Constrained Posterior Sampling: Time Series Generation with Hard Constraints
Authors: Sai Shankar Narasimhan
Shubhankar Agarwal
Litu Rout
Sanjay Shakkottai
Sandeep P. Chinchali
Venue: NeurIPS 2025 poster ()
Keywords: Time Series Generation
Posterior Sampling
Diffusion Models
Controlled Generation
Generative Models
Primary area: deep_learning
TL;DR: This paper proposes a constrained posterior sampling approach for time series generation with hard constraints.

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.

Constrained Posterior Sampling: Time Series Generation with Hard Constraints
Sai Shankar Narasimhan
Shubhankar Agarwal
Litu Rout
Sanjay Shakkottai
Sandeep Chinchali
Chandra Family Department of Electrical Computer Engineering,
The University of Texas at Austin, Austin, TX, 78712.
(nsaishankar, somi aogmiaaggrwal,aaa rout, sanjay, shakkottai, sandeepc) @utexas.eeeeeeeoo edu
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 nature. Consider, for example, generat- ing 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, introduce Constrained Pos- terior Sampling (CPS), Liffusion-basse 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 (~ 100) without requiring additional training. We provide theoretical justifications highlighting the impact of our projection step on sampling. Empirically, CPS outperforms state-of-the-ae 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.
Introduction
Realistic time series generation plays cru- cial role tasks such as "what-if" scenario analysis [1], stress-testing machine learn- ~
Correlated Stocks Stock Nase
Highest Price
Lowest Price
XYZ
Nonth
Septerbor
ing models [2, 3], and anonymizing private user data [4]. Current approaches leverage state-of-the-art generative models, includ- ing Diffusion Models (DMs) [1, 5, 6] and Generative Adversarial Networks (GANs) [7, 8J, sample from time series distribu- tions. Ensuring realism and high fidelity through constraint satisfaction remains key challenge in domain-specific time se- ries generation. For example, generating daily Heee--HighLLow-Cli for stock prices of an S&P 500 company requires the high and low values to bound the opening and closing prices (Figure l). Similarly, generating stock price time series with user-
CPS
Opening closing Price Highest Price Openinn Price >0 Lowest Price closing Price >= Lowest Price
Highest
Days
Mean(Closing Price) $35
Opening Price
Closing Price
Constraints
Figure 1: Our Proposed Constrained Posterior Sam- pling (CPS). CPS is novel diffusion-based sampling approach to generate time series samples that satisfy hard constraints. Here, CPS generates a daily stock price time series with natu- ral constraints such the bounds the opening and closing prices of the stock.
show example where
(OHLC) chart
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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specified volatility to stress-test trading strategies necessitates adherence to volatility constraints, as inaccuracies in the generated samples lead to inaccurate trrss-testing.
Large-scale foundation models have achieved remarkable progress in generative modeling tasks for language and vision [9. 10]. This generative power has sparked significant interest in addressing constraint satisfaction problems [11-15]. However, constraints are often challenging to define and verify in language and vision domains.
80
D5 Improvement (%)
60
DTW improvement
40
30
For instance, verifying the correctness of an image (e.g., a hand with 6 fingers)
30
20
is inherently difficult due prediction errors perception models. contrast, time series samples
10
10
be described through statistical features computed using well-defined functions, and these features can be imposed as constraints and verified accurately. This clarity in defining and verifying constraints makes the time series domain promising area for advancing re- search on constraint satisfaction problems. desired constrained time series generator exhibits the following properties:
Air Quality Traffic
Stocks
Air Quality Traffic
Stocks
Dataset
Dataset
Figure 2: CPS outperforms the best-performing base- lines on the Dynamic Time Warping (DTW) distance that measures the similarity between the real and the generated samples by 22% and the Discriminative Score (DS), which is the classification error of a model trained to differentiate real and synthetic data, by 70%.
Training-free approach include multiple constraints:
Training generative model for specific constraints, as in Loss-DiffTime [12], lacks scalability. For example, a model trained for mean constraints cannot easily adapt argmax constraints.
Independence from external realism enforcers: Prior works often involve projecting samples onto feasible defined by constraints, which can degrade sample quality. To mitigate this, previous approaches [12] employ external models alongside the generative model to enforce realism, necessitating additional training and complex sampling procedures.
syperparameterr-rre approach to constrained generation: The choice of guidance weights in guidance-based approaches significantly affects sample quality. However, optimizing these weights becomes computationally intractable when dealing with hundreds of constraints.
Given these requirements, propose Constrained Posterior Sampling (CPS), a novel algorithm for diffusion-based time series generation with desired constraints, as illustrated in Figure 1. CPS projects the posterior mean estimate onto the constraint set after each denoising update, leveraging off-the-shelff optimization algorithms and eliminating the need for training hyperparameter tuning to incorporate multiple constraints. CPS does not require external models enforee realism, subsequent denoising steps naturally mitigate the adverse effects of projection. Our main contributions are as follows:
1. We present Constrained Posterior Sampling (CPS), a scalable diffusion posterior sampling algo- rithm to generate time series samples that satisfy desired constraints with high sample quality. CPS efficiently handles large number of constraints (~ 100) without additional training.
2. We provide rigorous theoretical justification demonstrating the impact of CPS on traditional diffu- sion sampling. We also provide convergence analysis for well-studied settings (linear constraint sets with Gaussian prior data distribution), offering valuable insights for practice.
Through extensive experiments on six diverse real-world and simulated datasets covering finance, traffic, and environmental monitoring, we show that CPS outperforms state-of of-thh-rrt methods in sample quality, similarity, and constraint adherence metries, as shown in Figure
2 Preliminaries
Notations: We denote a time series sample by æ E RKXL Here, K and refer to number of channels and the horizon, respectively. A dataset is defined as D {x'...., ND where the superscript E (1....,Ny Np) refers the sample number, and Np is the total number of samples in the dataset. Pdata denotes the real time series data distribution. z² is the realization of the

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random vector X', where X .'....X ....XNND. with Paata is represented by Panta RRXx. + E. where J Paata integration operator over TR. Kx The notation ((, Σ) refers to the Gaussian distribution with mean j and covariance matrix Σ. Similarly, (( b) indicates the uniform distribution with non-zero density from a to b. ||. ||2 indicates the l2 norm in the case of a vector and the spectral norm in the case of a matrix. We denote the constraint set C as C = Ci 2.......... F..,CNee where Ne is the total number of constraints and n denotes intersection. Here, C, = { Sa (æ) S 0} with I V c4 E [l...., Nc). Amax(M) and Amia (min(M) (M) refer to the largest and the smallest
~ Pdata- The Probability Density Function (PDF) associated 1. Here, refers to the
fe: _RRxx - eigen values a square matrix M with rank denoted by ran)(M).
Example I: The stocks dataset has 6 channels (K = 6) with 96 timestamps in each channel (L = 96). The first channels represent the opening price (o), the highest price (h), the lowest price (I), and the closing price (c), and each timestamp represents day. The OHLC constraint is given by - h s O, c e-h h I 0, I o I O, and c I 0, i.e. the opening and closing prices must lie between the highest and the lowest prices. The mean equality constraint on the closing price is expressed as L (Ee= Cw) Me S and 44 - L (Eh=1 Cu Σ O, where Pe is the required mean and Cu refers to the value of the closing price for the timestamp
2.1 Diffusion Models
Diffusion models [16, 17] consist of two stochastic processes: (a) forward noising process and (b) reverse denoising process. The forward process is modeled by Markov process where sample 2o from the data distribution (c.g.. a clean image) is transformed into a sample žr from the standard Gaussian (N0, I) in T steps. Let O and I denote zero mean and identity covariance. The Markov process uses a Gaussian transition kernel governed by the diffusion coefficients co. ....... ãT, where Oe E [0, 1] and ūt-1 > G₁ V/ t E [1, T), with boundary conditions given by ag 1, 0. Formally, (2 20) is the PDF of the conditional Gaussian distribution at time t with mean 20,20 and covariance matrix (1 ã₂) T. The PDF for the marginal at t 0 is given by qu Pdata-
The reverse process transforms a noise sample 2T ~ No I) into a clean image from Platt A neural network is trained to align the marginals of the reverse process (=-x =x) with that of the forward process for all & [1. The neural network usually trained to approximate the mean of (0,1 26) for a fixed variance schedule. The resulting training objective becomes:
|lle <s e)112),
where ea (22,,)) t) is trained to estimate the noise from 24 and 2 Varz0 + ā with t~ 2 []. Interestingly, the training objective (Equation 1) is equivalent to that of deterministic sampling algorithm, DDIM [18], where the forward noising process is modeled by a non-Markon Markovian process, and the corresponding reverse update becomes:
2t-1 vār- (0 60) + V ā, of t) + o,E.
Here, Co) =iv1--aco(z the posterior mean E [2]],c,, and or controls the sampling
Vac process stochasticity induced by the noise term 7 ~ N(OO, I). From [18], we note that Equation 2 corresponds to the following reverse process:
(20 (0(2))60)) (z,,(=z-1 2t, 20(21;
if
(0, =1)
otherwise,
where (0 Re, 20 20( (0)) represents the PDF of the Gaussian distribution with mean vār-1& 60) + /I āi-1 of (2(,,))) and covariance matrix I.I. Similarly, po,init (20 (0) is the PDF of the Gaussian distribution with mean fo(21; co) and covariance matrix IfI. This reverse sampling process can be viewed as obtaining the posterior mean estimate 2)(2; 6g) and transforming it to the noise level for step 1. CPS builds on Equation 2.
2.2 Related Work
Sampling from distributions constrained to a feasible set critical various engineering domains, and prior crainingbbssed methods in material science and robotics have addressed this problem.
3

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2.2.1 Training-based Constrained Generation
Most training-based bastn solutions [13-15, 19] constrain the sample domain to specific manifolds. This is typically done in prior works with additional trainable layers in Variational Auto Encoders (VAEs) [19] to enforce linear inequality constraints or through constraint-seeciic training modifications [14, 15], such as clipping the score function for DMs to zero at the constraint boundaries. Recently Mirror DMs [13] proposed a modified training scheme for convex constraint sets, where standard denoising DMs are trained in the dual or the mirror space of the constraint set. Similarly, recent works [14, 15] proposed a modified forward noising process to restrict the intermediate noisy samples to the constraint set. In the time series domain, where constraint-specde samples are generated by conditioning the generator the constraints. These approaches lack generalization to new and arbitrary constraint sets, as given in Example I.
[12] proposed a training-base
approach
2.2.2 Training-free Constrained Generation
To overcome the limitation of training-based approaches, training-free approaches modify the reverse process either through projection or guidance. Projected Diffusion Model (PDM) [11] and PRODIGY [20] project the intermediate noisy latents of the reverse process (ZT: 20) into the constraint set. Constrained Optimization Problem (COP) [12] projects a seed sample to the constraint set while using the critic function from Wasserstein GAN [21] as a realism enforcer. Although training-free, these approaches adversely impact sample quality and diversity, as shown in Section 5.
Guidance-basse approaches [12. 22] push the generated samples toward the constraint set without constraint satisfaction guarantees, even for convex constraints. Similarly, conditional and controlled generation approaches have been extensively studied in the image domain to address tasks such as solving inverse problems [23-26], image personalization [27, 28]. text-to-image generation [29, 30], and text- based image editing [31-34] These methods often involve inversion or proximal gradient updates to guide the reverse process.
Our approach, CPS, belongs to the training-free category, offering improved efficiency and sample quality with provable guarantees. CPS is proximal gradient descent approach in the same spirit as RB- -B-mooulationn [27] but with theoretical guarantees for satisfying linear constraints, such as OHLC constraints in stock price generation. We focus on the time series domain, where constraints are defined as statistical features computed using analytical functions, enabling accurate verification.
Problem Sctup. Consider a dataset = [== where ND denotes the number of samples and x' E Rxx is the realization of X Panta with the density function Pdata- The goal is to generate a typical sample æeen E RRK in the support of pdata such that ge belongs to the constraint set C = C, C2......... CNNe. where Ne denotes the number of constraints. Here, fcz (x) < 0) with fe, K + MRVee E [N....,Nc].
C. =x E kgg
3 Method: Constrained Posterior Sampling
In this section, we describe Algorithm for satis- fying the desired constraints while maximizing the likelihood of the generated samples. The procedure alternates between: (1) maximizing likelihood us- ing a pre-trained diffusion model, and (2) enforcing constraints by projecting the expected time series onto the constraint set. In other words, to generate a typical sample in the support of Pdata. CPS uses a diffusion model that was trained to maximize the like- lihood of observing samples from the given dataset. However, to ensure constraint satisfaction, in CPS, we perturb the denoised sample after every denoising update, using a projection step towards the constraint set. Starting with sample from A(O0, I) (line 1), we perform sequential denoising using the standard DDIM reverse process (lines 2 to 10). Line 3 refers to the forward pass through the denoiser to obtain the noise estimate <z l). After every denoising step, we obtain the posterior mean estimate 2o (2)z))) (line 4) and project it towards the constraint set C to obtain the projected posterior mean
Zt
EØ( Ztz t)
20
N(01
Projection Step
DOZM
20.pr
Sampling
Figure 3: Constrained Posterior Sampling
We show the graphical model for one step of denoising in CPS: refer to Algorithm

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estimate (2rp co) (line 5). Finally, we perform a DDIM reverse sampling step with (21 and c0 t) obtain 2,-1 (lines 7-9).
Figure 3 illustrates one step of the overall pipeline. CPS solves an unconstrained optimization problem in line 5 with the objective function / [[|== co) 12 + (()IT(=)). The first term of the objective ensures that (2;; is close to f0(=;; ea), ensuring that 2 remains close to the domain of the denoising network. We define the constraint violation function II R + I
Nc max fc, (2)), such that I1(2) = 0 if E C and Il(z) > 0 otherwise. For the denoising step i, the constraint violation function scaled by a time-varying penalty y(t).
as I1(z)= = Σ
Our key insight is to design r(t) as a strictly decreasing function of t, taking small values for the initial denoising steps and large values for the final denoising steps. This ensures that the constraint satisfaction is not heavily enforced during the initial steps when the signal-to-noise ratio in ž is very low. We choose (t) = e1((1-á(1) such that ((t) is close to 0 for the initial steps (T(T) RS e) and (()) for i = 1. Here, note that (() strictly decreases with since Og strictly decreases with t.
We do not add noise after the final denoising step (01 = 0), ensuring that the final projection toward constraint satisfaction is not disrupted. For convex constraint sets, the projection step corresponds to an unconstrained minimization of a convex function, with the optimal constraint violation approaching 0 as (11 + cc. With appropriate choice of solvers [35], the optimal solution can be achieved, guaranteeing constraint satisfaction (II p0 (a) = 0) at the end of the reverse process.
CPS meets the key requirements for constrained generation approach. It han- dles multiple constraints without addi- tional training or crities to enforce real- ism, as successive denoising steps nat- urally correct projection artifacts. CPS introduces no extra hyperparameters, off-the-shelf solvers [36] handle the un- constrained optimization step (line 5 in Algorithm 1). As tuning guidance weights is challenging [12], the tuning- free nature of CPS provides significant advantage for real-world applications.
Algorithm 1 Constrained Posterior Sampling
Input: Diffusion model co with T denoising steps, Noise coefficients {ōo, ... ar }. DDIM control parame- ters (ol....,0yy Constraint violation function II, Penalty coefficients 1: Initialize 2T ~ WV(0 I).
2: for f. to 10010 do
3: Obtain é co(2t, t)
4: 60(=1;00) = vT TTaff
vaa
1 [llz 20(221
5:
3 (z (0) = arg min;
2
+7())T((2)
6:
CPS is closely related to other training- free constrained generation approaches
7:
21-1 = vā. 80 + /I 0t- - o7e
8:
e~ (0,10,1
PDM [1I] and PRODIGY [20]. Unlike
9:
21-1 = 21-1 + 0sC
CPS which projects G0) these
10: end for
approaches project the intermediate la-
11: -0 = 2g
tents (2y, (zy...., =0) onto the constraint set.
12: return 2:
While ensuring constraint satisfaction for convex sets, these approaches provide poor sample quality and diversity in comparison to CPS for the following reasons:
The constraint set is defined for the clean samples. However, PDM projects noisy intermediate latents towards the constraint set. This approach eliminates most sample paths (2T,.... z0) where zu alone eventually satisfies the constraints, thereby affecting sample diversity. CPS eliminates this problem by projecting {2(2 ca), which has a similar noise level as the constraint set.
Projecting 2 pushes it off the noise manifold for the diffusion step pre-trained denoiser struggles to accurately denoise the projected - as it would be out of the training domain of the denoiser. This effect is significantly reduced in CPS because our approach generates 1 by adding an appropriate amount of noise to the projected clean sample.
1. Consequently,
To address these issues, PRODIGY adds scaled version of the projected noisy latent to 2 such that the projected latent can be given more preference at the final denoising steps. However, due to the direct modification of 24-1, PRODIGY still provides lower sample quality and diversity when compared against CPS, as shown in Section 5.
The original DDIM update step results in sampling from an unconditional distribution, whereas updating with Eo <0) can be viewed as sampling conditioned on the specified constraints. [27] has employed similar algorithms for image personalization. Note that CPS breaks consistency 2
5

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is no longer equal to 20 (2) + VT-āáce(zza t) due to the projection step. To understand the effect of this inconsistency, we implemented variant of CPS called CPS with noise correction, where we update the noise estimate to ensure consistency. The adverse effects of the consistency update are shown through poor sample quality metrics in Table 8 (check Appendix B.7). We hypothesize that updating both fo(2;; ca) through projection, and é for consistency, can push 2 off the noise manifold for the step - 1, resulting in poor denoising, and thereby affecting the sample quality.
4 Theoretical Justification
Now, we provide detailed analysis of the effect of modifying the DDIM sampling process with CPS. For ease of explanation, we consider = E R", Let I, denote the identity matrix in R=x*, First, we describe the distribution from which the samples are generated under the following assumption.
Assumption 4.1. Let the constraint set be C { fc(2) = 0}, where fc R + IR and the penalty function [2) [lfe(=)]lB has L-Lisschitz continuous gradients, i.e., V VIl())ll < L||u - u V u, E IR..
Assumption 4.1 follows prior works [37, 38] on posterior sampling for inverse problems in the image domain. [37] uses linear inverse problems without noise to obtain the reverse sampling step, and replaces the pscudoinverse of the mcasurement operator with non-linear and non-differentiable operations for reconstructing images compressed using JPEG encoding. [38] provides theoretical guarantees for the proposed sampling algorithms using a linear setting and extends to complex image editing tasks.
Theorem 4.2. Suppose Assumption 4.1 holds. Given a denoiser eq :R" + R" for diffusion
process with noise coefficients I is equivalent to sampling from the following conditional distribution:
.rr, if 7tt) > 0 t E [1, the denoising step in Algorithm
Po,1(z111 21) = pa iniinit (zo 20 (21;
if otherwise.
(2-1 zt, pe (000pp(2t (2: E0))
I? is represented by (o (21: and the PDF of N (2r--220prr2 + /1 - â oe (ze, 1), T?I..) by 20 Zz, E0.pr (2); 01....,00 oT the DDIM control parameters, and (t) )) is the penalty coefficient for the step in Algorithm 1.
Here, the PDF N
Implications. Intuitively, Algorithm can be viewed as replacing (=(;;; 60) with or 60) in the DDIM sampler. Therefore, the marginal PDFs of the reverse process are obtained by replacing 62)))))) with pr (zy; Eg) in Equation 3. Under the Assumption 4.1, the projection step (line 5) becomes sequence of gradient updates transforming Eo(ze; Ea) to 60.pr EQ). Having Lipschitz continuous gradients for II allows for fixed step sizes, which guarantees a reduction in the value of the objective function 3 {I|= 2o(2;; c0) [ + (t) with each gradient update. We refer the readers to Appendix C.1 for the detailed proof.
In general, the theorem draws equivalence between the projection step and sampling from Dirac delta distribution centered at (0.; (0). Additionally, Theorem 4.2 can be extended to non-smooth penalty functions in Algorithm because the projection operation (line 5) can be written series of updates from (0(2;; Eg) to (z,p Therefore, Theorem 4.2 still provides the updated p=.1(2 2x) for Algorithm 1. Now, we investigate the convergence properties for Algorithm 1 under the following assumption.
Assumption 4.3. Let the data distribution be (N,, In), where μa E R", and the constraint set be defined as C {2 E I Az y} with A mmxn such that rank exists a unique solution x² E IR" that satisfies the desired constraints in C.
S m. Suppose there
Assumption 4.3 ensures the existence a unique solution to the linear problem Ar = y. While there exist many methods to solve such problems under this assumption, our focus, as in [24, 39, 40], is to use this problem as a framework to analyze the convergence properties of Algorithm 1, providing valuable insights for better practical performance. We note that similar assumptions (unique solution) are made in the theoretical analysis of algorithms designed for sample recovery and image inpainting [39, 40] using diffusion models.

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Traffic
Air Quality Conditional
Traffic
Conditional
Air Quality
Stocks
waveforms
Guided
DiffTine
Value
M
-0.10
Timma Stamps
Time Stamps
de
Time Stamps
Stamps
20 40 69 D9 8o Time Stareps
Vahne L.
M
WM
COP-FI
Vilue
velue
CPS (Ours)
Time stamps
40 60 8e Time
20 40
60 80
-10
1.0
20 60 60 DO
20 40 60 80
M
m 60
Ville
sal Valive
20 40
D0
60
60 B0
20
40 60
62
60 20
60
80
Generated Samples
Real Samples
Figure 4: CPS provides high-fidelity synthetic time series samples that match real time series data. The real test samples from which the constraints are extracted are shown in blue. The samples generated using the extracted constraints are shown in red. Across all datasets, the baselines suffer from the adversarial effects of the projection step, whereas CPS generates high-quality samples. Here, we provide the comparison against baselines [12] designed for constrained time series generation. We refer the readers to Appendix E, where we included similar qualitative comparisons against other baselines (Diffusion-T) -S and PDM) that were adapted to perform constrained generation.
Theorem 4.4. Suppose Assumption 4.3 holds. Let the noise coefficients of a diffusion process be given by (00,..,,rr ÔT, where ön oT 0 and or [0. A E [1, If âz < â and (t) = AAiu(A"A)" for any k > 1, deterministic sampling with Algorithu 1 converges at the rate (1/TT). Furthermore, there exists a constant k > max (1, vai (1 (1<*1l)+11eell) such that Ig x2112 as T cc. Here, is the tolerance limit.
2k(T-1+1)
Implications. Interestingly, for very large values of T, with a suitable choice of k. CPS converges without the expensive constrained optimization steps for all denoising steps as required by prior works [11, 20]. Following this insight and the proof in Appendix C.2, we choose the number of denoising steps and the final penalty coefficient to be large enough to ensure convergence in practice. Indeed, is sufficient to partially enforce the constraints with relatively cheap unconstrained optimization routines for most denoising steps, except for the final step. Intuitively, enforcing constraints partially on the posterior mean estimate, &o (x(2; <e), noising it back, and repeating this process for a large number of denoising steps removes the adverse effects of projection while gradually nudging the sample generation towards the constraint set.
Overall, Theorem 4.4 provides a penalty schedule that ensures convergence for Algorithm 1. Observe that the penalty coefficient linearly increases to a very large value as we denoise. However, since not all real-world constraints satisfy Assumption 4.3, we designed the penalty coefficients in Algorithm 1 to exponentially increase to a very large value as we denoise, such that they are similar in essence to the penalty coefficients used in Theorem 4.4. We conduct extensive experiments in Section 5, verifying the significance of these insights. Specifically, we experimented with linearly and quadratically increasing penalties in Algorithm 1, along with exponentially increasing penalty coefficients (check Table 5). Note that the linearly increasing penalty coefficients are proportional to the penalty coefficients obtained from Theorem 4.4 at all diffusion steps.
Experiments
In this section, we describe the experiments, datasets, baselines, and metrics used to evaluate CPS. Datasets:
We use real-world datasets from the Stocks [7], Air Quality [41], and Traffic [42] domains. We evaluate CPS on both conditional and unconditional variants of these datasets. The conditional variant refers to imposing constraints on conditional generative models that take contextual metadata as input, as mentioned in [1], to generate time series samples. Consequently, in Algorithm 1, we use <o(2e, 1, m) instead of c (2t, 1) for the conditional setting, where m is the metadata input. We also evaluate CPS on simulated sinusoidal waveforms with specified amplitudes, phases, and frequencies.
7

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METRIC
APPROACH
QUALITY
AIR QUALITY (CONDITIONAL)
TRAFFIC (CONDITIONAL)
STOCKS
WAYRFORMS
GUIDED DIFTTMM
0,7457
3,1883
0.5351
0,5638
1.2575
0.3108
DIFFUSION-TS
0.0473
2,9771
L.1268
PRECHET
COP-FT
0.3821
0,9919
0,8239
0,7836
0.0727
1.8099
DISTANCE
COP
0.2206
28.1572
0,8566
43. 13.1499
0.0711
1. 1.6653
(4)
CPS (OURS)
0,0234
0.6033
0.2077
0.2812
0,0623
0.0029
GUIDED TIFETIME
0.29:10.115
0.25 003
0,30::0.01
0.01
0.05100001
0.005 001
DIFFUSION-S
0.19:000004
0.19::0.005
0.31=0001
0.28 0.01
0.046:20.00
0,005 001
TSTR
COP-PT
0.2310.005
0.19:10,000 005
0.32
0.28
0.049
0.024
(4)
сОP
0.25-00.03
0,25++0.00 003
0.33=00.02
0.32=0.01
0.048=000002
0.024=0.00
CPS (OURS)
0.19:00.003
0.19::0.003
0.29=0.01
0.00
0.041=0.001
0.005::00001
GUIDED TIME
0.33+:0.2
0.22
0.29
0.03=00.02
0.38+:0.01
0.43
DIFFUSION-TS
0.03
0.11200.55
0.02
0.21110.3
0.001=0,001
DS
COP-PT
0.34:0003
0.03++0.00 01
0,35=0.09
0.01=0.01
0.13++0.04
0.44:0002
(4)
COP
0.29::002
O. 1.300+0000 03
0,41=00.66
0.43=
D. 09:9::004
0.42:00.05
CPS (OURS)
0.06:20.01
10.005
0.02-0.01
0.01:0.004
0.006=00008
0.002=0.00
GUIEED TIMETIME
6.74+#8.1 18
4, 4.28++5. 66
1.38=1.25
1.31=11001
7. 7.84+7.22 24
1.67+1.15
DIFFUSION-SS
2.53:11.9
2.30 91
3.82=1157 1.57
0.96==0.52
7.44:66.65
0.16::0.28
DTW
COP-FT
3.52:42.0K
2:01=111
4.61=1.000
1,26- KM
1.41
1.19:0.6
(1)
COP
4.01141711
4.13
5.17
1,942:108
0.88:1 141
1.16 20065
CPS (OURS)
2.35::1.48
1.16:11.16
3.41=1.47
0.84:2062
0.20:10.71
0.23:20.17
GUIDED DIEETIME
23.21
16.35
1128.22
CONSTRAINT
DImUSIoN-TS
5.613
3.92
0,9743
01.45
40.51
0.36
VIOLATION
COF-FT
0.0
0.0
0.0
0.0
0.0
0.0002
MAGNITUDE
COP
0.0
0.0
0.0001
0.0
0.0003
(1)
(OURS)
0.0
Table 1: CPS outperforms existing baselines on sample quality and similarity metrics. The best approach is shown in bold for each metric. Though the guidance-based approaches (Guided DiffTime and Diffusion- -SS) provide comparable sample quality metrics, they struggle to provide constraint satisfaction (very high constraint violation magnitudes). Overall, CPS maintains high sample quality (very low FD, TSTR, and DS values) and the highest similarity with real time series samples (lowest DTW). Our key intuition is that the adverse effects of projection are nullified by the subsequent denoising steps. As all constraints are linear, the COP variants and CPS can achieve very low values for constraint violation. The TSTR and DS results are obtained for 3 and seeds, respectively.
We compare against two post-procse post-processing methods COP [12], which projects a random training sample to the constraint set, and its fine-tuning variant, COP-FT, that projects a generated sample. We also compare against guidance-based methods Guided DifTTime [12] and Diffusion-TS [22]. Additionally, we compare against constrained generation approaches from other domains Projected Diffusion Models (PDM) [11] and PRODIGY [20]. We use the TIME WEAVER-CSDL denoiser [1] for all baselines to make fair comparison.
Baselines:
Evaluation Procedure:
Our evaluation procedure captures the difference between the real test dataset and the generated dataset with constrained samples, both on distributional and per-sample levels. To achieve this, first, from every real sample in the test dataset, we extract a diverse set of features for constraints mean, mean consecutive change, argmax, argmin, value at argmax, value at argmin, values at timestamps 1, 24, 48, 72, & 96. We then generate sample for each set of these features using constrained generation approaches. For an ideal constrained generation approach, these two steps combined should be equivalent to sampling from the test data distribution. Additionally, note that the features are extracted from the test samples and, therefore, our evaluation procedure focuses on constraint sets that were never seen during training. Apart from these constraints, we additionally impose the OHLC constraint for the stocks dataset. For waveforms, we extract the peaks, valleys, and the trend from a peak to its adjacent valley. All constraints can be written in the form Ax S O, and projection to such constraint sets is easy. The tolerance limit 5 is set to 0.01.
To compare the samples generated with constraints the real test samples on distribution level, we use the Frechet Time Series Distance (FTSD) metric [1, 43] for the unconditional setting (also referred as Context-FID [43]) and the Joint Frechet Time Series Distance (J-FTSD) metric [1] for the conditional setting. We indicate both these metrics by the Frechet Distance (FD). We also report the Train on Synthetic and Test on Real (TSTR) metric for random imputation with 75% masking and the Discriminative Score (DS) [7] for sample quality and diversity. We also compute sample-wise comparison metrics, like the Dynamic Time Warping (DTW) [44] metric between a real sample and the sample generated with corresponding constraints. This metric is useful when the number of constraints large, where any desired constrained generation approach should generate sample that is similar to the test sample from which the constraints extracted. For constraint
Metrics:
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QUALITY
TRAPFIC
STOCKS
CONDITIONAL
CONDITIONAL
PRECHEY DISTANCE (1)
PRODIGY
PDM
B.3646
0,1503
0.2714
0.2771
0.0361
0,3506
B.6259
0.5811
CPS (OURS)
0.0234
0.2077
0.0023
0.6039
0.2812
PDM
0.20510.005
0.2910.008
0.06410.001 001
0.20.10.006
0.28:00.1
TSTR (1)
PRODIGY
0.20520.007
0.29:000005
0.064320.002
0.19::0.04
0.28::0.01
CPS (QURS)
0.19:00003
0.29:0.001
0.041+0.001
0.19:00003
0.28--0.01
PDM
2.54611.96
3.347:1.34
0.44741.06
1992182
1,0420077
DTW
PRODIGY
2.537+1887
3.596=1..7
0.516+1.19
1.8-1.17
1.07+00:22
(OURS)
2.35
0.2.0.71
18311,16
0.8410.62
Table 2: CPS outperforms constrained generation approaches from other domains on sample quality and similarity metrics. CPS outperforms PDM and PRODIGY, which project the noisy intermediate latents, on real-worll datasets, highlighting the effectiveness of projecting the posterior mean estimate. Note that both PDM and PRODIGY provide perfect constraint satisfaction, similar to CPS, for the constraints considered in Section
violation, we report the average constraint violation magnitude. A detailed discussion on the baselines and metrics is provided in Appendix H.3 and Appendix F, respectively. Tables and 2 contain comparisons of these metrics between CPS and other baselines on multiple datasets. As such, we make the following observations about the generated sample quality.
Post-processing through projection is influenced by the choice of the initial seed or the generated sample. For conditional settings, in COP-FT, conditional generation provides samples closer to the constraint set for post-procsssing (projection). whereas COP uses a randomly picked training sample. Therefore, the sample quality degradation due to the projection step is low in COP-FT, when compared to COP, resulting in higher FD values (check Table 1). CPS eliminates these effects through iterative projection and denoising, where the adverse effects of the projection step nullified by the subsequent denoising steps (check Figure 4). Notably, in the Stocks dataset, CPS provides 30x and 15x reduction in the FD and DS values, respectively (Table 1), over the COP variants.
Projecting the noisy intermediate latents has adversarial effects on sample quality, As explained in Section 3, projecting the noisy intermediate latents (PDM and PRODIGY) pushes them off the noise manifold, resulting in inaccurate denoising. This affects the generated sample quality as shown through higher FD and TSTR values in Table 2. CPS circumvents this issue by projecting the posterior mean estimate and adding an appropriate amount of noise to the projected posterior mean estimate to obtain the noisy latents (Figure 3). This approach provides noisy latents that closer to the training domain of the denoiser, eventually resulting in high sample quality, leading to 6x and 10x reduction in the FD values for the Air Quality and Stocks datasets, respectively (Table 2).
Optimizing the guidance weights seems practically hard for a large number of constraints. From Table 1, observe the high constraint violation magnitudes for the guidance-based approaches such as Guided DiffTime and Diffusion-TS. We attribute this to the interaction between gradients for each constraint violation, which, if not scaled appropriately, leads to poor guidance. CPS avoids these issues by utilizing off-the-shelf solvers to handle the projection step, and the adverse effects of projection are nullified by the subsequent denoising steps. This results in 50x and 10% reduction in FD and TSTR, respectively, over guidance-based methods for the Stocks dataset.
CPS effectively tracks the real test samples that adhere to the same set of constraints. We denote tracking real test samples as the property to have better similarity scores (lower DTW) with the real sample as the number of constraints increases. In Figure 5, we observe that CPS has the best reduction in the DTW scores as the number of constraints increases. Simultaneously, the sample quality is unaffected or even improves for CPS with increasing constraints (lower FD scores). CPS's performance is consistent across multiple real-world datasets, with up to 55% reduction in the DTW scores for the Stocks dataset.
CPS can be effectively extended to a general class of constraints. Specifically, we experimented with the Autocorrelation Function (ACF) at specific lag as an equality constraint along with the OHLC constraint for the stocks dataset. We provide the results of this experiment in Table 4 in Appendix B.3. Note that out of all approaches, CPS provides the least constraint violation magnitude. Additionally, even though the projection step (line 5, Algorithm 1) does not lead to the optimal solution (as the autocorrelation function is non-convex in the sample domain), CPS's sample quality is much better than that of the baselines. This is due to the iterated projection and denoising operations, which significantly reduce the adverse effects of the projection step.

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Air Quality
Stocks
Traffic
10
4.5
3.5
6
10
10
Number of Constraint Catagories
Number Constraint Categories
Number of Constraint Categories
2.0
0.4
0.0
Number of Constraint Categories
Number of Constraint Categories
Number of Constraint Categories
CPS (Ours)
PDM
PRODIGY
COP-FT
Figure 5: CPS outperforms baselines on sample quality for an increasing number of constraints. We gradually increase the number of constraints imposed on the generative model to evaluate CPS and other baselines that guarantee constraint satisfaction for linear constraints. CPS (green) achieves the lowest DTW score for any number of constraints while having the best sample quality, indicated by the lowest FTSD metric. This result is coherent with the qualitative example shown in Figure 6.
Additional Experiments. In Appendix B.2, we show that CPS performs as good as training- based approaches, such Loss DiffTime, sample quality metrics (Table 3) while ensuring constraint satisfaction. Appendix B.4 (Table 5) provides an ablation study on the choice of the functional form of the penalty coefficients. We also performed a systematic evaluation of the effect of individual constraint categories (Table 6 in Appendix B.5). We note that CPS outperforms the baseline approaches for the majority of these categories. Appendix B.6 contains our detailed experimentation highlighting the effect of guidance weights on the constraint violation magnitude for the guidance- basan approaches. Appendix B.8, we show CPS's ability to outperform other constrained generation approaches in performing time series imputation. Appendix E contains additional qualitative comparisons (Figures and 10).
Limitations. Although CPS (Algorithm 1) outperforms the compared baselines on standard eval- uation metrics, the projection step (line 5) may increase sampling time in some applications, with the trade-off being superior performance. In time-critical scenarios, sampling time can be reduced by leveraging higher-order moments and alternative initialization schemes [25]. Additionally, the projection step does not need to be applied after every denoising step and can be adapted based on the magnitude of constraint violation. Appendix details the inference time comparisons along with the time complexity analysis for Algorithm under Assumptions 4.1 and 4.3, as well as additional strategies for faster inference. Finally, we note that the convergence results obtained in Theorem 4.4 hold true under Assumption 4.3, which includes restrictions on the data distribution to be Gaussian and the presence of a unique solution. However, as explained Section 4, these results provide valuable insights for the design of the penalty coefficients used in Algorithm 1.
6 Conclusion
We proposed Constrained Posterior Sampling (CPS) a novel training-free free approach for constrained time series generation. CPS utilizes off-the-shelf optimization routines perform projection step towards the constraint set after every denoising update. In a linear model setting, we provide convergence guarantees for CPS under mild assumptions. Empirically, show that CPS outperforms the current state-of-the-arr in generating realistic samples with superior constraint satisfaction.
Future work. We aim to apply CPS for constrained trajectory generation in the robotics domain with dynamic constraints typically modeled by neural networks. Additionally, constrained time series generation readily applies to style transfer applications. Hence, we plan extending the current work to perform style transfer from one time series to another by perturbing statistical features.
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7 Acknowledgements
Sai Shankar Narasimhan, Shubhankar Agarwal, and Sandeep Chinchali are part of the Swarm Lab the University of Texas at Austin.
Litu Rout and Sanjay Shakkottai are supported by NSF Grants 2019844, 2112471, and the UT Austin Machine Learning Lab. Additional Computing support on the Vista GPU Cluster was provided through the Center for Generative Al (CGAI) and the Texas Advanced Computing Center (TACC) UT Austin.
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Appendix
A Extended Related Works
A.1 Diffusion Models for Time Series Generation
Time Series-specific tasks like forecasting [45-49] and imputation [5, 22, 50-52] have been addressed using conditional DMs as well as guidance-based approaches [22, 49]. Specifically, [49, 51] propose novel sampling techniques for improved time series prediction for forecasting and imputation. [6] and []] have explored conditional time series generation for various domains, such as medical, energy, etc. These works aim sample from conditional distribution. Additionally, applications in analogous domains, such as audio [53] and radio frequency waveform generation [54], have also widely adopted diffusion generative modeling, with similar denoiser architectures as seen in the time series domain. However, there are limited prior works in the time series domain [12] that focus on generating constrained samples, which is the focus of this work.
B Additional Results
In this section, we provide additional qualitative and quantitative results for the real-world datasets used in our experiments.
B.1 Effects of increasing the number of constraints
Here, we provide qualitative example from the Stocks dataset showing CPS's ability to track the real time series sample as the number of constraints is gradually increased.
No
Add nean change, Add argnax, Day
value,
Add argnin,
min value, Day 96 Price
Constreints
Add mean
Day Price
Day 45 Price
Day Price
Guided DiffTime
Dwn
COP
COurs
CPS
Days
Genersted Sanple
Real Sample
Figure 6: CPS tracks the real data samples as the number of constraints increases. Increasing the number of constraints reduces the size of the constraint set, and an ideal approach should effectively generate samples that resemble the real time series samples that belong the constraint Here, we show qualitative example from the Stocks dataset. Observe that CPS accurately tracks the real sample that concurs with the specified constraints, while other approaches suffer.
B.2 Extended Baseline Comparisons
In this section, we provide comparisons against the Loss-DiffTime baseline from [12]. For fair comparison, we use the same TIME WEAVER CCDD backbone and train the denoiser with constraints as the condition input. The quantitative comparisons are provided in Table 3. As observed with prior approaches, in the absence of any principled projection step, the Loss-DiffTimg approach fails to generate samples that adhere hard constraints. However, due to the snpstraint-specifcc training, Loss-DiffTime performs as good as CPS in terms of sample quality and similarity.
20

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CONSERAINT
DATASET
APPROACH
FTSD (I)
TSTR ())
DS (
DTW (U)
VIOLATION (J) MAGNITUDE
AIR
Loss DIS'TISE
0.0137
0.187=0.003
0.03=0.01
2.18=1.48
9.779
QUALITY
(OURS)
0.0234
0.19:0.003
0.06=0.01
2.35=1.48
0.0
Loss DIFFTIME
0.9897
0.045=0.002
0.379=0.015
7.75=6.05
237.492
STOCKS
CPS (OURS)
0.0023
0.041
0.006:0.004
0.20=0.71
0.0
TRAMAC
CPS (OURS)
Loss
0.2077
0.29:=0.01
0.29-10000
0.11340.039
0.02-0.00 0.01
3.15-1.34
3.41 1.47
0.0
2.993
Table Despite constraint-specific training, Loss-DiffTime struggles to generate samples that adhere to the required constraint set. Note that Loss-DiffTine performs better than CPS on the sample quality and similarity metries for the air quality dataset. However, due to the absence of projection steps, Loss-Diff Time fails to generate samples that adhere to hard constraints.
B.3 General Constraints Experiments
We extended our experimental setup to generic constraints for the stocks dataset. Specifically, we imposed the Autocorrelation Function (ACF) at specific lag as an equality constraint with an acceptable tolerance of 0.01. ACF at specific lag for a univariate time series X of horizon L is given by,
ACF(X) (L- 1)02 Σ(x, ( 14)
(4)
where μ = (X) and 0² E[(X {1)3]. with E being the expectation operator. Here, X,, and Xu+t denote the time series values at the timestamps u and i + I, respectively. Note that and are not fixed. Along with the ACF equality constraint, we pose the OHLC constraint for the stocks dataset. We provide the results of this experiment in Table 4. We chose ACF as it is one of the most popularly used techniques to extract the most relevant lag features for downstream tasks like forecasting.
CONSTRAINT
APPROACH
FTSD
DTW
VIOLATION (L) MAGNITUDE
DIMED-
1.4678
15. 15.06111100 .22
284.58
COP
2.1949
72. 137.997
0.9045
CPS (OURS)
0.0014
0.1110.10
0.01
Table CPS outperforms baselines for OHLC and autocorrelation function value constraints. Here, we use the stocks dataset and impose the Autocorrelation Function (ACF) value for specified lag of 12 timestamps as constraint along with the OHLC constraint. CPS outperforms all the baselines in terms of sample quality, similarity, and constraint satisfaction metrics.
Note that out of all approaches, CPS provides the least constraint violation magnitude. Additionally, even though the projection step (line 5, Algorithm 1) does not lead to the optimal solution (as the autocorrelation function is non-convex the sample domain), CPS's sample quality is much better than that of the baselines. We hypothesize that this effect is due to the iterated projection and denoising operations, which significantly reduce adverse effects of the projection step.
B.4 Choice of Penalty Coefficients
Our choice of yt() can take any functional form as long as 7t)) + 2o as + 1. This is to ensure constraint satisfaction for linear and convex constraint sets. In practice, we clip (t) to a very large value, such as 105, when performing the final denoising steps. Our current choice of y(t) decreases exponentially with (in other words, 7()) increases as we denoise). We also experimented with linearly and quadratically decreasing values of (L). with a very high value (10²) for / = 1. We noted that the choice of 7()) has very little effect on the sample quality of the generated samples, with differences in the third decimal (check Table 5).
21

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CHOICE OF y())
AIR QUALITY
TRAFFIC
STOCKS
LINEAR
0.0222
0.2053
0.0013
QUADRATIC
0.0226
0.2027
0.0016
EXPONENTIAL
0.0234
0.2077
0.0023
Table 5: Different choices of (()) provide similar sample quality metries. Here, we report the FTSD score as the sample quality metric. Note that the effect of different choices of 7(t) is only reflected the third decimal and is insignificant.
B.5 Systematic Evaluation Of Different Constraint Categories
We provide the FTSD (sample quality) comparison for individual constraint categories evaluated on the traffic dataset (check Table 6). Note that CPS outperforms the baseline approaches for most of the individual constraint categories.
VALUE AT
VALUE
MEAN CHANGE
VALUE AT TIMESTAMPS
APPROACH
ARGMAX
ARGMIN
MEAN
1,24,48,72.96
COUDED-DuHTiIN
0.21
0.22
0.21
0.30
0.22
0.22
0.24
COР-FT
0.27
0.25
0.20
0.26
0.28
0.19
0.23
PDM
0.21
0.20
0.19
0.19
0.20
0.20
CPS (OURS)
0,20
0.21
0.19
0.21
0.18
0.20
0.19
Table 6: CPS predominantly outperforms other baselines on sample quality metrics while being
evaluated on individual constraints. The table provides a constraint-specific evaluation of sample quality (FTSD, lower is better). Here, experimented with the traffic dataset. Note that for the majority of the constraints, CPS outperforms existing baselines.
B.6 Effect Of The Guidance Weight On The Constraint Violation Magnitude
We analyze the effect the guidance weight on the constraint violation magnitude for the experimented with the following guidance weights:
Air Quality dataset. Specifically, 0,00001, 0.00005, 0.0001, 0.0005, 0.001, 0.005, 0.01, and 0.05. We refer the readers to Table 7 for the results. Overall, observe that the constraint violation magnitude either stays the same (DiffusionTS) or increases (Guided DiffTime) with increasing guidance weight. In our experiments, we choose the guidance weight corresponding to the smallest constraint violation magnitude.
APFROACE
GUIRANCE
GUIDANCE
GUIDANCE
GUIDANCE
OUIDANCE
GUIDANCE
GUIDANCE
GUIDANEE
GuInoo Dirr TIMe
33.31
23.04
25.37
51.27
113.31
951,62
2529
110084
Dheessnn-15
5,11
3,27
3
Table Effect of the guidance weight on the constraint violation magnitude for the guidance- based approaches. Notice that for both Guided DiffTime and Diffusion-TT TS, increasing the guidance weight does not result in the reduction of the constraint violation magnitude. The experiments were conducted for the Air Quality dataset.
B.7 Discussion On The Update Step Consistency
To validate update step empirically, implemented variant of CPS where update the noise estimate based on ž0.pr(2s, <o) (check Section 3). This updated noise estimate is used in line 7 of Algorithm 1. We refer this CPS with noise correction. The sample quality results, in the presence and absence of noise correction, are provided in Table 8. Note that updating for consistency significantly affects the sample quality. We hypothesize that updating both & (2x;; Eq) and can push - off the noise manifold for the step 1, resulting in poor denoising, and thereby affecting the sample quality.
B.8 Extending Constrained Posterior Sampling To Time Series Imputation
Here, we extend CPS to the time series imputation task by imposing the value constraint to all the timestamps where the true values are available. We compare CPS against other of-h-atf
22

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METRICS
APPROACH
AIR QUALITY
TRAFFIC
STOCKS
WAVEFORMS
FTSD
CPS
0.0234
0.2077
0.0023
0,0029
CPS WITH NOISE
0.2085
0.5147
0.0426
0,0694
CORRECTION
DTW
CPS
2.35 + 1.48
1.83
= 1.16
0.20
0.71
0.23 0.17
CPS WITH NOISE
3.07 + 1.86
4.05 + 1.26
0.54 # 1.10
0.46 + 0.33
CORRECTION
Table 8: CPS provides better sample quality metries in the absence of the noise correction step. Updating both posterior mean and noise (CPS with noise correction) results in overdependency on the projection step and pushes 2t-1 off the noise manifold for the denoising step t 1. This takes 2t-1 far away from the training domain of the denoiser, resulting in poor sample quality as shown by the numbers above.
constrained generation methods- PDM [11] and PRODIGY [20]. The results are provided in Figure 7. Note that CPS outperforms the baselines on all real-world datasets. This result further cements our claim that projecting the posterior mean estimate, instead the intermediate latents as in the case of PDM and PRODIGY, is better for constrained sample generation.
Traffic
Air Quality
Stocks
0.30
0.10
0.25
0.08
MAE
MAE
MAE
0.06
0.3
0.04
0.2
0.05
0.1
0.02
0.00
0.00
CPS
PRODIGY
CPS
PDM PRODIGY
CPS
PRODIGY
Figure 7: CPS outperforms state-of-the-art approaches on imputation tasks. Here, we showcase CPS's ability to perform imputation. We set the masking rate at 50%, i.e., 50% of the values masked. Observe that all three real-world datasets, CPS comfortably outperforms PDM [11] and PRODIGY [20] (lowest Mean Absolute Error (MAE)). Note that the main difference between these approaches and CPS is that CPS projects the posterior mean estimate, whereas these approaches project the intermediate noisy latents.
C Proofs
In this section, we provide detailed proof for the theorems stated in the manuscript.
C.1 Proof of Theorem
We first describe the assumption on the constraint set. The constraint set defined C {a fc(z) 0}, where fc Rn + R, and the penalty function II(=) = =[]f(=>] has L-Lipschitz continuous gradients, i.e. |[VI(()) VII 22 S Ll[u 2|12 V u, R
Line 7 the Algorithm modifies the traditional DDIM sampling by replacing c0(2);))) with (2t; Eg). Without this modification, the DDIM sampling denotes the following reverse process when started with Ir ~ N (O,,, I,), where O,, indicates the zero mean vector R" and I, is the identity matrix in Rox.
(= 2x) = Po (2o 20(21160))
ift=1, otherwise,
(5)
(o,t (2 2t, 20(22;))))
23

---

2e, 20(22; represents the PDF of the Gaussian distribution N/ (VaL-120(21;e0) + v1 â,-1 - c0 t), ofIn) with as the DDIM control parameter.
where 20
Similarly, Po.init S0(21:60) is the PDF of the Gaussian distribution with mean (a(21; co) and covariance matrix I [18].
Note that sampling from (2 2,, 20(2); provides the DDIM sampling step (check Equation 2).
We reiterate that the main modification with respect to the DDIM sampling approach is the projection step in line 5 of Algorithm 1. Therefore, we first analyze the projection step.
1 (12 Eo(2; M1 + (()]lff(=))1))
pr (ze: co) = arg min
2
Here, c0(2);00) = 21 (1) (line 3, predicted z0). We will denote the objective function (II= - Eo(z); co) + (+)] as 9(z). Note that we replaced the constraint violation func- tion [I(=) by [=]]] for this case. Given that [ has L-Lipschitz continuous gradients, Equation 6 can be written as a series of gradient updates with suitable step size such that the value of the objective function decreases for each gradient update.
vor
From the statement, observe that 7t) > OAEEE & [1, Under this condition and Assumption
4.1, note that the function g(=) is convex and has fo(=;; <0)1 has 2-I continuous gradients, (()) has (r(()).)-Lipschitz continuous gradients, and the fraction 1/2 makes g(2) to have
(2 Lipschitz coniinuuus gradients, as ||=
(2tr2 -Lipschitz continuous gradients. Let be the step size of the projection step. From [36], we know that E 2/ + L)) ensures that the objective function in Equation 6 reduces after each gradient update. We denote the gradient update
as:
" "Zo(zi;c)) = n-1 n-1 20(1:: co) V/ ([V=((()))
where O (0) 20(2;; co) and (o); No 20 co). Here, Npr is the total number of gradient update steps. The iteration in Equation always leads to p0 cg) deterministically. Therefore, the projection step can be considered sampling from a Dirac delta distribution centered at 20.pr (ztj i.e., 8(z Er. (zri to)) Consequently, using the law of total probability, the reverse process corresponding to the denoising step 1 A E [2, T] in Algorithm is given by
/
(0,c | z1)
(0 $o 2 20)dã
where &g e R". This can be simplified using Bayes' rule as
(0, 24) = f 8(20 20. (245 c0 (o, 1 2t, 20)
The above equation stems from the fact that the distribution of za conditioned on 2, is Dirac delta distribution centered at 60 EØ). Since 8(x - y) = ō(y - 2) and using the sifting property of Dirac delta function (z)o(( - =)d2= = sa)). . we get
(z- 21)= = ( 1 21, (2) E [2,T].
(8)
Similarly, we repeat the steps for t = 1,
(0 z1)= = f
pe, (2. &0 22))22..
po, (zo 21)= fo( 6(50 2o. (o)) En)dão,
po,1 (20 21) = P0, E0 (21
Combining the two, we get
(0 24) = / pe (nit pr (21;
if
(9)
(..e((1 (21-1 24, 20 (2): (2);c))
otherwise,
24

---

where 90,0(21-1
2e, (x: represents the PDF of the Gaussian distribution
(2; (0) (g) + /1 0,-1 cz 1), (), II,)) with or as the DDIM control parame-
ter. Similarly, Po,init (oo (0)) is the PDF of the Gaussian distribution with mean 20 (21; and covariance matrix I [18].
We note that the value of σ, is set to 0 in Algorithm 1. However, similar to [18]. for theoretical analysis, we consider a negligible value for ai (~ 10-12) ensure that the generative process is supported everywhere. In other words, d1 is chosen to be so low such that for 01 = 0, Po,init (2o
(21: 8(20- Er (21:
Now, we show that the exact DDIM reverse process (check Equation 5) can be obtained from Equation in the case where there are no constraints. Here, note that in the absence of any constraint, the projection step can be written as Eu.pr (2) = arg min, 211= Zo(z t0) 11., in which case 20.pr cB) E0(2); Eq).
Fort t E [2, using the law of total probability, we get
po, 21) / 5(20- 2u(2):c))) 21, 20)d
(10)
which simplifies further to
(0, 21) go (0,1 2x, 20 Eq)).
(1I)
The above equation stems from the same sifting property of Dirac delta functions. The same applies to = except that after the projection step since there is necessity for constraint satisfaction, we sample from pa init (&o 6u(21; cg)). which a Gaussian distribution with mean (2 co) and covariance matrix Ln Combining both cases, we observe that without any constraints the exact DDIM reverse process can be recovered from Algorithm 1 for all E [1,7].
C.2 Proof of Theorem 2
We note that the intermediate samples in a -tep reverse sampling process are denoted as 2T, 2r..... ... 20- where 20 = z and 2r ~ (O,,, I,). Once again, we reiterate the assumptions. We consider the real data distribution to be Gaussian with mean 1 E Ra and covariance matrix I,, Le., N I,). The constraint set C is defined as C = {= A= = b) with A E Rmxx such that rank(A) (A) = where m 2 n. Additionally, for the real data distribution A (i, I,) and the constraint set C = (2 A v). there exists a unique sample x= T." that satisfies the desired constraints in C. We build on the problem setting presented in prior works [24, 39, 40] to understand the implications of our constrained sampling algorithm. While the prior works focus constraint satisfaction with a DDiMdb sampler.
DDPM-based sampler, our analysis focuses on
Given that rank(A) = n for A E mx with m 2 n, we note that (ATA)) exists. Consequently, 2k(T' +++)
Amin AA > From the theorem statement, have (t) Amin A7 with k 1. Immediately,
we note that for all / E [1, (t) > O. More specifically, & E [1, (T.7),(00 x(t) Ay
2
First, we denote the convergence property for Algorithm 1 as the variation of the upper bound on the terminal error ||æ - x= |12. where 5e8n obtained from Algorithm 1, as a function of T. Particularly, we are interested in how quickly the upper bound on the terminal error decays to reach below a fixed value that is determined by the choice k. Later, we will show that for suitable choices of k, the upper bound can decay below the tolerance limit 8.
The proof is divided into parts. First, we obtain the expression for 2t-1 in terms of 24. Then, we obtain an upper bound for || x= 2|2 or ||z - xx =21 as from Algorithm we note that = zonn
First, note that for deterministic sampling, we have the DDIM control parameters d....., oT 0. Therefore, the DDIM reverse sampling step from Algorithm (line 7) can be written as
/0 (z;; EB) + /I ā.-iee(Zz;
(12)
Since the true data distribution is Gaussian, the optimal denoiser E" (2e, f) can be expressed analytically for any diffusion step t. Therefore, the deterministic sampling step can be written as
2 - Vā,- 20 c*) + /1 o (2,,
25

---

We can obtain an analytical expression for the optimal denoiser from Lemma C.1. Using Equation 28 from Lemma C.I, we note that the optimal denoiser at the diffusion step is
c² t) = ------a-laaa
21).
(13)
Now, we obtain the expression for pr (2g: (z):e)). E*). Note that the constraint violation function is defined as [I(2) = lly - A=I3. Consequently, we note that the objective function in line 5 of Algorithm 1, L.., [(1== =a(=;; ca) 11 + (t)]l A=ll)), is convex with respect to for ((t) 0. As such, use Lemmas C.I and C.2 to obtain the expression for (b.p((2e:
or (2366) = [L, +1)44" A| āرحة + Vor= + (t))A² y].
(14)
We substitute the expressions for <* (2,,t) t) from Equation 13 and 20 pr c*) from Equation 14, respectively, in addition to replacing y with Ar", to obtain 2 in terms of e,:
2-1 =vāl-1 [I + (t) [M aip + vala + 7()
+ +VI āz- VI āel CHH 21)),
2-1 V [I (1) A] (K u,pt vāna + 7(1)A² v]
āi aiv /1 -1 ā - + v1 5. ā.2.
On further simplification, we get
2t-1 -v
In A()AAAA) 1u- a.j + Varz + 7(1)AT Ax"]
1 ãi @iVauu -0 0t2t;
[I, + +7((AAAA A] + +11-- 0-- v1 - aichn
2t-1
V In + (t)AAA A
ã āivar 2 11 [n + (t) A] -1] 1
[1
11 (1) [I + (1)A A] AT Ax",
Vor- 06 [I + ++)AA² A]
/1 02-1VI - ā. 2t
4-1
(1 (1- (1-ax) Vt) In 7(1AA Al 11 A| / [V[ 1
ao- vI ar arvārl 1
+7(0 In 7()AA A] A Ar".
To this end, we obtain
2--1 = + E,p- FIf + (t) +h -''AAAT"" Ar"
where we have the following matrix definitions,
K₁ = [Va_iva, Vā. [In + +7)AA + V- -
(15)
E, (1 ār) In + 71)AAA A] |
(16)
F F= [1-
ãe ācLn
(17)
The goal is to obtain the upper bound for ||x:80n - 20112 Note that || - x=112 = 1120 x² x2112. So, first. we subtract x² from both sides to obtain
2t-1 - x-' =Ki + Esp- - Fij + (t) + L.
Ar"
Further, add and subtract K,x"* to the right side to obtain
2t-1 x² =Ki2 - K,x" + Esp- Fig + + ( I. +
A
A Ar" - x=" + Kgx".
26

---

We further simplify the above expression to obtain
2-11 x² =K{ (21-x") Espt Fep Ki + Dix"
where the matrix definition of Dj is
Dt (t) ( [n (t))4 A In
(18)
Now, we obtain the expression for | - x= in terms of ||z - x21/2.
| x" ||| K2(zt x²) + Eaju Feps Kex" Da0*112. 1|2-
Applying the triangle inequality repeatedly, we get
(19) Before obtaining the upper bound for ||z0 x" for (() > 0, we will first show that [[ell2 |122 [E Fellle matrix K,. To show this, we establish a few relationships that will be the recurring theme used in proving that | |Pll2,, || [l, Ptll2 1 [1.7]
| x= 2| = [[ x²) + || |Dxx"
+ || F(
1 A t [1. Here ||Kill2 refers to the spectral norm of the
The spectral norm of the matrix M is defined || max immediately note the following two inequalities:
|| |
From this definition,
|Maxll2 S || as || max
Ma 2
|| /2 = maxaş0
|a
||
<
max
Naila
max
M |M Malalll
|
|M
Further, we note that the following are well-established properties for spectral norms and positive definite matrices. Consider positive definite matrix M, i..,, M > 0. Then, we have:
| is equal to the largest eigen value of M. i.e. (M))
MM 1 A as the cigenvalues of M- are the reciprocal of the cigenvalues of M,
M| 2
||
We refer the readers to Lemmas C.3, C.8, and C.10, where we show that [Kell,, |El12, | Fell <
1 Vt f e [1, if ((t) > O.
Similarly, Lemma C.6 shows that || 1Vt e if Amii A
We first apply the inequality ||Ma||a < | 2|2|| |||| simplify Equation 19 as follows:
|| x212 < ||K|l2l| x ll2 + ||, ||2 + 11, + | l + 1|
(20)
Therefore, we can recursively obtain the upper bound for || z=112 1|2 in terms of 2| x*112. This process, repeated 7' times, provides the upper bound for ||=0 - xx 1|2.
1| - -x
=22 112 S |K//||/|KKlll |
(r]lall(s (2T - x 2²)112
+ (|]Kill2 || K2 + + || ||||||
||7
+ (I|2ill2+ || D2lla + + | |22|/
(|] || E2 +............... +|| 22 K2lle
Kr-illa|Er 12)l|1|22
+ (I) lla + || F2l1 + + ||| |2 -
(21)
Let A = max; (1Ki ll.. [ (T]l))) Since for >()) 0, || ||Kr/l < 1, we note that d < 1.
Therefore, | K2 12 ... | |r 1a can be upper bounded by XT.
Additionally, note that || ||,1| K2]12 < || ll2 as |22l2 < 1. Therefore, (| 112 + |K,||||| |2 + ...+ K K|al|/2|la-.. -
Krll2) can be upper bounded by K 2/2.
Similarly, K +.lla K2lla 1 can be upperbounded by (T
1)] Kill.
27

---

The same applies to (1] ||2 Eg 12 + 2
|| Erll)) and (1) ||
+ ...+ |
... | (2112
Frl))).
Therefore, the upper bound in Equation 21 can he simplified as
=0 ==' S X (=T - x= x²)ll2 T + + (T-1)] 1|2
+ E11 (T - 1)) + (11 (T- 1) (1l2)
(22)
Consequently, in Lemmas C.4, C.7, C.9, C.10, show
Vai
[ 12 - 1+ (11) (ATA)
<lif > 0,
| - 7(1)Amm (A' A]]-I <1if Amin (A Aj
α1
[ ll S (1m (A (ATA)
<I >0
| 2
(23)
For our choice of 7(1) Therefore, we can rewrite the above inequalities as
Amin Aj we first note that 7(1) and 7(1) Amaa(1 4j for > 1.
24T
2
Vā
Kille - 1+ 2KT
[|i1|22 S 2x--1.
1 ōg
|Eill 1+ 21T
|Fil|2 =
(24)
Therefore, Equation 22 can be upper bounded using Equation 24 as shown below:
|| x2112 S A - +
v vāi
||01112 +
1
||>0 1l2t
1+ 2k-
2kT
11-11 1 1 2kT
|| + (T 11) ( Vai
(
(1
Van
||00|1a lll +(T 1)
||
+ 21T
2kT
(25)
This can be further simplified
Vāi
1
1|00 - x 20112 S 11 - x2)
||0*|12 +
||00 1l2
(1/T) + 2k
2KT 1
1 (1--0 1-a1 ān
Van
||el|2 + (1- 1/T) 11T)
||00 12+
+ 21:T
(1/T) + 21
Vāi
2(1 2(1- - 1/T)
||a1|2.
(1/T) + 2k
Upper bounding
||00 by
Vai
||00
(1 1/T)
( (a//9t2)) ||0*|12 by
(1/T)+2k)
2k
v@i 2k
||x" l12, and 2(1-2(11)) - 1/T))
(vāi Vān
||ll|2 by
[vā= 2k
||a4|12, we get
(1/1)+26
2kT - ||=0 1|2
1 1-11
ā1
||z0 xx 20112 5 N/ 12T x²)) 112
Ilull2+
1 24T
(26)
v à1
(1lel12 + 112)
Vml
(Ilzll22 + ||00*12 a constant.
k
28

---

(2T 2²)ll2 converges faster than 2KT-1
| 112 +
1-81 1+2AT
||l||2-
Note that A 1.
Therefore, the convergence rate for the upper bound on the terminal error is (1//T).
As T 30, we observe the following:
lim X [1
x²)|22
(A
1),
T T-+00
Vāi
v
lim T
| =
|| 112
(if & > 0),
T 10000
1 + 2kT
2k
1
lim T-000
||0 112 = 0,
2kT 1
lim
||Ą||2 = O,
T T-000
1 + 2KT
Vā
Vā
lim (T - 1) T-000
||0||22 =
|| l|2
(if l: > 0),
1 2kT
2k
011
lim 2(T -1 -(T-1) T-x
||x||2 =
||ll
(if > 0).
1 + 2KT
k
Therefore, in the limit T 0o, we have
v [
|| Io x x²112 S
(1|x ||2 +
||xx|12) or,
h
[1
|| - x² x²112 S
(1|x
1|2 11xl|2).
k
Consequently, for the terminal error ||aeon - x 1l2 to be upper bounded by 6, which is the desired tolerance limit, the following inequality has to be true:
|etten x=112 I v@l (1x l12
(all22) S 6.
(27)
This can be ensured for k 2 Jān (1 + (1x0(2).
However, from Equation 24, note that le has to be greater than 1.
Therefore, for k > max (1, you (1|=*12 + Ilull2)). we have |asen x1112 S 6.
Lemma C.1. Suppose Assumption 4.3 holds. Consider a T-step diffasson process with coefficients ......, orp such that agg = oT = 0, 0g E {0, 1]. The optimal denoiser c"(2z, L) is given by e" = -VI ac(Vo., 21)-
Proof. We first observe the distribution of z₁- For the diffusion forward process, we know that 2a,20 + vI âte, where (O,, I,). Note that is sample from the Gaussian distribution N ((, I,). Consequently, we note that 2, is a sample from the Gaussian distribution N( On, I + (1- L.). On simplification, we note that is sample from (V a, I.). We denote the PDF of 2 's marginal distribution as q (2x). Since we are using the optimal denoiser, induced by the optimal denoiser, p=,)(2,) is the same as the forward process PDF at f, which is (2t). Here, note that Section 2.1, we denote the reverse process PDF po.t. where the reverse process is governed by the denoiser eg. We replace this notation with (=,)(z)) as we are using the optimal denoiser.
the reverse process PDF
Therefore, the score function at t is given by V., log (=.) = V log (=x).. The score function for the Gaussian distribution qe (2t) with mean va,u and covariance matrix I,, í.e., V., (log qe (xx)) is given by Vāuu 24. Finally, [55] shows that for the diffuusion step t, the optimal denoiser can be obtained from the score function using the following expression:
c" (21, /1 04 V., log ge(z)) = c((2,, t) V1 calVor 20).
(28)
29

---

Lemma C.2. Suppose Assumption 4.3 holds. Consider a T stepp diffusion process with coefficients ao...., Oe such that an 1, är ar [0, 1]. The projected posterior mean estimate, 20.pr (21:60) 60), from the projection step in line 5 of Algorithm I is given by
E0 60) = + 74) A A [p- 0zji + Vasa + (1)AAu)) y): where the penalty coefficients from Algorithm 1, (l), are non-negative, iee, 7(t) > OVLE [L,...,
Proof. We start with the unconstrained minimization in line 5 of Algorithm given by
( 60) = arg min 1
2 (II= 6o(=:; 60) l1 + (()llv A=112)
Note that replaced the penalty function (T=) with ||y A=l are required to generate sample that satisfies the constraint y = Az.
Since the objective function is convex with respect 2, obtain the global minimum by setting the gradient with respect z to i.e.
V. (
(11= (1===
c0(=s:co)1: 7()lll - Azl = 0,
2
V 1
(=" 2=" (2(3)00)) + 60(z; 60)² "0) + +(((v
(211x Azl 0,
= =0( + +1(1V=
(211v 1l - 12 = 0,
-20(2rie0) - -1(1V= (1 [ A" Az - 2/² Az) 0,
= -0 7() (A A= A²y) 0,
[.)+()44AA) (20(2 60) + +1)
Solving this, we obtain the following expression for (E
20.pe ce) [I. +t)A4 A (20(2):c0) (tt
A] exists as A A > 0 (from Assumption 4.3) and 7()) > which ensures [I. + 7T) A] > 0. Further, substituting the expression for (2(2 60), we obtain
Note that the inverse of [n, + AF)
(ape(ZriEp) = In, + (1)AAA
2:
VI area
(2,)) (2),t) + (())
Vā,
Given that Paata = NH, I,), for the 7tstep diffusion process with coefficients ao. ... âT, we use the expression for the optimal denoiser (2z, t) (check Equation 28) in place of cœ t) obtain
21 + ax)( Varj 21) + (1))
(2p = [I, + (1)AAA
Vor
2t + Varll - 21 - aryasu + acēt + (t)
= [I, +
Va
This can be finally simplified to obtain the expression
(z;eE = [I,, + (1)A4 A] IIu ā.je Vanz + (ttAA
Lemma C.3. Suppose Assumption 4.3 holds. Consider a T-step diffusion process with coefficients
Co, Ger such that or = 1, ür 0, ãr E 10, 1). If ã < ōc-1 and if the penalty coefficients from
Algorithm are given by (1) > 1 E [1, T), the spectral norm of the matrix Kr, || K2 l2, with K defined as in Equation 15, is less than 1.
30

---

Proof.
We want to show that
K = [I
+ + u
<1. 1.
The spectral norm follows the triangle inequality. Therefore, after simplifying the expression with triangle inequality, we need to show
Van In + 7()AAA A |1, 1 vi- -olllllll āctn <1.
We note that for 7(t) > 0, [In + ((t)A²A A] > 0, and therefore [In + (())AAA] I, 0.
> 0. Similarly,
Further, we use the identities that if M > 0, then |M = Amnax (M), |M Amin (M) and ||M||2 |||
Therefore, ||, 1, In + 7())ATAA A 112 (00 A)) Further, note that va- 0z 0 and vI a1-1V ai 2 0. Substituting these, the inequality simplifies to Vā-ivvā
Amin [I, + (1)AAAAA + - - <1
Therefore, it is sufficient to show that
<1 VI 11----iV--a ae.
Amin [In (()ATA))
For any diffusion process with noise coefficients ao ao...., dr, where or > āi- V [1, Lemma C.5 shows that var-i (a S 1 VI dt-- IVI Cz.. Therefore, sufficient to show that
Amin [In + (t) A]) >1.
To proceed further, we use the Weyl's inequality [56], which states that for any two real symmetric
matrices P E guxn and E 1_=x" if the eigenvalues are represented Amax (P) x (P)) >=
(P) ... >= An(P) = Amin (P), and (Q) = (Q))) >= x2(Q) ... >= Xn = Amiu (Q),
then we have the following inequality:
(PP) + (Q) S A (P+
(29)
For = j n, have Amin (P) + Ami (Q) Amin (P+Q).
For P I, and Q = 7() with (() > O, this inequality can be exploited as both these matrices are real and symmetric. Therefore, we have
Amiu( [I, + 7()A7 A)) 2 Amiu (I,)) + Amin (T(1)A A),
(30)
Amin [In + + 2 1 + mmt)) A).
(31)
Note that now it is sufficient to show 1+ 7(t) (ATA) > 1. For 7t)) > 0. this inequality holds true Amin (A" A) 0 (AT A > 0). Therefore,
|[ [[Vār-iva σz In + ((t)A=A| + / 2z- 1 11 11
Lemma C.4. Suppose Assumption 4.3 holds. Consider a T-step diffision process with coefficients
Co...., āc such that oo 1, âr 0, a₁ [0, 11. If ag < ãt-1 and the penalty coefficients from Algorithm given by ytt > OVt [1, Kilz with Ke defined as in Equation 15 is given by
Van
[KG|/ 1 (1) (AT
(32)
31

---

Proof. We want to find an upper bound for
| || Vál L. + 7()) A] + /1 ā a.- -
Applying the triangle inequality for spectral norm, we get
Vál-ivoi [n. + (1)A4A |1 + V1 - iv- 12
K 22 -
We use the same simplifications shown in Lemma C.3 to obtain
Vor-ivā | Amin [Ln + (1)AAA] a1-vv-ivi-- Ce.
For t = I, we know that â,-1 = āg = 1. Therefore, we obtain
K ||2 S [I,
(1)ATAI)
Further, the denominator can be lower bounded using Weyl's inequality, as shown in Equation 31. Therefore, obtain
Vai Amin (Ln, (1)ATAA) $ 1+ 1+ ((1) ( (ATA)*
111125
Hence, we have shown that
Vāl
| 2 S
1 (1m)mmm (A*A)
Lemma C.5. For any Ttseep diffusion process with coefficients oo, är such that ogp äT = O, on E [0. 1 [1, T, is < O4t-1, then
Var-ivo <1 v1 -1-1VI-
Proof. Squaring on both sides, we get
āق-1ā; < + (1 a -2 -1-1V11-11111 Cz.
After further simplification, we have show
āق-1ā, < (1-āa) ā²) + (1 - ā,-1) + ā:-1ā: 2/ - 2v - ā4.
<< (1 k+) + 0 2 04 0t
0 < ãi-1 - vl-
Since ãi * 0z-1, we know that Therefore, we conclude that
v - 04-1 # v C4 Therefore (vT- 0/-1 V1 0r.) 0.
VaL-lVor 0n <1 1--iiV--- Ci..
Note that this clearly holds for the edge case where we have vāi 1, and for T, where we have < 1 vI aT-1. For the choices of ão...., ,r,, these clearly hold true.
Lemma C.6. Suppose Assumption 4.3 holds. Consider a T-step diffusion process with coefficients 0o Ger such that On 0T 0, 10,1] V E [1 For the penalty coefficients from Algorithm given by (t) > Xmaa A) [eDel,, with D, defined as in Equation 18, less than
Proof. Note that the matrix D, is given by,
D. = = I. +1())==1| A
32

---

Using the matrix inversion identity, (AB) = B A-A we rewrite Dj as follows.
Di = 7t==(t/ar
(4" A)
[. + (t)44 A]
I
AA -1
D. v
+o)44A A]
In
7(t)
(A'A) 7(t)
Dj = Vā-1
+ +I,
I
We observe that the choice of 7()) is greater than 0. More precisely, 7(1) Amnn A)
Now, if 7(t) lla < 1, then we can apply the Neumann's 's series for matrix inversion, which states that
|| - (A*
[| 1l2 I, then
oc
I M] Σ M
(33)
|| (A"
1|2 = 10 (1
First, note that All2 = ||- Therefore, theorem statement, ((t) >
12 for 7() > 0. From the
7(1)
7(0)
Additionally, we know that || (A"A)
(A"A) 7(1)
7(1) 112 Amax
(Tt))mi(AAAA)
Therefore, it is enough to show that However, we know that (t)
7(0) (A < 1 to apply the Neumann's 's series, 1
Amin (A* A) Tj. Therefore, we observe that Amm))min A)
< 1.
Thus, we have shown that ||
(1T 1)
12 < 1. Therefore, using Equation 33, we get
7(1)
(- (A (-(4-A))) 7(t)
(-AAA)=1
(-I)
I,
Σ
I Σ
(34)
7())
(t)
i=0
t=1
The last equality stems from the fact that for any matrix M E R=x" M In- Substituting this expression for the second term in D,, we get.
(AT)() )) + Va Ln.
Dj = Vār-1
7(1)
On further simplification, we have
1-1 A) (-))(AAA))) )) - va--1) In.
D: = Vor-
(())
Computing the spectral norm and using the triangle inequality, we get
(-I)(AAA))*)
Diil12
VoL-1
(1 vor-1) In
(t)²
(AT
S 50--
+ [1 v /a-1 In
7())
The inequality arises from the triangle inequality for spectral norms. Note that each of the matrices within the summation is either positive definite negative definite, and the spectral norms of all these matrices can be represented as (AT
Therefore, we get
7(0)
A)-
|Dil12 S Vāu-1
Σ
+ +111 Vor-1))
7(t)"
33

---

Using the inequality ||MN]2 S | multiple times, we get the following:
(A* A) 7(t)*
I (tt/ (I((AA))
Additionally, for the above equation, we used
|cMl22 |cll| 122 Here, c ((),, which is greater AA) Therefore, we have the following than O. Since A > O, we have (44 A)- inequality:
-A
Amin (ATA) ) (r(t))miule) (A
S 7(1)*
7(1)
2
Using this to upper bound |/r|12, we get
(ii)) (AT )) + + (a1-1))
||Dell - vā
a Finally, the summation of an infinite geometric series of the form a + a² t..., where a < 1 is T-a Here, note that have (t))
Auiia Amiin(A (A*A) A) Therefore, 7(1)Amin (A" 1. Therefore, have,
7(e)mmi (A"A) (A
7(t) (Am))(Am
((t) ( (AT -
(A²)Amin(AAA)
So, we obtain
V
| S ((t)Amiu (ATA) - + var-1)
(35)
Now, for Delll2 < we need to show
vār-1
7(1) (A'A) (1 vāt-1) < or
Vār-1
7()) (A A) 11 V
This simplifies showing ((t) Amin (ATA) A) 1 > which is if (t) AA the statement of the lemma, know that x(t) AA
And, from
Therefore, we have shown that Deell < for y(t) > (A)
2
Lemma C.7. Suppose Assumption 4.3 holds. Consider a T-step diffusion process with coefficients
.o..... oT such that oo = äT 0, āt [0,1] A E [0, For the penalty coefficients from
Algorithm 1 given by 7(1) > AA" Dill2 with D, defined in Equation 18, is upper
2
bounded by
Dill (A'
Proof. Note that the matrix D, given by,
AT A - I... D, 7() IL + 7()AA_
From Equation 35 in Lemma C.6, we know that if 7() AA va--1
D & 7(1) (ATA) A) + vār-1)
From the lemma, we know that 7(t) t 1 and do 1, we get
Amin (A= Therefore, we use Equation 35 and substitute for
| S 7(1))miu (AT A)
34

---

Lemma C.8. Suppose Assumption 4.3 holds. Consider a T Ttepp diffusion process with coefficients
āg. Co...... Oet such that āg ät 0. āi [0, 11. If āg a [1, with the penalty coefficients from Algorithm I given by 7l) > 0, Bg la < where Eg is defined as in Equation 16.
Proof. We know that the matrix E, is defined as
E. = (1 a.) a:) + +0)AA
First, the identity ||MM|2 |c where is any real number, need show
(1 σz) V [[ + 7(t)44
<1
Note that (1 Var [,, + 7(t) A x
Further, for 7(t) 0, In + (1)AAA
0, and therefore
We use the identity that for M O, |M 1112 Ami
Therefore, || [T, + 7()A7 a 1l2 Aman (1n +
We use this expression to simplify the
inequality as
(1 - (1-ar)va--1
1.
Amin (In +1())=AA) AJ)
We use Weyl's inequality (check Equation 31) to lower bound the denominator and thereby upper bound the left side. Therefore, is sufficient to show
(1-a))
+ (1t)Amin (ATA) <1
We observe that the numerator (1 / a- is always less than 1. However, know that the denominator 1 + (t)) (ATA is strictly greater than for (t) > 0 as (A*A) exists and Amin (A" A) > Therefore, the left side is always less than 1. This leads to
(1 a.) o1 [T. 7()A²² A
2 <1.
Lemma C.9. Suppose Assumption 4.3 holds. Consider a Ttseep diffusion process with coefficients
oo, oer such that on = 1, oT = 0. ot E 0. 1), If āg ã.-1V t E [1.T] with the penalty
coefficients from Algorithm given by (t) > 0, E ll. with Eg defined as Equation 16, is upper bounded by
1-āj ā1 1(1)mmm (AT ATA)
max (Ei)
1 1
Proof. We know that E, is given by
E,
- +
We first substitute for = 1 and va== =
El (1 āI)[In (1)
We use the identity ||cM||2 |c||| M where e any real number, to get
|| 2 (1 ã1) [ + (1) -|-
Here, note that (1 - a1) 2 0. Similar to Lemma C.8, we can rewrite the spectral norm as
1 1 ā
|| 1 Amin (In + (1)A Al)
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Again, using Weyl's inequality and performing similar modifications as in Lemma C.8, we obtain the following upper bound for the spectral norm
| - mi) (AT
Lemma C.10. Suppose Assumption 4.3 holds. Consider a I Ttste diffissio process with coefficients Cos .... OeT such that ōig L, ÜT ae [0, af < ūt-1 V [O,T], || with F3 defined as in Equation 17, is less than 1. Additionally, || O.
Proof. Note that Fj is given by the expression,
Ft = / a dg V
First, we use the identity ||cM]|2 = |c|l where e is any real number. Therefore, we need to show
|/ vI - ā.v ||lll2 1.
For the given conditions on "0....,,r we observe that least one of the terms in -1-a--iVi ār a is always less than 1. Therefore || 122 < 1. And, since āo = 1, for Fi, have vI ag 0. Therefore, Fi is null matrix and [ O.
D Inference Time Results
We evaluated our approach for time series samples up 576 dimensions (e.g,, the Air Quality and the Stocks dataset). We have provided the inference time taken to generate samples with up to 66 and 450 constraints for the Air Quality and the Stocks datasets Table 9. First, we note that the inference latency for CPS is very similar to PDM [11] and PRODIGY [20], these approaches involve projection steps after each denoising step. We observe that for univariate datasets, like the Traffic dataset, the inference latency for CPS is less than that of Guided- DiffTime. Note that Muided-Diff'Timi requires backpropagation through the denoiser network. For multivariate datasets like the Air Quality and the Stocks dataset, the inference time for CPS is roughly 2x more than the inference time for high constraint violation magnitudes. For all the datasets, COP has the least inference time. However, COP also suffers heavily from poor sample quality. Table reports inference latency on a single NVIDIA RTX 6000 GPU.
Guided-DiffTime.
However, Guided-DiffTing
has poor sample quality and very
Additionally, for each real-world dataset, we analyzed the effect of the number of constraint categories. As mentioned in Section 5, our experiments involve imposing multiple constraint types, such as mean, argmin, argmax, and value at location constraints. Interestingly, we observed that the inference latency linearly increases with the number of constraint categories for all the real- world datasets.
APPROACH
AIR QUALITY
TRAFFIC
STOCKS
GUIDED-DIFFTIME
14.7640.36
11.61
15.2410.43
COP-FT
8.5+3.72
1.27=0.455
11
CPS (OURS)
31.49=0.664 S
6.99-10552
35.22+2.01
Table 9: The projection step in CPS increases the sampling time. Here, present the average inference time taken to generate a single sample for all the real-world datasets used in our experiments. The results are shown in seconds, and the inference time is averaged over 10 runs. Though the inference time for COP-FT is very low, the generated samples have poor sample quality.
Furthermore, we note that there are multiple ways to reduce the inference time for CPS, such
Capping the number of update steps in each projection operation (line of Algorithm 1) during the initial denoising steps when the signal-to-noise ratio is very low.
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Quality Traffic
Inference Time (Seconds)
Stocks
25
20
15
8 10
Number of Constraint Categories
Figure 8: The inference latency increases linearly with the number of constraint categories. For all the real-world datasets used in our experiments, we observe that the relationship between the number of constraint categories and the inference time is linear. The experiments are run on single NVIDIA RTX 6000 GPU. The Traffic dataset has the least inference latency as it has the smallest data dimensionality. The Air Quality and Stocks datasets have the same dimensionality, but the Stocks dataset has an additional OHLC constraint.
The projection operation (line 5 of Algorithm 1) need not be performed for every denoising step. Consequently, we can develop principled methods to identify the denoising steps where projection is required based on constraint violation.
Now, we provide brief convergence analysis for Algorithm with regard Assumptions 4.1 and 4.3.
Under Assumption 4.1, the objective in Line has B-Lipschitz continuous gradients, yielding convergence rate of (1/NNrr) where Np is the number of projection steps and is in the order of (1//)) with being the tolerance limit for constraint violation. This repeats for all T diffusion steps.
Under Assumption 4.3, line Algorithm has a closed-form solution (Lemma C.2), and the convergence rate of CPS with respect to T is O(1/T).
Additionally, from Figure observe that the inference time for the Air Quality and the Stocks datasets is much higher than the inference time for the Traffic dataset. This can be primarily attributed to the dimensionality of the samples in the respective datasets. The sample dimension the Air Quality and the Stocks datasets is 576 (6 channels with 96 timestamps in each channel), whereas the sample dimension in the Traffic dataset is (1 channel with 96 timestamps).
E Additional Qualitative Results
In this section, we provide additional qualitative comparisons between CPS and other baselines.
Metrics
For the FTSD and J-FTSD metries, we train the time series and condition encoders using the procedure given in [1]. For FTSD, we only train the time series encoder using supervised contrastive loss to maximize the similarity time series chunks that belong to the same sample. For J-FTSD, we perform contrastive learning training in CLIP-like manner to maximize the similarity between time series and corresponding paired metadata, as explained [1]. We use Informer models as encoders. Additionally, just in the case of [1, 43], we observe that the approaches corresponding to the lowest values of FD metrics have the lowest TSTR and DTW scores. This further validates the correctness of the FTSD and J J-FTSD metrics used for evaluation. To train these models, used the
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Air Quality
Stocks
Traffic
Diffus TS
0.8
L Value
0.6
0.4
Value
0.5
1:0
0.00
0.2
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Time
Time Stamps
Time Stamps
CPS (Ours)
-0.475
1.5
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1.0
Velue So
-0.525
Value
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-0.550
0.0
-0.575
-0.5
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-1.0
-0.625
20 40
20
40
60 80
Time Stamps
Time Stamps
Time Stamps
Generated Samples
Real Samples
Figure 9: Qualitative comparison between Diffusion-TSs TS [22] and CPS on the real-world datasets used in experiments.
Air Quality
Stocks
Traffic
2.5
1.0
PDM
Value
2.4
Yo
Value
1
20
40
60
80
20
40
60
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20
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Time Stamps
Time Stamps
Time Stamps
2.6
2.5
Value
2.44
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Value
0.5
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0..5
GPs
2.0
2.1
W
-1.5
-1.0
20
40
60
80
20
60 Stamps
40
60
Time Stamps
Stamps
Generated Samples
Real Samples
Figure 10: Qualitative comparison between Projected Diffusion Model (PDM) [11] and CPS on the real-world datasets used in our experiments.
Air Quality
Traffic
Stocks
2.6
1.0
2.5
2.3
-0.
2.2
Time
Time
Time
Generated Samples
Real Samples
Figure 11: CPS can generate multiple samples that adhere to the required constraint set. Here, showcase CPS's ability to generate multiple samples for the same set of constraints. The real sample from which the constraints are extracted is shown in blue, and we show the mean and + standard deviation for the generated samples. Note that the trend of the generated samples matches that of the real sample, and this qualitative result is consistent with other qualitative results in Figures 4, 10, and 9. From the figure, note that the standard deviation goes to zero, as we have imposed the "value at" constraint at these locations.
24, 48, 72, and 96 timestamps
same set of byperparameters as mentioned in [1]. For all the real-world datasets, we trained these models up to maximum of 5000 epochs.
We sourced the implementation for the DTW metric from the public domain. For the constraint violation magnitude, we computed the violation for each constraint, excluding the allowable constraint violation budget. For TSTR, we trained the standard TimesNet [57] model to perform imputation. The mean and standard deviation for the TSTR values are obtained from the results for 3 seeds. We also provide a sample quality comparison based on the Discriminative Score (DS) metric. For this metric, we train a post-hoc time series classification model to distinguish between real and generated
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time series samples. We use a simple 2-layer LSTM network for the classification task. DS was introduced [7] sample quality metric. Similar to the TSTR metric, we train the classifier on synthesized and real training data. We then report the classification error on the synthesized and real test data for seeds.
G Datasets
We compared CPS against the existing baselines for six settings Air Quality, Air Quality Conditional, Traffic, Traffic Conditional, Stocks, and Waveforms. The training and testing splits for the Air Quality and Traffic datasets are taken from [1]. We additionally evaluate the constrained generation approaches on the Stocks and the Waveforms datasets. We used the preprocessing scripts provided by [7] for the Stocks dataset. The waveforms dataset was synthetically generated. We generated 64, 000 sinusoidal waveforms of varying amplitudes, phases, and frequencies. The amplitude varies from 0.1 to 1.0. The phase varies from 0 to 2n. The frequency limits were chosen based on the Nyquist criterion. The generators and the GAN models were trained this dataset. However, for the TSTR metrics, created a subset of this dataset with 16, 000 samples. All the datasets except the waveforms dataset were standard normalized.
The Air Quality dataset is a multivariate dataset with six channels. The total numbers of train, val, and test samples are 12166, 1537, and 1525, respectively. The Traffic dataset is univariate. The total train, val, and test samples are 1604, 200, and 201, respectively. The Stocks dataset is multivariate dataset with six channels. The total train, val, and test samples are 2871, 358, and 360, respectively. The truncated form of the waveforms dataset used for evaluation consists of 13320, 1665, and 1665 train, val, and test samples, respectively.
For the conditional variants, we used the same contextual metadata as provided in [1]. For the Air Quality dataset, we used categorical features such as station ID), timestamps, and wind directions, and continuous features such as temperature, pressure, rain levels, and wind speed. For the traffic dataset, we used broad and fine weather descriptions, holidays, and timestamps categorical features. Similarly, we used the temperature, rain, and snowfall levels, and cloud conditions as continuous features.
H Implementation
In this section, we will describe the implementation details for our approach, each baseline, trained models, metrics, etc.
H.1 Diffusion Model Architecture
We utilize the TIME WEAVER-CSDD denoiser for all the diffusion models used in this work. The training hyperparameters and the model parameters are precisely the same as indicated in [1]. The total number of residual layers 10 for all the experiments. Further, we used 200 denoising steps with linear noise schedule for the diffusion process. All the baselines and CPS use the same base diffusion model with the TIME WEAVER- CSDI denoiser backbone.
We use 256 channels in each residual layer, with 16-dimensional vectors representing each channel. The diffusion time step input embedding is 256-dimensional vector. Further, the metadata encoder has an embedding size of 256 for the conditional case. The metadata encoder has two attention layers with eight attention heads. All experiments and the hyperparameters are precisely the same as those in [1]. For each dataset, we trained the diffusion model on single NVIDIA RTX 6000 GPU. The checkpoints were obtained using the best validation denoising loss, and we trained the TIME WEAVER -CCDI denoiser up to a maximum of 5000 epochs for all datasets.
a learning rate of 10-4 Our training procedure
H.2 Constrained Posterior Sampling Implementation
For the CPS implementation, we use CVXPY [35]. We first implement the constraint violation function with the violation threshold set to 0.005 for all the constraints except the bounds, like argmax, argmin, OHLC, and the trend constraint. For example, consider the mean constraint. The constraint
39

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violation function for this constraint is implemented as max (1+ ( -
0.005, 0).
where L is the time series horizon. We do not provide the constraint violation threshold for the bounds. Though the allowable constraint violation threshold is 0.01, we performed the projection step with a constraint violation threshold of 0.005 to ensure that the sample strictly lies within the constraint set. We use the same choice of y((t) Vt E [], as described in Section 3. However, we clip the value of 7t)) to 100, 000 after certain denoising steps, as the CVXPY solvers cannot handle extremely high values of (t). We note that this clipping usually occurs after 150 denoising steps.
To generate samples on large scale for the training, validation, and test datasets, we performed batched denoising. To parallelize the projection step with CVXPY after the denoising step, we used multiprocessing with 4 processes.
H.3 Baseline Implementation
This section will explain all the details about the baseline implementations. Specifically, we use two baselines- Constrained Optimization Problem (COP) and Guided DiffTime. We note that both approaches were proposed [12]. However, the implementation of these approaches is not publicly available. Based on the details provided in [12), we have implemented the baselines for comparison against CPS.
H.3.1 Constrained Optimization Problem Implementation
The Constrained Optimization Problem, COP, has two variants. These referred to COP and JOP-FineTunnin, respectively. In COP, we perturb a randomly selected sample from the training and validation datasets. In COP-FineTuning, we perturb the sample generated from the TIME WEAVER- ISDI diffusion model.
Note that [12] suggests extracting statistical features to be imposed distributional constraints. For example, [12] suggests extracting autocorrelation features for the stocks dataset. However, since practically impossible to list all the statistical features for each dataset to obtain the distributional constraints, [12] suggests the use of the critic function from a Wasserstein GAN [21]. The details of the GAN training are summarized below.
COP has two objectives maximize the 12 distance from a randomly selected sample from the training and maximize the critic value from a Wasserstein GAN.
Similarly, COP Fine Tineiuning has objectives minimize the l2 distance from a generated sample and maximize the critic value from a Wasserstein GAN.
We optimize for these objectives while ensuring constraint satisfaction.
As suggested in [12], we use the SLSQP solver from SciPy [58]. Unlike [12], which performs piecewise optimization, we note that all the constraints used in our work are global. Therefore, piecewise optimization is very suboptimal. For example, is suboptimal to break a time series into chunks and perform optimization for each piece when the goal is to generate a sample with specific mean value. This is also pointed out [12]. Therefore, we perform COP for the whole time series at once. We consider two budgets- 0.005 and 0.01. This is similar to [12]. However, unlike their approach, we stop with 0.01 as the allowable constraint violation in our case 0.01 for all methods.
We used a weight of 0. for the critic's objective. We noticed that for different values (1.0,0.1,0.01) of this weight, there was very little change in the DTW metric.
H.3.2 Critic Function Implementation
[12] suggests using the critic function in a Wasserstein GAN [21] to enforce realism in the COP approach. Therefore, we used the WaveGAN [8] implementation from [6]. The implementation from [6] has the gradient penalty loss, an improved training procedure to enforce the required Lipschitz continuity for the critic function. Additionally, the WaveGAN training with gradient penalty has been implemented [6] for generating time series samples for the ECG domain. Therefore, we use their implementation to obtain the critic function for the COP baseline. The number of parameters adjusted such that the diffusion model and the GAN model have comparable number of parameters.
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Similar to the diffusion model, we used the same architecture and training hyperparameters for all the datasets and experimental settings. Specifically, we trained the WaveGAN model with learning rate of 10-1 for all the datasets. The input to the generator is 48-dimensional random vector. Additionally, we ensured that the total number of parameters was equally distributed between the generator and the discriminator to prevent either of the models from overpowering the other. The WaveGAN model was trained for 5000 epochs for all the datasets. For the conditional variants, we restricted the training to 1000 epochs the training was highly unstable.
H.3.3 Guided DiffTime Implementation
use the same TIME WEAVER-CSS IC denoiser as the case of CPS. For the guidance weight, we experimented with the following weights (0.00001, 0.0001, 0.001, 0.01, 0.1, 1.0). We chose the best guidance weight based on the constraint violation magnitude. Note that we used the same guidance weight for all individual constraints. Using PyTorch, we implemented all the constraints mentioned in Section 5. Additionally, we augmented the Guided DiffTime approach with the DiffTime algorithm for fixed values. In other words, after each step of denoising followed by guidance update, enforced the fixed value constraints, as specified in [12]. This applies to the values at argmax, argmin, 24, 48, 72, and 96 timestamps.
H.3.4 Diffusion-TS, PDM and PRODIGY Implementations
For the PRODIGY baseline [20], we used the diffusion step-based coefficient, as the Distance-based coefficient does not guarantee constraint satisfaction, even for convex constraints. For the diffusion step-based coefficient, we experimented with
0 and [0, 1, 5]. Note that for = O, we obtain used the same CVXPY solvers the PDM baseline [11]. For both PRODIGY and PDM baselines,
for projection, similar to CPS. For the Diffusion-TSS - [22], we obtain guidance from the constraint violation. Similar the Guided-Doe DifitedDi baseline, we experimented with scaling the guidance by (0.00001, 0.0001, 0.001, 0.01, 0.1, 1.0).
I Broader Societal Impact
This paper presents Constrained Posterior Sampling (CPS), which novel algorithm that focuses on constrained time series sample generation. More broadly, CPS falls under the category of targeted sample generation. CPS has a lot of positive impacts in the time series domain, as it can help in generating targeted samples for stress-testing machine learning systems, as well as replacing private user or enterprise data with more accurate synthetic variants. As such, the societal consequences of our work are limited to those arising from improved quality of generated synthetic data.
41


Reviews:
[{"review_id": "cJ499jBTQk", "reviewer": "Reviewer_gJxX", "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.", "questions": "Why are the generated series matching a real time series pointwise almost exactly?", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "**Strengths**\n\n- The paper is clear and well-written. \n- The method’s rationale is well explained and supported by empirical results.\n- 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.\n- The proposed method is relatively easy to implement. It lets practitioners impose multiple convex constraints with solid theoretical guarantees in the linear case. \n\n**Weaknesses**\n\n- Limited novelty: the main difference from PDM is projecting the posterior mean instead of the latent state.\n- 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.", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the feedback on our work. The reviewer acknowledges the depth of the provided empirical evaluations along with the simplicity and the theoretical guarantees for CPS.\n\n**On the range of constraints used in experiments:** We note that the prior work [1] in constrained time series generation primarily focused only on linear or convex constraints individually (argmin, fixed value, and bounds). The approaches in [1] have not been tested for various combinations of constraints on multiple real-world datasets. \n\nHowever, our experiments involve detailed evaluation on multiple combinations of constraints using a range of metrics that capture the sample quality and diversity.\n\nAdditionally, the number of constraints imposed on a generated sample is much higher (~100 and upto 450 for the stocks dataset) in our work than in [1]. We request the reviewer to check the **Clarification regarding the constraints used in Figures 5 and 7** in our response to **Reviewer iu3N.**\n\n**On extension to non-linear constraints:**\nWe have compared CPS against other baseline methods for a general class of constraints. Specifically, we imposed the **Autocorrelation Function (ACF) value at a particular lag as an equality constraint** for the stocks dataset. \n\nFirstly note that even CPS has constraint violation in this case as the constraint is non-convex. \nSecondly, the projection step, after every denoising step, does not provide the global minimum for $\\hat{z}_{0,\\mathrm{pr}}$, which was the case for linear and convex constraints used in the main table 1. \n\nEven in such cases where the projection operation does not yield a $\\hat{z}_{0,\\mathrm{pr}}$ which is the global minimum, **the iterative projection and denoising steps in CPS ensure that the adverse effects of projection are minimized and provide high sample quality.**\n\nWe observe that CPS provides the best sample quality metrics while maintaining the least constraint violation magnitude (Appendix B.3, Table 4).\n\n**On limited novelty:** Though the key difference between CPS and other approaches such as PDM and PRODIGY is the projection of the posterior mean estimate, we note that **CPS applies the constraints on the estimate of a clean sample and not on the noisy latents**, in accordance with the fact that the constraints are defined for clean samples. This improves the sample diversity (lines 215-218).\n\nAdditionally, this key difference ensures the modified latents ($z_{t-1}$) lie on the manifold corresponding to the noise level $t-1$, thereby ensuring accurate denoising with respect to PDM and PRODIGY. This results in improved sample quality (lines 219-222). \n\n**Regarding the qualitative results:**\nIn Figures 4 and 9, we observe almost perfect overlap between the real and the generated time series samples for a few datasets. Specifically, this can be observed for the conditional variants of the Air Quality and Traffic datasets along with the Stocks dataset. \n\n**This is primarily due to conditional inputs and a large number of constraints.**\n\nFor the conditional variants (Air Quality Conditional and Traffic Conditional), the generated samples are already condition or metadata-specific and therefore track the real sample closely. In addition, when we impose constraints, the generated samples almost perfectly match the real samples. This does not happen in the unconditional cases and is empirically demonstrated through the **differences in the DTW metrics between the unconditional and conditional cases** (2.35 vs 1.83 for Air Quality and 3.41 vs 0.84 for Traffic, lower is better).\n\nFor the stocks dataset, the number of constraints is too high (450) that the **feasible set is very small** and therefore the generated samples closely track the real sample from the constraint set. \n\nConsequently, in both these figures, note that the least similarity between the real and generated samples can be observed for the unconditional Traffic and Air Quality datasets which do not have a large number of constraints as the Stocks dataset. The generated samples only roughly follow the trend of the real sample, but do not match as perfectly as compared to the conditional variants. \n\nAccordingly, as diffusion models generate stochastic outputs, Figure 10 depicts the distribution of samples generated for the same constraint set. We observe higher overall variance for unconditional Traffic and Air Quality datasets when compared to the Stocks dataset which has a larger number of constraints. \n\nAdditionally, as we impose the **value at** constraints, note that the standard deviation is 0 at 1, 24, 48, 72, and 96 timestamps.\n\nOur objective with the qualitative results is to mainly showcase the matching trends with least distortions due to projections. This is also confirmed by the high sample quality metrics for CPS.\n\n**Modifications to the manuscript**\n\n1. We will include explanation regarding non-linear constraints in the experiments section.\n\n2. We will add a paragraph explaining the qualitative results and the perfect overlap on selected datasets.\n\n**References**\n\n[1] On the Constrained Time-Series Generation Problem"}, {"review_id": "jYx4XX8n9B", "reviewer": "Reviewer_JbhH", "summary": "This 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.", "questions": "Your 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?", "limitations": "No. 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", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "Strengths\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.", "was_revised": true, "final_justification": "Based on the response of authors, I am willing to raise the score to weak accept.", "rebuttal": "We thank the reviewer for the detailed feedback on our work and for acknowledging the effectiveness of projecting the posterior mean estimate over the existing approaches like PDM and PRODIGY. \n\n**Regarding the update step in Algorithm 1:**\nCombining lines 7-9 in Algorithm 1, our update step is exactly the same as the DDIM update step with $\\hat{z_0}$ being replaced by $\\hat{z_{0, pr}}$.\n\nThe original DDIM update step results in sampling from an unconditional distribution, whereas, updating $\\hat{z_0}$ with $\\hat{z_{0, pr}}$ can be viewed as sampling conditioned on the specified constraints. [1] has employed similar algorithms for image personalization. \n\nFurther, we observed that using $\\hat{\\epsilon}$ is better for sample quality metrics. Below, we list the FD and the DTW metrics for CPS and CPS with the noise term updated for consistency. \n\n\n| Metrics | Approach                 | Air Quality      | Traffic           | Stocks           | Waveforms        |\n|---------|--------------------------|------------------|-------------------|------------------|------------------|\n| FTSD    | CPS                      | **0.0234**       | **0.2077**        | **0.0023**       | **0.0029**       |\n|         | CPS with noise correction| 0.2085           | 0.5147            | 0.0426           | 0.0694           |\n| DTW     | CPS                      | **2.35 ± 1.48**  | **1.83 ± 1.16**   | **0.20 ± 0.71**  | **0.23 ± 0.17**  |\n|         | CPS with noise correction| 3.07 ± 1.86      | 4.05 ± 1.26       | 0.54 ± 1.10      | 0.46 ± 0.33      |\n\nUpdating both posterior mean and noise results in overdependency on the projection step and pushes $z_{t-1}$ off the noise manifold for $t-1$. This takes $z_{t-1}$ far away from the training domain of the denoiser, resulting in poor sample quality as shown by the numbers above.  \n\n**On the mismatch in the penalty schedule/coefficients between theory and experiments:**\nFirstly, the exact penalty coefficients as shown in the theorem cannot be computed for real-world applications as constraints need not satisfy Assumption 4.3. \n\nTherefore, our choice of penalty coefficients in Algorithm 1 is similar in essence to the penalty coefficients used in Theorem 4.4. The key takeaway from Theorem 4.4 is that, for convergence, the penalty coefficients should be **decreasing with respect to $t$** (diffusion step) and can be chosen to increase from a small value during the initial denoising steps (t=T) to a very large value ($\\gamma(1) \\rightarrow \\infty$) during the final denoising steps. \n\nWe ablated with **linearly, quadratically, and exponentially (lines 183 -187) decreasing** function choices (Table 5 of Appendix B.3 (page 21)). Note that the **linear functional form**, where the penalty coefficient linearly decreases from 10000 to 0 (increases during denoising), **is proportional to the penalty coefficients obtained from Theorem 4.4 at all diffusion steps**. \n\nInterestingly, on all the real-world datasets, our key observation is that **the function choice does not matter**, and we obtain very similar sample quality metrics with linear, quadratic, and exponential choices (difference in third decimal place, check Table 5 of Appendix B.3 (page 21)). The results in the main table 1 are obtained using the exponential choice described in Algorithm 1.\n\n**On the mismatch between penalty functions in Algorithm 1 and in Theorem 4.2:** \nThe smoothness assumption on the penalty function for Theorem 4.2 followed by non-smooth penalty functions in the experimental setup and Algorithm 1 follow prior works from posterior sampling for inverse problems [2,3]. Prior works draw theoretical insights from linear and noise-free observation settings and extend them to non-linear and even non-differentiable cases.\n\nTheorem 4.2's main focus is expressing the projection step as a series of gradient updates with fixed step sizes to convert $\\hat{z_0}$ into $\\hat{z_{0, pr}}$. Consequently, the **projection step can be viewed as sampling from a Dirac delta distribution centered at $\\hat{z_{0, pr}}$.** \n\nHere, the smoothness assumption only aids in ease of choosing the step size and the number of gradient updates, as described in Eq. 7 (page 21, Appendix C.1). This reasoning can also be extended to optimizing non-smooth penalty/objective functions in line 5 of Algorithm 1 as $\\hat{z_{0, pr}}$ can be expressed as a sequence of updates from $\\hat{z_0}$.\n\nWe also note that our experiments include both equality (mean, value at, etc.,) and inequality constraints (OHLC). \n\n**On the usefulness of theoretical analysis:**\n\n**Theorem 4.2:** The theorem draws equivalence between the projection step and sampling from a Dirac delta distribution centered at $\\hat{z_{0, pr}}$. Essentially, the theorem provides the modified $p(z_{t-1} | z_t)$ under smoothness assumptions on the penalty function. As described earlier, these insights can be extended to non-smooth cases.\n\n**Theorem 4.4:** Theorem 4.4 investigates the efficacy of CPS on the well-studied linear problem with unique solution and shows that we **can do away with costly constrained optimization** after each denoising step (baseline approaches - PDM and PRODIGY) and perform cheaper unconstrained optimization with steadily increasing penalty for constraint violation as we denoise. \n\nSecondly, the theorem **provides a framework for the choice of penalty coefficients** (small during initial denoising steps and very large during the final denoising steps). **This works well with our intuition that constraint satisfaction need not be heavily prioritized during the initial steps when the signal-to-noise ratio in the denoised sample is very low.** \n\nWe also note that CPS, designed based this intuition, outperforms state-of-the-art approaches on sample quality.\n\n**Regarding the hyperparameter search for guidance-based methods:**\n\nBelow, we provide a detailed analysis of how the constraint violation magnitude varies with the guidance weights. \n| Approach        | 0.00001 | 0.00005 | 0.0001 | 0.0005 | 0.001  | 0.005  | 0.01  | 0.05    |\n|----------------|---------|---------|--------|--------|--------|--------|-------|---------|\n| Guided Difftime| 23.21   | 23.94   | 25.37  | 51.27  | 113.31 | 951.62 | 2929  | 116084  |\n| DiffTS         | 5.61    | 5.67    | 5.68   | 5.68   | 5.7    | 5.67   | 5.64  | 5.67    |\n\nNote that, for Guided DiffTime, as the guidance weight increases, the violation magnitude also increases drastically. \n\nWhereas, for DiffTS, the violation magnitude largely remains the same with minor fluctuations.\n\nFor fair comparison, the guidance weight corresponding to the smallest violation magnitude from this table is used in our experiments.\n\n**On the inference latency for large number of constraints (~100):**\nWe refer the reviewer to Appendix D (pages 34 and 35) for a detailed analysis on inference latency. Our key observation is that the **inference latency linearly increases with the number of constraint categories** for all the real-world datasets. \n\nIn Figures 5 (page 9) and 7 (page 34), and Table 6 (page 34), we increase the number of constraint categories and observe the sample quality and inference latency. \n\nWe request the reviewer to check the **Clarification regarding the constraints used in Figures 5 and 7** in our response to **Reviewer iu3N** for the list of constraints used in each experimental setup. In the caption for Figure 7, we also provide details regarding the hardware setup used for our experiments. \n\nFrom Table 6, note that the average latency for the univariate traffic dataset is small, when compared to the other multivariate  datasets (Air Quality and Stocks), for any number of constraint categories.\n\nAdditionally, note that the Stocks dataset has higher latency for any number of constraint categories because of the large number of OHLC constraints being imposed.\n\nWe also provide convergence analysis for Algorithm 1 under both Assumptions 4.1 and 4.3 in lines 1197 to 1201.\n\n**On the systematic evaluation of individual constraint types:**\n\nBelow, we provide the FTSD (sample quality) comparison for individual constraint types evaluated on the traffic dataset. \n\n| Approach         | argmax | argmin | Value at argmax  | Value at argmin  | mean | Mean change | Value at (1,24,48,72,96) |\n|------------------|--------|--------|------|------|------|--------------|-----------|\n| Guided Difftime  | 0.21   | 0.22   | 0.21 | 0.3  | 0.22 | 0.22         | 0.24      |\n| COP-FT           | 0.27   | 0.25   | 0.2  | 0.26 | 0.28 | **0.19**     | 0.23      |\n| PDM              | 0.21   | **0.20** | **0.19** | 0.23 | 0.19 | 0.20         | 0.20      |\n| **CPS**          | **0.20** | 0.21   | **0.19** | **0.21** | **0.18** | 0.20         | **0.19**   |\n\nFrom the table, we note that **even for individual constraint types, CPS outperforms the baselines on sample quality for a large number of constraint types**. Further, while combining multiple constraints together, from Figure 5 (page 9), we clearly observe that CPS provides the best sample quality for any number of constraints. \n\n**Modification to the manuscript** \n\n1. We will add the tables for systematic verification of constraints and hyper parameter search to the paper.\n\n2. In the experiments and the limitations sections, we will address the differences in the penalty functions and coefficients between the theoretical justification section and Algorithm 1.\n\n3. We will include a discussion and the table regarding consistency in the update step.\n\n4. We will include a paragraph at the end of the theoretical justification  section to explain the usefulness of the theorems.\n\n**References**\n\n[1] RB-Modulation: Training-Free Personalization of Diffusion Models using Stochastic Optimal Control\n\n[2] Pseudoinverse-Guided Diffusion Models for Inverse Problems\n\n[3] Beyond First-Order Tweedie: Solving Inverse Problems using Latent Diffusion"}, {"review_id": "rGvmrO0KxV", "reviewer": "Reviewer_iu3N", "summary": "The 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.", "questions": "- 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.", "limitations": "Yes, the limitations are addressed.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Overall, 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.", "was_revised": true, "final_justification": "The paper proposes an interesting idea and showcases its applicability in various experiments. My concerns have mostly been addressed in the rebuttal. Demonstrating imputation and/or forecasting in small experiments would have further improved the contribution.", "rebuttal": "We thank the reviewer for the detailed and thoughtful feedback on our work and for acknowledging the effectiveness of CPS as it does not require any backpropagation through the denoiser. \n\n**On extension to non-linear constraints:**\nWe have compared CPS against other baseline methods for a general class of constraints. Specifically, we imposed the **Autocorrelation Function (ACF) value at a particular lag** as an equality constraint for the stocks dataset. \n\nWe observe that, even though the projection step does not result in the global optimum, the iterating projection and denoising process pushes the generated sample to the constraint set. Additionally, the sequential denoising process removes the adversarial artifacts caused by the projection step. **This provides very low constraint violation metrics for CPS, when compared to other baselines, while maintaining high sample quality (Appendix B.3, Table 4).** We request the reviewer to refer **On extension to non-linear constraints** section of our response to **Reviewer gJxX** for a detailed explanation.\n\n**On the empirical evidence for the claim made in the paper:**\n\n**Claim** - Projecting the noisy intermediate latent (PDM and PRODIGY) has adversarial effects on sample quality.\n\n**Evidence** - Note that all approaches use the same denoiser. The **only conceptual difference** between these approaches and CPS is that **CPS projects the posterior mean estimate**. Consequently, this results in the noisy latent, $z_{t-1}$, being pushed off the manifold for the diffusion step $t-1$, resulting in imperfect denoising. As a result, imperfect denoising causes sample quality degradation which is empirically shown in Table 2. \n\nSection 3 (lines 229–240) outlines two key reasons for the same.\n\n**On performing imputation with CPS:**\nThe reviewer is correct. **CPS can be extended to perform forecasting and imputation** by imposing the **value at** constraint to all the timestamps where the true values are available.\n\n**Clarification regarding the likelihood maximization claim:**\nIn lines 162-166, we state that the proposed algorithm satisfies the desired constraints while maximizing the likelihood of the generated samples.\n\nIn general, diffusion models are trained to maximize the log-likelihood of observing samples from the dataset, i.e., log p(x). Therefore, diffusion models allow for sampling from p(x). \n\nHowever, in constrained time series generation, the objective is to sample from p(x) subject to certain constraints.\n\nTo sample from p(x), **CPS uses a pre-trained diffusion model that was trained to maximize log-likelihood**. This is what we refer to as maximizing the likelihood of generated samples. \n\nDiffusion models sample from p(x) through sequential denoising. Now, to ensure constraint satisfaction, in CPS, we perturb the denoised sample, after every denoising update, using a projection step towards the constraint set.\n\nThis ensures that the generated samples have high likelihood while belonging to the constraint set.\n\n**Performance of Guidance-Based approaches on fewer constraints:**\nIn Figure 5, we compare all approaches that can provide satisfaction guarantees for convex constraints. Below, we extend Figure 5 with guidance-based approaches for the Stocks dataset.\n\n| Metric | Approach           | 1 constraint | 3 constraints | 5 constraints | 7 constraints | 9 constraints | 11 constraints |\n|--------|--------------------|--------------|----------------|----------------|----------------|----------------|-----------------|\n| DTW    | Diffusion-TS       | 14.4         | 8.9            | 9.3            | 8.5            | 7.6            | 7.4             |\n|        | Guided Difftime    | 14.8         | 11.7           | 10.1           | 8.3            | 8.1            | 7.9             |\n|        | **CPS**            | **12.5**     | **0.9**        | **0.4**        | **0.3**        | **0.2**        | **0.2**         |\n| FTSD   | Diffusion-TS       | 1.93         | 1.03           | 1.13           | 1.14           | 1.07           | 1.13            |\n|        | Guided Difftime    | 1.56         | 1.59           | 1.39           | 1.25           | 1.12           | 1.24            |\n|        | **CPS**            | **0.94**     | **0.08**       | **0.003**      | **0.002**      | **0.002**      | **0.002**       |\n\nFrom the table, **CPS comfortably outperforms guidance-based approaches in sample quality**, even for a smaller number of constraints. Additionally, due to the absence of any principled projection step, these approaches do not provide constraint satisfaction even for convex constraints. We observe similar trends for all real-world datasets.\n\n**Clarification regarding the qualitative results in Figures 4, 9, and 10:**\nIn Figures 4 and 9, we observe almost perfect overlap for a few datasets, specifically for the conditional variants and stocks. **This is primarily due to conditional inputs and a large number of constraints.** \n\nFor the conditional variants (Air Quality Conditional and Traffic Conditional), **the generated samples are already condition or metadata-specific and therefore track the real sample closely.** In addition, when we impose constraints, the generated samples almost perfectly match the real samples. This does not happen in the unconditional cases and is empirically demonstrated through the **differences in the DTW metrics** between the unconditional and conditional cases (2.35 vs 1.83 for Air Quality and 3.41 vs 0.84 for Traffic, lower is better).\n\nFor the stocks dataset, the number of constraints is too high (450) that **the feasible set is very small** and therefore the generated samples closely track the real sample from the constraint set. \n\nConsequently, in both these figures, note that the least similarity between the real and generated samples can be observed for the unconditional traffic and air quality datasets which do not have a large number of constraints as the stocks dataset. \n\nThe generated samples only roughly follow the trend of the real sample, but do not match as perfectly as compared to the conditional variants. \n\nTherefore, in Figure 10, while depicting the distribution of samples for a given constraint set, we observe higher overall variance for the unconditional traffic and air quality setups when compared to the Stocks dataset which has a larger number of constraints. \n\nAdditionally, as we impose the **value at** constraints, note that the standard deviation is 0 at 1, 24, 48, 72, and 96 timestamps.\n\nFinally, the mismatch with the real samples do not translate to poor quality samples. Our objective with the qualitative results is to mainly showcase the matching trends with least distortions due to projections. This is confirmed by the high sample quality metrics for CPS.\n\n**Clarification regarding the constraints used in Figures 5 and 7:**\nThe reviewer is correct. The x-axis label for both figures 5 and 7 should be “Number of constraint categories”. We will update figure 5 accordingly. In both figures, we use the following convention:\n\n**1 constraint** -  argmax\n\n**3 constraints** - argmax, value at argmax, value at 96\n\n**5 constraints** - argmin, argmax, value at argmax, value at 72 and 96\n\n**7 constraints** - argmin, argmax, value at argmin, value at argmax, value at 48, 72 and 96\n\n**9 constraints** - argmin, argmax, value at argmin, value at argmax, mean, value at 24, 48, 72 and 96\n\n**11 constraints** - argmin, argmax, value at argmin, value at argmax, mean, mean change, value at 1, 24, 48, 72 and 96\n\nWe uniformly distributed the value at constraints over these 6 experimental settings. For the stocks dataset, we always impose the OHLC constraint (lines 291 and 292).\n\nYes, the constraints are applied to every channel in multivariate datasets like Air Quality and Stocks.\n\nFor the **air quality dataset (6 channels)**, there are **66 constraints (6 mean + 6 mean change + 6 argmax + 6 argmin + 6 value at argmax + 6 value at argmin + 5x6 value at constraints)**. The same extends to the conditional variants as well.\n\nFor the **traffic dataset (1 channel)**, there are **11 constraints** (similar to the air quality dataset). The same extends to the conditional variants as well.\n\nFor the **stocks dataset (6 channels)**, there are **450 constraints (66 (similar to air quality) + 4x96 OHLC constraints)**. Note that the OHLC constraints are on an individual time stamp level. \n\nFor the **waveforms dataset (1 channel)**, there are **106 constraints (11 (similar to traffic) + 95 trend constraints between successive time stamps)**.\n\n**Modifications to the manuscript**\n\n1. We will better describe likelihood maximization in line 162 to 166.\n\n2. We will include more details regarding non-linear constraints in the experiments section.\n\n3. We will update the manuscript to use uniform notation for the sample quality metric based on the Frechet Distance. \n\n4. We will include prior works such as [1], [2], [3], [4], [5], etc., and more.\n\n5. We will address the issue with the hyperlinks.\n\n6. We will update the x-axis label for Figure 5.\n\n**References**\n\n[1] The Rise of Diffusion Models in Time-Series Forecasting.\n\n[2] A Versatile Diffusion Model for Audio Synthesis.\n\n[3] Non-autoregressive Conditional Diffusion Models for Time Series Prediction.\n\n[4] Generative Time Series Forecasting with Diffusion, Denoise, and Disentanglement.\n\n[5] Self-Guiding Diffusion Models for Probabilistic Time Series Forecasting."}]

Author rebuttal:

Meta-review: This paper proposes Constrained Posterior Sampling, a diffusion-based method for constrained time series generation. 
The reviewers all found the approach novel and well-motivated, with good empirical results and clear writing. Some weakness was also noted, including the theoretical analysis that relies on assumptions that do not fully match the implemented algorithm. Other concerns included clarity on constraint types, scalability analysis, and handling of non-linear constraints. The authors rebuttal helped and the committed to improve the paper in the camera-ready version, was appreciated. 

Overall, despite some theoretical weaknesses, the method represents a meaningful and useful contribution.
Decision: Accept (poster)