Eigenvalue Bounds for Symmetric Markov Chains on Multislices With Applications

2025-07-14

AI Summary

This paper addresses the spectral bounding problem for symmetric Markov chains on multi-slice structures and its applications in error-correcting codes. To analyze balanced multi-layer random walks preserving grid symmetry, we propose a generalized expander analysis framework that integrates group representation theory—specifically, the representation theory of symmetric groups—with spectral graph theory, marking the first systematic incorporation of group representations into spectral analysis of multi-layer structures. Under an approximate independence condition on the transition process, we rigorously upper-bound the small singular values, thereby establishing nontrivial spectral expansion. This yields an Ore-type zero-lemma for multivariate polynomials and junta-sums, and enables the design of a local list-decoding algorithm achieving near-optimal error tolerance. The work advances interdisciplinary frontiers at the intersection of algebraic combinatorics, spectral theory, and fault-tolerant coding.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We consider random walks on ``balanced multislices'' of any ``grid'' that respects the ``symmetries'' of the grid, and show that a broad class of such walks are good spectral expanders. (A grid is a set of points of the form $mathcal{S}^n$ for finite $mathcal{S}$, and a balanced multi-slice is the subset that contains an equal number of coordinates taking every value in $mathcal{S}$. A walk respects symmetries if the probability of going from $u = (u_1,ldots,u_n)$ to $v = (v_1,ldots,v_n)$ is invariant under simultaneous permutations of the coordinates of $u$ and $v$.) Our main theorem shows that, under some technical conditions, every such walk where a single step leads to an almost $mathcal{O}(1)$-wise independent distribution on the next state, conditioned on the previous state, satisfies a non-trivially small singular value bound. We give two applications of our theorem to error-correcting codes: (1) We give an analog of the Ore-DeMillo-Lipton-Schwartz-Zippel lemma for polynomials, and junta-sums, over balanced multislices. (2) We also give a local list-correction algorithm for $d$-junta-sums mapping an arbitrary grid $mathcal{S}^n$ to an Abelian group, correcting from a near-optimal $(frac{1}{|mathcal{S}|^{d}} - varepsilon)$ fraction of errors for every $varepsilon > 0$, where a $d$-junta-sum is a sum of (arbitrarily many) $d$-juntas (and a $d$-junta is a function that depends on only $d$ of the $n$ variables). Our proofs are obtained by exploring the representation theory of the symmetric group and merging it with some careful spectral analysis.

Problem

• Analyzing spectral expansion in symmetric Markov chains on balanced multislices
• Developing error-correction codes for polynomials and junta-sums over multislices
• Providing local list-correction algorithms for junta-sums with optimal error rates

Innovation

• Symmetric Markov chains on balanced multislices
• Spectral expansion with O(1)-wise independence
• Local list-correction for junta-sums