Hidden Invariants in Data IEEE International Symposium on Information Theory July 1, 2003 Yokohama, Japan Roger Brockett Engineering and Applied Sciences Harvard University

Points we hope to Establish in the Talk 1. There are a great many different types of channels providing reliable transmission of data but some of the most widely used are seldom studied as such. 2. In formalizing these one needs to be open to a range of ideas coming from analysis, algebra and geometry. 3. For some real world problems involving digital computing and neural information processing, differential inclusions can be natural and effective in characterizing these. 4. There often exists a type of “matched filter” which is effective in decoding such signals. 5. It seems probable that there are many applications of these ideas.

Outline 1. Some spaces of functions 2. Associating topological invariants to functions 3. Simple “decoders” that extract topological invariants 4. More complex decoders 5. Conclusions

Part 1: Some function spaces that do not show up in books but perhaps should. The point of the next 12 slides is to introduce a particular way of characterizing spaces of functions. Our claim will be that not only does this characterization capture much of what is important from an engineering point of view but that it also leads to a effective way to bridge the gap between analog and digital points of view.

a trajectory a set of possible trajectories

larger valve of b smaller valve of b

band limited quasi-periodic annular

smaller valve of b larger valve of b

The Topological Space for Quantizers The forbidden disks (actually the “all bets off” region) correspond to the need for the signal to pass through threshold cleanly. Without this the output of the quantized can not be trusted.

Part 2: A few things from topology you might have felt it was safe to postpone learning about. Our purpose here is to set up the mathematics needed to provide a concise characterization of the significant properties of the signals introduced above.

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Illustrating oriented closed curves on different spaces “pair of pants” space “two eyes” space

Part 3: Matching Systems to the Signals Our goal here is to show that when the input space is suitably restricted and the system is suitably chosen, then the system responds only to the topological type of the input, ignoring all other details. This is important because it demonstrates an extreme form of robustness one normally associates with digital systems. To make the ideas as transparent as possible we focus on a example.

equilibria for dx/dt= -sin x

The enforcing a refractory period.

Part 4: The Bigger Picture and Beyond Counting is fine but what do we need to do these ideas extend to a full range of capabilities? What are the overarching principles?

An analogy with something that has worked before. A finite state machine as the”skeleton” of a differential equation

Conclusions 1.There are a great many different types of channels in use and some of the most widely used are seldom studied. 2. Defining these in mathematical terms seems to require new ideas. 3. Differential inclusions can be natural and effective characterizations of some function spaces showing up in computing and biology. 4. We illustrated robust “decoding” of signals in such spaces using a class of “matched filters” 5. It seems that many other applications of these ideas area possible.