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Average-Case Matrix Discrepancy: Asymptotics and Online Algorithms

Dmitriy Kunisky, Peiyuan Zhang

TL;DR

This work establishes average-case matrix discrepancy results for random symmetric matrices, extending the vector discrepancy paradigm to the matrix setting. It derives both lower and upper bounds that pinpoint the typical discrepancy in regimes where the number of matrices scales as $T \sim \Theta(n^2)$, notably showing $\Delta \asymp \Theta(n)$ for GOE-like data and $\Delta \asymp \Theta(\sqrt{rn})$ for low-rank Wishart-type models. It then introduces the Matrix Hyperbolic Cosine (MHC) online algorithm, and under matrix-anti-concentration (MACI) and unbiasedness conditions it achieves discrepancy bounds of order $\sqrt{rn}\,\log T$, bridging online discrepancy with average-case optimality in several regimes. The results illuminate non-commutative discrepancy phenomena, extend average-case vector discrepancy techniques to matrices via vectorization and hyperbolic-cosine potentials, and provide concrete algorithms and proofs for GOE and Wishart distributions with implications for online matrix balancing. Collectively, the paper advances understanding of average-case matrix discrepancy and offers robust online methods with provable performance in non-commutative settings.

Abstract

We study the operator norm discrepancy of i.i.d. random matrices, initiating the matrix-valued analog of a long line of work on the $\ell^{\infty}$ norm discrepancy of i.i.d. random vectors. First, using repurposed results on vector discrepancy and new first moment method calculations, we give upper and lower bounds on the discrepancy of random matrices. We treat i.i.d. matrices drawn from the Gaussian orthogonal ensemble (GOE) and low-rank Gaussian Wishart distributions. In both cases, for what turns out to be the "critical" number of $Θ(n^2)$ matrices of dimension $n \times n$, we identify the discrepancy up to constant factors. Second, we give a new analysis of the matrix hyperbolic cosine algorithm of Zouzias (2011), a matrix version of an online vector discrepancy algorithm of Spencer (1977) studied for average-case inputs by Bansal and Spencer (2020), for the case of i.i.d. random matrix inputs. We both give a general analysis and extract concrete bounds on the discrepancy achieved by this algorithm for matrices with independent entries (including GOE matrices) and Gaussian Wishart matrices.

Average-Case Matrix Discrepancy: Asymptotics and Online Algorithms

TL;DR

This work establishes average-case matrix discrepancy results for random symmetric matrices, extending the vector discrepancy paradigm to the matrix setting. It derives both lower and upper bounds that pinpoint the typical discrepancy in regimes where the number of matrices scales as , notably showing for GOE-like data and for low-rank Wishart-type models. It then introduces the Matrix Hyperbolic Cosine (MHC) online algorithm, and under matrix-anti-concentration (MACI) and unbiasedness conditions it achieves discrepancy bounds of order , bridging online discrepancy with average-case optimality in several regimes. The results illuminate non-commutative discrepancy phenomena, extend average-case vector discrepancy techniques to matrices via vectorization and hyperbolic-cosine potentials, and provide concrete algorithms and proofs for GOE and Wishart distributions with implications for online matrix balancing. Collectively, the paper advances understanding of average-case matrix discrepancy and offers robust online methods with provable performance in non-commutative settings.

Abstract

We study the operator norm discrepancy of i.i.d. random matrices, initiating the matrix-valued analog of a long line of work on the norm discrepancy of i.i.d. random vectors. First, using repurposed results on vector discrepancy and new first moment method calculations, we give upper and lower bounds on the discrepancy of random matrices. We treat i.i.d. matrices drawn from the Gaussian orthogonal ensemble (GOE) and low-rank Gaussian Wishart distributions. In both cases, for what turns out to be the "critical" number of matrices of dimension , we identify the discrepancy up to constant factors. Second, we give a new analysis of the matrix hyperbolic cosine algorithm of Zouzias (2011), a matrix version of an online vector discrepancy algorithm of Spencer (1977) studied for average-case inputs by Bansal and Spencer (2020), for the case of i.i.d. random matrix inputs. We both give a general analysis and extract concrete bounds on the discrepancy achieved by this algorithm for matrices with independent entries (including GOE matrices) and Gaussian Wishart matrices.
Paper Structure (22 sections, 26 theorems, 81 equations, 1 algorithm)

This paper contains 22 sections, 26 theorems, 81 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Notation
  3. Main Results: Discrepancy Asymptotics
  4. Main Results: Matrix Hyperbolic Cosine Algorithm
  5. Related Work
  6. Worst-case discrepancy
  7. Average-case discrepancy
  8. Online algorithms for discrepancy
  9. Handling non-commutativity
  10. Discrepancy Asymptotics
  11. Lower Bounds: First Moment Method and Scaling Heuristic
  12. Lower Bounds: Gramian Spectral Method
  13. Upper Bounds: Vectorization
  14. GOE Matrices: Proof of Theorem \ref{['thm:goe']}
  15. General Analysis: Proof of Theorem \ref{['thm:lb']}
  16. ...and 7 more sections

Key Result

Theorem 1.1

If $\|\bm v_i\|_{\infty} \leq 1$ for all $i \in [T]$, then $\Delta(\bm v_1, \dots, \bm v_T) \lesssim \sqrt{T \log(1 + \frac{n}{T})}$. Alternatively:

Theorems & Definitions (59)

  • Theorem 1.1: Spencer's "six deviations suffice" theorem Spencer-1985-SixDeviationsSpencer-1994-TenLecturesProbabilisticMethod
  • Remark 1.2: The case of many vectors
  • Theorem 1.3: Theorem 1 and Remark 2 of TMR-2020-BalancingGaussianVectors
  • Theorem 1.4: Section 3 of BS-2020-OnlineBalancingRandom, Section 2.1 of BJSS-2020-OnlineVectorBalancingDiscrepancy
  • Conjecture 1.5: Matrix Spencer conjecture Meka-2014-DiscrepancyMatrixSpencerZouzias-2011-MatrixHyperbolicCosine
  • Conjecture 1.6: Informal discrepancy asymptotics
  • Definition 1.7: Gaussian orthogonal ensemble
  • Proposition 1.8: GOE norm
  • Theorem 1.9: Asymptotics: GOE matrices
  • Theorem 1.10: Asymptotics: general distributions
  • ...and 49 more