Landmark Selection for Isometric Feature Mapping Based on Mixed-Integer Optimization

Abstract

Abstract. Isometric feature mapping (Isomap) demonstrated noteworthy performance for nonlinear dimensionality reduction in a wide range of application domains. To improve the scalability of the algorithm a fast variant, called Landmark Isomap (L-Isomap), has been proposed in which time-consuming computations are performed on a subset of points referred to as landmarks. In this paper we present a novel method for landmark selection to be framed within the L-Isomap procedure. It is based on a mixed-integer problem aimed at finding a set of landmarks which are representative of dense regions of points in the input space which mostly contain samples of the same class. The optimization model is solved by a heuristic algorithm based on Lagrangian relaxation with subgradient method. Computational experiments performed on benchmark data sets highlighted the effectiveness of the proposed landmark selection algorithm which, combined with L-Isomap, provided promising results in terms of classification accuracy and computational effort.

Homology-Preserving Dimensionality Reduction via Manifold Landmarking and Tearing

“…In addition, modification to existing landmark-based DR method such as L-Isomap can be done by introducing modifications to the distance matrices involving landmarks and its related spectral problems [59,68]. Statistically, manifold landmarking utilizes regression [61], mixed-integer optimization [46], active learning [74], and Gaussian processes [39].…”