 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2019/ii/paper-1/29j/e/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

Using the estimator from part (b) and the spectral decomposition of

X^{T} X

Y_{λ} = X V (Λ + λ I_{p})^{- 1} V^{T} X^{T} Y = U (Λ + λ I_{p})^{- 1} U^{T} Y .

Thus

Y_{λ} = U (Λ + λ I_{p})^{- 1} U^{T} Y = i = 1 \sum p \frac{Λ _{ii}}{Λ _{ii} + λ} w_{i} w_{i}^{T} Y .

Under the stated separation of scales, the shrinkage factor is approximately one for

i \leq q

and approximately zero for

i > q

Y_{λ} \approx \sum_{i = 1}^{q} w_{i} w_{i}^{T} Y,

Y

onto the span of the

q

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