On Sketching Quadratic Forms

We undertake a systematic study of sketching a quadratic form: given an n × n matrix A, create a succinct sketch sk(A) which can produce (without further access to A) a multiplicative (1 + ε)-approximation to xT Ax for any desired query x ∈ Rn . While a general matrix does not admit non-trivial sketches, positive semi-definite (PSD) matrices admit sketches of size Θ(ε−2 n), via the Johnson-Lindenstrauss lemma, achieving the “for each” guarantee, namely, for each query x, with a constant probability the sketch succeeds. (For the stronger “for all” guarantee, where the sketch succeeds for all x’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 of work culminating in that of Batson, Spielman, and Srivastava (SIAM Review, 2014), shows that by choosing and reweighting O(ε−2 n) edges in a g...

• Type: Journal
• Year: 2016
• Where: INNOVATIONS
• Authors: Alexandr Andoni, Jiecao Chen, Robert Krauthgamer, Bo Qin, David P. Woodruff, Qin Zhang
• Keywords: INNOVATIONS 2016