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Register a new userGaussianity and typicality in matrix distributional semantics
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Added by
Sanjaye Ramgoolam
Publication date
2019
fields
Informatics Engineering
and research's language is
English
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No Arabic abstract
Constructions in type-driven compositional distributional semantics associate large collections of matrices of size $D$ 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.
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Permutation invariant Gaussian matrix models were recently developed for applications in computational linguistics. A 5-parameter family of models was solved. In this paper, we use a representation theoretic approach to solve the general 13-parameter Gaussian model, which can be viewed as a zero-dimensional quantum field theory. We express the two linear and eleven quadratic terms in the action in terms of representation theoretic parameters. These parameters are coefficients of simple quadratic expressions in terms of appropriate linear combinations of the matrix variables transforming in specific irreducible representations of the symmetric group $S_D$ where $D$ is the size of the matrices. They allow the identification of constraints which ensure a convergent Gaussian measure and well-defined expectation values for polynomial functions of the random matrix at all orders. A graph-theoretic interpretation is known to allow the enumeration of permutation invariants of matrices at linear, quadratic and higher orders. We express the expectation values of all the quadratic graph-basis invariants and a selection of cubic and quartic invariants in terms of the representation theoretic parameters of the model.
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