Multi-view Clustering via

AI and Robotics

Nov 25, 2013 (4 years and 5 months ago)

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Multi
-
view Clustering via

Canonical Correlation Analysis

Kamalika

Chaudhuri

et al.
ICML

2009.

INTRODUCTION

ASSUMPTION IN MULTI
-
VIEW
PROBLEMS

The input variable (a real vector) can be partitioned into
two different views, where it is assumed that either view
of the input is sufficient to make accurate predictions
---

essentially the co
-
training assumption.

e.g.

Identity recognition with one view being a video stream and
the other an audio stream;

Web page classification where one view is the text and the

Object recognition with pictures from different camera angles;

A bilingual parallel corpus, with each view presented in one
language.

INTUITION IN MULTI
-
VIEW
PROBLEMS

Many multi
-
view learning algorithms force
agreement between the predictors based on either
view. (usually force the predictor on view 1 to equal
to the predictor based on view 2)

The complexity of the learning problem is reduced
by eliminating hypothesis from each view that do
not agree with each other.

BACKGROUND

CANONICAL CORRELATION ANALYSIS

CCA

is a way of measuring the linear relationship
between two multidimensional variables.

F
ind two basis vectors, one for
x
and one for
y
, such
that the correlations between the
projections

of the
variables onto these basis vectors are maximized.

CALCULATING CANONICAL
CORRELATIONS

Consider the total covariance matrix of random
variables
x

and
y

with zero mean:

The canonical correlations between
x

and
y

can be
found by solving the eigenvalue equations

RELATION TO OTHER LINEAR
SUBSPACE METHODS

Formulate the problems in one single eigenvalue
equation

PRINCIPAL COMPONENT ANALYSIS

The principal components are the eigenvectors of
the covariance matrix.

The projection of data onto the principal
components is an orthogonal transformation that
diagonalizes

the covariance matrix.

PARTIAL LEAST SQUARES

PLS

is basically the singular value decomposition
(
SVD
) of a between
-
sets covariance matrix.

In
PLS

regression, the principal vectors
corresponding to the largest principal values are
used as basis. A regression of
y

onto
x

is then
performed in this basis.

ALGORITHM

THE BASIC IDEA

Use
CCA

to project the data down to the subspace
spanned by the means to get an easier clustering
problem, then apply standard clustering algorithms
in this space.

When the data in at least one of the views is well
separated, this algorithm clusters correctly with high
probability.

A
L
GORITHM

Input: a set of samples
S,

the number of clusters
k

1.
Randomly partition
S

into two subsets
A

and
B

of
equal size.

2.
Let
C_12
(A) be the covariance matrix between
views 1 and 2, computed from the set
A
.
Compute the top
k
-
1

left singular vectors of
C_12
(A), and project the samples in
B

on the
subspace spanned by these vectors.

3.
clustering, K
-
means) to the projected examples in
view 1.

EXPERIMENTS

SPEAKER IDENTIFICATION

Dataset

41 speakers, speaking 10 sentences each

Audio features 1584 dimensions

Video feature 2394 dimensions

Method 1: use
PCA

project into 40 D

Method 2:
u
se
CCA

(after
PCA

into 100 D for images
and 1000 D for audios)

Cluster into 82 clusters (2 / speaker) using K
-
means

SPEAKER IDENTIFICATION

Evaluation

Conditional perplexity

= the mean # of speakers corresponding to each cluster

CLUSTERING WIKIPEDIA ARTICLES

Dataset

128 K Wikipedia articles, evaluated on 73 K articles that
belong to the 500 most frequent categories.

L

is a concatenation of ``to`` and
``from`` vectors.
L(
i
)

is the number of times the current
i
.

Text feature is a bag
-
of
-
words vector.

Methods: compared
PCA

and
CCA

Used a hierarchical clustering procedure, iteratively pick the
largest cluster, reduce the dimensionality using
PCA

or
CCA
,
and use k
-
means to break the cluster into smaller ones, until
reaching the total desired number of clusters.

RESULTS

THANK YOU

APPENDIX: A NOTE ON CORRELATION

Correlation between
x_i

and
x_i

is the covariance
normalized by the geometric mean of the
variances of
x_i

and
x_j

AFFINE TRANSFORMATIONS

An affine transformation is a map