Brian Hayes writes: "But note that the effective reduction in dimension works even when we don’t know which of the 360 dimensions can safely be ignored. That’s almost magical.
How commonplace is this phenomenon? Is it just a fluke, or confined to a narrow class of problems? The answer is not yet entirely clear, but a notion called “concentration of measure” offers a reason for optimism. It suggests that the high-dimension world is mostly a rather smooth and flat place, analogous to a high-gravity planet where it costs too much to create jagged alpine landscapes."
There is a related idea of "sparsity" of natural phenomena where the information has a much lower dimensionality than the overall dimensionality of the problem. And, there is a related approach to finding low dimensional solutions (about which the author himself wrote in the past): compressive sampling. It is interesting to note that one approach to compressive sampling is based on using random projections.
posted by Michael Korkin
July 5, 2011
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