The uncovering of hidden structures by Latent Semantic Analysis

Latent Semantic Analysis (LSA) is a well-known method for information retrieval. It has also been applied as a model of cognitive processing and word-meaning acquisition. This dual importance of LSA derives from its capacity to modulate the meaning of words by contexts, dealing successfully with polysemy and synonymy. The underlying reasons that make the method work are not clear enough. We propose that the method works because it detects an underlying block structure (the blocks corresponding to topics) in the term-by-document matrix. In real cases this block structure is hidden because of perturbations. We propose that the correct explanation for LSA must be searched in the structure of singular vectors rather than in the profile of singular values. Using the Perron-Frobenius theory we show that the presence of disjoint blocks of documents is marked by sign-homogeneous entries in the vectors corresponding to the documents of one block and zeros elsewhere. In the case of nearly disjoint blocks, perturbation theory shows that if the perturbations are small, the zeros in the leading vectors are replaced by small numbers (pseudo-zeros). Since the singular values of each block might be very different in magnitude, their order does not mirror the order of blocks. When the norms of the blocks are similar, LSA works fine, but we propose that when the topics have different sizes, the usual procedure of selecting the first k singular triplets (k being the number of blocks) should be replaced by a method that selects the perturbed Perron vectors for each block.