| Low-dimensional representations of sensory signals are key to solving many of the computational problems encountered in high-level vision. Principal Component Analysis (PCA) has been used in the past to derive practically useful compact representations for different classes of objects. One major objection to the applicability of PCA is that it invariably leads to global, nontopographic representations that are not amenable to further processing and are not biologically plausible. In this Thesis we present a new mathematical construction--Local Feature Analysis (LFA)--for deriving local topographic representations for any class of objects. The LFA representations are sparse-distributed and, hence, are effectively low-dimensional and retain all the advantages of the compact representations of PCA. Unlike the global eigenmodes, they give a description of objects in terms of statistically derived local features and their locations. Moreover, the LFA representation exposes partial, local symmetries, which are present in the ensemble, but are not naturally captured by PCA, which allows further reduction of the dimensionality of the representation. We illustrate the LFA theory by using it to extract local features for human faces.; We understand the preparation of object ensembles as breaking of global symmetries and show how to do it automatically, upon which we base a bootstrap mechanism for symmetry-breaking. We understand the localization of features as breaking of local symmetries, which define new ensembles--of features--whose representational modules are hierarchically interconnected, in a manner similar to the thalamo-cortical and cortico-cortical computational feedback loops, and serve as active blackboards.; We generalize LFA to the scale-translationally-symmetric ensembles of natural signals, with full PCA dimensionality, and show how to reduce it by the construction of multi-scale representations. We show that the sparsification step of LFA, when applied to 1-dimensional time-dependent infinite ensembles, results in representation and transmission of the signal with only on type of variables--the sparse set of timings of, otherwise identical, events; this is a key property of the sensory coding of signals with action potentials (spikes) in most biological systems.; Finally, we argue that LFA is applicable to all levels of sensory processing. |
