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arXiv:2504.02974v2 [math.ST] 21 May 2025E-variables for hypotheses generated by constraints Martin Larsson∗Aaditya Ramdas†Johannes Ruf‡ May 23, 2025 Abstract An e-variable for a family of distributions Pis a nonnegative random variable whose expected value under every distribution in Pis at most one. E-variables have r... | https://arxiv.org/abs/2504.02974v2 |
in some range; this is a constraint. In nonparametric settings , it is common to specify classes of distributions whose moments or supports are restricted in s ome way; these are also constraints. Thus, the classes Pconsidered in this paper are quite general, and special case s of such ‘constrained hypotheses’ have bee... | https://arxiv.org/abs/2504.02974v2 |
a function f∈ LholdsP-quasi-surely, abbreviated P-q.s., if the set where it fails isP-negligible. For subsets A,Bof a vector space, we write A−B={a−b:a∈A,b∈B}. The set of natural numbers is N={1,2,...}. 2 Hypotheses generated by constraints Definition 2.1. Aconstraint set is any nonempty set of functions Φ⊂ L. The eleme... | https://arxiv.org/abs/2504.02974v2 |
then for any functionf∈ C, sayf=g−hwithg∈cone(Φ) andh∈ LΦ p, we have/integraltext Xfdµ≤/integraltext Xgdµ≤0, and henceµ∈ C◦. ThusC◦=R+P, and the proof is complete. (ii): Since every e-variable is P-q.s. equal to a finite e-variable, and since LΦcontains all finite e-variables, the claim is immediate from (i). Remark 2.3.... | https://arxiv.org/abs/2504.02974v2 |
some π′∈Rd +andh′∈ LΦ p. We claim that g′has a version g=/summationtextd i=1πigifor someπ∈K. (Thatgis aversion ofg′means that the two are equal, P-q.s.) To see this, suppose supp(π)is redundant and letρ∈Rd +be as in the definition of redundant. Then there exists ε >0such that π′′=π′−ερbelongs to Rd +and satisfies supp(π′... | https://arxiv.org/abs/2504.02974v2 |
Theorem 3.1is the following generalization of a result of Clerico (2024a , Theorem 1). Here an e-variable his called admissible if whenever another e-variableh′satisfiesh′≥h,P-q.s., we actually have h′=h,P-q.s. (See also Section 7for further discussion of admissibility.) 6 Corollary 3.3. Every admissible e-variable is P... | https://arxiv.org/abs/2504.02974v2 |
the class of e-variables used i nWaudby-Smith and Ramdas (2024);Larsson et al. (2025);Orabona and Jun (2023);Clerico (2024b ). A minor variant of Example 3.4shows that without the boundedness assumption, there do not exist any nontrivial e-variables. Example 3.7. TakeX=Rand letPconsist of all distributions whose mean e... | https://arxiv.org/abs/2504.02974v2 |
any sub- ψdistributed random variable X satisfies the Chernoff tail bound P(X≥x)≤e−ψ∗(x)for allx. Indeed, if dom(ψ∗) =R there is nothing to prove. Otherwise, let ¯x <∞denote the right endpoint of the interval dom(ψ∗). If¯x /∈dom(ψ∗), thenψ∗(¯x) =∞and hence P(X≥¯x)≤e−ψ∗(¯x)= 0, so thatXis concentrated on dom(ψ∗). If¯x∈dom... | https://arxiv.org/abs/2504.02974v2 |
Krein–Šmulian theorem (see Appendix A.3). Unfortunately the Krein–Šmulian theorem cannot be applied directly, because LΦis not the dual of a Banach space. Instead, we first endow LΦwith a slightly weaker topology than σ(LΦ,MΦ), which allows us to embed it into a larger space that isthe dual of a Banach space. Checking c... | https://arxiv.org/abs/2504.02974v2 |
4.2) and is a subset of LΦ. We will show below that G −E′ +isσ(E′,E)-closed. (4.5) Once this has been done, the proof of Theorem 4.3(i) is completed as follows. Observe that LΦ∩(G −E′ +) =G −LΦ∩E′ +=G −LΦ +, where the equality LΦ∩E′ +=LΦ +holds because a functionf∈ LΦis nonnegative if and only if/integraltext Xfdµ≥0for... | https://arxiv.org/abs/2504.02974v2 |
that his an e-variable. Conversely, let hbe an e-variable and set f=h−1. Then by part (i)of the theorem, there is some π′∈ M+(Λ)such thatf(x)≤/integraltext Λgλ(x)π′(dλ)for allx∈dom(ψ∗). Since f≥ −1, the argument after ( 4.6) withr= 1yieldsπ′((0,λmax])≤1. Thus the measure π=π′(· ∩(0,λmax]) + (1−π′((0,λmax]))δ0belongs to... | https://arxiv.org/abs/2504.02974v2 |
then πis both a left and right Haar measure, and thus σ∗fπ=fπ. We are interested in describing the set of e-variables for th e hypothesis consisting of all Σ-invariant distributions, P={µ∈ M1:µ=σ∗µfor allσ∈Σ}. Such classes, or infinite-sample versions of them, have been studied in many recent works. For example, testing... | https://arxiv.org/abs/2504.02974v2 |
generating set forΣis a subset Σ0such that any σ∈Σcan be expressed as σ=σ1σ2···σn for somen∈Nandσ1,...,σn∈Σ0. Theorem 5.6. LetFbe a separating set for M, andΣ0a generating set for Σ. A distribution µbelongs to Pif and only if/integraltext X(f(σx)−f(x))µ(dx) = 0for allσ∈Σ0andf∈ F. In other words,Pis generated by the con... | https://arxiv.org/abs/2504.02974v2 |
the condition/integraltext Nfndµ≤0translates to the inequality 1−pn+pn(1−2n)≤0, or pn≥2−n. It follows that P(Φ)consists of the single measure µ=/summationtext x∈N2−xδx, and that E(Φ) consists of all nonnegative functions hsuch that/summationtext x∈Nh(x)2−x≤1. However,/integraltext Nf0dµ=−∞, soµ /∈ P(Φ0). ThusP(Φ0) =∅, ... | https://arxiv.org/abs/2504.02974v2 |
3). Definition 7.1. Acomplete class of e-variables for Pis a subset E′⊂ Esuch that every h∈ EisP-q.s. dominated by some h′∈ E′. A complete class E′isminimal if removing an e-variable (and all its P-versions) from E′renders the class non-complete. For statistical applications it suffices to work with complet e classes, and... | https://arxiv.org/abs/2504.02974v2 |
then it cannot be admissible. To show this, assume h(0)<1and lety∈(0,1]be such that h(y)<2−h(0); suchymust exist, because otherwise/integraltext1 0h(x)dx= 2−h(0)>1, contradicting the e-variable property. Then, the functiongwithg(y) = 2−h(0)andg=helsewhere is still an e-variable, and it strictly dominateshunder(δ0+δy)/2... | https://arxiv.org/abs/2504.02974v2 |
20 for real analytic functions (see e.g. Krantz and Parks (2002, Corollary 1.2.7)) implies that h′′=hon(−∞,¯x)and then by continuity on dom(ψ∗). It follows that h′=hondom(ψ∗), showing that his admissible. Consider now case (2). Then hλmax(x) = 0 for allx∈dom(ψ∗). Indeed, if λmax<∞, this follows from the fact that ψ(λ)→... | https://arxiv.org/abs/2504.02974v2 |
in ( 8.1) above is actually a P-supermartingale, i.e., a supermartingale under every P∈P. The e-process property then follows from the stopping theorem. It is now natural to ask whether every admissible e-process for Pis of the form (8.1), at least assuming the constraint qualification ( 3.2). Settling this question wou... | https://arxiv.org/abs/2504.02974v2 |
Theorem 11.8 of Willard (1970). Part (ii) follows from Theorem 11.5 and Theorem 17.4 of Willard (1970). We now specialize some of the above to Euclidean space Rd. The Heine–Borel theorem states that any closed and bounded subset of Rdis compact. Therefore, Theorem A.1(ii) implies that any bounded net in Rdhas a converg... | https://arxiv.org/abs/2504.02974v2 |
topology σ(E′,E)coming from the dual pair /an}bracketle{tE′,E/an}bracketri}ht with bilinear form /an}bracketle{tϕ,x/an}bracketri}ht=ϕ(x); see Section A.2. The following result plays a crucial role in Section 4. For a proof, see Theorem 12.1 in Conway (1990). Theorem A.3 (Krein–Šmulian) .Let(E,/bardbl·/bardbl)be a Banac... | https://arxiv.org/abs/2504.02974v2 |
Muriel Felipe Pérez-Ortiz, Tyron Lardy, Rianne de Heide, an d Peter D Grünwald. E-statistics, group invariance and anytime-valid testing. The Annals of Statistics , 52(4):1410–1432, 2024. 15 Aleksandr Podkopaev and Aaditya Ramdas. Sequential predic tive two-sample and indepen- dence testing. Advances in Neural Informat... | https://arxiv.org/abs/2504.02974v2 |
High-dimensional ridge regression with random features for non-identically distributed data with a variance profile Issa-Mbenard Dabo & J´ er´ emie Bigot Institut de math´ ematiques de Bordeaux & CNRS (UMR 5251) Universit´ e de Bordeaux, France April 7, 2025 Abstract The behavior of the random feature model in the high... | https://arxiv.org/abs/2504.03035v1 |
identically distributed. These random features are fixed during the training process, and they serve as a way of projecting the input data into a new space where linear methods can be applied more effectively. Importantly, the entries ofWare uncorrelated with the training data ( yi, xi)1≤i≤n, meaning that the random fe... | https://arxiv.org/abs/2504.03035v1 |
in a broader context involving variance profiles (a notion to be defined later) to model settings where the rows of the random matrix Xnare not necessarily iid. In order to achieve this goal, our approach strongly relies on an extension of the “linear-plus-chaos” approximation of the matrix Hrecently proposed in [DM24]... | https://arxiv.org/abs/2504.03035v1 |
direction we refer to [MM22] where the estimation of non-linear models using RF ridge regression has been studied the iid setting. The primary aim of this paper is then to evaluate the performance of RF ridge regression within the framework of the linear model stated above, by specifically analyzing the training risk E... | https://arxiv.org/abs/2504.03035v1 |
random matrices XnandW. To the contrary, the second set of asymptotic equivalents E□ train(λ) and E□ test(λ) only depend on the variance profiles Γ n= (γ2 ij) and ˜Γn= (˜γ2 ij). The construction of this second set of asymptotic equivalents heavily relies on results related to the notion of amalgamation over the diagona... | https://arxiv.org/abs/2504.03035v1 |
2 ... 1˜n0s⊤ K ∈Rn×pwith ˜ n=K˜n0, (1.7) 5 where 1n0denotes the vector of length n0with all entries equal to one. We then apply RF ridge regression with h(x) =x3for several values of λ. In all the numerical experiments reported in the paper, the entries of W′,X′andβare chosen as iid random variables sampled from t... | https://arxiv.org/abs/2504.03035v1 |
that the matrix H/√ncan be asymptotically approximated by the following “linear-plus-chaos” decomposition H♢ √n= Θ lin(h)◦WX⊤ n√np + Θ chaos(h)◦ZG √n. (1.10) In the above equation, WandXnare random matrices defined as W= Υ w◦WGandXn= Υ x◦XG, (1.11) 6 where WG,XG, and ZGare independent matrices, each made of iid entri... | https://arxiv.org/abs/2504.03035v1 |
3.1, the training error Etrain and the predictive error Etestcan be expressed as expectations involving traces of matrix polynomials depending on H, W, X n,˜H,˜X. Therefore, using Proposi- tion 1.1 and under the assumption (1.15) that Whas a constant variance profile, we prove that the asymptotic equivalents E♢ train a... | https://arxiv.org/abs/2504.03035v1 |
in Section 3. We first describe the asymptotic equivalents E♢ train andE♢ test. Then, we introduce free probability results that enable us to provide the second set of asymptotic equivalents, namely E□ train andE□ test. Section 4 is dedicated to the proof of these main results. The proof of the derivation of the expres... | https://arxiv.org/abs/2504.03035v1 |
a constant γ >0such that, for all n,˜n, m, p ≥1, the variance profiles of Xn and ˜X˜nadmit the factorizations Υx=U∗ xVxand˜Υx=˜U∗ x˜Vxwhere the columns of the matrices Ux∈Rr×n, Vx∈Rr×p,˜Ux∈R˜rטn,˜Vx∈R˜r×phave their Euclidean norm that is bounded by√γ. Note that Assumption 2.3 implies, by Cauchy-Schwarz inequality, tha... | https://arxiv.org/abs/2504.03035v1 |
(aij/√ N) is a GOE matrix (that is ANis symmetric and its coefficients ( aij)i≤jform a sequence of iid standard real Gaussian variables). Therefore, the matrix Lis interpreted as an additive deformation by CNof a GOE matrix with variance profile Υo2 Lwhich is the key point to derive the second set of asymptotic equival... | https://arxiv.org/abs/2504.03035v1 |
Q□: DN(C)+→DN(C)+, analytic in each variable, that solves the following fixed point equation Q□(Λ) = id4⊗∆ CN−Λ− R N Q□(Λ)−1 , (3.2) for any Λ∈DN(C)+, with RN(Λ) = diag i=1,...,N NX j=1 γ(L) ij2 NΛ(j, j) . Let γ(L) max2 = max i,j γ(L) ij2 ,0< δ < 1, and consider Λ∈DN(C)+satisfying ℑmΛ≥ γ(L) max2... | https://arxiv.org/abs/2504.03035v1 |
H♢,Xn, Q♢(λ) as precisely stated in Theorem 3.1 that we prove below. Proof of Theorem 3.1. Letz∈C\R+be a complex such that ℑm(z)>2 sup n≥n0{∥H⊤H n∥;∥H♢⊤H♢ n∥} for some sufficiently large n0. Under Assumption 2.2 and Assumption 2.3, we have that the entries of 14 the diagonal matrices Dlin(h) and Dchaos(h) appearing in ... | https://arxiv.org/abs/2504.03035v1 |
from [HS12][Theorem 5.8] that1√n∥Qp(z)∥F≤∥Qp(z)∥≤1 dist(z,R+)and1√n∥Q♢(z)∥F≤∥Q♢(z)∥≤ 1 dist(z,R+). Moreover, Xn√p F,nand Xn√p F,nare bounded, hence for each compact subset C⊂C\R+,fp is uniformly bounded on Cwith|fn(z)| ≤2K δC,where δCis the distance between CandR+. Then, by the normal family Theorem [Rud87][Theorem 14.... | https://arxiv.org/abs/2504.03035v1 |
np−W⊤W pW⊤H♢Q♢(−λ)Dlin(h)Xn n√p−Ip −X⊤ nDlin(h)Q♢(−λ)√n−X⊤ nDlin(h)Q♢(−λ)H♢⊤ nX⊤ nDlin(h)Q♢(−λ)H♢⊤W n√p−IpX⊤ nDlin(h)Q♢(−λ)Dlin(h)Xn n (4.2) As shown in Section 4.3, the first set of asymptotic equivalents E♢ train andE♢ testcan be expressed in terms of some sub-blocks of Q(Λλ). Therefore, it is now relevant to... | https://arxiv.org/abs/2504.03035v1 |
is the Stieltjes transform (1.22) of the Marchenko-distribution, which yields the expressions (3.4) and (3.5) of the training and predictive risks. 4.3.2 Second set of asymptotic equivalents when θlin(h)̸= 0 Suppose now that θlin(h)̸= 0. In this case, using the expression (4.2) of Q(Λλ) and Equation (4.3), it follows t... | https://arxiv.org/abs/2504.03035v1 |
triangle inequality with (4.7) and (4.5), to obtain that id4⊗∆[Q□′(˜Λz)−Q′(˜Λz)] F,n− →0 almost surely . (4.8) Since we have bounded ∥Q′(Λz)−Q′(Λz+iηNIN)∥F,nand id4⊗∆[Q□′(˜Λz)−Q′(Λ−z)] F,nindepen- dently from n, one can prove (4.4) and (4.8) for z∈C\R+as in the proof of Theorem 3.1, which completes the proof of Corolla... | https://arxiv.org/abs/2504.03035v1 |
compute E□ train andE□ test, it is needed to calculate the matrix Q□. Although we do not possess an explicit analytical expression for this matrix, it can be effectively approximated using a fixed- point algorithm to solve Equation (3.2). This algorithm plays a central role in our numerical experiments, as it enables u... | https://arxiv.org/abs/2504.03035v1 |
relevant when analyzing the resulting predictive risk. In agreement with the findings of [dSB20], Figure 3 confirms the presence of a double-descent curve whose shape depends on the value of c. Specifically, for high values of c, a pronounced peak in the predictive risk is observed around m=n, highlighting the classica... | https://arxiv.org/abs/2504.03035v1 |
DN(C)+→DN(C)−, analytic in each variable, that solves of the following fixed point equation: G□ MN(Λ) = ∆ Λ− R N G□ MN(Λ) −YN−1 , for any Λ∈DN(C)+, with RN(Λ) = diag i=1,...,NPN j=1γ2 N(i,j) NΛ(j, j) . Let γ2 max= max i,jγ2 N(i, j),0< δ < 1, and consider Λ∈DN(C)+satisfying ℑmΛ≥γmax 2√ 2 N(1−δ)!1/5 IN. Then, for... | https://arxiv.org/abs/2504.03035v1 |
that ΥL◦AN=X 1≤i≤j≤4Nγ(L) ij√ NaijFij, with Fij=Eij+Eji 1+δij=( Eij+Eji,ifi̸=j, Eii,ifi=j,with Ekl= (δukδvl)1≤u,v≤4N, for 1 ≤k, l≤4N. We directly obtain from this equation that Eh (ΥL◦AN) (L−Λ)−1i =X 1≤i≤j≤4Nγ(L) ij√ NFijEh aij(L−Λ)−1i . Since the entries of ANare gaussian , we obtain from Stein’s lemma A.5 that Eh (ΥL... | https://arxiv.org/abs/2504.03035v1 |
us denote by fthe following function f(M) =RN ∆ X i≤jmi,jFij+CN−Λ −1 . SinceRNis linear, one has that ∂ ∂x′ i,jf2 x′ i,j i,j =RN −(L−Λ)−1Fij(L−Λ)−1 =RN(−Q(Λ)FijQ(Λ)). Thus we obtain from Proposition A.1 (Gaussian Poincar´ e inequality) that Var(Dn(k, k))≤γ2 max nE X i≤j|RN(Q(Λ)FijQ(Λ)) ( k, k)... | https://arxiv.org/abs/2504.03035v1 |
N. The maps Λ 7→E[Q(Λ)], Λ 7→ΘNare analytic, thus Λ 7→Q□(˜Λ) and Λ 7→˜Λ are analytic. Then we obtain that for all Λ >0 and ˜Λ>0 ∥E[Q(Λ)]−Q□(Λ)∥ ≤ ∥ Q□(˜Λ)−Q□(Λ)∥+∥ΘN∥. (A.6) we obtain from Lemma A.1 that ∥Q□(˜Λ)−Q□(Λ)∥ ≤ ∥ (ℑm(Λ))−1∥∥(ℑm(˜Λ))−1∥∥Λ−˜Λ∥. (A.7) We have from Lemma A.3 that ∥Λ−˜Λ∥=∥RN(ΘN)∥ ≤γ2 max∥ΘN∥. More... | https://arxiv.org/abs/2504.03035v1 |
Theorem 3.2, thus one has a deterministic equivalent for this matrix, which is Q□(Λλ+iηnIN). Moreover we prove in the following lemma the Q(Λλ+iηnIN) is close from Q(Λλ) when ngoes to infinity so that Q□(Λλ+iηnIN) is a good approximation of Q(Λλ) in high dimension. Lemma A.10. Forλ >0the following limits hold almost su... | https://arxiv.org/abs/2504.03035v1 |
345, 1998. [BS10] Zhidong Bai and Jack W. Silverstein. Spectral analysis of large dimensional random matrices . Springer Series in Statistics. Springer, New York, second edition, 2010. 32 [CL22] Romain Couillet and Zhenyu Liao. Random matrix methods for machine learning . Cambridge University Press, 2022. [DM24] Issa D... | https://arxiv.org/abs/2504.03035v1 |
, 25(44):1–49, 2024. 33 [P´ ec19] S. P´ ech´ e. A note on the Pennington-Worah distribution. Electronic Communications in Proba- bility, 24(none):1 – 7, 2019. [PW17] Jeffrey Pennington and Pratik Worah. Nonlinear random matrix theory for deep learning. In I. Guyon, U. V. Luxburg, S. Bengio, H. Wallach, R. Fergus, S. Vi... | https://arxiv.org/abs/2504.03035v1 |
A LANCZOS-BASED ALGORITHMIC APPROACH FOR SPIKE DETECTION IN LARGE SAMPLE COVARIANCE MATRICES CHARBEL ABI YOUNES, XIUCAI DING, AND THOMAS TROGDON Abstract. We introduce a new approach for estimating the number of spikes in a general class of spiked covariance models without directly computing the eigenvalues of the samp... | https://arxiv.org/abs/2504.03066v1 |
ASD. Rather than 1Modern eigensolvers are very accurate for symmetric matrices. The errors here stem from using individual eigevalues that may fluctuate wildly, while averages of eigenvalues have much smaller variance. 2Due to instabilities in the iteration we use reorthogonalization [39]. 1arXiv:2504.03066v1 [math.ST]... | https://arxiv.org/abs/2504.03066v1 |
Im xijare independent centered random variables with variance p2Mq´1. We also assume that the random variables xijpossess arbitrarily high moments, meaning that for any fixed kPN, there exists a constant Cksuch that ´ E|xk ij|¯1{k ďCkM´1{2. (1.4) The assumption that (1.4) holds for all kPNmay be easily relaxed. For ins... | https://arxiv.org/abs/2504.03066v1 |
or the spiked ASD, using the eigenvalues of W, our approach employs Lanczos iterations for pW,bq with multiple vectors bsampled independently and uniformly from the hypersphere. Technically, a central component of our algorithmic approach and its analysis is the eigenvector empirical spectral distribution (VESD) and it... | https://arxiv.org/abs/2504.03066v1 |
applied until a steady state is reached, at which point the matrix Jnis extended by constants to be semi- infinite (we refer the reader to J0“L0L˚ 0in the proof of Theorem 5.5). From this semi-infinite Jacobi matrix J0, we define the spectral measure pµ0as [10], pm0pzq“e˚ 1pJ0´zq´1e1,e1“r1,0, . . .s˚, zPCzR, (1.12) whe... | https://arxiv.org/abs/2504.03066v1 |
the following two categories. ‚Detection based on the first few outlier eigenvalues. It is well known that if spikes exist, the corresponding outlier eigenvalues may undergo the so-called BBP transition [4,13]. In particular, if the spikes (i.e., rσiin (1.5)) exceed a certain threshold, the corresponding eigenvalues λi... | https://arxiv.org/abs/2504.03066v1 |
threshold in Monte Carlo simula- tions. In contrast, methods in the second category, which use all or nearly all eigenvalues, require at least a cost of order O pN3qto compute the spectrum of W. Our proposed method not only guarantees statistical efficiency and robustness under weaker separation assumptions but is also... | https://arxiv.org/abs/2504.03066v1 |
iteration at step nďNproduces a Jacobi matrix Jnand orthogonal vectors q1, . . . ,qn`1such that WQ n“QnJn`bn´1qn`1e˚ n, 7 where Qn“rq1,q2, . . . ,qns,q1“b{}b}and Jn“JnpW,bq“» ————–a0b0 b0a1... ......bn´2 bn´2an´1fi ffiffiffiffifl, a jPR, b ją0. (2.1) The columns of Qnform an orthonormal basis for the Krylov subspace sp... | https://arxiv.org/abs/2504.03066v1 |
A positive definite matrix W, a vector b, and a convergence criteria. Output: A lower-bidiagonal semi-infinite matrix pL. 1:Run the Lanczos iteration (Algorithm B.1) using the pair pW,bqwith 1!n!N1{6until the convergence criteria is satisfied. 2:Compute the Cholesky factors tpαiun´1 i“0,tpβiun´2 i“0using Algorithm B.2.... | https://arxiv.org/abs/2504.03066v1 |
6:forj“1 tok 7: Run Algorithm SR.2 on pLpbjqto compute1pγ˘andpmj 0pzq. 8:end for 9:Compute pm0pzq“1 kkÿ j“1pmj 0pzq, 10:returnpγ˘,pm0pzqandppmj 0pzqqk j“1. 10 Using Algorithm SR.3, one can draw sequences of bi,1ďiďkand obtain a robust estimator for the ASD of Win Algorithm P.2. The consistency of this estimator put for... | https://arxiv.org/abs/2504.03066v1 |
allow µto depend implicitly on a parameter Nbut require that pbe non-negative and constant (for sufficiently large N). Additionally, we assume that mini‰j|ci´cj|ěCγe´γN for all γą0, and that mint|a´b|,|a´cj|,|b´cj|uěτfor all j“1,2, . . . , p. (3) We associate a bounded open set Ω(independent of N) containingra, bsfor a... | https://arxiv.org/abs/2504.03066v1 |
resulting in the following continued fraction representation for thep1,1q-entry of the resolvent mµpzq“1 α2 0´z´α2 0β2 0´ m1pzq 1`β2 0m1pzq¯, m ipzq“1 α2 i´z´α2 iβ2 i´ mi`1pzq 1`β2 imi`1pzq¯,fori“1,2, . . . .(3.9) At first glance, this expansion may not appear to provide any new insights. But in the special case where ... | https://arxiv.org/abs/2504.03066v1 |
SR.2 to find the estimator pm0pzq“1 pα2 0´z´pα2 0pβ2 0ˆ pm1pzq 1`pβ2 0pm1pzq˙ wherepm1pzqsatisfies (3.13). Thus, the limit of pm0pzqexists and can be explicitly determined, yielding after simplification pm0pzq“´2z`ℓp1´c`z`?z´c`?z´c´q 2zppℓ`1qz`ℓpℓ´1`cqq. This expression shows that pm0pzqpotentially has poles at x0“ℓ`ℓc... | https://arxiv.org/abs/2504.03066v1 |
small constant 0ăτ1ă1, τ1ďσNďσN´1﨨¨ď σ1ďτ´1 1. (4.6) We further assume Σ0is such that ϱdMPis supported on a single bulk component supp µdMP“ rγ´, γ`sand that there exists δą0such that wpxq:“ϱdMPpxqpx´γ`q´1{2px´γ´q´1{2and 1{wpxqhave analytic extensions to tzPC: min xPrγ´,γ`s|x´z|ăδuthat are bounded above by a constant... | https://arxiv.org/abs/2504.03066v1 |
set of spectral parameters D“Dpτ, Mq:“tzPC`:|z|ěτ,|λ|ďτ´1, M´1`τďηďτ´1u. (4.13) Moreover, define the set D0as D0“D0pτ, Mq:“tzPC`:τďλďτ´1,0ăηďτ´1,distpλ,supp ϱdMPqěM´2{3`τu, (4.14) and the control parameter Ψpzq:“d ImmdMPpzq Mη`1 Mη1pzRD0q. (4.15) Importantly, we have for all zPDpτ, Mq, Ψpzq“OpM´τ{2q.We also note that I... | https://arxiv.org/abs/2504.03066v1 |
[14]. □ Denote the spectral parameter set rD“rDpτ, Nq“pDpτ, NqYD0pτ, NqqX" min 1ďiďr|z´fp´˜σ´1 iq|ěτ* , (4.24) where τis some small fixed constant. The following lemma presents a generalized version of the local laws applicable to the spiked model. 20 Lemma 4.11. Consider the eigenvectors tviuofΣand any unit determinis... | https://arxiv.org/abs/2504.03066v1 |
possibly random) version of µ. The following result from [14] establishes the relation between perturbed and unperturbed Jacobi matrices, as well as their Cholesky factors, asymptotically in terms of the difference of the Stieltjes transform mpz, µ´νq“ż Rpµ´νqpdxq x´z. Theorem 5.1. LetNbe a positive integer and suppose... | https://arxiv.org/abs/2504.03066v1 |
4.11) and (4.16), we have }mpz,pµb´µW,bq}L8pΓqă N´1{2η´1{2,where Γ“Γpηq“pr γ´´η, , γ``ηs`iηqYpr γ´´η, γ``ηs´iηq Ypγ``η`ir´η, ηsqYp γ´´η`ir´η, ηsq. The statement of the corollary follows from Theorem 5.1 by choosing η"N´1{3and noting qµb also satisfies Assumption 1 with overwhelming probability. □ Remark 5.4. It is impo... | https://arxiv.org/abs/2504.03066v1 |
now demonstrate that pm0pzqis approximated well by mpzq with overwhelming probability. Again using the second resolvent identity and the resolvent bounds, we have |mpzq´pm0pzq|ď1 Im2z}qLqL˚´L0L˚ 0}2. The right-hand side can be further bounded using }qLqL˚´L0L˚ 0}2ď}a0}8`2}b0}8, 25 where a0and b0are the vectors of diago... | https://arxiv.org/abs/2504.03066v1 |
0}}pJpqµbq´zq´1}ăN´1{2`δ, which implies that }pL0L˚ 0´zq´1}ăNδ, and, in particular, with overwhelming probability L0L˚ 0 has no elements of its spectrum within Ω. We have the second resolvent identity pL0L˚ 0´zq´1´pJpqµbq´zq´1“pL0L˚ 0´zq´1pJpqµbq´L0L˚ 0qpJpqµbq´zq´1, which implies }pL0L˚ 0´zq´1´pJpqµbq´zq´1}ăN´1{2`2δ, ... | https://arxiv.org/abs/2504.03066v1 |
simulations. Simulation 1. This example examines the performance of Algorithms P.2 and P.3 in the context of Johnstone’s spiked covariance model. We consider the sample covariance matrix W“1 MΣ1{2XX˚Σ1{2, (7.1) where Σ“diagp5,5,4.5, σ2, . . . , σ2q, σ2“1.5, (7.2) andXPRNˆMconsists of iid normal entries. We note that th... | https://arxiv.org/abs/2504.03066v1 |
vector ( k“1) in the detection algorithm. Each table entry represents the average over 200 samples, with the value in brackets denoting the probability of correctly detecting the true number of spikes. being relatively small near the right endpoint. Moreover, the convergence of the density follows the optimal rate of N... | https://arxiv.org/abs/2504.03066v1 |
in estimating the number of spikes across 100 realizations of the sample covariance matrix. In addition, we examine the average number of detected spikes and compare the computational efficiency of each method based on their average runtime. The results are summarized in Figures 6, 7 and Table 2. Figure 6 and Table 2 s... | https://arxiv.org/abs/2504.03066v1 |
the spikes of Σ are located at 7 , δ. See Figure 8 for a visualization of the ESD of Σ and the ASD of Wwhen δ“5. We approximate the ASD and detect the outliers of Wusing Algorithms P.2 and P.3 with k“100 vectors. Our ASD estimate closely matches the empirical distribution, accurately identifying both the location and n... | https://arxiv.org/abs/2504.03066v1 |
algorithm to compute the Cholesky decompo- sition of a Jacobi matrix (Algorithm B.2). A.2.Matrix Identities. We emphasize two well-known identities that are used in a key way. Lemma A.1 (Schur Complement) .Suppose p, qare nonnegative integers such that p`qą0, and suppose A, B, C, D are respectively pˆp,pˆq,qˆp,qˆqmatri... | https://arxiv.org/abs/2504.03066v1 |
computing the poles of the Stieltjes transform associated with per- turbations of Toeplitz Jacobi operators through connection coefficients, as described in [50]. An alternative method to approximate these poles, by truncating the semi-infinite matrix, is also pre- sented here. The first method, that of Olver & Webb, g... | https://arxiv.org/abs/2504.03066v1 |
rµpdλq“1 pCpλqµpdλq`rÿ i“1wiδλipλq. Although there is a clear connection between the poles of spzqand the roots of pCpλq, the condition wi‰0 introduces challenges in both computation and analysis. However, the Joukowski map Jpzq“1 2pz`z´1q, (C.9) can be used to overcome these issues. The Joukowski map is a conformal ma... | https://arxiv.org/abs/2504.03066v1 |
λ1, . . . , λ rof the Stieltjes transform associated with J. 1:SetJℓ“J1:ℓ,1:ℓwhere ℓis sufficiently large. 2:return Return the top eigenvalues tλiuˆr i“1ofJℓthat are larger than γ. assumptions, it can be shown that the finite-section method does not suffer from this issue and accurately captures the correct number of d... | https://arxiv.org/abs/2504.03066v1 |
[17] X. Ding and F. Yang. Spiked separable covariance matrices and principal components. Annals of Statistics , 49(2):1113–1138, may 2021. [18] X. Ding and F. Yang. Tracy-Widom distribution for heterogeneous gram matrices with applications in signal detection. IEEE Transactions on Information Theory , 68(10):6682–6715,... | https://arxiv.org/abs/2504.03066v1 |
Mathematics of the USSR-Sbornik , 1(4):457–483, 1967. [36] R. R. Nadakuditi and A. Edelman. Sample eigenvalue based detection of high-dimensional signals in white noise using relatively few samples. IEEE Transactions on Signal Processing , 56(7):2625–2638, 2008. [37] A. Onatski. Testing hypotheses about the number of f... | https://arxiv.org/abs/2504.03066v1 |
arXiv:2504.03097v1 [stat.ML] 4 Apr 2025A computational transition for detecting multivariate shu ffled linear regression by low-degree polynomials Zhangsong Li Peking University April 7, 2025 Abstract In this paper, we study the problem of multivariate shuffled linear reg ression, where the correspondence between predictor... | https://arxiv.org/abs/2504.03097v1 |
over the d∗mGrassmanian manifold {Q∈Rd∗m:Q⊤Q=Im}. Then the following results hold: (1) when m=o(d), there is evidence suggesting that all algorithms based on d egree-Dpolynomials fails to distinguish (X,Y)with two independent Gaussian matrices even with σ= 0, provided thatD4=o/parenleftbigd m/parenrightbig ; (2) when m... | https://arxiv.org/abs/2504.03097v1 |
computationall y efficient algorithms, in the sense that the best-known polynomial-time algorithms for a wide varie ty of high-dimensional inference prob- lems are captured by the low-degree class such as spectral me thods, approximate message passing and small subgraph counts; see e.g., [ Hop18,SW22,KWB22]. Furthermore,... | https://arxiv.org/abs/2504.03097v1 |
for th is detection problem under the low- degree polynomial framework, suggesting that (roughly spe aking)σ=O(1) is the separation of the computational “easy” and “hard” regime. Open problems. While we have characterized the computation transition for low-degree poly- nomial algorithms in the detection problem, the in... | https://arxiv.org/abs/2504.03097v1 |
|A| and #Ato denote its cardinality. For any two positive sequences {an}and{bn}, we write equivalently an=O(bn),bn= Ω(an), an/lessorsimilarbnandbn/greaterorsimilaranif there exists a positive absolute constant csuch that an/bn≤cholds for all n. We write an=o(bn),bn=ω(an),an≪bn, andbn≫anifan/bn→0 asn→ ∞. We write an= Θ(... | https://arxiv.org/abs/2504.03097v1 |
degree- O(1)polynomial that strongly separates PandQ. 3 Proof of Theorem 2.2 In this section we prove Theorem 2.2formally. In Subsection 3.1we provide a universal bound for the low-degree advantage between PandQfor alld,m∈N. In Subsections 3.2we prove Item (1) in the single-variate as preparations. In Subsections 3.3,3... | https://arxiv.org/abs/2504.03097v1 |
=/summationdisplay 0≤k≤D/summationdisplay α1,...,αk∈Nd β1,...,βk∈Nm 0<|αi|,|βj|≤DEQ∼ν/bracketleftBig/productdisplay 1≤i≤kΛαi,βi(Q)/bracketrightBig2 , where in equality we use that fix any σ∈Skthere is a one-to-one correspondence between (β1,...,β k) and (βσ(1),...,β σ(k)). This completes our proof. 3.2 Proof of Item (1)... | https://arxiv.org/abs/2504.03097v1 |
from ( 3.5). This yields the desired result since κ(Od,Om) = 1. We can now prove ( 3.15), thus finishing the proof of Theorem 2.2, Item (2). Proof of (3.15).We will calculate L(X,Y) directly. We divide our proof into two cases. Case 1:k≤mandk4=o(d/m). Note that given Y=BandXX⊤=A, the conditional distribution of Xunder b... | https://arxiv.org/abs/2504.03097v1 |
m=d,σ=ω(1) andD=o(σ). Still, based on ( 3.8), it suffices to show that for 1 ≤k≤Dwe have /summationdisplay (α1,...,αk):αi∈Nd (β1,...,βk):βi∈Nm 0<|αi|,|βj|≤DEQ∼ν/bracketleftBig/productdisplay 1≤i≤kΛαi,βi(Q)/bracketrightBig2 =o(D−1). (3.29) We will adopt the similar approach as in Subsection 3.3by relating the left hand si... | https://arxiv.org/abs/2504.03097v1 |
for helpful discussions. The aut hor thanks Shuyang Gong for helpful discussions on the forthcoming man uscript [ GWX25+ ]. The author thanks Hang Du for pointing him to the reference [ MM13]. This work is partially supported by National Key R&D program of China (No. 2023YFA1010103) and N SFC Key Program (Project No. 1... | https://arxiv.org/abs/2504.03097v1 |
have E/bracketleftBig e−tr(Z⊤AZ)/bracketrightBig =E/bracketleftBig e−tr(Z⊤U⊤AUZ)/bracketrightBig . Thus, without losing of generality we may assume that A= Diag(a1,...,ad) is a diagonal matrix. In this case, we get from direct calculation that E/bracketleftBig e−tr(Z⊤AZ)/bracketrightBig =/productdisplay 1≤i≤dE/bracketl... | https://arxiv.org/abs/2504.03097v1 |
high-dimensional geometry in random graphs. In Random Structures and Algorithms , 49(3):503–532, 2016. [CDGL24+] Guanyi Chen, Jian Ding, Shuyang Gong, and Zhangsong L i. A computational transition for de- tecting correlatedstochastic block models by low-degreepolynomia ls. arXiv preprint, arXiv:2409.00966 . [DCK20] Osm... | https://arxiv.org/abs/2504.03097v1 |
Science (FOCS) , pages 720–731. IEEE, 2017. 21 [HS17] Samuel B. Hopkins and David Steurer. Efficient Bayesian estim ation from few samples: community detection and related problems. In IEEE 58th Annual Symposium on Foundations of Computer Scien ce (FOCS), pages 379–390. IEEE, 2017. [HSS17] Daniel J. Hsu, Kevin Shi, and X... | https://arxiv.org/abs/2504.03097v1 |
L. Bookstein. The s oftassign Procrustes matching al- gorithm. In Biennial International Conference on Information Process ing in Medical Imaging , pages 29–42. Springer, 1997. 22 [SW22] Tselil Schramm and Alexander S. Wein. Computational barrier s to estimation from low-degree polynomials. In Annals of Statistics , 50... | https://arxiv.org/abs/2504.03097v1 |
Adaptive sparse variational approximations for Gaussian process regression Dennis Nieman dennis.nieman@hu-berlin.de Institut f¨ ur Mathematik Humboldt-Universit¨ at zu Berlin Botond Szab´ o botond.szabo@unibocconi.it Department of Decision Sciences, Bocconi Institute for Data Science and Analytics, Bocconi University, ... | https://arxiv.org/abs/2504.03321v1 |
by theory and empirical evidence. Although our approach can be more generally applied, we mainly focus on sparse vari- ational approximations for Gaussian process (GP) posteriors in the nonparametric regres- sion model. We consider as examples both the inducing variable approach proposed by Titsias (2009b) and a standa... | https://arxiv.org/abs/2504.03321v1 |
the nonparametric regression model are given. The theory is ap- plied in concrete examples in Section 4, including different variational methods and priors. Beside theoretical guarantees also numerical results are provided. All proofs and supporting lemmas are given in the subsequent appendix. 2 Model selection with va... | https://arxiv.org/abs/2504.03321v1 |
posterior Π( ·|x,y). We choose among these approximations the one which minimizes the KL-divergence or equivalently maximizes the ELBO, i.e., we estimate λas maximiser of λ7→ELBO( ˜Πλ(·|x,y)) = max Q∈QλELBO( Q). (4) For computational reasons and to keep the presentation of the results clean, we optimize the hyperparame... | https://arxiv.org/abs/2504.03321v1 |
this fully Bayesian approach. Since, to deal with the variational approach, we need tighter control on the tail behaviour of the hyper-posterior π(λ|x,y), we do not follow standard techniques (as in e.g. de Jonge and van Zanten (2010); Arbel et al. (2013)), but use an argument via empirical Bayes, as developed in Szab´... | https://arxiv.org/abs/2504.03321v1 |
in which we measure the contraction rate (recall the definition of ϵn(λ) in (9)). Loosely speaking, we compare the two norms for functions projected onto the basis φ1, . . . , φ Jγand handle the tail behaviour with the above upper bounds. We verify these assumptions in several examples of regularity classes and priors ... | https://arxiv.org/abs/2504.03321v1 |
the same rate of contraction as the hierarchical posterior. As seen in the examples below, Condition (19) is mild. It basically requires that there is at least one hyperparameter amongst the ”good ones” in Λn,0such that the variational class is sufficiently close in Kullback-Leibler divergence to the corresponding post... | https://arxiv.org/abs/2504.03321v1 |
However, this ap- proach requires the explicit knowledge of the eigenfunctions of the prior GPs, which are often not available analytically. The empirical version of this method is called the sample spectral features variational method, where the inducing variables are defined as uj=vT jf(x), where vjis the j-th princi... | https://arxiv.org/abs/2504.03321v1 |
deferred to Section B.1 in the Appendix. As an example of these theoretical findings we consider a simulated data set of n= 10,000 observations ( xi, yi) generated as xi∼i.i.d.unif(0 ,2π) and yi|xi∼ind.N(f0(xi),0.01), (22) 10 where f0(x) =∞X j=0(3j+ 1)−1.1φ3j+1(x−π) (23) and{φj}j∈Nform the real Fourier basis on L2(0,2π... | https://arxiv.org/abs/2504.03321v1 |
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