text string | source string |
|---|---|
theory of eigenvectors for random matrices with diverging spikes. Journal of the American Statistical Association 117 , 996– 1009. Fan, J., Y. Fan, X. Han, and J. Lv (2022b). SIMPLE: statistical inference on membership profiles in large networks. Journal of the Royal Statistical Society Series B 84 , 630–653. Fan, J., ... | https://arxiv.org/abs/2503.00640v1 |
Lei, J. and A. Rinaldo (2015). Consistency of spectral clustering in stochastic block models. The Annals of Statistics 43 (1), 215–237. Lubotzky, A., R. Phillips, and P. Sarnak (1988). Ramanujan graphs. Combinatorica 8 (3), 261–277. Merris, R. (1994). Laplacian matrices of graphs: a survey. Linear Algebra and Its Appli... | https://arxiv.org/abs/2503.00640v1 |
Latent Embeddings with Generalized Laplacian Matrices” Jianqing Fan, Yingying Fan, Jinchi Lv, Fan Yang, and Diwen Yu This Supplementary Material contains the proofs of Theorems 1–6 and Corollaries 1–3, as well as some propositions, key lemmas, additional technical details including some refined results under the networ... | https://arxiv.org/abs/2503.00640v1 |
the relationships between matrices X, W, eigenvalues bδk,δk, and their original values can be expressed as X→β2α nX q1−4α,W→β2α nW q1−4α,bδk→β2α nbδk q1−4α, δ k→β2α nδk q1−4α, (A.10) while the eigenvectors remain unchanged. Throughout the rest of this section, the notation X,W,bδk, and δkshould be understood as the res... | https://arxiv.org/abs/2503.00640v1 |
complex-valued deterministic matrix Υk(z) :=Υ(z)−Υ(z)V−k1 ∆−1 −k+VT −kΥ(z)V−kVT −kΥ(z) (A.18) with z∈C+and 1 ≤k≤K. For notational simplicity, we will drop the dependence on z whenever there is no confusion. By comparing (26) and (A.16), one can observe that the relationship between eΥandΥand the relationship between eΥ... | https://arxiv.org/abs/2503.00640v1 |
Theorem 9. Assume that Condition 1 and Assumption 2 are satisfied, and Kψn(δk)βn≲1,∥V∥max≪1 |δk|βn+ξ qβ2n(A.30) for1≤k≤K0. Then for each i∈[n], it holds w.h.p. that bvk(i) = (Λ i/Li)αvk(i) +1 tkLα iX j∈[n]WijΛ−α jvk(j) +O ∥V∥max √ K |δk|+Kξ q!1 |δk|βn+ξ qβ2n! +Oξ√n|δk|1 |δk|+ξ qβn ,(A.31) 6 where we choose the di... | https://arxiv.org/abs/2503.00640v1 |
define the diagonal random error matrix Eas E:= diag( E1,···,En) =L−Λ. (A.40) With the aid of Lemma 12, we can readily obtain the results in the two lemmas below. Lemma 3. Under Condition 1, there exist some constants C1, c1>0(depending on τ,λ, andC0) such that for all i∈[n], C−1 1≤Λi=β−1 n(θi+τi¯θ+λi/q2)≤C1β−1 n, (A.4... | https://arxiv.org/abs/2503.00640v1 |
events \ z∈S(C) |uT(R(z)−Υ(z))v| ≤C4logn q|z|2 , (A.52) \ z∈S(C) max i∈[n]|eT i(R(z)−Υ(z))v| ≤C4 |z|2ξ√n+ξ q∥v∥∞ , (A.53) \ z∈S(C) max i∈[n]|eT iΛ−αWΛ−α(R(z)−Υ(z))v| ≤C4 |z|2ξ√n+∥v∥∞ (A.54) hold with probability at least 1−n−D. Using the generalized QVE in (A.16) and the definition in (A.17), we can easily sh... | https://arxiv.org/abs/2503.00640v1 |
for G(i), i.e., uT G(i)(z)−G(i) [i](z) v ≲1 qβn|u| ∥G(i)v∥∞∧ ∥G(i) [i]v∥∞ . (A.63) By Schur’s complement formula, we have the resolvent identities collected in the lemma below. The reader can also refer to Lemma 3.4 in Erd˝ os et al. (2013) for proof. Lemma 8 (Resolvent identities) .The following resolvent identiti... | https://arxiv.org/abs/2503.00640v1 |
kΥ(z)V−k=O(|z|−3),(A.75) where in the second and third expressions above, O(|z|−3) denotes a matrix Eand a vector ε satisfying ∥E∥=O(|z|−3) and|ε|=O(|z|−3), respectively. Further, it follows from Theorems 14 and 15 that the estimates max i∈[n]|eT i(G(z)−Υ(z))vk|≲ξ |z|1√n|z|+∥vk∥∞ qβn , (A.76) max i∈[n]|eT i(G(z)−Υ(z)... | https://arxiv.org/abs/2503.00640v1 |
asymptotic bound above is understood implicitly for the absolute value of the quantity involved (for notational simplicity). For the deterministic term, with the aid of (A.55) and (A.75), we can 16 rewrite it as vT kΥ(bδk)V−k1 ∆−1 −k+VT −kΥ(bδk)V−kVT −kΥ(bδk)vk (A.90) =vT kΥ(tk)V−k1 ∆−1 −k+VT −kΥ(tk)V−kVT −kΥ(tk)vk+O|... | https://arxiv.org/abs/2503.00640v1 |
18 (A.78) and (A.89), it holds that w.h.p., vT k(Gk(z)−Υk(z))vk ≲ξ q|δk|ψn(δk) +√ Kξ qψn(δk) 1 |δk|3+√ K |δk|5+√ Kξ q|δk|ψn(δk)! ≲ξ q|δk|ψn(δk) 1 +K |δk|4(A.98) uniformly in z∈ Ck, where we have used (A.15) in the second step. With the aid of (A.55) and (A.83), we can deduce that uTΥ(z)vk=−uTvk/z+O(|z|−3),∥uTΥ(z)V−k∥... | https://arxiv.org/abs/2503.00640v1 |
3 and the Taylor expansion, it holds that w.h.p., (Li/Λi)−α= 1−α Λiβn1 qX jWij+τi nqX j,lWjl +Oξ2 q2β2n . From (A.213), it follows that w.h.p., X l∈[n]WilΛ−α lvk(l)≲ξ q∥vk∥∞+ξ√n. Then using Lemma 3 and the above two estimates, we can obtain that bvk(i) =vk(i)−α Λiβn1 qX j∈[n]Wij+τi nqX j,l∈[n]Wjl vk(i) +1 tkX j∈[... | https://arxiv.org/abs/2503.00640v1 |
we have used Theorem 16 in the second step above. Finally, recalling that Υ(z) +z−1=E1=O(|z|−3) by (A.55) and using (A.213), we can 23 deduce that Λα ieT iW Υ(tk) +t−1 k vk=eT iWΛ−αE1(tk)vk =X j∈[n]WijΛ−α i(E1(tk))jjvk(j)≲ξ |δk|31√n+1 q∥vk∥∞(A.120) with high probability. Therefore, a combination of (A.43), (A.119),... | https://arxiv.org/abs/2503.00640v1 |
these two scenarios separately in this proof. We first aim to prove part 2) of Theorem 11. Taking u=vkin (A.97), it follows from (A.42), (A.43), (A.46), (A.48), (A.55), and (A.78)–(A.82) that vT k(L/Λ)−αbvkbvT k(L/Λ)−αvk =δ−2 k(vT kΥ′ k(tk)vk)−1+1 2πiI CkvT k(Υ(z) +z−1A(z))vk (1 +δkvT kΥk(z)vk)2dz +OK |δk|4+Kξ2ψn(δk)2... | https://arxiv.org/abs/2503.00640v1 |
+−t−2 k δkvT kΥ′ k(tk)vkuTV−k1 ∆−1 −k−t−1 k ×VT −k −2αL−Λ Λ+α(2α+ 1)L−Λ Λ2 +t−1 k(L/Λ)−2αW(L/Λ)−2α+t−2 kW vk +−t−2 k δkvT kΥ′ k(tk)vkuT −2αL−Λ Λ+t−1 kW V−k1 ∆−1 −k−t−1 kVT −k × −2αL−Λ Λ+t−1 kW vk +O K |δk|3+Kξ3 q3β3n+K3/2ξψn(δk) qξψn(δk) q+1 |δk|2! .(A.138) Moreover, with the aid of (A.213), we have that w.h.... | https://arxiv.org/abs/2503.00640v1 |
2. C.9 Proof of Corollary 3 The proof for part 1) of Corollary 3 is still a simple application of the classical Lindeberg– Feller CLT, and thus, we omit the details there. It remains to prove part 2) of Corollary 3. Clearly, we need only to establish the CLT for α2 2vT kL−Λ Λ2 vk−1 2t2 kvT kW2vk. Such a term can be w... | https://arxiv.org/abs/2503.00640v1 |
max i,j,l∈[n]|fk(i, j, l)|≲1 q2β2n+1 |δk|2 ∥vk∥2 ∞,max i,j,l1,l2∈[n]|gk(i, j, l 1, l2)|≲∥vk∥2 ∞ nq2β2n(A.156) max i,j∈[n]|cij|≲1 q2β2n+1 |δk|2 ∥vk∥2 ∞, (A.157) Eb2 ij≲1 q2β2n+1 |δk|22 ∥vk∥4 ∞i+j n, (A.158) Eb3 ij≲1 q2β2n+1 |δk|23 ∥vk∥6 ∞i+j nq, (A.159) Eb4 ij≲1 q2β2n+1 |δk|24 ∥vk∥8 ∞i+j nq2+i2+j2 n2 . (A.16... | https://arxiv.org/abs/2503.00640v1 |
concludes the proof of Theorem 14. C.12 Proof of Theorem 15 Denote by E′:= (L/Λ)2α−I.By Lemma 4, we have that ∥E′∥≲β−1 nξ/q with high probability. Then it follows from Theorem 13 and Proposition 2 that w.h.p., uT(G(z)−Υ(z))v=uT(G−R)v+Ologn q|z|2 =zuTGE′Rv+Ologn q|z|2 =zuTΥE′Υv+Oξ q|z|2βn+ξ2 q2|z|β2n . (A.167) Let... | https://arxiv.org/abs/2503.00640v1 |
Proof of Proposition 1 The estimate (A.46) can be shown using the same arguments as in the proof of Lemma 4.3 in Erd˝ os et al. (2013). Note that (A.47) is a simple consequence of (A.46) by definition. Then in light of (A.43) and (A.44), we have that for T=∅,{i}, or{i, j}, ∥(L(T)/Λ(T))2α∥ ∼1 with ( c, ξ)-high probabili... | https://arxiv.org/abs/2503.00640v1 |
(A.59). Thus, we obtain (A.62). The estimate in (A.63) can be proved in a similar fashion, which concludes the proof of Lemma 7. D.8 Proof of Lemma 8 We will focus on proving the conclusion for G, since the proof for G[i]follows a similar approach. Additionally, the proof for Rcan be derived directly from Lemma 3.4 in ... | https://arxiv.org/abs/2503.00640v1 |
q|z|+ξ q2|z|βn max i∈[n]|Giv|+ξ√n|z|2+∥v∥∞ |z|, which together with the assumption of ξ≪q2|z|βnyields (A.67). Thus, an application of Lemma 7 and (A.67) leads to (A.68). Finally, applying (A.187) and (A.68) to (A.185), we can derive (A.69), which concludes the proof of Lemma 9. 44 E Additional technical details and ad... | https://arxiv.org/abs/2503.00640v1 |
i(G[j](z)−Υ(z))v| ≤C7(logn)1/2 |z|1√n|z|+1 qβn∥v∥∞ (A.198) hold with probability at least 1−n−D. Proposition 5 (Corresponding to Theorem 15) .Under the conditions of Theorem 13 and (A.189) , for each constant D > 0, there exists some constant C8>0such that for any 46 deterministic unit vectors uandv, the event \ z∈S... | https://arxiv.org/abs/2503.00640v1 |
any deterministic unit vector usuch that uTvk= 0, it holds w.h.p. that uT(L/Λ)−αbvk− A k=tkuTV−k1 tk−∆−kVT −k −2αL−Λ Λ+t−1 kW vk +wT −2αL−Λ Λ+t−1 kW vk+X l∈[K]\{k}tkuTvl tk−δlBk,l+Bw k +O K 1 |δk|2+(logn)1/2 qωn(δk)! 1 |δk|+(logn)1/2 qβn!! +O K3/2(logn)1/2 qωn(δk) (logn)1/2 qωn(δk) +1 |δk|2!! ,(A.210) 49 where we c... | https://arxiv.org/abs/2503.00640v1 |
di+τ¯d+λ:i∈[n] without the rescaling population parameters qand βn. The blue curves represent the KDEs for the empirical spiked eigenvalue corrected by estimate bAk, whereas the red curves stand for the target normal density. Both curves are centered with the asymptotic limit tk. The top right plot is due to relativel... | https://arxiv.org/abs/2503.00640v1 |
the fact that the normalized Laplacian matrix has a trivial largest eigenvalue at 1. 55 −0.05 0.00 0.050510152025alpha = 0.25, k = 2Density −0.003 −0.001 0.001 0.0030200400600alpha = 0.5, k = 2Density −5e−06 0e+00 5e−060100000 200000 300000alpha = 1, k = 2Density −2e−11 0e+00 2e−110e+00 4e+10 8e+10alpha = 2, k = 2Densi... | https://arxiv.org/abs/2503.00640v1 |
Powerful rank verification for multivariate Gaussian data with any covariance structure Anav Sood Stanford University March 1st, 2025 Abstract Upon observing n-dimensional multivariate Gaussian data, when can we infer that the largest K observations came from the largest Kmeans? When K= 1 and the covariance is isotropi... | https://arxiv.org/abs/2503.01065v1 |
also encompasses the problem of verifying that the machine learning model with the best performance on a challenge dataset is actually the best model. In this case, thenmodels’ observed average performances X∈Rnon the dataset obey a central limit theorem (provided that the challenge dataset is moderately large), i.e., ... | https://arxiv.org/abs/2503.01065v1 |
k∈S,ℓ̸∈S: ρij,kℓ<0Dij−1 ρij,kℓDkℓ ≤α, (1) then, conditional on S, the probability of making a false rejection is at most α. 2 Our next result, Theorem 2, helps us make sense of Theorem 1’s method. We prove an analog of Theorem 2 that applies whenever δ≥0 in Section 2.2. Theorem 2 (Understanding Gaussian rank verif... | https://arxiv.org/abs/2503.01065v1 |
rejects ∪i∈S,j̸∈SH0 ijif and only if the condition (2) from Theorem 2 is satisfied. There are a couple other notable situations where the conclusion of Corollary 1 applies. When K= 1 orK=n−1 and the Xihave a small amount of autocorrelation (i.e., Σ ij=σ2ρ|i−j|with|ρ| ≤1/2), the result of Corollary 1 still holds. It als... | https://arxiv.org/abs/2503.01065v1 |
Related work Gutmann and Maymin [1987] study our problem in the case that K= 1, δ= 0, and the data Xi∼ N(µi, σ2) are independent Gaussian samples with common variance. They show that drawing the inference mini∈Sµi−max j̸∈Sµj> δwhenever the two-sided difference-of-means test comparing the largest and second largest obse... | https://arxiv.org/abs/2503.01065v1 |
Proofs In this section, we prove more general versions of the results stated in Section 1. The more general result we are aiming to prove is stated clearly at the start of each proof. We will use Dδ ij=(Xi−Xj)−δ vij to denote the standardized distance between Xi−Xjandδ. Note that D0 ij=Dij, where Dijis the standardized... | https://arxiv.org/abs/2503.01065v1 |
X ℓ⇐⇒ ϵδ ij,kℓ>0, 3. If ρij,kℓ<0 then Xk> X ℓ⇐⇒ Dδ ij<−1 ρij,kℓϵδ ij,kℓ Ultimately, we see that S={1, . . . , K } ⇐⇒ Xk> X ℓfor all k≤Kandℓ > K ⇐⇒ Dδ ij∈ max k≤K, ℓ>K : ρij,kℓ>0−1 ρij,kℓϵδ ij,jk,min k≤K, ℓ>K : ρij,kℓ<0−1 ρij,kℓϵδ ij,jk and min k≤K, ℓ>K : ρij,kℓ=0ϵδ ij,kℓ>0. Essentially, the selection event S={1, . ... | https://arxiv.org/abs/2503.01065v1 |
have x > x −1 ρij,kℓD0 kℓ because ρij,kℓ≥0 and D0 kℓ≥0. The non-positiveness of the derivative and the fact that Dδ ij> D0 kℓimplies that 1−Φ(Dδ ij) 1−Φ(Dδ ij−1 ρij,kℓD0 kℓ)≤1−Φ(D0 kℓ) 1−Φ(D0 kℓ−1 ρij,kℓD0 kℓ) ≤1−Φ(D0 kℓ) 1−Φ(0) ≤2(1−Φ(D0 I0J0)). where we have that D0 kℓ−1 ρij,kℓD0 kℓ≤0 because1 ρij,kℓ≥1 and D0 kℓ≥0. 7... | https://arxiv.org/abs/2503.01065v1 |
the method of paired comparisons. Biometrika , 39(3/4):324–345, 1952. Shuenn-Ren Cheng and S Panchapakesan. Is the selected population the best?—location and scale parameter cases. Communications in Statistics—Theory and Methods , 38(10):1553–1560, 2009. Wei-Lin Chiang, Lianmin Zheng, Ying Sheng, Anastasios Nikolas Ang... | https://arxiv.org/abs/2503.01065v1 |
σ2> σ1andσ2> σ3, we can ensure that ρ12,1jis very negative for j >2. This will mean there is a lot of benefit to running Theorem 1’s full test in place of Theorem 2’s simpler approach. We instantiate this problem by setting n= 5,σ2 1= 1,σ2 2= 5,σ2 3= 0.1,µ1= 5,µ2= 3,µ3=µ4=µ5= 0. Over B= 10000 simulated trials run at le... | https://arxiv.org/abs/2503.01065v1 |
the numerator of Dδ 1jand also maximize its denominator, which suffices to establish our claim. 11 C Simultaneous approach For the sake of comparison, we derive a simultaneous inference approach for the problem of drawing the inference min i∈Sµi>max j̸∈Sµj. We base our approach off of that in Bofinger [1983, 1985] and ... | https://arxiv.org/abs/2503.01065v1 |
2006] and the fact that n−K≥n/2 per our assumption. As a consequence, q1−αmust grow at least on the order of√lognas well. This implies that, in the independent case, the HSD quantile (8) grows at least on the order of√lognalso. D Getting a confidence lower bound By inverting the test (3) for different values of δ(i.e.,... | https://arxiv.org/abs/2503.01065v1 |
PSEUDO-MAXIMUM LIKELIHOOD THEORY FOR HIGH-DIMENSIONAL RANK ONE INFERENCE CURTIS GRANT, AUKOSH JAGANNATH, AND JUSTIN KO Abstract. We develop a pseudo-likelihood theory for rank one matrix estimation problems in the high dimensional limit. We prove a variational principle for the limiting pseudo-maximum likeli- hood whic... | https://arxiv.org/abs/2503.01708v1 |
We seek here to close this gap. To this end, observe that many popular optimization based esti- mators for such problems, such as those mentioned above, can be interpreted as pseudo-likelihood methods [37]. In this paper, we provide a unified analysis of the performance of pseudo-likelihood methods. We develop a pseudo... | https://arxiv.org/abs/2503.01708v1 |
in a special regime called the “Nishimori Line” as a consequence of Bayes theorem [46]. Withthisinmind, itisnaturalthatoptimization-basedprocedureshavebeenlessunderstood: on the“Nishimoriline”thecorrespondingspinglassmodelisintheso-called“replicasymmetricphase”. While deeply challenging, this regime is comparatively si... | https://arxiv.org/abs/2503.01708v1 |
the pseudo-likelihood by gand the parameter set by Ω. Throughout we shall denote our pseudo maximum likelihood estimator by ˆxPMLE, and it is given by: ˆxPMLE := arg max x∈ΩN/summationdisplay i≤jg/parenleftig Yij,xixj√ N/parenrightig , (2.2) 3 which again may not be uniquely defined. We measure the performance of the... | https://arxiv.org/abs/2503.01708v1 |
the case of well- scored models. Our first main result is a variational formula for the asymptotic pseudo-maximum likelihood and corresponding characterization of the asymptotic performance of pseudo-maximum likelihood estimators. To this end, we need to define a corresponding Parisi-type functional. Let M([0,S])denote... | https://arxiv.org/abs/2503.01708v1 |
cosine similarity converges almost surely to m//radicalig sEQx2 0 We note here the following remark regarding the centreing in (2.15). Remark 2.1. The term/summationtext i≤jg(Y,0)does not depend on x, so it will not affect the pseudo- maximum likelihood estimator. However, these normalization terms need to be subtract... | https://arxiv.org/abs/2503.01708v1 |
setof limitpoints of SN(ˆxPMLE )andMN(ˆxPMLE ) is unique and given by C¯β={(x2 +,x+EQ(x0))}. In particular, CS(ˆxPMLE,x0)→EQ(x0) (EQ(x0)2)1/2 1 N∥ˆxPMLE∥2→x2 +. Evidently if ¯x0= 0thenCS(ˆxPMLE,x0) = 0and the estimator is useless. The case when β4<0is more delicate since there is the large spurious information induced ... | https://arxiv.org/abs/2503.01708v1 |
when multiplied byN3/2, i.e., the appropriate power of Nto counteract the expected score. Thus EP0[∂wg(Yij,0)] unfortunately remains inaccessible. To account for this, let us introduce a hyper-parameter α∈Rand define the corresponding score-corrected pseudo-likelihood by Lg N,α(Y,x) =/summationdisplay i≤jg/parenleftig... | https://arxiv.org/abs/2503.01708v1 |
task (g0,g1)with information parameters given by ¯βis strongly equivalent to the inference task (g¯β U,0,g¯β U,1). Remark 3.1. Theorem 3.1 simplifies greatly in the well scored case with β3>0. In this case g¯β U,1, may instead be taken to be −1 2(y−√β3w)2, with an appropriate normalization in g¯β U,0. 10 An important c... | https://arxiv.org/abs/2503.01708v1 |
lying in Cβ. Ifλ=σ√βLSthen the information parameters satisfy the Rao relation. Note that by Cauchy- Schwarz,βLS≤√β0.If, furthermore, βLS=√β0then information parameters are equal to those of the log-likelihood. In practice, we do not necessarily know that the data distribution under the null model has zero mean, and, a... | https://arxiv.org/abs/2503.01708v1 |
appearing within and outside of each group. Notice that when xi,xjtake the same sign, the probability is higher, 13 Figure 1. The cosine similarity in the spiked matrix problem with Rademacher latent variable and noise with mean 1solved using corrected and uncorrected least squares. A data matrix of size 2500×2500and t... | https://arxiv.org/abs/2503.01708v1 |
By Proposition 4.1 the model is well scored, and furthermore, an explicit computation shows the Fisher parameters for the gaussian equivalent are given by: β1=λp,β 2= 0,β3=λ2, and consequently by Proposition 4.2 the least-square estimator is completely uninformative pro- vided that the limiting empirical measure of x0,... | https://arxiv.org/abs/2503.01708v1 |
N =√β1√ N/summationdisplay ijgijxixj+Nβ2 2MN(x)2−Nβ3 4SN(x)2+β4N3 2(¯x)2+O(1) (6.1) where we recall that MNandSNdenote the normalized inner product and norm defined in (2.14) and¯x=1 N/summationtextN i=1xiis the sample mean. WeproveinAppendixAthattheasymptoticmaxmiumpseudolikelihoodisequaltotheonegiven by the maximum o... | https://arxiv.org/abs/2503.01708v1 |
6.1. By the triangle inequality, we have 1 N|Lg,ε N(S,M,v )−L¯β,ε N(S,M,v )|≤|1 NLg,ε N(S,M,v )−1 LFN(Lg,ε ;S,M,v )| +/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 LFN(Lg,ε ;S,M,v )−1 LFN(L¯β,ε;S,M,v )/vextendsingle/vextendsingle/vextendsingle/vextendsingle +|1 LFN(L¯β,ε;S,M,v )−1 NL¯β,ε N(S,M,v )|. The fir... | https://arxiv.org/abs/2503.01708v1 |
Bound: We then prove the matching lower bound FN,α(¯β,α,ε ;S,M,v )≥φβ,α(S,M,v ). This proof uses the cavity method, in this case called the Aizenman-Sims-Star scheme [4], and a perturbation of the posterior that forces the limiting overlap to satisfy the Ghirlanda–Guerra identities [36] and ultrametricity [68]. The pro... | https://arxiv.org/abs/2503.01708v1 |
N(x) =ψ¯β(s,m) (6.15) which follows from Lemma 6.3. In the notation above, we have defined ψ¯β(S,M ) =/braceleftigg supvψ¯β,α(S,M,v )ifα̸= 0 ψ¯β,0(S,M ) ifα= 0. to handle the cases for well-scored and corrected models simultaneously. Next, by concentration [3, Section 2.1] for every ε>0, lim N→∞1 NEmax x∈ΩNH¯β N(x) = ... | https://arxiv.org/abs/2503.01708v1 |
under the null-model we have that Yis Gaussian with mean β4and variance β1. A direct computation implies that the information parameters of (g¯β U,0,g¯β U,1)areβ1,β2,β3,β4. □ Acknowledgements. C.G. acknowledges the partial support of the Natural Sciences and Engi- neering Research Council of Canada Post-Graduate Schola... | https://arxiv.org/abs/2503.01708v1 |
2688. [17] Francesco Camilli, Pierluigi Contucci, and Emanuele Mingione, An inference problem in a mismatched setting: a spin-glass model with Mattis interaction , SciPost Phys. 12(2022), no. 4, Paper No. 125, 27. MR 4409513 [18] Philippe Carmona and Yueyun Hu, Universality in Sherrington-Kirkpatrick’s spin glass model... | https://arxiv.org/abs/2503.01708v1 |
(1984), no. 3, 681–700. [38] Francesco Guerra, Broken replica symmetry bounds in the mean field spin glass model , Communications in mathematical physics 233(2003), no. 1, 1–12. [39] Alice Guionnet, Justin Ko, Florent Krzakala, Pierre Mergny, and Lenka Zdeborová, Spectral phase transitions in non-linear Wigner spiked m... | https://arxiv.org/abs/2503.01708v1 |
[58] Marc Lelarge and Léo Miolane, Fundamental limits of symmetric low-rank matrix estimation , Conference on Learning Theory, PMLR, 2017, pp. 1297–1301. [59] Thibault Lesieur, Florent Krzakala, and Lenka Zdeborová, Mmse of probabilistic low-rank matrix estimation: Universality with respect to the output channel , 2015... | https://arxiv.org/abs/2503.01708v1 |
2022, pp. 1288–1293. [77] Sundeep Rangan and Alyson K Fletcher, Iterative estimation of constrained rank-one matrices in noise , 2012 IEEE International Symposium on Information Theory Proceedings, IEEE, 2012, pp. 1246–1250. [78] Galen Reeves, Information-theoretic limits for the matrix tensor product , IEEE Journal on... | https://arxiv.org/abs/2503.01708v1 |
)−FN(¯β;S,M,v )/vextendsingle/vextendsingle≤K√ N. Proof.The proof is in Section 3 from [40]. We highlight the key steps. Throughout we let Kde- note a universal constant that only depends on the supports ΩandΩ0, but not on the dimension N. Step 1 - Approximation by Third Order Terms: We first show that to leading order... | https://arxiv.org/abs/2503.01708v1 |
can be made precise using a standard approximate Gaussian integration by parts argument—as was applied, e.g., to prove universality for the SK model in [18] to conclude that |F(2) N(g;S,M,v )−F(3) N(g;S,M,v )|≤K(g,g0)√ N(A.4) where the constant K(g,g0)only depends on the quantities appearing in F0, all of which are uni... | https://arxiv.org/abs/2503.01708v1 |
The following proofs are stated in terms of a quantity called the Ruelle probability cascades [69, Chapter 2]. A quick summary of the notation is provided for convenience in Appendix H. Proposition B.1 (Large Deviation Upper Bound of the Free Energy) .There exists a universal finite constant Lsuch that for every S,M,v∈... | https://arxiv.org/abs/2503.01708v1 |
/integraldisplay Ωε(S,M)e/summationtext i≤N(β1Zi(α)xi+λx2 i+µxix0 i+ρxi)dP⊗N X(x) ≤eN(λ−λ′)S+N(µ−µ′)M+N(ρ−ρ′)v+NO(ε)/integraldisplay e/summationtext i≤N(β1Zi(α)xi+λ′x2 i+µ′xix0 i+ρ′xi)dP⊗N X(x). 30 We pause to introduce the notion of exposed points: (S,M,v )is said to be exposedif there exists (λ,µ,ρ )such that for eve... | https://arxiv.org/abs/2503.01708v1 |
below. Proposition B.2 (Lower Bound of the Free Energy) .For any real numbers β1,β2,β3, for any (S,M,v )∈C, for anyε>0, we have lim N→∞FN(¯β,ε;S,M,v )≥φ¯β(S,M,v ) +O(ε) Proof.The key ideas of the proof is similar to the ones used to derive the lower bound of the Sherrington–Kirkpatrick model. The approximation techniqu... | https://arxiv.org/abs/2503.01708v1 |
sequences define the density function µ(Q) =ζkforQk≤Q<Qk+1. LetvαdenotetheweightsoftheRuelleprobabilitycascadescorrespondingtothesequence(B.17). If (αℓ)ℓ≥1aresamplesfromtheRuelleprobabilitycascades,then P(α1∧α2≤t) =µ(t)byconstruction. This gives us an explicit way to construct the off-diagonal entries of the overlap ar... | https://arxiv.org/abs/2503.01708v1 |
LF(Lβ), and we shall do so by means of Γconvergence. For fixed 0≤t≤S,h,y∈Rwe define functionals FL(ζ,λ,µ )by: FL,S(ζ,µ,λ ;t,y,h ) =/braceleftigg ΦL ζ,λ,µ(t,y)ifζ=Lρ(t)dt +∞ otherwise, where ΦL ζ,λ,µis the weak solution to the Parisi PDE: /braceleftigg ∂tΦ +β1 4(∆Φ +Lρ(s)(∂yΦ)2) = 0 Φ(S,y) =1 Llog/integraltexteL(yx+λx... | https://arxiv.org/abs/2503.01708v1 |
)⊂Ω2δ(S′,M′,v′) Ωε(S′,M′,v′)⊂Ω2δ(S,M,v ), and hence it will suffice to bound the quantity E/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglemax Ω2δ(S,M,v )1 NHN−max Ωε(S,M,v )1 NHN/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. Letπ: Ω2δ(S,M,v )→Ωε(S,M,v )be the map that tak... | https://arxiv.org/abs/2503.01708v1 |
N→∞Emax x∈ΩNH¯β N N= sup (s,m,v )∈Cψ¯β(s,m,v ) Proof.For a lower bound note for any ε>0and any (S,M,v )one has: Emax x∈ΩN1 NHN(x)≥E max x∈Ωε(S,M,v )1 NHN(x), and taking N→∞and thenε→0we obtain: lim N→∞Emax x∈ΩN1 NHN(x)≥ψ¯β(S,M,v ), for any (S,M,v )∈C. In the case Ωis an interval, to prove an upper bound, let us define ... | https://arxiv.org/abs/2503.01708v1 |
2MN(x)2−Nβ3 4SN(x)2+N3/2β4 2¯x2+oN(1).(E.5) Notice that the last termN3/2β4 2¯x2is the leading order term. This leading order term does not depend on the unknown variable, but dictates the performance of the MLE. If β4>0, then the estimator must maximize this term, which is the statement of Theorem 2.3. Proof of Theore... | https://arxiv.org/abs/2503.01708v1 |
Proof of Theorem 2.5. This follows immediately by combining Lemma E.2 and (E.1). □ E.3.Proof of the Variational Formula for the Zero Score Model. For completeness, we also provide the proof for zero score models, which follows from a simple modification of the previous arguments for score biased score models. Proof of ... | https://arxiv.org/abs/2503.01708v1 |
mator of the original inference problem. To this end, given a model gand the corresponding Fisher score parameters ¯β, we define ˆ xg PMLE= arg max x∈ΩN/summationdisplay i≤jg/parenleftig Yij,xixj√ N/parenrightig , as was defined in (2.2). The following Lemma is a universality statement for the overlaps of the ground ... | https://arxiv.org/abs/2503.01708v1 |
by the maximizers ofψ¯βon the sets where ψ¯βis differentiable. Consider the function f(β1,β2,β3) = sup s,mψ¯β(S,M ), (F.3) and define the sets Dβ2={(β1,β2,β3)|∂β2fexists}andDβ3={(β1,β2,β3)|∂β3fexists}. We show that the characterization of the overlap for the gaussian equivalent is valid at points where f(β1,β2,β3)is di... | https://arxiv.org/abs/2503.01708v1 |
maximizing pairs by Danskin’s envelope theorem as was shown in part 3 of the proof of Lemma F.5. □ We end this section by showing that although in some cases the maximizers of ψβmay not be unique, the performance of the MLE is still characterized by the maximizers of ψ, as stated in Lemma 6.5. Proof of Lemma 6.5. We pr... | https://arxiv.org/abs/2503.01708v1 |
contains points of the form (s,0). □ 48 Appendix G.Coarse Equivalence of Pseudo Estimators In this section, we prove Theorem 3.2, using results proved in Sections A and F. Proof of Theorem 3.2. We first consider the case of well-scored models. Given two well-scored loglikelihood functions g1andg2, we let ¯β(g1)and ¯β(g... | https://arxiv.org/abs/2503.01708v1 |
by Nr, α1∧α2= min/braceleftig 0≤j≤r|α1 |1=α2 |1,...,α1 |j=α2 |j,α1 |j+1̸=α2 |j+1/bracerightig These averages with respect to the Ruelle probability cascades variable αcan be computed using the following recursive formulation from, for example [69, Theorem 2.9]. Lemma H.1 (Averages with Respect to the Ruelle Probabili... | https://arxiv.org/abs/2503.01708v1 |
In particular, for any η>0there exists a finite integer number K(η)so that these functionals can be approximated within ηby a continuous function of the finite array (xℓ·xℓ′)1≤ℓ,ℓ′≤K(η)uniformly over all possible choices of Gibbs measures Gand all y0limiting empirical distribution Q. Proof.The proof of this argument is... | https://arxiv.org/abs/2503.01708v1 |
JOURNAL OF L ATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 1 A Near Complete Nonasymptotic Generalization Theory For Multilayer Neural Networks: Beyond the Bias-Variance Tradeoff Hao Yu, Xiangyang Ji Abstract We propose a first near complete (that will make explicit sense in the main text) nonasymptotic generalization ... | https://arxiv.org/abs/2503.02129v1 |
and study the training dynamics of SGD in such limitation. The efficiency and power of this method is that it can be often casted as a stochastic ordinary differential equation or partial differential equation problem that forms a beautiful mathematical framework and so can leverage strong mathematical tools to solve t... | https://arxiv.org/abs/2503.02129v1 |
satisfactory generalization theory for multilayer neual networks should be, described in the next section. This paper tries to fill the gap between theory and practice by giving a quite general rigorous characterization of the generalization property of a general multilayer neural network for quite general problems inc... | https://arxiv.org/abs/2503.02129v1 |
knowledge, this result is new. 5)We show that the generalization upper bound can predict the double descent phenomenon. We don’t realize any other non-asymptotic bounds having such implication. The organization of this paper is as follows: In section 2 we discuss some relevant works. Section 3 is the notation statement... | https://arxiv.org/abs/2503.02129v1 |
generalization property analysis is therefore also tough. Since the discovery of double descent phenomenon [Belkin et al., 2019], there were many works trying to theoretically understand this under various assumptions. [Belkin et al., 2020, Hastie et al., 2022, Muthukumar et al., 2020] started detailed analysis for cla... | https://arxiv.org/abs/2503.02129v1 |
descent to increasing object manifold dimensionality. To the knowledge of the authors, an incomplete list for rigorous generalization theory for multilayer neural network is [Allen-Zhu et al., 2019, Bartlett et al., 2017, 2019, Gu et al., 2023, Jakubovitz et al., 2019, Neyshabur et al., 2017b, 2018, Tirumala, Wang and ... | https://arxiv.org/abs/2503.02129v1 |
true model into place and the corresponding empirical error, then, is not exactly computable from running experiments only . Through inspecting people’s experience in deep learning in recent decades, we propose here what we think a satisfactory generalization theory there should be. We call a set of generalization erro... | https://arxiv.org/abs/2503.02129v1 |
true models explicitly rather than ignoring it completely as in previous works, to ensure that a machine learning model learned from a predefined model class exhibits favorable empirical and generalization properties, a necessary condition is that this model class possesses the capability to approximate the underlying ... | https://arxiv.org/abs/2503.02129v1 |
elements are weights. The careful readers can notice that the bias terms in the usual appearance of deep neural network are actually absorbed into the weights. So this formulation doesn’t distinguish between normal weights and biases. Such form is very convenient and useful for our discussions in this paper. We callLth... | https://arxiv.org/abs/2503.02129v1 |
. . . , w1), its path enhanced scaled variation norm , denoted by ν(θ), is defined as the right hand side of Eq. (2): ν(θ) =mL−1X iL−1=1···m1X i1=1|aL iL−1wL−1 iL−1iL−2···w2 i2i1|∥w1 i1∥2 (8) Sometimes we will write ν(θ)asν(f)iffis of the form IV .2 for convenience. ForXΩL,...,Ω0;K, the analogous property is theorem IV... | https://arxiv.org/abs/2503.02129v1 |
expression that our assertion is correct. Lastly, since we change the network structure, the width vector ⃗ mshould be replaced by ⃗ m+1(addition applies element-wise). The dimension of the training data is d+ 1now. V. P ROBLEM SETUP The framework of the learning problem we discuss in this work is described as follows:... | https://arxiv.org/abs/2503.02129v1 |
equality requirement of Cauchy inequality to render the above proposition. Besides, there is another familiar regularization that leads to its equivalence to PeSV norm and makes PeSV norm a more valuable regularization to study with. Let us define the lpmax norm (or per-unit lPnorm) and its extension, called mixed lp,q... | https://arxiv.org/abs/2503.02129v1 |
in L2norm. All the proofs of the results in this section are deferred to supplemental information XV. The first L2approximation result we will present is theorem 3.6 from [Wojtowytsch et al., 2020]. theorem VI.1.LetPbe a probability measure with compact support spt(P)⊂Bd(R). Then for any L≥1, f∈WLand m∈N, there exists ... | https://arxiv.org/abs/2503.02129v1 |
every m∈Nandϵ >0, there exists melements g1, . . . , g m∈ G such that f−1 mmX i=1gi H≤R+ϵ√m(24) We now give an iterative version of the above theorem, which is a key to prove general Lcase theorem VI.2, and it seems new also. Before that, we need two lemmas on combinatoric expressions. lemma VI.2.1 .Assume n≥m, we have... | https://arxiv.org/abs/2503.02129v1 |
2023]. theorem VI.4.For any f∈W2, there exists a network g(·;θ)with depth 2and width Min the form of Eq. (IV .2) such that ν(θ)≤6∥f∥W2and ∥f−g(·;θ)∥L∞(Bd)≤C∥f∥W2;Bd M−(d+3)/(2d) (29) for some constant C≥0depending only on d. The proof of this theorem is based on geometric discrepancy theory, in particular, the follow... | https://arxiv.org/abs/2503.02129v1 |
WL (33) +max{12Dσ,2L+1cLL−1 σ√ d}(σ2 ϵ+∥f∗∥2 WL)r logn n) (34) for some constants C1, C2, C > 0. Asf∗is unknown, the strategy to prove the empirical error estimation is to leverage the approximation theory to find its ”proxy” in multilayer neural networks space, and then one can manage to compare estimator with this pr... | https://arxiv.org/abs/2503.02129v1 |
on overparametrised regime is stated as below. theorem VIII.1 .Under Conditions V .0.1,V .0.2 and V .0.3, as before, if H(⃗ m)≤q max{6Dσ,2LcLL−1 σ√ d} C1, then the regularized network estimator g(·;ˆθ)withλ=λ1≡max{6Dσ,2LcLL−1 σ√ d}satisfies ∥g(·;ˆθ)−f∗∥2 2≤C H2(⃗ m)∥f∗∥2 WL (35) +max{12Dσ,2L+1cLL−1 σ√ d}(σ2 ϵ+∥f∗∥2 WL... | https://arxiv.org/abs/2503.02129v1 |
above result. A main ingredient in the proof of theorem VIII.1 is functional concentration inequality after all, in our case, it is Talagrand concentration inequality, absorbed in the above lemma. Along with these ideas we give the proofs in supplemental information XV. IX. U NDERPARAMETRISED REGIME In this section, we... | https://arxiv.org/abs/2503.02129v1 |
generalization error bound in overparametrised regime, using Talagrand concentration inequality and lemma IX.1.1 to associate it with empirical error IX.1. We follow completely the same strategy as the proof of theorem 4 in [Wang and Lin, 2023], modifying appropriately according to lemma IX.0.1 and IX.1.1, referring th... | https://arxiv.org/abs/2503.02129v1 |
infinity when kapproaches infinity as well, the above analysis shows that something related to the double descent phenomenon may occur. Qualitatively, when mis small, one can bound the variance by something proportional to the number of weights (and the norm, of course). This bound is better than the bound depending on... | https://arxiv.org/abs/2503.02129v1 |
of common machine learning problems. On set of data (x1, y1),(x2, y2),···,(xn, yn)the empirical version of loss function is denoted by Ln(f, y) :=1 nPn i=1L(f(xi), yi) where x= (x1, x2,···, xn)andy= (y1, y2,···, yn). And by Cauchy inequality |Ln(f1, y1)− Ln(f2, y2)| ≤L0∥(f1, y1)− (f2, y2)∥nand|Ln,y(f1, y1)− Ln,y(f2, y2... | https://arxiv.org/abs/2503.02129v1 |
with probability at least 1−O(n−C2)for some constants C1, C2, C > 0. These two estimations are based on the analogy to lemma IX.1.1 and a key ingredient relying on local Rademacher complexity estimation for our general L, which are summarized as the following two results. lemma XI.4.1 .For any 0< γ < 1, defined by BF(γ... | https://arxiv.org/abs/2503.02129v1 |
same conditions with us, the regularized network estimator g(·;ˆθ)withλ=C1σϵp dlogn/n satisfies ||g(·;ˆθ)−f∗||2 n≤C( ||f∗||2 Sm−(d+3)/d+ (σ2 ϵ+||f∗||2 S)r dlogn n) (82) with probability at least 1−O(n−C2). Therefore, it is clear that, except for the approximation terms, the second term (variance term) has exactly the s... | https://arxiv.org/abs/2503.02129v1 |
number of research questions remain. First, what is the corresponding theory for more general loss functions and activation functions (without Lipschitzness, strongly convexity etc.). Second, what is the corresponding theory for other regularization terms, including implicit regularization like early stopping. Lastly, ... | https://arxiv.org/abs/2503.02129v1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.