Synthesising Counterfactual Explanations via Label-Conditional Gaussian Mixture Variational Autoencoders
Authors: Junqi Jiang
Francesco Leofante
Antonio Rago
Francesca Toni
Venue: ICLR 2026 Poster (2026)
Keywords: Counterfactual Explanations
Contrastive Explanations
Explainable AI
Trustworthy AI
Algorithmic Recourse
Primary area: interpretability and explainable AI

Abstract: Counterfactual explanations (CEs) provide recourse recommendations for individuals affected by algorithmic decisions. A key challenge is generating CEs that are robust against various perturbation types (e.g. input and model perturbations) while simultaneously satisfying other desirable properties. These include plausibility, ensuring CEs reside on the data manifold, and diversity, providing multiple distinct recourse options for single inputs. Existing methods, however, mostly struggle to address these multifaceted requirements in a unified, model-agnostic manner. We address these limitations by proposing a novel generative framework. First, we introduce the Label-conditional Gaussian Mixture Variational Autoencoder (L-GMVAE), a model trained to learn a structured latent space where each class label is represented by a set of Gaussian components with diverse, prototypical centroids. Building on this, we present LAPACE (LAtent PAth Counterfactual Explanations), a model-agnostic algorithm that synthesises entire paths of CE points by interpolating from inputs' latent representations to those learned latent centroids. This approach inherently ensures robustness to input changes, as all paths for a given target class converge to the same fixed centroids. Furthermore, the generated paths provide a spectrum of recourse options, allowing users to navigate the trade-off between proximity and plausibility while also encouraging robustness against model changes. In addition, user-specified actionability constraints can also be easily incorporated via lightweight gradient optimisation through the L-GMVAE's decoder. Comprehensive experiments show that LAPACE is computationally efficient and achieves competitive performance across eight quantitative metrics.

Published as a conference paper at ICLR 2026
SYNTHESISING COUNTERFACTUAL EXPLANATIONS VIA LABEL-CONDITIONAL GAUSSIAN MIXTURE VARIATIONAL AUTOENCODERS
Junqi Jiang 12, Francesco Leofante', Antonio Rago's, Francesea Toni! 'Imperia College London PJ.P. Morgan AI Research 'King's College London {junqi. jiang, f. I.leofante, a. a.rago f.toni coonieiimperinn aamperiil... ieeeeriallaaa ac. uk
ABSTRACT
Counterfactual explanations (CEs) provide recourse recommendations for indi- viduals affected by algorithmic decisions. A key challenge is generating CEs that are robust against various perturbation types (c.g. input and model perturba- tions) while simultaneously satisfying other desirable properties. These include plausibility, ensuring CEs reside the data manifold, and diversity, providing multiple distinct recourse options for single inputs. Existing methods, however, mostly struggle address these multifaccted requirements in a unified, model- agnostic manner. We address these limitations by proposing novel generative framework. First, we introduce the Label-conditional Gaussian Mixture Vari- ational Autoencoder (L-GMVAE), a model trained to learn structured latent space where each class label represented by a set of Gaussian components with diverse, prototypical centroids. Building on this, we present LAPACE (LAtent PAth Counterfactual Explanations), a model-agnostic algorithm that synthesises entire paths of CE points by interpolating from inputs' latent representations those learned latent centroids. This approach inherently ensures robustness to input changes, all paths for given target class converge to the same fixed cen- troids. Furthermore, the generated paths provide a spectrum of recourse options, allowing users to navigate the trade-off between proximity and plausibility while also encouraging robustness against model changes. In addition, user-specified actionability constraints can also be easily incorporated via lightweight gradient optimisation through the L-GMVAE's decoder. Comprehensive experiments show that LAPACE is computationally efficient and achieves competitive performance across eight quantitative metrics.
INTRODUCTION
Counterfactual Explanations (CEs) are a prominent method in explainable AI, illustrating the min- imal changes needed for an input to achieve different, more desirable prediction from a machine learning model (Wachter et al. 2017 Tolomei et al.) 2017 Dwivedi et al., 2023). CEs are espe- cially useful for providing algorithmic recourse, such as offering actionable suggestions to bank customer who was denied a loan by an algorithmic pocision-making system. An ideal CE should satisfy several properties: it must be valid (achieving the desired outcome), proximal (close to the original input), plausible (residing within the data manifold), and part of diverse set of options (we refer the reader to Guidotti| (2022); : Karimi et al. (2023) for recent overviews).
In addition to these standard properties, various forms of robusiness have been identified as crit- ical challenge (Upadhyay et al.) [021]] Dominguez-Olmmn Olmedo et al., 2022 Jiang et al.) 2023 2024a] Leofante & Wicker 2025). Research in this area has been fragmented, often addressing specific ro- bustness types in isolation or without considering their interplay with other key criteria (Jiang et al.) 2024b). This has resulted in lack of methods that can generate CEs integrating multiple properties, such as validity, plausibility, diversity, and multiple forms of robustness, within single framework. This paper introduces a novel approach to bridge this gap, with specific focus (i) encourag-
Work done while Imperial College London

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Published as a conference paper at ICLR 2026
Input space and the classifier's
Input space and the classifier's
decision boundary
L-GMVAF's latent space
decision boundary
Class
Claster:
Class
Claster
Clister
Chster
Encoder
Decoder
Class
Chiste:
Clusker
2
Class 0
Datasel points, we want find CEs for input in Class 0
Latent paths from the cheodingg relevant cluster eentroids (Gaussian means)
Lutent puths reconstructed in the input space, leading to CEs
Figure I: Illustration of LAPACE in binary classification. Given a dataset with trained classifier's predictions (Left), a L.-GVVAE first learned with latent clusters (Gaussian components), capturing the data distribution with the classifier's predictions. In this example, we have 6 Gaussian compo- nents (Middle, the coloured areas). Prediction label 0 (1) is associated with Clusters 0-2 (3-5). The cluster centroids (learned Gaussian mixture prior) for classes 0 and I are marked with crosses and check marks. Assuming we are computing CEs for negatively classified point (Left, purple star), LAPACE first performs linear interpolations linking the input's ' latent representation to each class cluster centroid (Middle, dashed lines). These paths are then decoded to the input space to obtain paths of points, where they terminate at the decoded class 1 cluster centroids (Right).
ing robustness to model changes, such that the recourse stays valid under model parameter changes (such as regular retraining), and (ii) guaranteeing robustness to input perturbations, which amounts to the stability of generated CEs against small perturbations applied the input (Artelt et al. [021 Leofante & Potyka, 2024). Indeed, if a method finds CEs that are readily invalidated under some model updates, or provides significantly different CEs for nearly identical inputs, the CEs are likely based on arbitrary model behaviour and algorithmic artefacts, rather than meaningful features. This particularly undermines CEs' trustworthiness for recourse purposes.
A recent state-of the-the-ad art method enhances robustness to input changes by leveraging diverse in- stances of nearest neighbours in the training dataset from the target class (Leofante & Potyka (024)) However, this method remains heuristic and cannot guarantee perfect stability given the randomness in its distance-based thresholding process, and risks exposing sensitive training data. We argue that more principled approach to robustness involves identifying a set of diverse, prototypical recourse points for the target class and then guiding all generated CEs to converge towards these points. While prior works have used generative models like standard variational autoencoders (VAEs), normalis- ing flows, or diffusion models with similar intuitions, their primary goal was to improve plausibility by obtaining realistic data manifold (Pawelczyk et al. 2020aa] Wielopolski et al. 2024 Na & Lee] 2025). A key limitation is that they are typically unconditional, ignoring the information encoded in classifier's predicted labels that readily available during recourse generation. As result, sam- pling with randomness, or complex algorithms such as latent space gradient optimisation, is needed to perform the CE search in the latent space, without guaranteeing their validity.
Based on these insights, we propose Label-conditional Gaussian Mixture VAE (L-GMVAE). a novel generative model for recourse. We adapt the standard GMVAE (Kingma et al. 2014 Dilokthanakul et al. 2016 Shu 2016) to be label-conditional (Sohn et al., 2015) by explicitly mapping each class label to a dedicated set of Gaussian clusters in the latent space. After training, the mean of each Gaussian component serves as a latent centroid. As these centroids are learned through the VAE's objective, their decodings act as diverse, plausible, and robust prototypes for their associated class, making them ideal targets for recourse.
Building on this model, we introduce LAPACE (LAtent PAth Counterfactual Explanations). a sim- ple yet effective method for generating paths describing smooth trajectories that connect inputs to their CEs. This method is illustrated in Figure | For any input, LAPACE finds a latent representation and generates multiple paths by linearly interpolating When decoded, these latent trajectories form paths of high-quality CEs in the input space. Because all paths for given target class converge to the same set of fixed prototypes, LAPACE achieves per-
each of the target class's 's latent centroids.
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feet robustness to input perturbations. Furthermore, providing paths of CEs allows users choose between solutions that are close to their original input and those that are more robust but further away. In addition, LAPACE is moodel-ggosttie, requiring only black-box access to the classifier's prediction function. The synthetic CEs also do not expose real dataset points.
Overall, we make the following contributions: (1) We propose L-GMVAE, new generative model tailored for generating high-quality CEs (Section (. (2) We introduce LAPACE, a novel model- agnostic CE generation method that uses the L-GMVAE
produce paths of CEs that are robust to input perturbations while also addressing validity, proximity, plausibility (our strong plausibility
also leads to robustness against model changes), and diversity (Section 3.2). (3) We show how user-
specified actionability constraints can be easily incorporated into LAPACE (Section 3.3). (4) We
conduct comprehensive experiments that validate the competitive performance of LAPACE across eight quantitative metrics (Section
2 RELATED WORK
Counterfactual explanations are typically computed using specialised algorithms with gradient
descent (Wachter et (017) or mixed integer programming (Mohammadi et al.] 202I) to address
validity and proximity. To enhance these explanations, plausibility is often enforced by aligning
CEs with the data manifold through nearest-neighhour approaches (Brughmans et al.] 2023 Poyi-
adzi et al., (2220) by using generative models (Mahajan et al.) [019 Pawelezyk et al. 2020a
Wielopolski et al. 2024| Na & Lee, 2025). Diversity can be incorporated through multi-objective
optimisation (Mothilal et al. 2020) Dandl et al. 2020) or creating variations in certain sampling process (Leofante & Potykal 2024). More recently, y. various forms of robustness have been identified as a critical property of CEs (Mishra al.
2021, Jiang et al., 2024b). These include robustness to input changes, ensuring similar inputs receive similar explanations (Artelt et al. [02I). and ro- bustness to model changes, which requires CEs to remain valid after model updates like retraining or parameter shifts (Upadhyay et al. 2021 Dutta et al.] [022] Jiang et al., [202c)). This is often encouraged by pushing the CE further away from the classifier's ' decision boundary into some more plausible regions (Pawelczyk et al.) 2020b). We refer the reader to recent surveys (Guidotti) 2022
Karimi et al. 2023 Laugel et al. 2023] Jiang et al., 2024b) for in-depth reviews. Unlike most existing methods which address these properties in isolation,
proposed method is designed to simultaneously address a comprehensive set of these properties within a single, unified framework.
Parallel to the CE research in the explainable AI literature which concerns revealing faithfully what changes would be needed for a given classifier to flip its prediction, there have been substantial research efforts on causal inference, that instead aim at causally characterising the underlying data
generation process (Pawlowski et al. 20200) Shen et al. 2022 Ribeiro et al. 2023 Monteiro et al.)
2023
Wu et al. 2025 Ribeiro al.] 2025). A causal model often explicitly considered, allowing counterfactual inferecce (Pearl) 2009) and thereby discovering causes behind certain observations
(Halpem) 2016). From cognitive science perspective, CEs have been shown serve complemen- tary goals for end-user perception compared to the causal explanations (Byrne) 2019), highlighting the inherent differences between the two fields. Our work follows the standard CE setups (causality- agnostic) introduced above, and incorporating causal constraints into the CE generation process (as studied by Karimi et al. (0200 202I)) would be exciting future work.
Learning latent space with clusters. Early approaches, such as (Xie et al. 2016), learn low- dimensional embeddings for clustering using an autoencoder structure but lacked a generative com- ponent for sampling new data. VAEs (Kingma & Welling 2014) provide a natural framework for this task by imposing a Gaussian prior on the latent space variables. The M2 model by Kingma et al. (214) extends this idea semi-supervised learning, formulating mixture VAE where each latent component is explicitly tied to a class label. Building on this, the Gaussian Mixture VAE (GM- VAE) (Dilokthanakul et al., 2016) introduces a categorical latent variable model cluster assign- ments, thus encouraging structured latent spaces. A simplified variant by Shu (016)) incorporates the assignment variable directly into the encoder, yielding more effective clustering. In parallel, Conditional VAEs (Sohn et al. 2013) generalise the framework by conditioning both inference and generation on auxiliary attributes. Our proposed L-GMVAE combines these perspectives: it learns Gaussian mixture-structure latent space while conditioning on predicted label information.
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3 LATENT PATH COUNTERFACTUAL EXPLANATIONS
In this section, we first present L-GMVAE, discussing how it is inherently suitable for synthesising CEs. Then, we present LAPACE, and show how actionability requirements can be accommodated via gradient updates through the decoder network.
General notation... For classification tasks, given an input x E X S Rd and a set of L discrete labels } = {l..... L), classification model is a function that maps input to label, i.e. M X + V. We denote the discrete prediction variable as y E V, and model predictions as M M() = y- Furthermore,
D f2² "e". 1i=......N where each y" E y is the ground --uttt label. For an input x and a desirable label y E EV.V V + M(),, counterfactual explanation is z' E X' such that (x') y', and x' is close to measured by some distance metric(s). 1.1 is commonly used to induce sparse changes (Wachter et al. 2017).
refer to classification dataset with N points
3.1
GMVAR
L-GMVAE. Throughout the CE generation pipeline, the classifier seeffirr-prdittedd label for the input and the desirable label for CEs are known. For computing CEs, it would therefore make sense explic- itly incorporate label information into the cluster assignments. Following GMVAE ((Shu) 2016)'s version, see Appendix [)). the latent variable z R² governs the data generation process. is regulated by a Gaussian Mixture with K components, with a discrete variable E C = {I.... ......... K} controlling the cluster assignment. In L-GMVAE, we partition the Gaussian clusters C using the possible L labels in ) as: C = C1 UC2 U... UCL where each C; C C, and for all C, C2 with i + and i,j E V. C, n nj = O. The most intuitive approach for partitioning is to uniformly assign K/L clusters for each class. Then, for a given class label y, its corresponding clusters are Cy.
The generative component of L.-MVVAE, p(x, c, = y), is characterised as:
p(z,c,2 1y) p(c 3) (a c) po(x 2)
(1a)
p(c y) 1/|C,/ if C.
(1b)
pa(= c) NM (((())
(Ic)
po(x 2) N( (z
(Id)
The inference model g(2, x, v) of the GMVAE is factorised as:
c(2,c x. v) = (c xr v) (s(= (x,,,))
where pe and g6 indicate that relevant components approximated via neural networks. We derive the evidence lower bound (ELBO) for this model, starting with the data log likelihood:
p(z, 2, v) dz log f log p(zly) = log / Σ
Σ
c x,
p(z, 2, c v)
q(2, c x. v) de
cEC,
CEC,
a(2,c| x. 1) log p(z, 2, v)
Σ
g(2,c| x. dz
(Jensen's Inequality)
cECy
E,((.e//.))) (log p(x, 2, c y) - log g(2,c x.
(3)
p(c y)
Po(= (z c) v)
= -g(,,
log go x.
+ log
+
log po x)
g(26
Reconstruction
KTT(()
(K((2)
-LLLOOLL-GMAA I-GMVAE
Model architecture and training. In practice, the L-GMVAE consists of two neural network com- ponents, the inference model ds and the generative model ps. matching the above definitions. The inference model takes x and the known y label as inputs, and outputs the probability distributions over the clusters. These are then concatenated together to predict 2. The generative model produces a reconstructed input using 2, and learns the Gaussian mixture prior (po(2 c)). The ELBO objective has three terms, Kullback-Leible (KI.) divergence for and regularisers, and reconstruction

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term. The KL terms are expressed analytically, while the reconstruction term is the MSE loss be- tween z and the reconstructed x'. The practical loss function is a negated, weighted sum of these ELBO terms. See Appendices B and C for more details of our trained L-GMVAE models.
Handling categorical data. Additionally, L-GMVAE can accommodate one-hot-encoded (OHE) categorical data by simply adding sigmoid function at the decoder's final layer dimensions which correspond to the categorical dimensions. During training. the reconstruction loss for these dimen- sions uses binary cross-entropy loss. During inference, rounding operation placed after the sigmoid output to determine an integer value of either O or 1 for each OHE dimension.
We next discuss three key desiderata of a well-optimised L-GMVAE which would make it ideal tool as a recourse generator, motivating LAPACE. A recourse task is typically instantiated on and trained classifier M. In our framework, L-GMVAE trained and tested using the predicted dataset by the classifier, {x², (x)) Ji=),, instead of the original dataset, in order for the explanations be faithful to M.
dataset {x', " Ji==,....N
Meaningful centroids. Each cluster centroid (the mean of its Gaussian prior) evolves into a repre- sentative point of its assigned class in the latent space. The synthetic point should also be represen- lative of the class after being decoded into the input space. This is enforced by two components the reconstruction loss ensures that the decoded centroid is valid class instance, while the KL of loss positions the centroid at the geometric centre of all data points assigned to that cluster.
This characteristic motivates using these reconstructed centroids as part of the CE generation method, because they guarantee the validity of the generated CEs. Furthermore, these points should be well within the data manifold. Therefore, if a classifier is updated, these prototypes are more likely to retain their predicted class, promoting plausibility and robustness to model changes for CES (Pawelczyk et al., 2020b). Finally, if we enforce that all CEs converge to the vicinity of the centroids of one class, it would also enhance the robustness of CEs to input changes. These proper- ties are all quantitatively measured in our experiments in Section
Diverse representations. The model learns varied prototypes for each class, implicitly encouraged by the two regularising terms. The KL of c penalises deviation from the uniform prior over a class's ' assigned clusters, promoting the use of all available clusters. Concurrently, the KL of a term incentivises separation, as each cluster learns a unique prior mean to best model the data routed to it. This encourages the reconstructed centroids to be distinetly different from each other. We would therefore have a diverse set of CEs (also quantitatively evaluated in experiments) if all prototypes were involved in CE generation.
Smooth latent manifold. As is standard in VAES, proximity in the latent space corresponds to perceptual similarity in the input space. The KL of and the reconstruction loss are responsible for this property, respectively regularising an organised latent space and forcing the decoder to be a smooth, continuous function to accurately reconstruct inputs from their latent representations. Therefore, closeness in the latent space would translate to closeness in the input space, which is evaluated as the proximity between the CE and the original input.
3.2 LATENT PATHS AND COUNTERFACTUAL EXPLANATIONS
Next, describe how L-GMVAE can be used to generate paths of counterfactual explanations leading to high-quality recourse. After learning. the L-GMVAE's latent cluster centroids are fixed and available through the optimised prior distri- bution (z c) by taking the mean of the Gaus- sian component, denoted as že; for each e; E C. We refer to the L-GM VAE's decoder reconstruc- tion functionality as Dec : Re + IRd. Then, for target counterfactual class y' and its associ- ated clusters Cy,, Dec(ze,) ) is the reconstructed cluster centroid for each çj E Cy. From the dis- cussions in Section 3.1 these Dec(z,,) points carry suitable properties for recourse purposes
Algorithm LAPACE
Require:
x, y. v', že, for Cj E Cy,, Enc, Dec, interpolation steps T {O,..., 1}.
1: Init: P + {h 2z t Enc(x,y)
for ej E CCy do
3:
path, +
4:
for 7 E T do
5:
22-beç,7 +- (1 T) + TEe;
6:
Add Dec(Zz-tof10 to path,
Add path,
return P
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because they are valid. plausible, robust, and diverse. Additionally, they do not risk exposing exist- ing data points given their synthetic nature.
LAPACE is described in Algorithm For imput with predicted label y = (xx and target class y' for its CE, we first obtain its latent representation &z via the Encoder of L-GMVAE. (Line 1). Then, latent paths are obtained from 2x each Zeg- Cj Cy' via linear interpolation, denoted as 2z- == (1 7) 2g + T Zez with finite steps 7 E [0, 1]. During the walk in the latent space, obtain paths of CE points as (Dec(zz-ve,r)) for each т value and each Gj Cy (Lines and 6). Points along the paths can then be tested with M to obtain their predicted labels. Once the L-GMVAE trained, LAPACE is simple and effective, eliminating the need for gradient-based optimisation in the latent space and avoiding a complex ppperparameter-tuningg process (as the step size is relatively trivial) for the CE generation process.
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Figure 2: Example CE paths found by LAPACE on MNIST dataset, for an input image of class
and target label of 7. This L-GMVAE has 3 Gaussian clusters per class. Each row is separate CE path, going from the reconstructions of the original input (the second image from the left, T = 0) to each reconstructed cluster centroid (the last image, when 1).
5 5 5
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Figure 3: Continued MNIST example for the second cluster (the middle row in Figure 2). A require- ment that a dash should not appear in the resulting CE images of class is enforced every image. For an input image size of 28 x 28, pixel values greater than a threshold of 0.01 at the 13th to 18th rows and the 8th to 14th columns (where the dash appears) are penalised via a loss function.
For intuitive visualisation, we show a CE example with image data on MNIST dataset (LeCun 1998) in Figure 2 Details of this trained L-GMVAE are in Appendix B We observe that the L- GMVAE learns diverse writing styles of digit 7 as the reconstructed latent cluster centroids. The reconstructed paths are also plausible evolutions from digits 5 to 7. Later points in the path (c.g., roughly when т 2 0.7) are more likely to be classified as the target class by a classifier. Section quantitatively demonstrate LAPACE's competitive performance tabular datasets.
3.3
ACCOMMODATING ACTIONABILITY CONSTRAINTS
LAPACE also supports actionability constraints defining whether and how portions of the input space (e.g. individual input features) could be modified. Formally, an actionability constraint has form x; bd a, where x; is the ith feature of an input x, D-= x, and a is a constant. To model the satisfaction of these constraints, we use differentiable function ( where g(x;) $ indicates that the constraint is satisfied. For example, for a feature 2; that must not be greater than or equal to a value a, the function is simply g(x;)) = I; a. Constraints on multiple features can be intuitively summed together.
At each step during the latent space interpolation, we evaluate whether the constraints are satisfied via g ((Dec(zn-cc.) )). If not, we iteratively correct 2g-te,:7 via gradient descent until the constraints are satisfied or up to maximum number of corrections Neoreett Zg-aey.7 -x-r0,77 V= where vi is a small learning rate and V is the gradient.
In our MNIST example (Figure the second cluster learns a dash for digit 7. If we decide that the dash is unwanted in our CE, paths, high pixel values at the dash positions are mitigated, including for the intermediate images.
enforce such constraint, illustrated by Figure 3 In the new
4 EVALUATION
In this section, we quantitatively evaluate the utility of L-GMVAE, and benchmark the quality of L LAPACE along the common desirable properties against state-of-the-art baselines.

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4.1 EXPERIMENT SETUP
Datasets and classifiers. We run our experiments on datasets which are commonly used in prior work (Karimi et al. 2023) and are advocated in the CE benchmarking library, CARLA (Pawelezyk)
et al. 2021). They are heloc credit (FICO, 2018), wine quality (Cortez et al. 2009
adulr income
(Becker & Kohavi 1996), and compas recidivism (Julia Angwin & Kirchner 2016 We split each
dataset into a train and test set for training classifiers. To demonstrate the model-agnostic nature
of our approach, we train a random forest and neural network for each dataset, which we use to compute the CEs. Characteristics of the datasets and models are summarised in Table !
Dataset
Classifier
Utility: Train on Real vs. Synthetic
Centroid Acc.
heloc
RF 73.05%
73.97% 71.07% 91.82%:=055% $9.99 +0.50%
100%
23=23+0
NN 73.58%
100%
wine
RF 75. 13%
89.70%=11.6%%
87.44%
100%
11=11+0
NN 76.95%
85.88%:0085% / 84.21 1.68%
100%
adult
RF 81.15%
93.82%110.94 88.11%%.11 1.3.81%
100%
13=6+7
NN 82. 22.04
88.83% 11.1%% 75.86% 13.311
100%
compas
RF 78.17%
90.79% 1. 85.0% 1.545
100%
7=3+4
NN 77.70%
12.26% 84. 80%00% 1.3266
100%
Table 1: Dataset, classifier details, and L-GMVAE utility evaluation. The first column reports the number of features (#continuous + #categorical) of each dataset. The test accuracies of classifiers are included. The Utility columns report the test accuracies of new classifiers following TSTR.
L-GMVAE training and evaluation. The train set further split into train and a test set for L- GMVAE training and evaluation. An L-GMVAE is obtained for each dataset-classifee pair, learning the classifier's predictions on the new training set. Each L-GMVAE has 5 clusters per class. See Appendix C for more details. Following the Train on Synthetic, Test on Real (TSTR) Protocol com- monly employed to evaluate the utility of generative models (Stoian et al. (2025), we sample new synthetic dataset from the L-GMVAE matching the size of the new train set. Then, we train new random forest classifiers respectively using the new train set and the new synthetic train set, and eval- uate their accuracy on the new test set. Well-optimised L-GMVAES should result in synthetic train set with a similar distribution to the original data, yielding a small gap between the test accuracies of the new classifiers, indicating good utility. Additionally, we evaluate whether the original classifier indeed classifies each reconstructed cluster centroid as its dedicated class. This is crucial for guar- anteeing that the CEs found by LAPACE are valid. The results are shown in Table T L-GMVAE shows promising utility (up to 2.8% accuracy gap) on continuous-only datasets. When more cate- gorical features are involved, larger gaps become apparent. Nonetheless, the centroid accuracies are consistently 100%. Also, as we will see by the plausibility evaluation in Section 4.2, the synthetic data from all trained L-GMVAEs remains realistic. These results indicate that our L-GMVAEs are suitable for recourse generation purposes.
CE evaluation procedure and metrics"] For each catsset-classifier pair, we obtain CE test set of 100 test points predicted with label O. We then use each method to generate CEs and evaluate on the following metrics. This process is repeated 5 times on different test sets. The procedures are in line with the existing literature. Validity: the portion of test points receiving valid CEs. Proximity: the average LI distance between a test point and its CE(s). Lower distances indicate that the recourse prescribed by the CE would be easier to achieve by the end user, therefore it is more preferred (Wachter et al., 2017). Plausibility: the average local outlier factor (lof) (Breunig et al. 2000) of each CE. Lof identifies outliers by measuring the local density deviation of data point with respect to its neighbours, and lower values indicate better proximities to realistic data, therefore better plausibility. Diversity: if CE algorithm finds multiple CEs per test input, diversity is calculated as the average pair-wise (L.I) distance between these CEs. Higher distances indicate that the CEs are more diverse. Robustness to Model Changes: we obtain 20 retrained classifiers using different selections of the dataset as the training set, and test on average how many CEs are valid on these new classifiers. Robustness to Inpur Changes evaluates the stability of generated CEs
'The evaluation metrics and baseline method implementations are adapted from CARLA library (Pawelczyk al. (022) and RobustX library (Jiagg et al. 2025). All experiments were performed on a linux machine with Intel Xeon w5-2455X CPU with 128GB RAM, and 2x24GB NVIDIA GeForce RTX 4090 GPUs.
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when small norm-bounded perturbations are applied to the test input. We perturb the test input 10 times with small noises, generate new CEs for each of the perturbations, and report the average distance between the new CEs and the original CE (or se--ste-d (Leofante & Potyka 2024) if the method supports finding multiple CEs per input). Smaller distances would mean better stability finding similar CEs for similar inputs.
Additionally, on the two continuous-only datasets, we manually specify some actionability con- straints (10 and 18 for wine and heloc) in the style of previous work in tabular data generation (Stoian & Giunchiglia 2025). Each constraint specifies either a range for a feature value or the relation between two features (one greater than another) for the resulting CE. We randomly select
constraints for each test input, which are then enforced on a path of CEs and check if there is at least
one CE satisfying all constraints while staying valid.
Method
Val.
Prox.
Plaus.
Div.
M Rob.
In. Rob.
NNCE (Brughmans et al.
2023
FACE PPyyiadz et at. 2020
RobXCE (Duita a 2022
DiCE (Mothiial el al. 2020
DRCF
(Leotante Potyka
2024
LAPACE Ours
Table 2:
The CE desirable properties addressed by each CE method in our experiments, indicates the property is explicitly taken into account (implicitly encouraged) by the method.
V(*)
Baselines. We include state-of-the-ar baselines covering the desirable properties of CEs, sum- marised in Table 2 Because LAPACE generates path of CE points, enable direct comparison with the baselines, we take CE points at three different locations in the path. LAPACE-Fire refers the first CE point which turns the prediction label, when moving from the original input to the reconstructed centroids. LAPACE-Last refers to the last points in each CE path - the reconstructed centroids. Then, LAPACE-Middle is generated by decoding the latent vector located at the midpoint of the linear path connecting the latent representations of the previous two points.
For the actionability evaluation, we compare our path CEs with two path- variants of FACE (Poyiadzi et al. 2020), an algorithm designed to obtain in-distribution CEs. FACE-Interpolae simply interpolates between the input and the FACE CE with 20 steps. FACE-Neighbours greedily selects the nearest neighbours from the training dataset, forming path of 20 points from the input to the FACE CE. For these two baselines, due to the lack of an explicit way to incorporate constraint requirements, we directly modify the feature values to satisfy them and observe whether any CEs stay valid. In contrast, LAPACE allows building the constraints into CE generation, as introduced in Section 3.3
To allow for direct comparison to the baselines, also directly modify the feature values in unconstrained LAPACE, generation (LAPACC-naive).
4.2 EVALUATION RESULTS
Benchmarking results. The benchmarking results are visualised in Figure 4 All included methods, apart from DICR, always find valid CFs. In terms of proximity, NNCE and DRCE consistently perform the best, with DiCE showing competitive performances on continuous-only datasets. Our methods, especially LAPACE-First, show comparable performances. Notably, LAPACE achieves the best plausibility results across all datasets and models, effectively managing the known tradeoff between plausibility and proximity. In terms of diversity, our methods outperform DiCE and DRCE on datasets with mixed data types but perform worse on the two continuous-onle datasets.
Our method demonstrates strong performance on two robustness metrics. LAPACE-Last achieves perfect (100%) robustness to model changes and is guaranteed to be perfectly robust to input changes, as it always generates the same cluster centroid as CE for any input. LAPACE-Laat also competitive when compared with the robust baselines. It matches the proximity of RobXCE (robust to model changes) while achieving better plausibility, diversity, and a more efficient runtime. Compared to DRCE (robust to input changes), LAPACE-Lasi has notably better robustness and plau- sibility. Among our method's 's variants, robustness to model changes steadily increases from First to Last. This is expected, as the CE moves from a point near the original input to more central point within the desired class. This shift improves plausibility and encourages that newly trained classifiers will likely predict it correctly. Finally, LAPACE is one of the fastest methods (second
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Validity
Proximity
Plausibliity
Diversity
Robusiness
Robusstress
holoc
12-
RF
0.99
10-
20
1.0
heloc
25
2.-
0.90
RF
0.5
1.0
wine
NN
1.00
adult
RF
0.25
adult
NN
0.25
0.82
campes
RF
25-
3.5
compas
0.2
NN
Figure 4: Quantitative evaluation of CEs generated by: NNCE, FACE, RobXCE, DiCE, DRCE, LAPACE-Fie LAPACE-Middle, LAPACE-Last. Each subplot is the quantitative comparison of all methods on one dataset-classifier combination and on one evaluation metric. The arrows following each metric indicate that higher lower values are considered better. For diversity, the first three methods find only one CE per input for which the evaluation metric cannot be computed. Therefore, they are assigned value of -1.
only to NNCE) because of its amortised inference nature. After one-time training cost for the main model (L-GMVAE), generating a counterfactual for any new input is very fast, requiring only few forward passes. It also offers key advantage over nearest heapreettneigghuuu-beee based methods (NNCE, RobXCE, DRCE by generating synthetic CEs, protecting the privacy of the original dataset.
Dataset
InCe-lnter
FACE-Greedy
LAPACE-Naive
LAPACE-constrained
heloc RF
0.92
0.96
1.00
1.00
heloc-NN
0.82
1.00
1.00
1.00
wine-RF RF
0.74
0.88
1.00
1.00
wine-NN
0.90
1.00
1.00
1.00
Table 3: Actionability evaluations portion of test inputs for which the CEs satisfy the actionability constraints while remaining valid.
Actionability evaluation. Table 3 reports the actionability results of LAPACE against two FACE variants. We used combination of constraints on 5 features per dataset, which is around 20% and 50% of total features on heloc and wine, respectively. Under these conditions, both LAPACE variants consistently found valid CEs that satisfied all constraints, outperforming the baselines.
More on the path points. To further examine the characteristics of all generated path points con- necting the input to LAPACE-Last centroids, we evaluate the classifier's predicted target class prob- abilities and the plausibility (LOF) of the paths. We present the results on heloc and wine in Figure [S] As T increases, the class probabilities grow consistently, validating that the learned latent space of the L-GMVAE faithfully aligns with the classifier's 's predictions. Note that the probabilities at smaller values are highly dependent on the random selection of test inputs; therefore, variance could be high there. LOF scores for entire paths, including those before crossing the decision boundary, are consistently low (near 1.0), indicating that these synthetic points lie well within the data manifold. The reconstructed cluster centroids (LAPACE-Last points) often exhibit the greatest plausibility with LOF below 1.0, confirming they are located in regions denser than their neighbours.
In summary, LAPACE generates multiple paths of CEs that allow users to choose between solutions with high proximity and those with high plausibility and robustness. Actionability constraints can also be reliably addressed. The method is fast to compute, guarantees perfect robustness to input changes, and delivers the best-balanced performance across all metrics. Furthermore, the very low standard deviation across multiple runs confirms the stability of our approach.

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Published as a conference paper at ICLR 2026
HELDC
Clas
1.10
0.2
0.0
HELOC
Clss
Scor
100
WIDE B7
Class
1.
WANE NN
Lor Scor
3.
0.2
Figure 5: Classifier-predictedd probability of the CE target class and plausibility evaluation for all path points with varying т. Each subfigure row shows the results for path obtained from each L-GMVAE cluster for the desirable class. Points with class probability greater than or equal to 0.5 (grey dotted line) are classified as the desirable class and could be used as CE points.
CONCLUSION
In this work, we introduce L-GMVAE, generative model inherently suitable for recourse, and LAPACE, novel, computationally efficient, and model-agnostic algorithm for generating CEs. LAPACE leverages the structured latent space of the L--MVAE to produce diverse paths of CEs that are plausible, thus are robust to model changes. By design, all paths for a given target class converge to the same set of prototypical points, guaranteeing robustness against input perturbations. We also demonstrate that actionability constraints can be easily incorporated. Our comparative experiments using eight quantitative metrics validate LAPACE's competitiveness in generating high-quality CEs.
LAPACE has limitations worth discussing. First, the guaranteed validity of CEs contingent upon successful L-GMVAE training. This requires validation step beyond monitoring the loss, where one must confirm that the decoded cluster centroids are correctly classified by the original classifier. Second, in line with other VAE-based methods, L-GMVAE is less effective datasets with a large number of categorical features, as is reflected utility results (Table T]. Nevertheless, the performance of our CEs on these heterogeneous datasets remains highly competitive.
This work opens several exciting avenues for future research. Establishing theoretical robustness guarantees CEs obtained leveraging L-GMVAE would be valuable (Jiang al., 2024b). Links to the synthetic tabular data generation literature (Stoian et al. 2025) are identified since it is how we view CE generation in this work. For example, dataset-level causal requirements between features (beyond user-specified actionability requirements) can be incorporated, possibly through adding specialised neural network layers into the training process of L-GMVAE (Stoian & Giunchiglia, 2025). Furthermore, the Gaussian mixture-regularisee latent space learned by our model offers powerful tool for extended forms counterfactual explainability. Advanced interpolation methods such
interpolation (Bhatia et al. 2019) could be tested for replacing our linear one. Instead of generating point- point-based CEs for one input at a time, the structured manifold could be used to derive region-based explanations, providing insights into the model's behaviour for entire sub- groups of the data (Rawal Lakkaraju, 2020 [eeleey et al. 2024). Future work could also investigate extending L-GMVVE for synthesising CEs in other data modalities.
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ACKNOWLEDGEMENTS
Jiang, Rago and Toni were partially funded by J.P. Morgan and by the Royal Academy of Engineer- ing under the Research Chairs and Senior Research Fellowships scheme. Leofante was supported by an Imperial College Research Fellowship. Rago and Toni were partially funded by the Euro- pean Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 101020934). Any views or opinions expressed herein are solely those of the authors listed.
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A GAUSSIAN MIXTURE VARIATIONAL AUTOENCODERS
GMVAE. Given input space X, the GMVAE assumes that the observed variable E X' is generated by a latent variable & E R governed by a Gaussian mixture prior which has K Gaussian components. A discrete variable E C {1...., K} controls the cluster assignment for the mixture. The generative process of GMVAE is defined as follows:
p(z, c) p(c) c) 12)
(4a)
p(c) = Cat(1/K)
(4b)
p( c) = [² (c))
(4c)
p( 2) = N( (x(2).
(4d)
The inference model g(2, c x)) of the GMVAE is factorised as:
((2,c x) = <(c x)) q(= [ x.
(5)
where q(c æ²) is categorical distribution and q(> 2, c), the approximate posterior, is Gaussian. Both are approximated using neural network models. Practically, the encoder takes in æ as an input, then predicts logits over the clusters, which then concatenated with x: (or its processed version) to predict the Gaussian parameters [J=(c). (c)) for sampling 2. The standard reparameterisation trick is also applied this step for differentiable training. Note that, unlike standard VAE, the prior distribution of z, p(z c), is also parameterised and has an associated learnable network component. This is updated via the KL divergence term (the second term Equation 6 below). The evidence lower bound (ELBO) is:
LELBO (y [log p(I,
(g(s,cl))[oog p(a, c) llog c(2,c x:)
((2, c (
(6)
p(c)
p(z cc
(l [log
+ log
+ log p(x [
Q x))
q(> x)
B L-GMVAE IN THE MNIST EXAMPLE
For illustration purposes, the L-GMVAE for the MNIST examples in Figures 2 and 3is trained on the original MNIST dataset, instead of using any classifier's predictions. As mentioned in Section 3.1 practically the L.GMVAE loss function is a weighted combination of the two KI. terms and the reconstruction term. For this particular model, the weights 1:1:1 (unweighted). This model is trained with input size of 28x28 images, label dimension of 10, 3 latent clusters for each label for the latent Gaussian model, learning rate of le-3 for the Adam optimiser and batch size of 1024, with early stopping based on the validation loss.
Our network architecture is implemented as follows. The encoder network has components. One takes in the input image and its one-hot-encoded label, produces a probability vector matching the cluster dimension through a 3-layer MLP component with ReLU activation and 512 hidden neurons, plus a softmax function. This is the q(c xr, component in Equation 2 and is useful for predicting the assigned cluster for an input once the L-GMVAE is trained. Another component, q(> ær c, y) in Equation 2 takes in the input images, label, and cluster information, to produce the mean and logvar for the corresponding latent variable z, through a similar 3-layer MLP with two output heads. At inference time, this is useful for obtaining the latent encoding of input.
The decoder network also has two components. One is responsible for the Gaussian mixture prior (Equation Ic), which simply takes in cluster information and outputs the mean and logvar of the latent variable via a linear layer. The second part implements the reconstruction functionality, map- ping latent vector back to the input space through 3-layer MLP with ReLU activation and 512 hidden neurons.
More implementation details can be found in the accompanying code repository.
Figure 6 visualises the reconstructed cluster centroids for each digit class. We observe that this L-GMVAE model recognises different prototypical writing styles of each digit.
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000
0
1
/
2
2 2
3
3 4 5
3
4 5 6
4
5
6 7
6
7
7
8
8
8
9
9
9
Figure 6: Reconstructed cluster centroids for the L-GMVAE model on MNIST dataset used in the examples.
C L-GMVAE IN THE CE EVALUATION
The architecture the I.-GMVAE models for Section 4 is identical to the one described in Appendix
B apart from different input and label sizes. The input sizes are decided by the dataset, and the label size matches the binary classification task. Each label is associated with five clusters. The training details summarised in Table The other details have been covered in the main text of the paper.
Input Size
Latent Size
Batch Size
Learning Rate
Loss Weights
heloc-RE RF
23
18
1024
le-3
0.1, 0.1,
heloc-NN
23
15
1024
le-3
0.1, 0.1,
RF
11
8
64
1e-3
0.1, 0.05, 1
wine-NN
8
16
3e-4
0.5, 0.3,
adult-RF
13
10
1024
le-3
0.4,
adult-NN
13
10
1024
le-3
0.4, 0.2,
compas-Re RF
5
512
5e-4
0.5, 0.3,
compas-NN
6
512
5e-4
0.5, 0.3,
Table 4: L-GMVAE training details. The loss weights are the weighting terms for KL(2), KL(2),
and reconstruction.
D THE USE OF LARGE LANGUAGE MODELS
Large language model tools were used in a lightweight way to help structure the presentation of some paragraph drafts in the introduction and conclusion, at early stages of paper writing. They are not involved in further paper refinement. They are also used to aid in debugging the code implemen- tations.
16


Reviews:
[{"review_id": "GAeKsrAPw1", "reviewer": "Reviewer_XwU6", "summary": "The paper addresses the problem of generating counterfactual explanations (CE) that remain robust to small changes in both model parameters and input data. It introduces the Label-Conditional Gaussian Mixture Variational Autoencoder (L-GMVAE), which represents each target class as a set of clusters following a gaussian mixture distribution. The centroids of these clusters act as robust prototypes for the target class. Building on this, the authors propose LAPACE (Latent Path Counterfactual Explanations), which generates CE by interpolating between an input’s latent representation and the centroids of the target class, yielding diverse and plausible counterfactual paths. The method is benchmarked against several common baselines and datasets, demonstrating improved robustness, plausibility, and diversity compared to existing approaches.", "questions": "- Line 170, there seems to be a typo, it should be uniformly assign $K/L$  cluster to each class.\n- Equation (3) probably has a typo, the LHS should be $p(x|y)$ instead of $p(x)$.\n- Line 249, \"Additionally, they do not risk exposing existing data points given their synthetic nature.\" I don't follow this, please clarify.", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "- The L-GMVAE formulation is a good contribution, offering a unified framework that simultaneously addresses key aspects of robustness, validity, and diversity in CE generation.\n\n- The experiments on L-GMVAE training and evaluation are well designed, providing convincing evidence that the learned latent structure captures the underlying data distribution effectively. This acts as an important sanity check, showing that the generative model produces samples closely resembling the true data, thereby validating its suitability for CE generation.\n\n- The empirical evaluation demonstrates that the proposed LAPACE method performs strongly across multiple datasets. Results indicate that it is competitive or outperforms existing baselines, particularly in robustness and plausibility, highlighting the effectiveness of the proposed approach.", "weaknesses": "- The qualitative MNIST experiments are rather synthetic and do not fully demonstrate the method’s generality. It would be great if the authors could include results on more challenging image or text datasets. Similarly, the authors could incorporate more high-dimensional tabular datasets as well.\n\n- The proposed approach for accommodating actionability constraints may risk moving latent representations away from the learned data manifold, potentially generating invalid counterfactuals. Some quantitative or qualitative assessment of this would be nice.\n\n- The argument that cluster centroids lie well within the data manifold and improve robustness to model updates feels somewhat ad hoc. While there is supporting empirical evidence, a stronger conceptual or theoretical justification would be better,\n\n- The presentation of results in Figure 4 is overly dense and makes it difficult to interpret. Simplifying the figure, or reorganizing the metrics for better readability, would help communicate the results more effectively.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "9EPLlrc40d", "reviewer": "Reviewer_uE7n", "summary": "This paper proposes to provide counterfactual explanation using deep generative models (VAE with gaussian mixtures to be exact), by interpolating on the latent space between different latent clusters corresponding to different conditions/perturbations.", "questions": "N/A", "limitations": "", "rating": 2, "confidence": 4, "soundness": 1, "presentation": 3, "contribution": 1, "strengths": "The paper is overall very easy to follow. Aside from a very minor clarity issue which I described in the next section, the methodology as well as mathematical formulation are generally laid out very clearly. The figures also gave a good demonstration of the key idea. The derivation of ELBO and it's underlying factorization are also expressed clearly.", "weaknesses": "I found the approach proposed in this work not well-justified, heuristic, and very much lack robustness if the authors decide to frame it under the counterfactual scope. The key concern is the blatant violation of the consistency assumption in causal inference, which is the fundamental backbone of the definition of counterfactual in Pearl's 3 layers of causality. To put it simply, the counterfactual needs to maintain the same exact exogenous noise as the factual, i.e. minimal changes under perturbation, i.e. anything that is not causality affected by the perturbation should remain exactly the same in the counterfactual. And there is nothing in this work's methodology that gives that kind of guarantee or at least that kind of encouragement. This aspect (usually called \"exogenous noise abduction\") is very important in deep counterfactual modeling, and I suggest the authors consult some literatures in counterfactual modeling [1,2,3] and especially the recent literatures [4,5] that specifically tackle this consistency issue, and made it more and more clear that VAE is not well-suited for deep counterfactual modeling [4,5]. To put it more simply in the context of this work, VAE essentially gives you these different latent clusters of points under different $c$, but you don't know which two points in two different clusters correspond to the same exogenous features. And when you do interpolation without knowing the optimal transport path, features not causally related to $c$ also change. To use MNIST for example, if you simply do interpolation from a point in a digit cluster to the center of another digit cluster, not only is the digit changed, the exogenous features (style of writing, thickness of writing, intensity of writing, etc.) could all change, which results in not minimal changes, which results in the wrong counterfactual explanation.\n\nWhat the authors are doing in this work is better described as interventional (layer 2 of causality), not counterfactual (layer 3 of causality). The metrics used in experiments are also predominantly interventional inference metrics. I suggest the authors consult latest literatures in counterfactual evaluation [6] for the more valuable evaluations of counterfactuals. As for dataset, I also suggest evaluation on Morpho-MNIST instead of MNIST, which is a true perturbation dataset where the counterfactual truths can be simulated and the preservation of exogenous features can be evaluated.\n\nMinor:\n\n1. Clarity of the main objective: I suggest the authors parameterize Eq. (3) to make it more clear which components are parameterized by neural nets and which ones are assumed to be known priors. I found the paragraph underneath it (L195-202) to be slightly vague and not sufficient. Canonically, I would assume the learnt encoder components to be $q_\\phi(c | x, y)$ and $q_\\phi(z | x, c, y)$ and learnt decoder components to be $p_\\theta(x | z)$, but in this case $p_\\theta(z | c)$ is also parameterized and it'd be good to make it clear.\n\n[1] Pawlowski, Nick, Daniel Coelho de Castro, and Ben Glocker. \"Deep structural causal models for tractable counterfactual inference.\" Advances in neural information processing systems 33 (2020): 857-869.\n\n[2] Shen, Xinwei, et al. \"Weakly supervised disentangled generative causal representation learning.\" Journal of Machine Learning Research 23.241 (2022): 1-55.\n\n[3] Ribeiro, Fabio De Sousa, et al. \"High Fidelity Image Counterfactuals with Probabilistic Causal Models.\" International Conference on Machine Learning. PMLR, 2023.\n\n[4] Wu, Yulun, Louis McConnell, and Claudia Iriondo. \"Counterfactual Generative Modeling with Variational Causal Inference.\" The Thirteenth International Conference on Learning Representations.\n\n[5] Ribeiro, Fabio De Sousa, Ainkaran Santhirasekaram, and Ben Glocker. \"Counterfactual Identifiability via Dynamic Optimal Transport.\" arXiv preprint arXiv:2510.08294 (2025).\n\n[6] Monteiro, Miguel, et al. \"Measuring axiomatic soundness of counterfactual image models.\" The Eleventh International Conference on Learning Representations.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "oJmItOvBTn", "reviewer": "Reviewer_HV4K", "summary": "This work presents a novel, model-agnostic approach for generating counterfactual explanations by constructing latent paths between factual instances and class-conditioned cluster centroids. These centroids are obtained using L-GMVAE, a label-conditional Gaussian Mixture VAE that learns multiple cluster centers per class. By interpolating in the latent space, the method allows users to control the trade-off between proximity and robustness when selecting counterfactuals. Robustness is particularly relevant in the current landscape of counterfactual explanations, where stability under input perturbations and model updates remains a key challenge. Additionally, the method supports actionability constraints by applying gradient-based adjustments whenever a generated sample violates a constraint.", "questions": "- How could the proposed method be extended to support global or group-level explanations, rather than generating counterfactuals for individual instances?\n- How does the number of clusters per class influence proximity, plausibility, and diversity of the generated counterfactuals? Is there a principled way to select this parameter?\n- What is the computational overhead introduced by enforcing actionability constraints? How does this affect the latency of counterfactual generation in practice?\n- Since generative models often struggle with datasets containing many categorical features, did you consider techniques such as dequantization to improve latent space learning for categorical variables?\n- The paper empirically demonstrates robustness, but does the method admit a formal robustness guarantee? If so, could you provide a more explicit formulation or theoretical justification?", "limitations": "", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "1. The paper is generally well written, and the proposed method is clearly described.\n2. The solution is model-agnostic and can be applied to both differentiable and non-differentiable models.\n3. The approach allows users to choose an appropriate trade-off between proximity, plausibility, and robustness.\n4. The authors introduce a novel L-GMVAE architecture that is specifically designed to support the proposed counterfactual generation method.", "weaknesses": "1. Evaluating plausibility using a single metric may be insufficient. Relying solely on Local Outlier Factor may not fully capture realism; complementary metrics (e.g., Isolation Forest, reconstruction likelihood, or classifier confidence margins) could provide a more comprehensive assessment.\n2. The experimental evaluation is relatively limited. Although several baselines are included, the number and diversity of datasets is small.\n3. Increasing the number of diverse counterfactuals requires retraining the L-GMVAE with a different number of clusters, which introduces additional computational overhead and slows down the overall counterfactual generation pipeline.\n4. The effectiveness of the method heavily depends on successful L-GMVAE training. When latent clusters do not align clearly with class structure (e.g., due to class imbalance or categorical noise), the guarantees regarding validity and robustness may break down.\n5. The visualizations in Figure 4 are dense and difficult to interpret. Presenting the results in table format will improve readability.\n6. The evaluation uses only 100 test samples per fold pair, which may be insufficient for more complex datasets such as Adult.\n7. The description of actionability constraints lacks precision. It would be useful to provide a clearer formal definition.\n8. The method performs less effectively on datasets with many categorical features. This limits applicability in common tabular domains.\n9. The work lacks an ablation study evaluating how the number of clusters per class affects diversity, robustness, and plausibility. Since this parameter controls core method behavior, such analysis would strengthen the empirical grounding.\n10. The experimental section does not include comparisons to recent plausibility-oriented generative counterfactual methods such as PPCEF or C-CHVAE. Including such baselines would provide a more meaningful reference for plausibility performance.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "1ryduamvKL", "reviewer": "Reviewer_7zY7", "summary": "A novel method to generate counterfactual explanations (CEs) is proposed. By learning a label-conditional VAE with Gaussian-mixture latent distribution (LGMVAE), the proposed method enables the generation of valid, plausible, close, robust, diverse and actionable CEs. The generation process only requires a few forward passes through the decoder of the LGMVAE (and one through its encoder) along an interpolation path, from the latent input to several possible cluster centroids, which correspond to the target class. Most costs are amortized through a single offline LGMVAE training. Extensive numerical experiments are provided.", "questions": "- How to better train a LGMVAE so that interpolation paths are realistic ? Regularization techniques to obtain a Euclidean latent space (where shortest paths are indeed straight lines) could be explored, e.g., regularizing the anisotropy of the decoder’s Jacobian.\n- Differently, how to produce geodesic interpolations (in the sense of prefering traversing high-density regions) with a normally-trained LGMVAE ?\n- A plot showing the evolution of the likelihood of the interpolated CE along the produced paths (as a function of \\tau) is lacking. Since this is not optimized for, we can expect these interpolation paths to traverse low-density regions of the latent space (and thus of the feature space, once decoded).\n- The predicted class (according to the model) along this path, as a function of \\tau, would also be interesting, since the validity of these decoded interpolated points as CEs is not guaranteed.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 4, "contribution": 3, "strengths": "- The proposed method is simple and effective.\n- Most desirable properties of CEs are addressed. \n- The method is clear and the paper is rather well-written.\n- The numerical experiment section provides comparison against multiple baselines.", "weaknesses": "- Several questions regarding the interpolation process itself remain unexplored: I detail this in the “Questions” part of the review.\n- Questions about the choice of target centroid are also unexplored. Indeed, although providing every interpolation path to every valid class centroid guarantees diverse explanations, one could wonder if smart target centroid choices can be made. For example, in Figure 1, choosing the centroid of cluster 4 would guarantee better proximity of the generated CEs than choosing that of cluster 3: such considerations could allow users to optimize for different criteria (in this example, a trade-off between diversity and proximity). \n- Typo: L/K should be K/L at line 169.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)