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Hilbert Space Theory

Complex random variables form a Hilbert space with inner product
⟨X,Y⟩=E[XY∗]\langle X, Y \rangle = \mathbb{E}\left[XY^*\right]
. If we have a random complex vector, then we can use Hilbert Theory in a more efficient manner by looking at the matrix of inner products. For simplicity, we will call this the “inner product” of two complex vectors.

Definition 1

Let the inner product between two random, complex vectors
Z1,Z2\boldsymbol{Z_1}, \boldsymbol{Z_2}
 be defined as
​
⟨Z1,Z2⟩=E[Z1Z2∗]\langle \boldsymbol{Z_1}, \boldsymbol{Z_2} \rangle = \mathbb{E}\left[\boldsymbol{Z_1}\boldsymbol{Z_2}^*\right]
​
The ij-th entry of the matrix is simply the scalar inner product
E[XiYj∗]\mathbb{E}\left[X_iY_j^*\right]
where
XiX_i
 and
YjY_j
 are the ith and jth entries of
X\boldsymbol{X}
 and
Y\boldsymbol{Y}
 respectively. This means the matrix is equivalent to the cross correlation
RXYR_{XY}
 between the two vectors. We can also specify the auto-correlation
RX=⟨X,X⟩R_X = \langle \boldsymbol{X}, \boldsymbol{X} \rangle
and auto-covariance
ΣX=⟨X−E[X],X−E[X]⟩\Sigma_X = \langle \boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right] , \boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right] \rangle
. One reason why we can think of this matrix as the inner product is because it also satisfies the properties of inner products. In particular, it is
  1. 1.
    Linear:
    ⟨α1V1+α2V2,u⟩=α1⟨V1,u⟩+α2⟨V2,u⟩\langle \alpha_1\boldsymbol{V_1}+\alpha_2\boldsymbol{V_2}, \boldsymbol{u} \rangle = \alpha_1\langle \boldsymbol{V_1}, \boldsymbol{u} \rangle + \alpha_2\langle \boldsymbol{V_2}, \boldsymbol{u} \rangle
    .
  2. 2.
    Reflexive:
    ⟨U,V⟩=⟨V,U⟩∗\langle \boldsymbol{U}, \boldsymbol{V} \rangle = \langle \boldsymbol{V}, \boldsymbol{U} \rangle ^*
    .
  3. 3.
    Non-degeneracy:
    ⟨V,V⟩=0⇔V=0\langle \boldsymbol{V}, \boldsymbol{V} \rangle = \boldsymbol{0} \Leftrightarrow \boldsymbol{V} = \boldsymbol{0}
    .
Since we are thinking of the matrix as an inner product, we can also think of the norm as a matrix.

Definition 2

The norm of a complex random vector is given by
∥Z∥2=⟨Z,Z⟩\|\boldsymbol{Z}\|^2 = \langle \boldsymbol{Z}, \boldsymbol{Z} \rangle
.
When thinking of inner products as matrices instead of scalars, we must rewrite the Hilbert Projection Theorem to use matrices instead.

Theorem 1 (Hilbert Projection Theorem)

The minimization problem
min⁡X^(Y)∥X^(Y)−X∥2\min_{\hat{\boldsymbol{X}}(\boldsymbol{Y})}\|\hat{\boldsymbol{X}}(\boldsymbol{Y}) - \boldsymbol{X}\|^2
 has a unique solution which is a linear function of
Y\boldsymbol{Y}
. The error is orthogonal to the linear subspace of
Y\boldsymbol{Y}
 (i.e
⟨X−X^,Y⟩=0\langle \boldsymbol{X} - \hat{\boldsymbol{X}}, \boldsymbol{Y} \rangle = \boldsymbol{0}
)
When we do a minimization over a matrix, we are minimizing it in a PSD sense, so for any other linear function
X′\boldsymbol{X}'
,
​
∥X−X^∥2⪯∥X−X′∥2.\|\boldsymbol{X}-\hat{\boldsymbol{X}}\|^2 \preceq \|\boldsymbol{X} - \boldsymbol{X}'\|^2.
​

Innovations

Suppose we have jointly distributed random variables
Y0,Y1,⋯ ,YnY_0, Y_1,\cdots,Y_n
. Ideally, we would be able to “de-correlate” them so each new vector
E0E_0
 captures the new information which is orthogonal to previous random vectors in the sequence. Since vectors of a Hilbert Space operate like vectors in
Rn\mathbb{R}^n
, we can simply do Gram-Schmidt on the
{Yi}i=0n\{Y_i\}_{i=0}^n
.

Definition 3

Given jointly distributed random vectors
{Yi}i=0n\{Y_i\}_{i=0}^n
 with
Li=span{Yj}j=0i\mathcal{L}_i = \text{span}\{Y_j\}_{j=0}^i
, the ith innovation
EiE_i
 is given by
​
Ei=Yi−proj(Yi∣Li−1)=Yi−∑j=0i−1⟨Yi,Ej⟩∥Ej∥2EjE_i = Y_i - \text{proj}(Y_i|\mathcal{L}_{i-1}) = Y_i - \sum_{j=0}^{i-1}\frac{\langle Y_i, E_j \rangle }{\|E_j\|^2}E_j
​
Innovations have two key properties.
  1. 1.
    ​
    ∀i≠j, ⟨Ei,Ej⟩=0\forall i\neq j,\ \langle E_i, E_j \rangle =0
    ​
  2. 2.
    ​
    ∀i, span{Yj}j=0i=span{Ej}j=0i\forall i,\ \text{span}\{Y_j\}_{j=0}^i = \text{span}\{E_j\}_{j=0}^i
    ​
We can also write innovations in terms of a matrix where
ε=AY\boldsymbol{\varepsilon} = A\boldsymbol{Y}
 where
ε=[E0E1⋯En]T\boldsymbol{\varepsilon} = \begin{bmatrix}E_0 & E_1 & \cdots & E_n\end{bmatrix}^T
 and
Y=[Y0Y1⋯Yn]T\boldsymbol{Y} = \begin{bmatrix}Y_0 & Y_1 & \cdots & Y_n\end{bmatrix}^T
. Since each
EiE_i
 only depends on the previous
YiY_i
, then A must be lower triangular, and because we need each
EiE_i
 to be mutually orthogonal,
RεR_{\varepsilon}
 should be diagonal.
Rε=ARYA∗R_{\varepsilon} = AR_YA^*
, so if
RY≻0R_Y \succ 0
, then we can use its unique LDL decomposition
RY=LDL∗R_Y = LDL^*
 and let
A=L−1A = L^{-1}
.
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