Denoising Path Dependence in Diffusion Models

• Denoising-path dependence is the phenomenon where the chosen reverse trajectory, token update order, or key-timestep schedule influences residual error accumulation and final output quality.
• Models like SPI-GAN and ShortDF illustrate that geometric straightening or shortest-path optimization can preserve image fidelity while reducing computational steps.
• Research in diffusion language models and DPCache demonstrates that local circulations, pseudo-joint gaps, and path-aware cost tensors offer actionable diagnostics for achieving order consistency and efficient sampling.

to=arxiv_search 北京赛车计划 无码不卡高清免费 code: {"query":"id:(Jeon et al., 2022) OR id:(Kim, 10 May 2026) OR id:(Cui et al., 26 Feb 2026) OR id:(Chen et al., 5 Mar 2025)","max_results":10,"sort_by":"submittedDate","sort_order":"descending"}ასუხარი to=arxiv_search code 大发时时彩是pecially: [{"arxiv_id":"(Kim, 10 May 2026)","title":"Path-Dependent Denoising: A Non-Conservative Field Perspective on Order Collapse in Diffusion LLMs","authors":["Borui Wang","Yongchao Zhou","Jingtao Zhan","Kexin Feng","Minyuan Li","Pengxiang Cheng","Jian-Yun Nie","Ji-Rong Wen"],"abstract":"Diffusion LLMs (DLMs) offer a structural alternative to autoregressive generation: denoising can update tokens in arbitrary orders or in parallel rather than along a fixed left-to-right chain. In practice, fast DLM decoding remains strongly order-sensitive and often drifts toward autoregressive-like trajectories. We trace this tension to compatibility. At each reverse-time step, a DLM provides local denoising conditionals over the unresolved tokens. Arbitrary-order denoising becomes well defined when these local conditionals compose into order-invariant pseudo-joints. We formalize this view by defining order-induced pseudo-joints and a local denoising circulation: the log-ratio between the two pseudo-joints obtained by swapping a pair of unresolved positions. This circulation is zero under compatible conditionals, and global order gaps decompose into sums of local circulations along adjacent swaps. We further separate incompatibility-driven path dependence from conditional-dependence error in parallel updates and from order-specific estimation error. The resulting framework provides inference-only diagnostics for testing when DLM decoding is genuinely order-free.","categories":["cs.LG","cs.CL"],"published":"2026-05-10","url":"http://arxiv.org/abs/([2605.09303](/papers/2605.09303))v1"},{"arxiv_id":"([2602.22654](/papers/2602.22654))","title":"Denoising as Path Planning: Training-Free Acceleration of Diffusion Models with DPCache","authors":["Sifan Tang","Feng Liu","Yuqi Fan","Yuxiang Wei","Weiyang Yuan","Qi Zhang","Xinggang Wang","Pengfei Zhu"],"abstract":"Diffusion models have demonstrated remarkable success in image and video generation, yet their practical deployment remains hindered by the substantial computational overhead of multi-step iterative sampling. Among acceleration strategies, caching-based methods offer a training-free and effective solution by reusing or predicting features across timesteps. However, existing approaches rely on fixed or locally adaptive schedules without considering the global structure of the denoising trajectory, often leading to error accumulation and visual artifacts. To overcome this limitation, we propose DPCache, a novel training-free acceleration framework that formulates diffusion sampling acceleration as a global path planning problem. DPCache constructs a Path-Aware Cost Tensor from a small calibration set to quantify the path-dependent error of skipping timesteps conditioned on the preceding key timestep. Leveraging this tensor, DPCache employs dynamic programming to select an optimal sequence of key timesteps that minimizes the total path cost while preserving trajectory fidelity. During inference, the model performs full computations only at these key timesteps, while intermediate outputs are efficiently predicted using cached features. Extensive experiments on DiT, FLUX, and HunyuanVideo demonstrate that DPCache achieves strong acceleration with minimal quality loss, outperforming prior acceleration methods by +0.031 ImageReward at 4.87× speedup and even surpassing the full-step baseline by +0.028 ImageReward at 3.54× speedup on FLUX, validating the effectiveness of our path-aware global scheduling framework.","categories":["cs.CV","cs.LG"],"published":"2026-02-26","url":"http://arxiv.org/abs/([2602.22654](/papers/2602.22654))v1"},{"arxiv_id":"([2503.03265](/papers/2503.03265))","title":"Optimizing for the Shortest Path in Denoising Diffusion Model","authors":["Dingsheng Hong","Jiachen Hu","Linjie Yang","Yudong Chen"],"abstract":"In this research, we propose a novel denoising diffusion model based on shortest-path modeling that optimizes residual propagation to enhance both denoising efficiency and quality. Drawing on Denoising Diffusion Implicit Models (DDIM) and insights from graph theory, our model, termed the Shortest Path Diffusion Model (ShortDF), treats the denoising process as a shortest-path problem aimed at minimizing reconstruction error. By optimizing the initial residuals, we improve the efficiency of the reverse diffusion process and the quality of the generated samples. Extensive experiments on multiple standard benchmarks demonstrate that ShortDF significantly reduces diffusion time (or steps) while enhancing the visual fidelity of generated samples compared to prior arts. This work, we suppose, paves the way for interactive diffusion-based applications and establishes a foundation for rapid data generation.","categories":["cs.CV"],"published":"2025-03-05","url":"http://arxiv.org/abs/([2503.03265](/papers/2503.03265))v1"},{"arxiv_id":"([2206.14464](/papers/2206.14464))","title":"SPI-GAN: Denoising Diffusion GANs with Straight-Path Interpolations","authors":["Daehoon Gwak","Minjun Choi","Sunghun Kim","Myeongjong Kang","Jong Chul Ye"],"abstract":"Score-based generative models (SGMs) show the state-of-the-art sampling quality and diversity. However, their training/sampling complexity is notoriously high due to the highly complicated forward/reverse processes, so they are not suitable for resource-limited settings. To solving this problem, learning a simpler process is gathering much attention currently. We present an enhanced GAN-based denoising method, called SPI-GAN, using our proposed straight-path interpolation definition. To this end, we propose a GAN architecture i) denoising through the straight-path and ii) characterized by a continuous mapping neural network for imitating the denoising path. This approach drastically reduces the sampling time while achieving as high sampling quality and diversity as SGMs. As a result, SPI-GAN is one of the best-balanced models among the sampling quality, diversity, and time for CIFAR-10, and CelebA-HQ-256.","categories":["cs.CV","cs.LG"],"published":"2022-06-29","url":"http://arxiv.org/abs/([2206.14464](/papers/2206.14464))v1"}] Need likely enough. Need maybe search for score-based generative modeling, DDIM? But can avoid referencing with ids if not in data? Yet can mention as named within data; but if mention specific works need citations perhaps. We can cite only provided ids, each section once max. Maybe no need search more.

Need write comprehensive encyclopedia article with intro and 5-7 numbered h2 sections. No heading at beginning. Need only article itself. Need factual claims from data. So avoid adding too many interpretive statements unless marked. Need define topic broadly spanning image diffusion, language diffusion, acceleration, shortest path, perhaps contrast that term across literatures. Since topic is "Denoising-path Dependence", central concept is across papers. We can synthesize from four relevant papers. Need numbered h2 sections with logical titles. Could do 6 sections:

1. Concept and scope
2. Denoising paths in continuous image generators
3. Order dependence in diffusion LLMs
4. Path planning and acceleration
5. Shortest-path optimization and path shaping
6. Diagnostics, implications, and limitations

Need introductory paragraph without heading define concept. Need sections numbered manually like "## 1. Conceptual foundation". Must avoid extraneous claims not in data. Need maybe mention two distinct meanings: geometric trajectory dependence in image generation; order/grouping dependence in DLMs; schedule dependence in acceleration. Need cite (Jeon et al., 2022, Kim, 10 May 2026, Cui et al., 26 Feb 2026, Chen et al., 5 Mar 2025).

Need include math notation from data. Must be precise. Since audience technical. Could include formulas for pseudo-joint, circulation, straight-path interpolation, PACT cost, ShortDF distance. Need ensure formulas verbatim or from data. That's allowed.

Need references at most once per section for a given paper. Can cite more than one paper per section. Let's craft carefully.

Need maybe mention common misconception: "order-free interface implies order-invariant decoding"—corrected by (Kim, 10 May 2026). Another misconception: "faithfully following reverse SDE is uniquely necessary"—SPI-GAN/ShortDF/DPCache suggest simplification can work; mark as "This suggests". Need objective tone.

Need maybe include a small table summarizing operational notions across settings. Table up to 3 columns. Could use 3 columns: setting, path object, dependence criterion. Since table content from data. Good.

Need be careful with claims like "first such formulation" absent. Avoid. For DPCache metrics: there is a confusing scaling difference. User says concrete claims must appear verbatim. Could cite exact numbers as in data. Need not mention ambiguous one maybe. But can say "outperforming prior acceleration methods by +0.031 ImageReward at 4.87× speedup and even surpassing the full-step baseline by +0.028 ImageReward at 3.54× speedup on FLUX" appears verbatim in abstract. Later details also say DPCache Reward 0.958 vs SpeCa 0.927. Could mention. Need not overdo.

Need same language English. Need markdown. No preface.

Let's ensure every concrete claim appears verbatim. Even section names are not claims.

Need maybe mention "local circulation behaves like curl in a non-conservative vector field". from data. Good.

Potential challenge: Should not use "Editor's term" unless coining helpful shorthand. Not needed.

Need maybe mention not just images and language but "denoising-path dependence is the phenomenon that output depends not only on endpoints but on sequence/order/grouping or trajectory". That comes from ShortDF and DLM paper. Fine.

Let's draft around 1200-1800 words maybe. Since comprehensive. We can be concise but thorough. Need sections numbered 1-6. Let's do intro + 6 sections.

Need citations with arXiv ids bracket. Use each paper once per section max.

Let's outline:

Intro paragraph: define as phenomenon where model behavior/output depends on chosen denoising trajectory; in image diffusion path is sequence of states/timesteps from noise to data; in DLM path is order/grouping of token updates; in acceleration path is selected key timesteps. Mention across SPI-GAN, DLM, DPCache, ShortDF.

Section 1. Formal object of a denoising path

• Explain in score-based models forward SDE/reverse SDE and DDPM chain.
• Table summarizing settings and path variable: image SGMs/SPI-GAN: \mathbf{i}(u) DLMs: permutation \pi / pseudo-joint Q^\pi cached acceleration: key timestep sequence \mathcal{T} Maybe mention path dependence = different path implies different induced distribution/error/cost. Cite (Jeon et al., 2022, Kim, 10 May 2026, Cui et al., 26 Feb 2026, Chen et al., 5 Mar 2025) maybe.

Section 2. Straightening geometric denoising trajectories

• SPI-GAN replaces reverse SDE path with straight-line interpolation i(u)=u x0 + (1-u)x_T.
• mapping network NODE, generator StyleGAN2 modified, time-dependent discriminator.
• NFE and results on CIFAR/CelebA. Cite (Jeon et al., 2022).
• Note implication: elaborate SDE path not strictly necessary; mark "This suggests". Need metrics exactly from data.
• mention balance between noisy and clean images.

Section 3. Order dependence in diffusion LLMs

• define local conditionals q, compatible if exists joint
• order-induced pseudo-joint Q^\pi
• order gap Delta
• local circulation C_ij, theorem and expected circulation equals KL.
• global decomposition by adjacent swaps.
• Three error sources: incompatibility/curl, conditional total correlation, order-specific estimation error.
• Bayes-optimal uniform masking zero curl.
• diagnostics and order collapse. Cite (Kim, 10 May 2026).

Section 4. Path-dependent scheduling in accelerated diffusion

• DPCache operationalizes path dependence in skipping timesteps with PACT tensor C[i,j,k] sum of L1 diffs of final layer features. Explain indices.
• objective Cost(T)=sum C[t_{m-1}, t_m, t_{m+1}], DP recurrence. mention O(KT^2), O(KT). fixed early M=3 maybe mention in experiments.
• Why 2D cost insufficient; Table 4 numbers 1.001/20.87, 1.007/21.65, 0.977/19.46.
• Main empirical results on FLUX, HunyuanVideo, DiT exact metrics perhaps brief. cite (Cui et al., 26 Feb 2026). Need ensure each number appears exactly from data.

Section 5. Shortest-path optimization of denoising

• ShortDF on DDIM deterministic reverse mapping sigma_n=0
• residuals, path residual, graph nodes/edges, edge(k,t), dist(x_t,t), relaxation condition, loss.
• empirical setup and metrics on CIFAR, CelebA, LSUN. mention exact FIDs and step comparisons.
• This suggests training can reshape path dependence.

Section 6. Cross-cutting implications and controversies

• Common misconception 1: order-free interface => order-invariant generation. corrected by DLM paper.
• Misconception 2: path simplification necessarily degrades quality. SPI-GAN, ShortDF, DPCache show simplified/optimized paths can preserve or improve metrics under specific settings.
• Distinguish dependence types: trajectory geometry, token-update order, schedule/path-conditioning under caching.
• Structural criteria / diagnostics: zero circulation reachable curl-free, pseudo-joint gaps, operator commutators, conditional TC proxies, path-aware cost tensors, wall-clock/NFE/FID etc. Need maybe avoid mixing not verbatim? All terms from data. Fine.
• Future directions cautiously: learned nonlinear paths, explicit joint potentials, adaptive parallelism, flow-matching/shortest-path diffusion ideas. But must mark as suggestion/plausible implication because not all directly proven. DLM and SPI-GAN mention. Use "The cited works point toward..." maybe okay because directly supported. But concrete future direction names from data. Good.

Need maybe mention that DPCache assumes content-agnostic trajectory shape, Markovian path dependence wrt last key, smooth feature evolution. Good.

Let's ensure citations no more than once/section/paper. Could cite each relevant paper once per section. Fine.

Need markdown table if using. Let's produce concise table in section 1.

Let's write with neutral tone, technical. Denoising-path dependence is the phenomenon that the behavior, induced distribution, or accumulated error of a denoising system depends on the particular path used to move from a noisy state toward a clean state. In diffusion-style image generation, the path is a trajectory through state space or through a timestep schedule; in diffusion LLMs, it is the order and grouping of token updates; in training-free acceleration, it is the sequence of key timesteps at which full computation is performed. Across these settings, the central question is not only whether denoising reaches the target distribution, but how the chosen trajectory, ordering, or schedule alters quality, diversity, stability, and computational cost (Jeon et al., 2022, Kim, 10 May 2026, Cui et al., 26 Feb 2026, Chen et al., 5 Mar 2025).

1. Formal object of the denoising path

In score-based generative models, the standard picture is a forward SDE that corrupts a data sample $\mathbf{x}_0$ into a noisy state $\mathbf{x}_T$, followed by a reverse-time denoising process. The induced denoising path is the sequence $\{\mathbf{x}_t\}_{t=0}^T$ or its discretization $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$. In DDPM-style formulations this becomes a Markov chain $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$, while in score-SDE formulations it is a numerical trajectory governed by the score field $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$ (Jeon et al., 2022).

In diffusion LLMs, the path is not a spatial trajectory in image space but an update order over unresolved coordinates. For a block $B$ of unresolved tokens and a permutation $\pi$, the order-induced pseudo-joint is

$$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$

Denoising-path dependence arises when different permutations produce different pseudo-joints, so the completed-sequence distribution depends on the order or grouping of denoising updates (Kim, 10 May 2026).

In training-free diffusion acceleration, the path is a schedule of key timesteps

$$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$

where the model is fully evaluated only at selected key steps and skipped steps are predicted from cached features. Here dependence means that the cost of a skip is conditioned on the previously computed anchors, not only on the current timestep interval (Cui et al., 26 Feb 2026).

In shortest-path formulations of diffusion, timesteps become graph nodes and denoising trajectories become graph paths. The output quality then depends on residual accumulation along the chosen path rather than only on the endpoints (Chen et al., 5 Mar 2025).

Setting	Path object	Dependence criterion
Score-based image generation	$\mathbf{x}_T$0 or timestep subsequence	Sample quality and NFE depend on the reverse trajectory
Diffusion LLMs	Permutation or blockwise update schedule	Pseudo-joints $\mathbf{x}_T$1 differ across orders
Caching-based acceleration	Key-timestep path $\mathbf{x}_T$2	Skip cost depends on preceding key timestep

2. Straight-path interpolation and geometric simplification

"SPI-GAN" (Jeon et al., 2022) studies denoising-path dependence by replacing the complicated reverse SDE trajectory with a straight-line interpolation in image space. Given a clean data sample $\mathbf{x}_T$3 and a noisy endpoint $\mathbf{x}_T$4, it defines

$\mathbf{x}_T$5

with $\mathbf{x}_T$6 and $\mathbf{x}_T$7. The model explicitly states, “We use a straight-path between $\mathbf{x}_T$8 and $\mathbf{x}_T$9 to learn the shortest path with the minimum Wasserstein distance” (Jeon et al., 2022).

This straight-path construction changes denoising-path dependence in three ways. First, it imposes geometric simplicity: the path is linear in pixel space rather than following nonlinear score dynamics. Second, it imposes temporal simplicity: $\{\mathbf{x}_t\}_{t=0}^T$0 is sampled uniformly and treated as a continuous time variable instead of a fixed discrete noise schedule. Third, it collapses sampling to a single generator evaluation at $\{\mathbf{x}_t\}_{t=0}^T$1, rather than a multi-step reverse process (Jeon et al., 2022).

The architecture is organized around this path. An embedding network computes $\{\mathbf{x}_t\}_{t=0}^T$2, and a Neural ODE mapping network evolves the latent along continuous time:

$\{\mathbf{x}_t\}_{t=0}^T$3

A modified StyleGAN2 generator produces $\{\mathbf{x}_t\}_{t=0}^T$4, and a time-dependent discriminator distinguishes real and fake pairs $\{\mathbf{x}_t\}_{t=0}^T$5 and $\{\mathbf{x}_t\}_{t=0}^T$6. The discriminator is therefore conditioned not only on image realism but on path position (Jeon et al., 2022).

The paper reports NFE $\{\mathbf{x}_t\}_{t=0}^T$7 for SPI-GAN, compared with NFE $\{\mathbf{x}_t\}_{t=0}^T$8 for DDPM, $\{\mathbf{x}_t\}_{t=0}^T$9 for Improved DDPM, $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$0 for Score SDE (VP/VE), $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$1 for LSGM, and $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$2 for DD-GAN with $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$3. On CIFAR-10, SPI-GAN reports IS $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$4, FID $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$5, Recall $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$6, and NFE $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$7; on CelebA-HQ-256 it reports FID $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$8, outperforming DD-GAN $\mathbf{x}_T,\mathbf{x}_{T-1},\dots,\mathbf{x}_0$9, Score SDE $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$0, LSGM $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$1, and UDM $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$2 (Jeon et al., 2022). The wall-clock comparison on CIFAR-10 with batch size 100 gives 0.04 s for SPI-GAN, 0.04 s for Diffusion-GAN (StyleGAN2), 0.04 s for StyleGAN2, and 0.36 s for DD-GAN (Jeon et al., 2022).

These results are specific to the reported datasets, but they suggest that the elaborate reverse SDE trajectory is not uniquely necessary for high-quality image synthesis. The paper further argues that the straight-path interpolation “maintains a better balance between noisy and clean images than the case where we sample following the SDE path,” linking path design to discriminator exposure and training stability (Jeon et al., 2022).

3. Order-induced path dependence in diffusion LLMs

"Path-Dependent Denoising: A Non-Conservative Field Perspective on Order Collapse in Diffusion LLMs" (Kim, 10 May 2026) gives a formal account of denoising-path dependence for discrete token generation. At a fixed reverse-time step $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$3, a diffusion LLM provides local conditionals $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$4 over unresolved coordinates. Order-free denoising would require that these local conditionals compose into the same distribution over completed blocks regardless of update order (Kim, 10 May 2026).

The paper adopts compatibility as the exact criterion. The family $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$5 is compatible if there exists a joint $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$6 whose conditional distributions match all these $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$7. Under compatibility, all order-induced pseudo-joints coincide by the chain rule. Under incompatibility, different permutations $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$8 produce different $p(\mathbf{x}_{t-1}\mid \mathbf{x}_t)$9, so denoising is order-dependent at the model level (Kim, 10 May 2026).

The local mechanism is quantified by local denoising circulation. For two unresolved positions $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$0 and tokens $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$1,

$\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$2

This is the log-ratio between the two pairwise pseudo-joints obtained by denoising $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$3 versus $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$4. The expected circulation under one order equals the KL divergence between the two pairwise pseudo-joints:

$\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$5

Zero circulation therefore means local order invariance; non-zero circulation means local non-integrability in the discrete field of log-conditionals (Kim, 10 May 2026).

The paper then shows that global order gaps decompose exactly into sums of local circulations along adjacent swaps. For two permutations $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$6 of a block $\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$7,

$\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$8

This gives a path-integral interpretation: the discrepancy between two denoising orders is the accumulated circulation along a path through permutation space (Kim, 10 May 2026).

The framework separates three sources of practical degradation. The first is incompatibility-driven path dependence, measured by curl or circulation. The second is conditional-dependence error in parallel updates, quantified in the Bayes-optimal case by the conditional total correlation

$\nabla_{\mathbf{x}} \log p_t(\mathbf{x})$9

The third is order-specific estimation error, where certain orders are favored because the model is more accurate on the contexts they visit. This decomposition is used to explain “order collapse,” namely the empirical drift of practical DLM decoding toward left-to-right or autoregressive-like trajectories (Kim, 10 May 2026).

A common misconception addressed directly by this formulation is that an “order-free” denoising interface implies order-invariant decoding semantics. The paper shows that this is true only when the local conditionals are compatible and circulation is zero on every reachable local square. It also provides inference-only diagnostics: local circulation measurement, pseudo-joint gap estimation on blocks, operator commutator diagnostics, conditional-dependence controls, order-specific error profiling, and parallelism stress tests (Kim, 10 May 2026).

4. Path-aware scheduling in accelerated diffusion

"Denoising as Path Planning: Training-Free Acceleration of Diffusion Models with DPCache" (Cui et al., 26 Feb 2026) studies denoising-path dependence at the level of timestep skipping and cached prediction. Its central claim is that the error at a skipped timestep is path-dependent: it depends on which previous timesteps were actually computed, how many and which intermediate steps are skipped, and in particular on the immediate predecessor key timestep (Cui et al., 26 Feb 2026).

This dependence is encoded in the Path-Aware Cost Tensor

$B$0

with entries $B$1 for $B$2. Using only the final layer feature map $B$3, the tensor is defined as

$B$4

The crucial conditioning is on both $B$5 and $B$6: the cost of skipping from key $B$7 to key $B$8 changes when the preceding key timestep $B$9 changes, because the cached anchors used for prediction are different (Cui et al., 26 Feb 2026).

Given a path of key timesteps

$\pi$0

the total path cost is

$\pi$1

and the optimal schedule is selected by dynamic programming. The recurrence

$\pi$2

shows where path dependence enters: the segment cost depends on the predecessor of the predecessor, not only on the current jump. The reported complexity is $\pi$3 time and $\pi$4 memory, and the first $\pi$5 timesteps are fixed as key steps in the experimental setting (Cui et al., 26 Feb 2026).

The ablation on PACT is explicit. Table 4 reports: 2D, non-cumulative gives ImageReward 1.001 and PSNR 20.87; 3D, cumulative gives ImageReward 1.007 and PSNR 21.65; 2D + cumulative degrades to ImageReward 0.977 and PSNR 19.46. The paper interprets this as evidence that cumulative cost without path conditioning misestimates the global cost landscape and leads dynamic programming to poor schedules (Cui et al., 26 Feb 2026).

The experimental claims are likewise path-specific. On FLUX.1-dev, DPCache reports +0.031 ImageReward at 4.87$\pi$6 speedup relative to prior acceleration methods and +0.028 ImageReward at 3.54$\pi$7 speedup relative to the full-step baseline. On HunyuanVideo at 4.75$\pi$8 speedup it reports VBench 80.23%, PSNR 21.04, and SSIM 0.6852; on DiT-XL/2 at 3.02$\pi$9 speedup with $$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$0 it reports FID 3.285 and sFID 5.063 (Cui et al., 26 Feb 2026).

The paper’s assumptions are also explicit: content-agnostic trajectory shape, Markovian path dependence with respect to the last key timestep, and smoothness of feature evolution. Its stated limitation is that it focuses on fidelity to the original sampling trajectory and can therefore faithfully propagate or amplify errors already present in the base model (Cui et al., 26 Feb 2026).

5. Shortest-path modeling of residual propagation

"Optimizing for the Shortest Path in Denoising Diffusion Model" (Chen et al., 5 Mar 2025) reframes denoising-path dependence as a shortest-path problem over DDIM trajectories. The model, ShortDF, uses the standard DDPM forward process and the deterministic DDIM reverse mapping with $$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$1:

$$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$2

The denoising path is an ordered sequence of timesteps

$$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$3

and the paper’s claim is that some such paths are better than others for a fixed number of steps (Chen et al., 5 Mar 2025).

The shortest-path construction is based on residual propagation. The distance from node $$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$4 to $$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$5 is

$$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$6

and for a pair $$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$7 the edge cost is

$$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$8

A Bellman-Ford-style relaxation condition is then imposed:

$$Q_{\theta,t}^\pi(x_B \mid x_S) = \prod_{m=1}^{|B|} q_{\theta,t}\bigl(x_{\pi_m}\mid x_S,x_{\pi_{<m}}\bigr).$$9

If the condition holds, going from $$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$0 to $$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$1 via $$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$2 is treated as a shorter path than the current direct path (Chen et al., 5 Mar 2025).

Training augments the standard noise-prediction loss with a shortest-path relaxation loss

$$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$3

where

$$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$4

and

$$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$5

The intended effect is to reduce the initial residuals at relatively late and noisy steps so that direct jumps to $$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$6 become competitive with longer DDIM paths (Chen et al., 5 Mar 2025).

The empirical comparisons are step-sensitive. On CIFAR-10, the paper reports 10-step DDIM FID $$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$7 versus 2-step ShortDF FID $$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$8, and at 10 steps reports gDDIM(DDPM) 4.17 versus ShortDF 3.75. On CelebA, at 10 steps it reports DDIM 10.59, DPM-solver++ 8.31, and ShortDF 5.00; at 20 steps it reports DDIM 6.35 and ShortDF 3.25. On LSUN Churches, at 10 steps it reports DDIM 21.52, PNDM 11.80, and ShortDF 11.78; at 50 steps it reports DDIM 7.65 and ShortDF 6.95 (Chen et al., 5 Mar 2025).

These results support a specific interpretation of denoising-path dependence: final quality depends on how residual errors accumulate along the chosen path, and training can reshape the model so that very short deterministic paths become nearly optimal in residual space (Chen et al., 5 Mar 2025).

6. Synthesis, diagnostics, and conceptual boundaries

Across these works, denoising-path dependence is not a single mechanism but a family of structurally related phenomena. In SPI-GAN, it concerns the geometry of the reverse trajectory between $$\mathcal{T} = \{t_1 = T > t_2 > \dots > t_K > t_{K+1}=0\},$$9 and $\mathbf{x}_T$00; in diffusion LLMs, it concerns the algebra of local conditionals under permutations and block updates; in DPCache, it concerns schedule-dependent skip error conditioned on cache history; in ShortDF, it concerns residual accumulation along DDIM subsequences (Jeon et al., 2022, Kim, 10 May 2026, Cui et al., 26 Feb 2026, Chen et al., 5 Mar 2025).

A second common misconception is that path simplification necessarily reduces quality. The image-generation papers do not support that as a general rule. SPI-GAN reports FID 3.01, Recall 0.66, and NFE 1 on CIFAR-10, plus FID 6.62 on CelebA-HQ-256 (Jeon et al., 2022). ShortDF reports stronger low-step FID than DDIM on CIFAR-10, CelebA, and LSUN Churches (Chen et al., 5 Mar 2025). DPCache reports improved speed-quality trade-offs through path-aware scheduling rather than learned path-following (Cui et al., 26 Feb 2026). This suggests that path choice is a design degree of freedom rather than a fixed consequence of the forward noising process.

The strongest exact structural criterion appears in the language-model setting: exact order consistency on a block is equivalent to zero circulation on every reachable local square under strictly positive conditionals (Kim, 10 May 2026). In the acceleration setting, the operational criterion is not compatibility but faithful path-cost modeling, implemented through $\mathbf{x}_T$01 and dynamic programming (Cui et al., 26 Feb 2026). In straight-path and shortest-path image models, the criterion is empirical: NFE, FID, IS, Recall, wall-clock time, and reconstruction-oriented residual behavior (Jeon et al., 2022, Chen et al., 5 Mar 2025).

The papers also delimit future research directions without resolving them. SPI-GAN explicitly does not experiment with learned or nonlinear paths and cites shortest path diffusion and flow-matching as related directions (Jeon et al., 2022). The DLM framework points toward architectures or training objectives for compatibility, explicit joint potentials $\mathbf{x}_T$02, commutator-aware schedulers, and adaptive parallelism (Kim, 10 May 2026). DPCache suggests that the cost landscape over $\mathbf{x}_T$03 can be treated as a discrete-time approximation to diffusion dynamics and that global planning can outperform local greediness (Cui et al., 26 Feb 2026). ShortDF suggests that training-time objectives can directly favor efficient paths rather than only better local denoisers (Chen et al., 5 Mar 2025).

Taken together, these results define denoising-path dependence as a central modeling variable. The path may be straightened, reordered, optimized, or globally planned, but it cannot in general be treated as a neutral implementation detail.