Beyond $L_2$: Generalizing Abductive Latent Explanations
to Diverse Prototype-Based Architectures

Jules Soria , Alban Grastien , Romain Xu-Darme ,
Julien Girard-Satabin , Zakaria Chihani , Daniela Cancila
Université Paris-Saclay, CEA, List, F-91120, Palaiseau, France
ECML PKDD 2026
Naples, Italy · September 7–11, 2026
Paper Code arXiv Zenodo

Abstract

Prototype-based neural networks are hailed as interpretable-by-design architectures. Recently, Abductive Latent Explanations (ALE) were introduced to provide formal, mathematically guaranteed explanations that leverage the intrinsic structure of these networks to ensure both predictive safety and human readability. ALEs rely on computing tight bounds on latent space distances to produce formal explanations.

However, existing ALE formulations are rigidly confined to Euclidean latent spaces. This leaves a critical gap: modern state-of-the-art architectures increasingly rely on non-Euclidean representations — spherical metrics, Gaussian densities, and dimensional projections — rendering current formal explanation methods incompatible. In this work, we generalize the ALE framework to support non-Euclidean prototype architectures.

For each geometric variant, we systematically derive how to either map the architecture to existing bounds or construct novel, architecture-specific bounding algorithms. We validate our theoretical constructions by computing subset-minimal formal explanations on fully trained image classifiers. By unifying these diverse models under a single formal framework, we enable the first rigorous, cross-architecture comparison of their interpretability.

Where Architectures Diverge from ProtoPNet

Every prototypical-parts network shares the same skeleton: an encoder turns an image into latent patches, each patch is compared to a set of learned prototypes, the per-patch scores are pooled into one activation per prototype, and a linear decision layer turns those activations into class logits. Architectures diverge in exactly three places along this pipeline — click the arrows (or dots) below to see what changes at each one.

See what happens, step by step, when an image is fed in.
encoder $f$
$D$
$H_1$
$W_1$
$z_l$
$p_1$
distance $d$
$p_2$
$d_2$
$p_3$
$d_3$
$p_m$
distance map
similarity map
$a_1$
$a_2$
$a_3$
$a_m$
pooling
⋮
⋮
⋮
⋮
✓ predicted
image space
$\mathbb{R}^{H_0\times W_0\times C_0}$
latent space
$\mathbb{R}^{H_1\times W_1\times D}$
prototype space
$\mathbb{R}^{D\times m}$
distance → similarity
$\mathbb{R}^{m\times H_1\times W_1}$
activation space
$\mathbb{R}^{m}$
class logits
$\mathbb{R}^{C}$
encoding layer $f$
prototype activation layer $a$
decision layer $h$
predictor function $\kappa$

1. Distance Function

How "closeness" between a latent patch and a prototype is first measured, before any conversion to a bounded score.

p z d

$d(z_l,p_j) = \|z_l-p_j\|_2$

VariantFormulaUsed by
Euclidean ($L_2$) $d(z_l,p_j) = \|z_l-p_j\|_2$ ProtoPNet, ProtoPool, ProtoGMM†, MGProto†, ProtoPFormer†
Cosine / spherical $d(z_l,p_j) = \arccos(z_l \cdot p_j)$ TesNet, Deformable ProtoPNet, PIP-Net, HyperPG†, NonParam
Gaussian $d(z_l,p_j) = \dfrac{\|z_l-\mu_j\|_2}{\sigma_j}$
prototype $\sim\mathcal{N}(\mu_j,\sigma_j^2 I)$		 ProtoGMM, MGProto, HyperPG, ProtoPFormer, Isotropic Gaussian ProtoPNet (ours)

† Gaussian-distributed prototypes are layered on top of an existing base metric rather than replacing it — that's why these four names also appear in the Gaussian row below.

Why not the raw log-likelihood? For the Gaussian variant, we bound distances by mapping back to the equivalent Euclidean radius (Sec. 4.4) instead of working directly with the negative log-likelihood — the Euclidean form keeps training smooth and fully differentiable.

Geometric Bounding Mechanisms

The original ALE framework bounds a latent patch's location using intersections of Euclidean hyperspheres. We show that a similar principle — shrink the set of locations consistent with known prototypes down to a single, tighter region — carries over to three very different geometries, plus one pooling-level extension that composes with any of them. Use the arrows (or dots) below to cycle through them.

Spherical Cap Intersection (Cosine Similarity)

Architectures such as TesNet use cosine similarity, i.e., a dot product on the unit hypersphere. Cosine similarity is not a distance metric, but the geodesic angular distance $d_\angle(x,y) = \arccos(x \cdot y)$ is. We approximate the intersection of two "spherical caps" around known prototypes by a single minimal-radius bounding cap $(\mathbf{p}_{int}, r_{int})$, which provably shrinks every time a new prototype is added:

$$ \underline{\text{sim}}_{\mathcal{E}}(z_l,p_k) = \cos\!\Big(\min\big(\pi,\ d_\angle(p_k,\mathbf{p}_{int})+r_{int}\big)\Big) $$ $$ \overline{\text{sim}}_{\mathcal{E}}(z_l,p_k) = \cos\!\Big(d_\angle(p_k,\mathbf{p}_{int})-r_{int}\Big) $$
d∠(pₖ,p₳ₙₜ) p₁ p₂ z pint pₖ
p₁ p₂ z pₖ θ (azimuth)φ (polar)

Figure 1 (try it): drag the sphere to rotate the view; use the $\theta,\varphi$ sliders to move $p_1$, $p_2$, $z$ or $p_k$. The two observed caps (blue, orange) are bounded by a single tighter cap (dashed red), giving a live similarity bound for $p_k$: […].

Experimental Results

We trained a ProtoPNet, PIP-Net, TesNet, and Isotropic Gaussian ProtoPNet (plus ProtoPool on CUB200) using the CaBRNet library, and computed subset-minimal ALEs for every applicable paradigm on three fine-grained datasets: Oxford Flowers 102, Oxford IIIT Pets, and CUB200 — select a dataset below to see its table. For each architecture, bold marks the best (smallest) result among its own ALE paradigms, and the overall best per column is additionally underlined. PIP-Net's non-negative, sparse decision head consistently yields the smallest and fastest explanations across all three datasets, while Gaussian and cosine-based spatial paradigms (Scaled/Spherical HIA) tighten the relative bound at the cost of runtime — and time out entirely on CUB-200, exposing a scalability limit as the number of classes grows.

ArchitectureAcc. (%)ALEAbs. SizeRel. Size (%)Time (s)
ProtoPNet82.8Top-k199 ± 25419.5 ± 24.90.327 ± 0.406
TI5273 ± 607532.3 ± 37.260.434 ± 61.289
HIA675 ± 3614.1 ± 2.211.781 ± 13.676
PIP-Net80.5Top-k135 ± 2917.6 ± 3.81.229 ± 1.413
Sparse103 ± 5513.5 ± 7.20.075 ± 0.089
Simplex250 ± 1030.2 ± 0.10.286 ± 0.069
TesNet81.9Top-k295 ± 16628.9 ± 16.30.341 ± 0.224
Cosine TI10438 ± 286264.0 ± 17.5202.580 ± 64.435
Spherical HIA187 ± 511.1 ± 0.34.491 ± 2.827
Gaussian72.8Top-k257 ± 23725.2 ± 23.20.651 ± 0.563
Scaled TI1022 ± 20356.3 ± 12.520.490 ± 44.649
Scaled HIA931 ± 17705.7 ± 10.8157.373 ± 312.566

Framework

Our models were trained using CaBRNet (Case-Based Reasoning Network), an open-source library designed for developing and evaluating prototype-based models.

CaBRNet Logo View CaBRNet Repository

BibTeX

@inproceedings{soria2026beyond,
  title={Beyond $L_2$: Generalizing abductive latent explanations to diverse prototype-based architectures},
  author={Soria, Jules and Grastien, Alban and Xu-Darme, Romain and Girard-Satabin, Julien and Chihani, Zakaria and Cancila, Daniela},
  booktitle={Joint European Conference on Machine Learning and Knowledge Discovery in Databases},
  year={2026},
  organization={Springer}
  }

Acknowledgements

This publication was made possible by the use of the FactoryIA supercomputer, financially supported by the Ile-De-France Regional Council. This work was also supported by the SAIF project, funded by the "France 2030" government investment plan managed by the French National Research Agency, under the reference ANR-23-PEIA-0006.

This is a follow-up to our AAAI 2026 paper, Formal Abductive Latent Explanations for Prototype-Based Networks.