Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Decremental $(1+ε)$-Approximate Maximum Eigenvector: Dynamic Power Method

Deeksha Adil, Thatchaphol Saranurak

TL;DR

This work develops a dynamic algorithm to maintain a $(1+\epsilon)$-approximate maximum eigenvalue and eigenvector of a PSD matrix undergoing decremental rank-one updates, achieving polylog-update efficiency under an oblivious adversary. The core method adapts the classical power method to a dynamic setting and is supported by a novel analysis that uses subspace-based potentials to bound the number of power-method executions. A key contribution is a conditional lower bound showing that an efficient adaptive-adversary algorithm with a natural property would imply breakthroughs in PSD-certification hardness, suggesting a separation between oblivious and adaptive models. The paper also connects these results to dynamic PSDPs and outlines open questions, including incremental updates and broader dynamic SDP frameworks.

Abstract

We present a dynamic algorithm for maintaining $(1+ε)$-approximate maximum eigenvector and eigenvalue of a positive semi-definite matrix $A$ undergoing \emph{decreasing} updates, i.e., updates which may only decrease eigenvalues. Given a vector $v$ updating $A\gets A-vv^{\top}$, our algorithm takes $\tilde{O}(\mathrm{nnz}(v))$ amortized update time, i.e., polylogarithmic per non-zeros in the update vector. Our technique is based on a novel analysis of the influential power method in the dynamic setting. The two previous sets of techniques have the following drawbacks (1) algebraic techniques can maintain exact solutions but their update time is at least polynomial per non-zeros, and (2) sketching techniques admit polylogarithmic update time but suffer from a crude additive approximation. Our algorithm exploits an oblivious adversary. Interestingly, we show that any algorithm with polylogarithmic update time per non-zeros that works against an adaptive adversary and satisfies an additional natural property would imply a breakthrough for checking psd-ness of matrices in $\tilde{O}(n^{2})$ time, instead of $O(n^ω)$ time.

Decremental $(1+ε)$-Approximate Maximum Eigenvector: Dynamic Power Method

TL;DR

This work develops a dynamic algorithm to maintain a -approximate maximum eigenvalue and eigenvector of a PSD matrix undergoing decremental rank-one updates, achieving polylog-update efficiency under an oblivious adversary. The core method adapts the classical power method to a dynamic setting and is supported by a novel analysis that uses subspace-based potentials to bound the number of power-method executions. A key contribution is a conditional lower bound showing that an efficient adaptive-adversary algorithm with a natural property would imply breakthroughs in PSD-certification hardness, suggesting a separation between oblivious and adaptive models. The paper also connects these results to dynamic PSDPs and outlines open questions, including incremental updates and broader dynamic SDP frameworks.

Abstract

We present a dynamic algorithm for maintaining -approximate maximum eigenvector and eigenvalue of a positive semi-definite matrix undergoing \emph{decreasing} updates, i.e., updates which may only decrease eigenvalues. Given a vector updating , our algorithm takes amortized update time, i.e., polylogarithmic per non-zeros in the update vector. Our technique is based on a novel analysis of the influential power method in the dynamic setting. The two previous sets of techniques have the following drawbacks (1) algebraic techniques can maintain exact solutions but their update time is at least polynomial per non-zeros, and (2) sketching techniques admit polylogarithmic update time but suffer from a crude additive approximation. Our algorithm exploits an oblivious adversary. Interestingly, we show that any algorithm with polylogarithmic update time per non-zeros that works against an adaptive adversary and satisfies an additional natural property would imply a breakthrough for checking psd-ness of matrices in time, instead of time.
Paper Structure (31 sections, 29 theorems, 89 equations, 4 algorithms)

This paper contains 31 sections, 29 theorems, 89 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Results
  3. Towards Separation between Oblivious and Adaptive adversaries
  4. Organization.
  5. Preliminaries
  6. Algorithms against an Oblivious Adversary
  7. Proof Overview
  8. Proof of the Main \ref{['thm:upper']} assuming the Key \ref{['lem:BoundW']}
  9. Correctness.
  10. Runtime.
  11. Proof of the Key Lemma \ref{['lem:BoundW']}: Few Executions of Line \ref{['algline:PM']}
  12. The Potentials.
  13. Proof of the Progress Lemma \ref{['lem:EigenspaceChange']}
  14. Notation.
  15. Case $q_{\nu}\geq 20\log n$:
  16. ...and 16 more sections

Key Result

Theorem 1.2

There is an algorithm for prob:dyn under a sequence of $T$ decreasing updates, that given parameters $n$, $\boldsymbol{\mathit{A}}_0$, and $\epsilon>1/n$ as input, with probability at least $1-1/n$ works against an oblivious adversary in total time,

Theorems & Definitions (56)

  • Theorem 1.2
  • Theorem 1.6
  • Definition 2.1: $\epsilon$-max span and dimension
  • Definition 2.2
  • Lemma 2.3: Chernoff Bound
  • Lemma 2.4: Norm of Gaussian Vector
  • proof
  • Lemma 2.5: Distribution of $\chi^2$ Variable
  • proof
  • Lemma 3.1
  • ...and 46 more