Geometry Induced Contraction Degradation and Stabilization of Learning Enabled Observers

Learned perception models are increasingly used as measurement maps within nonlinear observers, mapping high-dimensional sensory inputs to low-dimensional quantities for state estimation. The learned measurement Jacobian enters the observer error dynamics and can shrink Euclidean contraction margins under fixed gains, leading to a geometry-dependent degradation of stability.

The gist

The paper shows that learned measurement geometry enters the observer error dynamics and can rescale Euclidean contraction margins, which can eliminate exponential convergence guarantees under fixed gains.

How it works: Geometry-Dependent Contraction Degradation

1. The system is defined by state dynamics and a measurement model, where the latter is learned:

xt+1 = f(xt) + wt

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yt = hϕ(xt) + vt, where hϕ: R n → R m denotes the learned measurement model.

2. The key geometric quantity is the Jacobian of the learned measurement model:

> Jh(x):= ∂hϕ(x) / ∂x.

This Jacobian characterizes the local geometry of the learned representation, and its spatial variation reflects changes in representation sensitivity across the state space, modifying how measurement feedback enters observer dynamics.

3. The observer error dynamics are derived using averaged Jacobians:

> et+1 = Atet + dt,

where the error propagation matrix is defined as:

> At:= J¯f,t − αtK˜ J¯h,t.

4. Under fixed gains, the contraction factor ρt is bounded by a term dependent on measurement sensitivity:

> ρt ≤ Lf + α∥K˜∥∥J¯h,t∥.

This demonstrates that measurement sensitivity increases reduces the available contraction margin.

5. A critical threshold exists where stability cannot be certified:

> Scrit = 1 − Lf / (α∥K˜∥).

If the measurement sensitivity is such that its magnitude exceeds this threshold, "the sufficient Euclidean contraction condition ρt < 1 cannot be certified," potentially leading to the loss of fixed-gain exponential convergence guarantees.

How it works: Representation-Aware Stabilization

The paper introduces a representation-aware gain normalization to counteract this geometry-induced amplification without modifying the learned model.

1. The goal is to design a normalization that compensates for geometry-induced amplification using only local Jacobian information, treating the learned measurement model as a black box and requiring no retraining or architectural modification.

2. The proposed normalized gain is defined as:

> αt = β /∥K J˜h(ˆxt)∥ + ε,

where β > 0 and ε > 0.

3. This normalization restores a uniform contraction bound at the leading order, showing that geometry dependence is not intrinsic to the system dynamics, but arises from gain scaling in the observer correction term.

4. The resulting local contraction bound is shown to be:

> ρt ≤ Lf + β + ce∥et∥,

where ce = β / (ε∥K˜Lh). This restores a geometry-independent leading-order contraction certificate, with the remaining dependence appearing only through the Jacobian mismatch term, which vanishes as the estimation error decreases.

Validation and Consequences

The mechanism is validated through both scalar nonlinear examples and higher-dimensional dynamics.

1. Scalar Experiments: Using a specific model, the paper shows that increasing measurement sensitivity leads to unity, contraction-based stability is lost, while applying the normalization restores convergence by reducing peak contraction-rate values and increasing the fraction of time with µt < 0.

2. Robustness Under Disturbances: Monte Carlo simulations confirm that under high measurement sensitivity, fixed-gain observers frequently lose contraction and exhibit growing estimation error. The normalized observer maintains bounded error and consistent convergence across trials.

3. Real-Data Validation: Experiments on VisDrone (illumination variation) and UOT32 (underwater distortion) demonstrate that environmental variations produce substantial changes in measurement sensitivity, confirming that real-world conditions induce geometry-dependent variation, which the representation-aware normalization effectively mitigates.

4. Higher-Dimensional Dynamics: In the planar forced Duffing oscillator, the normalized observer "reduces peak contraction-rate values and increases the fraction of time with µt < 0," indicating improved local stability in a higher-dimensional setting.

Conclusion

The research establishes that learned measurement geometry directly influences observer stability under fixed gains, and that representation-aware normalization restores a uniform contraction margin without modifying the learned measurement model, providing a practical mechanism for improving robustness and reliability in learning-enabled observer architectures.

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REFERENCES

[1] G. Revach, N. Shlezinger, X. Ni et al., “Kalmannet: Neural network aided kalman filtering for partially known dynamics,” IEEE Transactions on Signal Processing, vol.

Based on the provided scientific paper, here are specific improvements that can be made to existing AI systems and what these improved systems could achieve:

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1. The core improvement is the implementation of a representation-aware gain normalization technique for learning-enabled observers. This technique treats the learned measurement model (e.g., a neural network output) as a black box and requires no retraining or architectural modification.

2. This normalization compensates for geometry-induced amplification in the observer error dynamics, which is caused by state-dependent Jacobians of the learned measurement function, thereby restoring a uniform Euclidean contraction margin without modifying the underlying learned model.

The improved AI system can achieve the following specific capabilities:

1. It will maintain guaranteed exponential convergence (or bounded error) even when using measurement models with highly variable or complex geometries (like those encountered in changing lighting or underwater environments).

2. It will operate robustly under fixed observer gains, eliminating the risk of certification failure that occurs when measurement sensitivity exceeds a certain threshold.

3. It will exhibit significantly improved tracking accuracy and robustness in real-world applications (e.g., drone tracking, autonomous underwater vehicle navigation) where environmental factors cause the underlying perception model's output sensitivity to change over time (e.g., illumination shifts or turbidity).

4. It will maintain a simplified observer structure compatible with standard architectures, avoiding the need for complex state-dependent metric calculations or retraining of the neural components.

Specifically, this system can:

1. Accurately track objects in low-visibility conditions (like underwater) by effectively compensating for measurement noise amplification caused by environmental distortion.

2. Provide reliable state estimation in dynamic terrestrial environments where visual features change rapidly due to illumination variations, ensuring convergence guarantees are met regardless of the instantaneous geometric sensitivity of the perception model.

3. Be deployed as a plug-and-play component within existing hybrid learning/model-based estimators (like Neural Kalman Filters) without requiring specialized retraining or architectural redesign of the learned measurement mapping itself.