Split (1)

train · 282k rows

text string	source string
TUERLINCKX , F. The shape of partial correlation matrices. Communications in Statistics-Theory and Methods 51 , 12 (2022), 4133–4150. [4] BARLOW , J., AND DEMMEL , J. Computing accurate eigensystems of scaled diagonally dominant matrices. SIAM Journal on Numerical Analysis 27 , 3 (1990), 762–791. [5] BECKERS , S., E BE...	https://arxiv.org/abs/2503.09194v2
National Academy of Sciences 45 , 5 (1959), 740–744. [22] POLONI , F. Positive matrix and diagonally dominant. MathOverflow, 2020. URL:https://mathoverflow.net/q/350334 (version: 2020-01-13). [23] REISACH , A., S EILER , C., AND WEICHWALD , S. Beware of the simulated dag! causal discovery benchmarks may be easy to game...	https://arxiv.org/abs/2503.09194v2
We use 𝑈to denote the set of unobserved (unmeasured) variables, 𝑂to denote the set of observed (measured) random variable, in short observable (s). 𝑢,𝑣,𝑖,𝑗,𝑘 , as well as integers and 𝑎,𝑏,𝑐,𝑑 , are used to index a vertex (node) in a graph. Let 𝑌represents a random variable vector, then 𝑌𝑢,𝑌𝑣,𝑌𝑖,𝑌𝑗re...	https://arxiv.org/abs/2503.09194v2
assertions. B.2 Problems of DAG representation when there exist unobserved confounder Causal sufficiency means no unmeasured (unobserved) common causes or the confounder takes only one value, which is usually violated in practice. As an example, in Figure 6 (reproducedfrom [ 29]),𝑈is an unobserved confounder in the DA...	https://arxiv.org/abs/2503.09194v2
Algorithm in [29]). DEFINITION 14 ( M-SEPARATION ).m-separation extends d- separation to the mixed graph: if there is no active path 𝜌𝑎(𝐴,𝐵\|𝑆), then𝑆m-separates 𝐴,𝐵 DEFINITION 15 (A NCESTRAL [10]).An ordinary (single edge) mixed graph without any undirected edge is ancestral if its directed part is acyclic, and...	https://arxiv.org/abs/2503.09194v2
The intersection of ADMGs and ancestral graph is called ancestral ADMG. 12 understanding solution space of causal discovery with unobserved confounding DEFINITION 17 (O-M ARKOV EQUIVALENT BETWEEN DAG S). Two DAGS𝐷1(𝑂,𝑈)and𝐷2(𝑂,𝑈)are O-Markov equivalent if they represent the same set of CI statements with respect ...	https://arxiv.org/abs/2503.09194v2
=𝑄(Σ′)−1𝑄𝑇(74) =(Σ11)−10 0([Σ/Σ11])−1 (75) while (Σ′)−1=Σ11Σ12 Σ21Σ22−1 =" [(Σ′)−1]11[(Σ′)−1]12 [(Σ′)−1]21([Σ/Σ11])−1# (76) So the congruent transform with 𝑄𝑇of(Σ′)−1as a precision matrix, the lower-right block ([Σ/Σ11])−1is preserved. COROLLARY 8.Let𝑋1,...,𝑋𝑝be multivariate Gaussian whose covariance matrix...	https://arxiv.org/abs/2503.09194v2
11Σ12=0 (101) The corresponding adjacency matrix 𝑊′′ 2is 𝑊′′ 2=𝑄−𝑇𝑊′𝑄𝑇=𝑄−𝑇𝑊𝑜Λ 0 0 𝑄𝑇(102) =𝐼−Σ−1 11Σ12 0𝐼 𝑊𝑜Λ 0 0 𝐼 0 Σ21Σ−1 11𝐼 (103) =𝑊𝑜+ΛΣ21Σ−1 11Λ 0 0 (104) □ REMARK 21.The𝑄12component in Equation (62)will mix𝜉into the second partition of 𝜖′′in Equation (97). Unless𝑄(equivalently ...	https://arxiv.org/abs/2503.09194v2
graph 𝐺 upon marginalization. Let one of such a DAG be 𝐷1(𝐺). Although 𝐷1(𝐺)and𝐷(𝐺)(the canonical DAG as defined above) satisfy the same set of polynomial constraints among the covariances of the observed variables, they could still differ as models. For instance, there could be certain polynomial inequalities (...	https://arxiv.org/abs/2503.09194v2
(13) and (14) can be equivalently reformulated via the implicit formulation in Equations (1)and (5), with the covariance matrix Ω, defined in Equation (4), having nonzero off-diagonal entries. PROOF . Define 𝜖′ 𝑌=Λ𝜉+𝜖𝑂. Then Ω′=E(𝜖′ 𝑌𝜖′ 𝑌𝑇) =E (Λ𝜉+𝜖𝑂)(Λ𝜉+𝜖𝑂)𝑇 =ΛE(𝜉𝜉𝑇)Λ𝑇+E(𝜖𝑂𝜖𝑇 𝑂). (121) By L...	https://arxiv.org/abs/2503.09194v2
path travels 𝑟1steps to reach vertex 𝑢and starting from 𝑡2a path lead to 𝑣in𝑟2steps. This is a trek from 𝑢to𝑣 with top𝑡1,𝑡2being confounded if 𝑡1≠𝑡2. The summation over all possible treks between 𝑢,𝑣is given byÍ∞ 𝑟1=0,𝑟2=0Í 𝑡1,𝑡2can be written as ∑︁ 𝑡𝑟𝑒𝑘(𝑢,𝑣;𝑘=(𝑡1,𝑡2)\|𝐷)∈𝑇(𝑢,𝑣\|𝐷)𝑚𝑡𝑟𝑒�...	https://arxiv.org/abs/2503.09194v2
2 with Δ𝑖(Ω)>1, we have 𝜆(Ω−1)<1. PROOF .According to [ 32], ifΔ𝑖(Ω)>0as in Definition 7, then ∥Ω−1∥∞≤max 1≤𝑖≤\|𝑉\|1 Δ𝑖(Ω), (182) where the matrix infinity norm for the precision matrix Ω−1of size \|𝑉\|is defined as ∥Ω−1∥∞=max 𝑥≠0∥Ω−1𝑥∥∞ ∥𝑥∥∞=max 1≤𝑖≤\|𝑉\|\|𝑉\|∑︁ 𝑗=1 [Ω−1]𝑖,𝑗 . (183) The Gershgorin disks for Ω−...	https://arxiv.org/abs/2503.09194v2
of the singular values of(𝐼−𝑊)𝑇. Let𝑣(𝑊,𝜆(𝑊))be eigenvector of 𝑊corresponding to eigenvalue 𝜆(𝑊), then (𝐼−𝑊)𝑣(𝑊,𝜆(𝑊))=(1−𝜆(𝑊))𝑣(𝑊,𝜆(𝑊)) (216) (𝐼−𝑊)−1𝑣(𝑊,𝜆(𝑊))=1 1−𝜆(𝑊)𝑣(𝑊,𝜆(𝑊)) (217) series decomposintion of (𝐼−𝑊)−1 Since det 𝑊𝑇−𝜆𝐼 =det (𝑊−𝜆𝐼)𝑇 =det(𝑊−𝜆𝐼) (218) and 𝜆(...	https://arxiv.org/abs/2503.09194v2
Low-Rank Graphon Estimation: Theory and Applications to Graphon Games Olga Klopp ESSEC Business School Fedor Noskov HSE University Abstract This paper tackles the challenge of estimating a low-rank graphon from sampled network data, employing a singular value thresholding (SVT) estimator to create a piecewise-constant ...	https://arxiv.org/abs/2503.09299v1
a sampled network emerges as a significant problem extensively explored in the literature [Chatterjee, 2015, Gao et al., 2015, Klopp et al., 2017, Klopp and Verzelen, 2019, Xu, 2018]. In connection to our motivating applica- tions, in this study, our focus lies on providing a low-rank estimator for the underlying unkno...	https://arxiv.org/abs/2503.09299v1
results on graphon estimation to the problem of targeted interventions in networks games and provide numerical experiments to illustrate our foundings. The proofs of all the results are given in the appendix. Notation Let [n] :={1, . . . , n }. Given two numbers a, b, denote their maximum (minimum) by a∨b(a∧b). We use ...	https://arxiv.org/abs/2503.09299v1
the results of Sections 4 can be easily extended to the model given by Qij=ρnW(ξi, ξj). 1Note that the expected number of neighbours of a node in the graph defined by Ais of the order ρnn, and hence the expected number of edges in the graph is ρnn2. Hence, the density of the graph is of the order ρnn2/n2=ρn. Thus, the ...	https://arxiv.org/abs/2503.09299v1
of player xis given by U(s(x), zW(x\|s), θ(x)) =−1 2s(x)2+s(x)·(γzW(x\|s) +θ(x)). (3) Here, γ >0 is the peer-effect parameter, zW(x\|s) is the local aggregate sensed by player xvia the graphon, defined as zW(x\|s) :=Z1 0W(x, y)s(y)dy, (4) andθ(x) is a parameter modeling heterogeneity among the players. We denote a graphon ...	https://arxiv.org/abs/2503.09299v1
be reformulated as a semidefinite program (SDP) with only two variables and a linear matrix inequality (LMI) con- straint involving a matrix of size r+ 2 (see [Parise and Ozdaglar, 2023, footnote 14]). This insight, together with Theorem 1, highlights the need for a graphon estimator that simultaneously satisfies both ...	https://arxiv.org/abs/2503.09299v1
Conjugate Gradient method. This procedure generates a monotone sequence L1, L2, . . .that converges to the optimal solution L(see [Gander et al., 1989, Reinsch, 1967]). However, each iteration of the algorithm may require O(n·\|E\|) operations, where \|E\|is the number of edges in A, since solving the linear equation I+L...	https://arxiv.org/abs/2503.09299v1
an absolute constant defined in Proposition 5 in Appendix J. Remark 3.In [Klopp and Verzelen, 2019], the authors consider the problem of estimating the ma- trixQin cut-norm and show that for ρn>1/n, there exists a constant Csuch that inf bQsup Q∈SBM(2 ,ρn)E∥Q−bQ∥□≥Crρn n, where SBM(2 , ρn) denotes a Stochastic Block Mo...	https://arxiv.org/abs/2503.09299v1
=X 1≤i,j≤kSijI{(x, y)∈Pi×Pj}. Note that networks generated from the SBM graphon defined above correspond to networks generated from the classic Stochastic Block Model (SBM) [Holland et al., 1983]. The following result provides a bound on the error and the rank of the estimator WbQλwhen Wis an SBM graphon. Theorem 3. Co...	https://arxiv.org/abs/2503.09299v1
and the coefficients are bounded by M. Since the functions ckare bounded, Wsatisfies the ( H,1)-H¨ older condition for some H.4Consequently, using Theorem 5, we derive the following result. Theorem 6. Consider a graphon Wthat is (M, r)-analytic. Assume that nρn≥CPr.5log(2n/δ). Then, there are constants CandC′depending ...	https://arxiv.org/abs/2503.09299v1
\|x1−x2\|1/2+\|y1−y2\|1/2, 14 where we used the triangle inequality and the fact that√ a+b≤√a+√ bfor any non-negative a, b. For each n∈ {20,120, . . . , 4920}, we sample a network Aand heterogeneity parameters θas follows:  Aij=Bern (ρnW(ξ(i), ξ(j))), i < j, θi=ξ2 (i),(11) where ρn=n−0.25andξ1, . . . , ξ nare samples fr...	https://arxiv.org/abs/2503.09299v1
Upper Left: difference between target functions for optimal interventions of a network on nvertices sampled from (1 ,1/2)-H¨ older graphon W1(x, y) =p \|x−y\|and interventions based on hard-thresholding estimator. Lower Left: Rank of the hard-thresholding estimator for a network on nvertices sampled from W1(x, y).Midlle:...	https://arxiv.org/abs/2503.09299v1
and Caines, 2019a] Gao, S. and Caines, P. E. (2019a). Optimal and approximate solutions to linear quadratic regulation of a class of graphon dynamical systems. In 2019 IEEE 58th Conference on Decision and Control (CDC) , pages 8359–8365. [Gao and Caines, 2019b] Gao, S. and Caines, P. E. (2019b). Spectral representation...	https://arxiv.org/abs/2503.09299v1
center 0. Thus, using Proposition 7 we get ∥ˆθ1−ˆθ2∥L2≤(1 +γ∥W1∥op)2∥R2(γ,W1)[θ+ˆθ1]− R2(γ,W2)[θ+ˆθ1]∥L2 ≤(1 +γ∥W1∥op)2· ∥R2(γ,W1)− R2(γ,W2)∥op· ∥θ+ˆθ1∥L2. (12) To bound the operator norm of the difference, we use decomposition R2(γ,W) =∞X i=0(i+ 1)γiWi. The right-hand side converges because γ∥Wi∥<1 for i= 1,2. Hence, ...	https://arxiv.org/abs/2503.09299v1
bound, with probability at least 1 −δ/2, it holds µ{ψ(x)̸=ϕ(x)} ≤k−1X a=1ta≤2k nlog4k δ+k−1X a=1r 2ea nlog4k δ. (15) The second term in the right-hand side can be bounded using the Cauchy-Schwartz inequality to obtain k−1X a=1r 2ea nlog4k δ≤vuut2k nk−1X a=1ealog4k δ. Assumption n≥2klog4k δimplies 2k nlog4k δ≤r 2k nlog4...	https://arxiv.org/abs/2503.09299v1
where the sum is taken over all adjacency matrix A. We introduce a new random variable θ= max{k\|ξkbelongs to the first comminity }, with the convention that θ= 0 if ξkbelongs to the second community. Subject to θ, the distributions P∆(· \|θ) and P0(· \|θ) are identical, since both of them are random graph models with the...	https://arxiv.org/abs/2503.09299v1
1 −δ/2 and applying Lemma 1 we obtain the result. ii)Upper bound on the rank. We start by recalling the following result from approximation theory. Lemma 2. Fixk∈N. Define ∆ij= [(i−1)/k, i/k )×[(j−1)/k, j/k )for1≤i, j≤k. Define ℓ=⌊α⌋. Then for any (H, α)-H¨ older graphon Wthere exists a function Pk: [0,1]2→Rsuch that 1...	https://arxiv.org/abs/2503.09299v1
rank 1, then Pkhas decompositionPℓ2p r=1vrgT r, where vr= (ϕr(ξi))n i=1,gr= (ψr(ξi))n i=1. We have, σℓ2p(Q)≤ ∥Q−ρnPp∥op≤ ∥Q−ρnPp∥F≤Mnρ n21−p. For any k≥ℓ2, by choosing p=k ℓ2 we can conclude that σk(Q)≤Mnρ n21−p≤4Mnρ n2−k ℓ2. 28 Since∥Q∥op≤nρn, there exists constant C(M, r) such that σk(Q)≤C(M, r)nρn2−kfor all positi...	https://arxiv.org/abs/2503.09299v1
graphon Wis (H,1)-H¨ older for H= max {4Mr−1,2r−1}, so, repeating the proof of Case 2, we conclude that, with probability at least 1 −δ, we have T(A,θ)(ˆθ)−T(A,θ)(ˆθ′)≤2α(1 +α∥A∥op/N)2(∥θ∥/√ N+√ B)2 (1−α·max{∥A∥op/N, ρ N∥A′∥op/nρn})5 × 15rρN nρn+C2(M, r)ρNlog(16 n/δ) n1/2! , and rank( cW)≤C2(M, r) log( nρn) by the co...	https://arxiv.org/abs/2503.09299v1
ij\|Xij\|. In our paper, we choose ϵ=3√ 2 4−1≈0.06. Then, (1+ ϵ)2√ 2 = 3. We denote the corresponding cϵbyCPr.5. J.2 Stability of variational inequality problem LetHbe a Hilbert space. Definition 4. We say that a function F:H→Hadmits the strong monotonicity property with constant α >0 if for each x1, x2we have ⟨F(x1)−F(x...	https://arxiv.org/abs/2503.09299v1
Zaiats, 2009]. Suppose that we have some family of ditributions Pθ, θ∈Θ,over a sample space Xparametrized by parametric space Θ. Let R: Θ×Θ→R+be a semidistance on Θ. Any function bθ:X → Θ is referred to as an estimator of the parameter θ. Given a sample Xdrawn from the distribution Pθ, the quantity R(bθ(X), θ) serves a...	https://arxiv.org/abs/2503.09299v1
Competing-risk Weibull survival model with multiple causes Kai Wanga,b, Yuqin Mua,b, Shenyi Zhanga, Zhengjun Zhangc,d, and Chengxiu Ling*,a aAcademy of Pharmacy, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China bDepartment of Mathematical Sciences, University of Liverpool, Liverpool, L693BX, UK cDepartment of...	https://arxiv.org/abs/2503.09310v1
risk analyses, including worldwide studies on HIV-positive patients (Breger et al., 2020), gastric adenocarcinoma patients (Xie et al., 2020), post-lung transplantation patients (Anderson et al., 2023), and all-cause mortality (Bishop et al., 2023; Dobson et al., 2023; Wijnen et al., 2022). In this paper, we propose a ...	https://arxiv.org/abs/2503.09310v1
should not be completely identical, otherwise unidentifiability will be introduced. The intercepts α1, . . . , α Lcontrol the relative importance of each risk, where a very large intercept indicates an approximately eliminated risk with the min-competing structure. Then, the survival probability of Tbecomes P{T > t}=P{...	https://arxiv.org/abs/2503.09310v1
by Jensen’s inequality when p(k\|θ,y)̸=p(k\|θ0,y). To this, we introduce an augmented variable K, denoting the dominating group for the uncensored samples, the complete likelihood for ( T, K) is LC(θ) =nY i=1 LY l=1f(Ti, Ki=l)I(Ki=l)!δi S(Ti)1−δi. (3.1) 6 Noting the explicit form of survival function S(t) specified in Eq...	https://arxiv.org/abs/2503.09310v1
penalizing the intercept αlin the model specified in Eq.(2.1) to the direction towards infinity. Due to the parametrization, the penalization on the group intercept would not penalize any intercept to exactly infinity and hence would not introduce group sparsity directly. With the lasso-type penalization, Ql(θ\|θ(m)) = ...	https://arxiv.org/abs/2503.09310v1
time Similar to the expected survival time in traditional parametric survival models, we compute the expectation under the min-survival structure. The conditional expectations are not the minimum of the expected values across the risks (groups) due to the heteroskedasticity. The expectation of Tis given by E{T}=Z∞ 0S(t...	https://arxiv.org/abs/2503.09310v1
of uncensored ( d) and censored ( c) samples for three examples and provides the estimated values of the mean and standard errors (SE) for σ, α, β . The results indicate a slight reduction in precision with increasing proportions of censorship. Nonetheless, all parameters are accurately estimated with relatively small ...	https://arxiv.org/abs/2503.09310v1
1 0.911 (0.039) 0.955 (0.085) −2.907 (0.071) 2.001 (0.053) - 0.934 (0.052) - - CF 2 1.071 (0.038) 1.400 (0.101) 2.011 (0.082) 2.007 (0.053) - - - - (n=d= 1500) CF 3 1.101 (0.048) 1.043 (0.129) −2.000 (0.101) 3.098 (0.070) 2.035 (0.088) - - - Example 2CF 1 1.025 (0.040) 1.011 (0.101) −2.977 (0.085) 1.967 (0.062) - 0.977...	https://arxiv.org/abs/2503.09310v1
the most common cause 17 of dementia (Knopman et al., 2021). Although the amyloid beta hypothesis is the predomi- nant explanation, the exact causes of Alzheimer’s disease remain poorly understood (Burns and Iliffe, 2009). Definitive diagnosis of Alzheimer’s disease is only possible through au- topsy, while clinical di...	https://arxiv.org/abs/2503.09310v1
and middle). Figure 2: Time-dependent ROCs (left, middle) for competing-Weibull, Cox, and Weibull models at 10 percentile survival time ( t= 357) and at 2 years ( t= 730). Estimated time- varying winning probability (right) of four competing groups for uncensored samples. Table 5 presents the estimated scale parameters...	https://arxiv.org/abs/2503.09310v1
model to the max-linear family. This model keeps the linear structure within each competing group, maintaining the inter- pretability of the Weibull model, and employs the minimum structure to allow non-Weibull survival time and competition across the factors. Our proposal also enables incorporation of 21 lasso or othe...	https://arxiv.org/abs/2503.09310v1
parsi- mony and goodness of fit, considering factors such as the number of variables, number of groups, overall accuracy, redundancy, and similar effects of variables, as proposed by (Liu et al., 2024) for logistic max-linear models. Additionally, an automatic selection algorithm is encouraged to explore possible confi...	https://arxiv.org/abs/2503.09310v1
−LX k=1Ti exp αk+X⊤ ikβk1/σk!! ≤δilog LX l=1T1/σl−1 i σlexp (αl+X⊤ ilβl)/σlexp −Ti exp αl+X⊤ ilβl1/σl!! =:δilog LX l=1ψ(Ti;σl, αl, Xil,βl)! . Note that ψ(t;σl, αl, Xil,βl), t > 0,is the Weibull density with scale parameter exp αl+X⊤ ilβl and shape parameter 1 /σl. Thus for 0 < t < 1, we have t−1> t1/σl−1, a...	https://arxiv.org/abs/2503.09310v1
Under assumption A3, the inequality is strict, and θ0is the unique maximizer of Q(θ) over Θ. Noting that Θ is compact (assumption A1) and Q(θ) is continuous, Lemma 6.3 follows. Lemma 6.4 (Extremum Consistency Theorem, Newey and McFadden (1994, Theorem 2.1)) . If there is a function Q(θ), such that 1.Q(θ)is uniquely max...	https://arxiv.org/abs/2503.09310v1
lS1/σl il LP l=1 1 TiσlS1/σl il− −1 Tiσ2 lS1/σl il2 LP l=1 1 TiσlS1/σl il2 ≤ 1 Tiσ3 lS1/σl il LP l=1 1 TiσlS1/σl il + −1 Tiσ2 lS1/σl il2 LP l=1 1 TiσlS1/σl il2 ≤2 σ2 l. Further, we deal with the second-order derivative w.r.t. βlandβk. Recalling Eqs.(6.4) and 31 (6.5), we have for l̸=k ∂2logg(Ti\|Xi;θ,1)...	https://arxiv.org/abs/2503.09310v1
calculation and the properties of Weibull density that E (logTi)2 ≤δiLX l=1Z∞ 0(logt)2ψ(t;σl, αl, Xil,βl)dt +(1−δi)LX l=1Z∞ 0(logt)2exp −t exp αl+X⊤ ilβl1/σl! dt <∞, 34 and we obtain Eq.(6.2). The next two lemmas are used for the proof of the asymptotic normality of the MLE for the parameters involved. Lemma 6.5. ...	https://arxiv.org/abs/2503.09310v1
=Z∞ M−h′(t)LP k=1ζk(t) LP k=1ζk(t)2+h(t)LP k=1(1 σk−1)ζk(t) t LP k=1ζk(t)2dt=−h(t) LP k=1ζk(t) ∞ M=h(M) LP k=1ζk(M). Thus, Z∞ Mh(t)dt≥h(M) LP k=1ζk(M) 1 +1 σ∗MLP k=1ζk(M) −1 ≥h(M) LP k=1ζk(M) 1−1 σ∗MLP k=1ζk(M)  using 1 /(1 +x)≥1−xforx >0. We complete the proof of Lemma 6.7. References Ahmad, F....	https://arxiv.org/abs/2503.09310v1
L. (1995). Analysis of competing risks survival data when some failure types are missing. Biometrika , 82(4):821–833. He, Y., Kim, S., Mao, L., and Ahn, K. W. (2022). Marginal semiparametric transfor- mation models for clustered multivariate competing risks data. Statistics in Medicine , 41(26):5349–5364. Heagerty, P. ...	https://arxiv.org/abs/2503.09310v1
Latouche, A. (2013). Regression modeling of the cumulative inci- dence function with missing causes of failure using pseudo-values. Statistics in medicine , 32(18):3206–3223. Moreno-Betancur, M., Sadaoui, H., Piffaretti, C., and Rey, G. (2017). Survival analysis with multiple causes of death: extending the competing ri...	https://arxiv.org/abs/2503.09310v1
arXiv:2503.09507v1 [math.ST] 12 Mar 2025Parameter estimation for the stochastic Burgers equation driven by white noise from local measurements Josef Jan´ ak Universit` a di Pavia, Italy Email: josefjanak@seznam.czEnrico Priola Universit` a di Pavia, Italy Email: enrico.priola@unipv.it March 13, 2025 Abstract For one di...	https://arxiv.org/abs/2503.09507v1
in space of the solution against some small “ob servational window”. That is represented by a function called kernel Kthat is scaled and localized around certain observation point x0∈Λ. The observational data then comes in the form of convolution /an}brack⌉tl⌉{tX(t),Kδ,x0/an}brack⌉tri}htL2(Λ)and the asymptotics is stud...	https://arxiv.org/abs/2503.09507v1
measu rements. The precise statement is formulated in Theorem 3.4. Based on the asymptotic result, we can deduce (data-driven) con ﬁdence in- tervalsfor ϑ(thatalsomatchthe developedtheory). We mentionthatnumerica l simulations for stochastic Burgers equations ( 1.1) driven by space-time white noise were already present...	https://arxiv.org/abs/2503.09507v1
equation ( 1.1) driven by additive space-time white noise. The initial value X0∈H3/2is supposed to be deterministic. By Theorem 14.2.4 in [ 14] (see also [ 12], [13] and Remark 2.3below), there exists a unique mild solution to the equation ( 1.1). 4 Namely, for any T >0, there exists an L2(Λ)-valued adapted process X= ...	https://arxiv.org/abs/2503.09507v1
all t≥0.In [13] they also deﬁne the usual stopping times: τn= inf{t≥0 :/bar⌈blXn(t)/bar⌈bl ≥n}, n≥1. (2.6) They prove that τn→ ∞,P-a.s.. Since Xn(t) =Xm(t),m≥n,t≤τnone can setX(t) =Xn(t),t≤τn, and obtain a mild solution to the initial Burgers equation. 2.3 The observation scheme As motivated in [ 2], [3], [23], we obse...	https://arxiv.org/abs/2503.09507v1
Iδ)−1Rδand (Iδ)−1Mδ vanish, as δ→0, and to prove asymptotic normality, we will show that δ−1(Iδ)−1Rδ→0, whileδ−1(Iδ)−1Mδconverges in distribution to a Gaussian random variable. To analyze these terms, we use the ‘splitting technique’ of the solut ion (i.e., we write X=¯X+/tildewideXas above) and we study separately the...	https://arxiv.org/abs/2503.09507v1
sum in (S.3) to converge. (Such result comes fr om Lemma S.2(ii) and it is not quite clear if Lemma S.1 is able to tame it even in the case γ >1/4.) Moreover, in the proof of Lemma S.8, it is not completely clear why the random variable /tildewideX(t,0) (that is basically /tildewideX(t,x0)) isG-measurable. The isonorma...	https://arxiv.org/abs/2503.09507v1
< s≤1, the operator −(−A)sis also the generator of a symmetric Markovian semigroup Ts tonL2(Λ) obtained by subordination of order sofTt. (For more on subordination like Tf t=/integraltext∞ 0Tsµf t(ds), see for instance [ 4].) The Dirichlet form associated to Ts tisEswith Dom( Es) = Dom/parenleftbig (−A)s/2/parenrightbi...	https://arxiv.org/abs/2503.09507v1
2.1.We deduce from Lemma 14.2.1 of [ 14] that the solu- tionX∈C([0,T];Hs),P-a.s., for s∈[0,1/2). For the stochastic convolution ¯X, we know that from Proposition 4.2, for the nonlinear part /tildewideX, it follows from Lemma4.5(iv). Recall that the initial condition X0∈H3/2. To get more spatial regularity for /tildewid...	https://arxiv.org/abs/2503.09507v1
weak solution Proof of Lemma 2.2.The proof is not diﬃcult, but rather lengthy. We give a sketch of proof. For the nonlinear part /tildewideXwe have by ( 2.1)P-a.s. /tildewideX(t) =Sϑ(t)X0+1 2/integraldisplayt 0Sϑ(t−s)∂x/parenleftBig (¯X(s)+/tildewideX(s))2/parenrightBig ds. (4.5) Now we work with ωﬁxed (i.e., we ﬁx ω∈Ω...	https://arxiv.org/abs/2503.09507v1
/tildewideXover¯X. The convergence δU3,δP→0 will be established by a speciﬁc representation of /tildewideX(t,x0). First, we establish upper bounds for relevant terms (c.f. Lemma S.3 in [2]). Lemma 4.7. For any small ε >0, uniformly in 0≤t≤T,k≥1,r≤1: (i)/vextendsingle/vextendsingle/vextendsingle/tildewideX∆ δ(t)/vextend...	https://arxiv.org/abs/2503.09507v1
main point of the proof will be to represent /tildewideX(t,x0),x0∈Λ,t∈[0,T] a.e., in the form /tildewideX(t,x0) =∞/summationdisplay i=1bi(t)νi, (4.13) wherebi∈L2([0,T]) are deterministic functions and ( νi)∞ i=1form an orthonor- mal basis in a separable space L2(Ω,F′) =L2(Ω,F′,P), with a σ-ﬁeldF′⊂ F that will be speciﬁ...	https://arxiv.org/abs/2503.09507v1
the solution Xto the Burgers equation veriﬁes, P-a.s., for any t∈[0,T],n≥1, X(t∧τn) =Xn(t∧τn) (τnis deﬁned in ( 2.6)). We deduce that X(t∧τn) isF′-measurable, t∈[0,T], n≥1. Passing to the limit, P-a.s. asn→ ∞(sinceτn→ ∞) we obtain that for eacht∈[0,T] theH-valued random variable X(t) isF′-measurable. Assertion (4.20) f...	https://arxiv.org/abs/2503.09507v1
E¯Iδ+δ2oP(δ−2) δ2E¯IδP→1, as well as δ2E¯IδP→T/bar⌈blK′/bar⌈bl2 L2(R) 2ϑ/bar⌈blK/bar⌈bl2 L2(R). Therefore, the ﬁrst factor in ( 4.25) converges to N(0,1) in distribution by the standard continuous martingale central limit theorem (e.g., [ 26], Theorem 1.19, or [27], Theorem 5.5.4), while the second factor assembles the...	https://arxiv.org/abs/2503.09507v1
Prob. Th. Rel. Fields 103, 143–163. [21] I. Iscoe, M. B. Marcus, D. McDonald, M. Talagrand, J. Zinn (19 90) Conti- nuity ofl2-valued Ornstein-Uhlenbeckprocesses, The Annals of Probability , 68–84. [22] K. Itˆ o, M. Nisio (1968).On the convergenceofsums ofindepe ndent Banach space valued random variables, Osaka J. Math ...	https://arxiv.org/abs/2503.09507v1
arXiv:2503.09583v1 [cs.LG] 12 Mar 2025Minimax Optimality of the Probability Flow ODE for Diﬀusion Models Changxiao Cai∗Gen Li† March 13, 2025 Abstract Score-based diﬀusion models have become a foundational par adigm for modern generative modeling, demonstrating exceptional capability in generating sampl es from complex...	https://arxiv.org/abs/2503.09583v1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 Theoretical guarantees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4 Analysis 12 4.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ...	https://arxiv.org/abs/2503.09583v1
B.1 Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 B.2 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 B.3 Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....	https://arxiv.org/abs/2503.09583v1
Chen et al. (2024a );Tang and Zhao (2024) for an overview of theoretical advances. At the core of diﬀusion models are two complementary process es: a forward process that progressively corrupts a sample from the target data distribution with Gau ssian noise, and a reverse process that aims to reverse the forward proces...	https://arxiv.org/abs/2503.09583v1
) and ODE-based samplers ( Chen et al. ,2023c;Li et al. ,2023) known to converge polynomially fast toward distributions whose total variation (TV) distance to the target scales proporti onally to the score estimation error. These results suggest that with accurate score estimates available, diﬀu sion models can, in pri...	https://arxiv.org/abs/2503.09583v1
truncated score estimator obt ains the (near-)minimax rate in TV distance for subgaussian distributions with β-Soblev-smooth densities with β≤2.Holk et al. (2024) investigated stochastic samplers in reﬂected diﬀusion models for constr ained generative modeling, and established the 3 (near-)minimax optimality in TV dist...	https://arxiv.org/abs/2503.09583v1
with β-Hölder smooth densities for β≤2 (see Section 2.2for rigorous deﬁnitions), we propose a smooth kernel-based regularized score estimator that simultaneously controls the L2score error and the associated Jacobian error. Building upo n a reﬁned convergence guarantee that characterizes the discretizat ion error of th...	https://arxiv.org/abs/2503.09583v1
TV distance) either replied onL∞-accurate estimates ( De Bortoli et al. , 2021;Albergo et al. ,2023), or exhibited exponential dependence ( De Bortoli ,2022;Block et al. ,2020). A breakthrough came from Lee et al. (2022), which provided the ﬁrst polynomial iteration complexity as- suming only L2-accurate score estimate...	https://arxiv.org/abs/2503.09583v1
provide a brief introduction to score-ba sed diﬀusion models. Forward process. The forward process begins with a sample X0∼p⋆ 0distributed according to some initial distribution p⋆ 0(typically chosen to match or closely approximate the targe t distribution p⋆) and progressively adds Gaussian noise via the following Mar...	https://arxiv.org/abs/2503.09583v1
scheme in ( 7) is fundamentally grounded in the probability ﬂow ODE for diﬀusion models. Indeed, the continuous process (Xsde t)t∈[0,T]deﬁned by the forward SDE ( 5), which can be interpreted as the continuum limit of the forwar d process (Xk)K k=0in (1), admits a reverse- time process (Yode t)t∈[0,T](Song et al. ,2020...	https://arxiv.org/abs/2503.09583v1
this condition accommodates many cases of interest and simp liﬁes analysis, it can exclude multi-modal distributions frequently found in real-world application s—where diﬀusion models have shown particular advantages over traditional methods like Langevin dynamic s. Indeed, a constant lower-bounded density on a compact...	https://arxiv.org/abs/2503.09583v1
score estima tor/hatwidest(·) :Rd→RdforZtas follows: /hatwidest(x):=∇/hatwidept(x) /hatwidept(x)ψ/parenleftbig /hatwidept(x);ηt/parenrightbig withηt:=logn n(2πt)d/2. (18) 9 Algorithm 1 Probability ﬂow ODE-based sampler 1:Input: training data{X(i)}n i=1sampled from the target distribution p⋆. 2:Set the learning rates {α...	https://arxiv.org/abs/2503.09583v1
random initi alization of YK), that is, (YK−1,...,Y 1)is purely deterministic given YK. 10 3.2 Theoretical guarantees In this section, we present the end-to-end sampling guarant ee of the proposed ODE-based sampler in Theo- rem1. A proof outline is provided in Section 4. Theorem 1. Suppose the target distribution p⋆sat...	https://arxiv.org/abs/2503.09583v1
Dou et al. (2024) where early stopping techniques are not needed to achieve the (nea r-)minimax optimal performance. •Analysis framework for ODE-based samplers. The theoretical framework we establish provides a foun- dation for analyzing the performance guarantees of various ODE-based samplers. For instance, it can be ...	https://arxiv.org/abs/2503.09583v1
t >0, we denote the density, score function, and associated Jacobia n matrix of Ztby pt(x):=pZt(x), s t(x):=sZt(x),andJt(x):=Jst(x). (24) 12 By Tweedie’s formula ( Efron,2011), the score function st(x)takes the form st(x) =1 tE/bracketleftbig Z0−Zt\|Zt=x/bracketrightbig . (25) The Jacobian matrix Jt(x)can be expressed a...	https://arxiv.org/abs/2503.09583v1
an arbitrary score estimator. The proof can be found in Appendix A.1. Theorem 2. Suppose that the number of iterations satisﬁes K/greaterorsimilard2(logK)5andKc2≥EX0∼p⋆ 0[/⌊a∇d⌊lX0/⌊a∇d⌊l2 2]for some absolute constant c2>0and that the learning rates are chosen according to (15)withc1/8−c2/2≥1. Then for any score estima...	https://arxiv.org/abs/2503.09583v1
in(20)satisfy ε2 sc/lessorsimilarC′ d n/braceleftbigg 1+σd/parenleftbigg τ−d/2∧Kc0d/2 dlogK/parenrightbigg/bracerightbigg (logn)d/2+1; (35a) εjcb/lessorsimilar/radicalbigg C′ d n/braceleftbigg 1+σd/2/parenleftbigg τ−d/4∧Kc0d/4 dlogK/parenrightbigg/bracerightbigg (logn)d/4+1+C′ dσd n/parenleftbigg τ−d/2∧Kc0d/2 dlogK/par...	https://arxiv.org/abs/2503.09583v1
( 12): sXk(x) =1√αkstk(x/√αk)andJsXk(x) =1 αkJtk(x/√αk). Leveraging this with our construction of the score estimato r/hatwidesXk=/hatwidestk(x/√αk)/√αkin (20) allows us to establish a direct correspondence between E/bracketleftbig /⌊a∇d⌊lJ/hatwidesXk(Xk)−JsXk(Xk)/⌊a∇d⌊l/bracketrightbig andE/bracketleftbig /⌊a∇d⌊lJ/hat...	https://arxiv.org/abs/2503.09583v1
Ec t(x)/parenrightbig /lessorsimilarn−10. (41) 17 Remark 5. Since the target density is allowed to be vanishingly small p⋆(x), the termpt(x)appearing on the numerator on the right-hand side of the above bounds is essential to obtaining sharp bounds. In addition, both /⌊a∇d⌊lst(x)/⌊a∇d⌊l2and/⌊a∇d⌊lHt(x)/⌊a∇d⌊lassociated...	https://arxiv.org/abs/2503.09583v1
selected τ=n−2/(d+2β)into Theorem 3, we can characterize the estimation errors εsc andεjcb(deﬁned in ( 33)) of our proposed score estimator {/hatwidesXk(·)}K k=1in (20): ε2 sc/lessorsimilarC′ d n/braceleftbigg 1+σd/parenleftbigg nd d+2β∧Kc0d 2 dlogK/parenrightbigg/bracerightbigg (logn)d 2+1/lessorsimilarC′ dσdn−2β d+2β...	https://arxiv.org/abs/2503.09583v1
Hong K ong Direct Grant for Research. References Albergo, M. S., Boﬃ, N. M., and Vanden-Eijnden, E. (2023). St ochastic interpolants: A unifying framework for ﬂows and diﬀusions. arXiv preprint arXiv:2303.08797 . Anderson, B. D. (1982). Reverse-time diﬀusion equation mod els.Stochastic Processes and their Applications ...	https://arxiv.org/abs/2503.09583v1
neural information processing systems , 34:8780–8794. Dou, Z., Kotekal, S., Xu, Z., and Zhou, H. H. (2024). From opti mal score matching to optimal sampling. arXiv preprint arXiv:2409.07032 . Durmus, A. and Moulines, É. (2019). High-dimensional Bayes ian inference via the unadjusted Langevin algorithm. Bernoulli , 25(4...	https://arxiv.org/abs/2503.09583v1
low-dimensio nal structures in score-based diﬀusion models. arXiv preprint arXiv:2405.14861 . Li, G. and Yan, Y. (2024b). O(d/T)convergence theory for diﬀusion probabilistic models unde r minimal assumptions. arXiv preprint arXiv:2409.18959 . Li, S., Chen, S., and Li, Q. (2024c). A good score does not lead to a good ge...	https://arxiv.org/abs/2503.09583v1
S. (2021). Solving in verse problems in medical imaging with score-based generative models. arXiv preprint arXiv:2111.08005 . Song, Y., Sohl-Dickstein, J., Kingma, D. P., Kumar, A., Ermo n, S., and Poole, B. (2020b). Score-based generative modeling through stochastic diﬀerential equat ions.arXiv preprint arXiv:2011.134...	https://arxiv.org/abs/2503.09583v1
addition, we deﬁne the pointwise score error and Jacobian error as ε2 sc,k(x):=/vextenddouble/vextenddouble/hatwidesXk(x)−sXk(x)/vextenddouble/vextenddouble2 2andεjcb,k(x):=/vextenddouble/vextenddoubleJ/hatwidesXk(x)−JsXk(x)/vextenddouble/vextenddouble. (49) Finally, for two functions f(K),g(K)>0, we usef(K)≪g(K)to mea...	https://arxiv.org/abs/2503.09583v1
expectation over the score estimator {/hatwidesXk}K k=1, we ﬁnd that E/bracketleftbig TV(pX1,pY1)/bracketrightbig /lessorsimilardlog4K K+√ dlog3/2K/radicaltp/radicalvertex/radicalvertex/radicalbt1 KK/summationdisplay k=1(1−αk)ε2 sc,k+dlogK KK/summationdisplay k=2(1−αk)εjcb,k =dlog4K K+√ dεsclog3/2K+dεjcblogK, where the...	https://arxiv.org/abs/2503.09583v1
As for the integral over the complement set of Etyp 2, we can derive /integraldisplay x0/∈Etyp 2pX0\|Xk(x0\|x)exp/parenleftbigg −(1−αk)/vextenddouble/vextenddoublex−√αkx0/vextenddouble/vextenddouble2 2 2(αk−αk)(1−αk)−/⌊a∇d⌊lu/⌊a∇d⌊l2 2−2u⊤/parenleftbig x−√αkx0/parenrightbig 2(αk−αk)/parenrightbigg dx0 (i) ≤∞/summationdis...	https://arxiv.org/abs/2503.09583v1
K/parenleftbiggα1 1−α1/parenrightbiggd/2 +1 c1logKk0/summationdisplay k=2αk−1−αk αk−1(1−αk−1)/parenleftbiggαk 1−αk/parenrightbiggd/2 +1 (ii) ≤1 K(1−α1)d/2+1 c1logKk0/summationdisplay k=22 αk(1−αk)/parenleftbiggαk 1−αk/parenrightbiggd/2/parenleftbig αk−1−αk/parenrightbig +1 =1 K(1−α1)d/2+2 c1logKk0/summationdisplay k=2α...	https://arxiv.org/abs/2503.09583v1
as cη≥2. This proves the claim in ( 74). Now, given the expression of /hatwidest(x), we can then decompose E/bracketleftBig/vextenddouble/vextenddouble/hatwidest(Zt)−st(Zt)/vextenddouble/vextenddouble2 2/bracketrightBig =/integraldisplay RdE/bracketleftBig/vextenddouble/vextenddouble/hatwidest(x)−st(x)/vextenddouble/ve...	https://arxiv.org/abs/2503.09583v1
(27a), one can bound /vextenddouble/vextenddouble/hatwidegt(x)/vextenddouble/vextenddouble 2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble1 ntn/summationdisplay i=1(Xi−x)ϕt(Xi−x)/vextenddouble/vextenddouble/vextenddouble/vextenddouble 2 ≤max i∈[n]1 t/vextenddouble/vextenddouble(Xi−x)ϕt(Xi−x)/vextenddouble/ve...	https://arxiv.org/abs/2503.09583v1
n(2πt)d/2t+1 t/radicalBigg logn n(2πt)d/21 n(2πt)d/2/parenrightbigg +/⌊a∇d⌊lHt(x)/⌊a∇d⌊l pt(x) ≍1 t√logn+/⌊a∇d⌊lHt(x)/⌊a∇d⌊l pt(x) where (i) uses ( 40c) in Lemma 3; (ii)pt(x)≤cηηton the setFc t; (iii) plugs in the values of ηtin (18). Thus, we ﬁnd that (I)/lessorsimilar/integraldisplay Fc t/parenleftbigg1 t+/⌊a∇d⌊lHt(x...	https://arxiv.org/abs/2503.09583v1
the standard Gaussian property and the change of variable z=y/√ t, one has: /integraldisplay Rdysϕt(y)dy=t\|s\|/2/integraldisplay Rdd/productdisplay i=1zsi iϕ1(z)dz≤Cst\|s\|/2 39 for some constant Csdepending on sif\|s\|is even, and the integral is zero for odd \|s\|. Similarly, the standard Gaussian property tells us /integra...	https://arxiv.org/abs/2503.09583v1

Subsets and Splits

No community queries yet

The top public SQL queries from the community will appear here once available.