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OPAL: Orthonormal Prototype Alignment Learning for Interpretable Image Classification

Ilán Carretero, Gustavo Jesús Angulo, Rocío del Amor, Valery Naranjo

arXiv:2608.30003v1cs.CV

TL;DR

Existing interpretable prototype models often require complex multi-stage training and auxiliary regularization. OPAL replaces unconstrained prototype learning with fixed orthonormal class subspaces and spatial competition, and reports leading interpretable performance across fine-grained benchmarks while retaining localized explanations.

  • Problem

    Existing interpretable prototype models rely on complex multi-stage pipelines and auxiliary regularization, while explanations must remain faithful to model decisions.

  • Method

    OPAL anchors each class in a predefined orthonormal subspace and uses competitive feature aggregation for single-stage, end-to-end interpretable classification.

  • Results

    OPAL attains leading performance among inherently interpretable models on most fine-grained datasets and consistently outperforms corresponding non-interpretable CNN backbones.

  • Takeaways & Limitations

    OPAL produces localized part-based explanations directly tied to model computation without auxiliary regularizers or multi-stage optimization schedules.

  • Takeaways & Limitations

    OPAL currently relies on convolutional backbones and uses a fixed number of non-shared prototypes per class, which can restrict extension across architectures and varying class complexity.

Abstract

from arXiv · show

Prototypical part-based models provide explainable predictions by comparing input regions to learned prototypes. However, current approaches are burdened by complex, multi-stage training pipelines and heavily rely on auxiliary regularization to prevent prototype collapse. To overcome these limitations, we introduce Orthonormal Prototype Alignment Learning (OPAL), a single-stage, end-to-end framework that simplifies interpretable classification. Our approach anchors the latent space using predefined orthonormal bases, embedding each class within a dedicated subspace spanned by fixed part-prototypes. To achieve precise part localization, OPAL enforces spatial competition across feature maps. This mechanism isolates sparse, discriminative regions, directing each prototype to consistently attend to the same semantic concept across different images. By framing classification as a direct representation alignment task, our method eliminates the need for auxiliary losses. Extensive experiments on fine-grained benchmarks demonstrate that OPAL outperforms both its non-interpretable counterparts and state-of-the-art part-prototype methods, delivering granular visual explanations by explicitly revealing the specific image regions driving every prediction. Code is available at https://github.com/ilancarretero/OPAL.

1 Introduction

OPAL addresses the complexity and weak faithfulness of prior interpretable prototype models by replacing unconstrained prototype learning with fixed orthonormal geometry and spatial competition. This yields a single-stage architecture that isolates discriminative regions and provides granular part-based explanations.

  • Prior interpretable models use unconstrained latent prototypes, while post-hoc saliency methods may not faithfully reflect the model’s decision process.
  • OPAL assigns each class to a dedicated orthonormal subspace, replacing free prototype discovery with structured representation alignment.
  • Channel-wise spatial competition prevents semantic overlap and isolates sparse, discriminative image regions without part-level annotations or bounding-box supervision.
  • OPAL provides a single-stage, end-to-end training pipeline without warm-up phases, alternating optimization, or auxiliary regularization losses.
  • Qualitative results indicate that prototype slots align with distinct semantic parts, producing granular and conceptually meaningful visual explanations.

2 Related work

Related work establishes the burden of multi-stage optimization and auxiliary losses in prototype-based interpretability, while geometric representation learning motivates OPAL’s fixed-space design. OPAL combines these ideas with spatial evidence aggregation for interpretable part-based decisions under image-level supervision.

  • ProtoPNet introduced case-based reasoning but required multi-stage training and auxiliary regularization to align latent prototypes with training patches.
  • PIP-Net improves semantic prototype grounding but requires a two-stage schedule and two auxiliary loss functions.
  • Geometric representation-learning methods show that normalized embeddings, angular objectives, and contrastive alignment can improve class-level structure.
  • Metric-learning approaches use fixed or proxy-based class representatives, supporting OPAL’s distance-based alignment to structured class geometry.
  • OPAL combines fixed geometric alignment with spatial evidence aggregation to obtain interpretable part-based decisions under image-level supervision.

3 Orthonormal Prototype Alignment Learning (OPAL)

OPAL maps convolutional features into a fixed orthonormal coordinate system, uses channel competition and max pooling to extract localized evidence, and trains by distance-based alignment to fixed class anchors.

  • 3.1 Preliminaries and Problem Formulation: OPAL addresses fine-grained classification using only global class labels while requiring localized evidence for each prediction.
  • 3.1 Preliminaries and Problem Formulation: A convolutional extractor and lightweight projection produce K = C · m channels that support structured part-based reasoning.
  • 3.2 Predefined Orthonormal Subspaces: Each class receives m coordinates forming a dedicated subspace orthogonal to every other class subspace.
  • 3.2 Predefined Orthonormal Subspaces: Fixed canonical part-prototypes are unit-norm and mutually orthogonal, creating maximally separated coordinate directions for part-based evidence.
  • 3.3 Spatial Competition for Part Localization: Channel-wise softmax forces channels to compete at each spatial location, encouraging distinct channels to specialize in separate discriminative regions.
  • 3.3 Spatial Competition for Part Localization: Global max pooling records one spatial maximizer per channel, and L_2 normalization forms an embedding whose coordinates retain localized evidence.
  • 3.4 Single-stage Optimization and Inference: Classification uses negative Euclidean distance to fixed orthonormal anchors, which is equivalent to cosine-similarity alignment for normalized vectors.
  • 3.4 Single-stage Optimization and Inference: End-to-end cross-entropy training keeps anchors fixed and avoids auxiliary clustering, projection steps, and additional regularization losses.

4 Experiments

OPAL is evaluated on five standard fine-grained benchmarks using their official train/test splits. The datasets cover birds, cars, pets, dogs, and flowers with category counts ranging from 37 to 200.

  • OPAL is evaluated on five fine-grained benchmarks: CUB-200-2011, Stanford Cars, Oxford-IIIT Pet, Stanford Dogs, and Oxford Flowers-102.
  • The benchmarks contain 200 bird species, 196 car models, 37 pet breeds, 120 dog breeds, and 102 flower categories.

Datasets.

OPAL is evaluated across diverse fine-grained benchmarks and standardized training settings, with five prototypes per class used by default.

  • Datasets: OPAL is evaluated on five fine-grained benchmarks using three convolutional backbone families at three model scales each.The backbones are ResNet-50/101/152, EfficientNetV2 S/M/L, and ConvNeXt Tiny/Small/Base.
  • Datasets: The default configuration fixes the number of prototypes per class to m = 5 across datasets and backbones.Sensitivity to the prototype count is analyzed separately.
  • Datasets: Models are trained for 30 epochs with Adam, a 1 × 10^-4 learning rate, and batch size 64.Images are resized to 224 × 224 and augmented with TrivialAugment.

Comparison with Inherently Interpretable Models.

On five fine-grained benchmarks, OPAL is compared with representative inherently interpretable methods and a ConvNeXt-Tiny baseline under backbone-aligned evaluation. It achieves the strongest overall benchmark pattern among interpretable models while also exceeding the corresponding CNN baseline on average.

  • Comparison with Inherently Interpretable Models: The comparison uses ConvNeXt-Tiny as the baseline backbone to enable direct backbone-aligned comparisons with recent prototype-based approaches.The evaluation spans five fine-grained datasets and representative interpretable baselines.
  • Comparison with Inherently Interpretable Models: OPAL achieves the best performance on 4 out of 5 datasets and ranks third on CARS behind TesNet and LucidPPN.Table 2 reports top-1 accuracy comparisons with eight interpretable methods and a non-interpretable counterpart.
  • Comparison with Inherently Interpretable Models: OPAL is the only method that consistently surpasses the corresponding CNN backbone across all benchmarks, averaging +1.56 absolute percentage points over the baseline.The baseline is ConvNeXt-Tiny in the main comparison.
  • Comparison with Inherently Interpretable Models: The results associate fixed, discriminative latent geometry with a single-stage pipeline that avoids auxiliary losses while improving interpretability and performance.This interpretation is stated as the paper’s explanation of the benchmark outcome.

Backbone Generalization and Scalability.

OPAL transfers across convolutional architectures and model capacities while preserving prototype-based visual explanations. Its prototypes provide localized, complementary evidence and show consistent semantic roles within classes.

  • Backbone Generalization and Scalability: 39 out of 45 backbone–dataset experiments improve or match the corresponding CNN baseline, with an average top-1 gain of +2.94 percentage points.The evaluation covers three backbone families and three model scales per family.
  • Backbone Generalization and Scalability: The largest observed OPAL drop is 1.0 percentage points, indicating limited performance trade-off across diverse convolutional architectures.This summarizes the few cases where OPAL does not outperform its baseline.
  • Qualitative Interpretability Analysis: With m = 5, OPAL prototypes activate on multiple distinct object regions, providing compact and complementary evidence across five fine-grained benchmarks.Colored boxes identify spatial maximizers, while patch grids show prototype-wise evidence from the displayed image and same-class images.
  • Qualitative Interpretability Analysis: Prototype evidence remains consistent within classes, with individual slots repeatedly attending to related semantic parts across different instances.Examples include ears for Abyssinian cats and eyes for Maltese dogs.
  • Backbone Generalization and Scalability: On CUB-200-2011, OPAL attains the second-best prototype consistency and stability results among prototype-based methods.Part-level annotations are available only for this dataset.

Architectural Component Analysis.

Ablations indicate that OPAL’s performance and interpretability depend on combining orthonormal anchoring, spatial competition, and max-based aggregation. Prototype-count sensitivity further shows that too few prototypes underperform, while excessive prototypes can become less semantically relevant.

  • Architectural Component Analysis: +4.07 percentage points in average top-1 accuracy result from enabling orthonormal anchors over matched non-orthogonal variants.The ablation identifies orthonormal anchoring as a central component.
  • Architectural Component Analysis: GMP without orthonormal anchoring produces a -9.86 percentage-point average top-1 drop relative to the baseline despite localized evidence.Pooling-based localization alone is therefore insufficient in the reported ablation.
  • Architectural Component Analysis: Reliable performance, prototype diversity, and localized interpretability require orthonormal anchoring together with spatial competition and max-based aggregation.Removing GMP weakens one-maximizer traceability, while removing CWS increases prototype redundancy.
  • Sensitivity to Prototype Count: m = 1 underperforms the ConvNeXt-Tiny baseline by -1.04 percentage points on average, whereas m ∈ {3,5,10} averages +1.25 percentage points over it.The sensitivity analysis keeps the remaining training protocol fixed.

Sensitivity to the Number of Prototypes

OPAL performs best with a moderate number of prototypes per class: too few limit representational capacity, while too many dilute discriminative evidence.

  • A moderate number of prototypes per class, m = 5, provides a robust operating point for interpretable predictions and performance.Across most benchmarks, both too few and excessive prototypes perform worse than the best setting.

5 Conclusion

OPAL replaces latent prototype learning with alignment in a fixed, class-structured orthonormal geometry and competitive feature aggregation. This design supports single-stage, end-to-end training while retaining localized, interpretable part-based explanations without auxiliary regularizers.

  • OPAL shifts prototype-based classification from learning latent prototypes to aligning representations within a fixed, class-structured geometry.Each class is anchored in a predefined orthonormal subspace.
  • Competitive feature aggregation yields localized evidence tied directly to the model computation.
  • OPAL enables single-stage, end-to-end training with a standard classification objective.
  • The framework preserves interpretable, part-based explanations without auxiliary regularizers or multi-stage optimization schedules.
  • OPAL currently relies on convolutional-backbone inductive biases and uses a fixed, non-shared number of prototypes per class.The authors identify vision-transformer extensions, adaptive allocation, and controlled prototype sharing as future directions.

Supplementary Material

The supplementary material formalizes OPAL’s fixed orthonormal geometry and block-uniform anchors, then evaluates prototype collapse, prototype-count behavior, distance choices, and computational overhead.

  • A.1 Formal Justification of the Block-uniform Anchor: For fixed in-block energy, the block-uniform anchor receives higher similarity than collapsed evidence for any m > 1.The maximum occurs when evidence is distributed uniformly across the class block.
  • A.2 Canonicity of the Prototype Basis: Under OPAL’s nonnegative orthogonality constraints, admissible prototype bases are permutations of the canonical basis.The learned 1 × 1 projection absorbs any such permutation.
  • Supplementary Qualitative Results: Increasing prototypes to m = 10 can make some slots attend to weakly relevant or background regions, whereas smaller settings generally retain meaningful localization.This supports a limited number of salient, discriminative regions per class.
  • D.1 Quantifying Prototype Collapse: Full OPAL reduces spatial prototype collapse to near-zero values across all five datasets, unlike the severe collapse observed without spatial competition.Without CWS and with the orthonormal basis fixed, SPC ranges from 72.1 to 93.6.
  • E.1 Alternative Distance Functions: L_2 distance reaches 85.6, 90.3, 92.8, 86.9, and 96.1 top-1 accuracy across the five benchmarks.Chebyshev distance performs substantially worse on all benchmarks.
  • E.2 Computational Efficiency: OPAL’s head averages 2.36% of backbone parameters and 0.53% of backbone GFLOPs across backbones and datasets.Its overhead is higher than a standard linear classifier but remains small relative to the feature extractor.

E.3 Prototype Consistency and Stability on CUB-200-2011

On CUB-200-2011, OPAL provides quantitatively strong prototype explanations while retaining a single-stage design and competitive classification accuracy. Its CWS and GMP mechanisms improve prototype consistency and stability, and OPAL remains competitive beyond fine-grained benchmarks.

  • Prototype quality: 50.5 consistency and 75.9 stability place OPAL second-best on both prototype-quality metrics on CUB-200-2011.The evaluation is limited to CUB because part-level annotations are available only there.
  • Training simplicity: OPAL combines competitive prototype quality with a simpler single-stage end-to-end pipeline than methods using external masks, warm-up stages, or auxiliary losses.LucidPPN achieves the highest consistency partly through external segmentation masks, while PIP-Net achieves the highest stability with warm-up and auxiliary regularization.
  • Ablation: 85.6 accuracy, 50.5 consistency, and 75.9 stability outperform the ablated variant by 0.6, 10.2, and 10.8 percentage points, respectively.Removing both CWS and GMP preserves accuracy more closely than it preserves prototype consistency and stability.
  • Beyond fine-grained benchmarks: OPAL remains competitive with ConvNeXt-Tiny on ImageNet-1K and Caltech-101, while improving accuracy on DTD-47 and SUN-397.The reported changes are -0.6 percentage points on IN-1K, +0.3 on CT-101, +2.8 on DTD-47, and +2.3 on SUN-397.
  • Qualitative explanations: Qualitative explanations show prototype slots aligning with distinct semantic parts across five fine-grained benchmarks.Colored boxes indicate the spatial maximizers selected by each prototype.
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