cs.LGMar 23, 2026
SaveShow Me What You Don't Know: Efficient Sampling from Invariant Sets for Model Validation
Organizations: Computer Vision and Learning Lab, Heidelberg University.
Abstract
The performance of machine learning models is determined by the quality of their learned features. They should be invariant under irrelevant data variation but sensitive to task-relevant details. To visualize whether this is the case, we propose a method to analyze feature extractors by sampling from their fibers -- equivalence classes defined by their invariances -- given an arbitrary representative. Unlike existing work where a dedicated generative model is trained for each feature detector, our algorithm is training-free and exploits a pretrained diffusion or flow-matching model as a prior. The fiber loss -- which penalizes mismatch in features -- guides the denoising process toward the desired equivalence class, via non-linear diffusion trajectory matching. This replaces days of training for invariance learning with a single guided generation procedure at comparable fidelity. Experiments on popular datasets (ImageNet, CheXpert) and model types (ResNet, DINO, BiomedClip) demonstrate that our framework can reveal invariances ranging from very desirable to concerning behaviour. For instance, we show how Qwen-2B places patients with situs inversus (heart on the right side) in the same fiber as typical anatomy.
Figures & tables
Figure 1: Sampling from the invariant set by guiding pretrained generative models. Top: Black arrows represent the ordinary denoising steps in a flow-matching model. Alternating black (FM) and red (correction) arrows indicate the guided trajectory to the dashed target fiber. Bottom: Given a case of ”situs inversus”, where the heart is located on the right side of the body, our invariance sampling – guided by the Qwen-2B vision model – finds several images that yield the same features but have the heart on the left side.
Table 1: Comparison of different methods for sampling from invariant sets. We give a “ ✓ ” if a method fulfills a property and a “ ” if a property is approximately fulfilled.
Figure 2: Invariance learning with conditional generative models. The fiber model is trained to generate images based on the subject model representations . Conditioning the fiber model on a target representation at inference, its samples should be invariant under the subject model. This can be verified using the fiber loss, which can also be used to supplement the training.
Figure 3: Invariance learning using a pretrained, unconditional diffusion model. At each step in the denoising process, the sample is extrapolated to the final step using Tweedie’s estimate. The fiber loss between this estimate and the target representation is then used to find a correction to the sample in each timestep. This requires no finetuning of the pretrained diffusion model.
Figure 4: Benchmark of different invariance learning methods and architectures on color MNIST. Fiber samples are evaluated by KL-divergence of background color distribution and the fiber loss (lower is better for both). Each colored line represents one method, with the marker size representing the strength of the regularization/guidance. NDTM and NDTM w/ OOD guide pretrained unconditional models, while other methods train conditional generators from scratch.
Figure 5: Samples from the fibers of DINOv2, drawn with a pretrained ImageNet diffusion model using NDTM. The first three columns show the original, nearest neighbor in the dataset and a version of the original augmented with DINOv2 training augmentations. The last three columns show different fiber samples, ordered by fiber loss.
Figure 6: Samples from the fibers of InceptionV3, drawn with a pretrained ImageNet diffusion model using NDTM. The first three columns show the original, nearest neighbor in the dataset and a version of the original augmented with DINOv2 training augmentations. The last three columns show different fiber samples, ordered by fiber loss.
| Subject Model | Fiber Model | TF FID | PT FID | Classifier top-5 accuracy |
| Baseline on ImageNet | 26.21 | - | ||
| DINOv2 | NDTM | 28.34 | 29.68 | |
| InceptionV3 | NDTM | 33.52 | 3.81 | |
Table 2: FIDs and top-5 accuracy of a pretrained ConvNeXt-L on fibers of DINOv2 and InceptionV3. We measure FID both with a TensorFlow (TF) and PyTorch (PT) Inception model. Baselines show the unconditional diffusion FID and classifier performance on ImageNet.
Figure 7: With samples from the cue conflict dataset as input, we use NDTM to draw samples from a pretrained ImageNet diffusion model that are invariant under the representation of DINOv2 ViT-B14 and pretrained ResNet50. From left to right: Original image (ground truth shape and texture label on the left side), augmented image, two invariant samples of DINOv2, two invariant samples of ResNet50. Below each image we show the label predicted by an ImageNet-trained ConvNeXt-L.
Figure 8: Label preference of ConvNeXt-L, pretrained on ImageNet, on the cue conflict dataset and invariant samples under DINOv2 and ResNet50. In blue and orange we show the amount of times the classifier predicts the shape and texture labels respectively. Cases where none of the two labels were predicted or both labels coincide are filtered out.
Figure 9: Originals with negative cardiomegaly diagnosis and samples of the fiber from BiomedClip (left) and ConvNeXt (right), obtained by our search for contradicting samples.
Figure 10: Classifier Agreement of a BiomedClip classifer with a ConvNeXt classifer on the CheXpert dataset. For images from the test set where both models get the correct prediction, we sample from the fiber of each model. We report the rate at which both models give the same label to the fiber sample.
Figure 11: Invariances of the Qwen-2B vision model on a ”situs inversus” image (shown on the left). The atypical anatomy does not appear in the selected samples from the fiber.
Appendix figures & tables11 assets
Supplementary material from the paper’s appendix.
Appendix
Figure 12: Sample refinement on a colorMNIST example. From left to right: Original image, invariant sample, sample refined with gradient descent in a VAE latent space and sample refined with gradient descent in pixel space. Below each image we show the difference of the decolorized image to the decolorized original. Note that even for non-zero difference the fiber loss may be small if they fall exactly in the invariances of the almost lossless autoencoder subject model, which is achieved by pixel level gradient descent. On the right we show the fiber loss improvement when applying sample refinement to fiber samples of a conditional normalizing flow fiber model on color MNIST.
Figure 13: Gaussian mixture distributions for sampling the color values for each channel to colorize MNIST.
| blocks per residual net | 3 |
| layers per block | 3 |
| layer size | 512 |
| latent dimension | 54 |
| activation function | SiLU |
| epochs | 700 |
| lr scheduler | one-cycle |
Table 3: Hyperparameters of the lossless VAE used for the color MNIST experiments.
Figure 14: Comparative plot of fiber loss (top left), KL-divergence (top right) and -distance (bottom) on the color distribution of the color MNIST benchmark. gives the weight of the fiber loss during training of the conditional models, while shows the guidance strength of NDTM. Standard deviations are taken across all sample sets and target embeddings.
| Training conditional model | Time per sample conditional model | Time per sample NDTM |
| min | 0.3 min | 1.8 min |
Table 4: Estimates of the time required to train and sample from a conditional fiber model vs. sampling via NDTM guidance on ImageNet. Numbers are based on sampling with 100 timesteps and optimization steps for NDTM on a NVIDIA A40 GPU.
| Subject Model | Dataset | invariant samples | nearest neighbor | augmented |
| DINOv2 | ImageNet | 273 4 | 1225 | 440 |
| Inception | ImageNet | 34.8 0.7 | 123.9 | 94.9 |
| DINOv2 | Cue Conflict | 487 3 | 1069 | 946 |
| ResNet | Cue Conflict | 38.1 0.7 | 74.9 | 101.3 |
| BiomedClip | CheXpert | 4.9 9.9 | 23.3 | - |
| ConvNext | CheXpert | 1.2 0.0 | 10.9 | - |
Table 5: Fiber losses of all subject models on large image datasets. As a comparison we provide the fiber losses of the nearest neighbors from the dataset as well as the original image with typical DINOv2 training augmentations.
Figure 15: From left to right: Original image, nearest neighbor and two random invariant samples of BiomedClip, nearest neighbor and two random invariant samples from ConvNeXt. Below each image we give the summed absolute difference in predicted probabilities across all classes.
Figure 16: ROC curves for each of the five classes, that the CheXpert classifiers were trained on. We show results for comparing the BiomedClip classifier to the ConvNeXt classifier. Top: ROC curves for originals, using the ground truth labels. Center: ROC curves for originals, using the opposite model to create target labels. Bottom: ROC curves for invariances sampled for each model, with target labels created by the opposite model.
Figure 17: ROC curves for each of the five classes, that the CheXpert classifiers were trained on. We show results for two identically trained ConvNeXt classifiers. Top: ROC curves for originals, using the ground truth labels. Center: ROC curves for originals, using the opposite model to create target labels. Bottom: ROC curves for invariances sampled for each model, with target labels created by the opposite model. Below the ROC curves we show the classifier agreement on invariant sets sampled for originals, where both models predict the correct label.
Figure 18: Random DINOv2 invariant samples on ImageNet without (middle panel), then with (lower panel) our modification to NDTM.
Figure 19: Color distribution resulting from coloring 5k MNIST digits at a fixed color (dashed line) and creating fiber samples with NDTM. The upper panel shows sampling without our modification, the lower panel with our modification. The solid black line shows the ground truth color distribution.