13,417 research outputs found
Deterministic Polynomial Time Algorithms for Matrix Completion Problems
We present new deterministic algorithms for several cases of the maximum rank
matrix completion problem (for short matrix completion), i.e. the problem of
assigning values to the variables in a given symbolic matrix as to maximize the
resulting matrix rank. Matrix completion belongs to the fundamental problems in
computational complexity with numerous important algorithmic applications,
among others, in computing dynamic transitive closures or multicast network
codings (Harvey et al SODA 2005, Harvey et al SODA 2006).
We design efficient deterministic algorithms for common generalizations of
the results of Lovasz and Geelen on this problem by allowing linear functions
in the entries of the input matrix such that the submatrices corresponding to
each variable have rank one. We present also a deterministic polynomial time
algorithm for finding the minimal number of generators of a given module
structure given by matrices. We establish further several hardness results
related to matrix algebras and modules. As a result we connect the classical
problem of polynomial identity testing with checking surjectivity (or
injectivity) between two given modules. One of the elements of our algorithm is
a construction of a greedy algorithm for finding a maximum rank element in the
more general setting of the problem. The proof methods used in this paper could
be also of independent interest.Comment: 14 pages, preliminar
Weakening Assumptions for Deterministic Subexponential Time Non-Singular Matrix Completion
In (Kabanets, Impagliazzo, 2004) it is shown how to decide the circuit
polynomial identity testing problem (CPIT) in deterministic subexponential
time, assuming hardness of some explicit multilinear polynomial family for
arithmetical circuits. In this paper, a special case of CPIT is considered,
namely low-degree non-singular matrix completion (NSMC). For this subclass of
problems it is shown how to obtain the same deterministic time bound, using a
weaker assumption in terms of determinantal complexity.
Hardness-randomness tradeoffs will also be shown in the converse direction,
in an effort to make progress on Valiant's VP versus VNP problem. To separate
VP and VNP, it is known to be sufficient to prove that the determinantal
complexity of the m-by-m permanent is . In this paper it is
shown, for an appropriate notion of explicitness, that the existence of an
explicit multilinear polynomial family with determinantal complexity
m^{\omega(\log m)}G_nO(n^{1/\sqrt{\log n}})G_nM(x)poly(n)ndet(M(x))$ is a multilinear polynomial
Fast, deterministic computation of the Hermite normal form and determinant of a polynomial matrix
Given a nonsingular matrix of univariate polynomials over a
field , we give fast and deterministic algorithms to compute its
determinant and its Hermite normal form. Our algorithms use
operations in ,
where is bounded from above by both the average of the degrees of the rows
and that of the columns of the matrix and is the exponent of matrix
multiplication. The soft- notation indicates that logarithmic factors in the
big- are omitted while the ceiling function indicates that the cost is
when . Our algorithms are based
on a fast and deterministic triangularization method for computing the diagonal
entries of the Hermite form of a nonsingular matrix.Comment: 34 pages, 3 algorithm
Learning probability distributions generated by finite-state machines
We review methods for inference of probability distributions generated by probabilistic automata and related models for sequence generation. We focus on methods that can be proved to learn in the inference
in the limit and PAC formal models. The methods we review are state merging and state splitting methods for probabilistic deterministic automata and the recently developed spectral method for nondeterministic probabilistic automata. In both cases, we derive them from a high-level algorithm described in terms of the Hankel matrix of the distribution to be learned, given as an oracle, and then describe how to adapt that algorithm to account for the error introduced by a finite sample.Peer ReviewedPostprint (author's final draft
A Characterization of Deterministic Sampling Patterns for Low-Rank Matrix Completion
Low-rank matrix completion (LRMC) problems arise in a wide variety of
applications. Previous theory mainly provides conditions for completion under
missing-at-random samplings. This paper studies deterministic conditions for
completion. An incomplete matrix is finitely rank- completable
if there are at most finitely many rank- matrices that agree with all its
observed entries. Finite completability is the tipping point in LRMC, as a few
additional samples of a finitely completable matrix guarantee its unique
completability. The main contribution of this paper is a deterministic sampling
condition for finite completability. We use this to also derive deterministic
sampling conditions for unique completability that can be efficiently verified.
We also show that under uniform random sampling schemes, these conditions are
satisfied with high probability if entries per column are
observed. These findings have several implications on LRMC regarding lower
bounds, sample and computational complexity, the role of coherence, adaptive
settings and the validation of any completion algorithm. We complement our
theoretical results with experiments that support our findings and motivate
future analysis of uncharted sampling regimes.Comment: This update corrects an error in version 2 of this paper, where we
erroneously assumed that columns with more than r+1 observed entries would
yield multiple independent constraint
- …