David Mumford

Brown University


Election Year: 1975
Primary Section: 32, Applied Mathematical Sciences
Secondary Section: 11, Mathematics
Membership Type: Member

Research Interests

I work in the area of machine and natural intelligence, following the statistical approach pioneered by Ulf Grenander, which we call "Pattern Theory." In this approach, thinking is modeled as statistical inference rather than logic and learning results from the accumulation of massive data from interactions with the world. My work concentrates on visual perception which seems to be more approachable than high-level thinking, more complex than auditory and tactile perception yet has been solved by three distinct classes of animals (cephalopods, birds, and mammals) so it can't be that hard! One aspect of this research is the construction of probability models for the variables of vision: the observed images, the shape, placement and illumination of objects, the texture of their surfaces etc. A second question is how to sample and estimate with these models, e.g., compute conditional means and modes. Many approaches involve computing with one or more samples from the distribution which evolve deterministically or stochastically, smoothly or with jumps. Sampling from the distribution can be viewed as feedback, i.e., prior knowledge of high level structures guides the reconstruction of an image. A third set of questions concerns how such statistical estimation may be performed in cortex, in neural nets with feedback. Perhaps the biggest open question here is whether the information being handled by neurons is contained only in their firing rates or in the precise timing and synchrony of their spikes in the full network.

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