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5
votes
MATLAB has a built-in function lsqnonneg() which is an implementation of the active set method described in the book "Solving Least Squares Problems" by Lawson and Hanson (1974) The \ solution of l …
answered Nov 11 '13 by Brian Borchers
5
votes
If $A$ is of full column rank and $A^{T}A$ is non-singular and well-conditioned, then you can compute the pseudoinverse as: $ A^{\dagger}=(A^{T}A)^{-1}A^{T} $. This will be faster than computing a …
answered Sep 5 by Brian Borchers
3
votes
If you have access to the MATLAB optimization toolbox then this can easily be done using the quadprog() function. You'd start by writing the objective in quadratic form as $ \| Ax - b \|_{2}^{2} = …
answered May 9 '14 by Brian Borchers
1
vote
Others have already supplied the two most likely answers to this question, but I'll add a bit of comparison and a way to help decide between the two approaches. I'd suggest either A primal-dual in …
answered Jul 23 '14 by Brian Borchers
1
vote
Neither of these approaches is recommended. Although pivoted Cholesky factorization can help with badly conditioned matrices, it ultimately won't help with a singular matrix. The modified Cholesky f …
answered May 29 by Brian Borchers
2
votes
Minimizing the 2-norm of $x$ among all least squares solutions is relatively easy to do- this is the pseudoinverse solution. It can be computed using either a rank revealing version of the QR factori …
answered Apr 2 '14 by Brian Borchers
1
vote
The notation here is very confusing, and that's probably the main cause of your difficulty in formulating this as a linear least squares problem. Introduce the notation $\mbox{vec}(x)$ for the $mp$ …
answered Apr 22 '14 by Brian Borchers
5
votes
In general, you can formulate this as a nonlinear least squares problem. If your values are known at points $(x_{i},y_{i})$, and the known values are $f_{i}$, then you can minimize $\min_{a,b,c,d,e} …
answered Dec 6 '17 by Brian Borchers
1
vote
This isn't a linear least squares problem as it is written. However, the problem does simplify down to a quadratic optimization problem with nonnegativity constraints and you can even write it as a l …
answered Feb 14 '14 by Brian Borchers
4
votes
If we let $\phi(x)=\sum_{i=1}^{m} F_{i}(x)^{2}$, we could compute $\nabla \phi(x)$ by finite difference approximation. However, it is generally smart to make use of the special structure of $\phi …
answered Mar 29 '18 by Brian Borchers
2
votes
If $N$ is on the order of 100,000 and $m$ is on the order of $100$, Then $J$ requires about 80 gigabytes to store in double precision and $V$ requires a trivial amount of storage. The product $M=JV$ …
answered Jun 30 '18 by Brian Borchers
1
vote
Your problem is still a linear least squares problem. You can write $\Psi(x)$ as $\Psi(x)=\| Hx - g \|_{2}^{2}$ where $H=\left[ \begin{array}{c} I \\ M \end{array} \right] $ and $g=\left[ \be …
answered Feb 24 by Brian Borchers
5
votes
How would you solve the problem if you didn't need to do the centering? Since $A$ is large and sparse, you'd probably pick an iterative method such as CGNE which depends on being able to perform ma …
answered Aug 25 '18 by Brian Borchers
2
votes
You can reformulate this problem as a conventional linear least squares problem as follows: First, write $YCZ$ as $YCZ=\sum_{i=1}^{p} \sum_{j=1}^{m} C_{i,j} (Y_{:,i}Z_{j,:})$ Next, define the $\mb …
answered Nov 26 '17 by Brian Borchers