The signal reconstruction of speech by KPCA

A new method for speech signal reconstruction is proposed by performing a nonlinear Kernel Principal Component Analysis (KPCA). By the use of kernel functions, one can efficiently compute principal components in high-dimensional feature spaces, and reconstruct vectors mapping from input space by those dominant principal components. As the reconstructed vectors is expressed in high dimensional feature space and they could not exist pre-image in input space. For finding pre-image, we use iteration method to approximate the pre-image. The experimental results using KPCA in data reconstruction and denoising in speech signal show that it had many potential advantages comparing with PCA. 1.PRINCIPLE Principal Component Analysis (PCA) is an orthogonal basis transformation. The new basis is founded by diagonalizing the centered covariance matrix of a data set, The coordinates in the Eigenvector basis are called principal components. The size of an Eigenvalue corresponding to an Eigenvector v of covariance matrix equals the amount of variance in the direction of v. Furthermore, the directions of the first n Eigenvectors corresponding to the biggest n Eigenvalues cover as much variance as possible by n orthogonal directions. In many applications they contain the most interesting information: for instance, in data compression, where we project onto the directions with biggest variance to retain as much information as possible, or in de-noising, where we deliberately drop directions with small variance. Assume that our data is mapped into feature space by nonlinear map ) ( : x x Φ > − Φ , ) ( , ), ( ), ( 2 1 l x x x Φ Φ Φ Λ , (1) And do PCA for the covariance matrix. ∑ = Φ Φ = l i T l 1 ) ( ) ( 1 i i x x C (2) We have to find Eigenvalues and Eigenvectors satisfying V C �V = . (3) This implies that we may consider the equivalent system. l j C i , , 1 ), ( ), ( Λ = > Φ >=< Φ < i j i j v x v x λ . (4) By defining > Φ Φ =< ) ( ), ( j i x x ij K , we get the expression