48,420 research outputs found
Innovation Pursuit: A New Approach to Subspace Clustering
In subspace clustering, a group of data points belonging to a union of
subspaces are assigned membership to their respective subspaces. This paper
presents a new approach dubbed Innovation Pursuit (iPursuit) to the problem of
subspace clustering using a new geometrical idea whereby subspaces are
identified based on their relative novelties. We present two frameworks in
which the idea of innovation pursuit is used to distinguish the subspaces.
Underlying the first framework is an iterative method that finds the subspaces
consecutively by solving a series of simple linear optimization problems, each
searching for a direction of innovation in the span of the data potentially
orthogonal to all subspaces except for the one to be identified in one step of
the algorithm. A detailed mathematical analysis is provided establishing
sufficient conditions for iPursuit to correctly cluster the data. The proposed
approach can provably yield exact clustering even when the subspaces have
significant intersections. It is shown that the complexity of the iterative
approach scales only linearly in the number of data points and subspaces, and
quadratically in the dimension of the subspaces. The second framework
integrates iPursuit with spectral clustering to yield a new variant of
spectral-clustering-based algorithms. The numerical simulations with both real
and synthetic data demonstrate that iPursuit can often outperform the
state-of-the-art subspace clustering algorithms, more so for subspaces with
significant intersections, and that it significantly improves the
state-of-the-art result for subspace-segmentation-based face clustering
Well-Pointed Coalgebras
For endofunctors of varieties preserving intersections, a new description of
the final coalgebra and the initial algebra is presented: the former consists
of all well-pointed coalgebras. These are the pointed coalgebras having no
proper subobject and no proper quotient. The initial algebra consists of all
well-pointed coalgebras that are well-founded in the sense of Osius and Taylor.
And initial algebras are precisely the final well-founded coalgebras. Finally,
the initial iterative algebra consists of all finite well-pointed coalgebras.
Numerous examples are discussed e.g. automata, graphs, and labeled transition
systems
A Coalgebraic View on Reachability
Coalgebras for an endofunctor provide a category-theoretic framework for
modeling a wide range of state-based systems of various types. We provide an
iterative construction of the reachable part of a given pointed coalgebra that
is inspired by and resembles the standard breadth-first search procedure to
compute the reachable part of a graph. We also study coalgebras in Kleisli
categories: for a functor extending a functor on the base category, we show
that the reachable part of a given pointed coalgebra can be computed in that
base category
How to estimate the number of self-avoiding walks over 10^100? Use random walks
Counting the number of N-step self-avoiding walks (SAWs) on a lattice is one
of the most difficult problems of enumerative combinatorics. Once we give up
calculating the exact number of them, however, we have a chance to apply
powerful computational methods of statistical mechanics to this problem. In
this paper, we develop a statistical enumeration method for SAWs using the
multicanonical Monte Carlo method. A key part of this method is to expand the
configuration space of SAWs to random walks, the exact number of which is
known. Using this method, we estimate a number of N-step SAWs on a square
lattice, c_N, up to N=256. The value of c_256 is 5.6(1)*10^108 (the number in
the parentheses is the statistical error of the last digit) and this is larger
than one googol (10^100).Comment: 5 pages, 3 figures, 1 table, to appear in proceedings of YSMSPIP in
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The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets
Given two nonempty and disjoint intersections of closed and convex subsets,
we look for a best approximation pair relative to them, i.e., a pair of points,
one in each intersection, attaining the minimum distance between the disjoint
intersections. We propose an iterative process based on projections onto the
subsets which generate the intersections. The process is inspired by the
Halpern-Lions-Wittmann-Bauschke algorithm and the classical alternating process
of Cheney and Goldstein, and its advantage is that there is no need to project
onto the intersections themselves, a task which can be rather demanding. We
prove that under certain conditions the two interlaced subsequences converge to
a best approximation pair. These conditions hold, in particular, when the space
is Euclidean and the subsets which generate the intersections are compact and
strictly convex. Our result extends the one of Aharoni, Censor and Jiang
["Finding a best approximation pair of points for two polyhedra", Computational
Optimization and Applications 71 (2018), 509-523] which considered the case of
finite-dimensional polyhedra.Comment: 25 pages, 1 figur
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