Jacobian Multiply Function with Linear Least Squares

Using a Jacobian multiply function, you can solve a least-squares problem of the form

$$\underset{x}{\min }\frac{1}{2}{\Vert C\cdot x-d\Vert }_2^2$$

such that lb ≤ x ≤ ub, for problems where C is very large, perhaps too large to be stored. For this technique, use the 'trust-region-reflective' algorithm.

For example, consider a problem where C is a 2n-by-n matrix based on a circulant matrix. The rows of C are shifts of a row vector v. This example has the row vector v with elements of the form $$(–1{)}^{k+1}/k$$:

$$v=[1,-1/2,1/3,-1/4,\dots ,-1/n]$$,

where the elements are cyclically shifted.

$$C=\left[\begin{array}{ccccc}1& -1/2& 1/3& ...& -1/n\\ -1/n& 1& -1/2& ...& 1/(n-1)\\ 1/(n-1)& -1/n& 1& ...& -1/(n-2)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ -1/2& 1/3& -1/4& ...& 1\\ 1& -1/2& 1/3& ...& -1/n\\ -1/n& 1& -1/2& ...& 1/(n-1)\\ 1/(n-1)& -1/n& 1& ...& -1/(n-2)\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ -1/2& 1/3& -1/4& ...& 1\end{array}\right].$$

This least-squares example considers the problem where

$$d=[n-1,n-2,\dots ,-n]$$,

and the constraints are $$-5\le x_i\le 5$$ for $$i=1,\dots ,n$$.

For large enough $$n$$, the dense matrix C does not fit into computer memory ($$n=10,000$$ is too large on one tested system).

A Jacobian multiply function has the following syntax.

w = jmfcn(Jinfo,Y,flag)

Jinfo is a matrix the same size as C, used as a preconditioner. If C is too large to fit into memory, Jinfo should be sparse. Y is a vector or matrix sized so that C*Y or C'*Y works as matrix multiplication. flag tells jmfcn which product to form:

• flag > 0 ⇒  w = C*Y

• flag < 0 ⇒  w = C'*Y

• flag = 0 ⇒  w = C'*C*Y

Because C is such a simply structured matrix, you can easily write a Jacobian multiply function in terms of the vector v, without forming C. Each row of C*Y is the product of a circularly shifted version of v times Y. Use circshift to circularly shift v.

To compute C*Y, compute v*Y to find the first row, then shift v and compute the second row, and so on.

To compute C'*Y, perform the same computation, but use a shifted version of temp, the vector formed from the first row of C':

temp = [fliplr(v),fliplr(v)];

temp = [circshift(temp,1,2),circshift(temp,1,2)]; % Now temp = C'(1,:)

To compute C'*C*Y, simply compute C*Y using shifts of v, and then compute C' times the result using shifts of fliplr(v).

The helper function lsqcirculant3 is a Jacobian multiply function that implements this procedure; it appears at the end of this example.

The dolsqJac3 helper function at the end of this example sets up the vector v and calls the solver lsqlin using the lsqcirculant3 Jacobian multiply function.

When n = 3000, C is an 18,000,000-element dense matrix. Determine the results of the dolsqJac3 function for n = 3000 at selected values of x, and display the output structure.

[x,resnorm,residual,exitflag,output] = dolsqJac3(3000);

Local minimum possible.

lsqlin stopped because the relative change in function value is less than the function tolerance.

disp(x(1))

    5.0000

disp(x(1500))

   -0.5201

disp(x(3000))

   -5.0000

disp(output)

         iterations: 16
          algorithm: 'trust-region-reflective'
      firstorderopt: 5.9351e-05
       cgiterations: 36
    constrviolation: []
       linearsolver: []
            message: 'Local minimum possible.↵↵lsqlin stopped because the relative change in function value is less than the function tolerance.'

Helper Functions

This code creates the lsqcirculant3 helper function.

function w = lsqcirculant3(Jinfo,Y,flag,v)
% This function computes the Jacobian multiply function
% for a 2n-by-n circulant matrix example.

if flag > 0
    w = Jpositive(Y);
elseif flag < 0
    w = Jnegative(Y);
else
    w = Jnegative(Jpositive(Y));
end

    function a = Jpositive(q)
        % Calculate C*q
        temp = v;

        a = zeros(size(q)); % Allocating the matrix a
        a = [a;a]; % The result is twice as tall as the input.

        for r = 1:size(a,1)
            a(r,:) = temp*q; % Compute the rth row
            temp = circshift(temp,1,2); % Shift the circulant
        end
    end

    function a = Jnegative(q)
        % Calculate C'*q
        temp = fliplr(v);
        temp = circshift(temp,1,2); % Shift the circulant for C'

        len = size(q,1)/2; % The returned vector is half as long
        % as the input vector.
        a = zeros(len,size(q,2)); % Allocating the matrix a

        for r = 1:len
            a(r,:) = [temp,temp]*q; % Compute the rth row
            temp = circshift(temp,1,2); % Shift the circulant
        end
    end
end

This code creates the dolsqJac3 helper function.

function [x,resnorm,residual,exitflag,output] = dolsqJac3(n)
%
r = 1:n-1; % Index for making vectors

v(n) = (-1)^(n+1)/n; % Allocating the vector v
v(r) =( -1).^(r+1)./r;

% Now C should be a 2n-by-n circulant matrix based on v,
% but it might be too large to fit into memory.

r = 1:2*n;
d(r) = n-r;

Jinfo = [speye(n);speye(n)]; % Sparse matrix for preconditioning
% This matrix is a required input for the solver;
% preconditioning is not used in this example.

% Pass the vector v so that it does not need to be
% computed in the Jacobian multiply function.
options = optimoptions('lsqlin','Algorithm','trust-region-reflective',...
    'JacobianMultiplyFcn',@(Jinfo,Y,flag)lsqcirculant3(Jinfo,Y,flag,v));

lb = -5*ones(1,n);
ub = 5*ones(1,n);

[x,resnorm,residual,exitflag,output] = ...
    lsqlin(Jinfo,d,[],[],[],[],lb,ub,[],options);
end