David Mumford

Pattern Theory started in the 70’s with the ideas of Ulf Grenander and his school at Brown. The aim is to analyze from a statistical point of view the patterns in all ‘signals' generated by the world, whether they be images, sounds, written text, DNA or protein strings, spike trains in neurons, time series of prices or weather, etc. Pattern theory proposes that the types of patterns – and the hidden variables needed to describe these patterns – found in one class of signals will often be found in the others and that their characteristic variability will be similar. The underlying idea is to find classes of stochastic models which can capture all the patterns that we see in nature, so that random samples from these models have the same ‘look and feel' as the samples from the world itself. Then the detection of patterns in noisy and ambiguous samples can be achieved by the use of Bayes's rule, a method that can be described as ‘analysis by synthesis'.

I recently gave an overview of this approach to modeling perception at the ICM02: “Pattern Theory: The Mathematics of Perception”. A book entitled “Pattern Theory Through Examples”, with Agnes Desolneux is in preparation.

Current research

A) Object recognition requires that you know when two shapes are ‘similar’. But what does similar mean? The mathematician says: make the set of all (two dimensional, three dimensional or higher) shapes into the points of an infinite-dimensional space and put a metric on this space. If shape means an open subset of Euclidean space with smooth boundary, then this space should be a Banach manifold, but a highly non-linear one. What structure does this space have? I am looking at various metrics and the associated completions of the space of shapes; the geodesics in these metrics and the curvature of the space; examples and applications to object recognition.

B) There is now a much better understanding of how to solve problems of perception on a computer: but do any of these ideas correspond to processes occurring in our brains? Standard modeling of cortex employs very primitive mechanisms – neural nets – but it is hard to see how these suffice for fast, robust perception. Big questions are: what is the role of feedback, how are concepts ‘bound’ on the fly to create a percept, how can more than one interpretation or hypothesis be entertained at once in the cortex? Radical ideas for doing this can be found in my recent paper with Tai Sing Lee “Hierarchical Bayesian Inference in Visual Cortex”.

Indra’s Pearls, with Caroline Series and Dave Wright, Cambridge Univ. Press

From the Introduction:

This is a book about serious mathematics, but one, which we have written primarily for non-mathematicians. It is an account of our exploration of a family of symmetrical but infinitely convoluted sets, part of the modern investigation of how chaos evolves from very simple rules, producing intricate complexity on every scale from the very large to the very small. In our case, two rules, each on its own producing a pair of spirals, are allowed to interact. Our scheme is not at all arbitrary; it forms parts of a century old mathematical dream, involving much of the deep mathematics of the 19th century, conceived by the great German geometer Felix Klein. With the aid of modern computers for rendering the results, the answers to our `What if...?' questions turned to be not only intellectually fascinating but also strikingly beautiful. Sometimes the outcome is simple, sometimes it is total disorder and sometimes -- and this is the most exciting case -- it has layer upon layer of structure teetering on the very brink of chaos. There is no religion in our book but we were amazed at how well our constructions reflected the ancient Buddhist metaphor of Indra's net. Mathematicians often use the word `beautiful' in talking about their proofs and ideas, but in this case our judgment has been confirmed by a number of unbiased and definitely non-mathematical people.

Most mathematics is accessible, as it were, only by crawling through a long tunnel in which you laboriously build up your vocabulary and skills as you abstract your understanding of the world. But the mathematics behind the figures we drew turned out not to need too much in the way of preliminaries. So long as you got through high school algebra with some confidence, everything we say should be understandable, given a bit of careful reading here and there. And if not, then browsing through the figures alone should give a sense of our journey. Our dream is that this book will reveal to a larger audience that mathematics is not alien, cold and remote but just a very human exploration of the patterns of the world, one which thrives on play and surprise and beauty.

8 See Indra’s home page (stills) with movies here.

Spring 2006: AM 18

Modeling the World with Mathematics:

clocks, waves, chaos and chance

Motivation:

The idea is widespread that we are ‘two cultures”, a mathematically literate minority of physical, computer and mathematical scientists, engineers and economists and an alienated majority with a sense that mathematics is abstruse, too hard to use or just plain “fuzzy”. The goal of this course is to explore some ways to bridge this gap. It is addressed to students who are not interested in or do not have the time for the conventional rigorous sequence of courses offered by the Mathematics and Applied Mathematics departments. It seeks to show these students something about what is exciting in mathematics and how it is relevant to their lives. The fundamental premise is that much of higher mathematics is not much more than simple arithmetic – if you use the power of modern computers to do a lot of it fast enough. In addition, the goal is to teach each new discovery in the context of applications which impact our everyday lives and to take a historical approach, using readings from the original books and articles in which an idea first appeared.

What do you need to know to take this course? A semester of high school calculus is ideal, but being confident with algebra is what is really needed. We will use the computer calculation tool, MatLab. This is an experimental course and the exact topics may be varied as we go along. Here is the syllabus as I see it now:

Part I: Clocks. The powerful idea of measuring time and space accurately.

The historical context – 13^th century, why did Europe change? Clocks, accounting, latitude and longitude. What is a clock? Simple harmonic motion, 1^st and 2^nd differences, discovering it yourself with strobes and a computer, its universality and formal expression. Planets and the way it really developed.

Part II: Waves. Understanding and harnessing wave-like phenomena is the core of modern technology

Music and Pythagoras, the 1D wave equation, superposition. Water waves and some of their exotic manifestations. Electro-magnetic waves, the transatlantic cable and all the technology that we use now. The 19^th century dream: the universe is nothing but a set of PDE’s.

Part III: Chaos. The real limits of predictability and computability

Lotka-Volterra: the dance of the predator and the prey, epidemics and the world of non-linear effects. Weather and the butterfly effect (Liapounov exponents), the Lorenz attractor and fractals. Gödel’s theorem, P vs. NP.

Part IV: Chance. Randomness can sometimes be useful

The Manhattan project and Dimitri Metropolis. Brownian motion and options pricing. IQ’s and heavy tails, p-values and fingerprints. Logic vs. probability/statistics as a model of thought.