Sparse Recovery Analysis of Preconditioned Frames via Convex Optimization

Orthogonal Matching Pursuit and Basis Pursuit are popular reconstruction algorithms for recovery of sparse signals. The exact recovery property of both the methods has a relation with the coherence of the underlying redundant dictionary, i.e. a frame. A frame with low coherence provides better guarantees for exact recovery. An equivalent formulation of the associated linear system is obtained via premultiplication by a non-singular matrix. In view of bounds that guarantee sparse recovery, it is very useful to generate the preconditioner in such way that the preconditioned frame has low coherence as compared to the original. In this paper, we discuss the impact of preconditioning on sparse recovery. Further, we formulate a convex optimization problem for designing the preconditioner that yields a frame with improved coherence. In addition to reducing coherence, we focus on designing well conditioned frames and numerically study the relationship between the condition number of the preconditioner and the coherence of the new frame. Alongside theoretical justifications, we demonstrate through simulations the efficacy of the preconditioner in reducing coherence as well as recovering sparse signals.Comment: 9 pages, 5 Figure

Social Game for Building Energy Efficiency: Utility Learning, Simulation, and Analysis

We describe a social game that we designed for encouraging energy efficient behavior amongst building occupants with the aim of reducing overall energy consumption in the building. Occupants vote for their desired lighting level and win points which are used in a lottery based on how far their vote is from the maximum setting. We assume that the occupants are utility maximizers and that their utility functions capture the tradeoff between winning points and their comfort level. We model the occupants as non-cooperative agents in a continuous game and we characterize their play using the Nash equilibrium concept. Using occupant voting data, we parameterize their utility functions and use a convex optimization problem to estimate the parameters. We simulate the game defined by the estimated utility functions and show that the estimated model for occupant behavior is a good predictor of their actual behavior. In addition, we show that due to the social game, there is a significant reduction in energy consumption

Sufficient conditions for the uniqueness of solution of the weighted norm minimization problem

Prior support constrained compressed sensing, achieved via the weighted norm minimization, has of late become popular due to its potential for applications. For the weighted norm minimization problem, $$min \|x\|_{p,w} \text{ subject to } y=Ax, \; p=0,1, \text{ and } w \in [0,1],$$ uniqueness results are known when $w=0,1$. Here, $\|x\|_{p,w}=w\|x_T\|_p+\|x_{T^c}\|_p, \; p=0,1$ with $T$ representing the partial support information. The work reported in this paper presents the conditions that ensure the uniqueness of the solution of this problem for general $w \in [0,1]$.Comment: