Efficient sketches for earth mover distance with applications
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Efficient Sketches for Earth-Mover Distance, with Applications. David Woodruff IBM 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 ,

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Efficient Sketches for Earth-Mover Distance, with Applications

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Efficient Sketches for Earth-Mover Distance, with Applications

David Woodruff

IBM 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

EMD(, ) = 6 + 3√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

2 Rk



≈ EMD(A,B)

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”]

Prior and New Results

Geometric representation of EMD:

Main Theorem

For any ε2(0,1), there exists a distribution over linear mappings F: R∆2!R∆εs.t. for multisets A,Bµ [∆]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.


  • Streaming:

  • Approximate nearest neighbor:

* N = number of points

* s = number of data points (multisets) to preprocess

α>1 free parameter

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| andS 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

Old Idea [Indyk ’07]

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

EEMD over [∆1/2]2

+ … +

Old Idea [Indyk ’07]

Solve EEMD in each of ¢ cells,

each a problem in [¢1/2]2

EMD over [∆]2


Old Idea [Indyk ’07]

Solve one additional

EEMD problem in [¢1/2]2


Should also scale edge

lengths 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

||M||1, X =


+ … +

For normed space X = (Rt, ||¢||X) and M2Xn, denote ||M||1,X = ∑i ||Mi||X.




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

μ: Xn!X(λlog n)O(1)

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

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

M2 contains most of mass





Proof Outline: Sum of Norms (cont.)

M = (M1,






  • Our approach:

    • Split into exponential levels:

      • Assume ||M||1, X· C

      • Sk = {i2[n] s.t. ||Mi||X2(Tk, 2Tk]}, Tk=C/2k

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

    • For each level k, subsample from [n] at a rate

      such that event Ek (“isolation” of level k)

      holds with probability proportional to |Sk|

    • Repeat experiment several times, count number of successes

M4, M7



M1, M3, M8, M9


M5, M10, Mn






Proof Outline: Event Ek

  • Ek$ “isolation” of level k:

    • Exactly one i 2Sk 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 collisions









Sketch and Recovery Algorithm


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

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

  • 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

Recovery algorithm:

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

  • Return ∑kTk · (ck/t) · (1/pk)


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

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||2X )1/2

Cascaded Norms [JW09]

  • No!

  • L2(L1), i.e., (∑i ||Mi||21)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!

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