Compressed matrix multiplication

Motivated by the problems of computing sample covariance matrices, and of transforming a collection of vectors to a basis where they are sparse, we present a simple algorithm that computes an approximation of the product of two <i>n</i>-by-<i>n</i> real matrices <i>A</i> and <i>B</i>. Let ||<i>AB</i>||_<i>F</i> denote the Frobenius norm of <i>AB</i>, and <i>b</i> be a parameter determining the time/accuracy trade-off. Given 2-wise independent hash functions <i>h</i>₁, <i>h</i>₂: [<i>n</i>] → [<i>b</i>], and <i>s</i>₁, <i>s</i>₂: [<i>n</i>] → {-1, +1} the algorithm works by first "compressing" the matrix product into the polynomial [EQUATION] Using FFT for polynomial multiplication, we can compute <i>c</i>₀,...,<i>c</i>_<i>b</i>-1 such that Σ_<i>i</i> <i>c</i>_<i>i</i><i>x</i>^<i>i</i> = (<i>p</i>(<i>x</i>) mod <i>x</i>^<i>b</i>) + (<i>p</i>(<i>x</i>) div <i>x</i>^<i>b</i>) in time <i>Õ</i>(<i>n</i>² + <i>nb</i>). An unbiased estimator of (<i>AB</i>)_<i>ij</i> with variance at most ||<i>AB</i>||²_<i>F</i>/<i>b</i> can then be computed as: [EQUATION] Our approach also leads to an algorithm for computing <i>AB</i> exactly, whp., in time <i>Õ</i>(<i>N</i> + <i>nb</i>) in the case where <i>A</i> and <i>B</i> have at most <i>N</i> nonzero entries, and <i>AB</i> has at most <i>b</i> nonzero entries. Also, we use error-correcting codes in a novel way to recover significant entries of <i>AB</i> in near-linear time.