Principal component analysis and matrix factorizations for learning (part 1) ding - icml 2005 tutorial - 2005

Principal Component Analysis and
Matrix Factorizations for Learning

Chris Ding
Lawrence Berkeley National Laboratory

Supported by Office of Science, U.S. Dept. of Energy

PCA & Matrix Factorizations for Learning, ICML 2005 Tutorial, Chris Ding 1

Many unsupervised learning methods
are closely related in a simple way
PCA NMF

K-means Spectral
clustering Clustering
Semi-supervised
classification

Indicator Matrix Semi-supervised
clustering
Quadratic Clustering
Outlier detection


Part 1.A.
Principal Component Analysis (PCA)
and
Singular Value Decomposition (SVD)

• Widely used in large number of different fields
• Most widely known as PCA (multivariate
statistics)
• SVD is the theoretical basis for PCA


Brief history
• PCA
– Draw a plane closest to data points (Pearson, 1901)
– Retain most variance (Hotelling, 1933)
• SVD
– Low-rank approximation (Eckart-Young, 1936)
– Practical application/Efficient Computation (Golub-
Kahan, 1965)
• Many generalizations


PCA and SVD

Data: n points in p-dim: X = ( x1 , x2 ,L, xn )
p
Covariance C = XX T = ∑ λk uk uk
T

k =1
r
Gram (kernel) matrix XTX = ∑
k =1
λk v k v k
T

Principal directions: u k Principal components: k v
(Principal axis,subspace) (projection on the subspace)
p
Underlying basis: SVD X = ∑k =1
σ k uk vk = UΣV T
T


Further Developments
SVD/PCA
• Principal Curves
• Independent Component Analysis
• Sparse SVD/PCA (many approaches)
• Mixture of Probabilistic PCA
• Generalization to exponential familty, max-margin
• Connection to K-means clustering
Kernel (inner-product)
• Kernel PCA


Methods of PCA Utilization
Principal components X = ( x1 , x2 ,L, xn )
(uncorrelated random variables):

uk = uk (1) ⋅ X 1 + L + uk (d ) ⋅ X d
p
Dimension reduction: X= ∑
k =1
σ k uk vk = UΣV T
T

Projection to low-dim ~ U = (u1 ,L, uk )
X =UT X
subspace

Sphereing the data ~
X = C −1 / 2 X = UΣ −1U T X
Transform data to N(0,1)

Applications of PCA/SVD
• Most popular in multivariate statistics
• Image processing, signal processing
• Physics: principal axis, diagonalization of
2nd tensor (mass)
• Climate: Empirical Orthogonal Functions
(EOF)
• Kalman filter. s ( t +1) = As ( t ) + E , P ( t +1) = AP (t ) AT
• Reduced order analysis


Applications of PCA/SVD

• PCA/SVD is as widely as Fast Fourier Transforms
– Both are spectral expansions
– FFT is more on Partial Differential Equations
– PCA/SVD is more on discrete (data) analysis
– PCA/SVD surpass FFT as computational sciences
further advance
• PCA/SVD
– Select combination of variables
– Dimension reduction
• An image has 104 pixels. True dimension is 20 !


PCA is a Matrix Factorization
(spectral/eigen decomposition)
Principal directions: U = (u1 , u2 ,L, uk )
Principal components: V = (v1 , v2 ,L, vk )
p
Covariance C = XX T = ∑k =1
λk uk uk =UΛU T
T

r
Kernel matrix XTX = ∑k =1
λk vk vk = VΛV T
T

p
Underlying basis: SVD X = ∑k =1
σ k uk vk = UΣV T
T


From PCA to spectral clustering
using generalized eigenvectors
Consider the kernel matrix: Wij = φ ( xi ),φ ( x j )

In Kernel PCA we compute eigenvector: Wv = λv

Generalized Eigenvector: Wq = λDq

D = diag (d1,L, dn ) di = ∑w j ij

This leads to Spectral Clustering !

Scale PCA ⇒ Spectral Clustering

PCA: W= ∑ k
vk λk vT
k

∑
1
~ 1
Scaled PCA: W = D W D2 = D
2
qk λk qT D
k
k =1
~ −1 −1 ~
W = D WD , wij = wij /(did j)
2 2 1/ 2

−1
qk = D vk scaled principal component
2


Scaled PCA on a Rectangle Matrix
⇒ Correspondence Analysis
~ −1 −1 ~
Re-scaling: P = Dr 2 PD 2 , pij = pij /( pi. pj.)1/ 2
c

~
Apply SVD on P Subtract trivial component

P − rc / p.. = Dr ∑ f k λk g Dc
T T
k r = ( p1.,L, pn. ) T

k =1
−1
fk = D u , gk = D v
2
−1
2 c = ( p.1,L, p.n ) T
r k c k

are scaled row and column principal
component (standard coordinates in CA)
(Zha, et al, CIKM 2001, Ding et al, PKDD2002)

Nonnegative Matrix Factorization

Data Matrix: n points in p-dim:
xi is an image,
X = ( x1 , x2 ,L, xn ) document,
webpage, etc

Decomposition
(low-rank approximation) X ≈ FG T

Nonnegative Matrices
X ij ≥ 0, Fij ≥ 0, Gij ≥ 0

F = ( f1 , f 2 , L, f k ) G = ( g1 , g 2 ,L, g k )

Solving NMF with multiplicative updating

J =|| X − FGT ||2 , F ≥ 0, G ≥ 0

Fix F, solve for G; Fix G, solve for F

Lee & Seung ( 2000) propose

( XG )ik ( X T F ) jk
Fik ← Fik G jk ← G jk
( FGT G )ik (GF T F ) jk


Matrix Factorization Summary
Symmetric Rectangle Matrix
(kernel matrix, graph) (contigency table, bipartite graph)

PCA: W = VΛV T X = UΣV T

Scaled PCA:
1
1
~ 1 ~ 1
W = D W D = D QΛQ D
2 2 T
X = Dr X Dc2 = Dr FΛG T Dc
2

NMF: W ≈ QQ T
X ≈ FG T


Indicator Matrix Quadratic Clustering

Unsigned Cluster indicator Matrix H=(h1, …, hK)
Kernel K-means clustering:

max Tr( H T WH ), s.t. H T H = I , H ≥ 0
H

K-means: W = XT X; Kernel K-means W = (< φ ( xi ),φ ( x j ) >)

Spectral clustering (normalized cut)

max Tr( H T WH ), s.t. H T DH = I , H ≥ 0
H

Difference between the two is the orthogonality of H

Indicator Matrix Quadratic Clustering
Additional features:

Semi-suerpvised classification: max Tr( H T WH + C T H )
H

Semi-supervised clustering: (A) must-link and (B) cannot-link constraints

max Tr( H T WH + αH T AH − βH T BH )
H

Outlier Detection: max Tr( H T WH ) allowing zero rows in H
H

Nonnegative Lagrangian Relaxation:
(WH )ik + Cik / 2
H ik ← H ik , α = H T WH + H T C.
( Hα )ik


Tutorial Outline
• PCA
– Recent developments on PCA/SVD
– Equivalence to K-means clustering
• Scaled PCA
– Laplacian matrix
– Spectral clustering
– Spectral ordering
• Nonnegative Matrix Factorization
– Equivalence to K-means clustering
– Holistic vs. Parts-based
• Indicator Matrix Quadratic Clustering
– Use Nonnegative Lagrangian Relaxtion
– Includes
• K-means and Spectral Clustering
• semi-supervised classification
• Semi-supervised clustering
• Outlier detection

Part 1.B.
Recent Developments on PCA and SVD
Principal Curves
Independent Component Analysis
Kernel PCA
Mixture of PCA (probabilistic PCA)
Sparse PCA/SVD
Semi-discrete, truncation, L1 constraint, Direct
sparsification
Column Partitioned Matrix Factorizations
2D-PCA/SVD
Equivalence to K-means clustering

PCA & Matrix Factorization for Learning, ICML 2005, Chris Ding 20

PCA and SVD

Data Matrix: X = ( x1 , x2 ,L, xn )
p
Covariance C = XX T = ∑ λk uk uk
T

k =1
r
Gram (kernel) matrix XTX = ∑
k =1
λk v k v k
T

Principal directions: u k Principal components: k v
(Principal axis,subspace) (projection on the subspace)
p
Underlying basis: SVD X = ∑σ u v T
k k k
k =1

Kernel PCA
xi → φ ( xi )
Kernel
K ij = φ ( xi ),φ ( x j ) PCA Component v
Feature extraction
v, φ ( x ) = ∑ i
vi φ ( xi ),φ ( x)

Indefinite Kernels
Generalization to graphs with nonnegative weights

(Scholkopf, Smola, Muller, 1996)


Mixture of PCA
• Data has local structures.
– Global PCA on all data is not useful
• Clustering PCA (Hinton et al):
– Using clustering to cluster data into clusters
– Perform PCA in each cluster
– No explicit generative model
• Probabilistic PCA (Tipping & Bishop)
– Latent variables
– Generative model (Gaussian)
– Mixture of Gaussians ⇒ mixture of PCA
– Adding Markov dynamics for latent variables (Linear
Gaussian Models)


Probabilistic PCA
Linear Gaussian Model

Latent variables S = ( s1 ,L, sn )

xi = Wsi + μ + ε , ε ~ N (0,σ ε I ) 2

Gaussian prior P( s) ~ N ( s0 , σ s I )
2

x ~ N (Ws0 ,σ ε I + σ sWW )
2 T

Linear Gaussian Model
si +1 = Asi + η , xi = Wsi + ε ,
(Tipping & Bishop, 1995; Roweis & Ghahramani, 1999)


Sparse PCA
• Compute a factorization X ≈ UV T
– U or V is sparse or both are sparse
• Why sparse?
– Variable selection (sparse U)
– When n >> d
– Storage saving
– Other new reasons?
• L1 and L2 constraints


Sparse PCA: Truncation and
Discretization
X ≈ UΣV T
• Sparsified SVD U = (u1 Luk ) V = (v1 Lvk )
– Compute {uk,vk} one at a time, truncate those entries
below a threshold.
– Recursively compute all pairs using deflation.
– (Zhang, Zha, Simon, 2002)
X ← X − σ uvT
• Semi-discrete decomposition
– U, V only contains {-1, 0, 1}
– Iterative algorithm to compute U,V using deflation
– (Kolda & O’leary, 1999)


Sparse PCA: L1 constraint
• LASSO (Tibshirani, 1996)
min || y − X T β ||2 , || β ||1 ≤ t
• SCoTLASS (Joliffe & Uddin, 2003)
max u T ( XX T )u T , || u ||1 ≤ t , u T uh = 0
• Least Angle Regression (Efron, et al 2004)
• Sparse PCA (Zou, Hastie, Tibshirani,2004)
n k k
min
α ,β
∑
i =1
|| xi − α β T xi ||2 +λ ∑
j =1
|| β j ||2 + ∑
j =1
λ1, j || β j ||1 , α T α = I

v j = β j / || β j ||

Sparse PCA: Direct Sparsification

• Sparse SVD with explicit sparsification
min || X − udvT ||F + nnz(u ) + nnz(v)
u ,v
– rank-one approximation (Zhang, Zha, Simon 2003)
– Minimize a bound
– deflation
• Direct sparse PCA, on covariance matrix S
u = max u T Su = max Tr( Suu T ) = max Tr( SU )
s.t. Tr(U ) = 1, nnz(U ) ≤ k 2 , U f 0, rank(U ) = 1
(D’Aspremont, Gharoui, Jordan,Lancriet, 2004)

Sparse PCA Summary
• Many different approaches
– Truncation, discretization
– L1 Constraint
– Direct sparsification
– Other approaches
• Sparse Matrix factorization in general
– L1 constraint
• Many questions
– Orthogonality
– Unique solution, global solution

PCA: Further Generalizations
• Generalization to Exponential Family
– (Collins, Dasgupta, Schapire, 2001)

• Maximum Margin Factorization (Srebro, Rennie, Jaakkola, 2004)
– Collaborative filtering
– Input Y is binary
– Hard margin Yia X ia ≥ 1, ∀ia ∈ S
– Soft margin
min || X ||Σ + c ∑ max(0,1 − Y
ia∈S
ia X ia )

X = UV T , || X ||= 1 (|| U ||2 + || V ||2 )
2 Fro Fro

Column Partitioned Matrix Factorizations

n n n
6 7 8 64748
4 14 2 64748
4k 4 n1 + L + nk = n
X = ( x1,L xn ) = ( x1 L xn1 , xn1 +1 L xn2 , L, xnk −1 +1 L xn )

(Zhang & Zha, 2001)
• Column Partitioned Data Matrix
• Partitions are generate by clustering (Dhillon & Modha, 2001)
• Centroid matrix U = (u1 Luk ) (Park, Jeon & Rosen, 2003)

– uk is centroid
– Fix U, compute V min || X − UV T ||2
F V = X T U (U T U ) −1
• Represent each partition by a SVD. k k
6 74
41 8 4l 8
6 74
– Pick leading Us to form U
U = (U1,LU l ) = (u11) Luk1) , L, u1l ) Lukl ) )
( ( ( (

– Fix U, compute V 1 l

(Castelli, Thomasian & Li 2003)
• Several other variations
(Zeimpekis & Gallopoulos, 2004)


Two-dimensional SVD
• Large number of data objects are 2-D: images, maps
• Standard method:
– convert (re-order) each image as a 1D vector
– collect all 1D vectors into a single (big) matrix
– apply SVD on the big matrix
• 2D-SVD is developed for 2D objects
– Extension of standard SVD
– Keeping the 2D characteristics
– Improves quality of low-dimensional approximation
– Reduces computation, storage


Linearize a 2D object into 1D object

⎡0.0⎤
⎢ 0.5⎥
⎢ ⎥
⎢0.7⎥
⎢10 ⎥
⎢. ⎥
⎢M ⎥
⎢ ⎥
⎢0.8⎥
⎢0.2⎥
⎢ ⎥
⎣0.0⎦
Pixel vector


SVD and 2D-SVD
SVD X = ( x1 , x2 ,L, xn )
Eigenvectors of XX T and X T X
X = UΣV T Σ =UT X V

2D-SVD { A} = { A1 , A2 ,L, An }
Eigenvectors of
F= ∑ i
( Ai − A )( Ai − A )T row-row covariance

G= ∑ i
( Ai − A )T ( Ai − A ) column-column cov

Ai = UM iV T M i = U Ai V
T


2D-SVD
{ A} = { A1 , A2 ,L, An } assume A =0
F = ∑ Ai Ai = ∑ λ u u
T T
row-row cov:
k k k
i
G = ∑ Ai Ai = ∑ ζ k uk u T
T
col-col cov: k
i k =1

Bilinear U = (u1 , u2 ,L, uk )
subspace V = (v1 , v2 ,L, vk ) M i = U Ai V
T

Ai = UM iV , i = 1,L, n T

Ai ∈ ℜr×c ,U ∈ ℜr×k ,V ∈ ℜc×k , M i ∈ ℜk×k

2D-SVD Error Analysis
p
SVD: min || X − UΣV T ||2 = ∑ σ i2
i = k +1

Ai ≈ LM i RT , Ai ∈ R r×c , L ∈ R r×k , R ∈ R c×k , M i ∈ R k×k
n c
min J1 = ∑i =1
|| Ai − LM i ||2 = ∑ζ
j = k +1
j

n r
min J 2 = ∑i =1
|| Ai − M i RT ||2 = ∑λ
j = k +1
j

n r c
min J 3 = ∑i =1
|| Ai − LM i RT ||2 ≅ ∑ λ + ∑ζ
j = k +1
j
j = k +1
j
n r
min J 4 = ∑i =1
|| Ai − LM i LT ||2 ≅ 2 ∑λ
j = k +1
j


Temperature maps (January over 100 years)

Reconstruction
Errors
SVD/2DSVD=1.1

Storages
SVD/2DSVD=8


Reconstructed image

SVD

2dSVD

SVD (K=15), storage 160560

2DSVD (K=15), storage 93060


2D-SVD Summary

• 2DSVD is extension of standard SVD
• Provides optimal solution for 4 representations for
2D images/maps
• Substantial improvements in storage, computation,
quality of reconstruction
• Capture 2D characteristics


Part 1.C.
K-means Clustering ⇔
Principal Component Analysis

(Equivalence between PCA and K-means)

40
PCA & Matrix Factorizations for Learning, ICML 2005 Tutorial, Chris Ding

K-means clustering

• Also called “isodata”, “vector quantization”
• Developed in 1960’s (Lloyd, MacQueen, Hatigan,
etc)
• Computationally Efficient (order-mN)
• Widely used in practice
– Benchmark to evaluate other algorithms

Given n points in m-dim: X = ( x1 , x2 ,L, xn ) T

K
K-means objective min J K = ∑∑
k =1 i∈C k
|| xi − ck ||2
41

PCA is equivalent to K-means

Continuous optimal solution for cluster
indicators in K-means clustering are
given by principal components.

Subspace spanned by K cluster centroids
is given by PCA subspace.

42

2-way K -means Clustering
Cluster membership ⎧+ n2 / n1n
⎪ if i ∈ C1
q (i ) = ⎨
indicator: ⎪− n1 / n2 n if i ∈ C2
⎩
nn ⎡ d (C1 , C2 ) d (C1 , C1 ) d (C2 , C2 ) ⎤
J K = n〈 x 〉 − J D ,
2
JD = 1 2 ⎢2 n n − 2
− 2 ⎥
n ⎣ 1 2 n1 n2 ⎦

Define distance matrix: D = (dij ), dij =|xi − x j|2
T ~ ~
J D = −q Dq = −q Dq = 2q ( X X )q = 2q Kq D = K
T T T T

min J K ⇒ max J D Solution is principal eigenvector v1of K

Clusters C1, C2 are determined by: C1 = {i | v1 (i ) < 0}, C2 = {i | v1 (i ) ≥ 0}
43

A simple illustration

44

DNA Gene Expression File for Leukemia

Using v1 , tissue
samples separated
into 2 clusters, 3
errors

Do one more K-
means, reduce to 1
error

45

Multi-way K-means Clustering

Unsigned Cluster membership indicators h1, …, hK:

C1 C2 C3

⎡1 0 0⎤
⎢1 ⎥
0 0⎥
⎢ = (h1 , h2 , h3 )
⎢0 1 0⎥
⎢ ⎥
⎣0 0 1⎦

46

K K

∑ ∑ ∑ ∑ ∑
1
JK = xi −
2
xiT x j = xi2 − hk X T Xhk
T
i n i , j∈C k i
k =1 k k =1

(Unsigned) Cluster indicators H=(h1, …, hK)

J K = ∑ xi2 − Tr ( H k X T XH k )
T

i
K

Regularized Relaxation Redundancy: ∑
k =1
n1/ 2 hk = e
k

Transform h1, …, hK to q1 - qk via orthogonal matrix T
(q1 ,..., qk ) = (h1 ,L, hk )T Qk = H kT q1 = e /n1/2
47


maxTr[Qk −1 ( X T X )Qk −1 ]
T
Qk −1 = (q2 ,..., qk )

Optimal solutions of q2 … qk are given by
principal components v2 … vk.
JK is bounded below by total variance minus
sum of K eigenvalues of covariance:
K −1
nx2 − ∑
k =1
λk < min J K < n x 2

48

Consistency: 2-way and K-way approaches

⎛ n2 / n − n1 / n ⎞
Orthogonal Transform: T =⎜ ⎟
⎜ n /n n2 / n ⎟
⎝ 1 ⎠

T transforms (h1, h2) to (q1,q2):

h1 = (1L1,0L0) , h2 = (0L0,1L1)
T T a=
n2
n1n

q2 = (a,L, a,−b,L,−b) T n1
q1 = (1L1) , T b=
n2 n

Recover the original 2-way cluster indicator
49

Test of Lower bounds of K-means clustering
| J opt − J LB |
J opt

Lower bound is within 0.6-1.5% of the optimal value
50

Cluster Subspace (spanned by K centroids)
= PCA Subspace
Given a data point x,

P= ∑k
T
ck c k project x into the cluster subspace

Centroid is given by ck = ∑ h (i) x = Xh
k
k i k

P= ∑c c
k
T
k k =X ∑h h
k
T
k k XT = X ∑v v
k
T
k k XT = ∑λ u u
k
T
k k k

PK −means = ∑k
λk u k u k
T
⇔ ∑
k
uk uk ≡ PPCA
T

PCA automatically project into cluster subspace
PCA is unsupervised version of LDA
51

Effectiveness of PCA Dimension Reduction

52

Kernel K-means Clustering
Kernal K-means objective: xi → φ ( xi )
K
φ
min J K = ∑∑
k =1 i∈Ck
|| φ ( xi ) − φ (ck ) ||2

K

∑ ∑ ∑
1
= | φ ( xi ) |2 − φ ( xi )T φ ( x j )
i
n
k =1 k i , j∈Ck

1 K
Kernal K-means max J K = ∑ ∑ φ ( xi ),φ ( x j )
φ

k =1 nk i , j∈Ck

53

Kernel K-means clustering
is equivalent to Kernal PCA

Continuous optimal solution for cluster
indicators are given by Kernal PCA components

Subspace spanned by K cluster centroids
are given by Kernal PCA principal subspace

54

Principal component analysis and matrix factorizations for learning (part 1) ding - icml 2005 tutorial - 2005

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