Neural networks are effectively gauge invariant, and you have a huge space of valid isomorphisms as far as possible "valid" layer orderings go, and if your network is overparameterized, the space of "good enough" approximations gets correspondingly larger. The good enough sets are a sort of fuzzy gauge quotient approximating some "ideal" function per layer or cluster or block (depending on your optimizer and architecture.)
https://arxiv.org/html/2506.13018v2 - Here's an interesting paper that can help inform how you might look at networks, especially in the context of lottery tickets, gauge quotients, permutations, and what gradient descent looks like in practice.
Kolmogorov Arnold Networks are better about exposing gauge symmetry and operating in that space, but aren't optimized for the hardware we have - mechinterp and other reasons might inspire new hardware, though. If you know what your layer function should look like, if it were ordered such that it resembled a smooth spline, you could initialize and freeze the weights of that layer, and force the rest of the network to learn within the context of your chosen ordering.
The number of "valid" configurations for a layer is large, especially if you have more neurons in the layer than you need, and the number of subsequent layer configurations is much larger than you'd think. The lottery ticket hypothesis is just circling that phenomenon without formalizing it - some surprisingly large percentage of possible configurations will approximate the function you want a network to learn. It doesn't necessarily gain you advantages in achieving the last 10% , and there could be counterproductive configurations that collapse before reaching an optimal configuration.
There are probably optimizer strategies that can exploit initializations of certain types, for different classes of activation functions, and achieve better performance for architectures - and all of those things are probably open to formalized methods based on existing number theory around gauge invariant systems and gauge quotients, with different layer configurations existing as points in gauge orbits in hyperdimensional spaces.
It'd be really cool if you could throw twice as many neurons as you need into a model, randomly initialize a bunch of times until you get a winning ticket, then distill the remainder down to your intended parameter count, and train from there as normal.
It's more complex with architectures like transformers, but you're not dealing with a combinatorial explosion with the LTH - more like a little combinatorial flash flood, and if you engineer around it, it can actually be exploited.
- you can solve neural networks in analytic form with a hodge star approach* [0]
- if you use a picture to set your initial weights for your nn, you can see visually how close or far your choice of optimizer is actually moving the weights - eg non-dualized optimizers look like they barely change things whereas dualized Muon changes the weights much more to the point you cannot recognize the originals [1]
*unfortunately, this is exponential in memory
[0] M. Pilanci — From Complexity to Clarity: Analytical Expressions of Deep Neural Network Weights via Clifford's Geometric Algebra and Convexity https://arxiv.org/abs/2309.16512
Wouldn't such local invariance tie in with flatness or shallowness of the minima ?
This would tie in with the observation that flat/shallow minimas are easier to find with stochastic gradient descent and such weights generalise better.
https://arxiv.org/html/2506.13018v2 - Here's an interesting paper that can help inform how you might look at networks, especially in the context of lottery tickets, gauge quotients, permutations, and what gradient descent looks like in practice.
Kolmogorov Arnold Networks are better about exposing gauge symmetry and operating in that space, but aren't optimized for the hardware we have - mechinterp and other reasons might inspire new hardware, though. If you know what your layer function should look like, if it were ordered such that it resembled a smooth spline, you could initialize and freeze the weights of that layer, and force the rest of the network to learn within the context of your chosen ordering.
The number of "valid" configurations for a layer is large, especially if you have more neurons in the layer than you need, and the number of subsequent layer configurations is much larger than you'd think. The lottery ticket hypothesis is just circling that phenomenon without formalizing it - some surprisingly large percentage of possible configurations will approximate the function you want a network to learn. It doesn't necessarily gain you advantages in achieving the last 10% , and there could be counterproductive configurations that collapse before reaching an optimal configuration.
There are probably optimizer strategies that can exploit initializations of certain types, for different classes of activation functions, and achieve better performance for architectures - and all of those things are probably open to formalized methods based on existing number theory around gauge invariant systems and gauge quotients, with different layer configurations existing as points in gauge orbits in hyperdimensional spaces.
It'd be really cool if you could throw twice as many neurons as you need into a model, randomly initialize a bunch of times until you get a winning ticket, then distill the remainder down to your intended parameter count, and train from there as normal.
It's more complex with architectures like transformers, but you're not dealing with a combinatorial explosion with the LTH - more like a little combinatorial flash flood, and if you engineer around it, it can actually be exploited.