Global law of conjugate kernel random matrices with heavy-tailed weights

• URL: http://arxiv.org/abs/2502.18428v1
• Date: Tue, 25 Feb 2025 18:22:58 GMT
• Title: Global law of conjugate kernel random matrices with heavy-tailed weights
• Authors: Alice Guionnet, Vanessa Piccolo,
• Abstract summary: We study the spectral behavior of the conjugate kernel random matrix $YYtop$, where $Y= f(WX)$ arises from a two-layer neural network model.<n>We show that heavy-tailed weights induce strong correlations between the entries of $Y$, leading to richer and fundamentally different spectral behavior compared to models with light-tailed weights.
• Score: 1.8416014644193066
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We study the asymptotic spectral behavior of the conjugate kernel random matrix $YY^\top$, where $Y= f(WX)$ arises from a two-layer neural network model. We consider the setting where $W$ and $X$ are both random rectangular matrices with i.i.d. entries, where the entries of $W$ follow a heavy-tailed distribution, while those of $X$ have light tails. Our assumptions on $W$ include a broad class of heavy-tailed distributions, such as symmetric $\alpha$-stable laws with $\alpha \in (0,2)$ and sparse matrices with $\mathcal{O}(1)$ nonzero entries per row. The activation function $f$, applied entrywise, is nonlinear, smooth, and odd. By computing the eigenvalue distribution of $YY^\top$ through its moments, we show that heavy-tailed weights induce strong correlations between the entries of $Y$, leading to richer and fundamentally different spectral behavior compared to models with light-tailed weights.

Related papers

• Distribution of lowest eigenvalue in $k$-body bosonic random matrix ensembles [0.8999666725996978]
  We numerically study the distribution of the lowest eigenvalue of finite many-boson systems with $k$-body interactions. The first four moments of the distribution of lowest eigenvalues have been analyzed as a function of the $q$ parameter.
  arXiv  Detail & Related papers  (2024-04-30T20:44:31Z)
• Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime [12.221087476416056]
  In this paper, we show that this phenomenon is universal, holding as soon as $X$ has i.i.d. entries with all finite moments. In the case of non-integer $ell$, the Marvcenko-Pastur term disappears.
  arXiv  Detail & Related papers  (2023-10-27T17:15:55Z)
• $L^1$ Estimation: On the Optimality of Linear Estimators [64.76492306585168]
  This work shows that the only prior distribution on $X$ that induces linearity in the conditional median is Gaussian. In particular, it is demonstrated that if the conditional distribution $P_X|Y=y$ is symmetric for all $y$, then $X$ must follow a Gaussian distribution.
  arXiv  Detail & Related papers  (2023-09-17T01:45:13Z)
• An Equivalence Principle for the Spectrum of Random Inner-Product Kernel Matrices with Polynomial Scalings [21.727073594338297]
  This study is motivated by applications in machine learning and statistics. We establish the weak limit of the empirical distribution of these random matrices in a scaling regime. Our results can be characterized as the free additive convolution between a Marchenko-Pastur law and a semicircle law.
  arXiv  Detail & Related papers  (2022-05-12T18:50:21Z)
• Random matrices in service of ML footprint: ternary random features with no performance loss [55.30329197651178]
  We show that the eigenspectrum of $bf K$ is independent of the distribution of the i.i.d. entries of $bf w$. We propose a novel random technique, called Ternary Random Feature (TRF) The computation of the proposed random features requires no multiplication and a factor of $b$ less bits for storage compared to classical random features.
  arXiv  Detail & Related papers  (2021-10-05T09:33:49Z)
• Deformed semicircle law and concentration of nonlinear random matrices for ultra-wide neural networks [29.03095282348978]
  We study the limiting spectral distributions of two empirical kernel matrices associated with $f(X)$. We show that random feature regression induced by the empirical kernel achieves the same performance as its limiting kernel regression under the ultra-wide regime.
  arXiv  Detail & Related papers  (2021-09-20T05:25:52Z)
• Spectral properties of sample covariance matrices arising from random matrices with independent non identically distributed columns [50.053491972003656]
  It was previously shown that the functionals $texttr(AR(z))$, for $R(z) = (frac1nXXT- zI_p)-1$ and $Ain mathcal M_p$ deterministic, have a standard deviation of order $O(|A|_* / sqrt n)$. Here, we show that $|mathbb E[R(z)] - tilde R(z)|_F
  arXiv  Detail & Related papers  (2021-09-06T14:21:43Z)
• Analysis of One-Hidden-Layer Neural Networks via the Resolvent Method [0.0]
  Motivated by random neural networks, we consider the random matrix $M = Y Yast$ with $Y = f(WX)$. We prove that the Stieltjes transform of the limiting spectral distribution satisfies a quartic self-consistent equation up to some error terms. In addition, we extend the previous results to the case of additive bias $Y=f(WX+B)$ with $B$ being an independent rank-one Gaussian random matrix.
  arXiv  Detail & Related papers  (2021-05-11T15:17:39Z)
• Sparse sketches with small inversion bias [79.77110958547695]
  Inversion bias arises when averaging estimates of quantities that depend on the inverse covariance. We develop a framework for analyzing inversion bias, based on our proposed concept of an $(epsilon,delta)$-unbiased estimator for random matrices. We show that when the sketching matrix $S$ is dense and has i.i.d. sub-gaussian entries, the estimator $(epsilon,delta)$-unbiased for $(Atop A)-1$ with a sketch of size $m=O(d+sqrt d/
  arXiv  Detail & Related papers  (2020-11-21T01:33:15Z)
• Linear Time Sinkhorn Divergences using Positive Features [51.50788603386766]
  Solving optimal transport with an entropic regularization requires computing a $ntimes n$ kernel matrix that is repeatedly applied to a vector. We propose to use instead ground costs of the form $c(x,y)=-logdotpvarphi(x)varphi(y)$ where $varphi$ is a map from the ground space onto the positive orthant $RRr_+$, with $rll n$.
  arXiv  Detail & Related papers  (2020-06-12T10:21:40Z)
• Curse of Dimensionality on Randomized Smoothing for Certifiable Robustness [151.67113334248464]
  We show that extending the smoothing technique to defend against other attack models can be challenging. We present experimental results on CIFAR to validate our theory.
  arXiv  Detail & Related papers  (2020-02-08T22:02:14Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.