads to the bound $\frac{\|\nabla V_{d}\|}{V_{d}}\geq\sqrt{\beta+\frac{4}{d-2}}\frac{1}{M_{d}^{(d-2)/2}}\,.$ (5.11) Our dimensional reduction argument leaves $\beta$ unfixed, but the value $\beta=0$ is distinguished for two reasons: first of all, it is the smallest value of $\beta$ such that $c_{d}$ remains non-negative in the limit $d\rightarrow\infty$. Secondly, the bound with $\beta=0$, $\frac{\|\nabla V_{d}\|}{V_{d}}\geq\sqrt{\frac{4}{d-2}}\frac{1}{M_{d}^{(d-2)/2}}\,,$ (5.12) has physical meaning: it is exactly the condition required for a theory of Einstein gravity coupled to a dilaton field to avoid accelerated expansion of the universe in $d$ dimensions [4]. Observational constraints rule out single-field quintessence models satisfying (5.12) [47]. Even if one entertains the possibility of fine-tuned, multi-field quintessence models [48, 49, 50], it is very hard to imagine that (5.12) could be satisfied at the maximum of the Higgs potential [51]. However, this bound may yet be satisfied in asymptotic limits of scalar field space, as is evidenced by many examples in string theory. For instance, toroidal compactifications of $O(16)\times O(16)$ heterotic string theory to $d$ dimensions satisfy the bound [4]: $\frac{\|\nabla V_{d}\|}{V_{d}}\geq\text{min}\left(2\sqrt{\frac{3d-5}{d-2}},\frac{4\sqrt{2}}{\sqrt{(10-d)(d-2)}}\right)\frac{1}{M_{d}^{(d-2)/2}}\,.$ (5.13) This bound implies (5.12), and it implies (5.11) in $d\geq 3$ provided $\beta\leq 4/7$. Next, we may consider the KKLT scenario [52] and the LVS scenario [53]. Both of these models have de Sitter critical points, so they violate (5.12) in the interior of scalar field space. However, as noted in [54], the KKLT potential behaves asymptotically as $V(\phi)\sim\exp\left(-\sqrt{6}\frac{\phi}{M_{4}}\right)\,,$ (5.14) and the LVS potential behaves asymptotically as $V(\phi)\sim\exp\left(-3\sqrt{\frac{3}{2}}\frac{\phi}{M_{4}}\right)\,.$ (5.15) Thus, in asymptotic limits of field space, both of these satisfy (5.12), and more generally the former satisfies (5.11) provided $\beta\leq 4$, whereas the latter requires $\beta\leq 23/2$. More examples of 4d theories satisfying (5.12) in asymptotic regions of scalar field space can be found in [4, 6, 45, 55, 56]. However, some of the string compactifications considered in [55, 56, 6] naively seem to violate (5.12). For instance, in Type IIA compactifications in the presence of O4-planes but no D4-brane sources, [55] derived a lower bound on $\|\nabla V\|M_{4}/V$ of $\sqrt{2/3}$, which if saturated would violate (5.12) while saturating the Transplackian Censorship Conjecture (TCC) bound [54]: $\frac{\|V^{\prime}\|}{V}\geq\frac{2}{\sqrt{(d-1)(d-2)}}\frac{1}{M_{d}^{(d-2)/2}}\,.$ (5.16) However, there is an important subtlety here: the lower bound in this example comes from considering only the volume modulus and the dilaton, so it is plausible that including other scalar fields could lead to consistency with (5.12) in asymptotic limits of scalar field space. Likewise, the lower bounds on $\|V^{\prime}\|M_{4}/V$ listed in Table 2 of [6] come from considering the derivative $V^{\prime}$ of the potential with respect to the geodesic distance along certain geodesics in scalar field space, not from considering the gradient $\nabla V$ of the potential with respect to _all_ scalar fields in the theory.bbbThe fact that the derivative with respect to geodesic distance $V^{\prime}$ may saturate the TCC bound (5.16) is crucial for the connection between the TCC and the Swampland Distance Conjecture (SDC) proposed in [6]. Once contributions to the gradient $\nabla V$ from additional scalar fields are included, it is plausible that our bound (5.12) may yet be satisfied.cccWe are very thankful to David Andriot for discussions on these points. It may sound like wishful thinking to hope that contributions to the gradient from additional scalar fields will always come to the rescue, ensuring consistency with (5.12). However, we already know of at least one simple example where this is precisely what happens: in heterotic string theory compactified to four dimensions, one has $\|\partial_{\rho}V\|M_{4}/V=\sqrt{2/3}$, where $\rho$ is the volume modulus of the compactification manifold (see Table 2 of [6]). This saturates the TCC bound (5.16) along a geodesic in the $\rho$ direction and naively violates our bound (5.12). However, the potential also depends asymptotically on the dilaton, $\|\partial_{\tau}V\|M_{4}/V=\sqrt{2}$. This additional contribution to the gradient leads to consistency with (5.12). Note, however, that while this example is a useful proof of principle, it does not represent the typical scenario. In general, it is quite difficult to determine the dependence of the potential on _every_ scalar field in the theory, so falsifying (5.12) in asymptotic regions of scalar field space is not easy to do.††footnotemark: Finally, note that (5.12) is incompatible with the condition (4.23) needed for eternal inflation, since $\gamma_{d}=1<\gamma_{d}^{\text{EI}}=(d+2)/4$. This is rather unsurprising from a physical perspective: (5.12) forbids accelerated expansion of the universe, which is obviously a prerequisite for eternal inflation. Dimensional reduction induces an exponential potential for the radion $\lambda$, $V_{d}=V_{D}\exp(\lambda/(d-2))$, so if the radion is not stabilized, this exponential potential alone will suffice to violate the condition (4.23) required for eternal inflation. Radion stabilization will not occur in asymptotic regions of scalar field space [57], so dimensional reduction will not lead to eternal inflation in such regions. ## 6 Conclusions We have seen that dimensional reduction distinguishes particular $O(1)$ constants appearing in the WGC, RFC, (A)dSDC, and dSC. In the case of the (A)dSDC, one mass scale that emerges is suggestively close to the neutrino mass scale, which may be related to the Higgs vev via the see-saw mechanism. It might be worthwhile to search for other swampland-related reasons for a connection between the neutrino mass/Higgs vev and the cosmological constant. In the case of the dSC, the bound we obtain precisely forbids accelerated expansion of the universe in Einstein-dilaton gravity. This is interesting in that it offers an alternative physical explanation to Trans-Planckian Censorship [54] as to why the dSC seems to hold universally in asymptotic limits of scalar field space: perhaps quantum gravity simply forbids accelerated expansion in such regions. This possibility merits further investigation. We have also worked out conditions for eternal inflation in $d\geq 4$ spacetime dimensions, generalizing previous bounds obtained in $d=4$. Models of inflation in more than four dimensions have not received very much attention, due to the obvious fact that such models are of little experimental interest. But given the recent, renewed interest in scalar field potentials and de Sitter vacua in quantum gravity, it may be worthwhile to temporarily ignore the question of experimental relevance and instead explore more general theories of inflation and cosmology in diverse dimensions, as well as their possible embeddings in string theory. It would be interesting to study de Sitter critical points of scalar field potentials in string compactifications to $d>4$ dimensions, to see if these critical points may support eternal inflation. We have not answered the most important question: to what extent are the bounds we have obtained actually satisfied in quantum gravities? It is clear that many string compactifications satisfy them, but it is also quite possible that these results suffer from a lamppost effect. Understanding the domain of validity of these conjectured bounds is a crucial task for the swampland program, though it may not be an easy one. ## Acknowledgements We thank David Andriot, Ben Heidenreich, Juan Maldacena, Liam McAllister, Georges Obied, Eran Palti, Matthew Reece, and Cumrun Vafa for useful discussions. We thank David Andriot, Ben Heidenreich, Liam McAllister, and Matthew Reece for comments on a draft of this paper. 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# The RR Lyrae Delay-Time Distribution: A Novel Perspective on Models of Old Stellar Populations Sumit K. Sarbadhicary Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA Mairead Heiger Pittsburgh Particle Physics, Astrophysics and Cosmology Center (PITT PACC), University of Pittsburgh, 3941 O’Hara St, Pittsburgh, PA 15260, USA Carles Badenes Pittsburgh Particle Physics, Astrophysics and Cosmology Center (PITT PACC), University of Pittsburgh, 3941 O’Hara St, Pittsburgh, PA 15260, USA Cecilia Mateu Departamento de Astronomía, Facultad de Ciencias, Universidad de la República, Iguá 4225, 14000, Montevideo, Uruguay Jeffrey Newman Pittsburgh Particle Physics, Astrophysics and Cosmology Center (PITT PACC), University of Pittsburgh, 3941 O’Hara St, Pittsburgh, PA 15260, USA Robin Ciardullo Department of Astronomy & Astrophysics, The Pennsylvania State University, University Park, PA 16802, USA Na’ama Hallakoun Department of Particle Physics and Astrophysics, Weizmann Institute of Science, Rehovot, 7610001, Israel Dan Maoz School of Physics and Astronomy, Tel-Aviv University, Tel- Aviv 6997801, Israel Laura Chomiuk Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA ###### Abstract The delay-time distribution (DTD) is the occurrence rate of a class of objects as a function of time after a hypothetical burst of star formation. DTDs are mainly used as a statistical test of stellar evolution scenarios for supernova progenitors, but they can be applied to many other classes of astronomical objects. We calculate the first DTD for RR Lyrae variables using 29,810 RR Lyrae from the OGLE-IV survey and a map of the stellar-age distribution (SAD) in the Large Magellanic Cloud (LMC). We find that $\sim 46\%$ of the OGLE-IV RR Lyrae are associated with delay-times older than 8 Gyr (main-sequence progenitor masses less than 1 M⊙), and consistent with existing constraints on their ages, but surprisingly about $51\%$ of RR Lyrae appear have delay times $1.2-8$ Gyr (main-sequence masses between $1-2$ M⊙ at LMC metallicity). This intermediate-age signal also persists outside the Bar-region where crowding is less of a concern, and we verified that without this signal, the spatial distribution of the OGLE-IV RR Lyrae is inconsistent with the SAD map of the LMC. Since an intermediate-age RR Lyrae channel is in tension with the lack of RR Lyrae in intermediate-age clusters (noting issues with small-number statistics), and the age-metallicity constraints of LMC stars, our DTD result possibly indicates that systematic uncertainties may still exist in SAD measurements of old-stellar populations, perhaps stemming from the construction methodology or the stellar evolution models used. We described tests to further investigate this issue. RR Lyrae variable stars (1410); Large Magellanic Cloud (903); Stellar populations (1622); Stellar evolution (1599); Stellar evolutionary models (2046); Horizontal branch (2048); Stellar ages (1581); Stellar pulsations (1625); Hertzsprung Russell diagram (725); ## 1 Introduction A detailed understanding of stellar evolution remains one of the most sought- after goals in astrophysics. Popular stellar evolution codes such as Geneva (Schaller et al., 1992; Schaerer et al., 1993), Y2 (Kim et al., 2002; Yi et al., 2003; Demarque et al., 2004), BaSTI (Pietrinferni et al., 2004, 2006; Hidalgo et al., 2018), Darthmouth (Dotter et al., 2008), PARSEC (Bressan et al., 2012; Chen et al., 2014) and MESA (Paxton et al., 2011, 2013, 2015, 2018) are powerful tools for interpreting observations of stellar populations. However, many essential topics in stellar evolution are still not well understood, and/or not properly taken into account in even the most state-of- the-art models. Examples of such topics include convection, mass loss and mass transfer, common envelope evolution, and binary interaction. Often, these three-dimensional phenomena are approximated by simplified parametric models tuned to specific observables and integrated into one-dimensional stellar evolution codes. These uncertainties limit our understanding of many important phases of stellar evolution, such as the horizontal branch, the asymptotic giant branch (AGB) and post-AGB phase, planetary nebulae, and supernovae (see discussions in Gallart et al., 2005; Conroy et al., 2009; Conroy, 2013). The delay-time distribution (DTD) is a promising method for testing stellar evolution models in complex stellar populations (Maoz & Mannucci, 2012; Maoz et al., 2014). The DTD is defined as the occurrence rate of a class of astronomical object as a function of time since a hypothetical brief burst of star formation; it is equivalent to the impulse response, or Green’s function. The DTD constrains the evolutionary timescale and formation efficiency of the object’s progenitors, and theoretical DTDs are common predictions of stellar population synthesis models (Mennekens et al., 2010; Nelemans et al., 2013; Toonen et al., 2013; Zapartas et al., 2017). Observationally, DTDs can be derived from surveys of objects, provided that the stellar age distributions (SADs) of their host galaxies are measured (Gal-Yam & Maoz, 2004; Totani et al., 2008; Maoz & Sharon, 2010; Maoz et al., 2011, 2012; Graur et al., 2014; Maoz & Graur, 2017; Friedmann & Maoz, 2018). More recently it was shown that DTDs can be a useful stellar evolution diagnostic in Local Group galaxies with high-quality observations of their resolved stellar populations (Badenes et al., 2010; Maoz & Badenes, 2010; Badenes et al., 2015). Using this approach, Badenes et al. (2015) measured the first DTD for planetary nebula, showing evidence of two distinct populations of planetary nebula progenitors: one with ages of 5–8 Gyrs, and another with ages of 35–800 Myrs. The key advantage of a DTD is that it constrains the evolutionary timescales for the progenitors of the entire population of objects in a galaxy, taking into account the variety of star-formation histories that these objects have evolved from. It can be a powerful tool for identifying or ruling out the presence of specific formation channels, measuring their efficiency, and identifying physical mechanisms that are not part of canonical progenitor models. Figure 1: The spatial distribution of LMC RR Lyrae (red dots) in the OGLE-IV sample overlaid on the spatial cells (blue) from the Harris & Zaritsky (2009) SAD map of the galaxy. Figure 2: $I$-band luminosity function of the OGLE-IV RR Lyrae. The cumulative histograms show the fraction of RR Lyrae dimmer than a certain $I$-band brightness. The luminosity function of the full RR Lyrae sample from Figure 1 is shown in black, and that of RR Lyrae in the ‘LMC503’ field of the OGLE-IV survey, a region with dense stellar crowding (Udalski et al., 2015), is shown in blue. The inset shows the location of the LMC503 region on an $r$-band continuum map (as reference) of the LMC Bar from the Magellanic Cloud Emission-Line Survey (MCELS, Winkler et al., 2005; Pellegrini et al., 2012). The $I$-band completeness limit of the LMC503 region is shown with the dashed line. In this paper, we will use a SAD map of the Large Magellanic Cloud (LMC) to derive the DTD of RR Lyrae stars— pulsating horizontal branch stars with periods between 0.2 and 1 day (Smith, 2004). We chose to test the DTD method on RR Lyrae for several reasons. Firstly, the sample size of RR Lyrae in the LMC is quite large (see Section 2.1), allowing us to measure a DTD with high significance. Secondly, there is strong evidence that RR Lyrae are mostly ancient stars, older than 10 Gyrs, given their pulsational properties (Smith, 2004; Marconi et al., 2015) and abundance in old globular clusters (Clement et al., 2001; Soszyński et al., 2014, 2016). Measuring an RR Lyrae DTD therefore provides a rigorous test of the DTD method for the recovery of progenitor age- distributions, as well as star-formation history measurements of old resolved stellar populations. Lastly, a DTD analysis provides an opportunity to test stellar evolution models of RR Lyrae in a completely new way. While there is consensus on the interpretation of RR Lyrae as ancient stars, the lower-limit on their ages has been somewhat unconstrained. For example, the absence of RR Lyrae in the SMC cluster Lindsey 1 ($t\approx 9$ Gyr), compared to their presence in NGC 121 ($t\approx$10–11 Gyr), is generally cited as evidence of a lower limit of 10 Gyr for progenitor age of RR Lyrae (Olszewski et al., 1996; Glatt et al., 2008). However, growing evidence of thin disk RR Lyrae in the Milky Way raises questions about whether this limit might be lower, and if an intermediate-age progenitor channel can exist (Layden, 1995; Zinn et al., 2019; Prudil et al., 2020). Additionally uncertainties may exist in the evolutionary models of RR Lyrae stars themselves: since RR Lyrae are horizontal-branch stars, their positions on the color-magnitude diagram depend on an unknown combination of factors (commonly known as the second-parameter phenomenon) like metallicity, age, mass-loss on the red-giant branch, stellar rotation, core structure, and chemical abundance (see Fusi Pecci & Bellazzini, 1997; Catelan, 2009; Dotter, 2013, for reviews). The paper is organized as follows: Section 2 briefly describes the two ingredients for calculating the RR Lyrae DTD – the OGLE-IV survey of RR Lyrae, and the Harris & Zaritsky (2009) SAD map of LMC, Section 3 describes the measurement of the RR Lyrae DTD from the OGLE-IV survey, Section 4 checks whether the measured DTD is consistent with the observed spatial distribution of RR Lyrae in the OGLE-IV survey, incompleteness in the SAD map and RR Lyrae in star clusters, Section 5 discusses the two possible interpretations of the DTD result – that RR Lyrae may have a previously-unidentified younger progenitor channel, or unknown systematics still exist in SAD measurements of resolved stellar populations older than a Gyr. ## 2 Ingredients for Calculating The DTD ### 2.1 OGLE-IV Sample of LMC RR Lyrae We use the Soszyński et al. (2016) catalog of 39,082 RR Lyrae stars from the Optical Gravitational Microlensing Experiment (OGLE- IV)111http://ogle.astrouw.edu.pl/ survey of the LMC (Udalski et al., 2015). These RR Lyrae were selected by the OGLE-IV pipeline from the full database of OGLE $I$-band light curves with periods between 0.2 and 1 day, and then further classified as fundamental (RRab), first-overtone (RRc), and mixed mode pulsators based on their periods, Fourier amplitudes, and light curve shapes. We excluded catalog entries that were flagged as Galactic RR Lyrae or eclipsing variables, objects with uncertain classification, and sources that fall outside the SAD maps. This produced a final sample of 29,810 RR Lyrae (Figure 1). The photometric completeness of the sample is quite high, as evidenced by the I-band luminosity function of RR Lyrae in the most crowded OGLE-IV field (LMC503, see Figure 2). The field has a completeness limit of $I\approx 20.5$ (Udalski et al., 2015), whereas the RR Lyrae sample in the field has a median magnitude $\bar{I}=18.82$ and standard deviation $\sigma_{I}=0.4$. The median magnitude of the RR Lyrae is nearly 4.2$\sigma_{I}$ above the completeness limit (almost 99.9$\%$ of the RR Lyrae have I-band magnitude above the completeness limit). Thus even in the most crowded region, the RR Lyrae sample can be considered photometrically complete. Also as seen in Figure 2, the I-band luminosity function of the full OGLE-IV sample inside the Harris & Zaritsky (2009) region (see Section 2.2) is also well above the completeness limit of the most crowded region. Such high completeness is important for measuring unbiased rates and DTDs (Maoz & Badenes, 2010; Badenes et al., 2010). ### 2.2 Stellar Age Distribution Map of LMC We use the SAD map of the LMC constructed by Harris & Zaritsky (2009) (hereafter, HZ09). This map provides the best-fit stellar mass formed as a function of lookback time in spatial cells resolving the central $8.5^{\circ}\times 7.5^{\circ}$ of the galaxy (Zaritsky et al., 2004). HZ09 also provides the associated 1$\sigma$ upper and lower limits to the SAD in each cell, which we will incorporate into the DTD uncertainties in Section 3.2. The SADs were calculated using data from the Magellanic Cloud Photometric Survey (MCPS) of nearly 4 million stars collected with the 1m Swope telescope, down to a completeness of V=20–21 mag (Zaritsky et al., 1997, 2004). The MCPS region was divided into 1376 cells, each measuring $24^{\prime}\times 24^{\prime}$, or $12^{\prime}\times 12^{\prime}$ if the field contained more than 25,000 stars. The contours of these cells are shown in Figure 1. SADs were derived using the StarFISH algorithm by fitting the color magnitude diagram in each cell with a linear combination of isochrones (Harris & Zaritsky, 2001). After accounting for extinction and photometric errors, each cell’s SAD was fit using 16 logarithmically-spaced bins spanning the ages between 4 Myr and 20 Gyr, and four metallicity bins ($Z=0.008$, 0.004, 0.0025, and 0.001). For ages younger than 100 Myr, a single metallicity of Z=0.008 was used because the different metallicity isochrones are almost indistinguishable. Note that while the lower limit on the age of RR Lyrae stars is generally quoted as 10 Gyr, the SAD map has a single indivisible age-bin of 8–12 Gyr, and so we refer to this lower limit as 8 Gyr in the rest of the paper. ## 3 The RR Lyrae Delay-Time Distribution ### 3.1 Method The RR Lyrae catalog and SAD maps of the LMC are used to estimate the RR Lyrae DTD using the non-parametric method described in Badenes et al. (2015) (hereafter, B15), although we improve on some aspects of it. The RR Lyrae DTD is the number of RR Lyrae formed per unit stellar mass as a function of the time-delay between star-formation and the RR Lyrae phase. The convolution of the DTD with the SAD in each spatial cell predicts the number of RR Lyrae that will be produced by the stellar population in that cell. For each SAD cell $i$, the number of RR Lyrae predicted ($\lambda_{i}$) is: $\lambda_{i}=\sum_{j=1}^{N}M_{ij}\left(\Psi T_{vis}\right)_{j}$ (1) where $M_{ij}$ is the stellar mass formed in age-bin $j$ and cell $i$, $\Psi$ is the RR Lyrae formation rate (RR Lyrae per year per unit M⊙), and Tvis is the duration of the RR Lyrae phase (both $\Psi$ and Tvis are function of age- bin). The widths of the age-bins, $j$, are selected using the methodology described in Appendix A and provide the best compromise between detection significance and temporal resolution. In this paper, we retain the notation of B15 and refer to the quantity $\left(\Psi T_{vis}\right)$ as the DTD (with units of RR Lyrae per unit M⊙). The DTD ($\Psi T_{vis}$) is determined by minimizing the difference between the predicted and observed number of RR Lyrae across spatial cells. This is carried out in a similar manner to B15 using the Markov Chain Monte Carlo (MCMC) solver emcee (Foreman-Mackey et al., 2013). We denote $\mathbf{N}=[N_{i}]$ as the vector representing the number of RR Lyrae in each spatial cell, and $\mathbf{\Psi T_{vis}}=[\left(\Psi T_{vis}\right)_{j}]$ as the vector of predicted number of RR Lyrae per stellar mass for each age-bin. The posterior is calculated as $p\left(\mathbf{\Psi T_{vis}}\|\mathbf{N}\right)\propto\mathcal{L}\left(\mathbf{N}\|\mathbf{\Psi T_{vis}}\right)\pi\left(\mathbf{\Psi T_{vis}}\right)$ (2) where $\mathcal{L}$ is the likelihood and $\pi$ is the prior. We assume $\mathcal{L}$ is either a product of Poisson or Gaussian probabilities depending on Ni, $\mathcal{L}\left(\mathbf{N}\|\mathbf{\Psi T_{vis}}\right)=\begin{cases}\displaystyle\prod_{i=1}^{K}\frac{\mathrm{e}^{-\lambda_{i}}\lambda_{i}^{N_{i}}}{N_{i}!}&\quad N_{i}\leq 25\\\ \displaystyle\prod_{i=1}^{K}\frac{1}{2\pi\lambda_{i}}\left(\frac{(\lambda_{i}-N_{i})^{2}}{2\lambda_{i}}\right)&\quad N_{i}>25\end{cases}$ (3) where $K$ is the number of SAD cells. Although Poisson distributions converge to Gaussian for large $N_{i}$, we specify the likelihoods separately to avoid computational issues. The prior is defined such that the $(\Psi T_{vis})_{j}$ values have uniform probabilities in log-space: $\pi\left(\mathbf{\Psi T_{vis}}\right)=\begin{cases}\mathbf{\left(\Psi T_{vis}\right)^{-1}}&\quad(\Psi T_{vis})_{j}>0\\\ 0&\quad(\Psi T_{vis})_{j}\leq 0\end{cases}$ (4) ### 3.2 Estimating Uncertainties We propagate the errors on the $M_{ij}$ values in the SAD map into uncertainties on $\mathbf{\Psi T_{vis}}$. B15 calculated DTDs for the best-fit SAD, the upper-limit on the SAD, and the lower-limit on the SAD. The differences between the best-fit and upper/lower limits were treated as the DTD’s 1$\sigma$ uncertainties. In this paper, we use an improved method of propagating SAD uncertainties into the DTD. We randomly generate 100 mock SADs, assuming $M_{ij}$ has the normally-distributed uncertainties given by HZ09, and calculate a DTD for each mock SAD. We combine the MCMC posterior chains from these 100 DTDs into a single chain, and estimate the $95\%$ credible interval on this chain using a highest posterior density criterion. We define the mode of the distribution minus the upper and lower-limits of this interval as our $2\sigma_{+}$ and $2\sigma_{-}$ confidence intervals, respectively. We define a “signal” detection in each bin $j$ of the DTD as a value of $\Psi T_{vis}$ in that bin that is $\geq$ $2\sigma_{-}$ above 0. Non-detections are presented as $2\sigma$ limits on the DTD signal in a particular age bin. ### 3.3 Sample Contribution per Age-bin We estimate the contribution of each age-bin to the total sample by multiplying the value of $\Psi T_{vis}$ with the total stellar mass formed in each age-bin. The contribution percentage will then have uncertainties due to both the DTD and the total stellar masses formed. We estimate these uncertainties with a Monte Carlo method. We draw DTD values from the recovered posterior probability distributions (called $\left(\Psi T_{vis}\right)_{j}$), multiply by the total stellar mass per age-bin in each randomized SAD map (called Mj), and get the number of RR Lyrae contributed by age-bin, $\lambda_{j}$. The contribution percentages (= $\lambda_{j}/\Sigma\lambda_{j}$) and their 1$\sigma$ uncertainties are listed in Table 1. Figure 3: The delay-time distribution, in units of number of RR Lyrae per M⊙, as a function of delay-time in Myr. The grey histogram represents the DTD for the full OGLE-IV sample, while the colored histograms are DTDs for RR Lyrae subtypes RRab and RRc Filled histograms represent bins with a statistically significant signal from the MCMC analysis. The error bars are 1$\sigma$ uncertainties that include uncertainties from the SAD maps. The arrows represent 2$\sigma$ upper limits in bins that do not have a statistically significant signal. At top, the progenitor mass-scales for two different metallicities, calculated using PARSEC, are shown for reference. ### 3.4 Results The DTD for the full OGLE-IV RR Lyrae sample and the two main RR Lyrae subtypes is shown in Figure 3, and the values are tabulated in Table 1. We detect signal in the DTD at a high significance ($>5\sigma$) for all age-bins older than 1.3 Gyrs. The detections with the highest significance are found in the 2–3, 5–8 and $>12$ Gyr bins. Most of this signal is contributed by the RRab stars, which are the most common subtype in the OGLE-IV sample. The RRc subclass contributes to the DTD mostly above 2 Gyr. Although RRc’s have been susceptible to confusion with other variable sources in time domain-surveys (Kinman & Brown, 2010; Mateu et al., 2012; Drake et al., 2014), Figure 3 shows that the total DTD is dominated by RRab objects in all age-bins, making it unlikely that the DTD is biased by sample contamination. The detected signal of the full DTD is relatively flat above 1.2 Gyr, with about $1-3$ RR Lyrae produced per 105 M⊙ of stellar mass formed. While there is signal in the DTD below 0.8 Gyrs, it falls below our 2$\sigma_{-}$ threshold, and we only show the 2$\sigma_{+}$ upper limit. Our detection of a DTD signal below 8 Gyrs is a surprising result. About $46\%$ of the LMC’s RR Lyrae stars are produced from populations older than 10 Gyr, the age range generally inferred for RR Lyrae in star clusters. But about $51\%$ of the DTD signal comes from progenitors with ages between 1.3 and 8 Gyr, and this result has high ($>5\sigma$) significance. If we re-calculate the DTD without assuming normal errors on $M_{ij}$ (i.e., using the same method as B15, see Appendix B), we still detect a strong signal below 8 Gyr, but with a total contribution of 41$\%$. A comparison of these timescales with those of the PARSEC222https://people.sissa.it/~sbressan/parsec.html. We use PARSEC because it is one of the latest stellar evolution codes with both publicly available main-sequence and horizontal branch tracks. We get similar results with the MIST evolutionary tracks of Choi et al. (2016). models for the onset of helium burning leads to the conclusion that RR Lyrae can arise from main-sequence progenitors as massive as $\sim 2$ M⊙ at LMC metallicity. Incidentally, this upper limit is similar to the mass at which stars transition between igniting He under degenerate conditions (the “He-flash”) to burning helium under stable, non-degenerate conditions (Bildsten et al., 2012; Mosser et al., 2014). We also tested the dependence of the DTD as a function of RR Lyrae star period and brightness by sub-dividing the OGLE-IV data by pulsation time (3 bins with $<0.45$ d, $0.45-0.58$ d, and $>0.58$ d) and $I$-band magnitude (3 bins with I$<19.2$, $19.2-19.4$ and $>19.4$). By sub- dividing the sample in this way, we ensured that each period and magnitude bin contained enough RR Lyrae for a robust measurement of their DTDs. We found that all the sub-samples have significant ($>2\sigma$) DTD signals in the range 1 to 8 Gyr, with no discernible trend in the shape of the DTD with magnitude or period. Although, in principle, it is possible to derive a metallicity-dependent DTD with the HZ09 SAD maps, we defer that study to a future work. However, it is well-known that LMC RR Lyrae are generally metal-poor, with [Fe/H]$<-0.5$ and a peak in the metallicity distribution function at [Fe/H]$\sim-1.5$ (Skowron et al., 2016). We can therefore check to see if there is DTD signal below 8 Gyrs under the assumption that OGLE-IV RR Lyrae are only produced by the HZ09 SAD in the two metal-poor bins, i.e., for $Z<0.0025$ or [Fe/H]$\lesssim-1.02$ (Bertelli et al., 1994). Star-formation in this metallicity range dominated the LMC SAD until $\sim 2$ Gyrs, as seen in Figure 4, and the resulting DTD is shown in Figure 4. Even if we restrict our analysis to the metal-poor SAD, the signal below 8 Gyrs persists. This implies that, even if we assume that all LMC RR Lyrae are produced by metal-poor stars, progenitors younger than 8 Gyrs are still needed to explain their distribution. Figure 4: (a) The global (summed over all SAD cells) stellar mass formed versus age ($M_{ij}$ in Eq (1)) with $Z<0.0025$ (red) and $Z>0.0025$ (blue) in the LMC as given by the HZ09 SAD. The black histogram shows the sum of the red and blue histograms. Overlapping regions appear in a darker shade. (b) The DTD measured from the OGLE-IV survey using only the SAD in the $Z<0.0025$ bin (grey; see Section 3.4 for details). For comparison, the DTD from Figure 3 is shown in red. Before delving into the implications for RR Lyrae studies, we carry out a more detailed analysis of the recovered DTD to assess the robustness of our results. Table 1: The RR Lyrae DTD calculations, with lifetimes, significance of detection and contribution to the OGLE-IV RR Lyrae sample considered in this study. Delay-Times \| DTD ($\Psi T_{\rm{vis}}$) \| Significance \| Contribution ---\|---\|---\|--- (Gyr) \| (N/105 M⊙) \| ($\sigma$) \| ( $\%$) $<$ 0.8 \| $<0.75$ \| $-$ \| $-$ 0.8–1.3 \| $2.12^{+0.61}_{-0.78}$ \| $3.8$ \| $2.9\pm 0.8$ 1.3–2.0 \| $1.36^{+0.2}_{-0.21}$ \| $5.6$ \| $8.4\pm 1.2$ 2.0–3.2 \| $2.05^{+0.11}_{-0.22}$ \| $20.5$ \| $20.7\pm 1.2$ 3.2–5.0 \| $1.10^{+0.18}_{-0.23}$ \| $5.6$ \| $6.0\pm 1.0$ 5.0–7.9 \| $2.17^{+0.19}_{-0.36}$ \| $11.6$ \| $15.6\pm 1.3$ 7.9–12.6 \| $1.08^{+0.17}_{-0.29}$ \| $6.8$ \| $7.5\pm 1.1$ 12.6–20.0 \| $1.18^{+0.07}_{-0.1}$ \| $18.5$ \| $39.0\pm 1.8$ ## 4 On the robustness of the recovered RR Lyrae DTD ### 4.1 Mock DTD Test The HZ09 SAD map of the LMC allows us to test whether the traditional DTD for RR Lyrae (i.e., one in which the progenitors are always older than 8 Gyr) is consistent with the spatial distribution of RR Lyrae observed by OGLE. This is equivalent to inverting the DTD recovery process. To do this, we generate mock RR Lyrae maps by convolving the SAD map with a DTD that is non-zero only in the 2 oldest age-bins, 8–12 and 12–20 Gyr. The total stellar mass that formed in these age-bins was $1.23\times 10^{9}$ M⊙. Assuming all 29,810 RR Lyrae were produced by progenitors in these age-bins results in a DTD of the form, $\Psi T_{vis}=\begin{cases}2.42\times 10^{-5}\mathrm{\ RRL\ M_{\odot}^{-1}}&\quad t\geq 8\ \mathrm{Gyrs}\\\ 0&\quad t<8\ \mathrm{Gyrs}\end{cases}$ (5) We generate 100 mock RR Lyrae maps using this DTD, where the number of RR Lyrae per cell, $N_{i}$ is drawn from the Poisson distribution in Equation (3) with $\lambda_{i}$ given by Equation (1). Figure 5 shows the DTD recovered from this analysis. Our MCMC solver correctly measures the input mock DTD with strong detections only in the two oldest age- bins. The younger bins have 2$\sigma$ upper limits that are almost an order of magnitude lower than the DTD recovered from the OGLE-IV sample. Moreover, the difference between populations of RR Lyrae with the OGLE-IV and mock DTDs can be seen visually in Figure 6. The measured OGLE-IV DTD predicts a distribution of RR Lyrae stars elongated along the LMC Bar and declining smoothly with radius (center bottom panel of Figure 6); this is very similar to what is seen in the OGLE-IV map (left panel). In contrast, our mock distribution from a uniformly old RR Lyrae progenitor population (Equation (5)) has more structure and leaves larger residuals in the difference map (top center and right panels of Figure 6). To identify significant regions of discrepancy in the OGLE-IV DTD and mock old DTD maps, we generate 104 maps of RR Lyrae, where in each map the number of RR Lyrae in each cell is drawn from a Poisson distribution with mean number of RR Lyrae in that cell given by the DTD according to Eq 1. We then estimate the mean and standard deviation of the number of RR Lyrae per cell, and identify cells with black squares in Figure 6 where the observed RR Lyrae is greater than 5 times the standard deviation from the mean. We find the mock old DTD map has a larger number of cells where the RR Lyrae number count is discrepant, and these cells are mostly located in the Bar region, with a few located outside (note that this is only for the purpose of visualizing the discrepancy; the actual DTD is constrained by the joint likelihood measured from _all_ the cells as in Eq 3.) Figure 5: Result of our mock DTD test in Section 4.1. The red histogram shows the DTD measured with the original OGLE-IV survey in Figure 3. The black histogram shows the mock input DTD defined in Eq 5. The blue histogram with arrows shows the DTD recovered from the mock input DTD and the SAD maps of HZ09. We conclude that if the OGLE-IV RR Lyrae had indeed been produced exclusively from the old ($t>8$ Gyr) stars of the HZ09 SAD map, our method would have recovered the correct DTD, without spurious signals at younger ages. The DTD signal we recover at ages between 0.8 and 8 Gyr must either be real, or an artifact produced by systematics in the HZ09 SADs. Next, we investigate the impact of these SAD systematics on the DTD more thoroughly. Figure 6: Comparison between the spatial distribution of RR Lyrae in OGLE-IV (left, large panel) and the distributions predicted by convolving the HZ09 SAD with the ‘mock’ old DTD (upper middle panel) and our recovered RR Lyrae DTD (lower middle panel). The right panels show “residuals”, or the difference between the observed distribution and DTD-predicted distribution of RR Lyrae for the ‘mock’ old DTD (top right) and measured DTD (bottom right). The black squares show the cells where the difference between predicted and observed number of RR Lyrae differ by 5$\sigma$ (see Section 4.1 for details) . ### 4.2 Effect of incompleteness and crowding in the SAD map We explored if any incompleteness in the MCPS photometry (from which the SAD was measured) could be driving the intermediate-age signal. The photometric completeness limit is around V=21 mag, with the Bar region being almost 1 mag shallower as a result of the stellar crowding. Because of this, StarFISH could reliably solve for the number of stars per age-bin only for ages younger than 4 Gyr (although we note that our intermediate-age DTD signal is detected younger than 4 Gyrs). Populations older than 4 Gyr were traced by their giant branch stars, and assigned by StarFISH to a single age-bin. To anchor the ‘shape’ of the SAD beyond 4 Gyrs, stellar ages were measured in a few isolated _HST_ -fields in the LMC Bar (Holtzman et al., 1999; Olsen, 1999; Smecker-Hane et al., 2002). As shown in Figure 7 of HZ09, the SADs of each _HST_ Bar field are characterized by star-formation at look-back times of 10 Gyrs and 5 Gyrs, with a period of quiescence in between. This consistency allowed HZ09 to adopt a common shape for the SAD in the rest of the Bar region. We first checked if an underestimation of the total mass formed at old stellar ages due to any photometric incompleteness could be driving the intermediate- age signal. Obtaining a complete census of old stellar mass is generally difficult without including infra-red data (Conroy, 2013), so we studied the changes to our DTD by manually changing the old SADs. We recalculated the RR Lyrae DTD with the same SAD map, except with each cell’s 8–12 Gyr and 12–20 Gyr stellar mass multiplied by factors of 2, 4 and 10. We find that regardless of how much mass is added, the DTDs still find signals for ages between 0.8 and 8 Gyrs with $>5\sigma$ confidence. Since the incompleteness in the SAD map is dominant in the Bar region, we also tried re-calculating the DTD without the LMC Bar (i.e., by removing the SAD cells and their RR Lyrae from our calculations). We show the RR Lyrae DTDs without the “Inner Bar” in Figure 7, and without both the “Inner” and “Outer Bar” in Figure 8 (the cells of these regions are defined according to Figure 6 in HZ09). The recovered DTDs outside the excluded regions are similar to the DTD obtained from the full OGLE-IV sample. For the case with both the Inner and Outer Bar removed, the number of RR Lyrae and the number of SAD cells are smaller, leading to a higher upper limit on the DTD older than $0.8$ Gyr, and a non-detection in the 3–5 Gyr bin. However, there is still significant signal at ages younger than 8 Gyr, and particularly below 4 Gyr where the ages are determined by the main-sequence turnoff in the MCPS photometry. It therefore appears unlikely that missing stellar masses at ages $>8$ Gyrs due to incompleteness, or crowding in the Bar region are the only factors driving the intermediate-age signal in the DTD. Figure 7: RR Lyrae DTD of the OGLE-IV survey, but excluding the SAD cells and RR Lyrae of the Inner Bar. The excluded cells of HZ09 are shown in the inset plot, overlaid on the $r$-band continuum map from MCELS. Figure 8: Same as Figure 7, but now excluding both the Inner and Outer Bar. Note that the excluded cells for both Inner and Outer Bar are based on Figure 6 of HZ09. ### 4.3 Comparison with RR Lyrae in LMC star clusters Figure 9: (_Left_): Comparison of ages of the 62 clusters in Baumgardt et al. (2013) older than 1 Gyr, and their $\Delta n=(n_{o}-n_{b})-n_{p}$, the difference between the background-subtracted number of RR Lyrae observed per cluster, and the number predicted by the DTD given the cluster mass and age (see Eq (6), (7) and Section 4.3 for details). Colors indicate the cluster mass. The inset zooms-in on clusters between 1–4 Gyr. (_Right_): $\Delta n$ vs metallicity for a subset of 29 clusters with metallicity information available in the literature. Colorbar indicates the cluster age. The inset zooms-in on the 1–4 Gyr clusters, which also happen to cluster in the metallicity range $-1\lesssim$ [Fe/H] $\lesssim-0.25$. Our measured DTD suggests that progenitors of RR Lyrae can be as young as $\sim$1 Gyr, which is much smaller than the lower limit on RR Lyrae age of 10 Gyrs inferred from star clusters (Olszewski et al., 1996). We therefore examine the number of OGLE-IV RR Lyrae in the LMC star clusters as a function of age, realizing that cluster membership can only be confirmed with spectroscopic and proper motion measurements that are beyond the scope of this work. To perform this test, we use the catalog of LMC clusters compiled by Baumgardt et al. (2013) and only include the 296 systems inside the HZ09 area. Baumgardt et al. (2013) compiled ages and masses for these clusters measured using either isochrone fitting or broadband spectral energy distribution fitting of data obtained by previous surveys (Pietrzynski & Udalski, 2000; Hunter et al., 2003; Mackey & Gilmore, 2003; de Grijs & Anders, 2006; Milone et al., 2009; Glatt et al., 2010; Popescu et al., 2012). We determine the cluster radii, $r_{c}$ from their major ($a$) and minor axes ($b$) reported in Bica et al. (2008) as $r_{c}=(a+b)/4$. Bica et al. (2008) notes that these are ‘apparent’ sizes (measured as far as the background limit in the images), but they most likely enclose the majority of the cluster mass. We also obtained chemical abundances for 29 clusters ([Fe/H], shown in Table 2 with references), and as we show later, this is sufficient for understanding the correlation of cluster RR Lyrae statistics with metallicity. Although there have been more recent studies of the LMC star cluster population (e.g. Nayak et al., 2016; Bitsakis et al., 2017; Piatti, 2017, 2018), the effects of field star contamination and asterisms on cluster identification in these studies are unclear. Since Baumgardt et al. (2013) compiles previously identified clusters above 5000 M⊙, we rely on this catalog for our analysis. Additionally, this catalog (to the best of the authors’ knowledge) is the only catalog of LMC clusters in the literature with both age _and_ mass estimates of the clusters—both being critical to our analysis. Out of the 296 clusters, we investigate the RR Lyrae statistics of 62 clusters that have ages above 1 Gyr in order to compare with the DTD. We define $n_{o}$ as the number of OGLE-IV RR Lyrae observed within a circle of radius $r_{c}$ centered on the cluster, and $n_{b}$ as the expected number of RR Lyrae within the cluster area but unassociated with the cluster (we refer to these as ‘background’ RR Lyrae). Assuming $N$ is the number of RR Lyrae within radii $2r_{c}$ and $4r_{c}$ of the cluster (we allow a buffer region of $r_{c}$ to account for uncertainty in the actual extent of the cluster), we define $n_{b}$ as, $n_{b}=\frac{N\pi r_{c}^{2}}{\pi(4r_{c})^{2}-\pi(2r_{c})^{2}}=\frac{N}{12}$ (6) Values for $n_{o}$ and $n_{b}$ are listed in Table 2 and shown in Figure 9. LMC globular clusters older than 10 Gyr clearly host populations of multiple RR Lyrae that exceed the background. The richest population is in NGC 1835 with over a 100 RR Lyrae, while NGC 2005, NGC 1928 and NGC 1939 have less than 10 RR Lyrae. In contrast the intermediate-age clusters (1–10 Gyr old) are generally lacking in RR Lyrae. Out of the 53 intermediate-age clusters, 42 have 0 RR Lyrae and only three clusters (NGC 1978, SL180 and HS190) have more than 1 RR Lyrae star after subtracting the background numbers (i.e. $n_{o}-n_{b}$). This is also the case in the Milky Way, where studies of variable stars in well known intermediate-age systems such as M67 (Pribulla et al., 2008), NGC 188 (Zhang et al., 2004), NGC 6791 (de Marchi et al., 2007) and NGC 6253 (Kaluzny et al., 2014) have found no RR Lyrae that are likely to be cluster members. In contrast, the Milky Way’s globular cluster system contains almost 2000 of such objects (Clement et al., 2001; Soszyński et al., 2014). While this deficit of RR Lyrae in intermediate-age clusters appears contradictory to the 1–8 Gyr signal in the DTD, we note that the intermediate- age clusters in the LMC are generally less massive than old globular clusters, and are therefore less likely to host relatively short-lived objects such as RR Lyrae stars. We quantify this with $n_{p}$, the expected number of RR Lyrae per cluster given our measured DTD $\Psi T_{vis}$, the mass $M_{c}$, and age $t_{c}$ of a cluster: $n_{p}=M_{c}(\Psi T_{vis})\|_{t=t_{c}}$ (7) For each cluster, we measure $\Delta n=(n_{o}-n_{b})-n_{p}$, i.e. the difference between the observed background-subtracted number of RR Lyrae ($n_{o}-n_{b}$) and the number predicted by the DTD. We estimate the uncertainty in $\Delta n$ in the following way: we create 106 random samples of the 62 clusters, each sample having the same $n_{o}$ per cluster as given in Table 2 but with $n_{b}$ and $n_{p}$ generated from a Poisson distribution with mean $n_{b}$ and $n_{p}$ values given in Table 2. For any cluster in these samples with a negative value of $n_{o}-n_{b}$, we set its $n_{o}-n_{b}=0$. The mean and standard deviation of this random sampling is taken to be the value and error of $\Delta n$. These $\Delta n$ are shown as a function of cluster age and metallicity in Figure 9. We see that $\Delta n$ is consistent with 0 within 2$\sigma$ for most intermediate-age clusters in the LMC. This is because the DTD predicts $\sim 1-3$ RR Lyrae per 105 M⊙ of stars, whereas the intermediate-age clusters have an average mass of $4.3\times 10^{4}$ M⊙, so the majority will have less than 1 RR Lyrae star. This caveat was also raised by Olszewski et al. (1996) in their study of the SMC: if the RR-Lyrae-rich cluster NGC 121 ($t\approx 12$ Gyr) were scaled down to the magnitude of the 9 Gyr system Lindsey 1 (which has no RR Lyrae), it would host less than 1 RR Lyrae star. The intermediate-age clusters of the Milky Way, including the ones listed above, have the same issue: all have masses less than $10^{4}$ M⊙ (Kaluzny & Udalski, 1992; Mermilliod, 2000; Piskunov et al., 2008; Grundahl et al., 2008; Elsanhoury et al., 2016; Kruijssen et al., 2019). Thus, the traditional lower limit on the age of the RR Lyrae progenitor population is subject to small number statistics. We note that the globular clusters host a diversity of RR Lyrae populations. NGC 1939, NGC 1916 and NGC 2005 are consistent with $\Delta n=0$ within their uncertainties, and their production rates $(n_{o}-n_{b})/M_{c}=(1.8-2.5)\times 10^{-5}$ RR Lyrae per M⊙ are similar to the measured DTD. The other old clusters, however, host RR Lyrae significantly in excess of their DTD- predicted $n_{p}$ (i.e. $\Delta n\gg 0$), and have an average production rate $\sim 10^{-4}$ RR Lyrae per M⊙, a factor $\sim 5$ times higher than the DTD. The high production rate in the old clusters may be an effect of their low metallicities, which is consistent with the differences in production rates of RR Lyrae in the halo versus the bulge and disk of our Galaxy (Layden, 1995; Dékány et al., 2018), although the large differences in $\Delta n$ for the globular clusters could also be an effect of the second parameter phenomenon in clusters (see Fusi Pecci & Bellazzini, 1997; Catelan, 2009; Dotter, 2013, for reviews). A more detailed assessment of the field and cluster DTDs can be done with a metallicity-dependent DTD that we reserve for future work. Table 2: LMC Clusters older than 1 Gyr in the Baumgardt et al. (2013) catalog and inside the Harris & Zaritsky (2009) region. Name \| Age (Gyr) \| [Fe/H] \| aRef \| Log Mass (M⊙) \| $n_{o}$ \| $n_{b}$ \| $n_{p}$ ---\|---\|---\|---\|---\|---\|---\|--- SL569 \| 1.2 $\pm$ 0.03 \| -0.32 $\pm$ 0.05 \| 6 \| 4.29 \| 0 \| 0 \| 0 KMK88-38 \| 1.26${}^{+0.15}_{-0.14}$ \| – \| – \| 3.71 \| 0 \| 0 \| 0 BSDL880 \| 1.26${}^{+0.15}_{-0.14}$ \| – \| – \| 3.83 \| 0 \| 0 \| 0 SL197 \| 1.26${}^{+0.25}_{-0.21}$ \| – \| – \| 3.93 \| 0 \| 0 \| 0 HS102 \| 1.26${}^{+0.15}_{-0.14}$ \| – \| – \| 3.87 \| 1 \| 0 \| 0 HS223A \| 1.26${}^{+0.15}_{-0.14}$ \| – \| – \| 4.02 \| 0 \| 0 \| 0 NGC1795 \| 1.29${}^{+0.16}_{-0.14}$ \| -0.23 \| 4 \| 4.36 \| 0 \| 0 \| 0 NGC1917 \| 1.29${}^{+0.16}_{-0.14}$ \| -0.21 \| 4 \| 4.42 \| 0 \| 1 \| 1 KMHK898 \| 1.32 $\pm$ 0.09 \| – \| – \| 4.16 \| 0 \| 0 \| 0 NGC1852 \| 1.32${}^{+0.16}_{-0.14}$ \| – \| – \| 4.51 \| 1 \| 0 \| 0 SL282 \| 1.35${}^{+0.35}_{-0.28}$ \| – \| – \| 3.71 \| 1 \| 0 \| 0 BSDL946 \| 1.35${}^{+0.2}_{-0.17}$ \| – \| – \| 4.17 \| 0 \| 0 \| 0 NGC2154 \| 1.41${}^{+0.17}_{-0.15}$ \| -0.56 \| 3 \| 4.57 \| 1 \| 0 \| 1 NGC1751 \| 1.41${}^{+0.17}_{-0.15}$ \| -0.44 $\pm$ 0.05 \| 6 \| 4.6 \| 0 \| 0 \| 1 SL151 \| 1.41${}^{+0.25}_{-0.21}$ \| – \| – \| 4.1 \| 0 \| 0 \| 0 HODGE7 \| 1.48${}^{+0.18}_{-0.16}$ \| – \| – \| 4.47 \| 0 \| 0 \| 0 NGC1846 \| 1.48${}^{+0.18}_{-0.16}$ \| -0.4 \| 1 \| 5.1 \| 1 \| 1 \| 2 NGC1783 \| 1.51${}^{+0.18}_{-0.16}$ \| -0.75 \| 3 \| 5.26 \| 0 \| 1 \| 2 NGC1806 \| 1.51${}^{+0.18}_{-0.16}$ \| -0.71 $\pm$ 0.23 \| 1 \| 5.01 \| 1 \| 1 \| 1 SL136 \| 1.55${}^{+0.45}_{-0.35}$ \| – \| – \| 4.39 \| 0 \| 0 \| 0 NGC2213 \| 1.58${}^{+0.46}_{-0.35}$ \| -0.7 $\pm$ 0.1 \| 1 \| 4.56 \| 0 \| 0 \| 0 SL180 \| 1.58${}^{+0.41}_{-0.33}$ \| – \| – \| 4.32 \| 3 \| 0 \| 0 H1 \| 1.58${}^{+0.41}_{-0.33}$ \| -0.29 \| 5 \| 4.84 \| 1 \| 2 \| 1 SL357 \| 1.58${}^{+0.32}_{-0.27}$ \| – \| – \| 4.75 \| 0 \| 1 \| 1 SL390 \| 1.58 $\pm$ 0.11 \| -0.4 \| 2 \| 4.48 \| 0 \| 1 \| 0 HS117 \| 1.58${}^{+0.41}_{-0.33}$ \| – \| – \| 4.15 \| 2 \| 1 \| 0 BSDL1102 \| 1.62${}^{+0.47}_{-0.36}$ \| – \| – \| 3.77 \| 0 \| 0 \| 0 SL66 \| 1.7${}^{+0.7}_{-0.5}$ \| – \| – \| 4.34 \| 0 \| 0 \| 0 NGC1652 \| 1.7${}^{+0.21}_{-0.18}$ \| -0.46 \| 3 \| 4.29 \| 0 \| 0 \| 0 BSDL2652 \| 1.74${}^{+0.5}_{-0.39}$ \| – \| – \| 3.73 \| 0 \| 0 \| 0 HS87 \| 1.78${}^{+0.22}_{-0.19}$ \| – \| – \| 3.98 \| 0 \| 0 \| 0 OGLE-LMC0114 \| 1.78${}^{+0.46}_{-0.37}$ \| – \| – \| 3.71 \| 0 \| 0 \| 0 HS37 \| 1.78${}^{+1.24}_{-0.73}$ \| – \| – \| 4.02 \| 1 \| 0 \| 0 HS177 \| 1.78${}^{+0.22}_{-0.19}$ \| – \| – \| 3.95 \| 0 \| 0 \| 0 H2 \| 1.78${}^{+0.46}_{-0.37}$ \| -0.38 \| 5 \| 4.98 \| 0 \| 1 \| 1 BSDL734 \| 1.78${}^{+0.22}_{-0.19}$ \| – \| – \| 3.8 \| 1 \| 0 \| 0 KMHK355 \| 1.86${}^{+0.96}_{-0.63}$ \| – \| – \| 3.7 \| 0 \| 0 \| 0 OGLE-LMC0531 \| 2.0${}^{+0.24}_{-0.22}$ \| – \| – \| 3.91 \| 0 \| 0 \| 0 NGC1978 \| 2.0${}^{+0.24}_{-0.22}$ \| -0.38 $\pm$ 0.07 \| 1 \| 5.33 \| 2 \| 0 \| 3 NGC1651 \| 2.0${}^{+0.46}_{-0.37}$ \| -0.53 $\pm$ 0.03 \| 1 \| 5.24 \| 0 \| 0 \| 2 SL629 \| 2.04${}^{+0.1}_{-0.09}$ \| – \| – \| 4.0 \| 0 \| 0 \| 0 KMHK1112 \| 2.4${}^{+0.69}_{-0.54}$ \| – \| – \| 3.78 \| 1 \| 0 \| 0 HS88 \| 2.45${}^{+1.09}_{-0.76}$ \| – \| – \| 4.1 \| 0 \| 0 \| 0 SL244 \| 2.69 $\pm$ 0.06 \| -0.7 $\pm$ 0.2 \| 1 \| 4.77 \| 0 \| 0 \| 1 SL150 \| 2.75 $\pm$ 0.06 \| – \| – \| 4.34 \| 0 \| 0 \| 0 HS190 \| 2.75${}^{+3.41}_{-1.52}$ \| – \| – \| 4.51 \| 2 \| 0 \| 1 BSDL2300 \| 2.95${}^{+0.21}_{-0.2}$ \| – \| – \| 4.7 \| 0 \| 0 \| 1 KMHK1188 \| 3.02${}^{+0.14}_{-0.14}$ \| – \| – \| 4.0 \| 0 \| 0 \| 0 H88-93 \| 3.02${}^{+4.06}_{-1.73}$ \| – \| – \| 3.85 \| 0 \| 0 \| 0 BSDL1334 \| 3.02${}^{+4.74}_{-1.85}$ \| -0.4 \| 2 \| 3.86 \| 0 \| 0 \| 0 SL663 \| 3.24${}^{+0.48}_{-0.42}$ \| -0.7 $\pm$ 0.1 \| 1 \| 5.23 \| 0 \| 0 \| 2 NGC2121 \| 3.24${}^{+0.48}_{-0.42}$ \| -0.4 $\pm$ 0.1 \| 1 \| 5.69 \| 0 \| 1 \| 5 NGC2155 \| 3.24${}^{+0.48}_{-0.42}$ \| -0.7 $\pm$ 0.1 \| 1 \| 4.9 \| 0 \| 0 \| 1 NGC1939 \| 13.49${}^{+3.49}_{-2.77}$ \| -2.1 $\pm$ 0.19 \| 1 \| 5.08 \| 4 \| 1 \| 1 NGC1928 \| 13.49${}^{+3.49}_{-2.77}$ \| -1.27 $\pm$ 0.14 \| 1 \| 4.87 \| 7 \| 1 \| 1 NGC1898 \| 14.13${}^{+2.47}_{-2.1}$ \| -1.37 $\pm$ 0.15 \| 1 \| 5.88 \| 38 \| 2 \| 9 NGC1786 \| 15.14${}^{+0.35}_{-0.34}$ \| -2.1 $\pm$ 0.3 \| 1 \| 5.57 \| 47 \| 1 \| 4 NGC1754 \| 15.49${}^{+2.29}_{-2.0}$ \| -1.42 $\pm$ 0.15 \| 1 \| 5.39 \| 32 \| 0 \| 3 NGC1916 \| 15.85${}^{+3.65}_{-2.97}$ \| -2.08 \| 4 \| 5.79 \| 14 \| 3 \| 7 NGC2005 \| 16.6${}^{+6.31}_{-4.57}$ \| -1.35 $\pm$ 0.15 \| 1 \| 5.49 \| 9 \| 2 \| 4 NGC1835 \| 16.6${}^{+2.9}_{-2.47}$ \| -1.62 $\pm$ 0.15 \| 1 \| 5.83 \| 105 \| 2 \| 8 NGC2019 \| 17.78${}^{+3.6}_{-2.99}$ \| -1.23 $\pm$ 0.15 \| 1 \| 5.68 \| 49 \| 1 \| 6 Note. — _a_ – References for [Fe/H] measurement. (1) Harris & Zaritsky (2009) (2) Palma et al. (2016), (3) Girardi & Marigo (2007) (4) Kontizas et al. (1993) (5) Olszewski et al. (1991) (6) Grocholski et al. (2006) * - These 2 clusters are listed as having ages $\sim 1.7$ Gyr in Baumgardt et al. (2013), but are more likely to be $\gtrsim 10$ Gyr based on HST WFPC2 observations (Mackey & Gilmore, 2004). ## 5 Implications of the RR Lyrae DTD If we assume the DTD measured from the OGLE-IV survey is correct, there are two possible interpretations of this result: 1) RR Lyrae can form from progenitors younger than 10 Gyr _in addition_ to the conventional route via older stars, and this result was undetected in previous studies due to various observational limitations, or 2) all OGLE-IV RR Lyrae are older than 10 Gyr stars, and our result is a product of significant (though not readily obvious) systematics in the age derivation of older stellar populations. We discuss both of these interpretations below. ### 5.1 Can LMC RR Lyrae have an intermediate-age channel? Since the DTD is generally regarded as a reflection of the progenitor age distribution (Maoz & Badenes, 2010; Badenes et al., 2015), it is tempting to consider that RR Lyrae in the LMC are being produced by an as-of-yet undiscovered intermediate-age progenitor channel between 1-8 Gyrs, in addition to the usual channel older than 8 Gyrs. However, looking at currently available evidence, the possibility of such an undiscovered intermediate-age channel appears to be questionable. Although the small-number statistics described in Section 4.3 is a factor, it is nevertheless true that ancient globular clusters host abundant and rich populations of RR Lyrae compared to the RR Lyrae-poor intermediate-age clusters, a feature easily explained by an exclusively old channel for RR Lyrae formation. Evidence that a small fraction of RR Lyrae may arise from progenitors only a few Gyrs old was recently obtained from thin-disk metal- rich RR Lyrae observed in the _Gaia_ data (Zinn et al., 2019; Prudil et al., 2020; Iorio & Belokurov, 2020), and from measurements of companion masses greater than 1 Msun in wide-orbit RR Lyrae binaries (Kervella et al., 2019a, b). The thin disk, metal-rich RR Lyrae population was speculated by Iorio & Belokurov (2020) to be manifestation of binary evolution pulsators (Karczmarek et al., 2017), which will register as younger stars. However, the existence of such an intermediate-age channel for LMC RR Lyrae (assuming single-stellar evolution) would be in tension with the age-metallicity relation of the LMC constrained by mutiple studies of field and cluster stars(Cole et al., 2005; Carrera et al., 2008; Rubele et al., 2012; Meschin et al., 2014). According to these studies, the LMC star-formation history between 2-8 Gyrs was associated with [Fe/H] between -0.4 and -1. In contrast, the LMC RR Lyrae are predominantly metal-poor, with [Fe/H]$<-1$ and peaking at [Fe/H]$\sim-1.5$, as confirmed by spectroscopic studies of field RR Lyrae (Gratton et al., 2004; Borissova et al., 2006), photometric light curves of RR Lyrae (Haschke et al., 2012; Wagner-Kaiser & Sarajedini, 2013; Skowron et al., 2016), and RR Lyrae- hosting globular clusters (Fig 9). Thus the LMC stellar population at ages below 8 Gyrs, where we measure significant signal in the DTD, is more metal- rich than the metallicity range measured for LMC RR Lyrae. ### 5.2 Systematic uncertainties in old SADs? The DTD represents an empirical connection between the RR Lyrae sample and the SAD map. Since the RR Lyrae sample is highly complete (Section 2.1) and has strong independent evidence of originating from old stars (Section 5.1), and since our DTD recovery method would have correctly recovered a purely old RR Lyrae DTD signal from this SAD map (Section 4.1), it may be possible that the intermediate-age signal in the DTD is indicative of some systematic uncertainty in measuring older stellar ages in the LMC. The source and magnitude of this uncertainty is not readily obvious. The global star- formation history measured by HZ09 is broadly consistent with the interaction history of the Magellanic Clouds derived from proper motion modeling (Lin et al., 1995; Zaritsky & Harris, 2004; Besla et al., 2007) and with the star formation and chemical enrichment history derived from star clusters (Chilingarian & Asa’d, 2018). As shown in Section 4.2, we verified that any incompleteness or statistical uncertainties in the MCPS photometry is unlikely to be affecting our DTD because: 1) we propagate the reported uncertainties in the SAD map into our DTDs; 2) the DTD retains signal below 8 Gyrs even when measured outside the crowded LMC Bar; 3) wiping out the DTD signal of younger progenitors would require an unreasonably large unseen stellar mass with age $>$8 Gyr; and 4) we directly detect a signal in the DTD below 4 Gyrs, where the main-sequence turnoff is detectable above the MCPS completeness limit. On the other hand, the SAD solutions per region depend on the overall methodology adopted by the study. An example of this can be found by comparing the Harris & Zaritsky (2004) SAD map of the SMC, which was derived using roughly the same methodology used for the LMC, with the Rubele et al. (2018) SAD map of SMC derived from the deeper VISTA near-infrared survey of the Magellanic System (VMC, Kerber et al., 2009; Cioni et al., 2011). The Harris & Zaritsky (2004) 2–3 Gyr SADs has a distinct ring pattern in the SMC, which they suspected was either the result of photometric incompleteness or systematic uncertainties in the photometric zero-point of the central SMC stars. In the VMC SAD, this ring pattern is absent in the 2–3 Gyr stellar population. Figure 11 of Rubele et al. (2018) shows that the global stellar mass formed at ages $>1$ Gyr in the SMC differ by more than a factor of 2 between Rubele et al. (2018) and Harris & Zaritsky (2004), and the bimodal star-formation history at 2.5 and 10 Gyrs found by Harris & Zaritsky (2004) is replaced with a single broad peak at 5 Gyr in Rubele et al. (2018). These systematic differences could stem from differences in IMF and distance assumed, the stellar isochrone models used, and/or age-metallicity binning. For example, the age-metallicity relation measured by the two studies diverges for stars older than 4 Gyrs; this is most likely because Rubele et al. (2018) uses metallicity bins that extend to lower abundances than Harris & Zaritsky (2004). Similar systematic differences in the LMC SADs may also exist. For example, the star-formation history of the Northern Void region measured by Meschin et al. (2014), using $V$\- and $I$-band photometry with the CTIO Blanco 4 m telescope, differs from that of HZ09 for ages younger than 4 Gyr. Partial estimates from upcoming SMASH data reveals a well-mixed SAD in the LMC for ages Another source of systematic uncertainty in the SADs may be due to the use of single-stellar evolution isochrone models. According to Moe et al. (2019), the close ($<10$ AU) binary fraction for Milky Way field stars of LMC metallicity is $\sim 30\%$ (compared to $\sim 20\%$ for Solar metallicity), so the influence of binary evolution physics in older populations may be non- negligible. Stanway & Eldridge (2018) have shown that models of integrated spectra and photometry of globular clusters and elliptical galaxies that include binary evolution physics yield age estimates that are different by a few Gyr compared to single-star models. A well-known observational manifestation of binary interaction in old stellar populations is the appearance of blue straggler stars, which are formed from the merger of $\sim 1M_{\odot}$ stars. Blue straggler stars can mimic younger stars in color- magnitude diagrams as they appear brighter and bluer than the main-sequence turnoff (Santana et al., 2016), and single stellar population models that correct for the presence of blue straggler stars yield globular cluster ages older by a few Gyrs (Fan & de Grijs, 2012). Correcting for blue stragglers however is non-trivial as the frequency of blue straggler stars is likely a function of stellar density (Weisz et al., 2014; Santana et al., 2013, 2016), and the contribution of blue straggler stars at $\sim$Gyr ages have been difficult to determine in composite stellar populations (Surot et al., 2019). Leaving binary evolution aside, there are uncertainties even in the physics of single-star evolution models that can affect age estimates. For example, Tayar et al. (2017) showed that the mixing length parameter—commonly used to approximate convection theory in 1D stellar models—appears to be correlated with metallicity, and if left unaccounted for when estimating ages from the giant branch (which is the case for ages $>4$ Gyr in the HZ09 maps), it can lead to age uncertainties up to a factor of 2. Any number of these reasons could distort the SAD solutions in a subset of the cells, and therefore the final DTD, which is derived from these data. ### 5.3 Caveats and Future work From our work, we have shown that DTD provides a new rigorous and quantitative test of SADs of Local Group galaxies, in addition to its original purpose as a stellar evolution diagnostic. Although it is possible that unknown sources in systematic uncertainties in the SAD map may be driving our DTD result, we would need further tests to verify the authenticity of this issue, which we will perform in subsequent papers. Estimating the precise form of the HZ09 SAD map that would be consistent with a purely old RR Lyrae DTD is non-trivial because of the large number of SAD parameters involved in this exercise (the stellar masses per age and metallicity bin per cell), and also because RR Lyrae can only reliably constrain the oldest ages. In addition, the coarse age and metallicity binning of the HZ09 map may be less than ideal for constraining the production rate of RR Lyrae if they are coming from a narrower range of ages and metallicities within each bin (as e.g. indicated in Section 4.3. We can however check if our DTD result persists when calculated with SADs derived from deeper photometric data with finer age and metallicity resolution, such as the upcoming SMASH star-formation histories (Ruiz-Lara et al., 2020), as well as in other Local Group galaxies like the SMC (Rubele et al., 2018), M31 (Williams et al., 2017) and Local Group dwarfs (Weisz et al., 2014). A more constraining test of the SADs at intermediate and younger ages can also be obtained by calculating DTDs of younger variables stars with well- constrained ages, such as $\delta$-Scutis (ages 1-3 Gyrs, Petersen & Christensen-Dalsgaard, 1996) and Classical Cepheids (ages 70-200 Myrs, Bono et al., 2005). We will pursue these in future papers. ## 6 Conclusions We have calculated the first delay time-distribution (DTD) of RR Lyrae stars using the large sample of LMC RR Lyrae from the OGLE-IV survey (Soszyński et al., 2016) and the LMC’s SAD map from Harris & Zaritsky (2009). Our DTD, shown in Figure 3 and Table 1, constrains the age-distribution of the full LMC RR Lyrae population, given the measured SAD of the LMC. The OGLE-IV RR Lyrae sample which overlaps the SAD map of HZ09 contains 29,810 objects, allowing us to recover a DTD with an unprecedented balance of age resolution and detection significance. We determined the DTD signal in each age-bin with an MCMC solver, and used a randomization technique to propagate uncertainties in the SAD map into the final DTD. Our measured RR Lyrae DTD has statistically significant ($>5\sigma$) power in all age-bins above 1.3 Gyrs, with about 51$\%$ of the RR Lyrae associated with ages between 1.3 and 8 Gyr, and only 46$\%$ with ages above 8 Gyr (the conventional lower limit to RR Lyrae age; note that while the lower limit quoted in the literature is 10 Gyr, the SAD map has a single indivisible age- bin of 8–12 Gyr, and so we refer to the lower limit as 8 Gyr in this paper). This would imply that the progenitors of RR Lyrae have zero-age main-sequence masses $\lesssim 2$ M⊙ at LMC metallicity, in contrast with existing constraints. We checked the DTD for possible sources of bias. Completeness of the RR Lyrae sample is probably not an issue based on their $I$-band luminosity function and the predominance of RRab (fundamental) pulsators, which are least susceptible to confusion with other types of variables. We also tested our DTD algorithm on fake RR Lyrae maps drawn from a DTD that assumes all RR Lyrae are older than 8 Gyrs, and found that our MCMC algorithm recovers this old DTD without any outlying detections at younger ages. The spatial distribution of RR Lyrae from a purely old DTD is also inconsistent with the spatial distribution of the OGLE-IV RR Lyrae. A possible caveat to our result is the incomplete photometry in the MCPS data and heavy crowding in the LMC central region, which limits the reliability of the SAD maps to ages younger than 4 Gyr; information about older populations are all based on _HST_ -derived SADs in a few narrow fields. However it is not readily obvious how this is producing an intermediate-age signal since: (1) we recover the DTD signal at ages younger than 8 Gyr even after excluding the Bar region, (2) we measure a DTD signal younger than 4 Gyr, and (3) we find that in order to affect the solution, the estimates of the LMC’s old stellar mass must be more than an order-of-magnitude greater than current measurements. The direct interpretation of our result would be that RR Lyrae have an intermediate-age progenitor channel of 1.3-8 Gyrs stars, in addition to its conventional route via ancient stars, but this possibility is in tension with existing constraints. Both in the Milky Way and Magellanic Clouds, RR Lyrae are abundantly hosted in ancient globular clusters, in contrast with intermediate-age clusters (although this conclusion is somewhat affected by small-number statistics due to the lower-masses of intermediate-age clusters). In addition, the 1.3-8 Gyr RR Lyrae population would likely have [Fe/H]$>$-1 based on existing constraints on the LMC age-metallicity relation, whereas the metallicity distribution of LMC RR Lyrae measured from spectroscopy and photometric light curves is in the range of [Fe/H]$<$-1 with a peak at [Fe/H]$\sim-1.5$. The other possibility of the intermediate-age DTD result is the presence of unknown systematics in the LMC SAD map. This is not obvious because the global star-formation history based on the SAD is consistent with the LMC–SMC–Milky Way interaction histories and the chemical enrichment history of the LMC derived from independent studies. However, comparison of the SMC SADs of Harris & Zaritsky (2004) and Rubele et al. (2018) shows that spatial solutions of the SAD maps can be influenced by the overall methodology adopted for their construction (e.g., assumptions about the IMF, the spatial size of cells, the age and metallicity bin size, and stellar isochrone models). Age estimation of old stellar populations from color-magnitude diagrams can also be affected by binary evolution processes, such as mass-transfer and mergers, as well as approximations in single-star evolution models. Any combination of these reasons could be skewing the SADs for old stellar populations, and this error could be propagating into the DTD results. We laid out further tests to investigate the physical nature of the systematics that are driving the intermediate-age signal in the DTD, such as revisiting the RR Lyrae DTD once LMC SADs from deeper photometric studies (e.g., SMASH) become available, and also measuring DTDs in SMC and dwarf galaxies, which have less crowded star fields and deep _HST_ -derived SADs. We will also continue to apply this technique to other types of variable stars, such as Cepheids and Delta Scutis, to probe other enigmatic phases of stellar evolution. ## 7 Acknowledgements We are grateful to Horace Smith for reading the manuscript and providing many helpful comments and insight on RR Lyrae observations and models. We also acknowledge the detailed feedback from our anonymous referee on the interpretation of RR Lyrae ages in the LMC and their relation with metallicity, and Dennis Zaritsky, Knut Olsen, Thomas Matheson, Benjamin Williams, J.J. Eldridge and Jay Strader for insightful discussions and feedback on this work. SKS, CB, and LC are grateful for the suppport of NSF grants AST-1412980 and AST-1907790. CM acknowledges support from the DGAPA/UNAM PAPIIT program grant IG100319 and from the ICC University of Barcelona visiting academic grants and thanks the _Gaia_ -UB team for hosting her during part of this research. CM also thanks the Polo de Desarrollo Universitario (PDU) en Ciencias Físicas at CURE-UdelaR (Rocha), for their hospitality. DM acknowledges support by grants from the Israel Science Foundation, the German Israeli Science Foundation and the European Research Council (ERC) under the European Union’s FP7 Programme, Grant No. 833031. This work made use of the publicly available OGLE-IV variable star catalog 333http://ogledb.astrouw.edu.pl/ ogle/OCVS/. This research has made use of NASA’s Astrophysics Data System, and the VizieR catalogue access tool, CDS, Strasbourg, France. The original description of the VizieR service was published in A&AS 143, 23. This work made use of the IPython package (Pérez & Granger, 2007), SciPy (Jones et al., 2001–), NumPy (van der Walt et al., 2011), matplotlib, a Python library for publication quality graphics (Hunter, 2007), and Astropy, a community-developed core Python package for Astronomy (Astropy Collaboration et al., 2013). 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We calculate DTDs for different age binnings using just the best-fit SAD (i.e., no randomized SADs) and the original OGLE-IV sample inside the SAD area. We show eight example binning schemes for our DTDs in Figure 10, which differ in how the young and old ages are binned. The four binning schemes in the bottom row have the smallest bin-sizes for ages $>0.8$ Gyr and varying bin- sizes for young ages. The schemes in the top row have coarser resolution in the oldest age-bins. More binning schemes similar to these are possible and can be tested, but these eight provide a general sense of the impact of binning. The DTDs measured in the detected bins vary at most by a factor of 2, and are generally around $10^{-5}$ RR Lyrae per M⊙. Statistical errors in the DTDs increase with the number of bins, because we are essentially increasing the number of fitting parameters. Binning schemes that retain the highest resolution in the oldest bins measure smaller values of BIC, irrespective of the binning in the younger age-bins. This implies that the RR Lyrae DTD has the strongest signal in the oldest bins, which is consistent with an older stellar origin of RR Lyrae. Apart from BIC, we also show the acceptance fraction of the MCMC solver, $a_{f}$ in Figure 10. This is the fraction of new steps accepted by the emcee walkers as the scan the multi-dimensional parameter space. While not a model selection statistic like BIC, values of $a_{f}$ = 0.2–0.5 indicate that the emcee algorithm is performing optimally (Foreman-Mackey et al., 2013). Values of $a_{f}$ which are too low indicate multiple peaks in the posterior space separated by valleys of “low probability”, while high values imply that the walkers are simply random-walking, with no regard for the target probability density. We note that while binning schemes in the second row have similar values of BIC, they have different $a_{f}$, with the native resolution of the SAD (last panel) having the smallest $a_{f}$. Based on these tests, we choose the binning scheme 7 which has the smallest value of BIC, and the largest value of $a_{f}$ = 0.28 in Figure 10. Figure 10: RR Lyrae DTDs calculated with different age binning. The scheme number is shown in the top right corner of each panel. The acceptance fraction $a_{f}$ of the MCMC solver and the value of the BIC test (BIC) are shown in each panel. Shaded grey regions mark age-bins with significant detections, while arrows show 2$\sigma$ upper limits. We use the binning scheme plotted at bottom right, as it has the smallest information loss, and also has the highest acceptance fraction. ## Appendix B Uncertainties using the B15 method When generating the randomized SADs, we assumed that the uncertainties of the stellar masses were normally distributed. However, this is only an approximation, and the underlying shape of the probability distribution of masses is unknown. We therefore also calculated the DTD uncertainties based on the method in B15, which can be treated as a more conservative estimate of the uncertainties. In B15, the 1 $\sigma$ uncertainty due to the SAD is equal to the difference between the DTDs for the best fit SAD, and DTDs for the 68$\%$ upper and lower limits on the SADs. This difference is added in quadrature to the statistical uncertainties in the best-fit DTD, and the total value is used for assessing detectability. The uncertainties using this method are larger, and as a result the DTD (Figure 11) is different from the DTD in Figure 3. Nevertheless, it still shows significant signal below 10 Gyrs, particularly in the age range 2-8 Gyrs. The signals in the other age-bins fall below the 2$\sigma$ limit. The total contribution below 10 Gyrs in this case is about 41.6$\%$. Our main science result,$-$ that the DTD has statistically significant signal below 10 Gyrs $-$ is unchanged, even with our most conservative estimates of the uncertainties. Figure 11: Same as Figure 3, but with uncertainties calculated using the method in Badenes et al. (2015), described in Section B.
# Spacetime Quantum Reference Frames and superpositions of proper times Flaminia Giacomini<EMAIL_ADDRESS>Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada ###### Abstract In general relativity, the description of spacetime relies on idealised rods and clocks, which identify a reference frame. In any concrete scenario, reference frames are associated to physical systems, which are ultimately quantum in nature. A relativistic description of the laws of physics hence needs to take into account such quantum reference frames (QRFs), through which spacetime can be given an operational meaning. Here, we introduce the notion of a spacetime quantum reference frame, associated to a quantum particle in spacetime. Such formulation has the advantage of treating space and time on equal footing, and of allowing us to describe the dynamical evolution of a set of quantum systems from the perspective of another quantum system, where the parameter in which the rest of the physical systems evolves coincides with the proper time of the particle taken as the QRF. Crucially, the proper times in two different QRFs are not related by a standard transformation, but they might be in a quantum superposition one with respect to the other. Concretely, we consider a system of $N$ relativistic quantum particles in a weak gravitational field, and introduce a timeless formulation in which the global state of the $N$ particles appears “frozen”, but the dynamical evolution is recovered in terms of relational quantities. The position and momentum Hilbert space of the particles is used to fix the QRF via a transformation to the local frame of the particle such that the metric is locally inertial at the origin of the QRF. The internal Hilbert space corresponds to the clock space, which keeps the proper time in the local frame of the particle. Thanks to this fully relational construction we show how the remaining particles evolve dynamically in the relational variables from the perspective of the QRF. The construction proposed here includes the Page- Wootters mechanism for non interacting clocks when the external degrees of freedom are neglected. Finally, we find that a quantum superposition of gravitational redshifts and a quantum superposition of special-relativistic time dilations can be observed in the QRF. ## 1 Introduction In both quantum theory and general relativity, spacetime is treated as an abstract entity. In quantum theory, spacetime is considered as a fixed, non- dynamical background providing an arena for physical phenomena. In general relativity, spacetime is a dynamical quantity, which is influenced by a change in the configuration of massive bodies. In both cases, spacetime coordinates acquire meaning via rods and clocks, which identify a reference frame. However, these rods and clocks, and hence reference frames, are typically treated as idealised classical systems. In this sense, the description of spacetime is not fully operational. Reference frames are very useful to specify the point of view from which observations are carried out. Although measurements are usually made in a specific reference frame, all physical laws are formulated in a way that is independent of the reference frame chosen, thanks to the principle of general covariance. This means that there is no preferred reference frame. The rods and clocks which specify the reference frames are physical systems, and ultimately quantum in nature. Hence, an operational formulation of the laws of physics requires being able to describe physical phenomena from the point of view of such quantum systems, which can be in a superposition or entangled with other physical systems. This is the main intuition behind the notion of Quantum Reference Frames (QRFs). QRFs have been studied in quantum information, quantum foundations, and quantum gravity since 1967. In the quantum information literature [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21], they have been used to overcome superselection rules, applied to quantum communication scenarios, and related to invariance properties and symmetries of quantum systems. In the quantum gravity literature, it has been conjectured that QRFs are needed to formulate a quantum theory of gravity [22, 23], especially when taking a relational perspective on spacetime [24]. In particular, a relational approach to QRFs is common to both the quantum gravity and quantum information approaches. Recently, Ref. [25] introduced a formalism to describe physics from the point of view of a specific QRF and transformations to change the description between different QRFs. Subsequent work developed this approach further in different contexts. Refs. [26, 27, 28] adopt a symmetry principle to describe a Galilean system in a fully relational way, and Refs. [29, 30] extend the formalism to the special relativistic regime, applying it to concrete problems in Relativistic Quantum Information. Other works [31, 32, 33] have focussed on the group properties of QRFs both in discrete and in continuous-variable systems. A time version of QRFs, also related to the Page- Wootters mechanism [34, 35], has been introduced in Ref. [36] for interacting clocks. Recently, a generalisation of the formalism has been applied to to quantum systems living on a superposition of gravitational fields, and showed that the Einstein Equivalence Principle can be extended to QRFs [37]. The work on QRFs is related to research on quantum clocks, which have been studied with a relational approach, both in the interacting and non- interacting case, in Refs. [38, 39, 40, 41, 42, 43]. In particular, Ref. [43] studied relativistic clocks by extending the Page-Wootters approach to a relativistic particle with internal and external degrees of freedom. A conceptually similar approach to QRFs, but with some important differences, is the quantum coordinate systems [44, 45]. All these works have studied QRFs either in space or in time, and separately considered the role of internal and external variables (such as, respectively, position or momentum and internal quantities acting as clocks) in identifying the QRF. So far, no unifying approach to QRFs in spacetime has been formulated. A spacetime formulation of QRFs is a crucial step towards a fully relational and covariant description of physics from the point of view of a quantum system, and could also provide a natural setting for a quantum approach to time. However, achieving such a formulation faces some important challenges, such as describing space and time in quantum theory on an equal footing. This task requires the incorporation of a time operator into the description of QRFs in space, and is conceptually related to the problem of time in quantum gravity [46, 47, 48]. The methodology and the scope of the present work are different to those employed in the canonical approaches to the problem of time. In this work, we overcome the difficulty of incorporating a time operator into QRFs and develop a model for spacetime QRFs. We introduce a timeless and fully relational formulation, which methodologically adopts techniques from different approaches to QRFs [25, 26, 27, 36, 37], unifying them by adopting aspects of Covariant Quantum Mechanics [49, 50, 51] and the Page-Wootters mechanism [34, 35]. In particular, we consider a set of $N$ relativistic quantum particles in a weak gravitational field, each of which has a quantum state living in the tensor product Hibert space of position/momentum and some internal “clock” Hilbert space (see Refs. [52, 43] for a similar analysis in a different context). In this timeless formulation, the global state of the $N$ particles is “frozen”, but the dynamical evolution is recovered in terms of the relational variables to one of the particles, which is chosen as the QRF. In order to obtain this relational description on the reduced set of $N-1$ particles, we build a transformation to some relational variables (e.g., relative positions) of the particle chosen as the QRF, and then fix the origin of the QRF at the location of this particle. The evolution of the remaining $N-1$ particles is parametrised in terms of the proper time of the particle serving as the QRF. Concretely, the proper time of each particle is encoded in its internal variables, which serve as quantum clocks. Hence, both the external and the internal variables play an important role in buiding the spacetime QRF: the former allow us to transform to the local frame of a quantum particle, whose spacetime position can be in a superposition or entangled with other systems, and the latter allow us to describe the dynamical evolution of the remaining particles in terms of the proper time in each QRF. Notice that, although proper time is a standard evolution parameter in each QRF, the proper times of two different quantum particles are not in a classical relation one with respect to the other, but can be in a quantum superposition. This joint use of the external variables to fix the QRF and the internal variables as clocks is one of the novel aspects of this work. In addition, we show that we can always find a transformation to the QRF of the particle such that the metric is locally inertial at the origin of the QRF. This is an instance of a Quantum Locally Inertial Frame (QLIF), as introduced in Ref. [37]. We find the relational dynamics of the remaining particles from the point of view of the chosen QRF, which coincides, in the appropriate limit, with the dynamical evolution predicted with different methods (for instance, see Refs. [53, 54, 55, 56]). The expression of this relational dynamics has the same functional form in any chosen QRF, and is thus symmetric under QRF transformations, as defined in Ref. [25]. Finally, we find that this description of QRFs in spacetime allows us to observe a _superposition of special relativistic time dilations_ and a _superposition of gravitational redshifts_ , which could be measured experimentally in the future. Overall, our model paves the way for a full extension of the formalism of QRFs to arbitrary spacetimes, and superpositions thereof. This is an important step to achieve a fully relational formulation of physics on a nonclassical spacetime, which is a key requirement to formulate a quantum theory of gravity. The structure of the paper is as follows. In Section 2, we introduce the model for spacetime QRFs: we start from a fully constrained description of the system, which is neutral to the specific perspective, and is encoded in a set of first-class constraints. We then show how to recover a relational description from the perspective of one of the particles. In Section 3, we show that this construction leads to an extension of the spacetime symmetries to the set of QRF transformations. This result generalises to the spacetime picture and to quantum clocks the notion of extended symmetry transformation introduced in Ref. [25]. In Section 4, we introduce a measurement model and show how the phenomenon of _superposition of special relativistic time dilation_ and _superposition of gravitational redshift arises_. For clarity, the technicalities are kept to the essential in the main text, however the relevant calculations are detailed in the Appendices, which are referred to where appropriate. ## 2 From a covariant timeless model to relational dynamics in a locally inertial quantum reference frame [scale=0.35]GlobalModel Figure 1: We consider a system of $N$ relativistic quantum particles in a weak gravitational field produced by a mass $M$. Each particle has a quantum state representing its position in the spacetime diagram, as well as a clock state (the hands of the clocks) which keeps the proper time in the particle’s frame. We introduce a timeless model, in which the global state of the $N$ particles and of the mass $M$ is “frozen”, and is described by an $N$-particle quantum state in spacetime. Concretely, this means that the quantum state of the external variables can be, for instance, in a quantum superposition of coordinate times $x^{0}$ and spatial coordinates $\mathbf{x}$. Classically, the position in spacetime and the velocity of a particle influence its proper time due to relativistic time dilation. When the particle is in a quantum superposition of positions or velocities, the proper time displayed by its internal clock is also in a quantum superposition from an external perspective. While, at the global level, the system does not evolve, the dynamical evolution is recovered in terms of the relational variables between one of the particles, which is chosen as the quantum reference frame (QRF), and the rest of the particles. To transform to the QRF of one of the particles, we first map the initial spacetime coordinates to the relational spacetime coordinates from the perspective of the particle chosen as the QRF. We then find a transformation which makes the metric locally inertial at the origin of the QRF. Finally, we use the proper time of the particle to parametrise the dynamics of the remaining particles from the perspective of the chosen QRF. In such QRF, the dynamical evolution of the remaining particles is described in terms of a Hamiltonian operator and is parametrised by the clock’s proper time, which is just a classical parameter in the clock’s rest frame. In this section, we introduce a fully relational model for $N$ quantum particles in spacetime, where * • Space and time are treated on equal footing, as in special and general relativity, also when the systems under study are quantum, and not classical; * • Time evolution emerges from internal correlations of physical clocks, which are described as living in the internal Hilbert spaces of the particles. The way in which the dynamical laws are recovered from the perspective of such clocks incorporates the Page-Wootters formulation [34, 35]; * • The formalism is fully relational, i.e., all external spacetime structure is eliminated. Furthermore, we show that it is possible to take the perspective of one of these quantum particles, and how the transformation to such QRF is constructed. We consider a system of N particles of mass $m_{I}$, for $I=1,\cdots,N$, each of which has a quantum state living in the tensor product of an external Hilbert space, representing its position in spacetime or momentum, and an internal Hilbert space (the clock), and a mass $M$ which is the source of the gravitational field, as illustrated in Fig. 1. The ideas we present in this paper are described in their simplest form in the time and radial components of a model with spherical symmetry. To simplify the presentation, we work in $1+1$ dimensions, which allows us to neglect the angular part of the model. We adopt a timeless formulation [34, 49, 50, 51, 35] describing the full state of the particles. The intuitive idea behind these formulations is that the dynamics of particles is an emergent phenomenon stemming from the relational degrees of freedom. Formally, this means that the state of the $N$ particles and of the mass $M$ does not evolve, but the dynamics is recovered by “conditioning” on one of the systems considered. The global state of the $N$ particles and of the mass $M$ — called the _physical state_ — is then a “frozen” (i.e., non-evolving) state $\ket{\Psi}_{ph}$ satisfying the relation $\hat{C}\ket{\Psi}_{ph}=0,$ (1) where $\hat{C}$ is a set of first-class constraints defined on the Hilbert space of the $N$ particles and of the mass $M$. This fully constrained model is not reference-frame specific, but corresponds to a neutral perspective. The set of constraints encodes both the dynamics of the particles in spacetime and the global symmetries (global translations in space and time) of the model. In this Section, we discuss in detail the form of the constraints and their physical meaning. Differently to other formulations for QRFs, we here describe both the quantum state associated to the position/momentum of the particle and the clock state, and they both play a role in the identification of the QRF. In particular, when one of the particles, say particle $1$, acts as a QRF, the external part of the quantum state is used to fix the QRF (in our case, the origin of the coordinate system in spacetime), while the internal part serves as a clock, ticking according to the proper time in each particle’s frame. We will see that imposing the constraints corresponds to removing absolute (or external) variables from the description, and that the relational variables, defined from the point of view of one of the physical systems considered, evolve dynamically in a non-trivial way. In particular, the presence of the internal clocks is crucial to recover the relational dynamics of the rest of the particles from the point of view of the particle chosen as the QRF. Throughout the paper, we use the following notation: Greek letters indicate spacetime labels, i.e., $\mu=0,1$, while capital Latin letters label the particles from $1$ to $N$. Vectors with no indices are, unless differently specified, two-vectors in $1+1$ spacetime, i.e., $v=(v^{0},\mathbf{v})$, where $v^{0}$ is the time component of the vector and $\mathbf{v}$ denotes the spatial component of the vector. We use the Einstein’s convention on all sums, unless explicitly stated. We take the particles to be special-relativistic and in a Newtonian field generated by a mass $M$. We describe the latter as the weak-field limit of a more general gravitational field, corresponding to a metric $\begin{split}&g_{00}=1+\frac{2\Phi(\mathbf{x}-\mathbf{x}_{M})}{c^{2}};\\\ &g_{01}=g_{10}=0;\\\ &g_{11}=-1,\end{split}$ (2) where $\Phi(\mathbf{x})$ is the Newtonian potential due to the mass $m_{M}$ of the system $M$ and $\|\Phi(\mathbf{x})\|/c^{2}\ll 1$ in the spacetime region considered. The external Hilbert space of each particle $I=1,\cdots,N$ is $\mathcal{H}_{I}\simeq L^{2}(\mathbb{R}^{2})$. The mass $M$ is also assigned a Hilbert space $\mathcal{H}_{M}\simeq L^{2}(\mathbb{R}^{2})$. The motion of any of the $N$ particles in this external, weak gravitational field sourced by the mass $M$ can be encoded in the expression $\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\hat{\omega}_{p}^{I},$ (3) where $[\hat{x}^{\mu}_{I},\hat{p}_{\nu}^{I}]=\mathrm{i}\hbar\delta^{\mu}_{\nu}$, $\hat{\omega}_{p}^{I}=\sqrt{m_{I}^{2}c^{2}+\hat{\mathbf{p}}^{2}_{I}}$ with $I=1,\cdots,N$, and $[\hat{x}^{\mu}_{M},\hat{p}_{\nu}^{M}]=\mathrm{i}\hbar\delta^{\mu}_{\nu}$. The constraints $\hat{C}_{I}$ of Eq. (3) straightforwardly follow from the quantisation of the (classical) general-relativistic dispersion relation $g^{\mu\nu}(x)p_{\mu}p_{\nu}-m^{2}c^{2}=0$ thanks to the weak-field approximation, which is crucial to avoid ordering ambiguities. Notice that, for simplicity, we assume that the state of the mass $M$ does not undergo any dynamical evolution in this initial description, hence we do not associate a dynamical constraint to it. In the following we show that the mass acquires a dynamical evolution as a result of taking the perspective of one of the particles. By applying Eq. (3) to a quantum state $\ket{\psi_{I}}_{ph}$ that solves the constraint, i.e., $\hat{C}_{I}\ket{\psi_{I}}_{ph}=0$ and conditioning on the time coordinate $\ket{x^{0}_{I}}_{I}$ via the procedure $\braket{x^{0}_{I}}{\hat{C}_{I}}{\psi_{I}}_{ph}=0$, it is possible to recover the Schrödinger equation for a quantum particle in the Newtonian field (see Appendix A for details). Hence, the dynamical evolution emerges from the entanglement, at the level of $\ket{\psi_{I}}_{ph}$, between the time and spatial degrees of freedom. We are working here in a perturbative regime in which the commutator $\left[\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})},\hat{\omega}_{p}^{I}\right]$ is negligible. This is equivalent to demanding that all terms which are at least of the order $\frac{\Phi(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})\hat{\mathbf{p}}^{2}_{I}}{m_{I}^{2}c^{4}}$ are negligible whenever they are applied to the quantum state of each particle $I$. Notice that we retain terms of the order $\left[\hat{\mathbf{p}}_{I}/(m_{I}c)\right]^{4}$, corresponding to special- relativistic corrections in the velocity of the particles. The constraint $\hat{C}_{I}$ encoding the dynamical evolution of the particle, which holds for each particle of the set we consider, is not yet cast in relational terms. The introduction, compared to the standard quantum- mechanical formalism, of the $\hat{x}_{I}^{0}$ and $\hat{p}_{0}^{I}$ operators, such that $[\hat{x}_{I}^{0},\hat{p}_{0}^{J}]=\mathrm{i}\hbar\delta_{I}^{J}$ is necessary to treat space and time on the same footing, as in the case of Covariant Quantum Mechanics [49, 50]. In order to enforce the relational character of the formalism, we need to eliminate the external structure. In Appendix B we review and compare different relational approaches. In the following, we show how to build a fully relational model for quantum particles in spacetime and in a weak gravitational field. Methodologically, this construction adopts the tools of QRFs, and unifies the framework for spatial QRFs [25, 26, 27] with time reference frames [36], the latter in the non-interacting case. The resulting model utilises techniques of Covariant Quantum Mechanics in order to recover a full spacetime covariance, and incorporates both external and internal degrees of freedom. In particular, we consider two constraints on top of the $N$ constraints of Eq. (3), one for each spacetime dimension, which encode the conservation of the total energy and the total momentum of our $N$-particle model. Thus, we write $\begin{split}&\hat{f}^{0}=\sum_{I=1}^{N}\left[\hat{p}^{I}_{0}+\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})\frac{\hat{H}_{I}}{c}\right]+\hat{p}_{0}^{M};\\\ &\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I}+\hat{\mathbf{p}}_{M},\end{split}$ (4) where $\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})=\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\left(1+\frac{\hat{\mathbf{p}}^{2}_{I}}{m_{I}^{2}c^{2}}\right)^{-1/2}$. The constraint $\hat{f}^{1}$ corresponds to the global translational invariance required in Refs. [26, 27]. The constraint $\hat{f}^{0}$ corresponds to the conservation of the total energy. Notice that particle $M$ counts towards the total energy and momentum balance because, although its state does not obey a dynamical constraint, it is a quantum system in its own right, with energy and momentum associated to it. Such quantum state in spacetime should be chosen so that it corresponds to a physical configuration of the mass $M$ which does not dynamically evolve in the initial coordinates. The constraint $\hat{f}^{0}$ is derived by summing all the contributions to the energies (up to a multiplicative factor of $c$) coming from the motion of the particles in spacetime, from the internal Hamiltonian $\hat{H}_{I}$, living in the Hilbert space $\mathcal{H}_{C_{I}}\simeq L^{2}(\mathbb{R})$ of the $N$ particles, and from the system $M$. In particular, the factor in front of the internal energy Hamiltonian corresponds to the relativistic time dilation from the rest frame to an arbitrary frame. The relation between the rest frame and an external frame to the particle is quantum, because the particles have a quantum state associated to the position/momentum, which can be in a quantum superposition of positions and velocities. QRF techniques justify operationally the change between the rest frame of a quantum system and an arbitrary frame [25, 29, 30]. The relation between the internal energies in the rest frame and in an arbitrary frame is encoded in the operator $\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})$, which coincides with the worldline element of particle $I$. In Appendix C we provide an explicit derivation of the operator $\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})$. The two constraints $\hat{f}^{0}$ and $\hat{f}^{1}$, being first-class constraints, generate a gauge transformation on the canonically conjugated variables. These constraints have the effect of reducing the number of degrees of freedom of the full system to obtain the dynamics of the relative positions between the particles. In the following we show that, by picking one of the particles as the preferred point of view from which we describe the dynamics of the remaining particles, the resulting dynamics can be seen as the dynamics of the $N-1$ particles and the mass $M$ from the point of view of the chosen particle. We then see that the introduction of the zero component of the coordinate and momentum operators not only allows us to treat the $N$-particle system in a covariant way, but allows us to generalise the symmetry principle of Refs. [26, 27] to all the spacetime components and to show the connection with the Page-Wootters inspired approach of Ref. [36]. When only the internal state of the particles is considered, the constraint $\hat{f}^{0}$ reduces to the Page- Wootters constraint of the clocks. We stress here that the constraint $\hat{f}^{0}$ can be equivalently seen as arising from a Page-Wootters construction, which emphasises the relational approach to time, or from a relational formulation of physics, which eliminates the background structure by enforcing that the total energy is zero. As a result of this construction, we have a fully constrained model describing the $N$ particle system, where the full constraint is $\hat{C}=\sum_{I=1}^{N}\mathcal{N}_{I}\hat{C}_{I}+z_{\mu}\hat{f}^{\mu},$ (5) where $\mathcal{N}_{I}$, with $I=1,\cdots,N$ and $z_{\mu}$, with $\mu=0,1$ are Lagrange multipliers. Notice that, to our order of approximation, all constraints are first-class, i.e., they at least weakly commute with each other. In the two cases $\left[\hat{C}_{I},\hat{f}^{1}\right]=0$ and $\left[\hat{f}^{0},\hat{f}^{1}\right]=0$, the relation holds exactly. For the commutator $\left[\hat{C}_{I},\hat{f}^{0}\right]=0$, the result can be easily obtained by remembering that we are working in a regime where $\left[g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M}),\hat{\omega}_{p}^{I}\right]=0$ for all $I=1,\cdots,N$. Notice that, thanks to the fact that all the constraints commute, it is possible, as we will show, to straightforwardly build a state that satisfies all of them. Here, we only consider non-interacting clocks in order to maintain the commutative property of the constraints. Had we allowed for gravitational interaction between the particles, we would have had non-commutative constraints (namely, second-class constraints), which require a different treatment to what is outlined here. We thus leave this question for future work. The physical state of Eq. (1) with the constraint $\hat{C}$ of Eq. (5) can equivalently be written as $\ket{\Psi}_{ph}\propto\int d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{I}\hat{C}_{I}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$ (6) where $\ket{\phi}$ is an arbitrary state that can be expanded in a basis as $\ket{\phi}=\int\Pi_{I}\left[d\mu(x_{I})dE_{I}\right]d^{2}x_{M}\phi(x_{1},\cdots,x_{N},x_{M},E_{1},\cdots,E_{N})\ket{x_{1},\cdots,x_{N},x_{M}}\ket{E_{1},\cdots,E_{N}},$ (7) with $d\mu(x_{I})=\sqrt{g_{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}d^{2}x_{I}$ being the covariant integration measure and $\ket{x_{I}}=\ket{x_{I}^{0},\mathbf{x}_{I}}$. The states $\ket{E_{I}}$ are the eigenstates of the internal hamiltonian of each particle $I$, such that $\hat{H}_{I}\ket{E_{I}}=E_{I}\ket{E_{I}}$. The state $\ket{\phi}$, in general, is not a solution of the constraint. However, the state that solves the constraint $\hat{C}\ket{\Psi}_{ph}=0$ can be obtained via the improper projection $\hat{P}\ket{\phi}=\ket{\Psi}_{ph}$ defined in Eq. (6), where $\hat{P}=\int d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{I}\hat{C}_{I}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}$. The procedure to obtain the state $\ket{\Psi}_{ph}$ involves a technical subtlety, due to the fact that $\hat{P}^{2}\neq\hat{P}$. For instance, in situations in which the spectrum of the constraint is continuous around zero, the solution the set of states satisfying the constraint is empty. However, this problem can be solved, e.g., by redefining the inner product in order to renormalise the state $\ket{\Psi}_{ph}$ (for details and different approaches to the solution see, e.g., [57, 58, 59, 60]). In order to take the perspective of a specific particle, e.g., particle $1$, we need to first map the phase-space observables to the relational observables to particle $1$, and then enforce that particle $1$ is in the origin of the reference frame via a projection operation. This procedure extends the technique introduced in Ref. [26, 27] to our model for $N$ relativistic particles in a weak gravitational field. Overall, we obtain the relational state from the perspective of particle $1$ to be $\ket{\psi}^{(1)}={}_{1}\bra{q_{1}=0}\hat{\mathcal{T}}_{1}\ket{\Psi}_{ph},$ (8) where we choose $\hat{\mathcal{T}}_{1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}^{0}-\hat{p}^{1}_{0}\right)},$ (9) and we have defined the positions after the transformation $\hat{\mathcal{T}}_{1}$ as $q_{I}$. The operator $\hat{\mathcal{T}}_{1}$ maps the initial positions to the relative positions to particle $1$. Its construction can be intuitively explained by distinguishing two parts of the operator. The two rightmost terms map the spacetime coordinates $\hat{x}_{\ell}$, with $\ell=2,\cdots,N,M$ to the relative coordinates to particle $1$, i.e., $\hat{x}_{\ell}-\hat{x}_{1}\mapsto\hat{x}_{\ell}$. The last term on the left, which is a function of the metric, sets the metric field to a locally inertial metric field at the location of particle $1$. In order to achieve this, the metric at the location of particle $1$ is mapped to the Minkowski metric, and the motion of the other particles is encoded in a new constraint $\hat{C}^{\prime}_{i}=\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\sqrt{\frac{g_{00}(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}}\hat{k}_{0}^{i}-\hat{\omega}_{k}^{i},$ (10) where $i=2,\cdots,N$, $\hat{q}_{i}$, $\hat{q}_{M}$ being the (spacetime) position operators of the particles relative to particle $1$ in the new coordinate system and $\hat{k}_{i}$, $\hat{k}_{M}$ the momentum operators that are canonically conjugated respectively to $\hat{q}_{i}$ and $\hat{q}_{M}$, i.e., $\left[\hat{q}_{i}^{\mu},\hat{k}_{\nu}^{j}\right]=\mathrm{i}\hbar\,\delta^{\mu}_{\nu}\,\delta_{ij}$ and $\left[\hat{q}_{M}^{\mu},\hat{k}_{\nu}^{M}\right]=\mathrm{i}\hbar\,\delta^{\mu}_{\nu}$ (see Appendix D for the complete action of the operator $\hat{\mathcal{T}}_{1}$ on the phase-space operators). We hence find that $g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})=\hat{\mathcal{T}}_{1}\frac{g_{00}(\hat{\mathbf{x}}_{i}-\hat{\mathbf{x}}_{M})}{g_{00}(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M})}\hat{\mathcal{T}}_{1}^{\dagger},$ (11) is the new metric field in the perspective of particle $1$. In summary, the transformation $\hat{\mathcal{T}}_{1}$ realises a transformation to the Quantum Locally Inertial Frame (QLIF) centred in particle $1$, in the spirit of Ref. [37]. A different choice of the operator $\hat{\mathcal{T}}_{1}$ is possible, and would correspond to a different set of relational variables (and coordinate system). By performing a lengthy calculation (which is detailed in Appendix E), it is possible to show that the state of Eq. (8) can be cast as $\ket{\psi}^{(1)}=\int d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$ (12) where $\ket{\tau_{1}}$ is the state of the internal clock $1$ such that, given the internal time operator $\hat{T}_{I}$, $\hat{T}_{I}\ket{\tau_{I}}=\tau_{I}\ket{\tau_{I}}$ and $\left[\hat{T}_{I},\hat{H}_{J}\right]=\mathrm{i}\hbar\delta_{IJ}$, the Hamiltonian $\hat{H}^{(1)}$ is $\hat{H}^{(1)}=\hat{\gamma}_{\Sigma p,1}\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left\\{c\hat{\omega}_{k}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right\\}+c\hat{\gamma}_{\Sigma k,1}\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}+m_{1}c^{2}\hat{\gamma}_{\Sigma k,1}^{2},$ (13) with $\hat{\gamma}_{\Sigma k,1}=\sqrt{1+\frac{\left(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}}$, $\hat{\gamma}_{i}=\sqrt{1+\frac{\hat{\mathbf{k}}_{i}^{2}}{m_{i}^{2}c^{2}}}$. Finally, the state $\ket{\psi^{(1)}_{0}}$ formally plays the role of an initial state, whose relation with the state $\ket{\phi}$ is given in Appendix E. The state of Eq. (12) can be interpreted as a “history state”, namely a state that associates to any arbitrary time eigenstate $\ket{\tau_{1}}$ read in the QRF of particle $1$ a state $\ket{\psi^{(1)}_{\tau_{1}}}=e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}$ of the remaining particles at the time $\tau_{1}$ read by the clock. A further projection on the state of the clock $1$, i.e., $\braket{\tau_{1}}{\psi^{(1)}}$, recovers the standard description of quantum mechanics. From the expression in Eq. (13), we see that the Hamiltonian of the particles $i=2,\cdots,N$ is affected by the presence of particle $1$. On the one hand, this is because the metric field at the location of the other particles, in the QLIF of particle $1$, is a quantum operator depending also on the initial value of the metric field at the location of particle $1$, as it is clear by looking at Eq. (11). This is an expression of the relational character of general relativity, extended to when we consider quantum particles as QRFs: the metric field in each QRF is defined purely in terms of relational quantities (operators) of the particles and the reference frame. On the other hand, there is also a contribution from the special-relativistic time-dilation operator. This contribution is due to the motion of particle $1$ and appears as the operator $\hat{\gamma}_{\Sigma k,1}$. The interpretation and the operational consequences of this result are discussed in Section 4. In this Section, we have outlined the method to reduce to the perspective of one of the particle in the general model that we have introduced. However, this model has a well-defined limit to the Galilean free-particle case, the special relativistic case, and the Newtonian case of slowly-moving particles. In all these cases, a similar procedure to what is described in this Section can be defined. All these limiting cases are detailed in Appendix F. ## 3 Extended symmetries and QRF changes The construction we have introduced in the previous section is compatible with an extended notion of spacetime symmetries. This means that every quantum particle can equally serve as a QRF, and that no preferred QRF is singled out. This is easy to see by noticing that the procedure we have followed to transform to the QLIF of particle $1$ can be repeated if we take another particle, i.e., particle $2$, as the QRF. In this case, we obtain the same result, but with all indices $1$ and $2$ swapped. The “history state” from the point of view of particle $2$ is then $\ket{\psi}^{(2)}=\int d\tau_{2}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(2)}\tau_{2}}\ket{\psi^{(2)}_{0}}\ket{\tau_{2}},$ (14) where all quantities are defined as in the previous section, and $\hat{H}^{(2)}=\hat{\gamma}_{\Sigma u,2}\sum_{k\neq 2}\sqrt{g^{\prime}_{00}(\hat{\mathbf{r}}_{k},\hat{\mathbf{r}}_{M})}\left\\{c\hat{\omega}_{u}^{k}+\hat{\gamma}_{u}^{-1}\hat{H}_{k}\right\\}+c\hat{\gamma}_{\Sigma u,2}\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\hat{u}_{0}^{M}+m_{2}c^{2}\hat{\gamma}_{\Sigma u,2}^{2},$ (15) with $\hat{\gamma}_{\Sigma u,2}=\sqrt{1+\frac{\left(\sum_{k\neq 2}\hat{\mathbf{u}}_{k}+\hat{\mathbf{u}}_{M}\right)^{2}}{m_{2}^{2}c^{2}}}$. Here, $\hat{r}_{k}$, with $k=1,3,\cdots,N$, and $\hat{r}_{M}$ are the position operators in the QRF of particle $2$, and $\hat{u}_{k}$, $\hat{u}_{M}$ their conjugate momenta. This construction then induces an invertible transformation from the reduced “history state” of Eq. (8) to the reduced “history state” of Eq. (14) which amounts to changing the QRF from the particle $1$ to the particle $2$. It is then clear by construction, and can also be checked by direct computation, that the state of Eq. (14) is obtained from Eq. (8) via the operation $\ket{\psi}^{(2)}={}_{2}\bra{r_{2}=0}\hat{\mathcal{T}}_{12}\ket{\psi}^{(1)}\otimes\ket{p_{1}=0}_{1},$ (16) where $\hat{\mathcal{T}}_{12}=\hat{\mathcal{T}}_{2}\hat{\mathcal{T}}_{1}^{\dagger}$ (see Appendix D for the action of this operator on the phase space operators). This construction parallels the one introduced in Ref. [26]. We have thus found that the Hamiltonian from the perspective of particle $1$ is mapped to a Hamiltonian that is form-invariant, but where all labels $1$ and $2$ are swapped, by a reversible QRF transformation. Thus, we have generalised the notion of an extended symmetry introduced in Ref. [25] to the full spacetime picture and quantum relativistic particles in a weak gravitational field. ## 4 Gravitational redshift and relativistic time-dilation from a QRF a) [scale=0.4]MeasurementSup b) [scale=0.4]SupTime Figure 2: We depict the particular situation in which particle $2$ ($C_{2}$ in the picture) evolves in a superposition of two semi-classical trajectories from the perspective of particle $1$ ($C_{1}$ in the picture). In the most general case, the state of particle $2$ from the point of view of particle $1$ is arbitrary. We describe a measurement performed at time $\tau_{2}^{}$ in the frame of clock $2$ from the point of view of $C_{1}$. a) The measurement, which is localised in time in the frame of $C_{2}$, appears delocalised in time in the frame $C_{1}$. In particular, it is performed in a superposition of proper times $\tau_{1}^{\prime}$ and $\tau_{1}^{\prime\prime}$ in the proper time of $C_{1}$. The two times $\tau_{1}^{\prime}$ and $\tau_{1}^{\prime\prime}$ are related to $\tau_{2}^{}$ by the expressions $\tau_{1}^{\prime}=\Delta_{12}^{-1}(W_{1})\tau_{2}^{}$ and $\tau_{1}^{\prime\prime}=\Delta_{12}^{-1}(W_{2})\tau_{2}^{}$, where $\Delta_{12}^{-1}(W_{i})$, $i=1,2$, encodes the special-relativistic time dilation or the gravitational redshift evaluated on the worldline $W_{i}$. b) In general, $C_{1}$ “sees” the clock $C_{2}$ as ticking in a superposition of times, depending on the state of the external degrees of freedom. Specifically, this effect can be understood as a superposition of special- relativistic time dilations, when the particles move at relativistic velocities in a Minkowski background, or as a superposition of gravitational redshifts, when the particles move slowly in a weak gravitational field. We now restrict our consideration to $N=2$ particles and a source mass $M$ and introduce the possibility of performing a measurement in the QRF of particle $2$. Analogously to the measurement procedure of Ref. [36], we modify the constraint $\hat{f}^{0}$ as $\begin{split}&\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\hat{\omega}_{p}^{I}\qquad\text{for}\qquad I=1,2;\\\ &\hat{f}_{Q}^{0}=\hat{p}^{1}_{0}+\hat{p}^{2}_{0}+\hat{p}_{0}^{M}+\Delta(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{1})\frac{\hat{H}_{1}}{c}+\Delta(\hat{\mathbf{x}}_{2}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{2})\left[\frac{\hat{H}_{2}}{c}+\delta(\hat{T}_{2}-\tau_{2})\frac{\hat{Q}_{2}}{c}\right];\\\ &\hat{f}^{1}=\hat{\mathbf{p}}_{1}+\hat{\mathbf{p}}_{2}+\hat{\mathbf{p}}_{M},\end{split}$ (17) where $\hat{Q}_{2}$ is an observable commuting with every constraint111Notice that this condition can be relaxed by adding an auxiliary quantum state as a measurement device. For the sake of simplicity, we do not consider this case.. The same procedure adopted in Section 2 can be used to calculate the history state, with the only _caveat_ that the operator $\hat{\mathcal{T}}_{1}$ needs to be modified to $\hat{\mathcal{T}}_{Q,1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}_{Q}^{0}-\hat{p}^{1}_{0}\right)}.$ (18) With a similar procedure to the one adopted in Section 2, we can calculate the physical state $\ket{\Psi}_{ph}$ and the history state $\ket{\psi}^{(1)}={}_{1}\bra{q_{1}=0}\hat{\mathcal{T}}_{Q,1}\ket{\Psi}_{ph}$. The calculations are detailed in Appendix G, and the resulting history state is $\ket{\psi}^{(1)}=\int d\tau_{1}\overleftarrow{T}\left\\{e^{-\frac{\mathrm{i}}{\hbar}\int_{0}^{\tau_{1}}ds\left[\hat{H}^{(1)}+\Delta_{12}\delta(\hat{T}_{2}+\Delta_{12}s-\tau_{2}^{})\hat{Q}_{2}\right]}\right\\}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$ (19) where $\overleftarrow{T}$ denotes the time-ordering operator, defined as $\overleftarrow{T}[\hat{O}(t_{1})\hat{O}(t_{2})]=\hat{O}(t_{1})\hat{O}(t_{2})$ if $t_{1}>t_{2}$ and $\overleftarrow{T}[\hat{O}(t_{1})\hat{O}(t_{2})]=\hat{O}(t_{2})\hat{O}(t_{2})$ if $t_{2}>t_{1}$ for $\hat{O}(t_{1})$, $\hat{O}(t_{2})$ being any two arbitrary operators. We have also defined $\Delta_{12}=\frac{\Delta(\hat{\mathbf{q}}_{2}-\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{2})}{\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\hat{\gamma}_{\Sigma k,1}^{-1}}=\sqrt{\frac{g_{00}(\hat{\mathbf{q}}_{2}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}}\frac{\hat{\gamma}_{2}^{-1}}{\hat{\gamma}_{\Sigma k,1}^{-1}},$ (20) as the “worldline operator” of particle $2$ in the perspective of clock $1$. Notice that the metric field in Eq. (20) coincides with the metric $g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})$ in the QLIF of particle $1$, which was derived in Section 2. The Hamiltonian $\hat{H}^{(1)}$ was defined in Eq. (13), and $\ket{\psi^{(1)}_{0}}$ is explicitly calculated in Appendix G. In order to understand how a measurement happening at time $\tau_{2}^{}$ in the frame of particle $2$ is seen in the QLIF of clock $1$, we restrict our attention to the Galilean, special relativistic, and Newtonian cases, detailed in Appendix F. We want to study the case in which the clock $2$ is an ideal clock, which corresponds to the requirement that the initial state $\ket{\psi_{0}^{(1)}}$ is sharp in the clock $2$ time variable. Mathematically, this condition can be expressed as $\psi_{0}^{(1)}(\mathbf{p}_{2},p_{M},t_{2})\propto\delta(t_{2}-t_{2}^{})\psi_{0}^{(1)}(\mathbf{p}_{2},p_{M})$. In the following, we choose $t_{2}^{}=0$, equivalent to the initial synchronisation of clocks $1$ and $2$. Our results are summarised in Fig. 2. In the Galilean case of Appendix F.1, there is no mass $M$ sourcing the gravitational field, and the special relativistic effects are not present. Hence, $\Delta_{12}$ reduces to the identity operator. We can rewrite the state of Eq. (19) as $\ket{\psi}^{(1)}=\int_{-\infty}^{\tau_{2}^{}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}}+\int^{\infty}_{\tau_{2}^{}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}(\tau_{1}-\tau_{2}^{})}e^{-\frac{\mathrm{i}}{\hbar}\hat{Q}_{2}}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{2}^{}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}}.$ (21) In this case we see that, in the QRF of clock $1$, the measurement happens at the same time $\tau_{2}^{}$ as it would in the QRF of clock $2$. Let us consider now the special-relativistic case described in Appendix F.2. In this case, we take into account the special relativistic effects, but we do not have a mass $M$ sourcing the gravitational field. The operator $\Delta_{12}$ reduces to $\Delta_{12}=\hat{\gamma}_{\Sigma k,1}/\hat{\gamma}_{2}\approx\sqrt{1+\frac{\hat{\mathbf{k}}_{2}^{2}}{m_{1}^{2}c^{2}}-\frac{\hat{\mathbf{k}}_{2}^{2}}{m_{2}^{2}c^{2}}}$. The history state is $\begin{split}\ket{\psi}^{(1)}=\int d\mathbf{k}_{2}dt_{2}&\left\\{\int_{-\infty}^{\tau_{k}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\psi_{0}^{(1)}(\mathbf{k}_{2},t_{2})\ket{\mathbf{k}_{2},t_{2}}+\right.\\\ +&\left.\int^{\infty}_{\tau_{k}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}(\tau_{1}-\tau_{k})}e^{-\frac{\mathrm{i}}{\hbar}\hat{Q}_{2}}\psi_{\tau_{k}}^{(1)}(\mathbf{k}_{2},t_{2})\ket{\mathbf{k}_{2},t_{2}}\right\\}\ket{\tau_{1}},\end{split}$ (22) where $\ket{\psi_{\tau_{k}}^{(1)}}=\int dt_{2}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{k}}\psi_{0}^{(1)}(\mathbf{k}_{2},t_{2})\ket{\mathbf{k}_{2},t_{2}}$ and $\tau_{k}=\Delta_{12}^{-1}(\mathbf{k}_{2})\tau_{2}^{}$. From the previous equation we see that the measurement happening at time $\tau_{2}^{}$ in the QRF of clock $2$ is seen by clock $1$ in a quantum superposition of special-relativistic time dilations. The relation to the usual time-dilation can easily be understood by comparison with the classical case by considering two particles, $1$ and $2$, which measure the same interval in coordinate time with their own proper time. In this case, the relation $dx^{0}=d\tau_{1}\gamma_{1}=d\tau_{2}\gamma_{2}$ is precisely the classical equivalent of what we found when we consider two clocks $1$ and $2$ in their own QRF. Here, for each value $\mathbf{k}_{2}$ of the momentum of particle $2$, there is a well-defined value of the time dilation, which coincides with the classical one. However, since the relational degrees of freedom between the QRF and particle $2$ are in a quantum superposition of momentum eigenstates, we obtain a coherent superposition of special-relativistic time dilations, which implies that clock $1$ sees the measurement as a delocalised event in time. These results are in agreement with those presented in Refs. [40, 43] in the case of a quantum relativistic particle with a sharp momentum, or in a superposition of two momenta222Formally, each sharp momentum state $\ket{p^{}}$ is described in Refs. [40, 43] by a gaussian wavepacket centred in the momentum $p^{}$, and following a semi-classical trajectory., and generalise them to arbitrary quantum states of the particles. Finally, we consider the Newtonian case described in Appendix F.3. In this case, we take into account the gravitational effects due to the presence of the mass $M$, but we do not consider the special-relativistic effects. Hence, $\Delta_{12}=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})}=\sqrt{\frac{g_{00}(\hat{\mathbf{q}}_{2}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}}$ and the history state can be written as $\begin{split}\ket{\psi}^{(1)}=\int d\mathbf{q}_{2}d^{2}q_{M}dt_{2}&\left\\{\int_{-\infty}^{\tau_{q}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\psi_{0}^{(1)}(\mathbf{q}_{2},q_{M},t_{2})\ket{\mathbf{q}_{2},q_{M},t_{2}}+\right.\\\ +&\left.\int^{\infty}_{\tau_{q}}d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}(\tau_{1}-\tau_{q})}e^{-\frac{\mathrm{i}}{\hbar}\hat{Q}_{2}}\psi_{\tau_{q}}^{(1)}(\mathbf{q}_{2},q_{M},t_{2})\ket{\mathbf{q}_{2},q_{M},t_{2}}\right\\}\ket{\tau_{1}},\end{split}$ (23) where $\ket{\psi_{\tau_{q}}^{(1)}}=\int dt_{2}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{q}}\psi_{0}^{(1)}(\mathbf{q}_{2},q_{M}t_{2})\ket{\mathbf{q}_{2},q_{M},t_{2}}$ and $\tau_{q}=\Delta_{12}^{-1}(\mathbf{q}_{2},\mathbf{q}_{M})\tau_{2}^{}$. Analogously to the special-relativistic case, we have that the measurement happening at time $\tau_{2}^{}$ in the QRF of clock $2$ is delocalised in the proper time of clock $1$. However, the effect in this case is a quantum superposition of gravitational redshifts, as can be easily seen by recalling the standard expression of the gravitational redshift $\tau_{obs}=\sqrt{\frac{g_{00}(x_{obs})}{g_{00}(x_{em})}}\tau_{em}$, where $\tau_{obs}$ and $x_{obs}$ are respectively the proper time and spacetime coordinates of the observer, and $\tau_{em}$ and $x_{em}$ are respectively the proper time and spacetime coordinates of the emitter. If we identify the observer with clock $1$ and the emitter with clock $2$, we recover the relation $\tau_{q}=\Delta_{12}^{-1}(\mathbf{q}_{2},\mathbf{q}_{M})\tau_{2}^{}$. Interestingly, this result can be equivalently obtained in the initial set of coordinates, where $\Delta_{12}=\sqrt{\frac{g_{00}(\hat{\mathbf{x}}_{2}-\hat{\mathbf{x}}_{M})}{g_{00}(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M})}}$, or knowing that it is always possible to transform to a QLIF where the metric is locally inertial at the location of clock $1$. In this case, it is enough to calculate $\Delta_{12}$ using the metric in the QLIF, i.e., $\Delta_{12}=\sqrt{\frac{g_{00}(x_{em})}{g_{00}(x_{obs})}}=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})}$, because $g_{00}(x_{obs})=1$ in the QLIF. The two effects discussed here, namely the _superposition of special- relativistic time dilations_ and the _superposition of gravitational redshifts_ are, in principle observable. A promising experimental system to carry out tests on the generalisation of these concepts could be atom interferometry [61]. An interesting consequence of the delocalisation of the measurement that we have described is that, if we fix some time $\tau_{1}=\tau_{1}^{}$, clock $1$ in general “sees” a superposition of the measurement having taken place or not. Hence, it seems that whether a measurement has occurred or not is a QRF- dependent feature. It is, however, an open question whether this would allow for a change of the causal relations. Previous work on QRFs [36] and on causal reference frames [62] suggests that the causal relations are preserved under change of time reference frame, but other approaches [44] might allow for a change of the causal relations. The time delocalisation of events has been studied in Ref. [36]. However, in that case the mechanism causing the delocalisation was the interaction between the internal degrees of freedom of the clocks, or the choice of a clock with an unsharp initial state. Here, the delocalisation happens due to the fact that the external degrees of freedom of the particle are in a quantum state, which is unsharp in the relevant basis (momentum in the special-relativistic case, and position in the Newtonian case). Other authors have also studied, with different techniques to the one that is developed here, the consequences of proper time running in a superposition due to the external degrees of freedom being in a quantum superposition of positions or momenta [53, 54, 56, 43, 63]. However, these results rely on an external and classical reference frame. Here, we generalise these effects to QRFs in a quantum relationship with each other, and show that they are purely due to the relation between the QRF and a physical system that is described. Furthermore, we show here that regardless of the state of the particle taken as a QRF in the global model, it is always possible to find a QLIF in which, locally, the metric field is flat and the particle is at rest. This result corroborates the generalisation of the Einstein Equivalence Principle of Ref. [37] and provides a concrete model which describes the QRF change including multiple particles. In particular, this model is suitable to describe an experimental test, such as an interferometric experiment [64], to verify the superposition of gravitational redshifts and the extension of the Einstein Equivalence Principle. It would be interesting to generalise this result to more general metric fields and to a superposition of metric fields as considered in Ref. [37], where only the dynamics of a single system was considered. ## 5 Discussion In this work, we have introduced the notion of a spacetime quantum reference frame, i.e., a reference frame in spacetime associated to a quantum system which can be in a superposition, or entangled from the point of view of another quantum system. We have developed a formulation to describe a set of relativistic quantum particles in a weak gravitational field from the perspective of such spacetime quantum reference frame. In order to achieve this, we developed a covariant formulation describing both the quantum state associated to the position/momentum of the particles and its internal state. The former is used to fix the spacetime quantum reference frame, and the latter is used as the internal clock of the quantum reference frame ticking according to its own proper time. This formulation allows us to compare the dynamical evolution in the proper time of different quantum clocks, when the state of their external variables (position or momentum) is in a quantum superposition from the perspective of one of them. We describe such clocks as being attached to a quantum particle evolving with a Hamiltonian operator, and not having a (semi)classical worldline. We find a transformation to the Quantum Locally Inertial Frame of such clocks, introduced in Ref. [37], and we describe a _superposition of special-relativistic time dilations_ and a _superposition of gravitational redshifts_ from the perspective of such spacetime quantum reference frames. We adopt a timeless formulation, according to which the global state of $N$ particles appears “frozen”, but the dynamical evolution is recovered in terms of relational quantities of the particles and the quantum reference frame. Formally, we achieve this by imposing a set of constraints on the model, some of which encode the free evolution of the particles and treat space and time on the same footing, and others impose total energy and momentum conservation. The model that we derive also includes the Page-Wootters mechanism for non- interacting clocks in Refs. [34, 35, 36] as a particular case, when the external degrees of freedom are neglected. This model is suitable to be extended to more general situations. Here, we have only considered a description in $1+1$ dimensions, and a more general model in $3+1$ dimensions will probably have to face technical difficulties such as those solved in Ref. [27]. For instance, it would be interesting to explore the possibility of imposing as a constraint the total angular momentum tensor in spacetime, i.e., $M_{\mu\nu}$, with $\mu,\nu=0,\cdots,3$, rather than just its spatial components, as done in Ref. [27]. Another possible generalisation involves the description of (gravitationally) interacting systems. This will likely require a different way of handling the constraints, which would no longer commute (and would thus be second-class constraints). This generalisation, if successful, would allow us to fully incorporate the interacting clocks model, i.e., the description of time reference frames [36] in the spacetime description of quantum reference frames. In addition, it would be interesting to generalise the model to arbitrary spacetimes and superpositions thereof and build the transformation to a Quantum Locally Inertial Frame as in Ref. [37], in the general case. Finally, we notice that the internal energies enter the model presented here in the total energy balance. This means that they are considered as part of the zero component of the total momentum. In other works, for instance in Refs. [53, 56, 65, 43], the internal energies contribute instead to the total mass, via the relativistic mass-energy equivalence. These two ways of considering the role of the internal energies, as the zero-component of a vector and as a relativistically invariant quantity are, in principle, different. However, the results presented here are in agreement, to our order of approximation, to those one obtains by considering the internal energy as part of the total mass via the mass-energy equivalence. It would be interesting to investigate the connection between these two methods further, in order to gain a deeper insight on the role of the internal degrees of freedom when both quantum and gravitational effects are relevant. At the interface of quantum theory and gravity, the notion of a classical spacetime is no longer adequate to describe physical phenomena. It is then crucial to generalise the formulation of the laws of physics to situations in which spacetime is non-classical. One of the main goals of the research on quantum reference frames is to generalise the laws of physics to when reference frames are associated to quantum systems. Here, we have taken a step further and we have associated a reference frame in spacetime to a quantum system. Generalisations of this approach to more general scenarios could pave the way for formulating physics on a quantum spacetime. I would like to thank Lin-Qing Chen, Thomas D. Galley, and Lee Smolin for useful discussions at the early stages of this work. I am also grateful to Achim Kempf, Lorenzo Maccone, and Alexander R. H. Smith for helpful comments on the draft. Research at Perimeter Institute is supported in part by the Government of Canada through the Department of Innovation, Science and Economic Development and by the Province of Ontario through the Ministry of Colleges and Universities. ## Appendix A Mass dispersion relation in a weak gravitational field In the Newtonian limit, the metric field is described as in Eq. (2), i.e., $\begin{split}&g_{00}=1+\frac{2\Phi(\mathbf{x}-\mathbf{x}_{M})}{c^{2}};\\\ &g_{01}=g_{10}=0;\\\ &g_{11}=-1,\end{split}$ (24) where $\Phi(\mathbf{x})$ is the Newtonian potential due to the mass $m_{M}$ of the system $M$ and $\|\Phi(\mathbf{x})\|/c^{2}\ll 1$ in the spacetime region considered. Hence, the general-relativistic dispersion relation $\tilde{C}_{I}=g^{\mu\nu}p_{\mu}p_{\nu}-m_{I}^{2}c^{2}$ of a classical particle $I$ with mass $m_{I}$ takes the form $\tilde{C}_{I}=g^{00}(\mathbf{x}_{I}-\mathbf{x}_{M})p_{0}^{I2}-\mathbf{p}_{I}^{2}-m_{I}^{2}c^{2}.$ (25) We impose the positive energy condition by enforcing that $p^{I}_{0}\geq 0$333Notice that this condition can only be enforced in the weak-field limit of the gravitational field, i.e., when $\|\Phi(\mathbf{x})\|/c^{2}\ll 1$.. By noting that $\tilde{C}_{I}$ can be decomposed as $\tilde{C}_{I}=\left[\sqrt{g^{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}p^{I}_{0}-\sqrt{\mathbf{p}_{I}^{2}+m_{I}^{2}c^{2}}\right]\left[\sqrt{g^{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}p^{I}_{0}+\sqrt{\mathbf{p}_{I}^{2}+m_{I}^{2}c^{2}}\right],$ (26) we see that, if the energies are to be positive, it is enough to consider the new dispersion relation $C_{I}=\sqrt{g^{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}p^{I}_{0}-\sqrt{\mathbf{p}_{I}^{2}+m_{I}^{2}c^{2}}.$ (27) Clearly, this relation only holds in the low-energy regime of the particle, hence it holds perturbatively in $\frac{\mathbf{p}_{I}^{2}}{m_{I}^{2}c^{2}}$. It is now straightforward to quantise Eq. (27) and obtain the constraint $\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\sqrt{\hat{\mathbf{p}}_{I}^{2}+m_{I}^{2}c^{2}}.$ (28) If we now define $\hat{\omega}^{I}_{p}=\sqrt{\hat{\mathbf{p}}_{I}^{2}+m_{I}^{2}c^{2}}$, we find the constraint $\hat{C}_{I}$ of Eq. (3). Notice that the weak-field approximation $\|\Phi(\mathbf{x})\|/c^{2}\ll 1$ played a crucial role here to avoid ordering ambiguities in the quantisation procedure, as a more general form of the metric field would also have position-dependent spatial components, which, when quantised, do not commute with their conjugate momentum. It is easy to show that the expression $\bra{x_{I}^{0}}\hat{C}_{I}\ket{\psi_{I}}_{ph}=0$, where $\ket{\psi_{I}}_{ph}$ is the quantum state of particle $I$ solving the constraint $\hat{C}_{I}$, is equivalent to $\mathrm{i}\hbar\frac{d}{dx_{I}^{0}}\ket{\psi_{I}(x^{0}_{I})}=\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{\omega}_{p}^{I}\ket{\psi_{I}(x^{0}_{I})},$ (29) where $\ket{\psi_{I}(x^{0}_{I})}=\braket{x_{I}^{0}}{\psi_{I}}_{ph}$. By defining $x^{0}_{I}=ct_{I}$ and expanding perturbatively, we find $\mathrm{i}\hbar\frac{d}{dt_{I}}\ket{\psi_{I}(t_{I})}=\left[m_{I}c^{2}+\frac{\hat{\mathbf{p}}_{I}^{2}}{2m_{I}}-\frac{\hat{\mathbf{p}}_{I}^{4}}{8m_{I}^{3}c^{2}}+m_{I}\Phi(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})\right]\ket{\psi_{I}(t_{I})},$ (30) where we do not consider higher-order terms. The equation above corresponds to Schrödinger equation of a particle in a Newtonian field, with first-order special-relativistic corrections. ## Appendix B Comparison between relational approaches In the literature, there are several, and in principle different, ways of eliminating an external, absolute structure. For instance, the Page-Wootters mechanism [34, 35] considers a set of clocks which, at the global level, satisfy a Hamiltonian constraint as in Eq. (1). Upon conditioning on the state of one of the clocks, one recovers the dynamics of the other clocks. This is usually interpreted as the internal perspective of the clock on which one conditions. The Page-Wootters mechanism was applied to interacting clocks in Refs. [40, 36]. In particular, in the case of gravitationally interacting clocks studied in Ref. [36] it was shown that the interaction term between the internal degrees of freedom of the clocks leads to a relative temporal localisation of events, due to the fact that different internal energy states contribute differently to the total mass of the system. In addition, the limits to the measurability of time studied in Ref. [65] could be recovered. In a different approach to relational dynamics, it was shown in Refs. [26, 27] that the formalism for QRFs introduced in Ref. [25] can be derived by imposing a symmetry principle on a set of $N$ particles, and specifically by imposing invariance under global translations and global rotations. This symmetry principle is reminiscent of the construction of Shape Dynamics [66], which also inspired the construction of a classical model in which a gravitational arrow of time emerges [67]. The above-mentioned techniques have been derived in different frameworks and are, in principle, different. However, a connection between different approaches to relational dynamics can be found, as shown in Refs. [41, 42] for the case of the Page-Wootters mechanism, relational Dirac observables, and quantum deparametrisation. ## Appendix C Worldline operator of a quantum relativistic particle in a weak gravitational field Let us consider a classical relativistic particle of mass $m$ in a weak gravitational field. The classical equivalent of the constraint considered in the main text (up to factors of order $O(c^{-2})$, is $C_{I}=\sqrt{g^{00}(\mathbf{x}_{I})}p_{0}^{I}-m_{I}c\sqrt{1+\frac{\mathbf{p}_{I}^{2}}{m^{2}_{I}c^{2}}},$ (31) where $\mathbf{p}_{I}^{2}$ is the spatial norm of the momentum. Via the Hamilton’s equations of motion we have $\dot{x}_{I}^{\mu}=\frac{dx_{I}^{\mu}}{d\tau}=\frac{\partial C_{I}}{\partial p^{I}_{\mu}}$, which yields $\begin{split}&\dot{x}_{I}^{0}=\sqrt{g^{00}(\mathbf{x}_{I})},\\\ &\dot{\mathbf{x}}_{I}=\frac{\mathbf{p}_{I}}{m_{I}c}\left(1+\frac{\mathbf{p}_{I}^{2}}{m_{I}^{2}c^{2}}\right)^{-1/2}.\end{split}$ (32) Hence, the line element is, to order $O(c^{-2})$, $\begin{split}ds&=\sqrt{g_{00}(\mathbf{x}_{I})-\frac{1}{c^{2}}\left(\frac{d\mathbf{x}_{I}}{d\tau}\right)^{2}\left(\frac{d\tau}{dx^{0}_{I}}\right)^{2}}dx^{0}_{I}=\\\ &=\sqrt{g_{00}(\mathbf{x}_{I})}\sqrt{1+\frac{\mathbf{p_{I}}^{2}}{m^{2}_{I}c^{2}}}dx_{I}^{0}.\end{split}$ (33) The quantisation of this expression is straightforward, given that in the regime we consider we have $\left[g_{00}(\hat{\mathbf{x}}),\sqrt{1+\frac{\hat{\mathbf{p}}_{I}^{2}}{m_{I}^{2}c^{2}}}\right]=0$. We then write the “worldline operator” as $\Delta(\hat{\mathbf{x}}_{I},\hat{\mathbf{p}}_{I})=\sqrt{g_{00}(\mathbf{\hat{x}}_{I})}\sqrt{1+\frac{\hat{\mathbf{p}}_{I}^{2}}{m_{I}^{2}c^{2}}}.$ (34) ## Appendix D Action of the $\mathcal{\hat{T}}_{1}$ operator on the position and momentum operators In this Appendix, we calculate the action of the operator $\mathcal{\hat{T}}_{1}$ on the phase space operators of the quantum particle in spacetime. We have $\begin{split}&\mathcal{\hat{T}}_{1}\hat{x}_{1}^{0}\mathcal{\hat{T}}_{1}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{q}_{1}^{0},\hskip 10000.0pt\\\ &\mathcal{\hat{T}}_{1}\hat{\mathbf{x}}_{1}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{q}}_{1},\\\ &\mathcal{\hat{T}}_{1}\hat{x}_{i}^{0}\mathcal{\hat{T}}_{1}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\left(\hat{q}_{i}^{0}+\hat{q}_{1}^{0}\right),\\\ &\mathcal{\hat{T}}_{1}\hat{\mathbf{x}}_{i}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{q}}_{i}+\hat{\mathbf{q}}_{1},\\\ &\mathcal{\hat{T}}_{1}\hat{x}_{M}^{0}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{q}_{M}^{0}+\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{q}_{1}^{0},\\\ &\mathcal{\hat{T}}_{1}\hat{\mathbf{x}}_{M}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{q}}_{M}+\hat{\mathbf{q}}_{1},\end{split}$ $\begin{split}&\mathcal{\hat{T}}_{1}\hat{p}^{1}_{0}\mathcal{\hat{T}}_{1}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\Big{(}\hat{k}^{1}_{0}-\sum_{i\neq 1}\hat{k}_{0}^{i}\Big{)}-\sum_{i\neq 1}\Delta(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{i})\frac{\hat{H}_{i}}{c}-\hat{k}_{0}^{M}-\Delta\Big{(}\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{1}-\sum_{i\neq 1}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M}\Big{)}\frac{\hat{H}_{1}}{c},\\\ &\mathcal{\hat{T}}_{1}\hat{\mathbf{p}}_{1}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{1}-\sum_{i\neq 1}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M},\\\ &\mathcal{\hat{T}}_{1}\hat{p}^{i}_{0}\mathcal{\hat{T}}_{1}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\hat{k}^{i}_{0},\\\ &\mathcal{\hat{T}}_{1}\hat{\mathbf{p}}_{i}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{i},\\\ &\mathcal{\hat{T}}_{1}\hat{p}^{M}_{0}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{k}^{M}_{0},\\\ &\mathcal{\hat{T}}_{1}\hat{\mathbf{p}}_{M}\mathcal{\hat{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{M},\end{split}$ (35) where $i=2,\cdots,N$. We can also define a new transformation $\mathcal{\hat{T}}_{2}$ to the relational variables of particle $2$ by swapping the labels $1$ and $2$ in the transformation $\mathcal{\hat{T}}_{1}$. We define the relational phase-space operators to particle $2$ as $\hat{r}_{j}$ (position operator) and $\hat{u}_{j}$ (momentum operator), for $j=1,3,\cdots,N,M$, and we obtain $\begin{split}&\mathcal{\hat{T}}_{2}\hat{x}_{2}^{0}\mathcal{\hat{T}}_{2}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\hat{r}_{2}^{0},\hskip 10000.0pt\\\ &\mathcal{\hat{T}}_{2}\hat{\mathbf{x}}_{2}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{r}}_{2},\\\ &\mathcal{\hat{T}}_{2}\hat{x}_{j}^{0}\mathcal{\hat{T}}_{2}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\left(\hat{r}_{j}^{0}+\hat{r}_{2}^{0}\right),\\\ &\mathcal{\hat{T}}_{2}\hat{\mathbf{x}}_{j}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{r}}_{j}+\hat{\mathbf{r}}_{2},\\\ &\mathcal{\hat{T}}_{2}\hat{x}_{M}^{0}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{r}_{M}^{0}+\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\hat{r}_{2}^{0},\\\ &\mathcal{\hat{T}}_{2}\hat{\mathbf{x}}_{M}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{r}}_{M}+\hat{\mathbf{r}}_{2},\\\ \end{split}$ $\begin{split}&\mathcal{\hat{T}}_{2}\hat{p}^{2}_{0}\mathcal{\hat{T}}_{2}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{r}}_{M})}\Big{(}\hat{u}^{2}_{0}-\sum_{j\neq 2}\hat{u}^{j}_{0}\Big{)}-\sum_{j\neq 2}\Delta(\hat{\mathbf{r}}_{j}-\hat{\mathbf{r}}_{M},\hat{\mathbf{u}}_{j})\frac{\hat{H}_{j}}{c}-\hat{u}_{0}^{M}-\Delta\Big{(}\hat{\mathbf{r}}_{M},\hat{\mathbf{u}}_{2}-\sum_{j\neq 2}\hat{\mathbf{u}}_{j}-\hat{\mathbf{u}}_{M}\Big{)}\frac{\hat{H}_{1}}{c}\\\ &\mathcal{\hat{T}}_{2}\hat{\mathbf{p}}_{2}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{u}}_{2}-\sum_{j\neq 2}\hat{\mathbf{u}}_{j}-\hat{\mathbf{u}}_{M}\\\ &\mathcal{\hat{T}}_{2}\hat{p}^{j}_{0}\mathcal{\hat{T}}_{2}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{r}}_{M})}\hat{u}^{j}_{0}\\\ &\mathcal{\hat{T}}_{2}\hat{\mathbf{p}}_{j}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{u}}_{j}\\\ &\mathcal{\hat{T}}_{2}\hat{p}^{M}_{0}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{u}^{M}_{0}\\\ &\mathcal{\hat{T}}_{2}\hat{\mathbf{p}}_{M}\mathcal{\hat{T}}_{2}^{\dagger}=\hat{\mathbf{u}}_{M},\end{split}$ (36) where $j=1,3,\cdots,N$. We can also transform from one set of relational phase-space operators to another, e.g., from particle $1$ to particle $2$. In order to do this, we define $\mathcal{\hat{T}}_{12}=\mathcal{\hat{T}}_{2}\mathcal{\hat{T}}_{1}^{\dagger}$ and find $\begin{split}&\mathcal{\hat{T}}_{12}\hat{q}_{1}^{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{\frac{g_{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g_{00}(\hat{\mathbf{r}}_{M})}}\left(\hat{r}_{1}^{0}+\hat{r}_{2}^{0}\right),\hskip 10000.0pt\\\ &\mathcal{\hat{T}}_{12}\hat{\mathbf{q}}_{1}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{r}}_{1}+\hat{\mathbf{r}}_{2},\\\ &\mathcal{\hat{T}}_{12}\hat{q}_{2}^{0}\mathcal{\hat{T}}_{12}^{\dagger}=-\sqrt{\frac{g_{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g_{00}(\hat{\mathbf{r}}_{M})}}\hat{r}_{1}^{0},\\\ &\mathcal{\hat{T}}_{12}\hat{\mathbf{q}}_{2}\mathcal{\hat{T}}_{12}^{\dagger}=-\hat{\mathbf{r}}_{1},\\\ &\mathcal{\hat{T}}_{12}\hat{q}_{\ell}^{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{\frac{g_{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g_{00}(\hat{\mathbf{r}}_{M})}}(\hat{r}_{\ell}^{0}-\hat{r}_{1}^{0}),\\\ &\mathcal{\hat{T}}_{12}\hat{\mathbf{q}}_{\ell}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{r}}_{\ell}-\hat{\mathbf{r}}_{1},\\\ &\mathcal{\hat{T}}_{12}\hat{q}_{M}^{0}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{r}_{M}^{0}-\sqrt{g^{00}(\hat{\mathbf{r}}_{M})}\hat{r}_{1}^{0},\\\ &\mathcal{\hat{T}}_{12}\hat{\mathbf{q}}_{M}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{r}}_{M}-\hat{\mathbf{r}}_{1},\end{split}$ $\begin{split}&\mathcal{\hat{T}}_{12}\hat{k}^{1}_{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{g^{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}\sqrt{g_{00}(\hat{\mathbf{r}}_{M})}\hat{u}^{2}_{0},\hskip 10000.0pt\\\ &\mathcal{\hat{T}}_{12}\hat{\mathbf{k}}_{1}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{u}}_{2},\\\ &\mathcal{\hat{T}}_{12}\hat{k}^{2}_{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{\frac{g^{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g^{00}(\hat{\mathbf{r}}_{M})}}\Big{(}\hat{u}^{2}_{0}-\sum_{j\neq 2}\hat{u}_{0}^{j}\Big{)}-\sqrt{g^{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}\left[\sum_{j\neq 2}\Delta_{j}\frac{\hat{H}_{j}}{c}+\hat{u}_{0}^{M}+\Delta_{\Sigma u,2}\frac{\hat{H}_{2}}{c}\right],\\\ &\mathcal{\hat{T}}_{12}\hat{\mathbf{k}}_{2}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{u}}_{2}-\sum_{j\neq 2}\hat{\mathbf{u}}_{j}-\hat{\mathbf{u}}_{M},\\\ &\mathcal{\hat{T}}_{12}\hat{k}^{\ell}_{0}\mathcal{\hat{T}}_{12}^{\dagger}=\sqrt{\frac{g^{00}(\hat{\mathbf{r}}_{1}-\hat{\mathbf{r}}_{M})}{g^{00}(\hat{\mathbf{r}}_{M})}}\hat{u}^{\ell}_{0},\\\ &\mathcal{\hat{T}}_{12}\hat{\mathbf{k}}_{\ell}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{u}}_{\ell},\\\ &\mathcal{\hat{T}}_{12}\hat{k}^{M}_{0}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{u}^{M}_{0},\\\ &\mathcal{\hat{T}}_{12}\hat{\mathbf{k}}_{M}\mathcal{\hat{T}}_{12}^{\dagger}=\hat{\mathbf{u}}_{M},\end{split}$ (37) where $\ell=3,\cdots,N$ and $\Delta_{j}=\Delta(\hat{\mathbf{r}}_{j}-\hat{\mathbf{r}}_{M},\hat{\mathbf{u}}_{j})$ and $\Delta_{\Sigma u,2}=\Delta\Big{(}\hat{\mathbf{r}}_{M},\hat{\mathbf{u}}_{2}-\sum_{j\neq 2}\hat{\mathbf{u}}_{j}-\hat{\mathbf{u}}_{M}\Big{)}$. Notice that the relevant relational operators are those of the particles $2,\cdots,N,M$ from the perspective of particle $1$, which are mapped into quantities which only depend on the operators of particle $2$ via $\hat{u}_{2}$, which is the transformed constraint, since $\hat{\mathcal{T}}_{2}\hat{f}^{0}\hat{\mathcal{T}}_{2}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{r}}_{M})}\hat{u}^{0}_{2}$ and $\hat{\mathcal{T}}_{2}\hat{f}^{1}\hat{\mathcal{T}}_{2}^{\dagger}=\hat{\mathbf{u}}_{2}$. ## Appendix E Explicit calculation of the “history state” in the frame of particle 1 We recover in this Appendix the dynamical evolution of the relational degrees of freedom from the state of Eq. (6) $\ket{\Psi}_{ph}\propto\int d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$ (38) where $\ket{\phi}$ is represented as $\ket{\phi}=\int\Pi_{I}\left[d\mu(x_{I})dE_{I}\right]d^{2}x_{M}\phi(x_{1},\cdots,x_{N},x_{M},E_{1},\cdots,E_{N})\ket{x_{1},\cdots,x_{N},x_{M}}\ket{E_{1},\cdots,E_{N}},$ (39) and $d\mu(x_{i})=\sqrt{g_{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}d^{2}x_{I}$ is the covariant integration measure. The set of constraints that we enforce in this model is $\begin{split}&\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\hat{\omega}_{p}^{I},\qquad\qquad I=1,\cdots,N\\\ &\hat{f}^{0}=\sum_{I=1}^{N}\left[\hat{p}^{I}_{0}+\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})\frac{\hat{H}_{I}}{c}\right]+\hat{p}_{0}^{M};\\\ &\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I}+\hat{\mathbf{p}}_{M},\end{split}$ (40) where $\Delta(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{I})=\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\left(1+\frac{\hat{\mathbf{p}}^{2}_{I}}{m_{I}^{2}c^{2}}\right)^{-1/2}$. In order to find the dynamical evolution, we define an operator $\hat{\mathcal{T}}_{1}$ which maps the observables to the relational observables from the perspective of particle $1$. We define the operator $\hat{\mathcal{T}}_{1}$ as $\hat{\mathcal{T}}_{1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}^{0}-\hat{p}^{1}_{0}\right)},$ (41) where $\hat{f}^{1}-\hat{\mathbf{p}}^{1}=\sum_{i}\hat{\mathbf{p}}_{i}+\hat{\mathbf{p}}_{M}$, $\hat{f}^{0}-\hat{p}^{1}_{0}=\sum_{i}\left[\hat{p}_{0}^{i}+\Delta(\hat{\mathbf{x}}_{i}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{i})\frac{\hat{H}_{i}}{c}\right]+\Delta(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{1})\frac{\hat{H}_{1}}{c}+\hat{p}_{0}^{M}$ and the lowercase latin letters label all particles except for the particle serving as the QRF, i.e., $i=2,\cdots,N$. The action of the operator $\hat{\mathcal{T}}_{1}$ on the constraints is $\begin{split}\hat{\mathcal{T}}_{1}\hat{f}^{0}\hat{\mathcal{T}}_{1}^{\dagger}=&\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\hat{k}^{1}_{0};\qquad\hat{\mathcal{T}}_{1}\hat{f}^{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{1};\qquad\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{C}^{\prime}_{i}\,\,\,i\neq 1;\\\ \hat{\mathcal{T}}_{1}\hat{C}_{1}\hat{\mathcal{T}}_{1}^{\dagger}=&\hat{k}_{0}^{1}-\left\\{\sum_{i}\left[\hat{k}_{0}^{i}+\Delta^{\prime}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{i})\frac{\hat{H}_{i}}{c}\right]+\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}\right\\}+\\\ &-m_{1}c\sqrt{1+\frac{\left(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}}-\left(1+\frac{\left(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}\right)^{-1/2}\frac{\hat{H}_{1}}{c},\end{split}$ (42) where $\hat{C}^{\prime}_{i}$ has been defined in the main text as $\hat{C}^{\prime}_{i}=\sqrt{g^{\prime 00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{i}-\hat{\omega}_{k}^{i}$, $\Delta^{\prime}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M},\hat{\mathbf{k}}_{i})=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left(1+\frac{\hat{\mathbf{k}}_{i}^{2}}{m_{i}^{2}c^{2}}\right)^{-1/2}$, and the transformed metric field is $g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})=\frac{g_{00}(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}.$ (43) The relational state from the perspective of system $1$ is then obtained as $\ket{\psi}^{(1)}=\bra{q_{1}=0}\hat{\mathcal{T}}_{1}\ket{\Psi}_{ph}$ By defining the internal time operator $\hat{T}_{I}$ such that $\left[\hat{T}_{I},\hat{H}_{J}\right]=i\hbar\delta_{IJ}$ and the state of the internal clock of each particle $\ket{\tau_{I}}$ as the one satisfying the relation $\hat{T}_{I}\ket{\tau_{I}}=\tau_{I}\ket{\tau_{I}}$, we find $\ket{\psi}^{(1)}=2\pi\hbar\left(1-\frac{\Phi(\mathbf{\hat{q}}_{M})}{c^{2}}\right)\int d^{N}\mathcal{N}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}^{\prime}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\frac{\hat{\gamma}_{\Sigma k,1}^{-1}}{c}\left(\hat{K}^{(1)}+\hat{H}_{1}\right)\mathcal{N}_{1}}\bra{q_{1}=0}\hat{\Pi}_{0}\mathcal{\hat{T}}_{1}\ket{\phi},$ (44) where $\hat{K}^{(1)}$ is the operator encoding the relational dynamics of particles $2,\cdots,N$ from the point of view of particle $1$, i.e., $\hat{K}^{(1)}=\hat{\gamma}_{\Sigma k,1}\left[\sum_{i}c\hat{k}_{0}^{i}+c\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}\right]+\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\frac{\hat{\gamma}_{i}^{-1}}{\hat{\gamma}_{\Sigma k,1}^{-1}}\hat{H}_{i}+m_{1}c^{2}\hat{\gamma}_{\Sigma k,1}^{2},$ (45) $\hat{\gamma}_{i}=\sqrt{1+\frac{\hat{\mathbf{k}}_{i}^{2}}{m_{i}^{2}c^{2}}}$, $\hat{\gamma}_{\Sigma k,1}=\sqrt{1+\frac{\left(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}}$, and $\hat{\Pi}_{0}=\ket{k_{1}=0}\bra{k_{1}=0}$. We now define $\ket{\phi_{/1}}=2\pi\hbar\left(1-\frac{\Phi(\mathbf{\hat{q}}_{M})}{c^{2}}\right)\bra{q_{1}=0}\hat{\Pi}_{0}\mathcal{\hat{T}}_{1}\ket{\phi}$ (46) and act with the operator $\hat{H}_{1}$ on the internal state of clock $1$. We then rewrite $\ket{\psi}^{(1)}=\int d\tau_{1}d^{N-1}\mathcal{N}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}^{\prime}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}}\ket{\tau_{1}},$ (47) where $\begin{split}\ket{\phi^{(1)}_{0}}=&\int dt_{1}\Pi_{i}[d^{2}k_{i}dE_{i}]d^{2}k_{M}e^{\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}t_{1}}\phi_{/1}(k_{2},\cdots,k_{N},k_{M},t_{1},E_{2},\cdots,E_{N})\times\\\ &\times\ket{k_{2},\cdots,k_{N},k_{M}}\ket{E_{2},\cdots,E_{N}}.\end{split}$ (48) Following a similar procedure to the one outlined above, it is possible solve all the constraints and to cast the previous expression as a “history state” in the sense of Refs. [36, 37] evolving unitarily as $\ket{\psi}^{(1)}=\int d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$ (49) where the Hamiltonian $\hat{H}^{(1)}$ is $\hat{H}^{(1)}=\hat{\gamma}_{\Sigma k,1}\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left\\{c\hat{\omega}_{k}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right\\}+c\hat{\gamma}_{\Sigma k,1}\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}+m_{1}c^{2}\hat{\gamma}_{\Sigma k,1}^{2},$ (50) and $\begin{split}\ket{\psi^{(1)}_{0}}=&\int\Pi_{i}\left(d\mu(q_{i})dq^{\prime 0}_{i}dE_{i}\sqrt{g^{\prime}_{00}(\mathbf{q}_{i},\mathbf{q}_{M})}\right)d^{2}q_{M}e^{\frac{\mathrm{i}}{\hbar}\sum_{i}(q^{\prime 0}_{i}-q^{0}_{i})\sqrt{g^{\prime}_{00}(\mathbf{q}_{i},\mathbf{q}_{M})}\hat{\omega}_{k}^{i}}\times\\\ &\times\phi^{(1)}_{0}\left(q_{2},\cdots,q_{N},q_{M},E_{2},\cdots,E_{N}\right)\ket{q^{\prime 0}_{2},\mathbf{q}_{2},\cdots,q^{\prime 0}_{N},\mathbf{q}_{N}}\ket{q_{M}}\ket{E_{2}\cdots E_{N}},\end{split}$ (51) which reproduces the result of the main text. The generalisation of this technique to general gravitational fields requires dealing with second-class constraints, and is thus beyond the scope of this work. ## Appendix F Limiting cases of the general $N$-particle model In the following, we are going to study some relevant limits of the model introduced in the main text. In particular, we will focus on a set of quantum free particles moving slowly compared to the speed of light (Galilean case), on a set of quantum special-relativistic particles, and on a set of quantum Galilean particles in an external weak gravitational field (Newtonian case). We remind the reader that operators and vectors with no indices are, unless differently specified, operators and two-vectors in spacetime, while the spatial component of operators and vectors is boldface. Greek letters label spacetime indices, capital Latin letters label all the particles, e.g., $I=1,\cdots,N$, and lowercase Latin letters label all the particles except the one serving as the QRF, e.g., $i=2,\cdots,N$. ### F.1 Galilean case When the particles move slowly compared to the speed of light and are free, the contraints introduced in the main text reduce to $\displaystyle\hat{C}_{I}=\hat{p}_{I}^{0}-\frac{\hat{\mathbf{p}}^{2}_{I}}{2m_{I}c},$ (52) $\displaystyle\hat{f}^{0}=\sum_{I=1}^{N}\left(\hat{p}_{I}^{0}+\frac{\hat{H}_{I}}{c}\right),$ (53) $\displaystyle\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I},$ (54) where here we have neglected particle $M$ because it plays no role. In addition, we eliminated the constant term $m_{I}c$ in the expansion of $\hat{C}_{I}$, because it simply amounts to a rescaling of the zero component of the momentum. We write the state in the Physical Hilbert space as $\ket{\Psi}_{ph}\propto\int d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$ (55) where $\ket{\phi}$ is expanded in momentum basis as $\ket{\phi}=\int\Pi_{I}\left[d^{2}p_{I}dE_{I}\right]\phi(p_{1},\cdots,p_{N},E_{1},\cdots,E_{N})\ket{p_{1},\cdots,p_{N}}\ket{E_{1},\cdots,E_{N}}.$ (56) In this case, the operator $\hat{\mathcal{T}}_{1}$ simplifies to $\hat{\mathcal{T}}_{1}=e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\sum_{i}\hat{\mathbf{p}}_{i}}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left[\sum_{i}\left(\hat{p}_{0}^{i}+\frac{\hat{H}_{i}}{c}\right)+\frac{\hat{H}_{1}}{c}\right]},$ (57) and the constraints are mapped to $\begin{split}&\hat{\mathcal{T}}_{1}\hat{f}^{0}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}^{1}_{0};\qquad\hat{\mathcal{T}}_{1}\hat{f}^{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{1};\qquad\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{C}_{i}\,\,\,i\neq 1;\\\ &\hat{\mathcal{T}}_{1}\hat{C}_{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}_{0}^{1}-\sum_{i}\left(\hat{k}_{0}^{i}+\frac{\hat{H}_{i}}{c}\right)-\frac{\hat{H}_{1}}{c}-\frac{\left(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i}\right)^{2}}{2m_{1}}.\end{split}$ (58) Defining the relational state from the perspective of particle $1$ to be $\ket{\psi^{(1)}}={}_{1}\bra{q_{1}=0}\mathcal{T}_{1}\ket{\Psi}_{ph}$, we find $\ket{\psi}^{(1)}=\int d\tau_{1}d^{N-1}\mathcal{N}\ket{\tau_{1}}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}},$ (59) where $\hat{K}^{(1)}=\sum_{i}\left(\hat{k}_{0}^{i}+\frac{\hat{H}_{i}}{c}\right)+\frac{\left(\sum_{i}\hat{\mathbf{k}}_{i}\right)^{2}}{2m_{1}},$ (60) and $\begin{split}&\ket{\phi^{(1)}_{0}}=\frac{1}{\sqrt{2\pi\hbar}}\int dt_{1}dE_{1}\Pi_{i}(d^{2}k_{i}dE_{i})e^{\frac{\mathrm{i}}{\hbar}(\hat{K}^{(1)}+E_{1})t_{1}}\phi\left(\bar{k}_{1},k_{2},\cdots,k_{N},E_{1},\cdots,E_{N}\right)\times\\\ &\hskip 56.9055pt\times\ket{k_{2}\cdots k_{N}}\ket{E_{2}\cdots E_{N}}.\end{split}$ (61) By solving the constraint we have rewritten the two-vector $\bar{k}_{1}=\left(-\sum_{i}\left(k_{0}^{i}+\frac{E_{i}}{c}\right)-\frac{E_{1}}{c},-\sum_{i}\mathbf{k}_{i}\right)$. Notice that we can easily recover the framework of standard quantum mechanics by integrating over the $N_{i}$ variables. After this operation, the state $\ket{\psi^{(1)}}$ can be rewritten as the “history state” $\ket{\psi}^{(1)}=\int d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$ (62) where $\displaystyle\hat{H}^{(1)}=\sum_{i}\frac{\hat{\mathbf{k}}_{i}^{2}}{2m_{i}}+\frac{(\sum_{{i}}\hat{\mathbf{k}}_{i})^{2}}{2m_{1}}+\sum_{i}\hat{H}_{i},$ $\displaystyle\begin{split}&\ket{\psi^{(1)}_{0}}=\int\Pi_{i}(d\mathbf{k}_{i}dE_{i})\phi^{(1)}_{0}\left(\frac{\mathbf{k}_{2}^{2}}{2m_{2}},\mathbf{k}_{2},\cdots,\frac{\mathbf{k}_{N}^{2}}{2m_{N}},\mathbf{k}_{N},E_{1},\cdots,E_{N}\right)\times\\\ &\hskip 56.9055pt\times\ket{\frac{\mathbf{k}_{2}^{2}}{2m_{2}},\mathbf{k}_{2},\cdots,\frac{\mathbf{k}_{N}^{2}}{2m_{N}},\mathbf{k}_{N}}\ket{E_{2}\cdots E_{N}}.\end{split}$ ### F.2 Special-relativistic case In this section, we consider a set of N special-relativistic particles moving freely. The set of constraints is $\displaystyle\hat{C}_{I}=\hat{p}_{I}^{0}-\hat{\omega}_{p}^{I},$ (63) $\displaystyle\hat{f}^{0}=\sum_{I=1}^{N}\left(\hat{p}_{I}^{0}+\hat{\gamma}_{I}^{-1}\frac{\hat{H}_{I}}{c}\right),$ (64) $\displaystyle\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I},$ (65) where we have defined $\hat{\omega}_{p}^{I}=m_{I}c^{2}\sqrt{1+\frac{\hat{\mathbf{p}}_{I}^{2}}{m_{I}^{2}c^{2}}}$ and $\hat{\gamma}_{I}=\sqrt{1+\frac{\hat{\mathbf{p}}_{I}^{2}}{m_{I}^{2}c^{2}}}$. We write the state in the Physical Hilbert space as $\ket{\Psi}_{ph}\propto\int d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$ (66) where $\ket{\phi}$ is expanded in momentum basis as $\ket{\phi}=\int\Pi_{I}\left[d^{2}p_{I}dE_{I}\right]\phi(p_{1},\cdots,p_{N},E_{1},\cdots,E_{N})\ket{p_{1},\cdots,p_{N}}\ket{E_{1},\cdots,E_{N}},$ (67) In this case, the trivialisation operator is $\hat{\mathcal{T}}_{1}=e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\sum_{i}\hat{\mathbf{p}}_{i}}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left[\sum_{i}\left(\hat{p}_{0}^{i}+\hat{\gamma}_{i}^{-1}\frac{\hat{H}_{i}}{c}\right)+\,\hat{\gamma}_{1}^{-1}\frac{\hat{H}_{1}}{c}\right]},$ (68) and the constraints are mapped to $\begin{split}&\hat{\mathcal{T}}_{1}\hat{f}^{0}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}^{1}_{0};\qquad\hat{\mathcal{T}}_{1}\hat{f}^{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{\mathbf{k}}_{1};\qquad\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{C}_{i}\,\,\,i\neq 1;\\\ &\hat{\mathcal{T}}_{1}\hat{C}_{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}_{0}^{1}-\sum_{i}\left(\hat{k}_{0}^{i}+\hat{\gamma}_{i}^{-1}\frac{\hat{H}_{i}}{c}\right)-\left(1+\frac{(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i})^{2}}{m_{1}^{2}c^{2}}\right)^{-1/2}\frac{\hat{H}_{1}}{c}-m_{1}c\sqrt{1+\frac{(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i})^{2}}{m_{1}^{2}c^{2}}}.\end{split}$ (69) Following analogous steps to the Galilean case, we define $\ket{\psi}^{(1)}=\bra{q_{1}=0}\hat{\mathcal{T}}_{1}\ket{\Psi}_{ph}$, where $\ket{\psi}^{(1)}=\int d\tau_{1}d^{N-1}\mathcal{N}\ket{\tau_{1}}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}},$ (70) where $\hat{K}^{(1)}=\hat{\gamma}_{\Sigma k,1}\left[\sum_{i}\left(c\hat{k}_{0}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right)+c\hat{\omega}_{\Sigma k}^{1}\right],$ (71) and $\begin{split}&\ket{\phi^{(1)}_{0}}=\frac{1}{\sqrt{2\pi\hbar}}\int\Pi_{i}(d^{2}k_{i}dE_{i})dt_{1}dE_{1}\gamma_{\Sigma k,1}e^{\frac{\mathrm{i}}{\hbar}(\hat{K}^{(1)}+E_{1})t_{1}}\phi\left(\bar{k}_{1},k_{2},\cdots,k_{N},E_{1},\cdots,E_{N}\right)\times\\\ &\hskip 56.9055pt\times\ket{k_{2}\cdots k_{N}}\ket{E_{2}\cdots E_{N}}.\end{split}$ (72) Here, we have defined $\hat{\gamma}_{\Sigma k,1}=\sqrt{1+\frac{(\sum_{i}\hat{\mathbf{k}}_{i})^{2}}{m_{1}^{2}c^{2}}}$ and $\hat{\omega}_{\Sigma k}^{1}=m_{1}c\sqrt{1+\frac{(\sum_{i}\hat{\mathbf{k}}_{i})^{2}}{m_{1}^{2}c^{2}}}$ and the two-vector $\bar{k}_{1}=\left(-\sum_{i}\left(k_{0}^{i}+\gamma_{i}^{-1}\frac{E_{i}}{c}\right)-\gamma_{\Sigma k,1}^{-1}\frac{E_{1}}{c},-\sum_{i}\mathbf{k}_{i}\right)$. It is again possible, with a similar procedure to the Galilean case, to recover the dynamical evolution of a quantum relativistic particle. We find $\ket{\psi}^{(1)}=\int d\tau_{1}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$ (73) where $\displaystyle\hat{H}^{(1)}=\hat{\gamma}_{\Sigma k,1}\left[\sum_{i}\left(c\hat{\omega}_{k}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right)+c\hat{\omega}_{\Sigma k}^{1}\right],$ $\displaystyle\begin{split}&\ket{\psi^{(1)}_{0}}=\int\Pi_{i}(d\mathbf{k}_{i}dE_{i})\phi^{(1)}_{0}\left(\omega_{k}^{2},\mathbf{k}_{2},\cdots,\omega_{k}^{N},\mathbf{k}_{N},E_{2},\cdots,E_{N}\right)\times\\\ &\hskip 56.9055pt\times\ket{\omega_{k}^{2},\mathbf{k}_{2},\cdots,\omega_{k}^{N},\mathbf{k}_{N}}\ket{E_{2}\cdots E_{N}}.\end{split}$ Note that the factor $\sqrt{1+\frac{\left(\sum_{i\neq 1}\hat{k}_{i}\right)^{2}}{m_{1}^{2}c^{2}}}$ multiplying the hamiltonian of the $N-1$ particles from the point of view of particle $1$ is the special- relativistic time dilation due to the fact that the QRF is moving in a superposition of special relativistic velocities. ### F.3 Newtonian case Finally, we consider the case in which a set of $N$ particles move in a Newtonian gravitational field produced by a mass $M$. The gravitational field in the Newtonian limit is the same as in the main text $\begin{split}&g_{00}=1+\frac{2\Phi(\mathbf{x}-\mathbf{x}_{M})}{c^{2}};\\\ &g_{01}=g_{10}=0;\\\ &g_{11}=-1,\end{split}$ (74) where $\Phi(\mathbf{x})$ is the Newtonian potential due to the mass $m_{M}$ of the system $M$ and $\|\Phi(\mathbf{x})\|/c^{2}\ll 1$ in the spacetime region considered. The set of constraints for this situation are $\begin{split}&\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-m_{I}c-\frac{\hat{\mathbf{p}}^{2}_{I}}{2m_{I}c}\qquad\text{for}\qquad I=1,\cdots,N;\\\ &\hat{f}^{0}=\sum_{I=1}^{N}\left[\hat{p}^{I}_{0}+\sqrt{g_{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\frac{\hat{H}_{I}}{c}\right]+\hat{p}_{0}^{M};\\\ &\hat{f}^{1}=\sum_{I=1}^{N}\hat{\mathbf{p}}_{I}+\hat{\mathbf{p}}_{M}.\end{split}$ (75) Notice that, since this is a special case of the case considered in the main text, the constraints are still first-class to our order of approximation. The state in the Physical Hilbert space is $\ket{\Psi}_{ph}\propto\int d^{N}\mathcal{N}d^{2}ze^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}_{i}}e^{\frac{\mathrm{i}}{\hbar}z_{\mu}\hat{f}^{\mu}}\ket{\phi},$ (76) where, in this case, it is convenient to expand $\ket{\phi}$ in position basis as $\ket{\phi}=\int\Pi_{I}\left[d\mu(x_{I})dE_{I}\right]d^{2}x_{M}\phi(x_{1},\cdots,x_{N},x_{M},E_{1},\cdots,E_{N})\ket{x_{1},\cdots,x_{N},x_{M}}\ket{E_{1},\cdots,E_{N}},$ (77) where $d\mu(x_{I})=\sqrt{g_{00}(\mathbf{x}_{I}-\mathbf{x}_{M})}d^{2}x_{I}$ is the covariant integration measure. We define the operator $\hat{\mathcal{T}}_{1}$ as $\hat{\mathcal{T}}_{1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}^{0}-\hat{p}^{1}_{0}\right)},$ (78) where in this case $\hat{f}^{1}-\hat{\mathbf{p}}^{1}=\sum_{i}\hat{\mathbf{p}}_{i}+\hat{\mathbf{p}}_{M}$ and $\hat{f}^{0}-\hat{p}^{1}_{0}=\sum_{i}\left[\hat{p}_{0}^{i}+\sqrt{g_{00}(\hat{\mathbf{x}}_{i}-\hat{\mathbf{x}}_{M})}\frac{\hat{H}_{i}}{c}\right]+\sqrt{g_{00}(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M})}\frac{\hat{H}_{1}}{c}+\hat{p}_{0}^{M}$. The action of the operator $\hat{\mathcal{T}}_{1}$ on the constraints is $\begin{split}&\hat{\mathcal{T}}_{1}\hat{f}^{0}\hat{\mathcal{T}}_{1}^{\dagger}=\sqrt{g_{00}(\hat{\mathbf{q}}_{M})}\hat{k}^{1}_{0};\qquad\hat{\mathcal{T}}_{1}\hat{f}^{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{\mathbf{k}}^{1};\qquad\hat{\mathcal{T}}_{1}\hat{C}_{i}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{C}^{\prime}_{i}\,\,\,i\neq 1;\\\ &\hat{\mathcal{T}}_{1}\hat{C}_{1}\hat{\mathcal{T}}_{1}^{\dagger}=\hat{k}_{0}^{1}-\left[\sum_{i}\left(\hat{k}_{0}^{i}+\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\frac{\hat{H}_{i}}{c}\right)+\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}\right]+\\\ &\hskip 56.9055pt- m_{1}c-\frac{(\hat{\mathbf{k}}_{1}-\sum_{i}\hat{\mathbf{k}}_{i}-\hat{\mathbf{k}}_{M})^{2}}{2m_{1}c}-\frac{\hat{H}_{1}}{c},\end{split}$ (79) where $\hat{C}^{\prime}_{i}=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\hat{k}^{i}_{0}-m_{i}c-\frac{\hat{\mathbf{k}}^{2}_{i}}{2m_{i}c}$ and $g^{\prime 00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})=\frac{g_{00}(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})}{g_{00}(\hat{\mathbf{q}}_{M})}$ as in the main text. Similarly to the other cases, the relational state from the perspective of system $1$ is $\ket{\psi}^{(1)}=\bra{q_{1}=0}\hat{\mathcal{T}}_{1}\ket{\Psi}_{ph}$. In order to find its explicit expression, we here define $\begin{split}\ket{\tilde{\phi}}=\hat{\mathcal{T}}_{1}\ket{\phi}=&\int d^{2}k_{1}\Pi_{i}\left[d\mu(q_{i})dE_{i}\right]dE_{1}d^{2}q_{M}\tilde{\phi}(k_{1},q_{2},\cdots,q_{N},q_{M},t_{1},E_{2},\cdots,E_{N})\times\\\ &\times\ket{k_{1},q_{2},\cdots,q_{N},q_{M}}\ket{t_{1}}\ket{E_{2},\cdots,E_{N}}.\end{split}$ (80) We then find $\ket{\psi}^{(1)}=\int d\tau_{1}d^{N-1}\mathcal{N}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}^{\prime}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}}\ket{\tau_{1}},$ (81) where $\begin{split}\ket{\phi^{(1)}_{0}}&=c\int dt_{1}\Pi_{i}[d\mu(q_{i})dE_{i}]d^{2}q_{M}\sqrt{g^{00}(\mathbf{q}_{M})}e^{\frac{\mathrm{i}}{\hbar}\hat{K}^{(1)}t_{1}}\times\\\ &\times\tilde{\phi}(k_{1}=0,q_{2},\cdots,q_{N},q_{M},t_{1},E_{2}\cdots,E_{N})\ket{q_{2},\cdots,q_{N},q_{M}}\ket{E_{2},\cdots,E_{N}},\end{split}$ (82) and $\hat{K}^{(1)}$ is the operator encoding the relational dynamics of particles $2,\cdots,N$ from the point of view of particle $1$, i.e., $\hat{K}^{(1)}=\sum_{i}c\hat{k}_{0}^{i}+c\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}+\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\hat{H}_{i}+m_{1}c^{2}+\frac{(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M})^{2}}{2m_{1}}.$ (83) With a similar procedure to the previous cases, we find that we can recover the usual dynamical evolution as $\ket{\psi}^{(1)}=\int d\tau_{1}\ket{\tau_{1}}e^{-\frac{\mathrm{i}}{\hbar}\hat{H}^{(1)}\tau_{1}}\ket{\psi^{(1)}_{0}},$ (84) where the relational Hamiltonian in the perspective of particle $1$ is $\hat{H}^{(1)}=\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}c\hat{k}_{0}^{M}+\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left[m_{i}c^{2}+\frac{\hat{\mathbf{k}}_{i}^{2}}{2m_{i}}+\hat{H}_{i}\right]+m_{1}c^{2}+\frac{(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M})^{2}}{2m_{1}},$ (85) and $\begin{split}&\ket{\psi^{(1)}_{0}}=\int\Pi_{i}\left[d\mu(q_{i})dq^{\prime 0}_{i}\sqrt{g^{\prime}_{00}(\mathbf{q}_{i}-\mathbf{q}_{M})}dE_{i}\right]d^{2}q_{M}e^{\frac{\mathrm{i}}{\hbar}\sum_{i}(q^{\prime 0}_{i}-q^{0}_{i})\sqrt{g^{\prime}_{00}(\mathbf{q}_{i}-\mathbf{q}_{M})}\left[m_{i}c+\frac{\hat{\mathbf{k}}_{i}^{2}}{2m_{i}c}\right]}\times\\\ &\times\phi^{(1)}_{0}\left(q_{2},\cdots,q_{N},q_{M},E_{2},\cdots,E_{N}\right)\ket{q^{\prime 0}_{2},\mathbf{q}_{2},\cdots,q^{\prime 0}_{N},\mathbf{q}_{N}}\ket{q_{M}}\ket{E_{2}\cdots E_{N}}.\end{split}$ (86) Notice that, if we expand the Hamiltonian to lowest order in the general relativistic corrections for the external degrees of freedom, while still allowing the internal degrees of freedom to have relativistic corrections, we obtain $\begin{split}\hat{H}^{(1)}=&\sum_{i}\left\\{m_{i}c^{2}+\frac{\hat{\mathbf{k}}_{i}^{2}}{2m_{i}}+m_{i}[\Phi(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})-\Phi(\hat{\mathbf{q}}_{M})]+\left[1+\frac{[\Phi(\hat{\mathbf{q}}_{i}-\hat{\mathbf{q}}_{M})-\Phi(\hat{\mathbf{q}}_{M})]}{c^{2}}\right]\hat{H}_{i}\right\\}+\\\ &+m_{1}c^{2}+\frac{(\sum_{i}\hat{\mathbf{k}}_{i}+\hat{\mathbf{k}}_{M})^{2}}{2m_{1}}+\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}c\hat{k}_{0}^{M}.\end{split}$ (87) As a result, we see that the potential in the QRF of particle $1$ is the difference between the gravitational potential between particle $i$ and the mass $M$ and particle $1$ and the mass $M$. This is the straightforward generalisation of the standard reference frame description. ## Appendix G Explicit calculation of the “history state” with a quantum measurement The calculation of the history state in the case with a measurement parallels the one outlined in Appendix E. With a fully analogous procedure, fixing $N=2$, and choosing the constraints of Eq. (17) $\begin{split}&\hat{C}_{I}=\sqrt{g^{00}(\hat{\mathbf{x}}_{I}-\hat{\mathbf{x}}_{M})}\hat{p}^{I}_{0}-\hat{\omega}_{p}^{I}\qquad\text{for}\qquad I=1,2;\\\ &\hat{f}_{Q}^{0}=\hat{p}^{1}_{0}+\hat{p}^{2}_{0}+\hat{p}_{0}^{M}+\Delta(\hat{\mathbf{x}}_{1}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{1})\frac{\hat{H}_{1}}{c}+\Delta(\hat{\mathbf{x}}_{2}-\hat{\mathbf{x}}_{M},\hat{\mathbf{p}}_{2})\left[\frac{\hat{H}_{2}}{c}+\delta(\hat{T}_{2}-\tau_{2})\frac{\hat{Q}_{2}}{c}\right];\\\ &\hat{f}^{1}=\hat{\mathbf{p}}_{1}+\hat{\mathbf{p}}_{2}+\hat{\mathbf{p}}_{M},\end{split}$ (88) where $\hat{Q}_{2}$ is an observable commuting with every constraint, and the operator $\hat{\mathcal{T}}_{Q,1}=e^{\frac{i}{\hbar}\frac{\log\sqrt{g_{00}(\hat{\mathbf{x}}_{M})}}{2}\sum_{I=1}^{N}(\hat{x}^{0}_{I}\hat{p}_{0}^{I}+\hat{p}_{0}^{I}\hat{x}^{0}_{I})}e^{\frac{\mathrm{i}}{\hbar}\hat{\mathbf{x}}_{1}\left(\hat{f}^{1}-\hat{\mathbf{p}}^{1}\right)}e^{\frac{\mathrm{i}}{\hbar}\hat{x}_{1}^{0}\left(\hat{f}_{Q}^{0}-\hat{p}^{1}_{0}\right)},$ (89) we find that the history state can be cast in the form $\ket{\psi}^{(1)}=\int d\tau_{1}d^{N-1}\mathcal{N}e^{\frac{\mathrm{i}}{\hbar}\mathcal{N}_{i}\hat{C}^{\prime}_{i}}e^{-\frac{\mathrm{i}}{\hbar}\hat{K}_{Q}^{(1)}\tau_{1}}\ket{\phi^{(1)}_{0}}\ket{\tau_{1}},$ (90) where $\hat{C}^{\prime}_{i}$ is the same as in the main text and in Appendix E. In this case, we have that $\hat{K}_{Q}^{(1)}=\hat{K}^{(1)}+\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})}\frac{\hat{\gamma}_{2}^{-1}}{\hat{\gamma}_{\Sigma k,1}^{-1}}\delta(\hat{T}_{2}-\tau_{2}^{})\hat{Q}_{2},$ (91) where $\hat{K}^{(1)}$ was defined in Eq. (45) for a general number of particles $N$ (which here we have fixed to $N=2$). We also recall that we have defined $\hat{\gamma}_{2}=\sqrt{1+\frac{\hat{\mathbf{k}}_{2}^{2}}{m_{2}^{2}c^{2}}}$ and $\hat{\gamma}_{\Sigma k,1}=\sqrt{1+\frac{\left(\hat{\mathbf{k}}_{2}+\hat{\mathbf{k}}_{M}\right)^{2}}{m_{1}^{2}c^{2}}}$. Finally, we also write $\ket{\phi^{(1)}_{0}}=c\int dt_{1}d^{2}k_{2}dt_{2}d^{2}k_{M}\gamma_{\Sigma k,1}e^{\frac{\mathrm{i}}{\hbar}\hat{K}_{Q}^{(1)}t_{1}}\phi_{/1}(k_{2},k_{M},t_{1},t_{2})\ket{k_{2},k_{M},t_{2}}$ (92) where the state $\ket{\phi_{/1}}$ has the same expression as in Eq. (46) with the operator $\hat{\mathcal{T}}_{Q,1}$ instead of $\hat{\mathcal{T}}_{1}$. We now consider the following equality, proved in Ref. [36], $e^{-\frac{\mathrm{i}}{\hbar}\alpha[\hat{H}_{I}+\hat{f}(\hat{T}_{I})]}=e^{-\frac{\mathrm{i}}{\hbar}\alpha\hat{H}_{I}}\overleftarrow{T}\left\\{e^{-\frac{\mathrm{i}}{\hbar}\int_{0}^{1}ds\alpha\hat{f}(\hat{T}_{I}+\alpha s)}\right\\},$ (93) where $[\hat{T}_{I},\hat{H}_{I}]=\mathrm{i}\hbar$ and $\hat{f}(\hat{T}_{I})$ is an operator-valued function, and we slightly generalise it to $e^{-\frac{\mathrm{i}}{\hbar}\alpha[\hat{K}_{R}+\hat{H}_{I}+\hat{f}(\hat{T}_{I})]}=e^{-\frac{\mathrm{i}}{\hbar}\alpha\hat{H}_{I}}\overleftarrow{T}\left\\{e^{-\frac{\mathrm{i}}{\hbar}\int_{0}^{1}ds\alpha\left[\hat{K}_{R}+\hat{f}(\hat{T}_{I}+\alpha s)\right]}\right\\},$ (94) where $\hat{K}_{R}$ is an arbitrary hermitian operator which does not necessarily commute with $\hat{f}(\hat{T}_{I})$. Then, the expression of Eq: (90) can be cast as $\ket{\psi}^{(1)}=\int d\tau_{1}\overleftarrow{T}\left\\{e^{-\frac{\mathrm{i}}{\hbar}\int_{0}^{\tau_{1}}ds\left[\hat{H}^{(1)}+\Delta_{12}\delta(\hat{T}_{2}+\Delta_{12}s-\tau_{2}^{})\hat{Q}_{2}\right]}\right\\}\ket{\psi^{(1)}_{0}}\ket{\tau_{1}},$ (95) where $\Delta_{12}=\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{2},\hat{\mathbf{q}}_{M})}\frac{\hat{\gamma}_{2}^{-1}}{\hat{\gamma}_{\Sigma k,1}^{-1}}$. The Hamiltonian $\hat{H}^{(1)}$ is the same as in the main text $\hat{H}^{(1)}=\hat{\gamma}_{\Sigma k,1}\sum_{i}\sqrt{g^{\prime}_{00}(\hat{\mathbf{q}}_{i},\hat{\mathbf{q}}_{M})}\left\\{c\hat{\omega}_{k}^{i}+\hat{\gamma}_{i}^{-1}\hat{H}_{i}\right\\}+c\hat{\gamma}_{\Sigma k,1}\sqrt{g^{00}(\hat{\mathbf{q}}_{M})}\hat{k}_{0}^{M}+m_{1}c^{2}\hat{\gamma}_{\Sigma k,1}^{2},$ (96) and $\begin{split}\ket{\psi^{(1)}_{0}}=&\int d\mu(q_{2})dq^{\prime 0}_{2}dt_{2}d^{2}q_{M}\sqrt{g^{\prime}_{00}(\mathbf{q}_{2}-\mathbf{q}_{M})}e^{\frac{\mathrm{i}}{\hbar}(q^{\prime 0}_{2}-q^{0}_{2})\sqrt{g^{\prime}_{00}(\mathbf{q}_{2}-\mathbf{q}_{M})}\hat{\omega}_{k}^{2}}\times\\\ &\times\phi^{(1)}_{0}\left(q_{2},q_{M},t_{2}\right)\ket{q^{\prime 0}_{2},\mathbf{q}_{2}}\ket{q_{M}}\ket{t_{2}},\end{split}$ (97) ## References Aharonov and Susskind [1967a] Yakir Aharonov and Leonard Susskind. Charge Superselection Rule. _Phys. Rev._ , 155:1428–1431, 1967a. doi: 10.1103/PhysRev.155.1428. 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figuret # Using an atom interferometer to infer gravitational entanglement generation Daniel Carney Joint Center for Quantum Information and Computer Science/Joint Quantum Institute, University of Maryland/NIST, College Park, MD 20742, USA Fermi National Accelerator Laboratory, Batavia, IL 60510, USA Holger Müller Department of Physics, University of California, Berkeley, CA 94720, USA Jacob M. Taylor Joint Center for Quantum Information and Computer Science/Joint Quantum Institute, University of Maryland/NIST, College Park, MD 20742, USA ###### Abstract If gravitational perturbations are quantized into gravitons in analogy with the electromagnetic field and photons, the resulting graviton interactions should lead to an entangling interaction between massive objects. We suggest a test of this prediction. To do this, we introduce the concept of interactive quantum information sensing. This novel sensing protocol is tailored to provable verification of weak dynamical entanglement generation between a pair of systems. We show that this protocol is highly robust to typical thermal noise sources. The sensitivity can moreover be increased both using an initial thermal state and/or an initial phase of entangling via a non-gravitational interaction. We outline a concrete implementation testing the ability of the gravitational field to generate entanglement between an atomic interferometer and mechanical oscillator. Preliminary numerical estimates suggest that near- term devices could feasibly be used to perform the experiment. ## I Introduction If a particle is in a superposition of two locations, will its gravitational field also be in a superposition, and can this field generate entanglement with another system? This foundational question feynman1971lectures ; page1981indirect has received considerable attention diosi1984gravitation ; penrose1996gravity ; kafri2015bounds ; bose2017spin ; Marletto:2017kzi ; Haine:2018bwu ; Chevalier:2020uvv ; Howl:2020isj ; Anastopoulos:2020cdp ; Matsumura:2020law ; carney2019tabletop . Proposed experimental tests to detect entanglement due to gravity based on Bell tests (or more generally, entanglement witnesses HORODECKI19961 ; terhal2000bell ) require performing measurements on both subsystems and are challenging in practice. As a result, there is still no direct experimental evidence as to whether gravitational interactions generate entanglement. Here, we propose a test that only requires observing a single subsystem feynman1963theory ; caldeira1983path ; joos1985emergence ; zurek2003decoherence . We show that, if an interaction (such as gravity) between two systems can cause both decoherence (collapse) and recoherence (revival) of a subsystem, then for restricted classes of systems the interaction is _necessarily_ capable of generating entanglement. We propose a concrete implementation based on atom interferometry kasevich1992measurement ; santarelli1999quantum ; gross2010nonlinear ; xu2019probing , in which an atom in a superpositon of being in one of two interferometer arms interacts with a low-frequency mechanical resonator lee2020new ; catano2020high ; the signal for entanglement-generation is a collapse and revival of the atomic interference fringes due to the periodic motion of the resonator. The experiment does not require preparing a non- classical state of the oscillator and can in fact be enhanced by placing the oscillator in a thermal state, which appears to make this experiment feasible with near-term devices. The relation of such an experiment to the quantization gravity is a subject of intense current study belenchia2018quantum ; Christodoulou:2018cmk ; Marshman:2019sne ; Galley:2020qsf . These experiments operate in a regime where the energy density (or equivalently, spacetime curvature), is far below the Planck scale $\rho\ll m_{\rm Pl}/\ell_{\rm Pl}^{3}\sim 10^{123}~{}{\rm eV}/{\rm cm^{3}}$. Thus the non-linearity of the gravitational interaction is very weak, and one can treat the metric $g_{\mu\nu}$ as a linear perturbation around flat spacetime. In this limit, one can quantize the gravitational perturbations (“gravitons”) in exact analogy with quantum electrodynamics; graviton exchange generates a two-body Newton potential operator $V_{N}=-\frac{G_{N}m_{1}m_{2}}{\|\mbox{\boldmath$x$}_{1}-\mbox{\boldmath$x$}_{2}\|}$ (1) between a pair of masses, just as photons generate the Coulomb potential Feynman:1963ax ; t1974one ; deser1974one ; Veltman:1975vx ; donoghue1994general ; burgess2004quantum . We review some standard demonstrations of this in appendix A. In equation (1), $\mbox{\boldmath$x$}_{1,2}$ are the position _operators_ on a pair of masses, and thus this interaction can generate entanglement. However, there are dissenting opinions penrose1996gravity ; howl2019exploring ; Tilloy:2019hxe ; Bruschi:2020xbm about whether gravity should be quantized in this way, and indeed one can produce models where classical gravitational interactions can arise but without generating entanglement Kibble:1979jn ; kafri2014classical ; Oppenheim:2018igd ; Kent:2020gov , providing substantial motivation to perform tests of (1). Figure 1: Implementation of the basic protocol using an atom interferometer and a suspended pendulum (see Section VI). A trapped atom (labeled A) is prepared some distance $L$ away from a mechanical resonator (B, here pictured as a pendulum). The atom is then put into a superposition of two different locations separated by $\ell$, effecting a Hadamard gate $H$. This generates a state-dependent force between the atoms and resonator, leading to motion in opposite directions for some time $\Delta t$. Finally, the atom state is recombined using the inverse Hadamard gate and measured to check for decoherence caused by the atom-mechanical interaction. When the resonator undergoes a complete period of motion, its state no longer depends upon the atoms and coherence is recovered for the interferometer. The ability to test such a weak entanglement signal relies entirely on our central technical result, a novel sensing protocol which we refer to as _interactive quantum information sensing_. This is a detection scheme tailored specifically to the verification of weak dynamical entanglement generation. The traditional methods to detect entanglement in bipartite systems $H_{A}\otimes H_{B}$ use non-local measurements HORODECKI19961 ; terhal2000bell , and can be very difficult in practice with noisy systems and weak entanglement. However, in the past two decades, more sophisticated methods have been developed to address these types of problems guhne2009entanglement ; pezze2018quantum . We suggest here a new protocol which relies on time-dependent measurements on a single subsystem. Within standard quantum mechanics, system $A$ will decohere—evolve from a pure to mixed state—if it becomes entangled with another system $B$ which is not measured feynman1963theory ; caldeira1983path ; joos1985emergence ; zurek2003decoherence . This loss of coherence can be observed via an interference measurement on $A$ alone. Simple decoherence could be explained by entanglement but also by, for example, random classical noise stern1990phase . However, if the same interaction can cause both decoherence and recoherence of $A$, in a manner controlled by $B$, then for certain classes of systems we prove that the interaction is necessarily capable of generating entanglement between subsystems $A$ and $B$. This protocol provides an indirect test of the quantum communication capabilities of the two systems, and is a limited probe of the family of quantum channels associated with the interaction between the two systems. The interplay between the information theoretic channel properties and the physical interaction provides our suggested nomenclature. We outline the interactive sensing protocol in sections II and III. We find the remarkable result that using an initial state at high temperature can _increase_ the sensitivity of the protocol, because it can increase the rate of entanglement generation and lead to a thermally-enhanced collapse and revival signal. In section IV we demonstrate that this conclusion is robust to typical sources of noise, essentially because the test does not involve producing large superpositions of the non-observed subsystem. In section V we show how to further enhance the protocol using pre-entangled initial conditions. Finally, we outline an experimental realization with gravitational entanglement generation between an atom interferometer and a mechanical oscillator in section VI, before concluding with a discussion of implications and loopholes in section VII. ## II Collapse and revival dynamics To begin, we illustrate the basic idea of the collapse-and-revival dynamics with an example. The setup is similar to electron spin echo envelope modulation rowan1965electron ; dikanov1992electron and the cavity QED experiments of Haroche _et al._ raimond2001manipulating . Consider an harmonic oscillator $B$ coupled to a two-state system $A$ through the Hamiltonian $H=\omega a^{\dagger}a+g(a+a^{\dagger})\sigma_{z}.$ (2) In section VI we give an implementation of this Hamiltonian where the oscillator $B$ is a mechanical resonator, the two-state system $A$ corresponds to an atom located in one of two spatial locations, and $g\ll\omega$ is set by the atom-oscillator gravitational interaction (1), so $g$ is proportional to Newton’s constant $G_{N}$. The essential idea is to do an interferometry measurement on the two-state system $A$ (the “control”) in the presence of system $B$ (the “target”). The key is the dynamical response of the target system $B$ to a superposition of $A$. (a) (b) (c) Figure 2: Equivalent circuit (left) and phase space (center) descriptions of the experiment, and schematic interferometric data output (right). In the circuit, the top line represents the atom and bottom line the resonator. The large box represents joint evolution of the trapped atom and the resonator, which can be decomposed into conditional displacement of the resonator, followed by free evolution and an inverse displacement operator. This sequence can be visualized in the phase space of the oscillator, where the solid and dashed lines represent the two oscillator evolutions conditioned on the two possible atomic locations. Interferometric measurement of the atom population will show rapid fringes with frequency $\omega_{\rm DC}$ due to any stray DC accelerations (e.g., due to electric fields, Earth’s gravity, or off-center location of the resonator or atom), modulated by an overall reduction and then increase due to the atom-resonator entanglement. Resonator motion over a full period leads to nominal full recovery of the fringes. To understand the entanglement dynamics generated by (2), it is useful to note that the time evolution operator can be re-written $U(t)=e^{-iHt}=D^{\dagger}\left(\sigma_{z}\lambda\right)e^{-i\omega a^{\dagger}at}D\left(\sigma_{z}\lambda\right)$ (3) up to an overall phase, where $D(\alpha)\equiv\exp\left\\{\alpha a^{\dagger}-\alpha^{}a\right\\}$ (4) is the usual displacement operator.111To see this, note that $D^{\dagger}(\alpha)a^{\dagger}aD(\alpha)=\|a+\alpha\|^{2}$ and expand the free evolution operator $e^{-i\omega a^{\dagger}at}$ in the middle of (3). Here and throughout, we will use the dimensionless quantity $\lambda\equiv\frac{g}{\omega}.$ (5) This is the length, measured in units of the zero-point length $x_{0}$, that the oscillator equilibrium is displaced under the force from the atom. This ratio will set the scale of all observables considered in this paper. Observing the collapse-and-revival can be done with a typical interferometric measurement. Consider starting the full system in its decoupled ground state $\ket{0}_{A}\otimes\ket{0}_{B}$. The interferometry experiment then proceeds by performing a Hadamard gate (or any other beamsplitter operation) on the two-state system $A$, $\ket{0}\to(\ket{0}+\ket{1})/\sqrt{2}$, evolving the joint system for some time $t$, performing the inverse Hadamard gate to recombine the two-level system, and then measuring its population. Mathematically, this proceeds as follows: $\displaystyle\begin{split}\ket{\psi}&=\ket{0}_{A}\otimes\ket{0}_{B}\\\ &\xrightarrow{H}\frac{\ket{0}_{A}+\ket{1}_{A}}{\sqrt{2}}\otimes\ket{0}_{B}\\\ &\xrightarrow{U_{\rm int}}\frac{\ket{0}_{A}\ket{\delta}_{B}+\ket{1}_{A}\ket{-\delta}_{B}}{\sqrt{2}}\\\ &\xrightarrow{H^{\dagger}}\ket{0}_{A}\frac{\ket{\delta}_{B}+\ket{-\delta}_{B}}{2}+\ket{1}_{A}\frac{\ket{\delta}_{B}-\ket{-\delta}_{B}}{2}.\end{split}$ (6) Here, the conditionally-evolved states of the oscillator are simply coherent states $\ket{\pm\delta}_{B}=D\left(\pm\lambda(e^{-i\omega t}-1)\right)\ket{0}.$ (7) If we now measure the two-state system $A$, we find for example that the probability of being in the $\ket{0}$ state is $P_{A}(0)=\frac{1}{2}+\frac{1}{2}{\rm Re}\braket{\delta}{-\delta}_{B}=\frac{1}{2}\left(1+e^{-8\lambda^{2}\sin^{2}(\omega t/2)}\right).$ (8) We see that the interference term is reduced, with a period set by the oscillator frequency $\omega$. In particular, at half-period we have a maximum reduction of the phase contrast, and after a full period the contrast is completely restored, as in Fig. 2. Before moving on, we mention for later use an alternative calculation of the same effect. Consider the Pauli lowering operator $\sigma_{-}=(\sigma_{x}-i\sigma_{y})/2$ on the two-level system. The expectation value $\braket{\sigma_{-}(t)}$ tracks the loss of phase contrast; we will refer to the absolute value as the interferometric visibility $V=\|\braket{\sigma_{-}}\|$. Using the time-evolution operator (3), we have $\displaystyle\begin{split}\sigma_{-}(t)&=U^{\dagger}(t)\sigma_{-}U(t)\\\ &=D^{\dagger}(-\lambda)e^{i\omega a^{\dagger}at}D(-\lambda)\sigma_{-}D^{\dagger}(\lambda)e^{-i\omega a^{\dagger}at}D(\lambda)\\\ &=\sigma_{-}D(2\lambda(1-e^{i\omega t})).\end{split}$ (9) This is easy to show by working with explicit components in the $\sigma_{z}$ basis, where $\sigma_{-}=\ket{1}\bra{0}$. With an oscillator initially in the ground state, this gives $\braket{\sigma_{-}(\pi/\omega)}=\braket{\sigma_{-}(0)}e^{-8\lambda^{2}},\ \ \ \braket{\sigma_{-}(2\pi/\omega)}=\braket{\sigma_{-}(0)}.$ (10) Here we see again the loss of phase contrast at half period followed by the revival at a full period. Up to this point, we have assumed that the oscillator was initialized in its ground state $\ket{0}$. In a realistic implementation–particularly one where the oscillator is a massive mechanical object–the oscillator will instead start in a mixed state, such as a thermal state, due to its coupling to an environment. Although one may be concerned that this would destroy the revival of coherence in the atom, it turns out that not only does the revival persist, but in fact the relative contrast between decoherence and revival is _enhanced_ so long as the thermalization time scale remains very long. That the revival persists is a consequence of the harmonic potential: after a full- period, the state of the oscillator must return to its initial condition. To see this, consider first the oscillator initialized to an arbitrary coherent state $\ket{\alpha}$. Using (9), we have $\displaystyle\begin{split}\braket{\alpha}{\sigma_{-}(t)}{\alpha}&=e^{-2\lambda[\alpha^{}(1-e^{i\omega t})-\alpha(1-e^{-i\omega t})]}\\\ &\times e^{-8\lambda^{2}\sin^{2}(\omega t/2)}\braket{\sigma_{-}(0)}.\end{split}$ (11) We see the complete revival after a full period, while at half period we now pick up a phase involving the initial oscillator momentum $p_{\alpha}=\alpha+\alpha^{}$. To obtain the thermal-state result, one can now average over the coherent states (i.e. use the oscillator density matrix $\rho_{\rm th}=\int d^{2}\alpha e^{-\|\alpha\|^{2}/\bar{n}}/(\pi\bar{n})\ket{\alpha}\bra{\alpha}$, with $\bar{n}$ the thermal phonon occupancy). The result for the qubit visibility is $V_{\rm th}(t)=\exp\left[-8\lambda^{2}(2\bar{n}+1)\sin^{2}(\omega t/2)\right].$ (12) In particular, we have $V_{\rm th}(2\pi/\omega)=1$, showing a full revival of the qubit coherence after a full oscillator period. On the other hand, at half-period, we have $V_{\rm th}(\pi/\omega)=\exp\left[-8\lambda^{2}\left(2\bar{n}+1\right)\right]$, an enhancement to the loss of visibility by a factor of $\bar{n}$. Thus, starting with a thermal state increases the contrast between the ‘dip’ of coherence halfway through oscillation and the recovery at full oscillation. The experiment is _easier_ with a hot oscillator. ## III Revival verifies entanglement generation As this example clearly shows, entanglement generation between two systems $A$ and $B$ can cause periodic collapse and revival of $A$’s wavefunction. The crucial question is then: does observation of this collapse and revival _necessarily_ require entanglement generation between $A$ and $B$? Our central result says that the answer is yes, under some particular assumptions. We characterize this with a theorem: ###### Theorem 1 Let $L$ be a channel on $H_{A}\otimes H_{B}$, where $H_{A}$ is a two-state system and $H_{B}$ is arbitrary. Assume that 1. (a) The channel $L$ generates time evolution, in a manner consistent with time translation invariance, thus obeying a semigroup composition law $L_{t\to t^{\prime\prime}}=L_{t\to t^{\prime}}L_{t^{\prime}\to t^{\prime\prime}}$ for all $t\leq t^{\prime}\leq t^{\prime\prime}$, 2. (b) The two-level subsystem $H_{A}$ has its populations preserved under the time evolution, $\sigma_{z}(t)=\sigma_{z}(0)$, and 3. (c) $L$ is a separable channel rains1999rigorous : all of its Krauss operators are simple products. In particular, this means that any initial separable (non- entangled) state evolves to a separable state: $\rho(t)=L_{t}[\rho(0)]$ is separable for all separable initial states $\rho(0)$. Then the visibility $V(t)=\|\braket{\sigma_{-}(t)}\|$ is a monotonic function of time. Here, we have modeled the time evolution of the $A$-$B$ system as a quantum channel $L$, a map on density matrices $\rho(t)=L_{t}[\rho(0)]$. For example, within standard quantum mechanics, the unitary evolution of the universe ($A,B$ and their environment $C$, including the experimentalist) generates such a channel for the reduced $A-B$ evolution. Suppose that we can experimentally convince ourselves that time-translation invariance in the form (a) and population condition (b) hold. Then the theorem says that if $L$ cannot generate entanglement (c), then the only possible evolution for the qubit $A$ is to have its interferometric visibility decay monotonically. Thus if we observe non-monotonic visibility like the oscillatory signal described above, we can conclude that the channel must be capable of generating entanglement. We note that non-entangling channels still allow for non-trivial interactions. For example, semiclassical gravity $G_{\mu\nu}=8\pi\braket{T_{\mu\nu}}$ (appropriately completed by a modified version of the Schrödinger equation) is of this form kafri2015bounds . On the other hand, the graviton model will produce an entangling channel. We now give a proof of this theorem. By assumption (a), there exists a generator $\mathcal{L}$ of $L_{t}$ of Lindblad form lindblad1976generators ; gorini1976completely : $\dot{\rho}=\mathcal{L}\rho=-i[H,\rho]-\sum_{j}\gamma_{j}\left[E_{j}^{\dagger}E_{j}\rho+\rho E_{j}^{\dagger}E_{j}-2E_{j}\rho E_{j}^{\dagger}\right].$ (13) These Lindblad operators $E_{j}$ are highly constrained by the separability assumption, because they cannot be used to generate $A-B$ entanglement. To make this precise, we write the channel in its Krauss representation $L[\rho]=\sum_{j\geq 0}L_{j}\rho L_{j}^{\dagger}$. Expanding for small times and comparing to (13), one finds that the Krauss operators $L_{j}$ take the form, to lowest order in $dt$, $L_{0}=1-iHdt+Kdt,\ \ L_{j}=E_{j}\sqrt{dt},\ \ K=-\frac{1}{2}\sum_{j>0}E_{j}^{\dagger}E_{j}.$ (14) See, for example, chapter 3 of preskill1998lecture . Now we invoke the separability criterion (c), which says that the Krauss operators for $j>0$ take the form of simple product operators, i.e. $E_{j}=A_{j}\otimes B_{j}$ rains1999rigorous . Furthermore, the separability of $L_{0}$ to order $dt$ means that $L_{0}=(1+A_{0}dt)\otimes(1+B_{0}dt)$ for some $A_{0},B_{0}$, and this can only be satisfied if both $H$ and $E_{j}^{\dagger}E_{j}$ can be written as sums of operators acting either on $\mathcal{H}_{A}$ _or_ $\mathcal{H}_{B}$. This in turn requires that for each $j>0$, either $A_{j}^{\dagger}A_{j}=1_{A}$ or $B_{j}^{\dagger}B_{j}=1_{B}$. Finally, we impose the requirement (b) that the atom populations are invariant. This means that $\dot{\sigma}_{z}=0$. The only possible non-trivial interaction term which satisfies these requirements is $E_{z}=\sigma_{z}\otimes B$, with $B$ any operator on $H_{B}$. We are then left with the the very simple form of the Lindblad generator: $\mathcal{L}\rho=-\gamma\left[B^{\dagger}B\rho+\rho B^{\dagger}B-2B\sigma_{z}\rho\sigma_{z}B^{\dagger}\right]+\mathcal{L}_{A}+\mathcal{L}_{B}.$ (15) Here $\mathcal{L}_{A(B)}$ are Lindblad operators (including Hamiltonians) acting only on $\mathcal{H}_{A(B)}$, and $\mathcal{L}_{A}(\sigma_{z})=0$. With this result for the channel’s structure, we can compute the time derivative of the interferometric visibility $V(t)=\|\braket{\sigma_{-}(t)}\|$. Since $[H,\sigma_{z}]=0$, the most general qubit Hamiltonian is a sum of $\sigma_{z}$ and the identity. We thus have, in the Heisenberg picture, $\displaystyle\begin{split}\Braket{\frac{d\sigma_{-}}{dt}}&=-i\braket{[H,\sigma_{-}]}+\gamma\left[\braket{E_{z}^{\dagger}\sigma_{-}E_{z}}-\frac{1}{2}\braket{\left\\{E_{z}^{\dagger}E_{z},\sigma_{-}\right\\}}\right]\\\ &=2(-i\omega_{0}-\gamma)\braket{\sigma_{-}},\end{split}$ (16) where the oscillatory term is generated by the qubit Hamiltonian. Taking the absolute value to compute the visibility $V=\|\braket{\sigma_{-}}\|$ removes the oscillating phase and we have $\frac{dV}{dt}=-2\gamma V,$ (17) so it is monotonically decreasing, as we set out to prove. ## IV Effects of noise during evolution The sensing protocol is subject to errors caused by random noise during the time evolution. In a typical realization, the dominant sources of this continuous noise will consist of thermal load on the oscillator and dephasing in the atomic system (from, e.g., background fields and gas weakly measuring the atomic position joos1985emergence ; gallis1990environmental ). These sources of noise can be modeled by a Lindblad evolution of the form $\dot{\rho}=-i[H,\rho]-\sum_{i}\frac{1}{2}\\{L_{i}^{\dagger}L_{i},\rho\\}-L_{i}\rho L_{i}^{\dagger},$ (18) where the error operators are $L_{i}\in\\{\sqrt{\bar{n}\gamma_{m}}a^{\dagger},\sqrt{(\bar{n}+1)\gamma_{m}}a,\sqrt{\gamma_{a}}\sigma_{z}\\}$. The decay rates of the oscillator and atom are $\gamma_{m},\gamma_{a}$, respectively, and $\bar{n}$ is the thermal phonon occupancy. This description should be accurate for times similar to or shorter than the damping time $1/\gamma_{m}$, and assuming only small changes over time in the mechanical frequency. It is possible to analytically solve for the atomic visibility $\eqref{visibility}$ in the presence of this noise, using an explicit Ohmic heating model where the bath is taken to be an infinite set of bosonic modes linearly coupled to the mechanical system. The same displacement-operator picture used in (3) generalizes to this linear bath (see appendix B). One finds that the visibility at half and full-period evolution is given by $\displaystyle\begin{split}V(\pi/\omega)&=\exp[-\pi\gamma_{a}/\omega]\exp[-8\lambda^{2}(2\bar{n}+1)]\\\ V(2\pi/\omega)&=\exp[-2\pi\gamma_{a}/\omega]\exp[-8\lambda^{2}(2\bar{n}+1)/Q].\end{split}$ (19) Here we have assumed the mechanical damping factor $Q=\omega/\gamma_{m}\gg 1$. This recovers the previous result for the visibility (12), up to an overall exponential damping from the atomic dephasing and small correction from mechanical heating. Neglecting atomic dephasing, the visibility at half period is exactly the same as (12), while at full period, for $Q\gg 1\gg\bar{n}\lambda^{2}$, we have $V(2\pi/\omega)\approx 1$, i.e. we have full recovery up to a correction at order $1/Q$. Thus, with a sufficiently high-$Q$ oscillator, and with atomic coherence times longer than the mechanical period $\gamma_{a}\lesssim\omega$, damping does not pose a substantial barrier to the experiment. Before moving on, we consider the effects of decoherence from another inevitable source: blackbody radiation of the oscillator. Here we are discussing position superpositions of the oscillator at distances of about $\lambda x_{0}$. With the sorts of experimental parameters we suggest later, this will be a length many orders of magnitude smaller than a typical blackbody photon wavelength (or ambient gas molecule’s de Broglie wavelength). Thus these interactions will be incapable of efficiently decohering the oscillator, because they are too long-wavelength to efficiently measure the oscillator’s position gallis1990environmental . ## V Protocol linear in the weak coupling Figure 3: Left: Experimental realization of the atomic system as a lattice interferometer. The lines marked “x” denote populations that do not interfere. Right: Some example implementations with one or more mechanical masses connected rigidly. Small black dots represent the atom. In each case, the mechanical system is restricted to oscillate along the $z$-axis. More masses enable a stronger gravitational coupling. A natural limiting case would be to use a toroidal mass. In the example with a single sphere, we have $R=\sqrt{L^{2}+(\ell/2)^{2}}$ and $\kappa=1$. Our basic observable (8) is quadratic in the ratio $\lambda=g/\omega$, which for a weak coupling is a small dimensionless number. Here we suggest a “boosted” method in which linear sensitivity can be achieved by first preparing an _entangled_ state of the atom-oscillator system (as demonstrated, for example, in karg2020light ; thomas2021entanglement ). Let $\lambda^{\prime}=g^{\prime}/\omega$, where $g$ is the coupling of interest (e.g., gravity) and $g^{\prime}$ is some other coupling. Consider performing a $\pi$ gate with the coupling $V_{\rm int}=(g+g^{\prime})\sigma_{z}x$. This will produce an initial entanglement set by displacement operators $D(\pm(\lambda+\lambda^{\prime}))$, as in equation (9). Turning off the non-gravitational $g^{\prime}$ coupling then leads to only a partial revival of the atomic signal at later times $t>\pi/\omega$. This leads to the visibility, for $t>\pi/\omega$, $V_{\rm b}(t)=\exp\left[-8(2\bar{n}+1)\left(\lambda^{\prime 2}+2\lambda\lambda^{\prime}\sin^{2}\frac{\omega t}{2}+\lambda^{2}\sin^{2}\frac{\omega t}{2}\right)\right].$ (20) A detailed calculation is given in appendix (C). For times $0<t<\pi/\omega$, the visibility is given by the previous result (12) but with $\lambda\to\lambda+\lambda^{\prime}$. The observable we are interested in is the difference in visibility at half- period and full-period: $\displaystyle\begin{split}\Delta V_{\rm b}&=V_{b}(2\pi/\omega)-V_{b}(\pi/\omega)\\\ &\approx\exp[-8(2\bar{n}+1)\lambda^{\prime 2}](1-16(2\bar{n}+1)\lambda^{\prime}\lambda+O(\lambda^{2})),\end{split}$ (21) assuming $\lambda\ll\lambda^{\prime}$. We see again that using an initially “hot” resonator increases the relative visibility. However, here the observable is _linear_ in the weak gravitational coupling $\lambda$. We note that if $\bar{n}$ or $\lambda^{\prime}$ are too large, the signal will be destroyed by the overall prefactor $e^{-8(2\bar{n}+1)\lambda^{\prime 2}}$. The optimal solution is to tune the non-gravitational coupling to satisfy $\lambda^{\prime}_{\rm opt}=1/\sqrt{8(2\bar{n}+1)}$, in which case the prefactor is order one, and the relative visibility is given roughly by $\Delta V_{b}\approx\sqrt{8(2\bar{n}+1)}\lambda$. Use of this boosted protocol substantially improves the viability of an experiment with a weak coupling $g$. We note that this protocol does not violate our assumptions about time- translation invariance in Theorem 1: once the extra $g^{\prime}$ coupling is turned off, the entire system proceeds in a time-independent fashion. ## VI Experimental implementation with atom interferometry Figure 4: Logarithm $\log_{10}$ of the visibility change $\Delta V$ as function of the logarithms of the hold time $\tau$ in seconds and the temperature $T$ in Kelvin for the unboosted scheme (left) and the boosted scheme (right). The plots assume $\ell=1\,$mm, $\rho=20$g/cm3, $m=m_{\rm Cs}$. We now show how to apply our sensing protocol to a test of quantum gravity. The idea is to realize the qubit in the Hamiltonian (2) as an optical-lattice atom interferometer xu2019probing with a hold-time $\tau$ and splitting $\ell$ between the matter wave packets. The majority of the interferometer time sees the atoms trapped in one of two different potential wells created by the lattice. The atom position thus becomes a two-state system with $\sigma_{z}$-eigenvalues corresponding to the two locations. The mechanical oscillator has a mass $M$ and fundamental frequency $\omega$. Expanding the Newtonian atom-oscillator potential (1), we then have the total Hamiltonian $H=\omega a^{\dagger}a-g\sigma_{z}(a+a^{\dagger}).$ (22) Here, $a,a^{\dagger}$ are oscillator operators, so the second term represents the position-position coupling. The coupling strength is $g=\kappa\frac{G_{N}mM\ell x_{0}}{\hbar R^{3}}$ (23) where $x_{0}=\sqrt{\hbar/2M\omega}$ is the ground state oscillator uncertainty, $R$ parametrizes the distance between the oscillator and atom, and $\kappa$ is a dimensionless number of order one which depends on the specific oscillator mass geometry (see Fig. 3). Technical challenge \| Examples \| Possible strategies ---\|---\|--- Non-gravitational interactions \| Van der Waals, stray fields, scattered laser light \| Superconducting shielding, place atoms in waveguide Xin2018fiber Mean field shift \| Parasitic atom-atom interactions leading to inhomogeneous dephasing Jannin2015Meanfield \| Spin-echo techniques Laudat (see also appendix E), fermionic atoms (e.g., Yb-171 or 173) McAlpine ; Niederriter Exponential decay of signal \| Atomic dephasing \| Interleaved differential measurement, e.g. by toggling the mass between near and far positions Jaffe2017subgrav Deviations from harmonicity \| Time-dependent oscillator frequency, anharmonic perturbations \| Keep effective temperature below nonlinear thresholds; change materials, mounting, or frequency Table 1: Some systematic effects and other perturbations expected in a realistic implementation. Loophole or pathology \| Typical sources \| Problematic behavior allowed \| Possible solutions ---\|---\|---\|--- Non-gravitational interactions between atom and oscillator \| Casimir/van der Waals interactions \| Can generate entanglement (reproduce the full desired signal), can generate extra noise \| Vary parameters (masses and distance) to check proper scaling with $V=G_{N}m_{1}m_{2}/r$ law Stationarity assumption on bath (and/or experimentalist) violated \| Explicit time-dependence introduced by experimentalist (e.g. spin-echo protocol); low-frequency noise (e.g., gravity gradients, seismic noise) \| Violates assumption of theorem in Sec. III. In principle, could mimic collapse and revival \| Adjust theorem to allow for bath relaxation timescale; experimentally verify Markovian nature of oscillator noise Non-locality \| Time of interaction for experiment is much longer than light-crossing time $T_{\rm int}\gg T_{com}$ \| Allows for non-local, hidden variable model explaining the entanglement (same as Bell test) \| Long baseline version? Table 2: Some loopholes and pathologies in our proposed test. The information sensing protocol requires generation of an initial state $\ket{0}+\ket{1}$. This can be generated, e.g., by a pair of Raman pulses separated by a free evolution time xu2019probing , by spin-dependent kicks Jaffe2018SDK , by optical lattice techniques Pagel2020lattice , or by rapidly splitting a single-well potential to the double-well. Measuring in the $\sigma_{z}$ basis at the end of the protocol corresponds to closing the atom interferometer and counting the atoms in the two output ports. To implement the “boosted” protocol of section V, we can use a number of non-gravitational interactions to generate the initial entanglement. For example, a hyperfine or Rydberg atomic state could be magnetically or optically coupled to the oscillator. Entanglement of this type has been recently demonstrated experimentally karg2020light ; thomas2021entanglement . Let us consider how we can obtain a visibility change that is large enough to be measured. In order to observe at least one full cycle of decay and revival, we choose $\omega=2\pi/\tau$ where $\tau$ is the atom hold-time. In this case, the visibility change is given by $\displaystyle\begin{split}\Delta V&=\frac{\pi G_{N}^{2}m^{2}\rho}{3\sqrt{2}\ell\omega^{3}\hbar}\left(8+\bar{n}\right)\xrightarrow[k_{\rm B}T/\omega\to\infty]{}\frac{\pi}{3\sqrt{2}}K^{2},\\\ \Delta V_{\rm b}&=2^{1/4}G_{N}m\sqrt{\frac{\pi\rho}{3\ell\omega^{3}\hbar}(8+\bar{n})}\xrightarrow[k_{\rm B}T/\omega\to\infty]{}\frac{2^{1/4}}{\sqrt{3}}K\end{split}$ (24) in the unboosted and boosted scheme respectively, where $\displaystyle\begin{split}K^{2}&=\frac{G_{N}^{2}m^{2}\rho k_{B}T}{\ell\omega^{4}\hbar^{2}}\\\ &\approx 1.04\times 10^{-14}\left(\frac{T}{300\,{\rm K}}\right)\left(\frac{\ell}{1\,{\rm mm}}\right)^{-1}\left(\frac{\tau}{10\,{\rm s}}\right)^{4}.\end{split}$ (25) Here we took a solid density $\rho=20~{}{\rm g/cm}^{3},$ cesium atoms $m=m_{\rm Cs}=133~{}{\rm amu}$, used the four-sphere configuration (Fig. 3) for definiteness, and maximized the coupling $g$ for a given splitting $\ell$ by choosing a sphere radius of $R_{s}=\ell/(\sqrt{8})$. Longer atomic interrogation times $\tau$ are preferable. This would require a correspondingly low-frequency oscillator, e.g. a mHz-scale torsional pendulum. While 20 s have been experimentally realized xu2019probing , 100 s may be a reasonable expectation for the future. Using a small matter-wave splitting $\ell$ is desirable, but subject to mechanical constraints. Choosing, e.g., $\ell=1\,$mm, $L=1/\sqrt{2}\,$ mm and $R_{s}=0.35\,$mm would leave about $0.15\,$ mm free space between the spheres. For $\tau=100\,$s and $T=300$ K we obtain $\Delta V\sim 10^{-10}$; but for the boosted scheme, it will be as large as $\Delta V=7\times 10^{-6}$ (see Fig. 4). At the standard quantum limit, this can be detected with $5-\sigma$ significance by running the experiment with $\sim 5\times 10^{11}\,$ atoms (see appendix D for details on noise scaling with many atoms). Assuming that the experiment has $10^{7}$ atoms per run, and each run takes 2 minutes, this will be possible in two months total run time. Remarkably, this suggests that the experiment may be feasible in the near future. A number of systematic effects and technical issues will need to be understood. We postpone detailed discussion to future work, but flag some likely issues and ways to handle them in Table 1. ## VII Implications, loopholes, and conclusions Our interactive information sensing protocol is a novel strategy for verification of dynamical entanglement generation. While a standard Bell-type test requires measurements on both parts of a bipartite system, our protocol can verify entanglement generation with only single-body measurements. Crucially, the test verifies the ability of an interaction channel to generate entanglement, without needing to directly verify the entanglement of the final state. However, it is important to note that this test is subject to loopholes. Some are analogous to those in standard Bell tests and others are particular to our proposal. We suggest a few of these in Table 2. In our view, the most important loophole stems from our time-translation invariance assumption, which we used to write the atom-oscillator dynamics in Lindblad form (13). Non-Markovian time dependence introduced by an experimentalist or Maxwell’s demon could, in principle, reproduce the observed collapse and revival dynamics. One way to improve the situation would be to reformulate the theorem to include some level of non-Markovianity, for example a bath relaxation time scale. A more robust option would be to prove experimentally that it is simply the Markovian thermalizing channel acting on the mechanical system. Methods for this include precision quantum thermometry purdy2017quantum , which can support the hypothesis of detailed balance. In any case, extending the results here beyond the strictly Markovian assumption will be a crucial next step. The central technical advances suggested here are the interactive sensing protocol and the use of atoms as a sensor. The key advantage of the periodic collapse-and-revival protocol is that it enables a huge enhancement with a thermal state of the mechanical system; understanding if this can be extended beyond the specific context here would be very interesting. While using trapped atoms is perhaps counter-intuitive since it decreases the strength of the signal (the Newton potential), we emphasize that the extremely long coherence lifetime and ability to generate spatially well-separated superpositions of the atoms lead to similar parametric scaling of the overall signal strength. We have shown how the interactive sensing protocol can be used to test the ability of the gravitational field to communicate quantum information. If the answer is yes, this would constitute the first direct evidence that the gravitational field itself is a quantum mechanical degree of freedom carney2019tabletop ; belenchia2018quantum ; Christodoulou:2018cmk ; Marshman:2019sne ; Galley:2020qsf . On the contrary, if the answer is negative, the existence of the graviton is ruled out carney2019tabletop . The simple estimates of section VI suggest that this experiment is feasible with realistic devices, even in the presence of noise. We will present a more detailed proposal and analysis of systematic effects in a future paper. ## Acknowledgements We thank Thomas Guff, Jack Harris, John Kitching, and Jess Riedel for discussions. H.M. has been supported in part by Jet Propulsion Laboratory under grant number 1652036, the Office of Naval Research under grant numbers GRANT12980618 and N00014-20-1-2656, as well as the National Science Foundation under grant number 1708160. ## Appendix A Newtonian entanglement from graviton exchange For completeness, we review here some standard arguments about the perturbative quantization of gravity and its relation to entanglement generation via the Newton potential (1). Our goal is to explain the standard logic by which one treats small fluctuations of the metric as a quantum field and uses this to make predictions in non-relativistic systems. We do not mean to say that this derivation somehow proves that this is the correct model of low-energy quantum gravity—on the contrary, determining if this is the correct set of predictions is a central objective of the experiment proposed in this paper. By far the most common and efficient method to compare a field theoretical description to the non-relativistic setting relevant to these experiments is to do a “matching” calculation. For example, one can compute scattering amplitudes in the field theory, compare these to the same amplitude computed in a potential scattering model, and thus obtain the effective non- relativistic potential. Since the scattering states form a complete basis for the Hilbert space (other than bound states), if these two calculations agree for all scattering states, we can conclude that the two descriptions are equivalent quantum-mechanically in the regime in which the calculations match. Figure 5: Feynman diagrams for single photon and graviton exchange, respectively. For example, consider electrodynamics: photons $A_{\mu}$ coupled to a Dirac fermion $\psi$ (e.g., the electron, with charge $e$). See section 4.8 of peskin2018introduction for a textbook treatment of what follows. The Lagrangian is $\mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\bar{\psi}\gamma^{\mu}(\partial_{\mu}-ieA_{\mu})\psi$ (26) with $F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$ the electromagnetic field tensor and $\gamma^{\mu}$ the usual Dirac matrices. With this Lagrangian we can perturbatively compute the elastic $e^{-}e^{-}\to e^{-}e^{-}$ scattering amplitude to lowest order in the charge using standard methods. See the Feynman diagram of figure 5. One finds222There is also a $u$-channel diagram coming from the indistinguishability of the two electrons, which will also appear in the potential scattering computation, but this is not important for our argument here so we drop it for simplicity. $\braket{\mbox{\boldmath$p$}_{1}^{\prime}\mbox{\boldmath$p$}_{2}^{\prime}}{T}{\mbox{\boldmath$p$}_{1}\mbox{\boldmath$p$}_{2}}=4\pi e^{2}\frac{N_{\rm EM}}{-t-i\epsilon}$ (27) where $t=-(p_{1}^{\prime}-p_{1})^{2}$ is the Lorentz-invariant four-momentum transfer, and the numerator is $N_{\rm EM}=\bar{u}(\mbox{\boldmath$p$}_{1}^{\prime})\gamma^{\mu}u(\mbox{\boldmath$p$}_{1})\bar{u}(\mbox{\boldmath$p$}_{2}^{\prime})\gamma_{\mu}u(\mbox{\boldmath$p$}_{2}).$ (28) Here $u,\bar{u}$ are Dirac spinors. In the non-relativistic limit where the rest masses dominate over the spatial momenta, the numerator reduces to $N_{\rm EM}\to 1$ and $t\to(\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1})^{2}$. In the center-of-mass frame, this means that the amplitude for the relative momentum $\mbox{\boldmath$p$}=(\mbox{\boldmath$p$}_{1}-\mbox{\boldmath$p$}_{2})/2$ to transition to $\mbox{\boldmath$p$}^{\prime}$ is given by $\braket{\mbox{\boldmath$p$}^{\prime}}{T}{\mbox{\boldmath$p$}}=\frac{4\pi e^{2}}{(\mbox{\boldmath$p$}^{\prime}-\mbox{\boldmath$p$})^{2}},$ (29) where we used conservation of the total momentum to write $\mbox{\boldmath$p$}_{1}^{\prime}-\mbox{\boldmath$p$}_{1}=\mbox{\boldmath$p$}^{\prime}-\mbox{\boldmath$p$}$. We can then compare this to the amplitude computed non-relativistically with a Hamiltonian description $H=H_{1}+H_{2}+V(r)$ (30) including a central potential $V(r)=V(\|\mbox{\boldmath$x$}_{1}-\mbox{\boldmath$x$}_{2}\|)$. In the center of mass frame, the first Born approximation gives $\braket{\mbox{\boldmath$p$}^{\prime}}{T}{\mbox{\boldmath$p$}}=\tilde{V}(\mbox{\boldmath$p$}^{\prime}-\mbox{\boldmath$p$}),$ (31) where $\tilde{V}$ is the Fourier transform of the potential. Comparing these expressions, we see that the effective potential is $\tilde{V}(\mbox{\boldmath$q$})=\frac{4\pi e^{2}}{\mbox{\boldmath$q$}^{2}},$ (32) or in position space, $V(\mbox{\boldmath$x$})=\int\frac{d^{3}\mbox{\boldmath$q$}}{(2\pi)^{3}}e^{i\mbox{\boldmath$q$}\cdot\mbox{\boldmath$x$}}\tilde{V}(\mbox{\boldmath$q$})=\frac{e^{2}}{\|\mbox{\boldmath$x$}\|},$ (33) recovering the usual Coulomb force. We emphasize that this line of argument is at the level of transition _amplitudes_ $\braket{\psi_{f}}{U(t_{f},t_{i})}{\psi_{i}}$, where $U$ means time evolution and we take $t_{i}\to-\infty,t_{f}\to+\infty$. Thus, the potential (33) is the potential operator in the full meaning of the term—it is a two-body operator and can entangle particles, and so forth. Indeed, entanglement generated by this Coulomb potential is exactly what underlies, for example, quantum information processing with chains of trapped ions in Coulomb crystals. The same type of matching calculation can be performed in gravity. The additional complication is that we need to work perturbatively around a background spacetime, and only quantize the perturbations. For an experiment well-localized in spacetime like ours, we can expand around a locally flat spacetime metric $\eta_{\mu\nu}$, by the equivalence principle.333A more accurate statement would be that we should expand around a Schwarzschild solution, with Schwarzschild mass given by the mass of Earth. This term would generate a constant background external potential and is inessential to the core quantum mechanics of the following argument. We write $g_{\mu\nu}=\eta_{\mu\nu}+\frac{h_{\mu\nu}}{m_{\rm Pl}}$ (34) where we scale out a factor of the Planck mass to give $h_{\mu\nu}$ dimensions of mass (i.e. the same dimensions as a canonical bosonic field in four dimensions). In this expansion, the Einstein-Hilbert Lagrangian becomes $\mathcal{L}=\frac{1}{2}\partial_{\alpha}h_{\mu\nu}\partial^{\alpha}h^{\mu\nu}+\cdots+\frac{1}{m_{\rm Pl}}h_{\mu\nu}T^{\mu\nu},$ (35) where the dots represent other similar second-derivative kinetic terms for the perturbations $h_{\mu\nu}$ and $T_{\mu\nu}$ is the matter stress tensor. These metric perturbations $h_{\mu\nu}$ can be quantized through the exact same procedure as the electromechanical potential $A_{\mu}$ Feynman:1963ax ; t1974one ; deser1974one ; Veltman:1975vx . There is a key difference—namely that the interaction is non-renormalizable—which means that we have an effective quantum field theory kadanoff1966scaling ; wilson1971renormalization ; weinberg1979phenomenological ; donoghue1994general ; burgess2004quantum which can only make reliable predictions at energy densities well below the Planck scale.444To be more precise: the non-renormalizable nature of the interaction means that we have to include all possible generally-covariant terms, in particular curvature-curvature couplings, in the action $S=m_{\rm Pl}^{2}\int d^{4}x\sqrt{-g}[R+m_{\rm Pl}^{-2}(c_{1}R^{2}+c_{2}R_{\mu\nu}R^{\mu\nu})+m_{\rm Pl}^{-4}c_{3}R^{3}+\cdots]$. Since this is an infinite series of terms each with an unknown constant coefficient $c_{i}$, the model becomes non-predictive once these $R^{n\geq 2}$ terms become important. In the regime of these experiments $R\ll m_{\rm Pl}^{2}$, so all of these terms are extraordinarily small and the dynamics is determined entirely by the first term $R$, i.e., the usual Einstein-Hilbert action. However, we are well within this limit in the kind of experiment envisaged here, as discussed in the introduction. Proceeding accordingly, the scattering of a pair of masses (here modelled as single-particle excitations of a massive spinless field) via gravitons is given by donoghue1994general ; burgess2004quantum $\braket{\mbox{\boldmath$p$}_{1}^{\prime}\mbox{\boldmath$p$}_{2}^{\prime}}{T}{\mbox{\boldmath$p$}_{1}\mbox{\boldmath$p$}_{2}}=\frac{4\pi}{m_{\rm Pl}^{2}}\frac{N_{\rm grav}}{-t-i\epsilon}.$ (36) The numerator is more complicated due to the tensorial nature of the interaction, $\displaystyle\begin{split}N_{\rm grav}&=2(p_{1}\cdot p_{1}^{\prime})(p_{2}\cdot p_{2}^{\prime})+2(p_{1}\cdot p_{2}^{\prime})(p_{1}^{\prime}\cdot p_{2})\\\ &\ \ \ +8(p_{1}\cdot p_{1}^{\prime}+m^{2})(p_{2}\cdot p_{2}^{\prime}+m^{2})\end{split}$ (37) but reduces in the non-relativistic limit to the simple value $N_{\rm grav}\to m^{2}$. Recognizing that $m_{\rm Pl}^{2}=1/G_{N}$ in terms of the Newton constant, we can compare this again to the Born approximation (31), and determine the effective potential $\tilde{V}(\mbox{\boldmath$q$})=\frac{4\pi G_{N}m^{2}}{\mbox{\boldmath$q$}^{2}}$ (38) which again is just $V(r)=G_{N}m^{2}/r$ in real space. In this way, we see that “graviton exchange” leads to the Newton potential operator (1) in the non-relativistic limit. Finally, we note that can one directly obtain a Hamiltonian operator for the field theory and read off the non-relativistic potential directly, without resorting to scattering or other matching calculations. In contrast to the gauge-invariant scattering amplitude approach, this is complicated by the gauge symmetries of the model (in both the electrodynamics and gravity cases). To see how this works, consider the electrodynamical Lagrangian (26). To perform the transformation from the Lagrangian to Hamiltonian we have to fix a gauge, say Coulomb gauge $\partial_{i}A^{i}=0$. This gauge leads to a second- class Dirac constraint $\partial_{i}F^{i0}=-J^{0}$, so that $\nabla^{2}A^{0}=-J^{0}$, i.e., the $A^{0}$ part of the potential is non- dynamical and simply fixed by the current $A^{0}(\mbox{\boldmath$x$},t)=-\int d^{3}\mbox{\boldmath$y$}\frac{J^{0}(\mbox{\boldmath$y$},t)}{\|\mbox{\boldmath$x$}-\mbox{\boldmath$y$}\|}.$ (39) Performing the Legendre transformation to obtain the Hamiltonian, one then finds a coupling $\displaystyle\begin{split}H_{\rm Coul}&=-\int d^{3}\mbox{\boldmath$x$}A^{0}(\mbox{\boldmath$x$})J^{0}(\mbox{\boldmath$x$})\\\ &=\int d^{3}\mbox{\boldmath$x$}d^{3}\mbox{\boldmath$y$}\frac{J^{0}(\mbox{\boldmath$x$})J^{0}(\mbox{\boldmath$y$})}{\|\mbox{\boldmath$x$}-\mbox{\boldmath$y$}\|},\end{split}$ (40) which is just the usual non-relativistic Coulomb interaction, since $J^{0}$ is the charge density. Again, everything here is at the level of operators. In the case of gravity, the exactly analogous calculation can be performed and one finds an instantaneous Newton interaction $H=m_{\rm Pl}^{-2}\int d^{3}\mbox{\boldmath$x$}d^{3}\mbox{\boldmath$y$}T^{00}(\mbox{\boldmath$x$})T^{00}(\mbox{\boldmath$y$})/{\|\mbox{\boldmath$x$}-\mbox{\boldmath$y$}\|}$. See chapter 8 of weinberg1995quantum for a thorough treatment of this kind of Hamiltonian approach in the case of electrodynamics, and e.g. anastopoulos2013master for the calculation in perturbative gravity. The fact that the field component responsible for the Coulomb/Newton interaction is non-dynamical (e.g., $A^{0}$ in the above example) has led some authors to argue that observing Newtonian entanglement would tell us nothing about the quantization of the “physical” (i.e., dynamical) degrees of freedom of the gravitational field; for a prototypical expression of this view, see Anastopoulos:2018drh . There are, however, strong arguments against this belenchia2018quantum ; Christodoulou:2018cmk ; Marshman:2019sne ; Galley:2020qsf , which essentially say that there is no consistent way to have both an entangling Newton interaction and non-quantized metric fluctuations. We anticipate substantial further debate on this topic, and will present a detailed discussion in a separate paper carneyprep . ## Appendix B Detailed calculation of oscillator noise To get a quantitative estimate of the effects of thermal loading on the oscillator, let us assume that we can safely neglect atomic dephasing $\gamma_{a}t\ll 1$ for the experimental timescale of interest. To develop an exact result, we will use an input-operator method, in which we include an explicit heat bath for the oscillator. The Lindblad system (18) can be derived through this method by tracing out the oscillator bath. See clerk2010introduction for a lucid review of this technique. Let $H_{0}$ denote the Hamiltonian for the oscillator _and its bath_. The total time evolution operator is then $U(t)=e^{-i(H_{0}+g\sigma_{z}x)t}=e^{-iH_{0}t}\hat{T}e^{-ig\sigma_{z}\int_{0}^{t}x_{I}(t^{\prime})dt^{\prime}}.$ (41) Here $x_{I}(t)$ is the oscillator position operator in the interaction picture and $\hat{T}$ is the time-ordering operator. For the case of a linear bath, such as that assumed in quantum optics or in quantum Brownian motion, we can explicitly find $x_{I}$. Writing $x=(a+a^{\dagger})/\sqrt{2}$, we have $\displaystyle\begin{split}a_{I}(t)&=\exp[-i(\omega+\gamma_{m}/2)t]a(0)\\\ &+\sqrt{\gamma_{m}}\int_{0}^{t}\exp[-i(\omega+\gamma_{m}/2)(t-t^{\prime})]a_{\rm in}(t^{\prime})dt^{\prime}\end{split}$ (42) where $a_{\rm in}(t)$ is the vacuum noise fluctuation operator, satisfying $[a_{\rm in}(t),a_{\rm in}^{\dagger}(t^{\prime})]=\delta(t-t^{\prime})$. Using the linearity of this expression and the Baker-Campbell-Hausdorf relation, we then have that $\displaystyle\begin{split}&\hat{T}\exp\left(-ig\sigma_{z}\int_{0}^{\tau}x_{I}(t)dt\right)\\\ &=\exp\left(-ig\sigma_{z}\int_{0}^{\tau}x_{I}(t)dt\right)\exp(-ig^{2}C(t))\end{split}$ (43) where $C(t)$ is a real, time-dependent number, arising from the non-commuting elements of $x_{I}(t)$. Having dispensed with the time-ordering, we can now explicitly perform the time integral (including a change of integration order in the $a_{\rm in}$ term). Dropping the $e^{-ig^{2}C(t)}$ phase, which will cancel out of our observable, we find the time evolution reduces to a simple product of displacement operators, one for the oscillator and one for each mode $a_{\rm in}(t^{\prime})$ for $0\leq t^{\prime}\leq t$, that is $U(t)=e^{-iH_{0}t}D_{a}[\sigma_{z}\alpha(t)]\prod_{0\leq t^{\prime}\leq t}D_{a_{in}(t^{\prime})}[\sigma_{z}\alpha_{\rm in}(t^{\prime})]$ (44) where $\displaystyle\begin{split}\alpha(t)&=\frac{ig}{i\omega-\gamma_{m}/2}(1-e^{(i\omega-\gamma_{m}/2)t}),\\\ \alpha_{\rm in}(t^{\prime})&=\frac{ig}{i\omega-\gamma_{m}/2}(1-e^{(i\omega-\gamma_{m}/2)(t-t^{\prime})}).\end{split}$ (45) Finally, we can evaluate our visibility $\sigma_{-}(t)=U^{\dagger}(t)\sigma_{-}U(t)$, assuming an initial thermal state for the oscillator and each bath mode and the $\ket{+}$ state for the atom. Using the same results for coherent states as above, one finds $\displaystyle\begin{split}\braket{\sigma_{-}}&=\braket{D_{a}[2\alpha(t)]}\prod_{t^{\prime}}\braket{D_{a_{\rm in}(t^{\prime})}[\alpha_{\rm in}(t^{\prime})]}\\\ &=\exp\left[-8\lambda^{2}(2\bar{n}+1)f(t)\right]\end{split}$ (46) with $\displaystyle\begin{split}f(t)&=\frac{\omega^{2}/4}{\omega^{2}+\gamma_{m}^{2}/4}\Big{(}2-2\cos(\omega t)e^{-\gamma_{m}t/2}+\gamma_{m}t\\\ &-\frac{8\gamma_{m}}{\omega}\sin(\omega t)e^{-\gamma_{m}t/2}+O(1/Q^{2})\Big{)}\end{split}$ (47) where $Q=\omega/\gamma_{m}$ is assumed much larger than one. In particular, at full and half-period this gives $\displaystyle\begin{split}V(\pi/\omega)&=\exp[-8\lambda^{2}(2\bar{n}+1)],\\\ V(2\pi/\omega)&=\exp[-8\lambda^{2}(2\bar{n}+1)/Q].\end{split}$ (48) Here we have assumed the mechanical damping factor $Q=\omega/\gamma_{m}\gg 1$. Re-inserting the exponential damping factor for atomic dephasing then reproduces the results in (19). In Fig. 6, we compare this analytical model with a numerical simulation of the Lindblad equation (13), showing excellent agreement. Figure 6: Examples of the signal of interest, the phase contrast $V=\|\braket{\sigma_{-}(t)}\|$, compared with its initial value $V(0)=1/2$. Left: Direct simulation of the Lindblad evolution (18) in blue, and we see good agreement with our analytic solution including noise (46) in orange, dashed. Normalizing all units to the oscillator frequency $\omega=1$, here we use values $g=10^{-2}$ for the gravitational coupling, $\gamma_{m}=5\times 10^{-3}$ for the mechanical damping, and $T=2$ for the temperature. (Numerical simulation with a much higher $T\gg\omega$ as discussed in the paper is infeasible due to restrictions on the oscillator Hilbert space dimension). Same parameters as left figure, but with an initial $\pi$-pulse using a non- gravitational coupling $g^{\prime}=10^{-1}$. The difference between the first collapse and revival is now much larger than in the un-boosted protocol, as predicted in (21). ## Appendix C Detailed calculation of the boosted protocol Here we give the full computation of the visibility in our entanglement- enhanced, “boosted” protocol of Section V. The total evolution is a product of two unitaries, one for the first half-period under the coupling $g+g^{\prime}$ and the second under only $g$. We write these as $\displaystyle\begin{split}U_{g+g^{\prime}}&=D^{\dagger}((\lambda+\lambda^{\prime})\sigma_{z})e^{-i\omega n}D((\lambda+\lambda^{\prime})\sigma_{z}),\\\ U_{g}(t)&=D^{\dagger}(\lambda\sigma_{z})e^{-i\omega n(t-\pi/\omega)}D(\lambda\sigma_{z}).\end{split}$ (49) With this notation, the visibility of the atom, given some initial coherent state $\ket{\alpha}$ for the oscillator, is given by (defining $\tilde{\lambda}=\lambda+\lambda^{\prime}$ for brevity) $\displaystyle\begin{split}V_{{\rm b},\alpha}(t)&=\braket{\alpha}{U^{\dagger}_{g+g^{\prime}}U_{g}(t)\sigma_{-}U_{g}(t)U_{g+g^{\prime}}(t)}{\alpha}\\\ &=\bra{\alpha}D^{\dagger}(-\tilde{\lambda})e^{i\omega n/\pi}D(-\tilde{\lambda})U^{\dagger}_{g}(t)\\\ &\ \ \ \times\sigma_{-}U_{g}(t)D^{\dagger}(\tilde{\lambda})e^{-i\omega n/\pi}D(\tilde{\lambda})\ket{\alpha}\\\ &=\braket{\alpha}{D(2\tilde{\lambda})e^{i\omega n/\pi}U^{\dagger}_{g}(t)\sigma_{-}U_{g}(t)e^{-i\omega n/\pi}D(2\tilde{\lambda})}{\alpha}\\\ &=\braket{\alpha}{D\left[2\tilde{\lambda}-\lambda(1+e^{i\omega t})\right]D\left[2\tilde{\lambda}-\lambda(1+e^{i\omega t})\right]}{\alpha}\\\ &=\bra{0}D(-\alpha)D\left[2\tilde{\lambda}-\lambda(1+e^{i\omega t})\right]\\\ &\ \ \ \times D\left[2\tilde{\lambda}-\lambda(1+e^{i\omega t})\right]D(\alpha)\ket{0}\\\ &=e^{\phi}\braket{\alpha-2\tilde{\lambda}+\lambda(1+e^{i\omega t})}{\alpha+2\tilde{\lambda}-\lambda(1+e^{i\omega t})}\\\ &=e^{2\phi}e^{-\|4\tilde{\lambda}^{2}-2\lambda(1+e^{i\omega t})\|^{2}/2}.\end{split}$ (50) To go from the second to third line, we inserted a pair of identity operators $1=e^{-i\omega n/\pi}e^{i\omega n/\pi}$ and used $e^{i\omega n/\pi}D(\tilde{\lambda})e^{-i\omega n/\pi}=D(-\tilde{\lambda})$. From the third to fourth we used the same trick and the more general time evolution $e^{i\omega nt}D(\tilde{\lambda})e^{-i\omega nt}=D(\tilde{\lambda}e^{i\omega t})$. In the last few lines the “phase” is $\phi=\alpha^{}(2\tilde{\lambda}-\lambda(1+e^{i\omega t}))/2+{\rm c.c.}.$ (51) Note that we got two factors of this: one in the fifth line, from the braiding relation $D(\alpha)D(\beta)=e^{(\alpha\beta^{}-\alpha^{}\beta)/2}$, and then another in the subsequent line from the inner product $\braket{\beta}{\alpha}=e^{-\|\beta-\alpha\|^{2}/2}e^{(\alpha\beta^{}-\alpha^{}\beta)/2}$. Notice also that the second exponential does not depend on the coherent state parameter $\alpha$. Thus we only need to average this phase term over the Glauber representation, which gives $\displaystyle\begin{split}&\int\frac{d^{2}\alpha}{\pi\bar{n}}e^{-\|\alpha\|^{2}/\bar{n}}e^{2\phi}\\\ &\ \ \ =\exp\left(8\lambda n(\lambda+2\lambda^{\prime})\cos(t\omega)-8n\left(\lambda^{2}+2\lambda\lambda^{\prime}+2\lambda^{\prime 2}\right)\right),\end{split}$ (52) where we used the explicit coefficient $\tilde{\lambda}=\lambda+\lambda^{\prime}$. Doing the same with the second term in (50), and simplifying the terms, we finally obtain $\displaystyle\begin{split}&V_{\rm b}(t)=\int\frac{d^{2}\alpha}{\pi\bar{n}}V_{{\rm b},\alpha}(t)\\\ &=\exp\left[-8(2\bar{n}+1)\left(\lambda^{\prime 2}+2\lambda\lambda^{\prime}\sin^{2}\frac{\omega t}{2}+\lambda^{2}\sin^{2}\frac{\omega t}{2}\right)\right],\end{split}$ (53) as quoted in (20). Note the limit $\lambda^{\prime}\to 0$ reproduces the basic, un-boosted protocol. We show the form of this visibility evolution in Fig. 6. ## Appendix D Using many atoms Figure 7: Spin-echo variant of the basic protocol. After a half-period of evolution, the two pathways through oscillator phase space are maximally distant. The atomic positions are then flipped ($X$ gate), followed by another half-period of evolution. This procedure can be repeated arbitrarily, leading to a net amplification of the basic protocol. The bottom-right figure shows the resulting conditioned paths of the oscillator through phase space, through one iteration. The sensitivity of the protocol can be substantially improved by moving from using a single atom to use a collection of atoms, as is typical in an atom interferometer kasevich1992measurement ; santarelli1999quantum ; gross2010nonlinear . For simplicity, we will take $g$ the same for all the atoms, though that is not necessary in practice. In that limit, we can define $J_{z}=\sum_{j}\sigma_{z}^{j}$ and $J_{-}=\sum_{j}\sigma_{-}^{j}$ as the collective variables that will enter. Consider the extension of (9) to the case of $N$ atoms prepared in the initial state $\ket{+++\cdots}$. The “observable” of interest is $\braket{J_{-}(t)}$. This is easiest to calculate term by term for each atom. The total time evolution operator, following the same logic as in (3), is $U(t)=\exp\left(-i\beta J_{z}^{2}\right)D^{\dagger}(\lambda J_{z})e^{i\omega a^{\dagger}at}D(\lambda J_{z}),$ (54) with $\beta=\frac{g^{2}t}{\omega}.$ (55) This total $J_{z}^{2}$ term is peculiar to the case of $N>1$ atoms; for $N=1$ it is just an overall phase which we dropped in (3). Here, however, it is a non-trivial operator, physically representing the ponderomotive squeezing of the spins due to the gravitational coupling with the oscillator. At time $t>0$ we have, for each $i=1,\ldots,N$, the operator evolution $\sigma_{-}^{i}(t)=U^{\dagger}(t)\sigma_{-}^{i}U(t).$ (56) If we define $\tilde{J}_{z}^{i}=J_{z}-\sigma_{z}^{i}$, we can see that we can write the $J_{z}^{2}$ contribution to (56) as $\exp\left(i\beta J_{z}^{2}\right)\sigma_{-}^{i}\exp\left(-i\beta J_{z}^{2}\right)=\exp\left(-2i\beta\tilde{J}_{z}\right)\sigma_{-}^{i}.$ (57) This prefactor then commutes with the rest of the operators in (56). Using this and the same basic logic as in (9), we find that all the $i\neq j$ spins will just give a phase proportional to $\tilde{J}_{z}$: $\sigma_{-}^{i}(t)=\sigma_{-}^{i}e^{-2i\beta\tilde{J}_{z}}D(-\lambda)D(2\lambda e^{i\omega t})D(-\lambda).$ (58) Acting on the initial state $\ket{0,+++\cdots}$ with the oscillator prepared in $\ket{0}$ and each atom in the $\ket{+}$ state, that is to say with $N$ unentangled atoms, we obtain $\braket{\sigma_{-}^{i}(t)}=\cos^{N-1}(2\beta)\braket{\sigma_{-}(t)}_{1}$ (59) where the term $\braket{\sigma_{-}(t)}_{1}$ means the answer with a single spin, as in equations (9) and (10). For $N\gg 1$ and $\beta\ll 1$ (recall $\beta=g^{2}t/\omega$, so this condition is certainly satisfied for us), we can Taylor expand the cosine and match it to an exponential for convenience, i.e., $\cos^{N-1}(2\beta)\approx e^{-2N\beta^{2}}$. Thus since $\braket{J_{-}}$ has $N$ terms of the form (59), we finally get $\braket{J_{-}(t)}=Ne^{-2Ng^{4}t^{2}/\omega^{2}}\braket{\sigma_{-}(t)}_{1}.$ (60) For example, at a half period and full period we then have $\displaystyle\begin{split}\braket{J_{-}(\pi/\omega)}&=Ne^{-2\pi^{2}N\lambda^{4}}e^{-8\lambda^{2}},\\\ \braket{J_{-}(2\pi/\omega)}&=Ne^{-4\pi^{2}N\lambda^{4}}.\end{split}$ (61) Note that this noisy phase is independent of the oscillator’s initial state, so for example we get the same answer if the oscillator begins in a thermal state. We see the basic $N$ enhancement to the signal here as the prefactor. The phase noise scales like $N\lambda^{4}$. For our particular implementation with parameters like those quoted in (25), we have $\lambda\sim 10^{-13}$, so for $N\sim 10^{10}$ atoms these phase noise exponentials are completely negligible. The overall $N$ factor here represents the usual $\sqrt{N}$ statistical enhancement in the signal-to-noise, assuming uncorrelated atom errors. The same calculation extends directly to the entanglement-enhanced, $g$-linear protocol (20). This is clear by the algebraic structure of the argument given above. Explicitly, we now have two time evolution operators of the form (54), one with a coupling $g+g^{\prime}$ from $t=-\pi/\omega$ to $t=0$, followed by another with only the gravitational $g$ coupling from $t=0$ onward. In an obvious notation we can write $\sigma^{i}_{-}(t)=U^{\dagger}_{g}(t)U^{\dagger}_{g+g^{\prime}}\sigma_{-}^{i}U_{g+g^{\prime}}U_{g}(t).$ (62) In these $U$ operators, we have the same phase-noise terms, namely $e^{i\beta J_{z}^{2}}$ in the $U_{g}$ and another factor $e^{i\beta^{\prime}J_{z}^{2}}$, with $\beta^{\prime}=-(g+g^{\prime})^{2}\pi/\omega$, from the $U_{g+g^{\prime}}$ factor. These depend only on the $J_{z}$ operator and thus commute with the other terms (displacement operators and free oscillator evolution) in $U_{g}$ and $U_{g+g^{\prime}}$. Thus we get an expression $\displaystyle\begin{split}\sigma^{i}_{-}(t)&\sim\exp(i(\beta+\beta^{\prime})J_{z}^{2})\sigma_{-}^{i}\exp(-i(\beta+\beta^{\prime})J_{z}^{2})\\\ &=\exp(-2i(\beta+\beta^{\prime})\tilde{J}_{z})\sigma_{-}^{i},\end{split}$ (63) just as in (57) except now with $\beta$ replaced by $\beta+\beta^{\prime}=\frac{g^{2}t}{\omega}-\frac{(g+g^{\prime})^{2}\pi}{\omega}.$ (64) In particular, all the terms other than these phase noise exponentials contribute as given by (20). The overall signal is still increased linearly in $N$ as in (61), times a negligible contribution from this ponderomotive squeezing noise. ## Appendix E Spin-echo version for faster physical oscillators The physical oscillator frequency is crucially important to the size of the observable effect. The interferometric contrast scales as a power of $\lambda=g/\omega$, so a low-frequency oscillator is ideal. However, in practice, using a very low-frequency (sub-Hz) oscillator would present substantial technical problems. This can be alleviated by using a high- frequency oscillator and a spin-echo like sequence to mimic the effect of a low-frequency oscillator. Specifically, after every $\pi/\omega$ half-period, we swap the two atom locations, i.e. perform a $\sigma_{x}$ operation (see Fig. 7). This produces the evolution $\displaystyle\begin{split}U&=\sigma_{x}e^{-iH\tau/2}\sigma_{x}e^{-iH\tau/2}\\\ &=e^{-i(H_{p}-V)\tau/2}e^{-i(H_{p}+V)\tau/2}\\\ &=D(\lambda\sigma_{z})e^{-iH_{p}\tau/2}D(-\lambda\sigma_{z})D(-\lambda\sigma_{z})e^{-iH_{p}\tau/2}D(\lambda\sigma_{z})\\\ &=D(\lambda\sigma_{z})D(2\lambda\sigma_{z})D(\lambda\sigma_{z})=D(4\lambda\sigma_{z})\end{split}$ (65) where we used $\tau=2\pi/\omega$ for the final line. With $N_{\pi}$ iterations of this, we produce the total evolution $U(N_{\pi})=D(4N_{\pi}\lambda\sigma_{z})$. Performing $N_{\pi}$ iterations, followed by a $\sigma_{x}$ operation, followed by $N_{\pi}$ further iterations, we recover the revival: $\displaystyle\begin{split}U_{\rm spin- echo}&=\sigma_{x}D(4N_{\pi}\lambda\sigma_{z})\sigma_{x}D(4N_{\pi}\lambda\sigma_{z})\\\ &=D(-4N_{\pi}\lambda\sigma_{z})D(4N_{\pi}\lambda\sigma_{z})\\\ &=\mathbb{I}.\end{split}$ (66) In this spin-echo style variant, the wavefunction overlap after a total time $t=N_{\pi}\tau$, i.e. to the halfway point, is given by $O=\exp\left[-32\frac{N_{\pi}^{2}g^{2}}{\omega^{2}}\right].$ (67) Thus we have an effect scaling like $g^{2}/\omega_{\rm eff}^{2}$, with the effective frequency $\omega_{\rm eff}=2\pi/t=\omega/N_{\pi}$. This means we can have an effectively slow oscillator (which is beneficial for the signal strength) while using a faster physical oscillator (beneficial for noise reasons), at the cost of having to perform some $\sigma_{x}$ operations on the two-state system. We note that application of this spin-echo protocol would violate the Markov assumption used to prove the theorem in section III. Adjusting that proof to accommodate this specific type of non-Markovian control would be necessary to draw the same conclusion, namely, that the gravitational entanglement in this protocol is necessarily due to the mass-atom entangling interaction. ## References * (1) R. P. Feynman, F. B. Morinigo, and W. G. Wagner, Lectures on Gravitation, 1962-63. California Institute of Technology, 1971. * (2) D. N. Page and C. Geilker, “Indirect evidence for quantum gravity,” Physical Review Letters 47 no. 14, (1981) 979. * (3) L. 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# Simple and maximally robust processes with no classical common-cause or direct-cause explanation Marcello Nery<EMAIL_ADDRESS>Departamento de Física, Universidade Federal de Minas Gerais, Av. Pres. Antonio Carlos 6627 - Belo Horizonte, MG, Brazil - 31270-901. Marco Túlio Quintino Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria Philippe Allard Guérin Faculty of Physics, University of Vienna, Boltzmanngasse 5, 1090 Vienna, Austria Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, A-1090 Vienna, Austria Perimeter Institute for Theoretical Physics, 31 Caroline St. N, Waterloo, Ontario, N2L 2Y5, Canada Thiago O. Maciel Departamento de Física, Universidade Federal de Santa Catarina, Florianópolis, SC, 88040-900, Brazil Universidade Federal do Rio de Janeiro, Caixa Postal 68528, Rio de Janeiro, RJ 21941-972, Brazil Reinaldo O. Vianna Departamento de Física, Universidade Federal de Minas Gerais, Av. Pres. Antonio Carlos 6627 - Belo Horizonte, MG, Brazil - 31270-901. (September 02, 2021) ###### Abstract Guided by the intuition of coherent superposition of causal relations, recent works presented quantum processes without classical common-cause and direct- cause explanation, that is, processes which cannot be written as probabilistic mixtures of quantum common-cause and quantum direct-cause relations (CCDC). In this work, we analyze the minimum requirements for a quantum process to fail to admit a CCDC explanation and present “simple” processes, which we prove to be the most robust ones against general noise. These simple processes can be realized by preparing a maximally entangled state and applying the identity quantum channel, thus not requiring an explicit coherent mixture of common- cause and direct-cause, exploiting the possibility of a process to have both relations simultaneously. We then prove that, although all bipartite direct- cause processes are bipartite separable operators, there exist bipartite separable processes which are not direct-cause. This shows that the problem of deciding whether a process is direct-cause _is not_ equivalent to entanglement certification and points out the limitations of entanglement methods to detect non-classical CCDC processes. We also present a semi-definite programming hierarchy that can detect and quantify the non-classical CCDC robustnesses of every non-classical CCDC process. Among other results, our numerical methods allow us to show that the simple processes presented here are likely to be also the maximally robust against white noise. Finally, we explore the equivalence between bipartite direct-cause processes and bipartite processes without quantum memory, to present a separable process which cannot be realized as a process without quantum memory. ## Introduction Common-cause and direct-cause relations are the building blocks of classical causal models, as a causal model of multiple variables consists of combinations of common-cause and direct-cause relations between them. Understanding the relationship between cause and effects is one of the fundamental goals of several physical theories and of statistical analysis [1]. Also, determining the causal relation behind a correlation of two objects is a fundamental problem in causal inference theory and plays a main role in topics such as hypothesis testing, social sciences, medicine, and machine learning [2, 3]. In order to analyze causality in quantum phenomena, recent works proposed new frameworks of theory of Bayesian inference [4] and causal modeling [5], where understanding common-cause and direct-cause relations in quantum mechanics plays a fundamental role. Quantum causal processes consist of a sequence of quantum operations and may be analyzed from different equivalent perspectives, such as sequential quantum operations via non-Markovian processes [6, 7, 8, 9], fragments of quantum circuits via quantum combs [10, 11], quantum channels with memory [12], and quantum strategies for playing an $n$-turn game [13]. When referring to causal modeling, a process which can be written as a probabilistic mixture of common- cause (CC) and direct-cause (DC) processes is said to admit a classical common-cause or direct-cause (CCDC) explanation, since the causal relations in such process could be simulated classically, by sampling from a probability distribution and then implementing either a common-cause or direct-cause process. In Ref. [14] the authors consider the different possibilities of combining CC and DC processes and, inspired by coherent mixture of quantum channels, the authors of Ref. [15] experimentally certify the existence of a quantum process with no classical CCDC explanation. Also, inspired by the _Quantum Switch_ [16, 17, 18], Ref. [19] presents a coherent superposition of common-cause and direct-cause processes which cannot be explained by a classical CCDC. In this work, we analyze quantum processes which cannot be decomposed as probabilistic mixtures of common-cause and direct-cause ones, hence admitting no classical CCDC explanation. We focus on the bipartite case, the simplest scenario where such processes can exist. In this simplest scenario, exploring the possibility of a process to have both CC and DC relations simultaneously, we present a process which can be simply realized by preparing a maximally entangled state and an identity channel, not needing an interpretation of it as a coherent mixture of causal relations. When attempting to work with the minimum non-trivial dimensions, we propose another process, which requires a controlled-NOT operation. We prove that these processes are the most robust ones against their worst possible noise, known as generalized noise. We also develop a semi-definite programming (SDP) numerical approach to quantify the non-classical CCDC property of a process, based on its robustness against white and general noise. Our numerical methods allow us to show that these simple processes presented here are also maximally robust against white noise when considering qubits, and to quantify the non-classical CCDC property in any bipartite ordered process. We also investigate the differences and similarities between quantum entanglement and processes without a direct-cause explanation, showing that there exist separable processes which are not direct-cause. This example points out the limitations of purely entanglement-based methods to detect non direct-cause processes and answers a conjecture first raised in Ref. [20]. Finally, we connect our results with a different related field by proving that for the bipartite case, processes without quantum memory [20] are equivalent to processes having a direct-cause decomposition. With that, we present a bipartite separable process which cannot be realized as a process without quantum memory and contribute to the understanding of the relation between entanglement and of quantum memory [21, 20, 22, 23, 24]. ## 1 Mathematical Preliminaries ### 1.1 The Choi-Jamiołkowski isomorphism As is standard in several branches of quantum information theory, we will make use of the Choi-Jamiołkowski isomorphism to represent linear operators and linear maps [25, 26, 27]. We first present the “pure” version of the isomorphism, which allows us to represent linear operators as bipartite vectors. Here $A_{I}$ and $A_{O}$ are finite-dimensional complex vector spaces, which will late be identified with Alice’s input and Alice’s output, respectively. ###### Definition 1.1 (“Pure” Choi-Jamiołkowski isomorphism). Let $\textrm{U}:(A_{I})\rightarrow(A_{O})$ be a linear transformation (which is not necessarily unitary). The Choi vector $\|{\textrm{U}}\rangle\rangle\in A_{I}\otimes A_{O}$ of the operator U is defined as $\|{\textrm{U}}\rangle\rangle:=\sum_{i}\|{i}\rangle\otimes(\textrm{U}\,\|{i}\rangle),$ (1) where $\\{\|{i}\rangle\\}_{i=0}^{d_{A_{I}}-1}$ is the computational basis for $A_{I}$ and $d_{A_{I}}$ is the dimension of $A_{I}$ In quantum information theory, linear transformations between operators are sometimes referred to as linear maps, or simply, as maps. In this paper, we identify linear maps with a single tilde. For instance, let $\mathcal{L}(A_{I})$ be the set of linear operators mapping $A_{I}$ to itself, $\widetilde{\Lambda}:\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})$ is a map transforming operators from $\mathcal{L}(A_{I})$ to $\mathcal{L}(A_{O})$. We now present the standard version of the Choi-Jamiołkowski isomorphism, which allows us to represent linear maps as bipartite operators. ###### Definition 1.2 (Choi-Jamiołkowski isomorphism). Let $\widetilde{\Lambda}:\mathcal{L}(A_{I})\rightarrow\mathcal{L}(A_{O})$ be a linear map and $\\{\|{i}\rangle\\}_{i=0}^{d_{A_{I}}-1}$ be the computational basis for $A_{I}$. The Choi operator $\Lambda\in\mathcal{L}(A_{I}\otimes A_{O})$ of the map $\widetilde{\Lambda}$ is defined as $\Lambda:=\sum_{ij}\|{i}\rangle\langle{j}\|\otimes\widetilde{\Lambda}(\|{i}\rangle\langle{j}\|).$ (2) A common equivalent way to define the Choi operator of a map $\widetilde{\Lambda}$ is given by the equation $\Lambda:=\left[\left(\widetilde{\mathds{1}}\otimes\widetilde{\Lambda}\right)(\|{\mathds{1}}\rangle\rangle\langle\langle{\mathds{1}}\|)\right],$ (3) where $\widetilde{\mathds{1}}:\mathcal{L}(A_{I})\rightarrow\mathcal{L}(A_{I})$ is an identity channel, and $\|{\mathds{1}}\rangle\rangle=\sum_{i=0}^{d_{A_{I}}-1}\|{ii}\rangle\in A_{I}\otimes A_{I}$. The action of the map $\widetilde{\Lambda}:\mathcal{L}(A_{I})\rightarrow\mathcal{L}(A_{O})$ on an operator $\rho\in\mathcal{L}(A_{I})$ can be obtained from the Choi operator $\Lambda$ by means of the relation $\widetilde{\Lambda}(\rho)=\mathrm{tr}_{A_{I}}\left[({\rho^{A_{I}}}^{T}\otimes\mathds{1}^{A_{O}})\cdot\Lambda\right],$ (4) where $\mathds{1}$ is the identity operator and $(\cdot)^{T}$ stands for the transposition in the computational basis. A linear map is a quantum channel iff it is completely positive (CP) and trace-preserving (TP). In the Choi-Jamiołkowski representation, a map $\widetilde{\Lambda}:\mathcal{L}(A_{I})\rightarrow\mathcal{L}(A_{O})$ is CP iff $\Lambda\succeq 0$, _i.e._ , $\Lambda$ is positive semi-definite, and TP iff $\mathrm{tr}_{A_{O}}(\Lambda)=\mathds{1}^{A_{I}}$. For the sake of clarity, some equations in this manuscript will explicitly identify the input and output Hilbert spaces of a map with a superscript in the symbol representing the Choi operator of the map. In the above situation, for example, the Choi operator is equivalently represented by $\Lambda^{A_{I}/A_{O}}$, indicating that the map $\widetilde{\Lambda}$ takes an operator from $\mathcal{L}(A_{I})$ to one in $\mathcal{L}(A_{O})$. ### 1.2 The link product operation First defined in Ref. [11], the link product is a useful mathematical tool to deal with composition of elements in a quantum circuit presented in the Choi- Jamiołkowski representation. ###### Definition 1.3 (Link product). Let $\Lambda_{1}\in\mathcal{L}(A\otimes A^{\prime})$ and $\Lambda_{2}\in\mathcal{L}(A^{\prime}\otimes A^{\prime\prime})$ be linear operators. The link product between $\Lambda_{1}$ and $\Lambda_{2}$ is defined as $\displaystyle\begin{split}\Lambda_{2}\Lambda_{1}:=\mathrm{tr}_{A^{\prime}}\Bigg{[}\bigg{(}\Lambda_{1}^{T_{A^{\prime}}}\ \otimes\mathds{1}^{A^{\prime\prime}}\bigg{)}\cdot\bigg{(}\mathds{1}^{A}\otimes\Lambda_{2}\bigg{)}\Bigg{]},\end{split}$ (5) where $\Lambda_{1}^{T_{A^{\prime}}}$ is the partial transpose of $\Lambda_{1}$ on the space $A^{\prime}$. As stated previously, the link product is useful for composing linear maps and quantum objects. If $\widetilde{\Lambda}_{1}:\mathcal{L}(A)\rightarrow\mathcal{L}(A^{\prime})$ and $\widetilde{\Lambda}_{2}:\mathcal{L}(A^{\prime})\rightarrow\mathcal{L}(A^{\prime\prime})$ are linear maps with Choi operators $\Lambda_{1}\in\mathcal{L}(A\otimes A^{\prime})$ and $\Lambda_{2}\in\mathcal{L}(A^{\prime}\otimes A^{\prime\prime})$, respectively, it can be shown that the Choi operator of the composition $\widetilde{\Phi}:=\widetilde{\Lambda}_{2}\circ\widetilde{\Lambda}_{1}:\mathcal{L}(A)\rightarrow\mathcal{L}(A^{\prime\prime})$ is $\Phi=\Lambda_{2}\Lambda_{1}.$ In particular, when $\rho\in\mathcal{L}(A)$, is a linear operator with no components on $A^{\prime}$, and $\Lambda\in\mathcal{L}(A\otimes A^{\prime})$, we have $\displaystyle\begin{split}\Lambda\rho:=&\mathrm{tr}_{A}\left[\left(\rho^{T_{A}}\ \otimes\mathds{1}^{A^{\prime}}\right)\cdot\left(\Lambda\right)\right]\\\ =&\widetilde{\Lambda}(\rho).\end{split}$ (6) Hence, we can use the link product to represent quantum operations being performed on quantum states. Also, if $\rho_{1}\in\mathcal{L}(A_{1})$ and $\rho_{2}\in\mathcal{L}(A_{2})$, acting on different spaces, we have $\rho_{1}\rho_{2}:=\rho_{1}\otimes\rho_{2}$. Therefore, the link product of independent systems is simply the tensor product. Additionally, when $\rho,M\in\mathcal{L}(A)$ act in the same linear space, the link product is given by $\rhoM:=\mathrm{tr}(M^{\textrm{T}}\cdot\rho)$, which is simply the trace of their product with an extra transposition. This form can also be used to write Born’s rule. ## 2 The CCDC scenario The main objects analyzed in this work are the bipartite ordered processes, which may be understood as a physical dynamics which flows from Alice to Bob. For all definitions in this section, we consider a scenario where Alice is a quantum physicist in a laboratory who can perform any quantum operation that transforms states from system $A_{I}$ (Alice input) to $A_{O}$ (Alice output), and Bob is a quantum physicist in a laboratory who can perform any quantum measurement on states defined on system $B_{I}$ (Bob input). ### 2.1 Markovian processes We start by presenting the definition of bipartite Markovian processes, which admits a direct-cause interpretation [6, 28, 29] and are also known in the literature as quantum processes with no memory [20], and cause-effect causal maps [14]. In a Markovian scenario, Alice receives a known quantum state $\rho\in\mathcal{L}(A_{I})$ which acts on her input system space. Alice can then perform an arbitrary quantum operation111Which may be a quantum channel (deterministic quantum operation) or a quantum instrument (probabilistic quantum operation).$\widetilde{\Lambda}:\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})$ to obtain a quantum state on system $A_{O}$, which will then be subjected to a known deterministic dynamics described by a channel $\widetilde{D}:\mathcal{L}(A_{O})\to\mathcal{L}(B_{I})$. Finally, the state $\widetilde{D}\Big{(}\widetilde{\Lambda}(\rho)\Big{)}\in\mathcal{L}(B_{I})$ arrives to Bob, who can perform an arbitrary measurement on it. See Fig. 1 for a circuit based pictorial illustration. A bipartite Markovian process can then be described by the known quantum state $\rho\in\mathcal{L}(A_{I})$ and the known dynamics represented by the channel $\widetilde{D}:\mathcal{L}(A_{O})\to\mathcal{L}(B_{I})$. In the Choi- Jamiołkowski representation, this can be conveniently described by $W_{\textrm{Markov}}:=\rho^{A_{I}}\otimes D^{A_{O}/B_{I}}$. Under this definition, for any quantum operation $\widetilde{\Lambda}:\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})$ performed by Alice, the quantum state arriving in Bob’s input space can be obtained via the link product as $\displaystyle\begin{split}W_{\textrm{Markov}}\Lambda^{A_{I}/A_{O}}&=\rho^{A_{I}}\otimes D^{A_{O}/B_{I}}\Lambda^{A_{I}/A_{O}}\\\ &=\rho^{A_{I}}D^{A_{O}/B_{I}}\Lambda^{A_{I}/A_{O}}\\\ &=D^{A_{O}/B_{I}}\Lambda^{A_{I}/A_{O}}\rho^{A_{I}}\\\ &=D^{A_{O}/B_{I}}\widetilde{\Lambda}(\rho^{A_{I}})\\\ &=\widetilde{D}\Big{(}\widetilde{\Lambda}(\rho)\Big{)}.\end{split}$ (7) Figure 1: Circuit representation of Markovian processes: Alice receives a fixed state $\rho\in\mathcal{L}(H_{A_{I}})$ on which she can perform an arbitrary quantum operation. After Alice’s operation, the system is subjected to a fixed dynamics described by a quantum channel $\widetilde{D}:\mathcal{L}(A_{O})\to\mathcal{L}(B_{I})$, then arriving at Bob’s input space laboratory. No auxiliary system is used in this circuit, which implies that correlations between the parties are only due to the direct communication from Alice to Bob. ###### Definition 2.1 (Markovian process). A linear operator $W_{\text{Markov}}\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is bipartite Markovian process if it can be written as $\displaystyle\begin{split}W_{\textrm{Markov}}:=&\,\rho^{A_{I}}D^{A_{O}/B_{I}},\\\ =&\,\rho^{A_{I}}\otimes D^{A_{O}/B_{I}},\end{split}$ (8) where $\rho^{A_{I}}$ is a quantum state i.e., $\rho\succeq 0$, $\mathrm{tr}(\rho)=1$ and $D^{A_{O}/B_{I}}$ is the Choi operator of a quantum channel from the output of Alice to Bob’s input i.e., $D\succeq 0$, $\mathrm{tr}_{B_{I}}(D)=\mathds{1}_{A_{O}}$. We denote the set of Markovian processes by $\mathcal{L}_{\textrm{Markov}}$. ### 2.2 Direct-cause processes In a Markovian process, all correlations between Alice and Bob admit a direct- cause interpretation, since no auxiliary system or environment is required at any point. In a scenario where Alice and Bob may share prior _classical_ correlations, it is natural to define direct-cause processes as any process which can be written as a probabilistic mixture of Markovian processes. Hence, the correlations arising from these processes can always be explained by classical mixture of direct-cause ones. ###### Definition 2.2 (Direct-cause process). A linear operator $W_{\text{DC}}\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is direct-cause if it is a classical mixture of Markovian processes, that is, $W_{\text{DC}}$ can be written as $\displaystyle W_{\textrm{DC}}:=$ $\displaystyle\sum_{i}p_{i}\rho_{i}^{A_{I}}D_{i}^{A_{O}/B_{I}}$ (9a) $\displaystyle=$ $\displaystyle\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}},\ p_{i}\in[0,1],$ (9b) where $\rho^{A_{I}}_{i}$ are quantum states, i.e., $\rho_{i}\succeq 0$ and $\mathrm{tr}(\rho_{i})=1\ \forall i$, whereas $D^{A_{O}/B_{I}}_{i}$ are Choi operators of quantum channels from the output of Alice to Bob’s input, i.e., $D_{i}\succeq 0$ and $\mathrm{tr}_{B_{I}}(D_{i})=\mathds{1}_{A_{O}}\ \forall i$. We denote the set of direct-cause processes by $\mathcal{L}_{\textrm{DC}}$. In the bipartite scenario, an ordered process is direct-cause if and only if it is a process without quantum memory, as defined and shown in Ref. [20]. See Appendix A for details. Although Ref. [19] analyzes tripartite ordered processes, when bipartite processes are considered, their definition is equivalent to the one presented in Refs. [14, 15], which is also equivalent to the one used here. We discuss this in more details in Section 6.2. #### 2.2.1 Quantum separability as an outer approximation for the set of DC processes A positive semi-definite linear operator $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is separable in the bipartition $A_{I}\|A_{O}B_{I}$ if there exist a probability distribution $p_{i}$ and positive semi-definite operators $\rho_{i}^{A_{I}}\succeq 0$ and $D_{i}^{A_{O}/B_{I}}\succeq 0$ such that $W=\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}}$. Which coincides with the definition of a DC process (Def. 2.2) without the trace normalization constraints. It follows directly from Eq. (9b) that every direct-cause process is separable in the bipartition $A_{I}\|A_{O}B_{I}$. Conversely, if a process is entangled in the bipartition $A_{I}\|A_{O}B_{I}$, such process cannot have a DC explanation. This allows us to employ techniques from entanglement theory to certify whether a process is not direct-cause. A simple outer approximation for the set of separable states is the set of states with positive partial transpose (PPT) [30, 31]. Also, it is known that the set of states with a PPT k-symmetric extension [32, 33] forms a hierarchy of sets which converges to the set of separable states when $k\to\infty$. For every fixed $k$, checking if a state has a PPT symmetric k-extension can be done via SDP. The precise definition of a PPT k-symmetric extension is provided in Appendix E. ###### Definition 2.3 (Separable outer approximation for DC processes). A bipartite ordered process $W$ belongs to $\mathcal{L}_{\text{DC}}^{\text{out},\text{PPT}_{k}}$ if $W$ has a PPT k-symmetric extension on the bipartition $A_{I}\|A_{O}B_{I}$. One could be tempted to assume that all process which are separable in $A_{I}\|A_{O}B_{I}$ bipartition have a direct-cause explanation. However, in Section 7, we prove that this is not the case by constructing an explicit example. This ensures that the problem of certifying if a given process is not direct-cause is not equivalent to certifying entanglement. ### 2.3 Common-cause processes Common-cause processes are those where the correlations are not due to communication from Alice to Bob, but only due to a bipartite quantum state $\rho\in\mathcal{L}(A_{I}\otimes B_{I})$ initially shared by them. In this way, a common-cause process does not involve any particular quantum dynamic and can be fully characterized by a fixed bipartite state. In order to establish a notation which dialogues to general scenarios which may have non- trivial dynamics between Alice and Bob, we describe common-cause processes by $W_{\textrm{CC}}=\rho^{A_{I}B_{I}}\otimes\mathds{1}^{A_{O}}$. ###### Definition 2.4 (Common-cause process). A linear operator $W_{\text{CC}}\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is a common-cause process if it can be written as $W_{\textrm{CC}}=\rho^{A_{I}B_{I}}\otimes\mathds{1}^{A_{O}},$ (10) where $\rho^{A_{I}B_{I}}$ is a quantum state shared between Alice and Bob, i.e., $\rho^{A_{I}B_{I}}\succeq 0$, $\mathrm{tr}(\rho^{A_{I}B_{I}})=1$, and $\mathds{1}^{A_{O}}$ is the identity operator acting on $A_{O}$. We denote the set of common-cause processes by $\mathcal{L}_{\textrm{CC}}$. ### 2.4 Classical CCDC processes Classical CCDC processes are processes which can be decomposed as a convex combination of a common-cause and a direct-cause ones. Such processes can always be understood as simple classical statistical mixtures between quantum common-cause and quantum direct-cause processes [14, 19]. ###### Definition 2.5 (Classical CCDC processes). A linear operator $W_{\text{CCDC}}\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is a classical CCDC process if it can be written as $W_{\textrm{CCDC}}:=pW_{\textrm{CC}}+(1-p)W_{\textrm{DC}},$ (11) where $W_{\textrm{CC}}\in\mathcal{L}_{\textrm{CC}}$, $W_{\textrm{DC}}\in\mathcal{L}_{\textrm{DC}}$, and $p\in[0,1]$. We denote the set of classical CCDC processes by $\mathcal{L}_{\textrm{CCDC}}$, which is the convex hull of $\mathcal{L}_{\textrm{CC}}\cup\mathcal{L}_{\textrm{DC}}$. ### 2.5 Bipartite ordered processes Previously, we have only considered processes which do not require any explicit use of auxiliary systems or environments. In a more general process, the initial state $\rho$ received by Alice may be defined in a larger space, and Alice’s lab can only operate on part of it. In order to tackle this situation, we now consider that the initial state is defined not only on Alice’s input system, but also in some auxiliary system with unconstrained finite dimension, that is, $\rho\in\mathcal{L}(A_{I}\otimes\textrm{aux})$. Now, Alice can only operate on the part of $\rho$ that enters her laboratory, that is, she can perform any operation $\widetilde{\Lambda}:\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})$, which will transform the initial state $\rho$ into222Here $\widetilde{\mathds{1}}^{\text{aux}}$ stands for the identity map on the auxiliary space, that is, for any operator $A\in\mathcal{L}(\textrm{aux})$ we have $\widetilde{\mathds{1}}^{\text{aux}}(A)=A$.$\widetilde{\Lambda}^{A_{I}\to A_{O}}\otimes\widetilde{\mathds{1}}^{\text{aux}}(\rho)$, which will be subjected to a global dynamics described by a quantum channel $\widetilde{D}:\mathcal{L}(A_{O}\otimes\textrm{aux})\to\mathcal{L}(B_{I})$. See Fig. 2 for a circuit based pictorial illustration. Figure 2: A circuit illustrating a bipartite ordered process $W=\rhoD$ where $\rho\in\mathcal{L}(A_{I}\otimes\textrm{aux})$ is a quantum state and $D$ is the Choi operator of a channel $\widetilde{D}:\mathcal{L}(A_{O}\otimes\textrm{aux})\rightarrow\mathcal{L}(B_{I})$. When a quantum channel $\widetilde{\Lambda}:\mathcal{L}(A_{I})\rightarrow\mathcal{L}(A_{O})$ with Choi operator $\Lambda$ is “plugged” into the process $W$, the state $\rho^{\prime}=W\Lambda\in\mathcal{L}(B_{I})$ is obtained. A bipartite ordered process can then be described by the known quantum state $\rho\in\mathcal{L}(A_{I}\otimes\textrm{aux})$ and the known dynamics represented by the channel $\widetilde{D}:\mathcal{L}(A_{O}\otimes\textrm{aux})\to\mathcal{L}(B_{I})$. In the Choi-Jamiołkowski representation, this can be conveniently described by $W:=\rho^{A_{I}\textrm{aux}}D^{A_{O}\textrm{aux}/B_{I}}$. Under this definition, for any quantum operation $\widetilde{\Lambda}:\mathcal{L}(A_{I})\to\mathcal{L}(A_{O})$ performed by Alice, the quantum state arriving in Bob’s input space can be obtained via the link product as $\displaystyle\begin{split}W\Lambda^{A_{I}/A_{O}}&=\rho^{A_{I}\textrm{aux}}D^{A_{O}\textrm{aux}/B_{I}}\Lambda^{A_{I}/A_{O}}\\\ &=D^{A_{O}/B_{I}}\Lambda^{A_{I}/A_{O}}\rho^{A_{I}\textrm{aux}}\\\ &=D^{A_{O}/B_{I}}\widetilde{\Lambda}^{A_{I}\to A_{O}}\otimes\widetilde{\mathds{1}}^{\text{aux}}\big{(}\rho^{A_{I}\textrm{aux}}\big{)}\\\ &=\widetilde{D}\Big{(}\widetilde{\Lambda}^{A_{I}\to A_{O}}\otimes\widetilde{\mathds{1}}^{\text{aux}}\big{(}\rho^{A_{I}\textrm{aux}}\big{)}\Big{)}.\end{split}$ (12) ###### Definition 2.6 (Bipartite ordered process). A linear operator $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is a bipartite ordered process if there exists a quantum state $\rho^{A_{I}\textrm{aux}}\in\mathcal{L}(A_{I}\otimes\textrm{aux})$,i.e., $\rho^{A_{I}\textrm{aux}}\succeq 0$, $\mathrm{tr}(\rho^{A_{I}\textrm{aux}})=1$ and a quantum channel $\widetilde{D}:\mathcal{L}(A_{I}\otimes\textrm{aux})\rightarrow\mathcal{L}(B_{I})$ with Choi operator $D^{A_{O}\textrm{aux}/B_{I}}$, i.e., $D^{A_{O}\textrm{aux}/B_{I}}\succeq 0$, $\mathrm{tr}_{B_{O}}(D^{A_{O}\textrm{aux}/B_{I}})=\mathds{1}_{A_{0},\textrm{aux}}$, such that $\displaystyle\begin{split}W:=&\rho^{A_{I}\textrm{aux}}D^{A_{O}\textrm{aux}/B_{I}}\\\ =&\mathrm{tr}_{\textrm{aux}}\Bigg{(}\left(\rho^{A_{I}\textrm{aux}}\otimes\mathds{1}^{A_{O}B_{I}}\right)^{T_{\textrm{aux}}}\\\ &\hskip 19.91684pt\cdot\left(\mathds{1}^{A_{I}}\otimes D^{A_{O}\textrm{aux}/B_{I}}\right)\Bigg{)}.\end{split}$ (13) We denote the set of bipartite ordered processes by $\mathcal{L}_{A\rightarrow B}$. Bipartite ordered process may be seen as quantum objects transforming a quantum operation into a quantum state333In Ref. [34, 11] the authors prove that under a set of assumptions, it can be proven that bipartite ordered process are indeed the most general method to transform arbitrary quantum operations into quantum states.and are also known in the literature as quantum non-Markovian processes [7], causal maps [14], quantum co-strategies [13], and ordered process matrices [35]. Also, ordered processes may also be seen as a particular instance of quantum channels with memory [12] and quantum combs [11]. In this general definition of bipartite ordered processes, Markovian processes form the particular case where there is no auxiliary space aux. Common-cause processes can also be represented as bipartite ordered processes. Indeed, if $W_{\textrm{CC}}=\rho^{A_{I}B_{I}}\otimes\mathds{1}^{A_{O}}$, we can set the auxiliary space $\mathcal{H}_{\text{aux}}$ as isomorphic to $\mathcal{H}_{B_{I}}$ and the initial state $\rho^{A_{I}\text{aux}}$ as isomorphic to $\rho^{A_{I}B_{I}}$. Then, one can set the channel $\widetilde{D}$ as an operation that sends the auxiliary system to $B_{I}$ and discard the system in $A_{O}$. This can be done formally via $\displaystyle W_{\textrm{CC}}$ $\displaystyle=\mathrm{tr}_{\textrm{aux}^{\prime}}\left[\rho^{A_{I}\textrm{aux}}\|{\textrm{U}_{\textrm{SWAP}}}\rangle\rangle\langle\langle{\textrm{U}_{\textrm{SWAP}}}\|^{A_{O}\textrm{aux}/B_{I}\textrm{aux}^{\prime}}\right]$ $\displaystyle=\rho^{A_{I}B_{I}}\otimes\mathds{1}^{A_{O}},$ (14) where $\textrm{U}_{\textrm{SWAP}}:=\sum_{ij}\|{ij}\rangle\langle{ji}\|$ is the swap operator. A circuit illustration of common-cause process as bipartite ordered ones is presented in Fig. 3. Figure 3: A general bipartite common-cause process can be represented by a circuit of this form. The shared quantum state between Alice and Bob comes from an initial correlated state between Alice and an auxiliary system. The state of the auxiliary system is exchanged with the system coming from the output of Alice and sent to Bob, and the auxiliary system is then discarded, which makes the output state of Alice to have no impact on the input of Bob. This makes only the input states of Alice and Bob to be relevant for the process, which is why the correlations between them stem from the common-cause correlations on $\rho$. With all the important processes defined, Fig. 4 shows the relationship between all the sets defined in this section. Figure 4: Illustrative representation of the sets in the CCDC scenario. The sets of common-cause processes ($\mathcal{L}_{\textrm{CC}}$) and of direct- cause processes ($\mathcal{L}_{\textrm{DC}}$) are proper subsets of the set of bipartite ordered process $\mathcal{L}_{A\rightarrow B}$, with a non-empty intersection between $\mathcal{L}_{\textrm{CC}}$ and $\mathcal{L}_{\textrm{DC}}$. Classical CCDC processes are all processes in the convex hull of $\mathcal{L}_{\textrm{CC}}\cup\mathcal{L}_{\textrm{DC}}$, represented by $\mathcal{L}_{\textrm{CCDC}}$. In addition to its definition, bipartite ordered processes admit a characterization in terms of linear and positive semi-definite constraints, which will be useful later in this work. If follows from direct inspection that all bipartite ordered process $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ respect $\displaystyle W$ $\displaystyle\succeq 0,$ (15a) $\displaystyle\mathrm{tr}_{B_{I}}(W)$ $\displaystyle=\sigma^{A_{I}}\otimes\mathds{1}^{A_{O}},$ (15b) where $\sigma^{A_{I}}$ is a quantum state. Interestingly, this condition is also sufficient [11]. For any linear operator $W$ respecting the conditions in Eq. (15), one can find a quantum state $\rho\in\mathcal{L}(A_{I}\otimes\textrm{aux})$ and a quantum channel $\widetilde{D}:\mathcal{L}(A_{O}\otimes\textrm{aux})\to\mathcal{L}(B_{I})$ such that $\rho^{A_{I}\text{aux}}D^{\text{aux}B_{I}/B_{O}}=W$. One possible construction is done by setting the auxiliary space aux to be isomorphic to $A_{I}$ and $\rho\in\mathcal{L}(A_{I}\otimes\textrm{aux})$ be be isomorphic to a purification of $\sigma^{A_{I}}$, for instance, $\rho:=\left(\mathds{1}^{A_{I}}\otimes\sqrt{\sigma^{\textrm{aux}}}^{T}\right)\;\|{\mathds{1}}\rangle\langle{\mathds{1}}\|^{A_{I}\text{aux}}\;\left(\mathds{1}^{A_{I}}\otimes\sqrt{\sigma^{\textrm{aux}}}^{T}\right).$ (16) Now, one can define a quantum channel via $\displaystyle\begin{split}D:=&\left(\sqrt{\sigma^{\textrm{aux}}}^{-1}\otimes\mathds{1}^{A_{O}B_{I}}\right)\\\ &W^{\text{aux}A_{O}B_{I}}\;\left(\sqrt{\sigma^{\textrm{aux}}}^{-1}\otimes\mathds{1}^{A_{O}B_{I}}\right),\end{split}$ (17) where $\sqrt{\sigma}$ is the unique positive semi-definite square root of $\sigma$ and $\sigma^{-1}$ is the Moore–Penrose inverse of $\sigma$, that is, the inverse of $\sigma$ on its range. In this way, direct inspection shows that $D$ is the Choi operator of a quantum channel and that $W=\rhoD$. Inspired by Ref. [36], we now present another characterization for bipartite ordered processes which will be useful for proving our results. A linear operator $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is a bipartite ordered process if and only if it respects $\displaystyle W$ $\displaystyle\succeq 0,$ (18a) $\displaystyle W$ $\displaystyle=L_{A\rightarrow B}(W),$ (18b) $\displaystyle\mathrm{tr}(W)$ $\displaystyle=d_{A_{O}},$ (18c) where $L_{A\rightarrow B}(W):=W+\prescript{}{A_{O}B_{I}}{W}-\prescript{}{B_{I}}{W},$ (19) with $\prescript{}{X}{(\cdot)}=\mathrm{tr}_{X}(\cdot)\otimes\frac{\mathds{1}^{X}}{d_{X}}$ and $L_{A\rightarrow B}(W)$ is the map projecting an operator $W$ into the linear space spanned by bipartite ordered processes. Eqs. (15) and (18) are equivalent conditions for an operator $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ to be a valid bipartite ordered process. In particular, conditions given by Eqs. (18) are rather useful for implementing our numerical methods, due to the use of the projector $L_{A\rightarrow B}(W)$. ## 3 Detecting and quantifying non-classical CCDC Inspired by the robustness of entanglement [37, 38], one can quantify the violation of a classical CCDC decomposition in a process $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ in terms of its generalized robustness of non-classical CCDC444Note that the definition of generalized robustness $R_{G}$ here has a one-to-one relation to the “non-classicality of causality” $\mathcal{C}$ presented in Ref.[19] via $R_{G}(W)=\frac{\mathcal{C}(W)}{1+\mathcal{C}(W)}$. [19] by $\displaystyle R_{G}(W):=\textrm{min}\hskip 5.69046pt$ $\displaystyle r$ subject to $\displaystyle(1-r)W+r\Omega\in\mathcal{L}_{\textrm{CCDC}},$ (20a) $\displaystyle 0\leq r\leq 1,$ (20b) $\displaystyle\Omega\in\mathcal{L}_{A\rightarrow B},$ (20c) where $\mathcal{L}_{A\rightarrow B}$ is the set of bipartite ordered processes defined in Eqs. (18). The generalized robustness $R_{G}(W)$ corresponds to how resistant is the non-classical CCDC property of the process $W$ against its worst possible noise. In Appendix B, we show that the generalized robustness of all bipartite processes is upper-bounded by $R_{G}(W)\leq 1-\frac{1}{d_{A_{I}}}$, which is attainable with the examples of non-classical CCDC processes we present in Section 5. Another quantification of non-classical CCDC is in terms of its _white-noise robustness_ , given by $\displaystyle R_{WN}(W):=\textrm{min}\hskip 5.69046pt$ $\displaystyle r$ subject to $\displaystyle(1-r)W+r\frac{\mathds{1}}{d_{A_{I}}d_{B_{I}}}\in\mathcal{L}_{\textrm{CCDC}},$ (21a) $\displaystyle 0\leq r\leq 1,$ (21b) where $\mathds{1}$ is the identity operator in $\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$. The value $R_{WN}(W)$ corresponds to how resistant is the non-classical CCDC property of the process $W$ against white noise. Figure 5 gives a representation of both types of robustness. In Appendix C, we show that the white noise robustness of all bipartite processes is upper-bounded by $R_{WN}(W)\leq 1-\frac{1}{d_{A_{I}}d_{A_{O}}d_{B_{I}}+1}$. Such bound is not tight, but is useful to demonstrate that the convex optimization problem of Eq.(21) respects strong duality (see Appendix D for more details). Figure 5: Representation of the problem of determining if a given process $W$ is non-classical CCDC. The above figure illustrates an ordered process operator $\Omega\in\mathcal{L}_{A\rightarrow B}$ such that the minimum convex combination of it with the process $W$ results in a classical CCDC process. ### 3.1 Witnessing non-classical CCDC processes A non-classical CCDC witness is a Hermitian operator $S\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ which respects $\mathrm{tr}(SW_{\text{CCDC}})\geq 0$ for every classical CCDC process $W_{\text{CCDC}}$ [19]. Non-classical CCDC witnesses that are not trivial are the ones having $\mathrm{tr}(SW)<0$ for some non-classical CCDC process $W$, as the main purpose of a witness is to certify that a given process is non- classical CCDC. Since the set of classical CCDC processes is closed and convex, similarly to entanglement [39], every non-classical CCDC process may be certified by a non-classical CCDC witness. Also, as proven in Appendices D and E, the convex optimization problems used to define generalized and white noise robustness of non-classical CCDC respect a strong duality condition and the dual formulation of our outer approximation can be used to explicitly construct non-classical CCDC witness. Similarly to entanglement witnesses, verifying if an arbitrary operator $S\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is a non-classical CCDC witness is likely to be computationally hard [40], as it would require, in principle, to verify that $\mathrm{tr}(SW_{\text{CCDC}})\geq 0$ for all classical CCDC processes $W_{\text{CCDC}}$. Despite the hardness of the general problem, we now provide simple sufficient (but not necessary) analytical conditions to ensure that $S$ is a non-classical CCDC witness. Any operator $S\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ respecting $\mathrm{tr}_{A_{O}}(S)\succeq 0$ and $S^{T_{A_{I}}}\succeq 0$ is a non- classical CCDC witness. This claim follows from the fact that every CC process respects $W_{\text{CC}}=\prescript{}{A_{O}}{W_{\text{CC}}}\succeq 0$ and every DC process respects $W_{\text{DC}}^{T{{}_{A_{I}}}}\succeq 0$. Hence, $\displaystyle\begin{split}\mathrm{tr}(SW_{\text{CCDC}})&=p\leavevmode\nobreak\ \mathrm{tr}(SW_{\text{CC}})+(1-p)\mathrm{tr}(SW_{\text{DC}})\\\ &=p\leavevmode\nobreak\ \mathrm{tr}(S\prescript{}{A_{O}}{W_{\text{CC}}})+(1-p)\mathrm{tr}(S^{T_{A_{I}}}W^{T_{A_{I}}}_{\text{DC}})\\\ &=p\leavevmode\nobreak\ \mathrm{tr}(\prescript{}{A_{O}}{S}W_{\text{CC}})+(1-p)\mathrm{tr}(S^{T_{A_{I}}}W^{T_{A_{I}}}_{\text{DC}})\\\ &\geq 0.\end{split}$ (22) ## 4 Beyond the entanglement approximation: SDP hierarchies for tight non- classical CCDC certification As stated in Sec. 2.2.1, the set of separable processes in the bipartition $A_{I}\|A_{O}B_{I}$ provides an outer approximation for the set of DC processes. As pointed in Ref. [19, 20], this provides a SDP approach that gives lower bounds for the non-DC and non-classical CCDC robustness of a bipartite ordered process. Indeed, since the relation $\mathcal{L}_{\text{DC}}\subseteq\mathcal{L}_{\text{DC}}^{\text{PPT}_{k}}$ holds for every natural number $k$, we can define the convex hull $\mathcal{L}_{\text{CCDC}}^{\text{out},\text{PPT}_{k}}:=\text{conv}(\mathcal{L}_{\text{CC}},\mathcal{L}_{\text{DC}}^{\text{out},\text{PPT}_{k}})$, and the quantity $\displaystyle R_{G}^{\text{low},\text{PPT}_{k}}(W):=\textrm{min}\hskip 5.69046pt$ $\displaystyle r$ subject to $\displaystyle(1-r)W+r\Omega\in\mathcal{L}_{\text{CCDC}}^{\text{out},\text{PPT}_{k}}$ (23a) $\displaystyle 0\leq r\leq 1,$ (23b) $\displaystyle\Omega\in\mathcal{L}_{A\rightarrow B},$ (23c) which gives a lower-bound for the actual generalized robustness, since the relation $R_{G}^{\text{low,PPT}_{k}}(W)\leq R_{G}(W)$ holds for every $W$. Analogously, we can also obtain a lower-bound for the white noise robustness by defining $R_{WN}^{\text{low,PPT}_{k}}(W)$. As we show in Sec. 7, approximating the set of DC processes by the set of separable process is rather limited. In particular, there exist processes which are separable in the bipartition $A_{I}\|A_{O}B_{I}$ but are non- classical CCDC. Hence, such processes would never be detected by a method solely based in quantum entanglement techniques. In this section, we overcome this problem by presenting two novel SDP hierarchies of sets which converge to the set of CCDC processes. Also, one hierarchy is based in an inner approximation, whereas the other constitutes an outer approximation. Hence, we can obtain a sequence of converging upper and lower bounds on the non- classical CCDC robustnesses. All the sets discussed in this section are illustrated in Fig. 6. Figure 6: Hierarchical relation between the set of processes separable in the bipartition $A_{I}\|A_{O}B_{I}$ ($\mathcal{L}_{\text{SEP}}^{A_{I}\|A_{O}B_{I}}$) and the set of direct-cause processes $\mathcal{L}_{\text{DC}}$. In the limit of large $k$, $\mathcal{L}_{\text{SEP}}^{\text{PPT}_{k}}$ converges to $\mathcal{L}_{\text{SEP}}^{A_{I}\|A_{O}B_{I}}$. By increasing the set of $N$ states, the inner approximation ($\mathcal{L}_{\text{DC}}^{\text{in},\mathcal{E}_{N}}$) converges to $\mathcal{L}_{\text{DC}}$. Similarly, by increasing the set of $N$ trace-one operators $\\{\hat{\rho}_{i=1}^{N}\\}_{i}$, the outer approximation $\mathcal{L}_{\text{DC}}^{\text{out},\hat{\mathcal{E}}_{N}}$ also converges to $\mathcal{L}_{\text{DC}}$. The point $W_{\text{SEP}}$ stands for the example of process which is separable in the bipartition $A_{I}\|A_{O}B_{I}$, but is non-classical CCDC, defined and discussed in Sec. 7. ### 4.1 Inner approximation for direct-cause processes The following inner approximation leads to a hierarchy that converges to the set of direct-cause processes and is inspired by the SDP approach for entanglement detection introduced in [41]. ###### Definition 4.1 (Inner approximation for the direct-cause set). Let $\mathcal{E}_{N}:=\\{\|{\psi_{i}}\rangle^{A_{I}}\\}_{i=1}^{N}$ be a fixed set of pure quantum states $\|{\psi_{i}}\rangle^{A_{I}}$. The set $\mathcal{L}_{\text{DC}}^{\text{in},\mathcal{E}_{N}}$ corresponds to the inner approximation of the set $\mathcal{L}_{\text{DC}}$ and is composed by every process $W$ that can be written as $W=\sum_{i}p_{i}\|{\psi_{i}}\rangle\langle{\psi_{i}}\|^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}},$ (24a) where $\sum_{i}p_{i}=1\;\text{ and }\;p_{i}\geq 0,\;\forall i,$ (24b) and $D_{i}^{A_{O}/B_{I}}$ are the Choi operators of quantum channels, i.e., $D_{i}^{A_{O}/B_{I}}\succeq 0\;\text{ and }\;\mathrm{tr}_{B_{I}}\big{(}D_{i}^{A_{O}/B_{I}}\big{)}=\mathds{1}^{A_{O}},\;\forall i.$ (24c) Note that in Def. 4.1, we imposed the partial trace quantum channel normalization condition $\mathrm{tr}_{B_{I}}\big{(}D_{i}^{A_{O}/B_{I}}\big{)}=\mathds{1}^{A_{O}}$, which does not appear in an entanglement-based approach where we only impose positivity and the full trace constraint. As we will see in the further sections, this corresponds to a fundamental difference, since the set of separable quantum states is not equivalent to the set of direct-cause processes. By construction, for any set of quantum states $\mathcal{E}_{N}$, we have the relation $\mathcal{L}_{\text{DC}}^{\text{in},\mathcal{E}_{N}}\subseteq\mathcal{L}_{\text{DC}}$. Also, if $\mathcal{E}_{\infty}$ is the set of all pure quantum states in $\mathcal{L}(A_{I})$, we have $\mathcal{L}_{\text{DC}}^{\text{in},\mathcal{E}_{\infty}}=\mathcal{L}_{\text{DC}}$. We can now define the convex hull $\mathcal{L}_{\text{CCDC}}^{\text{in},\mathcal{E}_{N}}:=\text{conv}(\mathcal{L}_{\text{CC}},\mathcal{L}_{\text{DC}}^{\text{in},\mathcal{E}_{N}})$, and the quantity $\displaystyle\begin{split}R_{G}^{\text{up,}\mathcal{E}_{N}}(W):=\textrm{min}\hskip 5.69046pt&r\\\ \textrm{subject to}\hskip 5.69046pt&(1-r)W+r\Omega\in\mathcal{L}_{\textrm{CCDC}}^{\text{in,}\mathcal{E}_{N}},\\\ &0\leq r\leq 1,\\\ &\Omega\in\mathcal{L}_{A\rightarrow B},\end{split}$ (25) which provides an upper-bound for the generalized robustness $R_{G}(W)$, since the relation $R_{G}(W)\leq R_{G}^{\text{up},\mathcal{E}_{N}}(W)$ holds for every bipartite ordered process $W$ and $\mathcal{E}_{N}$. As discussed in details in Appendix E, the quantity $R_{G}^{\text{up},\mathcal{E}_{N}}(W)$ can be evaluated by an SDP. Moreover, this SDP converges to the exact value of $R_{G}(W)$ when the set of states $\mathcal{E}_{N}$ contains all pure states in $\mathcal{L}(A_{I})$. Analogously, we can also obtain an upper-bound for the white noise robustness by defining $R_{WN}^{\text{up},\mathcal{E}_{N}}(W)$. ### 4.2 Outer approximation for direct-cause processes For the outer approximation for the set of direct-cause processes, we will make use of an outer approximation for the set of quantum states. Since the set of quantum states is convex, we can always construct an outer approximation given by a polytope, similarly to the strategy used to construct a polytopes providing outer approximations for the set of quantum measurements [42, 43, 44, 45, 46]. Let $\mathcal{S}_{d}$ be the set of quantum states with dimension $d$, there exists a set of linear operators $\hat{\rho}_{i}\in\mathcal{L}(\mathbb{C}_{d})$ that satisfy $\mathrm{tr}(\hat{\rho}_{i})=1$ and such that the convex hull of $\\{\hat{\rho}_{i}\\}_{i}$ contains $\mathcal{S}_{d}$. Note that the operators $\hat{\rho}_{i}$ are _not_ required to be positive semi-definite, hence they do not correspond to quantum states. For the moment, let us assume that we know a set of trace-one operators $\\{\hat{\rho}_{i=1}^{N}\\}_{i}$ such that $\text{conv}(\\{\hat{\rho}_{i}\\}_{i=1}^{N})\supseteq\mathcal{S}_{d}$. We can then use this finite set to construct an outer approximation for the set of DC processes, similarly to the inner approximation in Def. 4.1. ###### Definition 4.2 (Outer approximation for the direct-cause set). Let $\hat{\mathcal{E}}_{N}:=\\{\hat{\rho}_{i}^{A_{I}}\\}_{i=1}^{N}$ be a fixed set of linear operators with $\mathrm{tr}(\hat{\rho}_{i})=1\ \forall i$, such that the convex hull of $\hat{\mathcal{E}}_{N}$ contains the set of all quantum states acting on $A_{I}$. A bipartite ordered process $W$ is in the set $\mathcal{L}_{\text{DC}}^{\text{out},\hat{\mathcal{E}}_{N}}$ if it can be written as $W=\sum_{i}p_{i}\hat{\rho}_{i}^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}},$ (26a) where $\sum_{i}p_{i}=1\;\text{ and }\;p_{i}\geq 0\;\forall i,$ (26b) and $D_{i}^{A_{O}/B_{I}}$ are the Choi operators of quantum channels i.e. $D_{i}^{A_{O}/B_{I}}\succeq 0\;\text{ and }\;\mathrm{tr}_{B_{I}}\big{(}D_{i}^{A_{O}/B_{I}}\big{)}=\mathds{1}^{A_{O}}\;\forall i.$ (26c) By construction, we have that $\mathcal{L}_{\text{DC}}\subseteq\mathcal{L}_{\text{DC}}^{\text{out},\hat{\mathcal{E}}_{N}}$ and the set $\mathcal{L}_{\text{CCDC}}^{\text{out},\hat{\mathcal{E}}_{N}}:=\text{conv}(\mathcal{L}_{\text{CC}}\cup\mathcal{L}_{\text{DC}}^{\text{out},\hat{\mathcal{E}}_{N}})$ is an outer approximation for $\mathcal{L}_{\text{CCDC}}$. The generalized and white noise robustnesses are defined in a way analogous to what is done in Sec. 4.1, thus being denoted by $R_{G}^{\text{low},\hat{\mathcal{E}}_{N}}(W)$ and $R_{WN}^{\text{low},\hat{\mathcal{E}}_{N}}(W)$, and these are lower bounds for the robustness $R_{G}(W)$ and $R_{WN}(W)$ respectively. We now address the problem of finding sets of trace-one operators which contain the set of quantum states. Let us first start with the simple qubit case ($d=2$) where the set of quantum states can be faithfully represented in terms of the Bloch sphere [47]. For this case, we can construct a set of trace-one operators $\\{\hat{\rho}_{i}\\}_{i}$ such that $\text{conv}(\\{\hat{\rho}_{i}\\}_{i})\supseteq\mathcal{S}_{2}$ simply by finding a polyhedron that includes a sphere of unit radius. One method to find such polyhedron goes as following: 1. 1. Sort $N$ normalized vectors $\\{\vec{v_{i}}\\}_{i=1}^{N}$ in $\mathbb{R}^{3}$. These will be the vertices of a polyhedron inscribed by the Bloch sphere; 2. 2. Find the radius $r_{\text{in}}<1$ of the largest sphere inscribed by the polyhedron; 3. 3. Construct a polyhedron with vertices given by $\frac{\vec{v_{i}}}{r_{\text{in}}}$. This polyhedron includes the Bloch sphere. The last step is ensured by the symmetry of the problem, a sphere of radius $r_{\text{in}}$ is contained in the polytope with norm-one vertices $\\{\vec{v_{i}}\\}_{i=1}^{N}$ if and only if a sphere of radius one contains the polytope with vertices $\frac{\vec{v_{i}}}{r_{\text{in}}}$. Also, the radius of the largest sphere inscribed by a polytope (required in step 2) can be made by first finding the facet representation of this polytope with vertices $\\{\vec{v_{i}}\\}_{i=1}^{N}$, step which can be done via Fourier- Motzkin elimination and can be tackled with the aid of numerical packages such as lrs [48] or the Matlab code vert2lcon [49]. With the facet representation, the radius of the largest inscribed sphere can be found by evaluating the distance of the origin to the hyperplane represented by each facet. A concrete example for the qubit case can be found in the Appendix B: “A family of polyhedra”, of Ref. [50]. In such work, the authors provide a family of polyhedra parameterized by $n\in\mathbb{N}$ where the radius of the larger inscribed sphere is greater than or equals to $r_{n}:=\cos^{2}\big{(}\frac{\pi}{2n}\big{)}$ and if $n$ is an odd number, the number of vertices is given by $N=2n^{2}$. This family of polyhedra can be used for a systematic approach to the problem. For the general $d>3$, we start by pointing out that every convex set can be approximated by a polytope. Now, in order to obtain a concrete set of operators $\\{\hat{\rho}_{i}\\}_{i}$, we refer to the methods described in Appendix A: “Calculating shrinking factors”, of Ref. [43] (see also Red. [51]). In a nutshell, the method starts by sampling random pure (which are extremal) states $\|{\psi_{i}}\rangle\langle{\psi_{i}}\|$ in the set $\mathcal{S}_{d}$ and by obtaining the facet representation for the polytope associated to this set. Then, for any fixed “shrinking factor” $\eta$, we can verify whether a noisy quantum state $\eta\rho+(1-\eta)\mathds{1}/d$ can be written as a convex combination of $\\{\|{\psi_{i}}\rangle\langle{\psi_{i}}\|\\}_{i}$ by means of an SDP. If all $\eta$-noisy states can be written as convex combinations of $\\{\|{\psi_{i}}\rangle\langle{\psi_{i}}\|\\}_{i}$, the convex hull of operators is given by $\hat{\rho}_{i}:=\frac{1}{\eta}\|{\psi_{i}}\rangle\langle{\psi_{i}}\|+1-\frac{1}{\eta}\mathds{1}/d$. We observe that we can tighten the outer approximation presented in Def. 4.2 by imposing that the process $W$ should be separable in the bipartition $A_{I}\|A_{O}B_{I}$. Also, if we do this by means of the $PPT$ k-symmetric extension (as in Def. 2.3), the problem can still be tackled by means of an SDP. Set \| Robustness \| Requirements for set membership ---\|---\|--- $\mathcal{L}_{\text{DC}}^{\text{out,PPT}_{k}}$ \| $R^{\text{low,PPT}_{k}}$ \| $W$ is a valid bipartite ordered process and has a PPT $k$-symmetric extension. (This hierarchy converges to the set of separable processes, which is strictly larger than $\mathcal{L}_{\text{DC}}$) $\mathcal{L}_{\textrm{DC}}^{\text{out,}\hat{\mathcal{E}}_{N}}$ \| $R^{\text{low},\hat{\mathcal{E}}_{N}}$ \| Given a set of operators $\hat{\mathcal{E}}_{N}=\\{\hat{\rho}^{A_{I}}_{i}\\}_{i=1}^{k}$ forming an outer approximation for the set of quantum states, there exist quantum channels $D_{i}^{A_{O}/B_{I}}$ s.t. $W=\sum_{i}p_{i}\hat{\rho}^{A_{I}}_{i}\otimes D_{i}^{A_{O}/B_{I}}$. $\mathcal{L}_{\textrm{DC}}^{\text{in,}\mathcal{E}_{N}}$ \| $R^{\text{up},\mathcal{E}_{N}}$ \| Given a set of states $\mathcal{E}_{N}=\\{\rho^{A_{I}}_{i}\\}_{i=1}^{k}$, there exist quantum channels $D_{i}^{A_{O}/B_{I}}$ s.t. $W=\sum_{i}p_{i}{\rho}^{A_{I}}_{i}\otimes D_{i}^{A_{O}/B_{I}}$. Table 1: Summary of the sets used as approximation to the set of direct-cause processes, $\mathcal{L}_{\text{DC}}$, ordered from the outside to the inside of the set. The operators $D_{i}^{A_{O}/B_{I}}$ stands for Choi operators of quantum channels, $D_{i}^{A_{O}/B_{I}}\succeq 0$, $\mathrm{tr}_{B_{I}}(D_{i}^{A_{O}/B_{I}})=\mathds{1}$ and $\\{p_{i}\\}_{i}$ is a probability distribution. The operators $\hat{\rho}^{A_{I}}_{i}$ are not positive semi-definite, hence they are not quantum states, but the convex hull of the set $\\{\hat{\rho}^{A_{I}}_{i}\\}_{i=1}^{k}$ is a valid outer approximation for the set of quantum states. The precise definitions of these sets are given in Def. 2.3, Def. 4.2, and Def. 4.1. These approximations can be used for generating approximations for the set of classical CCDC processes, as $\mathcal{L}_{\text{CCDC}}=\text{conv}(\mathcal{L}_{\text{CC}}\cup\mathcal{L}_{\text{DC}})$. We present a summary of all the sets defined as approximations to the set $\mathcal{L}_{\text{DC}}$ in Table 1. For obtaining the corresponding inner or outer approximations for the set of classical CCDC processes $\mathcal{L}_{\text{CCDC}}$, it is necessarily to take the convex hull of $\mathcal{L}_{\text{CC}}$ with the chosen set for approximating $\mathcal{L}_{\text{DC}}$. Before finishing this section, we state that, throughout this work, whenever we calculate the non-classical CCDC robustnesses of the inner and outer approximations for $\mathcal{L}_{\text{CCDC}}$, the first lower bounds to be calculated are always provided by the outer approximation $\mathcal{L}_{\text{CCDC}}^{\text{out,PPT}_{k}}$. If the lower bounds obtained with such approximation does not match the upper bounds, then we take the outer approximation provided by $\mathcal{L}_{\text{CCDC}}^{\text{out,}\hat{\mathcal{E}}_{N}}$. ## 5 Simplest non-classical CCDC processes ### 5.1 The scenario where $d_{A_{I}}=d_{A_{O}}=d$ and $d_{B_{I}}=d^{2}$ We now present a “simple” process which has the highest generalized robustness in the scenario where $d_{A_{I}}=d_{A_{O}}=d$ and $d_{B_{I}}=d^{2}$. In this process, Alice shares a $d$-dimensional maximally entangled state $\|{\phi^{+}}\rangle=\frac{1}{\sqrt{d}}\sum_{i=0}^{d-1}\|{ii}\rangle$ with the auxiliary system, and the channel of communication to Bob (decoder) is composed of an identity channel, $D=\|{\mathds{1}}\rangle\rangle\langle\langle{\mathds{1}}\|^{\textrm{aux}A_{O}/B_{I}^{1}B_{I}^{2}}$ (see Fig. 7). This process is mathematically represented by $\displaystyle\begin{split}W_{ddd^{2}}&:=\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}\textrm{aux}}\|{\mathds{1}}\rangle\rangle\langle\langle{\mathds{1}}\|^{\textrm{aux}A_{O}/B_{I}^{1}B_{I}^{2}}\\\ &=\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}B_{I}^{1}}\otimes\|{\mathds{1}}\rangle\rangle\langle\langle{\mathds{1}}\|^{A_{O}/B_{I}^{2}}.\end{split}$ (27) Figure 7: Circuit representation of the process $W_{ddd^{2}}$. This process consists in the preparation of a maximally entangled state between Alice and an auxiliary system and an identity channel of communication to Bob. We argue that this process has a conceptually simple realization, since it only requires the preparation of a maximally entangled state and the identity channel. The process $W_{ddd^{2}}$ has the interesting property of transforming a quantum channel $\widetilde{\Lambda}:\mathcal{L}(A_{I})\rightarrow\mathcal{L}(A_{O})$ into a state which is proportional to its Choi operator $\Lambda=\sum_{ij}\|{i}\rangle\langle{j}\|\otimes\widetilde{\Lambda}(\|{i}\rangle\langle{j}\|)$, that is $\displaystyle\begin{split}W_{ddd^{2}}\Lambda&=\left(\widetilde{\mathds{1}}\otimes\widetilde{\Lambda}\right)\|{\phi^{+}}\rangle\langle{\phi^{+}}\|,\\\ &=\frac{\Lambda}{d}.\end{split}$ (28) Despite the simplicity, we now show that $W_{ddd^{2}}$ is non-classical CCDC and, moreover, it is the one with highest generalized robustness in the scenario where $d_{A_{I}}=d_{A_{O}}=d$ and $d_{B_{I}}=d^{2}$. ###### Theorem 5.1. The bipartite ordered process $W_{ddd^{2}}=\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}B_{I}^{1}}\otimes\|{\mathds{1}}\rangle\rangle\langle\langle{\mathds{1}}\|^{A_{O}/B_{I}^{2}}$ attains the maximum generalized robustness of all processes with the same dimensions. That is, $\displaystyle\begin{split}R_{G}(W_{ddd^{2}})=&\max_{W\in\mathcal{L_{A\rightarrow B}}}\left[R_{G}(W)\right]\\\ =&1-\frac{1}{d_{A_{I}}}\end{split}$ (29) ###### Proof. We start the proof by defining the operator $\displaystyle\begin{split}S:=&\mathds{1}^{A_{I}A_{O}B_{I}}-W_{ddd^{2}}\\\ =&\mathds{1}^{A_{I}A_{O}B_{I}}-\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}B_{I}^{1}}\otimes\|{\mathds{1}}\rangle\rangle\langle\langle{\mathds{1}}\|^{A_{O}/B_{I}^{2}}\end{split}$ (30) and showing that $S$ is a valid non-classical CCDC witness, _i.e._ , every CCDC process $W_{\text{CCDC}}$ satisfies $\mathrm{tr}(SW_{\text{CCDC}})\geq 0$. To ensure that $S$ is a witness, we start by pointing that $\displaystyle\begin{split}\mathrm{tr}_{A_{O}}S&=d\mathds{1}^{A_{I}B_{I}}-\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}B_{I}^{1}}\otimes\mathds{1}^{B_{I}^{2}}\\\ &\succeq 0,\end{split}$ (31) where the last inequality holds because the smallest eigenvalue of $\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}B_{I}^{1}}\otimes\mathds{1}^{B_{I}^{2}}$ is $1$. Now, note that $\displaystyle\begin{split}S^{T_{A_{I}}}&=\mathds{1}-\sum_{ij}\frac{\|{ij}\rangle\langle{ji}\|}{d}^{A_{I}B_{I}^{1}}\otimes\|{\mathds{1}}\rangle\rangle\langle\langle{\mathds{1}}\|^{A_{O}/B_{I}^{2}}\\\ &=\mathds{1}-\frac{\textrm{U}_{\textrm{SWAP}}^{A_{I}B_{I}^{1}}}{d}\otimes\|{\mathds{1}}\rangle\rangle\langle\langle{\mathds{1}}\|^{A_{O}/B_{I}^{2}}\\\ &\succeq 0,\end{split}$ (32) where the last inequality holds because the eigenvalues of $\textrm{U}_{\textrm{SWAP}}$ are $+1$ or $-1$. Conditions (31) and (32) together ensure that $S$ is a witness, as shown in Eqs. (22). Direct calculation shows that $\mathrm{tr}(SW_{ddd^{2}})=d-d^{2}=d(1-d)$ and that, for any bipartite ordered process $\Omega$, we have $\displaystyle\mathrm{tr}(S\Omega)$ $\displaystyle=\mathrm{tr}(\Omega)-\mathrm{tr}(\Omega\,W_{ddd^{2}})$ (33a) $\displaystyle\leq\mathrm{tr}(\Omega)=d.$ (33b) Since $S$ is a non-classical CCDC witness, if $(1-r)W_{ddd^{2}}+r\Omega$ is a CCDC process, it holds that ${\mathrm{tr}\left(S\left[(1-r)W_{ddd^{2}}+r\Omega\right]\right)\geq 0}$. By combining $\mathrm{tr}(SW_{ddd^{2}})=d(1-d)$ with the inequality (33b), we have $\displaystyle(1-r)d(1-d)+rd\geq 0,$ (33c) thus $r\geq 1-\frac{1}{d}$. We finish the proof by invoking Theorem B.3, which states that $r\leq 1-\frac{1}{d}$. ∎ We now analyze the robustness of $W_{ddd^{2}}$ against the white noise process. For that, we use the SDP formulation to numerically tackle the case where $d=2$ and $d=3$. For the inner approximation, we have used the set $\mathcal{L}_{\text{CCDC}}^{\text{in},\mathcal{E}_{N}}$ with ${\mathcal{E}_{N}}$ being a set with $N=10^{4}$ uniformly random pure states for $d=2$ and $N=200$ uniformly random pure states for $d=3$. For the outer approximation, it was enough to use the loose approximation $\mathcal{L}_{\text{CCDC}}^{\text{out,PPT}}$. The results obtained were $\displaystyle R_{WN}^{\text{up},\mathcal{E}_{10^{4}}}(W_{222^{2}})=R_{WN}^{\textrm{low,PPT}}(W_{222^{2}})=0.8421$ (34a) $\displaystyle R_{WN}^{\text{up},\mathcal{E}_{200}}(W_{333^{2}})=R_{WN}^{\textrm{low,PPT}}(W_{333^{2}})=0.9529,$ (34b) where equations hold up to numerical precision. Since the upper and lower- bound coincides, these values are the actual values of robustnesses. For $d=2$, we believe that $W_{222^{2}}$ is the process with maximum white noise robustness on its scenario. Our conjecture is based on a heuristic see- saw technique inspired by [52, 53], which suggests that the highest value of white noise robustness in the scenario where $d=2$ is $0.8421$. In a nutshell, our see-saw technique goes as follows: for a fixed scenario $d_{A_{I}}=d_{A_{O}}=d$ and $d_{B_{I}}=d^{2}$, we sample a random bipartite ordered process $W$. Then, we obtain its optimal non-classical CCDC witness $S$ from the dual formulation of the PPT white noise robustness (see Appendix E). We then find the bipartite ordered process which maximally violates the witness $S$, then finding its optimal witness after that. This process is re- iterated until it converges to a robustness value, which we expect to be considerably greater than the robustness of the initially sampled process $W$. The see-saw method is described in details in Appendix F. Moreover, in the scenario where $d=3$, our see-saw method could find a bipartite ordered process $W_{\text{max}}$ which has $R_{WN}^{\text{up},\mathcal{E}_{200}}(W_{\text{max}})=R_{WN}^{\text{low,PPT}}(W_{\text{max}})=0.9643>R_{WN}^{\text{up,}\mathcal{E}_{200}}(W_{333^{2}})$. This result shows the power of the see-saw method in finding processes which are robust against white noise. It also shows that, even though $W_{ddd^{2}}$ has maximum generalized robustness for every $d$, which we proved analytically, in the case where $d=3$, $W_{333^{2}}$ is not the most robust process against white noise. This leads us to conjecture that $W_{ddd^{2}}$ may be not the most robust process against white noise for $d>3$. Before finishing this section we mention that, following the same steps from the demonstration of Theorem 5.1, we can also obtain an analytical lower-bound for white noise robustness of $W_{ddd^{2}}$, which is $R_{WN}(W_{ddd^{2}})\geq\frac{d^{3}}{(d^{2}+1)(d+1)}.$ (35) Differently from the lower-bound presented for generalized robustness, the above inequality is not tight. For instance, when $d=2$, this lower-bound provides $R_{WN}(W_{ddd^{2}})\geq\frac{8}{15}\approx 0.5333$, which is considerably lower than $R_{WN}^{\text{low,PPT}}(W_{ddd^{2}})=0.8421$. However, it is interesting to point out that the above expression shows that $R_{WN}(W_{ddd^{2}})\rightarrow 1$ when $d\rightarrow\infty$. ### 5.2 The scenario with minimum dimensions: $d_{A_{I}}=d_{A_{O}}=d_{B_{I}}=2$ Figure 8: Circuit representation of the process $W_{222}$, constructed with the smallest possible space dimensions $d_{A_{I}}=d_{A_{O}}=d_{B_{I}}=2$. A maximally entangled state is initially shared between Alice and the auxiliary system. After Alice’s operation, a control-NOT is applied, then $\textrm{aux}^{\prime}$ is discarded. We now consider the scenario where $d_{A_{I}}=d_{A_{O}}=d_{B_{I}}=2$, that is, the minimum non-trivial dimensions. For this scenario, we propose the process composed by a two-qubit maximally entangled state between Alice and the auxiliary system, and the decoder channel from Alice output to Bob’s input being a control-NOT channel, where the auxiliary system is later discarded (see Fig. 8). Mathematically, such process is described by $\displaystyle\begin{split}W_{222}:=\ \mathrm{tr}_{\textrm{aux}^{\prime}}\Big{(}&\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}\textrm{aux}}\\\ &\|{\textrm{U}_{\textrm{CNOT}}}\rangle\rangle\langle\langle{\textrm{U}_{\textrm{CNOT}}}\|^{\textrm{aux}A_{O}/\textrm{aux}^{\prime}B_{I}}\Big{)}\end{split}$ (36) where $\|{\textrm{U}_{\textrm{CNOT}}}\rangle\rangle$ is the Choi vector of the control-NOT unitary gate $\textrm{U}_{\textrm{CNOT}}$, given by $\textrm{U}_{\textrm{CNOT}}=\left(\begin{matrix}1&0&0&0\\\ 0&1&0&0\\\ 0&0&0&1\\\ 0&0&1&0\\\ \end{matrix}\right).$ (37) Direct calculation shows that the process $W_{222}$ can also be written in terms of a un-normalized $GHZ$ state $\|{GHZ}\rangle\rangle:=\|{000}\rangle+\|{111}\rangle\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ : $W_{\textrm{222}}=\frac{1}{2}\left(\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|+\sigma_{X}^{A_{O}}\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|\sigma_{X}^{A_{O}}\right),$ (38) where $\sigma_{X}^{A_{O}}$ is the Pauli matrix $\sigma_{X}$ applied on subsystem $A_{O}$. The decomposition of Eq. (38) expresses $W_{222}$ as a probabilistic mixture of pure “GHZ-like” non-normalized states555Interestingly, one can verify that for every $\epsilon\in\left(0,\frac{1}{2}\right]$, the operator $W=\left(\frac{1}{2}+\epsilon\right)\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|+\left(\frac{1}{2}-\epsilon\right)\sigma_{X}^{A_{O}}\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|\sigma_{X}^{A_{O}}$ is outside $\mathcal{L}_{A\rightarrow B}$. This ensures that, as illustrated in Fig. 9, $W_{222}$ is on the boundary of $\mathcal{L}_{A\rightarrow B}$. and it will be useful to prove that $W_{222}$ attains the maximum generalized robustness value for its scenario. Another decomposition of the process $W_{222}$ is in terms of Pauli matrices: $\displaystyle\begin{split}W_{222}=\frac{1}{4}&\bigg{(}\mathds{1}^{A_{I}A_{O}B_{I}}+\sigma_{Z}^{A_{I}}\mathds{1}^{A_{O}}\sigma_{Z}^{B_{I}}\\\ &+\sigma_{X}^{A_{I}}\sigma_{X}^{A_{O}}\sigma_{X}^{B_{I}}-\sigma_{Y}^{A_{I}}\sigma_{X}^{A_{O}}\sigma_{Y}^{B_{I}}\bigg{)},\end{split}$ (39) with implicit tensor product between the operators. Figure 9: The process $W_{222}$ seen as a convex combination of two GHZ-type un-normalized states acting on $A_{I}$, $A_{O}$ and $B_{I}$. Due to the ordered process causal constraints, GHZ states do not lead to valid bipartite ordered processes, but such convex combination of these two GHZ-type results in the valid bipartite ordered process $W_{222}$. ###### Theorem 5.2. The bipartite ordered process $W_{222}$ attains the maximum generalized robustness of all processes with the same dimensions. That is, $\displaystyle R_{G}(W_{222})=$ $\displaystyle\max_{W\in\mathcal{L_{A\rightarrow B}}}\left[R_{G}(W)\right]$ (40a) $\displaystyle=$ $\displaystyle\frac{1}{2}.$ (40b) ###### Proof. The proof of this theorem follows similar steps to the proof of theorem 5.1. We start by defining the operator $\displaystyle S:=$ $\displaystyle\mathds{1}-2W_{222}$ (41a) $\displaystyle=$ $\displaystyle\mathds{1}-\left(\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|+\sigma_{x}^{A_{O}}\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|\sigma_{x}^{A_{O}}\right),$ (41b) and showing that $S$ is a non-classical CCDC witness. For this, we need to show that $\mathrm{tr}_{A_{O}}(S)\succeq 0$ and $S^{T_{A_{I}}}\succeq 0$, as shown in Eqs. (22). $\displaystyle\begin{split}\frac{1}{2}\mathrm{tr}_{A_{O}}(S)&=\mathds{1}^{A_{I}B_{I}}-\left(\|{00}\rangle\langle{00}\|+\|{11}\rangle\langle{11}\|\right)\\\ &=\|{01}\rangle\langle{01}\|+\|{10}\rangle\langle{10}\|\\\ &\succeq 0.\end{split}$ (42) Now, for the second condition, we have $\displaystyle\begin{split}S^{T_{A_{I}}}&=\mathds{1}-\left(\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|+\sigma_{x}^{A_{O}}\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|\sigma_{x}^{A_{O}}\right)^{T_{A_{I}}}\\\ &=\mathds{1}-\sum_{ij}\left(\|{jii}\rangle\langle{ijj}\|+\sigma_{x}^{A_{O}}\|{jii}\rangle\langle{ijj}\|\sigma_{x}^{A_{O}}\right)\\\ &\succeq 0,\end{split}$ (43) where the last inequality holds true because $\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|^{T_{A_{I}}}$ and $\left(\sigma_{x}^{A_{O}}\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|\sigma_{x}^{A_{O}}\right)^{T_{A_{I}}}$ have orthogonal support, i.e., $\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|^{T_{A_{I}}}\left(\sigma_{x}^{A_{O}}\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|\sigma_{x}^{A_{O}}\right)^{T_{A_{I}}}=0,$ (44) and the eigenvalues of $\sum_{ij}\|{jii}\rangle\langle{ijj}\|$ and $\sum_{ij}\sigma_{x}^{A_{O}}\|{jii}\rangle\langle{ijj}\|\sigma_{x}^{A_{O}}$ are $+1$ and $-1$. Direct calculation shows that $\mathrm{tr}(SW_{222})=-2$ and, for every process $\Omega$ in this scenario, we have $\mathrm{tr}(S\Omega)\leq 2$, by an argument analogous to Eq. (33b). Since $S$ is a non-classical CCDC witness, if $(1-r)W_{222}+r\Omega$ is a CCDC process, it holds that ${\mathrm{tr}\left(S\left[(1-r)W_{222}+r\Omega\right]\right)\geq 0}$. It is also true that $-2r+2(1-r)\geq 0,$ (45) thus $r\geq\frac{1}{2}$. We finish the proof by invoking Theorem B.3, which states that $r\leq\frac{1}{2}$ in this scenario. ∎ We also evaluated the robustness of $W_{222}$ against white noise, obtaining evidences that $W_{222}$ attains the maximal white noise robustness on its scenario. ###### Theorem 5.3. The white noise robustness of the bipartite ordered processes $W_{222}$ is $R_{WN}(W_{222})=\frac{2}{3}.$ (46) ###### Proof. We provide a lower-bound for $R_{WN}(W_{222})$ by using similar steps to the proof of theorem 5.2, starting with the non-classical CCDC witness $\displaystyle\begin{split}S:=&\mathds{1}-2W_{222}\\\ =&\mathds{1}-\left(\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|+\sigma_{X}^{A_{O}}\|{GHZ}\rangle\rangle\langle\langle{GHZ}\|\sigma_{X}^{A_{O}}\right).\end{split}$ (47) We have $\mathrm{tr}(SW_{222})=-2$ and $\mathrm{tr}(S)=2^{3}-4=4$. As $S$ is a non-classical CCDC witness, if $(1-r)W_{222}+r\Omega$ is a CCDC process, ${\mathrm{tr}\left(S\left[(1-r)W_{222}+r\frac{\mathds{1}}{4}\right]\right)\geq 0}$. So, it is true that $\displaystyle-2(1-r)+r\geq 0,$ (48) thus $r\geq\frac{2}{3}$. We now show, using techniques which are similar to the proof of Theorem B.3, that the above lower-bound can be attained. First notice that the qubit depolarizing channel $\widetilde{D}_{\eta}(\rho)=(1-\eta)\rho+\eta\mathrm{tr}(\rho)\frac{\mathds{1}}{2}$ is entanglement breaking when $\eta=\frac{2}{3}$ [54, 55]. Also, notice that, since $\mathrm{tr}_{A_{I}}(W_{222})=\frac{\mathds{1}^{A_{O}B_{I}}}{d_{B_{I}}}$, if we apply the depolarizing channel on the subspace $A_{I}$ of the process $W_{222}$, we obtain $\widetilde{D}_{\eta}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W_{222})=(1-\eta)W_{222}+\eta\frac{\mathds{1}}{4}$. Since $\widetilde{D}_{\eta}$ is entanglement breaking for $\eta=\frac{2}{3}$, lemma B.1 ensures that $(1-\frac{2}{3})W_{222}+\frac{2}{3}\frac{\mathds{1}}{4}$ is CCDC, thus concluding the proof. ∎ It is worth to mention that every witness introduced in the proofs of Theorems 5.1 and 5.2 are the optimal witnesses for these particular processes. Similarly to the scenario with $d_{A_{I}}=d_{A_{O}}=d$ and $d_{B_{I}}=d^{2}$, we implemented the see-saw algorithm that indicates that, in the scenario with $d_{A_{I}}=d_{A_{O}}=d_{B_{I}}=2$, the highest white noise robustness is exactly $\frac{2}{3}$. This suggests that $W_{222}$ has also maximum white noise robustness in its scenario. ## 6 Relation with previous research ### 6.1 Comparison with the non-classical CCDC process of Ref. [15] We now compare our proposed process with the non-classical CCDC process presented and experimentally implemented in Ref. [15]. The process consists of the preparation of a maximally entangled state $\|{\phi^{+}}\rangle$ shared between Alice and the auxiliary system, a partial swap channel from $\mathcal{L}(A_{O}\otimes\textrm{aux})$ to $\mathcal{L}(B_{I}\otimes\textrm{aux}^{\prime})$, which is simply a unitary composed by a coherent mixture of an identity and a SWAP gate, and a partial trace on the output of the auxiliary system. This process can be explicitly written as $\displaystyle\begin{split}W_{\textrm{MRSR}}=\ \mathrm{tr}_{\textrm{aux}^{\prime}}&\Big{(}\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}\textrm{aux}}\\\ &\|{\textrm{U}_{\textrm{PS}}}\rangle\rangle\langle\langle{\textrm{U}_{\textrm{PS}}}\|^{A_{O}\textrm{aux}/B_{I}\textrm{aux}^{\prime}}\Big{)},\end{split}$ (49) where $\|{\textrm{U}_{\textrm{PS}}}\rangle\rangle^{A_{O}\textrm{aux}/B_{I}\textrm{aux}^{\prime}}$ is the Choi vector of the unitary partial SWAP666 The authors from Ref. [15] use an equivalent way to represent the unitary partial SWAP gate, which is $\textrm{U}_{\textrm{PS}}^{A_{I}\textrm{aux}/B_{I}\textrm{aux}^{\prime}}=\frac{1}{\sqrt{2}}\left(\mathds{1}^{A_{O}\textrm{aux}/B_{I}\textrm{aux}^{\prime}}+i\mathds{1}^{A_{O}\textrm{aux}/\textrm{aux}^{\prime}B_{I}}\right).$ (50) In the second term, the identity channel exchanges the outputs $B_{I}$ and $\textrm{aux}^{\prime}$, in comparison to the identity channel in the first term. This part of the channel does exactly the same as $\textrm{U}_{\textrm{SWAP}}$ does, differing only by the fact that the output labels are not explicitly exchanged. gate $\textrm{U}_{\textrm{PS}}$, given by $\textrm{U}_{\textrm{PS}}=\frac{1}{\sqrt{2}}\left(\mathds{1}+i\,\textrm{U}_{\textrm{SWAP}}\right),$ (51) with $\textrm{U}_{\textrm{SWAP}}$ being the SWAP gate for qubits, given by $\textrm{U}_{\textrm{SWAP}}=\left(\begin{matrix}1&0&0&0\\\ 0&0&1&0\\\ 0&1&0&0\\\ 0&0&0&1\end{matrix}\right).$ (52) The label MRSR makes reference to the names MacLean, Ried, Spekkens and Resch, authors of Ref. [15]. Fig. 10 ilustrates the process $W_{\text{MRSR}}$. Figure 10: Circuit representation of the process $W_{\text{MRSR}}$ presented in Ref. [15]. A maximally entangled state is initially shared between Alice and the auxiliary system. After Alice’s operation, the partial-SWAP gate $\text{U}_{\text{PS}}$ is applied, then $\textrm{aux}^{\prime}$ is discarded. Using our numerical methods, we can evaluate the values of robustnesses for $W_{\textrm{MRSR}}$, which are $\displaystyle R_{G}^{\textrm{low,PPT}}(W_{\textrm{MRSR}})=R_{G}^{\text{up},\mathcal{E}_{10^{4}}}(W_{\textrm{MRSR}})=0.3506,$ (53a) $\displaystyle R_{WN}^{\textrm{low,PPT}}(W_{\textrm{MRSR}})=R_{WN}^{\text{up},\mathcal{E}_{10^{4}}}(W_{\textrm{MRSR}})=0.5000.$ (53b) When comparing the robustness values of $W_{\textrm{MRSR}}$ with $W_{\textrm{222}}$ (see Section 5.2 and Eqs. (53)), which is a process defined in the same scenario, we verify that $W_{\textrm{222}}$ is strictly more robust against both generalized and white noise than $W_{\text{MRSR}}$. For the case of the process $W_{222^{2}}$, we argue that the construction of $W_{222^{2}}$ is simpler than $W_{\textrm{MRSR}}$. Both processes require the preparation of a maximally entangled qubit state, but $W_{\textrm{MRSR}}$ requires the implementation of the control swap operation, which is a coherent superposition between the identity channel and the swap channel, while $W_{222^{2}}$ only requires the identity channel. ### 6.2 Comparison with the non-classical CCDC process of Ref. [19] In Ref. [19], the authors proposed to study non-classical CCDC by considering quantum superpositions of both relations. Their example is a the tripartite process ordered as $A\rightarrow B\rightarrow C$. A tripartite process ordered as $A\rightarrow B\rightarrow C$ is an operator $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I}\otimes B_{O}\otimes C_{I})$ which can be written as $W=\rho^{A_{I}\text{aux}}D_{A}^{\text{aux}A_{O}/B_{I}\text{aux'}}D_{B}^{\text{aux'}B_{O}/C_{I}},$ (54) where $D_{A}$ and $D_{B}$ are Choi operators of quantum channels. In a tripartite scenario, common-cause and direct-cause relations can be more complex than in the bipartite scenario. In the case of common-cause relations, more general situations are discussed in Refs. [56, 57]. However, following the definitions presented in Ref. [19], we restrict the attention to the same common-cause and direct-cause relations of the bipartite case, recovering them when taking $d_{B_{O}}{=}d_{C_{I}}{=}1$. The following definitions are taken considering the ones presented in Ref. [19]. A tripartite ordered process $W_{\text{CC}}$ is common-cause if $\mathrm{tr}_{B_{O}C_{I}}(W_{\text{CC}}):=d_{B_{O}}\rho^{A_{I}B_{I}}\otimes\mathds{1}^{A_{O}},$ (55) where $\rho^{A_{I}B_{I}}$ is a quantum state in $\mathcal{L}(A_{I}\otimes B_{I})$. A tripartite ordered process $W_{\text{DC}}$ is direct-cause if $\mathrm{tr}_{B_{O}C_{I}}(W_{\text{DC}}):=d_{B_{O}}\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}},$ (56) where $\rho_{i}^{A_{I}}$ are quantum states in $\mathcal{L}(A_{I})$ and $D_{i}^{A_{O}/B_{I}}$ are quantum channels from $A_{O}$ to $B_{I}$. A tripartite ordered process $W_{\text{CCDC}}$ is classical CCDC if it can be decomposed in a convex combination of a common-cause process and a direct- cause process. Note that when bipartite processes are considered, _i.e._ , the dimensions of $B_{O}$ and $C_{I}$ are equal to one, their definition are equivalent to the ones presented in section 2. The example of non-classical CCDC process presented by the authors is the operator777The state $\|{\phi^{+}}\rangle$ appearing in the expression of $W_{\text{FB}}$ is originally written, in Ref. [19], as generic bipartite state $\|{\psi}\rangle$. However, we verified that the numerical robustness results obtained in Ref. [19] are reproduced when $\|{\psi}\rangle$ is a maximally entangled state, which lead us to take $\|{\psi}\rangle=\|{\phi^{+}}\rangle$, as this state is used in every other process mentioned before in this work. $W_{\text{FB}}=\|{W_{\text{FB}}}\rangle\rangle\langle\langle{W_{\text{FB}}}\|$, where $\displaystyle\begin{split}\|{W_{\text{FB}}}\rangle\rangle=\frac{1}{\sqrt{2}}\Big{(}&\|{\phi^{+}}\rangle^{A_{I}B_{I}}\|{\mathds{1}}\rangle\rangle^{A_{O}/C_{I}^{1}}\|{\mathds{1}}\rangle\rangle^{B_{O}/C_{I}^{2}}\|{0}\rangle^{C_{I}^{3}}\\\ +&\|{\phi^{+}}\rangle^{A_{I}C_{I}^{1}}\|{\mathds{1}}\rangle\rangle^{A_{O}/B_{I}}\|{\mathds{1}}\rangle\rangle^{B_{O}/C_{I}^{2}}\|{1}\rangle^{C_{I}^{3}}\Big{)},\end{split}$ (57) The label FB makes reference to the names Feix and Brukner, authors of Ref. [19]. This process can be seen as a superposition of a common-cause and a direct- cause processes, as the first term corresponds to a common-cause process, whereas the second term corresponds to a direct-cause process. We can also represent $W_{\text{FB}}$ in terms of ordered quantum circuits888Indeed, every ordered quantum process can be represented in terms of ordered quantum circuits by concatenating quantum states and quantum operations [11]. as illustrated in Fig. 11. Figure 11: Circuit representation of the process $W_{\text{FB}}$ presented in Ref. [19]. A fixed qubit $\|{+}\rangle=\frac{1}{\sqrt{2}}(\|{0}\rangle+\|{1}\rangle)$ in $C_{I}^{3}$ controls the swap operation from the auxiliary system in $C_{I}^{2}$ and $A_{O}$ to $B_{I}$ and from $C_{I}^{2}$ and $B_{O}$ to $C_{I}^{1}$. As the control qubit is fixed, the swapping operation occurs in the same way in both channels. This combination of operations generate the pure non-classical CCDC process $W_{\text{FB}}$ In Ref. [19], the authors numerically obtained a lower-bound to the generalized robustness of $W_{\text{FB}}$, using the PPT outer approximation of DC processes, obtaining999Strictly speaking, the authors have evaluated quantity before mentioned here, the non-classicality of causality $\mathcal{C}$, which has a one-to-one relation with the generalized robustness via $R_{G}^{\text{low,PPT}}(W)=\frac{\mathcal{C}(W)}{1+\mathcal{C}(W)}$. $R_{G}^{\text{low,PPT}}(W_{\text{FB}})=0.1855.$ (58) Using our inner approximation method with $N=200$, we could obtain, up to numerical precision, the upper-bound $R_{G}^{\text{up,}\mathcal{E}_{200}}(W_{\text{FB}})=0.1855,$ (59) showing that $R_{G}(W_{\text{FB}})=0.1855$, up to numerical precision. For completeness, we have also computed the white noise robustness, obtaining $R_{WN}^{\text{low,PPT}}(W_{\text{FB}})=R_{WN}^{\text{up,}\mathcal{E}_{200}}(W_{\text{FB}})=0.3324.$ (60) ### 6.3 The role of coherent mixture of causal relations As discussed in this section, previous research has shown that one way to obtain processes with the non-classical CCDC property is by coherently superposing causal relations. For instance, the tripartite processes presented in Ref. [19] and discussed in Section 6.2 is constructed in a way to be a coherent superposition of a purely common-cause and a purely direct-cause processes. Also, the bipartite process presented in Ref. [15] is inspired by a coherent mixture of causal relations which is mathematically formalized by the application of the partial swap operation (see Eq. (50)). Differently from previous works, we have shown in this work that the connection between non-classical CCDC and coherent superpositions of causal relations may be more subtle than it seems at first glance. In particular, although the non-classical CCDC process $W_{ddd^{2}}$ presented in section 5 may be viewed as a superposition of a process which is initialized in $\|{00}\rangle$ with a process which is initialized in state $\|{11}\rangle$, it also admits a natural interpretation as a process with both common-cause and direct-cause relations simultaneously, without explicitly considering any superposition of causal relations. We then argue that the process $W_{ddd^{2}}$ does not need to be interpreted as a coherent superposition of causal relations. ## 7 Separable process without a direct-cause explanation As mentioned in previous sections, the definition of direct-cause processes (Def.2.2) reminds one of the definition of separable quantum states. More precisely, if we do not impose that the operators $D_{i}^{A_{O}/B_{I}}$ have to respect the quantum channel condition $\mathrm{tr}_{B_{I}}D_{i}^{A_{O}/B_{I}}=\mathds{1}^{A_{O}}$, equation (9b) is precisely the definition of a separable state on the bipartition $A_{I}\|A_{O}B_{I}$. We now show that, despite being related, these two definitions are not equivalent. Consider the process $\displaystyle\begin{split}W_{\text{SEP}}:=\frac{1}{2}\Big{(}&\|{0}\rangle\langle{0}\|^{A_{I}}\otimes\|{0}\rangle\langle{0}\|^{A_{O}}\otimes\|{0}\rangle\langle{0}\|^{B_{I}}\\\ +&\|{1}\rangle\langle{1}\|^{A_{I}}\otimes\|{0}\rangle\langle{0}\|^{A_{O}}\otimes\|{1}\rangle\langle{1}\|^{B_{I}}\\\ +&\|{+}\rangle\langle{+}\|^{A_{I}}\otimes\|{1}\rangle\langle{1}\|^{A_{O}}\otimes\|{+}\rangle\langle{+}\|^{B_{I}}\\\ +&\|{-}\rangle\langle{-}\|^{A_{I}}\otimes\|{1}\rangle\langle{1}\|^{A_{O}}\otimes\|{-}\rangle\langle{-}\|^{B_{I}}\Big{)},\end{split}$ (61) which is a separable operator by construction. First, notice that $\displaystyle\mathrm{tr}_{B_{I}}W_{\text{SEP}}=\frac{1}{2}\mathds{1}^{A_{I}}\otimes\mathds{1}^{A_{O}},$ (62) which shows that $W$ is a valid bipartite ordered process. The process $W_{\text{SEP}}$ can be physically realized by preparing a maximally entangled qubit state between $A_{I}$ and aux, and using a “decoder” described by $\displaystyle D^{A_{O}\textrm{aux}/B_{I}}$ $\displaystyle=\|{0}\rangle\langle{0}\|^{A_{O}}\otimes\left(\|{00}\rangle\langle{00}\|+\|{11}\rangle\langle{11}\|\right)^{\textrm{aux}B_{I}}$ $\displaystyle+\|{1}\rangle\langle{1}\|^{A_{O}}\otimes\left(\|{++}\rangle\langle{++}\|+\|{--}\rangle\langle{--}\|\right)^{\textrm{aux}B_{I}}.$ (63) In this way we have $\displaystyle W_{\text{SEP}}=\|{\phi^{+}}\rangle\langle{\phi^{+}}\|^{A_{I}\textrm{aux}}D^{A_{O}\textrm{aux}/B_{I}}.$ (64) Note that the channel $D^{A_{O}\textrm{aux}/B_{I}}$ can be implemented as follows: first, perform a computational basis measurement on $A_{O}$. If the outcome is $0$, perform a computational basis measurement on aux and send the output qubit to $B_{I}$. If the outcome is $1$, perform a measurement on aux in the $X$-basis instead. ###### Theorem 7.1. The bipartite ordered process $W_{\text{SEP}}$ is separable in the bipartition $A_{I}\|A_{O}B_{I}$, but is not direct-cause. ###### Proof. Equation (61) represents $W_{\text{SEP}}$ as a convex combination of product states, ensuring that $W_{\text{SEP}}$ is separable in the bipartition $A_{I}\|A_{O}B_{I}$. In order to show that $W_{\text{SEP}}$ is not a direct- cause process, let us assume that $W_{\text{SEP}}$ can be written as a convex combination $W_{\text{SEP}}=\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}}$, where $\rho_{i}^{A_{I}}$ are normalized quantum states and every $D_{i}^{A_{O}/B_{I}}$ satisfy $\mathrm{tr}_{A_{I}}D_{i}^{A_{O}/B_{I}}=\mathds{1}^{A_{O}}$. Note that each $\rho_{i}^{A_{I}}$ has non-trivial overlap with, at least, 3 out of the 4 states $\|{0}\rangle\langle{0}\|$, $\|{1}\rangle\langle{1}\|$, $\|{+}\rangle\langle{+}\|$,$\|{-}\rangle\langle{-}\|$. Indeed, let $\rho=\frac{1}{2}\left(\mathds{1}+\sum_{i}\alpha_{i}\sigma_{i}\right)$ be an arbitrary state where $\\{\alpha_{i}\\}$ are real numbers that satisfy $\sum_{i}\alpha_{i}^{2}\leq 1$ and $\sigma_{i}$ are Pauli matrices. Suppose that $\rho$ has zero overlap with some pure state $\|{\psi}\rangle\langle{\psi}\|=\frac{1}{2}\left(\mathds{1}+\sum_{i}\beta_{i}\sigma_{i}\right)$ with $\sum_{i}\beta_{i}^{2}=1$, then we have that $(\vec{\alpha},\vec{\beta})=-1$, where $(\cdot,\cdot)$ is the Euclidean inner product. By the Cauchy-Schwarz inequality, we have $1=(\vec{\alpha},\vec{\beta})^{2}\leq(\vec{\alpha},\vec{\alpha})(\vec{\beta},\vec{\beta})=(\vec{\alpha},\vec{\alpha}),$ (65) with equality if and only if $\vec{\alpha}$ is a multiple of $\vec{\beta}$. This shows that $\vec{\alpha}=-\vec{\beta}$ and therefore $\rho$ cannot be orthogonal to any other pure quantum state. Let us choose some fixed index $j$ in the sum and suppose, without lack of generality, that $\rho_{j}$ has non-zero overlap with $\|{1}\rangle,\|{+}\rangle,\|{-}\rangle$. Then, from the above definition of $W_{\text{SEP}}$, we calculate $\displaystyle\mathrm{tr}(\|{1}\rangle\langle{1}\|^{A_{I}}\otimes\|{0}\rangle\langle{0}\|^{A_{O}}\otimes\|{0}\rangle\langle{0}\|^{B_{I}}W_{\text{SEP}})=0,$ (66) From the decomposition $W_{\text{SEP}}=\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}}$, we get $\sum_{i}p_{i}\mathrm{tr}(\|{1}\rangle\langle{1}\|\rho_{i}^{A_{I}})\mathrm{tr}(\|{0}\rangle\langle{0}\|^{A_{O}}\otimes\|{0}\rangle\langle{0}\|^{B_{I}}D_{i}^{A_{O}/B_{I}})=0.$ (67) By positivity of $D^{A_{O}/B_{I}}$ and $\rho^{A_{I}}$, each term in the sum has to be zero. Since, by assumption, $\mathrm{tr}(\|{1}\rangle\langle{1}\|\rho_{j})\neq 0$, it must be the case that $D_{j}$ obeys $\mathrm{tr}(\|{0}\rangle\langle{0}\|^{A_{O}}\otimes\|{0}\rangle\langle{0}\|^{B_{I}}D_{j}^{A_{O}/B_{I}})=0.$ (68) Similarly, by calculating other projectors we get $\displaystyle\mathrm{tr}(\|{1}\rangle\langle{1}\|^{A_{O}}\otimes\|{-}\rangle\langle{-}\|^{B_{I}}D_{j}^{A_{O}/B_{I}})=0,$ (69a) $\displaystyle\mathrm{tr}(\|{1}\rangle\langle{1}\|^{A_{O}}\otimes\|{+}\rangle\langle{+}\|^{B_{I}}D_{j}^{A_{O}/B_{I}})=0.$ (69b) From this we get $\mathrm{tr}(\|{1}\rangle\langle{1}\|^{A_{O}}\otimes\mathds{1}^{B_{I}}D_{j}^{A_{O}/B_{I}})=0$, which means that $\mathrm{tr}_{B_{I}}D_{j}^{A_{O}/B_{I}}\neq\mathds{1}^{A_{O}}$. ∎ By construction, $W_{\text{SEP}}$ is separable in the bipartition $A_{I}\|A_{O}B_{I}$, implying that $W_{\text{SEP}}$ has a PPT $k$-symmetric extension for every $k\in\mathbb{N}$ and $R_{G}^{\text{low, PPT}_{k}}(W)=R_{WN}^{\text{low, PPT}_{k}}(W)=0$. This means that the non- classical CCDC property of $W_{\text{SEP}}$ cannot be certified by any purely entanglement-based criterion, such as the ones explored in Refs.[15, 19, 20]. Using the inner and outer approximations presented in Secs. 4.1 and 4.2 with the family of qubits presented in the Appendix B of Ref.[50] ($n=171$, which corresponds to $N=2171^{2}$ states), we obtain upper and lower bounds for the generalized and white noise robustnesses of $W_{\text{SEP}}$ $\displaystyle R_{G}^{\text{up},\mathcal{E}_{N}}(W_{\text{SEP}})=R_{G}^{\text{low},\hat{\mathcal{E}}_{N}}(W_{\text{SEP}})=0.1465,$ (70a) $\displaystyle R_{WN}^{\text{up},\mathcal{E}_{N}}(W_{\text{SEP}})=R_{WN}^{\text{low},\hat{\mathcal{E}}_{N}}(W_{\text{SEP}})=0.2930,$ (70b) with equality holding up to numerical precision. In Ref. [20], it was conjectured that the set of separable processes and the set of processes without quantum memory are not the same, the latter being a strict subset of the first. Since, the definitions of bipartite processes without quantum memory and bipartite direct-cause processes are equivalent (see Appendix A), we have then proven the conjecture presented in Ref. [20] by explicitly constructing the bipartite separable process $W_{\text{SEP}}$, which cannot be realized by processes without quantum memory. It is interesting to point that our numerical methods allowed us to obtain a relatively high robustness of $R_{WN}(W_{\text{SEP}})=0.2930$, but techniques exclusively based on entanglement would lead to the trivial lower bound $R^{\text{sep}}_{WN}(W_{\text{SEP}})\geq 0$. Since $R_{WN}(W_{\text{SEP}})=0.2930$ is considerably greater than zero, we see that the approximating the set of direct-cause processes by separable processes may lead to very unsatisfactory results. ## 8 Certifying non-classical CCDC on PPT processes In this section, we present an example of a non-classical CCDC process which has $R_{G}^{\text{low,PPT}}(W)=R_{WN}^{\text{low,PPT}}(W)=0$, _i.e._ , its non-classical CCDC property cannot be certified by the PPT approximation used in Refs. [15, 19]. Such a process can be obtained by exploiting a class of entangled states with positive partial transpose, presented in Ref. [58]. The class of states of our interest is $\displaystyle\begin{split}&\rho_{a}^{2\times 4}:=\\\ &\frac{1}{7a+1}\begin{bmatrix}a&0&0&0&0&a&0&0\\\ 0&a&0&0&0&0&a&0\\\ 0&0&a&0&0&0&0&a\\\ 0&0&0&a&0&0&0&0\\\ 0&0&0&0&\tfrac{1}{2}(1+a)&0&0&\tfrac{1}{2}\sqrt{1-a^{2}}\\\ a&0&0&0&0&a&0&0\\\ 0&a&0&0&0&0&a&0\\\ 0&0&a&0&\tfrac{1}{2}\sqrt{1-a^{2}}&0&0&\tfrac{1}{2}(1+a)\end{bmatrix},\end{split}$ (71) being entangled for $a\in(0,1)$ and separable for $a=0$ or $a=1$. We now set $a=\frac{1}{2}$ and use $\rho_{\frac{1}{2}}^{2\times 4}$ to define: $W_{\textrm{PPT}}:=d_{A_{O}}\cdot\rho_{\frac{1}{2}}^{2\times 4},$ (72) with $d_{A_{O}}=2$. $W_{\textrm{PPT}}$ is a valid bipartite ordered process, as it satisfies every condition from Eqs. (18) by direct inspection. Also, it has dimensions $d_{A_{I}}{=}d_{A_{O}}{=}d_{B_{I}}=2$ and has bound entanglement in the bipartition $A_{I}\|A_{O}B_{I}$. ‘We will now ensure that $W_{\textrm{PPT}}$ is a non-classical CCDC process by using a better approximation $\mathcal{L}_{\text{DC}}^{\text{out,PPT}_{k}}$. In particular, we set $k=2$, then we obtain $\displaystyle\begin{split}R_{G}^{\text{low,PPT}_{2}}(W_{\textrm{PPT}})&=0.0083,\\\ R_{WN}^{\text{low,PPT}_{2}}(W_{\textrm{PPT}})&=0.0230.\end{split}$ (73) For $k=3$, the robustnesses do not change, which indicates that using greater values of $k$ does not improve the values of generalized and white noise robustnesses. When using the inner and outer approximations presented in Secs. 4.1 and 4.2 with the family of qubits presented in the Appendix B of Ref.[50] ($n=171$, which corresponds to $N=2171^{2}$ states), up to numerical precision, we obtain $\begin{split}R_{G}^{\text{up},\mathcal{E}_{10^{4}}}(W_{\textrm{PPT}})&=R_{G}^{\text{low},\hat{\mathcal{E}}_{10^{4}}}(W_{\textrm{PPT}})=0.1085,\\\ R_{WN}^{\text{up},\mathcal{E}_{10^{4}}}(W_{\textrm{PPT}})&=R_{WN}^{\text{low},\hat{\mathcal{E}}_{10^{4}}}(W_{\textrm{PPT}})=0.2782.\end{split}$ (74) We verify that the lower bounds for the robustnesses obtained with the entanglement criterium in Eqs. (73) are rather loose in comparison to the actual robustnesses values from Eqs. (74). This example also illustrates the limitations of certifying non-classical CCDC solely based on entanglement criteria. ## 9 Summary of generalized and white noise robustnesses Process (Eq.) \| $R_{G}$ \| $R_{G}^{\text{low, PPT}_{k=2}}$ \| $R_{WN}$ \| $R_{WN}^{\text{low, PPT}_{k=2}}$ ---\|---\|---\|---\|--- $W_{333^{2}}$ (27) \| $\frac{2}{3}$ \| $\frac{2}{3}$ \| $0.9529$ \| $0.9529$ $W_{222^{2}}$ (27) \| $\frac{1}{2}$ \| $\frac{1}{2}$ \| $0.8421$ \| $0.8421$ $W_{222}$ (36) \| $\frac{1}{2}$ \| $\frac{1}{2}$ \| $\frac{2}{3}$ \| $\frac{2}{3}$ $W_{\text{MRSR}}$ (49) \| $0.3506$ \| $0.3506$ \| $0.5000$ \| $0.5000$ $W_{\text{FB}}$ (57) \| $0.1855$ \| $0.1855$ \| $0.3324$ \| $0.3324$ $W_{\text{SEP}}$ (61) \| $0.1465$ \| $0$ \| $0.2930$ \| $0$ $W_{\text{PPT}}$ (72) \| $0.1085$ \| $0.0083$ \| $0.2782$ \| $0.0230$ Table 2: Table presenting generalized and white noise robustnesses for every process analyzed in this work. Values represented in fractions were obtained by mathematical theorems and coincide with SDP optimization. Values with decimal digits were obtained only via SDP optimization, where our upper and lower bounds are identical up to $4$ decimals. The lower bounds obtained by the approximations of the CCDC set based on entanglement are the values provided in columns $R_{G}^{\text{low, PPT}_{k}}$ and $R_{WN}^{\text{low, PPT}_{k}}$. Lower-bounds that match the actual robustnesses are highlighted in green, while lower bounds that are rather different from the actual robustnesses are highlighted in red. We can observe that entanglement based criteria could never detect $W_{\text{SEP}}$ as a non-classical CCDC process, while the PPT $k$-symmetric extension bound for $W_{\text{PPT}}$ provides the loose lower bounds $R_{G}^{\text{low, PPT}_{k}}(W_{\text{PPT}})\geq 0.0083$ $R_{WN}^{\text{low, PPT}_{k}}(W_{\text{PPT}})\geq 0.0230$, which are values obtained both with $k=2$ and $k=3$. In previous sections, we presented several examples of non-classical CCDC processes with different values of generalized and white noise robustness. Table 2 summarizes the non-classical CCDC robustnesses of several processes presented in this work and compare the actual value of robustness with the values obtained with methods based on entanglement criteria. ## 10 Conclusions In this work, we have introduced a class of bipartite ordered processes, and a process with dimension of three qubits, which are maximally robust against general noise and very likely to be the most robust against white noise for the qubit case. This class of processes can be implemented by preparing a pair of maximally entangled states and an identity channel, admitting a natural interpretation of a process with both common-cause and direct-cause relations simultaneously. Hence, in contrast to previously known non-classical CCDC processes [14, 19], the class presented here does not require either the construction or the interpretation directly based on coherent superposition of causal relations. Several analytical results proved in this work employed general convex analysis arguments based on witness hyper-planes, combined with entanglement theory concepts, such as entanglement breaking channels. We believe that the techniques developed here may find applications in related problems. We have also presented a systematic semi-definite approach to characterize the set of non-classical CCDC processes. In particular, we provided a hierarchy of inner and outer approximations that converge to the set of classical CCDC processes, and an entanglement-based hierarchy which, despite not converging to the set of CCDC processes, provides us several tight and non-trivial bounds. In order to tackle situations where we could not prove the value of the highest robustness of a given scenario analytically, we constructed a heuristic see-saw method, which provided numerical evidence of the highest robustnesses attainable on such scenario, also providing a valid lower-bound for the value of the highest robustness. Finally, we have shown that, although all bipartite processes that are entangled in the bipartition $A_{I}\|A_{O}B_{I}$ do not have a direct-cause decomposition, the converse does not hold. Our proof consists in explicitly constructing a process which is separable on the bipartition $A_{I}\|A_{O}B_{I}$, but does not have a direct-cause decomposition. Since bipartite processes without quantum memory are equivalent to bipartite direct- cause ones, our results prove a conjecture first raised in Ref. [20] and contributes towards a better understanding of the particularities of quantum memory, spacial entanglement and temporal entanglement [21, 20, 22, 23, 24]. All SDP optimization problems presented in this manuscript were implemented using MATLAB, the convex optimization package Yalmip [59] and CVX [60], the solvers MOSEK, SeDuMi and SDPT3 [61, 62, 63], and the toolbox for quantum information QETLAB [64]. All our codes are available in the public repository [65] and can be freely used under the GNU Lesser General Public License v3.0. We are grateful to Mateus Araújo, Jessica Bavaresco, Rafael Chaves, Simon Milz and Philip Taranto for interesting discussions. MN, RV and TM acknowledge financial support by the Brazilian agencies INCT-IQ (National Institute of Science and Technology for Quantum Information), FAPEMIG, CNPq, CAPES and Instituto Serrapilheira. MTQ has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska- Curie grant agreement No 801110 and the Austrian Federal Ministry of Education, Science and Research (BMBWF). 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Quintino, Strict hierarchy between parallel, sequential, and indefinite-causal-order strategies for channel discrimination, (2020), arXiv:2011.08300 [quant-ph]. ## Appendix A Ordered processes without quantum memory are equivalent to direct-cause processes on the bipartite case We now present the definition of ordered (non-Markovian) processes without quantum memory, first introduced in Ref.[20]. ###### Definition A.1 (Process without quantum memory[20]). A linear operator $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is a bipartite ordered process without quantum memory if it can be written as $W=\rho_{\text{SEP}}^{A_{I}\textrm{aux}}D^{A_{O}\textrm{aux}/B_{I}},$ (75) where $\rho_{\text{SEP}}^{A_{I}\textrm{aux}}$ is a separable state and $D^{A_{O}\textrm{aux}/B_{I}}$ is the Choi operator of a quantum channel. We now show that all bipartite processes without quantum memory are bipartite direct-cause, vice-versa. This proof was first presented in Appendix A.3.2 of Ref. [20], but we reproduce it here for the sake of completeness. ###### Theorem A.1 (Appendix A.3.2 of Ref.[20]). A bipartite ordered process $W$ is a process without quantum memory if and only if $W$ is direct-cause. ###### Proof. We first show that every process without quantum memory is direct-cause. Since $\rho_{\text{SEP}}^{A_{I}\textrm{aux}}$ is separable, there exists $\rho_{i}^{A_{I}}$ and $\sigma_{i}^{\textrm{aux}}$ and some probabilities $p_{i}$ such that $\rho_{\text{SEP}}^{A_{I}\textrm{aux}}=\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes\sigma_{i}^{\textrm{aux}}$. Thus, we can write $\displaystyle\left(\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes\sigma_{i}^{\textrm{aux}}\right)D^{\textrm{aux}A_{O}/B_{I}}=\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}},$ (76) where $D_{i}^{A_{O}/B_{I}}:=\sigma_{i}^{\textrm{aux}}D^{\textrm{aux}A_{O}/B_{I}}$ are valid quantum channels, since $D_{i}^{A_{O}/B_{I}}\succeq 0$ and $\mathrm{tr}_{B_{I}}(D_{i}^{A_{O}/B_{I}})=\mathds{1}^{A_{O}}$. Now, we need to show that every direct-cause process is a process without quantum memory in the bipartite case. Let us assume that $W$ is direct-cause. Then $W$ can be written as $W=\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes D_{i}^{A_{O}/B_{I}}$ (77) for some states $\rho_{i}^{A_{I}}$ and channels $D_{i}^{A_{O}/B_{I}}$. Now, let us define the separable state $\sigma^{A_{I}\textrm{aux}}:=\sum_{i}p_{i}\rho_{i}^{A_{I}}\otimes\|{i}\rangle\langle{i}\|^{\textrm{aux}}$, and the quantum channel $D^{\textrm{aux}A_{O}/B_{I}}:=\sum_{i}\|{i}\rangle\langle{i}\|^{\textrm{aux}}\otimes D_{i}^{A_{O}/B_{I}}$. Direct calculation shows that $\sigma^{A_{I},\textrm{aux}}D^{\textrm{aux}A_{O}/B_{I}}=W$, ensuring that $W$ is a process without quantum memory. ∎ ## Appendix B A tight upper bound for the generalized robustness In this section, we prove that the generalized robustness of any process is upper-bounded by $R_{G}(W)\leq 1-\frac{1}{d_{A_{I}}}$. This bound is saturated by the processes $W_{ddd^{2}}$ (Eq. (27)) for every dimension $d$ and by $W_{222}$ (Eq. (36)). ###### Lemma B.1. Let $W$ be a bipartite ordered process. If $\widetilde{\Lambda}:\mathcal{L}(A_{I})\to\mathcal{L}(A_{I})$ is an entanglement breaking channel, the process $\widetilde{\Lambda}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)$ is direct-cause. ###### Proof. By definition, any bipartite ordered process can be written as $W=\rho^{A_{I}\textrm{aux}}D^{\textrm{aux}A_{O}/B_{I}}$. Since $\widetilde{\Lambda}$ is entanglement breaking, it holds that $\widetilde{\Lambda}^{A_{I}}\otimes\widetilde{\mathds{1}}^{\textrm{aux}}(\rho^{A_{I}\textrm{aux}})$ is a separable state. Therefore, we can use the same argument presented in the proof of theorem A.1 to ensure that $\widetilde{\Lambda}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)$ is a direct-cause process. ∎ ###### Lemma B.2. Let $\widetilde{\Lambda}:\mathbb{C}_{d}\to\mathbb{C}_{d}$ be a quantum channel, $\omega=e^{\frac{2\pi\sqrt{-1}}{d}}$, and $Z:=\sum_{i=0}^{d-1}\omega^{i}\|{i}\rangle\langle{i}\|$ be the $d$-dimensional clock operator. The channel $\widetilde{\Lambda}(\rho)=\frac{1}{d}\sum_{k=0}^{d-1}Z^{k}\rho Z^{-k}$ (78) is an entanglement breaking channel. ###### Proof. A necessary and sufficient condition [55] for $\widetilde{\Lambda}$ to be entanglement-breaking is that its Choi operator $\Lambda$ is separable between input and output spaces. We now show that $\Lambda$ is separable by $\displaystyle\Lambda:=$ $\displaystyle\sum_{ab}\|{a}\rangle\langle{b}\|\otimes\widetilde{\Lambda}(\|{a}\rangle\langle{b}\|)$ (79a) $\displaystyle=$ $\displaystyle\frac{1}{d}\sum_{abk}\|{a}\rangle\langle{b}\|\otimes Z^{k}(\|{a}\rangle\langle{b}\|)Z^{-k}$ (79b) $\displaystyle=$ $\displaystyle\frac{1}{d}\sum_{abkij}\|{a}\rangle\langle{b}\|\otimes\|{i}\rangle\langle{i}\|\omega^{ik}(\|{a}\rangle\langle{b}\|)\|{j}\rangle\langle{j}\|\omega^{-jk}$ (79c) $\displaystyle=$ $\displaystyle\frac{1}{d}\sum_{kij}\|{i}\rangle\langle{j}\|\otimes\|{i}\rangle\langle{j}\|\omega^{ik}\omega^{-jk}$ (79d) $\displaystyle=$ $\displaystyle\frac{1}{d}\sum_{ij}\|{i}\rangle\langle{j}\|\otimes\|{i}\rangle\langle{j}\|\sum_{k}\omega^{ik}\omega^{-jk}$ (79e) $\displaystyle=$ $\displaystyle\frac{1}{d}\sum_{ij}\|{i}\rangle\langle{j}\|\otimes\|{i}\rangle\langle{j}\|\sum_{k}\omega^{k(i-j)}$ (79f) $\displaystyle=$ $\displaystyle\frac{1}{d}\sum_{ij}\|{i}\rangle\langle{j}\|\otimes\|{i}\rangle\langle{j}\|d\delta_{ij}$ (79g) $\displaystyle=$ $\displaystyle\sum_{i}\|{i}\rangle\langle{i}\|\otimes\|{i}\rangle\langle{i}\|.$ (79h) ∎ ###### Theorem B.3. The generalized robustness of a bipartite ordered process $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is upper-bounded by $R_{G}(W)\leq 1-\frac{1}{d_{A_{I}}}$. ###### Proof. Let $\widetilde{\Lambda}:\mathcal{L}(A)\to\mathcal{L}(A)$ be the entanglement- breaking channel defined in Lemma B.2. Note that since $Z^{0}=\mathds{1}$, the action of $\widetilde{\Lambda}$ can be written as $\displaystyle\widetilde{\Lambda}(\rho)$ $\displaystyle=\frac{1}{d}\sum_{k=0}^{d-1}Z^{k}\rho Z^{-k}$ (80a) $\displaystyle=\frac{1}{d}\rho+\frac{1}{d}\sum_{k=1}^{d-1}Z^{k}\rho Z^{-k}$ (80b) $\displaystyle=\frac{1}{d}\rho+\left(1-\frac{1}{d}\right)\widetilde{\Lambda}_{\setminus}(\rho),$ (80c) where $\widetilde{\Lambda}_{\setminus}(\rho):=\frac{1}{d-1}\sum_{k=1}^{d-1}Z^{k}\rho Z^{-k}$ is a valid quantum channel. Lemma B.1 states that, for any bipartite ordered process $W$, the process $\widetilde{\Lambda}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)$ is direct-cause, thus being CCDC. Hence, by making use of Eq. (80a), we see that the resulting process $\frac{1}{d_{A_{I}}}W+\left(1-\frac{1}{d_{A_{I}}}\right)\widetilde{\Lambda}_{\setminus}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)$ (81) is CCDC, with $\widetilde{\Lambda}_{\setminus}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)$ being a valid bipartite ordered process. By analyzing the definition of generalized robustness presented in Eq. (20), we see that setting $\Omega=\widetilde{\Lambda}_{\setminus}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)$ ensures that the relation $R_{G}(W)\leq 1-\frac{1}{d_{A_{I}}}$ (82) holds for any bipartite ordered process $W$. ∎ ## Appendix C An upper bound for the white noise robustness In this section, we present an upper bound for the white noise robustness. Differently from the generalized robustness case, this bound is not tight, but is useful for proving the strong duality relation for the problem of evaluating the white noise robustness for non-classical CCDC processes in Section D. ###### Theorem C.1. The white noise robustness of a bipartite ordered process $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ is upper-bounded by $R_{WN}(W)\leq 1-\frac{1}{d_{A_{I}}d_{A_{O}}d_{B_{I}}+1}$. ###### Proof. The depolarizing channel $\widetilde{D}_{\eta}(\rho):=(1-\eta)\rho+\eta\frac{\mathds{1}}{d}$ is known to be entanglement breaking when $\eta\geq\frac{d}{d+1}$ [54, 55]. Hence, Lemma B.1 ensures that, for any bipartite ordered process $W\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$, the process $\widetilde{\Lambda}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)=\frac{1}{d_{A_{I}}+1}W+\frac{d_{A_{I}}}{d_{A_{I}}+1}\left(\frac{\mathds{1}^{A_{I}}}{d_{A_{I}}}\otimes\mathrm{tr}_{A_{I}}(W)\right)$ (83) is direct-cause. We now define the operator $W^{\prime}:=\frac{\mathds{1}^{A_{I}}}{d_{A_{I}}}\otimes\frac{\left(d_{A_{O}}\mathds{1}^{A_{O}B_{I}}-\mathrm{tr}_{A_{I}}(W)^{A_{O}B_{I}}\right)}{d_{A_{O}}d_{B_{I}}-1},$ (84) which is a valid direct-cause process by direct inspection. Taking a convex combination of $\widetilde{\Lambda}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)$ and $W^{\prime}$, we obtain $\displaystyle\begin{split}&q\widetilde{\Lambda}^{A_{I}}\otimes\widetilde{\mathds{1}}^{A_{O}B_{I}}(W)+(1-q)W^{\prime}\\\ =&q\left(\frac{1}{d_{A_{I}}+1}W+\frac{d_{A_{I}}}{d_{A_{I}}+1}\left(\frac{\mathds{1}^{A_{I}}}{d_{A_{I}}}\otimes\mathrm{tr}_{A_{I}}(W)\right)\right)+(1-q)\frac{\mathds{1}^{A_{I}}}{d_{A_{I}}}\otimes\frac{\left(d_{A_{O}}\mathds{1}^{A_{O}B_{I}}-\mathrm{tr}_{A_{I}}(W)^{A_{O}B_{I}}\right)}{d_{A_{O}}d_{B_{I}}-1}\\\ =&\frac{q}{d_{A_{I}}+1}W+\left(\frac{(1-q)d_{A_{O}}d_{B_{I}}}{d_{A_{O}}d_{B_{I}}-1}\right)\frac{\mathds{1}^{A_{I}}}{d_{A_{I}}}\otimes\frac{\mathds{1}^{A_{O}B_{I}}}{d_{B_{I}}}+\left(\frac{qd_{A_{I}}}{d_{A_{I}}+1}-\frac{(1-q)}{(d_{A_{O}}d_{B_{I}}-1)}\right)\frac{\mathds{1}^{A_{I}}}{d_{A_{I}}}\otimes\mathrm{tr}_{A_{I}}(W),\end{split}$ (85) which is direct-cause by construction. Now, by setting $q=\frac{d_{A_{I}}+1}{d_{A_{I}}d_{A_{O}}d_{B_{I}}+1}$, we obtain $\displaystyle\frac{qd_{A_{I}}}{d_{A_{I}}+1}-\frac{(1-q)}{(d_{A_{O}}d_{B_{I}}-1)}=0,$ (86) and the process $\frac{1}{d_{A_{I}}d_{A_{O}}d_{B_{I}}+1}W+\left(1-\frac{1}{d_{A_{I}}d_{A_{O}}d_{B_{I}}+1}\right)\frac{\mathds{1}}{d_{A_{I}}d_{B_{I}}}$ (87) is guaranteed to be direct-cause. ∎ ## Appendix D Strong duality for CCDC robustness problems ###### Theorem D.1. The convex optimization problems for non-classical CCDC generalized robustness (Eq. (20)) and non-classical CCDC white noise robustness (Eq. (21)) satisfy strong duality. ###### Proof. We recall that every convex optimization problem admitting a strictly feasible solution i.e., all equality constraints are satisfied, and all inequality constraints are strictly satisfied, necessarily satisfies strong duality (Slater condition [66]). From Theorem C.1 we see that for any value $1-\frac{1}{d_{A_{I}}d_{A_{O}}d_{B_{I}}+1}<r<1$, for any process $W$, the process $\Omega:=(1-r)W+r\frac{\mathds{1}}{d_{A_{I}}d_{B_{I}}}$ is a strictly feasible solution, ensuring that both robustness problems respect strong duality. ∎ ## Appendix E SDPs for CCDC separability In this section, we present the explicit forms of the SDPs mentioned in Section 3 to obtain the generalized and white noise robustnesses of a bipartite ordered process $W$. Consider the problem of obtaining the PPTk generalized robustness of a process $W$, _i.e._ , finding the robustnesses of a process against its worst noise $\Omega$, so the resulting combination process lies in $\mathcal{L}_{A\rightarrow B}^{\text{out,PPT}_{k}}$. This is represented by the following optimization program: $\displaystyle R_{G}^{\text{low,PPT}_{k}}(W):=\textrm{min}\hskip 5.69046ptr$ (88a) $\displaystyle\textrm{s.t.}\hskip 5.69046pt(1-r)W+r\Omega=qW_{\textrm{CC}}+(1-q)W_{\textrm{DC}}^{\text{PPT}_{k}},$ (88b) $\displaystyle\hskip 22.76228pt0\leq r\leq 1,$ (88c) $\displaystyle\hskip 22.76228pt0\leq q\leq 1,$ (88d) $\displaystyle\hskip 22.76228ptW_{\textrm{CC}}\in\mathcal{L}_{\text{CC}},\ W_{\textrm{DC}}^{\text{PPT}_{k}}\in\mathcal{L}_{\text{DC}}^{\text{out,PPT}_{k}},\ \Omega\in\mathcal{L}_{A\rightarrow B},$ (88e) with the process $W_{\text{DC}}^{\text{PPT}_{k}}$ having a $k$-symmetric PPT extension $W_{\text{DC}}^{A_{I}^{\otimes k}\|A_{O}B_{I}}\in\mathcal{L}(A_{I}^{\otimes k}\otimes A_{O}\otimes B_{I})$ [32, 33]. This means that there exists a positive semi-definite operator $W_{\text{DC}}^{A_{I}^{\otimes k}\|A_{O}B_{I}}$ such that $\displaystyle\mathrm{tr}_{A_{I}^{\otimes k-1}}(W_{\text{DC}}^{A_{I}^{\otimes k}\|A_{O}B_{I}})$ $\displaystyle=W_{\textrm{DC}}^{\text{PPT}_{k}}$ (89) $\displaystyle W_{\text{DC}}^{A_{I}^{\otimes k}\|A_{O}B_{I}}$ $\displaystyle=(P_{\text{sym}}^{A_{I}^{\otimes k}}\otimes\mathds{1}^{A_{O}B_{I}})W_{\text{DC}}^{A_{I}^{\otimes k}\|A_{O}B_{I}}(P_{\text{sym}}^{A_{I}^{\otimes k}}\otimes\mathds{1}^{A_{O}B_{I}})$ (90) with positive partial transposition on $A_{I}^{\otimes k}$, that is $\left(W_{\text{DC}}^{{A_{I}^{\otimes k}\|A_{O}B_{I}}}\right)^{T_{A_{I}}}\succeq 0,$ (91) and where $P_{\text{sym}}^{A_{I}^{\otimes k}}$ is the projector onto the symmetric subspace of $A_{I}^{\otimes k}$. We point that, in practice, instead of Eq. (90), it is more advantageous to impose the Bose k-symmetric extension condition, that is, to impose $W_{\text{DC}}^{A_{I}^{\otimes k}\|A_{O}B_{I}}=(P_{\text{sym}}^{A_{I}^{\otimes k}}\otimes\mathds{1}^{A_{O}B_{I}})W_{\text{DC}}^{A_{I}^{\otimes k}\|A_{O}B_{I}}.$ (92) This follows from the fact that, computationally, imposing the Bose k-symmetric extension condition instead of the k-symmetric extension condition does not add any complexity to the problem, but the Bose k-symmetric extension condition detects more entangled states than the standard k-symmetric one [67]. Since Eqs. (88) contains products of optimization variables, such as $qW_{\textrm{CC}}$, the optimization program above is not linear, hence not an SDP. This issue can be circumvented by rewriting the problem in the following equivalent form: $\displaystyle R_{G}^{\text{low, PPT}_{k}}(W):=\textrm{min}\hskip 5.69046pt\mathrm{tr}\left(\frac{\overline{\Omega}}{d_{A_{O}}}\right)$ (93a) $\displaystyle\textrm{s.t.}\hskip 5.69046pt\left(1-\mathrm{tr}\left(\frac{\overline{\Omega}}{d_{A_{O}}}\right)\right)W+\overline{\Omega}=\overline{W_{\textrm{CC}}}+\overline{W_{\textrm{DC}}},$ (93b) $\displaystyle\hskip 22.76228pt\overline{W_{\textrm{CC}}}=\rho^{A_{I}B_{I}}\otimes\mathds{1}^{A_{O}}$ (93c) $\displaystyle\hskip 19.91684pt\left(\overline{W_{\textrm{DC}}}^{{A_{I}^{\otimes k}\|A_{O}B_{I}}}\right)^{T_{A_{I}}}\succeq 0,$ (93d) $\displaystyle\hskip 22.76228pt\overline{W_{\textrm{DC}}}=L_{A\rightarrow B}(\overline{W_{\textrm{DC}}}),\ \overline{\Omega}=L_{A\rightarrow B}(\overline{\Omega}),$ (93e) $\displaystyle\hskip 22.76228pt\overline{\Omega},\ \overline{W_{\textrm{DC}}},\ \rho^{A_{I}B_{I}}\succeq 0.$ (93f) In the above SDP we have used the fact that valid processes should respect $\mathrm{tr}(W)=d_{A_{O}}$ to embed the scalar variables into the operators. This means that the new variables relate to the variables from Eqs. (88) with $\overline{\Omega}=r\Omega,\ \overline{W_{\textrm{CC}}}=qW_{\textrm{CC}},\ \overline{W_{\textrm{DC}}}=(1-q)W_{\textrm{DC}}^{\text{PPT}_{k}}.$ (94) Analogously, the same steps are applied to the PPT white noise robustness, thus being $\displaystyle R_{WN}^{\text{low,PPT}_{k}}(W):=\textrm{min}\hskip 5.69046ptr$ (95a) $\displaystyle\textrm{s.t.}\hskip 5.69046pt(1-r)W+r\frac{\mathds{1}^{A_{I}A_{O}B_{I}}}{d_{A_{I}}d_{B_{I}}}=\overline{W_{\textrm{CC}}}+\overline{W_{\textrm{DC}}},$ (95b) $\displaystyle\hskip 22.76228pt\overline{W_{\textrm{CC}}}=\rho^{A_{I}B_{I}}\otimes\mathds{1}^{A_{O}}$ (95c) $\displaystyle\hskip 19.91684pt\left(\overline{W_{\textrm{DC}}}^{{A_{I}^{\otimes k}\|A_{O}B_{I}}}\right)^{T_{A_{I}}}\succeq 0,$ (95d) $\displaystyle\hskip 22.76228pt\overline{W_{\textrm{DC}}},\ \rho^{A_{I}B_{I}}\succeq 0.$ (95e) From the Lagrangian of the SDPs, we can obtain their dual problems, which generate the optimal non-classical CCDC witnesses $S$ for the given process $W$ [66]. For instance, considering $k=1$, the dual form101010Strictly speaking, the dual objective function is to “maximize $-\mathrm{tr}(SW)$”, which results in $R(W)=\mathrm{tr}(SW)$. Here, we adopted a convention of changing the sign of $S$ and replacing this objective function by “minimize $\mathrm{tr}(SW)$”. With this convention, the operator $S$ is a non-classical CCDC witnesses as defined in Section 3.1, that is, it satisfies $\mathrm{tr}(SW_{\text{CCDC}})\geq 0$ for all classical CCDC processes $W_{\text{CCDC}}$. of the PPT generalized robustness is given by $\displaystyle\textrm{min}\hskip 5.69046pt\mathrm{tr}\left(SW\right)$ (96a) $\displaystyle\textrm{s.t.}\hskip 14.22636pt\mathds{1}^{A_{I}A_{O}B_{I}}\left(1+\mathrm{tr}(SW)\right)-d_{A_{O}}S\succeq 0$ (96b) $\displaystyle\hskip 28.45274ptS- S_{\text{DC}}+S_{\text{DC}}^{\perp}\succeq 0$ (96c) $\displaystyle\hskip 28.45274ptL_{A\rightarrow B}(S_{\text{DC}}^{\perp})=0,$ (96d) $\displaystyle\hskip 28.45274ptS_{\text{DC}}^{T_{A_{I}}}\succeq 0,$ (96e) $\displaystyle\hskip 28.45274pt\mathrm{tr}_{A_{O}}(S)\succeq 0,$ (96f) which is the one we use in our heuristic see-saw presented in Section F. The variable $S_{\text{DC}}^{\perp}$ in Eq. (96c) is associated with an orthogonal projection onto $\mathcal{L}_{A\rightarrow B}$. Also, the same methods presented in Section D to prove that the robustness optimization problem satisfies strong duality can be used to show that this upper bound problem also respects strong duality. We, then, have $R_{G}^{\text{low, PPT}}=-\mathrm{tr}(SW)$, when $S$ is the optimal witness for $W$. We can see that Eq. (96c) together with Eqs. (96e) and (96f) ensure that $S$ is a non-classical CCDC witness. Also, Eq. (96b) corresponds to the normalization condition, which determines that it is a witness for the generalized robustness measure. Analogously, the dual form of the PPT white noise robustness is given by $\displaystyle\textrm{min}\hskip 5.69046pt\mathrm{tr}\left(SW\right)$ (97a) $\displaystyle\textrm{s.t.}\hskip 14.22636pt\frac{\mathrm{tr}(S)}{d_{A_{I}}d_{B_{I}}}-\mathrm{tr}(SW)\leq 1,$ (97b) $\displaystyle\hskip 28.45274ptS- S_{\text{DC}}^{T_{A_{I}}}+S_{\text{DC}}^{\perp}\succeq 0$ (97c) $\displaystyle\hskip 28.45274ptL_{A\rightarrow B}(S_{\text{DC}}^{\perp})=0,$ (97d) $\displaystyle\hskip 28.45274ptS_{\text{DC}}\succeq 0,$ (97e) $\displaystyle\hskip 28.45274pt\mathrm{tr}_{A_{O}}(S)\succeq 0,$ (97f) where Eq. (97b) is the normalization condition for the witness $S$, which corresponds to the white noise robustness measure. A similar method can be used to characterize our inner approximation $\mathcal{L}_{\text{CCDC}}^{\text{in},\mathcal{E}_{N}}$. In this case we just need to absorb the probabilities $p_{i}$ of Eq. (93d) into the quantum channel $D_{i}$ to obtain $\overline{D_{i}}:=p_{i}D_{i}$. This allows us to replace the constraint of Eq. (93d) by $\displaystyle W_{\textrm{DC}}=\sum_{i=1}^{N}\|{\psi_{i}}\rangle\langle{\psi_{i}}\|^{A_{I}}\otimes\overline{D_{i}}^{A_{O}/B_{I}},$ (98a) $\displaystyle\overline{D_{i}}^{A_{O}/B_{I}}\succeq 0,$ (98b) $\displaystyle\prescript{}{B_{I}}{\overline{D_{i}}^{A_{O}/B_{I}}}=\prescript{}{A_{O}B_{I}}{\overline{D_{i}}^{A_{O}/B_{I}}},$ (98c) where $\\{\|{\psi_{i}}\rangle^{A_{I}}\\}$ is a set of random states in $\mathcal{L}(A_{I})$ and $\\{D_{i}^{A_{O}/B_{I}}\\}$ is a set of optimization variables in $\mathcal{L}(A_{O}\otimes B_{I})$. We can also obtain the dual form of the inner approximation problems of robustnesses from the Lagrangian, which results in a similar SDP to Eqs. (96) and Eqs. (97), but replacing Eqs. (96c) and (96e) by $\displaystyle\mathrm{tr}_{A_{I}}\left[S(\|{\psi_{i}}\rangle\langle{\psi_{i}}\|^{A_{I}}\otimes\mathds{1}^{A_{O}B_{I}})+\prescript{}{B_{I}}{S_{i}}-\prescript{}{A_{O}B_{I}}{S_{i}}\right]\succeq 0\ \forall i\in\mathbb{N},$ (99) where $\\{S_{i}\\}$ is a set of variables in $\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$. ## Appendix F Heuristic algorithm that seeks for the maximum robustness on a given scenario Given the dimensions $d_{A_{I}},d_{A_{O}},d_{B_{I}}$, what is the maximum robustness values $R_{G}(W)$ or $R_{WN}(W)$ that a process $W\in\mathcal{L}_{A\rightarrow B}$ can obtain? Inspired by the see-saw techniques of Refs. [52, 53], we now present a heuristic iterative method to seek for the highest robustness values for a given scenario. This method works either for the generalized or white noise robustness, which is why in the following we do not make explicit which robustness quantifier to work with. Our heuristic algorithm works as follows. First, we sample a bipartite ordered process $W_{1}$ using one of the methods we describe in Appendix G. Then, we perform the dual robustness problem for $W_{1}$ and obtain its optimal non- classical CCDC witness $S_{1}$, which gives $R(W_{1})=-\mathrm{tr}(S_{1}W_{1})$. Next, we find a process $W_{2}$ which maximally violates this first witness $S_{1}$, a problem that can be solved by the following SDP: $\displaystyle\textrm{min}\hskip 39.83368pt\mathrm{tr}(S_{1}W_{2})$ (100a) $\displaystyle\textrm{s.t.}\hskip 44.10185ptW_{2}\succeq 0,$ (100b) $\displaystyle\hskip 56.9055ptW_{2}=L_{A\rightarrow B}(W_{2}),$ (100c) $\displaystyle\hskip 56.9055pt\mathrm{tr}(W_{2})=d_{A_{O}}.$ (100d) Now we repeat the previous steps, that is, we evaluate the dual robustness program for $W_{2}$, and its optimal non-classical CCDC witness $S_{2}$, then find the process $W_{3}$ which maximally violates $S_{2}$. These steps are taken iteratively until some stopping criterion is satisfied. In our code, the stopping criterion used is $R(W_{i+1})-R(W_{i})\leq\epsilon=0.0001$. In the end of this procedure, we obtain a process $W$ which attains a non-classical CCDC robustness, providing a lower bound on the maximal value for the given scenario. In order to increase the confidence in this heuristic method, we perform this algorithm several times with for various initial processes $W_{1}$ randomly sampled from different manners. We have implemented this heuristic methods for three different scenarios: $d_{A_{I}}{=}d_{A_{O}}{=}d_{B_{I}}{=}2$, $d_{A_{I}}{=}d_{A_{O}}{=}2,d_{B_{I}}{=}4$, and $d_{A_{I}}{=}d_{A_{O}}{=}3,d_{B_{I}}{=}9$. For these cases, our see-saw algorithm led to the same value of robustness for several different random initial processes, suggesting that the heuristic method may have attained the global maximum. ## Appendix G Sampling random ordered process In this section, we describe the methods for sampling random processes used in this work. We remark that Ref. [68] presents a method for generating uniformly distributed random process matrices which are different from the ones considered here. ### Method 1 This method is similar to the technique for generating random quantum channels used in Ref. [69]. 1. 1. Sort a random density operator $\rho\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I})$ with the Hilbert-Schmidt measure, _i.e_ , sort a random pure quantum state with the Haar measure $\|{\psi}\rangle\langle{\psi}\|\in\mathcal{L}(A_{I}\otimes A_{O}\otimes B_{I}\otimes\textrm{aux})$ with $d_{\textrm{aux}}=d_{A_{I}}d_{A_{O}}d_{B_{I}}$, then trace out the auxiliary space; 2. 2. Project $\rho$ into the subspace of bipartite ordered process to obtain $\overline{W}=L_{A\rightarrow B}(\rho)$; 3. 3. Evaluate the minimum eigenvalue $\lambda_{\text{min}}$ of $\overline{W}$ and output the bipartite ordered process: $W=d_{A_{O}}\cdot\frac{\overline{W}-\lambda_{\text{min}}\mathds{1}^{d_{A_{I}}d_{A_{O}}d_{B_{I}}}}{\mathrm{tr}\left(\overline{W}-\lambda_{\text{min}}\mathds{1}^{d_{A_{I}}d_{A_{O}}d_{B_{I}}}\right)},$ (101) which is positive semi-definite by construction. ### Method 2 This method is good for generating random processes with high values of generalized and white noise robustnesses. However, it only works when $\frac{d_{A_{I}}d_{A_{O}}}{d_{B_{I}}}$ is an integer; 1. 1. Set $d_{\textrm{aux}}=d_{A_{I}}$ and sort a random pure state $\|{\psi}\rangle\langle{\psi}\|^{A_{I}\textrm{aux}}\in\mathcal{L}(A_{I}\otimes\textrm{aux})$ according to the Haar measure. 2. 2. Set $d_{\textrm{aux}^{\prime}}=\frac{d_{A_{O}}d_{\textrm{aux}}}{d_{B_{I}}}$ and sort a random unitary operator $\text{U}:A_{O}\otimes\textrm{aux}\to B_{I}\otimes\textrm{aux}^{\prime}$ according to the Haar measure. Then, obtain its Choi operator $\|{\text{U}}\rangle\rangle\langle\langle{\text{U}}\|^{A_{O}\textrm{aux}/B_{I}\textrm{aux}^{\prime}}$ and define the channel $D^{A_{O}\textrm{aux}/B_{I}}:=\mathrm{tr}_{\textrm{aux}^{\prime}}\left(\|{\text{U}}\rangle\rangle\langle\langle{\text{U}}\|^{A_{O}\textrm{aux}/B_{I}\textrm{aux}^{\prime}}\right)$; 3. 3. Output the operator: $\displaystyle W$ $\displaystyle=\rho^{A_{I}\textrm{aux}}D^{A_{O}\textrm{aux}/B_{I}},$ (102) which is a valid bipartite ordered process. ### Method 3 1. 1. Set $d_{\textrm{aux}}=d_{A_{I}}$ and sort a random pure state $\|{\psi}\rangle\langle{\psi}\|^{A_{I}\textrm{aux}}$ according to the Haar measure; 2. 2. Sort a random density operator $\rho\in\mathcal{L}(A_{O}\otimes\textrm{aux}\otimes B_{I})$ according to the Hilbert-Schmidt measure; 3. 3. Define $D^{A_{O}\textrm{aux}/B_{I}}=\left(\sigma^{-\frac{1}{2}}\otimes\mathds{1}^{B_{I}}\right)\rho\left(\sigma^{-\frac{1}{2}}\otimes\mathds{1}^{B_{I}}\right)$, with $\sigma=\mathrm{tr}_{B_{I}}(\rho)$. The resulting operator $D^{A_{O}\textrm{aux}/B_{I}}$ is the Choi operator of a channel $\widetilde{D}:\mathcal{L}(A_{I}\otimes\textrm{aux})\rightarrow\mathcal{L}(B_{I})$, as $D^{A_{O}\textrm{aux}/B_{I}}\succeq 0$ and $\mathrm{tr}_{B_{I}}(D^{A_{O}\textrm{aux}/B_{I}})=\mathds{1}^{A_{O}\textrm{aux}}$ by direct inspection; 4. 4. Output the operator $W=\|{\psi}\rangle\langle{\psi}\|^{A_{I}\textrm{aux}}*D^{A_{O}\textrm{aux}/B_{I}},$ (103) which is a valid bipartite ordered process.
# Impact of correlations and heavy-tails on quantum error correction B.D. Clader<EMAIL_ADDRESS>Colin J. Trout Jeff P. Barnes Kevin Schultz Gregory Quiroz Paraj Titum Johns Hopkins University Applied Physics Laboratory 11100 Johns Hopkins Road, Laurel, MD 20723, USA ###### Abstract We show that space- and time-correlated single-qubit rotation errors can lead to high-weight errors in a quantum circuit when the rotation angles are drawn from heavy-tailed distributions. This leads to a breakdown of quantum error correction, yielding reduced or in some cases no protection of the encoded logical qubits. While heavy-tailed phenomena are prevalent in the natural world, there is very little research as to whether noise with these statistics exist in current quantum processing devices. Furthermore, it is an open problem to develop tomographic or noise spectroscopy protocols that could test for the existence of noise with such statistics. These results suggest the need for quantum characterization methods that can reliably detect or reject the presence of such errors together with continued first-principles studies of the origins of space- and time-correlated noise in quantum processors. If such noise does exist, physical or control-based mitigation protocols must be developed to mitigate this noise as it would severely hinder the performance of fault-tolerant quantum computers. ## I Introduction The theory of fault tolerance and the associated threshold theorem demonstrate that if the physical error rate per gate can be lowered below some threshold, then one can perform quantum computation with arbitrary accuracy with polynomial overhead (see e.g. Shor (1995); Steane (1996); Aliferis _et al._ (2006); Aharonov and Ben-Or (2008); Terhal (2015)). The prevailing noise model for analyzing quantum error correcting codes is noise that manifests itself as bit flips and phase flips that are local in both time and space, meaning there are no spatial or temporal correlations Nielsen and Chuang (2010). However, it is well known that at least some spatial correlations are inevitable in a quantum system due to such physical effects as common baths shared amongst qubits or control-line crosstalk Harper _et al._ (2020). In addition, time- correlated noise is generally always present (e.g. $1/f^{\alpha}$ type noise in superconducting qubit system Bialczak _et al._ (2007); Bylander _et al._ (2011); Burnett _et al._ (2014); Paladino _et al._ (2014); Kumar _et al._ (2016); Burnett _et al._ (2019)). For these reasons, there have been a number of studies that have examined the impact of spatial and temporal noise correlations on quantum error correction (see e.g. Clemens _et al._ (2004); Klesse and Frank (2005); Terhal and Burkard (2005); Aharonov _et al._ (2006); Aliferis _et al._ (2006); Alicki _et al._ (2006); Novais and Baranger (2006); Novais _et al._ (2008); Shabani (2008)). Despite the proliferation of studies examining noise correlations, the complete impact on error correction is mixed. Some studies suggest that noise correlations are fairly detrimental Clemens _et al._ (2004); Klesse and Frank (2005); Alicki _et al._ (2006); Novais _et al._ (2008); Cafaro and Mancini (2010) while others suggest that most realistic models of correlated noise can still be handled via quantum error correction with a manageable overhead Clemens _et al._ (2004); Terhal and Burkard (2005); Aharonov _et al._ (2006); Aliferis _et al._ (2006); Novais and Baranger (2006); Shabani (2008). These disagreements arise due to the difficulty in analyzing and simulating a quantum error correction (QEC) code operating on realistic multi-level quantum systems interacting with a general open quantum system bath. Approximations are always required to create manageable calculations such as considering only the two-level subspace of the multi-level system and severe approximations as to the impact of the bath on the qubits. The most restrictive of these bath approximations is the assumption that it can be modeled using the Pauli error model where random bit flips and phase flips are inserted stochastically into the circuit. Despite the simple nature of this error model, it has proven highly useful in the general theory of fault-tolerant QEC and the development of new QEC codes. The standard theory of fault-tolerant QEC assumes that if a weight-one Pauli error occurs with probability $p$ then a weight-two Pauli error occurs with probability of order $p^{2}$ and so on. Under these assumptions, a QEC code will yield a logical error rate proportional to $p^{(d+1)/2}$ where $d$ is the code distance. Correlated Pauli errors are clearly detrimental and reduce the effectiveness of QEC. If a weight-two error occurs with an error rate proportional to $p$ then the effective distance of the code is reduced by 1 and so on. Generating these types of errors can happen if the interaction Hamiltonian contains entangling terms (sometimes called weight-two generators). These types of errors might arise from a common bath shared by all qubits that generates entanglement or residual coupling between qubits that cannot be effectively turned off to a sufficient level. It is generally assumed that qubits at best might share some portion of the bath, but that the correlations would be short range at worst. Unfortunately, models of open quantum systems with spatially correlated baths are not well studied, although some progress has been made Jeske and Cole (2013). Despite this, it is generally assumed that these higher weight errors can be effectively controlled and mitigated in future quantum systems. In this report, we analyze a related model of decoherence, but one that avoids the difficulties required when performing general open quantum system calculations. The errors are modeled as classical random variables within the Hamiltonian. This model is often called the semi-classical system bath model. It is an alternative model to a general open quantum system coupled to a quantum bath, and is considered a reasonable approximation in certain instances. In particular, the bath must be in thermal equilibrium, there is no back action on the environment from the qubits (i.e. the bath dephases instantly), and the bath is at infinite temperature yielding equal populations of the qubit states after long term decay Kubo (1963); Haken and Strobl (1973); Čápek (1993); Cheng and Silbey (2004); Gardiner (2004); Cheng and Silbey (2005); Van Kampen (1997). These assumptions apply directly to classical noise from the classical control system. The advantage to such a model from our perspective is that spatial and temporal correlations are fairly straightforward to model in this manner. There are known situations where this approximation breaks down and the quantum nature of the bath leads to observable signatures such as nontrivial phase evolution in addition to pure decoherence Paz-Silva _et al._ (2017). This can occur for example for baths at low temperature or systems that are strongly coupled. We do not consider these more general quantum bath models here, and leave that for future work. Using the semiclassical noise model allows us to demonstrate a rather surprising result. We show that single-qubit rotation errors arising from weight-one error generators can lead to high-weight errors in a quantum system when the noise is either spatially or temporally correlated and drawn from certain types of heavy-tailed distributions Bryson (1974). This results in a reduction in the effective distance of the QEC code that depends on the tails of the distribution. Various definitions of heavy-tailed (sometimes referred to as fat-tailed) distributions exist. Generally speaking they are distributions whose tails decay more slowly than exponential (e.g. as a power law) and they have undefined (or infinite) variance. They are well-studied in the quantitative finance literature Haas and Pigorsch (2009) as they are routinely used to model things such as exogenous shocks to financial markets (see e.g. Refs. Fernandez-Villaverde and Rubio-Ramirez (2007); Fagiolo _et al._ (2008); Mishkin (2011); Ascari _et al._ (2012)) or even the value of returns for asset prices recognized early on with the seminal work of Mandelbrot Mandelbrot (1963). Relatively little research has been conducted as to whether semi-classical noise with heavy-tailed distributions exists in quantum computing devices, but this not true for quantum systems in general. Events with heavy-tailed distributions have been discussed extensively in the context of physical models that generate $1/f^{\alpha}$ noise spectra. It has been shown that signals from systems with dynamics that adhere to families of point process models Niemann _et al._ (2013); Eliazar and Klafter (2010); Davidsen and Schuster (2002); Kaulakys _et al._ (2005); Lowen and Teich (1993); Lukovic and Grigolini (2008) and linear/nonlinear stochastic differential equations (SDEs) Kaulakys _et al._ (2013); Ruseckas and Kaulakys (2010) generate $1/f^{\alpha}$ spectra when the signal probability density functions (PDFs) obey heavy-tailed statistics. Various physical systems have exhibited signals with power-law statistics. Similar point process models have been utilized to describe the power-law behavior of fluorescent blinking in quantum dots Kuno _et al._ (2000); Shimizu _et al._ (2001); Pelton _et al._ (2004); Margolin _et al._ (2006); Pelton _et al._ (2007); Mahler _et al._ (2008); Frantsuzov _et al._ (2009, 2013) and single-molecule fluorescence of organic molecules Haase _et al._ (2004); Schuster _et al._ (2005); Hoogenboom _et al._ (2005); Yeow _et al._ (2006). Power-law behavior has been observed in trapping times for charge transport in amorphous semiconductors Lowen and Teich (1993); Dhariwal and Deoraj (1991); Tiedje and Rose (1981); Lowen and Teich (1992); Adler _et al._ (1985) and nanoscale electrodes Krapf (2013). The family of SDEs that exhibit power law behavior generate dynamics that violate the fluctuation dissipation and equipartition of energy theorems which has been observed in finite-dimensional spin-glasses Lobaskin and Kehrein (2006). Furthermore, spin glasses exhibit random couplings and relaxation rates that obey power-law behavior Klein (1968); Klein _et al._ (1976); Cizeau and Bouchaud (1993); Berkov (1996); Pickup _et al._ (2009); Neri _et al._ (2010). For quantum processors, non-Gaussian noise spectroscopy techniques have been developed Norris _et al._ (2016) and demonstrated experimentally Sung _et al._ (2019), but this approach only applies to noise with distributions tighter than Gaussian. The difficulty in developing characterization techniques for heavy-tailed distributions is that most quantum characterization techniques that seek to characterize the statistics of noise correlations rely on expanding the statistics of the noise into moments (or cumulants) Kubo (1962); Kampen (1974a, b). Since higher-order moments are undefined or infinite for heavy-tailed distributions these existing techniques do not apply. Our results show that a new characterization technique is needed to test for this type of harmful noise in quantum systems. If this type of noise is present, it is imperative that physical or control-based mitigation schemes are developed to reduce its effect as QEC will not suffice. ## II Impact of correlated noise on QEC: Analytic Model We begin our analysis by considering an analytic model of QEC where we consider noise arising from stochastic unitary rotation errors on the data qubits only in an $n$-qubit perfect code. This model is referred to as the code-capacity error model. Physically, these correlations would correspond to spatial correlations between qubits. The term perfect here refers to the fact that our analytical analysis assumes that a distance $d$ code can correct exactly $(d-1)/2$ errors on the encoded qubits and no more. Certain codes, such as surface codes, can correct certain types of high-weight errors that will break this assumption. Numerical simulations shown later demonstrate that our code-capacity model accurately predicts the behavior of fully fault- tolerant implementations of QEC with noise affecting data and ancilla qubits at all locations in the circuit. Our model starts by assuming that we have perfectly encoded an $n$-qubit logical state into a QEC code denoted as $\ket{\psi_{L}}$. We then apply a single-qubit rotation about an arbitrary axis to all of the data qubits in the encoded state yielding $\ket{\psi_{L}}\to\prod_{j=1}^{n}\left[\cos\left(\frac{\theta_{j}}{2}\right)\hat{I}-i\sin\left(\frac{\theta_{j}}{2}\right)\vec{v}\cdot\hat{\vec{\sigma}}^{(j)}\right]\ket{\psi_{L}},$ (1) where $\hat{I}$ is the identity matrix, $\vec{v}$ is an arbitrary unit three vector of real numbers specifying the direction of rotation, $\hat{\vec{\sigma}}^{(j)}$ is a three vector containing the Pauli $x,y$, and $z$ operators, and $\theta_{j}$ is the angle of rotation for qubit $j$. We consider two cases. In the uncorrelated case, each angle of rotation is drawn from a probability distribution and taken to be independent for a total of $n$ independent random variables. In the correlated case, we assume that a single angle is drawn from the same probability distribution and applied to each qubit. The average probability of a logical error is then given by $P_{unc}=1-\sum_{k=0}^{w}\binom{n}{k}\left\langle\cos^{2}\left(\frac{\theta}{2}\right)\right\rangle^{n-k}\left\langle\sin^{2}\left(\frac{\theta}{2}\right)\right\rangle^{k}$ (2a) $P_{cor}=1-\sum_{k=0}^{w}\binom{n}{k}\left\langle\cos^{2(n-k)}\left(\frac{\theta}{2}\right)\sin^{2k}\left(\frac{\theta}{2}\right)\right\rangle,$ (2b) where we have assumed that our error correcting code can correct all Pauli errors of weight $w=(d-1)/2$, $\theta$ is a random variable, and the angular brackets $\langle\cdot\rangle$ denote an ensemble average. After some algebraic manipulations, we can transform Eqs. (2) into $P_{unc}=1-\frac{1}{2^{n}}\sum_{k=0}^{w}\sum_{l=0}^{n-k}\sum_{m=0}^{k}\binom{n}{k}\binom{n-k}{l}\binom{k}{m}(-1)^{m-k}f(t=1)^{n-l-m}$ (3a) $P_{cor}=1-\frac{1}{2^{2n}}\sum_{k=0}^{w}\sum_{l=0}^{2(n-k)}\sum_{m=0}^{2k}\binom{n}{k}\binom{2(n-k)}{l}\binom{2k}{m}(-1)^{m-k}f(t=n-l-m),$ (3b) where $f(t)$ is the characteristic function of the probability distribution. In deriving Eqns. (3) we have assumed distributions that are symmetric about 0 such that the characteristic function is even $f(t)=f(-t)$. For a single physical qubit, we define the failure probability to be the probability that upon measurement we get either a bit-flip or phase-flip error. For a qubit rotated by an angle $\theta$, this is given by $P=\sin^{2}\left(\frac{\theta}{2}\right)$ (4) with the corresponding expectation value in terms of the characteristic function given by $\langle P\rangle\equiv P_{ph}=\frac{1}{2}\left[1-f(t=1)\right],$ (5) where $P_{ph}$ stands for the probability of a physical error occurring. Evaluation and comparison of Eqs. (3) and (5) allow us to examine the impact of various noise distributions by inserting the known characteristic function and computing the formula for given code sizes and comparing the logical error rate with the physical error rate. For the purpose of our analysis, the number of qubits in the code will be equal to the the Knill–Laflamme bound $n\geq 4w+1$ unless otherwise specified. Table 1: Comparison of the leading order term in a $\sigma\ll 1$ expansion of the failure probability for a physical qubit and various distance perfect codes with uncorrelated and correlate noise. The rows are for various $\nu=2r-1$ parameters in the Student’s $t$-distribution as given in Eq. (8). As $r$ increases the tails of the distribution are reduced, resulting in a tighter distribution. As this occurs, the reduction in effective code distance is pushed out to higher distances. As an example, for a $d=3$ code at $r=3$ the effective distances are equivalent for uncorrelated and correlated noise. We find that when $d\leq 2r-3$ the code distance is equal for correlated and uncorrelated noise, but when $d>2r-3$ the effective distance is reduced for correlated noise. For all code distances we set the number of qubits to $n=4w+1$, which is the minimum Knill-Laflamme bound. $r$ \| Physical \| $d=3$ \| $d=5$ \| $d=7$ \| $d=9$ ---\|---\|---\|---\|---\|--- \| \| Unc \| Cor \| Unc \| Cor \| Unc \| Cor \| Unc \| Cor 1 \| $\frac{1}{2}\sigma^{\phantom{1}}$ \| $\frac{5}{2}\sigma^{2}$ \| $\frac{35}{64}\sigma^{\phantom{1}}$ \| $\frac{21}{2}\sigma^{3}$ \| $\frac{9009}{16384}\sigma^{\phantom{1}}$ \| $\frac{715}{16}\sigma^{4}$ \| $\frac{1154725}{2097152}\sigma^{\phantom{1}}$ \| $\frac{1547}{8}\sigma^{5\phantom{1}}$ \| $\frac{591534125}{1073741824}\sigma^{\phantom{10}}$ 2 \| $\frac{3}{4}\sigma^{2}$ \| $\frac{45}{8}\sigma^{4}$ \| $\frac{175\sqrt{3}}{64}\sigma^{3}$ \| $\frac{567}{16}\sigma^{6}$ \| $\frac{27027\sqrt{3}}{16384}\sigma^{3}$ \| $\frac{57915}{256}\sigma^{8}$ \| $\frac{3002285\sqrt{3}}{2097152}\sigma^{3}$ \| $\frac{375921}{256}\sigma^{10}$ \| $\frac{1436582875\sqrt{3}}{1073741824}\sigma^{3\phantom{1}}$ 3 \| $\frac{5}{12}\sigma^{2}$ \| $\frac{125}{72}\sigma^{4}$ \| $\frac{125}{8}\sigma^{4}$ \| $\frac{875}{144}\sigma^{6}$ \| $\frac{705705\sqrt{5}}{16384}\sigma^{5}$ \| $\frac{446875}{20736}\sigma^{8}$ \| $\frac{345262775\sqrt{5}}{18874368}\sigma^{5}$ \| $\frac{4834375}{62208}\sigma^{10}$ \| $\frac{130729041625\sqrt{5}}{9663676416}\sigma^{5\phantom{1}}$ 4 \| $\frac{7}{20}\sigma^{2}$ \| $\frac{49}{40}\sigma^{4}$ \| $\frac{49}{8}\sigma^{4}$ \| $\frac{7203}{2000}\sigma^{6}$ \| $\frac{7203}{16}\sigma^{6}$ \| $\frac{343343}{32000}\sigma^{8}$ \| $\frac{113011411513\sqrt{7}}{94371840}\sigma^{7}$ \| $\frac{26000429}{800000}\sigma^{10}$ \| $\frac{3787119775075\sqrt{7}}{9663676416}\sigma^{7\phantom{1}}$ 5 \| $\frac{9}{28}\sigma^{2}$ \| $\frac{405}{392}\sigma^{4}$ \| $\frac{243}{56}\sigma^{4}$ \| $\frac{2187}{784}\sigma^{6}$ \| $\frac{2187}{16}\sigma^{6}$ \| $\frac{4691115}{614656}\sigma^{8}$ \| $\frac{4691115}{256}\sigma^{8}$ \| $\frac{13049829}{614656}\sigma^{10}$ \| $\frac{1086610719708657}{7516192768}\sigma^{9\phantom{1}}$ 6 \| $\frac{11}{36}\sigma^{2}$ \| $\frac{605}{648}\sigma^{4}$ \| $\frac{605}{168}\sigma^{4}$ \| $\frac{9317}{3888}\sigma^{6}$ \| $\frac{1331}{16}\sigma^{6}$ \| $\frac{10468315}{1679616}\sigma^{8}$ \| $\frac{10468315}{2304}\sigma^{8}$ \| $\frac{249145897}{15116544}\sigma^{10}$ \| $\frac{249145897}{256}\sigma^{10}$ Figure 1: Ratio of $P_{cor}/P_{unc}$ of the leading order term in powers of $\sigma$ in the error expansion of Eqs. (3) for correlated and uncorrelated Gaussian noise, assuming $\sigma\ll 1$. The labels on the data points correspond to the numerical value of the ratio. As the code distance increases, the ratio increases as $d!!$. ### II.1 Gaussian First, we examine the case where the noise is drawn from a Gaussian distribution with zero mean and standard deviation $\sigma$. The characteristic function of a Gaussian random variable with zero mean is $f(t;\sigma)=e^{-\frac{1}{2}\sigma^{2}t^{2}}.$ (6) To understand the impact on QEC, we study the low noise scaling of the physical and logical error rates with respect to the width parameter of the probability distribution. For the Gaussian distribution this implies taking a series expansion about $\sigma\to 0$ for Eqs. (3). Evaluation of the leading order terms for both correlated and uncorrelated noise reveals logical error rates that scale as $\sigma^{2(w+1)}$ where $w$ is the number of correctable errors for a code of distance $d=2w+1$. Thus, Gaussian correlated noise does not yield high-weight errors that might impact the code distance, however it does affect the series coefficient. To see this, we plot the ratio of the first non-zero term for correlated noise to uncorrelated noise in Fig. 1. As the distance of the code increases, the ratio of the series coefficient grows at a rate proportional to $d!!$ implying that the logical error rate is increased by a related factor for correlated noise. This implies that if correlated single-qubit rotation noise drawn from a Gaussian distribution is present, its impact may be minor for small codes, but grows as code distance increases. We note as an aside that the $\sigma\to 0$ approximation of Eqs. (2) must be done with care as the number of terms in the binomial expansion grows for larger codes. Thus the approximation must technically be in the regime where $\sigma\ll(1/d!!)^{2/(d+1)}$. These results agree with previously published work and imply that a threshold does not exist Clemens _et al._ (2004). Despite this, the code can still suppress noise and a pseudo-threshold does exist for each distance. For a code of distance 3, the leading order terms assuming $\sigma\ll 1$ are $\displaystyle P_{ph}$ $\displaystyle\approx\frac{\sigma^{2}}{4}$ (7) $\displaystyle P_{unc}$ $\displaystyle\approx\frac{5\sigma^{4}}{8}$ $\displaystyle P_{cor}$ $\displaystyle\approx\frac{15\sigma^{4}}{8}.$ The logical error rate for both correlated and uncorrelated noise is reduced by a squared factor of $\sigma$, while the pre-factor of the correlated noise logical error rate increases by a factor of three relative to the uncorrelated case. The quadratic reduction of the logical error rate suggests the code is behaving as expected by a distance $3$ code in both cases. ### II.2 Student’s t Next, we examine the Student’s $t$-distribution to see what impact varying the probability distribution of the random variable has on the physical and logical error rates. The Student’s $t$-distribution is an example of a heavy- tailed distribution with a discrete parameter $\nu$ that governs the heaviness of the tails of the distribution. This can be seen by considering the probability density function of the Student’s $t$-distribution $PDF_{t}(\theta;\nu,\sigma)=\frac{(\nu\sigma^{2})^{\frac{\nu}{2}}\Gamma\left(\frac{\nu+1}{2}\right)}{\sqrt{\pi}\Gamma\left(\frac{\nu}{2}\right)}\frac{1}{\left(\nu\sigma^{2}+\theta^{2}\right)^{\frac{\nu+1}{2}}}.$ (8) Here $\nu\geq 1$ is an integer corresponding to the number of degrees of freedom of the distribution and $\Gamma(x)$ is the gamma function. We will restrict our attention to cases where $\nu$ is odd. Taking $\nu=1$ gives exactly the Cauchy distribution, while larger values of $\nu$ tightens the tails of the probability distribution via this discrete parameter. In the limit of $\nu\to\infty$ the Gaussian distribution is recovered. The characteristic function of a Student’s $t$-distribution is $f(t;\sigma,\nu)=\frac{\sigma^{\frac{\nu}{2}}\nu^{\frac{\nu}{4}}\|t\|^{\frac{\nu}{2}}}{2^{\frac{\nu}{2}-1}\Gamma\left(\frac{\nu}{2}\right)}K_{\nu/2}\left(\sigma\sqrt{\nu}\|t\|\right),$ (9) where $K_{n}(x)$ is the modified Bessel function of the second kind. Again, we just report the approximate expressions in the $\sigma\ll 1$ limit. For additional ease of analysis, we take $\nu$ to be odd, but we lift this restriction in our numerical simulations. We display the results in Table 1 for the physical and logical errors rates with both correlated and uncorrelated noise, where we have replaced $\nu$ with $\nu=2r-1$ and $r\geq 1$ is a positive integer. For correlated Cauchy random noise ($r=1$) error correction provides no error suppression as the effective code distance is reduced to 1 for all code distances considered. We conjecture that this extends to arbitrary code distance. This implies that single-qubit rotations with correlated rotation angles drawn from a Cauchy distribution result in at least weight $(d+1)/2$ errors in the quantum circuit. As $r$ increases the tails of the distribution are tightened, and the reduction in the code distance is pushed out to higher distances. We find that when the distance is $d\leq 2r-3$, for positive $d$, the effective distance of the QEC is equivalent for the correlated and uncorrelated cases. However, once $d>2r-3$ the effective distance begins to be reduced for correlated noise with the maximum exponent on $\sigma$ appearing to be $2r-1$ for correlated noise. These results imply that correlated single-qubit noise drawn from the Student’s $t$-distribution yields higher-weight Pauli errors with the weight of the error related to the tail index of the distribution. Figure 2: The various plots correspond to the value of the error term proportional to $\sigma^{\alpha}$ for all valid values of $\alpha$ in the correlated noise error expansion, $P_{cor}$. The dashed line is the first order term of the physical error rate $P_{ph}=0.5$ of Eqs. (3). Code distance does not have a large impact as seen by the nearly indistinguishable curves. For all cases plotted, the logical error rate containing correlated noise from an alpha stable distribution will not provide protection of the encoded logical qubits as this distribution results in errors with weight of $(d+1)/2$ or greater expect at the exact point where $\alpha=2$, which corresponds to the Gaussian distribution. ### II.3 Lévy alpha-stable Finally, we consider the Lévy alpha-stable distributions. These are a family of probability distributions that contain the Gaussian and Cauchy distributions and allow for continuous interpolation between them. It is also useful as it is a stable distribution meaning that sums of independent random variates have the same distribution up to location and scale parameters. This allows us to efficiently model time correlations in our numerical simulations that we show in the next section. The probability density function for the stable distribution is not analytically expressible. However the characteristic function is expressible. We write a simplified form of the characteristic function where we take the skewness and location parameters to be zero yielding $f(t;\sigma,\alpha)=e^{-\|\sigma t\|^{\alpha}}.$ (10) The parameter $\alpha$ is the stability parameter and it lies in the range $(0,2]$. The stable distribution corresponds to a Gaussian when $\alpha=2$ and a Cauchy when $\alpha=1$, and allows for continuous interpolation between Gaussian and heavy-tailed distributions. The formulas are complicated for the general case, so we examine the simplest case for a distance 3 code. The leading order terms, assuming $\sigma\ll 1$, for the physical and logical error rates are $\displaystyle P_{ph}$ $\displaystyle\approx\frac{\sigma^{\alpha}}{2}$ (11) $\displaystyle P_{unc}$ $\displaystyle\approx\frac{5\sigma^{2\alpha}}{2}$ $\displaystyle P_{cor}$ $\displaystyle\approx\frac{1}{128}\left[5\left(2^{\alpha+2}\right)-5\left(3^{\alpha}\right)-5\left(4^{\alpha}\right)-5^{\alpha}+70\right]\sigma^{\alpha}$ $\displaystyle+\frac{1}{256}\left[5\left(-4^{\alpha+1}+9^{\alpha}+16^{\alpha}-14\right)+25^{\alpha}\right]\sigma^{2\alpha}.$ Only for exactly $\alpha=2$, which corresponds to a Gaussian distribution, does the lowest order term (the term of order $\sigma^{\alpha}$) of the correlated error rate disappear. Any $\alpha<2$ yields a term that is proportional to the physical error rate, causing the error correcting code to become ineffective against this type of noise. To examine larger code distances, we plot the value of the term proportional to $\sigma^{\alpha}$ for the correlated noise logical error rate in Fig. 2. The figure shows that for all code distances considered the logical error rate scaling is proportional to the physical error rate scaling. In other words, the QEC code offers no protection for this type of noise no matter how large of a code is used. Only exactly when $\alpha=2$ does the term dissapear and we recover protection of the encoded logical state as shown previously in the discussion on Gaussian noise. ## III Numerical Simulations Our analytic model has shown the detrimental effect of spatially correlated noise on the data qubits within a single error correction block, with correlated single-qubit (weight one) rotation errors leading to uncorrectable multi-qubit (high-weight) errors. However, determining fault tolerance also requires the accounting of errors within the time-length of a decoding block Aliferis _et al._ (2006). Therefore, we anticipate that time correlations of weight-one error generators would have an equally harmful impact on quantum error correction as they would lead to a breakdown of fault-tolerance due to high-weight errors occurring in time across the decoding boundaries. To study this, we use numerical simulations of low-distance surface codes to examine the case of time-correlated errors from heavy-tailed distributions. In addition to perfect correlations that we studied with our analytical model, we also generalize to the situation where the noise correlations are defined by a correlation function. Our simulations show that time-correlations do result in a similar breakdown as our analytical results predict for spatially-correlated noise on the data qubits only, but with non-trivial dependence on the correlation function of the noise. Because our errors are stochastic unitary errors, we must simulate the entire state-vector. This limits us to low-distance codes and for this report, we limit ourselves to just simulations of a distance-3 rotated surface code. We make use of the same simulation framework and circuits that we used in our study of random coherent errors Barnes _et al._ (2017). We model the errors as random unitary gates $U_{k}^{(\ell)}(\theta_{k}^{(\ell)})=\exp(-i\theta_{k}^{(\ell)}\sigma_{y}^{(\ell)})$ that create a rotation by a random angle $\theta_{k}^{(\ell)}$ (here drawn from Gaussian, Cauchy, Student’s t, and Lévy alpha-stable distributions) about the $Y$ axis at a circuit time location $k$ for qubit $\ell$. We insert these errors across all the qubits in the code (both data and ancilla) after every single gate in the circuit. To compute the performance of the code we start with a random initial state $\ket{\psi_{0}}=\cos\alpha\ket{0}+e^{i\beta}\sin\alpha\ket{1},$ (12) where $0\leq\alpha<2\pi$ and $0\leq\beta<2\pi$ are both uniform random variables. This random initialization covers the Bloch sphere, but it is not uniform. We define the fidelity to be $\mathcal{F}^{2}=\frac{1}{(2\pi)^{2}}\iint_{0}^{2\pi}d\alpha d\beta\int_{-\infty}^{\infty}d\theta p(\theta)\|\braket{\psi_{0}}{e^{-i\theta\sigma_{y}}}{\psi_{0}}\|^{2},$ (13) where $p(\theta)$ is the probability distribution for the error terms. This expression simply computes the overlap of the initial state and the final state and averages over the probability distributions. We note, as before, that this is not the standard definition of fidelity, since the distribution of $\alpha$ and $\beta$ is uniform this is not a Haar average over the Bloch sphere. This yields slightly different error rates for different errors on different axes, but this does not have any meaningful impact on our overall results. Using Eqs. (12) and (13), it is straightforward to analytically calculate the physical fidelity for the various probability distributions considered in this manuscript. It is $\mathcal{F}^{2}=\frac{5}{8}+\frac{3}{8}f(t=2)$ (14) where $f(t=2)$ is the characteristic function of the probability distribution and is given in Eqs. (6), (9), and (10) for the Gaussian, Student’s t, and Stable distributions respectively. To calculate the logical fidelity, we perfectly encode the random state defined in Eq. (12), simulate three rounds of faulty syndrome extraction with errors inserted after every location where a gate exists (a circuit-level noise model), and follow that by decoding and perfect correction. We conclude the simulation with a round of perfect error correction to correct any trailing errors. We estimate the logical fidelity by numerically estimating the integral given in Eq. (13) by computing the overlap of the final decoded logical state with the initial state and Monte Carlo sampling over the initial random states and error terms. We use bootstrap resampling to report the $95\%$ confidence intervals with $10^{3}$ samples with replacement used. Each data point is the result of $10^{7}$ independent trials. For arbitrary time correlations we leverage SchWARMA Schultz _et al._ (2020) to simulate time-correlated noise in quantum circuits. The SchWARMA method leverages a classical time-series modeling approach called autoregressive- moving-average (ARMA) models where the angle of rotation at circuit time location $k$ is an ARMA model $\theta_{k}^{(\ell)}=\underbrace{\sum_{i=1}^{p}a_{i}\theta_{k-i}^{(\ell)}}_{AR}+\underbrace{\sum_{j=0}^{q}b_{j}x_{k-j}^{(\ell)}}_{MA}\,.$ (15) The set $\\{a_{i}\\}$ defines the autoregressive portion of the model, and $\\{b_{j}\\}$ the moving-average portion with $p$ and $q+1$ elements of each set, respectively, and the $x_{k-j}^{(\ell)}$ are random variables drawn from the user-defined probability distribution. Because ARMA models require one to add random variates we must restrict ourselves to stable distributions when using this method to ensure that the probability distribution of the output model remains the same as the random variables. The Student’s $t$-distribution is not stable. Therefore, in that case we consider white noise (uncorrelated in time) and direct-current (DC) noise in which we draw a single random variable at the beginning of the quantum circuit for each qubit and we use that same angle at all subsequent times in the circuit. These two limits correspond to an ARMA model with $p=0,q=0$ and $p=0,q\to\infty$ respectively. For the other cases we can consider more general time correlations. For the purposes of this study, we interpolate between white noise and DC noise by considering exponential moving-averages (EMAs) where the terms $b_{j}=N\exp(-\ln(2)j/T_{h})$ where $N$ is chosen such that $\sum_{j}b_{j}=1$ and $T_{h}$ is the “half-life” of the moving average. We do not consider the AR portion in this paper, so $p=0$. We set each gate in the circuit to take a single unit of time, so the parameter $j$ corresponds exactly to the circuit depth to that point and we set the number of terms in the moving average to $q=10\lceil T_{h}\rceil$. The various syndrome extraction circuits for the surface code take anywhere from 2 to 6 time ticks in those units. ### III.1 Gaussian The results for Gaussian noise are plotted in Fig. 3 where we plot the logical infidelity versus the physical infidelity. There is no discernible difference between the DC and white noise case. Below the pseudo-threshold the logical error rate is reduced relative to the physical error rate and the reduction is quadratic as expected for a distance three code. This is all in agreement with our analytical results given in Eq. (7). Figure 3: Pseudo-threshold plots obtained from simulating the rotated distance-3 surface code with independent and time-correlated single-qubit rotation errors drawn from a Gaussian distribution. The SchWARMA simulations denoted with titles EMA $T_{h}$ interpolate between the white and DC noise cases by using an exponential moving average filter. The various curves are denoted by their color, but are indistinguishable on this scale. This demonstrates that time-correlated noise drawn from a Gaussian distribution has minimal impact to low distance QEC codes. The top and bottom dashed lines show slopes where the logical infidelity is proportional to the physical infidelity and the square of the physical infidelity respectively. Error bars are the 95% confidence intervals obtained from bootstrap resampling. They are cutoff at the lowest error rates for display purposes. Figure 4: Pseudo-threshold plots obtained from simulating the rotated distance-3 surface code with independent (top) and DC correlated (bottom) single-qubit rotation errors drawn from a Student’s $t$-distribution for multiple values of the tail index $\nu$ denoted in the legend. For the DC noise case, the curves move from top to bottom in the same order as the legend is displayed with $\nu=1$ having the highest logical error rate and $\nu=5$ the lowest. For white noise, the curves are indistinguishable on this scale. Odd values of $\nu$ correspond to the analytical calculations shown in Table 1. The numerical results agree with our analytical results presented in Tab. 1 with the slope of the correlated DC noise logical error rate varying as the tail index increases, while the slope of the uncorrelated white noise is not impacted. Once $\nu\geq 4$ the correlated logical infidelity scales as the square of the physical infidelity in agreement with our analytical predictions. The top and bottom dashed lines show slopes where the logical infidelity is proportional to the physical infidelity and the square of the physical infidelity respectively. Error bars are the 95% confidence intervals obtained from bootstrap resampling. They are cutoff at the lowest error rates for display purposes. ### III.2 Student’s t Next, we consider the Student’s $t$-distribution. We plot the simulation results in Fig. 4. We consider the white-noise and DC noise cases and vary the tail index $\nu$ for each simulation. The results are consistent with our analytical predictions. For white (uncorrelated) noise, the code suppresses the logical error rate and is relatively immune to different tail indices. Meanwhile, for DC noise (infinite time correlation), the slope of the logical to physical infidelity depends on the tail index. When $\nu=1$ the distributions have the fattest tails and the simulations show that the logical infidelity is proportional to the physical infidelity. This implies that the QEC code provides no protection for the encoded qubits. For $\nu=\\{2,3\\}$ the code offers some protection, but not full protection. Finally, when $\nu\geq 4$ the code offers full protection with the logical infidelity scaling quadratically with the physical infidelity. We note that our analytical results only considered odd values of $\nu$, so the appearance of full code protection was only predicted for $\nu\geq 5$ from our analytical results. ### III.3 Lévy alpha-stable Finally, we show simulations of correlated noise drawn from the Lévy alpha- stable distribution. Since the distribution is stable, we can use the SchWARMA formalism to simulate time-correlated noise that interpolates between the white noise and DC noise cases. We plot these results in Fig. 5. The results once again agree with the analytical results for the white-noise and DC-noise cases. For intermediate correlations times, enabled by our SchWARMA simulations, we observe that the slope of the logical error rate is related to the half-life of the exponential moving average. The time units of the half life correspond exactly to the gate ticks of the circuit. For the surface code, the syndrome extraction cycles take between 2 and 6 ticks (2-4 CNOT gates for the weight-two and weight-four operators and two rotation gates for the X syndrome), so the slope reduction occurs as the half-life approaches the syndrome cycle as one would expect. The noise begins to generate weight-two errors across syndrome boundaries. We have not examined any mitigation schemes in this paper, but our results suggest that if the noise correlations can be mitigated, through decoupling techniques as one example, that these effects can be reduced and fault-tolerant quantum computing could be realized even if heavy-tailed noise exists. Figure 5: Pseudo-threshold plots obtained from simulating the rotated distance-3 surface code with DC, time-correlated, and independent single-qubit rotation errors drawn from an alpha-stable distribution with $\alpha=1.5$. The curves move from top to bottom in the same order as the legend is displayed with DC noise having the highest logical error rate and White noise the lowest. The numerical results agree with our analytical results with the uncorrelated white-noise case yielding a pseudo-threshold. For DC noise, the code offers no protection with the logical error rate scaling proportionally to the physical error rate. The SchWARMA simulations denoted with titles EMA $T_{h}$ interpolate between the white and DC noise cases by using an exponential moving average filter. These simulations show that time-correlated noise drawn from a non-Gaussian heavy-tailed distribution can have a strongly detrimental impact to QEC. The top and bottom dashed lines show slopes where the logical infidelity is proportional to the physical infidelity and the square of the physical infidelity respectively. Error bars are the 95% confidence intervals obtained from bootstrap resampling. ## IV Conclusions We have presented analytical evidence that space- or time-correlated single- qubit noise drawn from heavy-tailed distributions can lead to high-weight errors in a quantum circuit. This can lead to a breakdown of quantum error correction via a reduced code distance (in some instances yielding no protection at all). The exact predictions depend upon the type of distribution used for the noise. For Gaussian noise, correlations cause a reduction in leading order coefficient of the pseudo-threshold, but the code can still suppress the noise. This leads to logical error rates scaling as expected with a slope proportional to $\sigma^{(d+1)/2}$ for a distance $d$ code. Meanwhile for noise with heavy-tailed behavior such as Cauchy, Student’s $t$, or Lévy alpha- stable distributions we find more interesting behavior. The quantum error correcting codes that we considered could not correct correlated noise drawn from Cauchy distribution. This leads to logical failure rates that scale proportionally to the physical error rate. There is no suppression relative to $\sigma$ for any code distance. The Student’s $t$-distribution allows us to interpolate between the Gaussian and Cauchy case via the tail-index parameter $r$. As $r$ increases, the error correcting code is able to suppress the logical error rate relative to the physical error rate with increasing power. Once the code distance $d\leq 2r-3$ the code achieves its expected error suppression. Finally, the Lévy alpha-stable distribution also allows us to interpolate between the Gaussian and Cauchy distributions, but since it is a stable distribution we can also consider arbitrary time correlations. Here we find that the correlation time of the noise has a direct impact on the slope of the logical error rate with longer correlation times leading to more reduction in error suppression ability. These results all reinforce the notion that the ability of quantum error correction to suppress noise is highly dependent on the physical noise model Cafaro and van Loock (2014); Barnes _et al._ (2017); Iyer and Poulin (2018). It is known that many complex physical systems have non-Gaussian noise and we have pointed to many examples in Sec. I of quantum systems where noise with heavy-tailed statistics is present Niemann _et al._ (2013); Eliazar and Klafter (2010); Davidsen and Schuster (2002); Kaulakys _et al._ (2005); Lowen and Teich (1993); Lukovic and Grigolini (2008); Kaulakys _et al._ (2013); Ruseckas and Kaulakys (2010); Kuno _et al._ (2000); Shimizu _et al._ (2001); Pelton _et al._ (2004); Margolin _et al._ (2006); Pelton _et al._ (2007); Mahler _et al._ (2008); Frantsuzov _et al._ (2009, 2013); Haase _et al._ (2004); Schuster _et al._ (2005); Hoogenboom _et al._ (2005); Yeow _et al._ (2006); Lowen and Teich (1993); Dhariwal and Deoraj (1991); Tiedje and Rose (1981); Lowen and Teich (1992); Adler _et al._ (1985); Krapf (2013); Lobaskin and Kehrein (2006); Klein (1968); Klein _et al._ (1976); Cizeau and Bouchaud (1993); Berkov (1996); Pickup _et al._ (2009); Neri _et al._ (2010); Sung _et al._ (2019). 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# Statistical physics through the lens of real-space mutual information Doruk Efe Gökmen Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Zohar Ringel Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem 9190401, Israel Sebastian D. Huber Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Maciej Koch-Janusz Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Department of Physics, University of Zurich, 8057 Zurich, Switzerland James Franck Institute, The University of Chicago, Chicago, Illinois 60637, USA ###### Abstract Identifying the relevant degrees of freedom in a complex physical system is a key stage in developing effective theories in and out of equilibrium. The celebrated renormalization group provides a framework for this, but its practical execution in unfamiliar systems is fraught with ad hoc choices, whereas machine learning approaches, though promising, lack formal interpretability. Here we present an algorithm employing state-of-art results in machine-learning-based estimation of information-theoretic quantities, overcoming these challenges, and use this advance to develop a new paradigm in identifying the most relevant operators describing properties of the system. We demonstrate this on an interacting model, where the emergent degrees of freedom are qualitatively different from the microscopic constituents. Our results push the boundary of formally interpretable applications of machine learning, conceptually paving the way towards automated theory building. Fundamental physical theories, in a reductionist spirit, are often formulated at the smallest scales, describing the interactions of elementary constituents. Nonetheless, the experimentally accessible features typically arise from their collective behaviour. Indeed, there exist profound examples of effective theories, e.g. classical hydrodynamics and thermodynamics, consistently describing complex phenomena in terms of a few macroscopic variables, without making any reference to individual particles. Bridging this scale gap to _derive_ the emergent macroscopic properties from microscopic models is a perpetual challenge. The renormalization group (RG),PhysicsPhysiqueFizika.2.263 ; Wilson1974 ; Wilson1975 ; Fisher1998 provides a powerful framework for this, associating physical theories at different length scales by iteratively coarse-graining configurations of local degrees of freedom (DOFs). The induced RG transformation acts as a telescope in the space of models, generating the RG flow, whose structure around the fixed point eventually reveals the relevant DOFs. They are the scaling operators, which determine the correlations, and thus the physical properties, at large scales. In practice, executing this program in real-space RG is very difficult. The accuracy of the procedure is improved by optimizing the coarse-graining to retain the highest real-space mutual information (RSMI),Koch-Janusz2018 ; optimalRSMI quantifying correlations to distant parts of the system. However, this still misses a crucial insight: any long-range information is due to the scaling operators and thus its optimal compression not only can serve as a better RG transformation, but should allow to extract all the operators themselves, without ever explicitly executing the RG flow. This was recently proven in part: in critical systems, the formal solutions to the RSMI compression problem are determined by the most relevant operators,Gordon2020 _in theory_ allowing to access them directly. Unfortunately, solving this mathematical problem is notoriously hard in a general setting.poole2019variational Figure 1: Extracting the relevant operators with RSMI NE. (a) The most relevant operators are _learnt_ as the optimal compressions of long-range information $I(\mathcal{H}:\mathcal{E})$ at each point in the phase diagram. The learnt maps can be associated to the physical operators by computing the correlators and extracting the scaling dimensions. (b) The architecture of RSMI-NE: the relevant operators are extracted via the transformations $\Lambda$ and discretizing step $\tau$. The long-range information which $\Lambda$ maximize is estimated by $f_{\Theta}$, all of which are parametrized by neural networks and co-trained together. Here, using state-of-art machine learning results in estimating mutual information,poole2019variational we overcome this challenge to develop a highly efficient algorithm extracting relevant operators of the theory from real-space configurations. In contrast to standard approaches no RG maps are iterated: scaling operators are _not_ constructed from the RG flow, but instead using their definition as dominant contributions to RSMI, _in a single step_. The RSMI neural estimator (RSMI-NE) returns them parametrized as neural networks, which can be assigned scaling dimensions and used in computations (see Fig. 1.a). Moreover, we empirically demonstrate the power of the method across the whole phase diagram, also away from criticality. In particular, the algorithm can, unsupervised, construct order parameters, locate phase transitions, and identify spatial correlations and symmetries for complex and large dimensional real-space data. Our findings, elevating the coarse-graining transformations to formal operators, give a new paradigm in investigating statistical systems, and a numerical toolbox to do so. An often raised criticism of the use of machine learning in physics is the lack of interpretability of the results.doi:10.1098/rsta.2016.0153 Particularly, the extent to which architecture- and training-dependent conclusions from machine learning relate to formal concepts in physical theories is unclear. RSMI-NE overcomes this challenge: its outputs are explicitly identified with the scaling operatorsGordon2020 on the lattice. Thus, in contrast to generic data-driven approaches, RSMI-NE executes a physical principle using machine learning tools to produce theoretically interpretable results. Below we give an overview of the general RSMI setup, introducing the probabilistic language of the coarse-graining optimisation. We then present the RSMI-NE algorithm, and the theoretical and numerical results in machine learning underlying its efficiency. We validate its capabilities on an interacting model, whose non-trivial RG flow was a subject of a detailed theoretical analysis.PhysRevLett.94.235702 ; PhysRevE.74.041124 We investigate the physical data contained in the ensemble of RSMI filters. We conclude with a discussion of further applications, most notably to non- equilibrium problems. Consider a system of classical DOFs in any dimension denoted by a collective random variable $\mathcal{X}$, whose physics is specified by a probability measure $p(x)$, either Gibbsian dictated by the energy of the realization $x$ of $\mathcal{X}$, or a generic non-equilibrium distribution. A coarse-graining (CG) rule $\mathcal{X}\to\mathcal{X}^{\prime}$ is defined as a conditional distribution $p_{\Lambda}(x^{\prime}\|x)$, determined by a set of parameters $\Lambda$ to be optimised. The coarse-graining is typically carried out on disjoint spatial blocks $\mathcal{V}_{i}\subset\mathcal{X}$, and it factorises: $p(x^{\prime}\|x)=\prod_{i}p_{\Lambda_{i}}(h_{i}\|v_{i})$, such that $\mathcal{X}=\bigcup_{i}\mathcal{V}_{i}$ and $\mathcal{X}^{\prime}=\bigcup_{i}\mathcal{H}_{i}$, with $p_{\Lambda_{i}}(h_{i}\|v_{i})$ the CG rule applied to block $i$. In translation invariant systems a fixed $\Lambda_{i}\equiv\Lambda$ suffices; with disorder each block can be individually optimised. The RSMI principle identifies CG rules extracting the most relevant long-range features as the ones retaining the most information shared by a block $\mathcal{V}\subset\mathcal{X}$ to be coarse-grained, and its distant environment $\mathcal{E}$,Koch-Janusz2018 ; optimalRSMI _i.e._ those that optimally _compress_ this information. The environment is separated from $\mathcal{V}$ by a shell of non-zero thickness constituting the buffer $\mathcal{B}$, and forms the remainder of the system (see Fig. 1.a). The “shared information” between the random variables $\mathcal{H}$ and $\mathcal{E}$ is given by the mutual information: $I_{\Lambda}(\mathcal{H}:\mathcal{E})=\sum_{h,e}p_{\Lambda}(e,h)\log\left(\frac{p_{\Lambda}(e,h)}{p_{\Lambda}(h)p(e)}\right),$ (1) where $p_{\Lambda}(e,h)$ and $p(h)$ are the marginal probability distributions of $p_{\Lambda}(h,x)=p_{\Lambda}(h\|v)p(x)$ obtained by summing over the DOFs in $\\{\mathcal{V}$, $\mathcal{B}\\}$ and $\\{\mathcal{V},\mathcal{B},\mathcal{E}\\}$, respectively. Finding such optimal coarse-graining requires thus maximizing $I_{\Lambda}$ as a function of parameters $\Lambda$. The conceptual importance of the buffer $\mathcal{B}$ cannot be overstated: it sets the length-scale filtering out contributions of short-range correlations between $\mathcal{V}$ and $\mathcal{E}$. Increasing its thickness $L_{\mathcal{B}}$ corresponds to growing the RG scale, preserving only the long-range physics. With an arbitrary fixed CG rule this can only be achieved in RG by iterating the coarse-graining, with all the ensuing difficulties, particularly amplifying the errors in the formulation of the rule. In our approach _the CG rules themselves contain long-range information_ , and are obtained in a single shot, by solving the $I_{\Lambda}$ optimization problem directly at large $L_{\mathcal{B}}$, at different points in the phase diagram. The optimization problem of Eq. (1) is, however, difficult, as estimating or maximizing mutual information is notoriously hard.poole2019variational This was a major weakness of the RSMI proposal,Koch-Janusz2018 hindering numerical and theoretical progress. We can now overcome this challenge. At the heart of our approach, encapsulated in the RSMI-NE algorithm, is a series of recent results combining mathematically rigorous variational bounds on mutual information doi:10.1002/cpa.3160360204 ; NIPS2003_2410 ; 5605355 with deep learning.belghazi2018mine ; poole2019variational A family of _differentiable_ lower bounds to $I_{\Lambda}$ is introduced, parametrised by neural networks $f_{\Theta}$ (see Fig. 1.b), which in the course of gradient descent training on the joint samples of $\mathcal{H}$ and $\mathcal{E}$ become accurate, and in the limit exact, estimators of $I_{\Lambda}$, see the Supplemental Material (SM).111See Supplemental Material for details about the dimer model, more technical details of the RSMI-NE algorithm and its dependence on length scales, where also Refs. oord2018representation, ; 899a65b4919f47c8a06d115df85dad11, ; Goodfellow-et-al-2016, ; NIPS2014_5449, ; gumbel1954, ; kingma2014adam, are included. The transformation $p_{\Lambda}(h\|v)$ feeding the coarse-grained variables into the estimator is also expressed by a neural network _ansatz_. We use the following composite architecture (see Fig. 1.b): $h=\tau\circ(\Lambda\cdot v).$ (2) Here $\Lambda$ become parameters of a convolutional neural network (CNN) applied to the configurations, and $\tau$ differentiably maps $\Lambda\cdot v$ into states of variable $h$ of pre-determined type (_e.g._ pseudo-binary spins). This last embedding step is both crucial,PhysRevX.10.031056 and algorithmically non-trivial.jang2016categorical We emphasize that while the CNN choice is motivated by convenience, any sufficiently expressive _ansatz_ can be used. Figure 2: RSMI analysis of the interacting dimer model. (a) RG flow of the model (see Eq.3) and representative configurations (top panel). (b) Total RSMI extracted with the optimal filters as a function of $T$ and its scaling with the buffer size. (c) Information extracted by the pristine staggered and plaquette filters at different T. (d) Samples of optimal filters obtained with RSMI-NE for different $T$ [columnar (C), plaquette (P1, P2) and staggered (S1, S2)]. (e) The average overlap of the optimal filters at $T$ with the pristine components. (f) The dimer symmetry breaking and plaquette order parameters extracted using the low-$T$ pristine filters. We have thus cast _both_ the CG rule and the lower-bound to the cost function it optimizes as differentiable neural networks. Next, we can chain them together (see Fig. 1.b), and _simultaneously_ optimize via stochastic gradient descent, improving the RSMI estimator and the CG rule in each pass. Note, that it is this numerical breakthrough which enables the exploration of new theoretical ideas and renders the RSMI algorithm a promising new approach to tackle open challenges in complex domains. We demonstrate this on the example of an interacting dimer model. This is an optimal test-bed for the illustration of our algorithm. First, a large class of classical statistical physics problems can be mapped to interacting dimer models.doi:10.1063/1.1703953 ; Blote_1982 ; Henley_1997 ; 10.2307/2692028 ; Kenyon_2002 ; Cimasoni_2007 ; PhysRevLett.101.155702 ; PhysRevB.80.045112 ; PhysRevB.80.134413 ; PhysRevLett.122.080601 Moreover, aspects of the quantum dimer modelPhysRevLett.61.2376 ; PhysRevB.69.224415 leave their footprint on the phase diagram.PhysRevLett.94.235702 ; PhysRevE.74.041124 Second, in the dimer model, the relevant low-energy degrees of freedom are profoundly different form the microscopic building blocks of the theory and change qualitatively throughout the phase diagram. Hence, the algorithm is presented a non-trivial task. The model is defined by the partition function $Z(T)=\sum_{\\{C\\}}\exp{(-E_{C}/T)}$ at a given temperature $T$ and the configurations $C$ involve binary-valued microscopic degrees of freedom, dimers, that sit on the edges of the square lattice. They obey the constraint of exactly one dimer being connected to every vertex. The energy $E_{C}=N_{C}(\|\|)+N_{C}(=)$ counts plaquettes covered by parallel dimers favoured by the interaction, see Fig. 2.a. The essence of this system is in the interplay of aligning interaction energy and entropic effects due to the non-local cooperation of local dimer covering constraints. At low $T$ the former facilitates a long-range order (LRO) crystallizing the system into one of the four translation symmetry breaking columnar states, see Fig. 2.a. With increasing $T$ the system undergoes a Berezinskii-Kosterlitz-Thouless (BKT) transition at $T_{\scriptscriptstyle\rm BKT}=0.65(1)$,PhysRevE.74.041124 entering a critical phase characterised by algebraic decay of correlations (also at $T\to\infty$) with exponents continuously changing with $T$. This is reflected in the effective continuum field theory, which, via the mapping of dimer configurations to height-field $\varphi(\mathbf{r})$fradkin_2013 (see also SM for the definition) is given by a sine-Gordon (SG) action:PhysRevE.74.041124 $S[\varphi(\mathbf{r})]=\int\mathrm{d}^{2}\mathbf{r}\left[\frac{g(T)}{2}\|\nabla\varphi(\mathbf{r})\|^{2}+V\cos\left(4\varphi(\mathbf{r})\right)\right].$ (3) The potential $V$ locks $\varphi(\mathbf{r})$ into four values corresponding to the columnar states. The stiffness $g(T)$ controls fluctuations of $\varphi(\mathbf{r})$: large $g(T)$ favours “flat” fields of high-entropy, low $g(T)$ allows large gradients corresponding to the staggered configurations, which are not suppressed in the algebraic phase. The RG flow is shown in Fig. 2.a: the $T<T_{\scriptscriptstyle\rm BKT}$ fixed point with finite $g(T)$ and $V\to\infty$ leaves energy minimisation as the sole relevant constraint; the _line_ of fixed points at $V=0$ at $T>T_{\scriptscriptstyle\rm BKT}$ indicates that the energetic interactions are irrelevant and exponents vary with $T$. The flow reveals the physical nature of the algebraic correlations: $\nabla\varphi(\mathbf{r})$ obeys Gauss’ law and so the fixed point theory is that of electrical fields. To showcase RSMI-NE, we input Monte Carlo samples of the model across the whole temperature range to the algorithm. For concreteness, we restrict the coarse-grained variables $\mathcal{H}$ to a two-component binary vector $\\{\pm 1,\pm 1\\}$ (the optimal dimensionality can be found systematicallyrsmine ). Hence, we are looking for a two component vector of filters $\Lambda_{1}$, $\Lambda_{2}$ determining how the visible region $\mathcal{V}$ is mapped onto $\mathcal{H}$. Optimizing the filters $\Lambda_{1}$, $\Lambda_{2}$ for all $T$ separately gives a comprehensive picture of the long-wavelength physics, culminating in the construction of the relevant operators on the lattice, as we now show. First, we find that already the curve $I_{\Lambda}(T)$, _i.e._ the amount of long-range information attained _with the optimal_ $\Lambda$, reveals the structure of the phase diagram (see Fig. 2.b). To wit, for $T<T_{\scriptscriptstyle\rm BKT}$ its value is constant and equal to $\log 4$. The information shared between distant parts of the system in the ordered phase is precisely which of the four columnar states they are in. Phase transitions are reflected by non-analyticities in $I_{\Lambda}(T)$ (_cf._ Wilms_2011, ; PhysRevE.87.022128, ). Moreover, the algebraic decay of $I_{\Lambda}(T)$ with the buffer size for $T>T_{\rm BKT}$ is indicative of a critical phase, see Fig. 2.b and Fig 6.c in Ref. rsmine, . This behaviour should also be contrasted with the exponential decay for the paramagnetic phase of 2D Ising model in Ref. rsmine, . Going beyond the mutual information and examining the filters $\Lambda(T)$ yields further insight about spatial correlations. As conjectured, the optimal CG rules depend on the tuning parameters of the system. In the high- and low- temperature limits, three classes of filters emerge: independent optimizations return exclusively sets of $\Lambda_{1,2}$ that correspond to columnar and plaquette at low temperatures, and staggered ones at high temperatures, see Fig. 2.d. We call these filters “pristine” as they reflect simple limiting cases. They are orthogonal to each other and represent independent degrees of freedom. The filters for intermediate temperatures and their overlap with the pristine ones is shown in Figs. 2.d and e, respectively. We first discuss in detail the individual filters $\Lambda_{1,2}$ in the different temperature regimes $T\to 0$, $T\sim T_{\scriptscriptstyle\rm BKT}$ and $T\gg T_{\scriptscriptstyle\rm BKT}$, and then explicitly match them with the RG- relevant operators of the continuum sine-Gordon theory. The pristine plaquette and columnar filters at $T\to 0$ break translation or rotation symmetry, respectively. Any pair of $\Lambda_{1,2}$ drawn out of these filters is a _bijection_ between the four ordered columnar states and the four states $(\pm 1,\pm 1)$ taken by the compressed degrees of freedom $\mathcal{H}$. This degeneracy of plaquette and columnar filters is lifted when the rotation symmetry is restored: the pristine columnar filter is not found above $T_{\scriptscriptstyle\rm BKT}$. Strikingly, its modulus acquires an expectation value for $T<T_{\scriptscriptstyle\rm BKT}$ (see Fig. 2.f). This filter is thus an order parameter _discovered_ by RSMI-NE, and is in fact equal to the dimer symmetry breaking (DSB) order parameter identified in Ref. PhysRevLett.94.235702, . The optimal CG rules around $T_{\scriptscriptstyle\rm BKT}$ hold yet further insights. Particularly, the plaquette filters give rise to a _putative_ plaquette order parameter (see Fig. 2.g). The corresponding regime where it attains a non-vanishing value does not survive in the thermodynamic limit. However, the non-zero expectation value at finite system sizes (see Fig. 2.f) reveals the importance of such plaquette correlations, which are stabilised in the quantum dimer model (QDM).PhysRevLett.61.2376 RSMI-NE indicates this without any prior insights about QDM, which inspired previous studies.PhysRevLett.94.235702 ; PhysRevE.74.041124 Finally, the critical phase $T>T_{\scriptscriptstyle\rm BKT}$ interpolates between pristine plaquette and staggered filters, due to the competition between the electric field operator and plaquette correlations in the finite system, as per Eq. (3). The value of RSMI attained with _fixed_ rules reflects this competition: the plaquette filter retains more information until well above $T_{\scriptscriptstyle\rm BKT}$, where the staggered one takes over as plaquette correlations dwindle (see Fig. 2.c). The staggered filters _are_ the electric fields _viz._ they define coarse-grained variables $E_{1,2}(\mathbf{r}):=\tau\circ\Lambda_{1,2}\left(\mathcal{V}(\mathbf{r})\right)$, which precisely target the operator $\nabla\varphi(\mathbf{r})$ (see SM). Table 1: Pairs of filters drawn from the columnar and plaquette coarse- graining rules unambiguously label each of the four columnar ground states. The mapping is given by the sign of the scalar product of the filter (blue=-1, red=+1) with a dimer configuration (1 for occupied, 0 for unoccupied link). As the columnar configurations correspond to uniform height field, the electric charge operators $\mathcal{O}_{n=1,2}$, acting on the height-field $\varphi$ also serve as order parameters for the columnar phase and they directly correspond to the RSMI-optimal filters at low $T$. The RSMI-NE finding the order parameters or the electric fields is no accident: the pristine $\Lambda$ filters define the relevant operators on the lattice. The considerable technical machinery behind this is the subject of Ref. Gordon2020, , here we show it using the field theory of the dimer model, also away from criticality. To wit, the columnar and the DSB order parameters in Fig. 2.f correspond to the relevant electric charge operators $\mathcal{O}_{n}(\varphi)=(\cos(n\varphi),\sin(n\varphi))$,PhysRevB.76.134514 for $n=\pm 1$ and $n=\pm 2$, respectively. This is seen explicitly, using the height-field map in Table 1 as a dictionary: $\displaystyle\left(\Lambda_{\rm P1},\Lambda_{\rm P2}\right)\circ\varphi$ $\displaystyle=\left(\cos(\varphi+3\pi/4),\sin(\varphi+3\pi/4)\right),$ (4) $\displaystyle\Lambda_{\rm C}\circ\varphi$ $\displaystyle=\cos(2\varphi),$ (5) where on the left dimer configurations (on which the $\Lambda$ act) mapped to height field value $\varphi$ are denoted by $\varphi$ itself. Though competing correlations, especially in finite-size systems, may result in mixing of the pristine components, they can be identified by applying standard machine learning tools _to the ensemble of filters_. Note that RSMI- NE is a stochastic algorithm, and through independent runs produces a distribution of optimal CG rules. Thus Fig. 2.d shows a sample of filters at each $T$, and in Fig. 2.e the overlap is averaged over the filter ensemble at each temperature. The distribution contains crucial information, _e.g._ the disappearance of the columnar filter above $T_{\scriptscriptstyle\rm BKT}$ signals the lifting of the columnar/plaquette degeneracy (consistent with the scaling dimensions of $\mathcal{O}_{n}$, which go as $n^{2}$), and restoration of the rotation symmetry. More concretely, representations of the broken symmetries can be identified in the distribution, whereas at high-$T$ it can be used to retrieve even the emergent $U(1)$ symmetry of the electrical field! See Ref. rsmine, for a more detailed discussion. We thus managed to automatically sequence the operators of the theory, returning their lattice representations which are modular, reusable and may be formally labelled by their scaling dimensions. Indeed, evaluating a correlator of two neural networks parametrized by the plaquette filters, we fit a scaling dimension of $1.00037$ at $T\rightarrow\infty$ (see SM), in excellent agreement with $1.0$ predicted for $\mathcal{O}_{1}$.PhysRevB.76.134514 This raises the remarkable prospect of building a complete effective theory from raw data using machine learning. Though the discussion centred around an equilibrium example in two dimensions, our procedure works in any dimension, can be adapted to disorder,PhysRevX.10.031056 and does not require the existence of a Hamiltonian, as it only uses probability distributions. While a formal understanding of this approach for non-equilibrium distributions, extending the results of Ref. Gordon2020, , is missing, in the companion paperrsmine we validated the concept on the example of lattice model with aggregation and chipping,PhysRevE.63.036114 for which RSMI-NE locates precisely the non- equilibrium phase transition. We believe complex systems, such as realized in _e.g._ active matterPhysRevE.58.4828 ; Bialek4786 or atmospheric phenomena,Peters2006 to be a natural arena where information-theoretic methods can be applied,Dewar_2003 and our conceptual and numerical advancements may provide new theoretical insights (see also Ref. Nir30234, ). The understanding of challenging higher dimensional interacting and quasiperiodic statistical systems PhysRevB.82.085114 ; PhysRevB.86.214414 ; quasiperiodic_spin_chains ; PhysRevX.10.011005 may also benefit from this new method. Code availability Source code for the RSMI-NE is available online at https://github.com/RSMI-NE/RSMI-NE. Acknowledgements M.K.-J. is grateful to F. Alet for his comments on the physics of the interacting dimer model. D.E.G., S.D.H., and M.K.-J. gratefully acknowledge financial support from the Swiss National Science Foundation and the NCCR QSIT, and the European Research Council under the Grant Agreement No. 771503 (TopMechMat), as well as from European Union’s Horizon 2020 programme under Marie Sklodowska-Curie Grant Agreement No. 896004 (COMPLEX ML). Z.R. acknowledges support from ISF grant 2250/19. Some of the computations were performed using the Leonhard cluster at ETH Zurich. This work was supported by a grant from the Swiss National Supercomputing Centre (CSCS) under project ID eth5b. Supplemental Material: Statistical physics through the lens of real-space mutual information ## Appendix A RSMI optimization Here we provide a description of the algorithmic components of the RSMI-NE, referencing the necessary definitions and results from machine learning and statistics. The estimation of of real-space mutual information (RSMI) is three to four orders of magnitude faster than in Ref. Koch-Janusz2018, ; allowing to explore large systems (we tested up to the size of 256x256), and the limit of large buffer, while showing superior convergence and stability. The typical run-times are _ca._ 30 seconds. This dramatic improvement in performance is due to very recent advances in estimating information theoretic quantities,belghazi2018mine ; poole2019variational and it, in turn, enables development of an entirely new approach to real-space renormalization based on RSMI, in particular enabling the extraction of relevant operators (see main text, and section C). ### A.1 Some properties of mutual information The Shannon mutual information (MI) of two random variables $X$ and $Y$ quantifies the amount of knowledge we gain about one of them, when observing the other. Formally it is defined as a difference of entropies: $I(X:Y):=H(X)-H(X\|Y)=H(X)+H(Y)-H(X,Y).$ (6) Typically RSMI can take values at most on the order of a few units of information when coarse-graining small blocks $\mathcal{V}$ that contain $N_{\mathcal{V}}$ individual degrees of freedom with a discrete alphabet of $n$ symbols. Indeed: $I_{\Lambda}(\mathcal{H}:\mathcal{E})\leq I(\mathcal{V}:\mathcal{E})=H(\mathcal{V})-H(\mathcal{V}\|\mathcal{E})\leq H(\mathcal{V})\leq N_{\mathcal{V}}\log n,$ (7) where, since $\Lambda$ compresses $\mathcal{V}$ into $\mathcal{H}$, the first step is the data-processing inequality, and the second inequality follows from the positive semi-definiteness of Shannon entropies. This is important, as MI estimation by variational lower-bounds can suffer from a bias-variance trade- off in the opposite regime, _i.e._ when the MI is large. An alternative expression for $I(X:Y)$ is in terms of the Kullback-Leibler (KL) divergence $D_{\rm KL}$:222The Kullback-Leibler divergence is a measure of distance (but formally not a metric) between the two probability distributions in its argument. $I(X:Y)=D_{\rm KL}\left[p(x,y)\|\|p(x)p(y)\right],$ (8) The Gibbs’ inequality $D_{KL}(p\|\|q)\geq 0$ predicates on a useful interpretation of MI: given $(X,Y)$ jointly distributed according to $p(x,y)$, $I(X:Y)$ measures the information lost when encoding $(X,Y)$ as a pair of independent random variables while they may not be so. This is $0$ if and only if $X$ and $Y$ are actually independent, _i.e._ $p(x,y)=p(x)p(y)$. ### A.2 Noise-contrastive lower-bound of mutual information The results in the text were obtained with the noise-contrastive lower-bound of MI (which we refer to as InfoNCE). Here we introduce it and briefly motivate its form. We begin by casting the MI as follows: $I(X:Y)=\mathbb{E}_{p(x,y)}\left[\log\frac{p(x\|y)}{p(x)}\right]=\mathbb{E}_{p(x,y)}\left[\log\frac{p(y\|x)}{p(y)}\right].$ To proceed, let the conditional probability distribution $q(x\|y)$ be a variational ansatz for the conditional probability distribution $p(x\|y)$ appearing in the definition of MI above. Consider the KL divergence between $p(x\|y)$ and $q(x\|y)$: $\displaystyle D_{\rm{KL}}(p(x\|y)\|\|q(x\|y))$ $\displaystyle=\mathbb{E}_{p(x\|y)}\left[\log\frac{p(x\|y)}{q(x\|y)}\right]$ $\displaystyle=\sum_{x}\frac{p(x,y)}{p(y)}\left(\log p(x\|y)-\log q(x\|y)\right),$ (9) where the joint and the marginal distributions are related by $p(x,y)=p(x\|y)p(y)=p(y\|x)p(x)$. The variational distribution $q(x\|y)$ serves to approximate $p(x\|y)$: the KL divergence vanishes for $q^{}(x\|y)=p(x\|y)$. Positive semi-definiteness of the KL divergence yields a lower-bound for $I(X:Y)$, known as the Barber-Agakov (BA) bound:NIPS2003_2410 $\displaystyle I(X:Y)\geq$ $\displaystyle\mathbb{E}_{p(x,y)}\left[\log\frac{q(x\|y)}{p(x)}\right]$ $\displaystyle=\mathbb{E}_{p(x,y)}\left[\log q(x\|y)\right]+H(X)=:I_{\rm BA}(X:Y),$ (10) where $H(X)$ is the entropy of the random variable $X$. $I_{\rm BA}$ is a functional of the marginal distribution $q(x\|y)$ $I_{\rm BA}(X:Y)=I_{\rm BA}(X:Y)[q(x\|y)].$ Since $D_{\rm KL}=0$ if and only if $q(x\|y)=p(x\|y)$, the BA bound is tight only for the optimal ansatz $I_{\rm BA}(X:Y)[q(x\|y)=p(x\|y)]=I(X:Y)$. MI being the KL divergence between the joint and the product of marginal distributions motivates the idea of circumventing the exhaustive modelling of $p(x\|y)$ by $q(x\|y)$, focusing instead on the correlations between the variables $X$ and $Y$. Consider then a variational ansatz $q(x\|y)$ constrained into an energy-based family of functions: $q(x\|y):=\frac{p(x)}{Z(y)}e^{f(x,y)},$ (11) with the partition function $Z(y):=\mathbb{E}_{p(x)}\left[e^{f(x,y)}\right]$. By forcing $q(x\|y)$ to this form, the complex correlations within the possibly high-dimensional data $X$ are stripped-out into the marginal distribution $p(x)$. The resulting lower-bounds are sensitive mainly to the variables’ interdependency. In other words, maximising the lower-bound of MI is rephrased as a search for a function $f(x,y)$ modelling the relationships (or shared information) between $X$ and $Y$ very well, ignoring the surplus relationships within $X$ or $Y$ (whose contribution to the MI are insignificant). As noted in Ref. poole2019variational, , this conceptual step underlies the leap in terms of practical efficiency gained in the corresponding energy-based MI lower-bounds _e.g._ by Nguyen, Wainwright and Jordan (NWJ)5605355 and tractable unnormalised BA (TUBA) bound.poole2019variational . Even so, such so-called single-sample bounds are known to suffer from the large variance of the resulting MI estimators.poole2019variational ; oord2018representation An improved approach is to divide a single batch of MC samples for the pair of random variables $(X,Y)$ into minibatches of K-fold replicated random variables $(X_{i},Y_{i})_{i=1}^{K}$, and to derive the corresponding “multi-sample” lower-bounds (note the confusing, but standard, dual usage of the term “sample”). These are obtained by taking the average of the single-sample bounds introduced above, and address the issue of large variance by means of noise-contrastive estimation (NCE)899a65b4919f47c8a06d115df85dad11 , as first proposed in the context of MI estimation in Ref. oord2018representation, . More concretely, a multi-sample (or replica) bound estimates $I(X_{1},Y)$, where $(X_{1},Y)\sim p(x_{1},y)$, given $K-1$ additional independent replicas for one of the random variables, say $X$ (drawn from the marginal distribution), which are gathered into a multidimensional variable: $X_{2:K}\sim\prod_{j=2}^{K}p(x_{j})$ Considering the average of $I_{\rm NWJ}(X_{i}:Y_{i})$ over the $K$ replica random variables such that $(X_{i},Y_{i})\sim p(x_{i},y_{i})$, _i.e._ with each $Y_{j}$ playing the role of $Y$ in turn, leads to the InfoNCE lower-bound of MI:poole2019variational $I(X:Y)\geq I_{\rm NCE}(X:Y):=\langle I_{\rm NWJ}(X:Y)\rangle=\frac{1}{K}\mathbb{E}_{\prod_{k=1}^{K}p(x_{k},y_{k})}\left[\sum_{j=1}^{K}\log\frac{e^{f(x_{j},y_{j})}}{\frac{1}{K}\sum_{i=1}^{K}e^{f(x_{i},y_{j})}}\right].$ (12) Maximisation of the InfoNCE lower-bound for RSMI is the method used in the main text, where maximisation is over the discriminator functions $f$. A key algorithmic idea here, originally introduced in Ref. belghazi2018mine, , is to parametrize them by neural networks $f_{\Theta}$ and instead optimize over the network parameters $\Theta$ using stochastic gradient descent. ### A.3 Deep neural network architecture for RSMI optimisation Our goal in using the lower-bounds derived above is to optimize the RSMI, i.e. MI between the random variable $\mathcal{H}$ representing the coarse-grained degrees of freedom in the block $\mathcal{V}$, and $\mathcal{E}$ representing the degrees of freedom in the environment. There exist several admissible multi-layer perceptron (MLP) architectures for the corresponding energy-based ansatz $f\equiv f_{\Theta}(h,e)$ (see above), with a set of variational parameters $\Theta$. Here, we opt for a separable form, such that: $f_{\Theta}(h,e)=v^{\rm T}(h)u(e),$ (13) where $v$ and $u$ are array-valued functions (here, neural networks, whose weights constitute $\Theta$) that depend only on hidden variables and the environment, respectively. The networks $v$ and $u$ independently map $\mathcal{H}$ and $\mathcal{E}$ to a so-called embedding space. The advantage of this choice is the ability to construct the elements of the scores matrix, storing the values of the ansatz $f_{\Theta}$ for all pairs of jointly and independently drawn samples, in $N$ passes of the MLP ($N$ passes for both $v$ and $u$ networks) for a sample dataset of size $N$. This is in contrast to the requirement of $N^{2}$ passes for all $N(N-1)$ independent samples and $N$ joint samples in a concatenated architecture $f_{\Theta}(h,e)=f_{\Theta}([h,e])$, where the data for the pair of variables are concatenated before they are fed to the MLP. Table 2: Details of architecture for the RSMI-estimation module of RSMI-NE. The multi-layer perceptron architecture expresses the variational ansatz $f_{\Theta}(h,e)=v^{\rm T}(h)u(e)$ for the RSMI lower-bound. For each of $v$ and $u$, we use a single fully-connected hidden layer, which is a tensor of shape (hidden dimension=32, embed dimension=8). $f_{\Theta}(h,e)$ is obtained by taking the inner product along the embedding axis. The neurons are activated by the ReLU function. We use the InfoNCE lower-bound, hence the baseline function is the constant $e$. lower-bound \| ansatz type \| $\\#$ layers \| hidden dim. \| embed dim. \| activation function ---\|---\|---\|---\|---\|--- InfoNCE \| separable \| 2 \| 32 \| 8 \| ReLU Tabulated in Tab.2 are the details of the network architecture for $f_{\Theta}(h,e)$ that we have used in this work. Nevertheless, we note that the results of the RSMI-NE are not very sensitive to a specific choice of these parameters. Specifically, we opted for two hidden layers each with 32 neurons fully-connected to the layer containing the $(\mathcal{H},\mathcal{E})$ data. The embedding dimension is 8. The neurons are activated by the rectified linear unit (ReLU) function (see, e.g. Ref. Goodfellow-et-al-2016, ). #### A.3.1 Gumbel-softmax reparametrisation trick for discretization of coarse-grained variables In the RSMI-NE architecture the coarse-grained variables $h$ are fed into the MI estimator. Since the MI value depends on what kind of distribution $h$ belongs to, we need to ensure this estimation step is not falsified by _e.g._ neglecting to force the output of the coarse-grainer into a discrete form, rather than a real number, if we decided $h$ to be Ising spins. The apparent problem is that discreteness of $h$ seems to spoil the differentiability of the whole setup. This is somewhat similar to the problem encountered in Variational Autoencoders (VAEs), which is solved there using the so-called _reparametrization trick_ , effectively allowing to only differentiate w.r.t. to the parameters of the latent space probability distribution. This is the intuition behind the solution to the issue in RSMI-NE, which goes under the name of _Gumbel-softmax reparametrisation trick_.NIPS2014_5449 Let $h$ be a categorical random variable which can be in one of the states $\\{i\\}_{i=1}^{N}$ with the set of probabilities $\\{\pi_{i}\\}_{i=1}^{N}$. Given $\\{g_{i}\\}_{i=1}^{N}$, random variables drawn from the Gumbel distribution gumbel1954 ; NIPS2014_5449 , we define a vector-valued random variable utilizing the softmax function, whose j-th component takes the form: ${\rm softmax}_{j,\epsilon}\left(\\{g_{i}+\log\pi_{i}\\}_{i=1}^{N}\right)=\frac{\exp\left[(\log\pi_{j}+g_{j})/\epsilon\right]}{\sum_{i=1}^{N}\exp\left[(\log\pi_{i}+g_{i})/\epsilon\right]},$ (14) where $\epsilon$ is the smearing parameter. For $\epsilon\rightarrow 0$ the softmax becomes the argmax function, mapping the argument vector $y=\\{g_{i}+\log\pi_{i}\\}_{i=1}^{N}$ into a $N$-component one-hot vector (one-hot encoding maps each of $N$ possible states $i$ of a discrete variable into a $N$-dimensional vector, with $1$ on $i$-th position, and zeros elsewhere) with some $k^{}$-th entry taking the value $1$, thereby marking $y_{k^{}}=\max y$. The resulting random variable is called a Gumbel-softmax random variable; it is only approximately (or pseudo-) discrete, for small enough $\epsilon$ (do not confuse with a discrete random variable defined by taking the maximum component of the softmax function). In practice though, the error coming from using a finite $\epsilon$ can be made comparable to machine precision. The samples from the Gumbel-softmax approximation of a certain categorical distribution $\\{\pi_{k}\\}_{k=1}^{N}$ are approximately one-hot vectors for small $\epsilon$. We anneal $\epsilon$ during the training, from $\epsilon_{\rm max}$ to $\epsilon_{\rm min}$, exponentially, with a decay exponent $r$. #### A.3.2 Network architecture for coarse-graining In Tab. 3 we tabulate the details of the architecture for the coarse-graining network we used for the 2D dimer model (and other examples in the companion paper Ref. rsmine, ). We stack a single layer convolutional neural network (CNN) (generally with multiple kernels, corresponding to different components of $\mathcal{H}$) and the Gumbel-softmax reparametrisation layer to embed the components of $\mathcal{H}$ into (pseudo-) binary variables. While we determine the relaxation parameter $r$ by experimentation and fix it for all models, we tune the initial value of the Gumbel-softmax temperature $\epsilon_{\rm max}$ according to the total number of iterations during training. Table 3: Architecture details of the coarse-graining module of RSMI-NE for the 2-d interacting dimer model on a square lattice. $L_{\mathcal{V}}$ \| 8 ---\|--- $L_{\mathcal{B}}$ \| $\\{2,4,6,8\\}$ $L_{\mathcal{E}}$ \| 4 number of components of $\mathcal{H}$ \| 2 embedding of $\mathcal{H}$ \| binary $(\epsilon_{\rm max}$, $\epsilon_{\rm min})$ for Gumbel-softmax (GS) \| $(0.75,0.1)$ GS annealing parameter $r$ \| $5\times 10^{-3}$ Representing the coarse-graining map using a CNN enables interpreting the weights of the network to be directly understood in terms of a generalized Kadanoff block-spin construct for real-space RG.Kadanoff:1976vx Nevertheless, the RSMI principle does not restrict the specific type of the variational ansatz for coarse-graining, and the CNN form is not imperative. ### A.4 Unsupervised learning scheme We now describe the (unsupervised) training, maximising our neural network ansatz $I_{\Lambda}(\mathcal{H}:\mathcal{E})[f_{\Theta}]$. The inputs of the RSMI-NE can be _e.g._ the Monte Carlo (MC) samples from the desired model (as in the main text), but equally well experimentally measured data. As usual in numerical investigations, ensuring good quality sampling is important. Since we use the InfoNCE bound, the sampling is divided into mini-batches, each containing $K$ samples. We separate in each sample the visible patch $\mathcal{V}$ and its environment $\mathcal{E}$, dismissing a finite buffer separating them. Then a single mini-batch is denoted by the multi-dimensional random variable $(v_{1:K},e_{1:K})=(v_{1},\cdots,v_{K},e_{1},\cdots,e_{K})$. Let $\Lambda^{s}$ and $\Theta^{s}$ be the network parameters for the coarse- graining, and the InfoNCE ansatz $f$, respectively. At the outset we initialize these as tensors containing random numbers. At each step $s$ of training, the samples in the mini-batch are used to coarse-grain the samples $v_{i}$ into $h_{i}[\Lambda^{s}]$ and compute the scores matrix $F_{ij}(\Theta^{s},\Lambda^{s})=f(h_{i}[\Lambda^{s}],e_{j};\Theta^{s})$ for the InfoNCE ansatz and the current values of the network parameters. Here $\Lambda^{s}$ and $\Theta^{s}$ denote value of the network parameters in $s$’th training step. In the scores matrix, the entries with $i=j$ denote the jointly drawn samples and the rest denote independently drawn samples for the coarse-grained degree of freedom and the environment. The ansatz for RSMI can be formulated in a way which fixes the allowed alphabet from which the coarse-grained degrees of freedom are drawn. In our implementation, this is done by choosing a layer $\tau$ to generate $\mathcal{H}$. For discrete-valued degrees of freedom we use the Gumbel- softmax layer with an annealing schedule as described above. More generally, one can specify the number of convolutional channels according to the symmetries of the system. The InfoNCE prediction (that of $p(h,e)$ being equal to $p(h)p(e)$ or not) for the mini-batch is computed via the scores matrix as (compare Eq. 12): $Q(h_{1:K},e_{1:K};\Theta^{s},\Lambda^{s})=\sum_{j=1}^{K}\frac{F_{jj}(\Theta^{s},\Lambda^{s})}{\sum_{ij=1}^{K}\exp{F_{ij}(\Theta^{s},\Lambda^{s})}}.$ (15) Then $\log Q(h_{1:K},e_{1:K};\Theta^{s},\Lambda^{s})+\log K$ gives our single mini-batch estimate of RSMI. The gradients of the mini-batch estimate of RSMI with respect to $\Lambda$ and $\Theta$ are then used for proposing the updated set of network parameters. More concretely, we use the adam optimizerkingma2014adam to perform stochastic gradient-ascent. We have found that using the same learning rate for both parameter sets $\Lambda$ and $\Theta$ leads to efficient training. We repeat this procedure over all mini-batches until all samples are fed to the network once. This constitutes one epoch of training. We repeat it for multiple epochs until convergence (see Ref. rsmine, for a more detailed discussion of convergence criteria), when we are left with an optimized coarse-graining filter represented by the final convolutional network parameters $\Lambda$, and an estimate of the RSMI given by a moving average of the time-series of mini-batch estimates. ## Appendix B Dependence on the buffer, block and system size Here we provide a more detailed discussion of the length scales set by the linear sizes of the buffer $\mathcal{B}$ and the block $\mathcal{V}$, respectively denoted by $L_{\mathcal{B}}$ and $L_{\mathcal{V}}$, and their physical interpretation as well as practical importance. The nature of the scales $L_{B}$ and $L_{\mathcal{V}}$ is different. $L_{B}$ sets the distinction between long- and short-range contributions to RSMI. Naively, one may assume $L_{B}\rightarrow\infty$ is necessary, but this is not entirely true. To determine the functional form of the operator in terms of local degrees of freedom, the scale $L_{B}$ needs only to be sufficiently large, so that correlations due to the _second-most_ relevant operator decayed to the point of being indistinguishable from statistical noise always present in the sampling, while those of the most relevant one are still discernable (assuming the leading operator is not degenerate). For a given total system size, further increasing $L_{B}$ cannot change the answer, as the most relevant operator has already “won”, and is the sole contributor to RSMI. In fact, with finite sampling, it would make the problem computationally harder, or impossible, as the leading correlations would be drowned by the noise. Moreover, even when $L_{B}$ is not large enough to perfectly distinguish leading from subleading contributions, _both of them_ can be extracted using the _filter ensemble_ analysis descrived in the companion manuscript Ref. rsmine, . Thus, in practice, already small buffers of a dozen sites are sufficient to find the correct operator content (but not necessarily their correct dimensions yet, see below). In use, one should systematically scan (increase) the size of $L_{B}$. The block $V$, on the other hand, is the support (in the mathematical sense) of the relevant operator/order parameter. Assuming this operator is _local_ , in the first instance $V$ needs only to be sufficiently large so that the appropriate function of the original degrees of freedom can be constructed. For example, the “plaquette” operators in the dimer model require a block twice smaller than what we used, and the plaquette pattern is repeated in the block. This effectively corresponds to computing _block-averages_ of the operator values, which produces quasi-continuous outputs even in completely discrete systems. The degree of smearing is controlled by $L_{\mathcal{V}}$. This is discussed in detail in Sec.III.D.1 and Sec.III.D.2 of the companion Ref.rsmine, , where we show it allows to recover continuous emergent symmetries. Note also that for _e.g._ experimental measurements of a quantity performed on a grid of an arbitrary spatial resolution, the relevant operator/order parameter may require a large support in such arbitrary units. As for $L_{B}$, one should then scan the block size, or set it based on prior knowledge. In typical cases, the size $V$ is very small though. Though $L_{B}$ and $L_{\mathcal{V}}$ need not be large to extract the operators, the above _does not imply_ that larger system sizes are superfluous. In smaller systems, certain operators may appear more relevant than others in parameter regimes (_e.g._ above a phase transition) where they would be already subleading, had the system been closer to the thermodynamic limit (see plaquette filters in Fig. 2c in the main text). The correctness and accuracy of determination of their scaling dimension from the correlator will be improved with the system size (and the number of samples). We emphasize, however, that RSMI-NE discovers the parametrized form, in terms of the original local degrees of freedom, of both the leading and subleading operators. This allows to “export” them to analyse their properties with other methods (even analytically, perhaps). We also expect that standard MC scaling analysis to simplify, as we already have _the correct operators_ and only need to compute their correlators. To wit, we fit for the charge operator $\mathcal{O}_{1}$ at $T\rightarrow\infty$ a dimension of 1.00037, in excellent agreement with the predicted value of $1.0$, already at system size of $64\times 64$. Thus, the method has very favorable properties, extracting parametrized leading and subleading operators on the lattice even from modest system sizes, but large systems and more samples allow to more accurately determine their properties, particularly the scaling dimensions. ## Appendix C Details of operator extraction in the interacting dimer model ### C.1 Dimer model: Height-field mapping of dimers on 2-d square lattice Figure 3: Height field mapping of the dimer model on a square lattice. a The height fields are defined on the plaquettes, and the electric fields related to their gradients are shown by the arrows on the bonds (see explanation in the text). The color-map shows the average height $\langle H(\mathbf{r}_{i})\rangle$ over the four plaquettes surrounding an even- sublattice site $i$ (denoted by green dots). Left: for a columnar ground-state configuration the height profile $\langle H(\mathbf{r})\rangle$ is _uniform_. Correspondingly, the average electric field vanishes over the whole configuration. Middle: upon flipping a pair of dimers around a plaquette, $\langle H(\mathbf{r})\rangle$ changes by one. Right: staggered configurations have a net electric field over the whole configuration (observe the orientation of the arrows). Correspondingly the height field has a _tilted_ profile. b $\varphi=\frac{\pi}{2}\langle H\rangle$ is interpreted as an angle and is directly related to the orientation of the dimer at the given site. Here we describe the classic mapping of the dimer models to height fields, as per _e.g._ Ref. fradkin_2013, . Since the 2-d square lattice is bipartite, we can map each dimer configuration $C(\mathbf{r}_{i})$ onto a unique height profile $H(\mathbf{r}_{i})$ living on the plaquettes of the lattice. Without loss of generality, we start by assigning a reference height $H(\mathbf{r}_{0})=1/2$ to the plaquette at $i=0$. The heights of the neighboring plaquettes are then determined by winding clockwise (counter- clockwise) around a vertex of the even (odd) sublattice, and changing the height by $-3$ if a dimer is crossed on the link connecting the two plaquettes, or changing by $+1$ otherwise, see Fig. 3. The fact that each site in the fully packed dimer model can have a single dimer with only one of the four orientations attached to it is reflected in the height profile as an invariance under a uniform shift by 4 of the entire profile: $H(\mathbf{r})\equiv H(\mathbf{r})+4$. This makes it natural to define the $2\pi$ periodic rescaled field $\varphi(\mathbf{r}_{i})=\frac{\pi}{2}H(\mathbf{r}_{i})$, which can be interpreted as the orientation angle of the dimer connected to a given site. As shown in Fig. 3, we can assign to each site in the odd sublattice the average height of the four plaquettes that surround it. This assignment takes four distinct values $\left\\{0,\frac{\pi}{2},\pi,\frac{3\pi}{2}\right\\}$, respectively corresponding to $\\{\text{right, up, left, down}\\}$ oriented dimers connected to the site. The mapping can also be realized in terms of the electric fields on the links, see Fig. 3. Imagine giving each dimer a fixed orientation from even to odd sites. Then the above mapping corresponds to assigning 3 units of electric field pointing along the orientation of the dimer (_i.e._ flowing from the even to the odd sublattice site), and 1 unit of electric field, flowing out along the three remaining unoccupied links connected to each odd site. It follows that for fully packed dimers these electric fields locally conserve the flux, _i.e._ have a vanishing lattice divergence $\nabla\cdot\mathbf{E}=0$, which is the Gauss’ law. Observe that the height can be treated as a scalar potential for this field: $\nabla\varphi=\mathbf{E}$ if we identify $\mathbf{E}=(E_{\|},E_{-})$, that is: the gradient of the height field in x-direction is the y-component of the electric field, and _vice versa_ (see Fig. 3). This height mapping serves as a basis for deriving the continuum effective field theory, ultimately giving the sine-Gordon action in the main text.PhysRevB.76.134514 ; fradkin_2013 ### C.2 Order parameters from RSMI-optimal coarse-graining filters As mentioned in the main text, any two-component vector of filters with the components drawn from among the columnar (C) and the two plaquette filters (P1, P2) defines a coarse-graining which deterministically encodes the four columnar ground-states without any ambiguity, as shown in Tab.1 in the main text. As all these filters recover two bits of information, they occur degenerately in the long-range ordered phase (LRO). In fact, the filters can be interpreted as components of an order parameter, which distinguish the four distinct sectors of the phase space arising due to the spontaneous breaking of the $C_{4}$ symmetry. More concretely, one can define a global two component _plaquette_ order parameter $(D_{1},D_{2})$: $D_{i}:=\mathbb{E}\left[\frac{1}{N_{\mathcal{V}}}\sum_{k}\tau\circ\left(\Lambda_{i}\cdot\mathcal{V}_{k}\right)\right],$ (16) using _e.g._ $\Lambda_{1}=\Lambda_{\rm C}$ or $\Lambda_{1}=\Lambda_{\rm P1}$, and $\Lambda_{2}=\Lambda_{\rm P2}$, where the sum inside the expectation is taken over all disjoint blocks $\mathcal{V}_{k}$ on the lattice, and $N_{\mathcal{V}}$ is the number of such blocks. The norm of this order parameter is plotted as order parameter $P$ in Fig. 2.f in the main text. The columnar filter $\Lambda_{\rm C}$ alone defines the dimer orientational symmetry breaking (DSB) order parameter, exactly equivalent to the one introduced by Alet _et al._PhysRevE.74.041124 (also plotted in Fig. 2.f in the main text): ${\rm DSB}:=\mathbb{E}\left[\sum_{k}\tau\circ\left(\Lambda_{\rm C}\cdot\mathcal{V}_{k}\right)\right].$ (17) Although both columnar and plaquette filters signal the columnar order with a non-vanishing expectation value, their behavior differs above the BKT transition temperature. As shown in Fig.2.f in the main text, while DSB decays quickly to $0$ for $T>T_{\scriptscriptstyle\rm BKT}$ even for small lattices the plaquette ordered parameter decays much more slowly. This distinction is readily understood. The lowest-lying excitations above the columnar ground states are plaquette flips, which cost two units of energy. However, the mappings defined by the $\Lambda_{\rm P1/P2}$ filters are invariant under subsets of plaquette flips in the configuration, and therefore the expectation stays non-zero for finite-size systems at $T>T_{\scriptscriptstyle\rm BKT}$ (in an infinite system the plaquette order parameter also decays to zero). ### C.3 Discovering the relevant operators ##### Plaquette/columnar filters, and the electric charge operators. Table 4: Scaling dimensions of the electric charge operators at the BKT transition, and at the free dimer point. electric charge operator \| $\mathcal{O}_{1}$ \| $\mathcal{O}_{2}$ ---\|---\|--- $T\to\infty$ \| $d_{1}=1$ \| $d_{2}=4$ $T=T_{\scriptscriptstyle\rm BKT}$ \| $d_{1}=\frac{1}{8}$ \| $d_{2}=\frac{1}{2}$ For $T\geq T_{\scriptscriptstyle\rm BKT}$, the temperature-dependent RG scaling dimensions of the $\mathcal{O}_{n=1,2}$ operators, given in Ref. PhysRevB.76.134514, , are proportional to $n^{2}$. We list the values they take at the BKT transition, and in the limit of infinite $T$ in Tab. 4. This field-theoretic result underlies the evolution of the RSMI-optimal filters with $T$. The orientational symmetry breaking and plaquette operators are degenerate in the long-range ordered phase; in the critical phase, however, the columnar order parameter corresponding to $n=2$ has the higher scaling dimension, therefore its correlations decay faster, and this is why we obtain the plaquette filters but not the columnar ones beyond the BKT point. We thus verify for the case of interacting fully-packed dimers that the RSMI-NE indeed finds the operators with the lowest-scaling dimension, _i.e._ the RG relevant operators. We emphasize that the extracted RSMI filters can be used exactly as one usually uses operators. For instance, their correlation functions can be evaluated numerically from the samples, to recover (fit) their scaling dimensions. Though above we gave an explicit mapping between the filters and the scaling operators of the dimer model, so it is superfluous in this case, in more complex scenarios, with systems not fully understood, this would be necessary. As an illustration, in Fig. 4 we evaluate and plot the correlation function (on a $128\times 128$ dimer system, utilizing size $4\times 4$ filters) of the plaquette operators in the limit $T\rightarrow\infty$, and verify accurately it does indeed decay as $r^{-2}$, _i.e._ the we extract the scaling dimension $d_{1}=1$ for the $\mathcal{O}_{1}$ operator, a predicted.PhysRevB.76.134514 The decay of correlation function of the columnar filters is much faster, consistent with the prediction of $r^{-8}$ (we do not fit the exponent, as the fast decay requires better sample statistics). Thus, RSMI-NE allows to gain a formal understanding of the long- distance properties of the system, unsupervised and without inputting any prior knowledge. Figure 4: Using RSMI filters to compute correlation functions and scaling dimensions: The plaquette and the columnar coarse-graining filters correspond to the discretized $\mathcal{O}_{1}$ and $\mathcal{O}_{2}$ operators, with scaling dimensions $d_{1}=1,d_{2}=4$ at $T\to\infty$. (a) The correlators obtained using these filters confirm this picture. We fit a power-law decay to the plaquette correlator (on a $128\times 128$ dimer system, using size $4\times 4$ filters), obtaining a best-fit of $2.00074$, in excellent agreement with the field theoretic result $2d_{1}=2$, thus extracting the scaling dimension. The columnar correlator is consistent with the theoretical prediction of $r^{-8}$, but the fast decay would necessitate larger sample size for an accurate fitting of the exponent. (b) More precisely, the correlation function for $\mathcal{O}_{1,2}$ on the height field $\varphi$ is directly computed from the MC samples as the correlator of the trained neural networks parametrizing the coarse-graining filters $\Lambda_{\rm P,C}$, evaluated at spatial separations $r$ on the dimer configurations $C$. ##### Staggered filters, electric fields. The local action of pristine staggered filters which are RSMI-optimal at $T\to\infty$ has the form $\displaystyle\Lambda_{\rm S+}\cdot\mathcal{V}(\mathbf{r})$ $\displaystyle=(-1)^{x+y}N_{\|}(\mathbf{r}),$ (18) $\displaystyle\Lambda_{\rm S-}\cdot\mathcal{V}(\mathbf{r})$ $\displaystyle=(-1)^{x+y+1}N_{-}(\mathbf{r}).$ (19) Note that here $\left(\Lambda_{\rm S+}:=\frac{\Lambda_{\rm S1}+\Lambda_{\rm S2}}{\sqrt{2}},\,\Lambda_{\rm S-}:=\frac{\Lambda_{\rm S1}-\Lambda_{\rm S2}}{\sqrt{2}}\right)$ corresponds to the orthogonal basis $(E_{x},E_{y})$ for the electrical field.333S1 and S2 refers to the pristine staggered filters introduced in the main text. On the other hand, any unitary rotation $R(\vartheta)(\Lambda_{\rm S+},\Lambda_{\rm S-})\sim\left(\cos(\vartheta)E_{x}-\sin(\vartheta)E_{y},\sin(\vartheta)E_{x}+\cos(\vartheta)E_{y}\right)$ recovers the same amount of RSMI, indicating the emergent $U(1)$ symmetry of the free dimer model. For the interested reader, we point Ref. rsmine, for more details. Ref. PhysRevB.76.134514, gives an expansion of the dimer densities on the lattice in terms of the observables of the height-field continuum model: $\displaystyle N_{\|}(\mathbf{r})=\frac{1}{4}+\frac{(-1)^{x+y+1}}{2\pi}\partial_{x}\varphi(\mathbf{r})+(-1)^{y}\sin\varphi(\mathbf{r})$ , (20) $\displaystyle N_{-}(\mathbf{r})=\frac{1}{4}+\frac{(-1)^{x+y}}{2\pi}\partial_{y}\varphi(\mathbf{r})+(-1)^{x}\cos\varphi(\mathbf{r})$ . (21) Here, $1/4$ is the average dimer density (due to the $C_{4}$ symmetry). The terms $(-1)^{y}\sin\varphi$ and $(-1)^{x}\cos\varphi$, respectively, select the vertical and horizontal dimers connected to point $\mathbf{r}$ (_cf._ Fig. 3.b). Substituting these in Eq. 18 and averaging over the degrees of freedom in the block $\mathcal{V}$ to which the filter is applied, we obtain the coarse- grained degrees of freedom: $\displaystyle\mathcal{H}_{1}\sim\sum_{\mathbf{r}\in\mathcal{V}}\Lambda_{\rm S+}\cdot\mathcal{V}(\mathbf{r})$ $\displaystyle=\sum_{\mathbf{r}\in\mathcal{V}}\left[\frac{(-1)^{x+y}}{4}+\frac{\partial_{x}\varphi(\mathbf{r})}{2\pi}+(-1)^{x}\sin\varphi(\mathbf{r})\right]=\sum_{\mathbf{r}\in\mathcal{V}}\left[\frac{\partial_{x}\varphi(\mathbf{r})}{2\pi}+(-1)^{x}\sin\varphi(\mathbf{r})\right],$ (22) $\displaystyle\mathcal{H}_{2}\sim\sum_{\mathbf{r}\in\mathcal{V}}\Lambda_{\rm S-}\cdot\mathcal{V}(\mathbf{r})$ $\displaystyle=\sum_{\mathbf{r}\in\mathcal{V}}\left[\frac{(-1)^{x+y}}{4}+\frac{\partial_{y}\varphi(\mathbf{r})}{2\pi}+(-1)^{y}\cos\varphi(\mathbf{r})\right]=\sum_{\mathbf{r}\in\mathcal{V}}\left[\frac{\partial_{y}\varphi(\mathbf{r})}{2\pi}+(-1)^{y}\cos\varphi(\mathbf{r})\right],$ (23) where the $\sim$ symbol is used to denote that the coarse-grained variable $\mathcal{H}$ is the result of applying the binary mapping $\tau$ to the right hand side. We see that the first term in the summand vanishes since there are an even number of links in the region $\mathcal{V}$. The last term probes the tilt of the height configuration either in the $x$ or in the $y$ direction. This is because the term $(-1)^{x}\sin\varphi(\mathbf{r})$ has a non-zero average only if the region $\mathcal{V}$ contains a vertical dimer with a missing parallel neighbor, in which case the height profile has a slope in the $x$ direction, while it vanishes for a uniform height field (similarly for the other component in the $y$ direction). Equivalently, only not-constant-in-x function $\varphi(\mathbf{r})$ survives averaging with $-1^{x}$ over the block, and so under the average we may replace $\sin\varphi(\mathbf{r})$ with $\partial_{x}\varphi(\mathbf{r})$. 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Present address: ]Centre for Engineered Quantum Systems, School of Mathematics and Physics, University of Queensland, QLD 4072, Australia # A Bright Source of Telecom Single Photons Based on Quantum Frequency Conversion Christopher L. Morrison<EMAIL_ADDRESS>Markus Rambach [ Zhe Xian Koong Francesco Graffitti Fiona Thorburn Ajoy K. Kar Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK Yong Ma College of Optoelectronic Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Suk-In Park Jin Dong Song Center for Opto-Electronic Materials and Devices Research, Korea Institute of Science and Technology, Seoul 02792, Republic of Korea Nick G. Stoltz Materials Department, University of California, Santa Barbara, California 93106, USA Dirk Bouwmeester Huygens-Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA Leiden, Netherlands Department of Physics, University of California, Santa Barbara, California 93106, USA Alessandro Fedrizzi Brian D. Gerardot Institute of Photonics and Quantum Sciences, School of Engineering and Physical Sciences, Heriot-Watt University, Edinburgh EH14 4AS, UK (August 27, 2024) ###### Abstract On-demand indistinguishable single photon sources are essential for quantum networking and communication. Semiconductor quantum dots are among the most promising candidates, but their typical emission wavelength renders them unsuitable for use in fibre networks. Here, we present quantum frequency conversion of near-infrared photons from a bright quantum dot to the telecommunication C-band, allowing integration with existing fibre architectures. We use a custom-built, tunable 2400 nm seed laser to convert single photons from 942 nm to 1550 nm in a difference frequency generation process. We achieve an end-to-end conversion efficiency of $\sim$35%, demonstrate count rates approaching 1 MHz at 1550 nm with $g^{\left(2\right)}\left(0\right)=0.04$, and achieve Hong-Ou-Mandel visibilities of 60%. We expect this scheme to be preferable to quantum dot sources directly emitting at telecom wavelengths for fibre based quantum networking. Semiconductor quantum dots (QDs) are a leading technology for bright, indistinguishable single-photon sources. QDs emitting in the 920 nm - 980 nm window have been used as a source of single photons with count rates upwards of 10 MHz, indistinguishability greater than 95%, and low multi-photon contributions on the order of 1% [1, 2, 3]. Yet, in order to be compatible with existing telecommunication technology, an ideal single-photon source should operate in the telecommunication C-band, around 1550 nm, where fibre loss is minimal. While there has been recent progress in producing QD sources that directly emit single photons in the C-band [4, 5, 6], achieving high- quality single-photon emission at high rates remains an open challenge, with the best-performing single photon sources to date being confined to near- infrared (NIR) [7] wavelengths. One route to bridge the gap to the C-band is quantum frequency conversion (QFC), converting single photons from a NIR QD to telecommunication wavelengths. Quantum frequency conversion is a nonlinear process where a single-photon input is mixed with a strong seed beam producing a single-photon output at either the sum or difference frequency. QFC can in principle be noise free and therefore preserve the quantum statistics of near-infrared emitters. Frequency conversion of QD sources has been demonstrated from 700 nm to the telecommunication O-band [8], and from 900 nm to the C-band [9, 10], culminating in the remote two-photon interference between independent downconverted QD sources [11]. Spin-photon entanglement between a 910 nm QD and a 1560-nm photon has been demonstrated using frequency conversion seeded by ultrafast pulses [12, 13]. QFC of QDs has also been demonstrated in nano- photonic circuits using four-wave mixing in silicon nitride [14]. However, a source that is simultaneously bright, pure and coherent has not been demonstrated in QDs emitting directly at telecom wavelengths. Here we demonstrate a frequency-converted InGaAs quantum dot source approaching 1 MHz count rates at 1550 nm, with $g^{\left(2\right)}\left(0\right)$ around 4% and HOM visibilities of 60%. Figure 1: Difference-frequency generation schematic. Blue lines indicate the optical path of the QD photons. Pink lines represent the path of the 2401 nm seed light. The polarisation of both beams is aligned to the extraordinary axis of the ppLN crystal with a quarter-wave plate (QWP) and half-wave plate (HWP). A 100 mm focal length lens is used in the QD beam path to mode match the seed beam at the waveguide facet. The green lines represent the converted 1550 nm light after the frequency conversion. The converted light is sent through two short pass filters at 2050 nm (SP 2050), a 1400 nm long pass filter (LP 1400) and a bandpass filter at 1550 nm (BP 1550) before being collected in a single mode fibre for detection. The inset shows the experimental layout used to produce the 2401 nm laser light for QFC. The seed laser is pumped by a commercial Thulium fibre laser (yellow lines). The pump beam polarisation is prepared with a HWP to reduce loss in the laser cavity due to Fresnel reflections. The light enters the cavity through a partially reflective curved mirror which acts as the input coupler at 1900 nm and a focusing mirror at 2401 nm. A longpass filter with a cutoff wavelength of 2000 nm (LP 2000) is placed after the output coupler to remove unabsorbed 1900 nm pump light. For difference frequency generation, energy conservation demands $\left(1/\lambda_{\text{in}}-1/\lambda_{\text{seed}}\right)^{-1}=\lambda_{\text{out}}$. Our 942 nm InGaAs QD source requires a seed wavelength of 2401 nm to generate output photons at 1550 nm. The seed beam is produced in a chromium doped zinc selenide (Cr:ZnSe) laser while the difference frequency generation occurs in a periodically poled lithium niobate (ppLN) waveguide. Near-infrared photons are generated by a single self-assembled InGaAs/GaAs QD coupled to a high quality ($Q\approx 4.4\times 10^{4}$) oxide-apertured micropillar cavity, as shown in Fig. 1. The QDs are embedded in a p-i-n diode structure [15, 16] which enables charge control and tuning of the QD emission to the cavity mode via the quantum-confined Stark effect. The sample is kept at a temperature of $4\,\mathrm{K}$ in a closed-cycle helium flow cryostat. A dark-field confocal microscope is used to excite and collect the scattered photons from the QD before filtering in a cross-polarisation scheme with a $\sim 10^{7}$ extinction ratio to suppress the excitation laser. Pulsed excitation of the QD is performed using a mode-locked titanium:sapphire laser with a repetition rate of 80.3 MHz and pulse duration of 10 ps. The QD output is then detected by superconducting nanowire single photon detectors (SNSPDs), with a nominal quantum efficiency of $\sim 90\,\%$ at 950 nm. We exploit a neutral exciton QD transition ($X^{0}$), resonantly coupled to the cavity with a Purcell factor of $\sim 4$ and an emission wavelength of 942.33 nm. The $T_{2}$ coherence time of the emission, measured using standard Fourier spectroscopy under $\pi$-pulse resonant excitation, shows $T_{2}=0.348\,(2)\,\mathrm{ns}$, corresponding to a linewidth of $915\,(5)\,\mathrm{MHz}$. This value is $\approx 1.5$ times larger than the transform-limited linewidth ($h/T_{1}=607\,\mathrm{MHz}$), with an independently measured lifetime of $T_{1}=0.2622\,(1)\,\mathrm{ns}$. The seed laser for the QFC stage consists of a z-cavity resonator with a Cr:ZnSe crystal, a gain medium with an emission spectrum spanning 1900-3300 nm [17, 18, 19], see inset in Fig. 1. This laser design allows for both continuous-wave [20] and mode-locked operation generating pulses as short as 43 fs [21]. Here, we operate with a narrowband CW seed to drive the QFC. Our 2401 nm laser system is pumped by a thulium-doped fibre laser (IPG Photonics TLR-20-LP) which has a maximum CW output power of 20 W at 1900 nm. The pump light is focused into the Cr:ZnSe crystal using a 100 mm $\text{CaF}_{2}$ lens. The cavity consists of a dichroic input coupler (50 mm radius of curvature (ROC), transmissive at 1900 nm reflective at 2400 nm), a gold mirror (50 mm ROC), two plane silver mirrors and the Cr:ZnSe crystal. The crystal is placed at Brewster angle to minimise losses due to reflection. A diffraction grating (450 lines/mm) is inserted into the cavity to control the emission wavelength. The output coupler (Layertec) has a transmission of 60% at 2401 nm, allowing a good trade-off between the cavity enhancement and available output power. Fig. 1 shows our difference-frequency generation (DFG) setup. The DFG, one special case of QFC, takes place in a 48 mm periodically-poled lithium niobate crystal (ppLN, NTT Electronics). The chip contains multiple ridge waveguides with poling periods ranging from 26.00 $\mu$m to 26.25 $\mu$m. These poling periods are designed for type-0 DFG from 942 nm to 1550 nm. Quarter- and half- wave plates are used to align the polarisation of the incoming single photons at 942 nm and the generated pump light at 2401 nm to the extraordinary axis of the crystal. Seed light from the laser is overlapped with single photons from the QD using a dichroic mirror (Omega Optical) and coupled into the waveguide using a NIR coated aspheric lens with a focal length of 11 mm. A NIR coated lens is used to match the beam size of the single photons to the seed beam and to compensate for the chromatic aberration of the aspheric lenses. The converted 1550 nm light is collimated with an 11 mm NIR-coated aspheric lens and sent towards a filtering stage. The filtering stage consists of two shortpass filters at 2050 nm ($>$OD 4), which are used to remove seed light impinging on the collection fibre; a longpass filter at 1400 nm ($>$OD 5) to remove weakly phase-matched second- harmonic generation from the seed beam and unconverted quantum dot light; and finally, a 2.8 nm full-width-at-half-maximum (FWHM) bandpass filter ($>$OD 4, BP 1550) to isolate the converted single photons. The converted 1550 nm light is collected into a single mode fibre with 86% coupling efficiency and sent to SNSPDs with a nominal quantum efficiency greater than 80%. The DFG conversion efficiency is characterised by sending CW coherent light from a 942 nm laser (Toptica DL Pro) into the QFC setup. For CW-seeded QFC, the conversion efficiency is almost independent of the temporal mode of the input light [22]. This allows characterisation with a CW beam despite the single photons’ decaying exponential wavepacket. In the case that we can treat the seed beam as unamplified, QFC is expressed as a beam-splitter Hamiltonian between two different frequency modes [23]. This means the measured conversion efficiency is independent of the input intensity. These two factors ensure the conversion efficiency measured with a low power $\left(500\text{ $\mu$W}\right)$ CW coherent field is equivalent to the single-photon conversion efficiency. Figure 2: Conversion efficiency of the difference frequency generation process as a function of seed power coupled into the waveguide. The power in the waveguide is determined by measuring the pump power after the waveguide and factoring out the loss through the c-coated aspheric collimation lens. The transmission through this lens is measured to be 64 % at the pump wavelength. The data is fitted with Eq. 1. Inset shows the signal to noise ratio for off- resonant excitation with a measured noise count rate of $12\,(1)$ Hz/mW. Figure 3: Photo-luminescence emission profile of the QD under 820 nm excitation shows cavity modes up to the $5^{\mathrm{th}}$ order. The QD is resonantly coupled to the 1st cavity mode. Inset shows the detected count rate as a function of the excitation power when the excitation laser is resonant to the 1st (resonant, blue) or 3rd (non-resonant, green) cavity mode. Figure 4: Characterisation of the single photon properties before (upper row) and after (lower row) QFC. (a) Time-resolved emission spectra under pulsed resonant excitation reveals an exponential decay which gives the emitter’s lifetime $T_{1}$ and a fast oscillation indicating the quantum beating between the fine-structure peaks of the neutral exciton emission, $\Delta_{\rm fss}=4.807\,(3)\,\mathrm{GHz}$. (b, c) Second-order intensity correlation histogram $g^{(2)}$, of the emitted photons under off-resonant (b) and resonant (c) excitation. The lack of coincidences in the central peak indicates the low probability of multi-photon emission. (d) Two-photon interference of consecutively scattered photons delayed by $12.5\,\mathrm{ns}$, prepared in cross ($g^{(2}_{\perp}$) and parallel ($g^{(2}_{\parallel}$) polarisations, under resonant $\pi$-pulse excitation. The extracted photon indistinguishability, given by ratio of zero-delay- coincidences from both configurations, along with the extracted values ($T_{1}$ and $g^{(2)}(0)$) from the fits (solid lines) are summarised in Table. 1. Figure 2 shows the internal conversion efficiency for 942 nm light, measured by comparing output 1550 nm light to the 942 nm coupled through the waveguide with the seed laser blocked. This factors out coupling losses into the waveguide which are measured to be $17\,\%$. The data is fitted with [24] $\eta=\eta_{max}\sin^{2}\left(\sqrt{\eta_{nor}P}\leavevmode\nobreak\ L\right),$ (1) where $\eta_{max}$ is the maximum possible conversion efficiency, $\eta_{nor}$ is the normalised conversion efficiency of the process, $P$ is the input power, and $L$ is the waveguide length. The fit gives a normalised conversion efficiency (to waveguide length in the limit of small pump powers [24]) is $\eta_{nor}=44\left(1\right)\,\%\text{ /}\left(\text{W cm}^{2}\right)$. The maximal external conversion efficiency, the ratio of photons collected into single-mode fibre after the conversion stage to the number of NIR photons impinging on the waveguide, $\eta_{max}=38\,\%$, leading to a maximum internal conversion efficiency of $56.7\left(4\right)\,\%$ when taking losses into account. This $\eta_{max}$ is higher than previously reported values for NIR QD frequency conversion to 1550 nm with similar waveguides [11]. We would like to highlight that we achieve SNRs $>250$ for all seed powers (inset Fig. 2), meaning that the noise contribution of the DFG process towards the converted single photons is minimal. We now compare the characteristics of the converted telecom photons to the QD NIR photons. We tune the QD into resonance with the first cavity mode, and excite either resonantly or non-resonantly into the third cavity mode; see Fig. 3 for spectral properties of the cavity under 820 nm excitation. The inset in Fig. 3 shows the detected count rates for these two excitation scenarios as a function of power: Rabi oscillations are observed for resonant excitation while a clear maximum is observed for non-resonant excitation. For non-resonant characterisation, the QD photons are spectrally filtered with a grating filter with a 30 GHz FWHM to suppress the excitation laser. We detect a count rate of $1.85(5)\leavevmode\nobreak\ \mathrm{MHz}$ at an excitation power of $6.8\,\mu\mathrm{W}$. The grating filter was removed when characterising the converted photons as low-loss bandpass filters were used at 1550 nm. For resonant driving, we optimize the excitation power to the $\pi$-pulse and detect a count rate of $1.46(4)\leavevmode\nobreak\ \mathrm{MHz}$. This value is slightly lower than for off-resonant excitation due to the presence of spectral fluctuations [25]. After QFC, the detected count rate at 1550 nm, for the off-resonant and on-resonant case is $856(18)\leavevmode\nobreak\ \mathrm{kHz}$ and $456(14)\leavevmode\nobreak\ \mathrm{kHz}$, respectively. Comparing the NIR and telecom counts under resonant excitation gives an end-to-end conversion efficiency of $\approx 35\,\%$, after accounting for the difference in the detection efficiency of both NIR ($\sim 90\%$) and telecom C-band ($\sim 80\%$) detectors. This agrees well with the measured loss budget through the optical components including the conversion efficiency. The difference in efficiency for off-resonant excitation is accounted for by the loss of the grating filter. Figure 4 shows the comparison between the performance of the QD signal before and after QFC. The lifetime measured under resonant excitation in Fig. 4(a) remains unchanged within experimental error after conversion. The oscillation in the time-resolved emission, indicative of the quantum beating of the $X^{0}$ fine-structure splitting, shows a frequency of $4.807\,(3)\,\mathrm{GHz}$. The equivalent oscillation after the QFC process is unchanged ($4.803\,(1)\,\mathrm{GHz}$), indicating that the CW-seeded frequency conversion preserves the temporal mode of the input photons. Next, we measure the second-order intensity correlation $g^{(2)}$ using a Hanbury-Brown and Twiss (HBT) interferometer. For a perfect single-photon source $g^{(2)}(0)=0$, indicating the absence of multi-photon emissions. Under off-resonant driving, Fig. 4(b), we observe a slight increase from $g^{(2)}(0)=0.045\,(0)$ to $g^{(2)}(0)=0.051\,(1)$ before and after the QFC process, respectively. We observe similar values under resonant driving, Fig. 4(c), demonstrating near-ideal single-photon emission with $g^{(2)}(0)=0.040\,(0)$ and $g^{(2)}(0)=0.043\,(1)$ before and after the QFC process, respectively. The slight increase in the normalized coincidences in the uncorrelated side peaks in the HBT histogram is due to blinking of the emitters, a common effect resulting from QD coupling to the solid-state charge environment [26]. The imperfection in $g^{(2)}(0)$ can be due to imperfect suppression of the cavity emission due to cavity feeding [27, 28, 29], slight imperfection in the wave-plate retarders used in our confocal microscope, and presence of multi-photon capture processes [30, 31]. Nevertheless, with a modest increase in $g^{(2)}(0)$ after the QFC process, we have demonstrated near background-free single photon frequency conversion from NIR to telecom C-band, with the photon number purity predominately limited by the quantum dot. To demonstrate that our QFC setup preserves photon coherence, we perform Hong- Ou-Mandel (HOM) interference between photons emitted from two consecutive excitation pulses. We use an unbalanced Mach-Zehnder interferometer with a delay of 12.5 ns to match photons temporally on a 50/50 beam splitter. We measure the coincidence counts for parallel and perpendicular polarised photons and evaluate the visibility as $\mathrm{V_{HOM}}=1-g^{(2)}_{\parallel}/g^{(2)}_{\perp}$. For a pair of indistinguishable photons, $\mathrm{V_{HOM}}=1$. For resonant excitation we achieve an interference visibility of $\mathrm{V_{HOM}}=0.88\,(1)$ before QFC. We calculate the single photon indistinguishability $M_{s}$ as $M_{s}=(\mathrm{V_{HOM}}+g^{(2)}(0))/(1-g^{(2)}(0))$ [32]. This gives an upper bound to the HOM visibility taking the finite $g^{\left(2\right)}\left(0\right)$ into account. Before conversion, $\rm M_{s}$ is equal to $0.95\,(1)$. After conversion we find the raw visibility and corrected indistinguishability to be $0.60\,(1)$ and $0.67\,(2)$ respectively. The results of lifetime, HBT and HOM measurements are summarised in Table 1. The reduced interference visibility originates in spectral instability introduced by fluctuating power in multiple longitudinal modes of the seed laser. The line width of the seed laser is around 4 GHz with a free spectral range estimated to be 177 MHz, corresponding to $\sim 22$ modes. Despite this, we show that our QFC setup indeed preserves the coherence of single photons as measured from non-classical two-photon interference which can be improved by increased control over the cavity dispersion and active stabilization of the cavity length. \| 942 nm \| 1550 nm ---\|---\|--- Lifetime $T_{1}\,(\mathrm{ns}$) \| 0.2622 (2) \| 0.2621 (2) Resonant count rate (kHz) \| 1,460 (40) \| 456 (14) Off-resonant count rate (kHz) \| 1,850 (50) \| 856 (18) Off-resonant $g^{(2)}(0)$ \| 0.045 (0) \| 0.051 (1) Resonant $g^{(2)}(0)$ \| 0.040 (0) \| 0.043 (1) Resonant $\rm V_{HOM}$ \| 0.88 (1) \| 0.60 (1) Resonant $M_{s}$ \| 0.95 (1) \| 0.67 (2) Table 1: Summary of the lifetime, count rate, $g^{(2)}(0)$ and indistinguishability $\rm V_{HOM}$ for converted and unconverted photons. Values are obtained from measurement results, illustrated in Fig. 4. The corrected photon indistinguishability $M_{S}$ is estimated based on the measured $g^{(2)}(0)$ and uncorrected $\rm V_{HOM}$ [32]. The error, given by the standard deviation from the fit, is included in brackets. Modest improvements to the current QFC system will allow us to improve the converted two-photon interference visibility to equal the unconverted visibility. The external conversion efficiency could be further improved with lower-loss filtering and improved mode matching between the single-mode fibre and the waveguide mode. We believe that this source will find applications in fibre-based quantum communication where a source of bright and highly pure single photons in the C-band is required. This can lead to demonstrations of various quantum communication protocols including measurement-device- independent quantum key distribution, teleportation and entanglement swapping between distant quantum nodes. ###### Acknowledgements. We acknowledge Pierre M. Petroff for his contribution to the sample design and fabrication. This work was supported by the EPSRC (Grants No. EP/L015110/1, No. EP/M013472/1, No. EP/P029892/1, EP/N002962/1 and EP/T001011/1), the ERC (Grant No. 725920), and the EU Horizon 2020 research and innovation program under Grant Agreement No. 820423. B. D. G. thanks the Royal Society for a Wolfson Merit Award and the Royal Academy of Engineering for a Chair in Emerging Technology. Y. M. acknowledges the support from Chongqing Research Program of Basic Research and Frontier Technology (No.cstc2016jcyjA0301). The authors in K.I.S.T. acknowledge the support from the KIST institutional program, the program of quantum sensor core technology through IITP and the IITP grant funded by the Korea government(MSIT) (No. 20190004340011001). ## References ## References * Tomm _et al._ [2020] N. Tomm, A. Javadi, N. O. Antoniadis, D. Najer, M. C. Löbl, A. R. Korsch, R. Schott, S. R. Valentin, A. D. Wieck, A. Ludwig, and R. J. 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††institutetext: Institut für Theoretische Physik und Astrophysik and Würzburg-Dresden Cluster of Excellence ct.qmat, Julius-Maximilians-Universität Würzburg, Am Hubland, 97074 Würzburg, Germany # Monodromy methods for torus conformal blocks and entanglement entropy at large central charge Marius Gerbershagen<EMAIL_ADDRESS> ###### Abstract We compute the entanglement entropy in a two dimensional conformal field theory at finite size and finite temperature in the large central charge limit via the replica trick. We first generalize the known monodromy method for the calculation of conformal blocks on the plane to the torus. Then, we derive a monodromy method for the zero-point conformal blocks of the replica partition function. We explain the differences between the two monodromy methods before applying them to the calculation of the entanglement entropy. We find that the contribution of the vacuum exchange dominates the entanglement entropy for a large class of CFTs, leading to universal results in agreement with holographic predictions from the RT formula. Moreover, we determine in which regime the replica partition function agrees with a correlation function of local twist operators on the torus. ## 1 Introduction Entanglement entropy is a measure for the amount of entanglement between two parts of a quantum system. It is defined as the von Neumann entropy of the reduced density matrix $\rho_{A}$ for a subsystem $A$. In general, the entanglement entropy depends on details of the theory and state in question such as the spectrum and operator content. However, certain universal features are common to all quantum field theories. For example, the leading order divergence in the UV cutoff usually scales with the area of the boundary of the subregion $A$ Bombelli:1986rw ; Srednicki:1993im . Conformal field theories in two dimensions admit more general universal features. In particular, the entanglement entropy of a single interval $A$ at zero temperature is given by Calabrese:2004eu $S_{A}=\frac{c}{3}\log(l/\epsilon_{\text{UV}}),$ (1) depending only on the central charge, irrespective of any other details such as the OPE coefficients or the spectrum of the theory. For subsystems $A$ consisting of multiple intervals, the entanglement entropy is no longer universal for any CFT. However, as shown in Hartman:2013mia , in the semiclassical large central charge limit and at zero temperature, the entanglement entropy again becomes universal for a large class of conformal field theories. These CFTs are characterized by a sparse spectrum of light operators and at most exponentially growing OPE coefficients. By using conformal transformations, the universal results of Calabrese:2004eu ; Hartman:2013mia translate to the case of either finite temperature or finite size. This publication is dedicated to the study of universal features of the entanglement entropy in a system with both finite size and finite temperature. The computational approach most commonly used to determine entanglement entropies is the replica trick. It is based on the calculation of the Rényi entropies $S_{A}^{(n)}=\frac{1}{1-n}\log{\mathrm{Tr}}\rho_{A}^{n},$ (2) via a partition function $Z_{n}$ on a higher genus Riemann surface ${\mathcal{R}}_{n}$ obtained by gluing $n$ copies of the complex plane cyclically together along the entangling interval $A$. This partition function is then mapped to a correlation function of twist operators, i.e. local operators with scaling dimension $h=\bar{h}=c/24(n-1/n)$ inserted at the endpoints of the entangling interval. For a subsystem $A$ consisting of $N$ disjoint intervals, the correlation function contains $2N$ twist operator insertions. Finally, the entanglement entropy is obtained by analytically continuing $n$ to the real numbers and taking the limit $n\to 1$. The universality of the entanglement entropy for a single interval follows immediately from the universality of the two-point function in any conformal field theory Calabrese:2004eu . For multiple intervals, the Rényi entropy is mapped to a higher-point correlation function of twist operators which decomposes into a sum over conformal blocks. The universality in the semiclassical limit observed in Hartman:2013mia is explained by the fact that only a single conformal block (the vacuum block) contributes to the entanglement entropy. More precisely, in the semiclassical limit of large central charge, the contribution of other conformal blocks is exponentially suppressed in the central charge, assuming the aforementioned restrictions on the theory, i.e. a sparse spectrum of light operators and at most exponentially growing OPE coefficients. In the case of a system with both finite size and finite temperature, the replica trick instructs us to calculate the partition function on a higher genus surface constructed by gluing $n$ copies of the torus along the entangling interval $A$. In the small interval limit, this partition function can be obtained analogously to the zero temperature case as a correlation function on the torus of two (local) twist operators inserted at the endpoints of the entangling interval (see e.g. Azeyanagi:2007bj ; Calabrese:2009qy ; Cardy:2014jwa ; Chen:2016lbu ). For large intervals, on the other hand, the replica partition function does not agree with a correlation function of local twist operators, as can be seen by the following argument. It is well known that for a pure state $\rho$, the entanglement entropy for $A$ is equal to the entanglement entropy of the complement $A^{c}$, $S_{A}=S_{A^{c}}$. However, for mixed states such as the thermal states described by the CFT on the torus, this property no longer holds. Since the correlation function of local twist operators contains no information about whether we compute the entanglement entropy for $A$ or for $A^{c}$ (the location of the branch cuts between the twist operators is not fixed by the twist correlator), it cannot give the correct answer on the torus111In the case of a free CFT, a discrepancy between the correlator of local twist operators and the higher genus partition function has been observed in Lokhande:2015zma ; Mukhi:2017rex based on previous work Headrick:2012fk ; Datta:2013hba ; Chen:2015cna . In particular, it was found in Lokhande:2015zma that the twist operator result is not modular covariant and violates Bose-Fermi equivalence.. This issue can be resolved for instance by defining non-local twist operators on the torus222We would like to thank the referee for pointing this out to us. as in Chen:2014hta , however we will not do this and instead phrase the calculation directly in terms of the replica partition function. Therefore, in the following the term “twist operator” will always refer to a local operator of scaling dimension $h=\bar{h}=c/24(n-1/n)$. A convenient way to calculate the entanglement entropy on the plane works by expanding the twist correlator in conformal blocks and calculating these using the well-known monodromy method first described in Zamolodchikov1987 . This monodromy method is derived as follows. By inserting a degenerate field into the correlation function, one obtains a differential equation for an auxiliary function $\Psi(z)$. This differential equations contains derivatives of the sought after conformal block as accessory parameters. The accessory parameters are fixed by demanding a certain monodromy of the solution $\Psi(z)$ around cycles which encircle a number of operator insertion points. Which insertion points are encircled depends on the channel in which the correlation function is expanded. We review this method in detail in sec. 2.1 before generalizing it to the case of finite temperature and finite size. A different perspective on this method was offered in Faulkner:2013yia , where it was related to a uniformization problem on the replica surface ${\mathcal{R}}_{n}$. In general, a compact Riemann surface $\Sigma$ can be obtained as the quotient of the complex plane by a subgroup of $PSL(2,{\mathbb{C}})$ Zograf_1988 . Thus, there exists a single valued map $w\to z$ from the complex plane to ${\mathcal{R}}_{n}$. This uniformization map is given as the quotient $w=\Psi_{1}(z)/\Psi_{2}(z)$ of two independent solutions of a differential equation. It turns out that this differential equation is equal to the one from the monodromy method for the conformal blocks of the twist correlator on the plane Faulkner:2013yia . Using the equivalence between the CFT partition function and the gravitational action in the dual AdS space, this yields a proof of the RT formula at zero temperature Faulkner:2013yia . We use similar arguments from the uniformization problem on the replica surface of the torus to determine a monodromy method for the zero- point conformal blocks of the replica partition function at finite temperature. This new monodromy method is closely connected to the monodromy method for the conformal blocks of the correlator of local twist operators on the torus, with the crucial difference being that the new method allows for choosing a larger set of cycles around which to impose the monodromy. These new cycles are necessary to reproduce the $S_{A}\neq S_{A^{c}}$ property for thermal states. In the context of the AdS/CFT correspondence, universal features of the entanglement entropy for holographic CFTs at large central charge are predicted by the Ryu-Takayanagi formula (RT formula for short) Ryu:2006bv . The RT formula states that the entanglement entropy of a subregion $A$ in the boundary field theory corresponds to the area of a minimal surface $\gamma_{A}$ in the bulk, anchored at $\partial A$ on the boundary of the AdS space. Apart from reproducing the universal results obtained in Calabrese:2004eu ; Hartman:2013mia , the RT formula also predicts interesting universal features of the entanglement entropy in the case of both finite temperature and finite size. In particular, there are two phase transitions Azeyanagi:2007bj . First, there is a Hawking-Page transition in the bulk from thermal AdS3 to the BTZ black hole phase as the temperature increases. This induces a corresponding phase transition in the entanglement entropy. Second, the entanglement entropy in the BTZ phase also shows a phase transition as the size of the entangling interval increases. We explain how these features appear from the CFT side. Related work on the entanglement entropy in conformal field theories at finite size and finite temperature includes Chen:2014hta ; Chen:2014ehg ; Chen:2014unl ; Barrella:2013wja ; Chen:2015kua . Chen:2014hta ; Chen:2014ehg ; Chen:2014unl is concerned with the entanglement entropy in various limits of high and low temperature or size of the entangling interval $A$, in which case universal results for arbitrary values of the central charge can be obtained. In Barrella:2013wja ; Chen:2015kua , the holographic entanglement entropy for a single entangling interval on the boundary of a thermal AdS3 space and the BTZ black hole was calculated using a monodromy method on the gravity side. The monodromy method derived from the CFT side in this publication will turn out to be equivalent to the monodromy method on the gravity side used in Barrella:2013wja ; Chen:2015kua . Related work on torus conformal blocks includes KashaniPoor:2012wb which derived the monodromy method for the special case of one-point Virasoro conformal blocks on the torus and Alkalaev:2016fok ; Alkalaev:2017bzx which performed explicit calculations of one- and two-point Virasoro conformal blocks in various limits including the semiclassical one which we study in this publication. Our paper is organized as follows. In sec. 2, we derive the monodromy methods used in this publication. After a review of the standard monodromy method for conformal blocks on the plane in sec. 2.1, we generalize to torus conformal blocks in sec. 2.2. Sec. 2.3 explains how to obtain a monodromy method for zero-point conformal blocks of the partition function on the replica surface and the differences between it and the monodromy method for conformal blocks on the torus. Following this, we apply the newly derived monodromy methods to the calculation of the entanglement entropy in sec. 3. Assuming that the vacuum exchange dominates the partition function on the higher genus Riemann surface, we find universal results in agreement with the RT formula. For the partition function on the replica surface, we find in particular agreement with the phase transition in the entanglement entropy at large interval size and high temperature. This feature cannot be reproduced from the correlator of local twist operators. We check the assumption on the dominance of the vacuum exchange numerically in sec. 3.4. Finally, we conclude with a brief discussion and outlook in sec. 4. ## 2 Monodromy methods This section contains an overview over the monodromy methods used in this publication. We start with a review of the standard monodromy method for conformal blocks on the plane, then turn to the case of conformal blocks on the torus and finally explain how to derive a monodromy method for zero-point blocks of the partition function on the replica surface relevant to the computation of entanglement entropy on the torus. ### 2.1 Conformal blocks on the plane In this section, we review the monodromy method for the calculation of four- point semiclassical conformal blocks on the plane first derived in Zamolodchikov1986 (see also Harlow:2011ny for a more detailed explanation). The starting point of the derivation is the correlation function of four primary fields ${\mathcal{O}}_{i}$ $\langle{\mathcal{O}}_{1}(z_{1},\bar{z}_{1}){\mathcal{O}}_{2}(z_{2},\bar{z}_{2}){\mathcal{O}}_{3}(z_{3},\bar{z}_{3}){\mathcal{O}}_{4}(z_{4},\bar{z}_{4})\rangle.$ (3) Let us parametrize the central charge as $c=1+6(b+1/b)^{2}$ and take the semiclassical limit $c\to\infty,b\to 0$ in which the conformal weights $h_{i}$ of the operators ${\mathcal{O}}_{i}$ as well as the internal conformal weight $h_{p}$ scale proportional to the central charge. In the correlation function (3), we insert a degenerate operator $\Psi(z,\bar{z})$ with conformal weight $h_{\Psi}=-1/2-3b^{2}/4\sim{\mathcal{O}}(c^{0})$ obeying $\left(\hat{L}_{-2}+\frac{1}{b^{2}}\hat{L}_{-1}^{2}\right)\Psi(z,\bar{z})=0$ (4) From the conformal Ward identities, this yields the following differential equation known as the decoupling equation, $\left[\frac{1}{b^{2}}\partial_{z}^{2}+\sum_{i}\left(\frac{h_{i}}{(z-z_{i})^{2}}+\frac{\partial_{z_{i}}}{z-z_{i}}\right)\right]\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle=0.$ (5) To get to the $s$-channel conformal block, we then insert the operator product expansion ${\mathcal{O}}_{1}(z_{1},\bar{z}_{1}){\mathcal{O}}_{2}(z_{2},\bar{z}_{2})=\sum_{p}C^{p}_{21}\sum_{k,\bar{k}}(z_{2}-z_{1})^{h_{p}-h_{1}-h_{2}+\|k\|}(\bar{z}_{2}-\bar{z}_{1})^{\bar{h}_{p}-\bar{h}_{1}-\bar{h}_{2}+\|\bar{k}\|}\beta^{pk}_{21}\beta^{p\bar{k}}_{21}{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}(z_{1},\bar{z}_{1})$ (6) into the correlation function which yields terms containing $\langle{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$. At large central charge, these terms can be approximated by $\langle{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle\approx\Psi_{p}\langle{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle,$ (7) where $\Psi_{p}$ is defined by $\Psi_{p}=\frac{\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle}{\langle{\mathcal{O}}_{p}{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle}.$ (8) This can be shown by employing the form of $\langle{\mathcal{O}}_{p}^{\\{k,\bar{k}\\}}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$ in terms of a string of differential operators ${\mathcal{L}}_{k_{i}},\bar{\mathcal{L}}_{k_{i}}$ acting on $\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$, where ${\mathcal{L}}_{-k_{i}}^{\Psi}=-\sum_{j=3,4,\Psi}\left(\frac{(1-k_{i})h_{j}}{(z_{j}-z_{1})^{k_{i}}}+\frac{1}{(z_{j}-z_{1})^{k_{i}-1}}\partial_{z_{j}}\right).$ (9) Now $\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$ scales as $e^{-c/6S_{\text{cl.}}}$ in the semiclassical limit where $S_{\text{cl.}}\sim{\mathcal{O}}(c^{0})$ while $\Psi_{p}\sim{\mathcal{O}}(c^{0})$ and $h_{\Psi}\sim{\mathcal{O}}(c^{0})$. Hence, we can neglect the derivatives acting on $\Psi_{p}$ and on the $h_{\Psi}$ term to obtain (7) in the leading order in $c$. Now, use a conformal transformation to send $z_{1}\to 0$, $z_{3}\to 1$, $z_{4}\to\infty$ and $z_{2}$ to the cross ratio $x$. This implies $\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle\approx\sum_{p}\Psi_{p}(z,x,\bar{x})C^{p}_{21}C^{p}_{43}{\mathcal{F}}^{p}_{12,34}(x)\bar{\mathcal{F}}^{p}_{12,34}(\bar{x}),$ (10) where ${\mathcal{F}}^{p}_{12,34}(x)$ is the desired conformal block which in the semiclassical limit scales as ${\mathcal{F}}^{p}_{12,34}(x)\sim e^{-\frac{c}{6}f_{\text{cl.}}(x)}$ as was conjectured in Zamolodchikov1986 and recently shown in Besken:2019jyw . The semiclassical conformal block $f_{\text{cl.}}$ depends only on the cross ratio $x$ and on $b^{2}h_{i},b^{2}h_{p}$. The decoupling equation (5) then yields at leading order in $c$ $\left[\partial_{z}^{2}+\sum_{i}\left(\frac{b^{2}h_{i}}{(z-z_{i})^{2}}-\frac{\partial_{z_{i}}f_{\text{cl.}}(x)}{z-z_{i}}\right)\right]\Psi_{p}=0.$ (11) There is one separate decoupling equation for each term in the sum over $p$ since generically, each term has a different monodromy and thus must vanish separately. All terms involving derivatives of $\Psi_{p}$ vanish to leading order due to $\Psi_{p}\sim{\mathcal{O}}(c^{0})$. From the expression for the cross ratio $x=\frac{(z_{1}-z_{2})(z_{4}-z_{3})}{(z_{4}-z_{2})(z_{1}-z_{3})}$, we obtain linear relations among the $\partial_{z_{i}}f_{\text{cl.}}$ $\sum_{i}\partial_{z_{i}}f_{\text{cl.}}=\sum_{i}(z_{i}\partial_{z_{i}}f_{\text{cl.}}-b^{2}h_{i})=\sum_{i}(z_{i}^{2}\partial_{z_{i}}f_{\text{cl.}}-2z_{i}b^{2}h_{i})=0.$ (12) These follow from $\partial_{z_{i}}f_{\text{cl.}}=(\frac{\partial x}{\partial z_{i}})\partial_{x}f_{\text{cl.}}$ and $\sum_{i}\frac{\partial x}{\partial z_{i}}=\sum_{i}\frac{\partial x}{\partial z_{i}}z_{i}=\sum_{i}\frac{\partial x}{\partial z_{i}}z_{i}^{2}=0$ as can easily be shown from the definition of $x$ and the conformal transformation properties of correlation functions of primary operators. This yields the final form of the decoupling equation, $\left[\partial_{z}^{2}+\frac{b^{2}h_{1}}{z^{2}}+\frac{b^{2}h_{2}}{(z-x)^{2}}+\frac{b^{2}h_{3}}{(z-1)^{2}}-\frac{b^{2}(h_{1}+h_{2}+h_{3}-h_{4})}{z(z-1)}+\frac{x(1-x)\partial_{x}f_{\text{cl.}}}{z(z-x)(z-1)}\right]\Psi_{p}=0.$ (13) To obtain $f_{\text{cl.}}$ from this equation, we use the fact that the solutions $\Psi_{p}$ must have a certain monodromy when $z$ is taken in a loop around $0,x$. This monodromy can be derived from the decoupling equation of $\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle$, $\biggl{[}\frac{1}{b^{2}}\partial_{z}^{2}+\left(\frac{h_{p}}{(z-z_{1})^{2}}+\frac{1}{z-z_{1}}\partial_{z_{1}}\right)+\sum_{i=3,4}\left(\frac{h_{i}}{(z-z_{i})^{2}}+\frac{1}{z-z_{i}}\partial_{z_{i}}\right)\biggr{]}\langle{\mathcal{O}}_{p}\Psi{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle=0.$ (14) As $z\to z_{1}$, the leading coefficient of the OPE between $\Psi$ and ${\mathcal{O}}_{p}$ is given by $(z-z_{1})^{\kappa}{\mathcal{O}}_{p}^{\prime}(z_{1})$ where $\kappa$ can be determined by inserting this coefficient into (14), $\displaystyle\biggl{[}$ $\displaystyle\frac{1}{b^{2}}\kappa(\kappa-1)(z-z_{1})^{\kappa-2}+\sum_{i=3,4}\left(\frac{h_{i}}{(z-z_{i})^{2}}+\frac{1}{z-z_{i}}\partial_{z_{i}}\right)(z-z_{1})^{\kappa}$ (15) $\displaystyle+h_{p}(z-z_{1})^{\kappa-2}-\kappa(z-z_{1})^{\kappa-2}\biggr{]}\langle{\mathcal{O}}_{p}^{\prime}{\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle=0.$ The leading contribution in $z\to z_{1}$ is given by $\biggl{[}\frac{1}{b^{2}}\kappa(\kappa-1)+h_{p}-\kappa\biggl{]}(z-z_{1})^{\kappa-2}=0.$ (16) Thus as $b^{2}\to 0$ and $z\to z_{1}$, $\kappa=\frac{1}{2}\left(1\pm\sqrt{1-4h_{p}b^{2}}\right)\text{\leavevmode\nobreak\ \leavevmode\nobreak\ and\leavevmode\nobreak\ \leavevmode\nobreak\ }\Psi_{p}\sim(z-z_{1})^{\frac{1}{2}(1\pm\sqrt{1-4h_{p}b^{2}})}.$ (17) Therefore, the monodromy matrix around $0,x$ is given by $M_{0,x}=\left(\begin{array}[]{cc}e^{i\pi(1+\sqrt{1-4h_{p}b^{2}})}&0\\\ 0&e^{i\pi(1-\sqrt{1-4h_{p}b^{2}})}\\\ \end{array}\right).$ (18) The trace of the monodromy matrix, which is independent of the basis in which the two solutions of (13) are decomposed, is given by ${\mathrm{Tr}}M_{0,x}=-2\cos\left(\pi\sqrt{1-4h_{p}b^{2}}\right).$ (19) Thus, the torus conformal block can be extracted from (13) by choosing $\partial_{x}f_{\text{cl.}}$ such that the monodromy of the solution $\Psi_{p}$ around a loop enclosing $z_{1}$ and $z_{2}$ is given by (19) 333The loop needs to enclose both $z_{1}$ and $z_{2}$ in order for the OPE between ${\mathcal{O}}_{1}(z_{1})$ and ${\mathcal{O}}_{2}(z_{2})$ to converge.. Finally, the conformal block is obtained by integrating the chosen $\partial_{x}f_{\text{cl.}}$. The four point conformal block in other channels is obtained from the same decoupling equation by imposing different monodromy conditions. For example, for the $t$-channel block we impose the monodromy condition around the insertion points of ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{3}$, ${\mathrm{Tr}}M_{1,x}=-2\cos\left(\pi\sqrt{1-4h_{p}b^{2}}\right)$. Higher point conformal blocks on the plane are computed from similar monodromy methods derived analogously to the four-point case. For $n$ point blocks, the decoupling contains $n-3$ independent derivatives fixed by $n-3$ monodromy conditions around the operator insertion points which are contracted in the OPE. ### 2.2 Conformal blocks on the torus We now continue with the derivation of a monodromy method for conformal blocks on the torus. The derivation of this monodromy method proceeds in a very similar way to the one on the plane. We illustrate the derivation using the two-point function on the torus $\langle{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\rangle_{\tau}={\mathrm{Tr}}[e^{2\pi i\tau(L_{0}-c/24)}e^{-2\pi i\bar{\tau}(\bar{L}_{0}-c/24)}{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})],$ (20) however conformal blocks for other correlation functions on the torus are obtained in a similar fashion, as we briefly discuss as the end of this section. The modular parameter of the torus is denoted by $\tau$, related to the inverse temperature by $\beta=-2\pi i\tau$. We also introduce the parameter $Q=e^{-\beta}=e^{2\pi i\tau},$ (21) written with an uppercase $Q$ instead of the standard lowercase $q$ to distinguish it from the conformal dimension $h_{q}$ of the internal index of the conformal block which we are about to derive. As on the plane, we insert the degenerate operator $\Psi(z,\bar{z})$ into (20). To derive the corresponding decoupling equation, we use the conformal Ward identity on the torus Eguchi:1986sb 444Note that Eguchi:1986sb uses a convention where correlation functions on the torus are normalized by the inverse of the partition function $Z(\tau)$ and thus the expression for the conformal Ward identity there contains an additional term $(2\pi i\partial_{\tau}Z(\tau))\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\rangle_{\tau}$., $\displaystyle\langle T(z)\prod_{i}{\mathcal{O}}_{i}(z_{i})\rangle_{\tau}=\left[\sum_{i}\bigl{(}h_{i}\left(\wp(z-z_{i})+2\eta_{1}\right)+\left(\zeta(z-z_{i})+2\eta_{1}z_{i}\right)\partial_{z_{i}}\bigr{)}+2\pi i\partial_{\tau}\right]\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\rangle_{\tau}.$ (22) Here, $z\sim z+1\sim z+\tau$ are the coordinates on the torus with modular parameter $\tau$ and $\wp(z),\zeta(z)$ denote Weierstraß elliptic functions with associated $\eta_{1}$ parameter (see App. A for more details on the Weierstraß functions). Using the definition of the Virasoro generators, $\hat{L}_{-n}\Psi(z)=\int\frac{dw}{2\pi i}\frac{1}{(w-z)^{n-1}}T(w)\Psi(z).$ (23) and the conformal Ward identity (22), we see that $\displaystyle\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\hat{L}_{-2}\Psi(z)\rangle_{\tau}=\biggl{[}$ $\displaystyle\sum_{i}\left(h_{i}(\wp(z-z_{i})+2\eta_{1})+(\zeta(z-z_{i})+2\eta_{1}z_{i})\partial_{z_{i}}\right)$ (24) $\displaystyle+2\eta_{1}z\partial_{z}+2h_{\Psi}\eta_{1}+2\pi i\partial_{\tau}\biggr{]}\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\Psi(z)\rangle_{\tau}$ and $\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\hat{L}_{-1}\Psi(z)\rangle_{\tau}=\partial_{z}\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\Psi(z)\rangle_{\tau}.$ (25) Thus $\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\Psi(z)\rangle_{\tau}$ obeys the decoupling equation $\displaystyle\biggl{[}$ $\displaystyle\frac{1}{b^{2}}\partial_{z}^{2}+\sum_{i}\left(h_{i}(\wp(z-z_{i})+2\eta_{1})+(\zeta(z-z_{i})+2\eta_{1}z_{i})\partial_{z_{i}}\right)$ (26) $\displaystyle+2\eta_{1}z\partial_{z}+2h_{\Psi}\eta_{1}+2\pi i\partial_{\tau}\biggr{]}\langle\prod_{i}{\mathcal{O}}_{i}(z_{i})\Psi(z)\rangle_{\tau}=0.$ To relate $\langle{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\Psi(z)\rangle_{\tau}$ to a conformal block, we decompose the trace over states into contributions from a primary ${\mathcal{O}}_{q}$ and its descendants and insert the appropriate OPE contractions. 12$p$$q$12$p$$q$ Figure 1: Conformal blocks for the two point function on the torus. Left: OPE channel, right: projection channel. For the two-point function, there are two possible channels (see fig. 1). The projection block is obtained by OPE contracting ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{q}$. On the other hand, for the OPE block we contract ${\mathcal{O}}_{2}$ and ${\mathcal{O}}_{1}$, $\displaystyle\langle{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\Psi(z)\rangle_{\tau}=\sum_{q}\sum_{l}Q^{h_{q}-c/24+\|l\|}\langle{\mathcal{O}}_{q}^{\\{l\\}}(z_{0}){\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\Psi(z){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)}$ (27) $\displaystyle=\sum_{p,q}C^{p}_{21}\sum_{k,l}(z_{2}-z_{1})^{h_{p}-h_{1}-h_{2}+\|k\|}Q^{h_{q}-c/24+\|l\|}\beta^{pk}_{21}\langle{\mathcal{O}}^{\\{l\\}}_{q}(z_{0}){\mathcal{O}}^{\\{k\\}}_{p}(z_{1})\Psi(z){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)},$ where (c.c.) denotes schematically the antiholomorphic parts of the expression and $z_{0}\to-i\infty$ while $z_{\infty}\to+i\infty$. We define $\Psi_{pq}=\frac{\langle{\mathcal{O}}_{q}(z_{0})\Psi(z){\mathcal{O}}_{p}(z_{1}){\mathcal{O}}_{q}(z_{\infty})\rangle}{\langle{\mathcal{O}}_{q}(z_{0}){\mathcal{O}}_{p}(z_{1}){\mathcal{O}}_{q}(z_{\infty})\rangle}.$ (28) As on the plane, in the large $c$ limit $\langle{\mathcal{O}}^{\\{l\\}}_{q}(z_{0}){\mathcal{O}}^{\\{k\\}}_{p}(z_{1})\Psi(z){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)}\approx\Psi_{pq}\langle{\mathcal{O}}^{\\{l\\}}_{q}(z_{0}){\mathcal{O}}^{\\{k\\}}_{p}(z_{1}){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)}$ (29) This yields $\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}\Psi\rangle_{\tau}\approx\sum_{p,q}C^{p}_{21}C^{q}_{pq}\Psi_{pq}{\mathcal{F}}_{21,pq}\bar{\mathcal{F}}_{21,pq},$ (30) where ${\mathcal{F}}_{21,pq}$ is the conformal block which we want to compute. Assuming that exponentiation of the conformal blocks in the semiclassical limit holds, ${\mathcal{F}}_{21,pq}\sim e^{-c/6\,f_{\text{cl.}}}$, and using that $\partial_{z_{1}}f_{\text{cl.}}=-\partial_{z_{2}}f_{\text{cl.}}$, we obtain $\biggl{[}\partial_{z}^{2}+\sum_{i=1,2}\left(b^{2}h_{i}(\wp(z-z_{i})+2\eta_{1})+\partial_{z_{2}}f_{\text{cl.}}(-1)^{i+1}(\zeta(z-z_{i})+2\eta_{1}z_{i})\right)-2\pi i\partial_{\tau}f_{\text{cl.}}\biggr{]}\Psi_{pq}=0.$ (31) From the definition of $\Psi_{pq}$ we derive the monodromy conditions in the same way as on the plane. For the OPE block, these are ${\mathrm{Tr}}M_{z_{1},z_{2}}=-2\cos(\pi\sqrt{1-4h_{p}b^{2}}),\leavevmode\nobreak\ {\mathrm{Tr}}M_{z_{0}}=-2\cos(\pi\sqrt{1-4h_{q}b^{2}}).$ (32) The subscripts of the monodromy matrices show around which cycles the monodromy is taken. The derivation for the projection block works analogously. Here we contract ${\mathcal{O}}_{q}(z_{0}){\mathcal{O}}_{2}(z_{2})$: $\displaystyle\langle{\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{2}(z_{2})\Psi(z)\rangle_{\tau}$ (33) $\displaystyle=\sum_{p,q}C^{p}_{2q}\sum_{k,l}(z_{2}-z_{0})^{h_{p}-h_{q}-h_{2}+\|k\|}Q^{h_{q}-c/24+\|l\|}\beta^{pk}_{2q}\langle{\mathcal{O}}^{\\{k\\}}_{p}(z_{0}){\mathcal{O}}_{1}(z_{1})\Psi(z){\mathcal{O}}^{\\{l\\}}_{q}(z_{\infty})\rangle\text{(c.c.)}.$ Using $\Psi_{pq}$ defined by $\Psi_{pq}=\frac{\langle{\mathcal{O}}_{p}(z_{0})\Psi(z){\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{q}(z_{\infty})\rangle}{\langle{\mathcal{O}}_{p}(z_{0}){\mathcal{O}}_{1}(z_{1}){\mathcal{O}}_{q}(z_{\infty})\rangle}$ (34) and related to the two point correlator by $\langle{\mathcal{O}}_{1}{\mathcal{O}}_{2}\Psi\rangle_{\tau}\approx\sum_{p,q}C^{p}_{2q}C^{q}_{1p}\Psi_{pq}{\mathcal{F}}_{2q,1p}\bar{\mathcal{F}}_{2q,1p}$ (35) we find the same decoupling equation (31). However, the monodromy conditions differ. They are given by ${\mathrm{Tr}}M_{z_{0},z_{2}}=-2\cos(\pi\sqrt{1-4h_{p}b^{2}}),\leavevmode\nobreak\ {\mathrm{Tr}}M_{z_{\infty}}=-2\cos(\pi\sqrt{1-4h_{q}b^{2}}).$ (36) To solve the decoupling equation, it is useful to perform a coordinate transformation $u=e^{-2\pi iz}$. Using the transformation of primary operators under conformal transformations as well as the series representations of the Weierstrass elliptic functions from app. A, the decoupling equation becomes $\displaystyle\biggl{[}$ $\displaystyle\partial_{u}^{2}+y(h_{2}-(1+y)\partial_{y}f_{\text{cl.}})\sum_{m=-\infty}^{\infty}\frac{Q^{m}}{u(u-Q^{m})(u-Q^{m}(1+y))}+\frac{1/4-Q\partial_{Q}f_{\text{cl.}}}{u^{2}}$ (37) $\displaystyle+h_{1}\sum_{m=-\infty}^{\infty}\frac{Q^{m}}{u(u-Q^{m})^{2}}+h_{2}\sum_{m=-\infty}^{\infty}\frac{Q^{m}(1+y)}{u(u-Q^{m}(1+y))^{2}}\biggr{]}\Psi_{pq}=0,$ where we have chosen w.l.o.g. $z_{1}=0$ and $e^{-2\pi iz_{2}}=1+y$. In these coordinates, the monodromy conditions become $\displaystyle{\mathrm{Tr}}M_{1,1+y}$ $\displaystyle=-2\cos(\pi\sqrt{1-4h_{p}b^{2}})$ $\displaystyle,\leavevmode\nobreak\ {\mathrm{Tr}}M_{0}=$ $\displaystyle-2\cos(\pi\sqrt{1-4h_{q}b^{2}})$ (OPE block) (38) $\displaystyle{\mathrm{Tr}}M_{0,1+y}$ $\displaystyle=-2\cos(\pi\sqrt{1-4h_{p}b^{2}})$ $\displaystyle,\leavevmode\nobreak\ {\mathrm{Tr}}M_{\infty}=$ $\displaystyle-2\cos(\pi\sqrt{1-4h_{q}b^{2}})$ (projection block) This representation of the decoupling equation is immediately applicable for the calculation of the OPE block, which is defined through a series expansion in $y$ and $Q$. Using this series expansion as well as a WKB approximation for large $h_{p},h_{q}$, (37) can be solved order by order. For example, to first order in $y$ and $Q$ we get $f_{\text{cl.}}^{\text{OPE}}=-b^{2}(h_{p}-h_{1}-h_{2})\log y-(b^{2}h_{q}-1/4)\log Q+\frac{1}{2}b^{2}(h_{p}+h_{2}-h_{1})y-b^{2}\frac{h_{p}^{2}}{2h_{q}}Q+...$ (39) The projection block, on the other hand, can be expanded in a series in $q_{1}=Q/(1+y)$ and $q_{2}=1+y$. The decoupling equation can then be solved in the same way as for the OPE block order by order in $q_{1}$ and $q_{2}$. For example, to first order in $q_{1}$ and $q_{2}$ we obtain $\displaystyle f_{\text{cl.}}^{\text{projection}}=$ $\displaystyle-(b^{2}(h_{p}-h_{2})-1/4)\log q_{2}-(b^{2}h_{q}-1/4)\log q_{1}$ (40) $\displaystyle-b^{2}\frac{(h_{1}-h_{p}+h_{q})(h_{2}-h_{p}+h_{q})}{2h_{q}}q_{1}-b^{2}\frac{(h_{1}+h_{p}-h_{q})(h_{2}+h_{p}-h_{q})}{2h_{p}}q_{2}+...$ We have checked that the results for both the OPE and the projection block are in agreement with the recursion formulas derived in Cho:2017oxl (see app. B for detailed expressions) as well as explicit calculations up to third order. It is clear that the above derivation can be easily generalized to other conformal blocks on the torus. The simplest case is the zero-point block on the torus, i.e. the Virasoro character. Performing a similar derivation as above or equivalently taking the limit $h_{1,2,p}\to 0$ in (37), we arrive at the following decoupling equation $\left[\partial_{u}^{2}+\frac{1/4-Q\partial_{Q}f_{\text{cl.}}}{u^{2}}\right]\Psi_{q}=0,$ (41) together with the monodromy condition ${\mathrm{Tr}}M_{0}=-2\cos(\pi\sqrt{1-4h_{q}b^{2}})$. In this case, we can give the full solution. The decoupling equation is solved by $\Psi_{q}=u^{1/2\pm\sqrt{Q\partial_{Q}f_{\text{cl.}}}}$, from which we obtain $f_{\text{cl.}}=(1/4-b^{2}h_{q})\log Q$ which correctly reproduces the leading order contribution in $c$ of the Virasoro character $\chi_{q}=\frac{1}{\eta(\tau)}Q^{h_{q}-(c-1)/24}\sim e^{-c/6f_{\text{cl.}}}$. For a general $n$-point conformal block, the decoupling equation is given by $\biggl{[}\partial_{z}^{2}+\sum_{i=1}^{n}\left(b^{2}h_{i}(\wp(z-z_{i})+2\eta_{1})+\partial_{z_{i}}f_{\text{cl.}}(\zeta(z-z_{i})+2\eta_{1}z_{i})\right)-2\pi i\partial_{\tau}f_{\text{cl.}}\biggr{]}\Psi=0,$ (42) and there are $n$ monodromy conditions around non-trivial cycles determined by the OPE contractions. By conformal transformations, the insertion point of one of the operators can be fixed, for example to $z_{1}\to 0$. Then, there are $n$ independent accessory parameters $\partial_{\tau}f_{\text{cl.}}$ and $\partial_{z_{i}}f_{\text{cl.}}$ for $i\geq 2$ fixed by these monodromy conditions. ### 2.3 Partition function on the replica surface We now turn to the computation of the partition function $Z_{n}$ on the replica surface ${\mathcal{R}}_{n}$. In general, the partition function on any higher genus Riemann surface can be expanded in zero-point conformal blocks, which can again be calculated via a monodromy method. This monodromy method can be derived by inserting the degenerate operator directly on the higher genus Riemann surface – in contrast to the last section, where we inserted it in a correlation function on the torus – and inserting projection operators in the appropriate places. The difficulty of this approach is of course that deriving the decoupling equation for an arbitrary Riemann surface is quite hard. However, we will see that things simplify for the special higher genus surface that we are interested in, that is the replica surface ${\mathcal{R}}_{n}$. Assuming that the dominant contribution to the partition function depends only on the temperature and size of the entangling interval and not on any other moduli of ${\mathcal{R}}_{n}$, we find the same decoupling equation as for the twist operator correlator on the torus. The difference to the last section lies in the monodromy conditions. The zero- point block on ${\mathcal{R}}_{n}$ admits more general monodromy conditions (corresponding to different channels) than the conformal block on the torus. One of these more general monodromy conditions will give the dominant contribution to the entanglement entropy for large intervals. Before deriving the decoupling equation on ${\mathcal{R}}_{n}$, we collect some facts about the topology and moduli of ${\mathcal{R}}_{n}$. For simplicity, we specialize again to the single interval case. The replica surface is given by $n$ copies of a torus with modular parameter $\tau$, joined at a branch cut along the entangling interval $A$. We use coordinates $z,\bar{z}$ to parametrize ${\mathcal{R}}_{n}$ with identifications $z\sim z+1$ and $z\sim z+\tau$. In these coordinates, ${\mathcal{R}}_{n}$ is described by a branched cover of the torus with branch points located at $z=z_{1,2}+k+l\tau$ for $k,l\in{\mathbb{Z}}$. Near the branch points, the covering map is given by $y^{n}\propto(z-z_{1}-k-l\tau)$ and $y^{n}\propto 1/(z-z_{2}-k-l\tau)$. The genus of ${\mathcal{R}}_{n}$ is then obtained by the Riemann-Hurwitz theorem. The ramification index at each branch point is equal to $n$, yielding $g=n$. Since the Euler characteristic is $\chi<0$, there are no conformal Killing vectors. This implies by the Riemann-Roch theorem that there exist $3(n-1)$ holomorphic quadratic differentials $\omega_{zz}^{(i)}$ parametrizing deformations of the complex structure of the Riemann surface. The $\omega_{zz}^{(i)}$ are meromorphic doubly periodic functions that are regular everywhere on the covering surface, i.e. $\omega_{yy}^{(i)}dy^{2}=\omega_{zz}^{(i)}\bigl{(}\frac{dz}{dy}\bigr{)}^{2}dz^{2}$ is non-singular for all $y$. Simple examples include $\omega_{zz}^{(1)}=\text{const.}$ which is trivially regular and doubly periodic as well as $\omega_{zz}^{(2)}=\zeta(z-z_{1})-\zeta(z-z_{2})+2\eta_{1}(z_{1}-z_{2})$. $\omega_{zz}^{(2)}$ is regular since near $z=z_{1}+k+l\tau$ we have $\omega_{yy}^{(2)}\propto y^{n-2}$ which is regular at $y=0$ for $n\geq 2$. Near $z=z_{2}+k+l\tau$, regularity can be shown in an analogous way. In fact, these two examples are the only ones relevant for the following arguments since they are the only ones that respect the ${\mathbb{Z}}_{n}$ replica symmetry permuting the different copies of the torus with each other555This can be seen as follows. The replica symmetry acts as $y\to ye^{2\pi i/n}$. Therefore, only the $\omega_{zz}\sim(z-z_{i})^{\alpha_{i}}$ with $\alpha_{i}\in{\mathbb{Z}}$, $i=1,2$ are invariant under this symmetry. The case $\alpha_{i}<-1$ is singular at $z=z_{i}$. $\alpha_{i}>0$ is singular at some other point since any non-constant elliptic function has at least two poles inside the fundamental parallelogram, which lead to singularities in $\omega_{yy}$. This leaves only $\alpha_{i}=0,-1$ which are the two examples described above.. The derivation of the decoupling equation on the replica surface then proceeds in a similar fashion as in the previous section. Assuming exponentiation of the zero-point block in the semiclassical limit, the conformal Ward identities for a general Riemann surface Eguchi:1986sb imply a decoupling equation of the form $\left[\partial_{z}^{2}+\langle T_{zz}\rangle+\sum_{i=1}^{n}\omega_{zz}^{(i)}\partial_{w_{i}}f_{\text{cl.}}\right]\Psi(z)=0,$ (43) where $w_{i}$ are the modular parameters associated to $\omega_{zz}^{(i)}$. $\langle T_{zz}\rangle$ is the expectation value of the energy momentum tensor. It can be derived along the lines of Faulkner:2013yia : $\langle T_{zz}\rangle$ transforms with a Schwarzian derivative, $\langle T_{yy}\rangle=\left(\frac{\partial z}{\partial y}\right)^{2}\langle T_{zz}\rangle+\frac{nc}{12}\\{z,y\\},$ (44) and $\langle T_{yy}\rangle$ must be regular. The Schwarzian derivative term comes with a $nc/12$ prefactor since the stress-energy tensor on the replica surface is given as the sum of the stress-energy tensors of the $n$ tori. Therefore, the Schwarzian for the transformation of the stress-energy tensor on the replica surface is given by the sum of $n$ identical Schwarzian terms with prefactor $c/12$. Together with the requirement that $\langle T_{zz}\rangle$ be doubly periodic, regularity of $\langle T_{yy}\rangle$ implies $\langle T_{zz}\rangle=\frac{c}{24}\left(n-\frac{1}{n}\right)\sum_{i}(\wp(z-z_{i})+2\eta_{1}).$ (45) The $1/(z-z_{i})^{2}$ poles in $\wp(z-z_{i})$ give a $1/y^{2}$ contribution to $\langle T_{yy}\rangle$ that cancels with the Schwarzian derivative term666For ease of comparison with the previous section we have also added a constant term $\frac{c}{24}\left(n-\frac{1}{n}\right)4\eta_{1}$ to $\langle T_{zz}\rangle$ which is not strictly necessary for regularity and could be absorbed into the prefactor of $\omega_{zz}^{(1)}$.. Letting the sum over $i$ in (43) run only over $i=1,2$, we recover the decoupling equation (31) for the twist correlator. Restricting the sum to this range means that we assume $\partial_{w_{i}}f_{\text{cl.}}=0$ for $i>2$, i.e. we assume that the result for the partition function on the replica surface does not depend on other moduli of the replica surface than the size of the torus $\tau$ and the length of the entangling interval $z_{2}-z_{1}$. $\langle(P_{p}{\mathcal{O}}_{1}{\mathcal{O}}_{2}P_{p}){\mathcal{O}}_{3}{\mathcal{O}}_{4}\rangle\sim$$4$$3$$p$$1$$2$$\sim$$P_{p}$${\mathcal{O}}_{1}$${\mathcal{O}}_{2}$${\mathcal{O}}_{3}$${\mathcal{O}}_{4}$$\langle{\mathcal{O}}_{1}(P_{p}{\mathcal{O}}_{2}{\mathcal{O}}_{3}P_{p}){\mathcal{O}}_{4}\rangle\sim$$4$$1$$p$$3$$2$$\sim$${\mathcal{O}}_{1}$$P_{p}$${\mathcal{O}}_{2}$${\mathcal{O}}_{3}$${\mathcal{O}}_{4}$ Figure 2: Inserting projection operators $P_{p}$ onto the Verma module of a primary ${\mathcal{O}}_{p}$ into a correlator yields the conformal block with internal weight $h_{p}$. However, as mentioned in the beginning of this section, the admitted monodromy conditions for the decoupling equation (43) are more general than those of (31) for the twist correlator. To see this, recall that conformal blocks of any correlation function can be obtained in two equivalent ways. Either we can perform OPE contractions between two or more operators and then keep only terms of particular primaries and their descendants in the OPE or equivalently we can insert projection operators onto the Verma modules of these primaries in the correlation function at appropriate places. The projectors of the latter approach can be thought of as non-local operators acting in a closed line around the operators whose OPE contractions are performed in the former approach (see fig. 2). For the zero-point block on an arbitrary higher genus Riemann surface, there are in general $3(n-1)$ projectors to be inserted corresponding to $3(n-1)$ monodromy conditions. However, as mentioned above we assume that the partition function on the higher genus Riemann surface depends only on two of the moduli and thus we consider only two of the $3(n-1)$ monodromy conditions. Which monodromy conditions are appropriate for the calculation of the entanglement entropy? For the conformal block, the monodromy conditions must be taken around the spatial circle and around $z_{1},z_{2}$ 777It is also possible to calculate the modular transformed block for the twist correlator, which we expect to be the dominant contribution at large temperature and small intervals. In this case, the monodromy conditions are taken around $z_{1},z_{2}$ and around the time circle.. On the other hand, for the zero- point block on the replica surface the prescription described in this section still leaves open the question of where to put the monodromy conditions – i.e. which channel to choose – in order to obtain the dominant contribution to the partition function from the vacuum block. Taking the limits of high and low temperature, it is clear that for small intervals one of the monodromy conditions must be taken around the spatial circle for low temperatures and the time circle for high temperatures, while the other monodromy condition must be imposed around the entangling interval $A$ between $z_{1}$ and $z_{2}$. For large intervals, the correct monodromy condition is obtained by reformulating the problem along the lines of Cardy:2014jwa ; Chen:2015kua . We separate the branch cut on the torus along $A$ yielding the replica surface into a branch cut along the full spatial circle and a branch cut in the opposite direction along $A^{c}$ (see fig. 3). We then impose trivial monodromy around $A^{c}$ to fix the dependence on the size of the entangling interval. For small temperatures, the monodromy condition around the spatial circle remains unchanged. However, for high temperatures the monodromy condition around the time circle is now transformed into a monodromy condition around a time circle of size $n\tau$, since the branch cut along the full spatial circle connects all $n$ replica copies together to effectively create a torus with modular parameter $n\tau$. $A$$A^{c}$ Figure 3: Branch cut structure for large intervals. We can decompose the branch cut along $A$ (denoted in red on the left) into a branch cut along the full spatial circle (denoted in red on the right) and a branch cut in the opposite direction along $A^{c}$ (denoted in blue on the right). Note again that it is perfectly valid to use any of the above monodromy conditions for all values of the temperature and entangling interval size. However, outside of the regimes of validity of the monodromy conditions described above, we don’t expect the vacuum block to give the dominant contribution in the semiclassical limit and thus the partition function in this case would be obtained by summing up all of the conformal blocks for different values of the dimensions of the exchanged operators. The cross-over point between the regimes must be determined by an analysis of these contributions from the exchange of non-identity operators. Let us also note that explicit calculations for the free fermion case support the arguments presented in this section with regards to the differences between twist correlators and partition functions on ${\mathcal{R}}_{n}$ and with regards to the monodromy condition for large intervals. Namely, in Lokhande:2015zma it was observed that the twist correlator on the torus does not give the correct answer for the entanglement entropy and in particular violates Bose-Fermi equivalence and modular covariance. This was traced back in Mukhi:2017rex to the way in which different spin structures of the replica surface ${\mathcal{R}}_{n}$ combine to give the total answer for the partition function $Z_{n}$. The replica surface, being composed of $n$ copies of a torus, has $2n$ nontrivial cycles around which the fermions have either periodic or antiperiodic boundary conditions. To calculate the total partition function $Z_{n}$ on the replica surface, it is necessary to sum over all possible spin structures each of which corresponds to a particular choice of boundary conditions around the nontrivial cycles of the replica surface. For small intervals, the replica surface essentially factorizes into $n$ unconnected torus copies. In this limit, it was shown in Mukhi:2017rex that $Z_{n}$ is given by an “uncorrelated” sum where the summation over spin structures is performed for each of the $n$ tori separately. For large intervals, $Z_{n}$ is given by a “correlated” sum where only one sum over spin structures is performed, i.e. we take equal boundary conditions around the time resp. space circles of each of the $n$ replica copies Mukhi:2017rex . In this case, the partition function $Z_{n}$ of the replica surface is essentially given by the partition function on a torus with modular parameter $n\tau$. ## 3 Entanglement entropy at large central charge In this section, we present the calculation of the entanglement entropy on the torus at large central charge. As explained in the previous section, the entanglement entropy can be obtained from the partition function $Z_{n}$ on the replica surface ${\mathcal{R}}_{n}$ which decomposes into zero-point conformal blocks. The claim we want to investigate is that at large central charge $c\to\infty$ and for $n\to 1$ the dominant contribution to $Z_{n}$ comes from the vacuum block with $h_{p}=h_{q}=0$. The derivation of this statement proceeds as follows. First, it is necessary to show that the semiclassical limit is well-defined not only for $h_{p,q}={\mathcal{O}}(c)$ but also for $h_{p,q}={\mathcal{O}}(c^{0})$. This means that for $h_{p,q}=\gamma c$ the limits $\lim_{c\to\infty}$ and $\lim_{\gamma\to 0}$ of the conformal block commute. A discussion of this point starting from the recursion relation for torus conformal blocks is relegated to app. B. In the next step we solve the decoupling equation (31) perturbatively in $\varepsilon=n-1$ up to first order. Imposing the monodromy conditions derived in sec. 2.2 then yields the conformal block from which we finally extract the entanglement entropy. We first consider the limits of high and low temperature as well as small and large entangling interval size in sec. 3.1. Each combination of these limits comes with different monodromy conditions as explained in the previous section. We find agreement with the known universal results in the limits where the torus degenerates into a cylinder. We then examine the conditions on the CFT spectrum and OPE coefficients under which these results extend to intermediate temperature and interval size regimes in sec. 3.2, leading to the conclusion that for holographic CFTs the results from sec. 3.1 are valid for all temperatures and interval sizes. Moreover, we consider the case of multiple intervals on the torus. Finally, we numerically check the assumption on the dominance of the vacuum block in sec. 3.4. ### 3.1 Limiting cases #### 3.1.1 Low temperature and small intervals In the low temperature limit $\beta\to\infty$ the torus degenerates into a cylinder with periodic space direction. For the cylinder, the entanglement entropy of a single interval can be obtained directly by mapping this cylinder to the plane and using the known formula for the entanglement entropy on the plane Calabrese:2004eu , $S_{A}=\frac{c}{3}\log\left(\sin(\pi(z_{2}-z_{1}))\right)+\text{const.}$ (46) To obtain the same result from the monodromy method, we series expand $\Psi_{pq}$ and $f_{\text{cl.}}$ in $n-1$: $\Psi_{pq}=\sum_{k}\Psi_{pq}^{(n)}(n-1)^{k}$ and $f_{\text{cl.}}=\sum_{k}f_{n}(n-1)^{k}$. The decoupling equation (31) at zeroth order in $n-1$ becomes $[\partial_{z}^{2}-2\pi i\partial_{\tau}f_{0}]\Psi_{pq}^{(0)}(z)=0.$ (47) This is solved by $\Psi^{(0)}_{pq}(z)=\exp(\pm\sqrt{2\pi i\partial_{\tau}f_{0}}z).$ (48) By a coordinate transformation $u=\exp(-2\pi iz)$, we obtain $\tilde{\Psi}_{pq}^{(0)}(u)=u^{h_{\Psi}}\Psi_{pq}^{(0)}(z=i\log(u)/(2\pi))$. Imposing trivial monodromy of $\tilde{\Psi}_{pq}^{(0)}(u)$ around $u=0$ is equivalent to antiperiodic monodromy conditions for $\Psi_{pq}^{(0)}(z)$ around the spatial circle of the torus, $\Psi_{pq}^{(0)}(z+1)=-\Psi_{pq}^{(0)}(z)$. As expected, this implies that $f_{0}$ is equal to the leading order in $c$ of the vacuum character on the torus, $f_{0}=\pi i\tau/2=-\beta/4\hskip 28.45274pt\Leftrightarrow e^{-c/6f_{0}}=e^{c/24\beta}=\left.\chi_{h=0}(\beta)\right\|_{c\to\infty}.$ (49) At first order in $n-1$, the decoupling equation is given by $[\partial_{z}^{2}-2\pi i\partial_{\tau}f_{0}]\Psi_{pq}^{(1)}(z)+m(z)\Psi_{pq}^{(0)}(z)=0,$ (50) yielding $\Psi_{pq}^{(1)}(z)=\frac{e^{-i\pi z}}{2\pi i}\int^{z}dx\,m(x)e^{i\pi x}\Psi_{pq}^{(0)}(x)-\frac{e^{i\pi z}}{2\pi i}\int^{z}dx\,m(x)e^{-i\pi x}\Psi_{pq}^{(0)}(x),$ (51) where $m(z)$ is given by $m(z)=\sum_{i}\left(\frac{1}{2}(\wp(z-z_{i})+2\eta_{1})+(-1)^{i+1}(\zeta(z-z_{i})+2\eta_{1}z_{i})\partial_{z_{2}}f_{1}\right)-2\pi i\partial_{\tau}f_{1}.$ (52) To compute the conformal block we impose trivial monodromy around $z_{1},z_{2}$ which is equivalent to the vanishing of $\oint_{z_{1},z_{2}}dx\,m(x)e^{\pm i\pi x}\Psi_{pq}^{(0)}(x).$ (53) This gives the $z_{1},z_{2}$ dependence of $f_{1}$, $f_{1}=\log(\sin(\pi(z_{2}-z_{1})))+C_{1}(\tau).$ (54) From trivial monodromy around $u=0$ we find the $\tau$ dependence to be $\partial_{\tau}f_{1}=0$, which implies that $C_{1}(\tau)=\text{const.}$ is independent of $\tau$. The antiholomorphic conformal block $\bar{f}_{1}$ gives the same result as the holomorphic one. Then the entanglement entropy is given by $S_{A}=\frac{c}{6}(f_{1}+\bar{f}_{1})$, in agreement with (46). In this limit, the OPE vacuum block of the twist correlator gives the same results, since it is given by the same monodromy method as the zero-point vacuum block of the replica partition function computed in this section. #### 3.1.2 Low temperature and large intervals In this limit, we demand trivial monodromy around the spatial circle (i.e. around $u=e^{-2\pi iz}=0$) as well as trivial monodromy around $A^{c}$ (i.e. around $z_{1},z_{2}-1$). This gives the same vacuum block and thus the same entanglement entropy (46) as in the small interval case at low temperature. Also in this case, the twist correlator gives the correct result. #### 3.1.3 High temperature and small intervals In the high temperature limit $\beta\to 0$ the torus again degenerates into a cylinder, now with periodic time direction. As in the low temperature case, the entanglement entropy of a single interval can be obtained by mapping this cylinder to the plane Calabrese:2004eu , $S_{A}=\frac{c}{3}\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}(z_{2}-z_{1})\right)\right)+\text{const.}$ (55) In the monodromy method, we impose trivial monodromy around the time circle and around $z_{1},z_{2}$. As in the low temperature limit, we solve the decoupling equation (31) in a series expansion around $n-1$. An analogous calculation as above yields $f_{\text{cl.}}=-\frac{\pi i}{2\tau}+\epsilon\log\left(\tau\sinh\left(\frac{\pi i}{\tau}(z_{2}-z_{1})\right)\right)+\text{const.}=f_{0}+\epsilon f_{1}$ (56) At zeroth order in $\epsilon$ we recover the leading order in $c$ of the vacuum character $\chi_{h=0}(\beta)=e^{\frac{c}{24}\frac{4\pi^{2}}{\beta}}=e^{-\frac{c}{6}(-\frac{\pi i}{2\tau})}$. The entanglement entropy given by the first order contribution, $S_{A}=\frac{c}{6}(f_{1}+\bar{f}_{1})$, agrees with the result from the cylinder (55). From the twist correlator point of view, we obtain the high temperature limit by a modular transformation $\tau\to-1/\tau$ from the low temperature result. The two point function on the torus transforms covariantly, $\langle{\mathcal{O}}_{1}(-z_{1}/\tau,-\bar{z}_{1}/\bar{\tau}){\mathcal{O}}_{2}(-z_{2}/\tau,-\bar{z}_{2}/\bar{\tau})\rangle_{-1/\tau}=(-\tau)^{h_{1}+h_{2}}(-\bar{\tau})^{\bar{h}_{1}+\bar{h}_{2}}\langle{\mathcal{O}}_{1}(z_{1},\bar{z}_{1}){\mathcal{O}}_{2}(z_{2},\bar{z}_{2})\rangle_{\tau}.$ (57) If the two point function of twist operators is dominated by a single conformal block, the conformal block for the modular transformed $\tau$ is obtained from the modular transformation properties of the correlation function as $f_{\text{cl.}}^{-1/\tau}(-z_{1}/\tau,-z_{2}/\tau)=-b^{2}(h_{1}+h_{2})\log(-\tau)+f_{\text{cl.}}^{\tau}(z_{1},z_{2}).$ (58) Therefore, we can immediately read off the high temperature behavior of the twist correlator from $\left.f_{\text{cl.}}^{\tau}(z_{1},z_{2})\right\|_{\tau\to\infty}=\left[f_{\text{cl.}}^{-1/\tau}(-z_{1}/\tau,-z_{2}/\tau)+b^{2}(h_{1}+h_{2})\log(-\tau)\right]_{\tau\to\infty}$ (59) For $h_{1}=h_{2}=1/2\epsilon$, $\left.f_{\text{cl.}}^{\tau}(z_{1},z_{2})\right\|_{\tau\to 0}=\frac{\pi i\tau}{2}+\epsilon\log(\sin(\pi(z_{2}-z_{1})))+\text{const.}$ we again obtain (56). Thus the twist correlator agrees with the replica partition in this limit. Let us note that is also possible to determine the high temperature expansion of twist correlator by applying the modular transformation to the monodromy conditions. At low temperatures, we demand trivial monodromy around the spatial circle of the torus at time. Since the modular transformation $\tau\to-1/\tau$ exchanges the time and space directions of the torus, we can obtain the high temperature behavior of the twist correlator by demanding trivial monodromy around the time circle of the torus, showing directly the equivalence between the twist correlator result and the replica partition function in the limit of high temperature and small entangling intervals. #### 3.1.4 High temperature and large intervals As explained in sec. 2.3, the correct monodromy condition of the zero-point block on the replica surface for large intervals imposes trivial monodromy around $A^{c}$ and $z\to z+n\tau$. To zeroth order in $n-1$ the decoupling equation (31) is solved by $\Psi_{pq}^{(0)}(z)=\exp(\pm\sqrt{2\pi i\partial_{\tau}f_{0}})$. Imposing $\Psi_{pq}^{(0)}(z+n\tau)=-\Psi_{pq}^{(0)}(z)$ yields $f_{0}=-\frac{\pi i}{2n^{2}\tau}.$ (60) To first order, the solution reads $\Psi_{pq}^{(1)}(z)=n\tau\frac{e^{-\frac{i\pi z}{n\tau}}}{2\pi i}\int^{z}dx\,m(x)e^{\frac{i\pi x}{n\tau}}\Psi_{pq}^{(0)}(x)-n\tau\frac{e^{\frac{i\pi z}{n\tau}}}{2\pi i}\int^{z}dx\,m(x)e^{-\frac{i\pi x}{n\tau}}\Psi_{pq}^{(0)}(x).$ (61) Imposing trivial monodromy around $A^{c}$ and expanding in $n-1$ we obtain $f_{\text{cl.}}=-\frac{\pi i}{2\tau}+(n-1)\left(\frac{i\pi}{\tau}+\log\left(\tau\sinh\left(\frac{i\pi}{\tau}(1-(z_{2}-z_{1}))\right)\right)\right)+\text{const.}$ (62) giving $S_{A}=\frac{c}{3}\left(\frac{i\pi}{\tau}+\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}(1-(z_{2}-z_{1}))\right)\right)\right)+\text{const.}$ (63) As expected from general arguments Azeyanagi:2007bj ; Cardy:2014jwa , the difference between the entanglement entropy for $A$ and for the complement $A^{c}$ in the limit of large interval size is given by the thermal entropy $S(\beta)=\beta^{2}(-\beta^{-1}\partial_{\beta}\log Z(\beta))=\frac{c}{3}\frac{2\pi^{2}}{\beta}=\frac{c}{3}\frac{i\pi}{\tau}$, using the partition function $Z(\beta\to 0)=\exp(\frac{c}{12}\frac{4\pi^{2}}{\beta})$ from the Cardy formula. The twist correlator, on the other hand, cannot reproduce this feature as is easy to see by applying the modular transformation argument from the last section which gives (56) in disagreement with (62). ### 3.2 Holographic CFTs We now argue that the results in the limits considered in the previous section are valid for all temperatures and interval sizes in holographic CFTs. This statement holds if the vacuum block computed in the previous section gives the dominant contribution to the partition function on the replica manifold ${\mathcal{R}}_{n}$ in the large central charge limit. Let us first consider the case $n=1$, i.e. the zeroth order in the $n-1$ expansion. In this order, the monodromy method calculates the vacuum character to leading order in $c$. If this vacuum character dominates the partition function, $Z(\beta)$ takes on the universal form $Z(\beta)=\left\\{\begin{array}[]{ll}\exp\bigl{(}\frac{c}{12}\beta\bigr{)}&,\leavevmode\nobreak\ \beta>2\pi\\\ \exp\bigl{(}\frac{c}{12}\frac{4\pi^{2}}{\beta}\bigr{)}&,\leavevmode\nobreak\ \beta<2\pi\end{array}\right.$ (64) for all temperatures. For consistency of the computation method, dominance of the conformal block to first order in $n-1$ also requires dominance of the zeroth order in $n-1$. It has been argued in Hartman:2014oaa that a partition function of the form (64) is characteristic of a holographic CFT. Thus it is a necessary condition that the CFT in question be holographic in order for the results of sec. 3.1 to hold at arbitrary temperatures. More explicit conditions on the CFT in question can be given following an argument for dominance of the vacuum block in the zero temperature case which goes as follows Hartman:2013mia . The arguments below hold for CFTs with OPE coefficients $C^{p}_{21}$ that grow at most exponentially with $c$ 888This is equivalent to the requirement that correlation functions obey the cluster decomposition principle and are smooth in a neighborhood of the point where multiple operator insertion points coincide Hartman:2013mia . and a density of states for light operators that does not grow with $c$. Here light operators mean operators with conformal weight $h,\bar{h}<h_{\text{gap}}$ where $h_{\text{gap}}$ is of the order of the central charge. In the large central charge limit, the conformal block expansion in the $s$-channel of the four- point twist correlator on the plane takes on the form $\langle\sigma_{n}(0)\sigma_{n}(x)\sigma_{n}(1)\sigma_{n}(\infty)\rangle=\sum_{p}C^{p}_{21}\exp\left(-\frac{c}{6}\bigl{(}f_{\text{cl.}}(h_{\sigma_{n}}/c,h_{p}/c,x)+\bar{f}_{\text{cl.}}(\bar{h}_{\sigma_{n}}/c,\bar{h}_{p}/c,\bar{x})\bigr{)}\right).$ (65) On account of the cluster decomposition principle, the vacuum exchange is the leading contribution in the $s$-channel around $x=0$ and in the $t$-channel around $x=1$. This implies that the contribution of the heavy operators with $h,\bar{h}>h_{\text{gap}}$ in the $s$-channel in some finite region around $x=0$ is suppressed exponentially and similarly for the $t$-channel around $x=1$. The sparse spectrum of the light operators allows ignoring the multiplicity factors for $h,\bar{h}<h_{\text{gap}}$. Consider first the scenario that the OPE coefficients grow subexponentially with $c$. In this case, we can also ignore the coefficient $C^{p}_{21}$ and the sum over the light operators in (65) is dominated exponentially in $e^{-c}$ by its largest term. If the semiclassical conformal block $f_{\text{cl.}}$ is a monotonically increasing function with $h_{p}/c$, this implies that the vacuum block with the lowest possible $h_{p}/c=0$ dominates. In the case that the OPE coefficients grow exponentially with $c$, the sum over the light operators gives $\int_{0}^{h_{\text{gap}}}dh_{p}d\bar{h}_{p}\exp\left(\frac{c}{6}\bigl{(}g(h_{p}/c,\bar{h}_{p}/c)-f_{\text{cl.}}(h_{\sigma_{n}}/c,h_{p}/c,x)-\bar{f}_{\text{cl.}}(\bar{h}_{\sigma_{n}}/c,\bar{h}_{p}/c,\bar{x})\bigr{)}\right),$ (66) where $g(h_{p}/c,\bar{h}_{p}/c)$ contains the contribution of the OPE coefficients and multiplicities. This integral is dominated either by the endpoints of the integration or by a saddle point. Near coincident points, the leading universal term in any correlation function is given by the vacuum exchange. Therefore, in a finite region around $x=0$ and $x=1$, the integral in (66) is dominated by the endpoint at $h_{p}=0$. However, now – unlike for subexponentially growing OPE coefficients – it is not possible to exclude saddle points dominating the integral (66) in some finite subset of $x\in[0,1]$. Examples of such saddle points have been found in Belin:2017nze for genus two partition functions computing the third Rényi entropy on the plane. In summary, the vacuum block dominates correlation functions in a finite region around coincident operator insertion points if the conformal block $f_{\text{cl.}}$ increases monotonically with the internal conformal weights $h_{p,q}$. In sec. 3.4, we present numerical evidence that this is indeed the case. Therefore, the results of sec. 3.1 are valid for all temperatures and interval sizes assuming subexponentially growing OPE coefficients and a sparse spectrum of light operators. For exponentially growing OPE coefficients, the results are still valid in a finite region around the respective limiting points. The points at which the different limits exchange dominance are obtained as follows. From the requirement of consistency of our results at $n=1$ with the partition function (64) we see that the low and high temperature regimes exchange dominance at the Hawking-Page phase transition point $\beta=2\pi$. The transition between small and large interval behavior in the high temperature regime is estimated by equating the results for the conformal block in the small and large interval regime. The dominant contribution comes from the smaller conformal block. Assuming the aforementioned restrictions on the CFT spectrum and OPE coefficients, the results match perfectly with the predictions from the RT formula Ryu:2006bv ; Azeyanagi:2007bj . There is also a direct way to implement the monodromy computation for the calculation of the entanglement entropy in the dual gravity theory, as was shown in Faulkner:2013yia at zero temperature. This was generalized to the finite temperature case in Barrella:2013wja ; Chen:2015kua . We obtain the same monodromy method as Barrella:2013wja ; Chen:2015kua from the CFT side. This clearly shows that our results are valid for holographic CFTs. Furthermore, the high temperature and large interval size result (62) agrees with a CFT calculation of the vacuum sector contribution to the entanglement entropy done in Chen:2015kua using complementary techniques to obtain the vacuum block. Specifically, the conformal block is obtained in Chen:2015kua by an explicit construction of the Virasoro generators and descendant states in the twisted sector of a ${\mathbb{Z}}_{n}$ orbifold. This allows for a series expansion of the Rényi entropy. Moreover, in the limit $n\to 1$, the authors of Chen:2015kua find that the leading order of the semiclassical vacuum block is given by an expression in terms of the four-point function of twist-operators on the plane, which gives the same conformal block (62) that we obtain using the monodromy method. Let us also note that the universal form of the partition function (64) in holographic theories can be derived from the same vacuum block dominance argument as the universal form of the entanglement entropy. The partition function can be expanded in zero-point blocks on the torus either in a low temperature expansion (zero-point blocks are Virasoro characters) or in a high temperature expansion (zero-point blocks are modular transformed Virasoro characters). From the known form of the Virasoro characters, the leading order contribution in the central charge of the characters is given by $\chi\sim Q^{h_{q}-c/24}$ where $Q=e^{2\pi i\tau}$ resp. $Q=e^{-2\pi i/\tau}$ in the low and high temperature expansions. Since $(-6/c)\log\chi$ is an increasing function of $h_{q}$, the same arguments as for the entanglement entropy given above apply to the partition function which is dominated by the vacuum character with $h_{q}=0$. ### 3.3 Multiple intervals The generalization to an entangling interval consisting of the union of multiple intervals, $A=[z_{1},z_{2}]\cup[z_{3},z_{4}]...[z_{2N-1},z_{2N}]$, is straightforward. The decoupling equation is given by $\left[\partial_{z}^{2}+\sum_{i=1}^{2N}\left(\frac{1}{4}(n-1/n)(\wp(z-z_{i})+2\eta_{1})-\partial_{z_{i}}f_{\text{cl.}}(\zeta(z-z_{i})+2\eta_{1}z_{i})\right)-2\pi i\partial_{\tau}f_{\text{cl.}}\right]\Psi(z)=0.$ (67) We impose trivial monodromy around $N$ pairs $(i,j)$ of interval endpoints $z_{i},z_{j}$ to fix the $\partial_{z_{i}}f_{\text{cl.}}$. The temperature dependence is fixed by demanding trivial monodromy around either the spatial circle, a time circle of size $\tau$ or a time circle of size $n\tau$ depending on the temperature and total entangling interval size $\|A\|=\sum_{i}\|z_{2i}-z_{2i-1}\|$. This yields * • low temperature: trivial monodromy for $z\to z+1$ $S_{A}=\frac{c}{3}\sum_{(i,j)}\log\left(\sin(\pi(z_{i}-z_{j}))\right)+\text{const.}$ (68) * • high temperature and small total interval size: trivial monodromy for $z\to z+\tau$ $S_{A}=\frac{c}{3}\sum_{(i,j)}\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}(z_{i}-z_{j})\right)\right)+\text{const.}$ (69) * • high temperature and large total interval size: trivial monodromy for $z\to z+n\tau$ $S_{A}=\frac{c}{3}\left(\frac{i\pi}{\tau}+\sum_{(i,j)}\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}(z_{i}-z_{j})\right)\right)\right)+\text{const.}$ (70) Which monodromy condition and which combination of pairs $(i,j)$ to take, i.e. in which channel to expand the conformal block, depends on the interval size. We expect the dominant contribution to come from the channel with the smallest $f_{\text{cl.}}$, in which case we find agreement with the RT formula. However, we caution that this argument depends on the vacuum block dominating the partition function, which we have checked only for a single interval. One particular interesting special case of the above calculation is the time dependence of the entanglement entropy between two intervals on opposite boundaries of a two-sided BTZ black hole. The two-sided BTZ black hole is dual to the thermofield double state. On the CFT side, the entanglement entropy is obtained by positioning the two intervals at a distance $\tau/2$ in the time coordinate on the torus. To obtain the time evolution of the entanglement entropy, it is necessary to perform an analytic continuation of the endpoints Hartman:2013qma , $\displaystyle z_{1}=\bar{z}_{1}=0$ $\displaystyle z_{2}=\bar{z}_{2}=L$ (71) $\displaystyle z_{3}=2t+L+\tau/2,\bar{z}_{3}=-2t+L+\bar{\tau}/2$ $\displaystyle z_{4}=2t+\tau/2,\bar{z}_{4}=-2t+\bar{\tau}/2,$ where $L$ is the size of the entangling interval taken to be equal on both boundaries and $t$ is the time coordinate at which both parts of the entangling interval are placed on the asymptotic boundaries of the wormhole999Note that this time coordinate has nothing to do with the euclidean time coordinate on the torus on which we calculate the finite temperature correlator in the euclidean CFT. (see Hartman:2013qma for more details on the setup). For small interval size $L$, there are two conformal blocks to consider. At early times, i.e. for small $t$, the dominant contribution comes from imposing trivial monodromy around $z_{1},z_{4}$ and $z_{2},z_{3}$, while at late times the vacuum block with trivial monodromy around $z_{1},z_{2}$ and $z_{3},z_{4}$ dominates. Taking into account that due to the analytic continuation $f_{\text{cl.}}\neq\bar{f}_{\text{cl.}}$, the corresponding entanglement entropy is given by $S_{A}=\frac{2c}{3}\log\left(\frac{\tau}{i\pi}\cosh\left(\frac{2\pi i}{\tau}t\right)\right)+\text{const.}$ (72) at early times and by $S_{A}=\frac{2c}{3}\log\left(\frac{\tau}{i\pi}\sinh\left(\frac{i\pi}{\tau}L\right)\right)+\text{const.}$ (73) at late times. This reproduces the phase transition in the dual RT surfaces from geodesics that connect the two boundaries through the interior of the two-sided black hole at early times to disconnected geodesics on opposite boundaries that do not enter the black hole interior at late times Hartman:2013qma . ### 3.4 Vacuum block dominance In this section, we provide numerical evidence that the vacuum block exponentially dominates the twist correlator in the large $c$ limit. For simplicity, we restrict to the single interval case. Assuming the same conditions on the spectrum and OPE coefficients of the CFT detailed in the last section, we need to show that the conformal block monotonically increases with the weight of the internal operators $h_{p,q}$. For the zero temperature case, this was done numerically in Hartman:2013mia for arbitrary $n$, giving evidence that the Rényi entropies are given by the vacuum conformal block contribution only. However, the calculation is much simpler if we restrict to $n$ close to one which implies $h_{p}/c\gg h_{i}/c\to 0$ for $i=1,2,3,4$. In this limit, the conformal block can be obtained in closed form from the monodromy calculation by a WKB expansion in $1/(h_{p}/c)$ Zamolodchikov1987 , $\frac{c}{6}f_{\text{cl.}}(0,h_{p}/c,x)=h_{p}\left(\pi\frac{K(1-x)}{K(x)}-\log 16\right),$ (74) where $K(x)$ is the complete elliptic integral of the first kind. Thus, $f_{\text{cl.}}$ is an increasing function of $h_{p}/c$ if $\pi K(1-x)/K(x)-\log 16>0$ which can easily be checked to be fulfilled for $x<1/2$. In fact, $f_{\text{cl.}}(0,h_{p}/c,x)-f_{\text{cl.}}(0,h_{p}/c,1-x)$ reaches its crossover point exactly at $x=1/2$, confirming that at this point dominance is exchanged from the $s$ to the $t$-channel block. Applying the same arguments as on the plane to the case of the torus, it is clear that the vacuum block dominates if the semiclassical block is an increasing function of $h_{p}$ and $h_{q}$. Without loss of generality, we take $z_{1}=0$ in the following. Restricting again to $n\approx 1$, we need to find the semiclassical block in the limit $h_{p,q}/c\gg h_{1,2}/c\to 0$. Unlike on the plane, however, $f_{\text{cl.}}$ can not easily be obtained in a closed form expression from the monodromy calculation in this limit101010The reason for this is that the solution of the decoupling equation on the torus takes on the schematic form of an integral over $\sqrt{A\partial_{z_{2}}f_{\text{cl.}}+B\partial_{\tau}f_{\text{cl.}}}$ for some functions $A$ and $B$, from which it is not easily possible to extract $\partial_{z_{2}}f_{\text{cl.}}$ and $\partial_{\tau}f_{\text{cl.}}$. The solution of the decoupling equation on the plane, on the other hand, is given by an integral over $\sqrt{A\partial_{x}f_{\text{cl.}}}$ from which $\partial_{x}f_{\text{cl.}}$ can be factored out immediately.. Thus, we apply a series expansion in $y=e^{-2\pi iz_{2}}-1$ and $Q=e^{2\pi i\tau}$ on top of the WKB approximation in $1/(h_{p,q}/c)$. While this yields a very precise numerical approximation to the true value of the conformal block if enough terms are included in the series, the expansion has a restricted domain of validity in the $y,Q$ plane. In particular, the series expansion in the cross- ratio $x$ of the four-point block on the plane converges for $\|x\|<1$ Zamolodchikov1987 , therefore we expect the series expansion of the two-point block on the torus to have a convergence radius of $\|y\|=1$ (the torus block reduces to the block on the plane in the limit $Q\to 0$). The numerics confirm this expectation. For $\|y\|>1$, we observe large fluctuations in the value of the conformal block as we include more terms in the series expansion. The numerics for the convergence radius in $Q$ is less clear, but also in this variable we observe large fluctuations close to $Q=e^{-2\pi}$. Thus, we can check the vacuum block dominance only in a restricted region around the origin in $y$, corresponding to small intervals. The restricted convergence radius in $Q$ is not limiting since above the Hawking-Page transition temperature given by $Q=e^{-2\pi}$, we expect the block in the high temperature expansion to dominate. The high temperature expansion of the conformal block is given by a series expansion in $e^{2\pi iz_{2}/\tau}-1$ and $e^{-2\pi i/\tau}$. In the $h_{p,q}/c\gg h_{1,2}/c$ limit, the series coefficients are equal to those of the low temperature expansion. Moreover, in the same limit at high temperatures and for $n\to 1$ the conformal block in the large interval limit (where the monodromy condition is taken around a time circle of size $n\tau$) is equivalent to the conformal block in the small interval limit with the replacement $z_{2}\to 1-z_{2}$. We show some plots of $\frac{\partial f_{\text{cl.}}}{\partial h_{p}}$ and $\frac{\partial f_{\text{cl.}}}{\partial h_{q}}$ in fig. 4 for small temperatures and values of $y$ and $Q$ inside the convergence radius. Figure 4: Derivative of the semiclassical conformal block $f_{\text{cl.}}$ w.r.t. $h_{p}$ (left) and $h_{q}$ (right) for $z_{2}=0.1$, $\beta=4\pi$ and different values of $h_{p}$ and $h_{q}$ in the range $[0,c/24]$. Note that the plotted value is greater than zero in all cases, showing that the semiclassical conformal block increases with $h_{p}$ and $h_{q}$. The series expansion used in this figure was truncated at order 10 in both $y$ and $Q$. The plots for the conformal block in the high temperature limit show no significant differences from the ones in the low temperature limit. We find in all cases that inside the convergence radius of the series expansion $\frac{\partial f_{\text{cl.}}}{\partial h_{p,q}}>0$, i.e. the assumption of vacuum block dominance is fulfilled While it is not possible to find an analytic continuation for the conformal block from a truncated series expansion, we can use a Padé approximant to get a heuristic approximation of the series outside its convergence radius. The Padé approximation works by replacing the truncated series expansion by a rational function whose Taylor expansion agrees with the series expansion up to the order in which the truncation was performed BakerGraves-Morris . In many cases, this approximation has a better radius of convergence than the original series expansion due to poles in the function limiting the radius of convergence of its Taylor expansion being taken into account in the Padé approximation. We plot the Padé approximant of $f_{\text{cl.}}$ in the special case $h_{p}=h_{q}$ for different orders of the denominator polynomial in fig. 5 depending on $z_{2}$ (the size of the entangling interval). Figure 5: Derivative of the Padé approximation of the semiclassical conformal block w.r.t. the internal conformal weight plotted against $z_{2}$. The upper plot shows the conformal block at $\beta=4\pi$ in the low temperature expansion and the lower plot the block at $\beta=\pi$ in the high temperature expansion. For plotting convenience we set $h_{p}=h_{q}$ which gives a derivative of $f_{\text{cl.}}$ that is constant for all $h_{p}$. As we increase the order of the denominator polynomial in the Padé approximation, the fluctuations in $f_{\text{cl.}}$ outside of the convergence radius of the series expansion decrease. Moreover, multiple Padé approximants converge to the same value outside of the convergence radius. The convergence radius $\|y\|=1$ is reached at $z_{2}=1/6$ in the upper plot and $z_{2}=\frac{\log(2)}{4\pi}=0.0552$ in the lower plot. The series expansion approximated by the Padé approximants in this figure was truncated at order 10 in both $y$ and $Q$. We find that different Padé approximants for $f_{\text{cl.}}$ converge to the same value and yield $\frac{\partial f_{\text{cl.}}}{\partial h_{p,q}}>0$ in a finite region outside of the convergence radius $\|y\|=1$ of the series expansion of $f_{\text{cl.}}$. While this is not a formal proof, it does indicate that vacuum block dominance holds also outside of the convergence radius. ## 4 Discussion Let us briefly summarize the main points investigated in this publication. In the first part in sec. 2, we took a detailed look at monodromy methods for the computation of arbitrary conformal blocks on the torus as well as zero-point conformal blocks on the special higher genus Riemann surface relevant to the calculation of the entanglement entropy via the replica trick. We then applied the derived monodromy methods to the calculation of the entanglement entropy in sec. 3. We found that for holographic CFTs, the vacuum conformal block dominates the partition function on the replica surface and therefore the entanglement entropy takes on a universal form in agreement with the RT formula. The main advancement compared to previous work on entanglement entropy for finite size and finite temperature systems at large central charge is as follows. First of all, we clarified the role of the correlation function of twist operators on the torus compared to the partition function $Z_{n}$ on the replica surface ${\mathcal{R}}_{n}$. The twist correlator only gives the correct results for small intervals or low temperature, while the replica partition function $Z_{n}$ must give the entanglement entropy in all cases by construction. The reason why the twist correlator agrees with $Z_{n}$ for small intervals or low temperature is evident from the monodromy method. The conformal blocks of the twist correlator obey the same monodromy method as the zero-point conformal blocks for $Z_{n}$ apart from the set of allowed monodromy conditions, which for the twist correlator blocks is a subset of those of the replica partition function. Secondly, our derivation of the monodromy method is based solely on CFT techniques and thus provides a non-trivial check of both the RT formula as well as previous calculations of Barrella:2013wja ; Chen:2015kua using the monodromy method on the gravity side. Moreover, the monodromy method presented here is derived from first principles, further strengthening the heuristic arguments that were used to justify the monodromy method in Barrella:2013wja ; Chen:2015kua . The derivation from the CFT side furthermore allows for a determination of the restrictions on the operator content and OPE coefficients that must be obeyed by the CFT in question in order for the vacuum block to give the dominant contribution and the RT results for the entanglement entropy to be valid. Perhaps unsurprisingly, these restrictions are equivalent to those imposed in Hartman:2013mia for the entanglement entropy at zero temperature. That is, the entanglement entropy for any CFT with large central charge, sparse spectrum of light operators and at most exponentially growing OPE coefficients agrees with the RT formula not only at zero temperature as was already known from the results of Hartman:2013mia , but also at finite temperature and finite size. In addition, the same arguments that imply the universal form of the entanglement entropy in these CFTs also imply a universal form of the partition function at all temperatures, another feature of holographic CFT investigated also in Hartman:2014oaa . We close with a short outlook on possible future directions. One interesting direction is an investigation in full generality of the issue under which conditions the twist correlator yields the same results as the replica partition function $Z_{n}$. Previous work in the free fermion case Lokhande:2015zma ; Mukhi:2017rex indicates that for short intervals the twist correlator is equivalent to the replica partition function, a result which we can corroborate for large $c$ CFTs. Does this hold in general? Other directions can be found in further applications of the monodromy method. Since conformal blocks are the basic building blocks of any correlation function, the derivation of the monodromy methods from first principles lays the groundwork for a number of applications. In particular, the monodromy methods found in this paper may prove useful in deriving recursion relations in the dimensions of the exchanged operators for torus conformal blocks with better convergence properties than the known recursion relations in the central charge found in Cho:2017oxl . Moreover, the monodromy methods also pave the way to the study of other information theoretic quantities closely connected to the entanglement entropy such as entanglement negativity Calabrese:2012nk ; Calabrese:2012ew ; Calabrese:2014yza , symmetry resolved entanglement Goldstein:2017bua ; Bonsignori:2019naz ; Murciano:2020lqq ; Capizzi:2020 ; Zhao:2020qmn or entwinement Balasubramanian:2014sra ; Lin:2016fqk ; Balasubramanian:2016xho ; Balasubramanian:2018ajb ; Erdmenger:2019lzr for finite temperature states, providing opportunities for new insights into the physics of the dual AdS black holes. We hope to be able to report on further results in this direction in the future. ##### Acknowledgements I would like to thank Johanna Erdmenger and Christian Northe for useful discussions. I acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy through Würzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter - ct.qmat (EXC 2147, project-id 390858490). ## Appendix A Conventions for elliptic functions This appendix contains an overview over conventions and useful identities for the Weierstraß elliptic functions used in the rest of the publication (see e.g. NIST:DLMF for more details about these functions). All elliptic functions are defined for a lattice $\Lambda$ generated by the identifications $z\sim z+1$ and $z\sim z+\tau$. The Weierstraß elliptic functions $\wp(z),\zeta(z)$ and $\sigma(z)$ are defined by $\displaystyle\wp(z)$ $\displaystyle=-\zeta^{\prime}(z)=\frac{1}{z^{2}}+\sum_{(m,n)\neq(0,0)}\left(\frac{1}{(z+n+m\tau)^{2}}-\frac{1}{(n+m\tau)^{2}}\right),$ (75) $\displaystyle\zeta(z)$ $\displaystyle=\frac{\sigma^{\prime}(z)}{\sigma(z)}=\frac{1}{z}+\sum_{(m,n)\neq(0,0)}\left(\frac{1}{(z+n+m\tau)}-\frac{1}{(n+m\tau)}+\frac{z}{(n+m\tau)^{2}}\right),$ $\displaystyle\sigma(z)$ $\displaystyle=z\prod_{(m,n)\neq(0,0)}\left(1-\frac{z}{n+m\tau}\right)\exp\left(\frac{z}{n+m\tau}+\frac{1}{2}\frac{z^{2}}{(n+m\tau)^{2}}\right).$ $\wp(z)$ is a true elliptic (i.e. doubly periodic) function while $\zeta(z)$ and $\sigma(z)$ are quasiperiodic: $\displaystyle\wp(z+1)=\wp(z+\tau)=\wp(z)$ (76) $\displaystyle\zeta(z+1)=\zeta(z)+2\eta_{1},$ $\displaystyle\zeta(z+\tau)=\zeta(z)+2\eta_{3}$ $\displaystyle\sigma(z+1)=-\exp(2\eta_{1}(z+1/2))\sigma(z),$ $\displaystyle\sigma(z+\tau)=-\exp(2\eta_{3}(z+\tau/2))\sigma(z),$ where $\eta_{1}=\zeta(1/2)$ and $\eta_{3}=\zeta(\tau/2)=\tau\eta_{1}-\pi i$. Another useful definition of $\wp(z)$ and $\zeta(z)$ is given by $\displaystyle\wp(z)+2\eta_{1}=(2\pi i)^{2}\sum_{m=-\infty}^{\infty}\frac{Q^{m}u}{(u-Q^{m})^{2}}=\left(\frac{2\pi i}{\tau}\right)^{2}\sum_{m=-\infty}^{\infty}\frac{\tilde{Q}^{m}\tilde{u}}{(\tilde{u}-\tilde{Q}^{m})^{2}}$ (77) $\displaystyle\zeta(z)-2\eta_{1}z=i\pi\sum_{m=-\infty}^{\infty}\frac{Q^{m}+u}{Q^{m}-u}=-\frac{i\pi}{\tau}\sum_{m=-\infty}^{\infty}\frac{\tilde{Q}^{m}+\tilde{u}}{\tilde{Q}^{m}-\tilde{u}},$ where $u=e^{-2\pi iz},Q=e^{2\pi i\tau}$ and $\tilde{u}=e^{2\pi iz/\tau}$, $\tilde{Q}=e^{-2\pi i/\tau}$. ## Appendix B Recursion relations for torus conformal blocks For completeness, this appendix shows the recursion formulas for the two-point conformal blocks on the torus following as a special case from the general method derived in Cho:2017oxl . For details of the derivation, see Cho:2017oxl and also Hadasz:2009db for the one-point torus block. For the OPE block, the conformal block is given by the following recursion scheme $\displaystyle{\mathcal{F}}^{\text{OPE}}_{21,pq}(h_{p},h_{q},c)$ $\displaystyle=U^{\text{OPE}}(h_{p},h_{q},c)$ (78) $\displaystyle-\sum_{r\geq 2,s\geq 1}$ $\displaystyle\frac{\partial c_{rs}(h_{q})}{\partial h_{q}}Q^{rs}\frac{A^{c_{rs}(h_{q})}_{rs}{P^{rs}_{c_{rs}(h_{q})}\biggl{[}\begin{array}[]{c}h_{p}\\\ h_{q}+rs\end{array}\biggr{]}}{P^{rs}_{c_{rs}(h_{q})}\biggl{[}\begin{array}[]{c}h_{p}\\\ h_{q}\end{array}\biggr{]}}}{c-c_{rs}(h_{q})}{\mathcal{F}}^{\text{OPE}}_{21,pq}(h_{q},h_{q}+rs,c_{rs}(h_{q}))$ $\displaystyle-\sum_{r\geq 2,s\geq 1}$ $\displaystyle\frac{\partial c_{rs}(h_{p})}{\partial h_{p}}y^{rs}\frac{A^{c_{rs}(h_{p})}_{rs}{P^{rs}_{c_{rs}(h_{p})}\biggl{[}\begin{array}[]{c}h_{q}\\\ h_{q}\end{array}\biggr{]}}{P^{rs}_{c_{rs}(h_{p})}\biggl{[}\begin{array}[]{c}h_{2}\\\ h_{1}\end{array}\biggr{]}}}{c-c_{rs}(h_{p})}{\mathcal{F}}^{\text{OPE}}_{21,pq}(h_{p}+rs,h_{q},c_{rs}(h_{p}))$ where the fusion polynomials are given by ${P^{rs}_{c}\biggl{[}\begin{array}[]{c}h_{1}\\\ h_{2}\end{array}\biggr{]}}=\prod_{m=1-r,m\in 1-r+2{\mathbb{N}}}^{r-1}\prod_{n=1-s,n\in 1-s+2{\mathbb{N}}}^{s-1}\frac{\lambda_{1}+\lambda_{2}+mb+nb^{-1}}{2}\frac{\lambda_{1}-\lambda_{2}+mb+nb^{-1}}{2}$ (79) with $\lambda_{i}=\sqrt{(b+b^{-1})^{2}-4h_{i}}$ while the prefactor is $A^{c}_{rs}=\frac{1}{2}\prod_{\stackrel{{\scriptstyle(m,n)=(1-r,1-s)}}{{(m,n)\neq(0,0),(r,s)}}}^{(r,s)}(mb+nb^{-1})^{-1}.$ (80) $c_{rs}$ denotes the value of the central charge where degenerate representations of the Virasoro algebra appear, $\displaystyle c_{rs}(h)=1+6(b_{rs}(h)+b_{rs}^{-1}(h))^{2},$ (81) $\displaystyle b_{rs}(h)^{2}=\frac{rs-1+2h+\sqrt{(r-s)^{2}+4(rs-1)h+4h^{2}}}{1-r^{2}}.$ The $c$-regular part $U$ is given by $U^{\text{OPE}}(h_{p},h_{q},c)=\left[\prod_{n=2}^{\infty}\frac{1}{1-Q^{n}}\right]\sum_{i,j\geq 0}Q^{i}y^{j}\frac{s_{ij}(h_{q},h_{p},h_{q})(1-h_{p}-j)_{j}(h_{p}+h_{1}-h_{2})_{j}}{i!(2h_{q})_{i}j!(2h_{p})_{j}},$ (82) where we define the rising and falling Pochhammer symbols by $(a)_{n}=\prod_{k=0}^{n-1}(a+k)\hskip 56.9055pt(a)^{(n)}=\prod_{k=0}^{n-1}(a-k)$ (83) and $\displaystyle s_{ij}(h_{1},h_{2},h_{3})=\langle h_{1}\|(L_{-1}^{i})^{\dagger}{\mathcal{O}}_{h_{2}}(1)L_{-1}^{j}\|h_{3}\rangle$ (84) $\displaystyle=\left\\{\begin{aligned} &\sum_{p=0}^{i}\biggl{(}\begin{array}[]{c}i\\\ p\end{array}\biggr{)}(j)^{(p)}(2h_{3}+j-1)^{(p)}(h_{1}+h_{2}-h_{3}-(j-p))_{i-p}(h_{3}+h_{2}-h_{1})_{j-p}\leavevmode\nobreak\ ,\leavevmode\nobreak\ j\geq i\\\ &\sum_{p=0}^{j}\biggl{(}\begin{array}[]{c}j\\\ p\end{array}\biggr{)}(i)^{(p)}(2h_{1}+i-1)^{(p)}(h_{3}+h_{2}-h_{1}-(i-p))_{j-p}(h_{1}+h_{2}-h_{3})_{i-p}\leavevmode\nobreak\ ,\leavevmode\nobreak\ i\geq j\end{aligned}\right.$ The recursion formula for the projection block is given by 111111The authors of Cho:2017oxl also claim to have found a different recursion formula in the conformal dimensions of the exchanged primaries $h_{p,q}$ for a class of conformal blocks termed the “necklace blocks” which include the projection block considered here as a special case. We cannot confirm this claim since the formula presented in Cho:2017oxl agrees neither with the recursion relation in the central charge derived in Cho:2017oxl nor with an explicit calculation in the first few orders of the series expansion in $q_{1},q_{2}$ for the projection block. $\displaystyle{\mathcal{F}}^{\text{projection}}_{2p,1q}(h_{p},h_{q},c)=$ $\displaystyle U^{\text{projection}}(h_{p},h_{q},c)$ (85) $\displaystyle-\sum_{r\geq 2,s\geq 1}$ $\displaystyle\frac{\partial c_{rs}(h_{q})}{\partial h_{q}}q_{1}^{rs}\frac{A^{c_{rs}(h_{q})}_{rs}{P^{rs}_{c_{rs}(h_{q})}\biggl{[}\begin{array}[]{c}h_{p}\\\ h_{1}\end{array}\biggr{]}}{P^{rs}_{c_{rs}(h_{q})}\biggl{[}\begin{array}[]{c}h_{p}\\\ h_{2}\end{array}\biggr{]}}}{c-c_{rs}(h_{q})}{\mathcal{F}}^{\text{projection}}_{2p,1q}(h_{p},h_{q}+rs,c_{rs}(h_{q}))$ $\displaystyle-\sum_{r\geq 2,s\geq 1}$ $\displaystyle\frac{\partial c_{rs}(h_{p})}{\partial h_{p}}q_{2}^{rs}\frac{A^{c_{rs}(h_{p})}_{rs}{P^{rs}_{c_{rs}(h_{p})}\biggl{[}\begin{array}[]{c}h_{q}\\\ h_{1}\end{array}\biggr{]}}{P^{rs}_{c_{rs}(h_{p})}\biggl{[}\begin{array}[]{c}h_{q}\\\ h_{2}\end{array}\biggr{]}}}{c-c_{rs}(h_{p})}{\mathcal{F}}^{\text{projection}}_{2p,1q}(h_{p}+rs,h_{q},c_{rs}(h_{p})),$ where the $c$-regular part is given by $U^{\text{projection}}(h_{p},h_{q},c)=\left[\prod_{n=2}^{\infty}\frac{1}{1-Q^{n}}\right]\sum_{i,j\geq 0}q_{1}^{i}q_{2}^{j}\frac{s_{ij}(h_{q},h_{2},h_{q})s_{ji}(h_{p},h_{1},h_{q})}{i!(2h_{q})_{i}j!(2h_{p})_{j}}.$ (86) By explicit calculation, it is easy to check in the first few orders of the series expansion that for $h_{1}=h_{2}$ and $h_{p,q}=\gamma c$, the limits $\lim_{\gamma\to 0}$ and $\lim_{c\to\infty}$ of the OPE block commute. We have checked this up to fourth order in $y,Q$. A more convenient proof is possible with a recursion relation in the conformal weights of the exchanged operators $h_{p,q}$, as derived for the conformal block on the plane in Zamolodchikov1987 . In fact, the singular parts proportional to $\sim 1/(h_{p,q}-h_{rs})$ of such a recursion relation are proportional to the singular parts $\sim 1/(c-c_{rs}(h_{p,q}))$ of the above recursion relations in $c$ Cho:2017oxl . Using these known singular parts, one can show that the limits $\gamma\to 0$ and $c\to\infty$ commute for the singular parts of this recursion relation to all orders, assuming the above conditions on $h_{1,2,p,q}$. Together, these calculations provide some evidence that the semiclassical limit is well-defined for the vacuum block on the torus derived in sec. 3. 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# Green function estimates on complements of low-dimensional uniformly rectifiable sets Guy David Guy David. Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France guy.david@universite-paris- saclay.fr , Joseph Feneuil Joseph Feneuil. Université Paris-Saclay, CNRS, Laboratoire de mathématiques d’Orsay, 91405, Orsay, France <EMAIL_ADDRESS>and Svitlana Mayboroda Svitlana Mayboroda. School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA <EMAIL_ADDRESS> ###### Abstract. It has been recently established in [DM2] that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship” degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators $L_{\beta,\gamma}=-\operatorname{div}D^{d+1+\gamma-n}\nabla$ associated to a domain $\Omega\subset\mathbb{R}^{n}$ with a uniformly rectifiable boundary $\Gamma$ of dimension $d<n-1$, the now usual distance to the boundary $D=D_{\beta}$ given by $D_{\beta}(X)^{-\beta}=\int_{\Gamma}\|X-y\|^{-d-\beta}d\sigma(y)$ for $X\in\Omega$, where $\beta>0$ and $\gamma\in(-1,1)$. In this paper we show that the Green function $G$ for $L_{\beta,\gamma}$, with pole at infinity, is well approximated by multiples of $D^{1-\gamma}$, in the sense that the function $\big{\|}D\nabla\big{(}\ln\big{(}\frac{G}{D^{1-\gamma}}\big{)}\big{)}\big{\|}^{2}$ satisfies a Carleson measure estimate on $\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical” distance function from [DEM]. Résumé. Dans [DM2] il est démontré que pour les domaines à bord uniformément rectifiable, la fonction de Green vérifie des estimations faibles de bonne approximation par des fonctions affines, avec une réciproque vraie dans certains cas encourageants. Ici on part de la rectifiabilité uniforme et on démontre les estimations fortes naturelles d’approximation de la fonction de Green, et aussi des solutions, par des applications affines (ou, de manière équivalente, des multiples de la distance au bord adoucie). L’étude inclut les analogues naturels du Laplacien dans les domaine dont la frontière est de grande co-dimension. On considère les opérateurs elliptiques $L_{\beta,\gamma}=\operatorname{div}D^{d+1+\gamma-n}\nabla$ associés à un domaine $\Omega\subset\mathbb{R}^{n}$ dont le bord $\Gamma$ est Ahlfors régulier et uniformément rectifiable de dimension $d<n-1$ et à la distance au bord maintenant usuelle $D=D_{\beta}$ définie par $D_{\beta}(X)^{-\beta}=\int_{\Gamma}\|X-y\|^{-d-\beta}d\sigma(y)$ pour $X\in\Omega$, où $\beta>0$ et $\gamma\in(-1,1)$ sont des paramètres et $\sigma$ une mesure Ahlfors régulière sur $\Gamma$. Les auteurs ont montré précédemment que la mesure elliptique associée à $L_{\beta,\gamma}$ est bien définie et est mutuellement absolument continue par rapport à $\sigma$, avec un poids de $A_{\infty}$. Ici on démontre que la fonction de Green $G$ avec pôle à l’infini associée à $L_{\beta,\gamma}$ est bien approchée par les multiples de $D$, au sens où la fonction $\big{\|}D\nabla\big{(}\ln\big{(}\frac{G}{D^{1-\gamma}}\big{)}\big{)}\big{\|}^{2}$ vérifie une condition de Carleson sur $\Omega$. Ces nouvelles estimations sont différentes en nature. Les estimations de [DM2] reposaient sur des arguments de compacité; ici on a besoin d’estimations plus précises, obtenues par intégration par parties et en utilisant les propriétés algébriques de la fonction $D_{\alpha}$ dans le cas“magique” de [DEM]. David was partially supported by the European Community H2020 grant GHAIA 777822, and the Simons Foundation grant 601941, GD. Mayboroda was supported in part by the NSF grant DMS 1839077 and the Simons foundation grant 563916, SM Key words/Mots clés. Uniformly rectifiable sets, degenerate elliptic operators, Estimates on Green functions, harmonic measure in higher codimension. AMS classification: 42B37, 31B25, 35J25, 35J70. ###### Contents 1. 1 Introduction 2. 2 Definitions and anterior results 3. 3 Proof of Theorem 2.21 ## 1\. Introduction Rectifiable sets are an important notion in geometric measure theory and the calculus of variation, in particular because the sets that minimize an energy often enter this category. In the past decades, many mathematicians worked on finding characterizations of rectifiability by properties apparently unrelated to geometric measure theory. In the early 90’s, the quantifiable version - uniform rectifiability - was introduced in [DS1, DS2] along with many characterizations in terms of geometry (such as big pieces of Lipschitz images, or using Peter Jones’ $\beta$ numbers) and in terms of singular integrals. Later, it was observed that uniformly rectifiable sets may be the right extension of Lipschitz graphs for elliptic boundary value problems, that is, if $\Omega\subset\mathbb{R}^{n}$ is an open set with uniformly rectifiable boundary and $\Omega$ provides enough access its the boundary, then we can control the oscillations of the harmonic functions in $\Omega$. It is even more noteworthy that a criterion for rectifiability can be obtained using harmonic functions. Indeed, Hofmann, Martell, and Uriarte-Tuero proved in [HM1, HMU] that under some conditions regarding the access to the boundary, $\partial\Omega$ is uniformly rectifiable if and only if the harmonic measure on $\partial\Omega$ is absolutely continuous in a quantitative way - called $A_{\infty}$ \- with respect to the surface measure (see also [AHMNT]). The optimal topological conditions in this regard have been identified in [AHMMT]. These were accompanied but a rich array of beautiful and difficult alternative characterizations, exploring Carleson estimates for the solutions, behavior of the singular integral operators, extensions to more general elliptic operators, to mention just a few. Here we do not aim to provide a survey of the related literature; the reader can consult, e.g., [DFM4] for a more detailed presentation of the literature. A weakness of the above theory is the fact that the harmonic measure on $\Gamma$ only makes sense when $\Gamma\subset\mathbb{R}^{n}$ is of dimension $d>n-2$, because lower dimensional sets have probability zero to be hit by a Brownian motion. As a consequence, rectifiability, a notion that exists for all integer dimensions, can only be characterized by means of the harmonic measure for sets $\Gamma\subset\mathbb{R}^{n}$ of dimension $d=n-1$. To overcome this obstacle, the authors of the present article developed a theory of degenerate elliptic operators. The idea was to define a ‘harmonic measure’ on a set $\Gamma$ of dimension $d<n-1$ by replacing the Laplacian by an operator on $\Omega:=\mathbb{R}^{n}\setminus\Gamma$ in the form $L=-\operatorname{div}w(x)\nabla$, where $w(x)$ goes to infinity at an appropriate rate when $x$ is approaching $\Gamma$, so that the corresponding ‘Brownian motion’ is attracted by $\Gamma$ and hits the boundary with probability one. The articles [DFM2, DFM5] set the elliptic theory for this. In particular an appropriate elliptic measure is constructed for a large class of sets of any dimension $d<n$ (or even mixed dimensions) and operators $L$ as above, with the usual nice properties such as the non-degeneracy and doubling properties for the harmonic measure, the Harnack inequality and the comparison principle for solutions, and estimates for the change of poles. Then we tested the relevancy of our new elliptic measure. Dahlberg proved in [Da] that the classical harmonic measure is absolutely continuous with respect to the surface measure, and even $A_{\infty}$, whenever the boundary is Lipschitz. In [DFM3] we assumed that $\Gamma$ is the graph of a Lipschitz function $\varphi:\mathbb{R}^{d}\to\mathbb{R}^{n-d}$ with small Lipschitz constant, and proved the same $A_{\infty}$ property for correctly chosen operators. Yet we had to be careful about the operator we picked because as showed in [DFM4], based on counterexamples from [CFK, MM], not every operator $L$ will work. We wanted an explicit operator, which could be constructed by a single method for any set of dimension $d$. We set our choice on (1.1) $L_{\beta,\gamma}:=-\operatorname{div}(D_{\beta})^{d+1+\gamma-n}\nabla,$ where $\beta>0$, $\gamma\in(-1,1)$, and $D_{\beta}$ is defined on $\Omega$ as (1.2) $D_{\beta}(X):=\left(\int_{\Gamma}\|X-y\|^{-d-\beta}d\sigma(y)\right)^{-1/\beta}.$ The quantity $D_{\beta}$ is equivalent to the distance to the boundary, i.e., there exists $C>0$ such that (1.3) $C^{-1}\,\mathrm{dist}(X,\Gamma)\leq D_{\beta}\leq C\,\mathrm{dist}(X,\Gamma)\qquad\text{ for }X\in\Omega,$ but the advantage of $D_{\beta}$ over $\,\mathrm{dist}(.,\Gamma)$ is to always be smooth in a certain quantitative way. When $\gamma=0$, the level of the degeneracy $D_{\beta}^{d+1-n}$ in the coefficients of the operator $L_{\beta,0}:=-\operatorname{div}(D_{\beta})^{d+1-n}\nabla$ makes it a perfect analogue of the Laplacian for the sets with a $d$-dimensional boundary when $d<n-1$. In particular, we proved that the harmonic measure associated to $L_{\beta,\gamma}$ is $A_{\infty}$ with respect to the $d$-dimensional Hausdorff measure. Moreover, recently, in two distinct papers [DM1, Fen1] we extended this result to the more general case where $\Gamma\subset\mathbb{R}^{n}$ is uniformly rectifiable. Studying these operators and the dimension of the support of the corresponding elliptic measure, we were naturally drawn to introducing a parameter $\gamma$ which seemingly unbalances the situation. Besides, for $d=n-1$, the operator $L_{\beta,\gamma}:=-\operatorname{div}(D_{\beta})^{\gamma}\nabla$ is the celebrated Caffarelli-Silvester extension of the fractional Laplacian operator (cf. [CS]). However, we were surprised to realize that the argument in [DFM3] extends to all $\gamma\in(-1,1)$ rather simply, and the generalization to uniformly rectifiable set is stated for any $\gamma\in(-1,1)$ in [Fen1]. The proof in [DM1] relies on geometric arguments, such as corona decompositions and the construction of sawtooth domains, and an extrapolation argument, which allow one to reduce to the case of small Lipschitz graphs. The proof of [Fen1] is substantially simpler and more direct, and is based on a trick unique to the case where the dimension of $\Gamma$ is at most $n-2$. ###### Theorem 1.4 ([DM1, Fen1]). Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly rectifiable set with $d<n-1$, and let $\sigma$ be an Ahlfors regular measure that satisfies (2.1). Take $\beta>0$, $\gamma\in(-1,1)$, define $L_{\beta,\gamma}$ as in (1.1), and construct the associated harmonic measure $\omega_{\beta,\gamma}^{X}$ as in Definition 2.20. Then $\omega_{\beta,\gamma}^{X}$ is $A_{\infty}$-absolutely continuous with respect to $\sigma$. This means that for every choice of $\epsilon\in(0,1)$, there exists $\delta\in(0,1)$, that depends only on $C_{\sigma}$, $C_{0}$, $\epsilon$, $n$, $d$, $\beta$, and $\gamma$, such that for each choice of $x\in\Gamma$, $r>0$, a Borel set $E\subset B(x,r)\cap\Gamma$, and a corkscrew point $X=A_{x,r}$ as in (2.3), (1.5) $\frac{\sigma(E)}{\sigma(B(x,r)\cap\Gamma)}<\delta\Rightarrow\frac{\omega^{X}_{\Omega,L}(E)}{\omega^{X}_{\Omega,L}(B(x,r)\cap\Gamma)}<\epsilon.$ It is known that in the present context where all our measures are doubling, the $A_{\infty}$ condition also implies that under the assumptions of (1.5), (1.6) $\frac{\omega^{X}_{\Omega,L}(E)}{\omega^{X}_{\Omega,L}(B(x,r)\cap\Gamma)}<\delta\Rightarrow\frac{\sigma(E)}{\sigma(B(x,r)\cap\Gamma)}<\epsilon.$ Observe that in the previous theorem, contrary to the case of co-dimension 1, i.e. when $d=n-1$, we don’t assume any topological condition. And that is perfectly natural, since the domain $\Omega:=\mathbb{R}^{n}\setminus\Gamma$ has a lot of paths and ample access to the boundary (see Lemmas 2.1 and 11.6 in [DFM2]). The next big objective would be to prove the reverse implication, meaning that if the harmonic measure on $\Gamma$ is $A_{\infty}$ with respect to the Hausdorff measure, then $\Gamma$ is uniformly rectifiable. Unfortunately (and surprisingly) this fails brutally when $d+\beta=n+2$ (see [DEM]), and although we expect this to be the only exception, the methods used to prove the converse in co-dimension 1 do not appear to be adaptable to the higher codimension case. The purpose of the present paper is to provide different estimates on the harmonic functions, which we hope will ultimately furnish one side of the desired criterion. Indeed, as established in [DM2], some weak bounds on the Green function are equivalent to the uniform rectifiability even in lower dimensional settings. The general idea is that instead of trying to characterize rectifiable sets using the harmonic measure, we would do so using the property that the Green function behaves like a distance to the boundary. This is not a surprising idea, as the Green functions and harmonic measure are deeply connected, and are both prominent in the analysis of the free boundary problems, and the proof of the properties of the harmonic measure in [DFM2, DFM5] heavily relies on a comparison between the harmonic measure and Green functions. Yet, the results we are about to prove are not known in the “classical” setting of domains with an $n-1$ dimensional boundary, except for a few simple situations and certainly not in the generality of the uniformly rectifiable sets which we attack in this paper. We will use the Green function with a pole at infinity, which is constructed in [DEM, Definition 6.2, Lemma 6.5] with the following properties. ###### Proposition 1.7 ([DEM]). Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly rectifiable set with $d<n-1$. Take $\beta>0$, $\gamma\in(-1,1)$, and construct $L_{\beta,\gamma}$ on $\Omega:=\mathbb{R}^{n}\setminus\Gamma$ as in (1.1). There exists a continuous function $G^{\infty}=G_{\beta,\gamma}^{\infty}$ on $\mathbb{R}^{n}$ such that 1. (i) $G^{\infty}>0$ on $\Omega$, 2. (ii) $G^{\infty}=0$ on $\Gamma$, 3. (iii) there exists a positive Borel measure $\omega_{\infty}=\omega^{\infty}_{\beta,\gamma}$ on $\Gamma$ such that $\int_{\Omega}\nabla G^{\infty}\cdot\nabla\varphi\,D_{\beta}^{d+1-n+\gamma}dx=\int_{\Gamma}\varphi(y)d\omega^{\infty}(y)\qquad\text{ for }\varphi\in C^{\infty}_{0}(\mathbb{R}^{n}).$ We call $G^{\infty}$ the Green function with pole at infinity, and $G^{\infty}$ is unique up to multiplication by a scalar constant. The Green function with pole at infinity for the Laplacian in $\mathbb{R}^{n}_{+}=\\{(x,t)\in\mathbb{R}^{n-1}\times(0,+\infty)\\}$ is $G^{\infty}(x,t)=t$. The one for the operator $-\operatorname{div}\|t\|^{d+1-n}$ in $\mathbb{R}^{n}\setminus\mathbb{R}^{d}=\\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}^{n-d},\,t\neq 0\\}$ is $G^{\infty}(x,t)=\|t\|$. There is a third case where $G^{\infty}$ can be computed, which is when $\Gamma$ is a $d$-Ahlfors regular set with $d<n-2$, and we choose the specific operator $L_{n-d-2,0}$; then one can easily compute that $G^{\infty}=D_{n-d-2}$. In [DEM] this is called the magic case. The rough idea of what follows is that, when $\gamma=0$, a good Green function at infinity should behave a bit like the distance to the boundary $\Gamma$, or like $D_{\beta}$ which is its smooth substitute. When $\gamma\neq 0$, similar homogeneity considerations lead us to expect that in the good cases, $G^{\infty}$ will behave like $D_{\beta}^{1-\gamma}$ instead. Here is our main result. ###### Theorem 1.8. Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly rectifiable set with $d<n-1$, and let $\sigma$ be an Ahlfors regular measure on $\Gamma$ that satisfies (2.1). For $\beta>0$ and $\gamma\in(-1,1)$, let $L_{\beta,\gamma}$ be as in (1.1) and define $G^{\infty}_{\beta,\gamma}$ as in Proposition 1.7. Then for any $\alpha>0$, there exists $C>0$ that depends only on $C_{\sigma}$, $C_{0}$, $\alpha$, $\beta$, $\gamma$, $d$, $n$ such that for any ball $B:=B(x,r)$ centered on $\Gamma$, one has (1.9) $\int_{B}\left\|\nabla\ln\left(\frac{G^{\infty}_{\beta,\gamma}}{D_{\alpha}^{1-\gamma}}\right)\right\|^{2}\,D_{\alpha}^{d+2-n}\,dX\leq C\sigma(B).$ Remarks: * • Remember that $G^{\infty}_{\beta,\gamma}$ is only defined up to a constant. Yet, the above theorem makes perfect sense, because the value of left-hand side of (1.9) does not change when we replace $G^{\infty}_{\beta,\gamma}$ by $KG^{\infty}_{\beta,\gamma}$, where $K$ is a positive constant. * • Theorem 1.8 is just the application of Theorem 2.21 to the Green function at infinity: we shall generalize the estimate of Theorem 1.8 to a large class of solutions that is interesting by itself. However, we decided to highlight the above statement, which was our true purpose in our search for characterizations of rectifiability. * • Here we decided to use the same measure $\sigma$ for the definitions of $L_{\beta,\gamma}$ and $D_{\alpha}$, but a minor modification of the proof would allow us to take two different Ahlfors regular measures $\sigma$ and $\widetilde{\sigma}$ to define $L_{\beta,\gamma}$ and and $D_{\alpha}$. * • As we have pointed out above, these results are not known for $d=n-1$ in the generality of the uniformly rectifiable sets. In the half-space one can see somewhat similar estimates in [FKP] and the bounds on the second derivatives of the Green function for a special class of operators were, in disguise, obtained in [HMT]. However, it is not clear how to deduce from [HMT] a co- dimension 1 analogue of (1.9), directly or using interpolation. In the present paper we only show that good geometric properties (the uniform rectifiability of $\Gamma$) imply precise approximation properties of $G^{\infty}$ by $D_{\alpha}$, and the proof will rely heavily on the $A_{\infty}$ property of the harmonic measure and some intermediate results in [Fen1] concerning the uniform rectifiability of $\Gamma$. We do not address the issue of the converse in this paper. Yet there are good reasons to believe that it may be easier to prove than for the absolute continuity of the harmonic measure. In a parallel paper [DM2], the authors study a less precise (weaker) approximation property of the Green function, and show that in some cases (but where $d>n-2$) it already implies the uniform rectifiability of $\Gamma$, while in the case of the present paper it implies another strange property of potentials defined on $\Gamma$. But we did not manage to prove that this strange property is impossible to obtain when $d+\alpha\neq n+2$. The reader should be aware that even though we think of the Green function estimate (1.9) as a possible alternative to the $A_{\infty}$ absolute continuity of the harmonic measure for the characterization of uniform rectifiability, we already know that in the general context of elliptic operator in the form $L=-\operatorname{div}A\nabla$, the $A_{\infty}$ absolute continuity (1.5) - or (1.6) - is not equivalent to (1.9). A counterexample in $\mathbb{R}^{n}_{+}$, that can be extended to any codimensions using the construction (4.6) in [DFM4], is given in the next lines. Denote the running point in $\mathbb{R}^{n}_{+}$ as $(x,r)\in\mathbb{R}^{n-1}\times(0,+\infty)$, and observe that in this case $D_{\alpha}(x,r)=c_{\alpha}r$. We set $b(r):=1/(2+\cos(r))$ and we construct $L_{b}:=-\operatorname{div}b(r)\nabla$. By uniqueness of the Green function with pole at infinity (Proposition 1.7), we have $G^{\infty}(x,r)=\int_{0}^{r}\frac{ds}{b(s)}=2r+\sin(r).$ On one hand, $r\leq G^{\infty}(x,r)\leq 3r$ for all $(x,r)\in\mathbb{R}^{n}_{+}$, and that is enough, by using a comparison principle with the harmonic measure, to show that the harmonic measure associated to $L$ is equivalent to the surface measure on $\mathbb{R}^{n-1}$ (see for instance Theorem 1.17 in [Fen2] for details), hence it is $A_{\infty}$ absolutely continuous with respect to the surface measure. On the other hand, we have $\begin{split}\left\|\nabla\ln\Big{(}\frac{G_{\infty}}{D_{\alpha}}\Big{)}\right\|=\left\|\frac{\nabla G^{\infty}}{G^{\infty}}-\frac{\nabla D_{\alpha}}{D_{\alpha}}\right\|=\left\|\frac{2+\cos(r)}{2r+\sin(r)}-\frac{1}{r}\right\|\geq\frac{1}{3r}\left\|\cos(r)-\frac{\sin(r)}{r}\right\|.\end{split}$ That is, for any ball $B(x,R)\subset\mathbb{R}^{n}$ centered on the boundary $\mathbb{R}^{n-1}$, we have $\begin{split}\int_{B(x,R)}\left\|\nabla\ln\Big{(}\frac{G_{\infty}}{D_{\alpha}}\Big{)}\right\|^{2}D_{\alpha}\,dX\geq c_{n}R^{n-1}\int_{0}^{R/2}\left\|\cos(r)-\frac{\sin(r)}{r}\right\|^{2}\frac{dr}{r}\geq c^{\prime}_{n}R^{n-1}\ln(R),\end{split}$ when $R$ is large (e.g. $R\geq 100$), and where $c_{n},c^{\prime}_{n}$ are constants that depends only on $n$. As a consequence, (1.9) fails for the elliptic operator $L_{b}$. This highlights the difference between Green function and harmonic measure estimates for general elliptic operators. Coming back to the Green function, the reader might wonder about the comparison with the results in [DM2]. It is not easy to describe the results in [DM2] avoiding technicalities, but roughly speaking, we only proved there the weak statement that the set where the Green function behaves like a distance to the boundary is a Carleson-prevalent set. This carries the structural information saying that there are a lot of points and a lot of scales where the desired estimate is true, but it carries no norm control. Respectively, the methods heavily rely on compactness arguments. In the present paper the arguments and the results are completely different, aiming at a strong norm control in the form of (1.9) and using a new idea that the “magical” distance function identified in [DEM], being an explicit solution to a certain PDE, can be effectively used in the integration-by-parts arguments (cf. [Fen1]). A good comparison is a familiar to many experts integration by parts with the weight $t$ which disappears when the Laplacian hits $t$, although the details in our case are necessarily considerably more involved. In the next section, we shall give the definitions that we skipped up to now to lighten the introduction, such as the definition of Ahlfors regular and uniformly rectifiable sets. We also introduce the results from [Fen1] which we will rely upon. The remainder of the article will be devoted to the proof of our main result. We shall use the notation $A(x)\lesssim B(x)$ when $A(x)\leq CB(x)$ and $C$ is a constant whose dependence into the various parameters will be either recalled or obvious from context. We also write $A(x)\approx B(x)$ when $A(x)\lesssim B(x)$ and $A(x)\gtrsim B(x)$. ## 2\. Definitions and anterior results For the rest of the article, we take $\Gamma\subset\mathbb{R}^{n}$ and $\Omega:=\mathbb{R}^{n}\setminus\Gamma$. We assume that $\Gamma$ is a $d$-Ahlfors regular set with $d<n-1$, that is $\Gamma$ is closed and there exists a measure $\sigma$ supported on $\Gamma$ and a constant $C_{\sigma}\geq 1$ such that (2.1) $C_{\sigma}^{-1}r^{d}\leq\sigma(B(x,r))\leq C_{\sigma}r^{d}\qquad\text{ for }x\in\Gamma,\ r>0.$ It is known that if the above property (2.1) is true for some measure $\sigma$, then it is also true when $\sigma$ is replaced by $\mathcal{H}^{d}\|_{\Gamma}$ \- the $d$-dimensional Hausdorff measure restricted to $\Gamma$. The Ahlfors regularity of $\Gamma$ and the low dimension $d<n-1$ are sufficient conditions to obtain the aforementioned equivalence (1.3). Indeed, Lemma 5.1 in [DFM3] gives us that (2.2) $C^{-1}\,\mathrm{dist}(X,\Gamma)\leq D_{\beta}\leq C\,\mathrm{dist}(X,\Gamma)\qquad\text{ for }X\in\Omega,$ where the constant $C>0$ above depends only on $\beta>0$, $C_{\sigma}$ and $n-d>1$. Moreover, Lemma 11.6 in [DFM2] entails the existence of a constant $C$ that depends only on $C_{\sigma}$ and $n-d>1$ such that for any $x\in\Gamma$ and $r>0$, we can find a point $A_{x,r}$ such that (2.3) $C^{-1}r\leq\,\mathrm{dist}(A_{x,r},\Gamma)\leq\|A_{x,r}-x\|\leq Cr.$ In other words, when $\Gamma$ is $d$-Ahlfors regular with $d<n-1$, its complement automatically satisfies the interior corkscrew condition. We shall also assume that $\Gamma$ is uniformly rectifiable. Equivalent definitions of uniform rectifiability were given in [DS1, DS2], and the reader may use their preferred one, but since we will only use the uniform rectifiability of $\Gamma$ via results from [Fen1] that rely on the summability properties of Tolsa’s $\alpha$-numbers, we will use these properties as a definition of uniform rectifiability. We need some notation first. We denote by $\Xi$ the set of affine $d$-dimensional planes in $\mathbb{R}^{n}$. Each plane $P\in\Xi$ is associated with a measure $\mu_{P}$, which is the restriction to $P$ of the $d$-dimensional Hausdorff measure (i.e. $\mu_{P}$ is the Lebesgue measure on the plane). A flat measure is a measure $\mu$ that can be written $\mu=c\mu_{P}$ where $c$ is a positive constant and $P\in\Xi$. The set of flat measure is called $\mathcal{F}$. We need the following variant of Wasserstein distances to quantify the difference between two measures, and then measure how far $\sigma$ is from flat measures. ###### Definition 2.4. For $x\in\mathbb{R}^{n}$ and $r>0$, denote by $Lip(x,r)$ the set of $1$-Lipschitz functions $f$ supported in $\overline{B(x,r)}$, that is the set of functions $f:\mathbb{R}^{n}\to\mathbb{R}$ such that $f(y)=0$ for $y\in\mathbb{R}^{n}\setminus B(x,r)$ and $\|f(y)-f(z)\|\leq\|y-z\|$ for $y,z\in\mathbb{R}^{n}$. The normalized Wasserstein distance in B(x,r) between two measures $\sigma$ and $\mu$ is (2.5) $\,\mathrm{dist}_{x,r}(\mu,\sigma)=r^{-d-1}\sup_{f\in Lip(x,r)}\Big{\|}\int fd\sigma-\int fd\mu\Big{\|}.$ The distance to flat measures is then defined by (2.6) $\alpha_{\sigma}(x,r)=\inf_{\mu\in{\mathcal{F}}}\,\mathrm{dist}_{x,r}(\mu,\sigma).$ One can easily check that when (2.1) holds, the quantity $\alpha_{\sigma}$ is uniformly bounded, i.e. there exists a constant $C$ that depends only on $d$, $n$, and $C_{\sigma}$ such that $\alpha_{\sigma}(x,r)\leq C$ for $x\in\Gamma$ and $r>0$. Let $\Gamma$ be a $d$-Ahlfors regular set, and $\sigma$ a measure that satisfies (2.1). Tolsa’s characterization of uniform rectifiability, Theorem 1.2 in [Tol], is as follows111Tolsa’s characterization of rectifiability in [Tol] is given with dyadic cubes but one can easily check that our bound (2.7) is equivalent to Tolsa’s one.: $\Gamma$ is uniformly rectifiable if and only if there exists a constant $C_{0}>0$ such that (2.7) $\int_{0}^{r}\int_{\Gamma\cap B(x,r)}\|\alpha_{\sigma}(y,s)\|^{2}\,d\sigma(y)\,\frac{ds}{s}\leq C_{0}\sigma(B(x,r))\qquad\text{ for }x\in\Gamma\text{ and }r>0.$ Here we will only use the fact that (2.7) holds when $\Gamma$ is uniformly rectifiable. That is, we will only use (2.7) and do not need to know other properties of uniformly rectifiable sets. The property (2.7) will allow us to obtain additional estimates on the smooth distance $D_{\beta}$. The presentation of those bounds will be easier after the following definition. ###### Definition 2.8. Let the function $f$ be defined on $\Omega$. We say that $f$ satisfies the Carleson measure condition when $f\in L^{\infty}(\Omega)$ and $\|f(X)\|^{2}\,\mathrm{dist}(X,\Gamma)^{d-n}dX$ is a Carleson measure, that is, (2.9) $\int_{B(x,r)}\|f(X)\|^{2}\,\mathrm{dist}(X,\Gamma)^{d-n}dX\leq C\sigma(B(x,r))$ for $x\in\Gamma$ and $r>0$, with a constant $C$ that does not depend on $x$ or $r$. Thus this is actually a quadratic Carleson condition. For short, we shall write $f\in CM$, or $f\in CM(C)$ when we want to refer to the constant in (2.9). Due to (2.2), we can replace $\,\mathrm{dist}(X,\Gamma)^{d-n}$ with $D_{\beta}^{d-n}(X)$, and even choose $\beta$ to fit our purposes; we shall often do this without additional explanations. We shall rely strongly on Lemma 1.24 in [Fen1], which says the following. ###### Lemma 2.10. Let $\Gamma$ be uniformly rectifiable, so that (2.1) and (2.7) hold. Let $\beta>0$. Then there exist a scalar function $b$ and a vector function $\mathcal{V}$, both defined on $\Omega$, such that (2.11) $\int_{\Gamma}\|X-y\|^{-n}(X-y)d\sigma(y)=(b\nabla D_{\beta}+\mathcal{V})D_{\beta}^{d+1-n}\qquad\text{ for }X\in\Omega$ and a constant $C_{1}$ that depends only on $C_{\sigma}$, $C_{0}$, $\beta$, $n$, and $d$, such that (2.12) $C_{1}^{-1}\leq b\leq C_{1},$ (2.13) $D_{\beta}\nabla b\in CM(C_{1}),$ (2.14) $\|\mathcal{V}\|\leq C_{1},$ and (2.15) $\mathcal{V}\in CM(C_{1}).$ Observe that the left-hand side of (2.11) is divergence free. So if we use (2.11) and we write the divergence free condition in weak terms, we obtain (2.16) $\int_{\Omega}(b\nabla D_{\beta}+\mathcal{V})\cdot\nabla\varphi\,D_{\beta}^{d+1-n}\,dX=0\qquad\text{ for }\varphi\in C^{\infty}_{0}(\Omega).$ We also need the following, which is Lemma 1.26 in [Fen1]. ###### Lemma 2.17. Let $\Gamma$ be uniformly rectifiable, i.e., assume that (2.1) and (2.7) hold. Let $\alpha,\beta>0$. Then $D_{\alpha}\nabla[D_{\beta}/D_{\alpha}]$ satisfies the Carleson measure condition with a constant that depends only on $C_{\sigma}$, $C_{0}$, $\alpha$, $\beta$, $n$, and $d$. We are now finished with the geometric background and turn to the elliptic theory. Pick $\beta>0$ and $\gamma\in(-1,1)$. We whall use the operator $L_{\beta,\gamma}$ constructed in (1.1); hence $L_{\beta,\gamma}$ enters the scope of the theory developed in [DFM5], and in particular there is an elliptic measure $\omega^{X}$ which we shall describe now. We first need a Hilbert space $W_{\gamma}$, which is the same as in [DFM5], Definition 3.1, but is more easily defined as (2.18) $W_{\gamma}=\big{\\{}u\in L^{1}_{loc}(\mathbb{R}^{n}),\\|u\\|_{\gamma}:=\int_{\Omega}\|\nabla u(X)\|^{2}\,\mathrm{dist}(X,\Gamma)^{d+1+\gamma-n}dX<+\infty\big{\\}};$ the equivalence between the two definition is proved as in [DFM2, Lemma 3.3 and Lemma 5.21]. Each $f\in W_{\gamma}$ has a trace on $\Gamma$, which lies in a corresponding Sobolev space $H_{\gamma}$ (which is equal to $H^{1/2}(\Gamma)$ when $\gamma=0$); then we denote by $W_{\gamma,0}$ the set of functions in $W_{\gamma}$ with zero trace; $W_{\gamma,0}$ is also the completion of $C^{\infty}_{0}(\Omega)$ under the norm $\\|.\\|_{\gamma}$. And for any open set $E\subset\mathbb{R}^{n}$, we write that $u\in W_{\gamma}(E)$ [respectively, $u\in W_{\gamma,0}(E)$] if $u\varphi\in W_{\gamma}$ [respectively, $u\varphi\in W_{\gamma,0}$] for any $\varphi\in C^{\infty}_{0}(E)$. Then there is a notion of weak solution for $L_{\beta,\gamma}=0$, which the reader may also find in [DFM5], such that in particular (2.19) $\int_{\Omega}(\nabla u\cdot\nabla\varphi)\,D_{\beta}^{d+1+\gamma-n}=0\qquad\text{ for }\varphi\in C^{\infty}_{0}(\Omega).$ when $u$ is a weak solution for $L_{\beta,\gamma}=0$ (in $\Omega$). Here and below, we remove the variable $X$ and the integration symbol $dX$ from the notation when they are not entirely needed; unless otherwise specified, all our integrals on $\Omega$ will be against the Lebesgue measure $dX$. With all this notation, the main properties of our elliptic measures $\omega^{X}$ are as follows. ###### Definition 2.20. For each $X\in\Omega$, we can define a unique probability measure $\omega^{X}:=\omega^{X}_{\beta,\gamma}$ on $\Gamma$ with the following properties. For any $g\in C_{0}(\Gamma)$ (i.e., continuous function on $\Gamma$ and compactly supported), the function $u_{g}$ defined as $u_{g}(X)=\int_{\Gamma}g(y)d\omega^{X}(y)$ is a weak solution to $L_{\beta,\gamma}$ and, if in addition $g$ lies in the Sobolev space $H_{\gamma}$, then $u_{g}\in W_{\gamma}$ and the trace of $u_{g}$ is equal to $g$. The space $H_{\gamma}\cap C_{0}(\Gamma)$ is dense in $C_{0}(\Gamma)$ (with the sup norm), so the last condition is our way of solving a Dirichlet problem. We are now ready to state the general version of our main theorem. ###### Theorem 2.21. Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly rectifiable set with $d<n-1$, and let $\sigma$ be an Ahlfors regular measure on $\Gamma$ that satisfies (2.1). For $\beta>0$ and $\gamma\in(-1,1)$, define $L_{\beta,\gamma}$ as in (1.1). Then for any $\alpha>0$, there exists $C>0$ that depends only on $C_{\sigma}$, $C_{0}$, $\alpha$, $\beta$, $\gamma$, $d$, $n$, such that for any ball $B:=B(x,r)$ centered on $\Gamma$ and any non- negative non identically zero weak solution $u$ of $L_{\beta,\gamma}u=0$ in $\Omega\cap 3B$ which lies in $W_{\gamma,0}(3B)$, one has (2.22) $\int_{B}\left\|\nabla\ln\left(\frac{u}{D_{\alpha}^{1-\gamma}}\right)\right\|^{2}\,D_{\alpha}^{d+2-n}\leq C\sigma(B).$ The proof of the Theorem will use the uniform rectifiability of $\Gamma$ via Theorem 1.4 and Lemma 2.10. The Lemma will be used to estimate the left-hand side of (2.22) via simple integration techniques, except for one more complicated term in the form $\int\left\|\nabla\ln\left(\frac{u}{D_{\alpha}^{1-\gamma}}\right)\right\|\|\nabla\phi_{B,\epsilon}\|\,D_{\alpha}^{d+1-n},$ where $\phi_{B,\epsilon}$ is a well chosen cut off function, which is related to the integral of the logarithm of the Poisson kernel and that will be estimated using Theorem 1.4, that is, the $A_{\infty}$ absolute continuity of the harmonic measure. ## 3\. Proof of Theorem 2.21 Let us recall ###### Lemma 3.1. Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly rectifiable set with $d<n-2$, and let $\sigma$ be an Ahlfors regular measure on $\Gamma$ that satisfies (2.1). Define $L_{\beta,\gamma}$ as in (1.1). Then there exist $C>0$ and $\theta\in(0,1]$, that depend only on $C_{\sigma}$, $C_{0}$, $\beta$, $\gamma$, $n$ and $d$, such that for each choice of $x\in\Gamma$, $r>0$, any Borel set $E\subset B(x,r)\cap\Gamma$, and any corkscrew point $X=A_{x,r}$ as in (2.3), one has (3.2) $\frac{\omega_{\beta,\gamma}^{X}(E)}{\omega_{\beta,\gamma}^{X}(B(x,r))}\leq C\left(\frac{\sigma(E)}{\sigma(B(x,r))}\right)^{\theta}$ and (3.3) $\frac{\sigma(E)}{\sigma(B(x,r))}\leq C\left(\frac{\omega_{\beta,\gamma}^{X}(E)}{\omega_{\beta,\gamma}^{X}(B(x,r))}\right)^{\theta}.$ Here and below, we assume implicitly that the constant $C$ in (2.3) (the definition of corkscrew points) is chosen to depend on $C_{\sigma}$, $n$ and $d$ only. If we allow a larger $C$ in (2.3), the constants in (3.2) and (3.3) depend on $C$ as well. Proof. The conditions (3.2) and (3.3) are another characterizations of the fact that $\omega^{X}\in A_{\infty}(\sigma)$, which is true by Theorem 1.4. The fact that (3.2)–(3.3) is implied by (1.6) can be found in [Ken, Theorem 1.4.13] and its proof in [CF, Lemma 5]. $\Box$ For our next result, we want to establish that the logarithm of the Poisson kernel, that is, $\ln(\frac{d\omega^{X}}{d\sigma})$, is integrable. We want a quantitative version, and moreover, we shall state this in a form that is more directly applicable when we need it (in the proof of Proposition 3.8). ###### Lemma 3.4. Let $\Gamma$, $\sigma$, and $L_{\beta,\gamma}$ as in Lemma 3.1. Take $B:=B(x,r)$, a ball centered on $\Gamma$, and $X=A_{x,r}$, a corkscrew point as in (2.3). If $\\{Q_{i}\\}_{i\in\mathcal{I}}$ is a finitely overlapping collection of Borel subsets of $B\cap\Gamma$, then (3.5) $\sum_{i\in\mathcal{I}}\left\|\ln\left(\frac{\omega_{\beta,\gamma}^{X}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\right)\right\|\sigma(Q_{i})\leq C\sigma(B),$ where $C$ depends only on $C_{\sigma}$, $C_{0}$, $\beta$, $\gamma$, $n$, $d$, and the maximal number of overlaps in the collection $\\{Q_{i}\\}_{i\in\mathcal{I}}$. Proof. We introduce for $k\in\mathbb{Z}$, $\mathcal{I}_{k}:=\left\\{i\in\mathcal{I},\,2^{k}\leq\frac{\omega_{\beta,\gamma}^{X}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\leq 2^{k+1}\right\\}.$ Then we define $E_{k}:=\bigcup_{i\in\mathcal{I}_{k}}Q_{i}.$ Since $\\{Q_{i}\\}$ is finitely overlapping, we have $\sum_{i\in\mathcal{I}_{k}}\sigma(Q_{i})\leq C\sigma(E_{k})$, and we can thus write (3.6) $\sum_{i\in\mathcal{I}}\ln\left(\frac{\omega_{\beta,\gamma}^{X}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\right)\sigma(Q_{i})\lesssim\sum_{k\in\mathbb{Z}}(\|k\|+1)\sigma(E_{k}).$ Yet, since $\\{Q_{i}\\}_{i\in\mathcal{I}_{k}}$ is a finitely overlapping covering of $E_{k}$, $\omega_{\beta,\gamma}^{X}(E_{k})\approx\sum_{i\in\mathcal{I}_{k}}\omega_{\beta,\gamma}^{X}(Q_{i})\approx 2^{k}\frac{\omega_{\beta,\gamma}^{X}(B)}{\sigma(B)}\sum_{i\in\mathcal{I}_{k}}\sigma(Q_{i})\approx 2^{k}\frac{\omega_{\beta,\gamma}^{X}(B)}{\sigma(B)}\sigma(E_{k}),$ which means that (3.7) $\frac{\omega_{\beta,\gamma}^{X}(E_{k})}{\sigma(E_{k})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\approx 2^{k}.$ The use of (3.7) in (3.2) leads to $\sigma(E_{k})\lesssim 2^{-k/(1-\theta)}\sigma(B)$ while the use of (3.7) in (3.3) gives $\sigma(E_{k})\lesssim 2^{k\theta/(1-\theta)}\sigma(B).$ We use the first of the last two estimate to bound $\sigma(E_{k})$ when $k\geq 0$ and the second one to bound $\sigma(E_{k})$ when $k$ is negative. Combined with (3.6), we deduce $\begin{split}\sum_{i\in\mathcal{I}}\ln\left(\frac{\omega_{\beta,\gamma}^{X}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(B)}{\omega_{\beta,\gamma}^{X}(B)}\right)\sigma(Q_{i})&\lesssim\sum_{k\geq 0}(k+1)2^{-\frac{k}{1-\theta}}\sigma(B)+\sum_{k<0}(1-k)2^{-\frac{k\theta}{1-\theta}}\sigma(B)\\\ &\lesssim\sigma(B).\end{split}$ The lemma follows. $\Box$ ###### Proposition 3.8. Let $\Gamma\subset\mathbb{R}^{n}$ be a $d$-Ahlfors regular uniformly rectifiable set with $d<n-2$, and let $\sigma$ be an Ahlfors regular measure that satisfies (2.1). Take $\beta>0$ and $\gamma\in(-1,1)$, define $L_{\beta,\gamma}$ as in (1.1). Then for any ball $B$ centered on $\Gamma$ and any non-negative non identically zero weak solution $u$ to $L_{\beta,\gamma}u=0$ in $3B\cap\Omega$, with $u\in W_{\gamma,0(3B)}$, (3.9) $\int_{B}\left\|\nabla\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\right)\right\|^{2}\,D_{\beta}^{d+2-n}\leq C\sigma(B).$ with a constant $C>0$ that depends only on $C_{\sigma}$, $C_{0}$, $\beta$, $\gamma$, $d$, and $n$. Observe that Proposition 3.8 is the special case of Theorem 2.21 where we take $\alpha=\beta$. We shall now check that conversely, Theorem 2.21 follows from Proposition 3.8 and Lemma 2.17. Proof of Theorem 2.21 from Proposition 3.8 with the help of Lemma 2.17. Let $\alpha>0$. Notice that $\begin{split}\left\|\nabla\ln\left(\frac{u}{D_{\alpha}^{1-\gamma}}\right)\right\|&=\left\|\nabla\left[\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\right)+(1-\gamma)\ln\left(\frac{D_{\beta}}{D_{\alpha}}\right)\right]\right\|\\\ &\leq\left\|\nabla\left[\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\right)\right]\right\|+(1-\gamma)\frac{D_{\alpha}}{D_{\beta}}\left\|\nabla\left[\dfrac{D_{\beta}}{D_{\alpha}}\right]\right\|.\end{split}$ So Proposition 3.8, Lemma 2.17, and (2.2) easily implies (3.9). $\Box$ Proof of Proposition 3.8. The proof of the lemma will be divided into 4 steps. The core step is step 4, where we use the properties of the solutions. Step 3 treats in advance the most complicated term that we met in Step 4. In this step, we will compare a solution $u$ to a Green function, then to the harmonic measure, and finally we use Lemma 3.4. In Step 2, we construct the finitely overlapping covering $\\{Q_{i}\\}_{i\in\mathcal{I}}$ that will be needed in order to invoke Lemma 3.4. At last, Step 1 introduces the cut-off function $\phi_{B,\epsilon}$ used to bound the left-hand side of (2.22) and shows that $D_{\beta}\nabla\phi_{B,\epsilon}$ satisfies the Carleson measure condition. Step 1: Introduction of the cut-off function $\phi_{B,\epsilon}$. Let $B=B(x,r)$ be a ball in $\mathbb{R}^{n}$ centered on the boundary $\Gamma$ and $\epsilon>0$ small. The proof of Theorem 2.21 is a local one and thus as usual will involve cut off functions. Take $\psi\in C^{\infty}_{0}(\mathbb{R})$ be such that $\psi\equiv 1$ on $[-1,1]$, $\psi$ is compactly supported in $(-2,2)$, $0\leq\psi\leq 1$, and $\|\psi^{\prime}\|\leq 2$. We define the function $\phi_{B,\epsilon}$ on $\Omega$ by (3.10) $\phi_{B,\epsilon}(X):=\psi\left(\frac{\,\mathrm{dist}(X,B)}{10\,\mathrm{dist}(X,\Gamma)}\right)\psi\left(\frac{2\,\mathrm{dist}(X,B)}{r}\right)\psi\left(\frac{\epsilon}{\,\mathrm{dist}(X,\Gamma)}\right).$ The support of $\phi_{B,\epsilon}$ is thus contained in (3.11) $E_{0}=\big{\\{}X\in 2B,\,\,\mathrm{dist}(X,B)\leq 20\,\mathrm{dist}(X,\Gamma)\text{ and }\,\mathrm{dist}(X,\Gamma)\geq\epsilon/2\big{\\}}.$ The gradient of $\phi_{B,\epsilon}$ comes from $3$ regions; the first one (associated to the first cut-off) is (3.12) $\displaystyle\big{\\{}X\in E_{0}\,,$ $\displaystyle 10\,\mathrm{dist}(X,\Gamma)\leq\,\mathrm{dist}(X,B)\leq 20\,\mathrm{dist}(X,\Gamma)\big{\\}}$ $\displaystyle\subset E_{1}:=\big{\\{}X\in 2B,\,10\,\mathrm{dist}(X,\Gamma)\leq\,\mathrm{dist}(X,B)\leq 20\,\mathrm{dist}(X,\Gamma)\big{\\}},$ the second one is (3.13) $\big{\\{}X\in E_{0}\,,\,r\leq 2\,\mathrm{dist}(X,B)\leq 2r\big{\\}}\subset E_{2}:=\big{\\{}X\in 2B,\,\,r/40\leq\,\mathrm{dist}(X,\Gamma)\leq 2r\big{\\}}$ because $\,\mathrm{dist}(X,\Gamma)\leq 2r$ when $X\in 2B$ and similarly $\,\mathrm{dist}(X,\Gamma)\geq\frac{1}{20}\,\mathrm{dist}(X,B)\geq\frac{r}{20}$, and the third region is contained in (3.14) $E_{3}:=\big{\\{}X\in 2B,\,\epsilon/2\leq\,\mathrm{dist}(X,\Gamma)\leq\epsilon\big{\\}}.$ Then it is easy to check that the gradient of $\phi_{B,\epsilon}$ satisfies (3.15) $\|\nabla\phi_{B,\epsilon}\|\leq\frac{100}{\,\mathrm{dist}(X,\Gamma)}\left[{\mathds{1}}_{E_{1}}+{\mathds{1}}_{E_{2}}+{\mathds{1}}_{E_{3}}\right].$ We claim that ${\mathds{1}}_{E_{1}}$, ${\mathds{1}}_{E_{2}}$, and ${\mathds{1}}_{E_{3}}$ all satisfy the Carleson measure condition. That is, we have (3.16) $\int_{B(y,s)}{\mathds{1}}_{E_{j}}^{2}\,\mathrm{dist}(X,\Gamma)^{d-n}\leq Cs^{d}$ for $1\leq j\leq 3$, $y\in\Gamma$, and $0<s<+\infty$. Of course the square can be removed, and it will follows from (3.15), (3.16) (as soon as we prove it), and (2.2) that for any $\beta>0$, $y\in\Gamma$ and $s\in(0,+\infty)$, we have (3.17) $\int_{B(y,s)}\|D_{\beta}\nabla\phi_{B,\epsilon}\|D_{\beta}^{d-n}+\int_{B(y,s)}\|D_{\beta}\nabla\phi_{B,\epsilon}\|^{2}D_{\beta}^{d-n}\leq Cs^{d},$ where the constant $C>0$ depends only on $\beta$, $n$, $d$, and $C_{\sigma}$. We will prove the claim (3.16) in the end of the next Step. Step 2: Construction of the collection $\\{Q_{i}\\}_{i\in\mathcal{I}}$. We would like to have a collection of boundary cubes $Q_{i}\subset\Gamma$ and the corresponding Whitney cubes $R_{i}\subset\Omega$ satisfying rather usual nice properties: bounded overlap, control of the size and and the distance to the boundary for $R_{i}$’s, reasonable placement of the corkscrew points. There are plenty of papers that present various versions of this construction, but here, in fact, we need something simpler than usual as we do not need to control the cones or the related Harnack chains. For these reasons, let us simply carry out the construction by hands. We need a family of Whitney cubes $\mathcal{W}$, as constructed in [Ste]. We record the basic properties of $\mathcal{W}$ that we shall need. The collection $\mathcal{W}$ is the family of maximal dyadic cubes $R\subset\Omega$ such that $20R\subset\Omega$, different cubes $R_{i}$ and $R_{j}$, $i\neq j$, in $\mathcal{W}$ have disjoint interiors (by maximality), their union covers $\Omega=\mathbb{R}^{n}\setminus\Gamma$, and for $Q\in{\mathcal{W}}$ (3.18) $20Q\subset\Omega\text{ but }60R\cap\Gamma\neq\emptyset.$ Moreover, given $R\in\mathcal{W}$, the number of cubes $\widetilde{R}\in\mathcal{W}$ such that (3.19) $\,\mathrm{dist}(R,\Gamma)\approx\,\mathrm{dist}(\widetilde{R},\Gamma)\ \text{ and }\,\,\mathrm{dist}(R,\widetilde{R})\lesssim\,\mathrm{dist}(R,\Gamma)$ is uniformly bounded by a constant that depends only on the dimensions and the constants involved in (3.19). Also, the cubes $\widetilde{R}\in{\mathcal{W}}$ such that $3R\cap 3\widetilde{R}$, all satisfy (3.19). We will use the cubes of ${\mathcal{W}}$ to cover the region $E_{1}\cup E_{2}\cup E_{3}$. Consider the subset ${\mathcal{W}}_{0}\subset{\mathcal{W}}$ of cubes $R\in{\mathcal{W}}$ that meet $E_{1}\cup E_{2}\cup E_{3}$, and label these cubes with a set $I$, so that ${\mathcal{W}}_{0}=\\{R_{i}\,;\,i\in I\\}$. We also want to associate a boundary ball $Q_{i}=\Gamma\cap B(x_{i},r_{i})$ to each $R_{i}$, $i\in I$. This is a classical thing to do, but we shall do it by hand to get a slightly better control. We shall choose the $Q_{i}$ so that (3.20) $Q_{i}\subset 3B\cap\Gamma$ (we can do this and this will be helpful because we shall consider solutions in $3B$), (3.21) $\\{Q_{i}\\}_{i\in\mathcal{I}}$ has bounded overlap, where the bound for the overlap depends only on $C_{\sigma}$, $n$, and $d$, and (3.22) $r_{i}\approx\,\mathrm{dist}(R_{i},\Gamma)\approx\,\mathrm{dist}(R_{i},x_{i}),$ also with constants depend only on $C_{\sigma}$, $n$, and $d$. As a consequence, we will be able to use any point $X_{i}\in R_{i}$ as a corkscrew point for the pair $(x_{i},Cr_{i})$ (see (2.3)). So let us construct the $Q_{i}$. First write $I=I_{1}\cup I_{2}\cup I_{3}$, a disjoint union where $I_{2}=\big{\\{}i\in I\,;\,R_{i}\cap E_{2}\neq\emptyset\big{\\}}$, then $I_{3}=\big{\\{}i\in I\setminus I_{2}\,;\,R_{i}\cap E_{3}\neq\emptyset\big{\\}}$, and finally $I_{1}=\big{\\{}i\in I\setminus(I_{2}\cup I_{3})\,;\,R_{i}\cap E_{1}\neq\emptyset\big{\\}}$. We start with $i\in I_{2}$. This is the simplest case because $E_{2}\subset 2B=B(x,2r)$, and $\,\mathrm{dist}(X,\Gamma)\geq r/40$ on $E_{2}$. By definition of ${\mathcal{W}}$, $I_{2}$ has at most $C$ elements (see near (3.19)), and for each $i\in I_{2}$ we take $x_{i}=x$ and $r_{i}=3r$ (and hence $Q_{i}=\Gamma\cap 3B$); the constraints (3.20), (3.21), (3.22) for $I_{2}$ are easily checked. Next consider $i\in I_{3}$; thus $R_{i}$ meets $E_{3}$, where $\varepsilon/2\leq\,\mathrm{dist}(X,\Gamma)\leq\varepsilon$. Pick $x_{i}\in\Gamma$ such that $\,\mathrm{dist}(x_{i},E_{3}\cap R_{i})=\,\mathrm{dist}(\Gamma,E_{3}\cap R_{i})$, and take $r_{i}=\varepsilon$. Then $Q_{i}\subset 3B$ if $\varepsilon$ is small enough, because $E_{3}\subset 2B$ and $\mathrm{diam}(R_{i})\approx\,\mathrm{dist}(R_{i},\Gamma)\approx\,\mathrm{dist}(E_{3}\cap R_{i},\Gamma)\approx\varepsilon$, (3.22) holds for the same reasons, and the $Q_{i}$, $i\in I_{3}$ have bounded overlap by the property near (3.19). Finally, for $i\in I_{1}$, take $X_{i}\in R_{i}\cap E_{1}$ and $x_{i}\in\Gamma$ such that $\|X_{i}-x_{i}\|=\,\mathrm{dist}(R_{i}\cap E_{1},\Gamma)$, and set $Q_{i}=\Gamma\cap B(x_{i},r_{i})$, with $r_{i}=\|X_{i}-x_{i}\|$. Since $X_{i}\in E_{1}$, the definition (3.12) yields (3.23) $\,\mathrm{dist}(X_{i},B)\leq 20\,\mathrm{dist}(X,\Gamma)=20\|X_{i}-x_{i}\|\leq 2\,\mathrm{dist}(X_{i},B)\leq 2r,$ and in particular $x_{i}\in\frac{5}{2}B$. Also, $X_{i}\notin E_{2}$ (because $i\notin I_{2}$), so $r_{i}=\,\mathrm{dist}(R_{i}\cap E_{1},\Gamma)\leq\,\mathrm{dist}(X_{i},\Gamma)\leq r/40$, and hence $Q_{i}\subset 3B$, as needed for (3.20). The Whitney property (3.22) holds essentially by definition (and because $R_{i}$ is a Whitney cube), so we just need to bound the overlap of the $Q_{i}$. Assume that $Q_{j}\cap Q_{i}\neq\emptyset$ for two indices $i,j\in\mathcal{I}_{1}$, and let $X_{i},X_{j},r_{i}$, and $r_{j}$ be as above. Let us first check that $r_{i}\approx r_{j}$. By (3.23) $\,\mathrm{dist}(x_{i},B)\leq\|X_{i}-x_{i}\|+\,\mathrm{dist}(X_{i},B)\leq 21\|X_{i}-x_{i}\|=21r_{i}$ and $\,\mathrm{dist}(x_{i},B)\geq\,\mathrm{dist}(X_{i},B)-\|X_{i}-x_{i}\|\leq 9\|X_{i}-x_{i}\|=9r_{i},$ which can be summarized as (3.24) $9r_{i}\leq\,\mathrm{dist}(x_{i},B)\leq 21r_{i}.$ Similarly, (3.25) $9r_{j}\leq\,\mathrm{dist}(x_{j},B)\leq 21r_{j}.$ We can assume without loss of generality that $\,\mathrm{dist}(x_{j},B)\leq\,\mathrm{dist}(x_{i},B)$, which, together with (3.24)-(3.25), leads to $9r_{j}\leq 21r_{i}$. Moreover, since $Q_{i}\cap Q_{j}\neq\emptyset$, $r_{i}+r_{j}\geq\|x_{j}-x_{i}\|\geq\,\mathrm{dist}(x_{i},B)-\,\mathrm{dist}(x_{j},B)\geq 9r_{i}-21r_{j},$ hence $22r_{j}\geq 8r_{i}$. Recall that $9r_{j}\leq 21r_{i}$, so the two radii are equivalent. The bounded overlap property follows, because $\|x_{j}-x_{i}\|\leq r_{i}+r_{j}$ when $Q_{i}\cap Q_{j}\neq\emptyset$. This completes our construction of Whitney cubes $R_{i}$ and associated surface balls $Q_{i}$, with the properties (3.20)–(3.22) (notice that the overlap constant for the $Q_{i}$, $i\in I$, is less than the sum of the overlap constants for the $I_{j}$). With this at hand, let us prove (3.16). This is now quite easy as for any $y\in\Gamma$ and $0<s<+\infty$ we have (3.26) $\int_{B(y,s)}{\mathds{1}}_{E_{1}\cup E_{2}\cup E_{3}}^{2}\,\mathrm{dist}(X,\Gamma)^{d-n}\\\ \leq\sum_{R_{i}:\,R_{i}\cap B(y,s)\neq\emptyset}\int_{R_{i}}\,\mathrm{dist}(X,\Gamma)^{d-n}\lesssim\sum_{R_{i}:\,R_{i}\cap B(y,s)\neq\emptyset}r_{i}^{d}\lesssim s^{d},$ using the bounded overlap property (3.21). Step 3: Estimates for the integral $S$. Take $u\in W_{\gamma,0}(3B)$, a weak solution to $L_{\beta,\gamma}u=0$ in $3B$. Let $X_{0}\in 2B$ be a corkscrew point for the boundary ball $2B\cap\Gamma$, and also choose another corkscrew point $X_{1}\in\Omega\cap 8B\setminus 4B$, so that $\,\mathrm{dist}(X_{0},\Gamma)\geq cr$ and $\,\mathrm{dist}(X_{1},\Gamma)\geq cr$ for a small constant $c$ that depends only on $C_{\sigma}$, $d$, and $n$. In this Step 3, we prove that (3.27) $S:=\int_{E_{1}\cup E_{2}\cup E_{3}}\left\|\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\frac{D_{\beta}^{1-\gamma}(X_{0})}{u(X_{0})}\right)\right\|D_{\beta}^{d-n}\leq Cr^{d},$ where $C$ depends only on $C_{\sigma}$, $C_{0}$, $\beta$, $\gamma$, $n$ and $d$. We shall use the comparison principle to compare $u$ to the Green function, and then estimate the Green function in terms of harmonic measure. We define the Green function $g$ on $\Omega\times\Omega$ as in [DFM5, Section 11]. The precise definition is not relevant for the present proof, and the properties that we care about are the fact that $X\to g(X,X_{1})$ lies in $W_{\gamma,0}(3B)$ and is a solution to $L_{\beta,\gamma}u=0$ in $3B$, and that $g(X,Y)=g(Y,X)$ (which is true because the operator $L_{\beta,\gamma}$ is selfadjoint). Theorem 15.64 in [DFM5] (the comparison theorem) yields that (3.28) $\frac{u(X)}{u(X_{0})}\approx\frac{g(X,X_{1})}{g(X_{0},X_{1})}\qquad\text{ for }X\in 2B,$ with constants that depend only on $C_{\sigma}$, $C_{0}$, $n$, $d$, $\beta$ and $\gamma$. Actually, Theorem 15.64 in [DFM5] requires the solutions (in our case $u$ and $g(.,X_{1})$) to be solutions in a larger ball $2KB\cap\Omega$, and not in only $3B\cap\Omega$. This condition is only needed because [DFM5] also allows sets $\Gamma$ of codimension $1$ or less, and then we need to ensure that we can connect every component of $2B\cap\Omega$ by Harnack chains that stays in $2KB$. Here $\Gamma$ is of dimension $d<n-1$, so $2B\cap\Omega$ is very well connected in the first place (see Lemma 2.1 in [DFM2]) , and assuming that $u$ and $g(.,X_{1})$ are solutions in $3B$ is enough. Using the fact that $g(X,Y)$ is symmetric, we deduce that (3.29) $\frac{u(X)}{u(X_{0})}\approx\frac{g(X_{1},X)}{g(X_{1},X_{0})}\qquad\text{ for }X\in 2B.$ Recall that $X_{0}$ is a corkscrew point associated to $2B$ and that if $X\in R_{i}$, then $\,\mathrm{dist}(X,\Gamma)\geq C^{-1}r_{i}$ and $\|X-x_{i}\|\leq Cr_{i}$ by (3.22), so $X$ can be used as a corkscrew point for the boundary ball $Q_{i}$. As a consequence, Lemma 15.28 in [DFM5], (where here $m(B\cap\Omega)$ is the mass of $B$ for the measure $\,\mathrm{dist}(X,\Gamma)^{d+1+\gamma-n}dX$ on $\Omega$, with $\gamma\in(-1,1)$, so $m(B\cap\Omega)\approx r^{d+1+\gamma}$; see the discussion in Section 3.2 of [DFM5]), and the doubling property of harmonic measure (Lemma 15.43 of [DFM5]) give that $g(X_{1},X_{0})\approx\,\mathrm{dist}(X_{0},\Gamma)^{1-d-\gamma}\omega^{X_{1}}(3B)$ and for each $i\in I$ $g(X_{1},X)\approx\,\mathrm{dist}(X,\Gamma)^{1-d-\gamma}\omega^{X_{1}}(Q_{i})\qquad\text{ for }X\in R_{i},$ where the constants depend only on $C_{\sigma}$, $C_{0}$, $n$, $d$, $\beta$ and $\gamma$. Using the equivalence (2.2) and the Ahlfors regularity of $\sigma$, we deduce that (3.30) $g(X_{1},X_{0})\approx D_{\beta}(X_{0})^{1-\gamma}\,\frac{\omega^{X_{1}}(3B)}{\sigma(3B)}$ and for each $i\in I$, (3.31) $g(X_{1},X)\approx D_{\beta}(X)^{1-\gamma}\,\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\qquad\text{ for }X\in R_{i}.$ We gather (3.29), (3.30), and (3.31) to obtain that, for every $i\in I$ and every $X\in R_{i}$, (3.32) $\frac{u(X)}{D_{\beta}^{1-\gamma}(X)}\frac{D_{\beta}^{1-\gamma}(X_{0})}{u(X_{0})}\approx\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)},$ where the constants depend only on $C_{\sigma}$, $C_{0}$, $n$, $d$, $\beta$ and $\gamma$. This immediately implies that for $i\in I$ and $X\in R_{i}$, $\left\|\ln\left(\frac{u(X)}{D_{\beta}^{1-\gamma}(X)}\frac{D_{\beta}^{1-\gamma}(X_{0})}{u(X_{0})}\right)\right\|\leq C+\left\|\ln\left(\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)}\right)\right\|$ and then (3.33) $\begin{split}S&\leq\sum_{i\in I}\int_{R_{i}}\left\|\ln\left(\frac{u}{D_{\beta}^{1-\gamma}}\frac{D_{\beta}^{1-\gamma}(X_{0})}{u(X_{0})}\right)\right\|D_{\beta}^{d-n}\\\ &\leq\sum_{i\in I}\left[C+\left\|\ln\left(\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)}\right)\right\|\right]\int_{R_{i}}D_{\beta}^{d-n}.\end{split}$ By(2.2) and the fact that $R_{i}\in{\mathcal{W}}$ is a Whitney cube with the property (3.22) (by construction), we have $D_{\beta}\approx r_{i}$ on $R_{i}$ and $\|R_{i}\|\lesssim r_{i}^{n}$. Hence $\int_{R_{i}}D_{\beta}^{d-n}\lesssim r_{i}^{d}\approx\sigma(Q_{i})$ by (2.1). Using this observation in (3.33), we infer that $\begin{split}S&\leq\sum_{i\in I}\left[C+\left\|\ln\left(\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)}\right)\right\|\right]\sigma(Q_{i})\\\ &=C\sum_{i\in I}\sigma(Q_{i})+\sum_{i\in I}\left\|\ln\left(\frac{\omega^{X_{1}}(Q_{i})}{\sigma(Q_{i})}\frac{\sigma(3B)}{\omega^{X_{1}}(3B)}\right)\right\|\sigma(Q_{i}).\end{split}$ The first term of the right-hand side is bounded by $C\sigma(3B)$ since the $Q_{i}$, $i\in I$, are contained in $\Gamma\cap 3B$ (by (3.20)) and have bounded overlap (by (3.21)). The second term is also less than $C\sigma(3B)$, by Lemma 3.4. We conclude that $S\lesssim\sigma(3B)\lesssim r^{d}$ by (2.1). The claim 3.27 follows. Step 4: Core of the proof. Let us turn to the main and last step of the proof. Set (3.34) $T:=\int_{\Omega}\bigg{\|}\nabla\ln\Big{(}\frac{u}{D_{\beta}^{1-\gamma}}\Big{)}\bigg{\|}^{2}\phi_{B,\epsilon}^{2}D_{\beta}^{d+2-n}.$ We aim to prove that (3.35) $T\leq Cr^{d}+Cr^{d/2}T^{1/2},$ where $C$ depends only on $C_{\sigma}$, $C_{0}$, $\beta$, $\gamma$, $n$ and $d$. The solution $u$ and the smooth distance $D_{\beta}$ are uniformly bounded from above and from below by a positive constant on the support of the cut-off function $\phi_{B,\epsilon}$. Thanks to this fact, the quantities of both sides of (3.35) are finite, and (3.35) self-improves into (3.36) $\int_{\Omega}\Big{\|}\nabla\ln\Big{(}\frac{u}{D_{\beta}^{1-\gamma}}\Big{)}\Big{\|}^{2}\phi_{B,\epsilon}^{2}D_{\beta}^{d+2-n}=T\leq Cr^{d},$ where $C>0$ depends on the same parameters as in (3.35). Once we are there, the left-hand side is uniformly bounded in $\epsilon$, so taking $\epsilon\to 0$ leads to the desired result (3.9). Keep in mind that (3.37) $\nabla\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}=\frac{\nabla u}{u}-\frac{\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}=\dfrac{D_{\beta}^{1-\gamma}\nabla u-u\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}u}.$ We shall use Lemma 2.10 to obtain the existence of a scalar function $b$ and a vector function $\mathcal{V}$ such that (3.38) $H_{n-d-1}:=\int_{\Gamma}\|X-y\|^{-n}(X-y)d\sigma(y)=(b\nabla D_{\beta}+\mathcal{V})D_{\beta}^{d+1-n}$ as in (2.11), with the bounds (2.12)-(2.15). Before we start to bound $T$, let us comment on $H_{n-d-1}$ and $\nabla D_{\beta}$. First, the vector function $H_{n-d-1}$ is smooth and divergence free in $\Omega$, and the formulation of this fact in the weak sense is that for any compactly supported $\varphi\in W^{1,1}(\Omega)$, (3.39) $\int_{\Omega}H_{n-d-1}\cdot\nabla\varphi=0.$ Set $H_{\alpha}(X):=\int_{\Gamma}\|X-y\|^{-d-1-\alpha}(X-y)d\sigma(y)$ for $\alpha>0$. Since $\|H_{\alpha}(X)\|\leq\int_{\Gamma}\|X-y\|^{-d-\alpha}d\sigma(y)=D_{\alpha}(X)^{-\alpha}$, (2.2) implies that (3.40) $\|H_{\alpha}\|\lesssim D_{\alpha}^{-\alpha}\lesssim D_{\beta}^{-\alpha}.$ Moreover, observe that $\nabla(D_{\beta}^{-\beta})=-(d+\beta)H_{\beta+1}$, directly by (1.2); since also $\nabla(D_{\beta}^{-\beta})=-\beta D_{\beta}^{-\beta-1}\nabla D_{\beta}$, we deduce from (3.40) that (3.41) $\|\nabla D_{\beta}\|\lesssim 1.$ We turn to the proof of (3.35). Write $\phi$ instead of $\phi_{B,\epsilon}$ to lighten the notation. By (2.12) and (3.37), $\begin{split}T\lesssim\int_{\Omega}\bigg{\|}\nabla\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{\|}^{2}\phi^{2}b\,D_{\beta}^{d+2-n}&=\int_{\Omega}\frac{b\nabla u}{u}\cdot\bigg{(}\dfrac{D_{\beta}^{1-\gamma}\nabla u-u\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}u}\bigg{)}\ \phi^{2}D_{\beta}^{d+2-n}\\\ &\hskip 56.9055pt-\int_{\Omega}\frac{b\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2}D_{\beta}^{d+2-n}\\\ &:=T_{1}-T_{2}.\end{split}$ Let us start with $T_{2}$. We simplify the factors $D_{\beta}$ and we invoke the relation (3.38) to get $\begin{split}T_{2}&=(\gamma-1)\int_{\Omega}\frac{b\nabla D_{\beta}}{D_{\beta}^{n-d-1}}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2}\\\ &=(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2}-(\gamma-1)\int_{\Omega}\mathcal{V}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2}D_{\beta}^{d+1-n}\\\ &:=T_{21}+T_{22}.\end{split}$ We use the Cauchy-Schwarz inequality to bound $T_{22}$, as follows: $\begin{split}\|T_{22}\|&\lesssim\Big{(}\int_{\Omega}\|\mathcal{V}\|^{2}\phi^{2}D_{\beta}^{d-n}\Big{)}^{\frac{1}{2}}\bigg{(}\int_{\Omega}\bigg{\|}\nabla\bigg{[}\ln\bigg{(}\frac{u}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\bigg{\|}^{2}\phi^{2}D_{\beta}^{d+2-n}\bigg{)}^{\frac{1}{2}}\\\ &\lesssim T^{1/2}\left(\int_{2B}\|\mathcal{V}\|^{2}D_{\beta}^{d-n}\right)^{\frac{1}{2}}\lesssim r^{d/2}T^{1/2}\end{split}$ by (2.15). This fits with (3.35). As for $T_{21}$, notice that its value will not be changed if we replace $u$ by $Ku$, where $K$ is a constant. Hence $T_{21}=(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\bigg{[}\ln\bigg{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{]}\phi^{2},$ where $K$ is a constant to be chosen later. We force $\phi^{2}$ into the gradient, then use the fact that $H_{n-d-1}$ is divergence free (see (3.39)), and obtain $\displaystyle T_{21}$ $\displaystyle=(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\Big{[}\phi^{2}\ln\Big{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\Big{)}\Big{]}-2(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\phi\ \phi\ln\Big{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\Big{)}$ $\displaystyle=0-2(\gamma-1)\int_{\Omega}H_{n-d-1}\cdot\nabla\phi\ \phi\ln\Big{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\Big{)}.$ Recall that $\|H_{n-d-1}\|\lesssim D_{\beta}^{d+1-n}$ by (3.40), and that $\|\nabla\phi\|\lesssim{\mathds{1}}_{E_{1}\cup E_{2}\cup E_{3}}/D_{\beta}$ by (3.15) and (2.2), so $\|T_{21}\|\lesssim\int_{E_{1}\cup E_{2}\cup E_{3}}\bigg{\|}\ln\bigg{(}\frac{Ku}{D_{\beta}^{1-\gamma}}\bigg{)}\bigg{\|}D_{\beta}^{d-n}.$ We choose $K=D_{\beta}^{1-\gamma}(X_{0})/u(X_{0})$, so that the right-hand side above is what we called $S$ in (3.27). Hence $T_{21}\lesssim r^{d}$ by (3.27), as needed for (3.35). We switch to the estimation of $T_{1}$. We want to use the fact that $u$ is a solution, and for this we write (3.42) $\begin{split}T_{1}&=\int_{\Omega}\nabla u\cdot\frac{D_{\beta}^{1-\gamma}}{u}\bigg{(}\dfrac{D_{\beta}^{1-\gamma}\nabla u-u\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}u}\bigg{)}\ b\phi^{2}D_{\beta}^{d+1+\gamma-n}\\\ &=-\int_{\Omega}\nabla u\cdot\nabla\bigg{[}\dfrac{D_{\beta}^{1-\gamma}}{u}\bigg{]}\ b\phi^{2}D_{\beta}^{d+1+\gamma-n}\\\ &=-\int_{\Omega}\nabla u\cdot\nabla\bigg{[}b\phi^{2}\dfrac{D_{\beta}^{1-\gamma}}{u}\bigg{]}\ D_{\beta}^{d+1+\gamma-n}+\int_{\Omega}\nabla u\cdot\nabla b\bigg{(}\phi^{2}\dfrac{D_{\beta}^{1-\gamma}}{u}\bigg{)}\ D_{\beta}^{d+1+\gamma-n}\\\ &\qquad+2\int_{\Omega}\nabla u\cdot\nabla\phi\bigg{(}\phi b\dfrac{D_{\beta}^{1-\gamma}}{u}\bigg{)}\ D_{\beta}^{d+1+\gamma-n}\\\ &:=T_{11}+T_{12}+T_{13}.\end{split}$ Notice that $T_{11}=0$ because $u$ is a weak solution to $L_{\beta,\gamma}u=0$ on $\Omega\cap 2B$ and $\phi^{2}\dfrac{D_{\beta}^{1-\gamma}}{u}$ lies in $W^{1,2}_{loc}(\Omega)$ and compactly supported in $\Omega$. The terms $T_{12}$ and $T_{13}$ are morally similar. In both case, we don’t like the terms with $u$ because we don’t know so much about it, so we use (3.37) to replace $u$ by the nice function $D_{\beta}^{1-\gamma}$, and the difference will be controlled with the help of $T^{1/2}$, the square root of our initial integral. We start with (3.43) $\begin{split}T_{13}&=2\int_{\Omega}\frac{\nabla u}{u}\cdot\nabla\phi\,\phi\,b\,D_{\beta}^{d+2-n}\\\ &=2\int_{\Omega}\left(\frac{\nabla u}{u}-\frac{\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\right)\cdot(\nabla\phi)\,\phi\,b\,D_{\beta}^{d+2-n}-2(\gamma-1)\int_{\Omega}\frac{b\nabla D_{\beta}}{D_{\beta}^{n-d-1}}\cdot(\nabla\phi)\phi\\\ &:=T_{131}+T_{132}.\end{split}$ We use (2.12) and the fact that $D_{\beta}\nabla\phi$ satisfies the Carleson measure property to get $\begin{split}\|T_{131}\|&\lesssim\Big{(}\int_{\Omega}\|\nabla\phi\|^{2}D_{\alpha}^{d+2-n}\Big{)}^{1/2}\bigg{(}\int_{\Omega}\bigg{\|}\frac{\nabla u}{u}-\frac{\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\bigg{\|}^{2}\phi^{2}D_{\alpha}^{d+2-n}\bigg{)}^{1/2}\\\ &\lesssim r^{d/2}\bigg{(}\int_{\Omega}\bigg{\|}\nabla\ln\bigg{(}\frac{u}{D_{\alpha}^{1-\gamma}}\bigg{)}\bigg{\|}^{2}\phi^{2}D_{\alpha}^{d+2-n}\bigg{)}^{1/2}\end{split}$ by (3.17), and then (3.37). The bound of $T_{131}$ that we just obtained appears in the right-hand side of (3.35), as desired. As for $T_{132}$, we invoke (3.41), (2.12), and then (3.17) to write $\begin{split}\|T_{132}\|&\lesssim\int_{2B}\|D_{\beta}\nabla\phi\|D_{\beta}^{n-d}\lesssim r^{d}\end{split}.$ Similarly to $T_{13}$, we treat $T_{12}$ as follows: (3.44) $\begin{split}T_{12}&=\int_{\Omega}\frac{\nabla u}{u}\cdot\nabla b\,\phi^{2}D_{\beta}^{d+2-n}\\\ &=\int_{\Omega}\bigg{(}\frac{\nabla u}{u}-\frac{\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\bigg{)}\cdot\nabla b\,\phi^{2}D_{\beta}^{d+2-n}-(\gamma-1)\int_{\Omega}\nabla D_{\beta}\cdot\nabla b\,\phi^{2}D_{\beta}^{d+1-n}\\\ &:=T_{121}+T_{122}.\end{split}$ Thanks to the definition (3.34) and then (2.13), (3.45) $\begin{split}\|T_{121}\|&\lesssim\bigg{(}\int_{\Omega}\bigg{\|}\frac{\nabla u}{u}-\frac{\nabla D_{\beta}^{1-\gamma}}{D_{\beta}^{1-\gamma}}\bigg{\|}^{2}\phi^{2}D_{\beta}^{d+2-n}\bigg{)}^{1/2}\Big{(}\int_{\Omega}\|\nabla b\|^{2}\phi^{2}D_{\beta}^{d+2-n}\Big{)}^{1/2}\\\ &\lesssim T^{1/2}\left(\int_{2B}\|D_{\beta}\nabla b\|^{2}D_{\beta}^{d-n}\right)^{\frac{1}{2}}\lesssim r^{d/2}T^{1/2}.\end{split}$ Now, we want to use (3.38) again, so we force the function $b$ to appear and we place all the remaining terms in the second gradient. Then $T_{122}$ becomes $\begin{split}T_{122}&=-(\gamma-1)\int_{\Omega}b\nabla D_{\beta}\cdot\frac{\nabla b}{b}\,\phi^{2}D_{\beta}^{d+1-n}\\\ &=-(\gamma-1)\int_{\Omega}b\nabla D_{\beta}\cdot\nabla[\phi^{2}\ln(b)]\,D_{\beta}^{d+1-n}-2(1-\gamma)\int_{\Omega}b\nabla D_{\beta}\cdot\nabla\phi\,\phi\ln(b)\,D_{\beta}^{d+1-n}\\\ &:=T_{1221}+T_{1222}.\end{split}$ We start with $T_{1222}$. Since $b\approx 1$, we have that $\|b\ln(b)\|\lesssim 1$. Besides, $\|\nabla D_{\beta}\|\lesssim 1$ by (3.41). Therefore, $\|T_{1222}\|\lesssim\int_{2B}\|D_{\beta}\nabla\phi\|D_{\beta}^{d-n},$ and then $\|T_{1222}\|\lesssim r^{d}$ by (3.17). We use (3.38) to write $T_{1221}$ as $\begin{split}T_{1221}&=(1-\gamma)\int_{\Omega}H_{n-d-1}\cdot\nabla[\phi^{2}\ln(b)]+(\gamma-1)\int_{\Omega}\mathcal{V}\cdot\nabla[\phi^{2}\ln(b)]\,D_{\beta}^{d+1-n}\\\ &=0+(\gamma-1)\int_{\Omega}\mathcal{V}\cdot\Big{[}2\phi\nabla\phi\ln(b)+\phi^{2}\frac{\nabla b}{b}\Big{]}D_{\beta}^{d+1-n}\end{split}$ by (3.39). Hence by (2.12) and then the Cauchy-Schwarz inequality, $\begin{split}\|T_{1221}\|&\lesssim\int_{2B}\|\mathcal{V}\|(\|\nabla\phi\|+\|\nabla b\|)D_{\beta}^{d+1-n}\\\ &\lesssim\left(\int_{2B}\|\mathcal{V}\|^{2}D_{\beta}^{n-d}\right)^{\frac{1}{2}}\left(\int_{2B}(\|D_{\beta}\nabla\phi\|^{2}+\|D_{\beta}\nabla b\|^{2})D_{\beta}^{d-n}\right)^{\frac{1}{2}}.\end{split}$ But since $\mathcal{V}$, $D_{\beta}\nabla b$, and $D_{\beta}\nabla\phi$ all satisfies the Carleson measure condition, we conclude that $\|T_{1221}\|\lesssim r^{d}$ as desired. We bounded each term derived from $T$ by either $r^{d}$ or $r^{d/2}T^{1/2}$, and consequently proved the claim (3.35). As was observed before, (3.9), Proposition 3.8, and then Theorems 2.21 and 1.8 follow. $\Box$ ## References * [AHMMT] J. Azzam, S. Hofmann, J.M. Martell, M. Mourgouglou, X. Tolsa. Harmonic measure and quantitative connectivity: geometric characterization of the $L^{p}$-solvability of the Dirichlet problem. 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# Predictive Control and Communication Co-Design via Two-Way Gaussian Process Regression and AoI-Aware Scheduling Abanoub M. Girgis, †Jihong Park, Mehdi Bennis, and ‡Mérouane Debbah A preliminary conference version of this work appeared in the proceedings of IEEE SPAWC-2020 [1].A. Girgis and M. Bennis are with the Centre for Wireless Communications, University of Oulu, 90014 Oulu, Finland (e-mail: <EMAIL_ADDRESS>mehdi.bennis@oulu.fi).†J. Park is with the School of Information Technology, Deakin University, Geelong, VIC 3220, Australia (e-mail: jihong.park@deakin.edu.au).‡Mérouane Debbah is with Université Paris- Saclay, CNRS, CentraleSupélec, 91190, Gif-sur-Yvette, France (e-mail: <EMAIL_ADDRESS>and the Lagrange Mathematical and Computing Research Center, 75007, Paris, France. ###### Abstract This article studies the joint problem of uplink-downlink scheduling and power allocation for controlling a large number of actuators that upload their states to remote controllers and download control actions over wireless links. To overcome the lack of wireless resources, we propose a machine learning- based solution, where only a fraction of actuators is controlled, while the rest of the actuators are actuated by locally predicting the missing state and/or action information using the previous uplink and/or downlink receptions via a Gaussian process regression (GPR). This GPR prediction credibility is determined using the age-of-information (AoI) of the latest reception. Moreover, the successful reception is affected by the transmission power, mandating a co-design of the communication and control operations. To this end, we formulate a network-wide minimization problem of the average AoI and transmission power under communication reliability and control stability constraints. To solve the problem, we propose a dynamic control algorithm using the Lyapunov drift-plus-penalty optimization framework. Numerical results corroborate that the proposed algorithm can stably control $2$x more number of actuators compared to an event-triggered scheduling baseline with Kalman filtering and frequency division multiple access, which is $18$x larger than a round-robin scheduling baseline. ###### Index Terms: Predictive control, age of information (AoI), Gaussian process regression (GPR), ultra-reliable and low-latency communication (URLLC), Beyond 5G (B5G), 6G. ## I Introduction Ultra-reliable and low-latency communication (URLLC) is a key enabler for ensuring the stability of wirelessly networked control systems in real-time [2, 3]. By physically decoupling sensors, actuators, and controllers, the control system can exploit the recent progress in the fifth generation (5G) connectivity [4], machine learning and edge computing [5], thereby spearheading many emerging applications ranging from large-scale smart industrial internet of things (IIoT) [6] to autonomous platooning [7]. The success of these application relies on addressing several fundamental challenges emanating from unstable and intermittent wireless connectivity, which incurs distorted and delayed control information receptions, degrading control stability. Wireless resource allocation and scheduling are thus instrumental in not only improving communication efficiency but also in guaranteeing control stability. Resource allocation and scheduling of control systems play a key role in the context of wireless networked control systems (WNCS). Round-robin scheduling, is a static scheduling in which each sensor/controller periodically transmits the state/action to a controller/actuator with a fixed transmission power and a predefined repeating order [8, 9]. The scheduling method in [8, 9] maintains stability for a small number of control systems, but fails to stabilize a large number of control systems since the scheduling decisions are not adaptive to the CSI, which hinder control performance. Dynamic round-robin scheduling in [10, 11] is a channel-aware scheduling adapting the transmission power to the CSI to ensure reliability of the transmitted signal and minimize the system energy. However, it does not guarantee stability for a large number of control systems, since the waiting period of these control systems to update the state/action information is proportional to the number of served control systems. Moreover, the scheduling decisions are not adaptive to the control state that results in wasting wireless resources. To efficiently utilize communication resources, the event-triggered scheduling suggested in [12, 13] is a dynamic scheduling method in which one control system with the largest state instability is scheduled at each time to transmit the state/action with fixed transmission power. The previously mentioned scheduling methods save wireless communication resources since the scheduling decisions are based on the control state but they are CSI-agnostic which may result in control performance degradation. In addition, these scheduling methods can not support a large number of control systems owing to the outdated control action dependent on the last received state. The control-aware communication scheduling suggested in [14] incorporates in the scheduling decisions both control and channel states to minimize the overall transmission time of the scheduled control systems. Therein, the scheduling decisions are based on communication reliability that is sensitive to the control state and system dynamics. However, in the absence of wireless communication, the outdated applied actions over ideal channels are based on the last received state, hindering control stability. While the previously mentioned scheduling methods in [8, 9, 11, 10, 12] are effective in small- scale control environments, as pointed out in [15], the communication and control designs in [8, 9, 11, 10, 12] are separated from each other, limiting their adoption for supporting a large number of control systems. This mandates a tight co-design of reliable communication and stable control operations for optimal resource allocation and scheduling. ### I-A Contributions Spurred by the aforementioned motivation, in this work we consider a scenario consisting of a large number of control systems connected through wireless links. To overcome the lack of wireless resources, we propose a GPR based solution in which only a fraction of control systems is controlled by communicating with the controller, while the remaining control systems are controlled by locally predicting the missing state/action via GPR, using the previous received state/action. For each control system, a controller calculates the action using a linear quadratic regulator (LQR) whose input, i.e., either the estimated state using a minimum mean square error (MMSE) estimator if a sensor-controller pair of a control system is scheduled in the uplink (UL) or the predicted state using GPR at the controller when it is not scheduled. Then, to regulate the control system, an actuator applies either the estimated action using an MMSE estimator if the controller-actuator pair of a control system is scheduled in the downlink (DL), or the predicted action using GPR at the actuator if it is not scheduled. The centralized scheduler shared among all control systems schedules at most one sensor-controller pair in the UL and one controller-actuator pair in the DL, based on both control and channel states. Here, control stability depends on the GPR prediction credibility [16] that is determined by the age of information (AoI) of the latest received signal, and the reliability of the received signal that is dictated by transmission power, highlighting the importance of the communication and control co-design. Given the proposed predictive control framework, we focus on the problem of jointly optimizing the UL-DL scheduling and power allocation so as to minimize the network-wide average AoI and transmission power while guaranteeing communication reliability and control stability. To solve the formulated non- convex stochastic optimization problem, we develop a dynamic control algorithm using Lyapunov optimization. Considering an inverted pendulum, numerical results demonstrate that the proposed scheduling method can stably support $2$x more control systems compared to the event-triggered scheduling with Kalman filtering and frequency division multiple access (FDMA), which is $18$x larger than a time-triggered scheduling baseline. Furthermore, the results show that the proposed predictive control algorithm is more communication efficient while achieving a faster control stability than the time-triggered and event-triggered control baselines, highlighting the effectiveness of the UL-DL decoupled scheduling and the use of two-way GPRs at both controller and actuator sides. The remainder of this paper is organized as follows. In Section II, we specify the WNCS architecture including the system models of the control, communication, and GPR based approach. In Section III, we formulate the communication, control, GPR based co-design optimization problem and propose the stability-aware scheduling algorithm by leveraging Lyapunov optimization framework to solve the co-design problem in Section IV. In Section V and Section VI, we present simulations results and conclude the paper. Figure 1: An illustration of $M=2$ WNCSs operated via both state/action measurement by remote sensors/controllers and state/actions prediction by GPR. ## II System Model ### II-A Wireless Networked Control System Architecture As depicted in Fig. 1, the WNCS architecture under study consists of a set $\mathcal{M}$ of $M$ independent linear _control systems_ over a shared wireless channel. Each control system comprises a _plant_ , a _sensor_ that measures the plant’s state, and _actuator_ that takes an action to control the plant’s state. The action is computed by a remote _controller_ based on the control and channel states. To this end, the plant’s state is received by the controller in the _uplink_ , and the controller’s action is received by the actuator in the _downlink_. To avoid interference under limited communication bandwidth, each UL or DL channel is allocated to a single sensor-controller pair or controller-actuator pair per unit time, respectively. The rest of the sensor-controller and controller-actuator pairs without receptions locally predict their missing control states and actions, respectively, based on their previously received information, to be detailed in Sec. II-D. To be specific, the plant’s state of control system $i\in\mathcal{M}$ at discrete control time $k\in\mathbb{Z}_{+}$ is denoted by $\mathbf{x}^{u}_{i,k}\in\mathbb{R}^{D}$. For a received action at an actuator $\mathbf{u}^{a}_{i,k}\in\mathbb{R}^{P}$ based on the computed action at controller $\mathbf{u}^{d}_{i,k}\in\mathbb{R}^{P}$, the state evolution of control system $i$ at time $k$ is described by the discrete-time linear time-invariant (LTI) system as follows: $\mathbf{x}^{u}_{i,k+1}=\mathbf{A}_{i}\mathbf{x}^{u}_{i,k}+\mathbf{B}_{i}\mathbf{u}^{a}_{i,k}+\mathbf{w}_{k},$ (1) where $\mathbf{A}_{i}\in\mathbb{R}^{D\times D}$ is a fixed state transition matrix of the $i$-th control system, $\mathbf{B}_{i}\in\mathbb{R}^{D\times P}$ is a fixed control action matrix of the $i$-th control system, and $\mathbf{w}_{k}\in\mathbb{R}^{D}$ is the plant noise at time $k$ which is independent and identically distributed (IID) Gaussian noise with zero mean and covariance matrix $\mathbf{W}$. Here, to avoid a non-trivial problem, $\mathbf{A}_{i}$ is assumed to be unstable, i.e., $\mathbf{A}_{i}$’s spectral radius $\rho(\mathbf{A}_{i})=\text{max}\\{\|\lambda_{1}(\mathbf{A}_{i})\|,\cdots,\|\lambda_{D}(\mathbf{A}_{i})\|\\}$ is larger than unity, where $\lambda_{D}(\mathbf{A}_{i})$ is the $D$-th eigenvalue of $\mathbf{A}_{i}$. This implies that the plant’s state infinitely grows over time unless a proper control action $\mathbf{u}^{a}_{i,k}$ is provided. To stabilize such control system, each time $k$, the following four phase operations are considered. 1. 1. Sensing and Uplink Transmission (at a sensor): A centralized scheduler located at the base station (BS) shared among all control systems decides which sensor-controller pair is scheduled to transmit and close its sensing loop based on both the channel and control states. Then, the scheduled sensor transmits its state to its controller over a wireless UL fading channel using analog uncoded communication to be elaborated in Sec. II-B. 2. 2. State Reception or Prediction (at a controller): If the sensor-controller pair is scheduled, the controller obtains the current estimated state using MMSE estimator, and predicts the next state via GPR to be discussed in Sec. II-D. Otherwise, the controller directly predicts the current and the next states based on the state history using GPR. The current predicted state by GPR is fed to the LQR to calculate the action unless the estimated state by the MMSE estimator is provided. The future predicted state is fed to the centralized scheduler to make the scheduling decisions. 3. 3. Action Computation and Downlink Transmission (at a controller): For a given control state, the controller computes the optimal action using LQR [17]. The controller transmits the computed action to the scheduled actuator over a wireless DL fading channel using analog uncoded communication to be elaborated in Sec. II-B. 4. 4. Action Reception or Prediction (at an actuator): If the controller-actuator pair is scheduled, the actuator obtains the current estimated action using a MMSE estimator, and predicts the next action via an action GPR to be discussed in Sec. II-D. Otherwise, the actuator directly predicts the current and next actions based on the action history using GPR. For a given action, the actuator takes an action and subsequently the plant’s state is updated according to the dynamics in (1). ### II-B State and Action Communications The UL state and DL action communications are elaborated, in terms of the received signal, signal-to-noise ratio (SNR), scheduling, and age-of- information (AoI) as follows. Noisy State and Action Receptions: At the control time slot $k$, the received signal $\mathbf{y}^{l}_{i,k}$ at the $l$-th communication at the receiver of control system $i$ is represented as $\displaystyle\mathbf{y}^{l}_{i,k}=\sqrt{P^{l}_{i,k}}\mathbf{C}_{i}\mathbf{H}^{l}_{i,k}\mathbf{q}^{l}_{i,k}+\mathbf{n}^{l}_{k},$ (2) where $l\in\\{u,d\\}$ represents a communication indicator between the transmitter-receiver pair in which $l=u$ refers to the UL state communication between the sensor-controller pair while $l=d$ refers to the DL action communication between the controller-actuator pair. The transmitted signal, in the UL state communication (i.e., $\mathbf{q}=\mathbf{x}$ and $l=u$), $\mathbf{x}^{u}_{i,k}=[x^{u}_{i,k}(1)\cdots x^{u}_{i,k}(D)]$ is the plant’s state transmitted by the sensor of a control system $i$ at time $k$ such that $\mathbb{E}\\{\|x^{u}_{i,k}(\iota)\|^{2}\\}=1,\forall\iota\in\\{1,\cdots,D\\}$. The transmitted signal, in the DL action communication (i.e., $\mathbf{q}=\mathbf{u}$, $l=d$), $\mathbf{u}^{d}_{i,k}=[u^{d}_{i,k}(1)\cdots u^{d}_{i,k}(P)]$ is the action transmitted by the controller to an actuator of a control system $i$ at time $k$ such that $\mathbb{E}\\{\|u^{d}_{i,k}(p)\|^{2}\\}=1,\forall p\in\\{1,\cdots,P\\}$. The matrix $\mathbf{H}^{l}_{i,k}\in\mathbb{R}^{\mathcal{F}\times\mathcal{F}}$ represents the wireless channel of the $l$-th communication between the transmitter-receiver pair of a control system $i$ at time $k$, and $\mathcal{F}\in\\{D,P\\}$ represents the dimensions of the transmitted state or action, respectively. The channel is modeled as a Rayleigh block fading which is static and flat-fading within either UL or DL transmission time. Channel statistics are perfectly known at the transmitters and receivers, and $P^{l}_{i,k}\in\left[0,P^{l}_{max}\right]$ is the transmission power of control system $i$ at time $k$ with total transmission power $P^{l}_{max}$. Lastly, $\mathbf{n}^{l}_{k}$ is the additive white Gaussian noise at the receiver with zero-mean and covariance matrix $\mathbb{E}\\{\mathbf{n}_{k}^{l^{T}}\mathbf{n}^{l}_{k}\\}=N_{0}\mathbf{I}_{\mathcal{F}}$, where $N_{0}$ is the measurement noise variance, and $\mathbf{I}_{\mathcal{F}}$ is the $\mathcal{F}\times\mathcal{F}$ identity matrix. The matrix $\mathbf{C}_{i}\in\mathbb{R}^{\mathcal{F}\times\mathcal{F}}$ is the observation matrix of the control system $i$ that equals the identity matrix in the DL action communication and equals the identity matrix in the UL state communication to characterize the full-state observations. The measurement noise $\mathbf{n}^{l}_{k}$ and the plant’s noise $\mathbf{w}_{k}$ in (1) are uncorrelated zero-mean Gaussian noise with unit variance. Hence, the SNR at the receiver of the $l$-th communication of a control system $i$ at time $k$ is given as $\displaystyle\text{SNR}^{l}_{i,k}=\frac{P^{l}_{i,k}\\|\mathbf{H}^{l}_{i,k}\\|^{2}}{N_{0}},$ (3) where SNR in (3) is equivalent to the signal-to-distortion ratio (SDR) as a result of the channel theoretical limit in which the rate-distortion function of the source equals to the channel capacity and the number of source samples is matched to the number of channels under analog uncoded communications [18]. In this work, we consider analog uncoded communications, in which the discrete-time continuous amplitude source samples are amplified and transmitted to a receiver over wireless channel. Compared to digital communications, analog communications are favorable for achieving low latency thanks to skipping encoding and decoding operations, at the cost of signal distortion during channel propagation [18]. Furthermore, analog uncoded communication is robust to channel conditions and performs well at different SNRs compared to digital communication that is sensitive to any degradation in channel conditions. To ensure reliable communication for control stability, the successful decoding of the transmitted signal is described by the indicator function $\mathbb{I}_{\\{\text{SNR}^{l}_{i,k}\geq\text{SNR}^{l}_{\text{th}}\\}}$ for a target SNR threshold $\text{SNR}^{l}_{\text{th}}$. Scheduling and AoI: At time $k$, the centralized scheduler located at the BS and shared among all control systems schedules at most one sensor-controller pair of control system $i$ in the UL state communications, and at most one controller-actuator pair of the control system $i$ in the DL action communications. Let $\alpha^{l}_{i,k}\in\\{0,1\\}$ be the scheduling variable of the $l$-th communication of the control system $i$ at time $k$, where $\alpha^{l}_{i,k}=1$ when the transmitter-receiver pair of the $l$-th communication of control system $i$ is scheduled at time $k$ and $\alpha^{l}_{i,k}=0$ otherwise. The freshness of the received information is measured using AoI, i.e., the number of elapsed time since the generation of the latest received information [19]. AoI is composed of the inter-arrival time that is defined as the time elapsed between two consecutive update generations and the service time defined as the transmission time of update information. In analog uncoded communications, AoI depends only on the inter-arrival time since the service time is deterministic based on channel bandwidth. Hence, the AoI of the $l$-th communication of the control system $i$ at the receiver linearly increases with time if it is not scheduled or its SNR is below a threshold. Formally, the AoI of the $l$-th communication of control system $i$ at the receiver is: $\displaystyle\beta^{l}_{i,k}=\left\\{\begin{array}[]{ll}1+\beta^{l}_{i,k-1},&\text{if}\;\xi^{l}_{i,k}=0,\\\ 1,&\text{o.w.s.},\end{array}\right.$ (6) where $\beta^{l}_{i,k}\in\mathbb{Z}_{++}$ is the AoI of the $l$-th communication of the control system $i$ at time $k$ at the receiver, and $\xi^{l}_{i,k}=\alpha^{l}_{i,k}\,\mathbb{I}_{\\{\text{SNR}^{l}_{i,k}\geq\text{SNR}^{l}_{th}\\}}$ is the transmission indicator variable of $l$-th communication of control system $i$ at time $k$ that depends on both scheduling variable and SNR indicator function. ### II-C State and Action Estimation Over Noisy Communication Links The UL received states and DL received actions are distorted by Rayleigh fading channels. The original signals are estimated using the MMSE estimator as detailed next. When one transmitter-receiver pair of the $l$-th communication of control system $i$ is scheduled, i.e., $\alpha_{i,k}^{l}=1$, the receiver applies the MMSE estimator to restore the original signal from the noisy received signal in (2). The resultant estimated signal $\bar{\mathbf{q}}^{l}_{i,k}$ is given as: $\displaystyle\bar{\mathbf{q}}^{l}_{i,k}=\mathbb{E}\\{\mathbf{q}^{l}_{i,k}\|\mathbf{y}^{l}_{i,k}\\}=\mathbf{G}^{l}_{i,k}\mathbf{y}^{l}_{i,k}=\mathbf{q}^{l}_{i,k}+\mathbf{v}^{l}_{i,k},$ (7) where $\mathbf{G}^{l}_{i,k}\in\mathbb{R}^{\mathcal{F}\times\mathcal{F}}$ is the linear MMSE matrix at the receiver of the $l$-th communication of the control system $i$ at time $k$ that minimizes the mean-squared error (MSE) between the original and estimated signals as $\displaystyle\mathbf{G}^{l}_{i,k}=\sqrt{P^{l}_{i,k}}\mathcal{S}_{q}\mathbf{H}^{l^{T}}_{i,k}\left(P^{l}_{i,k}\mathbf{H}^{l}_{i,k}\mathcal{S}_{q}\mathbf{H}^{l^{T}}_{i,k}+N_{0}\mathbf{I}_{\mathcal{F}}\right)^{-1}$ (8) The term $\mathbf{v}^{l}_{i,k}$ in (7) is the MMSE estimation error following a zero-mean Gaussian random vector with the covariance matrix $\mathbf{V}^{l}_{i,k}\in\mathbb{R}^{\mathcal{F}\times\mathcal{F}}$. Following [20], we assume that $\mathbf{q}^{l}_{i,k}$ follows a zero-mean Gaussian distribution with the covariance matrix $\mathcal{S}_{q}\in\mathbb{R}^{\mathcal{F}\times\mathcal{F}}$, then we have $\displaystyle\mathbf{V}^{l}_{i,k}$ $\displaystyle=\mathbb{E}\\{\mathbf{v}^{l}_{i,k}\mathbf{v}_{i,k}^{l^{T}}\\}=\mathbb{E}\Big{\\{}\left(\bar{\mathbf{q}}^{l}_{i,k}-\mathbf{q}^{l}_{i,k}\right)\left(\bar{\mathbf{q}}^{l}_{i,k}-\mathbf{q}^{l}_{i,k}\right)^{T}\Big{\\}}=\mathcal{S}_{q}-\mathbf{G}^{l}_{i,k}\sqrt{P^{l}_{i,k}}\mathbf{H}^{l}_{i,k}\mathcal{S}_{q}$ (9) ### II-D State and Action Prediction Without Communication When one transmitter-receiver pair of the $l$-th communication of the control system $i$ at time $k$ is not scheduled, i.e., $\alpha^{l}_{i,k}=0$, a receiver applies a number of parallel GPRs proportional to the missing signal dimensions to predict both the missing current signal and the next signal using the previously received signals. Each individual GPR learns the functional relationship $g\in\mathbb{R}$ between the control discrete-time $k^{\prime}\in\mathbb{Z}_{+}$ and each output of the received signal. This means that each output of the MMSE estimated signal $\bar{\mathbf{q}}^{l}_{i,k^{\prime}}$ in (7) is the state observation $\bar{\mathbf{x}}^{u}_{i,k^{\prime}}\in\mathbb{R}^{D}$ in the UL and the action $\bar{\mathbf{u}}^{d}_{i,k^{\prime}}\in\mathbb{R}^{P}$ in the DL. This is accomplished by learning a latent function of the following regression model $\bar{q}^{l}_{i,k^{\prime}}(j)=g_{j}(k^{\prime})+\epsilon,\;j\in\\{1,\cdots,\mathcal{F}\\},\;\forall i,\,l,\,k^{\prime}$, where $\bar{q}^{l}_{i,k^{\prime}}(j)\in\mathbb{R}$ is the $j$-th output of the estimated signal of the $l$-th communication of the control system $i$ at time $k^{\prime}$, $g_{j}$ is a $j$-th output latent function, and $\epsilon\sim\mathcal{N}\left(0,\sigma^{2}_{n}\right)$ is an IID Gaussian noise distribution with zero mean and variance $\sigma^{2}_{n}$ that accounts for the measurements or modeling errors [21]. Specifically, to predict the missing received signal $\bar{\mathbf{q}}^{l}_{i,k}$ of the $l$-th communication of the control system $i$ at test time $k$, we exploit $\mathcal{F}$ individual GPRs, where $\mathcal{F}$ represents the MMSE estimated signal dimensions, and feeding each individual GPR with a training set $\mathcal{D}^{l,j}_{i,n_{l}}$ of each output of the previous received signals associated with its observation time $k^{\prime}$, given as $\mathcal{D}^{l,j}_{i,n_{l}}=\\{(k^{\prime},\xi^{l}_{i,k^{\prime}}\,\bar{q}^{l}_{i,k^{\prime}}(j))\|\,j=1,\cdots,\mathcal{F},\;k^{\prime}=1,\cdots,n_{l},\;i=1,\cdots,M,\;l\in\\{u,d\\}\\}$. Here, $n_{l}=\sum_{k^{\prime}}\xi^{l}_{i,k^{\prime}}$ counts the number of received signals of the $l$-th communication until time $k^{\prime}$ in the training set of the control system $i$. Hence, the last time instant in which the transmitter of the $l$-th communication of the control system transmitted its observation to the receiver is given as $\tilde{n}_{l}=k^{\prime}-\beta^{l}_{i,k^{\prime}}+1$. It is obvious that a large value of AoI decreases the number of observations at the receiver that affects the signal prediction credibility at a particular level. In each individual GPR, according to the Gaussian process (GP) characteristics where any finite subset of random variables taken from a realization of a GP follows a joint Gaussian distribution, each $j$-th output latent function $g_{j}$ of the vector-valued latent function $\mathbf{g}(k)=\left[g_{1}(k)\cdots g_{\mathcal{F}}(k)\right]$ is assumed to follow a GP as $g_{j}\left(k\right)\sim\mathcal{GP}\left(m_{j}\left(k\right),\mathcal{R}_{j}(k,k^{\prime})\right)$, where $m_{j}\left(k\right)$ is the mean function of the $j$-th output of the missing received signal which is usually taken as zero without loss of generality [22], and $\mathcal{R}_{j}(k,k^{\prime})$ is the covariance function of the $j$-th output of the missing received signal between the outputs at time $k$ and $k^{\prime}$ that defines the correlation between the outputs according to the input times. It is noted that the stationary covariance function between the outputs is based on the difference between their corresponding input times $\|k-k^{\prime}\|$ in which the two outputs are strongly correlated if their corresponding input times are sufficiently close to each other. Since we focus on time-series data, we utilize information from previously received signals to describe the current data depending on the past observations. Hence, we use a squared exponential kernel function coupled with a periodic kernel function, to model the correlation between the outputs according to their temporal behaviours, as defined in [22] $\displaystyle\mathcal{R}(k,k^{\prime})=h_{q}^{2}\exp\left[\frac{-\left(k-k^{\prime}\right)^{2}}{2h^{2}_{k}}\right]+\exp\left\\{-2\sin^{2}\left[\nu\pi\left(k-k^{\prime}\right)\right]\right\\},$ (10) where the first term represents the stationary covariance function that depends on when the signal $\|k-k^{\prime}\|$ was received with $h_{k}$ and $h_{q}$ being hyperparameters representing the time-scaling and output-scaling of a squared exponential function, respectively, and the second term gives the periodicity with hyperparameter $\nu$ representing frequency. For a set of j-th output observations $\bar{\mathbf{q}}^{l}_{i}(j)=\\{\bar{q}^{l}_{i,1}(j),\cdots,\bar{q}^{l}_{i,n_{l}}(j)\\}^{T}$ and the associated observation times $\mathbf{k}^{\prime}=\\{1,\cdots,n_{l}\\}^{T}$, the joint distribution of the $j$-th output past observations $\bar{\mathbf{q}}^{l}_{i,k^{\prime}}(j)$ together with the $j$-th output $g_{j}(k)$ at test time $k$ is given as $\displaystyle\left[\begin{array}[]{c}\bar{\mathbf{q}}^{l}_{i}(j)\\\ g_{j}(k)\end{array}\right]\sim\mathcal{N}\left(\left[\begin{array}[]{c}\mathbf{0}\\\ 0\end{array}\right],\left[\begin{array}[]{cc}\mathbf{R}_{j}(\mathbf{k}^{\prime},\mathbf{k}^{\prime})&\mathbf{r}_{j}(\mathbf{k}^{\prime},k)\\\ \mathbf{r}_{j}(k,\mathbf{k}^{\prime})&\mathcal{R}_{j}(k,k)\end{array}\right]\right),$ (17) where $\mathcal{R}_{j}\left(k,k\right)\in\mathbb{R}$ is the prior covariance function of $j$-th output observation at a test time $k$, and $\mathbf{R}_{j}(\mathbf{k}^{\prime},\mathbf{k}^{\prime})\in\mathbb{R}^{n_{l}\times n_{l}}$ is the symmetric and positive semi-definite covariance matrix of $j$-th output past observations with the elements $\mathcal{R}_{j}\left({\mathbf{k}^{\prime}(a),\mathbf{k}^{\prime}(b)}\right)$ for $a,b=1,\cdots,n_{l}$. Following [21], we treat the prediction mean as the $j$-th output predicted signal $\hat{q}^{l}_{i,k}(j)$, the posterior distribution of $g_{j}\left(k\right)$ at test time $k$ based on the training set $\mathcal{D}^{l,j}_{i,n_{l}}$ can be analytically derived as $\displaystyle\text{Pr}\left(g_{j}\left(k\right)\|\mathcal{D}^{l,j}_{i,n_{l}},k,\mathbf{\Theta}_{j}\right)$ $\displaystyle\sim\mathcal{N}\left(\hat{q}^{l}_{i,k}(j),\sigma^{2}_{i,k}(j)\right).$ (18) Following [21], the $j$-th output prediction mean $\hat{q}^{l}_{i,k}\left(j\right)$, and the $j$-th output prediction variance $\sigma^{2}_{i,k}(j)$ are respectively given as $\displaystyle\hat{q}^{l}_{i,k}(j)$ $\displaystyle=\mathbf{r}_{j}(k,\mathbf{k}^{\prime})^{T}\mathbf{R}_{j}(\mathbf{k}^{\prime},\mathbf{k}^{\prime})^{-1}\bar{\mathbf{q}}^{l}_{i}(j)=q^{l}_{i,k}(j)+e^{l}_{i,k}(j),$ (19) $\displaystyle\sigma^{2}_{i,k}(j)$ $\displaystyle=\mathbb{E}\Big{\\{}\left(\hat{q}^{l}_{i,k}(j)-q^{l}_{i,k}(j)\right)\left(\hat{q}^{l}_{i,k}(j)-q^{l}_{i,k}(j)\right)^{T}\Big{\\}}=\mathbb{E}\\{e^{l}_{i,k}(j)e^{l^{T}}_{i,k}(j)\\}$ (20) $\displaystyle=\mathcal{R}_{j}(k,k)-\mathbf{r}_{j}(k,\mathbf{k}^{\prime})\mathbf{R}_{j}(\mathbf{k}^{\prime},\mathbf{k}^{\prime})^{-1}\mathbf{r}_{j}(k,\mathbf{k}^{\prime})^{T},$ where $\mathbf{r}_{j}(\mathbf{k}^{\prime},k)\in\mathbb{R}^{n_{l}\times 1}$ is $j$-th output observation covariance between the outputs at the $n_{l}$ observation times and a test time $k$, and the term $e^{l}_{i,k}(j)$ is $j$-th output prediction error defined as the difference between true and predicted outputs. Moreover, $\mathbf{\Theta}_{j}$ in (18) is the $j$-th output hyperparameters of the covariance function $\mathcal{R}$. Finally, the predicted signal at the receiver of the $l$-th communication of control system $i$ at time $k$ and its prediction error covariance matrix are $\displaystyle\hat{\mathbf{q}}^{l}_{i,k}$ $\displaystyle=\\{\hat{q}^{l}_{i,k}(1)\cdots\hat{q}^{l}_{i,k}(\mathcal{F})\\}^{T}=\mathbf{q}^{l}_{i,k}+\mathbf{e}^{l}_{i,k},$ (21) $\displaystyle\mathbf{\mathcal{J}}^{l}_{i,k}$ $\displaystyle=\left[\begin{array}[]{ccc}\sigma^{2}_{i,k}(1)&\cdots&0\\\ \vdots&\ddots&\vdots\\\ 0&\cdots&\sigma^{2}_{i,k}(\mathcal{F})\end{array}\right].$ (25) ### II-E Action Computation and Actuation By feeding the estimated or predicted state, the controller computes the action using LQR. Then, the actuator applies the estimated or predicted action to stabilize the state as detailed next. Action Computation After State Estimation/Prediction: For a given estimated state (i.e., MMSE output) in (7) or predicted state (i.e., GPR output) in (21), at time $k$, the state $\mathbf{x}_{i,k}^{c}$ available at the controller based on the UL transmission indicator variable is given as $\displaystyle\mathbf{x}_{i,k}^{c}=\xi^{u}_{i,k}\bar{\mathbf{x}}^{u}_{i,k}+\left(1-\xi^{u}_{i,k}\right)\hat{\mathbf{x}}^{u}_{i,k}.$ (26) The received state $\mathbf{x}_{i,k}^{c}$ is used in the LQR located at the controller, and the optimal action of a control system $i$ at time $k$ is given by the following linear feedback control law as $\displaystyle\mathbf{u}^{d}_{i,k}=-\mathbf{\Phi}_{i}\mathbf{x}^{c}_{i,k},$ (27) where $\mathbf{u}^{d}_{i,k}\in\mathbb{R}^{P}$ is the computed action at the controller, $\mathbf{\Phi}_{i}=\left(\mathbf{Z}^{u}+\mathbf{B}^{T}_{i}\mathbf{P}\mathbf{B}_{i}\right)^{-1}\mathbf{B}_{i}^{T}\mathbf{P}\mathbf{A}_{i}$ is the feedback gain matrix of the control system $i$, $\mathbf{Z}^{s}\in\mathbb{S}_{+}^{D\times D}$ is a positive semi-definite weight matrix of the state deviation cost, and $\mathbf{Z}^{u}\in\mathbb{S}_{++}^{P\times P}$ is a positive definite weight matrix of the action cost. The term $\mathbf{P}=\mathbf{A}_{i}^{T}\mathbf{P}\mathbf{A}_{i}-\mathbf{A}_{i}^{T}\mathbf{P}\mathbf{B}_{i}\left(\mathbf{B}^{T}_{i}\mathbf{P}\mathbf{B}_{i}+\mathbf{Z}^{u}\right)^{-1}\mathbf{B}_{i}^{T}\mathbf{P}\mathbf{A}_{i}+\mathbf{Z}^{s}$ is the unique positive definite matrix which satisfies the discrete-time algebraic Riccati equation (DARE). Then, the controller transmits the computed action $\mathbf{u}^{d}_{i,k}$ in (27) to an actuator of control system $i$ at time $k$ in the DL, if $\xi_{i,k}^{d}=1$, as discussed in Sec. II-B. Actuation After Action Estimation/Prediction: For a given estimated control action (i.e., MMSE output) in (7) or predicted action (i.e., GPR output) in (21), at time $k$, the action $\mathbf{u}_{i,k}^{a}$ available at the actuator based on the DL transmission indicator variable is given as $\displaystyle\mathbf{u}_{i,k}^{a}=\xi^{d}_{i,k}\,\bar{\mathbf{u}}^{d}_{i,k}+\left(1-\xi^{d}_{i,k}\right)\hat{\mathbf{u}}^{d}_{i,k}.$ (28) Note that the UL and DL transmission indicator variables are periodically generated by the centralized scheduler in which, within each unit control time duration, the UL state communication can be firstly activated for sensing the plant’s state based on the UL transmission indicator variable. Then, the DL action communication can be activated for actuation based on the DL transmission indicator variable. Consequently, for a given pair of UL and DL transmission indicator variables with (26) and (28), the actuator takes a control action that changes the plant’s state of the control system $i$ at time $k$ in (1) into four cases of state evolution as follows: $\displaystyle\mathbf{x}^{o}_{i,k+1}=\mathbf{A}_{i}\mathbf{x}^{u}_{i,k}-\mathbf{B}_{i}(\mathbf{\Phi}_{i}\hat{\mathbf{x}}^{u}_{i,k}+\mathbf{e}^{d}_{i,k})+\mathbf{w}_{k},\;$ $\displaystyle\text{if}\;\xi^{u}_{i,k}=0\;\text{and}\;\xi^{d}_{i,k}=0,\;\text{ (open-loop)},$ (29) $\displaystyle\mathbf{x}^{s}_{i,k+1}=\mathbf{A}_{i}\mathbf{x}^{u}_{i,k}-\mathbf{B}_{i}(\mathbf{\Phi}_{i}\bar{\mathbf{x}}^{u}_{i,k}+\mathbf{e}^{d}_{i,k})+\mathbf{w}_{k},\;$ $\displaystyle\text{if}\;\xi^{u}_{i,k}=1\;\text{and}\;\xi^{d}_{i,k}=0,\;\text{ (sensing-loop)},$ (30) $\displaystyle\mathbf{x}^{a}_{i,k+1}=\mathbf{A}_{i}\mathbf{x}^{u}_{i,k}-\mathbf{B}_{i}(\mathbf{\Phi}_{i}\hat{\mathbf{x}}^{u}_{i,k}+\mathbf{v}^{d}_{i,k})+\mathbf{w}_{k},\;$ $\displaystyle\text{if}\;\xi^{u}_{i,k}=0\;\text{and}\;\xi^{d}_{i,k}=1,\;\text{ (actuating-loop)},$ (31) $\displaystyle\mathbf{x}^{c}_{i,k+1}=\mathbf{A}_{i}\mathbf{x}^{u}_{i,k}-\mathbf{B}_{i}(\mathbf{\Phi}_{i}\bar{\mathbf{x}}^{u}_{i,k}+\mathbf{v}^{d}_{i,k})+\mathbf{w}_{k},\;$ $\displaystyle\text{if}\;\xi^{u}_{i,k}=1\;\text{and}\;\xi^{d}_{i,k}=1,\;\text{ (closed-loop)}.$ (32) Figure 2: Timing diagram of the control system. First diagram illustrates the UL and DL transmission indicator variables generated by scheduler, second diagram illustrates uniform sampling by a sensor, third diagram illustrates the received/predicted state and calculated action at a controller. Fourth diagram illustrates the received/predicted action at an actuator. Timing diagram: Based on the UL and DL transmission indicator variables within each unit control time duration, the timing diagram of a control system is illustrated in Fig. 2. The centralized scheduler shared among all control systems, within each unit control time duration, primarily transmits the UL and DL transmission indicator variables to the sensor, controller, and actuator nodes of all control systems. Then, the state is only transmitted by the sensor if the control system has a reliable UL communication and has valuable information affecting the control stability (i.e, $\xi^{u}_{i,k}=1$) which result in saving wireless communication resources. After that, LQR located at the controller computes the action based on the state available at the controller in (26) in which the predicted state, if $\xi^{u}_{i,k}=0$, is applied to the LQR. Lastly, the action is transmitted by the controller if has a reliable DL communication and valuable information affecting the control stability (i.e, $\xi^{d}_{i,k}=1$). This is a result of assuming that the UL and DL transmission indicator variables as being periodically transmitted by the centralized scheduler every control time duration, the controller periodically calculates the action depending on the available state, and the actuator periodically applies the action depending on the available action. Moreover the discrete-time control time $k$ equals the continuous-time control time duration unit $\Delta k$ comprising the UL and DL transmission times while ignoring the computational delay. ## III Communication Control Co-design ### III-A Control-constrained Problem Formulation Our objective is to minimize the total communication cost per control system subject to ensuring communication reliability and control stability. The total communication cost incorporates the AoI and transmission power since the AoI indirectly affects the control stability through the GPR prediction stability and wireless resources consumption, while transmit power affects communication reliability and energy consumption. Formally speaking, we have: $\displaystyle\mathcal{C}\left(\\{\bar{\beta}^{l}_{i}\\},\\{\bar{\hat{P}}^{l}_{i}\\}\right)=\omega_{\beta_{l}}\sum_{i=1}^{M}\mathcal{G}_{\beta}(\bar{\beta}^{l}_{i})+\omega_{P_{l}}\sum_{i=1}^{M}\mathcal{G}_{P}(\bar{\hat{P}}^{l}_{i}),\qquad\forall l\in\\{u,d\\},$ (33) where the non-decreasing concave functions $\mathcal{G}_{\beta}(\beta)=\log(1+\beta)$ and $\mathcal{G}_{P}(\hat{P})=\log(1+\hat{P})$ are proportionally fair cost functions of the AoI and the transmission power function for each control system, respectively [23]. The transmission power function that depends on the scheduling variable is given as $\hat{P}^{l}_{i,k}=\alpha^{l}_{i,k}P^{l}_{i,k}$, and the given positive weights $\omega_{\beta_{l}}$ and $\omega_{P_{l}}$ adjust the relative importance of the corresponding cost functions. Throughout this work, the following notation for the long-term time-averaged of any quantity $z$ is defined as $\bar{z}\triangleq\underset{K\to\infty}{\lim\sup}\frac{1}{K}\sum_{k=1}^{K}z$. In particular, $\bar{\beta}^{l}_{i}$ and $\bar{\hat{P}}^{l}_{i}$ are the long- term time-averaged of $\beta^{l}_{i}$ and $\hat{P}^{l}_{i}$, respectively. To evaluate control stability, we consider the quadratic Lyapunov function that measures the performance of each control system as a function of the state expressed as $\displaystyle\mathcal{L}(\mathbf{x}^{u}_{i,k})=\mathbf{x}^{u^{T}}_{i,k}\,\mathcal{Z}\,\mathbf{x}^{u}_{i,k},\qquad\forall\mathcal{Z}\in\mathbb{S}^{D}_{++},$ (34) where $\mathcal{Z}\in\mathbb{S}^{D}_{++}$ is a unique positive definite solution to the discrete Lyapunov equation $\mathbf{A}^{c^{T}}_{i}\mathcal{Z}+\mathcal{Z}\mathbf{A}^{c}_{i}=-\mathbb{I}_{D}$, and $\mathbf{A}^{c}_{i}$ is a closed-loop state transition matrix defined as $\mathbf{A}^{c}_{i}=\mathbf{A}_{i}-\mathbf{B}_{i}\mathbf{\Phi}_{i}$. Because the centralized scheduler has only access to the predicted state, the expected current value of $\mathcal{L}(\mathbf{x}^{u}_{i,k})$ is calculated in the following lemma. ###### Lemma 1. Given the predicted state $\hat{\mathbf{x}}^{u}_{i,k}$ and the state prediction error covariance matrix $\mathcal{J}^{u}_{i,k}$ at the controller, the expected current value of $\mathcal{L}(\mathbf{x}^{u}_{i,k})$ is given as $\displaystyle\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{u}_{i,k})\|\hat{\mathbf{x}}^{u}_{i,k}\right]=\\|\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\mathcal{Z}\,\mathcal{J}^{u}_{i,k}\right],\qquad\forall\;\mathcal{Z}\in\mathbb{S}^{D}_{++}.$ (35) ###### Proof. Please refer to Appendix.A ∎ Note that the expected current value of $\mathcal{L}(\mathbf{x}^{u}_{i,k})$ of the control system $i$ at time $k$ naturally grows as the predicted state and the prediction error get larger as a result of increasing AoI and/or the insufficiency of received observations number in the training set. The requirement of control stability is that the expected future value of $\mathcal{L}(\mathbf{x}^{u}_{i,k+1})$ should decrease at a given rate $\zeta_{i}\in(0,1]$ of its expected current value of $\mathcal{L}(\mathbf{x}^{u}_{i,k})$, which means the state of the control system is monotonically decreasing along trajectories, as $\displaystyle\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{u}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},\mathbf{H}^{d}_{i,k},P^{u}_{i,k},P^{d}_{i,k}\right]\leq\zeta_{i}\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{u}_{i,k})\|\hat{\mathbf{x}}^{u}_{i,k}\right],$ (36) where the expectation in the right hand side of (36) is with respect to the plant noise $\mathbf{w}_{k}$ in (1), the signal estimation error $\mathbf{v}^{l}_{i,k}$ defined in (9), and the signal prediction error $\mathbf{e}^{l}_{i,k}$ defined in (21). According to the objective function in (33) and the control stability constraint in (36), the control-constrained optimization problem can be formulated as follows: $\displaystyle\underset{\mathbf{a}_{k}^{l},\mathbf{P}_{k}^{l}}{(\mathcal{P}1)\quad\text{Minimize}}\mathcal{C}\left(\\{\bar{\beta}^{l}_{i}\\},\\{\bar{\hat{P}}^{l}_{i}\\}\right)$ (37a) $\displaystyle\hskip 40.0pt\text{subject to:}\quad 0\leq P^{l}_{i,k}\leq P^{l}_{max},\qquad\qquad\qquad\qquad\quad\;\;\forall\,l\in\\{u,d\\},i\in\mathcal{M},\,k,$ (37b) $\displaystyle\hskip 95.0pt\\|\mathbf{H}^{l}_{i,k}\\|^{2}\,P^{l}_{i,k}/N_{0}\geq\text{SNR}^{l}_{th},\qquad\qquad\quad\;\;\;\forall\,l\in\\{u,d\\},i\in\mathcal{M},\,k,$ (37c) $\displaystyle\hskip 95.0pt\alpha^{l}_{i,k}\in\\{0,1\\},\qquad\qquad\qquad\qquad\qquad\quad\;\forall\,l\in\\{u,d\\},i\in\mathcal{M},\,k,$ (37d) $\displaystyle\hskip 95.0pt\sum^{M}_{i=1}\alpha^{l}_{i,k}\leq 1,\qquad\qquad\qquad\qquad\qquad\quad\;\forall\,l\in\\{u,d\\},i\in\mathcal{M},\,k,$ (37e) $\displaystyle\hskip 50.0pt\eqref{Future_Lyapunov},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\forall\;i\in\mathcal{M},\,k,$ where $\mathbf{a}_{k}^{l}=\\{\alpha^{l}_{i,k}:\forall l\in\\{u,d\\},\;i\in\mathcal{M}\\}$ and $\mathbf{P}_{k}^{l}=\\{P^{l}_{i,k}:\forall l\in\\{u,d\\},\;i\in\mathcal{M}\\}$ are the UL-DL scheduling vector at time $k$, and the UL-DL transmission power vector at time $k$, respectively. The constraint in (37b) bounds the UL-DL transmission power allocation of a control system $i$ at time $k$ by the total available transmission power $P^{l}_{max}$, while the constraint in (37c) ensures the communication reliability that is based on SNR or SDR in analog uncoded communications. The constraints in (37d)-(37e) ensure at most one transmitter-receiver pair of a control system $i$ is scheduled at time $k$. The constraint in (36) ensures the state is decreasing along trajectories that satisfies the control stability. It is noted that control stability constraint in (36) is independent with communication constraints in (37b)-(37e). However, the control stability constraint is affected and determined by the communication and scheduling variables, hence the original control-constrained problem $\mathcal{P}1$ is rewritten after directly reflecting the communication control relationship of the constraint (36) in the following lemma. ###### Lemma 2. Given predicted state $\hat{\mathbf{x}}^{u}_{i,k}$, state prediction error covariance matrix $\mathcal{J}^{u}_{i,k}$, the predicted action $\hat{\mathbf{u}}^{d}_{i,k}$, the action prediction error covariance matrix $\mathcal{J}^{d}_{i,k}$, the channel between the sensor-controller pair $\mathbf{H}^{u}_{i,k}$, the channel between the controller-actuator pair $\mathbf{H}^{d}_{i,k}$, the UL transmission power $P^{u}_{i,k}$, and the DL transmission power $P^{d}_{i,k}$, the control stability constraint in (36) is satisfied IFF the following conditions on the transmission indicator variables hold, i.e., $\displaystyle\textstyle{\underset{K\to\infty}{\lim\sup}\frac{1}{K}\sum\limits_{k=1}^{K}\xi^{u}_{i,k}\geq\underset{K\to\infty}{\lim\sup}\frac{1}{K}\sum\limits_{k=1}^{K}\frac{\\|\left(\mathbf{A}^{c}_{i}-\zeta_{i}\mathbf{I}_{D}\right)\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\left(\mathbf{A}^{T}_{i}\mathcal{Z}\mathbf{A}_{i}-\zeta_{i}\mathcal{Z}\right)\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]}{\text{Tr}\left[\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)^{T}\mathcal{Z}\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)\mathcal{J}^{u}_{i,k}\right]-\text{Tr}\left[\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)^{T}\mathcal{Z}\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)\mathbf{V}^{u}_{i,k}\right]}},$ (38) $\displaystyle\textstyle{\underset{K\to\infty}{\lim\sup}\frac{1}{K}\sum\limits_{k=1}^{K}\xi^{d}_{i,k}\geq\underset{K\to\infty}{\lim\sup}\frac{1}{K}\sum\limits_{k=1}^{K}\frac{\\|\left(\mathbf{A}^{c}_{i}-\zeta_{i}\mathbf{I}_{D}\right)\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\left(\mathbf{A}^{T}_{i}\mathcal{Z}\mathbf{A}_{i}-\zeta_{i}\mathcal{Z}\right)\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]}{\text{Tr}\left[\mathbf{B}_{i}^{T}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]-\text{Tr}\left[\mathbf{B}_{i}^{T}\mathcal{Z}\mathbf{B}_{i}\mathbf{V}^{d}_{i,k}\right]}},$ (39) $\displaystyle\textstyle{\underset{K\to\infty}{\lim\sup}\frac{1}{K}\sum\limits_{k=1}^{K}\xi^{u}_{i,k}\xi^{d}_{i,k}\geq\underset{K\to\infty}{\lim\sup}\frac{1}{K}\sum\limits_{k=1}^{K}\frac{\\|\left(\mathbf{A}^{c}_{i}-\zeta_{i}\mathbf{I}_{D}\right)\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\left(\mathbf{A}^{T}_{i}\mathcal{Z}\mathbf{A}_{i}-\zeta_{i}\mathcal{Z}\right)\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]}{\text{Tr}\left[\left((\mathbf{B}_{i}\mathbf{\Phi}_{i})^{T}\mathcal{Z}(\mathbf{B}_{i}\mathbf{\Phi}_{i})\right)(\mathcal{J}^{u}_{i,k}-\mathbf{V}^{u}_{i,k})\right]+\text{Tr}\left[\mathbf{B}_{i}^{T}\mathcal{Z}\mathbf{B}_{i}(\mathcal{J}^{d}_{i,k}-\mathbf{V}^{d}_{i,k}\right]}}.$ (40) ###### Proof. Please refer to Appendix. B ∎ Note that the conditions on the UL and DL transmission indicator variables in (38) and (39), respectively, ensure the control stability constraint in (36) in a decoupled scheduling between the UL and DL communications based on the current predicted control and channel states, while the condition on both the UL and DL scheduling variables in (40) ensures control stability in a coupled scheduling between the UL and DL communications. Intuitively, the growth of AoI at a controller/actuator leads to an increasing state/action prediction error increasing due to an outdated training set. Therefore, the transmitter- receiver pair of a control system should be scheduled when it has a reliable communication link and the state/action prediction error is greater than the state/action estimation error to ensure control stability. The actuator is physically decoupled from the centralized scheduler and the controller which is co-located at BS, and the DL indicator variable at the centralized scheduler relies on the action prediction error at the actuator. Hence, the controller leverages another GPR, where the input of this GPR is the discrete-time associated with the generated action by LQR plus the action estimation error as $\bar{\mathbf{u}}^{d_{c}}_{i,k^{\prime}}=\mathbf{u}^{d}_{i,k^{\prime}}+\mathbf{v}^{d}_{i,k^{\prime}}$. As a result of the applied input to this GPR that yields a training set similar to the one at the actuator as $\mathcal{D}^{d_{c}}_{i,n_{d}}=\\{(k^{\prime},\xi^{d}_{i,k^{\prime}}\,\bar{\mathbf{u}}^{d_{c}}_{i,k^{\prime}})\|\,k^{\prime}=1,\cdots,n_{d},i=1,\cdots,M\\}$, we obtain the action prediction error similar to the one generated at actuator side. ### III-B Joint Communication and Control Problem According to the UL and DL transmission indicator variables constraints in (38)-(40) that result from the control stability constraint in (36) in problem $\mathcal{P}1$, problem $\mathcal{P}1$ is rewritten as $\displaystyle\underset{\mathbf{a}_{k}^{l},\mathbf{P}_{k}^{l}}{(\mathcal{P}2)\quad\text{Minimize}}\mathcal{C}\left(\\{\bar{\beta}^{l}_{i}\\},\\{\bar{\hat{P}}^{l}_{i}\\}\right)$ (41a) $\displaystyle\text{subject to:}\quad\bar{\alpha}^{u}_{i}\geq\bar{\mathcal{G}}_{lb}\left(\mathfrak{m}^{u}_{i,k}\right),\;\;\;\forall i\in\mathcal{M},$ (41b) $\displaystyle\hskip 45.0pt\bar{\alpha}^{d}_{i}\geq\bar{\mathcal{G}}_{lb}\left(\mathfrak{m}^{d}_{i,k}\right),\;\;\;\forall i\in\mathcal{M},$ (41c) $\displaystyle\hskip 45.0pt\overline{\alpha^{u}_{i}\alpha^{d}_{i}}\geq\bar{\mathcal{G}}_{lb}\left(\mathfrak{m}_{i,k}\right),\;\forall i\in\mathcal{M},$ (41d) $\displaystyle~{}\eqref{eqopt1e_1a}~{}-~{}\eqref{eqopt1e_1d},$ where $\bar{\alpha}^{u}_{i}$ and $\bar{\alpha}^{d}_{i}$ are the time-averaged of the UL and DL scheduling variables, respectively, $\mathfrak{m}^{u}_{i,k}$, $\mathfrak{m}^{d}_{i,k}$, $\mathfrak{m}_{i,k}$ are the lower-bound stability of the uplink, downlink, and coupling transmission indicator variables of a control system $i$ at time $k$ in (38), (39), and (40) respectively. $\bar{\mathcal{G}}_{lb}$ is the time-averaged of the lower-bound function $\mathcal{G}_{lb}$ that is defined as $\mathcal{G}_{lb}\left(.\right)=\max\left[\min\left(.,1\right),0\right]$ to ensure the feasibility of the scheduling constraints. The transmission indicator variable in (38)-(40) is only written as a function of the scheduling variables since the SNR indicator function is satisfied in (37c). The stochastic problem $\mathcal{P}2$ is a mixed-integer non-convex problem where the source of stochasticity is due to the observed channel and predicted state at each time $k$. Moreover, the scheduling decision constraint in (37d) not only depends on its own decision but on all others scheduling decisions. Hence, a dynamic control algorithm is proposed in the next section to find the optimal scheduling vector and the optimal transmission power vector of problem $\mathcal{P}2$ utilizing Lyapunov optimization framework. ## IV Dynamic Control Algorithm Using Lyapunov Optimization In this section, we propose a dynamic control algorithm using the stochastic Lyapunov optimization framework to solve problem $\mathcal{P}2$. However, the problem involves minimizing a weighted sum of non-decreasing concave functions of the time-averaged AoI and transmission power. Based on the dynamic stochastic optimization theory [24], it can be transformed into an equivalent problem that involves minimizing a time-averaged cost function of instantaneous AoI and transmission power. This transformation is achieved through the use of non-negative auxiliary variables $\gamma^{\beta^{l}}_{i,k}$ and $\gamma^{P^{l}}_{i,k}$ and corresponding virtual queues $Q^{\beta^{l}}_{i,k}$ and $Q^{P^{l}}_{i,k}$ with queue dynamics as $\displaystyle Q^{\beta^{l}}_{i,k+1}=\max\Big{\\{}Q^{\beta^{l}}_{i,k}-\gamma^{\beta^{l}}_{i,k},0\Big{\\}}+\beta^{l}_{i,k},\qquad\forall l\in\\{u,d\\},i\in\mathcal{M},k,$ (42a) $\displaystyle Q^{P^{l}}_{i,k+1}=\max\Big{\\{}Q^{P^{l}}_{i,k}-\gamma^{P^{l}}_{i,k},0\Big{\\}}+\hat{P}^{l}_{i,k},\qquad\forall l\in\\{u,d\\},i\in\mathcal{M},k,$ (42b) where $\hat{P}^{l}_{i,k}$ will be optimized at each time $k$. Then, the transformed problem is given as $\displaystyle\underset{\mathbf{a}_{k}^{l},\mathbf{P}_{k}^{l},\mathbf{r}^{\beta^{l}},\mathbf{r}^{P^{l}}}{(\mathcal{P}3)\quad\text{Minimize}}\overline{\mathcal{C}\left(\\{\gamma^{\beta^{l}}_{i,k}\\},\\{\gamma^{P^{l}}_{i,k}\\}\right)}$ (43a) $\displaystyle\text{subject to:}\qquad\bar{\beta}^{l}_{i}\leq\bar{\gamma}_{i}^{\beta^{l}},\qquad\qquad\quad\forall l\in\\{u,d\\},i\in\mathcal{M}$ (43b) $\displaystyle\bar{\hat{P}}^{l}_{i}\leq\bar{\gamma}_{i}^{P^{l}},\qquad\qquad\quad\forall l\in\\{u,d\\},i\in\mathcal{M}$ (43c) $\displaystyle 1\leq\gamma_{i,k}^{\beta^{l}}\leq B_{max},\qquad\forall l\in\\{u,d\\},i\in\mathcal{M},k$ (43d) $\displaystyle 0\leq\gamma_{i,k}^{P^{f}}\leq P^{l}_{max},\qquad\forall l\in\\{u,d\\},i\in\mathcal{M},k$ (43e) $\displaystyle~{}\eqref{eqopt1e_1a}-\eqref{eqopt1e_1d},~{}\eqref{eqopt1_1a}-\eqref{eqopt1_1c},$ where $\mathbf{r}_{k}^{\beta^{l}}=\\{\gamma^{\beta^{l}}_{i,k}:l\in\\{u,d\\},i\in\mathcal{M}\\}$ and $\mathbf{r}_{k}^{P^{l}}=\\{\gamma^{P^{l}}_{i,k}:l\in\\{u,d\\},i\in\mathcal{M}\\}$ are the vectors of the introduced auxiliary variables. The constraints in (43d) and (43e) are introduced to bound the auxiliary variables. These constraints can be satisfied by ensuring the stability of their virtual queues, since the lower-bound of these constraints can be viewed as the arrival rate of their virtual queues, while the upper-bound can be viewed as the service rate of such virtual queues [24]. Following [24], the problem $\mathcal{P}2$ and the transformed problem $\mathcal{P}3$ are equivalent in which the optimal solution of $\mathcal{P}3$ can be directly turned into an optimal solution of $\mathcal{P}2$. To handle UL and DL scheduling variables constraints in (41b)$-$(41d) associated with the control stability constraint in (36), the virtual queues $Q^{C^{l}}_{i,k}$ and $Q^{C}_{i,k}$ are introduced for all control systems whose dynamics are $\displaystyle Q^{C^{l}}_{i,k+1}=\max\\{Q^{C^{l}}_{i,k}-\alpha^{l}_{i,k},0\\}+\mathcal{G}_{lb}(m^{l}_{i,k}),\qquad\forall l\in\\{u,d\\},i\in\mathcal{M},k,$ (44a) $\displaystyle Q^{C}_{i,k+1}=\max\\{Q^{C}_{i,k}-\alpha^{u}_{i,k}\alpha^{d}_{i,k},0\\}+\mathcal{G}_{lb}(m_{i,k}),\qquad\qquad\;\;\;\forall i\in\mathcal{M},k,$ (44b) where $\alpha^{l}_{i,k}$ will be optimized at each time $k$. The constraints in (41b)$-$(41c) can be satisfied, if their virtual queues are mean-rate stable, i.e., their time-averaged arrival rate is not larger than its time- averaged service rate [24]. At this point, the dynamic stochastic optimization is applied to solve the transformed problem $\mathcal{P}3$, which minimizes a weighted sum of the time-averaged cost function of instantaneous AoI and transmission power subject to the virtual queues stability constraints and the original problem constraints in (37b)$-$(37e). In this regard, we define $\mathbf{Q}^{\beta^{l}}_{k}$, $\mathbf{Q}^{P^{l}}_{k}$, $\mathbf{Q}^{C}_{k}$, and $\mathbf{Q}^{C^{l}}_{k}$ as a vector of all virtual queues $Q^{\beta^{l}}_{i,k}$, $Q^{P^{l}}_{i,k}$, $Q^{C}_{i,k}$, and $Q^{C^{l}}_{i,k}$ for all control systems, respectively. We denote the combined queue vector of all virtual queues at time $k$ by ${\scriptstyle\mathcal{X}_{k}=\left[\mathbf{Q}^{\beta^{l}}_{k},\mathbf{Q}^{P^{l}}_{k},\mathbf{Q}^{C}_{k},\mathbf{Q}^{C^{l}}_{k}\right]}$, and express the conditional Lyapunov drift-plus-penalty as $\small\Delta\left(\mathcal{X}_{k}\right)=\mathbb{E}\left[\mathcal{L}\left(\mathcal{X}_{k+1}\right)-\mathcal{L}\left(\mathcal{X}_{k}\right)+V\mathcal{C}\left(\\{\gamma^{\beta^{l}}_{i,k}\\},\\{\gamma^{P^{l}}_{i,k}\\}\right)\Big{\|}\mathcal{X}_{k}\right],$ (45) where $\mathcal{L}\left(\mathcal{X}_{k}\right)$ is the quadratic Lyapunov function of $\mathcal{X}_{k}$ that measures the virtual queues congestion in a scalar metric and is defined as $\mathcal{L}\left(\mathcal{X}_{k}\right)=\frac{1}{2}\sum^{M}_{i=1}\Big{[}(Q^{\beta^{l}}_{i,k})^{2}+(Q^{P^{l}}_{i,k})^{2}+(Q^{C}_{i,k})^{2}+(Q^{C^{l}}_{i,k})^{2}\Big{]}$. $V\geq 0$ controls the trade-off between minimizing the objective function and stabilizing the virtual queues. Subsequently, plugging the inequalities $\left(\max\left[a-b,0\right]+c\right)^{2}\leq a^{2}+b^{2}+c^{2}-2a\left(b-c\right)$, $\forall a,b,c\geq 0$, $\left(\max\left(a,0\right)\right)^{2}\leq a^{2}$, and all virtual queue dynamics into (45), we derive $\displaystyle\eqref{eq46}$ $\displaystyle\leq{B}+\mathbb{E}\Big{[}\sum^{M}_{i=1}\left(V\omega_{\beta_{l}}\mathcal{G}_{\beta}\left(\gamma^{\beta^{l}}_{i,k}\right)-Q^{\beta^{l}}_{i,k}\gamma^{\beta^{l}}_{i,k}\right)\|\mathcal{X}_{k}\Big{]}+\mathbb{E}\Big{[}\sum^{M}_{i=1}\left(V\omega_{P_{l}}\mathcal{G}_{P}\left(\gamma^{P^{l}}_{i,k}\right)-Q^{P^{l}}_{i,k}\gamma^{P^{l}}_{i,k}\right)\|\mathcal{X}_{k}\Big{]}$ (46) $\displaystyle\qquad\,+\mathbb{E}\Big{[}\sum^{M}_{i=1}Q^{\beta^{l}}_{i,k}\beta^{l}_{i,k}\|\mathcal{X}_{k}\Big{]}+\mathbb{E}\Big{[}\sum^{M}_{i=1}Q^{P^{l}}_{i,k}\hat{P}^{l}_{i,k}\|\mathcal{X}_{k}\Big{]}-\mathbb{E}\Big{[}\sum^{M}_{i=1}Q^{C^{l}}_{i,k}\left(\alpha^{l}_{i,k}-\mathcal{G}_{lb}(m^{l}_{i,k})\right)\|\mathcal{X}_{k}\Big{]}$ $\displaystyle\qquad\,-\mathbb{E}\Big{[}\sum^{M}_{i=1}Q^{C}_{i,k}\left(\alpha^{u}_{i,k}\alpha^{d}_{i,k}-\mathcal{G}_{lb}(m_{i,k})\right)\|\mathcal{X}_{k}\Big{]}.$ The constant B details in (46) is omitted since it does not affect the system performance in the Lyapunov optimization. A solution to $\mathcal{P}3$ can be obtained by minimizing the upper-bound (46) at each time as $\displaystyle\underset{\hskip 25.0pt\mathbf{a}^{l},\mathbf{P}^{l},\mathbf{r}^{\beta^{l}},\mathbf{r}^{P^{l}}}{(\mathcal{P}4)\quad\text{Minimize}}\quad\sum^{M}_{i=1}\Bigg{[}\left(V\omega_{\beta_{l}}\mathcal{G}_{\beta}\left(\gamma^{\beta^{l}}_{i,k}\right)-Q^{\beta^{l}}_{i,k}\gamma^{\beta^{l}}_{i,k}\right)+\left(V\omega_{P_{l}}\mathcal{G}_{P}\left(\gamma^{P^{l}}_{i,k}\right)-Q^{P^{l}}_{i,k}\gamma^{P^{l}}_{i,k}\right)+Q^{\beta^{l}}_{i,k}\beta^{l}_{i,k}$ (47a) $\displaystyle\qquad\qquad\qquad\qquad\qquad+Q^{P^{l}}_{i,k}\hat{P}^{l}_{i,k}-Q^{C^{l}}_{i,k}\left(\alpha^{l}_{i,k}-\mathcal{G}_{lb}(m^{l}_{i,k})\right)-Q^{C}_{i,k}\left(\alpha^{u}_{i,k}\alpha^{d}_{i,k}-\mathcal{G}_{lb}(m_{i,k})\right)\Bigg{]}$ $\displaystyle\text{subject to:}\qquad~{}\eqref{eqopt1e_1a}-\eqref{eqopt1e_1d}~{}\text{and}~{}\eqref{eqopt2_1c}-\eqref{eqopt2_1d}.$ The optimality of problem $\mathcal{P}4$ is asymptotically approached by increasing $V$ [24]. Since the problem $\mathcal{P}4$ is of separable structure, which motivate us to determine the AoI auxiliary vector $\mathbf{r}^{\beta^{l}}$, transmission power auxiliary vector $\mathbf{r}^{P^{l}}$, scheduling vector $\mathbf{a}^{l}$, and transmission power vector $\mathbf{P}^{l}$ in an alternative optimization form. Hence, the overall minimization problem $\mathcal{P}4$ can be decomposed into two separate sub-problems that can be solved concurrently with the observation of the virtual queues, control, and channel states. #### IV-1 Auxiliary Variable Sub-Problems The first decomposed sub-problem is the AoI auxiliary sub-problem, while the second decomposed sub-problem is the transmission power sub-problem. Since the auxiliary variables of such problems are separated and independent among different control systems, their minimization sub-problems can be decoupled to be computed for each control system separately as the following convex problems $\displaystyle\underset{\hskip 45.0pt\gamma^{\beta^{l}}_{i,k}}{(\mathcal{P}4.1)\quad\text{Minimize}}\quad V\omega_{\beta_{l}}\mathcal{G}_{\beta}\left(\gamma^{\beta^{l}}_{i,k}\right)-Q^{\beta^{l}}_{i,k}\gamma^{\beta^{l}}_{i,k}$ (48a) $\displaystyle\hskip 20.0pt\text{subject to:}\quad\quad 1\leq\gamma_{i,k}^{\beta^{l}}\leq B_{max},$ (48b) $\displaystyle\underset{\hskip 45.0pt\gamma^{P^{l}}_{i,k}}{(\mathcal{P}4.2)\quad\text{Minimize}}\quad V\omega_{P_{l}}\mathcal{G}_{P}\left(\gamma^{P^{l}}_{i,k}\right)-Q^{P^{l}}_{i,k}\gamma^{P^{l}}_{i,k}$ (49a) $\displaystyle\hskip 20.0pt\text{subject to:}\quad\quad 0\leq\gamma_{i,k}^{P^{l}}\leq P^{l}_{max}.$ (49b) The optimal AoI auxiliary variables are obtained by differentiating the objective functions of these problems. Let $\mathcal{A}(\gamma^{\beta^{l}})=V\omega_{\beta_{l}}\log(1+\gamma^{\beta^{l}})-Q^{\beta^{l}}_{i,k}\gamma^{\beta^{l}}_{i,k}$ and $\gamma^{\beta^{l^{}}}_{i,k}$ denotes the solution of $\mathcal{A}(\gamma^{\beta^{l}})$ as $\acute{\mathcal{A}}(\gamma^{\beta^{l}})=\frac{V\omega_{\beta_{l}}}{\left(1+\gamma^{\beta^{l}}_{i,k}\right)}-Q^{\beta^{l}}_{i,k}=0$, the optimal AoI auxiliary variable of $\mathcal{P}4.1$ is given as $\small\gamma^{\beta^{l^{}}}_{i,k}=\min\left\\{\max\left\\{\frac{V\omega_{\beta_{l}}-Q^{\beta^{l}}_{i,k}}{Q^{\beta^{l}}_{i,k}},1\right\\},B_{max}\right\\},\qquad\forall l\in\\{u,d\\},i\in\mathcal{M},k.$ (50) Similarly, by letting $\mathcal{A}(\gamma^{P^{l}})=V\omega_{P^{l}}\log(1+\gamma^{P^{l}})\,-\,Q^{P^{l}}_{i,k}\gamma^{P^{l}}_{i,k}$ and $\gamma^{P^{l^{}}}_{i,k}$ denotes the solution of $\mathcal{A}(\gamma^{P^{l}})$ as $\acute{\mathcal{A}}(\gamma^{P^{l}})=\frac{V\omega_{P^{l}}}{\left(1+\gamma^{P^{l}}_{i,k}\right)}-Q^{P^{l}}_{i,k}=0$, the optimal transmission power auxiliary variable of $\mathcal{P}4.2$ is given as $\small\gamma^{P^{l^{}}}_{i,k}=\min\left\\{\max\left\\{\frac{V\omega_{P^{l}}-Q^{P^{l}}_{i,k}}{Q^{P^{l}}_{i,k}},0\right\\},P^{l}_{max}\right\\},\qquad\forall l\in\\{u,d\\},i\in\mathcal{M},k.$ (51) #### IV-2 Scheduling Decision and Transmission Power Sub-Problems The optimal UL-DL scheduling variables and the optimal UL-DL transmission power variables are obtained by minimizing the remaining terms of the objective function of problem $\mathcal{P}4$ at each time subject to the scheduling and transmission power constraints in (37b)$-$(37e), which is expressed as $\displaystyle\small\underset{\hskip 40.0pt\mathbf{a}^{l}_{i,k},\mathbf{P}^{l}_{i,k}}{(\mathcal{P}4.3)\quad\text{Minimize}}\;\sum^{M}_{i=1}\left[Q^{\beta^{l}}_{i,k}\beta^{l}_{i,k}+Q^{P^{l}}_{i,k}\,\hat{P}^{l}_{i,k}-Q^{C^{l}}_{i,k}\left(\alpha^{l}_{i,k}-\mathcal{G}_{lb}(m^{l}_{i,k})\right)-Q^{C}_{i,k}\left(\alpha^{u}_{i,k}\alpha^{d}_{i,k}-\mathcal{G}_{lb}(m_{i,k})\right)\right]$ (52a) $\displaystyle\text{subject to:}\quad~{}\eqref{eqopt1e_1a}-\eqref{eqopt1e_1d},$ which is a mixed-integer non-convex problem. Due to the complexity of exhaustive search for finding the optimal solution, we propose a low- complexity two-stage sequential optimization strategy to find a sub-optimal solution to the joint power allocation and scheduling assignment problem. This strategy firstly obtains the UL and DL transmission power variables, followed by the UL and DL scheduling variables. The optimal UL and DL transmission power for each control system, determined by solving the following power allocation problem $\displaystyle\underset{\hskip 15.0pt\mathbf{P}^{u}_{k},\mathbf{P}^{d}_{k}}{(\mathcal{P}4.4)\quad\text{Minimize}}\;\sum^{M}_{i=1}\alpha^{u}_{i,k}\left[\mathcal{Q}^{S_{1}}_{i,k}+Q^{P^{u}}_{i,k}P^{u}_{i,k}\right]+\alpha^{d}_{i,k}\left[\mathcal{Q}^{C_{1}}_{i,k}+Q^{P^{d}}_{i,k}P^{d}_{i,k}\right]-\alpha^{u}_{i,k}\alpha^{d}_{i,k}Q^{C}_{i,k}+\mathcal{Q}^{l_{1}}_{i,k},$ (53a) $\displaystyle\text{subject to:}\quad\frac{\text{SNR}^{u}_{th}\,{N_{0}}}{\\|\mathbf{H}^{u}_{i,k}\\|^{2}}\leq P^{u}_{i,k}\leq P^{u}_{max},\qquad\forall i\in\mathcal{M},k$ (53b) $\displaystyle\frac{\text{SNR}^{d}_{th}\,{N_{0}}}{\\|\mathbf{H}^{d}_{i,k}\\|^{2}}\leq P^{d}_{i,k}\leq P^{d}_{max},\qquad\forall i\in\mathcal{M},k,$ (53c) where $\mathcal{Q}^{S_{1}}_{i,k}=-Q^{\beta^{u}}_{i,k}\beta^{u}_{i,k-1}-Q^{C^{u}}_{i,k}$, $\mathcal{Q}^{C_{1}}_{i,k}=-Q^{\beta^{d}}_{i,k}\beta^{d}_{i,k-1}-Q^{C^{d}}_{i,k}$, and $\mathcal{Q}^{l_{1}}_{i,k}=Q^{C^{u}}_{i,k}\mathcal{G}_{lb}(m^{u}_{i,k})+Q^{C^{d}}_{i,k}\mathcal{G}_{lb}(m^{d}_{i,k})+Q^{C}_{i,k}\mathcal{G}_{lb}(m_{i,k})+Q_{i,k}^{\beta^{u}}\left(1+\beta^{u}_{i,k-1}\right)+Q_{i,k}^{\beta^{d}}\left(1+\beta^{d}_{i,k-1}\right)$ are the constant terms defined in the objective function of $\mathcal{P}4.4$. The above problem $\mathcal{P}4.4$ is a generalized min-weight problem, which can be decoupled into a series of independent sub-problems for each control system separately. Hence, the optimal UL-DL transmission power variables are given as $\small P^{l^{}}_{i,k}=\left\\{\begin{array}[]{cc}\frac{\text{SNR}^{l}_{th}N_{0}}{\\|\mathbf{H}^{l}_{i,k}\\|^{2}},&\text{if}\;Q^{P^{l}}_{i,k}\geq 0\\\ P^{l}_{max},&\mbox{if}\;Q^{P^{l}}_{i,k}<0.\end{array}\right.$ (54) Given the optimal UL and DL transmission power variables in (54), the optimal UL and DL scheduling variables for each control system that has a control state/action to transmit are obtained by solving the following scheduling assignment problem $\displaystyle\underset{\hskip 40.0pt\mathbf{a}^{u}_{k},\mathbf{a}^{d}_{k}}{(\mathcal{P}4.5)\quad\text{Minimize}}\;\sum^{M}_{i=1}\alpha^{u}_{i,k}\left[\mathcal{Q}^{S_{1}}_{i,k}+Q^{P^{u}}_{i,k}P^{u^{}}_{i,k}\right]+\alpha^{d}_{i,k}\left[\mathcal{Q}^{C_{1}}_{i,k}+Q^{P^{d}}_{i,k}P^{d^{}}_{i,k}\right]-\alpha^{u}_{i,k}\alpha^{d}_{i,k}Q^{C}_{i,k}+\mathcal{Q}^{l_{1}}_{i,k},$ (55a) $\displaystyle\text{subject to:}\quad~{}\eqref{eqopt1e_1c}-\eqref{eqopt1e_1d}.$ The optimal UL and DL scheduling variables are obtained as follows: $\small\alpha^{u^{}}_{i,k}\,\&\,\alpha^{d^{}}_{i,k}=\left\\{\begin{array}[]{cc}\alpha^{u}_{j_{1},k}=1,\;\alpha^{d}_{j_{2},k}=1,&\text{if}\;\mathbb{Q}^{1}_{j_{1},k}+\mathbb{Q}^{2}_{j_{2},k}<\mathbb{Q}^{1}_{j_{3},k}+\mathbb{Q}^{2}_{j3,k}+\mathbb{Q}^{3}_{j_{3},k}\\\ \alpha^{u}_{j_{3},k}=1,\;\alpha^{d}_{j_{3},k}=1,&\text{if}\;\mathbb{Q}^{1}_{j_{1},k}+\mathbb{Q}^{2}_{j_{2},k}>\mathbb{Q}^{1}_{j_{3},k}+\mathbb{Q}^{2}_{j_{3},k}+\mathbb{Q}^{3}_{j_{3},k}\\\ \alpha^{u}_{j,k}=0,\;\alpha^{d}_{j,k}=0,&\forall j\notin\\{j_{1}\,\&\,j_{2}\|\|j_{3}\,\\},\end{array}\right.$ (56) where $\mathbb{Q}_{i,k}^{1}=\mathcal{Q}^{S_{1}}_{i,k}+Q^{P^{u}}_{i,k}P^{u^{}}_{i,k}$, $\mathbb{Q}_{i,k}^{2}=\mathcal{Q}^{C_{1}}_{i,k}+Q^{P^{d}}_{i,k}P^{d^{}}_{i,k}$, and $\mathbb{Q}_{i,k}^{3}=-Q^{C}_{i,k}$ are the terms defined in the objective function of problem $\mathcal{P}4.5$. Moreover, $j_{1}=\operatorname{arg\,min}_{i\in\mathcal{M}}\mathbb{Q}^{1}_{i,k}$, $j_{2}=\operatorname{arg\,min}_{i\in\mathcal{M}}\mathbb{Q}^{2}_{i,k}$, and $j_{3}=\operatorname{arg\,min}_{i\in\mathcal{M}}(\mathbb{Q}^{1}_{i,k}+\mathbb{Q}^{2}_{i,k}+\mathbb{Q}^{3}_{i,k})$ are control system indices. ## V Simulation Results and Discussions In this section, the performance of the proposed stability-aware scheduling algorithm is investigated in an inverted-pendulum on a cart system with $M=2$, and $M=20$ inverted-pendulums, respectively. Each inverted-pendulum system is described by a four-dimensional state vector as $\mathbf{x}^{u}_{i,k}=\left[x_{i,k},\dot{x}_{i,k},\theta_{i,k},\dot{\theta}_{i,k}\right]$, where $x_{i,k}$ represents the cart’s position along the horizontal axis, $\dot{x}_{i,k}$ represents the cart’s velocity, $\theta_{i,k}$ represents the pendulum angle along the vertical axis, and $\dot{\theta}_{i,k}$ represents the pendulum’s angular velocity. The initial state of the control systems $i$ is $\mathbf{x}^{u}_{i,0}=\left[0\quad 0\quad 0.1\quad 0\right]^{T}$. The action $\mathbf{u}^{a}_{i,k}$ is the horizontal force applied on the linear cart. By applying a zeroth-order with a state sampling of $10$ms on the continuous dynamics of the inverted-pendulum system and linearizing around the pendulum up-position, i.e., $\theta_{i,k}=0$, we obtain the following discrete-time linear dynamics matrices [14], $\displaystyle\mathbf{A}_{i}=\left[\begin{array}[]{cccc}1&0&0&0\\\ 0&2.055&-0.722&4.828\\\ 0&0.023&0.91&0.037\\\ 0&0.677&-0.453&2.055\\\ \end{array}\right],\mathbf{B}_{i}=\left[\begin{array}[]{c}0.034\\\ 0.168\\\ 0.019\\\ 0.105\\\ \end{array}\right],$ (65) Since $\mathbf{A}_{i}$’s largest eigenvalues $\left\\{3.85,0.42,0.92,1.00\right\\}$ is greater than unity, the inverted- pendulum is unstable without an appropriate control action [20]. To stabilize the control system, the feedback gain matrix $\mathbf{\Phi}_{i}$ is calculated at the controller based on LQR in (27). The rest of the simulation parameters are $P^{l}_{max}=20\,\text{dBm}$, $N_{0}=-20\,\text{dBm}$, $\zeta_{i}=0.01$, $V=1000$, $\omega_{\beta}=1$, $\omega_{P}=1$, $h_{k}=1$, $h_{q}=1$, $\mu=1$, and $\sigma_{n}^{2}=0.01$. The performance of the proposed stability-aware scheduling method is compared versus five scheduling baselines. In Baseline $1$. (Round-Robin Scheduling), each sensor/controller periodically transmits its state/action over a wireless channel with fixed transmission power and a predefined repeating order [8, 9]. In Baseline $2$. (Opportunistic Scheduling), the sensor/controller is scheduled under favorable channel conditions. Otherwise, the controller/actuator applies the last received state/action [10, 11]. In Baseline $3$, (Event-triggered Scheduling without FDMA), one control system with the largest state discrepancy, i.e., the difference between the current predicted state using Kalman filtering and the previous received/predicted state is larger than a predefined threshold, is scheduled at each time to transmit the state/action with fixed transmission power [12, 13]. In Baseline $4$. (Event-triggered Scheduling with FDMA), each sensor/controller transmits its state/action with fixed transmission power based on its stability condition, i.e., difference between the current and previous states is less than a predefined threshold, using FDMA [25]. In Baseline $5$. (Ideal Control Scheduling), all control systems simultaneously transmit their states/actions with ultra-low latency and high reliability over perfect channels [9]. Results are collected over ten independent simulation runs, and each simulation is run for $K=90$ discrete-time steps. (a) Average Pendulum Angle with $M=2$. (b) Average Pendulum Angle with $M=20$. Figure 3: Average pendulum angle to the vertical center, i.e., control error with $M=2$ and $M=20$ pendulum angle systems using the proposed stability- aware, round-robin, opportunistic, event-triggered with and without FDMA. Average Control Error Vs. Control Systems. Fig. 3 illustrates the average pendulum angle to the vertical center of each control system, i.e., the average control error of each control system, during $90$ control time steps. As shown in Fig.3(a), the proposed and baseline scheduling methods, assuming a low number of control systems $\left(M=2\right)$, can keep all the pendulums upright. Moreover, the proposed stability-Aware with GPR and the event- triggered with FDMA keep both pendulums close to zero ensuring both pendulums have the same control performance. This is because the proposed solution adapts to both channel and control states, i.e., the control system is scheduled if it has a favorable channel condition and unstable control state needs to be stabilized. Meanwhile, the event-triggered with FDMA scheduling has approximately the same performance at the cost of wasting wireless communication resources by transmitting with a fixed transmission power and a high communication rate. The opportunistic scheduler without GPR has better control performance compared to the event-triggered without FDMA and the round-robin methods leveraging channel state in scheduling compared to the scheduling baselines. This in turn reflects the connection between the state estimation stability and control stability. Fig. 3(b) plots the average control error of each control system for the proposed and baseline scheduling methods. In large numbers of control systems the proposed stability-aware with GPR and the event-triggered with FDMA scheduling methods keep all pendulum upright, unlike the baselines. Unlike the scheduling baselines except the event-triggered with FDMA scheduling wherein at most one control system is scheduled each time due to the limited bandwidth, our proposed scheduling allows all control systems to operate simultaneously even without receiving either the current state or action, highlighting the effectiveness of GPRs at the controller and actuator thereby improving communication efficiency and control stability. Moreover, it maintains the GPR prediction credibility, achieving control stability. Meanwhile, the event-triggered with FDMA scheduling keeps some pendulums upright at the cost of frequent transmissions by equally dividing the available bandwidth between the control systems, such that each control system receives a fixed fraction $f_{i}$ of the total capacity $f_{i}=\text{BW}/M$, affecting transmission latency. (a) Communication rate of sensing link. (b) Communication rate of actuating link. Figure 4: Histogram of achieved communication rate in a large number of control system $\left(M=20\right)$ of the proposed stability-aware and event- triggered with FDMA throughout $90$ control time steps. Communication Rate Vs. Number of Control Systems. Fig.4 presents a histogram of the achieved communication rates for the sensing and actuating links for $M=20$ during $90$ control time steps. The sensing/actuating communication rate is defined as the number of times the sensing/actuating link of a control system is scheduled divided by the time interval as $n_{l}/K$. It is clear that the proposed approach achieves sensing communication rates concentrated in the range from $0$ to $0.4$ and actuating communication rates ranging from $0$ to $0.2$. On the other hand, the event-triggered with FDMA scheduling achieves wide sensing and actuating communication rates ranging from $0$ to $1$. The reason behind this result is that the stability-aware with GPR scheduling is adapting to both the channel and control states, and GPRs at the controller and actuator sides compensate for the missing received observations, hence improving the communication efficiency. Meanwhile, each control system, in the event-triggered with FDMA scheduling, only transmits its control state/action based on its control stability condition without taking into account channel states that result in increasing state/action estimation uncertainty from the adverse channel states, and in turn affecting control stability. Hence, each control system in the event-triggered with FDMA scheduling requires frequent transmissions to ensure control stability by applying appropriate action based on low communication uncertainty. Note that the range of the sensing communication rate is larger than that of the actuating communication rate since some control systems fail to transmit their states at the beginning affecting prediction credibility. However, the number of control systems that require frequent scheduling in the actuating link is larger than that of the sensing link which stems from the fast dynamics related to the inverted-pendulum system that require a quick appropriate action. (a) Single control system out of $M=2$. (b) Single control system out of $M=20$. Figure 5: Comparison of the state trajectory, controller AoI, and actuator AoI of a randomly chosen control system between the proposed stability-aware, round-robin, Opportunistic, event-triggered with and without FDMA, and ideal control system. State Trajectory and Controller/Actuator AoI Vs. Time. To dive deeper into the benefits of the proposed stability-aware with GPR scheduling, we present in Fig. 5 the state trajectory, controller AoI, and actuator AoI of a randomly chosen control system in low and large number control systems regimes. In Fig 5(a), the state trajectory of the proposed stability-aware with GPR scheduling and event-triggered with FDMA scheduling, in a low number of control systems, are extremely close to that of the ideal control system where their pendulums remain upright over time. Meanwhile, the state trajectory of the opportunistic without GPR scheduling is slightly better than that of the event-triggered without FDMA scheduling by keeping all pendulums upright over time due to the scheduled control system with a favorable channel state. Finally, the state trajectory of the round-robin without GPR scheduling slightly matches the desired state at the cost of a high communication rate and fixed transmission power. As shown in Fig. 5(a), the controller and actuator AoI of the proposed stability-aware scheduling equals one, i.e., the sensing and actuating links of a control system are synchronously scheduled, until $15$ control time steps. This is to guarantee that the received actions to action GPR has a low state/action estimation uncertainty that affects the GPR state/action prediction stability and control stability. Then, the actuator AoI starts increasing compared to the controller AoI that remains at value one until $45$ control time steps, i.e., the sensing link of a control system is scheduled until $45$ control time steps while the actuating link is not scheduled. The rationale behind this result is to schedule either the sensing link or/and actuating link of a control system with favorable channel condition to ensure the state/action prediction credibility. Hence, the transmitted action, when the sensing link of a control system is not scheduled, depends on a credible predicted state. This shows the impact of the decoupled scheduling between the UL and DL communications compared to the coupled scheduling in terms of improving communication efficiency by reducing sensing and actuating communication rates while guaranteeing control stability. For large numbers of control systems shown in Fig. 5(b), the control error of the stability-aware with GPR scheduling and event-triggered with FDMA scheduling exponentially decay over time compared to the scheduling baselines. Meanwhile, the other scheduling baselines are exponentially growing over time due to the accumulated control error in the absence of appropriate action. (a) Single control system out of $M=2$. (b) Single control system out of $M=20$. Figure 6: Comparison of the state trajectory, controller AoI, and actuator AoI of a randomly chosen system between the proposed stability-aware w. GPR using both direct and recursive approaches. Time-Series Direct and Recursive GPR Approaches. Fig. 6 illustrates the state trajectory, controller AoI, and actuator AoI of a randomly chosen control system in low and large number control systems and for two different time series GPR approaches. The recursive GPR approach uses the previous predicated and observed states/actions to predict the next control state/action, while the direct GPR approach only uses the previous observed states/actions to predict the next control state/action. It is interesting to observe that the state trajectory of the direct and recursive GPR approaches at the early phase are identical since the number of predicted states/actions in the training set of recursive GPR approach is smaller than the observed states/actions to predict future states/actions. However, when time increases, the state trajectory of the direct GPR approach outperforms the recursive GPR approach since the number of predicted states/actions in training set of the recursive GPR approach is greater than the observed states/actions. This in turn deteriorates the predication accuracy hindering control stability [26]. Moreover, the direct and recursive GPR approaches for a low number of control systems require frequent scheduling at the early phase (until $45$ control time steps in the sensing link), during which the amount of state observations ensures the state prediction credibility. Meanwhile, the recursive and direct GPR approaches for a low number of control systems require only $15$ control time steps in the actuating link, within which the amount of control actions ensures the action prediction stability. Fig. 6 highlights that the AoI increase deteriorates the control performance when increasing prediction uncertainty until the training set has a sufficient number of observations to ensure steady-state state/action prediction. Finally, Fig. 6 illustrates that the state/action prediction of the recursive GPR approach converges faster to the steady-state compared to the direct GPR approach. This affects the scheduling decision by increasing the controller and actuator AoIs of the recursive GPR approach at the cost of a low state/action prediction accuracy and control performance compared to the direct GPR approach. Figure 7: Total number of supported control systems using stability-aware with GPR, event-triggered with FDMA, round-robin, event-triggered without FDMA, and opportunistic scheduling. Number of Served Control Systems Vs. Time Interval. Fig. 7 presents the final capacity of the scheduled control systems over two different time intervals. We assume that a scheduling method successfully control several controls systems for all control systems within $\\|\theta_{i,k}\\|\leq 0.05$ error region for $10$ independent simulation runs. As observed in Fig. 7, the proposed approach has a significant impact compared to the baselines in terms of supporting a large number of control systems over time interval. The rationale behind this result is due to exploiting two GPRs at the controller and actuator sides and obtaining a sufficient number of observations in the GPR training sets increases. This in turn enhances the prediction credibility, communication efficiency, and control stability performance while supporting a large number of control systems. ## VI Conclusion In this work, we proposed a novel stability-aware scheduling algorithm based on the communication and control co-design exploiting analog uncoded communication and GPR based to reduce the required communication rate and ensure control stability through maintaining prediction stability. We performed extensive simulations for low-latency control systems to demonstrate the effectiveness of utilizing the GPR based approach, as well as, the effectiveness of decoupled scheduling between the UL and DL communications to support a large number of control systems compared to the scheduling baselines. ## Appendix A Proof of Lemma 1 Given the predicted state $\hat{\mathbf{x}}^{u}_{i,k}$ in (21) and the state prediction error covariance matrix $\mathcal{J}^{u}_{i,k}$ in (25), it holds that the expected current Lyapunov value of the system is $\displaystyle\mathbb{E}\big{[}\mathcal{L}(\mathbf{x}^{u}_{i,k})\|\hat{\mathbf{x}}^{u}_{i,k}\big{]}$ $\displaystyle=\mathbb{E}\big{[}\mathbf{x}^{u^{T}}_{i,k}\,\mathcal{Z}\,\mathbf{x}^{u}_{i,k}\|\hat{\mathbf{x}}^{u}_{i,k}\big{]}=\mathbb{E}\big{[}\left(\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{e}^{u}_{i,k}\right)^{T}\mathcal{Z}\left(\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{e}^{u}_{i,k}\right)\big{]}$ (66) $\displaystyle=\mathbb{E}\big{[}\hat{\mathbf{x}}^{u^{T}}_{i,k}\mathcal{Z}\hat{\mathbf{x}}^{u}_{i,k}-\hat{\mathbf{x}}^{u^{T}}_{i,k}\mathcal{Z}\mathbf{e}^{u}_{i,k}-\mathbf{e}^{u^{T}}_{i,k}\mathcal{Z}\hat{\mathbf{x}}^{u}_{i,k}+\mathbf{e}^{u^{T}}_{i,k}\mathcal{Z}\mathbf{e}^{u}_{i,k}\big{]},\qquad\quad\forall\;\mathbb{S}^{D}_{++}.$ (67) In (67), the first term is a constant as the expectation is taken w.r.t. the state prediction error $\mathbf{e}^{u}_{i,k}$, while the cross-terms such as $\mathbb{E}[\hat{\mathbf{x}}^{u^{T}}_{i,k}\mathcal{Z}\mathbf{e}^{u}_{i,k}]$ and $\mathbb{E}[\mathbf{e}^{u^{T}}_{i,k}\mathcal{Z}\hat{\mathbf{x}}^{u}_{i,k}]$ can be canceled since the predicted state $\hat{\mathbf{x}}^{u}_{i,k}$ and the state prediction error $\mathbf{e}^{u}_{i,k}$ are uncorrelated. Hence, the expected current value of $\mathcal{L}(\mathbf{x}_{i,k})$ is given as $\displaystyle\mathbb{E}\big{[}\mathcal{L}(\mathbf{x}^{u}_{i,k})\|\hat{\mathbf{x}}^{u}_{i,k}\big{]}$ $\displaystyle=\hat{\mathbf{x}}^{u^{T}}_{i,k}\mathcal{Z}\hat{\mathbf{x}}^{u}_{i,k}+\text{Tr}\left[\mathcal{Z}\;\mathbb{E}[\mathbf{e}^{u^{T}}_{i,k}\mathbf{e}^{u}_{i,k}]\right]=\\|\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\mathcal{Z}\;\mathcal{J}^{u}_{i,k}\right],\qquad\qquad\forall\;\mathbb{S}^{D}_{++}$ (68) where the last term in (68) is obtained via the expectation of the quadratic form ${\scriptstyle\mathbb{E}[\mathbf{e}^{u^{T}}_{i,k}\mathcal{Z}\mathbf{e}^{u}_{i,k}]}$ w.r.t the state estimation error, i.e, ${\scriptstyle\mathbb{E}[\mathbf{e}^{u^{T}}_{i,k}\mathcal{Z}\mathbf{e}^{u}_{i,k}]=(\mathbb{E}[\mathbf{e}^{u}_{i,k}])^{T}\mathcal{Z}(\mathbb{E}[\mathbf{e}^{u}_{i,k}])+\text{Tr}[\mathcal{Z}\;\mathcal{J}^{u}_{i,k}]}$. It is observed in (68) that the expected current Lyapunov value grows larger as the predicted state $\hat{\mathbf{x}}^{u}_{i,k}$ is near instability and/or the prediction error covariance matrix $\mathcal{J}^{u}_{i,k}$ is larger due to lack of sufficient observations. $\blacksquare$ ## Appendix B Proof of Lemma 2 As a result of the remote sensing-loop state evolution in (30) and the open- loop state evolution in (29), the expected future value of $\mathcal{L}(\mathbf{x}^{u}_{i,k+1})$ in (36) of the UL transmission is given as $\displaystyle\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{u}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\right]=$ $\displaystyle\xi^{u}_{i,k}\;\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{s}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\right]$ (69) $\displaystyle+\left(1-\xi^{u}_{i,k}\right)\;\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{o}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\right],$ For a given predicted state $\hat{\mathbf{x}}^{u}_{i,k}$, state prediction error covariance matrix $\mathcal{J}^{u}_{i,k}$, predicted action $\hat{\mathbf{u}}^{d}_{i,k}$, action prediction error covariance matrix $\mathcal{J}^{d}_{i,k}$, wireless UL channel $\mathbf{H}^{u}_{i,k}$, and UL transmission power $P^{u}_{i,k}$, the expected future value of $\mathcal{L}(\mathbf{x}^{u}_{i,k+1})$ of the UL transmission in (69) is given as $\displaystyle\mathbb{E}$ $\displaystyle\left[\mathcal{L}(\mathbf{x}^{u}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\right]$ (70) $\displaystyle\;\overset{(1)}{=}\xi^{u}_{i,k}\mathbb{E}\left[\mathbf{x}^{s^{T}}_{i,k+1}\mathcal{Z}\mathbf{x}^{s}_{i,k+1}\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\right]+\left(1-\xi^{u}_{i,k}\right)\;\mathbb{E}\left[\mathbf{x}^{o^{T}}_{i,k+1}\mathcal{Z}\mathbf{x}^{o}_{i,k+1}\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\right]$ $\displaystyle\overset{(2)}{=}\xi^{u}_{i,k}\mathbb{E}\left[\left(\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{A}^{c}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{\Phi}_{i}\mathbf{v}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{e}^{d}_{i,k}+\mathbf{w}_{k}\right)^{T}\mathcal{Z}\left(\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{A}^{c}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{\Phi}_{i}\mathbf{v}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{e}^{d}_{i,k}+\mathbf{w}_{k}\right)\right]$ $\displaystyle+(1-\xi^{u}_{i,k})\mathbb{E}\left[\left(\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{A}^{c}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{\Phi}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{e}^{d}_{i,k}+\mathbf{w}_{k}\right)^{T}\mathcal{Z}\left(\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{A}^{c}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{\Phi}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{e}^{d}_{i,k}+\mathbf{w}_{k}\right)\right]$ $\displaystyle\overset{(3)}{=}\xi^{u}_{i,k}\left[\\|\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\mathbf{A}^{c^{T}}_{i}\mathcal{Z}\mathbf{A}^{c}_{i}\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)^{T}\mathcal{Z}\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)\mathbf{V}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]\right]$ $\displaystyle+(1-\xi^{u}_{i,k})\left[\\|\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\mathbf{A}^{c^{T}}_{i}\mathcal{Z}\mathbf{A}^{c}_{i}\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)^{T}\mathcal{Z}\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]\right].$ Step $(1)$ is a result of using the quadratic Lyapunov function. The step $(2)$ holds when applying the remote sensing-loop and open-loop state evolution in (30) and (29), respectively. The step $(3)$ holds using the expectation in (70) with respect to the state estimation error $\mathbf{e}^{u}_{i,k}$, the action prediction error $\mathbf{e}^{d}_{i,k}$, and the plant noise $\mathbf{w}_{k}$. As a consequence of obtaining the expected current Lyapunov value in (68) and the expected future Lyapunov value of the UL transmission in (70), the control stability constraint in (36) for the UL transmission is $\displaystyle\xi^{u}_{i,k}\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{s}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\right]+(1-\xi^{u}_{i,k})\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{o}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\right]\leq\zeta_{i}\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{u}_{i,k})\|\hat{\mathbf{x}}^{u}_{i,k}\right]$ (71) $\displaystyle=\xi^{u}_{i,k}\left[\text{Tr}\left[(\mathbf{B}_{i}\mathbf{\Phi}_{i})^{T}\mathcal{Z}(\mathbf{B}_{i}\mathbf{\Phi}_{i})\mathbf{V}^{u}_{i,k}\right]-\text{Tr}\left[(\mathbf{B}_{i}\mathbf{\Phi}_{i})^{T}\mathcal{Z}(\mathbf{B}_{i}\mathbf{\Phi}_{i})\mathcal{J}^{u}_{i,k}\right]\right]+\\|\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\mathbf{A}^{T}_{i}\mathcal{Z}\mathbf{A}_{i}\mathcal{J}^{u}_{i,k}\right]$ $\displaystyle\qquad+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]\leq\zeta_{i}\\|\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\zeta_{i}\text{Tr}\left[\mathcal{Z}\mathcal{J}^{u}_{i,k}\right]$ After rearranging the terms in (LABEL:lemm2.3), the constraint on the UL transmission indicator variable is $\displaystyle\xi^{u}_{i,k}\geq\frac{\\|\left(\mathbf{A}^{c}_{i}-\zeta_{i}\mathbf{I}_{D}\right)\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\left(\mathbf{A}^{T}_{i}\mathcal{Z}\mathbf{A}_{i}-\zeta_{i}\mathcal{Z}\right)\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]}{\text{Tr}\left[\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)^{T}\mathcal{Z}\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)\mathcal{J}^{u}_{i,k}\right]-\text{Tr}\left[\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)^{T}\mathcal{Z}\left(\mathbf{B}_{i}\mathbf{\Phi}_{i}\right)\mathbf{V}^{u}_{i,k}\right]}.$ (72) To capture the overall state evolution of each control system in the UL over time interval of length $K$, according to the time-averaged Lyapunov [27], (53) is summed over time ${\scriptstyle k\in\\{0,\cdots,K-1\\}}$, then the result is divided by $K$ and taking the limit as time tends to infinity. This yields the UL transmission indicator variable constraint in (38). Similarly, we obtain the DL transmission indicator variable constraint in (39) according to the remote actuating-loop state evolution in (31) and open-loop state evolution in (29). Finally, the expected future Lyapunov value of the UL-DL coupled transmission is obtained using the closed-loop state evolution in (32) and the open-loop state evolution in (29) where $\displaystyle\mathbb{E}\Big{[}\mathcal{L}(\mathbf{x}^{u}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k},$ $\displaystyle\mathbf{H}^{d}_{i,k},P^{d}_{i,k}\Big{]}=\xi^{u}_{i,k}\;\xi^{d}_{i,k}\;\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{c}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k},\mathbf{H}^{d}_{i,k},P^{d}_{i,k}\right]$ (73) $\displaystyle+\left(1-\xi^{u}_{i,k}\;\xi^{d}_{i,k}\right)\;\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{o}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k},\mathbf{H}^{d}_{i,k},P^{d}_{i,k}\right],$ For a given predicted state $\hat{\mathbf{x}}^{u}_{i,k}$, the state prediction error covariance matrix $\mathcal{J}^{u}_{i,k}$, the predicted action $\hat{\mathbf{u}}^{d}_{i,k}$, the action prediction error covariance matrix $\mathcal{J}^{d}_{i,k}$, the wireless UL channel $\mathbf{H}^{u}_{i,k}$, the wireless DL channel $\mathbf{H}^{d}_{i,k}$, the UL transmission power $P^{u}_{i,k}$, and the DL transmission power $P^{d}_{i,k}$, the expected future value of $\mathcal{L}(\mathbf{x}^{u}_{i,k+1})$ of the UL-DL coupling transmission in (73) is given as $\displaystyle\mathbb{E}$ $\displaystyle\left[\mathcal{L}(\mathbf{x}^{u}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k},\mathbf{H}^{d}_{i,k},P^{d}_{i,k}\right]=\xi^{u}_{i,k}\;\xi^{d}_{i,k}\mathbb{E}\left[\mathbf{x}^{c^{T}}_{i,k+1}\mathcal{Z}\mathbf{x}^{c}_{i,k+1}\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k},\mathbf{H}^{d}_{i,k},P^{d}_{i,k}\right]$ (74) $\displaystyle\hskip 160.0pt+\left(1-\xi^{u}_{i,k}\;\xi^{d}_{i,k}\right)\;\mathbb{E}\left[\mathbf{x}^{o^{T}}_{i,k+1}\mathcal{Z}\mathbf{x}^{o}_{i,k+1}\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\mathbf{H}^{d}_{i,k},P^{d}_{i,k}\right]$ $\displaystyle=\xi^{u}_{i,k}\;\xi^{d}_{i,k}\mathbb{E}\left[\left(\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{A}^{c}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{\Phi}_{i}\mathbf{v}^{c}_{i,k}-\mathbf{B}_{i}\mathbf{v}^{d}_{i,k}+\mathbf{w}_{k}\right)^{T}\mathcal{Z}\left(\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{A}^{c}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{\Phi}_{i}\mathbf{v}^{c}_{i,k}-\mathbf{B}_{i}\mathbf{v}^{d}_{i,k}+\mathbf{w}_{k}\right)\right]$ $\displaystyle+(1-\xi^{u}_{i,k}\;\xi^{d}_{i,k})\mathbb{E}\left[\left(\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{A}^{c}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{\Phi}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{e}^{d}_{i,k}+\mathbf{w}_{k}\right)^{T}\mathcal{Z}\left(\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}-\mathbf{A}^{c}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{\Phi}_{i}\mathbf{e}^{u}_{i,k}-\mathbf{B}_{i}\mathbf{e}^{d}_{i,k}+\mathbf{w}_{k}\right)\right]$ $\displaystyle=\xi^{u}_{i,k}\;\xi^{d}_{i,k}\left[\\|\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\mathbf{A}^{c^{T}}_{i}\mathcal{Z}\mathbf{A}^{c}_{i}\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[(\mathbf{B}_{i}\mathbf{\Phi}_{i})^{T}\mathcal{Z}(\mathbf{B}_{i}\mathbf{\Phi}_{i})\mathbf{V}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathbf{V}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]\right]$ $\displaystyle+(1-\xi^{u}_{i,k}\;\xi^{d}_{i,k})\left[\\|\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\mathbf{A}^{c^{T}}_{i}\mathcal{Z}\mathbf{A}^{c}_{i}\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[(\mathbf{B}^{T}_{i}\mathbf{\Phi}_{i})^{T}\mathcal{Z}(\mathbf{B}^{T}_{i}\mathbf{\Phi}_{i})\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]\right]$ Given the expected current value of $\mathcal{L}(\mathbf{x}^{u}_{i,k})$ in (68) and future of $\mathcal{L}(\mathbf{x}^{u}_{i,k+1})$ of the UL-DL coupled transmission in (74), the control stability constraint in (36) for UL-DL coupled transmission is $\displaystyle\xi^{u}_{i,k}\;\xi^{d}_{i,k}\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{a}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\mathbf{H}^{d}_{i,k},P^{d}_{i,k}\right]+(1-\xi^{u}_{i,k}\;\xi^{d}_{i,k})\mathbb{E}\Big{[}\mathcal{L}(\mathbf{x}^{o}_{i,k+1})\|\hat{\mathbf{x}}^{u}_{i,k},\hat{\mathbf{u}}^{d}_{i,k},\mathbf{H}^{u}_{i,k},P^{u}_{i,k}\mathbf{H}^{d}_{i,k},P^{d}_{i,k}\Big{]}$ (75) $\displaystyle\leq\zeta_{i}\mathbb{E}\left[\mathcal{L}(\mathbf{x}^{u}_{i,k})\|\hat{\mathbf{x}}^{u}_{i,k}\right]=\xi^{u}_{i,k}\;\xi^{d}_{i,k}\Big{[}\text{Tr}\left[(\mathbf{B}_{i}\mathbf{\Phi}_{i})^{T}\mathcal{Z}(\mathbf{B}_{i}\mathbf{\Phi}_{i})\mathbf{V}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}_{i}^{T}\mathcal{Z}\mathbf{B}_{i}\mathbf{V}^{d}_{i,k}\right]-\text{Tr}\left[(\mathbf{B}_{i}\mathbf{\Phi}_{i})^{T}\mathcal{Z}(\mathbf{B}_{i}\mathbf{\Phi}_{i})\mathcal{J}^{u}_{i,k}\right]$ $\displaystyle-\text{Tr}\left[\mathbf{B}_{i}^{T}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]\Big{]}+\\|\mathbf{A}^{c}_{i}\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\mathbf{A}^{T}_{i}\mathcal{Z}\mathbf{A}_{i}\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]\leq\zeta_{i}\\|\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\zeta_{i}\text{Tr}\left[\mathcal{Z}\mathcal{J}^{u}_{i,k}\right]$ The UL-DL transmission indicator variable constraint after rearranging the terms in (LABEL:lemm2.11) is: $\displaystyle\xi^{u}_{i,k}\;\xi^{d}_{i,k}\geq\frac{\\|\left(\mathbf{A}^{c}_{i}-\zeta_{i}\mathbf{I}_{D}\right)\hat{\mathbf{x}}^{u}_{i,k}\\|_{\mathcal{Z}^{\frac{1}{2}}}^{2}+\text{Tr}\left[\left(\mathbf{A}^{T}_{i}\mathcal{Z}\mathbf{A}_{i}-\zeta_{i}\mathcal{Z}\right)\mathcal{J}^{u}_{i,k}\right]+\text{Tr}\left[\mathbf{B}^{T}_{i}\mathcal{Z}\mathbf{B}_{i}\mathcal{J}^{d}_{i,k}\right]+\text{Tr}\left[\mathcal{Z}\mathbf{W}\right]}{\text{Tr}\left[\left((\mathbf{B}_{i}\mathbf{\Phi}_{i})^{T}\mathcal{Z}(\mathbf{B}_{i}\mathbf{\Phi}_{i})\right)(\mathcal{J}^{u}_{i,k}-\mathbf{V}^{u}_{i,k})\right]+\text{Tr}\left[\mathbf{B}_{i}^{T}\mathcal{Z}\mathbf{B}_{i}(\mathcal{J}^{d}_{i,k}-\mathbf{V}^{d}_{i,k}\right]}$ (76) At last, we apply the time-averaged Lyapunov [27] to obtain the overall state evolution per control system in the UL-DL coupled transmission and UL-DL coupled transmission indicator variable constraint in (40). $\blacksquare$ ## References * [1] A. 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11institutetext: Berkeley Center for Theoretical Physics and Department of Physics, University of California, Berkeley, CA 94720, USA22institutetext: Stanford Institute for Theoretical Physics, Stanford University, Stanford, CA 94305, USA # Island Finder and Entropy Bound Raphael Bousso 2 and Arvin Shahbazi-Moghaddam<EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract Identifying an entanglement island requires exquisite control over the entropy of quantum fields, which is available only in toy models. Here we present a set of sufficient conditions that guarantee the existence of an island and place an upper bound on the entropy computed by the island rule. This is enough to derive the main features of the Page curve for an evaporating black hole in any spacetime dimension. Our argument makes use of Wall’s maximin formulation and the Quantum Focusing Conjecture. As a corollary, we derive a novel entropy bound. ## 1 Introduction ### 1.1 Entanglement Wedges and Islands The quantum-corrected Engelhardt:2014gca , covariant Hubeny:2007xt Ryu- Takayanagi Ryu:2006bv (RT) prescription computes the CFT entropy of a boundary region in terms of a dual asymptotically AdS bulk spacetime. Originally an ad hoc proposal, it follows under certain assumptions from a Euclidean gravitational path integral Lewkowycz:2013nqa ; Faulkner:2013ana ; Dong:2017xht ; Penington:2019kki ; Almheiri:2019qdq . This derivation implies that the RT prescription is not tied to the AdS/CFT correspondence but can be evaluated in any spacetime $M$. Indeed, RT yields the Page curve Page:1993wv for the entropy of the bulk radiation emitted by a black hole Penington:2019npb ; Almheiri:2019psf . The bulk state and geometry are treated semiclassically. In this approximation, the radiation is thermal Hawking:1974sw , and its von Neumann entropy $S(R)$ increases monotonically, implying information loss Hawking:1976ra . Using the same semiclassical solution, the RT proposal computes the radiation entropy differently, as the generalized entropy111For a partial Cauchy surface $X\subset M$, $S_{\rm gen}(X)=\text{Area}[\partial X]/4G\hbar+S(X)$, where $\partial$ denotes the boundary of a set, and $S(X)$ is the renormalized von Neumann entropy of the density operator of the quantum field theory state reduced to $X$. See the appendix in Ref. Bousso:2015mna for a detailed discussion of this quantity. of the entanglement wedge of the radiation, $E(R)$: $S(\mathbf{R})=S_{\rm gen}[E(R)]~{}.$ (1) The bold-face notation Almheiri:2019yqk distinguishes the (presumably correct) entropy computed by RT from the entropy $S(R)$ computed directly from the semiclassical radiation state (See Bousso:2019ykv ; Bousso:2020kmy for a proposal to reconcile the bold and unbold states.) The original RT prescription defines an entanglement wedge for regions on the conformal boundary of AdS. In the present context, $R$ is a bulk system, and the entanglement wedge must be defined as follows Bousso:2020kmy : 1. 1. $E(R)=I\cup R$, where $I\subset M$; 2. 2. $S_{\rm gen}(I\cup R)$ is stationary under any local variations of the boundary surface $\partial I$; 3. 3. among all such regions globally, $I$ yields the smallest $S_{\rm gen}(I\cup R)$. The above definitions apply if $R$ is a nongravitating system external to $M$; in asymptotically AdS geometries the radiation can be extracted into such a system Penington:2019npb ; Almheiri:2019psf . We now turn to the case where $R$ is a weakly gravitating region inside the spacetime. For example, $R$ may be a distant region occupied by Hawking radiation in an asymptotically flat or AdS spacetime (see Ref. Bousso:2019ykv for a detailed setup). Physically, one expects the RT prescription for a weakly gravitating region to reduce to that for a nongravitating system, Eq. (1), and we shall assume this here. $R$ resides in a weakly gravitating region far from any potential island $I$, so $\partial I\cap\bar{R}=\varnothing$, where an overbar denotes topological closure. As before, stationarity of $S_{\rm gen}$ is required only under variations of $\partial I$, not of $\partial R$. (This can be implemented in a path integral derivation Dong:2020uxp .) Thus, the definition of $E(R)$ is essentially unchanged. However, in the presence of gravity, it is simplest to work with the generalized entropy of the region $\mathbf{R}$ (a cutoff-independent quantity), so we add its boundary area to both sides of Eq. (1): $S_{\rm gen}(\mathbf{R})=S_{\rm gen}[E(R)]~{}.$ (2) It is easy to derive the Page curve, at least approximately, if one ignores condition 2. We will now summarize this incomplete argument, before discussing how it can be completed. Figure 1: Left: evaporating black hole; right: its Page curve. After the Page time, the semiclassical entropy $S(R)$ of the Hawking radiation in the asymptotic region $R$ exceeds the Bekenstein-Hawking entropy of the black hole, $A_{h}/4G\hbar$. The “Hawking partners” in the black hole interior purify $R$. (Dashed lines indicate entanglement.) Therefore, adjoining $\hat{I}$ to $R$ decreases the generalized entropy $S_{\rm gen}$. However, islands must have stationary $S_{\rm gen}$. Solving for this condition exceeds present analytic control over the entropy. The island finder presented here sidesteps this obstruction. The Page time $t_{\rm Page}$ is defined as the time when the black hole and radiation entropies are equal in the semiclassical analysis: $S[R(t_{\rm Page})]=\frac{A_{h}(t_{\rm Page})}{4G\hbar}~{}.$ (3) Let $\hat{I}(t)$ be the black hole interior at time $t$ (see Fig. 1). The Hawking “partners” in $\hat{I}(t)$ purify the radiation $R(t)$ emitted so far; hence $S_{\rm gen}[\hat{I}(t)\cup R]\approx\frac{A_{h}(t)}{4G\hbar}~{}.$ (4) Before the Page time, this is greater than $S(R)$ by definition, so $\hat{I}$ is not a viable island candidate; one finds that $I(t)=\varnothing$, $E(R)=R$, and $S(\mathbf{R})=S(R)$. This corresponds to the rising part of the Page curve, where it agrees with Hawking’s curve. But after the Page time, $S_{\rm gen}[\hat{I}(t)\cup R]<S(R)$ by Eqs. (3) and (4). Thus, the inclusion of an island $I(t)\approx\hat{I}(t)\neq\varnothing$ is favored, and we have $S(\mathbf{R})=S(I\cup R)\approx A_{h}/4G\hbar$. As the black hole evaporates and its horizon shinks, this yields the decaying part of the Page curve required by unitarity. The Page curve result has been extended to asymptotically flat spacetimes Gautason:2020tmk ; Hartman:2020swn , settings with two layers of holography Almheiri:2019hni ; Almheiri:2019psy , and eternal black holes Almheiri:2019yqk . Entanglement islands can also appear in cosmology, where their significance is less obvious Chen:2020tes ; Hartman:2020khs . ### 1.2 Summary and Outline Our brief summary of the Page curve result has a major gap. We explained why condition 3 (global minimization of $S_{\rm gen}$) favors inclusion of the black hole interior $\hat{I}(t)$ in $E(R)$ after the Page time. However, we did not show that condition 2 (local extremality) can be satisfied by some deformation of $\hat{I}(t)$ small enough to preserve condition 3. One way to fill this gap is to find $I(t)$ exactly, and to verify condition 2. However, explicit solutions of the quantum extremality condition have been found only in $1+1$ bulk dimensions Almheiri:2019psf ; Almheiri:2019hni ; Almheiri:2019yqk , or in toy models of higher-dimensional black holes Penington:2019npb . The difficulty lies in computing the von Neumann entropy $S(I\cup R)$. This depends on the detailed state of the dynamics of the radiation fields, including modes with nonzero angular momentum, and their interactions. Even free fields scatter nontrivially in a black hole background, placing an exact calculation out of reach. In Sec. 2, we develop an alternative way to ensure that condition 2 holds. We show that the existence of a suitable island $I$ follows from simple sufficient conditions that are easy to verify:222Ref. Penington:2019npb presents an elegant existence argument that establishes an island in the setting of an evaporating black hole. It makes use of properties of the event horizon and is inequivalent to the more general argument presented here. Let $I^{\prime}$ be a region that (i) satisfies condition 1, $S_{\rm gen}(I^{\prime}\cup R)<S(R)$, and suppose that (ii) the generalized entropy of $I^{\prime}\cup R$ increases under any small outward deformation of $I$, or decreases under any such deformation. Then there exists an island, $I\neq\varnothing$; and moreover $S(\mathbf{R})=S_{\rm gen}(I\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)~{}.$ (5) We will illustrate in a number of examples that finding a suitable $I^{\prime}$ is not difficult; in particular, it suffices to understand the scaling of corrections to simple models of the entropy. Moreover, the upper bound (5) is powerful enough to establish the main features of the Page curve for an evaporating black hole. In Sec. 3, we consider a different but related problem that yields to a similar analysis. We consider a spacetime and (internal or external) reference system $R$ in a pure state. We show that the true entropy $S(\mathbf{R})$ cannot exceed the generalized entropy of appropriate bulk regions. For example, if $R$ is external, and $M$ is an evaporating black hole spacetime, an upper bound on $S(\mathbf{R})$ is furnished by the generalized entropy of the bulk region that can be probed by an asymptotic observer (the black hole exterior). ## 2 Sufficient Conditions for Islands In this section we identify sufficient conditions for the existence of an island. In Sec. 2.1 we begin with the case of an external reference system, $R\cap M=\varnothing$. In Sec. 2.2 we allow $R$ to intersect with $M$. In Sec. 2.3 we consider various examples in which our conditions easily establish the presence of islands; in particular, we show that they suffice to derive the Page curve. ### 2.1 External Reference System Let $M$ be a semiclassical spacetime which together with a reference system $R$ is in a pure state. We take $R$ to be external to $M$. For definiteness, we assume that $R$ is nongravitating; otherwise, simply substitute $S\to S_{\rm gen}$ when the argument contains $R$. Figure 2: Island finder. Suppose that $I^{\prime}\cup R$ is quantum normal (top) or anti-normal (bottom). Then the generalized entropy of $I^{\prime}\cup R$ decreases along the dashed lines to $\tilde{I}^{\prime}\subset\Sigma$. An island $I$ with even smaller generalized entropy $S_{\rm gen}(I\cup R)$ must exist on the maximin Cauchy slice $\Sigma$. If $S_{\rm gen}(I^{\prime}\cup R)<S(R)$, the island cannot be empty. Suppose there exists a partial Cauchy surface $I^{\prime}\subset M$ satisfying the following conditions: $\displaystyle(i)$ $\displaystyle~{}~{}~{}S_{\text{gen}}(I^{\prime}\cup R)<S(R)~{};$ (6) $\displaystyle(ii)$ $\displaystyle\begin{cases}k^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\geq 0~{},~{}~{}\ell^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\leq 0~{};\\\ \text{~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}or}\\\ k^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\leq 0~{},~{}~{}\ell^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\geq 0~{},\end{cases}$ (7) where $\Theta$ is the quantum expansion one-form Engelhardt:2014gca ; Bousso:2015mna and $k$ and $\ell$ are the outward and inward future-directed null vectors normal to $\partial I^{\prime}$. Thus, for example, $k^{\mu}\Theta_{\mu}[I^{\prime}\cup R,y]$ is the rate of change of $S_{\rm gen}$, per unit transverse area and unit affine parameter length, as $I^{\prime}\cup R$ is deformed outward along the future-directed null geodesic orthogonal to $I^{\prime}$ at $y$. We drop the argument $y$ when an equation holds for all $y$. The first condition states that adjoining $I^{\prime}$ to $R$ decreases the entropy, even as the Bekenstein-Hawking entropy of $I^{\prime}$ is included. The second, Eq. (7), says that $I^{\prime}\cup R$ is quantum normal (its generalized entropy does not decrease under any small outward deformation of $I^{\prime}$) or quantum anti-normal (the opposite). We will now show that these conditions imply the existence of a non-empty island region $I$. Our proof uses the maximin construction of the HRT surface Wall:2012uf , which we assume extends to a quantum maximin prescription Akers:2019lzs : on every Cauchy surface of $M$, one finds the region $I^{\prime\prime}$ that minimizes $S_{\rm gen}(I^{\prime\prime}\cup R)$. (Note that $I^{\prime\prime}$ may be the empty set.) One then maximizes the same quantity over all Cauchy surfaces of $M$. The island $I$ is defined to be the region $I^{\prime\prime}$ that achieves this maximum. (This is expected to exist Wall:2012uf ; Akers:2019lzs .) Note that we define $I$ as an achronal region; hence it is non-unique. Similarly, the maximin Cauchy slice $\Sigma$ is non-unique. The relevant unique object is the domain of dependence $D(I)$. (We ignore the degenerate case where there are two islands with identical generalized entropy but different $D$.) Any other Cauchy slice of $D(I)$ will be an island if $I$ is, though not every Cauchy slice of $D(I)$ will be part of a maximin slice $\Sigma$. Our goal is to show that $I\neq\varnothing$. In the normal case, $k^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\geq 0$, we define $\tilde{I}^{\prime}\equiv D(I^{\prime})\cap\Sigma~{},$ (8) as the representative of $I^{\prime}$ on $\Sigma$. In the anti-normal case, $k^{\mu}\Theta_{\mu}[I^{\prime}\cup R]\leq 0$, we define the representative instead as $\tilde{I}^{\prime}\equiv J(I^{\prime})\cap\Sigma~{},$ (9) where $J$ denotes all points that can be reached from $I^{\prime}$ along a causal curve (the future and past of $I^{\prime}$). In either case, note that $\tilde{I}^{\prime}$ is obtained from $I^{\prime}$ by deforming along an orthogonal null congruence with initially negative quantum expansion. We assume the Quantum Focussing Conjecture (QFC) Bousso:2015mna , that the quantum expansion cannot increase along a null shape deformation. This implies that $S_{\rm gen}(\tilde{I}^{\prime}\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)~{}.$ (10) Since $\Sigma$ is the maximin Cauchy slice, $S_{\rm gen}(\tilde{I}^{\prime}\cup R)\geq S_{\rm gen}(I\cup R)~{}.$ (11) Combined with Eq. (10), this implies333This intermediate result is closely related to corollary 16b of Wall:2012uf . A simple application of our argument to asymptotically AdS spacetimes yields the following result: given a partial Cauchy slice $A$ on the asymptotic boundary of AdS, let $X$ be the RT surface associated to $A$ with homology slice $H$. Now, consider another surface $X^{\prime}$, homologous to $A$ with homology slice $H^{\prime}$. If $H^{\prime}$ is a quantum normal or anti-normal region, then $S_{\rm gen}(H)\leq S_{\rm gen}(H^{\prime})$. $S_{\rm gen}(I^{\prime}\cup R)\geq S_{\rm gen}(I\cup R)~{}.$ (12) Using the assumption in Eq. (6), we get $S_{\rm gen}(I\cup R)<S_{\rm gen}(R)~{}.$ (13) Therefore, we conclude that $I\neq\varnothing$. ### 2.2 Distant Reference System In Section 2.1, the reference system $R$ was external to the spacetime $M$. We will now allow a system that is wholly or partially inside the spacetime: $R\cap M\neq\varnothing$. In order to generalize our island finder to this setting, we shall require that there exists a partial Cauchy slice $I_{0}$ spacelike to $R$ such that $I_{0}\cup R$ is a quantum normal region with respect to deformations at $\partial I_{0}$. That is, we require $\displaystyle\bar{I_{0}}$ $\displaystyle\subset M-\overline{J(R)}~{},$ (14) $\displaystyle k^{\mu}\Theta_{\mu}[I_{0}\cup R]$ $\displaystyle\geq 0~{},$ (15) $\displaystyle\ell^{\mu}\Theta_{\mu}[I_{0}\cup R]$ $\displaystyle\leq 0~{}.$ (16) where as before an overbar denotes closure. As before, $\Theta_{\mu}[I_{0}\cup R]$ is the quantum expansion one-form, and $k^{\mu}$ and $\ell^{\mu}$ are future-directed null vectors fields in the normal bundle of $\partial I_{0}$ in the outward and inward directions respectively. For example, these conditions will be satisfied when $I_{0}$ is the interior of a sufficiently large, approximately round sphere in an asymptotically flat spacetime; and $R$ is the exterior of a slightly larger sphere concentric with the first, or any subsystem thereof (such as the Hawking radiation emitted by a black hole). Note that we do not require that $R$ be weakly gravitating, but in many examples of interest this will be the case. Also, we do not require that Eqs. (15) and (16) hold at $\partial R$. Now, suppose there exists a partial Cauchy slice $I^{\prime}\subset D(I_{0})$ satisfying the conditions (6) and (7). That is, $I^{\prime}\cup R$ is quantum normal or anti-normal, and adjoining $I^{\prime}$ to $R$ reduces the generalized entropy of $R$. Then there exists a non-empty quantum extremal region $\hat{I}\subset D(I_{0})$ satisfying $S_{\rm gen}(\hat{I}\cup R)<S_{\rm gen}(R)$. Note that this conclusion is weaker than in Sec. 2.1: $\hat{I}$ need not globally minimize $S_{\rm gen}(I\cup R)$ among all eligible regions, since we are restricting our search to a subset of $M$. However, the true entanglement wedge can only have lower entropy, so $S_{\rm gen}(\hat{I}\cup R)$ provides an upper bound. Note also that when $I^{\prime}\cup R$ is quantum normal, we can set $I_{0}=I^{\prime}$, so there is no need to identify a larger $I_{0}$ region. #### Proof, part I The proof strategy is similar to that in Section 2.1, except that we wish to restrict our search to the closed set $D(I_{0})$.444A maximin procedure restricted to entanglement wedges of AdS without reflecting boundary conditions was considered in Marolf:2019bgj . See also Appendix B of Brown:2019rox for a related discussion. We will first assume strict version of conditions (15) and (16): $\displaystyle k^{\mu}\Theta_{\mu}[I_{0}\cup R]$ $\displaystyle>0~{},$ (17) $\displaystyle\ell^{\mu}\Theta_{\mu}[I_{0}\cup R]$ $\displaystyle<0~{},$ (18) Later, we will demonstrate how to relax this assumption back to conditions (15) and (16). Let $\hat{I}$ be the maximin region of $D(I_{0})\cup R$, and let $\Sigma$ be a Cauchy surface of $D(I_{0})$ on which $\hat{I}$ minimizes the generalized entropy of $\hat{I}\cup R$ among all subregions of $\Sigma$. Without loss of generality, we may take $\hat{I}\subset\Sigma$: since any Cauchy surface of $D(\hat{I})$ is an equally good maximin region, we set $\hat{I}\to\Sigma\cap D(\hat{I})$. As in Ref. Wall:2012uf , we assume that $\hat{I}$ is stable: any nearby Cauchy surface $\Sigma^{\prime}$ obtained by infinitesimal deformation of $\Sigma$ contains a locally minimal region $\hat{I}^{\prime}$ infinitesimally close to $\hat{I}$ with $S_{\rm gen}(\hat{I}^{\prime}\cup R)\leq S_{\rm gen}(\hat{I}\cup R)$. It immediately follows from the analysis of Sec. 2.1 that $\displaystyle S_{\rm gen}(\hat{I}\cup R)<S_{\rm gen}(R)~{}.$ (19) We will now show that $\partial\hat{I}\cap\partial D(I_{0})=\varnothing$. This precludes the (unwanted) possibility that $I\cup R$ is maximin but not locally stationary. It follows that $\hat{I}\cup R$ is a quantum extremal region Wall:2012uf . $\partial D(I_{0})$ is the disjoint union of three sets: $\partial I_{0}$, and two null hypersurfaces $N_{+\ell}$ and $N_{-k}$ that lie in the future and past of $I_{0}$ respectively. The latter sets are generated by future and past-directed null geodesics orthogonal to $\partial I_{0}$ in the inward direction which end at caustics or self-intersections Akers:2017nrr . Let $\ell^{\mu}$ ($k^{\mu}$) be the normal vector field to $N_{+\ell}$ ($N_{-k}$) obtained by parallel propagation of $\ell^{\mu}\|_{\partial I_{0}}$ ($k^{\mu}\|_{\partial I_{0}}$). Let $\Sigma_{M}$ be a Cauchy surface of $M$ that intersects each null generator of $N_{+\ell}$ and $N_{-k}$ at most at one point. Let $\Sigma_{N}\equiv\Sigma_{M}\cap D(I_{0})$. By Eqs. (17), (18), and the QFC Bousso:2015mna , $\displaystyle k^{\mu}\Theta_{\mu}[\Sigma_{N}\cup R;p]$ $\displaystyle>0~{}~{}~{}\text{ for all }p\in N_{-k}\cap\partial I_{0}~{};$ (20) $\displaystyle\ell^{\mu}\Theta_{\mu}[\Sigma_{N}\cup R;p]$ $\displaystyle<0~{}~{}~{}\text{ for all }p\in N_{+\ell}\cap\partial I_{0}~{},$ (21) Suppose for contradiction that there exists a point $q\in\partial D(I_{0})\cap\partial\hat{I}$. The generator of $\partial D(I_{0})$ that contains $q$, and hence its tangent vector $\ell^{\mu}$ or $k^{\mu}$, will be normal to $\partial\hat{I}$ at $q$. (If $q\in\partial I_{0}$, this statement holds for either generator emanating from $q$.) Since $N_{+\ell}$ ($N_{-k}$) is nowhere to the past (future) of $\hat{I}$, theorem 1 in Ref. C:2013uza implies $\displaystyle k^{\mu}\Theta_{\mu}[\hat{I}\cup R;q]$ $\displaystyle>0~{}~{}~{}\text{ for }q\in N_{-k}\cap\partial I_{0}~{};$ (22) $\displaystyle\ell^{\mu}\Theta_{\mu}[\hat{I}\cup R;q]$ $\displaystyle<0~{}~{}~{}\text{ for }q\in N_{+\ell}\cap\partial I_{0}~{}.$ (23) Suppose first that $q\in\partial I_{0}$. In this case the above expansions imply that a small inward deformation of $\hat{I}$ will decrease the generalized entropy, in contradiction with the minimality of $S_{\rm gen}(\hat{I}\cup R)$ among all subregions of the maximin Cauchy surface $\Sigma$. We will now demonstrate this rigorously, using the notion of a surface- orthogonal exponential map Akers:2017nrr , $\exp_{K}~{}:~{}NK\to M~{},~{}~{}(p,v)\to c_{p,v}(1)~{},$ (24) which takes a vector $v$ in the normal bundle $NK$ of a smooth submanifold $K$ at the point $p$ to the point at affine distance 1 along the unique geodesic through $p$ with tangent vector $v$. We will use $\widetilde{\exp}$ to denote an exponential map in which the submanifold $\Sigma$ plays the role of $M$ in the above definition. Let $\tilde{v}^{\mu}(p)$ be a smooth inward-directed vector field in the normal bundle to $\partial\hat{I}$, viewed as a submanifold of $\Sigma$. We define the continuous one-parameter family of inward deformations of $\hat{I}$ as the regions $\displaystyle\hat{I}(\epsilon)=\text{int}\\{\widetilde{\exp}_{\partial\hat{I}}(p,\epsilon\tilde{v}^{\mu}(p)):p\in\partial\hat{I}\\}~{}.$ (25) Similarly, in the manifold $M$ we define $\displaystyle i(\epsilon)=\hat{I}\cap\text{Int}\\{\exp_{\partial\hat{I}}(p,\epsilon v^{\mu}(p)):p\in\partial\hat{I}\\}~{},$ (26) where $v$ is the push-forward of $\tilde{v}$ under the embedding map of $\Sigma$ into $M$, and $\text{Int}\,X$ denotes the spacetime region spacelike and interior to $X$. For sufficiently small $\epsilon$, both families are well-defined. Moreover, $S_{\rm gen}[\hat{I}(\epsilon)\cup R]=S_{\rm gen}[i(\epsilon)\cup R]+O(\epsilon^{2})~{}.$ (27) In $M$, the deformation profile $v^{\mu}\in N\partial\hat{I}$ can be decomposed as $\displaystyle v^{\mu}=-ak^{\mu}+b\ell^{\mu}~{},$ (28) where $a$ and $b$ are positive definite functions. Hence $\displaystyle\left.\frac{dS_{\rm gen}[\hat{I}(\epsilon)\cup R]}{d\epsilon}\right\|_{\epsilon=0}=\int dx\sqrt{h}~{}v^{\mu}\Theta_{\mu}[\hat{I}\cup R]~{},$ (29) where $h$ refers to the intrinsic metric of $\partial\hat{I}$. We now choose $\tilde{v}\equiv 0$ outside a $\delta$-neighborhood of $q$ in $\partial\hat{I}$. For small enough $\delta$, $v^{\mu}\Theta_{\mu}[\hat{I}\cup R]<0$ in the entire $\delta$-neighborhood, by Eqs. (22) and (23) and continuity. Hence, $\left.\frac{dS_{\rm gen}[\hat{I}(\epsilon)\cup R]}{d\epsilon}\right\|_{\epsilon=0}=\left.\frac{dS_{\rm gen}[i(\epsilon)\cup R]}{d\epsilon}\right\|_{\epsilon=0}<0~{}.$ (30) Hence, $\hat{I}$ does not minimize the generalized entropy on $\Sigma$. This contradicts our assumption that $\hat{I}$ is a maximin region. Therefore, no such point $q\in\partial\hat{I}\cap\partial I_{0}$ can exist: $\partial\hat{I}\cap\partial I_{0}=\varnothing~{}.$ (31) Suppose instead that $q\in\partial D(I_{0})-\partial I_{0}$. In this case, minimality of $S_{\rm gen}(\hat{I}\cup R)$ on $\Sigma$, together with Eq. (22) or (23), implies that a small deformation of $\Sigma$ into the interior of $D(I_{0})$ near $q$ will produce a Cauchy surface of $I_{0}$ whose minimal-$S_{\rm gen}$ region (together with $R$) has greater generalized entropy than $\hat{I}\cup R$. But this is impossible if $\hat{I}$ was constructed by the maximin procedure. Again we will now aim to make this argument rigorous. Figure 3: Maximin restricted to the domain of dependence (“wedge”) $D(I_{0})$ returns a region $\hat{I}$ on a maximin slice $\Sigma$. If $I_{0}\cup R$ is quantum normal, then $\partial\hat{I}$ cannot intersect $\partial D(I_{0})$ (dashed). Left: if $\hat{I}\cap\partial I_{0}\neq\varnothing$, then $S_{\rm gen}(\hat{I}(\epsilon)\cup R)<S_{\rm gen}(\hat{I}\cup R)$, contradicting the min of maximin. Right: if $\hat{I}\cap\partial D(I_{0})-\partial I_{0}$ then $S_{\rm gen}(\hat{I}(\epsilon)\cup R)>S_{\rm gen}(\hat{I}\cup R)$ on the deformed slice $\Sigma(\epsilon)$ violates the max of maximin. For definiteness, we assume that $q\in N_{+\ell}$. (If instead $q\in N_{-k}$, the time reverse of our argument applies.) In any open neighborhood $O(q)$, $\Sigma$ (and hence $\hat{I}$) must enter the interior, $O(q)\cap\Sigma\cap\text{int}[D(I_{0})]\neq\varnothing$, or else Eq. (23) would violate the minimality of $S_{\rm gen}(\hat{I}\cup R)$ on $\Sigma$. Hence the inward-directed vector field $t^{\mu}$ orthogonal to $\partial\hat{I}$ and tangent to $\hat{I}$ is spacelike in an open neighborhood of $q$ on $\partial\hat{I}$. Moreover, $\Sigma$ must contain the null generator segment of $N_{+\ell}$ connecting $\partial I_{0}$ to $q$, or $\Sigma$ would fail to be achronal. Let the achronal hypersurfaces $\Sigma(\epsilon)$ be a smooth one parameter deformation of $\Sigma$ such that $\Sigma(\epsilon)$ agrees with $\Sigma$ outside a $\delta$-neighborhood of $q$ (and everywhere for $\epsilon=0$). We also require that $\Sigma(\epsilon_{2})$ is nowhere to the future of $\Sigma(\epsilon_{1})$ if $\epsilon_{1}<\epsilon_{2}$. The stability assumption guarantees the existence of a smooth one-parameter family of regions $\hat{I}(\epsilon)$, each of which locally minimizes $S_{\rm gen}[\hat{I}(\epsilon)\cup R]$ on the corresponding $\Sigma(\epsilon)$. By the max step of maximin, $\frac{dS_{\rm gen}[\hat{I}(\epsilon)\cup R]}{d\epsilon}\leq 0~{}.$ (32) For small enough $\epsilon$, there exists an infinitesimal vector field $w^{\mu}$ in the normal bundle of $\partial\hat{I}$ in a neighborhood of $q$ such that $\partial\hat{I}(\epsilon)=\\{\exp_{\partial\hat{I}}(p,\epsilon w^{\mu}(p)+O(\epsilon^{2})):p\in\partial\hat{I}\\}~{}.$ (33) We have $\frac{dS_{\rm gen}[\hat{I}(\epsilon)\cup R]}{d\epsilon}=\int dx\sqrt{h}~{}w^{\mu}\Theta_{\mu}{[\hat{I}\cup R]}~{}.$ (34) Since $t^{\mu}$ is spacelike, it is linearly independent of $\ell^{\mu}$, so there exists a unique decomposition $w^{\mu}=ct^{\mu}+d\,(-\ell^{\mu})~{}.$ (35) with $c$ and $d$ nonnegative functions on $\partial\hat{I}$ that vanish outside an open neighborhood of $q$. Hence $\frac{S_{\rm gen}[\hat{I}\cup R]}{d\epsilon}=\int dx\sqrt{h}~{}[ct^{\mu}\Theta_{\mu}+d\,(-\ell^{\mu})\Theta_{\mu}]~{}.$ (36) The first term is positive-definite by the minimality of $S_{\rm gen}[\hat{I}\cup R]$ on $\Sigma$; the second is positive by Eq. (23). Hence $\frac{S_{\rm gen}[\hat{I}\cup R]}{d\epsilon}>0~{},$ (37) which contradicts Eq. (32). Therefore, $\partial\hat{I}\cap N_{+\ell}=\varnothing~{},$ (38) and a time-reversed argument implies $\partial\hat{I}\cap N_{-k}=\varnothing~{}.$ (39) Together with Eq. (31) this establishes that $\hat{I}\subset\text{int}[D(I_{0})]~{};$ (40) hence $\hat{I}$ is locally quantum extremal. #### Proof, part II We will now discuss what happens if we relax the conditions (17) and (18) to their non-strict versions (15) and (16). We will argue that while in this case $\hat{I}$ might not be contained in $\text{int}[D(I_{0})]$, an island candidate still exists, i.e., there exists $\hat{I}\subset D(I_{0})$ such that $\hat{I}\cup R$ is quantum extremal and $S_{\rm gen}(\hat{I}\cup R)<S_{\rm gen}(R)$. Let us start with the case where there exists at least a point $p\in\partial I_{0}$ where $k^{\mu}\Theta_{\mu}[I_{0}\cup R;p]>0$ and a point $q$ ($p=q$ is allowed) where $\ell^{\mu}\Theta_{\mu}[I_{0}\cup R;p]<0$. Then, $\partial\hat{I}$ cannot be a cross section of $N_{-k}$ ($N_{+\ell}$) because then by the QFC there would be a point $r$ in the cross section, along the same generator as $p$ ($q$), where $k^{\mu}\Theta_{\mu}[\hat{I}\cup R;r]>0$ ($\ell^{\mu}\Theta_{\mu}[\hat{I}\cup R;r]<0$). As discussed above, this contradicts maximin. If $\partial\hat{I}$ only partially coincides with $\partial D(I_{0})$ then there must exist a point $r$ in the boundary of the intersection set such that in any sufficiently small neighborhood of it $\partial\hat{I}$ and $N_{-k}$ ($N_{+\ell}$) do not coincide. In the $\hbar\to 0$ limit, Lemma B of C:2013uza implies that there exists a point $s$ in a neighborhood of $r$ such that $k^{\mu}\theta_{\mu}[\hat{I};s]>0$ ($\ell^{\mu}\theta_{\mu}[\hat{I};s]<0$). Here, we will assume that the Lemma B of C:2013uza continues to hold when we replace the classical expansion with the quantum expansion.555Our assumption is motivated by the semiclassical generalization of many similar conditions on the classical expansion Bousso:2015mna . Note that if the von Neumann entropy term in the quantum expansion is $O(G\hbar)$ while the classical expansion is $O(1)$, Lemma B trivially generalizes to the version with quantum expansions. We will then conclude that $k^{\mu}\Theta_{\mu}[\partial\hat{I}\cup R;s]>0$ ($\ell^{\mu}\Theta_{\mu}[\partial\hat{I}\cup R;s]<0$). However, $s\in\text{int}[D(I_{0})]$, so a non-zero quantum expansion at $s$ is not allowed by maximin. Next, we consider the case where $\ell^{\mu}\Theta_{\mu}[I_{0}\cup R]=0$ (The case with $k$ and $\ell$ exchanged is similar by time-reflection symmetry). By the previous arguments, $\partial\hat{I}$ cannot intersect $N_{-k}$. Also, by the quantum version of Lemma B, $\partial\hat{I}$ cannot intersect $N_{+\ell}$ only in part. We will therefore focus on discussing the case where $\partial\hat{I}$ is a cross section of $N_{+\ell}$. Associated with any cross section $L$ of $N_{+\ell}$, we can define a partial Cauchy slice $\Sigma_{L}$ of $D(I_{0})$ which intersects $N_{+\ell}$ at $L$. Let $L_{1}$ be the latest cross section of $N_{+\ell}$ such that any other cross section $L$ in the past of $L_{1}$ satisfies $\ell^{\mu}\Theta_{\mu}[\Sigma_{L}\cup R]=0$.666If such $L_{1}$ does not exists, then $N_{+\ell}$ is a semi-infinite stationary null hypersurface which, at least classically, by the no-hair theorem needs to be a semi- infinite portion of the horizon of a Kerr-Newman black hole. And since $k^{\mu}\Theta_{\mu}[I_{0}\cup R]\geq 0$, $\partial I_{0}$ needs to lie fully in the past of the bifurcation surface which provides us with a classical extremal surface on $N_{+\ell}$. We then expect to find a quantum extremal region $\hat{I}\cup R$ such that $\partial\hat{I}$ is either on or near this classical extremal surface. By the QFC, any cross section $L$ which is partly in the future of $L_{1}$ needs to contain at least a point $r$ at which $\ell^{\mu}\Theta_{\mu}[\Sigma_{L}\cup R;r]<0$. Hence, $\partial\hat{I}$ must be a cross section in the past of $L_{1}$. Furthermore, a maximin Cauchy slice $\Sigma$ corresponding to $\hat{I}$ cannot intersect the future of $L_{1}$ as it would violate the minimality of $S_{\rm gen}(\hat{I}\cup R)$. $\Sigma$ then has to leave $N_{+\ell}$ in the past of $L_{1}$ or on $L_{1}$. Let $L_{2}$ be the cross section at which $\Sigma$ leaves $N_{+\ell}$. Then, $k^{\mu}\Theta_{\mu}[\Sigma_{L_{2}}\cup R]\leq 0$ or else the min condition is violated. Since $k^{\mu}\Theta_{\mu}[I_{0}\cup R]\geq 0$, we expect that there exists a cross section $L_{3}$ between $\partial I_{0}$ and $L_{2}$ such that $k^{\mu}\Theta_{\mu}[\Sigma_{L_{3}}\cup R]=0$. In the classical limit in particular, we expect that the results of Andersson:2007gy showing a similar existence on spacelike Cauchy surfaces can be applied here by taking a limit of spatial Cauchy surfaces that approach $N_{+\ell}$. Together with stationarity along $N_{+\ell}$, this would imply that $\Sigma_{L_{3}}\cup R$ is quantum extremal. ### 2.3 Examples We will now present some examples where the above sufficient conditions (i) and (ii) provide an efficient diagnostic for the existence of islands. We will require no detailed calculations of matter entropy and its derivatives, nor will we be forced to assume special symmetries, low dimensions of spacetime, or adopt other toy models. Our sufficient conditions establish the existence of an island and its key properties, at the cost of not exactly locating the island. In a final example, we show that neither of the two sufficient conditions can be eliminated. #### Evaporating black hole after the Page time For concreteness, we pick asymptotically flat boundary conditions, though our conclusion will not depend on this choice. We furthermore assume that the black hole mostly radiates massless particles, as will be the case if its initial mass is sufficiently large. We consider an evaporating black hole formed in a pure state. At late times, the spacetime will be approximately spherically symmetric. Let $r=(A/4\pi)^{1/2}$ be the area radius of spheres. Near the horizon, the metric is well-approximated by $ds^{2}=-\left(1-\frac{r_{s}(v)}{r}\right)dv^{2}+2dv\,dr+r^{2}d\Omega^{2}~{},$ (41) Due to evaporation, $r_{s}(v)$ decreases slowly with retarded time $v$: $dr_{s}/dv\sim-O(G\hbar/r_{s}^{2})$. For $r\gg r_{s}$, the metric is well- approximated instead by the outgoing Vaidya metric Abdolrahimi:2016emo , but this will not be important in our analysis. Figure 4: Evaporating black hole after the Page time. Hawking radiation has accumulated in $R$. As shown on the left, the boundary of the past of $R$, denoted by $r_{0}(v)$, intersects the stretched horizon (shown in purple) at the sphere $A_{s}$, which together with the $A_{h}$ sphere on the event horizons reside at retarded time $v_{s}$. We consider candidate regions $I^{\prime}$ with boundary $\partial I^{\prime}$ on a causal horizon spacelike to $R$ (grey regions). The generalized second law implies that $S_{\rm gen}(I^{\prime}\cup R)$ increases under future outward deformations. For future inward deformations, quantum normalcy follows from the (trivial) classical normalcy in the dark grey subregion, which is chosen to keep quantum corrections to the expansion small. The $I^{\prime}$ that minimizes $S_{\rm gen}(I^{\prime}\cup R)$ subject to these restrictions is shown in pink. Its boundary is located a $\Delta v=r_{s}\log c_{1}$ to the future of $A_{h}$, as shown on the right. We show that it provides an extremely tight upper bound on the true entropy $S(\mathbf{R})=S_{\rm gen}(I\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)=A_{h}/4G\hbar+O(1)$. Let $u$ be retarded time on $\mathscr{I}^{+}$, and let $R$ be the portion of $\mathscr{I}^{+}$ given by $u\leq u_{0}$; see Fig. 4. $R$ is a reservoir that contains the Hawking radiation emitted until the time $u_{0}$. The boundary of the past of $R$ is given by $u=u_{0}$. We will be interested in the behavior of the metric only near the retarded time when this surface intersects the stretched horizon $r_{s}$, so it will be sufficient to set $r_{s}$ to that value and neglect its $v$-dependence from here on. Let $A_{s}=4\pi r_{s}^{2}$ be the area of the stretched horizon where it meets the past of $R$. Let $A_{h}=4\pi r_{h}^{2}$ be the area of the event horizon where it intersects the future of $A_{s}$; the areas satisfy $A_{s}=A_{h}+O(G\hbar)~{}.$ (42) We choose $u_{0}$ late enough so that $S(R)>A_{h}/4G\hbar+\log c_{1}$, where $\log c_{1}$ will be small in a sense made precise below. That is, $R$ extends to after the Page time, with a little room to spare. We seek an $I^{\prime}$ that satisfies our sufficient conditions while placing a tight bound on the entropy $S(\mathbf{R})$. Let $I^{\prime}(r,v)$ be a Cauchy slice of the interior of the sphere $(r,v)$. Since $S_{\rm gen}(I^{\prime}\cup R)$ is well-defined only for achronal $I^{\prime}\cup R$, we require $I^{\prime}(r,v)\subset M-J^{-}(R)$, or $u>u_{0}$. In the ingoing coordinates of Eq. (41), $u=u_{0}$ corresponds to a function $r_{0}(v)$, defined implicitly by $v=u_{0}+2r_{}(r_{0})$, where $r_{}(r)=r+r_{h}\log(\frac{r}{r_{h}}-1)$. Near the horizon, this satisfies $\Delta r_{0}(v)\equiv r_{0}(v)-r_{h}=r_{h}\exp\left(\frac{v-u_{0}}{2r_{h}}-1\right)~{}.$ (43) We thus require $r<r_{0}(v)$ for the boundary of $I^{\prime}$. Quantum normalcy of $I^{\prime}\cup R$ requires that the generalized entropy grows along both of the null directions away from $I^{\prime}$. Any future outward light-cone outside the horizon is a null surface of constant $u$ that reaches $\mathscr{I}^{+}$. Hence it is a causal horizon. The generalized second law of thermodynamics Bekenstein:1972tm ; Wall:2011hj applies to all causal horizons. It implies that the future outward quantum expansion at $I^{\prime}$ is positive if $\partial I^{\prime}$ is outside the horizon. (In more general settings that are not exactly spherically symmetric, we can accomplish the same goal by choosing $\partial I^{\prime}$ to be a cut of a causal horizon.) We thus require $r>r_{h}$. The past outward classical expansion is trivially positive: $\partial_{r}A=8\pi r$. Quantum normalcy follows if the quantum correction, $4G\hbar\,\partial_{r}S(I^{\prime}\cup R)$, is negligible, i.e., if $\partial_{r}S(I^{\prime}\cup R)\ll r/G\hbar$. Let $\Sigma$ be a global Cauchy slice containing $I^{\prime}\cup R$ and define $I^{\prime}_{c}=\Sigma-(I^{\prime}\cup R)$. By purity of the global quantum state, $S(I^{\prime}\cup R)=S(I^{\prime}_{c})$. Dimensional analysis dictates that the leading divergence in the renormalized entropy scales as $\partial_{\rho}S(I^{\prime}_{c})\sim O(1/\rho)~{},$ (44) where $\rho=r_{0}-r$ and we may assume $\rho\ll r_{0}$. To see this, suppose first that the only available scales are $r$ and $\rho$. Terms stronger than Eq. (44) would be of the form $\partial_{r}S(I^{\prime}_{c})\sim r^{n}/\rho^{n+1}$ with $n>0$ and positive coefficient. Such a term would imply that at fixed $\rho$, $dS/dr<0$, which is not physically sensible. In the presence of an additional mass scale $m\sim\hbar/\lambda$, an enhancement of Eq. (44) would have to take the form $\partial_{r}S(I^{\prime}_{c})\sim\lambda^{n}/\rho^{n+1}$, $n>0$. Formally, this is an enhancement for $\rho\ll\lambda$, but physically, a mass scale cannot have any physical effect in this UV regime. Hence, quantum normalcy is assured if we require $r_{0}-r>c_{1}\frac{G\hbar}{r}~{},~{}~{}c_{1}\gg 1~{}.$ (45) To summarize, we may consider any $I^{\prime}$ whose boundary is in the range $r_{h}\leq r\leq r_{0}(v)+c_{1}\frac{G\hbar}{r}~{},~{}~{}c_{1}\gg 1~{}.$ (46) We now minimize $S_{\rm gen}(I^{\prime}\cup R)$ over this range. By purity, $S_{\rm gen}(I^{\prime}\cup R)=4\pi r^{2}+4G\hbar S(I^{\prime}_{c})$. Along any ingoing light-cone, the classical area decreases rapidly and $\Delta r$ only increases as we go to smaller $r$, so we are driven to the smallest $r$ in the search space, the event horizon. Scanning in the other null direction, along the event horizon, $S_{\rm gen}(I^{\prime}_{c})$ will decrease towards the past, by the GSL. Hence we obtain the tightest upper bound on $S(\mathbf{R})$ by choosing $I^{\prime}$ to be the interior of the event horizon, as early as is possible while maintaining Eq. (45). With the boundary of $I^{\prime}$ on the event horizon, $r-r_{0}=\Delta r_{0}\propto\exp(v/2r_{h})$ by Eq. (43). Moreover, at $A_{h}$, we have $r_{0}=r_{s}$ and hence $\Delta r_{0}\sim O(G\hbar/r_{h})$. To grow this by the factor of $c_{1}$ demanded in Eq. (45), we must choose $v=v_{s}+r_{h}\log c_{1}$, where $v_{s}$ is the $v$-coordinate of $A_{s}$ and $A_{h}$. To summarize, the optimal choice of $I^{\prime}$ is $(r,v)=(r_{h},v_{s}+r_{h}\log c_{1})~{},~{}~{}c_{1}\gg 1~{}.$ (47) The true entropy $S(\mathbf{R})$ is upper bounded by $S_{\rm gen}(I^{\prime}\cup R)=\frac{\pi r^{2}}{G\hbar}+S(I^{\prime}_{c})=\frac{A_{h}}{4G\hbar}+O(\log c_{1})~{}.$ (48) Note that the $O(G\hbar)$ area difference between the event horizon and the stretched horizon is negligible. The $O(\log c_{1})$ term captures both the (negative) correction to the horizon area due to evaporation since $A_{h}$, and the (positive, and larger) correction due to the von Neumann entropy of $S(I^{\prime}_{c})$. We stress that this upper bound is quite tight. The correct $S(\mathbf{R})$ is given by Eq. (48) with $O(\log c_{1})$ replaced by $O(1)$. Recall that $c_{1}$ should be large enough to overcome any $O(1)$ coefficients that might enhance the von Neumann entropy in an exact calculation. But it is itself $O(1)$ in that sense, and $\log c_{1}$ is even smaller. In particular, we can always choose $\log c_{1}\ll\log\log(A_{h}/G\hbar)$ in the semiclassical limit. We also emphasize that exact spherical symmetry is not crucial; our argument only relies on the scaling behavior of the relevant terms. #### Recollapsing flat universe Our next Figure 5: Spatially flat radiation-dominated universe with negative cosmological constant, purified by a reference universe (thermal Minkowski space, right). If we choose a large enough reference region $R$ at $t_{\rm Mink}=0$, then the region $I^{\prime}$ at the turnaround time $t=0$ satisfies our sufficient conditions. Therefore an island $I$ must exist. example was studied in detail in Ref. Hartman:2020khs .777Ref. Hartman:2020khs considered a different but related question to ours: given a region $I$ in the cosmology, can one find a region $R$ in the reference spacetime such that $I$ is an island with respect to $R$. By contrast, we specify a reference region $R$ and use our sufficient conditions to establish the existence of an island for it. Consider a radiation-dominated, spatially flat Friedmann-Robertson- Walker (FRW) universe $M$ with cosmological constant $\Lambda$, purified by a thermal state on a Minkowski background $M_{R}$ without gravity. The metric of $M$ and $M_{R}$ is $\displaystyle ds^{2}$ $\displaystyle=-dt^{2}+a(t)^{2}(dr^{2}+r^{2}\,d\Omega^{2})~{},$ (49) $\displaystyle ds_{R}^{2}$ $\displaystyle=-dt_{R}^{2}+dr_{R}^{2}+r_{R}^{2}\,d\Omega_{R}^{2}~{}.$ (50) Without loss of generality, one can set the scale factor $a(0)=1$ at the turnaround time $t=0$, when $da/dt=0$ and hence $-\Lambda/8\pi G=\rho_{\rm rad}$. A thermofield double (TFD) state is constructed at $t=0$, $t_{R}=0$. A simple implication of the TFD state suffices for the purposes of our analysis. Consider two spatial regions, one in $M$ at $t=0$ and the other in $M_{R}$ at $t_{R}=0$. The von Neumann entropy of their union vanishes approximately, if they have the same spatial coordinates. If the regions are unequal, then the von Neumann entropy of their union will be given by the sum of the thermal entropy of the nonoverlap portions: $S=s_{\rm rad}(\hat{V}+\hat{V}_{R})~{},$ (51) where $\hat{V}$ and $\hat{V}_{R}$ are the volumes of the nonoverlap portions in $M$ and in $M_{R}$, and the entropy density is $s_{\rm rad}\sim\rho_{\rm rad}^{3/4}$. These statements receive corrections on scales below the thermal length scale, $\lambda\sim\rho_{\rm rad}^{-1/4}$. Now choose $R\subset M_{R}$ to be a ball of radius $r_{R}$ at $t_{R}=0$, and $I^{\prime}\subset M$ a ball of radius $r<r_{R}$ at $t=0$. By time symmetry around $t=t_{R}=0$, the region $I^{\prime}\cup R$ will be quantum normal or anti-normal. We have $S_{\rm gen}(I^{\prime}\cup R)-S(R)=\frac{\pi r^{2}}{G\hbar}-\frac{4\pi s_{\rm rad}}{3}r^{3}~{}.$ (52) To satisfy our second condition, this must be negative, so we require $r>r_{\rm crit}\equiv\frac{3}{4\pi Gs_{\rm rad}}$ (53) This condition on $I^{\prime}$ can be satisfied, and hence an island $I\subset M$ must exist, for a sufficiently large reference region, $r_{R}>r_{\rm crit}$. Going beyond spherical symmetry, we can choose $R$ to be any convex reference region of arbitrary shape in $M_{R}$, and let $I^{\prime}$ be the identical coordinate region in $M$. Then $I^{\prime}\cup R$ will be normal or anti- normal by convexity and time symmetry. Moreover, $S_{\rm gen}(I^{\prime}\cup R)-S(R)=\frac{A[\partial I^{\prime}]}{4G\hbar}-\left(V_{R}+O(A[\partial R]\lambda)\right)s_{\rm rad}$ (54) will be negative for any sufficiently large region of fixed shape. Any such references region must have an island $I$. Note that $I$ will not be the identical coordinate region to $R$, because of the $O(A[\partial I^{\prime}]\lambda)$ corrections to the von Neumann entropy. Moreover, in the nonspherical case, minimization of the area term will favor a more round shape for $I$ than for $R$. #### Bag of gold Figure 6: A time-symmetric Cauchy slice of a bag-of-gold geometry. The bag has large entropy (grey) compared to the area of its throat, and it purifies the reference system $R$. Then it is easy to find a (classically and quantum) anti-normal region $I^{\prime}$ (pink) such that $S_{\rm gen}(I^{\prime}\cup R)<S(R)$. Hence, there must exist an island $I$. $I$ is expected to be approximately the interior of the classically minimal surface labelled $r_{0}$. Next, we discuss a time-symmetric slice of a “bag-of-gold” geometry Marolf:2008tx , shown in Fig. 6. Its defining feature is the existence of an arbitrarily large volume of space behind a throat (a minimal area surface) of fixed area. To construct it, we glue the interior of a sphere of radius $r_{1}$ of a closed FRW universe, at the time of recollapse, $da/dt=0$, to the exterior of a sphere of the same size behind the bifurcation surface in a maximally extended Schwarzschild spacetime Horowitz:1983vn . The corresponding spatial metrics are: $\displaystyle ds_{\rm in}^{2}$ $\displaystyle=a(d\chi^{2}+\sin^{2}\chi d\Omega^{2})~{},~{}~{}$ $\displaystyle 0\leq\chi\leq\chi_{1}~{};$ (55) $\displaystyle ds_{\rm out}^{2}$ $\displaystyle=\left(1-\frac{r_{0}}{r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2}~{},~{}~{}$ $\displaystyle r_{0}\leq r\leq r_{1}~{},$ (56) where in the second line we omitted the portion of the Schwarzschild metric outside of the bifurcation surface as it will not be needed for the analysis below. Let $r_{0}$ be the radius of the throat, and let $\chi$ be the angle at which the metrics are glued. The gravitational constraints imply $\chi_{1}>\pi/2$ and $\displaystyle a\sin\chi_{1}$ $\displaystyle=r_{1}~{},$ (57) $\displaystyle a\sin^{3}\chi_{1}$ $\displaystyle=r_{0}~{}.$ (58) The Friedmann equation implies that the energy density in the bag is $\rho=\frac{1}{8\pi Ga^{2}}~{}.$ (59) Now suppose that the bag contains thermal photon radiation purified by a external reference system $R$. The entropy density in the bag is $s\sim\rho^{3/4}$, and hence $S(R)\sim(G\hbar)^{-3/4}a^{3/2}~{}.$ (60) Let $I^{\prime}$ be the interior of some sphere $r^{\prime}$ between the edge of the bag, $r_{1}$, and the throat, $r_{0}$; hence $S_{\rm gen}(I^{\prime}\cup R)\sim\frac{r^{\prime 2}}{G\hbar}~{}.$ (61) By time reversal symmetry, $I^{\prime}\cup R$ is quantum normal or anti- normal. Moreover, we can achieve $S(R)>S_{\rm gen}(I^{\prime}\cup R)$, by an arbitrarily large margin, by taking $a$ large while holding $r^{\prime}$ and $r_{0}$ fixed. (This will only increase $r_{1}$.) Hence, our conditions are satisfied, and a nontrivial island $I\subset M$ must exist. Importantly, this construction is insensitive to the spherical symmetry that we assumed for simplicity. It is also insensitive to the addition of perturbative matter near the throat. Such modifications can affect the precise position of the island, which may be very hard to determine. But so long as they are small enough, our sufficient conditions will hold, and they guarantee the existence of an island. #### Collapsing star (an example without islands) Figure 7: Collapse of a spherical star that is maximally entangled with a distant reservoir $R$. $I^{\prime}$ (pink) is the interior of a sphere surrounding the star at a time close to the singularity. Hence $\partial I^{\prime}$ has a small area, and condition (i) can be satisfied by an arbitrarily large margin. But $I^{\prime}$ is quantum trapped and so fails to satisfy condition (ii). Indeed, $R$ does not possess any island in this spacetime. To illustrate the importance of condition (ii), let us discuss a case for which condition (i), Eq. (6), is satisfied, but condition (ii), Eq. (7), is violated. Consider an Oppenheimer–Snyder spacetime: a black hole formed by the collapse of a “star,” modeled as a spherical, homogeneous ball of dust. Suppose that the star is in a maximally mixed state with entropy $S_{\rm star}$ and let $R$ be an early portion of $\mathscr{I}^{+}$ which contains only a purification of the star (and no Hawking radiation), giving $S(R)=S_{\rm star}$. We choose $I^{\prime}$ to be the interior of a sphere just outside of the star and very close to the singularity (see Fig. 7). Then $S_{\rm gen}(I^{\prime}\cup R)\approx A(\partial I^{\prime})/4G\hbar$. Picking $S_{\rm star}$ large with $A(\partial I^{\prime})$ held fixed, we can arrange for $\displaystyle 1\ll S_{\rm gen}(I^{\prime}\cup R)\ll S(R)~{}.$ (62) The first inequality ensures semiclassical control at $\partial I^{\prime}$. The second states that condition (i) is satisfied (by an arbitrarily large margin). However, $\partial I^{\prime}$ is a classically trapped surface, i.e. $\theta_{k}<0,\theta_{\ell}<0$. And since $\partial I^{\prime}$ is not close to $\partial J^{-}(R)$, we expect quantum corrections to be small: $\Theta_{\ell}=\theta_{\ell}+O(G\hbar)$. Condition (ii) is therefore violated. Indeed, there are no islands associated to $R$ in this spacetime. To see this, note that there are no classically extremal spheres. As in the previous example, near $\partial J^{-}(R)$, quantum corrections to $\theta_{\ell}$ can become large; but $\partial J^{-}(R)$ stays far from the horizon and so has large classical (and quantum) expansion everywhere. This example shows that condition (ii) is essential. So is condition (i), of course. For example, suppose we chose $R$ to be a later portion of $\mathscr{I}^{+}$. As before, $R$ contains only the purification of the star, but no Hawking radiation. Since $\partial J^{-}(R)$ gets close to the horizon, where $\Theta_{k}$ can vanish, there will be a quantum extremal region $\tilde{I}$ with $\Theta_{\ell}$=0. However, this region fails to be an island because $S_{\rm gen}(\tilde{I}\cup R)>S(R)$. ## 3 New Entropy Bound In this section, we will show that in a globally pure state, the entropy of a reference system $R$ cannot exceed the generalized entropy of suitable asymptotic regions. We consider an external reference system in Sec. 3.1, and we generalize to $R\subset M$ in Sec. 3.2. We discuss examples in Sec. 3.3. ### 3.1 External Reference System Given an external reference system $R$, an island $I\subset M$ in a semiclassical spacetime $M$ is defined as a region that is quantum extremal and homologous to $R$ (i.e., $\partial I\subset M$), such that $S_{\rm gen}(I\cup R)$ is minimal among all such regions. In section 2.1 we identified sufficient conditions for $I\neq\varnothing$. We will now employ similar techniques to derive an entropy bound on the exact entropy of the reference system, $S(\mathbf{R})$, assuming that this is computed by the “island formula”: $\displaystyle S(\mathbf{R})=S_{\rm gen}(I\cup R)~{}.$ (63) Following Ref. Almheiri:2019hni , we denote $\mathbf{R}$ in boldface when referring to the exact (nonperturbatively computed) state of the region. We write $R$ when referring to the semiclassically computed state. For simplicity, we will assume that the global quantum state is pure, $S(R\cup M)=0~{},$ (64) though generalizations can easily be considered. Let $I^{\prime}_{c}\subset{M}$ be any partial Cauchy slice of $M$ that is quantum normal or anti-normal: $\begin{cases}k^{\mu}\Theta_{\mu}[I^{\prime}_{c}]\geq 0~{},~{}~{}\ell^{\mu}\Theta_{\mu}[I^{\prime}_{c}]\leq 0~{};\\\ \text{or}\\\ k^{\mu}\Theta_{\mu}[I^{\prime}_{c}]\leq 0~{},~{}~{}\ell^{\mu}\Theta_{\mu}[I^{\prime}_{c}]\geq 0~{},\end{cases}$ (65) where $k$ and $\ell$ are the future-directed null vector fields orthogonal to $\partial I^{\prime}_{c}$. We also require that $I^{\prime}_{c}$ is “asymptotic,” though only in the weak sense that in the conformally compactified spacetime, $\partial\Sigma\subset\partial I^{\prime}_{c}~{},$ (66) where $\Sigma$ is a Cauchy slice of $M$. That is, $I^{\prime}_{c}$ must contain the asymptotic region of $M$, but it may extend deep into the interior of $M$. A simple example of a region $I^{\prime}_{c}$ that satisfies Eqs. (65) and (66) is the exterior of a sufficiently large approximately round sphere. Let $I^{\prime}$ be the complement of $I^{\prime}_{c}$ on some global Cauchy slice of $M$. By Eqs. (64) and (65), $I^{\prime}\cup R$ will be anti-normal or normal. By Eq. (66), $I^{\prime}$ is homologous to $R$. Our notation reflects the fact that $I^{\prime}$ shares these properties with the region denoted $I^{\prime}$ in Sec. 2.1. However, here we do not assume the inequality $S_{\rm gen}(I^{\prime}\cup R)<S(R)$, and hence we will not be guaranteed the existence of an island $I\neq\varnothing$. This does not affect the maximin analysis performed in Sec. 2.1: (anti-)normalcy of $I^{\prime}\cup R$ implies that the true island $I$ satisfies $S_{\rm gen}(I\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)~{},$ (67) regardless of whether $I$ is the empty set or not. Using Eqs. (63) and (64), we thus find the entropy bound $S(\mathbf{R})\leq S_{\rm gen}(I^{\prime}_{c})~{}.$ (68) ### 3.2 Distant Reference System The bound (68) generalizes to the case where $R\subset M$, subject to appropriate modifications. (It is easy to generalize further to the case where $R$ is partly internal to $M$ and partly an external system.) We shall assume that gravity is negligible in $R$, so that the notion of an exact state of $\mathbf{R}$ can be made precise. The island rule can then be adapted to compute the generalized entropy of $R$: $\displaystyle S_{\rm gen}(\mathbf{R})=S_{\rm gen}(I\cup R)~{},$ (69) where $I$ is an island (possibly the empty set), as described above. The relevant homology rule is $I\subset\mbox{int}[M-\overline{J(R)}]$, where $J$ denotes the union of the causal past and future. We again assume global purity, $S(M)=0$. To obtain a bound on $S_{\rm gen}(\mathbf{R})$, we consider a spatial region $I^{\prime}_{c}$ that satisfies the following conditions (see Fig. 8): * • For $I^{\prime}$ to be of the correct homology type, without directly referring to $I^{\prime}$,888Our goal is to formulate a bound in terms of quantities that are accessible to an asymptotic observer. we require that $I^{\prime}_{c}$ is adjacent to $R$ in $M$; and in the conformally compactified spacetime $\tilde{M}$, $I^{\prime}_{c}$ contains any conformal boundary portions not covered by $R$: $\partial I^{\prime}_{c}\supset\partial(\tilde{\Sigma}-\bar{R})~{},$ (70) where $\tilde{\Sigma}\supset R$ is a Cauchy slice of $\tilde{M}$, and $\bar{R}$ denotes the closure of $R$ in $\tilde{M}$. * • $I^{\prime}_{c}$ is quantum normal or anti-normal under shape deformations of its inner boundary in $M$, i.e., at $(\partial I^{\prime}_{c}-\partial R)\cap M$ . * • $I^{\prime}_{c}$ contains a region $I_{0,c}$ that is quantum anti-normal at $\partial I_{0,c}-\partial R$. (Normal is not allowed in this criterion.) Global purity implies that $I^{\prime}\cup R$ will be quantum normal or anti- normal at $\partial I^{\prime}$. It also guarantees quantum normalcy of $I_{0}\cup R$, where $I_{0}\equiv\Sigma-I_{0,c}-R$ and $\Sigma$ is a Cauchy surface that contains $I_{0,c}$ and $R$. By Sec. 2.2, the maximin procedure restricted to the wedge $D(I_{0})$ will return a region $\hat{I}\subset\mbox{int}[D(I_{0})]$ that satisfies the homology rule and has stationary $S_{\rm gen}(\hat{I}\cup R)$ under shape deformations at $\partial\hat{I}$. Figure 8: The entropy $S(\mathbf{R})$ of an external or distant reference system $R$ must be less than the generalized entropy of any region $I^{\prime}_{c}$ that is normal or anti-normal (blue). Note that $\hat{I}$ may be empty; and $S_{\rm gen}(\hat{I}\cup R)$ need not be globally minimal, since the true island $I$ may not be contained in $D(I_{0})$. However, we have $S_{\rm gen}(I\cup R)\leq S_{\rm gen}(\hat{I}\cup R)~{},$ (71) The quantum (anti-)normalcy of $I^{\prime}$ implies $S_{\rm gen}(\hat{I}\cup R)\leq S_{\rm gen}(I^{\prime}\cup R)~{}.$ (72) Using Eqs. (69) and global purity, we thus find the entropy bound $S_{\rm gen}(\mathbf{R})\leq S_{\rm gen}(I^{\prime}_{c})~{}.$ (73) Recall that $\partial I^{\prime}_{c}\supset\partial R$, so in any situation where the generalized entropy can be separated into a regularized entropy and Bekenstein-Hawking term, the area terms associated with $\partial R$ will cancel, and Eq. (73) reduces to Eq. (68). ### 3.3 Examples and Discussion The bound (73), and its external $R$ version (68), are powerful and versatile. They require knowledge only of $R$ and $I^{\prime}_{c}$, but not of the rest of the spacetime $M$. The only nontrivial condition, Eq. (65), can be easily verified. Often the quantum expansion is dominated by the classical expansion, so that it is easy to check whether $I^{\prime}_{c}$ is quantum (anti-)normal; yet the bound remains nontrivial. Figure 9: Not a Schwarzschild black hole. This could be a highly dynamical spacetime far from a stationary black hole solution. $I^{\prime}_{c}$ is is quantum anti-normal, so $S(\mathbf{R})\leq S_{\rm gen}(I^{\prime}_{c})$. In the left example, $R$ is on the conformal boundary. Quantum anti-normalcy follows from the GSL (future null congruence), and from the classical area theorem (past null congruence) if quantum corrections are small. In the right example, $R$ is inside the spacetime. Quantum anti-normalcy follows by the area theorem and smallness of corrections if $\partial I^{\prime}_{c}$ stays far enough from the horizon. For example, by the generalized second law, causal wedges of a boundary region must be quantum anti-normal.999Note that the causal wedge is not in general a domain of dependence. The region that is always quantum normal is the maximal Cauchy evolution of the causal wedge. This follows both for asymptotically anti-de Sitter and flat spacetimes. When $R$ is disjoint from the conformal completion of $M$, this ensures the quantum anti-normalcy of $I^{\prime}_{c}$. The requirement that $R$ and $I^{\prime}_{c}$ be spacelike separated prevents the application of the GSL when $R$ is part of the conformal completion of $M$. For instance, suppose that $R$ is a subset of $\mathscr{I}^{+}$. The GSL can still be applied to any future causal horizon associated to $\mathscr{I}^{+}$ regions including $R$, guaranteeing $k^{\mu}\Theta_{\mu}\geq 0$. However, all past horizons will intersect the past of $R$, blocking the application of the GSL to $I^{\prime}_{c}$ along them. To establish that $\ell^{\mu}\Theta_{\mu}\leq 0$, we can use the classical area law on past horizons, so long as quantum corrections to $\ell^{\mu}\theta_{\mu}$ is negligible (see Fig. 9, left). More generally, when $R$ is a subset of $M$, the classical area law ensures condition (65) if the quantum corrections to both expansions are suppressed (see right Fig. 9). This causal wedge method for finding $I^{\prime}_{c}$ suggests a nice physical interpretation of $I^{\prime}_{c}$ as a region that can be explored geometrically by asymptotic observers. The bound thus tells us that $S(\mathbf{R})$ cannot be greater than the generalized entropy of any causal wedge region (subject to quantum effects on the expansion remaining negligible). If $I^{\prime}_{c}$ is a whole Cauchy surface $\Sigma$ of $M$, this reduces to the trivial statement that $S(\mathbf{R})\leq S(\Sigma)$. (In this case, by purity, equality must hold.) But if $I^{\prime}_{c}$ has a boundary in $M$, the bound is nontrivial. Indeed, a quantum anti-normal causal wedge can reach very close ($O[(G\hbar)^{1/2}]$ distance) to a black hole horizon. 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# Spatio-temporal characterization of ultrashort vector pulses Apostolos Zdagkas Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom Venkatram Nalla Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom Nikitas Papasimakis Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom Nikolay I. Zheludev Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Southampton SO17 1BJ, United Kingdom Centre for Disruptive Photonic Technologies, School of Physical and Mathematical Sciences and The Photonics Institute, Nanyang Technological University, Singapore 637378, Singapore ###### Abstract Ultrafast vectorially polarized pulses have found many applications in information and energy transfer owing mainly to the presence of strong longitudinal components and their space-polarization non-separability. Due to their broad spectrum, such pulses often exhibit space-time couplings, which significantly affect the pulse propagation dynamics leading to reduced energy density or utilized to create new effects like a rotating or sliding wavefront at focus. Here, we present a new method for the spatio-temporal characterization of ultrashort cylindrical vector pulses based on a combination of spatially resolved Fourier transform spectroscopy and Mach- Zehnder interferometry. The method provides access to spatially resolved spectral amplitudes and phases of all polarization components of the pulse. We demonstrate the capabilities of the method by completely characterizing a $10$ fs radially polarized pulse from a Ti:sapphire laser at $800$ nm. ## I Introduction Space-time couplings (STCs) in propagating waves are defined as the dependence of the temporal properties of the electric field on the transverse spatial coordinates [1]. Mathematically they are revealed as the non-separability of the spatial and temporal terms of the electric field of a pulse into a product, $E(\mathbf{r},t)\neq f(\mathbf{r})g(t)$. They can significantly affect the energy density of ultrashort pulses at focus [2] since in most cases pulses possessing STCs are not transform limited. STCs also alter the propagation dynamics of ultrashort pulses. The latter has been utilized to create new effects like the lighthouse effect [3] where a tilted wavefront is transformed to a rotating wavefront at focus and the “sliding” or “flying” focus [4, 5] effect where STCs create a focal spot that locally travels with speed greater or lower than the speed of light in free space. A range of techniques have been demonstrated for the complete spatio-temporal characterization (retrieval of electric field amplitude and phase) of linearly polarized pulses [6]. Such approaches are typically based on scanning the transverse profile of the unknown pulse with a known reference pulse [7, 8, 9] or utilize concepts from wavefront characterization techniques [10, 11]. Moreover, self-referenced methods have been proposed, where a small part of the pulse under characterization is used as reference. In the latter, the reference is interfered with the unknown pulse and the unknown spectral phase is retrieved through spatially resolved Fourier transform spectroscopy [12]. A variation of the latter, named TERMITES [13], uses a slightly expanded replica of the unknown along with an iterative algorithm as a way to reduce the requirements of creating a very homogeneous reference pulse. However, none of the aforementioned methods can simultaneously characterize the spatially dependent polarization, intensity and phase that vector polarized pulses exhibit. Indeed, methods such as TERMITES are unsuitable for the characterization of cylindrical vector pulses (CVPs), for example radially polarized pulses, that exhibit polarization singularities at their center. Ultrafast radially polarized pulses have been generated in the femtosecond regime [14] and have been compressed down to few femtoseconds [15, 16]. Due to their broad spectrum, such pulses often exhibit space-time couplings (STC). Moreover, space-time couplings have been prescribed to them through a metamaterial converter [17]. Therefore, their complete characterization becomes of paramount importance for most applications of CVPs. The characterization of CVPs has been limited mainly to their spatially varying polarization profile. The temporal profile of such pulses has been characterized independently by standard approaches, such as FROG [18] and SPIDER [19]. However, such measurements are performed at a single spatial position of the cross section of the pulse, which is far from a complete characterization of the pulses that may exhibit STCs [1]. For example, theoretical calculations have shown that the Flying Donut pulse is isodiffracting [20], a property that leads to the spatial profiles of intensity for every frequency component of the pulse to scale along the trajectory of the pulse in the same way. Furthermore, since most ultrafast systems use pulse compressors based on prisms or gratings, in order to implement chirped pulse amplification (CPA) [21], the appearance of pulse front tilt (PFT) is very likely to occur from small misalignments [1, 22] leading to detrimental effects for the pulse duration and its energy density. Such space time couplings occur also in the case of ultrafast CVPs. For example, Fig. 1 illustrates the effect of pulse front tilt and pulse front curvature on the shape of a radially polarized pulse. PFT appears as a linearly varying delay of the pulse across a transverse direction while pulse front curvature is a quadratically varying delay from the center of the pulse to its edges. Therefore, the complete characterization of CVPs becomes of paramount importance for most applications. Only recently a new method has been utilized for the complete spatio-temporal characterization of a 100 fs vector pulse through a two-fold interferometer [23]. Although it was shown to successfully characterize spatially varying polarization gates, it is based on a point scan scheme with an optical fiber, rendering it impractical for the complete transverse spatial characterization of the pulse which would require millions of points to be scanned. Figure 1: Characteristic examples of space-time couplings in vector polarized pulses. Cross sections of the x component of the instantaneous electric field of a radially polarized pulse (a) in the absence of space-time couplings, (b) with pulse front tilt ($3\;\mathrm{fs/mm}$), and (c) with pulse front curvature ($3\;\mathrm{fs/mm^{2}}$). The insets to (a-c) present corresponding 3D illustrations in the form of isosurfaces. The pulse propagates along the positive z axis. In this work, we present a method for the complete spatio-temporal characterization of CVPs. The method is an extension of the TERMITES technique by means of a Mach-Zehnder interferometer and is termed TERMITES-MAZE (MAch- Zehnder Extended). It allows to fully characterize the spatial and temporal profile of all polarization components of ultrafast CVPs at the spatial resolution of a camera sensor. We illustrate the capabilities of the TERMITES- MAZE technique by applying it to the case of a $10$ fs radially polarized pulse centered at $800$ nm. We show that our approach can reveal the space- time couplings (e.g. pulse front tilt) across different planes for different polarization components of the pulse. We present a detailed description of the experimental implementation of TERMITES-MAZE, while the corresponding algorithm for the analysis of the experimental data is freely available under the BSD 3-clause licence as a python module [24]. As few cycle laser pulses with vectorial fields, orbital angular momentum (OAM) [25] and other forms of structured light [26] become common, the need for their characterization increases. The presence of STCs become even more important in such broadband pulses that can eventually alter their propagation properties [27] and their interaction with matter [28, 29] leading to the appearance of new effects. The tool provided here has the capability to characterize all these types of space-time-polarization “entanglement” that are increasingly studied [30] and thus has the potential to accelerate the research and emerging applications based on few cycle vector structured light. ## II Cylindrical vector pulses Cylindrical vector beams are solutions to Maxwell’s equation whose amplitude, phase and polarization are axially symmetric. They have been extensively studied and they are routinely generated and characterized [31]. Their unique properties have found numerous applications such as tight focusing [32], efficient particle trapping [33, 34], super-resolution microscopy [35], dense optical 3d data storage [36] and data encoding for optical communications [37] to name a few. Furthermore, their space polarization non-separability has been applied to extend the concepts and tools of quantum physics to “classically entangled” states [38, 39]. Cylindrical vector pulses are the pulsed version of the cylindrical vector beams. Although they offer more opportunities for applications, they are less studied due to the difficulty of their generation and characterization. They share all of the characteristics that make these light structures interesting with additional important features arising due to their pulsed nature. Their short duration renders them ideal for microprocessing of materials since they combine the reduced thermal damage with the increased efficiency and homogeneity of the radial polarization when used in micro-drilling [40]. In fact, single-cycle CVPs, termed ”Flying Donuts”, exist [41], for which applications in particle acceleration have been suggested. Moreover, such pulses exhibit a finite topological structure with a finite number of localized singularities [42], while their toroidal topology is ideal for engaging toroidal and anapole modes in matter [43, 44]. ## III Method A method for the complete characterization of a CVP should provide all the characteristics of the pulse like the polarization, spectral amplitude and spectral phase at each spatial position. The TERMITES-MAZE approach is capable of retrieving all these characteristics through an interference of a linearly polarized reference pulse with an unknown CVP. Firstly, the TERMITES method is applied for the characterization of the reference pulse and then a Mach- Zehnder interferometer is implemented for the characterization of the unknown. In this way only one polarization of the CVP is characterized at a time. Finally, since all the spatial information is captured, the complete vectorial shape of the pulse can be reconstructed. ### III.1 Experimental implementation The TERMITES-MAZE method is an extension of the TERMITES method and as such it involves two experiments. The first experiment is the characterization of a reference linearly polarized pulse using the TERMITES method and the second is a Mach-Zehnder interferometer in which an unknown pulse, in our case a CVP, and the reference pulse are interfered. In both cases, Fig. 2a and b, a $10$ fs Ti:Sa laser with peak frequency at $800$ nm (Spectra-Physics, Element PRO) is used to generate a linearly polarized pulse. The pulse then travels through a pulse shaper that uses a spatial light modulator (SLM) (Biophotonic solutions inc, MIIPS Box640). A polarizer after the pulse shaper is used to select the polarization and to control the intensity of certain frequencies that are set by the SLM. The pulse shaper is used to compress the ultrafast pulse to its Fourier limited duration after travelling though the many optical components that are required for the generation and characterization of the radially polarized pulse. Then a beam expander, doubles the beam width in order to fit the radius of a segmented waveplate that transforms a linear polarized beam to a radial polarized one. After the beam expander, a 50:50 beam splitter (BS1 in Fig.2) is used to generate two replicas of the pulse. Up to this point the setup is common to both experiments. In the following we describe the two experiments separately. Figure 2: Experimental implementation of TERMITES-MAZE. (a) Following the TERMITES technique, a Michelson interferometer setup with a convex mirror in one arm and a delay line in the other is used for the spatio-temporal characterization of a linearly polarized pulse (this pulse will be used as reference). The pulse compression part is shown in the lower part of the setup. (b) A Mach-Zehnder interferometer setup with the one arm carrying the reference pulse (as characterized in (a)) and the other hosting the setup for the generation of the CVP. A delay line is used to scan the unknown pulse with the reference. The generated images are then used to extract the spectral phase and frequency distribution of the unknown pulse. The interference pattern has the expected form of the interference of a linear and a radially polarized pulse. The intensity pattern is antisymmetric with respect to the center of the radially polarized pulse which is a manifestation of the radial polarization shown with yellow arrows. PS: pulse shaper, P: polarizer, BE: beam expander, BS: beam splitter, WP: $\lambda/2$ waveplate, DG: dispersive glass, PM: parabolic mirror, SHG: second harmonic generation crystal, L: lens, F: filter, FS: fiber spectrometer, CM: convex mirror and SWP: segmented waveplate. The TERMITES experimental setup consists of a simple Michelson interferometer and a digital camera (Thorlabs DCC1545M), see upper part of Fig. 2a. One arm of the interferometer includes a nanometre precision delay line with a scanning range of $20\mu m$. The other arm includes a convex mirror, whose purpose is to expand the unknown pulse. Hence the image on the camera sensor will be the result of a pulse interfering with an expanding replica of its central part. This expanding central part is considered smoother, in terms of spatio-temporal couplings, than the pulse itself since it is just a small part of the pulse stretched to fit a wider area. A polarizer is finally placed before the camera to increase the signal to noise ratio. The TERMITES algorithm reveals the spectral phase differences between every point of the pulse and its center. A simple temporal characterization of its central part is the only requirement for a complete spatio-temporal characterization. In our case the “Multiphoton Intrapulse Interference Phase Scan” (MIIPS) [45] technique is used for this purpose since it is already a part of the setup. More specifically, instead of simply characterizing the temporal shape of the pulse, we compress it. The compression will produce a Fourier limited pulse that is also required for the generation of a Fourier limited CVP. The compression part of the setup can be seen in the lower part of Fig. 2a. The pulse is focused at a second harmonic generation crystal. The generated spectral intensities are then recorded by a spectrometer (Ocean Optics USB2000+XR1-ES) and fed to the pulse shaper’s software where the MIIPS method is used for the compression. In the schematic of Fig. 2a, the dispersion compensation glass and the waveplate of the lower arm are used to create the same amount of dispersion with the waveplate of the upper arm, the two passes through the beam splitter and the polarizer. The mirrors have low group delay dispersion and more or less the same number of mirrors is used in the two arms. A good spatial resolution to resolve all the fringes and avoid $2\pi$ phase jumps across neighbouring pixels after the Fourier transform is required. Hence a careful choice of the convex mirror induced curvature and the camera sensor is necessary. The same is true for the step of the delay line. Since both spatial and temporal patterns are periodic effects, the experiments are designed to capture more than 10 samples per period, much more than the theoretical minimum according to the sampling theorem [46]. The above produces a reference (known) linearly polarized pulse. A simple interference of this reference pulse with an unknown more complex structured pulse, like a CVP, is sufficient to reveal the full spatial and temporal structure of the latter. Such an experimental setup is given in Fig. 2b. After the first beam splitter (BS1 in Fig. 2b) two replicas of the pulse are created that travel in two separate arms of a Mach-Zehnder interferometer. The right part of the interferometer carries the reference pulse, while the left arm is used to generate a CVP, which then propagates to the final beam splitter (BS2 in Fig. 2b) where the two pulses recombine and the interference takes place. In this case, the beam splitter has to be polarization independent, as the polarization of the unknown pulse is spatially dependent. As an example, here we characterize a radially polarized pulse that is generated from a segmented waveplate [32, 47, 16]. Our polarization transformer consist of 8 achromatic half-wave plates with the relative orientation of the fast axis at $\pm 11,\pm 34,\pm 56$ and $\pm 79$ degrees. The joints between the waveplates scatter part of the incident light and create diffraction patterns. For that, a $75\mu m$ pinhole is placed at the Fourier plane to spatially filter the pulse. An iris placed before the focusing mirror is used to select the size of the radial pulse that has to be smaller or equal to the reference. Any diffraction introduced by the iris will be also filtered by the pinhole. A half-waveplate is placed before the segmented waveplate to rotate the input polarization and hence select the output polarization that can be either radial or azimuthal. Finally the filtered pulse travels to a delay line that creates a variable delay between the CVP and the reference pulse. The Thorlabs TSGNF5 single axis piezoelectric stage along with a hollow retroreflector were used for the delay line. The interferogram is then captured by the camera. A polarizer before the camera is used not only for the increase of the signal to noise ratio but to characterize the two orthogonal polarizations of the CVP. The latter is achieved by rotating the polarizer at $\pm 45$ degrees with respect to the polarization of the reference pulse. Hence two measurements are required for the complete spatio-temporal characterization of the CVP. The delay between the polarizations is assumed to be zero because of the working principle of the achromatic segmented waveplate. Finally, the dispersion of the pulse must be considered carefully. Both paths need to have the same dispersion as the TERMITES part of the experiment, where the reference pulse was characterized. ### III.2 Analysis of the recorded data Figure 3: A schematic description of the required processing of the images used in the TERMITES algorithm. The data are acquired from the setup described in Fig. 2a. (a) A sequence of interference images captured by a digital camera for successive delays $\tau$ of the reference pulse. (b) The cross-correlation signal, $S(x,y,\tau)$, at a single pixel of the camera sensor. That is the measured intensity as a function of the delay. (c) The spectrum derived by a fast Fourier transform ($F_{T}$) performed on the cross-correlation signal of the pixel. (d) Unwrapped phase data (red dots) and the quadratic fitted surface (blue) close to the central wavelength ($\simeq 800$ nm). The fitted phase is removed from the total spatially depended phase to reveal the phase due to the STC. (e) The retrieved phase at $794$ nm after the application of the iterative algorithm. (f) The 3d reconstruction of the pulse in time and space through an inverse Fourier transform ($\tilde{F}_{T}$) of the retrieved amplitude and phase data. In this section we briefly discuss the TERMITES algorithm and we present the extra analysis steps needed for the characterization of CVPs. The TERMITES method is used for the spatiotemporal characterization of linearly polarized pulses up to a constant spectral phase and an ambiguity in the pulse front curvature. Our approach inherits these limitations. Figure 4: The final step of TERMITES-MAZE method for the characterization of a radially polarized pulse. (a) A simplified schematic of the experiment where the reference pulse from the previous part of the method is interfered with the unknown CVP. (b) A sequence of interference images captured by a digital camera for successive delays of the reference pulse.(c) The Fourier transformed cross-correlation signal close to the central wavelength of the pulse. The retrieved amplitude and phase of the x polarization of the radially polarized pulse at $794$ nm after the division and subtraction of the reference pulse’s amplitude and phase from the cross-correlation spectra. (d) The 3d reconstruction of the x polarization of the radially polarized pulse in time and space. In TERMITES, a pulse is interfered with a magnified replica of itself. A delay line is used to scan the unknown pulse with the reference. The interference pattern is then recorded by a digital camera and an image for each relative delay is stored in a computer, resulting in a 3D dataset (see Fig. 3a). For each pixel across all images an interferogram is captured which is the collected energy during the exposure period transferred to the sensor by the two pulses, as a function of the delay. This signal is proportional to the time integral of the intensity of the total electric field and can be written as $\displaystyle S(\mathbf{r},\tau)$ $\displaystyle=\int\|E_{\mathrm{R}}(\mathbf{r},t)+E(\mathbf{r},t-\tau)\|^{2}dt$ $\displaystyle=I(\mathbf{r})+I_{\mathrm{R}}(\mathbf{r})+\int[E_{\mathrm{R}}(\mathbf{r},t)E^{}(\mathbf{r},t-\tau)+E^{}_{\mathrm{R}}(\mathbf{r},t)E(\mathbf{r},t-\tau)]dt,$ (1) with the subscript “$\mathrm{R}$” denoting the reference (diverging) pulse and $\tau$ the delay. The signal on a central pixel is shown in Fig. 3b. A constant signal is observed when there is no overlap of the pulses, mathematically described by the first two terms of Eq. 1, and an interference signal when they overlap, mathematically described by the integral. The term $s(\mathbf{r},\tau)=\int E_{\mathrm{R}}(\mathbf{r},t)E^{}(\mathbf{r},t-\tau)dt$ is the cross- correlation of the two fields. Hence Eq. 1 takes the form $\displaystyle S(\mathbf{r},\tau)=I(\mathbf{r})+I_{\mathrm{R}}(\mathbf{r})+s(\mathbf{r},\tau)+s^{}(\mathbf{r},\tau).$ (2) Figure 5: (a) and (c) Retrieved amplitude of the horizontally and vertically polarized areas of a radially polarized pulse at $794$ nm respectively. (b) and (d) Retrieved phase at the same wavelength for the horizontal (vertical) polarization according to the TERMITES-MAZE algorithm. The spectral phase has a difference of $\pi$ between the left and right (top and bottom) areas as it is expected for a radially polarized pulse. A Fourier transform is then performed for every pixel and the frequency domain signal is now described by the equation $\displaystyle S(\mathbf{r},\omega)=F_{\mathrm{T}}[S(\mathbf{r},\tau)]=F_{\mathrm{T}}[I(\mathbf{r})+I_{\mathrm{R}}(\mathbf{r})]+s(\mathbf{r},\omega)+s^{}(\mathbf{r},-\omega),$ (3) with $F_{\mathrm{T}}$ denoting the time to frequency Fourier transform operator. The equation describes the presence of a zero frequency term, derived from the Fourier transform of the time stationary term, and two frequency signals with equal amplitude and at symmetric frequencies. Fig. 3c shows the results derived from the above analysis regarding a central pixel of the camera which corresponds to the central part of the pulse. A zero frequency term and two symmetric peaks with central wavelength at around $800$ nm, in accordance with the central wavelength of our laser, are formed by the Fourier transform. The result of the above procedure leads to two sets of images. One carries the intensity distribution at each frequency and the other the spectral phase. The negative frequencies can be derived by the complex conjugate of the positive and hence are redundant for the analysis. Additionally, the noise free spectrum is from about 700 to 900 nm. Only a few images are thus needed for the analysis and hence the processing time is reduced significantly without any loss of information. The Fourier transform of the cross-correlation of two signals, $E(\mathbf{r},t)$ and $E_{\mathrm{R}}(\mathbf{r},t)$, is equal to the product of their spectra with one of them being complex conjugate, as in the cross- correlation integrand. In our case we have $s(\mathbf{r},\omega)=E_{\mathrm{R}}(\mathbf{r},\omega)E^{}(\mathbf{r},\omega)$. The amplitude and phase of the signal that we are interested are thus given by $\displaystyle\|s(\mathbf{r},\omega)\|$ $\displaystyle=A_{\mathrm{R}}(\mathbf{r},\omega)A(\mathbf{r},\omega)$ (4) $\displaystyle\mathrm{arg}(s(\mathbf{r},\omega))$ $\displaystyle=\varphi_{\mathrm{R}}(\mathbf{r},\omega)-\varphi(\mathbf{r},\omega)$ (5) with $A$ being the amplitude of the spectrum and $\varphi$ the spectral phase. In the TERMITES algorithm, the spectral phase that is created from the convex mirror is removed by fitting a quadratic surface to the phase data, Fig. 3d and then an iterative algorithm is applied Fig. 3e until the retrieved spectral phase can correctly reproduce the known relationship between the pulse and its expanding replica [13]. An inverse Fourier transform then reconstructs the linearly polarized pulse in time and space, Fig. 3f. Following the characterization of the reference pulse the Mach-Zehnder experiment is performed. The analysis is the same up to Eq. 4 and 5. However, in this case the reference spectral amplitude and spectral phase are known, hence a simple division of the spectral amplitudes and a subtraction of the spectral phases gives the complete frequency domain representation of the unknown CVP. The procedure is then repeated for the orthogonal polarization and the results are combined. Figure 6: (a)-(b) $x-t$ cross-section of the x polarized electric field of a 10 fs radially polarized pulse and its isosurface plot at half of the maximum electric field value as it is reconstructed with the TERMITES-MAZE method. (c)-(d) $y-t$ cross-section of the y polarized electric field of the same pulse and its isosurface plot. Figure 7: (a-b) $x-t$ cross-section of the x polarized electric field of a 10 fs radially polarized pulse exhibiting PFT and its isosurface plot at half of the maximum electric field value as it is reconstructed with the TERMITES-MAZE method. (c-d) $y-t$ cross-section of the y polarized electric field of the same pulse and its isosurface plot. Due to the nature of the experiments, the pulses must be very well aligned. This is particularly important for the alignment between the two experiments. In order to achieve high precision, we perform the alignment with image processing. A fit of a two-dimensional elliptical Gaussian function to the intensity of the same pulse as captured by the sensor at the two different experiments, gives the position of its center, the FWHM at different orientations and the rotation angle. Since the pulse is the same, a comparison of these values provides the exact spatial shift of the pulse on the sensor. Then by cropping and flipping the corresponding images the two experiments are aligned with pixel size precision. Fig. 5 shows the spectral amplitude and spectral phase of the $x$ and $y$ polarizations of a radially polarized pulse at $794$ nm as it is derived by the analysis. Only areas with significant energy are considered for the phase maps. In these areas the pulse has a phase difference of $\pi$ between its left and right (or top-bottom) parts as it is expected for a CVP due to the polarization singularity in its center. The small deviation of the phase profile from the cylindrical symmetry is due to the existence of relatively weak space-time couplings, like pulse front tilt and curvature as it can be seen in Fig. 6. An inverse Fourier transform is used to reconstruct the 3 dimensional shape of the pulse in the time domain. Fig 6a-d show the $x-t$ and $y-t$ cross-sections for the $x$ and $y$ polarization respectively of a radially polarized pulse and their 3d reconstruction as an isosurface at half of the maximum amplitude. The pulse is 10 fs long and it is clear that apart from a small curvature and a small pulse front tilt in the $x$ direction, it does not have significant aberrations. In contrast, in Fig. 7 we show the reconstruction of a pulse exhibiting PFT in its $x$ direction. This is actually the initial pulse that was used for the experiment and the PFT indicates a misalignment of the pulse compression setup. The existence of the PFT offers an opportunity to apply the developed method for the characterization and correction of the latter. The measured PFT is found to be about 7 $\mathrm{fs/mm}$ and it was retrieved by fitting a paraboloid to the spatially depended delay of the pulse which is directly related to the spatially depended spectral phase provided by the analysis [48]. A way to quickly correct the tilt is to insert a wedge. The induced PFT due to a wedge can be calculated from the following equation [6] $\displaystyle\xi=\frac{\lambda_{0}\tan(\theta)}{c}\frac{\mathrm{d}n}{\mathrm{d}\lambda}$ (6) with $\lambda_{0}$ the central wavelength of the pulse, $c$ the speed of light, $n$ the refractive index and $\theta$ the angle of the wedge. Here, we have used a commercially available $5^{\circ}$ fused silica wedge plate in the path of the pulse. From Eq. 6 we calculate the PFT introduced by the wedge to be about 6 $\mathrm{fs/mm}$. The wedge was inserted in such a way that influenced and corrected the profile of the pulse along the horizontal axis, $x$ direction. The corrected pulse is shown in the Fig. 6 and exhibits a PFT of about 0.5 $\mathrm{fs/mm}$, which is close to the expected value from Eq. 6. The fact that the existence of the PFT on the generated CVP was discovered only during the development of the method highlights its practicality to any field employing such pulses. ## IV Conclusions In this paper we have presented a technique for the spatio-temporal characterization of CVPs and we demonstrated its capabilities with the complete spatio-temporal characterization of a 10 fs radially polarized pulse. The technique is a Mach-Zehnder interferometer extension to the TERMITES method that is necessary to overcome the limitation of the latter to characterize pulses with polarization singularities. A radially polarized pulse exhibiting pulse front tilt was also characterized to illustrate the capabilities of the method. In the current implementation, the method characterizes CVPs whose relative phase between the two orthogonal polarizations is a priory known and hence it is not measured. However, there are cases that this relative phase is not known and it is the important quantity. Such cases are pulses with dynamic polarization like polarization gates that are used for the generation of isolated attosecond pulses [49] or pulses with polarization singularities that are unknown to exist a priori. The TERMITES-MAZE method can be easily extended for the characterization of such pulses by altering the setup to use a reference pulse polarized at 45 degrees and interfere it with the x and y components of the unknown pulse on a polarizing beam splitter, similar to the POLLIWOG method [50]. The method is not limited by the bandwidth of the pulse since it is based on an interferometric setup. Apart from the characterization of CVPs it can be applied to ultrashort light structures possessing a number of polarization or phase singularities. The latter is of paramount importance to transfer the field of topological optics [51] from monochromatic beams to singular pulses [42]. 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# The Evolution of NGC 7465 as Revealed by its Molecular Gas Properties Lisa M. Young Physics Department, New Mexico Institute of Mining and Technology, 801 Leroy Place, Socorro, NM 87801, USA Adjunct Astronomer, National Radio Astronomy Observatory, Socorro, NM 87801, USA David S. Meier Physics Department, New Mexico Institute of Mining and Technology, 801 Leroy Place, Socorro, NM 87801, USA Adjunct Astronomer, National Radio Astronomy Observatory, Socorro, NM 87801, USA Martin Bureau Sub-department of Astrophysics, Department of Physics, University of Oxford, Denys Wilkinson Building, Keble Road, Oxford, OX1 3RH, UK Alison Crocker Department of Physics, Reed College, Portland, OR 97202, USA Timothy A. Davis School of Physics & Astronomy, Cardiff University, Queens Buildings, The Parade, Cardiff, CF24 3AA, UK Selçuk Topal Department of Physics, Van Yüzüncü Yıl University, Van 65080, Turkey (Received 26 November 2020; Revised 25 January 2021; Accepted 27 January 2021) ###### Abstract We present ALMA observations of CO isotopologues and high-density molecular tracers (HCN, HCO+, CN, etc.) in NGC 7465, an unusually gas-rich early-type galaxy that acquired its cold gas recently. In the inner 300 pc, the molecular gas kinematics are misaligned with respect to all other galaxy components; as the gas works its way inward it is torqued into polar orbits about the stellar kinematically-decoupled core (KDC), indicating that the stellar KDC is not related to the current gas accretion event. The galaxy also exhibits unusually high 12CO/13CO line ratios in its nucleus but typical 13CO/C18O ratios. Our calculations show that this result does not necessarily indicate an unusual [12CO/13CO] abundance ratio but rather that 12CO (1-0) is optically thin due to high temperatures and/or large linewidths associated with the inner decoupled, misaligned molecular structure. Line ratios of the higher-density tracers suggest that the densest phase of molecular gas in NGC 7465 has a lower density than is typical for nearby galaxies, possibly as a result of the recent gas accretion. All of the observed molecular properties of NGC 7465 are consistent with it having acquired its molecular (and atomic) gas from a spiral galaxy. Further detailed studies of the CO isotopologues in other early-type galaxies would be valuable for investigating the histories of those that may have acquired their gas from dwarfs. Finally, these ALMA data also show an unidentified line source that is probably a background galaxy similar to those found at $z=1-3$ in blind CO surveys. Early-type galaxies (429) – Interstellar molecules (849) – CO line emission (262) – Molecular gas (1073) – Galaxy evolution (594) ††facilities: ALMA††software: CASA (Emonts et al., 2019), Astropy (Astropy Collaboration et al., 2013, 2018) ## 1 Introduction The gas in early-type galaxies (ETGs) offers interesting clues into their evolution, as there can be a much stronger disconnect between the stars and the gas than there is in spiral galaxies. In spirals, the continuous star formation activity implies that there has been a continuous supply of cold gas over a Hubble time, and the stars and the gas have co-evolved in an uninterrupted symbiotic relationship. In contrast, in early-type galaxies much of the cold and warm gas is kinematically misaligned (even retrograde) with respect to the stars, such that there is little possibility the stars formed from that gas or the gas was expelled from those stars. We expect irregular gas kinematics in the outer parts of galaxies where the cold gas (especially HI) is more vulnerable to interactions and will retain signatures of disruption for many Gyr. But such misalignments are also present in the inner few kpc of early-type galaxies, where we find that 20% of the regular, relaxed CO disks have polar or retrograde rotation with respect to the stars in the host galaxy. Thus, as much as 30% to 50% of the molecular gas in early-type galaxies has been acquired recently from some external source such as an accreted dwarf galaxy (Davis et al., 2011). Similar results are found for ionized gas (Jin et al., 2016; Bryant et al., 2019). Of the remainder, some of the relaxed, prograde molecular gas may have been resident in its host early-type galaxy for a long time, through the galaxy’s transition to the red sequence (e.g. Davis & Bureau, 2016). Some of the molecular gas probably also originated in its current host, but made an extended detour through a hot phase in the galactic halo before condensing back into a cold phase (e.g. Russell et al., 2019; Davis et al., 2019). These varied histories also have implications for the physical properties of the gas. If the cold gas in an early-type galaxy was accreted in a minor merger or from a relatively pristine cold flow, the metallicity and isotopic abundance patterns in the molecular gas could reflect a significantly different star formation history and stellar initial mass function (IMF) from those that produced the current stellar populations in the galaxy (Davis & Young, 2019). On the other hand, if the gas has been resident in the galaxy for a long time, gradually mixing with the mass loss material from evolved stars in the galaxy, we would expect a much tighter correspondence between the metallicity and isotopic abundances of the stars and the gas. In this context we have been undertaking detailed studies of the chemical and physical properties of the molecular gas in early-type galaxies; these chemical and physical properties might also serve as indicators of the galaxies’ histories in a complementary manner to the information revealed by kinematics. Here we present Atacama Large Millimeter/submillimeter Array (ALMA) observations of the early-type galaxy NGC 7465, which is known to have recently accreted its cold gas (Section 2). We present 12CO (1-0) data at 0.′′8 (110 pc) resolution and 12CO, 13CO, C18O, and several other higher- density molecular tracers at 2′′ (280 pc) resolution (Section 3). We describe the millimeter continuum emission, compare it to the optical nebular emission line ratios, and estimate the ionized gas metallicity (Section 4). We also make comparisons of CO to stellar and ionized gas kinematics at matched resolution (Section 5). Quantitative analysis of the molecular line ratios, their spatial variations, and comparisons to other galaxy types (Sections 6 \- 11) reveal clues to the evolution of NGC 7465 and the broader context of galaxy evolution. ## 2 About NGC 7465 NGC 7465 was originally selected because it is a member of the ATLAS3D sample of early-type galaxies (Cappellari et al., 2011), so there is a wealth of information about its stellar and gas content, including two-dimensional maps of its stellar kinematics, ionized gas distribution and kinematics, stellar populations, star formation history, atomic gas, and molecular gas (Krajnović et al., 2011; Young et al., 2011; Serra et al., 2012; McDermid et al., 2015, and references therein). It is one of the more CO-bright members of the ATLAS3D sample, so there is extensive information on its 12CO J=1-0 emission. Its stellar mass is $\log(M_{\star}/M_{\odot})=10.4$ and its adopted distance is 29.3 Mpc (Cappellari et al., 2011). Though it is an early-type galaxy, NGC 7465 shows moderate levels of current star formation. Davis et al. (2014) estimated its total star formation rate from its 22µm WISE flux, giving 1.37 $\pm\ 0.5$ M⊙ yr-1 and a specific star formation rate SFR/M⋆ $\approx(5.5\pm 2)\times 10^{-11}$ yr-1. These rates are low when compared to nearby spirals, placing NGC 7465 below the “star formation main sequence” or SFR–M⋆ relation (e.g. Cluver et al., 2014) but high when compared to other early-type galaxies (Davis et al., 2014). Its depletion time (SFR/Mgas) is relatively long, at 7 Gyr. Its radio and far-IR continuum emission imply a radio/FIR $q$ ratio that is also consistent with star formation activity (Nyland et al., 2017). In addition to being CO-bright, NGC 7465 is notable for signatures of interactions with near neighbors NGC 7463 and NGC 7464 at projected separations $<$ 20 kpc. Though it is an early-type galaxy (an S0 galaxy and a fast rotator), it has faint blue outer arms at radii of 8 to 10 kpc (Figure 1). Its HI emission is highly disturbed and strongly misaligned with respect to the stellar body of the galaxy (Li & Seaquist, 1994; Serra et al., 2012). This kinematic evidence strongly suggests that the galaxy’s cold gas was recently acquired from some external source, and this accretion/interaction event probably drove the formation of the blue outer arms. The stellar kinematics show a kinematically-distinct core (Krajnović et al., 2011) which, as we will show, is probably not related to the most recent gas-transfer interaction. A deep $u$-band image from Duc et al. (2015) also shows recent star formation activity in an irregular spiral structure at radii $\approx$ 3′′ to 20′′ ($\approx$ 0.4 to 3 kpc). Figure 1: Top: color composite of the MATLAS $u,g,$ and $i$ images (Duc et al., 2015). Two neighboring galaxies are visible off the northwest corner of the image. Bottom: 12CO contours superposed on a $u-i$ color image (bluer colors are white). Contour levels are 0.0164 $\times$ (1, 3, 9, 27, 81) Jy bm-1 km s-1, where the lowest contour is the nominal column density sensitivity equivalent to a 5$\sigma$ signal in one channel of width 15 km s-1. The synthesized beam (0.′′8 $\times$ 0.′′7) is shown as a blue ellipse in the bottom left corner. A primary beam correction has been applied; in both panels the dotted circle indicates the half power point of the ALMA primary beam at 114 GHz. NGC 7465 is classified as a barred S0 galaxy by some (e.g. de Vaucouleurs et al., 1991), and indeed Figure 1 might be interpreted as a nearly face-on galaxy with a bar $\approx$ 1′ (8.5 kpc) in length. An alternative interpretation arising from the ATLAS3D dynamical analysis (Cappellari et al., 2013) is that the inner $r\lesssim 30$′′ of the galaxy is an edge-on axisymmetric fast rotator and the outer blue arms are transient extraplanar structures. The primary reason for this alternative interpretation is that the inner part of NGC 7465 does not show typical kinematic features of strongly barred early-type galaxies, which are (1) large misalignments between the photometric major axis of the bar and the kinematic major axis of the stars in that region, and (2) cylindrical rotation in the stellar kinematics, yielding parallel iso-velocity contours (Krajnović et al., 2011; Lablanche et al., 2012). Additional stellar kinematic data covering a larger field of view would be helpful in distinguishing between these alternatives. Finally, IRAM 30m observations of molecular lines in the ATLAS3D galaxies have shown that NGC 7465 has some uncommon integrated molecular line ratios (Crocker et al., 2012). Relative to 12CO, NGC 7465 has unusually faint 13CO and HCN, and it has a low HCN/HCO+ ratio. At higher resolution, these line ratios might reflect the influence of an active galactic nucleus on its surrounding medium. The galaxy contains a Seyfert or LINER nucleus (e.g. Gonçalves et al., 1999), a 5 GHz synchrotron point source (Nyland et al., 2016) and a Swift hard X-ray source (Baumgartner et al., 2013). Our new ALMA data, with additional lines and higher spatial resolution, provide a better perspective on how to interpret the molecular line ratios and connect them to the physical properties of the gas and to galaxy evolution. ## 3 Observations NGC 7465 has been observed by ALMA in several projects. Here we used the Band 3 observations of 13CO, C18O, and CS in project 2018.1.01253.S (December 2018); 12CO, HCN, C2H, and HCO+ in 2016.1.01119.S (November 2016); and 12CO and CN in 2018.1.01599.S (October 2018). We used the standard pipeline- calibrated raw data and carried out continuum subtraction, imaging, and cleaning ourselves. An additional round of phase self calibration was not found to offer any improvement in the images because of the relatively modest signal-to-noise ratios. For continuum subtraction we used zero-order or first- order fits to the line-free channels in the visibility domain; for imaging, we made a wide variety of images at varying channel widths and resolutions as necessary to optimize the resolution or to match resolutions for resolved line ratios. The times on source, along with fiducial (“natural” weighting) beam sizes and rms noise levels, are indicated in Table 1 for the detected spectral lines. Line emission in the data cubes was cleaned down to about the 1$\sigma$ level and primary beam corrections were applied. The 12CO integrated intensity images (e.g. Figure 1) were created by modestly smoothing the data cube in spatial and velocity dimensions and clipping on the smoothed cubes to create a velocity-dependent mask defining the volume with emission. These 12CO masks were then applied to the fainter lines for creating their integrated intensity images. Line fluxes reported in this paper include uncertainties which are only based on the thermal fluctuations in the data, representing signal-to-noise considerations; they do not include uncertainties in the absolute flux calibration scale, such as those due to secular variability in the calibrator sources. The flux calibration of ALMA data that have been processed with standard calibration procedures is usually accurate to $\approx$ 5% to 10% (e.g. Martín et al., 2019; Andrews et al., 2018; Remijan et al., 2019). Line ratios between data taken in different observing setups and different tunings, such as 12CO/13CO (see Table 1), have an additional uncertainty due to this absolute flux calibration. However, line ratios between data taken simultaneously using the same setup (e.g. 13CO/C18O or HCN/HCO+) do not have this additional uncertainty. In addition to the spectral lines mentioned above, we also have upper limits on CH3OH($2_{k}-1_{k}$), SiO(2-1; $v=0$), and HNCO$(4_{0,4}-3_{0,3})$. Furthermore, there is an unidentified line source in the vicinity of NGC 7465. It seems to be a galaxy at a much higher redshift, and it is described in more detail in Appendix A. Table 1: Fiducial data cube parameters for NGC 7465 Line \| Time \| Beam size \| rms \| $\Delta v$ ---\|---\|---\|---\|--- \| (sec) \| (′′) \| (mJy bm-1) \| (km s-1) 12CO(1-0) \| 1590 \| 1.60$\times$1.18 \| 1.59 \| 5.1 \| 11733 \| 0.81$\times$0.74 \| 0.53 \| 2.6 CN(1-0) \| 11733 \| 0.84$\times$0.76 \| 0.066 \| 45.0 13CO(1-0) \| 3810 \| 1.89$\times$1.81 \| 0.38 \| 13.3 C18O(1-0) \| 3810 \| 1.90$\times$1.82 \| 0.35 \| 13.3 CS(2-1) \| 3810 \| 2.12$\times$2.04 \| 0.29 \| 14.9 CH3OH($2_{k}-1_{k}$) \| 3810 \| 2.18$\times$2.06 \| 0.35 \| 15.1 HCO+(1-0) \| 8890 \| 1.70$\times$1.37 \| 0.26 \| 13.1 HCN(1-0) \| 8890 \| 1.71$\times$1.38 \| 0.26 \| 13.2 HNCO $(4_{0,4}-3_{0,3})$ \| 8890 \| 1.72$\times$1.39 \| 0.28 \| 13.3 C2H(N=1-0) \| 8890 \| 1.72$\times$1.38 \| 0.29 \| 13.4 SiO(2-1, $v=0$) \| 8890 \| 1.73$\times$1.39 \| 0.28 \| 13.5 Note. — Beam sizes and rms noise levels refer to images made with “natural” uv-weighting and the listed channel widths. In the case of the 12CO and CN observations, the channel widths in the last column are close to the best velocity resolutions allowed by the data. For the other lines, the listed channel widths are representative but not the best possible. At a distance of 29.3 Mpc, 1′′ corresponds to 140 pc. ## 4 Radio continuum and ionized gas As we have observations at similar angular resolutions covering a large range of frequencies from 86 to 115 GHz (with gaps), the data are suitable for imaging the continuum intensity and estimating the spectral index of any emission. NGC 7465 has two continuum point sources, one in the nucleus and one 5′′ south of the nucleus (Figure 2). The nuclear source has a falling spectral index; its flux densities measured at 86.15 and 107.9 GHz are 742 $\pm$ 12 $\mu$Jy and 616 $\pm$ 18 $\mu$Jy, respectively. Parametrizing the flux density as $S_{\nu}\propto\nu^{\alpha}$ therefore gives $\alpha=-0.83\pm 0.13.$ Similarly, using all of the available line-free frequencies in CASA’s multi- term, multi-frequency synthesis deconvolver (Rau & Cornwell, 2011), we find $\alpha=-0.59\pm 0.12.$ These estimates are consistent with each other and both clearly indicate synchrotron emission in the nucleus of the galaxy, which is not surprising given the other active galactic nucleus (AGN) indicators mentioned in Section 2. For comparison, Domínguez-Fernández et al. (2020) find that at 230 GHz the nuclear continuum source has a flux density of 900 $\pm$ 200 $\mu$Jy, which is broadly consistent with the flux densities measured here and suggests that in this mm regime the nuclear source’s spectrum is flattening as dust emission begins to dominate. The fainter continuum source has flux densities of 155 $\pm$ 12 $\mu$Jy and 245 $\pm$ 18 $\mu$Jy at 86.15 and 107.9 GHz, suggesting instead a rising spectral index consistent with dust emission. It appears to be associated with a region of recent star formation activity, as indicated by a bright blue source in the $u$ image, peaks in [O III] and H$\alpha$ emission (Ferruit et al., 2000), and a small peak in the molecular surface brightness. Moustakas & Kennicutt (2006) presented optical spectroscopy and measurements of the nebular emission line fluxes in NGC 7465. The nebular spectrum (from a region 2.′′5 square) has log([N II]/H$\alpha$) = -0.28 $\pm$ 0.03 and log([O III]/H$\beta$) = 0.44 $\pm$ 0.03, suggesting the ionization there is dominated by the AGN (Moustakas et al., 2010). The spectrum integrated over a much larger 50′′ $\times$ 60′′ rectangular region has log([N II]/H$\alpha$) = -0.39 $\pm$ 0.03 and log([O III]/H$\beta$) = 0.06 $\pm$ 0.03, consistent with star formation activity. The ATLAS3D data provide more spatial information at higher resolution, though only on the [O III]/H$\beta$ ratio; they show an even higher ratio of log([O III]/H$\beta$) = 0.62 in the nucleus, on the location of the nuclear radio synchrotron and hard X-ray sources (Figure 2). Lower ratios of log([O III]/H$\beta$) from $-0.25$ to $-0.3$ are seen in patches at radii of 3′′ to 10′′, and while this single ratio cannot conclusively identify the source of ionization (e.g. Sarzi et al., 2010), all the data are consistent with the interpretation that the current star formation activity in the interior of the galaxy is particularly concentrated in this annular region. The nebular emission line fluxes in Moustakas & Kennicutt (2006) also enable an estimate of the gas metallicity in NGC 7465. We use line fluxes integrated over the large 50′′ $\times$ 60′′ region; following the prescriptions outlined in Moustakas et al. (2010), the emission line ratios suggest a metallicity consistent with solar. We find 12 + log(O/H) = 8.38 $\pm$ 0.06 using the calibration of Pilyugin & Thuan (2005) and $\approx$ 9.0 using the calibration of Kobulnicky & Kewley (2004); each of these is consistent with its respective median of the SINGS galaxies (mostly spirals) in Moustakas et al. (2010). Figure 2: The [O III]/H$\beta$ ratio from the ATLAS3D project (Cappellari et al., 2011). Contours are the 3mm continuum emission, showing the two point sources. Contour levels are ($-3$, 3, 10, and 25) times the rms noise, which is 11 $\mu$Jy bm-1; the peak intensity is 590 $\mu$Jy bm-1. The beam size of the continuum emission is shown as the solid blue ellipse in the lower left corner. ## 5 Molecular Gas Distribution and Kinematics ### 5.1 Large-scale disk The 12CO emission from NGC 7465 (Figure 1) shows a disky structure consisting of irregular, flocculent spiral arms extending to radii of at least 20′′. Some emission is also detected beyond the half-power point of the ALMA primary beam, so it seems likely that there is significant molecular gas beyond the region that we are able to image in these datasets. This irregular disk has an axis ratio near 1, suggesting a low inclination, but its kinematic major axis is strongly twisted and asymmetric as might be expected of recently accreted gas (Figure 3). Although there is a large mismatch between the angular resolution of the HI and CO data, the CO velocity field on large scales is consistent with that of the HI (Li & Seaquist, 1994), suggesting that those two gas phases probably have a common origin. Section 2 discusses the fact that it is not clear whether NGC 7465 is a face- on barred galaxy or an edge-on axisymmetric galaxy with extraplanar arm/ring structures. In either case, steady-state models are likely to be misleading when trying to interpret this galaxy’s kinematics. The interactions with its neighbors mean that its potential is rapidly time-varying. Either these interactions, or the perturbations caused by a bar (if one is present), might explain the non-circular motions and possible gas inflows discussed in this section. In addition, we note that dense molecular gas is found over most of the extent of the elongated structure. In a bar model, this would imply that either dense gas is present beyond the inner Lindblad resonance (ILR), contrary to expectation (e.g. Athanassoula, 1992), or alternatively that the radius of the ILR is much larger than is usual in early-type disk galaxies and it extends to almost the end of the bar and thus corotation. Overall, we therefore argue that it is problematic to interpret the gas distribution and kinematics in NGC 7465 in terms of a relaxed/steady state early-type barred disk galaxy potential. ### 5.2 Inner kpc and decoupled core At radii $\lesssim$ 5′′ (where we have most of our molecular line detections) the ATLAS3D stellar isophotes are flattened and their major axis is aligned to the kinematically-decoupled stellar core (KDC; Figure 3, center panel). The KDC is both photometrically and kinematically misaligned by 150° with respect to stellar structures farther out. In this region the molecular gas forms a bright ridge about 10′′ long (1.4 kpc), oriented along a northeast-southwest axis. The ridge is misaligned by about 100° with respect to the stellar KDC and 120° with respect to the stellar kinematic and photometric axes at larger radii. The molecular gas shows a strong velocity gradient along the ridge, so that the most extreme CO velocities are found along the ridge, about 1′′ from the nucleus as defined by the radio continuum position. 12CO (2-1) emission at higher resolution (Domínguez-Fernández et al., 2020) shows a patchy spiral feature that can be followed inward to a radius of 2′′ and a smaller molecular ridge 2′′ long, with a saddle at the location of the AGN. But the smaller ridge observed in (2-1) emission maintains the same orientation as the more extended ridge we observe in the present data; thus, on scales of 1′′ (140 pc) the molecular ridge is still perpendicular to the stellar KDC. Davis et al. (2011) discussed the kinematics of ionized and molecular gas in early-type galaxies, and noted that the two phases agreed in all of the cases they studied at moderate ($\approx$ 1 kpc) resolution. That is also still true in NGC 7465 for radii $\gtrsim$ 2′′ (280 pc), but for smaller radii the kinematics of the ionized and molecular gas diverge. The ionized gas continues its inward twist such that at small radii the kinematic position angle of the ionized gas nearly matches that of the stellar KDC but is 45° offset from the molecular kinematic position angle. The molecular gas, on the other hand, maintains its fixed orientation nearly perpendicular to the stellar KDC. This divergence of the kinematics is suggestive of more complex noncircular motions or perhaps multiple ionization sources in the galaxy’s nucleus, such that the inner ionized gas is decoupled from the molecular gas. The ionized gas kinematics in the center of NGC 7465 may even be tracing an AGN-driven ionized outflow, and Appendix B presents a closer look at that evidence. The mismatch between molecular and stellar kinematics in the center of NGC 7465 also means that the stellar KDC has not formed out of the molecular gas currently present in the galaxy. It must represent a previous formation event, or perhaps a much earlier episode of an extended interaction. More detailed observations of the stellar populations in the KDC would be required to date its formation (e.g. Sarzi et al., 2016). Figure 3: CO, stellar, and ionized-gas velocity fields of NGC 7465. In the 12CO velocity field, the contours show the integrated CO intensity at 0.8′′$\times$ 0.7′′ resolution, with contour levels at 0.03, 0.3, and 1.0 Jy bm-1 km s-1. The CO beam size is shown as a gray ellipse in the bottom left corner. These CO data have slightly better angular resolution than the ground- based optical data, which are from Cappellari et al. (2011). In the stellar and ionized gas panels, the contours are stellar isophotes. ### 5.3 Other molecular species in the inner kpc The integrated intensity maps of the other molecular lines are presented in Figure 4. To avoid biases related to the different signal-to-noise ratios, a velocity-dependent mask tracing the 12CO emission is used for all of the other spectral lines. Their distributions can be broadly classified into two major groups. 12CO, HCN, HCO+, CN, and C2H are centrally concentrated and peaked on the nucleus of the galaxy, whereas 13CO and C18O show peaks a couple of arcseconds ($\approx$ 1 beam) north and east of the nucleus rather than on the nucleus itself. In 13CO, for example, the integrated intensity at the position of the nucleus is a factor of two lower than farther out along the molecular ridge. CS emission is relatively weak and while it appears to have a peak off the nucleus, the spatially-resolved ratios (Section 6) suggest that this appearance may be due to noise in the integrated intensity. Finally, HNCO, CH3OH and SiO emission are not detected in these data. Position-velocity slices through the data cubes (Figure 5) are consistent with those from Domínguez-Fernández et al. (2020), showing the presence of two kinematic structures in the central molecular gas of NGC 7465. At radii $<$ 1.′′3 we find a nuclear ring or bar (or possibly an outflow) that is most prominent in HCO+ and CN emission, also visible in HCN and 12CO emission, and extremely weak or absent in 13CO emission. Exterior to 1.′′3 the primary structure is the warped disk, where the 13CO emission is at its brightest, and there the 13CO emission follows the 12CO emission. The appearance of a Keplerian decline in the 12CO emission in Figure 5 is an artifact of the twist in the kinematic position angle, so that at large radii this slice is closer to the kinematic minor axis than the major axis. Figure 4: Integrated intensity images of multiple molecular species in the inner few kpc of NGC 7465. The position of the nuclear 3mm continuum source is marked with a cross, to guide the eye. These integrated intensities are computed using a velocity-dependent mask that follows the rotation of the gas, as defined by 12CO emission. Each line’s synthesized beam is indicated by the ellipse in the lower right corner, and the colors are scaled to the minimum and maximum of each image. Integrated line fluxes towards the nucleus are given in Table 2. For the fainter lines, a dotted white contour indicates pixels where the integrated intensity has a signal-to-noise ratio $>2.5.$ The corresponding 12CO contour is mostly outside this field of view. Figure 5: Position-velocity slices at a position angle of +42°, one beam wide. The slices cut through the nucleus and follow the molecular ridge, tracking the kinematic major axis at small radii (see Figure 3). Velocities are measured with respect to the systemic velocity, 1961 km s-1. Dotted lines at $\pm$ 1.′′3 demarcate the inner kinematic component with the strong velocity gradient. 12CO is plotted both at the highest available resolution and at a resolution matching the 13CO data. ## 6 Line strengths: CO isotopologues For the bright lines of NGC 7465, we can create line ratio maps from the integrated intensity images, as in Figure 6 for 12CO/13CO. For the fainter lines we measure integrated line fluxes and ratios by defining spatial regions (Figure 7), summing the cubes within the spatial regions to produce integrated spectra (e.g. Figure 8), and then summing the spectra over velocity ranges defined by the 12CO emission. This procedure ensures that the same data volume is used for both lines of a ratio, even if their signal-to-noise ratios are very different. Spatially-resolved line ratios measured in this manner are presented in Figures 9 and 10. Reference column densities for the nuclear spectrum in the optically thin and local thermodynamic equilibrium (LTE) approximations are listed in Table 2 and line fluxes for all of our defined regions are presented in Appendix C. For an alternate method of exploring spatial variations in line ratios, we also implemented a “shift and stack” technique. In this method all of the spectra within a given annulus (about the nuclear 3mm continuum peak) can be aligned in velocity, using the 12CO velocity field at each position to determine the velocity shift. Thus all of the expected signal can be concentrated in a narrow range of frequencies. However, because of the departures from azimuthal symmetry in the center of NGC 7465, ratios produced by this technique have significantly larger uncertainties than those derived from individual regions. Figure 6: 12CO/13CO integrated intensity ratio (computed in units of brightness temperature). Black contours show the 3mm continuum emission, and the angular resolution of the 12CO/13CO ratio image is shown by the gray ellipse in the lower left corner. Figure 7: Integrated intensities of 12CO (top) and 13CO (bottom). The angular resolution in each panel is shown with a gray ellipse in the lower left corner. Black and white polygons show the regions used for summing spectra, to investigate the spatial variations of the line ratios. The boundary of the nuclear region is dotted. A few white pixels in the lower portion of the top panel are locations where 12CO is not detected at this angular resolution. Numbers in the lower panel identify the regions for reference to the table in Appendix C. Figure 8: Spectra extracted from the nuclear region, which is 2′′ (1 beam) in diameter and centered on the 3mm continuum point source. The grey boxes mark the velocity range used for integrating at this position. In the C2H panel, the two dotted lines mark the centers of the two main fine structure blends and the velocity scale is calculated for the low frequency ($J=\frac{3}{2}-\frac{1}{2}$) blend. These nuclear spectra exhibit unusually weak 13CO emission. Figure 9: Line ratios are integrated over the regions shown in Figure 7; the horizontal error bars indicate the range of radii probed by each region. Ratios measured towards the nucleus are plotted as stars, and those measured over the entire region with detected 12CO emission are plotted at 10′′ and circled. The region with relatively low 12CO/HCN at 5.′′3 is centered on the fainter 3mm continuum source. For purposes of clarity, only the detections ($>3\sigma$) are plotted. Figure 10: Similar to Figure 9, for different ratios; symbols have the same meanings. The radial coordinates for some ratios have been slightly shifted for better visibility. Table 2: Line fluxes and reference column densities in the nucleus of NGC 7465 Species \| Line flux \| $\log N$ (20 K) \| $\log N$ (90 K) ---\|---\|---\|--- \| (Jy km s-1) \| (cm-2) \| (cm-2) 12CO \| 003.52 (0.04) \| 17.14 \| 17.69 13CO \| 000.096 (0.014) \| 15.65 \| 16.20 C18O \| $<$ 0.039 \| $<$ 15.26 \| 15.81 HCN \| 000.132 (0.012) \| 12.80 \| 13.37 HCO+ \| 000.303 (0.013) \| 13.39 \| 13.96 CN \| 000.226 (0.006) \| 13.61 \| 14.16 CS \| 000.041 (0.009) \| 13.02 \| 13.54 C2H \| 000.091 (0.014) \| 14.70 \| 15.27 CH3OH \| $<$ 0.035 \| $<$ 13.95 \| $<$ 14.57 SiO \| $<$ 0.040 \| $<$ 12.79 \| $<$ 13.33 HNCO \| $<$ 0.037 \| $<$ 13.49 \| $<$ 14.19 Note. — Line fluxes are measured in a circular region 2′′ (280 pc) in diameter, centered on the nuclear 3mm continuum source; column densities are averaged over this region. Statistical uncertainties of the line fluxes are listed in brackets, or the upper limits are quoted as 3 times the statistical uncertainty of the sum over the adopted velocity range. For 13CO the peak intensity is offset from the nucleus and a 2′′ region centered on the peak has a line flux of 0.015 Jy km s-1. Column densities are calculated for two representative temperatures in the optically thin LTE assumption. Current data do not constrain the optical depths of the HCN and HCO+ transitions, but CN and C2H are optically thin (Section 8) and even 12CO might be at this position (Section 11.1). NGC 7465 has relatively weak 13CO emission. Measured 12CO(1-0)/13CO(1-0) ratios are highest at 39 $\pm$ 9 in the nucleus, or 33 $\pm$ 5 when averaged over a circle of 1′′ radius, decreasing to $\approx$ 15 $\pm$ 2 at a radius of 1 kpc (see Figures 6 and 9).111All line ratios reported in this paper are computed using integrated fluxes expressed in temperature units (K km s-1). The large ratios in the center of NGC 7465 are higher than those typically found in nearby spirals, which tend to be in the range of 8–20 (especially on scales $>$ 1 kpc, e.g. Crocker et al., 2012; Cao et al., 2017; Cormier et al., 2018; Israel, 2020). For comparison, Figure 11 shows typical 12CO/13CO line ratios for galaxies of various types and illustrates that the high ratios in the nucleus of NGC 7465 are similar to those found in ULIRGs and advanced mergers such as Arp 220 and NGC 2623 (Brown & Wilson, 2019). Figure 11 also shows that the range of 12CO/13CO line ratios exhibited by early-type galaxies is a factor of 10, broader than the range exhibited by any other galaxy type. Other early-type galaxies known to have high 12CO/13CO ratios include NGC 1266 and UGC 09519 (Crocker et al., 2012); NGC 1266 is remarkable for having a strong AGN-powered molecular outflow (Alatalo et al., 2014), while UGC 09519 has a kinematically-misaligned HI disk that suggests it was recently acquired (Serra et al., 2014). NGC 7465 is thus consistent with the pattern that high 12CO/13CO ratios tend to be associated with major disturbances to the ISM. NGC 7465’s strong radial gradient in 12CO/13CO is also uncommon. It amounts to a factor of two over an unresolved length scale, such that the ratios at radii $<$ 140 pc are at least a factor of two higher than those outside and possibly more. For context, the spirals in Paglione et al. (2001), Cao et al. (2017), and Cormier et al. (2018) sometimes have enhancements of 13CO in their nuclei and sometimes deficits, but none of them are as extreme as in NGC 7465. Two barred lenticular galaxies are in fact similar to spirals in this respect (Topal et al., 2016), with typical 12CO/13CO ratios about 5 to 15, and with the higher ratios in the central kpc or so; they only exhibit modest gradients of a factor of two over kpc scales. Measurements at sub-kpc resolution sometimes show spatial gradients of the same magnitude as those in NGC 7465. Meier & Turner (2004) found variations of factors of 5 over $\approx$ 200 pc in the center of NGC 6946, but unlike NGC 7465, the spiral shows an enhancement of 13CO in its center. The strong deficit of 13CO in the center of NGC 7465 is more reminiscent of the nearby early-type galaxy Cen A, where McCoy et al. (2017) find 12CO/13CO $>$ 20 in the circumnuclear disk at radii $<$ 200 pc and 12CO/13CO reaching nearly to unity farther out in the arms. Figure 11: A compilation of observed 12CO/13CO line ratios, in the same spirit as Figure 1 of Zhang et al. (2018). Symbols in grey are dwarf galaxies, spirals, and ULIRGs; at the highest luminosities they are sub-mm galaxies at $z\approx 2-3.$ Data are drawn from Davis (2014), Jiménez-Donaire et al. (2017), Brown & Wilson (2019), Cormier et al. (2018), Henkel et al. (2014), Braine et al. (2017), and Heikkilä et al. (1999); see also the citations in Zhang et al. (2018). The large light error bars for the SMC and the LMC show the full range of ratios measured by Israel et al. (2003) at 12 pc resolution. Open grey circles are stacks of multiple galaxies (Méndez-Hernández et al., 2020). We use an improved FIR luminosity for IC 10 from Rémy-Ruyer et al. (2015). Line ratios of early-type galaxies (orange symbols) are compiled from Crocker et al. (2012) and Alatalo et al. (2015); Cen A is from McCoy et al. (2017). For NGC 7465 and Cen A (green and purple, respectively), the error bars show the full range of ratios measured in the galaxy. Measurements of 13CO/C18O in NGC 7465 are limited to a few spatial regions with detections of C18O (e.g. Figure 12). In a large spatial region defined by all of the detected 12CO emission, we have a tentative detection of C18O yielding 13CO/C18O = 3.0 $\pm$ 1.0. The bright molecular ridge has a ratio of 4.0 $\pm$ 1.2. Figure 13 shows that these ratios are similar to those of galaxies of comparable IR luminosity (e.g. Jiménez-Donaire et al., 2017). Further interpretation of the CO isotopologue line ratios is in Section 11. Figure 12: 13CO and C18O in NGC 7465. These spectra are from a region at the northeast end of the molecular ridge, where the 13CO emission is brightest. Gaussian fits are overlaid, though our analysis uses line ratios derived from simple sums rather than from the fits. Since the C18O line is so faint, its width and center velocity have been constrained to match those of 13CO in the fits. Figure 13: A compilation of observed 13CO/C18O line ratios, after Figure 1 of Zhang et al. (2018). For NGC 4526 (Young et al. in prep) and Cen A (McCoy et al., 2017), the error bars show the full range of ratios measured in the galaxy, with the symbol at the median ratio. In addition to the citations noted in Figure 11 and Zhang et al. (2018), this figure includes measurements by Johansson et al. (1994) and Wang et al. (2009). ## 7 High-Density tracers Figures 9 and 10 show that, as expected, the emission from HCN, HCO+, and C2H is enhanced relative to 12CO and 13CO in the center of NGC 7465. This pattern is common in nearby galaxies and is expected to be driven by higher gas densities in the centers (e.g. Jiménez-Donaire et al., 2019). We also observe radial trends in C2H/HCN and C2H/HCO+, in the sense that C2H is more strongly enhanced just outside of the nucleus than in the nucleus itself. HCN/HCO+ and 12CO/CS are effectively constant wherever we can measure them, but CS/HCN increases with radius. All of these results are discussed in greater detail below. ### 7.1 HCN and HCO+ Relative to 12CO, NGC 7465 has fairly typical HCN brightness compared to other nearby galaxies. Specifically, 12CO/HCN in Figure 9 ranges from 16 $\pm$ 1 to 50 $\pm$ 15\. These ratios are very similar to their analogs measured in the nearby spirals of the EMPIRE survey by Jiménez-Donaire et al. (2019) and in two other lenticulars by Topal et al. (2016). The EMPIRE data have somewhat lower linear resolution (1 to 2 kpc) than our measurements in NGC 7465 (240 pc) but in all of these cases the disks are well resolved. In contrast, the HCN in NGC 7465 is relatively faint compared to that in a larger sample of early-type galaxies; in the unresolved single-dish measurements of Crocker et al. (2012), NGC 7465 has the highest 12CO/HCN ratio of the sample. The unresolved measurements undoubtedly fold in some radial gradients, so it is difficult to make detailed comparisons at different scales and it would be worthwhile to obtain additional resolved 12CO/HCN measurements in early-type galaxies. The large difference in critical or effective density between CO and HCN implies that observed ratios like 12CO/HCN encode information about the density distribution of the molecular gas, or equivalently the relative proportions of low-density and high-density gas (e.g. $10^{2}$ – $10^{3}$ cm-3 vs. $10^{5}$ – 106 cm-3; Meier & Turner, 2012; Leroy et al., 2017). Thus, NGC 7465 has a mix of low/high-density gas which is typical of most nearby spirals, but it is relatively deficient in high-density gas compared to many other local early-type galaxies. The proportion of high-density gas is maximized in the nucleus but is also high towards the southern continuum source, where we find the second-lowest 12CO/HCN and 12CO/HCO+ ratios in the galaxy as well as some of the lowest reliably-measured 12CO/13CO ratios. This region must contain a concentration of dense gas with relatively high CO optical depth. Relative to 12CO, NGC 7465 is exceptionally bright in HCO+ emission when compared to nearby galaxies. We measure local 12CO/HCO+ ratios from $6.9\pm 0.3$ in the nucleus to $20.7\pm 2.9$ in the disk. In comparison, the spiral galaxies of the EMPIRE survey usually have 12CO/HCO+ $>25$ even in their nuclei (Jiménez-Donaire et al., 2019). Thus, the regions of NGC 7465 with the faintest HCO+, at radii $\approx 1$ kpc, still have brighter HCO+ than is common even in the nuclei of the spirals. It is also notable that the HCO+ emission in the nucleus of NGC 7465 has a dramatically different line shape than 12CO (Figure 8), which probably reflects differing spatial distributions. ### 7.2 HCN/HCO+ Because HCO+ is so bright, the HCN/HCO+ ratios in NGC 7465 are low; they are significantly $<1$ everywhere we are able to measure them, and this is unusual for nearby galaxies. We measure HCN/HCO+ = 0.44 $\pm$ 0.04 in the nucleus, with ratios elsewhere ranging up to 0.60 $\pm$ 0.11. In nearby galaxies it is much more common to find HCN/HCO+ $>$ 1 (e.g. Jiménez-Donaire et al., 2019; Topal et al., 2016). Ratios $<$ 1 are seen in restricted parts of a few spirals (e.g. parts of NGC 4254 and Maffei 2; Meier & Turner, 2012; Gallagher et al., 2018). Low ratios of HCN/HCO+ $\approx$ 0.5 are occasionally seen in other early-type galaxies (Crocker et al., 2012) and more commonly in ULIRGs and starburst galaxies (e.g. Baan et al., 2008; Krips et al., 2008; Privon et al., 2015; Sliwa & Downes, 2017). Extremely low HCN/HCO+ ratios $\lesssim$ 0.2 have been measured in dwarf galaxies like the LMC and IC 10 (Seale et al., 2012; Braine et al., 2017; Kepley et al., 2018; Anderson et al., 2014). In this respect, NGC 7465 has more in common with starburst or dwarf galaxies than with nearby spirals. The low HCN/HCO+ and HCN/CO ratios in NGC 7465 also place it firmly amongst the nuclear starburst galaxies and “composite” Seyferts in the HCN(1-0) diagnostic diagram of Kohno et al. (2001). Spatial variations in the HCN/HCO+ ratios of NGC 7465 do not significantly exceed the ratios’ mutual statistical uncertainties, so there is no compelling evidence for a radial gradient (Figure 10). Even though there is an AGN in NGC 7465, there is definitely no enhancement in HCN/HCO+ towards the nucleus, unless it is on length scales of tens of pc where it would be obscured by our spatial resolution. There is certainly nothing like the factor of 2 rise in HCN/HCO+ seen in the central 0.5 kpc of NGC 1068 (Viti et al., 2014). ### 7.3 Physical properties of the dense molecular gas Due to the difference in critical densities, the HCN/HCO+ line ratio is most commonly interpreted as an indicator of density in the high-density ($n\sim 10^{4}-10^{6}$ cm-3) portion of the molecular gas. However, for the lowest HCN/HCO+ line ratios in the LMC, Anderson et al. (2014) have appealed to a combination of low density and low gas-phase metallicity, with low metallicity contributing in two ways. First, lower heavy element abundances will produce lower column densities of these species and smaller optical depths in their transitions, which is necessary for the line ratio to deviate from 1. On top of that, lower metallicities are associated with lower [N/O] abundance ratios (van Zee & Haynes, 2006), which might manifest as lower abundances of HCN relative to HCO+ (Braine et al., 2017). NGC 7465 does not have a low metallicity (Section 4), so there is no reason to assume a low [N/O] ratio a priori. However, the observed line ratios in NGC 7465 do constrain the abundance ratio [HCN/HCO+] to be significantly smaller than that inferred for NGC 253 by Meier et al. (2015), which was [HCN/HCO+] $\approx$ 5\. In LTE, reproducing our observed line ratios of HCN/HCO+ = 0.44 $\pm$ 0.04 requires an abundance ratio [HCN/HCO+] $\leq 0.25$ (Table 2).222We adopt the convention that ratios written without brackets (e.g. HCN/HCO+) refer to measured line ratios whereas ratios inside square brackets (e.g. [HCN/HCO+]) refer to estimated molecular abundances after correcting for optical depth and excitation effects. In the more likely situation that LTE does not apply, we make calculations using the RADEX large velocity gradient code (van der Tak et al., 2007) to account for optical depth and excitation effects on the HCN/HCO+ line ratio; broad ranges of parameters will reproduce the data from NGC 7465 but they always require an abundance ratio [HCN/HCO+] $\lesssim 3,$ and a ratio of 5 is ruled out as it cannot reproduce the line ratios in NGC 7465. For an assumed [HCN/HCO+] = 3, relatively low densities $n_{\rm H_{2}}\leq 10^{4.3}$ cm-3 are required to reproduce the observed line ratios. Thus, even though the metallicity of NGC 7465 is not low enough to suggest unusually low N abundance, the observed HCN/HCO+ line ratios in NGC 7465 are best explained with a combination of relatively low-density gas and/or a relatively low HCN abundance (or enhanced HCO+). More specific estimates of the density and temperature will require additional transitions or isotopologues, but the inference of a low HCN abundance is robust to other variations in physical properties. Viewing the molecular gas as two “phases” – a more diffuse phase with densities $\sim 10^{2}$ to $10^{3}$ cm-3 traced by CO, and a denser phase $\gtrsim 10^{5}$ cm-3 traced by HCN and similar species – the strong spatial trends in diffuse/dense gas tracers suggest that the relative proportions of the two phases change with radius in NGC 7465. However, the internal properties of the denser phase do not seem to change, even though the properties in the more diffuse phase are dramatically varying (Section 11.1). ## 8 C2H and CN: photodissociation regions The C2H molecule is regarded as a tracer of photon-dominated or photodissociation regions (PDRs), as its formation is driven by the presence of C+, and it has been shown to be brighter on the illuminated sides of molecular clouds (e.g. Meier & Turner, 2005, 2012; Meier et al., 2015; García- Burillo et al., 2017). The two main fine structure blends in our data are $N=1-0$, $J=\frac{3}{2}-\frac{1}{2}$, with a rest frequency of 87.3169 GHz, and $N=1-0$, $J=\frac{1}{2}-\frac{1}{2}$, at 87.40199 GHz. They are separated by 292 km s-1, which is just greater than the velocity range covered by 12CO in NGC 7465, so they are not overlapping. The low-frequency component is detected in NGC 7465 but the high-frequency component is not; as the high- frequency component is a factor of 2.3 fainter in LTE in the optically thin limit, its nondetection is not surprising, but we can rule out line ratios $\approx$ 1 (the optically thick limit). Thus we cannot measure the C2H optical depth directly but the data are consistent with the transitions being optically thin. Further, the C2H line fluxes quoted in this paper refer only to the blend at 87.32 GHz. The C2H emission in NGC 7465 is relatively bright, when compared to HCN. We find C2H/HCN ratios ranging from 0.71 $\pm$ 0.13 in the nucleus to 1.0 $\pm$ 0.2 in the outer parts of the molecular ridge. These are significantly higher than the corresponding ratios in the central kpc of NGC 253, which range from 0.17 to 0.47 ($\pm$ 15%; Meier et al., 2015). The C2H/HCN ratios in NGC 7465 are also on the high side when compared to the set compiled by Martín et al. (2014) for active and starbursting galaxies. On the other hand, ratios of C2H/HCO+ in NGC 7465 are very similar to those in NGC 253 and other galaxies. We find 0.31 $\pm$ 0.05 to 0.50 $\pm$ 0.11 in NGC 7465, where Meier et al. find 0.20 to 0.57 ($\pm$ 15%) in NGC 253 and most of the sample galaxies in Martín et al. (2014) show a similar range. These results suggest that HCN emission in NGC 7465 is relatively faint because of moderate densities in the molecular gas; both C2H and HCO+ are relatively prominent because their critical densities are a factor of 10 lower than that of HCN. In addition to being bright, the C2H emission in NGC 7465 is also notable in that the radial trends of C2H/HCN and C2H/HCO+ in NGC 7465 are opposite to the trends in NGC 253. In the spiral those ratios peak at the nucleus; in other words, C2H is more strongly concentrated than HCN and HCO+ are. In NGC 7465 it is the opposite – C2H is less centrally concentrated than HCN and HCO+. The radial trend in NGC 7465 thus suggests enhanced C2H emission just outside of the nucleus, at radii 2′′ to 5′′ or 280 to 700 pc (as far out as we can detect C2H). If the C+ ionization is indeed dominated by star formation activity, this radial trend might be consistent with the ionized gas emission line ratios in Figure 2 (Section 4): the lowest [O III]/H$\beta$ ratios (most suggestive of star formation activity) occur slightly outside the nucleus at radii $\approx$ 3′′ to 10′′. Similarly, from the stellar population analysis of Krajnović et al. (2020), there is a suggestion that the smallest stellar ages occur at a radius of 2.′′5 rather than at larger or smaller radii. As another tracer of C+, we note that NGC 7465 is also detected in [C II] 158µm emission (Lapham et al., 2017), though the spatial resolution is not good enough to compare to the radial trends in our ALMA data. Many spiral galaxies are known to have significant central deficits of [C II] emission at radii $<$ 1 kpc, and many potential explanations have been proposed (e.g. Smith et al., 2017). Possibly, in some cases, the central [C II] deficit arises because an AGN’s hard radiation field drives the ionization balance towards C2+; in other cases, a softer radiation field (caused by old stellar populations) may be driving the ionization balance towards neutral carbon. At present we simply comment that the central C2H deficit in NGC 7465 (relative to HCN and HCO+) might be consistent with what is known about the ionization states of C in other galaxy nuclei. The CN(1-0) transition is also a PDR tracer (Boger & Sternberg, 2005), though it is expected to be even more strongly enhanced in X-ray dominated regions (XDRs) and may thus help to distinguish between PDRs and XDRs (Meijerink et al., 2007). It has fine structure components that appear in these data as blends with rest frequencies of about 113.17 GHz and 113.49 GHz; they are clearly separated in our data and both are clearly detected. The high- frequency blend is the brighter one and there is no evidence that their line ratio in NGC 7465 ever deviates from the theoretical optically thin limit of 2 (e.g. Meier et al., 2015). Thus, the CN emission in NGC 7465 is optically thin. For simplicity, we show only the brighter component in the figures and the line fluxes we quote refer only to it. The range of 12CO/CN ratios we measure in NGC 7465 is large, from $15.0\pm 0.4$ in the nucleus to $93\pm 14$ when averaged over the full 12CO emission region, and it is consistent with the wide range of ratios measured in other nearby galaxies (Wilson, 2018). CN/HCN ratios are consistent with 1 aside from one region, and this again is very typical of local galaxies (Ueda et al., 2017; Cicone et al., 2020). There is some evidence that CN is even more centrally concentrated in NGC 7465 than the other high-density tracers like HCN. The radial variations of 12CO/CN exceed those of all other 12CO ratios (Figure 9). The transition in line profile shapes, from double-peaked 12CO through flat-topped HCO+ to centrally- peaked CN (Figures 8 and 5), might thus suggest the presence of a high-density circumnuclear disk at $r\lesssim 100$ pc. CN might be particularly enhanced close to the AGN because of its X-ray sensitivity (Meijerink et al., 2007). Higher resolution data, especially for CN, would be valuable for testing this scenario. As a caveat we note that the current data do not show a measurable radial trend in CN/HCN (Figure 10), but such a trend could be obscured by the resolution and sensitivity of our data. We do find a systematic difference in the spatial distribution of the two PDR tracers, C2H and CN; C2H is less centrally concentrated than HCN whereas CN is equally or possibly more centrally concentrated than HCN. The differences between these two species might be consistent with an interpretation that C2H is a better tracer of star formation activity whereas CN is a better tracer of XDRs. ## 9 CS The CS molecule is also a high-density tracer, whose $J=2-1$ transition has a critical or effective density intermediate between those of HCN and HCO+ $J=1-0$ (e.g. Leroy et al., 2017). There is also some evidence that CS emission is enhanced in PDRs, where the radiation field is stronger (Lintott et al., 2005; Meier & Turner, 2005), which has motivated suggestions that CS emission traces massive star formation even better than HCN emission does (e.g. Bayet et al., 2009; Davis et al., 2013). NGC 7465 has 12CO/CS ratios that are fairly typical compared to those of other nearby galaxies. We find 12CO/CS = 62 $\pm$ 13 in the nucleus, with variations of no more than 25% in all of the regions where we detect CS. For comparison, the spirals in Gallagher et al. (2018) have typical ratios of 12CO/CS $\approx$ 90 in their central kpc, so NGC 7465 is brighter in CS than these. However, it is faint in CS compared to the other early-type galaxies in Davis et al. (2013), which have 12CO/CS in the range of 9 to 44. The spatially- unresolved (IRAM 30m) ratios in Davis et al. (2013) might also be biased high if the beam-filling factor of 12CO is higher than CS, so that resolved measurements of CS would probably increase the discrepancy between them and NGC 7465. CS/HCN, which should provide better insight (than 12CO/CS) into the properties of the dense molecular phase, is very similar in NGC 7465 to other nearby galaxies. We find 0.26 $\pm$ 0.06 in the nucleus of NGC 7465 and 0.57 $\pm$ 0.14 for the larger regions that extend to 0.8 kpc. These ratios are consistent with those in the spirals Maffei 2 and NGC 253 (0.2 to 1; Meier & Turner, 2012; Meier et al., 2015). The spirals in Gallagher et al. (2018) also typically have CS/HCN $\approx$ 0.1 to 0.3, with localized ratios up to 0.5 in the central 200 pc of NGC 3627 or the arms of NGC 4321. CS/HCN in NGC 7465 is also consistent with the single-dish measurements of some other early-type galaxies in Davis et al. (2013). It is notable that others of the early-type galaxies in that sample have highly unusual ratios of CS/HCN as large as 3, so that bright CS emission might be a feature of some early-type galaxies. In terms of radial trends, it is striking that the nucleus of NGC 7465 does not differ from the off-nuclear regions in terms of 12CO/CS; in contrast, there are strong radial trends of factors of 2–4 in 12CO/HCN, 12CO/HCO+, 12CO/C2H and 12CO/CN. We also find no radial variation of HCN/HCO+ but a factor of two in CS/HCN. If the density of the molecular gas were the only factor driving changes in those line ratios, we would expect HCN, HCO+, and CS to behave in qualitatively the same fashion as the critical density of CS is intermediate between those of HCO+ and HCN. Evidently, then, we require more than simple density variations to explain the behavior of these high-density tracers in NGC 7465. Davis et al. (2013) have also studied unresolved CS emission in a small sample of early-type galaxies. They compared it to [O III]/H$\beta$, which can serve as an indicator of whether the ionization in the optically-emitting ionized gas is dominated by AGN activity or star formation (see also Section 4). In their early-type galaxies, the global [O III]/H$\beta$ ratio is correlated with CS/HCN in the sense that galaxies whose ionization is more dominated by star formation (smaller [O III]/H$\beta$; softer radiation fields) have higher CS/HCN. NGC 7465 is consistent with this trend, both in terms of its integrated line ratios and its internal spatially-resolved behavior, for which we find higher CS/HCN and lower [O III]/H$\beta$ (and bluer optical colors) at $r\approx$ 0.5 to 1 kpc than towards the nucleus. NGC 7465 thus provides circumstantial evidence supporting the interpretation that CS emission is enhanced by PDR conditions. For the ionized gas metallicity estimates in Section 4 and the column density estimates in Table 2, NGC 7465 is also consistent with the suggestions of Bayet et al. (2012) and Davis et al. (2013) that CS/HCN could be used as an indicator of the metallicity of the molecular gas. Future work on spatially- resolved nebular emission line data might also be valuable to test whether the trend in Davis et al. (2013) also applies within individual galaxies. ## 10 Shock tracers Emission from SiO, CH3OH, and HNCO is commonly interpreted as tracing shocks in the ISM (e.g. Meier & Turner, 2012; Meier et al., 2015), as significant energy input in the form of shocks is required to liberate them from grain surfaces. These molecules are not detected in NGC 7465, with the most stringent limits being 12CO/(SiO, CH3OH, or HNCO) $>$ 83 near the center of NGC 7465. Ratios relative to 12CO are not particularly significant in terms of a physical interpretation, but here merely serve as an indication of the relative faintness of the lines compared to detections in other galaxies. For context, in NGC 253, Meier et al. (2015) found spatially-resolved ratios of 12CO/SiO and 12CO/HNCO $\approx$ 20 to 80; NGC 253 thus has brighter SiO and HNCO than NGC 7465, by at least a factor of 4 relative to 12CO. The circumnuclear disk of NGC 1068 has 12CO/SiO = 12.5 $\pm$ 1.5 (García-Burillo et al., 2010), which again is at least a factor of 7 brighter in SiO than NGC 7465. Finally, Topal et al. (2016) also measured 12CO/HNCO = 66 $\pm$ 5 and 78 $\pm$ 18 in the centers of the lenticulars NGC 4710 and NGC 5866. Thus, these two also have modestly brighter HNCO than NGC 7465. For CH3OH, Davis et al. (2013) detected a small sample of early-type galaxies with unresolved measurements of 12CO/CH3OH in the range 14 to 70. As we measure 12CO/CH3OH $>$ 83, we conclude that NGC 7465 is also relatively faint in CH3OH emission. Similarly, unresolved measurements of CH3OH/HCN reach as high as 1.5 to 2.2 in the early-type galaxies NGC 6014 and NGC 5866 (Davis et al., 2013), whereas the resolved upper limits in NGC 7465 are 0.2 to 0.8. In short, SiO, CH3OH, and HNCO are relatively weak in NGC 7465 compared to other galaxies that have been studied to date. The relatively weak emission from these shock tracers is a bit surprising, given the obvious disturbances in the galaxy (Sections 2 and 5). The molecular and HI disks are strongly warped and their kinematics are significantly misaligned with respect to those of the stars, so it is clear that the gas was recently acquired from an external source. We might have expected stronger emission from shock tracers as the gas disk is still in the process of settling; possibly the shocks are primarily occurring farther out in the galaxy, where our sensitivity is not currently good enough to detect the shock tracers. ## 11 Discussion: constraints on abundances and galaxy evolution ### 11.1 CO isotopic ratios in NGC 7465 Figure 11 shows that the center of NGC 7465 boasts an unsually high 12CO/13CO line ratio; it is reminiscent of the ratios commonly observed in LIRGs and ULIRGs rather than those in spirals and other early-type galaxies. It also constrains the [12CO/13CO] abundance in the nucleus of NGC 7465 to be $\geq$ 39 $\pm$ 9, which is consistent with most measurements in the disk of the Milky Way, in other nearby spirals, and in part of the LMC (e.g. Johansson et al., 1994; Meier et al., 2008; Romano et al., 2019, and references therein). However, it is incompatible with the very low abundance ratios of [12C/13C] = 9 ($\pm$ 2), 21 ($\pm$ 6), and 24 ($\pm$ 1) estimated in the centers of NGC 4945, NGC 253, and the Milky Way (Langer & Penzias, 1990; Tang et al., 2019; Martín et al., 2019). Thus, the chemical enrichment pattern in the central molecular gas of NGC 7465 differs from the pattern in these nearby spirals. The smaller 13C abundance in the center of NGC 7465 is plausibly connected with the clear signs of gas accretion, as the gas currently in the center of NGC 7465 was recently in a different galaxy and possibly in the outskirts of that galaxy. For context, the stellar dynamical analysis of Cappellari et al. (2013) implies a maximum circular rotation speed of 163 km s-1, so the orbital time at the outer edge of the molecular arms in our data (15′′ or 2.1 kpc) is 80 Myr. It may have taken a few orbital times for the gas to make its way inward to the center of the galaxy, and this timescale is comparable to or shorter than the timescale for significant 13C production (e.g. Romano et al., 2019). Low inferred optical depths in 12CO(1-0) (see below) also imply that both 13CO and C18O are optically thin and thus the measurements of [13CO/C18O] in NGC 7465 are 2.98 $\pm$ 0.99 and 4.0 $\pm$ 1.2. These measurements are made in overlapping regions so they are not independent. The optical depths of the CO transitions also convey information about the physical properties of the molecular gas, and if the [12CO/13CO] abundance ratio in the center of NGC 7465 is in the typical range of 40–60, then the 12CO emission is unusually optically thin. We thus consider several possible explanations for the unusually high 12CO/13CO line ratio in the center of NGC 7465 and its strong radial gradient. (1) Unusual isotopic abundances due to a burst of star formation? Away from the nucleus, the 12CO/13CO and 13CO/C18O line ratios in NGC 7465 are similar to those in nearby spirals. But a very recent burst of star formation in the center of the galaxy would affect the isotopic abundances, as 13C in particular comes from low-mass stars with long lifetimes and a region undergoing a starburst might therefore be deficient in 13C. Indeed, this effect probably contributes to the exceptionally high ratios of 12CO/13CO $\approx$ 90 in the advanced merger NGC 2623 (Brown & Wilson, 2019). In NGC 7465, although there is some current star formation activity in the central kpc (Sections 2 and 4), the star formation rate is modest compared to what we usually think of as starbursts. The luminosity-weighted mean stellar ages are younger in the center of the galaxy (Krajnović et al., 2020) but neither the ages themselves nor the radial gradient in stellar ages are unusual even for early-type galaxies. Ultimately a full multi-level isotopic abundance study would be necessary to resolve the question, but with current data the motivation for assuming unusual isotopic abundances in NGC 7465 is not compelling. (2) Unusual molecular abundances due to fractionation or photodissociation of the rarer isotopes? The photodissociation of CO and H2 molecules usually proceeds through absorption-driven excitation, meaning that the molecules can shield themselves from photodissociation if they have sufficient column densities to make the relevant UV transitions optically thick. 13CO and C18O molecules should then be found deep within the dark interior of a molecular cloud, in the same way (but more so) that 12CO molecules should be found deeper than H2 and there may be a skin of CO-dark H2 around a UV-illuminated cloud. However, this isotope-specific photodissociation is probably not the explanation for the high 12CO/13CO ratio in the center of NGC 7465. The C18O species should be even less abundant than 13CO (e.g. Meier et al., 2008; Romano et al., 2019). Thus, photodissociation effects predict that unusually large 12CO/13CO line ratios should also be accompanied by unusually large 13CO/C18O ratios, which we do not find in NGC 7465 (Figure 13). We further note that the center of NGC 7465 does not exhibit particularly unusual levels of photodissociation in general, as its [C II]/CO line ratio is typical for nearby spiral galaxies (Figure 14). Finally, Viti et al. (2020) argue that fractionation is unlikely to be a significant effect for the CO isotopologues in typical conditions, though it may be more important for other molecular species. Figure 14: Matched-resolution [C II] and 12CO intensities for early-type galaxies (circles; Lapham et al., 2017; Werner et al., 2014; Wilson et al., 2013; Mittal et al., 2011, 2012). For comparison, the greyscale shows a two- dimensional histogram of the same lines in the spiral galaxies of the KINGFISH and HERACLES surveys (Kennicutt et al., 2011; Leroy et al., 2009). The angular resolution of all these data is 12′′, which is $\lesssim$ 500 pc in the spirals but $\approx$ 2 kpc in the early-type galaxies. (3) Variations in the physical properties of the gas? The spatial variation of 12CO/13CO, with higher ratios in the central 100 pc of the galaxy (Figure 6), suggests that the optical depth of both molecular species is lower in the center. For typical abundances and conditions, the optical depth of 12CO (1-0) must change by a factor of a few such that it is $>1$ at 1 kpc and $<1$, possibly as low as 0.1, at 100 pc. Thus, while the 12CO column density estimates in Table 2 are underestimates, they are not as far off as usually assumed, and they are probably less than a factor of 2 too low. Further, the inferred variations of optical depth could be plausibly explained by radial variations of the temperature and/or density of the molecular gas. Bayet et al. (2013) used the $\mathrm{J}=1-0$ and $2-1$ transitions of 12CO and 13CO to constrain the densities and temperatures of the molecular gas in 18 early-type galaxies, including NGC 7465. They found the most probable conditions for NGC 7465 to be in the range $10^{4.5}$ to $10^{5.6}$ cm-3 and 80 to 150 K. Notably, these are the highest temperatures inferred for the sample of 18. These estimates are based on single-dish spectra, so they are spatially unresolved and dominated by the conditions at small radii where the emission is brightest. However, for these densities and temperatures, calculations using the RADEX code (van der Tak et al., 2007) suggest that increases in the temperature of a factor of 3 (e.g. from 30 K to 90 K) could account for the larger 12CO/13CO 1-0 ratio in the central 100 pc of NGC 7465. Density changes could have a similar effect on 12CO/13CO but would then also produce a factor of at least 2 variation in HCN/HCO+, which we have ruled out. Therefore, unless the CO-emitting gas is completely disconnected from the HCN- emitting gas, temperature variations are more likely than density variations to explain the 12CO/13CO gradient in NGC 7465. On the other hand, Bayet et al. (2013) did not make use of resolved kinematic information or the fact that the molecular line widths all increase dramatically towards the nucleus of NGC 7465, which can also affect the line ratios and is discussed in more detail below. (4) Lower opacity due to higher line width? It is striking that the highest 12CO/13CO ratios in NGC 7465 are coincident with the largest linewidths and strongest local velocity gradients in the galaxy (Figure 5; Appendix B). The 12CO line profiles in the disk at radii of 5′′ to 20′′ (0.7 to 2.8 kpc) have full widths at half maximum (FWHM) of 23.5 $\pm$ 5 km s-1, whereas the line profiles in the nucleus are much broader and double-peaked (Figure 8). At 0.′′8 (110 pc) resolution, the line profile toward the nucleus has a FWHM of 230 km s-1; that is 10 times larger than in the disk of the galaxy. And since the optical depth of a transition depends on the ratio of the total column density to the line width (e.g. Paglione et al., 2001; van der Tak et al., 2007), such a dramatic increase in the line width could produce a corresponding factor of 10 decrease in the optical depth and a rise in the 12CO/13CO ratio. With the present data it is not possible to distinguish whether the 12CO optical depths in NGC 7465 are more significantly affected by temperature changes or increases of the line widths. Some of the line profiles in the centers of the lenticulars NGC 4710 and NGC 5866 have similarly large linewidths to NGC 7465 but they do not have unusually large 12CO/13CO ratios (Topal et al., 2016). Thus, it’s not yet clear what the difference is between these cases and NGC 7465. Additional data on higher-energy transitions would help distinguish between the possibilities. For context, some authors have suggested that mechanical feedback from star formation increases local velocity dispersions in molecular gas, produces low- density, diffuse gas, and contributes to increases in 12CO/13CO ratios near regions of active star formation (e.g. Tan et al., 2011). But in most spirals the variations of linewidths and 12CO/13CO have smaller magnitude than we find. Conversely, other studies of early-type galaxies have found that dynamically regular, relaxed disks in high-density environments like the Virgo Cluster tend to have low 12CO/13CO ratios, and those could be due to a combination of relaxed kinematics and ram-pressure stripping of low-density gas (e.g. Crocker et al., 2012; Alatalo et al., 2015). AGN-driven outflows might also disturb their surrounding molecular gas, increasing its linewidths and its 12CO/13CO line ratios (e.g. NGC 1266, Crocker et al., 2012). Further inspection of the 12CO/13CO ratios in Figure 6 shows that the ratios are high (the optical depths are low) not just towards the AGN but also in a band stretching along the minor axis of the molecular ridge. In combination with the unusually misaligned gas kinematics in the center of the galaxy (Figure 3; Appendix B), this pattern in the 12CO/13CO ratio image suggests that there might be a modest-velocity ionized gas outflow from the AGN stirring up the gas along the minor axis of the molecular ridge. ### 11.2 Implications of the CO isotopic abundances for galaxy evolution Comparisons of the stellar and gas kinematics in early-type galaxies provide clear evidence that much – perhaps as much as half – of the gas in early-type galaxies has been acquired from some outside source, after the main stellar body was in place (Sarzi et al., 2006; Davis et al., 2011; Bryant et al., 2019). Additional detailed work on the gas/dust ratios and associations with morphological disturbances suggests that most of the “outside sources” are minor mergers with mass ratios $\gtrsim 10:1$ (Kaviraj et al., 2012; Davis et al., 2015). In some cases there are also additional clues about the origin of the gas from similarities or differences between the metallicities of the gas and the stars; for gas accreted in a minor merger, one might expect the metallicity of the accreted gas to be lower than the metallicity of the bulk of the stars (Griffith et al., 2019; Davis & Young, 2019). NGC 7465 clearly did acquire its cold gas from an external source, given the kinematic mismatches between the CO, HI, and the stars. But the gas probably did not come from a dwarf galaxy in a minor merger. The large HI content (9.5$\times 10^{9}$ M⊙; Serra et al., 2012) is most consistent with the gas coming from a large spiral galaxy. Indeed, there are several other similarly gas-rich galaxies still in its group (Li & Seaquist, 1994; Serra et al., 2012). Furthermore, the metallicity of the ionized gas in NGC 7465 is consistent with that of a spiral rather than a dwarf (Section 4). And finally, the inferred CO isotopic ratios in NGC 7465 are also consistent with it having captured its molecular gas from a spiral. Its 13CO/C18O ratios are entirely unremarkable for spirals; its nuclear 12CO/13CO line ratio is a factor of 2 to 3 higher than is common in spirals, but the measurement can be plausibly attributed to high temperatures and/or large linewidths rather than to abundance variations. For other early-type galaxies with smaller gas masses, CO isotopic ratios may serve as an important complement to metallicity for identifying the original host of their external-source cold gas. Or, indeed, for early-type galaxies with relaxed prograde gas, these indicators should help identify externally- sourced cold gas even when the kinematics are inconclusive. As 13C is a slow- release element with significant production in low- to intermediate-mass stars, it may retain signatures of an external origin longer than 12C or 18O, and possibly also longer than kinematic disturbances (e.g. if the gas is now located in the interior of the galaxy where dynamical timescales are short). Figures 11 and 13, which are motivated by the discussions in Taniguchi & Ohyama (1998), Zhang et al. (2018), and Méndez-Hernández et al. (2020), illustrate that ULIRGs and sub-mm galaxies tend to have high 12CO/13CO line ratios $(\gtrsim 20)$ and low 13CO/C18O ratios $(\lesssim 2)$. In fact, there also appears to be a strong correlation between these line ratios and the total FIR luminosity of a galaxy. The cause of such a correlation is not obvious; presumably the total FIR luminosity of a galaxy is a proxy for other properties that are more fundamental. Zhang et al. (2018) argued it is a proxy for the IMF, though Romano et al. (2019) cast doubt on that interpretation given the highly uncertain production rates of 18O. In any case, a thorough understanding of the chemical evolution of galaxies requires that we should be able to connect the sub-mm galaxies at $z\approx 2-3,$ at the highest luminosities in Figures 11 and 13, to their descendants in the local universe. It is usually assumed that the sub-mm galaxies have actual abundance ratios [13CO/C18O] $\approx$ 1 to 2, just like their observed line ratios, because their high 12CO/13CO line ratios indicate that 13CO and C18O should be optically thin (e.g. Danielson et al., 2013; Sliwa et al., 2017). Over time the 13C abundance should increase due to its production in low- to intermediate-mass stars. And as the sub-mm galaxies already have stellar masses typically $10^{10}$ to $10^{11.5}$ M⊙ at $z\approx 2-3$ (Boogaard et al., 2019), their modern-day descendants will be amongst the most massive local galaxies. But large galaxies in the local universe tend to have higher ratios of [13CO/C18O] or [13C/12C][16O/18O] around 5 to 6 (e.g. Martín et al., 2019; Harada et al., 2018; Romano et al., 2019). Whether the chemical evolution models can reproduce these kinds of variations on the appropriate timescales depends a great deal on their assumptions, particularly involving the production of the rare isotopes (Romano et al., 2019). Further isotopic observations of massive early-type galaxies, particularly those that might have retained their molecular gas through the cosmic star formation peak and through their transition to the red sequence, would be useful for testing the chemical evolution models. On a related note, Bayet et al. (2012) and Davis et al. (2013) have argued that the [HCN/CO] abundance ratio might be a useful indicator of $\alpha$-element enhancement in molecular gas. As this $\alpha$ enhancement is common in the stellar populations of early-type galaxies, studies of [HCN/CO] and [HCN/CS] abundance ratios (Section 9) could provide additional chemical clues to the evolution of early-type galaxies and the origin of their gas. ## 12 Summary We present spatially-resolved molecular line observations of NGC 7465, an unusually gas-rich early-type galaxy that recently acquired $\approx 10^{10}$ M⊙ of atomic and molecular gas in an interaction with another galaxy. Its disturbed kinematics indicate that the gas is still in the process of settling and migrating inward. We analyze ALMA observations of 12CO (1-0) at 0.′′8 resolution (110 pc) plus the 3mm lines of 13CO, C18O, HCO+, HCN, CN, C2H, and CS at $\approx$ 2′′ resolution (280 pc) and the continuum emission. Besides NGC 7465, the data reveal an unidentified line source with a relatively bright peak line flux density of 2.5 mJy; it is probably a galaxy at $z=0.2$ or 1.4. We find two 3mm continuum sources in NGC 7465; the brighter one is a nuclear synchrotron source associated with the AGN that is also detected in low- frequency radio and hard X-ray emission. The fainter one is associated with molecular clouds and star formation activity, traced by local peaks in 12CO, 13CO, HCN, HCO+, and CS (though not CN). The 12CO (1-0) distribution at 0.′′8 resolution (110 pc) shows a low- inclination flocculent disk with a strong kinematic position angle twist spiraling inwards to a linear ridge or bar-like feature where the highest velocities are found. In the inner 300 pc of the galaxy, the molecular gas kinematics are misaligned by $\approx$ 120° with respect to the large-scale stellar rotation, by $\approx$ 100° (in the other direction) with respect to the stellar kinematically-decoupled core, and by about 45° with respect to the ionized gas kinematics. We conclude that the prominent stellar kinematically- decoupled core did not form out of the molecular gas present now; it must have had its origin in a previous event. Furthermore, the complex misalignments may be signatures of outflows or other non-circular kinematics in ionized gas. Despite the dramatic kinematic misalignments, there is no evidence of enhanced emission from the shock tracers SiO, CH3OH, or HNCO. All of the detected molecules are centrally concentrated except for 13CO (and possibly C18O and CS). The distribution of 13CO has a prominent central dip of almost a factor of two in integrated line intensity and it peaks at about 300 pc from the nucleus. The central 13CO dip produces an unusually large nuclear 12CO/13CO line ratio, $39\pm 9$ at this resolution. This ratio is higher than those found in typical spirals and early-type galaxies; it is comparable to those measured in ULIRGs and $z\approx 2-3$ sub-mm galaxies. It also constrains the abundance ratio [12CO/13CO] $\geq 39\pm 9,$ which is typical of spiral disks but is higher than sometimes found in spiral nuclei. The isotopic ratios are thus consistent with the gas having been accreted from another spiral galaxy and transported rapidly (faster than the 13C enrichment timescale) to the nucleus of NGC 7465. The observed 13CO/C18O line ratio suggests [13CO/C18O] = 4.0 $\pm$ 1.2, which is also consistent with a spiral galaxy origin but is significantly higher than corresponding ratios estimated in ULIRGs/sub-mm galaxies and lower than ratios in Local Group dwarfs. Modest star formation activity is occurring in the center of NGC 7465 but there is no compelling reason to assume the intrinsic [12CO/13CO] abundance ratio is very different from the range of $40-60$ that is usually found in the disks of spirals. In this case the 12CO (1-0) emission from the nucleus of NGC 7465 must be unusually optically thin, perhaps even having $\tau<1.$ Such low optical depths are plausible because of (1) the high CO temperatures $\approx 100$ K inferred from single-dish data on multiple J-level transitions, and (2) the very large linewidth in the center of the galaxy. The nuclear spectrum ($r<140$ pc) has a FWHM of 250 km s-1, a factor of 10 larger than elsewhere in the galaxy, due to the strong velocity gradient in the central misaligned molecular structure (possibly an edge-on circumnuclear ring). HCN emission is relatively faint in NGC 7465 and HCO+ is relatively bright, yielding HCN/HCO+ ratios in the range of 0.4 to 0.6 everywhere we can measure – these ratios are a factor of two lower than typical for spiral galaxies. An assumption of LTE requires an intrinsic abundance ratio [HCN/HCO+] $<0.25$ but even for densities too low to approach LTE the data still require [HCN/HCO+] $<3,$ which is smaller than is found in some nearby spirals. We also find (based on previously published nebular line fluxes) roughly solar metallicity in the ionized gas outside of the nucleus, where ionization is not dominated by the AGN. Thus, even though the galaxy’s gas-phase metallicity does not suggest an intrinsically low N abundance, the observed HCN/HCO+ line ratios do still seem to require relatively low densities in the dense molecular phase (e.g. $n_{\rm H_{2}}\leq 10^{4.3}$ cm-3) and/or low HCN abundances. We find no measurable gradient in the HCN/HCO+ line ratio on scales of 100 pc to 1 kpc. Thus, even though there is an AGN in NGC 7465, it is not affecting the J=1-0 line ratios of the dense molecular tracers on those scales (in marked contrast to some other AGN). On the other hand, the proportion of dense relative to diffuse molecular gas is clearly changing with radius, as reflected by ratios like 12CO/HCN, 12CO/HCO+, and 12CO/CN. Thus the physical properties of the densest phase of the molecular gas do not appear to change with radius, even though the properties of the more diffuse phase must be changing to reproduce the 12CO/13CO variation. The CN emission from NGC 7465 is optically thin and unremarkable in its intensity. C2H is relatively bright, with C2H/HCN $\approx$ 1\. As C2H and HCO+ have quite similar critical densities, we infer that the relative brightness of C2H is also an indicator of relatively low densities in the “high-density” molecular phase. The intensity and spatial distribution of CS emission in NGC 7465 are consistent with previous suggestions that CS can be enhanced in regions of star formation activity, here traced by low [O III]/H$\beta$ ratios. All of the gas-phase data gathered about NGC 7465 to date – the gas content, kinematics, metallicity, and molecular and isotopic abundance patterns – are consistent with the interpretation that it acquired its gas recently from a large spiral galaxy rather than a dwarf galaxy. In the broader context, however, more work is needed on isotopic abundance ratios in early-type galaxies. Some of them are believed to have accreted their gas recently from a dwarf galaxy, while others may have retained small quantities of molecular gas from their previous lives as sub-mm galaxies at $z\approx 2-3,$ through the Universe’s peak star formation epoch, all the way down to the present day. More quantitative work is also required on modeling the isotopic abundance evolution of early-type galaxies, to test whether the current-day properties of these galaxies can be reproduced and what constraints they may impose on the evolutionary models. We thank Davor Krajnović for helpful discussions on the stellar kinematics of barred galaxies. This paper makes use of the following ALMA data: ADS/JAO.ALMA#2018.1.01253.S, ADS/JAO.ALMA#2016.1.01119.S, and ADS/JAO.ALMA#2018.1.01599.S. ALMA is a partnership of ESO (representing its member states), NSF (USA) and NINS (Japan), together with NRC (Canada), MOST and ASIAA (Taiwan), and KASI (Republic of Korea), in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO and NAOJ. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc. ## Appendix A A distant, unidentified source near NGC 7465 We find strong line emission and 3mm continuum emission from an unidentified source near (in projection) to NGC 7465. The line is centered at 97.67 GHz and its width is 0.14 GHz. The source is modestly resolved in these data, and its position is 23h 02m 02.s857, $+$15° 58′ 17.′′8 (ICRS). Figure 15 shows its location on a deep optical image from the MATLAS survey (Duc et al., 2015) and Figure 16 shows its spectrum. It also has a 3mm continuum flux density of (0.11 $\pm$ 0.01) mJy/beam and a slight suggestion of rotation along a northwest – southeast axis, but the distance between the image centroids in the extreme channels is only 0.′′35 so the rotation is not well resolved in these data. Although the source is superposed on the outer parts of NGC 7465, where there are bright blue patches associated with recent star formation activity, there is no optical source coincident with the 3mm emission. It certainly is not a part of NGC 7465 due to its large linewidth and the fact that its frequency does not match any known bright line. But as we only have one detected spectral line in the frequencies covered by these data, it is difficult to identify the line. If it is 12CO(1-0), it would be at $z=0.18.$ In this case 13CO(1-0) would be around 93.4 GHz, where we have no coverage, and HCN(1-0) would be beyond the low frequency end of ALMA Band 3. If the line is 12CO(2-1), it would be at $z=1.3604$, 13CO(2-1) would be at 93.3 GHz, and HCN(3-2) would be at 112.6 GHz. No line is apparent at 112.6 GHz but the data do not rule out typical HCN/12CO ratios like those seen in local galaxies. Similarly if the line is 12CO(3-2) at $z=2.5404,$ HCN(4-3) would be at 100.1 GHz but the limits on a nondetection there are not yet useful. Firm identifications of the redshift in this case will probably require searches for the corresponding higher-J transitions of 12CO in the higher ALMA bands or the lower-J transitions at the Jansky Very Large Array. Table 3: Flux, luminosity and mass estimates for the unidentified source Line ID \| $z$ \| Diameter \| Line width \| Mdyn \| Flux \| $L^{\prime}_{\mathrm{line}}$ \| Mmol \| Mmol/Mdyn ---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| (kpc) \| (km s-1) \| (M⊙) \| (Jy km s-1) \| (K km s-1 pc2) \| (M⊙) \| 12CO(1-0) \| 0.1802 \| 1.06 \| 409 \| 5.2$\times 10^{9}$/$\sin^{2}i$ \| 0.87 \| 1.4$\times 10^{9}$ \| 4.9$\times 10^{9}$ \| 0.95 $\sin^{2}i$ 12CO(2-1) \| 1.3604 \| 2.94 \| 217 \| 4.0$\times 10^{9}$/$\sin^{2}i$ \| 0.46 \| 1.1$\times 10^{10}$ \| 4.0$\times 10^{10}$ \| 10.0 $\sin^{2}i$ 12CO(3-2) \| 2.5404 \| 2.81 \| 115 \| 1.1$\times 10^{9}$/$\sin^{2}i$ \| 0.24 \| 8.1$\times 10^{9}$ \| 2.9$\times 10^{10}$ \| 27 $\sin^{2}i$ Note. — Calculations are made assuming a flat universe with H0 = 70 km s-1 Mpc-1, $\Omega_{m}=0.3$, and $\Omega_{\Lambda}=0.7$. The linear diameter is estimated from the emission centroids in the outermost channels. The line luminosity $L^{\prime}_{\mathrm{line}}$ is calculated as in Carilli & Walter (2013). Estimated luminosity conversions from J=2-1 or J=3-2 to J=1-0 are made using assumed excitations as in Boogaard et al. (2019) and the molecular mass (with He) is then estimated from the inferred 12CO(1-0) luminosity using a conversion factor $\alpha=3.6$ M⊙ (K km s-1 pc2)-1 (Boogaard et al., 2019). Table 3 lists estimated line fluxes, luminosities, and masses for the most probable line identifications. Assuming the object is in dynamical equilibrium, so that we can compare its inferred dynamical and molecular masses, it seems more likely that the line is 12CO(1-0) or 12CO(2-1) than any higher-J transition. The higher-J levels would require the source to be quite close to face-on in order to make the inferred dynamical mass larger than the molecular mass. The line profile is consistent with a rectangular or double- horned shape, suggesting it comes from a rotating disk with gas extending to the flat part of the rotation curve, so the dynamical equilibrium assumption is plausible by this measure. The source is very similar in integrated line flux and width to the typical 3mm 12CO lines detected by ALMA in a blind survey of the Hubble Ultra-Deep Field (HUDF; González-López et al., 2019; Walter et al., 2016). Its continuum is brighter than is typical, as the brightest 3mm continuum source in the HUDF has a flux density of 46 $\pm$ 7 $\mu$Jy whereas this one has 110 $\pm$ 10 $\mu$Jy. But overall, this source is compatible with being a similar object to those. Most of them are identified as 12CO(2-1) at redshifts of 1.0 to 1.5, based on associations with optical counterparts and occasionally detections of higher-J lines in higher ALMA bands. In fact, all of the 12CO detections in the HUDF have optical counterparts (González-López et al., 2019), and they are typically 0.01 to 1 $\mu$Jy at an observed wavelength of 1 $\mu$m (Boogaard et al., 2019). The ground-based optical data shown in Figure 15 cannot rule out the faintest of those typical optical counterparts. Figure 15: Contours show the line emission from the unidentified source; levels are $(0.2,0.5,0.8)\times 3.12\times 10^{5}$ (Jy bm-1) Hz. The background is the color composite of the MATLAS $u$, $g,$ and $i$ images (Duc et al., 2015). Figure 16: 3mm spectrum of the unidentified source. Primary beam corrections have been applied. ## Appendix B Possible evidence for an AGN-driven ionized outflow Figure 17 shows a side-by-side comparison of the inner structure of NGC 7465, in some indicators that might reveal a small AGN-driven ionized outflow. Ferruit et al. (2000) presented narrowband HST [O III] and H$\alpha$ imaging which showed enhanced [O III]/H$\alpha$ in a $r\lesssim 2$′′ region with a southeast-northwest elongation, consistent with the structure in the ATLAS3D [O III]/H$\beta$ data. This region of more strongly-ionized gas is expected to trace enhanced shocks and/or harder radiation fields. The velocity of the ionized gas shows a gradient along a similar kinematic axis of roughly $-$35°, as indicated by the ellipse in Figure 17, and the velocity dispersion in [O III] is also enhanced near the ends of the major axis of this ellipse. However, the velocity resolution of the [O III] kinematic data is not good enough to show any signatures of double-peaked line profiles. The 12CO kinematic and photometric position angles in this region show that the molecular gas predominantly traces a disk or a ring perpendicular to the suggested outflow axis. Thus, enhanced [O III]/H$\beta$ ratios along that axis might simply reflect lower opacities on that axis and higher opacities in the dusty molecular disk, rather than actual outflow. The enhanced 12CO velocity dispersion along the proposed outflow axis is probably then related to the kinematics of the molecular disk rather than the outflow, and enhanced 12CO/13CO line ratios along the proposed outflow axis might reflect higher temperatures or larger linewidths (Section 11.1). In short, the features observed here could be caused by an ionized outflow but are not definitive evidence; higher resolution spectrosocopy targeting the optical nebular lines would be useful in testing for the existence of an outflow. Figure 17: The inner structure of NGC 7465. The velocities and line ratios are the same data previously shown in Figures 2, 3, and 6; the panels on the right side also show the velocity dispersions in 12CO and [O III]. Beam sizes for the ALMA data are shown as ellipses in the lower right corners. The cross marks the location of the nuclear 3mm continuum source, and the dashed ellipse (6′′ $\times$ 4′′, 850 $\times$ 570 pc) roughly indicates the region where possible outflow signatures can be seen in the optical data. ## Appendix C Line fluxes in NGC 7465. Table C presents all the line fluxes measured in the regions used for the analysis of spatial variations in the molecular properties of NGC 7465. =5.5cm Table 4: Line fluxes in the defined regions of NGC 7465 Rgn \| Dist. \| Area \| 12CO \| 13CO \| C18O \| HCN \| HCO+ \| CN \| CS \| C2H \| CH3OH \| SiO \| HNCO ---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|---\|--- ID \| (′′) \| ($\square$′′) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) \| (Jy km s-1) 1 \| 0.5 \| 3.1 \| 3.515 (0.040) \| 0.096 (0.013) \| $<$ 0.039 \| 0.132 (0.012) \| 0.303 (0.013) \| 0.226 (0.006) \| 0.041 (0.009) \| 0.091 (0.014) \| $<$ 0.035 \| $<$ 0.040 \| $<$ 0.037 2 \| 2.3 \| 8.8 \| 6.877 (0.069) \| 0.325 (0.023) \| $<$ 0.061 \| 0.128 (0.022) \| 0.219 (0.018) \| 0.122 (0.015) \| 0.087 (0.018) \| 0.104 (0.022) \| $<$ 0.057 \| $<$ 0.066 \| $<$ 0.061 3 \| 1.9 \| 5.9 \| 4.852 (0.062) \| 0.186 (0.020) \| $<$ 0.050 \| 0.092 (0.019) \| 0.222 (0.018) \| 0.162 (0.008) \| $<$ 0.041 \| $<$ 0.065 \| $<$ 0.057 \| $<$ 0.053 \| $<$ 0.058 4 \| 3.4 \| 7.4 \| 3.787 (0.053) \| 0.182 (0.018) \| $<$ 0.054 \| 0.045 (0.014) \| 0.109 (0.015) \| 0.058 (0.007) \| $<$ 0.043 \| 0.048 (0.016) \| $<$ 0.042 \| $<$ 0.047 \| $<$ 0.050 5 \| 2.9 \| 57.2 \| 30.888 (0.277) \| 1.266 (0.093) \| 0.311 (0.091) \| 0.477 (0.070) \| 1.122 (0.070) \| 0.814 (0.035) \| 0.337 (0.068) \| 0.468 (0.086) \| $<$ 0.260 \| $<$ 0.234 \| $<$ 0.222 6 \| 5.8 \| 10.0 \| 2.163 (0.047) \| 0.130 (0.014) \| $<$ 0.041 \| $<$ 0.041 \| $<$ 0.034 \| $<$ 0.018 \| $<$ 0.034 \| $<$ 0.036 \| $<$ 0.027 \| $<$ 0.039 \| $<$ 0.032 7 \| 7.9 \| 15.6 \| 4.312 (0.091) \| 0.201 (0.029) \| $<$ 0.082 \| $<$ 0.057 \| $<$ 0.067 \| $<$ 0.037 \| $<$ 0.053 \| $<$ 0.069 \| $<$ 0.060 \| $<$ 0.064 \| $<$ 0.073 8 \| 5.3 \| 3.1 \| 0.598 (0.016) \| 0.039 (0.005) \| $<$ 0.013 \| 0.019 (0.005) \| 0.040 (0.005) \| $<$ 0.007 \| $<$ 0.013 \| $<$ 0.014 \| $<$ 0.015 \| $<$ 0.017 \| $<$ 0.018 9 \| 10.0 \| 759.8 \| 80.792 (0.866) \| 2.540 (0.279) \| 0.845 (0.265) \| $<$ 0.668 \| 1.328 (0.212) \| 0.843 (0.131) \| $<$ 0.643 \| $<$ 0.602 \| $<$ 0.659 \| $<$ 0.631 \| $<$ 0.745 Note. — Regions are identified by number in Figure 7. 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# Transfer-tensor description of memory effects in open-system dynamics and multi-time statistics Stefano Gherardini<EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy LENS, CNR-INO, and QSTAR, via N. Carrara 1, I-50019 Sesto Fiorentino, Italy INFN Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy Andrea Smirne<EMAIL_ADDRESS>Institute of Theoretical Physics, and IQST, Universität Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany Dipartimento di Fisica “Aldo Pontremoli", Università degli Studi di Milano, via Celoria 16, 20133 Milan, Italy Istituto Nazionale di Fisica Nucleare, Sezione di Milano, via Celoria 16, 20133 Milan, Italy Susana Huelga<EMAIL_ADDRESS>Institute of Theoretical Physics, and IQST, Universität Ulm, Albert-Einstein-Allee 11, 89069 Ulm, Germany Filippo Caruso <EMAIL_ADDRESS>Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino, Italy LENS, CNR-INO, and QSTAR, via N. Carrara 1, I-50019 Sesto Fiorentino, Italy ###### Abstract The non-Markovianity of an arbitrary open quantum system is analyzed in reference to the multi-time statistics given by its monitoring at discrete times. On the one hand, we exploit the hierarchy of inhomogeneous transfer tensors, which provides us with relevant information about the role of correlations between the system and the environment in the dynamics. The connection between the transfer-tensor hierarchy and the CP-divisibility property is then investigated, by showing to what extent quantum Markovianity can be linked to a description of the open-system dynamics by means of the composition of 1-step transfer tensors only. On the other hand, we introduce the set of stochastic transfer tensor transformations associated with local measurements on the open system at different times and conditioned on the measurement outcomes. The use of the transfer-tensor formalism accounts for different kinds of memory effects in the multi-time statistics and allows us to compare them on a similar footing with the memory effects present in non- monitored non-Markovian dynamics, as we illustrate on a spin-boson case study. ## I Introduction In a dynamical quantum system the interaction with the external environment typically leads to non-Markovian behaviours, which, broadly speaking, are associated with the presence of memory effects in the system evolution [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22]. Although by now many distinct definitions of quantum non-Markovianity have been introduced, two main pathways can be identified [23]. In one case, the focus is on the evolution of the open-system state at different times, as fixed, e.g., by the dynamical maps or the master equations that characterize the open-system dynamics [24, 25], while in the other case the focus is on the statistics associated with a sequence of local measurements (or other active interventions) performed on the open system at subsequent times. These two approaches to non-Markovianity are indeed inherently different, as the former addresses the predictions related with observables at a single time, while the latter concerns the multi-time statistics. In addition to such a difference, which already appears at the classical level [26, 27], the quantum nature of the system at hand implies a further, fundamental distinction between the single-time and the multi-time notions of non-Markovianity. Contrary to what happens classically, in quantum systems it is generally not possible, even in principle, to access the information associated with sequential measurements without disturbing the system that is being measured. Indeed, quantum measurements alter the current state of the system, as well as its correlations with the environment, which will later result in a modified evolution [7, 9, 29, 32, 28, 30, 34, 33, 31, 35]. As a consequence, the presence of intermediate measurements can bring along specific forms of memory, which combine in a non-trivial way with the memory strictly due to the system-environment interaction. In this paper, we investigate the relation between the memory effects appearing in an open quantum system dynamics and those associated with the multi-time statistics due to sequential measurements, by means of the transfer tensor (TT) method [36, 37, 38, 39]. We will show that the latter, which was introduced to treat efficiently the long-time dynamics of open quantum systems, also allows one to treat memory effects on the dynamics and the multi-time statistics on a similar footing. First, we prove to what extent the divisibility property of the dynamical maps, which is the defining property of quantum non-Markovianity according to [7, 4], is linked to the hierarchy of TTs also in the inhomogeneous case, i.e., beyond the time-translational invariant regime explored in the original paper [36]. Then, we extend the definition of the TTs to the situation where the open system is measured at subsequent times, via the corresponding conditional stochastic dynamics. This allows us to identify different forms of memory in the multi-time statistics, related with the interplay between the correlations and environmental-state transformations induced by, respectively, the interaction between the open system and the environment and the sequential measurements. After introducing proper quantifiers related with the $L_{2}$-norm of the multi-step (i.e., step greater than $1$) TTs, we compare in a case study the relevance of such memory effects for different kinds of measurements, including the case without any intermediate measurement. The rest of the paper is organized as follows. In Sec.II, we recall the main features of the TTs formalism that are relevant for our analysis, focusing in particular on the recursive construction of the TTs. In Sec.III, we prove that if only multi-step TTs are different from zero the open-system dynamics is divisible, while the converse statement does not hold, as we show by means of an explicit example. In Sec.IV, we first generalize the TTs formalism to the case of multi-time measurements, by introducing a family of TTs conditioned on the sequence of measurement outcomes, and then we exploit it to characterize the different kinds of memory effects associated with the multi-time statistics. Finally, the conclusions and outlooks of our analysis are discussed in Sec.V ## II Inhomogeneous transfer tensors hierarchy We begin by briefly recalling the TT formalism for the dynamics of an open quantum system [36] and the microscopic characterization of the TTs recently derived in [38], which allows to build up the complete TTs hierarchy and to connect it with the system-environment correlations. Such a construction will be extended to an evolution conditioned on the occurrence of sequential measurements in the following of the paper. ### II.1 General microscopic definition of TTs Let us consider a quantum system $\mathbb{S}$ in interaction with an environment $\mathbb{E}$. The dynamics of the composite system $\mathbb{S}+\mathbb{E}$ is governed by a possibly time-dependent Hamiltonian $H(t)=H_{0}(t)+H_{\rm int}$, where $H_{\rm int}$ takes into account the interaction of the system with the environment, while $H_{0}(t)=H_{\mathbb{S}}(t)+H_{\mathbb{E}}$ concerns the uncoupled time evolution of $\mathbb{S}$ and $\mathbb{E}$. The interaction Hamiltonian $H_{\rm int}$ is fixed but unknown, and $H_{\mathbb{S}}(t)$ can be time- dependent. Assuming an initial product state between the open system and the environment and a fixed initial state of $\mathbb{E}$, i.e., $\rho_{t_{0}}=\rho_{\mathbb{S},t_{0}}\otimes\rho_{\mathbb{E},t_{0}}$, the dynamics of $\mathbb{S}$ can be described by means of the formalism of quantum dynamical maps [24, 25, 8]. In fact, one can define a one parameter family of completely positive, trace-preserving (CPTP) dynamical maps $\left\\{\Phi(t,t_{0})\right\\}_{t\geq t_{0}}$, with $\Phi(t,t_{0}):\mathcal{S}(\mathcal{H}_{\mathbb{S}})\rightarrow\mathcal{S}(\mathcal{H}_{\mathbb{S}})$, where $\mathcal{S}(\mathcal{H}_{\mathbb{S}})$ denotes the sets of density operators (non-negative operators with unit trace) acting on $\mathcal{H}_{\mathbb{S}}$ and $\Phi(t,t_{0})$ is CPTP at any time $t$. Hereafter, the action of the dynamical maps of the system will be taken into account at the discrete time instants (not necessarily equally spaced) $t_{k}$, $k=1,\ldots,m$, and we will use the notation $\Phi(t_{k},t_{0})\equiv\Phi_{k}$ and $\rho_{\mathbb{S},k}=\rho_{\mathbb{S}}(t_{k})=\Phi_{k}[\rho_{\mathbb{S},0}].$ (1) This will make the comparison with the sequential-measurement scenario more transparent and will further allow us to directly apply the TT method [36]. The latter relies in fact on a family of maps, the TTs $T_{k,j}$, which are defined by the relation $T_{k,0}=\Phi_{k}-\sum_{j=1}^{k-1}T_{k,j}\Phi_{j},$ (2) equivalently expressed by $\Phi_{k}=\sum_{j=0}^{k-1}T_{k,j}\Phi_{j}.$ (3) In other terms, the state at the time instant $t_{k}$ is obtained by propagating the states at $t_{j}$ according to the equation $\rho_{\mathbb{S},k}=\sum_{j=0}^{k-1}T_{k,j}\rho_{\mathbb{S},j}$. We stress that, at variance with the original formulation, we are not assuming time- invariance; thus our analysis will also apply to time-dependent Hamiltonians and non-stationary initial environmental states [36]. A useful expression for generic, possibly inhomogeneous TTs was recently derived in [38]. Given the global unitary dynamics of the composite system $\mathbb{S}+\mathbb{E}$ from $t_{s}$ to $t_{k}$, i.e., $\mathcal{U}_{t_{s}:t_{k}}[\rho]\equiv U_{t_{s}:t_{k}}\rho\,U_{t_{s}:t_{k}}^{\dagger}$ with $U_{t_{s}:t_{k}}\equiv\mathcal{T}\exp(-i\int_{t_{s}}^{t_{k}}H(\tau)d\tau)$ ($\mathcal{T}$ is the time ordering operator), let us introduce the CPTP maps $\Gamma_{k\|k-n}:\mathcal{S}(\mathcal{H}_{\mathbb{S}})\rightarrow\mathcal{S}(\mathcal{H}_{\mathbb{S}})$ defined as $\Gamma_{k\|k-n}[\sigma_{\mathbb{S}}]\equiv{\rm Tr}_{\mathbb{E}}\left[\mathcal{U}_{t_{k-n}:t_{k}}[\sigma_{\mathbb{S}}\otimes\sigma_{\mathbb{E},k-n}]\right],$ (4) where ${\rm Tr}_{\mathbb{S}/\mathbb{E}}[\cdot]$ denotes the partial trace w.r.t. the Hilbert space of $\mathbb{S}$ or $\mathbb{E}$, $\sigma_{\mathbb{S}}$ denotes a density operator on $\mathcal{H}_{\mathbb{S}}$ and $\sigma_{\mathbb{E},k-n}$ is a time-dependent density operator on $\mathcal{H}_{\mathbb{E}}$. Here, we will focus on the case where $\sigma_{\mathbb{E},k-n}$ is the state of the environment at $t_{k-n}$. It can then be proven that the $n-$step TT $T_{k,k-n}$ is given by the recursive relation [38] $T_{k,k-n}=\Gamma_{k\|k-n}-\sum_{j=1}^{n-1}T_{k,k-j}\Gamma_{k-j\|k-n}$ (5) that allows one to reconstruct the whole hierarchy of TTs from the CPTP maps $\Gamma_{k\|k-n}$, as shown explicitly in [38]. In the next paragraph, we repeat the construction for the lowest orders of the hierarchy, both for the sake of illustration and since it will be useful in the following when we will move to the stochastic case. ### II.2 Reconstruction of the TTs hierarchy First, let us combine together the definition of the dynamical maps of the system at the discrete time instants $t_{k}$, $k=1,\ldots,n$, i.e., Eq. (1), and the recursive general expression of the TT transformation given by Eq.(3). In this way, the latter can be written via the following recursive relation: $\Phi_{k}=\underline{T}_{k}\underline{\Xi}_{k-1}[\rho_{\mathbb{S},0}],$ (6) where $\underline{T}_{k}\equiv\left(T_{k,0},T_{k,1},\ldots,T_{k,k-1}\right)$ and $\underline{\Xi}_{k-1}[\rho_{\mathbb{S},0}]\equiv\begin{pmatrix}\mathbbm{1}_{\mathbb{S}}[\rho_{\mathbb{S},0}]\\\ \Phi_{1}=\underline{T}_{1}\underline{\Xi}_{0}[\rho_{\mathbb{S},0}]\\\ \vdots\\\ \Phi_{k-1}=\underline{T}_{k-1}\underline{\Xi}_{k-2}[\rho_{\mathbb{S},0}]\end{pmatrix},$ which are valid for $n\geq 1$ ($\underline{T}_{0}=\underline{\Xi}_{-1}=\mathbbm{1}_{\mathbb{S}}[\rho_{\mathbb{S},0}]$, identity operator on $\mathbb{S}$). For the sake of clarity, we show the first terms of the recursive expansion of Eq. (6): $\begin{cases}\Phi_{0}=\mathbbm{1}_{\mathbb{S}}[\rho_{\mathbb{S},0}]\\\ \Phi_{1}=T_{1,0}\Phi_{0}=T_{1,0}[\rho_{\mathbb{S},0}]\\\ \Phi_{2}=T_{2,0}\Phi_{0}+T_{2,1}\Phi_{1}=\left(T_{2,0}+T_{2,1}T_{1,0}\right)[\rho_{\mathbb{S},0}]\\\ \Phi_{3}=T_{3,0}\Phi_{0}+T_{3,1}\Phi_{1}+T_{3,2}\Phi_{2}\\\ \,\,\,\,\,\,\,\,\,=\left(T_{3,0}+T_{3,1}T_{1,0}+T_{3,2}T_{2,0}+T_{3,2}T_{2,1}T_{1,0}\right)[\rho_{\mathbb{S},0}].\\\ \end{cases}$ (7) By combining Eqs. (1) and (6), we immediately find that $T_{1,0}[\rho_{\mathbb{S},0}]=\Phi_{1}[\rho_{\mathbb{S},0}],$ (8) so that $\rho_{\mathbb{S},1}=T_{1,0}[\rho_{\mathbb{S},0}]$. Then, by continuing the recursion, one has $\rho_{\mathbb{S},2}=T_{2,0}[\rho_{\mathbb{S},0}]+T_{2,1}[T_{1,0}[\rho_{\mathbb{S},0}]].$ Here, it is worth observing that $T_{2,1}[T_{1,0}[\rho_{\mathbb{S},0}]]$ returns the component of $\rho_{\mathbb{S},2}$ conditioned on finding the system in the states $\rho_{\mathbb{S},0}$ and $T_{1,0}[\rho_{\mathbb{S},0}]$ at the time instants $t_{0}$ and $t_{1}$ and no system-environment correlations were present. In fact, given the maps $\Gamma_{k\|k-n}$ defined in Eq. (4), the 1-step TTs $T_{k,k-1}$ satisfy $T_{k,k-1}=\Gamma_{k\|k-1}$ (9) as a direct consequence of Eq. (5) (for the trivial case $n=1$). In other terms, we have $T_{2,0}[\rho_{\mathbb{S},0}]=\rho_{\mathbb{S},2}-\Gamma_{2\|1}[\rho_{\mathbb{S},1}],$ (10) i.e., $T_{2,0}=\Phi_{2}-\Gamma_{2\|1}\Phi_{1}.$ (11) For the sake of brevity, from here on we will remove the subscript $\mathbb{S}$ from the notation; it will used (together with $\mathbb{E}$) only if necessary. Moving on to the level $k=3$ of the hierarchy, by applying Eq. (7) one has that the reduced density operator of $\mathbb{S}$ at $t_{3}$ is equal to $\displaystyle\rho_{3}$ $\displaystyle=$ $\displaystyle T_{3,0}[\rho_{0}]+T_{3,1}[T_{1,0}[\rho_{0}]]+T_{3,2}[T_{2,0}[\rho_{0}]]$ $\displaystyle+$ $\displaystyle T_{3,2}[T_{2,1}[T_{1,0}[\rho_{0}]]].$ Using the identity in Eq. (9) for $T_{3,2}$ and $T_{2,1}$, as well as Eq. (10), we have $\begin{cases}T_{3,2}[T_{2,0}[\rho_{0}]]=\Gamma_{3\|2}[\rho_{2}-\Gamma_{2\|1}[\rho_{1}]]\\\ T_{3,2}[T_{2,1}[T_{1,0}[\rho_{0}]]]=\Gamma_{3\|2}\Gamma_{2\|1}[\rho_{1}].\end{cases}$ (12) Moreover, $T_{3,1}$ can be expressed by exploiting Eq. (5) (for $n=2$) that along with Eq. (9) gives $T_{3,1}\equiv\Gamma_{3\|1}-\Gamma_{3\|2}\Gamma_{2\|1}.$ (13) Thus, $T_{3,1}$ can be written as a function of a term involving only one-step $\Gamma$s, i.e., $\Gamma_{3\|2}\Gamma_{2\|1}$, and of the $2-$step $\Gamma_{3\|1}$. By combining together all terms of Eqs. (12) and (13), one has $\rho_{3}=T_{3,0}[\rho_{0}]+\Gamma_{3\|1}[\rho_{1}]+\Gamma_{3\|2}[\rho_{2}]-\Gamma_{3\|2}\Gamma_{2\|1}[\rho_{1}],$ i.e., $T_{3,0}=\Phi_{3}-\Gamma_{3\|2}\Phi_{2}+(\Gamma_{3\|2}\Gamma_{2\|1}-\Gamma_{3\|1})\Phi_{1}.$ (14) As mentioned, the procedure can be generalized to any $k$ via Eq. (5), so that the whole hierarchy of TTs can be expressed in terms of the maps $\Gamma_{k\|k-n}$ [38] (indeed, $\Phi_{k}=\Gamma_{k\|0}$). Importantly, this construction reveals the influence of the correlations between the system and the environment into the dynamics of the open quantum system. In fact, the CPTP maps (4) generate a hierarchy of $1-$, $2-$,…,$n-$step TTs conditioned to the fact that the system passes through product states at the different steps taken into account. As we will show in the next section, this is naturally linked to the presence of memory effects in the open-system dynamics. ## III Transfer tensors and Markovianity of the dynamics Here, we derive a definite connection between a property of the TTs – the possibility to build up the whole hierarchy only in terms of $1-$step TTs – and the non-Markovianity of the dynamics. In particular, we rely on the definition introduced in [4], which identifies Markovian dynamics as those described by a family of CP-divisible dynamical maps. Adapting the original definition in [4] to the case of a discrete set of times, then the following definition can be stated: The (discrete) dynamics $\left\\{\Phi_{k}\right\\}_{k=1,\ldots,m}$ is Markovian when for any $k\geq j\geq 0$ there is a CPTP map $\mathcal{E}_{k,j}$ such that $\Phi_{k}=\mathcal{E}_{k,j}\Phi_{j}$. Hence, if we assume that $T_{k,k-n}=0$ for $n\geq 2$, then by applying recursively Eq. (3) we have $\Phi_{k}=T_{k,k-1}T_{k-1,k-2}\cdots T_{j+1,j}\Phi_{j}$ (15) for any $k\geq j\geq 0$. But then Eq. (9) implies $\Phi_{k}=\Gamma_{k\|k-1}\Gamma_{k-1\|k-2}...\Gamma_{j+1\|j}\Phi_{j}.$ (16) Any conditional map $\Gamma_{k\|j}$ is CPTP by construction, see Eq.(4), so that we can conclude that any map $\Phi_{k}$ can be decomposed as $\Phi_{k}=\mathcal{E}_{k,j}\Phi_{j}$, with $\mathcal{E}_{k,j}$ CPTP map: _If only one-step TTs are different from zero, the resulting (discrete-time) evolution is CP-divisible_. We further note that if we restrict to the case of equally-spaced time instants, $t_{k}=k\Delta$, and translational invariant TTs (i.e., $T_{k,k-n}=T_{n,0}$), then $T_{k,k-n}=0$ for $n\geq 2$ implies that the open- system dynamics is not only Markovian, but also a semigroup, i.e., $\Phi_{k}=(\Phi_{1})^{k}$, thus recovering what shown in [36]. The previous result, besides linking a property of the hierarchy of TTs to the Markovianity of the corresponding dynamics, provides us with an explicit illustration of one of the physical meanings of such property. In fact, from Eq.(16) we see that when one-step TTs are the only non-zero TTs, the dynamical maps can be obtained by using the maps $\Gamma$ only. System-environment correlations, despite being present as a consequence of the interaction term within the unitary operators, will not affect the reduced dynamics of the open system, so that at any time $t_{k-n}$ the actual global state can be effectively replaced by a product state $\rho_{\mathbb{S},k-n}\otimes\sigma_{\mathbb{E},k-n}$, see Eq.(4). Experimentally, Eq. (16) can be validated by independently measuring the maps $\\{\Phi_{k}\\}$ and the 1-step $\\{\Gamma_{k,k-1}\\}$, obtained by breaking the system-environment correlations at the time instants $t_{k}$. In particular, the former can be measured by means of quantum tomography processes, while the latter, e.g., by preparing a fresh copy of the system in the state $\sigma_{\mathbb{S},k}$ [13], which is known by virtue of the previously reconstructed map $\Phi_{k}$. The fact that only one-step TTs are non-zero is in general a stronger requirement than the CP-divisibility of the dynamics, as follows from the analysis of [23] (see also [40]) and the explicit example provided below. Indeed, having only one-step TTs different from zero is only a _sufficient criterion_ (and thus not necessary) for CP-divisible discrete-time evolutions. In this regard, let us consider the simplest case, namely $m=2$, whereby we recall that $m$ is the last (and greater) value of the index $k$ in the sequence of time instants $t_{k}$. Hence, the only non-trivial requirement for CP-divisibility is the existence of a CPTP map $\mathcal{E}_{2,1}$ obeying the relation $\mathcal{E}_{2,0}=\Phi_{2}=\mathcal{E}_{2,1}\Phi_{1}=\mathcal{E}_{2,1}T_{1,0}$ since by definition $T_{1,0}\equiv\Phi_{1}$. Now, according to the definition of the TT transformation, $\Phi_{2}$ is also equal to $\Phi_{2}=T_{2,0}+T_{2,1}T_{1,0}.$ Therefore, assuming that the inverse of $T_{1,0}$ (i.e., $(T_{1,0})^{-1}$) does exist, one has $\mathcal{E}_{2,1}=T_{2,0}\left(T_{1,0}\right)^{-1}+T_{2,1}.$ (17) What we will show is that $\mathcal{E}_{2,1}$ can be CPTP also if $T_{2,0}\neq 0$ and even if $\left(T_{1,0}\right)^{-1}$ is not be generally CPTP. In fact, consider the reduced dynamics of a two-level system, fixed by the unitary dynamics given by the global Hamiltonian $H=\sigma_{z}+H_{\mathbb{E}}+\sigma_{z}\otimes B$, where the free environmental Hamiltonian $H_{\mathbb{E}}$ and the interaction term commute: $[H_{\mathbb{E}},B]=0$. Now, also assume that the global state at the initial time $t_{0}=0$ is $\rho(0)=\rho_{\mathbb{S}}(0)\otimes\rho_{\mathbb{E}}(0)$, where the environmental state is stationary with respect to the free dynamics such that $[H_{\mathbb{E}},\rho_{\mathbb{E}}]=0$. As a consequence, on the one hand, the reduced dynamics of the open system is a pure dephasing, namely $\rho_{11}(t)=\rho_{11}(0)$ and $\rho_{10}(t)=k(t)\rho_{10}(0)$, where $\rho_{ij}(t)=\langle i\|\rho_{\mathbb{S}}(t)\|j\rangle$ ($\left\\{\lvert 1\rangle,\lvert 0\rangle\right\\}$ are the eigenvectors of $\sigma_{z}$) and $k(t)=e^{i\omega t}\mbox{tr}_{\mathbb{E}}[e^{2iBt}\rho_{\mathbb{E}}]$. On the other hand, the state of the environment will not evolve in time, also when interacting with the system, i.e., $\rho_{\mathbb{E}}(t)=\rho_{\mathbb{E}}(0)$. For any two times $t_{2}\geq t_{1}\geq 0$, we will then have $T_{2,1}=\Gamma_{2\|1}=\Phi_{1},$ where we used Eq.(9) and Eq.(4), respectively. As a result (see also Eq.(11)), $T_{2,0}=\Phi_{2}-\Phi^{2}_{1}.$ Specifically, using the matrix representation in the basis $\left\\{\lvert 1\rangle,\lvert 0\rangle\right\\}$, for a generic state $\rho_{\mathbb{S}}$ we have $\mathcal{E}_{2,1}[\rho_{\mathbb{S}}]=\begin{pmatrix}\rho_{11}&\frac{k(t_{2})}{k(t_{1})}\rho_{10}\\\ \frac{k^{}(t_{2})}{k^{}(t_{1})}\rho_{01}&\rho_{00}\end{pmatrix},$ which is positive definite for any $\rho_{\mathbb{S}}$ if and only if $\|k(t_{2})\|\leq\|k(t_{1})\|$. Note that this implies that $\mathcal{E}_{2,1}$ is also CPTP under the same condition, since for pure dephasing positivity and complete positivity coincide. On the other hand, $T_{2,0}[\rho_{\mathbb{S}}]=\begin{pmatrix}0&(k(t_{2})-k(t_{1})^{2})\rho_{10}\\\ (k^{}(t_{2})-k^{}(t_{1})^{2})\rho_{01}&0\end{pmatrix}$ that is equal to 0 for any $\rho_{\mathbb{S}}$ if and only if $k(t_{2})=k(t_{1})^{2}$. The latter condition implies but is not implied by the former (indeed, $\|k(t)\|\leq 1$), which proves our claim: $\mathcal{E}_{2,1}$ can be CPTP, and hence the dynamics CP-divisible, even though the $2$-step transfer tensor $T_{2,0}$ is different from zero. ## IV Transfer tensors and multi-time statistics In this paragraph, we move on to the second part of our investigation, where we consider an open system that, besides interacting with the environment, is measured at some discrete instants of time. We will first introduce a proper counterpart of the transfer tensors in such dynamical regime, and will then show how it can be used to account for the memory present in the multi-time statistics. Before that, let us specify the framework we refer to. Assume that the open quantum system $\mathbb{S}$ is monitored at the same $m$ instants of time where the discrete dynamics has been evaluated so far. In particular, we take a sequence of quantum measurements locally performed on $\mathbb{S}$ according to the observables $\mathcal{O}_{k}\equiv F_{\theta_{k}}\otimes I_{\mathbb{E}}$; $\\{\theta_{k}\\}$ is the set of the possible measurement outcomes, and $\\{F_{\theta_{k}}\\}$ denotes the set of positive semi-definite operators on $\mathcal{H}_{\mathbb{S}}$ satisfying the relation $\sum_{\theta_{k}}F_{\theta_{k}}=\mathbbm{1}_{\mathbb{S}}$ $\forall k$. The probability that the outcome $\theta_{k}$ associated with the measurement operator $F_{\theta_{k}}$ occurs is equal to ${\rm Tr}[\rho_{\mathbb{S},k}F_{\theta_{k}}]$, while the post-measurement state of $\mathbb{S}$ equals to $\widehat{\rho}_{\mathbb{S},k}\equiv\mathcal{M}_{\theta_{k}}[\rho_{\mathbb{S},k}]/{\rm Tr}[\mathcal{M}_{\theta_{k}}[\rho_{\mathbb{S},k}]]$, where we introduced the super-operator $\mathcal{M}_{\theta_{k}}[\rho_{\mathbb{S},k}]\equiv M_{\theta_{k}}\rho_{\mathbb{S},k}M^{\dagger}_{\theta_{k}}$ and $M_{\theta_{k}}$ fulfills the identity $F_{\theta_{k}}=M^{\dagger}_{\theta_{k}}M_{\theta_{k}}$. Note that $\rho_{\mathbb{S},k}$ ($\widehat{\rho}_{\mathbb{S},k}$) denotes the state of the system before (after) the measurement at the time step $k$. Importantly, a measurement of the open system affects not only its current state, but also its future evolution as a consequence of the change in the correlations between the system and the environment due to the measurement itself [9]. The very notion of dynamics of the open system becomes more subtle, since it cannot be clearly separated from the results of the sequential measurements. It is thus useful to introduce the notion of conditional dynamics, whereby the system admits a different dynamics for each sequence of measurement outcomes. Explicitly, the conditional dynamics of the system (for a given global evolution $\mathcal{U}_{t_{s}:t_{k}}$ and initial environmental state $\rho_{\mathbb{E},0}$) is fixed by the CP map $\displaystyle\widetilde{\Phi}^{\underline{\theta},\underline{t}}_{k}[\rho_{\mathbb{S},0}]$ $\displaystyle\equiv$ $\displaystyle{\rm Tr}_{\mathbb{E}}\left[\mathcal{U}_{t_{k-1}:t_{k}}(\mathcal{M}_{\theta_{k-1}}\otimes\mathbbm{1}_{\mathbb{E}})\cdots\right.$ (18) $\displaystyle\cdots$ $\displaystyle\left.(\mathcal{M}_{\theta_{1}}\otimes\mathbbm{1}_{\mathbb{E}})\,\mathcal{U}_{t_{0}:t_{1}}[\rho_{\mathbb{S},0}\otimes\rho_{\mathbb{E},0}]\right],$ depending on the time instants $\underline{t}\equiv(t_{1},\ldots,t_{k})$ as well as on the measurement outcomes $\underline{\theta}\equiv(\theta_{1},\ldots,\theta_{k-1})$. In other terms, $\widetilde{\Phi}^{\underline{\theta},\underline{t}}_{k}$ has to be understood as a stochastic map, so that we could effectively define different trajectories in the set of CP maps, each of them associated with a different sequence of measurement outcomes. In general $\widetilde{\Phi}^{\underline{\theta},\underline{t}}_{k}$ is not trace preserving. The joint probability distributions to get the measurement outcomes $\theta_{1},\theta_{2},\ldots,\theta_{k}$ at the time instants $t_{1},t_{2},\ldots,t_{k}$ is directly linked to the stochastic map $\widetilde{\Phi}^{\underline{\theta},\underline{t}}_{k}$ by the following relation: $q_{k}\equiv{\rm Prob}(\theta_{k},t_{k};\ldots;\theta_{1},t_{1})={\rm Tr}[\mathcal{M}_{\theta_{k}}\widetilde{\Phi}^{\underline{\theta},\underline{t}}_{k}[\rho_{\mathbb{S},0}]].$ (19) These quantities define the multi-time statistics associated to sequential measurements at different times and, as recalled in the introduction, suitable notions of Markovianity can be attributed to them [29, 23, 15, 13, 14, 28, 30, 34, 31]. ### IV.1 Conditional transfer tensors As anticipated, in order to characterize the conditional dynamics originated by different sequences of outcomes, we can introduce a stochastic version of the TTs. Let us denote them as $\widetilde{T}^{\underline{\theta},\underline{t}}_{k,j}$, with $k,j=1,\ldots,m$, explicitly pointing out their dependence on the instants and outcomes of the repeated measurements. The basic idea is to express the definition of the conditional dynamical map in a recursive fashion. As first step, note that the global state after the first measurement is proportional (via the normalization factor) to $\widehat{\rho}_{1}\sim(M_{\theta_{1}}\otimes I_{\mathbb{E}})\mathcal{U}_{t_{0}:t_{1}}[\rho_{0}](M_{\theta_{1}}^{\dagger}\otimes I_{\mathbb{E}})\equiv\mathcal{V}_{t_{0}:t_{1}}[\rho_{0}]$ (20) that also provides the formal definition of the map $\mathcal{V}[\rho]$. Accordingly, the stochastic quantum map of $\mathbb{S}$ at time $t_{1}$, i.e., $\widetilde{\Phi}_{1}$, can be implicitly linked to $\mathcal{V}_{t_{0}:t_{1}}[\rho_{0}]$ through the relation $\mathcal{M}_{\theta_{1}}\widetilde{\Phi}_{1}[\rho_{\mathbb{S},0}]\equiv{\rm Tr}_{\mathbb{E}}(\mathcal{V}_{t_{0}:t_{1}}[\rho_{0}]).$ (21) Analogously, the $k-$th dynamical map of $\mathbb{S}$ after a sequence of quantum measurements is defined by $\displaystyle\widehat{\rho}_{k}$ $\displaystyle\sim$ $\displaystyle(M_{\theta_{k}}\otimes I_{\mathbb{E}})\mathcal{U}_{t_{k-1}:t_{k}}[\widehat{\rho}_{k-1}](M_{\theta_{k}}^{\dagger}\otimes I_{\mathbb{E}})$ (22) $\displaystyle\equiv$ $\displaystyle\mathcal{V}_{t_{k-1}:t_{k}}[\widehat{\rho}_{k-1}]$ with $\mathcal{M}_{\theta_{k}}\widetilde{\Phi}_{k}[\rho_{\mathbb{S},0}]={\rm Tr}_{\mathbb{E}}(\mathcal{V}_{t_{k-1}:t_{k}}[\widehat{\rho}_{k-1}])$ (23) and $\widehat{\rho}_{0}\equiv\rho_{0}$. For the sake of clarity, we recall that, by definition, the CP maps $\widetilde{\Phi}_{k}$ are obtained by tracing the state of the composite system $\mathbb{S}+\mathbb{E}$ (w.r.t. the environment $\mathbb{E}$) just after applying the evolution superoperator $\mathcal{U}_{t_{k-1}:t_{k}}$. As done with the definition of transfer tensors linking together the quantum maps $\Phi_{k}$ of the system at time instants $t_{k}$, $k=1,\ldots,m$, we can thus introduce the stochastic transformations $\widetilde{T}^{\underline{\theta},\underline{t}}_{k,j}$ that relate the conditional dynamics of $\mathbb{S}$ after each measurement of the sequence. In particular, Eq. (23) has the same structure as the not monitored dynamics, with $\Phi_{k}$ replaced by $\mathcal{M}_{\theta_{k}}\widetilde{\Phi}_{k}$. Hence, we define the stochastic TTs via $\widetilde{\Phi}^{\underline{\theta},\underline{t}}_{k}=\sum_{j=0}^{k-1}\widetilde{T}^{\underline{\theta},\underline{t}}_{k,j}\mathcal{M}_{\theta_{j}}\widetilde{\Phi}^{\underline{\theta},\underline{t}}_{j}\,.$ (24) Note that if there are no measurements, $\mathcal{M}_{\theta_{k}}=\mathbbm{1}$ and $\widetilde{\Phi}^{\underline{\theta},\underline{t}}_{k}=\Phi_{k}$ (see Eq. (18)) with the result that the TTs $\widetilde{T}^{\underline{\theta},\underline{t}}_{k,j}$ are no longer conditional objects and can be identified with the $T_{k,j}$ since Eq. (24) reduces to Eq. (3). From now on, we will drop the label $(\cdot)^{\underline{\theta},\underline{t}}$, which is implied in all the expressions with a tilde. As before, by expanding Eq. (24) we can write the expression for the $k-$th dynamical map of $\mathbb{S}$ as a function of $\rho_{\mathbb{S},0}$, i.e., $\widetilde{\Phi}_{k}=\underline{\widetilde{T}}_{k}\underline{\mathcal{M}}_{\theta_{k-1}}\underline{\Upsilon}_{k-1}[\rho_{\mathbb{S},0}],$ (25) where $\displaystyle\underline{\widetilde{T}}_{k}$ $\displaystyle\equiv$ $\displaystyle\left(\widetilde{T}_{k,0},\widetilde{T}_{k,1},\ldots,\widetilde{T}_{k,k-1}\right)$ $\displaystyle\underline{\mathcal{M}}_{\theta_{k-1}}$ $\displaystyle\equiv$ $\displaystyle{\rm diag}\left(\mathcal{M}_{\theta_{0}},\mathcal{M}_{\theta_{1}},\ldots,\mathcal{M}_{\theta_{k-1}}\right)$ (26) and $\underline{\Upsilon}_{k-1}[\rho_{\mathbb{S},0}]\equiv\begin{pmatrix}\mathbbm{1}_{\mathbb{S}}[\rho_{\mathbb{S},0}]\\\ \widetilde{\Phi}_{1}=\underline{\widetilde{T}}_{1}\underline{\Upsilon}_{0}[\rho_{\mathbb{S},0}]\\\ \widetilde{\Phi}_{2}=\underline{\widetilde{T}}_{2}\underline{\mathcal{M}}_{\theta_{1}}\underline{\Upsilon}_{1}[\rho_{\mathbb{S},0}]\\\ \vdots\\\ \widetilde{\Phi}_{k-1}=\underline{\widetilde{T}}_{k-1}\underline{\mathcal{M}}_{\theta_{k-2}}\underline{\Upsilon}_{k-2}[\rho_{\mathbb{S},0}]\end{pmatrix}$ (27) with $\underline{\Upsilon}_{-1}$ and $\mathcal{M}_{\theta_{0}}$ equal to the identity operator $\mathbbm{1}_{\mathbb{S}}$ (the first measurement of the sequence, indeed, is not performed at time $t_{0}$, but at $t_{1}$) and $\underline{\Upsilon}_{0}=\underline{\Xi}_{0}=\Phi_{0}$. For the sake of clarity, we show also in this case the first terms of the recursive expansion of Eq. (25): $\begin{cases}\widetilde{\Phi}_{0}=\Phi_{0}=\mathbbm{1}_{\mathbb{S}}[\rho_{\mathbb{S},0}]\\\ \widetilde{\Phi}_{1}=\widetilde{T}_{1,0}\widetilde{\Phi}_{0}=\widetilde{T}_{1,0}[\rho_{\mathbb{S},0}]\\\ \widetilde{\Phi}_{2}=\widetilde{T}_{2,0}\widetilde{\Phi}_{0}+\widetilde{T}_{2,1}\mathcal{M}_{\theta_{1}}\widetilde{\Phi}_{1}=\left(\widetilde{T}_{2,0}+\widetilde{T}_{2,1}\mathcal{M}_{\theta_{1}}\widetilde{T}_{1,0}\right)[\rho_{\mathbb{S},0}]\\\ \widetilde{\Phi}_{3}=\widetilde{T}_{3,0}\widetilde{\Phi}_{0}+\widetilde{T}_{3,1}\mathcal{M}_{\theta_{1}}\widetilde{\Phi}_{1}+\widetilde{T}_{3,2}\mathcal{M}_{\theta_{2}}\widetilde{\Phi}_{2}\\\ \,\,\,\,\,\,\,\,\,=\left(\widetilde{T}_{3,0}+\widetilde{T}_{3,1}\mathcal{M}_{\theta_{1}}\widetilde{T}_{1,0}+\widetilde{T}_{3,2}\mathcal{M}_{\theta_{2}}\widetilde{T}_{2,0}\right.\\\ \,\,\,\,\,\,\,\,\left.+\,\widetilde{T}_{3,2}\mathcal{M}_{\theta_{2}}\widetilde{T}_{2,1}\mathcal{M}_{\theta_{1}}\widetilde{T}_{1,0}\right)[\rho_{\mathbb{S},0}].\end{cases}$ In addition, let us observe that also the multi-time statistics for $\mathbb{S}$, given by the joint probability distribution in Eq.(19), can be expressed by means of these recursive relations. Indeed, by using again Eq. (25), one has that $q_{k}={\rm Tr}\left(\mathcal{M}_{\theta_{k}}\underline{\widetilde{T}}_{k}\underline{\mathcal{M}}_{\theta_{k-1}}\underline{\Upsilon}_{k-1}[\rho_{\mathbb{S},0}]\right),$ (28) $\forall k=1,\ldots,m$, where the first terms of the recursion for $q_{k}$ are given by $\begin{cases}q_{0}={\rm Tr}\left(\rho_{\mathbb{S},0}\right)=1\\\ q_{1}={\rm Tr}\left(\mathcal{M}_{\theta_{1}}\widetilde{T}_{1,0}[\rho_{\mathbb{S},0}]\right)\\\ q_{2}={\rm Tr}\left(\mathcal{M}_{\theta_{2}}\widetilde{T}_{2,0}[\rho_{\mathbb{S},0}]\right)+{\rm Tr}\left(\mathcal{M}_{\theta_{2}}\widetilde{T}_{2,1}\mathcal{M}_{\theta_{1}}\widetilde{T}_{1,0}[\rho_{\mathbb{S},0}]\right)\\\ q_{3}={\rm Tr}\left(\mathcal{M}_{\theta_{3}}\widetilde{T}_{3,0}[\rho_{\mathbb{S},0}]\right)+{\rm Tr}\left(\mathcal{M}_{\theta_{3}}\widetilde{T}_{3,1}\mathcal{M}_{\theta_{1}}\widetilde{T}_{1,0}[\rho_{\mathbb{S},0}]\right)\\\ \,\,\,\,\,\,\,+\,\,{\rm Tr}\left(\mathcal{M}_{\theta_{3}}\widetilde{T}_{3,2}\mathcal{M}_{\theta_{2}}\widetilde{T}_{2,0}[\rho_{\mathbb{S},0}]\right)\\\ \,\,\,\,\,\,\,+\,\,{\rm Tr}\left(\mathcal{M}_{\theta_{3}}\widetilde{T}_{3,2}\mathcal{M}_{\theta_{2}}\widetilde{T}_{2,1}\mathcal{M}_{\theta_{1}}\widetilde{T}_{1,0}[\rho_{\mathbb{S},0}]\right).\end{cases}$ Finally, also the set $\\{\widetilde{T}\\}$ of stochastic TTs admits a hierarchic structure, very similar to that of Eq. (5), based on a generalization of the CPTP maps $\Gamma$s. The main difference is that now we need a map acting on the reduced dynamics of $\mathbb{S}$ at each $t_{k}$ after that a quantum measurement has been performed on the system. Thus, such map will depend on the resulting post-measurement environmental state. In particular, following the same construction as in [38], we get $\widetilde{T}_{k,k-n}=\widetilde{\Gamma}_{k\|k-n}-\sum_{j=1}^{n-1}\widetilde{T}_{k,k-j}\mathcal{M}_{\theta_{k-j}}\widetilde{\Gamma}_{k-j\|k-n},$ (29) where the CPTP maps $\widetilde{\Gamma}_{k\|k-n}[\sigma_{\mathbb{S}}]={\rm Tr}_{\mathbb{E}}[\mathcal{U}_{t_{k-n}:t_{k}}[\sigma_{\mathbb{S}}\otimes\widetilde{\sigma}_{\mathbb{E},k-n}]]$ (30) also include a dependence on the whole sequence of outcomes, fixing the environmental state $\widetilde{\sigma}_{\mathbb{E},k-n}$ (which motivates the use of the tilde). ### IV.2 Conditional transfer tensors and memory effects The relation in Eq. (29) allows us to bring forward the connection between TTs and CP-divisibility to the stochastic level that describes the effects sequential measurements. In particular, let us consider the situation where, for any sequence of outcomes, all the stochastic TTs $\widetilde{T}_{k,k-n}$ are equal to 0 for $n\geq 2$. The stochastic TTs hierarchy of Eq. (29) implies that each $1$-step transfer tensor $\widetilde{T}_{k,k-1}$ is equal to the corresponding $1$-step $\widetilde{\Gamma}$, i.e., $\widetilde{T}_{k,k-1}=\widetilde{\Gamma}_{k\|k-1}$, which by definition is completely positive. In addition, recursively applying $\widetilde{\Phi}_{k}=\sum_{j=0}^{k-1}\widetilde{T}_{k,j}\mathcal{M}_{\theta_{j}}\widetilde{\Phi}_{j}$, we get $\widetilde{\Phi}_{k}=\widetilde{T}_{k\|k-1}\mathcal{M}_{\theta_{k-1}}\widetilde{T}_{k-1\|k-2}\mathcal{M}_{\theta_{k-2}}\cdots\widetilde{T}_{j+1\|j}\mathcal{M}_{\theta_{j}}\widetilde{\Phi}_{j}$ (31) for any $k\geq j\geq 0$. Therefore, we have $\widetilde{\Phi}_{k}=\widetilde{\Gamma}_{k\|k-1}\mathcal{M}_{\theta_{k-1}}\widetilde{\Gamma}_{k-1\|k-2}\mathcal{M}_{\theta_{k-2}}\cdots\widetilde{\Gamma}_{j+1\|j}\mathcal{M}_{\theta_{j}}\widetilde{\Phi}_{j},$ (32) which implies that for any $k\geq j\geq 0$ there is a (conditional) CP map $\widetilde{\mathcal{E}}_{k,j}$ such that $\widetilde{\Phi}_{k}=\widetilde{\mathcal{E}}_{k,j}\widetilde{\Phi}_{j}$ i.e., the family of maps $\widetilde{\Phi}_{k}$ is CP-divisibile [41]. We conclude that, if all the stochastic TTs $\widetilde{T}_{k,k-n}$ are equal to 0 for $n\geq 2$, the corresponding conditional dynamics $\\{\widetilde{\Phi}_{k}\\}_{k=1,\ldots,m}$ is CP-divisible for any fixed sequence of outcomes. As one can directly check from the definition in Eq. (18), the condition in Eq. (32) is satisfied whenever the global state after the measurements is a product state, e.g., if the set $\\{\mathcal{M}_{\theta_{k}}\\}$ describe projective measurements of a non-degenerate system’s observables. This means that in these situations the conditional dynamics $\widetilde{\Phi}_{k}$ will be automatically CP-divisible, simply due to the kind of employed measurement. As we will see in the next paragraph by means of an explicit example, CP- divisibility of the conditional dynamics, and in particular the validity of Eq. (32), does not imply that only one-step conditional TTs are different from zero. The stochastic TTs $\widetilde{T}_{k,k-n}$ with $n\geq 2$ enclose the influence of the possible correlations resulting both from the system- environment interaction and the measurement on the open system up to a time step $j$ on the conditional dynamics at the later time step $k$. In general, such correlations depend on the whole sequence of measurement outcomes performed up to the $j-$th time instant, so that their influence on the subsequent conditional dynamics can be read as a signature of memory in the multi-time statistics defined by Eq. (19). However, even in the case where Eq. (32) holds, some memory can be present in the conditional dynamics, and hence in the multi-time statistics. In fact, the post-measurement environmental state $\widetilde{\sigma}_{\mathbb{E}}$, which defines the conditional maps $\widetilde{\Gamma}$ in Eq. (30), might (and in general will) depend on the sequence of outcomes. As we will see in the next paragraph, the proper description of such memory effects can require non-zero TTs $\widetilde{T}_{k,k-n}$ with $n\geq 2$. Finally, it is worth observing that, if in addition to the validity of Eq. (32) the dependence of $\widetilde{\sigma}_{\mathbb{E}}$ on the previous outcomes can be neglected (i.e., no memory of the sequence of measurements is left), Eq. (32) automatically implies the quantum regression theorem [42, 43, 44, 45, 46]. Essentially, the latter means that the whole multi-time statistics can be reconstructed only from the initial open-system state and open-system dynamical maps that are not conditioned on the sequence of outcomes. Similar considerations have been recently discussed also in [15]. ### IV.3 Spin-boson case study In this paragraph, we show an application of the previously introduced stochastic transfer tensors, with the aim to illustrate the capability of the method in giving a quantitative account of the memory effects both in the conditional dynamics and in the multi-time statistics. In addition, the use of TTs also allows to compare on a similar footing the memory effects in the presence of different kinds of measurements, or even no intermediate measurements at all. We consider a single spin in interaction with $5$ quantum harmonic oscillators and monitored by a sequence of $M=5$ quantum measurements at regular intervals $\Delta$ with $t_{k}=k\Delta$ and $k=1,\ldots,M$. The Hamiltonian $H$ of the composite system is given by $H=\frac{1}{2}\sigma^{z}_{\mathbb{S}}+\sum_{k=1}^{5}\omega_{\mathbb{E},k}b_{\mathbb{E},k}^{\dagger}b_{\mathbb{E},k}+\sum_{k=1}^{5}g_{k}(\sigma_{\mathbb{S}}^{-}b_{\mathbb{E},k}^{\dagger}+\sigma_{\mathbb{S}}^{+}b_{\mathbb{E},k})$ (33) where $\sigma^{z}$ is the Pauli matrix in the $z$ direction, $b_{\mathbb{E}}^{\dagger}$ and $b_{\mathbb{E}}$ denote respectively the creation and annihilation operators associated to each harmonic oscillator of $\mathbb{E}$, and $\sigma_{\mathbb{S}}^{+}$ and $\sigma_{\mathbb{S}}^{-}$ are the raising and lowering operators of the spin $\mathbb{S}$. Moreover, regarding the frequencies $\omega_{\mathbb{E},k}$ of the oscillators and the interaction couplings $g_{k}$, for the numerical simulations here performed we have chosen the values $\\{\omega_{\mathbb{E},k}\\}_{k=1}^{5}\approx(1.99,0.73,0.89,2.04,1.58)$ and $\\{g_{k}\\}_{k=1}^{5}\approx(1.67,1.32,2.15,2.70,1.07)$. All these values are expressed in units ensuring that $\hbar=1$. The former have been uniformly sampled from the interval $[0,5]$, while the latter are sampled from the probability distribution $\alpha g\exp(-g/\beta)$, with $\alpha=1$ and $\beta=2$. The parameters $\alpha$ and $\beta$ tune the intensity, respectively, of the interaction between $\mathbb{S}$ and $\mathbb{E}$ and of the value of $g$ that corresponds to the peak of the distribution. Note that this model accounts for both decoherence and dissipation in the dynamics of $\mathbb{S}$ due to the interaction with the external environment, yet with the form of the interaction preserving the total number of excitations. On the other hand, the excitations are generally not preserved by the action of the measurements. In our simulations, each measurement at $t=t_{k}$ is provided by the following POVM operators [47]: $F_{\pm}\equiv(1-\lambda)\|\pm\rangle\\!\langle\pm\|+\frac{\lambda}{2}\mathbbm{1}_{\mathbb{S}}$ (34) where $\|\pm\rangle\equiv(\|0\rangle\pm\|1\rangle)/\sqrt{2}$ denote the eigenstates of the Pauli matrix $\sigma^{x}$. This choice is motivated by the fact that, by changing the parameter $\lambda$, it is possible to analyze the case without measurements ($\lambda=1$), the one with projective measurements ($\lambda=0$) and all the intermediate cases ($0<\lambda<1$) corresponding to a partial collapse of the spin wave-function. The evolution of the composite system has been evaluated starting from an initial product state of the ground states of both $\mathbb{S}$ and $\mathbb{E}$, and taking the time interval $\Delta$ between measurements equal to 1, 2 and 3 (in natural units allowing for $\hbar=1$). Conversely, the hierarchy of transfer tensors has been derived by numerically implementing Eq. (29). The latter is based on the computation of the maps $\widetilde{\Gamma}_{k\|k-n}$, which are all obtained by propagating an initial product state composed by the reduced states of $\mathbb{S}$ and $\mathbb{E}$ after each possible quantum measurement at the time instants $t_{k}$. Figure 1: Spin-boson dynamics: Ground state population $\rho_{\mathbb{S}}^{(1,1)}$ (blues solid lines) and coherence terms $\|\rho_{\mathbb{S}}^{(1,2)}\|$ (red dash-dotted lines) of the single spin $\mathbb{S}$ as a function of time and $4$ distinct $\lambda$ values for $\Delta=1$ in natural units. Depending on the value of $\lambda$ that fixes the intermediate measurement operator, the coherence and population of the spin has a different behaviour. In particular, revivals occur for $\lambda=1$, while a more and more pronounced damping is observed for decreasing values of $\lambda$. Indeed, sudden changes in the population and coherence terms denote the action of the measurements, and they are more evident in the case of projective measurements ($\lambda=0$). Figure 2: Comparison between the $L_{2}$-norm of the stochastic TTs $\widetilde{T}_{k,0}$ (coloured dots) and the figure of merit $\mathcal{D}_{k}$ (circles, squares, diamonds and x-marks) for $k=2,\ldots,5$ (corresponding to the discrete time instants $t_{2},\ldots,t_{5}$) and $\lambda=0,0.25,0.5,1$ (identified, respectively, by the colour black, red, blue and magenta). The former quantifier takes into account the influence of the $k$-step TTs all referring to $t=t_{0}$, while the latter sums up the impact of all the involved $k-\ell$ steps TTs with $\ell=0,\ldots,k-2$ and they both account for the non-Markovianity of the implemented quantum dynamics. If all the stochastic TTs $\widetilde{T}_{k,k-n}$ are equal to 0 for $n\geq 2$, then both quantifiers are equal to zero (the opposite does not generally hold). In Fig. 1 we plot both the population and coherence terms of the single spin $\mathbb{S}$ as a function of time and $\lambda$ values ($\lambda\in\\{0,0.25,0.5,1\\}$) for $\Delta=1$ (all in natural units). As one can observe from the figure, depending on the value of $\lambda$ that corresponds to different intermediate measurement operators, the spin quantum coherence tends to decrease for smaller values of $\lambda$ or to be maintained oscillating over time otherwise. In the latter case, repeated “coherence revivals” are originated by the interaction of $\mathbb{S}$ with only the environment $\mathbb{E}$ composed of 5 quantum harmonic oscillators. Instead, in the former case also the external measurement apparatus plays a crucial role. Hence, from Fig. 1 one can clearly appreciate that monitoring the open quantum system $\mathbb{S}$ through a sequence of quantum measurements radically changes its reduced dynamics, as well as the corresponding non-Markovian behaviors. In particular, if the quantum coherence of $\mathbb{S}$ is damping over time, then also high-order step memory effects can be considered negligible. Otherwise, each state of $\mathbb{S}$ also depends on past contributions, even up to the initial instant in the extreme case. Let us now analyze more quantitatively these considerations by introducing proper quantifiers of non-Markovianity. In this regard, in Fig. 2 we plot two quantifiers of the impact of the stochastic $n$-step TTs, with $n\geq 2$, on the conditional dynamics of the system. In each panel, we consider different POVMs, defined as in Eq. (34) for $\lambda=0,0.25,0.5,1$ and identified respectively by the colours black, red, blue and magenta, as a function of the discrete time instants $t_{k}$ with $k=1,\ldots,5$. The three panels correspond to three different values of the time interval $\Delta$. The two quantifiers are the $L_{2}$-norm $\\|\cdot\\|_{2}$ of the stochastic transfer tensors $\widetilde{T}_{k,0}$ (coloured dots in the figure) and the figure of merit (circles, squares, diamonds and x-marks) $\mathcal{D}_{k}\equiv\frac{1}{k-2}\sum_{\ell=0}^{k-2}\\|\widetilde{T}_{k,\ell}\mathcal{M}_{\theta_{\ell}}\widetilde{\Phi}_{\ell}\\|_{2}.$ (35) Starting from the decomposition of the conditional map $\widetilde{\Phi}_{k}$ in correspondence of $t=t_{k}$ as given in Eq. (24), one can observe that the former quantifier focuses on the role of the $k$-step TT for $k\geq 2$ (note, indeed, that $\widetilde{\Phi}_{0}=\mathbbm{1}$, i.e., no measurement is performed at the initial time instant), while the latter sums up the influence of all the involved $k-\ell$ steps TTs with $k\geq 2$ and $\ell=0,\ldots,k-2$ (thus, the $1$-step TT is not considered). If all the stochastic TTs $\widetilde{T}_{k,k-n}$ are equal to 0 for $n\geq 2$, then both quantifiers are equal to zero; the opposite does not generally hold. Overall, from Fig. 2 we can observe that the quantities $\\|\widetilde{T}_{k,0}\\|_{2}$ and $\mathcal{D}_{k}$ are comparable; namely, in the considered parametric regime the $k$-th step TT provides with similar information on the memory effects influencing $\widetilde{\Phi}_{k}$, compared to the sum of all the contributions from the TTs with step larger than 1. They both indicate that non-Markovian behaviours, originated from the interaction of the spin both with the environment and the observer, become relevant for $t/\Delta\geq 3$ with all the considered values of $\Delta$. Moreover, both the non-Markovianity quantifiers reach their maximum values in the case where no intermediate measurement is performed (magenta dots and x-marks), corresponding in Fig. 1 to a nearly periodic evolution of the coherence term $\|\rho_{\mathbb{S}}^{(1,2)}\|$ over time. Thus, comparing Figs. 1 and 2, we can deduce that such periodic behaviours are originated by high-order step memory effects, presumably even from $t_{0}$. While without applying any measurement both quantifiers typically increase with time, a more pronounced non-monotonic behavior is observed in the presence of intermediate measurements, which reduce the memory effects caused by the interaction between $\mathbb{S}$ and $\mathbb{E}$. Although a decreased impact of multi-step TTs in the presence of intermediate measurements (corresponding to a decreased impact of system- environment correlations on the conditional dynamics) might have been expected, the quantifiers do not necessarily reach their minimum values in the case of projective measurements ($\lambda=0$). Rather, the minimum values are reached more often for $\lambda=0.25$ or $\lambda=0.5$ (the behavior for these two POVMs is quite similar). However, as anticipated, $n$-step TTs for $n\geq 2$ can be non-zero also in the case of projective measurements, i.e., when Eq. (32) holds. Such TTs account for memory effects due to the dependence of the post-measurement environmental state on the previous outcomes. Even more, the latter can exceed the memory effects – due to both system-environment correlations and changes in the environmental states – occurring in the case of non-projective measurements, as quantified by $\\|\widetilde{T}_{k,0}\\|_{2}$ and $\mathcal{D}_{k}$. Figure 3: $L_{2}$-norm of the difference between the left- and the right-and- side of equation (32), namely $\\|\widetilde{\Phi}_{k}-\widetilde{\Gamma}_{k\|k-1}\mathcal{M}_{\theta_{k-1}}\widetilde{\Gamma}_{k-1\|k-2}\mathcal{M}_{\theta_{k-2}}\cdots\widetilde{\Gamma}_{j+1\|j}\mathcal{M}_{\theta_{j}}\widetilde{\Phi}_{j}\\|_{2}$, computed for $\Delta=2$ and $\lambda=0,0.25,0.5,1$ (magenta dots, blue x-marks, red diamonds, black squares, respectively) in correspondence of the discrete time-instants $t_{k}=k\Delta$ with $k\in\\{2,3,4,5\\}$. This represents the $L_{2}$-norm of the difference between the actual stochastic map $\widetilde{\Phi}_{k}$ and the CP-divisible map provided by the product states of the dynamical evolution of the spin after each measurement. During the transient of the spin dynamics, Eq. (32) is violated as the value of $\lambda$ increases (thus, for less invasive intermediate measurements); however, such tendency can be reversed in case the spin approaches an equilibrium asymptotic state or a stable periodicity phase. In order to point out the memory effects originated by system-environment correlations, we consider the $L_{2}$-norm of the difference between the left- and the right-and-side of Eq. (32), namely between the actual stochastic map $\widetilde{\Phi}_{k}$ (in correspondence of the $k$-th time instant $t_{k}$) and the CP-divisible map $\widetilde{\Gamma}_{k\|k-1}\mathcal{M}_{\theta_{k-1}}\widetilde{\Gamma}_{k-1\|k-2}\mathcal{M}_{\theta_{k-2}}\cdots\widetilde{\Gamma}_{j+1\|j}\mathcal{M}_{\theta_{j}}\widetilde{\Phi}_{j}$ associated with the product states after each measurement. The results are plotted in Fig. 3. It is worth noting that Eq. (32) is satisfied only when projective measurements are performed, so that the composite system is actually in a product state after each measurement and the conditional dynamics of $\mathbb{S}$ can be fully described by means of the composition of $1$-step maps $\widetilde{\Gamma}$. In this case, the computation of the operators $\widetilde{\Gamma}_{k\|k-1}$ is enough by itself. However, as said, some $n$-step stochastic TTs with $n\geq 2$ can be still different from zero, despite overall their combination is canceled to ensure the validity of Eq. (32). As final remark, from Fig. 3 we can also observe that for $k=2,3,4$ the violation of Eq. (32) increases with the value of $\lambda$, i.e., when moving toward less invasive measurements; indeed, the largest violation is reached for the case without intermediate measurements. Moreover, quite interestingly, the situation is reversed in the correspondence of the last observed value of time, $t_{k}/\Delta=5$, for which $\lambda=0.25$ induces the largest violation, while the unconditional dynamics the smallest (apart, of course, from the case of projective measurements). The latter behaviour of the unconditional dynamics is usually originated by a tendency towards an equilibrium asymptotic state or to a stable periodicity phase. In our case study, being the environment composed by a finite number (five) of quantum harmonic oscillators, the simulated unconditional dynamics is practically periodic, with period approximately of $t=10$ (in natural units). ## V Conclusion and outlook In this paper, we have studied the non-Markovian dynamics of open quantum systems and their multi-time statistics by means of the transfer-tensor approach, which allowed us to treat the memory effects in the two different scenarios on a similar footing. After showing the connection between the hierarchy of the TTs and the divisibility of the dynamics, we have extended the definition of TTs to the case where the open system of interest undergoes quantum measurements at subsequent instants of time. We have thus introduced a stochastic family of TTs, depending on the sequence of measurement outcomes, and a related hierarchy that captures how the multi-time statistics is influenced by both the system-environment correlations and the dependence of the current environmental state on the previous outcomes. Finally, we have defined two quantifiers of multi-time memory effects, which rely directly on the various contributions to the hierarchy of TTs, and we have investigated their behavior in a paradigmatic case study, comparing different kinds of intermediate measurements, as well as the case with no monitoring at all. The precise relation between the memory effects present, respectively, in non- Markovian quantum dynamics and in their multi-time statistics remains still to be addressed. In particular, more quantitative and general results characterizing different kinds of multi-time statistics, such as those obeying the quantum-regression theorem, seem to be needed in order to distinguish the different sources of memory and to compare them with those leading to non- Markovian quantum dynamics. Hopefully, our results will provide useful insights to look for such rigorous connections. More in general, our analysis suggests the usefulness of the TT approach also to deal with multi-time statistics; an example in this direction might be to use the hierarchy of stochastic TTs to evaluate the non-Markovianity along the so-called most probable trajectory of the system, namely the trajectory originated with higher probability by a sequence of quantum measurements [48, 49]. In addition, the TT formalism addressed in this paper is also expected to provide a powerful tool to evaluate experimentally the degree of non-Markovianity [50] in monitored quantum dynamics [51, 52], as well as to test the validity of the quantum regression theorem or possible generalizations [53, 54, 11]. One further advantage of the presented approach would be to track memory effects also in the reduced state evolution of the environment $\mathbb{E}$. This feature might be useful in those quantum technology devices (e.g., even the commercial ones now available online as IBM or Rigetti [55, 56, 57]) in which the number of the environmental degrees of freedom is not much larger than the ones of the system $\mathbb{S}$ under analysis. Accordingly, in such a case, also the state of $\mathbb{E}$ is subjected to frequent and relevant changes due to the presence of $\mathbb{S}$, and the bi-directional exchange of information between the system and the environment likely leads to departures from Markovian dynamics and the quantum regression theorem. Finally, it is worth also mentioning how stochastic transfer tensors may find application in modeling the interaction between a quantum system $\mathbb{S}$ and one or more external thermal baths, giving rise to thermalization processes in the large- time limit. During the transient of the dynamics, indeed, $\mathbb{S}$ may exhibit non-Markovian behaviours and non-trivial memory effects. In this regard, if for example we resort to quantum collision models [58, 59], a thermal baths is assumed of being composed by a large number of small subsystems, such that, at discrete time instants, the dynamics between the quantum system and the bath take place through successive “collisions” provided by pairwise short interactions. Similar thermalization effects are also obtained by monitoring $\mathbb{S}$ through a sequence of projective measurements, in the limit of many measurements [60]. 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# De Giorgi’s inequality for the thresholding scheme with arbitrary mobilities and surface tensions Tim Laux Institut für angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany<EMAIL_ADDRESS>and Jona Lelmi Institut für angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany<EMAIL_ADDRESS> ###### Abstract. We provide a new convergence proof of the celebrated Merriman-Bence-Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoğlu and Otto and De Giorgi’s general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation. ## 1\. Introduction The thresholding scheme is a highly efficient computational scheme for multiphase mean curvature flow (MCF) which was originally introduced by Merriman, Bence, and Osher [25], [26]. The main motivation for MCF comes from metallurgy where it models the slow relaxation of grain boundaries in polycrystals [29]. Each ”phase” in our mathematical jargon corresponds to a grain, i.e., a region of homogeneous crystallographic orientation. The effective surface tension $\sigma_{ij}(\nu)$ and the mobility $\mu_{ij}(\nu)$ of a grain boundary depend on the mismatch between the lattices of the two adjacent grains $\Omega_{i}$ and $\Omega_{j}$ and on the relative orientation of the grain boundary, given by its normal vector $\nu$. It is well known that for small mismatch angles, the dependence on the normal can be neglected [30]. The effective evolution equations then read (1) $\displaystyle V_{ij}=-\mu_{ij}\sigma_{ij}H_{ij}\quad\text{along the grain boundary }\Sigma_{ij},$ where $V_{ij}$ and $H_{ij}$ denote the normal velocity and mean curvature of the grain boundary $\Sigma_{ij}=\partial\Omega_{i}\cap\partial\Omega_{j}$, respectively. These equations are coupled by the Herring angle condition (2) $\displaystyle\sigma_{ij}\nu_{ij}+\sigma_{jk}\nu_{jk}+\sigma_{ki}\nu_{ki}=0\quad\text{along triple junctions }\Sigma_{ij}\cap\Sigma_{jk},$ which is a balance-of-forces condition and simply states that triple junctions are in local equilibrium. Efficient numerical schemes allow to carry out large-scale simulations to give insight into relevant statistics like the average grain size or the grain boundary character distribution, as an alternative to studying corresponding mean field limits as in [4], [16]. The main obstruction to directly discretize the dynamics (1)–(2) are ubiquitous topological changes in the network of grain boundaries like for example the vanishing of grains. Thresholding instead naturally handles such topological changes. The scheme is a time discretization which alternates between the following two operations: _(i)_ convolution with a smooth kernel; _(ii)_ thresholding. The second step is a simple pointwise operation and also the first step can be implemented efficiently using the Fast Fourier Transform. One of the main objectives of our analysis is to rigorously justify this intriguingly simple scheme in the presence of such topological changes. The basis of our analysis is the underlying gradient-flow structure of (1)–(2), which means that the solution follows the steepest descent in an energy landscape. More precisely, the energy is the total interfacial area weighted by the surface tensions $\sigma_{ij}$, and the metric tensor is the $L^{2}$-product on normal velocities, weighted by the inverse mobilities $\frac{1}{\mu_{ij}}$. One can read off this structure from the inequality $\displaystyle\frac{d}{dt}\sum_{i,j=1}^{N}\sigma_{ij}\textup{Area}(\Sigma_{ij})=-\sum_{i,j=1}^{N}\frac{1}{\mu_{ij}}\int_{\Sigma_{ij}}V_{ij}^{2}\,dS\leq 0,$ which is valid for sufficiently regular solutions to (1)–(2). In the seminal work [8], Esedoğlu and Otto showed that the efficient thresholding scheme respects this gradient-flow structure as it may be viewed as a minimizing movements scheme in the sense of De Giorgi. More precisely, they show that each step in the scheme is equivalent to solving a variational problem of the form (3) $\displaystyle\min_{\chi}\left\\{\frac{1}{2h}d_{h}^{2}(\Sigma,\Sigma^{n-1})+E_{h}(\Sigma)\right\\},$ where $E_{h}(\Sigma)$ and $d_{h}(\Sigma,\Sigma^{n-1})$ are proxies for the total interfacial energy of the configuration $\Sigma$ and the distance of the configuration $\Sigma$ to the one at the previous time step $\Sigma^{n-1}$, respectively. Since the work of Jordan, Kinderlehrer, and Otto [14], the importance of the formerly often neglected metric in such gradient-flow structures has been widely appreciated. Also in the present work, the focus lies on the metric, which in the case of MCF is well-known to be completely degenerate [27]. This explains the proxy for the metric appearing in the related well-known minimizing movements scheme for MCF by Almgren, Taylor, and Wang, [1], and Luckhaus and Sturzenhecker [23]. This remarkable connection between the numerical scheme and the theory of gradient flows has the practical implication that it made clear how to generalize the algorithm to arbitrary surface tensions $\sigma_{ij}$. From the point of view of numerical analysis, (3) means that thresholding behaves like the _implicit_ Euler scheme and is therefore unconditionally stability. The variational interpretation of the thresholding scheme has of course implications for the analysis of the algorithm as well. It allowed Otto and one of the authors to prove convergence results in the multiphase setting [17], [18], which lies beyond the reach of the more classical viscosity approach based on the comparison principle implemented in [9], [3], [12]. Also in different frameworks, this variational viewpoint turned out to be useful, such as MCF in higher codimension [21] or the Muskat problem [13]. The only downside of the generalization [8] are the somewhat unnatural effective mobilities $\mu_{ij}=\frac{1}{\sigma_{ij}}$. Only recently, Salvador and Esedoğlu [31] have presented a strikingly simple way to incorporate a wide class of mobilities $\mu_{ij}$ as well. Their algorithm is based on the fact that although the same kernel appears in the energy and the metric, each term only uses certain properties of the kernel, which can be tuned independently: Starting from two Gaussian kernels $G_{\gamma}$ and $G_{\beta}$ of different width, they find a positive linear combination $K_{ij}=a_{ij}G_{\gamma}+b_{ij}G_{\beta}$, whose effective mobility and surface tension match the given $\mu_{ij}$ and $\sigma_{ij}$, respectively. It is remarkable that this algorithm retains the same simplicity and structure as the previous ones [26], [8]. We refer to Section 2 for the precise statement of the algorithm. In the present work, we prove the first convergence result for this new general scheme. We exploit the gradient-flow structure and show that under the natural assumption of energy convergence, any limit of thresholding satisfies De Giorgi’s inequality, a weak notion of multiphase mean curvature flow. This assumption is inspired by the fundamental work of Luckhaus-Sturzenhecker [23] and has appeared in the context of thresholding in [17], [18]. We expected it to hold true before the onset of singularities such as the vanishing of grains. Furthermore, at least in the simpler two-phase case, it can be verified for certain singularities [6], [5]. We would in fact expect this assumption to be true generically, which however seems to be a difficult problem in the multiphase case. The present work fits into the theory of general gradient flows even better than the two previous ones [17], [18] and crucially depends on De Giorgi’s abstract framework, cf. [2]. This research direction was initiated by Otto and the first author and appeared in the lecture notes [19]. There, De Giorgi’s inequality is derived for the simple model case of two phases. Here, we complete these ideas and use a careful localization argument to generalize this result to the multiphase case. A further particular novelty of our work is that for the first time, we prove the convergence of the new scheme for arbitrary mobilities [31]. Our proof rests on the fact that thresholding, like any minimizing movements scheme, satisfies a sharp energy-dissipation inequality of the form (4) $\displaystyle E_{h}(\Sigma^{h}(T))+\frac{1}{2}\int_{0}^{T}\left(\frac{1}{h^{2}}d_{h}^{2}(\Sigma^{h}(t),\Sigma^{h}(t-h))+\|\partial E_{h}\|^{2}(\tilde{\Sigma}^{h}(t))\right)dt\leq E_{h}(\Sigma(0)),$ where $\Sigma^{h}(t)$ denotes the piecewise constant interpolation in time of our approximation, $\tilde{\Sigma}^{h}(t)$ denotes another, intrinsic interpolation in terms of the variational scheme, cf. Lemma 3, and $\|\partial E_{h}\|$ is the metric slope of $E_{h}$, cf. (33). Our main goal is to pass to the limit in (4) and obtain the sharp energy- dissipation relation for the limit, which in the simple two-phase case formally reads (5) $\displaystyle\sigma\mathrm{Area}(\Sigma(T))+\frac{1}{2}\int_{0}^{T}\int_{\Sigma(t)}\left(\frac{1}{\mu}V^{2}+\sigma^{2}\mu H^{2}\right)dS\,dt\leq\sigma\mathrm{Area}(\Sigma(0)).$ To this end, one needs sharp lower bounds for the terms on the left-hand side of (4). While the proof of the lower bound on the metric slope of the energy (6) $\displaystyle\liminf_{h\downarrow 0}\int_{0}^{T}\|\partial E_{h}\|^{2}(\tilde{\Sigma}^{h}(t))\,dt\geq\sigma^{2}\mu\int_{0}^{T}\int_{\Sigma(t)}H^{2}dS\,dt$ is a straight-forward generalization of the argument in [19], the main novelty of the present work lies in the sharp lower bound for the distance-term of the form (7) $\displaystyle\liminf_{h\downarrow 0}\int_{0}^{T}\frac{1}{h^{2}}d_{h}^{2}(\Sigma^{h}(t),\Sigma^{h}(t-h))\,dt\geq\frac{1}{\mu}\int_{0}^{T}\int_{\Sigma(t)}V^{2}\,dS\,dt.$ This requires us to work on a mesoscopic time scale $\tau\sim\sqrt{h}$, which is much larger than the microscopic time-step size $h$ and which is natural in view of the parabolic nature of our problem. It is remarkable that De Giorgi’s inequality (5) in fact characterizes the solution of MCF under additional regularity assumptions. Indeed, if $\Sigma(t)$ evolves smoothly, this inequality can be rewritten as (8) $\displaystyle\frac{1}{2}\int_{0}^{T}\int_{\Sigma(t)}\sigma\Big{(}\frac{1}{\sqrt{\mu\sigma}}V+\sqrt{\sigma\mu}H\Big{)}^{2}dS\,dt\leq 0,$ and therefore $V=-\mu\sigma H$. For expository purpose, we focused here on the vanilla two-phase case. In the multiphase case, the resulting inequality implies both the PDEs (1) and the balance-of-forces conditions (2), cf. Remark 1. An optimal energy-dissipation relation like the one here also plays a crucial role in the recent weak-strong uniqueness result for multiphase mean curvature flow by Fischer, Hensel, Simon, and one of the authors [10]. There, a new dynamic analogue of calibrations is introduced and uniqueness is established in the following two steps: _(i)_ any strong solution is a calibrated flow and _(ii)_ every calibrated flow is unique in the class of weak solutions. De Giorgi’s general strategy we are implementing here is also related to the approaches by Sandier and Serfaty [32] and Mielke [28]. They provide sufficient conditions for gradient flows to converge in the same spirit as $\Gamma$-convergence of energy functionals, implies the convergence of minimizers. In the dynamic situation it is clear that one needs conditions on both energy and metric in order to verify such a convergence. There has been continuous interest in MCF in the mathematics literature, so we only point out some of the most relevant recent advances. We refer the interested reader to the introductions of [17] and [20] for further related references. The existence of global solutions to multiphase MCF has only been established recently by Kim and Tonegawa [15] who carefully adapt Brakke’s original construction and show in addition that phases do not vanish spontaneously. For the reader who wants to familiarize themselves with this topic, we recommend the recent notes [34]. Another approach to understanding the long-time behavior of MCF flow is to restart strong solutions after singular times. This amounts to solving the Cauchy problem with non-regular initial data, such as planar networks of curves with quadruple junctions. In this two-dimensional setting, this has been achieved by Ilmanen, Neves, and Schulze [11] by gluing in self-similarly expanding solutions for which it is possible to show that the initial condition is attained in some measure theoretic way. Most recently, using a similar approach of gluing in self- similar solutions, but also relying on blow-ups from geometric microlocal analysis, Lira, Mazzeo, Pluda, Saez [22] were able to construct such strong solutions, prove stronger convergence towards the initial (irregular) network of curves, and classify all such strong solutions. The rest of the paper is structured as follows. In Section 2 we recall the thresholding scheme for arbitrary mobilitites introduced in [31], show its connection to the abstract framework of gradient flows, and record the direct implications of this theory. We state and discuss our main results in Section 3. Section 4 contains the localization argument in space, which will play a crucial role in the proofs which are gathered in Section 5. Finally, in the short Appendix, we record some basic facts about thresholding. ## 2\. Setup and the modified thresholding scheme ### 2.1. The modified algorithm We start by describing the algorithm proposed by Salvador and Esedoğlu in [31]. Let the symmetric matrix $\mathbb{\sigma}=(\sigma_{ij})_{ij}\in\mathbf{R}^{N\times N}$ of surface tensions and the symmetric matrix $\mathbb{\mu}=(\mu_{ij})_{ij}$ of mobilities be given. In this work we define for notational convenience $\sigma_{ii}=\mu_{ii}=0$. Let $\gamma>\beta>0$ be given. Define the matrices $\mathbb{A}=(-a_{ij})_{ij}\in\mathbf{R}^{N\times N}$ and $\mathbb{B}=(-b_{ij})_{ij}\in\mathbf{R}^{N\times N}$ by (9) $\displaystyle a_{ij}=\frac{\sqrt{\pi}\sqrt{\gamma}}{\gamma-\beta}(\sigma_{ij}-\beta\mu_{ij}^{-1}),$ (10) $\displaystyle b_{ij}=\frac{\sqrt{\pi}\sqrt{\beta}}{\gamma-\beta}(-\sigma_{ij}+\gamma\mu_{ij}^{-1}),$ for $i\neq j$ and $a_{ii}=b_{ii}=0$. Then $a_{ij},b_{ij}$ are uniquely determined as solutions of the following linear system (11) $\begin{cases}\sigma_{ij}=\frac{a_{ij}\sqrt{\gamma}}{\sqrt{\pi}}+\frac{b_{ij}\sqrt{\beta}}{\sqrt{\pi}},\\\ \mu_{ij}^{-1}=\frac{a_{ij}}{\sqrt{\pi}\sqrt{\gamma}}+\frac{b_{ij}}{\sqrt{\pi}\sqrt{\beta}}.\end{cases}$ The algorithm introduced by Salvador and Esedoğlu is as follows. Let the time step size $h>0$ be fixed. Hereafter $G_{\gamma}^{h}:=G_{\gamma h}^{(d)}$ denotes the $d$-dimensional heat kernel (17) at time $\gamma h$. ###### Algorithm 1 (Modified thresholding scheme). Given the initial partition $\Omega_{1}^{0},...,\Omega_{N}^{0}$, to obtain the partition $\Omega_{1}^{n+1},...,\Omega_{N}^{n+1}$ at time $t=h(n+1)$ from the partition $\Omega_{1}^{n},...,\Omega_{N}^{n}$ at time $t=hn$ 1. (1) For any $i=1,...,N$ form the convolutions $\phi^{n}_{1,i}=G_{\gamma}^{h}\mathbf{1}_{\Omega_{i}^{n}},\ \phi^{n}_{2,i}=G_{\beta}^{h}\mathbf{1}_{\Omega_{i}^{n}}$ 2. (2) For any $i=1,...,N$ form the comparison functions $\psi^{n}_{i}=\sum_{j\neq i}a_{ij}\phi^{n}_{1,j}+b_{ij}\phi^{n}_{2,j}.$ 3. (3) Thresholding step, define $\Omega_{i}^{n+1}:=\left\\{x:\psi_{i}^{n}(x)<\min_{j\neq i}\psi_{j}^{n}(x)\right\\}.$ We will assume the following: (12) $\displaystyle\text{The coefficients}\ a_{ij},b_{ij}\ \text{satisfy the strict triangle inequality.}$ (13) $\displaystyle\text{The matrices}\ \mathbb{A}\ \text{and}\ \mathbb{B}\ \text{are positive definite on}\ (1,...,1)^{\perp}.$ In particular, for $v\in(1,...,1)^{\perp}$ we can define norms $\displaystyle\|v\|_{\mathbb{A}}^{2}=v\cdot\mathbb{A}v,\ \|v\|_{\mathbb{B}}^{2}=v\cdot\mathbb{B}v.$ Observe that condition (12) is always satisfied if we choose $\gamma$ large and $\beta$ small provided the surface tensions and the inverse of the mobilities satisfy the strict triangle inequality. Indeed define $\displaystyle m_{\sigma}=\min_{i,j,k}\\{\sigma_{ik}+\sigma_{kj}-\sigma_{ij}\\}\ \text{and}\ M_{\sigma}=\max_{i,j,k}\\{\sigma_{ik}+\sigma_{kj}-\sigma_{ij}\\},$ where $i,j,k$ range over all triples of distinct indices $1\leq i,j,k,\leq N$. Define $m_{\frac{1}{\mu}}$ and $M_{\frac{1}{\mu}}$ in a similar way. Then a computation shows that $a_{ij}$ and $b_{ij}$ satisfy the (strict) triangle inequality if (14) $\beta<\frac{m_{\sigma}}{M_{\frac{1}{\mu}}}\ \text{and}\ \gamma>\frac{M_{\sigma}}{m_{\frac{1}{\mu}}},$ which can always be achieved since $\gamma>\beta>0$ are arbitrary. For the second condition (13), we have the following result of Salvador and Esedoğlu [31]. ###### Lemma 1. Let the matrix $\sigma$ of the surface tensions and the matrix $\frac{1}{\mu}$ of the inverse mobilities (for the diagonal we set inverses to be zeros) be negative definite on $(1,...,1)^{\perp}$. Let $\gamma>\beta$ be such that (15) $\gamma>\frac{\min_{i=1,...,N-1}s_{i}}{\max_{i=1,...,N-1}m_{i}},\ \beta<\frac{\max_{i=1,...,N-1}s_{i}}{\min_{i=1,...,N-1}m_{i}}$ where $s_{i}$ and $m_{i}$ are the nonzero eigenvalues of $J\sigma J$ and $J\frac{1}{\mu}J$ respectively, where the matrix $J$ has components $J_{ij}=\delta_{ij}-\frac{1}{N}$. Then $\mathbb{A}$ and $\mathbb{B}$ are positive definite on $(1,...,1)^{\perp}$. In particular, if we choose $\gamma$ large enough and $\beta$ small enough, condition (13) on the matrices $\mathbb{A},\mathbb{B}$ is satisfied provided the matrices $\sigma$ and $\frac{1}{\mu}$ are negative definite on $1,...,1)^{\perp}$. By a classical result of Schoenberg [33] this is the case if and only if $\sqrt{\sigma_{ij}}$ and $1/\sqrt{\mu_{ij}}$ are $\ell^{2}$ embeddable. In particular, this holds for the choice of Read-Schockley surface tensions and equal mobilities. For $1\leq i\neq j\leq N$ define the kernels (16) $K_{ij}(z)=a_{ij}G_{\gamma}(z)+b_{ij}G_{\beta}(z)$ where, for a given $t>0$, we define $G^{(d)}_{t}$ as the heat kernel in $\mathbf{R}^{d}$, i.e., (17) $G^{(d)}_{t}(z)=\frac{e^{-\frac{\|z\|^{2}}{4t}}}{\sqrt{4\pi t}^{d}}.$ If the dimension $d$ is clear from the context, we suppress the superscript $(d)$ in (17). We recall here some basic properties of the heat kernel. (18) $\displaystyle G_{t}(z)>0\ \text{(non-negativity)},$ (19) $\displaystyle G_{t}(z)=G_{t}(Rz)\ \forall R\in O(d)\ \text{(symmetry)},$ (20) $\displaystyle G_{t}(z)=\frac{1}{\sqrt{t}^{d}}G_{1}\left(\frac{z}{\sqrt{t}}\right)\ \text{(scaling)},$ (21) $\displaystyle G_{t}G_{s}=G_{t+s}\ \text{(semigroup property)},$ (22) $\displaystyle G_{t}^{(d)}(z)=\prod_{i=1}^{d}G_{t}^{(1)}(z_{i})\ \text{(factorization property)}.$ We observe that the kernels $K_{ij}$ are positive, with positive Fourier transform $\hat{K}_{ij}$ provided $\gamma>\max_{i,j}\sigma_{i,j}\mu_{i,j}$ and $\beta<\min_{i,j}\sigma_{i,j}\mu_{i,j}$. In particular assuming 1. (1) $\sigma_{ij}$ and $\frac{1}{\mu_{ij}}$ satisfy the strict triangle inequality, 2. (2) $\sigma$ and $\frac{1}{\mu}$ are negative definite on $(1,...,1)^{\perp}$, we can always achieve the conditions posed on $\mathbb{A},\mathbb{B}$ and the positivity of the kernels $K_{ij}$ by choosing $\gamma$ large and $\beta$ small. Given any $h>0$ we define the scaled kernels (23) $K^{h}_{ij}(z)=\frac{1}{\sqrt{h}^{d}}K_{ij}(\frac{z}{\sqrt{h}}),$ then the first and the second step in Algorithm 1 may be compactly rewritten as follows $\displaystyle\psi_{i}^{n}=\sum_{j\neq i}K_{ij}^{h}\mathbf{1}_{\Omega_{j}^{n}}.$ For later use, we also introduce the kernel (24) $K(z)=\frac{1}{2}G_{\gamma}(z)+\frac{1}{2}G_{\beta}(z).$ ### 2.2. Connection to De Giorgi’s minimizing movements The first observation is that Algorithm 1 has a minimizing movements interpretation. To explain this, let us introduce the class $\mathcal{A}:=\left\\{\chi:[0,1)^{d}\to\\{0,1\\}^{N}\biggr{\rvert}\ \sum_{k=1}^{N}\chi_{k}=1\right\\}$ and its relaxation $\mathcal{M}:=\left\\{u:[0,1)^{d}\to[0,1]^{N}\biggr{\rvert}\ \sum_{k=1}^{N}u_{k}=1\right\\}.$ If $\chi\in\mathcal{A}\cap BV([0,1)^{d})^{N}$, then each of the sets $\Omega_{i}:=\\{\chi_{i}=1\\}$ is a set of finite perimeter. We denote by $\partial^{}\Omega_{i}$ the reduced boundary of the set $\Omega_{i}$, and for any pair $1\leq i\neq j\leq N$ we denote by $\Sigma_{ij}:=\partial^{}\Omega_{i}\cap\partial^{}\Omega_{j}$ the interface between the sets. For $u\in\mathcal{M}$ we define (25) $E(u):=\begin{cases}\sum_{i,j}\sigma_{ij}\mathcal{H}^{d-1}(\Sigma_{ij})\ &\text{if}\ u\in\mathcal{A}\cap BV([0,1)^{d})^{N}\\\ +\infty\ &\text{otherwise}.\end{cases}$ For $h>0$ fixed we define the approximate energy $E_{h}$ for $u\in\mathcal{M}$ (26) $E_{h}(u)=\sum_{i,j}\frac{1}{\sqrt{h}}\int_{[0,1)^{d}}u_{i}K_{ij}^{h}u_{j}dx.$ For $u,v\in\mathcal{M}$ and $h>0$ we also define the distance (27) $\displaystyle d_{h}^{2}(u,v)$ $\displaystyle:=-2hE_{h}(u-v)=-2\sqrt{h}\sum_{i,j}\int(u_{i}-v_{i})K_{ij}^{h}(u_{j}-v_{j})dx$ $\displaystyle=2\sqrt{h}\int\|G_{\gamma}^{h/2}(u-v)\|_{\mathbb{A}}^{2}+\|G_{\beta}^{h/2}(u-v)\|_{\mathbb{B}}^{2}\ dx,$ where we used the semigroup property (21) and the symmetry (19) to derive the last equality. ###### Lemma 2. The pair $(\mathcal{M},d_{h})$ is a compact metric space. The function $E_{h}$ is continuous with respect to $d_{h}$. For every $1\leq i\leq N$ and $n\in\mathbf{N}$ define $\chi_{i}^{n}=\mathbf{1}_{\Omega^{n}_{i}}$, where $\Omega_{1}^{n},...,\Omega_{N}^{n}$ are obtained from $\Omega_{1}^{n-1},...,\Omega_{N}^{n-1}$ by the thresholding scheme. Then $\chi^{n}$ minimizes (28) $\frac{1}{2h}d_{h}^{2}(u,\chi^{n-1})+E_{h}(u)\ \text{among all}\ u\in\mathcal{M}.$ ###### Proof. For $u,v\in\mathcal{M}$ definition (27) and the fact that $\mathbb{A}$ and $\mathbb{B}$ are positive definite imply that $d_{h}$ is a distance on $\mathcal{M}$. The fact that $(\mathcal{M},d_{h})$ is compact and $E_{h}$ is continuous is just a consequence of the fact that $d_{h}$ metrizes the weak convergence in $L^{2}$ on $\mathcal{M}$, the interested reader may find the details of the reasoning in [19]. We are thus left with showing that $\chi^{n}$ satisfies (28). For $u,v\in L^{2}([0,1)^{d})$ define $(u,v)=\frac{1}{\sqrt{h}}\sum_{i,j}\int u_{i}K^{h}_{ij}v_{j}dx,$ then by the symmetry (19) of the Gaussian kernel and by the symmetry of both matrices $\mathbb{A},\mathbb{B}$ it is not hard to show that $(\cdot,\cdot)$ is symmetric. In particular we can write for any $u\in\mathcal{M}$ $\displaystyle\frac{1}{2h}d_{h}^{2}(u,\chi^{n-1})+E_{h}(u)$ $\displaystyle=-E_{h}(u-\chi^{n-1})+E_{h}(u)$ $\displaystyle=-(u-\chi^{n-1},u-\chi^{n-1})+(u,u)$ $\displaystyle=2(\chi^{n-1},u)-(\chi^{n-1},\chi^{n-1}).$ Thus (28) is equivalent to the fact that $\chi^{n}$ minimizes $(\chi^{n-1},u)$ among all $u\in\mathcal{M}$. Since by (2) $(\chi^{n-1},u)=\int\sum_{i}u_{i}\psi^{n}_{i}dx,$ we see that $\chi^{n}$ minimizes the integrand pointwise, and thus it is a minimizer for the functional.∎ The previous lemma allows us to apply the general theory of gradient flows in [2] to this particular problem. We record the key statement for our purposes in the following lemma. ###### Lemma 3. Let $(\mathcal{M},d)$ be a compact metric space and $E:\mathcal{M}\to\mathbf{R}$ be continuous. Given $\chi^{0}\in\mathcal{M}$ and $h>0$ consider a sequence $\\{\chi^{n}\\}_{n\in\mathbf{N}}$ satisfying (29) $\chi^{n}\ \text{minimizes}\ \frac{1}{2h}d^{2}(u,\chi^{n-1})+E(u)\ \text{among all}\ u\in\mathcal{M}.$ Then we have for all $t\in\mathbf{N}h$ (30) $\displaystyle\begin{aligned} E(\chi(t))+\frac{1}{2}\int_{0}^{t}\left(\frac{1}{h^{2}}d^{2}(\chi(s+h),\chi(s))+\|\partial E\|^{2}(u(s))\right)ds\leq E(\chi^{0}).\end{aligned}$ Here $\chi(t)$ is the piecewise constant interpolation, $u(t)$ is another interpolation satisfying (31) $\displaystyle\begin{aligned} \int_{0}^{\infty}\frac{1}{2h^{2}}d^{2}(u(t),\chi(t))dt\leq E(\chi^{0}),\end{aligned}$ (32) $\displaystyle\begin{aligned} \\\ &E(u(t))\leq E(\chi(t))\ \text{for all}\ t\geq 0,\end{aligned}$ and $\|\partial E\|(u)$ is the metric slope defined by (33) $\|\partial E\|(u):=\lim_{d(u,v)\to 0}\frac{(E(u)-E(v))_{+}}{d(u,v)}\in[0,\infty].$ ## 3\. Statement of results Our main result is the convergence of the modified thresholding scheme to a weak notion of multiphase mean curvature flow. More precisely, given an initial partition $\\{\Omega_{1}^{0},...,\Omega_{N}^{0}\\}$ of $[0,1)^{d}$ encoded by $\chi^{0}:[0,1)^{d}\to\\{0,1\\}^{N}$ such that $\sum_{i}\chi_{i}^{0}=1$, define $\chi^{h}:[0,1)^{d}\times\mathbf{R}\to\\{0,1\\}^{N}$ by setting (34) $\displaystyle\begin{aligned} &\chi^{h}(t,x)=\chi^{0}(x)\ \text{for}\ t<h,\\\ &\chi^{h}(t,x)=\chi^{n}(x)\ \text{for}\ t\in[nh,(n+1)h)\ \text{for}\ n\in\mathbf{N}.\end{aligned}$ If $\chi^{0}$ is a function of bounded variation, we denote by $\Sigma_{ij}^{0}:=\partial^{}\Omega_{i}^{0}\cap\partial^{}\Omega_{j}^{0}$. Our main result is contained in the following theorem. ###### Theorem 1. Given $\chi^{0}\in\mathcal{A}$ and such that $\nabla\chi^{0}$ is a bounded measure and a sequence $h\downarrow 0$; let $\chi^{h}$ be defined by (34). Assume that there exists $\chi:[0,1)^{d}\times(0,T)\to[0,1]^{N}$ such that (35) $\chi^{h}\rightharpoonup\chi\ \text{in}\ L^{1}([0,1)^{d}\times(0,T)).$ Then $\chi\in\\{0,1\\}^{N}$ almost everywhere, $\sum_{i}\chi_{i}=1$ and $\chi\in L^{1}((0,T),BV([0,1)^{d}))^{N}$. If we assume that (36) $\limsup_{h\downarrow 0}\int_{0}^{T}E_{h}(\chi^{h}(t))dt\leq\sum_{ij}\sigma_{ij}\int_{0}^{T}\mathcal{H}^{d-1}(\Sigma_{ij}(t))dt,$ then $\chi$ is a De Giorgi solution in the sense of Definition 1 below. ###### Definition 1. Given $\chi^{0}\in\mathcal{A}$ and such that $\nabla\chi^{0}$ is a bounded measure, a map $\chi:[0,1)^{d}\times(0,T)\to\\{0,1\\}^{N}$ such that $\sum_{i}\chi_{i}=1$ and $\chi\in L^{1}((0,T),BV([0,1)^{d}))^{N}$ is called a De Giorgi solution to the multiphase mean curvature flow with surface tensions $\sigma_{ij}$ and mobilities $\mu_{ij}$ provided the following three facts hold: 1. (1) There exist $H_{ij}\in L^{2}(\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)dt)$ which are mean curvatures in the weak sense, i.e., such that for any test vector field $\xi\in C^{\infty}_{c}([0,1)^{d}\times(0,T))^{d}$ (37) $\displaystyle\sum_{i,j}\sigma_{ij}\int_{[0,1)^{d}\times(0,T)}(\nabla\cdot\xi-\nu_{ij}\cdot\nabla\xi\nu_{ij})\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$ $\displaystyle=-\sum_{i,j}\sigma_{ij}\int_{[0,1)^{d}\times(0,T)}H_{ij}\nu_{ij}\cdot\xi\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt.$ 2. (2) There exist normal velocities $V_{ij}\in L^{2}(\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt)$ with $\displaystyle\begin{aligned} \int_{[0,1)^{d}}\eta(t=0)\chi_{i}^{0}dx&+\int_{[0,1)^{d}\times(0,T)}\partial_{t}\eta\ \chi_{i}\ dxdt\\\ &+\sum_{k\neq i}\int_{[0,1)^{d}\times(0,T)}\eta V_{ik}\ \mathcal{H}^{d-1}_{\|\Sigma_{ik}(t)}(dx)dt=0\end{aligned}$ for all $\eta\in C^{\infty}_{c}([0,1)^{d}\times[0,T))$. 3. (3) De Giorgi’s inequality is satisfied, i.e. (38) $\displaystyle\begin{aligned} &\limsup_{\tau\downarrow 0}\frac{1}{\tau}\sum_{i,j}\sigma_{ij}\int_{(T-\tau,T)}\mathcal{H}^{d-1}(\Sigma_{ij}(t))dt\\\ &+\frac{1}{2}\sum_{i,j}\int_{[0,1)^{d}\times(0,T)}\left(\frac{V_{ij}^{2}}{\mu_{ij}}+\sigma_{ij}^{2}\mu_{ij}H_{ij}^{2}\right)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\leq\sum_{i,j}\sigma_{ij}\mathcal{H}^{d-1}(\Sigma^{0}_{ij}).\end{aligned}$ ###### Remark 1. Observe that inequality (38) together with the definition of the weak mean curvatures gives a notion of weak solution for the multiphase mean curvature flow incorporating both the dynamics $V_{ij}=-\sigma_{ij}\mu_{ij}H_{ij}$ and the Herring angle condition at triple junctions. Indeed if $\chi:[0,1)^{d}\times(0,T)\to\\{0,1\\}^{N}$ with $\sum_{i}\chi_{i}(t)=1$ is such that the sets $\Omega_{i}(t)=\\{\chi_{i}(\cdot,t)=1\\}$ meet along smooth interfaces $\Sigma_{ij}:=\partial\Omega_{i}\cap\partial\Omega_{j}$ which evolve smoothly and satisfy (37), (38) then 1. (1) The Herring angle condition at triple junctions is satisfied. Indeed by the divergence theorem on surfaces (see Theorem 11.8 and Remark 11.42 in [24]) for any $\xi\in C^{\infty}_{c}(\times[0,1)^{d})^{d}$ $\displaystyle\int_{\Sigma_{ij}(t)}(\nabla\cdot\xi-\nu_{ij}\cdot\nabla\xi\nu_{ij})\ \mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)=$ $\displaystyle-\int_{\Sigma_{ij}(t)}H_{ij}\nu_{ij}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)$ $\displaystyle+\int_{\partial\Sigma_{ij}(t)}\xi\cdot\nu_{ij}\mathcal{H}^{d-2}(dx).$ Thus (37) and $H_{ij}\in L^{2}(\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt)$ imply that $\displaystyle\sigma_{i_{1}i_{2}}\int_{\partial\Sigma_{i_{1}i_{2}}(t)}\xi\cdot\nu_{i_{1}i_{2}}\mathcal{H}^{d-2}(dx)$ $\displaystyle+\sigma_{i_{2}i_{3}}\int_{\partial\Sigma_{i_{2}i_{3}}(t)}\xi\cdot\nu_{i_{2}i_{3}}\mathcal{H}^{d-2}(dx)$ $\displaystyle+\sigma_{i_{3}i_{1}}\int_{\partial\Sigma_{i_{3}i_{1}}(t)}\xi\cdot\nu_{i_{3}i_{1}}\mathcal{H}^{d-2}(dx)=0,$ which forces $\sigma_{i_{1}i_{2}}\nu_{i_{1}i_{2}}+\sigma_{i_{2}i_{3}}\nu_{i_{2}i_{3}}+\sigma_{i_{3}i_{1}}\nu_{i_{3}i_{1}}=0$ at triple junctions. 2. (2) We have $V_{ij}=-\sigma_{ij}\mu_{ij}H_{ij}$ on $\Sigma_{ij}(t)$. Indeed in the smooth case inequality (38) reduces to $\displaystyle\sum_{i,j}\sigma_{ij}\int_{(0,T)}\frac{d}{dt}\mathcal{H}^{d-1}(\Sigma_{ij}(t))dt$ $\displaystyle+\frac{1}{2}\sum_{i,j}\int_{[0,1)^{d}\times(0,T)}\left(\frac{V_{ij}^{2}}{\mu_{ij}}+\sigma_{ij}^{2}\mu_{ij}H_{ij}^{2}\right)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\leq 0.$ If one recalls that $\frac{d}{dt}\mathcal{H}^{d-1}(\Sigma_{ij}(t))=\int_{[0,1)^{d}}V_{ij}H_{ij}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)$, after completing the square we arrive at $\displaystyle\sum_{i,j}\int_{[0,1)^{d}\times(0,T)}\left(\frac{V_{ij}}{\sqrt{\mu_{ij}}}+\sigma_{ij}\mu_{ij}H_{ij}\right)^{2}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\leq 0,$ which implies $V_{ij}=-\sigma_{ij}\mu_{ij}H_{ij}$. The following lemma establishes, next to a compactness statement, that our convergence can be localized in the space and time variables $x$ and $t$, but also in the variable $z$ appearing in the convolution. ###### Lemma 4. 1. (1) Let $\\{\chi^{h}\\}_{h\downarrow 0}$ be a sequence of $\\{0,1\\}^{N}$-valued functions on $(0,T)\times[0,1)^{d}$ that satisfies (39) $\limsup_{h\downarrow 0}\left(\operatorname{esssup}_{t\in(0,T)}E_{h}(\chi_{h}(t))+\int_{0}^{T}\frac{1}{2h^{2}}d_{h}^{2}(\chi_{h}(t),\chi_{h}(t-h))dt\right)<\infty$ and that is piecewise constant in time in the sense of (34). Such a sequence is compact in $L^{1}([0,1)^{d}\times(0,T))^{N}$ and any weak limit $\chi$ is such that $\chi\in L^{1}((0,T),BV([0,1)^{d}))^{N}$ with (40) $\sum_{i,j}\sigma_{ij}\int_{0}^{T}\mathcal{H}^{d-1}(\Sigma_{ij}(t))dt\leq\liminf_{h\downarrow 0}\int_{0}^{T}E_{h}(\chi_{h}(t))dt.$ 2. (2) Assume that $u^{h}$ is a sequence of $[0,1]^{N}$-valued functions with $\sum_{i}u_{i}^{h}=1$ such that (36) holds (with $\chi^{h}$ replaced by $u^{h}$) and such that $u^{h}\to\chi$ in $L^{1}([0,1)^{d}\times(0,T))^{N}$ holds. Assume also that (41) $\limsup_{h\downarrow 0}\operatorname{esssup}_{t\in(0,T)}E_{h}(u^{h}(t))<\infty.$ Then as measures on $\mathbf{R}^{d}\times[0,1)^{d}\times(0,T)$ we have the following weak convergences for any $i\neq j$ (42) $\displaystyle\begin{aligned} \frac{K_{ij}(z)}{\sqrt{h}}u_{i}^{h}(x,t)&u_{j}^{h}(x-\sqrt{h}z,t)dxdtdz\\\ &\rightharpoonup K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dtdz.\end{aligned}$ (43) $\displaystyle\begin{aligned} \frac{K_{ij}(z)}{\sqrt{h}}u_{i}^{h}(x-\sqrt{h}z,t)&u_{j}^{h}(x,t)dxdtdz\\\ &\rightharpoonup K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{-}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dtdz.\end{aligned}$ Here the convergence may be tested also with continuous functions which have polynomial growth in $z\in\mathbf{R}^{d}$. The next proposition is the main ingredient in the proof of Theorem 1. It establishes the sharp lower bound on the distance-term. ###### Proposition 1. Suppose that (35) and the conclusion of Lemma 4 (2) hold. Assume also that the left hand side of (45) is finite. Then for every $1\leq k\leq N$ there exists $V_{k}\in L^{2}(\|\nabla\chi_{k}\|dt)$ such that (44) $\partial_{t}\chi_{k}=V_{k}\|\nabla\chi_{k}\|dt$ in the sense of distributions. Given $i\neq j$, it holds that $V_{i}(x,t)=-V_{j}(x,t)$ on $\Sigma_{ij}(t)$ and if we define $V_{ij}(x,t):=V_{i}(x,t)\nu_{ij}(x,t)\|_{\Sigma_{ij}(t)}$ then we have (45) $\liminf_{h\downarrow 0}\int_{0}^{T}\frac{1}{h^{2}}d_{h}^{2}(\chi^{h}(t),\chi^{h}(t-h))dt\geq\sum_{i,j}\frac{1}{\mu_{ij}}\int_{0}^{T}\int_{\Sigma_{ij}(t)}\|V_{ij}(x,t)\|^{2}\mathcal{H}^{d-1}(dx)dt.$ The final ingredient is the analogous sharp lower bound for the metric slope. ###### Proposition 2. Suppose that the conclusion of Lemma 4 (2) holds and that (35) holds with $\chi^{h}$ replaced by $u^{h}$. Then for any $i\neq j$ there exists a mean curvature $H_{ij}\in L^{2}(\mathcal{H}^{d-1}_{\Sigma_{ij}(t)}(dx)dt)$ in the sense of (37). Moreover the following inequality is true: $\liminf_{h\downarrow 0}\int_{0}^{T}\|\partial E_{h}\|^{2}(u_{h}(t))dt\geq\sum_{i,j}\mu_{ij}\sigma_{ij}^{2}\int_{0}^{T}\int_{\Sigma_{ij}(t)}\|H_{ij}(x,t)\|^{2}\mathcal{H}^{d-1}(dx)dt.$ ## 4\. Construction of suitable partitions of unity In the sequel we will frequently want to localize on one of the interfaces. To do so, we need to construct a suitable family of balls on wich the behavior of the flow is split into two majority phases and several minority phases. Hereafter we will ignore the time variable and consider a map $\chi:[0,1)^{d}\to\\{0,1\\}^{N}$ such that $\chi\in BV([0,1)^{d},\mathbf{R}^{N})$, $\sum_{k}\chi_{k}=1$. Given $1\leq i<j\leq N$ we denote by $\partial^{}\Omega_{i}$ the reduced boundary of the set $\\{\chi_{i}=1\\}$ and by $\Sigma_{ij}=\partial^{}\Omega_{i}\cap\partial^{}\Omega_{j}$ the interface between phase $i$ and phase $j$. Given a real number $r>0$ and a natural number $n\in\mathbf{N}$ we define (46) $\mathcal{F}_{n}^{r}:=\left\\{B(x,nr\sqrt{d}):\ x\in r\mathbf{Z}^{d}\right\\}$ where the balls appearing in the definition are intended to be open. Observe that for any $n\geq 2$ and any $r>0$ the collection of balls in $\mathcal{F}_{n}^{r}$ is a covering of $\mathbf{R}^{d}$ with the property that any point $x\in\mathbf{R}^{d}$ lies in at most $c(n,d)$ distinct balls belonging to $\mathcal{F}^{r}_{n}$, where $0<c(n,d)\leq(2n)^{d}$ is a constant that depends on $n,d$ but not on $r$. Given numbers $1\leq l\neq p\leq N$ we define (47) $\mathcal{E}^{r}:=\left\\{B\in\mathcal{F}^{r}_{2}:\ B\cap\Sigma_{lp}\neq\emptyset,\ \frac{\mathcal{H}^{d-1}(\Sigma_{ij}\cap 2B)}{\omega_{d-1}(4r)^{d-1}}\leq\frac{1}{2^{d}},\ \\{i,j\\}\neq\\{l,p\\}\right\\}.$ Here $2B$ denotes the ball with center given by the center of $B$ and twice its radius. Given $l,p$ as above, denote by $\\{B_{m}^{r}\\}_{m\in\mathbf{N}}$ an enumeration of $\mathcal{E}^{r}$ and by $\\{\rho_{m}\\}_{m\in\mathbf{N}}$ a smooth partition of unity subordinate to $\\{B_{m}^{r}\\}_{m\in\mathbf{N}}$. ###### Lemma 5. Fix $1\leq l\neq p\leq N$. With the above construction the following two properties hold. 1. (1) For any $1\leq i\neq j\leq N$, $\\{i,j\\}\neq\\{l,p\\}$ and any $\eta\in L^{1}(\mathcal{H}^{d-1}_{\|\Sigma_{ij}})$ (48) $\lim_{r\downarrow 0}\sum_{m\in\mathbf{N}}\int_{B_{m}^{r}}\eta\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)=0.$ 2. (2) For any $\eta\in L^{1}(\mathcal{H}^{d-1}_{\|\Sigma_{lp}})$ (49) $\lim_{r\downarrow 0}\sum_{m\in\mathbf{N}}\int\rho_{m}\eta\mathcal{H}^{d-1}_{\|\Sigma_{lp}}(dx)=\int\eta\mathcal{H}^{d-1}_{\|\Sigma_{lp}}(dx).$ ## 5\. Proofs ###### Proof of Theorem 1. By Lemma 2, we can apply Lemma 3 on the metric space $(\mathcal{M},d_{h})$ so that we get inequality (30) with $(E,d,\chi,u)=(E_{h},d_{h},\chi^{h},u^{h})$. Our first observation is that (50) $\lim_{h\downarrow 0}E_{h}(\chi^{0})=\sum_{i,j}\sigma_{ij}\mathcal{H}^{d-1}(\Sigma_{ij}^{0}),$ which follows from the consistency, cf. Lemma 7 in the Appendix. Inequality (30) then yields that the sequence $\chi^{h}$ satisfies (39), so that Lemma 4 (1) applies to get that $\chi\in L^{1}((0,T),BV([0,1)^{d}))^{N}$, $\chi\in\\{0,1\\}^{N}$ a.e., $\sum_{i}\chi_{i}=1$ and, after extracting a subsequence, $\chi^{h}\to\chi$ in $L^{2}([0,1)^{d}\times(0,T))^{N}$. We claim that this implies $u^{h}\to\chi$ in $L^{2}([0,1)^{d}\times(0,T))^{N}$. To see this, observe that (31) implies (51) $\displaystyle\begin{aligned} hE_{h}(\chi^{0})&\geq-\int_{0}^{T}E_{h}(u_{h}(t)-\chi_{h}(t))dt\\\ &\geq C\frac{1}{\sqrt{h}}\sum_{i=1}^{N}\left(\int\|G_{\gamma}^{h/2}(u_{i}^{h}-\chi_{i}^{h})\|^{2}dxdt+\int\|G_{\beta}^{h/2}(u_{i}^{h}-\chi_{i}^{h})\|^{2}dxdt\right)\end{aligned}$ where $C$ is a constant which depends on $N,\mathbb{A},\mathbb{B}$ but not on $h$ and comes from the fact that all norms on $(1,...,1)^{\perp}$ are comparable. Inequality (51) clearly implies that $K^{h}u_{h}-K^{h}\chi_{h}$ converges to zero in $L^{2}$. Observe that inequality (LABEL:varEnergy) in particular yields (41) with $\chi^{h}$ replaced by $u^{h}$. Recalling (152) in the Appendix, we learn that $u^{h}-\chi^{h}$ converges to zero in $L^{2}$. This implies that we can apply Lemma 4 (2) both to the sequence $u^{h}$ and the sequence $\chi^{h}$. In particular, we may apply Proposition 1 for $\chi^{h}$ and Proposition 2 for $u^{h}$. Now the proof follows the same strategy as the one in the two-phase case in [19]. For the sake of completeness, we sketch the argument here. First of all, Lemma 3 gives inequality (30) for $(E_{h},d_{h},\chi^{h},u^{h})$, namely for $n\in\mathbf{N}$ (52) $\displaystyle\rho(nh)\leq E_{h}(\chi^{0}),$ where we set $\rho(t)=E_{h}(\chi^{h}(t))+\frac{1}{2}\int_{0}^{t}\left(\frac{1}{h^{2}}d_{h}^{2}(\chi^{h}(s+h),\chi^{h}(s))+\|\partial E_{h}(u^{h}(s))\|^{s}\right)ds$. Multiplying (52) by $\eta(nh)-\eta((n+1)h)$ for some non-increasing function $\eta\in C_{c}([0,T))$ we get $-\int\frac{d\eta}{dt}\rho dt\leq(\eta(0)+h\sup\left\|\frac{d\eta}{dt}\right\|)E_{h}(\chi^{0})$. As test function $\eta$, we now choose $\eta(t)=\max\\{\min\\{\frac{T-t}{\tau},1\\},0\\}$ and obtain (53) $\displaystyle\frac{1}{\tau}\int_{T-\tau}^{T}E_{h}(\chi^{h}(t))dt$ $\displaystyle+\frac{1}{2}\int_{0}^{T-\tau}\left(\frac{1}{h^{2}}d_{h}^{2}(\chi^{h}(t),\chi^{h}(t-h))+\|\partial E_{h}(u^{h}(t))\|^{2}\right)dt\leq(1+\frac{h}{\tau})E_{h}(\chi^{0}).$ Now it remains to pass to the limit as $h\downarrow 0$: to get (38) from inequality (53) one uses the lower semicontinuity (40) for the first left hand side term, the sharp bound (45) for the second left hand side term, the bound (2) for the last left hand side term and finally one uses the consistency Lemma 7 in the Appendix to treat the right hand side term. To get (38) it remains to pass to the limit in $\tau\downarrow 0$. ∎ ###### Proof of Lemma 4. Argument for (1). For the compactness, the arguments in [19] adapt to this setting with minor changes. The first observation is that, by inequality (152) in the Appendix, one needs to prove compactness in $L^{2}([0,1)^{d}\times(0,T))^{N}$ of $\\{K^{h}\chi^{h}\\}_{h\downarrow 0}$. For this, one just needs a modulus of continuity in time. I.e. it is sufficient to prove that there exists a constant $C>0$ independent of $h$ such that $I_{h}(s)\leq C\sqrt{s}$, where $I_{h}(s)=\int_{(s,T)\times[0,1)^{d}}\|\chi_{h}(x,t)-\chi_{h}(x,t-s)\|^{2}dxdt.$ This is can be done applying word by word the argument in [19] once we show the following: for any pair $\chi,\chi^{\prime}\in\mathcal{A}$ of admissible functions, we have (54) $\int\|\chi-\chi^{\prime}\|dx\leq\frac{C}{\sqrt{h}}d_{h}^{2}(\chi,\chi^{\prime})+C\sqrt{h}\left(E_{h}(\chi)+E_{h}(\chi^{\prime})\right).$ Here the constant $C$ depends on $N,\mathbb{A},\mathbb{B}$ but not on $h$. To prove (54) we proceed as follows: let $\mathbb{S}\in\mathbf{R}^{N\times N}$ be a symmetric matrix which is positive definite on $(1,...,1)^{\perp}$. Since any two norms on a finite dimensional space are comparable, there exists a constant $C>0$ depending on $\mathbb{S}$ and $N$ such that (55) $\|\chi-\chi^{\prime}\|\leq\|\chi-\chi^{\prime}\|^{2}\leq C\|\chi-\chi^{\prime}\|^{2}_{\mathbb{S}}$ where $\|\cdot\|_{\mathbb{S}}$ denotes the norm induced by $\mathbb{S}$. For a function $u\in\mathcal{M}$ write $(\tilde{K}^{h})u_{h}$ for the function $\left((\tilde{K}^{h})u_{h}\right)_{i}=\sum_{j\neq i}K_{ij}^{h}u_{h}^{j}.$ Then we calculate (56) $\displaystyle\begin{aligned} \|\chi-\chi^{\prime}\|^{2}_{\mathbb{S}}=-(\chi-\chi^{\prime})\cdot(\tilde{K}^{h})(\chi-\chi^{\prime})+(\chi-\chi^{\prime})(\mathbb{S}+(\tilde{K}^{h}))(\chi-\chi^{\prime}).\end{aligned}$ Select $\mathbb{S}=(s_{ij})$ where $s_{ij}=-\int K_{ij}(z)dz$. Then, by our assumption (13) $\mathbb{S}$ is positive definite on $(1,...,1)^{\perp}$ and after integration on $[0,1)^{d}$ identity (56) becomes $\int\|\chi-\chi^{\prime}\|^{2}_{\mathbb{S}}dx=\frac{1}{2\sqrt{h}}d_{h}^{2}(\chi,\chi^{\prime})+\int(\chi-\chi^{\prime})(\mathbb{S}+(\tilde{K}^{h}))(\chi-\chi^{\prime})dx.$ We now proceed to estimate the integral on the right hand side. By the choice of $\mathbb{S}$ and Jensen’s inequality we have (57) $\displaystyle\int(\chi-\chi^{\prime})(\mathbb{S}+(\tilde{K}^{h}))(\chi-\chi^{\prime})dx\leq C\int\|\mathbb{S}+(\tilde{K}^{h}))(\chi-\chi^{\prime})\|dx$ $\displaystyle\leq C\sum_{i,j}\int K^{h}_{ij}(z)\|(\chi_{j}-\chi_{j}^{\prime})(x-z)-(\chi_{j}-\chi_{j}^{\prime})(x)\|dxdz.$ Using the triangle inequality and (150) in the Appendix we can estimate the right hand side to obtain the following inequality (58) $\displaystyle\begin{aligned} \int(\chi-\chi^{\prime})(\mathbb{S}+(\tilde{K}^{h}))(\chi-\chi^{\prime})dx\end{aligned}$ $\displaystyle\begin{aligned} \leq C\sum_{i,j}&\left(\sum_{k\neq j}\int K^{h}_{ij}(z)\chi_{j}(x-z)\chi_{k}(x)dxdz\right.\\\ &\left.+\sum_{k\neq j}\int K^{h}_{ij}(z)\chi_{j}(x)\chi_{k}(x-z)dxdz\right.\\\ &\left.+\sum_{k\neq j}\int K^{h}_{ij}(z)\chi^{\prime}_{j}(x-z)\chi^{\prime}_{k}(x)dxdz\right.\\\ &\left.+\sum_{k\neq j}\int K^{h}_{ij}(z)\chi^{\prime}_{j}(x)\chi^{\prime}_{k}(x-z)dxdz\right).\end{aligned}$ Observing that there is a constant $C>0$ such that $K_{ij}\leq CK_{jk}$ we conclude that $\int(\chi-\chi^{\prime})(\mathbb{S}+(\tilde{K}^{h}))(\chi-\chi^{\prime})dx\leq C\sqrt{h}\left(E_{h}(\chi)+E_{h}(\chi^{\prime})\right).$ This proves (54) and closes the argument for the compactness. We also have to prove (40), but this follows from (42) with $u^{h}$ replaced by $\chi^{h}$ once we have shown that the limit $\chi$ is such that $\|\nabla\chi\|$ is a bounded measure, equiintegrable in time. This can be done with an argument similar to the one used in [19] for the two-phase case. Observe that this only requires the weaker assumption (41). Argument for (2). As mentioned in the previous paragraph, we already know that the limit $\chi$ is such that $\|\nabla\chi\|$ is a bounded measure, equiintegrable in time. We will prove (42). Then (43) easily follows by recalling that $\nu_{ij}=-\nu_{ji}$. A standard argument (to be found in [19]) which relies on the exponential decay of the kernel yields the fact that we can test convergences (42) with functions with at most polynomial growth in $z$ provided we already have the result for bounded and continuous test functions, thus we focus on this case. Let $\xi\in C_{b}(\mathbf{R}^{d}\times[0,1)^{d}\times(0,T))$ be a bounded and continuous function. To show (42) we aim at showing that (59) $\displaystyle\begin{aligned} \lim_{h\downarrow 0}\int\xi(z,x,t)\frac{K_{ij}(z)}{\sqrt{h}}&u_{i}^{h}(x,t)u_{j}^{h}(x-\sqrt{h}z,t)dxdtdz\\\ &=\int\xi(z,x,t)K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\Sigma_{ij}(t)}(dx)dtdz.\end{aligned}$ Upon splitting $\xi$ into the positive and the negative part, by linearity we may assume that $0\leq\xi\leq 1$. We can split (59) into the local lower bound (60) $\displaystyle\begin{aligned} \liminf_{h\downarrow 0}\int\xi(z,x,t)\frac{K_{ij}(z)}{\sqrt{h}}&u_{i}^{h}(x,t)u_{j}^{h}(x-\sqrt{h}z,t)dzdxdt\\\ &\geq\int\xi(z,x,t)K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\Sigma_{ij}(t)}(dx)dtdz.\end{aligned}$ and the global upper bound (61) $\displaystyle\begin{aligned} \liminf_{h\downarrow 0}\int\frac{K_{ij}(z)}{\sqrt{h}}&u_{i}^{h}(x,t)u_{j}^{h}(x-\sqrt{h}z,t)dzdxdt\\\ &\leq\int K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\Sigma_{ij}(t)}(dx)dtdz.\end{aligned}$ Indeed we can recover the limsup inequality in (59) by splitting $\xi=1-(1-\xi)$ and applying the local lower bound (60) to $1-\xi$. We first concentrate on the local lower bounds in the case where $u^{h}=\chi$, namely we will show (62) $\displaystyle\begin{aligned} \liminf_{h\downarrow 0}\int\xi(z,x,t)\frac{K_{ij}(z)}{\sqrt{h}}&\chi_{i}(x,t)\chi_{j}(x-\sqrt{h}z,t)dzdxdt\\\ &\geq\int\xi(z,x,t)K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\Sigma_{ij}(t)}(dx)dtdz.\end{aligned}$ By Fatou’s lemma the claim is reduced to showing that for a.e. point $t$ in time and every $z\in\mathbf{R}^{d}$ (63) $\displaystyle\liminf_{h\downarrow 0}\int\xi(z,x,t)\frac{K_{ij}(z)}{\sqrt{h}}$ $\displaystyle\chi^{i}(x,t)\chi^{j}(x-\sqrt{h}z,t)dx$ $\displaystyle\geq\int\xi(z,x,t)K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\Sigma_{ij}(t)}(dx).$ Fix a point $t$ such that $\chi(\cdot,t)\in BV([0,1)^{d},\\{0,1\\}^{N})$ and any $z\in\mathbf{R}^{d}$. In the sequel, we will drop those variables, so $\chi(x)=\chi(x,t)$, $\xi(x)=\xi(z,x,t)$. By approximation we may assume that $\xi\in C^{\infty}([0,1)^{d})$. Let $\rho_{m}$ be a partition of unity obtained by applying the construction of Section 4 to the function $\chi(x)$ on the interface $\Sigma_{ij}$. Then by Lemma 5 we have $\displaystyle\begin{aligned} &\int\xi(x)(\nu_{ij}(x)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\\\ &=\lim_{r\downarrow 0}\left(\sum_{m\in\mathbf{N}}\int\rho_{mij}(x)\xi(x)(\nu_{ij}(x)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\right)\end{aligned}$ $\displaystyle\begin{aligned} =\lim_{r\downarrow 0}\sum_{m\in\mathbf{N}}&\left(\int\rho_{mij}(x)\xi(x)(\nu_{i}(x)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}}(dx)\right.\\\ &\left.-\sum_{k\neq i,j}\int\rho_{mij}(x)\xi(x)(\nu_{ij}(x)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ik}}(dx)\right)\end{aligned}$ (64) $\displaystyle\begin{aligned} =\lim_{r\downarrow 0}\sum_{m\in\mathbf{N}}\int\rho_{mij}(x)\xi(x)(\nu_{i}(x)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}}(dx).\end{aligned}$ We now focus on estimating the argument of the last limit. Observe that $(\nu_{ij}(x)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}}(dx)=(\partial_{z}\chi_{i})_{+}$, thus by definition of positive part of a measure, given $\epsilon>0$ we can select, for any $m\in\mathbf{N}$, a function $\tilde{\xi}_{m}\in C^{1}_{c}(B_{m})$ such that $0\leq\tilde{\xi}_{m}\leq 1$ and such that (65) $\int\rho_{mij}\xi\tilde{\xi}_{m}\partial_{z}\chi_{i}+2^{-m}\epsilon\geq\int\rho_{mij}\xi(\nu_{i}\cdot z)_{+}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}}(dx).$ Let $\eta_{m}:=\rho_{mij}\xi\tilde{\xi}_{m}\in C^{1}_{c}(B_{m})$, then $\displaystyle\begin{aligned} \int\eta_{m}\partial_{z}\chi_{i}=-\int\partial_{z}\eta_{m}\chi_{i}dx\end{aligned}$ $\displaystyle\begin{aligned} =\lim_{h\downarrow 0}\int\frac{\eta_{m}(x+\sqrt{h}z)-\eta(x)}{\sqrt{h}}\chi_{i}(x)dx\end{aligned}$ $\displaystyle\begin{aligned} =\lim_{h\downarrow 0}\int\eta_{m}(x)\frac{\chi_{i}(x)-\chi_{i}(x-\sqrt{h}z)}{\sqrt{h}}dx\end{aligned}$ $\displaystyle\begin{aligned} \leq\liminf_{h\downarrow 0}\sum_{k\neq i}\int\eta_{m}(x)\frac{\chi_{i}(x)\chi_{k}(x-\sqrt{h}z)}{\sqrt{h}}dx\end{aligned}$ $\displaystyle\begin{aligned} \leq\liminf_{h\downarrow 0}&\int\eta_{m}(x)\frac{\chi_{i}(x)\chi_{j}(x-\sqrt{h}z)}{\sqrt{h}}dx\\\ &+\limsup_{h\downarrow 0}\sum_{k\neq i,j}\int\eta_{m}(x)\frac{\chi_{i}(x)\chi_{k}(x-\sqrt{h}z)}{\sqrt{h}}dx\end{aligned}$ $\displaystyle\begin{aligned} \leq\liminf_{h\downarrow 0}&\int\eta_{m}(x)\frac{\chi_{i}(x)\chi_{j}(x-\sqrt{h}z)}{\sqrt{h}}dx\\\ &+\sum_{k\neq i,j}\limsup_{h\downarrow 0}\int\eta_{m}(x)\frac{\chi_{i}(x)\chi_{k}(x-\sqrt{h}z)}{\sqrt{h}}dx.\end{aligned}$ Observe that for each $m\in\mathbf{N}$, using also the consistency Lemma 7 $\displaystyle\begin{aligned} \limsup_{h\downarrow 0}\int\eta_{m}(x)\frac{\chi_{i}(x)\chi_{k}(x-\sqrt{h}z)}{\sqrt{h}}dx\end{aligned}$ $\displaystyle\begin{aligned} \leq\limsup_{h\downarrow 0}\int\eta_{m}(x)\frac{\chi_{i}(x)\chi_{k}(x-\sqrt{h}z)+\chi_{i}(x-\sqrt{h}z)\chi_{j}(x)}{\sqrt{h}}dx\end{aligned}$ $\displaystyle\begin{aligned} =\int\eta_{m}(x)\|\nu_{ik}(x)\cdot z\|\mathcal{H}^{d-1}_{\|\Sigma_{ik}}(dx)\end{aligned}$ $\displaystyle\begin{aligned} \leq\|z\|\mathcal{H}^{d-1}(B_{mij}^{r}\cap\Sigma_{ik})\end{aligned}$ Thus we obtain $\displaystyle\begin{aligned} \int\eta_{m}\partial_{z}\chi_{i}\leq\liminf_{h\downarrow 0}&\int\eta_{m}(x)\frac{\chi_{i}(x)\chi_{j}(x-\sqrt{h}z)}{\sqrt{h}}dx\\\ &+\sum_{k\neq i,j}\|z\|\mathcal{H}^{d-1}(B_{mij}^{r}\cap\Sigma_{ik})\end{aligned}$ Inserting back into (64), recalling also Lemma 5 and the inequality (65), using Fatou’s lemma, the fact that $\rho_{mij}$ is a partition of unity and that $0\leq\tilde{\xi}_{m}\leq 1$ we obtain that $\displaystyle\int\xi(x)(\nu_{ij}(x)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\leq\liminf_{h\downarrow 0}$ $\displaystyle\int\xi(x)\frac{\chi_{i}(x)\chi_{j}(x-\sqrt{h}z)}{\sqrt{h}}dx+\epsilon$ and (62) follows letting $\epsilon$ go to zero. To derive inequality (60) we just apply Lemma 9 in the Appendix. To get the upper bound (61) we argue as follows. First of all recall Assumption (36) which says (66) $\int_{0}^{T}E_{h}(u^{h}(t))dt\to\int_{0}^{T}E(\chi(t))dt.$ Now, if we define $e^{ij}_{h}(u^{h})=\frac{1}{\sqrt{h}}\int_{0}^{T}\int u_{i}^{h}(t)K_{ij}^{h}u_{j}^{h}(t)dxdt$ we have that by (60) $\liminf_{h\downarrow 0}e_{h}^{ij}(u_{h})\geq e^{ij}(\chi)$, where $e^{ij}(\chi)$ is defined in the obvious way. Assume that there exists a pair $i,j$ such that $\limsup_{h\downarrow 0}e_{h}^{ij}(u^{h})>e^{ij}(\chi)$, then $\displaystyle\begin{aligned} \int_{0}^{T}E(\chi(t))dt&=\lim_{h\downarrow 0}\int_{0}^{T}E_{h}(u^{h}(t))dt\\\ &=\limsup_{h\downarrow 0}\int_{0}^{T}E_{h}(u^{h}(t))dt\\\ &\geq\sum_{(l,p)\neq(i,j)}\liminf_{h\downarrow 0}e_{h}^{lp}(u^{h})+\limsup_{h\downarrow 0}e_{h}^{ij}(u^{h})\\\ &>\int_{0}^{T}E(\chi(t))dt\end{aligned}$ which is a contradiction. Thus we have proved (61). ∎ ###### Proof of Proposition 1. Since we assume that the left hand side of (45) is finite, in view of (27), upon passing to a subsequence we may assume that, in the sense of distributions, the limit (67) $\lim_{h\downarrow 0}\frac{1}{h\sqrt{h}}\left(\left\|G^{h/2}_{\gamma}(\chi-\chi(\cdot-h))\right\|_{\mathbb{A}}^{2}+\left\|G^{h/2}_{\beta}(\chi-\chi(\cdot-h))\right\|_{\mathbb{B}}^{2}\right)=\omega$ exists as a finite positive measure on $[0,1)^{d}\times(0,T)$. Here we indicated with $\chi_{l}^{h}(\cdot-h)$ the time shift of function $\chi_{l}^{h}$. We denote by $\tau$ a small fraction of the characteristic spatial scale, namely $\tau=\alpha\sqrt{h}$ for some $\alpha>0$, which we think as a small number. Given $1\leq l\leq N$ we define (68) $\delta\chi^{h}_{l}:=\chi^{h}_{l}-\chi^{h}_{l}(\cdot-\tau).$ We divide the proof into two parts: first we show that the normal velocities exist, and afterwards we prove the sharp bound. But first, let us state two distributional inequalities that will be used later. Namely • In a distributional sense it holds that (69) $\limsup_{h\downarrow 0}-\frac{1}{\sqrt{h}}\sum_{i\neq j}\delta\chi_{i}K_{ij}^{h}\delta\chi_{j}\leq\alpha^{2}\omega.$ • There exists a constant $C>0$ such that for any $1\leq i\leq N$ and any $\theta\in\\{\gamma,\beta\\}$ in a distributional sense it holds that (70) $\limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}(\chi_{i}-\chi_{i}(\cdot-\tau))G_{\theta}^{h}(\chi_{i}-\chi_{i}(\cdot-\tau))\leq C\alpha^{2}\omega.$ We observe that it suffices to prove (69), then (70) follows immediately. Indeed recall that $\mathbb{A}$ and $\mathbb{B}$ are positive definite on $(1,...,1)^{\perp}$. In particular there exists a constant $C>0$ such that for any $v\in(1,...,1)^{\perp}$ one has $\|v\|^{2}_{\mathbb{A}}+\|v\|_{\mathbb{B}}^{2}\geq C\|v\|^{2}\geq Cv_{i}^{2}$ for any $i\in\\{1,...,N\\}$. Applying this to the vector $v=G_{\theta}^{h/2}\delta\chi_{i}$ one gets (71) $\|G_{\theta}^{h/2}\delta\chi_{i}\|^{2}\leq\frac{1}{C}\|G_{\theta}^{h/2}\delta\chi\|_{\mathbb{A}}^{2}+\|G_{\theta}^{h/2}\delta\chi\|_{\mathbb{B}}^{2}.$ The claim then follows from the definition of $\omega$, (69), the symmetry (19) and the semigroup property (21). Indeed it is sufficient to check that, in the sense of distributions (72) $\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\sum_{i\neq j}\delta\chi_{i}K_{ij}^{h}\delta\chi_{j}+\frac{1}{\sqrt{h}}\left(\|G_{\gamma}^{h/2}\delta\chi\|_{\mathbb{A}}^{2}+\|G_{\beta}^{h/2}\delta\chi\|_{\mathbb{B}}^{2}\right)=0.$ To this aim, pick a test function $\eta\in C^{\infty}_{c}([0,1)^{d}\times(0,T))$. Spelling out the definition of the norms $\|\cdot\|_{\mathbb{A}}$ and $\|\cdot\|_{\mathbb{B}}$, the claim is proved once we show that (73) $\displaystyle\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\sum_{i,j}a_{ij}\int\xi(\delta\chi_{i}G_{\gamma}^{h}\delta\chi_{j}-G_{\gamma}^{h/2}\delta\chi_{i}G_{\gamma}^{h/2}\delta\chi_{j})dxdt=0,$ and the same claim with $a_{ij}$, $\gamma$ replaced by $b_{ij},\beta$ respectively. We concentrate on (73). Clearly, we are done once we show that for any $i\neq j$ (74) $\displaystyle\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\int\xi(\delta\chi_{i}G_{\gamma}^{h}\delta\chi_{j}-G_{\gamma}^{h/2}\delta\chi_{i}G_{\gamma}^{h/2}\delta\chi_{j})dxdt=0.$ To show this, using the semigroup property (21) we rewrite the argument of the limit as (75) $-\frac{1}{\sqrt{h}}\int[\xi,G_{\gamma}^{h/2}](\delta\chi_{i})G_{\gamma}^{h/2}\delta\chi_{j}dxdt,$ and we observe that by the boundedness of the measures $\frac{1}{\sqrt{h}}\|G_{\gamma}^{h/2}\delta\chi\|^{2}_{\mathbb{A}}$ it suffices to show (76) $\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\int\|[\xi,G_{\gamma}^{h/2}](\delta\chi_{i})\|^{2}dxdt=0.$ To prove this, spelling out the integrand, using the Cauchy-Schwarz inequality and recalling the scaling (20) we observe that $\displaystyle\begin{aligned} \int\|[\xi,G_{\gamma}^{h/2}](\delta\chi_{i})\|^{2}dxdt\end{aligned}$ (77) $\displaystyle\begin{aligned} &\leq\int\left(\int\|\xi(x,t)-\xi(x-z,t)\|^{2}G_{\gamma}^{h/2}(z)dz\right)G_{\gamma}^{h/2}\|\delta\chi_{i}(x,t)\|^{2}dxdt\\\ &\leq\frac{h}{2}\sup\|\nabla\xi\|^{2}\int G_{\gamma}(z)\|z\|^{2}dz\int_{0}^{T}\int\|\delta\chi_{i}(x,t)\|^{2}dxdt.\end{aligned}$ Observe that by the compactness of $\chi^{h}$ in $L^{2}([0,1)^{d}\times(0,T))$, (LABEL:boundOh) is of order $h$, thus (76) indeed holds true. The proof of (69) is essentially already contained in the paper [19]. For the convenience of the reader we sketch the main ideas here. One reduces the claim to proving the following facts. (78) $\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\sum_{ij}\delta\chi_{i}K^{h}_{ij}\delta\chi_{j}-\frac{1}{\sqrt{h}}\left(\left\|G_{\gamma}^{h/2}\delta\chi\right\|_{\mathbb{A}}^{2}+\left\|G_{\beta}^{h/2}\delta\chi\right\|_{\mathbb{B}}^{2}\right)=0.$ (79) $\limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}\left\|G_{\gamma}^{h/2}\delta\chi\right\|_{\mathbb{A}}^{2}-\alpha^{2}\frac{1}{h\sqrt{h}}\left\|G_{\gamma}^{h/2}(\chi-\chi(\cdot-h))\right\|_{\mathbb{A}}^{2}\leq 0.$ (80) $\limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}\left\|G_{\beta}^{h/2}\delta\chi\right\|_{\mathbb{B}}^{2}-\alpha^{2}\frac{1}{h\sqrt{h}}\left\|G_{\beta}^{h/2}(\chi-\chi(\cdot-h))\right\|_{\mathbb{B}}^{2}\leq 0.$ Claim (78) was proved in the previous paragraph, while (79) and (80) are consequences of Jensen’s inequality in the time variable for the convex functions $\|\cdot\|_{\mathbb{A}}^{2}$ and $\|\cdot\|_{\mathbb{B}}^{2}$ respectively. More precisely, assume without loss of generality that $\tau=Nh$ for some $N\in\mathbf{N}$, then by a telescoping argument and Jensen’s inequality for $\|\cdot\|_{\mathbb{A}}^{2}$ we get $\displaystyle\begin{aligned} &\frac{1}{\sqrt{h}}\|G_{\gamma}^{h/2}\delta\chi\|_{\mathbb{A}}^{2}\\\ &\leq N\sum_{n=0}^{N-1}\frac{1}{\sqrt{h}}\|G_{\gamma}^{h/2}(\chi^{h}(\cdot- nh)-\chi^{h}(\cdot-(n+1)h))\|_{\mathbb{A}}^{2}.\end{aligned}$ Recalling that $N=\alpha/\sqrt{h}$ we can rewrite the right hand side as (81) $\frac{\alpha^{2}}{N}\sum_{n=0}^{N-1}\frac{1}{h\sqrt{h}}\|G_{\gamma}^{h/2}(\chi^{h}(\cdot- nh)-\chi^{h}(\cdot-(n+1)h))\|^{2}_{\mathbb{A}}.$ This is an average of time shifts of $\alpha^{2}\frac{1}{h\sqrt{h}}\|G_{\gamma}^{h/2}(\chi^{h}-\chi^{h}(\cdot-h))\|^{2}_{\mathbb{A}}$. Since $Nh=o(1)$ all these time shifts are small, thus the average has the same distributional limit as $\alpha^{2}\frac{1}{h\sqrt{h}}\|G_{\gamma}^{h/2}(\chi^{h}-\chi^{h}(\cdot-h))\|^{2}_{\mathbb{A}}$. This proves (79). The argument for (80) is similar. ### Existence of the normal velocities We now prove the existence of the normal velocities. Fix $1\leq i\leq N$ and observe that for $w\in\\{\gamma,\beta\\}$ we have (82) $\begin{split}\|\chi_{i}-\chi_{i}(-\tau)\|\leq&(\chi_{i}-\chi_{i}(-\tau))G_{w}^{h}(\chi_{i}-\chi_{i}(-\tau))+\|\chi_{i}-G_{w}^{h}\chi_{i}\|\\\ &+\|\chi_{i}(-\tau)-G_{w}^{h}\chi_{i}(-\tau)\|,\end{split}$ which follows simply by observing that $\|\chi_{i}-\chi_{i}(\cdot-\tau)\|=\|\chi_{i}-\chi_{i}(\cdot-\tau)\|^{2}=(\chi_{i}-\chi_{i}(\cdot-\tau)G_{w}^{h}(\chi_{i}-\chi_{i}(\cdot-\tau))+(\chi_{i}-\chi_{i}(\cdot-\tau))(\chi- G_{w}^{h}\chi)+(\chi_{i}(\cdot-\tau)-\chi_{i})(\chi_{i}(\cdot-\tau)-G_{w}^{h}\chi_{i}(\cdot-\tau))$. Using Jensen’s inequality and the elementary identity (150) in the Appendix we have (83) $\displaystyle\begin{aligned} \|\chi_{i}-G_{w}^{h}\chi_{i}\|&\leq\int G_{w}^{h}(z)\|\chi_{i}(x)-\chi_{i}(x-z)\|dz\\\ &=\int G_{w}^{h}(z)\chi_{i}(x)(1-\chi_{i}(x-z))dz+\int G_{w}^{h}(z)(1-\chi_{i}(x))\chi_{i}(x-z)dz\\\ &=\sum_{k\neq i}\int G_{w}^{h}(z)\chi_{i}(x)\chi_{k}(x-z)dz+\sum_{k\neq i}\int G_{w}^{h}(z)\chi_{k}(x)\chi_{i}(x-z)\|dz.\end{aligned}$ Now observe that by testing (42) with $G_{w}/K_{ij}$ (which is bounded, and thus admissible), we learn that (84) $\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\int G_{w}^{h}(z)\chi_{i}(x)\chi_{k}(x-z)dz=\int G_{w}(z)(\nu_{ik}(x,t)\cdot z)_{+}dz\mathcal{H}^{d-1}_{\|\Sigma_{ik}(t)}(dx)dt.$ Thus, if we divide (83) by $\sqrt{h}$ and let $h\downarrow 0$, using also (70) we obtain (85) $\begin{split}\alpha\|\partial_{t}\chi_{i}\|&\leq\liminf_{h\downarrow 0}\frac{\|\delta\chi_{i}\|}{\sqrt{h}}\\\ &\leq\limsup_{h\downarrow 0}\frac{\|\delta\chi_{i}\|}{\sqrt{h}}\\\ &\leq C\alpha^{2}\omega+C\mathcal{H}^{d-1}_{\partial^{}\Omega_{i}(t)}(dx)dt,\end{split}$ where $C$ is a constant which depends on $\gamma,\beta,N$, the mobilities and the surface tensions. If we divide by $\alpha$ and then let $\alpha\to 0$ we learn that $\|\partial_{t}\chi_{i}\|$ is absolutely continuous with respect to $\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)dt$. In particular, there exists $V_{i}\in L^{1}(\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)dt)$ which is the normal velocity of $\chi_{i}$ in the sense that $\partial_{t}\chi_{i}=V_{i}\|\nabla\chi_{i}\|$ in the sense of distributions. The optimal integrability $V_{i}\in L^{2}(\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)dt)$ will be shown in the second part of the proof. Let us record for later use that with a similar reasoning we actually obtain that $\limsup_{h}\frac{\|\delta\chi_{i}\|}{\sqrt{h}}$ is absolutely continuous with respect to $\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)dt$.Thus in particular inequality (85) holds with $\omega$ replaced by its absolutely continuous part with respect to $\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)dt$; calling this $\omega^{ac}_{i}$, it means (86) $\limsup_{h\downarrow 0}\frac{\|\delta\chi_{i}\|}{\sqrt{h}}\leq C\alpha^{2}\omega_{i}^{ac}+C\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)dt.$ ### Sharp Bound Before entering into the proof of the sharp bound, we need to prove the following property. For any $i\neq j$ we have that, in a distributional sense, the following holds (87) $\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\delta\chi_{i}^{+}K_{ij}^{h}\delta\chi_{j}^{+}=0=\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\delta\chi_{i}^{-}K_{ij}^{h}\delta\chi_{j}^{-}.$ We focus on the first limit, the second one being analogous. The first observation is that the limit (88) $\lambda:=\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\delta\chi_{i}^{+}K_{ij}^{h}\delta\chi_{j}^{+}$ is a nonnegative bounded measure, which is absolutely continous with respect to $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$. Indeed, spelling out the $z$ integral and using the fact that $\delta\chi_{i}^{+}=\chi_{i}(1-\chi_{i}(-\tau))$ we obtain (89) $\displaystyle\begin{aligned} \frac{1}{\sqrt{h}}\delta\chi_{i}^{+}K_{ij}^{h}\delta\chi_{j}^{+}=\frac{1}{\sqrt{h}}\int K_{ij}^{h}(z)\delta\chi_{i}^{+}(x,t)\delta\chi_{j}^{+}(x-z,t)dz\end{aligned}$ (90) $\displaystyle\begin{aligned} \leq\frac{1}{\sqrt{h}}\int K_{ij}^{h}(z)\chi_{i}(x,t)\chi_{j}(x-z,t)dz\end{aligned}$ which by (42) in Lemma 4, as $h\downarrow 0$, converges to (91) $\int K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$ which is absolutely continous with respect to $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$. Now, given $\nu_{0}\in\mathbf{S}^{d-1}$ we claim that (92) $\displaystyle\begin{aligned} \lambda\leq&\int_{\nu_{0}\cdot z\leq 0}K_{ij}(z)(\nu_{ij}\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\\\ &+\int_{\nu_{0}\cdot z\geq 0}K_{ij}(z)(\nu_{ij}\cdot z)_{-}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt.\end{aligned}$ To see this, let us denote momentarily the right-hand side of (88) (disintegrated in the $z$-variable) as $\lambda_{h}:=\chi_{i}(x,t)(1-\chi_{i})(x,t-\tau)K_{ij}^{h}(z)\chi_{i}(x-z,t)(1-\chi_{i})(x-z,t-\tau)$. Using the fact that $0\leq\chi_{i},\chi_{j}\leq 1$ and $\sum_{l}\chi_{l}=1$ we obtain the following inequalities (93) $\displaystyle\begin{aligned} \lambda_{h}\leq\chi_{i}(x,t)K_{ij}^{h}(z)\chi_{i}(x-z,t).\end{aligned}$ (94) $\displaystyle\begin{aligned} \lambda_{h}\leq&\chi_{j}(x,t-\tau)K_{ij}^{h}(z)\chi_{i}(x-z,t-\tau)\\\ &+C\sum_{k\neq i,j}K_{ij}^{h}(z)\left(\|\delta\chi_{k}\|(x,t)+\|\delta\chi_{k}\|(x-z,t)\right).\end{aligned}$ Here $C$ is a constant that does not depend on $h$. Using inequality (93) on the domain $\\{\nu_{0}\cdot z\leq 0\\}$ and inequality (94) on the domain $\\{\nu_{0}\cdot z\geq 0\\}$ we obtain $\displaystyle\begin{aligned} \mathllap{\lambda}\leq&\limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}\int_{\nu_{0}\cdot z\leq 0}\chi_{i}(x,t)K_{ij}^{h}(z)\chi_{i}(x-z,t)dz\\\ &+\limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}\int_{\nu_{0}\cdot z\geq 0}\chi_{j}(x,t-\tau)K_{ij}^{h}(z)\chi_{i}(x-z,t-\tau)dz\\\ &+C\sum_{k\neq i,j}\limsup_{h\downarrow 0}\left(\frac{1}{\sqrt{h}}\int K_{ij}^{h}(z)\|\delta\chi_{k}\|(x,t)dz+\frac{1}{\sqrt{h}}\int K_{ij}^{h}(z)\|\delta\chi_{k}\|(x-z,t)dz\right)\end{aligned}.$ Observe that for any $1\leq k\leq N$ we have (95) $\limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}\int K_{ij}^{h}(z)\|\delta\chi_{k}\|(x,t)dz=\frac{1}{\sqrt{h}}\int K_{ij}^{h}(z)\|\delta\chi_{k}\|(x-z,t)dz.$ This can be seen by showing that (96) $\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\int K_{ij}^{h}(z)\left(\|\delta\chi_{k}\|(x,t)-\|\delta\chi_{k}\|(x-z,t)\right)dz$ which can be shown to be true by testing with an admissible test function, and putting the spatial shift $z$ on it. Thus recalling (42) and (86), we obtain that (97) $\displaystyle\begin{aligned} \lambda\leq&\int_{\nu_{0}\cdot z\leq 0}K_{ij}(z)(\nu_{ij}\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\\\ &+\int_{\nu_{0}\cdot z\geq 0}K_{ij}(z)(\nu_{ij}\cdot z)_{-}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\\\ &+C\sum_{k\neq i,j}\alpha^{2}\omega_{k}^{ac}+\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt.\end{aligned}$ Since we already know that $\lambda$ is absolutely continous with respect to $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$, the same bound holds true if we replace the right hand side with its absolutely continuous part with respect to $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$. Observing that for $k\neq i,j$ by Lemma 6 in the Appendix the measures $\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt$ and $\mathcal{H}^{d-1}_{\|\partial^{}\Sigma_{ij}(t)}(dx)dt$ are mutually singular , this yields (92). Writing $\lambda=\theta(x,t)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$ for some $L^{1}(\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt)$-function $\theta$ we obtain that inequality (92) yields (98) $\displaystyle\begin{aligned} \theta(x,t)\leq\int_{\nu_{0}\cdot z\leq 0}&K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}dz\\\ &\int_{\nu_{0}\cdot z\geq 0}K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{-}dz\end{aligned}$ for every $\nu_{0}\in\mathbf{S}^{d-1}$ and $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$-a.e. $(x,t)\in[0,1)^{d}\times(0,T)$. By a separability argument, we see that the null set on which (98) does not hold can be chosen so that it is independent of the choice of $\nu_{0}$. If we select $\nu_{0}=\nu_{ij}(x,t)$ this yields $\theta\leq 0$ almost everywhere with respect to $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$. Since we already know that $\lambda$ is nonnegative this gives $\lambda=0$. Before getting the sharp bound, we also need to check that $V_{ij}$ is well defined, i.e. we need to prove that for any $i\neq j$ we have $V_{i}=-V_{j}$ a.e. with respect to $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$. To see this, we start by observing that if $\xi\in C^{\infty}_{c}([0,1)^{d}\times(0,T))$, thanks to the fact that $\sum_{k\neq i}\chi_{k}=1-\chi_{i}$, we get (99) $\displaystyle\begin{aligned} \int\xi V_{i}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)dt&=-\int\partial_{t}\xi\chi_{i}dxdt\\\ &=\sum_{k\neq i}\int\partial_{t}\xi\chi_{k}dxdt\\\ &=-\sum_{k\neq i}\int\xi V_{k}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt.\end{aligned}$ Choosing $\xi=f(t)g(x)$ for some $f\in C^{\infty}_{c}((0,T))$ and $g\in C^{\infty}([0,1)^{d})$, by a separability argument, we obtain that for a.e. $t$ and every $g\in C^{\infty}([0,1)^{d})$ (100) $\displaystyle\begin{aligned} \int gV_{i}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)&=-\sum_{k\neq i}\int gV_{k}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx).\end{aligned}$ Pick $t$ such that (100) holds. Let $g\in C^{\infty}([0,1)^{d})$ and let $\rho_{m}$ be a partition of unity obtained by the construction of Section 4 applied to the function $\chi(\cdot,t)$ on the interface $\Sigma_{ij}(t)$. Then (101) $\displaystyle\begin{aligned} \sum_{m\in\mathbf{N}}\int\rho_{m}gV_{i}\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)&=-\sum_{m\in\mathbf{N}}\sum_{k\neq i}\int\rho_{m}gV_{k}\mathcal{H}^{d-1}_{\partial^{}\Omega_{k}(t)}(dx).\end{aligned}$ Passing to the limit $r\downarrow 0$ in (101) we get by Lemma 5 that (102) $\displaystyle\begin{aligned} \int gV_{i}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)&=-\int gV_{j}\mathcal{H}^{d-1}_{\Sigma_{ij}(t)}(dx).\end{aligned}$ Since this identity holds for any $g\in C^{\infty}([0,1)^{d})$, a density argument gives $V_{i}(x,t)=-V_{j}(x,t)$ for $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}$-a.e. $x$. In other words (103) $\int\|V_{i}(x,t)+V_{j}(x,t)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)=0.$ Integrating in time yields that $V_{i}=-V_{j}$ a.e. with respect to $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$. We now proceed with the derivation of the sharp lower bound. Define $c_{ij}:=\int K_{ij}(z)dz$. Then we have (104) $\displaystyle\begin{aligned} c_{ij}(\|\delta\chi_{i}\|+\|\delta\chi_{j}\|)=c_{ij}(\delta\chi_{i}^{+}+\delta\chi_{j}^{-}+\delta\chi_{i}^{-}+\delta\chi_{j}^{+})\end{aligned}$ $\displaystyle\begin{aligned} =\frac{1}{2}&\left(\delta\chi_{i}^{+}K_{ij}^{h}(1-\delta\chi_{j}^{-})+(1-\delta\chi_{j}^{-})K_{ij}^{h}\delta\chi_{i}^{+}+\delta\chi_{j}^{-}K_{ij}^{h}(1-\delta\chi_{i}^{+})\right.\\\ &\left.+(1-\delta\chi_{i}^{+})K_{ij}^{h}\delta\chi_{j}^{-}+\delta\chi_{i}^{-}K_{ij}^{h}(1-\delta\chi_{j}^{+})+(1-\delta\chi_{j}^{+})K_{ij}^{h}\delta\chi_{i}^{-}\right.\\\ &\left.+\delta\chi_{j}^{+}K_{ij}^{h}(1-\delta\chi_{i}^{-})+(1-\delta\chi_{i}^{-})K_{ij}^{h}\delta\chi_{j}^{+}\right)+\left(\delta\chi_{i}^{+}K_{ij}^{h}\delta\chi_{j}^{-}\right.\\\ &\left.+\delta\chi_{j}^{-}K_{ij}^{h}\delta\chi_{i}^{+}+\delta\chi_{i}^{-}K_{ij}^{h}\delta\chi_{j}^{+}+\delta\chi_{j}^{+}K_{ij}^{h}\delta\chi_{i}^{-}\right)\end{aligned}$ Now we rewrite the terms in the second parenthesis using $-ab=a_{+}b_{-}+a_{-}b_{+}-a_{+}b_{+}-a_{-}b_{-}$ and then adding and subtracting the contributions of the minority phases we obtain (105) $\displaystyle\begin{aligned} c_{ij}(\|\delta\chi_{i}\|+\|\delta\chi_{j}\|)\leq\frac{1}{2}&\left(\delta\chi_{i}^{+}K_{ij}^{h}(1-\delta\chi_{j}^{-})+(1-\delta\chi_{j}^{-})K_{ij}^{h}\delta\chi_{i}^{+}+\delta\chi_{j}^{-}K_{ij}^{h}(1-\delta\chi_{i}^{+})\right.\\\ &\left.+(1-\delta\chi_{i}^{+})K_{ij}^{h}\delta\chi_{j}^{-}+\delta\chi_{i}^{-}K_{ij}^{h}(1-\delta\chi_{j}^{+})+(1-\delta\chi_{j}^{+})K_{ij}^{h}\delta\chi_{i}^{-}\right.\\\ &\left.+\delta\chi_{j}^{+}K_{ij}^{h}(1-\delta\chi_{i}^{-})+(1-\delta\chi_{i}^{-})K_{ij}^{h}\delta\chi_{j}^{+}\right)-\sum_{l,p}\delta\chi_{l}K_{lp}^{h}\delta\chi_{p}\\\ &+\delta\chi_{i}^{+}K_{ij}^{h}\delta\chi_{j}^{+}+\delta\chi_{i}^{-}K_{ij}^{h}\delta\chi_{j}^{-}+\delta\chi_{j}^{+}K_{ij}^{h}\delta\chi_{i}^{+}\\\ &+\delta\chi_{j}^{-}K_{ij}^{h}\delta\chi_{i}^{-}+\sum_{\\{l,p\\}\neq\\{i,j\\},\\{l,p\\}}\delta\chi_{l}K_{lp}^{h}\delta\chi_{p}.\end{aligned}$ Now the main idea is to split the integral of $K_{ij}$ in the definition of $c_{ij}$ into two parts. More precisely, by the symmetry (19), for any $\nu_{0}\in\mathbf{S}^{d-1}$ and any $V_{0}>0$ we have (106) $c_{ij}=2\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)dz+2\int_{\nu_{0}\cdot z>\alpha V_{0}}K_{ij}(z)dz.$ Substituting into (105) and dividing by $\sqrt{h}$ we obtain (107) $\displaystyle\begin{aligned} 2\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)dz\frac{(\|\delta\chi_{i}\|+\|\delta\chi_{j}\|)}{\sqrt{h}}\end{aligned}$ $\displaystyle\begin{aligned} =\frac{1}{2\sqrt{h}}&\bigg{(}\delta\chi_{i}^{+}K_{ij}^{h}(1-\delta\chi_{j}^{-})+(1-\delta\chi_{j}^{-})K_{ij}^{h}\delta\chi_{i}^{+}+\delta\chi_{j}^{-}K_{ij}^{h}(1-\delta\chi_{i}^{+})\\\ &+(1-\delta\chi_{i}^{+})K_{ij}^{h}\delta\chi_{j}^{-}+\delta\chi_{i}^{-}K_{ij}^{h}(1-\delta\chi_{j}^{+})+(1-\delta\chi_{j}^{+})K_{ij}^{h}\delta\chi_{i}^{-}\\\ &+\delta\chi_{j}^{+}K_{ij}^{h}(1-\delta\chi_{i}^{-})+(1-\delta\chi_{i}^{-})K_{ij}^{h}\delta\chi_{j}^{+}\\\ &-4\int_{\nu_{0}\cdot z>\alpha V_{0}}K_{ij}(z)dz(\|\delta\chi_{i}\|+\|\delta\chi_{j}\|)\\\ &-2\sum_{lp}\delta\chi_{l}K_{lp}^{h}\delta\chi_{p}+2\delta\chi_{i}^{+}K_{ij}^{h}\delta\chi_{j}^{+}+2\delta\chi_{i}^{-}K_{ij}^{h}\delta\chi_{j}^{-}\\\ &+2\delta\chi_{j}^{+}K_{ij}^{h}\delta\chi_{i}^{+}+2\delta\chi_{j}^{-}K_{ij}^{h}\delta\chi_{i}^{-}\\\ &+\sum_{(l,p)\neq(i,j),(l,p)\neq(j,i)}\delta\chi_{l}K_{lp}^{h}\delta\chi_{p}\bigg{)}.\end{aligned}$ We will be interested in bounding the $\liminf$ of the left hand side. Observe that the distributional limit of the last five terms is non-positive. Indeed, the limit of first four terms vanish distributionally by property (87), while the last term is bounded from above by $\sum_{(l,p)\neq(i,j),(l,p)\neq(j,i)}\delta\chi_{l}^{+}K_{lp}^{h}\delta\chi_{p}^{+}+\delta\chi_{l}^{-}K_{lp}^{h}\delta\chi_{p}^{-},$ which vanish distributionally by property (87). We thus obtain that the $\liminf$ of the left hand side of (107) is bounded from above by (108) $\displaystyle\begin{aligned} \liminf_{h\downarrow 0}\frac{1}{2\sqrt{h}}&\bigg{(}\delta\chi_{i}^{+}K_{ij}^{h}(1-\delta\chi_{j}^{-})+(1-\delta\chi_{j}^{-})K_{ij}^{h}\delta\chi_{i}^{+}+\delta\chi_{j}^{-}K_{ij}^{h}(1-\delta\chi_{i}^{+})\\\ &+(1-\delta\chi_{i}^{+})K_{ij}^{h}\delta\chi_{j}^{-}+\delta\chi_{i}^{-}K_{ij}^{h}(1-\delta\chi_{j}^{+})+(1-\delta\chi_{j}^{+})K_{ij}^{h}\delta\chi_{i}^{-}\\\ &+\delta\chi_{j}^{+}K_{ij}^{h}(1-\delta\chi_{i}^{-})+(1-\delta\chi_{i}^{-})K_{ij}^{h}\delta\chi_{j}^{+}\\\ &-4\int_{\nu_{0}\cdot z>\alpha V_{0}}K_{ij}(z)dz(\|\delta\chi_{i}\|+\|\delta\chi_{j}\|)-2\sum_{lp}\delta\chi_{l}K_{lp}^{h}\delta\chi_{p}\bigg{)}.\end{aligned}$ For the last term we use the sharp bound (69), relating this term to our dissipation measure $\omega$. We would like to get a good bound for the other terms. This cannot be done naively as before, since we want the bound to be sharp. We claim that (109) $\displaystyle\begin{aligned} \limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}&\left(\delta\chi_{i}^{+}K_{ij}^{h}(1-\delta\chi_{j}^{-})+(1-\delta\chi_{j}^{-})K_{ij}^{h}\delta\chi_{i}^{+}\delta\chi_{j}^{-}K_{ij}^{h}(1-\delta\chi_{i}^{+})\right.\\\ &\left.+(1-\delta\chi_{i}^{+})K_{ij}^{h}\delta\chi_{j}^{-}+\delta\chi_{i}^{-}K_{ij}^{h}(1-\delta\chi_{j}^{+})+(1-\delta\chi_{j}^{+})K_{ij}^{h}\delta\chi_{i}^{-}\right.\\\ &\left.+\delta\chi_{j}^{+}K_{ij}^{h}(1-\delta\chi_{i}^{-})+(1-\delta\chi_{i}^{-})K_{ij}^{h}\delta\chi_{j}^{+}\right.\\\ &\left.-4\int_{\nu_{0}\cdot z>\alpha V_{0}}K_{ij}(z)dz(\|\delta\chi_{i}\|+\|\delta\chi_{j}\|)\right)\end{aligned}$ $\displaystyle\begin{aligned} \leq 8\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|&dz\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\\\ &+C\sum_{k\neq i,j}(\alpha^{2}\omega^{ac}_{k}+\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt.\end{aligned}$ Here $C$ is a constant that depends on $\gamma,\beta,\mathbb{A},\mathbb{B}$, but not on $h$. Assume for the moment that (109) is true and let us conclude the argument in this case. Using (109) and (69) we obtain (110) $\displaystyle\begin{aligned} 2\liminf_{h\downarrow 0}\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)dz\frac{(\|\delta\chi_{i}\|+\|\delta\chi_{j}\|)}{\sqrt{h}}\end{aligned}$ $\displaystyle\begin{aligned} \leq\alpha^{2}\omega+4\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|&dz\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\\\ &+C\sum_{k\neq i,j}(\alpha^{2}\omega^{ac}_{k}+\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt\end{aligned}$ in the sense of distributions on $[0,1)^{d}\times(0,T)$. Observe also that the left hand side of (110) is an upper bound for $\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)dz(\|\partial_{t}\chi_{i}\|+\|\partial_{t}\chi_{j}\|)$, thus the inequality still holds true if the left hand side is replaced by this term. Remember that $\omega_{k}^{ac}$ is absolutely continuous with respect to $\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt$, thus there exist functions $W_{k}\in L^{1}(\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt)$ such that $\omega_{k}^{ac}=W_{k}(x,t)\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt$. We now disintegrate the measure $\omega$, i.e. we find a Borel family $\omega_{t},t\in(0,T)$, of positive measures on $[0,1)^{d}$ such that $\omega=\omega_{t}\otimes dt$. Having said this, it is not hard to see that (110) holds in a disintegrated version, i.e. we have for Lebesgue a.e. $t\in(0,T)$ (111) $\displaystyle\begin{aligned} 2\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)dz(\|V_{i}(x,t)\|\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{i}(t)}(dx)+\|V_{j}(x,t)\|\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{j}(t)}(dx))\end{aligned}$ $\displaystyle\begin{aligned} \leq\alpha^{2}\omega_{t}+4\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|&dz\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)\\\ &+C\sum_{k\neq i,j}(\alpha^{2}W_{k}(x,t)+1)\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx).\end{aligned}$ Here $\nu_{0}\in\mathbf{S}^{d-1}$ and $V_{0}\in(0,\infty)$ are arbitrary: indeed even if the set of points in time for which (111) holds is a priori dependent on $\nu_{0}$ and $V_{0}$, a standard separability argument allows us to conclude that we can get rid of this dependence. Fix a point $t$ in time such that (111) holds. In what follows, we drop the time variable $t$ which is fixed, so for example $V_{i}(x)=V_{i}(x,t)$, $\Sigma_{ij}=\Sigma_{ij}(t)$ and so on. Fix $\xi\in C([0,1)^{d})$, observe that by definition of $V_{ij}$ and by using the fact that $\Sigma_{ij}\subset\partial^{}\Omega_{i}\cap\partial^{}\Omega_{j}$ we have (112) $\displaystyle\begin{aligned} 4\alpha\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)dz\int_{[0,1)^{d}}\xi(x)\|V_{ij}(x)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\end{aligned}$ $\displaystyle\begin{aligned} \leq\alpha^{2}\int_{[0,1)^{d}}\xi(x)\omega_{t}(dx)&+4\int_{[0,1)^{d}}\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|dz\xi(x)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)\\\ &+C\sum_{k\neq i,j}\int_{[0,1)^{d}}\xi(x)(\alpha^{2}W_{k}(x,t)+1)\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx).\end{aligned}$ Let us relabel $\nu_{0}$, $V_{0}$ and $\xi$ to make clear that they may depend on the pair $i,j$. Thus $\nu_{0}^{ij}\in\mathbf{S}^{d-1}$, $V_{0}^{ij}\in(0,\infty)$ and $\xi_{ij}\in C([0,1)^{d})$ are arbitrary, and it holds (113) $\displaystyle\begin{aligned} 2\alpha\int_{0\leq\nu_{0}^{ij}\cdot z\leq\alpha V^{ij}_{0}}K_{ij}(z)dz\int_{[0,1)^{d}}\xi_{ij}(x)\|V_{ij}(x)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\end{aligned}$ $\displaystyle\begin{aligned} \leq\alpha^{2}\int_{[0,1)^{d}}\xi_{ij}(x)\omega_{t}(dx)&+4\int_{[0,1)^{d}}\int_{0\leq\nu_{0}^{ij}\cdot z\leq\alpha V_{0}^{ij}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|dz\xi_{ij}(x)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)\\\ &+C\sum_{k\neq i,j}\int_{[0,1)^{d}}\xi_{ij}(x)(\alpha^{2}W_{k}(x,t)+1)\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx).\end{aligned}$ Let $\\{\rho_{m}\\}$ be a partition of unity obtained using the construction of Section 4 applied to the function $\chi(\cdot,t)$ on the inferface $\Sigma_{ij}(t)$. Use the above inequality with $\xi_{ij}$ replaced by $\rho_{m}\xi_{ij}$ and sum over $m$ and $i,j$ to get (114) $\displaystyle\begin{aligned} \sum_{i<j}\sum_{m\in\mathbf{N}}\mathbf{LH}_{m}^{ij}\leq\sum_{i<j}\sum_{m\in\mathbf{N}}(\mathbf{I}_{m}^{ij}+\mathbf{II}_{m}^{ij}+\mathbf{III}_{m}^{ij})\end{aligned}$ where we have set $\displaystyle\begin{aligned} \mathllap{\mathbf{LH}_{m}^{ij}}=2\alpha\int_{0\leq\nu_{0}^{ij}\cdot z\leq\alpha V^{ij}_{0}}K_{ij}(z)dz\int_{[0,1)^{d}}\rho_{mij}(x)\xi_{ij}(x)\|V_{ij}(x)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx),\end{aligned}$ $\displaystyle\begin{aligned} \mathllap{\mathbf{I}^{ij}_{m}}=\alpha^{2}\int_{[0,1)^{d}}\xi_{ij}(x)\rho_{mij}(x)\omega_{t}(dx),\end{aligned}$ $\displaystyle\begin{aligned} \mathllap{\mathbf{II}^{ij}_{m}}=4\int_{[0,1)^{d}}\int_{0\leq\nu_{0}^{ij}\cdot z\leq\alpha V_{0}^{ij}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|dz\rho_{mij}(x)\xi_{ij}(x)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx),\end{aligned}$ $\displaystyle\begin{aligned} \mathllap{\mathbf{III}^{ij}_{m}}=C\sum_{k\neq i,j}\int_{[0,1)^{d}}\rho_{mij}(x)\xi_{ij}(x)(\alpha^{2}W_{k}(x,t)+1)\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx).\end{aligned}$ Observe that (115) $\sum_{i<j}\sum_{m\in\mathbf{N}}\mathbf{I}_{m}^{ij}\leq\sum_{i<j}\alpha^{2}\int_{[0,1)^{d}}\xi_{ij}(x)\omega_{t}(dx)$ because $\rho_{m}$ is a partition of unity. Moreover by Lemma 5 we get $\displaystyle\begin{aligned} \mathllap{\lim_{r\downarrow 0}\sum_{i<j}\sum_{m\in\mathbf{N}}\mathbf{LH}_{m}^{ij}}=2\alpha\int_{0\leq\nu_{0}^{ij}\cdot z\leq\alpha V^{ij}_{0}}K_{ij}(z)dz\int_{[0,1)^{d}}\xi_{ij}(x)\|V_{ij}(x)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx).\end{aligned}$ $\displaystyle\begin{aligned} \mathllap{\lim_{r\downarrow 0}\sum_{i<j}\sum_{m\in\mathbf{N}}\mathbf{II}_{m}^{ij}}=\sum_{i<j}4\int_{[0,1)^{d}}\int_{0\leq\nu_{0}^{ij}\cdot z\leq\alpha V_{0}^{ij}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|dz\xi_{ij}(x)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx).\end{aligned}$ $\displaystyle\begin{aligned} \mathllap{\lim_{r\downarrow 0}\sum_{i<j}\sum_{m\in\mathbf{N}}\mathbf{III}_{m}^{ij}}=0.\end{aligned}$ Putting things together we obtain that for any $\nu_{0}^{ij}\in\mathbf{S}^{d-1}$, any $V_{0}^{ij}\in(0,\infty)$, and any $\xi\in C([0,1)^{d})$ (116) $\displaystyle\begin{aligned} 2\alpha\sum_{i<j}\int_{0\leq\nu_{0}^{ij}\cdot z\leq\alpha V^{ij}_{0}}K_{ij}(z)dz\int_{[0,1)^{d}}\xi_{ij}(x)\|V_{ij}(x)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\end{aligned}$ $\displaystyle\begin{aligned} \leq&\sum_{i<j}\alpha^{2}\int_{[0,1)^{d}}\xi_{ij}(x)\omega_{t}(dx)\\\ &+4\sum_{i<j}\int_{[0,1)^{d}}\int_{0\leq\nu_{0}^{ij}\cdot z\leq\alpha V_{0}^{ij}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|dz\xi_{ij}(x)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx).\end{aligned}$ We now claim that by approximation the above inequality is valid for any simple function $\xi_{ij}\geq 0$. To see this, it is clear that we can concentrate on $\xi_{ij}=w_{ij}\mathbf{1}_{B_{ij}}$, where $B_{ij}\subset[0,1)^{d}$ are Borel and $w_{ij}\geq 0$. Observe that by the dominated convergence theorem, the family (117) $\mathcal{F}:=\left\\{B=\prod_{i<j}B_{ij}:\ B_{ij}\in\mathcal{B}([0,1)^{d})\ \text{s.t.}\ \forall w_{ij}\geq 0\ \text{ (\ref{inequalityToApr}) holds with}\ \xi_{ij}=w_{ij}\mathbf{1}_{B_{ij}}\right\\}$ is a monotone class. Thus by the monotone class theorem we just need to show that it contains all the products of open sets. But this is easy because given $B_{ij}\subset[0,1)^{d}$ open sets, we can always find sequences $\eta_{k}^{ij}$ of continuous functions with compact support such that $0\leq\eta_{k}^{ij}\leq\mathbf{1}_{B_{ij}}$ and such that $\eta_{k}^{ij}\to\mathbf{1}_{B_{ij}}$, thus the claim follows by the monotone convergence theorem. With this in place one can use an approximation argument to replace the vector $\nu_{0}^{ij}$ with the $\mathcal{H}^{d-1}$-measurable vector valued function $\nu_{ij}$ obtaining the following inequality: (118) $\displaystyle\begin{aligned} 2\alpha\sum_{i<j}\int_{[0,1)^{d}}\int_{0<\nu_{ij}(x)\cdot z<\alpha V^{ij}_{0}}K_{ij}(z)dz\xi_{ij}(x)\|V_{ij}(x)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\leq\end{aligned}$ $\displaystyle\begin{aligned} \leq&\sum_{i<j}\alpha^{2}\int_{[0,1)^{d}}\eta_{ij}(x)\omega_{t}(dx)\\\ &+4\sum_{i<j}\int_{[0,1)^{d}}\int_{0\leq\nu_{ij}\cdot z\leq\alpha V_{0}^{ij}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|dz\xi_{ij}(x)\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx).\end{aligned}$ Now divide by $\alpha^{2}$ and send $\alpha$ to zero. Record the following limits, which can be computed spelling out the definition of $K_{ij}$, and recalling the symmetry property (19) and the factorization property (22) for the heat kernel (119) $\displaystyle\begin{aligned} \lim_{\alpha\downarrow 0}\frac{1}{\alpha}\int_{0<\nu_{ij}(x)\cdot z<\alpha V_{0}^{ij}}K_{ij}(z)dz=\frac{V_{0}^{ij}}{\mu_{ij}}.\end{aligned}$ $\displaystyle\begin{aligned} \lim_{\alpha\downarrow 0}\frac{1}{\alpha^{2}}\int_{0\leq\nu_{ij}(x)\cdot z\leq\alpha V_{0}^{ij}}K_{ij}(z)\|\nu_{ij}(x)\cdot z\|dz=\frac{(V_{0}^{ij})^{2}}{4\mu_{ij}}.\end{aligned}$ Then if we insert back into (118) we obtain (120) $\displaystyle\sum_{i<j}\frac{2}{\mu_{ij}}\int_{[0,1)^{d}}V_{0}^{ij}\xi_{ij}(x)\|V_{ij}(x)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)$ (121) $\displaystyle\leq\int_{[0,1)^{d}}\omega_{t}(dx)+\sum_{i<j}\int_{[0,1)^{d}}\frac{(V_{0}^{ij})^{2}}{\mu_{ij}}\xi_{ij}(x)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx).$ Now given $M>0$ take sequences of simple functions (122) $s_{m}^{ij}=\sum_{k=1}^{p_{m}}w_{k}^{ij}\mathbf{1}_{B^{k}_{ij}}$ such that $s_{m}^{ij}\to\|V_{ij}\|\mathbf{1}_{\\{\|V_{ij}\|\leq M\\}}$ as $m\to+\infty$ monotonically almost everywhere with respect to $\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}$. We are assuming that $\\{B_{ij}^{k}\\}_{k=1,...,p_{m}}$ are disjoint and $\mathcal{H}^{d-1}$-measurable, with the property that $B_{ij}^{k_{1}}\cap B_{lr}^{k_{2}}=\emptyset$ if $\\{i,j\\}\neq\\{l,r\\}$. Choosing $V_{0}^{ij}=w_{k}^{ij}$, $\xi_{ij}=\mathbf{1}_{B^{k}_{ij}}$ in (120) and summing over $k$ we obtain (123) $\displaystyle\sum_{i<j}\frac{2}{\mu_{ij}}\int_{[0,1)^{d}}s_{m}^{ij}\|V_{ij}(x)\|\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)$ $\displaystyle\leq\int_{[0,1)^{d}}\omega_{t}(dx)+\sum_{i<j}\int_{[0,1)^{d}}\frac{(s_{m}^{ij})^{2}}{\mu_{ij}}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx).$ Taking the limit $m\to+\infty$, using the monotone convergence theorem we obtain (124) $\displaystyle\sum_{i<j}\frac{2}{\mu_{ij}}\int_{[0,1)^{d}}\|V_{ij}(x)\|^{2}\mathbf{1}_{\\{\|V_{ij}\|\leq M\\}}\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)$ $\displaystyle\leq\int_{[0,1)^{d}}\omega_{t}(dx)+\sum_{i<j}\int_{[0,1)^{d}}\frac{\|V_{ij}(x)\|^{2}}{\mu_{ij}}\mathbf{1}_{\\{\|V_{ij}\|\leq M\\}}\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)$ or, in other words, $\displaystyle\sum_{i<j}\frac{1}{\mu_{ij}}\int_{[0,1)^{d}}\|V_{ij}(x)\|^{2}\mathbf{1}_{\\{\|V_{ij}\|\leq M\\}}\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\leq\int_{[0,1)^{d}}\omega_{t}(dx).$ Recall that $\mu_{ij}=\mu_{ji}$, thus the inequality above may be rewritten as $\displaystyle\sum_{i,j}\frac{1}{2\mu_{ij}}\int_{[0,1)^{d}}\|V_{ij}(x)\|^{2}\mathbf{1}_{\\{\|V_{ij}\|\leq M\\}}\mathcal{H}^{d-1}_{\|\Sigma_{ij}}(dx)\leq\int_{[0,1)^{d}}\omega_{t}(dx).$ If we now integrate in time we learn by the monotone convergence theorem that $V_{ij}\in L^{2}(\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt)$ and that the sharp bound (45) is satisfied. ### Proof of (109) To prove (109) we proceed in several steps. First of all, we claim that the first eight terms may be substituted by (125) $\displaystyle 2\int_{\nu_{0}\cdot z\geq 0}K_{ij}^{h}(z)$ $\displaystyle\left(\|\delta\chi_{i}^{+}-\delta\chi_{j}^{-}(-z)\|+\|\delta\chi_{i}^{+}(-z)-\delta\chi_{j}^{-}\|\right.$ $\displaystyle\left.\|\delta\chi_{i}^{-}-\delta\chi_{j}^{+}(-z)\|+\|\delta\chi_{i}^{-}(-z)-\delta\chi_{j}^{+}\|\right)dz.$ To show this, observe that we may replace the implicit $z$-integrals in the convolution in the first eight terms by twice the integrals over the half space $\\{\nu_{0}\cdot z\geq 0\\}$ instead of $\mathbf{R}^{d}$. This is clearly true once we observe that (126) $\displaystyle\begin{aligned} \lim_{h\downarrow 0}\frac{1}{\sqrt{h}}&\left(\delta\chi_{i}^{+}\int_{\nu_{0}\cdot z\geq 0}K_{ij}^{h}(z)(1-\delta\chi_{j}^{-}(\cdot-z))dz\right.\\\ &\left.+(1-\delta\chi_{j}^{-})\int_{\nu_{0}\cdot z\geq 0}K_{ij}^{h}(z)\delta\chi_{i}^{+}(\cdot-z)dz\right)\end{aligned}$ $\displaystyle\begin{aligned} \mathllap{=}\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}&\left(\delta\chi_{i}^{+}\int_{\nu_{0}\cdot z\leq 0}K_{ij}^{h}(z)(1-\delta\chi_{j}^{-}(\cdot-z))dz\right.\\\ &\left.+(1-\delta\chi_{j}^{-})\int_{\nu_{0}\cdot z\leq 0}K_{ij}^{h}(z)\delta\chi_{i}^{+}(\cdot-z)dz\right)\end{aligned}$ and that similar identities hold exchanging the roles of $i,j$ and $+,-$ respectively. That (126) holds is not difficult to show. Indeed taking into account the fact that the kernel is even, the argument of the second limit is just a spatial shift of $z$ of the first one. The spatial shift may be put onto the test function, and thanks to the scaling of the kernel one can get the claim. We may thus substitute the first eight terms of the left hand side of (109) with twice the same terms with the integration with respect to $z$ on the half space $\\{\nu_{0}\cdot z\geq 0\\}$. If we rely again on the fact that $\delta\chi_{i}^{+}\in\\{0,1\\}$, by identity (150) in the Appendix we obtain (125), as claimed. Now we need two inequalities for the integrand. First note that the integrand is a second-order finite difference, we claim that (127) $\displaystyle\begin{aligned} \|\delta\chi_{i}^{+}-\delta\chi_{j}^{-}(\cdot-z)\|+\|\delta\chi_{i}^{+}(\cdot-z)-\delta\chi_{j}^{-}\|+\|\delta\chi_{i}^{-}-\delta\chi_{j}^{+}(\cdot-z)\|+\|\delta\chi_{i}^{-}(\cdot-z)-\delta\chi_{j}^{+}\|\end{aligned}$ $\displaystyle\begin{aligned} \leq\begin{cases}\|\delta\chi_{i}^{+}-\delta\chi_{i}^{+}(\cdot-z)\|+\|\delta\chi_{i}^{-}-\delta\chi_{i}^{-}(\cdot-z)\|+\|\delta\chi_{j}^{+}-\delta\chi_{j}^{+}(\cdot-z)\|+\|\delta\chi_{j}^{-}-\delta\chi_{j}^{-}(\cdot-z)\|\\\ +4\sum_{k\neq i,j}(\|\delta\chi_{k}\|+\delta\chi_{k}(\cdot-z)\|).\\\ \mathllap{}\\\ \|\delta\chi_{i}\|+\|\delta\chi_{i}(\cdot-z)\|+\|\delta\chi_{j}\|+\|\delta\chi_{j}(\cdot-z)\|.\end{cases}\end{aligned}$ The second one follows from the triangle inequality. To show the first one, observe that (128) $\displaystyle\begin{aligned} \mathllap{\|\delta\chi_{i}^{+}-\delta\chi_{j}^{-}(\cdot-z)\|}=&(1-\delta\chi_{i}^{+})\delta\chi_{j}^{-}(\cdot-z)+\delta\chi_{i}^{+}(1-\delta\chi_{j}^{-}(\cdot-z))\end{aligned}$ $\displaystyle\begin{aligned} \mathllap{}\leq&(1-\delta\chi_{i}^{+})\delta\chi_{i}^{+}(\cdot-z)+\sum_{k\neq i,j}\|\delta\chi_{k}(\cdot-z)\|+\delta\chi_{j}^{+}(1-\delta\chi_{j}^{+}(\cdot-z))\\\ &+\sum_{k\neq i,j}\|\delta\chi_{k}\|\end{aligned}$ and that similarly (129) $\displaystyle\begin{aligned} \mathllap{\|\delta\chi_{i}^{+}(\cdot-z)-\delta\chi_{j}^{-}\|}=&(1-\delta\chi_{i}(\cdot-z)^{+})\delta\chi_{j}^{-}+\delta\chi_{i}(\cdot-z)^{+}(1-\delta\chi_{j})\end{aligned}$ $\displaystyle\begin{aligned} \mathllap{}\leq&(1-\delta\chi_{i}(\cdot-z)^{+})\delta\chi_{i}^{-}+\sum_{k\neq i,j}\|\delta\chi_{k}\|+\delta\chi_{j}(\cdot-z)^{+}(1-\delta\chi_{j}^{-})\\\ &+\sum_{k\neq i,j}\|\delta\chi_{k}(\cdot-z)\|.\end{aligned}$ Summing up the two inequalities we get (130) $\displaystyle\|\delta\chi_{i}^{+}-\delta\chi_{j}^{-}(\cdot-z)\|+\|\delta\chi_{i}^{+}(\cdot-z)-\delta\chi_{j}^{-}\|\leq$ $\displaystyle\leq\|\delta\chi_{i}^{+}-\delta\chi_{i}^{+}(\cdot-z)\|+\|\delta\chi_{j}^{-}-\delta\chi_{j}^{-}(\cdot-z)\|+2\sum_{k\neq i,j}\|\delta\chi_{k}\|+\|\delta\chi_{k}(\cdot-z)\|.$ Similar bounds hold for the remaining terms in (127). We now split the integral (125) into the domains of integration $\\{0\leq\nu_{0}\cdot z\leq\alpha V_{0}\\}$ and $\\{\nu_{0}\cdot z>\alpha V_{0}\\}$. On the first one we use the first inequality in (127) for the integrand. Recalling identity (150) and inequality (151) in the Appendix we obtain, and using the fact that $\sum_{k}\chi_{k}=1$ $\displaystyle\begin{aligned} 2\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}^{h}(z)&\left(\|\delta\chi_{i}^{+}-\delta\chi_{j}^{-}(\cdot-z)\|+\|\delta\chi_{i}^{+}(\cdot-z)-\delta\chi_{j}^{-}\|\right.\\\ &\left.+\|\delta\chi_{i}^{-}-\delta\chi_{j}^{+}(\cdot-z)\|+\|\delta\chi_{i}^{-}(\cdot-z)-\delta\chi_{j}^{+}\|\right)dz\end{aligned}$ $\displaystyle\begin{aligned} \leq 2\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}^{h}(z)&\left(\|\chi_{i}-\chi_{i}(\cdot-z)\|+\|\chi_{i}(-\tau)-\chi_{i}(\cdot-\tau,\cdot-z)\|\right.\\\ &\left.\|\chi_{j}-\chi_{j}(\cdot-z)\|+\|\chi_{j}(\cdot-\tau)-\chi_{j}(\cdot-\tau,\cdot-z)\|\right.\\\ &\left.+8\sum_{k\neq i,j}\|\delta\chi_{k}\|+\|\delta\chi_{k}(\cdot-z)\|\right)dz\end{aligned}$ (131) $\displaystyle\begin{aligned} \leq 2\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}^{h}(z)&\left(\chi_{i}\chi_{j}(\cdot-z)+\chi_{i}(\cdot-z)\chi_{j}+\sum_{k\neq i,j}\chi_{i}\chi_{k}(\cdot-z)+\chi_{i}(\cdot-z)\chi_{k}\right.\\\ &\left.\chi_{i}(\cdot-\tau)\chi_{j}(\cdot-\tau,\cdot-z)+\chi_{i}(\cdot-\tau,\cdot-z)\chi_{j}(\cdot-\tau)\right.\\\ &\left.+\sum_{k\neq i,j}\chi_{i}(\cdot-\tau)\chi_{k}(\cdot-\tau,\cdot-z)+\chi_{i}(\cdot-\tau,\cdot-z)\chi_{k}(\cdot-\tau)\right.\\\ &\left.\chi_{j}\chi_{i}(\cdot-z)+\chi_{j}(\cdot-z)\chi_{i}+\sum_{k\neq i,j}\chi_{j}\chi_{k}(\cdot-z)+\chi_{j}(\cdot-z)\chi_{k}\right.\\\ &\left.\chi_{j}(\cdot-\tau)\chi_{i}(\cdot-\tau,\cdot-z)+\chi_{j}(\cdot-\tau,\cdot-z)\chi_{i}(\cdot-\tau)\right.\\\ &\left.+\sum_{k\neq i,j}\chi_{j}(\cdot-\tau)\chi_{k}(\cdot-\tau,\cdot-z)+\chi_{j}(\cdot-\tau,\cdot-z)\chi_{k}(\cdot-\tau)\right.\\\ &\left.+8\sum_{k\neq i,j}\|\delta\chi_{k}\|+\|\delta\chi_{k}(\cdot-z)\|\right)dz.\end{aligned}$ On the set $\\{\nu_{0}\cdot z>\alpha V_{0}\\}$ we use the second inequality in (127), obtaining (132) $\displaystyle\begin{aligned} 2\int_{\nu_{0}\cdot z>\alpha V_{0}}K_{ij}^{h}(z)&\left(\|\delta\chi_{i}^{+}-\delta\chi_{j}^{-}(\cdot-z)\|+\|\delta\chi_{i}^{+}(\cdot-z)-\delta\chi_{j}^{-}\|\right.\\\ &\left.+\|\delta\chi_{i}^{-}-\delta\chi_{j}^{+}(\cdot-z)\|+\|\delta\chi_{i}^{-}(\cdot-z)-\delta\chi_{j}^{+}\|\right)dz\end{aligned}$ $\displaystyle\begin{aligned} \leq 2\int_{\nu_{0}\cdot z>\alpha V_{0}}K_{ij}^{h}(z)(\|\delta\chi_{i}\|+\|\delta\chi_{i}(\cdot-z)\|+\|\delta\chi_{j}\|+\|\delta\chi_{j}(\cdot-z)\|)dz.\end{aligned}$ We now observe that for any $1\leq k\leq N$ we have, as we already observed in (96) (133) $\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}^{h}(z)(\|\delta\chi_{k}(\cdot-z)\|-\|\delta\chi_{k}\|)dz=0,$ thus in particular (134) $\displaystyle\limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}^{h}(z)\|\delta\chi_{k}(\cdot-z)\|dz$ $\displaystyle=\limsup_{h\downarrow 0}\frac{1}{\sqrt{h}}\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}^{h}(z)\|\delta\chi_{k}\|dz.$ By putting the time shift $\tau$ on the test function if is easy to check that the distributional limit of the terms of (131) which involve the shift $\tau$ have the same limit as the corresponding terms without the time shift. Thus recalling (86) and relying on (133) and (42) we obtain that inserting (131) and (132) into (125), the left hand side of (109) is bounded by $\displaystyle\begin{aligned} &8\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ij}(z)((\nu_{ij}(x,t)\cdot z)_{+}+(\nu_{ij}(x,t)\cdot z)_{-})\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\\\ &+C\sum_{k\neq i,j}\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ik}(z)((\nu_{ik}(x,t)\cdot z)_{+}+(\nu_{ik}(x,t)\cdot z)_{-})\mathcal{H}^{d-1}_{\|\Sigma_{ik}(t)}(dx)dt\\\ &+C\sum_{k\neq i,j}(\alpha^{2}\omega_{k}^{ac}+\mathcal{H}^{d-1}_{\|\partial^{}\Omega_{k}(t)}(dx)dt),\end{aligned}$ which clearly gives the claim once we realize that $\displaystyle\begin{aligned} &\int_{0\leq\nu_{0}\cdot z\leq\alpha V_{0}}K_{ik}(z)((\nu_{ik}(x)\cdot z)_{+}+(\nu_{ik}(x)\cdot z)_{-})\mathcal{H}^{d-1}_{\|\Sigma_{ik}(t)}(dx)dt\\\ &\leq 2\int_{\mathbf{R}^{d}}K_{ik}(z)\|z\|dz\leq C.\end{aligned}$ ∎ ###### Proof of Proposition 2. The proof is along the same lines as Proposition 2 in [19], where the claim is analized in the case of two phases. For the convenience of the reader, we outline the strategy of the full proof, providing details only for the required changes. The proof is split into several steps. step 1. The first observation is that for any $h>0$, any admissible $u\in\mathcal{M}$ and any smooth vector field $\xi$ we have the following lower bound for the metric slope, cf. (33) $\frac{1}{2}\|\partial E_{h}\|(u)\geq\delta E_{h}(u)_{\bullet}\xi-\frac{1}{2}\left(\delta d_{h}(\cdot,u)_{\bullet}\xi\right)^{2}.$ Here $\delta$ denotes the first variation, which is computed considering the curve $s\to u_{s}$ of configurations which solve the transport equations (135) $\begin{cases}\partial_{s}u_{i}^{s}+\xi\cdot\nabla u_{i}^{s}=0,\\\ u_{i}^{s}(\cdot,0)=u_{i}(\cdot),\end{cases}$ and by setting (136) $\delta E_{h}(u)_{\bullet}\xi:=\frac{d}{ds}_{\|{s=0}}E_{h}(u^{s})\ \text{and}\ \delta d_{h}(\cdot,u)_{\bullet}\xi:=\frac{d}{ds}_{\|{s=0}}d(u,u^{s}).$ step 2. The second observation is a representation formula for $\delta E_{h}(u)_{\bullet}\xi$. Namely (137) $\displaystyle\begin{aligned} \delta E_{h}(u)_{\bullet}\xi=\sum_{i,j}\frac{1}{\sqrt{h}}&\left(\int\nabla\cdot\xi u_{i}K_{ij}^{h}u_{j}dx+\int\nabla\cdot\xi u_{j}K_{ij}^{h}u_{i}dxdt\right.\\\ &\left.+\int[\xi,\nabla K_{ij}^{h}](u_{j})u_{i}dx\right).\end{aligned}$ Here $[\xi,\nabla K_{ij}^{h}]$ denotes the commutator obtained taking the convolution with $\nabla K_{ij}^{h}$ and multiplying by $\xi$. To check this formula one starts by assuming $u$ to be smooth and then an approximation argument gives the result for a general $u\in\mathcal{M}$. step 3. Representation for $\delta d_{h}(\cdot,u)_{\bullet}\xi$. One checks that $\displaystyle\begin{aligned} &\frac{1}{2}\left(\delta d_{h}(\cdot,u)_{\bullet}\xi\right)^{2}\\\ &=\frac{\sqrt{h}}{2}\sum_{i,j}\left(\int u_{i}\xi\cdot\nabla^{2}K_{ij}^{h}(\xi u_{j})dx+\int u_{j}\xi\cdot\nabla^{2}K_{ij}^{h}(\xi u_{i})dx\right.\\\ &\left.+\int u_{i}\nabla\cdot\xi\nabla K_{ij}^{h}(\xi u_{j})dx++\int u_{j}\nabla\cdot\xi\nabla K_{ij}^{h}(\xi u_{i})dx\right.\\\ &\left.-\int u_{i}\nabla\cdot\xi K_{ij}^{h}(u_{j}\nabla\cdot\xi)dx-\int u_{j}\nabla\cdot\xi K_{ij}^{h}(u_{i}\nabla\cdot\xi)dx\right.\\\ &\left.-\int\xi u_{i}\nabla K_{ij}^{h}(u_{j}\nabla\cdot\xi)dx-\int\xi u_{j}\nabla K_{ij}^{h}(u_{i}\nabla\cdot\xi)dx\right).\end{aligned}$ Once again this formula can be easily checked when $u$ is smooth, an approximation argument then gives the extension to the case $u\in\mathcal{M}$. step 4. Passage to the limit in $\delta E_{h}$. We claim that (138) $\lim_{h\downarrow 0}\int_{0}^{T}\delta E_{h}(u^{h}(t))_{\bullet}\xi dt=\sum_{i,j}\sigma_{ij}\int\left(\nabla\cdot\xi-\nu_{ij}\cdot\nabla\xi\nu_{ij}\right)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt.$ The proof is very similar to the two phases case, and relies on the weak convergence (42). Firstly, testing (42) with $\nabla\cdot\xi$ we get $\displaystyle\begin{aligned} &\lim_{h\downarrow 0}\sum_{i,j}\frac{1}{\sqrt{h}}\int\left(\nabla\cdot\xi u_{i}^{h}K_{ij}^{h}u_{j}^{h}+\nabla\cdot\xi u_{j}^{h}K_{ij}^{h}u_{i}^{h}\right)dxdt\\\ &=\sum_{i,j}2\sigma_{ij}\int\nabla\cdot\xi\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt.\end{aligned}$ For the term involving the commutator, one checks that $\displaystyle\lim_{h\downarrow 0}\left(\int[\xi,\nabla K_{ij}^{h}](u_{j}^{h})u_{i}^{h}dxdt-\int\nabla\xi z\cdot\nabla K_{ij}^{h}(z)u_{j}^{h}(x-z,t)u_{i}^{h}(x,t)dzdxdt\right)=0.$ With this in place, we observe that $\displaystyle\begin{aligned} &\int\nabla\xi z\cdot\nabla K_{ij}^{h}(z)u_{j}^{h}(x-z,t)u_{i}^{h}(x,t)dzdxdt\\\ &=\int\nabla\xi(x,t)z\cdot\nabla K_{ij}(z)(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\end{aligned}$ which can be seen by testing (42) with $\frac{\nabla\xi z\cdot\nabla K_{ij}(z)}{K_{ij}(z)}$ which is of polynomial growth in $z$. To conclude (138) we just need to show that for any symmetric matrix $A\in\mathbf{R}^{d\times d}$ and any unit vector $\nu$ we have $\int Az\cdot\nabla K_{ij}(z)(\nu\cdot z)_{+}dz=-\sigma_{ij}\left(\operatorname{tr}A+\nu\cdot A\nu\right).$ Using the definition of the kernel $K_{ij}$ it suffices to show that $\int Az\cdot\nabla G_{w}(z)(\nu\cdot z)_{+}dz=-\frac{\sqrt{w}}{\sqrt{\pi}}\left(\operatorname{tr}A+\nu\cdot A\nu\right)\ w\in\\{\gamma,\beta\\}.$ step 5. Passage to the limit in $\delta d_{h}(\cdot,u)\xi$. We claim that (139) $\displaystyle\begin{aligned} \lim_{h\downarrow 0}\frac{1}{2}\left(\delta d_{h}(\cdot,u^{h})_{\bullet}\xi\right)=\sum_{i,j}\frac{1}{2\mu_{ij}}\int(\xi\cdot\nu_{ij})^{2}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt.\end{aligned}$ To prove this, we observe that the terms which do not involve the Hessian $\nabla^{2}K_{ij}^{h}$ are all $O(\sqrt{h})$. For example, to prove that (140) $\sqrt{h}\int u_{i}^{h}\nabla\cdot\xi\nabla K_{ij}^{h}(\xi u_{j}^{h})dxdt=O(\sqrt{h}),$ spell out the integral in the convolution, use the fact that $\nabla K_{ij}^{h}=\frac{1}{\sqrt{h}^{d+1}}\nabla K_{ij}(\frac{z}{\sqrt{h}})$, use the fact that $\nabla\xi(x,t)\xi(x-\sqrt{h}z,t)$ is bounded and test (42) with $\nabla K_{ij}/K_{ij}$. The other terms can be treated similarly. For the terms involving the Hessian of the kernel, we split the claim into (141) $\displaystyle\begin{aligned} \lim_{h\downarrow 0}\sqrt{h}\int u_{i}^{h}(\xi\cdot\nabla^{2}K_{ij}^{h}u_{j})\xi dxdt=\frac{1}{2\mu_{ij}}\int(\xi\cdot\nu_{ij}(x,t))^{2}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt,\end{aligned}$ (142) $\displaystyle\begin{aligned} \sqrt{h}\int u_{i}^{h}\xi\cdot[\xi,\nabla^{2}K_{ij}^{h}](u_{j}^{h})dxdt=O(\sqrt{h}).\end{aligned}$ The proof of (142) is similar to the argument for (140). To prove identity (141) observe that by spelling out the $z$-integral, a change of variable and by testing (42) with $\frac{\xi(x,t)\cdot\nabla^{2}K_{ij}(z)\xi(x,t)}{K_{ij}(z)}$ we obtain $\displaystyle\begin{aligned} &\lim_{h\downarrow 0}\sqrt{h}\int u_{i}^{h}(\xi\cdot\nabla^{2}K_{ij}^{h}u^{j})\xi dxdt\\\ &=\int\xi\cdot\nabla^{2}K_{ij}(z)\xi(\nu_{ij}(x,t)\cdot z)_{+}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt.\end{aligned}$ Now identity (139) follows from the following formula: for any two vectors $\xi\in\mathbf{R}^{d}$ and $\nu\in\mathbf{S}^{d-1}$ we have (143) $\int\xi\cdot\nabla^{2}K_{ij}(z)\xi(\nu\cdot z)_{+}dz=\frac{1}{2\mu_{ij}}(\xi\cdot\nu)^{2}.$ To check (143), by relying on the definition of the kernels, we just need to show that for $w\in\\{\gamma,\beta\\}$ $\int\xi\cdot\nabla^{2}G_{w}(x)\xi(\nu\cdot z)_{+}dz=\frac{1}{2\sqrt{\pi w}}(\xi\cdot\nu)^{2}.$ Since the kernel is isotropic, we can reduce to the case $\xi=e_{1}$, thus we need to prove $\int\partial^{2}_{1}G_{w}(x)(\nu\cdot z)_{+}dz=\frac{1}{2\sqrt{\pi w}}\nu_{1}^{2}.$ This can be done after two integration by parts and observing that $\int_{\nu\cdot z=0}G_{w}(z)dz=\frac{1}{2\sqrt{\pi w}}.$ conclusion. By step 1 we have $\frac{1}{2}\int_{0}^{T}\|\partial E_{h}\|^{2}(u^{h})\ dt\geq\int_{0}^{T}\delta E_{h}(u^{h})_{\bullet}\xi dt-\frac{1}{2}\int_{0}^{T}\left(\delta d_{h}(\cdot,u^{h})_{\bullet}\xi\right)^{2}dt.$ Taking the liminf on the left hand side, using step 4 and step 5 we get that for any smooth vector field $\xi$ $\displaystyle\begin{aligned} \liminf_{h\downarrow 0}\frac{1}{2}\int_{0}^{T}\|\partial E_{h}\|^{2}(u^{h})dt\geq&\sum_{i,j}\left[\sigma_{ij}\int\left(\nabla\cdot\xi-\nu_{ij}\cdot\nabla\xi\nu_{ij}\right)\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\right.\\\ &\left.-\frac{1}{2\mu_{ij}}\int(\xi\cdot\nu_{ij})^{2}\ \mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\right].\end{aligned}$ Since the left hand side is bounded, the Riesz representation theorem for $L^{2}$ yields functions $H_{ij}\in L^{2}(\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt)$ such that $\sum_{i,j}\sigma_{ij}\int\left(\nabla\cdot\xi-\nu_{ij}\cdot\nabla\xi\nu_{ij}\right)\ \mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt=-\sum_{i,j}\sigma_{ij}\int H_{ij}\nu_{ij}\cdot\xi\ \mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt$ and such that for any $\xi\in L^{2}(\mathcal{H}^{d-1}_{\|\bigcup_{i,j}\Sigma_{ij}(t)}(dx)dt)$ $\displaystyle\begin{aligned} \liminf_{h\downarrow 0}\frac{1}{2}\int_{0}^{T}\|\partial E_{h}\|(u_{h})\ dt\geq\sum_{i,j}\bigg{(}&-\sigma_{ij}\int H_{ij}\nu_{ij}\cdot\xi\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\\\ &-\frac{1}{2\mu_{ij}}\int(\xi\cdot\nu_{ij})^{2}\mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt\bigg{)}.\end{aligned}$ Since the integration measures are mutually singular we can test with $\xi\in L^{2}(\mathcal{H}^{d-1}_{\|\bigcup_{i,j}\Sigma_{ij}(t)}(dx)dt)$ such that $\xi_{\|\Sigma_{ij}(t)}=-\mu_{ij}\sigma_{ij}H_{ij}\nu_{ij}$. This yields $\displaystyle\begin{aligned} \liminf_{h\downarrow 0}\frac{1}{2}\int_{0}^{T}\|\partial E_{h}\|^{2}(u_{h})\ dt\geq&\sum_{i,j}\frac{\sigma_{ij}^{2}\mu_{ij}}{2}\int H_{ij}^{2}\ \mathcal{H}^{d-1}_{\|\Sigma_{ij}(t)}(dx)dt.\end{aligned}$ ∎ ## 6\. Appendix ### 6.1. Proof of Lemma 5 Before giving the proof of this result, we need a simple technical lemma. ###### Lemma 6. Fix $1\leq l\neq p\leq N$. Then for any $1\leq i\neq j\leq N$ such that $\\{i,j\\}\neq\\{l,p\\}$ the interfaces $\Sigma_{ij}$ and $\Sigma_{lp}$ are disjoint. In particular for $\mathcal{H}^{d-1}$-a.e. $x\in\Sigma_{lp}$ we have that (144) $\lim_{r\to 0}\mathcal{H}^{d-1}(\Sigma_{ij}\cap B(x,r))=0$ ###### Proof. We first show that the interfaces $\Sigma_{ij}$ and $\Sigma_{lp}$ are disjoint. This follows immediately once we recall that every point in the reduced boundary of a set of finite perimeter has density $1/2$ (see [24], Corollary 15.8). Assume for example that $i\neq l,p$. Thus if $y\in\Sigma_{lp}$ we have (145) $\begin{split}&1\geq\limsup_{r}\frac{\|(\Omega_{l}\cup\Omega_{p}\cup\Omega_{i})\cap B(y,r)\|}{\omega_{d}r^{d}}\\\ &=\lim_{r}\frac{\|\Omega_{l}\cap B(y,r)\|}{\omega_{d}r^{d}}+\lim_{r}\frac{\|\Omega_{p}\cap B(y,r)\|}{\omega_{d}r^{d}}+\limsup_{r}\frac{\|\Omega_{i}\cap B(y,r)\|}{\omega_{d}r^{d}}\\\ &=1+\limsup_{r}\frac{\|\Omega_{i}\cap B(y,r)\|}{\omega_{d}r^{d}}\end{split}$ which says that $y$ has density zero in $\Omega_{i}$. The fact that (144) holds is now a consequence of the geneneral fact $\limsup_{r\downarrow 0}\frac{\mathcal{H}^{d-1}(\Sigma_{ij}\cap B(x,r))}{\omega_{d-1}r^{d-1}}=0$ for $\mathcal{H}^{d-1}$-a.e. $x\in\Sigma_{ij}^{c}$. ∎ ###### Proof of Lemma 5. The argument for (1) can be found in [17] in the case of two phases and without localization, i.e. with $\eta=1$ and $N=2$. For the sake of completeness, we provide the proof in our case. Upon splitting into the negative and positive part, we may assume $\eta\geq 0$. Clearly the only nonzero terms in the sum are those for wich $B_{m}^{r}\cap\Sigma_{ij}\neq\emptyset$. Fix such a ball: by definition there exists $y\in r\mathbf{Z}^{d}$ such that $B_{m}^{r}=B(y,2r\sqrt{d})$. If $x\in\Sigma_{ij}\cap B_{m}^{r}$ then we have that $B(x,2r\sqrt{d})\subset B(y,4r\sqrt{d})$, and by definition of $\mathcal{E}^{r}$ this yields $\mathcal{H}^{d-1}(B(x,2r\sqrt{d})\cap\Sigma_{ij})\leq\frac{\omega_{d-1}}{2^{d}}(4r)^{d-1}\sqrt{d}^{d-1}=\frac{\omega_{d-1}}{2}(2r)^{d-1}\sqrt{d}^{d-1}.$ Thus $x$ belongs to the set of points in $\Sigma_{ij}\cap B_{m}^{r}$ such that (146) $\frac{\mathcal{H}^{d-1}(B(x,2r\sqrt{d})\cap\Sigma_{ij})}{\omega_{d-1}(2r\sqrt{d})^{d-1}}\leq\frac{1}{2}.$ By De Giorgi’s structure theorem the approximate tangent plane exists at every point $x\in\Sigma_{ij}$, thus (146) cannot hold when $r$ is small enough: moreover every point $x\in\Sigma_{ij}$ is contained in at most $c(2,d)$ balls, this means that (147) $\sum_{m}\mathbf{1}_{\left\\{z\in B_{m}^{r}\cap\Sigma_{ij}:\ \frac{\mathcal{H}^{d-1}(B(x,2r\sqrt{d})\cap\Sigma_{ij})}{\omega_{d-1}(2r\sqrt{d})^{d-1}}\leq\frac{1}{2}\right\\}}(x)\eta(x)\leq c(2,d)\eta(x)$ and that the left hand side of (147) converges to zero pointwise. By the dominated convergence theorem we get our claim. Proof of (2). Upon splitting into the negative and positive part, we may assume $\eta\geq 0$. Given a point $x\in\Sigma_{lp}$, if $y\in r\mathbf{Z}^{d}$ is such that $x\in B(y,2r\sqrt{d})$, then $B(y,4r\sqrt{d})\subset B(x,6r\sqrt{d})$. Thus for any $1\leq i<j\leq N$ with $(i,j)\neq(l,p)$ we have $\displaystyle\mathcal{H}^{d-1}(B(y,4r\sqrt{d})\cap\Sigma_{ij})$ $\displaystyle\leq\mathcal{H}^{d-1}(B(x,6r\sqrt{d})\cap\Sigma_{ij})$ $\displaystyle\leq\frac{\omega_{d-1}}{2^{d}}(4r)^{d-1}\sqrt{d}^{d-1}$ provided $r$ is small enough, this follows from Lemma 6. Since $\mathcal{F}_{2}^{r}$ covers $\mathbf{R}^{d}$ we obtain that $x\in\bigcup_{m\in\mathbf{N}}B_{m}^{r}$ for all $r$ small enough. In other words $\lim_{r\downarrow 0}\sum_{m}\rho_{m}(x)\eta(x)=\eta(x)$ pointwise on $\Sigma_{lp}$, and the argument of the limit on the right hand side is dominated by $\eta$. Thus we may once again appeal to the dominated convergence theorem and conclude the proof. ∎ ### 6.2. Consistency and Monotonicity The following results are essetially contained in [8] and [17], indeed the proofs may be adapted because we are assuming that $a_{ij}$ and $b_{ij}$ satisfy the triangle inequality. ###### Lemma 7. Let $\chi\in L^{1}((0,T),BV([0,1)^{d})^{N})$ such that $\chi(\cdot,t)\in\mathcal{A}$ for a.e. $t$. Then $\lim_{h\downarrow 0}\int_{0}^{T}E_{h}(\chi)dt=\int_{0}^{T}E(\chi)dt.$ Even more is true: for any $g\in C^{\infty}([0,1)^{d})$ and any pair $1\leq i\neq j\leq N$ we have $\displaystyle\begin{aligned} &\lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\int_{0}^{T}\int g(x)(\chi_{i}(x,t)K_{ij}^{h}\chi_{j}(x,t)+\chi_{j}(x,t)K_{ij}^{h}\chi_{j}(x,t))dxdt\\\ &=\int g(x)K_{ij}(z)\|\nu_{ij}\cdot z\|dzdxdt.\end{aligned}$ ###### Lemma 8. For any $0<h\leq h_{0}$ we have $E_{h}(u)\geq\left(\frac{\sqrt{h_{0}}}{\sqrt{h}+\sqrt{h_{0}}}\right)^{d+1}E_{h_{0}}(u).$ ### 6.3. Improved convergence of the energies The following Lemma is an improvement of the convergence of the energies, the proof of this result is contained, with minor modifications, in the paper [17], Corollary 3.7. ###### Lemma 9. Let $u^{h}$ be a sequence of $[0,1]$-valued functions such that $u^{h}\to\chi$ in $L^{1}([0,1)^{d}\times(0,T))$ and (148) $\lim_{h\downarrow 0}\int_{0}^{T}E_{h}(u^{h}(t))dt=\int_{0}^{T}E(\chi(t))dt.$ Then we have that (149) $\displaystyle\begin{aligned} \lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\int G_{\gamma}^{h}(z)\|f^{\gamma}_{h}(z)-f^{\gamma}(z)\|dz=0,\end{aligned}$ $\displaystyle\begin{aligned} \lim_{h\downarrow 0}\frac{1}{\sqrt{h}}\int G_{\beta}^{h}(z)\|f^{\beta}_{h}(z)-f^{\beta}(z)\|dz=0.\end{aligned}$ Where we set $\displaystyle\begin{aligned} f_{h}^{\gamma}(z)=\sum_{i,j}a_{ij}\int u_{i}^{h}(x,t)u_{j}^{h}(x-z,t)dxdt,\quad f^{\gamma}(z)=\sum_{i,j}a_{ij}\int\chi_{i}(x,t)\chi_{j}(x-z,t)dxdt,\end{aligned}$ $\displaystyle\begin{aligned} f_{h}^{\beta}(z)=\sum_{i,j}b_{ij}\int u_{i}^{h}(x,t)u_{j}^{h}(x-z,t)dxdt,\quad f^{\beta}(z)=\sum_{i,j}b_{ij}\int\chi_{i}(x,t)\chi_{j}(x-z,t)dxdt.\end{aligned}$ ### 6.4. Some inequalities Here we gather some elementary inequalities which are used frequently. ###### Lemma 10. Let $a,b,a^{\prime},b^{\prime}\in\\{0,1\\}$, then the following inequalities hold. (150) $\displaystyle\begin{aligned} \|a-b\|=a(1-b)+b(1-a)\end{aligned}$ (151) $\displaystyle\begin{aligned} \|(a-a^{\prime})_{+}-(b-b^{\prime})_{+}\|&+\|(a-a^{\prime})_{-}-(b-b^{\prime})_{-}\|\\\ &\leq\|a-b\|+\|a^{\prime}-b^{\prime}\|\end{aligned}$ ###### Proof. The first identity follows by expanding $\|a-b\|=\|a-b\|^{2}$. The second one is proved in [19]. For the sake of completeness, we reproduce the proof here. There are two cases. In the first one we have $(a-a^{\prime})(b-b^{\prime})\geq 0$ and we may assume upon replacing $(a,a^{\prime},b,b^{\prime})$ with $(-a,-a^{\prime},-b,-b^{\prime})$ that $(a-a^{\prime})$ and $(b-b^{\prime})$ are non-negative. Then (151) reduces to $\|(a-a^{\prime})-(b-b^{\prime})\|\leq\|a-b\|+\|a^{\prime}-b^{\prime}\|.$ The second case is given by $(a-a^{\prime})(b-b^{\prime})\leq 0$. By an argument as before we may assume $(a-a^{\prime})\geq 0\geq(b-b^{\prime})$, thus (151) reduces to $(a-a^{\prime})+(b-b^{\prime})\leq\|a-b\|+\|a^{\prime}-b^{\prime}\|.$ ∎ ###### Lemma 11. There exists a constant $C>0$ depending only on $N,\mathbb{A},\mathbb{B}$ such that for any $v\in\mathcal{M}$ (152) $\int\|v-K^{h_{0}}v\|dx\leq C\sqrt{h_{0}}E_{h}(v)\ \text{for all}\ h_{0}\geq h.$ ###### Proof. The proof of (152) is aontained in the proof of Lemma 3 in [19] for the two phases case when $K^{h}$ is the scaled version of the Gaussian with variance $1$. 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# On the Cauchy problems associated to a ZK-KP-type family equations with a transversal fractional dispersion Jorge Morales P<EMAIL_ADDRESS>Departamento de Matemáticas, Universidad Nacional de Colombia, Sede–Bogotá and Félix H. Soriano M <EMAIL_ADDRESS>Departamento de Matemáticas, Universidad Nacional de Colombia, Sede–Bogotá ###### Abstract. In this paper we examine the well-posedness and ill-posedeness of the Cauchy problems associated with a family of equations of ZK-KP-type $\begin{cases}u_{t}=u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x},\cr u(0)=\psi\in Z\end{cases}$ in anisotropic Sobolev spaces, where $1\leq\alpha\leq 1$, $\mathscr{H}$ is the Hilbert transform and $D_{x}^{\alpha}$ is the fractional derivative, both with respect to $x$. ###### Key words and phrases: Cauchy Problem, Dispersive equations, Kadomtsev-Petviashvili equation, Zakharov-Kuznetsov equation, Local well-posedness, Ill-posedness, Anisotropic Sobolev spaces, Kato theory ###### 2010 Mathematics Subject Classification: 35Q53, 35Q35, 35A01 ## Introduction Nonlinear evolution equations play an important role in different areas of science and engineering. Some of them are worth mentioning: fluid mechanics, plasma physics, fiber optics, solid state physics, chemical kinetics, chemical physics and geochemistry, among others. From the study of their solutions, an attempt is made to understand the effects of dispersion, diffusion, reaction and convection associated with the models described by them. For example, the Korteweg-de Vries (KdV) equation $u_{t}=u_{xxx}+uu_{x}\qquad(x,t)\in\mathbb{R}^{2},$ (1) which models the behavior of water waves in shallow channels, has solitary waves as solutions that behave like particles, which is why Kruskal and Zabusky called them _solitons_ in their 1965 work ([37]). These solitons are stable, in the sense that if a solution of the KdV equation (equation (1)) that differs very little in shape from soliton-type solutions, at the beginning, its shape will maintain an aspect that will differ very little from the shape of a soliton-type solution through time (see [4] and [6]); in fact, these solutions eventually take the form of solitons (see [31]). From a practical point of view, the notion of soliton stability guarantees that, with meticulous care, in the laboratory we will be able to reproduce these phenomena, first observed by J. Scott Russell in 1834. The Benjamin-Bona-Mahony (BBM) equation $u_{t}+u_{x}+uu_{x}-u_{xxt}=0,$ (2) was introduced in [5] with the intention to modeling the propagation of long waves of small amplitude, where the dispersion effect is purely nonlinear. The way in which this was obtained, it was pursued to arrive at an equation equivalent to the equation KdV (1). It is interesting to note that despite this intention, from a purely mathematical point of view, these equations present significant and interesting differences. Other one-dimensional equations are, one, introduced independently by Benjamin in [3] and Ono in [30], $u_{t}+\mathscr{H}u_{xx}+uu_{x}=0.$ (3) which models the internal waves into deep stratified fluids, where $\mathscr{H}$ is the Hilbert transform. The another, is the regularized Benjamin-Ono (rBO) $u_{t}+u_{x}+uu_{x}+\mathscr{H}u_{xt}=0$ (4) where $u=u(x,t)$ is a real function, with $x,t\in\mathbb{R}$. This equation is a model for wave time evolution with large ridges at the interface between two immissible fluids. There are two-dimensional versions that extend the equations previously mentioned. In the case of the KdV equation we have the Kadomtsev-Petviashvilli (KP) equation, see [10], $(u_{t}+auu_{x}+u_{xxx})_{x}\pm u_{yy}=0,$ (5) that describes waves in thin films of high surface tension. Another one is the Zakharov-Kuznetsov (ZK) equation, see [38], $u_{t}=(u_{xx}+u_{yy})_{x}+uu_{x},$ (6) which arises in the study of the dynamics of geophysical fluids in isotropic sets (means in which the characteristics of the bodies do not depend on direction) and ionic acoustic waves in magnetic plasmas. As a two-dimensional extension of the Benjamin-Ono equation we consider the following family of equations $\left(u_{t}+u^{p}u_{x}+\mathscr{H}(u_{xx}+\alpha u_{yy})\right)_{x}-\gamma u_{yy}=0\quad p\in\mathbb{N}.$ (7) This is the model of weakly nonlinear dispersive long wave motion in a two- fluid system, where the interface is capillarized and the bottom fluid is infinitely deep (see [1], [2] and [18]). Another one version that has received some attention in recent literature is the ZK-BO equation, $u_{t}=(\mathscr{H}u_{x}+u_{yy})_{x}+uu_{x}.$ (8) To finish this introduction we will mention the equations that present a more general type of dispersion that include as particular cases those mentioned above. A case of these is the integro differential equation of Whitham $u_{t}+\alpha uu_{x}+ku_{x}=0.$ (9) This was introduced by Whitman in [36] to model the breakage of dispersive waves in water. It is clear that if $k=\delta+\delta^{\prime\prime}$ or $k=\mathrm{v.p.}\dfrac{1}{x}$ we have the aforementioned KdV and BO equations. There is also what Lannes and Saut called the fractional dispersion equation KdV, fKdV (see [22]) $u_{t}+\alpha uu_{x}+\partial_{x}D^{\alpha}u=0,$ (10) where $D=\sqrt{-\partial_{x}^{2}}$. In the papers [8], [19], [24] and [25] it is established the relationship between the parameter $\alpha$ and the arising of singularities or the global existence of solutions in their energy space, as well as how this behavior is related to the existence and stability of the solitary waves associated with these equations. It should be said that in this direction there are interesting open problems, although they can really be very difficult. At this time we also think it is important to mention the work of Kenig, Martel and Robbiano ([16]) that demonstrates the blow-up of solutions in the energy space of the generalized scattering equation BO (a slightly different version of the fKdV) $u_{t}+\|u\|^{2\alpha}u_{x}+\partial_{x}D^{\alpha}u=0,$ (11) when the initial condition is “larger” than the “shape” of the solitary wave associated with this equation. In the two-dimensional case we have the equation introduced by Lannes in [21] to model long waves of small amplitude in weakly transverse regimes $u_{t}+\mu\frac{3}{2}uu_{x}+c_{ww}(\sqrt{\mu}\|D^{\mu}\|)\left(1+\frac{D_{x}^{2}}{D_{y}^{2}}\right)^{\frac{1}{2}}u=0,$ (12) where $c_{ww}(k)=\left((1+\beta k^{2})\frac{\tanh(k)}{k}\right)^{\frac{1}{2}}\ \text{ and}\ \|D^{\mu}\|=\sqrt{D_{x}^{2}+\mu D_{y}^{2}}.$ In [22] the relation between the equations KP and FDKP is established. In fact, they highlight the difference in the dispersive character with respect to the parameter $\beta$. Furthermore, by limiting the parameter $\mu$, the equations KPII or KPI are obtained when $\beta=0$ or $\beta>0$ respectively. An interesting conjecture here is to regard the existence of solitary waves for values $\beta$ greater than $1/3$. In this paper we consider the ZK-KP type Cauchy problem $\begin{cases}u_{t}=u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy}+uu_{x},\cr u(0)=\psi\in Z\end{cases}$ (13) for $-1\leq\alpha\leq 1$, where $\mathscr{H}$ denotes the Hilbert transform in the $x$ variable defined by $\mathscr{H}(f)=\mathrm{p.v.}\frac{1}{\pi}\int\frac{f(y)}{x-y}\,dy\quad f\in H^{s}(\mathbb{R}),$ for each $f\in\mathcal{S}$, $D_{x}^{\alpha}$ is the homogeneous fractional derivative in $x$ variable defined by $\widehat{D_{x}^{\alpha}f}(\xi,\eta)=\|\xi\|^{\alpha}\hat{f}(\xi,\eta),$ and $Z$ is one of the Sobolev spaces $X^{s_{1},s_{2}}$, $\widehat{X}^{s_{1},s_{2}}$, $Y^{s_{1},s_{2}}$ and $\widehat{Y}^{s_{1},s_{2}}$, that we will specify in the notations. The cases $\alpha=1$ and $\alpha=-1$ are the Cauchy problems corresponding to the very popular ZK and KPI equations, respectively, that were mentioned earlier. If $u(x,y,t)$ is a solution to (13), then $u_{\lambda}$ given by $u_{\lambda}(x,y,t)=\lambda^{2}u(\lambda x,\lambda^{\frac{3-\alpha}{2}}y,\lambda^{3}t)$ is also a solution to (13) and $\left\\|u_{\lambda}(t)\right\\|_{\dot{H}^{s_{1},s_{2}}(\mathbb{R}^{2})}=\lambda^{s_{1}+\frac{3+\alpha}{4}}\left\\|u(t)\right\\|_{\dot{H}^{s_{1},s_{2}}(\mathbb{R}^{2})}$ with $s_{2}=\frac{2s_{1}}{3-\alpha}$. This suggests that the local well- posedness could be guaranteed in $H^{s_{1},\frac{2s_{1}}{3-\alpha}}(\mathbb{R}^{2})$ for $s_{1}\geq-\frac{3+\alpha}{4}$. In this work we propose to show the local well-posedness of the Cauchy problem (13) in the Sobolev spaces $Z$ mentioned above. For this purpose, we use Kato’s theory for quasilinear equations and the ideas introduced by Kenig [15] for the KP-I equation and developed by Linares, Pilod and Saut in [26] for the f-KPI and f-KPII equations. Specifically, it is making the use of the Strichartz estimate provided by the group generated by the homogeneous linear equation associated with (13), making use of energy estimates. We will also make some ill-posedness observations of this equation for $-1\leq\alpha<0$. For this we will use the ideas developed by Molinet, Saut and Tzvetkov in [29]. More precisely, we will show that the flow associated with the solutions of (13) is not of class $C^{2}$. This, in particular, implies that it cannot be applied the Picard iteration method to get a solution to the integral equation obtained by the Duhamel principle applied to (13). ## Notation 1. (1) $\mathcal{S}(\mathbb{R}^{2})=\mathcal{S}$ denotes the Schwartz space and $\mathcal{S}^{\prime}(\mathbb{R}^{2})=\mathcal{S}^{\prime}$ denotes its topological dual vector space, the tempered distributions. 2. (2) $H^{s}(\mathbb{R}^{2})=H^{s}$ is the $s^{th}$-order Sobolev space. 3. (3) For a variable or an operator $u$, we denote by $\langle u\rangle$ the expression $(1+u^{2})^{\frac{1}{2}}$. 4. (4) For $s_{1},s_{2}\in\mathbb{R}$, the anisotropic Sobolev space $H^{s_{1},s_{2}}(\mathbb{R}^{2})$ is defined by $H^{s_{1},s_{2}}(\mathbb{R}^{2})=\left\\{f\in\mathcal{S}^{\prime}\;\left\|\;\int_{\mathbb{R}^{2}}{(\langle\xi\rangle^{2s_{1}}+\langle\eta\rangle^{2s_{2}})\|\widehat{f}(\xi,\eta)\|^{2}d\xi d\eta}<\infty\right.\right\\}.$ The norm in this space is given by $\\|f\\|_{H^{s_{1},s_{2}}(\mathbb{R}^{2})}=\sqrt{\int_{\mathbb{R}^{2}}(\langle\xi\rangle^{2s_{1}}+\langle\eta\rangle^{2s_{2}})\|\widehat{f}(\xi,\eta)\|^{2}d\xi d\eta},$ for all $f$ in this space. When there is no risk of confusion, we denote this space by $H^{s_{1},s_{2}}$. Observe that $H^{s,s}=H^{s}$, and the immediately above norm is equivalent to the usually given in the literature. 5. (5) $D_{x}^{s}$, $D_{y}^{s}$, $J_{x}^{s}$, $J_{y}^{s}$ and $J^{s}$ denotes the operators defined, via Fourier transform, by $\begin{split}\widehat{D_{x}^{s}f}&=\|\xi\|^{s}\widehat{f},\\\ \widehat{D_{y}^{s}f}&=\|\eta\|^{s}\widehat{f},\\\ \widehat{J_{x}^{s}f}&=(1+\|\xi\|^{2})^{\frac{s}{2}}\widehat{f},\\\ \widehat{J_{y}^{s}f}&=(1+\|\eta\|^{2})^{\frac{s}{2}}\widehat{f}\quad\text{and}\\\ \widehat{J^{s}f}&=(1+\|\xi\|^{2}+\|\eta\|^{2})^{\frac{s}{2}}\widehat{f},\end{split}$ for any $f\in\mathcal{S}^{\prime}(\mathbb{R}^{2})$. 6. (6) For $s_{1},s_{2}\geq 0$, we denote by $X^{s_{1},s_{2}}(\mathbb{R}^{2})=X^{s_{1},s_{2}}$ the space $X^{s_{1},s_{2}}(\mathbb{R}^{2})=\\{f\in H^{s_{1},s_{2}}\;\|\;\partial_{x}^{-1}f\in H^{s_{1},s_{2}}\\}.$ The norm in this space is given by $\\|f\\|_{X^{s_{1},s_{2}}}^{2}=\\|f\\|_{H^{s_{1},s_{2}}}^{2}+\\|\partial^{-1}_{x}f\\|^{2}_{H^{s_{1},s_{2}}}.$ 7. (7) For $s_{1},s_{2}\geq 0$, we denote by $\widehat{X}^{s_{1},s_{2}}(\mathbb{R}^{2})=\widehat{X}^{s_{1},s_{2}}$ the space $\widehat{X}^{s_{1},s_{2}}(\mathbb{R}^{2})=\\{f\in H^{s_{1},s_{2}}\;\|\;\partial_{x}^{-1}f\in L^{2}\\}.$ The norm in this space is $\\|f\\|^{2}_{\widehat{X}^{s_{1},s_{2}}}=\\|f\\|^{2}_{H^{s_{1},s_{2}}}+\\|\partial^{-1}_{x}f\\|^{2}_{L^{2}}.$ 8. (8) For $s_{1},s_{2}\geq 0$, we denote by $X_{\alpha}^{s_{1},s_{2}}(\mathbb{R}^{2})=X_{\alpha}^{s_{1},s_{2}}$ the space $X_{\alpha}^{s_{1},s_{2}}(\mathbb{R}^{2})=\\{f\in H^{s_{1},s_{2}}\;\|\;\partial_{x}^{\frac{\alpha-1}{2}}f\in L^{2}\\}.$ The norm in this space is given by $\\|f\\|_{X^{s_{1},s_{2}}}^{2}=\\|f\\|_{H^{s_{1},s_{2}}}^{2}+\\|\partial^{-1}_{x}f\\|^{2}_{H^{s_{1},s_{2}}}.$ 9. (9) For $s_{1},s_{2}\geq 0$, we denote by $Y^{s_{1},s_{2}}(\mathbb{R}^{2})=Y^{s_{1},s_{2}}$ the space $Y^{s_{1},s_{2}}(\mathbb{R}^{2})=\\{f\in H^{s_{1},s_{2}}\;\|\;\partial_{x}^{-1}\partial_{y}f\in H^{s_{1},s_{2}}\\}.$ The norm in this space is $\\|f\\|^{2}_{Y^{s_{1},s_{2}}}=\\|f\\|^{2}_{H^{s_{1},s_{2}}}+\\|\partial^{-1}_{x}\partial_{y}f\\|^{2}_{H^{s_{1},s_{2}}}$ 10. (10) For $s_{1},s_{2}\geq 0$, we denote by $\widehat{Y}^{s_{1},s_{2}}(\mathbb{R}^{2})=\widehat{Y}^{s_{1},s_{2}}$ the space $\widehat{Y}^{s_{1},s_{2}}(\mathbb{R}^{2})=\\{f\in H^{s_{1},s_{2}}\;\|\;\partial_{x}^{-1}\partial_{y}f\in L^{2}\\}.$ The norm in this space is given by $\\|f\\|_{\widehat{Y}^{s_{1},s_{2}}}^{2}=\\|f\\|_{H^{s_{1},s_{2}}}^{2}+\\|\partial^{-1}_{x}\partial_{y}f\\|^{2}_{L^{2}}.$ 11. (11) We define $H^{\infty}(\mathbb{R}^{2})=\bigcap_{s_{1},s_{2}\geq 0}H^{s_{1},s_{2}}(\mathbb{R}^{2})$. Analogously $X^{\infty}$, $\widehat{X}^{\infty}$, $Y^{\infty}$ and $\widehat{Y}^{\infty}$. ## 1\. Preliminaries We start this section of preliminaries by observing that the spaces $X^{s_{1},s_{2}}$, $\widehat{X}^{s_{1},s_{2}}$, $X_{\alpha}^{s_{1},s_{2}}$, $Y^{s_{1},s_{2}}$ and $\widehat{Y}^{s_{1},s_{2}}$ are Hilbert spaces. Thanks to the next lemma, each of these spaces is dense in $L^{2}$ ###### Lemma 1.1. The space $\partial_{x}\mathcal{S}$ is dense in $L^{2}$. In general, it is dense also in $H^{s_{1},s_{2}}$, $X^{s_{1},s_{2}}$, $\widehat{X}^{s_{1},s_{2}}$, $X_{\alpha}^{s_{1},s_{2}}$, $Y^{s_{1},s_{2}}$ and $\widehat{Y}^{s_{1},s_{2}}$. ###### Proof. Take a non negative function $\phi\in C^{\infty}$ defined on the real numbers, identically zero on the interval $[-1/2,1/2]$ and identically $1$ out of $[-1,1]$. For any $\psi\in\mathcal{S}$ we define $\psi_{\lambda}$ using the equation $\widehat{\psi}_{\lambda}(\xi,\eta)=\phi(\lambda\xi)\widehat{\psi}(\xi,\eta),$ for all $(\xi,\eta)\in\mathbb{R}^{2}$. The Plancherel theorem allows us show that $\psi_{\lambda}$ converges to $\psi$ in $L^{2}$, as $\lambda\to\infty$. The same argument allows us to show that $\partial_{x}\mathcal{S}$ is dense in any of the spaces mentioned in the lemma statement. ∎ By a duality argument, we can conclude that $L^{2}$ is densely contained in the dual spaces of any of the spaces mentioned in the lemma. As consequence of the previously discussed, we can extend the operator $\partial^{3}_{x}-\mathscr{H}D_{x}^{\alpha}\partial^{2}_{y}$ to the entire $L^{2}$, with image in the $X^{3}$ dual, $(X^{3})^{}$. Indeed, $-\partial^{3}_{x}+\mathscr{H}D_{x}^{\alpha}\partial^{2}_{y}$ is bounded from $X^{3}$ to $L^{2}$. So its adjoint operator is bounded from $L^{2}$ to $(X^{3})^{}$. Thanks to the Fourier transform, it can be seen that this adjoint operator is an extension of $\partial^{3}_{x}-\mathscr{H}D_{x}^{\alpha}\partial^{2}_{y}$ to all $L^{2}$. As a corollary of above, we have the following lemma. ###### Lemma 1.2. Let $W_{\alpha}$ be the unitary group of operators generated by the operator $\partial^{3}_{x}-\mathscr{H}D_{x}^{\alpha}\partial^{2}_{yy}$ and let $f$ and $u$ be continuous functions from an open interval $I$ to $L^{2}$. Then $u=W_{\alpha}(t)\psi+\int_{0}^{t}W_{\alpha}(t-t^{\prime})f(t^{\prime})\,dt^{\prime}$ if, and only if $u$ has continuous derivative on $I$ with values in $(X^{3})^{}$, the dual space $X^{3}$, and $\partial_{t}u=\partial^{3}_{x}u-\mathscr{H}D_{x}^{\alpha}\partial^{2}_{y}u+f.$ (14) This lemma gives sense to the well-posedness results in $H^{s_{1},s_{2}}$, that we shall enunciate later, when $\alpha$ is negative. The next result is a technical lemma that we use later. ###### Lemma 1.3. Let us assume that $u$ and $f$ are continuous functions in the $L^{2}$ space and satisfy (14) in the sense described there. Then, $\frac{1}{2}\frac{d}{dt}\\|u\\|^{2}=(u,f).$ (15) ###### Proof. Assume that $u_{\lambda}=e^{\lambda(\bigtriangleup-D_{x}^{-1})}(u)$ and $f_{\lambda}$ is defined in the same way. $u_{\lambda}$ and $f_{\lambda}$ are in $X^{\infty}$ and they are uniformly convergent, as $\lambda$ tends to 0, on closed intervals to $u$ and $f$ in $L^{2}$. Also, they satisfy the equation (14). From the antisimmetry of the operator $\partial^{3}_{x}-\mathscr{H}D_{x}^{\alpha}\partial^{2}_{y}$, it is easy to see that $u_{\lambda}$ and $f_{\lambda}$ satisfy (15). When we make $\lambda$ tend to 0, we get the lemma. ∎ Next we shall state a set of results about the properties of the spaces with we work in this paper. Maybe one the most known of these results is the Sobolev lemma. Here we present a version for the $H^{s_{1},s_{2}}(\mathbb{R}^{2})$ spaces. ###### Lemma 1.4 (Sobolev). Let $s_{1}$ and $s_{2}$ be positive real numbers such that $\frac{1}{s_{1}}+\frac{1}{s_{2}}<2.$ Then, $H^{s_{1},s_{2}}(\mathbb{R}^{2})\subset C_{\infty}(\mathbb{R}^{2})$ (the set of continuous functions on $\mathbb{R}^{2}$ vanishing at infinity), with continuous embedding. ###### Proof. See [33] ∎ ###### Lemma 1.5. Let $1\leq p<q\leq\infty$ and $f\in L^{p}\cap L^{q}$. Then $f\in L^{r}$ for $r=\theta p+(1-\theta)q$, where $\theta\in(0,1)$, and we have $\left\\|f\right\\|_{L^{r}}^{r}\leq\left\\|f\right\\|_{L^{p}}^{\theta p}\left\\|f\right\\|_{L^{q}}^{(1-\theta)q}$ ###### Proof. The proof is immediate consequence of the Hölder inequality. ∎ ###### Lemma 1.6. If $s\in(0,n/2)$, then $H^{s}(\mathbb{R}^{n})$ is a continuous embedding in $L^{p}(\mathbb{R}^{n})$, with $p=\frac{2n}{n-2s}$, i.e. $s=n(\frac{1}{2}-\frac{1}{p})$. Furthermore, for $f\in H^{s}(\mathbb{R}^{n})$, $s\in(0,n/2)$ $\left\\|f\right\\|_{L^{p}(\mathbb{R}^{n})}\leq c_{n,s}\left\\|D^{s}f\right\\|_{L^{2}(\mathbb{R}^{n})}\leq c\left\\|f\right\\|_{H^{s}(\mathbb{R}^{n})}$ where $D^{l}f=(-\Delta)^{l/2}=((\|\xi\|)^{l}\hat{f})^{\vee}$ ###### Proof. See the Linares and Ponce book [27] page 48. ∎ ###### Lemma 1.7. Let $s_{1},s_{2}\in\mathbb{R}$ and assume that $D^{s_{1}}f\in L^{p}(\mathbb{R})$ and $D^{s_{2}}f\in L^{q}(\mathbb{R})$. Then, for all $\theta\in[0,1]$, $D^{s}f\in L^{r}$, and $\\|D^{s}f\\|_{L^{r}(\mathbb{R})}\leq C_{s}\\|D^{s_{1}}f\\|_{L^{p}(\mathbb{R})}^{\theta}\\|D^{s_{2}}f\\|_{L^{q}(\mathbb{R})}^{1-\theta}$ where $\theta=\dfrac{s_{2}-s}{s_{2}-s_{1}}$ and $\dfrac{1}{r}=\dfrac{\theta}{p}+\dfrac{1-\theta}{q}$. ###### Proof. A proof can be found in [28] ∎ The next estimate was proved by Kato and Ponce in [14] and it will be very useful later in this work. ###### Lemma 1.8. For $s>0$ and $1<p<\infty$, we have $\\|[J^{s},f]g\\|_{L^{p}(\mathbb{R}^{n})}\lesssim\\|\partial f\\|_{L^{\infty}(\mathbb{R}^{n})}\\|J^{s-1}g\\|_{L^{p}(\mathbb{R}^{n})}+\\|J^{s}f\\|_{L^{p}(\mathbb{R}^{n})}\\|g\\|_{L^{\infty}(\mathbb{R}^{n})},$ (16) for all $f$ and $g\in\mathcal{S}(\mathbb{R}^{n})$ ###### Corollary 1.9. For $s>0$ and $p\in(1\,\infty)$, $L_{s}^{p}\cap L^{\infty}$ is an algebra, also $\\|fg\\|_{L_{s}^{p}}\leq c(\\|f\\|_{\infty}\\|g\\|_{L_{s}^{p}}+\\|f\\|_{L_{s}^{p}}\\|g\\|_{\infty})$ (17) The following estimate for the commutator operator can be seen in [27] page 51. ###### Lemma 1.10. For $s>0$, we have $\left\\|\left[\partial_{x}^{s},g\right]f\right\\|_{L^{2}(\mathbb{R})}\lesssim\left\\|\partial_{x}g\right\\|_{L^{\infty}(\mathbb{R})}\left\\|\partial_{x}^{s-1}f\right\\|_{L^{2}(\mathbb{R})}+\left\\|\partial_{x}^{s}g\right\\|_{L^{2}(\mathbb{R})}\left\\|f\right\\|_{L^{\infty}(\mathbb{R})}$ (18) The next result is the Leibniz rule for fractional derivatives and it was proved by Kenig, Ponce and Vega in [17]. ###### Lemma 1.11. For $\alpha\in(0,1)$, we have $\left\\|D_{x}^{\alpha}(fg)\right\\|_{L^{p}(\mathbb{R})}\leq\left\\|D_{x}^{\alpha}(f)\right\\|_{L^{p_{1}}(\mathbb{R})}\left\\|g\right\\|_{L^{q_{1}}(\mathbb{R})}+\left\\|D_{x}^{\alpha}g\right\\|_{L^{p_{2}}(\mathbb{R})}\left\\|f\right\\|_{L^{q_{2}}(\mathbb{R})}$ (19) where $1<p_{1},p_{2},q_{1},q_{2}\leq\infty$ and satisfy $\frac{1}{p_{1}}+\frac{1}{q_{1}}=\frac{1}{p_{2}}+\frac{1}{q_{2}}=\frac{1}{p}$. ###### Lemma 1.12. Let $-1\leq\alpha\leq 1$ and $0\leq p\leq\frac{8}{1-\alpha}$ $\\|f\\|_{L^{p+2}(\mathbb{R}^{2})}^{p+2}\lesssim\\|f\\|_{L^{2}(\mathbb{R}^{2})}^{2-\frac{p(1-\alpha)}{4}}\\|\partial_{x}f\\|_{L^{2}(\mathbb{R}^{2})}^{\frac{p(3-\alpha)}{4}}\\|D^{\frac{\alpha-1}{2}}_{x}\partial_{y}f\\|_{L^{2}(\mathbb{R}^{2})}^{\frac{p}{2}}$ ###### Proof. First, let us prove the lemma for $p=p=\frac{8}{1-\alpha}$. From Lemmas 1.6 and 1.7 we have $\begin{split}\left\\|f\right\\|_{L^{p^{}+2}(\mathbb{R}^{2})}^{p^{}+2}&=\int_{\mathbb{R}^{2}}{\left\|f(x,y)\right\|^{p^{}+2}}dxdy\\\ &=\int_{\mathbb{R}}{\left\\|f(\cdot,y)\right\\|_{L^{p^{}+2}_{x}}^{p^{}+2}}dy\\\ &\leq C\int_{\mathbb{R}}{\left\\|D^{\frac{p^{}}{p^{}+2}}_{x}f(\cdot,y)\right\\|_{L^{2}_{x}}^{p^{}+2}}dy\\\ &\leq C\int_{\mathbb{R}}{\left\\|\partial_{x}f(\cdot,y)\right\\|_{L^{2}_{x}}^{2}\left\\|D^{\frac{p^{}-4}{2p^{}}}_{x}f(\cdot,y)\right\\|_{L^{2}_{x}}^{p^{}}}dy\\\ &\leq C\left\\|\partial_{x}f\right\\|_{L^{2}(\mathbb{R}^{2})}^{2}\sup_{y\in\mathbb{R}}{\left\\|D^{\frac{p^{}-4}{2p^{}}}_{x}f(x,y)\right\\|_{L^{2}_{x}}^{p^{}}}.\end{split}$ (20) On the other hand, for all $y\in\mathbb{R}$, $\begin{split}\left\\|D^{\frac{p^{}-4}{2p^{}}}_{x}f(\cdot,y)\right\\|_{L^{2}_{x}}^{2}&=\int_{\mathbb{R}}{\left\|D_{x}^{\frac{p^{}-4}{2p^{}}}f(x,y)\right\|^{2}}dx\\\ &=2\int_{\mathbb{R}}{\int_{-\infty}^{y}{D_{x}^{\frac{1+\alpha}{4}}f(x,\eta)D_{x}^{\frac{1+\alpha}{4}}\partial_{y}f(x,\eta)}}d\eta dx\\\ &=2\int_{-\infty}^{y}{\int_{\mathbb{R}}{D_{x}f(x,\eta)D_{x}^{\frac{\alpha-1}{2}}}\partial_{y}f(x,\eta)}dxd\eta\\\ &\leq 2\int_{-\infty}^{y}{\left\\|\partial_{x}f(\cdot,\eta)\right\\|_{L^{2}_{x}}\left\\|D^{\frac{\alpha-1}{2}}_{x}\partial_{y}f(\cdot,\eta)\right\\|_{L^{2}_{x}}}d\eta\\\ &\leq 2\left\\|\partial_{x}f\right\\|_{L^{2}(\mathbb{R}^{2})}\left\\|D^{\frac{\alpha-1}{2}}_{x}\partial_{y}f\right\\|_{L^{2}(\mathbb{R}^{2})}.\end{split}$ (21) Therefore, $\\|f\\|_{L^{p^{}+2}(\mathbb{R}^{2})}^{p^{}+2}\leq c\\|\partial_{x}f\\|_{L^{2}(\mathbb{R}^{2})}^{\frac{p^{}(3-\alpha)}{4}}\\|D^{\frac{\alpha-1}{2}}_{x}\partial_{y}f\\|_{L^{2}(\mathbb{R}^{2})}^{\frac{p^{}}{2}}$ (22) The inequality for $0<p<p^{}$ follows immediately from Lemma 1.5 and the last inequality. ∎ ### 1.1. Kato’s theory We will make a brief presentation of Kato’s theory described in [11]. With this it can be showed the well-posedness of the Cauchy problems associated to linear and quasilinear evolution equations. #### 1.1.1. Linear case Suppose that $X$ and $Y$ are reflexives Banach spaces with $Y\subseteq{X}$ in a dense and continuous way, and let $\\{A(t)\\}_{t\in[0,T]}$ be a operators family such that 1. (1) $A(t)\in{G}(X,1,\beta)$. In other words, $-A(t)$ generates a $C_{0}$-semigroup such that $\\|e^{-sA(t)}\\|\leq{e}^{\beta{s}},$ for all $s\in[0,\infty).$ 2. (2) There exists an isomorphism $S:Y\to{X}$ such that $SA(t)S^{-1}=A(t)+B(t)$, where $B(t)\in{B}(X),$ for $0\leq{t}\leq{T},$ $t\to{B}(t)x$ is strongly measurable, for each $x\in{X}$, and $t\to\\|B(t)\\|_{X}$ is integrable in $[0,T]$. 3. (3) $Y\subseteq{D}(A(t)),$ for $0\leq{t}\leq{T}$, and $t\rightarrow{A}(t)$ is strongly continuous from $[0,T]$ to $B(Y,X)$. ###### Theorem 1.13. Under the above conditions, there exists a operators family $\\{U(t,s)\\}_{0\leq{s}\leq{t}\leq{T}}$ such that: 1. (1) $U$ is strongly continuous from $\Delta\to{B}(X)$, where $\Delta=\\{(t,s):0\leq{s}\leq{t}\leq{T}\\}$. 2. (2) $U(t,s)U(s,r)=U(t,r)$ for $(t,s)$ and $(s,r)\in\Delta$, and $U(s,s)=I$. 3. (3) $U(t,s)Y\subset{Y}$ and $U$ is strongly continuous from $\Delta\to{B}(Y)$. 4. (4) $\dfrac{dU(t,s)}{dt}=-A(t)U(t,s),$ $\dfrac{dU(t,s)}{ds}=U(t,s)A(s),$ in the strong sense in $B(X,Y)$ space and are strongly continuous from $\Delta\to{B}(X,Y)$. The operators family $\\{U(t,s)\\}_{0\leq{s}\leq{t}\leq{T}}$ in the previous theorem is called _the evolution operators_ associated to ${A(t)}$. An immediate consequence from the last theorem is, for $\varphi\in{Y}$, $u(t)=U(t,s)\varphi$ is solution to the Cauchy problem $\displaystyle\frac{du}{dt}+A(t)u=0\qquad\text{ for }\quad s\leq{t}\leq{T},$ $\displaystyle u(s)=\varphi.$ Moreover, if $f\in{C}([0,T];X)\cap{L^{1}}([0,T];Y)$, then $u(t)=U(t,0)\varphi+\int_{0}^{t}U(t,s)f(s)ds$ if and only if $u\in{C}([0,T];Y)\cap{C}^{1}((0,T);X)$ and $\displaystyle\frac{du}{dt}+A(t)u$ $\displaystyle=f(t)\quad\text{ for }\quad 0\leq{t}\leq{T},$ $\displaystyle u(0)$ $\displaystyle=\varphi.$ #### 1.1.2. Quasilinear Case Let $X$ and $Y$ be reflexives Banach spaces, $Y\subseteq X$, with dense and continuous embedding. Let us consider the following problem $\begin{array}[]{ll}&\partial_{t}u+A(t,u)u=f(t,u)\in X,\ 0<t,\\\ &u(0)=u_{0}\in Y,\end{array}$ (23) where, for each $t$, $A(t,u)$ is a linear operator from $Y$ to $X$ and $f(t,u)$ is a function from $\mathbb{R}\times Y$ in $X$. Let us also consider the next conditions: $(X)$ There exists an isometric isomorphism $S$ from $Y$ to $X$. There exist $T_{0}>0$ and $W$ an open ball with $w_{0}$ as center such that: $(A_{1})$ For each $(t,y)\in[0,T_{0}]\times W$, the linear operator $A(t,y)$ belongs to $G(X,1,\beta)$, where $\beta$ is a positive real number. In the other words, $-A(t,y)$ generate a $C_{0}$ semigroup such that $\\|e^{-sA(t,y)}\\|_{\mathcal{B}(X)}\leq e^{\beta s},\ \text{for}\ s\in[0,\infty).$ Note that if $X$ is a Hilbert space, $A\in G(X,1,\beta)$ if, and only if, 1. a) $\langle Ay,y\rangle_{X}\geq-\beta\\|y\\|_{X}^{2}$ for all $y\in D(A)$, 2. b) $(A+\lambda)$ is onto for all $\lambda>\beta.$ (See [13] or [32]) $(A_{2})$ For each $(t,y)\in[0,T_{0}]\times W$ the operator $B(t,y)=[S,A(t,y)]S^{-1}\in\mathcal{B}(X)$ and is uniformly bounded, i.e., there exists $\lambda_{1}>0$ such that $\displaystyle\\|B(t,y)\\|_{\mathcal{B}(X)}\leq\lambda_{1}\ \ \text{for all}\ (t,y)\in[0,T_{0}]\times W,$ Furthermore, for some $\mu_{1}>0$, we have that, for all $y$ and $z\in W$, $\displaystyle\\|B(t,y)-B(t,z)\\|_{\mathcal{B}(X)}\leq\mu_{1}\\|y-z\\|_{Y}.$ $(A_{3})$ $Y\subseteq D(A(t,y))$, for each $(t,y)\in[0,T_{0}]\times W,$ (the restriction of $A(t,y)$ to $Y$ belongs to $\mathcal{B}(Y,X)$) and, for each $y\in W$ fix, $t\to A(t,y)$ is strongly continuous. Besides, for all $t\in[0,T_{0}]$ fix, it is satisfied the following Lipschitz condition, $\\|A(t,y)-A(t,z)\\|_{\mathcal{B}(Y,X)}\leq\mu_{2}\\|y-z\\|_{X},$ where $\mu_{2}\geq 0$ is constant. $(A_{4})$ $A(t,y)w_{0}\in Y$ for all $(t,y)\in[0,T]\times W$. Also, there exists a constant $\lambda_{2}$ such that $\\|A(t,y)w_{0}\\|_{Y}\leq\lambda_{2},\ \text{for all}\ (t,y)\in[0,T_{0}]\times W.$ $(f_{1})$ $f$ is a bounded function in $[0,T_{0}]\times W$ to $Y$, i.e., there exists $\lambda_{3}$ such that $\\|f(t,y)\\|_{Y}\leq\lambda_{3},\ \text{for all}\ (t,y)\in[0,T_{0}]\times W,$ Also, the function $t\in[0,T_{0}]\mapsto f(t,y)\in Y$ is continuous with respect to the topology of $X$ and for all $y$ and $z\in Y$ we have that $\displaystyle\\|f(t,y)-f(t,z)\\|_{X}\leq\mu_{3}\\|y-z\\|_{X},$ where $\mu_{3}\geq 0$ is a constant. ###### Theorem 1.14 (Kato). Assume that the conditions $(X),$ $(A_{1})-(A_{4})$ and $(f_{1})$ are satisfied. Given $u_{0}\in Y$, there exist $0<T<T_{0}$ and a unique $u\in C([0,T];Y)\cap C^{1}((0,T);X)$ solution to (23). Furthermore, the map $u_{0}\to u$ is continuous in the following sense: consider the sequence of Cauchy problems, $\displaystyle\partial_{t}u_{n}+A_{n}(t,u_{n})u_{n}=f_{n}(t,u_{n})\ \ t>0$ (24) $\displaystyle u_{n}(0)=u_{n_{0}}\ n\in\mathbb{N}.$ Suppose that the conditions $(X)$, $(A_{1})$–$(A_{4})$ and $(f_{1})$ are also satisfied for all $n\geq 0$ in (LABEL:dependencia), with the same $X,\ Y$ and $S$, and the correspondents $\beta$, $\lambda_{1}$–$\lambda_{3}$, $\mu_{2}$–$\mu_{3}$ can be chosen independent of $n$. Also, let us suppose that $\displaystyle\mathop{s\text{-}\lim}_{n\to\infty}A_{n}(t,w)$ $\displaystyle=A(t,w)\ \text{in }\ B(X,Y),$ $\displaystyle\mathop{s\text{-}\lim}_{n\to\infty}B_{n}(t,w)$ $\displaystyle=B(t,w)\ \text{in }\ B(X),$ $\displaystyle\lim_{n\to\infty}f_{n}(t,w)$ $\displaystyle=f(t,w)\ \text{in }\ Y,$ $\displaystyle\lim_{n\to\infty}u_{n_{0}}$ $\displaystyle=u_{0}\ \text{in }\ Y,$ where $s$-$\lim$ denotes the strong limit. Then, $T$ can be taken in such a way that $u_{n}\in C([0,T],Y)\cap C^{1}((0,T),X)$ and $\lim_{n\to\infty}\sup_{[0,T]}\\|u_{n}(t)-u(t)\\|_{Y}=0.$ A proof of this theorem can be found in [11] and [20]. ### 1.2. Other results ###### Proposition 1.15 (Kato’s inequality). Let $f\in H^{s}$, $s>2$, $\Lambda=(1-\Delta)^{1/2}$ and $M_{f}$ be the multiplication operator by $f$. Then, for $\|\tilde{t}\|,\|\tilde{s}\|\leq s-1$, $\Lambda^{-\tilde{s}}[\Lambda^{\tilde{s}+\tilde{t}+1},M_{f}]\Lambda^{-\tilde{t}}\in B(L^{2}(\mathbb{R}^{2}))$ and $\left\\|\Lambda^{-\tilde{s}}[\Lambda^{\tilde{s}+\tilde{t}+1},M_{f}]\Lambda^{-\tilde{t}}\right\\|_{{B}\left(L^{2}\left(\mathbb{R}^{2}\right)\right)}\leq c\left\\|\nabla f\right\\|_{H^{s-1}}.$ (25) ###### Proposition 1.16. Let $f:\mathbb{R}^{2}\to\mathbb{R}$ be a bounded continuous function such that $\partial_{x}f$ exists and is continuous and bounded. Then, if $A=f\partial_{x}$, $\langle A(u),u\rangle_{L^{2}}\geq-\frac{1}{2}\|\|\partial_{x}f\|\|_{L^{\infty}}\|\|u\|\|_{L^{2}}^{2},$ (26) for all $u\in D(A)$, $A+\lambda$ is onto, for all $\lambda>\frac{1}{2}\|\|f\|\|_{L^{\infty}}$. In particular, $A\in G\left(L^{2}\left(\mathbb{R}^{2}\right),1,\frac{1}{2}\|\|f\|\|_{L^{\infty}}\right)$. ###### Proof. The inequality (26) is obtained immediately after using the integration by parts. Let us see that $A+\lambda$ is onto, if $\lambda>\frac{1}{2}\|\|f\|\|_{L^{\infty}}$. Suppose that $\psi$ is such that $\langle(A+\lambda)(u),u\rangle_{L^{2}}=0$, for all $u\in D(A)$. Then $\psi\in D(A^{})\subseteq D(A)$. From (26), it follows that $0\geq\langle(a+\lambda)(u),u\rangle_{L^{2}}\geq(\lambda-\frac{1}{2}\|\|f\|\|_{L^{\infty}})\|\|\psi\|\|_{L^{2}}^{2}.$ Hence, $\psi=0$ and, therefore, $A+\lambda$ is onto. ∎ ## 2\. Local well-posedness in Sobolev spaces of $s^{th}$ order with $s>2$ In this section we examine the local well-posedness of the problem (13) in the Sobolev spaces $H^{s}$, $X^{s}$, $\widehat{X}_{\alpha}^{s}$, $Y^{s}$ and $\widehat{Y}^{s}$, for $s>2$. ### 2.1. Local well-posedness in $H^{s}(\mathbb{R}^{2})$ In this section, we will make use of Kato’s theory to show the local well- posedness of (13) in the $H^{s}$ spaces. More precisely we have the following theorem. ###### Theorem 2.1. Let $s$ and $\alpha$ be real numbers such that $s>2$ and $-1\leq\alpha\leq 1$. For $\psi\in H^{s}(\mathbb{R}^{2})$, there exist $T>0$, that depends only on $\\|\psi\\|_{H^{s}}$, and a unique $u\in C([0,T],H^{s}(\mathbb{R}^{2}))\cap$ $C^{1}([0,T],H^{s-3}(\mathbb{R}^{2})\cap(X^{3})^{})$ solution to the Cauchy problem (13) Moreover, the map $\psi\to u$ from $H^{s}$ to $C([0,T],H^{s}(\mathbb{R}^{2}))$ is continuous. ###### Proof. Let $W_{\alpha}(t)$ be the operators group defined by $W_{\alpha}(t)\psi=e^{t(\partial_{x}^{3}-\mathscr{H}\partial_{y}^{2})}\psi=\left(e^{-it(\xi^{3}+\operatorname{sgn}(\xi)\|\xi\|^{\alpha}\eta^{2})}\widehat{\psi}\right)^{\vee},$ for all $\psi\in H^{s}$. $u$ is solution to the problem (13) if and only if $v=W_{\alpha}(t)u$ is solution to the problem $\begin{cases}v_{t}+A(t,v)v=0\\\ v(0)=\psi,\end{cases}$ (27) where $A(t,v)=W_{\alpha}(t)(W_{\alpha}(-t)v)\partial_{x}W_{\alpha}(-t)$. Let us see that this last problem satisfies each condition of Kato’s theorem (Theorem 1.14). Let $X=L^{2}(\mathbb{R}^{2})$, $Y=H^{s}(\mathbb{R}^{2})$ and $S=\Lambda_{s}=J^{s}$. From Plancherel’s theorem, it is evident that $S$ is an isomorphism between $X$ and $Y$. With the following lemmas we show that the conditions $(A_{1})$-$(A_{4})$ are satisfied. ###### Lemma 2.2. $A(t,v)\in G(X,1,\beta(v))$, where $\beta(v)=\frac{1}{2}\sup\limits_{t}\\|\partial_{x}W_{\alpha}(t)v\\|_{L^{\infty}}$ ###### Proof. Since $\\{W_{\alpha}(-t)\\}$ is an strongly continuous unitary operators group and $u\in H^{s}(\mathbb{R}^{2})$, from Proposition 1.16, we obtain the result. ∎ ###### Lemma 2.3. For $S$ given as above, $SA(t,v)S^{-1}=A(t,v)+B(t,v),$ where $B(t,v)$ is a bounded operator in $L^{2}$, for all $t\in\mathbb{R}$ and all $v\in H^{s}$, and satisfies the inequalities $\displaystyle\|\|B(t,v)\|\|_{\mathcal{B}(L^{2})}$ $\displaystyle\leq\lambda(v)$ (28) $\displaystyle\|\|B(t,v)-B(t,v^{\prime})\|\|_{\mathcal{B}(L^{2})}$ $\displaystyle\leq\mu\|\|v-v^{\prime}\|\|_{H^{s}}$ (29) for $t\in\mathbb{R}$ and all $v$ and $v^{\prime}\in H^{s}(\mathbb{R}^{2})$. Where $\mu$ is a positive real number and $\lambda(v)=\sup\limits_{t}C_{s}\\|v\\|_{H^{s}}.$ ###### Proof. From Lemma 1.15, it follows that $[S,W_{\alpha}(-t)v]S^{-1}\in\mathcal{B}(L^{2})$ y $\\|[S,W_{\alpha}(-t)v]S^{-1}\\|_{\mathcal{B}(L^{2})}\leq C_{s}\\|v\\|_{H^{s}}.$ Therefore $B(t,v)\in\mathcal{B}(L^{2})$ and satisfies (28) Proceeding as before (29) can be shown. ∎ ###### Lemma 2.4. $H^{s}(\mathbb{R}^{2})\subset D(A(t,v))$ and $A(t,v)$ is a bounded operator from $Y=H^{s}(\mathbb{R}^{2})$ to $X=L^{2}(\mathbb{R}^{2})$ with $\\|A(t,v)\\|_{\mathcal{B}(Y,X)}\leq\\|v\\|_{H^{s}},$ for all $v\in Y$. Furthermore, the function $t\mapsto A(t,v)$ is strongly continuous from $\mathbb{R}$ to $\mathcal{B}(Y,X)$, for all $v\in H^{s}$. On the other hand, the function $v\mapsto A(t,v)$ satisfies the following Lipschitz condition $\\|A(t,v)-A(t,v^{\prime})\\|_{\mathcal{B}(Y,X)}\leq\\|v-v^{\prime}\\|_{X},$ where $\mu$ is as the lemma above. ###### Proof. Inasmuch as $\\{W_{\alpha}(t)\\}$ is a unitary group in $L^{2}$, from the definition of $A(t,v)$, it follows that $H^{s}\left(\mathbb{R}^{2}\right)\subset D(A(t,v))$. In fact, $\displaystyle\\|A(t,v)f\\|_{L^{2}}$ $\displaystyle=\\|W_{\alpha}(-t)v\partial_{x}W_{\alpha}(-t)f\\|_{L^{2}}$ $\displaystyle\leq C_{s}\\|v\\|_{H^{s}}\\|\partial_{x}f\\|_{L^{2}}$ $\displaystyle\leq\\|v\\|_{H^{s}}\\|f\\|_{H^{s}},$ for all $f\in H^{s}$. Now, for all $t,t^{\prime}\in\mathbb{R}$ and $f,v\in H^{s}$, we have $\displaystyle\\|A(t,v)f-A(t^{\prime},v)f\\|_{L^{2}}$ $\displaystyle\leq\\|\left(W_{\alpha}(t)-W_{\alpha}(t^{\prime})\right)W_{\alpha}(-t)v\partial_{x}W_{\alpha}(-t)f\\|_{L^{2}}+$ $\displaystyle+\\|(W_{\alpha}(-t)-W_{\alpha}(-t^{\prime}))v\partial_{x}W_{\alpha}(-t)f\\|_{L^{2}}+$ $\displaystyle+\\|W_{\alpha}(-t^{\prime})v\partial_{x}(W_{\alpha}(-t)-W_{\alpha}(-t^{\prime}))f\\|_{L^{2}}.$ Since the group $\\{W_{\alpha}(t)\\}_{t\in\mathbb{R}}$ is strongly continuous, $t\mapsto A(t,v)$ is strongly continuous from $\mathbb{R}$ to $\mathcal{B}(H^{s},L^{2})$. Finally, for any $t\in\mathbb{R}$, we have $\displaystyle\\|A(t,v)f-A(t,v^{\prime})f\\|_{L^{2}}$ $\displaystyle\leq\mu\\|W_{\alpha}(-t)v-W_{\alpha}(-t)v^{\prime}\\|_{L^{2}}\\|\partial_{x}W_{\alpha}(-t)f\\|_{L^{\infty}}$ $\displaystyle\leq\mu\\|v-v^{\prime}\\|\,\\|f\\|_{H^{s}},$ this ends the lemma proof. ∎ If we take the open ball $W$ of $v\in H^{s}$ such that $\\|v\\|_{H^{s}(\mathbb{R}^{2})}<R$, the preceding lemmas show that the Cauchy problem (13) satisfies the conditions of Theorem 1.14. Therefore, for each $\psi\in H^{s}(\mathbb{R}^{2})$, with $s>2$, there exists $T>0$, that depends on $\\|\psi\\|_{H^{s}}$, and a unique $v\in C([0,T],H^{s}(\mathbb{R}^{2}))\cap C^{1}([0,T],H^{s-1}(\mathbb{R}^{2}))$ solution to the problem (27). Moreover, the map $\psi\to v$ is continuous from $H^{s}(\mathbb{R}^{2})$ to $C([0,T],H^{s}(\mathbb{R}^{2}))$. Now, from the group $W_{\alpha}(t)$ properties it can be verified that $u(t)=W_{\alpha}(-t)v$ is solution to Cauchy problem (13) and satisfies all properties stated in the theorem. ∎ ###### Theorem 2.5. The existence time of the solution of the Cauchy problem (13) can be chosen independently of $s$ in the following sense: if $u\in C([0,T],H^{s}(\mathbb{R}^{2}))$ is the solution to (13) with $\psi\in H^{r}(\mathbb{R}^{2})$, for some $r>s$, then $u\in C([0,T],H^{r}(\mathbb{R}^{2}))$. In particular, if $\psi\in H^{\infty}(\mathbb{R}^{2})$, $u\in C([0,T],H^{\infty}(\mathbb{R}^{2}))$ ###### Proof. Let $r>s$, $u\in C([0,T],H^{r}(\mathbb{R}^{2}))$ solution to (13) and $v=W_{\alpha}(-t)u$. Suppose that $r\leq s+1$. If we apply $\partial_{x}^{2}$ on both sides of the differential equation (27), we arrive to the following linear evolution equation for $w(t)=\partial_{x}^{2}v(t)$ $\frac{dw}{dt}+A(t)w+B(t)w=0.$ (30) where $\displaystyle A(t)$ $\displaystyle=\partial_{x}W_{\alpha}(t)u(t)W_{\alpha}(-t)$ (31) and $\displaystyle B(t)$ $\displaystyle=2W_{\alpha}(t)u_{x}(t)W_{\alpha}(-t).$ (32) Since $v\in C\left([0,T];H^{s}\left(\mathbb{R}^{2}\right)\right)$, then $w\in C\left([0,T];H^{s-2}\left(\mathbb{R}^{2}\right)\right)$. Besides $w(0)=\psi_{xx}\in H^{r-2}\left(\mathbb{R}^{2}\right)$, because $\psi\in H^{r}\left(\mathbb{R}^{2}\right)$. It is needed to see that $w\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$. For this we shall prove that the Cauchy problem associated to the linear equation (30) is locally well-posed for $1-s\leq k\leq s-1$, for which we have the following lemma whose proof is similar to that of Lemma 3.1 in [12]. ###### Lemma 2.6. The family $\\{A(t)\\}_{0\leq t\leq T}$ has a unique family of evolution operators associated, $\\{U(t,\tau)\\}_{0\leq t\leq\tau\leq T}$, in the spaces $X=H^{h}$, $Y=H^{k}$, where $-s\leq h\leq s-2\quad 1-s\leq k\leq s-1\quad k+1\leq h.$ (33) In particular, $U(t,\tau):H^{r}\to H^{r}$ for $-s\leq s\leq s-1.$ Then, $w$ satisfies the equation $w(t)=U(t,0)\psi_{xx}+\displaystyle\int_{0}^{t}U(t,\tau)[-B(\tau)w(\tau)+f(\tau)]\,d\tau.$ (34) Since $\psi_{xx}\in H^{r-2},$ $B(t)$, given by (32), is an operators family in $H^{r-2}$ that is strongly continuous for $t$ in the interval $[0,T]$. From Lemma 2.6, the solution to (34) belongs to $C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$. In other words, $\partial_{x}^{2}u\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$. If $w_{1}(t)=\partial_{x}\partial_{y}v(t)$, we have $\frac{dw_{1}}{dt}+A(t)w_{1}+B_{1}(t)w_{1}=f_{1}(t),$ (35) where $\displaystyle B_{1}(t)$ $\displaystyle=\mathcal{W}(t)u_{x}(t)\mathcal{W}(-t)=\frac{1}{2}B(t),$ (36) and $\displaystyle f_{1}(t)$ $\displaystyle=-\mathcal{W}(t)\left(u_{xx}(t)u_{y}(t)\right).$ (37) As before, we have $w_{1}(t)=U(t,0)\psi_{xy}+\displaystyle\int_{0}^{t}U(t,\tau)(-B_{1}(\tau)w_{1}(\tau)+f_{1}(\tau))\,d\tau.$ (38) Since $u_{xx}\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$, $f_{1}\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$. Inasmuch as, also, $B_{1}(t)\in\mathbf{(}H^{r-2}\left(\mathbb{R}^{2}\right))$ is strongly continuous on the interval $[0,T]$. Arguing as before we have that $w_{1}\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$, or equivalently $u_{xy}\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$. Analogously, if $w_{2}(t)=\partial_{y}^{2}v(t)$, we have that $\frac{dw_{2}}{dt}+A(t)w_{2}=f_{2}(t),$ (39) where $f_{2}(t)=-2\mathcal{W}(t)(u_{xy}u_{y}(t).$ (40) Therefore, $\displaystyle w_{2}(t)=U(t,0)\psi_{yy}+\int_{0}^{t}U(t,\tau)f_{2}(\tau)\,d\tau.$ (41) Since $u_{xy}\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$, $f_{2}\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$. Repeating the argument above, we can conclude that $w_{2}\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$, or equivalently, $\partial_{y}^{2}u\in C\left([0,T];H^{r-2}\left(\mathbb{R}^{2}\right)\right)$. Hence, we have showed that if $s<r\leq s+1$ and $\psi\in H^{r}$, $u\in C\left([0,T];H^{r}\left(\mathbb{R}^{2}\right)\right)$. To see the case $r>s+1$, since $\psi\in H^{s^{\prime}}$, for $s^{\prime}<r$, we can use over and over again what we have proved so far, to obtain that $u\in C\left([0,T];H^{r}\left(\mathbb{R}^{2}\right)\right)$ ∎ ### 2.2. Local well-posedness in $X^{s}$, $\widehat{X}_{\alpha}^{s}$ and $Y^{s}$ Let us finish this section showing the local well-posedness of (13) in the spaces $X^{s}$, $\widehat{X}_{\alpha}^{s}$ and $Y^{s}$. ###### Theorem 2.7. Let $s$ and $\alpha$ be as in Theorem 2.1. Let $Z$ also be any of the spaces $X^{s}$, $\widehat{X}^{s}$, $\widehat{X}_{\alpha}^{s}$, $Y^{s}$ and $\widehat{Y}^{s}$. Then, if $\psi\in Z$ and $u\in C([0,T],H^{s})$ is solution to (13) with $u(0)=\psi$, $u\in C([0,T],Z)$. Moreover, $\psi\mapsto u$ is continuous from $Z$ to $C([0,T],Z)$ ###### Proof. Suppose that $\psi$ and $u$ are as in the hypothesis of the theorem. Then, from the fundamental calculus theorem, we have that $u(t)=W_{\alpha}(-t)\psi+\int_{0}^{t}W_{\alpha}(-t+\tau)\partial_{x}\left(\frac{u^{2}(\tau)}{2}\right)\,d\tau.$ So that, $\partial_{x}^{-1}u(t)=W_{\alpha}(-t)\partial_{x}^{-1}\psi+\int_{0}^{t}W_{\alpha}(-t+\tau)\left(\frac{u^{2}(\tau)}{2}\right)\,d\tau.$ From here it follows that (13) is locally well-posed in the spaces $\widehat{X}^{s}$ and $X^{s}$. On the other hand, $\partial_{x}^{-1}\partial_{y}u(t)=W_{\alpha}(-t)\partial_{x}^{-1}\partial_{y}\psi+\int_{0}^{t}W_{\alpha}(-t+\tau)(u\partial_{y}u)(\tau)\,d\tau.$ Then, (13) is locally well-posed in the space $\widehat{Y}^{s}$. To see the local well-posedness in $Y^{s}$ is slightly more complicated. Let $v$ be as in (27). Hence, $w=\partial_{x}^{-1}\partial_{y}v$ satisfies $\begin{cases}w_{t}+A(t,v)w=0\\\ w(0)=\partial_{x}^{-1}\partial_{y}\psi,\end{cases}$ (42) where $A(t,v)$ is as in (27). From linear case of Kato’s theory, more specifically from Theorem 1.14, when taking $X=L^{2}$ and $Y=H^{s}$, we have that, if $\partial_{x}^{-1}\partial_{y}\psi\in H^{s}$, $w$ is continuous on $t$ and depends continuously on this data. This shows the local well-posedness of the problem (13) in $Y^{s}$. Analogously we can show that (13) is local well-posedness in space $X_{\alpha}^{s}$. ∎ ## 3\. Local well-posedness in low regularity spaces In this section we examine the local well-posedness of equation (13) in $H^{s_{1},s_{2}}(\mathbb{R}^{2})$, for $-1\leq\alpha\leq 1$, $s_{1}>\frac{17}{12}-\frac{\alpha}{4}$ and $1<s_{2}\leq s_{1}$. We will use the dispersive properties of the group generated by the linear equation associated to (13). This is the same strategy used by Kenig in [15] for the KP-I equation and by Linares, Pilod and Saut in [26] for the f-KPI and f-KPII equations. ### 3.1. Linear estimates The linear Cauchy problem associated to the problem (13) is $\begin{cases}u_{t}=u_{xxx}-\mathscr{H}D_{x}^{\alpha}u_{yy},\cr u(0)=\psi.\end{cases}$ (43) where $-1\leq\alpha\leq 1$. The unitary group generated by this problem is $W_{\alpha}(t)\psi(x,y)=\left(e^{-it(\xi^{3}+\operatorname{sgn}(\xi)\left\|\xi\right\|^{\alpha}\eta^{2})}\widehat{\psi}\right)^{\vee}=S_{t}^{\alpha}\psi(x,y),$ (44) where $S_{t}^{\alpha}(x,y)=\int_{\mathbb{R}^{2}}{e^{-it(\xi^{3}+\operatorname{sgn}(\xi)\left\|\xi\right\|^{\alpha}\eta^{2})+ix\xi+iy\eta}}d\xi d\eta$ (45) We shall examine this group properties. ###### Lemma 3.1. Let $-1\leq\alpha\leq 1$ and $\frac{\alpha}{2}-1<\operatorname{Re}\beta<\frac{\alpha}{2}$. Then, $\left\\|D_{x}^{\beta}W_{\alpha}(t)\psi\right\\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim\left\|t\right\|^{-\frac{5+2\beta-\alpha}{6}}\left\\|\psi\right\\|_{L^{1}(\mathbb{R}^{2})}.$ (46) ###### Proof. Effectively, $\begin{split}D_{x}^{\beta}S_{t}^{\alpha}(x,y)&=\int_{\mathbb{R}^{2}}{\left\|\xi\right\|^{\beta}e^{-it(\xi^{3}+\operatorname{sgn}(\xi)\left\|\xi\right\|^{\alpha}\eta^{2})+ix\xi+iy\eta}}d\xi d\eta\\\ &=\int_{\mathbb{R}_{\xi}}{\left\|\xi\right\|^{\beta}e^{-it\xi^{3}+ix\xi}\int_{\mathbb{R}_{\eta}}{e^{-it\operatorname{sgn}(\xi)\left\|\xi\right\|^{\alpha}\eta^{2}+iy\eta}}}d\eta d\xi\\\ &=\int_{\mathbb{R}_{\xi}}{\left\|\xi\right\|^{\beta}e^{-it\xi^{3}+ix\xi+\frac{i\operatorname{sgn}(\xi)y^{2}}{4t\left\|\xi\right\|^{\alpha}}}\int_{\mathbb{R}_{\eta}}{e^{-i\left\|t\right\|\operatorname{sgn}(t\xi)\left\|\xi\right\|^{\alpha}\left[\eta-\frac{y}{2t\operatorname{sgn}(\xi)\left\|\xi\right\|^{\alpha}}\right]^{2}}}}d\eta d\xi\\\ &=\frac{\pi^{\frac{1}{2}}}{\left\|t\right\|^{\frac{1}{2}}}\int_{\mathbb{R}}{e^{-it\xi^{3}+ix\xi+\frac{i\operatorname{sgn}(\xi)y^{2}}{4t\left\|\xi\right\|^{\alpha}}-i\operatorname{sgn}(t\xi)\frac{\pi}{4}}}\left\|\xi\right\|^{\beta-\frac{\alpha}{2}}d\xi\\\ &=\frac{\pi^{\frac{1}{2}}}{\|t\|^{\frac{5+2\beta-\alpha}{6}}}\int_{\mathbb{R}}{e^{i\left(\theta^{3}-x\theta t^{-\frac{1}{3}}-\frac{\operatorname{sgn}(\theta)y^{2}}{4\|\theta\|^{\alpha}}\|t\|^{\frac{\beta}{3}-1}\right)}\|\theta\|^{\beta-\frac{\alpha}{2}}}d\theta.\end{split}$ (47) Let us see that the last integral is bounded. For this let us take a function $\chi$ defined on all $\mathbb{R}$, infinitely differentiable, with support in the interval $[-2,2]$ such that $\chi\equiv 1$ in the interval $[-1,1]$. So, $\int_{\mathbb{R}}{e^{i\left(\theta^{3}-x\theta t^{-\frac{1}{3}}-\frac{\operatorname{sgn}(\theta)y^{2}}{4\|\theta\|^{\alpha}}\|t\|^{\frac{\beta}{3}-1}\right)}\|\theta\|^{\beta-\frac{\alpha}{2}}}d\theta=\\\ =\int_{\mathbb{R}}{e^{i\left(\theta^{3}-x\theta t^{-\frac{1}{3}}-\frac{\operatorname{sgn}(\theta)y^{2}}{4\|\theta\|^{\alpha}}\|t\|^{\frac{\beta}{3}-1}\right)}\|\theta\|^{\beta-\frac{\alpha}{2}}}\chi(\theta)d\theta+\\\ +\int_{\mathbb{R}}{e^{i\left(\theta^{3}-x\theta t^{-\frac{1}{3}}-\frac{\operatorname{sgn}(\theta)y^{2}}{4\|\theta\|^{\alpha}}\|t\|^{\frac{\beta}{3}-1}\right)}\|\theta\|^{\beta-\frac{\alpha}{2}}}(1-\chi(\theta))d\theta.$ (48) Clearly the first integral on the right hand side is bounded. To see that the second integral on the right hand side is bounded, we will make use the Van der Corput lemma (see [27], Corollary 1.1). The second and third derivatives of phase function $\vartheta_{\alpha}(\theta)=\theta^{3}-x\theta t^{-\frac{1}{3}}-\frac{\operatorname{sgn}(\theta)y^{2}}{4\|\theta\|^{\alpha}}\|t\|^{\frac{\beta}{3}-1}$ in the integral are $\displaystyle\vartheta_{\alpha}^{\prime\prime}(\theta)=6\theta+\operatorname{sgn}(\theta)\alpha(\alpha+1)\frac{y^{2}\left\|t\right\|^{\frac{\beta}{3}-1}}{4}\left\|\theta\right\|^{-\alpha-2}$ and $\displaystyle\vartheta_{\alpha}^{\prime\prime\prime}(\theta)=6+\alpha(\alpha+1)(\alpha+2)\frac{y^{2}\left\|t\right\|^{\frac{\beta}{3}-1}}{4}\left\|\theta\right\|^{-\alpha-3}.$ It is easily verified that, for $\|\theta\|\geq 1$, $\|\vartheta_{\alpha}^{\prime\prime}(\theta)\|\geq 6$ when $-1\leq\alpha\leq 0$ and that $\|\vartheta_{\alpha}^{\prime\prime\prime}(\theta)\|\geq 6$ when $0\leq\alpha\leq 1$. Since the function $\theta\mapsto\|\theta\|^{\beta-\frac{\alpha}{2}}(1-\chi(\theta))$ is uniformly bounded and integrable on the set $\|\theta\|\geq 1$, it is verified that the second integral on the right side of (48) is bounded. Which verifies that the last integral in (47) is uniformly bounded. Therefore, $\\|D_{x}^{\beta}S_{t}^{\alpha}(x,y)\\|_{L^{\infty}(\mathbb{R}^{2})}\lesssim\|t\|^{\frac{5+2\beta-\alpha}{6}}$ The theorem follows immediately from Young’s inequality for convolution. ∎ ###### Remark 1. The last result coincides with that proved by Linares and Pastor in [23] for the ZK equation (case $\alpha=1$). The results of Saut in [34], for the KPI equation (case $\alpha=-1$), and of Lizarazo in his PhD thesis (see [28]) (case $\alpha=0$) are better estimates. ###### Corollary 3.2. Let $-1\leq\alpha\leq 1$, $0<\epsilon<1$ and $0\leq\theta\leq 1$. Then, $\left\\|D_{x}^{\theta\left(\frac{\alpha}{2}-\epsilon\right)}W_{\alpha}(t)\psi\right\\|_{L^{p}(\mathbb{R}^{2})}\leq\left\|t\right\|^{-\frac{\theta(5-2\epsilon)}{6}}\left\\|\psi\right\\|_{L^{p^{\prime}}(\mathbb{R}^{2})}$ (49) where $\frac{1}{p}+\frac{1}{p^{\prime}}=1$ and $p=\frac{2}{1-\theta}$. ###### Proof. Let $\psi\in L^{1}(\mathbb{R}^{2})\cap L^{2}(\mathbb{R}^{2})$. The corollary follows from Stein’s interpolation theorem for analytical operator families (see [35]). Let us set $z=\theta+i\gamma\in\mathbb{C}$. For each $z$ we define the operators $T^{z}$ by $T_{z}\psi=D_{x}^{z(\frac{\alpha}{2}-\epsilon)}W_{\alpha}(t)\psi.$ This family $\left\\{T_{z}\right\\}$ is an admissible family of operators. By Lemma 3.1, we have $\left\\|T_{1+i\gamma}\psi\right\\|_{L^{\infty}(\mathbb{R}^{2})}=\left\\|D_{x}^{(\frac{\alpha}{2}-\epsilon)(1+i\gamma)}W_{\alpha}(t)\psi\right\\|_{L^{\infty}(\mathbb{R}^{2})}\leq c\left\|t\right\|^{-(\frac{5-2\epsilon}{6})}\left\\|\psi\right\\|_{L^{1}(\mathbb{R}^{2})}.$ (50) Also, since $\left\\{W_{\alpha}(t)\right\\}_{t\in\mathbb{R}}$ is an strongly continuous group on the parameter $t$ in $L^{2}(\mathbb{R}^{2})$, we have $\left\\|T_{i\gamma}\psi\right\\|_{L^{2}(\mathbb{R}^{2})}=\left\\|D_{x}^{(\frac{\alpha}{2}-\epsilon)(i\gamma)}W_{\alpha}(t)\psi\right\\|_{L^{2}(\mathbb{R}^{2})}=\left\\|\psi\right\\|_{L^{2}(\mathbb{R}^{2})}.$ (51) From Stein’s interpolation theorem we obtain $\left\\|T_{\theta}\psi\right\\|_{L^{p}(\mathbb{R}^{2})}\leq c\left\|t\right\|^{-\theta(\frac{5-2\epsilon}{6})}\left\\|\psi\right\\|_{L^{p^{\prime}}(\mathbb{R}^{2})},$ (52) what was we wanted to prove. ∎ ###### Corollary 3.3. Let $-1\leq\alpha\leq 1$, $0<\epsilon<1$ and $0<\theta\leq 1$. Then, $\left\\|D_{x}^{\frac{12}{q\left(5-2\epsilon\right)}\left(\frac{\alpha}{2}-\epsilon\right)}\int_{-\infty}^{\infty}{W_{\alpha}(t-t^{\prime})F(t^{\prime})}dt^{\prime}\right\\|_{L_{T}^{q}L_{xy}^{p}}\lesssim\left\\|F\right\\|_{L_{T}^{q^{\prime}}L^{p^{\prime}}_{xy}},$ (53) where $\frac{1}{q}+\frac{1}{q^{\prime}}=\frac{1}{p}+\frac{1}{p^{\prime}}=1$, $p=\frac{2}{1-\theta}$, $\frac{2}{q}=\frac{\theta(5-2\epsilon)}{6}$ and $\frac{1}{p}+\frac{1}{q}=\frac{1}{2}-\frac{\theta(1+2\epsilon)}{12}$. ###### Proof. From Minkowski’s inequality, Corollary 3.2 and from the Hardy-Littlewood- Sobolev theorem, we have $\left\\|D_{x}^{\theta\left(\frac{\alpha}{2}-\epsilon\right)}\int_{-\infty}^{\infty}{W_{\alpha}(t-t^{\prime})F(t^{\prime})}dt^{\prime}\right\\|_{L_{t}^{q}L_{xy}^{p}}\\\ =\left\\|\left\\|\int_{-\infty}^{\infty}D_{x}^{\theta\left(\frac{\alpha}{2}-\epsilon\right)}W_{\alpha}(t-t^{\prime})F(\cdot,\cdot,t^{\prime})dt^{\prime}\right\\|_{L_{xy}^{p}}\right\\|_{L_{t}^{q}}\\\ \leq\left\\|\int_{-\infty}^{\infty}\left\\|D_{x}^{\theta\left(\frac{\alpha}{2}-\epsilon\right)}W_{\alpha}(t-t^{\prime})F(\cdot,\cdot,t^{\prime})\right\\|_{L_{xy}^{p}}dt^{\prime}\right\\|_{L_{t}^{q}}\\\ \lesssim\left\\|\int_{-\infty}^{\infty}{\left\|t-t^{\prime}\right\|^{-\frac{\theta(5-2\epsilon)}{6}}\left\\|F(\cdot,\cdot,t^{\prime})\right\\|_{L_{xy}^{p^{\prime}}}}dt^{\prime}\right\\|_{L_{t^{q}}}\\\ \lesssim\left\\|\left\\|F(\cdot,\cdot,t^{\prime})\right\\|_{L_{xy}^{p^{\prime}}}\right\\|_{L_{t^{q^{\prime}}}}=\left\\|F\right\\|_{L_{t}^{q^{\prime}}L_{xy}^{p^{\prime}}}.$ ∎ Using the Stein-Thomas argument (see [9]) we have. ###### Proposition 3.4. Let $\alpha$, $\epsilon$, $\theta$, $p$ and $q$ be as in the previous corollary. Then, $\left\\|D_{x}^{\frac{6}{q\left(5-2\epsilon\right)}\left(\frac{\alpha}{2}-\epsilon\right)}W_{\alpha}(t)\psi\right\\|_{L_{T}^{q}L_{xy}^{p}}\lesssim\left\\|\psi\right\\|_{L_{xy}^{2}}.$ (54) From the above we have the following two very useful corollaries in the proof of the local well-posedness in this section. ###### Corollary 3.5. For each $T>0$ and $0<\epsilon<1$, we have that $\left\\|W_{\alpha}(t)\psi\right\\|_{L_{T}^{2}L_{xy}^{\infty}}\lesssim T^{\frac{1+2\epsilon}{12}}\left\\|D_{x}^{\frac{1}{2}(\epsilon-\frac{\alpha}{2})}\psi\right\\|_{L_{xy}^{2}}.$ (55) ###### Proof. From Hölder’s inequality and Proposition 3.4 we have that $\begin{split}\left\\|W_{\alpha}(t)\psi\right\\|_{L_{T}^{2}L_{xy}^{\infty}}&\leq T^{\frac{1+2\epsilon}{12}}\left\\|W_{\alpha}(t)\psi\right\\|_{L_{T}^{\frac{12}{5-2\epsilon}}L_{xy}^{\infty}}\lesssim T^{\frac{1+2\epsilon}{12}}\left\\|D_{x}^{\frac{1}{2}(\epsilon-\frac{\alpha}{2})}\psi\right\\|_{L_{xy}^{2}}.\end{split}$ ∎ Just like in [15] and [26], we prove a refined Strichartz estimate for the solution to non-homogeneous linear problem $\partial_{t}w=w_{xxx}-\mathscr{H}D_{x}^{\alpha}w_{yy}+F.$ (56) So, we have the next lemma. ###### Lemma 3.6. Let $-1\leq\alpha\leq 1,0<\epsilon<1$ and $T>0$. If $w$ is solution to (56), then there exists $c_{\epsilon}>0$ such that $\left\\|\partial_{x}w\right\\|_{L_{T}^{1}L_{xy}^{\infty}}\leq c_{\epsilon}T^{\frac{7+2\epsilon}{12}}\left(\sup_{t\in[0,T]}{\left\\|J_{x}^{\frac{17}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}w\right\\|_{L_{xy}^{2}}}+\int_{0}^{T}\left\\|D_{x}^{\frac{5}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}F(t)\right\\|_{L_{xy}^{2}}dt\right).$ (57) ###### Proof. We will make use of a Littlewood-Paley type decomposition of $w$ on $x$ variable. For this, let $\varphi_{0},\varphi\in C^{\infty}_{0}$ with $\operatorname{supp}(\varphi_{0})=\left\\{\left\|\xi\right\|<2\right\\}$ and $\operatorname{supp}(\varphi)=\left\\{\frac{1}{2}<\left\|\xi\right\|<2\right\\}$ such that $\varphi_{0}(\xi)+\sum_{k=1}^{\infty}{\varphi(2^{-k}\xi)}=1$ for all $\xi\in\mathbb{R}$. Let us define $P_{k}w$ via Fourier transform by $\widehat{P_{0}w}(\xi,\eta)=\varphi_{0}(\xi)\widehat{w}(\xi,\eta)$ and $\widehat{P_{k}w}(\xi,\eta)=\varphi(2^{-k}\xi)\widehat{w}(\xi,\eta)$ for $k\geq 1$, in such a way that $w=\sum_{k=0}^{\infty}{P_{k}w}.$ Let us first make an estimate for $\left\\|\partial_{x}P_{0}w\right\\|_{L_{T}^{1}L_{xy}^{\infty}}$. Since $w$ satisfies (56), then $P_{0}w$ satisfies the integral equation when apply $P_{0}$ on both sides of the equation. So, from the Cauchy-Schwarz inequality and (55), it follows that $\begin{split}&\left\\|\partial_{x}P_{0}w\right\\|_{L_{T}^{1}L_{xy}^{\infty}}\leq\\\ &\leq\left\\|W_{\alpha}(t)\partial_{x}P_{0}w(0)\right\\|_{L_{T}^{1}L_{xy}^{\infty}}+\int_{0}^{t}\left\\|W_{\alpha}(t-t^{\prime})\partial_{x}P_{0}F(t^{\prime})\right\\|_{L_{T}^{1}L_{xy}^{\infty}}dt^{\prime}\\\ &\leq T^{\frac{1}{2}}\left(\left\\|W_{\alpha}(t)\partial_{x}P_{0}w(0)\right\\|_{L_{T}^{2}L_{xy}^{\infty}}+\int_{0}^{t}\left\\|W_{\alpha}(t-t^{\prime})\partial_{x}P_{0}F(t^{\prime})\right\\|_{L_{T}^{2}L_{xy}^{\infty}}dt^{\prime}\right)\\\ &\lesssim T^{\frac{1}{2}+\frac{1+2\epsilon}{12}}\left(\left\\|D_{x}^{\frac{1}{2}(\epsilon-\frac{\alpha}{2})}\partial_{x}P_{0}w(0)\right\\|_{L_{xy}^{2}}+\int_{0}^{t}\left\\|D_{x}^{\frac{1}{2}(\epsilon-\frac{\alpha}{2})}\partial_{x}P_{0}F(t^{\prime})\right\\|_{L_{xy}^{2}}dt^{\prime}\right)\\\ &\lesssim c_{\epsilon}T^{\frac{7+2\epsilon}{12}}\left(\left\\|J_{x}^{\frac{17}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}P_{0}w(0)\right\\|_{L_{xy}^{2}}+\int_{0}^{T}\left\\|D_{x}^{\frac{5}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}P_{0}F(t)\right\\|_{L_{xy}^{2}}dt\right).\end{split}$ (58) Now, let us estimate $\left\\|\partial_{x}P_{k}w\right\\|_{L_{T}^{1}L_{xy}^{\infty}}$, when $k\geq 1$. For this, we will make a suitable partition in time in such a way that allow us to control the localized frequencies associated to $x$ variable. So, let $\mathcal{P}=\left\\{a_{0},a_{1},...,a_{2^{k}}\right\\}$ be a partition from interval $[0,T]$ with $a_{j}=jT2^{-k}$, $j=0,1,\cdots,2^{k}$. We denote by $I_{j}$ the interval $[a_{j-1},a_{j}]$. Then, thanks to Young’s inequality, Cauchy-Schwarz’s inequality and the inequality $\left\\|(\chi_{\left\\{2^{k-1}<\left\|\xi\right\|<2^{k}\right\\}}\xi)^{\vee}\right\\|_{L^{1}(\mathbb{R})}\lesssim 2^{k},$ we have $\begin{split}\left\\|\partial_{x}P_{k}w\right\\|_{L_{T}^{1}L_{xy}^{\infty}}&\leq\left\\|(\chi_{\left\\{2^{k-1}<\left\|\xi\right\|<2^{k}\right\\}}(\xi)\left\|\xi\right\|)^{\vee}\right\\|_{L^{1}(\mathbb{R})}\left\\|P_{k}w\right\\|_{L_{T}^{1}L_{xy}^{\infty}}\\\ &\lesssim 2^{k}\sum_{j=1}^{2^{k}}\left\\|P_{k}w\right\\|_{L_{I_{j}}^{1}L_{xy}^{\infty}}\\\ &\lesssim(2^{k})\sum_{j=1}^{2^{k}}(a_{j}-a_{j-1})^{\frac{1}{2}}\left\\|P_{k}w\right\\|_{L_{I_{j}}^{2}L_{xy}^{\infty}}\\\ &=(2^{k}T)^{\frac{1}{2}}\sum_{j=1}^{2^{k}}\left\\|P_{k}w\right\\|_{L_{I_{j}}^{2}L_{xy}^{\infty}}.\end{split}$ (59) From Duhamel’s principle on each interval $[a_{j-1},a_{j}]$, we have that, for each $t\in[a_{j-1},a_{j}]$, $P_{k}w(t)=W_{\alpha}(t-a_{j})P_{k}w(\cdot,a_{j})+\int_{a_{j-1}}^{t}W_{\alpha}(t-t^{\prime})P_{k}F(t^{\prime})dt^{\prime}.$ By (59) and (55), we get $\begin{split}&\left\\|\partial_{x}P_{k}w\right\\|_{L_{T}^{1}L_{xy}^{\infty}}\lesssim(2^{k}T)^{\frac{1}{2}}\sum_{j=1}^{2^{k}}\left\\|P_{k}w\right\\|_{L_{[a_{j},b_{j}]}^{2}L_{xy}^{\infty}}\\\ &\lesssim(2^{k}T)^{\frac{1}{2}}\sum_{j=1}^{2^{k}}{\left(\left\\|W_{\alpha}(t-a_{j})P_{k}w(\cdot,a_{j})\right\\|_{L_{I_{j}}^{2}L_{xy}^{\infty}}+\int_{a_{j-1}}^{t}{\left\\|W_{\alpha}(t-t^{\prime})P_{k}F(t^{\prime})\right\\|_{L_{I_{j}}^{2}L_{xy}^{\infty}}}dt^{\prime}\right)}\\\ &\lesssim 2^{\frac{k(5+2\epsilon)}{12}}T^{\frac{7+2\epsilon}{12}}\sum_{j=1}^{2^{k}}{\left(\left\\|D_{x}^{\frac{1}{2}\left(\epsilon-\frac{\alpha}{2}\right)}P_{k}w(\cdot,a_{j})\right\\|_{L_{xy}^{2}}+\int_{I_{j}}{\left\\|D_{x}^{\frac{1}{2}\left(\epsilon-\frac{\alpha}{2}\right)}P_{k}F(t^{\prime})\right\\|_{L_{xy}^{2}}}dt^{\prime}\right)}\\\ &\lesssim T^{\frac{7+2\epsilon}{12}}\Bigg{\\{}2^{\frac{k(5+2\epsilon)}{12}}\sum_{j=1}^{2^{k}}{\left\\|D_{x}^{\frac{1}{2}\left(\epsilon-\frac{\alpha}{2}\right)}P_{k}w(\cdot,a_{j})\right\\|_{L_{xy}^{2}}}+\\\ &\phantom{T^{\frac{7+2\epsilon}{12}}\Bigg{\\{}2^{\frac{k(5+2\epsilon)}{12}}\sum_{j=1}^{2^{k}}\left\\|D_{x}^{\frac{1}{2}\left(\epsilon-\frac{\alpha}{2}\right)}P_{k}w\right.}+2^{\frac{k(5+2\epsilon)}{12}}\int_{0}^{T}{\left\\|D_{x}^{\frac{1}{2}\left(\epsilon-\frac{\alpha}{2}\right)}P_{k}F(t^{\prime})\right\\|_{L_{xy}^{2}}}dt^{\prime}\Bigg{\\}}\\\ &\lesssim c_{\epsilon}T^{\frac{7+2\epsilon}{12}}\left(\sup_{t\in[0,T]}{\left\\|D_{x}^{\frac{17}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}P_{k}w(t)\right\\|_{L_{xy}^{2}}}+\int_{0}^{T}{\left\\|D_{x}^{\frac{5}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}P_{k}F(t^{\prime})\right\\|_{L_{xy}^{2}}}dt^{\prime}\right).\end{split}$ (60) For the sake of the completeness, let us explain the inequality $2^{\frac{k(5+2\epsilon)}{12}}\sum_{j=1}^{2^{k}}{\left\\|D_{x}^{\frac{1}{2}\left(\epsilon-\frac{\alpha}{2}\right)}P_{k}w(\cdot,a_{j})\right\\|_{L_{xy}^{2}}}\leq\sup_{t\in[0,T]}{\left\\|D_{x}^{\frac{17}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}P_{k}w(t)\right\\|_{L_{xy}^{2}}},$ (61) what was used to prove the inequality above. Observe that, for each integer $j\leq 2^{k}$, we have $\left\\|D_{x}^{\frac{1}{2}\left(\epsilon-\frac{\alpha}{2}\right)}P_{k}w(\cdot,a_{j})\right\\|_{L_{xy}^{2}}\leq j^{-\frac{k(17+2\epsilon)}{12}}\left\\|D_{x}^{\frac{17}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}P_{k}w(\cdot,a_{j})\right\\|_{L_{xy}^{2}}.$ Summing on $j$ from $1$ to $2^{k}$ and considering $\sum_{j=1}^{2^{k}}j^{-\frac{k(17+2\epsilon)}{12}}\sim 2^{-\frac{k(5+2\epsilon)}{12}}$ it follows the inequality (61). (60) together with (58) show the present lemma. ∎ ### 3.2. Energy Estimate ###### Lemma 3.7. Let $-1\leq\alpha\leq 1$, $T>0$ and assume that $u\in C([0,T];H^{\infty}(\mathbb{R}^{2}))$ is a solution to the Cauchy problem (13). Then, there exists a positive constant $C$ such that, for $1\leq s_{2}\leq s_{1}$, $\left\\|u\right\\|_{L_{T}^{\infty}H_{xy}^{s_{1},s_{2}}}\leq\left\\|\psi\right\\|_{H^{s_{1},s_{2}}_{xy}}e^{{C(\left\\|\partial_{x}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}+\left\\|\partial_{y}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}})}}.$ (62) ###### Proof. Let $u$ as in the statement in lemma. Let us estimate first $\left\\|J_{x}^{s_{1}}u\right\\|_{L^{2}_{xy}}$. Operating with $J_{x}^{s_{1}}$ and then multiplying by $J_{x}^{s_{1}}u$ on both sides of the equation (13), and integrating with respect to $x,y$, we obtain $\begin{split}\frac{1}{2}\frac{d}{dt}\int_{\mathbb{R}^{2}}{{\left(J_{x}^{s_{1}}u\right)}^{2}dxdy}&=\int_{\mathbb{R}^{2}}{J_{x}^{s_{1}}(u\partial_{x}u)J_{x}^{s_{1}}u\,dxdy}\\\ &=\int_{\mathbb{R}^{2}}\left[J_{x}^{s_{1}},u\right]\partial_{x}uJ_{x}^{s_{1}}udxdy+\int_{\mathbb{R}^{2}}uJ_{x}^{s_{1}}\partial_{x}uJ_{x}^{s_{1}}u\,dxdy.\end{split}$ (63) From inequalities of Cauchy-Schwarz and Kato-Ponce (Lemma 1.8), applied in the $x$ variable, we have $\begin{split}\int_{\mathbb{R}^{2}}\left[J_{x}^{s_{1}},u\right]\partial_{x}uJ_{x}^{s_{1}}u\,dxdy&\leq\int_{\mathbb{R}_{y}}\left\\|\left[J_{x}^{s_{1}},u\right]\partial_{x}u\right\\|_{L_{x}^{2}}\left\\|J_{x}^{s_{1}}u\right\\|_{L_{x}^{2}}dy\\\ &\lesssim\int_{\mathbb{R}_{y}}\left\\|\partial_{x}u\right\\|_{L_{x}^{\infty}}\left\\|J_{x}^{s_{1}}u\right\\|^{2}_{L_{x}^{2}}dy\\\ &\leq\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\left\\|J_{x}^{s_{1}}u\right\\|^{2}_{L_{xy}^{2}}\leq\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\left\\|u\right\\|^{2}_{H_{xy}^{s_{1},s_{2}}}.\end{split}$ (64) On the other hand, $\int_{\mathbb{R}^{2}}uJ_{x}^{s_{1}}\partial_{x}uJ_{x}^{s_{1}}\,u\,dxdy=-\frac{1}{2}\int_{\mathbb{R}^{2}}\partial_{x}u\left(J_{x}^{s_{1}}u\right)^{2}dxdy\lesssim\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\left\\|J_{x}^{s_{1}}u\right\\|^{2}_{L_{xy}^{2}}.$ (65) By (63), (64) and (65), we have $\frac{d}{dt}\left\\|J_{x}^{s_{1}}u\right\\|^{2}_{L^{2}_{xy}}\lesssim\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\left\\|J_{x}^{s_{1}}u\right\\|^{2}_{L_{xy}^{2}}\lesssim\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\left\\|u\right\\|^{2}_{H_{xy}^{s_{1},s_{2}}}.$ (66) In a totally analogous way we shall estimate $\left\\|J_{y}^{s_{2}}u\right\\|_{L^{2}_{xy}}$. So, $\frac{1}{2}\frac{d}{dt}\left\\|J_{y}^{s_{2}}u\right\\|^{2}_{L^{2}_{xy}}=\int_{\mathbb{R}^{2}}\left[J_{y}^{s_{2}},u\right]\partial_{x}uJ_{y}^{s_{2}}u\,dxdy+\int_{\mathbb{R}^{2}}uJ_{y}^{s_{2}}\partial_{x}uJ_{y}^{s_{2}}u\,dxdy.$ (67) Proceeding as before, we have $\begin{split}&\int_{\mathbb{R}^{2}}\left[J_{y}^{s_{2}},u\right]\partial_{x}uJ_{y}^{s_{2}}u\,dxdy\leq\int_{\mathbb{R}_{x}}\left\\|\left[J_{y}^{s_{2}},u\right]\partial_{x}u\right\\|_{L_{y}^{2}}\left\\|J_{y}^{s_{2}}u\right\\|_{L_{y}^{2}}dx\lesssim\\\ &\lesssim\int_{\mathbb{R}_{x}}\left(\left\\|\partial_{y}u\right\\|_{L_{y}^{\infty}}\left\\|J_{y}^{s_{2}-1}\partial_{x}u\right\\|_{L^{2}_{y}}+\left\\|\partial_{x}u\right\\|_{L_{y}^{\infty}}\left\\|J_{y}^{s_{2}}\partial_{x}u\right\\|_{L^{2}_{y}}\right)\left\\|J_{y}^{s_{2}}u\right\\|_{L^{2}_{y}}dx\\\ &\leq\left\\|\partial_{y}u\right\\|_{L_{xy}^{\infty}}\left\\|J_{y}^{s_{2}-1}\partial_{x}u\right\\|_{L^{2}_{xy}}\left\\|J_{y}^{s_{2}}u\right\\|_{L^{2}_{xy}}+\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\left\\|J_{y}^{s_{2}}u\right\\|_{L^{2}_{xy}}^{2}\\\ &\leq\left\\|\partial_{y}u\right\\|_{L_{xy}^{\infty}}\left\\|J_{x}^{s_{2}}u\right\\|_{L^{2}_{xy}}^{\frac{1}{s_{2}}}\left\\|J_{y}^{s_{2}}u\right\\|^{\frac{2s_{2}-1}{s_{2}}}_{L^{2}_{xy}}+\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\left\\|J_{y}^{s_{2}}u\right\\|_{L^{2}_{xy}}^{2}\\\ &\lesssim\left(\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}+\left\\|\partial_{y}u\right\\|_{L_{xy}^{\infty}}\right)\left\\|u\right\\|_{H^{s_{1},s_{2}}_{xy}}^{2}.\end{split}$ (68) So, integrating by parts in the second term on the right side of the inequality (67) and from last inequality, we have $\frac{d}{dt}\left\\|J_{y}^{s_{2}}u\right\\|^{2}_{L^{2}_{xy}}\lesssim\left(\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}+\left\\|\partial_{y}u\right\\|_{L_{xy}^{\infty}}\right)\left\\|u\right\\|_{H^{s_{1},s_{2}}_{xy}}^{2}.$ (69) Gathering (66) and (69), we obtain $\frac{d}{dt}\left\\|u\right\\|^{2}_{H^{s_{1},s_{2}}_{xy}}\lesssim\left(\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}+\left\\|\partial_{y}u\right\\|_{L_{xy}^{\infty}}\right)\left\\|u\right\\|_{H^{s_{1},s_{2}}_{xy}}^{2}.$ (70) From Gronwall inequality we get the lemma. ∎ ### 3.3. A Strichartz estimate type The energy estimate suggests that we have to control the norms $\left\\|\partial_{x}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}$ and $\left\\|\partial_{y}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}$ in order to prove the local well-posedness of the Cauchy problem (13). The next lemma shows how we can control these norms. ###### Lemma 3.8. Let $-1\leq\alpha\leq 1$, $T>0$ and assume that $u\in C([0,T];H^{\infty}(\mathbb{R}^{2}))$ is a solution to initial value problem (13) with initial condition $\psi$. Then, for any $s_{1}>\frac{17}{12}-\frac{\alpha}{4}$ such that $\begin{cases}s_{2}>1,&\mbox{if }0\leq\alpha\leq 1,\\\ \frac{1}{s_{2}}-\frac{\alpha}{4s_{1}}<1,&\mbox{if }-1\leq\alpha\leq 0\end{cases}$ and $s_{1}\geq s_{2}$, there exist constants $C_{s_{1},s_{2}}$ and $k_{s_{1},s_{2}}\in(7/12,1)$ such that $f(T)=\left\\|u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}+\left\\|\partial_{x}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}+\left\\|\partial_{y}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}$ (71) satisfies $f(T)\leq C_{s_{1},s_{2}}T^{k_{s_{1},s_{2}}}(1+f(T))\left\\|u\right\\|_{L_{T}^{\infty}H^{s_{1},s_{2}}_{xy}}.$ (72) ###### Proof. First let us examine an estimate for $\left\\|\partial_{x}u\right\\|_{L^{1}_{T}L_{xy}^{\infty}}$. From the refined Strichartz estimate (57), we have $\left\\|\partial_{x}u\right\\|_{L^{1}_{T}L_{xy}^{\infty}}\leq c_{\epsilon}T^{\frac{7+2\epsilon}{12}}\left(\sup_{t\in[0,T]}{\left\\|J_{x}^{\frac{17}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}u\right\\|_{L^{2}_{xy}}}+\int_{0}^{T}\left\\|D_{x}^{\frac{5}{12}-\frac{\alpha}{4}+\frac{2\epsilon}{3}}(u\partial_{x}u)\right\\|_{L^{2}_{xy}}dt\right).$ (73) Let us choose $0<\epsilon_{1}<1$ in such a way that $\frac{5}{12}-\frac{\alpha}{4}+\frac{2\epsilon_{1}}{3}<1$ and ${\frac{17}{12}-\frac{\alpha}{4}+\frac{2\epsilon_{1}}{3}}<s_{1}$. So, the first term from the right side of (74) we have $\sup_{t\in[0,T]}{\left\\|J_{x}^{\frac{17}{12}-\frac{\alpha}{4}+\frac{2\epsilon_{1}}{3}}u\right\\|_{L^{2}_{xy}}}\leq\sup_{t\in[0,T]}{\left\\|J_{x}^{s_{1}}u\right\\|_{L^{2}_{xy}}}\leq\left\\|u\right\\|_{L^{\infty}_{T}H^{s_{1},s_{2}}_{xy}}.$ (74) For the second term on the right side of (74), from the Leibniz rule (equation (19)) in $x$, we have $\begin{split}&\int_{0}^{T}\left\\|D_{x}^{\frac{5}{12}-\frac{\alpha}{4}+\frac{2\epsilon_{1}}{3}}(u\partial_{x}u)\right\\|_{L^{2}_{xy}}dt\lesssim\\\ &\lesssim\int_{0}^{T}{\left\\|{{\left\\|D_{x}^{\frac{5}{12}-\frac{\alpha}{4}+\frac{2\epsilon_{1}}{3}}u\right\\|_{L^{2}_{x}}}{\left\\|\partial_{x}u\right\\|_{L_{x}^{\infty}}}+{\left\\|D_{x}^{\frac{5}{12}-\frac{\alpha}{4}+\frac{2\epsilon_{1}}{3}}\partial_{x}u\right\\|_{L^{2}_{x}}}{\left\\|u\right\\|_{L_{x}^{\infty}}}}\right\\|_{L^{2}_{y}}}dt\\\ &\lesssim\int_{0}^{T}{\left\\|{{\left\\|J_{x}^{s_{1}}u\right\\|_{L^{2}_{x}}}{\left\\|\partial_{x}u\right\\|_{L_{x}^{\infty}}}+{\left\\|J_{x}^{s_{1}}u\right\\|_{L^{2}_{x}}}{\left\\|u\right\\|_{L_{x}^{\infty}}}}\right\\|_{L^{2}_{y}}}dt\\\ &\lesssim\int_{0}^{T}\left(\left\\|u\right\\|_{L^{\infty}_{xy}}+\left\\|\partial_{x}u\right\\|_{L^{\infty}_{xy}}\right)\left\\|J_{x}^{s_{1}}u\right\\|_{L^{2}_{xy}}dt\\\ &\lesssim\left\\|u\right\\|_{L_{T}^{\infty}H^{s_{1},s_{2}}_{xy}}f(T)\end{split}$ (75) Then, from (75), (74) and (73) $\left\\|\partial_{x}u\right\\|_{L^{1}_{T}L_{xy}^{\infty}}\leq C_{1}(s_{1})T^{\frac{7+2\epsilon_{1}}{12}}(1+f(T))\left\\|u\right\\|_{L_{T}^{\infty}H^{s_{1},s_{2}}_{xy}}.$ (76) Let us estimate now $\left\\|u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}$. From Duhamel’s principle, Cauchy-Schwarz inequality in $t$ and Corollary 3.5, we have that $\begin{split}\left\\|u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}\leq\left\\|W_{\alpha}(t)\psi\right\\|_{L_{T}^{1}L_{xy}^{\infty}}+\int_{0}^{T}\left\\|W_{\alpha}(t-t^{\prime})(u\partial_{x}u)(t^{\prime})\right\\|_{L_{T}^{1}L_{xy}^{\infty}}dt^{\prime}\\\ \leq T^{\frac{1}{2}}\left(\left\\|W_{\alpha}(t)\psi\right\\|_{L_{T}^{2}L_{xy}^{\infty}}+\int_{0}^{T}\left\\|W_{\alpha}(t-t^{\prime})(u\partial_{x}u)(t^{\prime})\right\\|_{L_{T}^{2}L_{xy}^{\infty}}dt^{\prime}\right)\\\ \leq C_{2}T^{\frac{7+2\epsilon_{2}}{12}}\left(\left\\|D_{x}^{\frac{1}{2}(\epsilon_{2}-\frac{\alpha}{2})}\psi\right\\|_{L_{xy}^{2}}+\int_{0}^{T}\left\\|D_{x}^{\frac{1}{2}(\epsilon_{2}-\frac{\alpha}{2})}(u\partial_{x}u)(t^{\prime})\right\\|_{L_{xy}^{2}}dt^{\prime}\right),\end{split}$ (77) for some $\epsilon_{2}=\epsilon_{2}(s_{1})$ such that $0<\frac{1}{2}(\epsilon_{2}-\frac{\alpha}{2})$ and $\frac{1}{2}(\epsilon_{2}-\frac{\alpha}{2})+1<s_{1}$. The first term in the last line of equation above, thanks to the choice of $\epsilon_{2}$, is less than $\left\\|J_{x}^{s_{1}}\psi\right\\|_{L^{2}_{xy}}$. For the term within the integral, from the Leibniz rule in the $x$ variable, we have $\begin{split}\left\\|D_{x}^{{\frac{1}{2}(\epsilon_{2}-\frac{\alpha}{2})}}(u\partial_{x}u)\right\\|_{L_{xy}^{2}}&\leq\left\\|D_{x}^{\frac{1}{2}(\epsilon_{2}-\frac{\alpha}{2})}u\right\\|_{L_{xy}^{2}}\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}+\left\\|D_{x}^{\frac{1}{2}(\epsilon_{2}-\frac{\alpha}{2})}\partial_{x}u\right\\|_{L_{xy}^{2}}\left\\|u\right\\|_{L_{xy}^{\infty}}\\\ &\leq\left\\|J_{x}^{s_{1}}u\right\\|_{L_{xy}^{2}}\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}+\left\\|J_{x}^{s_{1}}u\right\\|_{L_{xy}^{2}}\left\\|u\right\\|_{L_{xy}^{\infty}}.\end{split}$ (78) So that $\begin{split}\left\\|u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}&\leq C_{2}T^{\frac{7+2\epsilon_{2}}{12}}\left(\left\\|J_{x}^{s_{1}}\psi\right\\|_{L^{2}_{xy}}+\int_{0}^{T}\left(\left\\|u\right\\|_{L_{xy}^{\infty}}+\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\right)\left\\|J_{x}^{s_{1}}u\right\\|_{L_{xy}^{2}}dt^{\prime}\right)\\\ &\leq C_{2}T^{\frac{7+2\epsilon_{2}}{12}}\left(\left\\|J_{x}^{s_{1}}\psi\right\\|_{L^{2}_{xy}}+\int_{0}^{T}\left(\left\\|u\right\\|_{L_{xy}^{\infty}}+\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\right)\left\\|J_{x}^{s_{1}}u\right\\|_{L_{xy}^{2}}dt^{\prime}\right)\\\ &\leq C_{2}T^{\frac{7+2\epsilon_{2}}{12}}(1+f(T))\left\\|u\right\\|_{L_{T}^{\infty}H^{s_{1},s_{2}}_{xy}}\end{split}$ (79) where $C_{2}=C_{2}(s_{1})$. Lastly, let us estimate $\left\\|\partial_{y}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}$. In the same way as we estimate the other two terms, we have that $\begin{split}\left\\|\partial_{y}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}\leq C_{3}T^{\frac{7+2\epsilon_{3}}{12}}\left(\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})}\partial_{y}\psi\right\\|_{L_{xy}^{2}}+\int_{0}^{T}\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})}\partial_{y}(u\partial_{x}u)\right\\|_{L_{xy}^{2}}dt\right),\end{split}$ (80) where we will choose $\epsilon_{3}$ later. For the term within the integral, from the commutator estimate (Lemma 1.10) in the $x$ variable, we obtain $\begin{split}\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})+1}(u\partial_{y}u)\right\\|_{L_{xy}^{2}}\leq\left\\|\left[D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})+1},u\right]\partial_{y}u\right\\|_{L_{xy}^{2}}+\left\\|uD_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})}\partial_{y}u\right\\|_{L_{xy}^{2}}\\\ \leq\left(\left\\|u\right\\|_{L_{xy}^{\infty}}+\left\\|\partial_{x}u\right\\|_{L_{xy}^{\infty}}\right)\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})}\partial_{y}u\right\\|_{L_{xy}^{2}}+\left\\|\partial_{y}u\right\\|_{L_{xy}^{\infty}}\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})+1}u\right\\|_{L_{xy}^{2}}.\end{split}$ (81) Let us, then, examine $\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})+1}u\right\\|_{L_{xy}^{2}}$ and $\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})}\partial_{y}u\right\\|_{L_{xy}^{2}}$. We will do this for two cases in $\alpha$. The first case is when $-1\leq\alpha\leq 0$. In this case $\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})>0$ for $0<\epsilon_{3}<1$. Choose $\epsilon_{3}$ in such a way that $\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})+1<s_{1}$ and $\frac{\epsilon_{3}}{2s_{1}}-\frac{\alpha}{4s_{1}}+\frac{1}{s_{2}}<1$, or equivalently $\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})\frac{s_{2}}{s_{2}-1}<s_{1}$. So, $\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})+1}u\right\\|_{L_{xy}^{2}}\leq\\|J_{x}^{s_{1}}u\\|_{L^{2}_{x,y}}\leq\\|u\\|_{H^{x_{1}.x_{2}}_{x,y}}$ (82) and $\begin{split}\left\\|D_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})}\partial_{y}u\right\\|_{L_{xy}^{2}}&\leq\left\\|J_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})}J_{y}u\right\\|_{L_{xy}^{2}}\leq\left\\|J_{x}^{\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})\frac{s_{2}}{s_{2}-1}}u\right\\|_{L_{xy}^{2}}^{\frac{s_{2}-1}{s_{2}}}\left\\|J_{y}^{s_{2}}u\right\\|_{L_{xy}^{2}}^{\frac{1}{s_{2}}}\\\ &\leq\left\\|J_{x}^{s_{1}}u\right\\|_{L_{xy}^{2}}^{\frac{s_{2}-1}{s_{2}}}\left\\|J_{y}^{s_{2}}u\right\\|_{L_{xy}^{2}}^{\frac{1}{s_{2}}}\leq\\|u\\|_{H_{x,y}^{s_{1},s_{2}}}\end{split}$ (83) The second case is when $0\leq\alpha\leq 1$. Choose, then, $\epsilon_{3}>\frac{\alpha}{2}$ such that also $\frac{1}{2}(\epsilon_{3}-\frac{\alpha}{2})+1<s_{1}$ and $\frac{\epsilon_{3}}{2s_{1}}-\frac{\alpha}{4s_{1}}+\frac{1}{s_{2}}<1$. So we are in the same situation of the first case, what leads us to the same two inequalities (82) and (83). This inequalities together to (81) and (80) allow us show that $\begin{split}\left\\|\partial_{y}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}\leq C_{3}T^{\frac{7+2\epsilon_{3}}{12}}\left(\left\\|\psi\right\\|_{H_{xy}^{s_{1},s_{2}}}+(1+f(T))\left\\|u\right\\|_{L_{T}^{\infty}H_{xy}^{s_{1},s_{2}}}\right).\end{split}$ (84) The theorem follows from (76), (79) and (84), taking $C_{s_{1},s_{2}}=C_{1}+C_{2}+C_{3}$ and $k_{s_{1},s_{2}}=\max_{i=1,2,3}{\frac{7+2\epsilon_{i}}{12}}$. ∎ ### 3.4. Local well-posedness We already have all the necessary ingredients to prove the following theorem. ###### Theorem 3.9. Let $-1<\alpha<1$ and $0<\epsilon<1$ be such that $s_{1}>\frac{17+4\epsilon-3\alpha}{12},\quad s_{2}\leq s_{1}\text{ and}\quad\begin{cases}s_{2}>1,&\text{ if }\alpha>0,\\\ \frac{5+4\epsilon-3\alpha}{12s_{1}}+\frac{1}{s_{2}}<1,&\text{ if }\alpha\leq 0.\end{cases}$ Then, for any $\psi\in H^{s_{1},s_{2}}(\mathbb{R}^{2})$, there exist a time $T=T(\left\\|\psi\right\\|_{H^{s_{1},s_{2}}})$ and a unique solution $u\in C([0,T]:H^{s_{1},s_{2}})$ to the Cauchy problem (13) such that $u,\partial_{x}u,\partial_{y}u\in L_{T}^{1}L_{xy}^{\infty}$. Furthermore, if $0<T^{\prime}<T$, there exists a neighborhood $\mathcal{V}$ of $\psi$ in $H^{s_{1},s_{2}}(\mathbb{R}^{2})$ such that $\psi\mapsto u(t)$ is continuous From the local well-posedness of (13) in $H^{\sigma}$ for $\sigma>2$ ($\sigma$ fix), for $\psi\in H^{\infty}(\mathbb{R}^{2})$, there exists a unique solution $u\in C([0,T^{}]:H^{\sigma}(\mathbb{R}^{2}))$, where $T^{}$ is the maximal existence time of the solution, such that if $T^{}<\infty$, then $\lim_{t\nearrow T^{}}\left\\|u(t)\right\\|_{H^{\sigma}}=\infty.$ (85) Thanks to the above we have the following a priori estimate that will be useful in the proof of the existence of the solution to problem (13). ###### Lemma 3.10. Let $2<\sigma$, $-1\leq\alpha\leq 1$ and $s_{1}$, $s_{2}$ be as in Theorem (3.9) such that $s_{i}\leq\sigma$, $i=1,2$. Let, also, $\psi\in H^{\sigma}$ and $u\in C([0,T^{});H^{\sigma}(\mathbb{R}^{2}))$ such that $u(0)=\psi$, where $T^{}$ is the maximal time of existence of $u$. Then, there exist $K_{0}=K_{0}(s_{1},s_{2})>0$ and $L_{s_{1},s_{2}}>0$ such that $T^{}>T$, where $(L_{s_{1},s_{2}}\left\\|\psi\right\\|_{H^{s_{1},s_{2}}}+1)^{\frac{12}{7}}T=1$, and $\begin{split}\left\\|u\right\\|_{L_{T}^{\infty}H^{s_{1},s_{2}}}&\leq 2\left\\|\psi\right\\|_{H^{s_{1},s_{2}}}\\\ f(T)=\left\\|u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}+\left\\|\partial_{x}u\right\\|&{}_{L_{T}^{1}L_{xy}^{\infty}}+\left\\|\partial_{y}u\right\\|_{L_{T}^{1}L_{xy}^{\infty}}\leq K_{0}\end{split}$ (86) ###### Proof. For $s_{1}$ and $s_{2}$ as in the statement of lemma, let $T_{0}=\sup_{\tilde{T}\in(0,T^{})}\left\\{\tilde{T}\,\|\,\left\\|u\right\\|_{L_{\tilde{T}}^{\infty}H^{s_{1},s_{2}}}\leq 2\left\\|\psi\right\\|_{H^{s_{1},s_{2}}}\right\\}$ (87) From the local well-posedness, this set is not empty. Let $C_{s_{1},s_{2}}$ be as in Lemma 3.8, $C$ be as in Lemma 3.7, $L_{s_{1},s_{2}}=2(2C-1)C_{s_{1},s_{2}}$ and $T=\frac{1}{(L_{s_{1},s_{2}}\\|\psi\\|_{H^{s_{1},s_{2}}}+1)^{\frac{12}{7}}}.$ Let us see that $T\leq T_{0}$. Suppose not. Thanks to Lemma 3.8 $f(T_{0})\leq C_{s_{1},s_{2}}T_{0}^{k_{s_{1},s_{2}}}(1+f(T_{0}))\left\\|u\right\\|_{L_{T_{0}}^{\infty}H^{s_{1},s_{2}}_{xy}}\leq C_{s_{1},s_{2}}T_{0}^{\frac{7}{12}}(1+f(T_{0}))\left\\|u\right\\|_{L_{T_{0}}^{\infty}H^{s_{1},s_{2}}_{xy}}$ (88) Since $\left\\|u\right\\|_{L_{T_{0}}^{\infty}H^{s_{1},s_{2}}}\leq 2\left\\|\psi\right\\|_{H^{s_{1},s_{2}}}$ and $T_{0}<T$ , $f(T_{0})\leq 2C_{s_{1},s_{2}}\frac{(1+f(T_{0}))}{L_{s}\left\\|\psi\right\\|_{H^{s_{1},s_{2}}}+1}\left\\|\psi\right\\|_{H^{s_{1},s_{2}}}$ or equivalently $(4CC_{s_{1},s_{2}}\left\\|\psi\right\\|_{H^{s_{1},s_{2}}}+1)f(T_{0})\leq 2C_{s_{1},s_{2}}\left\\|\psi\right\\|_{H^{s_{1},s_{2}}}$ Therefore, $f(T_{0})\leq\frac{1}{2C}$ By Lemma 3.7 we would have $\\|u(T_{0})\\|_{H^{s_{1},s_{2}}}\leq e^{\frac{1}{2}}\\|\psi\\|_{H^{s_{1},s_{2}}}<2\\|\psi\\|_{H^{s_{1},s_{2}}}.$ From the continuity of $u$, it follows that there exists $\tilde{T}>T_{0}$ such that $\\|u\\|_{L^{\infty}_{\tilde{T}}H^{s_{1},s_{2}}}\leq 2\\|\psi\\|_{H^{s_{1},s_{2}}},$ which contradicts the choice of $T_{0}$. Then, $T\leq T_{0}$. In particular, $\\|u\\|_{L^{\infty}_{T}H^{s_{1},s_{2}}}\leq 2\\|\psi\\|_{H^{s_{1},s_{2}}},$ and, repeating the above reasoning, from inequality (88), taking $T$ instead of $T_{0}$ we have that $f(T)\leq\frac{1}{2C}.$ This proves the lemma. ∎ ###### Corollary 3.11. Let $\psi$ and $T$ be as in the last lemma. If $\psi\in H^{\infty}$, then the solution $u$ to the Cauchy problem (13), with $u(0)=\psi$, belongs to the set $C([0,T];H^{\infty})$. ###### Proof. Let $T$ be as in the lemma above. From that lemma, for any number $\sigma$ that satisfies the condition established there, we have $u\in C([0,T];H^{\sigma})$. From here it follows the corollary. ∎ ###### Corollary 3.12. Let $R>0$ and $\psi\in H^{\infty}$ be such that $\\|\psi\\|_{H^{s_{1},s_{2}}}\leq R$. Then, there exists $T_{0}$ that depends on $R$ and $M$, constant that depends only on $s_{1},s_{2}$, such that $u\in C([0,T_{0}];H^{\infty})$ and $f(T_{0})=\left\\|u\right\\|_{L_{T_{0}}^{1}L_{xy}^{\infty}}+\left\\|\partial_{x}u\right\\|_{L_{T_{0}}^{1}L_{xy}^{\infty}}+\left\\|\partial_{y}u\right\\|_{L_{T_{0}}^{1}L_{xy}^{\infty}}\leq M.$ ###### Proof. If we set $T_{0}=1/{(L_{s_{1},s_{2}}R+1)^{\frac{12}{7}}}$, we have that $T_{0}\leq T$, $T$ as in the last lemma. From the proof of that lemma it follows that $f(T_{0})\leq 1/2C$, that does not depend on initial data, only on $s_{1}$ and $s_{2}$. Making $M=1/2C$ shows the corollary. ∎ #### 3.4.1. Proof of Theorem 3.9 ###### Lemma 3.13. Suppose that $\psi$ and $\phi\in H^{\infty}$ and that $u$ and $v\in C([0,T];H^{\infty})$ are the solutions to the problem (13) with initial conditions $\psi$ and $\phi$, respectively. Then, $\\|u-v\\|_{L^{2}}(T_{0}(R))\leq\\|\psi-\phi\\|_{L^{2}}e^{CM},$ where $T_{0}(R)$ and $M$ are as in Corollary 3.12, $C$ is a constant that depends only on $s_{1}$ and $s_{2}$, and $R$ is the maximum between the norm of $\phi$ and $\psi$ in the space $H^{s_{1},s_{2}}$. ###### Proof. The proof is analogous to obtaining the energy estimate. Let us see. Let $u$ and $v$ be as in statement of lemma. Then, $\partial_{t}(u-v)=\partial_{x}^{3}(u-v)-\mathcal{H}D^{\alpha}\partial_{y}^{2}(u-v)+\frac{1}{2}(u+v)\partial_{x}(u-v)+\frac{1}{2}(u-v)\partial_{x}(u+v).$ Multiplying by $u-v$ on both sides the equation and integrating by parts we have that $\frac{1}{2}\frac{d}{dt}\\|u-v\\|_{2}^{2}\leq\frac{1}{4}(\\|u_{x}\\|_{L^{\infty}}+\\|v_{x}\\|_{L^{\infty}})\\|u-v\\|_{2}^{2}.$ From the Gronwall Lemma and Corollary 3.12 it follows the lemma. ∎ Now, let $\psi\in H^{s_{1},s_{2}}$ and suppose that $\psi_{n}$ is a sequence of functions in $H^{\infty}$ that converges to $\psi$ in $H^{s_{1},s_{2}}$ . Taking $R=\sup_{n}\\|\psi_{n}\\|$, from lemma above, the solutions to (13) $u_{n}\in C([0,T_{0}];H^{\infty}(\mathbb{R}^{2}))$, with initial condition $\psi_{n}$, converge uniformly to a function $u$ in $C([0,T_{0}];L^{2}(\mathbb{R}^{2}))$. Moreover, from Corollary 3.12, the functions $u_{n}$ are uniformly bounded in $H^{s_{1},s_{2}}$. Therefore, from Banach-Alaoglu theorem, $u_{n}(t)$ have a subsequence that converges weakly in $H^{s_{1},s_{2}}$. From the uniform convergence of $u_{n}(t)$ to $u(t)$ in $L^{2}$, it follows that $u(t)\in H^{s_{1},s_{2}}$, for each $t\in[0,T_{0}]$. From the continuity of $u$ from $[0,T_{0}]$ to $L^{2}$ follows the weak continuity of $u$ from $[0,T_{0}]$ to $H^{s_{1},s_{2}}$. In particular, from uniform boundedness of the sequence $u_{n}(t)$ and Lemma 1.7, it follows that $u_{n}$ converges strong and uniformly in $H^{s^{\prime}_{1},s^{\prime}_{2}}$ to $u$, for any pair of non negative real numbers $s^{\prime}_{1},s^{\prime}_{2}$ strictly less than $s_{1},s_{2}$, respectively. Since each one of the functions $u_{n}$ satisfies the integral equation associated to (13), $w=W_{\alpha}(t)w(0)+\frac{1}{2}\int_{0}^{t}W_{\alpha}(t-t^{\prime})\partial_{x}(w^{2}(t^{\prime}))\,dt^{\prime},$ (89) in $H^{s^{\prime}_{1}-1,s^{\prime}_{2}-1}$, it follows that function $u$ satisfies this same integral equation in $H^{s^{\prime}_{1}-1,s^{\prime}_{2}-1}$, $1<s^{\prime}_{2}<s_{2}$. From Lemma 3.6, considering $17/12-\alpha/4<s_{1}^{\prime}<s_{1}$, follows that $\partial_{x}u\in{L^{1}_{T}L^{\infty}_{x,y}}$. In the same way, thanks to the inequalities (77) and (80), we obtain that $u$ and $\partial_{y}u\in{L^{1}_{T}L^{\infty}_{x,y}}$. So, ###### Lemma 3.14. Let $\psi\in H^{s_{1},s_{2}}(\mathbb{R}^{2})$. Then, there exists $T>0$ and a unique $u\in C([0,T];L^{2}(\mathbb{R}^{2}))\cap C_{w}([0,T];H^{s_{1},s_{2}}(\mathbb{R}^{2}))\cap C^{1}([0,T];H^{-3}\cap(X^{3}))$ solution to problem (13). Furthermore, $u$, $\partial_{x}u$, $\partial_{y}u\in{L^{1}_{T}L^{\infty}_{x,y}}$ and the map $\psi\mapsto u$ is Lipschitz continuous from $L^{2}(\mathbb{R}^{2})$ to $C([0,T];L^{2}(\mathbb{R}^{2}))$. ###### Proof. It only remains to prove that $u$ is unique and that the map $\psi\mapsto u$ is Lipschitz continuous from $L^{2}(\mathbb{R}^{2})$ to $C([0,T];L^{2}(\mathbb{R}^{2}))$. Let $u$ and $v$ be as in the statement of lemma. Since $u$ and $v$ are solutions to the integral equation, from Lemma 3.6, $u_{x}$ and $v_{x}\in{L^{1}_{T}L^{\infty}_{x,y}}$. Now proceed as in the lemma above, but we need Lemma (1.3), to obtain that $\frac{1}{2}\frac{d}{dt}\\|u-v\\|_{L^{2}}^{2}=\frac{1}{4}(\partial_{x}u+\partial_{x}v,(u-v)^{2})_{L^{2}}\leq(\\|u_{x}\\|_{L_{x,y}^{\infty}}+\\|v_{x}\\|_{L_{x,y}^{\infty}})\\|u-v\\|_{L^{2}}^{2}.$ Since $\\|u_{x}\\|_{L_{x,y}^{\infty}}+\\|v_{x}\\|_{L_{x,y}^{\infty}}$ is integrable and $\\|u-v\\|_{L^{2}}^{2}$ is continuous, from Gronwall lemma, it follows the theorem. ∎ ###### Lemma 3.15. For $\psi\in H^{s_{1},s_{2}}$ as in the lemma above, the solution $u$ to (13) described there is strongly continuous in $H^{s_{1},s_{2}}$ ###### Proof. Let $\psi$, $\psi_{n}$, $u_{n}$, $n=1,2,3,\cdots$, and $u$ be as before. Let, also, $M$, $T_{0}$ be as in Corollary 3.12 and $T\leq T_{0}$. It is clear that $f_{n}(T)=\\|u_{n}\\|_{L_{x,y}^{\infty}}+\\|\partial_{x}u_{n}\\|_{L_{x,y}^{\infty}}+\\|\partial_{y}u_{n}\\|_{L_{x,y}^{\infty}}$ satisfies that $f_{n}(T)\leq M,$ for all $n$. From Lemma 3.8 and Corollary 3.12, it follows that $f_{n}(T)\leq 2C_{s_{1},s_{2}}T^{k_{s_{1},s_{2}}}+(1+M)R.$ From Lemma 3.7, $\\|u_{n}(T)\\|_{H^{s_{1},s_{2}}}\leq\\|\psi_{n}\\|_{H^{s_{1},s_{2}}}e^{2C_{s_{1},s_{2}}T^{k_{s_{1},s_{2}}}(1+M)R},$ for all $n$. From the weak convergence of $u_{n}(T)$ to $u(T)$ in $H^{s_{1},s_{2}}$ and since $\psi_{n}$ converges to $\psi$ in $H^{s_{1},s_{2}}$, it follows that $\\|u(T)\\|_{H^{s_{1},s_{2}}}\leq\\|\psi\\|_{H^{s_{1},s_{2}}}e^{2C_{s_{1},s_{2}}T^{k_{s_{1},s_{2}}}(1+M)R}.$ From this last inequality and the weak continuity of $u$ in $H^{s_{1},s_{2}}$, it follows that $\\|\psi\\|_{H^{s_{1},s_{2}}}\leq\liminf_{T\to 0+}\\|u(T)\\|_{H^{s_{1},s_{2}}}\leq\limsup_{T\to 0+}\\|u(T)\\|_{H^{s_{1},s_{2}}}\leq\\|\psi\\|_{H^{s_{1},s_{2}}}.$ Then, $u$ is right strongly continuous at $0$ in the space $H^{s_{1},s_{2}}$. Since $u(-t,-x,-y)$ is also solution to the equation (13), we have that $u$ is also left strongly continuous at $0$ in the space $H^{s_{1},s_{2}}$. Now, for any $t^{}\in[0,T]$, $u(t+t^{})$ is also solution to the equation (13) with initial condition $u(t^{})$. Then from the unicity of the solution, $u$ also is strongly continuous at $t^{}$ in the space $H^{s_{1},s_{2}}$. This ends the proof. ∎ Now examine the continuity of the solutions to (13) with respect to initial data. For this purpose we will use a technique very useful an recurrently used in the literature related to the well-posedness of evolution equations. This technique is the Bona-Smith method of approximation introduced in [7]. Let us see. ###### Lemma 3.16. Let $\phi\in H^{s_{1},s_{2}}$, $s_{1}$ and $s_{2}$ be positive real numbers. For each $\tau>0$ define $\phi^{\tau}$ by $\phi^{\tau}(x,y)=\left(\widehat{\phi}(\xi,\eta)\exp\left(-\tau\left((1+\lvert\xi\rvert^{2})^{\frac{s_{1}}{2}}+(1+\lvert\eta\rvert^{2})^{\frac{s_{2}}{2}}\right)\right)\right)\text{\huge$\check{\ }$}(x,y).$ (90) Then $\lim_{\tau\to 0+}\\|\phi^{\tau}-\phi\\|_{H^{s_{1},s_{2}}}=0$ and there exists a constant $C=C(s)$ such that $\displaystyle\\|\phi^{\tau}\\|_{H^{s_{1}+1,s_{2}}}$ $\displaystyle\leq C\left(\frac{1}{\tau s_{1}}\right)^{\frac{1}{s_{1}}}\\|\phi\\|_{H^{s_{1},s_{2}}},$ (91) $\displaystyle\\|\phi^{\tau}\\|_{H^{s_{1},s_{2}+1}}$ $\displaystyle\leq C\left(\frac{1}{\tau s_{2}}\right)^{\frac{1}{s_{2}}}\\|\phi\\|_{H^{s_{1},s_{2}}}$ (92) and $\\|\phi^{\tau}-\phi^{\theta}\\|_{L^{2}}\leq C\lvert\tau-\theta\rvert\\|\phi\\|_{H^{s_{1},s_{2}}}$ (93) ###### Proposition 3.17. Let $R>0$, and assume that $\Lambda$ is a set and $\psi_{\lambda\in\Lambda}$ is a collection of functions in $H^{\infty}$ such that $\\|\psi_{\lambda}\\|_{H^{s_{1},s_{2}}}\leq R$, for all $\lambda\in\Lambda$. Also, let $\psi_{\lambda}^{\tau}$ be the approximations defined from $\psi_{\lambda}$ as in (90), and assume that $u_{\lambda}^{\tau}$ is the solution to (13) with initial condition $\psi_{\lambda}^{\tau}$, for all $\lambda\in\Lambda$. Suppose that $0\leq\theta<\tau$. Then, for $\nu>0$ $\left\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\right\\|^{2}_{H^{s_{1},s_{2}}}\leq C\left(\left\\|\psi_{\lambda}^{\tau}-\psi_{\lambda}^{\theta}\right\\|^{2}_{H^{s_{1},s_{2}}}+\tau^{\nu}\right).$ for all $\lambda\in\Lambda$. ###### Proof. It is evident that $u_{\lambda}^{\tau}(t)$ also is defined on $[0,T_{0}]$, for all $n$ and $\tau$. We will proceed as in the proof of Lemma 3.7. Then, $\displaystyle\frac{1}{2}\frac{d}{dt}\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}$ $\displaystyle- u_{\lambda}^{\theta})\\|_{L^{2}}^{2}=\int_{\mathbb{R}^{2}}J_{x}^{s_{1}}(u_{\lambda}^{\tau}\partial_{x}u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\partial_{x}u_{\lambda}^{\theta})J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy$ (94) $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}J^{s_{1}}_{x}\partial_{x}\left((u_{\lambda}^{\tau}+u_{\lambda}^{\theta})(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\right)J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}J^{s_{1}}_{x}\left((u_{\lambda}^{\tau}+u_{\lambda}^{\theta})\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\right)J^{s_{1}}_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy+$ $\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{2}}J_{x}^{s_{1}}\left(\partial_{x}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\right)J^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy.$ We estimate the last two terms that appear in the inequality above. From Lemma 1.8, for the first term, we have $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}J^{s_{1}}_{x}\big{(}(u_{\lambda}^{\tau}$ $\displaystyle+u_{\lambda}^{\theta})\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\big{)}J^{s_{1}}_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy\leq$ (95) $\displaystyle\leq$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}\left([J^{s_{1}}_{x},u_{\lambda}^{\tau}+u_{\lambda}^{\theta}]\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\right)J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy+$ $\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{2}}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})\partial_{x}J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy$ $\displaystyle\leq$ $\displaystyle C(\\|\partial_{x}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})\\|_{L^{\infty}}\\|J^{s_{1}-1}_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}\\|J^{s_{1}}_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}+$ $\displaystyle+\\|J^{s_{1}}_{x}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})\\|_{L^{2}}\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}\\|J^{s_{1}}_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}})-$ $\displaystyle-\frac{1}{2}\int_{\mathbb{R}^{2}}\partial_{x}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})(J^{s_{1}}_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta}))^{2}\,dxdy$ $\displaystyle\leq$ $\displaystyle C(\\|\partial_{x}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})\\|_{L^{\infty}}\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}+$ $\displaystyle+\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})\\|_{L^{2}}\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}})$ With the second term we proceed in the same way to obtain the following inequality $\frac{1}{2}\int_{\mathbb{R}^{2}}J^{s_{1}}_{x}\big{(}\partial_{x}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\big{)}J^{s_{1}}\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy\leq\\\ \leq C((\\|\partial_{x}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{x}u_{\lambda}^{\theta}\\|_{L^{\infty}})\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}+\\\ +\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})\\|_{L^{2}}\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}})$ (96) So, $\\|\psi_{\lambda}^{\tau}\\|_{H^{s_{1},s_{2}}}\leq R$, for all $\lambda$, from (94), (95), (96) and Lemma 3.10 we have $\frac{1}{2}\frac{d}{dt}\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}\leq C\big{(}(\\|\partial_{x}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{x}u_{\lambda}^{\theta}\\|_{L^{\infty}})\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}+\\\ +\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}\big{)}$ (97) On the other hand, $\displaystyle\frac{1}{2}\frac{d}{dt}\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}$ $\displaystyle- u_{\lambda}^{\theta})\\|_{L^{2}}^{2}=\int_{\mathbb{R}^{2}}J_{y}^{s_{2}}(u_{\lambda}^{\tau}\partial_{x}u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\partial_{x}u_{\lambda}^{\theta})J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy$ (98) $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}J^{s_{2}}_{y}\partial_{x}\left((u_{\lambda}^{\tau}+u_{\lambda}^{\theta})(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\right)J_{y}^{s_{2}}\partial_{(}u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy$ $\displaystyle=$ $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}J^{s_{2}}_{y}\left((u_{\lambda}^{\tau}+u_{\lambda}^{\tau})\partial_{x}(u_{\lambda}^{\theta}-u_{\lambda}^{\theta})\right)J^{s_{2}}_{y}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy+$ $\displaystyle+\frac{1}{2}\int_{\mathbb{R}^{2}}\left(J_{y}^{s_{2}}\partial_{x}(u_{\lambda}^{\tau}+u_{\lambda}^{\tau})(u_{\lambda}^{\theta}-u_{\lambda}^{\theta})\right)J^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy.$ Proceeding in the same way that allows us to obtain (95) and (96), it follows that $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}$ $\displaystyle J^{s_{2}}_{y}\big{(}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\big{)}J^{s_{2}}\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy\leq$ (99) $\displaystyle\leq$ $\displaystyle C\big{(}(\\|\partial_{y}u_{n}^{\tau}\\|_{L^{\infty}}+\\|\partial_{y}u_{n}^{\theta}\\|_{L^{\infty}})\times$ $\displaystyle\times(\\|J_{y}^{s_{2}-1}\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}+\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2})+$ $\displaystyle+\\|J_{y}^{s_{2}}(u_{n}^{\tau}+u_{\lambda}^{\theta})\\|_{L^{2}}\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}\big{)}$ $\displaystyle\leq$ $\displaystyle C\big{(}(\\|\partial_{y}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{y}u_{\lambda}^{\theta}\\|_{L^{\infty}})(\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}+\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2})+$ $\displaystyle+\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}\big{)}$ and $\displaystyle\frac{1}{2}\int_{\mathbb{R}^{2}}J^{s_{2}}_{y}\big{(}$ $\displaystyle\partial_{x}(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\big{)}J^{s_{2}}\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\,dxdy\leq$ (100) $\displaystyle\leq$ $\displaystyle C\big{(}(\\|\partial_{x}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{x}u_{\lambda}^{\theta}\\|_{L^{\infty}})\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}+$ $\displaystyle+\\|J_{y}^{s_{2}-1}\partial_{x}(u_{n}^{\tau}+u_{\lambda}^{\theta})\\|_{L^{2}}\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}\\|\partial_{y}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}})$ $\displaystyle\leq$ $\displaystyle C(\\|\partial_{x}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{x}u_{\lambda}^{\theta}\\|_{L^{\infty}})(\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}+\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2})+$ $\displaystyle+\\|\partial_{y}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}})$ Therefore, from (98), (99) and (100), we have $\frac{1}{2}\frac{d}{dt}\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}\leq\\\ \leq C(\\|\partial_{x}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{x}u_{\lambda}^{\theta}\\|_{L^{\infty}}+\\|\partial_{y}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{y}u_{\lambda}^{\theta}\\|_{L^{\infty}})\times\\\ \times(\\|J_{x}^{s_{1}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2}+\\|J_{y}^{s_{2}}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{2}}^{2})+\\\ +\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}+\\|\partial_{y}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}})$ (101) Gathering (97) and (101), we obtain $\frac{1}{2}\frac{d}{dt}\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\\|_{H^{s_{1},s_{2}}}^{2}\leq\\\ \leq C\big{(}(\\|\partial_{x}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{x}u_{\lambda}^{\theta}\\|_{L^{\infty}}+\\|\partial_{y}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{y}u_{\lambda}^{\theta}\\|_{L^{\infty}})\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\\|_{H^{s_{1},s_{2}}}^{2}+\\\ +\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}+\\|\partial_{y}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L^{\infty}}\big{)}$ (102) Integrating, it follows that $\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\\|_{H^{s_{1},s_{2}}}^{2}\leq\\|\psi_{\lambda}^{\tau}-\psi_{\lambda}^{\theta}\\|_{H^{s_{1},s_{2}}}^{2}+\\\ +C\int_{0}^{t}(\\|\partial_{x}u_{\lambda}^{\tau}\\|_{L^{\infty}}+\\|\partial_{x}u_{\lambda}^{\theta}\\|_{L^{\infty}})\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\\|_{H^{s_{1},s_{2}}}^{2}\,dt^{\prime}+\\\ +\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L_{T_{0}}^{1}L_{x,y}^{\infty}}+\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L_{T_{0}}^{1}L_{x,y}^{\infty}}$ (103) Let us estimate the last two terms of the last inequality. Observe that $u_{\lambda}^{\tau}-u_{\lambda}^{\theta}=W_{\alpha}(t)(\psi_{\lambda}^{\tau}-\psi_{\lambda}^{\theta})+\int_{0}^{t}W_{\alpha}(t-t^{\prime})\partial_{x}((u_{\lambda}^{\tau}+u_{\lambda}^{\theta})(u_{\lambda}^{\tau}-u_{\lambda}^{\theta}))(t^{\prime})\,dt^{\prime}.$ Therefore, proceeding as in the proof of Lemma 3.8, from refined the Strichartz estimate (57) and Lemma 3.13, we have $\displaystyle\\|\partial_{x}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L_{T_{0}}^{1}L_{x,y}^{\infty}}\leq$ $\displaystyle C(\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\\|_{L_{T_{0}}^{\infty}H_{x,y}^{s_{1}^{\prime},s_{2}^{\prime}}}+\\|(u_{\lambda}^{\tau}+u_{\lambda}^{\theta})(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L_{T_{0}}^{\infty}H_{x,y}^{s_{1}^{\prime},s_{2}^{\prime}}})$ $\displaystyle\leq$ $\displaystyle C(\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\\|_{L_{T_{0}}^{\infty}H_{x,y}^{s_{1}^{\prime},s_{2}^{\prime}}}+\\|u_{\lambda}^{\tau}+u_{\lambda}^{\theta}\\|_{L_{T_{0}}^{\infty}H_{x,y}^{s_{1}^{\prime},s_{2}^{\prime}}}\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L_{T_{0}}^{\infty}H_{x,y}^{s_{1}^{\prime},s_{2}^{\prime}}})$ $\displaystyle\leq$ $\displaystyle C\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\\|_{L_{T_{0}}^{\infty}H_{x,y}^{s_{1}^{\prime},s_{2}^{\prime}}}$ $\displaystyle\leq$ $\displaystyle C\\|\psi_{\lambda}^{\tau}-\psi_{\lambda}^{\theta}\\|_{L^{2}}^{\nu}\\|u_{\lambda}^{\tau}-u_{\lambda}^{\theta}\\|_{L_{T_{0}}^{\infty}H_{x,y}^{s_{1},s_{2}}}^{1-\nu}$ $\displaystyle\leq$ $\displaystyle C\tau^{\nu},$ where $s^{\prime}_{1}$, $s^{\prime}_{2}$ and $\nu$ are such that $1<s^{\prime}_{i}=\nu s_{i}<s_{i}$, $i=1,2$. In the same way, again, if we proceed as in the proof of Lemma 3.8, more precisely, in the way that we got (84), we also have $\\|\partial_{y}(u_{\lambda}^{\tau}-u_{\lambda}^{\theta})\\|_{L_{T_{0}}^{1}L_{x,y}^{\infty}}\leq C\tau^{\nu}$ From these last two inequalities with (103) and the Gronwall Lemma follows the proposition. ∎ ###### Corollary 3.18. Let $\psi_{\lambda}^{\tau}$ and $u_{\lambda}^{\tau}$ be as in the result above. Then, $\left\\{u_{\lambda}^{\tau}\right\\}_{\tau>0}$ converges uniformly in $t$ to $u_{\lambda}$, when $\tau\to 0+$. In other words, $\lim_{\tau\to 0+}\sup_{t\in[0,T]}\left\\|u_{\lambda}^{\tau}(t)-u_{\lambda}(t)\right\\|_{H^{s_{1},s_{2}}}=0.$ ###### Proof. Take $\theta=0$ in Proposition 3.17. ∎ ###### Theorem 3.19. In $H^{s_{1},s_{2}}$ the map $\psi\mapsto u$, where $u$ is solution to (13) with initial condition $\psi$, is continuous. More precisely, if $(\psi_{n})_{n\in\mathbb{N}}$ is a sequence such that $\psi_{n}\to\psi$ in $H^{s_{1},s_{2}}$ and if $u_{n}\in C([0,T_{0}];H^{s_{1},s_{2}})$ are the corresponding solutions to (13) with initial condition $\psi_{n}$, then $\lim_{n\to\infty}\sup_{t\in[0,T_{0}]}\left\\|u_{n}(t)-u(t)\right\\|_{s}=0$ ###### Proof. Let $\psi\in H^{s_{1},s_{2}}$ and $(\psi_{n})$ be a sequence in $H^{s_{1},s_{2}}$ that converges strongly to $\psi$ in this space. Let, also, $R=\max(\sup_{n}\\|\psi_{n}\\|_{H^{s_{1},s_{2}}},\\|\psi\\|_{H^{s_{1},s_{2}}})$. Now, let us take $\psi_{m,n}=e^{\frac{1}{m}\bigtriangleup}\psi_{n}$ and $\psi_{m}=e^{\frac{1}{m}\bigtriangleup}\psi$. Let $u_{n}$, $u$, $u_{m,n}$, $u_{m}$, $u^{\tau}_{n}$, $u^{\tau}$, $u^{\tau}_{m,n}$, $u^{\tau}_{m}$ be the corresponding solutions to (13) in $[0,T_{0}]$ with conditions $\psi_{n}$, $\psi$, $\psi_{m,n}$, $\psi_{m}$, $\psi^{\tau}_{n}$, $\psi^{\tau}$, $\psi^{\tau}_{m,n}$, $\psi^{\tau}_{m}$, respectively. Also, observe that $u_{n,m}$, $u_{m}$, $u^{\tau}_{m,n}$ and $u^{\tau}_{m}$ converge uniformly to $u_{n}$, $u$, $u^{\tau}_{n}$ and $u^{\tau}$ in the weak sense, as $m\to\infty$. Therefore, $\displaystyle\langle u_{n}-u,\varphi\rangle_{H^{s_{1},s_{2}}}=$ $\displaystyle\lim_{m\to\infty}\langle u_{m,n}-u_{m,n}^{\tau},\varphi\rangle_{H^{s_{1},s_{2}}}+\langle u_{m,n}^{\tau}-u_{m}^{\tau},\varphi\rangle_{H^{s_{1},s_{2}}}+$ $\displaystyle+\langle u_{m}^{\tau}-u_{m},\varphi\rangle_{H^{s_{1},s_{2}}}$ $\displaystyle=\lim_{m\to\infty}$ $\displaystyle\left[\langle u_{m,n}-u_{m,n}^{\tau},\varphi\rangle_{H^{s_{1},s_{2}}}+\langle u_{m,n}^{\tau}-u_{m},\varphi\rangle_{H^{s_{1},s_{2}}}\right]+$ $\displaystyle+\langle u_{n}^{\tau}-u^{\tau},\varphi\rangle_{H^{s_{1},s_{2}}}.$ On the other hand, Corollary 3.18 implies that, given $\epsilon>0$, there exists $\tau_{0}$ such that for $0<\tau\leq\tau_{0}$ $\lvert\langle u_{m,n}-u_{m,n}^{\tau},\varphi\rangle_{H^{s_{1},s_{2}}}+\langle u_{m,n}^{\tau}-u_{m},\varphi\rangle_{H^{s_{1},s_{2}}}\rvert\leq\epsilon\left\\|\varphi\right\\|_{H^{s_{1},s_{2}}},$ for all $m>0$. Then, $\displaystyle\lvert\langle u_{n}-u,\varphi\rangle_{H^{s_{1},s_{2}}}\rvert\leq\epsilon\left\\|\varphi\right\\|_{H^{s_{1},s_{2}}}+\left\\|u_{n}^{\tau}-u^{\tau}\right\\|_{H^{s_{1},s_{2}}}\left\\|\varphi\right\\|_{H^{s_{1},s_{2}}},$ for all $\varphi\in H^{s_{1},s_{2}}$. Therefore, $\left\\|u_{n}-u\right\\|_{H^{s_{1},s_{2}}}\leq\epsilon+\left\\|u_{n}^{\tau}-u^{\tau}\right\\|_{H^{s_{1},s_{2}}}.$ (104) Arguments similar to those used in Proposition 3.17 allow us show that, for $\tau$ small enough, $\left\\|u_{n}^{\tau}-u^{\tau}\right\\|_{H^{s_{1},s_{2}}}\leq C\left\\|\psi_{n}^{\tau}-\psi^{\tau}\right\\|_{H^{s_{1},s_{2}}}\tau^{-\frac{1}{s}}\leq\left\\|\psi_{n}-\psi\right\\|_{H^{s_{1},s_{2}}}\tau^{-\frac{1}{s}}.$ Then, fixing $\tau$ small enough, we can conclude from (104) that $\left\\|u_{n}-u\right\\|_{H^{s_{1},s_{2}}}\leq 2\epsilon,$ for $n$ large enough. ∎ Let us see now that the problem is locally well-posed in the spaces $X^{s_{1},s_{2}}$, $\widehat{X}^{s_{1},s_{2}}$ and $Y^{s_{1},s_{2}}$. Since the solution to (13) satisfy the integral equation $u=W_{\alpha}(t)\psi+\int_{0}^{t}W_{\alpha}(t-t^{\prime})(u\partial_{x}u)(t^{\prime})\,dt^{\prime},$ we have that $\partial_{x}^{-1}u$ and $\partial_{x}^{-1}\partial_{y}u$ satisfy the equations $\partial_{x}^{-1}u=W_{\alpha}(t)\partial_{x}^{-1}\psi+\int_{0}^{t}W_{\alpha}(t-t^{\prime})\left(\frac{u^{2}}{2}\right)(t^{\prime})\,dt^{\prime}$ and $\partial_{x}^{-1}\partial_{y}u=W_{\alpha}(t)\partial_{x}^{-1}\partial_{y}\psi+\int_{0}^{t}W_{\alpha}(t-t^{\prime})(u\partial_{y}u)(t^{\prime})\,dt^{\prime},$ respectively. From here and Theorem 3.9, for $s_{1}$ and $s_{2}$ as in that theorem, it follows that (13) is locally well-posed in the spaces $\widehat{X}^{s_{1},s_{2}}$, $X^{s_{1},s_{2}}$ and $\widehat{Y}^{s_{1},s_{2}}$. The case $Y^{s_{1},s_{2}}$ requires some additional effort. For $\psi\in Y^{\infty}$, the solution $u$ to (13), with initial condition $\psi$, belongs to $C([0,T_{0}];Y^{\infty})$. We can proceed as in the proof Lemma 3.7, to show that $\frac{1}{2}\frac{d}{dt}\\|\partial_{x}^{-1}\partial_{y}u\\|^{2}\leq C(\\|u_{x}\\|+\\|u_{y}\\|)\\|u\\|_{Y^{s_{1},s_{2}}}^{2}.$ This inequality with (70) allow us prove that $\frac{1}{2}\frac{d}{dt}\\|u\\|_{Y^{s_{1},s_{2}}}^{2}\leq C(\\|u_{x}\\|_{\infty}+\\|u_{y}\\|_{\infty})\\|u\\|_{Y^{s_{1},s_{2}}}^{2}.$ (105) Thanks to the Gronwall lemma we have the following generalization of Lemma 3.7, $\\|u\\|_{Y^{s}}\leq\\|\psi\\|_{Y^{s_{1},s_{2}}}e^{C(\\|u_{x}\\|_{L^{1}_{t}L_{x,y}^{\infty}}+\\|u_{y}\\|_{L^{1}_{t}L_{x,y}^{\infty}})}.$ Now, if $\psi$ is arbitrary, in the same way that we prove for the space $H^{s_{1},s_{2}}$, the solution $u\in C([0,T];Y^{s_{1},s_{2}})$. To see that the map $\psi\mapsto u$ is continuous from ${Y^{s_{1},s_{2}}}$ to $C([0,T];Y^{s_{1},s_{2}})$ we repeat the same argument of Bona-Smith that we use before. So, summarizing, we paraphrase Theorem (2.7) for the current situation. ###### Theorem 3.20. Let $s_{1},s_{2}$ and $\alpha$ be as in Theorem 3.9. Let, also, $Z$ any of the spaces $X^{s_{1},s_{2}}$, $\widehat{X}^{s_{1},s_{2}}$, $Y^{s_{1},s_{2}}$ and $\widehat{Y}^{s_{1},s_{2}}$. Then, if $\psi\in Z$ and $u\in C([0,T];H^{s_{1},s_{2}})$ is solution to (13) with $u(0)=\psi$, then $u\in C([0,T];Z)$. Moreover, $\psi\mapsto u$ is continuous from $Z$ to $C([0,T];Z)$ ## 4\. Remarks on ill-posedness of the equation (13) In this section we prove that the flow associated to the equation (13) is not of class $C^{2}$ for $-1\leq\alpha<0$. In particular, we have that we cannot apply Picard iterative process to solve the Duhamel equation associated to this equation. For this we will use the ideas given in [29] to prove that the flow associated with the KP-I equation is not of class $C^{2}$. ### 4.1. The flow associated to the problem (13) is not $C^{2}$ ###### Theorem 4.1. Let $(s_{1},s_{2})\in\mathbb{R}^{2}$ and $-1\leq\alpha<0$. Then, there does not exist $T>0$ such that (13) has a unique solution $u$ for all $\phi\in H^{s_{1}.s_{2}}$ and that the flow $S_{t}:\phi\mapsto u$ is not of class $C^{2}$ at $0$ from $H^{s_{1}.s_{2}}$ to $H^{s_{1}.s_{2}}$ ###### Proof. Let’s see first what we should show. For this we consider $u(\lambda,t)=S_{t}(\lambda\phi)$, $S_{t}$ the flow associated to the problem (13). So this solution satisfies the Duhamel equation associated to the equation (13), i. e., $u(\lambda,t)=\lambda W_{\alpha}(t)\phi-\int_{0}^{t}W_{\alpha}(t-t^{\prime})u(t^{\prime})u_{x}(t^{\prime})\,dt^{\prime}.$ (106) If the flow is twice differentiable around $0$ in $H^{s_{1}.s_{2}}$ then, thanks to the chain rule, $\partial_{\lambda}u(0,t)=W_{\alpha}(t)\phi,$ and $\partial_{\lambda}^{2}u(0,t)=-2\int_{0}^{t}W_{\alpha}(t-t^{\prime})W_{\alpha}(t^{\prime})\phi W_{\alpha}(t^{\prime})\phi_{x}\,dt^{\prime}.$ Which would imply that the map $\phi\mapsto\int_{0}^{t}W_{\alpha}(t-t^{\prime})W_{\alpha}(t^{\prime})\phi W_{\alpha}(t^{\prime})\phi_{x}\,dt^{\prime}$ would be a quadratic form coming from a continuous symmetric bilinear transformation in $H^{s_{1}.s_{2}}$, and that, in particular, for some fixed $C$ $\left\\|\int_{0}^{t}W_{\alpha}(t-t^{\prime})W_{\alpha}(t^{\prime})\phi W_{\alpha}(t^{\prime})\phi_{x}\,dt^{\prime}\right\\|_{H^{s_{1}.s_{2}}}\leq C\\|\phi\\|_{H^{s_{1}.s_{2}}}^{2}$ for all $\phi\in H^{s_{1}.s_{2}}$. So let’s show that this is not the case. To do this, suppose that this inequality is valid and consider the function $\phi$ defined via the Fourier transform by $\widehat{\phi}=\gamma^{-3/2}\mathbbm{1}_{D_{1}}+\gamma^{-3/2}N^{-s_{1}-\frac{3-\alpha}{2}s_{2}}\mathbbm{1}_{D_{2}},$ where $\gamma$ and $N$ are positive numbers such that $\gamma\ll 1$ and $N\gg 1$, $D_{1}$ and $D_{2}$ are the sets $D_{1}=\left[\frac{\gamma}{2},\gamma\right]\times\left[-\frac{\gamma^{2}}{6},\frac{\gamma^{2}}{6}\right]\ \text{and}\ D_{2}=\left[N,N+\gamma\right]\times\left[\sqrt{-\frac{3}{\alpha}}N^{\frac{3-\alpha}{2}},\sqrt{-\frac{3}{\alpha}}N^{\frac{3-\alpha}{2}}+\gamma^{2}\right]$ and $\mathbbm{1}_{D_{i}}$ are the characteristic functions of the sets $D_{i}$, $i=1,2$. $\\|\phi\\|_{H^{s_{1}.s_{2}}}\sim 1$ for whatever values that we take for $\gamma$ and $N$. Let us see that for a convenient choice in parameters of $\gamma$ and $N$, $\left\\|\int_{0}^{t}W_{\alpha}(t-t^{\prime})W_{\alpha}(t^{\prime})\phi W_{\alpha}(t^{\prime})\phi_{x}\,dt^{\prime}\right\\|_{H^{s_{1}.s_{2}}}$ (107) can be as large as we want. Calculating its Fourier transform, it follows that $\int_{0}^{t}W_{\alpha}(t-t^{\prime})W_{\alpha}(t^{\prime})\phi W_{\alpha}(t^{\prime})\phi_{x}\,dt^{\prime}$ (108) is $f_{1}+f_{2}+f_{3}$ where $\widehat{f}_{1}(t,\xi,\eta)=\frac{i\xi e^{it(\xi^{3}+\operatorname{sgn}(\xi)\|\xi\|^{\alpha}\eta^{2})}}{2\gamma^{3}}\int_{\begin{subarray}{c}(\xi_{1},\eta_{1})\in D_{1}\\\ (\xi-\xi_{1},\eta-\eta_{1})\in D_{1}\end{subarray}}\frac{e^{-it\chi(\xi,\xi_{1},\eta,\eta_{1})}-1}{\chi(\xi,\xi_{1},\eta,\eta_{1})}\,d\xi_{1}d\eta_{1},$ $\widehat{f}_{2}(t,\xi,\eta)=\frac{i\xi e^{it(\xi^{3}+\operatorname{sgn}(\xi)\|\xi\|^{\alpha}\eta^{2})}}{2\gamma^{3}N^{2(s_{1}+\frac{3-\alpha}{2}s_{2})}}\int_{\begin{subarray}{c}(\xi_{1},\eta_{1})\in D_{2}\\\ (\xi-\xi_{1},\eta-\eta_{1})\in D_{2}\end{subarray}}\frac{e^{-it\chi(\xi,\xi_{1},\eta,\eta_{1})}-1}{\chi(\xi,\xi_{1},\eta,\eta_{1})}\,d\xi_{1}d\eta_{1}$ and $\displaystyle\widehat{f}_{3}(t,\xi,\eta)=$ $\displaystyle\frac{i\xi e^{it(\xi^{3}+\operatorname{sgn}(\xi)\|\xi\|^{\alpha}\eta^{2})}}{2\gamma^{3}N^{s_{1}+\frac{3-\alpha}{2}s_{2}}}\int_{\begin{subarray}{c}(\xi_{1},\eta_{1})\in D_{2}\\\ (\xi-\xi_{1},\eta-\eta_{1})\in D_{1}\end{subarray}}\frac{e^{-it\chi(\xi,\xi_{1},\eta,\eta_{1})}-1}{\chi(\xi,\xi_{1},\eta,\eta_{1})}\,d\xi_{1}d\eta_{1}+$ $\displaystyle+\frac{i\xi e^{it(\xi^{3}+\operatorname{sgn}(\xi)\|\xi\|^{\alpha}\eta^{2})}}{2\gamma^{3}N^{s_{1}+\frac{3-\alpha}{2}s_{2}}}\int_{\begin{subarray}{c}(\xi_{1},\eta_{1})\in D_{1}\\\ (\xi-\xi_{1},\eta-\eta_{1})\in D_{2}\end{subarray}}\frac{e^{-it\chi(\xi,\xi_{1},\eta,\eta_{1})}-1}{\chi(\xi,\xi_{1},\eta,\eta_{1})}\,d\xi_{1}d\eta_{1},$ where $\chi$ is the resonant function $\begin{split}\chi&=\chi(\xi,\xi_{1},\eta,\eta_{1})=\vartheta(\xi,\eta)-\vartheta(\xi_{1},\eta_{1})-\vartheta(\xi-\xi_{1},\eta-\eta_{1})\\\ &=3\xi\xi_{1}(\xi-\xi_{1})+\operatorname{sgn}(\xi)\frac{\eta^{2}}{\|\xi\|^{\theta}}-\operatorname{sgn}(\xi_{1})\frac{\eta_{1}^{2}}{\|\xi_{1}\|^{\theta}}-\operatorname{sgn}(\xi-\xi_{1})\frac{(\eta-\eta_{1})^{2}}{\|\xi-\xi_{1}\|^{\theta}},\end{split}$ (109) where $\vartheta(\xi,\eta)=\xi^{3}+\operatorname{sgn}(\xi)\frac{\eta^{2}}{\|\xi\|^{\theta}},$ (110) is the phase function and $\theta=-\alpha$, which is between $0$ and $1$. Since in our case $\xi,\xi_{1}$ and $\xi-\xi_{1}$ are positive, we have $\chi=3\xi\xi_{1}(\xi-\xi_{1})+\frac{\eta^{2}}{\xi^{\theta}}-\frac{\eta_{1}^{2}}{\xi_{1}^{\theta}}-\frac{(\eta-\eta_{1})^{2}}{(\xi-\xi_{1})^{\theta}}$ (111) Observe that to estimate (107) it is enough estimate $\\|f_{3}(t)\\|_{H^{s_{1},s_{2}}}$, in fact, $\left\\|\int_{0}^{t}W_{\alpha}(t-t^{\prime})W_{\alpha}(t^{\prime})\phi W_{\alpha}(t^{\prime})\phi_{x}\,dt^{\prime}\right\\|_{H^{s_{1}.s_{2}}}\geq\\|f_{3}(t)\\|_{H^{s_{1},s_{2}}}.$ To continue we need the following lemma. ###### Lemma 4.2. Suppose that $(\xi_{1},\eta_{1})\in D_{1}\qquad\text{and}\qquad(\xi-\xi_{1},\eta-\eta_{1})\in D_{2}$ or $(\xi_{1},\eta_{1})\in D_{2}\qquad\text{and}\qquad(\xi-\xi_{1},\eta-\eta_{1})\in D_{1}.$ Then, $\|\chi(\xi,\xi_{1},\eta,\eta_{1})\|\lesssim\gamma^{2}N.$ ###### Proof. First let us calculate the $\eta$ such that $\chi(\xi,\xi_{1},\eta,\eta_{1})=0$. So, we have $[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]\xi_{1}^{\theta}\eta^{2}-2\xi^{\theta}\xi_{1}^{\theta}\eta_{1}\eta+[\xi_{1}^{\theta}+(\xi-\xi_{1})^{\theta}]\xi^{\theta}\eta_{1}^{2}-3\xi^{1+\theta}\xi_{1}^{1+\theta}(\xi-\xi_{1})^{1+\theta}=0$ (112) Then, we get $\eta=\frac{\xi^{\theta}\eta_{1}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]}\pm\sqrt{\frac{3\xi^{1+\theta}\xi_{1}(\xi-\xi_{1})^{1+\theta}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]}+\frac{\xi^{\theta}(\xi-\xi_{1})^{\theta}[\xi_{1}^{\theta}+(\xi-\xi_{1})^{\theta}-\xi^{\theta}]\eta_{1}^{2}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]^{2}\xi_{1}^{\theta}}}$ (113) Let $\eta$ be the smallest zero between the two calculated above. So, $\eta^{}-\eta_{1}=\frac{(\xi-\xi_{1})^{\theta}\eta_{1}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]}-\sqrt{\frac{3\xi^{1+\theta}\xi_{1}(\xi-\xi_{1})^{1+\theta}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]}+\frac{\xi^{\theta}(\xi-\xi_{1})^{\theta}[\xi_{1}^{\theta}+(\xi-\xi_{1})^{\theta}-\xi^{\theta}]\eta_{1}^{2}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]^{2}\xi_{1}^{\theta}}}.$ (114) Let $R=\frac{(\xi-\xi_{1})^{\theta}\eta_{1}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]}+\sqrt{\frac{3\xi^{1+\theta}\xi_{1}(\xi-\xi_{1})^{1+\theta}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]}+\frac{\xi^{\theta}(\xi-\xi_{1})^{\theta}[\xi_{1}^{\theta}+(\xi-\xi_{1})^{\theta}-\xi^{\theta}]\eta_{1}^{2}}{[\xi^{\theta}-(\xi-\xi_{1})^{\theta}]^{2}\xi_{1}^{\theta}}}.$ Then $\|\eta^{}-\eta_{1}\|=\dfrac{\frac{(\xi^{\theta}-\xi_{1}^{\theta})(\xi-\xi_{1})^{\theta}}{(\xi^{\theta}-(\xi-\xi_{1})^{\theta})\xi_{1}^{\theta}}{\left\|\eta_{1}^{2}-3\xi^{1+\theta}\xi_{1}^{1+\theta}g(\xi,\xi_{1})\right\|}}{R},$ (115) where $g(\xi,\xi_{1})=\begin{cases}\dfrac{\xi-\xi_{1}}{\xi^{\theta}-\xi_{1}^{\theta}}&\text{ if }\xi\neq\xi_{1}\\\\[14.22636pt] \frac{1}{\theta}\xi_{1}^{1-\theta}&\text{ in another case.}\end{cases}$ Now, since $\frac{1}{\theta}\xi_{1}^{1-\theta}\leq g(\xi,\xi_{1})\leq\frac{1}{\theta}\xi^{1-\theta}$ (116) and $R\geq\frac{(\xi-\xi_{1})^{\theta}\eta_{1}}{(\xi^{\theta}-(\xi-\xi_{1})^{\theta})},$ we have $\displaystyle\|\eta^{}-\eta_{1}\|$ $\displaystyle=\frac{\xi^{\theta}-\xi_{1}^{\theta}}{\xi_{1}^{\theta}}\frac{\left\|\eta_{1}^{2}-3\xi^{1+\theta}\xi_{1}^{1+\theta}g(\xi,\xi_{1})\right\|}{\eta_{1}}$ $\displaystyle\leq 3\frac{\xi^{\theta}-\xi_{1}^{\theta}}{\xi_{1}^{\theta}}\left\|\eta_{1}-\sqrt{3}\xi^{\frac{1+\theta}{2}}\xi_{1}^{\frac{1+\theta}{2}}\sqrt{g(\xi,\xi_{1})}\right\|$ $\displaystyle\leq 3\theta\frac{\xi-\xi_{1}}{\xi_{1}}\left\|\eta_{1}-\sqrt{\frac{3}{\theta}}\xi_{1}^{\frac{3+\theta}{2}}-\xi_{1}^{\frac{1+\theta}{2}}\left(\xi^{\frac{1+\theta}{2}}\sqrt{g(\xi,\xi_{1})}-\frac{1}{\sqrt{\theta}}\xi_{1}\right)\right\|,$ Remember that $\eta_{1}$ take values in $\left[\sqrt{\frac{3}{\theta}}N^{\frac{3+\theta}{2}},\sqrt{\frac{3}{\theta}}N^{\frac{3+\theta}{2}}+\gamma^{2}\right]$ and that $\xi_{1}$ in $[N,N+\gamma]$. Whence, thanks to the Taylor formula with remainder applied to the function $x\mapsto x^{\frac{3+\theta}{2}}$, $\sqrt{3}\xi_{1}^{\frac{3+\theta}{2}}\in\left[\sqrt{3}N^{\frac{3+\theta}{2}},\sqrt{3}N^{\frac{3+\theta}{2}}+\sqrt{3}\frac{3+\theta}{2}N^{\frac{1+\theta}{2}}\gamma+\sqrt{3}\frac{(3+\theta)(1+\theta)}{8}\gamma^{2}\right]$ and, moreover $\left\|\eta_{1}-\sqrt{3}\xi_{1}^{\frac{3+\theta}{2}}\right\|\leq 2\sqrt{3}N^{\frac{1+\theta}{2}}\gamma+\sqrt{3}\gamma^{2}.$ On the other hand from (116) $\xi^{\frac{1+\theta}{2}}\sqrt{g(\xi,\xi_{1})}-\frac{1}{\sqrt{\theta}}\xi_{1}\leq\frac{1}{\sqrt{\theta}}(\xi-\xi_{1})\leq\frac{1}{\sqrt{\theta}}\gamma.$ Therefore, $\|\eta^{}-\eta_{1}\|\leq 3\theta\frac{\gamma}{N}\left(2\sqrt{\frac{3}{\theta}}N^{\frac{1+\theta}{2}}\gamma+\sqrt{\frac{3}{\theta}}\gamma^{2}+\sqrt{\frac{3}{\theta}}(N^{\frac{1+\theta}{2}}+\gamma)\gamma\right)\leq 18\sqrt{\theta}\frac{\gamma^{2}}{N^{\frac{1-\theta}{2}}}.$ By the mean value theorem, there exists $\bar{\eta}\in[\eta,\eta^{}]$ such that $\displaystyle\chi(\xi,\xi_{1},\eta,\eta_{1})$ $\displaystyle=\chi(\xi,\xi_{1},\eta^{},\eta_{1})+(\eta-\eta^{})\partial_{\eta}\chi(\xi,\xi_{1},\bar{\eta},\eta_{1})$ $\displaystyle=-(\eta-\eta^{})\frac{2(\bar{\eta}(\xi^{\theta}-(\xi-\xi_{1})^{\theta})-\eta_{1}\xi^{\theta})}{\xi^{\theta}(\xi-\xi_{1})^{\theta}}$ $\displaystyle=-(\eta-\eta^{})\frac{2((\bar{\eta}-\eta_{1})\xi^{\theta}-(\xi-\xi_{1})^{\theta}\bar{\eta})}{\xi^{\theta}(\xi-\xi_{1})^{\theta}}$ $\displaystyle=-(\eta-\eta^{})2\left(\frac{\bar{\eta}-\eta_{1}}{(\xi-\xi_{1})^{\theta}}-\frac{\bar{\eta}}{\xi^{\theta}}\right).$ So, $\displaystyle\|\chi(\xi,\xi_{1},\eta,\eta_{1})\|$ $\displaystyle\lesssim\frac{\gamma^{2}}{N^{\frac{1-\theta}{2}}}\left(\frac{\gamma^{2}}{N^{\frac{1-\theta}{2}}\gamma^{\theta}}+\frac{N^{\frac{3+\theta}{2}}}{N^{\theta}}\right)$ $\displaystyle\lesssim\gamma^{2}N$ The lemma follows immediately observing that $\chi(\xi,\xi_{1},\eta,\eta_{1})=\chi(\xi,\xi-\xi_{1},\eta,\eta-\eta_{1}).$ ∎ Let us finish the proof of the theorem. Let us choose $\gamma$ and $N$ in such a way that $\gamma^{2}N=N^{-\varepsilon}$ for $\varepsilon\ll 1$. Thanks to the previous lemma we have that $\left\|\frac{e^{it\xi}-1}{\xi}\right\|=\|t\|+O(N^{-\epsilon})$ for $(\xi_{1},\eta_{1})\in D_{1}$ and $(\xi-\xi_{1},\eta-\eta_{1})\in D_{2}$ or $(\xi_{1},\eta_{1})\in D_{2}$ and $(\xi-\xi_{1},\eta-\eta_{1})\in D_{1}$. So $\\|f_{3}(t,\cdot,\cdot)\\|_{H^{s}}\gtrsim\frac{NN^{\frac{3+\theta}{2}s}\gamma^{3}\gamma^{\frac{3}{2}}}{N^{\frac{3+\theta}{2}s}\gamma^{3}}=\gamma^{\frac{3}{2}}N=N^{(1-3\varepsilon)/4}.$ This leads to a contradiction since $1\sim\\|\phi\\|_{H^{s}}^{2}\gtrsim\\|f_{3}(t,\cdot,\cdot)\\|_{H^{s}}.$ ∎ An immediate consequence of the previous theorem is the following theorem. ###### Theorem 4.3. For $(s_{1},s_{2})\in\mathbb{R}^{2}$, $-1\leq\alpha<0$ and a positive real number $T$, there does not exists a space $X_{T}$ continuously embedded in $C([-T,T];H^{s_{1},s_{2}})$ such that, for a fix constant $C$, $\\|W_{\alpha}(\cdot)\phi\\|_{X_{T}}\leq C\\|\phi\\|_{H^{s_{1},s_{2}}},$ (117) for all $\phi\in H^{s_{1},s_{2}}$, and $\left\\|\int_{0}^{t}W_{\alpha}(t-t^{\prime})(u(t^{\prime})u_{x}(t^{\prime}))\,dt^{\prime}\right\\|_{X_{T}}\leq C\\|u\\|_{X_{T}}^{2},$ (118) for all $u\in X_{T}$. Note that the estimates given in the theorem statement are necessary to prove the contraction properties of the operator $\Phi$ defined by $\Phi(u)=W_{\alpha}(t)\phi+\int_{0}^{t}W_{\alpha}(t-t^{\prime})(u(t^{\prime})u_{x}(t^{\prime}))\,dt^{\prime}.$ ###### Proof. Suppose that we have (117) and (118) for all $\phi\in H^{s_{1},s_{2}}$ and for all $u\in X_{T}$. Let $\phi\in H^{s_{1},s_{2}}$ and set $u(t)=W_{\alpha}(t)\phi$. Then, $\left\\|\int_{0}^{t}W_{\alpha}(t-t^{\prime})(W_{\alpha}(t^{\prime})\phi W_{\alpha}(t^{\prime})\phi_{x})\,dt^{\prime}\right\\|_{X_{T}}\leq C\\|W_{\alpha}(t)\phi\\|_{X_{T}}^{2}\leq\\|\phi\\|_{H^{s_{1},s_{2}}}^{2}.$ Since $X_{T}$ is continuously embedded in $C([-T,T];H^{s_{1},s_{2}})$ $\left\\|\int_{0}^{t}W_{\alpha}(t-t^{\prime})(W_{\alpha}(t^{\prime})\phi W_{\alpha}(t^{\prime})\phi_{x})\,dt^{\prime}\right\\|_{H^{s_{1},s_{2}}}\leq C\\|\phi\\|_{H^{s_{1},s_{2}}}^{2},$ which is contradictory with the previous theorem. This ends the proof. ∎ Another immediate corollary is the following theorem. ###### Theorem 4.4. The flow associated to the problem (13), for $-1\leq\alpha<0$, whose well- posedness was proved in the previous section, is not of class $C^{2}$. ## References [1] Ablowitz, M. J., and Clarkson, P. A. Solitons, nonlinear evolution equations and inverse scattering, vol. 149 of London Mathematical Society Lecture Note Series. 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# Bounds on mutual information of mixture data for classification tasks Yijun Ding James C. Wyant College of Optical Sciences University of Arizona Tucson, AZ, USA <EMAIL_ADDRESS>Amit Ashok James C. Wyant College of Optical Sciences and Department of Electrical Engineering University of Arizona Tucson, AZ, USA <EMAIL_ADDRESS> ###### Abstract The data for many classification problems, such as pattern and speech recognition, follow mixture distributions. To quantify the optimum performance for classification tasks, the Shannon mutual information is a natural information-theoretic metric, as it is directly related to the probability of error. The mutual information between mixture data and the class label does not have an analytical expression, nor any efficient computational algorithms. We introduce a variational upper bound, a lower bound, and three estimators, all employing pair-wise divergences between mixture components. We compare the new bounds and estimators with Monte Carlo stochastic sampling and bounds derived from entropy bounds. To conclude, we evaluate the performance of the bounds and estimators through numerical simulations. ###### Index Terms: Mixture distribution, classification, Shannon mutual information, bounds, estimation, mixed-pair ## I Introduction ### I-A Motivation We study the performance of classification tasks, where the goal is to infer the class label $C$ from sample data $\bm{x}$. The Shannon mutual information $\mathrm{I}(\bm{x};C)$ characterizes the reduction in the uncertainty of the class label $C$ with the knowledge of data $\bm{x}$ and provides a way to quantify the relevance of the data $\bm{x}$ with respect to the class label $C$. As the mutual information is related to probability of classification error ($\mathrm{P_{e}}$) through Fano’s inequality and other bounds [1, 2, 3], it has been widely used for feature selection [4, 5], learning [6], and quantifying task-specific information [7] for classification. Statistical mixture distributions such as Poisson, Wishart or Gaussian mixtures are frequently used in the fields of speech recognition [8], image retrieval [9], system evaluation [10], compressive sensing [11], distributed state estimation [12], hierarchical clustering [13] etc. As a practical example, consider a scenario in which the data $\bm{x}$ is measured with a noisy system, eg. Poisson noise in an photon-starved imaging system or Gaussian noise in a thermometer reading. If the actual scene (or temperature) has a class label, e.g. target present or not (the temperature is below freezing or not), then the mutual information $\mathrm{I}(\bm{x};C)$ describes with what confidence one can assign a class label $C$ to the noisy measurement data $\bm{x}$. The goal of this paper is to develop efficient methods to quantify the optimum performance of classification tasks, when the distribution of the data $\bm{x}$ for each given class label, $\mathrm{pr}(\bm{x}\|C)$, follows a known mixture distribution. As the mutual information $\mathrm{I}(\bm{x};C)$, which is commonly used to quantify task-specific information, does not admit an analytical expression for mixture data, we provide analytical expressions for bounds and estimators of $\mathrm{I}(\bm{x};C)$. ### I-B Problem Statement and Contributions We consider the data as a continuous random variable $\bm{x}$ and the class label as a discrete random variable $C$, where $C$ can be any integer in $[1,\Pi]$ and $\Pi$ is the number of classes. The bold symbol $\bm{x}$ emphasizes that $\bm{x}$ is a vector, which can be high-dimensional. We assume that, when restricted to any of the classes, the conditional differential entropy of $\bm{x}$ is well-defined, or in other words, $(\bm{x},C)$ is a good mixed-pair vector [14]. The mutual information between the data $\bm{x}$ and the class label $C$ can be defined as [15] $\begin{split}\mathrm{I}(\bm{x};C)&=\mathrm{KL}(\,\mathrm{pr}(\bm{x},C)\|\|\,\mathrm{Pr}(C)\cdot\mathrm{pr}(\bm{x})\,)\\\ &=\sum_{C}\int\mathrm{dx}\,\mathrm{pr}(\bm{x},C)\ln{\frac{\mathrm{pr}(\bm{x},C)}{\mathrm{Pr}(C)\cdot\mathrm{pr}(\bm{x})}}.\end{split}$ (1) When $\mathrm{pr}(\bm{x})$ is a mixture distribution with $N$ components, $\mathrm{pr}(\bm{x})=\sum_{i=1}^{N}w_{i}\,\mathrm{pr}_{i}(\bm{x}),$ (2) where $w_{i}$ is the weight of component $i$ ($w_{i}\geq 0$ and $\sum_{i}w_{i}=1$), and $\mathrm{pr}_{i}$ is the probability density of component $i$. The conditional distribution of the data, when the class label is $c$, also follows a mixture distribution. In this work, we propose new bounds and estimators of the Shannon mutual information between a mixture distribution $\bm{x}$ and its class label $C$. We provide a lower bound, a variational upper bound and three estimators of $\mathrm{I}(\bm{x};C)$, all based on pair-wise distances. We present closed- form expressions for the bounds and estimators. Furthermore, we use numerical simulations to compare the bounds and estimators to Monte Carlo (MC) simulations and a set of bounds derived from entropy bounds. ### I-C Related works Although estimation of conditional entropy and mutual information has been extensively studied [16, 17, 18, 19], research has focused on purely discrete or continuous data. Nair et al. [14] extended the definition of the joint entropy to mixed-pairs, which consists of one discrete variable and one continuous variable. Ross [20], Moon et al. [21] and Beknazaryan et al. [15] provided methods for estimating mutual information from samples of mixed-pairs based on nearest-neighbor or kernel estimator. Gao et al. [22] extended the definition of mutual information to the case that each random variable can have both discrete and continuous components through the Radon-Nikodym derivative. Here our goal is to study mutual information for mixed-pairs, where the data $\bm{x}$ is continuous and the class label $C$ is discrete. When the underlying distribution of the data is unknown, the mutual information can be approximated from samples with a number of density or likelihood-ratio estimators based on binning [23, 24], kernel methods[25, 26, 27], k-nearest-neighbor (kNN) distances [28, 29], or approximated Gaussianity (Edgeworth expansion [30]). To accommodate high dimensional data (such as image and text) or large datasets, Gao et al. [31] improved the kNN estimator with a local non-uniformity correction term; Jiao et al. [32] proposed a minimax estimator of entropy that achieves the optimal sample complexity; Belghazi et al. [33] presented a general purpose neural-network estimator; Poole et al. [34] provided a thorough review and several new bounds on mutual information that is capable to trade off bias for variance. However, when the underlying data distribution is known, the exact computation of mutual information is tractable only for a limited family of distributions [35, 36]. The mutual information for mixture distributions has no known closed-from expression [37, 38, 39]; hence MC sampling and numerical integration are often employed as unbiased estimators. MC sampling of sufficient accuracy is computationally intensive [40]. Numerical integration is limited to low-dimensional problems [41]. To reduce the computational requirement, deterministic approximations have been developed using merged Gaussian [8, 42], component-wise Taylor-series expansion [43], unscented transform [44] and pair-wise KL divergence between matched components [9]. The merged Gaussian and unscented transform estimators are biased, while the Taylor expansion method provides a trade-off between computational demands and accuracy. Two papers that have deeply inspired our work are [8] and [45]. Hersey et al. [8] proposed a variational upper bound and an estimator of the KL divergence between two Gaussian mixtures by pair-wise KL divergence. Hersey et al. [8] has shown empirically that the variational upper bound and estimator perform better than other deterministic approximations, such as merged Gaussian, unscented transform and matched components. Kolchinsky et al. [45] has bounded entropy of mixture distributions with pair-wise KL divergence and Chernoff-$\alpha$ ($C_{\alpha}$) divergence and demonstrated through numerical simulations that these bounds are tighter than other well-known existing bounds, such as the kernel-density estimator [41, 46] and the expected- likelihood-kernel estimator [47, 48, 49]. Our results are not obvious from either paper, as the calculation of $\mathrm{I}(\bm{x};C)$ involves a summation of multiple entropies or KL divergences. Instead of providing bounds for each term (entropy) in the summation, we directly bound and estimate the mutual information. ## II Main Results In this section, we provide three estimators of $\mathrm{I}(\bm{x};C)$ and a pair of lower and upper bounds. All bounds and estimators are based on pair- wise KL divergence and $C_{\alpha}$ divergence. Furthermore, we provide proofs of the lower and upper bounds. Before presenting our main results, we start with a few definitions. The marginal distribution on the class label $C$ is $\mathrm{Pr}(C=c)=P_{c}=\sum_{i\in\\{c\\}}w_{i}.$ (3) Note that $\sum_{c=1}^{\Pi}P_{c}=1$ and $\\{c\\}$ is the set of the components that have class label $C=c$. The conditional distribution of the data, when the class label is $c$, is given by $\mathrm{pr}(\bm{x}\|c)=\sum_{i\in\\{c\\}}\frac{w_{i}}{P_{c}}\,\mathrm{pr}_{i}(\bm{x}).$ (4) Expressing the marginal distribution in terms of the conditional distribution, we have $\mathrm{pr}(\bm{x})=\sum_{c}P_{c}\cdot\mathrm{pr}(\bm{x}\|c).$ (5) The joint distribution of the data and class label is $\mathrm{pr}(\bm{x},c)=P_{c}\cdot\mathrm{pr}(\bm{x}\|c)=\sum_{i\in\\{c\\}}w_{i}\mathrm{pr}_{i}(\bm{x})$ (6) ### II-A Pair-wise distances The Kullback-Leibler (KL) divergence is defined as $\mathrm{KL}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})=\int\mathrm{d\bm{x}}~{}\mathrm{pr}_{i}(\bm{x})\ln\frac{\mathrm{pr}_{i}(\bm{x})}{\mathrm{pr}_{j}(\bm{x})}.$ (7) The $C_{\alpha}$ divergence [50] between the two distribution $\mathrm{pr}_{i}(\bm{x})$ and $\mathrm{pr}_{j}(\bm{x})$ is defined as $\mathrm{C_{\alpha}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})=-\ln\int\mathrm{d\bm{x}}\,\mathrm{pr}_{i}^{\alpha}(\bm{x})\,\mathrm{pr}_{j}^{1-\alpha}(\bm{x}),$ (8) for real-valued $\alpha\in[0,1]$. More specifically, when $\alpha=1/2$, the Chernoff divergence is Bhattacharyaa distance. ### II-B Bounds and estimates of the mutual information We adopt the convention that $\ln 0=0$ and $\ln(0/0)=0$. An exact expression of the mutual information is $\mathrm{I}(\bm{x};C)=\mathrm{H}(C)-\sum_{i=1}^{N}w_{i}\mathrm{E}_{\mathrm{pr}_{i}}\left[\ln{\frac{\sum_{j=1}^{N}w_{j}\mathrm{pr}_{j}}{\sum_{k\in\\{C_{i}\\}}w_{k}\mathrm{pr}_{k}}}\right],$ (9) where $\\{C_{i}\\}$ is the set of component index that is in the same class with component $i$ and $\mathrm{E}_{\mathrm{pr}_{i}}[f]=\int\mathrm{dx}\,\mathrm{pr}_{i}(\bm{x})f(\bm{x})$ is the expectation of $f$ with respect to the probability density function $\mathrm{pr}_{i}$. Two approximations of $\mathrm{I}(\bm{x};C)$ are $\begin{split}\mathrm{{\hat{I}}_{C_{\alpha}}}(\bm{x};C)&=\mathrm{H}(C)-\sum_{i=1}^{N}w_{i}\ln{\frac{\sum_{j=1}^{N}w_{j}e^{-\mathrm{C_{\alpha}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})}}{\sum_{k\in\\{C_{i}\\}}w_{k}e^{-\mathrm{C_{\alpha}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{k})}}},\\\ \mathrm{{\hat{I}}_{KL}}(\bm{x};C)&=\mathrm{H}(C)-\sum_{i=1}^{N}w_{i}\ln{\frac{\sum_{j=1}^{N}w_{j}e^{-\mathrm{KL}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})}}{\sum_{k\in\\{C_{i}\\}}w_{k}e^{-\mathrm{KL}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{k})}}}.\end{split}$ (10) Another approximation of $\mathrm{I}(\bm{x};C)$ is $\begin{split}&\mathrm{{\hat{I}}_{KL\&C_{\alpha}}}(\bm{x};C)=\mathrm{H}(C)-\sum_{i=1}^{N}w_{i}\ln{\frac{\sum_{j=1}^{N}w_{j}e^{-D_{ij}}}{\sum_{k\in\\{C_{i}\\}}w_{k}e^{-D_{ik}}}},\\\ &\text{where}\quad\frac{1}{D_{ij}}=\frac{1}{2}\left(\frac{1}{\mathrm{KL}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})}+\frac{1}{{\mathrm{C_{\alpha}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})}}\right).\end{split}$ (11) As $D$ is a function of both KL and $C_{\alpha}$ divergences, we denote this estimator with the subscript ‘$KL\&C_{\alpha}$’. A lower bound on $\mathrm{I}(\bm{x};C)$ based on pair-wise $C_{\alpha}$ is $\begin{split}&\mathrm{{I}_{lb\\_{C_{\alpha}}}}=-\sum_{c=1}^{\Pi}P_{c}\ln\left[\sum_{c^{\prime}=1}^{\Pi}P_{c^{\prime}}\cdot\text{min}(1,Q_{cc^{\prime}})\right],\,\,where\\\ &Q_{cc^{\prime}}=\sum_{i\in\\{c\\}}\sum_{j\in\\{c^{\prime}\\}}\left(\frac{w_{i}}{P_{c}}\right)^{\alpha_{c}}\left(\frac{w_{j}}{P_{c^{\prime}}}\right)^{1-\alpha_{c}}e^{-C_{\alpha_{c}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})},\\\ \end{split}$ (12) and min$(\cdot)$ is the minimum value function. A variational upper bound on $\mathrm{I}(\bm{x},C)$ based on pair-wise KL is $\mathrm{{I}_{ub\\_KL}}=\mathrm{H}(C)-\sum_{m}\sum_{c=1}^{\Pi}\phi_{m,c}\ln\frac{\sum_{c^{\prime}=1}^{\Pi}\phi_{m,c^{\prime}}e^{-KL(\mathrm{pr}_{m,c},\mathrm{pr}_{m,c^{\prime}})}}{\phi_{m,c}},$ (13) where $\phi_{m,c}$ are the variational parameters and detailed explanation of the notations is given in Section II-D. ### II-C Proof of the lower bound We propose a lower bound on $\mathrm{I}(\bm{x};C)$ based on pair-wise $C_{\alpha}$ divergences. For ease of notation, we denote the conditional distribution $\mathrm{pr}(\bm{x}\|C=c)$ as $\mathrm{pr}_{c}$. We first make use of a derivation from [45] and [51] to bound $\mathrm{I}(\bm{x};C)$ with class- wise divergence $C_{\alpha}(\mathrm{pr}_{c},\mathrm{pr}_{c^{\prime}})$: $\begin{split}\mathrm{I}(\bm{x};C)=&\sum_{c}P_{c}\int\mathrm{dx}\,\mathrm{pr}_{c}\cdot\ln{\frac{\mathrm{pr}_{c}}{\mathrm{pr}(\bm{x})}}\\\ =&-\sum_{c}P_{c}\int\mathrm{dx}\,\mathrm{pr}_{c}\cdot\ln{\frac{\sum_{c^{\prime}}P_{c^{\prime}}\mathrm{pr}_{c^{\prime}}^{1-\alpha_{c}}}{\mathrm{pr}_{c}^{1-\alpha_{c}}}}\\\ &-\sum_{c}P_{c}\int\mathrm{dx}\,\mathrm{pr}_{c}\cdot\ln\frac{\mathrm{pr}(\bm{x})}{\mathrm{pr}_{c}^{\alpha_{c}}\sum_{c^{\prime}}P_{c^{\prime}}\mathrm{pr}_{c^{\prime}}^{1-\alpha_{c}}}\\\ \geq&-\sum_{c}P_{c}\ln{\sum_{c^{\prime}}P_{c^{\prime}}\int\mathrm{dx}\,\mathrm{pr}_{c}^{\alpha_{c}}\cdot\mathrm{pr}_{c^{\prime}}^{1-\alpha_{c}}}\\\ &-\ln\int\mathrm{dx}\,\sum_{c}P_{c}\mathrm{pr}_{c}^{1-\alpha_{c}}\cdot\frac{\mathrm{pr}(\bm{x})}{\sum_{c^{\prime}}P_{c^{\prime}}\mathrm{pr}_{c^{\prime}}^{1-\alpha_{c}}}\\\ \\\ =&-\sum_{c}P_{c}\ln\left[\sum_{c^{\prime}}P_{c^{\prime}}e^{-C_{\alpha_{c}}(\mathrm{pr}_{c}\|\|\mathrm{pr}_{c^{\prime}})}\right]\\\ \end{split}$ (14) This inequality follows from Jensen’s inequality and the convexity of function $\ln(x)$. The parameter $\alpha_{c}$, which is specific for a class $c$, can be any value in $[0,1]$. The class-wise $C_{\alpha}$ divergence has a minimum value of zero and the minimum is achieved when the two class has the same distribution. Furthermore, we can bound the $C_{\alpha}$ divergence through the subadditivity of the function $f(x)=x^{\alpha}$ when $0\leq\alpha\leq 1$. In other words, as $f(a+b)\leq f(a)+f(b)$ for $a\geq 0$ and $b\geq 0$, the $C_{\alpha}$ divergence between the conditional distributions $\mathrm{pr}_{c}$ and $\mathrm{pr}_{c^{\prime}}$ can be bounded by: $\begin{split}e&{}^{-C_{\alpha}(\mathrm{pr}_{c}\|\|\mathrm{pr}_{c^{\prime}})}=\int\mathrm{dx}\,\left[\sum_{i\in\\{c\\}}\frac{w_{i}}{P_{c}}\mathrm{pr}_{i}\right]^{\alpha}\left[\sum_{j\in\\{c^{\prime}\\}}\frac{w_{j}}{P_{c^{\prime}}}\mathrm{pr}_{j}\right]^{1-\alpha}\\\ &\leq\text{min}\left[1,\sum_{i\in\\{c\\}}\sum_{j\in\\{c^{\prime}\\}}\left(\frac{w_{i}}{P_{c}}\right)^{\alpha}\left(\frac{w_{j}}{P_{c^{\prime}}}\right)^{1-\alpha}e^{-C_{\alpha}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})}\right],\\\ &=\text{min}(1,Q_{cc^{\prime}}).\\\ \end{split}$ (15) Therefore, Equation 12 is a lower bound on $\mathrm{I}(\bm{x};C)$. The best possible lower bound can be obtained by finding the parameters $\alpha_{c}$ that maximize $\mathrm{{I}_{lb\\_{C_{\alpha}}}}$, which is equivalent to minimize $\sum_{c^{\prime}}P_{c^{\prime}}\text{min}(1,Q_{cc^{\prime}})$. In a special case when all components are symmetric and identical except the center location, eg. homoscedastic Gaussian mixture, $e^{-C_{\alpha}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})}$ achieves minimum value at $\alpha=1/2$ [45]. ### II-D Proof of the variational upper bound Here we propose a direct upper bound on the mutual information $\mathrm{I}(\bm{x};C)$ using a variational approach. The underlying idea is to match components from different classes. To pick one component from each class, there are $N_{1}\times N_{2}...\times N_{\Pi}$ combinations, where $N_{c}$ is the number of components in class $c$. Denote an integer $M=\prod_{c=1}^{\Pi}N_{c}$. A component $i$ in class $c$ can be split into $M/N_{c}$ components with each component corresponding to a component- combination in the other $\Pi-1$ classes. Mathematically speaking, we introduce the variational parameters $\phi_{ij}\geq 0$ satisfying the constraints $\sum_{j=1}^{M/N_{c}}\phi_{ij}=w_{i}$. Using the variational parameters, we can write the joint distribution as $\begin{split}\mathrm{pr}(\bm{x},c)=\sum_{i\in\\{c\\}}w_{i}\mathrm{pr}_{i}=\sum_{i\in\\{c\\}}\sum_{j=1}^{M/N_{c}}\phi_{ij}\mathrm{pr}_{i}.\end{split}$ (16) Note that the set $\\{c\\}$ has $N_{c}$ components. By rearranging indices $(i,j)$ into a vector $m$ of length $M$, we can simplify the joint distribution to $\mathrm{pr}(\bm{x},c)=\sum_{m=1}^{M}\phi_{m,c}\mathrm{pr}_{m,c}(\bm{x}),$ (17) where the subscript $c$ emphasizes that each class has a unique mapping from $(i,j)$ to $m$ and $\mathrm{pr}_{m,c}(\bm{x})$ equals to the corresponding $\mathrm{pr}_{i}(\bm{x})$. With this notation, the marginal distribution of the data $\bm{x}$ is $\mathrm{pr}(\bm{x})=\sum_{c=1}^{\Pi}\mathrm{pr}(\bm{x},c)=\sum_{c=1}^{\Pi}\sum_{m=1}^{M}\phi_{m,c}\mathrm{pr}_{m,c}(\bm{x}).$ (18) We further define a mini-batch $m$ as $b_{m}(\bm{x})=\sum_{c=1}^{\Pi}\phi_{m,c}\mathrm{pr}_{m,c}(\bm{x}).$ (19) Each mini-batch contains $\Pi$ components with one component from each class. With this definition, the marginal distribution of the data can be written as $\mathrm{pr}(\bm{x})=\sum_{m=1}^{M}b_{m}(\bm{x})$. The probability of a component in the $m^{th}$ batch is $P_{m}=\sum_{c}\phi_{m,c}$. The probability density function of the $m^{th}$ batch is $\mathrm{pr}(\bm{x}\|m)=b_{m}(\bm{x})/P_{m}$. Now we use Jensen’s inequality, or more specificly log-sum inequality [52], to bound $\mathrm{I}(\bm{x};C)$ by batch-conditional entropy, $\begin{split}&\mathrm{I}(\bm{x};C)=\mathrm{H}(C)+\sum_{c}\int\mathrm{dx}\,\mathrm{pr}(\bm{x},c)\cdot\ln{\frac{\mathrm{pr}(\bm{x},c)}{\mathrm{pr}(\bm{x})}}\\\ &=\mathrm{H}(C)+\sum_{c}\int\mathrm{dx}\left(\sum_{m}\phi_{m,c}\mathrm{pr}_{m,c}\right)\,\ln\frac{\sum_{m}\phi_{m,c}\mathrm{pr}_{m,c}}{\sum_{m}b_{m}}\\\ &\leq\mathrm{H}(C)+\sum_{c}\int\mathrm{dx}\sum_{m}\left(\phi_{m,c}\mathrm{pr}_{m,c}\ln\frac{\phi_{m,c}\mathrm{pr}_{m,c}}{b_{m}}\right)\\\ &=\mathrm{H}(C)+\mathrm{H}(\bm{x}\|m)-\mathrm{H}(C\|m)-\sum_{i=1}^{N}w_{i}\mathrm{H}_{i}(\bm{x})\\\ \end{split}$ (20) where $\mathrm{H}(C)=-\sum_{c}P_{c}\ln P_{c}$ is the entropy of the class label; $\mathrm{H}(\bm{x}\|m)=\sum_{m}P_{m}\mathrm{H}(\mathrm{pr}(\bm{x}\|m))$ is the batch-conditional entropy of the data; $\mathrm{H}(C\|m)=\sum_{m}P_{m}\mathrm{H}_{m}(C)$ is the batch-conditional entropy of the label, where $\mathrm{H}_{m}(C)$ is the entropy of the class label for batch $m$, and $\mathrm{H}_{i}(\bm{x})=\mathrm{H}(\mathrm{pr}_{i}(\bm{x}))$ is the entropy of the $i^{th}$ component. We can further bound the batch-conditional entropy with pair-wise KL divergence as $\begin{split}&\mathrm{I}(\bm{x};C)\leq\mathrm{H}(C)+\hat{\mathrm{H}}_{KL}(\bm{x}\|m)-\mathrm{H}(C\|m)-\sum_{i=1}^{N}w_{i}\mathrm{H}_{i}(\bm{x})\\\ &=\mathrm{H}(C)-\sum_{m}\sum_{c}\phi_{m,c}\ln\frac{\sum_{c^{\prime}}\phi_{m,c^{\prime}}e^{-KL(\mathrm{pr}_{m,c},\mathrm{pr}_{m,c^{\prime}})}}{\phi_{m,c}}\\\ &:=\mathrm{{I}_{ub\\_{KL}}}\end{split}$ (21) where $\hat{\mathrm{H}}_{KL}(\bm{x}\|m)$ is an upper bound of the batch- conditional entropy and the inequality has been proved in [45]. The tightest upper bound attainable through this method can be found by varying parameters $\phi_{m,c}$ to minimize $\mathrm{{I}_{ub\\_{KL}}}$. The minimization problem has been proved to be convex (see Appendix A). The upper bound $\mathrm{{I}_{ub\\_{KL}}}$ can be minimized iteratively by fixing the parameters $\phi_{m,c^{\prime}}$ (where $c^{\prime}\neq c$) and optimizing parameters $\phi_{m,c}$ under linear constraints. At each iteration step $\mathrm{{I}_{ub\\_{KL}}}$ is lowered, and the convergent is the tightest variational upper bound on the mutual information. Non-optimum variational parameters still provide upper bounds on $\mathrm{I}(\bm{x};C)$. There are $M\times\Pi$ variational parameters, $M\times\Pi$ non-equality constraints and $N$ equality constraints. When the number of classes or components is large, the minimization problem will be computationally intensive. A non-optimum solution that is similar to the matched bound [8, 53] can be obtained by dividing all components into max($N_{c}$) mini-batches by matching each component $i$ to one component in each class. Mathematically speaking, $\phi_{ij}=w_{i}$ for one pair of matched $(i,j)$ and $\phi_{ij}=0$ otherwise. To find the mini-batches, the Hungarian method [54, 55] for assignment problems can be applied. ## III Numerical simulations (a) (b) Figure 1: (a) The locations of the center of the components and the mixture distribution $\mathrm{pr}(\bm{x}\|c=2)$ when $\sigma=0.5$ (insert). (b) Estimates of $\mathrm{I}(\bm{x};C)$. In this section, we run numerical simulations and compare estimators on mutual information between mixture data and class labels. We consider a simple example of binary classification of mixture data, where the mixture components are two-dimensional homoscedastic Gaussians. The component centers are close to the class boundary and uniformly distributed along the boundary. The location of the component centers are plotted in the Figure 1(a), where the component centers are represented by a red star (class 1) or a yellow circle (class 2). Each class consists 100 two-dimensional Gaussian components with equal weights. The components have the same covariance matrix $\sigma^{2}I$, where $I$ is the identity matrix and $\sigma$ represents the size of the Gaussian components. The conditional distribution $\mathrm{pr}(\bm{x}\|c=2)$ is plotted in the insert of Figure 1(a) for $\sigma=0.5$. When $\sigma$ is larger, the components of the mixtures distribution are more connected; when $\sigma$ is smaller, the components are more isolated. Estimates of $\mathrm{I}(\bm{x};C)$ are calculated for varying $\sigma$. A pair of obvious bounds of $\mathrm{I}(\bm{x};C)$ are $[0,\mathrm{H}(C)]$, where $\mathrm{H}(C)$ is the entropy of the class label $\mathrm{Pr}(C)$. Another pair of upper and lower bounds of $\mathrm{I}(\bm{x};C)$ can be derived from bounds on mixture entropy as $\begin{split}\mathrm{{I}_{lb\\_2H}}&=\mathrm{H}_{lb}(\bm{x})-\mathrm{H}_{ub}(\bm{x}\|C)\\\ \mathrm{{I}_{ub\\_2H}}&=\mathrm{H}_{ub}(\bm{x})-\mathrm{H}_{lb}(\bm{x}\|C),\end{split}$ (22) where the upper and lower bound of entropy based on pair-wise KL and $C_{\alpha}$ divergences have been provided by [45]. These bounds on $\mathrm{I}(\bm{x},C)$ are based on two entropy bounds, hence the subscript ‘2H’. We evaluate the following estimates of $\mathrm{I}(\bm{x},C)$: 1. 1. The new variational upper bound and the new lower bound, $\mathrm{{I}_{ub\\_{KL}}}$ and $\mathrm{{I}_{lb\\_{C_{\alpha}}}}$, are plotted in dark red and blue solid lines, respectively. 2. 2. The estimates based on the pair-wise KL, $C_{\alpha}$ or D (a function of both KL and $C_{\alpha}$ divergences) are plotted in yellow, light blue and black dashed lines, respectively. 3. 3. The true mutual information, $\mathrm{I}(\bm{x},C)$, as estimated by MC sampling of the mixture model (grey solid line). 4. 4. The upper and lower bounds $\mathrm{{I}_{lb\\_2H}}$ and $\mathrm{{I}_{ub\\_2H}}$ are plotted in orange and green dot-dashed lines, respectively. The obvious bounds on $\mathrm{I}(\bm{x},C)$, which are $[0,H(C)]$, are also presented by an area in grey. The Monte-Carlo simulation results, which can serve as the benchmark, are calculated with $10^{6}$ samples. We use $\alpha=1/2$ in the calculation of $C_{\alpha}$ divergences, as it provides the optimum bounds for our example. We also present details of our implementation and results of two other scenarios in Appendix B. Our new upper bound and lower bound appear to be tighter than the bounds derived from entropy bounds over the range of $\sigma$ considered in our simulation. In Figure 1(b), where the estimates of $\mathrm{I}(\bm{x},C)$ are plotted, the results show that the blue and dark red solid lines are almost always within the area covered by the green and orange dot-dashed lines. The three estimates, $\mathrm{{\hat{I}}_{KL}}$, $\mathrm{{\hat{I}}_{C_{\alpha}}}$ and $\mathrm{{\hat{I}}_{KL\&C_{\alpha}}}$, all follow the trend of $\mathrm{I}(\bm{x},C)$. More specifically, $\mathrm{{\hat{I}}_{KL}}$ (yellow dashed line) follows the new variational upper bound $\mathrm{{I}_{ub\\_{KL}}}$ (deep red solid line) closely; $\mathrm{{\hat{I}}_{C_{\alpha}}}$ (blue dashed line) is a good estimator of $\mathrm{I}(\bm{x},C)$ (grey solid line); $\mathrm{{\hat{I}}_{KL\&C_{\alpha}}}$ (black dashed line) is another good estimator of $\mathrm{I}(\bm{x},C)$, as the black dashed line tracks the grey solid line closely. The differences between the three estimates and the $\mathrm{I}(\bm{x},C)$ calculated from MC simulation are plotted in Appendix B. ## IV Conclusion We provide closed-form bounds and approximations of mutual information between mixture data and class labels. The closed-form expressions are based on pair- wise distances, which are feasible to compute even for high-dimensional data. Based on numerical results, the new bounds we proposed are tighter than the bounds derived from bounds on entropy and the approximations serve as good surrogates for the true mutual information. ## Appendix A The Minimization of $\mathrm{{I}_{ub\\_KL}}$ The minimization problem of $\mathrm{{I}_{ub\\_KL}}$ by varying $\phi_{mc}$ is convex, which we prove in this section. The convexity of the minimization problem can be checked through the first and second-order derivatives. For ease of notation, we define $S_{m,c}=\sum_{c^{\prime}}\phi_{m,c^{\prime}}e^{-\mathrm{KL}(\mathrm{pr}_{m,c},\mathrm{pr}_{m,c^{\prime}})}$ and $E_{m,cc^{\prime}}=\exp({-\mathrm{KL}(\mathrm{pr}_{m,c},\mathrm{pr}_{m,c^{\prime}})})$. The first derivative of $\mathrm{{I}_{ub\\_{KL}}}$ is: $\frac{\partial{I}_{ub\\_{KL}}}{\partial\phi_{m,c}}=-\ln\left(\frac{S_{m,c}}{\phi_{m,c}}\right)-\frac{\phi_{m,c}}{S_{m,c}}-\sum_{c^{\prime}\neq c}\frac{\phi_{m,c^{\prime}}E_{m,c^{\prime}c}}{S_{m,c^{\prime}}}+1$ (23) The second derivative is: $\begin{split}H_{cc}&=\frac{\partial^{2}{I}_{ub\\_{KL}}}{(\partial\phi_{m,c})^{2}}\\\ &=\frac{(S_{m,c}-\phi_{m,c})^{2}}{(S_{m,c})^{2}\phi_{m,c}}+\sum_{c^{\prime}\neq c}\frac{\phi_{m,c^{\prime}}(E_{m,c^{\prime}c})^{2}}{(S_{m,c^{\prime}})^{2}}\\\ \end{split}$ (24) for the diagonal terms and $\begin{split}H_{cc^{\prime}}&=\frac{\partial^{2}{I}_{ub\\_{KL}}}{\partial\phi_{m,c}\partial\phi_{m,c^{\prime}}}\\\ &=\frac{\phi_{m,c}-S_{m,c}}{(S_{m,c})^{2}}E_{m,cc^{\prime}}+\frac{\phi_{m,c^{\prime}}-S_{m,c^{\prime}}}{(S_{m,c^{\prime}})^{2}}E_{m,c^{\prime}c},\\\ \end{split}$ (25) for $c^{\prime}\neq c$. For any given vector $\bm{\theta}$ of length $\Pi$, $\begin{split}\bm{\theta}^{T}H\bm{\theta}=&\sum_{c}\left(\theta_{c}^{2}H_{cc}+\sum_{c^{\prime}\neq c}\theta_{c}\theta_{c^{\prime}}H_{cc^{\prime}}\right)\\\ =&\sum_{c}\Bigg{[}\theta_{c}^{2}\frac{(S_{m,c}-\phi_{m,c})^{2}}{(S_{m,c})^{2}\phi_{m,c}}+\sum_{c^{\prime}\neq c}\theta_{c^{\prime}}^{2}\frac{\phi_{m,c}(E_{m,cc^{\prime}})^{2}}{(S_{m,c})^{2}}\\\ &\quad\quad+\sum_{c^{\prime}\neq c}2\theta_{c}\theta_{c^{\prime}}\frac{\phi_{m,c}-S_{m,c}}{(S_{m,c})^{2}}E_{m,cc^{\prime}}\Bigg{]}\\\ =&\sum_{c}\left[\frac{(S_{m,c}-\phi_{m,c})\theta_{c}}{S_{m,c}\sqrt{\phi_{m,c}}}-\sum_{c^{\prime}\neq c}\frac{\sqrt{\phi_{m,c}}E_{m,cc^{\prime}}\theta_{c^{\prime}}}{S_{m,c}}\right]^{2}\\\ \geq&0.\end{split}$ (26) Therefore, $\mathrm{{I}_{ub\\_{KL}}}$ is convex when $\phi_{m,c}$ are considered as the variables. ## Appendix B On the Numerical simulations This appendix provides more simulation results and the detailed expressions used in the numerical simulations. The additional results consider two different distributions of the component-center locations. We further present the difference between the the estimated and the true mutual information. Last but not least, the closed form expressions include the KL and $C_{\alpha}$ divergences between Gaussian components, the bounds on $\mathrm{I}(\bm{x};C)$ derived from entropy bounds, and the relation between Shannon mutual information and bounds on binary classification error ($\mathrm{P_{e}}$). ### B-A Numerical simulation results We consider three scenarios: (1) the component centers are uniformly distributed along the class boundary, (2) the component centers are bunched into one group, and (3) the component centers are bunched into several groups. Results on the first scenario has been presented in the Section III. In this section, we report on Scenarios 2 and 3. Illustration of the two scenarios are shown in Figure 2(a) and 3(a), respectively. (a) (b) Figure 2: Scenario 2, where the center of the components are bunched into one group, illustration (a), the mixture distribution $\mathrm{pr}(\bm{x}\|c=2)$ when $\sigma=0.5$ (insert), and estimates of $\mathrm{I}(\bm{x};C)$ (b). (a) (b) Figure 3: Scenario 3, where the center of the components are bunched into multiple groups, illustration (a), the mixture distribution $\mathrm{pr}(\bm{x}\|c=2)$ when $\sigma=0.5$ (insert), and estimators of $\mathrm{I}(\bm{x};C)$ (b). The results demonstrated in these two scenarios are similar to that of Scenario 1. To further demonstrate that our estimators are good surrogates for the true mutual information, we plot the difference between the three estimates and the true mutual information calculated from MC sampling in Figure 4. (a) (b) (c) Figure 4: $\mathrm{\hat{I}(\bm{x};C)-I(\bm{x};C)}$ for (a) Scenario 1, (b) Scenario 2 and (c) Scenario 3. The $\mathrm{I(\bm{;}C)}$ is calculated from MC simulations. ### B-B Closed form expressions for Gaussian mixtures Gaussian functions are often used as components in mixture distributions and have closed form expressions for pair-wise KL and $C_{\alpha}$ divergences. Denoting the difference in the means of two components as $\bm{\mu}_{ij}=\bm{\mu}_{i}-\bm{\mu}_{j}$ and $\Sigma_{\alpha,ij}=(1-\alpha)\Sigma_{i}+\alpha\Sigma_{j}$, the $C_{\alpha}$ divergence between two Gaussian components are $\begin{split}&\mathrm{C_{\alpha}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})=\frac{\alpha(1-\alpha)}{2}{\bm{\mu}^{T}_{ij}}\Sigma_{\alpha,ij}^{-1}\bm{\mu}_{ij}+\frac{1}{2}\ln\frac{\|\Sigma_{\alpha,ij}\|}{\|\Sigma_{i}\|^{{1-\alpha}}\|\Sigma_{j}\|^{{\alpha}}},\end{split}$ (27) where $\|\cdot\|$ is the determinant. The KL divergence between the same two Gaussian components are $\mathrm{KL}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})=\frac{1}{2}\left[{\bm{\mu}^{T}_{ij}}\Sigma_{j}^{-1}\bm{\mu}_{ij}+\ln\frac{\|\Sigma_{j}\|}{\|\Sigma_{i}\|}\right]+\frac{\text{tr}(\Sigma_{j}^{-1}\Sigma_{i})-d}{2},$ (28) where tr$(\cdot)$ is the trace of the matrix in the parenthesis and $d$ is the dimension of the data. When all mixture components have equal covariance matrices $\Sigma_{i}=\Sigma_{j}=\Sigma$, we can denote $\lambda_{ij}={\bm{\mu}^{T}_{ij}}\Sigma^{-1}\bm{\mu}_{ij}$ and have $\begin{split}\mathrm{C_{\alpha}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})&=\alpha(1-\alpha)\lambda_{ij}/2,\\\ \mathrm{KL}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})&=\lambda_{ij}/2.\end{split}$ (29) With these expressions, the bounds and estimates of $\mathrm{I}(\bm{x};C)$ have simple forms. ### B-C Expressions of $\mathrm{{I}_{lb\\_2H}}$ and $\mathrm{{I}_{ub\\_2H}}$ The detailed expression for the mutual information bounds derived from entropy bounds are: $\begin{split}\mathrm{{I}_{lb\\_2H}}&=\mathrm{H}(C)-\sum_{i=1}^{N}w_{i}\ln{\frac{\sum_{j=1}^{N}w_{j}e^{-\mathrm{C_{\alpha}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})}}{\sum_{k\in\\{C_{i}\\}}w_{k}e^{-\mathrm{KL}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{k})}}},\\\ \mathrm{{I}_{ub\\_2H}}&=\mathrm{H}(C)-\sum_{i=1}^{N}w_{i}\ln{\frac{\sum_{j=1}^{N}w_{j}e^{-\mathrm{KL}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{j})}}{\sum_{k\in\\{C_{i}\\}}w_{k}e^{-\mathrm{C_{\alpha}}(\mathrm{pr}_{i}\|\|\mathrm{pr}_{k})}}}.\end{split}$ (30) ### B-D Bounds on $\mathrm{P_{e}}$ for binary classification Bounds on $\mathrm{I}(\bm{x};C)$ can be used to calculate bounds on $\mathrm{P_{e}}$. The Fano’s inequality [1] provides a lower bound on $\mathrm{P_{e}}$ for binary classification, as following $\mathrm{P_{e}}\geq h_{b}^{-1}[\mathrm{H}(C)-\mathrm{I}(\mathbf{x};C)],$ (31) where $h_{b}(x)=-x\log_{2}(x)-(1-x)\log_{2}(1-x)$ is the binary entropy function, $h_{b}^{-1}(\cdot)$ is the inverse function of $h_{b}(\cdot)$. More specifically, one can calculate $P_{e}$ by placing the value $\mathrm{H}(C)-\mathrm{I}(\mathbf{x};C)$ on the left side of the binary entropy function and solving for $x$. A tight upper bound on binary classification error $\mathrm{P_{e}}$ has been reported recently [3], $\mathrm{P_{e}}\leq{\text{min}}\left\\{P_{min},\,f^{-1}[\mathrm{H}(C)-\mathrm{I}(\mathbf{x};C)]\right\\}:=\mathrm{\hat{P}_{e\\_ub}},$ (32) where $P_{min}$ is min$\\{P_{1},P_{2}\\}$, and $f(x)$ is a function defined by $f(x)=-P_{min}\log_{2}\frac{P_{min}}{x+P_{min}}-x\log_{2}\frac{x}{x+P_{min}},$ (33) and $f^{-1}(\cdot)$ is the inverse function of $f(\cdot)$. When $\mathrm{P_{e}}\ll 1$, $-\mathrm{P_{e}}(\log{\mathrm{P_{e}}}-\log{P_{min}})\lesssim\mathrm{H}(C)-\mathrm{I}(\mathbf{x};C)\lesssim-\mathrm{P_{e}}\log{\mathrm{P_{e}}}$. Therefore, $\mathrm{P_{e}}$ is on the same order of magnitude as $\mathrm{H}(C)-\mathrm{I}(\mathbf{x};C)$, when $\mathrm{P_{e}}\ll 1$. ### B-E $\mathrm{P_{e}}$ estimates When $\sigma$ is small, all estimates of $\mathrm{I}(\bm{x},C)$ converges to 1. To compare the estimates for $0.1>\sigma>0.01$, we calculate and present an estimate of $\mathrm{P_{e}}$ in this section. This estimate of $\mathrm{P_{e}}$ is the upper bound on $\mathrm{P_{e}}$ presented in the previous section. When a lower bound of $\mathrm{I}(\bm{x},C)$ is used in the calculation (blue solid line and green dashed line), the estimates of $\mathrm{P_{e}}$ are upper bounds on $\mathrm{P_{e}}$. The other lines in the $\mathrm{P_{e}}$ plots are neither upper bound nor lower bound. 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§ INTRODUCTION Supersymmetric field theories are invariant under a set of transformations taking bosons to fermions and vice versa. By definition, these supersymmetry transformations form a Lie superalgebra extending the algebra of Poincaré transformations to a super-Poincaré algebra. By construction, the fields in the theory are linear representations of the bosonic Poincaré subalgebra. Ideally, this supermultiplet of fields furnishes a (possibly reducible) linear representation of the full supersymmetry algebra. For this to be true, it is necessary for the commutator of two supersymmetries\|as realized on a component field\|to close onto the translations on that field (up to other bosonic symmetries and possibly gauge transformations). On- and Off-shell Supersymmetry However, it is generally the case that this requirement fails and—strictly speaking—the fields do not furnish a linear representation of the supersymmetry algebra. If the action is invariant under the fermionic transformations that extend the Poincaré symmetries to a consistent superalgebra, how can the fields fail to be a representation? An important observation in this context is that the failure of the supersymmetries to close on the fields is by terms that vanish if the equations of motion are imposed.[Such terms are called trivial symmetries. In a canonical treatment of symmetries, in which there is a Poisson bracket on the space of fields, these symmetries are canonical transformations generated by the action $S$ itself. They are trivial, in the sense that $\delta S \sim \{ S, S\}_{\mathrm {PB}} \equiv 0$.] In this context, the supersymmetry is said to close on-shell only, and the set is referred to as an on-shell supermultiplet. This should not be interpreted to mean that the component fields in the theory are required to be on-shell; only that the supersymmetries do not close on the off-shell realization. In the ideal case, this complication does not arise: Successive supersymmetry transformations close properly on the fields irrespective of any field equations, and the collection of fields furnishes a linear representation. When this fortuitous situation is realized, it is emphasized by referring to it as off-shell supersymmetry. When we compare these off-shell theories to the on-shell ones described previously, we notice that they avoid the obstruction to closure with extra fields: Roughly, the terms in the transformations of components that give rise to obstructions are replaced with new fields. Auxiliary Fields and the Off-shell Problem Under supersymmetry transformations, the new component fields transform into the obstruction. Such fields are called auxiliary fields. In two-derivative actions, they have algebraic equations of motion (no derivatives on them), so they do not contribute dynamical degrees of freedom. In principle, they can be integrated out of the path integral (or set to their classical values), but removing them changes the supersymmetry transformations which now close only on-shell. The problem of extending on-shell formulations of supersymmetric theories to off-shell supersymmetry by adding auxiliary fields is known as the off-shell problem. It is considered important for various reasons ranging from the purely mathematical representation theory of supersymmetry algebras to phenomenological supersymmetric model building. (In the case of most interest to us in this paper, we intend to use off-shell supersymmetry to study self-interactions in the form of higher-derivative corrections to the M-theory effective action.) A considerable amount of effort has been devoted to solving the off-shell problem. An immediate indication of the inadequacy of any step-wise approach attempting to add suitable component fields and transformations was provided by Siegel and Roček [1] (see also sec. 2.4.3 of [2]): For any non-real representation in theories with eight or more supersymmetries (e.g. hypermuliplets), any potential solution to this problem requires an infinite number of fields. The proof is essentially a counting argument comparing the dimensions of off-shell Fock representations of the supertranslation algebra to those of superfield representations. Manifest Supersymmetry Just as the translation part of the Poincaré algebra is represented on fields by coordinate derivatives, so too can the supersymmetries be geometrized to translations in fermionic directions. Introducing coordinates for these directions, we can consider superfields as functions of both the even and odd coordinates. The component fields of a supermultiplet then appear as coefficients in the Taylor expansion of the superfields in the odd variables. By construction, superfields are field representations of the super-Poincaré algebra (just as the component fields are so for the Poincaré subalgebra), and the supersymmetry is said to be manifest. However, these are generally (highly-)reducible representations, so we must reduce them in a way that is compatible with manifest supersymmetry. To this end, we can impose combinations of reality conditions, gauge symmetry, and multiplet-reducing constraints compatible with superspace. In the latter, we use the supertranslations generated by fermionic covariant derivatives $D$ to impose covariant constraints such as $D\Phi = 0$, $D^2 \Phi =0$, etc. These are often called shortening conditions, because they imply that $\Phi$ does not depend on some of the odd coordinates thereby shortening its Taylor expansion. The problem of finding irreducible representations of the super-Poincaré algebra thus becomes one of understanding the possible shortening conditions. This is non-trivial because the superalgebra implies that successive $D$s can generate bosonic derivatives $\{D, D\}\sim -2i \partial$, so imprudently-chosen constraints generate kinetic operators generalizing those of d'Alembert and Dirac. In this event, the multiplet is on-shell again in the sense described above. But note that in superspace it further implies the superfield equations of motion directly on the representation. In other words, we cannot take such a manifestly supersymmetric representation off-shell in any sense; the construction of an action principle using such a representation is moot. Much work in the superspace literature has been devoted to finding clever shortening conditions by enlarging the superspace with appropriate bosonic spaces. These naturally circumvent the counting arguments by introducing field dependence on bosonic coordinates geometrizing the R-symmetry transformations [3, 4, 2] or on bosonic ghost-like variables (e.g. Lorentz harmonics [5, 6] or pure spinor [7, 8, 9]). This approach can be generalized and systemized using supercosets [10]. Alternatively, we can attempt to work around the problem using a superspace with a smaller structure group than the maximal one. Examples of this include light-cone superspace [11] and its generalizations (e.g. [12, 13]). In these reductions, new shortening conditions can be defined that are not manifestly covariant under the original symmetry group, but also do not imply any dynamical equations. Such off-shell representations may then be combined into a supermultiplet representing the larger bosonic group linearly, albeit not manifestly. 11D, $N=1/8$ Superspace An attempt to classify the possible structure group reductions appropriate to 16 supercharges was made in [14]. In previous work, we applied a similar approach to eleven-dimensional supergravity based on the reduction $Spin(10,1)\to SL(2, \mathbf C) \times G_2$ [15, 16, 17, 18, 19, 20]. Besides being the most familiar and phenomenologically relevant, this superspace has the significant advantage of requiring only finitely-many auxiliary fields (and has the largest super-Poincaré symmetry with this property [1]). In this formulation the component fields of eleven-dimensional supergravity are embedded into a set of so-called prepotential superfields satisfying no constraints other than the reality and shortening conditions (i.e. chiral superfields) allowed off-shell. Besides the 11D frame $e_{\bm m}{}^{\bm a}$, 3-form $C_{\bm {mnp}}$, and gravitino $\psi_{\bm m}{}^{\bm \alpha}$, these superfields contribute a finite set of auxiliary fields guaranteeing that 4 of the 32 supersymmetries close off-shell on the set of physical + auxiliary fields. In this paper, we will construct superfields whose $\theta$-independent components are precisely the physical 11D gauge connections (i.e. the frame, 3-form, and gravitino). The manifest supersymmetry transformations of the component fields are effected by acting with the $SL(2, \mathbf C)$ superspace derivatives $D$ on these superfields. Besides mixing the bosonic gauge fields, this action generates the auxiliary fields in a form familiar from the standard component “tensor calculus”. We use this to explicitly identify the spectrum of auxiliary fields (or, more precisely, the superfields that have the auxiliary components as their $\theta$-independent term). The supergeometrical tensors of the eleven-dimensional theory (i.e. 4-form field strength, torsion, and curvature 2-form) may then be defined in the naïve way by taking the curls of the gauge connections. By construction, this spectrum of superfields forms a(n off-shell) representation of the reduced structure supergroup. As a result of this analysis, we derive the set of torsion constraints for the supergeometry with this reduced structure group. As it is traditionally done, this set is the starting point for solving the curved superspace Bianchi identities [21, 22, 23]. [It is known that imposing just the dimension zero torsion condition $T_{\bm\alpha \bm\beta}{}^{\bm c} \propto (\Gamma^{\bm c})_{\bm \alpha \bm \beta}$ is sufficient to put 11D superspace on-shell [24]. This observation allows one to postulate an “off-shell” formulation of 11D supergravity (or at least parametrize higher derivative corrections) by relaxing this constraint and re-solving the Bianchi identities [25]. However, there is some disagreement about this point [26, 27, 28]. Our result does not speak directly to this issue, because we attempt to keep only 4 of the 32 supersymmetries off-shell.] At the end of this lengthy and technical process (which we do not carry out in this paper), one finally arrives at the solution to the constraints which are the off-shell prepotentials. Amusingly, then, in the situation we described above and will follow in this paper, we have essentially started with the solution to a set of constraints we do not yet know, and can then derive these constraints by acting repeatedly with the superspace derivatives on the unconstrained fields. Since the prepotentials with which we start are unconstrained, this process is guaranteed to terminate without ever imposing a dynamical constraint. Along the way, all the physical potentials, field strengths, and Bianchi identities are derived. §.§ Results The component fields of eleven-dimensional supergravity consist of a frame $e_{\bm m}{}^{\bm a}$, a gauge 3-form $C_{\bm {mnp}}$, and a gravitino $\psi_{\bm m}^{\bm \alpha}$. Under linearized supersymmetry, they transform into each other as \begin{align} \label{E:LinSUSY} \delta_{\bm \epsilon} e_{\bm a }{}^{\bm b} &= \bar {\bm \epsilon} \Gamma^{\bm b} \psi_{\bm a} \cr \delta_{\bm \epsilon} C_{\bm {abc}} &= 3 \bar {\bm \epsilon} \Gamma_{[\bm {ab} }\psi_{\bm c]} \cr \delta_{\bm \epsilon} \psi_{\bm a}{}^{\bm \beta} &= -\tfrac12 \omega_{\bm a}{}^{\bm{bc}} (\Gamma_{\bm {bc}} \bm \epsilon)^{\bm \beta} - \tfrac1{6\cdot 4!} \left( 3 \Gamma^{\bm {bcde}}\Gamma_{\bm a }\bm \epsilon - \Gamma_{\bm a } \Gamma^{\bm {bcde}}\bm \epsilon \right) \bm G_{\bm {bcde}} \end{align} where $\bm G := d C$ is the 4-form field strength, and we have flattened all the indices with the background frame $\delta_{\bm m}^{\bm a}$. As a realization of supersymmetry, these transformations can only close on-shell: The degrees of freedom needed for the off-shell counting to match are missing (cf. table <ref>). \begin{align} {\renewcommand{\arraystretch}{1.7} %adds some padding \begin{array}{\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline \textrm{gauge field} & ~~~~\textrm{on-shell}~~~~ & \textrm{off-shell}\\ \hline \textrm{frame} & ~~\tfrac12 ({\textrm{D}}-1)({\textrm{D}}-2) -1 ~~ & \tfrac12 {\textrm{D}}({\textrm{D}}-1) \\ & 55 \\ \hline \textrm{gravitino} & 2^{\lfloor{\tfrac {\textrm{D}}2}\rfloor -1} \cdot ({\textrm{D}}-3) & ~~2^{\lfloor{\tfrac {\textrm{D}}2}\rfloor} \cdot ({\textrm{D}}-1)~~ \\ & 128 & 320 \\ \hline & {{\textrm{D}}-2\choose p} & {{\textrm{D}}-1\choose p} \\ & 120 \\ \hline \hline ~~\textrm{total (D=11)}~~ & {\color{PineGreen}128+128 } & {\color{Red}175+ 320} \\ \hline \end{array} \end{align} Counting of degrees of freedom in D dimensions relevant to Poincaré supergravity. (For conformal supergravity, some (gamma-)traces should be subtracted.) In this paper, we will explain how to fix this mismatch and give explicitly the modifications to these supersymmetry transformations (cf. eqs. <ref>, <ref>, and <ref>). To get a representation on which 4 of the 32 supersymmetries close off-shell, we decompose the 11D component fields under \begin{align} \label{E:StructureGroupReduction} \to Spin(3,1) \times Spin(7) \to Spin(3,1) \times G_2 = SL(2, \mathbf C)\times G_2 \end{align} For notational convenience, we will label spacetime, polarizations, indices, et cetera by $M$ for 11D, $X$ for the 4D part, and $Y$ for the 7D part. Under the decomposition of the tangent space, \begin{align} \begin{array}{cccclcc} e_{\bm m}{}^{\bm a} &\to & \frameXX_{m}{}^{a} ~,~ \frameYX_{i}{}^{a} ~,~ \frameXY_{m}{}^{j} ~,~ \frameYY_i{}^j &\textrm{with }& a, m = 0,1,2,3, \\ \psi_{\bm m}^{\bm \alpha}&\to & \grinoXX_{m}^{~\alpha} ~,~ \grinoXY_{m}^{~\alpha i} ~,~ \grinoYX_{j}^{~\alpha } ~,~ \grinoYY_{j}^{~\alpha i} ~,~ &\textrm{and }& \alpha = 1,2, \\ C_{\bm {mnp}} &\to & \CXXX_{mnp}~,~ \CXXY_{mn \, i} ~,~ \CXYY_{m \, ij}~,~ \CYYY_{ijk} &\textrm{and }& i,j = 1,\dots, 7 \end{array} \end{align} To avoid introducing too much notation, the $i, j$, …indices will be doing quadruple duty as 7D coordinate indices, 7D tangent space indices, $G_2$ indices, and as a label for 7 additional gravitino fields. (We collect the index definitions in table <ref>). \begin{align} {\renewcommand{\arraystretch}{1.5} %adds some padding \begin{array}{\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline \textrm{index} & ~~~~\textrm{range}~~~~ & \textrm{description} &~~~\textrm{label}~~~\\ \hline \bm m, \bm n, \dots & 0, \cdots , 10 & \textrm{11D coordinate} &\\ \bm a, \bm b, \dots & 0, \cdots , 10 & \textrm{11D tangent} &M\\ \bm \alpha, \bm \beta, \dots & 1, \cdots , 32 & \textrm{11D spinor} &\\ \hline m, n, \dots & 0, 1,2 , 3 & \textrm{4D coordinate} &\\ a, b, \dots & 0, 1,2 , 3 & \textrm{4D tangent} &X \\ ~~\alpha, \beta , \dots , \dt \alpha, \dt \beta \dots & 1, 2 & \textrm{4D spinor} &\\ \hline i,j,\dots & 1,\cdots , 7 & ~~\textrm{7-component label} ~~&Y\\ \hline \end{array} \end{align} Legend of indices used in this work. To avoid having even more indices, those in the various $\bm7$-dimensional representations ($GL(7)$ coordinate, $SO(7)$ tangent, $G_2$ representation, and label for seven gravitini), have all been identified. Using the $G_2$ structure to trade the 7D graviton polarizations $\gravitonYY_{ij}$ for a stable 3-form $F_{ijk}$, these components embed into a set of superfields called prepotentials [16], as summarized in table <ref>. The prepotential superfields are unconstrained superfields or constrained only to be real or chiral.[ Such constraints are innocuous: Besides factors of $\bar D^2$ arising from the variation of chiral fields, these fields are unconstrained as integration variables in a path integral. (For this reason, we will sometimes sloppily refer to them as unconstrained.)] The components that arise in the Taylor expansion of the fields are listed under the superfield. They have been separated into three groups: * Gauge fields transform canonically. Their (double) curls are field strength, torsion, and curvature 2-form components. Using table <ref>, they add up to $128\|128$ components on-shell, but off-shell (modulo gauge transformation), we find a mismatch between $175$ bosonic and $320$ fermionic components. * Auxiliary fields are gauge invariant, but do not propagate. Due to this, it is easily verified that they contribute $201$ bosonic and $56$ fermionic degrees of freedom all of which are off-shell. * Compensators suffer Stückelberg shifts under “pregauge” transformations. They do not appear in the spectrum of gauge(-invariant) superfields, so they do not contribute any degrees of freedom at all. The compensating fields are artifacts of the reduction of the structure supergroup implied by (<ref>). This reduction is crucial to the representation of the off-shell supersymmetries, but it has no geometrical eleven-dimensional meaning. (Presumably it has some pregeometrical meaning, but the off-shell pregeometry for 11D is precisely the thing we do not know.) This situation is precisely analogous to 4D, $N=1$ Poincaré supergravity itself, which is described in terms of conformal supergravity (which is not a physical symmetry) coupled to a scalar superfield called the conformal compensator and denoted $\G$ (<ref>) in this paper. Relatedly, the auxiliary fields that appear in the Taylor expansion of the prepotentials are—strictly speaking—not invariant under all of the pregeometrical symmetries. However, the parts of the pregeometrical transformations under which they are not covariant are again artifacts of the structure group reduction having no eleven-dimensional interpretation. Therefore, it is possible to covariantize these auxiliary components by mixing in some parts of the other prepotentials. This is done explicitly in section <ref>, but the details are not important here. \begin{align} {\renewcommand{\arraystretch}{1.5} %adds some padding \begin{array}{\|c\|c\|c\|c\|c\|c\|c\|c\|c\|} \hline & U^a & \Psi_i^\alpha & \mathcal V^i & X & \Sigma_{\alpha i} & V_{ij} & \Phi_{ijk} &~~\textrm{sdim}_{\mathbf R} ~~\\ ~\textrm{prepotential}~& \textrm{real} & \textrm{spinor} & ~~\textrm{real}~~ & ~~\textrm{real}~~ & ~~\textrm{chiral}~~ & \textrm{real} & \textrm{chiral}&\\ \hline \physical{\textrm{physical }} & ~~\gravitonXX_{ab} , \grinoXX_a{}^\beta ~~ & \grinoXY_a{}^{\beta j} & \gravitonXY_a^i & \CXXX_{abc} & \CXXY_{ab\,i} & ~~\CXYY_{a\, ij}, \spinorYY_{\alpha\, ij} ~~ & ~~\CYYY_{ijk}, \gravitonYY_{ij}, \spinorYYY_{\alpha\, ijk}~~&\\ % & 2\|2 & \\ \auxiliary{\textrm{auxiliary }} & \dX^a &~~ \l^i_\alpha, \y_i^a, \t_i{}^{\alpha \beta} ,\r_i^\alpha ~~ & \dY^i & \d & & \dYY_{ij} & \f_{ijk} &\\ &&&\surfeit^i_\alpha &\compX,\surfeit'_\alpha &\compY_i,\surfeit''_{\alpha i}&&& \\ \hline \textrm{on-shell} & 2\|2 & 0\|14 & 14\|0 & 0\|0 & 7\|0 & 42\|42 & 63\|70 & 128\|128 \\ %\textrm{off-shell} & 10\|12 & 98\|140 & 28\|0 & 2\|0 & 21\|0 & 84\|84 & 133\|140 & 376\|376\\ \textrm{off-shell} & (\physical{6}+\auxiliary{4})\|\physical{12} & \auxiliary{98}\|(\physical{84}+\auxiliary{56}) & (\physical{21}+\auxiliary{7})\|0 & (\physical{1}+\auxiliary{1})\|0 & \physical{21}\|0 & (\physical{63}+\auxiliary{21})\|\physical{84} & (\physical{63}+\auxiliary{70})\|\physical{140} & 376\|376\\ \hline \end{array} \end{align} The unconstrained “prepotential” superfields of the 11D, $N=1/8$ supergravity multiplet and their physical, auxiliary, and compensating component content. The components of the gravitino of the type $\psi_i{}^\beta$ and $\psi_i{}^{\beta j}$ are encoded in $\chi_{\alpha ij}$ and $\chi_{\alpha ijk}$ (<ref>). The real superdimension contributed by each superfield is computed modulo gauge transformation. The superdimension of the auxiliary field space adds up to $201\|56$. Supersymmety Transfomations Once the eleven-dimensional superfields corresponding to the components in table <ref> have been defined, we are in a position to compute their supersymmetry transformations. In superspace, supersymmetry transformations of component fields result from infinitesimal translations in the odd directions—in other words—from acting with the fermionic covariant derivatives. The result of this calculation is that the graviton transformation is unchanged from the on-shell rule: \begin{align} \label{E:SupersymmetryTransformationsGraviton} \delta_\epsilon \gravitonXX_{ab} &= 2i \epsilon \sigma_{(a} \bar \grinoXX_{b)} \cr \delta_\epsilon \gravitonXY_{a i} &= - i \epsilon\grinoXX_{a\,i} +i \epsilon \sigma_{a} \bar \grinoYX_{i} \cr \delta_\epsilon \gravitonYY_{ij} &= -2i \epsilon\grinoYY_{(i, j)} \end{align} ($SL(2, \mathbf C)$ spinor index contractions are implied.) The supersymmetry transformation of the 3-form acquires a correction by the dimension-1/2 auxiliary field $\l^i_\alpha$ of the gravitino multiplet (cf. table <ref>, or eq. <ref> for the explicit expression):[Some of the gravitino components can be shifted by the gauge-invariant dimension-1/2 auxiliary field $\l^i_\alpha$, and this type of correction could also have appeared in the graviton transformation. Because this field vanishes on-shell, this leads to an ambiguity in the definition of the off-shell gravitini. The definition of the gravitino we make below (see eq. <ref>) ensures that $\l^i_\alpha$ appears in only one place in the off-shell transformations (<ref>) and \begin{align} \label{E:SupersymmetryTransformationsThreeform} \delta_\epsilon \CXXX_{abc} &= -6 \epsilon \sigma_{[ab} \grinoXX_{c]} \cr \delta_\epsilon \CXXY_{ab\, i} &= - 4i \epsilon \sigma_{[a} \bar \grinoXY_{b] i} -2 \epsilon \sigma_{ab} \grinoYX_{i} \cr \delta_\epsilon \CXYY_{a\, ij} &= \varphi_{ijk} \epsilon \grinoXY_{a}{}^k -4i\delta_{k[i} \epsilon \sigma_{a} \bar \grinoYY_{j]}{}^k -\tfrac16 \varphi_{ijk} \epsilon \sigma_{a} \bar \l_{}^k \cr \delta_\epsilon \CYYY_{ijk} &= \varphi_{l[ij} \epsilon \grinoYY_{k]}{}^l \end{align} In these transformations, we define $\delta_\epsilon$ to include a Wess-Zumino gauge transformation\|that is, a field-dependent gauge transformation that involves unphysical terms in the superfield multiplet. Finally, we present the gravitino supersymmetry transformations. These allow us to identify the dimension-1 auxiliary fields needed to close the supersymmetry algebra. For legibility, we suppress the ubiquitous spin connection terms in addition to including compensating Wess-Zumino gauge transformations. (The complete expressions are presented in appendix <ref>.) \begin{align} \label{E:SupersymmetryTransformationsGravitino} \delta_\epsilon \grinoXX_a^\beta &\simeq - \tfrac i2 (\epsilon\eta_{ab} + \epsilon \sigma_{ab})^\beta \dX^b -\tfrac i2 (\bar \epsilon \bar \sigma_a )^{\beta} \R \\ \delta_\epsilon \grinoYX_i^\beta &\simeq -\tfrac i{12} (\epsilon \sigma^{ab})^\beta \t_{ab\, i} -\tfrac i2 \epsilon^\beta \dY_i - \tfrac i{12} (\bar \epsilon \bar \sigma_a )^\beta \Ga_i^{a} \cr \delta_\epsilon \grinoXY_{a}{}^{\beta j} -\tfrac{5i}{12} (\epsilon\eta_{ab} + \tfrac 85 \epsilon \sigma_{ab} )^\beta\, \bar \Ga^{bj} -i (\bar \epsilon\bar \sigma_a )^\beta \dY^j - \tfrac i6 \big[ 4 \eta_{ab} \bar \epsilon \bar \sigma_c + i \varepsilon_{abcd} \bar \epsilon \bar \sigma^d \big]^\beta \t^{jbc} \cr& -\tfrac14 \epsilon^\beta \big[ i \Ga^j_a -\y^j_a - \bar \y^j_a \big] \cr \delta_\epsilon \grinoYY_{i}{}^{\beta j} &\simeq -i(\epsilon \sigma^{ab})^\beta \pi_{\bm 14} \FSXXYY_{ab\,i}{}^j -i (\bar \epsilon \bar \sigma^a)^\beta \left[ \hmap_i^j(\FSXYYY_{a\, klm}) +\tfrac i{6}\varphi_i{}^{jk} ( i \Ga_{a k} -\y_{a k} \right] - \tfrac 12 \epsilon^\beta \Ztorsion_i{}^j \nonumber \end{align} In these expressions, only certain $G_2$ representations of the 4-form field strength $\FS G_{\bm{abcd}}$ and dimension-1 auxiliary fields appear explicitly. For the former, these are the $\bf{14}$-dimensional part of the $[2,2]$-form component $\FSXXYY_{ab\, ij}$, and the $\bf{27}+\bf{1}$ parts of the $[1,3]$-form component $\FSXYYY_{a\, ijk}$. (The explicit definitions of these projectors are given in appendix <ref>; cf. eqs. <ref> and <ref>.) For the latter, the invariants $\R$, $\Ga_i^a$, and $\Ztorsion_i{}^j$ appearing here are combinations of the more basic auxiliary fields $\d$, $_i^a$, $_ij$, $_ijk$ fields in table \ref{T:Components} and components of the 4-form field strength. These relations are given explicitly in section \ref{S:Auxiliary} (cf.\ eqs.\ \ref{E:R}, \ref{E:Ga}, and \ref{E:Ztorsion}). \paragraph{Outline} This concludes our description of the $376\|376$-component multiplet of eleven-dimensional supergravity furnishing a linear representation of the superalgebra with four real supersymmetries. In the remainder of the paper, we will explain the derivation of this result. In the process, we will provide explicit expressions for the gauge superfields, their field strengths, and relations between these under the action by the fermionic superspace derivatives. The results are organized as follows: In the next section, we give the steps starting from the basic prepotentials and arriving at the connection and auxiliary superfields. In section \ref{S:Dynamics}, we describe the dynamical equations that result from the off-shell action derived in [16]. These equations relate auxiliary fields to physical components, thereby demonstrating the relation between on- and off-shell field content. In section \ref{S:Supergeometry}, we consolidate the results for the 4-form, torsion, and curvature field strengths in terms of supergeometry, and conclude the body of the paper. What follows are four appendices with conventions beginning with basic results for 4D, $N=1$ superspace (app.\ \ref{S:Superspace}) and the essentials of $G_2$ geometry (app.\ \ref{S:G2}). This is followed by an appendix \ref{S:Pregauge} of intermediate calculations combining the prepotentials of section \ref{S:Prepotentials} into building blocks with simplified transformations that are used to define all the other fields throughout this work. Finally, we present the result of acting with fermionic superspace derivatives in all possible ways on the gravitino components in appendix \ref{S:GrinoDescendants}. This demonstrates the consistency of the scheme and generates the supersymmetry transformations of the eleven-dimensional gravitino components reported above in \eqref{E:SupersymmetryTransformationsGravitino}. \section{Prepotentials} \label{S:Prepotentials} We now begin the task of explicitly identifying all the elements described in the previous sections. As implied in table \ref{T:Components}, we embed the components into prepotential superfields \begin{align} \label{E:Prepotentials} U^a &\sim \cdots + (\theta \sigma^m \bar \theta) \frameXX_m{}^a + \theta^2 (\sigma^m \bar \theta)_\alpha \grinoXX_m{}^\alpha + \bar \theta^2 (\theta \sigma^m )_{\dt \alpha} \bar \grinoXX_m{}^{\dt \alpha } + \theta^2 \bar \theta^2 \dX^a \cr \Psi^{\alpha i} &\sim \cdots + (\theta \sigma^m \bar \theta) \grinoXY_m{}^{\alpha i} + \theta^2 (\bar \theta \bar \sigma_a)^\alpha \y_i^a + \bar \theta^2 (\theta \sigma^{ab})^\alpha \t_{ab i} + \theta^2 \bar \theta^2 \r^i_\alpha \cr \mathcal V^i &\sim \cdots + (\theta \sigma^m \bar \theta) \frameXY_m^i + \bar \theta^2 \theta^\alpha \surfeit_\alpha^i + \theta^2 \bar \theta_{\dt \alpha}\bar \surfeit^{\dt \alpha i} + \theta^2\bar \theta^2 \dY^i \cr X &\sim \cdots + \theta^2 \bar \compX + \bar \theta^2 \compX + \theta \sigma_m \bar \theta \mathcal \, \varepsilon^{mnpq} \CXXX_{npq} + \bar \theta^2 \theta^\alpha \surfeit_\alpha + \theta^2 \bar \theta_{\dt \alpha}\bar \surfeit^{\dt \alpha} + \theta^2\bar \theta^2 \d \cr \Sigma_i^\alpha &\sim \cdots + [\theta^\alpha \compY_i + (\theta \sigma^{mn})^\alpha \CXXY_{mn \, i} ] + \theta^2 \tilde \surfeit_i^\alpha \cr V_{ij} &\sim \cdots + (\theta \sigma^m \bar \theta) \CXYY_{m\, ij} + \bar \theta^2 \theta^\alpha \spinorYY_{\alpha \, ij} + \theta^2 \bar \theta_{\dt \alpha}\bar \spinorYY_{ij}^{\dt \alpha} + \theta^2\bar \theta^2 \dYY_{ij} \cr \Phi_{ijk} &\sim \CYYY_{ijk} + i \calibration_{ijk} + \theta^\alpha \spinorYYY_{\alpha\, ijk} + \theta^2 \f_{ijk} \end{align} We recall that, in addition to the {\color{PineGreen} gauge fields} transforming canonically, there are {\color{Orange} auxiliary fields} that do not propagate, and {\color{Red} compensating components} that shift under pregauge transformations so they do not appear in the spectrum. Our main result is a kind of inversion of this embedding: We will present explicit expressions for 11-dimensional superfields containing these gauge fields and auxiliary fields as the leading term in their Taylor expansions. (To avoid introducing yet more notation, we will use the same symbols to denote the component field and the superfield having that component as the leading term in its $θ$-expansion. This should cause no confusion, as henceforth we will always mean the superfield unless explicitly stated otherwise.) There are no analogous superfields for the compensating components, precisely because these have no lift to 11D. These superfields are complicated but explicit combinations of the prepotentials that transform only under the physical part of the pregauge transformations (by which we mean those corresponding to 11D diffeomorphisms, supersymmetry, Lorentz transformations, and 3-form gauge transformations). Concretely, the gauge fields must transform as 11D supergravity gauge fields, whereas the auxiliary fields must be invariant. The procedure to construct the eleven-dimensional superfields is the following: \begin{enumerate} \item We write the linearized transformations of the prepotentials \eqref{E:Prepotentials}. These are collected in appendix \ref{S:Pregauge}. They contain abelian transformations of the M-theory 3-form, a non-abelian gauging thereof under 7D diffeomorphisms, local 4D, $N=1$ superconformal transformations, and extensions thereof ({\it e.g.}\ extended supersymmetry transformations). \item The field strengths of the non-abelian tensor hierarchy (\ref{E:ATHXf}, \ref{E:KK}) are invariant under the linearized hierarchy transformations but not under the superconformal ones. So we make combinations that are invariant under one or two of these additional parameters using as few superspace derivatives as possible. \item These partial invariants are used to covariantize (with respect to the superconformal parameters) the 3-form components we already had from the non-abelian tensor hierarchy. This results in the eleven-dimensional 3-form superfield $\physical C_{\bm{abc}}$ \eqref{E:gauge3form} with leading Taylor component the physical 3-form in the on-shell spectrum. \item In the same way, we construct the eleven-dimensional frame superfield $\physical e_{\bm a}{}^{\bm b}$ \eqref{E:11Dframe} and gravitino superfield $\physical \psi_{\bm a}{}^{\bm \beta}$ \eqref{E:grino} from the partial invariants so that they transform as they should (\ref{E:translationParameter} and \ref{E:localSupersymmetry}). Taking curls defines the dimension-3/2 torsion $\FS{T}_{\bm {ab}}{}^{\bm \gamma}$ ({\it i.e.}\ the gravitino curvature) and dimension-2 curvature $\FS{R}_{\bm {ab}}{}^{\bm {cd}}$. \item Finally, we use the partial invariants to covariantize the auxiliary fields in table \ref{T:Components}. The resulting invariants are listed in equation \ref{E:AuxiliarySuperfields}. \end{enumerate} These steps are summarized in table \ref{T:Summary}. %%%% Tabel 2 \begin{table}[t] \begin{align} {\renewcommand{\arraystretch}{1.5} %adds some padding \begin{array}{\|ccccc\|} \hline ~~~\textrm{prepotential} ~~~ & ~~~\textrm{partial invariant} ~~~ & ~~~ \textrm{gauge field} ~~~ & ~~~ \textrm{auxiliary} ~~~ & ~~~\textrm{field strength} ~~~\\ \hline & \ast & \frameXX_a{}^b , \grinoXX_a{}^\beta & \dX^a & \Torsion_{ab}{}^{\gamma}, \Curvature_{ab}{}^{cd} \\ \Psi_i^\alpha & \X_i^a , \T_i & \frameYX_i{}^a, \grinoXY_b{}^{\alpha i} &\l_i^\alpha , \y_i^a , \Ga_i^a,\t_{ab\,i},\r_i^\alpha & \Torsion_{ab}{}^{\gamma k} , \Curvature_{aj}{}^{cd} \\ \mathcal V^i & \KK_\alpha^i & \frameXY_a{}^i, \grinoYX_i{}^{\beta} & \dY^i & \Curvature_{ab}{}^{ij} \\ X & \G , \S, \P & \CXXX_{abc} & \d, \R & \FSXXXX_{abcd}\\ \Sigma_i^\alpha & \T_i & \CXXY_{ab\, i} & \textrm{none} &\FSXYYY_{abc\, i}\\ & \hatW^\alpha_{ij} & \CXYY_{a\, ij}, \spinorYY^\alpha_{ij} & \dYY_{ij} &\FSXYYY_{ab\, ij} , \Curvature_{ab\, ij}\\ \Phi_{ijk} & \Z_{ijk} , \E_{ijkl} & \CYYY_{ijk}, \frameXX_i{}^j , \spinorYYY^\alpha_{ijk} & \m_{ijk} , \f_{ijk} & \FSXYYY_{a\, ijk}, \FSYYYY_{ijkl}, \Curvature_{aj\, kl}, \Curvature_{ij\, kl}\\ \hline \end{array} \end{align} \caption{\scriptsize Prepotentials and their derived quantities. All derived quantities contain admixtures of other prepotentials, so these identifications are approximate. (The assignment of curvature tensor components to any one field is similarly ambiguous.) Explicit formul\ae{} for partial invariants in terms of the prepotentials are deferred to appendix \ref{S:Pregauge}. (Hatted quantities are $\Psi$-corrected versions of the na\"ive hierarchy field strengths.) The dimension-$1, \tfrac32, 2$ components of the field strength $\FS{G}$, torsion $\Torsion$, and curvature $\Curvature$ are the curls of the gauge fields as described in section \ref{S:Connections}. \label{T:Summary} } %end scriptsize \end{table} \subsection{Eleven-dimensional Superfields} \label{S:11DSuperfields} In this section, we define superfields that have the property that their leading component corresponds to a field in eleven dimensions. These are either physical connections (and their curls) or auxiliary fields that do not propagate. \subsubsection{Connection Superfields} \label{S:Connections} We construct the superfields carrying the eleven-dimensional 3-form, frame, and gravitino as their leading component by combining the prepotentials \eqref{E:Prepotentials} and their derivatives into superfields that transform as connections. Much work goes into this construction in which successive combinations are made to have progressively simpler transformations under the gauge and local superconformal parameters. This is outlined in appendix \ref{S:Pregauge}, with the results presented here. \paragraph{M-theory 3-form} Under dimensional reduction, the eleven-dimensional 3-form splits up into a collection of $[p,q]$ forms embedded into superfields in [20]. From this, we may extract the superfields for these components, and covariantize them with respect to the local superconformal transformations. This results in the following set of superfields: \begin{subequations} \label{E:gauge3form} \begin{align} \hyperlabel{E:CXXX} \CXXX_{abc}&:= -\tfrac14\varepsilon_{abc\,{\delta \dt \delta} } \left( [D^\delta, \bar D^{\dt \delta}] X - (D^2 +\bar D^2) U^{\delta \dt \delta} \right) \\ \hyperlabel{E:CXXY} \CXXY_{\alpha \beta \, i} &:= -\tfrac i2 D_{(\alpha} \left[ \, \Sigma_{\beta) i} +\Psi_{\beta) i}\, \right] \\ \hyperlabel{E:CXYY} \CXYY_{\alpha \dt \alpha \, ij}&:= \tfrac12 [D_\alpha, \bar D_{\dt \alpha}] V_{ij} -\varphi_{ijk} \partial_{\alpha \dt \alpha} \mathcal V^k + \varphi_{ijk} \X_{\alpha \dt \alpha}^k \\ \hyperlabel{E:CYYY} \CYYY_{ijk} &:= \tfrac12(\Phi_{ijk} + \bar \Phi_{ijk}) + \tfrac12 \psi_{ijkl} \T^l \end{align} \end{subequations} The combinations $_i^a$ and $_i$ may be thought of as partial covariantizations of the $D̅Ψ$ and $DΨ$ parts of the gravitino superfields (by the derivatives $∂_i U^a$ and $∂_i X$, respectively). They are defined explicitly in \eqref{E:X} and \eqref{E:T}, but we will not need that result. Note, however that the corrections by ${U^a, Ψ_i^α, X}$ to all the na\"ive components of the gauge 3-form are needed to cancel the local superconformal transformations of the compensating fields. With this, the corrected superfields transform as gauge $[p,q]$-forms, so we may immediately define the field strength superfield $G_abcd = 4 ∂_[a C_bcd]$ corresponding to the M-theory 4-form invariant. (An explicit set of expressions for this in terms of tensor hierarchy invariants is given in \eqref{E:4formInvariants}.) The construction given here is typical of everything that follows. We will now present the analogous results for the frame and gravitino. \paragraph{Eleven-dimensional Frame} The eleven-dimensional frame is determined by starting with the na\"ive term in the $θ$-expansion \eqref{E:Prepotentials} of the prepotentials and adding terms so that the resulting superfields transform as $δe_a^b = ∂_a ξ^b -λ_a^b$. Up to an $SO(3,1) ×SO(7)$ gauge choice and a normalization we fix by matching to reference [23], this fixes the components of the frame to be \begin{subequations} \label{E:11Dframe} \begin{align} \hyperlabel{E:frameXX} \frameXX_{\un a}{}^{\un b} &:= -\tfrac12 [D_\alpha, \bar D_{\dt \alpha} ] \, U^{\un b} -\tfrac23 \delta_\alpha^\beta \delta_{\dt \alpha}^{\dt \beta} \S \\ \hyperlabel{E:frameXY} \frameXY_{\un a}{}^{j} &:= \tfrac12 [D_\alpha, \bar D_{\dt \alpha} ] \, \mathcal V^j \\ \hyperlabel{E:frameYX} \frameYX_i{}^{\un b} &:= - \tfrac12 [ \bar D^{\dt \beta} \Psi_i^\beta - D^\beta \bar \Psi_i^{\dt \beta} \\ \hyperlabel{E:frameYY} \frameYY_{i}{}^{j} &:= \tfrac14 \varphi^{j kl} \F_{ikl} - \tfrac1{36} \delta_i^j \varphi^{klm} \F_{klm} \end{align} \end{subequations} The translation and local Lorentz parameters are shared among these components, so they are also fixed. For the translations, this gives \begin{align} \label{E:translationParameter} \xi^{\un a} :=-i ( \bar D^{\dt \alpha} L^\alpha + D^\alpha \bar L^{\dt \alpha} ) \xi^i &:= \tfrac1{2} (\tau^i +\bar \tau^i ) -\tfrac1{2i} (\Omega^i - \bar \Omega^i) \end{align} We give the explicit expressions for the local Lorentz parameter \eqref{E:localLorentz} in appendix \ref{S:GrinoDescendants}. We see explicitly from these expressions that the 4D component is as expected, and the Kaluza-Klein component is the na\"ive one. We also see that the real part of the complex vector in $D̅ Ψ$ is the ``lower-left block'' of the Kaluza-Klein decomposition. This block is usually taken to be zero, but in superspace that is only true to lowest order in the $θ$-expansion. (The lowest component of $_i^b $ vanishes in Wess-Zumino gauge. See [29] for a complete discussion of this component.) We also see explicitly that the part $_i^j$ of the linearized frame lying along the seven-dimensional space is a particular component of the partial field strength $_ijk$ for the ``scalars'' of the tensor hierarchy. (This is a partial covariantization of the imaginary part of the chiral field $Φ_ijk$, see eq.\ \ref{E:F}.) The symmetrization of this frame defines the linearized eleven-dimensional metric $h_ab : = 2e_(a^cδ_b)c$. The Riemann tensor $R_ab^cd := ∂_[a ∂^[c h_b]^d]$ is the double curl of this. The linearized curvature 2-form $R_ab^cd := 2 ∂_[a ω_b]^cd$ is the curl of the spin connection. These curvatures agree component-wise when the spin connection is taken to be torsion-free, since $T_ab^c = 0$ implies $ω_c^ab := 12 ∂^[a h_c^b]$. \paragraph{Eleven-dimensional Gravitino} The definition of the eleven-dimensional gravitino suffers from various ambiguities. Below, we present what we consider the simplest one. To do so, it is helpful to first define the components (which appear also in table \ref{T:Components}) \begin{subequations} \label{E:spinor} \begin{align} \hyperlabel{E:spinorYY} \spinorYY_{ij}^\alpha &:= \W_{ij}^\alpha +2\partial_{[i} \Psi_{j]}^\alpha +i\varphi_{ijk} (\KK^{\alpha k} -\tfrac14 \bar D^2 \Psi^{\alpha k}) \\ \hyperlabel{E:spinorYYY} \spinorYYY_{ijk}^\alpha &:= D^\alpha \left( i\F_{ijk} +\tfrac 12 \psi_{ijkl} \T^l +3 \varphi_{l[ij} \partial_{k]} \mathcal V^l\right) \end{align} \end{subequations} The first of these may be understood as the covariantization (canceling the local superconformal shifts) of the 21 gaugini in the abelian vector multiplet prepotential $V_ij$ by the Kaluza-Klein gauge field $𝒱^i$ and the gravitino superfield $Ψ_i^α$. The second is the analogous thing for the 35 spin-1/2 fields in the chiral multiplet $Φ_ijk$. (The part of $Ψ_i^α$ that enters requires the further covariantization by $Σ_i^α$ into $$.) Together, they comprise 56 spin-1/2 fields from the 4D perspective [30]. Using various $G_2$ projectors given in appendix \ref{S:G2}, these components can be reorganized into the $56 = 7+ 49$ components $ ψ_i^β, ψ_i^βj$ of the linearized eleven-dimensional gravitino. Using this, we explicitly define the eleven-dimensional gravitino components as \begin{subequations} \label{E:grino} \begin{align} \hyperlabel{E:grinoXX} \grinoXX_{\un a}{}^\beta &:=-\tfrac{i}8 \bar D^2 D^{\beta} U_{\un a} +\tfrac i{3} \delta_\alpha^\beta \bar D_{\dt \alpha} \S \\ \hyperlabel{E:grinoYX} \grinoYX_i{}^\beta &:= \tfrac1{12}\l_i^\beta-\tfrac1{12}\left[ \varphi_i{}^{jk} \spinorYY^\beta_{jk} -\tfrac 1{6} \psi_i{}^{jkl} \spinorYYY^\beta_{jkl} \right] \\ \hyperlabel{E:grinoXY} \grinoXY_{\un a}{}^{\beta j} &:= 2 D^\beta \X_{\un a}^j -\tfrac23 \delta_\alpha^\beta \left[ 2D^\gamma \X^j_{\gamma \dt \alpha} + \bar D_{\dt \alpha} \T^j +2i \bar \KK^j_{\dt \alpha} \right] \\ \hyperlabel{E:grinoYY} \grinoYY_i{}^{\beta j} \hmap_i{}^j(\spinorYYY^\beta_{klm}) -\tfrac 1{36} \varphi_i{}^{jk} \psi_k{}^{ lmn} \spinorYYY_{lmn}^\beta -\pi_{\bm{14}} ( \spinorYY^\beta_i{}^j ) \end{align} \end{subequations} (In the last component, we see explicitly the $49 = 28 + 7 + 14$ structure (recall eq.\ \ref{E:hmap}) and how the original $21$ and $35$ are mixed together through $G_2$ representations to form the $7$ and $49$.) These components are defined by their gauge transformations \begin{align} \label{E:localSupersymmetry} \delta \physical \psi_{\bm a}{}^{\bm \beta} = \partial_{\bm a} \epsilon^{\bm \beta} \epsilon^\beta := -\tfrac14 \bar D^2 L^\beta \epsilon^{\beta j} := - 2 D^\beta \Omega^j \end{align} so that their curls $T_ab^γ$ are invariant; they are the dimension-3/2 torsion components. But this condition is ambiguous: As we will explain presently (cf.\ eq.\ \ref{E:grinEoM}), there is an invariant spinor superfield $ł^i_α$ of dimension 1/2 that can be added to three of the four components defined above. The ones we have defined here are as we originally derived them. Curiously, they also turn out to be the simplest ones in the sense that they imply that all the dimension-1/2 torsions vanish, as we will see in section \ref{S:Supergeometry}. \subsubsection{Auxiliary Superfields} \label{S:Auxiliary} We will now define superfields containing the auxiliary components as the leading term of their $θ$-expansion. These ``auxiliary superfields'' will be given the same name as their leading component to avoid a further proliferation of new notation. This should cause no confusion since, throughout this paper, we will always mean the superfield unless we explicitly state otherwise. %\paragraph{Auxiliary Superfields} The definition of the dimension-1/2 gravitini was ambiguous because there is a dimension-1/2 invariant \begin{subequations} \label{E:AuxiliarySuperfields} \begin{align} \hyperlabel{E:grinEoM} \l_i^\alpha &:= 2\bar D_{\dt \alpha} \X_i^{ \un a} -D^\alpha \T_i +2i \J_i^\alpha \end{align} This field is present in any Lagrangian theory of the gravitino multiplet $\Psi_i^\alpha$ where, as a component field, it forms a Lagrange multiplier pair with a dimension-3/2 spinor $\r_i^\alpha$ to be defined below. Because of this, and as we will see explicitly in section \ref{S:Dynamics}, this superfield is proportional to the $\Psi_i^\alpha$ equation of motion. There are many more dimension-1 auxiliary fields since most 4D, $N=1$ superfields contain one. The vector multiplets all contain a real auxiliary field we denote by $\auxiliary d$. There are 4 such real prepotentials: the conformal graviton $U^a$, the Kaluza-Klein field $\mathcal V^i$, the 3-form $X$, and the abelian vectors $V_{ij}$. Their $\auxiliary d$-components are covariantized as \begin{align} \hyperlabel{E:dX} \dX^{\un a} &:=\tfrac18 D^\beta \bar D^2 D_\beta U^{\un a} + \tfrac16 [ D^\alpha, \bar D^{\dt \alpha}] \S +\partial^{\un a} \P \\ \hyperlabel{E:d} \d -\tfrac 14 (D^2 + \bar D^2) \S \\ \hyperlabel{E:dY} \dY^i &:=\tfrac13 D^\alpha \KK_\alpha^i -\tfrac23 \partial^{a} \X_{a}^i +\tfrac i{12} (D^2 - \bar D^2) \T^i \\ \hyperlabel{E:dYY} \dYY_{ij}&:= -\tfrac1{32} \psi_{ij}{}^{kl} \big[ D^\alpha \spinorYY_{\alpha \, kl} +\bar D_{\dt \alpha} \bar \spinorYY_{kl}^{\dt \alpha } \big] -\tfrac16 \varphi_{ijk} \partial^k \S +\tfrac14 \hitchin_{ijk\, lmn}\partial^k \F^{lmn} \end{align} Another familiar 4D, $N=1$ auxiliary is that of the chiral scalar field, of which we have only one such representation $\Phi_{ijk}$. Covariantizing its auxiliary $\auxiliary f$-component, we find \begin{align} \hyperlabel{E:f} \f_{ijk} &:= D^2 \Z_{ijk} +\tfrac i3 (\hitchin^{-1})_{ijk\, lmn}\varepsilon^{lmnpqrs}\partial_p \Z_{qrs} \end{align} (Here $\Z_{ijk}$ is the partial covariantization of the scalars $\F_{ijk}$ defined in \eqref{E:Z}.) Finally, there are the additional auxiliary fields of the gravitino multiplet. This multiplet is reviewed in detail in appendix C of [16]. In addition to the dimension-1/2 auxiliary field \eqref{E:grinEoM}, there are two dimension-1 components \begin{align} \hyperlabel{E:y} \y^i_{\alpha \dt \alpha}&:= - \bar D_{\dt \alpha} \l_\alpha^i \\ \hyperlabel{E:t} \t_{\alpha \beta i} &:=\tfrac12 D_{(\alpha} \KK_{\beta) i} -\tfrac i4 \big[ D_{(\beta} \bar D^{\dt \alpha} +3 \bar D^{\dt \alpha} D_{(\beta} \big] \X_{\alpha) \dt \alpha i} \cr&= -\tfrac14 \bar D^2 D_{(\alpha} \Psi_{\beta) i} +\tfrac i4 \partial_i \big[ D_{(\beta} \bar D^{\dt \alpha} +3 \bar D^{\dt \alpha} D_{(\beta} \big] U_{\alpha)\dt \alpha} + \tfrac i2 \partial_{(\beta}{}^{\dt \alpha} \gravitonXY_{\alpha)\dt \alpha \end{align} comprising a complex vector and a 2-form, and the top component of $\Psi$ \begin{align} \hyperlabel{E:r} \r^i_\alpha&:=-\tfrac14 D^\beta \bar D^2 D_{(\alpha} \Psi_{\beta)}^i - i \partial^i \big[ D_\alpha \G +\tfrac12 \bar D^{\dt \alpha} D^2 U_{\un a} \big] \end{align} \end{subequations} completing the set of auxiliary fields with a dimension-3/2 spinor. The auxiliary fields defined in this section are important because they are part of a linear representation of the subalgebra generated by 4 of the 32 supersymmetries. Many of them appear explicitly in the supersymmetry transformation \eqref{E:SupersymmetryTransformationsGravitino} of the eleven-dimensional gravitino. To compare this rule to the known on-shell transformation \eqref{E:LinSUSY} and to prove that they do not contribute propagating degrees of freedom, we examine in section \ref{S:Dynamics} how they appear in the off-shell dynamics. \paragraph{Useful Alternative Composite Fields} The supersymmetry transformations of the gravitino are mixtures of the auxiliary fields and 4-form flux. The gauge-invariant combinations that appear naturally in the differential algebra are \begin{subequations} \begin{align} \hyperlabel{E:R} \R &:= -\tfrac 16 \bar D^2 \S = \tfrac i6 \bar D^2 \P = -\tfrac 1{12} \bar D^2 [ \bar \G + i \partial_{\un a} U^{\un a} ] = \tfrac13 \d + \tfrac i{6\cdot 4!} \varepsilon^{abcd} \FSXXXX_{abcd} \\ \hyperlabel{E:Ga} \Ga^{a}_i &:= -i\bar D^2 \X^{a}_i +\tfrac i2 \bar D \bar \sigma^a D \T_i \cr &= \tfrac 1{2i} \big[ \y_i^{a} - \bar \y_i^{a} \big] +\tfrac16 \psi_{ijkl} \FSXYYY^{a jkl} +i \tilde \FSXXXY_i^{a} \\ \hyperlabel{E:m} \m^{ijk} &:= \hitchin^{ijk\, lmn} D^2 \Z_{lmn} - \tfrac 13 \epsilon^{ijk lmnp}\partial_l \bar \Phi_{mnp} %\cr & = (\hitchin)^{ijk\, lmn} \f_{lmn}-\tfrac 1{12} \varepsilon^{ijklmnp} \FSYYYY_{lmnp} \\ \hyperlabel{E:Ztorsion} \Ztorsion_i{}^j - i \hmap_i{}^j ( \f_{lmn}) +\tfrac i{36}\varphi_i{}^{jk}\psi_k{}^{lmn} \f_{lmn} -8 \pi_{\bm{14}} \dYY_i{}^j \end{align} \end{subequations} The first of these is the chiral scalar field strength of old-minimal supergravity containing the complex auxiliary field, the spin-1/2 part of the gravitino, and the curvature scalar [21, 22, 23]. In the 3-form formulation of the theory employed here, the imaginary part of the auxiliary field is dualized to the 4-form component $_abcd$ [31, 32]. The second is a complex linear field $D̅^(β ^α)α_i = 0$. In fact, $_i^αα = 12i D̅^α _i^α$, which explains why it is this combination that features prominently in the gravitino supersymmetry transformations. Expanding it out explicitly, we see that it is related to the imaginary part of the gravitino auxiliary field $_i^a$. (Note, however, that the real part of $$ does appear in the transformation of $_a^βj$ \eqref{E:SupersymmetryTransformationsGravitino} cf.\ also eq.\ \ref{E:DgrinoXY}.) The third is a modification of the auxiliary field of the scalars \eqref{E:f} by a real field strength. It is an interesting alternative that is chiral.\footnote{ Contraction with the constant $\varepsilon_{i_1\dots i_7}$ symbol defines a chiral 4-form \begin{align} \bm E_{ijkl} &:= -\tfrac 1{12} \varepsilon_{ijklmnp} \bar \m^{mnp} = 4 \partial_{[i} \Phi_{jkl]} - \tfrac1{12} \varepsilon_{ijklmnp} \bar D^2 \left[ \hitchin^{mnp\,qrs} \bar \Z_{qrs}\right] \end{align} } %end footnote The last combination $_i^j$ of auxiliary fields appears in the supersymmetry transformation of $_i^βj$ (cf.\ eq.\ \ref{E:SupersymmetryTransformationsGravitino}). Equivalently, this final invariant is the component of the dimension-1 torsion $_αj^γk =4 δ_α^γ_j^k +⋯$ that is a singlet under the 4D part of the Lorentz group (see eq.\ \ref{E:TorsionY}). In the form presented, the decomposition $49 = 28 + 7 + 14$ is evident. Of these the $28 = 27 +1$ and $7$ are complex, whereas the $14$ is real. This agrees with the counting of the first-order invariants of the $G_2$ structure, with the real $49 = 27 + 1 + 7+14$ corresponding to the four torsion forms \eqref{E:TorsionForms} and the imaginary $35= 27 + 1+7$ to the analogous derivatives of the gauge 3-form (the derivative in the $14$ is not gauge invariant). Finally, we mention for completeness that additional invariants are possible that look new, but are actually linear combinations of the invariants defined above. For example, it is easy to check that the dimension-3/2 spinor $D̅^2 D^α_i$ is invariant. It is somewhat less trivial to show that it satisfies \begin{align} \tfrac {3i}{4} \bar D^2 D_\alpha \T_i &= \r_{\alpha i} + 2 (\sigma^{ab})_\alpha{}^\beta \partial_{[a}\grinoXX_{b] \beta i} -12 (\sigma^a)_{\alpha \dt \alpha} \partial_a \bar \grinoYX_i{}^{\dt \alpha} -\partial_i \bar \grinoXX_a{}^{\dt \alpha} \end{align} We conclude that this invariant is not new, since it is a combination of the previously-defined auxiliary field $_̊i^α$ \eqref{E:r} and curls of the gravitini \eqref{E:grinoXX} and \eqref{E:grinoYX}. (The choice of the dimension-3/2 spinor depends on how we order the derivatives $D$ and $D̅$ in the definition of the auxiliary component.) This conclusion was guaranteed by the component results of [16] where a Wess-Zumino gauge analysis was carried out to identify the spectrum: Any new invariant would have to correspond to a physical component field, but this would be in contradiction with the Wess-Zumino analysis since we already accounted for all of those fields. \section{Dynamics} \label{S:Dynamics} The prepotential superfields of tabel \ref{T:Components} are not constrained except for reality and chirality conditions. Therefore, we can write an action in terms of them to determine the dynamical equations. This was done at the linearized level in [16] (and to low order in a gravitino superfield expansion in [15]).\footnote{The gravitino and Kaluza-Klein field were treated non-linearly in reference [29] by including them into the superspace geometry. In this approach $\Psi_i^\alpha$ and $\mathcal V^i$ all appear in the connections alongside $U^a$.} Defining the variation $E_ϕ := δS/δϕ$ of that action with respect to the prepotential $ϕ$, we obtain the supercurrents \begin{subequations} \label{E:SuperEoM} \begin{align} \hyperlabel{E:EoMX} \EoMX{}_a %E_{U^{a}} &= - \dX_a +\tfrac1{18} \varphi^{ijk} \FSXYYY_{a\, ijk} \\ \hyperlabel{E:EoMgrino} \EoMgrino{}^i_\alpha &= \tfrac i4 \l_\alpha^i \\ \hyperlabel{E:EoMY} \EoMY{}_i &= - 3\dY_i +\tfrac16\varphi^{jkl} \FSYYYY_{ijkl} \\ \hyperlabel{E:EoM} \EoM &= -\tfrac23 \d +\tfrac 1{72}\varphi_{ijk}\hitchin^{ijk\, lmn} (\f_{lmn} +\bar \f_{lmn}) +\tfrac 1{48}\psi^{ijkl}\FSYYYY_{ijkl} \cr &= -\tfrac23 \d -\tfrac 1{96}\varphi_{ijk}(\m^{ijk} +\bar \m^{ijk}) +\tfrac 1{96}\psi^{ijkl}\FSYYYY_{ijkl} \\ \label{E:EoMSigma} \vanishing{E_{\Sigma} }{}_\alpha^i &= \tfrac i{16}\bar D^2 \l_\alpha^i \\ \hyperlabel{E:EoMYY} \EoMYY{}^{ij} &= 2\dYY^{ij} \\ \hyperlabel{E:EoMYYY} \EoMYYY^{ijk} \tfrac i{48} \hitchin^{ijk\, lmn} \bar \f_{lmn} % -\tfrac i{2^6 \cdot 3^2} \varepsilon^{ijklmnp} \FSYYYY_{lmnp} -\tfrac i{576} \varepsilon^{ijklmnp} \FSYYYY_{lmnp} +\tfrac i6 \varphi^{ijk} \R \cr \tfrac i{48} \bar \m^{ijk} +\tfrac i{12} \varphi^{ijk} \R \end{align} \end{subequations} These are in bold font because they are gauge-invariant, and we have colored them red to indicate that setting them to zero defines the superspace mass-shell condition.\footnote{This overloading of notion should not be confused with the compensating component fields, which are unrelated and have played no role since the earliest part of section \ref{S:Prepotentials}.} This is a useful distinction because, although {\em we will continue to work off-shell}, these equations are needed to prove the claim that auxiliary fields do not contribute any new degrees of freedom on-shell: The auxiliary fields differ from components of the physical 4-form field strength only by the superfields $E_∗$ that vanish on-shell. Explicitly, we can solve these equations \eqref{E:SuperEoM} for the auxiliary fields to find \begin{subequations} \begin{align} \dX_a &=\tfrac1{18} \varphi^{ijk} \FSXYYY_{a\, ijk} - \EoMX{}_a \\ \l_\alpha^i &= -4i \EoMgrino{}_\alpha^i \\ 3\dY_i&= \tfrac16\varphi^{jkl} \FSYYYY_{ijkl} - \EoMY{}_i \\ \tfrac12 \d -\tfrac 1{96}\psi^{ijkl}\FSYYYY_{ijkl} -\tfrac i2 \varphi_{ijk} ( \EoMYYY - \overline {\EoMYYY} )^{ijk} \\ \dYY_{ij} &= \tfrac12 \EoMYY{}_{kl} \\ \hitchin^{ijk\, lmn} \f_{lmn} &= \tfrac 1{36}\left[ \varphi^{ijk}(i \varepsilon^{abcd} \FSXXXX_{abcd} + \psi^{lmnp} \FSYYYY_{lmnp} ) +3 \varepsilon^{ijklmnp} \FSYYYY_{lmnp} \right] \cr +\tfrac{4i}3 \varphi^{ijk} \varphi_{lmn} ( \EoMYYY - \overline {\EoMYYY})^{lmn} -\tfrac 83\varphi^{ijk} \EoM \end{align} \end{subequations} The auxiliary fields not appearing here ${ _i^a, _i^αβ, _̊i^α}$ are those of the gravitino superfield. However, it is not difficult to show that \begin{align} \y_{i}^a&= -2i \bar D \bar \sigma^a \EoMgrino{}_i \cr \t_{\alpha \beta i} &= -\tfrac12 \varphi_i{}^{jk} \FSXXYY_{\alpha \beta jk} - D_{(\alpha}\EoMgrino{}_{\beta) i} \cr \r_i^\alpha -\tfrac 1{12} (\bar D\bar \sigma^a)^{\alpha} \left[ \psi_i{}^{jkl} \FSXYYY_{a\, jkl} -i \varepsilon_a{}^{bcd} \FSXXXY_{bcd\, i} \right] +2i \psi_i{}^{jkl} \partial_j \spinorYY_{kl}^\alpha - i \bar D_{\dt \alpha} D^\alpha \overline \EoMgrino^{\dt \alpha}_{ i} \end{align} It follows that $$ and $$ contribute nothing new on-shell. Similarly, up to equations of motion, $$̊ is a sum of three curls, each of which is separately an invariant of dimension 3/2—that is—they are components of the gravitino curl. Finally, we note that the action of the superspace derivatives on the field strength $\FS{G}=d\physical{C}$ (i.e. its supersymmetry transformation) generate gravitino curls by (<ref>), up to supersymmetry spacetime derivatives of $\sim D\l_i $. The latter can be rewritten in terms auxiliary fields and field strengths by (<ref>) and \begin{align} D_{(\alpha} \l_{\beta)}^i & = 4i \t_{\alpha \beta}{}^i + 2i \varphi^{ijk} \FSXXYY_{\alpha \beta jk} \\ D^\alpha \l_{\alpha}^i &= 6i \dY^i + 4\varphi^{ijk} \dYY_{jk} -\tfrac i6 \psi^{ijkl} \f_{jkl} \end{align} implying that supersymmetric derivatives of the dimension-1 field strength generate no new invariants. This concludes the demonstration that the $N=1/8$ off-shell component spectrum reduces correctly to the physical spectrum if the component equations of motion are imposed by taking $\color{Red}\bm E_\ast\| \to 0$. § TOWARD ELEVEN-DIMENSIONAL OFF-SHELL SUPERGEOMETRY The analysis that leads to the results of the previous sections also implies the constraints on the supergeometry needed to solve the superspace Bianchi identities. Such a top-down approach defines the superspace torsion and curvature through the commutator \begin{align} \label{E:TorsionCurvature11D} [ \nabla_{\bm A}, \nabla_{\bm B} ] = T_{\bm A \bm B}{}^{\bm C} \nabla_{\bm C} +\tfrac12 R_{\bm A \bm B}{}^{\bm c\bm d} M_{\bm c \bm d} \end{align} where $\nabla_{\bm A} = \nabla_{\bm \alpha} , \nabla_{\bm a}$ is the covariant superspace derivative of eleven-dimensional supergravity (${}_{\bm \alpha} = {}_1,\dots,{}_{32}$ and ${}_{\bm a} = {}_0,\dots , {}_{10}$). These are subject to the Bianchi identities that follow from the nested commutator identity \begin{align} \label{E:Jacobi11D} [[ \nabla_{\bm A}, \nabla_{\bm B} ] , \nabla_{\bm C} ] ] + \textrm{permutations}= 0 \end{align} The non-trivial part says that \begin{align} \label{E:BI11D} \nabla_{[\bm C} T_{\bm A \bm B]}{}^{\bm D} +T_{[\bm A \bm B]}{}^{\bm E} T_{\|\bm E\| \bm C] }{}^{\bm D} +\tfrac12 R_{\bm A \bm B}{}^{\bm c\bm d} (\Gamma_{\bm c \bm d})_{\bm C}{}^{\bm D} = 0 \end{align} These identities become non-trivial when covariant conditions are imposed on various components of the torsion and curvature. Some of these conditions are a matter of convention that follows from the possibility of shifting the definitions of the various connections (most importantly, the spin connection).[ This can be formalized in terms of first-order-flat $G$ structures [33, 34]. The obstruction theory for this is known as Spencer cohomology (cf. e.g. Schwachhöfer's article “Holonomy Groups and Algebras” in [35]). The Spencer cohomology for 11D supergravity was considered in [36].] For eleven-dimensional supergravity in superspace, these are \begin{align} \label{E:TorsionConstraints11D} T_{\bm \alpha \bm \beta}{}^{\bm \gamma} = 0 T_{\bm \alpha \bm \beta}{}^{\bm c} = -2i (\Gamma^{\bm c} )_{\bm \alpha \bm \beta} T_{\bm a\bm b}{}^{\bm c} = 0 \end{align} Others follow from the deformation of constraints defining representations of the flat space supersymmetry algebra (most notably, chiral). There is no known analog of this type of constraint in eleven-dimensional supergravity, presumably because there are no matter multiplets with spins $ < 3/2$ in this case. Historically, appropriate torsion constraints were often found by making an Ansatz and solving the Bianchi identities. Poor Ansätze over-constrain the torsions leading to trivial solutions or on-shell geometries. We can circumvent this problem by reducing the structure group so that there are fewer torsion constraints and more possibilities for deformed representation constraints. Of course, guessing the torsion constraints is complicated by the lack of symmetry. But now we notice that we already know the solution to the linearized constraints, even though we do not know the constraints! This is because in the case of off-shell supergeometry, the solution to the superspace Bianchi identities culminates in explicit expressions for the field strengths, torsions, and curvatures in terms of unconstrained prepotentials. We already know from the derivation of the correct spectrum [16], that these must be equivalent to the prepotentials (<ref>). Starting with these, then, it must be possible to derive the constraints from this solution directly. In principle, this can be done by acting repeatedly on the prepotentials with superspace derivatives in all possible ways to generate a graded space of superfields. In this large space, we will find (among many other things) invariants and relations between them. The invariants are the aforementioned field strengths, torsions, curvatures, and their derivatives, since these are the only invariants of a superspace by definition. The relations between them will then be the (reduced) Bianchi identities. In this paper, we have taken a shortcut by building the connections and covariantizing the auxiliary fields. The remaining invariants are the curls of the connections. The constraints can then be gleaned from the spectrum of invariants and the supersymmetry transformation of the connections. Off-shell 11D, $N=1/8$ Supergeometry Let us now specialize to the supergeometry appropriate to the eleven-dimensional superspace considered in this paper. Since only $N=1/8$ is manifest, we retain from the superspace covariant derivatives above only $\{\nabla_\alpha, \bar \nabla_{\dt \alpha}, \nabla_a, \nabla_i\}$. Similarly, we keep only those parts of the Bianchi identity (<ref>) with the implied free indices. However, the on-shell eleven-dimensional torsion (<ref>) is a tensor, so any non-vanishing component of it must be retained including $T_{\alpha\, \beta j}{}^k = -2i \varepsilon_{\alpha \beta} \delta_j^k $. The implication of this is that torsion components of the form $T_{\alpha \beta j}{}^{\ast}$ can appear through the quadratic terms in (<ref>). The remaining dimension-0 constraints of relevance from (<ref>) are \begin{align} T_{\alpha \beta}{}^c = 0 T_{\alpha \dt \beta}{}^c = -2i (\sigma^c)_{\alpha \dt \beta} T_{\alpha \dt \beta}{}^k = 0 \end{align} together with their conjugates. This is consistent with our finding that there are no invariants with dimension $<1/2$. Less obvious—but not difficult to verify—is that there are also no dimension-1/2 torsions with our choice of gravitino components: In this form, the gravitino auxiliary field appears only in the dimension-1/2 component \begin{align} \FS G_{\alpha b jk} = -\tfrac16 (\sigma_b \bar \l^i)_\alpha \varphi_{ijk} \end{align} of the super-4-form field strength. This leaves the dimension-1 conventional constraints. Since there is no bosonic eleven-dimensional torsion, we have chosen the constraint accordingly: $T_{\bm{ab}}{}^{\bm c} =0$ defines the 11D spin connection in terms of the frame as we have chosen it in section <ref>. We could now begin to solve the superspace Bianchi identities subject to these conditions. This would be useful as a direct method of deriving the field strengths, perhaps even beyond their linearized approximation. Instead, we observe that the dimension-1 torsion components can be deduced directly from the linearized gravitino transformations \begin{gather} \delta_\epsilon \psi_{\bm a}{}^{\bm \beta} \sim \delta_\epsilon E_{\bm a}{}^{\bm \beta} \sim -\epsilon D E_{\bm a}{}^{\bm \beta} \sim -\epsilon^{\gamma} \Torsion_{\gamma \bm a}{}^{\bm \beta} -\epsilon^{\dt \gamma} \Torsion_{\dt \gamma \bm a}{}^{\bm \beta} \end{gather} Collecting the relevant parts from (<ref>) (or app. <ref>), we find for the $\Torsion^{\gamma}$ part \begin{gather} \label{E:TorsionX} \Torsion_{\alpha b}{}^{\gamma} = \tfrac i2 [\delta^\gamma_\alpha \eta_{bc} + (\sigma_{bc})_\alpha {}^{\gamma}] \dX^c \Torsion_{\alpha b}{}^{\dt \gamma} = \tfrac 12 (\sigma_b)_{\alpha}{}^{\dt \gamma} \R \cr \Torsion_{\alpha j}{}^{\gamma} = \tfrac i2 \delta_\alpha^\gamma \dY_j +\tfrac i{12} (\sigma^{ab})_\alpha {}^\gamma \t_{abj} \Torsion_{\alpha j}{}^{\dt \gamma} = \tfrac i{12} (\sigma_a)_\alpha{}^{\dt \gamma} \bar \Ga_j^a \end{gather} The first line is the same as those of textbook treatments of 4D, $N=1$ superspace [21, 22, 23]. They are essentially the definitions of the $\dX^a$ and $\R$ auxiliary fields that contain the 4D part of the Ricci tensor and scalar curvature in their $\theta$-expansion. We see that for D $>4$, these are extended by $\dY_i$, $\t_{ab \,k}$, and $\bar \Ga_i^a$. The analogous dimension-1 torsion $\Torsion^{\gamma k}$ with $\bm 7$-valued spinor index decomposes as \begin{align} \label{E:TorsionY} \Torsion_{\alpha b}{}^{\gamma k} &= \tfrac i{12} [5 \delta_\alpha^\gamma \eta_{bc} + 8 (\sigma_{bc})_\alpha {}^\gamma ] \bar \Ga^{ck} + \tfrac i8 \delta_\alpha^\gamma \Ga_b^k -\tfrac 18 \delta_\alpha^\gamma [ \bar D \bar \sigma_b \l^k + D\sigma_b \bar \l{}^k] \cr \Torsion_{\alpha b}{}^{\dt \gamma k} &= i(\sigma_b)_\alpha{}^{\dt \gamma} \dY^k +\tfrac i6 (4 \eta_{bc} \sigma_d - i \epsilon_{bcde} \sigma^e)_\alpha {}^{\dt \gamma} \t^{kcd} , \cr \Torsion_{\alpha j}{}^{\gamma k} &= \tfrac12 \delta_\alpha^\gamma \Ztorsion_j{}^k + i (\sigma^{ab})_\alpha{}^\gamma \pi_{\bm {14}}\FSXXYY_{ab\, j}{}^k \cr \Torsion_{\alpha j}{}^{\dt \gamma k} &= i (\sigma^a)_\alpha{}^{\dt \gamma} [ \hmap_j{}^k(\FSXYYY_{a\,lmn}) -\tfrac 1{6} \varphi_j{}^{kl} \bar \Ga_{a l} - \tfrac 16 \varphi_j{}^{kl} D_\alpha \bar \l^{\dt \gamma}_l \end{align} The new structures include the real and imaginary parts of the descendant $\y \sim \bar D\l$ of the gravitino auxiliary (which vanish on-shell), various projections of the 4-form field strength, and the superspace covariantization $\Ztorsion_i{}^j$ of the $G_2$ torsion from (<ref>). The remaining torsion component was defined already at the end of section <ref> as the spacetime curl of the eleven-dimensional gravitino. The dimension-2 invariants (curvature) were also defined in that section. In summary, we have demonstrated that the only field strengths of the eleven-dimensional supergeometry with 4 off-shell supersymmetries are the dimension 1 4-form $\FS G_{\bm{abcd}}$, the dimension-3/2 torsion $\Torsion_{\bm {ab}}{}^{\bm \gamma}$, and the dimension-2 curvature $\Curvature_{\bm{ab}}{}^{\bm{cd}}$. These are all superfields with the implied component field strengths as their leading Taylor coefficient. In addition, we find the auxiliary superfields (<ref>) with $1/2\leq$ dimension $\leq 3/2$, which are reduced dynamically to the aforementioned physical spectrum by setting the supercurrents (<ref>) to zero. § EPILOGUE We have built up the linearized tensors of eleven-dimensional supergravity directly from the unconstrained prepotentials of this theory with the structure group reduced to $SO(3,1) \times G_2$. This gives a superfield for each physical component field of eleven-dimensional supergravity. In addition, the construction provides superfields embedding the auxiliary fields needed for a multiplet furnishing a linear representation with four off-shell supersymmetries manifest. The calculations are quite cumbersome, but the results are easy to understand: By reducing the structure supergroup of eleven-dimensional supergravity, it is possible to separate the torsion constraints of that supergeometry into a part that can be solved off-shell and a remainder that is responsible for putting the theory on-shell. Ignoring the latter, and solving the former by the usual method in a top-down approach, gives a collection of so-called reduced field strength superfields. These correspond to our collection of superspace invariants as expressed in (<ref>) and (<ref>). These invariants still satisfy a large set of identities expressing, for example, superspace derivatives of one in terms of another. Solving these identities results in explicit expressions for the reduced field strengths in terms of the unconstrained superfields (<ref>). The miraculous-looking series of fortunate events required in this calculation was guaranteed to have this outcome, because we had previously shown that the set of unconstrained superfields (<ref>) correctly describes eleven-dimensional supergravity [16]. Acting repeatedly with superspace derivatives could, therefore, never generate an equation of motion or an inconsistent constraint, but was guaranteed to construct invariants corresponding to their reduced field strengths, their reduced Bianchi identities, et cetera. What was unclear from the outset, was the expression of the torsions in terms of prepotentials, and, in particular, the torsion constraints needed to begin a top-down approach to the problem. With the supergeometry torsion constraints so clarified, we could now “start over” with the usual superspace Bianchi identity analysis. The advantage of doing this would be a more direct derivation of the super-torsion, super-curvature, and super-4-form. These can be used to define higher-derivative invariants directly that can be used to study deformations of the supergeometry. Alternatively, we can proceed to construct higher-derivative deformations of the action directly by making appropriate combinations of the tensors derived herein. (This is related to the approach above, but the methods used in the construction are different.) Of most direct interest to us are the higher-derivative invariants needed to deform the two-derivative action. For M-theory, these are the ones containing $R^4$ terms constructed from the curvature 2-form and those terms related to them by supersymmetry transformations. In our hybrid superspace, these are divided into groups related by the manifest supersymmetries which are, in turn, related by the non-manifest ones. We proceed by constructing $R^4$ invariants of the former that are closed as differential forms (as needed for terms like $\sim \int C \wedge R^4$) in superspace. To check that such an approach is feasible, we have carried out the analogous calculation in a five-dimensional version of this scenario in [37]. There the invariant is of the form $\int A \wedge R^2$ where $A$ is the gravi-photon. (Such an analysis was possible because enough of the 5D hybrid supergeometry had been worked out previously in [38].) A comparison of the 11D and 5D geometries reveals that the latter is embedded in the former. This implies that the results of the 5D analysis for $R^2$ can be extended to 11D, as we intend to demonstrate explicitly in future work. § ACKNOWLEDGEMENTS It is a pleasure to thank Jim Gates, Konstantinos Koutrolikos, and Warren Siegel for illuminating discussions. This work is partially supported by National Science Foundation grant PHY-1820921. § 4D, N=1 SUPERSPACE IDENTITIES In this appendix, we rederive the $D$-identities in flat space for Minkowski superspace $\mathbf R^{4\|4}$ in the conventions of reference [23] (which, for most practical calculations, agrees with [39]). The bosonic subspace is $\mathbf R^4$ with Minkowski metric (components $\eta_{ab}$ with ${}_{a,b} = {}_{0,1,2,3}$) and a compatible $SL(2, \mathbf C)$ structure (components $\varepsilon_{\alpha \beta}$ and conjugate $\varepsilon_{\dt\alpha \dt \beta}$ with (anti-)Weyl spinor indices ${}_{\alpha, \beta, \dt \alpha, \dt \beta} = {}_{1,2}$). These are related by the Pauli matrices $(\sigma_a)_{\alpha \dt \alpha}$ which may be interpreted as the components of isomorphisms $v^a \mapsto v_{\un a} :=v_{\alpha \dt \alpha}:= (\sigma_a)_{\alpha \dt \alpha} v^a$ between vectors and hermitian matrices acting on Weyl spinors. The operation is so pervasive in superspace calculations that we use this special notation throughout the paper without qualification. Using it, the compatibility of the Minkowski and $SL(2, \mathbf C)$ structures is expressed by the Fierz identity \begin{align} \label{E:Fierz} \eta_{\un a \un b} =-2 \varepsilon_{\alpha \beta} \varepsilon_{\dt \alpha \dt \beta} (\sigma_a)_{\alpha \dt \alpha} (\sigma_b)_{\beta \dt \beta} \eta^{ab} =-2 \varepsilon_{\alpha \beta} \varepsilon_{\dt \alpha \dt \beta} \end{align} $N=1$ algebra Here we recall some formulae which all follow from the fundamental definition \begin{align} \left\{ D_\alpha , \bar D_{\dot \alpha}\right\} = -2i \partial_{\un a} \end{align} First we define $\triangle_{\un a} := \left[ D_\alpha , \bar D_{\dot \alpha}\right]$ and find \begin{align} \triangle_{\un a} = \left\{ \begin{array}{r} 2i \partial_{\un a} + 2 D_\alpha\bar D_{\dot \alpha}\\ -2i \partial_{\un a} - 2\bar D_{\dot \alpha} D_\alpha \end{array} \right. \end{align} With three $D$s we have \begin{align} [ \bar D^2, D_\alpha]= 4i \partial_{\un a} \bar D^{\dot \alpha} ~~~&,~~~ [ D^2, \bar D_{\dot \alpha}]= -4i \partial_{\un a} D^{\alpha} \\ \label{E:DDD} \{ \bar D^2, D_\alpha \} = - 2 \bar D_{\dot \alpha}D_{\alpha}\bar D^{\dot \alpha} ~~~&,~~~ \{ D^2, \bar D_{\dot \alpha} \} = - 2 D^{\alpha}\bar D_{\dot \alpha}D_{\alpha} \\ \label{E:Diamond} D^\alpha (D_\alpha \bar D_{\dt \alpha} + 2\bar D_{\dt \alpha} D_\alpha ) &= -\bar D_{\dt \alpha} D^2 \\ (D_\alpha \bar D_{\dt \alpha} + 2\bar D_{\dt \alpha} D_\alpha) \bar D^{\dt \alpha} &= - \bar D^2 D_\alpha \\ (D_\alpha \bar D_{\dt \alpha} + 2\bar D_{\dt \alpha} D_\alpha) D^\beta \Psi_\beta &= D^2 \bar D_{\dt \alpha} \Psi_\alpha + 2 D^\beta \bar D_{\dt \alpha} D_{(\alpha} \Psi_{\beta)} \end{align} and usual ones with four \begin{align} \left\{ D^2, \bar D^2\right\}= 2D^\gamma\bar D^2 D_\gamma +16 \Box ~~~&,~~~ \left[ D^2, \bar D^2\right] = -4i \partial^{\un a} \triangle_{\un a} \end{align} \begin{align} D_{(\alpha} \bar D^2 D_{\beta)}= -2i \partial_{(\alpha}{}^{\dt \gamma} \triangle_{\beta) \dt \gamma} \bar D_{(\dt \alpha} D^2 \bar D_{\dt \beta)}= - 2i \partial^\gamma{}_{(\dt \alpha} \triangle_{\gamma \dt \beta)} \end{align} There are other useful identities involving commutators. First of all, there is the symmetric part of the product \begin{align} \label{CommSymm} \left\{ \triangle_{\un a}, \triangle_{\un b}\right\} &= -8 \partial_{\un a} \partial_{\un b} + 2 \varepsilon_{\alpha \beta} \varepsilon_{\dot \alpha \dot \beta} D^\gamma \bar D^2 D_\gamma \end{align} and the anti-symmetric part \begin{align} \label{CommAnti} \left[ \triangle_{\un a}, \triangle_{\un b}\right] &= 2\varepsilon_{\dot \alpha \dot \beta} D_{(\alpha} \bar D^2 D_{\beta)} - 2 \varepsilon_{\alpha \beta}\bar D_{(\dot \alpha} D^2 \bar D_{\dot \beta)} \cr &=- 4 i \varepsilon_{\dot \alpha \dot \beta}\partial_{(\alpha}{}^{\dot \gamma}\triangle_{\beta)\dot \gamma} + 4 i \varepsilon_{\alpha \beta}\partial^\gamma{}_{(\dot \alpha}\triangle_{ \gamma \dot \beta)} \end{align} This operator maps $V\mapsto \widetilde F_{ab}(V)$. The imaginary operator \begin{align} -4i \partial_{[{\un a}} \triangle_{{\un b}]} &= \varepsilon_{\dot \alpha \dot \beta} D_{(\alpha} \bar D^2 D_{\beta)} +\varepsilon_{\alpha \beta}\bar D_{(\dot \alpha} D^2 \bar D_{\dot \beta)}\cr &=-2i \varepsilon_{\dot \alpha \dot \beta}\partial_{(\alpha}{}^{\dot \gamma}\triangle_{\beta)\dot \gamma} -2i \varepsilon_{\alpha \beta}\partial^\gamma{}_{(\dot \alpha}\triangle_{ \gamma \dot \beta)} \end{align} maps $V\mapsto F_{ab}(V)$ (up to factors). Finally we point out that the dimension-3 operator \begin{align} D^\gamma \bar D^2 D_\gamma \triangle_{\un a} = -8 \partial^{\un b}\partial_{[{\un a}}\triangle_{{\un b}]} \end{align} maps $V$ to the component Maxwell equation. In this sense the operator $D^\gamma \bar D^2 D_\gamma$ maps $A_a \mapsto \partial^b F_{ab}(A)$. § LIE ALGEBRA MISCELLANEA In this appendix, we review some $G_2$ linear algebra [40, 41, 42, 43, 44]. §.§ Linear Algebra Set $Y = \mathbf R^7$ and let $\varphi \in \Lambda^3(Y)$ be a fixed constant 3-form. Define the map $s: \Lambda^3(Y) \to S^2(Y)$ from 3-forms to symmetric tensors by \begin{align} \label{E:smap} s(\omega)_{ij} := -\tfrac1{144} \varepsilon^{k_1\cdots k_7} \varphi_{i k_1k_2} \varphi_{j k_3k_4} \omega_{k_5k_6k_7} = - \tfrac1{24} \varphi_{i kl} \tilde \omega^{klmn} \varphi_{j mn} \end{align} Then $s(\varphi)_{ij}$ is a symmetric bilinear form. When this form is invertible, $\varphi$ is said to be stable. A stable 3-form on the tangent spaces of $Y$ reduces the structure group $GL(7) \to G_2$ so that $Y$ is a $G_2$-structure manifold. In this case, $s_{ij}(\varphi)$ is definite, and we may assume it to be positive-definite (by flipping $\varphi\to -\varphi$, if necessary). When this is true, we may take it to be [45] \begin{align} - \varphi= e^{123} + e^{145} +e^{167} + e^{246} -e^{257} - e^{347} \end{align} Then $s$ is normalized so that $s(\varphi)_{ij} = \delta_{ij}$. Define the Hodge dual $\psi : = \ast \varphi$. (Recall $\ast^2 = \mathrm {id}$.) These tensors satisfy the algebraic identities \begin{align} &{\psi}^{ijkl}{\psi}_{i'j'k'l} = 6\delta^{i}_{[i'} \delta^{j}_{j'}\delta^{k}_{k]} -9 \delta^{[i}_{[i'}{\psi}_{j'k']}{}^{jk]} \\ {\psi}^{ijkl} {\psi}_{ijk'l'} = 8 \delta_{[k'}^k \delta_{l']}^l - 2 {\psi}_{k'l'}{}^{kl} \varphi^{ijk} \varphi_{ij'k'} = 2\delta_{[j'}^j\delta_{k']}^k - {\psi}_{j'k'}{}^{jk} \label{E:contractions7projector} \\ \label{E:psiphi} &\psi^{ijk l} \varphi_{i'j' l} = 6 \delta^{[i}_{[i'} \varphi_{j']}{}^{jk]} \\ \label{E:phiphiclutch} &\varphi^{ijk} \varphi_{ijk'} = 6\delta_{k'}^k {\psi}^{ijkl} {\psi}_{ijkl'} = 24 \delta_{l'}^l \varphi_i{}^{lm} {\psi}_{jk lm} = - 4 \varphi_{ijk} \\ \label{E:3phis} &\epsilon^{lmnpqrs} \varphi_{imn}\varphi_{jpq}\varphi_{krs} = -48 \sqrt{g}\delta^l_{(i} g_{jk)} \\ \varphi_{ijk}\epsilon^{jkl mnpq} = 10 \psi^{[lmnp}\delta^{q]}_i \psi_{ijkl}\epsilon^{jkl mnpq} = 4! \varphi^{[mnp}\delta^{q]}_i \end{align} Under the reduction $GL(7)\to G_2$, the $\mathbf{21}$-dimensional space of 2-forms on $Y$ decomposes into $G_2$ representations as $\mathbf{21}=\mathbf{7}\oplus \mathbf{14}$. Similarly, the $\mathbf{35}$-dimensional space of 3-forms on $Y$ decomposes as $\mathbf{35}=\mathbf{1}\oplus\mathbf{7}\oplus \mathbf{27}$. For any $p$-form $\omega$, let $\omega_\mathbf i := \pi_{\bm {i}} \omega$ denote the projection to the $\mathbf{i}$-dimensional representation. Explicitly, for any 2-form $\eta$ and 3-form $\omega$, \begin{align} \label{E:14to7} \pi_{\bm 7} \eta_{ij} &= \left( \tfrac13\delta_i^k \delta_j^l - \tfrac16{\psi}_{ij}{}^{kl} \right) \eta_{kl} = \tfrac16 \varphi_{ijk}\varphi^{klm} \eta_{lm} \label{E:21to14} \pi_{\bm {14}} \eta_{ij} &= \left( \tfrac23\delta_i^k \delta_j^l +\tfrac16 {\psi}_{ij}{}^{kl} \right) \eta_{kl} \label{E:35to1} \pi_{\bm {1}} \omega_{ijk} &= \tfrac1{42} \varphi_{ijk} \varphi^{i'j'k'} \omega_{i'j'k'} \label{E:35to7} \pi_{\bm {7}} \omega_{ijk} &= \left( \tfrac14\delta_i^{i'}\delta_j^{j'}\delta_k^{k'} -\tfrac38 {\psi}_{[ij}{}^{i'j'}\delta_{k]}^{k'} -\tfrac1{24}\varphi_{ijk} \varphi^{i'j'k'}\right) \omega_{i'j'k'} = -\tfrac1{24} \psi_{ijkl}\psi^{lmnp} \omega_{mnp} \label{E:35to27} \pi_{\bm {27}} \omega_{ijk} &= \left( \tfrac34\delta_i^{i'}\delta_j^{j'}\delta_k^{k'} +\tfrac38 {\psi}_{[ij}{}^{i'j'}\delta_{k]}^{k'} +\tfrac1{56}\varphi_{ijk} \varphi^{i'j'k'}\right) \omega_{i'j'k'} \end{align} The $\bm 7$-projections of 2- and 3-forms play an important role in the gravitino analysis. We define for such projections the vectors fields[That is, for any $\eta \in \Lambda^2(Y)$ and $\omega \in \Lambda^3(Y)$, we are defining the vectors $\vec{\eta}$ and $\vec{\omega}$ on $Y$ such that \begin{align} \iota_{\vec{\eta}} \varphi = \pi_{\bm 7} \eta \iota_{\vec{\omega}} {\psi} = 2 \pi_{\bm 7} \omega \end{align} \begin{align} \label{E:7components2} \eta^i:=\tfrac16 \varphi^{ijk} \pi_{\bm 7} \eta_{jk} = \tfrac16 \varphi^{ijk} \eta_{jk} ~~~&\Leftrightarrow~~~ \pi_{\bm 7} \eta_{ij} = \varphi_{ijk} \eta^k \\ \label{E:7components3} \omega^i := \tfrac1{12} {\psi}^{ijkl} \pi_{\bm 7} \omega_{jkl}= \tfrac1{12}{\psi}^{ijkl} \omega_{jkl} \pi_{\bm 7} \omega_{ijk} = -\tfrac12 {\psi}_{ijkl} \omega^l \end{align} Note that this implies that there are conversion factors in squares \begin{align} \label{E:squares} (\eta^i)^2 = \tfrac16 (\pi_{\bm 7} \eta_{ij})^2 (\omega^i)^2 = \tfrac16 (\pi_{\bm 7} \omega_{ijk})^2 \end{align} The dual 4-form ${\psi}$ acts on 2-forms as ${\psi}_{ij}{}^{kl} \eta_{\bm 7 kl} = -4 \eta_{\bm 7 ij}$ and ${\psi}_{ij}{}^{kl} \eta_{\bm {14} kl} = 2\eta_{\bm {14} ij}$, or \begin{align} \label{E:2formSquare} {\psi}^{ij kl} \eta_{ij} \eta_{kl} = -4 \eta_{\bm 7 ij}^2 + 2 \eta_{\bm {14} ij}^2 = -24 (\eta^i)^2 + 2 \eta_{\bm {14} ij}^2 \end{align} Hitchin metric The co-calibration with raised indices $\psi^{ijkl} = \tfrac1{3!} \epsilon^{ijklmnp} \varphi_{mnp}$ acts as a(n indefinite) metric on the space of 2-forms on $Y$. There is also a natural (indefinite) metric on the space of 3-forms resulting from the second variation of the Hitchin functional with respect to $\varphi$ \begin{align} \hyperlabel{E:HitchinMetric} \hitchin^{ijk}{}_{lmn}&:= \tfrac12 \delta^{[i}_{[l}\delta^j_m\delta^{k]}_{n]} + \tfrac1{36} \varphi^{ijk}\varphi_{lmn} +\tfrac 34 \delta^{[k}_{[l} \psi^{ij]}{}_{mn]} \end{align} to which we will refer as the Hitchin metric.[This definition differs from that of our previous papers by a burdensome factor of 18.] In terms of projectors (<ref>)\|(<ref>), it is given by $\hitchin = - \tfrac43 \pi_{\bm {1}} -\pi_{\bm {7}} +\pi_{\bm {27}}$. Its partial contractions satisfy \begin{align} \varphi_{ijk} \hitchin^{jkl\, mnp} &= - \tfrac43\delta_i^{[l}\varphi^{mnp]} \\ \psi_{ijkl} \hitchin^{jkl\, mnp} &= - \psi_i{}^{mnp} \\ \varepsilon_{ijklqrs} \hitchin^{qrs\, mnp} \tfrac12 \varepsilon_{ijkl}{}^{mnp} +\tfrac16 \psi_{ijkl}\varphi^{mnp} +18 \delta_{[i}^{[m}\delta_{j}^{n} \varphi_{kl]}{}^{p]} \end{align} Torsion of the $G_2$ structure As we have reviewed, a stable 3-form $\varphi$ on a smooth 7-manifold $Y$ defines a Riemannian metric $g(\varphi)$ and, therefore, a unique compatible torsion-free connection $\nabla$. The exterior derivative of the calibration $\nabla_m \varphi_{ijk}$ need not vanish; without loss of generality, it can be parameterized as \begin{gather} \nabla_m \varphi_{ijk} = \psi_{ijkl}T^l_m \end{gather} in terms of a $7\times 7$ matrix called the torsion of the $G_2$ structure [43]. Under the $G_2$ action, this torsion decomposes into $\bm {49}=\bm {1}+\bm {7}+\bm {14}+\bm {27}$. Since the connection is torsion-free, antisymmetrizing all indices reduces the left-hand side to the exterior derivative of the calibration. This is a generic 4-form which may be expanded as \begin{align} d \varphi = \tau_0 \psi + 3\tau_1 \wedge \varphi + \ast \tau_3 \end{align} where $\tau_\mu$ is a $\mu$-form with $\mu=0,1,3$ in the $\bm {1}$-, $\bm {7}$-, and $\bm {27}$-dimensional representations, respectively. (Evidently the $14$-dimensional part drops out.) Similarly, the differential of the co-calibration can be expanded as \begin{align} d\psi = 4 \tau_1 \wedge \psi + \tau_2 \wedge \varphi \end{align} in terms of a 1-form and a 2-form corresponding to $\bm {7}$- and $\bm {14}$-dimensional representations. The 1-forms in these expansions are the same because $\psi^{ijkl} \partial_{[m} \psi_{ijkl]}$ is proportional to $\varphi^{ijk}\partial_{[m}\varphi_{ijk]}$. It follows that the $49$ components of the $G_2$ torsion $T_i^j$ may be expressed instead in terms of four $\mu$-forms with $\mu=0,1,2,3$. Explicitly [43] \begin{align} \label{E:Forms2Torsion} (\tau_0) &= \tfrac{4}7 T_k{}^k \cr (\tau_1)_i &= - \tfrac16 \varphi_{ij}{}^k T_k{}^j \cr (\tau_2)_{ij} &= -\tfrac43 T_{[i}{}^k\delta_{j]k} +\tfrac13 \psi_{ijk}{}^l T_l{}^k \cr (\tau_3)_{ijk} &= -\tfrac32 (T_{[i}{}^l+T^l{}_{[i})\varphi_{jk]l} +\tfrac37 \varphi_{ijk} T_l{}^l \end{align} Note that $\tau_2$ is in the $\bm {14}$ representation since it is $6\pi_{\bm {14}}$ acting on $T$ considered as a 2-form $T_{[i}{}^k\delta_{j]k}$, and $\tau_3$ is in the $\bm {27}$-dimensional one since the trace of the symmetric part $T_{(i}{}^k\delta_{j)k}$ is projected out. \begin{align} \label{E:Torsion2Forms} T_i{}^j = \tfrac1{4} \delta_i^j (\tau_0) +\varphi_i{}^{jk} (\tau_1)_k -\tfrac1{2} (\tau_2)_i{}^j -\tfrac1{4} \varphi^{jkl} (\tau_3)_{ikl} \end{align} Linearized Riemannian metric The map (<ref>) is $s_{ij}(\omega) = -\frac16 \ast( \varphi_i \wedge \varphi_j \wedge \omega)$, but we can use the identities (<ref>) to give alternative forms for it. For example, when $\omega$ is obtained by expanding a stable 3-form around the calibration, we obtain the symmetrical expression \begin{align} \label{E:smap2} s(\omega)_{ij} = -\tfrac1{3\cdot 144}\varepsilon^{k_1\cdots k_7} \left[ \omega_{i k_1k_2} \varphi_{j k_3k_4} \varphi_{k_5k_6k_7} +\varphi_{i k_1k_2} \omega_{j k_3k_4} \varphi_{k_5k_6k_7} +\varphi_{i k_1k_2} \varphi_{j k_3k_4} \omega_{k_5k_6k_7} \right] \end{align} This is the same map, because the first and third term together are symmetric and evaluate to the second term (with a factor of 2). Going the other direction, we can use (<ref>) to show that either form reduces to the more economical \begin{align} \label{E:smap3} s(\omega)_{ij} = \tfrac1{6} \varphi_{(i}{}^{kl}\omega_{j)kl} \end{align} In this form, it is also clear from (<ref>) and (<ref>) that $s$ projects out the $\bm 7$ component of $\omega$. \begin{align} \label{E:smapKer} s \circ \pi_{\bm 7}(\omega) \equiv 0 \end{align} This is the entire kernel, as we conclude by counting dimensions. Next, define the map $t: \Lambda^1 \otimes \Lambda^\ast_1 \to \Lambda^3$ from vector-valued 1-forms to 3-forms by \begin{align} \hyperlabel{E:tmap} \tmap(M)_{ijk} := \tfrac32 M_{[i}{}^l \varphi_{jk] l} \end{align} for any matrix $M \in \Lambda^1 \otimes \Lambda^\ast_1 $. This matrix space $\Lambda^1 \otimes \Lambda^\ast_1 \simeq \Lambda^2 \oplus S^2$ so its dimension is $\bm{49} = \bm{21} \oplus \bm{28}$ whereas the image is at most $\bm{35}$-dimensional. This suggests that the $\bm{14}$-dimensional projection of the anti-symmetric part of $M$ does not survive, and it is easily checked that $(\pi_{\bm {14}} \eta)_{[i}{}^l \varphi_{jk] l} \equiv 0$ with the projectors above. Similarly $(\pi_{\bm {7}} \eta)_{[i}{}^l \varphi_{jk] l} = \eta_{[i}{}^l \varphi_{jk] l}$ as required. Composing with $s$, we find $s\circ t (M) = - \tfrac13 M_{(i}{}^k\delta_{j)k} - \tfrac16 \delta_{ij} M_k{}^k $, so the symmetric part of $M$ does not come back to itself even up to a coefficient. This suggests we define a new map \begin{align} \hyperlabel{E:hmap} h_{ij} := 3 s_{ij} -\tfrac13 \delta_{ij} \delta^{kl} s_{kl} h \circ t (M) = M_{(i}{}^k\delta_{j)k} \end{align} Note that this map projects out the $\bm 7$ representation by (<ref>). The significance of it is that it defines the graviton as a metric fluctuation from the linearized calibration. § PREPOTENTIAL TRANSFORMATIONS AND PARTIAL INVARIANTS In this section, we collect the linearized transformations of the prepotentials (<ref>) found in [16]. They contain abelian transformations of the M-theory 3-form [20], a non-abelian gauging thereof under 7D diffeomorphisms [19], local 4D, $N=1$ superconformal transformations [21], and extensions thereof (gravitino transformations) [16]. Abelian Tensor Hierarchy Under abelian 2-form symmetry only the prepotentials corresponding to the components of the 3-form transform. Explicitly, [20] \begin{align} \delta_{ath} \Phi_{ijk} &= \\ \delta_{ath} V_{ij} &= \tfrac1{2i} \left( \Lambda_{ij} - \bar \Lambda_{ij} \right) -2 \partial_{[i} u_{j]} \\ \delta_{ath} \Sigma^\alpha_i &= -\tfrac14 \bar D^2 D^\alpha u_i + \partial_i \Upsilon^\alpha \\ \delta_{ath} X &= \tfrac1{2i} \left( D^\alpha \Upsilon_\alpha - \bar D_{\dt \alpha} \bar \Upsilon^{\dt \alpha}\right) \end{align} where $\Lambda_{ij}$ is chiral, $u_i$ is real, and $\Upsilon^\alpha$ is chiral. Together they define an eleven-dimensional 2-form. The field strengths of the abelian tensor hierarchy are \begin{align} \hyperlabel{E:G} \G &= -\tfrac14 \bar D^2 X \\ \hyperlabel{E:H} \H_i &= \tfrac1{2i} \left( D^\alpha \Sigma_{\alpha i} - \bar D_{\dt \alpha} \bar \Sigma^{\dt \alpha}_i\right) -\partial_i X \\ \hyperlabel{E:W} \W_{ij}^\alpha &= -\tfrac14 \bar D^2 D^\alpha V_{ij} +2\partial_{[i} \Sigma_{j]}^\alpha \\ \hyperlabel{E:F} \F_{ijk} &= \tfrac1{2i}\left( \Phi_{ijk} - \bar \Phi_{ijk} \right) -3\partial_{[i} V_{jk]} \\ \hyperlabel{E:E} \E_{ijkl} &= ~~4\partial_{[i} \Phi_{jkl]} \end{align} Note that $\G$, $\W$, and $\E$ are chiral and that $\H$ and $$ are real. It is almost manifest that these combinations are invariant under the linearized non-abelian gauge transformations \eqref{E:ATHXf}. \paragraph{Kaluza-Klein Gauge Field} The abelian tensor hierarchy was coupled to the non-abelian Kaluza-Klein field gauging the 7D diffeomorphisms in [19]. At the linearized level, this field is described by a real superfield $𝒱^i$ carrying a 7D vector index, and transforming as \begin{align} \delta_7 \mathcal V^i = \tfrac1{2i} (\tau^i -\bar \tau^i) \bar D_{\dt \alpha} \tau^i =0 \end{align} Its field strength \begin{align} \hyperlabel{E:KK} \KK_\alpha^i &= -\tfrac14 \bar D^2 D_\alpha \mathcal V^i \end{align} is invariant and chiral. \paragraph{Extended 4D Superconformal Transformations} The 4D, $N=1$ superconformal transformations are parameterized by a spinor parameter superfield $L^α$ [21]. The prepotential $U^a$ containing the conformal part of the 4D frame suffers the pregauge transformation \begin{align} \delta U^{\alpha \dt \alpha} := \bar D^{\dt \alpha} L^\alpha - D^\alpha \bar L^{\dt \alpha} \end{align} where $U^a = U^αα := (σ̅_a)^αα U^a$ is by definition the contraction of the vector index by the Pauli matrices (this invertible operation is the same as Feynman's slash but with the diacritical mark directly on the index instead of the field). In calculations, it is useful to note that $D̅ L$ appears instead of $L$ itself, so $L^α$ has a gauge-for-gauge ambiguity corresponding to a shift by a chiral spinor field. As explained in [46], this parameter must also enter the transformation of the gravitino superfield: \begin{align} \delta \Psi_i^{\alpha} = \Xi_i^\alpha + D^\alpha \Omega_i + 2i \partial_i L^\alpha \end{align} Here $Ξ$ is chiral and $Ω$ is complex and completely unconstrained. These two parameters are the conformal supersymmetry parameters of the matter gravitino multiplet [21, 47] (see also appendix C of [16] for a review of this multiplet). In [18], it was observed that the field $X$ in the tensor hierarchy carrying 4D polarizations of the 3-form, was also coupling exactly as the chiral conformal compensator of old-minimal supergravity [32]. This implies that it must transform in a specific way under the $L^α$ part of the superconformal transformations. It was also conjectured there that all the fields of the tensor hierarchy suffer similar compensating superconformal transformations. This is needed for the spectrum of component fields to match as summarized in table \ref{T:Components}. This claim was demonstrated explicitly in [16] where it was shown that under superconformal transformations, \begin{subequations} \begin{align} \delta_{sc} \mathcal V^i &= -\tfrac12( \Omega^i + \bar \Omega^i) \\ \delta_{sc} X &= D^\alpha L_\alpha + \bar D_{\dt \alpha} \bar L^{\dt \alpha} \\ \delta_{sc} \Sigma^\alpha_i &= -\Xi^\alpha_i \\ \delta_{sc} V_{ij} &= \tfrac1{2i} \varphi_{ijk} (\Omega^k - \bar \Omega^k) \\ \delta_{sc} \Phi_{ijk} &= -\tfrac i2 \psi_{ijkl} \bar D^2 \bar \Omega^l \end{align} \end{subequations} Crucially, many of these transformations are St\"uckelberg shifts---the hallmark of compensating fields [21]. For the field strengths of the non-abelian tensor hierarchy, this gives \begin{subequations} \begin{align} \delta_{sc} \KK_\alpha^i &= \tfrac18 \bar D^2 D_\alpha ( \Omega^i + \bar \Omega^i) \\ \delta_{sc} \G &= -\tfrac14 \bar D^2 D^\alpha L_\alpha \\ \delta_{sc} \H_{i}&= -\tfrac1{2i} \left( D^\alpha \Xi_{\alpha i} - \bar D_{\dt \alpha} \bar \Xi_i^{\dt \alpha} \right) - \partial_i \left( D^\alpha L_{\alpha} + \bar D_{\dt \alpha} \bar L^{\dt \alpha} \right) \\ \delta_{sc} \W^\alpha_{ij} &= \tfrac i8 \varphi_{ijk} \bar D^2 D^\alpha (\Omega^k-\bar\Omega^k) - 2 \partial_{[i} \Xi^\alpha_{j]} \\ \delta_{sc} \F_{ijk} &= -\tfrac14 \psi_{ijkl} \left( D^2 \Omega^l + \bar D^2 \bar \Omega^l \right) +\tfrac {3i}2 \varphi_{l[ij} \partial_{k]} (\Omega^l-\bar\Omega^l) \\ \delta_{sc} \E_{ijkl} &= - 2i \psi_{m[ijk} \partial_{l]}\bar D^2 \bar \Omega^m \end{align} \end{subequations} \paragraph{Partial Invariants} These transformations can be removed by combining them with the superfields $U^a$ and $Ψ_i^α$ requiring the compensation. We do this step-wise, first defining partially-invariant building blocks. A convenient set is \begin{subequations} \begin{align} \hyperlabel{E:X} \X_i^{\alpha \dt \alpha} &:= \tfrac 1{2i} ( \bar D^{\dt \alpha} \Psi_i^\alpha + D^\alpha \bar \Psi_i^{\dt \alpha} ) - \partial_i U^{\alpha \dt \alpha} \\ \hyperlabel{E:T} \T_i &:= \tfrac1{2i} ( D^\alpha \Psi_{\alpha i} - \bar D_{\dt \alpha} \bar \Psi_i^{\dt \alpha} ) + \H_i \\ \hyperlabel{E:hatW} \hatW_{ij}^\alpha &:= \W_{ij}^\alpha +2\partial_{[i} \Psi_{j]}^\alpha \\ \hyperlabel{E:J} \J_\alpha^i &:=\KK^i_\alpha - \tfrac i2\varphi^{i jk} \hatW_{\alpha jk} + \tfrac i{12} \psi^{i jkl} \spinorYYY_{\alpha jkl} \\ \hyperlabel{E:Z} \Z_{ijk} &:= \F_{ijk} -\tfrac i2 \psi_{ijkl} \T^l - 3i \varphi_{l[ij} \partial_{k]} \mathcal V^l \end{align} \end{subequations} These combinations are invariant under the transformations generated by the $L^α$ and $Ξ_i^α$ parameters: \begin{subequations} \begin{align} \delta \X_{\alpha \dt \alpha}^i &= \tfrac1{2i} \left( \bar D_{\dt \alpha} D_\alpha \Omega^i + D_\alpha \bar D_{\dt \alpha} \bar\Omega^i \right) \\ \delta \T^{i}&= \tfrac1{2i} \left( D^2 \Omega^i - \bar D^2 \bar \Omega^i \right) \\ \delta \hatW^\alpha_{ij} &= \tfrac i8 \varphi_{ijk} \bar D^2 D^\alpha (\Omega^k-\bar\Omega^k) + 2 \partial_{[i} D^\alpha \Omega_{j]} \\ \delta \J_\alpha^i &= \tfrac14 \bar D^2 D_\alpha \left( 2 \Omega^i -\bar \Omega^i\right) \\ \delta \Z_{ijk} &= - \tfrac 12 \psi_{ijkl} D^2 \Omega^l + 3i \varphi_{l[ij}\partial_{k]}( \Omega^l -i \bar \tau^l) \end{align} \end{subequations} The first of these is a partial covariantization of the vector component of $Ψ$ by the derivative of $U^a$. Alternatively, it is the partial covariantization of the derivative of $U^a$.\footnote{The other two combinations cannot be covariantized under the $L^\alpha$ transformation in this way, because there are no fields analogous to $U^a$ and $X$ transforming into $\bar D \Psi - D\bar \Psi$ and $DL - \bar D\bar L$.} The second is an analog of this for the scalar component of $Ψ$ or the derivative of the compensator $X$. $W$ is just the covariantization of $$ with respect to the $Ξ$ parameter. Next, $$ is a combination of the spin-1/2 fields in the $7$-dimensional representation of $G_2$: the KK-ino, the abelian gaugino, and the tensorino. Finally, $$ is the complex combination of scalars appearing in the tensorino $^α_ijk$ \eqref{E:spinorYYY} covariantized by the derivative of the KK prepotential $𝒱^i$. Note that it is the only combination transforming under the $τ^i$ parameter. Additionally, we define combinations \begin{subequations} \begin{align} \hyperlabel{E:S} \S &:=\tfrac12(\G+\bar \G) -\tfrac14 [D_\alpha \bar D_{\dt \alpha} ] U^{\un a} \\ \hyperlabel{E:P} \P &:=\tfrac 1{2i} (\G-\bar \G) + \partial_a U^a \end{align} \end{subequations} transforming {\em only} under $L^α$ \begin{subequations} \begin{align} \delta \S & = \tfrac38 ( D^\alpha \bar D^2 L_\alpha + \bar D_{\dt \alpha} D^2 \bar L^{\dt \alpha} ) = -\tfrac32 ( D^\alpha \varepsilon_\alpha + \bar D_{\dt \alpha} \bar \varepsilon^{\dt \alpha} ) \\ \label{E:NNMxF} \delta \P & = \tfrac i8 ( D^\alpha \bar D^2 L_\alpha - \bar D_{\dt \alpha} D^2 \bar L^{\dt \alpha} ) = \tfrac 1{2i} ( D^\alpha \varepsilon_\alpha - \bar D_{\dt \alpha} \bar \varepsilon^{\dt \alpha} ) \end{align} \end{subequations} These are essentially a slight rewriting of the 4D, $N=1$ chiral compensator $$ that transforms nicely (as new-minimal and virial compensators [48, 49, 50]). \paragraph{4-form Field Strength} Taking combinations of superspace derivatives of these partial invariants, we can construct superfields that are invariant at the linearized level under consideration (covariant at non-linear order). For example, since the 4D, $N=1$ supersymmetry parameter $ε^α= -14 D̅^2 L^α$ is chiral, the chiral superfield $= -16 D̅^2 $, is a dimension-1 invariant. In fact, it is the linearized chiral scalar curvature invariant of old-minimal supergravity that contains the complex auxiliary scalar as the leading component in a Taylor expansion and the four-dimensional curvature scalar at higher order [21, 22]. In the modification of old-minimal supergravity employed by eleven-dimensional supergravity, the prepotential of the chiral compensator is real. The effect of this is that one of the real components of the complex auxiliary field becomes (the 4D Hodge dual of) the 4-form curl of a gauge 3-form [31, 32]. Continuing to make other dimension-1 invariants in this way, we find (among other things) the following combinations \begin{subequations} \label{E:4formInvariants} \begin{align} \hyperlabel{E:FSXXXX} \FSXXXX_{abcd}&=3i \epsilon_{abcd} (\R - \bar \R) \\ \hyperlabel{E:FSXXXY} \FSXXXY_{abc\, i} &:= \tfrac14 \epsilon_{abc\un d} \left[ (D^2 +\bar D^2) \X_{i}^{\un d} -[D^\delta, \bar D^{\dt \delta}]\T_i \right] \\ \tilde \FSXXXY^{\un a}_{i} &:= \tfrac12 [D^\alpha , \bar D^{\dt \alpha}]\T_i - \tfrac12 (D^2 +\bar D^2) \X_{i}^{\un a} \\ \hyperlabel{E:FSXXYY} \FSXXYY_{\alpha \beta ij} &:= - \tfrac i2 D_{(\alpha} \hatW_{\beta)ij} + \tfrac12 \varphi_{ijk} \partial_{(\beta}{}^{\dt \alpha} \X_{\alpha) \dt \alpha}^k \\ \hyperlabel{E:FSXYYY} \FSXYYY_{\un a ijk} &:= \tfrac i2 \left[ \bar D_{\dt \alpha} \spinorYYY_{\alpha ijk} +D_\alpha \bar \spinorYYY_{\dt \alpha ijk}\right] -3 \varphi_{l [ij} \partial_{k]} \X_{\un a}^l \cr &= -\tfrac12 \left[ \bar D_{\dt \alpha} D_\alpha \Z_{ijk} -D_\alpha \bar D_{\dt \alpha} \bar \Z_{ijk} \right] -3 \varphi_{l [ij} \partial_{k]} \X_{\un a}^l \cr &=\tfrac12 [D_\alpha, \bar D_{\dt \alpha} ] \F_{ijk} + \tfrac12 \psi_{ijkl} \partial_{\un a} \T^l - 3 \varphi_{l[ij} \partial_{k]} ( \X_{\un a}^l - \partial_{\un a} \mathcal V^l) \\ \hyperlabel{E:FSYYYY} \FSYYYY_{ijkl} &:= 2 \partial_{[i} \left[ \Phi_{jkl]} + \bar \Phi_{jkl]} + \psi_{jkl] m}\T^m \right] \end{align} \end{subequations} It can be checked that the new superfields represent the dual of the seven 3-forms $_abc i = ϵ_abcd ^d_i$, the 21 2-forms $_abij = (σ_ab)_α^β_ijβ^α+$ h.c., the 35 1-forms and the 35 0-form field strengths. Instead of doing this, we explicitly construct the gauge 3-form in section \ref{S:11DSuperfields} (cf.\ eq.\ \ref{E:gauge3form}). Taking the bosonic curl of these components gives an alternative (equivalent) derivation of the dimension-1 4-form invariants. There, we also define the other eleven-dimensional connections (\ref{E:11Dframe} and \ref{E:grino}) and, through those, the torsion and curvature invariants. The off-shell spectrum is completed by auxiliary fields that arise by acting on the connections with supersymmetry transformations. \section{Gravitino Descendants} \label{S:GrinoDescendants} The manifest supersymmetry transformations act in superspace by translations in the fermionic directions. In this section, we present the result of acting with the fermionic derivatives on the gravitino superfields \eqref{E:grino}. The process that gives rise to this result is not algorithmic: The desired result is obtained only up to unknown field-dependent gauge transformation and spin connection terms, and in a form that is not easily parsed into 4-form terms and combinations of auxiliary fields. We present the solution to this superspace crossword puzzle as irreducible projections of $D$ and $D̅$ on each of the four gravitino polarizations: The 1-form index may lie along 4D (X) or 7D (Y), and the spinor may have a $G_2$-singlet index (X) or a $7$-dimensional one (Y). The fermionic derivatives of the gravitino with polarization XX may be decomposed as %%% \grinoXX descendants \begin{subequations} \begin{align} D^{(\gamma} \grinoXX_{\un a}{}^{\beta)} &=\omega_{\un a}{}^{\beta \gamma} -\tfrac i2 \delta_\alpha^{(\beta}\varepsilon^{\gamma)\delta} \dX_{\delta\dt \alpha} \\ D_{\beta} \grinoXX_{\un a}{}^{\beta} &= \partial_{\un a} N + i \dX_{\un a} \\ \bar D^{\dt \beta} \grinoXX_{\un a}{}^{\beta} &= i\delta_\alpha^{ \beta}\delta_{\dt \alpha}^{\dt \beta} \R \end{align} \end{subequations} The YX descendants are %%% \grinoYX descendants \begin{subequations} \begin{align} D^{(\gamma} \grinoYX_i{}^{\beta)} &= \omega_i{}^{\beta \gamma} - \tfrac i6 \t_i^{\beta \gamma} \\ D_{\beta} \grinoYX_{i}{}^{\beta} &=\partial_i N + i \dY_i \\ \bar D^{\dt \beta} \grinoYX_{i}{}^{\beta } &= -\tfrac i{12} \Ga_i^{\un b} \end{align} \end{subequations} The XY gravitino is messier: %%% \grinoXY descendants \begin{subequations} \begin{align} D^{(\gamma} \grinoXY_{\un a}{}^{\beta)j} -\tfrac{2i}3 \delta_\alpha^{(\beta}\varepsilon^{\gamma)\delta} \bar \Ga^j_{\delta \dt \alpha} \\ \label{E:DgrinoXY} D_{\beta} \grinoXY_{\un a}{}^{\beta j} \partial_{\un a} N^j + 2\varphi^j{}_{kl} \omega_{\un a}{}^{kl} +\tfrac i2\Ga^j_{\un a} +\tfrac{5i}6 \bar \Ga^j_{\un a} +\tfrac1{2} \big[ \bar D_{\dt \alpha} \l^j_{\alpha} - D_{\alpha} \bar \l^j_{\dt \alpha } \big] \\ \bar D^{\dt \beta} \grinoXY_{\un a}{}^{\beta j} \partial_{\un a} N^{\un b \,j} + 2 \omega_{\un a}{}^{\un b \,j} +2i \delta_{\dt \alpha}^{\dt \beta} \t^j_{\alpha}{}^{\beta} +\tfrac {2i}3 \delta_\alpha^\beta \bar \t^j_{\dt \alpha}{}^{\dt \beta} +2i \delta_\alpha^\beta \delta_{\dt \alpha}^{\dt \beta} \dY^j \end{align} \end{subequations} Finally, the YY part gives %%% \grinoYY descendants \begin{subequations} \begin{align} D^{(\gamma} \grinoYY_{i}{}^{\beta)j} &= - {2i} \pi_{\bm {14}}( \FSXXYY^{\gamma\beta}{}_i{}^j) \\ D_{\beta} \grinoYY_{i}{}^{\beta j} &= \partial_i N^j + 2\varphi^j{}_{kl} \, \omega_i{}^{kl} \\ \bar D^{\dt \beta} \grinoYY_{i}{}^{\beta j} \partial_{i} N^{\un b j} +2\omega_i{}^{\un b j} - i\gravitonYY_i{}^j (\FSXYYY^{\un b}{}_{klm} ) +\tfrac 1{6} \varphi_i{}^{jk}\left[ \bar D^{\dt \beta} \l_k^\beta +i\Ga_k^{\un b} \right] \end{align} \end{subequations} In this derivation, the field-dependent Wess-Zumino gauge transformations are \begin{align} N &:= \tfrac 13 \S-i\P \cr &:= \tfrac16 \psi^{ijkl} \left[ \F_{jkl} -\tfrac i2 \psi_{jklm}\T^m\right] \cr N^{\un a j} &= -\gravitonXY^{\un a j} -2i \X^{\un a j} -\bar D^{\dt \alpha} \Psi^{\alpha j} + D^{\alpha} \bar \Psi^{\dt \alpha j} \end{align} We will not need this explicit form, but it is important that the normalizations of the shared parameters agree across equations (and is an important technical aid in determining the correct decomposition of the descendants). The spin connections are given in terms of the frame fields as \begin{subequations} \begin{align} \omega_{\un c}{}^{\un a\un b}&=\partial^{[\un a} h^{\un b]}_{\un c} -\partial_{\un c} e^{[\un a\un b]} \\ \hyperlabel{E:spinYXX} \spinYXX_k{}^{\alpha \beta} &= -\tfrac14 \partial^{(\alpha}{}_{\dt \beta} \gravitonXY_k^{\beta) \dt \beta} -\tfrac18 \partial_k [D^{(\alpha} , \bar D_{\dt \beta} ] U^{\beta)\dt \beta} =\tfrac14 \varepsilon_{\dt \alpha \dt \beta} \left[ \partial^{[\un a} h_k^{\un b]} -\partial_k e^{[\un a\un b]} \right] \\ \omega_{\un c}{}^{\un a \,j} &= \tfrac12 \partial^{\un a} \gravitonXY_{\un c}^j -\tfrac12 \partial^j \gravitonXX_{\un c}^{\un a} \\ \omega_i{}^{\un b j} &=\tfrac12 \partial^{\un b} \gravitonYY_i{}^{j} -\tfrac12 \partial^j \gravitonXY_i{}^{\un b} \\ \omega_{\un c}{}^{ij} &= \partial^{[i} \gravitonXY_{\un c}^{j]} \\ \omega_k{}^{ij} &= \partial^{[i} \gravitonYY_k^{j]} \end{align} \end{subequations} These transform as $δω_c^ab = ∂_c λ^ab$ with local Lorentz parameters \begin{align} \label{E:localLorentz} \lambda_{\un a}{}^{\un b} &:=\tfrac 12\left[ \delta_{\dt \alpha}^{\dt \beta} D^{(\beta} \bar D^2 L_{\alpha)} - \delta_\alpha^\beta \bar D^{(\dt \beta} D^2 \bar L_{\dt \alpha)} \right] = -2 \left[ \delta_{\dt \alpha}^{\dt \beta} D^{(\beta} \epsilon_{\alpha)} - \delta_\alpha^\beta \bar D^{(\dt \beta} \bar \epsilon_{\dt \alpha)} \right] \cr \lambda_{\un a}{}^j &:= -\tfrac12 \left[ \bar D_{\dt \alpha} D_\alpha \Omega^j - D_\alpha \bar D_{\dt \alpha} \bar \Omega^j \right] \cr \lambda_i{}^{\un b} &:= - \delta_{ij} \varepsilon^{\beta\alpha} \varepsilon^{\dt \beta \dt \alpha} \lambda_{\un a}{}^j \cr \lambda_i{}^j &= \tfrac14 \varphi_i{}^{jk} ( D^2 \Omega_k + \bar D^2 \bar \Omega_k) +\tfrac i4 \psi_i{}^{jkl} \partial_k ( \Omega_l - \bar \Omega_l ) \end{align} In these equations $Ω^j$ should be interpreted as the quantity $Ω^j -i τ̅$. 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# Integrable hierarchy for the resolved conifold Murad Alim and Arpan Saha Fachbereich Mathematik, Universität Hamburg, Bundesstr. 55, 20146, Hamburg ###### Abstract. We provide a direct proof of a conjecture of Brini relating the Gromov–Witten theory of the resolved conifold to the Ablowitz–Ladik integrable hierarchy at the level of primaries. We use a functional representation of the Ablowitz–Ladik hierarchy as well as a difference equation for the Gromov–Witten potential. We express the solution of the difference equation as an analytic function which is a specialization of a Tau function put forward by Bridgeland in the study of wall-crossing phenomena of Donaldson–Thomas invariants. ## 1\. Introduction The study of integrable structures as the underlying organizing principle of Gromov–Witten (GW) theory has been a great source of insights and interactions between various areas of mathematics such as algebraic and symplectic geometry, the study of integrable systems, and mathematical physics. The KdV integrable hierarchy for GW theory of a point was put forward in the works of Witten [Wit91] and Kontsevich [Kon92], relating the intersection theory on the Deligne–Mumford compactification of the moduli space of curves to $2$d topological gravity. For $\mathbb{P}^{1}$, the relation of GW theory to the Toda integrable hierarchy was studied in [EHY95, EY94, EYY95, EHX97, Pan00]. In [Pan00], a conjectured difference equation for the GW potential implied a corresponding difference equation for Hurwitz numbers which was proved in [Oko00]. A general construction of an integrable hierarchy for a given Gromov–Witten theory was given by Dubrovin and Zhang in [DZ01], see also [Dub14] for an overview, yet explicit examples remain sparse, see also [ADK+06] for a relation between topological string theory and integrable hierarchies. For the Calabi–Yau threefold geometry given by the resolved conifold, Brini conjectured in [Bri12] that the equivariant descendent Gromov–Witten potential of the resolved conifold with anti-diagonal $\mathbb{C}^{\times}$ action coincides with the logarithm of a tau function of the Ablowitz–Ladik (AL) hierarchy introduced in [AL75] under a suitable identification of the variables. In particular, the conjecture implies that the _primary_ GW potential of the resolved conifold (i. e. without gravitational descendants) coincides with the restriction of a tau function of the AL hierarchy to the small phase space. In [Bri12], Brini provided a proof of the coincidence of the descendent Gromov–Witten potential with the logarithm of an AL tau function to genus $1$, and of the primary Gromov–Witten potential with the restriction of the logarithm of the tau function to the small phase space up to genus $2$. A proof of the conjecture in the stationary sector to all genera was subsequently given by Takasaki in [Tak13] using a melting crystal model of Gromov–Witten theory of the resolved conifold. This was essentially a two-step proof: First, the descendent potential was identified with a tau function of the $2$d Toda hierarchy. And then, it was shown that owing to a certain factorization property which the specific tau function in question possessed, it could be interpreted as a tau function of the relativistic Toda hierarchy, a reduction of the $2$d Toda hierarchy which was known to be equivalent to the Ablowitz–Ladik hierarchy. In complementary recent developments, Bridgeland put forward in [Bri19] a Riemann–Hilbert problem associated to wall-crossing of Donaldson–Thomas invariants. This was applied in [Bri20] to the resolved conifold, putting forward a Tau function as a solution to the Riemann–Hilbert problem, which in particular provides the Gromov–Witten potential as an asymptotic expansion. In this work, we give a proof of the identification of the primary Gromov–Witten potential for both the stationary and non-stationary sectors with an AL tau function to all genera. The main ingredients of our proof are two difference equations. One of these is obtained from a certain functional representation of the AL hierarchy due to Vekslerchik [Vek98, Vek02], while the other was proved by one of us in [Ali20] by adapting ideas which appeared in the study of the exact WKB method in [IKT18]. In particular, by combining this with the results of Bridgeland in [Bri20], we are able to obtain a closed-form expression for the tau function, thus providing a new, but perhaps expected, link between integrable hierarchies of GW theory and Bridgeland’s DT Riemann–Hilbert problem. The outline of this paper is as follows: In §2, we recall notions from the Gromov–Witten theory of the resolved conifold, introduce the difference equation proved in [Ali20], and construct a solution of it using Bridgeland’s Tau function [Bri20]. In §3, we recall Vekslerchik’s functional representation of the Ablowitz–Ladik hierarchy. In §4, we show how to take the dispersionless limit of the AL hierarchy in this setting and give a version of Dubrovin’s proof of the correspondence between the equivariant genus zero Gromov–Witten potential of the resolved conifold and the dispersionless AL hierarchy adapted to our setting. Finally, in §5, we bring all the ingredients together to extend a result of Brini. ### Acknowledgements We would like to thank Andrea Brini and Jörg Teschner for correspondence and comments on the draft. This work is supported through the DFG Emmy Noether grant AL 1407/2-1. ## 2\. Gromov–Witten potential and difference equation Gromov–Witten (GW) theory of a non-singular algebraic variety $X$ is concerned with the study of integrals over the moduli spaces of maps from Riemann surfaces into $X$. Let $X$ be a Calabi–Yau threefold. The GW potential of $X$ is the following formal power series: $F(\lambda,\boldsymbol{t})=\sum_{g\geq 0}\lambda^{2g-2}F^{g}(\boldsymbol{t})=\sum_{g\geq 0}\lambda^{2g-2}\sum_{\beta\in H_{2}(X,\mathbb{Z})}N^{g}_{\beta}\,q^{\beta}\,,$ (2.1) where $q^{\beta}:=\exp(2\pi i\langle\beta,\boldsymbol{t}\rangle)$ is a formal variable living in a suitable completion of the effective cone in the group ring of the small phase space $H_{2}(X,\mathbb{Z})$, $\lambda$ is a formal parameter corresponding to the topological string coupling, and $N^{g}_{\beta}$ are the GW invariants. The GW potential can be written as: $F=F_{\beta=0}+\tilde{F}\,,$ (2.2) where $F_{\beta=0}$ denotes the contribution from constant maps (the stationary sector) and $\tilde{F}$ the contribution from non-constant maps (the non-stationary sector). The constant-map contributions at genus 0 and 1 are $\boldsymbol{t}$ dependent and the higher-genus constant-map contributions take the universal form [FP00]: $F_{\beta=0}^{g}=\frac{(-1)^{g-1}\,\chi(X)\,B_{2g}\,B_{2g-2}}{4g(2g-2)\,(2g-2)!}\,,\quad g\geq 2\,,$ (2.3) where $\chi(X)$ is the topological Euler characteristic of $X$ and $B_{k}$ denotes the $k$-th Bernoulli number. This note is concerned with an equivariant version of the GW potential of the CY threefold given by the total space of a rank two bundle over the projective line: $\mathcal{O}(-1)\oplus\mathcal{O}(-1)\rightarrow\mathbb{P}^{1}\,.$ (2.4) This corresponds to the resolution of the conifold singularity in $\mathbb{C}^{4}$ and is known as the resolved conifold. Its small phase space is spanned by an identity element $\mathrm{id}$ and the Kähler class $\omega$ of the base $\mathbb{P}^{1}$. The associated coordinates shall be denoted $x:=\langle\mathrm{id},\boldsymbol{t}\rangle$ and $t:=\langle\omega,\boldsymbol{t}\rangle$ respectively. The (non-equivariant) GW potential for this geometry was determined in physics [GV98, GV99], and in mathematics [FP00] with the following outcome for the non-constant maps: $\tilde{F}^{0}=\mathrm{Li}_{3}(q)\,,\quad\tilde{F}^{g}=\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\mathrm{Li}_{3-2g}(q)\,,\quad g\geq 1\,,$ (2.5) where $q:=\exp(2\pi{\mathrm{i}}t)$ and the polylogarithm is defined by: $\mathrm{Li}_{s}(z)=\sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}\,,\quad s\in\mathbb{C}\,.$ (2.6) There is an anti-diagonal $\mathbb{C}^{\times}$ action on $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$, given by fibrewise $\mathbb{C}^{\times}$ action on the $\mathcal{O}(-1)$ with opposite characters. This $\mathbb{C}^{\times}$ action is especially interesting as it acts trivially on the canonical bundle, meaning that $\mathcal{O}(-1)\oplus\mathcal{O}(-1)$ is then _equivariantly_ Calabi–Yau. With respect to this action, one can define an equivariant analogue of Gromov–Witten theory where the ordinary cohomology ring of the resolved conifold is replaced by the equivariant cohomology. The equivariant Gromov–Witten invariants of rank $2$ bundles over curves were computed by Bryan and Pandharipande in [BP08]. The theory significantly simplifies in the case of the resolved conifold with anti-diagonal action: It is then just the contribution from constant maps which is affected. In particular, the genus zero Gromov–Witten potential becomes: $F_{\mathrm{ad}}^{0}=\frac{(2\pi)^{3}{\mathrm{i}}x^{2}t}{2\kappa^{2}}+\tilde{F}^{0},$ (2.7) where the $\mathrm{ad}$ in the subscript denotes that it is equivariant with respect to the _anti-diagonal_ action and $\kappa$ is the equivariant parameter. The following theorem was proved in [Ali20]: ###### Theorem 1. [Ali20] The contribution of the non-constant maps $\tilde{F}(\lambda,t)$ to the GW potential of the resolved conifold satisfies the following difference equation: $\tilde{F}(\lambda,t+\check{\lambda})-2\tilde{F}(\lambda,t)+\tilde{F}(\lambda,t-\check{\lambda})=\left(\frac{1}{2\pi}\frac{\partial}{\partial t}\right)^{2}\,\tilde{F}^{0}(t)\,,\quad\check{\lambda}=\frac{\lambda}{2\pi}\,.$ (2.8) We proceed by determining the solution of the difference equation. The latter already appears as a building block of the Tau function determined by Bridgeland in [Bri20] as a solution to a Riemann–Hilbert problem associated to wall-crossing of DT invariants of the resolved conifold. The special functions in [Bri20] involve the multiple sine functions which are defined using the Barnes multiple Gamma functions [Bar04]. For a variable $z\in\mathbb{C}$ and parameters $\omega_{1},\ldots,\omega_{r}\in\mathbb{C}^{\times}$, these are defined by: $\sin_{r}(z\,\|\,\omega_{1},\dots,\omega_{r}):=\Gamma_{r}(z\,\|\,\omega_{1},\dots,\omega_{r})\cdot\Gamma_{r}\left(\sum_{i=1}^{r}\omega_{i}-z\,\|\,\omega_{1},\dots,\omega_{r}\right)^{(-1)^{r}}.$ (2.9) For further definitions, see e. g. [Bri20, Rui00] and references therein. We will furthermore need the generalized Bernoulli polynomials, defined by the generating function: $\frac{x^{r}\,e^{zx}}{\prod_{i=1}^{r}(e^{\omega_{i}x}-1)}=\sum_{n=0}^{\infty}\frac{x^{n}}{n!}\,B_{r,n}(z\,\|\,\omega_{1},\,\dots,\omega_{r})\,.$ (2.10) Consider now the function $G(z\,\|\,\omega_{1},\omega_{2})$ of [Bri20, Sec. 4.2], defined by: $G(z\,\|\,\omega_{1},\omega_{2}):=\exp\left(\frac{\pi i}{6}\cdot B_{3,3}(z+\omega_{1}\,\|\,\omega_{1},\omega_{1},\omega_{2})\right)\cdot\sin_{3}(z+\omega_{1}\,\|\,\omega_{1},\omega_{2},\omega_{3}),$ (2.11) and define a function111The subscript np of the function stands for non- perturbative. In [Bri20], it is suggested that the Tau function of the DT Riemann–Hilbert problem gives a non-perturbative definition of the GW theory of the resolved conifold. The non-perturbative content of this function is discussed in a companion paper.: $F_{\mathrm{np}}(\lambda,t)=\log G(t\,\|\,\check{\lambda},1)\,.$ (2.12) We obtain the following: ###### Proposition 2. The function $F_{\mathrm{np}}(\lambda,t)$ is a solution of the difference equation (2.8). Moreover, $F_{\mathrm{np}}$ has an asymptotic expansion as $\lambda\rightarrow 0$ given by: $F_{\mathrm{np}}(\lambda,t)\sim\sum_{g=0}^{\infty}\lambda^{2g-2}\tilde{F}^{g}(t)\,,$ (2.13) where $\tilde{F}^{g}(t)$ are the non-constant parts of the conifold free energies defined in (2.5). The solution is furthermore unique given the above asymptotic expansion. ###### Proof. By [Bri20, Prop.4.2], the function $G(t\,\|\,\check{\lambda},1)$ is single valued for $\check{\lambda}\notin\mathbb{R}_{<0}$ and satisfies the difference equation: $\frac{G(t+\check{\lambda}\,\|\,\check{\lambda},1)}{G(t\,\|\,\check{\lambda},1)}=H(t+\check{\lambda}\,\|\,\check{\lambda},1)^{-1}\,,$ (2.14) where $H$ is denoted by $F$ in [Bri20] and is given by: $H(t\,\|\,\omega_{1},\omega_{2}):=\exp\left(-\frac{\pi i}{2}\cdot B_{2,2}(t\,\|\,\omega_{1},\omega_{2})\right)\cdot\sin_{2}(z\,\|\,\omega_{1},\omega_{2})\,.$ (2.15) This in turn satisfies the difference equation [Bri20, Prop 4.1]: $\frac{H(t+\omega_{1}\,\|\,\omega_{1},\omega_{2})}{H(t\,\|\,\omega_{1},\omega_{2})}=\frac{1}{1-x_{2}}\,,\quad x_{2}=\exp\left(\frac{2\pi it}{\omega_{2}}\right).$ (2.16) From (2.14) we obtain, by taking the logarithm: $\begin{split}&\log G(t+\check{\lambda}\,\|\,\check{\lambda},1)+\log G(t-\check{\lambda}\,\|\,\check{\lambda},1)-2\log G(t\,\|\,\check{\lambda},1)\\\ &=-\left(\log H(t+\check{\lambda}\,\|\,\check{\lambda},1)-\log H(t\,\|\,\check{\lambda},1)\right)=\log(1-q)\,,\quad q=\exp(2\pi it)\,.\end{split}$ (2.17) For the asymptotic expansion of $F_{\mathrm{np}}$, we use [Bri20, Prop 4.6], which in particular proves the asymptotic expansion of $G$ as $\omega_{2}\rightarrow 0$ to be: $\log G(z\,\|\,\omega_{2},\omega_{1})\sim\sum_{k\geq 0}\frac{(k-1)\cdot B_{k}\cdot\omega_{2}^{k-2}}{k!}\cdot\left(\frac{2\pi i}{\omega_{1}}\right)^{k-2}\textrm{Li}_{3-k}\left(\exp\left(\frac{2\pi iz}{\omega_{1}}\right)\right).$ (2.18) It follows that: $F_{\mathrm{np}}(\lambda,t)\sim\frac{1}{\lambda^{2}}\,\mathrm{Li}_{3}(q)+\sum_{g=1}^{\infty}\lambda^{2g-2}\frac{(-1)^{g-1}B_{2g}}{2g(2g-2)!}\,\textrm{Li}_{3-2g}(q)\,,$ (2.19) by noting that $B_{k}=0$ for $k>1$ odd. Finally, we prove the uniqueness of this solution given the asymptotic expansion (2.13). Let $F_{1}$ be a solution to the difference equation (2.8) satisfying (2.13). Then the difference $f(\lambda,t):=F_{1}(\lambda,t)-\log G(t\,\|\,\check{\lambda},1)$ satisfies the following equation: $f(\lambda,t+\check{\lambda})-f(\lambda,t)=f(\lambda,t)-f(\lambda,t-\check{\lambda})\,.$ (2.20) For any fixed value of $\lambda$, the function $f(\lambda,t+\check{\lambda})-f(\lambda,t)$ is periodic in $t$ with period $\check{\lambda}$. Thus, it must be of the form: $f(\lambda,t+\check{\lambda})-f(\lambda,t)=\sum_{k\in\mathbb{Z}}\lambda d_{k}(\lambda)\exp\left(\frac{2k\pi{\mathrm{i}}t}{\check{\lambda}}\right),$ (2.21) where $d_{k}(\lambda)$ are (possibly meromorphic) functions of $\lambda$. This can again be rewritten as: $\begin{split}&f(\lambda,t+\check{\lambda})-(t+\check{\lambda})\sum_{k\in\mathbb{Z}}d_{k}(\lambda)\exp\left(\frac{2k\pi{\mathrm{i}}(t+\check{\lambda})}{\check{\lambda}}\right)\\\ &=f(\lambda,t)-t\sum_{k\in\mathbb{Z}}d_{k}(\lambda)\exp\left(\frac{2k\pi{\mathrm{i}}t}{\check{\lambda}}\right).\end{split}$ (2.22) This gives another function which is periodic in $t$ with period $\check{\lambda}$. Thus, the most general $f(\lambda,t)$ satisfying this equation is: $f(\lambda,t)=\sum_{k\in\mathbb{Z}}(c_{k}(\lambda)+td_{k}(\lambda))\exp\left(\frac{2k\pi{\mathrm{i}}t}{\check{\lambda}}\right),$ (2.23) where $c_{k}(\lambda),d_{k}(\lambda)$ are (possibly meromorphic) functions of $\lambda$. Since $F_{1}(\lambda,t)$ and $\log G(t\,\|\,\check{\lambda},1)$ both satisfy (2.13) as $\lambda\rightarrow 0$, the functions $c_{k}(\lambda)$ and $d_{k}(\lambda)$ are forced to vanish. This implies $F_{1}(\lambda,t)=\log G(t\,\|\,\check{\lambda},1)$ and that the solution is indeed unique. ∎ ###### Corollary 3. The equivariant Gromov–Witten potential of the resolved conifold with anti- diagonal action is: $F_{\mathrm{ad}}(\lambda,t)=\frac{(2\pi)^{3}{\mathrm{i}}x^{2}t}{2\kappa^{2}\check{\lambda}^{2}}+\log G(t\,\|\,\check{\lambda},1)\,.$ (2.24) ## 3\. The Ablowitz–Ladik hierarchy The AL hierarchy [AL75] is a set of infinitely many non-linear differential- difference equations in two functions $a,b\colon\mathbb{Z}\rightarrow\mathbb{C}\llbracket z,\tilde{z}\rrbracket$, where $z$ and $\tilde{z}$ represent two countably infinite tuples of (complex) flow parameters $z_{i}$ and $\tilde{z}_{i}$ respectively. The simplest non- trivial flow equation in the hierarchy is the AL system: $\begin{split}{\mathrm{i}}\dot{a}&=\phantom{+}(\Lambda(a)+\Lambda^{-1}(a))(1-ab)\,,\\\ {\mathrm{i}}\dot{b}&=-(\Lambda(b)+\Lambda^{-1}(b))(1-ab)\,.\end{split}$ (3.1) Here the dot denotes derivative with respect to a linear combination of the flow parameters in the tuples $z$ and $\tilde{z}$, while $\Lambda$ denotes the shift operator whose action on any function $u$ on $\mathbb{Z}$ is given by $(\Lambda(u))(n)=u(n+1)\,.$ (3.2) One way of constructing the full AL hierarchy is by means of three tau functions $\sigma,\rho,\tau\colon\mathbb{Z}\rightarrow\mathbb{C}\llbracket z,\tilde{z}\rrbracket$ associated to a solution $(a,b)$ of the AL system. These are _defined_ by $a=\frac{\sigma}{\tau}\,,\quad b=\frac{\rho}{\tau}\,,\quad 1-ab=\frac{\Lambda(\tau)\Lambda^{-1}(\tau)}{\tau^{2}}\,.$ (3.3) Note that the tau functions are not independent but satisfy $\Lambda(\tau)\Lambda^{-1}(\tau)=\tau^{2}-\sigma\rho\,.$ (3.4) The AL hierarchy is then equivalent to the following system of Hirota bilinear equations (cf. Vekslerchik [Vek98, Vek02]): $\displaystyle\tau(z,\tilde{z})\tau(z+{\mathrm{i}}[\zeta],\tilde{z})-\rho(z,\tilde{z})\sigma(z+{\mathrm{i}}[\zeta],\tilde{z})$ $\displaystyle=\phantom{\zeta}\Lambda^{-1}(\tau)(z,\tilde{z})\Lambda(\tau)(z+{\mathrm{i}}[\zeta],\tilde{z})\,,$ (3.5a) $\displaystyle\tau(z,\tilde{z})\sigma(z+{\mathrm{i}}[\zeta],\tilde{z})-\sigma(z,\tilde{z})\tau(z+{\mathrm{i}}[\zeta],\tilde{z})$ $\displaystyle=\zeta\Lambda^{-1}(\tau)(z,\tilde{z})\Lambda(\sigma)(z+{\mathrm{i}}[\zeta],\tilde{z})\,,$ (3.5b) $\displaystyle\rho(z,\tilde{z})\tau(z+{\mathrm{i}}[\zeta],\tilde{z})-\tau(z,\tilde{z})\rho(z+{\mathrm{i}}[\zeta],\tilde{z})$ $\displaystyle=\zeta\Lambda^{-1}(\rho)(z,\tilde{z})\Lambda(\tau)(z+{\mathrm{i}}[\zeta],\tilde{z})\,,$ (3.5c) $\displaystyle\tau(z,\tilde{z})\tau(z,\tilde{z}+{\mathrm{i}}[\zeta])-\rho(z,\tilde{z})\sigma(z,\tilde{z}+{\mathrm{i}}[\zeta])$ $\displaystyle=\phantom{\zeta}\Lambda(\tau)(z,\tilde{z})\Lambda^{-1}(\tau)(z,\tilde{z}+{\mathrm{i}}[\zeta])\,,$ (3.5d) $\displaystyle\tau(z,\tilde{z})\sigma(z,\tilde{z}+{\mathrm{i}}[\zeta])-\sigma(z,\tilde{z})\tau(z,\tilde{z}+{\mathrm{i}}[\zeta])$ $\displaystyle=\zeta\Lambda(\tau)(z,\tilde{z})\Lambda^{-1}(\sigma)(z,\tilde{z}+{\mathrm{i}}[\zeta])\,,$ (3.5e) $\displaystyle\rho(z,\tilde{z})\tau(z,\tilde{z}+{\mathrm{i}}[\zeta])-\tau(z,\tilde{z})\rho(z,\tilde{z}+{\mathrm{i}}[\zeta])$ $\displaystyle=\zeta\Lambda(\rho)(z,\tilde{z})\Lambda^{-1}(\tau)(z,\tilde{z}+{\mathrm{i}}[\zeta])\,.$ (3.5f) In the above, $[\zeta]$ denotes the tuple: $[\zeta]:=\left(\zeta,\frac{\zeta^{2}}{2},\frac{\zeta^{3}}{3},\ldots\right).$ (3.6) The flows of the hierarchy are then obtained by expanding the above set of equations order by order in $\zeta$. In particular, the AL system is obtained by taking the flow along $\partial_{z_{1}}+\partial_{\tilde{z}_{1}}$. ## 4\. Dispersionless limit The dispersionless limit of any integrable hierarchy is a limit in which the contribution of higher-derivative terms (which cause dispersion) are suppressed. This is achieved by scaling the length and times (i. e. the flow parameters) by a factor $\check{\lambda}^{-1}$ inversely proportional to a dispersion parameter $\lambda$, and then taking the limit $\lambda\rightarrow 0$. Expanding the relevant flow equations as Taylor series in $\lambda$ about $\lambda=0$ then gives the small-dispersion expansion of the hierarchy. The most straightforward way to do this in the case of the AL hierarchy is to work at the level of tau functions. That is to say, we introduce tau functions $\sigma_{\lambda},\rho_{\lambda},\tau_{\lambda}\colon\mathbb{C}\rightarrow\mathbb{C}\llbracket z,\tilde{z}\rrbracket$ of a continuous complex variable $x$, parametrised by $\lambda$. (It is not a coincidence that the spatial coordinate is denoted with the same symbol as the group ring parameter associated to the identity class in $H_{2}(X,\mathbb{Z})$; we will later see that they can be identified.) The dependence of the tau functions on $\lambda$ is stipulated to take the following form: $\begin{split}\log\sigma_{\lambda}=\frac{\varpi_{\lambda}}{\check{\lambda}^{2}}+\frac{s_{\lambda}}{\check{\lambda}}+r_{\lambda}\,,\quad\log\rho_{\lambda}=\frac{\varpi_{\lambda}}{\check{\lambda}^{2}}-\frac{s_{\lambda}}{\check{\lambda}}+r_{\lambda}\,,\quad\log\tau_{\lambda}&=\frac{\varpi_{\lambda}}{\check{\lambda}^{2}}\,.\end{split}$ (4.1) Here, $\varpi_{\lambda},s_{\lambda},r_{\lambda}$ are analytic in $\lambda$. So, they have well-defined limits $\varpi,s,r$ as $\lambda\rightarrow 0$. We also introduce the operator $\Lambda_{\lambda}:=\exp(\check{\lambda}\partial_{x})$. The small-dispersion expansion of the AL hierarchy is then given by making the following replacements in (3.5): $\sigma\mapsto\sigma_{\lambda}\,,\quad\rho\mapsto\rho_{\lambda}\,,\quad\tau\mapsto\tau_{\lambda}\,,\quad\Lambda\mapsto\Lambda_{\lambda}\,,\quad[\zeta]\mapsto\check{\lambda}[\zeta]\,.$ (4.2) In the conjectured correspondence between GW theory and integrable hierarchies, the dispersion parameter $\lambda$ plays the role of the genus expansion parameter. In particular, the genus zero GW theory should correspond to the dispersionless limit of the associated hierarchy. In order to relate the dispersionless limit of the AL hierarchy to the resolved conifold, we will first identify a suitable set of dependent variables and describe the dynamical equations of the hierarchy in terms of them. ###### Proposition 4. The dispersionless limit of the AL hierarchy is given by the following system of equations in the functions $v=\partial_{x}s$ and $u=-\log(1-e^{2r})$: $\displaystyle D^{\zeta}_{z}v$ $\displaystyle={\mathrm{i}}\,\frac{\partial}{\partial x}\log\left(\frac{1-\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}{2}\right),$ (4.3a) $\displaystyle D^{\zeta}_{z}u$ $\displaystyle={\mathrm{i}}\,\frac{\partial}{\partial x}\log\left(\frac{1+\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}{2}\right),$ (4.3b) $\displaystyle D^{\zeta}_{\tilde{z}}v$ $\displaystyle={\mathrm{i}}\,\frac{\partial}{\partial x}\log\left(\frac{1-\zeta e^{-v}+\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}{2}\right),$ (4.3c) $\displaystyle D^{\zeta}_{\tilde{z}}u$ $\displaystyle=-{\mathrm{i}}\,\frac{\partial}{\partial x}\log\left(\frac{1+\zeta e^{-v}+\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}{2}\right),$ (4.3d) where $D^{\zeta}_{z}$ and $D^{\zeta}_{\tilde{z}}$ denote the operators: $D^{\zeta}_{z}=\sum_{j=1}^{\infty}\frac{\zeta^{j}}{j}\frac{\partial}{\partial z_{j}}\,,\quad D^{\zeta}_{\tilde{z}}=\sum_{j=1}^{\infty}\frac{\zeta^{j}}{j}\frac{\partial}{\partial\tilde{z}_{j}}\,.$ (4.4) ###### Proof. Making the replacements (4.2) in (3.5) and dividing equations (3.5a) and (3.5b) by the respective first terms on the left-hand side, we get: $\displaystyle 1-\left(\frac{\rho_{\lambda}}{\tau_{\lambda}}\right)(z,\tilde{z})\left(\frac{\sigma_{\lambda}}{\tau_{\lambda}}\right)(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})$ $\displaystyle\quad=\left(\frac{\Lambda^{-1}_{\lambda}(\tau_{\lambda})}{\tau_{\lambda}}\right)(z,\tilde{z})\left(\frac{\Lambda_{\lambda}(\tau_{\lambda})}{\tau_{\lambda}}\right)(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\,,$ (4.5a) $\displaystyle 1-\left(\frac{(\sigma_{\lambda}/\tau_{\lambda})(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{(\sigma_{\lambda}/\tau_{\lambda})(z,\tilde{z})}\right)^{-1}$ $\displaystyle\quad=\zeta\left(\frac{\Lambda^{-1}_{\lambda}(\tau_{\lambda})}{\tau_{\lambda}}\right)(z,\tilde{z})\left(\frac{\Lambda_{\lambda}(\tau_{\lambda})}{\tau_{\lambda}}\right)(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\left(\frac{\Lambda_{\lambda}(\sigma_{\lambda}/\tau_{\lambda})}{\sigma_{\lambda}/\tau_{\lambda}}\right)(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\,.$ (4.5b) Substituting (4.1) into the above then gives us: $\displaystyle 1-\exp\left(\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})-s_{\lambda}(z,\tilde{z}))}{\check{\lambda}}\right)\exp\left(r_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})+r_{\lambda}(z,\tilde{z})\right)$ $\displaystyle\quad=\exp\left(\frac{(\Lambda_{\lambda}^{-1}-1)(\varpi_{\lambda})(z,\tilde{z})+(\Lambda_{\lambda}-1)(\varpi_{\lambda})(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{\check{\lambda}^{2}}\right),$ (4.6a) $\displaystyle 1-\exp\left(-\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})-s_{\lambda}(z,\tilde{z}))}{\check{\lambda}}\right)\exp\left(-r_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})+r_{\lambda}(z,\tilde{z}))\right)$ $\displaystyle\quad=\zeta\exp\left(\frac{(\Lambda_{\lambda}^{-1}-1)(\varpi_{\lambda})(z,\tilde{z})+(\Lambda_{\lambda}-1)(\varpi_{\lambda})(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{\check{\lambda}^{2}}\right)$ $\displaystyle\quad\quad\times\exp\left((\Lambda_{\lambda}-1)\left(\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{\check{\lambda}}+\pi{\mathrm{i}}t_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\right)\right).$ (4.6b) For convenience, we now introduce the following notation: $\begin{split}X(z,\tilde{z})&=\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})-s_{\lambda}(z,\tilde{z}))}{\check{\lambda}}\,,\\\ Y(z,\tilde{z})&=\frac{r_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})-r_{\lambda}(z,\tilde{z})}{\check{\lambda}}\,,\\\ Z(z,\tilde{z})&=(\Lambda_{\lambda}-1)\left(\frac{s_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})}{\check{\lambda}}+r_{\lambda}(z+{\mathrm{i}}\check{\lambda}[\zeta],\tilde{z})\right).\end{split}$ (4.7) Eliminating $\varpi_{\lambda}$ from (4.6a) and (4.6b) gives us a quadratic equation in $e^{X(z,\tilde{z})}$: $\zeta e^{Z(z,\tilde{z})}e^{\check{\lambda}Y(z,\tilde{z})}e^{2r_{\lambda}(z,\tilde{z})}e^{2X(z,\tilde{z})}+\left(1-\zeta e^{Z(z,\tilde{z})}\right)e^{X(z,\tilde{z})}-e^{-\check{\lambda}Y(z,\tilde{z})}=0\,.$ (4.8) This has two solutions but only one such that $X(z,\tilde{z})$ has a well- defined limit as $\zeta\rightarrow 0$: $X(z,\tilde{z})=\log\left(\frac{-1+\zeta e^{Z(z,\tilde{z})}+\sqrt{(1-\zeta e^{Z(z,\tilde{z})})^{2}+4\zeta e^{Z(z,\tilde{z})}e^{2r_{\lambda}(z,\tilde{z})}}}{2\zeta e^{Z(z,\tilde{z})}e^{\check{\lambda}Y(z,\tilde{z})}e^{2r_{\lambda}(z,\tilde{z})}}\right).$ (4.9) In the dispersionless limit $\lambda\rightarrow 0$, this becomes: ${\mathrm{i}}D^{\zeta}_{z}s=\log\left(\frac{-1+\zeta e^{v}+\sqrt{(1-\zeta e^{v})^{2}+4\zeta e^{v}e^{2r}}}{2\zeta e^{v}e^{2r}}\right).$ (4.10) Differentiating with respect to $x$ on both sides and substituting $e^{2r}=1-e^{-u}$ then gives us: $D^{\zeta}_{z}v={\mathrm{i}}\,\frac{\partial}{\partial x}\log\left(\frac{-1+\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}{2\zeta e^{v}(1-e^{-u})}\right).$ (4.11) Equation (4.3a) then follows from the identity: $\frac{-1+\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}{2\zeta e^{v}(1-e^{-u})}=\frac{2}{1-\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\,.$ (4.12) We will now use what we have established so far to prove (4.3b). First, we note that (4.6a) in the limit $\lambda\rightarrow 0$ becomes: $1-\exp({\mathrm{i}}D^{\zeta}_{z}s)e^{2r}=\exp\left(\frac{\partial^{2}\varpi}{\partial x^{2}}+{\mathrm{i}}\frac{\partial D^{\zeta}_{z}\varpi}{\partial x}\right).$ (4.13) Setting $\zeta=0$ then gives us: $u=-\log(1-e^{2r})=-\frac{\partial^{2}\varpi}{\partial x^{2}}\,.$ (4.14) Substituting this back into (4.13), we obtain: $\frac{\partial D^{\zeta}_{z}\varpi}{\partial x}=-{\mathrm{i}}\log\left(\frac{1-\exp({\mathrm{i}}D^{\zeta}_{z}s)e^{2r}}{e^{-u}}\right).$ (4.15) Differentiating with respect to $x$ on both sides and substituting (4.10) and $e^{2r}=1-e^{-u}$ then gives $D^{\zeta}_{z}u={\mathrm{i}}\,\frac{\partial}{\partial x}\log\left(\frac{1+\zeta e^{v}-\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}{2\zeta e^{v}e^{-u}}\right).$ (4.16) Equation (4.3b) then follows from the identity: $\frac{1+\zeta e^{v}-\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}{2\zeta e^{v}e^{-u}}=\frac{2}{1+\zeta e^{v}+\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\,.$ (4.17) A similar argument involving (3.5d) and (3.5e) leads to (4.3c) and (4.3d). ∎ ###### Remark 5. By setting $\zeta=0$ in (4.6a) and taking the logarithm of both sides, we obtain a difference equation involving shifts in $x$: $\log(1-\exp(2r_{\lambda}))=\frac{\Lambda_{\lambda}(\varpi_{\lambda})-2\varpi_{\lambda}+\Lambda_{\lambda}^{-1}(\varpi_{\lambda})}{\check{\lambda}^{2}}\,.$ (4.18) ###### Remark 6. By applying $\zeta\partial_{\zeta}$ to (4.3) we get the following alternative form for the dispersionless AL hierarchy: $\displaystyle\Delta^{\zeta}_{z}v$ $\displaystyle=-{\mathrm{i}}\,\frac{\partial}{\partial x}\left(\frac{1+\zeta e^{v}}{2\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\right)$ $\displaystyle=-{\mathrm{i}}\,\frac{\partial^{2}}{\partial x\partial u}\tanh^{-1}\left(\frac{1+\zeta e^{v}}{\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\right),$ (4.19a) $\displaystyle\Delta^{\zeta}_{z}u$ $\displaystyle=\phantom{+}{\mathrm{i}}\,\frac{\partial}{\partial x}\left(\frac{1-\zeta e^{v}}{2\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\right)$ $\displaystyle=-{\mathrm{i}}\,\frac{\partial^{2}}{\partial x\partial v}\tanh^{-1}\left(\frac{1+\zeta e^{v}}{\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\right),$ (4.19b) $\displaystyle\Delta^{\zeta}_{\tilde{z}}v$ $\displaystyle=-{\mathrm{i}}\,\frac{\partial}{\partial x}\left(\frac{1+\zeta e^{-v}}{2\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}\right)$ $\displaystyle=-{\mathrm{i}}\frac{\partial^{2}}{\partial x\partial u}\tanh^{-1}\left(\frac{1+\zeta e^{-v}}{\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}\right),$ (4.19c) $\displaystyle\Delta^{\zeta}_{\tilde{z}}u$ $\displaystyle=-{\mathrm{i}}\,\frac{\partial}{\partial x}\left(\frac{1-\zeta e^{-v}}{2\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}\right)$ $\displaystyle=-{\mathrm{i}}\,\frac{\partial^{2}}{\partial x\partial v}\tanh^{-1}\left(\frac{1+\zeta e^{-v}}{\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}\right),$ (4.19d) where $\Delta^{\zeta}_{z}$ and $\Delta^{\zeta}_{\tilde{z}}$ denote the operators: $\Delta^{\zeta}_{z}=\sum_{j=1}^{\infty}\zeta^{j}\frac{\partial}{\partial z_{j}}\,,\quad\Delta^{\zeta}_{\tilde{z}}=\sum_{j=1}^{\infty}\zeta^{j}\frac{\partial}{\partial\tilde{z}_{j}}\,.$ (4.20) In particular, this is a dispersionless _Hamiltonian_ hierarchy with generating functions of Hamiltonian densities: $\begin{split}h(\zeta)&=-{\mathrm{i}}\tanh^{-1}\left(\frac{1+\zeta e^{v}}{\sqrt{(1+\zeta e^{v})^{2}-4\zeta e^{v}e^{-u}}}\right),\\\ \tilde{h}(\zeta)&=-{\mathrm{i}}\tanh^{-1}\left(\frac{1+\zeta e^{-v}}{\sqrt{(1+\zeta e^{-v})^{2}-4\zeta e^{-v}e^{-u}}}\right),\end{split}$ (4.21) and the following Poisson brackets: $\\{u(x),u(y)\\}=\\{v(x),v(y)\\}=0\,,\quad\\{u(x),v(y)\\}=\\{v(x),u(y)\\}=\delta^{\prime}(x-y)\,.$ (4.22) There are a number of interesting aspects of the Hamiltonian formulation of the dispersionless limit of the AL hierarchy. Firstly, the Hamiltonian densities depend only on $u$ and $v$ explicitly and not on their spatial derivatives. Secondly, the coefficients of $\delta^{\prime}(x)$ in the Poisson brackets form a constant symmetric matrix. A Hamiltonian hierarchy that has this structure (possibly after a suitable change of dependent variables) is said to be of _hydrodynamic type_. Such hierarchies were first defined and studied by Dubrovin and Novikov [DN84]. A key fact concerning Hamiltonian hierarchies of hydrodynamic type is that there is a natural such hierarchy called the _principal hierarchy_ associated to any (almost) Frobenius manifold, which conversely may be viewed as a differential-geometric structure on the initial data of such a hierarchy. The small phase space $H_{2}(X,\mathbb{Z})$ of a projective variety happens to carry the structure of a(n almost) Frobenius manifold, so this gives a rather general construction of a dispersionless integrable hierarchy associated to the (equivariant) genus zero GW theory of any projective variety. It was noted by Dubrovin in [Dub08] that the dispersionless limit of the AL hierarchy is contained within the principal hierarchy associated to the equivariant GW theory of the resolved conifold with anti-diagonal action (hereafter referred to as simply the principal hierarchy of the resolved conifold). We will now present a proof of this result in our notation to ensure that the dependent variables $u$ and $v$ that we have introduced are indeed the variables of the principal hierarchy identified by Dubrovin. ###### Proposition 7. The dispersionless limit of AL hierarchy with the initial conditions $v=2\pi{\mathrm{i}}x/\kappa$ and $u=-2\pi{\mathrm{i}}t$ is contained in the principal hierarchy of the resolved conifold with the following identification on the small phase space: $(2\pi)^{2}\varpi\|_{(z,\tilde{z})=(0,0)}=\left.\left(-\frac{uv^{2}}{2}+\tilde{F}^{0}\right)\right\|_{(z,\tilde{z})=(0,0)}.$ (4.23) ###### Proof. The principal hierarchy associated to a(n almost) Frobenius manifold $V$ is defined on the formal loop space $L(S^{1},V)$, whose elements are maps from $S^{1}=\mathbb{R}/\mathbb{Z}$ to $V$. In our case, $V$ is the small phase space $H_{2}(X,\mathbb{Z})$, while $w^{1}:=v$ and $w^{2}:=u$ can be interpreted as maps from $S^{1}$ to $V$. (Note that the map $w^{1}$ is distinguished.) Meanwhile, $x$ is interpreted as a coordinate on the universal cover $\mathbb{R}$ of $S^{1}$. A potential $\Phi$ on $V$ encodes a metric $\eta$ and an associative product $\star$ with identity $\partial_{w^{1}}$ on the tangent bundle $TV$: $\begin{split}\eta_{\mu\nu}:=\eta\left(\frac{\partial}{\partial w^{\mu}}\,,\frac{\partial}{\partial w^{\nu}}\right)&=\frac{\partial^{3}\Phi}{\partial w^{1}\partial w^{\mu}\partial w^{\nu}}\,,\\\ \eta\left(\frac{\partial}{\partial w^{\kappa}}\,,\frac{\partial}{\partial w^{\mu}}\star\frac{\partial}{\partial w^{\nu}}\right)&=\frac{\partial^{3}\Phi}{\partial w^{\kappa}\partial w^{\mu}\partial w^{\nu}}\,.\end{split}$ (4.24) We also have a Poisson bracket on $L(S^{1},V)$ in terms of the the inverse metric $\eta^{\mu\nu}$: $\\{w^{\mu}(x),w^{\nu}(y)\\}=\eta^{\mu\nu}\delta^{\prime}(x-y)-\eta^{\mu\kappa}\Gamma^{\nu}_{\kappa\gamma}\frac{\partial w^{\gamma}}{\partial x}\delta(x-y)\,,$ (4.25) where $\Gamma^{\nu}_{\kappa\gamma}$ are the Christoffel symbols of the metric $\eta$ and repeated indices are summed over. Comparing this with (4.22), we immediately obtain: $\eta_{11}=\eta_{22}=0,\quad\eta_{12}=\eta_{21}=1\,.$ (4.26) Thus, the most general form that the potential $\Phi$ can take is: $\Phi=\frac{uv^{2}}{2}+f(u)\,.$ (4.27) A consequence of the associativity of the product $\star$ is the existence of a pencil of flat torsion-free connections $\nabla^{\xi}$ parametrised by a parameter $\xi\in\mathbb{C}$: $\nabla^{\xi}_{\partial/\partial w^{\mu}}\frac{\partial}{\partial w^{\nu}}=\xi\,\frac{\partial}{\partial w^{\mu}}\star\frac{\partial}{\partial w^{\nu}}\,.$ (4.28) As these connections are flat, they admit local basis of parallel $1$-forms $\alpha_{1}(\xi)$ and $\alpha_{2}(\xi)$ analytic in $\xi$. And as these connections are torsion-free, the anti-symmetric parts of $\nabla^{\xi}\alpha_{1}$ and $\nabla^{\xi}\alpha_{2}$ are $\mathrm{d}\alpha_{1}$ and $\mathrm{d}\alpha_{2}$ (where $\mathrm{d}$ is the exterior derivative on $V$). Thus, $1$-forms $\alpha_{1}(\xi)$ and $\alpha_{2}(\xi)$ are closed, and so by Poincaré’s lemma, we can locally find functions $g_{1}(\xi)$ and $g_{2}(\xi)$ analytic in $\xi$ such that we have: $\alpha_{1}=\mathrm{d}g_{1}\,,\quad\alpha_{2}=\mathrm{d}g_{2}\,.$ (4.29) More explicitly, the functions $g_{1}$ and $g_{2}$ are solutions to the following system of equations: $\frac{\partial^{2}g}{\partial v^{2}}=\xi\,\frac{\partial g}{\partial v}\,,\quad\frac{\partial^{2}g}{\partial v\,\partial u}=\xi\,\frac{\partial g}{\partial u}\,,\quad\frac{\partial^{2}g}{\partial u^{2}}=\xi f^{\prime\prime\prime}(u)\,\frac{\partial g}{\partial v}\,.$ (4.30) The functions $g_{1}(\xi)$ and $g_{2}(\xi)$ are generating functions of Hamiltonian densities, which generate the principal hierarchy via the Poisson bracket (4.25). The generating functions $h(\zeta)$ and $\tilde{h}(\zeta)$ are related to these in a non-trivial way (see Remark 8). We can get around the need to obtain an explicit relationship between them by eliminating $\xi$ from the first and third equations in (4.30). This gives us the following equation that by virtue of its linearity is valid for any generating function of Hamiltonian densities: $\frac{\partial^{2}g}{\partial u^{2}}=f^{\prime\prime\prime}(u)\,\frac{\partial^{2}g}{\partial v^{2}}\,.$ (4.31) Substituting $g=h$ and $g=\tilde{h}$ as in (4.21) into the above, we obtain: $f^{\prime\prime\prime}(u)=\frac{1}{e^{u}-1}=\mathrm{Li}_{0}(e^{-u})\,.$ (4.32) This is solved by $f(u)=-\tilde{F}^{0}(t)=-\mathrm{Li}_{3}(e^{2\pi{\mathrm{i}}t})$ for $u=-2\pi{\mathrm{i}}t$. So the dispersionless AL hierarchy is contained in the principal hierarchy of the resolved conifold. The identification (4.23) meanwhile follows from (4.14) and the initial condition $v=2\pi{\mathrm{i}}x/\kappa$. ∎ The exponential $e^{\Phi}$ of the potential $\Phi$ of the (almost) Frobenius manifold will be henceforth called the _topological tau function_ of the principal hierarchy. ###### Remark 8. The first two equations of (4.30) imply that any solution $g$ necessarily has the form: $g=A(\xi,u)e^{\xi v}+B(\xi)\,,$ (4.33) where $A$ is a function independent of $v$ while $B$ is a function independent of $v$ and $u$. Since we are only interested in the flows of the Hamiltonians, we can without loss of generality set $B$ to be identically zero. Conversely, any function $g=A(\xi,u)e^{\xi v}$ which also satisfies (4.31) is a solution of (4.30). Now consider the Mellin transforms: $\begin{split}\int_{0}^{\infty}h\zeta^{-1-\xi}\mathrm{d}\zeta&=-{\mathrm{i}}e^{\xi v}\int_{0}^{\infty}\tanh^{-1}\left(\frac{1+\zeta}{\sqrt{(1+\zeta)^{2}-4\zeta e^{-u}}}\right)\zeta^{-1-\xi}\mathrm{d}\zeta\\\ &=\phantom{+}{\mathrm{i}}e^{\xi v}\int^{0}_{\infty}\,\,\tanh^{-1}\left(\frac{1+\zeta^{-1}}{\sqrt{(1+\zeta^{-1})^{2}-4\zeta^{-1}e^{-u}}}\right)\zeta^{1+\xi}\,\frac{\mathrm{d}\zeta}{\zeta^{2}}\\\ &=-{\mathrm{i}}e^{\xi v}\int_{0}^{\infty}\tanh^{-1}\left(\frac{\zeta+1}{\sqrt{(\zeta+1)^{2}-4\zeta e^{-u}}}\right)\zeta^{-1+\xi}\mathrm{d}\zeta\,,\\\ \int_{0}^{\infty}\tilde{h}\zeta^{-1+\xi}\mathrm{d}\zeta&=-{\mathrm{i}}e^{\xi v}\int_{0}^{\infty}\tanh^{-1}\left(\frac{1+\zeta}{\sqrt{(1+\zeta)^{2}-4\zeta e^{-u}}}\right)\zeta^{-1+\xi}\mathrm{d}\zeta\,.\end{split}$ (4.34) This is of the form $A(\xi,u)e^{\xi v}$. Moreover, being an integral transform of $h$ and $\tilde{h}$, it satisfies (4.31) with $f(u)=-\mathrm{Li}_{3}(e^{-u})$ as well. Hence, it must be a solution $g$ of (4.30). In other words, it is some constant linear combination of the Hamiltonian density generating functions $g_{1}$ and $g_{2}$ of the principal hierarchy of the resolved conifold. ## 5\. Identification of the Tau function In this section, we will extend the genus zero identification of the dispersionless AL tau function with the topological tau function of the principal hierarchy of the resolved conifold to an all-genera small-phase- space identification of the tau function of the small-dispersion expansion of the AL hierarchy with Bridgeland’s Tau function. The genus $2$ truncation of this identification was essentially the content of Theorem 1.4 in [Bri12] and it was proved by analyzing dispersive perturbations of the principal hierarchy. This is a computationally challenging task, however the control over expansion in the dispersive parameter offered by the functional representation of the AL hierarchies will allow us to sidestep these difficulties. Here is the gist of our approach. The solution of Prop. 2 is used to define initial conditions for the tau function of the AL hierarchy while the other difference equation in Remark 5 is used to show that these initial conditions reduce to those of the principal hierarchy of the resolved conifold in the dispersionless limit. We now prove our main theorem, which extends Theorem 1.4 in [Bri12] to all genera. ###### Theorem 9. There exists a solution to the small-dispersion expansion of the AL hierarchy satisfying the following conditions: 1. (1) The tau function $\tau_{\lambda}$ of the solution reduces at leading order in $\lambda$ to the topological tau function of the principal hierarchy of the resolved conifold as $\lambda\rightarrow 0$, 2. (2) The restriction of the tau function $\tau_{\lambda}$ of the solution to the small phase space coincides with the exponential of the equivariant GW potential of the resolved conifold with anti-diagonal action, namely: $\exp(F_{\mathrm{ad}}(\lambda,t))=\exp\bigg{(}\frac{(2\pi)^{3}{\mathrm{i}}x^{2}t}{2\kappa^{2}\check{\lambda}^{2}}\bigg{)}G(t\,\|\,\check{\lambda},1)\,,$ (5.1) to all orders in $\lambda$. ###### Proof. We impose the following initial conditions on $v_{\lambda}:=\check{\lambda}^{-1}(\Lambda_{\lambda}-1)s_{\lambda}$ and $\varpi_{\lambda}$: $v_{\lambda}\|_{(z,\tilde{z})=(0,0)}=\frac{2\pi{\mathrm{i}}x}{\kappa}\,,\quad(2\pi)^{2}\varpi_{\lambda}\|_{(z,\tilde{z})=(0,0)}=\frac{(2\pi)^{3}{\mathrm{i}}x^{2}t}{2\kappa^{2}}+\check{\lambda}^{2}\log G(t\,\|\,\check{\lambda},1)\,.$ (5.2) This entails condition (2). 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# LS-HDIB: A Large Scale Handwritten Document Image Binarization Dataset ###### Abstract Handwritten document image binarization is challenging due to high variability in the written content and complex background attributes such as page style, paper quality, stains, shadow gradients, and non-uniform illumination. While the traditional thresholding methods do not effectively generalize on such challenging real-world scenarios, deep learning-based methods have performed relatively well when provided with sufficient training data. However, the existing datasets are limited in size and diversity. This work proposes LS- HDIB - a large-scale handwritten document image binarization dataset containing over a million document images that span numerous real-world scenarios. Additionally, we introduce a novel technique that uses a combination of adaptive thresholding and seamless cloning methods to create the dataset with accurate ground truths. Through an extensive quantitative and qualitative evaluation over eight different deep learning based models, we demonstrate the enhancement in the performance of these models when trained on the LS-HDIB dataset and tested on unseen images. Index Terms— Document Image Binarization, Deep Learning, Adaptive Thresholding. ## 1 Introduction 11footnotetext: This work is supported by SERB IMPRINT-2 grant. Handwritten document image binarization is generally modeled as a classification problem in which intra-image pixels are assigned to either of the two classes: handwritten content (the foreground) or the background. Document image binarization has been an active research area for decades owing to its importance as an essential pre-processing step in facilitating several document image processing tasks such as optical character recognition, handwriting matching, document translation, document summarization, and changing the background. Fig. 1: A few samples of handwritten document images obtained from the proposed LS-HDIB dataset. Handwritten documents range from ancient documents, old legal records, and ledgers to music scores and handwritten bills. These documents often degrade over time and become difficult to comprehend. Common degradation scenarios include crumpled pages, poor foreground-background contrast, stains, paper aging, fainted characters, uneven illumination, and bleed/show-through. Furthermore, handwritten documents posses various page styles, i.e., grids, lines, staff annotation styles, and partially blank pages that increase the difficulty of segmenting the foreground content. The variability in the type and the thickness of strokes also increases the complexity. These challenges make handwritten document image binarization extremely difficult. Traditional methods like Otsu image segmentation [1] and adaptive thresholding [2] fail to address the aforementioned challenges completely. Since these algorithms utilize only the low-level features, they fail to capture the wide range of variability inherent in the handwritten documents, limiting their ability to distinguish the background from the foreground content. The high-level features can differentiate text pixels from background noises handling the degradations better. However, using them solely can cause loss of low-level information like character edges and contours, making it insufficient to address the binarization problem. Thus far, deep learning methods have shown promising results in segmenting foreground and background content [3, 4] by incorporating both low-level and high-level image features. Deep learning models rely heavily on a large amount of data for better generalization [5]. However, document image binarization lacks such a large and diverse dataset that covers numerous real-world scenarios. While the focus has been on designing new robust networks, little attention has been given to scale up the existing datasets. To address the aforementioned challenges and to generate a large scalable dataset, we propose a Large Scale Handwritten Document Image Binarization dataset (LS-HDIB) that contains over a million handwritten document images. We propose a simple and effective method for generating the LS-HDIB dataset with accurate ground truths containing segmented handwritten content (see Fig. 1). Interestingly, the proposed method requires no manual intervention for generating these segmented ground truths. The primary contributions of this work are: (i) A large scale dataset (LS-HDIB) containing over a million images with accurate ground truths for handwritten document image binarization. (ii) A scalable and efficient method to generate and extend the proposed dataset. ## 2 Related Work The standard approaches for document image binarization are classified into (i) global methods[1, 6] which use a single threshold value and (ii) local methods [7] which use adaptive threshold values for separating the foreground and the background content. While the global methods fail to handle complex degradations, the local methods are computationally expensive and driven mainly by manual parameter tuning. Several deep neural network architectures have been designed for document image binarization [8, 9, 10]. Researchers have used fully convolutional neural networks [11, 12], recurrent neural networks [13], encoder-decoder frameworks [9, 12], and generative adversarial networks [11, 14] to address document binarization. While the performance of these learning-based frameworks depends on the robustness of the network design, another aspect critical to their performance is the size and the versatility of the training data. The existing datasets [15, 16, 17, 18, 19, 20, 21, 22, 23, 24] do not completely span complex degradations, page styles, and illumination variations. Owing to a very large space spanned by the possible handwritten content and background variations, there is a need to develop a method to create a large and diverse dataset for handwritten document image binarization. In this work, we propose a simple yet effective method to create a large-sized dataset that can potentially circumvent the aforementioned limitations. ## 3 Method ### 3.1 Dataset Generation We propose a novel data generation technique that uses a combination of adaptive thresholding [2] and mixed gradient seamless cloning [25], as described in Fig. 2. We collect the images of a variety of handwritten content over a plain background and refer to these images as full-length document images $(I_{doc})$. We collected over $450$ full-length document images. Around $400$ images contain handwritten content in various forms such as alpha-numeric characters, electrical circuit diagrams, control system schematics, chemical molecular structures, and flow charts that are not present in the existing datasets. Further, to diversify the types, thickness, and styles of strokes, we obtained these images from $21$ different persons. Around $50$ document images were digitally created, using Google Translate, in $13$ different languages, including English, Urdu, Mandarin, Portuguese, Russian, French, Hindi, Telugu, Malayalam, Punjabi, Gujarati, Japanese, and Korean in $21$ different font styles of various sizes and colors. Fig. 2: Block schematic of the proposed method for generating LS-HDIB dataset. We apply the adaptive thresholding $(\mathcal{T})$ [2] on $(I_{doc})$ to obtain the segmented ground truth images $(I_{gt})$ such that $I_{gt}=\mathcal{T}(I_{ref})$. Next, we generate a total of $10,944$ unique content images $(I_{c})$ by cropping and augmenting multiple patches of size $480\times 480$ from the full-length document images $(I_{doc})$. Figure 3(a) shows a few sample images containing a variety of written content in different languages and font styles, including diagrams and texts from different subject domains. While the written content (foreground) associated with the generated ground truth remains unchanged, the background can vary depending on different page styles and degradation scenarios. We obtain multiple document images by merely changing the background with essentially the same ground truth. Moreover, this method can automatically generate ground truths saving hours of tedious manual annotation. To generate different backgrounds, we manually capture pages with a wide variety of page styles $(I_{p})$ and degradation effects $(I_{d})$ that are predominant in the real-world. We then use mixed gradient-based seamless cloning $(\mathcal{C})$ [25] to blend multiple patches of $(I_{p})$ and $(I_{d})$ to generate $20,484$ photorealistic background images $(I_{bg})$ such that $I_{bg}=\mathcal{C}(I_{p},I_{d})$. Datasets \| Page Styles \| Degradation Effects ---\|---\|--- \| Uniform --- ruled lines \| Non-uniform --- ruled lines \| Grid --- lines \| Staff notation --- lines \| Partially --- blank pages Plain page \| \| Shadow --- gradients \| Oily --- patches \| Ink --- bleed-through \| crumpled --- pages \| Non-uniform --- illumination \| Noisy --- background \| Liquid --- stains \| Poor foreground- --- background contrast \| Punched, stapled --- or torn pages DIBCO09 \| x \| x \| x \| x \| ✓ \| ✓ \| x \| x \| ✓ \| x \| x \| x \| ✓ \| ✓ \| x HDIBCO10 \| ✓ \| x \| x \| x \| x \| ✓ \| x \| ✓ \| ✓ \| x \| x \| ✓ \| ✓ \| ✓ \| x DIBCO11 \| x \| x \| x \| x \| x \| ✓ \| x \| x \| ✓ \| x \| x \| ✓ \| ✓ \| ✓ \| x HDIBCO12 \| ✓ \| x \| x \| x \| x \| ✓ \| x \| ✓ \| ✓ \| x \| ✓ \| ✓ \| ✓ \| ✓ \| x DIBCO13 \| x \| x \| x \| x \| x \| ✓ \| x \| ✓ \| ✓ \| ✓ \| x \| ✓ \| ✓ \| ✓ \| x HDIBCO14 \| x \| x \| x \| x \| x \| ✓ \| x \| x \| ✓ \| x \| x \| x \| ✓ \| ✓ \| x PHIBD12 \| ✓ \| x \| x \| x \| x \| ✓ \| x \| ✓ \| ✓ \| x \| x \| ✓ \| ✓ \| ✓ \| ✓ HDIBCO16 \| x \| x \| x \| x \| x \| ✓ \| x \| ✓ \| ✓ \| x \| ✓ \| ✓ \| ✓ \| ✓ \| x DIBCO17 \| x \| x \| x \| x \| x \| ✓ \| x \| ✓ \| ✓ \| x \| x \| ✓ \| ✓ \| x \| x DIBCO18 \| x \| x \| x \| x \| x \| ✓ \| x \| ✓ \| ✓ \| x \| x \| ✓ \| ✓ \| ✓ \| x Bickley Diary \| x \| x \| x \| x \| ✓ \| ✓ \| x \| ✓ \| x \| x \| x \| ✓ \| ✓ \| ✓ \| x Palm Leaf Manuscript \| x \| x \| x \| x \| ✓ \| ✓ \| ✓ \| ✓ \| x \| x \| x \| x \| ✓ \| ✓ \| ✓ LS-HDIB (Ours) \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| ✓ \| x Table 1: Comparison of page styles and degradation effects available across different publicly available datasets and LS-HDIB dataset. Fig. 3: A few sample images depicting different page styles available in LS-HDIB dataset. For a better visual understanding, we show multiple example images of different page styles and degradation effects in Fig. 3(a) and Fig. 3(b), respectively. Finally, we combine the content images $(I_{c})$ and the background images $(I_{bg})$ again using seamless cloning $(\mathcal{C})$ to generate the handwritten document images $(I_{in})$ such that $I_{in}=\mathcal{C}(I_{c},I_{bg})$. To generate the LS-HDIB dataset, we randomly sample $100$ background images $(I_{bg})$ for each of the $10,994$ handwritten content images $(I_{c})$. In this way, we obtain a total of $1.09$ million images in the LS-HDIB dataset. It is important to note that we can scale up the dataset size for up to $200$ times by considering all the background images instead of just $100$. With adaptive thresholding and mixed gradient-based seamless cloning, we have been able to generate accurate ground truths without the requirement of any manual annotation and generate a wide variety of degraded handwritten document images. In addition to the inherent scalability and ease of ground truth generation, the proposed dataset generation method is computationally less expensive compared to the other deep-learning-based generative methods such as the one proposed in [14]. As evident from Table 1, most of the publicly available datasets contain the written content only over plain pages. In contrast, the LS-HDIB dataset includes images with various page styles like ruled lines, gridlines, and partially blank pages that are evident in our day-to-day encounters. Further, they lack the document images with realistic degradations like crumpled pages, non-uniform illumination, and shadow gradients that are well incorporated in the proposed LS-HDIB dataset. These attributes enhance the diversity and the versatility of the proposed dataset. ### 3.2 Training Details We use eight widely used deep networks - DeepLabV3 [26], DeepLabV3+ [27], Feature Pyramid Networks (FPN) [28], LinkNet [29], Multi-scale Attention net (MANet) [30], Pyramid Attention Network (PAN) [31], Pyramid Scene Parsing Network (PSPN) [32], and U-Net [33] \- to understand the effectiveness of the proposed dataset for the handwritten document image binarization task. We have carefully chosen these networks as they collectively span the various deep learning based approaches [11, 12, 9, 14] that have been adopted thus far for the binarization task. Each of the eight networks is trained under three different training regimes. We follow the standard train, validation, and test split of $80\%$, $10\%$, and $10\%$, respectively, for each regime. (i) Regime 1: The deep models are trained only on the baseline dataset obtained by combining ten different publicly available datasets: DIBCO09 [15], HDIBCO10 [16], DIBCO11 [17], HDIBCO12[18], DIBCO13 [19], HDIBCO14 [20], PHIBD12 [24], HDIBCO16 [21], DIBCO17 [22], DIBCO18 [23]. However, even after combining ten different datasets, the size of the baseline dataset is relatively small (order of 1000 images). Therefore, we crop the full-length document images of the baseline dataset to the size $480\times 480$ with a stride of $240$ pixels and perform rotation ($90^{\circ},180^{\circ}$, and $270^{\circ}$) and horizontal flip to obtain around $6000$ images, $5000$ for training and $1000$ for testing. While the size of the baseline dataset can further be increased by reducing the stride value, this leads to greater overlap and high redundancy in the foreground content across different content images. (ii) Regime 2: Each deep model is trained only on the LS-HDIB training set. Although the LS-HDIB dataset has over 1 million images, we use only $5000$ images for training to have a fair performance comparison. (iii) Regime 3: We combine both the LS-HDIB and the baseline dataset, by randomly selecting $2500$ images from each dataset to train all the deep models on a total of $5000$ images. Regime 3 is targeted towards establishing the efficacy of augmenting the proposed dataset to the existing ones. The models are trained for a maximum of $20$ epochs with learning rate of $0.01$ using the Adam optimizer with default parameters. The training is performed on the NVIDIA RTX 2080 Ti GPU with the batch size of $8$. Loss Function. We use binary cross-entropy loss to train the segmentation models, as described in Equation 1. $\mathcal{L}=-I_{gt}\mathrm{log}\left(\mathcal{F}(I_{in})\right)-(1-I_{gt})\mathrm{log}\left(1-\mathcal{F}(I_{in})\right)$ (1) Here, $\mathcal{F}$ represents the functional form of deep model. Binary cross-entropy loss is found to be more effective than MSE loss for classification tasks [34]. Fig. 4: Qualitative result on (a) the LS-HDIB test set (b) Bickley Diary dataset, and (c) Palm Leaf Manuscript dataset. ## 4 Experimental Analysis We demonstrate the effectiveness of the proposed dataset for handwritten document image binarization through an extensive quantitative and qualitative analysis. We compare the performance of different deep models trained to observe how well the LS-HDIB dataset enhances the generalization capability of the deep models. We use four different datasets containing challenging scenarios for testing the deep models: the Bickley Diary dataset [35], the Palm Leaf Manuscript dataset [36], the DIBCO test set, and the LS-HDIB test set. We use three popular metrics to evaluate the network performance trained under different regimes: F-measure $(\mathrm{F_{score}})$ [37], pseudo F-measure $(\mathrm{PF_{score}})$ [37], and Peak Signal to Noise Ratio $(\mathrm{PSNR})$ [37]. As shown in Table 2, the $\mathrm{F_{score}}$, $\mathrm{PF_{score}}$, and $\mathrm{PSNR}$ is maximum over all the deep models trained under Regime 2 for LS-HDIB and the Bickley Diary dataset. Further, the performance under Regime 3 is better than that of Regime 1, indicating that augmenting the LS-HDIB dataset to the baseline dataset enhances network performance. Qualitatively, the foreground content of the image affected by liquid stains, noisy background, and poor foreground to background contrast is well recovered under Regime 2, as shown in Fig. 4(a) and 4(b). For the Palm Leaf dataset, the strokes in the estimated foreground content corresponding to Regime 1 (and 3) are relatively thicker when compared to those corresponding to Regime 2 across all the models (see Fig. 4(c)). Since the stroke widths are consistent with the ground truth, the $\mathrm{PSNR}$ continues to be higher for Regime 2. However, the $\mathrm{F_{score}}$ and $\mathrm{PF_{score}}$ of some models under Regime 2 and 3 are less than Regime 1. This is attributed to the presence of minor discontinuities in the strokes obtained under Regime 2 when compared to that of Regime 1 (see Fig. 4(c)). However, on average, the overall performance of each of the eight deep models is the highest when trained under Regime 2 across the three different test datasets, as evident from Table 2. Further, Fig. 4(c) depicts that models under Regime 1 and 3 fail to segment out the thread punched out through the document. However, almost every deep model trained on the LS-HDIB dataset (Regime 2) precisely identifies the appropriate foreground content even in the presence of such background artifacts. For the DIBCO test set, the performance under Regime 1 is better than Regime 2 (Table 2). This is inline with the expectation as the networks are trained on DIBCO train set itself. Interestingly, Regime 3 offers the best performance across different models and test sets indicating that the LS-HDIB dataset when augmented with the standard DIBCO datasets enhances the network performance. Metric \| Dataset \| DeepLabv3 \| DeepLabv3++ \| FPN \| LinkNet \| MANet \| PANet \| PSPNet \| U-Net ---\|---\|---\|---\|---\|---\|---\|---\|---\|--- \| \| 0.64 \| 0.71 \| 0.72 \| 0.71 \| 0.72 \| 0.72 \| 0.58 \| 0.74 \| \| 0.66 \| 0.77 \| 0.77 \| 0.82 \| 0.84 \| 0.74 \| 0.65 \| 0.83 \| Bickley Diary \| 0.60 \| 0.75 \| 0.74 \| 0.80 \| 0.82 \| 0.69 \| 0.60 \| 0.77 \| \| 0.49 \| 0.59 \| 0.58 \| 0.62 \| 0.62 \| 0.57 \| 0.47 \| 0.62 \| \| 0.48 \| 0.57 \| 0.59 \| 0.60 \| 0.62 \| 0.57 \| 0.48 \| 0.60 \| Palm Leaf \| 0.48 \| 0.58 \| 0.57 \| 0.61 \| 0.62 \| 0.57 \| 0.44 \| 0.61 \| \| 0.48 \| 0.68 \| 0.60 \| 0.66 \| 0.68 \| 0.56 \| 0.50 \| 0.67 \| \| 0.43 \| 0.57 \| 0.46 \| 0.58 \| 0.60 \| 0.41 \| 0.43 \| 0.58 \| DIBCO \| 0.52 \| 0.62 \| 0.61 \| 0.72 \| 0.71 \| 0.58 \| 0.46 \| 0.72 \| \| 0.44 \| 0.52 \| 0.54 \| 0.56 \| 0.57 \| 0.51 \| 0.42 \| 0.55 \| \| 0.54 \| 0.68 \| 0.68 \| 0.86 \| 0.87 \| 0.65 \| 0.53 \| 0.87 F-score \| LS-HDIB \| 0.53 \| 0.66 \| 0.67 \| 0.85 \| 0.85 \| 0.64 \| 0.52 \| 0.86 \| \| 0.64 \| 0.76 \| 0.75 \| 0.80 \| 0.79 \| 0.71 \| 0.59 \| 0.78 \| \| 0.66 \| 0.79 \| 0.79 \| 0.87 \| 0.89 \| 0.76 \| 0.66 \| 0.89 \| Bickley Diary \| 0.60 \| 0.77 \| 0.75 \| 0.82 \| 0.80 \| 0.73 \| 0.61 \| 0.82 \| \| 0.50 \| 0.60 \| 0.57 \| 0.61 \| 0.62 \| 0.58 \| 0.48 \| 0.63 \| \| 0.48 \| 0.58 \| 0.59 \| 0.63 \| 0.64 \| 0.57 \| 0.48 \| 0.62 \| Palm Leaf \| 0.48 \| 0.59 \| 0.58 \| 0.62 \| 0.64 \| 0.57 \| 0.44 \| 0.63 \| \| 0.49 \| 0.68 \| 0.60 \| 0.66 \| 0.68 \| 0.56 \| 0.50 \| 0.67 \| \| 0.41 \| 0.52 \| 0.51 \| 0.58 \| 0.51 \| 0.41 \| 0.43 \| 0.58 \| DIBCO \| 0.52 \| 0.62 \| 0.62 \| 0.73 \| 0.72 \| 0.59 \| 0.47 \| 0.72 \| \| 0.43 \| 0.51 \| 0.53 \| 0.56 \| 0.56 \| 0.51 \| 0.42 \| 0.54 \| \| 0.53 \| 0.67 \| 0.66 \| 0.87 \| 0.88 \| 0.64 \| 0.52 \| 0.88 PF-score \| LS-HDIB \| 0.52 \| 0.65 \| 0.66 \| 0.86 \| 0.86 \| 0.63 \| 0.52 \| 0.87 \| \| 10.07 \| 12.32 \| 12.11 \| 13.16 \| 12.83 \| 11.55 \| 9.61 \| 12.89 \| \| 10.06 \| 12.75 \| 12.69 \| 14.42 \| 14.82 \| 12.10 \| 10.01 \| 14.64 \| Bickley Diary \| 9.69 \| 12.40 \| 12.29 \| 13.41 \| 13.49 \| 11.64 \| 9.78 \| 13.24 \| \| 9.02 \| 10.53 \| 10.68 \| 11.04 \| 11.27 \| 10.41 \| 9.04 \| 11.11 \| \| 8.93 \| 11.57 \| 11.47 \| 12.45 \| 12.31 \| 11.12 \| 8.99 \| 12.41 \| Palm Leaf \| 9.19 \| 11.34 \| 11.36 \| 12.26 \| 12.24 \| 11.03 \| 9.36 \| 12.32 \| \| 8.26 \| 9.76 \| 9.63 \| 10.84 \| 11.13 \| 9.01 \| 7.90 \| 11.06 \| \| 6.75 \| 8.97 \| 8.74 \| 10.96 \| 9.54 \| 10.53 \| 6.56 \| 11.01 \| DIBCO \| 8.38 \| 10.47 \| 10.34 \| 12.54 \| 12.36 \| 9.74 \| 8.39 \| 12.56 \| \| 7.58 \| 9.29 \| 9.83 \| 9.63 \| 9.86 \| 9.26 \| 7.49 \| 9.05 \| \| 9.39 \| 12.15 \| 12.05 \| 16.99 \| 17.09 \| 11.58 \| 9.37 \| 17.29 PSNR \| LS-HDIB \| 9.11 \| 11.88 \| 12.08 \| 16.63 \| 16.74 \| 11.33 \| 9.10 \| 17.07 Table 2: Fscore, PFscore, and PSNR evaluated over eight deep models under three different regimes: Regime 1, Regime 2, and Regime 3 over test datasets. Fig. 5: Effect of varying dataset size on the model performance evaluated over the three test datasets. Given that nearly all the models have performed the best when trained on LS- HDIB dataset across all the three test datasets, we further investigate the effect of dataset size on the network performance. For LS-HDIB test dataset, the $\mathrm{F_{score}}$, $\mathrm{PF_{score}}$, and $\mathrm{PSNR}$ are observed to increase with the dataset size, as shown in Fig. 5. This indicates the requirement of a large-scale dataset to span complex real-world scenarios encountered in the handwritten document images. Fig. 5 shows that the performance over the Bickley Diary and Palm Leaf dataset peaks at the dataset size of $20$K and $5$K, respectively. Overall, we have established that the deep models are more robust to various degradation effects and page styles encountered in the real world when trained on the LS-HDIB dataset and necessitate the need for such a dataset. Note. Owing to the space constraints, we provide more qualitative results, detailed dataset statistics, training and validation logs for different models (for better selectivity), and the accompanying code on our website111https://kaustubh-sadekar.github.io/LS-HDIB/. ## 5 Conclusion We propose a large-scale dataset (LS-HDIB) for handwritten document image binarization and a simple yet effective method to generate it. When trained on the LS-HDIB dataset, different deep models can generalize better on unseen document images with a wide variety of degradations encountered in our day-to- day lives. This is possible due to the inherent diversity of the proposed dataset. Further, this work highlights that the fundamental image processing algorithms can be used as practical tools to support the existing deep- learning-based methods in producing significantly better results. 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# Nilpotent varieties in symmetric spaces and twisted affine Schubert varieties Jiuzu Hong Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, U.S.A<EMAIL_ADDRESS>and Korkeat Korkeathikhun Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3250, U.S.A<EMAIL_ADDRESS> ###### Abstract. We relate the geometry of Schubert varieties in twisted affine Grassmannian and the nilpotent varieties in symmetric spaces. This extends some results of Achar-Henderson in the twisted setting. We also get some applications to the geometry of the order 2 nilpotent varieties in certain classical symmetric spaces. ## 1\. Introduction Let $G$ be a reductive group over $\mathbb{C}$. Let $\mathcal{N}$ denote the nilpotent cone of the Lie algebra $\mathfrak{g}$ of $G$. Let $\operatorname{Gr}_{G}$ be the affine Grassmannian of $G$. Each spherical Schubert cell $\operatorname{Gr}_{\lambda}$ is parametrized by a dominant coweight $\lambda$. When $G=\mathrm{GL}_{n}$, Lusztig [Lu] defined an embedding from $\mathcal{N}$ to $\operatorname{Gr}_{G}$, and showed that each nilpotent variety in $\mathfrak{gl}_{n}$ can be openly embedded into certain affine Schubert variety $\overline{\operatorname{Gr}}_{\lambda}$. This embedding identifies the geometry of nilpotent varieties and certain affine Schubert varieties in type $A$. However, there is no direct generalization for general reductive groups. In [AH], Achar-Henderson took a different idea for a general algebraic simple group $G$. Let $\operatorname{Gr}_{0}^{-}$ be the opposite open Schubert cell in $\operatorname{Gr}_{G}$. One can naturally define a map $\pi:\operatorname{Gr}_{0}^{-}\to\mathfrak{g}$. Achar-Henderson showed that $\pi(\operatorname{Gr}_{0}^{-}\cap\operatorname{Gr}_{\lambda})$ is contained in $\mathcal{N}$ if and only if $\lambda$ is small in the sense of Broer [Br] and Reeder [Re], i.e. $\lambda\nsucceq 2\gamma_{0}$, where $\gamma_{0}$ is the highest short coroot of $G$. They also proved that $\pi:\operatorname{Gr}_{\rm sm}\cap\operatorname{Gr}_{0}^{-}\to\pi(\operatorname{Gr}_{\rm sm}\cap\operatorname{Gr}_{0}^{-})$ is a finite map whose fibers admits transitive $\mathbb{Z}/2\mathbb{Z}$-actions, where $\operatorname{Gr}_{\rm sm}$ is the union of all $\operatorname{Gr}_{\lambda}$ such that $\lambda$ is small. Moreover, with respect to $\pi$, Achar-Henderson [AH, AH2] related the geometric Satake correspondence and Springer correspondence. In this paper, we consider a twisted analogue, and we will extend some results of Achar-Henderson in [AH]. Let $\sigma$ be a diagram automorphism of order $2$, and let $\sigma$ act on the field $\mathcal{K}=\mathbb{C}((t))$ via $\sigma(t)=-t$ and $\sigma\|_{\mathbb{C}}={\rm Id}_{\mathbb{C}}$. Then, we may define a twisted affine Grassmannian $\mathcal{G}r:=G(\mathcal{K})^{\sigma}/G(\mathcal{O})^{\sigma}$, where $\mathcal{O}=\mathbb{C}[[t]]$. Each twisted Schubert cell $\mathcal{G}r_{\bar{\lambda}}$, i.e. a $G(\mathcal{O})^{\sigma}$-orbit, is parametrized by the image $\bar{\lambda}$ of a dominant coweight $\lambda$ in the coinvariant lattice $X_{}(T)_{\sigma}$ with respect to the induced action of $\sigma$, where $X_{}(T)$ is the coweight lattice of $G$. In fact, $X_{}(T)_{\sigma}$ can be regarded as the weight lattice of a reductive group $H:=(\check{G})^{\sigma}$, where $\check{G}$ is the Langlands dual group of $G$. Let $\mathcal{G}r_{0}^{-}$ be the opposite open Schubert cell in $\mathcal{G}r$. We may naturally define a map $\pi:\mathcal{G}r_{0}^{-}\to\mathfrak{p}$, where $\mathfrak{p}$ is the (-1)-eigenspace of $\sigma$ in $\mathfrak{g}$. Let $\mathcal{M}_{\bar{\lambda}}$ denote the intersection $\mathcal{G}r_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}$, which is a nonempty open subset of $\mathcal{G}r_{\bar{\lambda}}$. The following theorem is the main result of this paper, and it can follow from Proposition 2.3 in Section 2.1 and Theorem 4.2 in Section 4, based on case-by-case analysis. ###### Theorem 1.1. Assume that $G$ is of type $A_{\ell}$ or $D_{\ell+1}$. The image $\pi(\mathcal{M}_{\lambda})$ is contained in the nilpotent cone $\mathcal{N}_{\mathfrak{p}}$ of $\mathfrak{p}$, if and only if $\bar{\lambda}$ is a small dominant weight with respect to $H$. Actually, in Theorem 4.2 we describe precisely $\pi(\mathcal{M}_{\bar{\lambda}})$ as a union of nilpotent orbits in $\mathfrak{p}$ for each small $\bar{\lambda}$. In Theorem 4.5, Theorem 4.6, and Theorem 4.14, we also determine all small $\bar{\lambda}$ such that $\pi(\mathcal{M}_{\bar{\lambda}})$ is a nilpotent orbit and $\pi:\mathcal{M}_{\bar{\lambda}}\to\pi(\mathcal{M}_{\bar{\lambda}})$ is an isomorphism. Furthermore, We describe all fibers of $\pi:\mathcal{M}\to\pi(\mathcal{M})$ in Proposition 4.11 and Proposition 4.15, where $\mathcal{M}$ is the union of all small twisted Schubert cells. The fibers are closely related to anti-commuting nilpotent varieties for symmetric spaces. When $G$ is of type $A_{2\ell-1}$ (resp. $D_{\ell+1}$), the fiber $\pi^{-1}(0)$ is actually the minimal (resp. maximal) order 2 nilpotent variety in $\mathfrak{sp}_{2\ell}$ (resp. $\mathfrak{so}_{2\ell+1}$). This is a very different phenomenon with the untwisted setting in the work of Achar- Henderson [AH], and it actually makes the twisted setting more challenging. For general simple Lie algebra $\mathfrak{g}$ and general diagram automorphism $\sigma$, it was proved in [HLR, Appendix C] by Haines-Lourenço-Richarz that, when $\bar{\lambda}$ is quasi-miniscule and $\overline{\mathcal{O}}$ is the minimal nilpotent variety in $\mathfrak{p}$, the map $\pi:\overline{\mathcal{G}r}_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}\to\overline{\mathcal{O}}$ is an isomorphism. In fact, we have also obtained this result independently, cf. [Ko]. Also, under the same assumption as in Theorem 1.1, this isomorphism is a special case of our Theorem 4.5, Theorem 4.6, and Theorem 4.14. The geometric Satake correspondence for $\mathcal{G}r$ was proved by Zhu [Zh], and it exactly recovers the Tannakian group $H$. On the other hand, the Springer correspondence for symmetric spaces is more sophisticated than the usual Lie algebra setting, see a survey on this subject [Sh]. It would be interesting to relate these two pictures as was done in [AH, AH2]. Y. Li [Li] defined the symmetric space analogue called $\sigma$-quiver variety in the setting of Nakajima quiver variety, and he showed that certain $\sigma$-quiver variety can be identified with null-cone of symmetric spaces. It is an interesting question to investigate a connection between $\sigma$-quiver variety and twisted affine Grassmannian in the spirit of the work of Mirković- Vybornov [MV]. From Theorem 1.1, we can deduce some applications for the order 2 nilpotent varieties in classical symmetric spaces. Let $\langle,\rangle$ be a symmeric or symplectic non-degenerate bilinear form on a vector space $V$. Let $\mathcal{A}$ be the space of self-adjoint linear maps with respect to $\langle,\rangle$. We consider ${\rm Sp}_{2n}$-action on $\mathcal{A}$ when $\langle,\rangle$ is symplectic and $\dim V=2n$, and ${\rm SO}_{n}$-action when $\langle,\rangle$ is symmetric and $\dim V=n$. In Section 5, we obtain the following results. ###### Theorem 1.2. 1. (1) If $\langle,\rangle$ is symmetric and $\dim V$ is odd, then any order 2 nilpotent variety in $\mathcal{A}$ is normal. 2. (2) If $\langle,\rangle$ is symplectic, then there is a bijection of order 2 nilpotent varieties in $\mathfrak{so}_{2n+1}$ and in $\mathcal{A}$, such that they have the same cohomology of stalks of IC-sheaves. 3. (3) If $\langle,\rangle$ is symplectic, the smooth locus of any order 2 nilpotent variety in $\mathcal{A}$ is the open nilpotent orbit. It is known that when $\langle,\rangle$ is symplectic, any nilpotent variety in $\mathcal{A}$ is normal, but it is not always true when $\langle,\rangle$ is symmetric, cf. [Oh]. Using our methods, we can also prove that there is a bijection of order 2 nilpotent varieties in $\mathfrak{sp}_{2n}$ and in the space of symmetric $(2n+1)\times(2n+1)$ matrices, such that they have the same cohomlogy of stalks of IC-sheaves. This was already proved earlier by Chen- Vilonen-Xue [CVX] using different methods. Also, Part 3) of Theorem 1.2 is not true when $\langle,\rangle$ is symmetric and $\dim V$ is odd, see more detailed discussions in Section 5. Acknowledgments: This project grew out from a conversation with Yiqiang Li in March 2019. We would like to thank him for inspiring discussions. J. Hong is partially supported by NSF grant DMS-2001365. ## 2\. Notation and Preliminaries ### 2.1. Root datum Let $G$ be a simply-connected simple algebraic group over $\mathbb{C}$, and let $\mathfrak{g}$ be its Lie algebra. Let $\sigma$ be a diagram automorphism of $G$ of order $r$, preserving a maximal torus $T$ and a Borel subgroup $B$ containing $T$ in $G$. Then $G$ has a root datum $(X_{}(T),X^{}(T),\langle\cdot,\cdot\rangle,\check{\alpha}_{i},\alpha_{i},i\in I)$ with the action of $\sigma$, where • $X_{}(T)$ (resp. $X^{}(T)$ ) is the coweight (resp. weight) lattice; * • $I$ is the set of vertices of the Dynkin diagram of $G$; * • $\alpha_{i}$ (resp. $\check{\alpha}_{i}$) is the simple root (resp. coroot) for each $i\in I$; * • $\langle\cdot,\cdot\rangle:X_{}(T)\times X^{}(T)\to\mathbb{Z}$ is the perfect pairing. The automorphism $\sigma$ of this root datum satisfies * • $\sigma(\alpha_{i})=\alpha_{\sigma(i)}$ and $\sigma(\check{\alpha}_{i})=\check{\alpha}_{\sigma(i)}$; * • $\langle\sigma(\check{\lambda}),\sigma(\mu)\rangle=\langle\check{\lambda},\mu\rangle$ for any $\check{\lambda}\in X_{}(T)$ and $\mu\in X^{}(T)$. As a diagram autormophism on $G$, $\sigma$ also preserves a pinning with respect to $B$ and $T$, i.e. there exists root subgroups $x_{i},y_{i}$ associated to $\alpha_{i},-\alpha_{i}$ for each $i\in$, such that $\sigma(x_{i}(a))=x_{\sigma(i)}(a),\quad\sigma(y_{i}(a))=y_{\sigma(i)}(a),\quad\text{ for any }a\in\mathbb{C}.$ Let $I_{\sigma}$ be the set of $\sigma$-orbits in $I$. Denote $X^{}(T)^{\sigma}=\\{\lambda\in X^{}(T)\mid\sigma\lambda=\lambda\\}$ and $X_{}(T)_{\sigma}=X_{}(T)/(\mathrm{Id}-\sigma)X_{}(T)$. For each $\imath\in I_{\sigma}$, define $\gamma_{\imath}=\bar{\check{\alpha}}_{i}\in X_{}(T)_{\sigma}$ for any $i\in\imath$, and define $\check{\gamma}_{\imath}\in X^{}(T)^{\sigma}$ by $\check{\gamma}_{\imath}=\begin{cases}\sum_{i\in\imath}\alpha_{i}&\quad\text{if no pairs in }\imath\text{ is adjacent, }\\\ 2\sum_{i\in\imath}\alpha_{i}&\quad\text{if }\imath=\\{i,\sigma(i)\\}\text{ and }i\text{ and }\sigma(i)\text{ are adjacent,}\\\ \alpha_{i}&\quad\text{if }\imath=\\{i\\}.\end{cases}$ Let $\check{G}$ denote the Langlands dual group of $G$, and we still denote the induced diagram automorphism on $\check{G}$ by $\sigma$. Denoted by $H=(\check{G})^{\sigma}$ the $\sigma$-fixed subgroup of $\check{G}$. Then, $H$ has the root datum $(X^{}(T)^{\sigma},X_{}(T)_{\sigma},\check{\gamma_{\imath}},\gamma_{\imath},\imath\in I_{\sigma})$, cf. [HS, Section 2.2]. For $\bar{\lambda},\bar{\mu}\in X_{}(T)_{\sigma}$, define the partial order $\bar{\mu}\preceq\bar{\lambda}$ if $\bar{\lambda}-\bar{\mu}$ is a sum of positive roots of $H$. Let $X_{}(T)_{\sigma}^{+}$ be the set of dominant weight of $H$. In fact, $X_{}(T)_{\sigma}^{+}$ is the image of the quotient map $X_{}(T)^{+}\to X_{}(T)_{\sigma}$, where $X_{}(T)^{+}$ is the set of dominant weights of $G$. ### 2.2. Twisted affine Grassmannian Let $\sigma$ be a diagram automorphism of $G$ of order $r$. Let $\mathcal{O}$ denote the set of formal power series in $t$ with coefficients in $\mathbb{C}$ and denote $\mathcal{K}$ the set of Laurent series in $t$ with coefficients in $\mathbb{C}$. Denote the automorphism $\sigma$ of order $r$ on $\mathcal{K}$ and $\mathcal{O}$ given by $\sigma$ acts trivially on $\mathbb{C}$ and maps $t\to\epsilon t$ where we fix the primitive $r$-root of unity $\epsilon$. We consider the following twisted affine Grassmannian attached to $G$ and $\sigma$, $\mathcal{G}r_{G}=G(\mathcal{K})^{\sigma}/G(\mathcal{O})^{\sigma}.$ This space has been studied intensively in [BH, HR, PR, Ri]. The ramified geometric Satake correspondence [Zh] asserts that there is an equivalence between the category of spherical perverse sheaves on $\mathcal{G}r_{G}$ and the category of representations of the algebraic group $H=(\check{G})^{\sigma}$. If there is no confusion, we write $\mathcal{G}r$ for convenience. Let $e_{0}$ be the based point in $\mathcal{G}r$. For any $\lambda\in X_{}(T)$, we attach an element $t^{\lambda}\in T(\mathcal{K})$ naturally and define the norm $n^{\lambda}\in T(\mathcal{K})^{\sigma}$ of $t^{\lambda}$ by (1) $n^{\lambda}:=\prod_{i=0}^{r-1}\sigma^{i}(t^{\lambda})=\epsilon^{\sum_{i=1}^{r-1}i\sigma^{i}(\lambda)}t^{\sum\sigma^{i}\lambda}.$ Let $\bar{\lambda}$ be the image of $\lambda$ in $X_{}(T)_{\sigma}$. Set $e_{\bar{\lambda}}=n^{\lambda}\cdot e_{0}\in\mathcal{G}r$. Then $e_{\bar{\lambda}}$ only depends on $\bar{\lambda}$. Following [BH, Zh], $\mathcal{G}r$ admits the following Cartan decomposition (2) $\mathcal{G}r=\bigsqcup_{\bar{\lambda}\in X_{}(T)_{\sigma}^{+}}\mathcal{G}r_{\bar{\lambda}}$ where $\mathcal{G}r_{\bar{\lambda}}=G(\mathcal{O})^{\sigma}\cdot e_{\bar{\lambda}}$ is a Schubert cell. Let $\overline{\mathcal{G}r}_{\bar{\lambda}}$ be the closure of $\mathcal{G}r_{\bar{\lambda}}$. Then $\overline{\mathcal{G}r}_{\bar{\lambda}}=\bigsqcup_{\bar{\mu}\preceq\bar{\lambda}}\mathcal{G}r_{\bar{\mu}},$ and $\dim\overline{\mathcal{G}r}_{\bar{\lambda}}=\langle 2\rho,\bar{\lambda}\rangle,$ where $\rho$ is the half sum of all positive coroots of $H$. By abuse of notation, we still use $\sigma$ to denote the induced automorphism on $\mathfrak{g}$ of order $r$. Then there is a grading on $\mathfrak{g}$, $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}\oplus\cdots\oplus\mathfrak{g}_{r-1}$ where $\mathfrak{g}_{i}$ is the $\epsilon^{i}$-eigenspace. Set $\mathfrak{p}=\mathfrak{g}_{1}.$ Set $\mathcal{O}^{-}=\mathbb{C}[t^{-1}]$. Consider the evaluation map ${\rm ev}_{\infty}:G(\mathcal{O}^{-})\to G$. Let $G(\mathcal{O}^{-})_{0}$ denote its kernel. The map ${\rm ev}_{\infty}$ factors through $G(\mathbb{C}[t^{-1}]/(t^{-2}))\to G$. Note that the kernel of $G(\mathbb{C}[t^{-1}]/(t^{-2}))\to G$ is canonical identified with the vector space $\mathfrak{g}\otimes t$ with respect to the adjoint action of $G$ and $\sigma$. It induces a $G\rtimes\langle\sigma\rangle$-equivariant map $G(\mathcal{O}^{-})_{0}\to\mathfrak{g}\otimes t.$ Taking $\sigma$-invariance, we get a $K$-equivariant map (3) $G(\mathcal{O}^{-})_{0}^{\sigma}\to\mathfrak{p},$ where $K:=G^{\sigma}$. Note that $K$ is a connected simply-connected simple algebraic group, as $G$ is simply-connected. Set $\mathcal{G}r_{0}^{-}:=G(\mathcal{O}^{-})^{\sigma}\cdot e_{0}\simeq G(\mathcal{O}^{-})_{0}^{\sigma}$. Then $\mathcal{G}r_{0}^{-}$ is the open opposite Schubert cell in $\mathcal{G}r$. From (3), we have the following $G^{\sigma}$-equivariant map (4) $\pi:\mathcal{G}r_{0}^{-}\to\mathfrak{p}.$ ###### Lemma 2.1. $\mathcal{G}r_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}$ is nonempty for any $\lambda\in X_{}(T)_{\sigma}^{+}$. ###### Proof. It suffices to show that $G(\mathcal{O})^{\sigma}n^{\lambda}G(\mathcal{O})^{\sigma}\cap G(\mathcal{O}^{-})^{\sigma}$. Let $I$ be the Iwahori subgroup contained in $G(\mathcal{O})^{\sigma}$ and let $I^{-}$ be the opposite Iwahori subgroup contained in $G(\mathcal{O}^{-})^{\sigma}$. It suffices to show that $In^{\lambda}I\cap I^{-}\not=\emptyset$. Let $\mathcal{G}$ be the Kac-Moody group associated to the twisted loop group $G(\mathcal{K})^{\sigma}$, see a construction in [BH, p.14]. There is a projection map $p:\mathcal{G}\to G(\mathcal{K})^{\sigma}$. Let $\mathcal{I}$ (resp. $\mathcal{I}^{-}$) be the preimage of $I$ (resp. $I^{-}$) via the projection $p$. Then we are reduced to show that $\mathcal{I}w\mathcal{I}\cap\mathcal{I}^{-}\not=\emptyset$, where $w$ is an element of the Weyl group $\mathcal{W}$ of $\mathcal{G}$. This is true, since $\mathcal{I}s\mathcal{I}\cap\mathcal{I}^{-}\not=\emptyset$ for any simple reflection $s\in\mathcal{W}$, and for any $y\in\mathcal{W}$ and simple reflection $s$, $\mathcal{I}y\mathcal{I}s\mathcal{I}=\mathcal{I}ys\mathcal{I}$ if $\ell(ys)=\ell(y)+1$, cf. [Ku, 5.1.3 (d)]. ∎ Following [Br, Re, AH], an element $\bar{\lambda}$ of $X_{}(T)_{\sigma}^{+}$ is called small, if $\bar{\lambda}\nsucceq 2\gamma_{0}$, where $\gamma_{0}$ is the highest short root of $H$. The set of all small dominant weights is a lower order ideal of $X_{}(T)_{\sigma}^{+}$, i.e., if $\bar{\mu}\preceq\bar{\lambda}$ and $\bar{\lambda}$ is small, then $\bar{\mu}$ is also small. Let $\mathcal{G}r_{\mathrm{sm}}$ be the union of $\mathcal{G}r_{\bar{\lambda}}$ for small dominant weights $\bar{\lambda}$. Set $\mathcal{M}=\mathcal{G}r_{\mathrm{sm}}\cap\mathcal{G}r_{0}^{-}.$ For each small dominant weight $\bar{\lambda}$, set $\mathcal{M}_{\bar{\lambda}}=\mathcal{G}r_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}.$ Let $\mathcal{N}_{\mathfrak{p}}$ denote the nilpotent cone of $\mathfrak{p}$. We shall prove in Section 4 that $\pi(\mathcal{M})$ is contained $\mathcal{N}_{\mathfrak{p}}$, when $G$ is of type $A_{n}$ and $D_{n}$ and $\sigma$ is of order $2$. Recall that $\gamma_{0}$ is the highest short root of $H$. The following lemma is a twisted analogue of [AH, Lemma 3.3]. ###### Lemma 2.2. If $\sigma$ is a diagram automorphism of order $r$, then $\pi(\mathcal{G}r_{2\gamma_{0}}\cap\mathcal{G}r_{0}^{-})\nsubseteq\mathcal{N}_{\mathfrak{g}_{1}}$. ###### Proof. Let $X_{N}$ be the Dynkin diagram of $G$. Following [Ka, p.128-129], we choose the following root of $G$, ${\theta}_{0}=\begin{cases}{\alpha}_{1}+\cdots+{\alpha}_{2\ell-2},&\quad(X_{N},r)=(A_{2\ell-1},2);\\\ {\alpha}_{1}+\cdots+{\alpha}_{2\ell},&\quad(X_{N},r)=(A_{2\ell},2);\\\ {\alpha}_{1}+\cdots+{\alpha}_{\ell},&\quad(X_{N},r)=(D_{\ell+1},2);\\\ {\alpha}_{1}+{\alpha}_{2}+{\alpha}_{3},&\quad(X_{N},r)=(D_{4},3);\\\ {\alpha}_{1}+2{\alpha}_{2}+2{\alpha}_{3}+{\alpha}_{4}+{\alpha}_{5}+{\alpha}_{6},&\quad(X_{N},r)=(E_{6},2).\end{cases}$ where the label of simple roots ${\alpha}_{i}$ follows from [Ka, TABLE Fin, p.53]. Recall from the section 2.1 that for each $\iota\in I_{\sigma}$, we define simple roots of $H$, $\gamma_{\imath}=\bar{\check{\alpha}}_{i}\in X_{}(T)_{\sigma}$. Let $\check{\theta}_{0}$ be the coroot of ${\theta}_{0}$. Then, $\bar{\check{\theta}}_{0}=\begin{cases}\gamma_{0}&\quad\text{if }(X_{N},r)\not=(A_{2\ell},2)\\\ 2\gamma_{0}&\quad\text{if }(X_{N},r)=(A_{2\ell},2)\end{cases}.$ Suppose $(X_{N},r)\not=(A_{2\ell},2)$. Note that $\theta_{0}\in X^{}(T)$ and $\check{\theta}_{0}:\mathbb{C}^{\times}\to T$. Each $a\in\mathbb{C}^{\times}$ can be identified with $\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\in\mathrm{SL}_{2}.$ For each $i=0,...,r-1$, define a homomorphism $\phi_{\sigma^{i}(\theta_{0})}:\mathrm{SL}_{2}\to G$ given by $\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}\mapsto x_{\sigma^{i}(\theta_{0})}(a),\hskip 14.22636pt\begin{pmatrix}1&0\\\ a&1\end{pmatrix}\mapsto y_{\sigma^{i}(\theta_{0})}(a),\hskip 14.22636pt\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\mapsto\sigma^{i}(\check{\theta}_{0})(a).$ Let $\mathcal{S}$ be the product of $r$ copies of $\mathrm{SL}_{2}$. Then $\check{\theta}_{0}$ can be extended to $\phi:\mathcal{S}\to G$ given by $\phi(g_{0},...,g_{r-1})=\prod_{i=0}^{r-1}\phi_{\sigma^{i}(\theta_{0})}(g_{i}).$ This $\phi$ can extend scalar to $\mathcal{K}$. Abusing notation, define $\sigma:\prod_{i=1}^{r}(\operatorname{SL}_{2}(\mathbb{\mathcal{K}}))_{i}\to\prod_{i=1}^{r}(\operatorname{SL}_{2}(\mathbb{\mathcal{K}}))_{i}$ by $\sigma(g_{1}(t),g_{2}(t),...,g_{r}(t))=(g_{r}(\epsilon t),g_{1}(\epsilon t),...,g_{r-1}(\epsilon t)).$ There exists an isomorphism $\varphi:\mathrm{SL}_{2}(\mathcal{K})\to(\prod_{i=1}^{r}(\operatorname{SL}_{2}(\mathbb{\mathcal{K}}))_{i})^{\sigma}=\\{(g(t),g(\epsilon t),...,g(\epsilon^{r-1}t))\mid g(t)\in\operatorname{SL}_{2}(\mathcal{K})\\}.$ Hence $\phi\circ\varphi:\begin{pmatrix}t&0\\\ 0&t^{-1}\end{pmatrix}\mapsto\left(\begin{pmatrix}\epsilon^{i}t&0\\\ 0&(\epsilon^{i}t)^{-1}\end{pmatrix}\right)_{i=0,...,r-1}\mapsto\prod_{i=0}^{r-1}(\epsilon^{i}t)^{\sigma^{i}\check{\theta}_{0}}=n^{\check{\theta}_{0}}.$ Let $\mathfrak{s}$ be the product of $r$ copies of $\mathfrak{sl}_{2}$. Define $\sigma:\mathfrak{s}\to\mathfrak{s}$ by $\sigma(x_{1},,...,x_{r-1},x_{r})=(\epsilon x_{r},\epsilon x_{1},...,\epsilon x_{r-1}).$ Since $\sigma$ has order $r$, we have $\mathfrak{s}=\oplus_{i=0}^{r-1}\mathfrak{s}_{i}$ where $\mathfrak{s}_{i}$ is the eigenspace of eigenvalue $\epsilon^{i}$. Then $\mathfrak{s}_{1}=\\{(x,\epsilon x,...,\epsilon^{r-1}x)\mid x\in\mathfrak{sl}_{2}\\}\cong\mathfrak{sl}_{2}$. The derivative of $\phi$ is $\mathrm{d}\phi:\mathfrak{s}\to\mathfrak{g}$ which induces $\mathfrak{s}_{1}\to\mathfrak{g}_{1}$. Hence we have the map $\Psi:\mathfrak{sl}_{2}\to\mathfrak{g}_{1}$. Consider the matrix $g(t)\in\operatorname{SL}_{2}(\mathcal{O}^{-})$, $g(t)=\begin{pmatrix}1+t^{-1}&t^{-2}\\\ t^{-1}&1-t^{-1}+t^{-2}\end{pmatrix}=\begin{pmatrix}0&1\\\ -1&t^{2}-t+1\end{pmatrix}\begin{pmatrix}t^{2}&0\\\ 0&t^{-2}\end{pmatrix}\begin{pmatrix}1&0\\\ t^{2}+t&1\end{pmatrix}.$ Then $(\phi\circ\varphi)(g(t))\in G(\mathcal{O})^{\sigma}n^{2\check{\theta}_{0}}G(\mathcal{O})^{\sigma}$. Since $G$ is not type $A_{2l}$, $\bar{\check{\theta}}_{0}=\gamma_{0}$ and then $(\phi\circ\varphi)(g(t))\cdot e_{0}\in\mathcal{G}r_{2\gamma_{0}}\cap\mathcal{G}r_{G,0}^{-}$. We have the commutative diagram ${{\operatorname{Gr}_{\mathrm{SL}_{2},0}^{-}}}$${{\mathcal{G}r_{\mathcal{S},0}^{-}}}$${{\mathcal{G}r_{G,0}^{-}}}$${{\mathfrak{sl}_{2}}}$${{\mathfrak{s}_{1}}}$${{\mathfrak{g}_{1}}}$$\scriptstyle{g(t)\cdot L_{0}\mapsto\varphi(g(t))\cdot e_{0}}$$\scriptstyle{(g_{i}(t))_{i=0}^{r-1}\cdot e_{0}\mapsto\phi(g_{i}(t))_{i=0}^{r-1})\cdot e_{0}}$$\scriptstyle{\pi_{\mathrm{SL}_{2}}}$$\scriptstyle{\pi}$$\scriptstyle{x\mapsto(x,\epsilon x,...,\epsilon^{r-1}x)}$ where $\operatorname{Gr}_{\mathrm{SL}_{2},0}^{-}:=\mathrm{SL}_{2}(\mathcal{O}^{-})_{0}\cdot e_{0}\subset\operatorname{Gr}_{\mathrm{SL}_{2}}$, and ${\mathcal{G}r_{\mathcal{S},0}^{-}}$ is defined similarly. The commutativity follows from $\pi((\phi\circ\varphi)(g(t))\cdot e_{0})=\Psi(\pi_{\operatorname{SL}_{2}}(g(t)\cdot e_{0}))=\Psi\begin{pmatrix}1&0\\\ 1&-1\end{pmatrix},$ where the latter is not nilpotent. It follows that, $\pi(\mathcal{G}r_{2\gamma_{0}})\not\subseteq\mathcal{N}_{\mathfrak{p}}$. Suppose that $(X_{N},r)=(A_{2n},2)$. In this case, $\bar{\check{\theta}}_{0}=2\gamma_{0}$ and $\sigma(\check{\theta}_{0})={\check{\theta}_{0}}$. Then $\check{\theta}_{0}$ can be extended to $\phi:\mathrm{SL}_{2}\to G$ defined by $\begin{pmatrix}1&a\\\ 0&1\end{pmatrix}\mapsto x_{\theta_{0}}(a),\hskip 14.22636pt\begin{pmatrix}1&0\\\ a&1\end{pmatrix}\mapsto y_{\theta_{0}}(a),\hskip 14.22636pt\begin{pmatrix}a&0\\\ 0&a^{-1}\end{pmatrix}\mapsto\check{\theta}_{0}(a).$ $\phi$ can extend the scalar to $\mathcal{K}$. Define a group homomorphism $\sigma:\mathrm{SL}_{2}(\mathcal{K})\to\mathrm{SL}_{2}(\mathcal{K})$ by $\begin{pmatrix}a(t)&b(t)\\\ c(t)&d(t)\end{pmatrix}\mapsto\begin{pmatrix}a(-t)&-b(-t)\\\ -c(-t)&d(-t)\end{pmatrix}$ where $a(t)\in\mathcal{K}$. Then $\phi:\mathrm{SL}_{2}(\mathcal{K})\to G(\mathcal{K})$ is $\sigma$-equivariant. The induced homomorphism $\sigma:\mathfrak{sl}_{2}\to\mathfrak{sl}_{2}$ is given by $\begin{pmatrix}a&b\\\ c&-a\end{pmatrix}\mapsto\begin{pmatrix}a&-b\\\ -c&-a\end{pmatrix}.$ The derivative $\mathrm{d}\phi:\mathfrak{sl}_{2}\to\mathfrak{g}$ induces the map $\Psi:(\mathfrak{sl}_{2})_{1}\to\mathfrak{g}_{1}$. Similarly to the above arguement, we have the commutative diagram ${{\mathcal{G}r_{\mathrm{SL}_{2},0}^{-}}}$${{\mathcal{G}r_{\mathrm{G},0}^{-}}}$${{(\mathfrak{sl}_{2})_{1}}}$${{\mathfrak{g}_{1}}}$$\scriptstyle{\pi_{\mathrm{SL}_{2}}}$$\scriptstyle{\Psi}$$\scriptstyle{\pi}$ where ($\mathfrak{sl}_{2})_{1}$ is the eigenspace of eigenvalue $-1$ under $\sigma$. Now consider $g(t)\in\mathrm{SL}_{2}(\mathcal{O}^{-})^{\sigma}$ $g(t)=\begin{pmatrix}1&t^{-1}\\\ t^{-1}&1+t^{-2}\end{pmatrix}=\begin{pmatrix}0&t\\\ -t^{-1}&1+t^{2}\end{pmatrix}\begin{pmatrix}t^{2}&0\\\ 0&t^{-2}\end{pmatrix}\begin{pmatrix}1&0\\\ t&1\end{pmatrix}.$ Then $\phi(g(t))\in G(\mathcal{O})^{\sigma}n^{\check{\theta}_{0}}G(\mathcal{O})^{\sigma}$ and $\phi(g(t))\cdot e_{0}\in\mathcal{G}r_{2\gamma_{0}}\cap\mathcal{G}r_{0}^{-}$. The result follows from $\pi(\phi(g(t))\cdot e_{0})=\Psi(\pi_{\operatorname{SL}_{2}}(g(t)\cdot e_{0}))=\Psi\begin{pmatrix}0&1\\\ 1&0\end{pmatrix}$ where the latter is not nilpotent. It also follows that, $\pi(\mathcal{G}r_{2\gamma_{0}})\not\subseteq\mathcal{N}_{\mathfrak{p}}$. ∎ ###### Proposition 2.3. For $\bar{\lambda}\in X_{}(T)_{\sigma}^{+}$, if $\pi(\mathcal{G}r_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-})\subset\mathcal{N}_{\mathfrak{p}}$, then $\bar{\lambda}$ is small. ###### Proof. Since $\mathcal{G}r_{0}^{-}$ is an open subset of $\mathcal{G}r$, $\pi(\overline{\mathcal{G}r}_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-})\subset\mathcal{N}_{\mathfrak{p}}$. By Lemma 2.2, $\mathcal{G}r_{2\gamma_{0}}\nsubseteq\overline{\mathcal{G}r}_{\bar{\lambda}}$ which means $\bar{\lambda}\nsucceq 2\gamma_{0}$. ∎ Define the following anti-involution $\iota:G(\mathcal{K})\to G(\mathcal{K}),\hskip 14.22636ptg(t)\mapsto g(-t)^{-1}.$ It can be checked that $\iota$ commutes with $\sigma$, and $\iota$ preserves $G(\mathcal{K})^{\sigma},G(\mathcal{O})^{\sigma}$ and $K^{-}$. This induces the map $\iota:\mathcal{G}r_{0}^{-}\to\mathcal{G}r_{0}^{-},\hskip 14.22636ptg(t)\cdot e_{0}\mapsto g(-t)^{-1}\cdot e_{0}.$ The following lemma will be used in Section 4. ###### Lemma 2.4. For $\bar{\lambda}\in X_{}(T)_{\sigma}^{+}$, $\iota(\mathcal{M}_{\bar{\lambda}})\subset\mathcal{M}_{\bar{\lambda}}$. ###### Proof. It suffices to prove $\iota(n^{\lambda})\in\mathcal{M}_{\bar{\lambda}}$ for each $\lambda\in X_{}(T)^{+}$. $\displaystyle\iota(n^{\lambda})$ $\displaystyle=\iota(\epsilon^{\sigma\lambda+2\sigma^{2}\lambda+...+(r-1)\sigma^{r-1}\lambda}t^{\sum_{i=0}^{r-1}\sigma^{i}\lambda})$ $\displaystyle=\epsilon^{-(\sigma\lambda+2\sigma^{2}\lambda+...+(r-1)\sigma^{r-1}\lambda)}(-1)^{\sum_{i=0}^{r-1}\sigma^{i}\lambda}t^{-\sum_{i=0}^{r-1}\sigma^{i}\lambda}$ $\displaystyle=(-1)^{\sum_{i=0}^{r-1}\sigma^{i}\lambda}n^{-\lambda}.$ Since $(-1)^{\sum_{i=0}^{r-1}\sigma^{i}\lambda}$ is fixed by $\sigma$, $\iota(n^{\lambda})\in G(\mathcal{O})^{\sigma}n^{-\lambda}G(\mathcal{O})^{\sigma}$. Let $W$ be the Weyl group of $G$ with respect to the maximal torus $T$ and $\omega_{0}$ the longest element of $W$. We can choose a representative $\dot{\omega}_{0}\in G$ of $\omega_{0}$ such that $\sigma(\dot{\omega}_{0})=\dot{\omega}_{0}$, cf. [HS, Section 2.3]. When $G$ is of type $D_{2\ell}$ with $\ell\geq 2$, $w_{0}=-1$; otherwise, $w_{0}=-\sigma$ and $\sigma$ is of order $2$, cf. [Hu2, Ex 5, p.71]. If $w_{0}=-1$, it is easy to see that $n^{-\lambda}=w_{0}n^{\lambda}w_{0}^{-1}$. If $w_{0}=-\sigma$ and $\sigma$ has order 2, $n^{-\lambda}=(-1)^{-\sigma\lambda}t^{-(\lambda+\sigma\lambda)}=(-1)^{w_{0}\lambda}t^{w_{0}(\lambda+\sigma\lambda)}=w_{0}(-1)^{\lambda}t^{\lambda+\sigma\lambda}w_{0}^{-1}=w_{0}(-1)^{\lambda+\sigma\lambda}n^{\lambda}w_{0}^{-1}.$ In any case, $\iota(n^{\lambda})\in G(\mathcal{O})^{\sigma}n^{\lambda}G(\mathcal{O})^{\sigma}$. ∎ ## 3\. Nilpotent orbits in the space of self-adjoint maps In this section, we will review some facts on the nilpotent orbits in certain symmetric spaces. These results are known, cf. [Se]. We provide proofs here, as the proofs in [Se] are omitted. Let $B=\langle\cdot,\cdot\rangle$ be a nondegenerate symmetric or skew- symmetric bilinear form on a vector space $V=\mathbb{C}^{m}$ and $\mathcal{A}$ the set of self-adjoint linear maps under the bilinear form. In this section, we describe the classification of nilpotent orbits in the space $\mathcal{A}$ in Theorem 3.3, Theorem 3.4 and Theorem 3.6. The isometry group of the form $B$ is $I_{B}=\\{g\in\mathrm{GL}(V)\mid\langle gu,gv\rangle=\langle u,v\rangle\text{ for all }u,v\in V\\},$ whose Lie alegbra is (5) $\mathfrak{g}_{B}:=\\{X\in\mathfrak{sl}(V)\mid\langle Xu,v\rangle+\langle u,Xv\rangle=0\text{ for all }u,v\in V\\}.$ When $B$ is symplectic, $\dim V$ is even, $I_{B}\cong\mathrm{Sp}_{2n}$ and $\mathfrak{g}_{B}\simeq\mathfrak{sp}_{2n}$ where $m=2n$. When $B$ is symmetric, $I_{B}\cong\mathrm{O}_{m}$ and $\mathfrak{g}_{B}\cong\mathfrak{so}_{m}$. The group $I_{B}$ acts on the space of self-adjoint linear maps (6) $\mathcal{A}=\\{X\in\operatorname{End}(V)\mid\langle Xu,v\rangle=\langle u,Xv\rangle\text{ for all }u,v\in V\\}$ by conjugation. The orbit is called nilpotent if it is the orbit of a nilpotent element of $\mathcal{A}$. Let $\mathfrak{g}$ be a complex semisimple Lie algebra. Suppose that $\mathfrak{g}$ has $\mathbb{Z}_{m}$-grading $\mathfrak{g}=\bigoplus_{i\in\mathbb{Z}_{m}}\mathfrak{g}_{i}$ so that $[\mathfrak{g}_{k},\mathfrak{g}_{\ell}]\subset\mathfrak{g}_{k+\ell}$. We have the following graded version of Jacobson-Morozov theorem. ###### Lemma 3.1. Let $X$ be a nonzero nilpotent element in $\mathfrak{g}_{i}$. There exists an $\mathfrak{sl}_{2}$-triple $H,X,Y$ such that $H\in\mathfrak{g}_{0}$ and $Y\in\mathfrak{g}_{-i}$. ###### Proof. This Lemma follows from the usual Jacobson-Morozov Theorem, and the proof is similar to [EK, Lemma 1.1]. ∎ For each $A\in\mathfrak{sl}(V)$, as a linear map, we denote its adjoint by $A^{}$ under the form $B$. We define an involution $\sigma$ on $\mathfrak{g}=\mathfrak{sl}(V)$ by (7) $\sigma(A)=-A^{}.$ Then $\mathfrak{g}$ is the direct sum of eigenspaces, $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$. Thus $\mathfrak{g}_{0}=\mathfrak{g}_{B}$ and $\mathfrak{g}_{1}=\mathcal{A}$. Fix a nonzero nilpotent element $X\in\mathcal{A}$. By Lemma 3.1, there exists $Y\in\mathcal{A}$ and $H\in\mathfrak{g}_{B}$, such that $X,Y,H$ is an $\mathfrak{sl}_{2}$-triple. This induces a representation of $\mathfrak{sl}_{2}$ on $V$ and hence we have a decomposition (8) $V=\bigoplus_{r\geq 0}M(r)$ where $M(r)$ is a finite direct sum of irreducible representation of $\mathfrak{sl}_{2}$ of highest weight $r$. For $r\geq 0$, let $H(r)$ be the highest weight space in $M(r)$. Define a new bilinear form $(\cdot,\cdot)$ on $H(r)$ by $(u,v)_{r}=\langle u,Y^{r}v\rangle.$ ###### Lemma 3.2. For any $r\geq 0$, $(\cdot,\cdot)_{r}$ is symplectic (resp. symmetric) if $B$ is symplectic (resp. symmetric). ###### Proof. We assume $B$ is symplectic. The proof is similar when $B$ is symmetric. It is easy to see that $(\cdot,\cdot)_{r}$ is skew-symmetric. It remains to show that $(\cdot,\cdot)_{r}$ is nondegerate. Let $V_{r}$ be an $r$-weight space in $\mathbb{C}^{2n}$. For any $u\in V_{r}$, $v\in V_{s}$ with $s\neq-r$, $(r+s)\langle u,v\rangle=\langle ru,v\rangle+\langle u,sv\rangle=\langle Hu,v\rangle+\langle u,Hv\rangle=0$ This implies that $V_{r}$ and $V_{s}$ are $\langle\cdot,\cdot\rangle$-orthogonal. Let $W=\text{Span}\\{u\in V_{r}\mid u=Yv\text{ for some }v\in\mathbb{C}^{2n}\\}.$ It can be seen that $V_{r}=H(r)\oplus W$. For $u\in H(r)$ and $v\in W$, write $v=Yv^{\prime}$, $(u,v)_{r}=\langle u,Y^{r}v\rangle=\langle u,Y^{r+1}v^{\prime}\rangle=\langle Y^{r+1}u,v^{\prime}\rangle=0.$ Hence $H(r)$ is $(\cdot,\cdot)_{r}$-orthogonal to $W$. We claim that $\langle\cdot,\cdot\rangle:(Y^{r}\cdot H(r))\times H(r)\to\mathbb{C}$ is nondegenerate. Let $u=Y^{r}u^{\prime}\in Y^{r}\cdot H(r)$ be such that $\langle u,v\rangle=0$ for all $v\in H(r)$. For each $w\in\mathbb{C}^{2n}$, write $w=\sum_{s}w_{s}$ where each $w_{s}$ belongs to $V_{s}$. Since $u\in V_{-r}$, $\langle u,w_{s}\rangle=0$ for $s\neq r$. Write $w_{r}=w_{1}+w_{2}$ where $w_{1}\in H(r)$ and $w_{2}=Yw^{\prime}_{2}\in W$. By the assumption $\langle u,w_{1}\rangle=0$ and hence $\langle u,w_{r}\rangle=\langle u,w_{2}\rangle=\langle Y^{r}u^{\prime},Yw^{\prime}_{2}\rangle=\\-\langle Y^{r+1}u^{\prime},w^{\prime}_{2}\rangle=0.$ We obtain $\langle u,w\rangle=0$ for any $w$ and hence $u=0$. This claim implies that $(\cdot,\cdot)_{r}$ is nondegenerate. ∎ A partition of a positive integer is denoted by a tuple $[d_{1},d_{2},...,d_{k}]$ of positive integers. We use the exponent notation $[a_{1}^{i_{1}},...,a_{r}^{i_{r}}]$ to denote a partition where $a_{j}^{i_{j}}$ means there are $i_{j}$ copies of $a_{j}$. For example, $[3^{2},1^{4}]=[3,3,1,1,1,1]$ is a partition of 10. Put $r_{i}=\|\\{j\mid d_{j}=i\\}\|$ and $s_{i}=\|\\{j\mid d_{j}\geq i\\}\|$. In fact, each partition can be illustrated by Young diagram and then $s_{i}$ is the $i$-th part of the dual diagram. The following Theorem gives the parametrization of nilpotent $I_{B}$-orbits in $\mathcal{A}$. ###### Theorem 3.3. There exists one-to-one correspondences $\\{\text{nilpotent }\mathrm{Sp}_{2n}\text{-orbits in }\mathcal{A}\\}\leftrightarrow\left\\{\begin{array}[]{c}\text{partitions of }2n\text{ such that}\\\ \text{every part occurs with even multiplicity}\end{array}\right\\}.$ and $\\{\text{nilpotent }\mathrm{O}_{m}\text{-orbits in }\mathcal{A}\\}\leftrightarrow\\{\text{partitions of }m\\}.$ ###### Proof. The proof is similar to [CM, Lemma 5.1.17]. For the case that $B$ is symplectic, it suffices to show that any nilpotent element in $\mathcal{A}$ gives arise the partition of $2n$ such that every part occurs with even multiplicity. Given nilpoent $X\in\mathcal{A}$, a number of Jordan blocks of size $r+1$ equals to the multiplicity of $M(r)$ in $\mathbb{C}^{2n}$ which is exactly $\dim H(r)$. By Lemma 3.2, $\dim H(r)$ is even for every $r$. If $B$ is symmetric, there are no constraints on $\dim H(r)$ which means there are no conditions on partitions of $m$. ∎ ###### Theorem 3.4. There exists one-to-one correspondence $\\{\text{nilpotent }\mathrm{SO}_{2n+1}\text{-orbits in }\mathcal{A}\\}\leftrightarrow\left\\{\text{partitions of }2n+1\right\\}.$ ###### Proof. Since $\mathrm{O}_{2n+1}=\mathrm{SO}_{2n+1}\times\\{\pm I_{2n+1}\\}$, the orbits under $\mathrm{O}_{2n+1}$ and $\mathrm{SO}_{2n+1}$ coincide. The results immediately follows from Theorem 3.3 ∎ Consider the case that $B$ is symmetric and $m=2n$. Given nilpotent elements $X,X^{\prime}\in\mathcal{A}$ whose partitions are the same and have at least one odd part. Say that they are conjugated by an element $g\in\mathrm{O}_{2n}$. If $\det g=1$, we conclude that $X,X^{\prime}$ are in the same $\mathrm{SO}_{2n}$-orbits. Suppose that $\det g=-1$. We modify this $g$ so that it has determinant $1$. Note that either $X$ or $X^{\prime}$ gives the same decomposition (8). An odd part in the partition corresponds to an odd dimensional irreducible representation $S$ of $\mathfrak{sl}_{2}$ in $\mathbb{C}^{2n}$. We put $h=g$ except that $h(v)=-g(v)$ for $v\in S$. Therefore, $\det h=1$, and $X$ and $X^{\prime}$ are conjugated by $h$. If there is no odd parts, we need the following Lemma. ###### Lemma 3.5. Let $X$ be a nilpotent element in $\mathcal{A}$ whose partition contains only even parts, and $k\in\mathrm{O}_{2n}$ such that $k\cdot X=kXk^{-1}=X$. Then $\det k=1$. ###### Proof. Let $\mathrm{O}_{2n}^{X}$ be the stabilizer group of $O_{2n}$ at $X$. Then $k\in\mathrm{O}_{2n}^{X}$. By multiplicative Jordan decomposition, cf. [Bo, Theorem 4.4, p.83], let $k_{s}\in O_{2n}^{X}$ be the semisimple part of $k$. Then $\det k_{s}=\det k$. Hence we may assume that $k$ is semisimple. Let $\sigma$ be an automorphism on $\mathfrak{g}=\mathfrak{sl}(V)$ defined by (7). Then $\sigma$ commutes with ${\rm Ad}k$ on $\mathfrak{g}$, as $k\in\mathrm{O}_{2n}$. Then $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$, and we have the decomposition of $k$-stabilizers $\mathfrak{g}^{k}=\mathfrak{g}_{0}^{k}\oplus\mathfrak{g}_{1}^{k}$ where $\mathfrak{g}_{i}^{k}=\mathfrak{g}_{i}\cap\mathfrak{g}^{k}$. Since $\mathfrak{g}^{k}$ is reductive and $X\in\mathfrak{g}_{1}^{k}$, by Lemma 3.1, there exists an $\mathfrak{sl}_{2}$-triple $H,X,Y$ such that $X,Y\in\mathfrak{g}_{1}^{k}$ and $H\in\mathfrak{g}_{0}^{k}$ and hence we have the decomposition (8). It is easy to see that $k(M(r))\subset M(r)$, and also, $k$ stabilizes each weight space of $M(r)$. Recall that $\langle\cdot,\cdot\rangle$ is a nondegenerate symmetric form on $V=\mathbb{C}^{2n}$ and a form on $H(r)$ given by $(u,v)_{r}=\langle u,Y^{r}v\rangle$ is also symmetric for any $r\geq 0$. For any $u,v\in H(r)$, $(ku,kv)_{r}=\langle ku,Y^{r}kv\rangle=\langle ku,kY^{r}v\rangle=\langle u,Y^{r}v\rangle=(u,v)_{r}.$ Hence $k\big{\|}_{H(r)}\in\mathrm{O}(H(r))$ for any $r$. In particular $\det\left(k\big{\|}_{H(r)}\right)=\pm 1$. Let $M(r)_{\ell}$ be an $\ell$-weight space, and $L(r)$ the lowest weight space in $M(r)$. Observe that $X\big{\|}_{M(r)_{\ell}}$ is an isomorphism from $M(r)_{\ell}$ to $M(r)_{\ell+2}$ and the diagram ${{L(r)}}$${{M(r)_{\ell}}}$${{M(r)_{\ell+2}}}$${{H(r)}}$${{L(r)}}$${{M(r)_{\ell}}}$${{M(r)_{\ell+2}}}$${{H(r)}}$$\scriptstyle{X\big{\|}_{M(r)_{\ell}}}$$\scriptstyle{k\big{\|}_{M(r)_{\ell}}}$$\scriptstyle{k\big{\|}_{M(r)_{\ell+2}}}$$\scriptstyle{X\big{\|}_{M(r)_{\ell}}}$$\scriptstyle{k\big{\|}_{L(r)}}$$\scriptstyle{k\big{\|}_{H(r)}}$ commutes. Then $k\big{\|}_{M(r)_{\ell}}$ has the same determinant for all $\ell$. Since $X$ has only even parts, a number of weight spaces in $M(r)$ is even for each $r$. Then $\det k=\prod_{r}\det\left(k\big{\|}_{M(r)}\right)=\prod_{r}\prod_{\ell}\det\left(k\big{\|}_{M(r)_{\ell}}\right)=\prod_{r}\left(\det\left(k\big{\|}_{H(r)}\right)\right)^{\ell}=1$ as desired. ∎ Now suppose that $X,X^{\prime}\in\mathcal{A}$ have the same partition and contain only even parts. If $\det g=1$, they are in the same $\mathrm{SO}_{2n}$-orbits. Suppose that $\det g=-1$ and they are conjugated by another element $h\in\mathrm{SO}_{2n}$. Say $g\cdot X=X^{\prime}=h\cdot X$ and let $k=g^{-1}h$. Then $\det k=(\det g^{-1})(\det h)=-1$ but this contradicts to Lemma 3.5. In this case, it means $X,X^{\prime}$ are conjugated by an element in $\mathrm{O}_{2n}$ of determinant -1 only. We have the following theorem: ###### Theorem 3.6. Nilpotent $\mathrm{SO}_{2n}$-orbits in $\mathcal{A}$ are parametrized by partitions of $2n$ except that the partitions with only even parts correspond to two orbits. For each nilpotent element $X$ having the partition $[d_{1},d_{2},...,d_{k}]$, we denote the $I_{B}$-orbits of $X$ by $\mathcal{O}_{X}$,$\mathcal{O}_{[d_{1},d_{2},...,d_{k}]}$, or simply $[d_{1},d_{2},...,d_{k}]$. We are now ready to compute the dimension of nilpotent $I_{B}$-orbits. ###### Theorem 3.7. Let $X$ be a nilpotent element in $\mathcal{A}$. Then the dimension of $I_{B}$-orbit of $X$ is $\dim\mathcal{O}_{X}=\frac{1}{2}\left(m^{2}-\sum_{i}s_{i}^{2}\right).$ ###### Proof. Suppose that $B$ is symplectic on $\mathbb{C}^{m}$, $m=2n$. Recall that we have the decomposition (8). For each $Z\in\mathfrak{g}_{B}^{X}$, we consider how $Z$ sends $M(d)$. Suppose that $Z$ sends $M(d)$ to $M(e)$ for $d\neq e$. Since $Z$ and $X$ commute, by theory of representations of $\mathfrak{sl}_{2}$, $Z$ is uniquely determined by a linear map $L(d)\to M(e)$ where $L(d)$ is the lowest weight space in $M(d)$. In this case, $Z$ sends $L(e)$ to $M(d)$. Thus we can assume $d<e$. For $v\in L(d)$, $X^{d+1}v=0$ and then $ZX^{d+1}v=X^{d+1}Zv=0$ is the constraint of $Zv=0$. Note that $r_{d+1}=\dim L(d)$. Therefore the set of all linear maps from $L(d)$ to $M(e)$ forms a vector space of dimension $(d+1)r_{d+1}r_{e+1}$. Now consider the case $Z$ sends $M(d)$ to itself. We consider where $Z$ sends $L(d)$ to. Suppose that $Z$ sends $L(d)$ to $H(d)$. We define a new bilinear form $(\cdot,\cdot)_{d}$ on $L(d)$ given by $(u,v)_{d}=\langle u,Zv\rangle$. It can be checked $(\cdot,\cdot)_{d}$ is symmetric and completely determine the map $Z$. The set of all such $(\cdot,\cdot)_{d}$ forms a vector space of dimension $\frac{1}{2}r_{d+1}(r_{d+1}+1)$. If $Z$ sends $L(d)$ to $(d-2)$-weight space in $M(d)$, we define the new form by $(u,v)_{d-2}=\langle u,XZv\rangle$. Again, this form is symmetric and completely determine the map $Z$. Continue this process up to the case $Z$ sends $L(d)$ to itself. We obtain $\displaystyle\dim\mathfrak{k}^{X}$ $\displaystyle=\sum_{d\geq 0}\left[(d+1)\left(\sum_{e>d}r_{d+1}r_{e+1}\right)+\dfrac{d+1}{2}r_{d+1}(r_{d+1}+1)\right]$ $\displaystyle=\left[r_{1}(r_{2}+r_{3}+\cdots)+\dfrac{1}{2}r_{1}(r_{1}+1)\right]+\left[2r_{2}(r_{3}+r_{4}+\cdots)+\dfrac{2}{2}r_{2}(r_{2}+1)\right]$ $\displaystyle\hskip 56.9055pt+\left[3r_{3}(r_{4}+r_{5}+\cdots)+\dfrac{3}{2}r_{3}(r_{3}+1)\right]+\cdots$ $\displaystyle=\left[\dfrac{1}{2}r_{1}(r_{1}+2r_{2}+2r_{3}+...)+\dfrac{1}{2}r_{1}\right]+\left[\dfrac{2}{2}r_{2}(r_{2}+2r_{3}+2r_{4}+...)+\dfrac{2}{2}r_{2}\right]$ $\displaystyle\hskip 56.9055pt+\left[\dfrac{3}{2}r_{3}(r_{3}+2r_{4}+2r_{5}+...)+\dfrac{3}{2}r_{3}\right]+\cdots$ $\displaystyle=\left[\dfrac{1}{2}(s_{1}-s_{2})(s_{1}+s_{2})+\dfrac{1}{2}r_{1}\right]+\left[\dfrac{2}{2}(s_{2}-s_{3})(s_{2}+s_{3})+\dfrac{2}{2}r_{2}\right]$ $\displaystyle\hskip 56.9055pt+\left[\dfrac{3}{2}(s_{3}-s_{4})(s_{3}+s_{4})+\dfrac{3}{2}r_{3}\right]+\cdots$ $\displaystyle=\dfrac{1}{2}\sum_{i}s_{i}^{2}+\dfrac{1}{2}(r_{1}+2r_{2}+3r_{3}+\cdots)+\cdots$ $\displaystyle=\dfrac{1}{2}\sum_{i}s_{i}^{2}+\dfrac{1}{2}\sum_{i}s_{i}.$ $\displaystyle=n+\dfrac{1}{2}\sum_{i}s_{i}^{2}$ and hence $\dim\mathcal{O}_{X}=\dim\mathfrak{g}_{B}-\dim\mathfrak{g}_{B}^{X}=(2n^{2}+n)-\left(n+\dfrac{1}{2}\sum_{i}s_{i}^{2}\right)=\dfrac{m^{2}}{2}-\frac{1}{2}\sum_{i}s_{i}^{2}.$ If $B$ is symmetric, the arguement is similar except that the form $(u,v)_{d}$ is symplectic and hence the vector space consisting of such forms $(\cdot,\cdot)_{d}$ has dimension $\frac{1}{2}r_{d+1}(r_{d+1}-1)$. ∎ ###### Remark 3.8. The dimension of $I_{B}$-orbits can also be obtained from [Se, 3.1.c, 3.2.b], where the formulae are not uniform and the proofs are also omitted. The closure relation on the set of nilpotent orbits in $\mathcal{A}$ is given by $\mathcal{O}_{Y}\preceq\mathcal{O}_{X}\text{ if }\mathcal{O}_{Y}\subset\overline{\mathcal{O}}_{X}$ for nilpotent elements $X,Y\in\mathcal{A}$. Given two partitions $\bar{d}=[d_{1},...,d_{N}],\bar{f}=[f_{1},...,f_{N}]$ of $N$ (put some $d_{i},f_{i}=0$ if needed). We say that $\bar{d}$ dominates $\bar{f}$, denoted by $\bar{d}\succeq\bar{f}$ if $d_{1}\geq f_{1}$ $d_{1}+d_{2}\geq f_{1}+f_{2}$ $\vdots$ $d_{1}+...+d_{N}\geq f_{1}+...+f_{N}.$ ###### Theorem 3.9. Let $X,Y$ be nilpotent elements in $\mathcal{A}$ having partition $\bar{d},\bar{f}$, respectively. Then $\mathcal{O}_{\bar{d}}\succeq\mathcal{O}_{\bar{f}}$ if and only if $\bar{d}$ dominates $\bar{f}$. ###### Proof. See [Oh, Theorem 1]. ∎ For example, all nilpotent $\mathrm{Sp}_{10}$-orbits in $\mathcal{A}\subset\mathfrak{sl}_{10}$ are $\mathcal{O}_{[5^{2}]}\succeq\mathcal{O}_{[4^{2},1^{2}]}\succeq\mathcal{O}_{[3^{2},2^{2}]}\succeq\mathcal{O}_{[3^{2},1^{4}]}\succeq\mathcal{O}_{[2^{4},1^{2}]}\succeq\mathcal{O}_{[2^{2},1^{6}]}\succeq\mathcal{O}_{[1^{10}]}.$ The dimensions are 40, 36, 32, 28, 24, 16, 0, respectively. ## 4\. The connection between Schubert cells and nilpotent $K$-orbits The goal of this section is to show that for any small $\bar{\lambda}$, $\mathcal{M}_{\bar{\lambda}}$ is sent to the nilpotent cone $\mathcal{N}_{\mathfrak{p}}$ by the map $\pi$, and show that how each $\mathcal{M}_{\bar{\lambda}}$ is sent to nilpotent $K$-orbits in the $\mathcal{N}_{\mathfrak{p}}$. Theorem 4.2 describes the image $\pi(\mathcal{M}_{\bar{\lambda}})$ where the proofs are provided by case-by- case consideration in this section. Let $X_{N}$ be the type of Dynkin diagram of $G$ and $\sigma$ the diagram automorphism on $G$ of order $r$, denoted by the pair $(X_{N},r)$. We consider the cases $(X_{N},r)=(A_{2\ell},2),(A_{2\ell-1},2)$, and $(D_{\ell+1}2)$. Then $H=(\check{G})^{\sigma}$ is either of type $B_{\ell}$ or $C_{\ell}$. We make the following labelling for simple roots $\gamma_{i}$: (9) $\begin{cases}\gamma_{1}=\gamma_{\\{1,2\ell-1\\}},\dots,\gamma_{\ell-1}=\gamma_{\\{\ell-1,n+1\\}},\gamma_{n}=\gamma_{\\{\ell\\}}&\quad(X_{N},r)=(A_{2\ell-1},2);\\\ \gamma_{1}=\gamma_{\\{1,2\ell\\}},\dots,\gamma_{\ell-1}=\gamma_{\\{\ell-1,\ell+2\\}},\gamma_{\ell}=\gamma_{\\{\ell,\ell+1\\}}&\quad(X_{N},r)=(A_{2\ell},2);\\\ \gamma_{1}=\gamma_{\\{1\\}},\dots,\gamma_{\ell-1}=\gamma_{\\{\ell-1\\}},\gamma_{\ell}=\gamma_{\\{\ell,\ell+1\\}}&\quad(X_{N},r)=(D_{\ell+1},2).\\\ \end{cases}$ This labelling of vertices of type $B_{\ell}$ and $C_{\ell}$ agrees with the labelling in [Ka, TABLE Fin, p.53]. Then the highest short root $\gamma_{0}$ of $H$ can be described in the following table. $(X_{N},r)$ \| $G$ \| $H$ \| Simple roots of $H$ \| Highest short root $\gamma_{0}$ of $H$ ---\|---\|---\|---\|--- $(A_{2\ell},2)$ \| $\mathrm{SL}_{2\ell+1}$ \| $\mathrm{PSO}_{2\ell+1}$ \| $\gamma_{i}=\bar{\check{\alpha}}_{i}=\bar{\check{\alpha}}_{2\ell-i+1}$, $i=1,...,\ell$ \| $\gamma_{1}+\gamma_{2}+...+\gamma_{\ell}.$ $(A_{2\ell-1},2)$ \| $\mathrm{SL}_{2\ell}$ \| $\mathrm{PSp}_{2\ell}$ \| $\gamma_{i}=\bar{\check{\alpha}}_{i}=\bar{\check{\alpha}}_{2\ell-i}$, $i=1,...,\ell$ \| $\gamma_{1}+2\gamma_{2}+...+2\gamma_{\ell-1}+\gamma_{\ell}.$ $(D_{\ell+1},2)$ \| $\mathrm{Spin}_{2\ell+2}$ \| $\mathrm{PSO}_{2\ell+1}$ \| $\gamma_{i}=\bar{\check{\alpha}}_{i},i=1,...,\ell-1$ $\gamma_{\ell}=\bar{\check{\alpha}}_{\ell}=\bar{\check{\alpha}}_{\ell+1}$ \| $\gamma_{1}+\gamma_{2}+...+\gamma_{\ell}.$ From Section 2.1, we can identify the weight lattice of $H$ with $X_{}(T)_{\sigma}$, and the set of dominant weights of $H$ with $X_{}(T)^{+}_{\sigma}$. Then, from the construction of root system of classical Lie algebras given in [Hu2, §12], we can make the following identifications: $X_{}(T)_{\sigma}\cong\begin{cases}\mathbb{Z}^{\ell}&\text{if }H=B_{\ell};\\\ \\{(a_{1},...,a_{\ell})\in\mathbb{Z}^{\ell}\mid a_{1}+\cdots+a_{\ell}\in 2\mathbb{Z}\\}&\text{if }H=C_{\ell}\end{cases}$ and $X_{}(T)_{\sigma}^{+}\cong\\{(a_{1},...,a_{\ell})\in X_{}(T)_{\sigma}\mid a_{1}\geq\cdots\geq a_{\ell}\geq 0\\}$ for any cases of $H$. This identification preserves the relation on $X_{}(T)_{\sigma}^{+}$ and the dominance relation on $\\{(a_{1},...,a_{\ell})\in X_{}(T)_{\sigma}\mid a_{1}\geq\cdots\geq a_{\ell}\geq 0\\}$. In the Table 1, we can further make the following identifications for simple roots and fundamental weights of $H$. Those fundamental weights follows from [Hu2, Table 1., p.69]. $H$ \| Simple roots \| $\gamma_{0}$ \| Fundamental weights ---\|---\|---\|--- $B_{\ell}$ \| $\gamma_{1}=(1,-1,0,0,...,0)$ $\gamma_{2}=(0,1,-1,0,...,0)$ $\vdots$ $\gamma_{\ell-1}=(0,0,0,...,1,-1)$ $\gamma_{\ell}=(0,0,0,...,0,1)$ \| $(1,0,0,...,0,0)$ \| $\omega_{j}=(1^{j}0^{\ell-j})$, $j=0,...,\ell-1$ $\omega_{\ell}=(\dfrac{1}{2},\dfrac{1}{2},...,\dfrac{1}{2})$ $C_{\ell}$ \| $\gamma_{1}=(1,-1,0,0,...,0)$ $\gamma_{2}=(0,1,-1,0,...,0)$ $\vdots$ $\gamma_{\ell-1}=(0,0,0,...,1,-1)$ $\gamma_{\ell}=(0,0,0,...,0,2).$ \| $(1,1,0,...,0,0)$ \| $\omega_{j}=(1^{j}0^{\ell-j}),j=0,...,\ell$ Table 1. Simple roots, highest short root, and fundamental weights of $H$ in term of tuples The following lemma is well-known, cf.[AH]. We give a self-contained proof here. ###### Lemma 4.1. All small dominant weights of $H$ are 1. (1) $\omega_{j}=(1^{j}0^{\ell-j})$, $j=0,...,\ell-1$, $2\omega_{\ell}=(1,1,...,1)$, if $H$ has the type $B_{\ell}$. 2. (2) $\omega_{1}+\omega_{2j+1}=(21^{2j}0^{\ell-2j-1})$, $j=0,...,\lfloor\frac{\ell-1}{2}\rfloor$ $\omega_{2j}=(1^{2j}0^{\ell-2j})$, $j=0,...,\lfloor\frac{\ell}{2}\rfloor$, if $H$ has the type $C_{\ell}$. ###### Proof. Suppose that $H$ has the type $B_{\ell}$. The highest short root is $\gamma_{0}=(1,0,...,0)$. By definition, a dominant weight $(a_{1},...,a_{\ell})\in X_{}(T)_{\sigma}^{+}$ is small if and only if $(a_{1},...,a_{\ell})\nsucceq(2,0,...,0)$ which is equivalent to $a_{1}\leq 1$. This proves the first part. Now assume that $H$ has the type $C_{\ell}$. The highest short root is $\gamma_{0}=(1,1,...,0)$. Let $(a_{1},...,a_{\ell})\in X_{}(T)_{\sigma}^{+}$ be a small dominant weight. Then $(a_{1},...,a_{\ell})\nsucceq(2,2,...,0)$ and so $a_{1}\leq 2$. If $a_{1}=1$, then $(a_{1},...,a_{\ell})=(1^{2j}0^{\ell-2j})$. If $a_{1}=2$, then $a_{2}<2$ and hence $(a_{1},...,a_{\ell})=(21^{2j}0^{\ell-2j-1})$. ∎ Let $\bar{\mu}$ be the maximal element among all small dominant weights of $H$, then $\mathcal{G}r_{\mathrm{sm}}=\coprod_{{\bar{\lambda}}\preceq\bar{\mu},\\\ \bar{\lambda}\text{ small}}\mathcal{G}r_{\bar{\lambda}}=\overline{\mathcal{G}r}_{\bar{\mu}}.$ Since $\mathcal{G}r_{\mathrm{sm}}$ is irreducible and $\mathcal{M}$ is an open subset of $\mathcal{G}r_{sm}$, $\mathcal{M}$ is irreducible. The following theorem is the main result of this section. ###### Theorem 4.2. If $\bar{\lambda}$ is small, then $\pi(\mathcal{M}_{\bar{\lambda}})$ is contained in $\mathcal{N}_{\mathfrak{p}}$. Moreover, the image $\pi(\mathcal{M}_{\bar{\lambda}})$ can be described as the union of nilpotent orbits as the following table: $(X_{N},r)$ \| Small dominant weight $\overline{\lambda}$ of $H$ \| Orbits in $\pi(\mathcal{M}_{\bar{\lambda}})$ ---\|---\|--- $(A_{2\ell},2)$ \| $(1^{j}0^{\ell-j}),j=0,1,\dots,\ell$ \| $[2^{j}1^{2\ell-2j+1}]$ $(A_{2\ell-1},2)$ \| $(1^{2j}0^{\ell-2j}),j=0,1,\dots,\lfloor\frac{\ell}{2}\rfloor$ \| $[2^{2j}1^{2\ell-4j}]$ $(20^{\ell-1})$ \| $0,[2^{2}1^{2\ell-4}]$ $(21^{2}0^{\ell-3})$ \| $[2^{2}1^{2\ell-4}],[2^{4}1^{2\ell-8}],[3^{2}1^{2\ell-6}]$ $(21^{2j}0^{\ell-2j-1}),j=2,\dots,\lfloor\frac{\ell-3}{2}\rfloor$ \| $[2^{2j}1^{2\ell-4j}],[2^{2j+2}1^{2\ell-4j-4}],$ $[3^{2}2^{2j-2}1^{2\ell-4j-2}],[3^{2}2^{2j-4}1^{2\ell-4j+2}]$ $(21^{2\lfloor\frac{\ell-1}{2}\rfloor}0^{\ell-2\lfloor\frac{\ell-1}{2}\rfloor-1})$ \| $[2^{\ell-2}1^{4}],[2^{\ell}],$ $[3^{2}2^{\ell-4}1^{2}],[3^{2}2^{\ell-6}1^{4}]$ , if $\ell$ is even \| $[2^{\ell-1}1^{2}]$, $[3^{2}2^{\ell-3}],[3^{2}2^{\ell-5}1^{4}]$ , if $\ell$ is odd $(D_{\ell+1},2)$ \| $(1^{j}0^{\ell-j}),j=0,1,\dots,\ell$ \| $0$, if $j=0$ $0,[31^{2\ell-1}]$, if $j$ is even, $j\geq 2$ \| \| $[31^{2\ell-1}]$, if $j$ is odd where the nilpotent orbit $[a_{1}^{i_{1}},...,a_{r}^{i_{r}}]$ in the above table means empty set if the associated partition is invalid for some small $\ell$. This theorem follows from Theorem 4.5, Theorem 4.6, Theorem 4.10, and Theorem 4.14, which will be proved separately case by case. The partial orders of small dominant weights of $H$ are shown in the below picture, where the partial order is compatible with the height. ${{2\omega_{\ell}}}$${{\omega_{1}+\omega_{2\lfloor\frac{\ell-1}{2}\rfloor+1}}}$${{\omega_{\ell-1}}}$${{\omega_{1}+\omega_{2\lfloor\frac{\ell-1}{2}\rfloor-1}}}$${{\omega_{\ell-2}}}$${{\omega_{1}+\omega_{2\lfloor\frac{\ell-1}{2}\rfloor-3}}}$${{\omega_{2\lfloor\frac{\ell}{2}\rfloor}}}$${{\omega_{2\lfloor\frac{\ell}{2}\rfloor-2}}}$${{\omega_{1}+\omega_{3}}}$${{\omega_{2}}}$${{2\omega_{1}}}$${{\omega_{4}}}$${{\omega_{1}}}$${{\omega_{2}}}$${{0}}$${{0}}$${\text{Type }B_{\ell}}$${\text{Type }C_{\ell}}$ We first recall a crucial lemma from [AH, Lemma 4.3]. ###### Lemma 4.3. Let $g=\sum_{i=N}^{\infty}x_{i}t^{i}\in\mathrm{SL}_{n}(\mathcal{K})$, where $x_{N}\neq 0$. Let ${\lambda}=(a_{1},a_{2},...,a_{n})$ be a tuple of integers such that $a_{1}\geq a_{2}\cdots\geq a_{n}$ and $\sum a_{i}=0$, and $g(t)\in\mathrm{SL}_{n}(\mathcal{O})t^{{\lambda}}\mathrm{SL}_{n}(\mathcal{O})$. Then 1. (1) $N=a_{n}$. 2. (2) The rank of $x_{N}$ equals to the number of $j$ such that $a_{j}=a_{n}$. 3. (3) For any $s\geq 1$, $\operatorname{rk}\begin{pmatrix}x_{N}&x_{N+1}&\cdots&x_{N+s-2}&x_{N+s-1}\\\ 0&x_{N}&\cdots&x_{N+s-3}&x_{N+s-2}\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&\cdots&x_{N}&x_{N+1}\\\ 0&0&\cdots&0&x_{N}\end{pmatrix}=\sum_{j=1}^{n}\max\\{s-(a_{j}-a_{n}),0\\}.$ We have the following lemma for the twisted version. ###### Lemma 4.4. Let $g(t)=\sum_{i=N}^{\infty}x_{i}t^{i}\in G(\mathcal{K})^{\sigma}$ where $x_{i}\in\operatorname{Mat}_{m\times m}$, $x_{N}\neq 0$. Let $\bar{\lambda}=(a_{1},...,a_{\ell})\in X_{}(T)_{\sigma}^{+}$ be such that $g(t)\in G(\mathcal{O})^{\sigma}n^{\lambda}G(\mathcal{O})^{\sigma}$. Then 1. (1) $N=\begin{cases}-a_{1}&\text{if }(X_{N},r)=(A_{2\ell},2),(A_{2\ell-1},2);\\\ -2a_{1}&\text{if }(X_{N},r)=(D_{\ell+1},2).\end{cases}$ 2. (2) The rank of $x_{N}$ is equal to the number of $j$ such that $a_{j}=a_{1}$. ###### Proof. We can write $\bar{\lambda}=(a_{1},...,a_{\ell})=\sum_{i=1}^{\ell}\left(\sum_{j=1}^{i}a_{j}\right)\gamma_{i}$ where $\gamma_{i}$ are simple roots of $H$ as labelled by (9). We choose a representative $\lambda\in X_{}(T)$ of $\bar{\lambda}$ by $\lambda=\begin{cases}\sum_{i=1}^{\ell}\left(\sum_{j=1}^{i}a_{j}\right)\check{\alpha}_{i}&\text{if }(X_{N},r)=(A_{2\ell},2),(D_{\ell+1},2);\\\ \sum_{i=1}^{\ell-1}\left(\sum_{j=1}^{i}a_{j}\right)\check{\alpha}_{i}+\dfrac{1}{2}\left(\sum_{j=1}^{\ell}a_{j}\right)\check{\alpha}_{\ell}&\text{if }(X_{N},r)=(A_{2\ell-1},2)\\\ \end{cases}$ so that $\lambda+\sigma\lambda=\begin{cases}\sum_{i=1}^{\ell}\left(\sum_{j=1}^{i}a_{j}\right)\check{\alpha}_{i}+\sum_{i=\ell+1}^{2\ell}\left(\sum_{j=1}^{2\ell-i+1}a_{j}\right)\check{\alpha}_{i}&\text{if }(X_{N},r)=(A_{2\ell},2);\\\ \sum_{i=1}^{\ell}\left(\sum_{j=1}^{i}a_{j}\right)\check{\alpha}_{i}+\sum_{i=\ell+1}^{2\ell-1}\left(\sum_{j=1}^{2\ell-i}a_{j}\right)\check{\alpha}_{i}&\text{if }(X_{N},r)=(A_{2\ell-1},2);\\\ \sum_{i=1}^{\ell-1}\left(\sum_{j=1}^{i}2a_{j}\right)\check{\alpha}_{i}+\left(\sum_{j=1}^{\ell}a_{j}\right)(\check{\alpha}_{\ell}+\check{\alpha}_{\ell+1})&\text{if }(X_{N},r)=(D_{\ell+1},2).\\\ \end{cases}$ The simple coroots of ${G}$ are identified with tuples of integers through the construction of root system given from [Hu2, §12]. Let $\rho:G\to{\rm GL}(V)$ be the standard representation of $G$. We will determine the double ${\rm SL(V_{\mathcal{O}})}$-coset in ${\rm SL}(V_{\mathcal{K}})$ that $\rho(g(t))$ belongs to. If ${G}$ is of the type $A_{m}$, then $\check{\alpha}_{i}$, $i=1,...,m$, are identified with the following $(m+1)$-tuples $\check{\alpha}_{1}=(1,-1,0,0,...,0)$ $\check{\alpha}_{2}=(0,1,-1,0,...,0)$ $\vdots$ $\check{\alpha}_{m}=(0,0,0,...,1,-1)$ and hence, as the coweight of $\mathrm{SL}_{m}$, $\lambda+\sigma\lambda$ corresponds to the following tuples $\lambda+\sigma\lambda=\begin{cases}(a_{1},a_{2},...,a_{\ell},0,-a_{\ell},...,-a_{2},-a_{1})&\text{if }(X_{N},r)=(A_{2\ell},2);\\\ (a_{1},a_{2},...,a_{\ell},-a_{\ell},...,-a_{2},-a_{1})&\text{if }(X_{N},r)=(A_{2\ell-1},2).\end{cases}$ Assume that ${G}$ has the type $D_{\ell+1}$. Then $\check{\alpha}_{i}$, $i=1,...,\ell+1$, are identified with following $(\ell+1)$-tuples $\check{\alpha}_{1}=(1,-1,0,0,...,0)$ $\check{\alpha}_{2}=(0,1,-1,0,...,0)$ $\vdots$ $\check{\alpha}_{\ell}=(0,0,0,...,1,-1)$ $\check{\alpha}_{\ell+1}=(0,0,0,...,1,1).$ Then $\lambda+\sigma\lambda=(2a_{1},2a_{2},...,2a_{\ell},0)$ as the coweight of $G=\mathrm{Spin}_{2\ell+2}$. Choose an appropriate maximal torus and a positive root system in $G$. Composing with $\rho:G\to{\rm SL}_{2\ell+2}$, as the coweight of $\mathrm{SL}_{2\ell+2}$, $\lambda+\sigma\lambda$ corresponds to the following tuple $(2a_{1},2a_{2},...,2a_{\ell},0,0,-2a_{\ell},...,-2a_{2},-2a_{1}).$ We write $g(t)=A(t)n^{\lambda}B(t)$, where $n^{\lambda}$ is a norm of $t^{\lambda}$ defined by (1) and $A(t),B(t)\in G(\mathcal{O})^{\sigma}$. Hence $\rho(g(t))\in\mathrm{SL}_{m}(\mathcal{O})\rho(t^{\lambda+\sigma\lambda})\mathrm{SL}_{m}(\mathcal{O})$. By above descriptions of $\rho(t^{\lambda+\sigma\lambda})$, this lemma follows from Lemma 4.3. ∎ ### 4.1. Case $(X_{N},r)=(A_{2\ell},2)$ Let $\langle\cdot,\cdot\rangle$ be a nondegenerate symmetric bilinear form on $V=\mathbb{C}^{2\ell+1}$ whose matrix is $J=\begin{pmatrix}&&&&&&1\\\ &&&&&-1&\\\ &&&&1&&\\\ &&&-1&&&\\\ &&\reflectbox{$\ddots$}&&&&\\\ &-1&&&&&\\\ 1&&&&&&\end{pmatrix}.$ The diagram automorphism $\sigma$ on $\mathfrak{g}$ given by (7) becomes $\sigma(A)=-JA^{T}J^{-1}$ and the diagram automorphism $\sigma$ on $G$ is given by (10) $\sigma(A)=JA^{-T}J^{-1}.$ This $\sigma$ gives the decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ to 1 and -1 eigenspaces $\mathfrak{k}$ and $\mathfrak{p}$, respectively. Let $K:=(\mathrm{SL}_{2\ell+1})^{\sigma}=\mathrm{SO}_{2\ell+1}$. The classification of nilpotent $K$-orbits in $\mathfrak{p}$ and their dimensions follow from the Theorem 3.4 and 3.7. Set (11) $\mathcal{N}_{\mathfrak{p},2}=\\{x\in\mathcal{N}_{\mathfrak{p}}\,\|\,x^{2}=0\\}.$ ###### Theorem 4.5. $\pi$ maps $\mathcal{M}$ isomorphically onto $\mathcal{N}_{\mathfrak{p},2}$. Moreover, $\pi$ maps $\mathcal{M}_{(1^{j}0^{\ell-j})}$ isomorphically to $[2^{j}1^{2\ell-2j+1}]$. ###### Proof. Let $g(t)\cdot e_{0}\in\mathcal{M}$. Then $g(t)\cdot e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$ for some $j$. By Lemma 4.4, $g(t)=I+xt^{-1}$ for some $x\in\operatorname{Mat}_{2\ell+1\times 2\ell+1}$. By Lemma 2.4, $\iota(g(t)\cdot e_{0})\in\mathcal{M}_{(1^{j}0^{\ell-j})}$. Hence $\iota(g(t))=(I-xt^{-1})^{-1}=I+zt^{-1}$ for some $z\in\operatorname{Mat}_{2\ell+1\times 2\ell+1}$, and so $x^{2}=0$. Conversely, let $x\in\mathcal{N}_{\mathfrak{p}}$ be such that $x^{2}=0$. Then $I+xt^{-1}\in G(\mathcal{K})^{\sigma}$. By (2), $(I+xt^{-1})\cdot e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ for some $\bar{\lambda}=(a_{1},...,a_{\ell})\in X_{}(T)_{\sigma}^{+}$. By lemma 4.4, $\bar{\lambda}=(1^{j}0^{\ell-j})$. We have proved that $\mathcal{M}=\\{(I+xt^{-1})\cdot e_{0}\mid x\in\mathfrak{p},x^{2}=0\\}$. Let $(I+xt^{-1})\cdot e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-i})}$. Then $x^{2}=0$ and $x$ has the Jordan blocks of size at most 2. By Lemma 4.4, $\operatorname{rk}x=j$ and then $x$ has the partition $[2^{j}1^{2\ell-2j+1}]$. It is obvious that $\pi$ is injective. To prove surjectivity, let $x\in\mathcal{N}_{\mathfrak{p},2}$ having the partition $[2^{j}1^{2\ell-2j+1}]$. Then, $(I+xt^{-1})\cdot e_{0}\in\mathcal{M}$ and hence $(I+xt^{-1})\cdot e_{0}\in\mathcal{M}_{(1^{k}0^{\ell-k})}$ for some $k$. In fact, $k=j$ since $\pi((I+xt^{-1})\cdot e_{0})=x\in[2^{k}1^{2\ell-2k+1}].$ ∎ ### 4.2. Case $(X_{N},r)=(A_{2\ell-1},2)$ Let $\langle\cdot,\cdot\rangle$ be a symplectic bilinear form on $V=\mathbb{C}^{2\ell}$ whose matrix is $J=\begin{pmatrix}&&&&&&1\\\ &&&&&-1&\\\ &&&&1&&\\\ &&&-1&&&\\\ &&\reflectbox{$\ddots$}&&&&\\\ &1&&&&&\\\ -1&&&&&&\end{pmatrix}.$ The diagram automorphism $\sigma$ on $\mathfrak{g}$ given by (7) becomes $\sigma(A)=-JA^{T}J^{-1}$ and the action on $G$ is given by (12) $\sigma(A)=JA^{-T}J^{-1}.$ This $\sigma$ gives the decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$ to 1 and -1 eigenspaces $\mathfrak{k}$ and $\mathfrak{p}$, respectively. Let $K:=(\mathrm{SL}_{2\ell})^{\sigma}=\mathrm{Sp}_{2\ell}$. The classification of nilpotent $K$-orbits in $\mathfrak{p}$ and their dimensions follow from the Theorem 3.3 and 3.7. Define constructible sets $\mathcal{M}^{\prime}:=\bigcup_{j=0}^{\lfloor\frac{\ell}{2}\rfloor}\mathcal{M}_{(1^{2j}0^{\ell-2j})},\hskip 14.22636pt\mathcal{M}^{\prime\prime}:=\bigcup_{j=0}^{\lfloor\frac{\ell-1}{2}\rfloor}\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}.$ By Lemma 4.4, the element of $\mathcal{M}^{\prime}$ is of the form $(I+xt^{-1})\cdot e_{0}$ and the element of $\mathcal{M}^{\prime\prime}$ is of the form $(I+xt^{-1}+yt^{-2})\cdot e_{0}$. Let $\mathcal{N}_{\mathfrak{p},2}$ be the order 2 nilpotent cone defined as in (11). ###### Theorem 4.6. $\pi$ maps $\mathcal{M^{\prime}}$ isomorphically onto $\mathcal{N}_{\mathfrak{p},2}$. Moreover, $\pi$ maps $\mathcal{M}_{(1^{2j}0^{\ell-2j})}$ isomorphically to $[2^{2j}1^{2\ell-4j}]$. ###### Proof. The proof is the same as the proof of Theorem 4.5, where in this case we use Lemma 4.4 for $(A_{2\ell-1},2)$. ∎ Before we describe elements of $M^{\prime\prime}$, we need the following lemma. ###### Lemma 4.7. Let $g(t)\cdot e_{0}=(I+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}^{\prime\prime}$. Then 1. (1) $\iota(g(t))\neq g(t)$ 2. (2) If $g(t)\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}$, then $\operatorname{rk}\begin{pmatrix}y&x\\\ 0&y\end{pmatrix}=2j+2.$ ###### Proof. Note that $\iota(g(t))=g(t)$ if and only if $(I+xt^{-1}+yt^{-2})(I-xt^{-1}+yt^{-2})=I$ which is equivalent to $y=\dfrac{1}{2}x^{2}$ and $x^{4}=0$. Suppose that $\iota(g(t))=g(t)$. Observe that $\operatorname{rk}x^{3}\leq\operatorname{rk}x^{2}=\operatorname{rk}y=1$. If $\operatorname{rk}x^{3}=1$, then $\operatorname{rk}x^{4}=\operatorname{rk}x^{3}=1$ which is impossible. Hence $x^{3}=0$. Since $\operatorname{rk}x^{2}=1$, $x\in\mathfrak{p}$ is nilpotent having the partition $[32^{j}1^{2l-2j-3}]$ but this contradicts to Theorem 3.3. This proves the first part. Assume that $g(t)\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}$. We write $g(t)=At^{(2,1^{2j},0^{2\ell-4j-2},(-1)^{2j},-2)}B$ where $A=\sum_{i=0}A_{i}t^{i},B=\sum_{i=0}B_{i}t^{i}\in G(\mathcal{O})^{\sigma}$. In particular, $g(t)\in\mathrm{SL}_{2\ell}(\mathcal{O})t^{\check{\lambda}}\mathrm{SL}_{2\ell}(\mathcal{O})$ where $\check{\lambda}=(2,1^{2j},0^{2\ell-4j-2},(-1)^{2j},-2)$. By Lemma 4.3, $\operatorname{rk}\begin{pmatrix}y&x\\\ 0&y\end{pmatrix}=\sum_{j=1}^{2\ell}\max\\{-a_{j},0\\}=2j+2$ as desired. ∎ Let $g(t)=I+xt^{-1}+yt^{-2}$. By Lemma 2.4, $\iota(g(t))=I+x^{\prime}t^{-1}+y^{\prime}t^{-2}$ for some matrices $x^{\prime},y^{\prime}$. Hence $(I-xt^{-1}+yt^{-2})(I+x^{\prime}t^{-1}+y^{\prime}t^{-2})=I=(I+x^{\prime}t^{-1}+y^{\prime}t^{-2})((I-xt^{-1}+yt^{-2})$ which implies (13) $x^{\prime}=x,\hskip 7.11317ptx^{2}=y+y^{\prime},\hskip 7.11317ptxy=y^{\prime}x,\hskip 7.11317ptxy^{\prime}=yx,\hskip 7.11317ptyy^{\prime}=y^{\prime}y=0.$ By Lemma 4.4 and Lemma 4.7, $\operatorname{rk}y=\operatorname{rk}y^{\prime}=1$ and $y^{\prime}\neq y$. Since $\sigma(g(t))=g(t)$, $y^{\prime}=Jy^{T}J^{-1}$ which means that $y$ and $y^{\prime}$ are adjoint each other. We set $\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}:=\\{(I+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}\mid L=L^{\prime}\\},$ $\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}}:=\\{(I+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}\mid L\neq L^{\prime}\\}.$ The following lemma will be used in the proofs of Lemma 4.9 and Theorem 4.14. ###### Lemma 4.8. Let $(\cdot,\cdot)$ be a nondegenerate bilinear form on a vector space $V$ over a field $\mathbb{C}$ and let $T$ a linear map on $V$. Denote the adjoint of $T$ by $T^{}$. Assume that $\operatorname{Im}T=\operatorname{Im}T^{}$ and $\operatorname{rk}T=1$. Then $T$ is self-adjoint or skew-adjoint. ###### Proof. It is easy to see that $\ker T=(\operatorname{Im}T^{})^{\perp}$ and $\ker T^{}=(\operatorname{Im}T)^{\perp}$. Say that $\operatorname{Im}T=\mathbb{C}v$ and $\operatorname{Im}T^{}=\mathbb{C}v^{\prime}$ for some $v,v^{\prime}\in V$. Then $Tw=v$ for some $w\in V$. Since $\operatorname{Im}T=\operatorname{Im}T^{}$, we have $T^{}w=\lambda v$ for some $\lambda\in\mathbb{C}$. Let $u\in V$. Then $Tu=kv$ for some $k\in\mathbb{C}$. Since $T(u-kw)=0$, $u-kw\in\ker T=(\operatorname{Im}T^{})^{\perp}=(\operatorname{Im}T)^{\perp}=\ker T^{}$. Hence $T^{}u=T^{}(kw)=k\lambda v=\lambda Tu$. Since $u$ is arbitrary, $T^{}=\lambda T$. Consider $T+T^{}=(1+\lambda)T$. Then $(1+\lambda)T=T+T^{}=(T+T^{})^{}=(1+\lambda)T^{}.$ Therefore $1+\lambda=0$ or $T=T^{}$ which means that $T$ is skew-adjoint or self-adjoint. ∎ ###### Lemma 4.9. If $(I+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}$, then $y^{\prime}=-y$. ###### Proof. We know that $y\neq y^{\prime}$ are adjoint each other, they have the same images, and $\operatorname{rk}y=1$. By Lemma 4.8, $y$ is skew-adjoint, i.e., $y^{\prime}=-y$. ∎ ###### Theorem 4.10. 1. (1) If $\ell$ is even, then $\pi(\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}})=[2^{2j}1^{2\ell-4j}]\cup[2^{2j+2}1^{2\ell-4j-4}]$ for $j=0,1,...,\frac{\ell-2}{2}$. If $\ell$ is odd, then $\pi(\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}})=\begin{cases}[2^{2j}1^{2\ell-4j}]\cup[2^{2j+2}1^{2\ell-4j-4}]&\hskip 1.42271pt\text{if }\ell\geq 3,\,0\leq j\leq\frac{\ell-3}{2};\\\ [2^{\ell-1}1^{2}]&\hskip 1.42271pt\text{if }j=\frac{\ell-1}{2}.\end{cases}$ 2. (2) When $\ell\geq 3$, for $j=1,...,\lfloor\frac{\ell-1}{2}\rfloor$, we have $\pi(\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}})=\begin{cases}[3^{2}1^{2\ell-6}]&\hskip 1.42271pt\text{if }j=1;\\\ [3^{2}2^{2j-2}1^{2\ell-4j-2}]\cup[3^{2}2^{2j-4}1^{2\ell-4j+2}]&\hskip 1.42271pt\text{if }\ell\geq 4,\,2\leq j\leq\lfloor\frac{\ell-1}{2}\rfloor.\end{cases}$ Moreover, for any $\ell\geq 1$, $\mathcal{M}_{(20^{\ell-1})}^{\mathrm{II}}$ is empty. ###### Proof. Let $g(t)\cdot e_{0}=(I+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}$. By Lemma 4.7, $\operatorname{rk}x\leq\operatorname{rk}\begin{pmatrix}y&x\\\ 0&y\end{pmatrix}=2j+2\leq 2\operatorname{rk}y+\operatorname{rk}x=2+\operatorname{rk}x.$ Since $x^{2}=y+y^{\prime}=0$, $x$ is nilpotent whose partition is $[2^{2k}1^{2\ell-4k}]$ so that $\operatorname{rk}x=2k$. Hence $k=j$ or $j+1$. If $\ell$ is odd and $j=\frac{\ell-1}{2}$, then $k=j$. Let $E_{ij}\in\operatorname{Mat}_{2\ell\times 2\ell}$ be the matrix which has 1 at the entry $i,j$ and 0 elsewhere. For each $j=0,1,\dots,\lfloor\frac{\ell-1}{2}\rfloor$, let $x_{j}=\operatorname{diag}(0,J_{2},..,J_{2},0_{2\ell-4j-2},-J_{2},..,-J_{2},0)$ where there are $j$ blocks of $J_{2}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}$ and $j$ blocks of $-J_{2}$, and $0_{2\ell-4j-2}$ is the square zero matrix of size $2\ell-4j-2$. Then $x_{j}\in\mathfrak{p}$ is nilpotent and has the partition $[2^{2j}1^{2\ell-4j}]$. It is easy to check that $g(t)=I+x_{j}t^{-1}+E_{1,2\ell}t^{-2}\in G(\mathcal{K})^{\sigma}$ and $\iota(g(t))=I+x_{j}t^{-1}-E_{1,2\ell}t^{-2}$. By (2), $g(t)\cdot e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ for some $\bar{\lambda}=(a_{1},...,a_{\ell})\in X_{}(T)_{\sigma}^{+}$ with $a_{1}\geq a_{2}...\geq a_{\ell}\geq 0$ and $\sum a_{i}$ is even. By Lemma 4.4, since $\operatorname{rk}E_{1,2\ell}=1$, $\bar{\lambda}=(21^{2k}0^{\ell-2k-1})$ for some $k$. By Lemma 4.7, $2k+2=\operatorname{rk}\begin{pmatrix}E_{1,2\ell}&x_{j}\\\ 0&E_{1,2\ell}\end{pmatrix}=2j+2.$ Then $(I+x_{j}t^{-1}+E_{1,2\ell}t^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}$. For $j=0,1,\dots,\lfloor\frac{\ell-2}{2}\rfloor$, let $x^{\prime}_{j}=x_{j}+E_{1,2j+2}-E_{2\ell-2j-1,2\ell}.$ Then $x^{\prime}_{j}\in\mathfrak{p}$ is nilpotent and has the partition $[2^{2j+2}1^{2\ell-4j-4}]$. Similarly, one can show that $(I+x^{\prime}_{j}t^{-1}+E_{1,2\ell}t^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{I}}$. Since $\pi$ is $K$-invariant, this proves the first part. Now, we prove the second part. Let $g(t)\cdot e_{0}=(I+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}}$. Set $U=L+L^{\prime}$. Since $y\neq y^{\prime}$, $\dim U=2$ and $U=\operatorname{Im}x^{2}$. Assume that $L=\mathbb{C}v,L^{\prime}=\mathbb{C}v^{\prime}$. By (13), we have $xy=y^{\prime}x$ and $xy^{\prime}=yx$. Hence $xv=bv^{\prime}$ and $xv^{\prime}=av$ for some $a,b\in\mathbb{C},v,v^{\prime}\in\mathbb{C}^{2l}$. Then $x\big{\|}_{U}=\begin{pmatrix}0&a\\\ b&0\end{pmatrix},\hskip 14.22636ptx^{2}\big{\|}_{U}=\begin{pmatrix}ab&0\\\ 0&ab\end{pmatrix}$ Suppose that $ab\neq 0$. To show that $\langle\cdot,\cdot\rangle\big{\|}_{U\times U}$ is nondegenerate, let $u\in\mathbb{C}^{2\ell}$ be such that $\langle x^{2}u,x^{2}v\rangle=0$ for all $v\in\mathbb{C}^{2\ell}$. Since $x^{2}$ is self-adjoint, $\langle x^{4}u,v\rangle=0$ for all $v$. Therefore, $x^{4}u=0$ and so $x^{2}u\in\ker x^{2}\big{\|}_{U}=0$. Since $y,y^{\prime}$ are adjoint each other and $yy^{\prime}=0=y^{\prime}y$, $L^{\prime}\subset\ker y=(L^{\prime})^{\perp}$ and $L\subset\ker y^{\prime}=L^{\perp}$. This implies $\langle v,v\rangle=\langle v^{\prime},v^{\prime}\rangle=0$ and $\langle v,v^{\prime}\rangle\neq 0$. Observe that $ab\langle v,v^{\prime}\rangle=\langle v,x^{2}v^{\prime}\rangle=\langle xv,xv^{\prime}\rangle=\langle bv^{\prime},av\rangle=-ab\langle v,v^{\prime}\rangle$ which implies $ab=0$, a contradiction. This proves $x^{4}=0$. Since $x\in\mathfrak{p}$ and $\operatorname{rk}x^{2}=2$, by Theorem 3.3, $x$ is nilpotent having the partition $[3^{2}2^{2k}1^{2\ell-4k-6}]$. By Lemma 4.7, $\operatorname{rk}x\leq\operatorname{rk}\begin{pmatrix}y&x\\\ 0&y\end{pmatrix}=2j+2\leq 2\operatorname{rk}y+\operatorname{rk}x=2+\operatorname{rk}x.$ Since $\operatorname{rk}x=2k+4$, $k=j-1$ or $j-2$. Here we see that $j\neq 0$ and hence $\mathcal{M}_{(20^{\ell-1})}^{\mathrm{II}}$ is empty. When $j=1$, we see that $k=0$. Let $J_{3}=\begin{pmatrix}0&1&0\\\ 0&0&1\\\ 0&0&0\end{pmatrix},\hskip 5.69046ptJ_{2}=\begin{pmatrix}0&1\\\ 0&0\end{pmatrix}.$ For each $j=1,...,\lfloor\frac{\ell-1}{2}\rfloor$, let $x_{j-1}=\operatorname{diag}(J_{3},J_{2},...,J_{2},0_{2\ell-4j-2},-J_{2},...,-J_{2},-J_{3})$ where there are $j-1$ blocks of $J_{2}$, and $j-1$ blocks of $-J_{2}$. Then $x_{j-1}\in\mathfrak{p}$ is nilpotent having the partition $[3^{2}2^{2j-2}1^{2\ell-4j-2}]$. Note that $g(t):=1+x_{j-1}t^{-1}+E_{13}t^{-2}\in G(\mathcal{K})^{\sigma}$ and $\iota(g(t))=1+x_{j-1}t^{-1}+E_{2\ell-2,2\ell}t^{-2}$. By (2), $g(t)\cdot e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ for some $\bar{\lambda}=(a_{1},...,a_{\ell})\in X_{}(T)_{\sigma}^{+}$ with $a_{1}\geq a_{2}...\geq a_{\ell}\geq 0$ and $\sum a_{i}$ is even. By Lemma 4.4, since $\operatorname{rk}E_{13}=1$, $\bar{\lambda}=(21^{2k}0^{\ell-2k-1})$ for some $k$. By Lemma 4.7, $2k+2=\operatorname{rk}\begin{pmatrix}E_{13}&x_{j-1}\\\ 0&E_{13}\end{pmatrix}=2j+2.$ Then $(I+x_{j}t^{-1}+E_{13}t^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}}$. For $j=2,...,\lfloor\frac{\ell-1}{2}\rfloor$, let $x^{\prime}_{j-2}=\operatorname{diag}(0,J_{3},J_{2},...,J_{2},0_{2\ell-4j},-J_{2},...,-J_{2},-J_{3},0)$ where there are $j-2$ blocks of $J_{2}$, and $j-2$ blocks of $-J_{2}$. Then $x^{\prime}_{j-2}\in\mathfrak{p}$ is nilpotent having the partition $[3^{2}2^{2j-4}1^{2\ell-4j+2}]$. One can check that $h(t):=1+x^{\prime}_{j-2}t^{-1}+(E_{24}+E_{1,2\ell})t^{-2}\in G(\mathcal{K})^{\sigma}$ and $\iota(h(t))=1+x^{\prime}_{j-2}t^{-1}+(E_{2\ell-3,2\ell-1}-E_{1,2\ell})t^{-2}$. Similarly, $h(t)\cdot e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ where $\bar{\lambda}=(21^{2k}0^{\ell-2k-1})$ for some $k$. By Lemma 4.7, $2k+2=\operatorname{rk}\begin{pmatrix}E_{24}+E_{1,2\ell}&x^{\prime}_{j-2}\\\ 0&E_{24}+E_{1,2\ell}\end{pmatrix}=2j+2.$ Then $(I+x^{\prime}_{j-2}t^{-1}+(E_{24}+E_{1,2\ell})t^{-2})\cdot e_{0}\in\mathcal{M}_{(21^{2j}0^{\ell-2j-1})}^{\mathrm{II}}$. ∎ In the following proposition, we describe the fibers of $\pi:\mathcal{M}\to\pi(\mathcal{M})$. ###### Proposition 4.11. For any $x\in\pi(\mathcal{M})$, $\pi^{-1}(x)\cong\\{z\in\mathfrak{sp}_{2\ell}\mid xz+zx=0,z^{2}=0,\operatorname{rk}(z+\frac{1}{2}x^{2})\leq 1\\}.$ In particular, $\pi^{-1}(0)$ is isomorphic to the closure of nilpotent orbit $[21^{2\ell-2}]$ in $\mathfrak{sp}_{2\ell}$ and $\dim\pi^{-1}(0)=2\ell+1$. ###### Proof. Note that $(1+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}$ if and only if $\det(1+xt^{-1}+yt^{-2})=1,\operatorname{rk}y\leq 1$ and (14) $x^{T}J-Jx=0,\quad-x^{T}Jx+y^{T}J+Jy=0,\hskip 14.22636ptx^{T}Jy-y^{T}Jx=0,\hskip 14.22636pty^{T}Jy=0.$ Set $z=y-\frac{1}{2}x^{2}$, (14) is equivalent to (15) $x\in\mathfrak{p},\quad z\in\mathfrak{k},\quad xz+zx=0,\quad z^{2}=0.$ If $xz+zx=0$, then $\det(1+xt^{-1}+(z+\frac{1}{2}x^{2})t^{-2})=1$. Since $\pi^{-1}(0)=\\{(1+yt^{-2})\cdot e_{0}\mid y\in\mathfrak{sp}_{2\ell},y^{2}=0,\operatorname{rk}y\leq 1\\}$, the dimension, cf. [CM, Corollary 6.1.4], is $\dim\pi^{-1}(0)=\dim[21^{2\ell-2}]=(2\ell^{2}+\ell)-\frac{1}{2}((2\ell-1)^{2}+1^{2})-\frac{1}{2}(2\ell-2)=2\ell+1.$ ∎ ${{\mathcal{M}_{(21^{4})}^{\mathrm{I}}}}$${{\mathcal{M}_{(21^{4})}^{\mathrm{II}}}}$${{[3^{2}2^{2}]}}$${{\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{I}}}}$${{[3^{2}1^{4}]}}$${{\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{II}}}}$${{\mathcal{M}_{(1^{4}0)}}}$${{[2^{4}1^{2}]}}$${{\mathcal{M}_{(20^{4})}^{\mathrm{I}}}}$${{\mathcal{M}_{(20^{4})}^{\mathrm{II}}=\emptyset}}$${{[2^{2}1^{6}]}}$${{\mathcal{M}_{(1^{2}0^{3})}}}$${{0}}$${{0}}$$\scriptstyle{\sim}$$\scriptstyle{\sim}$ Figure 1. $\mathcal{M}_{\bar{\lambda}}$ for small dominant weight $\bar{\lambda}$ and their image under the map $\pi$ in type $(X_{N},r)=(A_{9},2)$ In [AH, Theorem 1.2], they proved that there are finitely many $G$-orbits in $\mathrm{Gr}_{\lambda}\cap\mathrm{Gr}_{0}^{-}$ for small dominant coweight $\lambda$. In the case of $(A_{2\ell},2)$, it is easy to see that $K$ acts transitively on $\mathcal{M}_{(1^{j}0^{\ell-j})}$ and hence there are finitely many $K$-orbits in $\mathcal{M}$. For the case $(A_{2\ell-1},2)$, it is not obvious to determine if there are finitely many $K$-orbits in $\mathcal{M}_{(21^{j}0^{\ell-j-1})}$. If $g(t)=1+xt^{-1}+(z+\frac{1}{2}x^{2})t^{-2}\in\mathcal{M}_{(21^{j}0^{\ell-j-1})}$, then $g(t)$ satisfies (15). If the action of $K$ on the following anti- commuting nilpotent variety $\\{(x,z)\in\mathcal{N}_{\mathfrak{p}}\times\mathcal{N}_{\mathfrak{k}}\mid xz+zx=0\\}$ by diagonal cojugation has finitely many orbits, then there are finitely many $K$-orbits in $\mathcal{M}_{(21^{j}0^{\ell-j-1})}$. ###### Example 4.12. Consider the case $(X_{N},r)=(A_{9},2)$ . In this case, $G=\mathrm{SL}_{10}$ and $\mathfrak{g}=\mathfrak{sl}_{10}=\mathfrak{k}\oplus\mathfrak{p}$. The diagram as shown in Figure 1 describes the image of $\mathcal{M}_{\bar{\lambda}}$ for each small dominant weight $\bar{\lambda}$. For instance, $\mathcal{M}_{(21^{2}0^{2})}$ consists of two parts, $\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{I}}$ and $\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{II}}$. By Theorem 4.10, $\pi(\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{I}})$ is precisely the union of two nilpotent orbits $[2^{4}1^{2}]$ and $[2^{2}1^{6}]$ in $\mathfrak{p}$ while $\pi(\mathcal{M}_{(21^{2}0^{2})}^{\mathrm{II}})$ is the single nilpotent orbit $[3^{2}1^{4}]$ in $\mathfrak{p}$. Since $(21^{2}0^{2})\succeq(1^{4}0)$, $\mathcal{M}_{(21^{2}0^{2})}\succeq\mathcal{M}_{(1^{4}0)}$. Similarly, $\mathcal{M}_{(21^{2}0^{2})}\succeq\mathcal{M}_{(20^{4})}$. According to the table in Theorem 4.2, the image of certain $\mathcal{M}_{\bar{\lambda}}$ is a union of 4 nilpotent orbits. It does not happen in this case since $\ell=5$ is not large enough. In the case of $(X_{N},r)=(A_{13},2)$, $\pi(\mathcal{M}_{(21^{4}0^{2})})$ is a union of nilpotent orbits $[2^{4}1^{6}],[2^{6}1^{2}],[3^{2}2^{2}1^{4}],[3^{2}1^{8}]$ in $\mathfrak{p}\subset\mathfrak{sl}_{14}$. ### 4.3. Case $(X_{N},r)=(D_{\ell+1},2)$ In this case, it is more convenient to work with $G=\mathrm{SO}_{2\ell+2}$ and $\sigma$ is a diagram automorphism on $G$. It is known that $G^{\sigma}\simeq\mathrm{SO}_{2\ell+1}\times\\{\pm I\\}$. Let $G(\mathcal{O})^{\sigma,\circ}$ denote the identity component of the group $G(\mathcal{O})^{\sigma}$. Then, the action of ${\rm Spin}_{2\ell+2}(\mathcal{O})^{\sigma}$ on the twisted affine Grassmanian $\mathcal{G}r$ of ${{\rm Spin}_{2n+2}}$ factors through $G(\mathcal{O})^{\sigma,\circ}$. Let $G(\mathcal{O}^{-})_{0}^{\sigma}$ be the kernel of the evaluation map $G(\mathcal{O}^{-})^{\sigma}\to G^{\sigma}$. The action of ${\rm Spin}_{2\ell+2}(\mathcal{O})_{0}^{-}$ on $\mathcal{G}r$ factors through $G(\mathcal{O}^{-})_{0}^{\sigma}$. Hence, the opposite open Schubert cell $\mathcal{G}r_{0}^{-}$ is a $G(\mathcal{O}^{-})_{0}^{\sigma}$-orbit. In fact, $\mathcal{G}r$ is naturally the neutral component of the twisted affine Grassmannian associated to $(G,\sigma)$, whose definition is a bit more sophisticated. We can realize the group $G$ as $\\{g\in{\rm SL}_{2\ell+2}\|gJg^{T}=J\\}$, and the Lie algebra of $G$ as $\mathfrak{g}=\mathfrak{so}_{2\ell+2}(J)=\\{x\in\mathfrak{gl}_{2n}\mid Jx+x^{T}J=0\\}$ where $J=\begin{pmatrix}&&&&&1\\\ &&&&1&\\\ &&&1&&\\\ &&\reflectbox{$\ddots$}&&&&\\\ &1&&&&\\\ 1&&&&&\end{pmatrix}.$ The diagram automorphism $\sigma$ of order 2 on $\mathfrak{g}$ can be given by $\sigma(x)=wxw$ where $w=\operatorname{diag}\left(I_{\ell},\begin{pmatrix}0&1\\\ 1&0\end{pmatrix},I_{\ell}\right).$ The diagram automorphism $\sigma$ on $G$ is also defined the same. We also have the decomposition $\mathfrak{g}=\mathfrak{k}\oplus\mathfrak{p}$. Let $K$ be the identity component of $G^{\sigma}$. $K$ has Lie algebra $\mathfrak{k}$ and acts on $\mathfrak{p}$ by conjugation. It can be checked that $J=A^{T}A$ where $A=\begin{pmatrix}\frac{1}{\sqrt{2}}&0&\cdots&\cdots&0&\frac{1}{\sqrt{2}}\\\ 0&\ddots&\cdots&\cdots&\reflectbox{$\ddots$}&0\\\ \vdots&\vdots&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\vdots&\vdots\\\ \vdots&\vdots&\frac{i}{\sqrt{2}}&-\frac{i}{\sqrt{2}}&\vdots&\vdots\\\ 0&\reflectbox{$\ddots$}&\cdots&\cdots&\ddots&0\\\ \frac{i}{\sqrt{2}}&0&\cdots&\cdots&0&-\frac{i}{\sqrt{2}}\end{pmatrix}.$ Another realization of $\mathfrak{so}_{2\ell+2}$ is $\mathfrak{so}_{2\ell+2}(I)=\\{x\in\mathfrak{gl}_{2\ell}\mid x+x^{T}=0\\}$. There exists an isomorphism from $\mathfrak{so}_{2\ell+2}(J)$ to $\mathfrak{so}_{2\ell+2}(I)$ given by $x\mapsto AxA^{-1}$. Under $\mathfrak{so}_{2\ell+2}(I)$, the diagram automorphism $\sigma_{0}$ is defined by $\sigma_{0}(x)=w_{0}xw_{0}$ where $w_{0}=(PA)w(PA)^{-1}=\operatorname{diag}(-1,1,1,...,1)$ and $P$ is some matrix of change of basis. ###### Proposition 4.13. 1. (1) If $x$ is a nonzero nilpotent element in $\mathfrak{p}$, then $x$ has the partition $[31^{2\ell-1}]$. 2. (2) There are exactly 2 nilpotent $K$-orbits in $\mathfrak{p}$: $\\{0\\}$ and $\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}.$ ###### Proof. Since $w_{0}xw_{0}=-x$, $x$ has the form $x=\begin{pmatrix}0&-u^{t}\\\ u&0\end{pmatrix}$ where $u\in\mathbb{C}^{2l+1}$ is a nonzero column vector. Then $x^{2}=\begin{pmatrix}-u^{t}u&0\\\ 0&-uu^{t}\end{pmatrix}.$ If $x^{2}=0$, then $uu^{t}=0$ which implies $u=0$, a contradiction. Since $\operatorname{rk}x=2$ and $x^{2}\neq 0$, $x$ has the partition $[31^{2n-1}]$. This proves the first part. The element of $K$ has the form $k=\begin{pmatrix}1&0\\\ 0&g\end{pmatrix}.$ where $g\in\mathrm{SO}_{2\ell+1}$ and $k$ acts on $x\in\mathfrak{p}$ by $k\cdot x=kxk^{-1}=\begin{pmatrix}1&-(gu)^{t}\\\ gu&0\end{pmatrix}.$ Hence the action of $K$ on $\mathfrak{p}$ is the same as the action of $\mathrm{SO}_{2\ell+1}$ on $\mathbb{C}^{2\ell+1}$. Note that for every $k\geq 3$, $x^{k}$ has the scalar $u^{T}u$ on every nonzero entry. Since $x$ is nilpotent, $u^{T}u=0$. The result immediately follows since the action of $\mathrm{SO}_{2\ell+1}$ on $\\{z\in\mathbb{C}^{2\ell+1}\mid z^{T}z=0\\}$ has two orbits. ∎ ###### Theorem 4.14. For $j=0,1,...,\ell$, we have $\pi(\mathcal{M}_{(1^{j}0^{\ell-j})})=\begin{cases}\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}&\quad\text{ if }j\text{ is odd};\\\ \\{0\\}&\quad\text{ if }j=0;\\\ \mathcal{N}_{\mathfrak{p}}&\quad\text{ if }j\text{ is even and }j\geq 2.\end{cases}.$ Moreover, $\mathcal{M}_{(10^{\ell-1})}=\\{(1+xt^{-1}+\frac{1}{2}x^{2}t^{-2})\cdot e_{0}\mid x\in\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}\\}$. Consequently, $\pi$ maps $\mathcal{M}_{(10^{\ell-1})}$ isomorphically onto $\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}$. ###### Proof. By Lemma 4.4, let $g(t)\cdot e_{0}=(1+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}.$ We work under $\mathfrak{so}_{2\ell+2}(I)$ and $\mathrm{SO}_{2\ell+2}(I)$. Since $g(t)$ is fixed by $\sigma_{0}$, $w_{0}xw_{0}=-x$ and $w_{0}yw_{0}=y$. Similarly to the proof of Proposition 4.13, $x,y$ are in the form (16) $x=\begin{pmatrix}0&-u^{T}\\\ u&0\end{pmatrix},\hskip 14.22636pty=\begin{pmatrix}y_{0}&0\\\ 0&D\end{pmatrix}$ where $u\in\mathbb{C}^{2\ell+1}$ is a column vector, $y_{0}\in\mathbb{C}$, and $D\in\operatorname{Mat}_{(\ell-1)\times(\ell-1)}$. Since $g^{T}g=I$, the following equations hold: (17) $x^{T}+x=0,\hskip 14.22636ptx^{T}x+y^{T}+y=0,\hskip 14.22636ptx^{T}y+y^{T}x=0,\hskip 14.22636pty^{T}y=0.$ Then $y_{0}=0$ and $u^{T}u=0$ which implies $x^{3}=0$. Hence $x\in\mathcal{N}_{\mathfrak{p}}$. Suppose that $j$ is odd and $x=0$. We have $y+y^{T}=0$ and $y^{T}y=0$. Then $y_{0}=0$ and $D$ is a nilpotent element of $\mathfrak{so}_{2\ell+1}$ with $D^{2}=0$. Since $\operatorname{rk}D=\operatorname{rk}y=j$, $D$ has the partition $[2^{j}1^{2\ell-2j+2}]$. This contradicts to the classification of nilpotent orbits of type B, [CM, Theorem 5.1.2]. Hence $x\neq 0$. Now, consider the matrix $x_{0}\in\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}$ defined by $x_{0}=\begin{pmatrix}0&\cdots&0&-1&-i\\\ \vdots&&&&\\\ 0&&&&\\\ 1&&&&\\\ i&&&&\end{pmatrix}.$ Let $N$ be a nilpotent element in $\mathfrak{so}_{2\ell-1}(I)$ having the partition $[2^{j-1}1^{2\ell-j+1}].$ Such a matrix $N$ exists in view of [CM, Theorem 5.1.2]. Then the matrices $x_{0}$ and $y_{0}:=\operatorname{diag}(0,...,0,N,0,0)+\dfrac{1}{2}x_{0}^{2}$ satisfy the relations in (17). Hence $g(t):=1+x_{0}t^{-1}+y_{0}t^{-2}\in G(\mathcal{O})^{\sigma}$. By (2), $g(t)\cdot e_{0}\in\mathcal{G}r_{\bar{\lambda}}$ for some $\bar{\lambda}=(a_{1},a_{2},...,a_{\ell})\in X_{}(T)_{\sigma}^{+}$ with $a_{1}\geq...\geq a_{\ell}$. Since $\operatorname{rk}y_{0}=j$, by Lemma 4.4, $\bar{\lambda}=(1^{j}0^{\ell-j})$ and hence $g(t)\cdot e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$. Since $\pi$ is $K$-equivariant, the first part is done. Suppose that $j$ is even. For each $j=0,2,4,...,2\lfloor\frac{\ell}{2}\rfloor$, consider the $\ell\times\ell$ matrix $\begin{pmatrix}&&1&&&&&\\\ &\reflectbox{$\ddots$}&&&&&&\\\ 1&&&&&&&\\\ &&&&&&&\\\ &&&&&&&\\\ &&&&&&&-1\\\ &&&&&&\reflectbox{$\ddots$}&\\\ &&&&&-1&&\end{pmatrix}$ where there are $\frac{j}{2}$ copies of each 1 and -1. Denote $z_{j}$ the square zero matrix of size $2\ell+2$ whose $\ell\times\ell$ submatrix on the right top is replaced by the above matrix. Now we work under $\mathfrak{so}_{2\ell+2}(J)$ and $\mathrm{SO}_{2\ell+2}(J)$. Since $wz_{j}w=z_{j}$ and $\operatorname{rk}z_{j}=j$, we have $(1+z_{j}t^{-2})\cdot e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$ and then $\pi((1+z_{j}t^{-2})\cdot e_{0})=0$. Let $x_{0}$ be the square zero matrix of size $2\ell+2$ whose $4\times 4$ submatrix at the center is replaced by $\begin{pmatrix}0&1&-1&0\\\ 0&0&0&1\\\ 0&0&0&-1\\\ 0&0&0&0\end{pmatrix}.$ Then $x_{0}\in\mathcal{N}_{\mathfrak{p}}\setminus\\{0\\}$, Set $y_{0}=\frac{1}{2}x_{0}^{2}+z_{j}$. Then $\operatorname{rk}y_{0}=j$. It can be checked that $x_{0},y_{0}$ satisfy $wx_{0}w=-x_{0},wy_{0}w=y_{0}$, and (18) $\displaystyle x_{0}^{T}J+Jx_{0}=0,\hskip 14.22636ptx_{0}^{T}Jx_{0}+y_{0}^{T}J+Jy_{0}=0,$ $\displaystyle x_{0}^{T}Jy_{0}+y_{0}^{T}Jx_{0}=0,\hskip 14.22636pty_{0}^{T}Jy_{0}=0.$ Hence $h(t):=1+x_{0}t^{-1}+y_{0}t^{-2}\in G(\mathcal{O})^{\sigma}$. Similarly, one can show that $h(t)\cdot e_{0}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$. This proves the second part. To prove the last part, let $x$ be a nonzero nilpotent element in $\mathfrak{p}$. Since $x$ has the partition $[31^{2\ell}]$, $\operatorname{rk}x^{2}=1$. It is easy to check that $(1+xt^{-1}+\frac{1}{2}x^{2}t^{-2})\cdot e_{0}\in\mathcal{M}_{(10^{\ell-1})}$. Conversely, let $g(t)\cdot e_{0}=(1+xt^{-1}+yt^{-2})\cdot e_{0}\in\mathcal{M}_{(10^{\ell-1})}$. Let $\iota(g(t))=1+xt^{-1}+y^{\prime}t^{-2}$. Since $g(t)=g(t)^{-T}=(\iota(g(-t)))^{T}$, $y=(y^{\prime})^{T}$. Then $y$ and $y^{\prime}$ are adjoint each other under the symmetric form whose matrix is $I$. Note that $\operatorname{rk}y=\operatorname{rk}y^{\prime}=1$. If $\operatorname{Im}y\neq\operatorname{Im}y^{\prime}$, then $\operatorname{rk}x^{2}=\operatorname{rk}y+\operatorname{rk}y^{\prime}=2$, a contradiction. Hence $\operatorname{Im}y=\operatorname{Im}y^{\prime}$. By Lemma 4.8, $y^{\prime}=y$ or $y^{\prime}=-y$. By (13), $x^{2}=y+y^{\prime}$ and hence $y^{\prime}=y$. By (17), $x^{T}+x=0$ and $x^{T}x+y^{T}+y=0$. Then $y+y^{\prime}=x^{2}=y+y^{T}$, so $y=y^{\prime}=y^{T}$. Therefore, $g(t)=1+xt^{-1}+yt^{-2}=1+xt^{-1}+\frac{1}{2}x^{2}t^{-2}.$ ∎ ###### Proposition 4.15. For $x\in\mathcal{N}_{\mathfrak{p}}$, write $x$ as in (16). Then $\pi^{-1}(x)\cong\\{D\in\mathfrak{so}_{2\ell+1}\mid Du=0,D^{2}=0\\}.$ In particular, $\pi^{-1}(0)$ is isomorphic to the maximal order 2 nilpotent variety in $\mathfrak{so}_{2\ell+1}$, and $\dim\pi^{-1}(0)=\begin{cases}\ell^{2}&\quad\text{if }\ell\text{ is even;}\\\ \ell^{2}-1&\quad\text{if }\ell\text{ is odd.}\\\ \end{cases}$ ###### Proof. Under the realization $\mathfrak{so}_{2\ell+2}(I)$ and $\mathrm{SO}_{2\ell+2}(I)$, and the diagram automorphism $\sigma_{0}$, we have that $1+xt^{-1}+yt^{-2}\in\mathcal{M}$ if and only if $w_{0}yw_{0}=y$ and the conditions (17) hold. Set $z=y-\frac{1}{2}x^{2}$, these conditions are equivalent to (19) $z=\begin{pmatrix}0&\\\ &D\end{pmatrix},\quad D\in\mathfrak{so}_{2\ell+1},\quad D^{2}=0,\quad Du=0.$ Hence $\pi^{-1}(0)\cong\\{D\in\mathfrak{so}_{2\ell+1}\mid D^{2}=0\\}$ which is $\overline{\mathcal{O}}_{[2^{k}1^{2\ell-2k+1}]}$ in $\mathfrak{so}_{2\ell+1}$ where $k$ is the maximal even integer. By the dimension formula, cf. [CM, Corollary 6.1.4], $\dim\pi^{-1}(0)=\begin{cases}\dim\mathcal{O}_{[2^{\ell}1]}=\ell^{2}&\quad\text{if }\ell\text{ is even;}\\\ \dim\mathcal{O}_{[2^{\ell-1}1^{3}]}=\ell^{2}-1&\quad\text{if }\ell\text{ is odd}\\\ \end{cases}$ as desired. ∎ Similar to the case $(A_{2\ell-1},2)$, it is not obvious to see if there are finitely many $K$-orbits in $\mathcal{M}_{(1^{j}0^{\ell-j})}$. If $g(t)=1+xt^{-1}+(z+\frac{1}{2}x^{2})t^{-2}\in\mathcal{M}_{(1^{j}0^{\ell-j})}$, then $g(t)$ satisfies (19). If the action of $K$ on the following anti- commuting nilpotent variety $\\{(x,z)\in\mathfrak{so}_{2\ell+2}(I)\times\mathfrak{so}_{2\ell+2}(I)\mid xz+zx=0,x,z\text{ nilpotent}\\}$ by diagonal cojugation has finitely many orbits, then there are finitely many $K$-orbits in $\mathcal{M}_{(1^{j}0^{\ell-j})}$. ## 5\. Applications In this section, we describe some applications to the geometry of order 2 nilpotent varieties in the certain classical symmetric spaces. Let $\langle,\rangle$ be a symmetric or symplectic non-degenerate bilinear form on a vector space $V$. Recall that $\mathcal{A}$ is the space of all self-adjoint linear maps with respect to $\langle,\rangle$. Set $\mathcal{N}_{\mathcal{A},2}$ denote the space of all nilpotent operators $x$ in $\mathcal{A}$ such that $x^{2}=0$. If $\langle,\rangle$ is symmetric and $\dim V=2n+1$, then ${\rm SO}_{2n+1}$-orbits in $\mathcal{N}_{\mathcal{A},2}$ are classified by the partitions $[2^{j}1^{2n+1-2j}]$ with $0\leq j\leq n$; if $\langle,\rangle$ is symplectic and $\dim V=2n$, then ${\rm Sp}_{2n}$-orbits in $\mathcal{N}_{\mathcal{A},2}$ are classified by the partitions $[2^{2j}1^{2n-2j}]$ with $0\leq j\leq\lfloor\frac{n}{2}\rfloor$. ###### Theorem 5.1. Assume that $\langle,\rangle$ is symplectic or symmetric and $\dim V$ is odd. Then any order 2 nilpotent variety in $\mathcal{A}$ is normal. ###### Proof. By Theorem 4.5 and Theorem 4.6, for any order 2 nilpotent variety $\overline{\mathcal{O}}$ in $\mathcal{A}$, $\overline{\mathcal{O}}$ is isomorphic to $\overline{\mathcal{M}}_{\bar{\lambda}}:=\overline{\mathcal{G}r}_{\bar{\lambda}}\cap\mathcal{G}r_{0}^{-}$ for a small dominant weight $\bar{\lambda}$ of $H$. Note that $\overline{\mathcal{M}}_{\bar{\lambda}}$ is an open subset of the twisted Schubert variety $\overline{\mathcal{G}r}_{\bar{\lambda}}$ and $\overline{\mathcal{G}r}_{\bar{\lambda}}$ is a normal variety. It follows that $\overline{\mathcal{O}}$ is also normal. ∎ In fact, when $\langle,\rangle$ is symplectic, any nilpotent variety in $\mathcal{A}$ is normal, see [Oh]. In loc.cit., Ohta also showed that not all nilpotent varieties are $\mathcal{N}_{\mathfrak{p}}$ is normal, when $\langle,\rangle$ is symmetric. When $\langle,\rangle$ is symmetric and $\dim V$ is odd, this theorem seems to be new. ###### Remark 5.2. Theorem 5.1 is true for any field $\mathrm{k}$ of characteristic $p>2$, as one can easily see that the classification theorem in Section 3 still holds for order 2 nilpotent orbits, and the arguments in Theorem 4.5, Theorem 4.6 applies as well. The same remark applies to the following Theorem 5.3 and Theorem 5.4 For any variety $X$, let ${\rm IC}_{X}$ denote the intersection cohomology sheaf on $X$. The perverse sheaf ${\rm IC}_{X}$ captures the singularity of the variety $X$. For any $x\in X$, we denote by $\mathscr{H}^{k}_{x}({\rm IC}_{X})$ the k-th cohomology of the stalk of ${\rm IC}_{X}$ at $x$. ###### Theorem 5.3. 1. (1) When $\langle,\rangle$ is symmetric and $\dim V=2n+1$, for any $0\leq j\leq n$, let $\mathcal{O}_{j}$ denote the nilpotent orbit in $\mathcal{A}$ associated to the partition $[2^{j}1^{2n+1-2j}]$ and let $\mathcal{O}^{\prime}_{j}$ denote the nilpotent orbit in $\mathfrak{sp}_{2n}$ associated to the partition $[2^{j}1^{2n-2j}]$, we have $\dim\mathcal{O}_{j}=\dim\mathcal{O}^{\prime}_{j}=j(2n+1-j).$ Moreover, for any $x\in\mathcal{O}_{[2^{i}1^{2n+1-2i}]}$ and $x^{\prime}\in\mathcal{O}^{\prime}_{[2^{i}1^{2n-2i}]}$, and for any $k\in\mathbb{Z}$, $\dim\mathscr{H}_{x}^{k}({\rm IC}_{\overline{\mathcal{O}}_{j}})=\dim\mathscr{H}_{x}^{k}({\rm IC}_{\overline{\mathcal{O}^{\prime}}_{j}}).$ 2. (2) When $\langle,\rangle$ is symplectic and $\dim V=2n$, for any $0\leq j\leq\lfloor\frac{n}{2}\rfloor$, let $\mathcal{O}_{2j}$ denote the nilpotent orbit in $\mathcal{A}$ associated to the partition $[2^{2j}1^{2n-4j}]$ and let $\mathcal{O}^{\prime}_{2j}$ denote the nilpotent orbit in $\mathfrak{so}_{2n+1}$ associated to the partition $[2^{2j}1^{2n+1-4j}]$, we have $\dim\mathcal{O}_{2j}=\dim\mathcal{O}^{\prime}_{2j}=4j(n-j),$ Moreover, for any integer $0\leq i\leq j$, $x\in\mathcal{O}_{2i}$, $x^{\prime}\in\mathcal{O}^{\prime}_{2i}$, and for any $k\in\mathbb{Z}$, we have $\dim\mathscr{H}_{x}^{k}({\rm IC}_{\overline{\mathcal{O}}_{2j}})=\dim\mathscr{H}_{x}^{k}({\rm IC}_{\overline{\mathcal{O}^{\prime}}_{2j}}).$ ###### Proof. We first prove part 1). By Theorem 3.7 and [CM, Corollary 6.1.4], it is easy to verify $\dim\mathcal{O}_{j}=\dim\mathcal{O}^{\prime}_{j}=j(2n+1-j)$. By Theorem 4.5, $\overline{\mathcal{O}}_{j}$ can be embedded into an open subset in the twisted affine Schubert variety $\overline{\mathcal{G}r}_{\omega_{j}}$ associated to $(\mathrm{SL}_{2n+1},\sigma)$. On the other hand, in view of [AH], $\overline{\mathcal{O}^{\prime}}_{j}$ can be embedded into the untwisted affine Schubert variety $\overline{\operatorname{Gr}}^{\omega_{j}}_{\mathrm{Sp}_{2n}}$ in the affine Grassmannian $\overline{\operatorname{Gr}}_{\mathrm{Sp}_{2n}}$ of $\mathrm{Sp}_{2n}$. Set $\mathcal{F}={\rm IC}_{\overline{\mathcal{O}}_{j}}[-\dim\overline{\mathcal{O}}_{j}],\quad\text{ and }\mathcal{F}^{\prime}={\rm IC}_{\overline{\mathcal{O}^{\prime}}_{j}}[-\dim\overline{\mathcal{O}^{\prime}}_{j}].$ By purity vanishing property of intersection cohomology sheaf of Schubert varieties (cf. [KL]), $\mathscr{H}_{x}^{k}(\mathcal{F})=\mathscr{H}_{x^{\prime}}^{k}(\mathcal{F}^{\prime})=0$ when $k$ is odd. Equivalently, $\mathscr{H}_{x}^{k}({\rm IC}_{\overline{\mathcal{O}}_{j}})=\mathscr{H}_{x}^{k}({\rm IC}_{\overline{\mathcal{O}^{\prime}}_{j}})=0$ for any odd integer $k$, as $\dim\overline{\mathcal{O}}_{j}=\dim\overline{\mathcal{O}^{\prime}}_{j}$ is even. Note that the affine Grassmannian $\operatorname{Gr}_{\mathrm{Sp}_{2n}}$ and the twisted affine Grassmannian $\mathcal{G}r_{{\rm SL}_{2n+1}}$ have the same underlying affine Weyl group. Applying the results in [KL], the polynomials $\sum\dim\mathscr{H}_{x}^{2k}(\mathcal{F})q^{k}$ and $\sum\mathscr{H}_{x}^{2k}(\mathcal{F}^{\prime})q^{k}$ are both equal to the same Kazhdan-Lusztig polynomial $P_{\omega_{i},\omega_{j}}(q)$ for the affine Weyl group of $\mathfrak{so}_{2n+1}$. It follows that $\dim\mathscr{H}_{x}^{k}({\rm IC}_{\overline{\mathcal{O}}_{j}})=\dim\mathscr{H}_{x}^{k}({\rm IC}_{\overline{\mathcal{O}^{\prime}}_{j}})$ for all even integer $k$. Alternatively, one can see these two polynomials are equal, as they both coincide with the jump polynomial of the Brylinsky-Kostant filtration on the irreducible representation $V_{\omega_{j}}$ of $H$, see [Bry, Zh]. For the second part of the theorem, the proof is almost the same, except that by Theorem 4.6, $\overline{\mathcal{O}}_{2j}$ can be openly embedded into the twisted affine Schubert variety $\overline{\mathcal{G}r}_{\omega_{2j}}$ associated to $(\mathrm{SL}_{2n},\sigma)$, and $\overline{\mathcal{O}}^{\prime}_{2j}$ can be openly embedded into the affine Schubert variety $\overline{\operatorname{Gr}}^{\omega_{2j}}_{\mathrm{Spin}_{2n+1}}$. ∎ Part 1) of this theorem was due to Chen-Xue-Vilonen [CVX] by different methods. This theorem shows that there is a natural bijection between order 2 nilpotent varieties in $\mathcal{A}$ and order 2 nilpotent varieties in its dual classical Lie algebras, such that they share similar geometry and singularities. However, under such bijection the associated two nilpotent varieties are not necessarily isomorphic. For example, when $\langle,\rangle$ is symmetric and $\dim V=2n+1$, $\overline{\mathcal{O}}_{[21^{2n-1}]}$ is the minimal nilpotent variety in $\mathcal{A}$. In fact $\overline{\mathcal{O}}_{[21^{2n-1}]}$ is smooth, since the quasi-miniscule twisted affine Schubert variety $\overline{\mathcal{G}r}_{\omega_{1}}$ is smooth, cf. [HR, Section 5.1]. On the other hand, the minimal nilpotent variety $\overline{\mathcal{O}^{\prime}}_{[21^{2n-2}]}$ in $\mathfrak{sp}_{2n}$ is not smooth, as the smooth locus of the quasi-miniscule affine Schubert variety $\overline{\operatorname{Gr}}_{\omega_{1}}$ is the open cell $\operatorname{Gr}_{\omega_{1}}$. We now describe another application. ###### Theorem 5.4. If $\langle,\rangle$ is symplectic, then the smooth locus of any order 2 nilpotent variety in $\mathcal{A}$ is the open nilpotent orbit. ###### Proof. Let $\overline{\mathcal{O}}$ be any order 2 nilpotent variety in $\mathcal{A}$. By Theorem 4.6, $\overline{\mathcal{O}}$ can be openly embedded into a twisted Schubert variety $\overline{\mathcal{G}r}_{\bar{\lambda}}$ with $\bar{\lambda}$ small, in the twisted affine Grassmannian $\mathcal{G}r_{\mathrm{SL}_{2n}}$. Then this theorem follows from [BH, Theorem 1.2]. ∎ ## References [AH] P. Achar and A. Henderson, Geometric Satake, Springer correspondence and small representations. Selecta Math. (N.S.) 19 (2013), no. 4, 949-986. * [AH2] P. Achar and A. Henderson and S. Riche, Geometric Satake, Springer correspondence, and small representations II . Represent. Theory 19 (2015), 94-166. * [BH] J. Hong and M. Besson, Smooth locus of twisted affine Schubert varieties and twisted affine Demazure modules, arXiv:2010.11357. * [Bo] A. Borel, Linear algebraic groups. Second edition. Graduate Texts in Mathematics, 126. Springer-Verlag, New York, 1991. * [Br] A. 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# The AL-Gaussian Distribution as the Descriptive Model for the Internal Proactive Inhibition in the Standard Stop Signal Task Mohsen Soltanifara, Michael Escobarb, Annie Dupuisc, Andre Chevrierd and Russell Schachare CONTACT Mohsen Soltanifar. Email: <EMAIL_ADDRESS>a,b,c Biostatistics Division, Dalla Lana School of Public Health, University of Toronto, Toronto, M5T 3M7, ON, Canada ; a,c,d,e Department of Psychiatry, The Hospital for Sick Children, 555 University Avenue, Toronto, M5G 1X8, ON, Canada ###### Abstract Measurements of response inhibition components of reactive inhibition and proactive inhibition within the stop-signal paradigm have been of particular interest to researchers since the 1980s. While frequentist nonparametric and Bayesian parametric methods have been proposed to precisely estimate the entire distribution of reactive inhibition, quantified by stop signal reaction times(SSRT), there is no method yet in the stop-signal task literature to precisely estimate the entire distribution of proactive inhibition. We introduce an Asymmetric Laplace Gaussian (ALG) model to describe the distribution of proactive inhibition. The proposed method is based on two assumptions of independent trial type(go/stop) reaction times and Ex-Gaussian (ExG) models. Results indicated that the four parametric, ALG model uniquely describes the proactive inhibition distribution and its key shape features; and, its hazard function is monotonically increasing, as are its three parametric ExG components. In conclusion, both response inhibition components can be uniquely modeled via variations of the four parametric ALG model described with their associated similar distributional features. ###### keywords: Proactive Inhibition, Reaction Times, Ex-Gaussian, Asymmetric Laplace Gaussian, Bayesian Parametric Approach, Hazard function. ## 1 Introduction Response inhibition refers to one’s ability to stop responses or impulses that just became inappropriate or unwanted within continually changing environments,[1]. This process’s importance lies in one’s being in continually changing conditions, which require new, updated courses of action,[2]. Some instances of response inhibition in daily life include braking while driving a vehicle into an intersection in reaction to a sudden traffic change, changing direction during a tennis game, and resisting an extra piece of pizza at a birthday party. Two paradigms have been proposed to study the lab setting’s response inhibition, [3]: the stop-signal task and the Go/No-go task. In the standard stop-signal task, as used in this study, the task consists of a two- choice, response time task called the “go task” and the “stop task.” The go task is the primary task in which the participants are asked to correctly press a right or left button, in response to stimulus presentation, an “X” or “O” on the computer screen. The stop task is the occasional, secondary task in which (with a probability of stop signal $p_{ss}$ ) the participants are presented with a stop signal alarm after a temporal delay; Participants are instructed to withhold their responses the ongoing go task. Successful response inhibition occurs when participants successfully withhold their response to the “X” or “O” on the screen in the stop task (Figure 1). Figure 1: The standard stop signal task with two inhibition components: proactive inhibition, reactive inhibition [4] Response inhibition has two distinctive temporal-dynamic components: reactive inhibition and proactive inhibition. Both components have been utilized within the standard stop-signal task, or its varieties, to discriminate different clinical groups, [5, 6]. We refer to reactive inhibition as the outright inhibition triggered by an external cause, while proactive inhibition is restraint of actions in preparation for stopping by external conditions, [7]. Each of these inhibition components have been quantified in distinctive methods as constant point estimate or distribution in the stop-signal task (SST) literature[1, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Reactive inhibition has been quantified as Stop Signal Reaction Times (SSRT) in the SST literature from both point estimation and distributional perspectives. Primary point estimations of the reactive inhibition up to now include the Crude SSRT, the Logan 1994 SSRT, [1], the Weighted SSRT, the Mixture SSRT, [8], and the time series-based SSRT,[9]. On the other hand, primary distributional estimations of the reactive inhibition include the Colonius’s nonparametric method,[10], the Bayesian Parametric Approach (BPA), [11, 12] — with two subtype methods: the Individual BPA (IBPA) and the Hierarchical BPA(HBPA) — and the mixture method, [13]. In the mixture method, the entire SST data is partitioned to type A SST data, trials following a go trial, and type B SST data, trials following a stop trial; The trial type weights are defined as $W_{A}^{c}=$proportion of type A stops within all stops, $W_{B}^{c}=1-W_{A}^{c}.$ Now, for $W_{A}\sim Bernoulli(W_{A}^{c}),W_{B}\sim Bernoulli(W_{B}^{c}),$ and fitted cluster type $SSRT_{A},SSRT_{B}$ distributions by the former two methods, the mixture index for the reactive inhibition is defined as: $\displaystyle SSRT_{Mixture}=^{d}W_{A}\times SSRT_{A}+W_{B}\times SSRT_{B}.$ (1) Note that for the case of parametric mixture SSRT, the cluster type components $SSRT_{A},$ and $SSRT_{B}$ may take a variety of proposed reaction time (RT) models, such as Ex-Gaussian (ExG), Ex-Wald, Wald, Gamma, Weibull, and lognormal, [14, 15, 16]. However, given the ExG model’s practical advantages to others, it is widely considered the parametric model for the reactive inhibition,[11, 12]. Proactive inhibition has been quantified in the SST literature merely as point estimation from different perspectives. The first such estimation is defined through a dynamic Bayesian model,[17]. Here, using the mixture assumption for the predictive distribution for the probability of the kth trial being a stop task $``p(Stop),”$ the reactive inhibition is defined as the following positive Pearson correlation called the ”Sequential Effect(SE)”: $\displaystyle SE$ $\displaystyle=$ $\displaystyle Corr(p(Stop),GORT).$ (2) The second estimation is based on the variation of differences of reaction times in a go trial (GORTs) in the associated arms of the modified standard stop signal task paradigm,[18, 19, 20, 21, 22, 23]. Generally, for two given probabilities of stop signals in two arms of the modified SST where $0\ll p_{ss}^{(1)}<p_{ss}^{(2)}\ll 1,$ the arm type proactive inhibition $\Delta_{AT}GORT$ is defined by as: $\displaystyle\Delta_{AT}GORT$ $\displaystyle=$ $\displaystyle mean(GORT_{(p_{ss}^{(2)})})-mean(GORT_{(p_{ss}^{(1)})}).$ (3) The last type of the point estimation of proactive inhibition is based on the differences of GORTs in the trial type clusters of the standard SST paradigm [8, 9](Figure.2.). Here, for the fixed stop signal probability (e.g.$p_{ss}=0.25$) and the type A GORTA and type B GORTB, the trial type proactive inhibition $\Delta_{TT}GORT$ is defined as: $\displaystyle\Delta_{TT}GORT$ $\displaystyle=$ $\displaystyle mean(GORT_{B})-mean(GORT_{A}).$ (4) Figure 2: Trial type point estimation of proactive inhibition in the standard stop signal task However, little information is available in the SST literature on the entire distribution of proactive inhibition and its key features. This subject is significant as measures of central tendencies, such as mean or median, are insufficient — and even unnecessary — to compare mostly skewed response inhibition distributions,[24]. Besides, masking prominent features of the proactive inhibitions by using central tendency measures may result in incorrect conclusions about their make-up. As an example, two different clinical groups may have the same mean of proactive inhibition, but the shape of their distributions may differ: one may be more positively skewed, or more leptokurtic, or possess a higher domain of variance. The methods mentioned earlier of the estimation of proactive inhibition do not allow for precise estimation and description of the appropriate models for the entire set of proactive inhibition distributions. This study proposes a four parametric, Asymmetric Laplace Gaussian (ALG) model for the entire proactive inhibition, given the assumption of independent trial type (go/stop) GORTs within the standard stop-signal task. The study outline is as follows. First, as in [8, 9] the overall SST data for each participant is partitioned to type A SST data and type B SST data. Second, using the Individual Bayesian Parametric Approach (IBPA) method [11, 12], the fitted ExG GORT mean posterior parameters are calculated for the cluster type SST data. Then, the distribution of proactive response inhibition is introduced as the difference of two independent fitted GORT ExG models, and it is shown that it has a four-parametric ALG distribution. Third, the descriptive statistics, shape statistics, and vital distributional properties of the ALG model, such as component decomposition, shape and tail behavior, and monotonic hazard function, are discussed. Finally, an empirical example is presented to manifest the above-presented results. Table 1 presents the summary of estimations methods presented in the literature, including this study given the type of inhibition component. Table 1: Summary of Estimation Methods of Inhibition Components [b] Estimation Inhibition Component Reactive Proactive Constant Index SSRT $\Delta GORT$,$SE$ Examples $SSRT_{Crude},SSRT_{Mixture},SSRT_{Weighted}$ $\Delta_{AT}GORT,\Delta_{TT}GORT$ $SSRT_{Logan1994},SSRT_{SS.Logan1994}$ Distribution Index SSRT $\Delta GORT$ Examples ExG,LN,Wald ALG Ex-Wald, Gamma ## 2 Materials & Methods ### 2.1 Data The study data have been previously described [25, 8, 9, 13]. Data were collected at the Ontario Science Center in Toronto, Canada, in 2009-2010. Included were 16,099 participants aged 6 to 17. The participants’ parents provided the required ethical consent for the SST experiment. Each participant completed the SST task, including four blocks of 24 trials with a total of 96 trials, including random 25% stop trials (24 stops) and 75% go trials(72 goes). The SST tracking algorithm was designed so that at the end of trials, each participant achieves a 50% probability of successful inhibition. ### 2.2 Participants The study participants are the same as those described in [9, 13]. Included here is a unique subsample of 44 participants with a mean age of 12.1 years, with 96 SST trials for each, and an almost balanced number of trial type stop trials (10-14 type B stop trials vs. 14-10 type A stop trials, respectively). This almost balanced number of trial type stop trials yields 30-42 type B go trials vs. 42-30 type A stop trials, respectively. ### 2.3 The SST Clusters The study Stop Signal Task clusters have been described before in [8, 9, 13]. Each participant’s SST data was partitioned to type A SST data, where a go trial preceded all trials, and type B SST data, all trials preceded by a stop trial. Hence, each participant has three types of SST data clusters: Type-A SST cluster (i.e., 56 trials), Type B SST cluster(i.e., 40 trials), and Type-S SST cluster (all 96 trials). Then, using IBPA, the parameters of the corresponding Ex-Gaussian (ExG) GORT’s parameters (i.e. $\theta_{A}=(\mu_{A},\sigma_{A},\tau_{A}),\theta_{B}=(\mu_{B},\sigma_{B},\tau_{B}),\theta_{S}=(\mu_{S},\sigma_{S},\tau_{S})$ were computed as described in the upcoming results section 3.2. ### 2.4 Preliminaries on Component Distributions The first two definitions provide the critical features of the components of our upcoming calculations, namely Ex-Gaussian distribution (ExG), [26], and the Asymmetric Laplace distribution (AL), [27]. ###### Definition 2.1. A random variable has an Ex-Gaussian (ExG) distribution with parameters $(\mu,\sigma,\tau)$ whenever it is considered as the sum of an independent normal random variable with parameters $(\mu,\sigma^{2})$ and an exponential random variable with parameter $\tau$: $\displaystyle ExG(\mu,\sigma,\tau)$ $\displaystyle=^{d}$ $\displaystyle N(\mu,\sigma^{2})\oplus Exp(\tau).$ (5) The density, the moment generating function, the nth cumulant $(n\geq 1),$ the variance, the skewness and the kurtosis of the ExG distribution are given by: $\displaystyle PDF$ $\displaystyle f_{ExG}(t\|\mu,\sigma,\tau)=\frac{1}{\tau}exp(\frac{\mu-t}{\tau}+\frac{\sigma^{2}}{2\tau^{2}})\Phi(\frac{\mu-t}{\sigma}-\frac{\sigma}{\tau}):\sigma,\tau>0,t\in\mathbb{R},$ $\displaystyle MGF$ $\displaystyle m_{ExG}(t)=(1-t\tau)^{-1}exp(\mu.t+\frac{\sigma^{2}}{2}t^{2}):t<\tau^{-1},$ $\displaystyle n^{th}Cumulant$ $\displaystyle\kappa_{n}^{ExG}=(n-1)!\tau^{n}+1_{n=1}(n)\mu+1_{n=2}(n)\sigma^{2}:1\leq n,$ $\displaystyle Mean$ $\displaystyle E(ExG)=\mu+\tau,$ $\displaystyle Variance$ $\displaystyle Var(ExG)=\sigma^{2}+\tau^{2},$ $\displaystyle Skewness$ $\displaystyle\gamma_{ExG}=2(1+\sigma^{2}\tau^{-2})^{-3/2},$ $\displaystyle Kurtosis$ $\displaystyle\kappa_{ExG}=3\frac{(1+2\sigma^{-2}\tau^{2}+3\sigma^{-4}\tau^{4})}{(1+\sigma^{-2}\tau^{2})^{2}}.$ (6) ###### Definition 2.2. A random variable has an Asymmetric Laplace (AL) distribution with parameters $(\alpha_{1},\alpha_{2})$ whenever it is considered as the difference of two independent exponential random variables with parameters $\alpha_{2}$, and $\alpha_{1}$, respectively: $\displaystyle AL(\alpha_{1},\alpha_{2})$ $\displaystyle=^{d}$ $\displaystyle Exp(\alpha_{2})\ominus Exp(\alpha_{1}).$ (7) The density, the moment generating function, the nth cumulant $(n\geq 1),$ the variance, the skewness and the kurtosis of the AL distribution are given by: $\displaystyle PDF$ $\displaystyle f_{AL}(t\|\alpha_{1},\alpha_{2})=\frac{exp(\frac{t}{\alpha_{1}})1_{(-\infty,0)}(t)+exp(\frac{-t}{\alpha_{2}})1_{[0,\infty)}(t)}{\alpha_{1}+\alpha_{2}}\ t\in\mathbb{R},$ $\displaystyle MGF$ $\displaystyle m_{AL}(t)=(1+(\alpha_{1}-\alpha_{2})t-\alpha_{1}\alpha_{2}.t^{2})^{-1}\ -\alpha_{1}^{-1}<t<\alpha_{2}^{-1},$ $\displaystyle n^{th}Cumulant$ $\displaystyle\kappa_{n}^{AL}=(n-1)!((-\alpha_{1})^{n}+(\alpha_{2})^{n})\ 1\leq n,$ $\displaystyle Mean$ $\displaystyle E(AL)=-(\alpha_{1}-\alpha_{2}),$ $\displaystyle Variance$ $\displaystyle Var(AL)=\alpha_{1}^{2}+\alpha_{2}^{2},$ $\displaystyle Skewness$ $\displaystyle\gamma_{AL}=-2(\alpha_{1}^{3}-\alpha_{2}^{3})\times(\alpha_{1}^{2}+\alpha_{2}^{2})^{-3/2},$ $\displaystyle Kurtosis$ $\displaystyle\kappa_{AL}=3(3\alpha_{1}^{4}+2\alpha_{1}^{2}\alpha_{2}^{2}+3\alpha_{2}^{4})\times(\alpha_{1}^{2}+\alpha_{2}^{2})^{-2}.$ (8) The convolution of two independent, AL random variables and Gaussian random variables, called ALG or Normal-Laplace (NL) random variables, has been of special attention in the literature [28, 29]. ###### Definition 2.3. A random variable has Asymmetric Laplace-Gaussian (ALG) distribution with parameters $(\alpha_{1},\alpha_{2},\mu,\sigma)$ whenever it is considered as the sum of two independent, Asymmetric Laplace random variables with parameters $(\alpha_{1},\alpha_{2})$, and a Normal random variable with parameters $(\mu,\sigma^{2})$, respectively: $\displaystyle ALG(\alpha_{1},\alpha_{2},\mu,\sigma)$ $\displaystyle=^{d}$ $\displaystyle AL(\alpha_{1},\alpha_{2})\oplus N(\mu,\sigma^{2}).$ (9) Note that since $AL(0^{+},\alpha_{2})=^{d}Exp(\alpha_{2}),$ it follows that $ALG(0^{+},\alpha_{2},\mu,\sigma)=^{d}ExG(\mu,\sigma,\alpha_{2})$. Consequently, the ExG model can be considered a special degenerate ALG model. Next, the following key Theorems equip us to propose the ALG distribution as the model for the proactive inhibition and compute the key descriptive and shape statistics of the ALG distribution in terms of its Laplacian and Gaussian components,[30]. ###### Theorem 2.4. Let $X,Y$ be two independent, real-valued random variables with finite, moment generating function $m_{X},m_{Y},$ and cumulant functions $\kappa,\kappa,$ respectively. Then, for some $s_{0}>0:$ $\displaystyle m_{X+Y}(t)=m_{X}(t)m_{Y}(t):\ \ (-s_{0}<t<s_{0}),$ (10) $\displaystyle\kappa_{X+Y}(t)=\kappa_{X}(t)+\kappa(t):\ \ \ (-s_{0}<t<s_{0}).$ (11) ###### Theorem 2.5. Let $X,Y$ be two real-valued random variables with finite moment generating functions $m_{X},m_{Y},$ respectively. Assume for some $s_{0}>0:\ m_{X}(t)=m_{Y}(t)\ (-s_{0}<t<s_{0}).$ Then, $X,Y$ have the same distribution. Finally, the last two theorems enable us to describe the behavior of the hazard function of the ALG model for the proactive inhibition, [31]. ###### Theorem 2.6. Let $X$ be a real-valued random variable with differentiable PDF $f_{X}$ and CDF $F_{X}$ such that $f_{X}(t)\rightarrow 0,F_{X}(t)\rightarrow 1\ as\ t\rightarrow\infty,$ and $-ln(f_{X}(t))$ is convex(concave). Then, the hazard function $h_{X}$ is increasing(decreasing). ###### Theorem 2.7. Let $X,Y$ be two independent, real-valued random variables with (strictly) increasing hazard functions $h_{X},h_{Y},$ respectively. Then, the hazard function of their sum, $h_{X+Y}$ is (strictly) increasing as well. ### 2.5 Proactive Inhibition Index Proactive inhibition was operationalized based on the standard stop signal task’s internal perspective,[9]. Here, for a given fixed stop signal probability (e.g., 0.25), type A GORT of $GORT_{A}$ (GORT for a trial after a go trial), and type B GORT of $GORT_{B}$ (GORT for a trial after a stop trial), the internal proactive inhibition is defined as: $\displaystyle\Delta GORT$ $\displaystyle=^{d}$ $\displaystyle GORT_{B}-GORT_{A}.$ (12) Note that there are two mathematical perspectives for the proactive inhibition: First, a model with two ExG components; Second, a model with Asymmetric Laplace (AL) and Gaussian components. Henceforward, it is understood within the given context which perspective is being discussed. ### 2.6 Statistical Analysis The statistical ALG model to describe the proactive inhibition distribution was presented using moment generating functions, [29]. Next, the ALG model’s descriptive and shape statistics were computed in terms of parameters of the cluster type ExG components,[30]. Finally, its hazard function behavior was theoretically inferred using its components’ parameters [31]. The ExG components of the presented statistical model were estimated using the IBPA method,[11, 12]. As in [13], each participant had three IBPA associated ExG parametric estimations $\theta_{A}=(\mu_{A},\sigma_{A},\tau_{A}),\theta_{B}=(\mu_{B},\sigma_{B},\tau_{B}),$ and $\theta_{S}=(\mu_{S},\sigma_{S},\tau_{S}),$ associated to type-A cluster SST data, type-B cluster SST data, and the entire SST data, respectively. These parameters were estimated as the posterior means of the associated following IBPA procedure with three chains, 5,000 burn-ins within 20,000 simulations in Bayesian Ex-Gaussian Estimation of Stop-Signal RT distributions (BEESTS) 2.0 software,[12]: Data \| Individual Priors ---\|--- $GORT\sim ExG(\mu_{go},\sigma_{go},\tau_{go})$ \| $SRRT\sim ExG(\mu_{go},\sigma_{go},\tau_{go},\mu_{stop},\sigma_{stop},\tau_{stop},SSD)I^{+}_{[1,1000]}$ \| $\mu_{go},\sigma_{go},\tau_{go}\sim U[10,2000]$ $SSRT\sim ExG(\mu_{go},\sigma_{go},\tau_{go},\mu_{stop},\sigma_{stop},\tau_{stop},SSD)I^{+}_{[1,1000]}$ \| $\mu_{go},\sigma_{go},\tau_{go}\sim U[10,2000]$ Two sets of comparisons were conducted using paired t-tests (DescTools, $R$ software version 4.0.0, [32]): First, a primary comparison between the cluster-type fitted parameter of the ExG distribution, the descriptive statistics, and the shape statistics; second, secondary comparisons between the ALG model descriptive and shape statistics and its associated cluster-type ExG components. ## 3 Results The results are divided into two subsections. In subsection 3.1, we explore the mathematical analysis of the proposed model for the proactive inhibition in the standard stop-signal task. This model includes a four parametric ALG for the proactive inhibition and its prominent distributional properties. In subsection 3.2, we present an empirical example of the case and discuss its various distributional features. ### 3.1 Mathematical Analysis #### 3.1.1 The Proactive Inhibition Distribution and its Parameters First of all, we propose a mathematical model for the proactive inhibition provided by the ALG: ###### Theorem 3.1. (The Main Result). The four parametric $ALG(\tau_{A},\tau_{B},\mu_{B}-\mu_{A},(\sigma_{B}^{2}+\sigma_{A}^{2})^{1/2})$ presents a model for the Internal Proactive Inhibition Index $\Delta GORT$ with trial type-related parameters $(\mu_{A},\sigma_{A},\tau_{A},\mu_{B},\sigma_{B},\tau_{B})$ in the Standard Stop Signal Task. As a corollary of Theorem 3.1, the probability density function $(f_{\Delta GORT})$ and the cumulative density function $(F_{\Delta GORT})$ of the ALG distribution for the Internal Proactive Inhibition Index are given by [28, 29],: $\displaystyle f_{\Delta GORT}(t)$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{A}+\tau_{B}}$ (13) $\displaystyle\times[$ $\displaystyle e^{(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{2\tau_{B}}(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{\tau_{B}}-2\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}))}\times(1-\Phi(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{\tau_{B}}-\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}))$ $\displaystyle+$ $\displaystyle e^{(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{2\tau_{A}}(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{\tau_{A}}+2\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}))}\times(1-\Phi(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{\tau_{A}}+\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}))]$ $\displaystyle\hskip 284.52756ptt\in\mathbb{R}$ and $\displaystyle F_{\Delta GORT}(t)$ $\displaystyle=$ $\displaystyle\frac{1}{\tau_{A}^{-1}+\tau_{B}^{-1}}\times[(\tau_{A}^{-1}+\tau_{B}^{-1})\Phi(\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}})$ (14) $\displaystyle-$ $\displaystyle\tau_{A}^{-1}e^{(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{2\tau_{B}}(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{\tau_{B}}-2\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}))}\times(1-\Phi(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{\tau_{B}}-\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}))$ $\displaystyle+$ $\displaystyle\tau_{B}^{-1}e^{(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{2\tau_{A}}(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{\tau_{A}}+2\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}))}\times(1-\Phi(\frac{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}{\tau_{A}}+\frac{t-(\mu_{B}-\mu_{A})}{\sqrt{\sigma_{B}^{2}+\sigma_{A}^{2}}}))]$ $\displaystyle\hskip 284.52756ptt\in\mathbb{R},$ respectively. Here $\Phi$ denotes the standard normal cumulative distribution function. Next, given trial type parameters, we estimate the descriptive and shape statistics for the proposed ALG model of the proactive inhibition: ###### Theorem 3.2. The descriptive statistics and the shape statistics of the Proactive Inhibition ALG distribution with trial type-related parameters $(\mu_{A},\sigma_{A},\tau_{A},\mu_{B},\sigma_{B},\tau_{B})$ in the standard stop signal task are given by: $\displaystyle n^{th}Cumulant$ $\displaystyle\kappa_{n}^{ALG}=(n-1)!((-\tau_{A})^{n}+\tau_{B}^{n})$ $\displaystyle\hskip 28.45274pt+1_{(n=1)}(n)(\mu_{B}-\mu_{A})+1_{(n=2)}(n)(\sigma_{B}^{2}+\sigma_{A}^{2}):\ \ 1\leq n$ $\displaystyle Mean$ $\displaystyle E(ALG)=\tau_{B}-\tau_{A}+\mu_{B}-\mu_{A},$ $\displaystyle Variance$ $\displaystyle Var(ALG)=\tau_{A}^{2}+\tau_{B}^{2}+\sigma_{A}^{2}+\sigma_{B}^{2},$ $\displaystyle Skewness$ $\displaystyle\gamma_{ALG}=\frac{2(\tau_{B}^{3}-\tau_{A}^{3})}{(\tau_{A}^{2}+\tau_{B}^{2}+\sigma_{A}^{2}+\sigma_{B}^{2})^{3/2}},$ $\displaystyle Kurtosis$ $\displaystyle\kappa_{ALG}=\frac{6(\tau_{B}^{4}+\tau_{A}^{4})}{(\tau_{A}^{2}+\tau_{B}^{2}+\sigma_{A}^{2}+\sigma_{B}^{2})^{2}}.$ (15) #### 3.1.2 The Proactive Inhibition’s Key ALG Distributional Properties In this section, we present key distributional properties for the ALG model for proactive inhibition including: (i) component decompositions in terms of trial type GORT; (ii) shape and tail behavior; and, (iii) behavior of the hazard function. ###### Theorem 3.3. (Component Decomposition). An ALG model for proactive inhibition emerges from uncountable pairs of trial type related GORT $(GORT_{A},GORT_{B})$ distributions. We remind the reader that Theorem3.3 presents a process to simulate a plausible four parametric ALG distribution for the proactive inhibition. ###### Theorem 3.4. (Shape and Tail Behavior). An ALG model for proactive inhibition has unimodal, generally asymmetric, infinite, differentiable density with extreme large values proportionate to the $Exp(1/\tau_{B})$ distribution. We remind the reader that contrary to the ALG model’s mean for proactive inhibition, there are no closed-form formulas for the mode and the median, respectively. Similar to the ExG model for reactive inhibition with increasing hazard function, we have: ###### Theorem 3.5. (Hazard Function’s Behavior). An ALG model for proactive inhibition has increasing hazard function. ### 3.2 The Empirical Example #### 3.2.1 The ALG Model for Proactive Inhibition This section presents an example of the empirical data for the theoretical results inferred in the previous section on the ALG model for proactive inhibition and its descriptive, shape, and hazard function’s key features. These results are based on the cluster type IBPA estimation of mean posterior ExG parameters $\theta=(\mu,\sigma,\tau)$ presented in Table 5(Appendix B). The ExG model for each of the two components of proactive inhibition has the following key features presented by Table 2. First, while type B $\mu,$, and $\sigma$ parameters are significantly larger than their type A counterparts, there is no difference for the $\tau$ parameter. In addition, the sample average proactive inhibition is 92.1 ms (95% CI = (69.4,114.9)). Second, both ExG components are positively skewed and leptokurtic. Finally, there is no significant difference between their trial-type skewness and their trial-type kurtosis, respectively. Figure3a presents the trial type ExG modeled components of the ALG model. Table 2: Descriptive and paired t-test [mean ($95\%CI$)] results for parameters, descriptive and shape statistics of fitted Ex-Gaussian distribution to cluster type GORT and AL-Gaussian distribution to $\Delta GORT,(n=44).$ [b] ExG model ALG model Cluster Comparison Cluster Type A Type B Type B vs. Type A Type S $\alpha_{1}$ - - - 104.2 - - - (90.4,117.9) $\alpha_{2}$ - - - 142.4 - - - (125.9,158.8) Parameter $\mu$ 478.8 532.8 53.9 53.9 (448.0,509.7) (498.6,566.9) (30.9,76.9) (30.9,76.9) $\sigma$ 109.9 133.1 23.2 179.2 (90.5,129.3) (108.4,157.8) (-0.1,46.4) (151.4,206.9) $\tau$ 104.2 142.4 38.2* - (90.4,117.9) (125.9,158.8) (19.6,56.8) - Mean 583.0 675.1 92.1* 92.1 (553.0,612.9) (633.8,716.4) (69.4,114.9) (69.4,114.9) Statistics St.D 160.6 202.4 41.8* 260.4 (143.5,177.8) (177.9,226.9) (25.9,57.6) (232.3,288.6) Skewness 0.787 0.918 0.131 0.186 (0.602,0.973) (0.751,1.085) (-0.113,0.375) (0.076,0.296) Kurtosis 4.966 5.300 0.334 1.153 (4.414,5.518) (4.790,5.808) (-0.397,1.064) (0.923,1.384) * • Notes: ∗p-value$<0.05$;∗∗p-value$<0.005$;∗∗∗p-value$<0.0005$. The ALG model for proactive inhibition has the following features, given cluster type parameter estimations. First, as a primary corollary of Theorem 3.2, the model is positively skewed whenever $\tau_{B}>\tau_{A}.$ The negatively skewed and symmetric cases hold whenever the strict greater inequality $>$ is replaced with $<$ and $=,$ respectively. According to appendix Table 5 data, all three cases exist (case 10: positive skew; case 16: symmetric; case 11: negative skew). Figure 3b presents all the mentioned three cases. Overall, the model is positively skewed given the results in Table 2. Second, as the second corollary of Theorem 3.2, the model is leptokurtic whenever $(2(\tau_{A}^{4}+\tau_{B}^{4}))^{1/2}-(\tau_{A}^{2}+\tau_{B}^{2})>\sigma_{A}^{2}+\sigma_{B}^{2}.$ The platykurtic and mesokurtic cases hold whenever the strict greater inequality $>$ is replaced with $<$ and $=,$ respectively. In particular, for the case, $\tau_{A}\approx\tau_{B},$ the model is always platykurtic. Overall, the model is platykurtic given results in Table 2. Third, while the ALG model’s standard deviation is larger than its two ExG’s components, its skewness and kurtosis are significantly smaller. Finally, the ALG model has a strictly increasing hazard function for various skewness cases, as mentioned in Theorem3.5 and presented in Figure 3c. Figure 3: The ALG density and its trial type ExG component densities; (b) The ALG density for the positively skewed, symmetric and negatively skewed cases; (c) The ALG hazard function for the positively skewed, symmetric and negatively skewed cases. #### 3.2.2 Proactive Inhibition ALG Model versus Reactive Inhibition ExG Model This section compares the ALG distribution of the proactive inhibition and the ExG distribution of the reactive inhibition in descriptive and shapes statistics. Table 3 presents the corresponding statistics for both models. As it is seen, the proactive ALG inhibition distribution has a significantly lower mean, lower skewness, lower kurtosis, and higher standard deviation than reactive inhibition ExG distribution. Also, while the proactive inhibition ALG distribution is platykurtic, the corresponding reactive ExG distribution is leptokurtic. However, the two distributions are both positively skewed. Overall, the two distributions are significantly distinctive. Table 3: Comparison of proactive inhibition ALG model versus reactive inhibition ExG model in terms of descriptive and shape statistics $(n=44).$ [b] Inhibition Reactive Proactive Proactive vs. Reactive Index $SSRT$ $\Delta GORT$ $\Delta GORT\ vs.\ SSRT$ Model ExG ALG ALG vs. ExG Mean 196.8 92.1 -104.6* (173.5,220.1) (69.4,114.9) (-140.6,-68.7) Statistics St.D 157.8 260.4 102.6* (139.4,176.2) (232.3,288.6) (71.8,133.6) Skewness 0.578 0.186 -0.401* (0.500,0.674) (0.076,0.296) (-0.540,-0.261) Kurtosis 4.231 1.153 -3.077* (3.998,4.465) (0.923,1.384) (-3.381,-2.775) * • Notes: ∗p-value$<0.05$;∗∗p-value$<0.005$;∗∗∗p-value$<0.0005$. ## 4 Discussion ### 4.1 Present Work This paper presents a four parametric model for the entire proactive inhibition distribution, the ALG model. This model is based on the two independent ExG components fitted to the trial type GORTs. Considering ExG models as a degenerate ALG model, this work indicates that a four parametric ALG model can model both response inhibition components. The proposed ALG model for proactive inhibition has several important aspects. First, the model is based on the independent assumption of GORTA and GORTB. Such speculation is warranted by the independent assignment of go trials and stop trials in the entire stop-signal task trials. Hence, via conditioning, their following associated type A go trial and type B go trials are independent, and so are their associated GORTs (i.e., GORTA and GORTB). Second, contrary to the point estimations of proactive inhibition in the form of mean [9], or correlation [17], it presents the entire distribution of the proactive inhibition. Third, the ALG model for proactive inhibition is entirely distinctive from the ExG model for reactive inhibition in terms of the mean, standard deviation, and kurtosis. This result provides more evidence on the distinction between proactive inhibition and reactive inhibition as a whole distribution. Fourth, similar to the ExG model for reactive inhibition, the ALG model for proactive inhibition is skewed to the right and has a monotonically increasing hazard function. Finally, the ALG model for proactive inhibition with its associated ExG modeled components is unique because other non-ExG RT models for its components (e.g., Gamma, Weibull, Lognormal, Wald, and Ex-Wald) do not yield to any known closed-form distribution for proactive inhibition. This limitation is easily verifiable by repeating the proof of Theorem 3.1 based on the uniqueness of moment generating functions for other non-ExG RT models for the trial type GORTs. The proposed ALG model for the proactive inhibition is estimated (or simulated) in two different methods. First, one may use the Bayesian or frequentist-based methods to fit the ExG parametric models to its trial-type components and then estimate the four parametric ALG model using Theorem 3.1, as done in Section 3.2. Second, one may subtract the trial type GORTs and fit the ALG model directly to the differenced GORT data using Maximum Likelihood (ML) or Expectation-Maximization (EM) algorithms [33]. The proposed ALG model uniquely distinguishes the reactive inhibition and proactive inhibition distribution in terms of vital distributional features. Table 4 presents an overall comparison for proactive inhibition and reactive inhibition in terms of the ALG model(with considering ExG as its particular case): Table 4: Comparison of proactive inhibition and reactive inhibition in terms of ALG model properties. [b] Inhibition Index $\\#$ Parameters $\\#$ Estimations Mean StD Skewness($+$) Kurtosis Hazard Proactive $\Delta GORT$ 4 2 lower higher lower platykurtic increasing Reactive $SSRT$ 3 1 higher lower higher leptokurtic increasing There are some limitations in the proposed ALG model for proactive inhibition. First, since GORTA data and GORTB data are unmatched, there is no way to calculate their correlation quantitatively. Hence, from a quantitative perspective, checking the validity of the assumption of independent GORTA and GORTB is difficult. Second, by its definition, proactive inhibition takes only non-negative values while the presented ALG model takes negative values. Third, similar to the ExG model for reactive inhibition, the ALG model for proactive inhibition has a monotonically increasing hazard function preventing it from being the best fitting model for the cases of proactive inhibition with peaked hazards. Finally, given the calculations’ structure of the ALG model parameters, based on those of ExG components, its parameters’ cognitive interpretations are highly dependent on its ExG components inheriting their constraints. ### 4.2 Future Work Future research should replicate the proposed approach in modeling the proactive inhibition distribution in this study in other different directions. This further work may include the following perspectives: First, one model should consider peaked hazard functions for the ALG model components to address RT data with such features. Second, there is a need to interpret the proposed ALG distribution parameters in terms of inhibition mechanisms in the brain and vise versa. Third, there is a lack of investigation comparing the proactive inhibition distribution in terms of the usual stochastic order, the descriptive and shape statistics across a spectrum of clinical groups such as ADHD, OCD, schizophrenia, and drug users. Finally, similar investigations on comparing the proactive inhibition distribution and its above key statistics are plausible in terms of the participants’ age. ### 4.3 Conclusion In conclusion, the ALG model provides a practical description of the proactive inhibition distribution that takes full advantage of its ExG components fitted for the trial type GORTs. It also offers a straightforward, computational analog of the proactive inhibition, comparable to the ExG model for reactive inhibition. Given the advantages of estimating the entire distribution of proactive inhibition over former point estimations, the researchers recommend considering the ALG model as the latest optimal choice to describe the distribution of proactive inhibition. Author Contributions: The authors contributed to the study in the following manner: Conceptualization, M.S.; methodology, M.S.; formal analysis, M.S.; validation, M.E, M.S., A.D and R.S.; investigation, M.E., M.S., A.D and R.S.; data curation, R.S.; writing—original draft preparation, M.S.; writing—review and editing, M.S.,M.E., A.D., A.C., and R.S.; All authors have read and agreed to the submitted version of the manuscript. ## Disclosure Statement Mohsen Soltanifar, Michael Escobar, Annie Dupuis and Andre Chevrier have no financial interests to disclose. Russell Schachar has equity in ehave and has been the on the Scientific Advisory Board(SAB) in Lilly and Highland Therapautic Inc, Toronto, Canada. ## Funding This work has no external funding. ## ORCID Mohsen Soltanifar: https://orcid.org/0000-0002-5989-0082 Michael Escobar: https://orcid.org/0000-0001-9055-4709 Annie Dupuis: https://orcid.org/0000-0002-8704-078X Andre Chevrier: https://orcid.org/0000-0002-4298-9529 Russell Schachar: https://orcid.org/0000-0002-2015-4395 ## Abbreviations $\begin{array}[]{ll}\text{ADHD}&\text{Attention Deficit Hyperactivity Disorder}\\\ \text{AL}&\text{Asymmetric Laplace distribution}\\\ \text{ALG}&\text{Asymmetric Laplace Gaussian distribution}\\\ \text{BEESTS}&\text{Bayesian Ex-Gaussian Estimation of Stop Signal RT distributions}\\\ \text{BPA}&\text{Bayesian Parametric Approach}\\\ \text{CDF}&\text{Cumulative Density Function}\\\ \text{EM}&\text{Expectation Maximization}\\\ \text{ExG}&\text{Ex-Gaussian distribution}\\\ \text{GORT}&\text{Reaction Time in a go trial}\\\ \text{GORTA}&\text{Reaction Time in a type A go trial}\\\ \text{GORTB}&\text{Reaction Time in a type B go trial}\\\ \text{HBPA}&\text{Hierarchical Bayesian Parametric Approach}\\\ \text{IBPA}&\text{Individual Bayesian Parametric Approach}\\\ \text{MGF}&\text{Moment Generating Function}\\\ \text{ML}&\text{Maximum Likelihood}\\\ \text{NL}&\text{Normal-Laplace distribution}\\\ \text{OCD}&\text{Obsessive Compulsive Disorder}\\\ \text{PDF}&\text{Prpobability Density Function}\\\ \text{SE}&\text{Sequential Effect}\\\ \text{SSD}&\text{Stop Signal Delay}\\\ \text{SRRT}&\text{Reaction Times in a failed stop trial}\\\ \text{SSRT}&\text{Stop Signal Reaction Times in a stop trial}\\\ \text{SSRTA}&\text{Stop Signal Reaction Times in a type A stop trial}\\\ \text{SSRTB}&\text{Stop Signal Reaction Times in a type B stop trial}\\\ \text{SST}&\text{Stop Signal Task }\\\ \text{$=^{d}$}&\text{Equality in distribution}\\\ \text{$\oplus$}&\text{Sum of independent random variables}\\\ \text{$\ominus$}&\text{Difference of independent random variables}\end{array}$ ## References * [1] Matzke, D.; Verbrugen, F; $\&$ Logan G.D. 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The Double Pareto-lognormal distribution-A New Parametric Model for Size Distributions. Communications in Statistics-Theory $\&$ Methods, 33: 1733-1753. Appendices ## Appendix A Proofs This appendix presents proofs for the new results presented in Section 3.1. ### A.1 Proof of Theorem 3.1. Proof. Method(1): Let $\Delta GORT=GORT_{B}-GORT_{A}$ where $GORT_{B},GORT_{A}$ are independent with ExG distribution with associated parameters $\theta_{B}=(\mu_{B},\sigma_{B},\tau_{B}),\theta_{A}=(\mu_{A},\sigma_{A},\tau_{A}),$ respectively. Then, by Definition 2.1, Definition2.2 and Theorem2.4 it follows that: $\displaystyle m_{\Delta GORT}(t)$ $\displaystyle=$ $\displaystyle m_{GORT_{B}}(t)\times m_{GORT_{A}}(-t)$ $\displaystyle=$ $\displaystyle m_{ExG(\theta_{B})}(t)\times m_{ExG(\theta_{A})}(-t)$ $\displaystyle=$ $\displaystyle((1-t\tau_{B})^{-1}exp(\mu_{B}.t+\frac{\sigma_{B}^{2}}{2}t^{2}))\times((1+t\tau_{A})^{-1}exp(-\mu_{A}.t+\frac{\sigma_{A}^{2}}{2}t^{2}))$ $\displaystyle=$ $\displaystyle((1-t\tau_{B})(1+t\tau_{A}))^{-1}exp((\mu_{B}-\mu_{A}).t+\frac{\sigma_{B}^{2}+\sigma_{A}^{2}}{2}t^{2})$ $\displaystyle=$ $\displaystyle(1+(\tau_{A}-\tau_{B})t-\tau_{A}\tau_{B}.t^{2})^{-1}exp((\mu_{B}-\mu_{A}).t+\frac{\sigma_{B}^{2}+\sigma_{A}^{2}}{2}t^{2})$ $\displaystyle=$ $\displaystyle m_{AL(\tau_{A},\tau_{B})}(t)\times m_{N(\mu_{B}-\mu_{A},\sigma_{B}^{2}+\sigma_{A}^{2})}(t):\ \ -\tau_{A}^{-1}<t<\tau_{B}^{-1}.$ Accordingly, by Theorem2.5 it follows that: $\Delta GORT=^{d}AL(\tau_{A},\tau_{B})\oplus N(\mu_{B}-\mu_{A},\sigma_{B}^{2}+\sigma_{A}^{2})=^{d}ALG(\tau_{A},\tau_{B},\mu_{B}-\mu_{A},(\sigma_{B}^{2}+\sigma_{A}^{2})^{1/2}).$ Method(2): Using Definition2.1, Definition2.2, Definition2.3 and equation(12) it follows that: $\displaystyle\Delta GORT$ $\displaystyle=^{d}$ $\displaystyle GORT_{B}\ominus GORT_{A}$ $\displaystyle=^{d}$ $\displaystyle ExG(\mu_{B},\sigma_{B},\tau_{B})\ominus ExG(\mu_{A},\sigma_{A},\tau_{A})$ $\displaystyle=^{d}$ $\displaystyle(N(\mu_{B},\sigma_{B}^{2})\oplus Exp(\tau_{B}))\ominus(N(\mu_{A},\sigma_{A}^{2})\oplus Exp(\tau_{A}))$ $\displaystyle=^{d}$ $\displaystyle(N(\mu_{B},\sigma_{B}^{2})\ominus N(\mu_{A},\sigma_{A}^{2}))\oplus(Exp(\tau_{B})\ominus Exp(\tau_{A})$ $\displaystyle=^{d}$ $\displaystyle N(\mu_{B}-\mu_{A},\sigma_{B}^{2}+\sigma_{A}^{2})\oplus AL(\tau_{A},\tau_{B})$ $\displaystyle=^{d}$ $\displaystyle AL(\tau_{A},\tau_{B})\oplus N(\mu_{B}-\mu_{A},\sigma_{B}^{2}+\sigma_{A}^{2})$ $\displaystyle=^{d}$ $\displaystyle ALG(\tau_{A},\tau_{B},\mu_{B}-\mu_{A},(\sigma_{B}^{2}+\sigma_{A}^{2})^{1/2}).$ $\Box$ ### A.2 Proof of Theorem 3.2. Proof. By Definition2.2 and Theorem2.4 for the nth cumulant of the ALG distribution with four parameters $(\alpha_{1},\alpha_{2},\mu,\sigma)$ we have: $\displaystyle\kappa_{n}^{ALG(\alpha_{1},\alpha_{2},\mu,\sigma)}$ $\displaystyle=$ $\displaystyle\kappa_{n}^{AL(\alpha_{1},\alpha_{2})}+\kappa_{n}^{N(\mu,\sigma^{2})}$ $\displaystyle=$ $\displaystyle((n-1)!((-\alpha_{1})^{n}+\alpha_{2}^{n}))+(1_{(n=1)}(n)\mu+1_{(n=2)}(n)\sigma^{2}):\ \ 1\leq n.$ Consequently, the descriptive and the shape statistics for the ALG distribution with four parameters $(\alpha_{1},\alpha_{2},\mu,\sigma)$, it follows that: $\displaystyle Mean$ $\displaystyle E(ALG)=\alpha_{2}-\alpha_{1}+\mu,$ $\displaystyle Variance$ $\displaystyle Var(ALG)=\alpha_{1}^{2}+\alpha_{2}^{2}+\sigma^{2},$ $\displaystyle Skewness$ $\displaystyle\gamma_{ALG}=\frac{2(\alpha_{2}^{3}-\alpha_{1}^{3})}{(\alpha_{1}^{2}+\alpha_{2}^{2}+\sigma^{2})^{3/2}},$ $\displaystyle Kurtosis$ $\displaystyle\kappa_{ALG}=\frac{6(\alpha_{1}^{4}+\alpha_{2}^{4})}{(\alpha_{1}^{2}+\alpha_{2}^{2}+\sigma^{2})^{2}}.$ Finally, the assertion follows for $\alpha_{1}=\tau_{A},\alpha_{2}=\tau_{B},\mu=\mu_{B}-\mu_{A},$ and $\sigma^{2}=\sigma_{B}^{2}+\sigma_{A}^{2}.$ $\Box$ ### A.3 Proof of Theorem 3.3. Proof. Given a four parametric $ALG(\alpha_{1},\alpha_{2},\mu,\sigma)$ for the proactive inhibition. Then, it can be written in the form: $ALG(\alpha_{1},\alpha_{2},\mu,\sigma)=\mu\oplus\sigma N(0,1)\oplus\alpha_{2}Exp_{2}(1)\ominus\alpha_{1}Exp_{1}(1)$ where the random variables $N(0,1),Exp_{i}(1)\sim Exp(1)(i=1,2)$ are mutually independent. Accordingly, there are uncountably many solutions $(\mu_{A},\sigma_{A},\tau_{A},\mu_{B},\sigma_{B},\tau_{B})$ for the following equations: $\tau_{A}=\alpha_{1},\tau_{B}=\alpha_{2},\mu_{B}-\mu_{A}=\mu,\sigma_{B}^{2}+\sigma_{A}^{2}=\sigma^{2}.$ $\Box$ ### A.4 Proof of Theorem 3.4. Proof. These are straightforward from the probability density function and [28]. $\Box$ ### A.5 Proof of Theorem 3.5. Proof. Let, $\Delta GORT=^{d}ALG(\alpha_{1},\alpha_{2},\mu,\sigma)=^{d}AL(\alpha_{1},\alpha_{2})\oplus N(\mu,\sigma^{2})$ where $(\alpha_{1},\alpha_{2},\mu,\sigma^{2})=(\tau_{A},\tau_{B},\mu_{B}-\mu_{A},\sigma_{A}^{2}+\sigma_{B}^{2}).$ For, $X=^{d}AL(\alpha_{1},\alpha_{2}),Y=^{d}N(\mu,\sigma^{2}),$ and two applications of Theorem2.6 for the following convex functions show that both components of the ALG distribution have increasing hazard functions: $\displaystyle-ln(f_{X}(t))$ $\displaystyle=$ $\displaystyle ln(\alpha_{1}+\alpha_{2})\times(\frac{-t}{\alpha_{1}})1_{(-\infty,0]}+\frac{t}{\alpha_{2}}1_{[0,\infty)}(t))\ \ -\infty<t<\infty,$ $\displaystyle-ln(f_{Y}(t))$ $\displaystyle=$ $\displaystyle ln(\sqrt{2\pi}\sigma)+\frac{(t-\mu)^{2}}{2\sigma^{2}}.\ \ -\infty<t<\infty$ Accordingly, by an application of the Theorem2.7, the plausible result follows. $\Box$ ## Appendix B ExG models parameters This appendix presents IBPA parameter estimations for cluster type ExG model. Table 5: Mean posterior Ex-Gaussian parameters estimations across trial types by IBPA (n = 44). [b] $\mu-$ parameter $\sigma-$ parameter $\tau-$ parameter $\\#$ $\mu_{S}$ $\mu_{A}$ $\mu_{B}$ $\sigma_{S}$ $\sigma_{A}$ $\sigma_{B}$ $\tau_{S}$ $\tau_{A}$ $\tau_{B}$ 1 357 350 372 32 35 14 86 96 68 2 637 599 732 175 170 132 47 48 68 3 469 484 411 60 57 42 76 73 111 4 597 567 631 163 165 96 66 69 149 5 640 618 608 156 133 58 47 62 121 6 452 431 469 108 106 64 66 65 135 7 689 668 664 136 130 146 47 60 115 8 665 609 660 145 91 237 51 103 120 9 543 484 640 166 151 118 120 147 157 10 470 468 483 56 59 52 98 87 156 11 414 399 597 46 37 118 177 168 80 12 557 534 597 132 128 146 53 58 123 13 550 538 564 137 133 98 55 38 190 14 319 318 365 307 295 370 170 137 264 15 421 416 437 61 56 90 138 142 149 16 358 342 389 61 57 61 48 56 57 17 594 599 561 130 130 133 78 62 196 18 467 397 747 229 190 299 127 131 159 19 426 426 424 67 75 50 102 103 110 20 423 449 504 62 74 129 122 65 169 21 521 519 487 144 157 96 91 97 125 22 397 346 463 87 58 110 94 132 101 23 540 525 588 80 78 79 94 88 128 24 592 571 529 176 136 304 46 69 180 25 577 459 602 165 70 244 69 181 124 26 562 555 694 79 75 160 172 154 148 27 446 436 541 71 60 166 240 236 233 28 486 476 629 82 64 196 172 155 151 29 414 363 391 133 66 213 62 115 111 30 486 484 541 87 86 146 141 127 181 31 546 502 656 137 118 157 90 100 137 32 436 421 462 107 109 90 72 81 88 33 452 454 458 38 46 40 156 156 165 34 404 422 408 105 109 42 95 72 92 35 470 549 595 230 200 298 207 171 136 36 429 400 448 116 139 95 158 163 245 37 521 497 507 89 130 68 112 125 222 38 284 271 321 40 37 53 100 108 91 39 424 432 416 57 55 87 70 52 131 40 419 418 476 52 53 196 145 148 105 41 533 537 517 105 151 93 72 35 159 42 388 445 497 53 145 206 145 57 116 43 506 467 539 97 82 100 66 96 78 44 842 824 822 175 341 165 34 95 320 * • Notes: $\mu_{S},\sigma_{S},\tau_{S}:$ ExG GORT parameters for single cluster SST data; $\mu_{A},\sigma_{A},\tau_{A}:$ ExG GORT parameters for type-A cluster SST data; $\mu_{B},\sigma_{B},\tau_{B}:$ ExG GORT parameters for type-B cluster SST data; IBPA: $\\#$Chains = 3; Simulations = 20,000; Burn-in = 5,000 (for all parameters).
# Hybrid Neural Pareto Front (HNPF): A Two-Stage Neural-Filter Approach for Pareto Front Extraction Gurpreet Singh † The University of Texas at Austin Soumyajit Gupta † Department of Computer Science Matthew Lease School of Information Clint Dawson Oden Institute for Computational Engineering and Sciences ###### Abstract Pareto solutions represent optimal frontiers for jointly optimizing multiple competing objective functions over the feasible set of solutions satisfying imposed constraints. Extracting a Pareto front is computationally challenging today with limited scalability and solution accuracy. Popular generic scalarization approaches do not always converge to a global optimum and can only return one solution point per run. Consequently, multiple runs of a scalarization problem are required to guarantee a Pareto front, where all instances must converge to their respective global optima. We propose a robust, low cost hybrid Pareto neural-filter (HNPF) optimization approach that is accurate and scales (compute space and time) with data dimensions, and the number of functions and constraints. A first-stage neural network first efficiently extracts a weak Pareto front, using Fritz-John conditions as the discriminator, with no assumptions of convexity on the objectives or constraints. A second-stage, low-cost Pareto filter then extracts the strong Pareto optimal subset from the weak front. Fritz-John conditions provide strong theoretical bounds on approximation error between the true and the network extracted weak Pareto front. Numerical experiments demonstrates the accuracy and efficiency of our approach. †††contributed equally to this work. ## 1 Introduction Multi-Objective Optimization (MOO) problems arise frequently across diverse fields such as engineering [22], finance [32], and supply chain management [33]. Such problems share the common requirement to satisfy multiple competing objectives under a set of constraints imposed by physical or economic limits. A Pareto optimal solution [28] for an MOO problem is defined as the solution point away from which no single objective can be improved without diminishing at least one other objective. A Pareto front is then defined as the set of all such optimal points that satisfy this definition. Since all solutions reflect optimal tradeoff points between competing objective functions, choosing between solutions depends on the user’s preferred tradeoff of objectives. Computing a Pareto solution to an MOO problem requires optimizing competing (often non-convex) objective functions under constraints. This optimization problem is quite challenging: the solution set to an MOO can seldom be formulated as a closed-form expression, and solving this for practical problems is compute intensive and often not feasible. Consequently, most research seeking practical solutions has focused on developing efficient approximations of the Pareto optimal solution set [8, 15, 14, 29]. The ability to accurately and efficiently compute a Pareto optimal solution set, with theoretical guarantees and interpretability, would have tremendous practical value. For example, imagine an online news search for ‘influential CEOs’ in which the user seeks results that are not only relevant, but also recent and reliable. In addition to optimizing for these three objective functions (information relevance, recency, and reliability), imagine the user further wishes to impose a demographic parity constraint on search results to ensure racial or gender parity (since implicit data biases might otherwise yield search result coverage skewed toward white males). The Pareto optimal solution set would not only satisfy this parity constraint, but further enable the user to vary the composition of optimal search results based on their tradeoff preference between getting more recent, breaking news (which may be less reliable) vs. getting more reliable news (which may be less recent). Our review of prior work reveals significant limitations: accuracy [5], compute time [30, 9, 26], and scalability, not to mention limited interpretability and verfiability. Recent methods for algorithmic fairness invoking Pareto optimality [1, 20, 23, 34, 35, 36] often suffer from inconsistent Pareto definitions and impractical assumptions of convex objective functions and constraints. Furthermore, a notable absence of benchmarks against known analytical forms makes it difficult to assess reported results, verify optimality, and A/B test alternative methods. This is in contrast to studies on computational methods [8, 14, 29, 15] in which such comparative benchmarking and verification is well established. Although accurate and verifiable, existing computational methods tend to generate Pareto points with low density (i.e. providing a coarser representation of the underlying Pareto front) and poor scalability, with compute times ranging from hours to days as data dimensionality increases. In this work, we propose a novel two-stage architecture called Hybrid Neural Pareto Front (HPNF) for inducing Pareto optimal solution sets. Stage 1 consists of an interpretable and robust neural network that extracts a weak Pareto solution manifold as the output, given a dataset as input. Following this, Stage 2 provides a low-cost Pareto filter. For the network loss function, we use a discriminator based on Fritz-John conditions [19] that accounts for multiple objectives and constraints. An approximate weak Pareto manifold is extracted as a weighted output of the softmax function from the last layer of the network. The softmax activation classifies weak Pareto vs. non-Pareto data points. Method \| \| Generates only --- Pareto points \| Generates --- Even Spread \| Ease --- of Use \| Efficient --- and Scalable Fair Pareto [34] \| ✗ \| ✗ \| ✗ \| ✗ mCHIM [14] \| ✓ \| ✗ \| ✗ \| ✗ PK [29] \| ✓ \| ✗ \| ✗ \| ✗ NBI [8] \| ✓ \| ✓ \| ✓ \| ✗ Our HNPF \| ✓ \| ✓ \| ✓ \| ✓ Table 1: HNPF vs. existing state-of-the-art methods. Our network architecture has few trainable parameters, making it robust to outliers and over-fitting. Furthermore, we empirically show computational efficiency vs. current state-of-the-art methods [8, 15, 29]. Our approach produces only Pareto points (no false positives) with an even spread and higher density than possible with existing approaches. Furthermore, our approach is scalable with both increasing dimensions of the input data, and the number of functions and constraints. Table 1 summarizes key properties of our HNPF approach vs. existing methods (see Section 2). Contributions. Our key contributions are as follows: 1. 1. A manifold solution strategy for weak Pareto front identification based on Fritz-John conditions as the discriminator. 2. 2. A robust neural network for approximating the weak solution manifold for both convex and non-convex scenarios. 3. 3. Design of a computationally efficient Pareto filter to extract the strong Pareto set, compared to existing Pareto filters. 4. 4. Compared to other neural Pareto approaches our method extracts only Pareto optimal points with an even spread. 5. 5. HNPF is computationally scalable as the dimension of variable space, or functions and constraints increases. 6. 6. The final layer of the neural net is fully interpretable in terms of extracting the efficient set of input data as a manifold. 7. 7. The approximate weak Pareto is bounded below by $0\leq\epsilon\leq 1$ w.r.t. the true manifold upon convergence. 8. 8. We will share our source code and benchmark datasets for reproducibility upon acceptance. ## 2 Related Work As noted earlier, since all Pareto optimal solutions reflect optimal tradeoff points between competing objective functions, choosing between solutions depends on the user’s preferred tradeoff of objectives. Prior work can be organized around four directions for managing user preferences: 1) No preference [37]: user preference criteria are not explicitly specified; 2) a priori [12]: preference criteria are explicitly specified before computation; 3) a posteriori [8]: preference criteria are explicitly specified after computation; and 4) Interactive methods [25]: preference criteria are continuously consulted to isolate one of the optimal solutions. ### 2.1 Generic and Enhanced Scalarization One common approach is to convert an MOO problem into a Single Objective Optimization (SOO) problem via scalarization. However, generic scalarization methods [1, 20, 23, 34, 35] suffer from various limitations. Firstly, these approaches can only extract one solution point at a time given that the minimization problem converges to the global optimum. However for practical applications, with non-convex objectives and constraints, ensuring global optimality is non-trivial. Secondly, multiple runs with different trade-off parameters must be performed in order to extract the weak Pareto solution set, resulting in substantial computational overhead [35]. Finally, the Pareto solution set can still form a non-convex manifold even when the objectives are convex [14] due to the presence of non-convex constraints (see Case III in Section 5). These challenges prove to be major obstacles in the deployment of scalarization approaches as a practical tool for Pareto set extraction. Generic scalarization should not be confused with enhanced scalarization approaches [8, 14, 29], whose strength lies in the specific localization of the objective space that allows treatment of non-convex functions and constraints. Although accurate and complete, enhanced scalarization approaches suffer from low computational scalability and low density of Pareto points on the solution manifold. For example, the $30$ dimensional benchmark in Section 5 Case V shows enhanced scalarization methods (mCHIM and PK) generating a Pareto set in approximately 18 hours. Enhanced scalarization methods fall under category (3) of a posteriori methods. One such enhanced approach to solve an MOO involves constructing a local linear or epsilon scalarization based SOO. These methods include Normal Boundary Intersection (NBI) [8], Normal Constraint (NC) [24], Successive Pareto Optimization [27], modified Convex Hull of Individual Minimum (mCHIM) [14] and Pirouz-Khorram (PK) [29]. NBI [8] produces an evenly distributed set of Pareto points given an evenly distributed set of weights. Furthermore, NBI produces Pareto points in the non-convex parts of the Pareto curve while being independent of the relative scales of the objective functions. It uses the concept of Convex Hull of Individual Minima (CHIM) to break down the boundary/hull into evenly spaced segments and then trace the weak Pareto points. As an improvement over the NBI method, mCHIM uses a quasi-normal procedure to update the aforementioned CHIM set iteratively, to obtain a strong Pareto set. PK [29], on the other hand, uses a local $\epsilon$-scalarization based strategy that searches for the Pareto front using controllable step-lengths in a restricted search region, thereby accounting for non-convexity. Gobbi et al. [15] proposed a framework using Fritz-John conditions [19] to obtain analytical solutions for convex functions and constraints with high point density. Note that, all of these aforementioned enhanced methods are guaranteed to converge to the Pareto front under their respective assumptions on the function property each method can handle. ### 2.2 Bayesian and Genetic Approaches Methods that are a priori (2) require a prior distribution or initial seed parameters to be specified beforehand. Examples include Bayesian [17, 4, 16] and Evolutionary [30, 9, 26] methods. Khan et al. [17]’s Bayesian method showed convergence to the Pareto front, but only under a linear setting, which is the strictest form of convexity. In recent Bayesian methods [4, 16], not only was convexity assumed, but even in actual convex cases significant error was still incurred. Deb et al. [9] introduced the Non-dominated Sorting Genetic Algorithm II (NSGA-II) algorithm that involves recombination, mutation and selection of a population representing the set of solutions points considered to be Pareto, each having one or more assigned objective values. The population is maintained to consist of diverse solutions, resulting in a set of non-dominated individuals that are expected to be near (not on) the real Pareto front. Other variants include NSGA-I [30] and NSGA-III [26]. However, convergence and reproducibility are not guaranteed with Genetic Algorithms, and significant hyper-parameter tuning is required. ### 2.3 Approaches in the Fairness Literature Pareto optimality is being increasingly pursued in classification and fairness research (e.g. see survey [5]). However, we are not aware of any work in this area providing verifiable solutions for benchmarking scenarios wherein the ground truth is known. Several works [1, 23] seek to balance classification accuracy vs. a no unnecessary harm notion of fairness relying upon convexity assumptions without justification. The Weighted Sum Method (WSM) [7], commonly used in fairness literature, is a linear scalarization approach to convert an MOO into an SOO using a convex combination of objective functions and constraints. However, this is viable only when the functions and constraints are also convex [14]. Valdivia et al. [34] present a group-Fairness based trade-off model for decision tree based classifiers using the aforementioned genetic algorithm NSGA-II, which has the same convergence and reproducibility issues mentioned earlier. In addition, their reported results violate fundamental definitions of Pareto optimality. Wei and Niethammer [35] provide the first neural architecture for Pareto front computation for Fairness vs. Accuracy on classification datasets. They rely upon a Chebyshev scalarization, which assumes that objective functions must sum up to a constant, but do not justify this assumption. In the Fair-Recommendation literature, Xiao et al. [36] seek to balance social welfare and group fairness for movie recommendations. They also propose a linear scalarization-based formulation which arrives at the true front for convex functions only. Lin et al. [20] claim Pareto optimality using KKT conditions, which is guaranteed to converge only if the functions and constraints are convex under linear scalarization. ### 2.4 Our Inspiration We draw inspiration from three seminal works: (1) Das and Dennis [8] proposed to break the functional domain boundary into uniform and evenly spaced segments (see CHIM in [8]) to identify weak Pareto points with guarantees. Motivated by this, we first identify the weak Pareto front using a robust neural network (Stage 1). (2) Messac et al. [24] proposed the first Pareto filter to obtain the set of strong Pareto points from the aforementioned weak Pareto set. The filter uses an all-pair comparison criterion to reject dominated points from the weak Pareto set. This filter motivates our low-cost Pareto filter (Stage 2) design, which avoids the expensive all-pair comparison using a plane search strategy. (3) Gobbi et al. [15] presented the matrix form of the Fritz-John conditions satisfying the existence of Pareto points. Although their approach is only valid for convex cases, we extend the Fritz- John matrix form as a discriminator to identify Pareto point even for non- convex cases. ## 3 Pareto Optimality A general multi-objective optimization problem can formulated as: $\displaystyle\underset{}{min}\quad F(x)$ $\displaystyle=(f_{1}(x),f_{2}(x),\ldots,f_{k}(x))$ (1) $\displaystyle\text{s.t.}\quad x\in S$ $\displaystyle=\\{x\in\mathbb{R}^{n}\|G(x)=(g_{1}(x),g_{2}(x),\ldots,g_{m}(x)\leq 0\\}$ in $n$ variables $(x_{1},\ldots,x_{n})$, $k$ objective functions $(f_{1},\ldots,f_{k})$, and $m$ constraint functions $(g_{1},\ldots,g_{m})$. Here, $S$ is the feasible set i.e. the set of input values $x$ that satisfy the constraints $G(x)$. For a multi-objective optimization problem there is typically no single global solution, and it is often necessary to determine a set of points that all fit a predetermined definition for an optimum. ### 3.1 Definitions We borrow and adapt existing definitions of Pareto optimality [28] from Marler and Arora [22]. Strongly Pareto Optimal: The Pareto optimal solution $x^{}$ for Eq. (1) satisfies the conditions: $\displaystyle\nexists x_{j}:f_{p}(x_{j})\leq f_{p}(x^{}),\quad\textrm{for}\quad p=1,2,\ldots,k$ $\displaystyle\exists l:f_{l}(x_{j})<f_{l}(x^{})$ (2) This states that for $x^{}\in X$ to be strongly Pareto optimal, there does not exist another $x_{j}\in X$, s.t. $f_{p}(x_{j})\leq f_{p}(x^{})$ for all functions $f_{p},\forall p\in[1,k]$ and $f_{l}(x_{j})<f_{l}(x^{})$ for at least one function $f_{l}$. Weakly Pareto Optimal: A weak Pareto point $\tilde{x}^{}$ satisfies: $\displaystyle\nexists x_{j}:f_{p}(x_{j})<f_{p}(\tilde{x}^{}),\quad\textrm{for}\quad p=1,2,\ldots,k$ (3) A point is weakly Pareto optimal if no other point exists that improves all of the objectives simultaneously. This is different from a strongly Pareto optimal point, s.t. no point exists, that improves at least one objective without detriment to other objectives. Efficient and Inefficient Points: A Pareto efficient point is defined on the domain $X$ as a point $x^{}$ iff no other point $x$ exists s.t. $F(x)\leq F(x^{})$ with at least one $f_{p}(x)<f_{p}(x^{})$. The point $x^{}$ is considered inefficient otherwise. Dominated and Non-Dominated Points: Dominated points are defined w.r.t. the objective function in the criterion space. The objective function vector $F(x^{})$ is non-dominated iff no other vector $F(x)$ exists s.t. $F(x)\leq F(x^{})$ with at least one $f_{p}(x)<f_{p}(x^{})$. The vector $F(x^{})$ is considered dominated otherwise. (a) Convex form (b) Non-Convex form Figure 1: Pareto optimal set under different objectives. Note that the red line corresponds to the min-min strong Pareto optimal front for both problems and the dashed line corresponds to the weak Pareto Front for the entire Feasible Set. Fig. 1 shows two separate MOOs for two competing functions. The feasible set of points are shown as gray shaded area with the Pareto boundary shown as dashed lines. For both problems, a joint minimization problem is considered, resulting in a Pareto optimal set (red curves) facing the origin. A different Pareto boundary can be obtained if a joint-maximization or a mixed min-max problem is considered. As shown in Fig. 1 (b), a strong Pareto optimal solution from a non-convex weak Pareto front can result in a discontinuous manifold. In the following sections, although we rely on dominated and non- dominated points for visualization purposes, the approximation errors and our algorithm rely primarily on efficient and inefficient points for computational purposes. ### 3.2 Fritz John Conditions Let the objective and constraint function in Eq. (1) be differentiable once at a decision vector $\tilde{x}^{}\in\mathcal{S}$. The Fritz-John necessary conditions for $\tilde{x}^{}$ to be weak Pareto optimal is that vectors must exists for $0\leq\lambda\in\mathbb{R}^{k}$, $0\leq\mu\in\mathbb{R}^{m}$ and $(\lambda,\mu)\neq(0,0)$ (not identically zero) s.t. the following holds: $\displaystyle\sum_{i=1}^{k}\lambda_{i}\nabla f_{i}(\tilde{x}^{})+\sum_{j=1}^{m}\mu_{j}\nabla g_{j}(\tilde{x}^{})=0$ (4) $\displaystyle\mu_{j}g_{j}(\tilde{x}^{})=0,\forall j=1,\ldots,m$ Gobbi et al. [15] presented an $L$ matrix form, comprising the gradients of the functions and constraints as follows: $\displaystyle L=\begin{bmatrix}\nabla F&\nabla G\\\ \mathbf{0}&G\end{bmatrix}\quad[(n+m)\times(k+m)]$ (5) $\displaystyle\nabla F_{n\times k}=[\nabla f_{1},\ldots,\nabla f_{k}]$ $\displaystyle\nabla G_{n\times m}=[\nabla g_{1},\ldots,\nabla g_{m}]$ $\displaystyle G_{m\times m}=diag(g_{1},\ldots,g_{m})$ The matrix equivalent of Fritz John Conditions for $x^{}$ to be Pareto optimal, is to show the existence of $\lambda\in\mathbb{R}^{k+m}$ in Eq. (4) such that: $\displaystyle L\cdot\delta=0\quad\text{s.t.}\quad L=L(\tilde{x}^{}),\mathbf{\delta}\geq 0,\mathbf{\delta}\neq 0$ (6) The non-trivial solution ($\mathbf{\delta}$ is not identically zero) for Eq. (6) is: $\displaystyle det(L^{T}L)=0$ (7) The weak Pareto front is characterized by the set of points such that matrix $L$ is low rank. This ensures that the points identified are either inside the feasible set or at boundaries dictated by the constraints. For e.g. if $\mu_{1}=0$ for any $\lambda_{i}$, then $\sum_{i}\lambda_{i}f_{i}=0$ must be satisfied for the corresponding internal point $x^{}$ to be Pareto. Similarly if $\mu_{1}\neq 0,\mu_{j\neq 1}=0$ in the aforementioned case, then $g_{1}=0$ holds true for the corresponding boundary point $x^{}$ to be Pareto. Note that all Pareto points satisfy $\nabla f_{i}=0$ for at least one $i$ whether they lie inside the feasible set or on the boundaries. This is to say that all points $x^{}$ are local optimizers for at least one $f_{i}$. The rank of the matrix $L^{T}L$ determines the dimension of the Pareto manifold. Furthermore, the necessary condition written as $det(L^{T}L)$ is independent of the preference parameters $\lambda_{i},\,\mu_{j}$. Eq. (7) now serves as a condensed discriminator to identify a weak Pareto front. In what follows, we use this matrix form of Fritz-John conditions to approximate the Pareto front using a robust, low-weight, neural network. ## 4 HNPF Framework In this section we lay out the details of the proposed computationally efficient hybrid two-stage architecture for Pareto set detection. ### 4.1 Stage 1: Neural Net for Weak Pareto Front The proposed neural network consists of feed-forward layers with tanh activation. There are three layers of dense connections with eight neurons each, to smoothly approximate the optimal solution manifold $M(X)$ as $\tilde{M}(X)$, shown in Fig. 2. The last layer of the network has two neurons with softmax activation for binary classification of Pareto vs. non-Pareto points. In other words, this layer approximates the separation manifold that distinguishes inefficient points from the weak Pareto points, in the feasible set $S$. Note that our network loss is representation driven, since the Fritz John discriminator (Eq. (7)) explicitly classifies points as being weak Pareto or not. The network accepts or rejects the input data points $X$ based on the Fritz-John discriminator described by the objective functions and constraints. The Fritz-John necessary conditions for weak Pareto optimality, as pointed out earlier, require that the $D=det(L(X)^{T}L(X))=0$. Therefore, $1-D$ and $D$ naturally provide us with binary labels for the softmax activated output layer. A binary cross entropy loss ensures that the distribution of the extracted manifold $\tilde{M}(X)$ matches the distribution of the weak Pareto front satisfying the Fritz-John conditions. The network architecture is purposely kept low weight for weak Pareto manifold extraction to provide robustness against outliers. Figure 2: Proposed Neural network for finding the weak Pareto points. The last layer performs Fritz John criteria enabled binary cross entropy classification of data points, using softmax activation, ensuring weak Pareto optimality. Error bound. For a user-prescribed relaxation margin $0\leq\epsilon\leq 1$, the approximation error between the network extracted manifold $\tilde{M}(\tilde{X})$ and the true solution $M(X^{})$ is bounded below by $\\|\tilde{M}(\tilde{X})-M(X^{})\\|_{2}\leq\epsilon$, upon convergence. See Appendix C for proof. ### 4.2 Stage 2: Pareto filter for Strong Pareto Set A Pareto filter is an algorithm that, given a set of weak Pareto points in objective space, retains a subset of non-dominated points. This corresponds to the strong Pareto set s.t. none of the points are dominated. In other words, the filter eliminates all dominated points from the given set. A state of the art Pareto filter, defined in Messac et al. [24], is used as a post-processing step to extract a strong Pareto set. However, note that this Pareto filter requires an all-pairs comparison, an $O(n!)$ calculation, and thus becomes computationally expensive as the point set grows. This necessitates that the number of weak Pareto points be small when using this filter. However, since Stage 1 of our approach generates weak Pareto points with high density, the filter proves to be quite expensive. We present an efficient Pareto filter algorithm for finding a strong Pareto set which is computationally scalable to arbitrary dimensions. The algorithm is based on a plane search strategy, inspired by Kd-Trees [3], well known for efficient data partitioning and storage. The compute complexity of our approach is determined by the number of competing functions while being linearly proportional to the number of points. The inputs to the algorithm (Alg. 1) are the number of functions $k$, their minimum and maximum bounds $f(min),f(max)$, discretization level $h$ of the function space and the weak Pareto points $P$. The output is the strong Pareto set $\\{x^{}\\}$. Following Alg. 1, we can estimate its time complexity. There are three nested for loops which carry the load. In the worst case that the points in the weak Pareto set are all strong Pareto, then the cardinality of the set $P$ remains unchanged. Let us also define $z=(f_{i}(max)-f_{i}(min))/h$ to denote the number of chunks into which the function space is divided. Thus, the worst case complexity of the proposed Pareto filter is $\mathbf{O(kzn)}$. For scenarios, where the strong Pareto set is a subset of the original weak set $P$, the complexity reduces in the factor guided by $n$. Algorithm 1 Pareto filter 1:Data $P=\\{\tilde{x}^{}\\}\in\mathbb{R}^{n}$ weak Pareto points 2:Input $f(min),f(max):$ bounds of each function $f_{i},\forall i\in k$ 3:Input $k:$ number of functions, $h:$ discretization level 4:for $i\in k$ do $\triangleright$ Loop over all functions 5: $level=f_{i}(min)$ 6: for $j\in(f_{i}(max)-f_{i}(min))/h)$ do $\triangleright$ Loop over all levels 7: $temp=\varnothing$ 8: for $p\in P$ do 9: if $level\leq f_{i}(p)<level+h$ then 10: $temp=temp\cup p$ 11: if $card(temp)>1$ then $\triangleright$ Cardinality of set 12: $x_{p}=min\,f_{q}(x),x\in temp,q=i+1$ $\triangleright$ Efficient point 13: $P=P\backslash(temp\backslash x_{p})$ $\triangleright$ Remove inefficient points 14: $level=level+h$ 15:Output: Strong Pareto set $x^{}=P$ (a) Weak Pareto Set (b) First pass (c) Second pass Figure 3: Illustration of the proposed Pareto Filter on a two-function scenario for visualization purposes It is easy to visualize the working of the proposed Pareto filter in Fig. 3. We start with the weak Pareto set $P$ (Fig. 3 (a)) for a non-convex form. The first pass over $f_{1}$ removes a set (segment DEF) of dominated Pareto points (Fig. 3 (b)). The leftover points in $P$ are then filtered again based on $f_{2}$ (Fig. 3 (c)), where the dominated points (segment ABC) as per $f_{2}$ are removed. Points surviving the filtering process belong to the strong Pareto set. ## 5 Results In this section, we present five numerical experiments for benchmarking and analysis (see Appendix A for two additional cases). These experiments address standard analytical forms, with increasing complexity and scale in the number of functions $(k)$, constraints $(m)$ and dimension of variables $(n)$. While cases may appear synthetic, they arise from practical physical domains in various engineering fields. We compare our results vs. those from two current state-of-the-art methods: mCHIM [14] and PK [29]. Sampling. Since we are only provided with objective functions and constraints, we must sample data points from the variable domain in order to generate candidates to test for optimality. Firstly, if there are any direct constraints on variable values, we consider this feasible domain for sampling, as in the benchmark cases. Secondly, lacking any prior knowledge of where the Pareto front may reside, we sample values random uniform distribution in the feasible variable domain. Objective functions evaluated at these points generate a quantized, topographic map of the function domain that is then used to identify optimal points. For each benchmark test case below, we generate 11K points from a random uniform distribution in the feasible variable domain, to serve as training data. The training-validation split is 90-10%. Once the manifold is learned by the network, we feed in 90k points within the permissible domain to plot the Pareto set for visualization. ### 5.1 Experimental Setup Experiments use an Nvidia 2060 RTX Super 8GB GPU, Intel Core i7-9700F 3.0GHz 8-core CPU and 16GB DDR4 memory. We use the Keras [6] library on a Tensorflow 2.0 backend with Python 3.7 to train the networks in this paper. For optimization, we use AdaMax [18] with parameters (lr=0.001) and $1000$ steps per epoch. While neural approaches often pre-initialize the network with layer-wise training [2], a strength of HPNF is that all network weights can be simply drawn from a uniform random distribution. Since the data domain is discrete, an exact zero might not be achievable. We therefore use a slightly relaxed criterion of $\epsilon=0.001$ as the classification margin. Any point above this value will be classified as weak Pareto. For all results, the extracted Pareto set (shaded red) overlaps the true Pareto set with an $\epsilon$ spread. Due to stochastic variation, neural network studies often report variance across several runs. However, the only approximation errors with our method lie in the extracted manifold over runs. Error bounds are given in Section 4.1 and Appendix C. Since the manifold remains constant across runs, the loss itself is the approximation error with a minimum achievable value of 0 at machine precision. With respect to this 0 over multiple runs, the loss function is the deviation from the true manifold. We thus do not report mean- variance across runs. Section 5.9 shows loss profiles. ### 5.2 Case I: n=2, k=2, m=2 This problem was originally proposed in [11]. Jointly minimize $\displaystyle f_{1}(x_{1},x_{2})=1-exp(-[(x_{1}-1/\sqrt{(}2))^{2}+(x_{2}-1/\sqrt{(}2))^{2}])$ $\displaystyle f_{2}(x_{1},x_{2})=1-exp(-[(x_{1}+1/\sqrt{(}2))^{2}+(x_{2}+1/\sqrt{(}2))^{2}])$ $\displaystyle\text{s.t.}\quad g_{1},g_{2}:-1/\sqrt{2}\leq x_{1},x_{2}\leq 1/\sqrt{2}$ The Pareto set can be computed by applying linear scalarization (WSM) for this problem since all the functions and constraints are convex. Gobbi et al. [15], NBI, mCHIM and PK are able to extract the Pareto solution set. Fig. 4 shows the solution from our proposed network with high point density, where we can visually verify (Fig. 4(b)) that the network approximated the Pareto manifold, validating that the Pareto points indeed closely satisfy $x_{1}=x_{2}$. (a) Function Domain (b) Variable Domain Figure 4: Pareto Front for Case I. Please refer to color plots for proper visualization for all following figures. ### 5.3 Case II: n=2, k=2, m=2 This problem was proposed in [14]. Jointly minimize $\displaystyle f_{1}(x_{1},x_{2})=x_{1}$ $\displaystyle f_{2}(x_{1},x_{2})=1+x_{2}^{2}-x_{1}-0.1sin3\pi x_{1}$ $\displaystyle\text{s.t.}\quad g_{1},g_{2}:0\leq x_{1}\leq 1,-2\leq x_{2}\leq 2$ Gobbi et al. [15] does not consider this scenario due to non-convexity of $f_{2}$. As shown in [14], WSM can only identify a subset of the Pareto optimal points. NBI, mCHIM and PK methods are able to identify points in this case with equal density. Fig. 5 shows the results from our model with high point density. It also satisfies closely the true Pareto manifold given by $0\leq x_{1}\leq 1,x_{2}=0$ in Fig. 5(b). (a) Function Domain (b) Variable Domain (c) WSM (d) NBI (e) mCHIM Figure 5: Pareto Front for Case II. Note the even spread of point HNPF produces. NBI produce an even spread while mCHIM cannot. WSM fails for non- convexity in function. ### 5.4 Case III: n=2, k=2, m=4 This problem was proposed in [31]. Jointly minimize $\displaystyle f_{1}(x_{1},x_{2})=x_{1}$ $\displaystyle f_{2}(x_{1},x_{2})=x_{2}$ $\displaystyle\text{s.t.}\quad g_{1}(x_{1},x_{2})=(x_{1}-0.5)^{2}+(x_{2}-0.5)^{2}\leq 0.5$ $\displaystyle g_{2}(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}-1-0.1\cos(16\arctan(\frac{x_{1}}{x_{2}}))\geq 0$ $\displaystyle g_{3},g_{4}:0\leq x_{1},x_{2}\leq\pi$ This form is convex in $f_{1},f_{2}$ but the non-convex constraints in $g_{1},g_{2}$ forces the Pareto front to be non-convex. While NBI (without Pareto filter) fails in this scenario, both mCHIM and PK extracts the non- dominated Pareto points with limited density $(\sim 40)$. HNPF extracts point with higher density (Fig. 6). Since the front is strongly affected by the constraints, Fig. 6(a) shows a sinusoidal weak Pareto front. To arrive at the non-dominated Pareto set, we then post-process this result using the efficient Pareto filter proposed in sub-section 4.2. The updated discontinuous set of non-dominated Pareto points can be seen in Fig. 6(b) following the visual explanation in Fig. 3. (a) Dominated (b) Non-Dominated (c) mCHIM Figure 6: Strong Pareto Front for Case III. All dominated points are removed from set after applying the Pareto filter. ### 5.5 Case IV: n=3, k=3, m=4 This problem was proposed in [14]. Jointly minimize $\displaystyle f_{1}(x_{1},x_{2},x_{3})=x_{1}$ $\displaystyle f_{2}(x_{1},x_{2},x_{3})=x_{2}$ $\displaystyle f_{3}(x_{1},x_{2},x_{3})=x_{3}$ $\displaystyle\text{s.t.}\quad g_{1}(x_{1},x_{2},x_{3})=(x_{1}-1)^{2}+(x_{2}-1)^{2}+(x_{3}-1)^{2}\leq 1.0$ $\displaystyle g_{2},g_{3},g_{4}:x_{1},x_{2},x_{3}\geq 0$ This form is convex in $f_{1},f_{2},f_{3}$ but the non-convex constraint in $g_{1}$ forces the Pareto front to be non-convex. The results using our method, as shown in Fig. 7, are in good agreement with mCHIM and PK methods with a higher point density. (a) Function Domain (b) Variable Domain Figure 7: Pareto Front for Case IV ### 5.6 Case V: n=30,k=2,m=30 This problem was proposed in [38]. Jointly minimize $\displaystyle f_{1}(x)=x_{1}+\frac{2}{\|J_{1}\|}\sum_{j\in J_{1}}y_{j}^{2}$ $\displaystyle f_{2}(x)=1-\sqrt{x_{1}}+\frac{2}{\|J_{2}\|}\sum_{j\in J_{2}}y_{j}^{2}$ $\displaystyle\text{s.t.}\quad g_{1},\ldots,g_{30}:0\leq x_{1}\leq 1,-1\leq x_{j}\leq 1,j=2,\ldots,m$ $\displaystyle J_{1}=\\{j\|j\,\textrm{is odd},2\leq j\leq m\\},J_{2}=\\{j\|j\,\textrm{is even},2\leq j\leq m\\}$ $\displaystyle y_{j}=\left\\{\begin{matrix}x_{j}-[0.3x_{1}^{2}\cos(24\pi x_{1}+\frac{4j\pi}{m})+0.6x_{1}]cos(6\pi x_{1}+\frac{j\pi}{m})\quad j\in J_{1}\\\ x_{j}-[0.3x_{1}^{2}\cos(24\pi x_{1}+\frac{4j\pi}{m})+0.6x_{1}]cos(6\pi x_{1}+\frac{j\pi}{m})\quad j\in J_{2}\end{matrix}\right.$ This form is non-convex in both $f_{1},f_{2}$. The dimension of the design variable space is $m=30$. The corresponding Pareto front is non-convex. The results using our method, as shown in Fig. 8, are in good agreement with mCHIM and PK methods (a) Function Domain (b) Variable Domain (c) mCHIM Figure 8: Pareto Front for Case V. Note the density difference between HNPF and mCHIM in the variable space. ### 5.7 Summary of Results Linear Scalarization, as in WSM [7], is a well-known approach for Pareto set detection in Fairness and Classification literature. This approach fails for all but one (Case I) of the cases presented above, since either the functions or constraints or both are non-convex. This raises serious concerns regarding validation in Fairness literature: whether the points in the solution set are Pareto optimal or not. If not, then all such works are generating points which are non-Pareto (weak, strong or otherwise) in any sense of the definitions posed in Section 3. Case III highlights the fallacies of such convexity assumptions, where in spite of functions being convex themselves, the analytical front is non-convex due to the interaction of the functions and non-convex constraints. \| HNPF \| mCHIM ---\|---\|--- Case \| Density \| Points \| Evals \| Density \| Points \| Evals Case II \| 1.83 \| 1648 \| 90K \| 4.39e-2 \| 33 \| 75,152 Case III \| 1.37 \| 1241 \| 90K \| 1.01e-2 \| 33 \| 328,375 Case IV \| 6.57 \| 5915 \| 90K \| 5.86e-3 \| 43 \| 733,752 Case V \| 0.20 \| 184 \| 90K \| 1.38e-6 \| 33 \| 2,379,459,895 Table 2: Pareto optimal point density % (ratio of #extracted optimal points to #function evaluations). HNPF finds many more optimal points with many fewer function evaluations. Case I has not been considered by mCHIM, hence left out. NBI [8] works for cases where the detected weak Pareto front consists of non- dominated points. Therefore, NBI generates correct solution sets in Cases I, II, IV, V with equal density of points on the Pareto front. In essence, applying the Pareto filter on the NBI generated solution set would resolve the discontinuous cases too. Gobbi [15], a Genetic Algorithm solution strategy, works for Case I. Their algorithm is developed only for cases where all the functions and constraints are convex. NBI, mCHIM,PK and HNPF produce only Pareto points, which is not guaranteed by WSM [7]. Additionally, HNPF generates Pareto points uniformly with high density, while state-of-the-art mCHIM [14] and PK [29], although accurate, limit themselves to low point density ($\sim 40$) with large computational overhead as the variable dimension scales. Table 2 shows a comparison between HNPF and mCHIM. See Appendix D for a similar comparison against PK. ### 5.8 Runtime Comparison Using numerical experiments, we previously verified that mCHIM, PK and our method always arrives at the correct results for all the considered cases. We now perform a compute time analysis against mCHIM and PK, to demonstrate improved performance using our proposed approach. The trajectories in Fig. 9 show the compute times for the high dimensional Case VII. Note that for mCHIM and PK, the timings are reported for dimensions n=30 and n=4, respectively. For our method, the runtimes are reported for Case VII with the variable space dimension ranging from $[2-30]$. Figure 9: Runtime of HNPF vs. mCHIM and PK, as the variable dimension increases. All methods have a linear increase in runtime with dimension, but HNPF scales much better. The reported runtime with two dimensional variable space might give the false notion that mCHIM and PK are more efficient than HNPF. However, as the variable dimension increases, both mCHIM and PK become far more expensive, as shown in Fig. 9. These methods also produce a low density of Pareto points $(\sim 40)$, while HNPF yields high density $(\sim 1k)$. Since both mCHIM and PK are based on enhanced scalarization, solving the resulting problem to extract Pareto points suffers from scaling issues. ### 5.9 Loss Profile We now briefly discuss the training process for the cases shown above. On an average, the network takes around $10/20/30$ epochs for the simple/moderate/hard cases as visualized in Fig. 10. Since the last layer of the network is classifying points as being weak Pareto or not, the runtime is dictated by the complexity of the curve in the design variable space. The more non-linear the solution manifold, the more training time is required to approximate it. Case II and IV both converge within 10 epochs although they lie in 2D and 3D space, respectively. Per Fig. 5 and 7 (b), the design variable space is convex and so the solution manifold is less complicated. Although in 2D variable space, Case III takes 20 epochs owing to the sinusoidal solution manifold. Case V converges in 30 epochs, the design space is 30 dimensional, hence the compute complexity increases due to the construction of a larger $L$ matrix. The validation loss curve lies below the training loss (but strictly at scale), suggesting that our low-weight network did not over/underfit. (a) Case II (b) Case III (c) Case IV (d) Case V Figure 10: Training (blue) and validation (orange) curves for 4 cases. An increase in the dimensions of the design variable space results in increased costs for constructing the L matrix. Consequently, the network takes more epochs to converge. ## 6 Modeler Interpretability Motivated by Lipton [21]’s definitions of model interpretability and trust, we adopt the persona of a modeler in assessing the interpretability of our model. In all of the problems above, the approximate manifold $\tilde{M}$ is described by the user specified loss function. If a domain specific analytical solution ($M(X^{})=0$) is known, then the approximate network generated solution set ($\tilde{M}(\tilde{X})=0$) can be verified by comparing $\tilde{X}$ and $X^{}$. Additionally, a domain-specific modeler can also compare the approximate manifold $\tilde{M}$ (see Fig. 11 for case III), at the last layer of the network, against the true manifold $M$ known from the analytical form. If the modeler is able to verify that the network classifies the correct (truly Pareto optimal) data points in the variable space as being Pareto optimal (high probability value), the trust in the network’s working is established. Figure 11: Classification boundary of weak Pareto points for Case III. The final layer assigns a probability score to each point in the variable space as being Pareto optimal or not. A modeler needs to examine the produced points/manifold against the known analytical form in the variable domain to verify the correctness of the network’s working. ## 7 An Application: Fair Search Imagine a new policy for predictive policing is under consideration, with various public arguments being published for and against adoption. By reviewing this body of arguments, one might arrive at an informed and balanced understanding of the issue and public debate surrounding it. This, in turn, could guide citizens or lawmakers in voting, or help a journalist to provide balanced reporting. Assume a search engine is used to find information that is both relevant and balanced. Specifically, assume we wish to maximize two objective functions computed on the set of retrieved search results: 1) relevance and 2) diversity (i.e., balanced inclusion of search results that are for and against the proposed policy). Note that our diversity target is specified here as a soft objective function to maximize rather than a hard constraint to rigidly enforce. Let us make a further independence assumption between relevance and diversity: knowing that a retrieved document is relevant does not provide any indication as to whether or not that document presents an argument for or against the proposed policy. Gao and Shah [13] present such a fair search problem as follows (albeit with a different motivating back story). Let $S$ denote the search result set of cardinality $\|S\|=s$. Assume each document $d_{i}$ has binary relevance $r_{i}\in\\{0,1\\}$ and group assignment $g_{i}\in\\{0,1\\}$ (i.e., for or against the policy, in our scenario), and that $r_{i}$ and $g_{i}$ are independent, as above. The optimization goal is dual maximization of the average relevance $f_{r}=\overline{r_{i}}=\frac{\sum r_{i}}{n}$ and the entropy $f_{g}=H(\frac{\sum g_{i}}{n})$ of the search result set, with entropy used as the measure of diversity (aka parity, balance, or group fairness). ### 7.1 Insights from Pareto Framing Gao and Shah [13] pose several questions, including “1. What are the possible relevance and fairness scores of a solution set $S$ (the solution space)?” and “2. What is the trade-of between fairness and relevance?” They define the solution space as the set of all possible subsets $S\subset D$ in document collection $D$ having cardinality $\|S\|=s$. They then proceed to investigate these questions via simulation: generating different subsets of search results and inspecting the empirical distributions of scores. In contrast, we suggest a conceptual Pareto framing provides more direct and informed answers. While we have emphasized generality of Pareto optimality under competing objectives with constraints, Pareto optimality is of course also applicable to simpler optimization problems, such as posed here. Firstly, since there are no constraints on the solution space, the feasible set spans the full range of $f_{r}\in[0,1]$ and $f_{g}\in[0,1]$. Secondly, since relevance and entropy are independent, they do not compete: maximizing one does not preclude maximizing the other, and each can be considered separately in turn. Relevance is maximized when all search results are relevant ($f_{r}=1$), while entropy is maximized when search results are evenly split across the two groups ($f_{g}=1$). This yields weak Pareto fronts at $f_{r}=1$ and $f_{g}=1$, with the only strong Pareto solution at the intersection of both fronts ($f_{r}=f_{g}=1$), when search results are all relevant and are evenly split across the two groups being represented. With regard to the probability of observing any given ($f_{r},f_{g})$ score for a given search result set $S$ (i.e., the chance of achieving the optimum $f_{r}=f_{g}=1$ or any other feasible point), since relevance and entropy are independent, their joint distribution is defined simply by the product of probability distribution functions (PDFs) for relevance $P(r_{i})=p_{r}$ and group membership $P(g_{i})=p_{g}$. In practice, we must induce these PDFs from data, but this is standard estimation and not unique to this particular problem setting. Moreover, this permits analytical analysis without simulation. Finally, whereas Gao and Shah [13]’s optimization problem is relatively easy (no constraints or competing objectives), one could easily introduce further complications. For example, imagine public opinion is highly skewed, such that nearly all relevant information supports one side of the argument. In this scenario, in order to get more balanced coverage of the minority position, we would need to include more non-relevant search results, forcing the user to sift through a larger result set to find relevant and balanced information. As a second example, given controversy surrounding predictive policing policies, one might like to constrain search results to enforce racial parity across authors of retrieved documents in the results set. In either case, our Pareto optimality framing would allow principled reasoning about the resulting solution spaces. ### 7.2 Another Test Case for HNPF As in Section 5, evaluating our HNPF approach on solutions with known analytical forms allows us to verify its accuracy. As discussed above, we know from first principles that the fair search problem considered here has weak Pareto fronts at $f_{r}=1$ and $f_{g}=1$, with the only strong Pareto solution at the intersection of both fronts ($f_{r}=f_{g}=1$). By running HNPF on this problem, we can verify it induces these expected fronts as another check on its correctness. Gao and Shah [13] simulate $r_{i}$ and $g_{i}$ values based on observed statistics of the YOW RSS Feed dataset [39]. They consider $n=48$ data points drawn from this dataset, with empirical $\hat{p_{r}}=0.56$ and and $\hat{p_{g}}=0.5$. Sampling from binomial distributions $R\sim B_{r}(n=48,p_{r}=0.56)$ and $G\sim B_{g}(n=48,p_{r}=0.50)$, in expectation we anticipate $R=(\sum^{n=48}r_{i})\approx 27$ relevant documents and an even split of $G=(\sum^{n=48}g_{i})=24$ across groups. Note that whereas average relevance $f_{r}$ is maximized when all documents are relevant, the entropy $f_{g}$ is maximized when documents are evenly split by group. The probability distributions above will thus naturally tend to yield near optimal entropy $f_{g}\approx 1$ (with $p_{g}=0.5$) but only mediocre average relevance $f_{g}\approx\overline{r_{i}}=0.56$. With $p_{r}=0.56$, the probability of maximum relevance $P(f_{r}=1)=P(R=48)\approx 10^{-60}$. (a) Function Domain (b) Variable Domain Figure 12: Weak Pareto Front for RSS dataset Figure 12 shows the weak Pareto front for (a) the function domain $(y=f_{r},x=f_{g})$ and (b) the variable domain $(x=R,y=G)$. As expected, we see the sample distribution lays roughly symmetrically about the expectation in both $R$ and $G$ in the variable domain, yielding near optimal entropy $f_{g}$ and mediocre average relevance $f_{r}$ in the function domain. Given this sample, the network correctly identifies the Pareto front for entropy at $f_{g}=1$, corresponding to $G=24$ in the variable domain. Note that this is still a weak front, where all points to the left are dominated by those to the right, hence the need for the Pareto filter to identify the non-dominated set. However, the network cannot identify the Pareto front for relevance at $f_{r}=1$ due to sample sparsity relative to the sampling distribution. As noted above, with $p_{r}=0.56$, $P(f_{r}=1)\approx 10^{-60}$, suggesting we would need $10^{60}$ samples to observe $f_{r}=1$. Our results above follow Gao and Shah [13] in sampling from the variable domain according to ($p_{r},p_{g}$). This makes sense if we want to explore the solution space via simulation, as they do. However, if our goal is actually to identify Pareto fronts (e.g., to measure how far a given solution is from optimality), we can instead probe the solution space far more efficiently by uniformly sampling the variable domain ($r,g\in[0,1]$). Appendix E presents these results. ## 8 Conclusion A hybrid, two-stage, neural-Pareto filter based optimization framework is presented for extracting the Pareto optimal solution set for multi-objective, constrained optimization problems. 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[2008] Qingfu Zhang, Aimin Zhou, Shizheng Zhao, Ponnuthurai Nagaratnam Suganthan, Wudong Liu, and Santosh Tiwari. Multiobjective optimization test instances for the cec 2009 special session and competition. 2008\. * Zhang [2005] Yi Zhang. Bayesian graphical models for adaptive information filtering, 2005. ## Appendix A Additional Cases ### A.1 Case VI: n=2, k=2, m=5 This problem was proposed in [10]. Jointly minimize $\displaystyle f_{1}(x_{1},x_{2})=x_{1}$ $\displaystyle f_{2}(x_{1},x_{2})=x_{2}$ $\displaystyle\text{s.t.}\quad g_{1}(x_{1},x_{2})=(x_{1}-0.5)^{2}+(x_{2}-0.5)^{2}\leq 0.5$ $\displaystyle g_{2}(x_{1},x_{2})=x_{1}^{2}+x_{2}^{2}-1-0.1\cos(16\arctan(\frac{x_{1}}{x_{2}}))\geq 0$ $\displaystyle g_{3}(x_{1},x_{2})=max(\|x_{1}-0.6\|,\|x_{2}-0.7\|)-0.2\geq 0$ $\displaystyle g_{4},g_{5}:0\leq x_{1},x_{2}\leq\pi$ This form is an extension of Case III, with an additional max boundary constraint $g_{3}$. Both mCHIM and PK computes the true Pareto front with limited density $n=40$ points. Fig. 13 (a) shows the weak Pareto front with dominated points. As before, after post-processing with the proposed Pareto filter we arrive at the Pareto set with non-dominated points shown in Fig. 13 (b). (a) Dominated (b) Non-Dominated (c) PK (d) NBI Figure 13: Strong Pareto Front for Case VI. Note that all the dominated points are removed from the set after application of Pareto filter. Also note the low density of points in PK and the even spread of points in NBI. Also note that PK is not able to remove some of the dominated points on the horizontal and vertical lines. ### A.2 Case VII: n=30, k=2, m=30 This problem was proposed in [29], albeit with a discrepancy222Although the normalization term proposed in $f(x)$ is $m-1$, it does not generate the curve reported in [29]. We were able to replicate the shown curve, in our experiments, by choosing a normalizing constant of $10000$.. Jointly minimize $\displaystyle f_{1}(x)=x_{1}$ $\displaystyle f_{2}(x)=f(x)\left(1-\left(\frac{f_{1}(x)}{f(x)}\right)^{0.5}-\frac{f_{1}(x)}{f(x)}sin(10\pi x_{1})\right)$ $\displaystyle\text{s.t.}\quad f(x)=1+\frac{9}{m-1}\sum_{i=2}^{n}x_{i}^{2}$ $\displaystyle g_{1},\ldots,g_{30}:x_{1}\in[0,1],x_{i}\in[-1,1],\forall i=2,\ldots,m$ This form is convex in $f_{1}$ and non-convex in $f_{2}$. The dimension of the design variable space is $m=30$. The corresponding Pareto front is non-convex. The results using our method, as shown in Fig. 14, are in good agreement with mCHIM and PK methods. Even in this high-dimensional setting, we obtain a weak Pareto front with high point density as shown in Fig. 14 (a). The distribution of the objective space in this setting is such that the entire space is the front itself. Hence, we cannot see the cyan points in Fig. 14 (a). As before, after post-processing with the proposed Pareto filter we arrive at the Pareto set with non-dominated points shown in Fig. 14 (b), where the dominated points are now visible. (a) Dominated (b) Non Dominated (c) PK (d) NBI Figure 14: Pareto Front for Case VII. Note that all the dominated points are removed from the set after application of Pareto filter. Also note the low density of points in PK and the even spread of points in NBI. ## Appendix B Working of Pareto filter The algorithm starts with the set of all weak Pareto points $p\in P$, which will be refined through the iterative process. The loop (line 4) iterates over all the functions $f_{i}$. It checks for set of dominating and non-dominating points for all discretization levels (line 6) and appends them to a temporary list (line 10). If multiple points do exist (line 11) for a given level (cardinality $>1$), then there certainly are dominated points. The non- dominated point is one which has the lowest function value for the next function $f_{q},q=i+1$. This (line 12) states that a point which seems non- dominated for a given function $f_{i}$ might be dominated for other functions $f_{r},r\in k,r\neq i$, but will be taken care of when iterating through function $f_{r}$. Once the non-dominated point has been found, it implies that all the other points are in fact dominated, hence should not be considered for further evaluation. They are rejected (line 13) from the active set $P=P\backslash(temp\backslash x_{p})$. The output is the set of strong Pareto points which were all non-dominated for every function $f_{i}$, and are essentially the points in the weak Pareto set $P$ that survived the filtering process (line 13) for every $f_{i}$. ## Appendix C Error Bound For a user-prescribed relaxation margin $0\leq\epsilon\leq 1$, the approximation error between the network extracted manifold $\tilde{M}(\tilde{X})$ and the true solution $M(X^{})$ is bounded below by $\\|\tilde{M}(\tilde{X})-M(X^{})\\|_{2}\leq\epsilon$. Assuming the $L$ matrix from Eq. (6) is square, we have $det(L)=0$. This form holds for some of the problems chosen in our numerical experiments (Cases I, II, III), where the number of functions and constraints are equal. From Leibnitz formula for determinants: $\displaystyle det(L)=det\Big{(}\begin{bmatrix}\nabla F&\nabla G\\\ \mathbf{0}&G\end{bmatrix}\Big{)}=det(\nabla F)det(G)=0$ Further assume that, $\displaystyle\|det(L(\tilde{x}))\|\leq\epsilon,\,\tilde{x}\neq x^{}$ (8) where $\epsilon>0$, and $x^{}$ and $\tilde{x}$ are the optimal points and the network generated approximate solution points, respectively. The Fritz John necessary conditions in Eq. (4) for weak Pareto optimality is: $\displaystyle det(L(x^{}))=0$ (9) Combining the assumption in Eq. (8) and Eq. (9), we have $\displaystyle\|det(L(\tilde{x}))-det(L(x^{}))\|\leq\epsilon$ (10) The solution manifold $M(X)=det(L(X))=0$ is weakly Pareto optimal. We assume a low precision manifold $M(\tilde{x})$ such that: $\displaystyle\\|M(\tilde{X})-M(X^{})\\|_{2}\leq\epsilon$ (11) When the network converges, Eq. (11) will hold for the network approximated $\tilde{M}(x)$. Here, $\tilde{x}\in\tilde{X}=\\{x\|\tilde{M}(x)=0\\}$ and $X^{}$ is the set of true optimal points such that $M(x^{})=0,\forall x\in X^{}$. Since we explicitly specify $\epsilon$ in our loss description, we know that the network generated solution is $\epsilon$ close to $M(\tilde{x})$ if: $\displaystyle\\|M(\tilde{X})-\tilde{M}(\tilde{X})\\|_{2}\leq C\epsilon,\quad 0\leq C\leq 1$ (12) The form in Eq. (12) implies that if we are able to find such a $C$, then we implicitly satisfy Eq. (11). Hence, $\displaystyle\\|\tilde{M}(\tilde{X})-M(X^{*})\\|_{2}\leq\epsilon$ (13) ## Appendix D Density Comparison \| HNPF \| PK ---\|---\|--- Case \| Density \| Points \| Evals \| Density \| Points \| Evals Case V \| 1.34 \| 1206 \| 90K \| 2.21e-4 \| 100 \| 45,126,324 Case VI \| 0.20 \| 180 \| 90K \| 1.32e-2 \| 151 \| 1,139,781 Case VII \| 6.57 \| 5915 \| 90K \| 9.29e-5 \| 101 \| 108,685,605 Table 3: Pareto optimal point density % (ratio of #extracted optimal points to #function evaluations). HNPF finds many more optimal points with many fewer function evaluations. ## Appendix E Verification Case for Fairness We now demonstrate the nature of the extracted Pareto front under a uniform distribution for the problem setting in Section 7.2. While the scenario presented in Gao and Shah [13] considered points being drawn from a Binomial distribution, it is highly improbable to reach a relevance value of 1. We therefore consider a unbiased uniform distribution, where we sample integers uniformly between $[0,48]\times[0,48]$ for both relevance ($R$) and entropy ($G$). (a) Function Domain (b) Variable Domain Figure 15: Weak Pareto Front for RSS dataset under a uniform setting. Note that now both min-min and the max-max solutions are achievable, as these points are visible to the network. In Fig. 15 we show the extracted weak Pareto front from the HNPF neural network. Under the uniform data distribution, the maximum value of achievable entropy corresponds to $G=24$. Similarly, the minimum achievable entropy value corresponds to both ($G=0,48$) which indicates that all documents belong to only one source. The neural network extracts the entire max-max weak Pareto front which when post-processed using the Pareto filter, results in the expected $(1,1)$ solution in the function domain. Note that the Stage-1 of HNPF is capable of extracting the entire weak Pareto front for any MOO problem. The middle red line in the variable domain corresponds to the top red line in the function domain. The two remaining red lines (top and bottom) in the variable domain collapses to the bottom in the function domain showing the weak Pareto front for a min-min MOO problem. Applying the Stage-2 Pareto filter now gives $(0,0)$ as the optimal min-min solution.
Data Analysis and Management (Computer Science and Engineering) June 2020 May, 2020 Victor LempitskyAssociate Professor, PhD Yury MalkovPhD TODOTODO, Skoltech # CNN with large memory layers Rasul Karimov This work is centred around the recently proposed product key memory structure [46], implemented for a number of computer vision applications. The memory structure can be regarded as a simple computation primitive suitable to be augmented to nearly all neural network architectures. The memory block allows implementing sparse access to memory with square root complexity scaling with respect to the memory capacity. The latter scaling is possible due to the incorporation of Cartesian product space decomposition of the key space for the nearest neighbour search. We have tested the memory layer on the classification, image reconstruction and relocalization problems and found that for some of those, the memory layers can provide significant speed/accuracy improvement with the high utilization of the key-value elements, while others require more careful fine-tuning and suffer from dying keys. To tackle the later problem we have introduced a simple technique of memory re-initialization which helps us to eliminate unused key-value pairs from the memory and engage them in training again. We have conducted various experiments and got improvements in speed and accuracy for classification and PoseNet relocalization models. We showed that the re-initialization has a huge impact on a toy example of randomly labeled data and observed some gains in performance on the image classification task. We have also demonstrated the generalization property perseverance of the large memory layers on the relocalization problem, while observing the spatial correlations between the images and the selected memory cells. ### Acknowledgments We wish to express our deep sense of gratitude and profound thanks to Karim Iskakov and all the engineers in Samsung AI Moscow who contributed to the project in one way or another. We are hugely indebted to Samsung Research Center for the provided resources that gave us the chance to implement the models and conduct the required experiments. ###### Contents 1. 1 Introduction 2. 2 Related work 3. 3 Methods 1. 3.1 Product key memory 1. 3.1.1 Design overview 2. 3.2 Re-initialization trick 1. 3.2.1 Overview 2. 3.2.2 Problem of dying keys 3. 3.2.3 Key re-sampling 4. 3.2.4 Re-sorting and key-value reinititalization 5. 3.2.5 Re-initialization complexity 3. 3.3 Classification pipeline 4. 3.4 Regression pipeline 5. 3.5 Image reconstruction pipeline 4. 4 Experiments 1. 4.1 Experiments on random labels 1. 4.1.1 Results on CIFAR-10 2. 4.1.2 Experiments on ImageNet 2. 4.2 Memory in PoseNet 1. 4.2.1 Analysis of the memory layer 3. 4.3 Image reconstruction experiments 4. 4.4 Other experiments 5. 5 Conclusion ###### List of Figures 1. 3.1 Overview of the product key-value memory design. Feature generator is the baseline image model which outputs the latent vector that is projected by the the query generator, normalized and divided into the two sub queries $q_{1}$/$q_{2}$ to work with product key values. As the result we have k selected keys and the corresponding values from $\mathcal{V}$. 2. 3.2 The illustration of the key re-initalization procedure. (a) We have 5 keys with three of which does not pass the threshold of being identified as the ”utilized keys”. We are removing those keys in the step (b). And in the step (c) we are initializing new keys sampled from existing ones and perturbed with an additive Gaussian noise ($\epsilon\sim\mathcal{N}(0,\sigma_{d}^{2})$). 3. 3.3 Bottleneck [30] augmented with a memory block $\mathcal{M}$. 4. 3.4 The overview of the modified squeeze-and-excitation block augmented with the memory layer. $F_{GB}$ is the function of global pooling which reduces the dimension of 3 dimensional feature map to the signle dimension, and $F_{add}$ is channel-wise addition of a resulting vector from memory to the original feature tensor. 5. 4.1 Results for the random data on the various models. The left graph is the plot for top-1 validation accuracy results, top-5 is pictured on the right. We can see that the model with no memory is not able to fit the data. Setting 8 heads for multi-head attention memory model, on the other hand, helps the model to easily fit the data. Re-initialization helps to get nearly full convergence with on only 1 head, i.e. multi-head mode disabled. Also, $h=1$ in the graph is the notation for the single head model. 6. 4.2 Relation between $\sigma_{d}$ hyper parameter of the Gaussian noise square variance parameter in re-initialization procedure and the values of train accuracy and loss for the random data. Setting higher $\sigma_{d}$ helps significantly, giving lower loss. 7. 4.3 Comparison of the top-1 scores, memory utilization and the inference speed for the model with no memory augmentation and three memory augmented models with the number of heads in the set of {1, 4, 8}. There are 20 runs for the each experiment with the different initial seeds. As it can be observed, there is no evident increase in the accuracy while the performance of the models with the heads is much worse compared to the baseline models. Inference values are calculated on GTX-1080Ti cards with fp32 mode, the results are approximate. 8. 4.4 We picture the key access distribution for each head. As we see there is a little perturbation in the distribution when we apply the random region crop in the input image. 9. 4.5 Change in values of the standard deviation of keys, memory usage, gradient of memory values, and key gradients during a training phase. 10. 4.6 The camera path and the four sample images captures in the given coordinates. We see the correlation between the traveled distance and the indices selected. 11. 4.7 Reconstruction results. Top row images in each section are the reconstruction results with the memory augmented autoencoder with mem_idx=2 and heads=8, the middle are the output of the baseline autoencoder while the final row are the input images. We see that there are some little details that are captured using the memory layer. ###### List of Tables 1. 4.1 Results for the modified SE blocks. WL is the table notation for the wide linear layer that replaces the memory layer, $d_{w}$ defines respectably the row and column of the projections matrices in MLP (with the row vector in the linear operator). Overall we see better results on Resnet-20 with the memory layer and with re-initialization trick we have superior memory utilization rate. Cosine similarity helps us to nearly reach the accuracy values of ResNet-100. Inference values are calculated on GTX-1080Ti cards with fp32 mode, the results are approximate. 2. 4.2 The results on the Resnet-50 and Imagenet. We have tested a number of the hyper parameters to find the best train strategy for the memory models. For now we dont see the clear picture on optimization issues. 3. 4.3 Results on the King’s College dataset, $\mathrm{trans}$ and $\mathrm{rot}$ are the translation and rotation errors respectively. We see a huge decrease in the loss on the train set for the networks with memory augmentation but also the increase in the inference time. Inference values calculated on the GTX-1080Ti with the batch size of 1. 4. 4.4 Reconstruction results after 1600 iterations. The memory utilization numbers are approximate since the amount of experiments conducted was low. The inference speed is calculated on GTX-1080Ti with batch size of 1. ## Chapter 1 Introduction With the huge development of deep learning, neural networks have made significant progress in various tasks such as image classification [44], speech recognition [26], and machine translation [74]. As it was shown in [72, 54], with sufficiently large data, increasing the capacity of neural networks could lead to far superior prediction accuracy. Therefore, scaling of both training and model size has been a central problem of deep learning field in recent years. For a typical neural model where the single input data sample depends on all the parameters of the network, therefore increasing both dataset and model sizes leads to a nearly quadratic surge in the training costs [69]. Previous works [15, 12] proposed several ideas of increasing the model size with no proportional increase in a computational complexity. Most of the proposed models rely on the sparse gating mechanism which determines whenever a particular node of the graph should be calculated during the forward pass of the network. This is the type of a branching problem with a discrete decision which is being solved with REINFORCE algorithm [75]. It was applied in [8] and gave good results on MNIST dataset and CIFAR-10 [43] with a reasonable speed up. Instead of REINFORCE estimator, one can also apply the ideas from [49, 34] by relaxing the discrete skipping decisions with reparametrization technique adoption. However, these approaches usually find sub-optimal solutions due to the approximation error introduced by the reparametrization trick [82, 1]. Other approaches rely on learning binary masks as the sparse $l_{0}$ regularization term for the final objective. Works like [53] employ a rectified sigmoid proposed in [48] to learn binary decision choices. Authors apply regularization during post-processing to quantize the weights of the network, but the idea could be used in the training phase too. Recently, the paper [58] on differentiable tabular data with neural networks has leveraged the entmax transformation [55] to learn ”hard” decisions for binary branching problem in decision trees. Though solving the issue of scalability, models still fall short in giving promising results due to the following challenges: * • GPUs are optimized to work faster with arithmetic tasks rather than branching. * • Batching reduces the batch sizes for conditionally activated chunks, therefore complicating parallelization. * • REINFORCE estimator has a large variance, making it hard to get a strong bias during train. There are some variance reduction tricks [28, 51] that try to solve the issue but most of them skew the bias-variance trade-off with no tuning on hyperparameters applied. * • Nearly all the methods suffer from neuron dying problem - if at some moment of the training a gate is not open for any input sample, this means it is highly unlikely it will be open at any further moment since the gate receives only zero gradient. A recent work [46] on the over-parametrized language models, on the other hand, rely on the key-value memory structures to scale the set of parameters in the neural network. Authors rely on the product key space decomposition for nearest neighbours to scale the networks with a little or no change in the performance and the memory consumption. These results encouraged us to research these methods in the computer vision applications. As an extension of [46], we augment the product key layer with the key-value re-initialization mechanism which allows to solve the dying neuron problem. The mechanism is based on re-initialization of dead or underutilized keys- values pairs using the information from more successful key-values pairs. ## Chapter 2 Related work Memory layers in neural models Memory augmented neural networks (MANNs) augment neural networks with external memory block which allows for explicit access with differentiable write and read operations. The memory is usually implemented as a key-value differentiable dictionary [59], array-structured memory such as Turing machine (NTM) [27], or recently proposed product-key memory layers [46]. Key-value memory architectures were analyzed extensively in deep learning literature. Mostly the models are based on the architectural designs described in [73, 78] and used mainly in natural language processing field (NLP) such as document reading and question answering tasks. The key-value paired structure can be seen as a generalization of how the huge context for question answering is stored in the memory. That makes key-value memory a natural choice for these tasks. And with the recent advancements in attention models (AM) [5, 80], it is becoming the predominant concept in natural language processing literature with the usage case in nearly every NLP task. The key-value design structure is a sparse version of attention models, and as previously described in [14, 7] the self-attention mechanism could be competitive in replacing convlolutions as a computation primitive for object detection and image classification. We hope to leverage and improve upon those ideas for computer vision applications. There were also some works in extending the key-value structure, [25] using unbound cache model to provide better modelling of infrequent words that occur in a recent context, including the out-of-vocabulary words. This is the evidence of interpretability of learned key-value structure which provides the linear interpolation between the values of the table. Other works [39] focus on the interpretability of memory blocks by linearly interpolating baseline language models (LM) with k-nearest neighbours (k-NN) models and assessing the generalization and memorization capability of the final models. Authors report increased perplexity in all of the experiments. Other approaches have successfully applied memory layers to store image features, more specifically for image captioning [13], image generation tasks [41], and video summarization [47]. Some neural network designs include non-differentiable memory layers with unsupervised updates. These models mostly rely on the architectural ideas of [89] and rely on contrastive loss to update the memory block. Authors of [89] have demonstrated the the efficiency of their memory block in few-shot and one-shot learning tasks, while [84] has shown the advantage of using the memory in style transfer [20] tasks with limited data. While in supervised approaches of memory usage where we are learning the mapping function between two spaces, in the unsupervised approach memory block is used for storing latent vectors with the ability to interpolate between them. This is the important property of memory blocks that is implicitly used in most of the models. Some works incorporate memory-like structure to store a bank of weights accessing them sparsely and using k-nearest neighbours to retrieve a set of indices that allows to encode the feature vector into the discrete code. There are some promising results in auto-regressive models [79] giving high-fidelity image reconstruction results in [63]. Authors argue that the discrete representation is a more natural fit for complex reasoning, planning and predictive learning. Moreover, memory layers were successfully incorporated in graph knowledge representation problems, with promising results in graph classification tasks [40]. Learning compact representations, ANN. Since the task of exact top-k neighbour search is expensive in practice for high dimensional embeddings, i.e. linear scale to the search set size, practitioners in the field usually resort to more scalable approximate nearest neighbours methods (ANN). Popular methods include Locality sensitive hashing (LSH) [22, 2] that relies on random space partitioning, graph methods like Navigable small world graphs (NSW) [50] and Hierarchical NSW (HNSW) [6] based on constructing the search algorithm by finding the clusters of data. Another important subset of methods utilize quantization [21, 18, 4] to reduce the memory consumption and speed up the distance calculations. Many of those methods exploit the idea of product decomposition, e.g. assumption that the space can be decomposed into a Cartesian product of low dimensional spaces. Product decomposition in neural models Most of this thesis is inspired by the work of [46] which are showing the efficiency of using product key memory layer in language modelling tasks. Here product key is a structure that allows more efficient scaling by scarifying some expressiveness of the model. Authors find that the language model augmented with memory with only 12 layers can outperform in accuracy a baseline transformer model with 24 layers while giving two times less inference time. The [46] didn’t however address the problem of dying keys other than by adding noise to the query (via batchnorm) and was focused solely on NLP applications. Product quantization in general has also been used in many computer vision applications, starting from scalable supervised [56] to semi-supervised [35] image retrieval tasks. There are some promising results [83] in face video generation with memory augmented Generative adversarial networks (GAN) [23]. Classification networks Huge chunk of work [76, 30, 45, 70] is done in designing the neural networks for the image classification problems. In our experiments we mainly focus on the ResNet-like [30] networks. Some recent work [77] demonstrated the SOTA results in ImageNet-2012 dataset [44] with the help of the reinforcement learning to tune the models and the whole architecture. Most of the existing neural networks for image classification rely on the convolutional block but there were some recent works suggesting the self- attention mechanism with promising results [7, 62]. ## Chapter 3 Methods ### 3.1 Product key memory #### 3.1.1 Design overview The overall pipeline of the differentiable product memory layer is similar to most of the key-value data structures that are augmented into neural network models [27, 78, 73]. More specifically, product memory design in our work is heavily inspired by previously proposed architecture in [46]. Here we build models upon this design to solve classification, regression, and reconstruction computer vision tasks. Higher view of the architecture is illustrated in Figure 3.1. The central idea is to augment baseline convolutional neural networks with sparse memory access. The input of the memory layer is the latent feature vector that describes the given input image. Depending on where we place the memory layer, the query can represent features like brightness gradients or colours with more complex patterns in later layers [87]. Therefore, the choice of memory access placement is important. Given the input query, memory block finds the distance scores by comparing it with all of the keys in the table and selecting the values associated with top-k distance scores. The scores are then used to produce the output $m(x)$ via a weighted sum over the values associated with the selected keys: $m(x)=\sum_{i\in\mathcal{I}}w_{i}v_{i}$ where $\mathcal{I}$ is the set of top-k indices by distance scores, $w_{i}$ are the scores, and $v_{i}$ are the values in the memory layer. Query generation. The memory block consists of the query generation network which is a learnable projection function $q:x\rightarrow q(x)\in\mathbb{R}^{d_{q}}$ mapping the d-dimensional input vector x into the $d_{q}$-dimensional query vector. Typical dimension sizes of the query vectors in our experiments are from 256 up to 2048. Also, since the keys are initialized in a fixed range, we follow [46] adding BatchNorm [33] layer on the top of the query network. This allows a better overlap between the distribution of keys and queries. And as in [46] we observe higher utilization of the memory with BatchNorm enabled. Key assignment We have a resulting query $q(x)$ that should be assigned to the closest keys with the selected metric for the distance. Let $\mathcal{K}=\\{k_{1},\ldots,k_{\|\mathcal{K}\|}\\}$ is the set of all keys, the set is composed of $\|\mathcal{K}\|$ $d_{q}$-dimensional vectors that are uniformly initialized in the $\mathbb{R}^{d_{q}}$ space. We can define the differentiable procedure to find a weighted sum over the value vectors (the memories associated with top-k keys). The sum is weighted by the distance scores between the subset of the top-k keys and the query value $q(x)$. Top-k procedure finds the most closest keys to the given query, i.e. maximization of the chosen similarity measure $d(\cdot,\cdot)$. The overall algorithm is: $\displaystyle\mathcal{I}=\mathcal{T}_{k}\left(d(q(x),k_{i})\right)$ $\displaystyle w=\operatorname{softmax}\left(\left\\{d(q(x),k_{i})\right\\}_{i\in\mathcal{I}}\right)$ $\displaystyle m(x)=\sum_{i\in\mathcal{I}}w_{i}v_{i}$ where $\mathcal{T}_{k}$ denotes the top-k operation which finds k largest values based on the similarity measure $d(\cdot,\cdot)$. $\mathcal{I}$ denotes the index set of the most similar keys to the query $q(x)$ and $w$ represents the normalized scores associated with the selected indices. The resulting value $m(x)$ is the sum of the selected values weighted by the normalized scores. As we see due to the summation over the normalized values in the $\operatorname{softmax}$ operation, the gradients can be calculated. Note that it is not possible to find the gradient for top-1 function, since there is no summation. Figure 3.1: Overview of the product key-value memory design. Feature generator is the baseline image model which outputs the latent vector that is projected by the the query generator, normalized and divided into the two sub queries $q_{1}$/$q_{2}$ to work with product key values. As the result we have k selected keys and the corresponding values from $\mathcal{V}$. Product keys We see that the bottleneck of the given procedure is the calculation of the $\mathcal{T}_{k}$ the operation which has linear complexity over the size of the key set $\mathcal{K}$, so it is hard to scale this procedure for large memory sizes. The remainder of operations are done for the reduced set of selected indices, e.g. the summation over top-k normalized weight values. To solve the performance issue, authors of [46] propose to represents the key set in the form of two independent sets of half dimension size $d_{q}/2$ vector sets $\mathcal{K}_{1}$ and $\mathcal{K}_{2}$ which constructs the Cartesian product set of resulting values with size $\|\mathcal{K}\|=\|\mathcal{K}_{1}\|\times\|\mathcal{K}_{2}\|$. The query vector should also splitted into two sub-queries $q_{1}$ and $q_{2}$ to work in each of the key sets. We then find the closest keys in both sets as: $\mathcal{I}_{\mathcal{K}_{1}}=\mathcal{T}_{k}\left(\left\\{d(q_{1}(x),k_{i}^{1})\right\\}_{i\in\\{1\ldots\|\mathcal{K}_{1}\|\\}}\right),\quad\mathcal{I}_{\mathcal{K}_{2}}=\mathcal{T}_{k}\left(\left\\{d(q_{2}(x),k_{j}^{(2)})\right\\}_{j\in\left\\{1\ldots\left\|\mathcal{K}_{2}\right\|\right\\}}\right)$ Then the two subsets of the keys associated with the index sets $\mathcal{I}_{\mathcal{K}_{1}}$ and $\mathcal{I}_{\mathcal{K}_{2}}$ are multiplied together to form a new Cartesian product set. We are applying the top-k operation on the newly created set and find the final subset of the top-k keys. Choice of distance Authors in [46] experiment with the inner product as the single similarity measure for the provided experiments. We observe that using cosine similarity not only provides us with better numbers in some experiments but also gives us control over the selection process of the keys. Since the dot product is proportional to the vector norm, the key vectors with the largest vector lengths will be selected in most of the cases, while low norm vectors may be completely ignored. This means that the distance measure captures the most popular candidates, the latter can skew the similarity metric. We balance the skew by introducing the hyperparameter $\alpha$ and raising the length to an exponent $\alpha<1$ to calculate the distance as: $d_{cos}(q,k,\alpha)=\|q\|^{\alpha}\|k\|^{\alpha}\cos(\theta)=\|q\|^{\alpha}\|k\|^{\alpha}\frac{q^{T}k}{\|q\|\cdot\|k\|}$ (3.1) Multi-head mode To make the model more expressive we are using the multi-head mechanism [80] which splits queries and keys into multiple chunks of vectors to be calculated independently. The similar calculations are conducted on each head and the results are concatenated. Due to the reduced dimension of each head, the overall computational complexity of the layer is similar to the single-head attention. Complexity Naive top-K key selection requires $\mathcal{K}$ comparisons of $d_{q}$ sized vectors, which gives us $\mathcal{O}\left(\|\mathcal{K}\|\times d_{\mathrm{q}}\right)$ operations overall. When using the product space $\mathcal{K}=K_{1}\times K_{2}$, we have two separate sets of keys for subspaces with significantly reduced carnality $\|K_{1}\|=\|K_{2}\|=\sqrt{\|\mathcal{K}\|}$. The overall complexity for the first step then is: $\mathcal{O}\left(\|K_{1}\|\times d_{q}/2+\|K_{2}\|\times d_{q}/2\right)=\mathcal{O}\left(\|K_{1}\|d_{q}\right)=\mathcal{O}\left(\sqrt{\mathcal{\|K\|}}d_{q}\right)$. The second step is performed on the reduced subset of $k\times k$ elements so it will require $\mathcal{O}\left(k^{2}\times d_{1}\right)$ operations. Therefore overall complexity is: $\mathcal{O}\left(\left(\sqrt{\|\mathcal{K}\|}+k^{2}\right)\times d_{\mathrm{q}}\right)$ ### 3.2 Re-initialization trick #### 3.2.1 Overview While conducting our initial experiments on random data, we have observed that a toy neural network augmented with memory block struggles to fit the data with no multi-head mode enabled even though the model should have had enough capacity to fit the whole dataset. By conducting some ablation study and literature review [3] we have concluded that the problem is due to the correct initialization of the memory layer. Additionally, authors in [81] suggest that most of the heads in the attention mechanism can be pruned without serious effect on the performance. To tackle the initialization issues we are introducing the re-initialization trick that dynamically initializes unused keys during the training phase. We are describing the whole procedure below. Figure 3.2: The illustration of the key re-initalization procedure. (a) We have 5 keys with three of which does not pass the threshold of being identified as the ”utilized keys”. We are removing those keys in the step (b). And in the step (c) we are initializing new keys sampled from existing ones and perturbed with an additive Gaussian noise ($\epsilon\sim\mathcal{N}(0,\sigma_{d}^{2})$). #### 3.2.2 Problem of dying keys Let’s assume that we are working with the dataset of size $\|\mathcal{D}\|$ which is equal to the number of values in the memory $\|\mathcal{M}\|$. We could assume that augmenting the neural network with the memory layer could lead to the full convergence, i.e. perfect accuracy, because of the one-to-one mapping between the input and the memory elements. We hovewer did not observe this in our experiments with random data (description is in the experiments section), and classification tasks. Instead, we discover continuously reduced cardinality of the selected key set at each iteration of the optimization, reaching some fixed value $\|\mathcal{K}^{\prime}\|$: $\displaystyle\|\mathcal{K}^{\prime}\|=\alpha\|\mathcal{K}\|,\,\alpha\ll 1$ (3.2) $\displaystyle\mathcal{K}^{\prime}=\left\\{k_{i}\in\mathcal{K}\|c_{i}>0,c_{i}\in\mathcal{C}\right\\}$ (3.3) where $K^{\prime}$ is the set of selected keys during the inference, $\mathcal{K}$ is the set of all keys, and $c_{i}$ is the utilization of the key $k_{i}\in\mathcal{K}$ summed for the whole dataset, i.e. number of times the key $k_{i}$ was selected. In the experiments we are not able to get full utilization of the selected keys and observed low final accuracy. We call this a problem of dying keys, when the optimizer is unable to pass gradients through certain key-value pair in the memory layer, leading us to the dead keys, useless for inference but still having a computational burden. #### 3.2.3 Key re-sampling To solve the problem we implement a simple trick of key re-initialization, which is being executed during the training phase at certain points. We observe that during training, the key utilization converges to some specific number, as it is given in Equation (3.2). We assume that the main reason for this is dying keys problem discussed in the previous section. For this reason, we are running the pipeline of key re-initialization when the utilization plateau is reached. Here we describe the algorithm for a single product space key subset but the algorithm is applicable for both of key subsets. Let $\mathcal{K}$ define the set of all the keys in memory and $\mathcal{K^{\prime}}$ is the subset of utilized keys where $\|\mathcal{K^{\prime}}\|\ll\|\mathcal{K}\|$. We also introduce the hyperparameter $k_{a}$ which will control how many keys should be re-sampled at the each call of our key initialization procedure. Then we have: $\displaystyle\mathcal{I}_{a}=\left\\{i\|i\sim U\\{0,\|\mathcal{K}^{\prime}\|\\}\right\\}$ $\displaystyle\mathcal{K}_{a}=\left\\{k_{i}\in\mathcal{K}^{\prime}\|i\in\mathcal{I}_{a}\right\\}+\epsilon,\epsilon\sim\mathcal{N}(0,\sigma_{n}^{2})$ $\displaystyle\mathcal{K}=\mathcal{K}_{a}\cup\mathcal{K}^{\prime}$ where $\mathcal{I}_{a}$ is the set of indices sampled uniformly from the used keys $\mathcal{K}^{\prime}$, $\mathcal{K}_{a}$ is the sampled set of utilized keys perturbed with guassian additive noise. We have an additional hyper parameter $\sigma_{n}$ that controls the noise variance of the re-initialized keys, i.e. the magnitude of difference between the original keys and re- initialized ones. Then the existing set of utilized keys are expanded by $\mathcal{K}_{a}$. The sampling mechanism we discussed above is very basic, but sampling more from the regions of high density/low density could potentially bring us more gain both in prediction accuracy and the compactness of the final representation. This, however, requires the re-initialization algorithm to be able to sample key points in the regions with higher density. Something like rejection sampling algorithms, i.e. Metroplolis-Hastings algorithm [66] could save us here, by defining the multimodal normal distribution and the utilization of the key values as the mean parameter. But because of the difficulty of tuning the rejection sampling algorithm, we plan to test those algorithm in the future and resorting to simple re-sorting discussed in the following section. #### 3.2.4 Re-sorting and key-value reinititalization To give more priority on the regions of high density during sampling, we are sorting the keys in the set by the utilization coefficients $c_{i}\in\mathcal{C}$, and adapting a naive thresholding logic to eliminate the least utilized keys by removing those with the values less than the hyper parameter $d_{k}$ Then the index set calculated is: $\displaystyle\mathcal{K}_{d_{k}}=\left\\{k_{i}\geq d_{k}\|k_{i}\in\mathcal{K}\right\\}$ $\displaystyle\mathcal{I}_{a}=\left\\{i\in\\{0..\|K\|\\}\|k_{i}\in\mathcal{K}_{d_{k}}\right\\}$ After we resample the keys by eliminating the least utilized, we to initialize new values $\mathcal{U}_{a}$ that will be mapped to the elements of the Cartesian product set of the new keys $\mathcal{K}_{a}$. Because of the set product, adding single key to the subset will add $\|\mathcal{K}_{a}\|\|\mathcal{K}\|$ new values into the memory. For each key from the first product set, we are initializing new values associated with the resulting keys concatenated with the given key from the first set and all the existing keys in the second set. The same applies to the values associated with the second product set. The overall algorithm for the re-sampling step is demonstrated in Algo 1. Algorithm 1 Re-initialization algorithm 1:existing set of keys $\mathcal{K}^{\\{0,1\\}}$, noise variance $\sigma_{n}$, utilization set $\mathcal{C}^{\\{0,1\\}}$, threshold parameter $d_{k}$, and the weight values $\mathcal{U}$ 2:Reinitialized set of keys $\mathcal{K}$, Reinitialized values $\mathcal{U}$ 3:function keysort($\mathcal{K},d_{k}$) 4: $\mathcal{K}_{d_{k}}=\left\\{k_{i}\geq d_{k}\|k_{i}\in\mathcal{K}\right\\}$ 5: return $\mathcal{K}_{d_{k}}$ 6:end function 7:for $j\in\\{0,1\\}$ do 8: $\mathcal{K}_{d_{k}}^{j}=\operatorname{keysort}(\mathcal{K}^{j},d_{k})$ 9: $\mathcal{U}=\mathcal{U}\setminus\left\\{\mathcal{U}_{i}\|i\in\left(\mathcal{K}^{j}\setminus K_{d_{k}}^{j}\right)\times\mathcal{K}^{(-j)}\right\\}$ 10: $\mathcal{I}_{a}^{j}\sim U\left\\{i\in\\{0..\|\mathcal{K}^{j}\|\\}\|k_{i}\geq\mathcal{K}^{j}_{d_{k}}\right\\}$ sample $a$ indices from the discrete distribution 11: $\mathcal{K}_{a}^{j}=\left\\{k_{i}\in\mathcal{K}^{j}\|i\in\mathcal{I}_{a}^{j}\right\\}+\epsilon,\epsilon\sim\mathcal{N}(0,\sigma_{n}^{2})$ 12: $\mathcal{K}^{j}=\mathcal{K}^{j}_{a}\cup\mathcal{K}^{j}_{d_{k}}$ 13: $\mathcal{U}^{\prime}=\left\\{u_{i}^{\prime}\sim U\|i\in[0..\|\mathcal{K}_{a}^{j}\|\|\mathcal{K}^{-j}\|]\right\\}$ 14: $\mathcal{U}^{j}=\mathcal{U}^{j}\cup\mathcal{U}_{a}^{j}$ We need to associate the indices of newly created values with keys 15:end for 16:$\textbf{return}\,\mathcal{K}^{\\{0,1\\}},\mathcal{U}$ #### 3.2.5 Re-initialization complexity Taking the constant complexity of random number generation we can assume that the index sampling from discrete distribution is also constant. Then the complexity of key generation for both subsets is $\mathcal{O}(d_{q}\times\|\mathcal{K}_{a}\|)$. The complexity for value re- sampling is $\mathcal{O}(\|\mathcal{K}_{a}\|\times\|\mathcal{K}\|\times d_{1})$ which in result give us the complexity of the whole procedure as: $\mathcal{O}(d_{q}\times\|\mathcal{K}_{a}\|)+\mathcal{O}(\|\mathcal{K}_{a}\|\times\|\mathcal{K}\|\times d_{v})$ where $d_{v}$ is the dimension of the memory value vectors. ### 3.3 Classification pipeline Figure 3.3: Bottleneck [30] augmented with a memory block $\mathcal{M}$. We are augmenting various types of the classification neural networks with the memory layer defined in the sections above. ResNet [30] is the baseline architecture for most of the experiments. The first idea is to augment the Bottleneck block with the memory layer. The memory is inserted after the ($3\times 3$) kernel size convolution layer. We could also add the memory access before the middle convolution layer but we didn’t find any differences between the two methods so we just stick with the first design. We are keeping the baseline high-level architecture the same by only replacing the single Bottleneck block with the augmented version. Replacing a single layer should be enough to observe the effect of the memory, while having only a single layer with relatively low spatial size allows less carrying about the efficiency of the layer implementation. Inspired by [32] we are also adding the memory access in squeeze-and- excitation (SE) like manner. SE is a computation block that can be built upon any transformation in the network. It consists of two parts. First, Squeeze which, given the feature map, captures the global spatial information and squeeze it into the channel descriptor. Authors use global average pooling with that goal. Second, Excitation, which employs the simple gating mechanism upon the projection of squeezed vector with a sigmoid activation. The projection is the bottleneck with two fully connected layers around the non- linearity. The bottleneck layer successfully limits the complexity of the SE block by introducing the dimensionality-reduction with ratio $r=d_{in}/d_{out}$, where $d_{in}$ is the dimension size of input vector and $d_{out}$ is the dimension size of the vector after the first projection in the bottleneck. We setup nearly the same design but with three main differences, first, we are replacing only one block instead of every/several blocks in [32] (fewer SE blocks give worse final score). This reduces the number of parameters to be stored in the memory and the overall FLOPS required in the inference. Second, channel-wise feature response is fed to the memory instead of the MLP with two fully-connected (FC) layers around the non-linearity. This design helps us to tackle the issues of large spatial shapes of the query input and therefore softens the overall performance drop. Finally, instead of re-scaling the values of the feature map with the gating output, we are simply adding the embedding pixel-wise, i.e. replacing multiplication by addition operation and adding the embedding to each pixel of the feature map. The overall model of memory augmented squeeze-and-excitation block is illustrated in Figure 3.4. Another option is to simply add the memory block as an additional layer between the existing ones. This way we still have the issues with large spatial shapes, especially for the earlier layers. We are testing this design type with the ImageNet dataset [44]. Figure 3.4: The overview of the modified squeeze-and-excitation block augmented with the memory layer. $F_{GB}$ is the function of global pooling which reduces the dimension of 3 dimensional feature map to the signle dimension, and $F_{add}$ is channel-wise addition of a resulting vector from memory to the original feature tensor. ### 3.4 Regression pipeline To test the capability of the memory layer to work on regression problems, we are also experimenting with the camera relocalization problem [38]. The task is to find the camera extrinsics, i.e. the camera’s location in the real world and its direction, from the single image. Inferring the camera extrinsics is a crucial task for mobile robotics, navigation, augmented reality. Authors of the PoseNet neural network [38] construct the simple neural model which consists of the backbone network and two additional layers that map the feature representation of the input into the pose and the direction values. First, it is the regression feature generation network as a basis for the architecture. For that purpose GoogleNet [76] is used, we are replacing it with ResNet [30] to conduct our experiments. The output of the backbone is fed to the fully connected layer, which is then regressing values for direction and orientation separately. Authors of the paper suggest to parametrize the direction with quaternions because of the overall simplicity compared to the rotational matrice, i.e. advantage in size: 4 scalars vs 9 in rotation matrix and speed since quaternion multiplication is much faster compared to a matrix- vector product used for rotation matrices. Additionally, since the rotation matrices $n\times n$ are the members of $SO(n)$ [37], they have the orthonormality property that should be preserved during the optimization, which is generally the hard problem. Since quaternion, q, is identical to -q, this leads us to the issue of non- injectivity of the rotation value. To solve it authors normalize a quaternion rotation to a unit length: $\mathcal{L}_{q}(I)=\left\\|\mathbf{q}-\frac{\hat{\mathbf{q}}}{\\|\hat{\mathbf{q}}\\|}\right\\|$ For position loss, $L_{2}$ Euclidean norm is used. Introducing scaling factor $\beta$, we can balance the overall loss, by keeping expected values of two losses approximately equal. We are not trying to tune the scaling factor in our experiments since it is not the main direction of this research, but we still experiment with a large grid of hyperparameters including various values for the scaling factor. The overall objective loss function is: $\operatorname{loss}(I)=\\|\hat{\mathbf{x}}-\mathbf{x}\\|_{2}+\beta\left\\|\hat{\mathbf{q}}-\frac{\mathbf{q}}{\\|\mathbf{q}\\|}\right\\|_{2}$ We are experimenting with memory block by replacing the fully connected layer before the final regressor of feature size 2048. Since the data size (King’s College [38]) on which the experiments are conducted is relatively small, we are constraining ourselves with setting the memory size to 1k/10k values. We also regularize the memory layer by augmenting weights with Dropout (multiplicative binomial noise) but find far worse results. ### 3.5 Image reconstruction pipeline To test the memory layers further we are working with an image reconstruction problem on the Imagenet-2012 [44] dataset. Image reconstruction is the type of dimensionality reduction problem to learn the projection function that could inject the given image into the latent representation in the compact manifold (data encoding) and then generate the image from the given latent. Autoencoder is a neural approach that helps us to tackle the problem in an unsupervised fashion. In the basic design of the autoencoders, we have two networks: encoder which maps the image into the small latent code and a decoder which generates the image from the code. We are experimenting with several autoencoder designs but stick to: DCGAN [60] generator as the decoder network and the encoder as the custom 2D neural network consisting of five ResNet blocks. The image latent is the 1024 dimensional vector. The architectural choice of the augmentation is described in the section about the classification pipeline. We are using the basic method of augmentation by inserting an additional memory layer in the decoder network.111More details on the architecture will be given in the released code. We observe that the location of the memory layer is important on how the memory is utilized on the train/validation sets and the final reconstruction results. ## Chapter 4 Experiments ### 4.1 Experiments on random labels The heart of our methodology to test the memory layers and re-initialization technique is a well-known variant of randomized test in non-parametric statistics [19, 88]. Drawing motivation from [88] we have conducted several experiments to test the ability of our memory layer to fit the randomly labelled set of data points. For this reason, we have prepared a simple data set with $N$ sample points. We are regulating the number of samples to much the memory size $M$. This is because our goal was having the one-to-one correspondence between the input data and the memory values, i.e. ideally overfitted memory layer. Sample points are the vectors uniformly generated in $\mathbb{R}^{8}$ space, i.e. points in 8 dimensional cube. There are a total of $m$ classes that are uniformly chosen for each data point. We have experimented with the data set of 100k data points with 10 classes, consequently setting $\|\mathcal{M}\|$ to 100k also. Architecturally we have limited ourselves with the simplest model design with two linear projections before and after the memory layer. It is the basic architecture we could think of with no convolutional neural networks involved. Moreover, we observe that using convolutional layers allows us to fit the model to the dataset ideally. There is some research on the connection between the multi-head self-attention and convolutional layers [14], so we have tried to avoid the ambiguity and focused on the fully connected layers as the projections in our network. Also to compare our key-value structure with classic dense layers, we have replaced memory access with very wide linear layers and point-wise non- linearity, i.e. ReLU, sigmoid. As it is described in [11], wide layer networks are a powerful memorizers, though in [86] authors are able to get great memorization for small ReLU networks also, with some initialization technique for SGD [64]. So it was interesting to see how the key-value structure memorization capability can be compared with the wide dense layers. We have used two fully connected layers with the ReLU in the middle. The weight matrix of the layers are set to project the 512-dimensional vector to the $\mathbb{R}^{15k}$ space and after applying the nonlinearity, acting as the discriminant function in the feature space divided by hyperplane decision surfaces, we are projecting the vector back to the space $\mathbb{R}^{512}$. This network of two projections and the nonlinearity in the middle is the approximation of our memory layer. This is because the k-nn function also acts as the discriminator function, more on this in [24] (Chapter 12). We have trained our models with an Adam optimizer [42], with an initial learning rate of $10^{-3}$, and $\beta_{1}=0.9$, $\beta_{2}=0.98$. The models were implemented in Pytorch [57]. For the memory values we have chosen the SparseEmbedding structure which calculates and stores only the sparse gradients in the backward. We have chosen the SparseAdam (implemented in Pytorch) to update the sparse gradients in the memory layer. Because of the sparse updates in the memory, we have multiplied the learning rate for the sparse parameters by the factor of 10. For key parameter update, we have used the same optimizer as for the dense parameters. Due to the usage of re- initialization trick and Adam optimizer which stores the values of past gradients and square gradients, these values should also be dynamically updated. The results for the models with memory blocks and wide dense layers compared in Figure 4.1. Figure 4.1: Results for the random data on the various models. The left graph is the plot for top-1 validation accuracy results, top-5 is pictured on the right. We can see that the model with no memory is not able to fit the data. Setting 8 heads for multi-head attention memory model, on the other hand, helps the model to easily fit the data. Re-initialization helps to get nearly full convergence with on only 1 head, i.e. multi-head mode disabled. Also, $h=1$ in the graph is the notation for the single head model. In our experiments, we varied the hyper-parameters of the memory model, such as memory size, number of heads, k parameter in top-k operator, etc. We provide the results only for {$k=10$, $N=M$} hyperparameter set with different values for the number of heads $h$ and the re-initialization trick enabled/disabled since other combinations contain no interest in these experiments. We observe that setting the number of heads to 8 gives us perfect fit to the data, i.e. full top-1 validation accuracy. As it is shown in Figure 4.1, replacing the memory layer with wide dense layer doesn’t help us with the accuracy. Lowering the number of heads, we see the declining accuracy in the validation. We speculate that this is caused by the poor initialization due to which the pair of the same keys could be selected for the two very close query vectors. Using uniform initialization to maximize the key coverage at the initial state of the model didn’t help us to resolve the issue as we have observed that the utilization of keys converged to some small subset. To overcome the problem, we have experimented with re-initialization trick that was introduced in the chapter above. As it is seen from Figure 4.1, re- initialization helps us to get nearly ideal validation accuracy even with a single head. We are setting $\epsilon_{d}=10^{-6}$ to get the results above. We haven’t experimented much with the special scheduler for the re- initialization trick, but early experiments showed that the frequency with which the re-initialization procedure is called and the number of added keys for each call can have the significant influence on the final accuracy we get. More experiments are required in this direction. We have conducted some additional experiments to see how the variance $\sigma_{d}$ parameter of the additive noise added to the re-initialized keys and the memory values affect the final accuracy. We are giving the results in in Figure 4.2 Figure 4.2: Relation between $\sigma_{d}$ hyper parameter of the Gaussian noise square variance parameter in re-initialization procedure and the values of train accuracy and loss for the random data. Setting higher $\sigma_{d}$ helps significantly, giving lower loss. #### 4.1.1 Results on CIFAR-10 We have implemented several architectural ideas to test the performance of memory augmented models on CIFAR-10 [43]. The first idea was to augment the bottleneck blocks [30] with the memory layer and replace the single bottleneck in the network with the modified block. We have also experimented with replacing multiple bottlenecks blocks but didn’t find anything reasonable to stick with it in the experiments because of the overall increased inference time we have observed. The logic behind the bottleneck augmentation is given in the chapter above. Here we describe the architectural choices we’ve made to incorporate the augmented bottleneck in the most optimal way possible taking into consideration the inference time and the final validation accuracy. The real hurdle during the experiments was the speed issues of the inference. It didn’t allow us to set up experiments with more broader set of models because of the time limitations and the general difficulty of running large grids rapidly for slower models. We were able to partly mitigate the issues by using a lower spatial size of the query input. Taking all this into the consideration, we have chosen the last layer to be augmented with the memory layer as it gave us the smallest spatial size possible in the ResNet-type network. We have abandoned the experiments with larger spatial sizes in classification experiments for CIFAR-10 since the balance between the performance and the accuracy wasn’t reasonable. But we still have conducted experiments with larger spatial sizes with the autoregressive models, the results are available in the sections below. We have chosen the ResNet-50 [30] to be the baseline network for the experimental models. The baseline consists of two projections and 16 Bottleneck blocks in the middle. We have added the memory layer in the 14th Bottleneck block and have illustrate the results in Figure 4.3. The training loop design described in [30] have been implemented with the SGD [65] optimizer, learning rate of $10^{-1}$ weight decay of $0.0001$ and momentum equal to $0.9$. Since the model contained the sparse parameters, we weren’t able to use the standard implementation of the SGD optimizer in PyTorch [57]. For that, we have implemented the SparseSGD optimizer with disabled weight decay. As for the momentum, to our knowledge, there is no mathematical ground of using it to accelerate sparse gradients, but we have still set it to 0.9 in all of our experiments. More information on the sparse SGD can be found here [17]. We have adopted the weight initialization as in [29] and the batch normalization (BN) [33]. The augmentation is the same as in [30] with 4-pixel padding on each side of the image and 32$\times$ 32 crop randomly sampled from the padded image and random horizontal flip. All of the experiments have been conducted with the memory size of 100k, top-k operator parameter k of 30 and no dropout on the retrieved indices applied. Setting the memory size to 100k we have two sets of product keys of size 100, $\|\mathcal{K}_{1}\|=\|\mathcal{K}_{2}\|=100$. Figure 4.3: Comparison of the top-1 scores, memory utilization and the inference speed for the model with no memory augmentation and three memory augmented models with the number of heads in the set of {1, 4, 8}. There are 20 runs for the each experiment with the different initial seeds. As it can be observed, there is no evident increase in the accuracy while the performance of the models with the heads is much worse compared to the baseline models. Inference values are calculated on GTX-1080Ti cards with fp32 mode, the results are approximate. We have calculated the distributions of top-1 scores for 20 runs with different seed values. As it is seen in the Figure 4.3 we weren’t able to gain any improvements in the accuracy scores with the memory augmentation, while the performance of the memory model, i.e. iterations per second in the train, decreases significantly and continues to decrease with the higher number of heads. We have also calculated the distributions of the memory utilization and observed that for larger heads we see the increase in the overall utilization. These findings mirror the results in [46]. Evaluation metrics for memory layer. As the simple evaluation metric of how well the memory is being utilized during the training phase, we have calculated the memory usage score which represents the fraction of accessed values $\\#\\{\mathcal{C}_{i}\neq 0\\}$, where $\mathcal{C}_{i}\in\mathcal{C}$ is the number of the times the key $\mathcal{K}_{i}$ is accessed summed for the whole validation set. Authors in [46] also use Kullback-Leibler (KL) [68] divergence between the distribution of summed values of the softmax results for the whole validation dataset with a uniform distribution. We have implemented the KL divergence metric in our experiments and found it giving more accurate numbers with the small changes of the real memory utilization. But in the given results here we have constrained our experiments to the first evaluation metric because of its simplicity and the numerical interpretability. So as we can see in Figure 4.3 the utilization of the memory is increasing with a larger number of heads. These findings were consistent during all the experiments with the classification networks. As the results failed on the BottleNeck blocks, we have changed the focus to other architectures. Since we had the problem with the performance due to the large spatial size, we have decided to limit ourselves with the image of spatial size $1\times 1$ as the input query for the memory layer. Therefore in the next experiments, we have leverage the architectural design of Squeeze- and-Excitation [32] with some changes that were described in the chapter above. For the experiments with the modified SE blocks, we have chosen the Resnet-20 as the baseline network. We have kept the training pipeline the same but modified the scheduler replacing it with the ReduceLRonPlateau111For eg. Pytorch implementation of ReduceLRonPlateau with the reduction factor of $0.1$. All the experiments with the memory layer enabled have been run with a memory size of 100k, top-k k parameter of 30 and no dropout on the selected indices. We have listed the most interesting results in the Table 4.1 \| accuracy, top1 \| inference, ms \| FLOPs \| utilization, % ---\|---\|---\|---\|--- ResNet-20 \| 92.73(92.46$\pm$0.15) \| 6.7ms \| $\sim$40.92M \| - SE-ResNet-20 \| 93.31(93.16$\pm$0.13) \| 7.4ms \| $\sim$41.49M \| - ResNet-110 \| 93.63(93.41$\pm$0.18) \| 25ms \| $\sim$254.98M \| - ResNet-20+WL, $d_{w}=15k$ \| 92.23 \| 7.2ms \| $\sim$43M \| - ResNet-20+Memory, scalar, h=8 \| 92.42 \| 19.5ms \| $\sim$41.05M \| 1-2% ResNet-20+Memory, cosine, $\alpha=0.9$, h=8 \| 92.45 \| 19.5ms \| $\sim$41.05M \| 1-2% ResNet-20+Memory/RI, scalar, h=8 \| 93.16 \| 19.5ms \| $\sim$41.05M \| 60-75% ResNet-20+Memory/RI, cosine, $\alpha=0.9$, h=4 \| 93.02 \| 12ms \| $\sim$41.05M \| 40-50% ResNet-20+Memory/RI, cosine, $\alpha=0.9$, h=8 \| 93.51(93.34$\pm$0.14) \| 19.5ms \| $\sim$41.05M \| 60-75% Table 4.1: Results for the modified SE blocks. WL is the table notation for the wide linear layer that replaces the memory layer, $d_{w}$ defines respectably the row and column of the projections matrices in MLP (with the row vector in the linear operator). Overall we see better results on Resnet-20 with the memory layer and with re-initialization trick we have superior memory utilization rate. Cosine similarity helps us to nearly reach the accuracy values of ResNet-100. Inference values are calculated on GTX-1080Ti cards with fp32 mode, the results are approximate. As we can see the re-initialization trick helps us with the utilization of the memory which in turn gives us better top-1 accuracy overall. We have also compared the memory block with the very wide MLP that consists of two large projections matrices and the pointwise nonlinearity in the middle. We are setting the row/column of two matrices to $d_{k}=15\text{k}$, meaning that we have two linear operators $W_{1}\in\mathbb{R}^{d_{k}\times d_{in}}$ and $W_{2}\in\mathbb{R}^{d_{in}\times d_{k}}$ that map the input vector $v\in\mathbb{R}^{d_{in}}$ to the $d_{k}$ dimensional vector, applies ReLU pointwise and project back to the vector $v^{\prime}\in\mathbb{R}^{d_{in}}$. We can see in Table 4.1 that adding the large MLP doesn’t affect the performance at the level compared to the memory layers. It is because the GPUs can easily parallelize the matrix multiplication while stumbling with the operations that require random access to the main memory[36] . We see this as the fundamental problem of the approach with the sparse memories. We haven’t conducted experiments with augmenting the ResNet-110 network with a memory layer because the goal of these experiments was to understand how the memory layer can help us with the very small networks to bit results of large ones. And since the inference speed of the small models was inferior compared to ResNet and SE-ResNet blocks we have changed our focus to different applications. But more experiments should be conducted to determine whenever ResNet-100+M results compares to the results of SE-ResNet-100 both in the final prediction scores and the performance. Analysis of memory layer. To find how good the introduced memory can generalize to the given images and overall get the better picture on how the properties of a convolutional neural network, e.g shift-invariance, are maintained with the memory augmentation, we have conducted more experiments in which we have randomly cropped the small region (4$\times$4) of a sample image from the validation and compared the accessed keys for the cropped and the original image. We see in Figure (4.4) that the small perturbation of the input data has insignificant affect on how the keys selected. Therefore we could assume that the generalization properties of the memory networks are maintained that could be crucial in other applications, e.g. pose regression on smaller datasets for which we have conducted additional experiments. Figure 4.4: We picture the key access distribution for each head. As we see there is a little perturbation in the distribution when we apply the random region crop in the input image. #### 4.1.2 Experiments on ImageNet We have conducted further experiments on the ImageNet-2012 dataset [67], assuming that the large size of the train set of the ImageNet would be more natural fit for our memory layer. The only issue was time limitations we had and the hard task of tuning the optimizer for the memory layer. Since it takes 90 epochs for the ImageNet to finish training with Resnet-50 and on NVIDIA M40 GPU it takes 14 days [85], the experiments with the size of 224$\times$224 weren’t reasonable. And since we have decided to increase the spatial size of the query input in the memory, the inference performance of the models plummeted. Therefore we have decided to resize the sizes of the images in train and validation to 64$\times$64 and run the pipeline. We kept the same augmentation pipeline as in [30]. First steps were to run the ResNet-50 augmented with the memory layer and compare it with the baseline results. The augmentation logic we have chosen for the ImageNet experiments were simpler. We have inserted the additional layer before the 44th layer of the network, where the image has the 7$\times$7 spatial size, this meant that the queries consisted of 49 feature vectors that are batched together to be fed to the memory layer. The memory size of the experiments was set to 256k, we have looked at the top-30 indices during the memory access and the batch size was set to 256. As the distance metric, we have chosen the cosine similarity with $\alpha=1$. We haven’t used dropout on the retrieved indices. SGD [65] was chosen as the optimizer with the initial learning rate of $10^{-1}$, weight decay for dense parameters of 0.0001 and momentum of 0.9. We haven’t set the weight decay for memory parameters because of the inferior results, more experiments should be conducted to find the reason for this. The results are given in 4.2. \| accuracy, top1 \| utilization \| FLOPs \| inference, ms ---\|---\|---\|---\|--- ResNet-50, 64$\times$64 \| 61.45 (61.24$\pm$0.12) \| - \| $\sim$338M \| 11ms ResNet-50+M, 64$\times$64, skip=True,heads=8 \| 61.61 \| $\sim$98% \| $\sim$382M \| 23ms ResNet-50+M, 64$\times$64, skip=True,heads=8, $\text{lr}_{key}$=$10^{2}$ \| 61.82 (61.59$\pm$0.17) \| $\sim$96% \| $\sim$382M \| 23ms ResNet-50+M, 64$\times$64, skip=True,heads=8, $\text{lr}_{key}$=$10^{3}$ \| 60.85 \| $\sim$86% \| $\sim$382M \| 23ms ResNet-50+M, 64$\times$64, skip=False,heads=8 \| 54.12 \| $\sim$15% \| $\sim$382M \| 23ms ResNet-50, 32$\times$32, skip=True \| 42.81 \| - \| $\sim$86.1M \| 6ms ResNet-50+M, 32$\times$32, skip=True,heads=8 \| 43.49 \| $\sim$98% \| $\sim$100M \| 12ms ResNet-50+M, 32$\times$32, skip=False,heads=8 \| 35.68 \| $\sim$18% \| $\sim$100M \| 12ms Table 4.2: The results on the Resnet-50 and Imagenet. We have tested a number of the hyper parameters to find the best train strategy for the memory models. For now we dont see the clear picture on optimization issues. We can see from the table that there is a small increase in the validation accuracy for the models augmented with the memory layer but the large drop in the performance (inference in the table). This is not a reasonable way of incorporating the memories with the classification models and that is why we have tried to analyze how the values in the memory were used in the inference and how did they change during the training phase. We hoped to find a way of increasing the accuracy of memory augmented models by tweaking the training pipeline. For that we have logged the gradients of the keys, memory values, memory utilization and standard deviation of the keys during the training phase. We have observed that for the activated residual connection on the memory layer, skip=True in Figure 4.5, gradients were overall higher both for memory and key values. The utilization of the skip=True was way higher reaching almost 100%, while the skip=False run plummeted to nearly 20%. What is most interesting is that the standard deviations of the keys in skip=True were not even during all training iterations. Our first assumption of the reason for this phenomena was the low learning keys for the key parameters. Further experiments are needed to tune the learning rate parameters. As a first step, we have conducted more experiments with super-convergence [71] to find the top value learning rate for key parameters in a single cycle train. We have observed that the super-convergence leaarning rate schedule reaches $10^{6}$ before the overall loss starts to increase. Figure 4.5: Change in values of the standard deviation of keys, memory usage, gradient of memory values, and key gradients during a training phase. We are not aware of all the underlining issues that do not allow us to get the learning rate in a reasonable range. Maybe setting the learning rate value to $10^{6}$ is logical too, but for now, we don’t know that yet. Also, we require the augmented models to get a way better final accuracy results taking into the consideration the performance issues of the memory blocks and the amount of the additional parameters introduced into the network, i.e. $\|\mathcal{M}\|\cdot(d_{q}+d_{v})$ where $d_{v}$ is the dimension of values in the memory. Because of this we stop our analysis here and acknowledge the need for more experiments. ### 4.2 Memory in PoseNet We have conducted some experiments on PoseNet [38] for 6-DOF camera relocalization. As it was mentioned in the previous chapter, authors of the paper have used the GoogleNet [76] the backbone for feature extraction. We have replaced it in a favour of ResNet-34 and run several models tuning the hyperparameter set, especially the scaling factor $\beta$. The Adam was the optimizer choice in these experiments. We have set the initial learning rate of $10^{-3}$ which decreases every 80 epoch. The weight decay for the dense parameters, i.e. all the parameters except the key and value vectors in the memory layer, was set to $2\times 10^{-4}$. In all the results listed, we set weight decay of memory layer to zero, Setting higher values for the weight decay was the plan of our initial experiments also, as we have hoped to provide some regularization for the values in the memory, but even the smallest weight decay failed to give us any reasonable results. We acknowledge that the additional work should be done here to find the reason behind this issue. We have set the memory size to 1k/10k and compared the results. Overall we have trained the models to the 250th epoch and have observed the plateau in the train loss. We have initiated the experiments on the King’s College outdoor relocalization [38] which is the dataset with the images from Cambridge landmark. There are overall 1220 images in the train set and nearly 350 in the validation set. The small train set size has discouraged us to apply larger memory sizes $\|\mathcal{M}\|$. Since the validation set is relatively large, we have assumed that the validation accuracy could give us an overview of how good the memory layers generalize to the dataset. For the augmentation part, we have resized the images to 256$\times$256 and applied a random crop of 224$\times$224, the same set of transformations have been done in the validation. We have set the batch size of the train set to 75 and run the experiments, the results are listed in Table 4.3. \| validation \| train loss \| inference \| FLOPs \| utiliazation ---\|---\|---\|---\|---\|--- $L_{r}(q_{1},q_{2})=\left\\|\mathbf{q}_{1}-\mathbf{q}_{2}\right\\|,q^{0}_{i}\in\mathbb{R}^{+}$ \| \| \| \| \| PN (PoseNet), $\beta=1$ \| 3.02m, 3.17∘ \| 37.5 \| 5.6ms \| $\sim$3B \| - PN, $\beta=0.1$ \| 1.57m, 5.73∘ \| 6.57 \| 5.6ms \| $\sim$3B \| - PN+LM, $\beta=1$ \| 3.24m, 3.43∘ \| 34.5 \| 5.8ms \| $\sim$(3B+10M) \| - PN+M, heads=1, $\beta=1$, $\|\mathcal{M}\|=1k$ \| 3.47m, 3.49∘ \| 10.81 \| 6ms \| $\sim$(3B+16k) \| 20-30% PN+M, heads=16, $\beta=1$, $\|\mathcal{M}\|=1k$, dp=0.3 \| 4.43m, 7.23∘ \| 14.27 \| 13ms \| $\sim$(3B+16k) \| 95-100% PN+M, heads=16, $\beta=1$, $\|\mathcal{M}\|=1k$ \| 2.19m, 2.99∘ \| 7.94 \| 13ms \| $\sim$(3B+16k) \| 95-100% PN+M, heads=16, $\beta=0.1$, $\|\mathcal{M}\|=1k$ \| 1.47m, 5.02∘ \| 1.76 \| 13ms \| $\sim$(3B+16k) \| 95-100% PN+M, heads=16, $\beta=0.1$, $\|\mathcal{M}\|=10k$ \| 1.44m, 4.98∘ \| 1.68 \| 15ms \| $\sim$(3B+50k) \| 50-55% $L_{r}(q_{1},q_{2})=\min\left\\{\left\\|\mathbf{q}_{1}-\mathbf{q}_{2}\right\\|,\left\\|\mathbf{q}_{1}+\mathbf{q}_{2}\right\\|\right\\}$ \| \| \| \| \| PN (PoseNet), $\beta=1$ \| 2.86m, 3.42∘ \| 33.1 \| 5.6ms \| $\sim$3B \| - PN, $\beta=0.1$ \| 1.59m, 5.36∘ \| 6.45 \| 5.6ms \| $\sim$3B \| - PN+M, heads=16, $\beta=1$, $\|\mathcal{M}\|=1k$ \| 1.87m, 2.53∘ \| 7.41 \| 13ms \| $\sim$(3B+16k) \| 80-95% PN+M, heads=16, $\beta=0.1$, $\|\mathcal{M}\|=1k$ \| 1.42m, 4.38∘ \| 1.63 \| 13ms \| $\sim$(3B+16k) \| 80-95% Table 4.3: Results on the King’s College dataset, $\mathrm{trans}$ and $\mathrm{rot}$ are the translation and rotation errors respectively. We see a huge decrease in the loss on the train set for the networks with memory augmentation but also the increase in the inference time. Inference values calculated on the GTX-1080Ti with the batch size of 1. We have compared the memory networks with the wide MLP layer that is defined as LM in the table. As in the classification experiments, the MLP layer consists of the two projection matrices and the nonlinearity between. First projections matrix maps the input vector $v_{in}\in\mathbb{R}^{d_{in}}$ to the $\mathbb{R}^{2k}$ then applying ReLU on the result we project the vector back to $\mathbb{R}^{d_{in}}$. We are using the residual on the MLP. As it can be seen from the table replacing the memory with MLP increases the train and the validation results both for rotation and positional losses. For now, we don’t understand why the replacement of the memory layer with the MLP can’t compete in the final score with the memory block augmentation. Overall we have seen the huge decrease in the train loss for memory models, while the validation loss, though decreased both for rotation and position loss, didn’t give us as a steep decrease in the value as we have expected. We assume that the more elaborate regularization technique could be applied here. But for now we have applied the naive dropout regularization on retrieved keys which didn’t give us any promising results (dp=0.3 for dropout rate in the Table 4.3). Though getting better numbers overall we are seeing the huge inference time increase for all the models augmented with the memory. #### 4.2.1 Analysis of the memory layer Figure 4.6: The camera path and the four sample images captures in the given coordinates. We see the correlation between the traveled distance and the indices selected. To get a better picture of how the memory layer is utilized in regressing the position coordinates, we have plotted the distribution of the accessed keys for each image. We have scattered 200 first images in the validation set by their x,y coordinates. We have set the colour for each point ranging it from 0 to $\|\mathcal{M}\|$. To calculate the colour for a particular image we have gathered the key indices that were used in the forward operation, averaged and rounded to the nearest integer. The results are given in Figure 4.6. We see that there is a correlation between the distance passed by the camera and the colours of the point, as they get darker with more distance, i.e. use lower key indices. We could assume that the memory can capture the spatial differences between the images and interpret them in the right manner. ### 4.3 Image reconstruction experiments We have conducted several experiments to test the performance of the memory layers in the reconstruction tasks. For that, we have constructed a naive encoder-decoder neural network with the memory augmentation in the decoder. The overall overview of the architecture is described above. We have experiment with the various types of memory placement: right after the latent vector (mem_idx=0), after the first layer in the decoder (mem_idx=1) and so on. We have used the Adam [42] optimizer with the initial learning rate of $10^{-3}$ and $\beta_{1}=0.9$, $\beta_{2}=0.98$. ImageNet samples were resized to 64 $\times$ 64 before training the model. We have chosen $L_{2}$ norm as the objective. Also, we used the memory size of 100k, k parameter top-k procedure of 30 and disabled dropout. The results are given in the Table 4.4. \| train loss \| validation loss \| inference, ms \| utilization, % ---\|---\|---\|---\|--- Baseline \| 0.092 \| 0.0076 \| 4.3ms \| - Baseline+M, heads=1, mem_idx=0 \| 0.0773 \| 0.0056 \| 4.3ms \| 0-10 Baseline+M, heads=4, mem_idx=0 \| 0.075 \| 0.0045 \| 5ms \| 10-20 Baseline+M, heads=8, mem_idx=0 \| 0.0734 \| 0.0044 \| 6.3ms \| 20-40 Baseline+M, heads=1, mem_idx=1 \| 0.0721 \| 0.0042 \| 12.1ms \| 40-50 Baseline+M, heads=4, mem_idx=1 \| 0.0693 \| 0.0037 \| 15.4ms \| 50-60 Baseline+M, heads=8, mem_idx=1 \| 0.0675 \| 0.0035 \| 17.7ms \| 60-65 Baseline+M, heads=1, mem_idx=2 \| 0.0628 \| 0.0026 \| 21.4ms \| 85-90 Baseline+M, heads=4, mem_idx=2 \| 0.0544 \| 0.0019 \| 26.7ms \| 90-100 Table 4.4: Reconstruction results after 1600 iterations. The memory utilization numbers are approximate since the amount of experiments conducted was low. The inference speed is calculated on GTX-1080Ti with batch size of 1. We see the steady decline in the train and validation losses with increasing the number of heads and the index of the layer of the decoder where the memory is being inserted, mem_idx. Utilization numbers increase which again supports the experiments we have conducted before. As the inference time giving us the degraded performance with mem_idx=2. This didn’t allow us to conduct more experiments with large spatial shapes of the input images. We include some reconstruction examples from the validation set in Figure 4.7. The overall pipeline and the more details on the final architecture will be given in the released code. For now, it is important to get an understating of the overall reconstruction improvements with the memory augmentation and if it is reasonable to be used with the performance issues in mind. Figure 4.7: Reconstruction results. Top row images in each section are the reconstruction results with the memory augmented autoencoder with mem_idx=2 and heads=8, the middle are the output of the baseline autoencoder while the final row are the input images. We see that there are some little details that are captured using the memory layer. ### 4.4 Other experiments We have also applied other experiments with memory usage in distillation [31], implicit shape modelling [10] and NERF (Representing Scenes as Neural Radiance Fields for View Synthesis) [52]. Overall, for now we can conclude that the large batch sizes of these models’ training pipelines, i.e. point coordinate samples for implicit modeling and the sampled rays for view synthesis in NERF, is the hurdle which won’t allow the memory to be used in the most efficient way, because of the difficulty of the random access parallelization with modern GPUs. Though we see some potential in knowledge distillation from very large models and more work should be conducted in this direction. ## Chapter 5 Conclusion This work analyzes the usage of product key-value memory in computer vision applications with a deep dive into the problems like image classification, regression (in the context of the camera relocalization, and the image reconstruction). We have found that for some of the problems the product key memory layer was able to provide significant speed/accuracy improvements with the high utilization of the key-value elements, while others require more careful fine-tuning and an efficient regularization strategy. We also find that the ”dying keys” affect the image classification problems. To help us tackle with it, we introduce a simple technique of memory re-initialization which helps us to eliminate ”unused” key-value pairs from the memory and cleverly re-initialize new keys that, with high probability, will be used in next iterations of training. We show that the re-initialization has a huge impact on a toy example of randomly labelled data and observe some gains in performance on the image classification tasks. In addition to the promising results in the experiments with the camera relocalization, we have also shown that the choice of the set of memory accessed indices in the inference depends on the spatial correlations of the input images. This signals us about the perseverance of the generalization property of the memory layer with no additional regularization required. Still, validation results didn’t meet our expectations and at this point, we could only assume that more work is required in defining more elaborate regularization strategies. 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# Hadamard Extensions and the Identification of Mixtures of Product Distributions Spencer L. Gordon Engineering and Applied Science, California Institute of Technology<EMAIL_ADDRESS>Leonard J. Schulman Engineering and Applied Science, California Institute of Technology<EMAIL_ADDRESS>Research supported in part by NSF grant CCF-1909972. ###### Abstract The Hadamard Extension of a matrix is the matrix consisting of all Hadamard products of subsets of its rows. This construction arises in the context of identifying a mixture of product distributions on binary random variables: full column rank of such extensions is a necessary ingredient of identification algorithms. We provide several results concerning when a Hadamard Extension has full column rank. ## 1 Introduction The Hadamard product for row vectors $u=(u_{1},\ldots,u_{k})$, $v=(v_{1},\ldots,v_{k})$ is the mapping $\odot:\mathbb{R}^{k}\times\mathbb{R}^{k}\to\mathbb{R}^{k}$ given by $\displaystyle u\odot v$ $\displaystyle:=(u_{1}v_{1},\ldots,u_{k}v_{k})$ The identity for this product is the all-ones vector $\mathbbm{1}$. We associate with vector $v$ the linear operator $v_{\odot}=\operatorname{diag}(v)$, a $k\times k$ diagonal matrix, so that $u\cdot v_{\odot}=v\odot u.$ Throughout this paper ${\mathbf{m}}$ is a real matrix with row set $[n]:=\\{1,\ldots,n\\}$ and column set $[k]$; write ${\mathbf{m}}_{i}$ for a row and ${\mathbf{m}}^{j}$ for a column. As a matter of notation, for a matrix $Q$ and nonempty sets $R$ of rows and $C$ of columns, let $Q\|_{R}^{C}$ be the restriction of $Q$ to those columns and rows (with either index omitted if all rows or columns are retained). ###### Definition 1. The Hadamard Extension of ${\mathbf{m}}$, written $\mathbb{H}({\mathbf{m}})$, is the $2^{n}\times k$ matrix with rows ${\mathbf{m}}_{S}$ for all $S\subseteq[n]$, where, for $S=\\{i_{1},\ldots,i_{\ell}\\}$, ${\mathbf{m}}_{S}={\mathbf{m}}_{i_{1}}\odot\cdots\odot{\mathbf{m}}_{i_{\ell}}$; equivalently ${\mathbf{m}}_{S}^{j}=\prod_{i\in S}{\mathbf{m}}_{i}^{j}$. (In particular ${\mathbf{m}}_{\emptyset}=\mathbbm{1}$.) This construction has arisen recently in learning theory [3, 8] where it is essential to source identification for a mixture of product distributions on binary random variables. We explain the connection further in Section 5. Motivated by this application, we are interested in the following two questions: (1) If $\mathbb{H}({\mathbf{m}})$ has full column rank, must there exist a subset $R$ of the rows, of bounded size, such that $\mathbb{H}({\mathbf{m}}\|_{R})$ has full column rank? (2) In each row of ${\mathbf{m}}$, assign distinct colors to the distinct real values. Is there a condition on the coloring that ensures $\mathbb{H}({\mathbf{m}})$ has full column rank? In answer to the first question we show in Section 2: ###### Theorem 2. If $\mathbb{H}({\mathbf{m}})$ has full column rank then there is a set $R$ of no more than $k-1$ of the rows of ${\mathbf{m}}$, such that $\mathbb{H}({\mathbf{m}}\|_{R})$ has full column rank. Considering the more combinatorial second question, observe that if ${\mathbf{m}}$ possesses two identical columns then the same is true of $\mathbb{H}({\mathbf{m}})$, and so it cannot be full rank. Extending this further, suppose there are three columns $C$ in which only one row $r$ has more than one color. Then $\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|^{C})$ is spanned by $\mathbbm{1}\|^{C}$ and $r\|^{C}$, so again $\mathbb{H}({\mathbf{m}})$ cannot be full rank. Motivated by these necessary conditions, set: ###### Definition 3. For a matrix $Q$ let $\text{NAE}(Q)$ be the set of nonconstant rows of $Q$ (NAE=“not all equal”); let ${\varepsilon}(Q\|^{C})=\|\text{NAE}(Q\|^{C})\|-\|C\|$; and let $\overline{{\varepsilon}}(Q)=\min_{C\neq\emptyset}{\varepsilon}(Q\|^{C})$. If $\overline{{\varepsilon}}(Q)\geq-1$ we say $Q$ satisfies the NAE condition. In answer to the second question we have the following: ###### Theorem 4. If ${\mathbf{m}}$ satisfies the NAE condition then 1. (a) There is a restriction of ${\mathbf{m}}$ to some $k-1$ rows $R$ such that $\overline{{\varepsilon}}({\mathbf{m}}\|_{R})=-1$. 2. (b) $\mathbb{H}({\mathbf{m}})$ is full column rank. (As a consequence also $\mathbb{H}({\mathbf{m}}\|_{R})$ is full column rank.) Apparently the only well-known example of the NAE condition is when ${\mathbf{m}}$ contains $k-1$ rows which are identical and whose entries are all distinct. Then the vectors ${\mathbf{m}}_{\emptyset},{\mathbf{m}}_{\\{1\\}},{\mathbf{m}}_{\\{1,2\\}},\ldots,{\mathbf{m}}_{\\{1,\ldots,k-1\\}}$ form a nonsingular Vandermonde matrix. This example shows that the bound of $k-1$ in (a) is best possible. For another example in which the NAE condition ensures that $\operatorname{rank}\mathbb{H}({\mathbf{m}})=k$, take the $(k-1)$-row matrix with ${\mathbf{m}}_{i}^{j}=1$ for $i\leq j$ and ${\mathbf{m}}_{i}^{j}=1/2$ for $i>j$. Here the NAE condition is only minimally satisfied, in that for every $\ell\leq k$ there are $\ell$ columns $C$ s.t. ${\varepsilon}({\mathbf{m}}\|^{C})=-1$. For $k>3$ the NAE condition is no longer necessary for $\mathbb{H}({\mathbf{m}})$ to have full column rank. E.g., for $k=2^{\ell}$, the $\ell\times k$ “Hamming matrix” ${\mathbf{m}}_{i}^{j}=(-1)^{j_{i}}$ where $j$ is an $\ell$-bit string $j=(j_{1},\ldots,j_{\ell})$, forms $\mathbb{H}({\mathbf{m}})=$ the Fourier transform for the group $(\mathbb{Z}/2)^{\ell}$ (often called a Hadamard matrix), which is invertible. Furthermore, almost all (in the sense of Lebesgue measure) $\lceil\lg k\rceil\times k$ matrices ${\mathbf{m}}$ form a full-rank $\mathbb{H}({\mathbf{m}})$. (This is because $\det\mathbb{H}({\mathbf{m}})$ is a polynomial in the entries of ${\mathbf{m}}$, and the previous example shows the polynomial is nonzero.) Despite this observation, the Vandermonde case, in which $k-1$ rows are required, is very typical, as it is what arises in $\mathbb{H}({\mathbf{m}})$ for a mixture model of observables $X_{i}$ that are iid conditional on a hidden variable. ## 2 Some Theory for Hadamard Products, and a Proof of Theorem 2 For $v\in\mathbb{R}^{k}$ and $U$ a subspace, extend the definition $v_{\odot}$ to $v_{\odot}(U)=\\{u\cdot v_{\odot}:u\in U\\}$ and introduce the notation $v_{\bar{\odot}}(U)=\operatorname{span}\\{U\cup v_{\odot}(U)\\}.$ We want to understand which subspaces $U$ are invariant under $v_{\bar{\odot}}$. Let $v$ have distinct values $\lambda_{1}>\ldots>\lambda_{\ell}$ for $\ell\leq k$. Let the polynomials $p_{v,i}$ ($i=1,\ldots,\ell$) of degree $\ell-1$ be the Lagrange interpolation polynomials for these values, so $p_{v,i}(\lambda_{j})=\delta_{ij}$ (Kronecker delta). Let $B(v)$ denote the partition of $[k]$ into blocks $B(v)_{(i)}=\\{j:v_{j}=\lambda_{i}\\}$. Let $V_{(i)}$ be the space spanned by the elementary basis vectors in $B(v)_{(i)}$, and $P_{(i)}$ the projection onto $V_{(i)}$ w.r.t. standard inner product. We have the matrix equation $p_{v,i}(v_{\odot})=P_{(i)}.$ The collection of all linear combinations of the matrices $P_{(i)}$ is a commutative algebra, the _$B(v)$ projection algebra_, which we denote $A_{B(v)}$. The identity of the algebra is $I=\sum P_{(i)}$. ###### Definition 5. A subspace of $\mathbb{R}^{k}$ respects $B(v)$ if it is spanned by vectors each of which lies in some $V_{(i)}$. For $U$ respecting $B(v)$ write $U=\operatorname{span}(\bigcup U_{(i)})$ for $U_{(i)}\subseteq V_{(i)}$. Let $D_{(i)}=(U_{(i)})^{\perp}\cap V_{(i)}$. Then $(U_{(i)})^{\perp}=D_{(i)}\oplus\bigoplus_{j\neq i}V_{(j)}$. ###### Lemma 6. Subspace $U^{\perp}$ respects $B(v)$ if $U$ does. ###### Proof. In general, $(\operatorname{span}(W\cup W^{\prime}))^{\perp}=W^{\perp}\cap W^{\prime\perp}$. So $U^{\perp}=\bigcap(U_{(i)})^{\perp}=\bigoplus D_{(i)}$. ∎ ###### Lemma 7. Subspace $U$ respects $B(v)$ iff $U=\bigoplus(P_{(i)}U)$. ###### Proof. ($\Leftarrow$): Because this gives an explicit representation of $U$ as a direct sum of subspaces each restricted to some $V_{(i)}$. ($\Rightarrow$): By definition $U$ is spanned by some collection of subspaces $V^{\prime}_{(i)}\subseteq V_{(i)}$; since these subspaces are necessarily orthogonal, $U=\bigoplus V^{\prime}_{(i)}$. Moreover, since $P_{(i)}$ annihilates $V_{(j)},j\neq i$, and is the identity on $V_{(i)}$, it follows that each $V^{\prime}_{(i)}=P_{(i)}U$. ∎ ###### Theorem 8. Subspace $U$ is invariant under $v_{\bar{\odot}}$ iff $U$ respects $B(v)$. ###### Proof. ($\Leftarrow$): It suffices to show $U^{\perp}$ is invariant under $v_{\bar{\odot}}$. By the previous lemma, it is equivalent to suppose that $U^{\perp}$ respects $B(v)$. So let $d\in U^{\perp}$ and write $d=\sum d_{i},d_{i}\in D_{(i)}$. Then $v\odot d_{i}=\lambda_{i}d_{i}\in D_{(i)}$. So $v\odot d=\sum v\odot d_{i}\in\bigoplus D_{(i)}=U^{\perp}$. ($\Rightarrow$): If $U=v_{\bar{\odot}}(U)$ then these also equal $v_{\bar{\odot}}(v_{\bar{\odot}}(U))$, etc., so $U$ is an invariant space of $A_{B(v)}$, meaning, $aU\subseteq U$ for any $a\in A_{B(v)}$. In particular for $a=P_{(i)}$. So $U\supseteq\bigoplus(P_{(i)}U)$. On the other hand, since $\sum P_{(i)}=I$, $U=(\sum P_{(i)})U\subseteq\bigoplus(P_{(i)}U)$. So $U=\bigoplus(P_{(i)}U)$. Now apply Lemma 7. ∎ The symbol $\subset$ is reserved for strict inclusion. ###### Lemma 9. If $S,T\subseteq[n]$ and $\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|_{S})\subset\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|_{S\cup T})$, then there is a row $t\in T$ such that $\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|_{S})\subset\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|_{S\cup\\{t\\}})$. ###### Proof. Without loss of generality $S,T$ are disjoint. Let $T^{\prime}\subseteq T$ be a smallest set s.t. $\exists S^{\prime}\subseteq S$ s.t. ${\mathbf{m}}_{S^{\prime}}\odot{\mathbf{m}}_{T^{\prime}}\notin\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}_{S})$. Select any $t\in T^{\prime}$ and write ${\mathbf{m}}_{S^{\prime}}\odot{\mathbf{m}}_{T^{\prime}}={\mathbf{m}}_{S^{\prime}}\odot{\mathbf{m}}_{T^{\prime}-\\{t\\}}\odot{\mathbf{m}}_{t}$. By minimality of $T^{\prime}$, ${\mathbf{m}}_{S^{\prime}}\odot{\mathbf{m}}_{T^{\prime}-\\{t\\}}\in\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}_{S})$. But then ${\mathbf{m}}_{S^{\prime}}\odot{\mathbf{m}}_{T^{\prime}}\in\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}_{S\cup\\{t\\}})$, so $\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|_{S})\subset\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|_{S\cup\\{t\\}})$. ∎ Theorem 2 is now a consequence of Lemma 9. ∎ It follows from Theorem 2 that we can check whether $\operatorname{rank}\mathbb{H}({\mathbf{m}})=k$ in time $O(n)^{k}$ by computing $\operatorname{rank}\mathbb{H}({\mathbf{m}}\|_{S})$ for each $S\in\binom{[n]}{k-1}$. ## 3 Combinatorics of the NAE Condition: Proof of Theorem 4 (a) Recall we are to show: 4 (a): If $\overline{{\varepsilon}}({\mathbf{m}})\geq-1$ then ${\mathbf{m}}$ has a restriction to some $k-1$ rows on which $\overline{{\varepsilon}}=-1$. ###### Proof. We induct on $k$. The (vacuous) base-case is $k=1$. For $k>1$, we induct on $n$, with base-case $n=k-1$. Supposing the Theorem fails for $k$, $k>1$, let ${\mathbf{m}}$ be a $k$-column counterexample with least $n$. Necessarily every row is in $\text{NAE}({\mathbf{m}})$, and $n>k-1\geq 1$. We will show ${\mathbf{m}}$ has a restriction ${\mathbf{m}}^{\prime}$ to $n-1$ rows, for which $\overline{{\varepsilon}}({\mathbf{m}}^{\prime})\geq-1$; this will imply a contradiction because, by minimality of ${\mathbf{m}}$, ${\mathbf{m}}^{\prime}$ has a restriction to $k-1$ rows on which $\overline{{\varepsilon}}=-1$. If $\overline{{\varepsilon}}({\mathbf{m}})\geq 0$ then we can remove any single row of ${\mathbf{m}}$ and still satisfy $\overline{{\varepsilon}}\geq-1$. Otherwise, $\overline{{\varepsilon}}({\mathbf{m}})=-1$, so there is a nonempty $S$ such that $\|\text{NAE}({\mathbf{m}}\|^{S})\|=\|S\|-1$; choose a largest such $S$. It cannot be that $S=[k]$ (as then $n=k-1$). Arrange the rows $\text{NAE}({\mathbf{m}}\|^{S})$ as the bottom $\|S\|-1$ rows of the matrix. As discussed earlier, for the NAE condition one may regard the distinct real values in each row of ${\mathbf{m}}$ simply as distinct colors; relabel the colors in each row above $\text{NAE}({\mathbf{m}}\|^{S})$ so the color above $S$ is called “white.” (There need be no consistency among the real numbers called white in different rows.) See Figure 1. Figure 1: Argument for Theorem 4 (a). Upper-left region is white. Entries $(t,f(t))$ are not white. Due to the maximality of $\|S\|$, there is no white rectangle on $\ell$ columns and $n-\|S\|-\ell+1$ rows inside ${\mathbf{m}}\|_{[n]-\text{NAE}({\mathbf{m}}\|^{S})}^{[k]-S}$ for any $\ell\geq 1$. That is to say, if we form a bipartite graph on right vertices corresponding to the columns $[k]-S$, and left vertices corresponding to the rows $[n]-\text{NAE}({\mathbf{m}}\|^{S})$, with non-white cells being edges, then any subset of the right vertices of size $\ell\geq 1$ has at least $\ell+1$ neighbors within the left vertices. By the induction on $k$ (since $S\neq\emptyset$), for the set of columns $[k]-S$ there is a set $R^{\prime\prime}$ of $k-\|S\|-1$ rows such that $\overline{{\varepsilon}}({\mathbf{m}}\|_{R^{\prime\prime}}^{[k]-S})=-1$. Together with the rows of $\text{NAE}({\mathbf{m}}\|^{S})$ this amounts to at most $k-2$ rows, so since $n\geq k$, we can find two rows outside this union; delete either one of them, leaving a matrix ${\mathbf{m}}^{\prime}$ with $n-1$ rows. This matrix has the rows $\text{NAE}({\mathbf{m}}\|^{S})$ at the bottom, and $n-\|S\|$ remaining rows which we call $R^{\prime}$. The lemma will follow by showing that $\overline{{\varepsilon}}({\mathbf{m}}^{\prime})\geq-1$. In ${\mathbf{m}}^{\prime}$, the induced bipartite graph on right vertices $[k]-S$ and left vertices $R^{\prime}$ has the property that any right subset of size $\ell\geq 1$ has a neighborhood of size at least $\ell$ in $R^{\prime}$. Applying Hall’s Marriage Theorem, there is an injective $f:[k]-S\to R^{\prime}$ employing only edges of the graph. Now consider any set of columns $T$, $T=T_{1}\cup T_{2},T_{1}\subseteq[k]-S,T_{2}\subseteq S$. We need to show that ${\varepsilon}({\mathbf{m}}\|^{T})\geq-1$. Let $R_{1}=\text{NAE}({\mathbf{m}}\|^{T_{1}})\cap R^{\prime\prime}$, $R_{2}=\text{NAE}({\mathbf{m}}\|^{T_{2}})\subseteq\text{NAE}({\mathbf{m}}\|^{S})$, and note that $\|R_{1}\|\geq\|T_{1}\|-1$, $\|R_{2}\|\geq\|T_{2}\|-1$. If $T_{2}=\emptyset$ we simply use $R_{1}$. Likewise if $T_{1}=\emptyset$, we use $R_{2}$. If both $T_{1}$ and $T_{2}$ are nonempty, $\text{NAE}({\mathbf{m}}\|^{T_{2}})\subseteq\text{NAE}({\mathbf{m}}\|^{S})$, and $\|\text{NAE}({\mathbf{m}}\|^{T_{2}})\|\geq\|T_{2}\|-1$. Now use the matching $f$. The set of rows $f(T_{1})$ lies in $R^{\prime}$ and is therefore disjoint from $\text{NAE}({\mathbf{m}}\|^{T_{2}})$. Moreover since $T_{2}\neq\emptyset$, every entry $(t,j)$ for $t\in T_{2},j\in R^{\prime}$ is white. On the other hand due to the construction of $f$, for every $t\in T_{1}$ the entry $(t,f(t))$ is non-white. Therefore every row in $f(T_{1})$ is in $\text{NAE}({\mathbf{m}}\|^{T_{1}\cup T_{2}})$. So $\|\text{NAE}({\mathbf{m}}\|^{T_{1}\cup T_{2}})\|\geq\|T_{2}\|-1+\|T_{1}\|$. Thus $\overline{{\varepsilon}}({\mathbf{m}}^{\prime})\geq-1$. ∎ ## 4 From NAE to Rank: Proof of Theorem 4 (b) Recall we are to show: 4 (b): $\mathbb{H}({\mathbf{m}})$ has full column rank if $\overline{{\varepsilon}}({\mathbf{m}})\geq-1$. ###### Proof. The case $k=1$ is trivial. Now suppose $k\geq 2$ and that Theorem 4 (b) holds for all $k^{\prime}<k$. Any constant rows of ${\mathbf{m}}$ affect neither the hypothesis nor the conclusion, so remove them, leaving ${\mathbf{m}}$ with at least $k-1$ rows. Now pick any set, $C$, of $k-1$ columns of ${\mathbf{m}}$. By Theorem 4 (a) there are some $k-2$ rows of ${\mathbf{m}}$, call them $R^{\prime}$, on which $\overline{{\varepsilon}}({\mathbf{m}}\|_{R^{\prime}}^{C})=-1$. Let $v$ be a row of ${\mathbf{m}}$ outside $R^{\prime}$. Call the rows of ${\mathbf{m}}$ apart from $v$, $R^{\prime\prime}$. Since $R^{\prime\prime}$ contains $R^{\prime}$, by induction $\dim\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|_{R^{\prime\prime}}^{C})=k-1$. Therefore $U:=\operatorname{Rowspace}\mathbb{H}({\mathbf{m}}\|_{R^{\prime\prime}})\subseteq\mathbb{R}^{k}$ is of dimension at least $k-1$. We claim now that $\dim U=k$. Suppose to the contrary that $\dim U=k-1$. If $v_{\odot}(U)\subseteq U$ then as proven earlier in Theorem 8, $U$ respects $B(v)$. Since $v$ is nonconstant, $B(v)$ is a partition of $[k]$ into $\ell\geq 2$ nonempty blocks $B(v)_{(i)}$, and $U=\bigoplus_{i=1}^{\ell}U_{(i)}$ with $U_{(i)}=P_{(i)}U_{(i)}$. So there is some $i_{0}$ for which $U_{(i_{0})}\subset V_{(i_{0})}$; specifically, $U_{(i)}=V_{(i)}$ for all $i\neq i_{0}$, and $\dim U_{(i_{0})}=\dim V_{(i_{0})}-1$. Since $\|B(v)_{(i_{0})}\|<k$, we know by induction that the rows of $\mathbb{H}({\mathbf{m}})$ span $V_{(i_{0})}$. Thus in fact $U=\mathbb{R}^{k}$. (Further detail for the last step: let $w\in\mathbb{R}^{k}$. Since the rows of $\mathbb{H}({\mathbf{m}})$ span $V_{(i_{0})}$, there is a $w^{\prime}\in\operatorname{Rowspace}\mathbb{H}({\mathbf{m}})$ s.t. $P_{(i_{0})}w^{\prime}=P_{(i_{0})}w$. Moreover since $U_{(i)}=V_{(i)}$ for all $i\neq i_{0}$, there is a $w^{\prime\prime}\in U$ s.t. $w^{\prime\prime}=(I-P_{(i_{0})})(w-w^{\prime})$. Then $w^{\prime}+w^{\prime\prime}\in\operatorname{Rowspace}\mathbb{H}({\mathbf{m}})$, and $w^{\prime}+w^{\prime\prime}=w$.) ∎ ## 5 Motivation Consider observable random variables $X_{1},\ldots,X_{n}$ that are statistically independent conditional on $H$, a hidden random variable $H$ supported on $\\{1,\ldots,k\\}$. (See causal diagram.) $\textstyle{H}$$\textstyle{X_{1}}$$\textstyle{X_{2}}$$\textstyle{\cdots}$$\textstyle{X_{i}}$$\textstyle{\cdots}$$\textstyle{X_{n}}$ The most fundamental case is that the $X_{i}$ are binary. Then we denote ${\mathbf{m}}_{i}^{j}=\Pr(X_{i}=1\|H=j)$. The model parameters are ${\mathbf{m}}$ along with a probability distribution (the mixture distribution) $\pi=(\pi_{1},\ldots,\pi_{k})$ on $H$. Finite mixture models were pioneered in the late 1800s in [13, 14]. The problem of learning such distributions has drawn a great deal of attention. For surveys see, e.g., [5, 17, 11, 12]. For some algorithmic papers on discrete $X_{i}$, see [9, 4, 7, 2, 6, 1, 15, 10, 3, 8]. The source identification problem is that of computing $({\mathbf{m}},\pi)$ from the joint statistics of the $X_{i}$. Put another way, the problem is to invert the multilinear moment map $\displaystyle\mu:({\mathbf{m}},\pi)$ $\displaystyle\to\mathbb{R}^{2^{[n]}}$ $\displaystyle\mu({\mathbf{m}},\pi)_{S}$ $\displaystyle=\Pr(X_{S}=1)\quad\text{ where }S\subseteq[n],\;X_{S}=\prod\nolimits_{i\in S}X_{i}$ $\displaystyle={\mathbf{m}}_{S}\cdot\pi^{\top}$ The last line shows the significance of $\mathbb{H}({\mathbf{m}})$ to mixture model identification, since ${\mathbf{m}}_{S}^{j}=\Pr(X_{S}=1\|H=j)$. ### Connection to $\operatorname{rank}\mathbb{H}({\mathbf{m}})$. In general $\mu$ is not injective (even allowing for permutation among the values of $H$). For instance it is clearly not injective if ${\mathbf{m}}$ has two identical columns (unless $\pi$ places no weight on those). More generally, and assuming all $\pi_{j}>0$, it cannot be injective unless $\mathbb{H}({\mathbf{m}})$ has full column rank. One sufficient condition for injectivity, due to [16], is that there be $2k-1$ “separated” observables $X_{i}$; $X_{i}$ is separated if all ${\mathbf{m}}_{i}^{j}$ are distinct, or in our terminology, if no color recurs in ${\mathbf{m}}_{i}$. (Further [8], one can lower bound the distance between $\mu({\mathbf{m}},\pi)$ and any $\mu({\mathbf{m}}^{\prime},\pi^{\prime})$ in terms of $\min_{i,j}\|{\mathbf{m}}_{i}^{j}-{\mathbf{m}}_{i}^{j^{\prime}}\|$ and the distance between $({\mathbf{m}},\pi)$ and $({\mathbf{m}}^{\prime},\pi^{\prime})$.) A weaker sufficient condition for injectivity of $\mu$, due to [8], is that for every $i\in[n]$ there exist two disjoint sets $A,B\subseteq[n]-\\{i\\}$ such that $\mathbb{H}({\mathbf{m}}\|_{A})$ and $\mathbb{H}({\mathbf{m}}\|_{B})$ have full column rank. (It is not known whether two disjoint such $A,B$ are strictly necessary, but the implied $n\leq 2k-1$ is in general best possible [15].) ## References * [1] A. Anandkumar, D. Hsu, and S. M. Kakade. A method of moments for mixture models and hidden Markov models. In Proc. 25th Ann. Conf. on Computational Learning Theory, pages 33.1–33.34, 2012. * [2] K. Chaudhuri and S. Rao. Learning mixtures of product distributions using correlations and independence. In Proc. 21st Ann. Conf. on Computational Learning Theory, pages 9–20, 2008. * [3] S. Chen and A. Moitra. Beyond the low-degree algorithm: mixtures of subcubes and their applications. In Proc. 51st Ann. ACM Symp. on Theory of Computing, pages 869–880, 2019. * [4] M. Cryan, L. Goldberg, and P. Goldberg. Evolutionary trees can be learned in polynomial time in the two state general Markov model. SIAM J. Comput., 31(2):375–397, 2001. Prev. FOCS ’98. * [5] B. S. Everitt and D. J. Hand. Mixtures of discrete distributions, pages 89–105. Springer Netherlands, Dordrecht, 1981. * [6] J. Feldman, R. O’Donnell, and R. A. Servedio. Learning mixtures of product distributions over discrete domains. SIAM J. Comput., 37(5):1536–1564, 2008. * [7] Y. Freund and Y. Mansour. Estimating a mixture of two product distributions. In Proc. 12th Ann. Conf. on Computational Learning Theory, pages 183–192, July 1999. * [8] S. L. Gordon, B. Mazaheri, Y. Rabani, and L. J. Schulman. Source identification for mixtures of product distributions. arXiv:2012.14540, 2020. * [9] M. Kearns, Y. Mansour, D. Ron, R. Rubinfeld, R. Schapire, and L. Sellie. On the learnability of discrete distributions. In Proc. 26th Ann. ACM Symp. on Theory of Computing, pages 273–282, 1994. * [10] J. Li, Y. Rabani, L. J. Schulman, and C. Swamy. Learning arbitrary statistical mixtures of discrete distributions. In Proc. 47th Ann. ACM Symp. on Theory of Computing, pages 743–752, 2015. * [11] B. G. Lindsay. Mixture models: theory, geometry and applications. In NSF-CBMS regional conference series in probability and statistics, pages i–163. JSTOR, 1995. * [12] G. J. McLachlan, S. X. Lee, and S. I. Rathnayake. Finite mixture models. Annual Review of Statistics and Its Application, 6(1):355–378, 2019\. * [13] S. Newcomb. A generalized theory of the combination of observations so as to obtain the best result. American Journal of Mathematics, 8(4):343–366, 1886. * [14] K. Pearson. Contributions to the mathematical theory of evolution III. Philosophical Transactions of the Royal Society of London (A.), 185:71–110, 1894. * [15] Y. Rabani, L. J. Schulman, and C. Swamy. Learning mixtures of arbitrary distributions over large discrete domains. In Proc. 5th Conf. on Innovations in Theoretical Computer Science, pages 207–224, 2014. * [16] B. Tahmasebi, S. A. Motahari, and M. A. Maddah-Ali. On the identifiability of finite mixtures of finite product measures. IEEE International Symposium on Information Theory (ISIT) 2018 and arXiv:1807.05444v1, 2018. * [17] D. M. Titterington, A. F. M. Smith, and U. E. Makov. Statistical Analysis of Finite Mixture Distributions. John Wiley and Sons, Inc., 1985.
11institutetext: imec-IDLab-UGent, Department of Electronics and Information Systems, Ghent University, Technologiepark-Zwijnaarde 126, 9052 Zwijnaarde, Belgium 22institutetext: IPEM-UGent, Department of Art History, Musicology and Theatre Studies, Ghent University, Miriam Makebaplein 1, 9000 Gent, Belgium 33institutetext: imec-mict-UGent, Department of Communication Sciences, Ghent University, Miriam Makebaplein 1, 9000 Gent, Belgium # Art and Science Interaction Lab A highly flexible and modular interaction science research facility Niels Van Kets§ 11 0001-5495-2240 Bart Moens§ 22 0002-1281-9406 Klaas Bombeke§ 33 0003-2056-1246 Wouter Durnez§ 33 0001-8045-8801 Pieter-Jan Maes 22 0002-9237-3298 Glenn Van Wallendael§ 11 0001-9530-3466 Lieven De Marez 33 0001-7716-4079 Marc Leman 22 0002-9780-2194 Peter Lambert§ 11 0001-5313-4158 ###### Abstract The Art and Science Interaction Lab (“ASIL”) is a unique, highly flexible and modular “interaction science” research facility to effectively bring, analyse and test experiences and interactions in mixed virtual/augmented contexts as well as to conduct research on next-gen immersive technologies. It brings together the expertise and creativity of engineers, performers, designers and scientists creating solutions and experiences shaping the lives of people. The lab is equipped with state-of-the-art visual, auditory and user-tracking equipment, fully synchronized and connected to a central backend. This synchronization allows for highly accurate multi-sensor measurements and analysis. ###### Keywords: Immersive Experience Virtual Reality User Testing ## 1 Introduction The Art and Science Interaction Lab (ASIL) team supports innovation in different key domains. Within these domains, the team focuses on interaction research in virtualized environments, unraveling complex user interactions and experiences in order to design and create novel applications and interfaces. The application domains span from smart home appliances, health, safety, smart public places to more artistic and creative applications. Furthermore, the lab infrastructure is used for fundamental research on virtual reality technologies (e.g. auralization, virtual acoustics, 6 degrees of freedom VR, multi-person VR…) and is dark fiber connected to three concert halls in Ghent. The team is an interdisciplinary consortium combining the expertise of three Ghent University research groups (IDLab, IPEM and mict) and has been co-funded under the medium-scale research infrastructure program governed by the Research Foundation Flanders (FWO). The combination of humanities, engineering, psychology and social sciences makes the ASIL a one of its kind research facility. Moreover, the research tracks in the ASIL target both industry and academia to deliver a unique interdisciplinary approach in measuring, analyzing and creating our next-generation appliances, interfaces and experiences. ## 2 Technical infrastructure The Art and Science Interaction lab is located in ”De Krook” building (Fig. 1) in the city center of Ghent. Next to the city of Ghent library, ”De Krook” also houses the three Ghent University research groups involved in the Art and Science Interaction Lab (IDLab, IPEM and mict), as well as imec, an R&D hub for nano- and digital technologies. Both IDLab and mict are affiliated imec research groups. The Lab is located in the sub-level floors of the building and has a volume of 10m x 10m and spans a height of two full floors ($\approx$ 6m). Figure 1: De Krook In view of the audiovisual applications envisioned in this room, the walls and ceiling all have been acoustically treated, delivering a RT60 Reverberation Time of 0.5s. The RT60 Reverberation Time describes how long it takes for sound to decay by 60dB in a room with a diffuse soundfield. By insulating the room, a reduction of approximately 2s was met. Furthermore, in order to deliver a highly flexible and modular research infrastructure, the lab has been equipped with a state of the art trussing system (Fig. 2, including 5 motorized trusses of 7m. This trussing system allows for a myriad of experimental setups. (a) (b) Figure 2: Trussing system In order to limit the length and amount of cables to be used in setting up new experiments, a large amount of patch boxes have been installed on both the fixed and moving trusses. All patch boxes deliver power, data, UTP, coaxial, DMX and XLR connections to several locations in the lab (Fig. 3a). Each of these patch-points are directly linked to the patch rack (Fig. 3b), from where connections towards the machine room or other patch points can be made. (a) (b) Figure 3: Patching infrastructure ## 3 Audiovisual equipment The Art and Science Interaction Lab is equipped with 80 individual speakers connected to a fully IP-based audio distribution system. The system is capable of delivering highly realistic spatial audio projection by making use of a dedicated audio wavefield processor. This audio system allows for accurate recreation and simulation of room acoustics. In terms of visual modalities, the lab is equipped with 2 fully untethered HTC Vive Pro Eye, 2 tethered HTC Vive Pro and 2 Microsoft Hololens version 2 devices, allowing free roaming spanning the full 10m x 10m area and allowing for multi person AR/VR. A 7m x 4m acoustically transparent projection screen in combination with a 12.000 lumen 4K projector delivers compelling and high- end immersive visualizations. Both the audio and visual systems are connected to a powerful state-of-the-art processing backend. ### 3.1 Audio equipment An immersive speaker system and a state-of-the-art sound processing backend allows compelling auditory experiences. Due to the acoustic insulation, sound reflections and deformations are vastly reduced in order to deliver the most accurate auditory stimuli to the listeners. #### 3.1.1 Equipment list * • 80 calibrated speakers (8 subs, 2 speaker rings and an overhead speaker set) * • Barco IOSONO [2] core wavefield synthesis system * • Fully IP-based Dante [1] audio infrastructure * • 10 amplifiers including built-in DSP for each individual speaker channel * • Highly flexible XLR and IP-audio patch possibilities * • Fully synced with 48kHz clock * • Software/frameworks: Ableton Live, Max/MSP, Ambisonics… #### 3.1.2 Applications: Figure 4: Audio Applications - Musical Performance The audio installation in the Art and Science Interaction Lab allows for accurate object-based projection of 3D audio in space by using wavefield synthesis. This enables the (re-)creation of immersive auditory environments, bringing a myriad of application and research opportunities such as the recreation of multi-track musical performances and the use of interactive sound objects that can dynamically move in space. Furthermore, such a system allows for the (re-)creation of real-life or simulated acoustic environments. This allows one to experience the acoustic properties of a certain location (e.g. concert hall, church, outdoors…) or the simulation of acoustic properties of future buildings or expositions. ### 3.2 Video and Mixed AR/VR equipment The visual installation caters towards multi-user applications, with the focus on usability and freedom of movement. State-of-the-art virtual and augmented reality headsets are readily available in the lab. Furthermore, a high- resolution projection system is capable of delivering compelling grouped or single experiences. #### 3.2.1 Equipment * • 2x HTC Vive Pro Eye untethered VR headsets with built-in eye tracking * • 2x HTC Vive Pro tethered VR headsets * • 2x Microsoft Hololens v2 AR glasses * • 7m x 4m acoustically transparent projection screen * • 12.000 lumens 4K projector with active stereo 3D and Extended Dynamic Range * • high-end rendering with latest NVIDIA GPU compute capabilities #### 3.2.2 Applications Figure 5: Visual applications The visual systems integrated in the Art and Science Interaction Lab allow for the delivery of immersive and interactive virtual and augmented reality experiences while maintaining the highest degree of free movement possible. The state-of-the-art processing backend allows for real-time rendering and visualization of complex 6 degrees of freedom experiences (6DoF). Secondly, the systems allow for multi-user VR and AR applications, allowing multiple users to interact naturally in a shared virtual environment. Integrated with the aforementioned positional audio and the motion capture system that will be described in the following section, these interactions can be of very high realism. Lastly, the high resolution, high dynamic range projection allows for immersive context creation without the use of a virtual reality headset, as well as delivering compelling visual experiences to larger groups. ### 3.3 Motion capture A dense Qualisys motion capture setup allows full body tracking of multiple users (up to 5) with an accuracy of $<$ 1mm3, on a 81m2 floor area and a 5m configurable volume height. This allows detailed tracking of (multi) user movement. Furthermore, real-time integration with both the audio and visualization solutions allow for interactive audiovisual experiences based on human movement. #### 3.3.1 Equipment * • Qualisys Oqus 7+ infrared cameras [3] (14 fixed+ 4 mobile units) * • Qualisys Miqus Video cameras [4] (4 fixed + 1 mobile unit) * • Synced with the Art and Science Interaction Lab 120Hz clock signal * • High-end real-time processing backend allowing for real-time skeleton tracking and streaming * • Tracking compatible with VR headsets #### 3.3.2 Applications Figure 6: Motion Capture - Qualisys Oqus 7+ IR camera The motion capture installation in the Art and Science Interaction Lab allows for highly accurate measurements of human behavior (motion) in interactive scenarios. The system allows for fine-granular motion tracking (e.g. finger tracking while playing an instrument, face expression tracking…) but also robust full body tracking of multiple users in the same area. The system is capable of mapping marker data onto skeletons in order to animate avatars in VR and in real-time. This allows for accurate real-time representations of users in a virtual (e.g. VR) space. Next to tracking users, the motion capture system allows for accurate 6 degrees-of-freedom tracking of objects. This by applying reflective markers to real-life objects. ## 4 Sensor equipment The Art and Science Interaction Lab is the go-to facility for interaction and user-experience research. The infrastructure provides a wide variety of synchronized sensors capable of measuring different aspects of an experience and is backed by a strong team of UX researchers. The interdisciplinary team of the Art and Science Interaction Lab can deliver a detailed unraveling of user experiences and interactions, even those where users are not aware of. #### 4.0.1 Equipment: * • 2x clinical grade untethered EEG headsets (64 channels with active electrodes), which can be used in combination with our wireless VR headsets. * • Eye trackers, both built into VR headsets and standalone. * • Skin conductance sensors * • Heart rate sensors * • EMG sensors * • Synced with the Art and Science Interaction Lab 120Hz clock signal #### 4.0.2 Applications: Figure 7: Sensor Applications - Physiological Analysis The available sensors in the Art and Science Interaction Lab and their integration and synchronization allow the measurement of a highly accurate physiological footprint of a user experience. By combining different sensors such as EEG, EMG, skin conductance… and the aforementioned motion capture system, physiological processes within human interactions and experiences can be unraveled on a granular scale. This analysis can be used to discover points of pain in a certain user-experience. Furthermore, in a more artistic setting, real-time sensor data can be used in order to create novel audiovisual experiences based on real-time biofeedback. ## 5 Interconnectivity The Art and Science Interaction Lab serves as a central hub towards different concert halls in the city center of Ghent. A 10Gbps dark fiber towards ’De Vooruit’, ’De Minard’ and the future ’Wintercircus’ allows for real-time shared experiences between the Art and Science Interaction lab and three important cultural venues in Ghent. Next to the cultural venues, the Art and Science Interaction Lab is also directly connected towards URGENT, the Ghent University Student radio, which also resides in De Krook. Lastly, the lab has a direct dark fiber link to an off-site back-up facility, delivering a secure and redundant data storage. #### 5.0.1 Applications: The direct link with the concert venues and URGENT radio allow for real-time streaming of multi-track audio between these facilities in order to enable the (re-)creation of a cultural experience in the Art and Science Interaction Lab as well as creating novel experiences based on real-time audio feeds. Furthermore, real-time motion capture allows for novel experiences where motion in the Art and Science Interaction Lab can be used in real-time in the cultural venues (e.g. the creation of virtual avatars or experiences). ## References * [1] Audinate: Audinate dante (2021), https://www.audinate.com * [2] Barco: Barco iosono (2019), https://www.barco.com/nl/product/iosono-core/specsheet * [3] Systems, Q.M.C.: Qualisys 5+, 6+ and 7+ (2018), https://cdn-content.qualisys.com/2018/11/PI_Camera-5-6-and-7.pdf * [4] Systems, Q.M.C.: Miqus video (2020), https://cdn-content.qualisys.com/2020/04/PI_Miqus_Video.pdf
# Robust and Efficient Single-Pixel Image Classification with Nonlinear Optics Santosh Kumar Department of Physics, Stevens Institute of Technology, Hoboken, NJ, 07030, USA Center for Quantum Science and Engineering, Stevens Institute of Technology, Hoboken, NJ, 07030, USA These authors contributed equally Ting Bu Department of Physics, Stevens Institute of Technology, Hoboken, NJ, 07030, USA Center for Quantum Science and Engineering, Stevens Institute of Technology, Hoboken, NJ, 07030, USA These authors contributed equally He Zhang Department of Physics, Stevens Institute of Technology, Hoboken, NJ, 07030, USA Center for Quantum Science and Engineering, Stevens Institute of Technology, Hoboken, NJ, 07030, USA Irwin Huang Department of Physics, Stevens Institute of Technology, Hoboken, NJ, 07030, USA Yuping Huang Department of Physics, Stevens Institute of Technology, Hoboken, NJ, 07030, USA Center for Quantum Science and Engineering, Stevens Institute of Technology, Hoboken, NJ, 07030, USA Corresponding author<EMAIL_ADDRESS> ###### Abstract We present a hybrid image classifier by mode-selective image upconversion, single pixel photodetection, and deep learning, aiming at fast processing a large number of pixels. It utilizes partial Fourier transform to extract the signature features of images in both the original and Fourier domains, thereby significantly increasing the classification accuracy and robustness. Tested on the MNIST handwritten digit images, it boosts the accuracy from 81.25% to 99.23%, and achieves an 83% accuracy for highly contaminated images whose signal-to-noise ratio is only -17 dB. Our approach could prove useful for fast lidar data processing, high resolution image recognition, occluded target identification, atmosphere monitoring, and so on. ††journal: ol Machine learning techniques based on deep neural networks (DNNs) have scored considerable success in image classification [1, 2], speech recognition [3], image generation [4], image reconstruction [5], and so on. They use a feed- forward multi-layer neural architecture to achieve high performance for complex operations [6, 7]. Despite notable success for their digital implementations, it remains a challenge to adopt them directly for embedded systems due to the high memory-demand, limited scalability, and excessive energy budget [8]. On the other hand, optical systems naturally offer extreme parallelism and multiplexing yet with little energy consumption and fast data- processing [9, 10, 11]. Thus far, optical Fourier transformation, diffraction, interference, and filtering have been exploited for optical pattern recognition [12, 13] and restoration [14], phase retrieval [15, 16], and optical information processing [17]. Recently, optically assisted image processing has become a vivid pursuit in the field of artificial intelligence [18], computer vision [19], robotics [20, 21], medical science [22], and so on. While most of these studies are based on linear optics, nonlinear optics is poised to enable even richer and more complex operations and lift our processing capability to yet another level. To this end, optical neural network with nonlinear activation functions was implemented for object identification and classification of ordered and disordered phases of Ising models [23]. Spontaneous parametric down-conversion was utilized for quantum- correlated pattern recognition with spatially structured photons [24]. Artificial neural networks based on nonlinear optics have shown advantages in reconstructing the amplitude and phase profiles of ultrashort pulses [25, 26] and neuromorphic computing [27, 28]. A super-Ising emulator was demonstrated with unprecedented four-body interaction using spatial light modulation and second-harmonic generation [29]. Recently, we have proposed and experimentally demonstrated a nonlinear-optics approach to pattern recognition with single-pixel imaging and a deep neural network [30]. It employs mode-selective image up-conversion to project a raw image with up to mega pixels onto a set of coherent spatial modes, whereby its signature features are extracted optically in a nonlinear manner. Our experimental results promise distinct applications in online classification of large-size images, fast lidar data analyses [31], complex pattern recognition, and so on. In this work, we continue to study the aforementioned hybrid machine learning technique, focusing on further improving the efficiency and effectiveness in the feature extraction. In our previous system [30], the mode-selective conversion is applied to the fully Fourier-transformed images, which samples the signature features in the Fourier domain, but overlooks some important identifying features in the original image domain. Here, we study a new approach where the upconversion is applied to the images undergoing only partial Fourier transform. While challenging to calculate partial Fourier transform numerically [32, 33], it is easily realized in the current system by displacing the nonlinear crystal from the focal point of the Fourier lens for the images. This simple adjustment allows both the original and Fourier features of the images to be sampled at the same time, leading to high classification accuracy using only a limited number of pump modes for the upconversion. Tested on the MNIST (Modified National Institute of Standards and Technology) handwritten digit images, a 40 mm crystal displacement significantly increases the accuracy from 81.25% to 99.23%. The same technique enables robust pattern recognition when the images are mixed with strong noise, where a high accuracy $\sim 83\%$ is achieved even when the signal-to- noise (SNR) is as low as -17 dB. All of our experimental results agree well with the simulated results, confirming this effect. Finally, we numerically compare two choices of the pump mode bases for the upconversion that can affect the feature extraction, showing better performance with the Laguerre Gaussian (LG) mode bases than the Hermite-Gaussian (HG) modes, likely due to a lower cross-talk for the former [34]. Our results highlight a viable approach to image recognition through feature extraction in both the original and Fourier domains by mode-selective image upconversion, for high accuracy, high efficiency, and exceptional robustness. Figure 1: Experimental setup for the feature extraction via frequency up- conversion. The PPLN crystal is moved away from the focal point to the focal lens by a distance $\Delta$, as shown in the lower right inset. SLM: Spatial Light Modulator, BS: Beamsplitter, WDM: Wavelength-division multiplexing, PPLN: Magnesium-doped Periodic Poled Lithium Niobate crystal; PM: power meter. The experimental setup for the present nonlinear optical pattern-recognition scheme is outlined in Fig. 1. The phase patterns of the Gaussian beams in signal and pump arms are modulated by two separate spatial light modulators (SLM). In the signal arm, SLM1 is used to upload phase patterns of input images while the SLM2 is applied to map phase patterns of spatial LG (or HG) modes on pump beam. The phase value is wrapped in the interval between 0 and $2\pi$ to express on the SLMs [35, 36]. The phase modulated two beams are then merged at a beam splitter (BS). A Fourier lens ($f$=200 mm) is used to focus the coupled beam in a temperature-stabilized periodic poled lithium niobate (PPLN) crystal for sum-frequency (SF) generation process. The power readings of the SF generation are extracted features from images, which are inputs to a deep neural network for digits recognition. Further details of our setup can be found in our previous work [30]. The difference between the current setup and the previous one is the position of the PPLN. As illustrated in the lower right inset in Fig. 1, $\Delta$ is the of the displacement of PPLN crystal towards the Fourier lens. In simulation, for each pair of the signal beam $E_{s}$ and pump beam $E_{p}$, their corresponding SF field $E_{f}$ is solved under the slowly-varying- envelope approximation as: $\displaystyle 2ik_{s}\partial_{z}E_{s}+(\partial_{x}^{2}+\partial_{y}^{2})E_{s}=-2\frac{\omega_{s}^{2}}{c^{2}}\chi^{(2)}E_{p}^{}E_{f}e^{i\triangle kz},$ (1) $\displaystyle 2ik_{p}\partial_{z}E_{p}+(\partial_{x}^{2}+\partial_{y}^{2})E_{p}=-2\frac{\omega_{p}^{2}}{c^{2}}\chi^{(2)}E_{s}^{}E_{f}e^{i\triangle kz},$ (2) $\displaystyle 2ik_{f}\partial_{z}E_{f}+(\partial_{x}^{2}+\partial_{y}^{2})E_{f}=-2\frac{\omega_{f}^{2}}{c^{2}}\chi^{(2)}E_{p}E_{s}e^{-i\triangle kz},$ (3) where $\omega_{s}$, $\omega_{p}$, and $\omega_{f}$ are the beam waists of signal, pump and SF light, respectively. $k_{s}={n_{s}\omega_{s}}/{c}$ , $k_{p}={n_{p}\omega_{p}}/{c}$ and $k_{f}={n_{f}\omega_{f}}/{c}$ are their wave numbers. $\Delta k=k_{s}+k_{p}-k_{f}-2\pi/\Lambda$ gives their phase mismatch, $\Lambda=19.36$ $\mu$m is the poling period of the PPLN. $z$ is the distance in propagation direction with $z=0$ being the focal point. $\chi^{(2)}$ is the second-order nonlinear susceptibility. We can use the standard split-step Fourier and adaptive step size methods [37] to numerically solve Eqs.(1)-(3) with an initial location $z=-\frac{L}{2}-\Delta$, where $L=1$ $cm$ is the length of the PPLN. Figure 2: Normalized confusion matrices for testing the performance of the hand-written digits with three different crystal positions. The top panels (a-c) are the simulated results using 40 LG pump modes with $\Delta=0$, 27 mm and 40 mm, respectively. The bottom panels (d-f) are the corresponding experimental results. In this paper, one of the most popular MNIST handwritten digit database is selected as our benchmark, which is a 10-class database including handwritten images representing digits from "0" to "9" [38]. A subset of the original database consisting of the first 200 handwritten images of each digit is applied as our database. These images in our database are shuffled and separated into training (1600 images) and testing (400 images) sets. The resolution of these 28$\times$28 images are increased to 400$\times$400 pixels to match with the input signal beam size and SLM pitch. The SF power readings of each handwritten image are normalized between 0 and 1, after collecting all results for our training and testing data sets. This normalization is different from our previous work in [30], where the upconversion takes place in the Fourier plane and data are normalized within each LG mode. While the original and Fourier planes offer equivalent descriptions [39], the amount of information that can be extracted using a small number of pump modes is quite different. In [30], the upconversion was performed in the Fourier domain, which resulted in similar SF power distributions for the first few pump modes. The signature features are contained in the higher order modes where the SF power is relatively low. To distinguish those digits therefore requires “amplifying” the SF power difference in the higher order modes, which was realized through SF power normalization by mode. While high performance is permissible, that method renders the system prone to noise and fluctuations. Also, after adding new training or testing data, the whole data set needs to be normalized again and new training process is warranted for the best performance. Figure 3: (a) An example of random noise mask with $\textrm{SNR}=-10$ dB. The zoomed windows are representing the noise clusters with cluster size S equal to the length of 10 SLM pixels (i.e. one noise pixel size, $\textrm{S}=10$ SLM pixels or 104 $\mu$m). (b) Experimental and simulated accuracy vs SNR. Blue and green dotted curves are the experimental and simulated accuracy for $\Delta$ = 27 mm, respectively, and red and orange dashed curves are for $\Delta$ = 40 mm. In contrast, when the upconversion is applied to partially Fourier transformed images, the signature features of the images are extracted in both the original and Fourier domains. They are readily manifested in SF power for the first few LG modes of high readings, resulting in distinct power distribution over the 40 LG modes. This eliminates the need to amplify the SF power for higher-order modes, and allows normalization for each image. It makes the system more robust and efficient, while avoiding the need of repeated normalization within each mode over the entire data set when new data are added. The structure of the deep neural network is same as the one in our previous work in [30], which is a combination of one convolutional layer and five fully connected layers. In the training process, the adaptive moment estimation (ADAM) gradient descent algorithm is selected to minimize the loss between predictions and their ground truth. The activation function softmax is applied to normalize the output values between 0 and 1 so that they can represent the probabilities of each class. Then the predicted class of each input image will be chosen as the one with the largest probability. Finally, the classification accuracy can be calculated as the fraction of correctly classified images. To optimally extract the spatial and Fourier features at the same time, we displace the PPLN crystal closer to the Fourier lens with its center away from the focal point by $\Delta$, as shown in Fig.1. While this leads to efficient and robust classification, a large $\Delta$ de-focuses the light and suppresses the SF generation. In light of this trade-off, we choose $\Delta=0$, 27 mm and 40 mm in this study. A larger $\Delta$ may be accommodated by using a higher pump power to compensate for the SF efficiency loss. Also, here we follow [30] to use a set of 40 LG modes, with $l\in[-2,2]$ and $p\in[0,7]$. Figure 2 lists the normalized confusion matrices using the 40 LG pump modes for features extraction. Figure 2(a-c) are the simulated results and (d-f) are the experimental results with $\Delta=0,$ 27 mm and 40 mm, respectively. When the crystal is centered at the Fourier plane with $\Delta$ = 0, the setup is the same with that in [30]. In this case, the simulated accuracy is only 77.25% and experimental accuracy is 81.25%. This discrepancy is likely attributed to the measurement errors in the upconverted SF power readings for high-order pump projection modes. When the crystal is displaced by $\Delta=27$ mm, both the simulated and experimental accuracy reached above 97% as shown in Fig. 2(b) and (e), respectively. This shows the advantage of feature extraction in both spatial and Fourier domains. Figure 2(c) and (f) show that the accuracy can be further increased over 99% with $\Delta=40$ mm. To test the robustness of our technique against noise, we next mix the images with uniformly distributed random noise. The noise level is quantified by the signal-to-noise ratio (SNR), defined as: $\textrm{SNR}=10\leavevmode\nobreak\ \textrm{log}_{10}(\sigma_{s}^{2}/\sigma_{n}^{2}),$ (4) where $\sigma_{s}^{2}$ and $\sigma_{n}^{2}$ are the variance of per-pixel phase values in the digit images and the added noise masks, respectively. Figure 3(a) shows an example of one uniformly distributed noise mask with SNR = -10 dB and cluster size $\textrm{S}=104$ $\mu$m, where the cluster size is the number of pixels in one noise cluster (which is 10 SLM pixels in this case). The white noise masks are added to the original handwritten digit images and are uploaded on SLM1 as the inputs. The results in Fig. 3(b) show that, with a larger SNR, the accuracy is higher and the difference is smaller between the simulated and experimental results. Similar to those in Fig. 2 for clear images, for the noise-contaminated images, both results reveal better performance with $\Delta$ = 40 mm than $\Delta$ = 27 mm. The advantage reaches 12.18% when the SNR is -15 dB. This again validates our approach for efficient and robust information extraction. Figure 4: The accuracy for different sets of LG modes with different azimuthal ($l$) and radial ($p$) indexes, with $\textrm{SNR}=-15$ dB and $\textrm{S}=52$ $\mu$m. In addition to SNR, the cluster size S is also an import factor to the classification accuracy. When the noise feature size is comparable with the image’s characteristic features, it is difficult to distinguish them in both original and Fourier domains. The simulated and experimental results with SNR=-15 dB and $\Delta$ = 40 mm in Fig. 3 (b) are 59% and 72.68%, respectively. However, when we reduce the cluster size by half ($\textrm{S}=52$ $\mu$m), the accuracy increases to 91.25% and 93.4% in simulation and experiment, respectively. Thus far, 40 LG modes {LG${}^{p}_{l}$} are chosen for the pump, with $l\in[-2,2]$ and $p\in[0,7]$, to achieve the accuracy for clear images while aiming at high efficiency. In practice, one may further optimize the mode bases for a particular application. To illustrate this opportunity, Fig. 4 plots the simulated accuracy using different sets of LG mode basis with $\textrm{SNR}=-15$ dB and $\textrm{S}=52$ $\mu$m. It shows that with more modes, the accuracy can be significantly higher when more than 40 LG modes are used. With 176 LG modes ($l\in[-5,5]$ and $p\in[0,15]$), the accuracy improved to 96% from 91%. On the other hand, when the number of LG modes is smaller than the 40 modes, the accuracy could drop quickly from 87% with $l\in[-2,2]$ and $p\in[0,4]$ to 48% with $l=0$ and $p\in[0,4]$). Figure 5: Simulated normalized confusion matrices using 49 HG pump modes with (a) $\Delta$ = 27 mm and (b) $\Delta$ = 40 mm. Lastly, we compare the choices of LG and HG pump modes to drive the upconversion for image classification. HG modes are another popular class of orthogonal mode basis which, unlike LG modes of rotational symmetric intensity, exhibit rectangular symmetry in intensity. Figure 5 shows the simulated results with the 49 lowest order HG modes, {HGmn} with $m$ and $n\in[0,7]$. The classification accuracy of the same data base is 88.5% with $\Delta=27$ mm and 96.75% with $\Delta=40$ $mm$, which are all inferior to the simulated results with 40 LG modes in Fig. 2(b) and (c). Therefore, LG modes are more efficient and suitable for the feature extractions of the handwritten digits than HG modes. Of course, in practice the optimum mode choice shall depend on the spatial characteristics of the target images. In summary, we have demonstrated a new approach to efficient and robust image classification using a hybrid system integrating nonlinear optics and a deep neural network. It combines mode selective frequency up-conversion and partial Fourier transformation to sample the signature features in the original and Fourier-transformed images through a single nonlinear optical stage. The partial Fourier transform is conveniently implemented by moving the up- conversion crystal away from the focal point. It significantly improves the classification accuracy for both clear and noise-contaminated images. Our results tap on an inherent advantage of coherent optics for complex machine learning tasks: the ease and flexibility of linear algebra operations for high volume data. The robustness of this approach against strong noise invites its future applications in challenging tasks such as identifying occluded targets, wide-field surveillance, and remote sensing under low visibility. Disclosures. 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IMPARTIAL GAMES WITH ENTAILING MOVES Urban<EMAIL_ADDRESS> School of Computing, National University of Singpore, Singapore Richard J<EMAIL_ADDRESS> Department of Mathematics and Statistics, Dalhousie University, Canada Carlos P. Santos333Partially supported by UID/MAT/04721/2019 strategic project<EMAIL_ADDRESS> Center for Functional Analysis, Linear Structures and Applications, University of Lisbon & ISEL--IPL ###### Abstract Combinatorial Game Theory has also been called ‘additive game theory’, whenever the analysis involves sums of independent game components. Such disjunctive sums invoke comparison between games, which allows abstract values to be assigned to them. However, there are rulesets with entailing moves that break the alternating play axiom and/or restrict the other player’s options within the disjunctive sum components. These situations are exemplified in the literature by a ruleset such as nimstring, a normal play variation of the classical children’s game dots & boxes, and top entails, an elegant ruleset introduced in the classical work Winning Ways, by Berlekamp Conway and Guy. Such rulesets fall outside the scope of the established normal play theory. Here, we axiomatize normal play via two new terminating games, ${\boldsymbol{\infty}}$ (Left wins) and $\overline{\infty}$ (Right wins), and a more general theory is achieved. We define affine impartial, which extends classical impartial games, and we analyze their algebra by extending the established Sprague-Grundy theory, with an accompanying minimum excluded rule. Solutions of nimstring and top entails are given to illustrate the theory. ## 1 Introduction Combinatorial Game Theory (CGT), as described in [1, 3, 5, 8], considers disjunctive sums of normal play games. In order to evaluate the outcome of a sum of such games, it suffices to analyze the components individually, and then add the individual values. However, some classical impartial rulesets, such as nimstring and top entails fall slightly outside the usual CGT axioms. In nimstring, certain moves require a player to play again, or carry-on, which is a violation of the alternating play axiom. And in top entails, certain moves enforce the next player to play in the same component, which violates the standard definition of a disjunctive sum. Thus, values of individual components is no longer a relevant measure, given the standard CGT axioms. The type of moves mentioned in this paragraph will be gathered under the term entailing moves.444Entailing means “involve something as a necessary or inevitable part or consequence”. The purpose of this paper is to extend impartial normal play games sufficiently to include games with entailing moves. While accomplishing this, we expand the classical Sprague-Grundy theory to fit this extension. We will rebuild the normal play axioms by using so-called terminating games, or infinities, ${\boldsymbol{\infty}}$ and $\overline{\infty}$. Here we focus on the impartial setting, and the general comprehensive theory for partizan games will appear in [6].555There are partizan rulesets, in the literature and in recreational play, with similar entailing and terminating moves. Probably the most prominent ones are the game of chess, and a version of go called atari go. These theories are called affine impartial and affine normal play respectively. Although we consider only impartial games in this paper, we will keep the players distinguished as Left and Right. In particular, Left wins if either player plays to ${\boldsymbol{\infty}}$, in any component, and Right wins in case of play to $\overline{\infty}$. Note that the normal play zero is restored by defining $0=\left\\{\overline{\infty}\\!\mid\\!{\boldsymbol{\infty}}\right\\}$, a first player losing position. It is well-known that, in classical Combinatorial Game Theory, the impartial values are nimbers. We will prove that there is exactly one more value modulo affine impartial, a game $\leftmoon$ , called the _moon_. This value was anticipated in the classical work Winning Ways, by Berlekamp, Conway and Guy. In [3], volume 2, page 398, one can read “A loony move is one that loses for a player, no matter what other components are.”. Before developing the theory, we illustrate how the infinities are used in the motivating rulesets, nimstring ([2, 3]) and top entails ([3], volume 2). Let us first briefly mention the organization of the paper. To facilitate the development of the new impartial theory, Section 2 considers the basic properties of unrestricted affine normal play, aiming for a game comparison result, Theorem 14.The affine impartial theory is developed in Section 3. The main result is Theorem 34 which shows that values in this extension are the nimbers plus one more value. Theorem 39 gives an algorithm to find the value of a given position, and notably, if there are no infinities in the options, then the nimbers are obtained by the usual mex-rule. We finish off with two case studies. In Section 4, we compute the value of an interesting nimstring position, anticipated in Section 1.1. In Section 5, we compute the values for top entail heaps of sizes $1$ through $12$, and Theorem 41 provides theoretical justification for computing top entails values. ### 1.1 The ruleset nimstring In nimstring, a player draws a line between two horizontally or vertically adjacent points, in a finite grid, not already joined by a line. If a player completes a $1\times 1$ square, then they must draw another line, and if they cannot do so, they lose. Figure 1 shows an example position, where no square can be completed in the next move. Later, through the new theory, we will see that the position $H$ equals $2$ modulo affine impartial. Figure 1: A nimstring position, $H$. In Figure 2, we show a position, $G$, with two options, one of which is an entailing move. Namely, if the the top bar is drawn, the next player continues, but if the middle bar is drawn, then the current player has to carry-on. Figure 2: A nimstring position, $G$, with its two options, a ‘double-box’ and an entailing carry-on position. When we develop the theory, we will see that the position $G$, to the left in Figure 2, is the abstract game $\displaystyle\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!0\right\\}\\!,0\\!\mid\\!\left\\{0\\!\mid\\!\overline{\infty}\right\\}\\!,0\right\\}.$ (1) The option 0 is obtained by drawing the top bar. The intuition for this is as follows: the player who moves inside the ‘double box’ loses, if $G$ is played in isolation, because they have to play again. If a player draws the middle bar in $G$, then they have to carry-on, and this is represented by the abstract option $\left\\{{\boldsymbol{\infty}}\\!\mid\\!0\right\\}$, if Left moved. There is an infinite urgency in this game: Right has to play here, or lose. And so, the effect is the desired: Left plays again, and alternating play is restored. Hence disjunctive sum play is also restored, within the affine impartial convention. Moreover, the Right option in this threat should be 0, because Left loses by playing this option if $G$ is played alone. If the sum is $G+H$, with $H$ as in Figure 1, then the next player wins, by playing this entailing middle bar in $G$. ### 1.2 The ruleset top entails Top entails is played on heaps of tokens. A player may either remove the top token from exactly one heap, or split a heap into two non-empty heaps. If the top token is removed from a heap, then the next move (in alternating play) must be played on the same heap. A heap with one token, say $H$, is a first player win, in any situation. Namely, a move in $H$ forces the opponent to play in the same heap, where no move remains. Note that the abstract game $H=\left\\{{\boldsymbol{\infty}}\\!\mid\\!\overline{\infty}\right\\}$ settles this behaviour. The player who moves first in $H$ wins independently of existence of other components. The point we wish to make here is that this abstract representation settles the problem of independecny of a heap of size one with other disjunctive sum components. Consider $G$, in Figure 3, a pile of size 3. There are two options, as depicted in Figures 4 and 5. Figure 3: A pile $G$ of top entails, of size 3. The option in Figure 4 splits $G$ into two piles and the next player’s options are unrestricted. By the terminating effect of playing in a heap of size one, this composite game should be equal to the game $H=\left\\{{\boldsymbol{\infty}}\\!\mid\\!\overline{\infty}\right\\}$. Figure 4: The game $G$ is split into two components. The option in Figure 5 is an entailing move, and the next player must continue in this component, even if other moves are available. Therefore, the game form of the entailing option in Figure 5 is $\left\\{{\boldsymbol{\infty}}\\!\mid\\!\bf 1+\bf 1,\bf 1_{\text{entail}}\right\\},$ if Left just moved, and where 1 denotes a heap of size one. The terminating threat forces Right to play here, instead of possibly using other options. Figure 5: An entailing option. Intuitively, either way of responding reduces to a game of the form $H=\left\\{{\boldsymbol{\infty}}\\!\mid\\!\overline{\infty}\right\\}$. In conclusion, the heap of size three should be equal to the game $H$, and disjunctive sum play has been restored. All this intuition will be rigorously justified in the coming complete theory for affine impartial play. It turns out that affine impartial games require only a small extension to the Sprague-Grundy theory. Namely, the game in (1), obtained from the nimstring position in Figure 2, equals the game $H=\left\\{\overline{\infty}\\!\mid\\!{\boldsymbol{\infty}}\right\\}$ in the previous paragraph, modulo affine impartial, and later we will devote the value ‘ $\leftmoon$ ’ to the equivalence class of such games. ## 2 Affine literal forms and order This section aims at Theorem 13, a comparison theorem for affine normal play that suffices for the purpose of this paper. We begin by defining the fundamental concepts for affine normal play, and we wait with the restriction to affine impartial until the next section. In classical Combinatorial Game Theory, the normal play forms, $\mathbb{Np}$, are recursively constructed from the empty set. The form $\\{\varnothing\,\|\,\varnothing\\}=0$ is the only form of day zero and the only form without options. The forms $\\{0\,\|\,\varnothing\\}=1$ , $\\{\varnothing\,\|\,0\\}=-1$, $\\{0\,\|\,0\\}=$, are born on day 1, and so on. The forms of affine normal play, denoted $\mathbb{Np}^{\infty}$, are recursively constructed from the games $\infty$ (infinity) and $\overline{\infty}$ (minus infinity) [6]. The forms $\infty$ and $\overline{\infty}$ are the only forms without options. The forms $\left\\{\overline{\infty}\\!\mid\\!{\boldsymbol{\infty}}\right\\}=0,\left\\{{\boldsymbol{\infty}}\\!\mid\\!\overline{\infty}\right\\}=\pm{\boldsymbol{\infty}},\left\\{{\boldsymbol{\infty}}\\!\mid\\!{\boldsymbol{\infty}}\right\\}$ and $\left\\{\overline{\infty}\\!\mid\\!\overline{\infty}\right\\}$ are born on day zero. And so on. The order of $\mathbb{Np}^{\infty}$ is defined in the standard way. Consider the four perfect play outcome classes $\mathscr{L}$ (Left wins), $\mathscr{N}$ (Next player wins), $\mathscr{P}$ (Previous player wins), and $\mathscr{R}$ (Right wins). From Left’s perspective, the first outcome is the best (she wins, regardless of playing first or second) and the fourth is the worst (she loses, regardless of whether playing first or second). On the other hand, regarding $\mathscr{N}$ and $\mathscr{P}$, the victory depends on playing first or second, so these outcomes are not comparable. These considerations explain the partial order in an ‘outcome diamond’: We write $G\in\mathscr{L}$, or equivalently $o(G)=\mathscr{L}$, if the outcome of $G\in\mathbb{Np}^{\infty}$ is Left wins, and so on. The evaluation of games in $\mathbb{Np}^{\infty}$ is based on the following axiomatic list: ###### Axiom 1 (Absorbing Nature of Infinities). The infinities satisfy 1. 1. $\infty\in\mathscr{L}$; 2. 2. $\overline{\infty}\in\mathscr{R}$; 3. 3. For all $X\in\mathbb{Np}^{\infty}\setminus\\{\overline{\infty}\\}$, $\infty+X=\infty$; 4. 4. For all $X\in\mathbb{Np}^{\infty}\setminus\\{{\boldsymbol{\infty}}\\}$, $\overline{\infty}+X=\overline{\infty}$; 5. 5. ‘$\infty+\overline{\infty}$’ is not defined. Addition of games is defined as usual, apart from items 3 and 4. The fifth item is natural in terms of perfect play, since if ${\boldsymbol{\infty}}$ appears, then $\overline{\infty}$ cannot appear and vice versa. The definitions of equality and partial order of games are based on the outcome diamond. ###### Definition 2 (Order and Equality of Games). Let $G,H\in\mathbb{Np}^{\infty}$. Then, $G\succcurlyeq H$ if, for every form $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$, $o(G+X)\geqslant o(H+X)$. Moreover $G=H$ if $G\succcurlyeq H$ and $H\succcurlyeq G$. Note that the exclusion of the infinities does not diminish the generality of the definition, but is necessary due to Axiom 5. As usual, we have the following observations. If $G=H$ then replacing $H$ by $G$ or $G$ by $H$ do not hurt the players under any circumstances. Similarly, if $G\succcurlyeq H$ then replacing $H$ by $G$ does not hurt Left, and replacing $G$ by $H$ does not hurt Right. ###### Theorem 3. Let $G\in\mathbb{Np}^{\infty}$. Then $\infty\succcurlyeq G$ and $G\succcurlyeq\overline{\infty}$. ###### Proof. If $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$ then, by Axiom 3, $\infty+X=\infty$. Hence, by Axiom 1, $o(\infty+X)=o(\infty)=\mathscr{L}$. Therefore, for every $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$, we have $o(\infty+X)\geqslant o(G+X)$, and so $\infty\succcurlyeq G$. Proving that $G\succcurlyeq\overline{\infty}$ is analogous. ∎ The concept of a check is fundamental to $\mathbb{Np}^{\infty}$. Indeed, this is an alternative, and perhaps more explicit, at least for those Chess playing readers, term for an entailing move, as seen in the Introduction. ###### Definition 4 (Check Games). Consider $G\in\mathbb{Np}^{\infty}$. If $\infty\in{G^{\mathcal{L}}}$ ($\overline{\infty}\in{G^{\mathcal{R}}}$) then $G$ is a _Left-check_ (_Right- check_). If $G$ is a Left-check or a Right-check then $G$ is a _check_. Denote by $G^{\mathrel{\mathop{L}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\rightarrow$}\vss}}}}$ ($G^{\mathrel{\mathop{R}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\leftarrow$}\vss}}}}$) a Left (Right) option of $G$ that is a Left-check (Right-check). Of course, all checks are asymmetric, apart from the ‘trivial check’, $\left\\{{\boldsymbol{\infty}}\\!\mid\\!\overline{\infty}\right\\}$. A player would not use this check, because the opponent ‘check mates’ by defending. ###### Definition 5 (Quiet Games). Let $G\in\mathbb{Np}^{\infty}$. If $G\neq{\boldsymbol{\infty}}$ ($G\neq\overline{\infty}$) and $G$ is not a Left-check (Right-check) then $G$ is _Left-quiet_ (_Right-quiet_). If $G$ is Left-quiet and Right-quiet then $G$ is _quiet_. ###### Definition 6 (Conway Forms and Games). A game $G\in\mathbb{Np}^{\infty}$ is a _Conway form_ if $G\not\in\\{\infty,\overline{\infty}\\}$, and $G$ has no checks as followers. Let $\mathbb{Np}^{\rm C}\subseteq\mathbb{Np}^{\infty}$ denote the substructure of Conway forms. A game $G\in\mathbb{Np}^{\infty}$ is a _Conway game_ if it equals a Conway form. ###### Example 7. The game $G=\left\\{\left\\{\overline{\infty}\\!\mid\\!\infty\right\\}\\!\mid\\!\left\\{\overline{\infty}\\!\mid\\!\infty\right\\}\right\\}=\left\\{0\\!\mid\\!0\right\\}=$ is a Conway form (no checks as followers). The game $G^{\prime}=\left\\{\left\\{\infty\\!\mid\\!\right\\}\\!\mid\\!\left\\{\\!\mid\\!\overline{\infty}\right\\}\right\\}$ is not a Conway form because there are checks as followers. However, later, we will see that $G^{\prime}=G$. Therefore, $G^{\prime}$ is a Conway game. In general, when we say form, we mean the literal form, and when we say game, we usually mean (any member in) the full equivalence class of games. When we write $G\in\mathbb{Np}^{\infty}$, we usually refer to the literal form, but the context may decide. Some classical theorems are still available in $\mathbb{Np}^{\infty}$. ###### Theorem 8 (Fundamental Theorem of Affine Normal Play). If $G\in\mathbb{Np}^{\infty}$ then $G\succcurlyeq 0$ if and only if $G\in\mathscr{L}\cup\mathscr{P}$. ###### Proof. Assume that $G\succcurlyeq 0$. We have $0\in\mathscr{P}$, and so, by order of outcomes, $G\in\mathscr{L}\cup\mathscr{P}$. Suppose now that $G\in\mathscr{L}\cup\mathscr{P}$. If $G=\infty$, by Theorem 3, $G\succcurlyeq 0$; hence, assume $G\neq\infty$. Let $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$. If, playing first, Left wins $X$ with the option $X^{L}$, then she also wins $G+X$ with the option $G+X^{L}$. Essentially, she mimics the strategy used when $X$ is played alone, answering locally when Right plays in $G$. Due to the assumption $G\in\mathscr{L}\cup\mathscr{P}$, this is a winning strategy for Left in $G+X$. If Left, playing second, wins $X$. Then, on $G+X$, she can respond to each of Right’s moves locally, with a winning move on the same component, because $G\in\mathscr{L}\cup\mathscr{P}$. Thus Left can win $G+X$ playing second. Therefore, $o(G+X)\geqslant o(X)$ and so, $G\succcurlyeq 0$. ∎ ###### Corollary 9 (Order-Outcome Bijection). If $G\in\mathbb{Np}^{\infty}$ then • $G\succ 0$ if and only if $G\in\mathscr{L}$; * • $G=0$ if and only if $G\in\mathscr{P}$; * • $G\operatorname{\\|}0$ if and only if $G\in\mathscr{N}$; * • $G\prec 0$ if and only if $G\in\mathscr{R}$. ###### Proof. The statement of Theorem 8 can equivalently be “$G\preccurlyeq 0$ if and only if $G\in\mathscr{R}\cup\mathscr{P}$”, so we can use that fact too. Suppose that $G\succ 0$. By Theorem 8, $G\in\mathscr{L}\cup\mathscr{P}$. But, we cannot have $G\in\mathscr{P}$, for otherwise $G\in\mathscr{R}\cup\mathscr{P}$ and $G\preccurlyeq 0$. Therefore, $G\in\mathscr{L}$. Conversely, suppose that $G\in\mathscr{L}$. By Theorem 8, we have $G\succcurlyeq 0$. But, we cannot have $G=0$, for otherwise $G\preccurlyeq 0$, and $G\in\mathscr{R}\cup\mathscr{P}$. Hence, $G\succ 0$. Thus, the first equivalence holds. The proof of the fourth equivalence is analogous. For the second equivalence, if $G=0$, then $G\succcurlyeq 0\;\wedge\;G\preccurlyeq 0$. So, $G\in(\mathscr{L}\cup\mathscr{P})\cap(\mathscr{R}\cup\mathscr{P})=\mathscr{P}$. The third equivalence is a consequence of eliminating all other possibilities. ∎ It is known that $\mathbb{Np}$ is a group. By Corollary 9 we may deduce that $\mathbb{Np}^{\infty}$ is only a monoid. Namely, if $G=\\{\infty\,\|\,0\\}$ then, for any $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$, $G+X\in\mathscr{L}\cup\mathscr{N}$ (playing first, Left wins). Hence, for all $X$, $G+X\neq 0$ and $G$ is non-invertible. Thus, in general, the comparison of $G$ with $H$ cannot be done by playing the game ‘$G-H$’, because, sometimes, ‘$-H$’ does not exist. However, the following theorem shows that not everything is lost. The _conjugate_ of a given game switches roles of the players. ###### Definition 10 (Conjugate). The conjugate of $G\in{\mathbb{Np}^{\infty}}$ is $\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}\,=\begin{cases}\overline{\infty},&\mbox{if $G=\infty$}\\\ \infty,&\mbox{if $G=\overline{\infty}$}\\\ \left\\{\mathrel{\mathop{{G^{\mathcal{R}}}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}\,\\!\mid\\!\;\mathrel{\mathop{{G^{\mathcal{L}}}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}\right\\},&\mbox{otherwise},\end{cases}$ where $\mathrel{\mathop{{G^{\mathcal{L}}}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$ denotes the set of literal forms $\mathrel{\mathop{G^{L}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$, for $G^{L}\in{G^{\mathcal{L}}}$, and similarly for ${G^{\mathcal{R}}}$. ###### Theorem 11. If $G\in\mathbb{Np}^{\infty}$ is a Conway game, then $G$ is invertible and $-G=\,\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$. ###### Proof. Suppose first that $G$ is a Conway form. If $G=0$ then $\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}=0$, and thetheorem holds. Otherwise, let us verify that $G+\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$ is a $\mathscr{P}$-position. If Left, playing first, chooses $G^{L}+\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$, because this game is not $\infty$ ($G$ is not a check), Right can answer with $G^{L}+\mathrel{\mathop{G^{L}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$ and, by induction, because $G^{L}$ is a Conway form with no checks as followers, that option is equal to zero. Because of that, by Corollary 9, that option is a $\mathscr{P}$-position, and Right wins. Analogous arguments work for the other options of the first player, and so, $G+\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$ is a $\mathscr{P}$-position. Again, by Corollary 9, $G+\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}=0$. Suppose now that $G$ is not a Conway form. Because it is a Conway game, by definition, it is equal to some $G^{\prime}\in\mathbb{Np}^{\rm C}$. The first paragraph proved that $G^{\prime}+\mathrel{\mathop{G^{\prime}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}=0$. Also, by symmetry, $\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$ is equal to $\mathrel{\mathop{G^{\prime}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$. Therefore, $G^{\prime}+\mathrel{\mathop{G^{\prime}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}=0$ implies $G+\mathrel{\mathop{G}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}=0$. ∎ ###### Lemma 12. Let $G,H\in\mathbb{Np}^{\infty}$, and let $J$ be an invertible form of $\mathbb{Np}^{\infty}$. Then $G\succcurlyeq H\Leftrightarrow G+J\succcurlyeq H+J.$ ###### Proof. ($\Rightarrow$) Consider any $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$ and let $X^{\prime}=J+X$. Since $J$ is invertible, $J$ is neither $\infty$ nor $\overline{\infty}$, and so, $X^{\prime}$ is neither $\infty$ nor $\overline{\infty}$. Definition of order implies $o(G+X^{\prime})\geqslant o(H+X^{\prime})$, that is $o(G+J+X))\geqslant o(H+J+X)$. Thus, the arbitrariness of $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$ implies $G+J\succcurlyeq H+J$. ($\Leftarrow$) Consider any $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$ and let $X^{\prime}=-J+X$ ($J$ is invertible, i.e. $-J$ exists and $J-J=0$). Since $-J$ is invertible, $-J$ is neither $\infty$ nor $\overline{\infty}$, and so, $X^{\prime}$ is neither $\infty$ nor $\overline{\infty}$. By definition of order, $o(G+J+X^{\prime})\geqslant o((H+J+X^{\prime})$, that is $o(G+J-J+X)\geqslant o(H+J-J+X)$. Hence, $o(G+X)\geqslant o(H+X)$, and so, given the arbitrariness of $X\in\mathbb{Np}^{\infty}\setminus\\{\infty,\overline{\infty}\\}$, $G\succcurlyeq H$. ∎ ###### Theorem 13. Let $G$ be any form of $\mathbb{Np}^{\infty}$ and suppose that $H$ is an invertible form of $\mathbb{Np}^{\infty}$. Then, $G\succcurlyeq H\Leftrightarrow G-H\in\mathscr{L}\cup\mathscr{P}\text{ and }G=H\Leftrightarrow G-H\in\mathscr{P}.$ ###### Proof. By Lemma 12, $G\succcurlyeq H\Leftrightarrow G-H\succcurlyeq H-H$. Therefore, we have $G\succcurlyeq H\Leftrightarrow G-H\succcurlyeq 0$. By Theorem 8, this is the same as $G\succcurlyeq H\Leftrightarrow G-H\in\mathscr{L}\cup\mathscr{P}$. Finally, $G=H\Leftrightarrow G-H\in\mathscr{P}$, by $G\succcurlyeq H\wedge H\succcurlyeq G$. ∎ ###### Theorem 14. Let $G$ be any form of $\mathbb{Np}^{\infty}$ and let $H\in\mathbb{Np}^{\infty}$ be a Conway game. Then * • $G\succcurlyeq H$ if and only if $G\,+\mathrel{\mathop{H}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}\,\in\mathscr{L}\cup\mathscr{P}$ * • $G=H$ if and only if $G\,+\mathrel{\mathop{H}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}\,\in\mathscr{P}$. ###### Proof. These are direct consequences of Theorems 11 and 13. ∎ In a follow up paper [6], where we study the full game space $\mathbb{Np}^{\infty}$, we provide a solution of the general case of $G\succcurlyeq H$. ## 3 Affine impartial theory In order to propose an extension of the Sprague-Grundy theory, we first define the concept of an affine impartial game.666In terms of ruleset: here ‘affine impartial’ is an abbreviation of affine normal play impartial, in the sense that if the player to move cannot complete their move they lose. Of course, rulesets like nimstring should be impartial. ###### Definition 15 (Symmetric Game). Consider a form $G\in{\mathbb{Np}^{\infty}}$. Then $G$ is _symmetric_ if $G\not\in\\{\infty,\overline{\infty}\\}$ and ${G^{\mathcal{R}}}=\;\mathrel{\mathop{{G^{\mathcal{L}}}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$. ###### Definition 16 (Affine Impartial). A form $G\in\mathbb{Np}^{\infty}$ is _affine impartial_ if it is symmetric and all quiet followers of $G$ are symmetric. The subset of affine impartial games is $\mathbb{Im}^{\boldsymbol{\infty}}\subset{\mathbb{Np}^{\infty}}$. Of course, a non-quiet game either has no option, or is a check, and so (unless a trivial check) is by definition asymmetric. But this is the only exception of symmetry in the world of affine impartial impartial games. It is easy to check that $\mathbb{Im}^{\boldsymbol{\infty}}$ satisfies the standard closure properties of combinatorial games, i.e. closure of taking options, addition, and conjugates. The following result must hold for any class of games that claims to be “impartial”. ###### Theorem 17 (Affine Impartial Outcomes). If $G$ is a symmetric form, then $G\in\mathscr{N}\cup\mathscr{P}$. ###### Proof. This proof uses a strategy-stealing argument. Suppose that $G\in\mathscr{L}$. Then Left wins $G$ playing first with some option $G^{L}$. Hence, by symmetry, Right wins $G$ playing first with $\mathrel{\mathop{G^{L}}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\curvearrowleftright$}\vss}}}$. That contradicts $G\in\mathscr{L}$. A similar argument holds against $G\in\mathscr{R}$. ∎ We want to restrict our analysis to $\mathbb{Im}^{\boldsymbol{\infty}}$. Therefore, we define equality modulo $\mathbb{Im}^{\boldsymbol{\infty}}$. ###### Definition 18 (Impartial Equality). Consider forms $G,H\in\mathbb{Im}^{\boldsymbol{\infty}}$. Then, $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}H$ if, for every form $X\in\mathbb{Im}^{\boldsymbol{\infty}}$, $o(G+X)=o(H+X)$. ###### Observation 19. Of course, $G=H$ in $\mathbb{Np}^{\infty}$ implies $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}H$. The opposite direction is not true. We can have $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}H$ and $G\neq H$ in $\mathbb{Np}^{\infty}$, if the there is no distinguishing game in $\mathbb{Im}^{\boldsymbol{\infty}}$. A simple example is $G=\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!0\right\\}\\!,0\\!\mid\\!\left\\{0\\!\mid\\!\overline{\infty}\right\\}\\!,0\right\\}$ and $H=\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!\right\\},\\!\mid\\!\left\\{\\!\mid\\!\overline{\infty}\right\\},\right\\}$. As we will see, these games are indistinguishable modulo $\mathbb{Im}^{\boldsymbol{\infty}}$. However, the game $X=\left\\{0\\!\mid\\!-1\right\\}$ distinguishes them in ${\mathbb{Np}^{\infty}}$; playing first, Left wins $G+X$, but loses $H+X$. It is easy to verify if a form in $\mathbb{Im}^{\boldsymbol{\infty}}$ equals a nimber. ###### Theorem 20 (Nimbers). Let $G\in\mathbb{Im}^{\boldsymbol{\infty}}$. Then, $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}n$ if and only if $G+n\in\mathscr{P}$. ###### Proof. Suppose that $G+n\in\mathscr{P}$. By Theorem 14, $G=n$ modulo $\mathbb{Np}^{\infty}$, and so $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}n$. Suppose now that $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}n$. By Theorem 17, $G+n\in\mathscr{N}\cup\mathscr{P}$, since impartiality is closed under addition. If $G+n\in\mathscr{N}$, then $G+n\in\mathscr{N}$ and $n+n\in\mathscr{P}$, contradicting $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}n$. Hence, $G+n\in\mathscr{P}$. ∎ ###### Notation 21. Let $\mathbb{nim}\subseteq\mathbb{Im}^{\boldsymbol{\infty}}$ denote the subset of affine impartial games that equal nimbers. It is well-known that, in classical Combinatorial Game Theory, the impartial values are nimbers. We will prove that there is exactly one more value modulo $\mathbb{Im}^{\boldsymbol{\infty}}$, a game $\leftmoon$ , called _moon_. In [3], volume 2, page 398, one can read “A loony move is one that loses for a player, no matter what other components are.”. The following general definition is motivated by that idea. ###### Definition 22 (Loony Game). A game $G\in{\mathbb{Np}^{\infty}}$ is _loony_ if, for all quiet $X\in{\mathbb{Np}^{\infty}}\cap(\mathscr{N}\cup\mathscr{P})$, $G+X\in\mathscr{N}$. Thus, in our interpretation, a ‘loony move’ exposes a loony game. There are no loony games in $\mathbb{Np}$. Suppose that $G\in{\mathbb{Np}}\cap(\mathscr{P}\cup\mathscr{L}\cup\mathscr{R})$ is a loony game. Of course, $G+0\in\mathscr{P}\cup\mathscr{L}\cup\mathscr{R}$ and that is a contradiction. Suppose that $G\in{\mathbb{Np}}\cap\mathscr{N}$ is a loony game. In that case, if $n$ is large enough, $G+\left\\{n\\!\mid\\!0\right\\}\in\mathscr{L}$, and that is a contradiction, since $\left\\{n\\!\mid\\!0\right\\}\in\mathscr{N}$ is quiet. There are loony games in $\mathbb{Np}^{\infty}$ . The obvious one is $\pm\infty=\left\\{{\boldsymbol{\infty}}\\!\mid\\!\overline{\infty}\right\\}$, but we can also have impartial quiet loony moves. Consider $G=\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!0\right\\}\\!,0\\!\mid\\!\left\\{0\\!\mid\\!\overline{\infty}\right\\}\\!,0\right\\}$ and a quiet $X\in\mathbb{Np}^{\infty}$ such that $X\in\mathscr{P}\cup\mathscr{N}$. If $X\in\mathscr{P}$, the first player wins moving to $X$. If $X\in\mathscr{N}$, the first player wins moving to $\\{\infty\,\|\,0\\}+X$ (Left) or to $\\{0\,\|\,\overline{\infty}\\}+X$ (Right). ###### Notation 23. The moon is the game form $\scalebox{1.1}{$\leftmoon$}=\left\\{{\boldsymbol{\infty}}\\!\mid\\!\overline{\infty}\right\\}$. When a player moves to $\scalebox{1.1}{$\leftmoon$}+X$, for any $X\in\mathscr{N}\cup\mathscr{P}$, he “goes to the moon” and loses. ###### Theorem 24 (Loony Uniqueness). All loony games are equal modulo $\mathbb{Im}^{\boldsymbol{\infty}}$. ###### Proof. Consider $G$ and $G^{\prime}$, two loony games. We know that all quiet $X\in\mathbb{Im}^{\boldsymbol{\infty}}$ belong to $\mathscr{N}\cup\mathscr{P}$. By definition of a loony game, we have $G+X\in\mathscr{N}$ and $G^{\prime}+X\in\mathscr{N}$. On the other hand, if $X\in\mathbb{Im}^{\boldsymbol{\infty}}$ is not quiet then $\infty\in X^{\mathcal{L}}$ and $\overline{\infty}\in X^{\mathcal{R}}$, and hence $G+X\in\mathscr{N}$ and $G^{\prime}+X\in\mathscr{N}$. In all cases, $o(G+X)=o(G^{\prime}+X)=\mathscr{N}$ and the theorem holds. ∎ ###### Observation 25. Two loony games may be different modulo ${\mathbb{Np}^{\infty}}$, but equal modulo $\mathbb{Im}^{\boldsymbol{\infty}}$. The games $\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!0\right\\}\\!,0\\!\mid\\!\left\\{0\\!\mid\\!\overline{\infty}\right\\}\\!,0\right\\}$ and $\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!2\right\\},0\\!\mid\\!\left\\{-2\\!\mid\\!\overline{\infty}\right\\},0\right\\}$ are loony. These games are different modulo $\mathbb{Np}^{\infty}$. Left, playing first, loses $\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!0\right\\}\\!,0\\!\mid\\!\left\\{0\\!\mid\\!\overline{\infty}\right\\}\\!,0\right\\}-1$ and wins $\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!2\right\\},0\\!\mid\\!\left\\{-2\\!\mid\\!\overline{\infty}\right\\},0\right\\}-1$. However, as will follow by theory developed here, one cannot distinguish these two games modulo $\mathbb{Im}^{\boldsymbol{\infty}}$. In order to prove an affine impartial minimum excluded rule, we separate the options into two classes. ###### Definition 26 (Immediate Nimbers). Let $G\in\mathbb{Im}^{\boldsymbol{\infty}}$. The set of $G$-_immediate nimbers_ , denoted $S_{G}$ is the set $S_{G}={G^{\mathcal{L}}}\cap\mathbb{nim}$. Not that, by symmetry, $S_{G}={G^{\mathcal{R}}}\cap\mathbb{nim}$, and note that $S_{\scalebox{0.9}{$\leftmoon$}}=\varnothing$. ###### Definition 27 (Protected Nimbers). Conisder a game form $G\in\mathbb{Im}^{\boldsymbol{\infty}}$. The set of $G$-_protected nimbers_ $P_{G}$ is 1. 1. $P_{G}=\mathbb{nim}$, if $\infty\in{G^{\mathcal{L}}}$; 2. 2. $P_{G}=\\{n:G^{\mathrel{\mathop{L}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\rightarrow$}\vss}}}}+n\in\mathscr{L},G^{\mathrel{\mathop{L}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\rightarrow$}\vss}}}}\in{G^{\mathcal{L}}}\\}$, otherwise. The second item says: if $\infty\not\in{G^{\mathcal{L}}}$ then $n\in P_{G}$ if there is a check $G^{\mathrel{\mathop{L}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\rightarrow$}\vss}}}}=\\{\infty\,\|\,G^{L\mathcal{R}}\\}\in{G^{\mathcal{L}}}$ such that Right, playing first, loses $G^{\mathrel{\mathop{L}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\rightarrow$}\vss}}}}+n$. That is, playing first, Left is protected against those nimbers in a disjunctive sum. Similar to Definition 26, we could have defined $P_{G}$ with respect to Right options, to obtain the same set. Note that $P_{\scalebox{0.9}{$\leftmoon$}}=\mathbb{nim}$. This statement holds for the literal form $\scalebox{1.1}{$\leftmoon$}=\pm\infty$. However, one can show that by using instead the form $\scalebox{1.1}{$\leftmoon$}=\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!0\right\\}\\!,0\\!\mid\\!\left\\{0\\!\mid\\!\overline{\infty}\right\\}\\!,0\right\\}$, as in (1), then $P_{\scalebox{0.9}{$\leftmoon$}}=\mathbb{nim}\setminus\\{0\\}$. The output of “protected” is sensitive to which form we choose. When the underlying game form is understood, we simply refer to the immediate and protected nimbers, respectively. ###### Example 28. Let $G\in\mathbb{Im}^{\boldsymbol{\infty}}$ be such that the Left options are $0$, $2$, and $\\{\infty\,\|\,\\{\,\|\,\overline{\infty}\\},0\\}$. Of course, $S_{G}=\\{0,2\\}$. On the other hand, playing first, Left can use the check to win $G+$. Because of that, $P_{G}=\\{\\}$. An important observation is that, although Left is protected against the nimber $$, Left cannot force a Left move to $$ in $G$. But if Right moves to 0, Left wins $G+$ anyway Sometimes, Right can manoeuvre Left’s eventual play to a nimber, or worse, via a sequence of ‘check upon check’. ###### Definition 29 (Manoeuvrable Form). A quiet form $G\in\mathbb{Im}^{\boldsymbol{\infty}}$ is _manoeuvrable_ if after each Left move that is neither a nimber nor $\overline{\infty}$, Right can force, with checks, a Left move to a nimber or a move by either player to $\overline{\infty}$. ###### Example 30. The form $G=\\{2,\\{\infty\,\|\,\\{0,4\,\|\,\overline{\infty}\\},0\\}\,\|\,2,\\{0,\\{\infty\,\|\,0,4\\}\|\,\overline{\infty}\\}\\}$ is manoeuvrable. If Left avoids the immediate nimber $2$, by checking, then Right can still force Left to move to one of the nimbers $0$ or $4$. ###### Lemma 31. If $G\in\mathbb{Im}^{\boldsymbol{\infty}}$ is manoeuvrable, then $P_{G}$ is finite. ###### Proof. After a Left first move in $G$, if needed, Right can force with checks a Left move to a nimber or a move by either player to $\overline{\infty}$. Let $C$ be the set of nimbers that can arise through this forcing strategy by Right. Then $C$ is finite, because we study short games. Let $n$ be a nimber such that, for all $m\in C$, we have $n>m$. In $G+n$, after a first check, say, to $G^{L}+n$, Right forces with checks a move by either player to $\overline{\infty}$ or a Left move to $m+n$ ($n>m$). In the second case, after the sequence, Right wins with a TweedleDee-TweedleDum move. Thus, Left can protect against at most a finite number of nimbers. That explains why $P_{G}$ is finite in case of manoeuvrable games. ∎ Let $\mathrm{mex}(X)$ denote the smallest nonnegative integer not in $X$. Let $\mathcal{G}$ denote the set of Sprague-Grundy values of a set of nimbers, i.e. if $S=\\{n_{i}\\}$, then $\mathcal{G}(S)=\\{n_{i}\\}$. ###### Lemma 32. If $G\in\mathbb{Im}^{\boldsymbol{\infty}}$ is manoeuvrable then $G$ equals the nimber $n$, where $n=\mathrm{mex}(\mathcal{G}(S_{G}\cup P_{G}))$. ###### Proof. By Lemma 31, we know that $S_{G}\cup P_{G}$ is finite. Let $n=\mathrm{mex}(\mathcal{G}(S_{G}\cup P_{G}))$. Let us argue that the game $G+n\in\mathscr{P}$. If the first player moves in $G$ to a nimber $m\in S_{G}$, because $n$ is excluded from $\mathcal{G}(S_{G})$, he loses. If the first player moves in $G$ to a quiet not nimber $G^{\prime}$, because $G^{\prime}$ is not a nimber, $G^{\prime}+n\in\mathscr{N}$ (Theorem 20), and the first player also loses. If the first player moves in $G$, giving a check, because $n$ is excluded from $\mathcal{G}(P_{G})$, he also loses. Finally, if the first player moves to $G+n^{\prime}$ ($n^{\prime}<n$), because $n$ is the minimum excluded from $\mathcal{G}(S_{G}\cup P_{G})$, he loses because the opponent has a direct TweedleDee-TweedleDum move or wins with a check. Hence, by Theorem 20, $G=n$. ∎ ###### Lemma 33. If $G,H\in\mathbb{Im}^{\boldsymbol{\infty}}$ are not nimbers, then $G+H\in\mathscr{N}$. ###### Proof. Consider $G,H\in\mathbb{Im}^{\boldsymbol{\infty}}\setminus\mathbb{nim}$. For a contradiction, assume that the sum of the birthdays, $b=b(G)+b(H)$, is the smallest possible such that $G+H\in\mathscr{P}$. Note that, by the assumptions on $G$ and $H$, $b>0$. Without loss of generality, we will analyze the move from $G+H$ to $G+H^{L}$. First, we prove two claims that concern local play in $G$ and $H$ respectively. Claim 1. Playing second in $G$, Left can avoid Left moves to nimbers and moves by either player to $\overline{\infty}$ until the first Right-quiet move. Proof of Claim 1. Suppose that Right, playing first in $G$, could force a Left move to a nimber or a move by either player to $\overline{\infty}$. If so, in $G+H$, by giving checks in $G$, Right could force some $G^{\mathrel{\mathop{R}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\leftarrow$}\vss}}}L\cdots\mathrel{\mathop{R}\limits^{\vbox to0.0pt{\kern-1.5pt\hbox{$\scriptstyle\leftarrow$}\vss}}}L}+H=n+H$ (Right’s turn) or a move by either player to $\overline{\infty}+H$. Of course, the second situation would be a victory for Right. Regarding the first case, at that moment, the position would be $n+H$. And, because $H$ is not a nimber, by Theorem 20, we would have $n+H\in\mathscr{N}$, which is a winning move for Right. In either case, Right, as first player, would win. That would contradict $G+H\in\mathscr{P}$. Claim 2. There is an $H^{L}$ such that Left can avoid Left moves to nimbers and moves by either player to $\overline{\infty}$, until the first Right-quiet move. Proof of Claim 2. This is exactly the same as saying that $H$ is non- manoeuvrable. If it was manoeuvrable, by Lemma 32, it would be a nimber, and we would have a contradiction again. Let us return to the move from $G+H$ to $G+H^{L}$. Because $G+H\in\mathscr{P}$, Right has a winning move from $G+H^{L}$. But, by Claims 1 and 2, Left can play such that, at any stage before a Right-quiet move, Right is moving on $g+h$, where $g$ is a follower of $G$ and $h$ is a follower of $H^{L}$, such that neither $g$ nor $h$ is a nimber. Either way, by assumption, there is a winning quiet Right-move $g^{R}+h$ or $g+h^{R}$. Since these are impartial games, we must have $g^{R}+h\in\mathscr{P}$ or $g+h^{R}\in\mathscr{P}$. But, because $h$ and $g$ are not nimbers, it follows by Theorem 20 that $g^{R}$ and $h^{R}$ are not nimbers. Therefore, we have $g^{R}+h\in\mathscr{P}$ or $g+h^{R}\in\mathscr{P}$ with both components not nimbers. But this contradicts the smallest birthday assumption. The result follows. ∎ ###### Theorem 34 (Affine Impartial Values). Every affine impartial form equals a nimber or the game $\scalebox{1.1}{$\leftmoon$}\pmod{\mathbb{Im}^{\boldsymbol{\infty}}}$. ###### Proof. Let $G\in\mathbb{Im}^{\boldsymbol{\infty}}$. If there is some $n$ such that $G+n\in\mathscr{P}$, then $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}n$, by Theorem 20. Suppose next that $G+n\in\mathscr{N}$, for all $n$, so that $G$ does not equal a nimber modulo $\mathbb{Im}^{\boldsymbol{\infty}}$. By Lemma 33, for all $X\in\mathbb{Im}^{\boldsymbol{\infty}}\setminus\mathbb{nim}$, we also have $G+X\in\mathscr{N}$. Hence, for all $X\in\mathbb{Im}^{\boldsymbol{\infty}}$, we have $G+X\in\mathscr{N}$, and therefore $G$ is a loony game. Because, by Theorem 24, all loony games are equal modulo $\mathbb{Im}^{\boldsymbol{\infty}}$, and $\leftmoon$ is a loony game, we have $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}\scalebox{1.1}{$\leftmoon$}$. ∎ ###### Observation 35. A form can be loony modulo $\mathbb{Im}^{\boldsymbol{\infty}}$ and not loony modulo ${\mathbb{Np}^{\infty}}$. An example is the form $G=\left\\{,\left\\{{\boldsymbol{\infty}}\\!\mid\\!\right\\}\\!\mid\\!,\left\\{\\!\mid\\!\overline{\infty}\right\\}\right\\}$. This game is not loony modulo ${\mathbb{Np}^{\infty}}$ because, if $X=\left\\{0\\!\mid\\!-1\right\\}\in\mathscr{N}$, playing first, Left loses $G+X$. However, $G=_{\mathbb{Im}^{\boldsymbol{\infty}}}\scalebox{1.1}{$\leftmoon$}$. This follows, by Theorem 34, since $G$ does not equal any nimber; if Right starts $G+n$, he wins, by an appropriate parity consideration. ###### Theorem 36. The game $\leftmoon$ is absorbing modulo $\mathbb{Im}^{\boldsymbol{\infty}}$, that is, $\scalebox{1.1}{$\leftmoon$}+Y=_{\mathbb{Im}^{\boldsymbol{\infty}}}\scalebox{1.1}{$\leftmoon$}$, for all $Y\in\mathbb{Im}^{\boldsymbol{\infty}}$. ###### Proof. Since $\scalebox{1.1}{$\leftmoon$}=\pm\infty$, regardless of what $X\in\mathbb{Im}^{\boldsymbol{\infty}}$ is, the first player wins both $\scalebox{1.1}{$\leftmoon$}+Y+X$ and $\scalebox{1.1}{$\leftmoon$}+X$. Therefore, by definition of equality of games, $\scalebox{1.1}{$\leftmoon$}+Y=_{\mathbb{Im}^{\boldsymbol{\infty}}}\scalebox{1.1}{$\leftmoon$}$. ∎ ###### Corollary 37. The game $\leftmoon$ is an idempotent modulo $\mathbb{Im}^{\boldsymbol{\infty}}$, that is, $\scalebox{1.1}{$\leftmoon$}+\scalebox{1.1}{$\leftmoon$}=_{\mathbb{Im}^{\boldsymbol{\infty}}}\scalebox{1.1}{$\leftmoon$}$. ###### Proof. This is a trivial consequence of Theorem 36. ∎ ###### Definition 38. The Sprague-Grundy value of the moon is $\mathcal{G}($ $\leftmoon$ $)=\infty$. The following theorem explains how the Sparague-Grundy value of $G\in\mathbb{Im}^{\boldsymbol{\infty}}$ is determined by the set $S_{G}\cup P_{G}$. ###### Theorem 39 (Affine Impartial Minimum Excluded Rule). Let $G\in\mathbb{Im}^{\boldsymbol{\infty}}$. We have the following possibilities: * • If $S_{G}\cup P_{G}=\mathbb{nim}$, then $G=\scalebox{1.1}{$\leftmoon$}$ and $\mathrm{mex}(\mathcal{G}(S_{G}\cup P_{G}))=\infty$; * • If $S_{G}\cup P_{G}\neq\mathbb{nim}$, then $G=\left(\mathrm{mex}(\mathcal{G}(S_{G}\cup P_{G}))\right)$. ###### Proof. If $S_{G}\cup P_{G}=\mathbb{nim}$, we have $G+n\in\mathscr{N}$ for all $n$. Because of that, $G$ is not a nimber and, by Theorem 34, $G=\scalebox{1.1}{$\leftmoon$}$. If $S_{G}\cup P_{G}\neq\mathbb{nim}$, we use the same argument of the proof of Lemma 32. ∎ ###### Corollary 40. If all the options of a game $G\in\mathbb{Im}^{\boldsymbol{\infty}}$ are quiet then $G$ is a nimber. ###### Proof. If all the options of a game $G\in\mathbb{Im}^{\boldsymbol{\infty}}$ are quiet, then $P_{G}=\varnothing$. Therefore, $S_{G}\cup P_{G}=S_{G}\neq\mathbb{nim}$ and, by Theorem 39, $G=\left(\mathrm{mex}(\mathcal{G}(S_{G}))\right)$. ∎ ## 4 Case study: nimstring In the introduction, we promised to show that the following component equals $2$. Study the positions: All (a), (b), (c), (d), and (e) are $\mathcal{P}$-positions. The game value of (f) is $\scalebox{1.1}{$\leftmoon$}=\left\\{\left\\{{\boldsymbol{\infty}}\\!\mid\\!0\right\\}\\!,0\\!\mid\\!\left\\{0\\!\mid\\!\overline{\infty}\right\\}\\!,0\right\\}$. Other positions that equal $\leftmoon$ are the following. In the position (l), the central horizontal move is the option (d), that is equal to $0$. The other options are (f) and (g), that are equal to $\leftmoon$ . Therefore, the literal form is $l=\left\\{0,\scalebox{1.1}{$\leftmoon$},\scalebox{1.1}{$\leftmoon$}\\!\mid\\!0,\scalebox{1.1}{$\leftmoon$},\scalebox{1.1}{$\leftmoon$}\right\\}$ with $S_{l}=\\{0\\}$, and $P_{l}=\varnothing$. Applying the affine impartial minimum excluded rule, we conclude that the position is $$. The position (m) is also equal to $$, i.e. $0+$. Now, we are ready for (n), a more complex situation. The literal form is $n=\\{h,i,k,\\{\infty\,\|\,l\\}\,\|\,h,i,k,\\{l\,\|\,\overline{\infty}\\}\\},$ that is, $n=\\{\leftmoon,\leftmoon,\leftmoon,\\{\infty\,\|\,\\}\,\|\,\leftmoon,\leftmoon,\leftmoon,\\{\,\|\,\overline{\infty}\\}\\}.$ Hence, $S_{n}=\varnothing$, and $P_{n}=\mathbb{nim}\setminus\\{\\}$. Applying the affine impartial minimum excluded rule, we conclude that the position is $$. Going back to the original question, we have the following. The form is $\\{\textbf{{\color[rgb]{0.00,1.00,0.00}n}},\textbf{{\color[rgb]{1.00,0.00,0.00}m}},\textbf{{\color[rgb]{0.00,0.07,1.00}c}}\,\|\,\textbf{{\color[rgb]{0.00,1.00,0.00}n}},\textbf{{\color[rgb]{1.00,0.00,0.00}m}},\textbf{{\color[rgb]{0.00,0.07,1.00}c}}\\}$, that is, $\\{,,0\,\|\,,,0\\}=2$. Here n represents a play of the top or bottom bar, m represents a play of some middle bar, and c represents play of the left line. ## 5 Case study: top entails We denote by $\boldsymbol{n}$ the literal form of a stack of size $n$. The literal form of the Left removal of the top coin from a stack of size $n$ is $\\{\infty\,\|\,(\boldsymbol{n-1})^{\mathcal{R}}\\}$ (and the symmetric from Right’s point of view). With that in mind, let us compute the first few values. First, we do it the tedious way, and then later after Theorem 41, we propose the slick recursive way for a few more values, in a table format. Of course, $\boldsymbol{0}=\\{\overline{\infty}\,\|\,\infty\\}$. The first player loses. Moreover, * $\boldsymbol{1}=\\{\\{\infty\,\|\,\boldsymbol{0}^{\mathcal{R}}\\}\,\|\,\\{\boldsymbol{0}^{\mathcal{L}}\,\|\,\overline{\infty}\\}\\}=\\{\\{\infty\,\|\,\infty\\}\,\|\,\\{\overline{\infty}\,\|\,\overline{\infty}\\}\\}$. Therefore, $S_{\boldsymbol{1}}=\varnothing$ and $P_{\boldsymbol{1}}=\mathbb{nim}$. Using the affine impartial minimum excluded rule, $\boldsymbol{1}=\scalebox{1.1}{$\leftmoon$}$. In the next step, for ease, we will use the form $\scalebox{1.1}{$\leftmoon$}=\pm\infty$. * $\boldsymbol{2}=\\{\boldsymbol{1}+\boldsymbol{1},\\{\infty\,\|\,\boldsymbol{1}^{\mathcal{R}}\\}\,\|\,\boldsymbol{1}+\boldsymbol{1},\\{\boldsymbol{1}^{\mathcal{L}}\,\|\,\overline{\infty}\\}\\}=\\{\scalebox{1.1}{$\leftmoon$},\\{\infty\,\|\,\overline{\infty}\\}\,\|\,\scalebox{1.1}{$\leftmoon$},\\{\infty\,\|\,\overline{\infty}\\}\\}$. Therefore, $S_{\boldsymbol{2}}=\varnothing$ and $P_{\boldsymbol{2}}=\varnothing$. Using the affine impartial minimum excluded rule, $\boldsymbol{2}=0$. * $\boldsymbol{3}=\\{\boldsymbol{1}+\boldsymbol{2},\\{\infty\,\|\,\boldsymbol{2}^{\mathcal{R}}\\}\,\|\,\boldsymbol{1}+\boldsymbol{2},\\{\boldsymbol{2}^{\mathcal{L}}\,\|\,\overline{\infty}\\}\\}$. This game is equal to $\\{\scalebox{1.1}{$\leftmoon$},\\{\infty\,\|\,\scalebox{1.1}{$\leftmoon$},\\{\infty\,\|\,\overline{\infty}\\}\\}\,\|\,\scalebox{1.1}{$\leftmoon$},\\{\scalebox{1.1}{$\leftmoon$},\\{\infty\,\|\,\overline{\infty}\\}\,\|\,\overline{\infty}\\}\\}$. Therefore, $S_{\boldsymbol{3}}=\varnothing$ and $P_{\boldsymbol{3}}=\mathbb{nim}$. Using the affine impartial minimum excluded rule, $\boldsymbol{3}=\scalebox{1.1}{$\leftmoon$}$. In the next step, for ease, we will use the form $\scalebox{1.1}{$\leftmoon$}=\pm\infty$. * $\boldsymbol{4}=\\{\boldsymbol{1}+\boldsymbol{3},\boldsymbol{2}+\boldsymbol{2},\\{\infty\,\|\,\boldsymbol{3}^{\mathcal{R}}\\}\,\|\,\boldsymbol{1}+\boldsymbol{3},\boldsymbol{2}+\boldsymbol{2},\\{\boldsymbol{3}^{\mathcal{L}}\,\|\,\overline{\infty}\\}\\}$. This game is equal to$\\{\scalebox{1.1}{$\leftmoon$},0,\\{\infty\,\|\,\overline{\infty}\\}\,\|\,\scalebox{1.1}{$\leftmoon$},0,\\{\infty\,\|\,\overline{\infty}\\}\\}$. Therefore, $S_{\boldsymbol{4}}=\\{0\\}$ and $P_{\boldsymbol{4}}=\varnothing$. Using the affine impartial minimum excluded rule, $\boldsymbol{4}=$. $\boldsymbol{5}=\\{\boldsymbol{1}+\boldsymbol{4},\boldsymbol{2}+\boldsymbol{3},\\{\infty\,\|\,\boldsymbol{4}^{\mathcal{R}}\\}\,\|\,\boldsymbol{1}+\boldsymbol{4},\boldsymbol{2}+\boldsymbol{3},\\{\boldsymbol{4}^{\mathcal{L}}\,\|\,\overline{\infty}\\}\\}$. This game is equal to $\\{\scalebox{1.1}{$\leftmoon$},\scalebox{1.1}{$\leftmoon$},\\{\infty\,\|\,0\\}\,\|\,\scalebox{1.1}{$\leftmoon$},\scalebox{1.1}{$\leftmoon$},\\{0\,\|\,\overline{\infty}\\}\\}$. Therefore, $S_{\boldsymbol{5}}=\varnothing$ and $P_{\boldsymbol{5}}=\mathbb{nim}\setminus\\{0\\}$. Using the affine impartial minimum excluded rule, $\boldsymbol{5}=0$. * $\boldsymbol{6}=\\{\boldsymbol{1}+\boldsymbol{5},\boldsymbol{2}+\boldsymbol{4},\boldsymbol{3}+\boldsymbol{3},\\{\infty\,\|\,\boldsymbol{5}^{\mathcal{R}}\\}\,\|\,\boldsymbol{1}+\boldsymbol{5},\boldsymbol{2}+\boldsymbol{4},\boldsymbol{3}+\boldsymbol{3},\\{\boldsymbol{5}^{\mathcal{L}}\,\|\,\overline{\infty}\\}\\}$. This game is equal to $\\{\scalebox{1.1}{$\leftmoon$},,\scalebox{1.1}{$\leftmoon$},\\{\infty\,\|\,\\{0\,\|\,\overline{\infty}\\}\\}\,\|\,\scalebox{1.1}{$\leftmoon$},,\scalebox{1.1}{$\leftmoon$},\\{\\{\infty\,\|\,0\\}\,\|\,\overline{\infty}\\}\\}$. Therefore, $S_{\boldsymbol{6}}=\\{\\}$ and $P_{\boldsymbol{6}}=\\{0\\}$. Using the affine impartial minimum excluded rule, $\boldsymbol{6}=2$. * $\boldsymbol{7}=\\{\boldsymbol{1}+\boldsymbol{6},\boldsymbol{2}+\boldsymbol{5},\boldsymbol{3}+\boldsymbol{4},\\{\infty\,\|\,\boldsymbol{6}^{\mathcal{R}}\\}\,\|\,\boldsymbol{1}+\boldsymbol{6},\boldsymbol{2}+\boldsymbol{5},\boldsymbol{3}+\boldsymbol{4},\\{\boldsymbol{6}^{\mathcal{L}}\,\|\,\overline{\infty}\\}\\}$. This game is equal to $\\{\scalebox{1.1}{$\leftmoon$},0,\scalebox{1.1}{$\leftmoon$},\\{\infty\,\|\,,\\{\\{\infty\,\|\,0\\}\,\|\,\overline{\infty}\\}\\}\,\|\,\scalebox{1.1}{$\leftmoon$},0,\scalebox{1.1}{$\leftmoon$},\\{,\\{\infty\,\|\,\\{0\,\|\,\overline{\infty}\\}\\}\,\|\,\overline{\infty}\\}\\}$. So, $S_{\boldsymbol{7}}=\\{0\\}$, $P_{\boldsymbol{6}}=\mathbb{nim}\setminus\\{0,\\}$, and with the affine impartial minimum excluded rule, $\boldsymbol{7}=$. Consider a stack of size $n$. We claim that an entailing move by Left does not protect her against an element in $S_{\boldsymbol{n-1}}$. To see this, let $m\in S_{\boldsymbol{n-1}}$. Moving in $\boldsymbol{n}+m$, if Left chooses $\\{\infty\,\|\,(\boldsymbol{n-1})\mathcal{{}^{R}}\\}+m$, Right answers $m+m$ and wins. On the other hand, we observe that an entailing move by Left does not protect her against the elements of $P_{\boldsymbol{n-1}}$. To see this, let $m$ be an element of $P_{\boldsymbol{n-1}}$. Moving in $\boldsymbol{n}+m$, if Left chooses $\\{\infty\,\|\,(\boldsymbol{n-1})\mathcal{{}^{R}}\\}+m$, because in $\boldsymbol{n-1}$, Right is protected against $m$, he has an entailing winning move in the first component. Therefore, we have the general recursion $P_{\boldsymbol{n}}=\mathbb{nim}\setminus(S_{\boldsymbol{n-1}}\cup P_{\boldsymbol{n-1}}).$ The set $S_{\boldsymbol{n}}$ is composed of the values of the positions of the form $\boldsymbol{\ell}+\boldsymbol{m}$, $\ell+m=n$, $\ell,m>0$, and disregarding any sum where $\leftmoon$ appears. Hence, the recurrence of top entails is as follows. ###### Theorem 41. The sets $P_{0}=S_{0}=\varnothing$, and for all $n>0$ $P_{\boldsymbol{n}}=\mathbb{nim}\setminus(S_{\boldsymbol{n-1}}\cup P_{\boldsymbol{n-1}}),$ $S_{\boldsymbol{n}}=\\{\mathcal{G}(\boldsymbol{\ell}+\boldsymbol{m}),\boldsymbol{\ell},\boldsymbol{m}\neq\scalebox{1.1}{$\leftmoon$}\\}$. ###### Proof. This is explained in the above paragraph. ∎ Now, we can fill a table in an easy way. $n$ \| $S_{\boldsymbol{n}}$ \| $P_{\boldsymbol{n}}$ \| $S_{\boldsymbol{n}}\cup P_{\boldsymbol{n}}$ \| $\mathcal{G}$-value (mex rule) ---\|---\|---\|---\|--- $0$ \| $\varnothing$ \| $\varnothing$ \| $\varnothing$ \| $0$ $1$ \| $\varnothing$ \| $\mathbb{nim}$ \| $\mathbb{nim}$ \| $\infty$ $2$ \| $\varnothing$ \| $\varnothing$ \| $\varnothing$ \| $0$ $3$ \| $\varnothing$ \| $\mathbb{nim}$ \| $\mathbb{nim}$ \| $\infty$ $4$ \| $\\{0\\}$ \| $\varnothing$ \| $\\{0\\}$ \| $1$ $5$ \| $\varnothing$ \| $\mathbb{nim}\backslash\\{0\\}$ \| $\mathbb{nim}\backslash\\{0\\}$ \| $0$ $6$ \| $\\{\\}$ \| $\\{0\\}$ \| $\\{0,\\}$ \| $2$ $7$ \| $\\{0\\}$ \| $\mathbb{nim}\backslash\\{0,\\}$ \| $\mathbb{nim}\backslash\\{\\}$ \| $1$ $8$ \| $\\{0,2\\}$ \| $\\{\\}$ \| $\\{0,,2\\}$ \| $3$ $9$ \| $\\{\\}$ \| $\mathbb{nim}\backslash\\{0,,2\\}$ \| $\mathbb{nim}\backslash\\{0,2\\}$ \| $0$ $10$ \| $\\{0,3\\}$ \| $\\{0,2\\}$ \| $\\{0,2,3\\}$ \| $1$ $11$ \| $\\{0,2\\}$ \| $\mathbb{nim}\backslash\\{0,2,3\\}$ \| $\mathbb{nim}\backslash\\{3\\}$ \| $3$ $12$ \| $\\{0,,2\\}$ \| $\\{3\\}$ \| $\\{0,,2,3\\}$ \| $4$ With the recursion, we know that $\boldsymbol{n}=\scalebox{1.1}{$\leftmoon$}$ if and only if $S_{\boldsymbol{n-1}}\cup P_{\boldsymbol{n-1}}\subseteq S_{\boldsymbol{n}}$. That happens for $n=2403$, $n=2505$, and $n=33243$, as mentioned in [9]. One of three possibilities must happen: a) A finite number of finite nimbers; b) A finite number of loony values; c) An infinite number of finite nimbers and an infinite number of loony values. However, it is an open problem to know what case happens. At the first Combinatorial Games Workshop at MSRI, John Conway proposed that an effort should be made to devise some game with entailing moves that is non- trivial, but (unlike top entails) susceptible to a complete analysis. All attempts which have been tried turn out to be not very interesting. As a sequel to this work, we are finalizing a paper [7] with a proposal of a ruleset to meet Conway’s suggestion. ## References [1] M. Albert, R. J. Nowakowski, D. Wolfe. Lessons in Play: An Introduction to Combinatorial Game Theory, A. K. Peters, 2007. * [2] E. R. Berlekamp. The Dots and Boxes Game: Sophisticated Child’s Play, A. K. Peter’s Ltd., 2000. * [3] E. R. Berlekamp, J. H. Conway, R. K. Guy. Winning Ways, Academic Press, London, 1982. * [4] E. R. Berlekamp, D. Wolfe. _Mathematical Go: Chilling Gets the Last Point_ , A. K. Peters, Ltd., 1994. * [5] J. H. Conway. On Numbers and Games, Academic Press, 1976. * [6] U. Larsson, R. J. Nowakowski, C. P. Santos. Combinatorial games with checks and terminating moves, preprint. * [7] U. Larsson, R. J. Nowakowski, C. P. Santos. Electric cables, preprint. * [8] A. N. Siegel. Combinatorial Game Theory, American Math. Soc., 2013. * [9] J. West. New Values for Top Entails, Games of No Chance, 29, MSRI Publications, 345–350, 1996.
# Better sampling in explanation methods can prevent dieselgate-like deception Domen Vreš, Marko Robnik-Šikonja University of Ljubljana, Faculty of Computer and Information Science Večna pot 113, 1000 Ljubljana, Slovenia <EMAIL_ADDRESS><EMAIL_ADDRESS> ###### Abstract Machine learning models are used in many sensitive areas where, besides predictive accuracy, their comprehensibility is also important. Interpretability of prediction models is necessary to determine their biases and causes of errors and is a prerequisite for users’ confidence. For complex state-of-the-art black-box models, post-hoc model-independent explanation techniques are an established solution. Popular and effective techniques, such as IME, LIME, and SHAP, use perturbation of instance features to explain individual predictions. Recently, Slack et al. (2020) put their robustness into question by showing that their outcomes can be manipulated due to poor perturbation sampling employed. This weakness would allow dieselgate type cheating of owners of sensitive models who could deceive inspection and hide potentially unethical or illegal biases existing in their predictive models. This could undermine public trust in machine learning models and give rise to legal restrictions on their use. We show that better sampling in these explanation methods prevents malicious manipulations. The proposed sampling uses data generators that learn the training set distribution and generate new perturbation instances much more similar to the training set. We show that the improved sampling increases the LIME and SHAP’s robustness, while the previously untested method IME is already the most robust of all. ## 1 Introduction Machine learning models are used in many areas where besides predictive performance, their comprehensibility is also important, e.g., in healthcare, legal domain, banking, insurance, consultancy, etc. Users in those areas often do not trust a machine learning model if they do not understand why it made a given decision. Some models, such as decision trees, linear regression, and naïve Bayes, are intrinsically easier to understand due to the simple representation used. However, complex models, mostly used in practice due to better accuracy, are incomprehensible and behave like black boxes, e.g., neural networks, support vector machines, random forests, and boosting. For these models, the area of explainable artificial intelligence (XAI) has developed post-hoc explanation methods that are model-independent and determine the importance of each feature for the predicted outcome. Frequently used methods of this type are IME (Štrumbelj & Kononenko, 2013), LIME (Ribeiro et al., 2016), and SHAP (Lundberg & Lee, 2017). To determine the features’ importance, these methods use perturbation sampling. Slack et al. (2020) recently noticed that the data distribution obtained in this way is significantly different from the original distribution of the training data as we illustrate in Figure 1a. They showed that this can be a serious weakness of these methods. The possibility to manipulate the post-hoc explanation methods is a critical problem for the ML community, as the reliability and robustness of explanation methods are essential for their use and public acceptance. These methods are used to interpret otherwise black-box models, help in debugging models, and reveal models’ biases, thereby establishing trust in their behavior. Non-robust explanation methods that can be manipulated can lead to catastrophic consequences, as explanations do not detect racist, sexist, or otherwise biased models if the model owner wants to hide these biases. This would enable dieselgate-like cheating where owners of sensitive prediction models could hide the socially, morally, or legally unacceptable biases present in their models. As the schema of the attack on explanation methods on Figure 1b shows, owners of prediction models could detect when their models are examined and return unbiased predictions in this case and biased predictions in normal use. This could have serious consequences in areas where predictive models’ reliability and fairness are essential, e.g., in healthcare or banking. Such weaknesses can undermine users’ trust in machine learning models in general and slow down technological progress. a) b) Figure 1: a) PCA based visualization of a part of the COMPAS dataset. The blue points show the original instances, and the red points represent instances generated with the perturbation sampling used in the LIME method. The distributions are notably different. b) The idea of the attack on explanation methods based on the difference of distributions. The attacker’s adversarial model contains both the biased and unbiased model. The decision function that is part of the cheating model decides if the instance is outside the original distribution (i.e. used only for explanation) or an actual instance. If the case of an actual instance, the result of the adversarial model is equal to the result of the biased model, otherwise it is equal to the result of the unbiased model. In this work, we propose to change the main perturbation-based explanation methods and make them more resistant to manipulation attempts. In our solution, the problematic perturbation-based sampling is replaced with more advanced sampling, which uses modern data generators that better capture the distribution of the training dataset. We test three generators, the RBF network based generator (Robnik-Šikonja, 2016), random forest-based generator, available in R library semiArtificial (Robnik-Šikonja, 2019), as well as the generator using variational autoencoders (Miok et al., 2019). We show that the modified gLIME and gSHAP methods are much more robust than their original versions. For the IME method, which previously was not analyzed, we show that it is already quite robust. We release the modified explanation methods under the open-source license111https://github.com/domenVres/Robust-LIME-SHAP-and- IME. In this work, we use the term robustness of the explanation method as a notion of resilience against adversarial attacks, i.e. as the ability of an explanation method to recognize the biased classifier in an adversary environment. This type of robustness could be more formally defined as the number of instances where the adversarial model’s bias was correctly recognized. We focus on the robustness concerning the attacks described in Slack et al. (2020). There are other notions of robustness in explanation methods; e.g., (Alvarez-Melis & Jaakkola, 2018) define the robustness of the explanations in the sense that similar inputs should give rise to similar explanations. The remainder of the paper is organized as follows. In Section 2, we present the necessary background and related work on explanation methods, attacks on them, and data generators. In Section 3, we propose a defense against the described weaknesses of explanation methods, and in Section 4, we empirically evaluate the proposed solution. In Section 5, we draw conclusions and present ideas for further work. ## 2 Background and related work In this section, we first briefly describe the background on post-hoc explanation methods and attacks on them, followed by data generators and related works on the robustness of explanation methods. ### 2.1 Post-hoc explanation methods The current state-of-the-art perturbation-based explanation methods, IME, LIME, and SHAP, explain predictions for individual instances. To form an explanation of a given instance, they measure the difference in prediction between the original instance and its neighboring instances, obtained with perturbation sampling. Using the generated instances, the LIME method builds a local interpretable model, e.g., a linear model. The SHAP and IME methods determine the impact of the features as Shapley values from the coalitional game theory (Shapley, 1988). In this way, they assure that the produced explanations obey the four Shapley fairness axioms (Štrumbelj & Kononenko, 2013). Due to the exponential time complexity of Shapley value calculation, both methods try to approximate them. The three methods are explained in detail in the above references, and a formal overview is presented in Appendix A, while below, we present a brief description. In our exposition of the explanation methods, we denote with $f$ the predictive model and with $x$ the instance we are explaining. Explanations of instances with the LIME method is obtained with an interpretable model $g$. The model $g$ has to be both locally accurate (so that it can obtain correct feature contributions) and simple (so that it is interpretable). Therefore in LIME, $g$ is a linear regression model trained on the instances sampled in the neighborhood of the explained instance $x$, which is first normalized.The perturbed explanation instances are generated by adding Gaussian noise to each feature of $x$ independently. Using linear models to determine the contributions of features, LIME is fast but may produce poor explanations for instances close to complex decision boundaries. SHAP estimates Shapley values using the approach of LIME (i.e. local linear models computed with the weighted least squares algorithm) but using a different neighborhood and loss function for local accuracy, which ensures that Shapley values are obtained. The sampling of explanation instances in SHAP relies on the distribution set $D$, obtained from the training set, by computing the centroids of k-means clusters. IME computes Shapley values using a pure sampling approach (no proxy with an interpretable model). Its sampling replaces part of the explained instance with existing values of selected features and then averages differences in the prediction of explained instance and samples. This makes the method slower but also potentially more robust as the method does not assume the shape of the dependency. ### 2.2 Attacks on explanations The idea of the attack on explanations presented in Slack et al. (2020) is that the attacker creates an adversarial (biased) model whose behavior it wants to hide from explanation methods, e.g., a racist model that does not grant credits to blacks. As Figure 1b illustrates, a part of the attacker’s adversarial model is an unbiased model, which, e.g., does not take disputed attributes such as race into account. The adversarial model tries to manipulate the explanation method by behaving like an unbiased model on perturbed instances used for the explanation – in this case, it uses the unbiased model’s output. On instances from the original distributions that are a part of normal model use, the adversarial model uses the biased model’s output. To distinguish between the original and perturbation-sampled data, the adversarial model contains a decision model that selects the right type of model (biased or unbiased) based on the predicted type of instances. The reason that the decision function can distinguish between the original data and sampled instances are the shortcomings of explanation methods that use the perturbation sampling (see Figure 1a). For that graph, we created new instances based on the COMPAS dataset (see Section 4) with the Gaussian noise as in the LIME explanation method. We mapped the original and sampled instances to 2D space using the PCA transformation. We can see that the sampled data is distributed differently than the original data. As the difference is revealed with PCA, other machine learning algorithms can also effectively distinguish between sampled and original data. The attack can be formally defined in the following way. The attacker has a biased model $b$ and a sample $S$ from the distribution $X_{dist}$. The attacker has no additional information about the distribution $X_{dist}$. The attacker’s goal is to produce an adversarial model $e$ that behaves like $b$ on the data from $X_{dist}$ but does not reveal $b$’s bias to the explanation methods. We assume that the attacker also has an unbiased model $\psi$ that hides the $f$’s bias. The attacker creates a decision model $d$, that should output $1$ if the input instance $x$ is from $X_{dist}$ and $0$ otherwise. The model $d$ is trained on $S$ and generated perturbation samples. The attacker creates the adversarial model $e$ with its output defined by the following equation: $e(x)=\begin{cases}b(x),&d(x)=1\\\ \psi(x),&d(x)=0\end{cases}\vspace{-5mm}$ (1) ### 2.3 Data generators We used three different data generators based on different algorithms, modeling the distribution of the training set: variational autoencoder with Monte Carlo dropout (Miok et al., 2019), RBF network (Robnik-Šikonja, 2016), and random forest ensemble (Robnik-Šikonja, 2019). In the remainder of the paper, we refer to the listed generators consecutively as MCD-VAE, rbfDataGen, and TreeEnsemble. Autoencoder (AE) consists of two neural networks called encoder and decoder. It aims to compress the input instances by passing them through the encoder and then reconstructing them to the original values with the decoder. Once the AE is trained, it can be used to generate new instances. Variational autoencoder (Doersch, 2016) is a special type of autoencoder, where the vectors $z$ in the latent dimension (output of the encoder and input of the decoder) are normally distributed. Encoder is therefore approximating the posterior distribution $p(z\|x)$, where we assume $p(z\|x)\sim\mathcal{N}(\mu_{x},\Sigma_{x}).$ The generator proposed by Miok et al. (2019) uses the Monte Carlo dropout (Gal & Ghahramani, 2016) on the trained decoder. The idea of this generator is to propagate the instance $x$ through the encoder to obtain its latent encoding $z$. This can be propagated many times through the decoder, obtaining every time a different result due to the Monte Carlo dropout but preserving similarity to the original instance $x$. The RBF network (Moody & Darken, 1989) uses Gaussian kernels as hidden layer units in a neural network. Once the network’s parameters are learned, the rbfDataGen generator (Robnik-Šikonja, 2016) can sample from the normal distributions, defined with obtained Gaussian kernels, to generate new instances. The TreeEnsemble generator (Robnik-Šikonja, 2019) builds a set of random trees (forest) that describe the data. When generating new instances, the generator traverses from the root to the leaves of a randomly chosen tree, setting values of features in the decision nodes on the way. When reaching a leaf, it assumes that it has captured the dependencies between features. Therefore, the remaining features can be generated independently according to the observed empirical distribution in this leaf. For each generated instance, all attributes can be generated in one leaf, or another tree can be randomly selected where unassigned feature values are filled in. By selecting different trees, different features are set in the interior nodes and leaves. ### 2.4 Related work on robustness of explanations The adversarial attacks on perturbation based explanation methods were proposed by Slack et al. (2020), who show that LIME and SHAP are vulnerable due to the perturbation based sampling used. We propose the solution to the exposed weakness in SHAP and IME based on better sampling using data generators adapted to the training set distribution. In general, the robustness of explanation methods has been so far poorly researched. There are claims that post-hoc explanation methods shall not be blindly trusted, as they can mislead users (deliberately or not) and disguise gender and racial discrimination (Lipton, 2016). Selbst & Barocas (2018) and Kroll et al. (2017) showed that even if a model is completely transparent, it is hard to detect and prevent bias due to the existence of correlated variables. Specifically, for deep neural networks and images, there exist adversarial attacks on saliency map based interpretation of predictions, which can hide the model’s bias (Dombrowski et al., 2019; Heo et al., 2019; Ghorbani et al., 2019). Dimanov et al. (2020) showed that a bias of a neural network could be hidden from post-hoc explanation methods by training a modified classifier that has similar performance to the original one, but the importance of the chosen feature is significantly lower. The kNN-based explanation method, proposed by Chakraborty et al. (2020), tries to overcomes the inadequate perturbation based sampling used in explanation methods by finding similar instances to the explained one in the training set instead of generating new samples. This solution is inadequate for realistic problems as the problem space is not dense enough to get reliable explanations. Our defense of current post-hoc methods is based on superior sampling, which has not yet been tried. Saito et al. (2020) use the neural CT- GAN model to generate more realistic samples for LIME and prevent the attacks described in Slack et al. (2020). We are not aware of any other defenses against the adversarial attacks on post-hoc explanations. ## 3 Robustness through better sampling We propose the defense against the adversarial attacks on explanation methods that replaces the problematic perturbation sampling with a better one, thereby making the explanation methods more robust. We want to generate the explanation data in such a way that the attacker cannot determine whether an instance is sampled or obtained from the original data. With an improved sampling, the adversarial model shown in Figure 1b shall not determine whether the instance $x$ was generated by the explanation method, or it is the original instance the model has to label. With a better data generator, the adversarial model cannot adjust its output properly and the deception, described in Section 2.2, becomes ineffective. The reason for the described weakness of LIME and SHAP is inadequate sampling used in these methods. Recall that LIME samples new instances by adding Gaussian noise to the normalized feature values. SHAP samples new instances from clusters obtained in advance with the k-means algorithm from the training data. Instead of using the Gaussian noise with LIME, we generate explanation samples for each instance with one of the three better data generators, MCD-VAE, rbfDataGen, or TreeEnsemble (see Section 2.3). We call the improved explanation methods gLIME and gSHAP (g stands for generator-based). Using better generators in the explanation methods, the decision function in the adversarial model will less likely determine which predicted instances are original and which are generated for the explanation purposes. Concerning gLIME, we generate data in the vicinity of the given instance using MCD-VAE, as the LIME method builds a local model. Using the TreeEnsemble and rbfDataGen generators, we do not generate data in the neighborhood of the given instance but leave it to the proximity measure of the LIME method to give higher weights to instances closer to the explained one. In SHAP, the perturbation sampling replaces the values of hidden features in the explained instance with the values from the distribution set $D$. The problem with this approach is that it ignores the dependencies between features. For example, in a simple dataset with two features, house size, and house price, let us assume that we hide the house price, but not the house size. These two features are not independent because the price of a house increases with its size. Suppose we are explaining an instance that represents a large house. Disregarding the dependency, using the sampled set $D$, SHAP creates several instances with a low price, as such instances appeared in the training set. In this way, the sampled set contains many instances with a small price assigned to a large house, from which the attacker can determine that these instances were created in perturbation sampling and serve only for the explanation. In the proposed gSHAP, using the MCD-VAE and TreeEnsemble generators, the distribution set $D$ is generated in the vicinity of the explained instance. In the sampled instances, some feature values of the explained instance are replaced with the generated values, but the well-informed generators consider dependencies between features detected in the original distribution. This will make the distribution of the generated instances very similar to the original distribution. In our example, the proposed approach generates new instances around the instance representing a large house, and most of these houses will be large. As the trained generators capture the original dataset’s dependencies, these instances will also have higher prices. This will make it difficult for the attacker to recognize the generated instances used in explanations. The advantage of generating the distribution set close to the observed instance is demonstrated in Appendix B. The rbfDataGen generator does not support the generation of instances around a given instance. Therefore, we generate the sampled set based on the whole training set and not for each instance separately (we have this option also for TreeEnsemble). This is worse than generating the distribution set around the explained instance but still better than generating it using the k-means sampling in SHAP. There are at least three advantages. First, the generated distribution set $D$ can be larger. The size of the k-means distribution set cannot be arbitrarily large because it is limited by the number of clusters in the training set. Second, the centroids of the clusters obtained by the k-means algorithm do not necessarily provide a good summary of the training set and may not be valid instances from training set distribution. They also do not capture well the density of instances (e.g., most of the data from the training set could be grouped in one cluster). Third, using the proposed generators, SHAP becomes easier to use compared to the k-means generator, where users have to determine the number of clusters, while the data generators we tested can be used in the default mode without parameters, ## 4 Evaluation To evaluate the proposed improvements in the explanation methods, we first present the used datasets in Section 4.1, followed by the experiments. In Section 4.2, we test the robustness of gLIME, gSHAP, and gIME against the adversarial models. To be more realistic, we equipped the adversarial models with the same improved data generators we used in the explanation methods. It is reasonable to assume that attackers could also use better data generators when preparing their decision function, making their attacks much stronger. As the evaluation metric for the success of deception, we use the proportion of instances where the adversarial model deceived the explanation methods so that they did not detect sensitive attributes as being important in the prediction models. In Section 4.3, we test if enhanced generators produce different explanations than the original ones. As the attacker might decide to employ deception only when it is really certain that the predicted instance is used inside the explanation method, we test different thresholds of the decision function $d$ from Equation 1 (currently set to $0.5$). We report on this analysis in Section 4.4. ### 4.1 Sensitive datasets prone to deception Following (Slack et al., 2020), we conducted our experiments on three data sets from domains where a biased classifier could pose a critical problem, such as granting a credit, predicting crime recidivism, and predicting the violent crime rate. The basic information on the data sets is presented in Table 1. The statistics were collected after removing the missing values from the data sets and before we encoded categorical features as one-hot-encoded vectors. Data set \| # inst. \| # features \| # categorical \| sensitive \| unrelated 1 \| unrelated 2 ---\|---\|---\|---\|---\|---\|--- COMPAS \| $6172$ \| $7$ \| $4$ \| race \| random1 \| random2 German \| $1000$ \| $25$ \| $15$ \| gender \| pctOfIncome \| / CC \| $1994$ \| $100$ \| $0$ \| racePctWhite \| random1 \| random2 Table 1: Basic information and sensitive features in the the used data sets. The target variable is not included in the number of features and is binary for all data sets. The pctOfIncome full name is loanRateAsPercentOfIncome. COMPAS (Correctional Offender Management Profiling for Alternative Sanctions) is a risk assessment used by the courts in some US states to determine the crime recurrence risk of a defendant. The dataset (Angwin et al., 2016)) includes criminal history, time in prison, demographic data (age, gender, race), and COMPAS risk assessment of the individual. The dataset contains data of 6,172 defendants from Broward Couty, Florida. The sensitive attribute in this dataset (the one on which the adversarial model will be biased) is race. African Americans, whom biased model associates with a high risk of recidivism, represent $51.4\%$ of instances from the data set. This set’s target variable is the COMPAS score, which is divided into two classes: a high and low risk. The majority class is the high risk, which represents $81.4\%$ of the instances. The German Credit dataset (German for the rest of the paper) from the UCI repository (Dua & Graff, 2019) includes financial (bank account information, loan history, loan application information, etc.) and demographic data (gender, age, marital status, etc.) for 1,000 loan applicants. A sensitive attribute in this data set is gender. Men, whom the biased model associates with a low-risk, represent $69\%$ of instances. The target variable is the loan risk assessment, divided into two classes: a good and a bad customer. The majority class is a good customer, which represents $70\%$ of instances. Communities and Crime (CC) data set (Redmond & Baveja, 2002) contains data about the rate of violent crime in US communities in $1994$. Each instance represents one community. The features are numerical and represent the percentage of the community’s population with a certain property or the average of population in the community. Features include socio-economic (e.g., education, house size, etc.) and demographic (race, age) data. The sensitive attribute is the percentage of the white race. The biased model links instances where whites’ percentage is above average to a low rate of violent crime. The target variable is the rate of violent crime divided into two classes: high and low. Both classes are equally represented in the data set. ### 4.2 Robustness of explanation methods To evaluate the robustness of explanation methods with added generators (i.e. gLIME, gSHAP, and gIME), we split the data into training and evaluation set in the ratio $90\%:10\%$. We used the same training set for the training of adversarial models and explanation methods. We encoded categorical features as one-hot-encoded vectors. We simulated the adversarial attack for every combination of generators used in explanation methods and adversarial models (except for method IME, where we did not use rbfDataGen, which cannot generate instances in the neighborhood of a given instance). When testing SHAP, we used two variants of the TreeEnsemble generator for explanation: generating new instances around the explained instance and generating new instances according to the whole training set distribution. In the LIME testing, we used only the whole training set variant of TreeEnsemble inside the explanation. In the IME testing, we only used the variant of Tree Ensemble that fills in the hidden values (called TEnsFillIn variant). For training the adversarial models, we used the whole training set based variant of TreeEnsemble, except for IME, where we used the TEnsFillIn variant. These choices reflect the capabilities of different explanation method and attempt to make both defense and attack realistic (as strong as possible). More details on training the decision model $d$ inside the adversarial model can be found in Appendix D. In all cases, the biased model $b$ (see Section 2.2) was a simple function depending only on the value of the sensitive feature. The unbiased model $\psi$ depended only on the values of unrelated features. The sensitive and unrelated features are shown on the right-hand side of Table 1. Features random1 and random2 were uniformly randomly generated from the $\\{0,1\\}$ set. On COMPAS and CC, we simulated two attacks, one with a single unrelated feature (the result of $\psi$ depends only on the value of the unrelated feature 1), and another one with two unrelated features. For every instance in the evaluation set, we recorded the most important feature according to the used explanation method (i.e. the feature with the largest absolute value of the contribution as determined by the explanation method). The results are shown as a heatmap in Figure 2. The green color means that the explanation method was deceived in less than $30\%$ of cases (i.e. the sensitive feature was recognized as the most important one in more than 70 % of the cases as the scale on the heatmap suggests), and the red means that it was deceived in more than $70\%$ of cases (the sensitive feature was recognized as the most important one in less than 30 % of the cases as the scale on the heatmap suggests). We consider deception successful if the sensitive feature was not recognized as the most important by the explanation method (the sensitive features are the only relevant features in biased models $b$). The gLIME method is more robust with the addition of rbfDataGen and TreeEnsemble than LIME and less robust with the addition of MCD-VAE. This suggests that parameters for MCD-VAE were not well-chosen, and this pattern repeats for other explanation methods. Both TreeEnsemble and rbfDataGen make gLIME considerably more robust on COMPAS and German datasets, but not on the CC dataset. We believe the reason for that is that features in CC are strongly interdependent, and many represent the same attribute as a fraction of value, e.g., we have the share of the white population, the share of the Asian population, and so on. This interdependence dictates that all fractions have to add up to $1$. The data generators are unlikely to capture such strong conditional dependencies, but the adversarial model’s decision function is likely to approximate it. The gSHAP method is most robust when using the TreeEnsemble generator, but it shows less robust behavior than the gLIME method. The reason for that could be in the feature value replacement strategy used by the SHAP and gSHAP methods, which change only some of the feature’s values with the help of the distribution set. This can lead to out-of-distribution instances if the instances in the distribution set are not close enough to the explained instance. With gLIME, we generate complete instances that are more likely to be in the distribution of the training set. IME is quite robust even with the perturbation sampling, as we expected. This suggests that IME is the most robust of all three tested explanation methods and shall be considered the chosen method in sensitive situations. The gIME results when using the TreeEnsemble generator (TEnsFillIn variant) are comparable to the original IME variant results. This suggests that sampling from a smaller data set, which represents the neighborhood of the explained instance, does not decrease the method’s robustness. Figure 2: The robustness results for gLIME (top), gSHAP (middle), and gIME (bottom). The graphs show the proportion of evaluation set instances, where the sensitive feature was recognized as the most important by the used explanation method. Rows represent the generators used for explanations. The column labels consist of the name of the data set on which the experiment was performed and the name of the generator used for training of the adversarial model. Compas2 and CC2 denote an attack with two independent features. Perturbation represents the original sampling used in LIME, SHAP, and IME, TEnsFillIn represents the TreeEnsemble variant where new instances are generated around the given one, and TreeEns represents the generation from the whole distribution. ### 4.3 Comparing explanations of original and modified methods We checked if improved data generators affect explanations in a non-adversary environment. We compared the explanations of original and modified methods for four different classification models on three different datasets. As a measure of differences, we use the mean of the absolute difference between the original and enhanced explanations over the whole evaluation set. Details of these experiments are reported in Appendix F, below we shortly summarize the results, contained in Table 4. The differences between original LIME and gLIME explanations are considerable (see the top part of Table 4). This is not surprising since LIME fits local linear models in its local explanations, which can strongly differ even for small perturbations of the model’s input. The differences in SHAP and IME explanations are small (the average MAD is almost negligible). We can conclude that explanations of gSHAP and gIME are not significantly different from SHAP and IME in the non-adversary environment. ### 4.4 Behavior of adversarial classifier with different prediction thresholds The attacker might be wary of being discovered by the regulator and decide to employ deception only when it is really certain that the predicted instance is used inside the explanation method. We express different levels of attacker’s conservatism by changing the decision function threshold $d$ from Equation 1, where it is currently set to $0.5$. We tested the robustness of modified explanation methods on the COMPAS dataset with adversarial models that use different certainty thresholds for deploying the biased classifier (simulating different levels of aggressiveness). For results and a more detailed description of the experiment, see Appendix G. Below, we shortly summarize the results. Even with different thresholds, gIME is still the most robust of the three explanation methods, and treeEnsemble still gives the best results as the data generator. The sensitive feature is recognized as the most important more often when the threshold is lower, which is expected as in this case the adversarial model behaves more aggressively. While the percentage of the instances on which the racist behavior of the adversarial model is recognized drops with higher values of the thresholds, it remains high enough to warn the regulator about the problematic behavior of the prediction model (especially in the case of gSHAP and gIME using treeEnsemble as data generator). We can conclude that the enhanced explanation methods remain robust enough, even with more conservative adversarial models. ## 5 Conclusions We presented the defense against adversarial attacks on explanation methods. The attacks exploit the shortcomings of perturbation sampling in post-hoc explanation methods. This sampling used in these methods produces instances too different from the original distribution of the training set. This allows unethical owners of biased prediction models to detect which instances are meant for explanation and label them in an unbiased way. We replaced the perturbation sampling with data generators that better capture the distribution of a given data set. This prevents the detection of instances used in explanation and disarms attackers. We have shown that the modified gLIME and gSHAP explanation methods, which use better data generators, are more robust than the original variants, while IME is already quite robust. The difference in explanation values between original and enhanced gSHAP and gIME is negligible, while for gLIME, it is considerable. Our preliminary results in Appendix C show that using the TreeEnsemble generator, the gIME method converges faster and requires from 30-50% fewer samples. The effectiveness of the proposed defense depends on the choice of the data generator and its parameters. While the TreeEnsemble generator turned out the most effective in our evaluation, in practice, several variants might need to be tested to get a robust explanation method. Inspecting authorities shall be aware of the need for good data generators and make access to training data of sensitive prediction models a legal requirement. Luckily, even a few non- deceived instances would be enough to raise the alarm about unethical models. This work opens a range of possibilities for further research. The proposed defense and attacks shall be tested on other data sets with a different number and types of features, with missing data, and on different types of problems such as text, images, and graphs. The work on useful generators shall be extended to find time-efficient generators with easy to set parameters and the ability to generate new instances in the vicinity of a given one. Generative adversarial networks (Goodfellow et al., 2014) may be a promising line of research. The TreeEnsemble generator, currently written in pure R, shall be rewritten in a more efficient programming language. The MCD-VAE generator shall be studied to allow automatic selection of reasonable parameters for a given dataset. In SHAP sampling, we could first fix the values of the features we want to keep, and the values of the others would be generated using the TreeEnsemble generator. #### Acknowledgments The work was partially supported by the Slovenian Research Agency (ARRS) core research programme P6-0411. This paper is supported by European Union’s Horizon 2020 research and innovation programme under grant agreement No 825153, project EMBEDDIA (Cross-Lingual Embeddings for Less-Represented Languages in European News Media). ## References * Alvarez-Melis & Jaakkola (2018) David Alvarez-Melis and Tommi S Jaakkola. 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Explaining prediction models and individual predictions with feature contributions. _Knowledge and Information Systems_ , 41:647–665, 2013\. ## Appendix A Details on post-hoc explanation methods For the sake of completeness, we present further details on the explanation methods LIME (Ribeiro et al., 2016), SHAP (Lundberg & Lee, 2017), and IME (Štrumbelj & Kononenko, 2013). Their complete description can be found in the above-stated references. In our exposition of the explanation methods, we denote with $f$ the predictive model, with $x$ the instance we are explaining, and with $n$ the number of features describing $x$. ### A.1 LIME Explanations of instances with the LIME method is obtained with an interpretable model $g$. The model $g$ has to be both locally accurate (so that it can obtain correct feature contributions) and simple (so that it is interpretable). The explanation of the instance $x$ for the predictive model $f$, obtained with the LIME method, is defined with the following equation: $\xi(x)=\operatorname*{arg\,min}_{g\in G}(\mathcal{L}(f,g,\pi_{x})+\Omega(g)),$ (2) where $\Omega(g)$ denotes the measure of complexity for interpretable model, $\pi_{x}(z)$ denotes the proximity measure between $x$ and generated instances $z$, $\mathcal{L}(f,g,\pi_{x})$ denotes the measure of local fidelity of interpretable model $g$ to the prediction model $f$, and $G$ denotes the set of interpretable models. We use the linear version of the LIME method, where $G$ represents the set of linear models. With $x^{\prime}$ we denote the normalized presentation of instance $x$, i.e. the numerical attributes have their mean set to 0 and variance to 1, and the categorical attributes contain value 1 if they have the same value as the exoplained instance and 0 otherwise. The proximity function $\pi_{x}$ is defined on the normalized instances (hence we use the notation $\pi_{x^{\prime}}$), and uses the exponential kernel: $\pi_{x^{\prime}}(z^{\prime})=e^{\frac{d(x^{\prime},z^{\prime})}{\sigma^{2}}}$, where $d(x^{\prime},z^{\prime})$ denotes a distance measure between $x^{\prime}$ and $z^{\prime}$. The local fidelity measure $\mathcal{L}$ from Equation 2 is defined as: $\mathcal{L}(f,g,\pi_{x^{\prime}})=\sum_{z^{\prime}\in\mathcal{Z}}\pi_{x^{\prime}}(z^{\prime})(f(z)-g(z^{\prime}))^{2},$ (3) where $\mathcal{Z}$ denotes the set of samples. In LIME, each generated sample is obtained by adding Gaussian noise to each feature of $x^{\prime}$ independently. Using linear models as the set of interpretable models, LIME is relatively fast but may produce poor explanations for instances close to complex decision boundaries. ### A.2 SHAP We refer to SHAP as the method called Kernel SHAP by (Lundberg & Lee, 2017). SHAP essentially estimates Shapley values using LIME’s approach, which means that the calculation of explanations is fast due to the use of local linear models computed with the weighted least squares algorithm. The explanation of the instance $x$ with the SHAP method are feature contributions $\phi_{i}$, $i=1,2,...,n$ that are coefficients of the linear model $g(x^{\prime})=\phi_{0}+\sum_{i=1}^{n}\phi_{i}\cdot x^{\prime}_{i}$, obtained with LIME, where $x^{\prime}_{i}\in\\{0,1\\}$ for $i\in\\{1,2,...,n\\}$. As LIME does not compute Shapley values, this property is assured with the proper selection of functions $\Omega(g),\pi_{x^{\prime}}(z^{\prime})$, and $\mathcal{L}(f,g,\pi_{x^{\prime}})$. Let us first define the function $h_{x}(z^{\prime})$ that maps from the $\\{0,1\\}^{n}$ to the original feature space. The function $h_{x}$ is defined implicitly with the equation $f(h_{x}(z^{\prime}))=E[f(z)\|z_{i}=x_{i}\ \forall i\in\\{j;z^{\prime}_{j}=1\\}]$. This is the expected value of the model $f$ when we do not know the values of features at indices where $z^{\prime}$ equals $0$ (these values are hidden). Functions $\Omega(g),\pi_{x^{\prime}}(z^{\prime})$ and $\mathcal{L}(f,g,\pi_{x^{\prime}})$ that enforce the computation of Shapley values are: $\Omega(g)=0,$ $\pi_{x^{\prime}}(z^{\prime})=\dfrac{n}{\binom{n}{\|z^{\prime}\|}\cdot\|z^{\prime}\|\cdot(n-\|z^{\prime}\|)},$ $\mathcal{L}(f,g,\pi_{x^{\prime}})=\sum_{z^{\prime}\in\mathcal{Z}}(f(h_{x}(z^{\prime}))-g(z^{\prime}))^{2}\cdot\pi_{x^{\prime}}(z^{\prime}),$ where $\|z^{\prime}\|$ denotes the number of nonzero features of $z^{\prime}$ and $\mathcal{Z}\subseteq 2^{\\{0,1\\}^{n}}$. The main purpose of the sampling in this method is to determine the value $f(h_{x}(z^{\prime}))$ because in general predictive models cannot work with hidden values. To determine $f(h_{x}(z^{\prime}))$, SHAP uses the distribution set $D$ which we obtain from the training set. For $D$, SHAP takes the centroids of clusters obtained by the k-means clustering algorithm on the training set. The number of clusters is set by the user. Value of $f(h_{x}(z^{\prime}))$ is determined by the following sampling: $f(h_{x}(z^{\prime}))=E[f(z)\|z_{i}=x_{i}\ \forall i\in\\{j;z^{\prime}_{j}=1\\}]=\dfrac{1}{\|D\|}\sum_{d\in D}f(x_{[x_{i}=d_{i},z^{\prime}_{i}=0]}),$ (4) where $x_{[x_{i}=d_{i},z^{\prime}_{i}=0]}$ denotes instance $x$ with features that are $0$ in $z^{\prime}$ being set to the feature values from $d$. ### A.3 IME The explanation of $x$ with the method IME are feature contributions $\phi_{i}$, $i=1,2,...,n$. Štrumbelj & Kononenko (2013) shoved that Shapley values are the only solution that takes into account contributions of interactions for every subset of features in a fair way. The $i$-th feature contribution can be calculated with the following expresiion: $\phi_{i}(x)=\dfrac{1}{n}\sum_{\pi\in S_{n}}\sum_{w\in\mathcal{X}}p(w)\cdot(f(w_{[w_{i}=x_{i},i\in Pre^{i}(\pi)\cup\\{i\\}]})-f(w_{[w_{i}=x_{i},i\in Pre^{i}(\pi)]})),$ (5) where $S_{n}$ denotes a group of permutations of $n$ elements, $\mathcal{X}$ denotes the training set instances, $Pre^{i}(\pi)$ represents the set of indices that precedes $i$ in the permutation $\pi$, i.e. $Pre^{i}(\pi)=\\{j;\pi(j)<\pi(i)\\}$). Let $p(w)$ denote the probability of the instance $w$ in $\mathcal{X}$ and let $w_{[formula]}$ denote the instance $w$ with some of its features values changed according to the formula. To calculate $\phi_{i}(x)$, we have to go through $\|\mathcal{X}\|\cdot n!$ iterations, which can be slow. Therefore, the method IME uses the following sampling. The sampling population of $i$-th feature is $V_{\pi,w}=(f(w_{[w_{i}=x_{i},i\in Pre^{i}(\pi)\cup\\{i\\}]})-f(w_{[w_{i}=x_{i},i\in Pre^{i}(\pi)]}))$ for every combination of the permutation $\pi$ and instance $w$. IME draws $m_{i}$ samples $V_{1},...,V_{m_{i}}$ at random with repetition. The estimate of $\phi_{i}(x)$ is defined with the equation: $\hat{\phi_{i}}=\dfrac{1}{m_{i}}\sum_{i=1}^{m_{i}}V_{i}.$ (6) Contrary to SHAP, IME does not use approximation with linear models, which compute all features’ contributions at once but has to compute the Shapley values by averaging over a large enough sample for each feature separately. This makes the method slower but also potentially more robust as the method does not assume the shape of the dependency in the space of normalized features. ## Appendix B Demonstration of better sampling in SHAP The graphs in Figure 3 show the PCA based 2D space of the evaluation part of the COMPAS dataset (see Section 4 for the dataset description). The left-hand side shows the SHAP-generated sampled instances using the k-means algorithm (14 clusters determined by the silhouette score (Rousseeuw (1987)). The sample produced with the MCD-VAE generator in gSHAP is shown on the right-hand side. This sample is much more similar to the original distribution compared to the SHAP sampling. \| ---\|--- Figure 3: Visual comparison of original and sampled distributions for the COMPASS dataset. The SHAP k-means based generator (left) produces instances less similar to the original data, compared to the MCD-VAE generator (right). ## Appendix C Improved IME convergence rate with the TreeEnsemble generator Preliminary, we tested how better generators affect the convergence rate of the IME explanation method. The preliminary results in Table 2 show a significant reduction in the number of needed samples and a slight increase in the error compared to the original perturbation sampling. Note that the error measure deployed is biased in favor of the perturbation sampling, which was used to determine the gold standard. This was determined with the sampling population’s variance, as described in Štrumbelj & Kononenko (2010). COMPAS dataset \| Error \| # samples \| ---\|---\|---\|--- Classifier \| Perturb. \| TEnsFillIn \| Perturb. \| TEnsFillIn \| Reduction % \| CA % Naive Bayes \| 0.0076 \| 0.0217 \| 18571 \| 11278 \| 39 \| 83 Linear SVM \| 0.0033 \| 0.0080 \| 8244 \| 4423 \| 46 \| 84 Random forest \| 0.0049 \| 0.0221 \| 45372 \| 26960 \| 41 \| 80 Neural network \| 0.0057 \| 0.0130 \| 16157 \| 8841 \| 45 \| 84 German dataset \| Error \| # samples \| ---\|---\|---\|--- Classifier \| Perturb. \| TEnsFillIn \| Perturb. \| TEnsFillIn \| Reduction % \| CA % Naive Bayes \| 0.0076 \| 0.0201 \| 56052 \| 39257 \| 30 \| 77 Linear SVM \| 0.0005 \| 0.0013 \| 3877 \| 2157 \| 44 \| 69 Random forest \| 0.0046 \| 0.0141 \| 92478 \| 66639 \| 28 \| 74 Neural network \| 0 \| 0 \| 0 \| 26 \| / \| 69 CC dataset \| Error \| # samples \| ---\|---\|---\|--- Classifier \| Perturb. \| TEnsFillIn \| Perturb. \| TEnsFillIn \| Reduction % \| CA % Naive Bayes \| 0.0028 \| 0.0046 \| 32910 \| 20117 \| 39 \| 70 Linear SVM \| 0.0009 \| 0.0048 \| 73324 \| 39098 \| 47 \| 62 Random forest \| 0.0032 \| 0.0045 \| 109958 \| 58852 \| 46 \| 79 Neural network \| 0.0012 \| 0.0061 \| 144183 \| 70020 \| 51 \| 72 Table 2: Comparison of original perturbation sampling and the TreeEnsemble generator with data fill-in inside the IME method. The results show average scores for the evaluation set. The column Perturb. presents the perturbation based sampling and TEnsFillIn presents the sampling using TreeEnsemble generator with missing parts of instances filled in. The Reduction column shows the reduction in the number of samples using the TEnsFillIn method compared to perturbations. The CA stands for the classification accuracy of the explained classifier on the evaluation set. ## Appendix D Training discriminator function of the attacker Details of training attacker’s decision models $d$ (see Figure 1b) is described in Algorithms 1, 2, and 3. We used a slightly different algorithm for each of the three explanation methods, LIME, SHAP, and IME, as each method uses a different sampling. Algorithms first create out-of-distribution instances by method-specific sampling. The training sets for decision models are created by labeling the created instances with $0$; the instances from sample $S$ (to which the attacker has access) from distribution $X_{dist}$ are labeled with $1$. Finally, the machine learning model dModel is trained on this training set and returned as $d$. In our experiments, we used random forest classifier as dModel and the training part of each evaluation dataset as $S$. Input: $S=\\{(x_{i})_{i=1}^{m}\\}$: training set, nSamples: number of generated instances for each instance $x_{i}\in D$, $gen$: data generator, dModel: machine learning algorithm Output: Classifier $d$ that outputs $1$ if its input $x$ is from $X_{dist}$ and $0$ otherwise $X\leftarrow\emptyset$ // Training set for dModel $gen.fit(S)$ // Train the data generator on $S$ for _$i=1$ to $m$_ do $X\leftarrow X\cup(x_{i},1)$ // Add an instance from distribution $G\leftarrow gen.newdata(nSamples,x_{i})$ // Generate nSamples new samples around $x_{i}$ for _$j=1$ to nSamples_ do // Add nSamples out of distribution instances $X\leftarrow X\cup(G[j],0)$ // Add $j$-th instance from set $G$ to $X$ end for end for $d\leftarrow dModel.fit(X)$ // Fit model dModel to set $X$ and save it in $d$ return $d$ Algorithm 1 Training of the decision model $d$, used by the attacker to distinguish between instances from distribution $X_{dist}$ and samples produced by explanation methods LIME or gLIME. Input: $S=\\{(x_{i})_{i=1}^{m}\\}$: training set, nSamples: number of generated instances for each instance $x_{i}\in D$, $k$: size of the generated distribution set, $gen$: data generator, dModel: machine learning algorithm Output: Classifier $d$ that outputs $1$ if its input $x$ is from $X_{dist}$ and $0$ otherwise $X\leftarrow\emptyset$ // Training set for dModel $gen.fit(S)$ // Train the data generator on $S$ if _$gen==KMeans$ or $gen==rbfDataGen$ or $gen==treeEnsemble$_ then $D\leftarrow gen.newdata(k)$ // Generate the distribution set with KMeans, rbfDataGen or treeEnsemble end if for _$i=1$ to nSamples_ do // Add nSamples out of distribution instances $x\leftarrow$ random instance from $S$ if _$gen==MCD-VAE$ or $gen==treeEnsembleFill$ _ then $w\leftarrow gen.newdata(1,x)$ // Generate an instance $w$ in the vicinity of $x$ end if else // KMeans, treeEnsemble or rbfDataGen $w\leftarrow$ take a random instance from $D$ end if $M\leftarrow$ choose a random subset of $\\{1,2,...,len(x)\\}$ // Choose random features $x[M]\leftarrow w[M]$ // Replace the values of chosen features in $x$ with values from $w$ as in SHAP method $X\leftarrow X\cup(x,0)$ // Add out of distribution instance end for for _$i=1$ to $m$_ do // Add instances from distribution $X\leftarrow X\cup(x_{i},1)$ end for $d\leftarrow dModel.fit(X)$ // Fit model dModel to set $X$ and save it in $d$ return $d$ Algorithm 2 Training of the decision model $d$, used by attacker to distinguish between instances from distribution $X_{dist}$ and samples produced by explanation methods SHAP or gSHAP. Input: $S=\\{(x_{i})_{i=1}^{m}\\}$: training set, nSamples: number of generated instances for each instance $x_{i}\in D$, $gen$: data generator, dModel: machine learning algorithm Output: Classifier $d$ that outputs $1$ if its input $x$ is from $X_{dist}$ and $0$ otherwise $X\leftarrow\emptyset$ // Training set for dModel $gen.fit(S)$ // Train the data generator on set $S$ for _$i=1$ to $m$_ do $X\leftarrow X\cup(x_{i},1)$ // Add an instance from distribution for _$j=1$ to nSamples_ do // Add nSamples out of distribution instances $w\leftarrow gen.newdata(1,x_{i})$ // Generate an instance $w$ in the vicinity of $x_{i}$ $b_{1}\leftarrow w$ // First out of distribution instance $b_{2}\leftarrow w$ // Second out of distribution instance $\pi\leftarrow$ choose a random permutation from $S_{len(x_{i})}$ // i.e. a random permutation of $x_{i}$’s features $idx\leftarrow$ choose a random number from $\\{1,2,...,len(x_{i})\\}$ $M_{1}\leftarrow\\{k\in\\{1,2,...,len(x_{i})\\},\pi(k)<\pi(idx)\\}$ // Features that precede $idx$ in permutation $\pi$ $M_{2}\leftarrow M_{1}\cup\\{idx\\}$ $b_{1}\leftarrow x_{i}[M_{1}]$ // Vector $b_{1}$ as in IME method (Štrumbelj & Kononenko (2013)) $b_{2}\leftarrow x_{i}[M_{2}]$ // Vector $b_{2}$ as in IME method (Štrumbelj & Kononenko (2013)) $X\leftarrow X\cup\\{(b_{1},0),(b_{2},0)\\}$ // Add out of distribution instances end for end for $d\leftarrow dModel.fit(X)$ // Fit model dModel to set $X$ and save it in $d$ return $d$ Algorithm 3 Training of the decision model $d$, used by attacker to distinguish between instances from distribution $X_{dist}$ and samples produced by explanation methods IME or gIME. ## Appendix E Heatmaps as tables We present the information contained in Figure 2 in a more detailed tabular form in Table 3. Table 3: The robustness results for gLIME (top table), gSHAP (middle table), and gIME (bottom table). The tables show the proportion of evaluation set instances, where the sensitive feature was recognized as the most important by the used explanation method. Columns represent the generators used for explanations. The row labels consist of the name of the dataset on which the experiment was performed and the name of the generator used for training of the adversarial model. Compas2 and CC2 denote attacks with two independent features. Perturbation represents the original sampling used in LIME, SHAP, and IME, TEnsFillIn represents a variant of the TreeEnsemble generator where new instances are generated around the given one, and TreeEns represents the generation from the whole distribution. ## Appendix F Comparing explanations of original and modified methods We check if improved data generators affect explanations in non-adversary environment. We split the dataset into training and evaluation set in the ratio $90\%:10\%$, and trained four classifiers from Python scikit-learn (Pedregosa et al. (2011)) library: Gaussian naive Bayes, linear SVC (SVM for classification), random forest, and neural network. We explained the predictions of each classifier on the evaluation set with every combination of explanation methods and generators used in the adversarial attack experiments. For instances in the evaluation set, we measured the mean absolute difference (MAD) of modified explanation methods, defined with the following equation: $\text{MAD}_{gen}(x)=\frac{1}{n}\sum_{i=1}^{n}\|\phi_{i}^{gen}(x)-\phi_{i}(x)\|,$ (7) where $\phi_{i}^{gen}(x)$ and $\phi_{i}(x)$ represent the explanations of $i$-th feature returned by the modified and original explanation method, respectively (recall that $n$ denotes the number of features in the data set). We experimented on three datasets. The COMPAS dataset is described in Section 4.1. In addition to that, we used synthetic dataset condInd from Robnik- Šikonja & Kononenko (2008), and Ionosphere dataset from UCI repository (Dua & Graff (2019)). Both datasets represent a binary classification problem. Apart from the target variable, condInd consists of $8$ binary features, while Ionosphere consists of $34$ numerical attributes. The condInd datasets contains $2000$ instances and Ionosphere contains $351$ instances. The results are shown in Table 4. The differences between original LIME and gLIME explanations are considerable (see the top table). This is not surprising since LIME fits local linear models in its local explanations, which can strongly differ even for small perturbations of the model’s input. SHAP and IME explanations are very similar (the average MAD is almost negligible). We can conclude that explanations of gSHAP and gIME are not significantly different from SHAP and IME in the non-adversary environment. Table 4: The differences in explanations between LIME and gLIME (top table), SHAP and gSHAP (middle table), and IME and gIME (bottom table). The tables show MAD and the variance of MAD on evaluation set for every combination of the explanation method and predictive model. Columns represent MAD and the variance of MAD on used datasets. The row labels consist of the used classifier and generator. ## Appendix G Adversarial classifiers with different prediction thresholds In a realistic scenario, the attacker might be wary of being discovered by the regulator and decide to employ deception only when it is really certain that the predicted instance is used inside the explanation method. In these circumstances, the number of times when the sensitive variable is used inside the model will also depend on how aggressively the biased classifier is used. This reflection leads to another metric for the defense’s success, which measures the proportion of instances when the sensitive variable is identified when it is being used, i.e. at different levels of conservatism. We express different levels of attacker’s conservatism by changing the threshold of the decision function $d$. The output of the adversarial model $e_{t}$, which uses threshold $t$ is now defined with the equation: $e_{t}(x)=\begin{cases}b(x),&d_{prob}(x)\geq t\\\ \psi(x),&d_{prob}(x)<t\end{cases},$ (8) where $d_{prob}(x)$ denotes the probability that $x$ is from distribution $X_{dist}$, returned by decision model $d$. Threshold $t$ therefore represents the required certainty of the attacker that given instance is from $X_{dist}$ before it deploys the biased classifier. Larger values of $t$ will result in biased classifier being deployed fewer times but with larger certainty that it has to be used. If we use $t=0.5$ in Equation (8), we get the adversarial model that behaves identically to the one, described with Equation (1) in Section 2.2. In this experiment, we used only the COMPAS dataset, which we split into the training and evaluation set in ratio $90\%:10\%$. We performed the same experiment as in section 4.2, but with different thresholds for the adversarial models. We used five different thresholds: $0.3,0.4,0.5,0.6$ and $0.7$. We first checked how different thresholds affect the deployment of the biased model. For each generator used in the adversarial model’s training and the certainty threshold used by the adversarial model, we measured the proportion of instances from the evaluation set, where the biased classifier was deployed. The results are shown in Table 5. As expected, the percentage of instances where the biased classifier is deployed decreases with increasing $t$. The biased classifier is deployed less frequently when using treeEnsemble and rbfDataGen in its training, which is also expected, as these two generators give the best results inside the explanation methods. This means that for these generators, it is more difficult for an attacker to distinguish between the actual instances and generated instances. The biased classifier is deployed the most often in the case of the LIME adversarial model and least often in the SHAP adversarial model. Next, we tested the proportion of instances where the sensitive feature (in case of COMPAS dataset this is race) was recognized as the most important feature by the explanation method on two different subsets of instances: instances from the evaluation set on which the biased classifier was deployed and the whole evaluation set. We measured these numbers for all combinations of thresholds and explanation methods. The results are shown in Tables 6 (gLIME), 7 (gSHAP) and 8 (gIME) for gLIME, gSHAP, and gIME, respectively. We can observe the same pattern as in section 4.2: gIME is still the most robust, and the treeEnsemble generator still gives the best results. The sensitive attribute is recognized as the most important feature more often when the decision threshold values are lower, which is expected as in these cases, the adversarial model behaves more aggressively. Whilst the percentage of the instances on which the racist behavior of the adversarial model is recognized drops with higher thresholds, it remains high enough to warn the regulator about the problematic behavior of the prediction model (especially in the case of gSHAP and gIME using treeEnsemble as data generator). From that, we can conclude that modified explanation methods remain robust enough, even with more conservative adversaries. Table 5: Proportions of instances in % of the evaluation set on which the biased classifier was deployed for adversarial LIME (top table), SHAP (middle table) and IME model (bottom table). Columns represent different threshold used for deploying the biased classifier. The rows represent the generator used in training of the adversarial attack. The labels ”One unrelated” and ”Two unrelated” represent the attacks with one or two unrelated features. Table 6: Percentages of instances where the sensitive feature (race) was recognized as the most important feature with gLIME for adversarial attacks with one (top table) or two unrelated features (bottom table). The columns labeled Biased pred. represent the results on instances on which the biased classifier was deployed, while the columns labeled All represent the results on the whole evaluation sets. The numbers above represent the used threshold. The row labels are in the form $<$explainer$>$_$<$adversarial$>$ where $<$explainer$>$ denotes the generator used in the explanation method and $<$adversarial$>$ denotes the generator used in the training of the adversarial model. Table 7: Percentage of instances where the sensitive attribute (race) was recognized as the most important feature with gSHAP for adversarial attacks with one (top table) or two unrelated features (bottom table). The columns labeled Biased pred. represent the results on instances on which the biased classifier was deployed, while columns labeled All represent the results on the whole evaluation sets. The numbers above represent the used threshold. The row labels are in the form $<$explainer$>$_$<$adversarial$>$ where $<$explainer$>$ denotes the generator used in the explanation method and $<$adversarial$>$ denotes the generator used in the training of the adversarial model. Table 8: Percentages of instances where the sensitive attribute (race) was recognized as the most important feature with gIME for adversarial attacks with one (top table) or two unrelated features (bottom table). The columns labeled Biased pred. represent the results on instances on which the biased classifier was deployed, while the columns labeled All represent the results on the whole evaluation sets. The numbers above represent the used threshold. The row labels are in the form $<$explainer$>$_$<$adversarial$>$ where $<$explainer$>$ denotes the generator used in the explanation method and $<$adversarial$>$ denotes the generator used in the training of the adversarial model.
# Unified Feature Extraction Framework based on Contrastive Learning Hongjie Zhang, Wenwen Qiang, Jinxin Zhang, and Ling Jing This work was supported by Next Generation Precision Aquaculture: R&D on intelligent measurement, control technology (No. 2017YFE0122100), the National Natural Science Foundation of China (Nos. 62076244, 12071024).H. Zhang is associated with the College of Information and Electrical Engineering, China Agricultural University, Beijing 100083, PR China; Beijing Engineering and Technology Research Center for Internet of Things in Agriculture, Beijing 100083, PR China. W. Qiang is associated with the University of Chinese Academy of Sciences, Beijing, PR China, in addition to the Science & Technology on Integrated Information System Laboratory, Institute of Software Chinese Academy of Sciences, Beijing, PR China.J. Zhang is associated with the College of Information and Electrical Engineering, China Agricultural University, Beijing 100083, PR China; National Innovation Center for Digital Fishery, China Agricultural University, Beijing 100083, PR China. L. Jing is associated with the College of Science, China Agricultural University, Beijing 100083, PR China; Key Laboratory of Agricultural Information Acquisition Technology, Ministry of Agriculture, Beijing 100083, PR China. (Corresponding author: Ling Jing, email<EMAIL_ADDRESS> ###### Abstract Feature extraction is an efficient approach for alleviating the issue of dimensionality in high-dimensional data. As a popular self-supervised learning method, contrastive learning has recently garnered considerable attention. In this study, we proposed a unified framework based on a new perspective of contrastive learning (CL) that is suitable for both unsupervised and supervised feature extraction. The proposed framework first constructed two CL graph for uniquely defining the positive and negative pairs. Subsequently, the projection matrix was determined by minimizing the contrastive loss function. In addition, the proposed framework considered both similar and dissimilar samples to unify unsupervised and supervised feature extraction. Moreover, we propose the three specific methods: unsupervised contrastive learning method, supervised contrastive learning method 1 ,and supervised contrastive learning method 2. Finally, the numerical experiments on five real datasets demonstrated the superior performance of the proposed framework in comparison to the existing methods. ###### Index Terms: feature extraction, dimension reduction, self-supervised learning, contrastive learning ## I Introduction Currently, high-dimensional data is widely used in pattern recognition and data mining[1, 2, 3], which wastes considerable time and cost apart from causing the problem known as “curse of dimensionality”[4]. Thus, feature extraction of data poses immense significance to solve this issue[5, 6, 7, 8]. Recently, self-supervised learning[9, 10, 11] has become a popular topic in the field of deep learning. Self-supervised learning is a method of unsupervised learning that uses data information to supervise itself by constructing positive and negative pairs. Self-supervised learning strives to learn more discriminative features to effectively bridge unsupervised and supervised learning. Consequently, contrastive learning[12, 13, 14, 15] has attracted extensive scholarly attention as the primary method of self- supervised learning. Tian et al. proposed contrastive multiview coding (CMC) to process multiview data[16]. First, CMC constructs the same samples in any two views as positive pairs and dissimilar samples as negative pairs, and subsequently optimizes a neural network framework by minimizing the contrastive loss function to maximize the similarity of the projected positive pairs. In addition, Chen et al. proposed a simple framework for contrastive learning (SimCLR) [17] to process the problem. First, it performs data enhancement to obtain distinct representations of the same sample to consider them as positive pairs, and thereafter, considers the representations of any two distinct samples as negative pairs. Finally, SimCLR optimizes the network framework by minimizing the contrastive loss, similar to CMC. However, for the problem of single view feature extraction, the existing models based on contrastive learning still have many defects. For example, data enhancement will increase the running time of the algorithm, and defining all different samples as negative pairs will lead to the separation of samples of the same class. Inspired by our prior research, we proposed a unified feature extraction framework based on contrastive learning (CL-UFEF) that is suitable for both unsupervised and supervised feature extraction for single-view data. The proposed framework, CL-UFEF, constructed two contrastive graphs (CLG) to establish a new approach for defining positive and negative pairs, instead of conducting data enhancement. Moreover, the contrastive loss function considers both similar and dissimilar samples based on the CLG to unify the aspects of unsupervised and supervised feature extraction. Furthermore, the effectiveness of the proposed framework was verified by three specific models, including the unsupervised contrastive learning method (u-CL), supervised contrastive learning method 1 (s-CL1), and supervised contrastive learning method 2 (s-CL2), respectively. The main contributions of this study are as follows: * • A unified feature extraction framework for single-view data based on contrastive learning (CL-UFEF) is proposed from a new perspective that is suitable for both unsupervised and supervised cases. * • CL-UFEF proposes a novel approach to define positive and negative pairs in contrastive learning. * • Based on CL-UFEF, we propose three specific models, including u-CL, s-CL1, and s-CL2, and the experiments on five real datasets proved the advantages of the proposed framework. The remainder of this article is organized as follows. The traditional feature extraction methods are briefly introduced in Section II. Subsequently, the development of the unified feature extraction framework (CL-UFEF) and the u-CL, s-CL1, and s-CL2 models are discussed in Section III. In addition, the extensive experiments conducted on several real-world datasets are presented in Section IV. Finally, the conclusions of the current study are detailed in Section V. ## II Related Work Recently, several feature extraction methods[18, 19, 20, 21, 22] have been proposed, which can be categorized into unsupervised[23], semi-supervised[24], and supervised[25] methods. In this study, we focused on unsupervised and supervised feature extraction. In the unsupervised feature extraction models, classical principal-component analysis (PCA)[18] provides a wide range of applicability and effectiveness, as it seeks the maximum variance of samples in the subspace, which is more conducive to subsequent classification, clustering, or other tasks. Nonetheless, PCA is a linear feature extraction method that is unsuitable for handling nonlinear data. With the development of manifold learning, numerous nonlinear feature extraction methods such as isometric feature mapping (ISOMAP)[26], Laplacian eigenmap (LE)[27], and local linear embedding (LLE)[28] have been proposed to solve this problem. However, these nonlinear feature extraction methods cannot be applied to new sample points because they directly obtain the low-dimensional representation of samples without using a projection matrix. Therefore, several nonlinear methods based on manifold assumptions have been modified into linearized versions. In particular, locality preserving projections (LPP)[29], neighborhood preserving embedding (NPE)[30], and isometric projection are considered as linearized LE, LLE, and ISOMAP, respectively. Although all the feature extraction methods preserve the manifold structures in subspace, they have various requirements for manifold learning. For instance, LPP obtains the neighbor graph of the original data in advance, and then the samples in the subspace maintain the same neighbor relationship. In contrast, NPE expects that the linear reconstruction relationship of the original space can be maintained between the samples and neighborhood samples after feature extraction. However, the above graph-based methods only considered the local structures of the data and neglected the global structures. Thus, sparsity preserving projections (SPP)[31], collaborative representation-based projections (CRP)[32], and low-rank preserving embedding (LRPE)[33] have been proposed to effectively solve this problem. SPP constructs a $l1$-graph with adaptive neighbors by utilizing the sparsity technique, whereas CRP aims to build a $l2$-graph by calculating the linear reconstruction coefficients of each sample based on the remaining samples; LRPE constructs a nuclear norm graph with adaptive neighbors using low-rank representation. However, the unsupervised graph-based methods closely projects the original similar samples in the subspace and neglects the dissimilar samples. Supervised feature extraction methods obtain more discriminant information using sample labels. For instance, linear discriminant analysis (LDA)[34] seeks an embedding transformation such that the within-class scatter is minimized and the between-class scatter is maximized. However, LDA, similar to PCA, is a linear feature extraction method as well. Consequently, it might deliver poor performance if the samples in a class form several separate clusters (i.e., multi-mode). Thus, researchers have proposed using local Fisher discriminant analysis (LFDA)[35] and marginal Fisher analysis (MFA)[36] based on GE and utilizing the approach of manifold learning in unsupervised feature extraction models. LFDA combines the ideas of LDA and LPP to locally construct the levels of within- and between-class scatters, which allows the LFDA to simultaneously achieve maximum preservation of the within- and between-class local structures at the same time. MFA is distinct from the LFDA as it considers the local structure within classes and constructs the local structure relationship between classes by accounting for the samples on the edges of various classes. However, MFA is limited by the problem of class isolation, i.e., not all samples of heterogeneous edges have local neighbor relationships. In this context, researchers have proposed multiple marginal Fisher analysis (MMFA)[37], which selects the nearest neighbor samples on all heterogeneous edges to construct the local relationship between classes. Sparsity preserving discriminant projections (SPDP)[38] is proposed based on SPP to maintain the sparse reconstruction coefficients of samples in the subspace. Note that these graph-based methods initially learn an affine graph using various measure metrics, and subsequently evaluate the projection based on graphs. Inspired by traditional graph-based methods, we proposed a unified feature extraction framework based on contrastive learning (CL-UFEF). Concretelly, compared with the previous models based on contrastive learning, CL-UFEF does not need data enhancement, and it constructs positive and negative pairs based on two contrastive learning graphs (CLG), which will make the similar samples in the subspace more clustered. Compared with the traditional graph-based models, CL-UFEF is suitable for both unsupervised and supervised feature extraction, and it considers similar and dissimilar samples in unsupervised and spervised learning based on the CLG. ## III Methodology Figure 1: Process of determining projection matrix $P$ using CL-UFEF. TABLE I: Notations and definitions. $X$ \| Training sample set ---\|--- $Y$ \| Set of training samples in a low-dimensional space $n$ \| Number of training samples $D$ \| Dimensionality of the samples in the original space $d$ \| Dimensionality of embedding features $c_{i}$ \| Labels of samples $x_{i}$ $C$ \| Number of classes $P$ \| Projection matrix $CLG$ \| Contrastive learning graph $G^{pos}$ \| Positive graph $G^{neg}$ \| Negative graph $S^{pos}$ \| Similarity matrix for positive pairs $S^{neg}$ \| Dissimilarity matrix of negative pairs $\sigma$ \| Positive parameter $k$ \| Number of neighbors $NK(x_{j})$ \| The $k$ nearest neighbors of $x_{j}$ $NK^{+}_{(x_{j})}$ \| The $k$ nearest neighbors of $x_{j}$ in the $c_{j}$ th class $\nabla L(P)$ \| Gradient of $L(P)$ with respect to $P$ $T$ \| Number of iterations In this section, a unified feature extraction framework is proposed based on contrastive learning (CL-UFEF) for both unsupervised and supervised feature extraction, including three specific cases: u-CL, s-CL1, and s-CL2. The projection matrix evaluation process using CL-UFEF is intuitively illustrated in Figure 1. This study considered the problem of unsupervised and supervised feature extraction. Let us mathematically formulate the feature extraction problem as follows. Feature extraction problem: Given a training sample set $X=[x_{1},x_{2},...,x_{n}]\in{R^{D\times n}}$, where $n$ and $D$ are the number of samples and features, respectively. In the supervised case, the class label $c_{i}$ of $x_{i}$ was provided, where $c_{i}\in\\{1,2,...,C\\}$ and $C$ represent the number of classes. The purpose of feature extraction is to find a projection matrix $P\in{R^{D\times d}}$ to derive the low- dimensional embedding $Y=[y_{1},y_{2},...,y_{n}]\in{R^{d\times n}}$ for $X$ calculated by $Y=P^{T}X$, where $d\ll D$. For convenience, the symbols used in this study are summarized in Table I. ### III-A Unified Feature Extraction Framework based on Contrastive Learning (CL-UFEF) ### Construct CLG The contrastive learning was applied by constructing the two contrastive learning graphs (CLG): a positive graph $G^{pos}=\\{X,{S^{pos}}\\}$ and a negative graph $G^{neg}=\\{X,{S^{neg}}\\}$, which define the positive and negative pairs, including the positive matrix $S^{pos}=(S^{pos}_{i,j})_{n\times n}$ and the negative matrix $S^{neg}=(S^{neg}_{i,j})_{n\times n}$ to measure the similarity of the positive pairs and the dissimilarity of the negative pairs. Morever, it should be note that the greater number of negative pairs strengthens the influence of the model based on contrastive learning[39]. In particular, compared with the previous methods based on ciontrastive learning, $G^{pos}$ and $G^{neg}$ define similar samples as positive pairs and dissimilar samples as negative pairs. Compared with the traditional graph- based methods, $G^{pos}$ and $G^{neg}$ are suitable for both supervised and unsupervised learning, and dissimilar samples are considered in unsupervised and spervised learning based on the $G^{neg}$. ### Model of CL-UFEF Contrastive learning is realized using the positive matrix $S^{pos}$ and negative matrix $S^{neg}$. We hope that the projections of the positive pair $x_{i}$ and $x_{j}$ with larger $S^{pos}_{i,j}$ should have greater similarity, and the projections of the negative pair $x_{i}$ and $x_{j}$ with larger $S^{neg}_{i,j}$ will have greater dissimilarity. Thus, the optimization problem of the CL-UFEF is proposed as follows: $\min_{P}L(P)=\sum_{i=1}^{n}-log\frac{\sum_{j=1}^{n}S^{pos}_{i,j}exp(SIM(P^{T}x_{i},P^{T}x_{j}))}{\sum_{j=1}^{n}S^{who}_{i,j}exp(SIM(P^{T}x_{i},P^{T}x_{j}))}$ (1) where， $SIM(P^{T}x_{i},P^{T},x_{j})=\frac{x_{i}^{T}PP^{T}x_{j}}{\\|P^{T}x_{i}\\|\\|P^{T}x_{j}\\|\sigma}$ (2) $\sigma$ is a positive parameter, and $S^{who}_{i,j}=S^{pos}_{i,j}+S^{neg}_{i,j}$ is the whole similarity matrix. Subsequently, based on the specific construction methods of $G^{pos}$ and $G^{neg}$, three special cases, including u-CL, s-CL1, and s-CL2 are proposed to verify the effectiveness of the proposed framework. Morever, some other construction methods of $G^{pos}$ and $G^{neg}$ are presented in APPENDIX A. ### III-B Unspervised contrastive learning method (u-CL) We use the k-nearest neighbor method to construct CLG, indicating that the local $k$ nearest neighbors were positive pairs whereas the non-nearest neighbors were negative pairs. In particular, the positive matrix $S^{pos}$ and negative matrix $S^{neg}$ can be defined as follows: $S^{pos}_{i,j}=\left\\{\begin{aligned} &exp(\frac{-\|\|x_{i}-x_{j}\|\|_{2}^{2}}{t}),&{\rm if}\,x_{i}\in NK(x_{j})\\\ &&{\rm or}\,x_{j}\in NK(x_{i});\\\ &0,&{\rm otherwise},\end{aligned}\right.$ (3) $S^{neg}_{i,j}=\left\\{\begin{aligned} &1,&{\rm if}\,x_{i}\notin NK(x_{j})\,{\rm and}\,x_{j}\notin NK(x_{i});\\\ &0,&{\rm otherwise}.\end{aligned}\right.$ (4) where $t$ is the thermal parameter used for adjusting the value range of the weight matrix $S^{pos}$, $NK(x_{j})$ represents the $k$ nearest neighbors of $x_{j}$ , and the parameter $k$ is a tuning parameter. The optimization problem of u-CL can be obtained by introducing (3) and (4) into CL-UFEF. ### III-C Spervised contrastive learning method 1 (s-CL1) We use class information samples to construct CLG, indicating that the within- class samples are positive pairs and between-class samples are negative pairs. In particular, the positive matrix $S^{pos}$ and negative matrix $S^{neg}$ can be defined as $S^{pos}_{i,j}=\left\\{\begin{aligned} &1,&{\rm if}\,c_{i}=c_{j};\\\ &0,&{\rm if}\,c_{i}\neq c_{j},\end{aligned}\right.$ (5) $S^{neg}_{i,j}=\left\\{\begin{aligned} &1,&{\rm if}\,c_{i}\neq c_{j};\\\ &0,&{\rm if}\,c_{i}=c_{j}.\end{aligned}\right.$ (6) Moreover, the optimization problem of s-CL1 can be obtained by introducing (5) and (6) into the CL-UFEF. ### III-D Spervised contrastive learning method 2 (s-CL2) We use class information and k nearest neighbors of samples to construct CLG, indicating that local $k$ nearest neighbors in the same class were positive pairs and the samples of other relationships were negative pairs. In particular, the positive matrix $S^{pos}$ and negative matrix $S^{neg}$ can be defined as $S^{pos}_{i,j}=\left\\{\begin{aligned} &exp(\frac{-\|\|x_{i}-x_{j}\|\|_{2}^{2}}{t}),&{\rm if}\,x_{i}\in NK^{+}(x_{j})\\\ &&{\rm or}\,x_{j}\in NK^{+}(x_{i});\\\ &0,&{\rm otherwise},\end{aligned}\right.$ (7) $S^{neg}_{i,j}=\left\\{\begin{aligned} &1,&{\rm if}\,x_{i}\notin NK^{+}(x_{j})\,{\rm and}\,x_{j}\notin NK^{+}(x_{i});\\\ &0,&{\rm otherwise}.\end{aligned}\right.$ (8) where $NK^{+}_{(x_{j})}$ represents the $k$ nearest neighbors of $x_{j}$ in the same class. The optimization problem of s-CL2 can be obtained by introducing (7) and (8) into CL-UFEF. ### III-E Optimization algorithm Algorithm 1 CL-UFEF Input: Data matrix: $X\in R^{D\times n}$, $\alpha,\beta_{1},\beta_{2},\epsilon,P_{0}$. $m_{0}=0$(Initialize $1^{st}$ moment vector) $v_{0}=0$(Initialize $2^{nd}$ moment vector) $t=0$ (Initialize time step) Output: Projection matrix $P$ while $P_{t}$ not converged do $t=t+1$$g_{t}=\bigtriangledown L(P_{t-1})$ is calculated using (9) (Obtain gradients with respect to the stochastic objective at time step $t$)$m_{t}=\beta_{1}\cdot m_{t-1}+(1-\beta_{1})\cdot g_{t}$ (Update biased first-moment estimate)$v_{t}=\beta_{2}\cdot v_{t-1}+(1-\beta_{2})\cdot g_{t}^{2}$ (Update biased second raw-moment estimate)$\hat{m}_{t}=m_{t}/(1-\beta_{1}^{t})$ (Compute bias-corrected first- moment estimate)$\hat{v}_{t}=v_{t}/(1-\beta_{2}^{t})$ (Compute bias-corrected second raw-moment estimate)$P_{t}=P_{t-1}-\alpha\cdot\hat{m}_{t}/(\sqrt{\hat{v}_{t}}+\epsilon)$ end while return result In this section, the optimization algorithm of CL-UFEF is presented (Algorithm 1), which uses the Adam optimizer[40] to solve the optimization problem for CL-UFEF. Adam is an advancement on the random gradient descent method and can rapidly yield accurate results. This method calculates the adaptive learning rate of various parameters based on the budget of the first and second moments of the gradient. The parameters $\alpha$, $\beta_{1}$, $\beta_{2}$, and $\epsilon$ represent the learning rate, the exponential decay rate of the first- and second-order moment estimation, and the parameter to prevent division by zero in the implementation, respectively. In addition, the gradient of the loss function $L(P)$ with respect to the projection matrix $P$ was obtained from (9). The convergence condition of Algorithm 1 was set as $L(P_{t})-L(P_{t-1})<0.001$, where $L(P_{t})$ and $L(P_{t-1})$ are the function values obtained after the $t$th and $t-1$th gradient descent, respectively. Therefore, the computational complexity of Algorithm 1 is primarily executed in the first step, and the complexity of the derivative of the objective function is $O(Tn^{3})$, where $T$ denotes the number of iterations. $\displaystyle\nabla L(P)=$ (9) $\displaystyle\sum_{i=1}^{n}\\{-\frac{\sum_{j=1}^{n}S^{who}_{i,j}exp(SIM(P^{T}x_{i},P^{T}x_{j}))}{\sum_{j=1}^{n}S^{pos}_{i,j}exp(SIM(P^{T}x_{i},P^{T}x_{j})}\cdot$ $\displaystyle[\sum_{j=1}^{n}S^{pos}_{i,j}exp(P^{T}x_{i},P^{T}x_{j})\nabla SIM(P^{T}x_{i},P^{T}x_{j})\cdot$ $\displaystyle\sum_{j=1}^{n}S^{who}_{i,j}exp(P^{T}x_{i},P^{T}x_{j})$ $\displaystyle-\sum_{j=1}^{n}S^{who}_{i,j}exp(P^{T}x_{i},P^{T}x_{j})\nabla SIM(P^{T}x_{i},P^{T}x_{j})\cdot$ $\displaystyle\sum_{j=1}^{n}S^{pos}_{i,j}exp(P^{T}x_{i},P^{T}x_{j})]/$ $\displaystyle[\sum_{j=1}^{n}S^{who}_{i,j}exp(P^{T}x_{i},P^{T}x_{j})]^{2}\\},$ where $\displaystyle\nabla SIM(P^{T}x_{i},P^{T}x_{j})=$ (10) $\displaystyle\\{(x_{i}{x_{j}}^{T}+x_{j}{x_{i}}^{T})P\cdot\\|P^{T}x_{i}\\|\\|P^{T}x_{j}\\|\sigma$ $\displaystyle-[({x_{i}}^{T}PP^{T}x_{i})^{-\frac{1}{2}}\cdot x_{i}{x_{i}}^{T}P\cdot\\|P^{T}x_{j}\\|\sigma$ $\displaystyle+({x_{j}}^{T}PP^{T}x_{j})^{-\frac{1}{2}}\cdot x_{j}{x_{j}}^{T}P\cdot\\|P^{T}x_{i}\\|\sigma]\cdot{x_{i}}^{T}PP^{T}x_{j}\\}/$ $\displaystyle{(\\|P^{T}x_{i}\\|\\|P^{T}x_{j}\\|\sigma)}^{2}$ The main computational complexity of each cycle in Algorithm 1 is the derivation of the loss function in the first step, which is $O(n^{2}(D^{2}d+Dd+D^{2}))$. Assuming that the Algorithm 1 performs a total of $T$ cycles when converging, the main computational complexity is $O(n^{2}T(D^{2}d+Dd+D^{2}))$. ## IV Experimental results Figure 2: Image samples used in experiments: (a) Multiple features data set. (b) Yale dataset. (c) COIL20 dataset. (d) MNIST data set. (e) USPS dataset. ### IV-A Data Descriptions and Experimental Setups In this study, the numerical experiments utilized five real datasets to demonstrate the performance advantages of the proposed CL-UFEF. Multiple Features dataset: This dataset was acquired from the UCI machine learning repository, comprising six distinct feature sets extracted from handwritten numbers from 0 to 9 with 200 patterns per class (i.e., 2000 patterns in ten classes). All the datasets were digitized in binary images. The six feature sets included fou (Fourier coefficients, 76 features), fac (profile correlations, 216 features), kar (Karhunen–Loeve coefficients, 64 features), pix (pixel averages in $2\times 3$ windows, 240 features), zer (Zernike moments, 47 features), and mor (morphological features, six features). Example images are presented in Figure 2 (a). Yale dataset: The dataset was created by Yale University Computer Vision and Control Center, containing data of $15$ individuals, wherein each person has 11 frontal images ($64\times 64$ pixels in size) captured under various lighting conditions. The images were edited to $50\times 40$ pixels with 256 Gy levels per pixel; examples are presented in Figure 2 (b). COIL20 dataset: The COIL20 dataset was created by Columbia University in 1996, containing 1440 images of 20 objects, wherein each object includes 72 images ($64\times 64$ pixels in size). In addition, each image was edited to $32\times 32$ pixels, and each pixel had 256 Gy levels; examples are presented in Figure 2 (c). MNIST dataset: This dataset contains 70,000 samples of $0--9$ digital images with a size of $28\times 28$. We randomly selected 2000 images as experimental data, uniformly rescaled all the images to a size of $16\times 16$, and used a feature vector of 256-level grayscale pixel values to represent each image. Examples are presented in Figure 2 (d). USPS dataset: This dataset contains 9298 samples of $0--9$ handwritten digital images, and the size of each image was adjusted to $16\times 16$. A total of 1800 images were randomly selected as experimental data, and each image was represented by a feature vector of 256-level grayscale pixel values. Examples are depicted in Figure 2 (e). In data processing, in order to shorten the running time, we first used PCA to reduce the dimensions of the Yale, COIL20, MNIST, and USPS datasets, and the details are listed in Table II. Thereafter, we separately standardized these five datasets to improve the convergence speed of the model. Finally, we compared u-CL, s-CL1, and s-CL2 with the traditional graph-based methods of LPP, FLPP, LDA, LFDA, and FDLPP and the method SimCLR beased on contrastive learning to verify the advantages of our proposed feature extraction framework. Note that the $k$-nearest neighbor classifier ($k$ = 1) was used in the experiment. Moreover, six samples of each class were randomly selected from the Yale, COIL20, and each feature set of multiple feature datasets for training, and the remaining data were used for testing. Furthermore, nine samples of each class were randomly selected from MNIST and USPS datasets for training, and the remaining data were used for testing. All the processes were repeated five times, and the final evaluation criteria constituted the average recognition accuracy and average recall rate of five repeated experiments. The calculation method of recognition accuracy and recall rate are shown in (11) and (12). The experiments were implemented using MATLAB R2018a on a computer with an Intel Core i5-9400 2.90 GHz CPU and Windows 10 operating system. ${\rm recognition\;accuracy}=\frac{T_{1}+...+T_{C}}{n}$ (11) ${\rm recall\;rate}=(\frac{T_{1}}{n_{1}}+...+\frac{T_{C}}{n_{C}})/C$ (12) where $T_{c},c=1,...,C$ is the count of true samples in $c$th class, $n_{c},i=1,...,C$ is the count of forecasting samples in $c$th class. TABLE II: Description of data sets. Datasets \| No. of Instances \| No. of Features \| No. of Classes \| No. of Features after PCA ---\|---\|---\|---\|--- fac \| 2000 \| 216 \| 10 \| - fou \| 2000 \| 76 \| 10 \| - kar \| 2000 \| 64 \| 10 \| - pix \| 2000 \| 240 \| 10 \| - zer \| 2000 \| 47 \| 10 \| - mor \| 2000 \| 6 \| 10 \| - Yale \| 165 \| 2000 \| 15 \| 100 COIL20 \| 1440 \| 1024 \| 20 \| 200 MNIST \| 2000 \| 256 \| 10 \| 100 USPS \| 1800 \| 256 \| 10 \| 100 ### IV-B Parameters Setting The performance of various feature extraction methods was evaluated by setting certain parameters in advance. First, the more appropriate default parameters for testing machine learning problems in Adam optimizer comprised $\alpha=0.001$, $\beta_{1}=0.9$, $\beta_{2}=0.999$, and $\epsilon=10^{−8}$. For all comparative algorithms, the search range of $k$ was set to $\\{2,4,6,8,10\\}$, whereas the range of $\sigma$ for CL-LPP, CL-LDA, and CL- LFDA was set as $\\{0.01,0.1,1,10,100,1000\\}$. In addition, the thermal parameter was calculated by $t=\\|x_{i}-x^{(7)}_{i}\\|\\|x_{j}-x^{(7)}_{j}\\|$, where $x^{(7)}_{i}$ and $x^{(7)}_{j}$ are the 7th nearest neighbors of $x_{i}$ and $x_{j}$, respectively. . ## V Conclusion In this study, we proposed a unified feature extraction framework based on contrastive learning (CL-UFEF) that is suitable for both unsupervised and supervised learning. The proposed method defined the positive and negative pairs from a new perspective and subsequently applied them to the field of feature extraction. Concretelly, compared with the previous models based on contrastive learning, CL-UFEF does not need data enhancement, and it constructs positive and negative pairs based on two contrastive learning graphs (CLG), which will make the similar samples in the subspace more clustered. 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# Ten Simple Rules for Attending Your First Conference Elizabeth Leininger Division of Natural Sciences, New College of Florida, USA Kelly Shaw Computer Science Department, Williams College, USA Niema Moshiri Computer Science & Engineering Department, University of California San Diego, USA Kelly Neiles Department of Chemistry & Biochemistry, St. Mary’s College of Maryland, USA Getiria Onsongo Mathematics, Statistics & Computer Science Department, Macalester College, USA Anna Ritz Corresponding Author: <EMAIL_ADDRESS>Biology Department, Reed College, USA ## Introduction Conferences are a mainstay of most scientific disciplines, where scientists of all career stages come together to share cutting-edge ideas and approaches. If you do research, chances are you will attend one or more of these meetings in your career. Conferences are a microcosm of their discipline, and while conferences offer different perspectives in different disciplines, they all offer experiences that range from a casual chat waiting in line for coffee to watching someone present their groundbreaking, hot-off-the-press research. Here, we share tips for trainees and their mentors. Our recommendations are based on our experiences of attending conferences and mentoring students to improve their conference experiences. As you head to your first scientific conference, these rules will help you navigate the conference environment and make the most of your experience. ### In-person Conferences: Scientific meetings have historically been in person, where all attendees travel to one location to share and learn about recent work in the field. In- person conferences provide extensive opportunities to meet and interact with other researchers. Our rules include considerations for all parts of attending an in-person meeting, including deciding whether to attend and present research, navigating travel and the conference schedule, and debriefing after the conference. ### Virtual Conferences: During the COVID-19 pandemic, many conferences have been and continue to be offered remotely [1, 2, 3]. Even before 2020, some conferences had already moved to a virtual format in response to climate change [4] and/or travel bans [5], noting the environmental, financial, and geopolitical challenges posed by in-person meetings. Virtual conferences are more inclusive than their in- person counterparts in many ways: more people can attend them from all over the world, virtual meetings are less costly and easier to organize, and attendees can participate from the comfort of their homes [3, 6, 7, 8]. Given that virtual conferences are becoming more commonplace [9, 10, 3, 7, 8], we include considerations for each rule when attending one. ### For Mentors: Academic mentorship takes many forms, including advising student research, teaching students in courses, and serving as a department chair or organizer for student activities. In many fields, graduate students mentored by a principal investigator (PI) are expected to disseminate their research at scientific meetings. Conference attendance is a valuable experience for undergraduate students, whether they are presenting research [11] or just beginning to learn about the field [12, 13]. As mentors, you might be working with undergraduates, graduate students, or other staff who may be new to conferences. We offer guidance on ways for you to help mentees navigate the subtleties and assumptions of your field. We have also developed a web portal (SI Web Portal) which contains far more information about these rules, tables of professional societies and conferences in different disciplines, and other resources that may come in handy for first-time conference attendees and their mentors. We encourage any reader to use, adapt, and contribute to these materials. ## Rule #1: Select a conference that aligns with your goals Why go to a conference, anyway? When you are deciding whether to attend a conference, consider the experience and how it will help you develop intellectually and professionally. There may be more than one answer! Conferences are a “meeting of minds” – a place for researchers to gather, present their research, give feedback on others’ research, engage in professional development, and network with one another. Conferences differ greatly in size and scope. While some conferences provide a large overview of an entire academic discipline with tens of thousands of attendees, other conferences focus on specific subdisciplines in an informal and personalized setting. Smaller conferences are excellent venues for networking, sharing your data informally, etc. Some conferences are national or international in scope, while others have a regional focus. All of these variables impact factors such as the conference location, scope of presented research, the cost of attendance, and more. If you are planning to attend a conference, you might also plan to present research. You and the others who have worked on the research would make this decision before the conference, often many months prior, and you typically submit materials to be considered for presenting. Consult with your mentor and colleagues in your research group about the benefits of and opportunities for presenting your work. In some fields, such as biology or chemistry, you might submit a short abstract for a poster presentation or a talk. In other fields, like computer science, you might submit a full paper that is peer-reviewed and published upon acceptance. Conventions vary by discipline; make sure to discuss norms in your field with your research mentor and all other authors of the work, who need to approve any submission. In addition to contributed talks and posters, conferences provide formal and informal opportunities for professional development and networking (See Rule #7). Review the professional development opportunities at the conference you are attending, and participate in opportunities that sound interesting and align with your goals. ### Virtual Conferences: Nearly every annual meeting in 2020 and 2021 has shifted to a virtual format due to the COVID-19 pandemic. Many traditionally in-person conferences aim to retain multiple goals of their conferences – a venue for presenting the latest work, networking, and professional development – in a virtual format. Alongside these legacy meetings, the virtual format has also spurred new grassroots virtual conferences [7]. For example, conferences such as Neuromatch and Black in Neuro are newer additions to Neuroscience conference offerings that originated as virtual experiences. Some virtual conferences offer more focused research areas than the larger meetings, since they are generally easier to organize and manage. They also have fewer barriers for attendance (allowing researchers from all over the world to attend), they are less wasteful (in terms of money and environmental impact), and they are overall more equitable than in-person conferences [2, 3]. Consider all of these aspects when deciding on a conference to attend. ### For Mentors: Empower students to attend conferences. Suggest conferences for your students that match their present goals/professional development needs and explain how they could benefit from attending these meetings. Be aware of socioeconomic privilege and implicit bias when recommending opportunities to trainees (e.g., recommend equally and equitably, help them find funding, and connect them to your professional network at conferences). ## Rule #2: Find others to foot the bill Organizing and running conferences cost money, and attendees usually pay to attend these events. While registration costs are often low for virtual conferences, in-person conferences can get expensive quickly due to additional costs associated with travel, room and food expenses, poster printing fees, childcare, appropriate clothing, etc. Therefore, it’s important to have a funding strategy to attend an in-person conference. Funding sources are often available to offset some (or even all) of your costs, and knowing where they exist early in the process will help you take advantage of these opportunities. First, some conferences offer discounted registration or travel grants for students, attendees from countries with traditionally fewer economic resources, etc. Some travel grants may include additional support for trainees with disabilities or trainees with small children. Other societies are establishing separate child care grants for trainees with families. Often these funding opportunities are made possible by professional societies, grants from organizations such as the National Science Foundation, or by sponsorship funds provided by industrial partners. An in-person conference might have volunteer opportunities where you are compensated with a waived registration fee or housing for working at the conference. Second, many academic institutions have funds intended to support student travel. Principal investigators may have a budget to support their students’ attendance at a conference. Research scholarships and/or travel awards may be available at the departmental or college level. Sometimes these opportunities are not widely advertised so it is always good to ask around. If you do not have a research mentor, asking the department chair may be appropriate to help you navigate your search. Another place to look is student clubs and organizations, which may have funding to support students presenting research in a field relevant to the group’s goals. Third, national societies, often the same societies that are organizing the conference, may provide support for student travel at the society, division, and regional chapter level. Some of these opportunities are dependent on student demographics such as being first-generation, a student from a historically underrepresented population, or a woman in the field. Information about these awards is generally not in a central location, so visit each level’s website to search for potential sources (the web portal includes a list of societies in STEM, many of which provide travel funds). An important note: many of these awards and funding opportunities for in- person conferences will often require that you front the money and then get reimbursed after the conference. This means you would need to put things like airfare, hotel reservations, and the like on your own credit card and then submit receipts for reimbursement. If this presents a hardship for you, reach out to your department for help. This is not the first time, nor will it be the last, that your department has encountered this problem so you should not hesitate to ask for assistance in navigating this process. Keep track of all itemized receipts for the conference, and ask the funding provider for details about reimbursement. ### Virtual Conferences: Virtual conferences generally cost less than their in-person counterparts [3], and many virtual conferences have reduced or waived the student fees associated with conference registration. If paying for a virtual conference registration is a hardship, there is no harm in asking the conference organizer if there is a way to attend. ### For Mentors: Funding support deadlines for conferences are often so far in advance that students, especially undergraduate students, may miss these opportunities. Help students identify and meet these deadlines. If your institution has funding sources for student travel but those sources are only available to those students who ask (or whose mentor’s ask), it might be time to suggest a conversation about equity of those opportunities. This policy may disproportionately disenfranchise students from historically underrepresented populations who are not as well versed in the hidden curriculum of academia and thus do not know to ask for these opportunities [14]. ## Rule #3: Know your logistics It is important to be aware of the Who, What, When, Where logistics (see Rule #1 for Why) to ensure a smooth conference-going experience. Such logistics will vary depending on if you choose to attend an in-person conference or a virtual conference, or even depending on the format of virtual conference you choose to attend. Who do you know will be at the conference? Housing and transportation costs for conferences can be expensive, complicated, and/or both. Consider traveling together with people you know and sharing housing costs by rooming with labmates, acquaintances, etc. Some conferences and field-specific societies offer roommate-matching services for those seeking conference roommates. You may also be traveling with family or dependents; check to see if the meeting has programming or resources for companions, childcare and lactation facilities, etc. as needed. What will you need to bring to the conference? It is usually preferable to pack light, and pack for the situations and weather you will encounter (see Rule #4). Double-check baggage policies for your mode of transportation, as policies vary drastically by carrier and fare. If you are presenting, make sure that your presentation materials travel safely with you. If you are giving a talk, make a few different backups of talk slides and media (store on a thumbdrive or in the cloud) in case your computer is lost, stolen, or broken. Posters can be transported in a poster tube or even printed on fabric to fold in your luggage. The poster can also be shipped to the venue or printed on-site if it is challenging to travel with it. When do you need to make key conference travel decisions? Chronologically, these include: writing and submitting a research abstract or paper, having your abstract or paper accepted, registering for the conference, making travel and housing arrangements, and actually traveling to the conference and presenting your work. Because abstract and paper submission deadlines can fall several months ahead of the meeting date, it is important to be aware of abstract deadlines for conferences in your field. When scheduling travel, make sure that you arrive before, and leave after, your scheduled presentation date/time. If your travel has multiple legs of transportation, give yourself enough time to make each connection. International travel may require visas, and the lead time and expense varies greatly depending on the origin and destination countries. Consider visas and passports in both your timeline and your budget, and talk with your mentor about whether an international conference is feasible. Conferences with an international scope will usually communicate information about visas and lead time. Where is the conference, and where will you stay while attending? Depending on the scope of the conference (see Rule #1), housing may be easily obtainable at the conference venue, or a mad scramble between all 30,000 attendees to find a room near the convention center. Conferences will typically contract guaranteed rates at nearby hotels; heed deadlines and sign up early if you possibly can. Be aware of ADA accessibility when booking if you require rooms with accessible features. Acquaint yourself with the transportation options to and within the conference host city. Again, conferences will usually send information and tips regarding transportation, housing, and the like. ### Virtual Conferences: The Who, What, When, Where logistics of virtual conferences are also extremely helpful to determine before the meeting. Identify Who you know at the meeting, and make plans to meet with them virtually during a break. Virtual conference platforms allow you to browse the list of presenters and/or conference attendees (see Rule #7). Know What you are expected to do in different sessions; if you plan to attend a virtual panel, for example, think about some questions you might want to ask the panelists. Familiarize yourself with When the conference activities are, and if any action is required for accessing the conference materials. The format of virtual conferences often differs from the time-limited experience of in- person conferences. For example, they may run longer to allow for a different pace of engagement, and they may have components that are synchronous (e.g. talks scheduled live for specific times) or asynchronous (pre-recorded talks for on-demand viewing). Some virtual conferences are offered completely asynchronously, making them more inclusive for attendees with care responsibilities or in time zones across the world [8]. Make sure to note conference dates and time zones of scheduled synchronous sessions as appropriate. If presenting at a virtual conference, pay very close attention to announcements regarding presentation logistics; you may need to upload your presentation before the formal start of the conference and check in at various times to field questions. Finally, be organized by knowing Where the conference activities are happening, often through a web interface or a collection of Zoom links. If you are unsure about the digital platform or need extra assistance, contact the organizer who will be able to help you get help before the conference. If the virtual conference requires high internet bandwidth, make sure your setup at home allows for participation. ### For Mentors: If you are also attending the meeting, make sure that your students are at the same point logistically as you (register for the conference together, book flights together, etc.) If you are funding the students, putting as much of the costs on your business card up-front reduces their financial burden. If you are not attending the meeting, reach out to your colleagues who will be there to help arrange housing and other connections for your students. ## Rule #4: Prepare for the environment A good rule of thumb is to dress as if you might meet your future employer at a conference (which you very well might do!). At the same time, wear something you are comfortable in. Different meetings and disciplines may have a variety of dress code standards. Attendees who are presenting may wear more formal clothes, but many people will dress as they do when they are at their home institutions. When you are packing for an in-person conference, check the weather before you go, since you may be outdoors (e.g., to reach the conference from the airport, to walk to find dinner, or participate in planned conference outings). Between moving around the conference venue, seeking out meals, and milling about poster sessions, you will spend a lot of time being active. Wear shoes and clothing that allow you to comfortably do all these things. If you have any questions about ADA accessibility at the conference, contact the conference venue (e.g., if it is a conference hall attached to a hotel) or contact the conference organizer (the conference website will contain this information). Some conferences have an app (See Rule #6) which may have a Q&A Forum. Finally, conference days can be long - bring a water bottle to stay hydrated, a light snack to sustain you between meals, and a comfortable bag to keep belongings. ### Virtual Conferences: You will likely be in the comfort of your own home during the conference. Realize it is easy to sit and stare at the screen for hours on end. Be sure to take breaks, stay hydrated, stretch, and focus your eyes away from your screen every so often. It is also a good idea to test your video, audio and background before the conference. Find a room with good lighting and make sure your background is not distracting. Some video conferencing platforms have video settings that will enhance your image if the lighting is low - try them out. Conference days can be extremely long in virtual platforms – sometimes longer than typical in-person schedules – so be sure to pace yourself (see Rule #6). ### For Mentors: Talk to your students about the level of formality at the meeting. If you know that an in-person conference will have outdoor events, mention that. If you are comfortable with sharing, tell your students the types of things you pack for conferences. ## Rule #5: Learn how to take in the science Science communication happens in a variety of ways at a conference. There are almost always formal and informal mechanisms for learning about and presenting science. Even within formal mechanisms, there is a range of how attendees communicate their research. Some speakers are invited to give keynote presentations, which often provide a perspective of the field or cover a broad range of projects. Other speakers are selected to give technical presentations, based on submitted abstracts or full papers. Both keynotes and technical presentations provide time for asking questions at the end, and attendees may line up in front of a microphone or raise their hand depending on the number of people. Finally, poster presentations offer a more interactive way to talk with researchers, where attendees walk up and chat with authors standing in front of their posters. But wait, there’s (often) more! Many conferences offer tutorials, workshops, or special sessions that are held either right before or right after the main conference, usually at the same location. These topics may be even more narrowly focused than those at the conference, or involve emerging concepts in the discipline. There also may be forums run by graduate/undergraduate student societies, designed for students. Note, though, some of these may require registration beyond the main conference. So, how do you actually “take in the science?” First, know that it is often impossible to see everything - the conference may have multiple conflicting sessions, the days are quite long, and you need to take time to rest and recharge. As you make a conference plan to prioritize what you want to see (see Rule #6), let the conference program be your guide. Find a way that works for you to take notes on what you learn throughout the conference. We, the authors, have developed note-taking strategies that involve carrying around a notebook, using Google Docs, writing in the conference program, taking notes on our smartphones, making sketchnotes, even sending ourselves emails. In addition to the science, take notes about whom you meet - you never know if you will meet them again or if you might want to connect with them after the conference. Find something that works for you. Some conferences may have rules about taking digital photos or videos of talks and posters, so be sure to check beforehand if you want to do either. At in-person conferences, there are a plethora of informal ways to interact with other researchers and learn about their science. Researchers appear to have a common need for caffeine, and coffee breaks are a staple of every conference. If you are not a coffee drinker, there are often other warm beverages and snacks on hand. Conferences may also offer breakfast or lunch, and each one is in an invitation to meet someone new (see Rule #7). Many conferences have a “big-ticket” event - an evening banquet, an outing to a tourist attraction, a social hosted by a sponsor, etc. If the conference has a big-ticket event, it is not one to miss! There may also be a career fair, special meetings for first-timers, or meet-ups for special interest or affinity groups. Take advantage of these to find other attendees with similar interests, backgrounds, and experiences (see Rule #8). ### Virtual Conferences: Virtual conferences have the enormous benefit in that the talks are almost always recorded (even when they are presented synchronously to all attendees). Further, sometimes conferences will rebroadcast talks 12 hours later for attendees in different time zones. The material from conferences often remains on the virtual platform after the conference concludes, allowing attendees to watch and rewatch talks at their leisure. In addition to the formal presentations, virtual conferences have developed strategies for the interactive conference portions (e.g., poster presentations, breaks, and social events). Some digital platforms offer mechanisms to ask questions and continue discussions after a talk has concluded. Cold Spring Harbor meetings, for example, have a Slack workspace with channels for each session to continue Q&A and networking. The chat functionality also allows researchers to talk to poster presenters throughout the entire meeting and send short messages about appreciating people’s work in a way that cannot happen at an in-person conference. Take advantage of the interactive parts of the conference that can help support your learning and professional development. ### For Mentors: Share the events you plan to attend with your students, and give advice about what talks or events will be most useful for them. Encourage your students to ask questions, either in the Q&A after a talk or less formally at the poster session. If you have students or colleagues who are presenting, encourage your students to attend those talks. ## Rule #6: Make a conference strategy The idea of a conference - days of uninterrupted learning about fascinating ideas and exchanging insights with other folks excited about the topics you are passionate about - might sound like a dream come true! The reality is that conferences can be exhausting if you do not have a plan for selectively attending activities that will provide you the most benefit and for practicing self-care. This is especially important for neurodivergent attendees who may get overwhelmed by the amount of information conveyed during a conference. It’s good to fashion a draft plan several weeks before the conference in which you prioritize the events you want to attend. Having a written (or app-built) plan with scheduled breaks gives you a solid framework that you can tweak on the fly as new opportunities appear. If you intend to participate in pre- or post-conference events, it is important to factor in extra self-care to compensate for the increasing duration. Before making a conference strategy, gather resources to help you make a plan. Conferences usually have a website with lots of information, and often release a schedule overview as the keynote and technical talks and posters are finalized. The conference may make use of an app, where attendees can find the schedule of events and connect with others, and/or you may receive a conference program. These are all great resources to have on hand as you make your strategy, and it is useful to skim the conference program before the meeting begins. How do you prioritize what to attend? First, it is good to attend keynote and panel sessions as they provide perspective into the wider concerns of your field and often are forward looking to emerging challenges. Second, definitely attend technical presentations related to your specific area of focus in order to know what research is being done and become part of that community of researchers. Reading papers or watching videos in advance and thinking of what questions you might like to ask about the work is a great way to prepare so that you can contribute to the discussion in a positive way. Third, the poster sessions are often short, so make sure you know which posters you want to visit while the presenter is there. Fourth, if the conference offers any first-time or new attendee events, plan on attending those as you will make some connections with other attendees that will make the conference more enjoyable and less lonely. Finally, attending the networking events (see Rule #7) helps you get to know your colleagues as individuals on a personal level (not all discussions are about the research) and also exchanging your research ideas. Working self-care into your plan is essential - but how? First, if you have a daily ritual such as exercising or going for a walk, sustain that ritual during conference days and decide when you will fit this activity in. If you are at an in-person conference, find out if the hotel has the equipment you need for exercising; many downtown gyms or community centers sell day passes. Second, recognize that your brain is going to need breaks between talks. If the conference is not in your native language, your brain may be working overtime to process the science. Determine time slots when the presentations are not of particular interest to you and plan on taking a walk, getting a coffee, or doing some other activity that will help you recharge. You will see lots of experienced conference attendees disappear for stretches of time for exactly this purpose. For many of us, traveling to in-person conferences is a way to experience new parts of the country or the world. It is fine to take a short break to experience local destinations such as a museum, park, cultural attraction, etc. ### Virtual Conferences: When attending virtually, it may be hard to devote entire days to attending the conference. Prioritize attending the synchronous events such as the keynotes and panel discussions, especially if they are not recorded for later viewing. Take advantage of videos posted for areas you are interested in and watch those ahead of time so you can choose to attend the sessions where you’re interested in the question and answer portion. ### For Mentors: Encourage your students to come up with a written plan and discuss it with them before the conference. Suggest specific presentations that would be good for them to attend and explain to them the importance of strategic downtime and self-care. ## Rule #7: Make new friends but keep the old; be ready to communicate Conferences offer a great opportunity to exchange ideas, network, and potentially form collaborations with other researchers. Networking opportunities can be organized or spontaneous. Organized networking events may include events such as socials, affinity group meetings, or mentorship opportunities that pair newcomers with established researchers in the field. Many researchers also make efforts to meet with current or prospective collaborators at meetings. In addition to organized events, lots of networking happens spontaneously – waiting in line for coffee or tea, at a poster presentation, etc. Take advantage of the opportunity to meet lots of people who are interested in sharing their science and forming new professional connections. Pre-plan a few people you would like to meet at the conference. A good way to find people is to look at the conference program, which contains presenter names and sometimes includes a full list of attendees (See Rule #6). Start with the research area you are interested in. Typically, the first author presents the work and will most likely be at the conference. Read their work before the conferences and prepare questions to start a conversation. If they have a research website, they will likely have a picture. Find out what they look like in advance. In-person conference attendees usually wear name badges which helps if you’re trying to meet someone whom you’ve never met in person. If they are giving a talk, try to attend; following up with the speaker after their talk is a good way to strike up a conversation with them. If you did not get a chance to talk with everyone you wanted to, you can always follow up with them over email (see Rule #10). You can use the conference to make it not feel like a cold email - tell them you attended their talk, saw their poster, etc., and ask any questions. Researchers delight in being asked informed and insightful questions about their work. In-person conferences have many opportunities for informal networking, but it still may feel awkward to strike up a conversation. If you are coming from a lab or institution with colleagues also attending the conference, agree to introduce one another to whomever you are speaking with. Also, most conferences have events just before the official program starts such as breakfast or socializing events in the evening. These are a great way to meet people in a more relaxed environment. If you are feeling overwhelmed or out of place, which is normal (see Rule #8), it is okay to spend time with people you already know. ### Virtual Conferences: Introducing yourself and having a one-on-one conversation can also be challenging in virtual environments. Networking still happens at virtual conferences, but the interactions are necessarily more intentional than striking up a conversation during a coffee break. Many virtual conference platforms offer ways to interact, and conferences organize social events to foster this networking. If the online conference platform has social events or research discussion groups, join in. Leave your camera on, if comfortable, and participate in the chat, if appropriate. Some virtual conference platforms have messaging capabilities, which you can use to have conversations with individuals or small groups. ### For Mentors: If you will also be attending an in-person conference, be intentional about introducing your students to your professional network. Invite them to join a group meal you plan on attending with people you already know and introduce them to colleagues. If you are talking to someone working in an area your student is interested in, and you see your student close by, be sure to introduce them. Pointing students towards posters that are relevant and interesting to them is also a great way to help attendees begin conversations with researchers. ## Rule #8: Prepare to (safely) get out of your comfort zone For most of us, meeting new people or joining a new community can be nerve- wracking or intimidating, even when you have a lot in common with people in the community. Things can be even more challenging if you come from a historically underrepresented community and do not see yourself represented at the conference. Academia, and by extension its conferences, is a traditional and elite institution whose diversity, or lack thereof, can make it a less than welcoming place at times. Imbalance in representation is gradually being acknowledged by assessing the demographics of invited speakers and society awardees [2, 15, 16, 17]. If you are feeling out of place, know you are not alone, and that it can take some time to feel comfortable. Try to meet new people and make new friends. You will likely see the same people if you attend the conference again. And, the more conferences you attend, the easier it gets. Some conferences host affinity group events, which is a good opportunity to meet and network with other attendees in a safe and welcoming space. Our web portal hosts a crowd-sourced list of affinity groups. Come up with a plan for how to talk about your interests and research. Have a 90 second elevator pitch prepared for your initial introduction and be prepared to switch into a slightly extended version of that pitch if the other person expresses interest. Because it is important to learn about the other person (people love good listeners), make sure to ask them about their research, including asking follow-up questions after their initial answers to show your interest. If the exchange goes well, create a note of their names and affiliations for use later. Also, if you see them later in the conference, acknowledge their existence with a smile or head nod. ### Virtual Conferences: An advantage of virtual conferences is you will be in your own space. Take advantage of this opportunity in a virtual setting to make yourself comfortable (See Rule #4). It is okay to turn off your video if you need to step away for a few minutes. ### For Mentors: It can be helpful for mentees to know they are not alone in feeling out of place. Normalize the experience of feeling nervous and out of place. Let them know conferences can be awkward and that is okay. If you are mentoring students from underrepresented groups, being culturally competent will help you better support them. Cultural competency is the ability to work effectively with people from different cultural backgrounds. In examining the role of cultural competence in a biology classroom, Tanner and Allen highlight the importance of cultural competence in creating an inclusive and welcoming environment for students from underrepresented groups [18]. A good place to start is finding out if your institution offers cultural competency training. If it does not, the Association of American Colleges & Universities offers cultural competency resources [19]. Finally, mentor your students – particularly marginalized students who may find academia and conferences less-than-welcoming – through the conference experience and their overall professional development with “compassion, advocacy, and support” [20]. Be aware that imposter syndrome stems from systematic bias, and calling out imposter syndrome sometimes can have the opposite effect for mentees [21]. Become involved in helping meetings achieve gender and racial balance in their selection of awardees and keynote speaker invitations [15]. See our web portal for more resources on cultural competency and improving representation at conferences. ## Rule #9: Take charge of your social interactions Conferences bring together many people from all over the world, and navigating a complex professional-yet-social environment can be challenging. All members of a scientific community have a responsibility to help make a welcoming environment and should in turn feel welcomed. As a conference attendee, you are a member of the scientific community. At the same time, we acknowledge that power imbalances may be prevalent at the conference, often reflective of career stage [22]. Many societies have established codes of conduct to which attendees must adhere [23], and the conference should include a contact if you witness or are subjected to troubling behavior. Above all, you should never feel unsafe or pressured to participate by anyone. Before you travel to in-person conferences, acquaint yourself with the code of conduct and have a contact person (e.g., your mentor, someone from your institution, or a friend) that you can reach out to. You may need to prepare yourself to adjust to the cultural norms of the conference location, if it is notably different from your academic environment. While there are social opportunities at in-person conferences, remember that these are professional events. ### Virtual Conferences: Social situations are different in a virtual platform. There may be fewer awkward conversations, but online interaction can pose its own problems. For example, you may see unwelcome visitors hack into presentations or observe uncourteous behavior from other attendees. You may also find some features (like the chat function or unmuted attendees) distracting during a presentation. When participating in virtual conferences, follow any instructions session chairs give about how to interact during presentations. As a general rule, make sure you follow online etiquette (such as muting yourself on Zoom unless speaking). Attendees should observe the same codes of conduct in a virtual setting that they would observe in an in-person conference. ### For Mentors: It is important that students have your contact information (or the information of someone trusted who will be at the meeting). Stay up to date on your organization’s code of conduct. If your society has not yet established a code of conduct (check on our web portal), encourage the leaders to do so and support their efforts [23]. ## Rule #10: Tie up the loose ends after the conference After attending the conference you will likely want to come home and collapse from all of the excitement, but wait…you have a few more things to do before you are done. These loose ends pertain to both in-person and virtual conferences. First and foremost, update any notes you have about the conference itself before you forget. Conferences can be a fast-paced blur, so make sure to record any feedback you get on your work at the conference, so that you have it when you are ready to make improvements. Updating your resume and/or curriculum vitae (CV) is another important step you should complete as quickly after the conference as possible. In addition to any posters or presentations you gave, you should also add any awards you received including travel funding awards. Depending on your career stage, you might also include other events or support you received - check with a mentor about what is appropriate. Next you should follow up with people you met at the conference. This helps to solidify the relationships you began at the conference. Email people you are interested in speaking with again and ask for an opportunity to meet. If you spoke with companies or potential job seekers, follow up with an email containing your resume and statement of interest. When you attend a conference you meet so many people that it is hard to remember everyone. By reaching out with a simple note or a LinkedIn invitation, you will be helping people to remember you which can lead to future collaborations and/or job opportunities! If you received a travel award (for an in-person conference, for example) that requires reimbursement you will need to carefully follow the instructions and/or rules of your funding agency. Reimbursements can take quite a while to process so the faster you get it done after the conference, the better. Pay special attention to rules regarding the submission of receipts (itemized receipts are required in some cases) and deadlines for submission (some agencies require reimbursement documents be submitted within a month of the conference ending). Make a note to yourself to follow up on reimbursement; if you do not hear back about your reimbursement within two weeks, go ahead and reach out to inquire about your paperwork (you can also ask your mentor if it has been longer than you were expecting). Finally, you should take time to write thank you notes or emails to anyone who supported your travel to the conference whether that be financially, in conference preparation, or in your research. This not only helps you to further strengthen relationships with those people, but also helps future students receive these awards by leaving a good impression with the awarders. ### For Mentors: Communicate your institution’s rules and requirements about reimbursements to your trainees, and tell your trainees who to contact for reimbursement paperwork (e.g. a department administrator). We suggest holding a paperwork session after the conference so that all travelers can come together and fill out paperwork, print and/or make copies of necessary materials, and ask questions of each other and you. If multiple travelers attended the same conference, then mentors can also submit all of this paperwork together to increase the chances of its timely completion. We sometimes also bring a giant box of thank you notes so that students can easily grab one to write a note of thanks to those who helped them along the way. If students run into bureaucratic difficulties with paperwork, help them get the issue resolved. ## Conclusion Scientific conferences are an amazing intellectual and professional opportunity, though attending one for the first time may be overwhelming. We hope that first-time conference attendees will feel more empowered and prepared to get the most out of their experience, be it virtual or in-person, with knowledge of these “unwritten rules.” Happy travels! ## Supporting information ### SI Web Portal The web portal includes additional details on tips for first time conference attendees and their mentors, tables of societies and scientific conferences, and affinity group organizations. https://sites.google.com/macalester.edu/simplerules/home. ## Acknowledgments We thank Erik Zornik and Andrew Bray for their early contributions to the conference advice template. This work is supported by the National Science Foundation (DBI-1750981 to AR). ## References * [1] Olena A. COVID-19 Ushers in the Future of Conferences. The Scientist. 2020. * [2] Sarabipour S. Research Culture: Virtual conferences raise standards for accessibility and interactions. Elife. 2020;9:e62668. * [3] Sarabipour S, Khan A, Seah YFS, Mwakilili AD, Mumoki FN, Sáez PJ, et al. Changing scientific meetings for the better. Nature Human Behaviour. 2021; p. 1–5. * [4] Ligozat AL, Névéol A, Daly B, Frenoux E. Ten simple rules to make your research more sustainable. PLoS Computational Biology. 2020;16(9):e1008148. * [5] Reardon S. How the latest US travel ban could affect science. Nature News. 2017;550(7674):17. * [6] Salomon D, Feldman MF. The future of conferences, today: Are virtual conferences a viable supplement to “live” conferences? EMBO reports. 2020;21(7):e50883. * [7] Rich S, Diaconescu AO, Griffiths JD, Lankarany M. Ten simple rules for creating a brand-new virtual academic meeting (even amid a pandemic). PLOS Computational Biology. 2020;16(12):1–9. doi:10.1371/journal.pcbi.1008485. * [8] Arnal A, Epifanio I, Gregori P, Martínez V. Ten Simple Rules for organizing a non-real-time web conference. PLOS Computational Biology. 2020;16(3):1–13. doi:10.1371/journal.pcbi.1007667. * [9] Gichora NN, Fatumo SA, Ngara MV, Chelbat N, Ramdayal K, Opap KB, et al. Ten simple rules for organizing a virtual conference-anywhere. PLoS Computational Biology. 2010;6(2):e1000650. * [10] Lortie CJ. Online conferences for better learning. Ecology and evolution. 2020;10(22):12442–12449. * [11] Mabrouk PA. Survey study investigating the significance of conference participation to undergraduate research students. Journal of Chemical Education. 2009;86(11):1335. * [12] Davis J, Alvarado C. Supporting undergraduates to make the most of conferences. ACM Inroads. 2017;8(3):32–35. * [13] Wright HM, Tamer NB. Can sending first and second year computing students to technical conferences help retention? In: Proceedings of the 50th ACM Technical Symposium on Computer Science Education; 2019. p. 56–62. * [14] Smith B. Mentoring at-risk students through the hidden curriculum of higher education. Lexington Books; 2013. * [15] Martin JL. Ten simple rules to achieve conference speaker gender balance. PLoS Computational Biology. 2014;10(11):e1003903. * [16] Le TT, Himmelstein DS, Anderson AAH, Gazzara MR, Greene CS. Analysis of ISCB honorees and keynotes reveals disparities. bioRxiv. 2020;. * [17] Shishkova E, Kwiecien NW, Hebert AS, Westphall MS, Prenni JE, Coon JJ. Gender diversity in a STEM subfield – analysis of a large scientific society and its annual conferences. Journal of The American Society for Mass Spectrometry. 2017;28(12):2523–2531. * [18] Tanner K, Allen D. Cultural competence in the college biology classroom. CBE – Life Sciences Education. 2007;6(4):251–258. * [19] Association of American Colleges & Universities. Cultural Competency Resources: Teaching to Increase Diversity and Equity in STEM (TIDES); 2021. Available from: https://www.aacu.org/tides/cultural-competency. * [20] Singleton KS, Tesfaye R, Dominguez EN, Dukes AJ. An open letter to past, current and future mentors of Black neuroscientists. Nature Reviews Neuroscience. 2020; p. 1–2. * [21] Tulshyan R, Burey JA. Stop Telling Women They Have Imposter Syndrome. Harvard Business Review. 2021;. * [22] Jackson L. The smiling philosopher: Emotional labor, gender, and harassment in conference spaces. Educational Philosophy and Theory. 2019;. * [23] Favaro B, Oester S, Cigliano JA, Cornick LA, Hind EJ, Parsons E, et al. Your science conference should have a code of conduct. Frontiers in Marine Science. 2016;3:103.
# A Cable Knot and BPS-Series John Chae Department of Physics and QMAP, UC Davis, 1 Shields Ave, Davis, CA, 95616, USA <EMAIL_ADDRESS> ###### Abstract A series invariant of a complement of a knot was introduced recently. The invariant for several prime knots up to ten crossings have been explicitly computed. We present the first example of a satellite knot, namely, a cable of the figure eight knot, which has more than ten crossings. This cable knot result provides nontrivial evidence for the conjectures for the series invariant and demonstrates the robustness of integrality of the quantum invariant under the cabling operation. Furthermore, we find interesting effects of the cabling on the series invariant. ###### CONTENTS 1. 1 Introduction 2. 2 Background 1. 2.1 Satellites 2. 2.2 Quantum Torus and Recursion Ideal 3. 3 Knot Polynomials 1. 3.1 The Colored Jones Polynomial 2. 3.2 The Alexander Polynomial 4. 4 The Recursion Relation 5. 5 An Expansion of a Knot Complement 6. 6 Effects of the Cabling 7. A The Definitions of the Operators ## 1 Introduction Inspired by the categorification of the Witten-Reshitikhin-Turaev invariant of a closed oriented 3-manifold [37, 30, 29] in [15, 16], a two variable series invariant $F_{K}(x,q)$ for a complement of a knot $M^{3}_{K}$ was introduced in [13]. Although its rigorous definition is yet to be found, it possesses various properties such as the Dehn surgery formula and the gluing formula. This knot invariant $F_{K}$ takes the form111Implicitly, there is a choice of group; originally, the group used is ${\rm SU}(2)$. $\displaystyle F_{K}(x,q)=\frac{1}{2}\sum_{\begin{subarray}{c}m\geq 1\\\ m\ \text{odd}\end{subarray}}^{\infty}\big{(}x^{m/2}-x^{-m/2}\big{)}f_{m}(q)\in\frac{1}{2^{c}}q^{\Delta}\mathbb{Z}\big{[}x^{\pm 1/2}\big{]}\big{[}\big{[}q^{\pm 1}\big{]}\big{]},$ (1) where $f_{m}(q)$ are Laurent series with integer coefficients222They can be polynomials for monic Alexander polynomial of $K$ (See Section 3.2), $c\in\mathbb{Z}_{+}$ and $\Delta\in\mathbb{Q}$. Moreover, $x$-variable is associated to the relative ${\rm Spin}^{c}\big{(}M^{3}_{K},T^{2}\big{)}$-structures, which is affinely isomorphic to $H^{2}\big{(}M^{3}_{K},T^{2};\mathbb{Z}\big{)}\cong H_{1}\big{(}M^{3}_{K};\mathbb{Z}\big{)}$; it has an infinite order, which is reflected as a series in $F_{K}$. The rational constant $\Delta$ was investigated in [14], which elucidated its intimate connection to the d-invariant (or the correction term) in certain versions of the Heegaard Floer homology ($HF^{\pm}$) for rational homology spheres. The physical interpretation of the integer coefficients in $f_{m}(q)$ are number of BPS states of 3d $\mathcal{N}=2$ supersymmetric quantum field theory on $M^{3}_{K}$ together with boundary conditions on $\partial M^{3}_{K}$. Furthermore, it was conjectured that $F_{K}$ also satisfies the Melvin–Morton–Rozansky conjecture [24, 31, 32] (proven in [1]): ###### Conjecture 1.1 ([13, Conjecture 1.5]). For a knot $K\subset$ $S^{3}$, the asymptotic expansion of the knot invariant $F_{K}\big{(}x,q={\rm e}^{\hbar}\big{)}$ about $\hbar=0$ coincides with the Melvin–Morton–Rozansky (MMR) expansion of the colored Jones polynomial in the large color limit: $\displaystyle\frac{F_{K}\big{(}x,q={\rm e}^{\hbar}\big{)}}{x^{1/2}-x^{-1/2}}=\sum_{r=0}^{\infty}\frac{P_{r}(x)}{\Delta_{K}(x)^{2r+1}}\hbar^{r},$ (2) where $x=e^{n\hbar}$ is fixed, n is the color of $K$, $P_{r}(x)\in\mathbb{Q}\big{[}x^{\pm 1}\big{]}$, $P_{0}(x)=1$ and $\Delta_{K}(x)$ is the (symmetrized) Alexander polynomial of $K$. Additionally, motivated by the quantum volume conjecture/AJ-conjecture [7, 11] (explained in Section 2.2), it was conjectured that $F_{K}$-series is q-holonomic: ###### Conjecture 1.2 ([13, Conjecture 1.6]). For any knot $K\subset$ $S^{3}$, the normalized series $f_{K}(x,q)$ satisfies a linear recursion relation generated by the quantum A-polynomial of $K$ $\hat{A}_{K}(q,\hat{x},\hat{y})$: $\displaystyle\hat{A}_{K}(q,\hat{x},\hat{y})f_{K}(x,q)=0,$ (3) where $f_{K}:=F_{K}(x,q)/\big{(}x^{1/2}-x^{-1/2}\big{)}$. The actions of $\hat{x}$ and $\hat{y}$ are $\hat{x}f_{K}(x,q)=xf_{K}(x,q)\qquad\hat{y}f_{K}(x,q)=f_{K}(xq,q).$ For examples of the $F_{K}$-series, several prime knots up to ten crossings have been analyzed [13, 27, 28]. They include the torus knots, the figure eight knot in [13], and $\bm{5_{1}}$ in [28]. Positive braid knots ($\bm{10_{139}}$, $\bm{10_{152}}$), strongly quasipositive braids knots ($\text{m}(\bm{10_{145}}),\bm{10_{154}},\bm{10_{161}}$), double twist knots (m($\bm{5_{2}}$), m($\bm{7_{3}}$), m($\bm{7_{4}}$)), and a few more prime knots (m($\bm{7_{5}}$), m($\bm{8_{15}}$)) were examined in [27]. In this paper we verify the above conjectures by computing the $F_{K}$-series for $(9,2)$-cabling of the figure eight knot and we compare our result to that of the figure eight knot. Furthermore, we conjecture about the $F_{K}$-series for a family of a cable knot of the figure eight. The rest of the paper is organized as follows. In Section 2 we review the satellite operation on a knot and the recursion ideal of the quantum torus. In Section 3 we analyze knot polynomials of the cable knot of the figure eight. In Section 4 we derive the recursion relation for the cable knot. Then we deduce $\hbar$ expansion from the recursion in Section 5. Finally in Section 6 consequences of the cabling operation are discussed and we propose conjectures about a family of a cable knot. Acknowledgments. I would like to thank Sergei Gukov, Thang L$\hat{\text{e}}$ and Laura Starkston for helpful conversations. I am grateful to Ciprian Manolescu for valuable suggestions on a draft of this paper. I am also grateful to Colin Adams for valuable comments. ## 2 Background ### 2.1 Satellites The satellite operation consists of a pattern knot P in the interior of the solid torus $S^{1}\times D^{2}$, a companion knot $K^{\prime}$ in the $S^{3}$ and an canonical identification $h_{K^{\prime}}$ $h_{K^{\prime}}:S^{1}\times D^{2}\longrightarrow\nu(K^{\prime})\subset S^{3},$ (4) where $\nu(K^{\prime})$ is the tubular neighborhood of $K^{\prime}$. Figure 1: A pattern knot $P$ (left), companion $K^{\prime}$ (center) and satellite knot $P(K^{\prime})$ (right). A well-known example of satellite knots is a cable knot $h_{K^{\prime}}(P)=C_{(r,s)}(K^{\prime})$ that is obtained by choosing P to be the $(r,s)$-torus knot pushed into the interior of the $S^{1}\times D^{2}$. This map $h_{K^{\prime}}$ has been investigated in [23, 25, 26]. ### 2.2 Quantum Torus and Recursion Ideal Let $\mathcal{T}$ be a quantum torus $\mathcal{T}:=\mathbb{C}[t^{\pm 1}]\left\langle M^{\pm 1},L^{\pm 1}\right\rangle/(LM-t^{2}ML).$ The generators of the noncommutative ring $\mathcal{T}$ acts on a set of discrete functions, which are colored Jones polynomials $J_{K,n}\in\mathbb{Z}[t^{\pm 1}]$ in our context, as $MJ_{K,n}=t^{2n}J_{K,n}\qquad LJ_{K,n}=J_{K,n+1}.$ The recursion(annihilator) ideal $\mathcal{A}_{K}$ of $J_{K,n}$ is the left ideal $\mathcal{A}_{K}$ in $\mathcal{T}$ consisting of operators that annihilates $J_{K,n}$: $\mathcal{A}_{J_{K,n}}:=\left\\{\alpha_{K}\in\mathcal{T}\,\|\,\alpha_{K}J_{K,n}=0\right\\}.$ As it turns out that $\mathcal{A}_{K}$ is not a principal ideal in general. However, by adding inverse polynomials of t and M to $\mathcal{T}$ [7], we obtain a principal ideal domain $\tilde{\mathcal{T}}$ $\tilde{\mathcal{T}}:=\left\\{\sum_{j\in\mathbb{Z}}a_{j}(M)L^{j}\Big{\|}\,a_{j}(M)\in\mathbb{C}[t^{\pm 1}](M),\,a_{j}=\text{almost always}\quad 0\right\\}$ Using $\tilde{\mathcal{T}}$ we get a principal ideal $\tilde{\mathcal{A}_{K}}:=\tilde{\mathcal{T}}\mathcal{A}_{K}$ generated by a single polynomial $\hat{A}_{K}$ $\hat{A}_{K}(t,M,L)=\sum_{j=0}^{d}a_{j}(t,M)L^{j}.$ This $\hat{A}_{K}$ polynomial is a noncommutative deformation of a classical A-polynomial of a knot [3] (see also [4]). Alternative approaches to obtain $\hat{A}_{K}(t,M,L)$ are by quantizing the classical A-polynomial curve using a twisted Alexander polynomial or applying the topological recursion [17]. A conjecture called AJ conjecture/quantum volume conjecture was proposed in [7, 11] via different approaches: ###### Conjecture 2.1. For any knot $K\subset$ $S^{3}$, $\hat{A}_{K}(t=-1,L,M)$ reduces to the (classical) A-polynomial curve $A_{K}(L,M)$ up to a solely M-dependent overall factor. In other words, $J_{K,n}(t)$ satisfies a linear recursion relation generated by $\hat{A}_{K}(t,M,L)$. This property of $J_{K,n}$ is often called q-holonomic [9]. The conjecture was confirmed for a variety of knots [5, 7, 8, 10, 19, 22, 36, 34]. ## 3 Knot Polynomials In this section we will analyze the colored Jones polynomial and the Alexander polynomial of a cable knot to show that the former satisfies the MMR expansion and the latter is monic. Furthermore, the MMR expansion enables us to read off the initial condition that is needed in Section 5. ### 3.1 The Colored Jones Polynomial For (r,2)-cabling of the figure eight knot $\bm{4_{1}}$, we set $P=T(r,2)$ and $K^{\prime}=\bm{4_{1}}$ in (4). The cabling formula for an unnormalized $\mathfrak{sl}_{2}(\mathbb{C})$ colored Jones polynomial of a $(r,2)$-cabling of a knot $K^{\prime}$ in $S^{3}$ is [35] $\tilde{J}_{C_{(r,2)}(K^{\prime}),n}(q)=q^{-\frac{r}{2}\left(n^{2}-1\right)}\sum_{w=1}^{n}(-1)^{r(n-w)}q^{\frac{r}{2}w(w-1)}\tilde{J}_{K^{\prime},\,(2w-1)}(q),\quad\quad\|r\|>8\quad\text{and odd}.$ Figure 2: (r,2)-cable of the figure eight knot. Its application to $K=C_{(9,2)}(\bm{4_{1}})$333This cabling parameters correspond to $\bm{9_{1}}$ for the pattern knot. We assume 0-framing for $\bm{4_{1}}$., whose diagram has 25 crossings, is $\displaystyle\tilde{J}_{K,n}(q)$ $\displaystyle=q^{-\frac{9}{2}\left(n^{2}-1\right)}\sum_{w=1}^{n}\left[(-1)^{(n-w)}q^{\frac{9}{2}w(w-1)}[2w-1]\sum_{r=0}^{2w-2}\prod_{k=1}^{r}\left(-q^{-k}-q^{k}+q^{1-2w}+q^{2w-1}\right)\right].$ Using the (0-framed) unknot $U$ value together with $q=t^{4}$ $J_{U,n}(t)=\frac{t^{2n}-t^{-2n}}{t^{2}-t^{-2}},$ the first few unknot normalized polynomials $J_{K,n}(q)$ are $\displaystyle J_{K,1}(q)=1$ $\displaystyle J_{K,2}(q)=q^{2}-q+\frac{1}{q^{4}}+\frac{1}{q^{6}}-\frac{1}{q^{7}}+\frac{1}{q^{8}}-\frac{1}{q^{9}}+\frac{1}{q^{12}}-\frac{1}{q^{13}}$ $\displaystyle J_{K,3}(q)$ $\displaystyle=q^{12}-q^{11}-q^{10}+q^{9}-q^{8}+q^{7}+q^{6}-q^{5}+q^{2}-1+\frac{1}{q^{8}}+\frac{1}{q^{11}}-\frac{1}{q^{13}}+\frac{1}{q^{14}}-\frac{1}{q^{16}}+\frac{1}{q^{17}}$ $\displaystyle-\frac{1}{q^{18}}-\frac{1}{q^{19}}+\frac{2}{q^{20}}-\frac{1}{q^{21}}+\frac{1}{q^{23}}-\frac{1}{q^{24}}+\frac{1}{q^{25}}+\frac{1}{q^{26}}-\frac{2}{q^{27}}-\frac{1}{q^{28}}+\frac{1}{q^{29}}-\frac{1}{q^{30}}+\frac{2}{q^{32}}-\frac{1}{q^{33}}$ $\displaystyle-\frac{1}{q^{34}}+\frac{1}{q^{35}}$ Their $\hbar$ series are $\displaystyle J_{K,n}(e^{\hbar})$ $\displaystyle=1+\left(6-6n^{2}\right)\hbar^{2}+\left(-42+42n^{2}\right)\hbar^{3}+\left(\frac{801}{2}-462n^{2}+\frac{123}{2}n^{4}\right)\hbar^{4}$ (5) $\displaystyle+\left(-\frac{8451}{2}+5173n^{2}-\frac{1895}{2}n^{4}\right)\hbar^{5}+\left(\frac{3111491}{60}-\frac{132779}{2}n^{2}+14986n^{4}-\frac{27281}{60}n^{6}\right)\hbar^{6}$ $\displaystyle+\left(-\frac{14631401}{20}+\frac{19399417}{20}n^{2}-\frac{3028829}{12}n^{4}+\frac{840097}{60}n^{6}\right)\hbar^{7}$ $\displaystyle+\left(\frac{39069313501}{3360}-\frac{950122877}{60}n^{2}+\frac{54585517}{12}n^{4}-\frac{1725671}{5}n^{6}+\frac{13273763}{3360}n^{8}\right)\hbar^{8}+\cdots$ We see that, at each $\hbar$ order, the degree of the polynomial in n is at most the order of $\hbar$, which is an equivalent characterization of the MMR expansion of the colored Jones polynomial of a knot. Secondly, as a consequence of the cabling, odd powers of $\hbar$ appear in the expansion, which are absent in the case of the figure eight knot [13]. Moreover, the coefficient polynomials for the odd $\hbar$-powers have one lower degree whereas the degree of the polynomials are the same for the even $\hbar$-powers. Hence they are unaffected by the cabling operation. ### 3.2 The Alexander Polynomial The cabling formula for the Alexander Polynomial of a knot $K$ is [18] $\Delta_{C_{(p,q)}(K)}(t)=\Delta_{K}(t^{p})\Delta_{T_{(p,q)}}(t),\quad 2\leq p<\|q\|\quad\text{gcd}(p,q)=1,$ where $\Delta(t)$ is the symmetrized Alexander polynomial and $T_{(p,q)}$ is the (p,q) torus knot. Note that our convention for the parameters of the torus knot are switched (i.e. $p\equiv 2,q\equiv r$). Applying the above formula to $C_{(9,2)}(\bm{4_{1}})$, we get $\displaystyle\Delta_{C_{(9,2)}{(\bm{4_{1}})}}(x)$ $\displaystyle=\Delta_{\bm{4_{1}}}(x^{2})\Delta_{T_{(2,9)}}(x)$ $\displaystyle=-x^{6}-\frac{1}{x^{6}}+x^{5}+\frac{1}{x^{5}}+2x^{4}+\frac{2}{x^{4}}-2x^{3}-\frac{2}{x^{3}}+x^{2}+\frac{1}{x^{2}}-x-\frac{1}{x}+1.$ From this Alexander polynomial its symmetric expansion about $x=0$ (in x) and $x=\infty$ (in $1/x$) in the limit of $\hbar\rightarrow 0$ can be computed. $\displaystyle\lim_{q\rightarrow 1}2F_{K}(x,q)$ $\displaystyle=2\,\text{s.e}\left(\frac{x^{1/2}-x^{-1/2}}{\Delta_{K}(x)}\right)$ $\displaystyle=x^{11/2}-\frac{1}{x^{11/2}}+2x^{15/2}-\frac{2}{x^{15/2}}+5x^{19/2}-\frac{5}{x^{19/2}}+13x^{23/2}-\frac{13}{x^{23/2}}$ $\displaystyle+34x^{27/2}-\frac{34}{x^{27/2}}-x^{29/2}+\frac{1}{x^{29/2}}+89x^{31/2}-\frac{89}{x^{31/2}}-2x^{33/2}+\frac{2}{x^{33/2}}$ $\displaystyle+233x^{35/2}-\frac{233}{x^{35/2}}-5x^{37/2}+\frac{5}{x^{37/2}}+610x^{39/2}-\frac{610}{x^{39/2}}+\cdots\in\mathbb{Z}\left[\left[x^{\pm 1/2}\right]\right]$ (6) The coefficients in the expansions are integers and hence the Alexander polynomial is monic, which is a necessary condition for $f_{m}(q)$’s in (1) to be polynomials. ## 4 The Recursion Relation The quantum (or noncommutative) A-polynomial of a class of cable knot $C_{(r,2)}(\bm{4_{1}})$ in $S^{3}$ having minimal L-degree is given by [33] $\hat{A}_{K}(t,M,L)=(L-1)B(t,M)^{-1}Q(t,M,L)\left(M^{r}L+t^{-2r}M^{-r}\right)\in\tilde{\mathcal{A}}_{K}$ (7) where $Q(t,M,L)=Q_{2}(t,M)L^{2}+Q_{1}(t,M)L+Q_{0}(t,M),\quad B(t,M):=\sum_{j=0}^{2}c_{j}b(t,t^{2j+2}M^{2})$ $b(t,M)=\frac{M(1+t^{4}M^{2})(-1+t^{4}M^{4})(-t^{2}+t^{14}M^{4})}{t^{2}-t^{-2}}$ $c_{0}=\hat{P}_{0}(t,t^{4}M^{2})\hat{P}_{1}(t,t^{6}M^{2}),\quad c_{1}=-\hat{P}_{1}(t,t^{2}M^{2})\hat{P}_{1}(t,t^{6}M^{2}),\quad c_{2}=\hat{P}_{1}(t,t^{2}M^{2})\hat{P}_{2}(t,t^{4}M^{2}).$ The definitions of the operators $\hat{P}_{i}$ are written in Appendix A. For $K=C_{(9,2)}(\bm{4_{1}})$, applying (7) to $f_{K}(x,q)$ together with $x=q^{n}$ yields $\alpha(x,q)F_{K}(x,q)+\beta(x,q)F_{K}(xq,q)+\gamma(x,q)F_{K}(xq^{2},q)+\delta(x,q)F_{K}(xq^{3},q)+F_{K}(xq^{4},q)=0,$ (8) where $\alpha,\beta,\gamma,\delta$ functions and their $\hbar$ series are documented in [2]. From (8) we find the recursion relation for $f_{m}$. $\displaystyle f_{m+98}\,(q)$ $\displaystyle=\frac{-1}{q^{\frac{109+m}{2}}\left(1-q^{\frac{87+m}{2}}\right)}\Big{[}t_{2}\,f_{m+94}+t_{4}\,f_{m+90}+t_{6}\,f_{m+86}+t_{8}\,f_{m+82}+t_{9}\,f_{m+80}+t_{10}\,f_{m+78}$ (9) $\displaystyle+t_{11}\,f_{m+76}+t_{12}\,f_{m+74}+t_{13}\,f_{m+72}+t_{14}\,f_{m+70}+t_{15}\,f_{m+68}+t_{16}\,f_{m+66}+t_{17}\,f_{m+64}$ $\displaystyle+t_{18}\,f_{m+62}+t_{19}\,f_{m+60}+t_{20}\,f_{m+58}+t_{21}\,f_{m+56}+t_{22}\,f_{m+54}+t_{23}\,f_{m+52}+t_{24}\,f_{m+50}$ $\displaystyle+t_{25}\,f_{m+48}+t_{26}\,f_{m+46}+t_{27}\,f_{m+44}+t_{28}\,f_{m+42}+t_{29}\,f_{m+40}+t_{30}\,f_{m+38}+t_{31}\,f_{m+36}$ $\displaystyle+t_{32}\,f_{m+34}+t_{33}\,f_{m+32}+t_{34}\,f_{m+30}+t_{35}\,f_{m+28}+t_{36}\,f_{m+26}+t_{37}\,f_{m+24}+t_{38}\,f_{m+22}$ $\displaystyle+t_{39}\,f_{m+20}+t_{40}\,f_{m+18}+t_{41}\,f_{m+16}+t_{43}\,f_{m+12}+t_{45}\,f_{m+8}+t_{47}\,f_{m+4}+t_{49}\,f_{m}\vphantom{1}\Big{]}\in\mathbb{Z}[q^{\pm 1}]$ where $t_{v}=t_{v}(q,q^{m})$’s are listed in [2]. Using this recursion and the initial data documented in [2], $F_{K}(x,q)$ can be obtained to any desired order in x. ## 5 An Expansion of a Knot Complement We next compute a series expansion of the $F_{K}$ of complement of the cable knot $K$. Specifically, a straightforward computation from (8) yields an ordinary differential equation(ODE) for $P_{m}(x)$ at each $\hbar$ order. Using the initial conditions for the ODEs obtained from (5) $P_{1}(1)=0,\quad P_{2}(1)=6,\quad P_{3}(1)=-42,\quad P_{4}(1)=\frac{801}{2},\quad P_{5}(1)=-\frac{8451}{2},\quad\cdots$ we find that $\displaystyle P_{1}(x)$ $\displaystyle=\,5x^{12}+\frac{5}{x^{12}}-10x^{11}-\frac{10}{x^{11}}-13x^{10}-\frac{13}{x^{10}}+36x^{9}+\frac{36}{x^{9}}-10x^{8}-\frac{10}{x^{8}}-16x^{7}-\frac{16}{x^{7}}+15x^{6}$ $\displaystyle+\frac{15}{x^{6}}-14x^{5}-\frac{14}{x^{5}}+16x^{4}+\frac{16}{x^{4}}-18x^{3}-\frac{18}{x^{3}}+19x^{2}+\frac{19}{x^{2}}-20x-\frac{20}{x}+20$ $\displaystyle\newline P_{2}(x)$ $\displaystyle=\,\frac{25x^{24}}{2}+\frac{25}{2x^{24}}-50x^{23}-\frac{50}{x^{23}}-14x^{22}-\frac{14}{x^{22}}+306x^{21}+\frac{306}{x^{21}}-\frac{641x^{20}}{2}-\frac{641}{2x^{20}}-448x^{19}$ $\displaystyle-\frac{448}{x^{19}}+\frac{2011x^{18}}{2}+\frac{2011}{2x^{18}}-358x^{17}-\frac{358}{x^{17}}-522x^{16}-\frac{522}{x^{16}}+612x^{15}+\frac{612}{x^{15}}-\frac{589x^{14}}{2}$ $\displaystyle-\frac{589}{2x^{14}}+508x^{13}+\frac{508}{x^{13}}-\frac{3325x^{12}}{2}-\frac{3325}{2x^{12}}+1648x^{11}+\frac{1648}{x^{11}}+1538x^{10}+\frac{1538}{x^{10}}-3932x^{9}$ $\displaystyle-\frac{3932}{x^{9}}+1574x^{8}+\frac{1574}{x^{8}}+1670x^{7}+\frac{1670}{x^{7}}-1798x^{6}-\frac{1798}{x^{6}}+396x^{5}+\frac{396}{x^{5}}-\frac{1521x^{4}}{2}$ $\displaystyle-\frac{1521}{2x^{4}}+4082x^{3}+\frac{4082}{x^{3}}-\frac{6541x^{2}}{2}-\frac{6541}{2x^{2}}-8334x-\frac{8334}{x}+16831.$ The subsequent $P_{m}$’s are documented in [2]. Substituting them into (2) results in $\displaystyle 2F(x,e^{\hbar})$ $\displaystyle=\Big{(}x^{11/2}-\frac{1}{x^{11/2}}+2x^{15/2}-\frac{2}{x^{15/2}}+5x^{19/2}-\frac{5}{x^{19/2}}+13x^{23/2}-\frac{13}{x^{23/2}}+34x^{27/2}-\frac{34}{x^{27/2}}$ $\displaystyle-x^{29/2}+\frac{1}{x^{29/2}}+89x^{31/2}-\frac{89}{x^{31/2}}-2x^{33/2}+\frac{2}{x^{33/2}}+233x^{35/2}-\frac{233}{x^{35/2}}-5x^{37/2}$ $\displaystyle+\frac{5}{x^{37/2}}+\cdots\Big{)}$ $\displaystyle+\hbar\,\Big{(}5x^{11/2}-\frac{5}{x^{11/2}}+12x^{15/2}-\frac{12}{x^{15/2}}+35x^{19/2}-\frac{35}{x^{19/2}}+104x^{23/2}-\frac{104}{x^{23/2}}+306x^{27/2}$ $\displaystyle-\frac{306}{x^{27/2}}-15x^{29/2}+\frac{15}{x^{29/2}}+890x^{31/2}-\frac{890}{x^{31/2}}-36x^{33/2}+\frac{36}{x^{33/2}}+2563x^{35/2}-\frac{2563}{x^{35/2}}$ $\displaystyle-105x^{37/2}+\frac{105}{x^{37/2}}+\cdots\Big{)}$ $\displaystyle+\hbar^{2}\,\Big{(}\frac{25}{2}x^{11/2}-\frac{25}{2}\frac{1}{x^{11/2}}+36x^{15/2}-\frac{36}{x^{15/2}}+\frac{247}{2}x^{19/2}-\frac{247}{2}\frac{1}{x^{19/2}}+426x^{23/2}-\frac{426}{x^{23/2}}$ $\displaystyle+1441x^{27/2}-\frac{1441}{x^{27/2}}-\frac{225}{2}x^{29/2}+\frac{225}{2}\frac{1}{x^{29/2}}+4781x^{31/2}-\frac{4781}{x^{31/2}}-324x^{33/2}+\frac{324}{x^{33/2}}$ $\displaystyle+2563x^{35/2}-\frac{2563}{x^{35/2}}-\frac{2207}{2}x^{37/2}+\frac{2207}{2}\frac{1}{x^{37/2}}+\cdots\Big{)}.$ Comparing to the series of the figure eight knot [13], we notice that every order of $\hbar$ appears in the above series whereas the series corresponding to the figure eight knot consists of only even powers of $\hbar$ (i.e. $P_{i}(x)=0$ for i odd). This difference is an effect of the torus knot whose expansion involve all powers of $\hbar$ [13]. Furthermore, the x-terms begin from $m=11$ instead of $m=1$ and there are gaps in their powers. Specifically, $x^{\pm 13/2},x^{\pm 17/2},x^{\pm 21/2}$ and $x^{\pm 25/2}$ are absent. This is a consequence of the structure of (6). A distinctive feature of the cable knot is that from $x^{\pm 29/2}$ the coefficients are negative. Moreover, the positive and the negative coefficients alternate from that x-power for all $\hbar$ powers. These differences persist in the higher $\hbar$-orders, which are recorded in [2]. We will see these differences in a manifest way in the next section. ## 6 Effects of the Cabling Since the initial data plays a core role in the recursion relation method, we discuss their features for the cable knot and then propose conjectures about them, which can be a useful guide for finding initial data for a family of the cable knots. In the initial data (see [2]) for the recursion relation (9), we notice several differences from that of the figure eight knot [13]. Before discussing them, let us begin with the properties of the $F_{K}$ that are preserved by the cabling. The initial data consists of an odd number of terms and power of q increases by one between every consecutive terms in a fixed $f_{m}$ for all m’s, which are also true for $f_{99}$ and $f_{101}$. Additionally, the reflection symmetry of coefficients is retained up to $f_{43}$ for positive coefficients and up to $f_{61}$ for the negative ones but of course, those $f_{m}$’s do not have the complete amphichiral structure. These invariant properties are a remnant feature of the amphichiral property of the figure eight knot. A difference is that the nonzero initial data begins from $f_{11}$ and the gaps between the powers of x is four up to $x^{27/2}$, which is in the accordance with $f_{m}$’s. These features are direct consequences of the symmetric expansion of the Alexander polynomial of the cable knot (6). In the case of the figure eight its coefficient functions start from $f_{1}$ and there are no such gaps. Another distinctive difference is that $f_{m}$’s containing negative coefficients appear from $m=29$. Moreover, the positive and negative coefficient $f_{m}$’s alternate from $f_{27}$ (i.e. positive coefficients for $f_{27}$, $f_{31}$, $\ldots$ and negative coefficients for $f_{29}$, $f_{33}$, $\ldots$). Furthermore, from $f_{47}$ the reflection symmetry of the positive coefficients in the appropriate $f_{m}$’s is broken. This phenomenon also occurs for the negative coefficient $f_{m}$’s from $m=65$. Breaking of the symmetry is expected since the cable knot of the figure eight is not amphichiral. The next difference is that the largest power of q in the positive coefficient $f_{m}$’s for $m\geq 15$, the powers increase by $2,2,3,3,4,4,\ldots,11,11$. For the negative coefficient case, the changes are $4,4,5,5,6,6,\ldots,11,11$ from $m=33$. The smallest powers of $f_{m}$’s having positive coefficients exhibit their changes as $0,0,-1,-1,-2,-2,-3,-3,\ldots$ and for those with negative coefficients the pattern is $2,2,1,1,0,0,-1,-1,-2,-2,\ldots$. An universal feature of the negative coefficient $f_{m}$’s in the initial data is that their coefficient modulo sign is determined by the positive coefficient $f_{m}$’s. For example, the absolute value of the coefficients of $f_{29}$ is same as that of $f_{11}$; $f_{33}$’s coefficients come from that of $f_{15}$ up to sign and so forth. Hence coefficients of $f_{m}$ having negative coefficients are determined by $f_{m-18}$. In fact, this peculiar coefficient correlation also exists in the non-initial data $f_{101}$ whose coefficients are correlated with that of $f_{83}$. ###### Conjecture 6.1. For a class of a cable knot of the figure eight $K_{r}=C_{(r,2)}(\bm{4_{1}})\subset S^{3}$, $r>8$ and odd having monic Alexander polynomial, the coefficient functions $\left\\{f_{m}(q)\in\mathbb{Z}[q^{\pm 1}]\right\\}$ in $F_{K_{r}}(x,q)$ can be classified into two (disjoint) subsets: one of them consists of elements having all positive coefficients $\left\\{f^{+}_{t}(q)\right\\}_{t\in I^{+}}$ and the other subset contains elements whose coefficients are all negative $\left\\{f^{-}_{w}(q)\right\\}_{w\in I^{-}}$. Furthermore, for every element in $\left\\{f^{-}_{w}(q)\right\\}$, its coefficients coincide with that of an element in $\left\\{f^{+}_{t}(q)\right\\}$ up to sign. ###### Conjecture 6.2. For the family of knots in Conjecture 6.1, every nonzero element in $\left\\{f_{m}(q)\right\\}$ consists of an odd number of terms and power of q increases(decreases) by one between every consecutive terms of the element. Moreover, coefficients of an element $f^{-}_{v}(q)\in\left\\{f^{-}_{w}(q)\right\\}$ agree with coefficients of $f^{+}_{v-2r}(q)\in\left\\{f^{+}_{t}(q)\right\\}$ up to sign for all $v\in I^{-}$. ## Appendix A The Definitions of the Operators We list the definitions of the operators in the $\hat{A}$-polynomial (7) in Section 4. $\displaystyle Q_{2}(t,M)$ $\displaystyle=\,\hat{P}_{2}(t,t^{4}M^{2})\,\hat{P}_{1}(t,t^{2}M^{2})\,\hat{P}_{0}(t,t^{6}M^{2})$ $\displaystyle Q_{1}(t,M)$ $\displaystyle=\,\hat{P}_{0}(t,t^{4}M^{2})\,\hat{P}_{1}(t,t^{6}M^{2})\,\hat{P}_{2}(t,t^{2}M^{2})-\hat{P}_{1}(t,t^{6}M^{2})\,\hat{P}_{1}(t,t^{2}M^{2})\,\hat{P}_{1}(t,t^{4}M^{2})$ $\displaystyle+\hat{P}_{2}(t,t^{4}M^{2})\,\hat{P}_{1}(t,t^{2}M^{2})\,\hat{P}_{0}(t,t^{6}M^{2})$ $\displaystyle Q_{0}(t,M)$ $\displaystyle=\,\hat{P}_{0}(t,t^{4}M^{2})\,\hat{P}_{1}(t,t^{6}M^{2})\,\hat{P}_{0}(t,t^{2}M^{2}),$ $\displaystyle\hat{P}_{0}(t,M):$ $\displaystyle=\,t^{6}M^{4}(-1+t^{12}M^{4})$ $\displaystyle\hat{P}_{1}(t,M):$ $\displaystyle=\,-(-1+t^{4}M^{2})(1+t^{4}M^{2})\left(1-t^{4}M^{2}-t^{4}M^{4}-t^{12}M^{4}-t^{12}M^{4}-t^{12}M^{6}+t^{16}M^{8}\right)$ $\displaystyle\hat{P}_{2}(t,M):$ $\displaystyle=\,t^{10}M^{4}(-1+t^{4}M^{4}).$ ## References * [1] Bar-Natan D., Garoufalidis S., On the Melvin–Morton–Rozansky conjecture, Invent. 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# Tracking Short-Term Temporal Linguistic Dynamics to Characterize Candidate Therapeutics for COVID-19 in the CORD-19 Corpus James Powell<EMAIL_ADDRESS>0000-0002-3517-7485 Los Alamos National LaboratoryP.O. Box 1663Los AlamosNew MexicoUSA87545 and Kari Sentz <EMAIL_ADDRESS>Los Alamos National LaboratoryP.O. Box 1663Los AlamosNew MexicoUSA87545 ###### Abstract. Scientific literature tends to grow as a function of funding and interest in a given field. Mining such literature can reveal trends that may not be immediately apparent. The CORD-19 corpus represents a growing corpus of scientific literature associated with COVID-19. We examined the intersection of a set of candidate therapeutics identified in a drug-repurposing study with temporal instances of the CORD-19 corpus to determine if it was possible to find and measure changes associated with them over time. We propose that the techniques we used could form the basis of a tool to pre-screen new candidate therapeutics early in the research process. ††conference: SenSys 2020: 18th ACM Conference on Embedded Networked Sensor Systems; November 16-19, 2020; Yokohama, Japan ## 1\. Introduction Diachronic word analysis is Natural Language Processing (NLP) technique for characterizing the evolution of words over time. Often used for historical linguistic studies, it can also be applied to scientific literature (Tshitoyan et al., 2019) and can reveal early evidence of scientific discoveries before they become widely known. Drug-repurposing studies aim to identify existing drugs that might be useful in treating other diseases. The availability of large amounts of data about drugs and infectious agents such as viruses has enabled such studies to be performed in-silico. In early 2020, a number of repurposing studies were undertaken to identify potential treatments for COVID-19. The CORD-19 corpus (Wang et al., 2020) was established in March 2020 as a repository for research related to SARS-COV-2 and other coronaviruses. It aggregates content from PubMed, bioRxiv, medRxiv, and other sources, and it is updated with new publications on a regular basis. Figure 1 illustrates the growth of CORD-19 through mid-2020. Figure 1. Weekly growth of the CORD-19 corpus Using CORD-19, we conducted a diachronic survey of candidate therapeutics identified in one of the more exhaustive drug re-purposing studies conducted to date for COVID-19, undertaken in February 2020 at Oak Ridge National Laboratory. This study, detailed in (Smith and Smith, 2020), analyzed drugs in the SWEETLEAD database for potential antiviral properties. The study produced a dataset identifying over 9,000 existing approved drugs and supplements as potential candidate therapeutics for COVID-19. Most importantly for our purposes, this dataset included commonly used drug or supplement names for each candidate. Our survey considered the following questions: * • How many candidate therapeutics appear in CORD-19? * • Do references to the candidates change over time? ## 2\. Materials and Methods Using temporal snapshots of CORD-19 spanning March 13 to June 30, we computed frequency and semantic representations for each candidate therapeutic found in the corpus. For each temporal instance, we computed TF/IDF (Term Frequency/Inverse Document Frequency) score for each candidate, a common metric to evaluate the relative importance of terms in a corpus. Figure 2. TF/IDF and cosine embedding distances To perform semantic analysis, we first computed diachronic word embeddings for each temporal instance of the corpus. These embeddings were aligned with one another to ensure that terms from each temporal instance were comparable. The technique we used is based on TWEC [3]. It uses a negative sampling optimization of softmax to maximize the probability that a set of words surrounding word $w_{k}$ are representative of its context in time ($C^{t}$), when multiplied by the mean of atemporal word embedding vectors from $u$ (the compass) for the same set of context words around $w_{k}$. $\mathnormal{\max_{\mathbf{C}^{t}}\log P({w_{k}}\|\gamma({w_{k}}))=\sigma(\vec{u}_{k}\cdot\vec{c}{\,{}^{t}_{\gamma(w_{k})}})}$ Since the TWEC embedding model did not account for phrases, we incoporated an additional step to indentify them. Phrases (including drug names) were then specially encoded to allow them to be treated like words. Figure 2 shows TF/IDF verses the mean embedding distance to the compass for candidate therapeutics. Because diachronic embedding instances were aligned with one another, we were able to isolate a given candidate and visualize its semantic trajectory over time (Figure 3). As the trajectory is based on nearest neighbors at a given time, subtle changes in semantic associations become apparent (Stewart et al., 2017). Figure 3. Semantic trajectory of ’acetazolamide’ (rank -2.4). Nodes along the path represent the candidate embedding vector and its two closest terms at time $t$ Semantic trajectory of ’acetazolamide’ (rank -2.4). Nodes along the path represent the candidate embedding vector and its two closest terms at time $t$ ## 3\. Results and Analysis We detected 14% (1267) of the candidate therapeutics in CORD-19 at 3/13, increasing to 26% (2361) by 6/30. For candidates detected in multiple adjacent temporal instances of the corpus, we were able measure their changes over time. We found that many candidates exhibited increases in frequency, and stable or strengthening semantic associations. However, given the nature of this corpus, we suspected some would exhibit other kinds of change over time. We found that some candidate therapeutics exhibited different patterns of semantic associations. Using heatmap visualizations as described in (Xu and Crestani, 2017), we can illustrate two additional recurring patterns of behavior. Some candidates exhibited weakening semantic associations over time (Figure 4), while others exhibited an abrupt persistent shift to a different pattern (Figure 5). Additionally, we found that these changes were not strongly correlated with changes to a target’s frequency scores. Figure 4. Example of weakening semantic associations for the candidate therapeutic ivermectin Figure 5. Example of disrupted semantic associations for the candidate therapeutic acetazolamide ## 4\. Conclusion Our diachronic survey of candidate therapeutics for COVID-19 in the CORD-19 corpus found that some exhibited weakening or abrupt changes of semantic associations. We speculate that this could be related to the publication of new research that positively or negatively affected consideration of a candidate therapeutic as a treatment for COVID-19. Future work will investigate how to detect and quantify these patterns, and to determine if there are any correlations between a target’s rank and magnitude of change. ## References * (1) * Smith and Smith (2020) Micholas Smith and Jeremy C. Smith. 2020. Repurposing Therapeutics for Covid-19. https://doi.org/10.26434/chemrxiv.11871402.v3 * Stewart et al. (2017) Ian Stewart, Dustin Arendt, Eric Bell, and Svitlana Volkova. 2017. Measuring, predicting and visualizing short-term change in word representation and usage in vkontakte social network. _arXiv preprint arXiv:1703.07012_ (2017). * Tshitoyan et al. (2019) Vahe Tshitoyan, John Dagdelen, Leigh Weston, Alexander Dunn, Ziqin Rong, Olga Kononova, Kristin A. Persson, Gerbrand Ceder, and Anubhav Jain. 2019. Unsupervised word embeddings capture latent knowledge from materials science literature. _Nature_ 571 (july 2019), 95–98. https://doi.org/10.1038/s41586-019-1335-8 * Wang et al. (2020) Lucy Lu Wang, Kyle Lo, Yoganand Chandrasekhar, Russell Reas, Jiangjiang Yang, Darrin Eide, Kathryn Funk, Rodney Kinney, Ziyang Liu, William Merrill, et al. 2020\. CORD-19: The Covid-19 Open Research Dataset. arXiv:2004.10706v2 * Xu and Crestani (2017) Zaikun Xu and Fabio Crestani. 2017. Temporal Semantic Analysis and Visualization of Words. In _Proceedings of the 8th Italian Information Retrieval Workshop, Lugano, Switzerland, June 05-07, 2017_ _(CEUR Workshop Proceedings)_ , Vol. 1911. CEUR-WS.org, 52–62. http://ceur-ws.org/Vol-1911/9.pdf
# Damage detection in operational wind turbine blades using a new approach based on machine learning Kartik Chandrasekhar<EMAIL_ADDRESS>Nevena Stevanovic add2 Elizabeth J. Cross add1 Nikolaos Dervilis add1 Keith Worden add1 Dynamics Research Group, Department of Mechanical Engineering, University of Sheffield, Mappin Street, Sheffield, S1 3JD, UK. Siemens Gamesa Renewable Energy A/S, Borupvej 16, 7330 Brande, Denmark ###### Abstract The application of reliable structural health monitoring (SHM) technologies to operational wind turbine blades is a challenging task, due to the uncertain nature of the environments they operate in. In this paper, a novel SHM methodology, which uses Gaussian Processes (GPs) is proposed. The methodology takes advantage of the fact that the blades on a turbine are nominally identical in structural properties and encounter the same environmental and operational variables (EOVs). The properties of interest are the first edgewise frequencies of the blades. The GPs are used to predict the edge frequencies of one blade given that of another, after these relationships between the pairs of blades have been learned when the blades are in a healthy state. In using this approach, the proposed SHM methodology is able to identify when the blades start behaving differently from one another over time. To validate the concept, the proposed SHM system is applied to real onshore wind turbine blade data, where some form of damage was known to have taken place. X-bar control chart analysis of the residual errors between the GP predictions and actual frequencies show that the system successfully identified early onset of damage as early as six months before it was identified and remedied. ###### keywords: structural health monitoring , wind turbine blades , machine learning , Gaussian processes , SCADA ## 1 Introduction Wind turbines rely on a number of structurally-critical components that may directly affect their power generation capabilities. Of note are the tower, the gearbox and generator (for variable-speed wind turbines), the magnetic drive (for direct-drive wind turbines), various bearings (such as main bearings), and perhaps most importantly, the blades [1]. Blades are among the most expensive components - depending on their size; their manufacturing costs range between 10% and 20% of total manufacturing costs [2]. In recent years, blades have progressively become larger, to harvest more energy, and maximise power production. At the same time, since larger blades are associated with larger weight penalties, blade designs have continually been modified to keep the penalties low. These compromises have led to more flexible blades, and therefore, lower safety margins [3, 4]. Cracks and delaminations are reported to be the most common form of damage in blades [3]; these typically occur close to the blade root since this is the region where the blades experience the highest bending strains due to turbine rotation; this is especially true when taking large blades into consideration. Towards the end of the design lifetime of the blades, damage mechanisms become accentuated during excessive levels of winds, lightning strikes, and ice accumulation (which leads to rotor imbalance problems). It is extremely important to avoid critical blade failures, because when damaged blades liberate, they have the potential to damage not only the turbines they were attached to, but also other turbines in their vicinity [5]. There have also been heavy investments in the offshore wind industry, whereby wind turbines have increasingly been deployed on various seas and oceans, where wind conditions are favourable for energy production [6]. However, this freedom has come at a cost of ease of maintenance, since it is significantly more expensive to send technicians to inspect offshore structures via expensive transportation [7]. For all these reasons, Structural Health Monitoring (SHM) technologies have become appealing to both wind turbine manufacturers and operators alike. The use of SHM technologies has largely been driven by a number of factors: 1. 1. minimisation of downtime, 2. 2. maximisation of reliability, 3. 3. minimisation of operation and maintenance costs, and 4. 4. extension of design lifetimes of monitored structures. Currently, most of the focus in SHM research on small-to-medium sized blades has been in laboratory environments, where conditions are controlled. Additionally, researchers have had the luxury of multiple sensors and data acquisition systems with high sampling rates. Signals captured by these sensors include strains [8, 9], velocities [10], accelerations [11], acoustic emissions [8, 12, 13], guided waves [14, 15], thermographic images [16], and digital image correlation (DIC) images [17], to name a few. This type of monitoring has facilitated high-fidelity diagnostics (indication and location of damage), as well as prognostics (remaining life prediction). There have been numerous methods and procedures to aid in damage indication and localisation; these include changes in stiffness [18, 19] and modal properties [20], principal component analysis [9], image processing of thermographic [16] and DIC images [17], etc. Machine learning algorithms, such as artificial neural networks, have also been used previously [14]. The above-mentioned research subjects are extremely valuable, as they illustrate various routes to tackle the ultimate goal of having a reliable SHM system. However, operational wind turbines around the world face challenges that cannot be replicated in laboratory environments. Due to this issue - at least for the time being - certain methods can seem to be impractical. Some of the challenges are listed below. * 1. Firstly, there are significant costs involved in applying numerous sensors with high data-acquisition rates; these include the costs of the sensors themselves, as well as costs associated with data storage and processing. One must also consider the fact that there are hundreds of thousands of wind turbines spread across the world. * 2. Furthermore, monitoring and diagnostic systems need to be designed to be robust to the heavily nonstationary conditions that wind turbines face. Nonstationarity primarily stems from constantly-changing wind and loading conditions (gusts, turbulence, etc.), and taking these into account is therefore a challenging task. * 3. Damage localisation is another challenge in operational wind turbines, since this is a function of excitation frequency and number of sensors, as well as data-acquisition rates and sensor placement. There are additional limitations as well; for example, one cannot place piezoelectric sensors along the length of the blades due to the risks associated with electromagnetic interference and lightning strikes. * 4. Diagnostic methods also need to find a balance between true and false positives, and this would involve a cost-benefit analysis. For example, in a laboratory, a 3% false positive rate may be deemed acceptable, but in the real world, if all turbines are stopped for 3% of the time, costs associated with downtime will have to be borne by turbine operators. On the other hand, if it does turn out that a damage indicator is correct, and the damage becomes irreparable, costs associated with blade replacement can also be considerable. This article proposes methodologies that address some of these challenges. As mentioned above, when turbines are operating, the effects of environmental and operational variables cause significant nonstationarity; this is especially true of the frequency response of the blades. For example, high wind speeds can cause faster rotation of the turbines, which lead to stiffer blades and higher natural frequencies; higher temperatures lead to less stiff blades, hence lowering the natural frequencies. The methodologies proposed in this article take advantage of the fact that the EOVs act simultaneously on all three blades. Therefore, when the monitored structural frequencies are viewed relative to each other, the complex nonstationarities are transformed into a simpler, and more explicable space. As a result of this transformation, it is possible to learn relationships between these monitored variables (as well as other EOVs). These relationships are learnt, in pairs, using probabilistic regression in the form of Gaussian Processes (GPs). The GPs are then used for predictive purposes, and the residuals between the actual signals and predicted signals can be used as an informative damage indicator. The SHM scheme proposed is summarised in the next section. This is followed by a detailed review of the theory behind GPs. The article is complemented by three case studies of onshore wind turbines, where the proposed methodologies were applied, and would have successfully identified damage in advance of critical failure. It should be noted that the versatility of this methodology means that they can also be applied to offshore wind turbines. ## 2 Methodology and Theory Figure 1 illustrates a summary of the SHM system that is presented in this article. Figure 1: Summary of proposed SHM system As shown in Figure 1, the SHM system capability is divided into four categories: data acquisition, pre-processing, machine learning, and diagnostics and actions; these are discussed in detail next. ### 2.1 Data acquisition Acceleration sensors, and sensors that measure the environmental and operational variables (EOVs) measure the respective signals. The edge frequencies of each blade (i.e. the frequencies of the first bending mode in the blade edgewise direction) are estimated from the acceleration signals. In this paper, the machine learning techniques are applied to Supervisory Control and Data Acquisition (SCADA) data. For readers unfamiliar with the SCADA system, it is a utility that acquires data from sensors mounted on a structure, and sends them to a central computer for monitoring and control purposes. The data statistics are logged at regular intervals, and these are typically ten-minute averages [21], which implies that the SCADA data has a low-pass filter applied to it. The variables used from the SCADA data include: 1. 1. edge frequencies of each blade (in Hz), 2. 2. ambient temperature (in °C), 3. 3. power output (in W), 4. 4. generator rotation speed (in RPM), and 5. 5. pitch angle (in °). Prior to data logging into the SCADA system, the individual signals are usually sampled at a rate of 10 Hz (in some cases, higher sampling is also possible). The aim of this work is to detect long term damage, which typically occurs over the course of several weeks or months. Therefore, although the data in the SCADA system have a low-pass filter applied to them, they are still capable of identifying the long term degradation in the blade stiffness, and hence the edge frequency. The benefit of using SCADA data over the higher- resolution data is that it saves processing time quite considerably. ### 2.2 Pre-processing In this part of the system, SCADA data from the respective channels were downloaded from a number of wind turbines whose blades were known to have incurred damage. These time-series were then filtered, and processed to remove outliers within the data. Outliers are quite a common occurrence in SCADA data, and typically occur in the data logging process. Furthermore, since the time-series are logged in various dedicated channels, occasionally, it is common for data to not be recorded at certain times in a given channel, whilst data in other channels are being logged. It is possible to resample the data such that all data channels have identical sampling rates. However, this requires some form of interpolation, which could result in misleading information. Therefore, it was decided that only data that had common timestamps would used for the remainder of the processing that followed. Note that all the data used for the purposes of this article have been normalised prior to the application of the machine learning and signal processing methods. This is performed according to, $x_{norm}=\frac{x-\mu}{\sigma}$ (1) where the $x$ are the data being normalised, $\mu$ is the mean, and $\sigma$ is the standard deviation of selected training data. The selection of training data will be discussed shortly, but it is the same training data that will be used in the training of the Gaussian Processes (GPs). There are two primary reasons for normalising the data. * 1. To avoid numerical instabilities that may occur in various computations in GP inference. * 2. To avoid disclosing confidential information regarding blade properties. Next, the various features were analysed to assess whether they contributed to the observed properties of the blade (i.e. the edge frequencies). Figure 2 shows a representative example of how the estimated edge frequencies of each blade correlate to each other. Note that the three blades on the turbine have been labelled A, B, and C. (a) (b) (c) Figure 2: Correlation between normalised edge frequencies for: (a) Blade A vs. Blade B, (b) Blade B vs. Blade C, and (c) Blade C vs. Blade A. Based on Figure 2, it is clear to see that these frequencies correlate well with each other in a linear fashion. Over time, as the blades age, the correlations generally worsen, and when damage occurs, the relationships generally break down. Therefore, the primary motivation behind the SHM system was to evaluate when the correlations worsened and/or broke down. It should be noted here that for the examples of correlations shown in Figure 2, simple methods such as Bayesian linear regression would be sufficient rather than the more complicated GPs. However, as shall be seen later, the correlations between the edge frequencies of each blade during normal condition are typically not as linear and noise-free on other wind turbines (see Figure 12). Since GPs are more robust to such scenarios (since the functional forms need not be predetermined), they are the chosen method for this work. Apart from the study of correlations between the respective edge frequencies, correlations between other EOVs and the edge frequencies of each blade were also studied. Figures 3 (a) to (d) illustrate exemplar relationships between the EOVs (ambient temperature, power output, generator rotation speed, and pitch angle, respectively) with respect to the estimated frequencies. (a) (b) (c) (d) Figure 3: Correlation between normalised edge frequencies and normalised: (a) ambient temperature, (b) power output, (c) generator rotation speed, and (d) pitch angle. As is indicated in Figure 3, it was found that, out of the EOVs considered, only the ambient temperature had any systematic influence on the estimated frequencies. When the estimated frequencies and temperatures are viewed over time, it is evident that the cyclical nature of the estimated frequencies occur due to the seasonal variations, as shown in Figure 4. (a) (b) Figure 4: Time series of normalised: (a) frequency, and (b) ambient temperature. The use of these EOVs (other than ambient temperature) in the machine learning tasks that follow were expected to lead to less accurate GP predictions since they do not contribute to the observed behaviour of the edge frequencies. Due to this fact, only the edge frequencies and ambient temperature features were selected for further processing. In the next section, the theory of Gaussian processes (GPs) is provided. ### 2.3 Gaussian Processes Gaussian processes (GPs) implement a nonparametric probabilistic regression. They are best understood in terms of a function space viewpoint [22]. Let $\boldsymbol{X}$ be an $N\times d$ matrix of inputs $\\{\boldsymbol{x}_{1},\boldsymbol{x}_{2},\cdots,\boldsymbol{x}_{N}\\}$, with $N$ observations and $d$ variables (also referred to as dimensions), $f(\boldsymbol{X})$ denote the underlying functions of the GP, $\\{f(\boldsymbol{x}_{1}),f(\boldsymbol{x}_{2}),\cdots,f(\boldsymbol{x}_{N})\\}$, and $\boldsymbol{y}$ be the output vector, which is actually observed, and can be corrupted by noise, $\boldsymbol{y}=f(\boldsymbol{X})+\boldsymbol{\varepsilon}$ (2) where $\boldsymbol{\varepsilon}$ is a zero mean Gaussian noise process, with variance ${\sigma_{N}}^{2}$. Note that for the sake of conciseness, $f(\boldsymbol{X})$ will henceforth simply be denoted as $\boldsymbol{f}$. In the Bayesian approach, the prior belief of the form of the GP, $\boldsymbol{f}$ is utilised along with the likelihood function of the observed data to calculate its posterior distribution. This can be expressed as, $\text{posterior}=\frac{\text{likelihood}\times\text{prior}}{\text{marginal likelihood}}$ (3) $p(\boldsymbol{f}\|\boldsymbol{y},\boldsymbol{X})=\frac{p(\boldsymbol{y}\|\boldsymbol{X},\boldsymbol{f})p(\boldsymbol{f}\|\boldsymbol{X})}{p(\boldsymbol{y}\|\boldsymbol{X})}=\frac{p(\boldsymbol{y}\|\boldsymbol{X},\boldsymbol{f})p(\boldsymbol{f}\|\boldsymbol{X})}{\int{p(\boldsymbol{y}\|\boldsymbol{X},\boldsymbol{f})p(\boldsymbol{f}\|\boldsymbol{X})d\boldsymbol{f}}}$ (4) where $p(\boldsymbol{f}\|\boldsymbol{y},\boldsymbol{X})$ is the posterior distribution (probability of functions given the outputs and inputs, $p(\boldsymbol{y}\|\boldsymbol{X},\boldsymbol{f})$ is the likelihood function (probability of outputs, given inputs and functions), $p(\boldsymbol{f}\|\boldsymbol{X})$ is the prior distribution (probability of the functions, given inputs), and $p(\boldsymbol{y}\|\boldsymbol{X})$ is the marginal distribution (probability of outputs, given only the inputs). $p(\boldsymbol{y}\|\boldsymbol{X})$ is independent of the functions because it is integrated over the entire functional space of $\boldsymbol{f}$. In the framework of GPs, functions between inputs and outputs can be completely defined by a mean function, $m(\boldsymbol{x})$, and a covariance function, $k(\boldsymbol{x}_{p},\boldsymbol{x}_{q})$, and these are in turn defined by hyperparameters [22]. Note that the subscripts are used to distinguish one input vector from another. This definition is analogous to a Gaussian distribution, which can be fully described using a mean and a variance. GPs therefore provide a distribution of functions between inputs and outputs. The mean and covariance functions are expressed as, $m(\boldsymbol{x})=E[f(\boldsymbol{x})]$ (5) $k(\boldsymbol{x}_{p},\boldsymbol{x}_{q})=E[(f(\boldsymbol{x}_{p})-m(\boldsymbol{x}_{p}))(f(\boldsymbol{x}_{q})-m(\boldsymbol{x}_{q}))]$ (6) The covariance function is a vital ingredient of the GP, as it characterises the general properties of the GP. Typically, when $\boldsymbol{x}_{p}$ and $\boldsymbol{x}_{q}$ are almost identical, $k(\boldsymbol{x}_{p},\boldsymbol{x}_{q})$ would yield high values (high correlation), and as they grow in distance, the values reduce in magnitude (low correlation). Due to this, when predictions are performed, new inputs that geometrically lie close to previously observed inputs will have a greater influence than those that lie far away. #### 2.3.1 Covariance functions The choice of covariance functions is one of fundamental importance. Depending on the problem being solved, there are numerous covariance functions (also referred to as kernels in the literature) available to use. Examples include squared-exponential, Matérn, $\gamma$-exponential, rational quadratic, and the Bayesian linear covariance functions [22]. These functions can be combined using basic algebra (addition, multiplication, etc.) to form advanced covariance functions. In the work presented in this article, only the squared- exponential and Bayesian linear covariance functions are used, and hence the discussion will be focussed on these. The squared-exponential covariance function is defined as, $k_{SE}(\boldsymbol{x}_{p},\boldsymbol{x}_{q})={\sigma_{f}}^{2}\exp\bigg{(}-\frac{(\boldsymbol{x}_{p}-\boldsymbol{x}_{q})^{T}(\boldsymbol{x}_{p}-\boldsymbol{x}_{q})}{2\lambda^{2}}\bigg{)}$ (7) where ${\sigma_{f}}^{2}$ is a hyperparameter that represents the signal variance, and $\lambda$ is a length-scale hyperparameter (which controls how smooth the function appears). Note that $(\boldsymbol{x}_{p}-\boldsymbol{x}_{q})^{T}(\boldsymbol{x}_{p}-\boldsymbol{x}_{q})$ is simply the squared Euclidean distance between input points. The Bayesian linear covariance function is defined as, $k_{BL}(\boldsymbol{x}_{p},\boldsymbol{x}_{q})={\sigma_{0}}^{2}\boldsymbol{x}_{p}\cdot\boldsymbol{x}_{q}$ (8) where ${\sigma_{0}}^{2}$ is also a hyperparameter that represents the signal variance. This covariance function is classed as a dot-product covariance function, for obvious reasons. Finally, input noise can also be taken into account, i.e. when observations are noisy. This is defined via, $k_{N}(\boldsymbol{x}_{p},\boldsymbol{x}_{q})={\sigma_{N}}^{2}\delta(\boldsymbol{x}_{p},\boldsymbol{x}_{q})={\sigma_{N}}^{2}\boldsymbol{I}$ (9) where ${\sigma_{N}}^{2}$ is a hyperparameter that represents the variance of the noise process and $\delta$ is the Kronecker-delta function, which can simply be viewed as the identity matrix, $\boldsymbol{I}$. For the GPs associated with this work, a sum of the above covariance functions was considered to explain the observed data. There are three distinct covariance matrices associated with the predictive GPs: 1. 1. $\boldsymbol{K_{\theta}(X,X)}$: Covariance matrix between training data only. For the sake of convenience, this will simply be denoted $\boldsymbol{K_{\theta}}$. $\boldsymbol{K_{\theta}}=k_{SE}(\boldsymbol{x}_{p},\boldsymbol{x}_{q})+k_{BL}(\boldsymbol{x}_{p},\boldsymbol{x}_{q})+k_{N}(\boldsymbol{x}_{p},\boldsymbol{x}_{q})$ (10) 2. 2. $\boldsymbol{K_{}(X,X^{})}$: Cross covariance matrix between training and test data. The asterisk superscript denotes test data. Note that $\boldsymbol{K_{}(X^{},X)}$ can also be calculated separately, but it is simply the transpose of $\boldsymbol{K_{}(X,X^{})}$. For conciseness, $\boldsymbol{K_{}(X,X^{})}$ will be denoted $\boldsymbol{K_{}}$. $\boldsymbol{K_{}}=k_{SE}(\boldsymbol{x}_{p},\boldsymbol{x}_{q}^{\boldsymbol{}})+k_{BL}(\boldsymbol{x}_{p},\boldsymbol{x}_{q}^{\boldsymbol{}})$ (11) 3. 3. $\boldsymbol{K_{*}(X^{},X^{})}$: Covariance matrix between test data only. This will simply be denoted $\boldsymbol{K_{}}$. $\boldsymbol{K_{}}=k_{SE}(\boldsymbol{x}_{p}^{\boldsymbol{}},\boldsymbol{x}_{q}^{\boldsymbol{}})+k_{BL}(\boldsymbol{x}_{p}^{\boldsymbol{}},\boldsymbol{x}_{q}^{\boldsymbol{}})$ (12) It should be noted that in the latter two covariance functions, the noise term is not added for the prediction of the noise-free underlying function, $\boldsymbol{f^{}}$. Now that the background of GPs has been introduced, it is necessary to discuss how they can be used in regression problems. Matters will be elaborated using the terms in the Bayes’ theorem: the prior, the likelihood, and the marginal likelihood. The marginal likelihood is utilised in the training process, where the hyperparameters that define the GP are learnt. #### 2.3.2 The prior distribution The prior to the functions, $\boldsymbol{f}$ has the form, $\boldsymbol{f}\sim\mathcal{GP}(0,\boldsymbol{K})$ (13) where the 0 indicates a zero-mean function. $\boldsymbol{K}$ is now the covariance matrix, without the noise term. The benefit of incorporating a mean function is that in the absence of training data in a given region of the input space, the prediction of the GP will approach the mean trend that is defined by the mean function. This is because it specifies a belief in the relationship between inputs and outputs having a certain functional form. It is noteworthy that it is not necessary to limit the form of the GP using a mean. This is because, in practice, data is typically de-trended, and so, mean trends are already removed [23]. In any case, the prior distribution gets updated in each step of the training process, and so the mean function can be set to zero. However, this does not imply that the output of the GP will have zero mean, and is hence not a limiting issue. It has a distribution, $p(\boldsymbol{f}\|\boldsymbol{X})=\mathcal{N}(0,\boldsymbol{K})$ (14) #### 2.3.3 The likelihood distribution $\boldsymbol{f}$ is a realisation of the underlying GP with inputs, $\boldsymbol{X}$. Because $\boldsymbol{y}$ is simply $\boldsymbol{f}$ with additive Gaussian noise ($\mathcal{N}(0,{\sigma_{N}}^{2})$), the likelihood distribution can be stated as, $p(\boldsymbol{y}\|\boldsymbol{X},\boldsymbol{f})=\mathcal{N}(\boldsymbol{f},{\sigma_{N}}^{2}\boldsymbol{I})$ (15) #### 2.3.4 The marginal likelihood and hyperparameter learning The denominator in equation (4) is important in estimating (or learning) the hyperparameters. It is also normally distributed with the form, $p(\boldsymbol{y}\|\boldsymbol{X})=\mathcal{N}(0,\boldsymbol{K_{\theta}})$ (16) where $\boldsymbol{K_{\theta}}$ is the covariance matrix with additive noise, i.e. $\boldsymbol{K_{\theta}}=\boldsymbol{K}+{\sigma_{N}}^{2}\boldsymbol{I}$. The marginal likelihood is usually expressed in terms of its logarithmic transformation: the log marginal likelihood [22], $\mathcal{L}=\log{}p(\boldsymbol{y}\|\boldsymbol{X})=-\frac{1}{2}\boldsymbol{y}^{T}\boldsymbol{K_{\theta}}^{-1}\boldsymbol{y}-\frac{1}{2}\log{}\|\boldsymbol{K_{\theta}}\|-\frac{N}{2}\log{}(2\pi)$ (17) where $N$ is the number of training points, and the other definitions are consistent with the preceding discussion. When the hyperparameters of the covariance functions are being learnt, the goal is to maximise the marginal likelihood with respect to the hyperparameters. Alternatively, the negative log marginal likelihood can also be minimised. If the hyperparameters of the covariance functions can be combined in a tuple, $\boldsymbol{\varphi}=\\{\sigma_{f},\lambda,\sigma_{0},\sigma_{N}\\}$, the partial derivative of the negative log marginal likelihood can be expressed as, $\frac{\partial}{\partial\boldsymbol{\varphi}_{i}}(-\mathcal{L})=\frac{1}{2}tr\bigg{[}\boldsymbol{K_{\theta}}^{-1}\frac{\partial\boldsymbol{K_{\theta}}}{\partial\boldsymbol{\varphi}_{i}}\bigg{]}-\frac{1}{2}\boldsymbol{y}^{T}\boldsymbol{K_{\theta}}^{-1}\frac{\partial\boldsymbol{K_{\theta}}}{\partial\boldsymbol{\varphi}_{i}}\boldsymbol{K_{\theta}}^{-1}\boldsymbol{y}$ (18) where $tr$ is the trace operator (i.e. sum of the diagonal elements of the matrix). In practice, the negative log marginal likelihood of GPs may have several local minima. Therefore, solving this problem becomes one of numerical optimisation. There are several routes one can take, for example gradient descent, differential evolution, Broyden-Fletcher-Goldfarb-Shanno algorithm, etc. There are no right or wrong choices in the optimisation routines, but each optimiser can have advantages and disadvantages associated with them. In this work, the Nelder-Mead simplex algorithm was used as the optimiser [24]. #### 2.3.5 Posterior distribution The posterior distribution can be obtained by combining equations (14), (15) and (16), and using Bayes’ theorem - equation (3); this is given by, $p(\boldsymbol{f}\|\boldsymbol{X},\boldsymbol{y})=\mathcal{N}(\boldsymbol{K}^{T}\boldsymbol{K_{\theta}}^{-1}\boldsymbol{y},\boldsymbol{K}-\boldsymbol{K}^{T}\boldsymbol{K_{\theta}}^{-1}\boldsymbol{K})$ (19) #### 2.3.6 Predictions using GPs Thus far, the necessary ingredients of the GPs have been shown. The next task is to use these for the main goal of the GPs: prediction. Let $\boldsymbol{x^{}}$ denote a test input vector, (i.e. a matrix which is now going to be used to predict the outputs). The corresponding noise-free test output is $\boldsymbol{f^{}}$. Now, the observed (noisy) training outputs and the noise-free test outputs are jointly Gaussian distributed according to the prior distribution, and can be expressed as, $\begin{bmatrix}\boldsymbol{y}\\\ \boldsymbol{f^{}}\end{bmatrix}=\mathcal{N}\begin{pmatrix}0,\begin{bmatrix}\boldsymbol{K_{\theta}}&\boldsymbol{K_{}}\\\ \boldsymbol{K_{}}^{T}&\boldsymbol{K_{}}\end{bmatrix}\end{pmatrix}$ (20) where the respective covariance matrices were defined in equations (10), (11), and (12). The corresponding conditional distribution can be expressed as, $p(\boldsymbol{f^{}}\|\boldsymbol{X},\boldsymbol{y},\boldsymbol{x^{}})=\mathcal{N}(\boldsymbol{\bar{f^{}}},cov(\boldsymbol{f^{}}))$ (21) where the predictive mean, $\boldsymbol{\bar{f^{}}}$ of the GP can be calculated using, $\boldsymbol{\bar{f^{}}}=\boldsymbol{K_{}}^{T}\boldsymbol{K_{\theta}}^{-1}\boldsymbol{y}$ (22) and the predictive covariance, $cov(\boldsymbol{f^{}})$ of the GP can be calculated using, $cov(\boldsymbol{f^{}})=\boldsymbol{K_{*}}-\boldsymbol{K_{}}^{T}\boldsymbol{K_{\theta}}^{-1}\boldsymbol{K_{}}$ (23) The diagonal of the matrix given by equation (23) is then used to give the uncertainties (in terms of variance) in prediction. The noise variance can be added to the predictive variance to take uncertainty related to noisy test outputs into consideration, and hence instead of finding the predictive distribution of $\boldsymbol{f^{}}$, the predictive distribution of $\boldsymbol{y^{}}$ can be calculated. Note that this does not affect the predictive mean, only the predictive covariance. ### 2.4 Application of machine learning Figure 5 provides an overview of the flow of data in the machine learning phase of the SHM scheme, which includes training, optimisation, and Gaussian Process (GP) predictions. Figure 5: Flow of data in the proposed methodology. Figure 5 shows that there are a total of three GPs that perform the predictions. However, before any predictions could be performed, the first task was to learn the hyperparameters, $\varphi$ of the models that controlled their predictive capabilities - and this entailed a training process. Here, a subset of the selected feature data were selected for the training. When choosing the training data, it was important to consider only data when the blades were in a normal (healthy) condition. However, the conditions of the blades were not known beforehand. It was therefore decided that training data would only be selected from the first two years of the turbine operation, assuming that the blades were in a normal condition throughout this period. Within this period of two years, 2500 data points were randomly selected from a uniform distribution. Choosing this two year time period for training ensured that the selected data spanned the entire normal operating range of the blades. The random selection ensured that there was no bias to any specific operating condition in the training phase. It should be noted that more training data would increase the computational costs of the GPs, since computational costs are $\mathcal{O}(N^{3})$, where $N$ is the number of training points. There are sparse methods available to reduce the computational load associated with training, but these were not employed in this work. The estimated frequencies, $f$ of one blade (e.g. for blade A, $f_{A}$) were combined with the ambient temperatures, $T$ to form a matrix of the training inputs. Referring to Section 2.3, this is the matrix, $\boldsymbol{X}$. The estimated frequencies of another blade (e.g. for blade B, $f_{B}$) formed the training outputs. This is the vector, $\boldsymbol{y}$. in Section 2.3. These training data were used to learn the hyperparameters ($\varphi$) for the respective GPs via the Nelder-Mead method. During the learning phase, the covariance matrices, equations (10 \- 12) were evaluated several times, which is why it was stated that the computational costs of GPs increases significantly with the number of training points. The hyperparameters selected corresponded to those that yielded the largest maximum log marginal likelihood, equation (17). The Nelder-Mead algorithm identifies this by minimising the negative log marginal likelihood, i.e. the negative of equation (17). These hyperparameters control the generalisation properties of the GPs, i.e. how the GPs adapt to the nature of the problem being tested. The asterisk symbols, ∗ shown in Figure 5 are used to denote test data, i.e. data not included in the training set. The test data, $f^{}$ and $T^{}$ were combined in a matrix (referred to as the matrix, $\boldsymbol{X^{}}$ in Section 2.3), and were then used in the various GPs (GPAB, GPBC, and GPCA), using the respective hyperparameters ($\varphi_{AB}$, $\varphi_{BC}$, and $\varphi_{CA}$), to evaluate the predicted frequencies, $\bar{f_{B}}$, $\bar{f_{C}}$, and $\bar{f_{A}}$, respectively. The bar symbol ( $\bar{}$ ) indicates a prediction, rather than an actual estimate of the frequencies. The predictions are referred to as $\boldsymbol{f^{}}$ in Section 2.3, and are calculated using equation (22). To elucidate, 1. in GPAB, ${f_{A}}^{}$ and $T^{}$ were used to predict $\bar{f_{B}}$ using hyperparameters $\varphi_{AB}$, * 2. in GPBC, ${f_{B}}^{}$ and $T^{}$ were used to predict $\bar{f_{C}}$ using hyperparameters $\varphi_{BC}$, and * 3. in GPCA, ${f_{C}}^{}$ and $T^{}$ were used to predict $\bar{f_{A}}$ using hyperparameters $\varphi_{CA}$. ### 2.5 Diagnostics and actions Figure 6 shows the tasks involved in the diagnostics and actions section. Figure 6: Diagnostics of GP residuals. Following the frequency predictions for each blade via the respective GPs, residual errors were calculated between the predicted frequencies and the actual estimations (for example, the GP residual error for blade B, $re_{B}$ was defined as the difference between $\bar{f_{B}}$ and ${f_{B}}^{}$). The assessment of whether the blades were healthy or not was performed using X-bar control charts. In the X-bar control charts, averages of the GP residuals were taken at regular time intervals. For the purposes of this work, averages were taken regularly over a 28 day interval. During a brief period that followed the GP training phase detailed in the previous section, 3$\sigma$ thresholds were also calculated. These were evaluated over a period of 6 months. The calculated averages of the X-bar control chart were then compared with the 3$\sigma$ thresholds, and if the averages exceeded the thresholds, it was concluded that the blades were no longer in a normal condition. During the development of this SHM system, maintenance actions were not taken. In practice, once the GP residuals surpass the thresholds of the X-bar control charts, warnings would be generated before alarms occur. In such a scenario, maintenance technicians would be sent to inspect the blades, they would assess the level of damage, and finally would decide whether to repair or replace the turbines. ### 2.6 Summary The following summarises the SHM system as detailed in the preceding sections. 1. 1. Data acquisition. 2. 2. Data cleansing (removal of outliers). 3. 3. Feature selection. 4. 4. GP training (identification of hyperparameters). 5. 5. GP prediction. 6. 6. Calculation of residual errors. 7. 7. X-bar control chart analysis of GP residuals. ## 3 Case studies Now that the diagnostic methodology and theory have been described in detail, it is important to illustrate its performance. Prior to the results and discussion of the case studies, the detailed steps of the employed machine learning methodology are stated. ### 3.1 Detailed implementation In this section, a detailed description is given/reiterated regarding the implementation of the machine learning methods described thus far, referring to the seven steps in Section 2.6. That is, 1. 1. Data acquisition: In the following sections, three sets of data are introduced. The data are either synthesised or acquired directly from operational wind turbines. In the synthesised case, it is acknowledged that the data does not reflect the true behaviour of blades. However, the method of the data synthesis is described in detail to allow readers to reproduce them and apply the machine learning methodology. In the two cases where the data is acquired from operational wind turbines, a blade on each turbine shown was known to be damaged, and as a result, it either needed to be repaired or replaced. Damage can occur at various locations across the length of the blade, and the types of damage in blades are predominantly cracks or delaminations. In some cases, both occur at the same time. Due to confidentiality issues, however, the specific damage types and locations are not disclosed here. 2. 2. Data cleansing (removal of outliers): In the case of the synthesised data, outliers are not included in the data, hence this process is not conducted. In the case of the real data, the standard deviations of the dataset are calculated, and data points outside 3$\sigma$ of the data are iteratively removed. This process is repeated until the difference in standard deviation between subsequent iterations is less than 0.1. This value is empirically identified to ensure that outliers due to blade damage are not removed. Note that when an outlier is removed from one of the data channels, data from the other channels with the same timestamp (which may not necessarily be outliers) are also removed. 3. 3. Feature selection: The main features of interest are the edge frequencies of the blades. It is shown in Example 3 below that the inclusion of temperature as a feature improves the performance of the GPs. 4. 4. GP training (identification of hyperparameters): For each example shown, 2500 training points are selected from a period when the blades on the turbine are considered ‘healthy’. In the case of data from the real turbines, this was chosen to be the first two years of turbine operation. The training points, indexed by the data point numbers, are sampled uniformly via that index. Standard normalisation, equation (1) is applied to the entire dataset using the mean and standard deviation values of the training data points. Next, the training input and output edge frequency data are prepared into three pairs (that is, $f_{A}$ and $f_{B}$, $f_{B}$ and $f_{C}$, and $f_{C}$ and $f_{A}$; temperature, $T$ is also included for the training inputs). The hyperparameters, $\phi_{AB}$, $\phi_{BC}$, and $\phi_{CA}$, are then identified by minimising the negative log marginal likelihood, which is equivalent to maximising log marginal likelihood, equation (17). The Nelder- Mead algorithm is used to accomplish this. 5. 5. GP prediction: The identified hyperparameters for each GP are used with the rest of the edge frequency and temperature data (${f_{A}}^{}$, ${f_{B}}^{}$, ${f_{C}}^{}$, and $T^{}$) are used to evaluate the predicted edge frequencies, $\bar{f_{A}}$, $\bar{f_{B}}$, and $\bar{f_{C}}$ using equation (22). Note that for relatively large datasets such as those used in this work, computer memory can easily be used up since the covariance matrices would be too large. For this reason, the GP predictions are performed in smaller chunks (1000 points at a time in this case). 6. 6. Calculation of residual errors: The GP residuals are calculated by subtracting the actual edge frequency data from the predicted edge frequencies. That is, $\displaystyle re_{A}$ $\displaystyle=\bar{f_{A}}-{f_{A}}^{}$ (24) $\displaystyle re_{B}$ $\displaystyle=\bar{f_{B}}-{f_{B}}^{}$ $\displaystyle re_{C}$ $\displaystyle=\bar{f_{C}}-{f_{C}}^{}$ 7. 7. X-bar control chart analysis of GP residuals: In this step, the X-bar control charts of the residuals are calculated. This is performed by calculating averages of the residual errors over periods spanning 28 days. As the GPs are trained over the first two year period on the examples shown from real wind turbines, it is important that the thresholds for the X-bar control chart analyses are not evaluated over the same period. This is because, generally, residual errors are low over training data since optimisation routines aim to minimise these errors as much as possible when identifying the hyperparameters. Thus, such practices can potentially lead to stringent thresholds that may increase false alarm rates. For this reason, a six-month period immediately after the initial two-year training period is chosen. As the standard deviation of the errors during this period may be occasionally large, the following technique is used to calculate robust thresholds. For each of the residual error datasets over this six-month period, small subsets are randomly selected several times (in this work, 20 times). For each subset $i$, the mean $\mu_{i}$ and standard deviation $\sigma_{i}$ values are calculated. Finally, averages of the mean and standard deviation data are calculated, $\bar{\mu}$ and $\bar{\sigma}$, respectively. The thresholds, $thr$ are then calculated using, $thr=\bar{\mu}\pm 3\bar{\sigma}$ (25) The residual errors for each blade are then compared to the upper and lower thresholds for each blade. Once the residual errors exceed these thresholds, the blade is assumed to be damaged. ### 3.2 Example 1: Synthesised data In this example, a demonstration of the algorithm is described using some synthesised data. It must be noted that due to confidentiality agreements in effect, signals simulating the actual blade physics (and subsequent frequency estimations of the blades) cannot be reported. Instead, the edge frequency data of one blade is synthesised using a non-zero mean normal distribution. The edge frequency data of the second blade is set to be equal to that of the first blade with some additional noise. Finally, the third blade initially has the same edge frequencies of the first blade, but ‘deteriorates’ over time by means of a negative linear gradient. That is, $f_{A}=\mathcal{N}(\mu_{A},\sigma^{2})$ (26) $f_{B}=f_{A}+\mathcal{N}(0,\sigma^{2})$ (27) $f_{C}=f_{A}+d_{C}+\mathcal{N}(0,\sigma^{2})$ (28) where $d_{C}$ is a vector of zeros followed by a line with a negative gradient. That is, $d_{C}=\\{\boldsymbol{0},m\boldsymbol{x}+c\\}$ (29) In this work, 360000 data points were generated. Only the first half of the data points of Blade C are considered ‘healthy’. The length of the vector of zeros in equation (29) is therefore 180000. The rest of the vector $d_{C}$, which is indexed by the data point number, $\boldsymbol{x}$ (i.e. 180001 to 360000) has a gradient, $m$, of $-1\times 10^{-7}$ and a y-intercept value, $c$ of 0.018. Whilst not realistic, $\mu_{A}$ is set to 10 and $\sigma^{2}$ has a value of $0.01$. Figure 7 illustrates the synthesised edge frequencies of the three blades. (a) (b) (c) Figure 7: Synthesised edge frequency data for: (a) Blade A, (b) Blade B, and (c) Blade C Referring to Section 2.6, the above description explains the data acquisition and feature selection processes. That is, the data acquired is As outliers do not exist on the synthesised data,t covers the first three The methodology, as described in Section 2, is applied. Figure 8 illustrates the residual errors between the actual and predicted frequencies. Note that due to data normalisation (which is required for numerical stability, as discussed in Section 2), the residual errors are scaled. It is clear that the ‘damage’ is identified as the residual errors of blades A and C diverge on the midway point. The residuals of blade B do not diverge since they do not rely on data from the damaged blade C (actual or predicted). Note that in reality, the dynamics of healthy blades may also be affected by the presence of a damaged blade, and hence it may not always be possible to conclude which blade contains damage. (a) (b) (c) Figure 8: Residual errors between synthesised and predicted data for: (a) Blade A, (b) Blade B, and (c) Blade C ### 3.3 Example 2: Real turbine data The correlation plots of this turbine were illustrated in Figures 2 and 3. In this example, one of the blades on the turbine was found to have some form of damage midway through Year 7 of turbine operation. Figure 9 illustrates the comparisons between the actual frequency estimations and the GP predictions for each blade. Note that on the following figures, the period between the vertical red lines illustrate the training period for the GPs, the period between the second vertical red line and the vertical green line illustrate the period the $3\sigma$ thresholds were calculated, and the vertical black lines indicate the date the damage was identified. (a) (b) (c) Figure 9: Comparison between actual estimated normalised edge frequencies and GP predictions for: (a) Blade A, (b) Blade B, and (c) Blade C in Site A. Figures 10 (a) - (c) illustrate the residual errors corresponding to the results shown in Figures 9 (a) - (c), respectively. (a) (b) (c) Figure 10: Residual errors between actual estimated normalised edge frequencies and GP predictions for: (a) Blade A, (b) Blade B, and (c) Blade C in Site A. From Figures 9 and 10, it is clear that the GP predictions become less accurate as the date of damage approaches. As damage levels increase, and the properties of the blades change, the inputs to the GPs become geometrically further away than the inputs used during the training phase. Since machine learning methods in general, and GPs in particular, should not be used for extrapolation, the predictions become poorer, and hence, the residual errors grow. On closer inspection of Figure 10, it can be seen that just prior to the sharp increases in the GP residuals, there are small mean shifts that occur. The corresponding X-bar control charts for Site A are shown in Figure 11. The averages were completed once every four weeks. (a) (b) (c) Figure 11: X-bar control charts of GP residuals for: (a) Blade A, (b) Blade B, and (c) Blade C in Site A. The horizontal black lines indicate the calculated thresholds of the X-bar control charts. On analysing these charts in Figure 11, it can be seen that the GP residuals begin to indicate that the blade properties have changed roughly 6 months before the blade was remedied/replaced. It is noteworthy that following the identification of damage, and the corresponding remedial action, there is a break down in the correlation between the pairs of blades. This is because the mechanical properties of the repaired/replaced blade are now different. Hence, whilst the residuals approach the original thresholds, they do not settle within them. This fact indicates that every time a remedial activity takes place, retraining is essential. ### 3.4 Example 3: Real turbine data The correlation plots for this turbine are shown in Figure 12. Note that the temperature scales shown in these plots are normalised. In this example, all three correlation plots are noisy, although there is the expected positive correlation. The temperature effects are noticeable - at lower temperatures, when the blades are stiffer, the edge frequencies are higher, whereas at higher temperatures, the opposite is true. (a) (b) (c) Figure 12: Correlation between normalised edge frequencies for: (a) Blade A vs. Blade B, (b) Blade B vs. Blade C, and (c) Blade C vs. Blade A in Site B. Figure 12 illustrates how the GP predictions vary with the input edge frequencies and temperature. It is interesting to note how the GPs have identified a hidden structure within the noisy data - there are various bands of predictions across the range of temperatures. These bands are both linear (captured by the linear kernel) and nonlinear (captured by the squared- exponential kernel). (a) (b) (c) Figure 13: Correlation between normalised edge frequencies for: (a) Blade A vs. Blade B, (b) Blade B vs. Blade C, and (c) Blade C vs. Blade A in Site B. Figures 14 and 15 compare the estimated and predicted edge frequencies. Once again, the seasonal variations are captured well. However, the amplitudes of the residual errors are high compared to those seen in Example 2, which is expected given the noisy estimates and correlations. The residuals in Figure 15 (a) and (b) indicate that an event occurred that changed the structural properties, and correlations between the blades around Year 4 as shown by the change in structure of the residuals. As the damage event approaches, there is a monotonic change in the residuals in Figure 15. (a) (b) (c) Figure 14: Comparison between actual estimated normalised edge frequencies, and GP predictions for: (a) Blade A, (b) Blade B, and (c) Blade C in Site B. (a) (b) (c) Figure 15: Residual errors between actual estimated normalised edge frequencies, and GP predictions for: (a) Blade A, (b) Blade B, and (c) Blade C in Site B. Figure 16 shows the results from this final case study. Once again, the 3$\sigma$ thresholds were exceeded in advance, this time only 3 months before the remedial activities to fix the blade took place. Furthermore, the amplitude of the threshold exceedance is not as pronounced as that seen in Example 2. This is primarily attributed to the noisy features used. (a) (b) (c) Figure 16: X-bar control charts of GP residuals for: (a) Blade A, (b) Blade B, and (c) Blade C in Site B. ## 4 Conclusions In this article, a new diagnostic methodology for operational wind turbine blades has been proposed. Gaussian Processes (GPs) are used to predict the edge frequencies of one blade, given the edge frequencies of another blade and the ambient temperature. This may be viewed as another way of applying the Johansen procedure to cointegrate variables, as shown in [25]. The premise is that as long as no damage has taken place on the blade, the relationship between the edge frequencies of the respective blades should remain consistent. However, when damage does take place, the predictive capabilities of the GPs would be affected, and the predictions would not be accurate. Hence, when damage takes place, the GP residuals (the difference between the actual and predicted edge frequencies) would grow in amplitude, and would hence act as informative damage indicators. X-bar control charts, along with 3$\sigma$ thresholds, were used to indicate whether the blades were healthy or not. A case study using synthesised data was used to illustrate the algorithm. Two other case studies from real turbines, where one of the blades was reported to sustain damage, were also shown to illustrate the diagnostic capabilities of the methodology proposed in this article. The results showed the successful implementation of this method whereby the respective damages were detected in advance of critical failure. The key outcomes of this work are: * 1. Although the individual edge frequency time series of the blades are nonstationary, there exists a common and explicable trend between them, which can be learned. * 2. Due to the noisy estimation of the edge frequencies, there may not be a one- to-one mapping between the edge-frequency estimates of the pairs of blades. However, as long as the nominal trend of this correlation is captured by the GPs, the residual errors give sufficient information regarding the onset of damage as long as these frequencies being monitored are sensitive to the damage type and location. * 3. The addition of ambient temperature as a feature helps in the predictive capabilities of the GPs, especially when the edge-frequency estimates are noisy. * 4. 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11institutetext: Ning Ge 22institutetext: School of Software, Beihang University, Beijing, China 22email<EMAIL_ADDRESS> _She was also with the Key Laboratory of Safety-Critical Software (Nanjing University of Aeronautics and Astronautics), Ministry of Industry and Information Technology._ 33institutetext: Guanghao Li 44institutetext: School of Software, Beihang University, Beijing, China 44email<EMAIL_ADDRESS>55institutetext: Li Zhang 66institutetext: School of Computer Science and Engineering, Beihang University, Beijing, China 66email<EMAIL_ADDRESS> _Li Zhang is the corresponding author._ 77institutetext: Yi Liu88institutetext: School of Computer Science and Engineering, Beihang University, Beijing, China 88email<EMAIL_ADDRESS> # Failure Prediction in Production Line Based on Federated Learning: An Empirical Study Ning Ge Guanghao Li Li Zhang Yi Liu (Received: date / Accepted: date) ###### Abstract Data protection across organizations is limiting the application of centralized learning (CL) techniques. Federated learning (FL) enables multiple participants to build a learning model without sharing data. Nevertheless, there is very few research works on FL in intelligent manufacturing. This paper presents the results of an empirical study on failure prediction in the production line based on FL. This paper (1) designs Federated Support Vector Machine (FedSVM) and Federated Random Forest (FedRF) algorithms for the horizontal FL and vertical FL scenarios, respectively; (2) proposes an experiment process for evaluating the effectiveness between the FL and CL algorithms; (3) finds that the performance of FL and CL are not significantly different on the global testing data, on the random partial testing data, and on the estimated unknown Bosch data, respectively. The fact that the testing data is heterogeneous enhances our findings. Our study reveals that FL can replace CL for failure prediction. ###### Keywords: Empirical study Federated learning Failure prediction Production line Manufacturing Bosch dataset ## 1 Introduction Artificial intelligence (AI) is one of the core techniques in the fourth industrial revolution Zhong et al. (2017); Li et al. (2017). For instance, AI techniques have been employed to improve the early detection or fault prediction within production lines Tao et al. (2018); Kusiak (2017) in intelligent manufacturing (IM). A prominent achievement is that in 2012, Intel saved $3 million in manufacturing costs through the use of predictive analytics to prioritize its silicon chip inspections Ronen and Burns . To this end, the Bosch company has published a dataset of product quality prediction on the data analysis competition platform Kaggle111https://www.kaggle.com/c/bosch-production-line-performance. The dataset reflects the relevant parameters and equipment operation of each product during the production process, hoping to reduce defective products. By using this dataset, several studies have proposed product quality prediction methods based on centralized learning (CL) algorithms Carbery et al. (2019, 2018); Zhang et al. (2016); Khoza and Grobler (2019); Kotenko et al. (2019); Mangal and Kumar (2016); Hebert (2016); Maurya (2016); Huang et al. (2019b); Moldovan et al. (2019); Liu et al. (2020d). The results of these work revealed two important facts. First, data preprocessing on the Bosch dataset has a significant impact on the prediction result. Second, among the CL algorithms, the Random Forest algorithm performed better than others when time-series features are excluded Zhang et al. (2016); Khoza and Grobler (2019) and the Long Short Term Memory (LSTM) network can improve the prediction result when time-series features are included Huang et al. (2019b). CL methods usually require that clients upload data to the server. The server trains an integrated model based on the shared data. However, in real-world scenarios, private data belongs to multiple independent organizations and cannot be shared with others. Therefore, data sharing has arisen to be an obstacle to the application of CL methods. As UC Berkeley pointed out in a report published in 2017, learning from private data is one of the challenges encountered by the current AI field Stoica et al. (2017). In response to the above problem, Google proposed the concept of federated learning (FL) in 2016 to decentralize the collaborative learning McMahan et al. (2016). FL trains models across multiple decentralized devices holding local private data samples, without exchanging or sharing these samples. Clients then encrypt and transmit the trained model or parameters back to the server to build an integrated model. On the one hand, FL addresses critical issues such as data privacy and data security Yang et al. (2019a); Li et al. (2020a). On the other hand, as FL trains model locally without removing the private data features, the data samples can be more complete than CL. FL has received widespread attention recently and has been applied in many fields. For example, Google launched a project on the prediction of mobile keyboard input in 2018 and obtained better prediction results using FL Hard et al. (2018). Intel started to support the hardware architecture for FL222https://www.intel.com/content/www/us/en/artificial- intelligence/posts/federated-learning-for-medical-imaging.html. FL has also been applied to various fields like medical systems Sheller et al. (2018b); Brisimi et al. (2018); Boughorbel et al. (2019b); Huang et al. (2019a), Internet finance Suzumura et al. (2019); Yang et al. (2019b), smart city Samarakoon et al. (2019); Saputra et al. (2019); Liu et al. (2020b), edge computing Bakopoulou et al. (2019), IOT Nguyen et al. (2019); Chen et al. (2020), Cyber Physical Systems (CPS) Mowla et al. (2019); Aussel et al. (2020), etc. In the field of IM, the manufacturing process often crosses enterprises, workshops, or even production lines. Due to data privacy and security, the application of CL is becoming the bottleneck. The studies on the failure prediction in the production line have been conducted under the assumption that all data are shared. Under the premise of data protection between production lines, these methods become unfeasible. Nevertheless, according to our investigation Liu et al. (2020c), applying FL to the field of IM is still in its infancy, and there are very few research works on it. Although FL has achieved success in many fields, whether it can achieve similar performance and replace CL in real-world manufacturing is still an open question. Accordingly, this paper bridges this gap by reporting an empirical study of defective product prediction in the production line by comparing FL with CL. This study is the first attempt to design FL algorithms for failure prediction in the production line. In addressing our research goal, we construct a horizontal FL (HFL) scenario and a vertical FL (VFL) scenario, respectively. In the former, we compared the Federated Support Vector Machine (FedSVM) and the Support Vector Machine (SVM) algorithms. In the latter, we compare the Federated Random Forest (FedRF) and the Random Forest (RF) algorithms. Each group of contrast experiment focuses on four research questions (RQs). * $\circ$ _RQ1_ , we are interested in whether FL can replace CL on the whole Bosch testing data. Hence, we ask whether the average performance of FedSVM (FedRF resp.) is similar to that of SVM (RF resp.) on the whole Bosch testing data? * $\circ$ _RQ2_ , we are interested in whether FL can replace CL on part of the given Bosch testing data. Therefore, we ask whether the average performance of FedSVM (FedRF resp.) is similar to that of SVM (RF resp.) on random partial Bosch testing data? * $\circ$ _RQ3_ , if FL is applied to predicting defective products on other unknown Bosch data, whether FL can replace CL. Hence, we ask whether the average performance of FedSVM (FedRF resp.) is similar to that of SVM (RF resp.) on the estimated unknown Bosch testing data? * $\circ$ _RQ4_ , we want to know whether the Bosch testing data is heterogeneous or not. If the data is heterogeneous and the answers of RQ1-RQ3 are positive, it means that the selected FL algorithms are robust. Therefore, we ask, is there local heterogeneity within the given testing data? Answers to these questions are helpful for manufacturing as they provide insights into how to evaluate an FL algorithm can achieve similar effects as a CL one in a real-world application, and then replace it. To answer these questions, this paper makes methodological, substantive, and theoretical contributions to the literature on failure prediction in the production line. First, in terms of algorithm design, to conduct empirical research on the failure prediction based on HFL and VFL, we first select SVM and RF from previous works as CL baseline because both algorithms performed better than others in the previous works. Then, we design FedSVM and FedRF algorithms for our problem based on existing works Bakopoulou et al. (2019); Liu et al. (2020a). In terms of novelty in the algorithm design, our work improved FedRF by introducing optimal feature selection and pruning steps. Second, we propose a set of experimental methodology to compare the average performance of FL and CL in manufacturing from three aspects: on the whole testing data, on the random partial testing data, and on the estimated unknown testing data. To compare the performance of FL and CL on the estimated unknown Bosch data, we follow the process of Measurement Systems Analysis (MSA) to fit a Markov process model of prediction error $\mathtt{M_{F}}$ ($\mathtt{M_{C}}$ resp.) based on FL (CL resp.) and then compare the two fitting models. The experimental methods also assess the heterogeneity of the testing data to enhance the results of RQ1-RQ3 on whether FL can replace CL. Last, our empirical research shows that our FL solution is not significantly different from the CL solution for failure prediction, on the whole, on the random partial, and on estimated unknown Bosch testing data. FL can replace CL in the application of failure prediction within the manufacturing process. In the remainder of this paper, we first introduce the background knowledge of our empirical research and related work in Section 2, present the designed HFL and VFL algorithms in Section 3, and outline the experimental methodology in Section 4. Section 5 presents the results of our empirical study. Section 6 discusses the threat of validity. Section 7 discusses whether FL can replace CL. Section 8 concludes this paper and looks forward to future work. ## 2 Background and Related Work This section introduces previous works on Bosch product quality prediction based on the CL algorithms and discusses the necessity of conducting empirical research on the FL algorithms for this problem; and then, briefly explains the principles of federated learning. ### 2.1 Failure Prediction in Bosch Production Line Bosch is one of the worldwide leading manufacturing companies. It ensures high quality of the production by monitoring its parts in the manufacturing processes. Because Bosch records detailed data for each step on the assembly lines, they can apply advanced techniques to improve the manufacturing processes. To this end, Bosch has published a dataset on the Kaggle competition platform to predict internal failures by thousands of measurements and tests made for each component along the assembly line. Some studies have analyzed the dataset and carried out approaches to predicting product quality based on CL algorithms excluding time-series features Carbery et al. (2019, 2018); Zhang et al. (2016); Khoza and Grobler (2019); Kotenko et al. (2019); Mangal and Kumar (2016); Hebert (2016); Maurya (2016) or including time-series features Huang et al. (2019b); Moldovan et al. (2019); Liu et al. (2020d), as summarized in Table 1. These works were conducted based on a set of learning methods, including Logistic Regression (LR), Gradient Boosting Machine (GBM), Random Forest (RF), Gradient Boosted Trees (GBT), Naive Bayes (NB), Bayesian Network (BN), K-Nearest Neighbors (KNN), Support Vector Machines (SVM), Multilayer Perceptron Classifier (MPC), Majority Voting (MV), Decision Tree (DT), Statistical Process Control (SPC), etc. Table 1: Research Works on Failure Prediction in Bosch Production Line Objectives \| Ref \| Learning Method \| Contribution ---\|---\|---\|--- Improving predictive model without time-series features \| Carbery et al. (2018) \| XGBoost, BN \| BN model performs well in failure prediction. Zhang et al. (2016) \| RF, Gradient Boosting, LR, NB, DT \| RF performs better on different clusters. Khoza and Grobler (2019) \| RF, SVM, NB, SPC \| RF outperforms other models. Kotenko et al. (2019) \| SVM, KNN, Perceptron, LR, DT, MV \| SVM and MV outperform other models. Mangal and Kumar (2016) \| LR, Extra Trees Classifier, RF, XGBoost \| XGBoost and RF outperform other methods. Hebert (2016) \| LR, RF, XGBoost \| RF and XGBoost can properly identify conditions leading to failure events. Maurya (2016) \| XGBoost \| Optimize MCC by using GBM as a base classifier. Improving predictive model with time-series features \| Huang et al. (2019b) \| Ontology-based LSTM neural network \| Ontology-based LSTM neural network yields a better performance Moldovan et al. (2019) \| RF, GBT, NB, KNN, SVM, and MPC \| The LSTM RNN model outperform others. Liu et al. (2020d) \| SP-LSTM models \| The A-Bi-SP-LSTM model outperforms other models. Carbery et al. Carbery et al. (2019) conducted a systematic feature analysis on the Bosch dataset and used BN to predict product quality Carbery et al. (2018). Zhang et al. Zhang et al. (2016) weakened data heterogeneity through clustering, and then applied RF, Boosting, LR, NB, DT to each cluster. They showed that RF performed better than other algorithms. Based on Zhang et al. (2016), Khoza et al. Khoza and Grobler (2019) compared RF, NB, SVM, and SPC. They also showed that RF performed better. Kotenko et al. Kotenko et al. (2019) used SVM, KNN, LR, Perceptron, DT, and MV and showed that SVM achieved relatively higher prediction accuracy. Mangal et al. Mangal and Kumar (2016) conducted a visual analysis of the three types of data: categorical, numeric, and time-series features on the Bosch dataset, and used LR, Extra Trees Classifier, RF, XGBoost for quality prediction. Hebert Hebert (2016) found that RF and XGBoost could properly identify conditions that lead to failure events. Maurya Maurya (2016) optimized MCC measure by using GBM as a base classifier. The studies Moldovan et al. (2019); Huang et al. (2019b); Liu et al. (2020d) have considered time-series features. Huang et al. Huang et al. (2019b) constructed an LSTM network model based on the time-series features of the data, which has great enlightening significance for other researchers. Moldovan et al. Moldovan et al. (2019) found that the LSTM RNN model outperformed other machine learning models. Liu et al. Liu et al. (2020d) proposed an end-to-end unified quality prediction framework to capture temporal interactions among the features of different processes. They showed that the A-Bi-SP-LSTM model outperforms the existing data-driven methods. Despite these studies on learning features for failure prediction, they have been all conducted under the assumption that all data are shared. Under the premise of data protection between production lines, these methods become unpractical. An open question is whether there exist some FL algorithms that can replace CL methods. Thus, our study is a first attempt to design FL algorithms for failure prediction in the assembly line. ### 2.2 Overview of Federated Learning According to the relationship between the datasets provided by the clients, we usually classify FL into HFL, VFL, and federated transfer learning (TFL) Kairouz et al. (2019). This article focuses on the HFL and VFL scenarios. In HFL, data are partitioned by user IDs or device IDs. For example, in the context of manufacturing, clients A and B are two independent factories having the same production line structure and machine configuration. The data features in each production line are roughly the same, while the product IDs on their production lines do not overlap. VFL is applicable to the cases that two datasets share the same sample ID space but differ in feature space. For example, production lines A and B are independent, but one belongs to the upstream, and the other belongs to the downstream of the entire production line. Their product IDs are likely to be the same, which guarantees the intersection of their product space. However, since both A and B record part of manufacturing behavior, their feature spaces are very different. In this paper, we design FedSVM as the HFL model, and design FedRF as the VFL model, respectively, to compare with centralized SVM and RF models. ## 3 Federated SVM and Federated RF This section introduces the design of the federation learning algorithms in this work. Through the investigation in Sect. 2, we selected SVM and RF models, which performed well on the Bosch dataset, as the baseline. In the HFL scenario, we reuse the existing FedSVM algorithm to compare it with SVM. In the VFL scenario, we improve the existing Federated Forest algorithm Liu et al. (2020a) to compare with RF. ### 3.1 Federated SVM As a supervised learning algorithm, SVM is suitable for classification and regression analysis. The effectiveness of SVM mainly depends on how to select 1) the kernel, 2) the kernel’s parameters, and 3) the soft margin parameter. In this work, we use FedSVM with a linear kernel. SVM specifies parameters and intercepts, which can be directly weighted. In FL, the SVM models generated on different clients can be integrated by averaging the parameters and intercepts to meet the need of the server. The FedSVM algorithm is first proposed in Bakopoulou et al. (2019) for mobile packet classification. We have adapted their FedSVM to our failure prediction problem. Its training process is shown in Fig. 1. The algorithm of the client and server is given by Algo. 1 and Algo. 2, respectively. The main steps of FedSVM are explained hereafter. Figure 1: The Training Process of FedSVM * $\circ$ S1. Parameter initialization: The server ($S$) notifies each client ($C_{i}$) to train models using the local data with random value of parameters and intercepts ($w_{0}$). * $\circ$ S2. Local training: $C_{i}$ trains the local SVM model independently. * $\circ$ S3. Return local SVM parameters: $C_{i}$ returns locally trained parameters and intercepts ($w^{i}_{t}$) in this round to $S$. * $\circ$ S4. Compute global SVM parameters: $S$ averages the trained values in two successive round $(\sum w^{i}_{t}+w_{t-1})/2$ as the value $w_{t}$ in the global model $\mathtt{M_{G}}$. * $\circ$ S5. Send global SVM parameter: $S$ sends the global value $w_{t}$ in the $\mathtt{M_{G}}$ to $C_{i}$ to train the model of next round. * $\circ$ S6. Repeat until convergence: Repeat steps S2 - S5 after a specified number of iterations until convergence. The global model in the last round is sent to all clients as the training result. Algorithm 1 FedSVM - Client 1:Data set $D_{i}$ on client $i\ (i=1\ to\ k)$ 2:Local SVM model on client $i$ 3:Initialize $w_{0}^{i}$ 4:$w_{1}^{i}\leftarrow$ SVM($w_{0}^{i}$, $D_{i}$) $\triangleright$ S2 5:Send $w_{1}^{i}$ to server $\triangleright$ S3 6:for $epoch\ t=2\ to\ N$ do $\triangleright$ S6 7: Receive global model $w_{t-1}$ $\triangleright$ S5 8: $w_{t}^{i}\leftarrow$ SVM($w_{t-1}$, $D_{i}$) $\triangleright$ S2 9: Send $w_{t}^{i}$ to server $\triangleright$ S3 10:end for Algorithm 2 FedSVM - Server 1:Local SVM model 2:Global SVM model 3:Inform all clients start training $\triangleright$ S1 4:Receive $w_{1}^{i}$ from all clients 5:$w_{1}\leftarrow\sum_{i=1}^{k}\frac{w_{1}^{i}}{k}$ $\triangleright$ S4 6:Send $w_{1}$ to all clients $\triangleright$ S5 7:for $epoch\ t=2\ to\ N$ do $\triangleright$ S6 8: Receive $w_{t}^{i}$ from all clients 9: $w_{t}\leftarrow\frac{1}{2}(w_{t-1}+\sum_{i=1}^{k}\frac{w_{t}^{i}}{k})$ $\triangleright$ S4 10: Send $w_{t}$ to all clients $\triangleright$ S5 11:end for ### 3.2 Federated Random Forest The Federated Forest algorithm was first proposed by Liu et al. in 2020 Liu et al. (2020a). The random forest can be regarded as an integrated implementation of the decision tree. The core steps of the decision tree (calculating the purity of each feature, represented by Gini coefficients) only involve all data in a single feature and the classification label. Each client in VFL can do this on its local data. In our work, we have optimized the federated random forest algorithm in Liu et al. (2020a) to get better accuracy and efficiency by improving the steps of optimal feature selection and pruning. The improved method is illustrated hereafter. Optimal feature selection: The optimal feature is the one with the smallest Gini coefficient according to the CART method in the decision tree. For continuous variables, it is necessary to try to divide all possible values when calculating the Gini coefficient. Here we reduce the calculation steps of this function and improve the efficiency of the algorithm. Pruning: There is no explicit pruning process in Liu et al. (2020a). Here we add a pre-pruning function, which enhances robustness and improves the efficiency of the overall algorithm. When the decision tree finds that the new node will not improve the accuracy, it will change this node to a leaf node. The training process of our improved FedRF algorithm is given in Fig. 2. The algorithm of client and server is given by Algo. 3 and Algo. 4, respectively. The main steps in its training process are explained hereafter. Figure 2: The Training Process of FedRF * $\circ$ S1. Select initial information: The server ($S$) randomly selects the feature subset $\mathtt{F^{\prime}}$, the sample ID subset $\mathtt{D^{\prime}}$ and test set $\mathtt{T}$, and notifies them to each client ($C_{i}$). * $\circ$ S2. Local training and pruning: $C_{i}$ knows which features have been selected, and does not know the number of features selected globally. $C_{i}$ calculates the Gini coefficient of each feature in $\mathtt{F^{\prime}}$. Because the feature is only on the client, the client also knows the classification labels of all samples, so the Gini coefficient calculated by the client represents the global Gini coefficient. $C_{i}$ obtains the optimal division feature and corresponding Gini coefficient, and retain the division threshold of this feature. $C_{i}$ also checks whether the accuracy of the decision tree is improved on this optimal feature partition, and if there is no improvement, it judges that pre-pruning is needed. * $\circ$ S3. Upload local Gini coefficient: $C_{i}$ uploads the Gini coefficient and pre-pruning information to $S$. * $\circ$ S4. Select global optimal features: $S$ selects the globally optimal feature. * $\circ$ S5. Send optimal global features: If pruning is not required, $S$ notifies the corresponding client to return the ID division result; otherwise, $S$ notifies all clients to perform pruning processing to form a leaf node. * $\circ$ S6. Send local decision: When pruning is not required, the notified client returns the result of dividing the dataset with this feature (the ID set that falls into the left and right subtrees). * $\circ$ S7. Broadcast global decision: $S$ notifies other clients of the ID division and forms the first node of the decision tree. $C_{i}$ knows the results of each ID division, but only knows the existence of local features in the global tree. $S$ knows the structure of the entire tree and the corresponding feature names of each node. $S$ and $C_{i}$ recursively establish new nodes and their left and right subtrees according to the current $\mathtt{F^{\prime}}$ and $\mathtt{D^{\prime}}$ until they form leaf nodes. * $\circ$ S8. Repeat until N tree built: Iteratively build N decision trees to form a random forest. Algorithm 3 FedRF - Client 1:Data set $D$, feature set $F_{i}$ on client $i\ (i=1\ to\ k)$ 2:Local FedRF model on $Ci$ 3:for $epoch\ t=1\ to\ N$ do 4: Receive $F_{i}^{{}^{\prime}}\subseteq F_{i}$, $D^{{}^{\prime}}\subseteq D$, $T=D-D^{{}^{\prime}}$ from $S$ 5: function TreeBuild($D^{{}^{\prime}},F_{i}^{{}^{\prime}},T$) 6: if all samples in $D^{{}^{\prime}}$ have the same label then 7: return $leaf\ node$ 8: end if 9: $BestF_{i},GiniPara_{i},threshold_{i}\leftarrow$ Gini($D^{{}^{\prime}},F_{i}^{{}^{\prime}}$) 10: if $BestF_{i}$ cannot improve the accuracy on $T$ then 11: Send $needPruning_{i}$ to $S$ $\triangleright$ S2 12: end if 13: Send $BestF_{i},GiniPara_{i}$ to $S$ $\triangleright$ S3 14: if Receive ${}^{\prime}Prune^{\prime}$ message from $S$ then 15: return $leaf\ node$ 16: else if Receive ${}^{\prime}Success^{\prime}$ message from $S$ then 17: Split $D^{{}^{\prime}}$ with $BestF_{i}$, get $D_{l}^{{}^{\prime}},D_{r}^{{}^{\prime}},T_{l},T_{r}$ 18: Send $D_{l}^{{}^{\prime}},D_{r}^{{}^{\prime}},T_{l},T_{r}$ to $S$ $\triangleright$ S6 19: leftTree $\leftarrow$ TreeBuild($D_{l}^{{}^{\prime}},F_{i}^{{}^{\prime}},T_{l}$) 20: rightTree $\leftarrow$ TreeBuild($D_{r}^{{}^{\prime}},F_{i}^{{}^{\prime}},T_{r}$) $\triangleright$ S7 21: else 22: Receive $D_{l}^{{}^{\prime}},D_{r}^{{}^{\prime}},T_{l},T_{r}$ from $S$ 23: leftTree $\leftarrow$ TreeBuild($D_{l}^{{}^{\prime}},F_{i}^{{}^{\prime}},T_{l}$) 24: rightTree $\leftarrow$ TreeBuild($D_{r}^{{}^{\prime}},F_{i}^{{}^{\prime}},T_{r}$) $\triangleright$ S7 25: end if 26: return tree node 27: end function 28: Add this decision tree to forest 29:end for$\triangleright$ S8 Algorithm 4 FedRF - Server 1:Sample ID, feature names on all clients 2:Global FedForest model 3:for $epoch\ t=1\ to\ N$ do 4: for $i=1\ to\ k$ do 5: Send $F_{i}^{{}^{\prime}}\subseteq F_{i}$, $D^{{}^{\prime}}\subseteq D$, $T=D-D^{{}^{\prime}}$ to $C_{i}$ $\triangleright$ S1 6: end for 7: function TreeBuild($D^{{}^{\prime}},T$) 8: if all samples in $D^{{}^{\prime}}$ have the same label then 9: return $leaf\ node$ 10: end if 11: Receive $BestF_{i}^{k},GiniPara_{i}^{k},needPruning_{i}^{k}$ $\triangleright$ S3 12: $BestF_{s}=Max(BestF_{i=1}^{k},key=GiniPara_{i})$ $\triangleright$ S4 13: if $needPruning_{s}==True$ then 14: Send ${}^{\prime}Prune^{\prime}$ message to $C_{i}$ 15: return $leaf\ node$ 16: end if 17: Send ${}^{\prime}Success^{\prime}$ message to $C$ $\triangleright$ S5 18: Receive $D_{l}^{{}^{\prime}},D_{r}^{{}^{\prime}},T_{l},T_{r}$ from $C_{i}$ 19: Send $D_{l}^{{}^{\prime}},D_{r}^{{}^{\prime}},T_{l},T_{r}$ to other $C_{i}$ 20: leftTree $\leftarrow$ TreeBuild($D_{l}^{{}^{\prime}},T_{l}$) 21: rightTree $\leftarrow$ TreeBuild($D_{r}^{{}^{\prime}},T_{r}$) 22: return tree node $\triangleright$ S7 23: end function 24: Add this decision tree to forest 25:end for$\triangleright$ S8 The prediction process is described as follows. $C_{i}$ analyzes the new data ID according to the stored tree. For each sample, when an unknown node is encountered, the sample enters the left and right subtrees. When the encountered node is known (that means, the dividing feature of this node belongs to this client’s dataset), the sample enters the corresponding subtree according to the threshold of the feature. The final data that falls on the leaf node _l_ of the global tree _t_ is the intersection of all client data that falls on this leaf node. We have experimented to evaluate the improvement of FedRF. In this experiment, we constructed a VFL scenario with two clients sharing data features using the Bosch dataset. There are 50 independent features in each client, which are extracted after performing the principal component analysis (PCA) from different production lines and the same 11154 samples. The experiment results are shown in Table 2. It can be seen that we get better prediction results than the federated random forest algorithm proposed in Liu et al. (2020a). Table 2: Improvement of FedRF Algo. \| ACC \| PRE \| F1 \| MCC \| AUC ---\|---\|---\|---\|---\|--- FedForest Liu et al. (2020a) \| 0.552 \| 0.493 \| 0.550 \| 0.118 \| 0.559 FedRF (Ours) \| 0.825 \| 0.802 \| 0.883 \| 0.593 \| 0.866 ## 4 Methodology ### 4.1 Experiment Overview and Research questions This paper studies whether there is a big difference between the effectiveness of FL and CL algorithms for failure prediction on the production line, and then whether FL algorithms can replace CL algorithms on this problem. To this end, we construct two production scenarios. One is HFL, where we compare FedSVM and SVM. The other is VFL, where we compare the effectiveness of FedRF and RF. We design four research questions (RQ), each of which concerns both HFL and VFL. The logical association between the four RQs is depicted, as shown in Fig. 3. Figure 3: Experiment Overview and Research Questions RQ1: On the whole Bosch testing dataset, is there a significant difference between FedSVM and SVM? We ask the same question for FedRF and RF. RQ1 is aimed at comparing the average performance of FL and CL algorithms on the whole Bosch testing dataset. If the difference between the value of each measurement is within the threshold $\delta$ ( $\delta=0.1$), it can be considered that there is no significant difference between this pair of algorithms on the Bosch testing dataset. RQ2: On the random partial Bosch testing dataset, is there a significant difference between FedSVM and SVM? We ask the same question for FedRF and RF. RQ2 is aimed at comparing the average performance of the FL and CL algorithms on the random partial Bosch testing dataset that contains consecutive time- series samples. If the prediction results have no significant difference, it means that the pair of algorithms perform similarly on the random partial testing data. RQ3: On the estimated unknown Bosch data, is there a significant difference between FedSVM and SVM? We ask the same question for FedRF and RF. RQ3 is aimed at comparing the average performance of FL and CL algorithms on the estimated unknown Bosch data. To answer this RQ, we have designed a method to compare the error distribution of FL and CL algorithm on the estimated unknown data based on the given data, as shown in Fig. 4. This method consists of four main steps (S1-S4), explained hereafter. Figure 4: Comparing FL and CL algorithms on the Estimated Unknown Bosch Data S1. Construct prediction table. We first construct a prediction table containing the ground truth (GT) label, the prediction results (Pre) of CL and FL algorithms of products ordered by timestamp (ts). S2. Construct timed state sequence. The states represent the prediction error compared to the GT labels: 1) _hit_ state: GT positive + Pre positive or GT negative + Pre negative; 2) _miss_ state: GT positive + Pre negative; 3) _mistake_ state: GT negative + Pre positive. S3. Fit Markov models. We fit Markov model using the timed state sequence, and obtain $\mathtt{M_{FL}}$ and $\mathtt{M_{CL}}$, respectively. S4. Compare FL and CL Markov models. If the difference between the parameters of $\mathtt{M_{FL}}$ and $\mathtt{M_{CL}}$ is within the threshold $\delta$ ($\delta=0.1$), the difference between FL and CL predictive models will maintain similar on estimated unknown Bosch data if the production line structure, manufacturing process, and quality control methods are not changed. RQ4: Is Bosch testing data heterogeneous? And what is the impact of data heterogeneity on the results of RQ1-RQ3? RQ4 analyzes the impact of data heterogeneity on the results of RQ1-RQ3. To evaluate the heterogeneity of data, we first obtain N groups of randomly selected continuous samples $\mathtt{D^{i}}$, and then uses the GT labels as states (GT positive state and GT negative state) to construct the state transition sequence $\mathtt{S^{i}_{GT}}$. Then we fit the Markov model $\mathtt{M^{i}_{GT}}$ using $\mathtt{S^{i}_{GT}}$, and perform DBSCAN clustering Ester et al. (1996) on the parameter matrix of $\mathtt{M^{i}_{GT}}$. If the clustering result shows multiple different clusters, it means that the data heterogeneity is strong. DBSCAN calculates the distance between two parameter matrices by converting the two matrices into a one-dimensional vector and calculating the angle between the two vectors. If the included angle is less than the specified neighborhood distance threshold, the two-parameter matrices can be regarded as on the same cluster. Based on the results of heterogeneity, the conclusions of RQ1-RQ3 (_Y_ means there is no significant difference between FL and CL, _N_ means there is significant difference between FL and CL) will be analyzed from the following four aspects. * $\circ$ If the data heterogeneity is strong, and the answers to RQ1-RQ3 are _Y_ , then the conclusion will be further strengthened. In our manufacturing scenario, the FL and CL algorithms can obtain similar prediction results and can replace each other. * $\circ$ If the data heterogeneity is strong and some answer in RQ1-RQ3 is _N_ , it means that data heterogeneity may be one of the reasons for disturbing the conclusions. * $\circ$ If the data heterogeneity is weak and the answers to RQ1-RQ3 are _Y_ , it can be concluded that the FL algorithm can replace the CL one under the premise of data homogeneity. * $\circ$ If the data heterogeneity is weak and some answer in RQ1-RQ3 is _N_ , it means that the FL algorithm cannot replace CL even with homogeneous data. ### 4.2 Measurements MSA is one of the commonly used analysis methods in manufacturing Montgomery (2007). The learning-based failure prediction method can be seen as a product quality measurement system. Taking the Bosch company’s product quality testing results as a benchmark, we can analyze the FL and CL algorithms through the MSA method. The evaluation measurements in MSA mainly include Accuracy (ACC), Precision (PRE), and Stability. On this basis, F1, AUC, and Matthew’s Correlation Coefficient (MCC) are also involved. The six measurements we use in the empirical study are provided in Table 3, where P represents the number of GT positive cases, N is the number of GT negative cases, TP (hit) is true positive, TN (correct rejection) is true negative, FP (false alarm) is false positive, and FN (miss) is false negative. Table 3: Measurements Measurement \| Formula ---\|--- ACC \| $\mathtt{\dfrac{TP+TN}{TP+FN+TN+FP}}$ PRE \| $\mathtt{\dfrac{TP}{TP+FP}}$ F1 \| $\mathtt{\dfrac{2TP}{2TP+FP+FN}}$ MCC \| $\mathtt{\dfrac{TP\times TN-FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}}$ AUC \| The area enclosed by the ROC curve and the reference line Stability \| The tendency of accuracy for N groups of random selected continuous data group. MCC is suitable for evaluating the prediction results of unbalanced data. AUC is a commonly used two-category evaluation method, and its value is the area enclosed by the ROC curve and the reference line. The x-axis of a ROC curve is the false positive rate, and the y-axis of a ROC curve is the true positive rate. It shows the relationship between clinical sensitivity and specificity for every possible cut-off. It is a graph with: The x-axis showing 1-specificity ($\mathtt{=FP/(FP+TN)}$) The y-axis showing sensitivity ($\mathtt{=TP/(TP+FN)}$). The range of AUC value is between 0 and 1. Stability characterizes the ability of the measurement system to maintain a constant performance within a certain time range. We divide the test set into N groups (N = 10) on average according to the time-series and calculate the accuracy of these N groups to analyze the stability of the target algorithm. ### 4.3 Bosch Dataset The Bosch dataset is one of the largest public manufacturing datasets (14.3Gb) on Kaggle. The Bosch dataset consists of a training set (1184687 samples) and a test set (1183748 samples). Each sample has three types of features: categorical feature, numeric feature, and date feature, and a two-category classification label indicating the faulty status of a product. The date feature gives the time stamp of each site through which the product passes. Based on the date feature, it is possible to analyze the timed behavior. Existing studies have constructed LSTM neural network models based on time-series information Huang et al. (2019b); Moldovan et al. (2019); Liu et al. (2020d). Since we focus on empirical research on FL without time information, the date features are excluded from the scope of our study. Nevertheless, the time-series information can still provide time-continuity data for empirical research on the estimated unknown data. The data features are shown in Table 4. There are 1184687 products in the dataset, including 1177808 positive samples (Pos) and 6879 negative ones (Neg). In the feature space, the dataset has 968 features. These features are collected from 51 workstations (S1-S51) on 4 production lines L0-L3 that contain 168, 513, 42, and 245 features, respectively. L1_S24_F1695 indicates that the feature No.1695 was observed at the No.24 workstation of the L1 production line. The label Response represents whether the final product is qualified (label = 0) or not (label = 1). From the perspective of data items, only a few items (0.58%) in the dataset result are qualified, and the data is extremely unbalanced. The data is also extremely sparse, with only a few workstations collecting data from most products Carbery et al. (2019). Table 4: Bosch dataset Sample Num (1184687) \| Features Num (968) ---\|--- Positive \| Negative \| L0 \| L1 \| L2 \| L3 1177808 \| 6879 \| 168 \| 513 \| 42 \| 245 ### 4.4 Experiment Scenarios of HFL and VFL To compare FedSVM and SVM on the testing data, we form four workshops A, B, C, and D that do not share manufacturing data but have the same structure and configuration of the production line. Under the premise that the product features are aligned, they contain different samples, and each sample has a unique ID. Each workshop can be regarded as an independent client. Table 5 shows the data used in this experiment. The number of samples on the 4 clients is basically the same. In the experiment, the data set is randomly distributed to all clients on average, each of which possesses 3000 positive and 1500 negative samples. The four workshops have 713 common features of data. Table 5: Data Description of HFL Scenario Sample Num. \| Positive \| Negative \| Feature Num. ---\|---\|---\|--- Total \| 12000 \| 6000 \| 713 selected Per client \| 3000 \| 1500 To compare FedRF and RF on the given testing data, it is assumed that the first three production lines (L0, L1, L2) of the Bosch dataset belong to the same organization O1, and the L3 production line belongs to another organization O2. O1 and O2 are independent. Since the samples in O1 and O2 have the same ID and different characteristics, VFL is applied. The data set used in the FedRF experiment is shown in Table 6. We extract the features belonging to L0, L1, and L2 as a group, and the features belonging to L3 as another group. Then we perform principal component analysis (PCA) to reduce the feature dimension in O1/O2 from 723/245 to 22/22, respectively. According to the overall analysis of PCA results, the first 22 dimensions can represent more than 95% of the variance, so reducing to 22 dimensions, respectively, which will not cause too much information loss Zhang et al. (2016). The positive and negative sample numbers are 12000 and 6000, respectively. Table 6: Data Description for VFL Scenario Sample Num \| Positive \| Negative \| Feature Num ---\|---\|---\|--- Total \| 12000 \| 6000 \| 44 (F1 - F44) Client O1 \| 22 (F1 - F22) Client O2 \| 22 (F23 - F44) ## 5 Experiment Results Our experiments investigate and answer four research questions. ### 5.1 RQ1: Comparison on Bosch Testing Data #### 5.1.1 FedSVM vs. SVM Table 7 shows the experimental results. To ensure that the data set and the algorithm kernel are consistent, the baseline is the result of FedSVM when the number of clients is set to 1, ie., FedSVM degenerates into SVM. Under this premise, the results of FedSVM and SVM are compared. When analyzing the stability, the experiment data is ordered by time-series and then divided into 10 groups on average. The measurements of each group are calculated. The overall results of stability are shown in Fig. 5. The average value of stability is above 0.7 for FedSVM and SVM. The variance of the stability is given as Stability (Var) in Table 7. Table 7: Experiment Results of RQ1 on FedSVM/SVM Algo. \| ACC \| PRE \| F1 \| MCC \| AUC \| Stab. (Var) ---\|---\|---\|---\|---\|---\|--- FedSVM \| 0.825 \| 0.859 \| 0.869 \| 0.607 \| 0.807 \| 0.005 SVM \| 0.859 \| 0.828 \| 0.903 \| 0.690 \| 0.902 \| 0.003 Diff \| -0.034 \| 0.031 \| -0.046 \| -0.083 \| -0.095 \| 0.002 Figure 5: FedSVM vs. SVM on Stability The value difference of ACC, Precision, F1, MCC, AUC, and Stability (Var) between FedSVM and SVM are all within a threshold value of 0.1, respectively. Therefore, the conclusion of RQ1 on FedSVM and SVM is that _the prediction results of FedSVM and SVM on the Bosch testing data are not significantly different_. In Figure 5, FedSVM performs slightly better than SVM on two test groups, namely 1 and 7. Similar results are also shown for the Precision measure in Table 7. This effect is only shown in the HFL scenario, where the clients contribute local data features to the central server. If the amount of data on each client is small, FL’s effect can be better than CL because FL expands the number of IID data samples in the HFL scenario. #### 5.1.2 FedRF vs. RF The experimental results are shown in Table 8. FedRF degenerates into RF by combining two clients so that the effect of FedRF and RF can be compared under the premise of ensuring the consistency of the testing data and the algorithm kernel. When analyzing the stability, the test set is divided into 10 groups on average according to the time-series, and the measurements of each group are calculated. The variance of the stability is given in Table 8, and the overall results of stability are shown in Fig. 6. Table 8: Experiment Results of RQ1 on FedRF/RF Algo. \| ACC \| PRE \| F1 \| MCC \| AUC \| Stab. (Var) ---\|---\|---\|---\|---\|---\|--- FedRF \| 0.843 \| 0.808 \| 0.894 \| 0.659 \| 0.902 \| 0.004 RF \| 0.868 \| 0.836 \| 0.909 \| 0.713 \| 0.912 \| 0.002 Diff \| -0.025 \| -0.028 \| -0.015 \| -0.054 \| -0.010 \| 0.002 Figure 6: FedRF vs. RF on Stability The value difference of ACC, Precision, F1, MCC, AUC, and Stability (Var) on each group are all within 0.1. Therefore, _the prediction results of FedRF and RF on the Bosch testing data are not significantly different._ ### 5.2 RQ2: Comparison on Random Partial Bosch Testing Data In order to compare the average difference between the FL and CL algorithms in randomly sampling partial data, we randomly select N groups (N=100) of testing samples. Each group contains L time-continuous samples. The starting point S of the group is randomly set. L is randomly generated from the interval [300, 1000]. #### 5.2.1 FedSVM vs. SVM The line charts of the prediction results of FedSVM and SVM on 100 groups are shown in Fig. 7. In all charts, the abscissa is the serial number of the 100 groups. The ordinate is the value of the measurement on each group. The solid blue line and red dot-dash line represent the prediction result of FedSVM and SVM, respectively. Through the charts, we can see the range and difference of the FedSVM and SVM on the prediction results of each group. (a) ACC Values of FedSVM and SVM (b) PRE Values of FedSVM and SVM (c) F1 Values of FedSVM and SVM (d) MCC Values of FedSVM and SVM (e) AUC Values of FedSVM and SVM Figure 7: FedSVM vs. SVM on Random Partial Data We draw a histogram to visualize the average difference between FedSVM and SVM on each measurement for 100 groups of data, as shown in Fig. 8. Based on all results, the difference values in ACC, PRE, and F1 are all within 0.1. The difference values in MCC and AUC are mostly within 0.2. It is thus validated that _the performance of FedSVM and SVM on random partial Bosch testing data is not significantly different_. (a) ACC Difference (b) PRE Difference (c) F1 Difference (d) MCC Difference (e) AUC Difference Figure 8: Difference of FedSVM and SVM on Random Partial Testing Data #### 5.2.2 FedRF vs. RF The line charts of FedRF and RF on 100 groups of samples are shown in Fig. 9. In all line charts, the abscissa is the number of groups. The ordinate is the value of the measurements on each group. The solid blue line and red dot-dash line represent the prediction result of the FedRF and RF, respectively. (a) ACC Values of FedRF and RF (b) PRE Values of FedRF and RF (c) F1 Values of FedRF and RF (d) MCC Values of FedRF and RF (e) AUC Values of FedRF and RF Figure 9: FedRF vs. RF on Random Partial Data We draw a histogram of the differences between FedRF and RF on each measurement for 100 groups of data, as shown in Fig. 10. Based on all results, the difference values in ACC, PRE, F1, and AUC are all within 0.1. The difference values in MCC are within 0.2, and the MCC values of over 80% groups are within 0.1. It is thus validated that _the performance of FedRF and RF on the random partial Bosch testing data is not significantly different._ (a) ACC Difference (b) PRE Difference (c) F1 Difference (d) MCC Difference (e) AUC Difference Figure 10: Difference of FedRF and RF on Random Partial Testing Data ### 5.3 RQ3: Comparison on Estimated Unknown Bosch Data This experiment is used to analyze whether the FL and CL algorithms can maintain no significant difference on the estimated unknown Bosch data. This experiment is conducted following the method presented in Sect. 4. #### 5.3.1 FedSVM vs. SVM We first constructed a timed state sequence indicating the prediction error using FedSVM and SVM, respectively, based on which their Markov models are fitted. The parameters of two Markov models are given in Table LABEL:tab:RQ3-SVM_resa and LABEL:tab:RQ3-SVM_resa. The value difference between each pair of parameters is calculated in Table LABEL:tab:RQ3-SVM_resc. The average difference value between each pair of parameters is 0.054. The maximum difference value is 0.096. It means that there exists no significant difference between the two Markov models, and the prediction error between FedSVM and SVM are almost equally distributed on the estimated unknown Bosch data. Therefore, _the performance of FedSVM and SVM in the subsequent Bosch production was not significantly different._ Table 9: Experiment Results of RQ3 on FedSVM/SVM State \| Hit \| Miss \| Mistake ---\|---\|---\|--- Hit \| 0.882 \| 0.113 \| 0.005 Miss \| 0.715 \| 0.280 \| 0.005 Mistake \| 0.813 \| 0.062 \| 0.125 (a) Markov Model Parameters of Prediction Error using FedSVM \| Hit \| Miss \| Mistake ---\|---\|---\|--- Hit \| 0.851 \| 0.077 \| 0.072 Miss \| 0.690 \| 0.225 \| 0.085 Mistake \| 0.717 \| 0.125 \| 0.158 (b) Markov Model Parameters of Prediction Error using SVM \| Hit \| Miss \| Mistake ---\|---\|---\|--- Hit \| 0.031 \| 0.036 \| 0.067 Miss \| 0.025 \| 0.055 \| 0.080 Mistake \| 0.096 \| 0.063 \| 0.033 (c) Comparison of Two Markov Model Parameters #### 5.3.2 FedRF vs. RF We first constructed a timed state sequence indicating the prediction error using FedRF and RF, respectively, based on which their Markov models are fitted. The parameters of two Markov models are given in Table LABEL:tab:RQ3-RF_resa and LABEL:tab:RQ3-RF_resa. The value difference between each pair of parameters is calculated in Table LABEL:tab:RQ3-RF_resc. The average difference value between each pair of parameters is 0.035. The maximum difference value is 0.100. It means that there exists no significant difference between the two Markov models, and the prediction error between FedRF and RF are equally distributed on the estimated unknown Bosch data. Therefore, _the performance of FedRF and RF in the subsequent Bosch production was not significantly different._ Table 10: Experiment Results of RQ3 on FedRF/RF \| Hit \| Miss \| Mistake ---\|---\|---\|--- Hit \| 0.869 \| 0.130 \| 0.001 Miss \| 0.701 \| 0.299 \| 0.000 Mistake \| 1.000 \| 0.000 \| 0.000 (a) Markov Model Parameters of Prediction Error using FedRF \| Hit \| Miss \| Mistake ---\|---\|---\|--- Hit \| 0.888 \| 0.109 \| 0.003 Miss \| 0.735 \| 0.263 \| 0.002 Mistake \| 0.900 \| 0.100 \| 0.000 (b) Markov Model Parameters of Prediction Error using RF \| Hit \| Miss \| Mistake ---\|---\|---\|--- Hit \| 0.019 \| 0.021 \| 0.002 Miss \| 0.034 \| 0.036 \| 0.002 Mistake \| 0.100 \| 0.100 \| 0.000 (c) Comparison of Two Markov Model Parameters ### 5.4 RQ4: Heterogeneity of Bosch Testing Data This experiment is used to evaluate the heterogeneity of the testing data used in the experiments of RQ1-RQ3. We follow the method proposed in Sect. 4 to experiment. One hundred groups of time-ordered consecutive samples were randomly selected. According to the GT label of the sample, the qualified sample is set to state S0, and the unqualified sample is set to state S1, forming a state sequence. For the state sequence of each group of samples, the Markov model of k=1 step and k=2 steps are established, respectively. The DBSCAN algorithm is used to cluster and analyze all the 1-step and 2-step parameter matrices. If the clustering results show multiple different clusters, it can explain that there is heterogeneity within the testing data. #### 5.4.1 Heterogeneity of the Testing Data used in FedSVM The experimental results are shown in Table 11. There are outliers in the test results, so the sum of the samples in each cluster is less than 100. It can be seen that the testing data is divided into 2 clusters. The difference between clusters is no less than the distance threshold, which means that the data used in the experiment of FedSVM is heterogeneous. It enforces the conclusion that _FedSVM and SVM have no significant difference on heterogeneous manufacturing data for the problem of failure prediction_. Table 11: Experiment Results of RQ4 on FedSVM Step \| Clusters \| Distance threshold (∘) \| Contour factor \| Samples / cluster ---\|---\|---\|---\|--- k = 1 \| 2 \| 5 \| 0.456 \| 92 / 6 k = 2 \| 2 \| 8 \| 0.421 \| 89 / 6 According to the experiment results and the statement of RQ4 in Section 4.1, the data heterogeneity in the VFL scenario is relatively strong and the answer to RQ1-RQ3 are positive. This means in our VFL manufacturing scenarios, the FL and CL algorithms can obtain similar prediction results and can thus replace each other under the premise of data heterogeneity. #### 5.4.2 Heterogeneity of the Testing Data used in FedRF The experimental results are shown in Table 12. It can be seen that the testing data is divided into 2 clusters. The difference between clusters is no less than the distance threshold, which means that the data used in the experiment of FedRF is heterogeneous. It enforces the conclusion that _FedRF and RF have no significant difference on heterogeneous manufacturing data for the problem of failure prediction_. Table 12: Experiment Results of RQ4 on FedRF Step \| Clusters \| Distance threshold (∘) \| Contour factor \| Samples / cluster ---\|---\|---\|---\|--- k = 1 \| 2 \| 4 \| 0.524 \| 94 / 6 k = 2 \| 2 \| 8 \| 0.473 \| 78 / 22 According to the experiment results and the statement of RQ4 in Section 4.1, the data heterogeneity in the HFL scenario is relatively strong and the answer to RQ1-RQ3 are positive. This means in our HFL manufacturing scenarios, the FL and CL algorithms can obtain similar prediction results and can thus replace each other under the premise of data heterogeneity. ## 6 Threats to Validity Following common guidelines for empirical studies Runeson and Höst (2009); Yin (2017), we discuss threats to the validity of our study. Threats to internal validity are mainly concerned with uncontrolled factors. These factors may impact the results and reduce their creditability. In this work, the main threat to internal validity is potential defects in the implementation of our own algorithm and the reimplementation of other algorithms (mainly the federated SVM and federated random forest algorithms). To reduce this threat, we reimplement the baseline techniques by following their papers. We carefully review all our code and experiment scripts by several developers to ensure their correctness. However, there is always a small possibility of defects, which introduces risk to the result’s correctness. Threats to external validity are mainly concerned with whether the performance of our techniques can still hold in other experimental settings. In this work, the main threat to external validity is the impact of data preprocessing method. As we mentioned in Sect. 1, the existing works revealed that the data preprocessing on the Bosch dataset has a significant impact on the prediction result. In our work, we have applied PCA to find the principle features. For example, in the VFL scenario, according to the PCA analysis, only 22 features are kept in each client. But, these features represent more than 95% of the variance of data. Another threat to external validity is the experimental methodology we designed. In order to answer whether the FL algorithm can replace the CL algorithm or not, we conduct experiment on the whole testing data, on the partial random testing data, and on the estimated unknown data, respectively. This method is aimed at simulating possible data set used in manufacturing. Therefore, the answer tends to be complete. The first threat to construct validity is the suitability of our evaluation metrics. To reduce this risk, we follow the same measurements (ACC, precision, F1, MCC, AUC, stability) as used in other works on failure prediction in production line Carbery et al. (2019, 2018); Zhang et al. (2016); Khoza and Grobler (2019); Kotenko et al. (2019); Mangal and Kumar (2016); Hebert (2016); Maurya (2016); Huang et al. (2019b); Moldovan et al. (2019); Liu et al. (2020d) to measure the quality of the learning model, which reduces the risk of construction validity. Another threat to construct validity is the threshold $\delta$ ( $\delta=0.1$ or $\delta=0.2$) used in the experiment. In the industry, the threshold value is set according to the production needs. We have performed a hypothesis test for the difference between the two approaches’ performances to explain that the difference between FL and CL is within the threshold. The original hypothesis H0 is difference $>\delta$, and the alternative hypothesis H1 is difference $<\delta$, where the difference = CL – FL on the same random partial testing data group. Suppose $\alpha$ = 0.05. The p-value of the evaluation metrics for FedSVM vs. SVM and FedForest vs. RForest is shown in Table 13 and Table 14, respectively. As $alpha>$ p-value for all metrics, the alternative hypothesis H1 is accepted, which means that the difference between CL and FL is less than the threshold $\delta$ ( $\delta=0.1$ or $\delta=0.2$). Table 13: P-value of FedSVM vs. SVM ($\alpha$ = 0.05) Metrics \| Threshold \| Statistics \| p-value ---\|---\|---\|--- ACC \| 0.1 \| -22.48877 \| 5.44616e-41 Precision \| 0.1 \| -83.57313 \| 6.33511e-94 F1 \| 0.1 \| -17.48523 \| 2.45813e-32 MCC \| 0.2 \| -20.44888 \| 1.27870e-37 AUC \| 0.2 \| -26.63278 \| 3.08126e-47 stability \| 0.1 \| -5.40692 \| 0.00021 Table 14: P-value of FedForest vs. RForest ($\alpha$ = 0.05) Metrics \| Threshold \| Statistics \| p-value ---\|---\|---\|--- ACC \| 0.1 \| -34.44843 \| 3.16706e-57 Precision \| 0.1 \| -48.40227 \| 5.04976e-71 F1 \| 0.1 \| -75.16218 \| 1.95987e-89 MCC \| 0.2 \| -12.90324 \| 3.11933e-23 AUC \| 0.2 \| -16.77851 \| 5.26331e-31 stability \| 0.1 \| -9.83780 \| 2.04964e-06 ## 7 Discussion Due to data privacy and security, we are facing a bottleneck when applying CL in the industry. In this work, we have adopted two FL algorithms to predict failure in the production line. According to our investigation of existing results of this problem, SVM and RF performed better on Bosch data among CL algorithms that did not consider time series. Therefore, we choose Federated- SVM and Federated-RF in the study. We expect that both will perform as well as the CL algorithms. The results showed that our intuition is not wrong on the given dataset. An interesting question is whether FL can achieve the same or similar prediction results as CL. If yes, what are the conditions (e.g., the the dataset’s size)? Otherwise, what are the guidelines on when we can use FL in place of CL? As our study is limited to the Bosch dataset from the manufacturing industry, we have conducted a literature investigation to answer this question. According to existing studies Kairouz et al. (2019); Duan et al. (2019), FL’s effect is mainly related to the data distribution and data amount. Two important facts are summarized as follows. 1. 1. When the data is independent and identically distributed, the learning results of FL and CL are similar. For instance, the work Lu et al. (2019); Chen et al. (2020); Sozinov et al. (2018); Olivia et al. (2019) showed that when the training data is independent and identically distributed (IID), the difference between FL and CL is within 3%. If the amount of data on each client is small, FL’s effect is better than CL because FL expands the number of IID data samples Ickin et al. (2019); Bakopoulou et al. (2019); Guha et al. (2019); Xinle et al. (2019); Shaoqi et al. (2020); Suzumura et al. (2019). 2. 2. When the training data is unevenly distributed, FL may not achieve the same effect as CL Sheller et al. (2018a); Hu et al. (2018); Feng et al. (2020); Dianbo et al. (2018); Luca and M (2019); Chen et al. (2020); Boughorbel et al. (2019a); Sozinov et al. (2018). The review article Kairouz et al. (2019) claimed that if the amount of data on a client is small, the CL model may be better than the FL model trained using data on multi-clients in the case of uneven distribution of training data. Besides, some works have shown that the effect of FL is also related to encryption algorithms R et al. (2019); Tao et al. (2020); Feng et al. (2020); Olivia et al. (2019); Lu et al. (2020); Li et al. (2019, 2020b). If the encryption strength increases, the information loss on the data will be worse. So that the effect of FL decreases. The trade-off between data privacy protection and the accuracy of the trained model is still inevitable. It is still considered an open question of how the effect of FL is related to other factors. ## 8 Conclusion There is increasing attention for federated learning in the manufacturing industry, but few works have studied how federated learning methods perform in practice. In this work, we have conducted an empirical study of comparing FL and CL methods for problem of the failure prediction in the production line. We constructed a horizontal and vertical federated learning scenario based on the Bosch dataset. We implemented FedSVM to compare with SVM in the HFL scenario and designed FedRF to compare it with RF in the VFL scenario. We also designed an experiment process for evaluating the effectiveness of FL and CL algorithms, which can be reused in future studies. Our results reveal that FedSVM and FedRF can replace SVM and RF, respectively, for failure prediction in the Bosch production line. Because the CL algorithm can be replaced by the FL one (1) on the global testing dataset; (2) on the random partial testing dataset; (3) on the estimated unknown Bosch data. The value of each evaluation metric for FL and CL algorithms differ within 0.1. Moreover, the fact that the testing data is heterogeneous enhances the above three conclusions. The results of our empirical study reveal that FL can replace CL in some applications and FL is a desirable way of protecting data in the manufacturing industry. More FL techniques can also be investigated in other manufacturing applications. The studies Moldovan et al. (2019); Huang et al. (2019b); Liu et al. (2020d) have considered time-series features and showed that the Long Short Term Memory (LSTM) network can improve the prediction result when time- series features are included Huang et al. (2019b). A future work is to design federated LSTM models to compare with centralized LSTM model, and study whether FL can replace CL within this context. ###### Acknowledgements. 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