Cost function for assessing the quality of low-dimensional manifolds

Dimensionality reduction and manifold learning techniques are used in numerous disciplines to find low-dimensional manifolds in complex systems with many degrees of freedom. This approach allows for a substantial reduction in the number of parameters needed to visualize, describe and predict complex systems, but some topological properties of low-dimensional manifolds can hinder their practical application. Here, we present a quantitative metric for characterizing the quality of low-dimensional manifolds. The metric collects information about variance in dependent variable values happening across multiple spatial scales on a manifold. It can characterize two topological aspects in particular: non-uniqueness and feature sizes in low-dimensional parameterizations. Our metric is scalable with manifold dimensionality and can work across different linear and nonlinear dimensionality reduction techniques. Moreover, dependent variables that are most relevant in modeling can be selected a priori and the manifold topology can be assessed for those variables specifically. Using the metric as a cost function in optimization algorithms, we show that optimized low-dimensional manifolds can be found. The metric can be also used to tune the hyper-parameters of reduction techniques and to select appropriate data preprocessing strategies to obtain improved manifolds. Our approach provides a way to quantify and automate decisions that need to be made prior to applying a reduction technique.
info:eu-repo/semantics/published