For any Y0 ∈ ¯B the full tangent space BY0, the space corresponding to subspace
perturbations, is naturally partitioned as BY0 = SY0 ⊕ NY0, where SY0 is the space
tangent to ¯B and NY0 is the orthogonal complement in BY0 (see figure 7). Using parallel
translation, a similarly partitioned basis for BY˜ at the unconstrained MLE ˜Y may be
formed. Using this S-N frame an alternate form of the MLE criterion (637) is developed. After developing this form we consider a new solution method when the submanifold is defined by a shift-invariant constraint condition.
Constraint Submanifold using an S-N Frame: Examples
Before considering the full problem on the Basis manifold we discuss an example set in a 3-dimensional Euclidean space with the primary objective of illuminating the submanifold membership constraint condition (636). In order to make the ideas more
easily transferable we denote this Euclidan space by B and generic points in the space by the symbol Y . The submanifold of interest, denoted ¯B, is a two-dimensional plane surface.
Consider a point Y0 ∈ ¯B and let n be normal to ¯B and let a and s be an orthogonal
pair tangent to ¯B. Define the 2-dimensional space spanned by n and s as BY0 and the
1-dimensional space spanned by a as AY0 (the full 3-dimensional tangent space, denoted
TY0B is then TY0B = AY0 ⊕ BY0).
Let ˜Y denote a point not on submanifold ¯B that lies in plane defined by BY0. The
submanifold membership condition Y ∈ ¯B can be stated in terms of ˜Y and the dis- placement vector ∆ as ( ˜Y + ∆) ∈ ¯B. The set of vectors ∆ that satisfy this condition is denoted ¯V and is defined as
¯
VY˜ = {∆ ∈ BY˜ such that ( ˜Y + ∆) ∈ ¯B} (638)
(compare to (634)).
By translating the set of basis vectors a, s, n from Y0 to ˜Y we establish a basis for
the vector space BY˜. Since the space is flat these vectors can be translated to the point
˜
Y in the usual rigid way (i.e.; no rotations). Representing ∆ with respect to this basis creates a new coordinate system with origin at ˜Y .
The vector with head at ˜Y and tail at Y0 is given by ∆ = (Ye 0 − ˜Y ). Since by
construction Y0 and ˜Y lie in the same BY0 plane this vector can be represented in the
S-N basis for BY˜ as ∆ = ss0+ nˆn, where
ˆ
n = h∆, ni ,e (639)
s0 = h∆, si .e (640)
The 3-tuple (α = 0, s = s0, n = ˆn) are the normal coordinates of Y0 with respect to the
Since, by construction, s at ˜Y is parallel to the plane ¯B, elements in the set ¯VY˜ are
given by
∆ = ss + nˆn (641)
where ˆn is given by (639) and s is a free parameter.
IfY ∈ ¯b B then Y = ˜b Y + (sˆs + nˆn) for ˆn fixed and s chosen appropriately. The
components (0, ˆs, ˆn) are the normal coordinates ofY with respect to the s, n basis at ˜b Y .
The point Y can, of course, also be expressed in terms of a coordinate system at withb
origin Y0. Referenced to Y0 the coordinates ofY are (0, ˆb s − s0, 0).
The distance between ˜Y and Y is equal to the length of the tangent vector ∆ and, for the s, n basis, we have
distB( ˜Y , Y ) = k∆k = q
ksk2+ kˆnk2 . (642)
From this expression it is evident that the point of ¯B nearest to ˜Y , denoted YM D where
the subscript stands for minimum distance, has normal coordinates (0, ˆb) so that
YM D = ˜Y + (nˆn) . (643)
Constraint Submanifold using an S-N Frame: Basis Manifold
The development of the submanifold membership condition for the manifold case proceeds in same way outlined above with a few modifications. Recall that for any point Y0 ∈ ¯B, the tangent space BY0 of the ambient manifold B partitions as
BY0 = SY0 ⊕ NY0 (644)
where NY0 is normal to the submanifold tangent plane TY0B = A¯ Y0 ⊕ SY0. Let sk and
nkbe an orthonormal basis set for SY0 and NY0, respectively.
In the Euclidean space example this basis defined at Y0 was translated in the usual
case this simple translation is replaced by its generalization, termed parallel translation. Denoting the parallel translation operation as τ we can define a set of q vectors at ˜Y by
(sk)Y˜ ≡ τ ((sk)Y0) , k = 1 · · · q . (645)
Denote the subspace of BY˜ spanned by this set as SY˜ and its orthogonal complete as NY˜
so that
BY˜ = SY˜ ⊕ NY˜ . (646)
Let (nk)Y˜ be an orthonormal basis for NY˜. When it is clear from the context the sub-
script indicating the base point of the vector is suppressed for convenience. Using the orthonormal S-N basis for BY˜ a vector in this space expands as
∆ = q X k=1 sksk+ Nb−q X k=1 nknk. (647)
Given the two points ˜Y and Y0the tangent vector between them is defined by∆ =e
exp−1˜
Y Y0 and the components, denoted (s0, ˆn) are found using (639) and (640). These
components (0, s0, ˆn) are the normal coordinates of Y0 with respect to the given basis
for BY˜.
The set of tangent vectors ∆ ∈ BY˜ that satisfy the membership condition expY˜ ∆ ∈
¯
B can be expressed in terms of this S-N basis and the ˆn component defined above. Since SY˜ is parallel to SY0 the set of tangent vectors ¯VY˜ (641) have the form
∆Y˜ = q X k=1 sksk+ Nb−q X k=1 nknˆk (648)
where ˆn is fixed by (639) and where s is a real free q-tuple.
The geodesic distance between the unconstrained MLE ˜Y and nearby elements of Y ∈ ¯B is given by the magnitude of the tangent vector between them. Using (647) this magnitude is given in terms of the constraint-satisfying pair (s, ˆn) as
distB( ˜Y , Y ) = k∆k = q
The point on ¯B that is the minimum distance from ˜Y , denoted YM D, occurs for s = 0
and is YM D = expY˜∆ where
∆ =
N n X
k=1
nknˆk . (650)
Alternate Real Representations and the set ¯VY˜
Suppose that a partitioned set of orthonormal S-N basis vectors for BY˜ are given by
sk= ˜Y⊥Skand nk= ˜Y⊥Nk. Using these basis vectors to represent the vector ∆ = ˜Y⊥B
as in (647) yields B ≡ q X k=1 Sksk+ Nb−q X k=1 Nknk. (651)
Vectorizing both sides of this equation yields
b = Ess + Enn (652)
where b = vec(B) and
Es= h vec(S1) · · · vec(Sq) i , (653) En = h vec(N1) · · · vec(N(Nb−q)) i . (654)
From (648) the set of complex ¯Nb-tuples that satisfy the membership constraint condi-
tion is given by
b = bn+ Ess (655)
where
bn= Ennˆ (656)
and s is a free real q-tuple.
Constrained MLE in terms of b
Substituting the constraint-satisfying form for b (655) into the constrained MLE criterion (637) yields an unconstrained minimization in s
ˆ s = min s Re (bn+ Ess)HΛebb(bn+ Ess) . (657)
Expanding the quadratic form on the right hand side yields ˆ s = min s Ls|YM D + s tL sss + Re(bHnΛebbbn) (658)
where the submatrix Fss(see (471)) is
Lss ≡ Re(EsHΛebbEs) (659)
and
Ls|YM D ≡ 2Re(E
H
s Λebbbn) . (660)
This term (660) corresponds to the first derivative of L with respect to the subspace coordinate variable s evaluated at the point YM D (643). This is discussed more fully in
appendix C.6. Since the last term on the right hand side of is Re(bHnΛebbbn) ≥ 0 and
indepdendent of s the solution to (657) is given by
ˆ s = min s (s t(L s|YM D + Lsss)) (661) which is ˆ s = L−1ssLs|YM D . (662)
Substituting this result for ˆs into (655) yields ˆb = b
n+ Ess .ˆ (663)
The MLE for Y ∈ ¯B denotedY then is given byb
b
Y = expY˜ ∆ˆ (664)
where the MLE tangent vector is
b