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Clustering

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How are we doing on the pass sequence? Pretty good! We can now automatically learn the features needed to track both people But, it sucks that we need to hand-label the coordinates of both men in 30 frames and hand- label the 2 classes for the white-shirt tracker Same set of weights used for all hidden units

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Unsupervised learning Goal: Learn a machine or model that explains the data using a predefined set of assumptions about how the explanations can work There isnt any labeled data, just input patterns (ie, instead of x,t -pairs, we have only x s) Examples of unsupervised learning –Clustering (eg, k-means clustering) –Dimensionality reduction (eg, PCA) –Data compression methods (eg, ZIP, JPEG, MPEG) –Generative models & Bayesian networks (tomorrow)

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A simple example of unsupervised learning Our explanation of the data is that each training case is equal to an unknown number plus zero-mean Gaussian noise with unknown variance 2 Unsupervised learning –Determine and 2, and –For each training case, determine the noise signal Noise signal for training case x n

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A simple example of unsupervised learning What can this model be used for? Outlier detection / novelty detection –Given a new input x N+1, define p as follows (see figure) If x N+1 < let p = z=- x N+1 N (z|, 2 )dz and otherwise let p = z=x N+1 N (z|, 2 )dz –Then, we can say that the new input x N+1 is an outlier (unusual) if p < 0.01 Preference prediction –Given a set of new inputs, we can rank them according to probability x N+1

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Problem Training data Gaussian estimated from training data This test case has higher density than any of the training cases, but is quite probably an outlier

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K -means clustering Data and initial (random) means Set each mean to the average of its data Assign each point to its nearest mean Iteration 1 Iteration 2 Iteration 3 Iteration 4 and convergence

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K -means clustering Suppose x 1,…,x N are real-valued or vector-valued Goal: Learn K means and assign each training case to a mean Denote the means by 1,…, K Use one-of-K encoding for the assignments: r nk = 1 if x n is assigned to k, and r nk = 0 otherwise The goal of k-means clustering is to find the r s and s that minimize the following cost function: Generally, finding an exact solution takes time that is exponential in N

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K -means clustering Note that given the s we can efficiently find the best r s, and given the r s, we can efficiently find the best s Heres the algorithm: Pick initial means randomly or cleverly Loop until convergence –Assign each training case to its nearest mean: –Set each mean to the average of its training cases: Each step minimizes J wrt the r s or the s, but the procedure is not guaranteed to find the minimum of J

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Example Data and initial (random) means Set each mean to the average of its data Assign each point to its nearest mean Iteration 1 Iteration 2 Iteration 3 Iteration 4 and convergence Iteration

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Example: Color quantization Consider each image pixel as a 3-D vector (RGB) and use K -means clustering to find K prototype colors Now, each pixel in the image can be stored using log 2 (K) bits, with some loss in color quality 1 bit per pixel24 bits per pixel1.6 bits per pixel3.3 bits per pixel

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EM for a mixture of Gaussians Initialization: Pick s, s, s randomly but validly E Step: For each training case, we need q(z) = p(z|x) = p(x|z)p(z) / ( z p(x|z)p(z)) Defining = q(z nk =1 ), we need to actually compute: M Step: Do it in the log-domain! Recall: For labeled data, (z nk )=z nk

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EM for mixture of Gaussians: E step c z 1 = 0.5, 2 = 0.5, Images from data set

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c =1 Images from data set z=z= c =2 P(c|z) c 0.52 0.48 1 = 0.5, 2 = 0.5, EM for mixture of Gaussians: E step

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Images from data set z=z= c c =1 c =2 P(c|z) 0.51 0.49 1 = 0.5, 2 = 0.5, EM for mixture of Gaussians: E step

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Images from data set z=z= c c =1 c =2 P(c|z) 0.48 0.52 1 = 0.5, 2 = 0.5, EM for mixture of Gaussians: E step

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Images from data set z=z= c c =1 c =2 P(c|z) 0.43 0.57 1 = 0.5, 2 = 0.5, EM for mixture of Gaussians: E step

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c 1 = 0.5, 2 = 0.5, z Set 1 to the average of zP(c =1 |z) Set 2 to the average of zP(c =2 |z) EM for mixture of Gaussians: M step

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c 1 = 0.5, 2 = 0.5, z Set 1 to the average of zP(c =1 |z) Set 2 to the average of zP(c =2 |z) EM for mixture of Gaussians: M step

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c 1 = 0.5, 2 = 0.5, z Set 1 to the average of diag((z- 1 ) T (z- 1 ))P(c =1 |z) Set 2 to the average of diag((z- 2 ) T (z- 2 ))P(c =2 |z) EM for mixture of Gaussians: M step

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c 1 = 0.5, 2 = 0.5, z Set 1 to the average of diag((z- 1 ) T (z- 1 ))P(c =1 |z) Set 2 to the average of diag((z- 2 ) T (z- 2 ))P(c =2 |z) EM for mixture of Gaussians: M step

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… after iterating to convergence: c z 1 = 0.6, 2 = 0.4,

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Non-vector-space clustering For K-means clustering and EM, The data must lie in a vector space, where Euclidean distance is a natural measure of similarity –Some methods (such as mixtures of Gaussians) allow each cluster to stretch and rotate its data, but these methods are still essentially based on Euclidean distance There can be advantages to using kernels k(x i,x k ) to measure similarity and then making predictions using training cases Now, we will study this approach for clustering, using s(i,k) to denote the similarity of x i to x k (these are like kernels, but not necessarily formally the same)

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K -centers clustering (aka K -medoids clustering and K -medians clustering)

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Randomly choose initial exemplars, (data centers) K -centers clustering (aka K -medoids clustering and K -medians clustering)

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Assign data points to nearest centers K -centers clustering (aka K -medoids clustering and K -medians clustering)

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For each cluster, pick best new center K -centers clustering (aka K -medoids clustering and K -medians clustering)

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For each cluster, pick best new center

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K -centers clustering (aka K -medoids clustering and K -medians clustering) Assign data points to nearest centers

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Assign data points to nearest centers K -centers clustering (aka K -medoids clustering and K -medians clustering) Convergence: Final set of exemplars (centers)

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The K -centers clustering algorithm Denote the index of the center that x i is currently assigned to by c i ( c i i indicates that x i is a center) Initialization: Randomly pick K points and set c k = k Loop until convergence –For all i s.t. c i i set c i argmax k:c k =k s(i,k) –For all k s.t. c k k Compute k new argmax i:c i =k ( j:c j =k s(j,i) ) For all i s.t. c i k set c i k new This is similar to K-means clustering, except that the means are always data points

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Handwritten digit clustering and recognition

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Example: Clustering 3000 Buffalo digits Similarity: s(i,k) = - || x i - x k || 2 K=10 K=20 K=40 K=80

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The effect of random initialization Squared error Affinity propagation

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k-centers clustering How good is the solution?

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Solution that minimizes distances

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Affinity propagation (Frey & Dueck, Science, Feb 2007)

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Affinity propagation

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Solution that minimizes distances

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How does affinity propagation work? The sum-product or max-sum algorithms (loopy belief propagation) can be used to approximately maximize the k-centers objective function (more on this tomorrow) Result – Affinity Propagation: Data points exchange responsibilities and availabilities Sending responsibilities Candidate exemplar k r(i,k) Data instance i Competing candidate exemplar k a(i,k) Sending availabilities Candidate exemplar k a(i,k) Data instance i Supporting data instance i r(i,k)

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Sending responsibilities Candidate exemplar k r(i,k) Data instance i Competing candidate exemplar k a(i,k) Sending availabilities Candidate exemplar k a(i,k) Data instance i Supporting data instance i r(i,k) Making decisions:

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Message damping Unstable dynamics are avoided in practice by damping messages: r(i,k)* = r(i,k) + (1- ) r(i,k) old a(i,k)* = a(i,k) + (1- ) a(i,k) old Default: = 0.5

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MATLAB implementation (from http://www.psi.toronto.edu/affinitypropagation)

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Document summarization using affinity propagation s(sentence i,sentence k) = - Number of bits needed to encode the words in sentence i using the words in sentence k and a global dictionary Preference( sentence k ) = - Number of bits needed to encode the words in sentence k using only the global dictionary 1) Affinity propagation identifies exemplars by recursively sending real-valued messages between pairs of data points. 2) The number of identified exemplars (number of clusters) is influenced by the values of the input preferences, but also emerges from the message-passing procedure. 3) The availability is set to the self- responsibility plus the sum of the positive responsibilities the candidate exemplar receives from other points. 4) For different numbers of clusters, the reconstruction errors achieved by affinity propagation and k-centers clustering are compared.

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Questions?

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How are we doing on the pass sequence? Can clustering be used to automatically learn the two modes of tracking for the man in the white shirt? Maybe, but this system is getting too complex! Is there any simple way to put the pieces together…?

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