First provide a summary of the paper, and then address the following criteria: Quality, clarity, originality and significance. Summary: The authors re-explain regularization in optimization problems as a constraint of the type the parameters ${\bf w}$ must belong to the convex set $O$ where the convex set O is obtained as the convex hull of all the points of the form $g.v$ where $v$ is some fix vector, $g$ an element from a group and $.$ is a (linear) group action of element $g$ on vector $v$. More concretely, their main contributions are as follows. For example, the ball associated to the L1 norm can be explained as the convex hull of the points obtained by flipping the sign and permuting the components of the vector $(1,0,0,..,0)$; (B) they show that given a seed $v$ and a group action associated to a group $G$, the notion of $w$ is a member of the convex set $O_G(v)$ can be seen as $v$ is smaller than $w$ under a pre-order; (C) they show that if $-v$ belongs to convex set $O$ then $O$ can be seen as the ball of an atomic norm (as defined in Chandra et al.); (D) they show that the L1-sorted norm equals the dual of the norm associated to the signed-pertumation orbitope; (E) they show how to reinterpret the main steps of conditional and projected gradient algorithms in the language of orbitopes and give a procedure to compute projections onto orbitopes. Quality: There are no technical mistakes in the paper.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of non-zero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and low-rank sparse bilinear regression as potential applications.

Based on a new atomic norm, we propose a new convex formulation for sparse matrix factorization problems in which the number of non-zero elements of the factors is assumed fixed and known. The formulation counts sparse PCA with multiple factors, subspace clustering and low-rank sparse bilinear regression as potential applications.

Low rank matrix completion plays a fundamental role in collaborative filtering applications, the key idea being that the variables lie in a smaller subspace than the ambient space. Often, additional information about the variables is known, and it is reasonable to assume that incorporating this information will lead to better predictions. We tackle the problem of matrix completion when pairwise relationships among variables are known, via a graph. We formulate and derive a highly efficient, conjugate gradient based alternating minimization scheme that solves optimizations with over 55 million observations up to 2 orders of magnitude faster than state-of-the-art (stochastic) gradient-descent based methods. On the theoretical front, we show that such methods generalize weighted nuclear norm formulations, and derive statistical consistency guarantees.

For structured estimation problems with atomic norms, recent advances in the literature express sample complexity and estimation error bounds in terms of certain geometric measures, in particular Gaussian width of the unit norm ball, Gaussian width of a spherical cap induced by a tangent cone, and a restricted norm compatibility constant. However, given an atomic norm, bounding these geometric measures can be difficult. In this paper, we present general upper bounds for such geometric measures, which only require simple information of the atomic norm under consideration, and we establish tightness of these bounds by providing the corresponding lower bounds. We show applications of our analysis to certain atomic norms, especially k-support norm, for which existing result is incomplete.

The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2)-orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3)-orbitope. We describe a tractable spectrahedral outer approximation of the SO(3)-orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3)-invariant dictionaries, both on synthetic and on real world data.

Tensor completion is a core machine learning algorithm used in recommender systems and other domains with missing data. While the matrix case is well-understood, theoretical results for tensor problems are limited, particularly when the sampling patterns are deterministic. Here we bound the generalization error of the solutions of two tensor completion methods, Poisson loss and atomic norm minimization, providing tighter bounds in terms of the target tensor rank. If the ground-truth tensor is order $t$ with CP-rank $r$, the dependence on $r$ is improved from $r^{2(t-1)(t^2-t-1)}$ in arXiv:1910.10692 to $r^{2(t-1)(3t-5)}$. The error in our bounds is deterministically controlled by the spectral gap of the sampling sparsity pattern. We also prove several new properties for the atomic tensor norm, reducing the rank dependence from $r^{3t-3}$ in arXiv:1711.04965 to $r^{3t-5}$ under random sampling schemes. A limitation is that atomic norm minimization, while theoretically interesting, leads to inefficient algorithms. However, numerical experiments illustrate the dependence of the reconstruction error on the spectral gap for the practical max-quasinorm, ridge penalty, and Poisson loss minimization algorithms. This view through the spectral gap is a promising window for further study of tensor algorithms.

Joint radar-communications (JRC) has emerged as a promising technology for efficiently using the limited electromagnetic spectrum. In JRC applications such as secure military receivers, often the radar and communications signals are overlaid in the received signal. In these passive listening outposts, the signals and channels of both radar and communications are unknown to the receiver. The ill-posed problem of recovering all signal and channel parameters from the overlaid signal is terms as dual-blind deconvolution (DBD). In this work, we investigate a more challenging version of DBD with a multi-antenna receiver. We model the radar and communications channels with a few (sparse) continuous-valued parameters such as time delays, Doppler velocities, and directions-of-arrival (DoAs). To solve this highly ill-posed DBD, we propose to minimize the sum of multivariate atomic norms (SoMAN) that depends on the unknown parameters. To this end, we devise an exact semidefinite program using theories of positive hyperoctant trigonometric polynomials (PhTP). Our theoretical analyses show that the minimum number of samples and antennas required for perfect recovery is logarithmically dependent on the maximum of the number of radar targets and communications paths rather than their sum. We show that our approach is easily generalized to include several practical issues such as gain/phase errors and additive noise. Numerical experiments show the exact parameter recovery for different JRC