Friday, April 6th, 2007
3:30pm
Wean Hall 7220
Uncertainty Principles, Extractors, and Explicit Embeddings of L2 into L1
Piotr Indyk (MIT)
The area of geometric functional analysis is concerned with studying properties of geometric (normed) spaces. A typical question in the area is: given two geometric spaces X, Y, is there an embedding F of X into Y so that F distorts the distances between any pair of points by only a constant factor? One of the classic results of this type is a constant-distortion embedding of an n-dimensional space equipped with L2 norm, into an O(n)-dimensional space equipped with L1 norm (also known as Dvoretzky’s theorem for L1). Embeddings have many interesting applications, in computer science and beyond.
A ubiquitous tool for constructing such embeddings is the probabilistic method: the mapping is chosen at random, and one shows that it “works” with high probability. Unfortunately, this approach does not yield a concrete (or explicit) construction of an embedding. A folklore open problem has been to obtain explicit constructions with parameters that (almost) match the non-constructive bounds. However, the progress on this front has been somewhat limited. For example, the best known explicit construction of an embedding of an n-dimensional L2 space into an m-dimensional L1 space guarantees only m=O(n^2).
In this talk I will show an explicit construction of an embedding of an n-dimensional L2 space into L1 space of dimension n^{1+o(1)}. The construction utilizes two tools: discrete uncertainty principles (introduced by Donoho and Stark in the area of digital signal processing) and randomness extractors. If time allows, I will also show some related constructions, with application to signal sketching and compressed sensing.
