Signal Decomposition Using Masked Proximal Operators

We consider the well-studied problem of decomposing a vector time series signal into components with different characteristics, such as smooth, periodic, nonnegative, or sparse. We describe a simple and general framework in which the components are defined by loss functions (which include constraints), and the signal decomposition is carried out by minimizing the sum of losses of the components (subject to the constraints). When each loss function is the negative log-likelihood of a density for the signal component, this framework coincides with maximum a posteriori probability (MAP) estimation; but it also includes many other interesting cases. Summarizing and clarifying prior results, we give two distributed optimization methods for computing the decomposition, which find the optimal decomposition when the component class loss functions are convex, and are good heuristics when they are not. Both methods require only the masked proximal operator of each of the component loss functions, a generalization of the well-known proximal operator that handles missing entries in its argument. Both methods are distributed, i.e., handle each component separately. We derive tractable methods for evaluating the masked proximal operators of some loss functions that, to our knowledge, have not appeared in the literature.Comment: The manuscript has 61 pages, 22 figures and 2 tables. Also hosted at https://web.stanford.edu/~boyd/papers/sig_decomp_mprox.html. For code, see https://github.com/cvxgrp/signal-decompositio

Witnessed entanglement and the geometric measure of quantum discord

We establish relations between geometric quantum discord and entanglement quantifiers obtained by means of optimal witness operators. In particular, we prove a relation between negativity and geometric discord in the Hilbert-Schmidt norm, which is slightly different from a previous conjectured one [Phys. Rev. A 84, 052110 (2011)].We also show that, redefining the geometric discord with the trace norm, better bounds can be obtained. We illustrate our results numerically.Comment: 8 pages + 3 figures. Revised version with erratum for PRA 86, 024302 (2012). Simplified proof that discord is bounded by entanglement in any nor

ShapeFit and ShapeKick for Robust, Scalable Structure from Motion

We introduce a new method for location recovery from pair-wise directions that leverages an efficient convex program that comes with exact recovery guarantees, even in the presence of adversarial outliers. When pairwise directions represent scaled relative positions between pairs of views (estimated for instance with epipolar geometry) our method can be used for location recovery, that is the determination of relative pose up to a single unknown scale. For this task, our method yields performance comparable to the state-of-the-art with an order of magnitude speed-up. Our proposed numerical framework is flexible in that it accommodates other approaches to location recovery and can be used to speed up other methods. These properties are demonstrated by extensively testing against state-of-the-art methods for location recovery on 13 large, irregular collections of images of real scenes in addition to simulated data with ground truth

Multiobjective H_2/H_∞-optimal control via finite dimensional Q-parametrization and linear matrix inequalities

The problem of multiobjective H_2/H_∞ optimal controller design is reviewed. There is as yet no exact solution to this problem. We present a method based on that proposed by Scherer (1995). The problem is formulated as a convex semidefinite program (SDP) using the LMI formulation of the H_2 and H_∞ norms. Suboptimal solutions are computed using finite dimensional Q-parametrization. The objective value of the suboptimal Q's converges to the true optimum as the dimension of and is increased. State space representations are presented which are the analog of those given by Khargonekar and Rotea (1991) for the H_2 case. A simple example computed using finite impulse response Qs is presented

Enhancing Sparsity by Reweighted ℓ(1) Minimization

It is now well understood that (1) it is possible to reconstruct sparse signals exactly from what appear to be highly incomplete sets of linear measurements and (2) that this can be done by constrained ℓ1 minimization. In this paper, we study a novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery. The algorithm consists of solving a sequence of weighted ℓ1-minimization problems where the weights used for the next iteration are computed from the value of the current solution. We present a series of experiments demonstrating the remarkable performance and broad applicability of this algorithm in the areas of sparse signal recovery, statistical estimation, error correction and image processing. Interestingly, superior gains are also achieved when our method is applied to recover signals with assumed near-sparsity in overcomplete representations—not by reweighting the ℓ1 norm of the coefficient sequence as is common, but by reweighting the ℓ1 norm of the transformed object. An immediate consequence is the possibility of highly efficient data acquisition protocols by improving on a technique known as Compressive Sensing

Inevitability of Phase-locking in a Charge Pump Phase Lock Loop using Deductive Verification

Phase-locking in a charge pump (CP) phase lock loop (PLL) is said to be inevitable if all possible states of the CP PLL eventually converge to the equilibrium, where the input and output phases are in lock and the node voltages vanish. We verify this property for a CP PLL using deductive verification. We split this complex property into two sub-properties defined in two disjoint subsets of the state space. We deductively verify the first property using multiple Lyapunov certificates for hybrid systems, and use the Escape certificate for the verification of the second property. Construction of deductive certificates involves positivity check of polynomial inequalities (which is an NP-Hard problem), so we use the sound but incomplete Sum of Squares (SOS) relaxation algorithm to provide a numerical solution