A characterization of deterministic sampling patterns for low-rank matrix completion

Pimentel-Alarcón, Daniel L.; Boston, Nigel; Nowak, Robert

doi:10.1109/allerton.2015.7447128

Cited by 29 publications

(115 citation statements)

References 9 publications

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“…Unfortunately none of these algorithms succeeded at this task. This supports theoretical results, which show that even when non-adaptive LRMC is theoretically possible using only the minimum required r + 1 samples per column, this may be computationally prohibitive [24].…”

Section: Methodssupporting

confidence: 89%

“…In addition, Algorithm 1 operates with as little as r + 1 samples per column (the minimum required, as X is rank-r, so observing columns with at least r + 1 entries is necessary for completion [24]). Therefore, Algorithm 1 works even on the minimal sampling regime.…”

Section: What We Gain By Being Adaptivementioning

confidence: 99%

“…It is known that O(log d) random samples per column are necessary to guarantee that LRMC is possible, but completing a matrix with such few random samples may be computationally prohibitive, as it may require solving a complex polynomial system of equations [24].…”

Section: What We Gain By Being Adaptivementioning

confidence: 99%

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Adaptive strategy for restricted-sampling noisy low-rank matrix completion

Pimentel-Alarcón

Nowak

2015

2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

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In this paper we propose a novel adaptive algorithm that provably performs low-rank matrix completion (LRMC) from restricted sets of observations, under ideal or noisy measurements, in lieu of coherence assumptions, with minimal sampling rates and optimal computational complexity. We discuss the main advantages of the adaptive setting of LRMC, and complement our theoretical analysis with experiments, illustrating the effectiveness of our algorithm.

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Section: Methodssupporting

confidence: 89%

Section: What We Gain By Being Adaptivementioning

confidence: 99%

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Adaptive strategy for restricted-sampling noisy low-rank matrix completion

Pimentel-Alarcón

Nowak

2015

2015 IEEE 6th International Workshop on Computational Advances in Multi-Sensor Adaptive Processing (CAMSAP)

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“…, x G in X . In other words, given a dataset of N incomplete measurements {A gi x i } N i=1 , it is possible to build X by trying all the possible combinations of G samples 5 and keeping only the points v which are the solutions of (14).…”

Section: Uniqueness Of Low-dimensional Modelsmentioning

confidence: 99%

“…Proof. If R Ag = R A g for all g = g , then the system in (14) will have multiple solutions for any choice of x 1 , . .…”

Section: Uniqueness Of Low-dimensional Modelsmentioning

confidence: 99%

Unsupervised Learning From Incomplete Measurements for Inverse Problems

Tachella¹,

Chen²,

Davies³

2022

Preprint

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In many real-world settings, only incomplete measurement data are available which can pose a problem for learning. Unsupervised learning of the signal model using a fixed incomplete measurement process is impossible in general, as there is no information in the nullspace of the measurement operator. This limitation can be overcome by using measurements from multiple operators. While this idea has been successfully applied in various applications, a precise characterization of the conditions for learning is still lacking. In this paper, we fill this gap by presenting necessary and sufficient conditions for learning the signal model which indicate the interplay between the number of distinct measurement operators G, the number of measurements per operator m, the dimension of the model k and the dimension of the signals n.In particular, we show that generically unsupervised learning is possible if each operator obtains at least m > k + n/G measurements. Our results are agnostic of the learning algorithm and have implications in a wide range of practical algorithms, from low-rank matrix recovery to deep neural networks.

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GLIMPS: A Greedy Mixed-Integer Approach for Super Robust Matched Subspace Detection

Rahman

Pimentel-Alarcón

2019

2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)

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Due to diverse nature of data acquisition and modern applications, many contemporary problems involve high dimensional datum x ∈ R d whose entries often lie in a union of subspaces and the goal is to find out which entries of x match with a particular subspace U, classically called matched subspace detection. Consequently, entries that match with one subspace are considered as inliers w.r.t the subspace while all other entries are considered as outliers. Proportion of outliers relative to each subspace varies based on the degree of coordinates from subspaces. This problem is a combinatorial NP-hard in nature and has been immensely studied in recent years. Existing approaches can solve the problem when outliers are sparse. However, if outliers are abundant or in other words if x contains coordinates from a fair amount of subspaces, this problem can't be solved with acceptable accuracy or within a reasonable amount of time. This paper proposes a twostage approach called Greedy Linear Integer Mixed Programmed Selector (GLIMPS) for this abundant-outliers setting, which combines a greedy algorithm and mixed integer formulation and can tolerate over 80% outliers, outperforming the state-of-the-art.

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A characterization of deterministic sampling patterns for low-rank matrix completion

Cited by 29 publications

References 9 publications

Adaptive strategy for restricted-sampling noisy low-rank matrix completion

Adaptive strategy for restricted-sampling noisy low-rank matrix completion

Unsupervised Learning From Incomplete Measurements for Inverse Problems

GLIMPS: A Greedy Mixed-Integer Approach for Super Robust Matched Subspace Detection

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