36,192 research outputs found
Infinite time decidable equivalence relation theory
We introduce an analog of the theory of Borel equivalence relations in which
we study equivalence relations that are decidable by an infinite time Turing
machine. The Borel reductions are replaced by the more general class of
infinite time computable functions. Many basic aspects of the classical theory
remain intact, with the added bonus that it becomes sensible to study some
special equivalence relations whose complexity is beyond Borel or even
analytic. We also introduce an infinite time generalization of the countable
Borel equivalence relations, a key subclass of the Borel equivalence relations,
and again show that several key properties carry over to the larger class.
Lastly, we collect together several results from the literature regarding Borel
reducibility which apply also to absolutely Delta_1^2 reductions, and hence to
the infinite time computable reductions.Comment: 30 pages, 3 figure
Towards Machine Wald
The past century has seen a steady increase in the need of estimating and
predicting complex systems and making (possibly critical) decisions with
limited information. Although computers have made possible the numerical
evaluation of sophisticated statistical models, these models are still designed
\emph{by humans} because there is currently no known recipe or algorithm for
dividing the design of a statistical model into a sequence of arithmetic
operations. Indeed enabling computers to \emph{think} as \emph{humans} have the
ability to do when faced with uncertainty is challenging in several major ways:
(1) Finding optimal statistical models remains to be formulated as a well posed
problem when information on the system of interest is incomplete and comes in
the form of a complex combination of sample data, partial knowledge of
constitutive relations and a limited description of the distribution of input
random variables. (2) The space of admissible scenarios along with the space of
relevant information, assumptions, and/or beliefs, tend to be infinite
dimensional, whereas calculus on a computer is necessarily discrete and finite.
With this purpose, this paper explores the foundations of a rigorous framework
for the scientific computation of optimal statistical estimators/models and
reviews their connections with Decision Theory, Machine Learning, Bayesian
Inference, Stochastic Optimization, Robust Optimization, Optimal Uncertainty
Quantification and Information Based Complexity.Comment: 37 page
Time and spectral domain relative entropy: A new approach to multivariate spectral estimation
The concept of spectral relative entropy rate is introduced for jointly
stationary Gaussian processes. Using classical information-theoretic results,
we establish a remarkable connection between time and spectral domain relative
entropy rates. This naturally leads to a new spectral estimation technique
where a multivariate version of the Itakura-Saito distance is employed}. It may
be viewed as an extension of the approach, called THREE, introduced by Byrnes,
Georgiou and Lindquist in 2000 which, in turn, followed in the footsteps of the
Burg-Jaynes Maximum Entropy Method. Spectral estimation is here recast in the
form of a constrained spectrum approximation problem where the distance is
equal to the processes relative entropy rate. The corresponding solution
entails a complexity upper bound which improves on the one so far available in
the multichannel framework. Indeed, it is equal to the one featured by THREE in
the scalar case. The solution is computed via a globally convergent matricial
Newton-type algorithm. Simulations suggest the effectiveness of the new
technique in tackling multivariate spectral estimation tasks, especially in the
case of short data records.Comment: 32 pages, submitted for publicatio
Computational Complexity of the Interleaving Distance
The interleaving distance is arguably the most prominent distance measure in
topological data analysis. In this paper, we provide bounds on the
computational complexity of determining the interleaving distance in several
settings. We show that the interleaving distance is NP-hard to compute for
persistence modules valued in the category of vector spaces. In the specific
setting of multidimensional persistent homology we show that the problem is at
least as hard as a matrix invertibility problem. Furthermore, this allows us to
conclude that the interleaving distance of interval decomposable modules
depends on the characteristic of the field. Persistence modules valued in the
category of sets are also studied. As a corollary, we obtain that the
isomorphism problem for Reeb graphs is graph isomorphism complete.Comment: Discussion related to the characteristic of the field added. Paper
accepted to the 34th International Symposium on Computational Geometr
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