Probabilistic clustering of high dimensional norms. Assaf Naor Princeton University SODA 17

Transcription

1 Probabilistic clustering of high dimensional norms Assaf Naor Princeton University SODA 17

2 Partitions of metric spaces Let (X; d X ) be a metric space and a partition of X. P

3 Given a point x 2 X; P(x) is the unique cluster in which contains x. P

4 Given a point x 2 X; x P(x) is the unique cluster in which contains x. P

5 Given a point x 2 X; P(x) P(x) is the unique cluster in which contains x. P

6 Given a point x 2 X; x P(x) is the unique cluster in which contains x. P

7 Given a point x 2 X; P(x) P(x) is the unique cluster in which contains x. P

8 Given, the partition is bounded if all the clusters of have diameter at most > 0 P P : 8 x 2 X; diam X P(x) := max u;v2p(x) d X(u; v) 6 :

9 Given, the partition is bounded if all the clusters of have diameter at most > 0 P P : x 2 X; diam X P(x) := max u;v2p(x) d X(u; v) 6 :

10 Random partitions - Key goal in several areas of computer science and mathematics: use a bounded partition to simplify the metric space.

11 Random partitions - Key goal in several areas of computer science and mathematics: use a bounded partition to simplify the metric space. - The partition should mimic the coarse geometric structure (at distance scale ) in some meaningful way.

12 Random partitions - Key goal in several areas of computer science and mathematics: use a bounded partition to simplify the metric space. - The partition should mimic the coarse geometric structure (at distance scale ) in some meaningful way. - Regions near boundaries should be thin.

13 Random partitions - Key goal in several areas of computer science and mathematics: use a bounded partition to simplify the metric space. - The partition should mimic the coarse geometric structure (at distance scale ) in some meaningful way. - Regions near boundaries should be thin. - Quite paradoxical, but randomness helps here

14 Separating random partitions Definition (Bartal, 1996): Suppose that is a metric space and ¾; > 0: P (X; d X ) A distribution over bounded random partitions of X is said to be ¾ separating if 8 x; y 2 X; P P(x) 6= P(y) 6 ¾ d X(x; y): (Implicit in several early works, variety of applications: Leighton- Rao [1988], Auerbuch-Peleg [1990], Linial-Saks [1991], Alon- Karp-Peleg-West [1991], Klein-Plotkin-Rao [1993], Rao [1999].)

15 Modulus of separated decomposability Denote by SEP(X) the minimum ¾ > 0such that for every > 0 there is a ¾ separating distribution over bounded random partitions of (X; d X ): Note: we are ignoring here technical measurability issues that are important for mathematical applications in the infinite setting. For TCS purposes, it suffices to deal with random partitions of finite subsets of X.

16 Theorem (Bartal, 1996): If jxj = n SEP(X). log n: then Goal of present work: to study SEP(X) for finite dimensional normed spaces X (and subsets thereof). Originated in Peleg-Reshef [1998], followed by important work of Charikar-Chekuri-Goel-Guha- Plotkin [1998].

17 Sharp a priori bounds Theorem: Suppose that X is an n-dimensional normed space. Then p n. SEP(X). n: The upper bound follows from [CCGGP98]. The lower bound hasn t been noticed before: it follows from a theorem of Bourgain-Szarek (1988) that is a consequence of the Bourgain- Tzafriri restricted invertibility principle (1987).

18 p n. SEP(X). n: Both bounds are asymptotically sharp, as shown in [CCGGP98]. In fact, it is proved there that SEP(`n2) ³ p n and SEP(`n1) ³ n: For p 2 [1; 1) and x = (x 1 ; : : : ; x n ) 2 R n ; µ X n 1 kxk`n p := jx j j p p kxk`n 1 := j=1 max jx jj: j2f1;:::;ng

19 In [CCGGP98], Charikar-Chekuri-Goel-Guha-Plotkin asserted that ( n SEP(`np) 1 p if 1 6 p 6 2; ³ n 1 1 p if 2 6 p 6 1: The upper bound on SEP(`np) in the above equivalence is valid as stated for all 1 6 p 6 1;

20 In [CCGGP98], Charikar-Chekuri-Goel-Guha-Plotkin asserted that ( n SEP(`np) 1 p if 1 6 p 6 2; ³ n 1 1 p if 2 6 p 6 1: The upper bound on SEP(`np) in the above equivalence is valid as stated for all 1 6 p 6 1; but we show here that the matching lower bound is incorrect when 2 < p 6 1:

21 In [CCGGP98], Charikar-Chekuri-Goel-Guha-Plotkin asserted that ( n SEP(`np) 1 p if 1 6 p 6 2; ³ n 1 1 p if 2 6 p 6 1: The upper bound on SEP(`np) in the above equivalence is valid as stated for all 1 6 p 6 1; but we show here that the matching lower bound is incorrect when 2 < p 6 1: Thus, in particular, we obtain an asymptotically better probabilistic clustering of, say, `n1:

22 Theorem: For every p 2 [2; 1]; SEP(`np). p n minfp; log ng: In particular, the previous best known bound when p = 1 was SEP(`n1). n (and this was asserted in [CCGGP98] to be sharp), but here we show that actually p n. SEP(`n 1 ). p n log n:

23 The source of the error in [CCGGP98] was that it relied on unpublished work of Indyk (1998) that was not published since then; we confirmed with Indyk as well as with some of the authors of [CCGGP98] that there is indeed a flaw in the (unpublished) work of Indyk that was cited. There is no flaw in the proof of [CCGGP98] in the range p 2 [1; 2]; i.e., p 2 [1; 2] =) SEP(`np) ³ n 1 p :

24 Refined probabilistic partitions for sparse or rapidly decaying vectors For n 2 N and k 2 f1; : : : ; ng denote by (`np) 6k the subset of R n consisting of all of those vectors with at most k nonzero entries, equipped with the `np metric. Theorem: For every p > 1 we have SEP r ³ (`np) 6k. k maxf 1 p ; 1 2g n log + minfp; log ng: k

25 The special case p = 2 becomes 8 k 2 f1; : : : ; ng; SEP (`n2 ) 6k. rk log ³ en k : A curious aspect of this bound is that despite the fact that it is a statement about Euclidean geometry, our proof involves non-euclidean geometric considerations. Specifically, the ubiquitous iterative ball partitioning method is applied to balls in `np with p = 1 + log(n=k):

26 Mixed-metric random partitions Theorem: For every p 2 [1; 1] and > 0 there exists a distribution P over random partitions of with the following properties. 1) 8 x 2 R n ; diam`n p P(x) 6 : 2) For every x; y 2 R n ; R n P P(x) 6= P(y). n 1 p p minfp; log ng kx yk`n 2 :

27 In particular, the special case p = 2 shows that one can obtain a random partition of R n into clusters of `n1 diameter at most yet with the exponentially stronger Euclidean separation property 8 x; y 2 R n ; P P(x) 6= P(y) p log n. kx yk`n2 :

28 Iterative ball partitioning method Karger-Motwani-Sudan (1998), Charikar-Chekuri-Goel-Guha-Plotkin (1998), Calinescu-Karloff-Rabani (2001). Iteratively remove balls of radius =2 at i.i.d. points in the normed space X. centered B X = fx 2 X : kxk X 6 1g:

29 = x B X:

30 = x B X:

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38 Theorem: Let k k X be a norm on and let P be the random partition that is obtained using iterative ball partitioning where the underlying balls are balls of radius =2 in the norm k k X : Then (by design) diam X P(x) 6 for all x 2 R n and for every x; y 2 R n we have P P(x) 6= P(y). vol n 1 Proj(x y)?(b X ) vol n (B X ) R n kx yk`n 2 : Sharp when the right hand side is < 1 (using SchmuckenschlÄager [1992]).

39 Extremal hyperplane projections The previously stated theorems about random partitions of `np follow from this general theorem in combination with the evaluation of the extremal volumes of hyperplane projections of the unit ball of `np that were obtained by Barthe-N. (2002).

40 Extremal hyperplane projections Theorem (Barthe-N., 2002): For every a 2 R n r f0g; the following function is increasing in p. p 7! vol n 1 Proja?(B`n ) p : ) vol n 1 (B`n 1 p When when p > 2; the above ratio attains its maximum a = (1; 1; : : : ; 1):

41 The need to use an auxiliary metric `n1; In the special case of if one applies iterative ball partitioning using balls in the intrinsic metric (which are in this case simply axis-parallel hypercubes [ =2; =2] n ), then one obtains a separation modulus of n. In other words, one cannot obtain our better estimate SEP(`n1). p n log n using the intrinsic metric of the space that we wish to partition!

42 The need to use an auxiliary metric Our bound follows by applying this procedure using balls in the metric that is induced from `nlog n: The metrics on and `nlog n are O(1)-equivalent (the balls in `nlog n are rounded cubes ). But the corresponding volumes change drastically, which allows our theorem to yield a better (almost sharp) bound on SEP(`n1): `n1

43 Further applications Solution of longstanding open problems on the extension of Lipschitz functions. Improved probabilistic partitions of the Schatten-von Neumann trace classes S n p and their subset consisting of all the matrices of rank at most k (N.-Schechtman, forthcoming); improved Lipschitz extension theorems for S n p : New volumetric stability theorems. Several additional results in full journal version.