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Matrix Denoising with Doubly Heteroscedastic Noise:
Fundamental Limits and Optimal Spectral Methods
Yihan Zhang
Institute of Science and Technology Austria
[email protected]
Marco Mondelli
Institute of Science and Technology Austria
[email protected]
Abstract
We study the matrix denoising problem of estimating the singular vectors of a
rank-1 signal corrupted by noise with both column and row correlations. Existing
works are either unable to pinpoint the exact asymptotic estimation error or, when | [
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works are either unable to pinpoint the exact asymptotic estimation error or, when
they do so, the resulting approaches (e.g., based on whitening or singular value
shrinkage) remain vastly suboptimal. On top of this, most of the literature has
focused on the special case of estimating the left singular vector of the signal
when the noise only possesses row correlation (one-sided heteroscedasticity). In
contrast, our work establishes the information-theoretic and algorithmic limits of | [
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] |
contrast, our work establishes the information-theoretic and algorithmic limits of
matrix denoising with doubly heteroscedastic noise. We characterize the exact
asymptotic minimum mean square error, and design a novel spectral estimator
with rigorous optimality guarantees: under a technical condition, it attains positive
correlation with the signals whenever information-theoretically possible and, for
one-sided heteroscedasticity, it also achieves the Bayes-optimal error. Numerical | [
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one-sided heteroscedasticity, it also achieves the Bayes-optimal error. Numerical
experiments demonstrate the significant advantage of our theoretically principled
method with the state of the art. The proofs draw connections with statistical
physics and approximate message passing, departing drastically from standard
random matrix theory techniques.
1
Introduction
Matrix denoising is a central primitive in statistics and machine learning, and the problem is to | [
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] |
Matrix denoising is a central primitive in statistics and machine learning, and the problem is to
recover a signal X ∈ Rn×d from an observation A = X + W corrupted by additive noise W. This
finds applications across multiple domains of sciences, e.g., imaging [21, 60], biology [13, 42] and
astronomy [67, 5]. When X has low rank and W i.i.d. entries, A is the standard model for principal
component analysis, typically referred to as the Johnstone spiked covariance model [38]. When | [
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] |
component analysis, typically referred to as the Johnstone spiked covariance model [38]. When
n, d are both large and proportional, which corresponds to the most sample-efficient regime, its
Bayes-optimal limits are well understood [48], and it has been established how to achieve them
efficiently [53]. Minimax/non-asymptotic guarantees are also available in special cases, such as
sparse PCA [17], Gaussian mixtures [69] and certain joint scalings of (n, d) [54]. | [
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sparse PCA [17], Gaussian mixtures [69] and certain joint scalings of (n, d) [54].
However, in most applications, noise is highly structured and correlated, thereby calling for more
realistic assumptions on W than having i.i.d. entries. A recent line of work addresses this concern
by studying matrix denoising with heteroscedastic noise [1, 66, 29, 40, 23], resting on two basic
ideas: whitening and singular value shrinkage. Whitening refers to multiplying the data matrix by | [
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ideas: whitening and singular value shrinkage. Whitening refers to multiplying the data matrix by
the square root of the inverse covariance, in order to reduce the model to one with i.i.d. noise; and
singular value shrinkage retains the singular vectors of the data while deflating the singular values to
correct for the noise. Though the exact asymptotic performance of these algorithms has been derived
[66, 29, 40, 23], their optimality is yet to be determined from a Bayesian standpoint. In fact, we will | [
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] |
prove that whitening and shrinkage are not the correct way to approach Bayes optimality.
Preprint. Under review.
arXiv:2405.13912v1 [math.ST] 22 May 2024
Main contributions.
We focus on the prototypical model A = X + W, where X = λ
nu∗v∗⊤ is a
rank-1 signal, λ is the signal-to-noise ratio (SNR), and W = Ξ1/2�
WΣ1/2 is doubly heterogeneous
noise. Here u∗, v∗ follow i.i.d. priors; �
W contains i.i.d. Gaussian entries; the covariance matrices | [
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] |
W contains i.i.d. Gaussian entries; the covariance matrices
Ξ, Σ capture column and row correlations; and we consider the typical high-dimensional regime in
which n, d are both large and proportional. Our main results are summarized below.
1. We design an efficient spectral estimator to recover u∗, v∗, and we provide a precise asymp-
totic analysis of its performance, see Theorem 5.1. This estimator is given by the top singular
vectors of a matrix obtained by carefully pre-processing A, see (5.3). | [
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] |
vectors of a matrix obtained by carefully pre-processing A, see (5.3).
2. When the priors of u∗, v∗ are standard Gaussian, we show in Corollary 5.2 that the spectral
estimator above is optimal in the following sense: (i) under a technical condition, it achieves
the optimal weak recovery threshold, namely its mean square error is non-trivial as soon
as this is information-theoretically possible; (ii) it achieves the Bayes-optimal error for | [
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as this is information-theoretically possible; (ii) it achieves the Bayes-optimal error for
u∗ (resp. v∗) when Ξ (resp. Σ) is the identity. These optimality guarantees follow from
rigorously obtaining the asymptotic minimum mean square error (MMSE) for the estimation
of the whitened signals Ξ−1/2u∗ and Σ−1/2v∗, see Theorem 4.2.
Our spectral estimator only involves matrix multiplication and computing principal singular vectors. | [
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