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A Hamilton-Jacobi-based proximal operator.

ABSTRACT: First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are known for only limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton-Jacobi (HJ) equations, heat equations, and Monte Carlo sampling. In particular, HJ-Prox smoothly approximates the Moreau envelope and its gradient. The smoothness can be adjusted to act as a denoiser. Our approach applies even when functions are accessible only by (possibly noisy) black box samples. We show that HJ-Prox is effective numerically via several examples.

SUBMITTER: Osher S

PROVIDER: S-EPMC10083605 | biostudies-literature | 2023 Apr

REPOSITORIES: biostudies-literature

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A Hamilton-Jacobi-based proximal operator.

Osher Stanley S Heaton Howard H Wu Fung Samy S

Proceedings of the National Academy of Sciences of the United States of America 20230329 14

First-order optimization algorithms are widely used today. Two standard building blocks in these algorithms are proximal operators (proximals) and gradients. Although gradients can be computed for a wide array of functions, explicit proximal formulas are known for only limited classes of functions. We provide an algorithm, HJ-Prox, for accurately approximating such proximals. This is derived from a collection of relations between proximals, Moreau envelopes, Hamilton-Jacobi (HJ) equations, heat ...[more]

PMID: 36989305

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Project description:IntroductionUnobtrusive metrics that can auto-assess performance during clinical procedures are of value. Three approaches to deriving wearable technology-based metrics are explored: (1) eye tracking, (2) psychophysiological measurements [e.g. electrodermal activity (EDA)] and (3) arm and hand movement via accelerometry. We also measure attentional capacity by tasking the operator with an additional task to track an unrelated object during the procedure.MethodsTwo aspects of performance are measured: (1) using eye gaze and psychophysiology metrics and (2) measuring attentional capacity via an additional unrelated task (to monitor a visual stimulus/playing cards). The aim was to identify metrics that can be used to automatically discriminate between levels of performance or at least between novices and experts. The study was conducted using two groups: (1) novice operators and (2) expert operators. Both groups made two attempts at a coronary angiography procedure using a full-physics virtual reality simulator. Participants wore eye tracking glasses and an E4 wearable wristband. Areas of interest were defined to track visual attention on display screens, including: (1) X-ray, (2) vital signs, (3) instruments and (4) the stimulus screen (for measuring attentional capacity).ResultsExperts provided greater dwell time (63% vs 42%, p = 0.03) and fixations (50% vs 34%, p = 0.04) on display screens. They also provided greater dwell time (11% vs 5%, p = 0.006) and fixations (9% vs 4%, p = 0.007) when selecting instruments. The experts' performance for tracking the unrelated object during the visual stimulus task negatively correlated with total errors (r = - 0.95, p = 0.0009). Experts also had a higher standard deviation of EDA (2.52 µS vs 0.89 µS, p = 0.04).ConclusionsEye tracking metrics may help discriminate between a novice and expert operator, by showing that experts maintain greater visual attention on the display screens. In addition, the visual stimulus study shows that an unrelated task can measure attentional capacity. Trial registration This work is registered through clinicaltrials.gov, a service of the U.S. National Health Institute, and is identified by the trial reference: NCT02928796.

Dataset Information

A Hamilton-Jacobi-based proximal operator.

Publications

A Hamilton-Jacobi-based proximal operator.

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