Learning Strictly Orthogonal p-Order Nonnegative Laplacian Embedding via
Smoothed Iterative Reweighted Method
Abstract
Laplacian Embedding (LE) is a powerful method to
reveal the intrinsic geometry of high-dimensional
data by using graphs. Imposing the orthogonal and
nonnegative constraints onto the LE objective has
proved to be effective to avoid degenerate and negative solutions, which, though, are challenging to
achieve simultaneously because they are nonlinear
and nonconvex. In addition, recent studies have
shown that using the p-th order of the `2-norm distances in LE can find the best solution for clustering and promote the robustness of the embedding
model against outliers, although this makes the optimization objective nonsmooth and difficult to ef-
ficiently solve in general. In this work, we study
LE that uses the p-th order of the `2-norm distances
and satisfies both orthogonal and nonnegative constraints. We introduce a novel smoothed iterative
reweighted method to tackle this challenging optimization problem and rigorously analyze its convergence. We demonstrate the effectiveness and
potential of our proposed method by extensive empirical studies on both synthetic and real data sets