Sistemas de Valoración

The unitary group U (N ), the collection of all N × N unitary matrices, may be viewed a submanifold of the complex space CN ×N, the set of all N × N complex matrices. Manifolds such as U (N ) and the usual 2-sphere are examples of Rieman- nian manifolds, general curved spaces with smoothly verifying metric functions. The submanifold inherits an inner product or metric from the parent manifold and this estab- lishes the intrinsic geometry of the space, most importantly geodesics (see chapter ?? and [3]). The Steifel and Basis manifolds likewise inherit their metric functions from their formulations as a quotient manifolds on U (N ).

Estimation problems where the parameter space is a Riemannian manifold differ from the more usual parameter estimation in two important ways. In the typical esti- mation problem (e.g; angle and Doppler) a fixed set of coordinates for the parameter space is naturally provided. In contrast, for the general manifold case, no set of intrinsic or preferred coordinates are given. This is the situation with position estimation on the Euclidean flat plane, for example, where no particular set of coordinates are naturally provided and Cartesian and polar are equally valid choices. Estimators and evaluators are free to choose among these or any others. The second difference with the usual es- timation problem is that more general curved Riemannian manifolds such as the sphere are not themselves vector spaces and so the vector space concepts implicitly used in classical Cram´er-Rao bounds (CRB) approaches are not immediately applicable.

In these situations, a universally meaningful estimation error measure is the dis- tance between the estimate and the true location. Distance is an invariant or intrinsic quantity dependent on the given metric but independent of any particular choice of co- ordinates (e.g., the 2-D mean square positional error on the sphere).

In [2] a methodology is established for computing the CRB when the parameter space is an arbitrary Riemannian Quotient manifolds. The frequently encountered exam- ple of estimating an unknown subspace is analyzed and the author gives intrinsic CRBs in the form of matrix inequalities relating the covariance of estimators and the Fisher information. The classical bounds developed for Euclidean spaces are generalized to Riemannian Quotient manifolds, such as B, via the exponential map, i.e., geodesics em- anating from the estimate to points in the parameter space3. In section 10 the approach detailed in [2] is used to compute the full FIM for the Basis manifold formulation of the unconstrained problem (342). The subspace submatrix of the FIM for this case is shown to be identical in form to that given in [2] for the Grassmann manifold formulation. The

3_{“Just as with classical bounds, the unbiased intrinsic CRBs depend asymptotically only on the Fisher}

information and do not depend in any nontrivial way on the choice of measurement units (e.g., degree versus radians) [2].

rotation submatrix of the FIM which appears in the Basis manifold formulation (but absent in the Grassmann case) is also computed here.

In section 3.8 the FIM for the constrained version of (342) is developed and closed form results are provided for the important special case when the signal matrix H(θ) that defines the constraint has a modal form.

Exponential Map and Normal Coordinates

Geodesics on general curved Riemannian manifolds are defined as distance min- imizings curves between points. For the special case of the standard Euclidean (flat) manifold E , a geodesic emanating from the point Y0 in the direction of the vector ∆ is

the straight line Y (t) = Y0+ t · ∆. The vector ∆ is an element of a vector space with

origin at Y0. This vector space is denoted TY0E where “T ” is read as “tangent” and Y0is

the origin or base which corresponds to the “head” of the vector.

For the unitary group manifold U (N ), a geodesic emanating from the point X0 in

the direction ∆ = X0W (W anti-Hermitian) is the curve given by exponential map

X(t) = X0expm(tW ) . (351)

Note that

∆ ≡ dX(t)

dt |t=0 = X0W (352)

so that the description of ∆ as a tangent is consistent with intuition. The collection of all such tangents through the point X0defines a vector spaced termed the tangent space.

This space is denoted as TX0U (N ) and the tangent vector ∆ is an element of this space.

The short-hand notation for the above exponential map at the point X0with tangent

vector ∆ = X0· W (351) is4

X(t) = expX0(∆ · t) . (353)

4_{Note that for W anti-Hermitian Q(t) = expm(tW ) is a unitary matrix so that X(t) = X} 0Q(t) is

The exponential or geodesic map plays a central role in the development of the intrinsic CRB for U (N ) and related quotient manifolds. For the B manifold, a quotient manifold of U (N ), geodesics are naturally defined by constraining the set of tangent vectors to a subspace of TX0U (N ), termed the horizontal space, that corresponds to TX0B. Elements

of this subspace have the same general form as with (352), with the constraint expressed through the matrix W in a partitioned form as

W = " A −BH B 0 # . (354)

The exponential map can be expressed in a more concrete form if we select only the first p columns of (351) by right multiplying by the selection matrix IN,p. Writing

the general unitary matrix X in partitioned form as X =h Y Y⊥ i

, where Y denotes the first p columns we have

Y (t) =h Y0 Y0⊥ i

exp(tW )IN,p (355)

where Y (t) = X(t)IN,p. The tangent vector

∆ ≡ dY (t) dt |t=0 = h Y0 Y0⊥ i W · IN,p= Y0A + Y0⊥B (356)

is as an element of the tangent space TYB. This space is evidently partitioned as the

direct sum

TYB = AY ⊕ BY (357)

where elements of AY and BY have the general form Y A and Y⊥B respectively. The

exponential map on B defined by (355) and (354) is denoted in short hand as Y (t) = expY0t∆ where ∆ ∈ TY0B has the form (356).

Denoting the basis vectors for AY and BY as ak and bk, respectively, an arbitrary

∆ expands as ∆ = Na X k=1 akαk+ Nb X k=1 bkβk (358)

where α and β are the real components of ∆ with respect to the given basis.5

The exponential map Y = expY0∆ which maps Y0 to Y , based on ∆, is invertible

(for small ∆ this map is well-approximated numerically by Y0+ ∆). Given an arbitrary

point Y near Y0 this inverse exponential map, which assigns a tangent vector ∆ ∈ TY0B

to the pair Y0, Y1, is denoted as ∆ = exp−1Y0Y . This defines the difference vector between

the points, and is the manifold generalization of the difference vector ∆ = Y − Y0in the

usual flat space. If ∆ = exp−1_Y0Y is expressed with respect to a given basis as (358) then

we see the parameters α, β provide an alternate means of labeling the point Y . These parameters are the so-called normal coordinates of Y . The exponential map and its inverse provide the transformation between extrinsic Y coordinates and intrinsic (α, β) normal coordinates. Since the choice of basis vectors is not unique so neither are the normal coordinates.

Functions defined on the product manifold (350) may be expressed in terms of any convenient set of coordinates. Thus the log-likelihood function L(Y, Λ) may al- ternately be written as L(α, β, Λ). This situation is analogous to the alternate coordi- nate representations of points on the usual sphere S2 in 3-space. An arbitrary point P ∈ S2 _{may be represented in extrinsic (x, y, z) Cartesian coordinates or by latitude-}

longitude (u, v) coordinates with the systems related by the invertible transformation (x, y, z) = h(u, v). A real function on S2 is the mapping L: S2 → R and its value at the point P is L(P). For the two above coordinate systems the value is expressed as L(P) = LY(x, y, z) = Lϕ(u, v) (see section A.1).

The distance between two points on a Riemannian Manifold is the path length of geodesic curve between them. It was shown in section 2.2.7 that this length is equal to the magnitude of the tangent vector ∆ = exp−1_Y0Y so that

5_{We use the word vector in the abstract sense, not as a N × 1 column matrix; this is sometimes}

distB(Y1, Y0) ≡ k∆k = q

kαk2_{+ kβk}2 _{. (359)}

Estimation Problem

We consider an estimation problem on the product manifold B × D based on mea- surements in M = CN × · · · × CN_{, with the probability density function of the mea-}

surement given a parameter (Y ∈ B, λ ∈ D) denoted as

f (:, Y, λ) : M → R . (360)

Let

L(:, Z) : B × D → R (361)

be the associated log likelihood function L(Y, λ; Z) = logf (Z; Y, λ) where Z ∈ M. The measurement data matrix Z is a random sample whose probability density function is shaped by the unknown parameters Y and Λ. An estimator of Y , denoted

ˆ

Y , is a mapping from the probability space M to the parameter space B, that is, ˆY : M → B or ˆY (Z). Similarly a generic estimator of Λ is denotedΛ and is given by anb

analogous mapping ˆΛ : M → D.

The estimation error in the general manifold case is the tangent vector ˆ∆(Z) ≡ exp−1_Y Y (Z). Going forward for conciseness we writeb Y to meanb Y (Z) and ˆb ∆ to mean

ˆ

∆(Z). By definition, ˆ∆ ∈ TYB, and given a partitioned orthonormal basis for this space

the estimation error vector expands as ˆ ∆ = Na X k=1 akαˆk+ Nb X k=1 bkβˆk. (362)

For conciseness we again write ˆα, ˆβ to mean the stochastic quantities ˆα(Z), ˆβ(Z). For unbiased estimators the error covariance is given by the tensor product E[ ˆ∆ ⊗ ˆ

basis, this covariance matrix is C = " Cαα Cβα Cαβ Cββ # = E   " ˆ α ˆ β # " ˆ α ˆ β #T = E " ˆ α ˆαT _{β ˆ}ˆ_αT ˆ α ˆβT _{β ˆ}ˆ_βT # (363)

where for convenience Cαα ≡ E[ ˆα ˆαt], etc. (see section A.1 in particular (A.89)).

Denoting the orthogonal projector onto BY < TYB as PB we have PB∆ =ˆ PNb

k=1bkβˆkand so

E[kPB∆kˆ 2] = E h

k ˆβk2 i= trCββ (364)

where we have used the standard vector identity kβk2 = tr (ββt_).