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Compact Data Representations and their Applications

Compact Data Representations and their Applications. Moses Charikar Princeton University. Lots and lots of data. AT&T Information about who calls whom What information can be got from this data ? Network router Sees high speed stream of packets

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Compact Data Representations and their Applications

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  1. Compact Data Representations and their Applications Moses Charikar Princeton University

  2. Lots and lots of data • AT&T • Information about who calls whom • What information can be got from this data ? • Network router • Sees high speed stream of packets • Detect DOS attacks ? fair resource allocation ?

  3. Lots and lots of data • Google search engine • About 3 billion web pages • Many many queries every day • How to efficiently process data ? • Eliminate near duplicate web pages • Query log analysis

  4. sketch complex object 0 1 0 1 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 Sketching Paradigm • Construct compact representation (sketch) of data such that • Interesting functions of data can be computed from compact representation estimated

  5. Why care about compact representations ? • Practical motivations • Algorithmic techniques for massive data sets • Compact representations lead to reduced space, time requirements • Make impractical tasks feasible • Theoretical Motivations • Interesting mathematical problems • Connections to many areas of research

  6. Questions • What is the data ? • What functions do we want to compute on the data ? • How do we estimate functions on the sketches ? • Different considerations arise from different combinations of answers • Compact representation schemes are functions of the requirements

  7. What is the data ? • Sets, vectors, points in Euclidean space, points in a metric space, vertices of a graph. • Mathematical representation of objects (e.g. documents, images, customer profiles, queries).

  8. What functions do we want to compute on the data ? • Local functions : pairs of objectse.g. distance between objects • Sketch of each object, such that function can be estimated from pairs of sketches • Global functions : entire data sete.g. statistical properties of data • Sketch of entire data set, ability to update, combine sketches

  9. Local functions: distance/similarity • Distance is a general metric, i.e satisfies triangle inequality • Normed spacex = (x1, x2, …, xd) y = (y1, y2, …, yd) • Other special metrics (e.g. Earth Mover Distance)

  10. Estimating distance from sketches • Arbitrary function of sketches • Information theory, communication complexity question. • Sketches are points in normed space • Embedding original distance function in normed space. [Bourgain ’85] [Linial,London,Rabinovich ’94] • Original metric is (same) normed space • Original data points are high dimensional • Sketches are points low dimensions • Dimension reduction in normed spaces[Johnson Lindenstrauss ’84]

  11. Streaming algorithms • Perform computation in one (or constant) pass(es) over data using a small amount of storage space • Availability of sketch function facilitates streaming algorithm • Additional requirements - sketch should allow: • Update to incorporate new data items • Combination of sketches for different data sets storage input

  12. Global functions • Statistical properties of entire data set • Frequency moments • Sortedness of data • Set membership • Size of join of relations • Histogram representation • Most frequent items in data set • Clustering of data

  13. Goals • Glimpse of compact representation techniques in the sketching and streaming domains. • Basic ideas, no messy details

  14. Talk Outline • Classical techniques: spectral methods • Dimension reduction • Similarity preserving hash functions • sketching vector norms • sketching Earth Mover Distance (EMD)

  15. Talk Outline • Dimension reduction • Similarity preserving hash functions • sketching vector norms • sketching Earth Mover Distance (EMD)

  16. f http://humanities.ucsd.edu/courses/kuchtahum4/pix/earth.jpg http://www.physast.uga.edu/~jss/1010/ch10/earth.jpg Low Distortion Embeddings • Given metric spaces (X1,d1) & (X2,d2),embedding f: X1 X2 has distortion D if ratio of distances changes by at most D • “Dimension Reduction” – • Original space high dimensional • Make target space be of “low” dimension, while maintaining small distortion

  17. Dimension Reduction in L2 • n points in Euclidean space (L2 norm) can be mapped down to O((log n)/2) dimensions with distortion at most 1+.[Johnson Lindenstrauss ‘84] • Two interesting properties: • Linear mapping • Oblivious – choice of linear mapping does not depend on point set • Quite simple [JL84, FM88, IM98, DG99, Ach01]: Even a random +1/-1 matrix works… • Many applications…

  18. Dimension reduction for L1 • [C,Sahai ‘02]Linear embeddings are not good for dimension reduction in L1 • There exist O(n) points in L1in n dimensions, such that any linear mapping with distortion  needs n/2dimensions

  19. Dimension reduction for L1 • [C, Brinkman ‘03]Strong lower boundsfor dimension reduction in L1 • There exist n points in L1, such that anyembedding with constant distortion  needs n1/2dimensions • Simpler proof by [Lee,Naor ’04] • Does not rule out other sketching techniques

  20. Talk Outline • Dimension reduction • Similarity preserving hash functions • sketching vector norms • sketching Earth Mover Distance (EMD)

  21. sketch complex object 0 1 0 1 1 0 0 1 1 0 0 0 1 0 1 1 0 0 1 0 Similarity Preserving Hash Functions • Similarity function sim(x,y), distance d(x,y) • Family of hash functions F with probability distribution such that

  22. Applications • Compact representation scheme for estimating similarity • Approximate nearest neighbor search [Indyk,Motwani ’98] [Kushilevitz,Ostrovsky,Rabani ‘98]

  23. Sketching Set Similarity:Minwise Independent Permutations [Broder,Manasse,Glassman,Zweig ‘97] [Broder,C,Frieze,Mitzenmacher ‘98]

  24. Other similarity functions ? [C’02] • Necessary conditions for existence of similarity preserving hash functions. • SPH does not exist for Dice coefficient and Overlap coefficient. • SPH schemes from rounding algorithms • Hash function for vectors based on random hyperplane rounding.

  25. Existence of SPH schemes • sim(x,y) admits an SPH scheme if family of hash functions F such that

  26. Theorem: If sim(x,y) admits an SPH scheme then 1-sim(x,y) satisfies triangle inequality. Proof:

  27. Non-existence of SPH

  28. Stronger Condition Theorem: If sim(x,y) admits an SPH scheme then (1+sim(x,y) )/2 has an SPH scheme with hash functions mapping objects to {0,1}. Theorem: If sim(x,y) admits an SPH scheme then 1-sim(x,y) is isometrically embeddable in the Hamming cube.

  29. For n vectors, random hyperplane can be chosen using O(log2 n) random bits.[Indyk], [Engebretson,Indyk,O’Donnell] • Alternate similarity measure for sets

  30. Random Hyperplane Rounding based SPH • Collection of vectors • Pick random hyperplane through origin (normal ) • [Goemans,Williamson]

  31. Sketching L1 • Design sketch for vectors to estimate L1 norm • Hash function to distinguish between small and large distances [KOR ’98] • Map L1 to Hamming space • Bit vectors a=(a1,a2,…,an) and b=(b1,b2,…,bn) • Distinguish between distances  (1-ε)n/k versus  (1+ε)n/k • XOR random set of k bits • Pr[h(a)=h(b)] differs by constant in two cases

  32. Sketching L1 via stable distributions • a=(a1,a2,…,an) and b=(b1,b2,…,bn) • Sketching L2 • f(a) = Σi ai Xi f(b) = Σi bi XiXi independent Gaussian • f(a)-f(b) has Gaussian distribution scaled by |a-b|2 • Form many coordinates, estimate |a-b|2 by taking L2 norm • Sketching L1 • f(a) = Σi ai Xi f(b) = Σi bi XiXi independent Cauchy distributed • f(a)-f(b) has Cauchy distribution scaled by |a-b|1 • Form many coordinates, estimate |a-b|1 by taking median[Indyk ’00] -- streaming applications

  33. Compact Representations in Streaming • Statistical properties of data streams • Distinct elements • Frequency moments, norm estimation • Frequent items

  34. Frequency Moments[Alon, Matias, Szegedy ’99] • Stream consists of elements from {1,2,…,n} • mi = number of times i occurs • Frequency moment • F0 = number of distinct elements • F1 = size of stream • F2 =

  35. Overall Scheme • Design estimator (i.e. random variable) with the right expectation • If estimator is tightly concentrated, maintain number of independent copies of estimator E1, E2, …, Er • Obtain estimate E from E1, E2, …, Er • Within (1+) with probability 1-

  36. Randomness • Design estimator assuming perfect hash functions, as much randomness as needed • Too much space required to explicitly store such a hash function • Fix later by showing that limited randomness suffices

  37. Distinct Elements • Estimate the number of distinct elements in a data stream • “Brute Force solution”: Maintain list of distinct elements seen so far • (n) storage • Can we do better ?

  38. Distinct Elements[Flajolet, Martin ’83] • Pick a random hash function h:[n]  [0,1] • Saythere are k distinct elements • Then minimum value of h over k distinct elements is around 1/k • Apply h() to every element of data stream; maintain minimum value • Estimator = 1/minimum

  39. (Idealized) Analysis • Assume perfectly random hash function h:[n]  [0,1] • S: set of k elements of [n] • X = min aS { h(a) } • E[X] = 1/(k+1) • Var[X] = O(1/k2) • Mean of O(1/2) independent estimators is within (1+) of 1/k with constant probability

  40. Analysis • [Alon,Matias,Szegedy]Analysis goes through with pairwise independent hash functionh(x) = ax+b • 2 approximation • O(log n) space • Many improvements[Bar-Yossef,Jayram,Kumar,Sivakumar,Trevisan]

  41. Estimating F2 • F2 = • “Brute force solution”: Maintain counters for all distinct elements • Sampling ? • n1/2space

  42. Estimating F2[Alon,Matias,Szegedy] • Pick a random hash functionh:[n]  {+1,-1} • hi = h(i) • Z = • Z initially 0, add hievery time you see i • Estimator X = Z2

  43. Analyzing the F2 estimator

  44. Analyzing the F2 estimator • Median of means gives good estimator

  45. Properties of F2 estimator • “sketch” of data stream that allows computation of • Linear function of mi • Can be added, subtracted • Given two streams, frequencies mi , ni • E[(Z1-Z2)2] = • Estimate L2 norm of difference • How about L1 norm ? Lp norm ?

  46. Talk Outline • Similarity preserving hash functions • Similarity estimation • Statistical properties of data streams • Distinct elements • Frequency moments, norm estimation • Frequent items

  47. Variants of F2 estimator[Alon, Gibbons, Matias, Szegedy] • Estimate join size of two relations(m1,m2,…) (n1,n2,…) • Variance may be too high

  48. In Conclusion • Simple powerful ideas at the heart of several algorithmic techniques for large data sets • “Sketches” of data tailored to applications • Many interesting research questions

  49. Content-Based Similarity Search Moses Charikar Princeton University Joint work with: Qin Lv, William Josephson, Zhe Wang, Perry Cook, Matthew Hoffman, Kai Li

  50. Motivation • Massive amounts of feature-rich digital data • Audio, video, digital photos, scientific sensor data • Noisy, high-dimensional • Traditional file systems/search tools inadequate • Exact match • Keyword-based search • Annotations • Need content-based similarity search

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