The two talks will discuss a notion of "deviation from flatness" called Beta. If we are given a set K in Euclidean space, we select a cube Q of sidelength L and renormalize K intersect Q to scale 1. We then find the deviation of that set from a best fitting line and call the associated number Beta(Q). There is a multiscale theory of these Beta Numbers that is useful in problems (e.g. those related to Traveling Salesman Problems = TSP) where the critical dimension is one. Examples include TSP, singular integrals on curves or sets, analytic capacity, and estimates on Green's functions or harmonic measure. Instead of using this definition one could approximate by best fitting hyperplanes of a fixed dimension D. Another version is to Define Beta(Q) by starting with a probability distribution P and renormalize Q to be the unit cube . Simultaneously one renormalizes P (restricted to Q) to get a new probability distribution P' on the unit square. One then performs classical least squares analysis to P', where one looks for the best fitting hyperplane of dimension D. On the numerical level this corresponds to doing a multiscale version of the Singular Value Decomposition for a distribution. This theory comes with theorems related to TSP, but with estimates for Volumes of approximating "Manifolds" of any dimension D. One of the features of this analysis is that it gives a fast (essentially as fast as SVD) method for preprocessing data. These algorithms avoid the curse of dimensionality: complexity growing exponentially with the dimension of the ambient space. All of these problems are closely related in spirit to Wavelet Analysis, and there even exists a "Dictionary" between the two that allows for translation of both concepts and theorems.
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