49,592 research outputs found
Parameter Learning of Logic Programs for Symbolic-Statistical Modeling
We propose a logical/mathematical framework for statistical parameter
learning of parameterized logic programs, i.e. definite clause programs
containing probabilistic facts with a parameterized distribution. It extends
the traditional least Herbrand model semantics in logic programming to
distribution semantics, possible world semantics with a probability
distribution which is unconditionally applicable to arbitrary logic programs
including ones for HMMs, PCFGs and Bayesian networks. We also propose a new EM
algorithm, the graphical EM algorithm, that runs for a class of parameterized
logic programs representing sequential decision processes where each decision
is exclusive and independent. It runs on a new data structure called support
graphs describing the logical relationship between observations and their
explanations, and learns parameters by computing inside and outside probability
generalized for logic programs. The complexity analysis shows that when
combined with OLDT search for all explanations for observations, the graphical
EM algorithm, despite its generality, has the same time complexity as existing
EM algorithms, i.e. the Baum-Welch algorithm for HMMs, the Inside-Outside
algorithm for PCFGs, and the one for singly connected Bayesian networks that
have been developed independently in each research field. Learning experiments
with PCFGs using two corpora of moderate size indicate that the graphical EM
algorithm can significantly outperform the Inside-Outside algorithm
Regression on fixed-rank positive semidefinite matrices: a Riemannian approach
The paper addresses the problem of learning a regression model parameterized
by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear
nature of the search space and on scalability to high-dimensional problems. The
mathematical developments rely on the theory of gradient descent algorithms
adapted to the Riemannian geometry that underlies the set of fixed-rank
positive semidefinite matrices. In contrast with previous contributions in the
literature, no restrictions are imposed on the range space of the learned
matrix. The resulting algorithms maintain a linear complexity in the problem
size and enjoy important invariance properties. We apply the proposed
algorithms to the problem of learning a distance function parameterized by a
positive semidefinite matrix. Good performance is observed on classical
benchmarks
Consistency-Checking Problems: A Gateway to Parameterized Sample Complexity
Recently, Brand, Ganian and Simonov introduced a parameterized refinement of
the classical PAC-learning sample complexity framework. A crucial outcome of
their investigation is that for a very wide range of learning problems, there
is a direct and provable correspondence between fixed-parameter
PAC-learnability (in the sample complexity setting) and the fixed-parameter
tractability of a corresponding "consistency checking" search problem (in the
setting of computational complexity). The latter can be seen as generalizations
of classical search problems where instead of receiving a single instance, one
receives multiple yes- and no-examples and is tasked with finding a solution
which is consistent with the provided examples.
Apart from a few initial results, consistency checking problems are almost
entirely unexplored from a parameterized complexity perspective. In this
article, we provide an overview of these problems and their connection to
parameterized sample complexity, with the primary aim of facilitating further
research in this direction. Afterwards, we establish the fixed-parameter
(in)-tractability for some of the arguably most natural consistency checking
problems on graphs, and show that their complexity-theoretic behavior is
surprisingly very different from that of classical decision problems. Our new
results cover consistency checking variants of problems as diverse as (k-)Path,
Matching, 2-Coloring, Independent Set and Dominating Set, among others
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