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  1. This work considers a super-resolution framework for overcomplete tensor decomposition. Specifically, we view tensor decomposition as a super-resolution problem of recovering a sum of Dirac measures on the sphere and solve it by minimizing a continuous analog of the ℓ1 norm on the space of measures. The optimal value of this optimization defines the tensor nuclear norm. Similar to the separation condition in the super-resolution problem, by explicitly constructing a dual certificate, we develop incoherence conditions of the tensor factors so that they form the unique optimal solution of the continuous analog of ℓ1 norm minimization. Remarkably, the derived incoherence conditions are satisfied with high probability by random tensor factors uniformly distributed on the sphere, implying global identifiability of random tensor factors. 
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    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. 
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    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. 
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