What are the basic principles of representation and computation in the nervous system? Cognitive scientists have argued for a theory based upon compositionality, which refers to the evident ability of brains to represent objects, scenes, thoughts and actions in a hierarchical structure. I am studying a mathematical formulation for compositionality, and the implications of this formulation for interpreting neural activity patterns and for building computer vision systems.

Biography

Dr. Geman is a James Manning Professor of Applied Mathematics in the Division of Applied Mathematics, Brown University. He graduated from the Massachusetts Institute of Technology in 1977, having completed his thesis on stochastic differential equations with smooth mixing processes. His advisors were Herman Chernoff and Frank Kozin.
Professor Geman's current research focuses on the mathematical formulation for compositionality, and the implications of this formulation for interpreting neural activity patterns and for building computer vision systems.

Research Description

Compositional Vision

Compositionality refers to the ability of humans to represent entities as hierarchies of reusable parts. The parts themselves are meaningful entities and are reusable in a near-infinite assortment of meaningful combinations. Compositional hierarchies can be fitted with a probability distribution and used as prior models in a Bayesian scene interpretation system.


Statistical Analysis of Neurophysiological Data

The statistical analysis of neuronal data in awake animals presents unique challenges. The status of tens of thousands of pre-synaptic neurons, not directly influenced by the experimental paradigm, is largely out of the control of the experimenter. Time-honored assumptions about "repeated" samples are untenable. These observations essentially preclude the gathering of statistical evidence for a lack of precision in the cortical micro circuitry, but do not preclude collecting statistical evidence against any given limitation on precision or repeatability. Statistical methods are being devised to support the systematic search for fine-temporal structure in stable multi-unit recordings.


Statistical Analysis of Rare Events in the Markets

When it comes to the prices of stocks and other securities, it seems that rare events are never rare enough. But they are too rare for meaningful statistical study. In order to test financial models of price fluctuations, focused on excursions, the issue of small samples can be side stepped by declaring an event "rare" if it is unusual relative to the interval of observation. Every interval has its own rare events, by fiat, and in fact as many as we need. Different classes of models have different invariants to the timings of these "rare" events. These invariants open the door to combinatorial-type hypothesis tests, under which many of the usual models do not hold up very well. The evidence is for very rapidly changing dynamics.


Statistical Analysis of Natural Images

Take a digital photo of a natural outdoor scene. For simplicity, convert the photo from color to black and white. The photo can be reduced, or scaled, to make a new (smaller) picture, say half the size in both dimensions. In comparison to the original picture, the new picture is of a scene in which each of the original objects, and in fact every imaged point, has been relocated twice as far from the camera. This "stretching" is artificial in that it does not correspond to any movement of the camera in the real world. Yet the picture looks perfectly normal, and the local spatial statistical structure (e.g. the distribution of values of horizontal or vertical derivatives) is largely indistinguishable from the local spatial statistical structure of the original. "Images of natural scenes are scale invariant." The source of scale invariance in natural images is an enduring mystery.

Neural Representation and Neural Modeling

We can imagine our house or apartment with the furniture rearranged, the walls repainted, and the floors resurfaced or re-covered. We can rehearse a tennis stroke, review a favorite hike, replay a favorite melody, or recall a celebrated speech "in our mind's eye," without moving a limb or receiving a sound. It is a mistake to model cortical function without acknowledging the cortical capacity for manipulating structured representations and simulating elaborate motor actions and perceptual stimuli.

It is tempting to model networks of neurons as networks of integrate-and-fire units, but integration is linear and overwhelming evidence demonstrates the highly nonlinear, and in fact space and time-dependent nature, of dendritic processing. An argument can be made that these nonlinearities, by their nature, promote a rich and local-correlative structure, as anticipated by Abeles, von der Malsburg and others, within the microcircuits. These spatio-temporal patterns, with their correlation-induced topologies, would be good candidates for the basic units of cognitive processing.

Awards

Highest Honors in Physics, University of Michigan 1971
Presidential Young Investigator Award 1984–1989
Fellow, Institute of Mathematical Statistics 1984
Elected, International Statistical Institute, 1991
Philip J. Bray Award for Excellence in Teaching in the Physical Sciences, Brown University, 2001
Elected, National Academy of Sciences, 2011
ISI Highly Cited Researcher
1986 International Congress of Mathematicians, invited lecture
1997 Rietz Lecture, Institute of Mathematical Statistics.
2001 Hotelling Memorial Lectures, University of North Carolina
2007 Jim Press Endowed Research Lecture, University of California, Riverside
2008 Bahadur Memorial Lectures, University of Chicago
2009 The Movies, the Markets, and Mathematics. Lunch briefing on Capitol Hill on behalf of the American Mathematical Society
2011 Plenary Speaker, Annual Meeting of the Taiwan Mathematical Society

Affiliations

Please see curriculum vita.

Funded Research

Please see curriculum vita

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