Cross-Pseudo Supervision (CPS) Overview

• CPS is a semi-supervised learning approach that uses two parallel networks to exchange pseudo-labels, leveraging unlabeled data effectively.
• The methodology applies mutual supervision with cross-entropy loss and network perturbations, achieving mIoU improvements of 2–4% in semantic segmentation tasks.
• Practical adaptations of CPS extend its use to 3D medical imaging, geospatial analysis, and action recognition, highlighting its versatility across domains.

Searching arXiv for core CPS papers and closely related variants to ground the article in the supplied literature. Cross-Pseudo Supervision (CPS) is a semi-supervised learning approach in which two predictive networks are trained in parallel and use each other’s predictions on the same input as pseudo-label supervision. In the canonical formulation for semantic segmentation, the pseudo one-hot label map output from one perturbed segmentation network is used to supervise the other segmentation network with the standard cross-entropy loss, and vice versa; the method therefore combines consistency regularization with the use of unlabeled data as additional training signal (Chen et al., 2021). Subsequent work preserves this cross-supervisory principle while altering the network family, the pseudo-label filtering rule, the loss weighting schedule, or the domain of application.

1. Canonical formulation

The original CPS formulation addresses semi-supervised semantic segmentation with two parallel segmentation networks of identical architecture but different initialization. Both networks receive the same input image, produce confidence maps after softmax, and convert those predictions to pseudo-labels by pixel-wise argmax followed by one-hot encoding. The pseudo-labels produced by one network supervise the other network, and the supervision is bidirectional (Chen et al., 2021).

In the standard segmentation setting, both networks are fully updated during training. This differs from Mean Teacher- or FixMatch-type schemes in which only one network is generally updated. The diversity required for effective cross supervision is introduced through network perturbation, specifically different parameter initialization; in the described implementation, both backbones are initialized from the same ImageNet pre-trained weights, while the segmentation head layers are initialized randomly and independently. Standard augmentation such as random flipping, multi-scale, and cropping is applied, and the framework has been instantiated with architectures such as DeepLabV3+ and HRNet (Chen et al., 2021).

The central mechanism serves two roles. First, it encourages high similarity between the predictions of two perturbed networks for the same input image. Second, it expands training data by assigning pseudo pixel-wise labels to unlabeled images, so that unlabeled samples can participate in optimization as if they carried supervision. The method is therefore both a consistency-regularization scheme and a pseudo-labeling scheme (Chen et al., 2021).

2. Objective functions and supervision dynamics

Let the two networks be indexed by $1$ and $2$, with probability maps $\mathbf{p}_{1i}$ and $\mathbf{p}_{2i}$ at pixel $i$, and let $\mathbf{y}_{1i}$ and $\mathbf{y}_{2i}$ denote the corresponding one-hot pseudo-labels generated by argmax. The supervised loss on labeled data is the standard pixel-wise cross-entropy applied to both branches: $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$

For unlabeled data, cross-pseudo supervision exchanges the peer prediction as target: $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$ A similar term can also be defined on labeled data, yielding $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$, and the total objective is

$2$0

This formulation makes the peer network’s discrete prediction the effective supervisory signal for the current network (Chen et al., 2021).

Later CPS variants preserved the same template while changing the loss family. In 3D-CPS for abdominal organ segmentation, the per-sample Dice + Cross-Entropy loss is used for both supervised and CPS terms, and the cross-pseudo loss is applied on both labeled and unlabeled data. In that implementation, the weighting coefficient $2$1 starts at $2$2, increases linearly with each epoch up to $2$3, and is then kept fixed, with the stated aim of preventing the model from low-quality pseudo-labels in the early training process (Huang et al., 2022).

This training dynamic addresses a recurrent issue in CPS systems: pseudo-label quality is often weakest early in optimization. Several later methods are built around this point. The data therefore supports a narrow but important interpretation: CPS is simple in its base objective, but practical performance often depends on how aggressively pseudo-label supervision is introduced, filtered, weighted, or smoothed.

3. Established behavior in semi-supervised semantic segmentation

In its original semantic-segmentation setting, CPS was evaluated on Cityscapes and PASCAL VOC 2012 under low-label protocols such as $2$4, $2$5, $2$6, and $2$7 labeled partitions. The reported Cityscapes results with DeepLabV3+ (ResNet-101) show baseline versus CPS versus CPS with CutMix as follows: $2$8 for $2$9, $\mathbf{p}_{1i}$0 for $\mathbf{p}_{1i}$1, $\mathbf{p}_{1i}$2 for $\mathbf{p}_{1i}$3, and $\mathbf{p}_{1i}$4 for $\mathbf{p}_{1i}$5. On PASCAL VOC 2012 with the same backbone, the corresponding sequence is $\mathbf{p}_{1i}$6 for $\mathbf{p}_{1i}$7, $\mathbf{p}_{1i}$8 for $\mathbf{p}_{1i}$9, $\mathbf{p}_{2i}$0 for $\mathbf{p}_{2i}$1, and $\mathbf{p}_{2i}$2 for $\mathbf{p}_{2i}$3 (Chen et al., 2021).

The reported interpretation is that CPS improves mIoU by $\mathbf{p}_{2i}$4-$\mathbf{p}_{2i}$5 over strong baselines and other semi-supervised methods, especially when fewer labels are available, and that CutMix provides further gains, particularly in lower-label settings. The same study also reports that dual-network CPS substantially outperforms single-network pseudo-supervision: on VOC 2012, ResNet-50, $\mathbf{p}_{2i}$6 partition, with CutMix, the single-network result is $\mathbf{p}_{2i}$7 and CPS reaches $\mathbf{p}_{2i}$8 (Chen et al., 2021).

These results anchored CPS as a strong segmentation baseline, but they also motivated a family of generalizations. One line extended the number of interacting networks, another altered the pseudo-label type or quality control, and several domain-specific adaptations modified the loss to match 3D volumes, sparse labels, or noisy annotations.

4. Medical-image adaptations

Medical-image segmentation has been a major site of CPS adaptation because annotation is time-consuming, laborious, and expensive. In 3D-CPS, the base architecture is nnU-Net: two parallel nnU-Net models, $\mathbf{p}_{2i}$9 and $i$0, of identical architecture but different random initializations are instantiated, each with its own optimizer. Standard nnU-Net preprocessing requires foreground masks for intensity statistics; for unlabeled data, the intensity distribution is instead computed using all voxels rather than only foreground voxels. The 3D nnU-Net version processes patches resized to shape $i$1 with typical values $i$2, uses a 6-layer encoder-decoder with residual blocks, and does not use model ensembling in this setup (Huang et al., 2022).

The reported 3D-CPS training protocol uses batch size $i$3 for $i$4 and $i$5 for $i$6, $i$7 epochs, SGD, and ReduceLROnPlateau. No model ensembling or test time augmentation was used for test submissions due to resource constraints. On the MICCAI FLARE2022 validation set of $i$8 cases, the 3D nnU-Net baseline reaches mean DSC $i$9 and mean NSD $\mathbf{y}_{1i}$0, whereas 3D nnU-Net + CPS reaches mean DSC $\mathbf{y}_{1i}$1 and mean NSD $\mathbf{y}_{1i}$2. The abstract reports an average DSC of $\mathbf{y}_{1i}$3 and an average NSD of $\mathbf{y}_{1i}$4 on the FLARE2022 validation set (Huang et al., 2022).

C$\mathbf{y}_{1i}$5PS introduced a different modification: Conditional Cross Pseudo Supervision (CCPS), in which cross pseudo supervision is conditioned on a given class label. The framework uses a multi-class regular network (RNet) and a class-conditioned binary network (CNet), both based on 3D U-Net. Context-awareness is introduced by sampling two overlapping patches and computing the CCPS loss only on the overlapping region; context-averaged pseudo-labels are formed from the overlapping predictions. After a warmup period, Hard Organ Learning restricts conditional labels to hard organs. On BCV, with $\mathbf{y}_{1i}$6 labeled and $\mathbf{y}_{1i}$7 unlabeled CTs for training, baseline supervised performance is DSC $\mathbf{y}_{1i}$8, CPS reaches $\mathbf{y}_{1i}$9, and C$\mathbf{y}_{2i}$0PS reaches $\mathbf{y}_{2i}$1; on MMWHS, with $\mathbf{y}_{2i}$2 labeled and $\mathbf{y}_{2i}$3 unlabeled CTs for training, the corresponding DSC values are $\mathbf{y}_{2i}$4, $\mathbf{y}_{2i}$5, and $\mathbf{y}_{2i}$6 (Liu et al., 2023).

Diff-CL altered CPS more radically by pairing a Diffusion Segmentation (DS) network with a Convolutional Segmentation (CS) network. In this framework, the DS branch receives the noisy version of real or pseudo labels as input, while the CS branch processes images conventionally. The CPS mechanism is bidirectional: the DS branch is supervised by pseudo-labels from the CS branch, and the CS branch is supervised by pseudo-labels from the DS branch. The paper emphasizes model diversity relative to standard CPS, stating that standard CPS usually employs two identical or architecturally similar CNNs, whereas Diff-CL uses two fundamentally different models with different inductive biases. In the pancreas dataset with $\mathbf{y}_{2i}$7 labeled data, the ablation sequence is Dice $\mathbf{y}_{2i}$8 for only supervised, $\mathbf{y}_{2i}$9 for $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$0CPS, $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$1 for $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$2CPS+HFM, and $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$3 for the full method (Guo et al., 12 Mar 2025).

5. Generalizations across models, supervision regimes, and domains

The conceptual core of CPS—peer-generated pseudo-labels used as supervision—has been generalized in several directions. One direct extension is n-CPS, which replaces two subnetworks with $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$4 simultaneously trained subnetworks. In n-CPS, every network is supervised by the one-hot pseudo-labels of every other network, and the loss is averaged by $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$5 so that the total CPS loss per network remains balanced regardless of $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$6. Unlike original CPS, which uses only one network’s output at test time, n-CPS studies ensembling strategies such as max-confidence voting and soft voting. Reported results include Pascal VOC 2012 with ResNet-101, $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$7 supervised: CPS+CutMix $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$8 and n-CPS-sv+CutMix $$\mathcal{L}_s = \frac{1}{|\mathcal{D}^l|}\sum_{\mathsf{X} \in \mathcal{D}^l} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left(\ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}^*_{1i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}^*_{2i})\right).$$9; and Cityscapes with ResNet-50, $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$0 supervised: CPS $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$1 and n-CPS-sv+CutMix $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$2 (Filipiak et al., 2021).

A different generalization is cross-model pseudo-labeling with heterogeneous architectures. In action recognition, Cross-Model Pseudo-Labeling (CMPL) uses a primary backbone and a lightweight auxiliary network that predict pseudo-labels for each other. For Kinetics-400 with only $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$3 labeled data and RGB modality, supervised 3D-ResNet50 gives $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$4 Top-1, FixMatch gives $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$5, and CMPL gives $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$6. On UCF-101 with $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$7 labeled data, the sequence is $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$8, $$\mathcal{L}_{cps}^u = \frac{1}{|\mathcal{D}^u|}\sum_{\mathsf{X} \in \mathcal{D}^u} \frac{1}{W \times H} \sum_{i=0}^{W\times H} \left( \ell_{ce}(\mathbf{p}_{1i}, \mathbf{y}_{2i}) + \ell_{ce}(\mathbf{p}_{2i}, \mathbf{y}_{1i}) \right).$$9, and $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$0 (Xu et al., 2021).

In semi-supervised audio-visual source localization, Cross Pseudo-Labeling (XPL) retains two models but replaces hard one-hot pseudo-labels with soft pseudo-labels plus sharpening, adds pseudo-label exponential moving average, and uses curriculum data selection based on Pearson correlation between the two models’ prediction maps. The stated goal is to mitigate bias accumulation, noise sensitivity, and instability associated with vanilla hard pseudo-labels (Guo et al., 2024).

Geospatial segmentation introduced another domain-specific modification. For sparsely labelled land-use and land-cover images, the proposed framework uses two identical DeepLabV3+ (EfficientNet backbone) networks and replaces ordinary supervised loss with a combination of Weighted Hausdorff Erosion loss and weighted cross-entropy; the CPS term also uses weighted cross-entropy with class weights $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$1, and the unsupervised coefficient $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$2 is ramped from $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$3 to $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$4 using a sigmoid schedule. At recall threshold $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$5, the reported CPS results are Trees $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$6, Buildings $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$7, Water $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$8, and Roads $$\mathcal{L}_{cps}=\mathcal{L}_{cps}^l+\mathcal{L}_{cps}^u$$9, compared with markedly lower supervised DeepLabV3+ and UNet baselines (Dixit et al., 2024).

Variant	Distinguishing CPS mechanism	Representative reported result
n-CPS	$2$00 subnetworks, all-versus-all pseudo supervision, ensemble inference	VOC 2012, R101, $2$01: $2$02 with CutMix
CMPL	Primary backbone and lightweight auxiliary network exchange pseudo-labels	Kinetics-400, $2$03 labels: $2$04 Top-1
XPL	Cross-refine with soft pseudo-labels, sharpening, PL-EMA, curriculum selection	Reported as state-of-the-art AVSL
Geospatial CPS	Weighted Hausdorff Erosion + weighted CE + ramp-up	Recall $2$05 for Trees/Buildings/Water/Roads

These adaptations suggest that CPS is best understood as a supervision pattern rather than a fixed architecture. The invariant ingredient is mutual pseudo-label exchange; the variable ingredients are the model family, the pseudo-label representation, and the mechanism used to suppress or exploit disagreement.

6. Relation to adjacent frameworks, misconceptions, and known limitations

A recurrent misconception is to equate CPS with ordinary self-training. The supplied literature repeatedly distinguishes the two. In the original segmentation formulation, CPS uses two independently initialized networks and cross supervision rather than self-generated pseudo-labels, and this is presented as a way to reduce confirmation bias relative to single-network pseudo-labeling (Chen et al., 2021). CMPL states the distinction even more sharply: pseudo-labels are never self-generated; they are always cross-model (Xu et al., 2021).

A second misconception is that CPS is intrinsically limited to two identical CNNs. That description is accurate for the standard formulation, but later work generalizes it in several directions. Diff-CL explicitly contrasts itself with standard CPS by combining diffusion and convolution segmentation networks rather than two architecturally similar CNNs (Guo et al., 12 Mar 2025), and n-CPS generalizes the bidirectional case to arbitrary $2$06 (Filipiak et al., 2021).

The principal limitation identified across the literature is pseudo-label noise. The core CPS paper notes that pseudo-label quality improves as training progresses, but later studies make the failure mode explicit. 3D-CPS linearly increases the semi-supervised weight to prevent harm from low-quality pseudo-labels in early training (Huang et al., 2022). XPL argues that vanilla hard pseudo-labels suffer from bias accumulation, noise sensitivity, and instability, and responds with sharpening, EMA, and curriculum selection (Guo et al., 2024). C$2$07PS restricts supervision to confident and context-consistent overlapping regions and adds hard-organ specialization (Liu et al., 2023). Geospatial CPS adds class weighting and Hausdorff-oriented supervision to cope with sparse and inaccurate labels (Dixit et al., 2024).

Another limitation is computational cost. Standard CPS doubles the number of predictive networks. USCS was proposed partly because ensemble-model training multiplies the cost of computation resources and decreases training efficiency; it replaces two full models with a multi-input multi-output segmentation model and reports savings of $2$08 on parameters and $2$09 on calculations relative to CPS while achieving state-of-the-art performance (Zhang et al., 2022). This indicates that the supervisory principle of CPS can be preserved even when the literal two-network implementation is altered.

Several related frameworks can be read as critiques or extensions of CPS rather than direct instances of it. CLS generalizes pseudo-labeling by adding complementary labels for negative learning, entropy-based soft reweighting, and cross-supervision between two peer networks; its authors describe it as extending and differing from CPS by using both positive and negative artificial labels (Yao et al., 2022). CPCL modifies dual-network pseudo supervision by separating agreement and disagreement into intersection supervision for a conservative branch and union supervision for a progressive branch, with dynamic confidence-based reweighting (Fan et al., 2022). CroSel is conceptually akin to CPS in that two models select pseudo-labels for each other, but it is tailored to partial-label learning with candidate label sets, memory-bank-based stability criteria, and co-mix regularization rather than ordinary unlabeled data (Tian et al., 2023).

Taken together, these developments define CPS as a broad research paradigm in semi-supervised learning: mutual pseudo-label exchange between multiple predictors, typically introduced to exploit unlabeled data while reducing the self-confirmation problems of single-model pseudo-labeling. The later literature does not abandon this premise; it repeatedly restructures it around confidence masking, conditional supervision, uncertainty weighting, architectural diversity, or larger ensembles.