An Efficient Implementation for Bayesian Manifold Regularization Method

When applying the Bayesian manifold regularization method to function estimation problem with manifold constraints, the direct implementation has computational complexity
$\mathcal{O}(N^{3})$
, where
$N$
is the number of input-output data measurements. This becomes particularly costly when
$N$
is large. In this paper, we propose a more efficient implementation based on the Kalman filter and smoother using a state-space model realization of the underlying Gaussian process. Moreover, we explore the sequentially semi-separable structure of the Laplacian matrix and the posterior covariance matrix. Our proposed implementation has computational complexity
$\mathcal{O}(N)$
and thus can be applied to large data problems. We exemplify the effectiveness of our proposed implementation through numerical simulations.