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PAID: Pairwise Angular-Invariant Decomposition for Continual Test-Time Adaptation
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Proposes PAID, a continual test-time adaptation method that decomposes ViT weights into magnitude and direction, freezes the direction, and adapts magnitudes plus a Householder-chained orthogonal rotation so pairwise angular structure is preserved.
Problem
Continual test-time adaptation (CTTA) of pre-trained models to shifting corruptions, where existing methods mostly exploit target data and ignore domain-invariant priors in pre-trained weights, while suffering catastrophic forgetting and error accumulation.
Approach
Identify the pairwise angular structure of neuron weight directions as domain-invariant, and adapt only magnitudes plus a learnable orthogonal matrix (chained Householder reflections) applied to frozen directions, so that adaptation is a global rotation that preserves pairwise angles.
Key results
- On ImageNet-to-ImageNet-C (ViT-Base, severity 5), PAID reaches 42.2% mean error vs. 42.5% for C-MAE and 43.4% for ViDA, versus 55.8% for the source model.
- On CIFAR10-to-CIFAR10-C, PAID reaches 11.0% mean error vs. 12.6% for C-MAE; on CIFAR100-to-CIFAR100-C it reaches 24.9% vs. 26.4% for C-MAE.
- On Cityscapes-to-ACDC (3-round average, SegFormer-B5), PAID achieves 62.2% mIoU vs. 61.9% for ViDA, a +5.5 gain over source.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use PAID instead of C-MAE | ViT-Base CTTA · ImageNet-to-ImageNet-C | mean classification error rate: 42.5 -> 42.2 | Helped |
| use PAID instead of C-MAE | ViT-Base CTTA · CIFAR100-to-CIFAR100-C | mean classification error rate: 26.4 -> 24.9 | Helped |
| use PAID instead of C-MAE | ViT-Base CTTA · CIFAR10-to-CIFAR10-C | mean classification error rate: 12.6 -> 11.0 | Helped |
| use PAID instead of ViDA | Segformer-B5 CTTA · Cityscapes-to-ACDC (3-round average) | mean mIoU: 61.9 -> 62.2 | Helped |
| add magnitude updates to orthogonal rotation of frozen directions (ablation) | PAID · ImageNet-to-ImageNet-C | mean classification error rate: 44.1 -> 42.2 | Helped |
| use orthogonal rotation instead of independently updating directions (ablation) | magnitude-and-direction adaptation · ImageNet-to-ImageNet-C | mean classification error rate: 52.2 -> 42.2 | Helped |
| add magnitude updates to direction-only adaptation (ablation) | unconstrained direction updates · ImageNet-to-ImageNet-C | mean classification error rate: 51.3 -> 52.2 | Hurt |
| inject orthogonal updates into all linear layers rather than qkvo (ablation) | PAID · ImageNet-to-ImageNet-C | mean classification error rate: 45.9 -> 42.2 | Helped |
| increase Householder-chain coefficient (ablation) | PAID · ImageNet-to-ImageNet-C | classification performance | Mixed |
| use 500 source images to pre-compute source statistics (ablation) | PAID · ImageNet-to-ImageNet-C | classification performance | No clear effect |
| decrease test-time batch size (ablation) | PAID and comparison methods · CTTA benchmarks | classification performance | Hurt |
| cycle through corruption domains for 10 rounds instead of 1 (ablation) | PAID · ImageNet-C | mean classification error rate: 42.2 -> 41.6 | Helped |
Surprising findings
- Directly adjusting the direction matrix (allowing independent per-vector rotation) degrades performance relative to magnitude-only adaptation (52.2% and 51.3% vs. 46.0%).
- Standard LoRA and DoRA adapters perform worse than PAID in the same setting (49.7% and 48.1% mean error on ImageNet-C).
Limitations
- The central hypothesis that source angular structure generalizes across domains has limited validation and unclear boundaries under extreme shifts.
- Hyperparameters (learning rate, orthogonal coefficient r, loss coefficient lambda) need per-dataset tuning.
- Performance degrades at small test batch sizes (below about 4).
Experimental setup
Datasets: ImageNet-C CIFAR10-C CIFAR100-C Cityscapes ACDC
Benchmarks: ImageNet-to-ImageNet-C CIFAR100-to-CIFAR100-C CIFAR10-to-CIFAR10-C Cityscapes-to-ACDC
Metrics: classification error rate (%) mIoU (%) gain over source
Baselines: Source Pseudo-labeling TENT CoTTA DePT VDP SAR RoTTA EcoTTA ViDA C-MAE LoRA
Builds on: DoRA OFT CoTTA TENT ViDA
Closest papers by meaning
- Stationary Latent Weight Inference for Unreliable Observations from Online Test-Time Adaptation (ICML 2024 · similarity 0.87)
- A Versatile Framework for Continual Test-Time Domain Adaptation: Balancing Discriminability and Generalizability (CVPR 2024 · similarity 0.86)
- Backpropagation-Free Test-Time Adaptation via Probabilistic Gaussian Alignment (NeurIPS 2025 · similarity 0.86)
- ViDA: Homeostatic Visual Domain Adapter for Continual Test Time Adaptation (ICLR 2024 · similarity 0.86)
- Bayesian Weight Enhancement with Steady-State Adaptation for Test-time Adaptation in Dynamic Environments (ICML 2025 · similarity 0.85)
- DPCore: Dynamic Prompt Coreset for Continual Test-Time Adaptation (ICML 2025 · similarity 0.85)
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