This work studies the combinatorial optimization problem of finding an optimal core tensor shape, also called multilinear rank, for a size-constrained Tucker decomposition. We give an algorithm with provable approximation guarantees for its reconstruction error via connections to higher-order singular values. Specifically, we introduce a novel Tucker packing problem, which we prove is NP-hard, and give a polynomial-time approximation scheme based on a reduction to the 2-dimensional knapsack problem with a matroid constraint. We also generalize our techniques to tree tensor network decompositions. We implement our algorithm using an integer programming solver, and show that its solution quality is competitive with (and sometimes better than) the greedy algorithm that uses the true Tucker decomposition loss at each step, while also running up to 1000x faster.

翻译：本文研究在大小约束下寻找最优核心张量形状（也称多线性秩）的组合优化问题，用于Tucker分解。我们提出一种基于高阶奇异值关联的算法，能够为该问题的重构误差提供可证明的近似保证。具体而言，我们引入一种新型的Tucker打包问题，证明其为NP难问题，并给出基于拟阵约束下二维背包问题归约的多项式时间近似方案。同时，我们将技术推广至树张量网络分解。通过整数规划求解器实现该算法后，我们证明其解的质量与每步使用真实Tucker分解损失的贪心算法相当（有时更优），且运行速度提升高达1000倍。