- The paper introduces a delta-position regression approach comparing MLP and Kolmogorov-Arnold Networks to tackle error accumulation in IMU odometry.
- The paper shows that KAN achieves a 44% lower final drift than MLP while using 6.9 times fewer parameters, at the cost of longer training times.
- The paper demonstrates that learnable B-spline activations in KAN enable stable long-term trajectory reconstruction and robust error compensation.
Delta-Position Regression for IMU Odometry: Evaluation of MLP and Kolmogorov-Arnold Networks
Problem Setting and Motivation
The paper addresses the challenge of accurate position estimation in GPS-denied environments via IMU-only odometry. The underlying difficulty is error accumulation due to double integration of sensor noise and biases, yielding drift that scales quadratically or cubically over time in classical INS mechanization. While model-based sensor fusion with exteroceptive sensors can mitigate this, it requires detailed sensor characterization and environment-specific tuning, with performance degradation in mismatched or dynamic conditions.
The recent paradigm of learning-based odometry, particularly delta-position regression, offers a sensor-model-agnostic framework. By learning the mapping from windows of raw IMU measurements to incremental displacement, these methods sidestep the catastrophic drift encountered with direct position regression, constraining error growth via local modeling and subsequent numerical integration.
Methods: MLP and KAN Architectures
This work compares two neural architectures for the delta-position mapping: a baseline deep MLP and the Kolmogorov-Arnold Network (KAN), a recent theoretically-motivated model.
- MLP Baseline: A feedforward network with three hidden layers (256, 128, 64 units), BatchNorm, ReLU, and Dropout, parameterized by 57,859 weights. The input is a vectorized 10×6 (60D) IMU window.
- KAN: Implements learnable univariate B-spline edge functions plus SiLU residuals along fully-connected edges, leveraging the Kolmogorov-Arnold representation theorem for compact, highly-expressive function approximation. The network width is [60,7,5,3] with cubic B-splines, amounting to 8,444 parameters—a 6.9x reduction versus MLP.
Both methods optimize mean squared error over delta-position regression targets (50 ms increments) and reconstruct global trajectories via simple integration. Training, validation, and test splits are chronologically divided from the EuRoC MAV V1_01_easy indoor flight sequence, with rigorous timestamp alignment and normalization to prevent data leakage.
Experimental Evaluation and Key Results
The primary evaluation criterion is not only per-step accuracy (dRMSE, dMAE), but the cumulative drift observed after long-term integration—quantified by median (P50) and 90th percentile (P90) drift, and final drift at the sequence terminus.
Quantitative results:
| Model |
Params |
dRMSE (m) |
dMAE (m) |
P50 (m) |
P90 (m) |
Final Drift (m) |
| MLP |
57,859 |
0.023 |
0.021 |
5.18 |
10.06 |
17.23 |
| KAN |
8,444 |
0.022 |
0.019 |
3.22 |
6.96 |
9.61 |
A salient finding is that while per-window prediction accuracy is nearly identical (dRMSE differs by 0.001 m), KAN achieves a 44% lower final cumulative drift than MLP (9.61 m vs. 17.23 m), with substantially reduced error propagation as evidenced by both P50 and P90 metrics.
The analysis suggests the MLP's step errors are correlated such that biases accumulate over time, worsening global drift. In contrast, the KAN model, via its flexible edge-wise B-spline parameterization, produces partially self-compensating errors, leading to more stable long-horizon estimation. Notably, this advantage is realized with an order-of-magnitude reduction in parameter count, underscoring the representational efficiency of KANs.
However, these benefits are accompanied by increased training cost: KAN requires roughly 10x longer wall-clock training time (460s vs. 41s for MLP under the same protocol), reflecting the computational demand of learning B-spline activations and potentially less optimized implementations.
Implications and Theoretical Significance
The results empirically validate the hypothesis that learnable univariate activations, as instantiated by KANs, confer advantages beyond parameter efficiency. Specifically, they improve the temporal structure of incremental prediction errors, reducing correlated bias drift in cumulative tasks like inertial odometry.
From an applied perspective, KANs present a compelling approach for resource-constrained platforms and edge deployment, where memory footprint and long-term estimation stability are critical. Moreover, the interpretability analysis (see paper for details) indicates that the learned B-spline responses in KANs reflect physical inductive biases aligned with IMU sensor characteristics, a property lacking in black-box MLPs.
On the theoretical front, this work extends the application domain of KANs from symbolic regression and scientific computing to sequential high-frequency time-series regression. It provides evidence that Kolmogorov-Arnold style univariate decomposition can structure solutions to mitigate the intrinsic error accumulation plaguing integration-based estimation systems.
Future Research Directions
Open questions remain regarding the generalization of KANs over varying window lengths, sampling frequencies, and more challenging flight regimes. Real-world deployment scenarios, transfer to other IMU devices or vehicles, and domain adaptation are yet to be explored. Additionally, integration with uncertainty modeling, e.g., Bayesian KANs or calibration-aware training, and real-time performance optimization (including inference latency) are promising avenues.
A deeper study of the mechanisms by which KAN activations counteract drift—potentially via explicit decorrelation of prediction residuals—could yield broader strategies for other physical integration-based estimation problems beyond odometry.
Conclusion
This comparative study establishes that Kolmogorov-Arnold Networks, despite their reduced parameter budget and higher training complexity, offer pronounced advantages in mitigating cumulative drift over standard MLPs in delta-position IMU odometry. Their architectural distinctiveness, grounded in learnable spline-based transformations, enables stable long-term trajectory reconstruction—a property of practical significance for autonomous robotics in sensor-degraded environments. Further investigation into the operational envelope and broader applicability of KANs in time-series estimation is warranted.