 Thank you so much for the invitation. Can you hear me? Okay. Okay. Great. So thank you so much. And it's very nice to see familiar faces again. I hope we can see each other in person soon. So I'm going to talk about a paper that we just posted in the archive like two days ago. It's a very simple paper where we are characterizing all gauge invariant functions with the goal of designing machine learning models that are expressive and easy to optimize and work with with them. So the paper is called scalars are universal gauging body and machine learning structure like classical physics and my collaborators are David hall case or a Fisher, which yellow and then blooms meet at NYU. So, as I said, the goal is to parameterize functions that arise from physics that obey classical physical symmetries like rotation invariance or equivariance party and boost in any dimension. So basically I'm saying I'm working with functions that take a collection of vectors in RV and give, for instance, a scalar are which is the function is invariant, meaning that if I apply if I act by an element of a group on my input vectors, then the function doesn't change. That's what I mean with invariant and I also want to parametrize equivariant functions so meaning that if I act on all the inputs by some say rotation, then the output of the function rotates accordingly. So this will be useful for a m body simulations and in particular this the Kate and David are in astrophysics. So the goal that we are looking into is cosmological simulations. And so the groups that we are taking into consideration here are like the classical groups are going to group rotations, which are SOV translations and then the clean group that concerns translations and rotations. The Lawrence group, which is the group that comes from special relativity, which basically is like on the orthogonal group, but you have a different transformation in the in the time space and the pointer a group that that's Lawrence transformations and translations and and also we we look to parametrize permutation invariant functions, so meaning that if I change the ordering of the inputs, the function doesn't change. So for instance, if you're doing particle simulations, then you don't care what order of the particles you give to your system. And then then they actually the groups act as you expect on your vector so the rotations are by duplication the translations. So the translations we're going to think that only act on some position vectors and don't act on other vectors like velocities and and that's it. And so for instance one example of translate of an invariant function is the total mechanical energy of the system that's something that you can write in this form for like masses positions and velocity, you can write it in in this way. And basically this total mechanical energy, as you may imagine, is invariant with respect to the translations and permutations of the particles and rotations. And basically, basically this is called like scalar function, it needs to have these properties in order for them to consider to be scalar. Anyway, so we are not going to consider other actions by groups that are interesting for for machine learning so for instance one of the typical groups that what people care about is the is the is the permutation group acting by conjugation. So in graph neural networks, so say you're given a graph by by its adjacency matrix and you want to find an embedding of the graph. And so what you want is that if you relabel the notes, then the, the embedding relabels accordingly. So that is saying that if I act by this group of permutations by conjugation, then the in the output, it, it permeates accordingly. So that's like the relabeling of the of the node equal to to that. We're not going to parameterize these these actions but I'm going to explain you how people do it in general. And basically what we propose is a different approach that may be simpler and maybe more expressive. So, the classical approach that people use to parameterizing body functions, and to design neural network architectures, depending on different groups are given for instance by the paper of RISC condor in 2018 called and body networks where they basically propose a way of parameterizing these functions based on irreducible representations, and there's also some some work by my own and collaborators and they have several papers on the design of in body and an equity body and networks. So the idea is the following. One approach that that they propose is based on what it would be like a generalization of a field forward neural network. So we have an input that is a vector. And then we're going to apply a linear map that is going to be equivariant, and then a non linear activation function that needs to be consistent with the action of the group. And then another linear function etc. So the linear functions that we are going to apply at each layer in the case of these architectures what they propose is that they actually going to take a tensor where you act by the group in each of the dimensions of the tensor like as a tensor action. So if you have a tensor like this, then you will apply the transformation in each of the dimensions of the tension in the tensor in the same way. And so this, and this is going to be linear and equivariant. That's the approach there. So linear and equivariant but extended to tensor so basically it's kind of like thinking of what would the polynomial functions will look like in a way. And so the approach that people use is based on introducing representation. So basically if the group is G, they consider raw representation of the group. And then they observe that having a linear equivalent map between these two spaces is equivalent to having a map between two groups of representations in the way in this way. So basically that the linear map composed with the representation of the group of the group object is is equal to the composition of that way, the linear map. So, so then once that we understand that understanding the linear equivalent maps is the same as understanding the maps between the representations, then what they do is they parametrize the maps between representations using the irreducible representations, because it's easy to do in in the case in the case that if you have a map between two irreducible representations is either the identity or zero or like a multiple of the identity or zero so basically if you can take your, your say say you want to find a basis to express all the linear equivalent maps that that for for a group action, then what you can do is take the group, look at the irreducible representations of that group, and then parametrize your all your linear equivalent functions in terms of the, of these irreducible representations. So in order to do that, then you will need to give in a representation, you will need to be able to express it in a satellite with this building blocks of irreducible representations. And typically, in our case, this, this, this representation that we have is of the has a form of a of a tensor of of a representation just because of the form of the linear maps that I gave you. So this identification here is typically in this in this neural network architectures, given by this class Gordon coefficients, like the, there is a way to decompose this product representation in irreducible for some groups, using this class Gordon coefficients, which are known for so three, but are not known nor implemented for groups of the greater equal down five, for instance. So for instance this approach has very nice mathematical properties and has been implemented there's some papers that actually implement this approach. So, for instance, a paper by Dean, my role in 2021 that proved that these approach universally approximate all, all SO3 equivalent functions. So other approaches, this is not the only approach to to produce a variant functions. There, there's a paper by a while and there's a line of work by taco coin where they propose other ways of designing a guardian convolutions. And the, there is this paper by Mark me things here also at NYU, and, and under Gordon Wilson, where they show how to express equivariance equivalent maps by giving a set of constraints so you can look at all the maps, satisfying the constraints that the equivariance constraints give you basically and they have a library that implements this and it's very nice. So our approach is going to be much simpler and and a little bit, and it will work for for any dimension. So, and this is, is like very simple mathematical property that is known from like the, from the 1900s, which is the, the characterization of invariant functions. So the idea is that, if you have an invariant function of n vectors in RV. This is a function is invariant if and only if it's a function of the inner products of the inputs. That is like the first fundamental theorem for for for all the. And this is also true for other groups of for instance for the Lawrence group. This characterization is true, but then it wouldn't be for the, for the typical inner product but for the Lawrence inner product, which is this one, which is not really an inner product but it works like an inner product, because it's not positive. The proof of this is very simple. Basically, if you have your vectors, you can construct the function, the matrix M which is the outer product or like the inner product matrix between the columns of B and B just the inner products. And then if you have the matrix of inner products, then you can do the Cholesky the composition and this the composition is unique up to orthogonal matrices. So if you're given the inner products, then you can recover the vectors up to the orthogonal matrices, which is basically the orbit of the group that you're acting by so that's, that's what you're doing. So in the case when you're looking at SOD, this characterization has a more, a little bit more is a little bit more complicated because of like something that I may mention a little bit later but it has more less same form but it's a little bit more complicated. So in physics point of view, this basically says that all is this is equivalent to say that all scalars can be written in nice transformation location. So just like taking inner products or things. And one may say, okay, but you're taking a function that had like n and d vectors and now you're giving me that is a function of like n square inner products so like if you're going to use these two parametric functions, then you are being very inefficient. And what I claim is that you can use the low rank low rank matrix completion theory or like literature to design some subspace of of inner products or subs like like subsets of inner products that allow you to reconstruct the the function from like the entire you can reconstruct all the inner products by just a subset. So basically you can say it's a sub is a function of a subset of the inner products and not only inner products. And then the design of the sampling procedure is something that I'm interested in and like how it would work in practice say it's restricted in this matrix to a subset of inner product. Okay, so that's for invariant functions and I said nothing new. But, but now we're going to go into like equivariant functions which is the ones that we care about in in practice. So something that is quite simple observation is that if we are restrict if you if we look at all equivariant functions that are vector functions. These can be parameterized as a linear combination of your inputs times where the coefficient functions are all the invariant functions. So basically functions of inner products, the proof of this is actually very simple. And so then this tells you that that just do only, you can write the idea you can write your in your equivalent function as a linear combination of your inputs, where each of the coefficient functions are an invariant scalar function. And actually we can prove that if this function is a polynomial, then these coefficients can be chosen to be polynomials, though we don't have a boundary degree of what of what these polynomials need to be. And, but maybe that's something simple that I didn't, I didn't figure out, but also we have examples where this function is continuous, and we cannot choose this coefficient functions to be continuous so there may be some something. Not that nice going on. But this, this parameter station works in any dimension, and we hope that this is going to be some some way to parametrize invariant functions that you use for physics, and does not require you to know the reducible representations. And so extending to the Euclidean group so translation invariance is trivial because it's just saying that if you if you have a function that is translation invariant of some vector inputs you can think of them as like a function of the differences of the inputs, right so So for instance, you is a function of the each of the vectors minus the center of mass, for instance, or like the vectors minus the first vector or something like that. So, including that is very easy. And then the next point is how do you characterize permutation invariance in this setting. So we know that if you have a permutation like all the invariant function we can use the formulation that we have from the previous slide. And, and, and then. Ah, I forgot to show you something from the previous slide that it was under the fold. Sorry. Okay, so we have an example for this that I think is useful for physics and I think it gives you like a nice perspective of what's going on. And how, how long am I going, am I out of time. Okay, I finished it to me. So basically the idea is that using the, the, the fact that the cross product you can. So the cross product is a is a is a is an example of a function that is not is something that you don't imagine that you can write as a linear combination of inputs. I mean, it's something that that is not actually the cross product is equivariant SOD equivariant, but will say so three if we have two vectors in our three, the, the cross product isn't is so three equivariant but it's not a linear combination of the inputs. So that's where the difference with those three lies. And, and so you imagine that if you write the electromagnetic force law in this form, then, then this, you may, may imagine that you cannot, you're not able to write the same combination of inputs. But actually there's like, if you, if you rewrite this tensor product as like these are like in this form. Then you can rewrite everything and see that actually the electromagnetic force can be written as a linear combination of the inputs where the linear combination functions are a little bit more annoying in the sense that in this expression over here. You can factor out your test particle, like it is kind of like a mean, like a field formulation. And in this formulation, you don't have a field formulation anymore. You cannot factor out the test particle. So it may be that actually the formulation, this formulation is not that I mean, it has disadvantages that's the point. And then, finally, for permutation invariance, I wanted to say that we can choose that all the functions, the coefficient functions are the same. And they will take the, when they multiply when we're not coefficient of the vector bi, then, then is the fun is a function of bi and everything else where is permutation invariant with respect to the last inputs. And these things are easy to implement using message passing neural networks because of like the way they're structured. I don't have any numerics to show you at this time, but I'm just going to tell you that the summary is that we provide a simple characterization of a word and functions that are based on ancient summation notation and classical invariant theory. And the goal is to design expressive graphic network architectures to address machine learning problems while avoiding the use of irreducible representations that are not always known how to implement. And we have a couple of open problems regarding incorporating multi scale information and design a subset of permutation invariance scalars that. So the scalars that we are using are not permutation invariant. If you remember in this, in this formulation that we have here but maybe we can also use some formation by and scalars are universally expressed. Thank you so much and I apologize if I went over.