Diffusion Crossover: Defining Evolutionary Recombination in Diffusion Models via Noise Sequence Interpolation

Abstract: Interactive Evolutionary Computation (IEC) provides a powerful framework for optimizing subjective criteria such as human preferences and aesthetics, yet it suffers from a fundamental limitation: in high-dimensional generative representations, defining crossover in a semantically consistent manner is difficult, often leading to a mutation-dominated search. In this work, we explicitly define crossover in diffusion models. We propose Diffusion crossover, which formulates evolutionary recombination as step-wise interpolation of noise sequences in the reverse process of Denoising Diffusion Probabilistic Models (DDPMs). By applying spherical linear interpolation (Slerp) to the noise sequences associated with selected parent images, the proposed method generates offspring that inherit characteristics from both parents while preserving the geometric structure of the diffusion process. Furthermore, controlling the time-step range of interpolation enables a principled trade-off between diversity (exploration) and convergence (exploitation). Experimental results using PCA analysis and perceptual similarity metrics (LPIPS) demonstrate that Diffusion crossover produces perceptually smooth and semantically consistent transitions between parent images. Qualitative interactive evolution experiments further confirm that the proposed method effectively supports human-in-the-loop image exploration. These findings suggest a new perspective: diffusion models are not only powerful generators, but also structured evolutionary search spaces in which recombination can be explicitly defined and controlled.

• The paper introduces Diffusion Crossover, a recombination method that uses step-wise spherical interpolation of DDPM noise sequences to generate offspring from two selected image parents.
• Experiments on MNIST and ModelNet40 show that interpolation produces perceptually continuous parent-to-parent transitions, with the mixing parameter controlling inheritance and statistically significant LPIPS trends in most trials.
• The method makes diffusion search more evolutionary by using interpolation duration to balance exploration and exploitation, although high sampling latency, limited user testing, and untested latent-diffusion extensions remain important constraints.

Motivation and problem statement

Interactive Evolutionary Computation (IEC) optimizes subjective criteria—user preferences, aesthetics—that resist explicit fitness functions, but its effectiveness in image domains is constrained by the difficulty of defining a semantically meaningful crossover operator in high-dimensional generative representations. Prior IGA approaches built on GANs or VAEs treat latent vectors as genotypes and realize crossover via vector interpolation or arithmetic, but GAN training instability and autoencoder synthesis quality limit their utility. When applied to diffusion models, existing recombination schemes operate only on static inputs: prompt/seed mixing [Kobayashi et al.], latent-space Voronoi partitioning requiring user-specified regions, or pixel-space blending aimed at prompt optimization. None of these intervene in the generative trajectory itself, so diffusion-based evolutionary search remains effectively mutation-dominated.

This paper addresses that gap by proposing Diffusion crossover, which defines recombination as step-wise spherical linear interpolation (Slerp) of the noise sequences injected during the reverse process of a DDPM. The work builds on the theoretical observation of Zhang et al. that mutation, selection, and recombination are embedded in diffusion dynamics, extending that correspondence from numerical optimization to human-in-the-loop image exploration.

Method

The genotype is defined as the initial noise $\bm{x}_T$ together with the sequence $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$ of Gaussian noise vectors added after each denoising step—not the noise predicted by the U-Net $\epsilon_\theta$. Given two user-selected parents with sequences $\boldsymbol{Z}^{(A)}$ and $\boldsymbol{Z}^{(B)}$, offspring are generated by Slerp between the parent noise vectors at each time step:

$$\boldsymbol{z}^{(\lambda)}_t = \frac{\sin((1-\lambda)\theta)}{\sin\theta}\boldsymbol{z}^{(A)}_t + \frac{\sin(\lambda\theta)}{\sin\theta}\boldsymbol{z}^{(B)}_t$$

with $\theta$ the angle between the flattened vectors and $\lambda \sim \mathcal{U}(0,1)$ per offspring. The initial noises $\bm{x}_T^{(A)}$, $\bm{x}_T^{(B)}$ are also Slerp-interpolated. Interpolation is applied only for the first $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$0 steps of the reverse process; beyond that, fresh Gaussian noise is sampled, playing the role of mutation. Because early reverse-diffusion steps determine global structure while later steps refine detail, increasing $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$1 over generations shifts the search from exploration toward exploitation, providing a principled diversity–convergence knob grounded in the coarse-to-fine structure of DDPM sampling.

The choice of Slerp over naive spatial splicing is motivated by the authors' observation that noise coordinates do not map locally to image structure; global operations on the full noise vector are required for semantic integration.

Experimental design

Two single-class datasets were used: MNIST digit "5" ($Z = \{\bm{z}_1, \dots, \bm{z}_T\}$2, 5,421 training images) and ModelNet40 sofas rendered at $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$3 (600 training images). Class-specific DDPMs with residual-block-only U-Nets were trained for 3,000 epochs (AdamW, Optuna-tuned hyperparameters), with checkpoint selection by lowest CMMD. Four experiments were conducted: (1) PCA analysis of intermediate images along standard vs. interpolated trajectories; (2) LPIPS-based verification of perceptual continuity as $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$4 varies from 0.1 to 0.9 at fixed $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$5; (3) effect of $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$6 on diversity measured by average pairwise LPIPS at fixed $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$7; and (4) a qualitative interactive evolution run targeting a "thick bold 5" (MNIST) and a "cushioned L-shaped sofa" (ModelNet), with the authors acting as users.

Results

Trajectory geometry (Experiment 1). PCA projection shows that images generated from interpolated noise lie at intermediate positions between the two parent trajectories at every time step, varying continuously with $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$8. This indicates that noise-space interpolation preserves the geometric structure of the diffusion process rather than producing out-of-distribution artifacts—a necessary condition for crossover to be well-defined in this genotype space.

Perceptual continuity (Experiment 2). LPIPS distance to parent A increases monotonically with $Z = \{\bm{z}_1, \dots, \bm{z}_T\}$9 while distance to parent B decreases, validated by Spearman rank correlations significant at $\epsilon_\theta$0 in nearly all runs—for example, $\epsilon_\theta$1 (Ref. A) and $\epsilon_\theta$2 (Ref. B) across most ModelNet runs. The sole exception is Run 1 on MNIST ($\epsilon_\theta$3 / $\epsilon_\theta$4, not significant), which the authors attribute to the domain gap between handwritten digits and LPIPS's natural-image training distribution. Offspring at intermediate $\epsilon_\theta$5 do not collapse and preserve object-level geometric consistency (e.g., sofa structure). The practical implication is that $\epsilon_\theta$6 acts as a predictable semantic mixing ratio, giving the user continuous control over inheritance from each parent.

Diversity control (Experiment 3). Diversity (average pairwise LPIPS among offspring) decreases monotonically as $\epsilon_\theta$7 increases, with significant negative correlations in all MNIST runs ($\epsilon_\theta$8) and all but one ModelNet run ($\epsilon_\theta$9). This confirms that the interpolation duration operationalizes the exploration–exploitation trade-off: shorter interpolated segments leave more stochastic noise injection (mutation), yielding more diverse populations.

Interactive feasibility (Experiment 4). In qualitative runs, populations converged from high-diversity initial generations to the target concepts by Gen 6 (MNIST) and Gen 7 (ModelNet). Global geometric features were robustly inherited, but fine-grained textures such as cushioning details showed greater variability—an expected consequence of interpolating during early denoising steps, which predominantly shape global structure.

Limitations and open questions

The paper is candid about several constraints. The interactive evaluation was a feasibility study with the authors as users; no third-party user study or statistical measures of satisfaction and convergence speed were performed. Sampling latency is substantial—approximately 40–50 seconds per population due to $\boldsymbol{Z}^{(A)}$0 DDPM steps plus the overhead of storing and interpolating full-resolution noise sequences—which challenges real-time interaction; the authors argue the method is compatible with accelerated samplers and Latent Diffusion Models, where compressed latent spaces would reduce noise-sequence dimensionality, but neither extension was tested. The claim that spatial splicing would fail rests on indirect evidence (e.g., flipped L-shapes appearing under related operations) rather than an explicit ablation. Additionally, model selection via CMMD conflates generation fidelity with IEC-relevant properties such as navigability and search efficiency; hyperparameter optimization accounting for these factors remains open. Finally, fine-grained control over localized features under early-step interpolation is unresolved.

Conclusion

This paper supplies what prior diffusion-based evolutionary image generation lacked: an explicitly defined crossover operator grounded in the generative process itself. By treating the stochastic noise sequence of DDPM sampling as the genotype and applying norm-preserving Slerp across its time steps, Diffusion crossover yields offspring whose perceptual distance to each parent varies monotonically and predictably with $\boldsymbol{Z}^{(A)}$1, while the interpolation duration provides a statistically validated control over population diversity. The empirical support comes from small, single-class datasets and a self-conducted qualitative study, so claims of general applicability—particularly to latent diffusion models and real-time interaction—remain to be substantiated. Within those bounds, the result establishes diffusion models as structured evolutionary search spaces in which both crossover and mutation can be separately specified and controlled.