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 @ 4:34 PM
About once a month at Sigma Xi headquarters, we liven up the lunch hour with an American Scientist Pizza Lunch talk. In these informal lectures, scientists describe new research to nonscientists. The series is light on jargon but heavy on solid science. Each Pizza Lunch offers an in-depth look at its subject, whether it's bedbugs or the smart grid. Click below to read about and download these talks -- and to subscribe!
JSTOR, the online academic archive, now contains complete back issues of American Scientist from its inception in 1913 (as Sigma Xi Quarterly) through 2005.
The table of contents for each issue is freely available to all users; those with institutional access can read each complete issue.
View the full collection here.