An implicit function learning approach for parametric modal regression

Abstract

For multi-valued functions---such as when the conditional distribution on targets given the inputs is multi-modal---standard regression approaches are not always desirable because they provide the conditional mean. Modal regression algorithms address this issue by instead finding the conditional mode(s). Most, however, are nonparametric approaches and so can be difficult to scale. Further, parametric approximators, like neural networks, facilitate learning complex relationships between inputs and targets. In this work, we propose a parametric modal regression algorithm. We use the implicit function theorem to develop an objective, for learning a joint function over inputs and targets. We empirically demonstrate on several synthetic problems that our method (i) can learn multi-valued functions and produce the conditional modes, (ii) scales well to high-dimensional inputs, and (iii) can even be more effective for certain uni-modal problems, particularly for high-frequency functions. We demonstrate that our method is competitive in a real-world modal regression problem and two regular regression datasets.