118,070 research outputs found
Fixed-Rank Approximation of a Positive-Semidefinite Matrix from Streaming Data
Several important applications, such as streaming PCA and semidefinite
programming, involve a large-scale positive-semidefinite (psd) matrix that is
presented as a sequence of linear updates. Because of storage limitations, it
may only be possible to retain a sketch of the psd matrix. This paper develops
a new algorithm for fixed-rank psd approximation from a sketch. The approach
combines the Nystrom approximation with a novel mechanism for rank truncation.
Theoretical analysis establishes that the proposed method can achieve any
prescribed relative error in the Schatten 1-norm and that it exploits the
spectral decay of the input matrix. Computer experiments show that the proposed
method dominates alternative techniques for fixed-rank psd matrix approximation
across a wide range of examples
Algorithms for Provisioning Queries and Analytics
Provisioning is a technique for avoiding repeated expensive computations in
what-if analysis. Given a query, an analyst formulates hypotheticals, each
retaining some of the tuples of a database instance, possibly overlapping, and
she wishes to answer the query under scenarios, where a scenario is defined by
a subset of the hypotheticals that are "turned on". We say that a query admits
compact provisioning if given any database instance and any hypotheticals,
one can create a poly-size (in ) sketch that can then be used to answer the
query under any of the possible scenarios without accessing the
original instance.
In this paper, we focus on provisioning complex queries that combine
relational algebra (the logical component), grouping, and statistics/analytics
(the numerical component). We first show that queries that compute quantiles or
linear regression (as well as simpler queries that compute count and
sum/average of positive values) can be compactly provisioned to provide
(multiplicative) approximate answers to an arbitrary precision. In contrast,
exact provisioning for each of these statistics requires the sketch size to be
exponential in . We then establish that for any complex query whose logical
component is a positive relational algebra query, as long as the numerical
component can be compactly provisioned, the complex query itself can be
compactly provisioned. On the other hand, introducing negation or recursion in
the logical component again requires the sketch size to be exponential in .
While our positive results use algorithms that do not access the original
instance after a scenario is known, we prove our lower bounds even for the case
when, knowing the scenario, limited access to the instance is allowed
Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix
This paper studies how to sketch element-wise functions of low-rank matrices.
Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we
aim for finding an approximated low-rank representation of the (possibly
high-rank) matrix [f(Aij)]. To this end, we propose an efficient
sketching-based algorithm whose complexity is significantly lower than the
number of entries of A, i.e., it runs without accessing all entries of [f(Aij)]
explicitly. The main idea underlying our method is to combine a polynomial
approximation of f with the existing tensor sketch scheme for approximating
monomials of entries of A. To balance the errors of the two approximation
components in an optimal manner, we propose a novel regression formula to find
polynomial coefficients given A and f. In particular, we utilize a
coreset-based regression with a rigorous approximation guarantee. Finally, we
demonstrate the applicability and superiority of the proposed scheme under
various machine learning tasks
Random projections for Bayesian regression
This article deals with random projections applied as a data reduction
technique for Bayesian regression analysis. We show sufficient conditions under
which the entire -dimensional distribution is approximately preserved under
random projections by reducing the number of data points from to in the case . Under mild
assumptions, we prove that evaluating a Gaussian likelihood function based on
the projected data instead of the original data yields a
-approximation in terms of the Wasserstein
distance. Our main result shows that the posterior distribution of Bayesian
linear regression is approximated up to a small error depending on only an
-fraction of its defining parameters. This holds when using
arbitrary Gaussian priors or the degenerate case of uniform distributions over
for . Our empirical evaluations involve different
simulated settings of Bayesian linear regression. Our experiments underline
that the proposed method is able to recover the regression model up to small
error while considerably reducing the total running time
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
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