Dimension reduction for finite trees in L1 James R. Lee Mohammad Moharrami University of Washington Arnaud De Mesmay École Normale Supérieure TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAA

dimension reduction in Lp Given an n-point subset X µ Rd, find a mapping such that for all x, y 2 X, n = size of X k = target dimension D = distortion Dimension reduction as “geometric information theory”

the case p=2 When p=2, the Johnson-Lindenstrauss transform gives, for every n-point subset X µ Rd and " > 0, Applications to… Statistics over data streams Nearest-neighbor search Compressed sensing Quantum information theory Machine learning

dimension reduction in L1 Natural to consider is p=1. n = size of X k = target dimension D = distortion History: - Caratheodory’s theorem yields D=1 and - [Schechtman’87, Bourgain-Lindenstrauss-Milman’89, Talagrand’90] Linear mappings (sampling + reweighting) yield D · 1+" and - [Batson-Spielman-Srivastava’09, Newman-Rabinovich’10] Sparsification techniques yield D · 1+" and

the Brinkman-Charikar lower bound There are n-point subsets such that distortion D requires [Brinkman-Karagiozova-L 07] Lower bound tight for these spaces Very technical argument based on LP-duality. [L-Naor’04]: One-page argument based on uniform convexity.

Brinkman-Charikar and ACNN lower bounds. more lower bounds [Andoni-Charikar-Neiman-Nguyen’11]: There are n-point subsets such that distortion 1+" requires [Regev’11]: Simple, elegant, information-theoretic proof of both the Brinkman-Charikar and ACNN lower bounds. Low-dimensional embedding ) encoding scheme

the simplest of L1 objects A tree metric is a graph theoretic tree T=(V, E) together with non-negative lengths on the edges Easy to embed isometrically into RE equipped with the L1 norm.

dimension reduction for trees in L1 Charikar and Sahai (2002) showed that for trees one can achieve A. Gupta improved this to In 2003 in Princeton with Gupta and Talwar, we asked: Is possible? even for complete binary trees?

dimension reduction for trees in L1 Theorem: For every n-point tree metric, one can achieve and (Can get for “symmetric” trees.) Complete binary tree using local lemma Schulman’s tree codes Complete binary tree using re-randomization Extension to general trees

dimension reduction for the complete binary tree Every edge gets B bits ) target dimension = B log2n Choose edge labels uniformly at random. Nodes at tree distance have probability to get labels with hamming distance

dimension reduction for the complete binary tree Every edge gets B bits ) target dimension = B log2n Choose edge labels uniformly at random. Siblings have probability 2-B to have the same label, yet there are n/2 of them.

Lovász Local Lemma Pairs at distance L have probability to be “good” Number of dependent “distance L” events is LLL + sum over levels ) good embedding

Schulman’s tree codes LLL argument difficult to extend to arbitrary trees. Same as construction of Schulman’96: Tree codes for interactive communication

re-randomization Random isometry: For every level on the right, exchange 0’s and 1’s with probability half (independently for each level)

re-randomization Pairs at distance L have probability to be “good” Number of pairs at distance L is

extension to general trees Unfortunately, the general case is technical (paper is 50 pages) Obstacles: General trees do not have O(log n) depth Use “topological depth” of Matousek. How many coordinates to change per edge, and by what magnitude? Multi-scale entropy functional

? open problems Coding/dimension reduction: Extend/make explicit the connection between L1 dimension reduction and information theory. Close the gap: For distortion 10, is the right target dimension ? or Other Lp norms: Nothing non-trivial is known for