How to get the gradient of function from vectors to scalars?

2
votes

Given a function f:R^600 -> R f(x)=0.5*norm(Ax-b)^2, where A is a 400x600 matrix, b is 400x1, and both of them are given.

How can I get the gradient(f) at some given x0 in MATLAB?

m=400
n=600
A=randn(m,n)
b=randn(m,1)

syms x
f= 0.5*norm(A*x-b)^2
gradient(f,x)

However it does not work because it seems it regard x as scalar, not a vector.

1
matlabgradient

1 Answers

1
votes

Given the function:

f(x) = \frac{1}{2}\|Ax-b\|^2

the gradient is just:

Ax-b

Check this for further details.

Therefore, your MATLAB implementation is:

g = A'*(A*x-b);

where g is the gradient of your function for a given x.

Note that A, b and x are numerical, not symbolic.

Proof

f(x) = \frac{1}{2}\|Ax-b\|^2

Symbolic approach

You can use the following code:

N = 3;

A = [1 0 1;
     2 1 -1;
     3 1 0];

b = [1;
     2;
     3];

x = sym('x', [N, 1]);

f = 0.5*norm(A*x-b)^2;

g = sym('g', [N, 1]); % g is the gradient of f
for i=1:N
    g(i) = diff(f,x(i));
end