This paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing, e.g. sparse deconvolution. This paper aims to induce sparsity more strongly than L1 norm regularization, while avoiding non-convex optimization. For this purpose, this paper describes the design and use of non-convex penalty functions (regularizers) constrained so as to ensure the convexity of the total cost function, F, to be minimized. The method is based on parametric penalty functions, the parameters of which are constrained to ensure convexity of F. It is shown that optimal parameters can be obtained by semidefinite programming (SDP). This maximally sparse convex (MSC) approach yields maximally non-convex sparsity-inducing penalty functions constrained such that the total cost function, F, is convex. It is demonstrated that iterative MSC (IMSC) can yield solutions substantially more sparse than the standard convex sparsity-inducing approach, i.e., L1 norm minimization.

Abstract—This paper addresses the suppression of transient artifacts in signals, e.g., biomedical time series. To that end, we dis-tinguish two types of artifact signals. We define “Type 1 ” artifacts as spikes and sharp, brief waves that adhere to a baseline value of zero. We define “Type 2 ” artifacts as comprising approximate step discontinuities. We model a Type 1 artifact as being sparse and having a sparse time-derivative, and a Type 2 artifact as having a sparse time-derivative. We model the observed time series as the sum of a low-pass signal (e.g., a background trend), an artifact signal of each type, and a white Gaussian stochastic process. To jointly estimate the components of the signal model, we formulate a sparse optimization problem and develop a rapidly converging, computationally efficient iterative algorithm denoted TARA (“transient artifact reduction algorithm”). The effectiveness of the approach is illustrated using near infrared spectroscopic time-series data. Index Terms—Measurement artifact, artifact rejection, sparse optimization, wavelet, low-pass filter, total variation, lasso, fused lasso. I.

This paper addresses the problem of smoothing data with ad-ditive step discontinuities. The problem formulation is based on least square polynomial approximation and total variation denoising. In earlier work, an ADMM algorithm was proposed to minimize a suitably defined sparsity-promoting cost function. In this paper, an algorithm is derived using the majorization-minimization optimiza-tion procedure. The new algorithm converges faster and, unlike the ADMM algorithm, has no parameters that need to be set. The pro-posed algorithm is formulated so as to utilize fast solvers for banded systems for high computational efficiency. This paper also gives op-timality conditions so that the optimality of a result produced by the numerical algorithm can be readily validated. 1.

Background- This paper addresses the problem of detecting sleep spindles and K-complexes in human sleep EEG. Sleep spindles and K-complexes aid in classifying stage 2 NREM human sleep. New Method- We propose a non-linear model for the EEG, consisting of a transient, low-frequency, and an oscillatory component. The transient component captures the non-oscillatory transients in the EEG. The oscillatory component admits a sparse time-frequency representation. Using a convex objective function, this paper presents a fast non-linear optimization algorithm to estimate the components in the proposed signal model. The low-frequency and oscillatory components are used to detect K-complexes and sleep spindles respectively. Results and comparison with other methods- The performance of the proposed method is evaluated using an online EEG database. The F1 scores for the spindle detection averaged 0.70 ± 0.03 and the F1 scores for the K-complex detection averaged 0.57 ± 0.02. The Matthews Correlation Coefficient and Cohen’s Kappa values were in a range similar to the F1 scores for both the sleep spindle and K-complex detection. The F1 scores for the proposed method are higher than existing detection algorithms. Conclusions- Comparable run-times and better detection results than traditional detection algorithms suggests that the proposed method is promising for the practical detection of sleep spindles and K-complexes.

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This paper jointly addresses the problems of chromatogram baseline correction and noise reduction. The proposed approach is based on modeling the series of chromatogram peaks as sparse with sparse derivatives, and on modeling the baseline as a low-pass signal. A convex optimization problem is formulated so as to encapsulate these non-parametric models. To account for the positivity of chromatogram peaks, an asymmetric penalty function is utilized. A robust, computationally efficient, iterative algorithm is developed that is guaranteed to converge to the unique optimal solution. The approach, termed Baseline Estimation And Denoising with Sparsity (BEADS), is evaluated and compared with two state-of-the-art methods using both simulated and real chromatogram data.