Total Least Squares Step

Problem

Compute the total least squares solution for A*x = b (with A of shape n x 2) from the smallest right singular vector v of the augmented matrix [A | b]:

x = -v[0:2] / v[2]

Return None if v[2] is 0 within 1e-12, since the problem is then not solvable by TLS.

Hint

TLS minimizes the orthogonal distance to the data and reduces to the last singular vector's null direction.

Test cases

case 1

in: [[[1,0],[0,1],[1,1]],[1,2,3],[-0.4082482904638631,-0.816496580927726,0.4082482904638631]]

expected: [1,2]

case 2

in: [[[1,0],[0,1]],[2,3],[-0.5345224838248488,-0.8017837257372732,0.2672612419124244]]

expected: [2,3]

case 3

in: [[[1,0],[0,1]],[1,1],[1,0,0]]

expected: null

case 4

in: [[[1,1],[1,2],[1,3]],[1,2,3],[0,-0.7071067811865475,0.7071067811865475]]

expected: [0,1]

Starter codePython

def tls_step(A, b, v_smallest):
    # Your code here
    pass

4 test cases · run your code against them in the editor