  Geometric Sensitive Hashing functions, a family of Local Sensitive Hashing
functions, are neural network models that learn class-specific manifold
geometry in supervised learning. However, given a set of supervised learning
tasks, understanding the manifold geometries that can represent each task and
the kinds of relationships between the tasks based on them has received little
attention. We explore a formalization of this question by considering a
generative process where each task is associated with a high-dimensional
manifold, which can be done in brain-like models with neuromodulatory systems.
Following this formulation, we define \emph{Task-specific Geometric Sensitive
Hashing~(T-GSH)} and show that a randomly weighted neural network with a
neuromodulation system can realize this function.
