SciPost Submission Page
High-dimensional random landscapes: from typical to large deviations
by Valentina Ros
Submission summary
| Authors (as registered SciPost users): | Valentina Ros |
| Submission information | |
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| Preprint Link: | scipost_202502_00002v1 (pdf) |
| Date submitted: | 2025-02-03 12:24 |
| Submitted by: | Ros, Valentina |
| Submitted to: | SciPost Physics Lecture Notes |
| for consideration in Collection: |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We discuss tools and concepts that emerge when studying high-dimensional random landscapes, i.e., random functions on high-dimensional spaces. As an example, we con- sider a high-dimensional inference problem in two forms: matrix denoising (Case 1) and tensor denoising (Case 2). We show how to map the inference problem onto the opti- mization problem of a high-dimensional landscape, which exhibits distinct geometrical properties in the two Cases. We discuss methods for characterizing typical realizations of these landscapes and their optimization through local dynamics. We conclude by highlighting connections between the landscape problem and Large Deviation Theory.
Current status:
Reports on this Submission
Report #1 by Bertrand Lacroix-A-Chez-Toine (Referee 1) on 2025-3-20 (Invited Report)
Report
In these lecture notes, the author considers the subject of high-dimensional random landscapes and illustrates the general theory on inference problems. This tool can be applied to understand a number of complex/disordered systems arising in subjects ranging from economy, computer science to statistical physics and bayesian inference.
These notes describe this topic in a very pedagogic and well-written way. They refer to very recent and top-level literature in this subject without getting too technical. It also provides illustrations and hands-on exercises for students to get familiar with the content and some technical aspects.
I highly recommend these lecture notes for publication in SciPost Physics Lecture Notes once the small list of changes below are implemented.
Requested changes
Find below a list of typos/comments:
1 - p6: "This properties allows us to draw" should be "This property allows us to draw"
2 - p12: "The Riemannian gradient on the hypersphere is and (N − 1)-dimensional vector" should be "The Riemannian gradient on the hypersphere is an (N − 1)-dimensional vector"
3 - p17: "by computing its the large-N expansion" should be "by computing its large-N expansion"
4 - p28: "An “hard" inference problem: noisy tensors" should be "A “hard" inference problem: noisy tensors"
5 - p30: In the section "In the case of quadratic landscape p = 2, the RMT results imply that:", it seems that point (ii) and (iii) are incompatible:
If the variable N (ε)/N "converges to its average", its limiting distribution is a delta function which seems incompatible with the fact that it has a "well-defined limit when N →∞"
This point should be clarified in the revised version
6 - p37: "One can check that that there are values" should be "One can check that there are values"
7 - p40: "These stationary points are marginally stabile" should be "These stationary points are marginally stable"
8 - p40: "stationary points at the equator have no isolate eigenvalues" should be "stationary points at the equator have no isolated eigenvalues"
9 - p42: A limit t\to\infty should appear in Eq. (120) (after the limit N\to \infty is taken)
10 - p42: "An paradigmatic solution" should be "A paradigmatic solution"
Recommendation
Publish (surpasses expectations and criteria for this Journal; among top 10%)