# 4. Method of Steepest Descent - PowerPoint PPT Presentation

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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 .

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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)

• 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=QQH (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:

• 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.