Timeline for High-dimensional uniform distribution
Current License: CC BY-SA 4.0
29 events
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| Sep 15, 2022 at 6:16 | comment | added | Pluviophile | @IosifPinelis I guess the product measure in this case is the result of this particular type of constraints. | |
| Sep 15, 2022 at 4:25 | comment | added | Iosif Pinelis | Previous comment continued: Perhaps, this has something to do with the intuition that the uniform distribution on the compact Riemann manifold is approximately a product measure in some sense (?). | |
| Sep 15, 2022 at 4:25 | comment | added | Iosif Pinelis | @Pluviophile : Alas, I still don't see a connection. On the one hand, we are talking about the entropy of distributions on a multidimensional compact Riemann manifold -- without restrictions on the distributions, so that the uniform one is the maximizer. On the other hand, about the entropy of distributions on the entire real line -- with affine restrictions on the distributions, and then a maximizer (or any distribution) cannot possibly be uniform. | |
| Sep 14, 2022 at 20:02 | comment | added | Pluviophile | @IosifPinelis I guess it has something to do with the fact that the uniform distribution is the least informative given the constraints $\frac{1}{n}\sum_i f_a(x_i)=m_a$, while the maximum entropy law is the least informative given the constraints $\mathbb E[f_a(X)]=m_a$. | |
| Sep 14, 2022 at 19:30 | comment | added | Iosif Pinelis | @Pluviophile : Oh, that was clear to me. But is that more than a mere coincidence? | |
| Sep 14, 2022 at 18:29 | history | edited | Pluviophile | CC BY-SA 4.0 |
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| Sep 14, 2022 at 18:28 | comment | added | Pluviophile | @MartinHairer Thank you for your comment. I found out that paper while trying to add more information to the question. I'll take a deeper look at it. | |
| Sep 14, 2022 at 18:25 | comment | added | Pluviophile | @IosifPinelis I added a new line in the question under the equation (2). | |
| Sep 14, 2022 at 18:21 | history | edited | Pluviophile | CC BY-SA 4.0 |
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| Sep 14, 2022 at 17:17 | comment | added | Iosif Pinelis | @Pluviophile : Thank you for this additional information. However, I still don't see a connection to maximizing entropy. Can you say more on that -- preferably in mathematical terms (as I know next to nothing in statistical physics)? | |
| Sep 14, 2022 at 17:02 | comment | added | Martin Hairer | By analogy with Sourav's result, it would be natural to conjecture that you're again asymptotically i.i.d. normal, but that one of the $x_i$'s will typically be about $(mn)^{1/3}$. | |
| Sep 14, 2022 at 15:05 | comment | added | Pluviophile | @IosifPinelis I've updated the question. | |
| Sep 14, 2022 at 15:04 | history | edited | Pluviophile | CC BY-SA 4.0 |
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| Sep 14, 2022 at 15:02 | comment | added | Anthony Quas | @IosifPinelis: this is just intuition on my part. | |
| Sep 14, 2022 at 12:06 | comment | added | Iosif Pinelis | @Pluviophile : How to show that your posted question is equivalent to the problem of maximizing entropy over random variables $X$ with $EX^2=1$ and $EX^3=m$ (as you seem to assert)? Is it something easy, or do you have a reference for this? | |
| Sep 14, 2022 at 10:04 | comment | added | Pluviophile | @AnthonyQuas Yes but such random variable does not exist. | |
| Sep 14, 2022 at 9:50 | comment | added | Anthony Quas | I guess the answer should be that the distribution is obtained by maximizing entropy over random variables satisfying $\mathbb E(X^2)=1$ and $\mathbb E(X^3)=m$? | |
| Sep 14, 2022 at 4:52 | comment | added | Pluviophile | Yes $m$ is fixed. | |
| Sep 14, 2022 at 4:51 | comment | added | Iosif Pinelis | Is $m$ fixed? I think the answer will depend significantly on whether $m$ is $0$ or not. Do you want to consider both cases? | |
| Sep 14, 2022 at 4:49 | history | edited | Pluviophile | CC BY-SA 4.0 |
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| Sep 14, 2022 at 4:48 | comment | added | Pluviophile | Another important thing I haven't mentioned is that in the quadratic and quartic case, the physical heuristic predicts that $X_i$ are asymptotically i.i.d, which means the joint distribution $(X_1,\dots,X_k)$ converges in law to $P_{X_1}^{\otimes k}$ for any finite $k$. So understand the marginal is the key to understand this high-dimensional uniform distribution. | |
| Sep 14, 2022 at 4:36 | comment | added | Pluviophile | @IosifPinelis This question is somewhat related to the principle of maximum entropy wikiwand.com/en/Principle_of_maximum_entropy. The cubic constraint corresponds to the following case wikiwand.com/en/Maximum_entropy_probability_distribution#/… | |
| Sep 14, 2022 at 4:27 | comment | added | Pluviophile | @DanielAsimov Yes. My background is physics and my way to visualize the uniform distribution on a compact manifold in $ \mathbb R^n$ is 1) choosing a box that contains the manifold 2) choose uniformly randomly a very large number of points inside this box 3) only keep the points that are very close to the manifold | |
| Sep 14, 2022 at 0:42 | comment | added | Daniel Asimov | A uniform distribution on a compact manifold makes sense once a Riemannian metric has been defined on it. I presume that the metric here is the one inherited from euclidean space — is that right? | |
| Sep 13, 2022 at 19:46 | comment | added | Iosif Pinelis | Thank you for your response. Also, do you mind sharing how this problem arises? | |
| Sep 13, 2022 at 19:14 | comment | added | Pluviophile | It's the same thing as uniform distribution on a compact manifold. The result for quartic case is an easy consequence of the equivalence between canonical and micro-canonical ensemble in statistical physics, although this equivalence does not hold for all kind of energy function, and there is no general math theorem about it. If we blindly apply this physical heuristic, we get the marginal $P(x) \propto e^{ax^3+bx^2}$ which is not well defined on $\mathbb R$ | |
| Sep 13, 2022 at 18:53 | comment | added | Iosif Pinelis | How do you define this uniform distribution? How do you do the quartic case? Why is it significantly different from the cubic one? | |
| Sep 13, 2022 at 16:46 | history | edited | Pluviophile | CC BY-SA 4.0 |
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| Sep 13, 2022 at 16:07 | history | asked | Pluviophile | CC BY-SA 4.0 |