Improved dependence on coherence in eigenvector and eigenvalue estimation error bounds

TL;DR

The paper addresses eigenvector and eigenvalue estimation in low-rank signal-plus-noise models with coherence constraints. It introduces new high-order noise concentration bounds and a unit-vector decomposition to reduce dependence on coherence μ, achieving coherence-free results in regimes where noise variance and tail behavior satisfy a subexponential condition. The main contributions include coherence-free perturbation bounds for rank-one and rank-r settings, and numerical evidence across Gaussian denoising, matrix completion, and network estimation that validate the improved rates. The results enhance the practical applicability of spectral methods in coherent signal settings and align with known minimax limits up to logarithmic factors under Gaussian-like noise.

Abstract

Spectral estimators are fundamental in lowrank matrix models and arise throughout machine learning and statistics, with applications including network analysis, matrix completion and PCA. These estimators aim to recover the leading eigenvalues and eigenvectors of an unknown signal matrix observed subject to noise. While extensive research has addressed the statistical accuracy of spectral estimators under a variety of conditions, most previous work has assumed that the signal eigenvectors are incoherent with respect to the standard basis. This assumption typically arises because of suboptimal dependence on coherence in one or more concentration inequalities. Using a new matrix concentration result that may be of independent interest, we establish estimation error bounds for eigenvector and eigenvalue recovery whose dependence on coherence significantly improves upon prior work. Our results imply that coherence-free bounds can be achieved when the standard deviation of the noise is comparable to its Orlicz 1-norm (i.e., its subexponential norm). This matches known minimax lower bounds under Gaussian noise up to logarithmic factors.

Figures (7)

• Figure 1: Eigenvalue estimation error by the sample eigenvalue of $\boldsymbol{M}$ as a function of $n$. Different colors represent different levels of coherence $\mu \in \{O(1), O(n^{1/4}), O(n^{1/2}), O(n)\}$. Each point represents the average error over 300 independent trials. Shaded bands indicate $95\%$ bootstrap CIs for the fitted lines.
• Figure 2: Eigenvalue estimation error as a function of $n$. Different colors represent varying levels of $\mu \in \{O(1), O(n^{0.1}), O(n^{0.2}), O(n^{0.3})\}$. The dashed lines correspond to the predicted rate from Equation \ref{['eq:mc-rate']}. Shaded bands indicate $95\%$ bootstrap CIs.
• Figure 3: Eigenvalue estimation error as a function of $n$. Different colors represent different levels of coherence $\mu \in \{O(1), O(n^{1/3}), O(n^{2/5})\}$. The dashed lines indicate the predicted rate from Equation \ref{['eq:net-rate']}. Shaded bands indicate $95\%$ bootstrap CIs.
• Figure 4: A graph created following Rule \ref{['rule:(i)']} when $k = 2$ and $p = 4$.
• Figure 5: A graph $G^{\text{new}}$ created following Rule \ref{['rule:(ii)']} when $p = 2$ from the graph $G$ in Figure \ref{['fig:example']}. The left plot shows the labels included in nodes of $G^{\text{new}}$ (which corresponds to connected components of $G$, see Figure \ref{['fig:example']}). The right plot displays a possible coloring of $G^{\text{new}}$ (see Section \ref{['sec:coloring-scheme']} below).

Theorems & Definitions (66)

• Remark 1
• Remark 2
• Example 1: Low-rank matrix completion
• Example 2: Low-rank networks
• Lemma 1
• Theorem 1: Neumann trick
• Theorem 2
• Corollary 1
• Theorem 3
• Lemma 2