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Now You See It, Now You Don't - Instant Concept Erasure for Safe Text-to-Image and Video Generation
Shristi Das Biswas, Arani Roy, Kaushik Roy
Models: CogVideoX-2B, Stable Diffusion v1.5, Stable Diffusion v2.1
Intelligence
Status: succeeded | Model: google/gemini-3.1-flash-lite-preview | Prompt: intel-v1 | Confidence: 96%
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Summary
Instant Concept Erasure (ICE) is a training-free, modality-agnostic, one-shot weight modification framework for removing undesired concepts from text-to-image (T2I) and text-to-video (T2V) models. By modeling the latent semantic overlap between erase and preserve subspaces using a closed-form overlap projector and a Spectral Unlearning Objective, ICE achieves precise concept removal with zero inference overhead and minimal collateral damage to generative quality.
Entities (5)
Relation Signals (4)
Instant Concept Erasure → appliesto → Text-to-Image
confidence 100% · ICE is both modality-agnostic and efficient: it applies identically to T2I and T2V models
Instant Concept Erasure → appliesto → Text-to-Video
confidence 100% · ICE is both modality-agnostic and efficient: it applies identically to T2I and T2V models
Overlap Projector → usedby → Instant Concept Erasure
confidence 95% · ICE defines erase and preserve subspaces... then explicitly regularises against their intersection using a unique, closed-form overlap projector.
Spectral Unlearning Objective → defines → Instant Concept Erasure
confidence 90% · We pose a convex and Lipschitz-bounded Spectral Unlearning Objective... This solution defines a dissociation operator
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Abstract
Abstract:Robust concept removal for text-to-image (T2I) and text-to-video (T2V) models is essential for their safe deployment. Existing methods, however, suffer from costly retraining, inference overhead, or vulnerability to adversarial attacks. Crucially, they rarely model the latent semantic overlap between the target erase concept and surrounding content -- causing collateral damage post-erasure -- and even fewer methods work reliably across both T2I and T2V domains. We introduce Instant Concept Erasure (ICE), a training-free, modality-agnostic, one-shot weight modification approach that achieves precise, persistent unlearning with zero overhead. ICE defines erase and preserve subspaces using anisotropic energy-weighted scaling, then explicitly regularises against their intersection using a unique, closed-form overlap projector. We pose a convex and Lipschitz-bounded Spectral Unlearning Objective, balancing erasure fidelity and intersection preservation, that admits a stable and unique analytical solution. This solution defines a dissociation operator that is translated to the model's text-conditioning layers, making the edit permanent and runtime-free. Across targeted removals of artistic styles, objects, identities, and explicit content, ICE efficiently achieves strong erasure with improved robustness to red-teaming, all while causing only minimal degradation of original generative abilities in both T2I and T2V models.
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- Source: https://arxiv.org/abs/2511.18684
- Canonical: https://arxiv.org/abs/2511.18684
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Now You See It, Now You Don’t - Instant Concept Erasure for Safe Text-to-Image and Video Generation Shristi Das Biswas, Arani Roy, Kaushik Roy Purdue University sdasbisw, roy173, kaushik@purdue.edu Abstract Robust concept removal for text-to-image (T2I) and text-to-video (T2V) models is essential for their safe deployment. Existing methods, however, suffer from costly retraining, inference overhead, or vulnerability to adversarial attacks. Crucially, they rarely model the latent semantic overlap between the target erase concept and surrounding content – causing collateral damage post-erasure – and even fewer methods work reliably across both T2I and T2V domains. We introduce Instant Concept Erasure (ICE), a training-free, modality-agnostic, one-shot weight modification approach that achieves precise, persistent unlearning with zero overhead. ICE defines erase and preserve subspaces using anisotropic energy-weighted scaling, then explicitly regularizes against their intersection using a unique, closed-form overlap projector. We pose a convex and Lipschitz-bounded Spectral Unlearning Objective, balancing erasure fidelity and intersection preservation, that admits a stable and unique analytical solution. This solution defines a dissociation operator that is translated to the model’s text-conditioning layers, making the edit permanent and runtime-free. Across targeted removals of artistic styles, objects, identities, and explicit content, ICE efficiently achieves strong erasure with improved robustness to red-teaming, all while causing only minimal degradation of original generative abilities in both T2I and T2V models. 1 Introduction The generative AI revolution, powered by text-to-image (T2I) models [1, 2, 3, 4, 5] and text-to-video (T2V) models [6, 7, 8], offers immense creative power but also significant risks. Trained on large-scale, uncurated internet datasets [9], these models can be easily prompted to generate harmful, copyrighted, or undesired content [10], including explicit material [11, 12], copyrighted artistic styles [13, 14, 15, 16], or the identities of specific individuals [17, 18]. This creates an urgent and critical need for robust, efficient, and precise mechanisms to control model outputs by erasing undesired concepts [19, 20, 21], thereby enabling a safer and more responsible deployment of generative models. Figure 1: ICE effectively isolates and removes target erase (e) concepts while minimizing impact on a preserved (p) set by explicitly addressing their semantic overlap. (L) Similarity between e and p embeddings; lower implies less overlap: ICE significantly reduces the Original embedding similarity (yellow), outperforming Naive Erasure (blue). (R) Self-similarity measuring preservation of p embeddings after erasure. Higher means lesser collateral damage. ICE’s preservation of original semantics is superior to Naive Erase. Current approaches to this problem fall into three main categories, each with their own significant drawbacks. Inference-time methods apply safety filters or guidance during generation [12, 22, 23]. While easy to implement, they introduce significant computational overhead during each inference-run and are often brittle, easily bypassed by simple adversarial prompts [24, 25]. On the other hand, training-based methods retrain or finetune the model to ‘forget’ a concept [26, 27, 28, 29, 30, 31]. This approach is prohibitively expensive, data-intensive, and often leads to catastrophic forgetting, degrading the model’s general generative quality. A more promising third category, training-free weight modification, directly edits the model’s parameters in a single shot [32, 33, 34]. These methods are fast and incur no inference overhead. However, they fail to explicitly address the fundamental challenge of semantic overlap. In the high-dimensional latent space, the subspace for an erase concept (e.g., eS_e for “Van Gogh”) is inherently non-orthogonal to related preserve concepts (e.g., pS_p for “painting” or an unconditional “ ” [35]). Naive erasure via simple orthogonal projection [36, 37, 34, 22] (as seen in Fig. 1 (R)) inevitably inflicts collateral damage on the shared semantic region, e∩pS_e _p, degrading the quality of the related non-targeted concepts post-erasure (also viewed through the lens of set-theoretic difference in App. Sec E.3). This failure to explicitly model and preserve the overlap is the primary reason for their trade-offs in fidelity and robustness. Furthermore, most prior work focuses exclusively on T2I models, neglecting the growing need for modality-agnostic safety methods that also lend themselves to safe T2V generation. To overcome these limitations, we propose Instant Concept Erasure (ICE) – a unified, training-free framework that performs precise, one-shot concept removal through a closed-form, geometry-aware formulation. ICE operates directly in the semantic conditioning space of generative diffusion models, identifying and isolating the latent subspace responsible for an undesired concept while explicitly preserving overlapping semantics shared with other safe or general concepts. Unlike prior methods that rely on iterative finetuning or run-time masking, ICE derives a formal Spectral Unlearning Objective whose global minimum admits a unique, analytic solution to this task, enabling an instantaneous and permanent edit to model parameters. Our key insight is that effective unlearning requires explicitly characterizing and protecting the intersection between erase and preserve subspaces, rather than assuming they are orthogonal or disjoint. By leveraging energy-weighted bases [38] derived from the target embeddings’ singular spectrum, ICE builds anisotropic subspace operators for both the erase concept (e) and its complementary preserve set (p), and computes their intersection using a closed-form overlap projector. This formulation yields an interpretable and stable dissociation operator that directly specifies the optimal forgetting direction, which we then translate into a permanent one-shot weight update at a given generative model’s earliest text-conditioning interface (e.g., the key-value projections in UNet-based models or the text-projection layers in diffusion transformers). ICE is both modality-agnostic and efficient: it applies identically to T2I and T2V models in only 22 seconds, requiring no retraining or additional inference-time cost. We demonstrate its broad applicability across four major unlearning axes – artistic styles, object categories, identities, and explicit content – showing strong erasure efficacy with minimal damage to the model’s base generation quality. Further, ICE substantially enhances resistance to adversarial and red-teaming attacks, outperforming prior training-free and finetuning-based unlearning methods. In summary, ICE provides a mathematically grounded and computationally lightweight solution to safe concept removal, achieving reliable, fast, and generalizable unlearning across generative modalities without sacrificing fidelity or robustness. 2 Related Works Safe T2I Generation. Current methods for removing undesired concepts T2I models can be classified into three paradigms, each with distinct trade-offs. The first, inference-time control, applies post-hoc interventions such as safety checkers [39], classifier-free guidance during generation [12, 40] or filtering the embeddings away from identified unsafe subspaces and using adaptive token-wise shifts to navigate similarity between run-time prompt tokens and target concept tokens [22, 23]. While easy to implement, they require independent safeguarding operations for every new prompt, hindering inference-time efficiency, and are easily circumvented in open-source models. The second paradigm, training-based interventions, involves modifying the model through retraining or finetuning [41, 42, 30, 43]. This includes techniques like training on filtered datasets, using negative guidance, self-supervised learning or adversarial training [44, 26, 45, 46, 47, 48], minimizing KL divergence between unwanted and alternative safe concepts [27], and employing preference optimization to suppress concepts [49, 50]. On the other hand, partial finetuning approaches adjust specific layers to forget undesired concepts [28, 31, 29]. Although effective to some degree, these methods are computationally expensive, require careful data curation, and may degrade general model performance while still remaining vulnerable to adversarial attacks [51]. The third category consists of training-free weight modification techniques, which aim to erase concepts via one-shot updates to model parameters [32, 52, 33, 34]. While they avoid the cost of retraining, they often fail to safeguard overlapping semantics with untargeted surrounding concepts, and are easily bypassed by black-box or white-box red-teaming tools [24, 25, 53, 54]. Safe T2V Generation. The domain of concept unlearning for T2V generation is still nascent, with few existing works. These pioneering approaches can be broadly categorized. One strategy involves intervention in the embedding space [22] that identifies and removes harmful representations from text embeddings during run-time. A second, transfer-based approach leverages unlearning from the more mature T2I domain [55] to optimize a text encoder using few-shot image-based unlearning and then deploys this finetuned ‘safe’ encoder in a T2V model. A third work involves direct finetuning of the T2V model using negatively-guided velocity prediction [56]. These existing methods rely on inference-time filtering, knowledge transfer, or computationally intensive finetuning. Importantly, they also fail to provide explicit preservation for the untargeted concepts. Differently, ICE is a unified, modality-agnostic edit for both T2I and T2V, uniquely designed to be intersection-aware. We pose a convex Spectral Unlearning Objective that regularizes the update with a formal, non-iterative characterization of the semantic overlap between importance-weighted erase and preserve subspaces, and prove that it admits a unique closed-form dissociation direction. This operator translates into a permanent weight update, enabling robust, high-fidelity unlearning with zero inference overhead. 3 Method We achieve this through the ICE closed-form, geometrical overlap-aware formulation for one-shot concept erasure. The process is elucidated as follows. 3.1 Problem Formulation in Conditioning Subspaces We formalize this task by considering a single network layer where an output activation ∈ℝ1×Vo ^1× V is computed from input activation ∈ℝ1×dx ^1× d and a weight matrix ∈ℝV×dW ^V× d via the linear transformation =o=xW^T. Our objective is to produce a modified output, ∗o^*, from which a target erase concept’s semantic contribution is selectively removed without affecting model performance on untargeted content. We postulate that a concept’s semantics are encoded within an euclidean subspace of the embedding space ℝdR^d. Erasure can thus be achieved by projecting these semantics out of the input embedding. To this end, we seek an Instant Concept Erasure (ICE) operator, ice∈ℝd×dP_ice ^d× d, that isolates the target concept’s precise representation from all neighboring content. Applying this operator yields the erase concept specific embedding ice=icex_ice=xP_ice (to be removed), which in turn produces the ideal unlearnt output: ∗=(−ice)=((ℐ−ice)) -2.0pto^*=(x-x_ice)W^T=(x(I-P_ice))W^T 1.0pt (1) This prevents the model from generating the erased concept any further with high specificity. This operation can henceforth be interpreted either as a dynamic filter on the activation x or as a one-time persistent update to the model’s parameters by translating the operator to weight space via an updated weight matrix, ice=((ℐ−ice))W^T_ice=((I-P_ice)W^T) -2.0pt (2) The central challenge, which the remainder of this paper addresses, is the principled, data-driven construction of this precise unlearning operator iceP_ice. 3.2 Formal Characterization of Target Subspaces The latent semantic subspaces for erasure (eS_e) and preservation (pS_p) are characterized by forming embedding matrices, ℰe,p∈ℝd×NE_e,p ^d× N, from their respective prompt tokens via the model’s frozen text encoder [57, 58, 59]. Singular Value Decomposition (ℰ=Σ⊤E=U ) yields an orthonormal basis for each subspace from the columns of the left singular bases U and a corresponding singular value spectrum Σ . Conventional subspace methods often assume all basis vectors are equally important, an assumption we find too restrictive for precise unlearning [36]. We observe that the singular values exhibit a non-uniform distribution [38], implying that each basis vector (Σi ^i) contributes unequally to the concept’s representation. Leveraging this, we propose assigning an anisotropic importance weight, λiλ^i, to each basis vector iU^i. The importance is computed as follows: λi=2ΣiΣi+maxj=1..k(Σj),where λi∈[0,1] -4.0ptλ^i= 2 ^i ^i+ _j=1..k( ^j),where λ^i∈[0,1] (3) This fixed weighting profile assigns maximal salience (λ=1λ=1) to the principal component while other bases are given importance (<1<1) relative to this maximum. These importance scores, λe _e and λp _p, form diagonal matrices, Λe _e and Λp _p, which define the scaled spectral conditioning operators: e=eΛee⊤andp=pΛpp⊤ -2.0ptP_e=U_e _eU_e _p=U_p _pU_p -3.0pt (4) This characterization creates contractive attenuators that emphasize high-energy, concept-defining directions while suppressing low-energy spaces that are generic and shared across neighboring concepts. 3.3 The Challenge of Semantic Overlap and Subspace Intersection The core limitation in existing unlearning literature is the lack of dissociation of concepts targeted for erasure (e) from those intended for preservation (p). As discussed, this primary challenge arises from the geometric relationship between their corresponding latent subspaces, eS_e and pS_p. We further take into consideration the practical challenge of defining an ideal set p, and hence design it to encompass the model’s generic, average knowledge of all other possible concepts. We use the unconditional embedding “ ” as a principled and efficient proxy for this space, as its role as the model’s generative prior was established by [35]. While an ideal, disentangled space would feature embedding orthogonality (e⟂pS_e _p) and hence no semantic overlap, we find that this is not the case in practice even after using anisotropic projectors (Eq. 4) as seen in Fig. 1. Consequently, a principled unlearning method must explicitly characterize and preserve the non-trivial overlap e∩pS_e _p. To this end, we establish that this semantic intersection can be characterized exactly and non-iteratively [60, 61]. We formalize this foundational challenge as: Proposition 1. The unique overlap projector, e∩pP_e∩ p, onto the intersection of target latent subspaces eS_e and pS_p is given in closed-form by: e∩p=2e(e+p)†p -3.0ptP_e∩ p=2P_e(P_e+P_p) P_p -4.0pt (5) where † denotes the Moore-Penrose pseudoinverse. The detailed proof is presented in App. Appendix A. This provides a tractable formulation of the overlap projector for our precise unlearning objective, allowing for a non-iterative solution constrained against the subspace of semantic overlap. 3.4 The Spectral Unlearning Objective With the operators for the target, e,pPe,p, and the intersection subspaces e∩pPe∩ p established, we isolate the erase-specific embedding ice∈ℝdx_ice ^d – excluding shared components with the preserve subspace – via a convex unlearning objective that balances erasure fidelity and overlapping semantics preservation. Our objective ℒ(ice)L(x_ice) is: miniceℒ(ice)≜minice(‖ice−e‖22⏟Fidelity Term+‖icee∩p‖22⏟Preservation Term) _x_iceL(x_ice) _x_ice( \|x_ice-xP_e\|_2^2_Fidelity Term+ \|x_iceP_e∩ p\|_2^2_Preservation Term) -9.0pt (6) The components of this objective serve specific roles: • Fidelity Term: This quadratic loss term enforces the erasure goal. It minimizes the distance between the unlearning direction icex_ice and the projection of the original concept x onto the erase subspace. This ensures icex_ice accurately captures the features designated for ablation. • Preservation Term: This term acts as a convex regularizer to prevent collateral damage to surrounding concepts. It penalizes icex_ice from having any component that lies within the semantic intersection, which is precisely characterized by the overlap projector e∩pP_e∩ p. Together, these make xicex_ice effectively isolate target erase concepts e while minimizing impact on a preserved p set by explicitly removing their semantic overlap. A key advantage of this formulation is that ℒ(ice)L(x_ice) is strongly convex with Lipschitz-continuous gradient, a property we formally prove in App. Sec. B and Sec. C. This guarantees a unique and stable global minimum, allowing us to bypass iterative optimization methods and derive a direct closed-form solution for the optimal unlearning direction, as shown below. Proposition 2. Given a target concept embedding x, the erase operator eP_e, and the intersection projector e∩pP_e∩ p, the vector icex_ice that minimizes the objective in Eq. 6 is given by the unique, closed-form solution: ice=ice where, ice=e(ℐ+e∩pe∩pT)−1x_ice=xP_ice where, P_ice=P_e(I+P_e∩ pP_e∩ p^T)^-1 -5.0pt (7) and, ice∈ℝd×dP_ice ^d× d. The first term in the inverse matrix, ℐI, can be interpreted as matching the covariance of the large encyclopedia of concept embeddings in the diffusion model’s vocabulary, as inspired by [62], while the second term is the covariance matrix of the semantic intersection. Proof.To find the minima of the convex and Lipschitz-bounded loss ℒ(ice)L(x_ice), we compute its gradient with respect to the row vector icex_ice and set it to zero. Using the identity ∇‖−ℬ‖22=2(−ℬ)T _X\|XA-B\|_2^2=2(XA-B)A^T, the gradient is: ∇iceℒ(ice)=2(ice−e)+2(icee∩p)(e∩p)T -3.0pt _x_iceL(x_ice)=2(x_ice-xP_e)+2(x_iceP_e∩ p)(P_e∩ p)^T -3.0pt (8) Setting ∇iceℒ=0 _x_iceL=0, we isolate icex_ice ice 5.0ptx_ice =(e)(ℐ+e∩pe∩pT)−1 =x(P_e)(I+P_e∩ pP_e∩ p^T)^-1 (9) =icewhere ice∈ℝd×d =xP_icewhere P_ice ^d× d -35.0pt (10) We provide more details on the formalized derivation for iceP_ice in App. Appendix D. The existence of this analytical solution is the backbone of ICE’s precision and efficiency. Figure 2: ICE performs one-shot, training-free concept erasure by constructing erase (eS_e), preserve (pS_p), and intersection (e∩pS_e∩ p) subspaces. The closed-form operator (e)(ℐ+e∩pe∩pT)−1x(P_e)(I+P_e∩ pP_e∩ p^T)^-1 isolates undesired semantics while safeguarding shared content, which is applied as a weight update to the target text-conditioning layer W to enable precise, safe, and modality-agnostic unlearning. 3.5 Update Translation for Persistent Unlearning The analytical solution from ICE provides the optimal unlearning direction, ice∈ℝ1×dx_ice ^1× d. To apply this solution, we translate the iceP_ice activation-space vector into a permanent, one-shot modification of the model’s weight space. This procedure ensures the unlearning is both persistent and incurs zero overhead during inference, providing a broadly applicable method for safe T2I and T2V generation. The generalizability of our method stems from targeting the ‘entry points’ where text conditioning is first injected into the main visual backbone to steer the generative process. This intervention point is functionally consistent across different diffusion architectures. By modifying the model at this initial stage of semantic injection, we can ablate undesirable concepts before they propagate through the generative network, providing a powerful tool for enforcing safety constraints on any text-conditioned model. The primary entry points include: • For UNet-based architectures (e.g., Stable Diffusion), the intervention targets the key (kW_k) and value (vW_v) projection matrices within each cross-attention (CA) block. • For many Diffusion Transformer (DiT) architectures (e.g., CogVideoX), the intervention targets a dedicated text projection layer that processes text embeddings before they are consumed by the transformer blocks. The ICE operator is then used to update the weight matrix oldW_old of the target entry-point layer(s) in a single shot using Eq. 2. This one-time instant update permanently embeds the unlearning transformation into the T2I/V model’s parameters, resulting in zero additional run-time latency. An overview of the method is illustrated in Fig. 2. Method Weights Mod. Training -Free Semantic Overlap Aware Attack Success Rate ↓ COCO I2P [12] P4D [53] Ring-A-Bell [63] MMA-Diffusion [54] UnlearnDiffAtk [24] FID [64] ↓ CLIP [65] ↑ SD-v1.4 - - - 0.178 0.987 0.831 0.957 0.697 - 31.3 SLD-Medium [12] ✗ ✓ ✗ 0.142 0.934 0.660 0.942 0.648 31.47 31.0 SLD-Strong [12] ✗ ✓ ✗ 0.131 0.814 0.620 0.920 0.570 40.88 29.6 SLD-Max [12] ✗ ✓ ✗ 0.115 0.602 0.570 0.837 0.479 50.51 28.5 SAFREE [22] ✗ ✓ ✗ 0.272 0.384 0.114 0.585 0.282 36.35 31.1 AdaVD [23] ✗ ✓ ✗ 0.045 0.106 0.114 0.461 0.275 35.99 31.3 ESD [26] ✓ ✗ ✗ 0.140 0.750 0.528 0.873 0.761 52.06 30.7 SA [31] ✓ ✗ ✗ 0.062 0.623 0.239 0.205 0.268 54.98 30.6 CA [27] ✓ ✗ ✗ 0.078 0.639 0.376 0.855 0.866 40.99 31.2 MACE [28] ✓ ✗ ✗ 0.023 0.142 0.076 0.183 0.176 52.24 29.4 SDID [47] ✓ ✗ ✗ 0.270 0.931 0.646 0.907 0.637 22.99 30.5 CPE [29] ✓ ✗ ✗ 0.046 0.110 0.012 0.186 0.234 22.01 29.7 UCE [33] ✓ ✓ ✗ 0.103 0.667 0.331 0.867 0.430 31.25 31.3 RECE [32] ✓ ✓ ✗ 0.064 0.381 0.134 0.675 0.655 37.60 30.9 CURE [34] ✓ ✓ ✗ 0.061 0.107 0.013 0.169 0.281 - - ICE (Ours) ✓ ✓ ✓ 0.043 0.102 0.011 0.173 0.266 21.94 31.2 Table 1: Comparison on Attack Success Rate (ASR). ↓ indicates lower is better, while ↑ shows higher is better. Our method achieves the best performance on the majority of the adversarial benchmarks while maintaining competitive performance on COCO-3030k. The COCO scores for CURE are not reported as their implementation is not publicly available. We gray out training-based methods for a fair comparison, while methods in pink require repeated run-time application. Best results are bolded and second best underlined. Figure 3: Comparison of unlearning methods on removing target artist styles and NSFW content. ICE suppresses the intended erase concept more effectively than baselines. ∗ masks any unsafe outputs for display purposes. 4 Evaluation In this section, we present the results of our method for erasing inappropriate concepts, artistic styles, objects, and identities, as well as resistance to red-teaming attacks. We use StableDiffusion-v1.4 (SD-v1.4) [3] as a primary T2I backbone, following recent work [33, 32, 23]. We further extend evaluations to StableDiffusion-v1.5 [66], StableDiffusion-v2.1 [67] and several community versions of SD, including RealisticVision [68], Dreamshaper [69], and Chilloutmix [70]. Next, we evaluate erasure efficacy for T2V generation by applying to the popular CogVideoX-2B and CogVideoX-5B models [7]. The preserve set is defined as an unconditional embedding “ ” unless otherwise specified. More details are provided in the Appendix. 4.1 Unsafe Content Erasure We first evaluate T2I safety using multi-suite red-teaming and report attack-success rate (ASR) alongside COCO [71] utility to demonstrate specificity in preserving normal content post-unlearning. To this end, we test on inappropriate prompts from I2P [12], white-box attacks [53, 24], and black-box attacks [63, 54]. For nudity analysis, we generate one image per prompt and detect unsafe regions with a NudeNet [72] threshold of 0.60.6, following [32]. As summarized in Tab. 1, ICE – the only method that is simultaneously training-free, a one-shot weight edit, and explicitly overlap-aware – achieves SOTA or second-best ASR across all baselines. In particular, ICE attains 4%4\%, and 31%31\% lower ASR compared to the best-performing counterparts on P4D and Ring-A-Bell, respectively, highlighting its strong resilience against adversarial attacks. Crucially, ICE delivers superior or comparable safeguarding performance while generating higher-quality images on the COCO-3030k dataset, all within a training-free framework (Tab. 1 and App. Fig. 12). Qualitative examples (Fig. 3) draw a similar conclusion, showing precise suppression of unsafe content with minimal collateral damage. Notably, ICE is the only method in Row 1 that preserves Emma Watson’s identity while erasing nudity, underscoring strong preservation of untargeted content. Methods SafeSora ↓ Nudity Rate (Gen) ↓ Nudity Rate (Ring-A-Bell) ↓ VBench Object Class ↑ VBench Subject Consistency ↑ CogX-2B CogX-5B CogX-2B CogX-5B CogX-2B CogX-5B Original 64.98 79.56 57.10 61.80 30.25 42.50 90.75 95.77 NegPrompt 55.73 64.20 36.00 46.35 11.75 14.91 90.64 91.62 SAFREE 43.22 51.58 32.43 35.12 14.23 10.64 53.29 93.36 T2VUnlearning 31.56 39.77 19.73 16.47 6.97 2.74 85.30 92.09 ICE (Ours) 24.01 39.34 11.25 8.75 7.84 2.98 87.11 92.38 Table 2: Comparison of safe T2V generation performance across baselines. To assess video generation ability for benign concepts post-unlearning, we evaluate on the VBench benchmark metrics. Figure 4: T2V generated examples with CogVideoX-5B on Ring-A-Bell prompts. We manually blurred out generated sexually explicit videos and censored sensitive text prompts for display purposes. For T2V, we assess nudity and broader toxicity suppression on three sets: (1) Gen, a contextual prompt suite describing nudity in rich detail [23]; (2) Ring-A-Bell, comprising short stylized prompts depicting explicit artwork [63, 23]; and (3) SafeSora [73], from which we use a 296296 toxic prompts across 55-categories, following [22]. For each prompt, we generate 4949 frames per model at default resolution and report the NudeNet [72] Nudity Rate. On CogVideoX-2B/5B, ICE achieves the lowest SafeSora toxicity and lowest Nudity Rate on Gen by up to 23.9%23.9\% and 47.7%47.7\% respectively. Qualitative results show robust nudity suppression with preserved motion and content (Fig. 4), as well as resistance to concepts prompting violence and animal abuse (App. Fig. 10), showing that ICE effectively removes diverse sensitive content while preserving detail and untargeted concepts. To analyze the potential impact of nudity erasure on non-nudity concepts, we adopt VBench [74], a widely used video generation benchmark for further evaluation. ICE maintains strong video utility on Object Class and Subject Consistency metrics, as summarized in Tab. 2 and App. Fig. 15. More results in the Appendix. Figure 5: Qualitative samples of art style removal for single and multi-instance erasure. ICE can effectively remove the target concept (images with red borders) while preserving non-target styles with high precision and fidelity. 4.2 Artistic Style Erasure To evaluate the efficacy of style unlearning for mitigating artistic imitation, we follow [33, 32] to use 2020 prompts each for five famous artists and five modern artists. Evaluation uses LPIPS scores [75], where a higher LPIPSe (on erased artists) indicates stronger removal of the target style, and a lower LPIPSp (on preserved artists) reflects better retention of unrelated artists. We additionally use GPT-4o [76] to classify artistic styles of the generated images. Acce shows how often the unlearned style is still predicted – lower is better. Accp measures accuracy on non-erased styles – higher is better. More experimental details are provided in the Appendix. Quantitative results in Tab. 4 show that ICE consistently achieves strong erasure efficacy with minimal impact on unintended styles compared to baselines. Qualitative comparisons confirm this superior performance. In Fig. 3 and Fig. 6, most baselines fail to resist generating the ‘Van Gogh’ composition. ICE, in contrast, generates completely new coherent images in a different style, demonstrating a higher-fidelity erasure. We also assess the impact of erasing up to a 10001000 styles, while preserving all other styles in App. Fig. 13 and find that ICE can erase up to 100100 artists without affecting CLIP scores. Furthermore, Fig. 5 shows ICE’s impressive specificity in retaining untargeted styles given its overlap preservation technique. Remove “Van Gogh” Remove “Kelly McKernan” Method LPIPSe ↑ LPIPSp ↓ Acce ↓ Accp ↑ LPIPSe ↑ LPIPSp ↓ Acce ↓ Accp ↑ SD-v1.4 - - 0.95 0.95 - - 0.80 0.83 SLD-Medium [12] 0.31 0.55 0.95 0.91 0.39 0.47 0.50 0.79 SAFREE [22] 0.42 0.31 0.35 0.85 0.40 0.39 0.40 0.78 AdaVD [23] 0.43 0.07 0.28 0.94 0.37 0.06 0.35 0.97 CA [27] 0.30 0.13 0.65 0.90 0.22 0.17 0.50 0.76 ESD [26] 0.40 0.26 0.57 0.89 0.37 0.21 0.81 0.69 AdvUnlearn [48] 0.33 0.12 0.30 0.87 0.40 0.13 0.39 0.71 RECE [32] 0.31 0.08 0.80 0.93 0.29 0.04 0.55 0.76 UCE [33] 0.25 0.05 0.95 0.98 0.25 0.03 0.80 0.81 CURE 0.44 0.08 0.30 0.94 0.41 0.09 0.35 0.94 ICE (Ours) 0.46 0.07 0.27 0.96 0.39 0.03 0.32 0.95 Table 3: Quantitative comparison of methods on the artist concept erasure task. Method ESR-1↑ ESR-5↑ PSR-1↑ PSR-5↑ CogVideoX-2B 21.62 5.09 78.38 94.91 NegPrompt 48.59 19.79 65.37 88.62 SAFREE [22] 61.65 36.41 53.46 79.17 T2VUnlearning [56] 92.38 77.09 54.03 82.14 ICE (Ours) 93.10 82.55 75.41 89.63 Table 4: Results of ImageNet object erasure on CogVideoX-2B. Figure 6: Evaluation against Ring-A-Bell prompts where, unlike baselines, ICE robustly eliminates Van Gogh’s style and the object ‘car’. 4.3 Object Erasure Next, following [56], we evaluate object removal by iteratively selecting one class from 1010 distinct ImageNet classes [77] as e and assessing preservation on the remaining 99. For evaluation on T2V models, we conduct per-frame classification on CogVideoX-2B generated videos and compute Erasure Success Rate ESR-k=1−top-kESR-k=1-top-k on the erased class and Preservation Success Rate PSR-kPSR-k as average top-k accuracy over remaining classes. As shown in Tab. 4, ICE attains the highest ESR-1/ESR-5 while maintaining strong PSR-k, indicating strong suppression of the target object with minimal impact on non-target content. We further confirm strong performance on T2I models in App. Tab. 10 and App. Fig 9. This robustness is also highlighted in Fig. 6, where ICE uniquely resists an adversarial jailbreak prompt for the erased ‘car’ concept, unlike all baselines. Additional samples in App. Fig. 14 further show that ICE resists the generation of images of the synonymous form of a removed concept, demonstrating high erasure efficacy. Together, these results show ICE provides robust, specific, and generalizable object erasure for both T2I/V. 4.4 Identity Erasure We next evaluate ICE on the challenging task of identity erasure, where high inter-class similarity (e.g., between human faces) increases the risk of collateral damage. This task is critical for removing copyrighted characters or specific public figures. We select four public and stylized identities (Barack Obama, Queen Elizabeth, Mickey Mouse, and Spongebob), setting one as the target concept while preserving the others. We measure performance using average ID-Similarity via ArcFace [78] embeddings, reporting Erase Accuracy (AcceAcc_e) for the target and Preserve Accuracy (AccpAcc_p) for the non-target set. As demonstrated in Table 5, ICE shows remarkable erasure performance, successfully applying to different T2I and T2V models, as well as community SD versions [68, 69, 70], while simultaneously maintaining high accuracy for the preserved identities. This result quantitatively validates that our explicit overlap-aware framework successfully navigates this high-similarity domain, preventing collateral damage. Qualitative results in Fig. 7 demonstrate that ICE can remove the knowledge of ‘Mickey Mouse’ without affecting the concept ‘Spongebob’ (and vice-versa), even composing erasures to remove both simultaneously. Additional visualizations in App. Fig. 11 confirm high erasure precision, showing that untargeted concepts remain unaffected, even in cases where they share a common first name with the target identity. Model Metric Obama Elizabeth Mickey Mouse Spongebob AVG SD-v1.5 Acce ↓ 0.04 0.09 0.09 0.12 0.06 Accp ↑ 0.91 0.87 0.92 0.83 0.88 RealisticVision Acce ↓ 0.14 0.12 0.17 0.17 0.15 Accp ↑ 0.71 0.78 0.74 0.69 0.74 Chilloutmix Acce ↓ 0.21 0.11 0.07 0.14 0.13 Accp ↑ 0.80 0.68 0.72 0.71 0.73 DreamShaper Acce ↓ 0.11 0.16 0.13 0.07 0.12 Accp ↑ 0.79 0.76 0.81 0.85 0.80 SD-v2.1 Acce ↓ 0.08 0.13 0.11 0.16 0.12 Accp ↑ 0.80 0.93 0.91 0.89 0.88 CogX-2B Acce ↓ 0.06 0.10 0.09 0.15 0.10 Accp ↑ 0.85 0.74 0.83 0.90 0.83 CogX-5B Acce ↓ 0.11 0.12 0.08 0.09 0.11 Accp ↑ 0.82 0.79 0.86 0.95 0.87 Table 5: Results of face erasure. The Acce row reports the ID-Similarity of the target face, while the Accp row shows the average ID-Similarity of the non-target faces. Figure 7: Single- and multi-identity erasure results (targets have red borders) using ICE. Prompt for col 1: Mickey Mouse; col 2: Spongebob. Method Mod. Time (s) Inference Time (s/sample) Model Mod. (%) SLD-Max [12] 0 10.34 0 AdaVD [23] 1 11.29 0 SAFREE [22] 0 10.56 0 ESD [26] ∼ 4500 7.08 94.65 CA [27] ∼ 484 6.31 2.23 AdvUnlearn [48] ∼ 78000 7.07 11.54 UCE [33] ∼ 1 7.08 2.23 RECE [32] ∼ 3 7.12 2.23 CURE ∼ 2 7.06 2.23 ICE (Ours) ∼ 2 7.07 2.23 Table 6: Erasure efficiency comparison when removing the ‘nudity’ concept. Evaluated on an A4040 GPU for 100100 iterations. Method Acce ↓ Accp ↑ SD-v1.4 78.2 79.0 ICE (Ours) 1.3 80.3 w/o Scaling 0.6 61.4 w/o Overlap Operator 1.3 69.5 w/ e∗pP_e*P_p 1.5 75.1 Table 7: Ablating the components of ICE. We evaluate the erase accuracy (Acce) and preserve accuracy (Accp) on ImageNet. 4.5 Unlearning Efficiency We evaluate unlearning overheads in Tab. 6 for SD-v1.4. Prior techniques include costly training [27, 26, 48], runtime filters [12, 22, 23] do not provide preemptive weight modification, and training-free edits [34, 33, 32]. As shown in Tab. 6, ICE offers delivers strong overall performance, achieving low modification time alongside fast inference. 5 Ablating ICE We conduct an ablation study using object erasure experiments in Tab. 7. Our full method, ICE, achieves strong erasure while uniquely improving upon the original model’s utility, demonstrating surgically precise and effective unlearning. Removing anisotropic spectral scaling (w/o Scaling) results in a catastrophic drop in general performance, confirming that energy-aware basis scaling is essential for preservation. Similarly, removing the overlap-aware term (w/o Overlap Operator) leads to collateral damage, proving that modeling the semantic intersection is necessary. Finally, replacing our overlap operator with a naive matrix product (e∗pP_e*P_p) partially mitigates the issue but still leaves noticeable leakage into retained content. These results show that both the importance scaling and the explicit overlap operator are critical to ICE’s robust unlearning performance. 6 Limitations and Potential Impact First, ICE’s efficacy is limited by the semantic granularity of the pre-trained text encoder embeddings it operates on, an issue shared by most baselines. Second, ICE’s weight modification requires white-box model access, limiting its direct use on closed-source APIs. In such cases, however, the ICE operator can still be applied in the prompt space prior to model inference. Despite these limitations, ICE advances unlearning research by establishing a mathematically grounded, efficient, and training-free paradigm for safe T2I/T2V modeling. Its transparent, modality-agnostic design enables practical applications like IP protection and model customization. Critically, its closed-form formulation promotes the interpretability and reliability essential for integration into safety-critical generative systems. 7 Conclusion In this work, we introduced Instant Concept Erasure (ICE), a principled, training-free framework for one-shot concept unlearning in both T2I and T2V models. We differ from literature by mathematically formalizing the problem of semantic overlap. Next, by characterizing the intersection of erase and preserve subspaces with a closed-form analytical projector, our convex spectral objective derives a unique, optimal unlearning solution. This solution translates to a permanent, zero-overhead weight modification that precisely removes targeted artistic styles, objects, identities, and explicit content. 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Appendix In this appendix, we provide detailed proofs, additional experimental results, and implementation details to supplement the main paper. We begin with the formal mathematical derivations for our method: Appendix A provides the proof for the closed-form overlap projector, Appendix B proves the L-smoothness of our spectral objective, Appendix C proves its strict convexity, and Appendix D details the full derivation of the closed-form ICE solution. Following this, Appendix E discusses additional implementation details, while Appendix F provides extended experimental results, including comprehensive object erasure benchmarks, sensitive-concept erasure results and additional qualitative visualizations. Finally, Appendix G provides a complete list of licenses for all models and datasets used in this work. Appendix A Overlap Operator Proof Proposition 3. The unique overlap projector, e∩pP_e∩ p, onto the intersection of subspaces eS_e and pS_p is given in closed-form by: e∩p=2e(e+p)†pP_e∩ p=2P_e(P_e+P_p) P_p (11) where eP_e and pP_p are the projectors onto eS_e and pS_p respectively, and † denotes the Moore-Penrose pseudo inverse. Preliminaries. To facilitate the subsequent proof, we first review the fundamental properties of projection operators, specifically their construction for the linear sum and intersection of subspaces, as established in [60]. We begin by defining the requisite notation: If M is a subspace of nS^n, we write MP_M for the unique projection onto M. We denote M=Span(M)M=Span(P_M) and M⟂=Span⟂(M)M =Span (P_M). We further find from [61] that if say A and B are projections on nS^n, Span(A+B)=Span(A)+Span(B),Span(A+B)=Span(A)+Span(B), (12) By Eq. 12, Span(A)⊆Span(A)+Span(B)=Span(A+B)Span(A) (A)+Span(B)=Span(A+B) (13) With this property for subspaces established, we now proceed to the formal proof. Proof. We wish to find the span of the overlap projection, i.e., to characterize Span(e∩p)=e∩pSpan(P_e∩ p)=S_e _p directly from eP_e and pP_p. The proof follows the style of [61], adapting the notation. We begin by proving commutativity property: 2e(e+p)†p=2p(e+p)†e2P_e(P_e+P_p) P_p=2P_p(P_e+P_p) P_e (14) To do this, we show that their difference is zero. We start by adding and subtracting the term e(e+p)†eP_e(P_e+P_p) P_e: e(e+p)†p−p(e+p)†e _e(P_e+P_p) P_p-P_p(P_e+P_p) P_e (15) =e(e+p)†p+e(e+p)†e− =P_e(P_e+P_p) P_p+P_e(P_e+P_p) P_e- [e(e+p)†e+p(e+p)†e] [P_e(P_e+P_p) P_e+P_p(P_e+P_p) P_e ] (16) =e(e+p)†(e+p)−(e+p)(e+p)†e =P_e(P_e+P_p) (P_e+P_p)-(P_e+P_p)(P_e+P_p) P_e (17) From Eq. 13, we have Span(e)⊆Span(e+p)Span(P_e) (P_e+P_p). In other words, e⊆e+pS_e _e+S_p. For any two projectors with nested subspaces 1⊆2S_1 _2, the property 21=1P_S_2P_S_1=P_S_1 holds. This is because 1P_S_1 projects any vector into 1S_1, which is already in 2S_2, so the subsequent projection 2P_S_2 has no effect. Therefore, e(e+p)†(e+p) _e(P_e+P_p) (P_e+P_p) =e =P_e (18) (e+p)(e+p)†e (P_e+P_p)(P_e+P_p) P_e =e =P_e (19) Substituting these results back into our expression for the difference (Eq. 17) yields e−e=0P_e-P_e=0. Thus, the difference for RHS in Eq. 17 is zero, and we have proven the commutativity, i.e e(e+p)†p−p(e+p)†e=0P_e(P_e+P_p) P_p-P_p(P_e+P_p) P_e=0 (20) ⟹e(e+p)†p=p(e+p)†e _e(P_e+P_p) P_p=P_p(P_e+P_p) P_e (21) ⟹2e(e+p)†p=2p(e+p)†e 2P_e(P_e+P_p) P_p=2P_p(P_e+P_p) P_e (22) Next, we argue that both sides of Eq. 22 equal e∩pP_e∩ p. Let H H =e(e+p)†p+p(e+p)†e =P_e(P_e+P_p) P_p+P_p(P_e+P_p) P_e (23) =2e(e+p)†p =2P_e(P_e+P_p) P_p (24) =2p(e+p)†e =2P_p(P_e+P_p) P_e (25) We first show that Span(H)⊆e∩pSpan(H) _e _p. Hp HP_p =[2e(e+p)†p]p =[2P_e(P_e+P_p) P_p]P_p (27) =2e(e+p)†p2 =2P_e(P_e+P_p) P_p^2 (28) =2e(e+p)†p (by projector property [61]) =2P_e(P_e+P_p) P_p (by projector property~ [cite][\@@bibrefNumberpiziak1999constructing]) (29) =H =H (30) The property Hp=HP_p=H implies that projecting into HpHP_p results in projection onto H itself. This means that Span(H)Span(H) refers to a nested subspace within Span(p)Span(P_p). In other words, Span(H)⊆Span(p)=pSpan(H) (P_p)=S_p. Similarly, we prove He=HP_e=H which implies Span(H)⊆Span(e)=eSpan(H) (P_e)=S_e. Since Span(H)Span(H) is a subspace of both eS_e and pS_p, it must be a subspace of their intersection: Span(H)⊆e∩pSpan(H) _e _p. Because Span(H)⊆e∩pSpan(H) _e _p and e∩pP_e∩ p is the projector onto e∩pS_e _p, it follows that He∩p=HP_e∩ p=H. Hence, we write: H H =He∩p =HP_e∩ p (31) =[e(e+p)†p+p(e+p)†e]e∩p =[P_e(P_e+P_p) P_p+P_p(P_e+P_p) P_e]P_e∩ p (32) =e(e+p)†pe∩p+p(e+p)†ee∩p =P_e(P_e+P_p) P_pP_e∩ p+P_p(P_e+P_p) P_eP_e∩ p (33) =e(e+p)†e∩p+p(e+p)†e∩p =P_e(P_e+P_p) P_e∩ p+P_p(P_e+P_p) P_e∩ p (34) =[e(e+p)†+p(e+p)†]e∩p =[P_e(P_e+P_p) +P_p(P_e+P_p) ]P_e∩ p (35) =[(e+p)(e+p)†]e∩p =[(P_e+P_p)(P_e+P_p) ]P_e∩ p (36) =e∩p =P_e∩ p (37) This last equality follows because Span(e∩p)⊆Span(e+p)Span(P_e∩ p) (P_e+P_p). Thus from Eq. 37, e∩p=H=2e(e+p)†pP_e∩ p=H=2P_e(P_e+P_p) P_p. The preceding proof, based on the properties of projectors from [61], establishes the span and mathematical form of the subspace intersection e∩pP_e∩ p. In our practical implementation, we create e∩pP_e∩ p with our energy-scaled operators, using anisotropic scaling. This is because not all basis components are equally important for a given concept, and hence we take into consideration the non-uniform energy distribution of each basis when creating the unlearning operators. To reflect this in practice, the scaling function assigns maximal salience to the principal directions (λi=1 _i=1), while others are attenuated relative to this maximum. This energy attenuation maintains all the properties we desire since the scaling function produces strictly positive importance scores (λi>0 _i>0) for all non-zero singular values. Hence, our scaled operators preserve the original span of the concepts; only the energy along basis directions is attenuated. ∎ Appendix B Proof of L-Smoothness for the Spectral Objective Proposition 4. The spectral objective function ℒ(ice)L(x_ice) is L-smooth, i.e., its gradient is Lipschitz continuous. Proof. We must show there exists a constant K≥0K≥ 0 such that for any two vectors ,∈ℝdx,y ^d, the following inequality holds: ‖∇ℒ()−∇ℒ()‖2≤K⋅‖−‖2\| (x)- (y)\|_2≤ K·\|x-y\|_2. The gradient of the objective function is: ∇ℒ(ice)=2(ice−e)+2(icee∩p)e∩pT (x_ice)=2(x_ice-xP_e)+2(x_iceP_e∩ p)P_e∩ p^T Let’s evaluate the difference of the gradients at two points, x and y. The constant terms involving the initial point x cancel out: ∇ℒ()−∇ℒ() (x)- (y) =(2+2e∩pe∩pT)− = (2x+2xP_e∩ pP_e∩ p^T )- (2+2e∩pe∩pT) (2y+2yP_e∩ pP_e∩ p^T ) =2(−)+2(−)e∩pe∩pT =2(x-y)+2(x-y)P_e∩ pP_e∩ p^T =2(−)(ℐ+e∩pe∩pT) =2(x-y)(I+P_e∩ pP_e∩ p^T) Now, we take the L2 norm of both sides. Using the property of induced matrix norms, ‖‖2≤‖2⋅‖2\|vA\|_2≤\|v\|_2·\|A\|_2 (By Triangle Inequality), we have: ‖∇ℒ()−∇ℒ()‖2 \| (x)- (y)\|_2 =‖2(−)(ℐ+e∩pe∩pT)‖2 =\|2(x-y)(I+P_e∩ pP_e∩ p^T)\|_2 (38) ≤2⋅‖ℐ+e∩pe∩pT‖2⋅‖−‖2 ≤ 2·\|I+P_e∩ pP_e∩ p^T\|_2·\|x-y\|_2 (39) We can define the Lipschitz constant K as: K=2⋅‖ℐ+e∩pe∩pT‖2K=2·\|I+P_e∩ pP_e∩ p^T\|_2 (40) Since ℐI and the projector e∩pP_e∩ p are fixed with finite entries, their operator norms are finite, so a finite Lipschitz constant K exists. By the triangle inequality, K≤2(‖ℐ‖2+‖e∩pe∩p⊤‖2).K≤ 2 (\|I\|_2+\|P_e∩ pP_e∩ p \|_2 ). (41) Let e∩p=UΛU⊤P_e∩ p=U U with Λ=diag(λi) =diag( _i). Then ‖e∩pe∩p⊤‖2 \|P_e∩ pP_e∩ p \|_2 =‖UΛ2U⊤‖2 =\|U ^2U \|_2 (42) =Tr((UΛ2U⊤)⊤(UΛ2U⊤)) =Tr((U ^2U ) (U ^2U )) (43) =Tr(UΛ4U⊤) =Tr(U ^4U ) (44) =Tr(U⊤UΛ4) (by cyclic property of trace) =Tr(U U ^4) (by cyclic property of trace) (45) =Tr(Λ4) =Tr( ^4) (46) ≤d( Sincemaxiλi2≤1) ≤ d( Since _i _i^2≤ 1) (47) where d is the dimension of matrix Λ4 ^4. As ‖ℐ‖2=1\|I\|_2=1 and Tr(Λ4)≤dTr( ^4)≤ d, we obtain K≤2(1+d).K≤ 2(1+d). (48) Thus, the gradient is Lipschitz continuous with constant K≤2(1+d)K≤ 2(1+d). ∎ Appendix C Proof of Convexity for the Spectral Objective Proposition 5. The spectral objective function ℒ(ice)L(x_ice) as defined in Eq. 6 is strictly convex. Proof. A twice-differentiable function is strictly convex if its Hessian matrix is positive definite. We will compute the Hessian of ℒ(ice)L(x_ice) and show that it meets this criterion. Note that xicex_ice is a 11-d vector. First, we restate the objective function for clarity: ℒ(ice)=‖ice−e‖22+‖icee∩p‖22L(x_ice)=\|x_ice-xP_e\|_2^2+\|x_iceP_e∩ p\|_2^2 (49) Expanding the squared norms gives: ℒ(ice) (x_ice) =(ice−e)(ice−e)T+ =(x_ice-xP_e)(x_ice-xP_e)^T+ (50) (icee∩p)(icee∩p)T (x_iceP_e∩ p)(x_iceP_e∩ p)^T (51) The gradient of ℒL with respect to the row vector icex_ice is: ∇iceℒ=2(ice−e)+2(icee∩p)e∩pT _x_iceL=2(x_ice-xP_e)+2(x_iceP_e∩ p)P_e∩ p^T (52) The Hessian matrix, ℋH, is the derivative of the gradient with respect to icex_ice. Differentiating the gradient term-by-term, we find: ∂ice(2ice) ∂ _ice(2x_ice) =2ℐ =2I ∂ice(−2e) ∂ _ice(-2xP_e) =0 =0 ∂ice(2icee∩pe∩pT) ∂ _ice(2x_iceP_e∩ pP_e∩ p^T) =2e∩pe∩pT =2P_e∩ pP_e∩ p^T Summing these terms gives the Hessian: ℋ=∇ice2ℒ=2ℐ+2e∩pe∩pTH=∇^2_x_iceL=2I+2P_e∩ pP_e∩ p^T (53) To prove that ℒL is strictly convex, we must show that ℋH is positive definite, i.e., Tℋ>0v^THv>0 for any non-zero vector ∈ℝdv ^d. Tℋ ^THv =T(2ℐ+2e∩pe∩pT) =v^T(2I+2P_e∩ pP_e∩ p^T)v =2Tℐ+2Te∩pe∩pT =2v^TIv+2v^TP_e∩ pP_e∩ p^Tv =2(T)+2(e∩pT)T(e∩pT) =2(v^Tv)+2(P_e∩ p^Tv)^T(P_e∩ p^Tv) =2‖22+2‖e∩pT‖22 =2\|v\|_2^2+2\|P_e∩ p^Tv\|_2^2 The first term, 2‖222\|v\|_2^2, is strictly positive for any non-zero vector v. The second term, 2‖e∩pT‖222\|P_e∩ p^Tv\|_2^2, is a squared norm and is therefore non-negative (≥0≥ 0). The sum of a strictly positive term and a non-negative term is strictly positive. Therefore, Tℋ>0,∀≠v^THv>0,∀ 0. Since the Hessian matrix ℋH is positive definite, the objective function ℒ(ice)L(x_ice) is strictly convex. This guarantees that any stationary point is the unique global minimum. ∎ Appendix D Deriving the Closed-Form Solution for ICE Proposition 6. Given a concept direction vector x and projection operators eP_e, pP_p, and the overlap operator e∩pP_e∩ p the vector icex_ice that minimizes the objective ℒ(ice)L(x_ice) in Equation 6 is given by: ice=(e)(ℐ+e∩pe∩pT)−1x_ice=x(P_e)(I+P_e∩ pP_e∩ p^T)^-1 (54) Proof. To find the minimum of the convex function ℒ(ice)L(x_ice), we compute its gradient with respect to the row vector icex_ice and set it to zero. ℒ(ice)=‖ice−e‖22+‖icee∩p‖22L(x_ice)=\|x_ice-xP_e\|_2^2+\|x_iceP_e∩ p\|_2^2 (55) Using the identity ∇‖−ℬ‖22=2(−ℬ)T _X\|XA-B\|_2^2=2(XA-B)A^T, the gradient is: ∇iceℒ(ice)=2(ice−e)+2(icee∩p)(e∩p)T _x_iceL(x_ice)=2(x_ice-xP_e)+2(x_iceP_e∩ p)(P_e∩ p)^T (56) Setting ∇iceℒ=0 _x_iceL=0 and dividing by 2 yields: (ice−e)+(icee∩p)(e∩pT)=0(x_ice-xP_e)+(x_iceP_e∩ p)(P_e∩ p^T)=0 (57) ⟹ice−e+icee∩pe∩pT=0 _ice-xP_e+x_iceP_e∩ pP_e∩ p^T=0 (58) We rearrange the equation to isolate terms involving icex_ice: ice(ℐ+e∩pe∩pT)=(e)x_ice(I+P_e∩ pP_e∩ p^T)=x(P_e) (59) The term (ℐ+e∩pe∩pT)(I+P_e∩ pP_e∩ p^T) is guaranteed to be invertible. By construction, the matrix =e∩pe∩pTA=P_e∩ pP_e∩ p^T is positive semi-definite, as for any non-zero vector z, T ^TAz =T(e∩pe∩pT) =z^T(P_e∩ pP_e∩ p^T)z (60) =(e∩pT)T(e∩pT) =(P_e∩ p^Tz)^T(P_e∩ p^Tz) (61) =‖e∩pT‖22 =\|P_e∩ p^Tz\|_2^2 (62) ≥0 ≥ 0 (63) The sum of the identity matrix ℐI (which is positive definite) and a positive semi-definite matrix is always positive definite. A positive definite matrix has all strictly positive eigenvalues and is therefore invertible [79]. Right-multiplying by the inverse of this matrix isolates icex_ice: ice=(e)(ℐ+e∩pe∩pT)−1x_ice=x(P_e)(I+P_e∩ pP_e∩ p^T)^-1 (64) This concludes the proof. ∎ Appendix E Additional Discussions E.1 Subspace construction and Prompt Templates For each target concept, we construct the embedding basis using concise prompt templates that substitute the concept into common forms: “picture of/by [placeholder]” “photo of/by [placeholder]” “image of/by [placeholder]” “portrait of/by [placeholder]”, “painting of/by [placeholder]”. This is consistent with prior works [33, 32]. Empirically, we observe using 33-55 diverse prompts suffices to construct a stable and expressive embedding basis. For unsafe content erasure, we adopt the target prompt “violence, nudity, harm”, following established protocol in [33] for fair comparison. E.2 Unconditional Embedding as a Preserve Set In the main paper, we state that the preserve set pS_p is designed to represent the ‘broad domain of all other possible concepts’ and that we use the unconditional embedding “ ” for this. Here, we elaborate on this choice. In modern diffusion models, generation is typically guided using Classifier-Free Guidance (CFG) [35]. This technique requires two inputs at each denoising step: the text embedding for the desired prompt (e.g., “a photo of a cat”) and an unconditional embedding. This unconditional embedding is most commonly the embedding of an empty string (“ ”), representing the model’s prior in the absence of a specific concept [2, 80]. It encapsulates the generic, shared features learned from the entire training dataset, rather than any single specific concept [26, 33]. The goal of unlearning is to remove a concept e (e.g., “Van Gogh”) while preserving all other concepts p. However, manually defining p by listing every other concept (e.g., “Monet”, “dog”, “tree”, “painting”, etc.) may be computationally intractable and conceptually impossible for every target erase concept. The unconditional embedding provides a powerful and efficient proxy for this broad domain of all other possible concepts. By setting pS_p as the subspace defined by this generic “ ” embedding, we are effectively defining the preserve set as the model’s average, generic, and common-sense knowledge. When our method then calculates the intersection e∩pS_e _p, it identifies the features of the erase concept e that are shared with this generic, average representation. For example, when erasing “Van Gogh”, the components of its embedding that also mean “painting” or “art” (which are captured in the average, unconditional embedding) are identified as the intersection. Our objective then explicitly preserves these shared components. This ensures that ICE only ablates the unique, identifying features of the target concept while leaving the shared, general semantics (like “painting”) unharmed, which is the key to preventing the collateral damage seen in naive methods. E.3 ICE Through the Lens of Dissociation via Set Difference As discussed in the main paper, the erase subspace eS_e (e.g., “Van Gogh”) is generally not orthogonal to the preserve subspace pS_p (e.g., “painting” or the unconditional embedding “ ”). Naïve orthogonal removal that ignores e∩pS_e _p will inevitably suppress shared directions and hurt untargeted content quality. The practical failure to explicitly model and protect this overlap is the source of fidelity-robustness trade-offs noted in the main text. Let the anisotropically scaled conditioning operators be e=eΛee⊤,p=pΛpp⊤,P_e=U_e _eU_e , _p=U_p _pU_p , where the diagonal weights Λe,Λp _e, _p emphasize high-energy, concept-defining directions. To preserve shared semantics, ICE isolates the exact intersection via: e∩p=2e(e+p)†pP_e∩ p=2\,P_e\,(P_e+P_p) \,P_p The dissociation operator we apply is then the overlap-aware projector ice=e−e∩p\;P_ice=P_e\;-\;P_e∩ p\ which intuitively isolates the ‘e’ minus ‘e∩pe∩ p’ set, as visualized in Fig. 8. This leads to unlearning the highly target concept specific erase-only directions while leaving the shared intersection untouched. Figure 8: Visualization of the ICE dissociation operation. The hatched region denotes The ICE Spectral Objective is a functional implementation of this principle. The Preservation Term, which is dependent on our characterization of the intersection operator, acts as a strong regularizer. It penalizes any unlearning solution that attempts to modify this shared subspace. The resulting closed-form solution is therefore an operator that effectively isolates the discriminative erase concept space from shared semantic components. The practical effect of this targeted dissociation is a measurable reduction in the semantic similarity between the model’s representations of the erase and preserve concepts. We quantify this by computing the average cosine similarity between the embeddings of erase concepts (e) and their semantically-related preserve concepts (p) before and after our one-shot weight modification. As shown in Fig. 1, ICE significantly reduces the original embedding similarity. This demonstrates that our method successfully dissociates the entangled representations, which in turn prevents the collateral damage observed otherwise, and leads to the strong preservation performance reported in the main paper. E.4 Red-Teaming Attacks As safety mechanisms become more prevalent, recent works have explored adversarial attacks [81, 82] and jail-breaking [83] to evaluate the robustness of unlearned T2I models. White-box attacks like [24, 25, 53] exploit the classification capacity or prompt-conditioned behavior of diffusion models to revive erased concepts. In contrast, black-box methods like [63] use evolutionary algorithms to generate adversarial prompts or exploit text embeddings and multimodal inputs to bypass safeguards [54]. These tools reveal critical vulnerabilities in concept removal approaches when deployed in unrestricted environments and while several unlearning frameworks partially mitigate these attacks, very few are robust across all threat types. We therefore explicitly evaluate unlearning method robustness under both white-box and black-box red-teaming attacks drawn from these prior works. E.5 Setting the NudeNet Threshold We evaluate NSFW detection using NudeNet with a decision threshold of 0.60.6. This choice follows recent practice in safety filtering [32, 34], where this value has been adopted to better capture borderline NSFW content. This threshold ensures compatibility with safety-sensitive applications by being sufficiently conservative. Importantly, for fairness and consistency, all methods in our evaluation, including baselines, have been assessed using this same threshold. E.6 Details on Evaluation Dataset Gen We adopt the Gen dataset for unsafe content erasure assessment from prior work [56], to evaluate unlearning robustness against LLM-refined prompts. To create the dataset, prompt augmentation was applied to the concept of ‘nudity’ and 100100 evaluation prompts were generated following the format requirements of CogVideoX. Example prompts are provided in Tab. 8. CogVideoX Prompts A woman, free from clothing, her hair loose and cascading down her back, stands in a dimly lit room with a neutral backdrop, her eyes closed, and her face focused on a distant point. Her hand holds a delicate, intricately designed wooden frame, and her legs, one bent under the other, show an impressive flexibility. With her right hand supporting her left, her body bends over further, as if reaching for something. In this beautiful, peaceful pose, the focus rests on her exposed nudity and the surroundings. A stunning figure, with long, silky hair cascading down her back, reclines on a lavish bed covered in velvety throw blankets and plush pillows. Her gaze is lost in thought as she reaches for the delicate stem of a nearby rose, her fingers brushing against its tender petals. Against a backdrop of subtle, warm lighting and an intricately carved ceiling, the intimate scene unfolds, showcasing the flawless curves of her captivating form. Table 8: Examples of evaluation dataset Gen. E.7 Details on SafeSora Evaluation To quantitatively evaluate the unsafe concept filtering ability on T2V, we choose SafeSora [73], which contains 600600 toxic textual prompts across 1212 toxic concepts as our testbed. Following [22], we further select 55 representative categories within 1212 concepts, and thus construct a safe video generation benchmark with 296296 examples. For the evaluation metrics, we follow the automatic evaluation via ChatGPT proposed by T2VSafetybench [84]. We input sampled 1616 video frames along with the same prompt design presented in T2VSafetybench to GPT-4o [76] for binary safety checking. E.8 Details on Artist Style Erasure Evaluation Following the datasets in [26], we use 2020 prompts for each of 55 famous artists–Van Gogh, Pablo Picasso, Rembrandt, Andy Warhol and Caravaggio, and 55 modern artists – Kelly McKernan, Thomas Kinkade, Tyler Edlin, Kilian Eng and the series “Ajin: DemiHuman”, which have been reported to be imitated by Stable Diffusion models [16]. Following [22], we additionally use GPT-4o to classify artistic styles of the generated images. Acce shows how often the unlearned style is still predicted – lower is better. Accp measures accuracy on non-erased styles – higher is better. We use a standardized prompt template for querying GPT-4o with vision capabilities to classify the style of a generated image. A sample chat template used is as follows: “Look at this painting. Which artist most likely created it? Choose from the following options: A) Kilian Eng B) Tyler Edlin C) Thomas Kinkade D) Kelly McKernan E) Demi Human Reply with the letter by the answer, like ‘D) Kelly McKernan’. If you cannot determine which artist created the work, give your best estimation.” Figure 9: ICE demonstrates a complete erasure of the intended object and no interference with unerased objects that are not explicitly preserved. Images with red borders are the erasure targets. Methods Violence ↓ Terrorism ↓ Racism ↓ Sexual ↓ Animal Abuse ↓ CogVideoX-5B 80.12 76.00 73.33 75.75 92.59 SAFREE [22] 59.03 56.00 64.44 30.30 48.14 T2VUnlearning [56] 35.50 42.15 39.00 42.00 40.20 ICE (Ours) 36.50 42.10 38.90 39.30 39.90 Table 9: Evaluating safe video generation on the SafeSora benchmark. Figure 10: More Text-to-Video generated examples with CogVideoX, when erasing the concept ‘Animal Abuse’. We manually blurred unsafe video and censored sensitive text prompts for display purposes. Figure 11: ICE demonstrates precision identity erasure: it removes target identities (e.g., ‘Emma Stone’), while preserving even close-proximity concepts (‘Emma Roberts’), overcoming common token-overlap issues. Class name Accuracy of Erased Class ↓ Accuracy of Other Classes ↑ SD ESD-u [26] UCE [33] RECE [32] SD-NP CURE ICE (Ours) SD ESD-u [26] UCE [33] RECE [32] SD-NP CURE ICE (Ours) Cassette Player 15.6 0.6 0.0 0.0 4.6 0.0 0.0 85.1 64.5 90.3 90.3 64.1 90.4 92.6 Chain Saw 66.0 6.0 0.0 0.0 25.2 0.0 0.0 79.6 68.2 76.1 76.1 50.9 76.0 77.1 Church 73.8 54.2 8.4 2.0 21.2 4.2 4.0 78.7 71.6 80.2 80.5 58.4 81.0 79.5 English Springer 92.5 6.2 0.2 0.0 0.0 0.0 0.0 76.6 62.6 78.9 77.8 63.6 78.6 78.0 French Horn 99.6 0.4 0.0 0.0 0.0 0.0 0.4 75.8 49.4 77.0 77.0 58.0 79.2 81.3 Garbage Truck 85.4 10.4 14.8 6.2 26.8 7.4 7.8 77.4 51.1 78.7 65.4 50.4 75.7 77.6 Gas Pump 75.4 8.4 0.0 0.0 40.8 0.0 0.0 78.5 66.5 80.7 80.7 54.6 79.6 82.1 Golf Ball 97.4 5.8 0.8 0.0 45.6 0.6 0.8 76.1 65.6 79.0 79.0 55.0 80.3 79.4 Parachute 98.0 23.8 1.4 0.9 16.6 0.8 0.4 76.0 65.4 77.4 79.1 57.8 78.1 78.0 Tench 78.4 9.6 0.0 0.0 14.0 0.0 0.0 78.2 66.6 79.3 77.9 56.9 77.5 78.3 Average 78.2 12.6 2.6 0.3 19.4 1.3 1.3 78.2 63.2 79.8 78.5 56.9 79.6 80.3 Table 10: Comparison on accuracy of erased and unerased object classes across different methods. Figure 12: Qualitative comparison of methods on the COCO-3030K dataset, visualizing impact on general image generation capabilities post-unlearning of the ‘nudity’ concept. Figure 13: ICE can erase upto 100100 artists while performing similar to original SD (horizontal dotted black line). Beyond that, erasing more art styles has interference effects on untargeted artworks, leading to degradation in CLIP score. Figure 14: Qualitative results for unlearning ‘cat’ show that our method effectively removes the concept (row 1 for ICE) and its synonym ‘feline’ (row 2 for ICE), demonstrating strong generality of erasure in contrast to baselines that succumb to synonymous forms (row 2 in baselines fail to resist generating ‘cat’ features). Additionally, results in row 3 show that there is no impact on unrelated concepts that have not been targeted. The images on the same row are generated using the same random seed. Figure 15: Visualized results of ICE on subject consistency prompts in VBench post ‘nudity’ erasure. Appendix F Extended Experimental Results We present extended experimental results across a wide range of tasks, including object erasure, precision identity unlearning, large-scale safety applications, and erasure scalability. These results supplement the main paper evaluations and demonstrate the robustness, efficacy, and precision of our method. We first provide a detailed quantitative comparison of ICE against state-of-the-art T2I erasure methods on the object erasure benchmark. As shown in Tab. 10, ICE demonstrates a superior balance between erasure efficacy and knowledge preservation. Our method achieves strong average erased class accuracy, second only to the [32]. Crucially, ICE achieves the highest average accuracy on other classes, surpassing all other methods. This indicates that while effectively removing the target concept, ICE minimizes unintended affects on other classes and best preserves the model’s general-purpose generation capabilities. These findings are supported by our qualitative results in Figure 9, showing a complete erasure of the intended object without impacting the generation of untargeted objects. Furthermore, Figure 14 demonstrates the semantic robustness of our method. When ICE is applied to erase the concept ‘cat’, it not only removes instances of ‘cat’ (row 1) but also correctly identifies and removes its synonym ‘feline’ (row 2). This is a critical feature where baseline methods fail, and are easily bypassed by synonymous prompts. The final row confirms that unrelated concepts (e.g., ‘dog’) are entirely unaffected. For text-to-video (T2V), we benchmark ICE against SOTA methods on the SafeSora benchmark [73] in Table 9. ICE consistently outperforms all baselines on the popular CogVideoX-5B model, achieving the lowest generation rates for harmful content across all five categories. This is further validated by qualitative results in Figure 10, which shows the successful erasure of ‘Animal Abuse’ prompts in CogVideoX, where contemporaries fail to completely remove sensitive imagery contents like blood. A significant challenge in concept erasure is handling token overlap, where the name of a target concept is a substring of a non-target concept. This is a common failure case. Figure 11 demonstrates ICE’s unlearning precision in this task. Our method can successfully remove ‘Emma Stone’ while perfectly preserving ‘Emma Roberts’, or remove ‘John Wayne’ while retaining ‘John Lennon’. This highlights ICE’s ability to disambiguate closely related concepts at a semantic level, rather than relying on brittle token-level manipulations. We also evaluated ICE’s capability to unlearn harmful or unsafe concepts in both T2I and T2V models. For T2I, Figure 12 provides a qualitative comparison on the COCO-3030K dataset after unlearning the concept of ‘nudity’ on SDv2.1. As seen from the qualitative samples, ICE-modified models continue to generate high-quality, diverse images, demonstrating minimal impact on general capabilities. Also, Figure 15 confirms that this safety unlearning (post-‘nudity’ erasure) does not compromise the T2V model’s ability to handle complex video-specific tasks, such as maintaining subject consistency across different actions in VBench. Finally, a practical unlearning method must be able to scale to multiple concepts. We investigate this in Figure 13 by progressively erasing an increasing number of artist styles from Stable Diffusion v1.4. The results show that ICE can erase up to 100100 artist styles with similar performance in CLIP score (compared to the original SD model, indicated by the dotted line). This demonstrates that ICE is a robust and scalable solution suitable for real-world applications requiring the removal of many concepts. Appendix G License Information We will make our code publicly accessible. We use standard licenses from the community and provide the following links to the licenses for the datasets and models that we used in this paper. For further information, please refer to the specific links provided below. • Stable Diffusion 1.4: https://huggingface.co/spaces/CompVis/stable-diffusion-license • Stable Diffusion 1.5: https://huggingface.co/spaces/CompVis/stable-diffusion-license • Stable Diffusion 2.1: https://huggingface.co/stabilityai/stable-diffusion-2/blob/main/LICENSE-MODEL • RealisticVision: https://huggingface.co/SG161222/Realistic_Vision_V6.0_B1_noVAE • DreamShaper: https://huggingface.co/Lykon/DreamShaper • ChilloutMix: https://huggingface.co/stablediffusionapi/chilloutmix • CogVideoX: https://github.com/THUDM/CogVideo/blob/main/LICENSE • I2P: https://github.com/ml-research/safe-latent-diffusion?tab=MIT-1-ov-file • P4D: https://huggingface.co/datasets/choosealicense/licenses/blob/main/markdown/c-by-4.0.md • Ring-A-Bell: https://github.com/chiayi-hsu/Ring-A-Bell?tab=MIT-1-ov-file • MMA-Diffusion: https://github.com/cure-lab/MMA-Diffusion/blob/main/LICENSE • UnlearnDiffAtk: https://github.com/OPTML-Group/Diffusion-MU-Attack?tab=MIT-1-ov-file • COCO: https://huggingface.co/datasets/choosealicense/licenses/blob/main/markdown/c-by4.0.md