Rank-Sparsity Incoherence for Matrix Decomposition

Suppose we are given a matrix that is formed by adding an unknown sparse matrix to an unknown low-rank matrix. Our goal is to decompose the given matrix into its sparse and low-rank components. Such a problem arises in a number of applications in model and system identification, and is NP-hard in general. In this paper we consider a convex optimization formulation to splitting the specified matrix into its components, by minimizing a linear combination of the $\ell_1$ norm and the nuclear norm of the components. We develop a notion of \emph{rank-sparsity incoherence}, expressed as an uncertainty principle between the sparsity pattern of a matrix and its row and column spaces, and use it to characterize both fundamental identifiability as well as (deterministic) sufficient conditions for exact recovery. Our analysis is geometric in nature, with the tangent spaces to the algebraic varieties of sparse and low-rank matrices playing a prominent role. When the sparse and low-rank matrices are drawn from certain natural random ensembles, we show that the sufficient conditions for exact recovery are satisfied with high probability. We conclude with simulation results on synthetic matrix decomposition problems

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Kolmogorov Width of Discrete Linear Spaces: an Approach to Matrix Rigidity

A square matrix V is called rigid if every matrix V\u27 obtained by altering a small number of entries of $V$ has sufficiently high rank. While random matrices are rigid with high probability, no explicit constructions of rigid matrices are known to date. Obtaining such explicit matrices would have major implications in computational complexity theory. One approach to establishing rigidity of a matrix V is to come up with a property that is satisfied by any collection of vectors arising from a low-dimensional space, but is not satisfied by the rows of V even after alterations. In this paper we propose such a candidate property that has the potential of establishing rigidity of combinatorial design matrices over the field F_2. Stated informally, we conjecture that under a suitable embedding of F_2^n into R^n, vectors arising from a low dimensional F_2-linear space always have somewhat small Kolmogorov width, i.e., admit a non-trivial simultaneous approximation by a low dimensional Euclidean space. This implies rigidity of combinatorial designs, as their rows do not admit such an approximation even after alterations. Our main technical contribution is a collection of results establishing weaker forms and special cases of the conjecture above

Shadows and intersections: stability and new proofs

We give a short new proof of a version of the Kruskal-Katona theorem due to Lov\'asz. Our method can be extended to a stability result, describing the approximate structure of configurations that are close to being extremal, which answers a question of Mubayi. This in turn leads to another combinatorial proof of a stability theorem for intersecting families, which was originally obtained by Friedgut using spectral techniques and then sharpened by Keevash and Mubayi by means of a purely combinatorial result of Frankl. We also give an algebraic perspective on these problems, giving yet another proof of intersection stability that relies on expansion of a certain Cayley graph of the symmetric group, and an algebraic generalisation of Lov\'asz's theorem that answers a question of Frankl and Tokushige.Comment: 18 page

Aeronautical Engineering: A special bibliography with indexes, supplement 72, July 1976

This bibliography lists 184 reports, articles, and other documents introduced into the NASA scientific and technical information system in June 1976

Design and optical characterization of gallium arsenide aluminum arsenide material system reflective modulators for mid-infrared free space optical applications using solid-source molecular beam epitaxy

With the ever-growing usage of free space optical communication implementations, new innovations are currently being made to help improve the quality of transmission of these systems. One particular method employed to help improve transmission efficiency of optical links is shifting the transmission wavelength into the mid-infrared spectrum. Studies have shown sufficient increase in atmospheric transmission at and around mid-infrared wavelengths (near 3-5 mm). In order to successfully implement such systems at these wavelengths, devices must first be designed that are capable of optical communication operation at such wavelengths. One such device common in modern free space optical systems is the reflective modulator. This device minimizes the pointing and tracking associated with establishing free space optical connections. In this dissertation, a free space optical reflective modulator is designed using Gallium Arsenide and Aluminum Arsenide (GaAs/AlAs) to operate at midinfrared transmission wavelengths. The reflective modulator consists of multiple quantum well modulator (QWM) atop of a distributed Bragg reflector (DBR). The physical device characteristics are analyzed and the device functionality evaluated using optical characterization techniques

Inference And Learning: Computational Difficulty And Efficiency

In this thesis, we mainly investigate two collections of problems: statistical network inference and model selection in regression. The common feature shared by these two types of problems is that they typically exhibit an interesting phenomenon in terms of computational difficulty and efficiency. For statistical network inference, our goal is to infer the network structure based on a noisy observation of the network. Statistically, we model the network as generated from the structural information with the presence of noise, for example, planted submatrix model (for bipartite weighted graph), stochastic block model, and Watts-Strogatz model. As the relative amount of ``signal-to-noise\u27\u27 varies, the problems exhibit different stages of computational difficulty. On the theoretical side, we investigate these stages through characterizing the transition thresholds on the ``signal-to-noise\u27\u27 ratio, for the aforementioned models. On the methodological side, we provide new computationally efficient procedures to reconstruct the network structure for each model. For model selection in regression, our goal is to learn a ``good\u27\u27 model based on a certain model class from the observed data sequences (feature and response pairs), when the model can be misspecified. More concretely, we study two model selection problems: to learn from general classes of functions based on i.i.d. data with minimal assumptions, and to select from the sparse linear model class based on possibly adversarially chosen data in a sequential fashion. We develop new theoretical and algorithmic tools beyond empirical risk minimization to study these problems from a learning theory point of view

Satellite power system: Concept development and evaluation program. Volume 3: Power transmission and reception. Technical summary and assessment

Efforts in the DOE/NASA concept development and evaluation program are discussed for the solar power satellite power transmission and reception system. A technical summary is provided together with a summary of system assessment activities. System options and system definition drivers are described. Major system assessment activities were in support of the reference system definition, solid state system studies, critical technology supporting investigations, and various system and subsystem tradeoffs. These activities are described together with reference system updates and alternative concepts for each of the subsystem areas. Conclusions reached as a result of the numerous analytical and experimental evaluations are presented. Remaining issues for a possible follow-on program are identified