bibkey: skalski2025level authors: Tomasz Skalski, Tomasz Stroiński year: 2025 title: Level sets and maximum likelihood estimation for the Ising model doi: 10.48550/arXiv.2511.20925 url: https://arxiv.org/abs/2511.20925v1 claim: The paper bounds the smallest set of uniqueness u(k,q) for the nonnegative cone of the Walsh space B^k_q on the cube {-1,+1}^k and conjectures u(k,2) = k + 1. strata_touched:

• D5/S3/Combinatorics/IsingUniquenessSets license: citation-only triage: anchor

Level sets and maximum likelihood estimation for the Ising model

Skalski and Stroiński study the existence of maximum likelihood estimators in the discrete exponential family of the Ising model through sets of uniqueness, following Bogdan, Bosy and Skalski. On X = {-1,+1}^k they set r_j(x) = x_j, w_L(x) = ∏_{j∈L} r_j(x) and, for 1 ≤ q ≤ k,

B^k_q = Lin{w_L : L ⊂ {1,…,k} and |L| ≤ q}.

A subset U ⊂ X is a set of uniqueness for (B^k_q)_+, the nonnegative functions of B^k_q, when φ = 0 is the only such function vanishing on U, and u(k,q) is the size of the smallest one. Lemma rem:three states that the points with exactly one coordinate +1, together with (+1,…,+1), form a set of uniqueness for (B^k_2)_+; the closing theorem of the section on extremal sizes lists log k + ½ log log k + O(1) ≤ u(k,2) ≤ k+1 and u(k,k) = 2^k, and the section ends with the conjecture

u(k,2)=k+1, i.e., for every k there are no sets of uniqueness having at most k elements.

At k = 2 the space B^2_2 is all of ℝ^X, so u(2,2) = 4, as the listed u(k,k) = 2^k also gives; the upper bound k + 1 and Lemma rem:three hold from k = 3 on.

Verified locator

• DOI: https://doi.org/10.48550/arXiv.2511.20925
• URL: https://arxiv.org/abs/2511.20925v1
• Version and location: arXiv:2511.20925v1 (2025-11-25), section on sets of uniqueness and Rademacher functions, Lemma rem:three, and the closing theorem and conjecture of the section on extremal sizes of sets of uniqueness.