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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

Opaque networks and post-hoc explanations that may not faithfully reflect decisions leave a need for inherently interpretable image classification. OPAL aligns representations to fixed, class-specific orthonormal subspaces with competitive feature aggregation in a single-stage framework; it performs strongly across benchmarks, surpassing corresponding CNN backbones and leading interpretable models on most datasets.

  • Problem

    Deep networks are difficult to interpret, while post-hoc explanations may not faithfully reflect their decisions, motivating inherently interpretable architectures.

    Source: §1 · pp. 1–2

  • Method

    OPAL aligns representations within fixed, class-specific orthonormal subspaces and uses competitive feature aggregation for single-stage, end-to-end interpretable classification.

    Source: §1 · p. 3, §5 · pp. 14–15

  • Results

    OPAL improves or matches CNN counterparts in 39/45 backbone–dataset experiments, gaining +2.94 percentage points in top-1 accuracy; it leads interpretable models on 4/5 datasets and averages +1.56 percentage points over baseline.

    Source: Backbone Generalization and Scalability · p. 10, Comparison with Inherently… · p. 10

  • Takeaways & Limitations

    OPAL provides localized, part-based explanations tied to model computation while avoiding auxiliary regularizers and multi-stage optimization.

    Source: §5 · pp. 14–15

    OPAL currently relies on convolutional backbones, and extending it to vision transformers requires token-level competition and aggregation mechanisms.

    Source: §5 · p. 15

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 replaces freely learned prototypes with fixed orthonormal class subspaces and spatial competition, aiming to simplify interpretable classification while producing localized part explanations. The introduction reports leading performance among interpretable models on most tested datasets and consistent gains over corresponding CNN backbones.

1 Introduction

  • ProtoPNet-style models compare local image regions with learned prototypes, but moving prototypes complicate optimization and prior methods use multi-stage training and auxiliary losses to prevent collapse.The introduction also argues that post-hoc saliency explanations may not faithfully reflect model decisions, motivating interpretable architectures enforced by design.Source: §1 · pp. 1–2
  • OPAL assigns each class a predefined orthonormal subspace, replacing unconstrained prototype discovery with alignment in a fixed geometry that separates prototype structure from representation learning.This frames classification as direct alignment in a stable coordinate system.Source: §1 · p. 3, Figure 1 · p. 3
  • Channel-wise normalization and global feature selection make feature maps compete spatially, preventing distinct prototypes from claiming the same region and inducing sparse, discriminative parts without part annotations.The method contrasts this winner-takes-all mechanism with soft attention maps and auxiliary concentration losses.Source: §1 · pp. 3–4
  • The fixed orthogonal output space and spatial competition support single-stage end-to-end training without warm-up or alternating optimization, while eliminating auxiliary separation, clustering, and diversity losses.The introduction attributes semantic sparsity and isolation of discriminative image regions to these design choices.Source: §1 · p. 4
  • OPAL leads inherently interpretable models on most of five fine-grained benchmarks and consistently outperforms corresponding non-interpretable CNN backbones across architectures and capacities.Qualitative results indicate that prototype slots align with distinct semantic parts and provide granular visual explanations.Source: §1 · p. 4

2 Related work

Prior interpretable classifiers address prototype redundancy and semantic grounding but often add training stages, auxiliary objectives, or external dependencies. OPAL builds on geometric representation-learning ideas while fixing class geometry in orthonormal subspaces and combining it with competitive spatial evidence aggregation.

2.1 Interpretable Image Classification

  • ProtoPNet introduced case-based reasoning, while later methods addressed prototype redundancy but often retained complex training objectives and auxiliary-loss tuning.ProtoTree uses hierarchical prototypes; ProtoPShare and ProtoPool use merging or differentiable assignment; TesNet uses class-specific subspaces.Source: §2.1 · p. 4
  • PIP-Net grounds prototypes in semantic patches but uses two-stage training; MCPNet uses abstract concepts, and LucidPPN depends on external part-discovery pretraining.PIP-Net uses self-supervised part discovery and two auxiliary losses; MCPNet’s abstract concepts make localized regions harder to distinguish, while LucidPPN relies on PDiscoNet.Source: §2.1 · pp. 4–5
  • Post-hoc methods Tan et al. and EPIC explain frozen classifiers without changing predictions, but cannot reshape their learned representations; OPAL jointly learns representation and explanation end to end.OPAL removes prior ante-hoc methods’ structural and procedural dependencies while targeting semantically meaningful interpretability.Source: §2.1 · p. 5

2.2 Representation Learning and Metric Alignment

  • Normalized-space angular and contrastive objectives motivate OPAL’s structured geometry, while proxy methods and fixed classifiers [13] inform its alignment approach.OPAL departs from proxy formulations by fixing class geometry a priori through orthonormal subspaces and anchors, coupled with competitive spatial evidence aggregation.Source: §2.2 · pp. 5–6

3 Orthonormal Prototype Alignment Learning (OPAL)

OPAL learns interpretable fine-grained classifiers from image-level labels by aligning representations to fixed orthonormal class subspaces and tying prototype evidence to localized image regions.

3.1 Preliminaries and Problem Formulation

  • The task is supervised fine-grained image classification with global class labels only; the model must predict labels and provide localized evidence without part or region-level annotations.Source: §3.1 · p. 6

3.2 Predefined Orthonormal Subspaces

  • OPAL assigns each class a dedicated orthogonal subspace with fixed, mutually orthogonal prototype directions and a unit class anchor.The block-uniform anchor favors distributing evidence across the correct class's prototype slots.Source: §3.2 · p. 7

3.3 Spatial Competition for Part Localization

  • Channel-wise softmax makes channels compete at each spatial location, encouraging distinct discriminative regions, while max pooling ties each embedding coordinate to a single evidence patch.A 1 × 1 projection maps backbone features to prototype-aligned channels, and the normalized channel activations sum to one at every location.Source: §3.3 · pp. 7–8

3.4 Single-stage Optimization and Inference

  • OPAL trains end-to-end in one stage by classifying with distances to fixed class anchors and cross-entropy, without auxiliary clustering, projection steps, or regularization losses.At inference, the predicted class selects its prototype channels, whose pooled evidence locations are mapped back to input resolution as explanation patches.Source: §3.4 · pp. 8–9

4 Experiments

The experiments evaluate OPAL across five fine-grained image benchmarks using each dataset’s official train/test split.

4.1 Experimental Setup

  • OPAL is evaluated on five fine-grained benchmarks—CUB-200-2011, Stanford Cars, Oxford-IIIT Pet, Stanford Dogs, and Oxford Flowers-102 —using official train/test splits.Additional dataset details and preprocessing steps are provided in supplementary Sec. B.1.Source: Datasets · p. 9

Datasets.

OPAL is evaluated with three CNN backbone families at three scales each, using a fixed five prototypes per class and a shared 30-epoch training protocol.

Datasets.

  • OPAL is evaluated with ResNet-50/101/152, EfficientNetV2 S/M/L, and ConvNeXt Tiny/Small/Base backbones.Images are resized to 224 × 224 and augmented with TrivialAugment.Source: Datasets · p. 9
  • All settings use five prototypes per class, with sensitivity to this choice analyzed in Sec. 4.4.Source: Datasets · p. 9
  • Models are trained for 30 epochs with Adam at a learning rate of 1 × 10^-4 and batch size 64.Experiments use an NVIDIA A100 (40GB), repeat across three random seeds, and report averages and absolute percentage-point differences from matched CNN backbones.Source: Datasets · p. 9

Comparison with Inherently Interpretable Models.

OPAL performs strongly against inherently interpretable models, leading on four of five benchmarks and consistently outperforming its CNN backbone. The evaluation also examines ProtoTree’s dataset-specific low performance and OPAL’s transfer across backbones and model scales.

Comparison with Inherently Interpretable Models

  • ProtoTree shows low performance on DOGS and FLOWER, a pattern also observed on those datasets in [33] and on another fine-grained benchmark in [46].The table caption reports this as context for the comparison.Source: Table 2 · p. 10
  • The comparison uses ConvNeXt-Tiny as its baseline backbone, matching recent prototype-based approaches for backbone-aligned side-by-side evaluation.The evaluation spans five datasets and representative interpretable baselines.Source: Comparison with Inherently… · p. 10
  • OPAL leads on 4 of 5 datasets, ranks third on CARS behind TesNet and LucidPPN, and alone beats its CNN backbone on every benchmark, averaging +1.56 percentage points.Table 2 compares top-1 accuracy on five fine-grained benchmarks; its caption does not provide the individual cell values.Source: Comparison with Inherently… · p. 10, Table 2 · p. 10
  • Tab. 3 tests OPAL as a plug-in module across three convolutional backbone families and three model scales per family, against each matching CNN backbone under the same protocol.This setup isolates OPAL’s effect from backbone-specific design choices and tests transfer across architectural families and capacities.Source: Backbone Generalization and Scalability · p. 10

Backbone Generalization and Scalability.

Across backbone–dataset experiments, OPAL generally preserves or improves top-1 accuracy, while its prototypes provide localized, complementary evidence and show consistent semantic roles across images.

Backbone Generalization and Scalability.

  • OPAL improves or matches its non-interpretable counterpart in 39 out of 45 experiments, gains +2.94 absolute percentage points in average top-1 accuracy, and has a largest drop of 1.0 points.The comparison spans diverse convolutional architectures, indicating broad robustness without a substantial performance trade-off for interpretability.Source: Backbone Generalization and Scalability · p. 10, Table 3 · p. 11

4.3 Qualitative Interpretability Analysis

  • Across five fine-grained datasets, OPAL’s m = 5 prototypes provide complementary, localized evidence by activating on multiple distinct object regions in each image.The examples illustrate part-level evidence useful when fine-grained class distinctions depend on subtle attributes rather than one salient region.Source: §4.3 · p. 11, Figure 3 · p. 12
  • Across images within a class, individual prototypes repeatedly activate on related parts; on CUB-200-2011, OPAL achieves the second-best prototype consistency and stability results among prototype-based methods.The qualitative examples suggest stable semantic roles, such as ears for Abyssinian cats and eyes for Maltese dogs.Source: §4.3 · pp. 11–12

4.4 Ablation Studies

  • The ablation selectively varies projection, competitive normalization, pooling, and orthonormal anchoring; anchor configurations require projection because anchors occupy a K = C · m space.The 1 × 1 projection matches channel dimensionality to that space.Source: Architectural Component Analysis · p. 12

Architectural Component Analysis.

The ablation finds that orthonormal anchors and localized aggregation work best together: pooling-based localization alone is insufficient. It also identifies prototype-count settings that improve average accuracy and explains how removing components compromises interpretability.

Architectural Component Analysis

  • Orth. improves top-1 accuracy by +4.07 percentage points over matched non-orthogonal variants, while GMP without Orth. drops 9.86 points versus baseline.These results indicate that localized aggregation benefits from the fixed class-structured geometry induced by orthonormal anchoring.Source: Architectural Component Analysis · pp. 12–13, Table 4 · p. 13
  • Removing GMP breaks the one-maximizer-per-channel link to a unique image region, while removing CWS weakens spatial competition and increases prototype redundancy.The findings indicate that reliable diversity and localized interpretability require orthonormal anchoring, spatial competition, and max-based aggregation together.Source: Architectural Component Analysis · p. 13
  • With m = 1, OPAL underperforms the ConvNeXt-Tiny baseline by -1.04 percentage points on average; m = 3, 5, or 10 yields +1.25 points averaged across those settings.The remaining training protocol is held fixed while varying m across 1, 3, 5, and 10 prototypes per class.Source: Sensitivity to the Number of Prototypes · p. 14, Figure 4 · p. 14

Sensitivity to the Number of Prototypes

A moderate number of prototypes per class (m = 5) is a robust operating point: too few limit representational capacity, while too many can dilute discriminative evidence and slightly reduce accuracy.

Sensitivity to the Number of Prototypes

  • Too few prototypes limit representational capacity, whereas excessive prototypes can dilute discriminative evidence; m = 5 per class is a robust operating point.The paper reports that this setting enables interpretable predictions while improving performance over the non-interpretable counterpart.Source: Sensitivity to the Number of Prototypes · p. 14

5 Conclusion … B Additional Experimental Setup

OPAL aligns representations to fixed, class-structured orthonormal subspaces and uses competitive feature aggregation to localize evidence, enabling end-to-end training without auxiliary regularizers or multi-stage optimization. The appendix formalizes why its anchor favors distributed in-class evidence and why its canonical prototype basis represents the only admissible family up to permutation.

5 Conclusion

  • OPAL anchors each class in a predefined orthonormal subspace and uses competitive feature aggregation to produce localized evidence tied directly to model computation.This shifts prototype-based classification from learning latent prototypes to aligning representations within fixed, class-structured geometry.Source: §5 · p. 14
  • With a standard classification objective, OPAL trains end to end in one stage while preserving part-based explanations without auxiliary regularizers or multi-stage schedules.Source: §5 · p. 15
  • OPAL’s convolutional-backbone formulation relies on locality and hierarchical features; future work must address transformer token-level competition, adaptive prototype allocation, and controlled cross-class sharing.The current model fixes the number of prototypes per class and does not share prototypes across classes.Source: §5 · p. 15

A.1 Formal Justification of the Block-uniform Anchor

  • For fixed in-block energy, the class-anchor similarity is maximized exactly when evidence is uniform across the block’s m slots, and for m > 1 exceeds that of one-slot collapse.When m = 1, distributed and collapsed evidence are identical.Source: §A.1 · pp. 20–21
  • Because embeddings and anchors are unit-norm, this similarity advantage means OPAL assigns a larger correct-class logit to distributed than collapsed in-block evidence at the same energy.Source: §A.1 · p. 21

A.2 Canonicity of the Prototype Basis

  • Under OPAL’s orthonormality and coordinate-wise non-negativity requirements, admissible prototype bases are permutations of the canonical basis.Non-negative orthogonal vectors have disjoint supports, which in K coordinates for K unit vectors forces each support to be a single coordinate.Source: §A.2 · p. 21
  • Any such permutation is absorbed as a row reordering in the learned 1 × 1 projection, making the block-ordered canonical basis a natural representative.Source: §A.2 · p. 21

B.1 Dataset Details and Preprocessing … E.2 Computational Efficiency

Across the supplied analyses, OPAL’s prototypes show semantic consistency and spatial diversity, while distance-function choices and the prototype head’s computational overhead affect performance and cost. The experiments also specify a consistent evaluation and preprocessing protocol across five fine-grained benchmarks.

B.1 Dataset Details and Preprocessing; B.2 Implementation Details

  • Experiments use five fine-grained benchmarks and official train/test partitions, with a consistent end-to-end training and evaluation protocol.Images are resized to 224×224; CUB-200-2011 and Stanford Dogs are cropped to object boxes for offline preprocessing, but training and inference use no localized supervision. Models are fine-tuned for 30 epochs with Adam, learning rate 1 × 10^-4, and batch size 64; final checkpoints are selected by validation cross-entropy and reported as test top-1 accuracy, averaged over three seeds.Source: §B.1 · p. 22, §B.2 · p. 22

C.1 Additional Prototype-Based Explanations

  • Across five datasets, prototypes avoid collapse, attend to distinct meaningful regions, and often retrieve similar evidence for different test images of the same class.These visualizations suggest consistent, semantically meaningful prototype patterns at the class level.Source: §C.1 · p. 23

C.2 Qualitative Effect of the Number of Prototypes per Class m

  • For m = 1, 3, or 5, prototypes generally attend to meaningful regions; at m = 10, a small subset attends to weakly relevant or background regions.This qualitative trend is consistent with the main paper’s finding that moderate m values offer the more favorable trade-off between capacity and discriminative focus.Source: §C.2 · p. 23

D.1 Quantifying Prototype Collapse

  • The Spatial Prototype Collapse score measures overlap among class-specific prototypes’ spatial maxima: 0 indicates distinct locations and 1 indicates complete collapse.The reported score averages SPC over the test set, with lower values indicating greater spatial diversity.Source: §D.1 · p. 23

Proj. GMP CWS Orth. CUB CARS PETS DOGS FLOWER

  • Without channel-wise softmax, spatial collapse is severe across all five benchmarks; full OPAL reduces collapse to near-zero, while channel-wise softmax and the orthonormal basis together yield diverse evidence regions.Without channel-wise softmax, prototypes often repeat one or two dominant regions; with it, they typically cover distinct, semantically meaningful object regions.Source: Proj. GMP CWS Orth. CUB CARS PETS DOGS… · p. 24

E.1 Alternative Distance Functions; CUB CARS PETS DOGS FLOWER

  • L1, L2, and L3 distances provide the strongest performance across datasets, whereas Chebyshev performs substantially worse and KL divergence and tropical projective distance also underperform those Lp alternatives.Under OPAL’s normalized formulation, Euclidean and cosine comparisons induce the same ordering; full-vector discrepancy measures align better with the method than maximum-coordinate discrepancy or distributional and ratio-based measures.Source: §E.1 · p. 24, CUB CARS PETS DOGS FLOWER · p. 25

E.2 Computational Efficiency

  • OPAL’s head averages 2.36% of backbone parameters and 0.53% of backbone GFLOPs, with overhead remaining small relative to the feature extractor.Across configurations, the head accounts for 0.20%–8.72% of backbone parameters and 0.02%–2.43% of GFLOPs; its learned 1×1 projection is the main source of overhead, which grows with class count and prototypes per class.Source: §E.2 · pp. 25–26, Table C · p. 26

E.3 Prototype Consistency and Stability on CUB-200-2011 … CUB CARS PETS DOGS FLOWER

Across the reported benchmarks, OPAL delivers strong classification results, while CUB-only evaluations show competitive prototype consistency and stability and reveal how its localization mechanisms affect interpretability. Additional results test transfer beyond fine-grained datasets and report variability across seeds and backbones.

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

  • On CUB-200-2011, OPAL ranks second on both metrics, scoring 50.5 for consistency and 75.9 for stability; this evaluation is limited to the only benchmark with part annotations.Consistency measures repeated association with an annotated object part, while stability measures preservation under Gaussian random-noise perturbations.Source: §E.3 · pp. 26–27, Table D · p. 27
  • LucidPPN leads consistency using PDiscoNet-generated masks, while PIP-Net leads stability with a prototype warm-up and auxiliary losses, unlike OPAL’s single-stage pipeline.The paper notes that these competing designs add reliance on an external model or training and tuning complexity.Source: §E.3 · p. 27
  • Removing CWS and GMP changes CUB accuracy from 85.6 to 85.0, consistency from 50.5 to 40.3, and stability from 75.9 to 65.1.The ablation indicates that spatial competition and max-based aggregation preserve localized, nonredundant prototype evidence even when accuracy changes only slightly.Source: §E.3 · pp. 27–28

E.4 Scalability and Generality Beyond Fine-Grained Benchmarks

  • Beyond fine-grained tasks, OPAL scores 76.1 on IN-1K, 97.2 on CT-101, 76.0 on DTD-47, and 62.6 on SUN-397, respectively, versus ConvNeXt-Tiny’s 76.7, 96.9, 73.2, and 60.3.Thus, it is competitive on IN-1K and CT-101 and improves on DTD-47 and SUN-397.Source: §E.4 · p. 28

F.1 Main-Paper Tables with Mean ± Standard Deviation Across Three Seeds

  • Expanded main-paper results report mean accuracy and standard deviations across three random seeds, while prior-publication results retain their reported values and may lack standard deviations.Tables G–J provide the expanded quantitative results; the stated reporting practice follows prior work.Source: §F.1 · p. 28, CUB CARS PETS DOGS FLOWER · pp. 32–33

CUB CARS PETS DOGS FLOWER

  • Additional qualitative explanations show prototype-selected spatial maximizers across five fine-grained benchmarks, including configurations with m ∈ {1, 3, 5, 10}.The patch-grid examples include regions from the displayed test image and other images of the same predicted class.Source: CUB CARS PETS DOGS FLOWER · pp. 29–30, Figure C · p. 31
  • OPAL scores 85.6±0.2, 90.3±0.1, 92.8±0.2, 86.9±0.2, and 96.1±0.3, versus ConvNext-Tiny’s 84.2±0.1, 87.8±0.2, 91.0±0.5, 85.3±0.3, and 95.6±0.1, respectively, higher in all five columns.The reported small values are standard deviations across three random seeds.Source: CUB CARS PETS DOGS FLOWER · p. 32
  • Expanded backbone results include OPAL’s 80.8±0.1 and 84.3±0.1 with ResNet-50, and 86.9±0.1 and 89.4±0.2 with EfficientNet-V2 L, among five benchmark scores per backbone.These rows are reported alongside each backbone’s baseline results.Source: CUB CARS PETS DOGS FLOWER · p. 32

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