Spherical Linear Interpolation (SLERP)

• SLERP is a geometric interpolation method that defines the shortest path between two points on a hypersphere while preserving constant norm and statistical consistency.
• It uses a trigonometric formulation to compute interpolated points that remain on-manifold, avoiding the unrealistic artifacts seen in linear approaches.
• SLERP is widely applied in generative modeling, diffusion processes, and quaternion rotations, ensuring smooth transformations and physically meaningful outputs.

Spherical Linear Interpolation (SLERP) is a fundamental geometric technique for interpolating between two points on the surface of a hypersphere, most commonly used in high-dimensional feature, latent, or rotation spaces encountered across a range of computational, geometric, and generative modeling applications in machine learning, computer vision, molecular simulation, and biometrics. SLERP produces interpolated points that respect both the original manifold geometry and the probabilistic properties of the distributions commonly used in deep learning latent spaces, notably in the context of priors that concentrate on spheres (e.g., high-dimensional Gaussian distributions).

1. Mathematical Formulation and Geometric Properties

SLERP operates between two points (vectors) on the hypersphere in a way that traces the shortest path (a geodesic) between them, preserving constant speed and norm throughout interpolation. The standard form, given two non-collinear vectors $z_1, z_2 \in \mathbb{R}^d$ and interpolation coefficient $\alpha \in [0,1]$, is:

$$z_{\text{interpolated}}(\alpha) = \frac{\sin((1-\alpha)\omega)}{\sin \omega} z_1 + \frac{\sin(\alpha \omega)}{\sin \omega} z_2$$

where

$$\omega = \arccos\left( \frac{z_1 \cdot z_2}{\|z_1\| \|z_2\|} \right )$$

At $\alpha=0$, SLERP returns $z_1$; at $\alpha=1$, $z_2$; and for $0 < \alpha < 1$, the formula traces the geodesic on the sphere's surface between the two points. This contrasts with linear interpolation (LERP), which forms a straight chord between endpoints and generally does not remain on the hypersphere, and may result in off-manifold, low-probability, or semantically unrealistic intermediate representations in high-dimensional contexts (White, 2016, Gaikwad et al., 3 Aug 2025).

2. Rationale and Statistical Advantages

High-dimensional latent spaces in generative models such as VAEs, GANs, and diffusion models are often defined with Gaussian or uniform priors, causing their samples to cluster on (or near) the surface of a high-dimensional hypersphere. Linear interpolation between such latent samples traverses the sphere's interior, producing intermediate vectors with lower norm, which are highly improbable under the data distribution and can yield degraded, blurry, or unrealistic decoded instances (White, 2016, Bodin et al., 16 Aug 2024). SLERP circumvents this by maintaining the norm, thus keeping interpolated points within the region of maximal prior density on the manifold. When the prior is strictly isotropic Gaussian or the latent vectors are normalized, this alignment becomes particularly critical for downstream fidelity of generative sampling.

3. Domain-Specific Implementations

A. Generative Models and Latent Space Manipulation

In VAEs, GANs, and diffusion-based generative models, SLERP provides a principled approach for morphing between samples or conditions by interpolating in latent space (White, 2016, Gaikwad et al., 3 Aug 2025, Zheng et al., 13 Mar 2024). For instance, in CVAE-driven microstructure prediction, SLERP is applied between latent representations corresponding to images with smallest and largest morphological feature sizes at a fixed alloy composition, generating smooth trajectories that emulate temporal microstructural coarsening (cf. phase-field models). Decoding these interpolated latents produces plausible microstructural evolutions matching physical expectations in both visual appearance and quantitative structure metrics (Gaikwad et al., 3 Aug 2025). Analogously, SLERP enables smooth pose or attribute transitions in face synthesis and attribute traversal tasks.

B. Diffusion Model Interpolation

SLERP is widely adopted for interpolating images within diffusion models. Images are encoded into noise (latent) space, interpolated with SLERP, and then decoded by the reverse process. This method ensures traversing the hypersphere consistent with the model's prior expectations (Zheng et al., 13 Mar 2024). However, applying SLERP directly to natural images that do not arise from the model’s own generative process can introduce artifacts due to invalid noise encoding, necessitating corrections such as those in NoiseDiffusion, which employs noise space corrections and controls to restore statistical alignment (Zheng et al., 13 Mar 2024). This suggests the necessity of noise-level and distributional matching for SLERP’s effectiveness in generative interpolations.

C. Embedding and Multimodal Retrieval

In composed image retrieval tasks, SLERP is used to combine image and text embeddings residing on a unit hypersphere, yielding an intermediate point representing a semantic blend of the modalities. This approach, in tandem with strategies such as Text-Anchored-Tuning for modality alignment, outperforms token-projection or naive linear combinations by keeping the composed embedding on-manifold and in direct correspondence with the structure of pre-trained joint embedding spaces (e.g., CLIP, BLIP) (Jang et al., 1 May 2024).

D. Rotational and Geometric Applications

Originally introduced for quaternion interpolation in 3D graphics, SLERP interpolates between rotational orientations by following the shortest path on the 3-sphere ($\mathbb{S}^3$) of unit quaternions. In molecular simulation, hybrid approaches combine linear interpolation for cell parameters with SLERP for molecular or fragment orientations, enabling the generation of physically realistic and collision-free inter-structural transition paths (e.g., for Nudged Elastic Band calculations of polymorph transitions and solid-state reactions) (Goncharova et al., 14 Oct 2024). SLERP preserves bond geometry and fragments’ physical properties, and ensures initial pathways are viable for subsequent energy minimization without manual correction.

4. Limitations and Generalizations

While SLERP adequately solves norm preservation and path smoothness for two-point, zero-mean, unit-covariance Gaussian cases, its applicability diminishes in settings with non-zero mean, non-identity covariance, or linear combinations involving more than two vectors. Recent research has introduced Latent Optimal Linear combinations (LOL, also denoted COG) as a generalization that allows arbitrary affine combinations while ensuring the output matches the desired Gaussian statistics, subsuming SLERP as a special case (Bodin et al., 16 Aug 2024). LOL/COG maintains full distributional alignment regardless of the combination weights or seed count, which is crucial for centroid formation, subspace projections, or composite attribute manipulations in generative models.

5. Empirical Assessment and Benchmarking

SLERP’s superiority over LERP and cubic interpolation is consistently validated experimentally:

• In latent space interpolation for generative models, SLERP generates sharper and more realistic outputs, with consistent visual quality and physical plausibility, avoiding the "tent-pole" effect wherein LERP midpoints are far from the prior shell (White, 2016, Gaikwad et al., 3 Aug 2025).
• In microstructure evolution emulation, SLERP-derived sequences from CVAE-latent space match both the qualitative morphology and quantitative progression (feature areas, autocorrelation) of ground-truth simulations, whereas LERP and cubic alternatives fail to preserve such smoothness and regularity (Gaikwad et al., 3 Aug 2025).
• In NEB calculations, SLERP ensures physically valid, collision-free, and automatically generated initial paths, improving convergence rates and obviating manual intervention in structure preparation; performance is robust compared to linear-only interpolation, which regularly yields infeasible candidates (Goncharova et al., 14 Oct 2024).
• In composed retrieval and multimodal embedding, SLERP, combined with textual or visual modality tuning, delivers state-of-the-art performance with minimal training overhead, highlighting its efficiency in high-level semantic tasks (Jang et al., 1 May 2024).

6. Summary Table: SLERP and Related Interpolation Methods

Method	Manifold/Distribution Guarantee	Composite Support	Common Use Cases
LERP	No (straight line, off-manifold)	Arbitrary	Rarely justified, may cause artifacts
Cubic Spline	No (off-manifold, high-order polynomials)	Label/interp. only	Label interpolation, not latent
SLERP	Yes (on hypersphere, 2-point)	Pairwise	Pairwise interpolation, norm matching
LOL/COG	Yes (full Gaussian moment/covariance matching)	Arbitrary	General affine combination of latents

7. Practical Implications and Future Directions

SLERP remains essential for problems demanding smooth, physically meaningful, and statistically valid interpolation between high-dimensional vectors that inhabit a spherical manifold—ranging from deep generative modeling to geometric simulation and representation fusion. However, for complex operations beyond simple pairwise morphing or in cases involving non-standard priors or latent distributions, methods like LOL/COG are now advocated (Bodin et al., 16 Aug 2024). Ongoing research is addressing SLERP’s extension to more general distributions, its combination with domain-specific correction schemes (e.g., in diffusion-based image interpolation), and its integration into dynamic and multi-agent geometric systems (e.g., via Kuramoto-model-based interpolation on $\mathbb{S}^3$ (Kapić et al., 2021)).

SLERP’s efficacy, broad applicability, and robust theoretical foundations establish it as a central algorithmic primitive in modern scientific and machine learning workflows. Its principled respect for latent or embedding space geometry justifies its default usage absent explicit deviations from the assumptions under which it is asymptotically optimal.