1,498 research outputs found
Improving compressed sensing with the diamond norm
In low-rank matrix recovery, one aims to reconstruct a low-rank matrix from a
minimal number of linear measurements. Within the paradigm of compressed
sensing, this is made computationally efficient by minimizing the nuclear norm
as a convex surrogate for rank.
In this work, we identify an improved regularizer based on the so-called
diamond norm, a concept imported from quantum information theory. We show that
-for a class of matrices saturating a certain norm inequality- the descent cone
of the diamond norm is contained in that of the nuclear norm. This suggests
superior reconstruction properties for these matrices. We explicitly
characterize this set of matrices. Moreover, we demonstrate numerically that
the diamond norm indeed outperforms the nuclear norm in a number of relevant
applications: These include signal analysis tasks such as blind matrix
deconvolution or the retrieval of certain unitary basis changes, as well as the
quantum information problem of process tomography with random measurements.
The diamond norm is defined for matrices that can be interpreted as order-4
tensors and it turns out that the above condition depends crucially on that
tensorial structure. In this sense, this work touches on an aspect of the
notoriously difficult tensor completion problem.Comment: 25 pages + Appendix, 7 Figures, published versio
On the reconstruction of planar lattice-convex sets from the covariogram
A finite subset of is said to be lattice-convex if is
the intersection of with a convex set. The covariogram of
is the function associating to each u \in
\integer^d the cardinality of . Daurat, G\'erard, and Nivat and
independently Gardner, Gronchi, and Zong raised the problem on the
reconstruction of lattice-convex sets from . We provide a partial
positive answer to this problem by showing that for and under mild extra
assumptions, determines up to translations and reflections. As a
complement to the theorem on reconstruction we also extend the known
counterexamples (i.e., planar lattice-convex sets which are not
reconstructible, up to translations and reflections) to an infinite family of
counterexamples.Comment: accepted in Discrete and Computational Geometr
Characterizing the dual mixed volume via additive functionals
Integral representations are obtained of positive additive functionals on
finite products of the space of continuous functions (or of bounded Borel
functions) on a compact Hausdorff space. These are shown to yield
characterizations of the dual mixed volume, the fundamental concept in the dual
Brunn-Minkowski theory. The characterizations are shown to be best possible in
the sense that none of the assumptions can be omitted. The results obtained are
in the spirit of a similar characterization of the mixed volume in the
classical Brunn-Minkowski theory, obtained recently by Milman and Schneider,
but the methods employed are completely different
A reflection approach to the broken ray transform
We reduce the broken ray transform on some Riemannian manifolds (with
corners) to the geodesic ray transform on another manifold, which is obtained
from the original one by reflection. We give examples of this idea and present
injectivity results for the broken ray transform using corresponding earlier
results for the geodesic ray transform. Examples of manifolds where the broken
ray transform is injective include Euclidean cones and parts of the spheres
. In addition, we introduce the periodic broken ray transform and use the
reflection argument to produce examples of manifolds where it is injective. We
also give counterexamples to both periodic and nonperiodic cases. The broken
ray transform arises in Calder\'on's problem with partial data, and we give
implications of our results for this application.Comment: 29 pages, 6 figures; final versio
Euler-Bessel and Euler-Fourier Transforms
We consider a topological integral transform of Bessel (concentric
isospectral sets) type and Fourier (hyperplane isospectral sets) type, using
the Euler characteristic as a measure. These transforms convert constructible
\zed-valued functions to continuous -valued functions over a vector
space. Core contributions include: the definition of the topological Bessel
transform; a relationship in terms of the logarithmic blowup of the topological
Fourier transform; and a novel Morse index formula for the transforms. We then
apply the theory to problems of target reconstruction from enumerative sensor
data, including localization and shape discrimination. This last application
utilizes an extension of spatially variant apodization (SVA) to mitigate
sidelobe phenomena
The inverse moment problem for convex polytopes
The goal of this paper is to present a general and novel approach for the
reconstruction of any convex d-dimensional polytope P, from knowledge of its
moments. In particular, we show that the vertices of an N-vertex polytope in
R^d can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to
an unknown polynomial measure od degree D) in d+1 distinct generic directions.
Our approach is based on the collection of moment formulas due to Brion,
Lawrence, Khovanskii-Pukhikov, and Barvinok that arise in the discrete geometry
of polytopes, and what variously known as Prony's method, or Vandermonde
factorization of finite rank Hankel matrices.Comment: LaTeX2e, 24 pages including 1 appendi
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