[Paper Review] Lazy Kronecker Product

Zhao Song|arXiv (Cornell University)|Mar 19, 2026

Complexity and Algorithms in Graphs0 citations

TL;DR

The paper generalizes the lazy update regime from dynamic matrix products to dynamic Kronecker products, presenting an algorithm with specified amortized update and worst-case query times, and establishing conditional lower bounds under the tensor MV conjecture.

ABSTRACT

In this paper, we show how to generalize the lazy update regime from dynamic matrix product [Cohen, Lee, Song STOC 2019, JACM 2021] to dynamic kronecker product. We provide an algorithm that uses $n^{ω( \lceil k/2 ceil, \lfloor k/2 floor, a )-a}$ amortized update time and $ n^{ω( \lceil(k-s)/2 ceil, \lfloor (k-s)/2 floor,a )}$ worst case query time for dynamic kronecker product problem. Unless tensor MV conjecture is false, there is no algorithm that can use both $n^{ω( \lceil k/2 ceil, \lfloor k/2 floor, a )-a-Ω(1)}$ amortized update time, and $ n^{ω( \lceil(k-s)/2 ceil, \lfloor (k-s)/2 floor,a )-Ω(1)}$ worst case query time.

Motivation & Objective

• Motivate and formalize the dynamic Kronecker product problem as a generalization of the dynamic matrix product setting.
• Develop an algorithm achieving specified amortized update and worst-case query times for the dynamic Kronecker product.
• Analyze the computational limits under the Tensor MV conjecture and establish hardness results.
• Offer insights by reducing Tensor Hinted MV to dynamic tensor multiplication to derive conditional lower bounds.

Proposed method

• Define the dynamic Kronecker product problem with rank-1 updates and multi-dimensional queries.
• Maintain a tensor and low-rank components to enable efficient updates and queries.
• Compute the full and low-rank parts of queries using Hadamard products and diagonal/diagonal-like transforms.
• Show how updating after every K iterations yields an amortized update time of n^{omega(ceil(k/2), floor(k/2), a) - a} and a worst-case query time of n^{omega(ceil((k-s)/2), floor((k-s)/2), a)}.
• Use a two-step tensor decomposition (forming B and C) and a final matrix product to bound update costs, leveraging n-omega time for matrix multiplication abstractions.
• Present a hardness theorem by contradiction via a reduction from Tensor Hinted MV to dynamic tensor multiplication.

Experimental results

Research questions

• RQ1 Can dynamic Kronecker product updates be processed with amortized time close to n^{omega(ceil(k/2), floor(k/2), a) - a} and queries in worst-case time close to n^{omega(ceil((k-s)/2), floor((k-s)/2), a)}?
• RQ2 What conditional lower bounds arise from Tensor Hinted MV conjectures for simultaneous improvements in amortized updates and query times?
• RQ3 How does the proposed approach relate to and extend the dynamic matrix product framework?

Key findings

• An algorithm achieving amortized update time O(n^{omega(ceil(k/2), floor(k/2), a) - a}) for dynamic Kronecker products.
• A worst-case query time bound of O(n^{omega(ceil((k-s)/2), floor((k-s)/2), a)}) for outputs of the queried subtensor.
• A reduction-based hardness result showing that, unless the Tensor MV conjecture is false, one cannot attain simultaneous improvements to both amortized update and worst-case query times by more than a constant factor in the exponent.
• Clarifies how output subtensors are formed and how the low-rank and full parts contribute to overall query cost.
• Connects dynamic Kronecker product performance to the broader Tensor MV framework and prior dynamic matrix product results.