Dimensionality Reduction with Adaptive Graph 1

Graph-based dimensionality reduction (DR) methods have 6 been applied successfully in many practical problems such as face 7 recognition, where graph plays a crucial role with the aim of modeling the 8 data distribution or structure. However, the ideal graph is difficult to be 9 known in practice. Usually, one needs to construct graph empirically 10 according to various motivations, priors or assumptions, which is 11 independent of the subsequent DR mapping calculation. In this paper, we 12 instead attempt to learn a graph closely linking to DR process, and 13 propose an algorithm called Dimensionality Reduction with Adaptive 14 Graph (DRAG), whose idea is to, during seeking projection matrix, 15 simultaneously learn a graph in the neighborhood of a pre-specified one. 16 Moreover, the pre-specified graph is treated as a noisy observation of the 17 true one, and the square Frobenius divergence is used to measure their 18 difference in the objective function. As a result, we achieve an elegant 19 graph update formula which naturally fuses the original and transformed 20 data information. In particular, the optimal graph is shown to be a 21 weighted sum of the pre-defined graph in the original space and a new 22 graph depending on transformed space. Empirical results on several face 23 datasets demonstrate the effectiveness of the proposed algorithm. 24