## Learning over Sets using Kernel Principal Angles

**
Lior Wolf, Amnon Shashua**; 4(Oct):913-931, 2003.

### Abstract

We consider the problem of learning with instances defined over a space of
sets of vectors. We derive a new positive definite kernel *f*(*A*,*B*) defined over
pairs of matrices *A*,*B* based on the concept of principal angles between two
linear subspaces. We show that the principal angles can be recovered
using only inner-products between pairs of column vectors of the input
matrices thereby allowing the original column vectors of *A*,*B* to be
mapped onto arbitrarily high-dimensional feature spaces.

We demonstrate the usage of the matrix-based kernel function *f*(*A*,*B*)
with experiments on two visual tasks. The first task
is the discrimination of "irregular" motion trajectory of an individual or a
group of individuals in a video sequence. We use the SVM approach
using *f*(*A*,*B*) where an input matrix
represents the motion trajectory of a group of individuals over a
certain (fixed) time frame. We show that the classification
(irregular versus regular) greatly outperforms the
conventional representation where all the trajectories form a single
vector. The second application is the visual recognition of faces from
input video sequences representing head motion and facial expressions
where *f*(*A*,*B*) is used to compare two image sequences.