Centrala begrepp
The optimal uncertainty ellipsoids are centered around the conditional mean and shaped as the conditional covariance matrix under the assumption of jointly Gaussian data. For more practical cases, a differentiable optimization approach using a neural network can approximately compute the optimal ellipsoids with less storage and fewer computations at inference time, leading to accurate yet smaller ellipsoids.
Sammanfattning
The paper considers the problem of learning uncertainty regions for parameter estimation problems. The regions are ellipsoids that minimize the average volumes subject to a prescribed coverage probability.
In the simplistic Gaussian setting, the authors prove that the optimal ellipsoid is centered around the conditional mean and shaped as the conditional covariance matrix. For more practical cases, they propose a differentiable optimization approach using a neural network called LMVE to approximately compute the optimal ellipsoids.
LMVE combines nearest-neighbor approaches, covariance estimation, and conformal prediction to generate ellipsoids with minimal average volume and prescribed coverage probability. It significantly reduces memory and computation resources compared to existing methods, while improving accuracy. The authors demonstrate the advantages of LMVE on four real-world localization datasets.
The key steps of LMVE are:
- Initialization: Use an existing baseline method to obtain approximate labels and train the network to approximate them.
- Training: Optimize a Lagrange penalized form of the original problem, balancing the coverage and volume.
- Calibration: Use conformal prediction to rescale the ellipsoids and ensure the desired coverage probability.
The experiments show that LMVE outperforms existing methods in terms of average ellipsoid volume while maintaining the required coverage levels. It also has lower computational complexity and memory requirements.
Statistik
The paper does not provide specific numerical data, but rather focuses on the theoretical analysis and the proposed LMVE framework. The experimental results are presented in the form of comparative performance metrics across different real-world localization datasets.
Citat
"The optimal argument is E(μ, F^-1_χ^2_n(η) · Σ) where F^-1_χ^2_n(·) is the inverse chi-square cdf with n degrees of freedom, and the optimal value is F^-1_χ^2_n(η)^n · Vol(Σ)."
"The optimal solution to (MVE) is μ(x) = E[y|x] and C(x) = κ(x) · E[(y-μ(x))(y-μ(x))^T] where κ(x) > 0 is a scaling factor that satisfies the MVE coverage constraint."