Uncertainty Principles , Extractors , and Explicit Embeddings of L 2 into L 1 Piotr Indyk

The area of geometric functional analysis1 is concerned with studying the properties of geometric (normed) spaces. A typical question in the area is: for two spaces X and Y , equipped with norms ‖·‖X and ‖·‖Y , under which conditions is there an embedding F : X → Y such that for any p, q ∈ X, we have ‖p − q‖X ≤ ‖F (p) − F (q)‖Y ≤ C‖p − q‖Y for some constant2 C ≥ 1 ? A ubiquitous tool for constructing such embeddings is the probabilistic method: the mapping is chosen at random from some distribution, and one shows that it “works” with high probability. Unfortunately, this approach does not lead to a concrete (or explicit) example of an embedding. A prototypical problem in the area is that of embedding ln 2 into l m 1 . It is known [FLM77] that there exist embeddings with both the distortion and the “dimension blowup” m/n bounded by a constant. However, the proof of that theorem is probabilistic: one shows that a “random” mapping preserves the distance between a fixed pair of points with high probability, and concludes that the same holds for any pair of points if the aforementioned probability is high enough. The problem of finding an explicit mapping with similar properties has been a subject of several studies (see Table 1). For constant distortion the best known result, attributed to Rudin [Rud60], guarantees m = O(n2); see also [LLR94] for an alternative proof. The problem of finding an explicit embedding with “low” distortion and dimension blowup has been posed, e.g., in [JS01] (Section 2.2), or in [Mil00] (Problem 8), or in [Mat04] (Problem 2.1).