[Paper Review] On Sampling from Ising Models with Spectral Constraints

Andreas Galanis, Alkis Kalavasis+1 more|arXiv (Cornell University)|Jan 1, 2024

Markov Chains and Monte Carlo Methods1 citations

TL;DR

This paper confirms Kunisky's conjecture that sampling from Ising models with spectral gap γ > 1 is NP-hard, even within exponential approximation factors. By introducing mild antiferromagnetic couplings to ferromagnetic random regular graphs, the authors exploit bimodal distributions to reduce to inapproximability results for antiferromagnetic systems, establishing strong NP-hardness for both approximate sampling and counting under spectral constraints.

ABSTRACT

We consider the problem of sampling from the Ising model when the underlying interaction matrix has eigenvalues lying within an interval of length $γ$. Recent work in this setting has shown various algorithmic results that apply roughly when $γ< 1$, notably with nearly-linear running times based on the classical Glauber dynamics. However, the optimality of the range of $γ$ was not clear since previous inapproximability results developed for the antiferromagnetic case (where the matrix has entries $\leq 0$) apply only for $γ>2$. To this end, Kunisky (SODA'24) recently provided evidence that the problem becomes hard already when $γ>1$ based on the low-degree hardness for an inference problem on random matrices. Based on this, he conjectured that sampling from the Ising model in the same range of $γ$ is NP-hard. Here we confirm this conjecture, complementing in particular the known algorithmic results by showing NP-hardness results for approximately counting and sampling when $γ>1$, with strong inapproximability guarantees; we also obtain a more refined hardness result for matrices where only a constant number of entries per row are allowed to be non-zero. The main observation in our reductions is that, for $γ>1$, Glauber dynamics mixes slowly when the interactions are all positive (ferromagnetic) for the complete and random regular graphs, due to a bimodality in the underlying distribution. While ferromagnetic interactions typically preclude NP-hardness results, here we work around this by introducing in an appropriate way mild antiferromagnetism, keeping the spectrum roughly within the same range. This allows us to exploit the bimodality of the aforementioned graphs and show the target NP-hardness by adapting suitably previous inapproximability techniques developed for antiferromagnetic systems.

Motivation & Objective

• To resolve the open question of whether Ising model sampling remains NP-hard when the spectral gap γ exceeds 1, closing the gap between known algorithmic results and inapproximability bounds.
• To provide a formal proof of NP-hardness for approximate sampling and counting in Ising models under spectral constraints, particularly when γ > 1.
• To overcome the typical barrier that ferromagnetic interactions prevent NP-hardness by introducing controlled antiferromagnetic couplings while preserving spectral bounds.
• To extend existing inapproximability techniques from antiferromagnetic systems to the broader spectral regime, using graph gadgets with bimodal behavior.
• To establish tight computational complexity bounds for Glauber dynamics and general-purpose Ising model sampling under spectral constraints.

Proposed method

• Construct a d-regular random graph gadget with ferromagnetic interactions (β > 0) and a small antiferromagnetic perturbation (−η) to break symmetry while preserving spectral range.
• Use spectral theory to bound the eigenvalue spread of the interaction matrix: λ_max(J) − λ_min(J) ≤ β(λ_d−1 + 2√(d−2)) + ε, where λ_d−1 = (d−1) + 2√(d−2).
• Leverage the bimodal distribution of spin configurations in the gadget (due to symmetry and large size) to ensure equal probability of positive and negative global phases.
• Condition the spin distribution on the global phase to show it approximates a product distribution with small error (1 ± ε), enabling reduction to MaxCut.
• Reduce from MaxCut to the BoundedSpectralIsing problem by connecting gadgets via antiferromagnetic edges, ensuring the overall interaction matrix has spectral gap < γ.
• Apply known inapproximability techniques from antiferromagnetic systems to show that approximating the partition function within an exponential factor is NP-hard.

Experimental results

Research questions

• RQ1 Is sampling from the Ising model NP-hard when the spectral gap γ exceeds 1, despite recent nearly-linear time algorithms for γ < 1?
• RQ2 Can the NP-hardness of approximate sampling and counting be established in the spectral regime γ > 1, where prior inapproximability results only applied for γ > 2?
• RQ3 How can ferromagnetic systems—typically not NP-hard—be adapted to yield NP-hardness under spectral constraints?
• RQ4 Can bimodal behavior in random regular graphs be exploited to construct a reduction from MaxCut to Ising models with bounded spectral gap?
• RQ5 What is the tightest possible spectral constraint γ for which NP-hardness of Ising model sampling holds?

Key findings

• The paper confirms Kunisky’s conjecture: sampling from the Ising model is NP-hard when the spectral gap γ > 1, even within exponential approximation factors.
• For any d ≥ 4, the problem BoundedSpectralIsing(d, γ) is NP-hard when γ > (1/2)ln(1 + 2/(d−3)) · (d−1 + 2√(d−2)), providing a tight threshold for NP-hardness.
• The authors construct a d-regular graph with ferromagnetic interactions and a small antiferromagnetic perturbation that preserves spectral bounds while inducing bimodal spin distributions.
• The conditional distribution of spins on a subset S of vertices, given the global phase, is shown to be (1±ε)-close to a product distribution, enabling robust reductions.
• The spectral gap of the full interaction matrix is bounded by γ using Weyr–Wielandt and Weyl’s inequalities, ensuring the instance lies within the required spectral constraint.
• The construction generalizes to d ≥ 3 with minor modifications, suggesting the hardness threshold could be tightened to γ > β_d λ_d for d-regular graphs, where β_d and λ_d are defined in terms of d.