12,912 research outputs found
Approximating Hereditary Discrepancy via Small Width Ellipsoids
The Discrepancy of a hypergraph is the minimum attainable value, over
two-colorings of its vertices, of the maximum absolute imbalance of any
hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum
discrepancy of a restriction of the hypergraph to a subset of its vertices, is
a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related
the natural extension of this quantity to matrices to rounding algorithms for
linear programs, and gave a determinant based lower bound on the hereditary
discrepancy. Matousek (2011) showed that this bound is tight up to a
polylogarithmic factor, leaving open the question of actually computing this
bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time
-approximation to hereditary discrepancy, as a by-product
of their work in differential privacy. In this paper, we give a direct simple
-approximation algorithm for this problem. We show that up to
this approximation factor, the hereditary discrepancy of a matrix is
characterized by the optimal value of simple geometric convex program that
seeks to minimize the largest norm of any point in a ellipsoid
containing the columns of . This characterization promises to be a useful
tool in discrepancy theory
Stable super-resolution limit and smallest singular value of restricted Fourier matrices
Super-resolution refers to the process of recovering the locations and
amplitudes of a collection of point sources, represented as a discrete measure,
given of its noisy low-frequency Fourier coefficients. The recovery
process is highly sensitive to noise whenever the distance between the
two closest point sources is less than . This paper studies the {\it
fundamental difficulty of super-resolution} and the {\it performance guarantees
of a subspace method called MUSIC} in the regime that .
The most important quantity in our theory is the minimum singular value of
the Vandermonde matrix whose nodes are specified by the source locations. Under
the assumption that the nodes are closely spaced within several well-separated
clumps, we derive a sharp and non-asymptotic lower bound for this quantity. Our
estimate is given as a weighted sum, where each term only depends on
the configuration of each individual clump. This implies that, as the noise
increases, the super-resolution capability of MUSIC degrades according to a
power law where the exponent depends on the cardinality of the largest clump.
Numerical experiments validate our theoretical bounds for the minimum singular
value and the resolution limit of MUSIC.
When there are point sources located on a grid with spacing , the
fundamental difficulty of super-resolution can be quantitatively characterized
by a min-max error, which is the reconstruction error incurred by the best
possible algorithm in the worst-case scenario. We show that the min-max error
is closely related to the minimum singular value of Vandermonde matrices, and
we provide a non-asymptotic and sharp estimate for the min-max error, where the
dominant term is .Comment: 47 pages, 8 figure
Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of L-fuzzy mathematical morphology
Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted.
The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing
Screening Rules for Convex Problems
We propose a new framework for deriving screening rules for convex
optimization problems. Our approach covers a large class of constrained and
penalized optimization formulations, and works in two steps. First, given any
approximate point, the structure of the objective function and the duality gap
is used to gather information on the optimal solution. In the second step, this
information is used to produce screening rules, i.e. safely identifying
unimportant weight variables of the optimal solution. Our general framework
leads to a large variety of useful existing as well as new screening rules for
many applications. For example, we provide new screening rules for general
simplex and -constrained problems, Elastic Net, squared-loss Support
Vector Machines, minimum enclosing ball, as well as structured norm regularized
problems, such as group lasso
Conditions for Existence of Dual Certificates in Rank-One Semidefinite Problems
Several signal recovery tasks can be relaxed into semidefinite programs with
rank-one minimizers. A common technique for proving these programs succeed is
to construct a dual certificate. Unfortunately, dual certificates may not exist
under some formulations of semidefinite programs. In order to put problems into
a form where dual certificate arguments are possible, it is important to
develop conditions under which the certificates exist. In this paper, we
provide an example where dual certificates do not exist. We then present a
completeness condition under which they are guaranteed to exist. For programs
that do not satisfy the completeness condition, we present a completion process
which produces an equivalent program that does satisfy the condition. The
important message of this paper is that dual certificates may not exist for
semidefinite programs that involve orthogonal measurements with respect to
positive-semidefinite matrices. Such measurements can interact with the
positive-semidefinite constraint in a way that implies additional linear
measurements. If these additional measurements are not included in the problem
formulation, then dual certificates may fail to exist. As an illustration, we
present a semidefinite relaxation for the task of finding the sparsest element
in a subspace. One formulation of this program does not admit dual
certificates. The completion process produces an equivalent formulation which
does admit dual certificates
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