jl-face-splitting-product-identity

IN premise — summaries/2026/08/24/wiki-JohnsonE28093Lindenstrauss_lemma-chunk-3.md

Created 2026-08-24T17:11:14+00:00

The face-splitting product satisfies the computational identity (C ∙ D)(x ⊗ y) = Cx ∘ Dy (element-wise Hadamard product), enabling fast application to tensor-structured vectors by decomposing into independent small projections plus a Hadamard product.

Summary

This means that when you need to apply a combined operator to a large tensor-structured vector, you do not have to build and multiply the full giant matrix. Instead, you can split the work into two small, independent operations on the component vectors and then simply multiply the results element by element, which is dramatically faster in practice.