  Consider an ill-posed inverse problem of estimating causal factors from
observations, one of which is known to lie near some (un- known)
low-dimensional, non-linear manifold expressed by a predefined Mercer-kernel.
Solving this problem requires simultaneous estimation of these factors and
learning the low-dimensional representation for them. In this work, we
introduce a novel non-linear dimensionality regulariza- tion technique for
solving such problems without pre-training. We re-formulate Kernel-PCA as an
energy minimization problem in which low dimensionality constraints are
introduced as regularization terms in the energy. To the best of our knowledge,
ours is the first at- tempt to create a dimensionality regularizer in the KPCA
framework. Our approach relies on robustly penalizing the rank of the recovered
fac- tors directly in the implicit feature space to create their
low-dimensional approximations in closed form. Our approach performs robust
KPCA in the presence of missing data and noise. We demonstrate state-of-the-art
results on predicting missing entries in the standard oil flow dataset.
Additionally, we evaluate our method on the challenging problem of Non-Rigid
Structure from Motion and our approach delivers promising results on CMU mocap
dataset despite the presence of significant occlusions and noise.
