Efficient Sketches for Earth-Mover Distance, with Applications

David WoodruffIBM Almaden

Joint work with Alexandr Andoni, Khanh Do Ba, and Piotr Indyk

(Planar) Earth-Mover Distance

• For multisets A, B of points in [∆]2, |A|=|B|=N,

i.e., min cost of perfect matching between A and B

AaBA

aaBA )(min),EMD(:

EMD( , ) = 6 + 3√2 2

Geometric Representation of EMD

• Map A, B to k-dimensional vectors F(A), F(B)– Image space of F “simple,” e.g., k small– Can estimate EMD(A,B) from F(A), F(B) via some

efficient recovery algorithm E

F2 Rk

E ≈ EMD(A,B)

3

Geometric Representation of EMD: Motivation

• Visual search and recognition:– Approximate nearest neighbor under EMD

• Reduces to approximate NN under simpler distances• Has been applied to fast image search and recognition in

large collections of images [Indyk-Thaper’03, Grauman-Darrell’05, Lazebnik-Schmid-Ponce’06]

• Data streaming computation:– Estimating the EMD between two point sets given as a

stream• Need mapping F to be linear: adding new point a to A

translates to adding F(a) to F(A)• Important open problem in streaming [“Kanpur List ’06”]

4

Our result Non-norm O(∆ε) O(1/ε)

Prior and New Results

Paper Recovery Dimension Approx.

[Charikar’02, Indyk-Thaper’03] ℓ1 O(∆2) O(log ∆)

[Naor-Schechtman’06] ℓ1 Any Ω(log1/2 ∆)

Main TheoremFor any ε 2 (0,1), there exists a distribution over linear mappings F: R∆2 ! R∆ε s.t. for multisets A,B µ [∆]2 of equal size, we can produce an O(1/ε)-approximation to EMD(A,B) from F(A), F(B) with probability 2/3.

5

Geometric representation of EMD:

Implications

• Streaming:

• Approximate nearest neighbor:

Paper Space Approximation

[Indyk’04] logO(1)(∆N) O(log ∆)

Our result ∆ε logO(1)(∆N) O(1/ε)

Paper Space Query time Approximation

[Andoni-Indyk-Krauthgamer’09]

s2+ε 2∆1/α

∆O(1) sε O((α/ε) loglog s)

Our result 2∆ε log(s∆N)

O(1)

(∆ log(s∆N))O(1) O(1/ε)

* N = number of points

* s = number of data points (multisets) to preprocess α>1 free parameter 6

Proof Outline• Old [Agarwal-Varadarajan’04, Indyk’07]:

– Extend EMD to EEMD which:• Handles sets of unequal size |A| · |B| in a grid of side-length k• EEMD(A,B) = min|S|=|A| and S µ B EMD(A,S) + k¢|B\S|

• Is induced by a norm ||¢||EEMD, i.e., EEMD(A,B) = ||Â(A) – Â(B)||EEMD, where Â(A)2 R∆2 is the characteristic vector of A

– Decomposition of EEMD into weighted sum of small EEMD’s• O(1/ε) distortion

• New:– Linear sketching of “sum-norms”

EMD over [∆]2

EEMD over [∆ε]2

+ + … +

EEMD over [∆ε]2 EEMD over [∆ε]2

∆O(1) terms

7

Old Idea [Indyk ’07]

EEMD over [∆1/2]2

EEMD over [∆ε]2

+ + … +

EEMD over [∆ε]2 EEMD over [∆ε]2

∆O(1) terms

EMD over [∆]2

EMD over [∆]2

+ … +

EEMD over [∆1/2]2

Old Idea [Indyk ’07]EMD over [∆]2

Solve EEMD in each of ¢ cells,each a problem in [¢1/2]2

2

Old Idea [Indyk ’07]

2

Solve one additionalEEMD problem in [¢1/2]2

Should also scale edgelengths by ¢1/2

Old Idea [Indyk ’07]• Total cost is the sum of the two phases

• Algorithm outputs a matching, so its cost is at least the EMD cost

• Indyk shows that if we put a random shift of the [¢1/2]2 grid on top of the [¢]2 grid, algorithm’s cost is at most a constant factor times the true EMD cost

• Recursive application gives multiple [¢ε]2 grids on top of each other, and results in O(1/ε)-approximation

Main New Technical Theorem

12

Given C > 0 and λ > 0, if C/λ · ||M||1, X · C, there is a distribution over linear mappings

μ: Xn ! X(λ log n)O(1)

such that we can produce an O(1)-approximation to ||M||1,X from μ(M) w.h.p.

For normed space X = (Rt, ||¢||X) and M 2 Xn, denote ||M||1,X = ∑i ||Mi||X.

+ + … +

||M1||X ||M2||X ||Mn||X

||M||1, X =

Proof Outline: Sum of Norms• First attempt:

– Sample (uniformly) a few Mi’s to compute ||Mi||X

– Problem: sum could be concentrated in 1 block

• Second attempt:– Sample Mi w/probability proportional to ||Mi||X [Indyk’07]– Problem: how to do online?– Techniques from [JW09, MW10]?

• Need to sample/retrieve blocks, not just individual coordinates

M1 M2 M3 Mn……

M2 contains most of mass

13

Proof Outline: Sum of Norms (cont.)

• Our approach:– Split into exponential levels:

• Assume ||M||1, X · C

• Sk = i2[n] s.t. ||Mi||X 2 (Tk, 2Tk], Tk=C/2k

• Suffices to estimate |Sk| for each level k. How?

– For each level k, subsample from [n] at a ratesuch that event Ek (“isolation” of level k)

holds with probability proportional to |Sk|

– Repeat experiment several times, count number of successes

S1

1S2

S3

Sℓ

…M2M4, M7

M1, M3, M8, M9

M5, M10, Mn

M = (M1, M2, …, Mn)

M:

Subsample:

Ek?

Y N 14

Proof Outline: Event Ek

• Ek $ “isolation” of level k:– Exactly one i 2 Sk gets subsampled– Nothing from Sk’ for k’<k

• Verification of trial success/failure– Hash subsampled elements

• Each cell maintains vector sum of subsampled Mi’s that hash there

– Ek holds roughly (we “accept”) when:• 1 cell has X-norm in (0.9Tk, 2.1Tk]• All other cells have X-norm ≤ 0.9Tk

– Check fails only if:• Elements from lighter levels contribute a lot to 1 cell• Elements from heavier levels subsampled and collide

– Both unlikely if hash table big enough– Under-estimates |Sk|. If |Sk| > 2k/polylog(n), gives O(1)-approximation – Remark: triangle inequality of norm gives control over impact of collisions15

∑ ∑ ∑ ∑

Subsample:

M1 M4 M5 M6 M9 M11 Mn–1

Sketch and Recovery Algorithm

– For each level k, count number ck of “accepting” hash tables

– Return ∑k Tk · (ck/t) · (1/pk)

Recovery algorithm:

Sketch:– For each level k, create t hash tables– For each hash table:

• Subsample from [n], including each i2[n] w.p. pk = 2-k

• Each cell maintains sum of Mi’s that hash to it

16

- For every k, the estimator under-estimates |Sk|

- If |Sk| > 2k/polylog n, the estimator is (|Sk|)

- For every k, the estimator under-estimates |Sk|

- If |Sk| > 2k/polylog n, the estimator is (|Sk|)

EMD Wrapup

• We achieve a linear embedding of EMD– with constant distortion, namely O(1/ε),– into a space of strongly sublinear dimension, namely ∆ε.

• Open problems:– Getting (1+ε)-approximation / proving impossibility– Reducing dimension to logO(1)∆ / proving lower bound

17

What We Did

• We showed that in a data stream, one can sketch ||M||1,X = ∑i ||Mi||X with space about the space complexity of computing (or sketching) ||¢||X

• This quantity is known as a cascaded norm, written as L1(X)

• Cascaded norms have many applications [CM, JW]

• Can we generalize this? E.g., what about L2(X), i.e., (∑i ||Mi||2

X )1/2

Cascaded Norms [JW09]• No!

• L2(L1), i.e., (∑i ||Mi||2 1)1/2, requires (n1/2) space, where n is

the number of different i, but sketching complexity of L1 is O(log n)

• More generally, for p ¸ 1, Lp(L1), i.e., (∑i ||Mi||p 1)1/p is £(n1-1/p) space

• So, L1(X) is very special

Thank You!