I liked the article and I hope that I can understand it more with some study. I think the following sentence in the article is wrong "Applying Noether's theorem gives us three conserved quantities—one for each degree of freedom in our group of transformations—which turn out to be horizontal, vertical, and angular momentum.” I think the correct statement is "Applying Noether's theorem gives us three conserved quantiti…
Where is Noether's principle in machine learning?
31–40 of 81 posts
Re: Where is Noether's principle in machine learning?
#32This is one of those links where just seeing the title sets you off, thinking about the implications. I'm going to have to spend more time digesting the article, but one thing that jumps out at me, and maybe it's answered in the article and I don't understand it, is the role of time. Generally in physics, you're talking about a quantity being conserved over time, and I'm not sure what plays the role of time when you'…
In physics, the conserved quantity isn't always time. Invariance over time translation is specifically conservation of energy. Invariance over spatial translation is conservation of momentum, invariance over spatial rotation is conservation of conservation of angular momentum, invariance of electromagnetic field is conservation of current, and invariance of wave function phase is conservation of charge. I think the a…
Conservation of energy absolutely still holds, but entropy is not conserved so the process is irreversible. If your model doesn't include heat, then discrete energy won't be conserved in a process that produces heat, but that's your modeling choice, not a statement about physics. It is common to model such processes using a dissipation potential.
Re: Where is Noether's principle in machine learning?
#33As a complete amateur I was wondering if it could be possible to use that property of light ("to always choose the most optimal route") to solve the traveling salesman problem (and the whole class of those problems as a consequence). Maybe not with an algorithmic approach, but rather some smart implementation of the machine itself.
https://www.microsoft.com/en-us/research/uploads/prod/2023/0...
Re: Where is Noether's principle in machine learning?
#34Softmax gives rise to translation symmetry, batch normalization to scale symmetry, homogeneous activations to rescale symmetry. Each of those induce their own learning invariants through training.
Re: Where is Noether's principle in machine learning?
#35I liked the article and I hope that I can understand it more with some study. I think the following sentence in the article is wrong "Applying Noether's theorem gives us three conserved quantities—one for each degree of freedom in our group of transformations—which turn out to be horizontal, vertical, and angular momentum.” I think the correct statement is "Applying Noether's theorem gives us three conserved quantiti…
Re: Where is Noether's principle in machine learning?
#36Earlier quoted context omitted.
In physics, the conserved quantity isn't always time. Invariance over time translation is specifically conservation of energy. Invariance over spatial translation is conservation of momentum, invariance over spatial rotation is conservation of conservation of angular momentum, invariance of electromagnetic field is conservation of current, and invariance of wave function phase is conservation of charge. I think the a…
Is there any way to deduce which invariance gives which conservation? I mean for example: how can you tell that time invariance is the one paired with conservation of energy? Why is e.g. time invariance not paired with momentum, current, or anything else, but specifically energy? I know that I can remember momentum is paired with translation simply because there's both the angular momentum and the non-angular momentu…
When deriving the conservation of energy from Noether's theorem you basically say that your Lagrangian (which is just a set of equations that describes a physical system) is invariant over time. When you do that you automatically get that energy is conserved. Each invariant produces a conserved quantity as explained in parent comment when you apple a specific transformation that is supposed to not change the system (i.e remain invariant).
Now in doing this you're also invoking the principle of least action (by using Lagrangians to describe the state of a physical system) but that is a separate topic.
Re: Where is Noether's principle in machine learning?
#37This is one of those links where just seeing the title sets you off, thinking about the implications. I'm going to have to spend more time digesting the article, but one thing that jumps out at me, and maybe it's answered in the article and I don't understand it, is the role of time. Generally in physics, you're talking about a quantity being conserved over time, and I'm not sure what plays the role of time when you'…
In physics, the conserved quantity isn't always time. Invariance over time translation is specifically conservation of energy. Invariance over spatial translation is conservation of momentum, invariance over spatial rotation is conservation of conservation of angular momentum, invariance of electromagnetic field is conservation of current, and invariance of wave function phase is conservation of charge. I think the a…
So they are not approximations, but are just terribly difficult calculations, no?
Maybe I'm misunderstanding your point, but this should be true regardless of our philosophy of physics correct?
Re: Where is Noether's principle in machine learning?
#38Earlier quoted context omitted.
In physics, the conserved quantity isn't always time. Invariance over time translation is specifically conservation of energy. Invariance over spatial translation is conservation of momentum, invariance over spatial rotation is conservation of conservation of angular momentum, invariance of electromagnetic field is conservation of current, and invariance of wave function phase is conservation of charge. I think the a…
> It's trivially easy to find a real-world situation where conservation of energy does not hold (any system with friction, which is basically all of them) Conservation of energy absolutely still holds, but entropy is not conserved so the process is irreversible. If your model doesn't include heat, then discrete energy won't be conserved in a process that produces heat, but that's your modeling choice, not a statement…
Basically this is a restatement of Box's Aphorism ("All models are wrong, but some are useful") or the ideas in Thomas Kuhn's "The Structure of Scientific Revolutions". The goal of science is to from concrete observations to abstract principles which ideally will accurately predict the value of future concrete observations. In many cases, you can do this. But not all. There is always messy data that doesn't fit into neat, simple, general laws. Usually the messy data is just ignored, because it can't be predicted and is assumed to average out or generally be irrelevant in the end. But sometimes the messy outliers bite you, or someone comes up with a new way to handle them elegantly, and then you get a paradigm shift.
And this has implications for understanding what machine learning is or why it's important. Few people would think that a model linking background color to likeliness to click on ads is a fundamental physical quality, but Google had one 15+ years ago, and it was pretty accurate, and made them a bunch of money. Or similarly, most people wouldn't think of a model of the English language as being a fundamental physical quality, but that's exactly what an LLM is, and they're pretty useful too.
Re: Where is Noether's principle in machine learning?
#39This is one of those links where just seeing the title sets you off, thinking about the implications. I'm going to have to spend more time digesting the article, but one thing that jumps out at me, and maybe it's answered in the article and I don't understand it, is the role of time. Generally in physics, you're talking about a quantity being conserved over time, and I'm not sure what plays the role of time when you'…
I found the thinking of William Sidis to be particularly thought provoking perspective on Noether's benchmark work, in his paper The Animate and the Inanimate he posits--at a high level--that life is a "reversal of the second law of thermodynamics"; not that the 2nd law is a physical symmetry, but a mental one in an existence where energy reversibly flows between positive and negative states.
Indeed, when considering machine learning, I think it's quite interesting to consider how the organizing of information/knowledge done during training in some real way mirrors the energy-creating information interred in the mind of Maxwell's demon.
When taking into account the possible transitive benefits of knowledge organized via machine learning, and its attendant oracle through application, it's easy to see a world where this results in a net entropy loss, the creation of a previously non-existent energy gradient.
In my mind this has interesting implications for Fermi's paradox as it seems to imply the inevitibility of the organization of information. Taken further into my own personal dogma, I think it's inevitable that we create--what we would consider--a sentient being as I believe this is the cycle of our own origin in the larger evolutionary timeline.
Re: Where is Noether's principle in machine learning?
#40I liked the article and I hope that I can understand it more with some study. I think the following sentence in the article is wrong "Applying Noether's theorem gives us three conserved quantities—one for each degree of freedom in our group of transformations—which turn out to be horizontal, vertical, and angular momentum.” I think the correct statement is "Applying Noether's theorem gives us three conserved quantiti…
More generically in 3 dimensions a transformation with 3 translational 2 rotational and 1 time independence would provide conservation of 3 momenta 2 angular momenta and 1 energy.