FlowCF: Sparse Counterfactual Explanations for Mixed-Type Tabular Data using Flow Matching

Emmanouil Panagiotou; Eirini Ntoutsi

cs.LGOct 6, 2026

FlowCF: Sparse Counterfactual Explanations for Mixed-Type Tabular Data using Flow Matching

Authors: Emmanouil Panagiotou, Eirini Ntoutsi

Organizations: Freie Universität Berlin · University of the Bundeswehr Munich

Abstract

In the field of Explainable AI (XAI), counterfactual (CF) explanations interpret a model's decision by suggesting the changes to the input that would lead to a more favourable outcome. To be useful in practice, such an explanation should change few features and change them as little as possible, properties known as sparsity and proximity. We observe that existing methods remain limited in this respect, especially for numerical features, whether they are model-agnostic and amortised, or gradient-based with full access to the model. In this paper, we propose FlowCF, a model-agnostic generative method that frames CF generation as sparse transport from the factual to the target class. We solve this transport with flow matching, which we extend to mixed feature types with a novel mixed flow operator, and exploit the resulting geometry to optimise for sparsity through a gating network that minimises the number of features the transport changes. Extensive experiments on six benchmark datasets demonstrate that FlowCF produces the best numerical sparsity and proximity, changing 29% of the numerical features where the best baseline changes 89%, at 70% smaller displacement, while remaining comparable on the other desiderata.

Figures & tables

	model- agnostic	amortised	exact numerical sparsity	no black-box queries at inference
Wachter	✗	✗	✗	✗
DiCE	✗	✗	✗	✗
TABCF	✗	✗	✗	✗
REVISE	✗	✗	✗	✗
CCHVAE	✓	✗	✗	✗
DiCoFlex	✓	✓	✗	✗

Table 1: Desired properties of existing counterfactual methods compared to FlowCF.

Figure 1: Counterfactuals generated by our method FlowCF vs DiCoFlex, the closest baseline method, on the two-moons dataset. FlowCF achieves exact numerical sparsity, changing a single feature when possible, while all counterfactuals sampled by DiCoFlex change both features.

Figure 2: FlowCF’s three training steps on samples from the 2D two-moons dataset. Green marks a learned displacement that changed one feature, red both. Step 1 produces valid CFs but changes both features. Step 2 achieves better sparsity making three of the four CFs axis-aligned. Step 3 further improves proximity while preserving validity and sparsity.

Dataset	Training size	#Features $\mathbf{(Num/Cat)}$	Classification task
Adult	32,561	4 / 8	Income above or below $50K
Bank marketing	20,002	7 / 9	Subscription to a term deposit
Credit default	27,000	14 / 9	Default on the next payment
Lending Club	50,000	8 / 4	Loan fully paid or charged off
MAGIC	9,510	10 / 0	Gamma versus hadron air shower
HTRU2	8,949	8 / 0	Pulsar versus spurious candidate

Table 2: Dataset characteristics.

	Validity (%) $\uparrow$	Sparsity (%) $\downarrow$		Proximity $\downarrow$	Top 2 $\uparrow$
		Cat	Num	Num
Wachter	0.863	-	1.000	1.113	0
DiCE	1.000	0.145	0.893	0.957	15
REVISE	0.717	0.172	0.990	0.819	6
CCHVAE	0.900	0.196	1.000	0.763	6
TABCF	0.926	0.458	0.898	1.164	6

Table 3: Results averaged over all datasets. Top 2 counts best or runner-up finishes in Table 4 .

	Validity (%) $\uparrow$	Sparsity (%) $\downarrow$		Proximity $\downarrow$	Validity (%) $\uparrow$	Sparsity (%) $\downarrow$		Proximity $\downarrow$
		Cat	Num	Num		Cat	Num	Num
Adult					Bank marketing
Wachter	0.948	-	1.000	0.778	0.803	-	1.000	1.953
DiCE	1.000	0.011	0.771	2.643	1.000	0.164	0.822	1.347
REVISE	0.879	0.196	0.993	0.965	0.418	0.177	0.945	1.032
CCHVAE	0.784	0.213	1.000	0.685	0.934	0.216	1.000	0.591

Table 4: Detailed results on each dataset. Bold is best, underlined second best. Categorical sparsity does not apply to the continuous-only datasets MAGIC and HTRU2.

Figure 3: Counterfactuals generated by all methods for two CelebA-HQ instances, targeting the male and young attributes. The factual image is marked in green, and each image is annotated with the black-box’s probability $P$ of the target class and the number of changed axes $|\Delta_{\text{axes}}|$ . The bar plot gives the per-axis displacement in standard deviations, with our method FlowCF hatched.

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FCx: An algorithm for finding Feasible Counterfactual Explanations

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National and Kapodistrian University of Athens, Department of Informatics and Telecommunications, Greece · Athens University of Economics and Business, Department of Informatics, Greece

Counterfactual Explanation Counterfactuals Explainable AI Methods Causal Inferences Loss Function Absolute Pose

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Counterfactual (CF) explanations identify changes that alter an input's classification. While existing methods produce realistic and low-cost CFs, they often fail to ensure feasibility, by suggesting non-constructive modifications or incompatible with future changes (e.g., changing an individual's race to secure a job offer). We introduce a refinement of CF explanations that explicitly enforces feasibility. Our approach is the first to efficiently generate CFs that are realistic, low-cost and feasible. We accommodate both hard feasible constraints, specified by domain knowledge users, and soft feasible constraints, inferred automatically via causal inference from the dataset. Our method, Feasible Counterfactual Explanations (FCx), is based on a modified Variational Autoencoder (VAE) optimized with a multi-factor loss function. We measure the cost of a change based on the absolute change in values (proximity) as well as the number of features changed (sparsity) while realism is measured based on the LOF for density estimation, guaranteeing that CFs reside in densely populated regions. Extensive experiments on four public datasets show that our approach matches state-of-the-art performance across multiple metrics while guaranteeing feasibility.

Abstract

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