GNMR: A provable one-line algorithm for low rank matrix recovery

Zilber, Pini; Nadler, Boaz

doi:10.48550/arxiv.2106.12933

Cited by 2 publications

(6 citation statements)

References 25 publications

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“…By combining assumption (7) and Theorem 3.3, the sensing operator A satisfies a min{d 1 , d 2 }-RIP with a constant δ ≤ 1/2. Hence, as in the proof of Lemma 4.2 in [ZN22], the minimal nonzero singular value of…”

Section: D2 Proof Of Proposition 52mentioning

confidence: 72%

“…Let K = ker L (Ut,Vt) . By combining our RIP guarantee (Theorem 3.3) with the second part of Lemma 4.4 in [ZN22] which holds due to our Lemma D.1,…”

Section: D1 a Computationally Efficient Way To Find The Minimal Norm ...mentioning

confidence: 96%

“…Third, we propose a simple Gauss-Newton based method to solve the IMC problem, that is both fast and enjoys strong recovery guarantees. Our algorithm, named GNIMC (Gauss-Newton IMC), is an adaptation of the GNMR algorithm [ZN22] to IMC. At each iteration, GNIMC solves a least squares problem; yet, its periteration complexity is of the same order as gradient descent.…”

Section: Algorithmmentioning

confidence: 99%

“…In this section, we describe an adaptation of the GNMR algorithm [ZN22] to IMC, and present recovery guarantees for it. Consider the factorized objective (2).…”

Section: Gnimc Algorithmmentioning

confidence: 99%

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Inductive Matrix Completion: No Bad Local Minima and a Fast Algorithm

Zilber¹,

Nadler²

2022

Preprint

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The inductive matrix completion (IMC) problem is to recover a low rank matrix from few observed entries while incorporating prior knowledge about its row and column subspaces. In this work, we make three contributions to the IMC problem: (i) we prove that under suitable conditions, the IMC optimization landscape has no bad local minima; (ii) we derive a simple scheme with theoretical guarantees to estimate the rank of the unknown matrix; and (iii) we propose GNIMC, a simple Gauss-Newton based method to solve the IMC problem, analyze its runtime and derive recovery guarantees for it. The guarantees for GNIMC are sharper in several aspects than those available for other methods, including a quadratic convergence rate, fewer required observed entries and stability to errors or deviations from low-rank. Empirically, given entries observed uniformly at random, GNIMC recovers the underlying matrix substantially faster than several competing methods.

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Section: D2 Proof Of Proposition 52mentioning

confidence: 72%

“…Let K = ker L (Ut,Vt) . By combining our RIP guarantee (Theorem 3.3) with the second part of Lemma 4.4 in [ZN22] which holds due to our Lemma D.1,…”

Section: D1 a Computationally Efficient Way To Find The Minimal Norm ...mentioning

confidence: 96%

Section: Algorithmmentioning

confidence: 99%

“…In this section, we describe an adaptation of the GNMR algorithm [ZN22] to IMC, and present recovery guarantees for it. Consider the factorized objective (2).…”

Section: Gnimc Algorithmmentioning

confidence: 99%

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Inductive Matrix Completion: No Bad Local Minima and a Fast Algorithm

Zilber¹,

Nadler²

2022

Preprint

Self Cite

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“…It is still a challenge to recover missing gene expression values more effectively in scRNA-seq data. Previous studies ( Narayanamurthy et al, 2019 ; Nguyen et al, 2019 ; Kummerle and Verdun, 2021 ; Zilber and Nadler, 2021 ) have shown that the low-rank matrix can recover missing values based on a few observable entries due to its low-rank structure. Considering this, we apply low-rank matrix completion to missing value imputation in scRNA-seq data.…”

Section: Introductionmentioning

confidence: 99%

Missing Value Imputation With Low-Rank Matrix Completion in Single-Cell RNA-Seq Data by Considering Cell Heterogeneity

Huang

Ye²,

Li³

et al. 2022

Front. Genet.

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Single-cell RNA-sequencing (scRNA-seq) technologies enable the measurements of gene expressions in individual cells, which is helpful for exploring cancer heterogeneity and precision medicine. However, various technical noises lead to false zero values (missing gene expression values) in scRNA-seq data, termed as dropout events. These zero values complicate the analysis of cell patterns, which affects the high-precision analysis of intra-tumor heterogeneity. Recovering missing gene expression values is still a major obstacle in the scRNA-seq data analysis. In this study, taking the cell heterogeneity into consideration, we develop a novel method, called single cell Gauss–Newton Gene expression Imputation (scGNGI), to impute the scRNA-seq expression matrices by using a low-rank matrix completion. The obtained experimental results on the simulated datasets and real scRNA-seq datasets show that scGNGI can more effectively impute the missing values for scRNA-seq gene expression and improve the down-stream analysis compared to other state-of-the-art methods. Moreover, we show that the proposed method can better preserve gene expression variability among cells. Overall, this study helps explore the complex biological system and precision medicine in scRNA-seq data.

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GNMR: A provable one-line algorithm for low rank matrix recovery

Cited by 2 publications

References 25 publications

Inductive Matrix Completion: No Bad Local Minima and a Fast Algorithm

Inductive Matrix Completion: No Bad Local Minima and a Fast Algorithm

Missing Value Imputation With Low-Rank Matrix Completion in Single-Cell RNA-Seq Data by Considering Cell Heterogeneity

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