Decoupled-PolyGCN: Graph Convolution Insights

• The paper introduces Decoupled-PolyGCN, isolating graph diffusion via polynomial operators to explicitly control smoothing and receptive fields.
• It details multiple formulations including simplified GCN, APPNP, and continuous-time diffusion approximations that mitigate over-smoothing and label noise.
• The approach extends to adaptive pseudo-label training, positive/negative coupling analysis, and layer-wise optimization for scalable, application-specific implementations.

Searching arXiv for the cited Decoupled-PolyGCN-related papers to ground the article. Decoupled-PolyGCN denotes a class of graph convolutional formulations in which graph propagation is represented as a polynomial operator and is separated from at least one other component of learning, most commonly feature transformation. In the cited literature, this family includes decoupled GCNs of the form $\hat{Y}=\bar{A}f_\theta(X)$ or $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$, where $\bar{A}$ is a graph diffusion operator; polynomial spectral constructions that explicitly separate positive and negative activation; continuous-time diffusion schemes that decouple the diffusion horizon from the number of propagation steps; and training procedures that decouple feature aggregation from feature transformation (Dong et al., 2020, Wen et al., 2024, Wang et al., 2021, You et al., 2020). In a broader application-specific sense, related decoupled graph filters have also been used to model heterogeneous multi-hop wireless interference through separate self and neighbor channels (Tarzjani et al., 15 Oct 2025).

1. Canonical formulation and operator families

The starting point is the contrast between the original GCN layer and its decoupled counterparts. In the coupled form, each layer performs transformation and neighborhood aggregation together,

$H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$

where $S$ is a normalized adjacency. Decoupling separates these roles: a standard neural network first computes node-wise logits or normalized scores $f_\theta(X)$, and a graph operator then propagates them,

$$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$

For undirected graphs with adjacency $A\in\mathbb{R}^{n\times n}$ and degree $D=\mathrm{diag}(d_1,\dots,d_n)$, the normalizations used in this literature include the symmetric normalization $S=D^{-1/2}AD^{-1/2}$, the row normalization $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$0, and the self-looped normalization $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$1 with $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$2 and $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$3 (Dong et al., 2020).

Within this decoupled template, two canonical instances are emphasized. Simplified GCN uses a single $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$4-step propagation,

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$5

which is equivalent to $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$6 with $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$7 and $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$8. APPNP first computes logits $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$9 and then applies personalized PageRank propagation,

$\bar{A}$0

followed by $\bar{A}$1. Unrolling the recurrence yields a polynomial diffusion

$\bar{A}$2

This establishes the general “PolyGCN” view:

$\bar{A}$3

with APPNP coefficients $\bar{A}$4 for $\bar{A}$5 and $\bar{A}$6, while SGCN is the degenerate case $\bar{A}$7 and $\bar{A}$8 for $\bar{A}$9 (Dong et al., 2020).

This formulation is significant because it isolates graph propagation as an explicit operator. It also makes the role of polynomial coefficients transparent: they determine how much mass is assigned to different hop distances and therefore control smoothing, receptive field, and the extent to which deep propagation is attenuated or amplified.

2. Equivalence to label propagation and adaptive pseudo-label training

A central theoretical result is that, under cross-entropy training, a decoupled GCN is equivalent to a two-step procedure consisting of label propagation followed by supervised training on propagated pseudo-labels (Dong et al., 2020). The first step propagates observed labels $H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$0 on the graph, typically with clamping so that labeled nodes remain fixed. In compact polynomial form,

$H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$1

The second step trains a standard neural classifier $H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$2 on $H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$3 using weighted cross-entropy against these soft pseudo-labels. In the static Propagation then Training formulation,

$H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$4

while the general form allows weights $H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$5 that depend on graph structure and model outputs.

The equivalence is shown at the gradient level. For the decoupled GCN objective

$H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$6

the gradient matches a weighted pseudo-label objective,

$H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$7

with

$H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$8

Here, $H^{(k+1)}=\sigma\!\big(SH^{(k)}W^{(k)}\big),$9 is the structure-aware component and $S$0 is the model-aware component. The weights satisfy $S$1 for each labeled node $S$2, so every labeled source contributes equal total mass (Dong et al., 2020).

This characterization explains several observed behaviors. Robustness to structure noise and over-smoothing arises because polynomial diffusion $S$3 smooths predictions while the model-aware factor downweights pseudo-labels that disagree with the classifier; APPNP and PTS degrade slower than standard GCN as noisy edges increase. Sensitivity to label noise follows from the per-source normalization: noisy labels inject equal mass regardless of reliability. Sensitivity to initialization follows because early $S$4 predictions determine $S$5; APPNP shows larger variance across random initializations than PTS, and removing normalization in PTD increases sensitivity further (Dong et al., 2020).

The proposed correction is Propagation Then Training Adaptively (PTA). PTA removes per-source normalization and introduces an epoch-adaptive exponent,

$S$6

with loss

$S$7

In concise matrix form,

$S$8

where $S$9. Early training uses small $f_\theta(X)$0 and relies primarily on the structure-aware term; later training increases the model-aware influence. On CITESEER, CORA_ML, PUBMED, and MS_Academic, PTA outperforms APPNP with statistically significant margins, including 85.90% versus 85.07% on CORA_ML with $f_\theta(X)$1. Its reported per-epoch time is approximately 3.3 ms versus 30–35 ms for APPNP, with total time 10–17 s versus 49–134 s; the fast PTA(F) variant is approximately 43 times faster per epoch than APPNP while remaining competitive in accuracy (Dong et al., 2020).

The practical implication is that decoupled polynomial propagation is not merely a feature-smoothing device. In this formulation, it is a pseudo-label generator whose training signal is modulated jointly by graph structure and model confidence.

3. Positive and negative coupling analysis and the decoupled spectral basis

A second major line of work reframes Decoupled-PolyGCN through Positive and Negative Coupling Analysis (PNCA), which defines separate positive and negative activations and uses them to redesign polynomial graph filters (Wen et al., 2024). For an undirected unweighted graph with symmetric normalized Laplacian $f_\theta(X)$2 and node features $f_\theta(X)$3, the $f_\theta(X)$4-step node activation of node $f_\theta(X)$5 is

$f_\theta(X)$6

with $f_\theta(X)$7. This activation is positive iff $f_\theta(X)$8 and all neighbor coefficients are nonnegative with at least one strictly positive; otherwise it is negative. At graph level, an activation is $f_\theta(X)$9, and a graph activation is positive when $$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$0 on edges in a graph with self-loops (Wen et al., 2024).

PNCA furnishes explicit positive and negative operators for polynomial spectral filters:

$$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$1

$$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$2

The mixed effect is their weighted fusion,

$$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$3

From the message propagation perspective, node activation is written as

$$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$4

where $$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$5 weights low-frequency smoothing signals and $$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$6 weights high-frequency, edge-enhancing signals. Under the label smoothness measure

$$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$7

positive activation decreases $$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$8 while negative activation increases it, corresponding respectively to homophily-promoting smoothing and heterophily-emphasizing contrast (Wen et al., 2024).

The proposed decoupled basis is therefore

$$\hat{Y}=\bar{A}f_\theta(X), \qquad \hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X)).$$9

and the associated GSCNet layer is

$A\in\mathbb{R}^{n\times n}$0

This decoupling removes the entanglement present in bases such as $A\in\mathbb{R}^{n\times n}$1, where a single coefficient scales both positive and negative effects simultaneously (Wen et al., 2024).

The reported empirical picture is consistent with the theory. Positive activation through $A\in\mathbb{R}^{n\times n}$2 excels on homophily graphs; negative activation through $A\in\mathbb{R}^{n\times n}$3 excels on heterophily graphs; and the mixed decoupled basis outperforms both across diverse graphs. GSCNet shows the least accuracy degradation as propagation depth increases compared with GCN, JKNet, and BernNet. Its sensitivity to polynomial order is asymmetric: on homophily graphs accuracy is relatively insensitive, whereas on heterophily graphs performance benefits when $A\in\mathbb{R}^{n\times n}$4 and $A\in\mathbb{R}^{n\times n}$5 are similar. The paper reports Micro-F1 or accuracy with 95% confidence intervals including 89.26 ± 0.38 on Cora, 80.92 ± 0.56 on Citeseer, 91.16 ± 0.34 on PubMed, 96.22 ± 0.98 on Texas, 94.59 ± 1.31 on Cornell, 42.68 ± 1.41 on Actor, 72.09 ± 1.64 on Chameleon, and 82.64 ± 0.43 on Penn94 (Wen et al., 2024).

Computationally, the forward pass can be implemented by sparse recurrences on $A\in\mathbb{R}^{n\times n}$6 and $A\in\mathbb{R}^{n\times n}$7, avoiding dense powers. The per-layer time complexity is

$A\in\mathbb{R}^{n\times n}$8

with memory $A\in\mathbb{R}^{n\times n}$9. Recommended search ranges in the paper are $D=\mathrm{diag}(d_1,\dots,d_n)$0, Adam optimization, and initialization of all polynomial coefficients $D=\mathrm{diag}(d_1,\dots,d_n)$1 to 1 (Wen et al., 2024).

4. Continuous-time diffusion and decoupling terminal time from propagation depth

A third interpretation views Decoupled-PolyGCN as a numerical approximation to continuous graph diffusion rather than as a purely discrete $D=\mathrm{diag}(d_1,\dots,d_n)$2-hop filter (Wang et al., 2021). For a graph with self-looped normalized adjacency $D=\mathrm{diag}(d_1,\dots,d_n)$3 and Laplacian $D=\mathrm{diag}(d_1,\dots,d_n)$4, the continuous-time diffusion equation is

$D=\mathrm{diag}(d_1,\dots,d_n)$5

with solution

$D=\mathrm{diag}(d_1,\dots,d_n)$6

If $D=\mathrm{diag}(d_1,\dots,d_n)$7, then $D=\mathrm{diag}(d_1,\dots,d_n)$8, so each Laplacian eigencomponent is filtered by $D=\mathrm{diag}(d_1,\dots,d_n)$9 (Wang et al., 2021).

This viewpoint clarifies why standard linear GCN propagation fails to benefit from large depth. SGC uses

$S=D^{-1/2}AD^{-1/2}$0

which corresponds to a Forward Euler discretization with step size $S=D^{-1/2}AD^{-1/2}$1 and terminal time $S=D^{-1/2}AD^{-1/2}$2. As $S=D^{-1/2}AD^{-1/2}$3 increases, the terminal time increases as well, so higher-frequency components vanish and features converge to a non-informative equilibrium. The paper’s Theorem 1 formalizes this over-smoothing mechanism, while Theorem 2 shows that Euler discretization error for fixed terminal time decreases as $S=D^{-1/2}AD^{-1/2}$4, but in SGC the effective $S=D^{-1/2}AD^{-1/2}$5 causes the error bound to grow exponentially in $S=D^{-1/2}AD^{-1/2}$6 (Wang et al., 2021).

Decoupled Graph Convolution (DGC) separates these quantities by fixing terminal time $S=D^{-1/2}AD^{-1/2}$7 and setting $S=D^{-1/2}AD^{-1/2}$8. The Euler version uses

$S=D^{-1/2}AD^{-1/2}$9

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$00

This is explicitly polynomial:

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$01

With $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$02,

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$03

and the scalar spectral response becomes

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$04

as $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$05, matching the heat kernel $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$06. The RK4 variant yields numerical error of order $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$07 (Wang et al., 2021).

The empirical results reported for this decoupled diffusion view include semi-supervised accuracies of 83.3 versus 81.0 on Cora, 73.3 versus 71.9 on Citeseer, and 80.3 versus 78.9 on PubMed for DGC versus SGC. In fully supervised node classification, DGC attains 88.2 on Cora, 78.7 on Citeseer, and 89.4 on PubMed. On Reddit, DGC reaches 95.8% versus 94.9% for SGC, while GCN is out-of-memory. Reported timings on PubMed include 65.3 ms for SGC with $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$08, approximately the same for DGC with $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$09, 225.0 ms for DGC with $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$10, and 17.0 s for GCN (Wang et al., 2021).

The conceptual consequence is precise: in this setting, polynomial degree controls numerical precision, not diffusion strength. The diffusion strength is set by the terminal time $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$11.

5. Decoupled training, layer-wise optimization, and scalable polynomial implementations

Decoupling can also be realized at the training level. L-GCN factorizes each layer into feature aggregation and feature transformation,

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$12

and then trains only the feature transformation part with mini-batches after caching the aggregated features (You et al., 2020). For the $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$13-th layer, the optimization problem is

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$14

where intermediate $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$15 is discarded after each layer and the last-layer classifier is retained.

This procedure avoids backpropagation through expanding neighborhoods. The operational sequence is fixed: compute $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$16, perform feature aggregation once to obtain $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$17, train the single-layer feature transformation on mini-batches, cache $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$18, and continue to the next layer. The same logic extends naturally to polynomial filters by precomputing

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$19

and then training either a linear mixture

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$20

or an MLP on $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$21 (You et al., 2020).

The claimed computational advantages are substantial. L-GCN training memory is $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$22 because only the mini-batch of a single-layer perceptron is active. The paper emphasizes near dataset-size-independent GPU memory during training. The learned extension $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$23-GCN introduces a controller modeled as an MDP, with action “stop layer $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$24” or “continue,” state composed of current loss, layer index, and controller hidden state, and reward

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$25

The controller is trained by REINFORCE and can reduce per-layer training without manual stopping rules (You et al., 2020).

The theoretical analysis is framed in terms of graph isomorphism and WL expressivity. Theorem 5 states that if a given architecture can be conventionally trained to achieve WL-level capacity under the paper’s injectivity conditions, then layer-wise training achieves the same capacity. Theorem 6 states that, under the stated distinguishability assumption, capacity is monotonically non-decreasing with depth (You et al., 2020).

Reported experiments show large speedups with competitive F1. On Reddit, GraphSAGE obtains F1 = 93.4 with time 998 s and GPU 4343M, VRGCN obtains F1 = 96.0 with time 201 s and GPU 1271M, L-GCN obtains F1 = 94.2 with time 44 s and GPU 621M, and $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$26-GCN obtains F1 = 94.0 with time 34 s and GPU 635M. On Amazon-3M, VRGCN reports F1 = 88.3 with time 2165 s and GPU 625M, L-GCN reports F1 = 88.4 with time 203 s and GPU 601M, and $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$27-GCN reports F1 = 88.4 with time 125 s and GPU 613M (You et al., 2020).

For Decoupled-PolyGCN, this establishes a distinct notion of decoupling: the polynomial graph operator may be fixed or precomputed, while optimization is concentrated on a comparatively small transformation head.

6. Application-specific decoupling, interpretability, and boundary conditions

An application-specific decoupled graph convolution for heterogeneous multi-hop p-CSMA networks further illustrates how decoupling can be aligned with domain structure (Tarzjani et al., 15 Oct 2025). In this setting, the network is an undirected conflict graph $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$28 with heterogeneous access probabilities $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$29, and the throughput of node $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$30 is

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$31

The paper argues that standard symmetric-normalization GCN fails because it conflates direct interference and cascading multihop interference, averages away additive suppression, and mixes self and neighbor signals before projection. Its reported test error for a standard GCN is MAE = 0.0495 and NMAE = 63.94% (Tarzjani et al., 15 Oct 2025).

The proposed D-GCN explicitly separates self transmission from neighbor interference. At depth $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$32,

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$33

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$34

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$35

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$36

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$37

The output head is

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$38

The architecture uses 8 D-GCN layers with 64 hidden units, followed by a 2-layer MLP $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$39, AdamW with learning rate 0.001 and weight decay $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$40, ReduceLROnPlateau with factor 0.5 and patience 5, and gradient clipping with max-norm 1.0 (Tarzjani et al., 15 Oct 2025).

Although the paper does not explicitly mention “PolyGCN,” ChebNet, or SGC, it places D-GCN in the broader context of polynomial filters by noting that stacked layers induce multi-hop information flow analogous to powers of $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$41. Its distinguishing properties are decoupled self and neighbor channels, per-neighbor per-layer attention without softmax normalization, and nonlinear aggregation aligned with interference physics. On the reported test set with $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$42 and single feature $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$43, D-GCN achieves MAE = 0.0026 and NMAE = 3.30%, compared with GINE at 4.70%, GIN at 21.35%, GraphSAGE at 23.72%, and standard GCN at 63.94%. Inference complexity is

$\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$44

with approximately 0.7 ms runtimes on the tested device. In gradient-based network optimization, D-GCN achieves within 1% of the theoretical optimum on a 3-node chain and essentially identical utility to exact optimization on a 10-node network, with a reported 13,621 times speedup over the exact Markov-chain procedure in the 10-node case (Tarzjani et al., 15 Oct 2025).

Across the literature, the boundary conditions of Decoupled-PolyGCN are explicit. The label-propagation equivalence assumes an undirected graph, symmetric normalized adjacency, polynomial $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$45, one-hot labels for labeled nodes, clamping during LP, and cross-entropy loss (Dong et al., 2020). The diffusion analysis assumes undirected graphs and symmetric Laplacians, with self-loops often improving convergence (Wang et al., 2021). PNCA is built on the normalized Laplacian spectrum in $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$46 and separates low-pass and high-pass activation through $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$47 and $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$48 (Wen et al., 2024). Layer-wise decoupled training presumes that cached aggregated features remain valid during optimization, which is incompatible with time-varying graphs unless re-aggregation is performed (You et al., 2020). The wireless D-GCN assumes saturated traffic, fixed transmission duration $\hat{Y}=\mathrm{softmax}(\bar{A}f_\theta(X))$49, and binary conflict graphs (Tarzjani et al., 15 Oct 2025).

Taken together, these results show that Decoupled-PolyGCN is not a single architecture but a technically coherent family of graph models in which the graph filter is made explicit and one or more entangled mechanisms are separated: propagation from transformation, pseudo-label generation from classifier fitting, positive from negative spectral activation, diffusion horizon from discretization depth, or feature aggregation from feature transformation during training.