Regularized Pre-image Estimation for Kernel PCA De-noising

Abstract

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, preimage 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.

A local linear algorithm for calculating the pre-image in kernel PCA

“…So the pre-image can be extracted by the inner products from the represented in this basis [6]. Trine Julie Abrahamsen and Lars Kai Hansen proposed the pre-image method is more stable than other algorithm, which performed the Tikhonov regularization in input space and used the sparse reconstruction by Lasso regularization [7]. This paper proposes a new pre-image approach by calculating the locally optimal linear fits.…”