CS6716 Pattern Recognition

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1 CS6716 Pattern Recognition Prototype Methods Aaron Bobick School of Interactive Computing

2 Administrivia Problem 2b was extended to March 25. Done? PS3 will be out this real soon (tonight) due April 10. Today chapter 13 of the Hastie book. (Save SVMs for after break) Slides from Sam Brown (who took them from Vipin Kumar), Jia Lee,

3 Outline K-means as the prototypical prototype method LVQ as a better version of K-means K-NN revisited Comparison to Bayes error Problem with dimensionality Discriminant Adaptive K-NN method

4 K-means (no classes yet) (R-means???) Assume a known number of clusters R (yes really, K is taken for number of classes). Start with some initial set of centers Repeat until tired: 1. For each center, find the training points closets to it (its cluster) 2. Re-estimate the center based upon the cluster.

5 K-means (with classes) Assume a known number of clusters R prototypes per class. Apply K-means to each class of data separately. Assign class label to each of the K*R prototypes (see?) Classify each new instance xx ii according to the nearest prototype label Start with some initial set of centers Repeat until tired: 1. For each center, find the training points closets to it (its cluster) 2. Re-estimate the center based upon the cluster.

6 Fixing K-Means The prototypes are not influenced by the other classes and can end up near the class boundary. Fix this by adjusting the prototypes by considering the training data: Kohonen s Learned Vector Quantization

7 Learned Vector Quantization 1. Asssume set of inputs xx ii and labels yy ii. Start from a set of initial prototypes with classes assigned. Denote the MM prototypes by ZZ = {zz 1,, zz MM } and their associated classes by CC zz mm, mm = 1, 2,, MM > The initial prototypes can be provided by k-means. 2. Sweep through the training samples and update zz mm after visiting each sample: 1. Suppose xx ii is assigned to the mm ttt prototype zz mm by the nearest neighbor rule: xx ii zz mm xx ii zz mm, mm mm, 1 mm MM 2. If yy ii = CC zz mm, move zz mm towards the training example. zz mm zz mm + εε xx ii zz mm where εε is the learning rate 3. Otherwise move zz mm away the training example. zz mm zz mm εε xx ii zz mm 3. Step 2 can be repeated a number of times.

8 Learned Vector Quantization

9 Some experiments Synthetic data 3-classes generated by mixture of Gaussians Bayesian boundaries known

10 Some experiments Real data: Use the diabetes data set Use prototypes obtained by k-means as initial prototypes. Use LVQ with εε = 0.1. Results obtained after 1, 2, and 5 passes are shown. Classification is not guaranteed to improve after adjusting prototypes. One pass with a small E usually helps. But don t over do it. Comments: Fine tuning often helps: Select initial prototypes. Adjust learning rate E. Read the package documents for details.

11 Diabetes k-means vslvq results (0) Error rate: 27.74% (1) Error rate: 27.61% (2) Error rate: 27.86% (5) Error rate: 32.37%

12 Is LVQ really any better Classification is not guaranteed to improve after adjusting prototypes. According to Jia Lee: One pass with a small εε usually helps. But don t over do it. Learning rate can be important Problem according to Hastie: the problem with [LVQ] methods is that they are defined by algorithms, rather than optimization of some fixed criteria; this makes it difficult to understand their properties

13 K-NN again???? We discussed K-NN in the overview of supervised learning and especially with respect to bias and variance. Now think about prototype/memory methods generally. Next time, maybe I ll do less early and more here

14 Why nearest neighbor? Used to classify objects based on closest training examples in the feature space Top 10 Data Mining Algorithm ICDM paper December 2007 A simple but sophisticated approach to classification? 14

15 K-Nearest Neighbor Compute the distance between two points: Euclidean distance: dd pp, qq = pp ii qq ii 2 Hamming distance (symbol overlap metric) bat (distance = 1) toned (distance = 3) cat roses Determine the class from nearest neighbor list Take the majority vote of class labels among the k-nearest neighbors Can be weighted by factor ww = 1/ (dd 2 ) Scaling issues can be a problem e.g. weight and height in very different units. 16

16 K-Nearest Neighbor different K k = 1: Belongs to square class? k = 3: Belongs to triangle class k = 7: Belongs to square class Choosing the value of k: If k is too small, sensitive to noise points If k is too large, neighborhood may include points from other classes Choose an odd value for k, to eliminate ties 17

17 K-Nearest Neighbor advantages Simple technique that is easily implemented by lazy computer scientists Building the model is cheap (but using it less so). Extremely flexible classification scheme Well suited for Multi-modal classes Records with multiple or ambigious class labels Can sometimes be the best method Michihiro Kuramochi and George Karypis, Gene Classification using Expression Profiles: A Feasibility Study, International Journal on Artificial Intelligence Tools. Vol. 14, No. 4, pp , 2005 K nearest neighbor outperformed SVM for protein function prediction using expression profiles 18

18 K-Nearest Neighbor advantages (more) Error rate of 1-NN in asymptotically at most twice that of Bayes error rate (Cover & Hart paper (1967) ) First way to see: Assume an underlying known (Bayes) densities and some variance added to that. Then the error rate on a sample point is just due to Bayes error plus the variance of one point. In 1-NN as NN the test point matches some training point. But the error from the Bayes rule is due to variances of both that training point and the new target point. Second way. At some point xx let kk be the dominant class and pp kk xx is the true conditional probability for class kk. Then:

19 K-Nearest Neighbor advantages (more) (cont) For K=2, pp 1 xx = 1 pp 2 xx so: 1-NN error = 2 pp kk xx 1 pp kk xx < [2 1 pp kk xx ] = 2*Bayes rate ie the 1-NN error rate is no worse than twice Bayes. See Duda and Hart for more complete proof.

20 A nice example: LANDSAT Pixel and 8 neighbors => 36 dimensions. 7 Possible labels Use 5-NN

21 A nice example: LANDSAT (1994)

22 A nice extension Problem: recognize digits. Input 16x16 gray values, so xx: R 256

23 Addressing systematic variation Consider slight rotations of the image, one connected to the other: That s just some curve in the 256 dimensional space. You might think you d have all these examples, but that s hard if multiple such variations, e.g. rotation and scale. To address this you could find the NN in terms of these curves. But that s painful.

24 Tangent method Instead, since only want small variations, just do a local tangent. Not perfect, but pretty good?

25 Tangent method Only need to find closest distance between two lines in d-dimensional space. Very good results but too slow to compute. Runtime matters for post office! Remember: boosted Neural Nest and SVM now around 0.8%

26 Why K>1 is usually a good idea

27 Cross-validation wants neighbors

28 K-Nearest Neighbor disadvantages Classifying unknown records are relatively expensive Requires distance computation of k-nearest neighbors Computationally intensive, especially when the size of the training set grows You can fix this you re computer people More fundamental: Problems in high dimensions Accuracy can be severely degraded by the presence of noisy or irrelevant features Hastie and Tibshirani, KDD 95 to the rescue: Discriminative Adaptive Nearest Neighbor 29

29 The dimensionality problem The size of the neighborhood we need to get K neighbors grows annoyingly. In Hastie: Why?

30 See, even at 1:30am I can do math Consider the unit cube in p dimensions centered at the origin. Assume a sphere with radius R about the origin. Assume it has a volume VV RR The probability that the distance d to the closest point of the N points is greater than R: NN Pr dd > RR = 1 VV RR Median => Pr dd > RR = 1 = 1 VV 2 RR NN So: ( 1 2 )1 NN= 1 VV RR VV RR = 1 ( 1 2 )1 NN If VV RR = vv pp RR pp vv pp RR PP = 1 ( 1 2 )1 NN => RR = vv pp 1 pp NN 1 pp

31 Neighborhood size goes up pretty fast

32 An motivating example Suppose two dimensions. But that classes only vary in x:? Class 1 Class 2 Devise a method that does this. Basically the neighborhood should be anisotropic. 33

33 Finding the right stretch (and dimensions) Goal: Adjust distance metric locally, so that the resulting neighborhoods stretch out in directions for which the class probabilities don t change much. Method: At each query point, a neighborhood of say 50 points is formed. Class probabilities are NOT assumed constant in this neighborhood. This neighborhood is used only to decide how to define the adapted metric. After the metric is decided, a normal k-nearest neighbor rule is applied to classify the query. The metric changes with the query.

34 Discriminant adaptive nearest neighbor classification (DANN) DANN: Discriminant sensitive to the set of classes Adaptive Capability of being able to adapt or adjust to fit the situation Nearest Neighbor classification based on a locality metric selected by the majority of adjacent neighbor s class DANN uses local linear discriminant analysis to estimate an effective metric for computing neighborhoods. DANN posterior probabilities tend to be more homogeneous in the modified neighborhoods. 35

35 DISCRIMINANT ADAPTIVE NEAREST NEIGHBOR CLASSIFICATION (DANN)? Class 1 Class 2 Using k -NN, we misclassify by crossing the boundary between classes. Standard linear discriminants extend infinitely in any direction. This is dangerous to local classification. 36

36 DISCRIMINANT ADAPTIVE NEAREST NEIGHBOR CLASSIFICATION (DANN)? Class 1 Class 2 DANN utilizes a small tuning parameter to shrink or stretch neighborhoods. 37

37 DISCRIMINANT ADAPTIVE NEAREST NEIGHBOR CLASSIFICATION (DANN)? The process of tuning can be done iteratively allowing stretching in all axis 38

38 DISCRIMINANT ADAPTIVE NEAREST NEIGHBOR CLASSIFICATION (DANN) 1. Initialize Σ to be the identity II 2. Given a test point xx 0 spread out a nearest neighborhood of K M points in the Σ metric around the xx 0 : DD xx, xx 0 = xx xx 0 TT Σ (xx xx 0 ) 3. Calculate the weighted within and between sum of squares matrices W and B using the points in the neighborhood (Remember LDA?) W = kk 1 WW kk, WW kk within class covariance And B is between class: Define a new metric: 1.Iterate steps 1, 2, and 3. 2.At completion, use the metric for k-nearest neighbor classification at the test point x o. 39

39 The magic What does WW 1 do? (W stands for something beyond within ) What is BB? It is the between matrix for the sphered (whitened)-data. It typically has some (many) zero eigenvalues. (Why?) If you left B alone, you would completely ignore distance in the zero-eigenvalue directions. I.E. you d use data arbitrarily far away. The εεεε term prevents that.

40 DISCRIMINANT ADAPTIVE NEAREST NEIGHBOR CLASSIFICATION (DANN) The DANN procedure has a number of adjustable tuning parameters: K M The number of nearest neighbors in the neighborhood N for estimation of the metric. K The number of neighbors in the final nearest neighbor rule. ε the softening parameter in the metric. Similar to Evolutionary Strategies Adjusts search space over a fitness landscape to find optimal solution. 41

41

42 Global approach to DANN

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