Psychophysics examines the connections between external physical inputs and internal mental processes. The scientific community’s early efforts to research how people see color are a great example. Scientists have always been fascinated by the visual perception of color. They have worked to understand how we perceive color and have made rudimentary mathematical calculations to try and quantify human perception. Ron Kimmel, from Technion-Israel Institute of Technology, explores the mechanisms of color perception by manifold learning and the development of fundamental cognitive geometries.

Young, Maxwell, Helmholtz, and subsequently Schrödinger’s research led to the incisive discovery that human color perception is 3D, although most birds and dinosaurs likely had higher-dimensional color perceptions and most mammals shared a lower-dimensional space for color (or absence thereof).

The early empirical study of color vision conducted through manifold learning represents a great advancement in our capacity to simulate human behavior and utilize this knowledge to our advantage. Multidimensional scaling (MDS), one of the analysis procedures used to make this crucial insight, is connected to the well-known principal component analysis machinery that is frequently employed in large data representation and for which several contemporary extensions exist.

Geometry modeling of image formation indeed led researchers to the introduction of a new manifold that marries the color line element with the image coordinates, giving rise to a 2D manifold (the image) embedded in a 5D space, where three of these dimensions are an exact result of our understanding of color perception.

Ron Kimmel

The next issue the academic community was working to tackle was the automated identification and classification of the information in a picture after human perception of color was thoroughly known. A discipline known as robot vision, computer vision, or image understanding was born as a result of this endeavor. Recent developments have significantly altered how subjects like computer vision are approached due to the idea of so-called “deep learning” without “understanding.” Introducing the model of Yair et al., Kimmel argues that in this instance, the learning does not fit the present definition of deep learning; rather, it falls under the category of methods known as manifold learning, or geometric reasoning.

A couple of centuries after the manifold learning way of deciphering the psychophysical modeling of color perception, the model of Yair et al. comes in a timely fashion, introducing novel concepts and ideas by which some understanding could still be extracted from the learned objective.

Ron Kimmel

By simple observation, it is possible to determine that each of the four drawings in row 1 of the picture above represents a phase portrait of a sequence including a Hopf bifurcation (a crucial location where a system’s stability changes and a periodic solution emerges). One might even be able to determine the overall pattern of the series when considering the four photographs collectively.

A key cognitive as well as scientific difficulty is the extraction of models from data (in a sense, the “knowledge” of the physical principles causing the data). Yair et al. demonstrate a geometric/analytic learning approach that may generate concise descriptions of unidentified nonlinear dynamical systems that rely on parameters. This is achieved by the data-driven identification of practical intrinsic-state parameters and variables that allow one to experimentally simulate the underlying dynamics.

In their study, Yair et al. employ geometric manifold learning strategies to discover compact representations of empirical observations of physical events using a geometry called diffusion maps. In other words, kernels that are specified using a few basis functions represent the distances between occurrences. This method appears to smoothly solve the difficulty of efficiently capturing the behavior of dynamical systems when the sole common denominator is the time axis. It remains to be seen whether this revolutionary approach to applying manifold learning to dynamical systems will result in technologically creative discoveries like those that come from understanding how people see color.

*From understanding of color perception to dynamical systems by manifold learning.* Ron Kimmel

Published: September 2017

DOI: 10.1073/pnas.1713161114

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*Reconstruction of normal forms by learning informed observation geometries from data. *Or Yair, Ronen Talmon, Ronald R. Coifman, Ioannis G. Kevrekidis

Published: September 2017

DOI: 10.1073/pnas.1620045114