Solving Separable Least Square Problem - Isy

In summary, the conversation discusses solving a LSQ problem involving a known matrix, known vector, and unknown vector and scalar. It is suggested to solve the problem directly using a pseudo inverse, but there is uncertainty about isolating a from x. The idea of using a separable least square scheme is mentioned, but it is noted that there may be two possible solutions. The conversation concludes by discussing the possibility of reformulating the problem with constrained LSQ.
  • #1
isolde_isy
2
0
Hi I have to solve the following LSQ problem:
min(||Aax-b||2)

where
A is a known matrix,
b is a know vector
x is an unknows vector
a is an unknown scalar

I can solve directly via pseudo inverse
ax=inv(A'A)A'b
but how can I isolate a from x?

Could Separable least square a scheme to follow...
Thnaks in advance
Isy :smile:
 
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  • #2
Unless you have some particularly property (for example the magnitude of x = 1), you can't "separate" a from x. Your solution is a vector.
 
  • #3
So it can be reformulate the problem in lsq constrained
min(||Aax-b||2)
subject to ||x||=1

Right?
Thanks Isy
 
  • #4
isolde_isy said:
So it can be reformulate the problem in lsq constrained
min(||Aax-b||2)
subject to ||x||=1

Right?
Thanks Isy
Yes - but note that there would (for real vector spaces) be two possibilities, since a and -a, associated with x and -x would satisfy.
 
  • #5


Hi Isy,

Thank you for reaching out with your question. The least squares problem you have described is a common problem in many scientific fields. In order to solve it, you can use the method of separable least squares.

This method involves separating the unknown vector x into two parts: x1 and x2. x1 represents the part of x that is multiplied by the unknown scalar a, while x2 represents the remaining part of x.

By doing this, you can rewrite the least squares problem as:

min(||A(x1+a*x2)-b||2)

This can then be solved using the pseudo inverse method as you mentioned, by solving for x1 and x2 separately. Once you have obtained the values for x1 and x2, you can then solve for a using the equation:

a = (x1'Ax2)/(x2'Ax2)

I hope this helps you solve your least squares problem. Best of luck in your research!

Sincerely,
 

1. What is a separable least square problem?

A separable least square problem is a type of optimization problem that involves finding the best fit for a set of data points by minimizing the sum of squared errors. It is called "separable" because the problem can be broken down into smaller, independent sub-problems.

2. How is a separable least square problem solved?

To solve a separable least square problem, the data points are first separated into smaller groups. Then, each group is treated as an independent least square problem and solved using standard least square methods. The solutions from each group are then combined to find the overall best fit for the entire set of data points.

3. What are the benefits of solving a separable least square problem?

Solving a separable least square problem allows for more efficient computation and can often provide a better fit for the data compared to solving the problem as a whole. Additionally, it can help identify any outliers or anomalies in the data that may affect the overall fit.

4. Is there any special software or tools needed to solve a separable least square problem?

No, a separable least square problem can be solved using standard least square methods and does not require any special software or tools. However, there are some software packages that offer specific algorithms or functions for solving separable least square problems.

5. Can a separable least square problem be applied to any type of data?

Yes, a separable least square problem can be applied to different types of data, including numerical, categorical, and mixed data. However, the data should be well-structured and have a clear relationship between the independent and dependent variables for the problem to be solvable.

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