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

Assessing Ontological Arguments

Part I argues that ontological arguments, like other classical proofs of the existence of God, are parts of larger arguments in which they are embedded. These larger arguments include reasons supporting the proofs’ premises and responses to them, and to the proofs’ claims to validity and non-circularity, since, in the final analysis, our assessment of the proofs will express our best judgment of the cumulative force of all the considerations bearing on their overall adequacy. Part II illustrates these points by examining contemporary defences of, and attacks on, one of the ontological argument’s central premises, namely, that God’s existence is logically possible

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

Restricted strong convexity and weighted matrix completion: Optimal bounds with noise

We consider the matrix completion problem under a form of row/column weighted entrywise sampling, including the case of uniform entrywise sampling as a special case. We analyze the associated random observation operator, and prove that with high probability, it satisfies a form of restricted strong convexity with respect to weighted Frobenius norm. Using this property, we obtain as corollaries a number of error bounds on matrix completion in the weighted Frobenius norm under noisy sampling and for both exact and near low-rank matrices. Our results are based on measures of the "spikiness" and "low-rankness" of matrices that are less restrictive than the incoherence conditions imposed in previous work. Our technique involves an $M$-estimator that includes controls on both the rank and spikiness of the solution, and we establish non-asymptotic error bounds in weighted Frobenius norm for recovering matrices lying with $\ell_q$-"balls" of bounded spikiness. Using information-theoretic methods, we show that no algorithm can achieve better estimates (up to a logarithmic factor) over these same sets, showing that our conditions on matrices and associated rates are essentially optimal