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Tensor Hinted Mv Conjectures

Zhao Song

TL;DR

The paper generalizes the Hinted Mv conjecture to tensor inputs, presenting Tensor Hinted Mv in Type I and Type II formulations with structured inputs and tensor operators. It leverages a tensor-trick to convert a k-way interaction into a Hadamard product of per-mode computations, expressed as $ ( \oslash_{j=1}^k P_j )^T ( \oslash_{j=1}^k V_j ) = \odot_{j=1}^k ( P_j^T V_j ) $. The authors propose two phase-based algorithmic strategies, provide polynomial-time considerations for the tensor setting, and state conjectured phase-wise lower bounds that scale with $k$; the $k=1$ case reduces to the original conjecture. A concise algebraic justification shows that the tensor construction satisfies $C = ⊙_{ℓ=1}^k C_{ℓ}$ when $C_{ℓ} = A_{ℓ}^T B_{ℓ}$, supporting the tensor-trick framework.

Abstract

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.

Tensor Hinted Mv Conjectures

TL;DR

The paper generalizes the Hinted Mv conjecture to tensor inputs, presenting Tensor Hinted Mv in Type I and Type II formulations with structured inputs and tensor operators. It leverages a tensor-trick to convert a k-way interaction into a Hadamard product of per-mode computations, expressed as . The authors propose two phase-based algorithmic strategies, provide polynomial-time considerations for the tensor setting, and state conjectured phase-wise lower bounds that scale with ; the case reduces to the original conjecture. A concise algebraic justification shows that the tensor construction satisfies when , supporting the tensor-trick framework.

Abstract

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.
Paper Structure (1 section, 1 theorem)

This paper contains 1 section, 1 theorem.

Table of Contents

  1. Introduction

Key Result

Lemma 3

For each $\ell \in [k]$, we define $A_{\ell} \in \mathbb{R}^{n_{\ell} \times d_a}$. Let $n:=\prod_{\ell=1}^k n_{\ell}$. Let $A := ( \oslash_{\ell=1}^k A_{\ell} ) \in \mathbb{R}^{ n \times d_a}$. Let $B := ( \oslash_{\ell=1}^k B_{\ell} ) \in \mathbb{R}^{ n \times d_b}$. We define $C \in \mathbb{R}^{d

Theorems & Definitions (6)

  • Definition 1: A Tensor Version of Hinted Mv, Type I
  • Conjecture 2
  • Lemma 3
  • proof
  • Definition 4: A Tensor Version of Hinted Mv, Type II
  • Conjecture 5