This paper describes two new methods for modeling the manifolds of dig
itized images of handwritten digits. The models allow a priori informa
tion about the structure of the manifolds to be combined with empirica
l data. Accurate modeling of the manifolds allows digits to be discrim
inated using the relative probability densities under the alternative
models. One of the methods is grounded in principal components analysi
s, the other in factor analysis. Both methods are based on locally lin
ear low-dimensional approximations to the underlying data manifold. Li
nks with other methods that model the manifold are discussed.