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4. Method of Steepest Descent

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4. Method of Steepest Descent

- There are two problems associated with the Wiener filtering in practical applications.
- The matrix inversion operation is difficult to implement.
- The R and P may not may easy to estimate.

- To overcome the first problem, we may solve the Wiener solution iteratively.
- Consider a optimization problem.
- A simplest procedure to solve the optimization problem iteratively is called the method of steepest descent.

- Observation: The gradient of J(w0) corresponds to a direction that has a largest slope at w0.

- Method of steepest descent (SD):
- Initial guess : w(0)
- Compute the gradient vector
- update w
- Repeat the process (i=i+1)

- The parameter is called the step size. It controls the rate of the convergence.

- Graphical interpretation:

- Project to the xy plane:

- Recall that for the Wiener filtering problem.
- Thus, we can use the SD method to solve the Wiener filtering problem.
- For real signals, we have

- For complex signals, we have
- The the update equation for the SD method is
- The SD method is a recursive algorithm. It is subject to the possibility of unstable.

- Let
- The weight update equation can be written as
- Since R is the correlation matrix, R=QQH (QHQ=I).
- Let v(i)=QHc(i). Then

- For the k-th component of v(i), we have
- Thus, for vk(i) to converge, it is necessary that
- Since all eigenvalues are nonnegative,
- To ensure every mode is convergent, we have

- Thus, if the step size satisfies the condition,
- The time constant (measuring the convergence speed).
- Since v(i)=QH[w(i)-wopt], we have

- For the i-th component of w(i), we have
- Let a be the time constant of wi(i). Then
- As we can see that the convergence speed is limited by min. However, we can adjust the step size such that the mode corresponding to max converges fast.
- We conclude that the factor control the rate of convergence is the eigenvalue spread (max/min). The smaller the eigenvalue spread, the faster the convergence rate we can achieve.

- The MSE can be analyzed similarly.
- If the step size is properly chosen,
- The curve by plotting J[w(i)] versus i is called the learning curve. The time constant associated with the k-th mode is

- Eigenvectors/eigenvalues of R:
- Thus, J(w) is a paraboloid. If we cut the paraboloid with planes parallel to w plane [J(w)=constant]. We obtain concentric ellipses.
- Let c=w-wopt. Then, vHRv=-Jmin and J=2Rc. Note that J is normal to cHRc. The principle axis of an ellipsoid passes the origin (c=0) and is normal to vTRv. If cp is a principle axis, it must satisfy
- Thus, the eigenvectors of R define the principle axes of the error surface.

- Geometrical interpretation:

c1

v1

c0

v0

- The eigenvalues of R give the second derivative of the error surface r.w.t. the principle axes of J=c (what does this mean?).
- Thus, if the eigenvalue spread is larger, the shape of the ellipsoid is more peculiar.
- Note that
- If we can translate and rotate the coordinates of w (to v), components of weights can be decoupled. As a matter of fact, we can use a different step size for different mode. This can have a fastest convergence rate.

- Recall the weight update equation.
- Let rk=(1-k). Thus, for the k-th mode, the convergence condition is then -1<ck<1.

- Weight convergence:

underdamped

overdamped

- Example: identification of AR parameters

d(n)=u(n)

- Convergence trajectory:

- Four eigenvalue spreads:

- continue:

- In terms of w(n):

- Newton’s method
- Newton’s method is primarily a method for finding zeros of a equation.

- Finding the minimum of a function g(x) means solve the equation g’(x)=0. This leads to the searching algorithm

- For the Wiener filtering problem, we have
- Thus, Newton’s method is then
- As shown, Newton’s method do not proceed in the gradient direction. Introducing the step size, we have

Or,

- Convergence properties:
- Thus, Newton’s method will converge if
- Properties
- Convergence of Newton’s method is same for every mode and doesn’t dependent on the eigenvalue spread of R.
- The computation is more intensive (require R-1).
- For nonquadratic cost function, Newton’s method is easy to become unstable.

- Question: if we know R-1, we can directly find wopt. Why do we have to use Newton’s method?
- Reason:
- Exact R-1 may not be necessary. Some efficient methods can be applied to find an approximated of R-1. This is specially true when the input is time-variant.

- In general, straightforward Newton’s method is seldom used. Only the concept is adopted.