163,575 research outputs found
On choosing and bounding probability metrics
When studying convergence of measures, an important issue is the choice of
probability metric. In this review, we provide a summary and some new results
concerning bounds among ten important probability metrics/distances that are
used by statisticians and probabilists. We focus on these metrics because they
are either well-known, commonly used, or admit practical bounding techniques.
We summarize these relationships in a handy reference diagram, and also give
examples to show how rates of convergence can depend on the metric chosen.Comment: To appear, International Statistical Review. Related work at
http://www.math.hmc.edu/~su/papers.htm
Sets of bounded discrepancy for multi-dimensional irrational rotation
We study bounded remainder sets with respect to an irrational rotation of the
-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who
characterized the intervals with bounded remainder in dimension one.
First we extend to several dimensions the Hecke-Ostrowski result by
constructing a class of -dimensional parallelepipeds of bounded remainder.
Then we characterize the Riemann measurable bounded remainder sets in terms of
"equidecomposability" to such a parallelepiped. By constructing invariants with
respect to this equidecomposition, we derive explicit conditions for a polytope
to be a bounded remainder set. In particular this yields a characterization of
the convex bounded remainder polygons in two dimensions. The approach is used
to obtain several other results as well.Comment: To appear in Geometric And Functional Analysi
Unsupervised edge map scoring: a statistical complexity approach
We propose a new Statistical Complexity Measure (SCM) to qualify edge maps
without Ground Truth (GT) knowledge. The measure is the product of two indices,
an \emph{Equilibrium} index obtained by projecting the edge map
into a family of edge patterns, and an \emph{Entropy} index ,
defined as a function of the Kolmogorov Smirnov (KS) statistic.
This new measure can be used for performance characterization which includes:
(i)~the specific evaluation of an algorithm (intra-technique process) in order
to identify its best parameters, and (ii)~the comparison of different
algorithms (inter-technique process) in order to classify them according to
their quality.
Results made over images of the South Florida and Berkeley databases show
that our approach significantly improves over Pratt's Figure of Merit (PFoM)
which is the objective reference-based edge map evaluation standard, as it
takes into account more features in its evaluation
A function space framework for structural total variation regularization with applications in inverse problems
In this work, we introduce a function space setting for a wide class of
structural/weighted total variation (TV) regularization methods motivated by
their applications in inverse problems. In particular, we consider a
regularizer that is the appropriate lower semi-continuous envelope (relaxation)
of a suitable total variation type functional initially defined for
sufficiently smooth functions. We study examples where this relaxation can be
expressed explicitly, and we also provide refinements for weighted total
variation for a wide range of weights. Since an integral characterization of
the relaxation in function space is, in general, not always available, we show
that, for a rather general linear inverse problems setting, instead of the
classical Tikhonov regularization problem, one can equivalently solve a
saddle-point problem where no a priori knowledge of an explicit formulation of
the structural TV functional is needed. In particular, motivated by concrete
applications, we deduce corresponding results for linear inverse problems with
norm and Poisson log-likelihood data discrepancy terms. Finally, we provide
proof-of-concept numerical examples where we solve the saddle-point problem for
weighted TV denoising as well as for MR guided PET image reconstruction
Computing Probabilistic Bisimilarity Distances for Probabilistic Automata
The probabilistic bisimilarity distance of Deng et al. has been proposed as a
robust quantitative generalization of Segala and Lynch's probabilistic
bisimilarity for probabilistic automata. In this paper, we present a
characterization of the bisimilarity distance as the solution of a simple
stochastic game. The characterization gives us an algorithm to compute the
distances by applying Condon's simple policy iteration on these games. The
correctness of Condon's approach, however, relies on the assumption that the
games are stopping. Our games may be non-stopping in general, yet we are able
to prove termination for this extended class of games. Already other algorithms
have been proposed in the literature to compute these distances, with
complexity in and \textbf{PPAD}. Despite the
theoretical relevance, these algorithms are inefficient in practice. To the
best of our knowledge, our algorithm is the first practical solution.
The characterization of the probabilistic bisimilarity distance mentioned
above crucially uses a dual presentation of the Hausdorff distance due to
M\'emoli. As an additional contribution, in this paper we show that M\'emoli's
result can be used also to prove that the bisimilarity distance bounds the
difference in the maximal (or minimal) probability of two states to satisfying
arbitrary -regular properties, expressed, eg., as LTL formulas
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