4,619 research outputs found
A Context-theoretic Framework for Compositionality in Distributional Semantics
Techniques in which words are represented as vectors have proved useful in
many applications in computational linguistics, however there is currently no
general semantic formalism for representing meaning in terms of vectors. We
present a framework for natural language semantics in which words, phrases and
sentences are all represented as vectors, based on a theoretical analysis which
assumes that meaning is determined by context.
In the theoretical analysis, we define a corpus model as a mathematical
abstraction of a text corpus. The meaning of a string of words is assumed to be
a vector representing the contexts in which it occurs in the corpus model.
Based on this assumption, we can show that the vector representations of words
can be considered as elements of an algebra over a field. We note that in
applications of vector spaces to representing meanings of words there is an
underlying lattice structure; we interpret the partial ordering of the lattice
as describing entailment between meanings. We also define the context-theoretic
probability of a string, and, based on this and the lattice structure, a degree
of entailment between strings.
We relate the framework to existing methods of composing vector-based
representations of meaning, and show that our approach generalises many of
these, including vector addition, component-wise multiplication, and the tensor
product.Comment: Submitted to Computational Linguistics on 20th January 2010 for
revie
Belief propagation in monoidal categories
We discuss a categorical version of the celebrated belief propagation
algorithm. This provides a way to prove that some algorithms which are known or
suspected to be analogous, are actually identical when formulated generically.
It also highlights the computational point of view in monoidal categories.Comment: In Proceedings QPL 2014, arXiv:1412.810
Solving multivariate polynomial systems and an invariant from commutative algebra
The complexity of computing the solutions of a system of multivariate
polynomial equations by means of Gr\"obner bases computations is upper bounded
by a function of the solving degree. In this paper, we discuss how to
rigorously estimate the solving degree of a system, focusing on systems arising
within public-key cryptography. In particular, we show that it is upper bounded
by, and often equal to, the Castelnuovo Mumford regularity of the ideal
generated by the homogenization of the equations of the system, or by the
equations themselves in case they are homogeneous. We discuss the underlying
commutative algebra and clarify under which assumptions the commonly used
results hold. In particular, we discuss the assumption of being in generic
coordinates (often required for bounds obtained following this type of
approach) and prove that systems that contain the field equations or their fake
Weil descent are in generic coordinates. We also compare the notion of solving
degree with that of degree of regularity, which is commonly used in the
literature. We complement the paper with some examples of bounds obtained
following the strategy that we describe
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