Nonlinear component analysis as a kernel eigenvalue problem
Neural Computation
Kernel PCA and de-noising in feature spaces
Proceedings of the 1998 conference on Advances in neural information processing systems II
A Tutorial on Support Vector Machines for Pattern Recognition
Data Mining and Knowledge Discovery
A Database for Handwritten Text Recognition Research
IEEE Transactions on Pattern Analysis and Machine Intelligence
Robust De-noising by Kernel PCA
ICANN '02 Proceedings of the International Conference on Artificial Neural Networks
Mustererkennung 1998, 20. DAGM-Symposium
Iterative Kernel Principal Component Analysis for Image Modeling
IEEE Transactions on Pattern Analysis and Machine Intelligence
Regularized Locality Preserving Learning of Pre-Image Problem in Kernel Principal Component Analysis
ICPR '06 Proceedings of the 18th International Conference on Pattern Recognition - Volume 02
KPCA denoising and the pre-image problem revisited
Digital Signal Processing
Input space versus feature space in kernel-based methods
IEEE Transactions on Neural Networks
The pre-image problem in kernel methods
IEEE Transactions on Neural Networks
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The main challenge in de-noising by kernel Principal Component Analysis (PCA) is the mapping of de-noised feature space points back into input space, also referred to as "the pre-image problem". Since the feature space mapping is typically not bijective, pre-image estimation is inherently illposed. As a consequence the most widely used estimation schemes lack stability. A common way to stabilize such estimates is by augmenting the cost function by a suitable constraint on the solution values. For de-noising applications we here propose Tikhonov input space distance regularization as a stabilizer for pre-image estimation, or sparse reconstruction by Lasso regularization in cases where the main objective is to improve the visual simplicity. We perform extensive experiments on the USPS digit modeling problem to evaluate the stability of three widely used pre-image estimators. We show that the previous methods lack stability in the is non-linear regime, however, by applying our proposed input space distance regularizer the estimates are stabilized with a limited sacrifice in terms of de-noising efficiency. Furthermore, we show how sparse reconstruction can lead to improved visual quality of the estimated pre-image.