Robust principal component analysis by self-organizing rules based on statistical physics approach

Abstract

This paper applies statistical physics to the problem of robust principal component analysis (PCA). The commonly used PCA learning rules are first related to energy functions. These functions are generalized by adding a binary decision field with a given prior distribution so that outliers in the data are dealt with explicitly in order to make PCA robust. Each of the generalized energy functions is then used to define a Gibbs distribution from which a marginal distribution is obtained by summing over the binary decision field. The marginal distribution defines an effective energy function, from which self-organizing rules have been developed for robust PCA. Under the presence of outliers, both the standard PCA methods and the existing self-organizing PCA rules studied in the literature of neural networks perform quite poorly. By contrast, the robust rules proposed here resist outliers well and perform excellently for fulfilling various PCA-like tasks such as obtaining the first principal component vector, the first k principal component vectors, and directly finding the subspace spanned by the first k vector principal component vectors without solving for each vector individually. Comparative experiments have been made, and the results show that the authors' robust rules improve the performances of the existing PCA algorithms significantly when outliers are present.