HOSVD-Based Algorithm for Weighted Tensor Completion

Matrix completion, the problem of completing missing entries in a data matrix with low-dimensional structure (such as rank), has seen many fruitful approaches and analyses. Tensor completion is the tensor analog that attempts to impute missing tensor entries from similar low-rank type assumptions. In this paper, we study the tensor completion problem when the sampling pattern is deterministic and possibly non-uniform. We first propose an efficient weighted Higher Order Singular Value Decomposition (HOSVD) algorithm for the recovery of the underlying low-rank tensor from noisy observations and then derive the error bounds under a properly weighted metric. Additionally, the efficiency and accuracy of our algorithm are both tested using synthetic and real datasets in numerical simulations.

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Cancer drug discovery as a low rank tensor completion problem

10.1101/2021.03.08.434311 ◽

2021 ◽

Author(s):

Vasanth S. Murali ◽

Didem Ağaç Çobanoğlu ◽

Michael Hsieh ◽

Meyer Zinn ◽

Venkat S. Malladi ◽

...

Keyword(s):

Drug Discovery ◽

Ex Vivo ◽

Human Tumor ◽

Low Rank ◽

Cancer Drug ◽

Tensor Completion ◽

Cancer Drug Discovery ◽

Completion Problem ◽

Rank Tensor ◽

Public Data

AbstractThe heterogeneity of cancer necessitates developing a multitude of targeted therapies. We propose the view that cancer drug discovery is a low rank tensor completion problem. We implement this vision by using heterogeneous public data to construct a tensor of drug-target-disease associations. We show the validity of this approach computationally by simulations, and experimentally by testing drug candidates. Specifically, we show that a novel drug candidate, SU11652, controls melanoma tumor growth, including BRAFWT melanoma. Independently, we show that another molecule, TC-E 5008, controls tumor proliferation on ex vivo ER+ human breast cancer. Most importantly, we identify these chemicals with only a few computationally selected experiments as opposed to brute-force screens. The efficiency of our approach enables use of ex vivo human tumor assays as a primary screening tool. We provide a web server, the Cancer Vulnerability Explorer (accessible at https://cavu.biohpc.swmed.edu), to facilitate the use of our methodology.

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Robust Matrix Completion By Exploiting Dynamic Low-Dimensional Structures

10.21203/rs.3.rs-420556/v1 ◽

2021 ◽

Author(s):

Ren Wang ◽

Pengzhi Gao ◽

Meng Wang

Keyword(s):

Matrix Completion ◽

Synthetic Data ◽

Original Data ◽

Low Rank ◽

Temporal Correlations ◽

Completion Problem ◽

Reconstruction Performance ◽

Matrix Completion Problem ◽

Recovery Error ◽

Low Dimensional

Abstract This paper studies the robust matrix completion problem for time-varying models. Leveraging the low-rank property and the temporal information of the data, we develop novel methods to recover the original data from partially observed and corrupted measurements. We show that the reconstruction performance can be improved if one further leverages the information of the sparse corruptions in addition to the temporal correlations among a sequence of matrices. The dynamic robust matrix completion problem is formulated as a nonconvex optimization problem, and the recovery error is quantified analytically and proved to decay in the same order as that of the state-of-the-art method when there is no corruption. A fast iterative algorithm with convergence guarantee to the stationary point is proposed to solve the nonconvex problem. Experiments on synthetic data and real video dataset demonstrate the effectiveness of our method.

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Nonconvex Low-Rank Tensor Completion from Noisy Data

Operations Research ◽

10.1287/opre.2021.2106 ◽

2021 ◽

Author(s):

Changxiao Cai ◽

Gen Li ◽

H. Vincent Poor ◽

Yuxin Chen

Keyword(s):

Large Scale ◽

Linear Time ◽

Practical Interest ◽

Optimal Estimation ◽

Low Rank ◽

Estimation Accuracy ◽

Tensor Completion ◽

Minimal Sample ◽

Completion Problem ◽

Rank Tensor

This paper investigates a problem of broad practical interest, namely, the reconstruction of a large-dimensional low-rank tensor from highly incomplete and randomly corrupted observations of its entries. Although a number of papers have been dedicated to this tensor completion problem, prior algorithms either are computationally too expensive for large-scale applications or come with suboptimal statistical performance. Motivated by this, we propose a fast two-stage nonconvex algorithm—a gradient method following a rough initialization—that achieves the best of both worlds: optimal statistical accuracy and computational efficiency. Specifically, the proposed algorithm provably completes the tensor and retrieves all low-rank factors within nearly linear time, while at the same time enjoying near-optimal statistical guarantees (i.e., minimal sample complexity and optimal estimation accuracy). The insights conveyed through our analysis of nonconvex optimization might have implications for a broader family of tensor reconstruction problems beyond tensor completion.

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