R3MC: A Riemannian three-factor algorithm for low-rank matrix completion

Mishra, Bamdev; Sepulchre, Rodolphe

doi:10.1109/cdc.2014.7039534

Cited by 53 publications

(73 citation statements)

References 21 publications

(60 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This convergence issue has been previously noticed in batch matrix completion algorithms, and several algorithms have been presented which alter the optimization on the Grassmann manifold in order to take into account the non-isotropic scaling of the space by incorporating the singular values into the optimization [9], [10]. These algorithms have demonstrated improved performance on ill-conditioned matrices, but are limited to the batch setting.…”

Section: The Isvd Formulation Of Grousementioning

confidence: 99%

“…Since the seminal results of [6], [7], many algorithms have been developed for low-rank matrix completion [1], [5], [9], [10], [11], [13]. However, the low-dimensional structure found in real data is rarely well-behaved: singular values of large data matrices often drop off in such a way that it is not obvious at what point we are distinguishing signal from noise.…”

Section: Introductionmentioning

confidence: 99%

“…In these scenarios, the suite of existing matrix completion algorithms all struggle to find the true low-rank component, both with regards to achieving low error and with regards to the number of algorithm iterations it takes to get a good result. Recently, several algorithms have been proposed which improve performance for matrices with large condition numbers [10], [9], but these algorithms still have difficulty for extremely illconditioned problems. Furthermore, these algorithms are batch and cannot easily be used for streaming data.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Online completion of Ill-conditioned low-rank matrices

Kennedy

Taylor

Balzano

2014

2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP)

View full text Add to dashboard Cite

Abstract-We consider the problem of online completion of illconditioned low-rank matrices. While many matrix completion algorithms have been proposed recently, they often struggle with ill-conditioned matrices and take a long time to converge. In this paper, we present a new algorithm called Polar Incremental Matrix Completion (PIMC) to address this problem. Our method is based on the GROUSE algorithm, and we show how a polar decomposition can be used to maintain an estimate of the singular value matrix to better deal with ill-conditioned problems. The method is also online, allowing it to be applied to streaming data. We evaluate our algorithm on both synthetic data and a real "structure from motion" dataset from the computer vision community, and show that PIMC outperforms similar methods.

show abstract

Section: The Isvd Formulation Of Grousementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Online completion of Ill-conditioned low-rank matrices

Kennedy

Taylor

Balzano

2014

2014 IEEE Global Conference on Signal and Information Processing (GlobalSIP)

View full text Add to dashboard Cite

show abstract

“…In the RTR algorithm of Mishra et al [194], and in GRIP, the matrix rank is gradually increased until a satisfactory solution is reached, but the meaning of "satisfactory" has to be clari ied. Below we outline a simple "cross-validation" procedure for determining a suitable matrix rank, k, when solving (1.1).…”

Section: The Training Set the Probe Set And Cross-validationmentioning

confidence: 99%

“…The stepsize β ℓ is de ined in the following way: 52) where the sequence {α ℓ } is generated by the rule α 0 = 1 and A Riemannian trust-region algorithm (RTR) for solving (10.38) is proposed in Mishra et al [194]. Here the matrix of unknowns is factorized in the form X = U SV T where U ∈ R m×k and V ∈ R n×k have orthonormal columns and S ∈ R k×k is a symmetric positive semide inite matrix.…”

Section: Regularized Nuclear Norm Problems: Soft-impute Fpc Apgl Amentioning

confidence: 99%

Imputing Missing Entries of a Data Matrix: A Review

Dax¹

2014

JAC

View full text Add to dashboard Cite

The problem of imputing missing entries of a data matrix is easy to state: Some entries of the matrix are unknown and we want to assign "appropriate values" to these entries. The need for solving such problems arises in several applications, ranging from traditional ields to modern ones. Typical examples of traditional ields are statistical analysis of incomplete survey data, business reports, operations management, psychometrika, meteorology and hydrology. Modern applications arise in machine learning, data mining, DNA microarrays data, chemometrics, computer vision, recommender systems and collaborative iltering. The problem is highly interesting and challenging. Many ingenious algorithms have been proposed, and there is vast literature on imputing techniques. Yet, most of the papers consider the imputing problem within the context of a speci ic application or a speci ic approach. The current survey provides a broad view of the problem, one that exposes the large variety of imputing methods. Old and new methods are examined with focus on the ideas behind the methods. It is illustrated how simple ideas evolve into highly sophisticated algorithms. The review enables us to identify a number of basic concepts and tools which are shared by several imputing methods. Equivalence theorems that we prove reveal surprising relations between apparently different methods.

show abstract

Sub‐sampled adaptive trust region method on Riemannian manifolds

Zhao,

Yan,

Zhu

2024

Numerical Linear Algebra App

View full text Add to dashboard Cite

We consider the problem of large‐scale finite‐sum minimization on Riemannian manifold. We develop a sub‐sampled adaptive trust region method on Riemannian manifolds. Based on inexact information, we adopt adaptive techniques to flexibly adjust the trust region radius in our method. We present the iteration complexity is when the algorithm attains an ‐second‐order stationary point, which matches the result on trust region method. Numerical results for PCA on Grassmann manifold and low‐rank matrix completion are reported to demonstrate the effectiveness of the proposed Riemannian method.

show abstract

R3MC: A Riemannian three-factor algorithm for low-rank matrix completion

Cited by 53 publications

References 21 publications

Online completion of Ill-conditioned low-rank matrices

Online completion of Ill-conditioned low-rank matrices

Imputing Missing Entries of a Data Matrix: A Review

Sub‐sampled adaptive trust region method on Riemannian manifolds

Contact Info

Product

Resources

About