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# Hilbert Space Theory

Complex random variables form a Hilbert space with inner product
$\langle X, Y \rangle = \mathbb{E}\left[XY^*\right]$
. 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
$\boldsymbol{Z_1}, \boldsymbol{Z_2}$
be defined as
$\langle \boldsymbol{Z_1}, \boldsymbol{Z_2} \rangle = \mathbb{E}\left[\boldsymbol{Z_1}\boldsymbol{Z_2}^*\right]$
The ij-th entry of the matrix is simply the scalar inner product
$\mathbb{E}\left[X_iY_j^*\right]$
where
$X_i$
and
$Y_j$
are the ith and jth entries of
$\boldsymbol{X}$
and
$\boldsymbol{Y}$
respectively. This means the matrix is equivalent to the cross correlation
$R_{XY}$
between the two vectors. We can also specify the auto-correlation
$R_X = \langle \boldsymbol{X}, \boldsymbol{X} \rangle$
and auto-covariance
$\Sigma_X = \langle \boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right] , \boldsymbol{X} - \mathbb{E}\left[\boldsymbol{X}\right] \rangle$
. 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:
$\langle \alpha_1\boldsymbol{V_1}+\alpha_2\boldsymbol{V_2}, \boldsymbol{u} \rangle = \alpha_1\langle \boldsymbol{V_1}, \boldsymbol{u} \rangle + \alpha_2\langle \boldsymbol{V_2}, \boldsymbol{u} \rangle$
.
2. 2.
Reflexive:
$\langle \boldsymbol{U}, \boldsymbol{V} \rangle = \langle \boldsymbol{V}, \boldsymbol{U} \rangle ^*$
.
3. 3.
Non-degeneracy:
$\langle \boldsymbol{V}, \boldsymbol{V} \rangle = \boldsymbol{0} \Leftrightarrow \boldsymbol{V} = \boldsymbol{0}$
.
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
$\|\boldsymbol{Z}\|^2 = \langle \boldsymbol{Z}, \boldsymbol{Z} \rangle$
.
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
$\min_{\hat{\boldsymbol{X}}(\boldsymbol{Y})}\|\hat{\boldsymbol{X}}(\boldsymbol{Y}) - \boldsymbol{X}\|^2$
has a unique solution which is a linear function of
$\boldsymbol{Y}$
. The error is orthogonal to the linear subspace of
$\boldsymbol{Y}$
(i.e
$\langle \boldsymbol{X} - \hat{\boldsymbol{X}}, \boldsymbol{Y} \rangle = \boldsymbol{0}$
)
When we do a minimization over a matrix, we are minimizing it in a PSD sense, so for any other linear function
$\boldsymbol{X}'$
,
$\|\boldsymbol{X}-\hat{\boldsymbol{X}}\|^2 \preceq \|\boldsymbol{X} - \boldsymbol{X}'\|^2.$

## Innovations

Suppose we have jointly distributed random variables
$Y_0, Y_1,\cdots,Y_n$
. Ideally, we would be able to “de-correlate” them so each new vector
$E_0$
captures the new information which is orthogonal to previous random vectors in the sequence. Since vectors of a Hilbert Space operate like vectors in
$\mathbb{R}^n$
, we can simply do Gram-Schmidt on the
$\{Y_i\}_{i=0}^n$
.

#### Definition 3

Given jointly distributed random vectors
$\{Y_i\}_{i=0}^n$
with
$\mathcal{L}_i = \text{span}\{Y_j\}_{j=0}^i$
, the ith innovation
$E_i$
is given by
$E_i = Y_i - \text{proj}(Y_i|\mathcal{L}_{i-1}) = Y_i - \sum_{j=0}^{i-1}\frac{\langle Y_i, E_j \rangle }{\|E_j\|^2}E_j$
Innovations have two key properties.
1. 1.
$\forall i\neq j,\ \langle E_i, E_j \rangle =0$
2. 2.
$\forall i,\ \text{span}\{Y_j\}_{j=0}^i = \text{span}\{E_j\}_{j=0}^i$
We can also write innovations in terms of a matrix where
$\boldsymbol{\varepsilon} = A\boldsymbol{Y}$
where
$\boldsymbol{\varepsilon} = \begin{bmatrix}E_0 & E_1 & \cdots & E_n\end{bmatrix}^T$
and
$\boldsymbol{Y} = \begin{bmatrix}Y_0 & Y_1 & \cdots & Y_n\end{bmatrix}^T$
. Since each
$E_i$
only depends on the previous
$Y_i$
, then A must be lower triangular, and because we need each
$E_i$
to be mutually orthogonal,
$R_{\varepsilon}$
should be diagonal.
$R_{\varepsilon} = AR_YA^*$
, so if
$R_Y \succ 0$
, then we can use its unique LDL decomposition
$R_Y = LDL^*$
and let
$A = L^{-1}$
.