In this paper we consider the problem of recovering a low-rank Tucker approximation to a massive tensor based solely on structured random compressive measurements. Crucially, the proposed random measurement ensembles are both designed to be compactly represented (i.e., low-memory), and can also be efficiently computed in one-pass over the tensor. Thus, the proposed compressive sensing approach may be used to produce a low-rank factorization of a huge tensor that is too large to store in memory with a total memory footprint on the order of the much smaller desired low-rank factorization. In addition, the compressive sensing recovery algorithm itself (which takes the compressive measurements as input, and then outputs a low-rank factorization) also runs in a time which principally depends only on the size of the sought factorization, making its runtime sub-linear in the size of the large tensor one is approximating. Finally, unlike prior works related to (streaming) algorithms for low-rank tensor approximation from such compressive measurements, we present a unified analysis of both Kronecker and Khatri-Rao structured measurement ensembles culminating in error guarantees comparing the error of our recovery algorithm's approximation of the input tensor to the best possible low-rank Tucker approximation error achievable for the tensor by any possible algorithm. We further include an empirical study of the proposed approach that verifies our theoretical findings and explores various trade-offs of parameters of interest.

翻译：本文研究基于结构化随机压缩测量从大规模张量中恢复低秩Tucker逼近的问题。关键在于，所提出的随机测量集成设计既具有紧凑表示（即低内存），又能在单次遍历张量时高效计算。因此，该压缩感知方法可用于生成超大规模张量的低秩分解——该张量因体积过大无法存入内存，而整体内存开销仅相当于远小于原张量的目标低秩分解规模。此外，压缩感知恢复算法（输入压缩测量值并输出低秩分解）的运行时间主要取决于待求分解的规模，使其时间复杂度对于被逼近的大型张量呈亚线性关系。最后，与现有基于压缩测量的低秩张量逼近（流式）算法不同，本文对Kronecker结构与Khatri-Rao结构测量集成进行统一分析，给出了恢复算法逼近输入张量的误差与任何算法对同一张量所能达到的最佳低秩Tucker逼近误差之间的理论保证。进一步通过实证研究验证理论发现，并探讨了相关参数间的权衡。