Selective Suppression with Delta Vector (SSDV)

• Selective Suppression with Delta Vector (SSDV) is a framework that uses targeted delta vectors to attenuate unwanted vector components while preserving core structure across multiple domains.
• In text-to-image diffusion, SSDV modifies token embeddings via calculated delta vectors to suppress persistent undesired features and enhance output quality.
• In fluid dynamics and vector–tensor gravity, SSDV improves numerical stability and localizes problematic modes, leading to significant error reduction and reliable physical modeling.

Selective Suppression with Delta Vector (SSDV) refers to a set of mathematically principled mechanisms for selectively attenuating particular vector components or features—be they latent representations, field perturbations, or interface forces—so as to suppress undesired behavior while preserving main or background structure. The SSDV framework appears in three distinct scientific domains: controllable text-to-image generative modeling, fluid dynamics with immersed boundaries, and cosmological vector–tensor gravity theories. In each context, selective suppression is achieved by constructing and strategically applying carefully defined delta vectors, which act as agents of targeted normalization, localization, or screening.

1. Mathematical Formulation in Text-to-Image Diffusion

In diffusion-based text-to-image synthesis, such as CLIP-guided latent diffusion, certain prompt tokens have embeddings strongly entangled with specific semantic or visual content, resulting in persistent unwanted features (e.g., “mustache” co-occurring with “Charlie Chaplin”). SSDV (Koh et al., 14 Aug 2025) introduces a principled translation of text embeddings to modulate the model’s inductive bias toward such content.

Given a prompt $P$ with token embeddings $$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$, let $e_w$ denote the embedding(s) of a word $w$ whose associated content is to be modified, and $C$ the negative concept. The SSDV constructs a delta vector

$$\delta_0 = \mathrm{MeanPool}(\xi(\tau(C))) \in \mathbb{R}^{1 \times d}$$

where $\xi$ is the text encoder. This delta is broadcast as $\delta \in \mathbb{R}^{n_w \times d}$. The word embedding is translated:

$e_w^* = e_w + \alpha \delta$

with $\alpha < 0$ for suppression. Cross-attention blocks are modified by introducing shifted tokens into both keys and values:

• Key side: $$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$0 localizes the negative concept via attention heat-map.
• Value side: $$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$1 suppresses the value’s contribution where $$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$2 is likely.

The final cross-attention operation becomes:

$$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$3

Zero-shot SSDV requires only the CLIP embedding of $$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$4, with no additional training; for highly personalized models, $$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$5 can be optimized via a two-term loss focusing on latent reconstruction (local to a mask $$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$6) and attention localization.

2. SSDV in Fluid Interfaces: Discrete Delta Vector in Immersed Boundary Methods

In computational fluid dynamics, the immersed boundary method (IBM) uses a delta function to interpolate forces from a moving interface onto a fixed Eulerian mesh. Traditional approaches use a scalar kernel, which fails to preserve irrotationality and causes artificial "parasitic currents" near high-density-ratio interfaces.

The SSDV scheme (Ong et al., 2018) reconstructs the discrete delta vector as the gradient of a discrete Heaviside function $$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$7:

$$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$8

For a finite-volume grid, this reduces to

$$\hat e = [e_0, \dots, e_{M}] \in \mathbb{R}^{M \times d}$$9

$e_w$0

with $e_w$1 encoding local surface-tension forces. This construction enforces $e_w$2 in the discrete sense, eliminating spurious rotational modes and restoring numerical stability across moving boundaries. Markers are redistributed via B-spline fitting to suppress high-frequency curvature noise.

3. Vector–Tensor Gravity: Radial Mode Suppression via Vainshtein Mechanism

In nonlinear gravity, generalized Proca theories introduce massive vector fields with higher-order derivatives and direct matter coupling. Here, SSDV describes the selective suppression of the problematic longitudinal ("delta vector") mode $e_w$3, while preserving the temporal component $e_w$4, which remains cosmologically relevant (Nakamura et al., 2017). The screening results from nonlinearities (Galileon terms) in the action:

$e_w$5

Static spherically symmetric solutions show the radial component $e_w$6 vanishes quickly inside the Vainshtein radius $e_w$7:

$e_w$8

for strong coupling, rendering fifth-force effects locally negligible, while $e_w$9 is free to influence cosmology.

4. Algorithmic and Implementation Details

In T2I diffusion (SSDV (Koh et al., 14 Aug 2025)), the main workflow is:

1. Extract $w$0 as the mean CLIP embedding of $w$1.
2. Tokenize $w$2, identify $w$3.
3. Optionally optimize $w$4 for personalized models (using losses on denoised latents and attention maps).
4. Insert shifted $w$5 and $w$6 into attention inputs.
5. Run the denoising loop using the modified attention; for finer control, use binary masks derived from attention map differences.

In IBM for fluids (Ong et al., 2018), SSDV replaces kernel products with discrete gradients of $w$7 for face-centered force interpolation, followed by standard fractional-step projection; B-spline fitting redistributes markers each timestep.

5. Experimental and Empirical Validation

Diffusion models (Koh et al., 14 Aug 2025):

• On the SEP-Benchmark (10 prompt-content pairs, 100 seeds), SSDV outperforms negative-prompt, P2P, SEGA, SuppressEOT, and Inst-Inpaint on all main metrics. CLIP scores drop from 17.6 to 15.9, DetScore from 0.314 to 0.113, user preference up to 86%. In DreamBooth personalized models, zero-shot SSDV beats baselines in 5/6 settings, and optimized-$w$8 achieves 73% preference.

Immersed Boundary (Ong et al., 2018):

• SSDV achieves an order-of-magnitude reduction in $w$9 error for 2D droplet pressure jump (from $C$0 to $C$1 at $C$2), eliminates parasitic circulation, stabilizes long-term integration, and matches analytic periods and terminal velocities to within 1%.

Gravity (Nakamura et al., 2017):

• For $C$3, the suppression $C$4 satisfies Solar System constraints for coupling $C$5 and field values $C$6, with post-Newtonian corrections within experimental bounds.

6. Application Scope, Limitations, and Extensions

Text-to-Image (T2I): SSDV is lightweight, zero-shot deployable, and optionally optimizable. It requires explicit specification of $C$7; automatic extraction of entangled sub-concepts remains open. Current local blending heuristics impose faint residuals; more rigorous joint image–text blending is proposed. Extension to multiple negatives or softer attributes is deferred.

Immersed Boundary: SSDV applies when irrotational force transfer is critical, especially at high density ratios. The main limitation is increased computational overhead due to B-spline and Heaviside computation.

Gravity: SSDV is effective only in the presence of Galileon-type non-linearities; certain higher-order (quintic) interactions lack regular solutions for the longitudinal mode. Cosmological and black hole implications derive from the unsuppressed $C$8, which remains subject to observational constraints.

7. Cross-Disciplinary Perspective

While the term “Selective Suppression with Delta Vector” refers to distinct mathematical instantiations across fields, its core motif—surgical attenuation of a well-defined undesired vector mode without globally degrading the system—demonstrates notable convergent evolution in machine learning, computational physics, and theory of gravity. Each application leverages the interplay between local and global structure through competitive normalization, geometrically-motivated transformations, or non-linear field equations to achieve precise suppression while retaining essential semantic, physical, or dynamical character.