Kronecker sum covariance models for spatio-temporal data

TL;DR

This work develops graphical-model-style inference for matrix-variate data with measurement errors by introducing a non-separable Kronecker sum covariance model $A \oplus B = A \otimes I_n + I_m \otimes B$ and observing $X = X_0 + W$, with ${\rm vec}(X) \sim N(0, A \oplus B)$. Identifiability is achieved by fixing ${\rm tr}(A)$ and estimating inverse covariances $\Theta = A^{-1}$ and $\Phi = B^{-1}$ via a corrected Lasso in a network of nodewise regressions, without assuming i.i.d. or Gaussian data. The paper establishes convergence rates under subgaussian assumptions and Restricted Eigenvalue-type conditions, and provides a PSD-correction option, situating the KS approach relative to Kronecker-product models. Practically, the framework enables reliable inference of two-way dependencies in high-dimensional matrix data with measurement errors across domains such as neuroscience and genetics.

Abstract

In this paper, we study the subgaussian matrix variate model, where we observe the matrix variate data $X$ which consists of a signal matrix $X_0$ and a noise matrix $W$. More specifically, we study a subgaussian model using the Kronecker sum covariance as in Rudelson and Zhou (2017). Let $Z_1, Z_2$ be independent copies of a subgaussian random matrix $Z =(Z_{ij})$, where $Z_{ij}, \forall i, j$ are independent mean 0, unit variance, subgaussian random variables with bounded $ψ_2$ norm. We use $X \sim \mathcal{M}_{n,m}(0, A \oplus B)$ to denote the subgaussian random matrix $X_{n \times m}$ which is generated using: $$ X = Z_1 A^{1/2} + B^{1/2} Z_2. $$ In this covariance model, the first component $A \otimes I_n$ describes the covariance of the signal $X_0 = Z_1 A^{1/2}$, which is an ${n \times m}$ random design matrix with independent subgaussian row vectors, and the other component $I_m \otimes B$ describes the covariance for the noise matrix $W =B^{1/2} Z_2$, which contains independent subgaussian column vectors $w^1, \ldots, w^m$, independent of $X_0$. This leads to a non-separable class of models for the observation $X$, which we denote by $X \sim \mathcal{M}_{n,m}(0, A \oplus B)$ throughout this paper. Our method on inverse covariance estimation corresponds to the proposal in Yuan (2010) and Loh and Wainwright (2012), only now dropping the i.i.d. or Gaussian assumptions. We present the statistical rates of convergence.

Theorems & Definitions (16)

• Proposition 2.1
• Proposition 2.2
• Theorem 3.1
• Remark 3.2
• Lemma 3.3
• Lemma 3.4
• Definition 3.5
• Theorem 4.1
• Lemma 4.2
• Definition B.1