3 research outputs found
Subset Selection for Matrices with Fixed Blocks
Subset selection for matrices is the task of extracting a column sub-matrix
from a given matrix with such that the
pseudoinverse of the sampled matrix has as small Frobenius or spectral norm as
possible. In this paper, we consider a more general problem of subset selection
for matrices that allows a block to be fixed at the beginning. Under this
setting, we provide a deterministic method for selecting a column sub-matrix
from . We also present a bound for both the Frobenius and spectral norms of
the pseudoinverse of the sampled matrix, showing that the bound is
asymptotically optimal. The main technology for proving this result is the
interlacing families of polynomials developed by Marcus, Spielman, and
Srivastava. This idea also results in a deterministic greedy selection
algorithm that produces the sub-matrix promised by our result
Polynomial Time Algorithms for Dual Volume Sampling
We study dual volume sampling, a method for selecting k columns from an n x m
short and wide matrix (n <= k <= m) such that the probability of selection is
proportional to the volume spanned by the rows of the induced submatrix. This
method was proposed by Avron and Boutsidis (2013), who showed it to be a
promising method for column subset selection and its multiple applications.
However, its wider adoption has been hampered by the lack of polynomial time
sampling algorithms. We remove this hindrance by developing an exact
(randomized) polynomial time sampling algorithm as well as its derandomization.
Thereafter, we study dual volume sampling via the theory of real stable
polynomials and prove that its distribution satisfies the "Strong Rayleigh"
property. This result has numerous consequences, including a provably
fast-mixing Markov chain sampler that makes dual volume sampling much more
attractive to practitioners. This sampler is closely related to classical
algorithms for popular experimental design methods that are to date lacking
theoretical analysis but are known to empirically work well
Faster Subset Selection for Matrices and Applications
We study subset selection for matrices defined as follows: given a matrix
\matX \in \R^{n \times m} () and an oversampling parameter (), select a subset of columns from \matX such that the
pseudo-inverse of the subsampled matrix has as smallest norm as possible. In
this work, we focus on the Frobenius and the spectral matrix norms. We describe
several novel (deterministic and randomized) approximation algorithms for this
problem with approximation bounds that are optimal up to constant factors.
Additionally, we show that the combinatorial problem of finding a low-stretch
spanning tree in an undirected graph corresponds to subset selection, and
discuss various implications of this reduction.Comment: To appear in SIAM Journal on Matrix Analysis and Application