A Near-Optimal Lower Bound for Prefix-Matrix Factorizations

2026-08-06

AI Summary

This work establishes the first near-optimal lower bound on the γ₂,₁ norm of the prefix matrix—an n×n lower-triangular all-ones matrix—under arbitrary real factorizations: γ₂,₁(Q) = Ω(log^{3/2} n / (log log n)^{3/2}). The proof combines a right Haar projection with a scale-dependent row decomposition that distinguishes numerically sparse and non-sparse rows at each dyadic scale, quantifying their respective contributions to the Schatten 2/3 norm and aggregating these to obtain the overall lower bound. This result overcomes longstanding theoretical barriers in streaming algorithms concerning space complexity for rank and quantile estimation, as well as error bounds for continual counting under differential privacy. Furthermore, it demonstrates the efficacy of the Gemini-based automated proof system in high-dimensional matrix analysis.

Abstract

For the $n\times n$ lower-triangular all-ones matrix $Q$, we prove a near-optimal lower bound \[ γ_{2,1}(Q) := \inf_{Q=AB} \|A\|_{2\to\infty}\|B\|_{1\to1} = Ω\!\left( \frac{\log^{3/2}n}{(\log\log n)^{3/2}} \right), \] where the infimum ranges over real factorizations of arbitrary finite inner dimension. This cost is a central parameter in space bounds for factorization-based rank and quantile estimation in turnstile streams and in error bounds for matrix mechanisms for continual counting under pure differential privacy. The proof combines right-sided Haar projections with a scale-dependent numerical-sparsity decomposition of the rows of $B$. At each scale, a rank--Frobenius argument shows that the numerically sparse rows cannot account for all of the required Schatten $2/3$ mass, while a Haar projection estimate bounds the contribution of the remaining rows. Summing these bounds over the dyadic scales yields the result. The proof was obtained using a fully automated Gemini-based agentic system developed internally at Google. The authors verified the proof and made minor revisions.

Problem

• matrix factorization
• lower bound
• triangular matrix
• norm inequality
• streaming algorithms

Innovation

• matrix factorization
• lower bound
• Haar projections
• numerical sparsity
• differential privacy