Fisher Kernels and Probabilistic Latent Semantic Models

Tasks that rely on semantic content of documents, notably Information Retrieval and Document Classification, can benefit from a good account of document context, i.e. the semantic association between documents. To this effect, the scheme of latent semantics blends individual words appearing throughout a document collection into latent topics, thus providing a way to handle documents that is less constrained than the conventional approach by the mere appearance of such or such word. Probabilistic latent semantic models take the matter further by providing assumptions on how the documents observed in the collection would have been generated. This allows derivation of inference algorithms that can fit the model parameters to the observed document collection; with their values set, these parameters can then be used to compute the similarities between documents. The Fisher kernels, similarity functions rooted in information geometry, constitute good candidates to measure the similarity between documents in the framework of probabilistic latent semantic models. In this context, we study the use of Fisher kernels for the Probabilistic Latent Semantic Indexing (PLSI) model. By thoroughly analysing the generative process of PLSI, we derive the proper Fisher kernel for PLSI and expose the hypotheses that relate former work to this kernel. In particular, we confirm that the Fisher information matrix (FIM) should not be approximated by the identity in the case of PLSI. We also study the impact on the performances of the Fisher kernel of the contribution of the latent topics and the one of the distribution of words among the topics; eventually, we provide empirical evidence, and theoretical arguments, showing that the Fisher kernel originally published by Hofmann, corrected to account for FIM, is the best of the PLSI Fisher kernels. It can compete with the strong BM25 baseline, and even significantly outperforms it when documents sharing few words must be matched. We further study of PLSI document similarities by applying the Language model approach. This approach shuns the usual IR paradigm that considers documents and queries to be of a similar nature. Instead, they consider documents as being representative of language models, and use probabilistic tools to determine which of these models would have generated the query with highest probability. Using this scheme in the framework of PLSI provides a way to bypass the issue of query representation, which constitutes one of the specific challenges of PLSI. We find the Language model approach to perform as well as the best of the Fisher kernels when enough latent categories are provided. Eventually, we propose a new probabilistic latent semantic model consisting in a mixture of Smoothed Dirichlet distributions which, by better modeling word burstiness, provides a more realistic model of empirical observations on real document collections than the usually used Multinomials.

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