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Abstract 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 (i.e., a sketch). 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. 

Temporal text data, such as news articles or Twitter feeds, often comprises a mixture of long-lasting trends and transient topics. Effective topic modeling strategies should detect both types and clearly locate them in time. We first demonstrate that nonnegative CANDECOMP/PARAFAC decomposition (NCPD) can automatically identify topics of variable persistence. We then introduce sparseness-constrained NCPD (S-NCPD) and its online variant to control the duration of the detected topics more effectively and efficiently, along with theoretical analysis of the proposed algorithms. Through an extensive study on both semi-synthetic and real-world datasets, we find that our S-NCPD and its online variant can identify both short- and long-lasting temporal topics in a quantifiable and controlled manner, which traditional topic modeling methods are unable to achieve. Additionally, the online variant of S-NCPD shows a faster reduction in reconstruction error and results in more coherent topics compared to S-NCPD, thus achieving both computational efficiency and quality of the resulting topics. Our findings indicate that S-NCPD and its online variant are effective tools for detecting and controlling the duration of topics in temporal text data, providing valuable insights into both persistent and transient trends.