A Unifying Framework of Concept-based Explainable AI with Completeness Guarantees

Vojtěch Kůr; Adam Kukučka; Tomáš Brázdil; Vít Musil

cs.LGSep 28, 2026

A Unifying Framework of Concept-based Explainable AI with Completeness Guarantees

Authors: Vojtěch Kůr, Adam Kukučka, Tomáš Brázdil, Vít Musil

Organizations: Masaryk University, Faculty of Informatics, Brno, Czechia

Abstract

Concept-based explanations describe neural network predictions through human-understandable properties of inputs called concepts. The field encompasses approaches that differ in how they define and represent concepts and connect them to model predictions. We introduce a theoretical framework that describes these approaches in a common mathematical language and supports a shared analysis of their properties. For concept discovery, which identifies concepts automatically within a latent space of a trained model, we employ a concept autoencoder view. An encoder extracts concept representations from the model's latent space, and a decoder uses them to reconstruct the original latent representation. The autoencoder's reconstruction error measures how accurately its decoder recovers the original latent representation. We revisit model completeness: how well the concepts can reproduce the model's outputs. We show that model incompleteness of the concepts can be bounded by the autoencoder's reconstruction error. The autoencoder view also provides a common way to define individual concept attributions, which measure each concept's contribution to a prediction. We establish when these attributions sum to the model's prediction, and bound the discrepancy otherwise, thus providing attribution completeness guarantees.

Figures & tables

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Method	$\mathcal{Z}$	$\mathcal{V}_{i}$	$\pi_{i}(z)$	$\sigma_{i}$	$D$	$\gamma$ affine
Explainable by design
CBM, probabilities	$[0,1]^{k}$	$[0,1]$	$z_{i}$	$\id$	$\id$	if $g$
CBM, logits	$\mathbb{R}^{k}$	$\mathbb{R}$	$z_{i}$	$s$	$\id$	if $g$
PCBM, LF-CBM	$\mathbb{R}^{k}$	$\mathbb{R}$	$z_{i}$	–	$\id$	yes
CEM	$\prod_{i}\mathbb{R}^{2m}$	$\mathbb{R}^{2m}$	$z_{i}$	$s(\langle w,v\rangle+\beta)$	$\id$	no
Concept detection

Appendix

Table 1: Instantiations of concept-based methods in our framework: CBMs ( Koh et al., 2020 ) , PCBM ( Yuksekgonul et al., 2023 ) , LF-CBM ( Oikarinen et al., 2023 ) , CEM ( Espinosa Zarlenga et al., 2022 ) , TCAV ( Kim et al., 2018 ) , CAR ( Crabbé & Schaar, 2022 ) , ACE ( Ghorbani et al., 2019 ) , CRAFT ( Fel et al., 2023b ) , ICE ( Zhang et al., 2021 ) , SAE ( Bricken et al., 2023 ) , and MCD ( Vielhaben et al., 2023 ) . Columns give the latent space

\mathcal{Z}

, the concept representation space

\mathcal{V}_{i}

, the probe

\pi_{i}

, the activation function

\sigma_{i}

, the decoder

D

, and whether the decoded head

\gamma=g\circ D

is affine; “if

g

” means that

\gamma

is affine whenever

g

is. A dash marks a component the method does not specify.

s

is the logistic sigmoid,

U

is a dictionary with concept directions

u_{i}

in its rows, NMF encoders compute nonnegative least-squares coefficients with

U

fixed, and

Q_{i}

projects channels onto the

i

-th MCD subspace, with the residual subspace included as a concept.

Method	$k$	$\rho_{\fiderror}$	$\alpha$	$\kappa$	$\hat{\rho}_{\modelerror}$
	5	0.106	0.270	0.394	0.967
PCA	25	0.103	0.266	0.386	0.927
	50	0.099	0.261	0.381	0.939
	5	0.106	0.269	0.395	0.975
NMF	25	0.102	0.264	0.389	0.960
	50	0.101	0.263	0.385	0.964

Appendix

Table 2: Tightness of the fidelity and model completeness bounds, mean over three seeds; the standard deviation over seeds is at most

0.012

for

\hat{\rho}_{\modelerror}

and at most

0.002

for the other ratios.

\rho_{\fiderror}=\alpha\kappa

is the tightness of , with alignment

\alpha

and spatial coherence

\kappa

from .

\hat{\rho}_{\modelerror}

upper-bounds the tightness of .

Boundary	$k$	Bound	Ins.	Occ.	G $\times$ I	Ins.	Occ.	G $\times$ I
			$\adderror(A,\varphi)$			$\rho_{\atterror}\times 10^{3}$
	5	2281	1.31	1.39	0.10	3.0	3.0	2.9
layer4	25	2381	2.34	2.92	0.06	2.9	1.9	2.4
	50	2749	3.49	4.20	0.06	2.7	1.4	2.0
	5	2593	0.78	0.87	0.25	3.0	2.8	2.9
penultimate	25	1909	1.70	3.32	0.13	3.5	2.2	2.9

Appendix

Table 3: Tightness of the attribution bound for the nonlinear autoencoder, mean over three seeds. Bound is

\fiderror(A)+M\sqrt{\mathbb{E}\bigl[\|C(x)\|_{2}^{4}\bigr]}

with

\fiderror

on the explained scores, shared by insertion (Ins.), occlusion (Occ.), and gradient-times-input (G

\times

I), and

\rho_{\atterror}

is the tightness of the attribution bound obtained by combining with . The bound, and with it

\rho_{\atterror}

, varies across seeds by up to

40\%

because

M

is estimated by sampling, whereas

\adderror

varies by at most

0.4

. For the four methods with affine decoders,

\adderror<2\cdot 10^{-5}

and

\rho_{\atterror}=1

for all three attribution functions.

Explore similar work

Concept-Based Abductive and Contrastive Explanations for Behaviors of Vision Models
May 7, 2026Ronaldo Canizales, Divya Gopinath, Corina Păsăreanu +1Explainability Explainable AI Methods
Colorado State University, Fort Collins CO, USA · KBR Inc., NASA Ames, Moffett Field CA, USA · Carnegie Mellon University, Pittsburgh PA, USA
ConceptSMILE: Auditing the Trustworthiness of Concept-Based Explainable AI
Jul 10, 2026Mohadeseh Mollapour, Koorosh Aslansefat, Zeinab Dehghani +3Explainable Artificial Intelligence Trustworthy Artificial Intelligence
School of Computer Science University of Hull Hull, United Kingdom · PhD Researcher (Engineering), WMG University of Warwick Coventry, United Kingdom · School of Computing Newcastle University Newcastle upon Tyne, United Kingdom
A Unifying Framework for Unsupervised Concept Extraction
Apr 27, 2026Chandler Squires, Pradeep RavikumarUnsupervised Unified Framework
Machine Learning Department, CMU

May 7, 2026, cs.LGJ/K move · Enter open · S save

Concept-Based Abductive and Contrastive Explanations for Behaviors of Vision Models

Ronaldo Canizales, Divya Gopinath, Corina Păsăreanu, Ravi Mangal

Colorado State University, Fort Collins CO, USA · KBR Inc., NASA Ames, Moffett Field CA, USA · Carnegie Mellon University, Pittsburgh PA, USA

Explainability Explainable AI Methods Vision Foundation Models Concept Bottleneck Models Abductive Reasoning Agent Behavior Modeling+1

Open paper AskPDF arXiv

Concept-based explanations offer a promising approach for explaining the predictions of deep neural networks in terms of high-level, human-understandable concepts. However, existing methods either do not establish a causal connection between the concepts and model predictions or are limited in expressivity and only able to infer causal explanations involving single concepts. At the same time, the parallel line of work on formal abductive and contrastive explanations computes the minimal set of input features causally relevant for model outcomes but only considers low-level features such as pixels. Merging these two threads, in this work, we propose the notion of concept-based abductive and contrastive explanations that capture the minimal sets of high-level concepts causally relevant for model outcomes. We then present a family of algorithms that enumerate all minimal explanations while using concept erasure procedures to establish causal relationships. By appropriately aggregating such explanations, we are not only able to understand model predictions on individual images but also on collections of images where the model exhibits a user-specified, common behavior. We evaluate our approach on multiple models, datasets, and behaviors, and demonstrate its effectiveness in computing helpful, user-friendly explanations.

Abstract

Explore similar work

Concept-Based Abductive and Contrastive Explanations for Behaviors of Vision Models

ConceptSMILE: Auditing the Trustworthiness of Concept-Based Explainable AI

A Unifying Framework for Unsupervised Concept Extraction