Title: Constrained Posterior Sampling: Time Series Generation with Hard Constraints
Venue: NeurIPS 2025
Accepted: yes
Keywords: Time Series Generation

Posterior Sampling

Diffusion Models

Controlled Generation

Generative Models
Abstract: Generating realistic time series samples is crucial for stress-testing models and protecting user privacy by using synthetic data. In engineering and safety-critical applications, these samples must meet certain hard constraints that are domain-specific or naturally imposed by physics or nature. Consider, for example, generating electricity demand patterns with constraints on peak demand times. This can be used to stress-test the functioning of power grids during adverse weather conditions. Existing approaches for generating constrained time series are either not scalable or degrade sample quality. To address these challenges, we introduce Constrained Posterior Sampling (CPS), a diffusion-based sampling algorithm that aims to project the posterior mean estimate into the constraint set after each denoising update. Notably, CPS scales to a large number of constraints ($\sim100$) without requiring additional training. We provide theoretical justifications highlighting the impact of our projection step on sampling. Empirically, CPS outperforms state-of-the-art methods in sample quality and similarity to real time series by around 70\% and 22\%, respectively, on real-world stocks, traffic, and air quality datasets.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes Constrained Posterior Sampling (CPS) to enable constrained time series generation. The method is based on the DDIM framework but enforces constraints via projections and solves an unconstrained problem in each denoising step. By defining a time-varying penalty approaching infinity, the constraints are enforced for the final steps. CPS does not require specific training and can be applied during inference. Further, unlike guidance-based approaches, it does not require additional hyperparameter tuning but uses off-the-shelf solvers.\n\nSTRENGTHS AND WEAKNESSES:\nOverall, I like the paper, and it makes valuable contributions. The major weaknesses I see are the simplicity of the method, increased runtime (see Appendix D), and unclear experimental setup. In detail:\n\n\n### Strengths\n\n- Unlike other approaches, CPS enforces constraints in the data space, i.e., at $t=0$. This seems a more natural approach and allows more flexibility in the latent steps, i.e., $t>0$, and helps to stay in the latent manifolds during generation (L206-222)\n- CPS does not require backpropagation through the network. This might be useful in specific settings, i.e., black box settings or non-differentiable constraints.\n- The approach is theoretically motivated and analyzed (Sec. 4)\n- Experiments with various constraints (e.g., mean, max, min, specific values, ...). Setting specific values is an interesting setup, especially as it should include forecasting and imputation as a special case.\n- The background and methodology are mostly clear, and the notation is introduced adequately. In general, the paper is nicely written.\n\n### Weaknesses\n\n- The paper only investigates linear constraints.\n- There are some claims lacking empirical evidence. E.g., \"projecting the noisy intermediate latents (PDM and PRODIGY) pushes them off the noise manifold\" and \"This approach provides noisy latents that are closer to the training domain of the denoiser\"\n- It is unclear which constraints are used in the experiments. L286-287 mentions: \"mean, mean consecutive change, argmax, argmin, value at argmax, value at argmin, values at timestamps 1, 24, 48, 72, & 96\". Which of these are omitted in Figure 5? Shouldn't the stocks dataset have more constraints? Why is the x-axis labeled \"Number of constraints\" while Figure 7 in the appendix is labeled \"Number of constraint categories\"? I assume that the constraints apply to each dimension for multivariate datasets. As this is crucial information to understand the empirical results, this should be clarified.\n- \"Optimizing the guidance weights seems practically hard for a large number of constraints\". Why are the guidance-based approaches not included in Figure 5? It is unclear how they compare for few constraints.\n\n### Minors\n\n- Hyperlinks do not work\n- Context-FID has three different names across the paper: Context-FID, (Joint) Frechet Time Series Distance, Frechet Distance.\n- The extended related work for time series generation (A.1) should be more thorough.\n\nQUESTIONS:\n- In L162-166, the paper claims that the algorithm maximizes the likelihood of the generated samples. Can you elaborate on what this means?\n- In Figures 4 and 9, the generated samples almost perfectly align with the real samples. Why do multiple samples, i.e., samples with standard deviations, look worse (see Figure 10)? There are multiple timesteps where the real samples are not within the confidence interval.\n- Can CPS be extended to non-linear constraints?\n- Can you perform imputation using CPS by constraining values to match observations?\n\nI am willing to increase my score if the weaknesses are addressed and if my questions are answered satisfactorily.\n\nLIMITATIONS:\nYes, the limitations are addressed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 2, "DATE": "2025-06-24"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper tackles generating time series that satisfy hard constraints (for applications like stress-testing and synthetic data generation. The authors propose Constrained Posterior Sampling (CPS), a training-free diffusion-based approach.\nThe key innovation is projecting the posterior mean estimate onto constraint sets after each denoising step, rather than projecting noisy intermediate latents zₜ as in prior methods (PDM, PRODIGY). The algorithm alternates between standard DDIM denoising and constraint projection using optimization solvers, with stronger constraint enforcement in later steps.\nThe authors provide theoretical analysis under simplified assumptions (Gaussian data, linear constraints) and evaluate against guidance-based methods, post-processing approaches, and other projection methods across stocks, traffic, and air quality datasets.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n\nThe core insight to project posterior mean estimates rather than noisy latents is novel and well-motivated. This addresses limitations of existing projection-based methods (PDM, PRODIGY) in a principled way.\n\nThe approach addresses an important problem with real applications. The training-free nature and elimination of hyperparameter tuning represent practical advantages over existing methods.\n\nThe experimental evaluation is sound with appropriate baselines, multiple evaluation metrics, and fair comparisons using the same diffusion backbone. Results show substantial improvements across diverse datasets.\n\nThe intuition that projecting clean estimates preserves the noise manifold structure while enforcing constraints is technically strong.\n\nWeaknesses\n\nThe theoretical analysis assumes smooth penalty functions (Π(z) = ||fC(z)||₂²) but the algorithm uses non-smooth penalties (Π(z) = Σ max(0, fcᵢ(z))). This fundamental mismatch raises questions about value of the main theoretical contributions. The theory analyzes equality constraints while practice focuses on inequality constraints.\n\nAlgorithm 1 uses original noise predictions εθ(zₜ,t) with projected clean estimates ẑ₀,pr, violating the diffusion model relationship zₜ = √ε̄ₜ·z₀ + √(1-ε̄ₜ)·ε. This inconsistency is neither acknowledged nor justified.\n\nThe relationship between theoretical guarantees and practical performance is unclear. Theorem 4.2 is essentially tautological, and the restrictive assumptions (Gaussian data, optimal denoiser) limit practical relevance of the convergence analysis.\n\nNo systematic evaluation of constraint types, computational scalability beyond limited evidence, or design guidance for practitioners. The claim of handling \"~100 constraints\" lacks systematic verification.\n\nThe penalty schedule derived in theory differs from implementation without explanation.\n\nWhile the practical contribution is valuable, the theoretical analysis provides little insight due to assumption violations and oversimplified settings that don't apply to the actual algorithm.\n \nLimited hyperparameter search for baseline methods (only 6 guidance weight values) potentially understates competitor performance. Insufficient analysis of constraint diversity and scaling behavior.\n\nThe paper presents a valuable practical contribution undermined by theoretical claims that don't apply to the actual method and missing systematic analysis needed for this type of venue.\n\nQUESTIONS:\nYour theory assumes smooth penalties Π(z) = ||fC(z)||₂² but Algorithm 1 uses non-smooth Π(z) = Σ max(0, fcᵢ(z)). Can you either provide analysis for the non-smooth case or explicitly acknowledge this limitation? Also, why use original noise εθ(zₜ,t) with projected ẑ₀,pr instead of updating for consistency?\n\nYou test various constraints but lack systematic analysis of which types work well/poorly. Can you provide a constraint taxonomy with performance guidance and design principles for practitioners?\n\nYour \"~100 constraints\" claim lacks systematic verification. Can you provide scaling curves (inference time/memory vs constraint count), computational complexity analysis, and practical upper limits?\n\nGuidance methods only tested with 6 hyperparameter values and showed high constraint violations. Did you perform sufficient hyperparameter search to ensure fair comparison?\n\n Theory derives penalty = 2k(T-t+1)/σₘᵢₙ(AᵀA) but implementation uses e^(1/(1-ε̄ₜ)-1). Can you explain this difference and justify your choice?\n\nLIMITATIONS:\nNo. The authors mention computational overhead and provide some mitigation strategies, but fail to address several key limitations identified in this review:\n\nSome missing Limitations:\n\n- Theory-practice disconnect where theoretical guarantees don't apply to the actual algorithm\n- Algorithmic inconsistencies (noise term mismatch) that may affect performance\n- Limited understanding of which constraint types are suitable for the approach\n- Lack of systematic scalability analysis", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-26"}

comments: SUMMARY:
The paper presents a new diffusion-based algorithm for generating time series satisfying linear constraints. Without retraining the denoiser, it only alters the reverse sampler by enforcing all constraints on the posterior mean at every denoising step. Compared to previous methods, it better preserves sample quality while satisfying the constraints.

STRENGTHS AND WEAKNESSES:
**Strengths**

- The paper is clear and well-written. 
- The method’s rationale is well explained and supported by empirical results.
- The empirical evaluation is sound and extensive: compares to representatives of different paradigms (guidance, post-processing, and Projected Diffusion Models (PDM)); uses multiple metrics, visualises samples, and measures cost and performance versus the number of constraints.
- The proposed method is relatively easy to implement. It lets practitioners impose multiple convex constraints with solid theoretical guarantees in the linear case. 

**Weaknesses**

- Limited novelty: the main difference from PDM is projecting the posterior mean instead of the latent state.
- Narrow scope: focuses on convex constraints. The appendix’s non-convex autocorrelation example (fixed lag $k$) is interesting but its discussion (lines 852–853) is too vague.

QUESTIONS:
Why are the generated series matching a real time series pointwise almost exactly?

LIMITATIONS:
yes
TITLE: Official Review
CLARITY: 3
IMPACT: 2
ORIGINALITY: 2
DATE: 2025-06-30