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Discriminative features extracted from the sparse coding model have been shown to perform well for classification. Recent deep learning architectures have further improved reconstruction in inverse problems by considering new dense priors learned from data. We propose a novel dense and sparse coding model that integrates both representation capability and discriminative features. The model studies the problem of recovering a dense vector x and a sparse vector u given measurement of the form y = Ax+Bu. Our first analysis relies on a geometric condition, specifically the minimal angle between the spanning subspaces of matrices A and B, which ensures a unique solution to the model. The second analysis shows that, under some conditions on A and B, a convex program recovers the dense and sparse components. We validate the effectiveness of the model on simulated data and propose a dense and sparse autoencoder (DenSaE) tailored to learning the dictionaries from the dense and sparse model. We demonstrate that (i) DenSaE denoises natural images better than architectures derived from the sparse coding model (Bu), (ii) in the presence of noise, training the biases in the latter amounts to implicitly learning the Ax + Bu model, (iii) A and B capture low- and high-frequency contents, respectively, and (iv) compared to the sparse coding model, DenSaE offers a balance between discriminative power and representation. 

The identification of interesting substructures within jets is an important tool for searching for new physics and probing the Standard Model at colliders. Many of these substructure tools have previously been shown to take the form of optimal transport problems, in particular the Energy Mover’s Distance (EMD). In this work, we show that the EMD is in fact the natural structure for comparing collider events, which accounts for its recent success in understanding event and jet substructure. We then present a Shape Hunting Algorithm using Parameterized Energy Reconstruction (Shaper), which is a general framework for defining and computing shape-based observables. Shaper generalizes N-jettiness from point clusters to any extended, parametrizable shape. This is accomplished by efficiently minimizing the EMD between events and parameterized manifolds of energy flows representing idealized shapes, implemented using the dual-potential Sinkhorn approximation of the Wasserstein metric. We show how the geometric language of observables as manifolds can be used to define novel observables with built-in infrared-and-collinear safety. We demonstrate the efficacy of the Shaper framework by performing empirical jet substructure studies using several examples of new shape-based observables. 

We propose K-Deep Simplex (KDS) which, given a set of data points, learns a dictionary comprising synthetic landmarks, along with representation coefficients supported on a simplex. KDS employs a local weighted ℓ1 penalty that encourages each data point to represent itself as a convex combination of nearby landmarks. We solve the proposed optimization program using alternating minimization and design an efficient, interpretable autoencoder using algorithm unrolling. We theoretically analyze the proposed program by relating the weighted ℓ1 penalty in KDS to a weighted ℓ0 program. Assuming that the data are generated from a Delaunay triangulation, we prove the equivalence of the weighted ℓ1 and weighted ℓ0 programs. We further show the stability of the representation coefficients under mild geometrical assumptions. If the representation coefficients are fixed, we prove that the sub-problem of minimizing over the dictionary yields a unique solution. Further, we show that low-dimensional representations can be efficiently obtained from the covariance of the coefficient matrix. Experiments show that the algorithm is highly efficient and performs competitively on synthetic and real data sets.