On Sketching Quadratic Forms
Alexandr Andoni, Jiecao Chen, Robert Krauthgamer, Bo Qin, David P. Woodruff, Qin Zhang

TL;DR
This paper studies efficient sketching methods for quadratic forms, especially for graph Laplacians, providing optimal bounds for 'for all' and improved sketches for 'for each' guarantees.
Contribution
It establishes optimal bounds for 'for all' quadratic form sketches of graph Laplacians and introduces improved 'for each' sketches with near-matching lower bounds.
Findings
Optimality of Batson et al.'s bound for 'for all' guarantees.
New sketches of size ~O(ε^{-1} n) bits for 'for each' guarantees on cut queries.
Extended results to symmetric diagonally dominant matrices.
Abstract
We undertake a systematic study of sketching a quadratic form: given an matrix , create a succinct sketch which can produce (without further access to ) a multiplicative -approximation to for any desired query . While a general matrix does not admit non-trivial sketches, positive semi-definite (PSD) matrices admit sketches of size , via the Johnson-Lindenstrauss lemma, achieving the "for each" guarantee, namely, for each query , with a constant probability the sketch succeeds. (For the stronger "for all" guarantee, where the sketch succeeds for all 's simultaneously, again there are no non-trivial sketches.) We design significantly better sketches for the important subclass of graph Laplacian matrices, which we also extend to symmetric diagonally dominant matrices. A sequence…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComplexity and Algorithms in Graphs · Stochastic Gradient Optimization Techniques · Optimization and Search Problems
