Publication Date

Fall 8-2026

Conference/Sponsorship/Institution

Forty-Second Annual Conference on Uncertainty in Artificial Intelligence

Description

TensorSketch by Pham and Pagh [2013], Kar and Karnick [2012] provides efficient sketching algorithms for highdimensional polynomial kernels $\vec{x}^{\otimes p} \in \R^{d^p}$. Kar and Karnick [2012] uses dense Johnson-Lindenstrauss (JL)-type projections with computational cost $O(pDd)$, whereas Pham and Pagh [2013] extends the sparse CountSketch [Charikar et al., 2024] algorithm, yielding a faster algorithm for high-dimensional sparse inputs with running time $O\left(p(\nnz{\vec{x}} + D \log D)\right)$. However, the variance of both estimators grows exponentially with the polynomial degree $p$, scaling as $3^{p}/D$, where $D$ denotes the sketch dimension. Recent work by Wacker et al. [2023] showed that using complex-valued distribution reduces this dependence to $2^{p}/D$ for the approach of Kar and Karnick [2012]. However, their method relies on dense JL-type projections with computational cost $O(pDd)$ and does not extend to the algorithm of Pham and Pagh [2013].

In this work, we introduce a simple variant of TensorSketch [Pham and Pagh, 2013] that achieves the same variance bound as Wacker et al. [2023], while retaining its advantage of the input-sparsity running time. We validate our results with supporting experiments on synthetic and real-world datasets.

Type

Conference Paper

Department

Mathematics

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