1,466 research outputs found
Mean-payoff Automaton Expressions
Quantitative languages are an extension of boolean languages that assign to
each word a real number. Mean-payoff automata are finite automata with
numerical weights on transitions that assign to each infinite path the long-run
average of the transition weights. When the mode of branching of the automaton
is deterministic, nondeterministic, or alternating, the corresponding class of
quantitative languages is not robust as it is not closed under the pointwise
operations of max, min, sum, and numerical complement. Nondeterministic and
alternating mean-payoff automata are not decidable either, as the quantitative
generalization of the problems of universality and language inclusion is
undecidable.
We introduce a new class of quantitative languages, defined by mean-payoff
automaton expressions, which is robust and decidable: it is closed under the
four pointwise operations, and we show that all decision problems are decidable
for this class. Mean-payoff automaton expressions subsume deterministic
mean-payoff automata, and we show that they have expressive power incomparable
to nondeterministic and alternating mean-payoff automata. We also present for
the first time an algorithm to compute distance between two quantitative
languages, and in our case the quantitative languages are given as mean-payoff
automaton expressions
Bayes and maximum likelihood for -Wasserstein deconvolution of Laplace mixtures
We consider the problem of recovering a distribution function on the real
line from observations additively contaminated with errors following the
standard Laplace distribution. Assuming that the latent distribution is
completely unknown leads to a nonparametric deconvolution problem. We begin by
studying the rates of convergence relative to the -norm and the Hellinger
metric for the direct problem of estimating the sampling density, which is a
mixture of Laplace densities with a possibly unbounded set of locations: the
rate of convergence for the Bayes' density estimator corresponding to a
Dirichlet process prior over the space of all mixing distributions on the real
line matches, up to a logarithmic factor, with the rate
for the maximum likelihood estimator. Then, appealing to an inversion
inequality translating the -norm and the Hellinger distance between
general kernel mixtures, with a kernel density having polynomially decaying
Fourier transform, into any -Wasserstein distance, , between the
corresponding mixing distributions, provided their Laplace transforms are
finite in some neighborhood of zero, we derive the rates of convergence in the
-Wasserstein metric for the Bayes' and maximum likelihood estimators of
the mixing distribution. Merging in the -Wasserstein distance between
Bayes and maximum likelihood follows as a by-product, along with an assessment
on the stochastic order of the discrepancy between the two estimation
procedures
On Order Types of Random Point Sets
A simple method to produce a random order type is to take the order type of a
random point set. We conjecture that many probability distributions on order
types defined in this way are heavily concentrated and therefore sample
inefficiently the space of order types. We present two results on this
question. First, we study experimentally the bias in the order types of
random points chosen uniformly and independently in a square, for up to
. Second, we study algorithms for determining the order type of a point set
in terms of the number of coordinate bits they require to know. We give an
algorithm that requires on average bits to determine the
order type of , and show that any algorithm requires at least bits. This implies that the concentration conjecture cannot
be proven by an "efficient encoding" argument
Techniques to explore time-related correlation in large datasets
The next generation of database management and computing systems will be significantly complex with data distributed both in functionality and operation. The complexity arises, at least in part, due to data types involved and types of information request rendered by the database user. Time sequence databases are generated in many practical applications. Detecting similar sequences and subsequences within these databases is an important research area and has generated lot of interest recently. Previous studies in this area have concentrated on calculating similitude between (sub)sequences of equal sizes. The question of unequal sized (sub)sequence comparison to report similitude has been an open problem for some time. The problem is an important and non-trivial one. In this dissertation, we propose a solution to the problem of finding sequences, in a database of unequal sized sequences, that are similar to a given query sequence. A paradigm to search pairs of similar, equal and unequal sized, subsequences within a pair of sequences is also presented. We put forward new approaches for sequence time-scale reduction, feature aggregation and object recognition. To make the search of similar sequences efficient, we propose an indexing technique to index the unequal-sized sequence database. We also introduce a unique indexing technique to index identified subsequences within a reference sequence. This index is subsequently employed to report similar pairs of subsequences, when presented with a query sequence. We present several experimental results and also compare the proposed framework with previous work in this area
Latent Semantic Learning with Structured Sparse Representation for Human Action Recognition
This paper proposes a novel latent semantic learning method for extracting
high-level features (i.e. latent semantics) from a large vocabulary of abundant
mid-level features (i.e. visual keywords) with structured sparse
representation, which can help to bridge the semantic gap in the challenging
task of human action recognition. To discover the manifold structure of
midlevel features, we develop a spectral embedding approach to latent semantic
learning based on L1-graph, without the need to tune any parameter for graph
construction as a key step of manifold learning. More importantly, we construct
the L1-graph with structured sparse representation, which can be obtained by
structured sparse coding with its structured sparsity ensured by novel L1-norm
hypergraph regularization over mid-level features. In the new embedding space,
we learn latent semantics automatically from abundant mid-level features
through spectral clustering. The learnt latent semantics can be readily used
for human action recognition with SVM by defining a histogram intersection
kernel. Different from the traditional latent semantic analysis based on topic
models, our latent semantic learning method can explore the manifold structure
of mid-level features in both L1-graph construction and spectral embedding,
which results in compact but discriminative high-level features. The
experimental results on the commonly used KTH action dataset and unconstrained
YouTube action dataset show the superior performance of our method.Comment: The short version of this paper appears in ICCV 201
Sharp Bounds on Davenport-Schinzel Sequences of Every Order
One of the longest-standing open problems in computational geometry is to
bound the lower envelope of univariate functions, each pair of which
crosses at most times, for some fixed . This problem is known to be
equivalent to bounding the length of an order- Davenport-Schinzel sequence,
namely a sequence over an -letter alphabet that avoids alternating
subsequences of the form with length
. These sequences were introduced by Davenport and Schinzel in 1965 to
model a certain problem in differential equations and have since been applied
to bounding the running times of geometric algorithms, data structures, and the
combinatorial complexity of geometric arrangements.
Let be the maximum length of an order- DS sequence over
letters. What is asymptotically? This question has been answered
satisfactorily (by Hart and Sharir, Agarwal, Sharir, and Shor, Klazar, and
Nivasch) when is even or . However, since the work of Agarwal,
Sharir, and Shor in the mid-1980s there has been a persistent gap in our
understanding of the odd orders.
In this work we effectively close the problem by establishing sharp bounds on
Davenport-Schinzel sequences of every order . Our results reveal that,
contrary to one's intuition, behaves essentially like
when is odd. This refutes conjectures due to Alon et al.
(2008) and Nivasch (2010).Comment: A 10-page extended abstract will appear in the Proceedings of the
Symposium on Computational Geometry, 201
The brick polytope of a sorting network
The associahedron is a polytope whose graph is the graph of flips on
triangulations of a convex polygon. Pseudotriangulations and
multitriangulations generalize triangulations in two different ways, which have
been unified by Pilaud and Pocchiola in their study of flip graphs on
pseudoline arrangements with contacts supported by a given sorting network.
In this paper, we construct the brick polytope of a sorting network, obtained
as the convex hull of the brick vectors associated to each pseudoline
arrangement supported by the network. We combinatorially characterize the
vertices of this polytope, describe its faces, and decompose it as a Minkowski
sum of matroid polytopes.
Our brick polytopes include Hohlweg and Lange's many realizations of the
associahedron, which arise as brick polytopes for certain well-chosen sorting
networks. We furthermore discuss the brick polytopes of sorting networks
supporting pseudoline arrangements which correspond to multitriangulations of
convex polygons: our polytopes only realize subgraphs of the flip graphs on
multitriangulations and they cannot appear as projections of a hypothetical
multiassociahedron.Comment: 36 pages, 25 figures; Version 2 refers to the recent generalization
of our results to spherical subword complexes on finite Coxeter groups
(http://arxiv.org/abs/1111.3349
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