Nimishakavi, Madhav, Jawanpuria, Pratik, Mishra, Bamdev

One of the popular approaches for low-rank tensor completion is to use the latent trace norm as a low-rank regularizer. However, most of the existing works learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm which helps to learn a non-sparse combination of tensors. We develop a dual framework for solving the problem of latent trace norm regularized low-rank tensor completion. In this framework, we first show a novel characterization of the solution space with a novel factorization, and then, propose two scalable optimization formulations. The problems are shown to lie on a Cartesian product of Riemannian spectrahedron manifolds. We exploit the versatile Riemannian optimization framework for proposing computationally efficient trust-region algorithms. The experiments show the good performance of the proposed algorithms on several real-world data sets in different applications.

Nimishakavi, Madhav, Jawanpuria, Pratik Kumar, Mishra, Bamdev

One of the popular approaches for low-rank tensor completion is to use the latent trace norm regularization. However, most existing works in this direction learn a sparse combination of tensors. In this work, we fill this gap by proposing a variant of the latent trace norm that helps in learning a non-sparse combination of tensors. We develop a dual framework for solving the low-rank tensor completion problem. We first show a novel characterization of the dual solution space with an interesting factorization of the optimal solution. Overall, the optimal solution is shown to lie on a Cartesian product of Riemannian manifolds. Furthermore, we exploit the versatile Riemannian optimization framework for proposing computationally efficient trust region algorithm. The experiments illustrate the efficacy of the proposed algorithm on several real-world datasets across applications.

Nimishakavi, Madhav, Jawanpuria, Pratik Kumar, Mishra, Bamdev

Boumal, Nicolas, Mishra, Bamdev, Absil, P. -A., Sepulchre, Rodolphe

Optimization on manifolds is a rapidly developing branch of nonlinear optimization. Its focus is on problems where the smooth geometry of the search space can be leveraged to design efficient numerical algorithms. In particular, optimization on manifolds is well-suited to deal with rank and orthogonality constraints. Such structured constraints appear pervasively in machine learning applications, including low-rank matrix completion, sensor network localization, camera network registration, independent component analysis, metric learning, dimensionality reduction and so on. The Manopt toolbox, available at www.manopt.org, is a user-friendly, documented piece of software dedicated to simplify experimenting with state of the art Riemannian optimization algorithms. We aim particularly at reaching practitioners outside our field.

Mishra, B., Meyer, G., Sepulchre, R.

This paper addresses the problem of low-rank distance matrix completion. This problem amounts to recover the missing entries of a distance matrix when the dimension of the data embedding space is possibly unknown but small compared to the number of considered data points. The focus is on high-dimensional problems. We recast the considered problem into an optimization problem over the set of low-rank positive semidefinite matrices and propose two efficient algorithms for low-rank distance matrix completion. In addition, we propose a strategy to determine the dimension of the embedding space. The resulting algorithms scale to high-dimensional problems and monotonically converge to a global solution of the problem. Finally, numerical experiments illustrate the good performance of the proposed algorithms on benchmarks.