Nonnegative matrix factorization with Hessian regularizer

Because of the underlying data structure preserved by the manifold regularization term, the Nonnegative matrix factorization (NMF) with manifold regularizer demonstrates an advantage over the variants of NMF for many data analysis tasks. Currently, the Laplacian regularizer is commonly used as the smooth operator to preserve the locality of data space. However, with the Laplacian regularizer, coding vectors are biased to a constant, which leads to a lack of extrapolating power. Thus, the locality of data space cannot be preserved, as would be expected. To address this drawback, a novel variant of NMF, namely HsNMF, is proposed, where the Hessian regularization term is incorporated into the traditional NMF framework. Because Hessian Energy favors the functions whose values vary linearly with respect to the geodesics of the data manifold, the local structure of data space is more effectively preserved. Clustering and classification experimental results on real-world image datasets demonstrate that our proposed NMF is superior to the variants of NMF based on Laplacian Embedding.