 region. And I count here the number of critical points. I do that in rather large detail. For the moment the formula is only written here for the minima. So this is the number of critical points of index zero. That is the number of local minima whose value is below you and who are in this band, right? So this thing has a certain, this limit exists and it is given by a function which we will call, okay, which will be the int of another function, if of w smaller than z of an f of u and w. So we have an explicit expression for this. And let me try to explain how, so the explicit expression for f is horrendous. But let me try to explain what it is, where it comes from. So of course we do the same thing. Now remember our function phi is our old Gaussian function which is centered and we have added now a deterministic function and so now the function is not centered. So, but I can still start with the same, in the same way. If I want to compute this thing, I can apply Katz-Rice formula. And so this with critical zero here, u z, I just have to do the same thing except here instead of integrating on the sphere, I integrate on this region that we call it a sub z and I have to compute the same thing except that here in this density, which was the density of the, remember, maybe I didn't remember that, this was the density of the distribution of the gradient, right? So now this gradient before was, so it's a Gaussian variable, but before it was a centered Gaussian variable. Now it's a centered Gaussian, it's a non-centered Gaussian variable and its mean is given by the spike. So the spike is hidden in this thing but otherwise we can start with the same computation. So if you start with the same computation, you have to compute here again the distribution of this Hessian, okay? And now instead of being a GOE, what you find is that this Hessian is the law of a spike GOE as I was describing before. So through Katz-Rice formula, you reduce this thing to the case of the usual PCA, right? The usual BBP transition, okay? And so exactly as before when we were using the large, so because of that, I won't be too precise. Now using the large deviation principle given by Milen Maida for the spike PCA, for the spike, I'm sorry, so it's not called spike but modified Wigner ensemble, that's how it's called. But it's the same thing. The vocabulary in the random matrix word is modified GOE. For the modified GOE, then one can prove the theorem and find this function, the rate function which is given over there, which I call capital F of U and W. Okay, so that's the strategy and which proved that in fact, in the end, this question of a random field reduces to the question of the spike PCA, except that the, now it's okay, I'm not sure. Offer, where's offer? Like what? Five, ten? Fifteen, great. So then fifteen, I can give you a formula. But maybe before, maybe give you, I hesitated doing the formula because it was fully bored, but before doing that, let me explain what we can get from this formula. So first, this is very preliminary because we here have only the first moment. So in order to be more sure about what this means, we will need the same thing that Subak did when there was no spike and being able to understand the second moment. But what do we see at the level of the first moment? So before explaining that, let me explain what you can do. So instead of doing the complexity here, what if we study this problem, instead of doing this problem at zero temperature, we do that at positive temperature. That is like a physicist. So we could again introduce the Gibbs measure as before, the same one, except my function is now as change. It has a spike here and the free energy. So you may want to understand the limit of log Zn over n. Try to understand this, right, which is now a function of the spike, n of p. So that, what you want to do if you want to do that as a physicist and then maybe let the temperature go to zero. So this is done in a work by myself and Giulio Biroli and Chiara Camarotta. And let me explain what happens. So maybe how you do that, that's not a very ... I mean if you look at it like this, it's not such a hard problem from the point of view of physics. What you have to do is you first start with the usual spin glass, no spike, but instead of doing it on the whole sphere, you do it on one parallel, right? Just one. So again here you have, that's not such a hard problem, because in fact this is exactly equivalent to studying the spin glass model with a magnetic field. By a Legendre transform, it's the same thing, right? Fixing the coordinate or fixing the field. So by a Legendre transform you can understand everything that happens for this problem on a given thing from what you know for the spin glass model in a field. And what do you know there? What do you obtain? For this question you have the following thing. When you look at when your parallel is on the equator, then at low temperature you have a one RSB phase. You have this huge complexity. But when your parallel becomes smaller and smaller because you come closer to the North Pole, at some point just for a question of volume, you lose this one replic asymmetric breaking. You lose the complexity because your thing becomes too small. And so above a certain parallel, you will have, at zero temperature you will have replic asymmetry, okay? So that's an easy thing. Once you have this, now you want to add this function, which is just a function of this coordinate. So it's not too hard. You can work and then you can prove that in this context here is what happens. I said that in words. So you, of course, on the equator you have this roughness and this very large, because this is very large, you have this very large number of local minima as was described before. On the equator here you don't feel the spike at all. So here it's a pure usual piece of spin model. And it's rough and complex as before. One RSB if you want at low temperature. But then if you have no spike, of course this dominates. But when you begin to increase the spike, what you will see for a while, the equator will still dominate the Gibbs measure. But at a certain moment there will be a parallel here, above here, which will dominate this one. So the Gibbs measure will be concentrated here instead of here, which is good news again for the question we have, which means the Gibbs measure feels the spike, right? So here you already have recovered a part of this signal. And in fact, and this is rough. Again it's one RSB. You have an exponentially large number of local minima on this thing if you want. And if you increase even more the spike, then this thing will come much closer to the north pole. And it will be, so here it will be one RSB. And if you increase even more the spike, it will be replic asymmetric here, which means essentially non-complex near the north pole. So that's the approach through, let's say the physics approach, the Gibbs measure approach. But now you can look at temperature zero directly and look at this function. And you find, of course, the same. You can find the equivalent of this statement. So now I cannot escape giving you the function. So this, and this with that I will log of 1 minus z plus 1 half of log, or maybe put the first this one because this is always there, minus z minus 1 over 2p alpha square p square z to the 2p minus 1 plus 1 half plus, I'm sorry, minus a function g of alpha eta of z square root of p over 2p minus 1 u plus alpha z to the p. And here eta of z is just p square root of pp minus 1 over 2z to the p minus 2. And the function g, so at this point of, and g of, let's say, theta and v. So at this point, this function, this part are not very interesting thing. The rate function, the large deviation for the spiked model is, of course, hidden in this g, which is given now. So it's the function I call L0 plus p minus 2 over 2p v square if v is smaller than negative square root of 2. Yeah, here my normalization is such that the 2 has become a square root of 2. And that is if, okay, if that and is 0 minus v plus p minus 2 over 2p times square root of 2 square, if v is larger than something negative square root of 2. That is if, this is if theta is smaller than 1 over square root of 2. That is you don't feel the spike. And if theta is larger than square root of 2, then, okay, the function is, you replace essentially, it's more complicated, but replace L0 by what I call L theta before. The rate function given by a MITA plus other things. I don't want to get too much into that. This is really painful. So let me explain, so you have an explicit formula. And now you can ask yourself the same question. And you find the same thing. That is, if, or at least I hope we find the same thing, if we have to check that the thresholds are the same or so. So if the spike is smaller than a certain alpha 1, then essentially the, so the local minima, you have an exponentially large number of local minima on the equator and nothing else. If alpha is between alpha 1, a certain alpha 1 and a certain alpha 2, then you have this phase that I was describing. That is, you have an exponentially large number of local minima on the equator, but you have even more, I mean, local minima on this, on a certain parallel here. And this height depends on the alpha. So in this thing, the model is complex, exponentially complex. Here too, here the local minima are essentially not related to the spike. So here most of the local minima are on the equator. And when alpha is larger than alpha 2, then the model is not complex. That is, this function, the complexity function that I call f was zero. I mean, this int of f was zero. So then this doesn't really tell you that you have a unique minimum, but it's compatible with the idea that you have one unique minimum very close to the north pole. And this regime, in this regime it will be clear that solving the maximum likelihood estimator will be simple. So here it's clear, in this regime it's clear that you can construct a consistent estimator because your maximum likelihood estimator will converge to the north pole, whereas in this regime, whatever you do, you will not find the north pole. In this regime, what this tells you, and so this becomes interesting and it will stop with that, this regime here alpha 1 and alpha 2 correspond to something which is really important in computer science and statistics, I mean, this kind of statistics, which is the computational statistical gap. So this happens in other models, for instance, random constraint satisfaction problems, where you have two different thresholds, one at which, so to come back to what I was saying at the beginning, one at which you can do detection and one at which you can do recovery, which are, so here in this regime, when alpha is there, what this will tell you, I mean, not really now because we only control the first moment, but what this indicates is that the information theory problem is solvable. You can find something correlated with the spike, but the algorithmic aspect of it, how do you find it? This is way more difficult and the reason is the following. So in this regime, your Gibbs measure, your minima, the important minima are here, but if you start from a random, so you try to find this north pole, you start from a random point, which naturally, when you take a random point on the sphere, it will be on the equator, right, because the equator has all the mass, but now whatever you try to apply is an algorithm to find your minimum or maximum, which will be here. First, you have to escape this very large equator, which, first, it's difficult anthropically because the equator is big, but also it's difficult because in the equator, you still have all these rough landscape, so escaping that is very long. And for the moment, in fact, there is a conjecture about how much time it takes to go from here to there, but that's just a conjecture. But there is one context in which we can do something, which is the following. Here I was asking the hardest possible question in statistics, which is, I give you one instance of something plus noise and I ask you, can you find the something? Now, if you are not that, I mean, difficult, you could ask, you could say instead, so typically these things are images. Instead of giving one instance, I give you K instance, right? So I give you time to learn a bit. K instance in which, of course, the spike will be the same, yeah. But the realization of the noise may be different, right? So you have always the same N, but you have now K instances and you try to find. So if you do that, then there are natural algorithms to do that, which are, for instance, stochastic gradient descent. And when K is large enough, so this is also a work with the same co-authors here, Giulio Biorio and Chiara Camarota. No, I'm sorry, in fact, I misspelled her name. It's Chiara, but it's Camarota. And there, in this context, when you apply a stochastic gradient descent, you can see really what happens. If you look, so let me say that in words, because this is a long paper, if you look, so you start from the equator and you follow the gradient given by this function, phi, for every new instance that I give you and I give you K instances. So, of course, you have to decide what is the time, the length of your time step, et cetera. But when you do that properly, what you see, if you look at the distance from your point to the spike, the thing you want to find, then when K is large, of course, you have to say how large, what you find is that this distance converges, in fact, to a diffusion, it's a dimension one thing, it converges to a diffusion process in one dimension, which is in the potential, which has this shape. This is 0, this is 1, 0 corresponds to overlap 0 with the north pole, that is, to being on the equator, 1 corresponds to being on the north pole, and it has a, it's like a diffusion in a potential which is like this. So, which means that you have, in order to escape the equator, you have to climb a barrier, which takes a lot of time. But then, in the end, you will end up here, and of course, this coordinate here corresponds to this Z. So, you will end up essentially there, but first, you have to do that, and the time to do that, of course, depends on how many, depends on K, how many instances you've been shown, right? But in some sense, this, I mean, this is nice, but it's cheating a bit, because we are, the problem that data scientists, statisticians, want really to solve is this one. I'll give you one instance, find the spike, okay? Yes, of course, when K becomes very large, this Z goes to, this thing goes to 1. Yeah, yeah, but you need a lot to do that, before you can know where the center is. So, yeah, the reason it's so hard, of course, to escape the equator, of course, here's what practitioners would say, they would say, for instance, we can do this with a hot start. That is, you have very vague information that then also the point you're looking at is inside this region. So, if you start with a hot start, then if you start here, then of course, you go, right? But in general, you don't know, of course, that and so why is it hard to escape this? It's because, of course, here, the drift, the gradient of your function in this direction is very small, because if you call X1 this coordinate, the function is X1 to the p. So, when you compute the gradient, it's very much zero, at X1 it calls zero, not only it, but it's second derivative too. And so, it's extremely hard to use this gradient to escape this. Yes. But I haven't, okay, I can answer yes to your question. That is, yes, we can. But in fact, we can in the two, you know, either on the, we can compute these things on the Gibbs measure, if not on the physics approach and let the temperature go to zero, we can compute them directly with the complexity, the geometric approach. Somebody has to work to prove that they are the same, but they should be. I'm sorry, I didn't, yeah, yeah. So, this can be done too. This is extremely difficult. So, in the case where you have no spike for just the polynomials, so that's the first step in this direction has been taken by Eliran, who computed the second moment and then could prove some form of concentration, right, around the mean, as I explained. You, and this was enough, in fact, for him to then understand what was happening. This was an information about zero temperature and he could climb to higher temperature, still very low, but higher temperature. Now, do, for instance, understanding the fluctuations would, that would be serious. But, but the, there are interesting information that you can get there at the level of convergence of processes. Elian, Soobag and Ophelia, for instance, looked at the following question. In the case where there's no spike, you know what the minimum should be. It should be this, what I call this negative E0, properly rescale. But now you could ask what is really, that's a deterministic number. What is the process of point which are near the minimum? And so that's what they found and they found that it was, because the second moment method worked here nicely, they could find that it was a Gumbel process. So, exactly as if this, even though those variables on the sphere are very correlated, their extreme values are the same as for Gaussian uncorrelated. They are a Gumbel process whose intensity they can, they could compute. So, some of these problems are solvable. For instance, what I just said here with respect to the spike model, what is the behavior of the landscape on this thing? For instance, this is, I don't know, but probably doable if somebody is as courageous as Elian, I'm not fair where.