OPAL: Orthonormal Prototype Alignment Learning for Interpretable Image Classification

The problem

The cost of learning prototypes

Part-prototype models are interpretable by design: a prediction is grounded in localized visual evidence. Learning those prototypes is what makes them expensive, because representation learning and prototype alignment have to converge jointly.

A moving target Prototypes are free parameters inside a latent space that is still changing, so the model must chase an alignment that keeps shifting under it.

Multi-stage pipelines Warm-up phases, prototype projection onto real patches and alternating optimization are the usual price of keeping that alignment under control.

Auxiliary regularizers Separation, clustering and diversity losses are added to prevent prototype collapse, and each one brings dataset-specific hyperparameters to tune.

OPAL: fix the geometry Assign every class a dedicated orthonormal subspace before training. Interpretability becomes alignment inside a known coordinate system, not a search for it.

Abstract

Interpretability as representation alignment

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.

Key idea

From prototype discovery to structured alignment

Prior work searches for prototypes in an unconstrained latent space, where evidence from different classes can overlap. OPAL preassigns the geometry instead: each class \(c\) spans its own orthonormal subspace \(\mathcal{S}_c \subset \mathbb{R}^{K}\), and the only thing left to learn is the representation that aligns to it.

Two latent spaces side by side. On the left, an unconstrained space where prototypes of three classes overlap. On the right, OPAL's geometry-constrained space, where each class occupies a mutually orthogonal subspace. — Figure 1. **Conceptual overview.** In an unconstrained latent space (left), prototypes are learned jointly with the features they are matched against and class evidence may share directions. OPAL imposes a geometry-constrained space (right) in which the class subspaces are mutually orthogonal, \(\mathcal{S}_c \perp \mathcal{S}_{c'}\). This structure encourages the \(m\) prototype coordinates within \(\mathcal{S}_c\) to capture distinct semantic attributes, and each of them maps back to a concrete image region.

Fix the directions, not the features. Part-prototypes are preassigned to canonical orthonormal axes and never updated, which removes the moving-target coupling between prototype optimization and representation learning.
Orthogonality structures parts and classes at once. Each class spans \(m\) orthonormal prototype directions, and the class subspaces are mutually orthogonal, so separation is a property of the design rather than of a loss term.
One objective is enough. Training is cross-entropy over distances to the fixed anchors: no warm-up, no projection step, no alternating optimization, no auxiliary regularizers.

Table comparing a standard CNN and nine inherently interpretable models across four properties: single-stage training, auxiliary-loss free, external-model free, and interpretable by design. OPAL is the only interpretable method satisfying all four. — Table 1. **Where OPAL sits.** Three training properties a standard CNN has for free, plus interpretability by design. Every prior part-prototype method gives up at least one of them, most often single-stage training. OPAL is the only interpretable model in the comparison that keeps all four.

Method

OPAL: Orthonormal Prototype Alignment Learning

A backbone feature map is projected onto \(K = C \cdot m\) prototype coordinates. Channels compete for every spatial location, each surviving channel contributes exactly one image region, and the resulting embedding is classified by its distance to fixed orthonormal class anchors.

Figure 2. The full forward pass. A convolutional backbone \(f_\theta\) produces \(F(x) \in \mathbb{R}^{D \times H \times W}\), projected to \(G(x) \in \mathbb{R}^{K \times H \times W}\). Spatial competition (CWS) followed by global max pooling and \(L_2\) normalization yields the embedding \(z \in \mathbb{R}^{K}\), which is classified by distance to the fixed anchors \(\{a_c\}_{c=1}^{C}\). For the predicted class, the channels of its block map their spatial maximizers back to the input by linear coordinate rescaling — the explanation is the same computation as the prediction, not a second pass.

Project

A \(1\times1\) convolution maps the backbone feature map \(F \in \mathbb{R}^{D\times H\times W}\) to \(G \in \mathbb{R}^{K\times H\times W}\) with \(K = C\cdot m\), so every class \(c\) owns a block \(B_c\) of \(m\) prototype slots.
Compete

A channel-wise softmax normalizes across channels at each location, so \(\sum_{k} A_k(u,v) = 1\) everywhere and a single image region cannot support several prototypes at once.

\[ A_k(u,v) \;=\; \frac{\exp\!\big(G_k(u,v)\big)}{\sum_{j=1}^{K} \exp\!\big(G_j(u,v)\big)} \]
Localize

Global max pooling keeps one response per channel together with its maximizer, and \(L_2\) normalization gives \(z = s / \lVert s \rVert_2\). Each coordinate of \(z\) is therefore sourced from exactly one spatial location, whose image patch is the evidence behind that coordinate.

\[ s_k \;=\; \max_{(u,v)\in\Omega} A_k(u,v), \qquad (u_k^{*}, v_k^{*}) \;=\; \arg\!\max_{(u,v)\in\Omega} A_k(u,v) \]
Align

Each class is represented by a fixed block-uniform anchor \(a_c\), unit-norm and orthogonal to every other anchor. Logits are negative distances, and training is plain cross-entropy over them: since \(z\) and \(a_c\) are both unit-norm, this is equivalent to maximizing cosine similarity on the hypersphere.

\[ a_c \;=\; \frac{1}{\sqrt{m}}\sum_{j=1}^{m} p_{c,j}, \qquad \ell_c \;=\; -\lVert z - a_c \rVert_2 \]
Explain by construction

For the predicted class \(\hat{y}\), the \(m\) channels of block \(B_{\hat{y}}\) each contribute one maximizer. Rescaling those coordinates from the \(H\times W\) grid back to the \(H_0\times W_0\) input resolution yields the part-prototype evidence, with no attribution method in between.

Quantitative results

Accuracy, generalization and efficiency

Top-1 accuracy on CUB-200-2011, Stanford Cars, Oxford-IIIT Pet, Stanford Dogs and Oxford Flowers-102. All numbers are means over three seeds under an identical protocol; arrows are absolute percentage-point differences with respect to the corresponding CNN backbone.

Comparison with inherently interpretable models

Top-1 accuracy of OPAL against eight inherently interpretable models and a ConvNeXt-Tiny baseline on CUB, CARS, PETS, DOGS and FLOWER. — Table 2. **Against a shared backbone.** All methods share the ConvNeXt-Tiny backbone reported in the first row, so the arrows measure what interpretability costs each design.

Leading performance. OPAL ranks first on 4 of the 5 benchmarks and remains competitive on CARS, where it is third behind TesNet and LucidPPN.
No accuracy trade-off. It is the only method that stays above its own ConvNeXt-Tiny backbone on all five datasets, by \(+1.56\) points on average, while every competitor loses accuracy on at least two of them.

Backbone generalization and scalability

Three convolutional families at three scales each — ResNet-50/101/152, EfficientNet-V2 S/M/L and ConvNeXt Tiny/Small/Base — are trained in their standard form and with the OPAL head under the same protocol, which isolates the contribution of the head from backbone-specific design choices.

Accuracy of nine backbones from the ResNet, EfficientNet-V2 and ConvNeXt families, each in standard form and with the OPAL head, across the five fine-grained datasets. — Table 3a. **OPAL as a drop-in head.** Each shaded row adds the OPAL head to the backbone directly above it. Arrows are differences with respect to that backbone.

Backbone-agnostic gains. OPAL improves or matches the non-interpretable counterpart in 39 of 45 backbone–dataset settings, by \(+2.94\) points on average.
Robust in the failure cases. Where it does not improve, the largest single drop across all 45 settings is 1.0 point.

Computational overhead

The OPAL head replaces the linear classifier with a \(1\times1\) projection to \(K = C\cdot m\) channels, so its cost is dataset-dependent and grows with the number of classes. The question is whether that cost is material next to the feature extractor.

Parameter count and GFLOPs of nine backbones, next to the additional parameters and GFLOPs of the standard linear head and of the OPAL head, averaged over the five datasets. — Table 3b. **Backbone cost versus head cost.** Backbone cost on the left; on the right, what the classification head adds on top, averaged over the five datasets. Unshaded rows are the standard linear head, shaded rows the OPAL head.

Cheap to add. Averaged over backbones and datasets, the OPAL head accounts for 2.36% of the backbone parameters and 0.53% of its GFLOPs, so the feature extractor remains the dominant cost.

Qualitative analysis

What the model actually looks at

Every explanation below is read directly off the forward pass: for the predicted class, each of its \(m = 5\) prototype channels contributes the single image location that produced its pooled response. Nothing is recomputed and no attribution method is involved.

Prototype-based explanations

Two test images per dataset for CUB, CARS, PETS, DOGS and FLOWER, each annotated with five coloured boxes marking the prototype maximizers, followed by grids of the corresponding image patches drawn from other images of the same class. — Figure 3. **Prototype-based explanations on five benchmarks (\(m = 5\)).** Colored boxes mark the spatial maximizer selected by each prototype. In the patch grids below each image, the first row is the evidence extracted from that image and the remaining rows come from other images of the same predicted class.

Complementary evidence. The \(m\) slots spread over distinct discriminative parts of the object instead of concentrating on one salient region — which is what fine-grained recognition needs, since class separation often depends on subtle part-level attributes.
Consistent semantics. The same slot repeatedly selects the same part across different images of a class: ears for Abyssinian cats in PETS, eyes for Maltese dogs in DOGS.

Understanding prototype diversity

Two design choices control what the explanations look like: the channel-wise softmax, which makes prototypes compete for spatial locations, and \(m\), which sets how many parts each class is allowed to use.

Comparison of OPAL explanations with and without channel-wise softmax across the five datasets. Without it, prototype boxes pile onto the same one or two regions. — Figure 5a. **Effect of channel-wise softmax on prototype diversity.** Without competition, prototype slots converge onto one or two dominant regions and the evidence patches become redundant; with it, they cover distinct parts of the object.

Explanations for m equal to 1, 3, 5 and 10 prototypes per class across the five datasets, showing that at m equal to 10 some slots start selecting background. — Figure 5b. **Effect of the number of prototypes per class.** \(m \in \{3,5\}\) captures meaningful parts. At \(m = 10\) most slots still land on informative regions, but a subset begins to attach to weakly informative or background areas — each class only contains so many salient, discriminative regions.

Competition prevents collapse. Measured by the Spatial Prototype Collapse score, the variant without channel-wise softmax reaches 72.1–93.6 across datasets, while full OPAL stays at 4.9 or below.
Granularity is controlled by \(m\). A single parameter trades representational capacity against discriminative focus, and its qualitative effect mirrors the accuracy trend measured below.

Quantifying prototype quality

CUB-200-2011 is the only one of the five benchmarks with part-level annotations, so prototype consistency — whether a prototype keeps selecting the same annotated body part across images of its class — and stability — whether that association survives input noise — are reported there.

Prototype consistency and stability on CUB-200-2011 for six prior part-prototype methods and OPAL. — Table 5. **Prototype consistency and stability on CUB-200-2011.** Higher is better; bold is best, underlined second best.

High prototype quality. OPAL is second best on both metrics, at 50.5 consistency and 75.9 stability.
Competitive without specialized machinery. The two per-metric leaders rely on it: LucidPPN on an external part-discovery model for initial prototype alignment, PIP-Net on a self-supervised prototype warm-up stage and auxiliary losses. OPAL needs neither.

Understanding design choices

What each component contributes

The \(1\times1\) projection (Proj.), global max pooling (GMP), channel-wise softmax (CWS) and orthonormal anchors (Orth.) are enabled one combination at a time. All crosses is the plain convolutional baseline; all checks is full OPAL.

Architectural ablation

Geometry is the key ingredient. Enabling orthonormal anchoring improves the matched non-orthogonal variants by \(+4.07\) points on average.
Localization alone is not enough. Configurations that pool with GMP but without Orth. fall 9.86 points below the baseline while still producing perfectly localized evidence.
The components are complementary. Some ablated variants win on isolated datasets, but they forfeit what OPAL is built for: dropping GMP breaks the one-maximizer-per-channel correspondence, dropping CWS removes the competition that keeps prototypes distinct.

Sensitivity to the number of prototypes per class

Six line plots of accuracy against the number of prototypes per class for CUB, CARS, PETS, DOGS, FLOWER and their average, each with the ConvNeXt-Tiny baseline as a dashed line. — Figure 4. **Accuracy against \(m\).** OPAL evaluated with \(m \in \{1,3,5,10\}\) on each dataset and on their average, with the ConvNeXt-Tiny baseline as a dashed line and the rest of the training protocol fixed.

One slot per class is insufficient. \(m = 1\) underperforms the baseline by 1.04 points on average.
Moderate budgets work best. \(m \in \{3,5,10\}\) all sit above the baseline, by \(+1.25\) points on average; too many slots dilute the discriminative evidence, which makes \(m = 5\) the operating point used throughout.

Take-home message

Fix the prototype directions, learn the representation

Orthonormal anchoring replaces the pipeline. Preassigning the class geometry makes single-stage, auxiliary-loss-free interpretable learning possible: one objective, one training run, image-level labels only.
Interpretability without an accuracy penalty. OPAL yields diverse, localized evidence while improving or matching its non-interpretable counterpart in 39 of 45 backbone–dataset settings.

Take-away results

+1.56points over the ConvNeXt-Tiny backbone, averaged over five benchmarks

39/45backbone–dataset settings improved or matched, across nine backbones

2.36%of backbone parameters added by the OPAL head, and 0.53% of its GFLOPs

1 stageend-to-end training, cross-entropy only, image-level labels only

Resources

Paper, code, poster and video

PaperarXiv preprint Codegithub.com/ilancarretero/OPAL PosterECCV 2026 · 1400 × 1000 mm VideoConference presentation

Citation

If OPAL is useful in your work, please cite it

@inproceedings{carretero2026opal,
  title     = {OPAL: Orthonormal Prototype Alignment Learning for Interpretable Image Classification},
  author    = {Carretero, Il\'an and Angulo, Gustavo Jes\'us and del Amor, Roc\'io and Naranjo, Valery},
  booktitle = {European Conference on Computer Vision (ECCV)},
  year      = {2026}
}