Tensor Representation of Topographically Organized Semantic Spaces

Human brains seem to represent categories of objects and actions as locations in a continuous semantic space across the cortical surface that reflects the similarity among categories. This vision of the semantic organization of information in the brain, suggested by recent experimental findings, is in harmony with the well-known topographically organized somatotopic, retinotopic, and tonotopic maps in the cerebral cortex. Here we show that these topographies can be operationally represented with context-dependent associative memories. In these models, the input vectors and, eventually also, the associated output vectors are multiplied by context vectors via the Kronecker tensor product, which allows a spatial organization of memories. Input and output tensor contexts localize matrices of semantic categories into a neural layer or slice and, at the same time, direct the flow of information arriving at the layer to a specific address, and then forward the output information toward the corresponding targets. Given a neural topographic pattern, the tensor representation will place a set of associative matrix memories within a topographic regionalized host matrix in such way that they reproduce the empirical pattern of patches in the actual neural layer. Progressive approximations to this goal are accomplished by avoiding excessive overlap of memories or the existence of empty regions within the host matrix.