Bounds and algorithms for subspace estimation on Riemannian quotient submanifolds

Subspace estimation appears in a wide variety of signal processing applications such as radar, communications, and underwater acoustics, often as a prelude to high resolution parameter estimation. As with any estimation problem the availability of statistical benchmarks on estimator accuracy is key to developing and understanding algorithm performance. The parameter space in general subspace/basis estimation problems is naturally described as a Riemannian quotient manifold. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the properties of Euclidean spaces. This identification permits the well-developed tools of differential geometry to be brought to bear on the analysis of subspace/basis estimation problems. Classical Cramér-Rao bounds (CRB), originally formulated for standard Euclidean spaces, have recently been generalized to Riemannian quotient manifolds and the particular case of the standard real linear statistical model with the Grassmann manifold (set of subspaces) as the parameter space has been analyzed. The present works applies this differential geometric approach to the analysis of the complex linear model estimation problem. We consider both the general unconstrained signal model and, most importantly, the parametrically constrained signal model. First, we show that the appropriate parameter space for the most general unconstrained signal model is a modified Stiefel manifold, termed here the Basis manifold. Elements of the Basis manifold are semi-unitary matrices grouped into equivalence classes dictated by the model invariances. Using this formulation we derive the full Fisher Information Matrix (FIM) and, from this, intrinsic CRBs on the both subspace and rotation estimation accuracy. Among the corollaries that flow from this Basis manifold formulation is a CRB on the individual columns of the eigen vector matrix, say Y (an N × p semi-unitary matrix), of the true signal covariance. The eigen decomposition of the sample covariance matrix (SCM) yields estimate of Y; the analytical expression for the corresponding estimation error covariance is well known from statistics. We show the relationship between this existing expression and the CRB developed here using the geometric formulation. Secondly, we consider situation when the signal matrix is constrained, and in particular, the important special case when the constraint arises due to some underlying parametric model. Here the set of semi-unitary matrices representing the signal matrix are constrained to a submanifold of the Basis manifold. In this geometric approach these underlying parameters are viewed as a set of natural coordinates on the submanifold; the set of values that describe the related N × p matrix representation is an alternate (extrinsic coordinate representation. Using this differential geometric framework, we derive the FIM with respect to a set of normal coordinates, and the resultant intrinsic CRB on subspace and rotational estimation accuracies for this constrained model. The relationship between this new bound on constrained subspace estimation accuracy and the existing bounds developed for system parameters (i.e., natural coordinates) is established and its relevance to the estimation problem is explored in detail. The CRB development for the constrained model naturally suggests an asymptotically ML estimation approach that leverages the estimate given by the standard EVD of the SCM (ML for the unconstrained signal matrix assumption). This estimate, referenced to an algorithmically convenient extrinsic coordinate set, may be transformed to the natural coordinates defined by the underlying parametric model. By the maximum likelihood (ML) invariance principle, this estimation approach yields an asymptotically ML estimator of the desired parameters. An efficient implementation of this general theoretical approach is developed for the important special case of the uniform multi-dimensional array and complex exponential waveform model. Utilizing the shift-invariant properties that define the constraint in this setting, we derive a closed-formed estimator of the signal matrix Y. This new estimation approach is applied to the challenging 2-D Direction-of-Arrival (DOA) estimation problem and a set specific scenarios drawn from the literature are evaluated to demonstrate performance