Nonlinear Functional Output Regression: a Dictionary Approach

In a large number of fields such as biomedical signal processing, speech and acoustics and climate science, data consists of a high number of simultaneous or sequential measurements of different aspects of the same phenomenon. Such data is inherently high dimensional, however it contains strong within-observation correlations and smoothness patterns which can be utilized in the learning process. A possible way to do so is to represent those observations as functions rather than vectors, opening the door to Functional Data Analysis (FDA) Ramsay & Silverman (2005), a research area that has recently attracted a growing interest due to the ubiquity of embedded devices and sensor data. In practice, FDA relies on the assumption that the sampling rate at which data are collected is high enough to get functional observations. Of special interest is the general problem of functional-output regression in which the output variable to regress is a function and no specific assumption is made on the input variable that can can be of any type, including functions. While functional linear model have received a great deal of attention--see Ramsay & Silverman (2005), Morris (2015) and references therein--nonlinear ones have been less studied.