On finite representations of infinite problems

A very nice theme that has emerged from research on perception, during the past couple of decades, is the idea that the apparent complexity of a variety of things ranging from images to natural language may be described as synthesis from a much smaller basis. These kinds of generative models are the bread and butter of one of the major branches of statistical machine learning today. One also finds interesting biological evidence, e.g., the famous basis set of Olshausen and Field – based on the nice observation that the natural statistics of images in the world induces the lower dimensional representation.

Where is the action/decision equivalent of these? To be sure, people have applied all of the heavy machinery, ranging from Bayesian Reinforcement Learning to compression of value functions. But, somehow, the payoff from this seems ambiguous at best. We don’t really have robots that can go off and perform wildly different tasks using a nice clean basis of policies that are quickly composed.

One obvious distinction, that often doesn’t get discussed except by people with deep background, is that the perception problem has the singular advantage of admitting a relatively well posed notion of goodness/penalty, which makes statistics well posed too. I want to reconstruct some aspect of the image I see – this was the raison d’etre of a whole variety of approximation theory people in applied math before the vision folks showed up asking for it! On the other hand, describing all possible decision making problems in a finite setting requires a much more elaborate notion of invariance. Could it be that the final answer to this question will demand more recent mathematical tools, perhaps even topology and the mathematics behind game theory?

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