High-Magnetization Sampling at Low Temperatures: Ising Models and Bayesian Sparse Linear Regression
Syamantak Kumar, Purnamrita Sarkar, Kevin Tian, Yusong Zhu
cs.DS · cs.LG · math.PR · math.ST · stat.ML
Quick verdict
Sparsity-aware frameworks yield any-temperature SK slice samplers and k^1.5 spike-and-slab complexity; solid theory, near-AT-line result partly conditional.
Summary
This theory paper develops frameworks for exploiting sparsity in high-dimensional sampling, applied to the Sherrington-Kirkpatrick (SK) spin glass and Bayesian sparse linear regression (BSLR). The first main result gives a polynomial-time sampler (the down-up walk) for the SK model restricted to fixed-magnetization slices, at any inverse temperature and under arbitrary external fields, whenever sparsity satisfies k <= c_beta*d; runtime is eO-tilde(d^2 + dk^2*max(1, beta*||h||_inf)). Combined with an annealing scheme estimating normalizing constants, this yields full-space SK samplers at arbitrarily low temperatures under strong fields, reaching field strengths within a constant factor (sqrt(2)(1+o(1))) of the Almeida-Thouless line, and within 1+o(1) if the signs of the mean magnetization are given — improving polynomially over [BAR26]. The second main result improves spike-and-slab posterior sampling measurement complexity from n >~ k^3 log^3 d [KSTZ25] to n >~ k^1.5 log^2 d + k log^3 d at any signal-to-noise ratio, with runtime linear in n*d. Techniques combine a sparse Dobrushin condition (path coupling), spectral mixing via the trickle-down theorem on rank-one-perturbed link graphs, shelling decompositions for sparse operator norms, and Schur-complement analysis of the support posterior. Limitations: the near-AT-line sampler is conditional on a sign oracle; the k^1.5 bottleneck is admitted to be inherent to the approach; and no empirical validation is provided, as expected for a purely theoretical contribution.
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Assessment at a glance
4
risk flags identified
● high 0
● medium 1
● low 3
5 strengths · 5 weaknesses · 5 author questions
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Key claims under review
- The canonical down-up walk samples fixed-magnetization SK models in polynomial time at any inverse temperature beta > 0 under arbitrary external fields whenever k <= c_beta*d, with runtime eO-tilde(d^2 + d*k^2*max(1, beta*||h||_inf)) (Theorems 1/4).
- Combining the slice sampler with annealing yields polynomial-time samplers for the full SK model at arbitrarily low temperatures under strong fields, within sqrt(2)(1+o(1)) of the Almeida-Thouless line unconditionally and within 1+o(1) given mean-sign oracle access (Theorems 6, 7), improving polynomially over [BAR26].
- Spike-and-slab posterior sampling at any signal-to-noise ratio admits a polynomial-time algorithm given n >~ k^1.5*log^2 d + k*log^3 d Gaussian measurements, a quadratic improvement in k over [KSTZ25]'s k^3*log^3 d (Theorems 2/5).
- A sparsity-aware toolkit — the sparse Dobrushin condition and spectral trickle-down on rank-one-perturbed link graphs — provides a common analytical engine across Ising, Hopfield, and BSLR applications, with the k^1.5 barrier attributed to worst-case union bounds.
Strengths
Unified, extensible analytical toolkit
The sparse Dobrushin condition (Definition 1, Theorem 3) and the spectral trickle-down framework (Lemmas 6-11) are developed abstractly over arbitrary universes, then instantiated on SK (Corollaries 3, Theorem 4), Gaussian Hopfield models (Corollary 4), bounded-magnetization models (Corollary 5), and the BSLR support posterior (Section 6). Lemma 10 formally shows the spectral framework subsumes the Dobrushin one.
Qualitative breakthrough on the temperature-sparsity tradeoff
Theorem 4 tolerates k = O(d/(beta_bar^2 log(beta_bar))) at arbitrary beta, whereas prior fixed-magnetization results [CDKP22, KPPY25] require sparsity decaying exponentially in beta (k < d*exp(-Delta*beta + O(1)), Appendix B), capping them at beta ~ log d. Appendix B derives this comparison honestly, including that their own framework applied to bounded-degree models only gives beta = O(1/k).
Careful quantification against the AT line
State-of-the-art, practical BSLR guarantee
Visible proof granularity
Weaknesses
Headline AT-line result partly conditional
Admitted method-inherent bottleneck in BSLR
Polynomial but heavy runtimes with existential constants
Comparison threshold for [BAR26] inferred, not quoted
No empirical component
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Scope & Claims
This is a pure theory paper (cs.DS/cs.LG/math.PR/math.ST) with two coupled contributions. First, it develops frameworks that exploit sparsity — formalized as restrictions to Hamming slices X_k^d — and applies them to the Sherrington-Kirkpatrick (SK) model. Theorem 1 (formal: Theorem 4) states that the canonical down-up walk samples fixed-magnetization SK in polynomial time at any inverse temperature beta > 0, under arbitrary external fields, for k <= c_beta*d, with runtime eO-tilde(d^2 + d*k^2*max(1, beta*||h||_inf)). Via an annealing scheme (Section 5) that estimates slice normalizing constants, this converts into full-space SK samplers at arbitrarily low temperature under strong fields: Theorem 6 reaches field h within sqrt(2)(1+o(1)) of the Almeida-Thouless line (Lemma 23 derives h_AT(beta) = sqrt(2)*beta*sqrt(log beta)*(1+O(1/log beta))), and Theorem 7 reaches 1+o(1) given oracle access to tau* = sign(E_pi[x]). Second, Theorem 2 (formal: Theorem 5, statement in a sampling gap) improves spike-and-slab posterior sampling from [KSTZ25]'s n >~ k^3 log^3 d measurements to n >~ k^1.5 log^2 d + k log^3 d at any SNR. Claims are stated as informal/formal pairs and the abstract does not overstate what is proven.
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Frequently asked questions
What does this paper prove about sampling the Sherrington-Kirkpatrick model?
It gives polynomial-time samplers for the SK spin glass restricted to fixed-magnetization slices (exactly k of d spins positive) at any inverse temperature, provided k is at most a temperature-dependent constant fraction of d. Combined with an annealing step that estimates normalizing constants, this also yields samplers for the full SK model at arbitrarily low temperatures under a sufficiently strong external field.
What is the Almeida-Thouless line and why is it a benchmark here?
The AT line is a predicted phase boundary in the (inverse temperature, field strength) plane of the SK model, separating a replica-symmetric regime from a replica-symmetry-breaking one. This paper's samplers work at field strengths within a constant factor (about 1.414) of the line unconditionally, and within a 1+o(1) factor if the signs of the mean magnetization are provided — meaning the algorithm covers essentially the entire provably replica-symmetric region at low temperature.
How does this paper improve Bayesian sparse linear regression posterior sampling?
For the canonical spike-and-slab posterior, the prior best sampler (KSTZ25) needed about k-cubed times polylog(d) Gaussian measurements to sample in polynomial time at any signal-to-noise ratio. This paper improves the k-dependence quadratically to about k^1.5 times log-squared d plus k times log-cubed d, with runtime linear in the input size n*d. Whether k*log d measurements suffice remains open.
Is code available for the proposed algorithms?
No code or data availability statement appears in the paper text; it is a theory contribution. The two main algorithms (the down-up walk and an annealing-based bounded-magnetization sampler) are given in pseudocode with per-iteration runtime analysis, so small-scale implementations are straightforward.
About this review
This is an AI-generated preliminary review by Evolve.Pub, produced from the openly available text of the paper. It is not peer review and does not represent the authors' views. The canonical record of the paper is at the publisher link above.
- The provided text is a sampled excerpt (head and tail of ~120k chars with marked gaps): portions of Sections 1.2, 3.1, the Theorem 4 statement, Sections 5.1-5.2, 6.3 (including the formal statement of Theorem 5), and Appendix A.2-A.3, plus the AI Disclosure section (TOC p. 44), fell in sampling gaps and were not reviewed.
- The provided excerpt contains no code or data availability statement; whether one exists in skipped pages was not verified.
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