40 research outputs found
Forchheimer Model for Non-Darcy Flow in Porous Media and Fractures
Imperial Users onl
Hybrid VCSPs with crisp and conservative valued templates
A constraint satisfaction problem (CSP) is a problem of computing a
homomorphism between two relational
structures. Analyzing its complexity has been a very fruitful research
direction, especially for fixed template CSPs, denoted , in
which the right side structure is fixed and the left side
structure is unconstrained.
Recently, the hybrid setting, written ,
where both sides are restricted simultaneously, attracted some attention. It
assumes that is taken from a class of relational structures
that additionally is closed under inverse homomorphisms. The last
property allows to exploit algebraic tools that have been developed for fixed
template CSPs. The key concept that connects hybrid CSPs with fixed-template
CSPs is the so called "lifted language". Namely, this is a constraint language
that can be constructed from an input . The
tractability of that language for any input is a
necessary condition for the tractability of the hybrid problem.
In the first part we investigate templates for which the
latter condition is not only necessary, but also is sufficient. We call such
templates widely tractable. For this purpose, we construct from
a new finite relational structure and define
as a class of structures homomorphic to . We
prove that wide tractability is equivalent to the tractability of
. Our proof is based on the key observation
that is homomorphic to if and only if the core of
is preserved by a Siggers polymorphism. Analogous
result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706
Context Vectors are Reflections of Word Vectors in Half the Dimensions
This paper takes a step towards theoretical analysis of the relationship
between word embeddings and context embeddings in models such as word2vec. We
start from basic probabilistic assumptions on the nature of word vectors,
context vectors, and text generation. These assumptions are well supported
either empirically or theoretically by the existing literature. Next, we show
that under these assumptions the widely-used word-word PMI matrix is
approximately a random symmetric Gaussian ensemble. This, in turn, implies that
context vectors are reflections of word vectors in approximately half the
dimensions. As a direct application of our result, we suggest a theoretically
grounded way of tying weights in the SGNS model
The algebraic structure of the densification and the sparsification tasks for CSPs
The tractability of certain CSPs for dense or sparse instances is known from
the 90s. Recently, the densification and the sparsification of CSPs were
formulated as computational tasks and the systematical study of their
computational complexity was initiated.
We approach this problem by introducing the densification operator, i.e. the
closure operator that, given an instance of a CSP, outputs all constraints that
are satisfied by all of its solutions. According to the Galois theory of
closure operators, any such operator is related to a certain implicational
system (or, a functional dependency) . We are specifically interested
in those classes of fixed-template CSPs, parameterized by constraint languages
, for which the size of an implicational system is a
polynomial in the number of variables . We show that in the Boolean case,
is of polynomial size if and only if is of bounded width. For
such languages, can be computed in log-space or in a logarithmic time
with a polynomial number of processors. Given an implicational system ,
the densification task is equivalent to the computation of the closure of input
constraints. The sparsification task is equivalent to the computation of the
minimal key. This leads to -algorithm for
the sparsification task where is the number of non-redundant
sparsifications of an original CSP.
Finally, we give a complete classification of constraint languages over the
Boolean domain for which the densification problem is tractable
Computing a partition function of a generalized pattern-based energy over a semiring
Valued constraint satisfaction problems with ordered variables (VCSPO) are a
special case of Valued CSPs in which variables are totally ordered and soft
constraints are imposed on tuples of variables that do not violate the order.
We study a restriction of VCSPO, in which soft constraints are imposed on a
segment of adjacent variables and a constraint language consists of
-valued characteristic functions of predicates. This kind of
potentials generalizes the so-called pattern-based potentials, which were
applied in many tasks of structured prediction.
For a constraint language we introduce a closure operator, , and give examples of constraint
languages for which is small. If all predicates in
are cartesian products, we show that the minimization of a generalized
pattern-based potential (or, the computation of its partition function) can be
made in
time, where is a set of variables, is a domain set. If, additionally,
only non-positive weights of constraints are allowed, the complexity of the
minimization task drops to where is the
arity of . For a general language and non-positive
weights, the minimization task can be carried out in time.
We argue that in many natural cases is of moderate
size, though in the worst case can blow up and
depend exponentially on