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

cs.LGAug 17, 2026

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

Authors: Jules Soria, Alban Grastien, Romain Xu-Darme, Julien Girard-Satabin, Zakaria Chihani, Daniela Cancila

Organizations: Université Paris-Saclay, CEA, List, F-91120, Palaiseau, France

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 - such as 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.

Figures & tables

Name	Similarity	Gaussian	Pooling	Reference
ProtoPNet	Euclidean	$\times$	Max	[ 7 ]
ProtoPool	Euclidean	$\times$	Focal	[ 37 ]
TesNet	Cosine	$\times$	Max	[ 43 ]
Def. ProtoPNet	Cosine	$\times$	Max	[ 11 ]
PIP-Net	Cosine $\star$	$\times$	Max	[ 28 ]
ProtoGMM	Euclidean	✓	Max	[ 27 ]

Table 1: Evolution of prototype-part network modules. Inspired by Table 1 of [ 19 ] .

Panel of Figure 1 — Figure 1: Spherical Cap Intersection Approximation. The blue spherical cap is defined by $\mathbf{p}_{1}$ as its center and its radius by the distance to $\mathbf{z}_{l}$ . Similarly, the orange spherical cap is defined by $\mathbf{p}_{2}$ as its center and its radius by the distance to $\mathbf{z}_{l}$ . Their intersection is approximated by the dashed red spherical with center $\mathbf{p}_{int}$ , who also has its radius equal to its distance to $\mathbf{z}_{l}$ . With that intersection computed, we have tighter bounds on the angular distance between an unknown prototype $\mathbf{p}_{k}$ and the latent patch $\mathbf{z}$ .

Panel of Figure 2 — Figure 2: Gaussian Hypersphere Intersection Approximation. Each prototype $\mathbf{p}_{j}$ is defined by its anchor $\mu_{j}$ and its variance $\sigma_{j}$ (represented as a layer of the same color). In this figure, $\mathbf{z}_{l}$ is at distance $2$ to $\bm{\mu_{1}}$ and distance $3$ to $\bm{\mu_{2}}$ . The red dotted circle encompasses the hypersphere that approximates the intersection between the two hyperspheres in the Euclidean metric space.

Architecture	Acc. (%)	ALE	Abs. Size	Rel. Size (%)	Time (s)
ProtoPNet	$82.8$	Top- $k$	$\mathbf{199}\pm 254$	$19.5\pm 24.9$	$\mathbf{0.327}\pm 0.406$
		TI	$5273\pm 6075$	$32.3\pm 37.2$	$60.434\pm 61.289$
		HIA	$675\pm 361$	$\mathbf{4.1}\pm 2.2$	$11.781\pm 13.676$
PIP-Net	$80.5$	Top- $k$	$135\pm 29$	$17.6\pm 3.8$	$1.229\pm 1.413$
		Sparse	$\underline{\mathbf{103}}\pm 55$	$13.5\pm 7.2$	$\underline{\mathbf{0.075}}\pm 0.089$
		Simplex	$250\pm 103$	$\underline{\mathbf{0.2}}\pm 0.1$	$0.286\pm 0.069$

Table 2: Comparison of explanation size and computing time on Oxford Flowers 102.

Architecture	Acc. (%)	ALE	Abs. Size	Rel. Size (%)	Time (s)
ProtoPNet	$89.3$	Top- $k$	$\mathbf{99}\pm 72$	$26.6\pm 19.4$	$\mathbf{0.101}\pm 0.06$
		TI	$7869\pm 6537$	$43.4\pm 36.1$	$105.805\pm 83.69$
		HIA	$733\pm 204$	$\mathbf{4.0}\pm 1.1$	$3.098\pm 1.49$
PIP-Net	$88.7$	Top- $k$	$131\pm 13$	$17.0\pm 1.7$	$1.586\pm 0.231$
		Sparse	$\mathbf{30}\pm 16$	$4.0\pm 2.0$	$\mathbf{0.076}\pm 0.038$
		Simplex	$190\pm 30$	$\underline{\mathbf{0.1}}\pm 0.0$	$0.252\pm 0.040$

Table 3: Comparison of explanation size and computing time on Oxford IIIT Pets.

Architecture	Acc. (%)	ALE	Abs. Size	Rel. Size (%)	Time (s)
ProtoPNet	$83.9$	Top- $k$	$\mathbf{314}\pm 218$	$15.7\pm 10.9$	$\mathbf{0.997}\pm 0.732$
		TI	$1950\pm 5860$	$2.0\pm 6.0$	$213.616\pm 666.449$
		HIA	$1349\pm 3769$	$\mathbf{1.4}\pm 3.8$	$2446.927\pm 11363.445$
ProtoPool	$82.5$	Top- $k$	$\mathbf{67}\pm 44$	$30.5\pm 20.3$	$\underline{\mathbf{0.195}}\pm 0.111$
		TI	$893\pm 522$	$45.5\pm 26.6$	$7.502\pm 3.505$
		HIA	$272\pm 315$	$\mathbf{13.8}\pm 16.1$	$7.666\pm 12.973$

Table 4: Comparison of explanation size and computing time on CUB200.

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