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and Schrab (2023) and proofs in Albert et al. (2022) to upper bound the first probability under the sufficient condition on L2-norms. A slight modification of Albert et al. (2022, Lemma 1) yields that a sufficient condition for the first probability in the above display to be less than β/16 is dHSIC2 k(PX1,...,Xd)≥p Va... | https://arxiv.org/abs/2503.18721v2 |
depend on α, β, R, M, d Y, dZ. Following the proofs of Theorem 4.4 along with the sensitivity result for UdHSIC in Lemma B.1, we may arrive at the point where the type II error of ϕu dpdHSIC is upper bounded as E[1−ϕu dpdHSIC ]≤P UdHSIC≤C1 n√λ0+C2 nλ0ξϵ,δ +15 16β We then use the proofs of Albert et al. (2022, Theorem... | https://arxiv.org/abs/2503.18721v2 |
in their kth component for some k∈[n]. Denote the resampled test statistics computed on eXφ1n, . . . ,eXφBnbyeT1, . . . ,eTB. We write Ti=T(Xφin),r1−α=r1−α(Xn;{φi, ζi}B i=1) ander1−α=r1−α(eXn;{φi, ζi}B i=1). Having this notation, first note that Ti≥eTi−∆Tfor all i∈[B] and thus 1−α≤1 BBX i=11(Ti+ 2∆ Tξϵ,δζi≤r1−α)≤1 BBX ... | https://arxiv.org/abs/2503.18721v2 |
q⟩Hkfor all f∈ Fk. Then we have dHSIC k(PX1,...,Xd)2= sup f∈Fk EPX1,...,X d[f(X1, . . . , Xd)]−E⊗d j=1PXj[f(X1, . . . , Xd)]2 = sup f∈Fk⟨f, µ p−µq⟩Hk2 =∥µp−µq∥2 Hk =⟨µp, µp⟩Hk+⟨µq, µq⟩Hk−2⟨µp, µq⟩Hk =EPX1,...,X dk(X, X′) +E⊗d j=1PXjk(X, X′)−2EPX1,...,X d,⊗d j=1PXjk(X, X′), where the subscripts of the expectation ... | https://arxiv.org/abs/2503.18721v2 |
has ⌊n/(d+ 1)⌋independent Rademacher random variables. To elaborate it clearly, we focus on the first group, which is introduced by discussing the element “1” belongs to which set. Specifically, we categorize the proper subsets ( S1, S2) of [d] into three distinct classes: the first class denoted with superscript “ a” ... | https://arxiv.org/abs/2503.18721v2 |
π1 m1 kandXj π1 m2 kfork= 1, . . . ,⌊n/(d+ 1)⌋and j∈[d], with corresponding Rademacher random variables εj 1, . . . , εj ⌊n/(d+1)⌋, denoted as εj. Similar to the discussion above, the original statistic is equal in distribution to the sum of the U-statistics, where the kernel functions are the product of the original k... | https://arxiv.org/abs/2503.18721v2 |
generality, we suppose that 1 ∈Sc 1∩S2and 2 ∈S1∩Sc 2. Thus the kernel with Rademacher r.v.s is Y j∈Sc 1∩S2εj i1Y j∈S1∩Sc 2εj i2Y j∈Sc 1∩Sc 2εj i1εj i2gπ,M(S1, S2;i1, i2) =ε1 i1ε2 i2Y j∈Sc 1∩S2\{1}εj i1Y j∈S1∩Sc 2\{2}εj i2Y j∈Sc 1∩Sc 2εj i1εj i2gπ,M(S1, S2;i1, i2) =:ε1 i1ε2 i2gb,ε π,M(S1, S2;i1, i2;ε3, . . . ,εd), and t... | https://arxiv.org/abs/2503.18721v2 |
il, Xl πl jl) −2 nd+1X (i,j1,...,jd)∈jn d+1dY l=1kl(Xl πl i, Xl πl jl). Then the difference between the U-statistic and the squared V-statistic can be bounded as |Uπ,dHSIC−\dHSIC2(Xπ n)| ≤ (n−2)! n!X (i,j)∈in 2dY l=1kl(Xl πl i, Xl πl j)−1 n2X (i,j)∈jn 2dY l=1kl(Xl πl i, Xl πl j) + (n−2d)! n!X (i1,j1,...,id,jd)∈in 2ddY ... | https://arxiv.org/abs/2503.18721v2 |
verify that (II) ≲p K0/nto complete the proof. (II)≤1 nnX j=1Esup f∈Fk 1 nd−1X jn d−1f(X1 i1, . . . , Xd−1 id−1, Xd j)−E⊗d j=1PXj[f(X1, . . . , Xd)] 63 =Esup f∈Fk 1 nd−1X jn−1 d−1f(X1 i1, . . . , Xd−1 id−1, Xd n) +X jn d−1\jn−1 d−1f(X1 i1, . . . , Xd−1 id−1, Xd n) −E⊗d j=1PXj[f(X1, . . . , Xd)] ≤Esup f∈Fk 1 nd−1X jn−... | https://arxiv.org/abs/2503.18721v2 |
Confidence set for mixture order selection Alessandro Casa1and Davide Ferrari1 1Faculty of Economics and Management, Free University of Bozen-Bolzano Abstract A fundamental challenge in the application of finite mixture models is selecting the number of mixture components, also known as order. Traditional approaches re... | https://arxiv.org/abs/2503.18790v1 |
uncertainty: when data are highly informative, the MSCS is small, converging toward a single best model in the most extreme case; while with limited or noisy data, it expands, indicating greater uncertainty in model selection. As the sample size grows, the MSCS captures the true order with high probability, providing a... | https://arxiv.org/abs/2503.18790v1 |
ˆkcomponents. The alternative hypothesis is that the model with kcomponents is further from the truth compared to the selected model. To this end, we use the following Vuong type likelihood ratio statistic (Vuong, 1989): LRT∗(ˆk,k) :=LRT(ˆk,k)−δn(ˆk,k) := 2/parenleftig ℓ(ˆθk)−ℓ(ˆθˆk)/parenrightig −δn(ˆk,k), (3) where... | https://arxiv.org/abs/2503.18790v1 |
at the maximum likelihood estimators. The penalty term δn(ˆk,k)in (3) plays a crucial role in balancing model complexity and goodness of fit in the likelihood ratio test. In particular, it corrects for the increased flexibility of models with a larger number of components, thus preventing overfitting. When comparing mo... | https://arxiv.org/abs/2503.18790v1 |
number of components, even when the initial selection fails to identify the correct model. 4 Numerical examples To empirically evaluate the finite-sample performance of the proposed approach, we gen- erateni.i.d. samples from univariate Gaussian mixture densities employed in Casa et al. (2020). The density curves are i... | https://arxiv.org/abs/2503.18790v1 |
97.2 (4.1) 97.0 (3.7) 98.6 (4.5) ˆk=k0(%) 53.6 2.2 3.0 26.0 2500.10Coverage (%) 99.6 (1.9) 70.0 (2.4) 69.8 (2.4) 96.8 (2.4) ˆk=k0(%) 94.2 4.6 5.8 73.8 0.05Coverage (%) 99.8 (2.4) 86.0 (3.0) 83.6 (2.9) 97.8 (3.0) ˆk=k0(%) 93.6 4.0 3.8 69.6 0.01Coverage (%) 99.6 (3.4) 97.0 (4.1) 95.6 (3.8) 99.4 (4.4) ˆk=k0(%) 92.4 5.0 4.... | https://arxiv.org/abs/2503.18790v1 |
would enhance its applicability but re- quires the derivation of novel distributional results or reliance on resampling methods to 11 approximate the null distribution of the likelihood ratio test. References Aitkin, M. (2001). Likelihood and bayesian analysis of mixtures. Statistical Modelling , 1(4):287–304. Bouveyro... | https://arxiv.org/abs/2503.18790v1 |
arXiv:2503.18801v1 [math.OC] 24 Mar 2025Benign landscapes for synchronization on spheres via normalized Laplacian matrices Andrew D. McRae∗ March 25, 2025 Abstract We study the nonconvex optimization landscapes of synchron ization problems on spheres. First, wepresentnewresultsforthestatistical problemof synchronizatio... | https://arxiv.org/abs/2503.18801v1 |
observe Rij≈zizjfor certain (unordered) pairs ( i,j). The problem ofestimating zfrom such measurements is called Z2synchronization (due to the identification of {±1}under multiplication with the two-element group Z2) and arises in applications such as graph clustering (where the elements of zrepresent graph vertex label... | https://arxiv.org/abs/2503.18801v1 |
larger rneeds to be for these results to apply. In this work, we prove that, for the above problems, the smallest c ontinuous relaxation ( r= 2) is sufficient to obtain asymptotic exact recovery all the way up to th e optimal threshold. This shows that optimization over ≈nvariables suffices to recover zexactly whenever it ... | https://arxiv.org/abs/2503.18801v1 |
not transfer in a straightforward way from one class of network to another. Can we unify these results under a common approach? 1IfAis the adjacency matrix of a graph G, we may say that Gis globally synchronizing. 2To make this fully rigorous, we also need the fact that the obj ective function and constraint set are an... | https://arxiv.org/abs/2503.18801v1 |
of xon the diagonal. ddiag( A) is the diagonal matrix with the same diagonal as Abut with the remaining elements set to zero. tr( A) is the trace (sum of the diagonal elements) of A. Re(x) denotes the real part of complex x, applied elementwise if x is a vector or matrix. We write a/lessorsimilarb(equivalently, b/great... | https://arxiv.org/abs/2503.18801v1 |
is still nonconvex, but it turns out that, when r >n 2, the landscape is benign for any C[19, Cor. 5.11]: for every second-order critical point Y, X=YYTis a global optimum of ( 3). 4This optimality analysis hinges on knowing the global optim um beforehand. The applications we consider allow for exact recovery/synchroni... | https://arxiv.org/abs/2503.18801v1 |
matrix C′=C+αA, whereα≥0 is some scale parameter, and Ais the adjacency matrix of a cycle graph. For sufficiently large α, the landscape becomes arbitrarily close to that of synchronization on a cycle grap h, which is known [ 23] to have spurious local optima. This resembles the observation of Ling et al. [ 10] that addi... | https://arxiv.org/abs/2503.18801v1 |
σ≤/radicalbiggn (2+ǫ)logn, then, with probability →1asn→ ∞, for allr≥2, every second-order critical point Yof(1)with cost matrix Csatisfies YYT=zzT. 7 As shownbyBandeira[ 28], the threshold/radicalBig n 2lognis optimal forit to be possible to recover zex- actly by the maximum likelihood estimator, and the semidefinite re... | https://arxiv.org/abs/2503.18801v1 |
condition o f Theorem 3.2to δ≥/radicalBigg 2(1+ǫ)logn np (again,ǫ >0 may be different). In the case p= 1, this resolves the conjecture of Bandeira [ 29, Conj. 9]. This is the optimal threshold for exact recovery of z(or, in the language of the Kuramoto oscillator,forthenetworktosynchronizeevenlocally)[ 30], [31]. Aswith... | https://arxiv.org/abs/2503.18801v1 |
unweighted, regular graph Gwith adjacency matrixAsuch that λn(L) λ2(L)=λn(L) λ2(L)≤2+ǫ, andGis not globally synchronizing. The counterexamples proving the second part are provided by The orem3.5below. In the rest of this section, we primarily focus on the case where Ais a graph adjacency matrix, and, therefore, LandLar... | https://arxiv.org/abs/2503.18801v1 |
by a more circuitous route, th is is also an implication of our Theorem 3.2). However, we can say more. Theorem 3.4, combined with a more particular result of [ 38] (in particular, their Thm. 1.3 and following discussion), recovers the main result of Jain et al. [ 39] (which was conjectured by [ 13]): that, for large e... | https://arxiv.org/abs/2503.18801v1 |
vertex degree dis at least 600. Theorem 3.4allows us to improve this requirement to d≥35. Indeed, for any ǫ >0,d-regular Ramanujan graphs and (with high probability) sufficiently lar ge d-regular random graphs (see [ 42]) satisfy /vextenddouble/vextenddouble/vextenddouble/vextenddoubleA−d n11T/vextenddouble/vextenddouble... | https://arxiv.org/abs/2503.18801v1 |
for the true normalized Laplacian LDis the main result of this section. Qualitatively similar results for random graphs were shown by Chung and Radcliffe [ 47] (whose proofs inspired ours), but those results are not quite su fficient for our appli- cations. Theorem 4.1. LetD= ddiag( CzzT), and assume dmin:= min iDii>0. Se... | https://arxiv.org/abs/2503.18801v1 |
that, for any ǫ′>0, with probability →1 asn→ ∞, dmin=n+σmin i(W1)i ≥n−σ/radicalbig (2+ǫ′)nlogn. If we take ǫ′< ǫ, then the assumption σ≤/radicalBig n (2+ǫ)lognimplies dmin≥ǫ′′n for some constant ǫ′′>0 depending on ǫ,ǫ′. We then obtain δC=2¯d (dmin∧¯d)2/bardblC−C/bardblℓ2/lessorsimilar1 (ǫ′′)2nσ√n/lessorsimilar1 (ǫ′′)2√... | https://arxiv.org/abs/2503.18801v1 |
sin/parenleftbigπ n/parenrightbig. To calculate λn(L) = max mHL[m], we need to know what value(s) of mto consider. First, we restrict ourselves to k≥0.2n. The precise ratio is not important for now; it suffices to assume that kis proportional to nwhile including the “interesting” parameter regimes covered by Theorem 3.5.... | https://arxiv.org/abs/2503.18801v1 |
change and rescaling. The zero eigen value from /tildewideL1= 0 corresponds to a trivial global shift of the angles. 18 enoughnandk≥0.3n)λn(L) λ2(L)=HL[2] HL[1], we see thatλn(L) λ2(L)>2 implies λ2(/tildewideL) =H/tildewideL[1]>0, in which case the twisted state is a stable equilibrium. Thus we obtain part 1of Theorem ... | https://arxiv.org/abs/2503.18801v1 |
≤ /an}bracketle{tL,(YYT)◦2/an}bracketri}ht−2ρ/an}bracketle{tL,diag(Yv)YYT/an}bracketri}ht =/an}bracketle{tL,(YYT−ρ[(Yv)1T+1(Yv)T])◦(YYT)/an}bracketri}ht =/an}bracketle{tL,[WWT−ρ211T]◦YYT/an}bracketri}ht. This implies (r−1+ρ2)/an}bracketle{tL,YYT/an}bracketri}ht ≤ /an}bracketle{tL,(WWT)◦(YYT)/bracehtipupleft/bracehtipdo... | https://arxiv.org/abs/2503.18801v1 |
(23) (again scaling by 2 ρ) gives (2r−1)/an}bracketle{tL,YY∗/an}bracketri}ht ≤ /an}bracketle{tL,Re(YY∗)◦(YY∗)/an}bracketri}ht−2ρ/an}bracketle{tL,diag(Re( Yv))YY∗/an}bracketri}ht =/an}bracketle{tL,Re(YY∗−ρ(Yv1∗+1(Yv)∗))◦(YY∗)/an}bracketri}ht =/an}bracketle{tL,Re(WW∗−ρ211∗)◦(YY∗)/an}bracketri}ht, from which we obtain the... | https://arxiv.org/abs/2503.18801v1 |
A. Townsend, “Sufficiently d ense Kuramoto networks are globally synchronizing,” Chaos, vol. 31, no. 7, 2021. [16] C. De Vita, J. F. Bonder, and P. Groisman, “The energy landsca pe of the Kuramoto model in random geometric graphs in a circle,” SIAM J. Appl. Dyn. Syst. , vol. 24, no. 1, pp. 1–15, 2025. [17] D. Sclosa, “Ku... | https://arxiv.org/abs/2503.18801v1 |
A. S. Bandeira, and G. Hall, “Exact recovery in the st ochastic block model,” IEEE Trans. Inf. Theory , vol. 62, no. 1, pp. 471–487, 2016. [36] E. Mossel, J. Neeman, and A. Sly, “Consistency thresholds for the planted bisection model,” inProc. ACM Symp. Theory Comput. (STOC) , Portland, OR, USA, Jun. 2015. [37] B. Haje... | https://arxiv.org/abs/2503.18801v1 |
AN IMPROVED CENTRAL LIMIT THEOREM FOR THE EMPIRICAL SLICED WASSERSTEIN DISTANCE David Rodríguez-Vítores∗ Universidad de Valladolid and IMUVa Valladolid, Spain Eustasio del Barrio Tellado Universidad de Valladolid and IMUVa Valladolid, Spain Jean-Michel Loubes Institut de Mathématiques de Toulouse, INRIA Toulouse, Franc... | https://arxiv.org/abs/2503.18831v1 |
distance presents a number of challenges for large-scale problems. The first challenge is the computational complexity. Let PnandQmdenote the empirical measures built from nandmindependent and identically distributed observations from PandQ, respectively. If n=m, computing the Wasserstein distance between one-dimension... | https://arxiv.org/abs/2503.18831v1 |
alternative distance metrics have been proposed, aiming to retain the favorable properties of the Wasserstein distance while mitigating its computational and statistical challenges. Notably, entropically regularized optimal transport, introduced by [ Cut13 ], adds a convex regularization term to the original problem, r... | https://arxiv.org/abs/2503.18831v1 |
θ∈Sd−1. For p >1andP̸=Q, distributional limits are obtained using the extended functional delta method, under the assumption that the probability measures are compactly supported with convex support. Additionally, for p= 1, distributional limits under mild assumptions are derived for both P̸=Qand P=Q. The assumptions i... | https://arxiv.org/abs/2503.18831v1 |
this computational challenge, and Theorem 5.1 establishes a central limit theorem for this approximation, providing a flexible Gaussian limit that adapts to the projection regime (τ= limk k+n∈[0,1]). To our knowledge, the existing results in the literature offer only probability bounds on the approximation error [ XH22... | https://arxiv.org/abs/2503.18831v1 |
ϕc(y) = inf x∈Rd ∥x−y∥p−ϕ(x) . Therefore, the theory of c-concavity plays a fundamental role, for which a thorough exposition can be found in [ GM96 ] and [ Vil03 ]. In particular, we say thatϕ:Rd→[−∞,∞)isc-concave if there exists ψ:Rd→[−∞,∞)such that ϕ=ψc. Obviously, ϕcisc-concave, and it is easy to see that ϕcc≥ϕ, ... | https://arxiv.org/abs/2503.18831v1 |
map θ7→ϕθ x0(⟨θ, x⟩). However, this definition cannot be extended to empirical potentials ϕθ n. Ifsupp(Qθ)is not lower bounded (resp. upper bounded), and ⟨θ, x0⟩<min i=1,...,n⟨θ, X i⟩ resp.⟨θ, x0⟩>max i=1,...,n⟨θ, X i⟩ then every c-concave optimal potential ϕθ nverifies ϕθ n(⟨θ, x0⟩) =−∞. To check this, consider an o... | https://arxiv.org/abs/2503.18831v1 |
2.1, ϕθ X1,ϕθ n,X 1andϕθ n,m,X 1are well defined up to a set of P-probability zero, which can be ignored, since it does not play any role in the expected values. Proposition 2.2. Letq≥1. Under the same assumptions as in Proposition 2.1, if P, Q have finite moments of order pq, then E ∥ϕθ X1∥q Lq(σ×P) ∨E ∥(ϕθ X1)c∥q ... | https://arxiv.org/abs/2503.18831v1 |
optimal plans rather than optimal maps, offering a simpler method to eliminate smoothness assumptions in Q, without relying on the approximation argument of Corollary 4.3 in [dBGSL24]. Proposition 3.1 (Variance bound ).Letp >1andP, Q∈ P2p(Rd). Given independent random variables X1, X′ 1 with probability law PandYwith p... | https://arxiv.org/abs/2503.18831v1 |
variance is given by v2 P,Q:= VarZ Sd−1ϕθ x0(X)dσ(θ) =Z Sd−1Z Sd−1CovP(ϕθ x0, ϕη x0)dσ(θ)dσ(η). CovP(ϕθ x0, ϕη x0)does not depend on the particular choice of the constant. Under the assumptions of Theorem 3.2, given any c-concave optimal transport potential ϕθfrom PθtoQθ, there exists aθ∈Rsuch that ϕθ=ϕθ x0+aθin int(... | https://arxiv.org/abs/2503.18831v1 |
int(supp(P))is connected. Then, if we further assume that the density of Pis continuous and positive in int(supp(P)), it follows that fθ(F−1 θ(t)) is positive and continuous in (0,1)for every θ∈Sd−1. 9 An improved CLT for the empirical sliced Wasserstein Theorem 4.1 and Corollary 4.2 enhance the existing results in the... | https://arxiv.org/abs/2503.18831v1 |
of the one-sample setting in Theorems 3.3 and 4.1, and k=k(n)→ ∞ asn→ ∞ , such that k/(k+n)→τ∈[0,1], then r kn k+n SWp p,k(Pn, Q)−SWp p(P, Q) ⇝N 0,(1−τ)w2 P,Q+τv2 P,Q . Moreover, if the assumptions of the two sample-setting are also verified and k=k(n)→ ∞ asn→ ∞ such that k/(k+nm n+m)→τ∈[0,1], then s knm n+m k+nm n... | https://arxiv.org/abs/2503.18831v1 |
(32) it follows that T∆ n,m,k⇝N 0,1 under the null hypothesis H0:SWp p(P, Q) = ∆ . This provides an effective tool to perform inference on the value SWp p(P, Q), which is valid for any value of τ∈[0,1]. However, for inferential purposes, the large number of projections regime ( τ= 1), is particularly relevant. Taking... | https://arxiv.org/abs/2503.18831v1 |
the American Mathematical Society , 261(1259), 2019. [BRPP15] Nicolas Bonneel, Julien Rabin, Gabriel Peyré, and Hanspeter Pfister. Sliced and Radon Wasserstein barycenters of measures. Journal of Mathematical Imaging and Vision , 51(1):22–45, 2015. [CCHM86] Miklos Csörgo, Sandor Csorgo, Lajos Horvath, and David M. Maso... | https://arxiv.org/abs/2503.18831v1 |
Systems , volume 32. Curran Associates, Inc., 2019. [MNW24] Tudor Manole and Jonathan Niles-Weed. Sharp convergence rates for empirical optimal transport with smooth costs. The Annals of Applied Probability , 34(1B):1108 – 1135, 2024. [NWR22] Jonathan Niles-Weed and Philippe Rigollet. Estimation of Wasserstein distance... | https://arxiv.org/abs/2503.18831v1 |
negligible boundary. The following lemma summarizes key regularity properties of c-concave optimal transport potentials used in the proof of Proposition 2.1. Assertions (1), (2), and (3) are just an adaptation of Theorem 3.3 and Proposition C.4 in [ GM96 ] for d= 1, but these properties hold more generally. Lemma A.2. ... | https://arxiv.org/abs/2503.18831v1 |
constants aθ n(which depend on ω∈Ω0) such that ϕθn X1−aθ n→ϕθ X1in the sense of uniform convergence in the compacts of int(supp(Pθ)). Since ⟨θ, X 1⟩ ∈int(supp(Pθ)), and ϕθn X1(⟨θ, X 1⟩) = ϕθ X1(⟨θ, X 1⟩) = 0 , then aθ n→0, and we can conclude that |ϕθn X1−ϕθ X1| ≤ |(ϕθn X1−aθ n)−ϕθ X1|+|aθ n| →0 uniformly in the compac... | https://arxiv.org/abs/2503.18831v1 |
(35) and (37), it follows that for every (s, t)∈∂cϕθ n,X 1, ϕθ n,X 1(s) =|s−t|p−(ϕθ n,X 1)c(t)≥ϕθ n,X 1(⟨θ, X 1⟩) +|s−t|p− |t− ⟨θ, X 1⟩|p(38) For every (s, t)∈∂cϕθ n,X 1, combining (36) and (38) and using inequality (57), ϕθ n,X 1(s)−ϕθ n,X 1(⟨θ, X 1⟩) ≤2Cp |s|p+|t|p+|⟨θ, X 1⟩|p+|tθ n,X 1|p . (39) Similarly, for ever... | https://arxiv.org/abs/2503.18831v1 |
the empirical sliced Wasserstein For each θ∈Sd−1, consider the following construction. First, define the random map τ′ θ:{1, . . . , n } → { 1, . . . , n } that maps each index ito the position of ⟨θ, X′ i⟩in the ordered statistic associated with the variables {⟨θ, X′ i⟩}n i=1. If there are kequal values in the ordered... | https://arxiv.org/abs/2503.18831v1 |
Yrepresents a random variable independent of X, withL(Y) =Q. Combined with (48), E (Zn−Z′ n)2 + ≤p2 n2E ∥X1−X′ 1∥2p1 pE ∥X1−Y∥2pp−1 p. (49) Finally, Efron-Stein inequality (19) allows us to conclude (a). Two-sample setting. LetZ=SWp p(Pn, Qm), where Qmis the empirical distribution associated with mi.i.d. random v... | https://arxiv.org/abs/2503.18831v1 |
uniform integrability of n2(Zn−Z′ n)2 +, for which it suffices to prove uniform integrability of the square of the right-hand side in (45). To do this, we provide a uniform bound on the 1 +γmoment. Applying Hölder’s inequality withq1=pandq2=p p−1, E ∥X1−X′ 1∥Z Sd−1Z R|⟨θ, X 1⟩ −t|p−1dQθ τ′ θ(1)(y)dσ(θ)2(1+γ) ≤E ∥... | https://arxiv.org/abs/2503.18831v1 |
moments, and therefore convergence in W2(see Lemma 8.3 in [BF81]). Applying the triangle inequality, we deduce W2 L√n SWp p(Pn, Q)−E SWp p(Pn, Q) , N(0, v2 P,Q) →0. C Proofs of Section 4 C.1 Weak convergence of weighted integrals of the quantile process Our main result in this section is based on the next lemma,... | https://arxiv.org/abs/2503.18831v1 |
in (0, ϵ], forϵsmall enough. (IV) limϵ→0Rϵ 0f(F−1(t))w(t/λ)t1/2|r(t)|dt= 0for every λ >1ifwis non-decreasing, and for every 0< λ < 1 ifwis non-increasing. Given a fixed value of λin(0,1)or(1,∞), forϵsmall enough, (0, ϵ/λ)⊂(0,1/2). Thus, for any of such ϵ, Zϵ 0f(F−1(t))w(t/λ)t1/2|r(t)|dt=Zϵ 01 f(F−1(t/λ))t1/2|h(t)|dt= ≤... | https://arxiv.org/abs/2503.18831v1 |
Q∈ P(R), with distribution functions F, G , respectively. If P, Q have finite moments of order β(p−1)(1 + γ), then EZ1 0 h′ p(F−1 n(t)−G−1(t))−h′ p(F−1(t)−G−1(t)) βdt →0, (60) EZ1 0 h′ p(F−1 n(t)−G−1 m(t))−h′ p(F−1(t)−G−1(t)) βdt →0. (61) Proof. From the inequality (57), it follows that 1 pCβ β h′ p(F−1 n(t)−G−1(t)... | https://arxiv.org/abs/2503.18831v1 |
to see that the first term vanishes. Thus, it suffices to show EZ1 0G−1(t)√n(F−1 n(t)−F−1(t))dt →0. (70) Under the assumptions of Theorem 4.1, we can apply Lemma C.1 with h=G−1to conclude Z1 0G−1(t)√n(F−1 n(t)−F−1(t))dt⇝Z1 0G−1(t) f(F−1(t))B(t)dt . The limiting variable is an integral of the Brownian bridge B, it fol... | https://arxiv.org/abs/2503.18831v1 |
n(t)−G−1 m(t)√n F−1 n(t)−F−1(t) dt →0. This can be concluded as in the one-sample case, using now the second part of Lemma C.3. C.4 Proof of Theorem 4.1 We aim to prove (21), which can be rewritten asZ Sd−1√n E Wp p(Pθ n, Qθ) −Wp p(Pθ, Qθ) dσ(θ)→0 Given that SJα(1+γ)(P)<∞, then Jα(1+γ)(Pθ)<∞for almost every θ∈S... | https://arxiv.org/abs/2503.18831v1 |
given by independent variables X1. . . , X nandΘ1, . . . , Θk. W2 2(L(Sn,k),L(Sk))≤E 1√ kkX i=1Zn(Θi)−EΘ(Zn(Θi))− Z(Θi)−EΘ(Z(Θi))2! =1 kkX i=1E Zn(Θi)−Z(Θi)−EΘ(Zn(Θi)−Z(Θi))2 ≤2E Zn(Θ)−Z(Θ)2 = 2E Wp p(PΘ n, QΘ)−Wp p(PΘ, QΘ)2 (76) From the bound (68) and the triangle inequality, for every θ∈Sd−1, Wp p(Pθ... | https://arxiv.org/abs/2503.18831v1 |
p(PΘi n, QΘi)2 +Z Sd−1Wp p(Pθ, Qθ)dσ(θ)2 (79) We will show convergence in probability to 0 of each term. For (78), this follows from a straightforward application of the strong law of large numbers. For (77), the convergence follows from Markov’s inequality and E 1 kkX i=1W2p p(PΘi n, QΘi)−1 kkX i=1W2p p(PΘi, QΘi) ≤... | https://arxiv.org/abs/2503.18831v1 |
E |h(Θ1, X1)|2(1+γ) =∥ϕθ x0∥2(1+γ) L2(1+γ)(σ×P)<∞. (80) For the term (C), consider the following decomposition 1 k2nkX i,j=1nX l=1ϕΘi x0(Xl)ϕΘj x0(Xl) = 30 An improved CLT for the empirical sliced Wasserstein =1 k 1 knkX i=1nX l=1ϕΘi x0(Xl)2! +2 k 2 k2 1 n 1 k 2X 1≤i<j≤knX l=1ϕΘi x0(Xl)ϕΘj x0(Xl)! The bound (80), ... | https://arxiv.org/abs/2503.18831v1 |
≤C1+γ E ϕΘ1 n,X 1(X2) 2(1+γ) +E ϕΘ1 X1(X2) 2(1+γ) =C1+γn−1 n∥ϕθ n,X 1∥2(1+γ) L2(1+γ)(σ×Pn)+∥ϕθ X1∥2(1+γ) L2(1+γ)(σ×P) <∞, An analogous argument proves uniform integrability of the second sequence, which allow us to finish the proof. Two-sample setting: By symmetry, it suffices to see ˆv2 Pn,Qm→Pv2 P,Q. The proo... | https://arxiv.org/abs/2503.18831v1 |
Calibration Bands for Mean Estimates within the Exponential Dispersion Family Lukasz Delong∗Selim Gatti†Mario V. W¨ uthrich‡ Version of March 25, 2025 Abstract A statistical model is said to be calibrated if the resulting mean estimates perfectly match the true means of the underlying responses. Aiming for calibration ... | https://arxiv.org/abs/2503.18896v1 |
binary observations, Hosmer–Lemeshow [14] derive a χ2-test by binning observations over disjoint intervals, whereas Gneiting–Resin [12] propose a bootstrap approach to test for auto-calibration in this binary setup. We take a different approach in this paper. Our goal is to construct calibration bands for mean estimate... | https://arxiv.org/abs/2503.18896v1 |
construction of the calibration bands to regression modeling and in Section 7, we introduce the auto-calibration property and provide conditions under which this property is equivalent to calibration. Moreover, we derive in the same section statistical tests that enable to test for calibration and auto-calibration of a... | https://arxiv.org/abs/2503.18896v1 |
and the σ-finite measures νithat define the supports of the responses Yi. Assumption 3.1. We assume that the effective domain has a non-empty interior ˚Θand that theσ-finite measures νiare not a single point mass. This assumption excludes any trivial case of the EDF and implies that the effective domain Θ is a (possibl... | https://arxiv.org/abs/2503.18896v1 |
For fixed y∈R,v > 0,φ > 0 and cumulant function κ, the stochastic ordering result in Proposition 3.4 implies that the functions µ∈κ′(˚Θ)7→F(y;h(µ), v, φ, κ (·)), and µ∈κ′(˚Θ)7→F∗(y;h(µ), v, φ, κ (·)), are non-increasing in µ. This observation leads to the construction of the bounds on the mean in the next proposition. ... | https://arxiv.org/abs/2503.18896v1 |
ical parameters fulfill θ1≤ ··· ≤ θn. This construction makes use of sets of ordered pairs J ⊆ { 1, . . . , n }2that we define as sets satisfying (j, k)∈ J =⇒j≤k. 6 By considering the union of the events where the true means fail to lie within the constructed lower and upper bounds in Proposition 3.7, we obtain P µj≤u... | https://arxiv.org/abs/2503.18896v1 |
to a low value for δand 7 vice versa. This creates a trade-off situation and there is thus no optimal choice for the set of ordered pairs in general. In their construction of calibration bands for the binary case, Dimitriadis et al. [7] suggest to use a slightly modified version of Jfullthat we call Jdistinct={(j, k)∈ ... | https://arxiv.org/abs/2503.18896v1 |
uδ(Zj:k, vj:k, φ, κ(·)) requires the use of a root-finding algorithm. For some members of the EDF, these bounds can be calculated in closed form and we give the resulting expressions for the binomial, Poisson, negative binomial, gamma and normal cases in this section. These expressions are derived using closed form cha... | https://arxiv.org/abs/2503.18896v1 |
the lower and upper bounds Lα Y,iandUα Y,iare defined in (4.3)-(4.4). These bounds can be explicitly expressed in the following three cases using the weighted partial sums Zj:kand aggregated volumes vj:kin(3.5)-(3.6): •Binomial case. The lower and upper bounds are given by Lα Y,i= sup (j,k)∈J:µi≥µkqB(δ;vj:kZj:k/φ,1 +vj... | https://arxiv.org/abs/2503.18896v1 |
and vice versa. The resulting weighted partial sums Zj:kare thus scaled sums of N(µi/σ2 i,1/σ2 i) random variables. Since the normal distribution has the nice property that any weighted sum of independent normal responses is again normal, other weights could in principle be chosen as, for example, ˜vj:k=kX i=j1 σiand ˜... | https://arxiv.org/abs/2503.18896v1 |
given volumes vi>0, as well as a dispersion parameter φ >0 and a cumulant function κ that do not depend on i. We denote the support of the features XibyXand call it the feature space . The goal of a regression on the mean is to estimate the true mean function µ∗:X → κ′(˚Θ),x7→κ′(θ(x)), (6.1) where the map θ:X → ˚Θ is u... | https://arxiv.org/abs/2503.18896v1 |
[0,1],Q(ω, A) =E[1A|X1, . . . ,Xn](ω), is a regular conditional probability of P, given the features X1, . . . ,Xn. This assumption fails to hold in general, and we refer to Section 3.2 of Rao–Swift [23] for necessary conditions ensuring the existence of this regular conditional probability. Moreover, we emphasize that... | https://arxiv.org/abs/2503.18896v1 |
level 1 −α∈(0,1). We emphasize, however, that the conditional probability bound in Theorem 6.3 is stronger than this inequality as it holds for a.e. fixed and known realization of the features ( Xi)n i=1, i.e., when only the responses ( Yi)n i=1are random. As the mean function µ∗ π:X → Rwas assumed to be a version of t... | https://arxiv.org/abs/2503.18896v1 |
following procedure in order to graphically determine the decision induced by the statistical test. First, the calibration band can be plotted against the ranking function, which re- sults in two non-decreasing step functions delimiting the band. Then, the pairs ( π(x),bµ(x))x∈X can be drawn in the same plot for all fe... | https://arxiv.org/abs/2503.18896v1 |
order to construct statistical tests for the auto-calibration of a given regression function bµ:X →R, we assume that Assumption 7.2 holds. That is, the regression function manages to correctly provide the ordering of the true mean function. Interestingly, we can show that under this assumption, calibration is equivalen... | https://arxiv.org/abs/2503.18896v1 |
now consider numerical examples where we construct calibration bands on the mean of given responses. Our goal is first to highlight the impact of different factors on the resulting calibration bands, as the choice of the confidence level and the set of ordered pairs, or the influence of binning observations. Then, we s... | https://arxiv.org/abs/2503.18896v1 |
Yi)n i=1for various confidence levels. As expected, the calibration bands get narrower as the value of αincreases. However, we point out that the value of the confidence level seems not to lead to significant impacts on the width of the calibration band. That is, the constructed bands are not very sensitive to the conf... | https://arxiv.org/abs/2503.18896v1 |
specific locations. That is, for both restricted sets of ordered pairs we consider in this example, our results show that constructing bands using smaller sets of ordered pairs than Jfullis computationally more efficient, and these smaller sets lead to suitable calibration bands as long as their size is not too small. ... | https://arxiv.org/abs/2503.18896v1 |
the set of ordered pairs Jfullto be small. In fact, the ratio Φ−1 0.05 20002+2000 Φ−1 0.05 502+50= 1.355 hints that the band constructed using 2000 bins should be approximately 1.355 wider than the band constructed using only 50 bins, see (5.10) and (5.11). However, this is is not the case in Figure 5 due to the ro... | https://arxiv.org/abs/2503.18896v1 |
French car drivers. We follow Listing 13.1 in W¨ uthrich–Merz [28] in order to clean the data, leading to a portfolio of n= 678 ,007 insurance policies and 26 ,383 claims1. For each policy 1 ≤i≤n, the resulting dataset provides 1The cleaned dataset can be downloaded under https://people.math.ethz.ch/ ~wueth/Lecture/ fr... | https://arxiv.org/abs/2503.18896v1 |
5.2.4 in W¨ uthrich–Merz [28]. 25 ranking the responses ( Yi)n i=1according to their mean estimates and using a weighted quantile binning. This procedure leads to the new volumes ˜vl=nX i=1vi1{bµPoi(xi)∈Il}∈[70.7,72.64],for 1≤l≤L. The same binning applied to the mean estimates ( bµPoi(xi))n i=1allows us to derive the m... | https://arxiv.org/abs/2503.18896v1 |
Figure 7, where we draw the same calibration band as above, i.e., we assume again the ranking function to be π(·) =bµPoi(·) in order to construct the band. Figure 7: Calibration plot of the regression function bµPoi rec:X → (0,∞) on the log scale. The calibration band is constructed using π(·) =bµPoi(·) as a ranking fu... | https://arxiv.org/abs/2503.18896v1 |
the number of rejections of the calibration of ( bµi)n i=1at a confidence level 1 −α= 0.95, are summarized in Table 5. They should be compared to the plots on page 12 of W¨ uthrich [27] as the mean estimates ( bµi)n i=1are calibrated if and only if they are auto-calibrated for the shifts considered in this example. Not... | https://arxiv.org/abs/2503.18896v1 |
986 /1000 Local shift of level l= 13 - 769 /1000 1000 /1000 Local shift of level l= 15 - 203 /1000 917 /1000 Table 6: Power of the performed statistical tests with confidence level 1 −α= 0.95. interesting tool to reduce this size while using all the observations is to bin those observations. As a result, the bands get ... | https://arxiv.org/abs/2503.18896v1 |
J.M., Brunk, H.D. (1972). Statistical inference under order restrictions : the theory and application of isotonic regression . London: Wiley. [2] Barndorff-Nielsen, O. (2014). Information and exponential families: In statistical theory . New York: Wiley. [3] Delong, L., W¨ uthrich, M. V. (2024). Isotonic regression for... | https://arxiv.org/abs/2503.18896v1 |
Scandinavian Journal of Statistics 42, 471-484. [27] W¨ uthrich, M.V. (2024). Auto-calibration tests for discrete finite regression functions. arXiv : 2408.05993. [28] W¨ uthrich, M.V., Merz, M. (2023). Statistical foundations of actuarial learning and its applications. Cham: Springer. [29] W¨ uthrich, M.V., Ziegel, J.... | https://arxiv.org/abs/2503.18896v1 |
a union bound argument that P µj≤uδ(Zj:k, vj:k, φ, κ(·)) and µk≥lδ(Zj:k, vj:k, φ, κ(·)) for all ( j, k)∈ J ≥1−2|J |δ. Due to the ordering assumed in (2.2), the above inequality can be rewritten as P sup (j,k)∈J:θi≥θklδ(Zj:k, vj:k, φ, κ(·))≤µi≤ inf (j,k)∈J:θi≤θjuδ(Zj:k, vj:k, φ, κ(·)) for all i∈ {1, . . . , n }! ≥1−2|... | https://arxiv.org/abs/2503.18896v1 |
β) denote the distribution and the δ-quantile of a Beta( α, β) random variable, respectively. This shows the claim for the lower bound. Similarly, the result for the upper bound in (5.7) follows from F(l;µ, v/φ )≥δ⇐⇒ 1−Ip(1 +l, v/φ )≥δ, ⇐⇒ 1−GB(p; 1 +l, v/φ )≥δ, ⇐⇒ p≤qB(1−δ; 1 +l, v/φ ), ⇐⇒µ 1 +µ≤qB(1−δ; 1 +l, v/φ ), ⇐... | https://arxiv.org/abs/2503.18896v1 |
, n }o , (A.1) that lies in Fas all the random variables involved in its definition are measurable. Instead of proving the existence of the uniform calibration band in (6.6) holding simultaneously for all x∈ X, we first prove that for a.e. realization (xi)n i=1of the features ( Xi)n i=1, we have Q(xi)n i=1 Lα π,(Yi,Xi... | https://arxiv.org/abs/2503.18896v1 |
arXiv:2503.19069v2 [math.ST] 18 May 2025Detecting Arbitrary Planted Subgraphs in Random Graphs Dor Elimelech Wasim Huleihel May 20, 2025 Abstract The problems of detecting and recovering planted structure s/subgraphs in Erd˝ os- R´ enyi random graphs, have received significant attention o ver the past three decades, lea... | https://arxiv.org/abs/2503.19069v2 |
un- derstanding statistical limits. However, computational feasibility h as often been overlooked, despite its growing importance as modern datasets continue to exp and. Recent research, e.g., [BR13 ,MW15b ,CLR17 ,CX16 ,HS17 ,HB18 ,GJW20 ,BHK+16,ZK16 ,LKZ15 ,HWX15a , BPW18 ,BBH18 ,BBH19a ,BB20 ], and many references th... | https://arxiv.org/abs/2503.19069v2 |
establish a unified framework for detecting arbitrary planted subgraphs can be found in the literature. Such an attempt already appeared in [ABBDL10 ], albeit for a slightly different model. However, for general structures, t he lower and upper bounds provided therein are loose. More recently, [ Hul22 ] proposed the gene... | https://arxiv.org/abs/2503.19069v2 |
ion of|e(Γ∩Γ′)|is intricate and strongly depends on Γ. For instance, when Γ is a k-clique, we have |e(Γ∩Γ′)|d=/parenleftbigH 2/parenrightbig , whereH∼Hypergeometric (n,k,k ). On the other hand, if Γ is a k-path, the distribution of|e(Γ∩Γ′)|is significantly more complex, implicit, and dominated by a certain Marko v chain... | https://arxiv.org/abs/2503.19069v2 |
sharp phase transition as a function of q. The rest of this paper is organized as follows. In Section 2, we introduce the problem setup and provide some necessary preliminaries. Section 3presents our main results for the various regimes, including the dense regime, the sparse regime, and the critical regime. Section 4a... | https://arxiv.org/abs/2503.19069v2 |
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