Dimension-Decomposed Learning (DiD-L) in Geometric Control

• Dimension-Decomposed Learning (DiD-L) is an online learning framework that decomposes high-dimensional disturbance identification into axis-aligned, low-dimensional slices for efficient control.
• It employs shallow neural networks and Lyapunov-based adaptive laws to achieve rapid disturbance rejection and near-exponential convergence in both SO(3) and SE(3) settings.
• By integrating Sliced Adaptive-Neuro Mapping (SANM) with geometric controllers, DiD-L supports high-frequency embedded implementation validated through real-world quadrotor flight tests.

Dimension-Decomposed Learning (DiD-L) is an online learning principle introduced for disturbance identification in quadrotor geometric control, first in geometric attitude control on $SO(3)$ and then in full-state control on $SE(3)$. Its defining move is to axially “slice” a high-dimensional identification map into multiple low-dimensional submappings, or slices, so that a complex high-dimensional problem is decomposed into simple low-dimensional tasks addressed by shallow neural networks and adaptive laws. The concrete module proposed in both papers is Sliced Adaptive-Neuro Mapping (SANM), whose parameters are updated online via Lyapunov-based adaptation without pre-training and without the persistent excitation condition (Gao et al., 20 Aug 2025, Gao, 3 Oct 2025).

1. Origin, motivation, and scope

DiD-L was proposed against a specific control-theoretic diagnosis: high-dimensional online disturbance-identification maps are underfit by lightweight online shallow neural networks, whereas deeper or more opaque models are harder to justify analytically and harder to deploy at high frequency on embedded platforms. In the attitude-control paper, this appears as a disturbance-identification problem for rotational dynamics on $SO(3)$; in the later $SE(3)$ paper, the same logic is extended to simultaneous estimation of pseudo-model features and translational and rotational disturbances from a 12-dimensional tracking error. In both cases, decomposition is presented as a structural remedy rather than a post hoc regularizer: the mapping itself is reorganized so that each slice corresponds to one physical axis and one low-dimensional local feature set (Gao et al., 20 Aug 2025, Gao, 3 Oct 2025).

The framework is therefore narrower than a generic slogan about modular learning. In the cited control papers, “dimension-decomposed” refers to axis-aligned decomposition of disturbance and uncertainty identification. The immediate target is not latent disentanglement, objective decomposition, or multitask parameter sharing, but online compensation of translational and rotational uncertainties inside a geometric controller. This concrete scope is important because the theoretical claims—Lyapunov-based adaptation, bounded estimation errors, and exponential or near-exponential convergence—are all derived for that control setting.

2. Mathematical decomposition principle

In the $SE(3)$ formulation, the full-state error is

$$\mathbf{E}\triangleq \big( \bm{e}_{\bm{x}}^{\top}, \bm{e}_{\bm{v}}^{\top}, \bm{e}_{\bm{R}}^{\top}, \bm{e}_{\bm{\Omega}}^{\top} \big)^{\top}\in\mathbb{R}^{12},$$

and a monolithic adaptive-neuro estimator is written as

$$(\bm{\bar{\Theta}^{\text{rec}}},\bm{\bar{\Phi}}) \triangleq \bm{\mathcal{S}_{AN}(\bm{\mathrm{w}_d},\mathbf{E})} :\mathbb{R}^6\times\mathcal{C}\to\mathbb{R}^6\times\mathbb{R}^6.$$

DiD-L replaces this with

$$\bigoplus_{j=1}^{3} \Big\{ \big( \bm{\bar{m}^{[j]}}, \bm{\bar{J}^{[j]}}, \bm{\bar{\phi}_{\bm{x}}^{[j]}}, \bm{\bar{\phi}_{\bm{R}}^{[j]}} \big) := \bm{\mathcal{S}^{[j]}_{AN} \big( \bm{\mathrm{F}_d}^{[j]}, \bm{\mathrm{M}_d}^{[j]}, \bm{\mathcal{E}_{\bm{x} j}}, (\bm{e}_{\bm{R}}^{[j]},\bm{e}_{\bm{\Omega}}^{[j]}) \big)} \Big\},$$

with

$$\bm{\mathcal{E}_{\bm{x} j}} := \left( \bm{e}_{\bm{x}}^{[j]}, \bm{e}_{\bm{v}}^{[j]} \right)^{\top}\in\mathbb{R}^{2}.$$

The paper explicitly states that the original 12-dimensional target feature is decomposed into twelve sliced 1-dimensional features (Gao, 3 Oct 2025).

In the $SO(3)$ formulation, the corresponding slice map is

$SE(3)$0

Here the original 6-dimensional target feature is decomposed into six sliced 1-dimensional features. This specialization makes clear that “dimension” in DiD-L refers to axis-aligned disturbance and model features, not to a latent disentangling of abstract representation coordinates (Gao et al., 20 Aug 2025).

The decomposition is physically structured. In the full $SE(3)$1 case, each slice receives one translational command component, one rotational command component, one translational error pair, and one rotational error pair, then returns one pseudo mass estimate, one pseudo inertia estimate, one translational disturbance estimate, and one rotational disturbance estimate for that axis. In the attitude-only case, each slice receives $SE(3)$2, $SE(3)$3, and $SE(3)$4, then returns the inertia and disturbance features for axis $SE(3)$5. The intended effect is that each local approximator solves a substantially simpler problem than a single multi-input, multi-output network.

3. SANM: slice-wise adaptive-neuro estimation and controller integration

SANM combines adaptive laws for pseudo-model features with slice-wise RBF networks for disturbance features. For $SE(3)$6 and $SE(3)$7,

$SE(3)$8

with $SE(3)$9 and $SO(3)$0. The Gaussian basis vector is built from

$SO(3)$1

The neural weights are updated online according to

$SO(3)$2

while pseudo-mass and pseudo-inertia features are updated by piecewise-projected reciprocal-form adaptive laws (Gao, 3 Oct 2025).

The slice outputs enter the controller axis by axis. For translation,

$SO(3)$3

and for rotation,

$SO(3)$4

In Scenario 1 the rigid-body coupling term is explicitly canceled when $SO(3)$5 is known, whereas in Scenario 2 it is absorbed into the unknown rotational disturbance (Gao, 3 Oct 2025).

In the attitude-only paper, SANM is presented as three shallow RBF networks of type $SO(3)$6-$SO(3)$7-$SO(3)$8, one per rotational axis, paired with three adaptive-law slices for the inertia features. The controller therefore learns at the acceleration level rather than through a single multi-input, multi-output disturbance approximator, and the slice outputs are recombined into the final compensation term used by the geometric controller (Gao et al., 20 Aug 2025).

4. Stability theory and analytical guarantees

The attitude-control formulation proves an almost-global result on $SO(3)$9. With initial $SE(3)$0, the rotational error state $SE(3)$1 exponentially converges into an arbitrarily small ball $SE(3)$2, while $SE(3)$3 remain uniformly bounded, despite time-varying disturbance moments and unknown inertia tensor. The attraction domain excludes the singular points corresponding to exactly $SE(3)$4 rotation mismatch, which is the standard topological obstruction on $SE(3)$5 (Gao et al., 20 Aug 2025).

The later $SE(3)$6 paper frames the global claim as Near-Exponential Stability (NES): a solution $SE(3)$7 is NES around $SE(3)$8 if

$SE(3)$9

with $SE(3)$0. Under coupled position-attitude control compensated by SANM, all state errors $SE(3)$1 are locally near-exponentially stable around zero and all estimation errors remain uniformly bounded, despite time-varying multidimensional disturbances and model uncertainties. The derived bound

$SE(3)$2

is obtained from a composite Lyapunov function that couples translational slice errors, rotational slice errors, reciprocal parameter errors, and neural-weight errors (Gao, 3 Oct 2025).

These guarantees do not assert exact parameter convergence without persistent excitation. Instead, they establish bounded estimation errors and exponentially fast convergence of the state errors to an arbitrarily small residual ball. This suggests that DiD-L is designed around closed-loop stability and disturbance rejection rather than classical identification consistency.

5. Embedded implementation and empirical status

The attitude-control paper ties DiD-L to embedded deployment. SANM was implemented in C++ on an STM32H750 ARM 480 MHz flight control unit, integrated into ArduPilot 4.6, and run online at 400 Hz. The real-time module uses three RBF networks with $SE(3)$3 hidden neurons each, so one SANM execution requires only $SE(3)$4 evaluations of expf() for Gaussian activations. The paper reports both ArduPilot SITL plus Gazebo Harmonic experiments and real-world flight tests; in the supplied comparisons, the baseline without SANM crashes under the disturbance, whereas SANM maintains stability and reduces the rotational errors (Gao et al., 20 Aug 2025).

The $SE(3)$5 extension is validated through real-time simulation experiments in Gazebo. There the controller estimates translational disturbances, rotational disturbances, mass uncertainty, inertia uncertainty, and, in the unknown-inertia scenario, internal rotational coupling uncertainty. The abstract emphasizes that the learning laws operate without pre-training or persistent excitation and that the closed-loop full state is arbitrarily close to exponential stability despite multidimensional time-varying disturbances and model uncertainties (Gao, 3 Oct 2025).

Taken together, these reports position DiD-L as a deployment-oriented online control method, not only as a decomposition heuristic. The practical emphasis is consistent across both papers: shallow slice-wise models, explicit adaptive laws, and controller-level integration are treated as necessary conditions for real-time operation and for Lyapunov-based interpretability.

6. Relation to other decomposition-based paradigms

A broader reading suggests that DiD-L belongs to a larger family of decomposition-based learning schemes, but the exact mechanism varies sharply across domains.

Work	What is decomposed	Relation to DiD-L
DEMI (Sordoni et al., 2021)	mutual-information estimation over subviews/components	objective-level, information-theoretic decomposition over progressively informative components rather than literal representation dimensions
DiD (Zhang et al., 5 Feb 2025)	semantic differences induced by perturbing latent dimensions	directly learns semantic differences rather than the statistical independence of latent variables
DecL (Samdani et al., 2012)	subsets of output coordinates during structured learning	decomposes training-time inference into local output subspaces or coordinate neighborhoods around gold outputs
DIPL (Weitekamp et al., 2021)	procedural skill learning into how, where, and when	decomposes functional dimensions of procedural competence rather than numerical axes
SDPC (Zhang et al., 2023)	continuous action dimensions into independently discretized per-dimension subspaces	factorized action decisions with a shared continuous critic
HDSD (Qin et al., 8 May 2026)	general and task-specific subspaces plus singular components	shared/private subspace decomposition with SVD-based task allocation

These works show that “decomposition” in contemporary arXiv usage is heterogeneous. In DEMI the decomposition is over subviews or progressively informative components; in DecL it is over local output subspaces; in SDPC it is over action dimensions with a shared critic; in HDSD it is over general and task-specific subspaces plus singular directions; and in DiD it is over semantic variation patterns rather than merely independent latents. DiD-L in the quadrotor papers is narrower and more concrete: it is an axis-aligned online identification framework whose slices are physically tied to translational or rotational channels.

Two misconceptions follow. First, DiD-L is not synonymous with disentanglement: the quadrotor papers do not attempt to assign abstract semantic factors to latent coordinates. Second, DiD-L is not simply “using multiple small models”; the decomposition is coupled to controller design, reciprocal-form adaptation, and Lyapunov proofs. A broader interpretation suggests a family resemblance across decomposition-based methods, but the exact name “Dimension-Decomposed Learning” is tied, in the supplied literature, to the SANM-based quadrotor-control formulation.