CHOQOLATE: Organizing Concept Bottleneck Latent Spaces with Choquet Integrals

Rémi Kazmierczak; Johanne Cohen; Marianne Clausel

cs.CVSep 29, 2026

CHOQOLATE: Organizing Concept Bottleneck Latent Spaces with Choquet Integrals

Authors: Rémi Kazmierczak, Johanne Cohen, Marianne Clausel

Organizations: CRAN, Université de Lorraine, CNRS, Nancy, France · LISN, Université Paris-Saclay, CNRS, Orsay, France

Abstract

Concept Bottleneck Models (CBMs) built on vision-language models such as CLIP represent a latent space as human-understandable concepts. These representations are unfaithful: related concepts are entangled, so individual scores do not reflect their intended meaning. We propose CHOQOLATE, an interpretable-by-design layer based on 2-additive Choquet integrals, which merges correlated concepts into compact nodes. Across four datasets, CHOQOLATE achieves a favorable accuracy-interpretability trade-off, with weight-sparse and semantically coherent nodes. A closed-form gradient derivation, backed by experiments, explains why Choquet layers drive this organization without explicit supervision. Choquet weights also map directly to Shapley values, which enables test-time intervention. On standard bias-mitigation benchmarks, suppressing spurious concepts after training performs on par with methods that require group annotations or retraining, while needing neither.

Figures & tables

Panel of Figure 1 — Figure 1: Concept redundancy motivates CHOQOLATE’s structured design. (a) In a standard network, every input concept connects to each hidden node, mixing concepts indiscriminately; CHOQOLATE instead connects each node to a small group of correlated concepts, yielding sparse and interpretable nodes. (b) Visualizing a 3-concept bottleneck ( hair , muzzle , tail ) on Cats/Dogs/Cars reveals strongly correlated axes, with samples spreading along a shared diagonal rather than across independent concept dimensions.

Figure 2: Principle of CHOQOLATE. Choquet layers guide the latent space toward merging semantically similar concepts, producing sparse and interpretable nodes, visualized as word clouds from closed-form Shapley values.

Figure 3: Accuracy vs. organization-quality metrics across datasets. Each column corresponds to one dataset (Cats/Dogs/Cars, MonumAI, COCO, CUB), each row to a quality metric. Each point is a method averaged over twenty runs, with ellipses indicating variance bounds.

Panel of Figure 4 — Figure 4: Global explanations on Cats/Dogs/Cars. Word size reflects concept importance for the node. CHOQOLATE (right) yields sparser, more coherent word clouds than the PCBM baseline (left).

Panel of Figure 5 — Figure 5: Training dynamics of Shapley attributions on COCO. Per-concept attributions (Eq. ( 3 )) across training epochs, in the two nodes where the tracked concepts receive weight.

	Waterbirds		CelebA		Properties
Method	Acc. $\uparrow$	Worst group $\uparrow$	Acc. $\uparrow$	Worst group $\uparrow$	FT free	Annot. free	Test time
Baseline	$84.56\pm 1.04$	$46.51\pm 2.87$	$89.21\pm 0.28$	$23.89\pm 2.72$
Concept removal (oracle)	$88.42\pm 0.61$	$60.28\pm 3.49$	$91.81\pm 0.15$	$32.22\pm 1.45$	✓	✓
CHOQ-PCR (ours)	$\mathbf{89.52\pm 1.12}$	$\mathbf{68.26\pm 10.14}$	$\mathbf{92.44\pm 1.53}$	$37.78\pm 11.99$	✓	✓	✓
Ortho-Proj	$88.52\pm 1.08$	$58.82\pm 3.68$	$91.65\pm 0.67$	$32.00\pm 3.33$		✓	✓
Ortho-Cali	$88.61\pm 0.33$	$64.14\pm 2.11$	$89.44\pm 0.47$	$11.44\pm 4.18$		✓	✓

Table 1: Accuracy (%) of CHOQ-PCR versus debiasing baselines. Baseline : standard training. Concept removal (oracle) : ground-truth spurious concepts removed. CHOQ-PCR (ours) : concept-weight nullification on the biased model at test time. FT-free : no fine-tuning; Annot.-free : no group annotations; Test-time : applied at test time. Best per column in bold.

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Dataset	Classes	Concepts
Cats-Dogs-Cars	cats, dogs, cars	engine, artifact wing, animal wing, stern, tail, locomotive, arm, hair, wheel, chain wheel, handlebar, hand, headlight, saddle, body, bodywork, beak, head, eye, foot, leg, neck, torso, cap, license plate, door, mirror, window, ear, muzzle, horn, nose, hoof, mouth, eyebrow, plant, pot, coach, screen
MonumAI	Baroque, Gothic, Hispanic-Muslim, Renaissance	horseshoe arch, lobed arch, pointed arch, ogee arch, trefoil arch, serliana, solomonic column, pinnacle gothic, porthole, broken pediment, rounded arch, flat arch, segmental pediment, triangular pediment, lintelled doorway
COCO	shopping and dining, workplace, home or hotel, transportation, sports and leisure, cultural	person, backpack, umbrella, handbag, tie, suitcase, bicycle, car, motorcycle, airplane, bus, train, truck, boat, traffic light, fire hydrant, stop sign, parking meter, bench, bird, cat, dog, horse, sheep, cow, elephant, bear, zebra, giraffe, frisbee, skis, snowboard, sports ball, kite, baseball bat, baseball glove, skateboard, surfboard, tennis racket, bottle, wine glass, cup, fork, knife, spoon, bowl, banana, apple, sandwich, orange, broccoli, carrot, hot dog, pizza, donut, cake, chair, couch, potted plant, bed, dining table, toilet, tv, laptop, mouse, remote, keyboard, cell phone, microwave, oven, toaster, sink, refrigerator, book, clock, vase, scissors, teddy bear, hair drier, toothbrush
CUB-200-2011	200 fine-grained bird species (e.g., Black-footed Albatross, Indigo Bunting, Brown Pelican, Winter Wren) ; full list at https://github.com/yossigandelsman/clip_text_span/blob/main/utils/cub_classes.py	a short, conical bill; a large, conical bill; a stout, conical bill; a long, thin bill; a long, curved bill; a thin, curved bill; a hooked bill; a long, hooked bill; a short, hooked beak; a short, stubby bill; a short, blunt beak; a long, pointed bill; a long, straight beak; a long, thick bill; a large bill; a small bill; a black bill; a yellow bill; a pink bill; a large, orange bill; a dark, glossy bill; a yellow bill with a red spot; a red spot on the beak; a black cap; a black hood; a black cap and bib; a brown cap; a rusty-brown cap; a red cap on the head; a gray crown; a yellow crown; a white crown; a rust-colored cap and nape; a brown cap and white eyebrows; a crest on the head; a black crest on the head; a red crest on the head; a pointy crest on the head; a black mask over the eyes; a black mask across the face; a black eye stripe; a black line through the eye; a black stripe on the head; a pale stripe above the eye; a white eyebrow; a white stripe over the eye; a black head; a black head and neck; a black face; a gray head; a red head; a blue head; a green head; a yellow head; a black and white striped head; a violet ear patch; all black plumage; all-white plumage; a black and white plumage; a gray and white plumage; a brown and white plumage; a black and orange plumage; spotted plumage; a streaked brown plumage; a brownish-gray plumage; grayish-brown plumage; a brown body; a gray body; a gray plumage; a dark coloration; a reddish-brown body; a grayish-brown body; a grayish-white body; a blue-gray body; a bright yellow body; a bright green body; a green body; iridescent plumage; iridescent green plumage; iridescent black feathers; iridescent blue-green back; a glossy, black plumage; bright golden-yellow plumage; a vibrant blue plumage; a Scarlet-red body; a bright red breast; a rusty-red breast; a bright orange breast; a rosy breast; a pinkish breast; a buffy breast; a white breast; a gray breast; a brown breast; a yellow breast; a black breast; a streaked breast; a black band across the breast; a black crescent on the breast; dark streaks on the breast; a ruby-red throat; a red throat; a yellow throat; a white throat; a white chin and throat; a yellow throat and breast; a orange breast and belly; a white belly; a yellow belly; a pale belly; a red belly; white underparts; grayish-white underparts; a white underbelly; a light-colored belly; a white rump; a white patch on the wing; a brown back; a black back; a gray back; a grayish-brown back; a reddish-brown back; a brownish back; a dark gray back; a green back; a blue back; iridescent blue-green back; blue upperparts; brown upperparts; gray upperparts; dark blue-black upperparts; olive-green upperparts; pale blue-gray upperparts; greenish upperparts; olive-gray upperparts; a streaked back; a gray back with black streaks; a brown back with dark streaks; black wingtips; dark wingtips; white wing bars; two white bars on the wings; black markings on the wings; white patches on the wings; white stripes on the wings; black wings with white bars; brown wings with white bars; black wings with orange bars; orange and black wings; blue wings with black bars; blue wings; brown wings; black and white wings; gray wings; blue-grey wings and back; long, blue-gray wings; long, pointed wings; long, narrow wings; long, tapered wings; long, dark wings; large wingspan; a forked tail; a long, deeply forked tail; a long tail; a short tail; a long, narrow tail; a long, pointed tail; a short, notched tail; a long, flowing tail; a tail with a distinct V-shape; a black tail; a white tail; a long, black tail; a black tail with white sides; a white tail with a black tip; a black and white barred tail; a tail with white bars; a small bird; a small, plump bird; a medium-sized bird; a large bird; a large, stocky body; a large, stocky bird; a slender body; a long, slender body; a plump body; a stocky body; a round body; a small, compact body; long legs; short legs; webbed feet; large webbed feet; long, webbed feet; orange legs; orange legs and feet; red legs; black legs; black legs and feet; gray legs; yellow legs and feet; pale pink legs; greenish-yellow legs; long, black legs; a long neck; a long, graceful neck; a long, slender neck; a long, curved neck; a white collar around the neck; a black ring around the neck; a black neck and head; red eyes; yellow eyes; orange eyes; blue eyes; a red ring around the eye; a white eye ring; orange or yellow eyes; shiny black eyes; a seabird; a swift, direct flight; a swift, acrobatic flight; shy and secretive behavior; a loud, melodious song; a loud, harsh cry; a raucous call

Appendix

Table A1: Classes and supervised concepts for each dataset. Cats-Dogs-Cars uses 39 concepts, MonumAI 15, COCO 80 (the standard COCO object categories), and CUB-200-2011 226 concepts.

Dataset	Classes	Concepts
Waterbirds	landbird, waterbird	Non-spurious: hooked bill, conical bill, thin bill, long bill, curved bill, pointed bill, black cap, colored crown, eye stripe, face mask, crested head, all-black plumage, all-white plumage, iridescent plumage, streaked plumage, spotted plumage, colored breast, white belly, pale underparts, banded breast, wing bars, wing patches, dark wingtips, long wings, pointed wings, forked tail, long tail, short tail, barred tail, small body, large body, stocky body, slender body, long legs, webbed feet, colored legs, long neck, neck collar, streaked back, dark back, colored back, soaring flight, large wingspan, secretive behavior, red eyes, yellow eyes, eye ring, melodious song
		Spurious: green, sea, river, lake, grass, rocks, trees, sky, blue
CelebA	not blond, blond	Non-spurious: blond hair, black hair, brown hair, gray hair, red hair, dark hair, light hair, straight hair, wavy hair, curly hair, long hair, short hair, bangs, bald head, oval face, round face, chubby cheeks, double chin, pale skin, rosy cheeks, high cheekbones, pointy nose, big nose, big lips, arched eyebrows, bushy eyebrows, narrow eyes, bags under eyes, wearing glasses, smiling, mouth open, young face, old face, attractive, receding hairline
		Spurious: feminine face, masculine face, heavy makeup, lipstick, wearing earrings, wearing hat, wearing necklace, wearing necktie, mustache, goatee, beard, sideburns

Appendix

Table A2: Classes and supervised concepts for the spurious-correlation benchmarks. Concepts are split into non-spurious (core, target-predictive attributes) and spurious (attributes correlated with the spurious factor, namely background for Waterbirds and gender for CelebA). Waterbirds uses 48 non-spurious and 9 spurious concepts; CelebA uses 35 non-spurious and 12 spurious concepts.

Architecture	CDC	MonumAI	COCO	CUB-200
PCBM	$\mathbf{96.70\pm 0.72}$	$58.12\pm 8.06$	$53.49\pm 5.13$	$18.59\pm 11.38$
PSCBM	$73.96\pm 3.40$	$40.10\pm 9.12$	$56.93\pm 0.62$	$\mathbf{35.84\pm 12.26}$
Sparse AE	$80.19\pm 4.49$	$47.16\pm 4.72$	$54.46\pm 0.33$	$27.32\pm 1.47$
SLR-AVD	$90.37\pm 17.70$	$54.57\pm 10.03$	$54.28\pm 4.90$	$19.41\pm 10.01$
CHOQOLATE	$95.12\pm 0.27$	$\mathbf{68.94\pm 1.18}$	$\mathbf{64.15\pm 0.25}$	$34.20\pm 1.56$

Appendix

Table A3: Accuracy (%) on four classification benchmarks. The best result per dataset is in bold. Results are averaged over twenty runs,

\pm

denoting standard deviations.

	Cats/Dogs/Cars		MonumAI		COCO		CUB-200
Architecture	Attr. Gini $\uparrow$	Node Coh. $\uparrow$	Attr. Gini $\uparrow$	Node Coh. $\uparrow$	Attr. Gini $\uparrow$	Node Coh. $\uparrow$	Attr. Gini $\uparrow$	Node Coh. $\uparrow$
PCBM	$0.386\pm 0.021$	$0.549\pm 0.011$	$0.349\pm 0.034$	$0.478\pm 0.021$	$0.404\pm 0.086$	$0.408\pm 0.005$	$0.482\pm 0.088$	$0.303\pm 0.016$
PSCBM	$0.394\pm 0.034$	$0.612\pm 0.059$	$0.394\pm 0.032$	$\mathbf{0.583\pm 0.052}$	$0.407\pm 0.010$	$0.233\pm 0.010$	$0.390\pm 0.007$	$0.385\pm 0.029$
Sparse AE	$0.398\pm 0.013$	$0.558\pm 0.009$	$0.345\pm 0.020$	$0.511\pm 0.013$	$0.422\pm 0.019$	$0.399\pm 0.004$	$0.433\pm 0.006$	$0.303\pm 0.002$
SLR-AVD	$0.394\pm 0.031$	$0.550\pm 0.012$	$0.356\pm 0.031$	$0.478\pm 0.025$	$0.425\pm 0.061$	$0.408\pm 0.005$	$0.321\pm 0.068$	$0.289\pm 0.009$
CHOQOLATE	$\mathbf{0.924\pm 0.012}$	$\mathbf{0.680\pm 0.020}$	$\mathbf{0.701\pm 0.076}$	$0.487\pm 0.036$	$\mathbf{0.779\pm 0.017}$	$\mathbf{0.409\pm 0.006}$	$\mathbf{0.982\pm 0.006}$	$\mathbf{0.490\pm 0.051}$

Appendix

Table A4: Latent space quality metrics across four benchmarks. The best result per column is in bold.

\uparrow

higher is better. Results averaged among twenty runs,

\pm

corresponding to standard deviations.

	Cats/Dogs/Cars		MonumAI		COCO		CUB-200
Architecture	HSIC $\uparrow$	CKNNA $\uparrow$	HSIC $\uparrow$	CKNNA $\uparrow$	HSIC $\uparrow$	CKNNA $\uparrow$	HSIC $\uparrow$	CKNNA $\uparrow$
PCBM	$0.588\pm 0.130$	$0.389\pm 0.262$	$0.540\pm 0.180$	$0.331\pm 0.272$	$0.088\pm 0.051$	$0.000\pm 0.000$	$0.745\pm 0.116$	$0.328\pm 0.220$
PSCBM	$0.779\pm 0.009$	$0.485\pm 0.112$	$0.874\pm 0.015$	$0.557\pm 0.035$	$0.499\pm 0.013$	$0.142\pm 0.041$	$0.938\pm 0.011$	$0.553\pm 0.132$
Sparse AE	$0.945\pm 0.008$	$1.084\pm 0.173$	$0.845\pm 0.035$	$0.710\pm 0.359$	$0.769\pm 0.105$	$0.082\pm 0.080$	$0.887\pm 0.010$	$1.113\pm 0.149$
SLR-AVD	$0.490\pm 0.209$	$0.279\pm 0.301$	$0.460\pm 0.205$	$0.204\pm 0.233$	$0.097\pm 0.040$	$0.000\pm 0.000$	$0.725\pm 0.220$	$0.394\pm 0.316$
CHOQOLATE	$0.924\pm 0.035$	$0.739\pm 0.151$	$0.927\pm 0.024$	$0.708\pm 0.125$	$0.950\pm 0.014$	$0.666\pm 0.139$	$0.706\pm 0.047$	$0.130\pm 0.074$

Appendix

Table A5: CKNNA and HSIC across architectures and datasets. Both metrics quantify the amount of information shared between the CLIP-score input and the learned latent representation. High values indicate that the representation preserves the input structure but do not, on their own, indicate good interpretability or concept organization (see text).

Figure A4: Effect of the number of latent nodes

N

on CHOQOLATE on COCO. Accuracy vs. Attribution Gini (left) and Node Coherence (right), for

N\in\{4,8,12,16,18\}

(one value per row). Each point is a method averaged over five runs, with ellipses indicating variance bounds.

Figure A5: Global explanations for PCBM. Dataset: MonumAI. The size of each word reflects the importance of the corresponding concept for the node.

Figure A6: Global explanations for CHOQOLATE. Dataset: MonumAI. The size of each word reflects the importance of the corresponding concept for the node.

Figure A7: Global explanations for the PCBM. Dataset: COCO. The size of each word reflects the importance of the corresponding concept for the node.

Figure A8: Global explanations for CHOQOLATE. Dataset: COCO. The size of each word reflects the importance of the corresponding concept for the node.

		Attr. Gini		Node Coh.
Dataset	Baseline	$\Delta$	$p$	$\Delta$	$p$
Cats/Dogs/Cars	PCBM	$1.37$	$<0.001$	$0.23$	$<0.001$
	PSCBM	$1.35$	$<0.001$	$0.12$	$<0.001$
	Sparse AE	$1.33$	$<0.001$	$0.21$	$<0.001$
	SLR-AVD	$1.35$	$<0.001$	$0.23$	$<0.001$
MonumAI	PCBM	$0.99$	$<0.001$	$0.01$	$0.231$

Appendix

Table A9: Mann Whitney U-test significance of the CHOQOLATE vs. baseline separation.

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