Optimality

286 V. J. Ribeiro, R. H. Riedi and R. G. Baraniuk
we have
(7.29)
cov(Vγk, E)
cov(Vγ, E) =
⎛
� �1/2 var(Vγk)
var(Vγ)
�
var(Vγk)
=: ξγ,k≥
var(Vγ)
⎝ ϱ2 γk
� 1/2
+ var(Wγk)
var(Vγ)
ϱ 2 γk
From (7.30) we see that ξγ,k is not a function of E.
Denote the covariance between Vγ and leaf node vector L = [ℓi]∈Λγk(n) as
Θγ,L = [cov(Vγ, ℓi)] T . Then (7.30) gives
(7.30) Θγk,L = ξγ,kΘγ,L.
From (4.2) we have
(7.31) E(Vγ|L) = var(Vγ)−ϕ(γ, L)
where ϕ(γ, L) = ΘT γ,LQ−1 L Θγ,L. Note that ϕ(γ, L)≥0 since Q −1
L is positive semidefinite.
Using (7.30) we similarly get
(7.32) E(Vγk|L) = var(Vγk)−
.
ϕ(γ, L)
ξ2 .
γ,k
From (7.31) and (7.32) we see thatE(Vγ|L) andE(Vγk|L) are both minimized over
L∈Λγk(n) by the same leaf vector that maximizes ϕ(γ, L). This proves Claim (2).
Claim (3): µγ,γk(n) is a positive, non-decreasing, and discrete-concave function
of n,∀k, γ.
We start at a node γ at one scale from the bottom of the tree and then move up
the tree.
Initial Condition: Note that Vγk is a leaf node. From (2.1) and (??) we obtain
(7.33) E(Vγ|Vγk) = var(Vγ)−
(ϱγkvar(Vγ)) 2
var(Vγk)
⎞
⎠
≤ var(Vγ).
For our choice of γ, µγ,γk(1) corresponds toE(Vγ|Vγk) −1 and µγ,γk(0) corresponds
to 1/var(Vγ). Thus from (7.33), µγ,γk(n) is positive, non-decreasing, and discreteconcave
(trivially since n takes only two values here).
Induction Step: Given that µγ,γk(n) is a positive, non-decreasing, and discreteconcave
function of n for k = 1, . . . , Pγ, we prove the same when γ is replaced by
γ↑. Without loss of generality choose k such that (γ↑)k = γ. From (3.11), (3.13),
(7.31), (7.32) and Claim (2), we have for L∈Lγ(n)
(7.34)
µγ(n) =
µγ↑,k(n) =
1
var(Vγ) ·
1
var(Vγ↑) ·
1
1− ϕ(γ,L)
1−
var(Vγ)
1
ϕ(γ,L)
, and
ξ 2
γ↑,k var(Vγ↑)
From (7.26), the assumption that µγ,γk(n)∀k is a positive, non-decreasing, and
discrete-concave function of n, and Lemma 3.1 we have that µγ(n) is a nondecreasing
and discrete-concave function of n. Note that by definition (see (3.11))
.
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