452 research outputs found
Approximation and Streaming Algorithms for Projective Clustering via Random Projections
Let be a set of points in . In the projective
clustering problem, given and norm , we have to
compute a set of -dimensional flats such that is minimized; here
represents the (Euclidean) distance of to the closest flat in
. We let denote the minimal value and interpret
to be . When and
and , the problem corresponds to the -median, -mean and the
-center clustering problems respectively.
For every , and , we show that the
orthogonal projection of onto a randomly chosen flat of dimension
will -approximate
. This result combines the concepts of geometric coresets and
subspace embeddings based on the Johnson-Lindenstrauss Lemma. As a consequence,
an orthogonal projection of to an dimensional randomly chosen subspace
-approximates projective clusterings for every and
simultaneously. Note that the dimension of this subspace is independent of the
number of clusters~.
Using this dimension reduction result, we obtain new approximation and
streaming algorithms for projective clustering problems. For example, given a
stream of points, we show how to compute an -approximate
projective clustering for every and simultaneously using only
space. Compared to
standard streaming algorithms with space requirement, our approach
is a significant improvement when the number of input points and their
dimensions are of the same order of magnitude.Comment: Canadian Conference on Computational Geometry (CCCG 2015
Counting tropical elliptic plane curves with fixed j-invariant
In complex algebraic geometry, the problem of enumerating plane elliptic
curves of given degree with fixed complex structure has been solved by
R.Pandharipande using Gromov-Witten theory. In this article we treat the
tropical analogue of this problem, the determination of the number of tropical
elliptic plane curves of degree d and fixed ``tropical j-invariant''
interpolating an appropriate number of points in general position. We show that
this number is independent of the position of the points and the value of the
j-invariant and that it coincides with the number of complex elliptic curves.
The result can be used to simplify Mikhalkin's algorithm to count curves via
lattice paths in the case of rational plane curves.Comment: 34 pages; minor changes to match the published versio
Polynomial-Sized Topological Approximations Using The Permutahedron
Classical methods to model topological properties of point clouds, such as
the Vietoris-Rips complex, suffer from the combinatorial explosion of complex
sizes. We propose a novel technique to approximate a multi-scale filtration of
the Rips complex with improved bounds for size: precisely, for points in
, we obtain a -approximation with at most simplices of dimension or lower. In conjunction with dimension
reduction techniques, our approach yields a -approximation of size for Rips filtrations on arbitrary metric
spaces. This result stems from high-dimensional lattice geometry and exploits
properties of the permutahedral lattice, a well-studied structure in discrete
geometry.
Building on the same geometric concept, we also present a lower bound result
on the size of an approximate filtration: we construct a point set for which
every -approximation of the \v{C}ech filtration has to contain
features, provided that for .Comment: 24 pages, 1 figur
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