Papers › Tight Dimensionality Reduction for Sketching Low Degree Polynomial Kernels
Tight Dimensionality Reduction for Sketching Low Degree Polynomial Kernels
Michela Meister, Tamas Sarlos, David Woodruff
We revisit the classic randomized sketch of a tensor product of q vectors xᵢ∈ℝⁿ. The i-th coordinate (Sx)ᵢ of the sketch is equal to ∏_(j = 1)^q ⟨u^(i, j), xʲ ⟩/ √(m), where u^(i,j) are independent random sign vectors. Kar and Karnick (JMLR, 2012) show that if the sketching dimension m = Ω(ϵ⁻² C_Ω² log(1/δ)), where C_Ω is a certain property of the point set Ω one wants to sketch, then with probability 1-δ, Sx₂ = (1±ϵ)x₂ for all x∈Ω. However, in their analysis C_Ω² can be as large as Θ(n^(2q)), even for a set Ω of O(1) vectors x. We give a new analysis of this sketch, providing nearly optimal bounds. Namely, we show an upper bound of m = Θ(ϵ⁻² log(n/δ) + ϵ⁻¹ log^q(n/δ) ), which by composing with CountSketch, can be improved to Θ(ϵ⁻²log(1/(δϵ)) + ϵ⁻¹ log^q (1/(δϵ)). For the important case of q = 2 and δ= 1/(n), this shows that m = Θ(ϵ⁻² log(n) + ϵ⁻¹ log²(n)), demonstrating that the ϵ⁻² and log²(n) terms do not multiply each other. We also show a nearly matching lower bound of m = Ω(⁻² log(1/(δ)) + ⁻¹ log^q(1/(δ))). In a number of applications, one has |Ω| = (n) and in this case our bounds are optimal up to a constant factor. This is the first high probability sketch for tensor products that has optimal sketch size and can be implemented in m ·∑ᵢ₌₁^q nnz(xᵢ) time, where nnz(xᵢ) is the number of non-zero entries of xᵢ. Lastly, we empirically compare our sketch to other sketches for tensor products, and give a novel application to compressing neural networks.
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