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We study the low-rank phase retrieval problem, where the objective is to recover a sequence of signals (typically images) given the magnitude of linear measurements of those signals. Existing solutions involve recovering a matrix constructed by vectorizing and stacking each image. These solutions model this matrix to be low-rank and leverage the low-rank property to decrease the sample complexity required for accurate recovery. However, when the number of available measurements is more limited, these low-rank matrix models can often fail. We propose an algorithm called Tucker-Structured Phase Retrieval (TSPR) that models the sequence of images as a tensor rather than a matrix that we factorize using the Tucker decomposition. This factorization reduces the number of parameters that need to be estimated, allowing for a more accurate reconstruction. We demonstrate the effectiveness of our approach on real video datasets under several different measurement models.
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Fast and low-memory compressive sensing algorithms for low tucker-rank tensor approximation from streamed measurements
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.
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- Award ID(s):
- 2106472
- PAR ID:
- 10592410
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Numerical Algorithms
- ISSN:
- 1017-1398
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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