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.

Eigenvector localization refers to the situation when most of the components
of an eigenvector are zero or near-zero. This phenomenon has been observed on
eigenvectors associated with extremal eigenvalues, and in many of those cases
it can be meaningfully interpreted in terms of "structural heterogeneities" in
the data.

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.

Recent work in theoretical computer science and scientific computing has
focused on nearly-linear-time algorithms for solving systems of linear
equations. While introducing several novel theoretical perspectives, this work
has yet to lead to practical algorithms. In an effort to bridge this gap, we
describe in this paper two related results. Our first and main result is a
simple algorithm to approximate the solution to a set of linear equations
defined by a Laplacian (for a graph $G$ with $n$ nodes and $m \le n^2$ edges)
constraint matrix.