3 research outputs found
Total Least Squares Regression in Input Sparsity Time
In the total least squares problem, one is given an matrix ,
and an matrix , and one seeks to "correct" both and ,
obtaining matrices and , so that there exists an
satisfying the equation . Typically the problem is
overconstrained, meaning that . The cost of the solution
is given by . We
give an algorithm for finding a solution to the linear system
for which the cost
is at most a multiplicative factor times the optimal cost, up to
an additive error that may be an arbitrarily small function of .
Importantly, our running time is , where for a matrix ,
denotes its number of non-zero entries. Importantly, our
running time does not directly depend on the large parameter . As total
least squares regression is known to be solvable via low rank approximation, a
natural approach is to invoke fast algorithms for approximate low rank
approximation, obtaining matrices and from this low rank
approximation, and then solving for so that . However,
existing algorithms do not apply since in total least squares the rank of the
low rank approximation needs to be , and so the running time of known
methods would be at least . In contrast, we are able to achieve a much
faster running time for finding by never explicitly forming the equation
, but instead solving for an which is a solution to an
implicit such equation. Finally, we generalize our algorithm to the total least
squares problem with regularization
A near-optimal algorithm for approximating the John Ellipsoid
We develop a simple and efficient algorithm for approximating the John
Ellipsoid of a symmetric polytope. Our algorithm is near optimal in the sense
that our time complexity matches the current best verification algorithm. We
also provide the MATLAB code for further research.Comment: COLT 201
Sketching Transformed Matrices with Applications to Natural Language Processing
Suppose we are given a large matrix that cannot be stored in
memory but is in a disk or is presented in a data stream. However, we need to
compute a matrix decomposition of the entry-wisely transformed matrix,
for some function . Is it possible to do it in a space
efficient way? Many machine learning applications indeed need to deal with such
large transformed matrices, for example word embedding method in NLP needs to
work with the pointwise mutual information (PMI) matrix, while the entrywise
transformation makes it difficult to apply known linear algebraic tools.
Existing approaches for this problem either need to store the whole matrix and
perform the entry-wise transformation afterwards, which is space consuming or
infeasible, or need to redesign the learning method, which is application
specific and requires substantial remodeling.
In this paper, we first propose a space-efficient sketching algorithm for
computing the product of a given small matrix with the transformed matrix. It
works for a general family of transformations with provable small error bounds
and thus can be used as a primitive in downstream learning tasks. We then apply
this primitive to a concrete application: low-rank approximation. We show that
our approach obtains small error and is efficient in both space and time. We
complement our theoretical results with experiments on synthetic and real data.Comment: AISTATS 202