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Zhang, Shaoqing Ren, and Jian Sun. Delving deep into rectifiers: Surpassing human-level performance on imagenet classification. In Proceedings of the IEEE international conference on computer vision , pages 1026–1034, 2015. Addison Howard, Eunbyung Park, and Wendy Kan. Imagenet object localization challenge. https: //k... | https://arxiv.org/abs/2502.07369v1 |
Therefore, the kernel ridge regression estimator of ηusing the representation ϕ(X)is given by αλ=arg min α∈HϕE[Y−α(X)]2+λ∥α∥2 Hϕ= Σ−λ ϕI∗ ϕη. Similarly, we can show that βλ=arg min β∈HψE[Y−β(X)]2+λ∥β∥2 Hψ= Σ−λ ψI∗ ψη. Now, αλ(x′) =Z η(x)h Σ−λ ϕKϕ(·, x)i (x′)dPX(x) =Z η(x)D Σ−λ ϕKϕ(·, x), Kϕ(·, x′)E HϕdPX(x) =Z η(x) Σ−λ... | https://arxiv.org/abs/2502.07369v1 |
j=1constitute orthonormal bases of HfandHg, respectively. The Mercer decompositions of the kernels KfandKgare given by, Kf(x, x′) =∞X i=1µf ief i(x)ef i(x′) and Kg(x, x′) =∞X j=1µg jeg j(x)eg j(x′). Note that, I(f)(x, x′) = Σ−λ 2 fKf(·, x),Σ−λ 2 fKf(·, x′) Hf=D Kf(·, x),Σ−λ fKf(·, x′)E Hf =∞X i=1µf i µf i+λef i(x)ef i(... | https://arxiv.org/abs/2502.07369v1 |
eigenfunctions. Consequently, (6) can be now written as I(f) =I(g) ⇐⇒∞X i=1µf i µf i+λef i(·)ef i(·) =∞X i=1µg i µg i+λeg i(·)eg i(·). (17) 19 Taking the L2(PX)inner product of both the RHS and LHS of (17)with ef itwice, we have that, for any i, µf i µf i+λ=µg i µg i+λ ⇐⇒µf i=µg i. Therefore, we must have that the inte... | https://arxiv.org/abs/2502.07369v1 |
ϕ−Σ−λ ϕ L∞(Hϕ)≤κ ˆΣ−λ ϕ−Σ−λ ϕ L2(Hϕ). Similarly, we have that ˆAij,ψ−Aij,ψ = D Kψ(·, Xi), ˆΣ−λ ψ−Σ−λ ψ Kψ(·, Xj)E Hψ ≤κ ˆΣ−λ ψ−Σ−λ ψ L∞(Hψ)≤κ ˆΣ−λ ψ−Σ−λ ψ L2(Hψ). 21 Note that, ˆΣ−λ ϕ−Σ−λ ϕ L∞(Hϕ) = ˆΣϕ+λI−1 (Σϕ+λI) (Σϕ+λI)−1− ˆΣϕ+λI−1 ˆΣϕ+λI (Σϕ+λI)−1 L∞(Hϕ) = ˆΣ−λ ϕh (Σϕ+λI)− ˆΣϕ+λIi Σ−λ ϕ L∞(Hϕ) ≤ ˆΣ−λ ϕ ... | https://arxiv.org/abs/2502.07369v1 |
B Additional Experiments In this appendix, we provide additional experimental results. B.1 MNIST experiments Training details We have already described the architectures of the 50 ReLU networks we trained for experiments using the MNIST dataset in Section 5.1. We used the uniform Kaiming initialization He et al. (2015)... | https://arxiv.org/abs/2502.07369v1 |
defined in Section 5.1 and the pairwise distances between the representations using the following distances - CCA, linear CKA, nonlinear CKA with Gaussian RBF kernel, GULP and UKP with Gaussian RBF kernel. 26 When (λ= 10−2, σ= 10−1)and(λ= 1, σ= 10−1), we observe from Fig. 5 that the pairwise UKP distance is positively ... | https://arxiv.org/abs/2502.07369v1 |
convnext_tiny •Other Architectures (5 models): alexnet, googlenet, inception, mnasnet, vgg16 . The penultimate layer dimensions for these networks, corresponding to the representation sizes, vary from 400 to 4096 depending on the architecture. Each model processes input data as 3-channel RGB images, with each channel h... | https://arxiv.org/abs/2502.07369v1 |
tuning parameters for the UKP distance, especially corresponding to the 4 major groups of architectures ResNets, EfficientNets, MobileNets and ConvNeXts. We also perform an agglomerative (bottom-up) hierarchical clustering of the representations based on the pairwise UKP distances and obtain the corresponding dendrogra... | https://arxiv.org/abs/2502.07369v1 |
interest will be used. For instance, consider an image classification task where the model’s predictions should remain unaffected by image rotations. In this case, we can incorporate this inductive bias into the UKP pseudometric by selecting a rotationally invariant kernel, such as the Gaussian RBF kernel, as the kerne... | https://arxiv.org/abs/2502.07369v1 |
Beyond Benign Overfitting in Nadaraya-Watson Interpolators Daniel Barzilai* Guy Kornowski* Ohad Shamir Weizmann Institute of Science {daniel.barzilai,guy.kornowski,ohad.shamir }@weizmann.ac.il May 23, 2025 Abstract In recent years, there has been much interest in understanding the generalization behavior of inter- pola... | https://arxiv.org/abs/2502.07480v2 |
for kernel regression and shallow ReLU networks [Manoj and Srebro, 2023, Kornowski et al., 2024, Joshi et al., 2024, Li and Lin, 2024, Barzilai and Shamir, 2024, Medvedev et al., 2024, Cheng et al., 2024a]. We note that one classical example of tempered overfitting is 1-nearest neighbor, which asymptotically achieves a... | https://arxiv.org/abs/2502.07480v2 |
the nature of the inconsistency for β̸=d? Is 2 the overfitting tempered, or in fact catastrophic? As our main contribution, we answer these questions and prove the following asymmetric behavior: Theorem 1.2 (Main results, informal) .For any dimension d∈Nand noise level p∈(0,1 2), the following hold asymptotically as m→... | https://arxiv.org/abs/2502.07480v2 |
a more general discussion on so called “kernel rules”, see [Devroye et al., 2013, Chapter 10]. In more recent works, Belkin et al. [2019b] derived non-asymptotic rates showing consistency under a slight variation of the kernel. Rad- hakrishnan et al. [2023], Eilers et al. [2024] showed that in certain cases, neural net... | https://arxiv.org/abs/2502.07480v2 |
which interpolating with kernels is in fact catastrophic , meaning that the excess error is lower bounded by a constant which is independent of the noise level, leading to substantial risk even in the presence of very little noise [Kornowski et al., 2024, Joshi et al., 2024, Medvedev et al., 2024, Cheng et al., 2024a].... | https://arxiv.org/abs/2502.07480v2 |
Under the described setting with noise level p∈(0,1 2), we say that: •ˆhβexhibits benign overfitting if L(ˆhβ) = 0 ; • Else, ˆhβexhibits tempered overfitting if L(ˆhβ)scales monotonically with p: there exists φ: [0,1]→ [0,1]non-decreasing, continuous with φ(0) = 0 , so that L(ˆhβ)≤φ(p); •ˆhβexhibits catastrophic overfi... | https://arxiv.org/abs/2502.07480v2 |
prediction at x. Denote by y(1), . . . , y (m)the labels ordered according to the distance of their corresponding datapoints, namely x−x(1) ≤ x−x(2) ≤ ··· ≤ x−x(m) . By analyzing the distribution of distances from the sample to x, for datapoints suffi- ciently close to xwe can jointly approximate the random variables b... | https://arxiv.org/abs/2502.07480v2 |
dintgenerally can only be estimated, it suggests a potential practical implication: Setting βto an over-estimate ofdintis less harmful than under-estimating it, as the former leads to tempered overfitting whereas the latter may lead to catastrophic overfitting. This is further supported by our experiments in Section 5.... | https://arxiv.org/abs/2502.07480v2 |
to argue thatPi j=1Ej≈ i, and (2)follows from showing k≈cm. 9 To show that T2is sufficiently large, we use the fact that ∥x−xi∥ ≤ ∥ x∥+∥xi∥ ≤5 4, and that i:∥xi∥ ≥3 4 ≈(1−c)m≥1 2mwith high probability to obtain T2=X i:∥xi∥≥3 41−2p ∥x−xi∥β≥(1−2p) xi:∥xi∥ ≥3 4 ·4 5β ≳m·4 5β . Lastly, we show that T3is asymptotic... | https://arxiv.org/abs/2502.07480v2 |
Figure 4, the same asymmetric phenomenon holds in which overly large βare more forgiving than overly small β, especially in low noise regimes. The main difference between the first and second experiment is that the optimal “benign” exponent in the second case is β= 2, matching the intrinsic dimension of the sphere, eve... | https://arxiv.org/abs/2502.07480v2 |
data interpolation contradict statis- tical optimality? In The 22nd International Conference on Artificial Intelligence and Statistics , pages 1611–1619. PMLR, 2019b. Mikhail Belkin, Daniel Hsu, and Ji Xu. Two models of double descent for weak features. SIAM Journal on Mathematics of Data Science , 2(4):1167–1180, 2020... | https://arxiv.org/abs/2502.07480v2 |
2022. 13 Naren Sarayu Manoj and Nathan Srebro. Interpolation learning with minimum description length. arXiv preprint arXiv:2302.07263 , 2023. Marko Medvedev, Gal Vardi, and Nathan Srebro. Overfitting behaviour of gaussian kernel ridgeless regres- sion: Varying bandwidth or dimensionality. In The Thirty-eighth Annual C... | https://arxiv.org/abs/2502.07480v2 |
ACM/IMS Journal of Data Science , 1, 2023. 15 A Notation and Order Statistics We start by introducing some notation that we will use throughout the proofs to follow. We denote X1≍X2 to abbreviate X1= Θ( X2),X1≲X2to abbreviate X1=O(X2)andX1≳X2to abbreviate X1= Ω(X2). Throughout the proofs we let α:=β/d, and abbreviate h... | https://arxiv.org/abs/2502.07480v2 |
1−p). Furthermore, given k∈N, we assume mis sufficiently large so that the knearest neighbors of xall lie in B(x, δ)with probability at least 1−exp(−k). Under this likely event, we decompose mX i=1yi Wi=X i:W(i)≤δyi Wi+X i:W(i)>δyi Wid=kX i=1y(i) F−1(U(i)) |{z } (I)+|Sx∩B(x,δ)|X i=k+1y(i) F−1(U(i)) | {z } (II)+X i:W(i)... | https://arxiv.org/abs/2502.07480v2 |
S[h(x)̸=f∗(x)|Ak x, Bk x]·Pr S[Bk x] + Pr S[h(x)̸=f∗(x)|Ak x,¬Bk x]·Pr S[¬Bk x] ≤ c1exp(−c2k) + exp( −cαk1−1 α) ·1 + 1·kp ≤Cαlogα α−1(1/p) log( p), where the last inequality follows by our assignment of k. Since this is true for any x, it is also true in expectation over x, thus completing the proof of the upper boun... | https://arxiv.org/abs/2502.07480v2 |
Fix some xwith∥x∥< r, we will show that for sufficiently large m, with high probability xwill be misclassified as +1. To that end, we decompose mX i=1yi ∥x−xi∥β=X i:∥xi∥≤ryi ∥x−xi∥β+X i:∥xi∥≥3ryi ∥x−xi∥β =X i:∥xi∥≤ryi ∥x−xi∥β+X i:∥xi∥≥3r1−2p ∥x−xi∥β+X i:∥xi∥≥3ryi−1 + 2 p ∥x−xi∥β ≥ −X i:∥xi∥≤r1 ∥x−xi∥β | {z } =:T1+X i:∥... | https://arxiv.org/abs/2502.07480v2 |
∥xi∥ − ∥x∥ ≥2r, and thusyi ∥x−xi∥αdare bounded as yi ∥x−xi∥αd ≤1 (2r)αd. We thus apply Hoeffding’s Inequality (cf. Vershynin, 2018, Theorem 2.2.6) yielding that for any t≥0 Pr X i:∥xi∥≥3ryi ∥x−xi∥αd−X i:∥xi∥≥3r1−2p ∥x−xi∥αd ≥t ≤2 exp −t2(2r)2αd 2(m−km) . In particular, we have that with probability at least 1−2 ... | https://arxiv.org/abs/2502.07480v2 |
CHEAP PERMUTATION TESTING BYCARLES DOMINGO -ENRICH1,a, RAAZ DWIVEDI2,c,AND LESTER MACKEY1,b 1Microsoft Research New England,acarlesd@microsoft.com;blmackey@microsoft.com 2Cornell Tech,cdwivedi@cornell.edu Permutation tests are a popular choice for distinguishing distributions and testing independence, due to their exac... | https://arxiv.org/abs/2502.07672v2 |
random Fourier features, Wilcoxon-Mann-Whitney. 1arXiv:2502.07672v2 [math.ST] 25 Mar 2025 2 establish the minimax rate optimality of cheap testing for a variety of standard testing prob- lems. We complement this theory in Sec. 6 with experiments demonstrating the benefits of cheap testing over standard maximum mean dis... | https://arxiv.org/abs/2502.07672v2 |
works do not analyze the power of their approximate tests, and, in each case, the approximation sacrifices finite-sample validity. Sequential permutation testing [see, e.g., 5, 17, 55, 18] reduces the total number of per- mutations required for a powerful test by processing permuted statistics sequentially and applying... | https://arxiv.org/abs/2502.07672v2 |
Tsatisfies T(Y,Z) = (bP×bP)ϕn1+ (bP×bQ)ϕn1,n2+ (bQ×bQ)ϕn2. (2) An important example of a QTS the canonical homogeneity U-statistic . DEFINITION 2 (Homogeneity U-statistic). Given samples YandZof size n1andn2 respectively, we call U(Y,Z), orUn1,n2for short, a homogeneity U-statistic if Un1,n2=1 (n1)2(n2)2P (i1,i2)∈in1 2... | https://arxiv.org/abs/2502.07672v2 |
of a test (1). In this section, we review standard (Monte Carlo) permutation tests for homogeneity and independence and introduce new cheap permutation tests that reduce the runtime and memory demands. The results of this section are summarized in Tab. 1. 3.1. Permutation tests of homogeneity. LetX≜(Xi)n i=1= (Y1,...,Y... | https://arxiv.org/abs/2502.07672v2 |
sufficient statistics using Θ(cϕn2)time and Θ(s2)memory 4fori,j∈[s]andw∈ {n1,n2,(n1,n2)}doΦw ij←P a∈Ii,b∈Ijϕw(Xa,Xb); 5// Compute original and permuted test statistics using Θ(Bs2)elementary operations 6forb= 0,1,2,...,Bdo 7 π←identity permutation if b= 0else uniform permutation of [s] 8 Tb←1 n2 1Ps1 i,j=1Φn1 π(i)π(j)+... | https://arxiv.org/abs/2502.07672v2 |
we find that the overhead of cheap permutation is negligible whenever Bs2≪cϕn2, and we will see in Secs. 4 and 6 that a slow-growing or constant setting of ssuffices to maintain power while substantially reducing runtime. 3.5. Finite-sample exactness. We conclude this section by noting that the cheap permu- tation test... | https://arxiv.org/abs/2502.07672v2 |
and thresholds). We have γn1,n2,s= Θ(γn1,n2) whenever s≥3q 24(1−α⋆)(1+Ω(1)) βα⋆ρn1n2andγn1,n2,s=γn1,n2(1 +o(1)) whenever s=ω(n2 n1). The separation threshold γn1,n2for standard permutation testing is a refined, quantitative version of the one derived in [29, Thm. 4.1]. Here, we keep track of all numerical factors and, ... | https://arxiv.org/abs/2502.07672v2 |
Ω 3q 1 βα⋆ bin size requirement of Cor. 1. 4.2. Power guarantees for independence testing. For the independence V-statistic of Def. 3, we introduce the shorthand Xi≜(Yi,Zi)and define a symmetrized version of hin, hin(x1,x2,x3,x4)≜1 4!P (i1,i2,i3,i4)∈i4 4hin(xi1,xi2,xi3,xi4), thevariance components ψ′ 1≜Var(E[hin(X1,X... | https://arxiv.org/abs/2502.07672v2 |
(12 α⋆β−2)(32 n+2304 ns+460800 ns2) ϵn+q ˜ξwild(18432 s+3686400 s2) √nq 9α⋆β 12−2α⋆β−√ 32+4q (12 α⋆β−2)ψ′ 24147200 s (3n−q (12 α⋆β−2)32n)1/2 . REMARK 5 (Equivalent MMD separation rates and thresholds). We have ϵn,s= Θ( ϵn) foranychoice of sandϵn,s=ϵn(1 +o(1)) whenever s=ω(1). The proof of Cor. 2 can be found in App. ... | https://arxiv.org/abs/2502.07672v2 |
T=E[T(X)]: Φn(β) =q (1 β−1)Var( T(X)),ΨX(α) =q (1 α−1)Var( T1|X) +E[T1|X], CHEAP PERMUTATION TESTING 13 Ψn(α,β) =q (1 αβ−1 β)E[Var( T1|X)]when E[T1|X] = 0 almost surely, and (24) Ψn(α,β) =q (2 αβ−2 β)E[Var( T1|X)] +E[T1] +q (2 β−1)Var( E[T1|X])otherwise. (25) 5. Minimax optimality of cheap testing. In this section, we ... | https://arxiv.org/abs/2502.07672v2 |
a universal constant Candα⋆≜α 2e β 31/⌊α(B+1)⌋. REMARK 8 (Cheap optimality: discrete L2independence). The cheap test of Prop. 5 achieves the minimax optimal separation rate of order b1/4 (2)/√n[29, Prop. 5.4] for any choice of s. 5.4. Hölder independence testing. Our final result, proved in App. J.4, shows that cheap ... | https://arxiv.org/abs/2502.07672v2 |
a quadratic test statistic (Def. 1) and hence is amenable to cheap testing. 6.1.1. Cheap vs. standard MMD testing. Fig. 1 (left) compares cheap MMD testing to standard MMD permutation testing as a function of the total sample size n=n1+n2and bin count s. We find that s= 128 bins suffice to closely match the power of st... | https://arxiv.org/abs/2502.07672v2 |
s=8s=16s=32s=256s=1024s=2048s=4096s=8192 Standard Cheap n=16384 FIG4.Power vs. runtime for cheap and standard Wilcoxon-Mann-Whitney tests of homogeneity as total sample sizen=n1+n2and bin count svary. 6.1.4. Cheap vs. standard WMW testing. Fig. 4 (left) compares cheap WMW testing to standard WMW permutation testing as ... | https://arxiv.org/abs/2502.07672v2 |
to exploit bin-wise sufficient statistics to avoid the standard sample-size-dependent overhead of permutation testing. While this work has focused primarily on quadratic test CHEAP PERMUTATION TESTING 19 512 1024 1536 2048 3072 4096 Sample size n0.20.40.60.81.0Power Standard HSIC Cheap s=128 Cheap s=64 Asymp. cross-HSI... | https://arxiv.org/abs/2502.07672v2 |
probabilita. In Atti del Congresso Internazionale dei Matematici: Bologna del 3 al 10 de settembre di 1928 47–60. [8] C HUNG , J. and F RASER , D. (1958). Randomization Tests for a Multivariate Two-Sample Problem. Journal of the American Statistical Association 729–735. [9] C HUNG , E. and R OMANO , J. P. (2013). Exact... | https://arxiv.org/abs/2502.07672v2 |
Berlin, Heidelberg. [23] G RETTON , A., F UKUMIZU , K., T EO, C., S ONG , L., S CHÖLKOPF , B. and S MOLA , A. (2007). A Kernel Statistical Test of Independence. In Advances in Neural Information Processing Systems 20. [24] G RETTON , A., B ORGWARDT , K. M., R ASCH , M. J., S CHÖLKOPF , B. and S MOLA , A. (2012). A Kern... | https://arxiv.org/abs/2502.07672v2 |
Systems 347588–7597. Curran Associates, Inc. [41] M OHRI , M., R OSTAMIZADEH , A. and T ALWALKAR , A. (2012). Foundations of Machine Learning . The MIT Press. [42] N EUHAUS , G. (1993). Conditional rank tests for the two-sample problem under random censorship. The Annals of Statistics 1760–1779. [43] P ANDA , S., S HEN... | https://arxiv.org/abs/2502.07672v2 |
Maximum Mean Discrepancy. InInternational Conference on Learning Representations . 22 [58] S ZEKELY , G. and R IZZO , M. (2004). Testing for equal distributions in high dimension. InterStat 5. [59] S ZÉKELY , G. J., R IZZO , M. L. and B AKIROV , N. K. (2007). Measuring and testing dependence by corre- lation of distanc... | https://arxiv.org/abs/2502.07672v2 |
V-statistics . . . . . . . . . . . 44 E.2 Proof of Lem. E.2: Conditional variance of permuted independence V-statistics 44 E.3 Proof of Lem. E.3: Mean of permuted independence V-statistics . . . . . . . 48 F Proof of Cor. 2: Power of cheap independence: finite variance, PD . . . . . . . . . 51 F.1 Proof of Lem. F.1: E[... | https://arxiv.org/abs/2502.07672v2 |
discrete L2independence . . . . . . 68 J.4 Proof of Prop. 6: Power of cheap testing: Hölder L2independence . . . . . . 69 K Supplementary experiment details . . . . . . . . . . . . . . . . . . . . . . . . . . 69 K.1 Selection of the number of permutations B. . . . . . . . . . . . . . . . . . . 70 K.2 Cheap permutation ... | https://arxiv.org/abs/2502.07672v2 |
frame a cheap independence V-statistic permutation test as a standard independence V-statistic per- mutation test operating on bins of datapoints. APPENDIX B: Proof of Thm. 1: Power of cheap homogeneity: finite variance We will establish both claims using the refined two moments method (Cor. 3) with the auxiliary seque... | https://arxiv.org/abs/2502.07672v2 |
2,(i′ 1,i′ 2)∈in1 2,(j′ 1,j′ 2)∈in2 2}, I1≜{i∈Itotal:s(i)≤1},and Ic 1={i∈Itotal:s(i)>1}. By the argument of [29, Thm. 4.1], we have E[Var( Uπ,n n1,n2|X)]≤˜ψY Z,2|Ic 1| (n1)2 (2)(n2)2 (2), (39) where ˜ψY Z,2is the maximum over i∈Ic 1of E E[hho(Xπi1,Xπi2;Xπn1+j1,Xπn1+j2)hho(Xπi′ 1,Xπi′ 2;Xπn1+j′ 1,Xπn1+j′ 2)|X] , andhh... | https://arxiv.org/abs/2502.07672v2 |
ℓ2) ×hho(X(πi′ 1) k′ 1,X(πi′ 2) k′ 2;X(πs1+j′ 1) ℓ′ 1,X(πs1+j′ 2) ℓ′ 2) X ,(42) where we define K≜(k1,k2,ℓ1,ℓ2,k′ 1,k′ 2,ℓ′ 1,ℓ′ 2), Jtotal≜{1,...,m }8,J1≜{K∈Jtotal:s(K)≤1},and Jc 1={K∈Jtotal:s(K)>1}. Each non-zero summand in the final expression of (42) has indices satisfying 1.i∈K2, with, for the indices in ithat a... | https://arxiv.org/abs/2502.07672v2 |
cases with (Y,Y′,Y′′,Z,Z′,Z′′)∼P×P×P×Q×Q×Qgiven (X˜πi1,π): •˜πi2≤n1,˜πn1+j1> n1,˜πn1+j2> n1or analogous configurations: Then, E[hho(X˜πi1,X˜πi2;X˜πn1+j1,X˜πn1+j2)|X˜πi1,π] =E[hho(X˜πi1,Y;Z,Z′)|X˜πi1,π]. •˜πi2≤n1,˜πn1+j1≤n1,˜πn1+j2> n1or analogous configurations: Then, E[hho(X˜πi1,X˜πi2;X˜πn1+j1,X˜πn1+j2)|X˜πi1,π] =E[hh... | https://arxiv.org/abs/2502.07672v2 |
k(P,Q)and the inequalities q 201s2 4n2ψY Z,2+12s nmax( ψY,1,ψZ,1)≤q 201s2 4n2ψY Z,2+q 12s nmax( ξQ,ξP)MMD k(P,Q) CHEAP PERMUTATION TESTING 33 and (43) justified by Lem. C.1 and the triangle inequality, we find that (10) holds whenever ax2−bx−c≥0forx= MMD k(P,Q), a = 1−1/f, b=q 3−β β 4ξQ n1+4ξP n2 +1 fq 12s nmax( ξQ,ξP... | https://arxiv.org/abs/2502.07672v2 |
with independent components satisfying E[eλXi]≤eλ2ν2/2for some ν≥0, allλ∈R, and all i∈[n]. LetA∈Rn×nbe a symmetric matrix such that aii= 0fori∈[n]. Then, there exists a universal constant c>3/20such that for any t >0, Pr(X⊤AX > t )≤exp(−cmin{t2 ν4∥A∥2 F,t ν2∥A∥op}). (48) Equivalently, for any δ∈(0,1), with probability ... | https://arxiv.org/abs/2502.07672v2 |
concatenation of YandZ, and let ˜Yand˜Zbe independent draws from P andQrespectively. By the triangle inequality, we have the following almost sure inequalities for each k∈[n]: |MMD k(P,Q)−MMD k(bP,bQ)| ≤MMD k(P,bP) + MMD k(Q,bQ)≜∆(X) |∆(X)−E[∆(X)|X−k]| ≤( 1 n1E[MMD k(δYk,δ˜Y)|Yk] ifk≤n1 1 n2E[MMD k(δZk−n1,δ˜Z)|Zk−n1]ot... | https://arxiv.org/abs/2502.07672v2 |
n1,n2is the cheaply permuted V-statistic (29). This in turn will imply (46) as Vπ,s n1,n2≥ Uπ,s n1,n2by the same argument (44) used to show that Vn1,n2≥Un1,n2. We follow a structure similar to that in the proof of [29, Thm. 6.1]. Our first step is to iden- tify a second random variable that, conditioned on X, has the s... | https://arxiv.org/abs/2502.07672v2 |
≤ P k∈Gζk E 1k∈B|π,k∈ G − P k∈Fζk E 1k∈C|π,k∈ F2+s1. 2⊔denotes a disjoint union 40 Letc1=E 1k∈B|π,k∈ G ,c2=E 1k∈C|π,k∈ F . By Hoeffding’s inequality [27, Thm. 2] and the union bound Pr |c1P k∈Gζk+c2P k∈Fζk| ≥t|π ≤2exp −t2 2(c2 1|G|+c2 2|F|) ≤2exp −t2 2s1 . Therefore, with probability at least 1−δ′′, E... | https://arxiv.org/abs/2502.07672v2 |
method (Cor. 3) with the aux- iliary sequence (Vπb,s n)B b=1of cheaply-permuted independence V-statistics (32). By Cor. 3 with option (25), it suffices to bound Var(Vn),E[Var( Vπ,s n|X)],Var(E[Vπ,s n|X]), and E[Vπ,s n]. These quantities are bounded in Lems. E.1, E.2, and E.3 with proofs in Apps. E.1, E.2, and E.3. LEMM... | https://arxiv.org/abs/2502.07672v2 |
2,i′ 3,i′ 4)∈Ktotal:|{i1,i2,i3,i4} ∩ {i′ 1,i′ 2,i′ 3,i′ 4}|>1}. Then, Var(Vn) =E[V2 n]−E[Vn]2 =1 n8P (i1,...,i 4,i′ 1,...,i′ 4)∈Ktotal E[hin(Xi1,Xi2,Xi3,Xi4)hin(Xi′ 1,Xi′ 2,Xi′ 3,Xi′ 4)] −E[hin(Xi1,Xi2,Xi3,Xi4)]E[hin(Xi′ 1,Xi′ 2,Xi′ 3,Xi′ 4)] = (I) + (II) + (III), where (I)is the summation over (i1,...,i′ 4)∈K1,(II)i... | https://arxiv.org/abs/2502.07672v2 |
2 by Cauchy-Schwarz. Thus, (II)is upper-bounded by ψ′ 2|K2\¯K2| n8+˜ψ′ 1|¯K2| n8, and we find that Var(Vπ,n n)≤ψ′ 2|(K1∪K2)c|+|K2\¯K2| n8 +˜ψ′ 1|¯K2| n8≤960ψ′ 2 n2+32˜ψ′ 1 n. Finally, by Jensen’s inequality, Var(E[Vπ,n n|X]) =E[(E[Vπ,n n|X])2]−(E[E[Vπ,n n|X]])2 ≤E[E[(Vπ,n n)2|X]]−(E[E[Vπ,n n|X]])2= Var( Vπ,n n). E.2.2.... | https://arxiv.org/abs/2502.07672v2 |
when K∈(L1∪L2)c, so the number of such summands is bounded by 144m6. Thus, we have that (II′)is upper-bounded by ˜ψ′ 1|¯K2|m7 n8+˜ψ′ 1|˜K2|·3m7 n8+˜ψ′ 1|K2\(¯K2∪˜K2)|·16m7 n8 +ψ′ 2|˜K2|·m6 n8+ψ′ 2|K2\(¯K2∪˜K2)|·|(L1∪L2)c| n8 ≤˜ψ′ 1 32s7m7 n8+384s6·3m7 n8+11520 s5·16m7 n8 +ψ′ 2 384s6m6 n8+11520 s5·144m6 n8 . Similarly... | https://arxiv.org/abs/2502.07672v2 |
that for any joint distribution over the Unif ({0,1})variables ϵi1,ϵi2,ϵi3,ϵi4, we have E[ ϵi1+ϵi4 ϵi2+ϵi3 ]≤2. Indeed, by Cauchy-Schwarz, E ϵi1+ϵi4 ϵi2+ϵi3 ≤E ϵi1+ϵi421/2E ϵi2+ϵi321/2, andE ϵi1+ϵi42 is maximized when ϵi1=ϵi4almost surely, with maximum value 0×1 2+ 4×1 2= 2. Thus, E ϵi1+ϵi4 ϵi... | https://arxiv.org/abs/2502.07672v2 |
4U| {z } (II)+8s(s−2)(s−4) s4×3 8U| {z } (III) +s4−s(s−2)(s−4)(s−6)−12s(s−2)(s−4) s4 ×1 2U| {z } (IV)+10n2−44n+48 n3˜ξ|{z} (V) =1 4 1 +s2+s−4 s3 U+10n2−44n+48 n3˜ξ. APPENDIX F: Proof of Cor. 2: Power of cheap independence: finite variance, PD To establish the standard testing claim, we will show that (18) implies the... | https://arxiv.org/abs/2502.07672v2 |
follows from the observation that R k(x,·)d(Π−P×Q)(x) k= MMD k(Π,P×Q), and that E[ R k(x,·)d(δX1+ Π−δY1×Q−P×δZ1)(x) 2 k] =E[RR k(x,x′)d(δX1+ Π−δY1×Q−P×δZ1)(x)d(δX1+ Π−δY1×Q−P×δZ1)(x′)] =E[E[hin(X1,X2,X3,X4)|X1]] =ξ. F.3. Proof of Lem. F.3: ˜ψ′ 1bound. By the definition (14) of ˜ψ′ 1, we can write ˜ψ′ 1=E hin,Y(Y1,Y2,Y... | https://arxiv.org/abs/2502.07672v2 |
k(Π,P×Q)−2MMD k(bΠ,bP×bQ)|<ΛV(β). Since|a2−b2|=|a−b||b+a| ≤ |a−b|(|b−a|+2|a|)for all real a,bthe result follows. G.2. Alternative representation of the independence V-statistic. We next define an alternative, equivalent way to represent the independence V-statistic (Def. 3) and its permuted versions, due to [23]. We wi... | https://arxiv.org/abs/2502.07672v2 |
G.4. LEMMA G.6 (A(i)complementary quantile bounds). Instantiate the assumptions and notation of Lem. G.5, and fix any i∈[4]. With probability at least 1−δ, A(i)≤Di+Ciq log(1 /δ) 2for (C1,C2,C3,C4) = (10√ Ks√m+2√ Ks√n,6√ Ks√m,14√ Ks√n,6√ Ks√n). Our fourth lemma, proved in App. G.8, uses the estimates of Lem. G.6 to veri... | https://arxiv.org/abs/2502.07672v2 |
+P×Q)k∥k + MMD k(Π,P×Q) ∥(µj−Π +P×Q)k∥k+∥(˜µi−Π +P×Q)k∥k .(83) Plugging q=swap s(j)andr=swap s(i)into (79), (80), (81), (82), we find that ˜bi=1 4s2Ps j=1 Rijji m2+Rij(swaps(j))i m2−Rijj(swaps(i)) m2−Rij(swaps(j))(swaps(i)) m2 +Rjiij m2+Rji(swaps(i))j m2−Rjii(swaps(j)) m2−Rji(swaps(i))(swaps(j)) m2 =1 4s2 Ps j=1(δ... | https://arxiv.org/abs/2502.07672v2 |
k+∥(µswaps(i)−Π +P×Q)k∥2 k ∥(˜µ−Π +P×Q)k∥2 k + MMD k(Π,P×Q)2 2∥(µ−Π +P×Q)k∥2 k+ 2∥(˜µ−Π +P×Q)k∥2 k +∥(˜µi−Π +P×Q)k∥2 k+∥(˜µswaps(i)−Π +P×Q)k∥2 k +∥(µi−Π +P×Q)k∥2 k+∥(µswaps(i)−Π +P×Q)k∥2 k =12(A(2))2(A(3))2 s+12(A(1))2(A(4))2 s + 12MMD2 k(Π,P×Q) (A(1))2+ (A(2))2+ (A(3))2+ (A(4))2 . G.6. Proof of Lem. G.5: A(i)mea... | https://arxiv.org/abs/2502.07672v2 |
s+2A(3)MMD k(Π,P×Q)√s ∥¯A∥2 F≤(A(1))2(A(2))2 4s4 +MMD2 k(Π,P×Q)((A(1))2+(A(2))2) 4s3 , Tr(¯A)≤A(1)A(2) 2s2+MMD k(Π,P×Q)(A(1)+A(2)) 4s3/2 ,and ¯b 2 2≤3(A(2))2(A(3))2+(A(1))2(A(4))2 4s3 +3MMD2 k(Π,P×Q)((A(1))2+(A(2))2+(A(3))2+(A(4))2) 4s2 . Hence, if we choose δ′=δ′′=δ′′′=α/3, then, by the union bound and the triangle i... | https://arxiv.org/abs/2502.07672v2 |
2)≤β/3forA2≜{ΨX(α⋆)≤Ψn(α⋆,β/3)}. Since ∆(X)rejects when T(X)> T(bα), the acceptance probability is upper-bounded by Pr(T(X)≤T(bα)) = Pr( T(X)≤T(bα),A1∩ A 2) + Pr( T(X)≤T(bα),Ac 1∪ Ac 2) ≤Pr(T(X)≤Ψn(α⋆,β/3)) + Pr( Ac 1∪ Ac 2) ≤Pr(T(X)≤ T − Φn(β 3)) + Pr( Ac 1) + Pr( Ac 2)≤β, where the final two inequalities used the uni... | https://arxiv.org/abs/2502.07672v2 |
3=Zi′ 4,i′ 3̸=i′ 4 −I Zi′ 1=Zi′ 3,i′ 1̸=i′ 3 −I Zi′ 2=Zi′ 2,i′ 3̸=i′ 4 (ii) ≤16max E[I(Y1=Y2)I(Zi=Zj)] : (i,j)∈ {(1,2),(1,3),(3,4)} CHEAP PERMUTATION TESTING 69 (iii)= 16max E[I(Y1=Y2,A1̸=A2)I(Zi=Zj,Bi̸=Bj)] : (i,j)∈ {(1,2),(1,3),(3,4)} = 16max E[gY((Y1,A1),(Y2,A2))2gZ((Zi,Bi),(Zj,Bj))2] : (i,j)∈ {(1,2),(1,3)... | https://arxiv.org/abs/2502.07672v2 |
feature function fj, and cheap permutation can be carried out in Θ(Bsr) additional time with Θ(sr)memory, as described in Alg. K.1. CHEAP PERMUTATION TESTING 71 Algorithm K.1: Cheap homogeneity testing with feature kernels Input: Samples (Yi)n1 i=1,(Zi)n2 i=1, bin count s, feature maps f= (fj)r j=1(88), level α, permut... | https://arxiv.org/abs/2502.07672v2 |
Sharp Anti-Concentration Inequalities for Extremum Statistics via Copulas Matias D. Cattaneo1Ricardo P. Masini2*William G. Underwood3 February 12, 2025 Abstract We derive sharp upper and lower bounds for the pointwise concentration function of the maximum statistic of didentically distributed real-valued random variabl... | https://arxiv.org/abs/2502.07699v1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 5.2 Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 5.3 Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 5.4 Lemma 3 . . . . . . . . . . . . . . . . . . . . .... | https://arxiv.org/abs/2502.07699v1 |
Gaussian random vector with a non-singular covariance matrix. Their proof leveraged the fact that conditioning on components preserves joint Gaussianity, and the resulting anti-concentration inequality was used to establish a conditional multiplier central limit theorem in a high-dimensional regime. A related approach ... | https://arxiv.org/abs/2502.07699v1 |
1, presented in Section 5.1, relies only on basic properties of copulas and their diagonal sections. A similar copula-based approach was taken by Frank et al. (1987), who obtained optimal upper and lower bounds for the distribution function of the sum (and other combinations) of several random variables, under arbitrar... | https://arxiv.org/abs/2502.07699v1 |
the copula associated with ( X1, . . . , X d) only through its diagonal section, as formalized in Definition 1. Definition 1. Letd∈N. A function ∆ : [0 ,1]→[0,1]is ad-dimensional copula diagonal if there exists a d-dimensional copula C: [0,1]d→[0,1]with ∆(u) =C(u, . . . , u )for all u∈[0,1]. Lemma 1 below gives a chara... | https://arxiv.org/abs/2502.07699v1 |
exists (X1, . . . , X d)withXi∼ N(0, σ2)fori∈[d]such that sup x∈RP x <max i∈[d]Xi≤x+ε ≥dε σϕε σ ∧Φε σ ≥dε σe−1/2 √ 2π∧1 2≥dε 5σ∧1 2. Compare Example 1 with Nazarov’s inequality (Nazarov, 2003; see also Chernozhukov et al., 2017b, for a detailed proof) which, under the assumption of joint Gaussianity, obtains a bo... | https://arxiv.org/abs/2502.07699v1 |
copula with d= 2. Figure 1: Top: the two d-dimensional copula diagonals (7)and(8)constructed to prove (3)and(4) respectively in Theorem 1. For the upper bound (a), the increment over ( u, u+δ] is maximized, while for the lower bound (b) it is minimized. Bottom: contour plots for possible two-dimensional (d= 2) copulas ... | https://arxiv.org/abs/2502.07699v1 |
it is necessary to restrict the class of admissible copulas. For example, as discussed in Section 2, in the setting where ( X1, . . . , X d) follows a multivariate Gaussian law, Nazarov’s inequality can be applied to the maximum statistic and produces a bound (5) with a square root-logarithmic dependence on the dimensi... | https://arxiv.org/abs/2502.07699v1 |
decreasing function ψ: [0,1]→[0,∞] with ψ(0) =∞andψ(1) = 0 satisfying C(x1, . . . , x d) =ψ−1 dX i=1ψ(xi)! , (9) for all ( x1, . . . , x d)∈[0,1]d. The function ψis known as the generator ofC. Since our focus is on high-dimensional phenomena, we consider only strict Archimedean generators with completely monotone inver... | https://arxiv.org/abs/2502.07699v1 |
cases where Fadmits a decreasing Lebesgue density f, the dimension- dependence of an anti-concentration bound derived using Theorem 2 is determined by the quantity H(x):=h(x)∧ d·f(x) , where h(x):=f(x)/ 1−F(x) is the hazard function (or inverse Mills ratio) associated with F. Typically, if his an increasing function... | https://arxiv.org/abs/2502.07699v1 |
d), with P ∥X−T∥∞> ε ≤p(ε) for some decreasing function p: [0,∞)→[0,1], where ∥x∥∞:=max i∈[d]|xi|. Typically, either one knows the law of Texplicitly, or can draw samples from it. Inference proceeds by choosing a significance level α∈(0,1) and computing a quantile qα:=inf q∈R:P max i∈[d]Ti≤q ≥1−α . It is straightf... | https://arxiv.org/abs/2502.07699v1 |
(2022) provides sufficient conditions for a coupling betweenPn i=1X(i)and an Rd-valued random vector Twith conditional distribution T|L∼ N(0,Σ), where Σ:=nX i=1 LVar g(i)|L LT+ Var ε(i) . We now impose some further conditions on the law of T; in particular, we ensure that the copula associated with Tis the indepe... | https://arxiv.org/abs/2502.07699v1 |
given in the main paper. 5.1 Theorem 1 Before proving Theorem 1, we first establish the result in the special case where the common law is the standard uniform distribution, in Lemma 4. For clarity, we write Ui∼ Ufor each i∈[d]. Lemma 4. Letd∈N. For u∈[0,1]andδ∈[0,1−u], max P∈Pd(U)P u <max i∈[d]Ui≤u+δ = (dδ)∧(u+δ), (... | https://arxiv.org/abs/2502.07699v1 |
0∨ 1−F(x)−d 1−F(x+ε) . 5.2 Theorem 2 Proof of Theorem 2. Let ∆ be the copula diagonal section associated with the law of ( X1, . . . , X d), and suppose it is a convex function on [0 ,1]. Since ∆(1) = 1, for any u∈[0,1] and δ∈[0,1−u], ∆(u+δ) = ∆1−u−δ 1−u·u+δ 1−u·1 ≤1−u−δ 1−u∆(u) +δ 1−u. Combining this with the fac... | https://arxiv.org/abs/2502.07699v1 |
copula has inverse generator ψ−1(x) =−1 θlog 1− 1−e−θ e−x forθ >0 (Nelsen, 2006, Example 4.24). With x∈(0,∞) and setting a= 1−e−θ ∈(0,1), ψ−1′(x) =−1 θae−x 1−ae−x<0, Ψ(x) =ex−a edx−a, Ψ′(x) =(1−d)e(d+1)x−aex+adedx edx−a2≤0, since by convexity, edx≤d−1 de(d+1)x+1 dex. Thus Ψ is non-increasing. 15 Finally, th... | https://arxiv.org/abs/2502.07699v1 |
Further, fis decreasing on [0 ,∞) while his increasing on [0 ,∞). Note that for d≥2, we have h(x) =d·f(x) if and only if x= log log d, and if d= 1 then d·f(x)≤1. So for x≥0, P x <max i∈[d]Xi≤x+ε ≤εf(x)1 1−F(x)∧d ≤ε h(x)∧ d·f(x) ≤ε 1 + log d . Details for Example 12. We assume that the scale parameter is λ= 1; ... | https://arxiv.org/abs/2502.07699v1 |
<max j∈[d]Ykj≤x+ε ≤KX k1=1···KX kd=1 dY j=1pkj! ε minj∈[d]σkjp 2 logd+ 1 =ε √2 logd+ 1 σdY j=1 KX k=1pk! =ε σp 2 logd+ 1 . We remark that the inequality minj∈[d]σkj≥σis essentially optimal in the regime where the dimension dis much larger than the number of original components K, since they differ in only (K−1)do... | https://arxiv.org/abs/2502.07699v1 |
Probability Theory and Related Fields , 143:219–238. Arakelian, V. and Karlis, D. (2014). Clustering dependencies via mixtures of copulas. Communica- tions in Statistics–Simulation and Computation , 43(7):1644–1661. Bakshi, A., Diakonikolas, I., Hopkins, S. B., Kane, D., Karmalkar, S., and Kothari, P. K. (2020). Outlie... | https://arxiv.org/abs/2502.07699v1 |
Training in Mathematics Workshop in Applied Probability, Indian Institute of Technology Bombay. Kuchibhotla, A. K., Mukherjee, S., and Banerjee, D. (2021). High-dimensional CLT: Improvements, non-uniform extensions and large deviations. Bernoulli , 27(1):192 – 217. L´ evy, P. (1954). Th´ eorie de l’Addition Des Variabl... | https://arxiv.org/abs/2502.07699v1 |
Alexander Mangulad Christgau Model-free Methods for Event History Analysis and Efficient Adjustment phd thesis this thesis has been submitted to the phd school of the faculty of science, University of Copenhagen Department of Mathematical Sciences University of Copenhagen August 2024arXiv:2502.07906v1 [stat.ME] 11 Feb ... | https://arxiv.org/abs/2502.07906v1 |
of all, I want to extend my deepest thanks to my entire family, especially Far and Mor, for their unwavering encouragement and support throughout my life. You have my deepest gratitude and love. To my grandparents, who never saw the end of this adventure, but whose inquisitive minds and kind hearts shaped me profoundly... | https://arxiv.org/abs/2502.07906v1 |
Det tredje afsnit vedrører justering for kovariater, hvilket er en metode til at estimere effekten af en behandling ved at tage højde for observeret confounding. Vi udvikler en generel teori, der kan beskrive justering for enhver delmængde af information i kovari- aterne. Vi identificerer den optimale information at ju... | https://arxiv.org/abs/2502.07906v1 |
. . . . . . . . . . . . . . . . . . 2 1.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Nonparametric conditional local independence testing 15 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.2 The Local Covariance Measure . . . . . . . ... | https://arxiv.org/abs/2502.07906v1 |
DOPE 95 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.2 Generalized adjustment concepts . . . . . . . . . . . . . . . . . . . . . . . 99 3.3 Efficiency bounds for adjusted means . . . . . . . . . . . . . . . . . . . . . 101 3.4 Estimation based on outcome-adapted representa... | https://arxiv.org/abs/2502.07906v1 |
. . . . . . . . . . . . . . . . . . . . . . . . . . 164 4.B Multiplicative hazards models . . . . . . . . . . . . . . . . . . . . . . . . . 175 Bibliography 179 xii 1 Introduction This thesis delves into a range of statistical challenges, including event history analysis, statistical efficiency, hypothesis testing, eff... | https://arxiv.org/abs/2502.07906v1 |
data-generating process is often more complicated than desired, this leads to a difficult trade-off between model misspecification due to simplicity, and impracti- cality due to complexity. Model misspecification may lead to bias in the results of the analysis, but a potentially more serious problem is the lack of inte... | https://arxiv.org/abs/2502.07906v1 |
the distribu- tionP. This estimand unambiguously defines a target parameter without reference to a (semi)parametric model, and the conditions for being well-defined are clear from the expression, unlike the model-based estimand τmb.2 To see how τmfis a generalization of τmb, consider a distribution PPPthat follows thep... | https://arxiv.org/abs/2502.07906v1 |
to the central limit theorem (CLT), the expectation operator is pointwise?n-approximated by the empirical mean operator, denoted Pnr¨sand given by PnrfpZqs–1 nřn i“1fpZiq, for any function fPL2pPq. Distributional quantities that are used to compute the target estimand, but are not of primary interest, such as mP, are c... | https://arxiv.org/abs/2502.07906v1 |
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