  A neural population responding to multiple appearances of a single object
defines a manifold in the neural response space. The ability to classify such
manifolds is of interest, as object recognition and other computational tasks
require a response that is insensitive to variability within a manifold. Linear
classification of object manifolds was previously studied for max-margin
classifiers. Soft-margin classifiers are a larger class of algorithms and
provide an additional regularization parameter used in applications to optimize
performance outside the training set by balancing between making fewer training
errors and learning more robust classifiers. Here we develop a mean-field
theory describing the behavior of soft-margin classifiers applied to object
manifolds. Analyzing manifolds with increasing complexity, from points through
spheres to general manifolds, a mean-field theory describes the expected value
of the linear classifier's norm, as well as the distribution of fields and
slack variables. By analyzing the robustness of the learned classification to
noise, we can predict the probability of classification errors and their
dependence on regularization, demonstrating a finite optimal choice. The theory
describes a previously unknown phase transition, corresponding to the
disappearance of a non-trivial solution, thus providing a soft version of the
well-known classification capacity of max-margin classifiers.
