Tensor Hinted Mv Conjectures

Brand, Nanongkai, and Saranurak introduced a conjecture known as the Hinted Mv Conjecture. Although it was originally formulated for the matrix case, we generalize it here to the tensor setting.

💡 Research Summary

The paper “Tensor Hinted Mv Conjectures” extends the Hinted Mv conjecture originally introduced by Brand, Nanongkai, and Saranurak (BNS19) from matrices to higher‑order tensors and investigates the algorithmic consequences of this generalization. The authors first formalize two tensor‑based problem variants, called Type I and Type II, each consisting of three phases: (1) a preprocessing phase that receives a collection of dense matrices and a dense matrix M, (2) a sparsity‑constrained phase that receives either a set of sparse matrices {P₁,…,P_k} (Type I) or a diagonal tensor P with at most n^τ non‑zero entries (Type II), and (3) a query phase that asks for a specific column or entry of the product involving all inputs.

A central technical tool is Lemma 3, the “tensor‑trick”, which shows that the transpose of the tensor product of the sparse matrices multiplied by the tensor product of the V‑matrices can be rewritten as a Hadamard (entry‑wise) product of k ordinary matrix products:
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