State-dependent delay differential equations for subspace clustering

We developed a neural network architecture — projective adaptive resonance theory (PART) — to detect low-dimensional patterns in a high-dimensional data set. This theory has been applied for gene filtering and cancer diagnosis, neural spiking trains clustering, ontology construction, and stock associations detection. The key feature of a PART network is a hidden layer which incorporates a selective output signal mechanism (SOS) that calculates the similarity between the output of a given input neuron with the corresponding component of the template of a candidate cluster neuron and allows the signal to be transmitted to the cluster neuron only when the similarity measure is sufficiently large. This clustering architecture was recently refined to incorporate adaptive transmission delays and signal transmission information loss (PART-D). The resultant selective SOS is based on the assumption that the signal transmission velocity between input processing neurons and clustering neurons is proportional to the similarity between the input pattern and the feature vector of the clustering neuron. The mathematical model governing the evolution of the signal transmission delay (the short-term memory traces and the long-term memory traces) represents a new class of delay differential equations, where the evolution of the delay is described by a nonlinear differential equation involving the aforementioned similarity measure. This talk will describe the PART-D architecture and the associated delay differential systems, and discuss future directions how statedependent delay differential equations can be used to design algorithms for clustering in skewed subspaces or sub-manifolds of the data space.