Metric Embedding of Task-Specific Similarity. joint work with Trevor Darrell (MIT)

Transcription

1 Metric Embedding of Task-Specific Similarity Greg Shakhnarovich Brown University joint work with Trevor Darrell (MIT) August 9, 2006

2 Task-specific similarity A toy example:

3 Task-specific similarity A toy example:

4 Task-specific similarity A toy example: Norm Angle

5 The problem Learn similarity from examples of what is similar [or not]. Binary similarity: for x, y X S(x, y) = { + if x and y are similar, if they are dissimilar. Two goals in mind: Similarity detection: judge whether two entities are similar. Similarity search: given a query entity and a database, find examples in a database that are similar to the query. Our approach: learn an embedding of the data into a space where similarity corresponds to a simple distance. 2

6 Task-specific similarity Articulated pose: 3

7 Task-specific similarity Visual similarity of image patches: 4

8 Related prior work Learning parametrized distance metric [Xing et al; Roweis], such as Mahalanobis distances. Lots of work on low-dimensional graph embedding (MDS, LLE,... ) - but unclear how to generalize without relying on distance in X. Embedding known distance [Athitsos et al]: assumes known distance, uses embedding to approximate/speed up. DistBoost [Hertz et al]: learning distance for classification / clustering setup. Locality sensitive hashing: fast similarity search. 5

9 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls: 6

10 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls: 0 6

11 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls: 0 6

12 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls: 0 0 6

13 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls: 0 0 6

14 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls: 0 0 6

15 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls: 0 0 6

16 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls: 0 0 6

17 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls:

18 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls:

19 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls:

20 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls:

21 Locality Sensitive Hashing [Indyk et al] Algorithm for finding a (ɛ, r)-neighbor of x 0 with high probability in sublinear time O ( N /(+ɛ)). Index the data by l random hash functions, and only search the union of the l buckets where the query falls:

22 Locality sensitive hashing [Indyk et al] A family H of functions is locality sensitive if P h U[H] (h(x 0 ) = h(x) x 0 x r) p, P h U[H] (h(x 0 ) = h(x) x 0 x R) p 2. Uses the gap between TP and FP rates; amplified by concatenating functions into a hash key. Projections on random lines are locality sensitive w.r.t. L p norms, p 2 [Gionis et al, Datar et al]. 7

23 How is this relevant? LSH is excellent if L p is all we want. L p may not be a suitable proxy for S: we may Waste lots of bits on irrelevant features; Miss pairs similar under S but not close w.r.t. L p If we know what S is, may be able to analytically design embedding of X into L space [Thaper&Indyk, Grauman&Darrell]. We will instead learn LSH-style binary functions that fit S as conveyed by examples. 8

24 Our approach Given: pairs of similar [and pairs of dissimilar] examples, based on the hidden binary similarity S. Two related tasks: Similarity judgment: S(x, y) =? Given a query x 0 ; need to find {x i : S(x i, x 0 ) = +}. Our solution to both problems: a similarity sensitive embedding H(x) = [α h (x),..., α M h M (x)] ; We will learn h m (x) {0, } and α m. 9

25 Desired embedding properties Embedding H(x) = [α h (x),..., α M h M (x)]: Rely on L (= weighted Hamming) distance H(x ) H(x 2 ) = M m= α m h m (x ) h m (x 2 ) H is similarity sensitive: for some R, want high P x,x 2 p(x)( H(x ) H(x 2 ) R S(x, x 2 ) = +), low P x,x 2 p(x)( H(x ) H(x 2 ) R S(x, x 2 ) = ). H(x) H(y) is a proxy for S. 0

26 Projection-based classifiers For a projection f : X R, consider, for some T R, h(x; f, T ) = { if f(x) T 0 if f(x) > T. This defines a simple similarity classifier of pairs: c(x, y; f, T ) = + h(x; f, T ) = h(y; f, T ) q y z b a h(x; f, T ) = 0 T h(x; f, T ) = f(x) c(q, y; f, T ) = c(q, z; f, T ) = +, c(q, a; f, T ) = c(q, b; f, T ) =.

27 Selecting the threshold Algorithm for selecting the threshold based on similar/dissimilar pairs: For a moment, we focus on N positive pairs only. Sort the 2N values of f(x). Need to check at most 2N + values of T, and count the number of pairs that are dissected (a bad event). f(x) 2

28 Selecting the threshold Algorithm for selecting the threshold based on similar/dissimilar pairs: For a moment, we focus on N positive pairs only. Sort the 2N values of f(x). Need to check at most 2N + values of T, and count the number of pairs that are dissected (a bad event). 0 f(x) 2

29 Selecting the threshold Algorithm for selecting the threshold based on similar/dissimilar pairs: For a moment, we focus on N positive pairs only. Sort the 2N values of f(x). Need to check at most 2N + values of T, and count the number of pairs that are dissected (a bad event). 0 f(x) 2

30 Selecting the threshold Algorithm for selecting the threshold based on similar/dissimilar pairs: For a moment, we focus on N positive pairs only. Sort the 2N values of f(x). Need to check at most 2N + values of T, and count the number of pairs that are dissected (a bad event) f(x) 2

31 Selecting the threshold Also need to consider negative examples (dissimilar pairs), and estimate the gap: true positive rate TP minus false positive rate FP. POS NEG f(x) 3

32 Selecting the threshold Also need to consider negative examples (dissimilar pairs), and estimate the gap: true positive rate TP minus false positive rate FP POS NEG f(x)

33 Optimization of h(x; f, T ) Objective (TP FP gap): argmax f,t N + i= N c(x + i, x+ i ) j= c(x j, x j ) Parametric projection families, e.g. f(x) = j θ jx p (d j )xp 2 (d 2 j ) Soft versions of h and c make the gap differentiable w.r.t. θ, T : h(x; f, T ) = + exp (f(x) T )) c(x, y) = 4 (h(x) /2) (h(y) /2) 4

34 Ensemble classifiers A weighted embedding H(x) = [α h (x),..., α M h M (x)]. Each h m (x) defines a classifier c m. Together they form an ensemble classifier of similarity: C(x, y) = sgn ( M m= α m c m (x, y) We will construct the ensemble of bit-valued hs by a greedy algorithm based on AdaBoost (operting on the corresponding cs). ). 5

35 Boosting [Schapire et al] Assumes access to weak learner that can at every iteration m produce a classifier c m better than chance. Main idea: maintain a distribution W m (i) on the training examples, and update it according to the prediction of c m : If c m (x i ) is correct, then W m+ (i) goes down; Otherwise, W m+ (i) increases ( steering c m+ towards it.) Our examples are pairs, weak classifiers are thresholded projections. To evaluate threshold, we will calculate the weight of separated pairs, rather than count them. 6

36 BoostPro Given pairs (x () i, x (2) i ) labeled with l i = S(x () i, x (2) i ): : Initialize weights W (i), to uniform. 2: for all m =,..., M do 3: Find f, T = argmax f,t r m (f, T ) using gradient descent on r m (f, T ) = N i= W m (i)l i c m (x () i, x (2) i ). 4: Set h m h(x; f, T ). 5: Set α m (see Boosting papers.) 6: Update weights: W m+ (i) W m (i) exp ( ) l i c m (x () i, x (2) i ). 7

37 Similarity is a rare event In many domains: vast majority of possible pairs are negative. People s poses, image patches, documents,... A reasonable approximation of the sampling process: Independently draw x, y from the data distribution. f(x), f(y) drawn from p(f(x)). Label (x, y) negative. 8

38 Similarity is a rare event In many domains: vast majority of possible pairs are negative. People s poses, image patches, documents,... A reasonable approximation of the sampling process: Independently draw x, y from the data distribution. f(x), f(y) drawn from p(f(x)). Label (x, y) negative. T f( x) 8

39 Similarity is a rare event In many domains: vast majority of possible pairs are negative. People s poses, image patches, documents,... A reasonable approximation of the sampling process: Independently draw x, y from the data distribution. f(x), f(y) drawn from p(f(x)). Label (x, y) negative. π T ( π ) T π T = Pr(f(x) T ) T f( x) 8

40 Similarity is a rare event In many domains: vast majority of possible pairs are negative. People s poses, image patches, documents,... A reasonable approximation of the sampling process: Independently draw x, y from the data distribution. f(x), f(y) drawn from p(f(x)). Label (x, y) negative. 2 π T ( π T ) T f( x) π T = Pr(f(x) T ) 8

41 Similarity is a rare event In many domains: vast majority of possible pairs are negative. People s poses, image patches, documents,... A reasonable approximation of the sampling process: Independently draw x, y from the data distribution. f(x), f(y) drawn from p(f(x)). Label (x, y) negative. 2 2 π T ( π T ) T f( x) π T = Pr(f(x) T ) 8

42 Semi-supervised setup Given similar pairs and unlabeled x,..., x N. Estimate TP rate as before. FP rate: Estimate the CDF of f(x); let π T = ˆP (f(x) T ). Then FP = π 2 T + ( π T ) 2. Note: this means FP /2 [Ke et al]. 9

43 BoostPro in a semi-supervised setup Normal boosting assumes positive and negative examples. Intuition: each unlabeled example x i represents all possible pairs (x i, y); those are, w.h.p., negative under our assumption. Two distributions: W m (j) on positive pairs, as before. S m (i) on unlabeled (single) examples. Instead of c m (x () i, x (2) i ) use the expectation E y [c m (x i, y)] to calculate the objective and set the weights. 20

44 Semi-supervised boosting: details Probability of misclassifying a negative (x j, y): P j = h(x j ; f, T )π + ( h(x j ; f, T ))( π). E y [c(x j, y)] = P j (+) + ( P j ) ( ) = 2P j. Modified boosting objective: r = N p i= W (i)c(x () i, x (2) i ) N j= S(j)E y [c(x j, y)] = N p i= W (i)c(x () i, x (2) i ) N j= S j (2P j ). 2

45 Results: Toy problems M =00, trained on 600 unlabeled points +,000 similar pairs 22

46 Results: Toy problems 23

47 Results: UCI data sets Data Set L PSH BoostPro M MPG ± ± ± ± 20 CPU ± ± ± ± 48 Housing ± ± ± ± 28 Abalone ± ± ± ± 8 Census ± ± ± ± 0 Test error on regression benchmark data from UCI/Delve. Mean ± std. deviation of MSE using locally-weighted regression. The last column shows the values of M (dimension of embedding H). Similarity defined in terms of target function values. 24

48 Results: pose retrieval Input 3 nearest neighbors in H H built with semi-supervised BoostPro, on 200,000 examples; M =

49 Results: pose retrieval Recall for k-nn retrieval. For each value of k, the fraction of true k-nn w.r.t. pose within the k-nn w.r.t. an image-based similarity measure is plotted. Black dotted line: L on EDH. Red dash-dot: PSH. Blue solid: BoostPro, M =

50 Visual similarity of image patches Define two patches to be similar under any rotation and mild shift (± 4 pixels). Should be covered by many reasonable similarity definitions. 27

51 Descriptor : sparse overcomplete code Generative model of patches [Olshausen&Field] Very unstable under transformation, hence L is not a good proxy for similarity. With BoostPro, improvement in area under ROC from 0.56 to

52 Descriptor 2: SIFT Scale-Invariant Feature Transform [Lowe] Histogram of gradient magnitude and orientation within the region, normalized by orientation and at an appropriate scale. Discriminative (can not generate a patch). Designed specifically to make two similar patches match. 29

53 Results Area under ROC curve: L on SIFT L on H(SIFT)

54 Conclusions It is beneficial to learn similarity directly for the task rather than rely on the default distance. Key property of our approach: similarity search reduced to L neighbor search (thus can be done in sublinear time.) Most useful for: Regression and multi-label classification; When large amounts of labeled data are available; When L p distance is not a good proxy for similarity. 3

55 Open questions What kinds of similarity concepts can we learn? How do we explore the space of projections more efficiently? Factorization of similarity. Combining unsupervised and supervised similarity learning for image regions. 32

56 Questions?.. 33