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Signal-Plus-Noise Decomposition of Nonlinear Spiked Random Matrix Models

Behrad Moniri, Hamed Hassani

TL;DR

This work extends spiked random matrix theory to nonlinear, element-wise transformations by analyzing $\mathbf{Y}=\frac{1}{\sqrt{n}}f(\mathbf{W}+\lambda\sqrt{n}\boldsymbol{x}\boldsymbol{x}^\top)$ with general, non-Gaussian noise $\mathbf{W}$. It proves a signal-plus-noise decomposition: $\mathbf{Y}=\frac{1}{\sqrt{n}}f(\mathbf{W})+\sum_{k=1}^{\ell}\mathbf{H}_k+o_{\mathbb{P}}(1)$, where $\mathbf{H}_k$ are rank-$\leq$ rank$(\mathbb{E}f^{(k)}(\mathbf{W}))$ components, and shows phase transitions as the signal exponent $\alpha$ grows, yielding up to $\ell$ outliers. The decomposition enables precise analysis of two problems: signed signal recovery, where sign identifiability depends on odd/even derivatives of $f$ and can require larger $\lambda$, and transformed stochastic block models, where community detection exhibits regime-dependent outliers and thresholds. Numerical experiments corroborate the theory, illustrating emergent outliers and alignment patterns that generalize spectral methods to nonlinear observations. The results offer a broad, non-Gaussian, non-identically distributed framework with potential implications for understanding feature learning in shallow neural networks.

Abstract

In this paper, we study a nonlinear spiked random matrix model where a nonlinear function is applied element-wise to a noise matrix perturbed by a rank-one signal. We establish a signal-plus-noise decomposition for this model and identify precise phase transitions in the structure of the signal components at critical thresholds of signal strength. To demonstrate the applicability of this decomposition, we then utilize it to study new phenomena in the problems of signed signal recovery in nonlinear models and community detection in transformed stochastic block models. Finally, we validate our results through a series of numerical simulations.

Signal-Plus-Noise Decomposition of Nonlinear Spiked Random Matrix Models

TL;DR

This work extends spiked random matrix theory to nonlinear, element-wise transformations by analyzing with general, non-Gaussian noise . It proves a signal-plus-noise decomposition: , where are rank- rank components, and shows phase transitions as the signal exponent grows, yielding up to outliers. The decomposition enables precise analysis of two problems: signed signal recovery, where sign identifiability depends on odd/even derivatives of and can require larger , and transformed stochastic block models, where community detection exhibits regime-dependent outliers and thresholds. Numerical experiments corroborate the theory, illustrating emergent outliers and alignment patterns that generalize spectral methods to nonlinear observations. The results offer a broad, non-Gaussian, non-identically distributed framework with potential implications for understanding feature learning in shallow neural networks.

Abstract

In this paper, we study a nonlinear spiked random matrix model where a nonlinear function is applied element-wise to a noise matrix perturbed by a rank-one signal. We establish a signal-plus-noise decomposition for this model and identify precise phase transitions in the structure of the signal components at critical thresholds of signal strength. To demonstrate the applicability of this decomposition, we then utilize it to study new phenomena in the problems of signed signal recovery in nonlinear models and community detection in transformed stochastic block models. Finally, we validate our results through a series of numerical simulations.
Paper Structure (36 sections, 10 theorems, 107 equations, 4 figures)

This paper contains 36 sections, 10 theorems, 107 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Examplar Applications
  3. Transformed Stochastic Block Models.
  4. Signed Signal Recovery.
  5. Other Examples.
  6. Contributions
  7. Prior Works on Nonlinear Spiked Random Matrix Models
  8. Notations
  9. Review of Spiked Wigner Models
  10. Signal-Plus-Noise Decomposition
  11. Applications
  12. Signed Signal Recovery
  13. Transformed Stochastic Block Model
  14. Experiments
  15. 5.1. Signed Signal Recovery.
  16. ...and 21 more sections

Key Result

Theorem 2.2

Let $\lambda_1, {\bm{u}}_1$ be the leading eigenvalue and its corresponding eigenvector of the matrix $\mathbf{Y}_{\rm Lin}$. Assuming that $\Vert{\bm{x}}\Vert_2 \overset{\mathrm{P}}{\to} 1$, if $\lambda < \sigma_w$, we have:

Figures (4)

  • Figure 1: Emergence of outlying eigenvalues of $\mathbf{Y}$ with eigenvectors aligned to $\mathbf{1}$ and the signal vector $\boldsymbol{\zeta}$ in different regimes of the signal strength exponent $\alpha$ when (a) $I_e < I_o$ and (b) $I_o < I_e$.
  • Figure 2: Position of the top and the second top eigenvalue of $\mathbf{Y}$ for the setting in Section \ref{['sec:experiment-signed']}.1 with signal strength $\lambda = cn^{1/4}$ as a function of $c$ for different values of $n$.
  • Figure 3: The correlation of the top eigenvector ${\bm{u}}_1$ of $\mathbf{Y}$, and $\mathbf{1}$ and the correlation of the second eigenvector ${\bm{u}}_2$ of $\mathbf{Y}$ and $\boldsymbol{\zeta}$ in the setting of Section \ref{['sec:experiment-signed']}.1 with signal strength $\lambda = cn^{1/4}$ as a function of $c$ for different values of $n$.
  • Figure 4: (Left) the position of the second top eigenvlaue of $\mathbf{Y}$ in the setting of Section 5.2. The curve is unchanged for different values of $n$. (Right) Histogram of the eigenvalues of $\mathbf{Y}$ in the setting of Section 5.2. The eigenvalues consist of a bulk of eigenvalues that stick together resulting from the noise component, and two outliers corresponding to signal terms.

Theorems & Definitions (16)

  • Definition 2.1
  • Theorem 2.2: BBP Phase Transition
  • Theorem 3.3
  • Proposition 4.1
  • Definition 4.3
  • Theorem 4.4
  • Theorem 4.5
  • Proposition 4.6
  • Proposition 4.7
  • Definition 4.8
  • ...and 6 more