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Molecular dynamic (MD) simulations are used to probe molecular systems in regimes not accessible to physical experiments. A common goal of these simulations is to compute the power spectral density (PSD) of some component of the system such as particle velocity. In certain MD simulations, only a few time locations are observed, which makes it difficult to estimate the autocorrelation and PSD. This work develops a novel nuclear norm minimization-based method for this type of sub-sampled data, based on a parametric representation of the PSD as the sum of Lorentzians. We show results on both synthetic data and a test system of methanol. 

Given a set of 3D to 2D putative matches, labeling the correspondences as inliers or outliers plays a critical role in a wide range of computer vision applications including the Perspective-n-Point (PnP) and object recognition. In this paper, we study a more generalized problem which allows the matches to belong to multiple objects with distinct poses. We propose a deep architecture to simultaneously label the correspondences as inliers or outliers and classify the inliers into multiple objects. Specifically, we discretize the 3D rotation space into twenty convex cones based on the facets of a regular icosahedron. For each facet, a facet classifier is trained to predict the probability of a correspondence being an inlier for a pose whose rotation normal vector points towards this facet. An efficient RANSAC-based post-processing algorithm is also proposed to further process the prediction results and detect the objects. Experiments demonstrate that our method is very efficient compared to existing methods and is capable of simultaneously labeling and classifying the inliers of multiple objects with high precision. 

Abstract We study the ubiquitous super-resolution problem, in which one aims at localizing positive point sources in an image, blurred by the point spread function of the imaging device. To recover the point sources, we propose to solve a convex feasibility program, which simply finds a non-negative Borel measure that agrees with the observations collected by the imaging device. In the absence of imaging noise, we show that solving this convex program uniquely retrieves the point sources, provided that the imaging device collects enough observations. This result holds true if the point spread function of the imaging device can be decomposed into horizontal and vertical components and if the translations of these components form a Chebyshev system, i.e., a system of continuous functions that loosely behave like algebraic polynomials. Building upon the recent results for one-dimensional signals, we prove that this super-resolution algorithm is stable, in the generalized Wasserstein metric, to model mismatch (i.e., when the image is not sparse) and to additive imaging noise. In particular, the recovery error depends on the noise level and how well the image can be approximated with well-separated point sources. As an example, we verify these claims for the important case of a Gaussian point spread function. The proofs rely on the construction of novel interpolating polynomials—which are the main technical contribution of this paper—and partially resolve the question raised in Schiebinger et al. (2017, Inf. Inference, 7, 1–30) about the extension of the standard machinery to higher dimensions. 

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.