Disentangled Concept-Residual Models: Bridging the Interpretability–Performance Gap for Incomplete Concept Sets
Authors: Renos Zabounidis
Ini Oguntola
Konghao Zhao
Joseph Campbell
Woojun Kim
Simon Stepputtis
Katia P. Sycara
Venue: Accepted by TMLR ()

Abstract: Deploying AI in high-stakes settings requires models that are not only accurate but also interpretable and amenable to human oversight. Concept Bottleneck Models (CBMs) support these goals by structuring predictions around human-understandable concepts, enabling interpretability and post-hoc human intervenability. However, CBMs rely on a ‘complete’ concept set, requiring practitioners to define and label enough concepts to match the predictive power of black-box models. To relax this requirement, prior work introduced residual connections that bypass the concept layer and recover information missing from an incomplete concept set. While effective in bridging the performance gap, these residuals can redundantly encode concept information, a phenomenon we term \textbf{concept-residual overlap}. In this work, we investigate the effects of concept-residual overlap and evaluate strategies to mitigate it. We (1) define metrics to quantify the extent of concept-residual overlap in CRMs; (2) introduce complementary metrics to evaluate how this overlap impacts interpretability, concept importance, and the effectiveness of concept-based interventions; and (3) present \textbf{Disentangled Concept-Residual Models (D-CRMs)}, a general class of CRMs designed to mitigate this issue. Within this class, we propose a novel disentanglement approach based on minimizing mutual information (MI). Using CelebA, CIFAR100, AA2, CUB, and OAI, we show that standard CRMs exhibit significant concept-residual overlap, and that reducing this overlap with MI-based D-CRMs restores key properties of CBMs, including interpretability, functional reliance on concepts, and intervention robustness, without sacrificing predictive performance.

Published in Transactions on Machine Learning Research (01/2026)
Disentangled Concept-Residual Models: Bridging the Inter- pretabilty-Peerormmacc Gap for Incomplete Concept Sets
Renos Zabounidis
remosz@is.cmmu
School of Computer Science, Cornegie Mellon Universtiy
Ini Oguntola
toguntol8andreww.o www.e./wcmccc......
School of Computer Science, Curnegie Mellon University
Konghao Zhao
konghaoz&Busc.chh edu
Department of Computer Science, University of Southern California
Joseph Campbell
Srecamp@parduer edu
Department of Computer Science, Purduc University
Woojun Kim
woojunkSSendrew.cmet
School of Computer Science, Cornegie Mellon University
Simon Stepputtis
stepputtis@ut. edu
Department Mechanical Engineering, Virginia Tech
Katia Sycara
sycora www.ddrwwwwcc..coom edu
School of Computer Science, Carnegie Mellon University
Reviewed OpenReview: https://openrevveet net/ pormff id= NKgNN zuDaō
Abstract
Deploying AI in high-stakes settings requires models that are not only accurate but also in- terpretable and amenable to human oversight. Concept Bottleneck Models (CBMs) support these goals by structuring predictions around human-understanddlle concepts, enabling in- terpretability via steerability and post-hoc human oversight. However, CBMs rely on 'complete' concept set, requiring practitioners to define and label enough concepts to match the predictive power of black-box models. To relax this requirement, prior work introduced residual connections that bypass the concept layer and recover information missing from an incomplete concept set. While effective in bridging the performance gap, these residuals can redundantly encode concept information, phenomenon we term concept-residual overlap. In this work, investigate the effects of concept-residhal overlap and evaluate strategies to mitigate it. We (1) define metrics to quantify the extent of concept-residual overlap in CRMs; (2) introduce complementary metrics to evaluate how this overlap impacts interpretability by reducing steerability to concept interventions; and (3) present Disen- tangled Concept-Residual Models (D-CRMs), a general class of CRMs designed to mitigate this issuc. Within this class, we propose novel disentanglement approach based on minimizing mutual information (MI). Using CelebA, CIFAR100, AA2,, CUB, and OAI, show that standard CRMs exhibit significant concept-residual overlap, and that reduc- ing this overlap with MI-based D-CRMs restores the key properties of CBMs that make them interpretable, namely functional reliance on concepts and robust intervention behav- ior, without sacrificing predictive performance.

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Published in Transactions on Machine Learning Research (01/2026)
Disentangled Concept-Residual Model (D-CRM)
Input Sample
Concepts
Label Prediction y
Plopoy
Golden Retriever
Predicted Label logits
Label
Embedding Network g
Medium Size
Label Predictor
Pypostergeeii
Cocker Spanicl
Ltarget (ū,
Vocal
Setter
Predicted
Concept
Desgle
Concept logits
Loss Function
Comnceps (é, c)
Residual /
Disentanglement
Residual
Loss Function
T
Lis (r,
Figure 1: Overview of the Disentangled Concept-Residual Model (D-CRM). Given an input sample, the embedding network predicts concept logits and residual representation. The predicted concepts are super- vised using ground-truth labels and, together with the residual, are passed to the label predictor to produce class logits. A disentanglement objective encourages the residual to encode information complementary the concepts, while label loss ensures accurate classification.
1 Introduction
State-of-the-art deep learning models have achieved remarkable performance in complex tasks, surpassing humans areas such as computer vision (Han al.] [2022). speoch recognition (Radford al. (023) and strategic game playing (Silver et al., 2016; Brown & Sandholm, 2018). However, despite their high accuracy, the underlying decision-making processes of these models remain opaque to humans, leading to their characterization black bores. This opacity poses significant risks when deep learning is deployed in safety-critical applications, such as medical imaging (Salahuddin et al. 2022), where the issued diagnosis must be grounded in observed evidence.
Concept Bottleneck Models (CBMs) (Koh et al., 2020) enhance model interpretability by explicitly pre- dicting intermediate, semantically meaningful concepts, which subsequently inform downstream predictions. These human--ndderstanddall concepts facilitate clear insights into model decisions, enabling to trace predictions back to interpretable features. However, CBMs rely heavily predefined comprehensive concept sets and sufficient labeled data, making them challenging to apply broadly in practice.
To address these limitations, Concept Residual Models (CRMs) (Yuksekgonul et al. 2023 Sawada & Naka- mura 2022 Zabounidis et al. 2023 Zhang et al.) 2023 Espinosa Zarlenga et al. 2022) introduce an auxiliary residual layer that captures additional lask-relevant. information beyond the predefined concepts. While this improves downstream performance, it creates challenge: residual layers can encode redundant concept information. This redundancy causes the downstream network to preferentially rely the opaque residual representation rather than the interpretable concept features. We term this phenomenon concept-residut overlap, and show it undermines intervention effectiveness and therefore interpretability originally sought by CBMs.
In this paper, we propose a comprehensive approach to tackle concept-residual overlap by enhancing disen- tanglement between concept and residual representations. We refer to this class of models as Disentangled Concept-Residual Models (D-CRMs). Central to our approach is novel training objective based on min- imizing an upper bound of mutual information (MI) between concept and residual representations, which addresses the root cause of reduced interpretability: models no longer steerable via concepts when residuals redundantly encode concept information.
Our contributions can be summarized as follows:
We propose mutual information as measure for concept-residual overlop and analyze how redundant encoding of concept information across concept and residual pathways fundamentally impairs intervention effectiveness and therefore interpretability.
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We introduce and emphasize new model stecrability metrics random concept interventions and random residual interventions that complement traditional performance evaluations, providing deeper insights into model steerability and residual reliance.
We propose a novel mutual information minimization objective that reduces concept-residual overlap by 61- 81 % while maintaining performance within 3% of baseline CRMs, achieving 3-57x x better concept importance preservation and intervention accuracy gains up to 16%.
We empirically validate our approach across five diverse datasets, demonstrating that MI-based D-CRMs achieve near-complete overlap suppression on challenging datasets and maintain 98% counterfactual rea- soning accuracy compared to 25% for standard CRMs. This significantly broadens the applicability and practical utility of interpretable concepl-residual models.
2
Background
Concept learning approaches a prediction task from inputs E X to outputs E Y by introducing a low- dimensional intermediate representation E Z that is in some way designed to correspond to semantically meaningful concepts (Lampert et al. 2009 Kumar et al.) 2009 Kim et al.) 2018 Koh et al.) 2020 Chen
et al. 2020 Typically these concepts are specified as a collection of ground truth values e representing couman-interpretabbee properties of input (for example, the color of bird's feathers), and are used during training learn suitable representation z.
For example, in the dog classification task shown in Figure 2 model might be trained to first predict concepts like 'medium size' or 'brown fur' before predicting the final class, 'duck tolling retriever'. This structure is the foundation of Concept Bottleneck Models (CBMs).
2.1 Concept Bottleneck Models
CBMs aim to learn an intermediate concept repre- sentation = è that directly predicts ground truth concept values (Koh et al. 2020). The model is de- composed into two components: concept encoder g + C supervised ues, and predictor f : C Y supervised on the prediction task.
Imput Sample
Concoptu e
Label Prediction
Euppy
Embolddngg Network
Dusinot Retriever
[hoesterent
Predicter
Cocker Spaniel
Setter
ground truth concept val-
Beagle
Observa
Exstesation
Ceseaver Bacople
This approach has the advantage of being directly interpretable and straightforward to intervene over. For instance, in Figure 2 the model incorrectly pre- dicts 'cocker spaniel' because it failed to recognize the 'medium size' concept. An expert can manually correct this single concept prediction, which should ideally correct the final prediction to 'duck tolling retriever'. This ability to intervene is a key advan- tage of CBMs.
Interone
Concepts
Lebel Prediction
Golden Retreeeer
Rutriever
Predictor
Cooker Spanie
Human Expert
Vocal
Softor
Figure 2: Demonstration of test-time concept inter- vention mislabeled sample. The input image is duck tolling retriever, but the model predicts cocker spaniel due to incorrectly predicting the "medium size" concept as false. Manually correcting the concept to true should ideally update the downstrremm prediction; however, if concept-resichal overlap exists, the resid- ual may still encode conflicting size information, pre- venting the prediction from updating. Our proposed D-CRMs mitigate this by enforcing disentanglement between concepts and residuals.
Despite their benefits, vanilla CBMs suffer from two main limitations: concept. leakage and the bottle- neck assumption.
Concept Leakage Concept Icakage occurs when extraneous information is encoded in the concept layer itself. This compromises interpretability and roduces intervention performance, as the down- stream network relies on leaked features that inter- ventions inadvertently remove. Mahinpei et al. were
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the first to name and empirically demonstrate this phenomenon in standard CBMs, showing how it degrades both interventionability and interpretability (Mahinpei et al. 2021 Margeloiu et al. 2021 l.
Limited Capacity and the Bottleneck Assumption CBMs rely on the bottleneck assumption, that the intermediate concept representation is sufficient to accurately predict the target label. This assumes that the provided set of ground truth concepts captures all information relevant to the downstream task. In our dog example, concepts like 'medium size' and 'brown fur' might not be enough to distinguish a 'duck tolling retriever' from 'golden retriever'. The model might need information about 'fur texture' (e.g., wavy vs. straight), which may not be in the predefined concept set. When the concept set is incomplete, the model may struggle to make accurate predictions, which limits the practical utility of CBMs.
2.2 Concept- Residual Models
Concept-Residual Models (CRMs) are an extension of CBMs that mitigate the bottleneck assumption by additionally including a residual r E Re in the intermediate representation = = (2, to encode extra information relevant to the downstream prediction task (Sawada & Nakamura 2022). The model is now decomposed into three components: a concept encoder g X + C, a residual encoder h : X + R, and a predictor f C R Y. In practice, many of the layers in 9 and h are shared. The residual can, in theory, learn the missing information about 'fur texture' to solve the classification task.
Concept-Residual Overlap By introducing residual pathway, CRMs relax the bottleneck constraint. However, this flexibility introduces the risk of concept-residual overlap, where the same concept is re- dundantly encoded in both the concept layer g(x) and the residual layer h(z).
Returning to D the residual, in addition to learning about 'fur texture', might also learn the 'medium size' concept. Now, when an expert intervenes and corrects the 'medium size' concept in the concept layer, the model might ignore this correction because the residual still contains the incorrect information about the dog's size. This creates ambiguity in the model's information flow and undermines the effectivencss of interventions. This ambiguily is the central problem we address in this paper. It undermines faithful attribution and intervention effectiveness, as modifying a concept may have little to no effect on the output if the residual retains the original, conflicting information.
3 Related Work
CBM variants Concept Bottleneck Models (CBMs), like those described by Koh et al. (2020) predict down- stream tasks using an intermediary bottleneck layer trained with supervised losses, Bahadori & Heckerman 22200) extended CBMs to denoise the concepts. ProbCBM models uncertainty in concept prediction and provides explanations based on concept predictions and their corresponding uncertainty (Kim el al.) 2023). Shin et al. (2023) develop various strategies for selecting which concepts maximally improve intervention effectiveness. Chauhan et al. (2023) similarly develop method combining concept prediction uncortainty and influence of the concept on the final prediction to improve intervention effectiveness. CBMs offer inter- pretability advantages but are limited by the need for complete concept sets during training. The challenge lies in creating a fully representative set of concepts for domain. When concepts are not expressive enough to capture all the needed information for the task, performance suffers.
Label-Free CBMs Recent advances CBMs aim to reduce reliance on human-labeled concepts by automat- ing concept discovery, redueing the assumption that concept labels exist for each dataset of interest. Label- Free CBM methods instead rely on foundation models to label concepts. Post-hoc CBMs (PCBMs) Yuuk- sekgomul et al. 2023) transform existing models into CBMs by leveraging concept activation vectors (CAVs) to discover and represent concepts. Label-Free CBMs (Oikarinen et al. 2023) take a different approach by learning à projection matrix that maximizes cosine similarity between the projection matrix output and CLIP scores, without requiring explicit concept labels. Both methods relax the requirement of labels, but still use a separate LLM to propose concept sets Vision-Language-Giun Guided CBMs (VLG-CBMs) (Srivastaval ltt al.) (024)) and Discover-then-Name CBMs (Rao et al.] (2024) automate concept generation and naming using multimodal models and latent, concept discovery, respectively.
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Every label-free CBM method relies foundation models to generate concept labels. As result, these methods do not have access to ground-truth labels and therefore cannot evaluate optimize for the ability to intervene on concept predictions. Since intervention is the primary mechanism through which concept leakage has been empirically demonstrated in CBMs, it remains unclear to what extent concept Icakage compromises the interpretability of label-free approaches (Havasi et al., 2022). For example, Label-Free CBMs (Oikarinen et al. 2023) attempt to filter out uninterpretable concepts by requiring a minimum correlation threshold (40%) between the predicted concept activations and CLIP similarity scores for a corresponding concept prompt. However, this filtering does not eliminate the possibility of leakage: concept that aligns with CLIP embedding at 40% correlation may still represent latent entanglement of multiple factors, or may not faithfully reflect the semantics of the intended concept at all. Since CLIP itself is not guaranteed to produce disentangled or interpretable concepl. representations, high correlation with its outputs may not be a sufficient condition for interpretability or faithful concept separation.
Our work addresses these limitations by integrating human-labelos concepts to ensure reliability, while utiliz- ing additional residual capacily to capture information not represented in the labeled concepts. This hybrid approach combines the strengths of human oversight and automated discovery, enabling both interpretability and precision across wide range of applications. By making stronger assumptions about label availabil- ity, we provide stronger guarantees regarding model transparency, disentanglement, and responsiveness to human intervention.
Concept-Residual Approaches Several works augment CBMs with residuals, with some recognizing that concept-residual overlap can degrade intervention effectiveness. Yüksekgöntil et al. add a post-hoc residual predictor to CBMs, modeling y f(cxx) + g(r), where g(x)) captures input-level information and can be removed for interpretability YYuksekgonul et al. 2023). However, when g(x) is active, it's unclear whether predictions still rely on concepts, and the additive form cannot capture nonlinear concept-residual interactions. Sawada et al. propose a similar residual augmentation but do not evaluate leakage (Sawada & Nakamura 2022.. Zabounidis et al. and Ismail ct al. both demonstrate that orthhogonality-basee decorrelation improves intervention performance in CRMs and generative CBMs respectively, confirming that overlap undermines steerability (Zabounidis et al. 2023 [smail et al.) 2024). However, neither work quantifies overlap systematically evaluates which measure of overlap best predicts intervention effectiveness. Zhang al.'s DCBMs use mutual information to trace concept-label alignment but do not isolate residual contributions (Zhang et al.] 2023). Oguntola et al. reduce intra-concept. leakage via mutual information mininization in RL agents, improving rewards but not addressing interpretability under intervention (Oguntola et al. 2023
Introducing Concept Embedding Models (CEMs), Espinosa Zarlenga et al. (2022 2023 relax the strict bottleneck of CBMs by embedding each concept into a high-dimensional, continuous latent space. This im- proves representational flexibility and predictive performance, especially when the concept set is incomplete noisy. However, the flexibility comes at the cost of interpretability: each concept embedding inherently mixes semantic content with residual information, making it difficult to reason about the causal role of indi- vidual concepts or to isolate their influence via counterfactual interventions. We provide further commentary in Appendix A.I
Despite these efforts, no prior work systematically evaluates which measure of overlap best predicts inter- vention offectivencss. We address this gap by proposing MI as a principled mcasure, comparing it against decorrelation-basee approaches, and demonstrating that MI-based disentanglement more effectively preserves steerability.
4 Method
We introduce Disentangled Concept Residual Models (D-CRM), a unifying class of models designed to improve the interpretability and intervention fidelity of CRMs by minimizing concept-residual overlap. This framework encompasses a range of existing methods such as IlerNorm (Zabounidis et al., 2023) and EYE (Wagg et al., 2022)- - that aim to reduce redundancy between concept and residual representations. Within this general mothodology, we propose two explicit techniques: (1) a decorrelation loss that penal- izes linear dependencies between concept and residual representations, and (2) mutual information (MI) minimization objective that enforces statistical independence. These techniques target different forms of
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overlap, with MI minimization providing more general approach by capturing both linear and nonlinear dependencies.
4.1 Mitigating CRM Overlap with Decorrelation
We introduce targeted decorrelation loss to reduce overlap between concept and residual representations in CRMss. Unlike prior methods that enforce full orthogonality across concept and residual vectors, our approach selectively penalizes only the linear dependencies between the two, preserving any natural correlations within the concept space.
Formally, given mini-batch of concept representations E R and residual representations R E g we define the covaricoe matrix as:
Coven 1 C R.
(1)
b
Our decorrelation loss penalizes the squared dff-giagnnl entries:
Liccore
Σ (Covea)
(2)
encouraging decorrelation between the concept and residual pathways while leaving intra-group correlations intact. This largeted approach contrasts with techniques such as ZCA whilening, used in prior work to
promote concept interpretability or improve intervention performance (Chen et al. 2020: Zabounidis et al,, 2023). Those methods docorrolate all dimensions-including between concepts-by applying a global transfor- mation:
X' DA-
(3)
where X contains concatenated concept and residual vectors. While this enforces orthogonality, it also removes meaningful correlations among concept dimensions. By contrast, our loss focuses only the interaction between concept and residual spaces. This makes it better suited for scenarios where concept relationships are semantically meaningful and should be preserved. Still, this approach only removes linear dependencies.
An alternative approach is adversarial decorvelation, which uses gradient reversal to prevent the residual from encoding concept information (Ganin & Lempitsky 2015). A concept classifier qa : R is trained to predict concepts from residuals, while the residual encoder h is trained to maximize the classifier's loss via a gradient reversal layer. This adversarial game encourages the encoder to produce residuals that are uninformative about concepts. Unlike the decorrelation loss, adversarial decorrelation can capture nonlinear dependencies, though it requires careful tuning to balance the adversary and main objective.
4.2 Minimizing Mutual Information
To more generally quantify dependence between residual and concept representations, we use mutual infor- mation (MI) between residuals r = h((z) and ground truth concepts c:
[(err) = H(c) - H(c r) = DRI(p((,r) p(r))
(4)
where H()) denotes entropy and DKL is the Kullback- Leibler divergence. Mutual information measures how much knowing r reduces uncertainty about e: higher values indicate stronger dependence, while I(cr)) corresponds to statistical independence, i.e., p(c r) p(c), meaning the residual contains no information about the concepts. We estimate T((r)) using the Contrastive Log-ratio Upper Bound (CLUB) (Cheng al., 2020), which learns variational approximation go(e (c r) to the true conditional distribution. The estimator is trained by solving:
min - -E Ep(er) [log (c r)]
(5)

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Once trained, this estimator provides differentiable approximation of mutual information. We then freeze 0 and use the estimated rc; regularizer in the D-CRM training objective. Because is differentiable, gradients can propagate through it into the residual encoder h, enabling effective minimization of concept- residual overlap.
4.3 Semi-Independent Training Objective
D-CRMs are trained to balance accurate concept & larget prediction with disentanglement between learned concepts and residuals. The model consists of: (1) a concept encoder g + č, (2) a residual encoder X is the input, è = g(x) are predicted concepts,
X - r, and (3) a prodictor f R + 5. where h(x) is the residual representation, and = f(c,r) r) is the model's final prediction using ground truth
concepts and learned residuals
Gradient-blocked supervision. Following Havasi al. 2022 supervise using ground truth concepts c, while preventing gradient flow from the target loss into g. This avoids concept leakage while maintaining alignment with c. The residual encoder h, however, remains fully trainable with respect to the target loss, enabling it to encode task-relevant information.
Training losses. The model is trained by minimizing a weighted combination of loss terms over the dataset distribution D:
min ((nowwDD a Looncept (g(x), + Lrarger (f ((r)),) ty Edisentangle (h(z),,) g.h.f
(6)
Concept supervision
Prediction with fixed
Disentanglement
where a, 7 E 1220 control the trade-off between concept fidelity, predictive performance, and representation disentanglement.
Disentanglement via mutual information. Because the CLUB estimate depends on learned variational distribution (r c), we must optimize both the model parameters and the estimator jointly. This results in nested optimization problem: c. Specifically, we solve:
min a Cconcepe e) + BB Lrarger h(c)(( 8) + cc 0*),
(7)
g.h.f
where 6* is obtained by solving the estimator subproblem:
8+ arg min -Ep(er) [log qe(c r)].
(8)
8
In practice, we alternate updates by taking a single gradient step on to improve the mutual information estimate, and a single gradient step on the model parameters (g, f) using (CLuB(r, c; as fixed, dif- ferentiable regularizer. This prevents the mutual information estimate from becoming stale r evolves, ensuring that gradients used to penalize overlap remain aligned with the current residual distribution.
5 Quantifying Concept- Overlap and CRM Performance
While our mutual information formulation confirms the presence of concept-residual overlap, it does not indicate how this overlap affects the model's use of concept during target prediction. To address this, we introduce intervention- based metrics that measure the model's reliance on concepts versus residuals. These metrics help assess whether concepts meaningfully influence the model's output, or are ignored in favor of the residual as overlap incrcases.
5.1 Intervention- Based Metrics
Intervention-based metrics directly assess how changes concept or residual inputs affect the predictions of the target model. We intervene on all concepts simultancously, not individual concepts, which ensures that label-relevant concepts drive observable effects while irrelevant concepts contribute signal.
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Positive Concept Interventions We construct modified input by setting all concepts to their ground- truth values: cluby (x: curue (z). Accuracy under positive interventions is defined as:
ACCpus B [1 (s), = =
(9)
High poC indicates that concept updates are correctly propagated to model outputs. However, when pre- dicted concepts already correct, ACCpos provides limited information about concept reliance, motivating the need for random interventions.
Random Concept Interventions We randomly replace all concepts with values from other training ex- amples:
2 ~ (o
IjED, d + c.ru
(10)
and measure:
ACCneg = B( [1 inty (z), (2),)(x))) = v)]
(11)
If the model truly relies concepts, randomizing them should degrade performance when the original predictions were correct. In CBMs, all predictive information flows through the concept layer, so randomizing concepts leads to random accuracy. For CRMs, where the target model has access to a residual, the degree of degradation reveals how much the CRM actually relies on concept representations for prediction. These two metrics are complementary: ACCpes measures sensitivity to correction when concepts are wrong, while ACCneg measures reliance when concepts are already correct.
Residual Interventions To test the model's reliance the residual layer, replace h(z) with residual from another sample:
M(x) ~ (h(x(0) 15 Ej
(12)
and define residual intervention accuracy:
ACCres = (a,y)~ [1 (f(c(z),h(x)) =y]
(13)
High ACCres indicates that concepts alone suffice for accurate target classification.
5.2 Feature Attribution with DeepLIFT
We assoss the influence of each concept using DeepLIFT (Shrikumar et al. 2017 which assigns contribution scores Cae,Af quantifying how the deviation of concept from reference cfel contributes to the deviation of the model logit fy (for class y) from its reference output fy such that:
de CSCAS + Σ CARAA = Afy-
(14)
1=1
We average CaC,A, over the validation set to estimate global concept importance. Baselines ofef are set to 0.5 for binary concepts and to the empirical mean for continuous ones. Attribution patterns reveal the extent to which the model relies on concepts vs. residuals, and how this balance shifts under decorrelation.
6 Experiments
In this sontion, we evaluate CRMs by exploring the following questions:
Q1: Cosccpt-Residual Overlap: Does concept-rosidual overlap exist CRMs, and how does change with increasing residual capacity? Can MI-based regularization suppress this overlap more effectively than existing decorrelation methods?
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Q2: Concept Importance: As residual dimension-ani thus concept-residual overlap inppeases, how does this affect the functional importance of concept representations in CRMs? Does MI-based disentangle- ment better preserve reliance on the concept layer compared to alternative methods?
Qa: Impact on Baseline Performance: How does MI minimization's constraint on concept-residual overlap affect task performance compared to other CRM disentanglement techniques and CFMs?
Q3b: Intervention Efficacy: Does MI minimization's reduction of concept-residual overlap enhance CRMs' responsiveness to test-time concept interventions compared to alternative approaches?
Datasets and Tasks We analyze five vision tasks: (1) CUB, bird classification task with the 112- concept subset (of 312 original attributes) from Koh et al. (020)) predicting 200 species (Wah et al., 2011
(2) OAI, using ten ordinal variables to predict four KL. grades knee osteoarthritis severity (Nevitt et al. 2006,, (3) CIFAR-100, with 20 superclasses as concepts where each image belongs to one of five associated classes per superclass (Krizhevsky al.) 2009), (4) CelebA using 6 of concept annotations with veclor product creating a 256-class task (Espinosa Zarlenga et al.) 2022 2023), and (5) AA2, where we use a 6-concept subset to create an incomplete task allowing only 38 of 50 classes to be uniquely distinguished (Xian el al.] 2019). Complete datasets (CUB, OAI) contain sufficient concepts to predict the task, while incomplete ones (CIFAR-100, CelebA, AA2) require additional information for high task performance.
Models We evaluate mutual information based Concept Residual Models against other disentanglement techniques including decorrelation loss, iterative normalization (Huang et al., 2019). and expert yielded estimates (EYE) regularization (Wang et al. 2022). We compare these approaches with baseline CBMs and CEMs using identical architectures, averaging metrics over five random initializations across challenging image classification and medical datasets. To match previous papers, we use ResNet18 pre-trained on ImageNet as the backbone for CIFAR, AA2, and OAI; ResNet34 for CelebA; and InceptionV3 for CUB, The output of the feature extractor is split into concept and residual layers. The target network consists of one layer for CIFAR, CelebA, and CUB, and three layers for OAL. ieniiindependent training is used for all datasets except for CelebA, where disentangled intervention aware training is used for all D-CRMs. CEMS are trained using intervention aware training with hyper parameters supplied in Espinosa Zarlenga et al.) (2022). Further details on Disentangled Intervention Aware Training can be found in Section D.2 and in our code included in the supplementary material.
Incomplete Concept Sets
Complete Concept Sets
CIFAR
100
CelebA
CUB
OAI
Method
F14
F1,4
MI4
MI,4
F14
F1
MIJ
MI
F14
F1,4
MI]
MI,4
RMSE
RMSE,4
MII
MI,
Latent
0.62
1.00
1.51
1.00
0.87
0.75
2.69
0.99
0.29
1.00
1.26
0.89
0.99
0.96
9.49
0.93
Decorr.
0.61
0.98
1.45
1.00
0.87
0.62
2.65
0.88
0.26
1.00
0.78
0.89
1.18
0.86
6.54
0.86
TterNorm
0.62
0.91
1.39
1.00
0.80
0.98
2.22
0.98
0.25
0.89
0.77
1.00
1.00
0.82
8.54
0.96
EYE
0.61
0.98
1.17
1.00
0.84
0.95
2.39
0.93
0.28
0.93
0.77
0.93
0.96
0.95
9.15
0.93
Adv. Decorr.
0.33
0.64
0.87
0.75
0.84
0.97
2.23
0.97
0.22
0.96
0.75
0.86
0.85
0.96
8.83
1.00
MI
0.12
0.50
0.70
0.19
0.78
0.99
1.50
0.22
0.75
0.70
0.82
1.34
0.25
1.25
0.39
Random
0.05
0.471
0.15
1.30
Table 1: Summary of concept-residual overlap across disentanglemenn strategies. F1-Score and Mutual In- formation (MI) are reported at a fixed high residual dimensionality (16 for all datasots, 64 for CUB), with lower values indicating better disentanglementt To assess monotonic trends, we compute Spearman's rank correlation coefficient (p) between residual dimensionality (powers of 2 from to 64) and the corresponding F1/MI values; these are shown in subscripted columns. Bold values denote statistically significant improve- ments (p 0.01) in the primary metric. Italicized p values indicate evidence of a significant monotonic relationship (p 2 0.01). Random baselines are reported for F1 only.
7 Results
Across five datasets encompassing both complete and incomplete concept scenarios, we demonstrate that standard CRMs exhibit. significant concept-residual overlap (F1-scores: 0.29-0.87, MI scores: 1.17-2.69).
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Our proposed MI-based D-CRMs reduce this overlap by 61-81% while maintaining performance within 3% of baseline CRMs. This overlap reduction yields substantial improvements in intervention effectiveness (up to 16% on OAI) and concept importance preservation (3-57x), validating our core hypothesis.
7.1 Suncept-Residure Overlap in Standard CRMs (RQ1)
Our first rescarch question asks whether concept-residual overlap cxists in CRMs and how effectively MI- based regularization can suppress it. Our hypothesis is that as the residual layer's capacity increases, it encodes more concept-related information, undermining model interpretability. We find that standard CRMs exhibit significant overlap that increases monotonically with residual capacity, while MI-based D- CRMs achieve near-complete suppression on 2/4 tested datasets and substantial reduction all others, outperforming all alternative disentanglement methods.
We systematically evaluate concept-residual overlap by training CRMss with residual dimensions rang- ing from to 64 (powers of 2) and measuring both F1-Score (ability to prodict concepts from residu- als) and Mutual Information between concepts and residuals. To validate monotonic relationships, compute Spearman's rank correlation coefficients (p) between residual dimension and overlap metrics.
Concept Prediction F1 Score vs Concept Importance
0.7
Concept P Score
latent residual decormiltted Lesidual (r=0.94)
0.6
iter_nom
0.5
eye tr=0,90)
adverserial decorrelation (r=0.86)
0.4
mi, residulllll tr-O.
random (r=0.77)
0.3
0.2
Doolmeeismarrizs our findings at high residual ca- pacities, with correlation coefficients quantifying the strength of monotonic trends.
0.1
0.0
0.00
0.05
0.10
0.15
0.20
Shapley Value
Standard CRMss exhibit strong concept- residual overlap. Latent residual CRMs show strong positive correlations between residual size and overlap across all datasets: F1-Score correla- tions of = 1.00 (CIFAR-100, CUB), = 0.75 (CelebA), and p == 0.96 (OAI). Mutual information correlations are similarly high: p = 1.00 (CIFAR-
Figure 3: F1-Scores for each concept plotted against the concepts' estimated Shapley value for a residual dimension of 64 for the CUB dataset. Results show the residual encodes more information on concepts that are found to be important to the task prediction.
100), P 0.99 (CelebA), p 0.89 (CUB), and p p= 0.93 (OAI). These results provide strong evidence that residual capacity directly enables concept infor- mation leakage in standard CRMss
Residuals selectively encode task-relevant concepts. Figure 3 reveals that standard CRMs exhibit strong correlations (R² > 0.80) between concept importance (SHAP values) and residual encoding strength (F1-scores), significantly exceeding random baseline correlations (p < 0.01). This demonst that overlap is not random but systematically biased toward task-critical concepts, making the interpretability problem even more severe.
MI-based D-CRMs achieve superior overlap reduction. As shown in Table 1 MT D-CRMs consis- tently achieve the lowest overlap across all datasets and metrics. The improvements are substantial: Fl-score reductions of 81% (CIFAR-100), 10% (CelebA), 24% (CUB) relative to the next-best method, and RMSE improvements of 34% (OAI). All improvements are statistically significant (p < 0.01). Importantly, MI D-CRMs reduce the correlation between concept importance and residual encoding R² 0.63, which is not significantly different from the random baseline (R²² 0.59), indicating more uniform disentanglementt
Near-complete suppression achieved on CIFAR-100 and OAI. On these datasets, MI D-CRMs break the monotonic relationship between residual size and overlap: Spearman correlations become non-significant (CIFAR-100: = 0.50, p = 0.21; OAI: == 0.25, p = 0.58). Furthermore, Fl-scores of 0.12 (CIFAR-100) and RMSE of 1.34 (OAT) are statistically indistinguishable from random baselines of 0.05 and 1.30 respectively (p < 0.01), indicating near-complete disentanglement.
Partial but significant reduction on CelebA and CUB. While MI D-CRMs do not fully eliminate overlap these datasets (F1-scores of 0.78 and 0.22 remain above random baselines), they still achieve the
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Accuracy Degradation from Random Concept Intervention
Cifar
AA2
CelebA
CUB
OAI
Accurscy
Digi Degrastatonn
0.40
0.30
0.20
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Residual Dim
Residual Dim
Res iual Dim
Residual Dim
Residual Dim
Accuracy Degradation from Random Residual Intervention
OAI
Accuracy
0.50
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0.50
0.25
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0.00
32 64
16
32
64
16
64
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Residual
Residual Dim
Res Wesiitall Dim
Residual
Residual Dim
Concept importance
Cancept Importance (DeepLift Shap)
Cifar 100
AA2
CelebA
CUB
QAI
2.00-
75
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Residual
Residual
Residual Dim
Residual
Latent Residual
Decomelated Residual
Iterative Norm
EYE
Adversarial Decomelation
MI Residual
Figure 4: Test-time intervention results averaged over 5 seeds. Top: Random concept interventions; higher degradation indicates stronger concept dependence. Middle: Random residual interventions; lower degra-
dation suggests reduced residual reliance. Bottom: DcepLIFT SHAP concept importance; higher values indicate greater concept influence.
best performance among all tested methods, with statistically significant improvements in both F Fl-score and MI metrics. CelebA presents a particularly challenging case where all methods, including MI, show uniform decline patterns, though MI consistently outperforms alternatives across all residual dimensions.
7.2 Impact of Overlap on Concept Importance (RQ2)
Having established that concept-residual overlap exists, we next investigate how this overlap affects the functional importance of concepts in CRM decision-making. Our analysis reveals that overlap progressively diminishes concept importance as residual capacity increases, with models increasingly relying on opaque residual pathways. However, MI-based D-CRMs preserve concept importance three times better than stan- dard CRMs, maintaining interpretable decision-making even at high residual dimensions.
We assess concept importance through three complementary metrics: random concept interventions (mea- suring performance degradation when concepts are corrupted), SHAP-based attribution scores, and random residual interventions (measuring residual dependence).
MI D-CRMs maintain significantly stronger concept reliance. Under random concept interventions (Figure top), MT D-CRMs exhibit substantially larger accuracy degradations than competing methods, indicating preserved concept dependence. At residual dimension 64: CLFAR-100 shows 40.1% degradation (vs. 0.7% for latent residual), CUB shows 57.4% degradation (vs. 4.4%), and CelebA shows 22.0% degrada- tion (vs. 6.7%). This 6-57x difference demonstrates that MT D-CRMs successfully prevent the model from bypassing the concept layer.
SHAP-based concept attribution reveals preserved reliance. Figure (bottom) shows that MI D- CRMs maintain higher concept attribution scores across increasing residual dimensions. On CIFAR-100, MI preserves a SHAP score of 1.01 at dimension 6v-over 3x higher than latent residual (0.33) and decorrelated residual (0.34). This pattern holds across CUB (0.18 vs 0.12) and CelebA datasets. ANOVA confirms statistical significance (p 6 x 10 50 across all residual dimensions >4.
Random residual interventions reveal reduced dependence on residual features. Random residual interventions (Figure 1 middle) reveal that MI D-CRMs maintain lower dependence on residual fealures. On complete concept datasets where residuals should be unnecessary: OAI shows <1% accuracy reduction for MI vs. 26% for latent residual; CUB shows 31.6% degradation for MI vs. 74.9% for latent residual. On incomplete concept datasets where some residual dependence is expected, MI still shows reduced reliance: CIF (38.6% 56.6%), CelebA (17.1% 27.6%).
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Incomplete Concept Sets
Complete Concept Sets
CIFAR 100
AA22
CelebA
CUB
OAI
Method
C++
C+ 1
C++
C f
B
C f
Bottleneck
0.11
0.2
0.28
0.79
0.24
0.27
0.75
0.99
0.69
0.96
Latent
0.60
0.66
0.42
0.90
0.35
0.70
0.79
0.93
0.72
0.78
Decorrelation.
0.60
0.67
0.42
0.90
0.35
0.70
0.79
0.94
0.72
0.82
Iterative Normalization
0.60
0.67
0.42
0.90
0.35
0.71
0.79
0.96
0.71
0.90
EYE
0.59
0.61
0.42
0.90
0.37
0.67
0.77
0.95
0.72
0.82
Concept Embedding Model
0.44
0.70
0.33
0.89
0.24
0.67
0.62
0.99
Adversarial Decorr.
0.56
0.68
0.40
0.91
0.36
0.72
0.74
0.89
0.65
0.89
Mutual Information
0.58
0.72
0.40
0.91
0.35
0.73
0.76
0.96
0.67
0.94
Table 2:
Classification accuracy comparison: baseline (B) and positive concept. interventions (C+). Bold
values indicate statistically significant best performance (p 0.05). MI consistently achieves superior inter- vention performance despite modest baseline costs.
Disentanglement Preserves Counterfactual Reasoning. Using the AA2 datasel where 'white' is the only distinguishing feature between brown and polar bears, we test counterfactual reasoning by intervening on brown bear images to set 'white"=true. Figure 5 shows that while CBMs achieve perfect 100% coun- terfactual accuracy, standard CRMs fail inereasingly as residual dimension grows (dropping to 25% at dimension 64). MI D-CRMs maintain robust counterfactual performance (98% accuracy), demonstrating that disentanglement preserves concept-level reasoning capabilities.
7.3 Performance Trade-offs and Intervention Efficacy (RQ3a & RQ3b)
Our final research questions examine whether MI- based disentanglement affects baseline task per- formance and intervention effectiveness. We find that while MI-based disentanglement incurs mod- est baseline accuracy drop (1-3%), it substantially improves intervention efficacy, achieving the high- est positive concept intervention accuracy across 4/5
Counterfactual: Intervene white concept should change brown bear prediction to Polar Bear
Accuracy
datasets with improvements up to 16%. These re- sults validate that the practical benefits of reduced concept-residual overlap outweigh the minimal per- formance trade-offs.
0.60
Latent Residual
Decorreisted Residual
Ihor Norm
Eye
Adversarial Decorrelation
Acceptable performance tradc-offs for inter- pretability gains. Table that MI D-CRMs achieve baseline accuracies within 1-5% of standard CRMs across all datasets. Specifically: CIFAR-100 (0.58 vs 0.60), AA2 (0.40 vs 0.42), CUB (0.76 vs 0.79), and OAT (0.67 vs 0.72). Only CelebA main- tains equivalent performance (0.35). These mod- est costs consistent with other interpretability methods and represent acceptable trade-offs given the substantial interpretability benefits.
0.00
16
32
Residual Dimension
Figure 5: Counterfactual intervention accuracy on the 'white' concept for brown bear images in the AA2 dataset, testing whether models update predictions to polar bear when the distinguishing concept is modi- fied.
Superior intervention efficacy validates practical utility. MI D-CRMs achieve the highest positive concept intervention accuracy on 4/5 datasets, with substantial improvements: CIFAR-100 (0.72 vs 0.66, +6%), AA2 (0.91 vs 0.90, +1%), CelobA (0.73 vs 0.70, +3%), and OAI (0.94 vs 0.78, +16%). These im- provements demonstrate that reduced concept-residual overlap directly translates enhanced suman-mode interaction capabilities, validating the core premise that disentanglement improves practical interpretability.
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Comparison with Concept Embedding Models. CEMs, which use more complex training procedures, show larger baseline performance drops (0.44 vs 0.58 on CIFAR-100) while achieving comparable interven- tion performance (0.70 vs 0.72). This suggests MI D-CRMs provide superior balance between baseline performance and intervention effectiveness.
8
Conclusion
Our study systematically demonstrates that standard Concept-Residual Models (CRMs) suffer from concept- residual overlap, a critical flaw where the residual pathway redundantly encodes concept information. We show not only that this overlap increases monotonically with residual capacity, but that it is systematically biased, with the residual preferentially encoding concepts most critical to the downstream task (Figure 3 This phenomenon fundamentally undermines the core goals of concept-based models: interpretability and reliable human-in-Che-loop intervention.
To address this, we introduced Disentangled Concept- Models (D-CRMs) and a novel mutual infor- mation (MI) minimization objective. Our experiments across five diverse datascts validate this approach, showing that MI-based D-CRMs reduce concept-residual overlap by up to 81% and achieve near-complete disentanglement on challenging datasets like CIFAR-100 and OAL. This disentanglement restores the modelé functional reliance on the interpretable concept pathway. As shown in Section 7.2 MI D-CRMs preserve con- cept importance up to x7x better than standard CRMs under random concept interventions and maintain 98% accuracy in counterfactual reasoning tasks where standard CRMs fail (dropping to 25% accuracy).
Crucially, these gains in interpretability and robustness do not come at a significant cost to performance. Our MI-based D-CRMs maintain baseline accuracy within 3% of standard CRMs while substantially im- proving intervention efficacy by up to 16% (Table 2 By successfully balancing predictive performance with interpretability, MI-based D-CRMs represent significant step forward. They are a practical and effective solution for building more reliable and transparent concept-based models, broadening their applicability in high-stakes domains where both accuracy and human oversight are paramount.
Limitations and Future Work While promising, our approach has limitations that suggest clear direc- tions for future work. The MI minimization does not completely eliminate overlap on all datasets, and it introduces a modest performance trade-off. Furthermore, the scalability of our method to massive, ImageNet- scale datasets remains an open question. Future work could explore more powerful disentanglement tech- niques to address the remaining overlap and develop methods to close the performance gap. Finally, creating large-scale concept benchmarks would be a significant contribution the field, allowing for more rigorous testing of the next generation of interpretable models.
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A Additional Background
A.1 Concept Embedding Models
Concept Embedding Models (CEMs) relax the strict bottleneck constraint imposed by CBMs by learning high-dimennsioall, continuous embeddings for each concept, rather than forcing predictions to rely solely on discrete, humsa-inkerppettbble concept labels. Specifically, Concept Embedding Models (CEMs) learn high- dimensional concept embeddings through a concept encoder g : Rd + R&&m that maps input X to concept embeddings, where m is the embedding dimension for each of the k concepts (Espinosa Zarlenga et al. 2022) 2023). For cach concept G,, CEMs learn two semantic embeddings: & and é representing the positive and negative states, respectively. The final concept embedding & is computed as a weighted mixture:
é = Paa + (1--p0 D.)0
(15)
where P = (Wale,,,, + b.) is the prodicted probability that the concept is active, calculated using a shared scoring function 8 : IR2m + [0, 1]. The downstream model ʃ : Ipmm + maps these concept embeddings to the target prediction.
By embedding concepts in continnous latent space and combining them in soft, learnable manner, CEMs provide greater representational fiexibility than CBMs. This allows them to capture subtle complex variations in concept. instantiations that may be lost in a discrete bottleneck. As result, CEMs effectively relax the bottleneck assumption by enabling the model to represent and use more nuanced information for prediction, even when the concept set is incomplete or ambiguous. However, this added flexibility comes at the cost of interpretability and counterfactual reasoning. Since each embedding combines concept activation with implicit residual structure, modifying a concept affects both its reprosentation and its interactions with the rest of the embedding space. This entanglement makes it difficult to isolate the causal impact of individual concepts to reason about how concept changes alone influence model predictions.
B Dataset Details
CIFAR-100 The CIFAR- 100 dataset is uniquely utilized by employing its 20 superclasses as primary con- cepts (Krizhevsky et al. 2009). Given superclass, an image can belong to any of the five associated classes, indicating that knowledge of the superclass alone is insufficient for precise classification, as the concept space is incomplete. The challenge lies in predicting one of 100 classes, which requires the residual to encode the conditional distribution of possible classes given a superclass. This setup makes CIFAR-100 100 an ideal bench- mark for evaluating methods that can disentangle the residual and concopt layers while proserving valuable information both.
CelebA For CelebA, we follow Espinosa Zarlenga et al. (2022), selecting the 8 most balanced attributes all, a8] out of cach image's 's 40 binary attributes, as defined by how close their distributions are to a6] as concepts annotations for each sample. To create scenario where full concept annotations are not available, each image in CelebA receives a label that matches the decimal value of the binary vector [al, ... a8), leading to a total of l = 2 = 256 classes. It is important to note that the concept annotations are partial, as attributes a7 and a8, crucial for the downstreem task, are omitted from the concept set. To enhance training efficiency and resource use, the size of the CclobA dataset is decrcased by selecting every 12th sample through random subsampling and reducing each image's resolution to (3, 64, 64). Consequently, this process yields approximately 16,900 RGB images, which are then divided into training, validation, and test sets following a conventional 70%-10%-20% distribution.
random uniform binary distribution, and use attributes [al,
UCSt Birds (CUB): The CUB dataset is a rich resource for fine-grained visual categorization, particularly in bird species recognition (Welinder et al.) 2010). It contains images of 200 bird species, each annotated with detailed attributes like feather color, beak shape, and wing patterns. These concepts are learned via bulti-binary classification (Chen et al. 2020). Bottleneck Models trained on CUB rely heavily the use of majority voting (Chen et al. 2020) Chen et al. explain that the provided concepts are noisy,
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Published in Transactions on Machine Learning Research (01/2026)
and thus to make them effective for CBMs they employ majority voting: if more than 50% of a downstream class has a particular concept in the data, then we set all that downstream task to have that concept.
OAI (Osteoarthritis Initiative): The OAI dataset, comprising 36,369 knee X-ray data points, focuses on individuals at risk of knee osteoarthritis. It inchudes both radiological and clinical data, with the primary task being to predict the Kellgren-Tawrence grade (KT.G), a four-level ordinal scale used by radiologists to assess osteoarthritis severity. Higher KLG scores indicate more severe disease. The dataset utilizes ten ordinal variables concepts for analysis, encompassing joint space narrowing, bone spurs, calcification, and other clinical features. These variables, essential for evaluating osteoarthritis severity, align with the preprocessing techniques employed by Pierson et al. (2021..
C Experiment Details
C.1 Model Architecture
We used pretrained imagenet models as our backbones. For CUB, we use inception v3 (Szegedy et al. 2016), following Chen et al. 2020). For CIFAR and OAT, we use resnet18 (He et al. 2016). For CelebA, we follow Espinosa Zarlenga et al. (202)) and use resnet34. The target network is one layer for CIFAR, ColebA and CUB, and three for OAI, with each hidden layer being 50 neurons. Specific hyperparameters and detailed configurations for each dataset can be found in the configs/ directory of the provided codebase.
C.2 Training Scheme
We apply intervention-awwrre training only to the **CelebA** ** dataset, where we observe consistent improve- ments in both baseline accuracy and intervention performance (typically 1-2%) compared to standard inde- pendent training. For **CUB** **OAI** and **CIFAR-100** we do not use intervention-aware training, empirical evaluation shows no statistically significant benefit over semi-independent training.
Concept Embedding Models (CEMs) We employ intervention-wwar training and maintain the origi- nal hyperparameters as specified in the source implementation. However, a key distinction in our approach is the omission of concept groups. Instead of using hundreds of concept embeddings for the CUB dataset as in the original implementation, we reduce this to 28 core concepts. This simplification likely explains the lower baseline performance we observe on CUB compared to the results reported in the original paper. Specifie hyperparameters and detailed configurations for each dataset be found in the configs/ directory of the provided codebase.
D Additional Methodology Details
D.1 EYE Regularization
Concept-Credde Credible models propose EYE regularization, which targets the dependencies features cor- related with but not within concept space C (Wang et al. 2022). Let 0, and 0. be the parameters for the linear transformation from the concept and residual layers. The combined penalty function, J(O2,, ([]) = 1011 + // + 10 The EYE penalty imposes a stricter penalty on 6, relative to tc: permitting a greater norm for 0g and thus promoting reliance on the concept space C.
While EYE regularization does not explicitly constrain the residual representations themselves, it creates implicit regularization effect through backpropagation. By differentially penalizing how the residual layer's outputs are transformed in the downstream network (0c), the gradients flowing back to the residual encoder will favor learning representations that are complementary to, rather than redundant with, the concept space. This occurs because the network optimizes to make efficient use of the less-penalized concept pathway, incentivizing the residual encoder to capture only the information that cannot be effectively encoded through concepts. However, this implicit approach to disentanglement may not be as robust as methods that directly constrain the statistical independence between concept and residual representations, such mutual information minimization.
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Published in Transactions on Machine Learning Research (01/2026)
D.2 Disentangled Intervention aware Training
For CelebA, extend our semi-inddependent training objective with disentangled intervention-aware train- ing, following the framework introduced in Intervention-Awant Concept Embedding Models (IntCEM) (Es- pinosa Zarlenga et al. 2023). As in the semi-independent setting, the model comprises: (1) a concept encoder 9 : X + è, (2) a residual encoder h X + r, and (3) a predictor f C x R + 9, where æ E A, = h(x), and the concept input to the predictor is an intervened version ē of g(x)).
Intervention-based supervision. During training, we sample stochastic initial intervention mask (0)) (ernoulli(piat) and trajectory length T Unif((1,.... mmx)). A sequence of interventions is applied to the predicted concepts è = g(x), replacing selected dimensions with their corresponding ground truth values c, producing an intervened vector ē(z, c( T). This process partially supervises the concept space while encouraging robustness to interventions.
Training losses.
The final training objective becomes:
[a Coonceps ()(x), +B Liargot Granzotf(((z (T)
min B(a,ag)D g.h.f
1 Bern(pmt) U-Uniff II..,,mm
Concept supervision
Teajectory-waree prediction
+ Ldisentangle (h(2),
(16)
Disentanglement
where ē(τ, (,),,,, denotes T'step intervention trajectory that replaces progressively more components of g(x) with corresponding values from c.
In practice, we sample a single ( and per training step, with Pint = 0.25 and Tiax linearly annealed from 2 6 during early training.
E
Further Discussion
E.1 CEM Baseline Performance
As discussed in Section 7.3 CEMs exhibit consistently lower baseline performance across all of our tested datasots. This reflects a fundamental trado-off in CEMs, which incorporate an intervention-aware mechanism that can be tuned to prioritize either baseline intervention performance. Their behavior is governed by several hyperparameters (e.g., Aroll, Acoucept), allowing practitioners to emphasize other. In our experiments, we adopt the parameters reported in prior work that maximize intervention performance for fair comparison to our work, aligning with the goal of this paper to evaluate and improve model intervenability.
objective over the
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Reviews:
[{"review_id": "HccEEn4pGc", "reviewer": "Reviewer_6jG2", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a variant of concept-residual models (CRM) that reduces “concept-residual overlap” that they call disentangled-CRM (D-CRM). This attempts to minimize an estimate of the mutual information between concept and residual via a learned variational approximation that is jointly trained with the main model. Experiments show that this removes concept-residual overlap while maintaining performance.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Yes, the paper provides a good summary of related work, clear explanation of the proposed method, and sufficient experimentation on a few datasets that clearly illustrates how concept-residual overlap is an issue with CRMs but not D-CRMs with the proposed mutual information objective.", "audience_yn": "Yes", "audience_explanation": "Yes, this seems to be a clear improvement over CRMs in terms of interpretability by preventing the model from pushing all information into the residuals.", "requested_changes": "Minor: the DeepLift reference is broken in section 5.2.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "OBiD8TWvVj", "reviewer": "Reviewer_YoDQ", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper addresses the problem of concept–residual overlap in Concept-Residual Models (CRMs), where residual pathways can encode redundant concept information and undermine interpretability. The authors propose to measure overlap via probes and mutual information, evaluate reliance with intervention-based metrics, and reduce overlap using cross-covariance decorrelation and a CLUB-based MI regularizer. Experiments across multiple datasets suggest that residuals indeed contain concept information, and that their methods reduce overlap while preserving predictive accuracy.\n\n**Strengths:**\n- Clearly motivates why residuals can undermine CBM interpretability\n- Proposes two disentanglement objectives that are technically simple and empirically effective\n- Evaluates on multiple datasets with residual dimension sweeps\n\n**Weaknesses:**\n- The novelty of the identified problem is overstated. The paper often reads as if the authors are the first to identify this problem, but it was already discussed in prior work, e.g. [1] introduces an orthogonality loss for this purpose. \n- The “formal definition” of overlap (see contributions) is not formal in the mathematical sense; it is operationalized via probes and MI bounds.\n- Since it partly operationalises concept overlap via probes, it misses adversarial decorrelation as a baseline. \n- The interventions are interesting but seem to have limitations in practice: If predicted concepts are already close to ground-truth, ACC_pos adds almost nothing. Small gains could be misinterpreted as “label ignores concepts” when in fact nothing changed. Similarly, some concepts may not affect the label at all. Intervening on them yields no effect, but this does not prove residual reliance.\n\n[1] A. A. Ismail, J. Adebayo, H. C. Bravo, S. Ra, and K. Cho, ‘Concept Bottleneck Generative Models’, in The Twelfth International Conference on Learning Representations, 2024.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors demonstrate that residual representations contain information correlated with concepts and provide evidence that MI-based regularizers can reduce this correlation and, in some settings, improve steerability. This constitutes meaningful empirical support for their central claim that residual channels undermine interpretability. At the same time, the scope of what is demonstrated is narrower than how some claims are phrased. The definition of overlap relies on probe accuracy and CLUB-based mutual information estimates, which establish statistical dependence but not necessarily semantic or causal redundancy. Similarly, the intervention results show that modifying concepts or residuals can affect predictions, but because interventions are applied to random concepts, they sometimes operate on already-correct or label-irrelevant dimensions, making the observed effects difficult to interpret as evidence that the model is systematically relying on residuals over concepts.", "audience_yn": "Yes", "audience_explanation": "Yes, researchers working on interpretability, concept bottleneck models, and disentangled representations would likely be interested in the paper’s findings on residual-concept overlap and methods to mitigate it.", "requested_changes": "Critical: \n- Acknowledge that residual–concept overlap is a recognized problem, with prior work already trying to address it (e.g., orthogonality loss)\n- The current operationalization via probes and MI estimates is useful but does not constitute a formal definition of overlap; the paper should avoid overstating this\n- Respond to concerns about ambiguity of interventions (see above) \n\nOptional:\n- Compare against adversarial decorrelation as a baseline", "broader_impact_concerns": "No.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "wbFCWtPyL2", "reviewer": "Reviewer_vKDv", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes Disentangled Concept-Residual Models to mitigate the problem of concept–residual overlap in Concept Residual Models. The authors formally define and measure this overlap, introduce new interpretability metrics, and present a mutual information minimization approach that reduces overlap while preserving accuracy. Experiments on multiple datasets (CUB, CIFAR-100, CelebA, AA2, OAI) demonstrate incremental interpretability, robustness of interventions, and strong predictive performance. Strengths include clearly demonstrated motivation, a well-structured paper organization, and comprehensive experiments. Weaknesses lie in limited baseline comparisons, lack of human evaluation, and some minor presentation issues.", "claims_and_evidence_yn": "No", "claims_explanation": "- Novel interpretability metrics. The paper introduces random concept and random residual intervention experiments. These are clearly defined and implemented in the evaluation section, effectively highlighting differences in prediction reliance between standard CRMs and the proposed D-CRMs\n\n- Mutual information minimization objective. The disentanglement objective (Eq. 6–8) is formally defined and validated in ablation studies. Results show a 61–81% reduction in concept–residual overlap across datasets\n\n- Comprehensive empirical evaluation. The model is tested on five diverse datasets (CUB, CIFAR-100, CelebA, AA2, OAI), covering both complete and incomplete concept sets. Experimental results demonstrate that D-CRMs restore interpretability and intervention reliability.\n\n- Tradeoff between performance and interpretability. The paper carefully quantifies baseline accuracy costs (typically ≤3% compared to CRMs), demonstrating that the modest reduction in raw accuracy is outweighed by substantial gains in interpretability and human-intervention reliability", "audience_yn": "Yes", "audience_explanation": "The paper introduces D-CRMs as a practical solution to concept–residual overlap, a major limitation of Concept Bottleneck and Residual Models. This contribution is especially relevant for the XAI community as it improves interpretability, supports more reliable human interventions, and narrows the gap between performance and transparency. The method is rigorous, combining a mutual information minimization objective with newly designed interpretability metrics.", "requested_changes": "- More baseline comparisons. The experimental validation does not include comparisons against more recent methods that address interpretability or disentanglement. For example:\n \n  - Concept Realignment for CBMs (Nishad et al., ECCV 2024) use post-intervention to realign correlated concepts to boost intervention efficacy.\n\n  - Vision-Language Guided CBMs (Srivastava et al., NeurIPS 2024) that leverages multimodal signals to improve interpretability.\n\n\n- Scale on larger dataset. All experiments are conducted on small- to medium-scale datasets (CUB, CIFAR-100, CelebA, AA2, OAI). It remains unclear whether D-CRMs scale effectively to larger, more complex datasets. The authors should provide an additional experiment on a larger benchmark (e.g., ImageNet subset with defined attributes).\n\n- Human Evaluation. Although the paper proposes new interpretability metrics, it does not include any human evaluation to confirm whether the disentangled concepts align with human understanding. This omission limits the practical relevance of the work for explainable AI. The authors should consider adding a small-scale user study (e.g., a survey asking participants to judge concept clarity and usefulness).\n\n- Minor formatting/presentation issue.\n\n  - inconsistent dash vs. hyphen, e.g., \"concept–residual overlap\" and \"concept-residual overlap\".\n\n  - Fig.4, overlapped vertical label (row 2 & 3).\n\n  - The term baseline performance costs (Section 7.3) might be clearer as baseline accuracy drop or performance trade-off.\n\n  - Yuksekgonul et al. is cited twice (conference and arXiv preprint).\n\n  - Some venues are abbreviated (ICML, ECCV) while others are written in full (European Conference on Computer Vision).", "broader_impact_concerns": "No broader impact", "additional_comments": "N/A", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)