Decremental (1+ϵ)-Approximate Maximum Eigenvector: Dynamic Power Method
arxiv(2024)
摘要
We present a dynamic algorithm for maintaining (1+ϵ)-approximate
maximum eigenvector and eigenvalue of a positive semi-definite matrix A
undergoing decreasing updates, i.e., updates which may only decrease
eigenvalues. Given a vector v updating A A-vv^⊤, our algorithm
takes Õ(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 Õ(n^2) time,
instead of O(n^ω) time.
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