  Manifold learning has been successfully applied to a variety of medical
imaging problems. Its use in real-time applications requires fast projection
onto the low-dimensional space. To this end, out-of-sample extensions are
applied by constructing an interpolation function that maps from the input
space to the low-dimensional manifold. Commonly used approaches such as the
Nystr\"{o}m extension and kernel ridge regression require using all training
points. We propose an interpolation function that only depends on a small
subset of the input training data. Consequently, in the testing phase each new
point only needs to be compared against a small number of input training data
in order to project the point onto the low-dimensional space. We interpret our
method as an out-of-sample extension that approximates kernel ridge regression.
Our method involves solving a simple convex optimization problem and has the
attractive property of guaranteeing an upper bound on the approximation error,
which is crucial for medical applications. Tuning this error bound controls the
sparsity of the resulting interpolation function. We illustrate our method in
two clinical applications that require fast mapping of input images onto a
low-dimensional space.
