Discussion: Latent variable graphical model selection via convex optimization

Discussion of "Latent variable graphical model selection via convex optimization" by Venkat Chandrasekaran, Pablo A. Parrilo and Alan S. Willsky [arXiv:1008.1290].Comment: Published in at http://dx.doi.org/10.1214/12-AOS981 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

The Spiritual Senses in Western Spirituality and the Analytic Philosophy of Religion

The doctrine of the spiritual senses has played a significant role in the history of Roman Catholic and Eastern Orthodox spirituality. What has been largely unremarked is that the doctrine also played a significant role in classical Protestant thought, and that analogous concepts can be found in Indian theism. In spite of the doctrine’s significance, however, the only analytic philosopher to consider it has been Nelson Pike. I will argue that his treatment is inadequate, show how the development of the doctrine in Puritan thought and spirituality fills a serious lacuna in Pike’s treatment, and conclude with some suggestions as to where the discussion should go nex

Sharp thresholds for high-dimensional and noisy recovery of sparsity

The problem of consistently estimating the sparsity pattern of a vector \betastar \in \real^\mdim based on observations contaminated by noise arises in various contexts, including subset selection in regression, structure estimation in graphical models, sparse approximation, and signal denoising. We analyze the behavior of $\ell_1$-constrained quadratic programming (QP), also referred to as the Lasso, for recovering the sparsity pattern. Our main result is to establish a sharp relation between the problem dimension \mdim, the number \spindex of non-zero elements in \betastar, and the number of observations \numobs that are required for reliable recovery. For a broad class of Gaussian ensembles satisfying mutual incoherence conditions, we establish existence and compute explicit values of thresholds \ThreshLow and \ThreshUp with the following properties: for any $\epsilon > 0$, if \numobs > 2 (\ThreshUp + \epsilon) \log (\mdim - \spindex) + \spindex + 1, then the Lasso succeeds in recovering the sparsity pattern with probability converging to one for large problems, whereas for \numobs < 2 (\ThreshLow - \epsilon) \log (\mdim - \spindex) + \spindex + 1, then the probability of successful recovery converges to zero. For the special case of the uniform Gaussian ensemble, we show that \ThreshLow = \ThreshUp = 1, so that the threshold is sharp and exactly determined.Comment: Appeared as Technical Report 708, Department of Statistics, UC Berkele

Asymptotic silence-breaking singularities

We discuss three complementary aspects of scalar curvature singularities: asymptotic causal properties, asymptotic Ricci and Weyl curvature, and asymptotic spatial properties. We divide scalar curvature singularities into two classes: so-called asymptotically silent singularities and non-generic singularities that break asymptotic silence. The emphasis in this paper is on the latter class which have not been previously discussed. We illustrate the above aspects and concepts by describing the singularities of a number of representative explicit perfect fluid solutions.Comment: 25 pages, 6 figure

Randomized Sketches of Convex Programs with Sharp Guarantees

Random projection (RP) is a classical technique for reducing storage and computational costs. We analyze RP-based approximations of convex programs, in which the original optimization problem is approximated by the solution of a lower-dimensional problem. Such dimensionality reduction is essential in computation-limited settings, since the complexity of general convex programming can be quite high (e.g., cubic for quadratic programs, and substantially higher for semidefinite programs). In addition to computational savings, random projection is also useful for reducing memory usage, and has useful properties for privacy-sensitive optimization. We prove that the approximation ratio of this procedure can be bounded in terms of the geometry of constraint set. For a broad class of random projections, including those based on various sub-Gaussian distributions as well as randomized Hadamard and Fourier transforms, the data matrix defining the cost function can be projected down to the statistical dimension of the tangent cone of the constraints at the original solution, which is often substantially smaller than the original dimension. We illustrate consequences of our theory for various cases, including unconstrained and $\ell_1$-constrained least squares, support vector machines, low-rank matrix estimation, and discuss implications on privacy-sensitive optimization and some connections with de-noising and compressed sensing