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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

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\hat y = f(x)

such that

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\hat y \approx y.

In particular, we will focus on linear estimates where

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\hat y=f(x):=x^T A,

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

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e=y-\hat y

and the squared error is the L2-norm of the error, that is,

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\left\|e\right\|^2 = e^T e

and its mean can be written as

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E\left[\left\|e\right\|^2\right].