Data recovery on a manifold from linear samples: theory and computation

Cai, Jian-Feng; Rong, Yi; Wang, Yang; Xu, Zhiqiang

doi:10.4310/amsa.2018.v3.n1.a11

Cited by 3 publications

(1 citation statement)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Evidently as long as A is injective on the set of matrices of rank at most r, X will be the unique solution to (3). Indeed, it has been shown that if A consists of m ě 4nr´4r 2 generic measurement matrices, then A will be injective; see [14,106] for more details. Despite this, the rank minimization problem is known to be NP-hard and computationally intractable since it is an extension of the 0 -minimization problem in compressed sensing [42,23].…”

Section: Convex Approach: Nuclear Norm Minimizationmentioning

confidence: 99%

Exploiting the Structure Effectively and Efficiently in Low-Rank Matrix Recovery

Cai

2018

Handbook of Numerical Analysis

Self Cite

View full text Add to dashboard Cite

Low rank model arises from a wide range of applications, including machine learning, signal processing, computer algebra, computer vision, and imaging science. Low rank matrix recovery is about reconstructing a low rank matrix from incomplete measurements. In this survey we review recent developments on low rank matrix recovery, focusing on three typical scenarios: matrix sensing, matrix completion and phase retrieval. An overview of effective and efficient approaches for the problem is given, including nuclear norm minimization, projected gradient descent based on matrix factorization, and Riemannian optimization based on the embedded manifold of low rank matrices. Numerical recipes of different approaches are emphasized while accompanied by the corresponding theoretical recovery guarantees. This is an ill-posed problem without assuming any structure on X since there are more unknowns than equations. However, noticing that the number of degrees of freedom in an n by n rank r matrix is p2n´rqr [100] which can be much smaller than n 2 provided r is small, it is reasonable to expect to reconstruct a low rank matrix from fewer than n 2 measurements.Authors are listed alphabetically.

show abstract

Section: Convex Approach: Nuclear Norm Minimizationmentioning

confidence: 99%

Exploiting the Structure Effectively and Efficiently in Low-Rank Matrix Recovery

Cai

2018

Handbook of Numerical Analysis

Self Cite

View full text Add to dashboard Cite

show abstract

Homomorphic sensing of subspace arrangements

Peng

Tsakiris

2021

Applied and Computational Harmonic Analysis

View full text Add to dashboard Cite

Homomorphic Sensing of Subspace Arrangements

Peng

Tsakiris

2020

Preprint

View full text Add to dashboard Cite

Homomorphic sensing is a recent algebraic-geometric framework that studies the unique recovery of points in a linear subspace from their images under a given collection of linear transformations. It has been successful in interpreting such a recovery in the case of permutations composed by coordinate projections, an important instance in applications known as unlabeled sensing, which models data that are out of order and have missing values. In this paper we make several fundamental contributions. First, we extend the homomorphic sensing framework from a single subspace to a subspace arrangement. Second, when specialized to a single subspace the new conditions are simpler and tighter. Third, as a natural consequence of our main theorem we obtain in a unified way recovery conditions for real phase retrieval, typically known via diverse techniques in the literature, as well as novel conditions for sparse and unsigned versions of linear regression without correspondences and unlabeled sensing. Finally, we prove that the homomorphic sensing property is locally stable to noise.

show abstract

Data recovery on a manifold from linear samples: theory and computation

Cited by 3 publications

References 30 publications

Exploiting the Structure Effectively and Efficiently in Low-Rank Matrix Recovery

Exploiting the Structure Effectively and Efficiently in Low-Rank Matrix Recovery

Homomorphic sensing of subspace arrangements

Homomorphic Sensing of Subspace Arrangements

Contact Info

Product

Resources

About