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On spectrum of sample covariance matrices from large tensor vectors

Wangjun Yuan

Abstract

In this paper, we investigate the limiting empirical spectral distribution (LSD) of sums of independent rank-one $k$-fold tensor products of $n$-dimensional vectors as $k,n \to \infty$. Assuming that the base vectors are complex random variables with unit modular, we show that the LSD is the Marčenko-Pastur law. Comparing with the existing results, our limiting setting allows $k$ to grow much faster than $n$. Consequently, we obtain the necessary and sufficient conditions for Marčenko-Pastur law to serve as the LSD of our matrix model. Our approach is based on the moment method.

On spectrum of sample covariance matrices from large tensor vectors

Abstract

In this paper, we investigate the limiting empirical spectral distribution (LSD) of sums of independent rank-one -fold tensor products of -dimensional vectors as . Assuming that the base vectors are complex random variables with unit modular, we show that the LSD is the Marčenko-Pastur law. Comparing with the existing results, our limiting setting allows to grow much faster than . Consequently, we obtain the necessary and sufficient conditions for Marčenko-Pastur law to serve as the LSD of our matrix model. Our approach is based on the moment method.
Paper Structure (9 sections, 12 theorems, 46 equations, 7 figures)

This paper contains 9 sections, 12 theorems, 46 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Graph combinatorics
  3. Preliminaries
  4. Characterization of $\mathcal{C}_{s,p}^{(1)}$
  5. Paired graph
  6. Convergence of spectral moments
  7. Proof of Theorem \ref{['Thm-main']}
  8. Proof of Theorem \ref{['Thm-main2']}
  9. Proof of Corollary \ref{['Coro']}

Key Result

Theorem 1.1

Let $M_{n,k,m}$ be in eq:matrix with $|\xi_1|=1$. Suppose that for all $q \in \mathbb N$, Assume that eq-def-ratio holds. Then for any fixed $p \in \mathbb N_+$, we have Here, $\mathrm {deg}_t(\alpha)$ is the frequency of $t$ in the sequence $\alpha$ and is given by eq:deg_t, and $\mathcal{C}_{s,p}^{(1)}$ is a set of sequences that is defined in Lemma lem-Bai.

Figures (7)

  • Figure 1:
  • Figure 2:
  • Figure 3:
  • Figure 4: $g(i,\alpha)$ with $p=4$ and $\alpha=i=(1,2,1,2)$.
  • Figure 5: Case 1-Split an $i$ vertex to cancel multiple pairs of edges.
  • ...and 2 more figures

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Definition 1
  • Theorem 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • ...and 10 more