4 research outputs found
A tight lower bound instance for k-means++ in constant dimension
The k-means++ seeding algorithm is one of the most popular algorithms that is
used for finding the initial centers when using the k-means heuristic. The
algorithm is a simple sampling procedure and can be described as follows: Pick
the first center randomly from the given points. For , pick a point to
be the center with probability proportional to the square of the
Euclidean distance of this point to the closest previously chosen
centers.
The k-means++ seeding algorithm is not only simple and fast but also gives an
approximation in expectation as shown by Arthur and Vassilvitskii.
There are datasets on which this seeding algorithm gives an approximation
factor of in expectation. However, it is not clear from these
results if the algorithm achieves good approximation factor with reasonably
high probability (say ). Brunsch and R\"{o}glin gave a dataset where
the k-means++ seeding algorithm achieves an approximation ratio
with probability that is exponentially small in . However, this and all
other known lower-bound examples are high dimensional. So, an open problem was
to understand the behavior of the algorithm on low dimensional datasets. In
this work, we give a simple two dimensional dataset on which the seeding
algorithm achieves an approximation ratio with probability
exponentially small in . This solves open problems posed by Mahajan et al.
and by Brunsch and R\"{o}glin.Comment: To appear in TAMC 2014. arXiv admin note: text overlap with
arXiv:1306.420
The Complexity of the k-means Method
The k-means method is a widely used technique for clustering points in Euclidean space. While it is extremely fast in practice, its worst-case running time is exponential in the number of data points. We prove that the k-means method can implicitly solve PSPACE-complete problems, providing a complexity-theoretic explanation for its worst-case running time. Our result parallels recent work on the complexity of the simplex method for linear programming
Information geometry
This Special Issue of the journal Entropy, titled āInformation Geometry Iā, contains a collection of 17 papers concerning the foundations and applications of information geometry. Based on a geometrical interpretation of probability, information geometry has become a rich mathematical field employing the methods of differential geometry. It has numerous applications to data science, physics, and neuroscience. Presenting original research, yet written in an accessible, tutorial style, this collection of papers will be useful for scientists who are new to the field, while providing an excellent reference for the more experienced researcher. Several papers are written by authorities in the field, and topics cover the foundations of information geometry, as well as applications to statistics, Bayesian inference, machine learning, complex systems, physics, and neuroscience