24 research outputs found
Least squares approximations of measures via geometric condition numbers
For a probability measure on a real separable Hilbert space, we are
interested in "volume-based" approximations of the d-dimensional least squares
error of it, i.e., least squares error with respect to a best fit d-dimensional
affine subspace. Such approximations are given by averaging real-valued
multivariate functions which are typically scalings of squared (d+1)-volumes of
(d+1)-simplices. Specifically, we show that such averages are comparable to the
square of the d-dimensional least squares error of that measure, where the
comparison depends on a simple quantitative geometric property of it. This
result is a higher dimensional generalization of the elementary fact that the
double integral of the squared distances between points is proportional to the
variance of measure. We relate our work to two recent algorithms, one for
clustering affine subspaces and the other for Monte-Carlo SVD based on volume
sampling
Approximation Algorithms for Bregman Co-clustering and Tensor Clustering
In the past few years powerful generalizations to the Euclidean k-means
problem have been made, such as Bregman clustering [7], co-clustering (i.e.,
simultaneous clustering of rows and columns of an input matrix) [9,18], and
tensor clustering [8,34]. Like k-means, these more general problems also suffer
from the NP-hardness of the associated optimization. Researchers have developed
approximation algorithms of varying degrees of sophistication for k-means,
k-medians, and more recently also for Bregman clustering [2]. However, there
seem to be no approximation algorithms for Bregman co- and tensor clustering.
In this paper we derive the first (to our knowledge) guaranteed methods for
these increasingly important clustering settings. Going beyond Bregman
divergences, we also prove an approximation factor for tensor clustering with
arbitrary separable metrics. Through extensive experiments we evaluate the
characteristics of our method, and show that it also has practical impact.Comment: 18 pages; improved metric cas
How Much and When Do We Need Higher-order Information in Hypergraphs? A Case Study on Hyperedge Prediction
Hypergraphs provide a natural way of representing group relations, whose
complexity motivates an extensive array of prior work to adopt some form of
abstraction and simplification of higher-order interactions. However, the
following question has yet to be addressed: How much abstraction of group
interactions is sufficient in solving a hypergraph task, and how different such
results become across datasets? This question, if properly answered, provides a
useful engineering guideline on how to trade off between complexity and
accuracy of solving a downstream task. To this end, we propose a method of
incrementally representing group interactions using a notion of n-projected
graph whose accumulation contains information on up to n-way interactions, and
quantify the accuracy of solving a task as n grows for various datasets. As a
downstream task, we consider hyperedge prediction, an extension of link
prediction, which is a canonical task for evaluating graph models. Through
experiments on 15 real-world datasets, we draw the following messages: (a)
Diminishing returns: small n is enough to achieve accuracy comparable with
near-perfect approximations, (b) Troubleshooter: as the task becomes more
challenging, larger n brings more benefit, and (c) Irreducibility: datasets
whose pairwise interactions do not tell much about higher-order interactions
lose much accuracy when reduced to pairwise abstractions