Non-linear Dimensionality Regularizer for Solving Inverse Problems

Ravi Garg, Anders Eriksson, Ian Reid

Abstract:

Consider an ill-posed inverse problem of estimating causal factors from observations, one of which is known to lie near some (unknown) 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 regularization 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 attempt to create a dimensionality regularizer in the KPCA framework. Our approach relies on robustly penalizing the rank of the recovered factors 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.

TL;DR: Predicting causal factors of an inverse problem which lie near unknown low-dimensional non-linear manifold defined by a mercer kernel.

Conflicts: adelaide.edu.au, qut.edu.au, qmul.ac.uk, ox.ac.uk

Keywords: Computer vision, Optimization, Structured prediction