Unsupervised Learning Techniques Class 07, 1 March 2006 Andrea Caponnetto

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1 Unsupervised Learning Techniques Class 07, 1 March 2006 Andrea Caponnetto

2 About this class Goal To introduce some methods for unsupervised learning: Gaussian Mixtures, K-Means, ISOMAP, HLLE, Laplacian Eigenmaps.

3 Unsupervised learning Only u i.i.d. samples drawn on X from the unknown marginal distribution p(x) {x 1, x 2,..., x u }. The goal is to infer properties of this probability density. In low-dimension many nonparametric methods allow direct estimation of p(x) itself. Owing to the curse of dimensionality, this methods fail in high dimension. One must settle for estimation of crude global models.

4 Unsupervised learning (cont.) Different types of simple descriptive statistics that characterize aspects of p(x) mixture modelling representation of p(x) by a mixture of simple densities representing different types or classes of observations [eg. Gaussian mixtures] combinatorial clustering attempt to find multiple regions of X that contain modes of X [eg. K-Means] dimensionality reduction attempt to identify low-dimensional manifolds in X that represent high data density [eg. ISOMAP,HLLE, Laplacian Eigenmaps] manifold learning attempt to determine very specific geometrical or topological invariants of p(x) [eg. Homology learning]

5 Limited formalization With supervised and semi-supervised learning there is a clear measure of effectiveness of different methods. The expected loss of various estimators I[f S ] can be estimated on validation set. In the context of unsupervised learning, it is difficult to find such a direct measure of success. This situation has led to proliferation of proposed methods.

6 Mixture Modelling Assumption that data is i.i.d. sampled from some probability distribution p(x). p(x) is modelled as a mixture of component density functions, each component corresponding to a cluster or mode. The free parameters of the model are fit to the data by maximum likelihood.

7 Gaussian Mixtures We first choose a parametric model P θ for the unknown density p(x), hence maximize the likelihood of our data relative to the parameters θ. Example: two-component gaussian mixture model with parameters The model: θ = (π, µ 1,Σ 1, µ 2,Σ 2 ). P θ (x) = (1 π)g Σ1 (x µ 1 ) + πg Σ2 (x µ 2 ) Maximize the log-likelihood l(θ {x 1,..., x u }) = u i=1 log P θ (x i )

8 The EM algorithm Maximization of l(θ {x 1,..., x u }) is a difficult problem. Iterative maximization strategies, as the EM algorithm, can be used in practice to get local maxima. 1. Expectation: compute the responsibilities γ i = πg Σ2 (x i µ 2 ) (1 π)g Σ1 (x i µ 1 ) + πg Σ2 (x i µ 2 ) 2. Maximization: compute means and variances µ 2 = i γ ix i i, Σ 2 = γ i(x i µ 2 )(x i µ 2 ) T, etc i γ i and the mixing probability π = 1 u 3. Iterate until convergence i γ i. i γ i

9 Combinatorial Clustering Algorithms in this class work on the data without any reference to an underlying probability model. The goal is assigning each data point x i to a cluster k belonging a predefined set {1,2,..., K} C(i) = k, i = 1,2,..., u The optimal encoder C (i) minimizes the overall dissimilarities d(x i, x j ) between points x i, x j assigned to the same cluster W(C) = 1 2 K k=1 C(i)=k C(j)=k d(x i, x j ) The simplest choice for the dissimilarity d(, ) is the squared Euclidean distance in X

10 Combinatorial Clustering (cont.) The minimization of the within-cluster point scatter W(C) is straightforward in principle, but... the number of distinct assignments grows exponentially with the number of data points u S(u, K) = 1 K! already S(19,4) 10 10! K k=1 ( 1) K k( K k ) k u In practice, clustering algorithms look for good suboptimal solutions. Most popular algorithms are based on iterative descent strategies. Convergence to local optima.

11 K-Means If d(x i, x j ) = x i x j 2, introducing the mean vectors x k associated to the k-th cluster, the within-cluster point scatter W(C) can be rewritten as W(C) = 1 2 K k=1 C(i)=k C(j)=k x i x j 2 = K k=1 C(i)=k x i x k 2. Exploiting this representation one can easily verify that the optimal encoder C is the solution of the enlarged minimization problem min C,(m 1,...,m K ) K k=1 C(i)=k x i m k 2.

12 K-Means (cont.) K-Means attempts the minimization of the enlarged problem by an iterative alternating procedure. Each step 1 and 2 reduces the objective function, so convergence is assured. 1. minimization with respect to (m 1,..., m K ), getting m k = x k 2. minimization with respect to C, getting 3. do until C does not change C(i) = arg min 1 k K x i m k One should compare solutions derived from different initial random means, and choose best local minimum.

13 Voronoi tessellation

14 Dimensionality reduction Often reducing the dimensionality of a problem is an effective preliminary step toward the actual solution of a regression or classification problem. We look for a mapping Φ from the high dimensional space IR D to the low dimensional space IR d which preserves some relevant geometrical structure of our problem.

15 Dimensionality reduction

16 Principal Component Analysis (PCA) Trying to approximate data {x 1,..., x u } in IR D by a d- dimensional hyperplane H = {c + Vθ θ IR d } c vector in IR D, θ coordinates vector in IR d and V = (v 1,...,v d ), D d matrix with {v i } orthonormal system of vectors in IR D. Problem: find H which minimizes sum of squared distances of data points x i from H H = argmin H u i=1 x i P H (x i ) 2

17 Linear approximation

18 PCA: the algorithm 1. center data points: u i=1 x i = 0 2. define u D matrix X = (x 1,..., x u ) T 3. construct singular value decomposition X = UΣW T D D matrix W = (w 1,...,w D ), with {w i } right eigenvectors of X u D matrix U = (u 1,...,u D ), with {u i } left eigenvectors of X D D matrix Σ = diag(σ 1,..., σ D ), with σ 1 σ 2 σ D 0 singular eigenvectors of X 4. answer: V = (w 1,...,w d )

19 Sketch of proof Rewrite the minimization problem u min x i c Vθ i 2 c,v,{θ i } i=1 Centering and minimizing with respect to c and θ i gives c = 0, θ i = V T x i Plugging into the minimization problem u arg min x i VV T x i 2 = argmax V V i=1 = argmax V u x T i VV T x i i=1 d vj T X T Xv j hence (v 1,...,v d ) are the first d eigenvectors of X T X: (w 1,...,w d ) j=1

20 Mercer s Theorem Consider the pd kernel K(x, x ) on X X, and the probability distribution p(x) on X. Define the integral operator L K (L K f)(x) = X K(x, x )f(x )dp(x ). Mercer s Theorem states that K(x, x ) = i λ i φ i (x)φ i (x ) where (λ i, φ i ) i is the eigensystem of L K.

21 Feature Map From Mercer s Theorem, the mapping Φ defined over X Φ(x) = ( λ 1 φ 1 (x), λ 2 φ 2 (x),...) is such that K(x, x ) = Φ(x) T Φ(x). K(x, x ) can be interpreted as the dot product in the feature space. given a mapping of X into an Euclidean space, we can construct a pd kernel X X.

22 Kernelization Algorithms that depend on the data, only through the dot products x T i x j, can be easily kernelized: 1. Choose pd kernel K(, ) 2. Replace x T i x j with K(x i, x j ) Example: PCA can be kernelized computing the eigenvectors of the matrix M ij = K(x i, x j ) instead of those of the matrix X T X.

23 ISOMAP Assumption: the support of the marginal distribution p(x) is a convex region of IR d (our manifold M) isometrically embedded in IR D. Goal: constructing a map Φ : IR D IR d which transforms geodesic distances in M into Euclidean distances in IR d Construction 1: approximate the matrix d M of pairwise geodesic distances between data points, estimating the shortest distances d ij over the neighborhood graph. Tenenbaum, et al, 00

24 Construction 2: compute the u u kernel matrix K = 1 2 HDH, H = I 1 u 11T, with 1 the u-dimensional column vector (1,1,...,1), and D the matrix of squared distances, that is: D ij = d 2 ij. Result: let (λ a,u a ) u a=1 be the eigensystem of K. The embedding Φ, of {x i } u i=1 Φ(x i ) = ( λ 1 (u 1 ) i, λ 2 (u 2 ) i,..., λ d (u d ) i ), is the isometry we were looking for.

25 ISOMAP global isometry

26 Explaining ISOMAP Firstly, we have to verify that the matrix K is a genuine pd kernel on the data points. 1. Symmetry: since both H and D are symmetric, K = 1 2 HT DH, hence K T = K. 2. Positivity: Note that, by assumption, there exist vectors {φ i } u i=1, such that d ij = φ i φ j. For all c = (c 1,..., c u ), defining ing c = c 1 u ui=1 c i 1, we get [D ij = φ i φ j 2 ] = 1 2 c T Kc = 1 2 (Hc)T D(Hc) = 1 2 c T Dc [ i c i = 0] = ( i ij c i (φt i φ i + φ T j φ j 2φ T i φ j)c j c i φ i) T ( i c i φ i) 0.

27 Explaining ISOMAP (cont.) We must prove that the pd kernel K ij induces the correct pairwise distances d ij between data points d 2 ij = K ii + K jj 2K ij. This can be verified by direct computation. By Mercer s Theorem, the feature map Φ 0 (x i ) = ( λ 1 (u 1 ) i, λ 2 (u 2 ) i,..., λ u (u u ) i ), is an isometry. If the manifold M is d-dimensional, λ a = 0 for a > d, and we can use the truncated mapping Φ.

28 Hessian Locally Linear Embedding (HLLE) ISOMAP outputs an embedding of the data points {x i } u i=1 into IR d, attempting to preserve pairwise distances on the underlying manifold M. The method gives guarantees of convergence if M is isometric to a convex region in IR d. Convexity is a very strong hypothesis. Typically, linear combinations of images are not reasonable images! HLLE gives guarantees of convergence while relaxing the convexity hypothesis. Hessian Eigenmaps; Donoho, Grimes 03

29 HLLE local isometry

30 Core idea of HLLE For every point x M and system of coordinates (ξ 1,..., ξ d ) on its tangent space, the Hessian at x of a function f : M IR, is the matrix of second derivatives (H f (x)) ij = ξ i ξ j f(x), i, j = 1,..., d The core idea of HLLE is that the null space of the quadratic form H(f) = M ij (H f (x)) 2 ij is independent of the choice of local coordinates ξ i. The null space of H is the d-dimensional linear space spanned by the global cartesian coordinates

31 Computing the Hessian In order to implement this idea, HLLE has to evaluate the quadratic form H using the data points x i. 1. construct proxies for the tangent spaces using the k- nearest neighborhood graph 2. implement a finite differences scheme to evaluate second derivatives 3. compute eigensystem of approximation of H. Use d eigenvectors with smallest eigenvalues as embedding coordinates.

32 Local Linear Neighborhood

33 Laplacian based methods Unsupervised methods based on the eigensystem of the Laplacian on the neighborhood graph with weights W ij. Dimensionality Reduction: consider the solutions of the eigenvector problem (0 = λ 0 λ 1 λ u 1 ) Lf a = λ a Df a where D = diag(d 11,..., D uu ). The considered embedding into the d-dimensional Euclidean space is Φ(x i ) = ((f 1 ) i,...,(f d ) i ). Spectral Clustering: use sign of components (f 1 ) j to define two clusters: connection to min cut problem. Belkin, Niyogi, 02