Dual Space Calibration for Model & Sensor Fusion

• Dual Space Calibration (DSC) is a framework that harmonizes weight and feature spaces to correct discrepancies in merged models, sensor alignments, and EEG adaptations.
• It combines a data-free weight space calibration (WSC) with a fine-tuned feature space calibration (FSC) to restore optimal magnitude consistency and improve task performance.
• DSC methodologies, including dual-quaternion sensor calibration and dual-stage BCI alignment, offer robust, real-time operation with minimal data and significant accuracy gains.

Dual Space Calibration (DSC) refers to a family of calibration methodologies and frameworks that target consistency—across both weight and feature (or representation) spaces, or across dual geometric domains—between complex systems and their components. Most prominently, DSC emerges in post-merging neural network calibration for model fusion in machine learning, dual-quaternion–based sensor extrinsic calibration, and dual-stage alignment for rapid calibration in brain-computer interfaces. The central objective is to resolve the discrepancies introduced by merging, sensor misalignment, or subject variance by leveraging both the parameter space and the induced feature or data space, often yielding SOTA improvements in accuracy, robustness, and adaptation efficiency.

1. Model Merging: Dual Space Calibration in Magnitude Alignment

In the context of neural network model merging, Dual Space Calibration (DSC) is a post-merging, plug-and-play algorithmic pipeline designed to correct layer-wise magnitude deviations that arise during parameter fusion or pruning-based merging of specialized, fine-tuned models. DSC systematically restores the magnitude of both task vectors (in weight space) and their induced features (in activation space) to match the statistics of the source expert models, rectifying the magnitude distortion effects left by merging and ensuring the merged network recovers optimal task-specific behavior—importantly, without further training or labeled data (Li et al., 22 Dec 2025).

1.1 Theoretical Foundation

• Magnitude Deviation: Merging operations such as Task Arithmetic or TRIM pruning perturb the L₂-norms of task vectors, which, under local network linearity, induce proportional shifts in feature-space magnitudes. Even equal-norm task vectors can yield degraded feature magnitudes due to misalignment with layer Jacobians.
• Sensitivity: The influence of magnitude deviation on end-task performance is highly layer-dependent. Layers with high sensitivity—notably the so-called "magnitude-sensitive layers"—are particularly susceptible to performance losses from naive or global scaling.
• Calibration Target: Optimal performance is empirically and theoretically (Theorem 3) attained by matching the per-layer feature norm of the merged model to those of the respective specialized models.

1.2 Methodological Architecture

DSC fuses two principal axes of calibration:

• Weight Space Calibration (WSC): Direct rescaling of layer task vectors places the merged vector onto a hyperellipsoid defined by the principal expert axes, using only model weights. No data are required.
• Feature Space Calibration (FSC): Fine correction is performed by scaling the merged model’s activations so that, on a small pool of unlabelled task-specific data, the average feature norm aligns with the corresponding source expert norms.
• DSC Pipeline: DSC first applies WSC to the merged weights, then applies FSC on the resulting activations, enforcing calibration in both spaces sequentially.

Summary Table: DSC Key Elements in Model Merging

Calibration Mode	Space	Data Dependency	Key Operation
WSC	Weight ($\theta$)	Data-free	Rescales merged task vectors to match expert norms via hyperellipsoid constraint
FSC	Feature ($h^l$)	Few unlabelled samples	Scales activations so per-layer feature norms agree with expert targets
DSC (WSC+FSC)	Both	Minimal (few data)	Sequentially applies WSC then FSC for optimal correction

Empirical findings confirm that DSC achieves superior performance across computer vision (e.g., ViT-B/32 backbone: +4.3% gain averaged over 8 datasets) and natural language processing (e.g., Llama-2-7B: +8.0% gain), outperforming stand-alone WSC or FSC (Li et al., 22 Dec 2025).

2. Dual-Quaternion-Based Sensor Calibration

In geometric robotics and multi-sensor systems, "Dual Space Calibration" also denotes methods that employ dual quaternions (DQ) for online, extrinsic sensor calibration from per-sensor ego-motion (Horn et al., 2021). Here, the DSC paradigm refers not to model magnitude calibration, but rather to solving for the rigid 6-DOF transform between sensors by formulating the problem in the dual quaternion domain.

2.1 Problem Formulation and Solver Duality

• Dual quaternions encode both rotation and translation in a compact algebraic structure.
• The core calibration constraint is posed as a quadratic cost: for two sensors $A,B$ with relative motions, the extrinsic calibration DQ $q_T$ minimizes

$$J(\mathbf q) = \sum_{i=1}^n \eta_i\: \mathbf q^\top Q_{w,i} \mathbf q$$

under unit DQ constraints and, if applicable, planar-motion priors.

• Two solution strategies are provided:
  • A certifiably globally optimal SDP, guaranteeing globality via duality gap examination.
  • A warm-started fast local solver (e.g., SQP) with runtime verification of globality, yielding real-time performance (planar: ≈1.8 ms/frame; full 3D: a few ms/frame).

2.2 Empirical Performance

Extensive benchmarking on RGB-D, lidar-stereo, and simulated trajectories confirms that the dual-quaternion DSC method achieves SOTA accuracy (translation <1.2 cm, rotation <1.1°), with robust global optimality certification and minimal latency (see table below) (Horn et al., 2021).

Method	Translation Error	Rotation Error	Time per Frame
DSC fast (SQP+certify)	1.16 cm	1.06°	6.2 ms
DSC global (SDP)	1.16 cm	1.06°	43.9 ms

A plausible implication is that DSC, as dual-quaternion calibration, is suitable for real-time, online estimation in deployed multi-sensor platforms where dual-space geometric consistency is critical.

3. Dual-Stage Alignment for EEG-Based Brain-Computer Interfaces

While not explicitly named "Dual Space Calibration," dual-stage calibration frameworks appear in online adaptation for BCI, where rapid calibration is achieved by alignment in both raw (data) and representation (feature) spaces (Duan et al., 23 Sep 2025).

3.1 Two-Stage Alignment

• Stage 1: Euclidean alignment whitens incoming EEG signals per trial, reducing channel covariance drift.
• Stage 2: Exponential moving average updates of BatchNorm statistics in the feature extractor network correct for representation-level shifts.
• Combined with a self-supervised loss, this dual-stage process enables plug-and-play fast calibration operational within single-trial latency.

3.2 Gains and Generality

Versus prior TTA and domain adaptation baselines, the DSAS algorithm yields consistent accuracy gains across five public datasets and seven backbone decoders. Notably, SSVEP accuracy improves by +4.9% and motor imagery by +3.6% (mean gain), with adaptation requiring only a single trial (Duan et al., 23 Sep 2025).

4. Experimental and Empirical Insights

Empirical analyses in (Li et al., 22 Dec 2025) and (Horn et al., 2021) emphasize:

• Model Merging: Ablation confirms that the combined DSC pipeline consistently outperforms either WSC or FSC alone, both on average and across diverse tasks. Only 1 unlabeled sample per task yields >90% of the full calibration gain.
• Sensor Calibration: Online DSC sustains certified global optimality with primal-dual convergence; after initial convergence, local solvers suffice for all new data.
• Sensitivity and Robustness: Model merging DSC is robust to hyperparameter value (α for identifying sensitive layers), model backbone, and variance in coefficient estimation owing to the data-free core of WSC.

5. Methodological Limitations and Scope

DSC in the model merging regime presupposes access to the weight vectors from specialized source models and, for FSC, at least one unlabeled sample per task. Data-free operation is possible with WSC alone, albeit with smaller gains. Extensions to fully supervised or unsupervised target domains are not evaluated in the referenced works.

For dual-quaternion-based sensor calibration, DSC is inherently restricted to platforms equipped with sufficient ego-motion estimates; in practice, accuracy depends on the trajectory ensemble and noise.

DSAS-style dual-stage approaches in BCI are universally applicable to deep EEG decoders, with little dependence on batch size; however, the self-supervised calibration loss assumes the availability of validation accuracy as a pseudo-label prior (Duan et al., 23 Sep 2025).

6. Significance and Broader Impact

Dual Space Calibration, in its prominent forms, addresses key limitations in model fusion, online system alignment, and physiological signal adaptation by jointly leveraging (a) parameter/weight space adjustment and (b) feature/data space correction. The hybrid approach achieves calibration efficacy unattainable by either space alone, with broad implications for scalable model modularization, resilient sensor fusion, and adaptive neural systems. Consistent multi-domain accuracy improvements, minimal data/compute requirements, and robust real-time operation position DSC as a foundational tool in modern machine learning, robotics, and computational neuroscience (Li et al., 22 Dec 2025, Horn et al., 2021, Duan et al., 23 Sep 2025).