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METHOD OF STEEPEST DESCENT. Mean Square Error (Revisited). For a transversal filter (of length M), the output is written as and the error term wrt. a certain desired response is . Mean Square Error (Revisited). Following these terms, the MSE criterion is defined as - PowerPoint PPT Presentation
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METHOD OF STEEPEST DESCENT
Week 5
Mean Square Error (Revisited)
For a transversal filter (of length M), the output is written as
and the error term wrt. a certain desired response is
Mean Square Error (Revisited) Following these terms, the MSE criterion is defined as
Substituting e(n) and manupulating the expression, we get
where
Quadratic in w !
Mean Square Error (Revisited) For notational simplicity, express MSE in terms of vector/matrices
where
Mean Square Error (Revisited)
We found that the solution (optimum filter coef.s wo) is given by the Wiener-Hopf eqn.s
Inversion of R can be very costly. J(w) is quadratic in w → convex in w → for wo,
Surface has a single minimum and it is global, then
Can we reach to wo, i.e. with a less demanding algorithm?
Basic Idea of the Method of Steepest Descent
Can we find wo in an iterative manner?
Basic Idea of the Method of Steepest Descent
Starting from w(0), generate a sequence {w(n)} with the property
Many sequences can be found following different rules.
Method of steepest descent generates points using the gradient Gradient of J at point w, i.e. gives the direction at which
the function increases most. Then gives the direction at which the function
decreases most. Release a tiny ball on the surface of J → it follows negative
gradient of the surface.
Basic Idea of the Method of Steepest Descent
For notational simplicity, let
, then going in the direction given by the negative gradient
How far should we go in –g → defined by the step size param. μ Optimum step size can be obtained by line search - difficult Generally a constant step size is taken for simplicity.
Then, at each step improvement in J is (from Taylor series expansion)
Application of SD to Wiener Filter For w(n)
From the theory of Wiener Filter we know that
Then the update eqn. Becomes
which defines a feedback connection.
Convergence Analysis Feedback → may cause stability problems under certain conditions.
Depends on The step size, μ The autocorrelation matrix, R
Does SD converge? Under which conditions? What is the rate of convergence?
We may use the canonical representation.
Let the weight-error vector be
then the update eqn. becomes
Convergence Analysis Let
be the eigendecomposition of R.
Then
Using QQH=I
Apply the change of coordinates
Then, the update eqn. becomes
Convergence Analysis We know that Λ is diagonal, then the k-th natural mode is
or, with the initial values vk(0), we have
Note the geometric series
Convergence Analysis Obviously for stability
or, simply
Geometric series results in an exponentially decaying curve with time constant τk, where letting
or
Convergence Analysis We have
but
We know that Q is composed of the eigenvectors of R, then
or
Each filter coefficient decays exponentially. The overall rate of convergence is limited by the slowest and fastest
modes
then
Convergence Analysis For small step size
What is v(0)? The initial value v(0) is
For simplicity assume that w(0)=0, then
Convergence Analysis Transient behaviour:
From the canonical form we know that
then
As long as the upper limit on the step size parameter μ is satisfied, regardless of the initial point
Convergence Analysis The progress of J(n) for n=0,1,... is called the learning curve.
The learning curve of the steepest-descent algorithm consists of a sum of exponentials, each of which corresponds to a natural mode of the problem.
# natural modes = # filter taps
Example
A predictor with 2 taps (w1(n) and w2(n)) is used to find the params. of the AR process
Examine the transient behaviour for Fixed step size, varying eigenvalue spread Fixed eigenvalue spread, varying step size.
σv2 is adjusted so that σu
2=1.
Example The AR process:
Two eigenmodes
Condition number (Eigenvalue Spread):
Example (Experiment 1)
Experiment 1: Keep the step size fixed at
Change the eigenvalue spread
Example (Experiment 1)
Example (Experiment 2)
Keep the eigenvalue spread fixed at
Change the step size (μmax=1.1)
Example (Experiment 2)
Depending on the value of μ, the learning curve can be Overdamped, moves smoothly to the min. ((very) small μ) Underdamped, oscillates towards the min. (large μ< μmax) Critically damped Generally rate of convergence is slow for the first two.
Observations SD is a ‘deterministic’ algorithm, i.e. we assume that
R and p are known exactly. In practice they can only be estimated
Sample average?
Can have high computational complexity.
SD is a local search algorithm, but for Wiener filtering, the cost surface is convex (quadratic) convergence is guaranteed as long as μ< μmax is satisfied.
Observations The origin of SD comes from the Taylor series expansion (as many
other local search optimization algorithms)
Convergence can we very slow. To speed up the process, second term can also be included as in
the Newton’s Method
High computational complexity (inversion), numerical stability problems.
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