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Spectral norm bound for the product of random Fourier-Walsh matrices

Libin Zhu, Damek Davis, Dmitriy Drusvyatskiy, Maryam Fazel

TL;DR

The paper studies the spectral properties of products of random Fourier-Walsh matrices by analyzing $M= A_1^{\top}(A_2A_2^{\top})\cdots(A_mA_m^{\top})A_{m+1}$ with $A_i=X_{{\mathcal S}_i}\mathrm{Diag}(w^{(i)})$. Using a moment-method framework coupled with a detailed combinatorial decomposition (partitioning indices, feature collapse, and subscripts relabeling), the authors derive conditions under which the operator norm of the expectation, $\|\mathbb{E}M\|_{op}$, remains controlled and tends to zero as the ambient dimension $d$ grows, provided degree bounds $|S_i|\le p_i$, trivial intersections among the ${\mathcal S}_i$, and small weights $w^{(i)}=O_d(n^{-1/2}\wedge d^{-p_i/2})$. The main result shows $\|\mathbb{E}M\|_{op} = O_d(d^{p_j})\|w^{(1)}\|_\infty\|w^{(m+1)}\|_\infty$ for any $j$ with ${\mathcal S}_j$ distinct from ${\mathcal S}_1$ and ${\mathcal S}_{m+1}$, revealing a polynomial-in-$d$ regime where the norm vanishes when $d^{p_j}=o(n)$. The paper further extends the analysis to more general unions of set families and discusses relaxing assumptions, broadening applicability to random Boolean matrices and Fourier-analytic spectral questions. This work links Fourier analysis of Boolean functions with random-matrix techniques to establish precise spectral bounds for dependent random matrix products with potential implications for learning theory and spectral analysis of Boolean-derived data.

Abstract

We consider matrix products of the form $A_1(A_2A_2)^\top\ldots(A_{m}A_{m}^\top)A_{m+1}$, where $A_i$ are normalized random Fourier-Walsh matrices. We identify an interesting polynomial scaling regime when the operator norm of the expected matrix product tends to zero as the dimension tends to infinity.

Spectral norm bound for the product of random Fourier-Walsh matrices

TL;DR

The paper studies the spectral properties of products of random Fourier-Walsh matrices by analyzing with . Using a moment-method framework coupled with a detailed combinatorial decomposition (partitioning indices, feature collapse, and subscripts relabeling), the authors derive conditions under which the operator norm of the expectation, , remains controlled and tends to zero as the ambient dimension grows, provided degree bounds , trivial intersections among the , and small weights . The main result shows for any with distinct from and , revealing a polynomial-in- regime where the norm vanishes when . The paper further extends the analysis to more general unions of set families and discusses relaxing assumptions, broadening applicability to random Boolean matrices and Fourier-analytic spectral questions. This work links Fourier analysis of Boolean functions with random-matrix techniques to establish precise spectral bounds for dependent random matrix products with potential implications for learning theory and spectral analysis of Boolean-derived data.

Abstract

We consider matrix products of the form , where are normalized random Fourier-Walsh matrices. We identify an interesting polynomial scaling regime when the operator norm of the expected matrix product tends to zero as the dimension tends to infinity.

Paper Structure

This paper contains 13 sections, 7 theorems, 82 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Related literature
  3. Notation
  4. Preliminary results
  5. Proof of Theorem \ref{['lemma:feature_product_tech']}
  6. Partition of indices.
  7. Feature collapse.
  8. Subscripts relabeling.
  9. Iterative summation over feature indices.
  10. Bounding $V_s$.
  11. Completing the proof.
  12. Relaxing assumptions in Theorem \ref{['lemma:feature_product_tech']}
  13. Proof of Equality \ref{['eqn:intro_eqn_moment']}

Key Result

Theorem 1

Fix collections of sets ${\mathcal{S}}_1,\ldots,{\mathcal{S}}_{m+1} \subseteq 2^{[d]}$ and weights $w^{(i)}\in {\mathbb{R}}^{{\mathcal{S}}_i}_{+}$. Define the matrix product where $A_i=X_{\mathcal{S}_i} {\mathsf{Diag}}(w^{(i)})$ are the scaled Fourier-Walsh matrices. Assume that the following regularity conditions hold for all indices $i,j\in [m+1]$: Then the estimate holds for any index $j \in

Figures (2)

  • Figure 1: An illustration of $K$.
  • Figure 2: An illustration of weighted feature.

Theorems & Definitions (19)

  • Theorem 1: Main result
  • Lemma 1: Binary matrices I
  • proof
  • Corollary 1: Binary matrices II
  • proof
  • Proposition 1: Products of Fourier-Walsh monomials
  • proof
  • Proposition 2: Sums of Fourier-Walsh polynomials
  • proof
  • Proposition 3: Weighted sums of Fourier-Walsh polynomials
  • ...and 9 more