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Concept# Generative model

Summary

In statistical classification, two main approaches are called the generative approach and the discriminative approach. These compute classifiers by different approaches, differing in the degree of statistical modelling. Terminology is inconsistent, but three major types can be distinguished, following :
# A generative model is a statistical model of the joint probability distribution P(X, Y) on given observable variable X and target variable Y;

# A discriminative model is a model of the conditional probability P(Y\mid X = x) of the target Y, given an observation x; and

# Classifiers computed without using a probability model are also referred to loosely as "discriminative".

The distinction between these last two classes is not consistently made; refers to these three classes as generative learning, conditional learning, and discriminative learning, but only distinguish two classes, calling them generative classifiers (joint distribution) and discriminative

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We discuss some properties of generative models for word embeddings. Namely, (Arora et al., 2016) proposed a latent discourse model implying the concentration of the partition function of the word vectors. This concentration phenomenon led to an asymptotic linear relation between the pointwise mutual information (PMI) of pairs of words and the scalar product of their vectors. Here, we first revisit this concentration phenomenon and prove it under slightly weaker assumptions, for a set of random vectors symmetrically distributed around the origin. Second, we empirically evaluate the relation between PMI and scalar products of word vectors satisfying the concentration property. Our empirical results indicate that, in practice, this relation does not hold with arbitrarily small error. This observation is further supported by two theoretical results: (i) the error cannot be exactly zero because the corresponding shifted PMI matrix cannot be positive semidefinite; (ii) under mild assumptions, there exist pairs of words for which the error cannot be close to zero. We deduce that either natural language does not follow the assumptions of the considered generative model, or the current word vector generation methods do not allow the construction of the hypothesized word embeddings.

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The underlying geometrical structure of the latent space in deep generative models is in most cases not Euclidean, which may lead to biases when comparing interpolation capabilities of two models. Smoothness and plausibility of linear interpolations in latent space are associated with the quality of the underlying generative model. In this paper, we show that not all such interpolations are comparable as they can deviate arbitrarily from the shortest interpolation curve given by the geodesic. This deviation is revealed by computing curve lengths with the pull-back metric of the generative model, finding shorter curves than the straight line between endpoints, and measuring a non-zero relative length improvement on this straight line. This leads to a strategy to compare linear interpolations across two generative models. We also show the effect and importance of choosing an appropriate output space for computing shorter curves. For this computation we derive an extension of the pull-back metric.

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