  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 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.
