You Can't Have It Both Ways: Concept Entanglement Limits Diffusion Model Unlearning

Yian Wang; Ali Ebrahimpour-Boroojeny; Hari Sundaram; Varun Chandrasekaran

cs.AISep 28, 2026

You Can't Have It Both Ways: Concept Entanglement Limits Diffusion Model Unlearning

Authors: Yian Wang, Ali Ebrahimpour-Boroojeny, Hari Sundaram, Varun Chandrasekaran

Organizations: University of Illinois Urbana-Champaign Urbana, IL, USA

Abstract

Concept unlearning in text-to-image diffusion models aims to suppress a target concept (e.g., \texttt{horse}) while preserving related but distinct content (e.g., \texttt{donkey}), yet existing methods either leak under indirect prompts or visibly degrade other concepts. We show that these failure modes stem from the geometry of concept representations rather than from any particular algorithm. Formalizing concepts as activation-space regions, we prove that the overlap between a target and other concepts lower-bounds the damage any robust erasure must inflict on them, with the trade-off scaling linearly in the degree of overlap. Across thirteen unlearning methods, including methods designed to preserve non-target concepts, no method achieves both strong erasure and strong neighbor preservation: STEREO nearly eliminates indirect leakage but cuts neighbor generation by more than 75%, while sparse inference-time methods preserve neighbors but leak. Damage increases with our overlap measure, monotonically so for STEREO; the $κ$ -scaling reproduces on SDXL, and neighbor-selective damage recurs on FLUX. Perfect unlearning is the wrong target for entangled concepts; methods should be evaluated on the Pareto frontier our theorem establishes.

Figures & tables

Figure 1: Concepts occupy coherent, partially overlapping regions in activation space. Top-left: UMAP projection of cross-attention activations at mid_block.attentions.0 (step 25/50) for 10 concepts in Stable Diffusion v1.4 (visualization pool; full evaluation pool in Section I.3 ); each point is one prompt and ellipses show covariance contours. Semantically related concepts ( horse / pony / donkey ) cluster together, while isolated concepts ( castle ) are well-separated, at the bottom left of the figure. Bottom-left: Estimated entanglement coefficient

\hat{\kappa}

(fraction of the target’s activation region shared with semantic neighbors; see Definition 4.1 ). The Neighbors column lists representative discovered neighbors; full per-target neighbor sets are reported in Appendix C . Animal subtypes have high

\hat{\kappa}

and dense neighbor sets; castle has low

\hat{\kappa}

despite a comparable number of architectural neighbors. Ellipses are

(\mathbf{z}-\boldsymbol{\mu}_{c})^{\top}\boldsymbol{\Sigma}_{c}^{-1}(\mathbf{z}-\boldsymbol{\mu}_{c})\leq 4

for the mean and covariance of each concept’s projected points; they are visualization aids only and are not used to compute

\hat{\kappa}

, which is estimated in the original activation space.

Panel of Figure 2 — Figure 2: Concept entanglement manifests as two coupled effects. Each row erases horse and is probed with three prompt types: Direct (mentions “ horse ”), Indirect (uses correlated context like jockey but omits “ horse ”), and Neighbor (semantically similar concept donkey ). ESD leaks horses through indirect prompts but preserves donkey; STEREO suppresses leakage but destroys donkey. We formalize this duality in Section 4 .

		Erasure		Retention
Method	Type	UA $\uparrow$	IRR $\downarrow$	IRA $\uparrow$	NP $\uparrow$	CP $\uparrow$	DamageGap $\downarrow$
Original	–	$0.0\pm 0.0$	$58.7\pm 38.1$	$70.45\pm 12.6$	$100.0\pm 0.0$	$100.0\pm 0.0$	–
UCE	Closed-form	$80.6\pm 18.4$	$10.5\pm 23.4$	$67.1\pm 12.4$	$88.6\pm 25.1$	$99.3\pm 3.1$	$10.7\pm 25.4$
RECE	Closed-form	$99.4\pm 0.9$	$3.0\pm 4.0$	$43.3\pm 12.1$	$56.1\pm 10.6$	$93.5\pm 2.0$	$37.4\pm 10.2$
ESD	FT	$80.7\pm 11.1$	$20.9\pm 14.2$	$59.2\pm 12.7$	$80.6\pm 13.5$	$97.0\pm 1.6$	$16.4\pm 14.5$
SalUn	FT	$93.7\pm 5.1$	$12.8\pm 11.5$	$46.5\pm 8.5$	$58.4\pm 13.0$	$90.2\pm 8.9$	$31.8\pm 13.1$

Table 1: Averaged evaluation of unlearning methods on the robustness-retention trade-off. Results are averaged over the five target concepts ( dog , bear , horse , cat , castle ) with

\hat{\kappa}\in[{0.27},{0.89}]

. Cells report mean

\pm

sample standard deviation across concepts. UA: Unlearning Accuracy; IRR: Indirect Recovery Rate; IRA/CP: In-domain/Cross-domain Retain Accuracy; NP: Neighbor Preservation (Eq. ( 27 )); DamageGap: composite retention-damage summary. The Original row reports baseline rates for the pretrained

\mathcal{D}_{\theta}

. A few runs use non-standard image counts; see the per-concept table captions in Appendix J . † SAeUron is averaged over four concepts ( dog , bear , horse , cat ); its released sparse autoencoder features do not cover castle (see Section J.5 ).

Figure 3: Neighbor preservation scales inversely with

\hat{\kappa}

. Left: NP for each (method, concept) pair among the seven original methods; dashed lines connect each method’s points in order of

\hat{\kappa}

and serve as visual guides, not fitted trends. Right: STEREO’s NP and in-domain retention ratio across the five concepts (

r_{s}=-1.0

for both).

Appendix figures & tables23 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 4: Pairwise centroid cosine distance between concept activation regions at mid_block.attentions.0 , ordered by hierarchical clustering. Block structure in the matrix corresponds to the neighborhoods discovered by the thresholding procedure described above and to the $\hat{\kappa}$ values reported in Figure 1 .

Target	$\hat{\kappa}$	$\|\mathcal{N}(c_{u})\|$	$\rho_{c_{u}}$	Discovered neighbors
dog	0.89	5	0.13	puppy, labrador, husky, beagle, retriever
bear	0.82	5	0.14	grizzly, panda, polar bear, koala, cub
horse	0.81	7	0.15	pony, mare, donkey, stallion, foal, zebra, mule
cat	0.77	5	0.15	kitten, lynx, leopard, tiger, cheetah
castle	0.27	6	0.18	fortress, palace, citadel, keep, manor, cathedral

Appendix

Table 2: Per-target

\hat{\kappa}

estimation results. For each target

c_{u}

, we report the estimated entanglement coefficient

\hat{\kappa}

, the number of discovered neighbors within the centroid-distance threshold

\delta_{c_{u}}

, the fattening radius

\rho_{c_{u}}

(median intra-concept cosine distance), and the discovered neighbor set. castle has a discovered neighborhood of comparable size to the animal targets but

\hat{\kappa}

roughly

3\times

lower, isolating semantic density rather than neighborhood cardinality as the driver of

\hat{\kappa}

Layer	castle	cat	horse	bear	dog	$r_{s}$ vs. mid	Animal std
down_blocks.2.attentions.0	0.34	0.77	0.86	0.92	0.92	$+0.97$	0.061
down_blocks.2.attentions.1	0.26	0.71	0.74	0.87	0.85	$+0.90$	0.069
mid_block.attentions.0	0.27	0.77	0.81	0.82	0.89	—	0.043
up_blocks.1.attentions.0	0.09	0.59	0.48	0.88	0.79	$+0.80$	0.158
up_blocks.1.attentions.1	0.39	0.36	0.61	0.91	0.76	$+0.80$	0.203
up_blocks.2.attentions.0	0.00	0.78	0.63	0.89	0.88	$+0.80$	0.105

Appendix

Table 3:

\hat{\kappa}

across six candidate probe layers. All five alternative layers produce a

\hat{\kappa}

ranking strongly consistent with mid-block (

r_{s}\geq 0.80

). Mid-block additionally yields the tightest cluster of animal-subtype

\hat{\kappa}

values (std

=0.043

, vs. 0.061 to 0.203 elsewhere), making it the layer at which the “animal-subtype shared-manifold” structure is most coherently expressed.

Concept	Step 10	Step 25	Step 40
castle	0.77	0.27	0.20
cat	0.81	0.77	0.74
horse	0.80	0.81	0.84
bear	0.81	0.82	0.81
dog	0.95	0.89	0.86
sunglasses	0.99	0.99	0.99

Appendix

Table 4:

\hat{\kappa}

at three denoising steps ( mid_block.attentions.0 ).

Figure 5: Mean activation displacement

\|\boldsymbol{\Phi}_{\hat{\theta}}(p)-\boldsymbol{\Phi}_{\theta}(p)\|_{2}

for six unlearning methods erasing horse . Target prompts (blue) confirm Assumption 4.5 : erasure requires measurable displacement. Neighbor prompts (orange) are displaced more than control prompts (green) for all methods, consistent with the propagation mechanism in Theorem 4.6 .

Figure 6: Lipschitz validation via prompt variation. Each point is a (base, variant) prompt pair; the

x

-axis shows activation distance and the

y

-axis shows the change in CLIP-based concept probability. Points are colored by semantic distance from the base prompt (green: synonyms; blue: scene changes; orange: related concepts; red: unrelated concepts). All points lie below the empirical bound

\hat{L}=0.174

(dotted line), supporting Assumption 4.4 .

Figure 7: Context-only prompts remain close to target prompts in CLIP text-embedding space. PCA projection of pooled prompts used for the horse case study. One group explicitly contains the concept token, while the other omits it but retains correlated context words (e.g., jockey , saddle , racetrack ). The proximity between these groups illustrates why token-level erasure can remain vulnerable to indirect contextual prompts.

Category	Concepts
Targets	horse , cat , dog , bear , castle
Equine	pony , donkey , zebra , mustang , mare , foal , stallion , mule , bison , camel
Feline	kitten , tiger , lion , leopard , panther , cheetah , lynx , jaguar , cougar , bobcat
Canid	puppy , fox , coyote , jackal , dingo , hyena , husky , labrador , beagle , retriever
Ursine	grizzly , polar bear , panda , cub , koala , black bear , brown bear , teddy bear , sloth bear , sun bear
Architectural	fortress , tower , cathedral , palace , keep , citadel , watchtower , stronghold , fortified wall , manor

Appendix

Table 5: The

65

-concept classification vocabulary used for all CLIP-based metric evaluations in Section 5 . The vocabulary is the union of the five targets, their respective fine-grained subtype candidates spanning four animal category and one architectural category, and ten out-of-domain controls. See Table 2 for the per-target subset

\mathcal{N}(c)

that survives the

\delta_{c}

-filter and forms the averaging set for NP.

Target	TPR (%) $\uparrow$	FPR (%) $\downarrow$
dog	90	0.25
cat	100	0.25
bear	96	1.00
horse	99	1.25
castle	100	0.00
Average	97.0	0.55

Appendix

Table 6: CLIP classifier reliability on labeled images.

Panel of Figure 8 — Figure 8: Per-concept operating points of the seven original methods. Each point is one (method, concept) pair from Appendix J ; color encodes method family and marker encodes the target concept. The shaded region marks strong erasure together with strong neighbor preservation (UA $>90$ , NP $>80$ ). No point for the four high- $\hat{\kappa}$ animal targets enters it; the only points that do are castle ( $\hat{\kappa}=0.27$ ) under SEOT and AdvUnlearn, consistent with the $\kappa$ -scaled bound of Theorem 4.6 . SAeUron has no castle point ( Section J.5 ).

Method	Type	UA $\uparrow$	IRR $\downarrow$	IRA $\uparrow$	NP $\uparrow$	CP $\uparrow$	DamageGap $\downarrow$
		Erasure		Retention
Original	–	0	27.0	78.92	100	100	–
ESD	FT	75.0	17.5	74.15	93.51	96.0	2.48
SalUn	FT	90.0	8.0	50.92	58.01	76.0	17.98
EDiff	FT	62.0	23.5	66.61	80.29	92.99	12.70
AdvUnlearn	Robust FT	96.0	20.0	74.30	63.05	99.0	35.94

Appendix

Table 7: Per-concept evaluation for dog (

\hat{\kappa}=0.89

). Discovered neighbors include puppy , labrador , beagle , husky , and retriever (full list in Section I.1 ). As the highest-

\hat{\kappa}

concept in our evaluation, dog exhibits the most severe robustness-retention trade-off: robust fine-tuning methods (STEREO, AdvUnlearn) achieve high UA but cause substantial IRA and NP degradation.

Method	Type	UA $\uparrow$	IRR $\downarrow$	IRA $\uparrow$	NP $\uparrow$	CP $\uparrow$	DamageGap $\downarrow$
		Erasure		Retention
Original	–	0	85.5	62.75	100	100	–
ESD	FT	94.5	9.0	44.00	58.02	95.0	36.97
SalUn	FT	99.5	5.5	51.51	53.89	89.0	35.10
EDiff	FT	96.0	6.0	38.61	56.75	94.0	37.24
AdvUnlearn	Robust FT	99.0	1.0	42.46	67.37	100	32.62

Appendix

Table 8: Per-concept evaluation for bear (

\hat{\kappa}=0.82

). Discovered neighbors include grizzly , panda , polar bear , koala , and cub (full list in Section I.1 ). Image counts differ for two runs: SalUn uses 272 neighbor images and SEOT uses 462 neighbor images.

Method	Type	UA $\uparrow$	IRR $\downarrow$	IRA $\uparrow$	NP $\uparrow$	CP $\uparrow$	DamageGap $\downarrow$
		Erasure		Retention
Original	–	0	75.5	71.23	100	100	–
ESD	FT	80.0	33.5	62.92	81.12	97.0	15.87
SalUn	FT	95.0	28.5	40.92	39.42	91.99	52.57
EDiff	FT	81.0	31.0	63.84	76.48	94.0	17.51
AdvUnlearn	Robust FT	97.5	23.0	65.38	71.42	100	28.57

Appendix

Table 9: Per-concept evaluation for horse (

\hat{\kappa}=0.81

). Discovered neighbors include pony , mare , donkey , zebra , stallion , foal and mule (full list in Section I.1 ). Image counts differ for two runs: Original uses approximately 480 neighbor and control images, and SEOT uses 58 direct images.

		Erasure		Retention
Method	Type	UA $\uparrow$	IRR $\downarrow$	IRA $\uparrow$	NP $\uparrow$	CP $\uparrow$	DamageGap $\downarrow$
Original	–	0	96.0	85.50	100	100	–
ESD	FT	66.0	38.0	66.92	87.35	98.0	10.65
SalUn	FT	87.0	21.0	34.53	73.04	100	26.95
EDiff	FT	76.0	24.0	62.30	65.99	100	34.00
AdvUnlearn	Robust FT	100.0	0.0	50.76	72.40	98.0	25.60

Appendix

Table 10: Per-concept evaluation for cat (

\hat{\kappa}=0.77

). Discovered neighbors include kitten , lynx , leopard , tiger , and cheetah (full list in Section I.1 ). The original baseline IRA =

85.50

reflects high CLIP-evaluator accuracy on cat’s in-domain object set after correcting for kitten/cat label confusion (see Appendix I ). Image counts differ for two runs: Original uses approximately 480 neighbor and control images, and SEOT uses 72 direct images.

Method	Type	UA $\uparrow$	IRR $\downarrow$	IRA $\uparrow$	NP $\uparrow$	CP $\uparrow$	DamageGap $\downarrow$
		Erasure		Retention
Original	–	0	9.5	53.84	100	100	–
ESD	FT	88.0	6.5	48.15	83.03	99.0	15.98
SalUn	FT	97.0	1.0	54.76	67.58	94.0	26.41
EDiff	FT	82.0	2.5	46.46	87.35	99.0	11.65
AdvUnlearn	Robust FT	100.0	1.0	44.00	83.10	100	16.89

Appendix

Table 11: Per-concept evaluation for castle (

\hat{\kappa}=0.27

). Discovered neighbors include fortress , palace , citadel , keep , cathedral and manor (full list in Section I.1 ). SAeUron’s released sparse autoencoder features do not cover castle ; we attempted to substitute its architectures feature, but the resulting ablation did not yield meaningful suppression of castle generation. Image counts differ for two runs: Original uses approximately 480 neighbor and control images, and SEOT uses 60 direct and 100 indirect images.

Target	$\hat{\kappa}$	$\|\mathcal{N}(c_{u})\|$	Discovered neighbors
snake	0.37	7	anaconda, boa, cobra, garter snake, king snake, python, rattlesnake
fish	0.52	8	bass, cod, eel, herring, mackerel, salmon, trout, tuna
tree	0.63	6	cedar, cypress, maple, oak, pine, spruce
butterfly	0.66	7	birdwing, dragonfly, fritillary, monarch butterfly, moth, red admiral, swallowtail
car	0.83	7	convertible, coupe, hatchback, minivan, sedan, suv, van

Appendix

Table 12: Discovered neighbors for the five additional targets. As for the original targets ( Table 2 ), neighborhood size does not determine

\hat{\kappa}

: fish has the most neighbors but an intermediate

\hat{\kappa}

Concept	Method	$\hat{\kappa}$	UA (%) $\uparrow$	IRR (%) $\downarrow$	IRA (%) $\uparrow$	NP (%) $\uparrow$	CP (%) $\uparrow$	DamageGap (%) $\downarrow$
Snake	Original	0.37	9.2	6.0	31.1	100.0	100.0	0.0
	EraseDiff	0.37	88.1	8.6	19.1	81.2	93.6	12.4
	ESD	0.37	97.9	6.8	20.6	81.3	98.1	16.8
	SalUn	0.37	100.0	0.0	5.5	72.4	86.2	13.8
	STEREO	0.37	100.0	0.0	14.8	25.1	96.0	71.1
Fish	Original	0.52	6.2	11.0	40.5	100.0	100.0	0.0

Appendix

Table 13: Results for the five additional target concepts. Original denotes the pretrained model before unlearning. All evaluation metrics are percentages. Higher UA, IRA, NP, and CP are better; lower IRR and DamageGap are better. DamageGap is the difference between control preservation (CP) and neighbor preservation (NP).

Method	castle	cat	horse	bear	dog	$r_{s}$
	( $\hat{\kappa}{=}0.27$ )	( $0.77$ )	( $0.81$ )	( $0.82$ )	( $0.89$ )	vs. $\hat{\kappa}$
ESD	128.4	194.8	213.6	221.0	235.4	$+1.00$
STEREO	0 75.9	234.2	233.4	240.9	257.0	$+0.90$
AdvUnlearn	0 71.6	110.5	150.9	123.2	123.7	$+0.70$
EDiff	0 67.8	163.0	178.1	172.1	175.6	$+0.70$
SalUn	0 88.1	220.7	215.6	220.4	215.3	$+0.10$

Appendix

Table 14: Mean activation displacement at neighbor prompts. Each cell reports

\Delta_{\text{neighbor}}

L_{2}

activation distance, averaged over

200

neighbor prompts.

r_{s}

is computed across the five concepts. Note that

n=5

concepts limits the statistical power of significance testing on individual methods; we report

r_{s}

as a descriptive monotonicity measure. The consistency of positive correlations across all five methods (range

+0.10

+1.00

; four of five at

\geq 0.70

) provides aggregate evidence for the

\kappa

-scaling predicted by Theorem 4.6 .

Method	castle	cat	horse	bear	dog	Spread
STEREO	0.0035	0.0032	0.0034	0.0033	0.0034	7.8%
EDiff	0.0039	0.0046	0.0045	0.0047	0.0050	23.6%
SalUn	0.0030	0.0034	0.0037	0.0036	0.0040	29.5%
ESD	0.0021	0.0039	0.0037	0.0036	0.0037	53.4%
AdvUnlearn	0.0037	0.0068	0.0053	0.0065	0.0070	57.1%

Appendix

Table 15: Per-concept

L_{\text{tight}}

values.

L_{\text{tight}}=\kappa(1-\varepsilon)/(\Delta_{\text{neighbor}}+c_{0})

is the Lipschitz constant value that places Theorem 4.6 ’s bound exactly at the observed displacement. If a method operates at the bound,

L_{\text{tight}}

should be approximately constant across concepts. STEREO is the only method exhibiting this consistency (

7.8\%

spread), supporting its operation at the bound’s saturated-erasure regime.

Concept	$\hat{\kappa}$ (CLIP text)	$\hat{\kappa}$ (denoiser)
horse	1.00	0.72
pony	1.00	0.00
donkey	1.00	0.00
deer	1.00	0.00
dog	1.00	0.60
cat	1.00	0.00

Appendix

Table 16:

\hat{\kappa}

for the ten-concept set in SDXL’s CLIP text-embedding space and in its denoiser cross-attention space.

Figure 9: UMAP projection of SDXL cross-attention activations for eight concepts, with covariance ellipses. As in SD v1.4 ( Figure 1 ), related animal concepts ( horse / pony / donkey , dog / cat ) overlap, while castle lies apart from the animal clusters. Castle’s architectural neighbors are not shown; its entanglement with them is

\hat{\kappa}_{\text{SDXL}}=0.72

( Table 17 ).

Concept	$\hat{\kappa}_{\text{SDXL}}$	UA $\uparrow$	NP $\uparrow$	DamageGap $\downarrow$
cat	0.65	99	90.9	9.1
castle	0.72	91	85.0	15.0
dog	0.92	77	77.1	22.9

Appendix

Table 17: ESD on SDXL. NP decreases monotonically with

\hat{\kappa}_{\text{SDXL}}

Concept	UA $\uparrow$	IRR $\downarrow$	NP $\uparrow$	CP $\uparrow$	DamageGap $\downarrow$
Base ( dog )	8.2	26.5	100.0	100.0	0.0
dog	78.0	8.5	35.6	96.9	61.3
cat	83.0	9.5	37.8	95.3	57.6
castle	85.0	20.0	40.5	98.9	58.3

Appendix

Table 18: ESD on FLUX (percent).

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