2 research outputs found
Gaussianity and typicality in matrix distributional semantics
Constructions in type-driven compositional distributional semantics associate
large collections of matrices of size to linguistic corpora. We develop the
proposal of analysing the statistical characteristics of this data in the
framework of permutation invariant matrix models. The observables in this
framework are permutation invariant polynomial functions of the matrix entries,
which correspond to directed graphs. Using the general 13-parameter permutation
invariant Gaussian matrix models recently solved, we find, using a dataset of
matrices constructed via standard techniques in distributional semantics, that
the expectation values of a large class of cubic and quartic observables show
high gaussianity at levels between 90 to 99 percent. Beyond expectation values,
which are averages over words, the dataset allows the computation of standard
deviations for each observable, which can be viewed as a measure of typicality
for each observable. There is a wide range of magnitudes in the measures of
typicality. The permutation invariant matrix models, considered as functions of
random couplings, give a very good prediction of the magnitude of the
typicality for different observables. We find evidence that observables with
similar matrix model characteristics of Gaussianity and typicality also have
high degrees of correlation between the ranked lists of words associated to
these observables.Comment: 38 pages, 11 figure
Permutation invariant Gaussian 2-matrix models
We construct the general permutation invariant Gaussian 2-matrix model for
matrices of arbitrary size . The parameters of the model are given in terms
of variables defined using the representation theory of the symmetric group
. A correspondence is established between the permutation invariant
polynomial functions of the matrix variables (the observables of the model) and
directed colored graphs, which sheds light on stability properties in the large
counting of these invariants. The refined counting of the graphs is given
in terms of double cosets involving permutation groups defined by the local
structure of the graphs. Linear and quadratic observables are transformed to an
representation theoretic basis and are used to define the convergent
Gaussian measure. The perturbative rules for the computation of expectation
values of graph-basis observables of any degree are given in terms of the
representation theoretic parameters. Explicit results for a number of
observables of degree up to four are given along with a Sage programme that
computes general expectation values.Comment: 80 pages, 11 figure