Shape, Symmetries, and Structure: The Changing Role of Mathematics in Machine Learning Research
Henry Kvinge asks "What is the Role of Mathematics in Modern Machine Learning?"
I’m very excited to share this piece by Henry Kvinge about the overlap between machine learning research and pure mathematical domains like topology, algebra, and geometry. Henry, who is an AI researcher and mathematician at Pacific Northwest National Laboratory, makes a very compelling case that simply scaling existing methods is not all we need and gives a fascinating tour of some recent work applying these domains to ML. He concludes that instead of seeing scale-driven progress and empirical breakthroughs as a challenge to mathematical theory, mathematicians should embrace these developments as opportunities to develop new tools that can deepen our theoretical understanding. — Cole
Thanks for this article, which strikes one as cogent and well-informed. I should like to append a few notes on the subjects addressed.
As to manifolds, we find the following passage in Riemann's groundbreaking work on the foundations of geometry:
So few and far between are the occasions for forming notions whose specializations make up a continuous manifold, that the only simple notions whose specializations form a multiply extended manifold are the positions of perceived objects and colors. More frequent occasions for the creation and development of these notions occur first in the higher mathematic.
~Riemann
Weyl also explored the geometry of color:
Thus the colors with their various qualities and intensities fulfill the axioms of vector geometry if addition is interpreted as mixing; consequently, projective geometry applies to the color qualities.
~Weyl
Recall that the dualities which figure so prominently in contemporary math and physics got their start in projective geometry.
Weyl expands on Riemann:
The characteristic of an n-dimensional manifold is that each of the elements composing it (in our examples, single points [...] colors, tones) may be specified by the giving of n quantities, the "coordinates," which are continuous functions within the manifold.
Concerning group theory:
Consider the field of the data of sense — a field of universal interest — and fundamental. We are here in the domain of sights and sounds and motions among other things [...] Do the colors constitute a group?[ ...] Let us pass from colors to figures or shapes — to figures or shapes, I mean, of physical or material objects — rocks, chairs, trees, animals and the like — as known to sense perception [...] And what of sounds — sensations of sound? Are sounds combinable? Is the result always a sound or is it sometimes silence? If we agree to regard silence as a species of sound — as the zero of sound — has the system of sounds the property of a group?
~Keyser
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A blue sphere is symmetric (or invariant) under translations, rotations, and reflections. The shape remains the same. And so does its color.
These kinds of symmetries are encoded in the QM "action," which takes us to the Lagrangian and Hamiltonian formulations of physics.
Traditional approaches to machine vision generally fail to acknowledge a fundamental problem with physical theory:
If you ask a physicist what is his idea of yellow light, he will tell you that it is transversal electromagnetic waves of wavelength in the neighborhood of 590 millimicrons. If you ask him: But where does yellow come in? he will say: In my picture not at all, but these kinds of vibrations, when they hit the retina of a healthy eye, give the person whose eye it is the sensation of yellow.
~Schrödinger
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Schrödinger thought that the vector space of colors was incomplete because those vectors have no inverse operation. But notice that a photon of a given color, when superposed upon another photon of the same color, but 180 degrees out of phase, gives us zero color. And similarly with sound.
Of course, once we introduce the phase of a photon, we have already taken a big step toward gauge theory.
In his day, Maxwell was the foremost authority on color science:
When a beam of light falls on the human eye, certain sensations are produced, from which the possessor of that organ judges of the color and luminance of the light. Now, though everyone experiences these sensations and though they are the foundation of all the phenomena of sight, yet, on account of their absolute simplicity, they are incapable of analysis, and can never become in themselves objects of thought. If we attempt to discover them, we must do so by artificial means and our reasonings on them must be guided by some theory.
~Maxwell
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What to do with objects of "absolute simplicity"?
All mathematical theories must begin with undefined elements, in order to avoid an infinite regression of definitions.
What happens when we make sounds and colors into the elements of a formal theory T?
Then all objects of T are either elements or are composed of elements.
Just so, perhaps, all objects of perception are either colors & sounds & etc., or are composed of such.
Then, too, we cannot define these elements in re: simpler objects because then these elements would not be elements.
That's kind of interesting. Fundamental feautures of perception would seem to be captured by basic logic.