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Abstract One of the classical approaches for estimating the frequencies and damping factors in a spectrally sparse signal is the MUltiple SIgnal Classification (MUSIC) algorithm, which exploits the low-rank structure of an autocorrelation matrix. Low-rank matrices have also received considerable attention recently in the context of optimization algorithms with partial observations, and nuclear norm minimization (NNM) has been widely used as a popular heuristic of rank minimization for low-rank matrix recovery problems. On the other hand, it has been shown that NNM can be viewed as a special case of atomic norm minimization (ANM), which has achieved great success in solving line spectrum estimation problems. However, as far as we know, the general ANM (not NNM) considered in many existing works can only handle frequency estimation in undamped sinusoids. In this work, we aim to fill this gap and deal with damped spectrally sparse signal recovery problems. In particular, inspired by the dual analysis used in ANM, we offer a novel optimization-based perspective on the classical MUSIC algorithm and propose an algorithm for spectral estimation that involves searching for the peaks of the dual polynomial corresponding to a certain NNM problem, and we show that this algorithm is in fact equivalent to MUSIC itself. Building on this connection, we also extend the classical MUSIC algorithm to the missing data case. We provide exact recovery guarantees for our proposed algorithms and quantify how the sample complexity depends on the true spectral parameters. In particular, we provide a parameter-specific recovery bound for low-rank matrix recovery of jointly sparse signals rather than use certain incoherence properties as in existing literature. Simulation results also indicate that the proposed algorithms significantly outperform some relevant existing methods (e.g., ANM) in frequency estimation of damped exponentials. 

Imaging photoplethysmography (iPPG) could greatly improve driver safety systems by enabling capabilities ranging from identifying driver fatigue to unobtrusive early heart failure detection. Unfortunately, the driving context poses unique challenges to iPPG, including illumination and motion. First, drastic illumination variations present during driving can overwhelm the small intensity-based iPPG signals. Second, significant driver head motion during driving, as well as camera motion (e.g., vibration) make it challenging to recover iPPG signals. To address these two challenges, we present two innovations. First, we demonstrate that we can reduce most outside light variations using narrow-band near-infrared (NIR) video recordings and obtain reliable heart rate estimates. Second, we present a novel optimization algorithm, which we call AutoSparsePPG, that leverages the quasi-periodicity of iPPG signals and achieves better performance than the state-of-the-art methods. In addition, we release the first publicly available driving dataset that contains both NIR and RGB video recordings of a passenger's face with simultaneous ground truth pulse oximeter recordings.