Tensor completion in hierarchical tensor representations

Compressed sensing extends from the recovery of sparse vectors from undersampled measurements via efficient algorithms to the recovery of matrices of low rank from incomplete information. Here we consider a further extension to the reconstruction of tensors of low multi-linear rank in recently introduced hierarchical tensor formats from a small number of measurements. Hierarchical tensors are a flexible generalization of the well-known Tucker representation, which have the advantage that the number of degrees of freedom of a low rank tensor does not scale exponentially with the order of the tensor. While corresponding tensor decompositions can be computed efficiently via successive applications of (matrix) singular value decompositions, some important properties of the singular value decomposition do not extend from the matrix to the tensor case. This results in major computational and theoretical difficulties in designing and analyzing algorithms for low rank tensor recovery. For instance, a canonical analogue of the tensor nuclear norm is NP-hard to compute in general, which is in stark contrast to the matrix case. In this book chapter we consider versions of iterative hard thresholding schemes adapted to hierarchical tensor formats. A variant builds on methods from Riemannian optimization and uses a retraction mapping from the tangent space of the manifold of low rank tensors back to this manifold. We provide first partial convergence results based on a tensor version of the restricted isometry property (TRIP) of the measurement map. Moreover, an estimate of the number of measurements is provided that ensures the TRIP of a given tensor rank with high probability for Gaussian measurement maps.Comment: revised version, to be published in Compressed Sensing and Its Applications (edited by H. Boche, R. Calderbank, G. Kutyniok, J. Vybiral

Low-Rank Representation for Incomplete Data

Low-rank matrix recovery (LRMR) has been becoming an increasingly popular technique for analyzing data with missing entries, gross corruptions, and outliers. As a significant component of LRMR, the model of low-rank representation (LRR) seeks the lowest-rank representation among all samples and it is robust for recovering subspace structures. This paper attempts to solve the problem of LRR with partially observed entries. Firstly, we construct a nonconvex minimization by taking the low rankness, robustness, and incompletion into consideration. Then we employ the technique of augmented Lagrange multipliers to solve the proposed program. Finally, experimental results on synthetic and real-world datasets validate the feasibility and effectiveness of the proposed method

Low-rank tensor completion by Riemannian optimization

In tensor completion, the goal is to fill in missing entries of a partially known tensor under a low-rank constraint. We propose a new algorithm that performs Riemannian optimization techniques on the manifold of tensors of fixed multilinear rank. More specifically, a variant of the nonlinear conjugate gradient method is developed. Paying particular attention to efficient implementation, our algorithm scales linearly in the size of the tensor. Examples with synthetic data demonstrate good recovery even if the vast majority of the entries are unknown. We illustrate the use of the developed algorithm for the recovery of multidimensional images and for the approximation of multivariate functions

Role and task allocation framework for Multi-Robot Collaboration with latent knowledge estimation

In this work a novel framework for modeling role and task allocation in Cooperative Heterogeneous Multi-Robot Systems (CHMRSs) is presented. This framework encodes a CHMRS as a set of multidimensional relational structures (MDRSs). This set of structure defines collaborative tasks through both temporal and spatial relations between processes of heterogeneous robots. These relations are enriched with tensors which allow for geometrical reasoning about collaborative tasks. A learning schema is also proposed in order to derive the components of each MDRS. According to this schema, the components are learnt from data reporting the situated history of the processes executed by the team of robots. Data are organized as a multirobot collaboration treebank (MRCT) in order to support learning. Moreover, a generative approach, based on a probabilistic model, is combined together with nonnegative tensor decomposition (NTD) for both building the tensors and estimating latent knowledge. Preliminary evaluation of the performance of this framework is performed in simulation with three heterogeneous robots, namely, two Unmanned Ground Vehicles (UGVs) and one Unmanned Aerial Vehicle (UAV)