Learning from the Past: Euclid Meets Big Data!?
Participant Profile
Masahiro Yukawa
Masahiro Yukawa
The World Cup and the Olympics. ICT has made it possible to watch games and competitions held in faraway countries in real time. In the medical field as well, people living in areas without nearby hospitals can now receive diagnoses from specialists remotely. Advances in sensor network technology have enabled high-precision sensing of environments around the world. Recently, it has become clear that big data analysis can even yield new insights that humans could not discover empirically. The progress of science and technology in modern society can only be described as remarkable.
Would you believe that these scientific technologies are actually based on academic disciplines established in ancient times? I would like to show with a practical example that this is not entirely nonsense. Let's look at Figure 1. Starting from an arbitrary point, we draw a perpendicular to line A. Next, from the point where the perpendicular was dropped, we draw a perpendicular to line B. You can see that by performing this operation of dropping perpendiculars alternately on the two lines, the intersection of the two lines can be found. This is a simple illustration of the principle of the method of alternating projections, proposed by John von Neumann, a leading researcher of the 20th century. The various concepts of geometry, starting with the "right angle" that we use unconsciously when drawing a perpendicular, are based on the system of geometry compiled by Euclid in his book *Elements*. Euclidean geometry is linked to algebraic equations by Cartesian coordinates (x-y coordinates), described in the book *Discourse on the Method* by the 17th-century philosopher and mathematician René Descartes. For example, let's consider a system of three linear equations with three variables: x, y, and z. The set of x, y, and z that satisfies each equation is known to form a plane, and the intersection of the three planes corresponding to the three equations is the solution to the system of equations (if it exists). In fact, the principle of the method of alternating projections has been generalized to three or more finite planes (called hyperplanes in four or more dimensions) and has been applied to various engineering problems, including image restoration, as a numerical method for solving high-dimensional systems of linear equations. I have heard that von Neumann got the idea for the method of alternating projections while observing the path of light between two mirrors arranged like lines A and B, but if Cartesian coordinates had not been discovered, would it have led to a method for solving systems of linear equations? The idea of the method of alternating projections has also been applied to the learning identification method (Nagumo and Noda 1967), a standard algorithm in adaptive signal processing, and supports various scientific technologies, including echo cancellers and noise suppression in mobile phones (see Figure 2).
Now, I have discussed how the two-dimensional principle of alternating projections is generalized to multiple dimensions. Let's take it one step further and move on to infinite-dimensional function spaces. Hilbert studied the case of an infinite number of variables (more precisely, the space of all square-summable sequences), and the combined knowledge of Fréchet, Riesz, Fischer, von Neumann, Stone, and others completed the fundamental theory of Hilbert spaces. Through abstraction, vectors that were previously tuples of real numbers (e.g., (x, y, z)) are extended to a variety of objects, including functions and random variables. Von Neumann's discussion of the convergence theorem for the method of alternating projections is set in a Hilbert space, and the principle illustrated in Figure 1 can also be applied in function spaces (spaces where functions are vectors). Since the beginning of this century, a method extending the learning identification method (mentioned above) to function spaces has been proposed, leading to dramatic progress in the research of adaptive estimation methods for nonlinear functions [1]. This is expected to advance science and technology in a wide range of fields, from online analysis of big data to online estimation of time-series data (such as solar power generation and financial data). Furthermore, the convergence theorem for the method of alternating projections has been extended from "two planes" to "a finite number of planes," and from "a finite number of planes" to "a finite number of (more general) convex sets." More recently, Yamada and Ogura achieved an extension from a finite number of convex sets to an infinite number, providing a unified convergence analysis for the learning identification method and other high-performance adaptive algorithms.
Thus, it is an undeniable fact that the rigorous system of geometry described in *Euclid's Elements* forms the foundation of modern science and technology. Modern signal processing engineering continues to evolve, using functional analysis, including Hilbert space theory, and tools such as convex analysis based on convexity [2]. As this example shows, it seems that not only pursuing the latest science and technology but also turning our eyes to the intellectual heritage left by our predecessors and mastering "scholarship" can lead to opening up new horizons. Looking around the world, researchers with backgrounds in mathematics, physics, and other fields are entering signal processing engineering one after another to explore applications in science and technology, contributing to the advancement of the field by leveraging their unique strengths. For researchers who wish to apply the knowledge of mathematical sciences to people's lives, signal processing engineering appears to be the ideal ground. Finally, it should be noted that Riemannian geometry (a type of non-Euclidean geometry), which played a central role in Einstein's general theory of relativity, has also developed into information geometry [3], and research into its applications in science and technology is underway.
References:
2. Isao Yamada, *Functional Analysis for Engineering* (Tokyo: Surikogakusha (Science-sha), 2009).
3. Shun-ichi Amari, *New Developments in Information Geometry* (Tokyo: Science-sha, 2014).