How far are two symmetric matrices from commuting? With an application to object characterisation and identification in metal detection

TL;DR

The paper tackles the challenge of comparing rotational data when eigenvectors are ill-conditioned by introducing semi-metrics $|d_E^{\pm}(A,B)|$ and $d_C(A,B)$ that approximate the Riemannian distance on $SO(n)$ without requiring eigenvectors. It develops small-angle approximations $d_{E,\theta}$ and $d_{C,\theta}$ based on spectral data (eigenvalues) and commutators, with proven rotational invariance and robustness to finite-precision effects. The framework is then applied to complex symmetric rank-two MPT descriptions in metal detection, including a rigorous analysis of regularisation and discretisation errors and a Bayesian classification demonstration that these new features improve object identification. Overall, the work provides theoretically grounded, computation-friendly distance measures that preserve orientation-invariance and improve reliability when eigenvalues are close, enabling more robust characterisation of objects from MPT data in practical, noisy settings.

Abstract

Examining the extent to which measurements of rotation matrices are close to each other is challenging due measurement noise. To overcome this, data is typically smoothed and Riemannian and Euclidean metrics are applied. However, if rotation matrices are not directly measured and are instead formed by eigenvectors of measured symmetric matrices, this can be problematic if the associated eigenvalues are close. In this work, we propose novel semi-metrics that can be used to approximate the Riemannian metric for small angles. Our new results do not require eigenvector information and are beneficial for measured datasets. There are also issues when using comparing rotational data arising from computational simulations and it is important that the impact of the approximations on the computed outputs is properly assessed to ensure that the approximations made and the finite precision arithmetic are not unduly polluting the results. In this work, we examine data arising from object characterisation in metal detection using the complex symmetric rank two magnetic polarizability tensor (MPT) description, we rigorously analyse the effects of our numerical approximations and apply our new approximate measures of distance to the commutator of the real and imaginary parts of the MPT to this application. Our new approximate measures of distance provide additional feature information, which is invariant of the object orientation, to aid with object identification using machine learning classifiers. We present Bayesian classification examples to demonstrate the success of our approach.

Figures (17)

• Figure 1: Two symmetric matrices $A$ and $B$ with distinct eigenvalues showing $(a)$ applied and predicted angles and $(b)$ relative error of angle prediction.
• Figure 2: Two symmetric matrices $A$ and $B$ where $A$ has two closely spaced eigenvalues and $B$ has distinct eigenvalues showing the predicted angles as a function of $\epsilon$.
• Figure 3: Irregular polyhedron: Illustration of the geometry.
• Figure 4: Homogeneous irregular polyhedron: Converged eigenvalue spectral signatures $(a)$$\lambda_i( {\mathcal{R}})$, $(b)$$\lambda_i( \mathcal{I} )$ and $(c)$$\lambda_i( \tilde{\mathcal{R}} )$.
• Figure 5: Homogeneous irregular polyhedron: $(a)$$\| {\mathcal{Z}} \|$, $(b)$$\| {\mathcal{Z}}^{(0)} \|$ and $(c)$$\|\tilde{\mathcal{Z}} \|$ and the corresponding error indicators $(a)$$\left \| {\mathcal{Z}} - {\mathcal{Z}}^\varepsilon \right \|\approx \Delta ( \| {\mathcal{Z}}\| )$, $(b)$$\left \| {\mathcal{Z}}^{(0)} - {\mathcal{Z}}^{(0),\varepsilon} \right \| \approx \Delta ( \| {\mathcal{Z}}^{(0)} \| )$, and $(c)$$\left \| \tilde{\mathcal{Z}} - \tilde{\mathcal{Z}}^\varepsilon \right \| \approx \Delta ( \| \tilde{\mathcal{Z}}\| )$.

Theorems & Definitions (21)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Remark 3.3
• Lemma 3.4
• proof
• Lemma 3.5
• proof
• Lemma 3.6