24 research outputs found
Phase Retrieval for Sparse Signals: Uniqueness Conditions
In a variety of fields, in particular those involving imaging and optics, we
often measure signals whose phase is missing or has been irremediably
distorted. Phase retrieval attempts the recovery of the phase information of a
signal from the magnitude of its Fourier transform to enable the reconstruction
of the original signal. A fundamental question then is: "Under which conditions
can we uniquely recover the signal of interest from its measured magnitudes?"
In this paper, we assume the measured signal to be sparse. This is a natural
assumption in many applications, such as X-ray crystallography, speckle imaging
and blind channel estimation. In this work, we derive a sufficient condition
for the uniqueness of the solution of the phase retrieval (PR) problem for both
discrete and continuous domains, and for one and multi-dimensional domains.
More precisely, we show that there is a strong connection between PR and the
turnpike problem, a classic combinatorial problem. We also prove that the
existence of collisions in the autocorrelation function of the signal may
preclude the uniqueness of the solution of PR. Then, assuming the absence of
collisions, we prove that the solution is almost surely unique on 1-dimensional
domains. Finally, we extend this result to multi-dimensional signals by solving
a set of 1-dimensional problems. We show that the solution of the
multi-dimensional problem is unique when the autocorrelation function has no
collisions, significantly improving upon a previously known result.Comment: submitted to IEEE TI
Homometric sets in trees
Let denote a simple graph with the vertex set and the edge
set . The profile of a vertex set denotes the multiset of
pairwise distances between the vertices of . Two disjoint subsets of
are \emph{homometric}, if their profiles are the same. If is a tree on
vertices we prove that its vertex sets contains a pair of disjoint homometric
subsets of size at least . Previously it was known that such a
pair of size at least roughly exists. We get a better result in case
of haircomb trees, in which we are able to find a pair of disjoint homometric
sets of size at least for a constant
On Conditions for Uniqueness in Sparse Phase Retrieval
The phase retrieval problem has a long history and is an important problem in
many areas of optics. Theoretical understanding of phase retrieval is still
limited and fundamental questions such as uniqueness and stability of the
recovered solution are not yet fully understood. This paper provides several
additions to the theoretical understanding of sparse phase retrieval. In
particular we show that if the measurement ensemble can be chosen freely, as
few as 4k-1 phaseless measurements suffice to guarantee uniqueness of a
k-sparse M-dimensional real solution. We also prove that 2(k^2-k+1) Fourier
magnitude measurements are sufficient under rather general conditions
Gromov-Monge quasi-metrics and distance distributions
Applications in data science, shape analysis and object classification
frequently require maps between metric spaces which preserve geometry as
faithfully as possible. In this paper, we combine the Monge formulation of
optimal transport with the Gromov-Hausdorff distance construction to define a
measure of the minimum amount of geometric distortion required to map one
metric measure space onto another. We show that the resulting quantity, called
Gromov-Monge distance, defines an extended quasi-metric on the space of
isomorphism classes of metric measure spaces and that it can be promoted to a
true metric on certain subclasses of mm-spaces. We also give precise
comparisons between Gromov-Monge distance and several other metrics which have
appeared previously, such as the Gromov-Wasserstein metric and the continuous
Procrustes metric of Lipman, Al-Aifari and Daubechies. Finally, we derive
polynomial-time computable lower bounds for Gromov-Monge distance. These lower
bounds are expressed in terms of distance distributions, which are classical
invariants of metric measure spaces summarizing the volume growth of metric
balls. In the second half of the paper, which may be of independent interest,
we study the discriminative power of these lower bounds for simple subclasses
of metric measure spaces. We first consider the case of planar curves, where we
give a counterexample to the Curve Histogram Conjecture of Brinkman and Olver.
Our results on plane curves are then generalized to higher dimensional
manifolds, where we prove some sphere characterization theorems for the
distance distribution invariant. Finally, we consider several inverse problems
on recovering a metric graph from a collection of localized versions of
distance distributions. Results are derived by establishing connections with
concepts from the fields of computational geometry and topological data
analysis.Comment: Version 2: Added many new results and improved expositio
Multi-Reference Alignment for sparse signals, Uniform Uncertainty Principles and the Beltway Problem
Motivated by cutting-edge applications like cryo-electron microscopy
(cryo-EM), the Multi-Reference Alignment (MRA) model entails the learning of an
unknown signal from repeated measurements of its images under the latent action
of a group of isometries and additive noise of magnitude . Despite
significant interest, a clear picture for understanding rates of estimation in
this model has emerged only recently, particularly in the high-noise regime
that is highly relevant in applications. Recent investigations
have revealed a remarkable asymptotic sample complexity of order for
certain signals whose Fourier transforms have full support, in stark contrast
to the traditional that arise in regular models. Often prohibitively
large in practice, these results have prompted the investigation of variations
around the MRA model where better sample complexity may be achieved. In this
paper, we show that \emph{sparse} signals exhibit an intermediate
sample complexity even in the classical MRA model. Our results explore and
exploit connections of the MRA estimation problem with two classical topics in
applied mathematics: the \textit{beltway problem} from combinatorial
optimization, and \textit{uniform uncertainty principles} from harmonic
analysis
Coarse and bi-Lipschitz embeddability of subspaces of the Gromov-Hausdorff space into Hilbert spaces
In this paper, we discuss the embeddability of subspaces of the
Gromov-Hausdorff space, which consists of isometry classes of compact metric
spaces endowed with the Gromov-Hausdorff distance, into Hilbert spaces. These
embeddings are particularly valuable for applications to topological data
analysis. We prove that its subspace consisting of metric spaces with at most n
points has asymptotic dimension . Thus, there exists a coarse
embedding of that space into a Hilbert space. On the contrary, if the number of
points is not bounded, then the subspace cannot be coarsely embedded into any
uniformly convex Banach space and so, in particular, into any Hilbert space.
Furthermore, we prove that, even if we restrict to finite metric spaces whose
diameter is bounded by some constant, the subspace still cannot be bi-Lipschitz
embedded into any finite-dimensional Hilbert space. We obtain both
non-embeddability results by finding obstructions to coarse and bi-Lipschitz
embeddings in families of isometry classes of finite subsets of the real line
endowed with the Euclidean-Hausdorff distance
The spread of finite and infinite groups
It is well known that every finite simple group has a generating pair.
Moreover, Guralnick and Kantor proved that every finite simple group has the
stronger property, known as -generation, that every nontrivial
element is contained in a generating pair. Much more recently, this result has
been generalised in three different directions, which form the basis of this
survey article. First, we look at some stronger forms of
-generation that the finite simple groups satisfy, which are
described in terms of spread and uniform domination. Next, we discuss the
recent classification of the finite -generated groups. Finally, we
turn our attention to infinite groups, focusing on the recent discovery that
the finitely presented simple groups of Thompson are also
-generated, as are many of their generalisations. Throughout the
article we pose open questions in this area, and we highlight connections with
other areas of group theory.Comment: 38 pages; survey article based on my lecture at Groups St Andrews
202
Generalized Shapes and Point Sets Correspondence and Registration
The theory of shapes, as proposed by David Kendall, is concerned with sets of labeled points in the Euclidean space Rd that define a shape regardless of trans- lations, rotations and dilatations. We present here a method that extends the theory of shapes, where, in this case, we use the term generalized shape for structures of unlabeled points. By using the distribution of distances between the points in a set we are able to define the existence of generalized shapes and to infer the computation of the correspondences and the orthogonal transformation between two elements of the same generalized shape equivalence class. This study is oriented to solve the registration of large set of landmarks or point sets derived from medical images but may be employed in other fields such as computer vision or biological morphometry
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Statistical and Computational Aspects of Learning with Complex Structure
The recent explosion of data that is routinely collected has led scientists to contemplate more and more sophisticated structural assumptions. Understanding how to harness and exploit such structure is key to improving the prediction accuracy of various statistical procedures. The ultimate goal of this line of research is to develop a set of tools that leverage underlying complex structures to pool information across observations and ultimately improve statistical accuracy as well as computational efficiency of the deployed methods. The workshop focused on recent developments in regression and matrix estimation under various complex constraints such as physical, computational, privacy, sparsity or robustness. Optimal-transport based techniques for geometric data analysis were also a main topic of the workshop