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Randomstrasse101: Open Problems of 2024

Afonso S. Bandeira, Anastasia Kireeva, Antoine Maillard, Almut Rödder

TL;DR

This paper presents a curated collection of open problems from Randomstrasse101 (2024) across probability, computation, combinatorics, and statistics, serving as a stable reference for reference and study. It surveys core questions such as the matrix Spencer conjecture and its special cases, global synchronization thresholds, ellipsoid-fitting transitions, multi-frequency synchronization, Tensor PCA with Kikuchi methods, discrepancy bounds, and SK model mixing. The work highlights both established results and conjectures, connecting disparate techniques from matrix concentration, spectral methods, stochastic localization, and convex relaxations, while outlining concrete thresholds and regimes where progress is possible. The compilation aims to stimulate rigor and progress in understanding the computational and statistical boundaries of these open problems, with implications for both theory and applied algorithm design.

Abstract

$\texttt{Randomstrasse101}$ is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2024, with the goal of easing academic referencing. The blog can currently be accessed at $\texttt{randomstrasse101.math.ethz.ch}$.

Randomstrasse101: Open Problems of 2024

TL;DR

This paper presents a curated collection of open problems from Randomstrasse101 (2024) across probability, computation, combinatorics, and statistics, serving as a stable reference for reference and study. It surveys core questions such as the matrix Spencer conjecture and its special cases, global synchronization thresholds, ellipsoid-fitting transitions, multi-frequency synchronization, Tensor PCA with Kikuchi methods, discrepancy bounds, and SK model mixing. The work highlights both established results and conjectures, connecting disparate techniques from matrix concentration, spectral methods, stochastic localization, and convex relaxations, while outlining concrete thresholds and regimes where progress is possible. The compilation aims to stimulate rigor and progress in understanding the computational and statistical boundaries of these open problems, with implications for both theory and applied algorithm design.

Abstract

is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2024, with the goal of easing academic referencing. The blog can currently be accessed at .
Paper Structure (7 sections, 2 theorems, 28 equations, 1 figure)

This paper contains 7 sections, 2 theorems, 28 equations, 1 figure.

Table of Contents

  1. Welcome $\&$ Matrix Discrepancy (ASB)
  2. Global Synchronization (ASB)
  3. Fitting ellipsoids to random points (AM)
  4. Detection in Multi-Frequency synchronization (AK)
  5. Tensor PCA and the Kikuchi algorithm (ASB)
  6. Did just a couple of deviations suffice all along? (ASB)
  7. Sampling from the Sherrington-Kirkpatrick Model (AR)

Key Result

Theorem 3.1

Let $n, d \geq 1$ and $x_1, \cdots, x_n \sim \mathcal{N}(0,\mathrm{I}_d/d)$. There is $\delta > 0$ such that:

Figures (1)

  • Figure 1: Fitting Gaussian random points $x_i \sim \mathcal{N}(0, \mathrm{I}_d/d)$ to an ellipsoid. Notice that the unit sphere itself is close to being a fit by simple concentration of measure: a random $x \sim \mathcal{N}(0, \mathrm{I}_d/d)$ has (with high probability) distance $\mathcal{O}(1/\sqrt{d})$ to it.

Theorems & Definitions (15)

  • Conjecture 1: Matrix Spencer
  • Conjecture 2: Group Spencer
  • Definition 2.1
  • Conjecture 3: Globally Synchronizing Regular Graphs
  • Conjecture 4: Global Synchrony with negative edges
  • Conjecture 5: Density threshold for Global Synchrony
  • Conjecture 6: Ellipsoid fitting
  • Theorem 3.1: tulsiani2023ellipsoidhsieh2023ellipsoidbandeira2024fitting
  • Theorem 3.2: maillard2023exact
  • Conjecture 9: Kikuchi Spectral Threshold
  • ...and 5 more