416 research outputs found
A Sparse Johnson--Lindenstrauss Transform
Dimension reduction is a key algorithmic tool with many applications
including nearest-neighbor search, compressed sensing and linear algebra in the
streaming model. In this work we obtain a {\em sparse} version of the
fundamental tool in dimension reduction --- the Johnson--Lindenstrauss
transform. Using hashing and local densification, we construct a sparse
projection matrix with just non-zero entries
per column. We also show a matching lower bound on the sparsity for a large
class of projection matrices. Our bounds are somewhat surprising, given the
known lower bounds of both on the number of rows
of any projection matrix and on the sparsity of projection matrices generated
by natural constructions.
Using this, we achieve an update time per
non-zero element for a -approximate projection, thereby
substantially outperforming the update time
required by prior approaches. A variant of our method offers the same
guarantees for sparse vectors, yet its worst case running time
matches the best approach of Ailon and Liberty.Comment: 10 pages, conference version
Dimensionality Reduction for k-Means Clustering and Low Rank Approximation
We show how to approximate a data matrix with a much smaller
sketch that can be used to solve a general class of
constrained k-rank approximation problems to within error.
Importantly, this class of problems includes -means clustering and
unconstrained low rank approximation (i.e. principal component analysis). By
reducing data points to just dimensions, our methods generically
accelerate any exact, approximate, or heuristic algorithm for these ubiquitous
problems.
For -means dimensionality reduction, we provide relative
error results for many common sketching techniques, including random row
projection, column selection, and approximate SVD. For approximate principal
component analysis, we give a simple alternative to known algorithms that has
applications in the streaming setting. Additionally, we extend recent work on
column-based matrix reconstruction, giving column subsets that not only `cover'
a good subspace for \bv{A}, but can be used directly to compute this
subspace.
Finally, for -means clustering, we show how to achieve a
approximation by Johnson-Lindenstrauss projecting data points to just dimensions. This gives the first result that leverages the
specific structure of -means to achieve dimension independent of input size
and sublinear in
Random forests with random projections of the output space for high dimensional multi-label classification
We adapt the idea of random projections applied to the output space, so as to
enhance tree-based ensemble methods in the context of multi-label
classification. We show how learning time complexity can be reduced without
affecting computational complexity and accuracy of predictions. We also show
that random output space projections may be used in order to reach different
bias-variance tradeoffs, over a broad panel of benchmark problems, and that
this may lead to improved accuracy while reducing significantly the
computational burden of the learning stage
Random Projections For Large-Scale Regression
Fitting linear regression models can be computationally very expensive in
large-scale data analysis tasks if the sample size and the number of variables
are very large. Random projections are extensively used as a dimension
reduction tool in machine learning and statistics. We discuss the applications
of random projections in linear regression problems, developed to decrease
computational costs, and give an overview of the theoretical guarantees of the
generalization error. It can be shown that the combination of random
projections with least squares regression leads to similar recovery as ridge
regression and principal component regression. We also discuss possible
improvements when averaging over multiple random projections, an approach that
lends itself easily to parallel implementation.Comment: 13 pages, 3 Figure
Four lectures on probabilistic methods for data science
Methods of high-dimensional probability play a central role in applications
for statistics, signal processing theoretical computer science and related
fields. These lectures present a sample of particularly useful tools of
high-dimensional probability, focusing on the classical and matrix Bernstein's
inequality and the uniform matrix deviation inequality. We illustrate these
tools with applications for dimension reduction, network analysis, covariance
estimation, matrix completion and sparse signal recovery. The lectures are
geared towards beginning graduate students who have taken a rigorous course in
probability but may not have any experience in data science applications.Comment: Lectures given at 2016 PCMI Graduate Summer School in Mathematics of
Data. Some typos, inaccuracies fixe
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