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Problem definition

In speech processing and elsewhere, a frequently appearing task is to make a prediction of an unknown vector y from available observation vectors x. Specifically, we want to have an estimate \( \hat y = f(x) \) such that \( \hat y \approx y. \) In particular, we will focus on linear estimates where \( \hat y=f(x):=A^T x, \) and where A is a matrix of parameters.

The minimum mean square estimate (MMSE)

Suppose we want to minimise the squared error of our estimate on average. The estimation error is \( e=y-\hat y \) and the squared error is the L2-norm of the error, that is, \( \left\|e\right\|^2 = e^T e \) and its mean can be written as the expectation \( E\left[\left\|e\right\|^2\right] = E\left[\left\|y-\hat y\right\|^2\right] = E\left[\left\|y-A^T x\right\|^2\right]. \) Formally, the minimum mean square problem can then be written as

\[ \min_A\, E\left[\left\|y-A^T x\right\|^2\right]. \]

This can in generally not be directly implemented because we have the abstract expectation-operation in the middle.

(Advanced derivation begins) To get a computational model, first note that the error expectation can be written in terms of the mean of a sample of vector ek as

\[ E\left[\left\|e\right\|^2\right] \approx \frac1N \sum_{k=1}^N \left\|e_k\right\|^2 = \frac1N {\mathrm{tr}}(E^T E), \]

where \( E=\left[e_1,\,e_2,\dotsc,e_N\right] \) and tr() is the matrix trace. To minimize the error energy expectation, we can then set its derivative to zero

\[ 0 = \frac{\partial}{\partial A} \frac1N {\mathrm{tr}}(E^T E) = \frac1N\frac{\partial}{\partial A} {\mathrm{tr}}((Y-A^TX)^T (Y-A^TX)) = \frac1N(Y-A^T X)X^T \]

where the observation matrix is \( X=\left[x_1,\,x_2,\dotsc,x_N\right] \) and the desired output matrix is \( Y=\left[y_1,\,y_2,\dotsc,y_N\right] \) . (End of advanced derivation)

It follows that the optimal weight matrix A can be solved as

\[ \boxed{A = \left(XX^T\right)^{-1}XY^T = X^\dagger Y^T}, \]

where the superscript \( \dagger \) denotes the Moore-Penrose pseudo-inverse.

Estimates with a mean parameter

Suppose that instead of an estimate \( \hat y=A^T x \) , we want to include a mean vector in the estimate as \( \hat y=A^T x + \mu \) . While it is possible to derive all of the above equations for this modified model, it is easier to rewrite the model into a similar form as above with

\[ \hat y=A^T x + \mu = \begin{bmatrix} \mu^T \\ A^T \end{bmatrix} \begin{bmatrix} 1 \\ x \end{bmatrix} := A'^T x'. \]

That is, we can extend x by a single 1, (the observation X similarly with a row of constant 1s), and extend A to include the mean vector. With this modifications, the above Moore-Penrose pseudo-inverse can again be used to solve the modified model.

Estimates with linear equality constraints

(Advanced derivation begins)

In practical situations we often have also linear constraints, such as \( A^T C = B \) .


(Advanced derivation ends)

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