5,153 research outputs found
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 -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 , 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 -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 -"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
- …