Invertible nonlinear dimensionality reduction via joint dictionary learning

This paper proposes an invertible nonlinear dimensionality reduction method via jointly learning dictionaries in both the original high dimensional data space and its low dimensional representation space. We construct an appropriate cost function, which preserves inner products of data representations in the low dimensional space. We employ a conjugate gradient algorithm on smooth manifold to minimize the cost function. By numerical experiments in image processing, our proposed method provides competitive and robust performance in image compression and recovery, even on heavily corrupted data. In other words, it can also be considered as an alternative approach to compressed sensing. While our approach can outperform compressed sensing in task-driven learning problems, such as data visualization.