959 research outputs found
Intrinsic Volumes of Polyhedral Cones: A combinatorial perspective
The theory of intrinsic volumes of convex cones has recently found striking
applications in areas such as convex optimization and compressive sensing. This
article provides a self-contained account of the combinatorial theory of
intrinsic volumes for polyhedral cones. Direct derivations of the General
Steiner formula, the conic analogues of the Brianchon-Gram-Euler and the
Gauss-Bonnet relations, and the Principal Kinematic Formula are given. In
addition, a connection between the characteristic polynomial of a hyperplane
arrangement and the intrinsic volumes of the regions of the arrangement, due to
Klivans and Swartz, is generalized and some applications are presented.Comment: Survey, 23 page
Gordon's inequality and condition numbers in conic optimization
The probabilistic analysis of condition numbers has traditionally been
approached from different angles; one is based on Smale's program in complexity
theory and features integral geometry, while the other is motivated by
geometric functional analysis and makes use of the theory of Gaussian
processes. In this note we explore connections between the two approaches in
the context of the biconic homogeneous feasiblity problem and the condition
numbers motivated by conic optimization theory. Key tools in the analysis are
Slepian's and Gordon's comparision inequalities for Gaussian processes,
interpreted as monotonicity properties of moment functionals, and their
interplay with ideas from conic integral geometry
Lower Bounds on the Bounded Coefficient Complexity of Bilinear Maps
We prove lower bounds of order for both the problem to multiply
polynomials of degree , and to divide polynomials with remainder, in the
model of bounded coefficient arithmetic circuits over the complex numbers.
These lower bounds are optimal up to order of magnitude. The proof uses a
recent idea of R. Raz [Proc. 34th STOC 2002] proposed for matrix
multiplication. It reduces the linear problem to multiply a random circulant
matrix with a vector to the bilinear problem of cyclic convolution. We treat
the arising linear problem by extending J. Morgenstern's bound [J. ACM 20, pp.
305-306, 1973] in a unitarily invariant way. This establishes a new lower bound
on the bounded coefficient complexity of linear forms in terms of the singular
values of the corresponding matrix. In addition, we extend these lower bounds
for linear and bilinear maps to a model of circuits that allows a restricted
number of unbounded scalar multiplications.Comment: 19 page
Effective Condition Number Bounds for Convex Regularization
We derive bounds relating Renegar's condition number to quantities that
govern the statistical performance of convex regularization in settings that
include the -analysis setting. Using results from conic integral
geometry, we show that the bounds can be made to depend only on a random
projection, or restriction, of the analysis operator to a lower dimensional
space, and can still be effective if these operators are ill-conditioned. As an
application, we get new bounds for the undersampling phase transition of
composite convex regularizers. Key tools in the analysis are Slepian's
inequality and the kinematic formula from integral geometry.Comment: 17 pages, 4 figures . arXiv admin note: text overlap with
arXiv:1408.301
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We show that recent results on randomized dimension reduction schemes that
exploit structural properties of data can be applied in the context of
persistent homology. In the spirit of compressed sensing, the dimension
reduction is determined by the Gaussian width of a structure associated to the
data set, rather than its size, and such a reduction can be computed
efficiently. We further relate the Gaussian width to the doubling dimension of
a finite metric space, which appears in the study of the complexity of other
methods for approximating persistent homology. We can therefore literally
replace the ambient dimension by an intrinsic notion of dimension related to
the structure of the data.Comment: 20 page
On the Error in Phase Transition Computations for Compressed Sensing
Evaluating the statistical dimension is a common tool to determine the
asymptotic phase transition in compressed sensing problems with Gaussian
ensemble. Unfortunately, the exact evaluation of the statistical dimension is
very difficult and it has become standard to replace it with an upper-bound. To
ensure that this technique is suitable, [1] has introduced an upper-bound on
the gap between the statistical dimension and its approximation. In this work,
we first show that the error bound in [1] in some low-dimensional models such
as total variation and analysis minimization becomes poorly large.
Next, we develop a new error bound which significantly improves the estimation
gap compared to [1]. In particular, unlike the bound in [1] that is not
applicable to settings with overcomplete dictionaries, our bound exhibits a
decaying behavior in such cases
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