Manifold learning by preserving distance orders

• A novel manifold learning method which preserves pairwise distance order relations in the projected space.
• Theoretical formulation of the constrained optimization problem extending classical MDS-based mean-squared error minimization.
• A new performance metric that involves number of preserved distance orders.
• Proposed method also provides the relation between distances in original and lower dimensional spaces.

Nonlinear dimensionality reduction is essential for the analysis and the interpretation of high dimensional data sets. In this manuscript, we propose a distance order preserving manifold learning algorithm that extends the basic mean-squared error cost function used mainly in multidimensional scaling (MDS)-based methods. We develop a constrained optimization problem by assuming explicit constraints on the order of distances in the low-dimensional space. In this optimization problem, as a generalization of MDS, instead of forcing a linear relationship between the distances in the high-dimensional original and low-dimensional projection space, we learn a non-decreasing relation approximated by radial basis functions. We compare the proposed method with existing manifold learning algorithms using synthetic datasets based on the commonly used residual variance and proposed percentage of violated distance orders metrics. We also perform experiments on a retinal image dataset used in Retinopathy of Prematurity (ROP) diagnosis.