Structured Sparsity Promoting Functions

Shen, Lixin; Suter, Bruce W.; Tripp, Erin E.

doi:10.1007/s10957-019-01565-0

Cited by 20 publications

(26 citation statements)

References 26 publications

(55 reference statements)

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“…This indicates that the operator prox λg is nearly unbiased for large values [7,11], which supports the use of g in applications to replace the ℓ 0 norm. We are not aware of any existing work quantitatively explaining it in this way.…”

Section: The Proximity Operators Of G and Fsupporting

confidence: 59%

“…Figure 1 (b) depicts prox λg for √ λ > ǫ with (λ, ǫ) = (3, 1), corresponding to Proposition 2. In both situations, prox λg (z) = {0} for z in a neighborhood of the origin, thus g is a sparsity promoting function as defined in [11].…”

Section: The Proximity Operators Of G and Fmentioning

confidence: 99%

“…As a consequence, it also implies x (k) > 0 for all k ≥ 0. To show the convergence of the sequence {x (k) }, we compare the values of x (0) and x (1) from (11) in order to infer the monotonicity of the sequence based on Lemma 7. To this end, our discussion is conducted for two cases: (i) z > λ ǫ or z = λ ǫ and √ λ > ǫ; and (ii) z = λ ǫ and √ λ ≤ ǫ.…”

Section: The Proximity Operators Of G and Fmentioning

confidence: 99%

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The Proximity Operator of the Log-Sum Penalty

Prater-Bennette¹,

Shen²,

Tripp³

2021

Preprint

Self Cite

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The log-sum penalty is often adopted as a replacement for the ℓ 0 pseudo-norm in compressive sensing and low-rank optimization. The hard-thresholding operator, i.e., the proximity operator of the ℓ 0 penalty, plays an essential role in applications; similarly, we require an efficient method for evaluating the proximity operator of the log-sum penalty. Due to the nonconvexity of this function, its proximity operator is commonly computed through the iteratively reweighted ℓ 1 method, which replaces the log-sum term with its first-order approximation. This paper reports that the proximity operator of the log-sum penalty actually has an explicit expression. With it, we show that the iteratively reweighted ℓ 1 solution disagrees with the true proximity operator of the log-sum penalty in certain regions. As a by-product, the iteratively reweighted ℓ 1 solution is precisely characterized in terms of the chosen initialization. We also give the explicit form of the proximity operator for the composition of the log-sum penalty with the singular value function, as seen in low-rank applications. These results should be useful in the development of efficient and accurate algorithms for optimization problems involving the log-sum penalty.

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Section: The Proximity Operators Of G and Fsupporting

confidence: 59%

Section: The Proximity Operators Of G and Fmentioning

confidence: 99%

Section: The Proximity Operators Of G and Fmentioning

confidence: 99%

See 1 more Smart Citation

The Proximity Operator of the Log-Sum Penalty

Prater-Bennette¹,

Shen²,

Tripp³

2021

Preprint

Self Cite

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show abstract

“…This CNC approach to sparse regularization has been used in machine fault detection [7,52]. The technique exploits the properties of strongly convex and weakly convex functions [31,46]. The flexibility and effectiveness of the CNC approach depends on the construction of non-trivial (i.e., nonseparable) convex functions.…”

Section: Related Workmentioning

confidence: 99%

Non-convex Total Variation Regularization for Convex Denoising of Signals

et al. 2020

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Total variation (TV) signal denoising is a popular nonlinear filtering method to estimate piecewise constant signals corrupted by additive white Gaussian noise. Following a 'convex non-convex' strategy, recent papers have introduced non-convex regularizers for signal denoising that preserve the convexity of the cost function to be minimized. In this paper, we propose a non-convex TV regularizer, defined using concepts from convex analysis, that unifies, generalizes, and improves upon these regularizers. In particular, we use the generalized Moreau envelope which, unlike the usual Moreau envelope, incorporates a matrix parameter. We describe a novel approach to set the matrix parameter which is essential for realizing the improvement we demonstrate. Additionally, we describe a new set of algorithms for non-convex TV denoising that elucidate the relationship among them and which build upon fast exact algorithms for classical TV denoising. 1 Introduction Piecewise constant signals arise in numerous fields such as physics, biology, and medicine [29]. These signals are often corrupted by additive noise which should be suppressed. Conventional linear time-invariant (LTI) filters

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“…In functional compression, functions themselves can also be exploited. There exist functions with special structures, such as sparsity promoting functions [31], symmetric functions, type sensitive and threshold functions [32]. One can also exploit a function's surjectivity.…”

Section: Related Workmentioning

confidence: 99%

How to Distribute Computation in Networks

Malak¹,

Cohen²,

Médard³

2020

IEEE INFOCOM 2020 - IEEE Conference on Computer Communications

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In network function computation is as a means to reduce the required communication flow in terms of number of bits transmitted per source symbol. However, the rate region for the function computation problem in general topologies is an open problem, and has only been considered under certain restrictive assumptions (e.g. tree networks, linear functions, etc.). In this paper, we propose a new perspective for distributing computation, and formulate a flow-based delay cost minimization problem that jointly captures the costs of communications and computation. We introduce the notion of entropic surjectivity as a measure to determine how sparse the function is and to understand the limits of computation. Exploiting Little's law for stationary systems, we provide a connection between this new notion and the computation processing factor that reflects the proportion of flow that requires communications. This connection gives us an understanding of how much a node (in isolation) should compute to communicate the desired function within the network without putting any assumptions on the topology. Our analysis characterizes the functions only via their entropic surjectivity, and provides insight into how to distribute computation. We numerically test our technique for search, MapReduce, and classification tasks, and infer for each task how sensitive the processing factor to the entropic surjectivity is.

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Structured Sparsity Promoting Functions

Cited by 20 publications

References 26 publications

The Proximity Operator of the Log-Sum Penalty

The Proximity Operator of the Log-Sum Penalty

Non-convex Total Variation Regularization for Convex Denoising of Signals

How to Distribute Computation in Networks

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