 Okay, so let's start. Hello, everyone. Hello, everyone who is here. Welcome to another seminar. Today we have a dorm manager of MIT, and he will talk about an invariance principle for the multi slice, and hopefully we will learn some applications as well. My name is Jacob and thank you everybody who has come. So, indeed, I'm going to tell you about an invariance principle for the multi slice and discuss some applications of it. So before we get into anything in the title, let me tell you what is invariance or one example of invariance. So, for example, for this example, you're familiar with me, this is the central limit theorem. So this is the following. So suppose we have x1 to xn random variables. That are independent. They have been zero variance one. And it's a more conditions. So I'll call it reasonable. I can therefore explore the something like that. Then in that case, if I take the sum of these random variables and divided by the standard deviation of the summer. And I get random variable that is behaving very similarly to a standard Gaussian. So this is the standard center limit theorem. So why am I claiming that this is an invariance principle? Well, so another way to state this theorem is to say, okay, I know that a Gaussian I can write it as a sum of Gaussians independent ones. So this is the standard deviation. So do you want to GM or IRD Gaussians. So in other words, now there is some syntactic meaning to this means that if I look at this polynomial p of z1 to ZN, which sums up its inputs and divide by square root of n. The distribution of this polynomial does not really depend on the input distribution that we plug into it. As long as the input distribution consists of independent random variables with the same mean and variance, then the value distribution is kind of the same. The value distribution of the Z1 to ZN is invariant under the input distribution. Again, as long as you are you have independence, mean zero, let's say, and variance one. So, so this is a very kind of suggestive way to write this claim that we all know now the central limit theorem. And it turns out that this or other generalizations of it is something that is very important in a PCP and in general in TCS. And this is something that I'm going to tell you about next. So back in 2005, there was this very interesting paper of a court, a kinder, a Muslim or donor that showed a reduction from unigames to Max cut. And that paper. Halfly speaking, showed that if you're willing to assume this unigames conjecture, then the Gomez Williamson algorithm for Max cut is optimal. Except that they couldn't quite prove it. They had some analytical conjecture. And in order to prove this conjecture, this invariance basically that I'm going to tell you about now was developed. So what is this invariance principle saying, this is the following theorem. But earlier, we had this polynomial p of zero and to the end, which was of very special form. And now we're going to say that the state same statement holds for more general class of polynomials. So feeling suppose we have a polynomial. P of the one to the end, which takes the form. So it's summation over some variables. So this set of variables is never too large. This is mostly to think of this constant. As times the monomial s. So, so far this is a general low degree polynomial. And want to say that if we plug in random variables to this polynomial, the value distribution is invariant, whether you plug into it past minus ones or garter's. So for a moment, you can see that, for example, if you take this polynomial to be just the first coordinate, then this is a bad example. This is not invariant polynomial in the sense. So you need to say that no coordinate appears in too many of them or not like it's this polynomial is not a single coordinate or something like that. So this is a technical condition. It's called. So you can think about it as no variables appear in too many of the monomials. But formally speaking, what you say is that if you sum up the square of all of the monomials that contain some variable I then this is never too large. So you can think about it that the polynomial is kind of four spread around these variables. Then invariance holds. That is, for example, if you plug into it, random plus minus ones, the value distribution is very similar to if you plug into it options. So this is the invariance principle of a should have given a citation. And yeah, so as I said, the motivation to prove this invariance principle was to prove this reduction between any games in Max cut. But after this result has been proven, it was used to prove many, many other things. And this includes results in DCS one very prominent one by the side of the vendor that I'm going to mention next, but also outside DCS for example, you can use this type of machinery to prove a lot of results in extremal combinatorics. So yeah, so it will be fair to say that it is a very important tool. So this has many applications. And one prominent one is other vendors theorem that I'm going to explain next. So, so this is a CSP seminar so I assume that most of the audience is familiar with what this theorem says but nevertheless I'm going to say what it is hopefully we're not making any mistake, but if I do, please correct me. So this result says, okay, suppose that I have some constraints satisfaction program. So suppose I have some pretty good. And now I consider the concept a second, the constraint satisfaction problem defined using me. Namely, I have a set of variables. Let's not call it because let's call it something else. And then I have a bunch of constraints of the form p of AI. I want to be equal to one. So this is what an instance of CSP is. This is an instance, let's call it fire. Now I have this instance. And suppose that I give you this instance and I tell you that this instance is 99% satisfiable. But I don't tell you, of course, the solution. How well can you perform efficiently. And what sort of harness result can you pull forward. So few of them. So this is what says that once I fix the speed. And I fix up to a better than zero. So if I give you an instance of CSP. And I promise to you that it is 99% satisfiable. Then, okay. Then there exists some s, which is a number. Then if I give you this instance, then maybe you cannot do 99% but you can do at least as function of the constraints. And what's more remarkable is that this is not some super duper algorithm. This is something very simple. This is achieved by a semi definite problem. But also, you can do at least s using an SDP program. And secondly, you cannot do better than s. And of course we have this aspects here, which is assuming the unigames connection. So I'm not going to explain what you do see is in this talk, but this is a, you can assume that it's cool so you can eliminate this line if you want. And yes, this is a very kind of nice. The economy result because it says that you can do us but you cannot do better than this. So, you know, you, this result really nails down the complexity of approximation for all CSP SP, but he does so only when the promise is that you are 99% satisfied. And the main question that motivated and one of the questions that motivated our current work was trying to extend this result to the regime where I promised you that the instance is fully satisfiable. So, what happens for XM equals zero. So I'm going to refer to this as a perfect openness. So at first I this may seem to be okay so this is a kind of the same problem with different parameters. Who cares nothing interesting happens. And but no, so there are some problems with complexity. There is a very sharply between the case that epsilon is equal to zero and epsilon, which is very very small but not equal to zero. So, there are CSP is was complexity changes dramatically. For example, this linearity CSP, which you can think of from F2 to the three to zero and the output when XYZ is one, even only if x plus y plus zero. So you can observe that if I give you a system of such equations, then you can do what we learned in the bachelors you can do Gaussian elimination, and indeed find a satisfying assignment. But if I don't promise you that it's fully satisfiable, if it's only 99% satisfiable, then you're in very better. You cannot do anything non trivial. So, now we can ask ourselves. Okay, so we have this very nice result of your agenda and it doesn't work for epsilon equals zero. So why is that is it because the statement is false. So first of all, the statement is false because if you look at this linearity predicate, there are integrity gaps for STP algorithms and actually for some squares algorithms that show that they cannot do linear algebra. So the statement is certainly false. But then you can start asking yourself. Okay, so, so what happens how do we try to prove some analog of this theorem for satisfiable assignments. So, there are several obstacles, let's call it, proving that the vendor type theorem for the couple things. So one of them is what I promised not to talk about the reduction starts from UGC. So this is a problem that inherently as imperfect complainers. So as long as you follow the things that people have been doing for 2030 years, there is no chance that you will end up with something with perfect completeness. So you need to do something else. So all of these analytical techniques that were employed in the proof of argument, it's very, they really use the fact that you can apply some extra noise and you can smoothen things out because it only adds a little bit of to the epsilon. So, and more generally this whole paradigm, we're in, first of all, you construct the dictatorship test and then you transfer it. This is also relies heavily on being able to apply noise on some functions. So, so really there are several obstacles that are terrible. So what is even the statement. So, yeah, so these are three obstacles. So, I'm going to tell you that there is indeed a substitute for UGC, which we do not know to be hard of course but it seems reasonable to be harder, which has perfect completeness. So this is something which is called rich to do on games. So I don't want to elaborate on it too much so I don't want to define it in particular. But this is something that is stronger than UGC, but at the same time it's a bit trickier to work with. So you cannot take all of the analytical machinery that exists over product spaces and apply it as easy. So we need to do some adjustments. And it turns out that something that is some domain that replaces all these hypercubes that is convenient to work with is the movie slice, which takes me to the topic of the current talk. Yeah, so I'm next going to define this movie slice business. But let me, before I define it, let me tell you why and what we do. So the why is this movie slice is a good replacement for product spaces, once you work with this rich to do on games conduction. So, if you're willing to take it and it works very nicely but then all of the analytical machinery that exists over product spaces does not carry over automatically. And what we do in this work is we show that indeed you can recover in the sense that you need some new techniques in order to do it, but you can recover some classical results from product spaces and indeed put some sort of invariance principle between this multi slice and product spaces. In this way, you can indeed extend some of these elements to work with rich to do on games and in particular we get some harness results using this connection. So if there are no questions, let me next define what the multi slices and tell you a bit more of what we're doing this work. Before I tell you about the movie slice and all of this. So let me just mention that this general question of obtaining an analog of other vendors for epsilon equals zero. So this is kind of a very big problem because in particular it includes in it the dichotomy condition which was just proven a couple of years ago that distinguishes between satisfiable not satisfiable. And here the task is satisfiable and what can you do, approximation wise. So, so we're far from solving it completely but this sort of motivates us to consider these elements that I'm going to talk about next. Okay. So what is the movie slice. So we have a parameter and so this you should think of the alphabet size, kind of the alphabet size of your CSP. So in particular should think of it as a constant. That's a m equals two or three. This will do. Then we have a no. This is very loud. And then we have m numbers k zero. m minus one. So this is a vector of numbers. And these sum up to n. So now the movie slice with this vector K of dimension and is the set of all strings in m to then. So, if I finished here this will be part of space. But you need to say you need to satisfy some budget conditions. So for every alphabet element for every being a number. The number of coordinates in X that are equal to this be is exactly k sub b. This is the movie slice. So, for our work and for this talk, the K the vector K that we consider has to be at least somewhat balanced. So, for us, we'll need K sub b to not be too close to an and not too close to zero. And really nothing really changes if you change it within this range. So let's just say that K sub b. They're all the same and they're all equal to an over m. It wouldn't change anything. So this is the movie slice. And the rough question that to ask yourself is how different is the movie slice from them to them. I'm sorry to ask a question is this K zero K I because it's the index looks a lot like zero. Yeah, so. Yeah. So came be is equal to an over m for all me. Yeah, sorry. Okay, and in the definition. It's also some K be right. So, okay. So, the question is how different are the multi slides with the uniform distribution and into them with the uniform distribution. You know, so on the one hand, of course, these two spaces are completely different right one of them is kind of flat and the other one is kind of large. They're not closing total variation distance or anything like that. But in another sense, if we sample an X, according to the uniform distribution in M to them. Then we do know that the number of eyes such that, let's say, Xi is equal to, I don't know, zero would indeed be an over m up to some literate of error. So on the one hand, they're completely different. On the other hand, they seem similar. So what can you say about the similarity. So is there an invariance principle between the two. So what do you mean by that. I mean that suppose that I give you a function over the multi slides. Can you find a function over the product space, such that the value distribution of effects is similar to the value distribution of F field. So here X is uniform over the multi slides and Z is uniform over M to them. So this is what I mean by any balance principle. Another question has been considered previously by these two works FKMW and FM. So they did consider this question, but only in the case that M is equal to two. So this is known as the slice. So what this work showed is that indeed you have invariance principle. As long as your function is not too high degree to not too high here is little off squad of them. And so this is a very nice result and it's kind of tight. And what's not so nice about this works is that the techniques are very, very specialized to the case that M is equal to two. And roughly speaking what they do is they look at the space of functions, real valued function over the multi slides. And this is not a product space so you don't have the nice basis of a character that we all know and love. So instead they come up with some different basis, which is not too bad, which we which is can still make some computations work. On this basis, they construct this function F field, which comes out very naturally and then they work very hard to prove that the value distribution is similar. But all of this is very tailored for MS equal to two. And once you go, try to go to M equal to three or four. It may be possible. I don't know. But it certainly will get a lot messier because you need to find the basis and make all of these things work. So, this is not what we do this work. So, so this work. We showed that actually this theorem holds for every constant M. In fact, we do not use any specific basis for functions. Instead, we appeal to symmetry base arguments that maybe I'll say a few words about. So here's what we move. So this is a basic version of our influence. So we show that if you have a function on the movie slice of the great most time, so most of them. So now I want to formalize this notion of being close a bit more delicately. So suppose they have this function. And then I can find a coupling between those spaces. So here, by coupling, I mean, this is a joint distribution of two random variables will marginally access uniform over the multi slice and these uniform over M to down. And you can find a function f t over and put down such that f x is close to a f deal of Z. So on average, when I sample x, they are called into this coupling, and they look at the difference squared. This is very, very small. So. So there are additional properties of this f t that we do need for applications, but this is sort of the main one. And I also want to say that this function f t. It's actually very natural. It's not something that comes off, you know, thin air. So what would be the first attempt for f t. So we already said that, okay, we want to evaluate f t at x. So we know that x is close to the multi slice in the sense that we're only squad of an off in each alphabet symbol. So we can try to define f x to be f deal of x to be f of Z, where Z is the closest point in the multi slice. So intuitively, this seems right, because the distance, the hamming distance between X and Z, is it more certain, the degree is it more certain. So maybe it works. And this is a bit too brutal. And this doesn't work in general, but something very similar to that works. So what does work is, you take some sort of average over the points that are close to X. So what does work is to evaluate f till text, you do a sampling according to the coupling that if you're in promises you of FZ condition on X being X. This is really very natural function to try and indeed we show that this works. But in order to show that this works, we cannot use any of the standard for analysis machinery because this is again, not a product space. And so instead we managed to convert. So this is not how do we convert this problem into some problem of Eigen values of some operator. And we use the symmetries of the movie slice and in particular the fact that it's an axon it in order to calculate its Eigen values. So, maybe I'll say a few more, more later, but let me just say that the analysis and users. And the symmetries and the movie slice under the action of fashion. And in particular, some basic representation theory of SM. Yeah, so as stated, this theorem only works for low degree polynomials, but you can actually extend it beyond that you can prove it for functions that look like low degree polynomials in the sense that they're small mass and on the high degrees and also a bit beyond that, but I don't want to state it because it's a bit too technical. So let me mention two concrete applications for CSPs that we have for this imbalance principle. And so what we do here is, so as I said, there is this connection between dictatorship tests and harness approximation results. And what we do here is we show that indeed you can use this movie slice business in order to carry several dictatorship tests into harness results. So I'll tell you about two of them. So the first one is about the largest gap a CSP can have. So it says that a summing this substitute for you to see the resist an infinite number of hours. So it's nothing mysterious. It just like powers of two or powers of two minus one, but I don't want to write it. And for each one of these hours, you have a CSP PR such that for every epsilon greater than zero, given an instance CSP PR and it's empty hard to distinguish between first of all the case where the instance is fully satisfiable. And secondly, the case where it's very not satisfying. So the value of psi is at most, I think what you get is two hours plus one or two to the hour. So, so here the problem is what is the largest that you can get between this one and this guy. So, right, because the CSP is to do the inputs and at least one of them is accepting the very worst you can do is one versus one of the two to the hour. So we of course don't know how to get that. But what we can show is that assuming this to one games thing, we can get this gap, which is the best that is known. And the second corollary is about coloring. So it says again that assuming it's reached one business for data greater than zero. If I give you a graph, it's empty hard to distinguish in the cases that first of all, the three global. And secondly, that G does not even have independence of size data. So unconditionally, we don't know much about coloring. So I think that the best that we know is that if you are that is empty out to distinguish between a graph that is three global and a graph that is five or six, I don't remember the exact number. But yeah, so, and this is a very hard problem. But if you're willing to assume this conjecture, then our understanding is much better. So any questions about it. I assume, just a quick technical question. I assume in density of Delta in this independent sense. Functional size. Okay. Thanks for a really nice talk. Are there any questions. Hi. So, what is this coupling between in the invariant theorem. Is it some natural natural coupling on. Yeah. So, it's something very natural. So what you can do is you can, you can sample statistics of something that sample according to them to them. And then for each symbol, so if you if you chose less than another. You can keep it as as that. All right, so you can sample statistics. Let's say that it says that we have L zero one zero zeros to L minus one. Equal it equal to L minus one. Then you take the minimum between L, I am K I and sample this jointly. So, so I so we have X, which is in the movie slice and then we have Z. And for that. Okay, so let's say that L zero is smaller than K zero, then you pick a subset of size L zero, and you put zeros in both X and Z. So you do that. If L zero is larger than K zero, you just say, okay, I pick another M. So this gives you almost all of all of X. And then you need to fill up the gaps. And this would just be sort of an coordinates in which they differ. So the point is that X and Z, they agree on almost all of the coordinates, except on one of those quote event fraction in expectation. Yeah. Another question. Maybe I will have one in the meantime. Yeah, there was there was some trouble with like approximate approximation hardness of not all equal. Can you, can you actually provide some good result for them. Yes, that's a good question. So I don't know. So is the issue there is that there is no dictatorship test right. There is a dictatorship test that that is pretty good. That is as good as it's supposed to be. But nobody was able to actually turn it into. I see. Yeah, so that, so I wasn't actually aware of this. So that would be good. And it may be the case that you can convert it into a harnesses out using this. Yeah, so somehow in my mind. If you pretend that you need games as perfect of witness. Then you can make the reduction work. Then you'd be able to use these things you know that to make the reduction work. But of course this is all play pretend. So essentially like saying that intuitively, if, if you could pretend that any games have perfect completeness and to approve words with that, then it's very likely to work with your framework as well. Yeah, yeah, that's exactly okay. Yeah, that's sort of the way we got to it. Yeah, yeah. Yeah. Okay, I see. Thanks. Any more questions. Yeah, thanks again. Thanks again for a really cool talk. And thanks. Thank all of you for coming. And I think we will see each other in a month. No recording now. You can stop recording. Yes.