Discussion Overview
The discussion revolves around solving a separable least squares problem involving a known matrix A, a known vector b, an unknown vector x, and an unknown scalar a. Participants explore methods to isolate the scalar a from the vector x within the context of least squares optimization.
Discussion Character
- Technical explanation
- Mathematical reasoning
- Debate/contested
Main Points Raised
- One participant proposes using the pseudo-inverse method to solve for x but questions how to isolate a from x.
- Another participant suggests that without specific properties of x, such as its magnitude being 1, it is not possible to separate a from x, indicating that the solution is inherently a vector.
- A later reply proposes reformulating the problem as a constrained least squares problem, suggesting the minimization of ||Aax-b||2 subject to the constraint ||x||=1.
- It is noted that this reformulation would lead to two possible solutions for a, corresponding to both a and -a, which would be associated with x and -x.
Areas of Agreement / Disagreement
Participants express differing views on the feasibility of isolating a from x, with some suggesting constraints may allow for a reformulation while others maintain that separation is not possible without specific conditions.
Contextual Notes
The discussion highlights limitations regarding the assumptions needed for isolating a from x and the implications of the constraints on the solution space.
