A Random Walk in 10 Dimensions (2021)

Summary

Physics in high dimensions is becoming the norm in modern dynamics. It is not only that string theory operates in ten dimensions (plus one for time), but virtually every complex dynamical system is described and analyzed within state spaces of high dimensionality. Population dynamics, for instance, may describe hundreds or thousands of different species, each of whose time-varying populations define a separate axis in a high-dimensional space. Coupled mechanical systems likewise may have hundreds or thousands (or more) of degrees of freedom that are described in high-dimensional phase space. In high-dimensional landscapes, mountain ridges are much more common than mountain peaks. This has profound consequences for the evolution of life, the dynamics of complex systems, and the power of machine learning. For these reasons, as physics students today are being increasingly exposed to the challenges and problems of high-dimensional dynamics, it is important to build tools they can use to give them an intuitive feeling for the highly unintuitive behavior of systems in high-D. Within the rapidly-developing field of machine learning, which often deals with landscapes (loss functions or objective functions) in high dimensions that need to be minimized, high dimensions are usually referred to in the negative as “The Curse of Dimensionality”. Dimensionality might be viewed as a curse for several reasons. First, it is almost impossible to visualize data in dimensions higher than d = 4 (the fourth dimension can sometimes be visualized using colors or time series). Second, too many degrees of freedom create too many variables to fit or model, leading to the classic problem of overfitting. Put simply, there is an absurdly large amount of room in high dimensions. Third, our intuition about relationships among areas and volumes are highly biased by our low-dimensional 3D experiences, causing us to have serious misconceptions about geometric objects in high-dimensional spaces. Physi...