Performance Analysis of a Manifold Learning Algorithm in Dimension Reduction

We consider the performance of local tangent space alignment (Zhang and Zha, 2004), one of several manifold learning algorithms which has been proposed as a dimension reduction method, when errors are present in the observations. Matrix perturbation theory is applied to obtain a worst-case upper bound on the deviation of the solution, which is an invariant subspace. Although we only prove this result for one algorithm, we anticipate that analogous results are derivable for several others due to their strong similarities. Our result clears a conceptual barrier in applying manifold learning algorithms to noisy data. It characterizes the situations under which these manifold learning algorithms are effective tools for dimension reduction.