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h(T). Even though this term is lower-order in T, but nonetheless it merits a mention, as here h(T) 28 coexists with the inverse clipping rate mapping, hinv, being evaluated at a “critical point” 1 +C/c. Since the inverse mapping hinvwill grow fast when hgrows slowly, this term can practically speaking become influentia... | https://arxiv.org/abs/2502.17427v1 |
variances and tells a similar story: the variance ClipOGDSCis smaller and decreases faster compared to that of ClipOGD0. As seen in the bottom row, the Neyman regret of ClipOGD0stays away from zero, whereas the regret of ClipOGDSCshrinks toward zero or remains lower throughout. This pattern suggests that ClipOGDSCconve... | https://arxiv.org/abs/2502.17427v1 |
EXPONENTIAL DIMENSIONAL DEPENDENCE IN HIGH-DIMENSIONAL HERMITE METHOD OF MOMENTS ANDREAS BASSE-O’CONNOR˚& DAVID KRAMER-BANG: Abstract. In this paper, we show exponential dimensional dependence for the Hermite method of moments as a statistical test for Gaussianity in the case of i.i.d. Gaussian variables, by constructi... | https://arxiv.org/abs/2502.17431v1 |
the proof of Theorem 1.1, we will show both upper and lower bound for the three metrics: Kolmogorov-Smirnov distance dK, hyper-rectangle distance dRand convex distance dConRd. All three of these metrics can be considered as a generalisation of the Kolmogorov metric on R. The metrics are defined as follows: dMpX,Yq:“sup... | https://arxiv.org/abs/2502.17431v1 |
the claim of Theorem 1.1. By (3.2), it follows that dRpSn,ZqědKpSn,d, Zdq, where Sn,d(resp. Zd) is the last entry in the vector Sn(resp. Z. Hence, due to this and Remark 1.2, we can simulate the one-dimensional quantity?ndRpSn,d, Zdqfor some large n“60.000 and check if the simulation follows an exponential function as ... | https://arxiv.org/abs/2502.17431v1 |
2πn´n e¯n ,for all nPN. Proof. Recall that dKis a weaker metric than dR, and hence (3.2) dRpSn,ZqědKpSn,Zqě#dKpSn,d, Zdq, ifdis even , dKpSn,d´1, Zd´1q,ifdis odd . Here Sn,d(resp. Sn,d´1) is the last (resp. penultimate) element of Snand likewise Zd(resp. Zd´1) is the last (resp. penultimate) element in Z. Since the arg... | https://arxiv.org/abs/2502.17431v1 |
main results, we recall the following fourth-moment bound on the Hermite polynomials from [1, Lem. 5.2]. EXPONENTIAL DIMENSIONAL DEPENDENCE IN HIGH-DIMENSIONAL HERMITE METHOD OF MOMENTS 6 Lemma 3.2 ([1, Lem. 5.2]) .Letℓě2,Hℓbe the ℓ’th Hermite polynomial and G„Np0,1q. Then, there exists a uniform constant cą0, such tha... | https://arxiv.org/abs/2502.17431v1 |
PREPRINT 1 Estimating Time Delays between Signals under Mixed Noise Influence with Novel Cross- and Bispectral Methods Tin Jurhar1,2, Franziska Pellegrini3,4, Ana I. Nu ˜nes del Toro4,5, Tilman Stephani6, Guido Nolte7, Stefan Haufe2,3,4,5,∗ Abstract —A common problem to signal processing are biases introduced by correl... | https://arxiv.org/abs/2502.17474v1 |
each method under controlled signal and noise conditions. Using this framework, we simulate time-delayed time series. We assess conventional phase-slope and bispectrum based TDE approaches alongside the proposed phase-periodicity and bispectral antisymmetrization protocols in unmixed and mixed noise environments. Addit... | https://arxiv.org/abs/2502.17474v1 |
literature [1], [4], [5] and has more recently resurfaced in applications of TDE in neuroscience [6], pipe leakage detection [7], and acoustics [8]. In neuroscience, making functional inferences from non- invasive electrophysiological data is especially difficult. In- dividual source currents are passively conducted th... | https://arxiv.org/abs/2502.17474v1 |
vation window. This makes cross-correlational TDE suitable for unmixed noise environments. In the mixed noise environment, cross-correlational TDE is unreliable. Here we expect to observe two interactions, namely the time-delayed interaction of signals and the instantaneous interaction of noise. In the Supplement (Eq. ... | https://arxiv.org/abs/2502.17474v1 |
time series Xs, Ysand their Fourier transforms FX, FY. IfX, Y are segmented, we can define the cross-spectrum as SXY(f) =D FX(f)F∗ Y(f)E =D rXrYei(φX−φY)E (11) [5], where ⟨.⟩denotes the expectation across Nsegments, and (*) the complex conjugate. The phase (difference) spectrum [18] PXY(f) =∠(SXY(f)) (12) isolates phas... | https://arxiv.org/abs/2502.17474v1 |
compromise the amplitude and thus also the slope of the phase spectrum. This biases the overall estimate. Again omitting the dependence of PXYand Fx,yon frequency, we can write PXY , mixed =∠(α2β⟨FxF∗ y⟩. . . + (1−α)2(θ2⟨|FnX|2 |{z} ≥0⟩+θ1⟨|FnY|2 |{z} ≥0⟩).(17) The exact effect of mixed noise on the phase spectrum will... | https://arxiv.org/abs/2502.17474v1 |
f2)−∠BXXX (f1, f2))) (23) IM2(f1, f2) = exp i ∠BXY X(f1, f2). . . −1 2(∠BXXX (f1, f2) +∠BY Y Y(f1, f2))!! (24) IM3(f1, f2) =BXY X(f1, f2) BXXX (f1, f2)(25) IM4(f1, f2) =|BXY X(f1, f2)|IM2(f1, f2)p |BXXX (f1, f2)| |BY Y Y(f1, f2)|.(26) From hXY(ρ), a delay estimate τbispec. is derived: τbispec. = arg max ρhXY(ρ). (27) T... | https://arxiv.org/abs/2502.17474v1 |
and it is in turn not expected that the bispectral hologram will consistently have its maximum at an underlying delay τ. How bispectral TDE is influenced by mixed noise precisely is yet to be outlined. G. Robust Cross-Spectrum based TDE using Phase Periodic- ity In Section I-E, it is discussed that the phase-slope dela... | https://arxiv.org/abs/2502.17474v1 |
the the difference between the cross-bispectrum BXY Z(f1, f2)and a permutation of it with respect to any pair of channel indices. In their original paper, [3] show that the ASB cannot originate from independent sources by demonstrating that it vanishes for any linear combination of independent sources. A nonzero ASB th... | https://arxiv.org/abs/2502.17474v1 |
we sample the noise either from a standard normal ( N(0,1)) or an exponential ( Exp(λ= 1) ) distribution. The signal component is drawn from the same exponential distribution, but independent of the noise. The observed time series are then assembled according to Eq. (3), where we choose β= 1andθ1,2=.7. Time delays esti... | https://arxiv.org/abs/2502.17474v1 |
for each SNR and each combination of signal and noise components to obtain a mean absolute error (MAE). We ultimately compare the MAE of trials which have been filtered according to our confidence criterion to those of the unfiltered dataset. We also investigate differences in accuracy across different TDE protocols. 5... | https://arxiv.org/abs/2502.17474v1 |
decreasing to near zero at SNR α=.3. In the mixed noise setting (red lines), ASB has a clear advantage over its unsymmetrized counterpart: whereas the MAE remained at around 500 ms for conventional bispectral TDE until SNR 8 PREPRINT noise: exponential no filter applied bootstrap filter applied no filter applied bootst... | https://arxiv.org/abs/2502.17474v1 |
segment bounds [−100; 100] (dF= 2008 ). JURHAR et al. : ESTIMATING TIME DELAYS BETWEEN SIGNALS UNDER MIXED NOISE INFLUENCE WITH NOVEL CROSS- AND BISPECTRAL METHODS 9 α=.5, the MAE decreases to near zero as early as for α=.2 for the antisymmetrized counterpart, thus even improving upon its result achieved in the unmixed... | https://arxiv.org/abs/2502.17474v1 |
signal and low SNR, α= 0 andα= 0.2. These distributions are summarized in the bottom panel of Figure 3. In the case where no signal is present at all (SNR α= 0), TDE outputs of all protocols are evenly distributed (top two rows). In the unmixed noise case, both bispectrum-basedapproaches and the phase-periodicity based... | https://arxiv.org/abs/2502.17474v1 |
white noise from a standard normal ( N(0,1)), and applying (i) a 1 Hz highpass-, (ii) a 45 10 PREPRINT Hz lowpass-, and (iii) a [8,13]Hz Butterworth filter. The peri- odic and non-periodic components are then power-normalized and added together. We again combine this modeled noise with a signal component identical to t... | https://arxiv.org/abs/2502.17474v1 |
filter applied -- phase slope － phase period.-- bispec., regular － ASB 𝞪 = 0.2 𝞪 = 0unmixed auto -correlated mixed auto -correlated 𝜏(true) 𝜏(true) 𝜏(true) 𝜏(true) 𝜏(true) 𝜏(true) 𝜏(true) 𝜏(true)𝜏(est.) 𝜏(est.) 𝜏(est.) 𝜏(est.)bispectrum TDE ASB TDE bispectrum TDE ASB TDE bispectrum TDE ASB TDE bispectrum ... | https://arxiv.org/abs/2502.17474v1 |
data points at the top and bottom of each phase- periodicity distribution plot. The effect is observed throughout the entire range of possible values, and does not seem to depend of the magnitude of the underlying delay. IV. TDE INELECTROPHYSIOLOGICAL DATA In the final experiment, we explore the viability of the pro- p... | https://arxiv.org/abs/2502.17474v1 |
bandpass-filtering at [30, 200] Hz, and re-referencing to the average of all channels. Eye movement artefacts were removed using independent component analysis [27]. CNAP recordings were highpass- filtered at 70 Hz and additionally notch-filtered at [48, 52] and [148, 152] Hz. 3) TDE Analysis: We limit the present anal... | https://arxiv.org/abs/2502.17474v1 |
supplement. The analysis was performed twice, once on data segments time-locked to the stimulus onset (left panel of Figure 8), and once where segmentation was arbitrary (right panel). For subject S12 and stimulus-locked segments, we found that the medians of estimates consistently range between 10.4 and 12.8 ms (corre... | https://arxiv.org/abs/2502.17474v1 |
. VALUES ARE GIVEN AS MEDIAN (INTERQUARTILE RANGE )AS ESTIMATED FROM 500 BOOTSTRAPP ITERATIONS IN UNITS OF MILLISECONDS . Stimulus-locked Segmentation ASB TDE Phase-period. TDE F4 CP4 P4 F4 CP4 P4 S12 11.2 (3.2) 10.4 (0.8) 10.4 (0.8) 12 (24.4) 12(0.8) 12.8 (1.6) S17 30.8 (52.8) 14.4 (44) 12(0.8) 22.4 (46.8) 18.4 (44.8)... | https://arxiv.org/abs/2502.17474v1 |
reduced Gaussian suppression in the direct FFT approximation of the bispectrum, we do not observe suppression of Gaussian sources at all. While the mechanisms behind this effect remain to be clarified, we presume that, while the absolute value of the bispectrum converges the zero, the phase information used for TDE ret... | https://arxiv.org/abs/2502.17474v1 |
the mixed noise setting. E. Viability of TDE Analysis on EEG Data Within a small exemplary dataset, we were able to quantify time delays between peripheral median nerve and cortical electrode recordings in two out of three subjects with our ASB TDE protocol and one out of three subjects with a phase- periodicity based ... | https://arxiv.org/abs/2502.17474v1 |
τ. (35) A bandwidth greater than 2lperiod needs to be available ac- cordingly in order to resolve a desired minimum delay τ. In contrast, such a hard limit on bandwidth does not exist for bispectrum based TDE. Here, delay information is ob- tained by integrating across the whole frequency spectrum, potentially with fre... | https://arxiv.org/abs/2502.17474v1 |
independent sources. Our methods perform best when combined with a bootstrapping procedure to filter out low-confidence estimates and can achieve close to zero-error accuracy. These results are independent of the statistical properties of the underlying signal and noise components that were tested, suggesting that our ... | https://arxiv.org/abs/2502.17474v1 |
coupled brain sources: Distinguishing true from spurious interaction,” Advances in Neural Information Processing Systems , vol. c, pp. 1027–1034, 2005. [12] G. Nolte, A. Ziehe, V . V . Nikulin, A. Schl ¨ogl, N. Kr ¨amer, T. Brismar, and K. R. M ¨uller, “Robustly estimating the flow direction of information in complex p... | https://arxiv.org/abs/2502.17474v1 |
2020. [26] J. Zhang, X. Yin, L. Zhao, A. C. Evans, L. Song, B. Xie, H. Li, C. Luo, and J. Wang, “Regional alterations in cortical thickness and white matter integrity in amyotrophic lateral sclerosis,” Journal of Neurology , vol. 261, no. 2, pp. 412–421, 2014. [27] A. Delorme and S. Makeig, “Eeglab: an open source tool... | https://arxiv.org/abs/2502.17474v1 |
=∠(BXY X(f1, f2))−1 2(∠(BXXX (f1, f2) +∠(BY Y Y(f1, f2))) =⟨φX(f1) +φY(f2)−φX(f1+f2)⟩ −1 2(⟨φX(f1) +φX(f2)−φX(f1+f2)⟩+⟨φY(f1) +φY(f2)−φY(f1+f2)⟩) =⟨φX(f1) +φY(f2)−φX(f1+f2)⟩ −1 2⟨φX(f1) +φX(f2)−φX(f1+f2)⟩ −1 2⟨φY(f1) +φY(f2)−φY(f1+f2)⟩ =⟨1 2φX(f1) +1 2φY(f2)−1 2φX(f1+f2)−1 2φX(f2)−1 2φY(f1)−1 2φY(f1+f2)⟩ =1 2⟨φX(f1)−φY... | https://arxiv.org/abs/2502.17474v1 |
indices, i.e., (FnXFnX),(FnYFnY). What remains are expressions in[.]with unequal indices only. When expanding this last expression, the final sum must contain only terms with three factors where one index always differs from the other two. By taking the expectation of each term separately, we can factor out the standal... | https://arxiv.org/abs/2502.17474v1 |
arXiv:2502.17671v2 [math.ST] 16 Mar 2025OPTIMAL RECOVERY MEETS MINIMAX ESTIMATION BYRONALD DEVORE1,a, ROBERT D. N OWAK2,d, RAHUL PARHI3,e, GUERGANA PETROVA1,b,AND JONATHAN W. S IEGEL1,c 1Department of Mathematics, Texas A&M University ,ardevore@tamu.edu ;bgpetrova@tamu.edu ;cjwsiegel@tamu.edu 2Department of Electrical ... | https://arxiv.org/abs/2502.17671v2 |
func- tions (see § 2for the definition and properties of these spaces). The unit b all ofBs τ(Lp(Ω)) MSC2020 subject classifications :Primary 62G05, 62C20; secondary 41A30, 41A63. Keywords and phrases: Besov spaces, minimax estimation, nonparametric regressi on, optimal recovery. 1 2 compactly embeds into C(Ω)if and only... | https://arxiv.org/abs/2502.17671v2 |
f∈Khas a wavelet decomposition and the membership in the Besov spaces can be exactly described by a suitable weighted ℓp-norm on these coefficients. One then uses the data observations to compute empirical noi sy wavelet coefficients and thresholds these coefficients to obtain ˆf. An issue that needs to be addressed in th... | https://arxiv.org/abs/2502.17671v2 |
literature in our setting is that the minimax risk sati sfies (12) c(σ)m−s 2s+d≤Rm(K;σ)q≤C(σ)m−s 2s+d, m≥1, for every fixed σ >0. This does not capture the effect of the noise level in the est imation error rate and instead, hides the effect of noise in the const antsc(σ),C(σ)>0. In fact, prior to the present paper, the ... | https://arxiv.org/abs/2502.17671v2 |
model classes in Hilbert spaces defined b y elliptical constraints. In [20,22,23], estimation of sparse vectors has been considered. To the b est of our knowl- edge, our results are the first NLA minimax rates which hold fo r general Besov classes with error measured in general Lq-norms. Our NLA minimax rate provides a r... | https://arxiv.org/abs/2502.17671v2 |
algorithm uses piecewise polynomial s and a thresholding proce- dure. Although the use of piecewise polynomials for nonpara metric function estimation is not new (see, e.g., [ 2,3,6,28,32]), our specific algorithm, to the best of our knowledge, has not been studied before. Finally, we want to mention that the re has bee... | https://arxiv.org/abs/2502.17671v2 |
. In other words, these spaces get smaller as τgets smaller, and thus all of these spaces are contained in Bs ∞(Lp(Ω)) oncesandpare fixed. The effect of τin the definition of the Besov spaces is subtle. In this paper, the space Bs ∞(Lp(Ω)) will OPTIMAL RECOVERY MEETS MINIMAX ESTIMATION 7 be the most important case when p... | https://arxiv.org/abs/2502.17671v2 |
near best Lp(I)approximation is also near best on larger cubes Jcontaining Iand larger values ¯p≥p. We shall use these facts throughout this paper. 2.2. Polynomial Norms. All norms on the finite dimensional space Prare equivalent. In what follows, we need good bounds on the constants that appea r when comparing norms. W... | https://arxiv.org/abs/2502.17671v2 |
considered as a subspace of the Hilbert spaceL2(µI). IfNd>ρandq0≤q≤∞ , then the following (quasi-)norms of a polynomial Q=/summationtextρ j=1βjQI,jare equivalent with constants of equivalency depending onl y onr,dand q0>0but not depending on Norq: (i)/bardblQ/bardbl∗ Lq(I); (ii)/bardbl(Q(zj))/bardbl∗ ℓq(ΛI):=N−d/q/bard... | https://arxiv.org/abs/2502.17671v2 |
We have the representation (54) /tildewideTk:=/summationdisplay I∈Dk ρ/summationdisplay j=1c∗ I,jQI,j χI, where (55) c∗ I,j:=cI,j+ηI,j, I∈Dk,0≤k≤n−r, j=1,...,ρ, are the noisy observation of the true cI,j’s, polluted by the additive Gaussian N(0,σ2 I,j)noise ηI,jwith variance (56) 0≤σ2 I,j≤Cρ2−(n−k)dσ2, I∈Dk, j=1,..... | https://arxiv.org/abs/2502.17671v2 |
q:=/parenleftBigg 1 LL/summationdisplay i=1|vi|q/parenrightBigg1/q . Utilizing that νkandˆνkare vectors of length Lk=ρ2kd, we rewrite the bound ( 68) as (72) /bardblTk−ˆTk/bardblLq(Ω)≤C/bardblνk−ˆνk/bardbl∗ q. We want to define ˆνkto make ( 72) small. Putting ( 72) together with ( 67) results in (73) /bardblf−ˆf/bardblL... | https://arxiv.org/abs/2502.17671v2 |
q=F1/q. THEOREM 4.3. For any increasing, convex function φonR+:= [0,∞)withφ(0)=0 and any1≤q <∞, we have (85) P(/bardblξλ/bardbl∗ q≥T)≤˜σ−1/integraldisplay∞ λ/2φ(xq) φ(Tq)e−x2/2˜σ2dx, T > 0. PROOF . First note that from the convexity of φand Jensen’s inequality, we have (86) φ(F)≤1 LL/summationdisplay j=1φ(fi), OPTIMAL ... | https://arxiv.org/abs/2502.17671v2 |
bound for (95) E∗:=n−r/summationdisplay k=0/bardblνk−ˆνk/bardbl∗ q, whereνkare defined in ( 49) andˆνk= thresh λk(ν∗ k)are obtained by thresholding the ob- served coefficients, i.e., the entries in the vector ν∗ k=νk+η∗ k. Here, the thresh λkis per- formed coordinatewise, with parameters λkspecified later in ( 103). Notic... | https://arxiv.org/abs/2502.17671v2 |
to s how that (109)n−r/summationdisplay k=0P(/bardblηλk/bardbl∗ q>tkε)≤Ce−ctα, t≥1, where we choose (110) tk:=¯ct/braceleftBigg 2δ(k−k∗),0≤k≤k∗, 2δ(k∗−k), k∗<k≤n−r, with0<δ <d/2, and define ¯cso that that/summationtextn−r k=0tk=t. Note that ¯c≥c(δ)>0, wherec(δ) is a constant depending only δ. The subsequent constants in... | https://arxiv.org/abs/2502.17671v2 |
complete and self-contained argument which unifies both regimes. We rest rict our presentation to the case where the parameters p,q,s are in the primary case (see ( 11)). We fix these parameters for the remainder of this section. We also fix m≥1and the variance σ2assumed on the noise vectors. We let the data sites X={xi∈Ω... | https://arxiv.org/abs/2502.17671v2 |
length 1/n, wherenis given by ( 138). We define the functions (140) φi(x)=γn−sφ(n(x−zi)), i=1,...,P, whereziis the bottom left corner of the cube Qiand the normalizing constant γdefined momentarily. Thus, each φiis a rescaling of φto the cube Qiand each φihasL∞(Ω)norm equal toγn−s. To construct our collection of function... | https://arxiv.org/abs/2502.17671v2 |
we have /bardblfi−A(y)/bardblLq(Ω)≥/tildewidec0 2ε. It follows that (156) E/bardblfi−A(˜y(fi))/bardblLq(Ω)≥µy(fi),σ(Bc i)·/tildewidec0 2ε>/tildewidec0 4ε. This completes the proof of ( 136) and thereby proves Theorem 1.3. /square 7. Proof of Theorem 1.4.The lower bound in Theorem 1.4follows from Theorem 1.3. To prove t... | https://arxiv.org/abs/2502.17671v2 |
I: Piecewise constant functions. Journal of Machine Learning Research 6. [3] B INEV , P., C OHEN , A., D AHMEN , W. and D EVORE, R. (2007). Universal algorithms for learning theory. Part II: Piecewise polynomial functions. Constructive Approximation 26127–152. [4] B ONITO , A., D EVORE, R., P ETROVA , G. and S IEGEL , ... | https://arxiv.org/abs/2502.17671v2 |
ical Models . Cambridge Series in Statistical and Probabilistic Mathema tics. Cambridge University Press. [22] G UO, Y., W ENG, H. and M ALEKI , A. (2024). Signal-to-Noise Ratio Aware Minimaxity and Hig her-Order Asymptotics. IEEE Transactions on Information Theory 703538-3566. [23] G UO, Y., G HOSH , S., W ENG, H. and... | https://arxiv.org/abs/2502.17671v2 |
these equivalences can be chosen to depend only on the dimension of the space Prandq0and not on q,IorN, once Nd>ρ. The fact that the constants do not depend on Iis a simple matter of rescaling which we do not discuss further. So in proving the lemma, we can assu me thatI=Ω=(0 ,1)d. It is well known, see for instance (3... | https://arxiv.org/abs/2502.17671v2 |
Rd1 2σ2/parenleftbig /bardblx/bardbl2−/bardblx−y/bardbl2/parenrightbig e−/bardblx−y/bardbl2/2σ2dx =1 (2πσ2)m/2/integraldisplay Rd−ln/parenleftbiggexp(−/bardblx/bardbl2/2σ2) exp(−/bardblx−y/bardbl2/2σ2)/parenrightbigg e−/bardblx−y/bardbl2/2σ2dx. (167) 28 We divide the last integral into integrals over the sets BandBcand... | https://arxiv.org/abs/2502.17671v2 |
Learning Density Evolution from Snapshot Data Rentian Yao∗1, Atsushi Nitanda†2,3, Xiaohui Chen‡4, and Yun Yang§5 1Department of Mathematics, University of British Columbia 2CFAR and IHPC, Agency for Science, Technology and Research (A ⋆STAR) 3College of Computing and Data Science, Nanyang Technological University 4Depa... | https://arxiv.org/abs/2502.17738v1 |
, R∗ tmfrom their associated noisy temporal marginal snapshots bµtj=1 NPN i=1δXi tj:j∈[m] . For this purpose, there are two central questions: (i) Can we build a statistically efficient estimator with sample complexity that recovers certain conventional nonparametric density estimation approaches and meanwhile sheds l... | https://arxiv.org/abs/2502.17738v1 |
efficient solution to our current problem (Chizat et al., 2022; Yao et al., 2024a; Zhu and Chen, 2025). In this work, we design a new algorithm by fully harnessing the joint convexity of FN,min the linear structure (cf. ahead Definition 1). Our key idea is to combine the gradient descent in the linear geometry with res... | https://arxiv.org/abs/2502.17738v1 |
continuous-time density flow map t7→R∗ tin real-world applications such as single-cell data analysis (Klein et al., 2015; Macosko et al., 2015), our rate O(max{m−1 2, m−1 6N−1 3}) (up to poly-log factor) in the bottom row of Table 1 is particularly relevant in scenarios with limited sample availability, a common constr... | https://arxiv.org/abs/2502.17738v1 |
state space X) under the assumption that the process Zfollows a particular class of SDEs. More precisely in our setting, they considered the special case with σ= 0 (noiseless setting) and dZt=∇Ψ(t, Zt) dt+τdBt, (4) where Ψ : [0 , T]×X → Ris an unknown potential, Btis the standard reversible Brownian motion on X, and τ ... | https://arxiv.org/abs/2502.17738v1 |
approximate the Wasserstein gradient flow of FN,m(bRt1, . . . ,bRtm). Nonetheless, since the smooth functional is not geodesically convex in the Wasserstein space, simulated annealing on the sampling step size has to be incorporated to yield a notably slow logarithmic convergence rate of O(log log k logk) in the k-th i... | https://arxiv.org/abs/2502.17738v1 |
section, we will determine the statistical sample complexity of the E-NPMLE on both fixed design and the density flow map. We first briefly review the background of the entropic optimal transport problem. 2.1 Background: entropic optimal transport The entropic optimal transport (EOT) cost between two absolutely continu... | https://arxiv.org/abs/2502.17738v1 |
that τis known in our theoretical analysis. Remark 2 (Extreme case m= 1: connection with unregularized NPMLE) .When only m= 1snapshot is available, the problem reduces to estimating the marginal distribution of all samples at a single time point, which aligns with the definition of the (unregularized) NPMLE problem (Ki... | https://arxiv.org/abs/2502.17738v1 |
above inequality implies that Z1 0d2 H(Kσ∗R∗ t,Kσ∗bRt) dt≲max{δ2 N,m, m−1}≲maxn1 m,1 N2/3m1/3o logmd+1(11) holds with probability at least 1−2e−Nδ2 N,m 2∆m. Remark 4 (Time discretization error) .The upper bound (10) demonstrates the impact of estimation error and time discretization error when estimating the density ... | https://arxiv.org/abs/2502.17738v1 |
(continuous-time) KL divergence gradient flow to minimize a jointly linearly convex functional on the probability space—the functional value converges to its minimum at a polynomial rate. Proposition 4. For a multivariate functional F:Pr(X)⊗m→R, its KL divergence gradient flow ρ(t) = ρ1(t), . . . , ρ m(t) is defined ... | https://arxiv.org/abs/2502.17738v1 |
constants Lj≥0, then for any ρ∈Pr 2(X)⊗mwe have min 0≤k≤K−1F(ρk)− F(ρ)≤DKL(ρ∥ρ0) η1+···+ηK+PK k=1η2 kPm j=1L2 j 2(η1+···+ηK). (17) We remark that the first term in (17) arises from the inherent properties of using KL divergence gradient descent to minimize a convex functional, while the second term in (17) represents t... | https://arxiv.org/abs/2502.17738v1 |
from the distribution ρk j. Assuming ρ0 jis the uniform distribution over X, iterative application of the updating formula (19) yields ρk j(yj)∝exp −kX l=1h ηlY l<l′≤k(1−τηl′)i Vj yj;ρl−1 . (21) To sample from such a distribution, the unadjusted Langevin algorithm (ULA, Dalalyan, 2017; Wibisono, 2018) is a popular ... | https://arxiv.org/abs/2502.17738v1 |
size of CKLGD {ηk}∞ k=1and the coefficients of the extra quadratic terms {αk}∞ k=1are positive and satisfy 11 Algorithm 2 Inexact CKLGD for minimizing the reduced objective functional FN,m Require: observations {Xi tj:i∈[N], j∈[m]}; number of particles B; number of iterations K; number of iterations for sampling {nk:k∈... | https://arxiv.org/abs/2502.17738v1 |
and its proof). Remark 11 (Uniformly bounded first variation) .Due to the presence of the negative self-entropy term in the reduced objective functional FN,m, its first variationδFN,m δρjcannot be uniformly bounded. Therefore, Theorem 5 for optimizing a generic jointly linearly convex functional is not directly applica... | https://arxiv.org/abs/2502.17738v1 |
the SDE. Figure 3: Reduced objective functional value FN,m(ρ)− FN,m(bρ) in the log scale versus the total number of iterations. The experiment is conducted five times independently with different observations and initializa- tions. The MFLD algorithm exhibits a slower decay rate of reduced objective functional values (... | https://arxiv.org/abs/2502.17738v1 |
rigorously summarized by a uniform laws of large number in the following lemma. A proof of this lemma is deferred to Appendix B.3. We highlight that the proof is highly nontrivial, with additional discussion provided at the end of this subsection. 15 Lemma 7. LetCHP:= 12 + 34 .5 logC2 σ+1 2and define the event A:= sup... | https://arxiv.org/abs/2502.17738v1 |
the local sub-Gaussian behavior of sub-exponential random variables, leading to a sharper convergence rate. Lastly, in order to derive the phase transition phenomenon in Theorem 1, a careful estimation of the covering number for the involved function class is required. For our specific context, where the function R dep... | https://arxiv.org/abs/2502.17738v1 |
B1, . . . , B m≥0 are some universal constants defined in Lemma 16. We refer to step 1 in the proof provided in Appendix C.2 for more details. Control the optimization error. We can directly apply the convexity of FN,mto derive FN,m(eρk)− FN,m(ρ)≤mX j=1Z Vj(yj;eρk) +τlogeρk j(yj) d[eρk j−ρj]. By adopting a stability ar... | https://arxiv.org/abs/2502.17738v1 |
Both and Remy Kusters. Temporal normalizing flows. arXiv preprint arXiv:1912.09092 , 2019. Jonah Botvinick-Greenhouse, Yunan Yang, and Romit Maulik. Generative modeling of time-dependent densities via optimal transport and projection pursuit. Chaos: An Interdisciplinary Journal of Nonlinear Science , 33(10), 2023. Nawa... | https://arxiv.org/abs/2502.17738v1 |
Adrian Veres, Victor Li, Leonid Peshkin, David A Weitz, and Marc W Kirschner. Droplet barcoding for single-cell transcriptomics applied to embryonic stem cells. Cell, 161(5):1187–1201, 2015. Roger Koenker and Ivan Mizera. Convex optimization, shape constraints, compound decisions, and empirical Bayes rules. Journal of ... | https://arxiv.org/abs/2502.17738v1 |
Sujayam Saha and Adityanand Guntuboyina. On the nonparametric maximum likelihood estimator for Gaussian location mixture densities with application to Gaussian denoising. The Annals of Statistics , 48 (2):738–762, 2020. Geoffrey Schiebinger, Jian Shu, Marcin Tabaka, Brian Cleary, Vidya Subramanian, Aryeh Solomon, Joshu... | https://arxiv.org/abs/2502.17738v1 |
Zhu and Xiaohui Chen. Convergence analysis of the wasserstein proximal algorithm beyond geodesic convexity, January 2025. URL https://arxiv.org/abs/2501.14993 . 22 Supplementary Materials: Appendix This appendix provides technical details of the theoretical results presented in the main paper. The appendix is structure... | https://arxiv.org/abs/2502.17738v1 |
2013; van Handel, 2014; Vershynin, 2018; Wainwright, 2019). The following definition of Orlicz norm characterizes the tail of a random variable. Generally, the sample mean of a group of i.i.d. random variables with finite Orlicz norm is closed to the population mean. 23 Definition 3 (Orlicz norm) .Forα≥1, define the fu... | https://arxiv.org/abs/2502.17738v1 |
≤ −mX j=1(tj+1−tj)d2 H Kσ∗R∗ tj Kσ∗R∗ tj+Kσ∗Rtj 2 =−∥gR−gR∗∥2 L2m. Recall that we have the modified basic inequality −mX j=1tj+1−tj NNX i=1loggbR(tj, Xi tj)≤λτ 4 DKL(R∗∥Wτ)−DKL(bR∥Wτ) . Case 2.1: τDKL(bR∥Wτ)≤2τDKL(R∗∥Wτ).In this case, we have −mX j=1tj+1−tj NNX i=1loggbR(tj, Xi tj)≤λτ 4DKL(bR∗∥Wτ). Therefore, we ha... | https://arxiv.org/abs/2502.17738v1 |
have logN η, KσR·(·) :R∈P(Ω), τD KL(R∥Wτ)≤2E ,∥ · ∥L∞(K×X) ≲min η−2,|K| · log1 ηd+1 . Proof. Step 1: construction of projection map. LetM∈Z+be an integer to be decided later, and T1< ···< T NIbe a rI-covering of K⊂I= [0,1] (not necessarily in K). For any j∈[NI], define the map Ij M:P(Ω)→R(2M+1)dby Ij M(R) :=Z Td... | https://arxiv.org/abs/2502.17738v1 |
k∈Zde−∥x−y−2πk∥2 2σ2 RTj(y)−Rrec j(y) Vol(d y) ≤ Z TdX k∈Zde−∥x−y−2πk∥2 2σ2RTj(y) Vol(d y)−lX s=1ws jX k∈Zde−∥x−xj,s−2πk∥2 2σ2 + lX s=1ws jX k∈Zde−∥x−xj,s−2πk∥2 2σ2 −lX s=1ws jX k∈Zde−∥x−vj,s−2πk∥2 2σ2 + lX s=1ws jX k∈Zde−∥x−vj,s−2πk∥2 2σ2 −lX s=1βj,sX k∈Zde−∥x−vj,s−2πk∥2 2σ2 =:J1+J2+J3. To control J3, note that J3≤l... | https://arxiv.org/abs/2502.17738v1 |
t∈[0,1],x∈XKσ∗Rt(x)< C σ for every R∈P(Ω). So, we have sup i∈[N],j∈[m],R∈GR(r) tj+1−tj N loggR(tj, Xi tj)−EloggR(tj, Xi tj) ≤2∥∆m∥ Nsup t∈[0,1],x∈X|loggR(t, x)|=∥∆m∥ Nlog sup t∈[0,1],x∈XKσ∗Rt(x) 2Kσ∗R∗ t(x)+1 2 ≤∥∆m∥ NlogC2 σ+ 1 2. Then by Talagrand’s inequality (Theorem 3, Massart, 2000), we have P SN,m(r)≥2ESN,m... | https://arxiv.org/abs/2502.17738v1 |
letGk⊂ GR(r) such that its cardinality |Gk|=N2(2−k,GR(r)); we specify G0={R∗}, which is possible when τDKL(R∗∥Wτ)≤u. Furthermore, define πk(R)∈Gksuch that ∥gR−gπk(R)∥L2m≤2−krand∥gR−gπk(R)∥L∞m≤2−k·√ 2Cσ. Now, for a fixed K∈Z+to be decided later and R∈ GR(r), define RK=πK(R), and Rk−1=πk−1(Rk) for k=K, K−1, . . . , 0. Th... | https://arxiv.org/abs/2502.17738v1 |
To bound the second term in (B.8), we have Z1 2 0r min 2(sr)−2, m · log2 srd+1 ds =Z1 r√ 2 m 0√m log2 srd+1 2ds+√ 2Z1 2 1 r√ 2 m1 sr· log2 srd+1 2ds =:J21+J22. 35 To estimate J21, using the change of variable formula with s=1 vrq 2 myields J21=√ 2 rZ∞ 1 log√ 2m+ log vd+1 2v−2dv≲(logm)d+1 2 r. ForJ22, the integ... | https://arxiv.org/abs/2502.17738v1 |
mandNis large enough. Combining all the above pieces yields P(Ac)≤2e−Nδ2 N,m 2∆m. 37 B.4 Proof of Theorem 3 Note that mX j=1(tj+1−tj)d2 H Kσ∗bRtj,Kσ∗R∗ tj −Z1 0d2 H Kσ∗bRt,Kσ∗R∗ t dt ≤Zt1 0d2 H Kσ∗bRt,Kσ∗R∗ t dt+mX j=1Ztj+1 tj d2 H Kσ∗bRt,Kσ∗R∗ t dt−d2 H Kσ∗bRtj,Kσ∗R∗ tj dt. The first term is upper bounded by... | https://arxiv.org/abs/2502.17738v1 |
associated with νs= (νs,1, . . . , ν s,m). By Lemma 16 and Pinsker’s inequality, we have rX s=0mX j=1Z XVj(yj;νs) d[νs,j−νs+1,j]≤rX s=0mX j=1Bj∥νs,j−νs+1,j∥L1(X)≤rX s=0mX j=1Bjq 2DKL(νs+1,j∥νs,j). Note that we also have rX s=0Z Xlogνs,j(yj) d[νs,j−νs+1,j] =H(ν0,j)−H(νr+1,j) +rX s=0DKL(νs+1,j∥νs,j), where H(νj) =R νjlog... | https://arxiv.org/abs/2502.17738v1 |
FN,m(bρk)− FN,m(ρ) ≤KX k=1τηk+1 H(bρk)−H(eρk+1) −KX k=1(1−τηk+1)DKL(eρk+1∥eρk) +KX k=1mX j=1ηk+1Z XVj(yj;bρk) d[eρk j−eρk+1 j]−KX k=1αkηk+1Z ∥y∥2deρk+1 +K+1X k=1αkηkZ ∥y∥2dρ+KX k=1ηk+1R1(k) + 2∥B∥ℓ2(m)KX k=1ηk+1δ1 2 k −U∗ 1+H(ρ) +mX j=1η1Z XVj(yj;bρ0) dρj. To control the first term, we have the following lemma. The ... | https://arxiv.org/abs/2502.17738v1 |
that H(eρk)≥C3logαk. Lemma 19. Assume that {ηk}∞ k=1and{αk}∞ k=1are two positive sequences that satisfy •limk→∞ηk= 0andP kηk=∞; •limk→∞αk−1−αk ηkαk= 0; •{αkeτ(η1+···+ηk)}∞ k=1is increasing when kis large enough and converge to ∞, Then, there are constants C4, C′ 4>0such that C′ 4αk τ<kX l=1h αlηlY l<l′≤k(1−τηl′)i ≥Pk l... | https://arxiv.org/abs/2502.17738v1 |
q) +2 Vol(X) Cε2. Proof. Letµs=p+s(p′−p) and νs=q+s(q′−q). Then, we have d2 H(p, q)−d2 H(p′, q′) = 2Zp p′q′−√pqdx= 2Z h1(x)−h0(x) dx, where we define hs(x) =p µs(x)νs(x) for simplicity. Note that hsis smooth with respect to s∈[0,1]. By mean value theorem, there exists ξxbetween h1(x) and h0(x), such that h1(x)−h0(x) =∂... | https://arxiv.org/abs/2502.17738v1 |
that|∂2 tfs1+ξx,s1(x)| ≤C. Therefore, we have DKL(ρs2∥ρs1)≤Zs2−s1 0Z X˙ρs1+ [6 ˙q2 t−2q2 t]t+C 2t2dxdt =Zs2−s1 04t+CVol(X) 2t2dt = 2(s2−s1)2+CVol(X)(s2−s1)3. Here, the constant Cmay change from lines to lines. Thus, we have shown (D.2). Now, let us take si=it∗/(r+ 1) for i= 0,1, . . . , r + 1, and we have inf r,µ0,...,... | https://arxiv.org/abs/2502.17738v1 |
εk+ε−1 k)e4∥B∥ℓ∞(m) τ δk+√2δk∥B∥ℓ2(m) τ+εk(1 +εk)d 2mX j=1e2Bj τi . D.5 Proof of Lemma 13 To control J1, note that J1= Vol( Td)· Z TdX k∈Zde−∥x−y−2πk∥2 2σ2 d RTj(y)−Rrec j(y) = Vol( Td)· X k∈Zde−2π2∥k∥2 σ2Z Tde− ∥x−y∥2 2σ2−2π σ2k⊤(x−y) d RTj(y)−Rrec j(y) ≤Vol(Td)· X k∈Zde−2π2∥k∥2 σ2Z Tdh e− ∥x−y∥2 2σ2−2π σ2k⊤(x... | https://arxiv.org/abs/2502.17738v1 |
A Unified Framework for Semiparametrically Efficient Semi-Supervised Learning Zichun Xu∗, Daniela Witten†∗, Ali Shojaie∗† March 20, 2025 Abstract We consider statistical inference under a semi-supervised setting where we have access to both a labeled dataset consisting of pairs {Xi, Yi}n i=1and an unlabeled dataset {Xi... | https://arxiv.org/abs/2502.17741v2 |
three settings We now introduce three settings that will be used throughout this paper. 1. In the supervised setting , only labeled data Lnare available. Supervised estimators can be written as ˆθn(Ln). 2. In the ordinary semi-supervised (OSS) setting , both labeled data Lnand unlabeled data UNare available. OSS estima... | https://arxiv.org/abs/2502.17741v2 |
extended this result to the high-dimensional setting and developed an approach for bias-corrected in- ference. Both Chakrabortty and Cai [2018] and Azriel et al. [2022] considered semi-supervised linear 2 regression, and proposed asymptotically normal estimators with improved efficiency over the super- vised ordinary l... | https://arxiv.org/abs/2502.17741v2 |
findings in the context of specific examples: M- estimation, U-statistics, and the estimation of average treatment effect. Numerical experiments and concluding remarks are in Sections 8 and 9, respectively. Proofs of theoretical results can be found in the Supplement. 2 Overview of semiparametric efficiency theory We p... | https://arxiv.org/abs/2502.17741v2 |
space leads to an easier inferential problem, and thus a lower efficiency bound. We let ϕ∗ η∗(x) :=E φ∗ η∗(Z)|X=x (3) denote the conditional efficient influence function. Our first result establishes the semiparametric efficiency bound under the ISS setting. Theorem 3.1. Suppose the efficient influence function of θ(... | https://arxiv.org/abs/2502.17741v2 |
supervised estimator. Under minimal assumptions, the safe estimator is always at least as efficient as the initial supervised estimator. By contrast, under a stronger set of assumptions, the efficient estimator achieves the efficiency lower bound (5) under the ISS setting. We first provide some intuition behind the two... | https://arxiv.org/abs/2502.17741v2 |
→ Rdbe a square-integrable function such that Eh ∥g(X)∥2i <∞andVar [g(X)]is non-singular, and let g0(x)be its centered version (10). 6 Then, for Bgin(13),ˆθsafe n,P∗ Xdefined in (12) is a regular and asymptotically linear estimator of θ∗with influence function φη∗(z)−Bgg0(x), and asymptotic variance Var φη∗(Z)−Bgg0(X)... | https://arxiv.org/abs/2502.17741v2 |
g Kn(x)⊤⊤satisfies (16) and inf Kn{λmin(Var [ GKn(X)])}>0. LetG0 Kn(x)be the centered version of GKn(x)as in (17). Ifα >dim(X),Kn→ ∞ ,Knρ(˜ηn, η∗)→0, andζ2 n n→0, then ˆθeff. n,P∗ Xin(19) is a regular and asymptotically linear estimator of θ∗with influence function φη∗(z)−ϕη∗(x) under the ISS setting. Moreover, its as... | https://arxiv.org/abs/2502.17741v2 |
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