Past exam of the mathematical and Computational Biology course of the University of Cambridge / 2024 / Paper 1 / B3 / a

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- Solution a

Solution

Choose the diagonal matrix

W = diag (w_{1}, \dots, w_{m})

. Multiplying a vector by

W

scales its

i

th component by

w_{i}

. Therefore

∥ Wb - W A x ∥^{2} = i = 1 \sum m w_{i}^{2} (b_{i} - (A x)_{i})^{2} .

The Euclidean norm objective for ordinary linear least squares on

(W A, Wb)

is exactly the weighted least squares objective, so the sets of minimizing vectors coincide. Zero weights simply discard the corresponding residuals.

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