TLS problem

Consider the simples case Ax ≈ b, A = [α,0]^T, b = [0,β]^T.
The plots below show the classical regression setting:
Each circle corresponds to one line of Ax ≈ b:
the vertical dimension corresponds to the measured data (observations, i.e. b),
the horizontal dimension to the measurement points (model, i.e. A).
Note that A^Tb = 0, therefore the only senseful solution in all the cases below is the zero solution.

1. TLS solution exists and it is unique

Given α = 1, β = 0.4.
The minimal correction has ||[g,E]||_F = 0.4. The TLS solution is x_TLS = 0.
The optimal approximation (in TLS sense) is when the red is line identical with horizontal axis.

2. TLS solution exists and it nonunique

Given α = 1, β = 1.
All the cases represent TLS solutions, all the corrections have ||[g,E]||_F = 1.
The TLS solution minimal in 2-norm is x_TLS = 0.
It corresponds to the case when the red line is identical with horizontal axis.

3. TLS solution does not exist

Given α = 0.4, β = 1.
There is no TLS solution, the infimal correction has ||[g,E]||_F = 0.4, the corresponding x has ||x||₂ → ∞.
The red line identical with horizontal axis corresponds to x = 0.
This is a solution to a constraint TLS problem, also called "nongeneric solution".

Download software

Here is tls.m file.