dimensional norms Assaf Naor Princeton University SODA17 - - PowerPoint PPT Presentation

Probabilistic clustering of high dimensional norms

Assaf Naor Princeton University SODA’17

Partitions of metric spaces

Let be a metric space and a partition

• f X.

(X; dX)

P

Given a point is the unique cluster in which contains x.

P(x)

P x 2 X;

Given a point is the unique cluster in which contains x.

P(x)

P x 2 X;

x

Given a point is the unique cluster in which contains x.

P(x)

P x 2 X;

P(x)

Given a point is the unique cluster in which contains x.

P(x)

P x 2 X;

x

Given a point is the unique cluster in which contains x.

P(x)

P x 2 X;

P(x)

Given , the partition is bounded if all the clusters of have diameter at most

8x 2 X; diamX ¡ P(x) ¢ := max

u;v2P(x) dX(u;v) 6 ¢:

¢ > 0

P P ¢:

¢¡

Given , the partition is bounded if all the clusters of have diameter at most

8x 2 X; diamX ¡ P(x) ¢ := max

u;v2P(x) dX(u;v) 6 ¢:

¢ > 0

P P ¢:

¢¡

6 ¢ 6 ¢ 6 ¢ 6 ¢ 6 ¢ 6 ¢

Random partitions

• Key goal in several areas of computer science

and mathematics: use a bounded partition to “simplify” the metric space.

¢¡

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.

¢¡

¢

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.”

¢¡

¢

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…

¢¡

¢

Separating random partitions

Definition (Bartal, 1996): Suppose that is a metric space and A distribution over bounded random partitions of X is said to be separating if

(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].)

(X; dX)

¾;¢ > 0: P ¢¡

¾¡

8x;y 2 X; P £ P(x) 6= P(y) ¤ 6 ¾ ¢dX(x;y):

Modulus of separated decomposability

Denote by the minimum such that for every there is a separating distribution over bounded random partitions of 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.

SEP(X)

¾ > 0 ¢ > 0

¾¡

¢¡

(X;dX):

Theorem (Bartal, 1996): If then Goal of present work: to study 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].

jXj = n

SEP(X) . logn: SEP(X)

Sharp a priori bounds

Theorem: Suppose that X is an n-dimensional normed space. Then 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).

pn . SEP(X) . n:

Both bounds are asymptotically sharp, as shown in [CCGGP98]. In fact, it is proved there that For and

pn . SEP(X) . n: SEP(`n

2) ³ pn

and SEP(`n

1) ³ n:

p 2 [1;1) x = (x1;:::;xn) 2 Rn;

kxk`n

p :=

µ

n

X

j=1

jxjjp ¶ 1

p

kxk`n

1 :=

max

j2f1;:::;ng jxjj:

In [CCGGP98], Charikar-Chekuri-Goel-Guha-Plotkin asserted that The upper bound on in the above equivalence is valid as stated for all SEP(`n

p) ³

( n

1 p

if 1 6 p 6 2; n1¡ 1

p

if 2 6 p 6 1:

SEP(`n

p)

1 6 p 6 1;

In [CCGGP98], Charikar-Chekuri-Goel-Guha-Plotkin asserted that The upper bound on in the above equivalence is valid as stated for all but we show here that the matching lower bound is incorrect when SEP(`n

p) ³

( n

1 p

if 1 6 p 6 2; n1¡ 1

p

if 2 6 p 6 1:

SEP(`n

p)

1 6 p 6 1; 2 < p 6 1:

In [CCGGP98], Charikar-Chekuri-Goel-Guha-Plotkin asserted that The upper bound on in the above equivalence is valid as stated for all but we show here that the matching lower bound is incorrect when Thus, in particular, we obtain an asymptotically better probabilistic clustering of, say, SEP(`n

p) ³

( n

1 p

if 1 6 p 6 2; n1¡ 1

p

if 2 6 p 6 1:

SEP(`n

p)

1 6 p 6 1; 2 < p 6 1: `n

1:

Theorem: For every In particular, the previous best known bound when was (and this was asserted in [CCGGP98] to be sharp), but here we show that actually

p 2 [2; 1]; SEP(`n

p) .

p nminfp;logng:

p = 1

SEP(`n

1) . n

pn . SEP(`n

1) .

p nlogn:

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

• f [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 i.e.,

p 2 [1; 2]; p 2 [1; 2] = ) SEP(`n

p) ³ n

1 p:

Refined probabilistic partitions for sparse or rapidly decaying vectors

For and denote by the subset of consisting of all of those vectors with at most k nonzero entries, equipped with the metric. Theorem: For every we have

n 2 N k 2 f1;:::;ng (`n

p)6k

Rn

`n

p

p > 1

SEP ¡ (`n

p)6k

¢ . kmaxf 1

p ; 1 2g

r log ³n k ´ + minfp; log ng:

The special case becomes 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 with

p = 2

8 k 2 f1; : : : ; ng; SEP ¡ (`n

2)6k

¢ . r k log ³en k ´ :

`n

p

p = 1 + log(n=k):

Mixed-metric random partitions

Theorem: For every and there exists a distribution over random partitions of with the following properties. 1) 2) For every

p 2 [1; 1] ¢ > 0

P Rn

8x 2 Rn; diam`n

p

¡ P(x) ¢ 6 ¢:

P £ P(x) 6= P(y) ¤ . n

1 p p

minfp; log ng ¢ ¢ kx ¡ yk`n

2 :

x;y 2 Rn;

In particular, the special case shows that

• ne can obtain a random partition of into

clusters of diameter at most yet with the exponentially stronger Euclidean separation property

p = 2 Rn `n

1

¢

8x; y 2 Rn; P £ P(x) 6= P(y) ¤ . plog n ¢ ¢ kx ¡ yk`n

2 :

Iterative ball partitioning method

Karger-Motwani-Sudan (1998), Charikar-Chekuri-Goel-Guha-Plotkin (1998), Calinescu-Karloff-Rabani (2001). Iteratively remove balls of radius centered at i.i.d. points in the normed space X.

¢=2 BX = fx 2 X : kxkX 6 1g:

= x1 + ¢ 2 BX:

= x2 + ¢ 2 BX:

Theorem: Let be a norm on and let be the random partition that is obtained using iterative ball partitioning where the underlying balls are balls of radius in the norm Then (by design) for all and for every we have

P

k ¢ kX

Rn

¢=2 k ¢ kX: diamX ¡ P(x) ¢ 6 ¢ x 2 Rn

P £ P(x) 6= P(y) ¤ . voln¡1 ¡ Proj(x¡y)?(BX) ¢ ¢voln(BX) ¢ kx ¡ yk`n

2 :

x;y 2 Rn

Sharp when the right hand side is < 1 (using SchmuckenschlÄ ager [1992]).

Extremal hyperplane projections

The previously stated theorems about random partitions of follow from this general theorem in combination with the evaluation of the extremal volumes of hyperplane projections

• f the unit ball of that were obtained by

Barthe-N. (2002).

`n

p

`n

p

Extremal hyperplane projections

Theorem (Barthe-N., 2002): For every the following function is increasing in p. When the above ratio attains its maximum when

a 2 Rn r f0g;

p 7! voln¡1 ¡ Proja?(B`n

p )

¢ voln¡1(B`n¡1

p

) :

p > 2; a = (1;1;:::;1):

The need to use an auxiliary metric

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 ), then one obtains a separation modulus of n. In other words, one cannot obtain our better estimate using the intrinsic metric of the space that we wish to partition!

`n

1;

[¡¢=2;¢=2]n SEP(`n

1) .

p nlogn

The need to use an auxiliary metric

Our bound follows by applying this procedure using balls in the metric that is induced from The metrics on and are O(1)-equivalent (the balls in are “rounded cubes”). But the corresponding volumes change drastically, which allows our theorem to yield a better (almost sharp) bound on

`n

log n:

`n

1

`n

log n

`n

log n

SEP(`n

1):

Further applications

• Solution of longstanding open problems on the

extension of Lipschitz functions.

• Improved probabilistic partitions of the

Schatten-von Neumann trace classes and their subset consisting of all the matrices of rank at most k (N.-Schechtman, forthcoming); improved Lipschitz extension theorems for

• New volumetric stability theorems.
• Several additional results in full journal version.

Sn

p

Sn

p: