Estimation error guarantees for Poisson denoising with sparse and structured dictionary models

Soni, Akshay; Haupt, Jarvis

doi:10.1109/isit.2014.6875184

Cited by 16 publications

(21 citation statements)

References 27 publications

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“…Thus our notion that the denoising problem becomes easier as the rate parameter decreases is intuitive and is consistent with classical analyses. On this note, we briefly mention recent efforts which do not make assumptions on the minimum rate of the underlying Poisson processes; for matrix estimation tasks as here [15], and for sparse vector estimation from Poisson-distributed compressive observations [29].…”

Section: Poisson-distributed Observationsmentioning

confidence: 99%

Minimax Lower Bounds for Noisy Matrix Completion Under Sparse Factor Models

Sambasivan

Haupt

2018

IEEE Trans. Inform. Theory

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This paper examines fundamental error characteristics for a general class of matrix completion problems, where the matrix of interest is a product of two a priori unknown matrices, one of which is sparse, and the observations are noisy. Our main contributions come in the form of minimax lower bounds for the expected per-element squared error for this problem under several common noise models. Specifically, we analyze scenarios where the corruptions are characterized by additive Gaussian noise or additive heavier-tailed (Laplace) noise, Poisson-distributed observations, and highly-quantized (e.g., one-bit) observations, as instances of our general result. Our results establish that the error bounds derived in (Soni et al., 2016) for complexity-regularized maximum likelihood estimators achieve, up to multiplicative constants and logarithmic factors, the minimax error rates in each of these noise scenarios, provided that the nominal number of observations is large enough, and the sparse factor has (on an average) at least one non-zero per column. Index TermsMatrix completion, dictionary learning, minimax lower bounds I. INTRODUCTIONThe matrix completion problem involves imputing the missing values of a matrix from an incomplete, and possibly noisy sampling of its entries. In general, without making any assumption about the entries of the matrix, the matrix completion problem is ill-posed and it is impossible to recover the matrix uniquely. However, if the matrix to be recovered has some intrinsic structure (e.g., low rank structure), it is possible to design algorithms that exactly estimate the missing entries. Indeed, the performance low-rank matrix completion methods have been extensively studied in noiseless settings [1]-[5], in noisy settings where the observations are affected by additive noise [6]-[12], and in settings where the observations are non-linear (e.g., highly-quantized or Poisson distributed observation) functions of the underlying matrix entry (see, [13]-[15]). Recent works which explore robust recovery of low-rank matrices under malicious sparse corruptions include [16]-[19].A notable advantage of using low-rank models is that the estimation strategies involved in completing such matrices can be cast into efficient convex methods which are well-understood and suitable to analyses. The

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Section: Poisson-distributed Observationsmentioning

confidence: 99%

Minimax Lower Bounds for Noisy Matrix Completion Under Sparse Factor Models

Sambasivan

Haupt

2018

IEEE Trans. Inform. Theory

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“…where D * ∈ R n 1 ×r , A * ∈ R r×n 2 and r ≪ min(n1, n2). Slight modifications of this formulation also allows for explicit control and facilitates imposing additional constraints like positivity [5], sparsity [6], or even tree-sparsity [7] on the factors. In terms of the well-known user-item recommendation problem [8]: X * pertains to the rating matrix, D * and A * denote the matrices of user and item latent-factors, respectively.…”

Section: Introductionmentioning

confidence: 99%

Noisy inductive matrix completion under sparse factor models

Soni

Chevalier

Jain

2017

2017 IEEE International Symposium on Information Theory (ISIT)

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Inductive Matrix Completion (IMC) is an important class of matrix completion problems that allows direct inclusion of available features to enhance estimation capabilities. These models have found applications in personalized recommendation systems, multilabel learning, dictionary learning, etc. This paper examines a general class of noisy matrix completion tasks where the underlying matrix is following an IMC model i.e., it is formed by a mixing matrix (a priori unknown) sandwiched between two known feature matrices. The mixing matrix here is assumed to be well approximated by the product of two sparse matrices-referred here to as "sparse factor models." We leverage the main theorem of [1] and extend it to provide theoretical error bounds for the sparsity-regularized maximum likelihood estimators for the class of problems discussed in this paper. The main result is general in the sense that it can be used to derive error bounds for various noise models. In this paper, we instantiate our main result for the case of Gaussian noise and provide corresponding error bounds in terms of squared loss.

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“…This is called as blind compressed sensing, and has been explored in the Gaussian noise case in works such as [8,9,10]. There also exists a substantial body of literature on inference of data-adaptive dictionaries from Poisson-corrupted images, e.g., [11,12,13,14,15]. Besides incorporating a criterion to encourage signal sparsity in the inferred dictionary, these techniques are based on a Poisson maximum likelihood framework.…”

Section: Introduction and Relation To Prior Workmentioning

confidence: 99%

Dictionary learning for Poisson compressed sensing

Patil

Velmurugan

Rajwade

2016

2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

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Imaging techniques involve counting of photons striking a detector. Due to fluctuations in the counting process, the measured photon counts are known to be corrupted by Poisson noise. In this paper, we propose a blind dictionary learning framework for the reconstruction of photographic image data from Poisson corrupted measurements acquired by a compressive camera. We exploit the inherent non-negativity of the data by modeling the dictionary as well as the sparse dictionary coefficients as non-negative entities, and infer these directly from the compressed measurements in a Poisson maximum likelihood framework. We experimentally demonstrate the advantage of this in situ dictionary learning over commonly used sparsifying bases such as DCT or wavelets, especially on color images.Index Terms-Poisson compressed sensing, non-negative sparse coding, dictionary learning

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Estimation error guarantees for Poisson denoising with sparse and structured dictionary models

Cited by 16 publications

References 27 publications

Minimax Lower Bounds for Noisy Matrix Completion Under Sparse Factor Models

Minimax Lower Bounds for Noisy Matrix Completion Under Sparse Factor Models

Noisy inductive matrix completion under sparse factor models

Dictionary learning for Poisson compressed sensing

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