SLERP-Based Quaternion Noise Scheduler

• The paper demonstrates that using SLERP interpolation along unit-quaternion geodesics preserves rotational integrity by maintaining the unit-norm constraint.
• It contrasts additive and exponential-map SLERP formulations, highlighting numerical stability near small and π angles to avoid drift and NaNs.
• The scheduler integrates into flow matching and diffusion models, achieving significant speedup and improved accuracy in protein generation and robotic manipulation.

Searching arXiv for the cited papers and closely related work on quaternion/rotation scheduling and rectified flow. arXiv search query: id:([2502.14637](/papers/2502.14637)) OR id:([2509.19696](/papers/2509.19696)) OR "Rectified Quaternion Flow" OR "SLERP-based quaternion noise scheduler" A SLERP-based quaternion noise scheduler is a mechanism for injecting, parameterizing, or integrating rotational uncertainty on $SO(3)$ by moving between unit quaternions along geodesics on $S^3$ rather than by additive perturbation in Euclidean coordinates. In the formulations reported for protein backbone generation and contact-rich manipulation, the scheduler uses spherical linear interpolation (SLERP) or its exponential-map form to preserve the unit-norm constraint, maintain valid rotations at intermediate times, and define time-indexed rotational noise or flow trajectories compatible with learned generative dynamics (Yue et al., 20 Feb 2025, Geiger et al., 24 Sep 2025). The term spans at least two closely related constructions: a flow-matching formulation in which quaternion trajectories are built by $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$, and a diffusion formulation in which rotational corruption is defined by $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$ with a monotonically increasing schedule $\mu_t\in[0,1]$.

1. Geometric basis on $S^3$ and $SO(3)$

A unit quaternion $q\in\mathbb{R}^4$, $\|q\|_2=1$, double-covers $SO(3)$ (Yue et al., 20 Feb 2025). This representation is central to SLERP-based scheduling because the interpolation is performed directly on the 3-sphere rather than in an ambient unconstrained vector space. In the protein-generation formulation, the exponential map is written as $S^3$0, where $S^3$1 is axis–angle, while the logarithm map retrieves $S^3$2 (Yue et al., 20 Feb 2025). Quaternion multiplication $S^3$3 implements group composition, and $S^3$4.

The geometric motivation is explicit in the manipulation setting: Gaussian perturbations in $S^3$5 do not respect the unit-norm constraint of quaternions and can easily push samples off the 3-sphere, breaking the group structure of $S^3$6 (Geiger et al., 24 Sep 2025). By contrast, SLERP between two unit quaternions moves along the great-circle geodesic on $S^3$7, preserving $S^3$8 at every step and ensuring that intermediate orientations remain valid rotations. This makes quaternion geodesics a natural substrate for forward corruption, reverse denoising, and continuous-time flow integration.

A plausible implication is that the scheduler is not merely a numerical convenience; it is the component that aligns the stochastic or deterministic time parameter with the intrinsic geometry of rotation space. Both reported systems treat rotational evolution as manifold-valued transport rather than coefficient-wise noise injection.

2. Canonical SLERP forms and their numerical properties

Two SLERP parameterizations appear in the source material. The manipulation paper gives the standard additive form

$S^3$9

and uses it for forward diffusion from a clean quaternion $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$0 to a random noise quaternion $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$1 (Geiger et al., 24 Sep 2025). At $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$2, $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$3; at $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$4, $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$5.

The protein-generation paper presents the standard additive SLERP only “for reference”:

$$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$6

and identifies its numerical weakness: it is “numerically unstable when $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$7 or $$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$8” (Yue et al., 20 Feb 2025).

The same paper therefore adopts an exponential-map SLERP:

$$q_t = q_0 \otimes \exp\!\big[t \cdot \log(q_0^{-1}\otimes q_1)\big]$$9

Because $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$0 implies $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$1, this form exactly traces the great-circle geodesic at constant speed (Yue et al., 20 Feb 2025). The stated numerical properties are specific: it “never divides by $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$2, remains stable for small or near-$$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$3 angles, and automatically stays on $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$4.” In the reported ablations, “SLERP-Exp vs. matrix-geodesic or additive SLERP” is “more stable when $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$5 or $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$6 (no NaNs, no renormalization)” and yields “$$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$7 faster quaternion ops vs. $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$8 matrix exp + log” (Yue et al., 20 Feb 2025).

These two formulations are mathematically aligned but operationally distinct. One uses coefficient interpolation with explicit numerical safeguards and renormalization; the other rewrites the path in Lie-group form and treats numerical stability as a direct consequence of the parameterization.

3. Time scheduling and noise parameterization

The scheduler is the time-dependent rule that determines how far the quaternion has moved from signal toward noise, or from noise toward sample, at each step. In the diffusion-based manipulation formulation, a standard discrete variance schedule $$q_t=\operatorname{SLERP}(q_0,q_{\text{noise}};\mu_t)$$9 is introduced with $\mu_t\in[0,1]$0 and $\mu_t\in[0,1]$1, and one “natural choice” is

$\mu_t\in[0,1]$2

Noise is then injected by geodesic interpolation:

$\mu_t\in[0,1]$3

The scheduler is therefore a monotonically increasing path from the clean quaternion to a fully random orientation (Geiger et al., 24 Sep 2025).

In ReQFlow, the time variable appears in both training and inference. During training, $\mu_t\in[0,1]$4; during inference, a discrete grid $\mu_t\in[0,1]$5, $\mu_t\in[0,1]$6, is used (Yue et al., 20 Feb 2025). For rotations, the paper introduces an “exponential step–size scheduler” to concentrate sampling near $\mu_t\in[0,1]$7:

$\mu_t\in[0,1]$8

and replaces

$\mu_t\in[0,1]$9

The instantaneous angular velocity becomes

$S^3$0

The reported effect is explicit: the scheduler “concentrates ‘effort’ near $S^3$1, allowing designability $S^3$2 with $S^3$3 steps,” whereas “without it, $S^3$4 needed for similar quality” (Yue et al., 20 Feb 2025).

The two schedules reflect different generative semantics. In diffusion, $S^3$5 indexes corruption severity within a forward noising process. In quaternion flow matching, $S^3$6 warps traversal speed along a paired geodesic and thereby reshapes the effective distribution of integration effort over time. This suggests that “noise scheduler” in this domain names a broader class of time warps on manifold-valued rotational paths, not only forward-noise variances in the diffusion sense.

4. Role in flow matching and diffusion models

In ReQFlow, the scheduler is embedded in a deterministic flow-matching objective defined jointly over translations and rotations. The endpoints are sampled as $S^3$7 and $S^3$8, the empirical distribution of backbone rotations, and the paired path has constant angular velocity

$S^3$9

The interpolation is

$SO(3)$0

with derivative

$SO(3)$1

A base $SO(3)$2-equivariant network $SO(3)$3 predicts endpoint $SO(3)$4 from $SO(3)$5, and the instantaneous velocity targets are

$SO(3)$6

$SO(3)$7

The loss is

$SO(3)$8

The learned field is interpreted as an ODE,

$SO(3)$9

integrated from $q\in\mathbb{R}^4$0 to $q\in\mathbb{R}^4$1 (Yue et al., 20 Feb 2025).

In the manipulation paper, the scheduler is part of a Transformer-based diffusion model with cross-attention to external wrenches (Geiger et al., 24 Sep 2025). At each forward step, the clean quaternion token is replaced with $q\in\mathbb{R}^4$2 produced by the SLERP scheduler, while the translation token is corrupted analogously with Gaussian noise. The combined 7D token $q\in\mathbb{R}^4$3 is embedded together with learned timestep embeddings; external wrenches are encoded as context tokens and attend via multi-head cross-attention into the trajectory tokens. During training, the network $q\in\mathbb{R}^4$4 predicts the quaternion “noise” element in a log-quaternion or axis-angle representation (3D) to avoid outputting raw 4D quaternions. At inference, the learned reverse-process updates quaternions via a small SLERP step, ensuring output remains on $q\in\mathbb{R}^4$5.

The common structural role is that the scheduler defines the admissible trajectory of quaternion states seen by the model. In one case it parameterizes supervised geodesic transport between coupled endpoints; in the other it parameterizes the corruption ladder that a denoiser must invert.

5. Numerical safeguards, implementation, and sampling conventions

Implementation details differ by formulation, but both are organized around preserving valid rotations throughout the computational graph. In the diffusion setting, the forward-noise pseudocode computes $q\in\mathbb{R}^4$6, uses a small-angle branch when $q\in\mathbb{R}^4$7, and renormalizes with $q\in\mathbb{R}^4$8 “to avoid drift” (Geiger et al., 24 Sep 2025). The paper explicitly identifies “clamping the dot-product into $q\in\mathbb{R}^4$9 before arccos and renormalizing after interpolation” as numerical safeguards. Quaternions are stored in robot-base frame with scalar-first convention $\|q\|_2=1$0 and are “renormalized after every attention block to mitigate drift.”

The random noise quaternion in that formulation is sampled uniformly on $\|q\|_2=1$1 by drawing a unit axis $\|q\|_2=1$2 uniformly, drawing an angle $\|q\|_2=1$3 with density proportional to $\|q\|_2=1$4—or “simply $\|q\|_2=1$5 and re-weight, or use a known unit-quaternion sampler”—and then setting

$\|q\|_2=1$6

This sampling rule supplies the terminal orientation for the SLERP corruption path (Geiger et al., 24 Sep 2025).

ReQFlow’s implementation uses the same geometric primitives but avoids the additive SLERP denominator. Its training loop samples $\|q\|_2=1$7 and $\|q\|_2=1$8, forms

$\|q\|_2=1$9

and computes endpoint-based velocity predictions (Yue et al., 20 Feb 2025). In the ODE Euler inference pseudocode, the update is

$SO(3)$0

with $SO(3)$1. The paper states that the same discrete grid $SO(3)$2 and exponential scheduler are retained during rectification.

A common misconception is that quaternion scheduling automatically eliminates all numerical issues. The reported evidence is narrower. Standard additive SLERP still requires clamping, small-angle handling, and renormalization in the diffusion implementation (Geiger et al., 24 Sep 2025), whereas the exponential-map form in ReQFlow is specifically described as stable near $SO(3)$3 and $SO(3)$4 and as automatically staying on $SO(3)$5 (Yue et al., 20 Feb 2025). The stability claim therefore depends on the precise SLERP parameterization.

6. Rectification, empirical behavior, and domain-specific outcomes

In ReQFlow, the scheduler is coupled to a rectification procedure that fine-tunes the learned model on its own generated pairs. The stated motivation is that “in a vanilla flow, different training pairs yield crossing trajectories, raising transport cost and slowing mixing,” while rectified flow “refits the same model to its own generated pairs to enforce non-crossing/geodesic alignment and reduce overall cost” (Yue et al., 20 Feb 2025). The procedure is three-step: sample noisy $SO(3)$6, run the trained QFlow model for $SO(3)$7 inference steps to obtain $SO(3)$8, pair these endpoints as new training data, and fine-tune $SO(3)$9 with exactly the same quaternion flow matching loss. The source reports a “Theorem (Marginal Preservation)” and a “Theorem (Cost Reduction)” for the rectified coupling.

The empirical effects reported for this SLERP-based rotational scheduler are domain-specific but substantial. In protein backbone generation, “ReQFlow reaches the same designability with far fewer steps (e.g. 50 vs. 500), roughly 10× speedup” (Yue et al., 20 Feb 2025). Additional ablations state: “QFlow (no rectification) at $S^3$00: designable fraction $S^3$01,” “ReQFlow at $S^3$02: fraction increases to $S^3$03 (PDB),” and “At $S^3$04: ReQFlow $S^3$05 vs. QFlow $S^3$06.” For long-chain generation, “ReQFlow remains $S^3$07 designable up to $S^3$08, while baselines collapse.” Inference speed is reported as “$S^3$09 on A100” for a 300-residue backbone, “37× faster than RFdiffusion ($S^3$10), 62× faster than Genie2 ($S^3$11).” The abstract states that ReQFlow is “37x faster than RFDiffusion and 63x faster than Genie2 when generating a backbone of length 300” (Yue et al., 20 Feb 2025).

In contact-rich manipulation, replacing naive additive Gaussian corruption of quaternion coefficients with the SLERP-based scheduler is reported to “maintain geometric consistency and avoid bias off the manifold” (Geiger et al., 24 Sep 2025). On the parkour dataset, the Transformer-Diffusion Model achieved “$S^3$12 mean angular error $S^3$13 and $S^3$14 axis-deviation $S^3$15 (with $S^3$16 positional error).” A combined parkour + therapy dataset improved rotational error to “$S^3$17 ($S^3$18) and $S^3$19 ($S^3$20).” Ablations replacing SLERP with unconstrained Gaussian noise on $S^3$21 “increased angular error by more than an order of magnitude and destabilized the learned reverse chain.” The broader system is reported to have achieved “30/30 success rates for cylindrical, square, and star peg insertions without any peg-specific demonstrations in the training data set,” together with sub-millimeter positional accuracy, sub-degree rotational accuracy, and real-time torque control on a KUKA LBR iiwa robot (Geiger et al., 24 Sep 2025).

Taken together, these results establish a consistent pattern: SLERP-based quaternion scheduling is used where rotational fidelity must be preserved under generative time evolution, and its practical value is tied to manifold consistency, stable interpolation near difficult angular regimes, and improved efficiency or accuracy when compared with unconstrained rotational noise models.