Collective spin operators for symmetric multi-quDit (namely, identical $D$-level atom) systems generate a U$(D)$ symmetry. We explore generalizations to arbitrary $D$ of SU(2)-spin coherent states and their adaptation to parity (multicomponent Schrod
inger cats), together with multi-mode extensions of NOON states. We write level, one- and two-quDit reduced density matrices of symmetric $N$-quDit states, expressed in the last two cases in terms of collective U$(D)$-spin operator expectation values. Then we evaluate level and particle entanglement for symmetric multi-quDit states with linear and von Neumann entropies of the corresponding reduced density matrices. In particular, we analyze the numerical and variational ground state of Lipkin-Meshkov-Glick models of $3$-level identical atoms. We also propose an extension of the concept of SU(2) spin squeezing to SU$(D)$ and relate it to pairwise $D$-level atom entanglement. Squeezing parameters and entanglement entropies are good markers that characterize the different quantum phases, and their corresponding critical points, that take place in these interacting $D$-level atom models.
We address the problem of efficient sparse fixed-rank (S-FR) matrix decomposition, i.e., splitting a corrupted matrix $M$ into an uncorrupted matrix $L$ of rank $r$ and a sparse matrix of outliers $S$. Fixed-rank constraints are usually imposed by th
e physical restrictions of the system under study. Here we propose a method to perform accurate and very efficient S-FR decomposition that is more suitable for large-scale problems than existing approaches. Our method is a grateful combination of geometrical and algebraical techniques, which avoids the bottleneck caused by the Truncated SVD (TSVD). Instead, a polar factorization is used to exploit the manifold structure of fixed-rank problems as the product of two Stiefel and an SPD manifold, leading to a better convergence and stability. Then, closed-form projectors help to speed up each iteration of the method. We introduce a novel and fast projector for the $text{SPD}$ manifold and a proof of its validity. Further acceleration is achieved using a Nystrom scheme. Extensive experiments with synthetic and real data in the context of robust photometric stereo and spectral clustering show that our proposals outperform the state of the art.
In this work, we address the problem of outlier detection for robust motion estimation by using modern sparse-low-rank decompositions, i.e., Robust PCA-like methods, to impose global rank constraints. Robust decompositions have shown to be good at sp
litting a corrupted matrix into an uncorrupted low-rank matrix and a sparse matrix, containing outliers. However, this process only works when matrices have relatively low rank with respect to their ambient space, a property not met in motion estimation problems. As a solution, we propose to exploit the partial information present in the decomposition to decide which matches are outliers. We provide evidences showing that even when it is not possible to recover an uncorrupted low-rank matrix, the resulting information can be exploited for outlier detection. To this end we propose the Robust Decomposition with Constrained Rank (RD-CR), a proximal gradient based method that enforces the rank constraints inherent to motion estimation. We also present a general framework to perform robust estimation for stereo Visual Odometry, based on our RD-CR and a simple but effective compressed optimization method that achieves high performance. Our evaluation on synthetic data and on the KITTI dataset demonstrates the applicability of our approach in complex scenarios and it yields state-of-the-art performance.
