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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. To get a computational model, we can approximate the expectation with the mean over desired outputs yk and observations xk as

\[ E\left[\left\|y-A^T x\right\|^2\right] \approx \frac1N \sum_{k=1}^N \left\|y_k-A^T x_k\right\|^2 = \frac1N \sum_{k=1}^N \]

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