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an antisymmetric function F: Ω2→Rsuch that f(X) =E[F(X, X′)|X], and g(x):=1 2E[|f(X)−f(X′)||F(X, X′)||X=x]≤V (3) for all x∈Ω. Then for all λ≥0we have P[f(X)≥λ]≤e−λ2/2V. The proof of the above theorem, which we adapt in the following section, uses the differential inequality m′(θ)≤V θ·m(θ) to bound the moment generating... | https://arxiv.org/abs/2503.07571v1 |
the function F(x, x′) =T−1X t=0E[f(Xt)−f(X′ t)|X0=x, X 0=x′], forTchosen appropriately. Now, we no longer have E[F(X, X′)|X=x] =f(x), but instead we have E[F(X, X′)|X=x] =f(x)−Pf(x) +Pf(x)−P2f(x) +··· +PT−1f(x)−PTf(x) =f(x)−PTf(x). If the chain mixes rapidly within the metastable well starting from x, and fhas expectat... | https://arxiv.org/abs/2503.07571v1 |
E[F(X, X′)|X]eθf(X) X∈Λ, X′/∈Λi ≤(M+δ)eθM, using condition (ii) again. We get the same bound for the term involving eθf(X′)using the tower property onX′instead, since we also have E[F(X, X′)|X′=x′]≤M+δfor any x∈Λ+by the same conditions as before (plus the antisymmetry of F), and so we can conclude that (6) = (8) ≤ε 1−ε... | https://arxiv.org/abs/2503.07571v1 |
of a subexponential form for initial values of λ, rather than subgaussian as above. Additionally, we present in Section 7 some alternative approaches to obtaining initially subgaussian concentration for observables with ∥v∥1= Θ( n2), one based on an argument in [Cha05] and another based on a separate theorem of [BBL22]... | https://arxiv.org/abs/2503.07571v1 |
in this case. Conditions (iv) and (v) are more involved. For (iv), as previously derived in Section 4.2, we have E[F(X, X′)|X=x] =f(x)−PTf(x), and so we need to show that PTf(x) is small for a suitable choice of T, when x∈Λ+. Since Eµ[f] = 0, we have |PTf(x)| ≤ ∥v∥1·dTV(δxPT, µ), where dTVdenotes the total variation di... | https://arxiv.org/abs/2503.07571v1 |
B□ η. We still use µto denote the ERGM measure conditioned on B□ η. At a high level, both the rapid mixing of the ERGM at high temperature and the metastable mixing at low temperature follow from the existence of large attracting sets within which contraction properties arise under the monotone coupling. To discuss the... | https://arxiv.org/abs/2503.07571v1 |
0∈Γεwith X0⪯X′ 0, since Γ εis an interval in the Hamming order. However, thisdoes not imply that the contraction holds for arbitrary X0, X′ 0∈Γε, as it is not a priori clear that the set Γ εis connected, with respect to the nearest-neighbor structure of the Hamming cube. This will be remedied in the next section, where... | https://arxiv.org/abs/2503.07571v1 |
applies, so we simply state the result here. Lemma 5.7. For all η >0, the following holds for all small enough δ >0. Let (Xt)be standard (uncondi- tioned) ERGM Glauber dynamics, but started at X0∼ G(n, p∗+δ). Then the entire trajectory up to time T stays within B□ η/2(p∗)with probability at least 1−Te−Ωδ(n). The above ... | https://arxiv.org/abs/2503.07571v1 |
here we return to using ( Xt) and ( X′ t) for the conditioned Glauber dynamics, i.e. the one which has stationary distribution µ(conditioned onB□ η). The reader should keep this in mind when cross-referencing the inputs from the previous subsection while reading the proof of the following proposition. Proposition 5.9 (... | https://arxiv.org/abs/2503.07571v1 |
us assume that Ξ⊆Γε∩B□ η/2from now on. LetX∼µandX′be obtained from Xby one step of the µ-Glauber dynamics. Then P[X′/∈Ξ|X∈Ξ] =P[X∈Ξ, X′/∈Ξ] P[X∈Ξ] ≤P[X′/∈Ξ] P[X∈Ξ] ≤e−Ω(n) 1−e−Ω(n) ≤e−Ω(n). Now, since p∗/∈ {0,1}as can be checked by examining the derivative of Lβ, ifε >0 is small enough, then for all x∈Γε∩B□ η/2and all ... | https://arxiv.org/abs/2503.07571v1 |
X′ 0=X′, using the uniform coupling between the Bernoulli random variables determining Xi(ei) and X′ i(ei). Additionaly, sample Zby setting Z(ei) = 1 with probability p∗+δ, using again the uniform coupling between the Bernoulli random variables. First, since φβis continuous and φβ(p∗) =p∗, there is some δ′>0 such that ... | https://arxiv.org/abs/2503.07571v1 |
t≤Twith probability at least 1 −e−Ω(n). LetBdenote the bad event that one of the above conditions fail, i.e. some XtorYtis not in ¯Λ for some t∈[0, T]. Then P[B]≤e−Ω(n). Now, if Xt, Yt∈¯Λ, since ¯Λ⊆Γε∩B□ η/2, we have contraction, although not necessarily in dH. Instead, we have contraction in d¯Λby a path coupling argu... | https://arxiv.org/abs/2503.07571v1 |
all of our applications, and it seems that the full power of Theorem 5.1, using Theorem 4.2 is necessary in general. Instead, in the present section, we give a simple argument to obtain exponential concentration at scale nfor large linear observables, using our original concentration result, Theorem 5.1. This works by ... | https://arxiv.org/abs/2503.07571v1 |
not hold for the measure conditioned on the ball B□ η. In the case where |Mβ|= 1, the conditioned measure is very close to the full measure by Theorem 3.2, and we can simply apply the FKG inequality of the full measure. However, when there is phase coexistence, for each p∗∈Uβthe measure conditioned on B□ ηmight be quit... | https://arxiv.org/abs/2503.07571v1 |
edge and e′is any edge sharing a vertex with e. The above implies that |Covµ[X(e), X(e′)]|≲1 n(18) whenever eande′share a vertex. Now consider the sum of alledge variables X(e1) +···+X(eN), where N= n 2 . Now using Proposition 5.14, we have Varµ"NX i=1X(ei)# ≲n2, and by expanding this as before we obtain Varµ"NX i=1X(... | https://arxiv.org/abs/2503.07571v1 |
it is conceptually simpler. Consider the sum S(X) =mX i=1X(ei 1)···X(ei j), where {ei ℓ: 1≤i≤m,1≤ℓ≤j}is some enumeration of a collection of pairwise vertex-disjoint edges. We can find exactly ⌊n 2⌋such edges, and so since jis fixed, we can ensure that the number m of summands in Sis≳n. Since S(X) isv-Lipschitz with ∥v∥... | https://arxiv.org/abs/2503.07571v1 |
for each fixed ithe edges ei 1, . . . , ei jspan a forest in Knof the same type as F, and all ei ℓwith ℓ∈R(across all i) are incident on a single vertex, with no other pairs of edges ei ℓandei′ ℓ′sharing a vertex if i̸=i′andℓorℓ′is not in R(see Figure 11, left). By the same analysis as in the previous case, we can obta... | https://arxiv.org/abs/2503.07571v1 |
πon{0,1}([n] 2)as follows: W1(µ, π):= inf X∼µ Y∼πX e∈([n] 2)P[X(e)̸=Y(e)], where the infimum is taken over all couplings of µandπ, instantiated by random variables X∼µandY∼π on the same probability space, whose joint distribution is being optimized over. The following theorem is the analog of the subcritical result [GN... | https://arxiv.org/abs/2503.07571v1 |
at least 1 −O(n−cα), we have |∂eH(X)−Eµ[∂eH(X)]| ≤αr logn n. (23) Now we need to estimate Eµ[∂eH(X)]. For this, we first expand the definition of NGi(X, e) to see that Eµ[NGi(X, e)] =X σ:Vi→[n] e={σu,σv}for some{u,v}∈EiEµ Y {u,v}∈Ei {σu,σv}̸=eX({σu, σv}) . (24) 37 In the case where |Uβ|= 1, we may invoke part (... | https://arxiv.org/abs/2503.07571v1 |
edges not sharing a vertex. Our proof also has the benefit of not relying on the FKG inequality. This result was informally stated in the introduction as Theorem 1.3. 38 Theorem 6.6. Letp∗∈Uβand let µdenote the ERGM measure conditioned on B□ η(p∗), for some small enough η >0. Suppose that we have a sequence of collecti... | https://arxiv.org/abs/2503.07571v1 |
it is a sum of at most 2kterms of the form Covµ" X(e),ℓY i=1X(eji)# . But all of these terms are ≲1 nin absolute value, by the proof of Proposition 6.2, part (b) (recall that the proof involved showing condition (20)). This finishes the proof. 39 Using Proposition 6.7 as well as Theorem 5.1, we can prove the following ... | https://arxiv.org/abs/2503.07571v1 |
. . , X (e′ l). This yields Eµh ˜X(e′ l+1) +···+˜X(e′ ⌊n/2⌋)m X(e′ 1), . . . , X (e′ l)i ≲nm/2, (30) where we replace the ⌊n/2⌋byn−1 in case (b). We can then expand the left-hand side into terms of the form Eµh ˜X(ej1)c1···˜X(ejL)cL X(e′ 1), . . . , X (e′ l)i , where c1≥ ··· ≥ cL≥1 and c1+···+cL=m, and consider the t... | https://arxiv.org/abs/2503.07571v1 |
o(1). Finally, by Proposition 6.2 and Theorem 3.2 again, we have Eµ k/2Y j=1˜X(ej)2 →(p∗(1−p∗))k/2, meaning that the limiting moments of Snare 0 when kis odd, and equal to k 2, . . . , 21 (k/2)!=k! 2k/2(k/2)!= (k−1)!! when kis even. These are the moments of N(0,1), and so we obtain Theorem 6.6. 7 Alternative appr... | https://arxiv.org/abs/2503.07571v1 |
in terms of the total edge count, since more local behavior is relevant for the ERGM Hamiltonian. So, while the above inequality provides subgaussian concentration for some modification of the edge count, it is not clear how to extract useful information about the edge count itself, and it seems that the inequality (32... | https://arxiv.org/abs/2503.07571v1 |
x′∈ΩP(x, x′)TX t=0(Ex[f(Xt)]−Ex′[f(X′ t)])2≲1 n2X eTX t=0E dH(Xt, X′ t) X0=x, X′ 0=x⊕e2. Here x⊕edenotes the configuration xbut with the state of the edge eswitched from 0 to 1 or vice versa. Now by a similar argument as before, since x∈Λ we have the same contraction of Hamming distance for all reasonable times Twhen... | https://arxiv.org/abs/2503.07571v1 |
where the same edge is chosen to be resampled in both graphs. Regardless of which edge was chosen, f(X′ 1) = 0 still, and if any edge other than e1is selected to be resampled, then f(X1) = 0 as well. However, if e1is selected and X1(e1) is updated to 1, then f(X1)−f(X′ 1) = 1−0 = 1. All this means that|E[f(X1)]−E[f(X′ ... | https://arxiv.org/abs/2503.07571v1 |
topological barrier to optimizing over random structures. Proceedings of the National Academy of Sciences , 118(41):e2108492118, 2021. [GN24] Shirshendu Ganguly and Kyeongsik Nam. Sub-critical exponential random graphs: concentra- tion of measure and some applications. Transactions of the American Mathematical Society ... | https://arxiv.org/abs/2503.07571v1 |
Personalized Convolutional Dictionary Learning of Physiological Time Series Axel Roques Samuel Gruffaz Kyurae Kim Centre Borelli, Laboratoire GBCM, Thales AVSCentre Borelli, ENS Paris-SaclayUniversity of Pennsylvania Alain O. Durmus Laurent Oudre CMAP, CNRS, Ecole polytechnique Centre Borelli, ENS Paris-Saclay Abstract... | https://arxiv.org/abs/2503.07687v1 |
individual-level structure, treating it as noise. Conversely, applying CDL to each individual independently (hereafter referred to as IndCDL for “individual-level CDL”) precludes the learning of thearXiv:2503.07687v1 [stat.ML] 10 Mar 2025 Personalized CDL of Physiological Time Series 1 21 2 Figure 1: Illustration of Pe... | https://arxiv.org/abs/2503.07687v1 |
(Section 2.2). •We present a meta-algorithm for solving the Per- CDL problem (Section 2.3). The algorithm is modular in design, offering the flexibility to com- bine various components from conventional CDL (Garcia-Cardona and Wohlberg, 2018; Truong and Moreau, 2024). A federated learning, fully paral- lelizable varian... | https://arxiv.org/abs/2503.07687v1 |
by run- ning CDL S-times (IndCDL) or, alternatively, learn a single dictionary that is shared across the whole pop- ulation (PopCDL). 2.2 Personalized Convolutional Dictionary Learning To accommodate inter-individual variability within the CDL framework, we propose to learn population- level atoms along with individual... | https://arxiv.org/abs/2503.07687v1 |
the infimum of the objective is substituted with its lower bound, 0 (Loizou et al., 2021). This can be run in parallel onSprocessors. When updating the dictionary given AandZ(PerCDU step), the transformation fmay introduce non-linearities that prohibit the use of con- ventional CDU update schemes, a problem shared with... | https://arxiv.org/abs/2503.07687v1 |
any trans- formation function appropriate for the task at hand can be used. This section will provide a family of transformation functions that are well suited for, but not exclusive to, human physiology data. Time Warping Transformation. Let us consider a continuous signal s: [0,1]→RP. For t∈[0,1], the “time-warped” v... | https://arxiv.org/abs/2503.07687v1 |
of D/W allow for greater flexibil- ity, while lower values increase the granularity of the transformation. As illustrated in Figure 1, this class of transformation is general enough to recover patholog- ical gait cycles from a healthy one. Additional exper- iments on the sensitivity of the transformations with respect ... | https://arxiv.org/abs/2503.07687v1 |
mixed-effects model if and only if Pis Gaussian (Nie, 2007). In H1,P needn’t be Gaussian, e.g., it can be a multimodal dis- tribution. Therefore, H1 can be seen as a refinement of Condition 2 by Nie (2007). Lemma 1. The assumption H1 is met in the two fol- lowing cases: •Time warping transformations: For any a∈Θ⊂ RM,L(... | https://arxiv.org/abs/2503.07687v1 |
barycenter of the individual atoms found in each signal. While we also considered a barycenter computed with the Soft- DTW metric, this led to worse results. Thus, we only consider the Euclidean barycenter to ensure a fair com- parison against PopCDL and PerCDL. The distance between the obtained common structures and t... | https://arxiv.org/abs/2503.07687v1 |
10 meters, while an inertial measurement unit measured their foot angular velocity in the sagittal plane. Par- ticipants were divided into three equal-sized groups of sizen= 50 (Healthy, Orthopedic, and Neurological) according to their health status. The common dictio- nary was initialized with the gait cycle extracted... | https://arxiv.org/abs/2503.07687v1 |
Convolutive Non- Negative Matrix Factorization (cNMF or cNMFsc, the latter imposing sparseness constraints; Smaragdis, 2006; O’Grady and Pearlmutter, 2008; Wang et al., 2009), which has been employed in speech separa- tion and the analysis of articulatory movement primi- tives during speech production (Ramanarayanan et... | https://arxiv.org/abs/2503.07687v1 |
work introduced Personalized Convolutional Dic- tionary Learning (PerCDL), a framework for learning interpretable representations of time-series datasets that captures both global structures and local vari- ations around these commonalities. A meta-algorithm was proposed and its theoretical performance was eval- uated.... | https://arxiv.org/abs/2503.07687v1 |
with inertial measurement units. Sensors , 18(11):4033, 2018. Tristan Dot, Flavien Quijoux, Laurent Oudre, Ali´ enor Vienne-Jumeau, Albane Moreau, Pierre-Paul Vidal, and Damien Ricard. Non-linear template-based ap- proach for the study of locomotion. Sensors , 20(7): 1939, 2020. Cyril Voisard, Nicolas De l’Escalopier, ... | https://arxiv.org/abs/2503.07687v1 |
Yann Cun, et al. Learning convolutional feature hier- archies for visual recognition. Advances in neural information processing systems , 23, 2010. Stephen Boyd, Neal Parikh, Eric Chu, Borja Peleato, Jonathan Eckstein, et al. Distributed optimization and statistical learning via the alternating direction method of mult... | https://arxiv.org/abs/2503.07687v1 |
non-negative matrix factorisation with a sparseness constraint. Neurocomputing , 72(1-3):88–101, 2008. Wenwu Wang, Andrzej Cichocki, and Jonathon A Chambers. A multiplicative algorithm for convo- lutive non-negative matrix factorization based on squared euclidean distance. IEEE Transactions on Signal Processing , 57(7)... | https://arxiv.org/abs/2503.07687v1 |
and algorithms presented, check if you include: (a) A clear description of the mathematical set- ting, assumptions, algorithm, and/or model. Yes. A clear description of the mathemat- ical setting, assumptions and algorithm are presented throughout Section 2. (b) An analysis of the properties and complexity (time, space... | https://arxiv.org/abs/2503.07687v1 |
a URL, if applicable. Yes. The authors code for the methods discussed in the paper will be available via a URL. (d) Information about consent from data providers/curators. Not Applicable . (e) Discussion of sensible content if applicable, e.g., personally identifiable information or of- fensive content. Not Applicable ... | https://arxiv.org/abs/2503.07687v1 |
(T⊤ ψa,σTψa,σx)⊤x =E (Tψa,σx)⊤Tψa,σx . (13) We have E (Tψa,σx)⊤Tψa,σx =PL i=1E [Tψa,σx]2 i . For any i∈[L], denoting by Ul=xl+ (Lψa(i/L)− l)(xl+1−xl), E [Tψa,σx]2 i =E LX l=1wl(ψa, σ, i)Ul!2 (14) =E LX l=1LX m=1wl(ψa, σ, i)wm(ψa, σ, i)UlUm! (15) =LX l=1LX m=1E(wl(ψa, σ, i)wm(ψa, σ, i)UlUm)| {z } =∆k,m,i.... | https://arxiv.org/abs/2503.07687v1 |
such that for any S≥˜S,ΣS,p=PS s=1psL(as ∗)⊤L(as ∗)is invertible. Then, for any S≥˜S, the MLE of the common atom ϕ∗is ˆϕmle= (Σ S,p)−1YΣ, Y Σ=SX s=1psX j=1L(as ∗)⊤[j]ys[j], (28) and thus we have ˆϕmle∼ N ϕ∗,(ΣS,p)−1 . Moreover, ˆϕmleconverges toward ϕ∗at a rate of O 1/ρqPS s=1ps in probability. Proof. First, by (Ja... | https://arxiv.org/abs/2503.07687v1 |
on the Local Servers by a gradient descent on both the individual parameters asand the personalized dictionaries ˜ϕs kat the same time. The common dictionary can then be updated as ϕk←PS s=1˜ϕs k/Sfor any k∈[K]. In order to avoid over-adaptation of the personalized dictionaries, the individual parameters should be init... | https://arxiv.org/abs/2503.07687v1 |
details the actual computation times observed on a personal laptops (MacBook Pro, M3 Max chip) for a subset of the experiments presented in this section. 1. Synthetic dataset ( K= 2,N= 500, nsteps = 5): (a)S= 2; IndCDL: 0 .29s; PopCDL: 0 .17s; PerCDL: 5 .86s. (b)S= 16: IndCDL: 2 .20s; PopCDL: 0 .32s; PerCDL: 7 .33s. (c... | https://arxiv.org/abs/2503.07687v1 |
10. Optimization parameters: nsteps = 250, step size of 10−3. Finding the Personalization Parameters. In this section, the IPU step of PerCDL was evaluated. PerCDL was initialized with the true activations and common atoms, which remained fixed for the duration of the experiment. Figure 8 presents the results of the op... | https://arxiv.org/abs/2503.07687v1 |
Gaussian noise. Figure 11(A) presents the evolution of the distance to the personalized atoms as a function of the signal to Gaussian noise ratio (SNR). Two distance metrics were tested: the Euclidean distance and the DTW. Only the latter was included to ensure a fair comparison between methods despite potential discre... | https://arxiv.org/abs/2503.07687v1 |
margin set relatively high ( τtol= 100 samples or roughly one second) as the precise beginning of a gait cycle is an ambiguous concept. The number of true positives, i.e., the number of correctly identified gait cycles, the number of false negatives, i.e., the number of gait cycles that were undetected, and the number ... | https://arxiv.org/abs/2503.07687v1 |
a similar analysis as the one presented in Section 3.2 with IndCDL. Initialization was identical to that of PerCDL. Figure 13 presents the group variability analysis of the atoms identified. A striking result is the presence of “parasitic variability” at the start and end of each atom. This variability is uninformative... | https://arxiv.org/abs/2503.07687v1 |
Atoms . The personalized atom corresponds to the personalized atom learned by PerCDL for the given subject. The cohort personalization mean corresponds to the atom obtained after averaging all personalized atoms for the given pathology group. The distance between the true cycles and the personalized atom is smaller tha... | https://arxiv.org/abs/2503.07687v1 |
meaningful information from the dataset. To that end, we tested whether each signal could be classified into its corresponding pathology superclass (5-class classification task) using only the personalization matrix. The parameters matrix was separated into train and test sets (70% −30% split). A simple SVM ( scikit-le... | https://arxiv.org/abs/2503.07687v1 |
The Hidden Toll of COVID-19 on Opioid Mortality in Georgia: A Bayesian Excess Opioid Mortality Analysis Cyen J Peterkina,∗, Lance A Wallera, Emily N Petersona aDepartment of Biostatistics and Bioinformatics, Emory Rollins School of Public Health, Atlanta, GA, USA Abstract COVID-19 has had a large scale negative impact ... | https://arxiv.org/abs/2503.07918v2 |
Konstantinoudis et al., 2023). Development of robust and granular methodologies to monitor small-area geographical-temporal trends in opioid mortal- ity during a crisis period has large-scale implications for informing public health policy, resource allocation, and public health response strategies (Jon Wakefield, 2007... | https://arxiv.org/abs/2503.07918v2 |
pre-pandemic, which is crucial for evaluating excess opioid mortality rates (Rossen et al., 2020; Wang et al., 2022). The model framework employs a spatio-temporal disease mapping approach, which is commonly used to investigate granular geographical-temporal variations of opioid mortality burden (Kline D and Hepler SA,... | https://arxiv.org/abs/2503.07918v2 |
of opioid-related deaths during pre-pandemic years. In 2020, the number of deaths increased with a significant spike in March. Between 2020 and 2022, the number of deaths continues to rise over time, not reverting to pre-COVID-19 levels. 6090120150180 0 20 40 60 MonthsTotal Opioid OverdosesYears 2018 2019 2020 2021 202... | https://arxiv.org/abs/2503.07918v2 |
B. Lawson, 2013; Knorr-Held and Julian Besag, 1998; Parker PA et al., 2024). Given the nature of our observed county-level opioid mortality count data, which contains an excess of zero values particularly for smaller counties, we model observed counts us- ing a Zero-Inflated Poisson (ZIP) model in Eq. 1, which accounts... | https://arxiv.org/abs/2503.07918v2 |
temporal trend in county iis similar to the average temporal trends of its neighbors (Knorr-Held, 2000). Using the complete process model, we obtain estimates of county-month specific log-transformed relative risks for years 2018-2022 including those months in which data has been excluded (during 5 COVID months), i.e.,... | https://arxiv.org/abs/2503.07918v2 |
a total 32,000 saved posterior samples. Standard diagnostic checks using traceplots were used to check for convergence (de Valpine et al., 2018; Gelman A. et al., 2013; Vehtari et al., 2017; Gabry et al., 2019). 5. Results 5.1. Simulation Results Predictive performance results illustrate the BOEM model can robustly cap... | https://arxiv.org/abs/2503.07918v2 |
areas. 8 DEKALB FULTONGWINNETT DEKALB FULTONGWINNETTDEKALB FULTONGWINNETT DEKALB FULTONGWINNETTDEKALB FULTONGWINNETT 2021 20222018 2019 2020 87°W86°W85°W84°W83°W82°W81°W87°W86°W85°W84°W83°W82°W81°W87°W86°W85°W84°W83°W82°W81°W31°N32°N33°N34°N35°N 31°N32°N33°N34°N35°N LongitudeLatitude −3−2−1012345678910Excess DeathsExce... | https://arxiv.org/abs/2503.07918v2 |
of -0.29 (95% CI: -0.57, 0) in July 2018. In Gwinnett County (Population: 927,337 - 975,353), the observed monthly opioid-related death rates per 100,000 range between 0 and 1.4 for years 2018-2019. The monthly opioid death rate increased to 1.84 in October 2022 showing a similar increasing trend in opioid deaths in ye... | https://arxiv.org/abs/2503.07918v2 |
from 3 in February to 10 in March, illustrating the sen- sitivity of small and moderate-sized counties to changes in monthly death counts. In Clayton County (Population: 289,197 - 297,623), the observed monthly opioid-related death rates range between 0 and 1.03 for 2018–2019. The observed monthly death rate increased ... | https://arxiv.org/abs/2503.07918v2 |
zeroes across the complete time series. In contrast to Toombs County, there are spikes in the observed death rate both pre and post the onset of COVID-19. The largest observed death rate 3.18 occurred in August 2018. The predicted opioid mortality rate ranged from 0.49 (95% CI: 0.26, 1.43) in February 2018 to 0.62 (95%... | https://arxiv.org/abs/2503.07918v2 |
and reactions to death counts, which vary based on the county’s population size. In large counties, findings showed that they experi- enced an increase in their excess death rates post-COVID-19. In moderate size counties, larger variability is observed in their excess death rates due to the impact of deaths on their sm... | https://arxiv.org/abs/2503.07918v2 |
D.B., Vehtari, A., Rubin, D.B., 2013. Bayesian Data Analysis. Third ed., Chapman & Hall/ CRC texts in statistical science. Georgia Department of Public Health, 2024. Drug surveillance. URL: https://dph.georgia.gov/ epidemiology/drug-surveillance. accessed: 2025-01-08. Georgia Department of Public Health (GADPH), 2021. ... | https://arxiv.org/abs/2503.07918v2 |
U., J ¨ockel, K.H., 2020. Excess mortality due to COVID-19 in Germany. Journal of Infection 81, 797– 801. URL: https://linkinghub.elsevier.com/retrieve/pii/S016344532030596X, doi: 10.1016/ j.jinf.2020.09.012 . Stokes, A.C., Lundberg, D.J., Elo, I.T., Hempstead, K., Bor, J., Preston, S.H., 2021. COVID-19 and excess mort... | https://arxiv.org/abs/2503.07918v2 |
representation of the BOEM hierarchical model. Shaded rectangles denote observed data quantities, and circles denote latent variables (shaded circles for global hyper-parameters). Solid arrows denote stochastic dependency. Boxes group quantities by indices, i.e., (1) Bottom box contains observed population data, strati... | https://arxiv.org/abs/2503.07918v2 |
number of excess opioid deaths defined as the difference between the ob- served number of deaths after the onset of Covid-19 yi,mand the true number of deaths generated under the assumed true data generating assumption without a change due to COVID-19 ˜ yi,m, i.e., ˜χi,m=yi,m−˜yi,mwhich captures the true number of exce... | https://arxiv.org/abs/2503.07918v2 |
arXiv:2503.08031v1 [cs.LG] 11 Mar 2025Empirical Error Estimates for Graph Sparsification Siyao Wang Miles E. Lopes University of California, Davis University of California, Davis Abstract Graph sparsification is a well-established technique for accelerating graph-based learn- ing algorithms, which uses edge sampling to a... | https://arxiv.org/abs/2503.08031v1 |
Making matters worse, such results typically involve unknown parameters or unspecified constants. Consequently, it can be infeasible to use theoretical error bounds in a way that is practical on a problem- specific basis. Based on the issues just discussed, we propose to Empirical Error Estimates for Graph Sparsification ... | https://arxiv.org/abs/2503.08031v1 |
Gand Las fixed unknown parameters. Error functionals. To measure how well ˆLap- proximates L, we will consider a variety of scalar- valued error functionals, denoted ψ(ˆL,L). For ex- ample,ψcould correspond to error in the Frobenius normψ(ˆL,L)=/parall⟩l.al⟪1ˆL−L/parall⟩l.al⟪1For operator (spectral) norm ψ(ˆL,L)=/parall... | https://arxiv.org/abs/2503.08031v1 |
we adapt resampling methods to our setting. In particular, for certain applications, we leverage a specialized type of resampling known as a “double bootstrap” (Chernick, 2011; Hall, 2013). In many classical statistical problems, it is known that a double bootstrap can substantially improve upon more basic bootstrap me... | https://arxiv.org/abs/2503.08031v1 |
terms of Q1,...,Q N, and hence, there are generally no explicit formulas for comput- ingϕ(Q1,...,Q N). Nevertheless, an approximation ˆ ϕ to the function ϕcan also be developed via bootstrap sampling, andapproximatesamplesof ζcanbedefined Empirical Error Estimates for Graph Sparsification asζ∗=ˆϕ(Q∗ 1,...,Q∗ N). From an ... | https://arxiv.org/abs/2503.08031v1 |
of “cut query” vectors C⊂{0,1}n, which is specific to the user’s task. Moreover, there is a well-established line of research on using edge sampling to efficiently approximate the cut val- ues of large or dense graphs (Bencz´ ur and Karger, 1996, 2015; Andoni et al., 2016; Arora and Upadhyay, 2019). Hence, this amounts to... | https://arxiv.org/abs/2503.08031v1 |
clustering (von Luxburg, 2007), which uses Laplacian eigenvectors to construct low-dimensional representations of data that allow clusters to be distinguished more effectively. Because the Laplacians in spectral clustering tend to be dense, sparsification has been advocated as a way to im- prove computational efficiency (C... | https://arxiv.org/abs/2503.08031v1 |
al., 2024).) Meanwhile, it is crucial to recognize that Algorithms 1 and 2do not require any additional access to G, since they only rely on the samples used to produce ˆL. Hence, when the communication cost to access Gis high, it is less likely that error estimation will be a bottleneck. High parallelism. Another fact... | https://arxiv.org/abs/2503.08031v1 |
words, the error estimation is accelerated because it is faster to run Algorithms 1 and 2 when there are N0 sampled edges, rather than N1. This process of “fore- casting”N1is based on an easily implemented type of extrapolation that is well established in the bootstrap literature (Bickel and Yahav, 1988), and is detail... | https://arxiv.org/abs/2503.08031v1 |
will also show empirically that Algorithm 2 can work well for natural graphs when all cuts in Cnare drawn uni- formly at random and /divid⟩s.al⟪0Cn/divid⟩s.al⟪0=n. Error estimates for the Frobenius norm. Our next result provides a guarantee on the performance of Algorithm 1 when ψ(ˆLn,Ln)=/parall⟩l.al⟪1ˆLn−Ln/parall⟩l.... | https://arxiv.org/abs/2503.08031v1 |
93.7 ER 89.2 94.0 90.9 95.2 92.5 95.9 88.4 93.5 AER 89.6 94.0 89.0 94.2 92.0 95.3 89.4 94.2 From each of the graphs mentioned above, we con- structed a corresponding graph Gby randomly sam- plingn=2,000 vertices according to their degrees (without replacement) and retaining all edges among the sampled vertices. The res... | https://arxiv.org/abs/2503.08031v1 |
the previous experiments were lim- ited by a number of factors, such as the need to perform thousands of Monte Carlo trials, and com- pute ground truth errors involving unsparsified Lapla- cians. To assess the computational efficiency of er- ror estimation, it is of interest to consider a run- time experiment involving a ... | https://arxiv.org/abs/2503.08031v1 |
the inner loop still being run sequentially, the error estimation process would only increase the runtime of the workflow by at most(7/slash.l⟩f⟪8)/slash.l⟩f⟪25=3.5%. Code. The code for Algorithms 1 and 2 is available at the repository https://github.com/sy-wwww/Error-Estimates-Graph-Sp arsifica 5 CONCLUSION Due to the ... | https://arxiv.org/abs/2503.08031v1 |
In 23rd International Conference on Pattern Recognition , pages 2301–2306. Chen, J.,Sun, H., Woodruff, D., andZhang,Q.(2016). Communication-optimal distributed clustering. Ad- vances in Neural Information Processing Systems . Chen, L., Kyng, R., Liu, Y. P., Peng, R., Gutenberg, M. P., and Sachdeva, S. (2022). Maximum flo... | https://arxiv.org/abs/2503.08031v1 |
matrices and sketching. Bernoulli , 29(1):428–450. Lopes, M. E., Wang, S., and Mahoney, M. W. (2019). A bootstrap method for error estimation in ran- domized matrix multiplication. Journal of Machine Learning Research , 20(39):1–40. Lopes, M. E., Wu, S., and Lee, T. C. (2020b). Mea- suring the algorithmic convergence o... | https://arxiv.org/abs/2503.08031v1 |
Error Estimates for Graph Sparsification Zhang, Z., Lee, S., and Dobriban, E. (2023). A frame- work for statistical inference via randomized algo- rithms.arXiv:2307.11255 . Zheng, C., Zong, B., Cheng, W., Song, D., Ni, J., Yu, W., Chen, H., and Wang, W. (2020). Ro- bust graph representation learning via neural spar- sifi... | https://arxiv.org/abs/2503.08031v1 |
of ˆLusing both ER and AER sampling, and employed the method outlined in Section 2.2 to construct simultaneous CIs forλ1(L)≤⋯≤λ15(L). Table 2 shows the observed simultaneous coverage probabilities P(⋂15 j=1{λj(L)∈ˆIj}) based on desired confidence levels of 90% and 95%, which were compu ted by averaging over the 1000 tri... | https://arxiv.org/abs/2503.08031v1 |
entries of ˆLshould have a 1/slash.l⟩f⟪√ Nscaling with respect to N. We refer to (Bickel and Yahav, 1988) for further background on the use o f extrapolation rules to reduce the cost of bootstrapping. In all cases, the average of ˆ q1−α(N)over all 1000 trials is plotted in Figures 2-6 as a function of Nusing a solid li... | https://arxiv.org/abs/2503.08031v1 |
( Citations )(Rossi and Ahmed, 2015): This graph represents the co-citatio n of scientific papers from arXiv’s high energy physics-theory (HEP-TH) section, involvin g 22,908 vertices and 2,444,798 edges. An edge between two papers means that both papers have been cited by a common third paper (Kunegis, 2013). •C2000-9 (... | https://arxiv.org/abs/2503.08031v1 |
laptop with approximately 16 GB of RAM, 8 physical co res, and 16 logical cores. D Notation and conventions in proofs TheLqnorm of a scalar random variable Vis denoted as/parall⟩l.al⟪1V/parall⟩l.al⟪1Lq=(E(/divid⟩s.al⟪0V/divid⟩s.al⟪0q))1/slash.leftq. For any random object V, we useL(V)to refer to its distribution, while... | https://arxiv.org/abs/2503.08031v1 |
vector drawn from N(0,R), whereRij=sij√siisjj, and define M(G)=max 1≤i≤/divides.alt0C/divides.alt0/divid⟩s.al⟪0Gi/divid⟩s.al⟪0. Also, let ˆG=(ˆG1,...ˆG/divides.alt0C/divides.alt0)be a random vector that is drawn N(0,ˆR)conditionally on Q1,...,Q N, where ˆRij=ˆsij/radical.alt1 ˆsiiˆsjj, and define M(ˆG)=max 1≤i≤/divides.a... | https://arxiv.org/abs/2503.08031v1 |
in Lemma G.2 imply dK/par⟩nl⟩f⟪.al⟪2L(M(G)),L(M(ˆG)/divid⟩s.al⟪0Q)/par⟩nrigh⟪.al⟪1≲/par⟩nl⟩f⟪.al⟪1/parall⟩l.al⟪1ˆR−R/parall⟩l.al⟪1∞log(/divid⟩s.al⟪0C/divid⟩s.al⟪0)2/par⟩nrigh⟪.al⟪21/slash.left2 =oP(1). Tohandlethe firsttermontherightsideof (13), wewillfollowthe arg umentusedinderiving (8). Thecalculation in (7) yields E... | https://arxiv.org/abs/2503.08031v1 |
that the limit dK/par⟩nl⟩f⟪.al⟪1L(T∗/divid⟩s.al⟪0Q),L(Z)/par⟩nrigh⟪.al⟪1/leftrightline →0 holds almost surely as n→∞alongn∈J′. The key ingredient for doing this is to show that if ˆd∈Rncontains the diagonal entries of ˆL, then/parall⟩l.al⟪1ˆd/parall⟩l.al⟪1∞/slash.l⟩f⟪/parall⟩l.al⟪1ˆd/parall⟩l.al⟪12=oP(1)holds asn→∞, wh... | https://arxiv.org/abs/2503.08031v1 |
2−L)ii+2/summation.disp 1≤i<j≤n(D1D⊺ 1−L)ij(D2D⊺ 2−L)ij. Define the collection of ordered pairs J={(i,i′)/divid⟩s.al⟪01≤i≤i′≤n}. For each i∈J, define ϕi(ˆe1)=1{i=i′∈ˆe1}−√ 2⋅1{i<i′,ˆe1={i,i′}} φi(ˆe1)=ϕi(ˆe1)−E(ϕi(ˆe1)).(23) Empirical Error Estimates for Graph Sparsification It can be checked that E(ϕi(ˆe1))=Lii′/par⟩nl⟩f... | https://arxiv.org/abs/2503.08031v1 |
conditions in Theorem 2 hold, then E(ϕi(ˆe1)ϕj(ˆe1))=⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩Liii=i′=j=j′ −Lij′i=i′≠j=j′ −2Lij′i=j<i′=j′ √ 2Li′j′i=i′=j<j′,j=j′=i<i′ √ 2Liji=i′=j′>j,j=j′=i′>i 0 otherwise. Proof.Observe that ϕi(ˆe1)ϕj(ˆe1)=1{i=i′∈ˆe1,j=j′∈ˆe1}+2⋅1{i<i′,ˆe1={i,i′},j<j′,ˆe1={j,j′}} −√ 2⋅1{i=i′∈ˆe1,j<j′,ˆe1={j,j′}}−√ ... | https://arxiv.org/abs/2503.08031v1 |
Using the bound var (X+Y)≤2var(X)+2var(Y)for generic random variables XandY, we have var(ˆσ2)≲var/par⟩nl⟩f⟪.al⟪41 BB /summation.disp b=1/par⟩nl⟩f⟪.al⟪1ξ∗ b/par⟩nrigh⟪.al⟪12/par⟩nrigh⟪.al⟪4+var/par⟩nl⟩f⟪.al⟪41 B2/summation.disp 1≤b≠b′≤Bξ∗ bξ∗ b′/par⟩nrigh⟪.al⟪4. Using the fact that ξ∗ 1,...,ξ∗ Bare conditionally i.i.d. ... | https://arxiv.org/abs/2503.08031v1 |
2⟩2⟨D∗ 1,D∗ 3⟩2/divid⟩s.al⟪0Q/par⟩nrigh⟪.al⟪1=1 N3N /summation.disp i,j,k=1/par⟩nl⟩f⟪.al⟪1D⊺ iDj/par⟩nrigh⟪.al⟪12/par⟩nl⟩f⟪.al⟪1D⊺ iDk/par⟩nrigh⟪.al⟪12 E/par⟩nl⟩f⟪.al⟪1⟨D∗ 1,D∗ 2⟩4/divid⟩s.al⟪0Q/par⟩nrigh⟪.al⟪1=1 N2/summation.disp 1≤i≠j≤N/par⟩nl⟩f⟪.al⟪1D⊺ iDj/par⟩nrigh⟪.al⟪14+16 N.(42) To bound E/par⟩nl⟩f⟪.al⟪1var(/par... | https://arxiv.org/abs/2503.08031v1 |
1,V∗ 2⟫/divid⟩s.al⟪0/divid⟩s.al⟪0Q/par⟩nrigh⟪.al⟪1≤4 N2tr(ˆL2) (52) hold almost surely. Combining above results with Lemma F.14 gives E(var(γ∗ 11/divid⟩s.al⟪0Q))≲N4E/par⟩nl⟩f⟪.al⟪2tr(ˆL2)2 N8/par⟩nrigh⟪.al⟪2 ≲tr(L2)2 N4.(53) To analyze E(var(γ∗ 12/divid⟩s.al⟪0Q)), the definition of γ∗ 12gives var(γ∗ 12/divid⟩s.al⟪0Q)=/s... | https://arxiv.org/abs/2503.08031v1 |
which completes the proof. G Background results Lemma G.1 (Chernozhuokov et al. (2022), Theorem 2.1) .LetX1,...,X nbe independent random vectors in Rp such that E(Xij)=0for alli=1,...nandj=1,...,p. Let(G1,...,G p)be a Gaussian random vector in Rp with mean 0 and covariance matrix1 n∑n i=1E(XiX⊺ i), and define M(G)=max1≤... | https://arxiv.org/abs/2503.08031v1 |
Generation and Balancing Capacity in Future Electric Power Systems - Scenario Analysis Using Bayesian Networks Seppo Borenius a*, Pekka Kekolahti a, Petri Mähönena, Matti Lehtonena a Aalto University, School of Electrical Engineering , PO Box15500, FI -00076 AALTO, Finlan d * Corresponding autho r, email addresses: sep... | https://arxiv.org/abs/2503.08232v1 |
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