We introduce a new approach for applying sampling-based sketches to two and three mode tensors. We illustrate our technique to construct sketches for the classical problems of ₀ sampling and producing ₁ embeddings. In both settings we achieve sketches that can be applied to a rank one tensor in (Rᵈ)⊗ q (for $q=2,3$) in time scaling with d rather than d² or d³. Our main idea is a particular sampling construction based on fast convolution which allows us to quickly compute sums over sufficiently random subsets of tensor entries.
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Swartworth et al. (2024) studied this question.
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