Visualizing High-Dimensional Functions with Dense Maps

Multivariate functions have a central place in the development of techniques present many domains, such as machine learning and optimization research. However, only a few visual techniques exist to help users understand such multivariate problems, especially in the case of functions that depend on complex algorithms and variable constraints. In this paper, we propose a technique that enables the visualization of high-dimensional surfaces defined by such multivariate functions using a two-dimensional pixel map. We demonstrate two variants of it, OptMap, focused on optimization problems, and RegSurf, focused on regression problems in machine learning. Both our techniques are simple to implement, computationally efficient, and generic with respect to the nature of the high-dimensional data they address. We show how the two techniques can be used to visually explore a wide variety of optimization and regression problems.