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

2024-02-27

AI Summary

This paper studies the dynamic maintenance of a $(1+varepsilon)$-multiplicative approximation to the top eigenvalue and eigenvector of a positive semidefinite matrix under decremental updates of the form $A leftarrow A - vv^ op$. We propose the first dynamic power method algorithm, integrating matrix perturbation analysis with amortized analysis to achieve $ ilde{O}(mathrm{nnz}(v))$ amortized update time per non-zero entry—i.e., polylogarithmic time per non-zero element—under the oblivious adversary assumption. Our work is the first to rigorously adapt the power method to dynamic settings, providing formal theoretical guarantees. It bridges a fundamental gap: unlike algebraic methods (which incur high computational overhead) and sketching-based approaches (which only yield additive error), our algorithm simultaneously achieves polylogarithmic efficiency and multiplicative spectral approximation. This yields the first solution for spectral maintenance under decremental updates that is both highly efficient and equipped with strong, multiplicative approximation guarantees.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Security and Privacy: Large-scale security measurements

Abstract

We present a dynamic algorithm for maintaining $(1+epsilon)$-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 $Agets A-vv^{ op}$, our algorithm takes $ ilde{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 $ ilde{O}(n^{2})$ time, instead of $O(n^{omega})$ time.

Problem

• Maintaining
• Semi-definite Matrix
• Approximate Eigenvalue

Innovation

• Dynamic Algorithm
• Maximum Eigenvector Approximation
• Polylogarithmic Update Time