MLCC 2015 Dimensionality Reduction and PCA

MLCC 2015 Dimensionality Reduction and PCA Lorenzo Rosasco UNIGE-MIT-IIT June 25, 2015

Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component PCA and Singular Value Decomposition Kernel PCA MLCC 2015 2

Dimensionality Reduction In many practical applications it is of interest to reduce the dimensionality of the data: data visualization data exploration: for investigating the effective dimensionality of the data MLCC 2015 3

Dimensionality Reduction (cont.) This problem of dimensionality reduction can be seen as the problem of defining a map M : X = R D R k, k D, according to some suitable criterion. MLCC 2015 4

Dimensionality Reduction (cont.) This problem of dimensionality reduction can be seen as the problem of defining a map M : X = R D R k, k D, according to some suitable criterion. In the following data reconstruction will be our guiding principle. MLCC 2015 5

Principal Component Analysis PCA is arguably the most popular dimensionality reduction procedure. MLCC 2015 6

Principal Component Analysis PCA is arguably the most popular dimensionality reduction procedure. It is a data driven procedure that given an unsupervised sample S = (x 1,..., x n ) derive a dimensionality reduction defined by a linear map M. MLCC 2015 7

Dimensionality Reduction by Reconstruction Recall that, if w R D, w = 1, then (w T x)w is the orthogonal projection of x on w MLCC 2015 9

Dimensionality Reduction by Reconstruction Recall that, if w R D, w = 1, then (w T x)w is the orthogonal projection of x on w MLCC 2015 10

Dimensionality Reduction by Reconstruction (cont.) First, consider k = 1. The associated reconstruction error is x (w T x)w 2 (that is how much we lose by projecting x along the direction w) MLCC 2015 11

Dimensionality Reduction by Reconstruction (cont.) Let S D 1 = {w R D w = 1} is the sphere in D dimensions. Consider the empirical reconstruction minimization problem, 1 min w S D 1 n n x i (w T x i )w 2. The solution p to the above problem is called the first principal component of the data MLCC 2015 13

An Equivalent Formulation A direct computation shows that x i (w T x i )w 2 = x i (w T x i ) 2 MLCC 2015 14

An Equivalent Formulation A direct computation shows that x i (w T x i )w 2 = x i (w T x i ) 2 Then, problem is equivalent to 1 min w S D 1 n n x i (w T x i )w 2 1 max w S D 1 n n (w T x i ) 2 MLCC 2015 15

Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component PCA and Singular Value Decomposition Kernel PCA MLCC 2015 16

Reconstruction and Variance Assume the data to be centered, x = 1 n x i = 0, then we can interpret the term (w T x) 2 as the variance of x in the direction w. MLCC 2015 17

Reconstruction and Variance Assume the data to be centered, x = 1 n x i = 0, then we can interpret the term (w T x) 2 as the variance of x in the direction w. The first PC can be seen as the direction along which the data have maximum variance. 1 n max (w T x i ) 2 w S D 1 n MLCC 2015 18

Centering If the data are not centered, we should consider 1 max w S D 1 n n (w T (x i x)) 2 (1) equivalent to with x c = x x. 1 max w S D 1 n n (w T x c i) 2 MLCC 2015 19

Centering and Reconstruction If we consider the effect of centering to reconstruction it is easy to see that we get where 1 min w,b S D 1 n n x i ((w T (x i b))w + b) 2 ((w T (x i b))w + b is an affine (rather than an orthogonal) projection MLCC 2015 20

Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component PCA and Singular Value Decomposition Kernel PCA MLCC 2015 21

PCA as an Eigenproblem A further manipulation shows that PCA corresponds to an eigenvalue problem. MLCC 2015 22

PCA as an Eigenproblem A further manipulation shows that PCA corresponds to an eigenvalue problem. Using the symmetry of the inner product, 1 n n (w T x i ) 2 = 1 n n w T x i w T x i = 1 n n w T x i x T i w = w T ( 1 n n x i x T i )w MLCC 2015 23

PCA as an Eigenproblem (cont.) We make two observations: The ( covariance ) matrix C n = 1 n n XT n X n is symmetric and positive semi-definite. MLCC 2015 25

PCA as an Eigenproblem (cont.) We make two observations: The ( covariance ) matrix C n = 1 n n XT n X n is symmetric and positive semi-definite. The objective function of PCA can be written as the so called Rayleigh quotient. w T C n w w T w Note that, if C n u = λu then ut C nu u T u = λ, since u is normalized. MLCC 2015 27

PCA as an Eigenproblem (cont.) Computing the first principal component of the data reduces to computing the biggest eigenvalue of the covariance and the corresponding eigenvector. C n u = λu, C n = 1 n n Xn T X n MLCC 2015 29

Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component PCA and Singular Value Decomposition Kernel PCA MLCC 2015 30

Beyond the First Principal Component We discuss how to consider more than one principle component (k > 1) M : X = R D R k, k D The idea is simply to iterate the previous reasoning MLCC 2015 31

Residual Reconstruction The idea is to consider the one dimensional projection that can best reconstruct the residuals r i = x i (p T x i )p i MLCC 2015 32

Residual Reconstruction The idea is to consider the one dimensional projection that can best reconstruct the residuals r i = x i (p T x i )p i An associated minimization problem is given by 1 min w S D 1,w p n (note: the constraint w p) n r i (w T r i )w 2. MLCC 2015 33

Residual Reconstruction (cont.) Note that for all i = 1,..., n, since w p r i (w T r i )w 2 = r i 2 (w T r i ) 2 = r i 2 (w T x i ) 2 MLCC 2015 34

Residual Reconstruction (cont.) Note that for all i = 1,..., n, since w p r i (w T r i )w 2 = r i 2 (w T r i ) 2 = r i 2 (w T x i ) 2 Then, we can consider the following equivalent problem 1 max w S D 1,w p n n (w T x i ) 2 = w T C n w. MLCC 2015 35

PCA as an Eigenproblem 1 max w S D 1,w p n n (w T x i ) 2 = w T C n w. Again, we have to minimize the Rayleigh quotient of the covariance matrix with the extra constraint w p MLCC 2015 36

PCA as an Eigenproblem 1 max w S D 1,w p n n (w T x i ) 2 = w T C n w. Again, we have to minimize the Rayleigh quotient of the covariance matrix with the extra constraint w p Similarly to before, it can be proved that the solution of the above problem is given by the second eigenvector of C n, and the corresponding eigenvalue. MLCC 2015 37

PCA as an Eigenproblem (cont.) C n u = λu, C n = 1 n n x i x T i The reasoning generalizes to more than two components: computation of k principal components reduces to finding k eigenvalues and eigenvectors of C n. MLCC 2015 38

Remarks Computational complexity roughly O(kD 2 ) (complexity of forming C n is O(nD 2 )). If we have n points in D dimensions and n D can we compute PCA in less than O(nD 2 )? MLCC 2015 39

Remarks Computational complexity roughly O(kD 2 ) (complexity of forming C n is O(nD 2 )). If we have n points in D dimensions and n D can we compute PCA in less than O(nD 2 )? The dimensionality reduction induced by PCA is a linear projection. Can we generalize PCA to non linear dimensionality reduction? MLCC 2015 40

Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component PCA and Singular Value Decomposition Kernel PCA MLCC 2015 41

Singular Value Decomposition Consider the data matrix X n, its singular value decomposition is given by where: X n = UΣV T U is a n by k orthogonal matrix, V is a D by k orthogonal matrix, Σ is a diagonal matrix such that Σ i,i = λ i, i = 1,..., k and k min{n, D}. The columns of U and the columns of V are the left and right singular vectors and the diagonal entries of Σ the singular values. MLCC 2015 42

Singular Value Decomposition (cont.) The SVD can be equivalently described by the equations 1 C n p j = λ j p j, n K nu j = λ j u j, X n p j = 1 λ j u j, n XT n u j = λ j p j, for j = 1,..., d and where C n = 1 n XT n X n and 1 n K n = 1 n X nx T n MLCC 2015 43

PCA and Singular Value Decomposition If n p we can consider the following procedure: form the matrix K n, which is O(Dn 2 ) find the first k eigenvectors of K n, which is O(kn 2 ) compute the principal components using p j = 1 λj X T n u j = 1 λj n x i u i j, j = 1,..., d where u = (u 1,..., u n ), This is O(knD) if we consider k principal components. MLCC 2015 44

Outline PCA & Reconstruction PCA and Maximum Variance PCA and Associated Eigenproblem Beyond the First Principal Component PCA and Singular Value Decomposition Kernel PCA MLCC 2015 45

Beyond Linear Dimensionality Reduction? By considering PCA we are implicitly assuming the data to lie on a linear subspace... MLCC 2015 46

Beyond Linear Dimensionality Reduction? By considering PCA we are implicitly assuming the data to lie on a linear subspace......it is easy to think of situations where this assumption might violated. MLCC 2015 47

From SVD to KPCA Using SVD the projection of a point x on a principal component p j, for j = 1,..., d, is (M(x)) j = x T p j = 1 λj x T X T n u j = 1 λj n x T x i u i j, Recall 1 C n p j = λ j p j, n K nu j = λ j u j, X n p j = 1 λ j u j, n XT n u j = λ j p j, MLCC 2015 49

PCA and Feature Maps (M(x)) j = 1 λj What if consider a non linear feature-map Φ : X F, before performing PCA? n x T x i u i j, MLCC 2015 50

PCA and Feature Maps (M(x)) j = 1 λj What if consider a non linear feature-map Φ : X F, before performing PCA? n x T x i u i j, (M(x)) j = Φ(x) T p j = 1 λj n where K n σ j = σ j u j and (K n ) i,j = Φ(x) T Φ(x j ). Φ(x) T Φ(x i )u i j, MLCC 2015 51

Kernel PCA (M(x)) j = Φ(x) T p j = 1 λj n Φ(x) T Φ(x i )u i j, If the feature map is defined by a positive definite kernel K : X X R, then (M(x)) j = 1 λj n where K n σ j = σ j u j and (K n ) i,j = K(x i, x j ). K(x, x i )u i j, MLCC 2015 52

Wrapping Up In this class we introduced PCA as a basic tool for dimensionality reduction. We discussed computational aspect and extensions to non linear dimensionality reduction (KPCA) MLCC 2015 53

Next Class In the next class, beyond dimensionality reduction, we ask how we can devise interpretable data models, and discuss a class of methods based on the concept of sparsity. MLCC 2015 54