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unique. This holds under multinomial as well as Poisson sampling. In what follows, we sometimes denote the vector of the MLE ˆ µby ˆµ(b;x) to explicitly show the dependence on sufficient statistics bandx. We apply this rule to the expected counts µand the UMVUE ˜ µdiscussed below. Some components of the MLE become zero i... | https://arxiv.org/abs/2502.00812v2 |
modelMAsatisfies the estimating equation of the MLE (7), that is, A˜µ=b (13) for a sample of the vector of sufficient statistics b. Proof.Applying the annihilator (8) to the A-hypergeometric polynomial ZA(b;x) and using the contiguity relation (11), we have /summationdisplay j∈[m]aijxjZA(b−aj;x) =biZA(b;x), i∈[d], which i... | https://arxiv.org/abs/2502.00812v2 |
he cliques, and the separator is empty. Theorem 2.9. The UMVUE and the MLE of a log-linear model coincides if and only if the model is a decomposable graphical model. More over, the MLE is rational. Proof.(if) Lemma 4.21 of [11] gives the normalization constant of a de- composable graphical model. We can represent it a... | https://arxiv.org/abs/2502.00812v2 |
the boundary of the polyhe dral cone generated by the column vectors of A, the MLE coincides with the UMVUE and vanishes exactly without bias. In practical terms, we obtain Proposition 2.13. The approximate algorithm obtained from Algorithm 1 by replacing Steps 2 and 3 with Steps 2’ and 3’ is a Markov chain on the Mark... | https://arxiv.org/abs/2502.00812v2 |
monomials of the holo- nomic ideal of rank r, where∂u(·)is a monomial in ∂j,j∈[m]. The standard monomials provide thebasisofthesolutions ofthe A-hypergeometric system. The system of equations (17) can be obtained via a normal form with respect to a Gr¨ obner basis of the holonomic ideal (see, e.g., [18], Theorem 1.4 .2... | https://arxiv.org/abs/2502.00812v2 |
the distribution of contingency tables. We estima ted the distribution of chi-square values under the conditional distribution (3) of contingency tables, that is, p(z) :=P(χ2(U) =z|AU=b), z≥0, (19) bytheempiricaldistribution ˆ pMofcontingencytablestakenfromtheMetropo- lis chain and the empirical distribution ˆ pDof con... | https://arxiv.org/abs/2502.00812v2 |
the burn- in and 18 Table 1: The total variation distances between the empirical distrib utions of chi-square values generated by the Metropolis algorithm and tha t by the direct sampling algorithm (exact and approximate for the independe nce and non-independence models, respectively). (burn-in, length) Model s(0,103) ... | https://arxiv.org/abs/2502.00812v2 |
using the Metropolis algorithm (including the burn- in), the time it took to draw the tables of the number of the ESS using the di- rect sampling algorithm, and one table using the direct sampling algorit hm. For the independence model, the direct sampling algorithm was unifor mly efficient than the Metropolis algorithm.... | https://arxiv.org/abs/2502.00812v2 |
distribution. Forthe Metropolis chain of four pairs of theburn-in an d length: (103,104), (104,104), (105,104) and (105,105), respectively, we computed the empirical distribution ˆ pMof the chi-square values and calculated the total variation distance (20). Here, we set the expected value of each c ell tos. We show the... | https://arxiv.org/abs/2502.00812v2 |
the bias of the MLE of the expected count, and in terms of the speed of computing the MLE, Newton’s method can be f aster than the generalized iterative proportional scaling. However, the se are typi- caltasksinstatistical analysiswhenapplying analgorithmtoeachmo deland are beyond the scope of this paper, which aims to... | https://arxiv.org/abs/2502.00812v2 |
Minimax Optimality of Classical Scaling Under General Noise Conditions Siddharth Vishwanath1and Ery Arias-Castro1,2 1Department of Mathematics, University of California, San Diego 2Halıcıo˘ glu Data Science Institute, University of California, San Diego Abstract We establish the consistency of classical scaling under a... | https://arxiv.org/abs/2502.00947v1 |
consider a loss function of the form L(bX,X) = min g∈G(p) bX−g(X) †, (2) as a metric to assess the quality of recovery in classical scaling, where ∥·∥†is a suitable norm on the space of n×pconfiguration matrices. The loss in (2) is often referred to as the reconstruction error of the embedding. Some common choices for ... | https://arxiv.org/abs/2502.00947v1 |
the authors don’t explicitly make the connection to the work of Javanmard and Montanari, 2013), they show that the output from the same semidefinite relaxation is consistent in the Lrmseloss. Classical scaling with noisy dissimilarities. To the best of our knowledge, detailed statistical analyses of classical scaling i... | https://arxiv.org/abs/2502.00947v1 |
2), the root mean-squared loss Lrmse(Corollary 2), and uniform convergence under the L2→∞ loss (Theorem 3). The results hold for both fixed and random configurations X—where the rows of Xare sampled iidfrom a sufficiently regular distribution (Corollary 3). In relation to previous work, the results make use of probabil... | https://arxiv.org/abs/2502.00947v1 |
an) and bn≳an), if there exists a constant C > 0 such that |an| ≤C|bn|for all n > N C, andbn=o(an) (equivalently, bn≪an) if lim n|bn/an|= 0. Table 2 in Appendix B provides a summary of the notation used in this work. 4 2 Setting Given an input dissimilarity matrix, the objective of classical multidimensional scaling is... | https://arxiv.org/abs/2502.00947v1 |
standard matrix concentration bounds (e.g., Vershynin, 2018), assumption ( A1) is satisfied with high probability if the rows of Xare sampled iidfrom a sufficiently regular probability distribution on Rp, as described in the following lemma. The proof can be found in Section 7.1. 2The condition Ψ(0 ,0) = 0 ensures that... | https://arxiv.org/abs/2502.00947v1 |
noise models satisfying these conditions are listed in Table 1. The additive and absolute additive models in Settings 1 and 3 correspond to the noise models in Li et al. (2020, Sections 2.1 &2.2). In Section 3, we analyze the performance of classical scaling under these assumptions. 6 3 Performance Analysis of Classica... | https://arxiv.org/abs/2502.00947v1 |
of the additive noise model in Table 1 where E= Ξ, Tao and Vu (2010, Theorem 1.6) show that fluctuations of order O(√n) forE∥E∥2depend on the fourth moment M4appearing in (12). Although (11) by itself doesn’t provide any guarantees for the embedded configuration, it will be the main workhorse in establishing the consis... | https://arxiv.org/abs/2502.00947v1 |
same parametric convergence rate as in (15). 8 In many scenarios, it is crucial to have a uniform bound that controls the worst-case deviation of the estimated configuration from the true configuration. To this end, we consider the ℓ2→∞-operator norm, which provides a finer control on the reconstruction error of each r... | https://arxiv.org/abs/2502.00947v1 |
κ, ϖ )such that for all n >N(r, q, σ, κ, ϖ ), Lrmse(bX,X)≲σκ√n 1 +o(1) and L2→∞(bX,X)≲c(κ, ϖ)σr logn n 1 +o(1) with probability greater than 1−O(n−2+n−r). Collectively, the results in Theorem 2 and Theorem 3 show that, given a noisy dissimilarity matrix, classical scaling is statistically consistent, with essential... | https://arxiv.org/abs/2502.00947v1 |
Corollary 2, the lower bound in Theorem 4 matches the upper bound up to constant factors. Importantly, the result shows that the rate of convergence for classical scaling in Corollary 2 is minimax optimal. Moreover, the minimum sample sizes N2(σ, κ)≍N4(σ, κ)≍σ2κ4also match up to absolute constants. The proof of Theorem... | https://arxiv.org/abs/2502.00947v1 |
κ]. The noisy dissimilarity matrix is then generated under the additive noise model: D= ∆(Xκ) + Ξ ,where ξij∼tq, q∈ {3,5,7}. As discussed in Section 2, the noise satisfies assumptions ( A1)–(A4) only for q= 5,7, while for q= 3, the fourth moment does not exist. For the embedding bX=CS(D, p) obtained using classical sca... | https://arxiv.org/abs/2502.00947v1 |
limitation concerns the assumptions on the noise distribution. As demonstrated in Section 5, the performance of classical scaling deteriorates significantly when the noise lacks a finite fourth moment. While some prior work has touched on robustness in this context (e.g., Cayton and Dasgupta, 2006; Mandanas and Kotropo... | https://arxiv.org/abs/2502.00947v1 |
with probability greater than 1 −2n−2. Similarly, on the event that (20) holds, since ∥Σ−1∥2≤κ2we have bΣ−1 2≤ Σ−1 2+ bΣ−1−Σ−1 2≲ Σ−1 2(1 +αn), and it follows that for all n >N0, spHX√n = bΣ−1 −1 2 2≳ Σ−1 −1 2 1 +αn!1 2 ≳1 κ√1 +αn≥1 κ(1 +αn). (22) with probability greater than 1 −2n−2. Combining (21) and (22), it fol... | https://arxiv.org/abs/2502.00947v1 |
A3), we have E[g2 ijε2 ij|δij] =E[ε2 ij|δij]≤σ2andE[g4 ijε4 ij|δij] =E[ε4 ij|δij]≤σ4. (26) Since the above bounds hold uniformly over ∆ = ( δij), there exists an absolute constant C1>0 such that E∥E −E(E)∥2≲E∆n EΞ,G ∥G◦ E∥2|∆o ≤C1σ√n. (27) Plugging (27) back into (23) and using the bound on 1 , we get E∥Dc−∆c∥2≲σ√n. ... | https://arxiv.org/abs/2502.00947v1 |
have ∥bX−g(X)∥F≤√p· ∥bX−g(X)∥2. Therefore, from Theorem 2, it follows that Lrmse(bX,X) = min g∈G(p)1 n∥bX−g(X)∥2 F1/2 ≤p nmin g∈G(p)∥bX−g(X)∥2 21/2 ≲σκrp n, (33) with probability greater than 1 −O(n−2+n−r) for all n >N2(r, q, σ, κ ). ■ 7.6 Proof of Theorem 3 ForbX=bUbΛ1/2=CS(D, p),HXQ⊤=UΛ1/2=CS(∆, p) and for X=X⊤1/... | https://arxiv.org/abs/2502.00947v1 |
with probability greater than 1 −O(n−2+n−r). Since the above bound holds uniformly over X∈X(κ, ϖ), it follows that 3≲σ(κ+ϖ)√nplognpwith probability greater than 1 −O(n−2+n−r). Plugging this back into the bound for 1 and combining with 2 , it follows that with probability greater than 1 −O(n−2+n−r), ∥R4∥2→∞≲c(κ, ϖ)σr pl... | https://arxiv.org/abs/2502.00947v1 |
Forη∈(0,1) and v∈Rpwith∥v∥= 1, let Ybe a collection of 2mconfiguration matrices given by Y≡Y(X, η, v) = Y(τ)∈Rn×p:Y(τ) =X+ηω(τ)v⊤, τ∈ {0,1}m . (45) Intuitively, each Y∈Yis a centered configuration such that Lrmse(Y,X) is large but ∥∆(Y)−∆(X)∥Fis sufficiently small. Moreover, when η=o(1) is chosen to be sufficiently sm... | https://arxiv.org/abs/2502.00947v1 |
lower bound for the right hand side of (50), we will need the following result for testing multiple hypotheses using Le Cam’s convex hull method. 4It suffices to take C= max {8,32c2}. 25 Lemma 5 (Adaptation of Yu, 1997, Lemma 1) .Letbθbe an estimator of θ(P)forP∈ P, taking values in a pseudometric space (Θ, ρ), and sup... | https://arxiv.org/abs/2502.00947v1 |
exp o(1) ≤1 +o(1), we have TV(P∗ 0,P∗)≤s exp1 2logn−logn + exp o(1) −1 =s 1√n+o(1) = o(1). (56) Moreover, by noting that γ+ϖ=2κ κ2+1+ϖ≤2(1 κ+ϖ), from Lemma 6 (ii) we also have L2→∞(Xk,X)≥η 2=σ 2√ C(γ+ϖ)r logn n≳σκ 1 +κϖr logn n. (57) Therefore, using (56) and (57) in Lemma 5, it follows that inf bXsup X∈XP∗ X( L2... | https://arxiv.org/abs/2502.00947v1 |
covariance and precision matrices along subspaces. Electronic Journal of Statistics 15 , 554–588. Klochkov, Y. and N. Zhivotovskiy (2020). Uniform Hanson-Wright type concentration inequalities for un- bounded entries via the entropy method. Electronic Journal of Probability 25 (22), 1–30. Kroshnin, A., E. Stepanov, and... | https://arxiv.org/abs/2502.00947v1 |
(5500), 2319–2323. Torgerson, W. S. (1952). Multidimensional scaling: I. Theory and method. Psychometrika 17 (4), 401–419. Torgerson, W. S. (1958). Theory and Methods of Scaling . Johhn Wiley & Sons. 29 Tsybakov, A. B. (2008). Introduction to Nonparametric Estimation (1st ed.). Springer Publishing Company, Incorporated... | https://arxiv.org/abs/2502.00947v1 |
for all t >4E∥Y∥2, P ∥Y∥2> t ≤4P ∥G◦(Y−P)∥2>t 4 . (59) Proof. The proof is similar to the proof of Proposition 3.1.24 of Gin´ e and Nickl, 2016. Let Y′be an iidcopy ofYwithY′⊥ ⊥Y, and let PYand (resp. PY′) be the probability measure associated with Y(resp. Y′) on the sample space Ω (resp. Ω′). For the unit m-sphere... | https://arxiv.org/abs/2502.00947v1 |
and van Handel (2016). Proposition 2. Suppose Y= (Yij)is a symmetric n×nrandom matrix with independent, symmetric, zero mean entries with max i,j∥Yij∥Lq≤σforq >4. Then, for any ε∈(0,1 2]there exists a universal constant cε>0such that for all 0< r≤(q−4)/2and for all t >0, with probability greater than 1−e−t−n−r, ∥Y∥2≤2(... | https://arxiv.org/abs/2502.00947v1 |
jYijzj∈Rp, is the sum of nindependent, zero mean random vectors. Step 1: Symmetrization. ForV={v∈Rp:∥v∥= 1}, we can write ∥(YZ)i,∗∥as the supremum of an em- pirical process indexed over V, given by ∥(YZ)i,∗∥= supv∈V|P j∈[n]Yijz⊤ jv|.Forγ2:= supv∈Vmax jE[(Yijz⊤ jv)2] and for all u≥p 2nγ2, using the symmetrization lemma ... | https://arxiv.org/abs/2502.00947v1 |
X jYijzj ≳σ(κ+ϖ)p np(t+ log n+ log p) +Kϖ(t+ log n+ log p) ≤4n 2e−t−logn+nσq/Kq ≤8 e−t+n2σq/Kq . Similar to (63) in the proof of Proposition 2, setting K=σn(r+2) qfor any 0 < r < (q−4)/2, it follows that n2Kq/σq=n−r, and, therefore, for all t >0 with probability greater than 1 −8(e−t+n−r) we have ∥YZ∥2→∞≲σ(κ+ϖ)p n... | https://arxiv.org/abs/2502.00947v1 |
proofs for lower bounding the minimax risk in Section 7. We begin with the following which establishes that any norm ∥·∥†induces a pseudometric ming∈G(p)∥Y−g(X)∥†onRn×p. 36 Lemma 10. Let∥·∥†be any norm on Rn×pand let ρ:Rn×p×Rn×p→R≥0be given by ρ(X,Y) = min g∈G(p)∥X−g(Y)∥†. Then, ρis a pseudometric on Rn×p. Proof. By de... | https://arxiv.org/abs/2502.00947v1 |
configuration matrices, respectively. We will establish the lower bound by showing that Lrmse Y,Y′ ≳∥Y′Y′⊤−YY⊤∥F n(γ+η)≳γη√n·p dH(τ, τ′) n(γ+η). (75) Note that the right hand side of (75) simplifies to the desired lower bound in Lemma 4 (iii). Proof of the first inequality in (75). By an application of the triangle i... | https://arxiv.org/abs/2502.00947v1 |
∥Xk∥2=η2 2, where the second equality follows from the fact that v(k) =Xk/∥Xk∥and the final inequality follows from the fact that Q=Ipis the optimal alignment for (79). Since the inequality above holds for all rigid transformations, plugging this back into (78) and minimizing over G(p) gives L2→∞(Xk,X) = min g∈G(p)∥Xk−... | https://arxiv.org/abs/2502.00947v1 |
kΓ(X,Xk,Xk) +n(n−1) n2max k̸=ℓΓ(X,Xk,Xℓ) ≤1 nexp1 σ2max k∥∆(Xk)−∆(X)∥2 F + exp1 σ2max k̸=ℓD ∆(Xk)−∆(X),∆(Xℓ)−∆(X)E F = exp1 σ2max k∥∆(Xk)−∆(X)∥2 F−logn + exp1 σ2max k̸=ℓD ∆(Xk)−∆(X),∆(Xℓ)−∆(X)E F . Substituting the bound on 1 back into (81) gives us the final bound on TV(P∗ 0,P∗)2. ■ B Toolkit The following sec... | https://arxiv.org/abs/2502.00947v1 |
λp−1 ≤κ2 n 1−Kεκ2√n n−1 . For sufficiently large n,Kεκ2/√n <1, and using the identity (1 −x)−1= 1 + x+o(x) for|x|<1 implies that bΛ−1 2≤κ2 n 1 +Kεκ2 √n+o1√n =κ2 n 1 +o(1) ≤2κ2 n. ♢ 44 Proof of Lemma 14 (ii).Let Θ( bU, U) be the p×pdiagonal matrix where each diagonal entry is the principal angle between the cor... | https://arxiv.org/abs/2502.00947v1 |
On the Private Estimation of Smooth Transport Maps Cl´ement Lalanne1Franck Iutzeler1Jean-Michel Loubes2 1Julien Chhor3 Abstract Estimating optimal transport maps between two distributions from respective samples is an impor- tant element for many machine learning methods. To do so, rather than extending discrete transp... | https://arxiv.org/abs/2502.01168v1 |
. . , X ni.i.d.∼P and Y1, . . . , Y ni.i.d.∼Q such that the Xis and the Yis are mutually independent.1In this scenario, the problem (1)cannot be solved directly to ob- tain an optimal transport map; instead, it must be estimated using the available samples. To do so, a crude approach would be to replace PandQwith their... | https://arxiv.org/abs/2502.01168v1 |
& Duchi, 2014; Diakonikolas et al., 2015; Karwa & Vadhan, 2018; Bun et al., 2019; 2021; Kamath et al., 2019;Biswas et al., 2020; Kamath et al., 2020; Acharya et al., 2021; Lalanne, 2023; Aden-Ali et al., 2021; Cai et al., 2019; Brown et al., 2021; Cai et al., 2019; Kamath et al., 2022a; Lalanne et al., 2023c;d; Lalanne... | https://arxiv.org/abs/2502.01168v1 |
set of functions from a space StoCthat are k times continuously differentiable, and C∞(S)is defined as ∩k∈NCk(S). Whenever applicable, ∇and∇2are used to refer to gradient and Hessian operators, respectively. For a subset Sof a normed vector space, |S|refers to supx∈S∥x∥ for the inherited norm. For any set S,# (S)denote... | https://arxiv.org/abs/2502.01168v1 |
item), and have a stable support (third item). In addition, we will assume that the optimal transport map be- longs to the set Tα(R)of admissible transport maps whose H¨older norm of order αis bounded by R. 2.2. Empirical Semi-Dual H¨utter & Rigollet (2021) observed that the objective in (2) consists of the sum of an e... | https://arxiv.org/abs/2502.01168v1 |
if P∈ M andT0∈ Tα, the estimator ˆTJ:= ∇ˆfJdefined above achieves this rate up to polylogarithmic factors. Here, the constants also hide a dependence on R. 2.4. Towards the Analysis of a Private Estimator In this paper, we extend this approach to the context of differential privacy. We derive a private estimator ˆfpriv... | https://arxiv.org/abs/2502.01168v1 |
use this mechanism as a building block to construct a private estimator of ˆfJ. 3.2. Noisy semi-dual estimator In this section, we construct our private estimator, building on the empirical semi-dual one presented above. The first step is to provide an upper bound on the sensitiv- ity of ˆS(f|X1:n, Y1:n), which is esta... | https://arxiv.org/abs/2502.01168v1 |
Rigollet, 2021). Theorem 4.3. LetP∈ M, T0∈ Tα. By choosing δappro- priately, the estimator ˆTprivhas a utility satisfying EZ ∥T0−ˆTpriv∥2dP ≲R22−2Jα+J2J(d−2)ln(n) n+1 n+2Jdln(nϵ) nϵ(24) forJlarger than a sufficiently large constant that also depend on R. Proof. See Appendix E.3 Ignoring poly-logarithmic terms and tre... | https://arxiv.org/abs/2502.01168v1 |
now only store the value of the potential and its (approximate) dual on a grid, we need to clip the points to this grid. We thus denote by X(grid) 1:nandY(grid) 1:nthe modified versions of the datasets X1:nandY1:nwhere each point is replaced with its closest neighbor in ˜Ω(grid). Finally, if one wishes to apply Lemma 3... | https://arxiv.org/abs/2502.01168v1 |
of this estimator and derived a lower bound on the difficulty of the statistical prob- lem. Finally, we proposed an adaptation of this theoretical estimator that is implementable in practice. This work opens up several research directions. One of them is the quest for an optimal private mechanism, and another one is th... | https://arxiv.org/abs/2502.01168v1 |
Conference on Artificial Intelligence and Statistics, AISTATS 2018, 9-11 April 2018, Playa Blanca, Lanzarote, Canary Islands, Spain , volume 84 of Proceedings of Ma- chine Learning Research , pp. 1771–1780. PMLR, 2018. URL http://proceedings.mlr.press/ v84/alvarez-melis18a.html . Arjovsky, M., Chintala, S., and Bottou,... | https://arxiv.org/abs/2502.01168v1 |
besov ellipsoids. CoRR , abs/1903.01927, 2019. URL http://arxiv.org/ abs/1903.01927 .Cai, T. T., Wang, Y ., and Zhang, L. The cost of privacy: Optimal rates of convergence for parameter estimation with differential privacy. CoRR , abs/1902.04495, 2019. URL http://arxiv.org/abs/1902.04495 . Ca˜nas, G. D. and Rosasco, L.... | https://arxiv.org/abs/2502.01168v1 |
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op- timal transport for cortical surface reconstruction. In The Twelfth International Conference on Learning Repre- sentations, ICLR 2024, Vienna, Austria, May 7-11, 2024 . OpenReview.net, 2024. URL https://openreview. net/forum?id=gxhRR8vUQb . Loukides, G., Denny, J. C., and Malin, B. A. The disclosure of diagnosis co... | https://arxiv.org/abs/2502.01168v1 |
21-27, 2024 . OpenReview.net, 2024. URL https: //openreview.net/forum?id=VZsxhPpu9T . Rakotomamonjy, A. and Ralaivola, L. Differentially private sliced wasserstein distance. In Meila, M. and Zhang, T. (eds.), Proceedings of the 38th International Con- ference on Machine Learning, ICML 2021, 18-24 July 2021, Virtual Eve... | https://arxiv.org/abs/2502.01168v1 |
Wallach, H. M., Fergus, R., Vishwanathan, S. V . N., and Garnett, R. (eds.), Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, December 4-9, 2017, Long Beach, CA, USA , pp. 2647–2658, 2017. URL https://proceedings. neurips.cc/paper/2017/hash/ 253f7b5d... | https://arxiv.org/abs/2502.01168v1 |
Details on Wavelet Decomposition In this article, we adopt the same conventions for functional spaces as in (H ¨utter & Rigollet, 2021). In particular, we encourage the reader to consult Appendix B in (H ¨utter & Rigollet, 2021) for formal definitions of the functional spaces considered in this paper. A notable differe... | https://arxiv.org/abs/2502.01168v1 |
of ¯f. We now turn to controlling the error of the semi-dual. We have ˆS(ˆfs) =sˆS(ˆfpriv) + (1 −s)ˆS(¯f) ≤s ˆS(ˆfpriv|X1:n, Y1:n)−ˆS(ˆf|X1:n, Y1:n) +sˆS(ˆf) + (1 −s)ˆS(¯f) ≤U+ˆS(¯f)(47) where the last inequality comes from the optimality of ˆffor the empirical semi-dual problem. Hence, S0(ˆfs)≤S0(¯f) +U+ 2 sup f∈FJ(... | https://arxiv.org/abs/2502.01168v1 |
D.2. Proof of Lemma 3.3 Let(X1:n, Y1:n)∼(X′ 1:n, Y′ 1:n)be two neighboring datasets. By definition, they contain the same data points except for one individual. Suppose first that Y′ 1:n=Y′ 1:nand that, for some i∈[n], the datasets X1:nandX′ 1:ndiffer in the i-th coordinate only. Then the term2∥f∥∞ nis obtained by noti... | https://arxiv.org/abs/2502.01168v1 |
Maps Since CJ,Mis aδ-covering of VJ∩ X(2M)with respect to ∥ · ∥∞, it is also a 2δ-covering of VJ∩ X(2M)with respect to ˆS(·|X1:n, Y1:n)by Lemma 4.1. Thus, ˆS(ˆfpriv|X1:n, Y1:n)−ˆS(ˆfJ|X1:n, Y1:n) = min i∈{1,...,#(CJ,M)} ˆS(fi|D) +32M2∨36d nϵLi −ˆS(ˆfJ|X1:n, Y1:n) ≤ min i∈{1,...,#(CJ,M)}n ˆS(fi|X1:n, Y1:n)−ˆS(ˆfJ|X1:n... | https://arxiv.org/abs/2502.01168v1 |
ensures that∀θ∈ {0,1}N,∀x∈˜Ω,1 M⪯ ∇2ϕθ(x)⪯M. 24 On the Private Estimation of Smooth Transport Maps •(Stability of the support) Let θ∈ {0,1}N. It follows from (80) that, when h≤1 m+1, one can take asmall enough to ensure that ∀i,∇ϕθ B∞ pi,h 2 ⊂B∞(pi, h). (83) Furthermore, since outside the balls of the form B∞ pi,... | https://arxiv.org/abs/2502.01168v1 |
over a finite family only. We verify that this family of statistics forms a packing for the pseudo metric defined below (with some overlap in the notation) d(Sθ1, Sθ2)2:=d(θ1, θ2)2:=Z ∥∇ϕθ1− ∇ϕθ2∥2dP (95) Lemma F.1. For any θ1, θ2∈ {0,1}N, d(Sθ1, Sθ2)2≳dham(θ1, θ2)h2α+d. (96) 26 On the Private Estimation of Smooth Tran... | https://arxiv.org/abs/2502.01168v1 |
TV (Qθ1, Qθ2)≲hα−1+dNX i=11θ(i) 1̸=θ(i) 2 =dham(θ1, θ2)hα−1+d.(108) Private distributional tests and private Assouad method. Lemma F.3 (Assouad’s Lemma) .Assume that there exists τ >0such that for any θ1, θ2∈ {0,1}N, d(Sθ1, Sθ2)2≳dham(θ1, θ2)τ. (109) LetˆTbe any estimator and define ˆθ= argminθ∈{0,1}Nd(ˆT,∇ϕθ). Then it... | https://arxiv.org/abs/2502.01168v1 |
. , Y′ n))̸= 1) | {z } =1! =EC e−ϵdham((X1,...,X n,Y1,...,Y n),(X′ 1,...,X′ n,Y′ 1,...,Y′ n)) =Eθ1,...,θi−1,θi+1...,θN E(X1,...,X n,Y1,...,Y n)(X′ 1,...,X′n,Y′ 1,...,Y′n) e−ϵdham((X1,...,X n,Y1,...,Y n),(X′ 1,...,X′ n,Y′ 1,...,Y′ n))! Jensen ≥Eθ1,...,θi−1,θi+1...,θN e−ϵE(X1,...,X n,Y1,...,Yn)(X′ 1,...,X′n,Y′ 1,...... | https://arxiv.org/abs/2502.01168v1 |
Uniform mean estimation for monotonic processes Eugenio Clericoa, Hamish E Flynna, Patrick Rebeschinib aUniversitat Pompeu Fabra, Barcelona, Spain bDepartment of Statistics, University of Oxford, Oxford, UK Abstract We consider the problem of deriving uniform confidence bands for the mean of a monotonic stochastic proc... | https://arxiv.org/abs/2502.01244v1 |
use a refined continuous union bound of PAC-Bayesian inspiration (Alquier, 2024; Jang et al., 2023), which fundamentally leverages the monotonicity of the process. Email addresses: eugenio.clerico@gmail.com (Eugenio Clerico), hamishedward.flynn@upf.edu (Hamish E Flynn), patrick.rebeschini@stats.ox.ac.uk (Patrick Rebesc... | https://arxiv.org/abs/2502.01244v1 |
seen as a particular instance of algorithmic mean estimation based on e-values (Ramdas et al., 2023). Recently, Clerico (2024a) showed that the coin-betting framework is in fact optimal among e-value based approaches. The idea of combining coin-betting with continuous union bounds (a.k.a. PAC-Bayes) has already been pr... | https://arxiv.org/abs/2502.01244v1 |
sgn( αt) as a bet on the sign of ctand|αt|as the size of the bet. The constraint αt∈[−1/b,1/a] ensures that the wealth Mtis always non-negative. If the initial wealth of the gambler is one, then the wealth after Trounds is MT=QT t=1(1 +αtct).3The gambler’s aim is to accumulate wealth as quickly as possible, and is form... | https://arxiv.org/abs/2502.01244v1 |
betting strategy used, this inequality can be leveraged to construct a confidence sequence for F(y) that does not depend on αt(y). For C≥0 (possibly infinite), we define the upper, ψ−1 T,+, and lower, ψ−1 T,−, inverses (w.r.t. µ) ofψT(y, µ) as ψ−1 T,+(y, C) = sup µ∈[0,1] :ψT(y, µ)≤C , ψ−1 T,−(y, C) = inf µ∈[0,1] :ψT(... | https://arxiv.org/abs/2502.01244v1 |
the necessary limits ( y→0 and/or y→1) of F◦ι(which exist due to monotonicity). One can then translate a uniform guarantee for ˜Finto a uniform guarantee for F. Proposition 1. LetY= [0,1]. With probability at least 1−δ, uniformly on T≥1and on y0∈ Y, we have F(y0)∈" sup y−∈[0,y0]ψ−1 T,− y−,log2√ T (y0−y−)δ ,inf y+∈[y0... | https://arxiv.org/abs/2502.01244v1 |
TPT t=1f(Xt, y). Moreover, this holds with equality whenever ( f(Xt, y))t≥1is a binary sequence. From (6), it follows that the inverses of ψTand kl satisfy ψ−1 T,+(y, C)≤kl−1 +(ˆFT(y), C/T ) and ψ−1 T,−(y, C)≥kl−1 −(ˆFT(y), C/T ). Therefore, from Proposition 1, with probability at least 1 −δ, uniformly for all T≥1 and ... | https://arxiv.org/abs/2502.01244v1 |
probability. We refer to Section 7 for some further remarks on asymptotic rates, in view of the law of the iterated logarithm. 5. Application to CDF estimation We return to the CDF estimation problem described in the introduction, where f(x, y) =I{x≤y}. For simplicity, we assume that Pis supported in [0 ,1].11Since eac... | https://arxiv.org/abs/2502.01244v1 |
to 0 and 1. However, as the dataset gets larger, this last method performs slightly better than ours for values where F(y) is far from 0 and 1. This is however expected, as the non-adaptive confidence band has an asymptotic width that matches the law of the iterated logarithm (p T−1log log T), while ours cannot shrink ... | https://arxiv.org/abs/2502.01244v1 |
T). However, this means that our confidence bands shrink at best at a rate ofp T−1logT(where the variance is non-zero), rather than the fastest possible rate ofp T−1log log T. To match the rate of the law of the iterated logarithm, one could use the betting algorithm from Section IV of Orabona and Jun (2023), though th... | https://arxiv.org/abs/2502.01244v1 |
PMLR. pp. 2240–2264. Larsson, M., Ramdas, A., Ruf, J., 2024. The numeraire e-variable and reverse information projection. arXiv:2402.18810 . Massart, P., 1990. The tight constant in the dvoretzky-kiefer-wolfowitz inequality. The annals of Probability , 1269–1283. Nguyen, D., Pathak, R., Angelopoulos, A., Bates, S., Jor... | https://arxiv.org/abs/2502.01244v1 |
NECESSARY AND SUFFICIENT CONDITIONS FOR CONVERGENCE IN DISTRIBUTION OF QUANTILE AND P-P PROCESSES IN L1(0,1) BRENDAN K. BEARE AND TETSUYA KAJI Abstract. We establish a necessary and sufficient condition for the quantile process based on iid sampling to converge in distribution in L1(0,1). The condition is that the quan... | https://arxiv.org/abs/2502.01254v2 |
distribution in L1(0,1). Any Qwhich is not locally absolutely contin- uous but satisfies (1.1) will do the job. A very simple example is obtained by choosing Qto be the quantile function for the Bernoulli distribution with success probability p∈(0,1). An example in which Qis continuous but not locally absolutely contin... | https://arxiv.org/abs/2502.01254v2 |
special construction to ameliorate troublesome behavior of Qnear zero and one, is undertaken in del Barrio, Gin´ e, and Utzet (2005). Theorem 4.6(i) therein establishes that the following conditions are collectively sufficient for√n(Qn−Q) to converge in distribution in L2(0,1): Qis locally absolutely continuous with de... | https://arxiv.org/abs/2502.01254v2 |
of Unis therefore critical to the equivalence of (1.12) and (1.13). Our proof that the quantile process converges in distribution in L1(0,1) if and only ifQhas Property Q does not rely on any special construction of the quantile process. Nor does the uniform quantile process play an important role. To prove the suffici... | https://arxiv.org/abs/2502.01254v2 |
of (0 ,1), nondecreasing in a neighborhood of zero and nonincreasing in a neighborhood of one, and satisfies Z1 0(u(1−u))−1exp −δ(u(1−u))−1w(u)2 du <∞for all δ >0. (1.14) Theorem 3.1 in Aly, Cs¨ org˝ o, and Horv´ ath (1987) establishes that if FandGare con- tinuous and admit continuous positive densities fandgon thei... | https://arxiv.org/abs/2502.01254v2 |
to hold when the norm on L1(0,1) is strengthened to the uniform norm. It is unnecessary to explic- itly require a ˇCibisov-O’Reilly condition to be satisfied, or to specify that the one-sided limits of the derivative of F(˜Q) at zero and one must be finite, because the pooling of samples already guarantees a uniformly ... | https://arxiv.org/abs/2502.01254v2 |
adopt the general approach of Pollard (1984) toward convergence in distribution, wherein mea- surability with respect to the ball σ-algebra is used to handle technical complications associated with nonseparable spaces. We specialize the setting from that of a metric space to that of a normed space, which suffices for o... | https://arxiv.org/abs/2502.01254v2 |
for instance, by the random selection of multinomial weights for the Efron bootstrap. A bootstrap counterpart to θnought to be influenced by both sampling uncertainty and the random variation inherent to bootstrapping, so it is natural to view it as a map θ∗ n:¯Ω→V. The following definition provides two ways to make pr... | https://arxiv.org/abs/2502.01254v2 |
that quasi-Hadamard differentiability does not require the norm controlling the behavior of each sequence ( hn) to be defined at each point in the domain off. See Beutner and Z¨ ahle (2010) for an extended discussion of the practical relevance of this distinction, focusing on applications involving weighted uniform nor... | https://arxiv.org/abs/2502.01254v2 |
We say that the conditions for quasi-Hadamard differentiability are very mild because in Section 4 we will use Theorem 3.1 to establish that Property Q is sufficient for convergence in distribution of the quantile process in L1(0,1), and then establish separately that Property Q is also necessary for such convergence. ... | https://arxiv.org/abs/2502.01254v2 |
prove that Property Q is sufficient for convergence in distribution of the quantile process in L1(0,1). Proof of Theorem 3.1. Before establishing the quasi-Hadamard differentiability of ϕwe will verify that h(Q)q∈L1(0,1) for every h∈D1(−∞,∞), and that h7→ −h(Q)qis continuous and linear as a map from D1(−∞,∞) into L1(0,... | https://arxiv.org/abs/2502.01254v2 |
1(v+tng(v)≥u)]dQ(v) du ≤t−1 nZ1 0Z1 0| 1(v+tngn(v)≥u)− 1(v+tng(v)≥u)|dudQ(v) ≤t−1 nZ1 0Z∞ −∞| 1(v+tngn(v)≥u)− 1(v+tng(v)≥u)|dudQ(v) =Z1 0|gn(v)−g(v)|dQ(v), where the first equality follows from (3.6) and (3.7), and the first inequality is obtained by applying Fubini’s theorem. Now we apply Lemma B.2 to the last integra... | https://arxiv.org/abs/2502.01254v2 |
and define Ac ϵ:= (0,1)\Aϵ. The bound (3.15) implies that Z1 0χn(u, v)dv≥tn√ϵfor all ϵ∈(0,1/2), n≥Nϵ,1∨Nϵ,2, u∈Aϵ. (3.20) We therefore deduce from the definition of χn,ϵthat Z1 0χn,ϵ(u, v)dv= 1 for all ϵ∈(0,1/2), n≥Nϵ,1∨Nϵ,2, u∈Aϵ. (3.21) We now return to showing that ∥ζn+gq∥1→0. Let ϵbe a point in (0 ,1/2) and let nbe... | https://arxiv.org/abs/2502.01254v2 |
v)|fϵ(v)−fϵ(u)|dvdu. From (3.18) we have Z1 0Z Aϵχn,ϵ(u, v)|fϵ(v)−f(v)|dudv≤ϵ−1/2Z1 0|g(v)||fϵ(v)−f(v)|dv≤√ϵsup v∈(0,1)|g(v)|, and from (3.21) we have Z AϵZ1 0χn,ϵ(u, v)|fϵ(u)−f(u)|dvdu=Z Ac|fϵ(v)−f(v)|dv≤ϵ. ThusZ Aϵ|ζn,ϵ(u) +g(u)q(u)|du≤√ϵsup u∈(0,1)|g(u)|+ϵ +Z AϵZ1 0χn,ϵ(u, v)|fϵ(v)−fϵ(u)|dvdu. (3.35) The inequalitie... | https://arxiv.org/abs/2502.01254v2 |
because resampling procedures are not the primary focus of this article, we confine attention to the Efron bootstrap. For each n∈N, letWn,1, . . . , W n,nbe random variables on Ω′which take values only in {0,1, . . . , n }and whose sum is equal to n. Each n-tuple ( Wn,1, . . . , W n,n) is assumed to have the multinomia... | https://arxiv.org/abs/2502.01254v2 |
OF QUANTILE AND P-P PROCESSES 21 4.2.Sufficiency of Property Q. In this section we establish the following result. Theorem 4.1. Assume that Qhas Property Q, and let qbe a density for Q. Let B: Ω×[0,1]→Rbe a Q-integrable Brownian bridge, and let Q: Ω→L1(0,1)be defined by Q(u) =−q(u)B(u)for all u∈(0,1). Then√n(Qn−Q)is an... | https://arxiv.org/abs/2502.01254v2 |
Mora-Corral (2024) that ( F∗ n) is almost surely a bootstrap version of ( Fn) w.r.t. the convergence in (4.3). That the latter property of ( F∗ n) implies the former may be shown by applying the portmanteau theorem as above. □ Proof of Theorem 4.1. We assume that the random variables X1, X2, . . .take values only in [α... | https://arxiv.org/abs/2502.01254v2 |
(4.5) to obtain 24 BRENDAN K. BEARE AND TETSUYA KAJI ZF−1(b)∧G−1(b) F−1(a)∨G−1(a)|F(x)−G(x)|dx =Z∞ −∞Z1 01(a≤F(x)< b) 1(a≤G(x)< b)| 1(F(x)≥u)− 1(G(x)≥u)|dudx. For each x∈Randu∈(0,1) such that 1(G(x)≥u)̸= 1(F(x)≥u), one of the two values F(x) and G(x) is no less than u, and the other less than u. Thus the inequalities a... | https://arxiv.org/abs/2502.01254v2 |
Since Qis monotone, it has at most countably many discontinuities. Thus we may choose an increasing sequence ( ak) of continuity points of Qconverging to a, and a decreasing sequence ( bk) of continuity points of Qconverging tob. We have ER1 0|Z(u)|du <∞as a consequence of (4.12). This justifies using the 26 BRENDAN K.... | https://arxiv.org/abs/2502.01254v2 |
we have regarded the cdf of a random variable to be a map from Rinto [0,1], and the corresponding quantile function to be a map from (0 ,1) into R. We will maintain this convention for GandQ; i.e. we have G:R→[0,1] and Q: (0,1)→R. It will be more convenient to adopt a different convention for FandF−1. We will regard th... | https://arxiv.org/abs/2502.01254v2 |
G∗ n:¯Ω→D(−∞,∞) be the maps defined by F∗ n(z) =1 nnX i=1Wn,i 1(Xi≤z) for all z∈[−∞,∞] and G∗ n(z) =1 nnX i=1Vn,i 1(Yi≤z) for all z∈R. We regard the generalized inverse Q∗ n:=G∗−1 nto be a map from ¯Ω into L1(0,1). The maps F∗ n,G∗ n, and Q∗ nmay be understood to be Efron bootstrap counterparts to Fn, Gn, and Qn. For e... | https://arxiv.org/abs/2502.01254v2 |
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