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=E[Φ(B)]for any Borel set B∈X. The k-th order factorial moment measure MΦ(k)ofΦis the mean measure of the k-th factorial power of Φ, i.e., of the point process Φ(k)defined as: Φ(k):≠=X (j1,...,jk)δ(Xj1,...,X jk), where the symbol ̸=over the summation means that the sum is extended over all pairwise distinct indexes. T... | https://arxiv.org/abs/2502.10257v2 |
k-th factorial moment measure M(k) Φisσ-finite, it is 7 possible to construct the family of k-th order Palm distributions {Px Φ}x∈Xk, and the generic probability measure Px Φcan be interpreted as the distribution of Φgiven that x= (x1, . . . , x k) are atoms of Φ. Again, by removing the trivial atoms (x1, . . . , x k),... | https://arxiv.org/abs/2502.10257v2 |
x∗ 1, . . . , x∗ k. One can recover the associated feature allocation Fn= (Bn,1, . . . , B n,Kn), where the set Bn,ℓcontains the indexes of the individuals exhibiting theℓ-th feature, with label x∗ ℓ, asℓ= 1, . . . , k. Feature allocation models describe probability distributions on the object Fn. Remarkably, extended ... | https://arxiv.org/abs/2502.10257v2 |
have Zi(X)<∞a.s. is the following: Z X×(0,1](1−s)MΨ(dxds) =∞. See Proposition S1 for the formal statement and proof. Notably, if Ψis a Poisson process with (infinite) mean measure νor a mixed Poisson MP(ν, fγ), then the conditionR X×(0,1]sν(dxds)< ∞is both necessary and sufficient for Zi(X)<∞a.s., as stated in Proposit... | https://arxiv.org/abs/2502.10257v2 |
dependsonthepreviouslyobservedfeatures x∗viathereducedPalmversionof Ψ, thusallowing interacting feature labels. To conclude the general Bayesian analysis of the extended feature models in (3), we provide the predictive distribution of the next observation, conditionally on the available sample, which easily follows fro... | https://arxiv.org/abs/2502.10257v2 |
having a product form efpf(Battiston et al., 2018; Ghilotti et al., 2024), such predictions may also depend on the number of distinct features k. 4.1 Sufficientness postulates We start by characterizing the class of extended feature allocation models for which the law of Z′ n+1depends on the initial sample Zonly throug... | https://arxiv.org/abs/2502.10257v2 |
Φ! y. Theorem 4 fully characterizes the class of crmpriors analyzed in James (2017). Moreover, Theorem 5 characterizes a broad class of prior distributions that includes, among others, all feature models having a product form efpf(Battiston et al., 2018), as well as the stable beta scaled processes analyzed in Camerlen... | https://arxiv.org/abs/2502.10257v2 |
variable, and G0a diffuse measure on X. Let Ψ =P j≥1δ(Xj,Sj)be such that Ψ|C∼PP Cρ(Cs) 1(0,1](s)ds G0(dx) . Then, µ=P j≥1SjδXjis a scaled process. The construction outlined in Proposition 1 is available for any scaled process, as introduced at the beginning of the section. It becomes clear that Ψis a mixed Poisson pr... | https://arxiv.org/abs/2502.10257v2 |
(5), where in this case µ′is acrmwith Lévy intensity measure ρn(ds|x)G0(dx). The weights S∗ ℓ’s of previously 17 observed features are independent random variables, and independent of µ′, with marginal density fS∗ ℓ(ds)∝smℓ(1−s)n−mℓρ(ds|x∗ ℓ), asℓ= 1, . . . , k. (iii) The predictive distribution of a future observation... | https://arxiv.org/abs/2502.10257v2 |
1, . . . , k. We emphasize that the process Z′ n+1in (10) depends on the initial sample Zonly through the sample size nand the number of distinct features k, which appear in the mixing law f˜γ, and on no additional sampling information, as expected from Theorem 5. Finally, if the kernel ρdoes not depend on the location... | https://arxiv.org/abs/2502.10257v2 |
not depend on G0. We further specialize this case to two choices of M∼qM. IfM∼Poi(λ), then Ψis a Poisson process with finite intensity measure. In this case, Ψ′ is a Poisson process with finite intensity λρn(ds)G0(dx), having M′∼Poi(ρn(0,1])points. In addition, Z′ n+1is a Poisson process with finite intensity, having K... | https://arxiv.org/abs/2502.10257v2 |
S∗ ℓ’s are independent random variables, further independent of µ′, with marginal density fS∗ ℓ(ds)∝smℓ(1−s)n−mℓH(ds|x∗ ℓ), asℓ= 1, . . . , k. Moreover, µ′in(5)can be represented asµ′=PM′ j=1S′ jδX′ j, where the X′ j’s are the atoms of a point process ξ′onXspecified by the Laplace functional Lξ′(f) =Lξ! x∗n f−logR (0,1... | https://arxiv.org/abs/2502.10257v2 |
and as discussed at the end of Section 4, in general, scaled processes yield predictive distributions for the newly discovered features which depend on the sampling 23 information through n, kand the frequency spectrum m1, . . . , m k. However, outside the case of stable beta scaled processes, the resulting expressions... | https://arxiv.org/abs/2502.10257v2 |
tree-specific probabilities of observing tree jin any survey. We also indicate the point process containing all the locations by ξ=P j≥1δXj. We further assume that the Sj’s are i.i.d. beta random variables with common parameters (a, b), and ξis adppon a rectangular region R⊂R2, as in Section 5.4. Indeed, it is well und... | https://arxiv.org/abs/2502.10257v2 |
exceeding a pre-specified threshold. An alternative strategy would focus on approximating the Poisson-binomial distribution via Le Cam’s theorem (Steele, 1994), i.e., approximating the law of ξ! x(X)with a Poisson distribution with parameterP k≥1λ∗ k. Since the sum of eigenvalues is equal to the trace of Kx, i.e.,R RKx... | https://arxiv.org/abs/2502.10257v2 |
model via the empirical Bayes approach described in Section 6.1. For completeness, Section S7 in the supplementary material provides the corresponding analyses under an oracle scenario, where all hyperparameters are fixed at their true values. Figures 1 and 2 highlight the key features of our predictions: in estimating... | https://arxiv.org/abs/2502.10257v2 |
trees. Note that the plots have different color scales. We analyze the spruces dataset from the R package spatstat , which contains the spatial locations of 134 Norwegian spruce trees in a natural forest stand in Saxony, Germany. These tree locations are represented as the point configuration ξ0. We assign an i.i.d. ma... | https://arxiv.org/abs/2502.10257v2 |
et al., 2017), by imposing conditions on the probability of observing a new species andtheprobabilityofre-observingaspeciesrecordedinthesample. Bycontrast, ourpostulates focus only on the distribution of new features and characterize broad classes of priors. It is then natural to wonder if, by adding further conditions... | https://arxiv.org/abs/2502.10257v2 |
F., B. Błaszczyszyn, and M. Karray (2020). Random measures, point processes, and stochastic geometry. HAL preprint available at https://hal.inria.fr/hal-02460214/ . Battiston, M., S. Favaro, D. M. Roy, and Y. W. Teh (2018). A characterization of product-form exchangeable feature probability functions. The Annals of App... | https://arxiv.org/abs/2502.10257v2 |
85 (5), 1357–1391. Fortini, S. and S. Petrone (2012). Predictive construction of priors in Bayesian nonparametrics. Brazilian Journal of Probability and Statistics 26 (4), 423 – 449. Franzolini, B., M. De Iorio, and J. Eriksson (2023). Conditional partial exchangeability: a probabilistic framework for multi-view cluste... | https://arxiv.org/abs/2502.10257v2 |
process models and sta- tistical inference. Journal of the Royal Statistical Society: Series B (Statistical Methodol- ogy) 77(4), 853–877. Lavancier, F. and E. Rubak (2023, October). On simulation of continuous determinantal point processes. Statistics and Computing 33 (45). Publisher Copyright: ©2023, The Author(s). L... | https://arxiv.org/abs/2502.10257v2 |
Notes-Monograph Series , 1–34. Pitman, J. and M. Yor (1997). The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. The Annals of Probability 25 (2), 855 – 900. Prünster, I. (2002). Random probability measures derived from increasing additive processes and their application to Bayesian sta... | https://arxiv.org/abs/2502.10257v2 |
feature allocation models, which are referenced in Remark 1 and throughout the paper. Section S2 provides additional results on the class of mixed binomial processes. Section S3 recalls a key formula for working with Palm distributions, namely the Campbell-Little-Mecke (clm) formula, as well as a fundamental characteri... | https://arxiv.org/abs/2502.10257v2 |
Zi(X)<∞a.s. if and only ifR X×(0,1]sν(dxds)<∞. Proof.We start by proving the statement for Ψdistributed as a Poisson process with mean measure ν. As discussed in Remark 1, the conditionR X×(0,1]sMΨ(dxds)<∞is sufficient forZi(X)<∞a.s.. Under the Poisson assumption for Ψ, this sufficient condition writes as R X×(0,1]sν(d... | https://arxiv.org/abs/2502.10257v2 |
is a mixed binomial process MB(ν, q(k) ˜M)where q(k) ˜M(m) =(m+k)! E{M(k)}m!qM(m+k). 43 Proof.Let us focus on the reduced Palm distribution of order k= 1. LetLΦbe the Laplace functional of Φ. By (Baccelli et al., 2020, Proposition 3.2.1), for any measurable functions f, g:X→R+, the Palm distribution of a point process ... | https://arxiv.org/abs/2502.10257v2 |
primary technical tool used in the proof of Theorems 1 and 2, namely the Campbell-Little-Mecke formula. This formula can be viewed as an extension of Fubini’s theorem to the case when both expectation and the integration are taken with respect to a point process. Lemma S2 (Campbell-Little-Mecke ( clm) formula) .LetΦbe ... | https://arxiv.org/abs/2502.10257v2 |
(i) =⇒(ii). Conversely, to prove the opposite implication, we need some preliminary lemmas, which are of independent interest. We start by recalling (Kallenberg, 1973, Lemma 5.1). Given the point process Φand the set C∈X, we indicate with ΦCthe restriction of ΦonC, i.e., ΦC(B) = Φ( B∩C), for any B∈X. Moreover, let Nden... | https://arxiv.org/abs/2502.10257v2 |
consider ¯kpoints in C, denoted with ¯x1, . . . , ¯x¯k, and let Bj= d¯xj,j≤¯k, be a neighborhood of ¯xj, such that Bi∩Bj=∅, 49 for any i, j≤¯k, i̸=j. Finally, define Bk=C\ ∪¯k j=1d¯xjandz= (z1, . . . , z ¯k,0)such that P¯k j=1zj=n−1. Then, the factorization in (S11) gives P¯k\ j=1(Φ−δη)(d¯xj) =zj,(Φ−δη)(C\ ∪¯k j=1d¯xj... | https://arxiv.org/abs/2502.10257v2 |
i.e., Equation (S13) when f= 0. Thus, the posterior Laplace functional in (S12) boils down to E e−µ(f)|Z=z =R (0,1]kEn e−Ψ! x∗,s(tf−nlog(1−t))oQk ℓ=1e−sℓf(x∗ ℓ)smℓ ℓ(1−sℓ)n−mℓρ(k)(ds|x∗) R (0,1]kEn eΨ! x∗,s(nlog(1−t))oQk ℓ=1smℓ ℓ(1−sℓ)n−mℓρ(k)(ds|x∗). To conclude the proof of Theorem 2, we need to show that the right-... | https://arxiv.org/abs/2502.10257v2 |
Φ! x1doesnotdependon x1(LemmaS3) anditisstillamixedPoisson(seetheproofoftheinverseimplicationofLemmaS3). Conversely, ifΦis a mixed binomial process, then Φ! x1is still a mixed binomial process which does not depend on x1(see Proposition S4). Therefore, in both cases, Φ! (x1,x2)= Φ! x1! x2does not depend on(x1, x2)and... | https://arxiv.org/abs/2502.10257v2 |
laws of S∗ ℓ|γdo not depend on γ, then the S∗ ℓ’s and Ψ′|γare independent. Second, the posterior distribution ofγis obtained from the likelihood in point (i) of Corollary 1 and the prior fγ, thus γ|Z∼f˜γ, with f˜γ(dγ)∝e−γφnγkfγ(dγ). The thesis in point (ii) follows. 54 Proof of point (iii) of Corollary 2 . To describe ... | https://arxiv.org/abs/2502.10257v2 |
The marginal distribution of Zis recovered from Theorem 1 as follows. Since Ψis an independently marked process with ground process ξand mark kernel H, then from (Baccelli et al., 2020, Proposition 3.2.14), the reduced Palm version Ψ! x,sis still an independently marked process, with ground process ξ! xand mark kernel ... | https://arxiv.org/abs/2502.10257v2 |
k, while it does not depend on the labels x∗ 1, . . . , x∗ k. S7 Additional details about the synthetic scenarios We present here the analysis of the synthetic scenarios from Section 6.2 under the oracle strat- egy, which assumes knowledge of the true values of all hyperparameters used to generate the data. First, we e... | https://arxiv.org/abs/2502.10257v2 |
given location. Left plot: the mean measure of ξ′. Right plot: the mean measure of ξ! x∗. The red dots represent the observed trees in the sample. The black crosses indicate the unseen trees. Note that the color scales of the two plots are different. 0.0 0.2 0.4 0.6 0.8 1.00.00.20.40.60.81.0Unseen trees and M/prime, n=... | https://arxiv.org/abs/2502.10257v2 |
arXiv:2502.10317v1 [stat.ME] 14 Feb 2025Biometrika (yyyy), vol, num, p. 1 Advance Access publication on dd mon yyyy Printed in Great Britain A Mechanistic Framework for Collider Detection in Observational Data B/y.pc S. PURKAYASTHA Department of Biostatistics and Health Data Science, Unive rsity of Pittsburgh, Pittsbur... | https://arxiv.org/abs/2502.10317v1 |
with collider variables is that, unlike adjustment for a mediator (denoted by the chain triad ) or confounder (given by the common ancestor triad), which clarifies causal effects, adjustment for a collider obscures causal effects by inducing bias in the (/u1D44B,/u1D44C)relation ( MacKinnon & Lamp ,2021;Holmberg & Anderse... | https://arxiv.org/abs/2502.10317v1 |
propose potential future work in Section 6. Biometrika style 3 1. F/o.pc/r.pc/m.pc/u.pc/l.pc/a.pc/t.pc/i.pc/o.pc/n.pc 1.1. Basic Information Theoretic Concepts In preparation for the exposition of conditional generativ e exposure models, we introduce some important information theoretic concepts. For the generat ive ex... | https://arxiv.org/abs/2502.10317v1 |
study the pathway in (ii), given by /u1D44B→/u1D44D|/u1D44C=/u1D466using the pointwise asymmetry coefficient between/u1D44Band/u1D44Dgiven/u1D44C=/u1D466, as defined below. D/e.pc/f.pc/i.pc/n.pc/i.pc/t.pc/i.pc/o.pc/n.pc 3. Let/u1D454/u1D466∈Kbe a generative function in ( 4) for fixed/u1D466∈Y. Then, we define the pointwise ... | https://arxiv.org/abs/2502.10317v1 |
any/u1D466∈Ythat satisfies the following condition:/uni222B.dsp X/u1D466log/parenleftbig|∇/u1D454/u1D466(/u1D465)|/parenrightbig/u1D453(/u1D465|/u1D466)/u1D451/u1D465=|X/u1D466|−1/uni222B.dsp X/u1D466log/parenleftbig|∇/u1D454/u1D466(/u1D465)|/parenrightbig/u1D451/u1D465. Remark 3. To interpret proposition 1, consider th... | https://arxiv.org/abs/2502.10317v1 |
asymmetry coefficients /u1D436/u1D44C→/u1D44D|/u1D465and/u1D436/u1D44B→/u1D44D|/u1D466using the sample{(/u1D44B/u1D456,/u1D44C/u1D456,/u1D44D/u1D456),1≤/u1D456≤/u1D45B}. For ease of exposition in this section, we will suppress the s ubscript and let /u1D453denote the marginal density of /u1D44B. Let/u1D453(/u1D466|/u1D46... | https://arxiv.org/abs/2502.10317v1 |
.(8) Similarly, we obtain ˆ/u1D43B(/u1D44D|/u1D44B=/u1D465)similarly using{(/u1D44B1,/u1D44D1),...,(/u1D44B/u1D45B,/u1D44D/u1D45B)}. The following theo- rem establishes consistency of ˆ/u1D43B(/u1D44C|/u1D465)for/u1D465∈X. The consistency of both ˆ/u1D436/u1D44B→/u1D44D|/u1D466for/u1D466∈Yand ˆ/u1D436/u1D44C→/u1D44D... | https://arxiv.org/abs/2502.10317v1 |
(see assumptions A1and 10 S. P/u.pc/r.pc/k.pc/a.pc/y.pc/a.pc/s.pc/t.pc/h.pc/a.pc /e.pc/t.pc /a.pc/l.pc. A2), we have |/u1D436/u1D44C→/u1D44D|/u1D465/u1D452 /u1D4571−/u1D436/u1D44C→/u1D44D|/u1D465min|</u1D7161, |/u1D436/u1D44B→/u1D44D|/u1D466/u1D452 /u1D4572−/u1D436/u1D44B→/u1D44D|/u1D466min|</u1D7162, for some/u1D7161,... | https://arxiv.org/abs/2502.10317v1 |
b y our method as well as the two competing methods under consideration. The mean Hamming di stance over the /u1D45F=250 iterations are reported in Table 1for sample size /u1D45B=500. Our method accurately captures the /u1D44B→/u1D44D←/u1D44C more accurately with lower mean Hamming distance scores tha n both Peter-Clar... | https://arxiv.org/abs/2502.10317v1 |
our conditional asymmetry coefficient (CAC), the Peter-Clar k al- gorithm (PC), and the Hill-Climbing algorithm (HC). Within each column, we report the mean Hamming distance when the correla - tion between /u1D44Band/u1D44Cis/u1D70C=0.00,(/u1D70C=0.05),[/u1D70C=0.10]. /u1D454/u1D44B→/u1D44D/u1D454/u1D44C→/u1D44D CAC PC H... | https://arxiv.org/abs/2502.10317v1 |
are found to form collider str uctures with both systolic and diastolic blood pressure under contracting an d expanding dynamics respectively; see Table 3. Systolic Blood Pressure As An Epigenetic Collider: Nine genes were selected as candidate genes for our study based on the selection criteria outlined earli er. They... | https://arxiv.org/abs/2502.10317v1 |
(e.g., hypertension, cardiovascular ris k). In summary, these new findings confer an added sense of directionality in the study of blood pressu re variation and epigenetic biomarkers, paving the way for future advancements in genetic risk asses sments and even therapeutic targets. 6. L/i.pc/m.pc/i.pc/t.pc/a.pc/t.pc/i.pc... | https://arxiv.org/abs/2502.10317v1 |
following conditions are true. 1./u1D453(/u1D466|/u1D465)log(/u1D453(/u1D466|/u1D465))is uniformly integrable for all /u1D465∈X, 2./u1D453(/u1D467|/u1D465)log(/u1D453(/u1D467|/u1D465))is uniformly integrable for all /u1D465∈X, 3./u1D453(/u1D465|/u1D466)log(/u1D453(/u1D465|/u1D466))is uniformly integrable for all /u1D46... | https://arxiv.org/abs/2502.10317v1 |
R. ,G/u.pc/i.pc/l.pc/l.pc/e.pc/n.pc, J. A. , R/i.pc/a.pc/t.pc, H. S. ,T/r.pc/e.pc/v.pc/a.pc/n.pc/i.pc/o.pc/n.pc, S. J. ,H/a.pc/l.pc/l.pc, P. ,J/u.pc/n.pc/k.pc/i.pc/n.pc/s.pc, H. ,F/l.pc/i.pc/c.pc/e.pc/k.pc, P. ,B/u.pc/r.pc/d.pc/e.pc/t.pc/t.pc, T. ,H/i.pc/n.pc/d.pc/o.pc/r.pc/f.pc/f.pc, L. A. ,C/u.pc/n.pc/n.pc/i.pc/n.pc/... | https://arxiv.org/abs/2502.10317v1 |
of european ancestry yields improved polygenic risk scores for blood pressure traits. Nature genetics , 1–14. K/o.pc/u.pc, M. ,L/i.pc, X. ,S/h.pc/a.pc/o.pc, X. ,G/r.pc/u.pc/n.pc/d.pc/b.pc/e.pc/r.pc/g.pc, E. ,W/a.pc/n.pc/g.pc, X. ,M/a.pc, H. ,H/e.pc/i.pc/a.pc/n.pc/z.pc/a.pc, Y. ,M/a.pc/r.pc/t.pc/i.pc/n.pc/e.pc/z.pc, J. ... | https://arxiv.org/abs/2502.10317v1 |
. MIT Press. S/p.pc/i.pc/r.pc/t.pc/e.pc/s.pc, P. L. &Z/h.pc/a.pc/n.pc/g.pc, K. (2016). Causal discovery and inference: concepts and recen t methodological advances. Applied Informatics 3. T/h.pc/e.pc I/n.pc/t.pc/e.pc/r.pc/n.pc/a.pc/t.pc/i.pc/o.pc/n.pc/a.pc/l.pc C/o.pc/n.pc/s.pc/o.pc/r.pc/t.pc/i.pc/u.pc/m.pc /f.pc/o.pc/... | https://arxiv.org/abs/2502.10317v1 |
Dimension-free Score Matching and Time Bootstrapping for Diffusion Models Syamantak Kumar∗Dheeraj Nagaraj†Purnamrita Sarkar‡ February 17, 2025 Abstract Diffusion models generate samples by estimating the score function of the target distribution at various noise levels. The model is trained using samples drawn from the... | https://arxiv.org/abs/2502.10354v1 |
Carlo (MCMC) algorithms, which have access to the underlying density, diffusion models can only access mi.i.d. samples from the target distribution. These models are trained by ‘score matching’, where a neural network is parametrized to learn the score function of the noised target distribution at various noise levels.... | https://arxiv.org/abs/2502.10354v1 |
be used with the discretization analyses such as those presented in [BDBDD24, CCL+22, LLT23] to theoretically analyze the quality of samples generated by the model. Learning from dependent data: Learning with data from a markov trajectory has been ex- plored in literature in the context of system identification, time s... | https://arxiv.org/abs/2502.10354v1 |
We consider the joint DSM objective to be: ˆL(f) :=1 mNmX i=1X t∈T f(t, x(i) t) +z(i) t σ2 t 2 2. (4) Intuitively, optimizing (4) represents a regression task with noisy labels. There are two primary sources of noise in this setup. The first comes from (3), since the targets, −zt/σ2 t, conditioned on the data point, xt... | https://arxiv.org/abs/2502.10354v1 |
1−δ, X i∈[m],j∈[N]γj ˆf tj, x(i) tj −s tjx(i) tj 2 2 m≲ϵ2 Remark 1. The sample complexity in Theorem 1 depends on the smoothness parameter Land onlog(B). Observe that Bdepends logarithmically on d, thus leading to a nearly dimension-free result, i.e. log log ddependence. This is in stark contrast to existing result... | https://arxiv.org/abs/2502.10354v1 |
. , S k, where each subset Sicontains timesteps of the form tj= ∆( i+nk)forn∈N. Define γ′ j:=k∆for all jin any subset Si. Then, there exists at least one subset Sisuch that: X j∈Siγ′ jExtj ˆf(tj, xtj)−s(tj, xtj) 2 2 ≲ϵ2, with probability at least 1−δ. 7 The subsets Siallow for a much coarser discretization with diffe... | https://arxiv.org/abs/2502.10354v1 |
to [GPPX24]. Lemma 2. Forf∈ H, lety(i) t:=−z(i) t σ2 tand L(f) :=X i∈[m],j∈[N]γj f tj, x(i) tj −s tj, x(i) tj 2 2 m, 8 Hf:=X i∈[m],j∈[N]γj m f tj, x(i) tj −s tj, x(i) tj , y(i) tj−s tj, x(i) tj . Ifs∈ Hthen for ˆf= arg inf f∈HˆL(f), we have L(ˆf)≤Hˆf, (7) where ˆLis defined in (4). We define ˆfas the minimize... | https://arxiv.org/abs/2502.10354v1 |
along with Giand∆. Lemma 5 (Variance bound for martingale difference sequence) .Consider the martingale difference sequence Ri,kand the predictable sequence Gi,k+1with respect to the filtration Fi,kfrom Lemma 22. Define ∆ := tN−k+1−tN−k. Then, Eh R2 i,k|Fi,k−1i ≤ν2 i,kwhere ν2 i,k= 0, ifk= 0, C(L∆2+ ∆ + L2∆)e2tN−... | https://arxiv.org/abs/2502.10354v1 |
xt)∥2≥ −˜Ω(L√ dk∆). Exploiting this property over a carefully selected range of kvalues allows us to relate ℓ∞andℓ2 norm bounds as we show in the following Lemma. Lemma 7. Under Assumption 1, with probability 1−δ, for a universal constant C >0the following holds uniformly for every f∈ H: sup i∈[m] j∈[N] f tj, xtj −... | https://arxiv.org/abs/2502.10354v1 |
.)fort′< t. Instead as we move along the trajectory, we learn score estimates ˆst. Therefore, we plug in ˆst′in˜ytinstead of the true score function, s(t′, .). This in-turn induces a bias at the cost of a reduced variance, which we trade-off using the parameter, αt, to achieve a better ℓ2-error of the score estimate. O... | https://arxiv.org/abs/2502.10354v1 |
the forefront of generative models with applications rang- ing from image to audio and video generation. To our knowledge, this is the first work , which estab- lishes (nearly) dimension-free sample complexity bounds for learning score functions across noise levels. We show that a mild assumption of time-regularity can... | https://arxiv.org/abs/2502.10354v1 |
diffusion language models.Advances in Neural Information Processing Systems , 36, 2024. [GPPX24] Shivam Gupta, Aditya Parulekar, Eric Price, and Zhiyang Xun. Improved sample complexity bounds for diffusion model training. In The Thirty-eighth Annual Confer- ence on Neural Information Processing Systems , 2024. 15 [HD05... | https://arxiv.org/abs/2502.10354v1 |
Robust covariance estimation under l_4- l_2 norm equivalence. 2020. [NWB+20] Dheeraj Nagaraj, Xian Wu, Guy Bresler, Prateek Jain, and Praneeth Netrapalli. Least squares regression with markovian data: Fundamental limits and algorithms. Advances in neural information processing systems , 33:16666–16676, 2020. [OAS23] Ka... | https://arxiv.org/abs/2502.10354v1 |
results which will be useful in subsequent proofs. 2. Section C provides variance calculation for the martingale decomposition. 3. Section B analyzes concentration properties for martingales with bounded variance and sub- Gaussianity, which may be of independent interest. 4. Section D analyzes convergence of the empiri... | https://arxiv.org/abs/2502.10354v1 |
independent of xt′. Let yt,t′:=e−(t−t′)s(t, xt)−s(t′, e(t−t′)xt). Then, ∥yt,t′∥=∥e−(t−t′)st(xt)−s(t′, e(t−t′)xt)∥ =∥E s t′, x′ t |xt −s(t′, e(t−t′)xt)∥ = Eh st′ e(t−t′)(xt−zt,t′) −s(t′, e(t−t′)xt)|xti ≤et−t′LE zt,t′ 2|xt Note that since zt,t′∼ N 0, σ2 t−t′I , E" exp zt,t′ 2 2 4σ2 t−t′d!# ≤2,using Lemma 9 Ther... | https://arxiv.org/abs/2502.10354v1 |
2β2dx = (2β2K)k+ 2kZ∞ x0x2k−1eKe−x2 2β2dx ≤(2β2K)k+ 2kZ∞ x0x2k−1e−(x−x0)2 2β2dx ≤(2β2K)k+ 22k−1kZ∞ x0(x2k−1 0+ (x−x0)2k−1)e−(x−x0)2 2β2dx 24 Inthesecondstepwehaveusedthefactthatwhenever x≥x0, wemusthave K−x2 2β2≤ −(x−x0)2 2β2. In the third step we have used the fact that x2k−1≤22k−2[(x−x0)2k−1+x2k−1 0]whenever x≥x0. A ... | https://arxiv.org/abs/2502.10354v1 |
¯λ∈ΛBsuch that ¯λ/λmin∈[1 e, e]and such that the event ∩n i=1Ai(¯λ)holds. Thus, we have: {Mn> Ceλ∗X iν2 i∥Gi∥2+eα λmin} ∩ E 4⊆ {Mn> Ceλ minX iν2 i∥Gi∥2+eα λmin} ∩ E 4 ⊆ {Mn> C¯λX iν2 i∥Gi∥2+α ¯λ} ∩ E 4={Mn> C¯λX iν2 i∥Gi∥2+α ¯λ} ∩ E 4∩n i=1Ai(¯λ) ⊆ E4∩ ∪λ∈ΛB{λMn> Cλ2nX i=1ν2 i∥Gi∥2+α} ∩n i=1Ai(λ)! (15) Similarly, under... | https://arxiv.org/abs/2502.10354v1 |
0|x(i) tj−N+1]. Plugging this into the display equation above, we conclude the result. We connect the quantity Hdefined above to the quantity Hfrelated to the excess risk. 28 Lemma 20. Lety(i) t:=−z(i) t σ2 t,f∈ Hand Hf:=X i∈[m],j∈[N]γjD f tj, x(i) tj −s tj, x(i) tj , y(i) tj−s tj, x(i) tjE m Suppose we pick ζ=s−... | https://arxiv.org/abs/2502.10354v1 |
(27) Now, note that xt=e−tx0+et′−tzt′+zt,t′. Taking y0=e−tx0+zt,t′, we have: xt=y0+et′−tzt′. Therefore, applying the second order Tweedie’s formula again, we must have: e2(t′−t)E[zt′z⊤ t′|xt] =e4(t′−t)σ4 t′s(t, xt)s(t, xt)⊤+e4(t′−t)σ4 t′ht(xt) +e2(t′−t)σ2 t′I 30 That is : E[zt′z⊤ t′|xt] =e2(t′−t)σ4 t′s(t, xt)s(t, xt)⊤+... | https://arxiv.org/abs/2502.10354v1 |
xτi+1 |xτii |xτj, xτj+1i s τj, xτj −e−(τj+1−τi)s τj+1, xτj+1⊤ |xt = 0 Therefore, E s t′, xt′ −e−(t−t′)s(t, xt) s t′, xt′ −e−(t−t′)s(t, xt)⊤ |xt op = E"B−1X i=0c2 i s(τi, xτi)−e−(τi+1−τi)s τi+1, xτi+1 s(τi, xτi)−e−(τi+1−τi)s τi+1, xτi+1⊤ |xt# op = E"B−1X i=0c2 iE s(τi, xτi)−e−(τi+1−τi)s τi+1... | https://arxiv.org/abs/2502.10354v1 |
xt′,λ , leverages the Lipschitz continuity of the mollified Hessian and can be analyzed by conditioning on xt. The second term, which represents the deviation between the original and mollified Hessians, requires a finer analysis that draws upon Lusin’s theorem, as developed further in Lemma 29. The decomposition allo... | https://arxiv.org/abs/2502.10354v1 |
arises from the deviation between the Hessian and its mollified counterpart. To achieve this, the interval [t′, t]is partitioned into smaller subintervals τ0, τ1, . . . , τ B, allowing the analysis to proceed incrementally. The lemma exploits the uniform continuity of ht(x)onGγto tightly control this difference using t... | https://arxiv.org/abs/2502.10354v1 |
contributions and leveraging smoothing techniques, we arrive at a sharp variance bound that scales with the parameters ∆(the interval size) and L(the bound on the Hessian) The final result, formalized in Lemma 30, also uses the second-order Tweedie formula to handle the special case of the last time step ( k=N) in the ... | https://arxiv.org/abs/2502.10354v1 |
compact convex set, we let F↑Rdto conclude the following: E s t′, xt′ −e−(t−t′)s(t, xt) s t′, xt′ −e−(t−t′)s(t, xt)⊤ |xt op=O L2∆ (36) 39 Using Lemma 24, we have Eh (E[x0|xt]−E[x0|xt′]) (E[x0|xt]−E[x0|xt′])⊤|xti op ≤2e2t σ4 t−t′ht(xt) +σ2 t−t′Id op + 2e2t′σ4 t′ E s t′, xt′ −e−(t−t′)s(t, xt) s t′, ... | https://arxiv.org/abs/2502.10354v1 |
hold: X i∈[m],k∈[N]E[R2 i,k|Fi,k−1]≤C∆3(L∆ + 1 + L2) N 1−e−2∆+1 (1−e−2∆)2X i∈[m]NX j=1 ζ(tj, x(i) tj) 2 and max sup i∈[m]βi,Np Wi,N∥¯Gi∥,sup i∈[m] k∈[N−1]βi,kp Wi,k∥¯Gi,k+1∥ ≤C(L+ 1)√ ∆ log(1 ∆)r dsup i,kWi,ksup i,k∥ζ(tk, x(i) tk)∥ 41 Proof.Define g2 0:= (L∆2+ ∆ + L2∆). Applying Lemma 30, we conclude: X i∈[m],k∈... | https://arxiv.org/abs/2502.10354v1 |
i,ke2tN−k+1∥Gi,k+1∥,sup i∥¯Gi∥)≥CeB log(2d)(L+ 1)√ m∆dsupi,kpWi,k This implies that there exists i, ksuch that ∥ζ(tk, x(i) tk)∥ σ2 tk≥CeB Nlog(2d)(L+ 1)√ m∆dsupi,kpWi,k We then conclude the result using the fact that σ2 tk≥c0∆ Lemma34. Assume N∆>1,∆< c0for some universal constant c0. Assume tj= ∆jandγj= ∆. Letα >1andB∈... | https://arxiv.org/abs/2502.10354v1 |
Lemma 2 and the Cauchy-Schwarz inequality, L ˆf ≤X i∈[m],j∈[N]γjD ˆf tj, x(i) tj −s tj, x(i) tj , y(i) tj−s tj, x(i) tjE m≤r L ˆf bL(s) which completes the first part of the proof. Next, we have bL(s) =X i∈[m],j∈[N]γj s tj, x(i) tj −y(i) tj 2 2 m Clearly, since y(i) tis marginally Gaussian , we conclude tha... | https://arxiv.org/abs/2502.10354v1 |
x(i∗) tj 2 2 ≥X j:|j−k∗|∆≤∆2/3c2 0 2 f tk∗, x(i∗) tk∗ −s tk∗, x(i∗) tk∗ 2 2−C2L2d|j−k∗|∆ log(Nm δ) This implies the following inequality from which we can conclude the result. X i∈[m],j∈[N] f tj, x(i) tj −s tj, x(i) tj 2 2≥c2 0 2∆1/3 f tk∗, x(i∗) tk∗ −s tk∗, x(i∗) tk∗ 2 2−2C2L2d∆1/3log(Nm δ) Theorem 1 (E... | https://arxiv.org/abs/2502.10354v1 |
Therefore, under the events (A1(ˆf)∩ D ∩ C )∪ A 2(ˆf), the guarantee for L(ˆf)stated in the theorem holds. It now remains to show that P (A1(ˆf)∩ D ∩ C )∪ A 2(ˆf) ≥1−δ. We begin with Equation (46): 1−δ 4≤P(A1(ˆf)∪ A 2(ˆf)∪ A 3(ˆf)) ≤P(A1(ˆf)∪ A 2(ˆf)) +P(A3(ˆf))≤P(A1(ˆf)∪ A 2(ˆf)) +δ 2,by applying Equation (47) =P((A... | https://arxiv.org/abs/2502.10354v1 |
m The result then follows by setting the RHS smaller by ϵ2. Theorem 5 (Accelerated Inference) .Under the same assumptions as Theorem 2, partition the timesteps {tj= ∆ j}j∈[N]into kdisjoint subsets S1, S2, . . . , S k, where each subset Sicontains timesteps of the form tj= ∆( i+mk)form∈N. Define γ′ j:=k∆for all jin any ... | https://arxiv.org/abs/2502.10354v1 |
σ2 t′ xt′ xt = 0 Therefore, E ht,t′|xt = 0. Let vt,t′:=st(xt)−αs(t′, xt′)andrt,t′:=zt σ2 t−αzt′ σ2 t′. First consider rt,t′. We have using (1), zt=e−(t−t′)zt′+zt,t′where zt,t′∼ N(0, σ2 t−t′). Then, rt,t′=zt σ2 t−αzt′ σ2 t′=e−∆zt′+zt,t′ σ2 t−αzt′ σ2 t′=e−∆ σ2 t−α σ2 t′ zt′+zt,t′ σ2 t(53) 54 Next, for vt,t′again us... | https://arxiv.org/abs/2502.10354v1 |
for the first few timesteps (t≤3) and 1epoch thereafter. We plot the squared error of the learned score matrix, ˆAtagainst the true score matrix, Atat all timesteps. In the second experiment, we move away from the Gaussian density, which is unimodal, to a Gaussian Mixture model (GMM), which is multimodal. We fix the di... | https://arxiv.org/abs/2502.10354v1 |
A new and flexible class of sharp asymptotic time-uniform confidence sequences Felix Gnettner∗, Claudia Kirch† Abstract Confidence sequences are anytime-valid analogues of classical confidence intervals that do not suffer from multiplicity issues under optional continuation of the data collection. As in classical stati... | https://arxiv.org/abs/2502.10380v1 |
2.8) prove that Ct(m;α) =bµt±bσt˜bt,m(α) is a sharp asymptotic confidence sequence for the mean of iid random variables ( Xi)i∈Nwith finite variance σ2 X>0, where bµt,bσ2 tare the sample mean and sample variance based on the first tobservations. Their ˜bt,m(α) has a specific shape inherited from a boundary crossing res... | https://arxiv.org/abs/2502.10380v1 |
(A1) lim sup s→0sγ1·ρ(s)<∞for some γ1∈[0,1/2), (A2) lim sup s→∞s1−γ2·ρ(s)<∞for some γ2∈[0,1/2). Typically, the endpoint eρwill be equal to infinity, corresponding to an open-end procedure where data is collected possibly forever, while for a finite endpoint at most ⌊m eρ⌋data points are collected. The following theorem... | https://arxiv.org/abs/2502.10380v1 |
H(ℓ) 1(µ) :µX< µ (2) with corresponding sequential one-sided tests φ(r) t(m;µ, α) =1n bµt−µ≥bσt·c(o) α(ρ)·bt(m;ρ)o , φ(ℓ) t(m;µ, α) =1n bµt−µ≤ −bσt·c(o) α(ρ)·bt(m;ρ)o , (3) where the critical values c(o) α(ρ) are chosen as the (1 −α)-quantile of supy>0(ρ(y)·W(y)) for a standard Wiener process ( W(y))y∈(0,∞). Unlike for... | https://arxiv.org/abs/2502.10380v1 |
the asymptotic variance is given by the so-called long- run variance taking the autocovariances of all lags into account; see Aue and Kirch (2024, Section 3.2) for more details. Proof of Theorem 4.1. By the assumptions on the sequence of variance estimators, possibly replacing 1/bσtby 1/bσt·1{bσt>0}, it holds (V1) sup ... | https://arxiv.org/abs/2502.10380v1 |
1−x D= sup x∈(0,1)|B(x)| xγ1(1−x)γ2, where the distributional equality follows from Cs¨ org˝ o and R´ ev´ esz (1981, Equation (1.4.5)). Remark 4.4. The statements of Theorem 4.1 and Proposition 4.3 remain true if all absolute values of quantities are replaced by the original quantities without absolute values. The pro... | https://arxiv.org/abs/2502.10380v1 |
arXiv:2212.14411v5 , 2024. doi: 10. 48550/arXiv.2212.14411. O. Chochola. Robust Monitoring Procedures for Dependent Data . PhD thesis, Charles University Prague, 2013. C.-S. J. Chu, M. Stinchcombe, and H. White. Monitoring structural change. Econometrica , 64(5): 1045–1065, 1996. ISSN 00129682, 14680262. doi: 10.2307/2... | https://arxiv.org/abs/2502.10380v1 |
Forecasting time series with constraints Nathan Doum `eche NATHAN .DOUMECHE @SORBONNE -UNIVERSITE .FR Sorbonne University, EDF R&D Francis Bach FRANCIS .BACH @INRIA .FR Inria, ENS, PSL Research University Eloi Bedek ELOI .BEDEK @EDF.FR EDF R&D G´erard Biau GERARD .BIAU @SORBONNE -UNIVERSITE .FR Sorbonne University, IUF... | https://arxiv.org/abs/2502.10485v1 |
the features, thereby constraining the shape of the regression function. Another example of weak constraint appears in the context of spatiotemporal time series with hierarchical forecasting. Here, the goal is to combine regional forecasts into a global forecast by enforcing that the global forecast must be equal to th... | https://arxiv.org/abs/2502.10485v1 |
of distin- guishing between two categories of constraints: shape constraints, which limit the set of admissible functions, and learning constraints, which introduce an initial bias during parameter optimization. In Section 3, we explore shape constraints and illustrate their relevance using the example of electricity d... | https://arxiv.org/abs/2502.10485v1 |
dimensions, which are not necessarily square. The matrix Mencodes a regularization penalty, which may include hyperpa- rameters to be tuned through validation, as we will see in several examples. Explicit formula for the empirical risk minimizer: WeaKL. The following proposition shows how to compute the exact minimizer... | https://arxiv.org/abs/2502.10485v1 |
k⟩/2))⊤ −m≤k1,...,kd1≤m, where the Fourier frequencies are truncated at m≥0. This map leverages the expressiveness of the Fourier basis to capture complex patterns in the data. Thus, for theℓ-th component of fθ, we consider the Fourier decomposition fℓ θ(x) =X ∥k∥∞≤mθℓ,kexp(−i⟨x, k⟩/2), which can approximate any functi... | https://arxiv.org/abs/2502.10485v1 |
that it is related to the matrix Φtof (4) by Φ∗Φ=Pn j=1Φ∗ tjΦtj=Pn j=1ϕ1(Xtj)ϕ1(Xtj)∗. Additive model: Additive WeaKL. The additive model constraint assumes that f⋆(x1, . . . , x d1) =Pd1 ℓ=1g⋆ ℓ(xℓ), where g⋆ ℓ:R→Rare univariate functions. This constraint is widely used in data science, both in classical statistical m... | https://arxiv.org/abs/2502.10485v1 |
the behavior of Ymay change rapidly following extreme events, resulting in structural breaks. A notable example is the shift in electricity demand during the COVID-19 lockdowns, as illustrated in use case 1. To provide a clear mathematical framework, we assume that the distribution of (X, Y)follows an additive model th... | https://arxiv.org/abs/2502.10485v1 |
horizons, is particularly valuable for making operational decisions in both the power industry and electricity markets. Although the cost of forecasting errors is difficult to quantify, a 1%reduction in error is estimated to save utilities several hundred thousand USD per gigawatt of peak demand (Hong and Fan, 2016). R... | https://arxiv.org/abs/2502.10485v1 |
goal here is to forecast the French electricity demand during the energy crisis in the winter of 2022-2023. Following the war in Ukraine and maintenance problems at nuclear power plants, electricity prices reached an all-time high at the end of the summer of 2022. In this context, French electricity de- mand decreased ... | https://arxiv.org/abs/2502.10485v1 |
However, this choice ensures that the validation period contains a structural break, making it as similar as possible to the test period. Next, the functions h0, . . . , h 10in (7) are trained on a period starting from 1July2020 , and updated online. The results are summarized in Table 2. The errors and their standard ... | https://arxiv.org/abs/2502.10485v1 |
also competitive with state-of-the-art techniques in terms of both optimization efficiency (they can run on GPUs) and performance (measured by MAPE and RMSE). 4 Learning constraints 4.1 Mathematical formulation Section 3 focused on imposing constraints on the shape of the regression function f⋆. In contrast, the goal o... | https://arxiv.org/abs/2502.10485v1 |
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