The Role of Principal Angles in Subspace Classification

Abstract--Subspace models play an important role in a wide range of signal processing tasks, and this paper explores how the pairwise geometry of subspaces influences the probability of misclassification. When the mismatch between the signal and the model is vanishingly small, the probability of misclassification is determined by the product of the sines of the principal angles between subspaces. When the mismatch is more significant, the probability of misclassification is determined by the sum of the squares of the sines of the principal angles. Reliability of classification is derived in terms of the distribution of signal energy across principal vectors. Larger principal angles lead to smaller classification error, motivating a linear transform that optimizes principal angles. The transform presented here (TRAIT) preserves some specific characteristic of each individual class, and this approach is shown to be complementary to a previously developed transform (LRT) that enlarges inter-class distance while suppressing intra-class dispersion. Theoretical results are supported by demonstration of superior classification accuracy on synthetic and measured data even in the presence of significant model mismatch. IGNALS that are nominally high dimensional often exhibit a low dimensional geometric structure. For example, fixed-pose images of human faces are recorded using more than 1000 pixels, but can be represented by a 9-dimensional harmonic subspace [1]. Motion trajectories of a rigid body might be recorded by hundreds of sensors, but must intrinsically be represented by a 4-dimensional subspace [2]. There are many more examples where a low-dimensional subspace model captures intrinsic geometric structure, ranging from user ratings in a recommendation system [3] to signals emitted by multiple sources impinging at an antenna array [4]. The subspace geometry has assisted tasks of interest to both signal processing [5], [6] and machine learning communities [7], [8]. It can be used to approximate a nonlinear manifold by fitting mixture components to local patches of the manifold [5], [9], hence providing a high fidelity representation of a wide variety of signal geometries.