9 research outputs found
Algorithmic and Statistical Perspectives on Large-Scale Data Analysis
In recent years, ideas from statistics and scientific computing have begun to
interact in increasingly sophisticated and fruitful ways with ideas from
computer science and the theory of algorithms to aid in the development of
improved worst-case algorithms that are useful for large-scale scientific and
Internet data analysis problems. In this chapter, I will describe two recent
examples---one having to do with selecting good columns or features from a (DNA
Single Nucleotide Polymorphism) data matrix, and the other having to do with
selecting good clusters or communities from a data graph (representing a social
or information network)---that drew on ideas from both areas and that may serve
as a model for exploiting complementary algorithmic and statistical
perspectives in order to solve applied large-scale data analysis problems.Comment: 33 pages. To appear in Uwe Naumann and Olaf Schenk, editors,
"Combinatorial Scientific Computing," Chapman and Hall/CRC Press, 201
Approximate Computation and Implicit Regularization for Very Large-scale Data Analysis
Database theory and database practice are typically the domain of computer
scientists who adopt what may be termed an algorithmic perspective on their
data. This perspective is very different than the more statistical perspective
adopted by statisticians, scientific computers, machine learners, and other who
work on what may be broadly termed statistical data analysis. In this article,
I will address fundamental aspects of this algorithmic-statistical disconnect,
with an eye to bridging the gap between these two very different approaches. A
concept that lies at the heart of this disconnect is that of statistical
regularization, a notion that has to do with how robust is the output of an
algorithm to the noise properties of the input data. Although it is nearly
completely absent from computer science, which historically has taken the input
data as given and modeled algorithms discretely, regularization in one form or
another is central to nearly every application domain that applies algorithms
to noisy data. By using several case studies, I will illustrate, both
theoretically and empirically, the nonobvious fact that approximate
computation, in and of itself, can implicitly lead to statistical
regularization. This and other recent work suggests that, by exploiting in a
more principled way the statistical properties implicit in worst-case
algorithms, one can in many cases satisfy the bicriteria of having algorithms
that are scalable to very large-scale databases and that also have good
inferential or predictive properties.Comment: To appear in the Proceedings of the 2012 ACM Symposium on Principles
of Database Systems (PODS 2012
Revisiting the Nystrom Method for Improved Large-Scale Machine Learning
We reconsider randomized algorithms for the low-rank approximation of
symmetric positive semi-definite (SPSD) matrices such as Laplacian and kernel
matrices that arise in data analysis and machine learning applications. Our
main results consist of an empirical evaluation of the performance quality and
running time of sampling and projection methods on a diverse suite of SPSD
matrices. Our results highlight complementary aspects of sampling versus
projection methods; they characterize the effects of common data preprocessing
steps on the performance of these algorithms; and they point to important
differences between uniform sampling and nonuniform sampling methods based on
leverage scores. In addition, our empirical results illustrate that existing
theory is so weak that it does not provide even a qualitative guide to
practice. Thus, we complement our empirical results with a suite of worst-case
theoretical bounds for both random sampling and random projection methods.
These bounds are qualitatively superior to existing bounds---e.g. improved
additive-error bounds for spectral and Frobenius norm error and relative-error
bounds for trace norm error---and they point to future directions to make these
algorithms useful in even larger-scale machine learning applications.Comment: 60 pages, 15 color figures; updated proof of Frobenius norm bounds,
added comparison to projection-based low-rank approximations, and an analysis
of the power method applied to SPSD sketche
A Training Set Subsampling Strategy for the Reduced Basis Method
We present a subsampling strategy for the offline stage of the Reduced Basis
Method. The approach is aimed at bringing down the considerable offline costs
associated with using a finely-sampled training set. The proposed algorithm
exploits the potential of the pivoted QR decomposition and the discrete
empirical interpolation method to identify important parameter samples. It
consists of two stages. In the first stage, we construct a low-fidelity
approximation to the solution manifold over a fine training set. Then, for the
available low-fidelity snapshots of the output variable, we apply the pivoted
QR decomposition or the discrete empirical interpolation method to identify a
set of sparse sampling locations in the parameter domain. These points reveal
the structure of the parametric dependence of the output variable. The second
stage proceeds with a subsampled training set containing a by far smaller
number of parameters than the initial training set. Different subsampling
strategies inspired from recent variants of the empirical interpolation method
are also considered. Tests on benchmark examples justify the new approach and
show its potential to substantially speed up the offline stage of the Reduced
Basis Method, while generating reliable reduced-order models.Comment: 31 pages, 10 figures, 6 table
Topics in Matrix Sampling Algorithms
We study three fundamental problems of Linear Algebra, lying in the heart of
various Machine Learning applications, namely: 1)"Low-rank Column-based Matrix
Approximation". We are given a matrix A and a target rank k. The goal is to
select a subset of columns of A and, by using only these columns, compute a
rank k approximation to A that is as good as the rank k approximation that
would have been obtained by using all the columns; 2) "Coreset Construction in
Least-Squares Regression". We are given a matrix A and a vector b. Consider the
(over-constrained) least-squares problem of minimizing ||Ax-b||, over all
vectors x in D. The domain D represents the constraints on the solution and can
be arbitrary. The goal is to select a subset of the rows of A and b and, by
using only these rows, find a solution vector that is as good as the solution
vector that would have been obtained by using all the rows; 3) "Feature
Selection in K-means Clustering". We are given a set of points described with
respect to a large number of features. The goal is to select a subset of the
features and, by using only this subset, obtain a k-partition of the points
that is as good as the partition that would have been obtained by using all the
features. We present novel algorithms for all three problems mentioned above.
Our results can be viewed as follow-up research to a line of work known as
"Matrix Sampling Algorithms". [Frieze, Kanna, Vempala, 1998] presented the
first such algorithm for the Low-rank Matrix Approximation problem. Since then,
such algorithms have been developed for several other problems, e.g. Graph
Sparsification and Linear Equation Solving. Our contributions to this line of
research are: (i) improved algorithms for Low-rank Matrix Approximation and
Regression (ii) algorithms for a new problem domain (K-means Clustering).Comment: PhD Thesis, 150 page