  When modeling multivariate data, one might have an extra parameter of
contextual information that could be used to treat some observations as more
similar to others. For example, images of faces can vary by age, and one would
expect the face of a 40 year old to be more similar to the face of a 30 year
old than to a baby face. We introduce a novel manifold approximation method,
parameterized principal component analysis (PPCA) that models data with linear
subspaces that change continuously according to the extra parameter of
contextual information (e.g. age), instead of ad-hoc atlases. Special care has
been taken in the loss function and the optimization method to encourage
smoothly changing subspaces across the parameter values. The approach ensures
that each observation's projection will share information with observations
that have similar parameter values, but not with observations that have large
parameter differences. We tested PPCA on artificial data based on known, smooth
functions of an added parameter, as well as on three real datasets with
different types of parameters. We compared PPCA to PCA, sparse PCA and to
independent principal component analysis (IPCA), which groups observations by
their parameter values and projects each group using PCA with no sharing of
information for different groups. PPCA recovers the known functions with less
error and projects the datasets' test set observations with consistently less
reconstruction error than IPCA does. In some cases where the manifold is truly
nonlinear, PCA outperforms all the other manifold approximation methods
compared.
