136,747 research outputs found
A statistical perspective of sampling scores for linear regression
In this paper, we consider a statistical problem of learning a linear model
from noisy samples. Existing work has focused on approximating the least
squares solution by using leverage-based scores as an importance sampling
distribution. However, no finite sample statistical guarantees and no
computationally efficient optimal sampling strategies have been proposed. To
evaluate the statistical properties of different sampling strategies, we
propose a simple yet effective estimator, which is easy for theoretical
analysis and is useful in multitask linear regression. We derive the exact mean
square error of the proposed estimator for any given sampling scores. Based on
minimizing the mean square error, we propose the optimal sampling scores for
both estimator and predictor, and show that they are influenced by the
noise-to-signal ratio. Numerical simulations match the theoretical analysis
well
A Statistical Perspective on Algorithmic Leveraging
One popular method for dealing with large-scale data sets is sampling. For
example, by using the empirical statistical leverage scores as an importance
sampling distribution, the method of algorithmic leveraging samples and
rescales rows/columns of data matrices to reduce the data size before
performing computations on the subproblem. This method has been successful in
improving computational efficiency of algorithms for matrix problems such as
least-squares approximation, least absolute deviations approximation, and
low-rank matrix approximation. Existing work has focused on algorithmic issues
such as worst-case running times and numerical issues associated with providing
high-quality implementations, but none of it addresses statistical aspects of
this method.
In this paper, we provide a simple yet effective framework to evaluate the
statistical properties of algorithmic leveraging in the context of estimating
parameters in a linear regression model with a fixed number of predictors. We
show that from the statistical perspective of bias and variance, neither
leverage-based sampling nor uniform sampling dominates the other. This result
is particularly striking, given the well-known result that, from the
algorithmic perspective of worst-case analysis, leverage-based sampling
provides uniformly superior worst-case algorithmic results, when compared with
uniform sampling. Based on these theoretical results, we propose and analyze
two new leveraging algorithms. A detailed empirical evaluation of existing
leverage-based methods as well as these two new methods is carried out on both
synthetic and real data sets. The empirical results indicate that our theory is
a good predictor of practical performance of existing and new leverage-based
algorithms and that the new algorithms achieve improved performance.Comment: 44 pages, 17 figure
A Statistical Perspective on Randomized Sketching for Ordinary Least-Squares
We consider statistical as well as algorithmic aspects of solving large-scale
least-squares (LS) problems using randomized sketching algorithms. For a LS
problem with input data , sketching algorithms use a sketching matrix, with . Then, rather than solving the LS problem using the
full data , sketching algorithms solve the LS problem using only the
sketched data . Prior work has typically adopted an algorithmic
perspective, in that it has made no statistical assumptions on the input
and , and instead it has been assumed that the data are fixed and
worst-case (WC). Prior results show that, when using sketching matrices such as
random projections and leverage-score sampling algorithms, with ,
the WC error is the same as solving the original problem, up to a small
constant. From a statistical perspective, we typically consider the
mean-squared error performance of randomized sketching algorithms, when data
are generated according to a statistical model , where is a noise process. We provide a rigorous
comparison of both perspectives leading to insights on how they differ. To do
this, we first develop a framework for assessing algorithmic and statistical
aspects of randomized sketching methods. We then consider the statistical
prediction efficiency (PE) and the statistical residual efficiency (RE) of the
sketched LS estimator; and we use our framework to provide upper bounds for
several types of random projection and random sampling sketching algorithms.
Among other results, we show that the RE can be upper bounded when while the PE typically requires the sample size to be substantially
larger. Lower bounds developed in subsequent results show that our upper bounds
on PE can not be improved.Comment: 27 pages, 5 figure
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
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