Ankit Parekh

Convex 1-D Total Variation Denoising with Non-convex Regularization

Total variation (TV) denoising is an effective noise suppression method when the derivative of the underlying signal is known to be sparse. TV denoising is defined in terms of a convex optimization problem involving a quadratic data fidelity term and a convex regularization term. A non-convex regularizer can promote sparsity more strongly, but generally leads to a non-convex optimization problem with non-optimal local minima. This letter proposes the use of a non-convex regularizer constrained so that the total objective function to be minimized maintains its convexity. Conditions for a non-convex regularizer are given that ensure the total TV denoising objective function is convex. An efficient algorithm is given for the resulting problem.

Enhanced Low-Rank Matrix Approximation

This letter proposes to estimate low-rank matrices by formulating a convex optimization problem with nonconvex regularization. We employ parameterized nonconvex penalty functions to estimate the nonzero singular values more accurately than the nuclear norm. A closed-form solution for the global optimum of the proposed objective function (sum of data fidelity and the nonconvex regularizer) is also derived. The solution reduces to singular value thresholding method as a special case. The proposed method is demonstrated for image denoising.

Convex Denoising using Non-Convex Tight Frame Regularization

This letter considers the problem of signal denoising using a sparse tight-frame analysis prior. The l1 norm has been extensively used as a regularizer to promote sparsity; however, it tends to under-estimate non-zero values of the underlying signal. To more accurately estimate non-zero values, we propose the use of a non-convex regularizer, chosen so as to ensure convexity of the objective function. The convexity of the objective function is ensured by constraining the parameter of the non-convex penalty. We use ADMM to obtain a solution and show how to guarantee that ADMM converges to the global optimum of the objective function. We illustrate the proposed method for 1D and 2D signal denoising.

Convex fused lasso denoising with non-convex regularization and its use for pulse detection

We propose a convex formulation of the fused lasso signal approximation problem consisting of non-convex penalty functions. The fused lasso signal model aims to estimate a sparse piecewise constant signal from a noisy observation. Originally, the ℓ1 norm was used as a sparsity-inducing convex penalty function for the fused lasso signal approximation problem. However, the ℓ1 norm underestimates signal values. Non-convex sparsity-inducing penalty functions better estimate signal values. In this paper, we show how to ensure the convexity of the fused lasso signal approximation problem with non-convex penalty functions. We further derive a computationally efficient algorithm using the majorization-minimization technique. We apply the proposed fused lasso method for the detection of pulses.

Detection of K-complexes and sleep spindles (DETOKS) using sparse optimization

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 Cohens 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.

Effects of aging on slow-wave sleep dynamics and human spatial navigational memory consolidation.

The consolidation of spatial navigational memory during sleep is supported by electrophysiological and behavioral evidence. The features of sleep that mediate this ability may change with aging, as percentage of slow-wave sleep is canonically thought to decrease with age, and slow waves are thought to help orchestrate hippocampal-neocortical dialog that supports systems level consolidation. In this study, groups of younger and older subjects performed timed trials before and after polysomnographically recorded sleep on a 3D spatial maze navigational task. Although younger subjects performed better than older subjects at baseline, both groups showed similar improvement across presleep trials. However, younger subjects experienced significant improvement in maze performance during sleep that was not observed in older subjects, without differences in morning psychomotor vigilance between groups. Older subjects had sleep quality marked by decreased amount of slow-wave sleep and increased fragmentation of slow-wave sleep, resulting in decreased slow-wave activity. Across all subjects, frontal slow-wave activity was positively correlated with both overnight change in maze performance and medial prefrontal cortical volume, illuminating a potential neuroanatomical substrate for slow-wave activity changes with aging and underscoring the importance of slow-wave activity in sleep-dependent spatial navigational memory consolidation.

Obstructive sleep apnea severity affects amyloid burden in cognitively normal elderly a longitudinal study

Rationale: Recent evidence suggests that obstructive sleep apnea (OSA) may be a risk factor for developing mild cognitive impairment and Alzheimers disease. However, how sleep apnea affects longitudinal risk for Alzheimers disease is less well understood. Objectives: To test the hypothesis that there is an association between severity of OSA and longitudinal increase in amyloid burden in cognitively normal elderly. Methods: Data were derived from a 2‐year prospective longitudinal study that sampled community‐dwelling healthy cognitively normal elderly. Subjects were healthy volunteers between the ages of 55 and 90, were nondepressed, and had a consensus clinical diagnosis of cognitively normal. Cerebrospinal fluid amyloid &bgr; was measured using ELISA. Subjects received Pittsburgh compound B positron emission tomography scans following standardized procedures. Monitoring of OSA was completed using a home sleep recording device. Measurements and Main Results: We found that severity of OSA indices (AHIall [F1,88 = 4.26; P < 0.05] and AHI4% [F1,87 = 4.36; P < 0.05]) were associated with annual rate of change of cerebrospinal fluid amyloid &bgr;42 using linear regression after adjusting for age, sex, body mass index, and apolipoprotein E4 status. AHIall and AHI4% were not associated with increases in ADPiB‐mask (Alzheimers disease vulnerable regions of interest Pittsburg compound B positron emission tomography mask) most likely because of the small sample size, although there was a trend for AHIall (F1,28 = 2.96, P = 0.09; and F1,28 = 2.32, not significant, respectively). Conclusions: In a sample of cognitively normal elderly, OSA was associated with markers of increased amyloid burden over the 2‐year follow‐up. Sleep fragmentation and/or intermittent hypoxia from OSA are likely candidate mechanisms. If confirmed, clinical interventions for OSA may be useful in preventing amyloid build‐up in cognitively normal elderly.

Sleep spindle detection using time-frequency sparsity

This paper proposes an EEG processor for sleep spindle detection algorithms. It non-linearly separates the raw EEG signal into non-oscillatory transient and sustained rhythmic oscillation components using long and short windows for the short-time Fourier transform. The processor utilizes the fact that sleep spindles can be sparsely represented via the inverse of a short-time Fourier transform. Five sleep spindle detectors were tested on the EEG database with and without the proposed EEG processor. We achieved an improvement of 13.3% in the by-sample F1 score, and 13.9% in the by-sample Matthews Correlation Coefficient score of these algorithms when the processed EEG was used for spindle detection. The processor was able to improve the scores by reducing the number of false positive spindles and increasing the number of true positive spindles detected.

Improved sparse low-rank matrix estimation

We consider estimating simultaneously sparse and low-rank matrices from their noisy observations.We use non-convex penalty functions that are parameterized to ensure strict convexity of the overall objective function.An ADMM based algorithm is derived with guaranteed global convergence.Several examples are shown to emphasize the benefit of using the proposed method over convex regularized sparse low-rank matrix estimation method. We address the problem of estimating a sparse low-rank matrix from its noisy observation. We propose an objective function consisting of a data-fidelity term and two parameterized non-convex penalty functions. Further, we show how to set the parameters of the non-convex penalty functions, in order to ensure that the objective function is strictly convex. The proposed objective function better estimates sparse low-rank matrices than a convex method which utilizes the sum of the nuclear norm and the 1 norm. We derive an algorithm (as an instance of ADMM) to solve the proposed problem, and guarantee its convergence provided the scalar augmented Lagrangian parameter is set appropriately. We demonstrate the proposed method for denoising an audio signal and an adjacency matrix representing protein interactions in the Escherichia coli bacteria.

Multichannel sleep spindle detection using sparse low-rank optimization

BACKGROUND Automated single-channel spindle detectors, for human sleep EEG, are blind to the presence of spindles in other recorded channels unlike visual annotation by a human expert. NEW METHOD We propose a multichannel spindle detection method that aims to detect global and local spindle activity in human sleep EEG. Using a non-linear signal model, which assumes the input EEG to be the sum of a transient and an oscillatory component, we propose a multichannel transient separation algorithm. Consecutive overlapping blocks of the multichannel oscillatory component are assumed to be low-rank whereas the transient component is assumed to be piecewise constant with a zero baseline. The estimated oscillatory component is used in conjunction with a bandpass filter and the Teager operator for detecting sleep spindles. RESULTS AND COMPARISON WITH OTHER METHODS The proposed method is applied to two publicly available databases and compared with 7 existing single-channel automated detectors. F1 scores for the proposed spindle detection method averaged 0.66 (0.02) and 0.62 (0.06) for the two databases, respectively. For an overnight 6 channel EEG signal, the proposed algorithm takes about 4min to detect sleep spindles simultaneously across all channels with a single setting of corresponding algorithmic parameters. CONCLUSIONS The proposed method attempts to mimic and utilize, for better spindle detection, a particular human expert behavior where the decision to mark a spindle event may be subconsciously influenced by the presence of a spindle in EEG channels other than the central channel visible on a digital screen.