Elastic Net Hypergraph Learning for Image Clustering and Semi-supervised Classification

Graph model is emerging as a very effective tool for learning the complex structures and relationships hidden in data. Generally, the critical purpose of graph-oriented learning algorithms is to construct an informative graph for image clustering and classification tasks. In addition to the classical $K$-nearest-neighbor and $r$-neighborhood methods for graph construction, $l_1$-graph and its variants are emerging methods for finding the neighboring samples of a center datum, where the corresponding ingoing edge weights are simultaneously derived by the sparse reconstruction coefficients of the remaining samples. However, the pair-wise links of $l_1$-graph are not capable of capturing the high order relationships between the center datum and its prominent data in sparse reconstruction. Meanwhile, from the perspective of variable selection, the $l_1$ norm sparse constraint, regarded as a LASSO model, tends to select only one datum from a group of data that are highly correlated and ignore the others. To simultaneously cope with these drawbacks, we propose a new elastic net hypergraph learning model, which consists of two steps. In the first step, the Robust Matrix Elastic Net model is constructed to find the canonically related samples in a somewhat greedy way, achieving the grouping effect by adding the $l_2$ penalty to the $l_1$ constraint. In the second step, hypergraph is used to represent the high order relationships between each datum and its prominent samples by regarding them as a hyperedge. Subsequently, hypergraph Laplacian matrix is constructed for further analysis. New hypergraph learning algorithms, including unsupervised clustering and multi-class semi-supervised classification, are then derived. Extensive experiments on face and handwriting databases demonstrate the effectiveness of the proposed method.