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In this paper, we study the problem of approximately computing the product oftwo real matrices. In particular, we analyze a dimensionality-reduction-basedapproximation algorithm due to Sarlos [1], introducing the notion of nuclearrank as the ratio of the nuclear norm over the spectral norm. The presentedbound has improved dependence with respect to the approximation error (ascompared to previous approaches), whereas the subspace -- on which we projectthe input matrices -- has dimensions proportional to the maximum of theirnuclear rank and it is independent of the input dimensions. In addition, weprovide an application of this result to linear low-dimensional embeddings.Namely, we show that any Euclidean point-set with bounded nuclear rank isamenable to projection onto number of dimensions that is independent of theinput dimensionality, while achieving additive error guarantees.

Approximate Matrix Multiplication with Application to Linear Embeddings