We consider the problem of estimating inverse temperature parameter $β$ of an $n$-dimensional truncated Ising model using a single sample. Given a graph $G = (V,E)$ with $n$ vertices, a truncated Ising model is a probability distribution over the $n$-dimensional hypercube $\{-1,1\}^n$ where each configuration $\mathbfσ$ is constrained to lie in a truncation set $S \subseteq \{-1,1\}^n$ and has probability $\Pr(\mathbfσ) \propto \exp(β\mathbfσ^\top A\mathbfσ)$ with $A$ being the adjacency matrix of $G$. We adopt the recent setting of [Galanis et al. SODA'24], where the truncation set $S$ can be expressed as the set of satisfying assignments of a $k$-SAT formula. Given a single sample $\mathbfσ$ from a truncated Ising model, with inverse parameter $β^*$, underlying graph $G$ of bounded degree $Δ$ and $S$ being expressed as the set of satisfying assignments of a $k$-SAT formula, we design in nearly $O(n)$ time an estimator $\hatβ$ that is $O(Δ^3/\sqrt{n})$-consistent with the true parameter $β^*$ for $k \gtrsim \log(d^2k)Δ^3.$ Our estimator is based on the maximization of the pseudolikelihood, a notion that has received extensive analysis for various probabilistic models without [Chatterjee, Annals of Statistics '07] or with truncation [Galanis et al. SODA '24]. Our approach generalizes recent techniques from [Daskalakis et al. STOC '19, Galanis et al. SODA '24], to confront the more challenging setting of the truncated Ising model.

翻译：我们考虑使用单样本估计$n$维截断伊辛模型逆温度参数$β$的问题。给定具有$n$个顶点的图$G = (V,E)$，截断伊辛模型是在$n$维超立方体$\{-1,1\}^n$上的概率分布，其中每个构型$\mathbfσ$被约束在截断集$S \subseteq \{-1,1\}^n$内，且具有概率$\Pr(\mathbfσ) \propto \exp(β\mathbfσ^\top A\mathbfσ)$，其中$A$为$G$的邻接矩阵。我们采用[Galanis et al. SODA'24]的最新设定，其中截断集$S$可表示为$k$-SAT公式的可满足赋值集合。给定来自截断伊辛模型的单样本$\mathbfσ$，其逆参数为$β^*$，基础图$G$具有有界度$Δ$，且$S$表示为$k$-SAT公式的可满足赋值集合，我们在近$O(n)$时间内设计了估计量$\hatβ$，当$k \gtrsim \log(d^2k)Δ^3$时，该估计量与真实参数$β^*$具有$O(Δ^3/\sqrt{n})$一致性。我们的估计量基于伪似然最大化方法——这一概念已在多种概率模型中（无论是否经过截断处理）得到广泛分析[Chatterjee, Annals of Statistics '07; Galanis et al. SODA '24]。我们的方法推广了[Daskalakis et al. STOC '19, Galanis et al. SODA '24]的最新技术，以应对截断伊辛模型这一更具挑战性的设定。