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For every $λ > 0$ , the Tikhonov objective is strictly convex and has the unique minimiser $(A^{} A + λ I)^{- 1} A^{} b$

Statement

Let $F \in {R, C}$ , let $A \in M_{m \times n} (F)$ , let $b \in F^{m}$ , and let $λ \in R$ satisfy $λ > 0$ . Then the Tikhonov objective

$J_{λ} (x) = ∥ A x - b ∥_{2}^{2} + λ ∥ x ∥_{2}^{2}$

is strictly convex and has the unique minimiser

$x_{λ} = (A^{*} A + λ I)^{- 1} A^{*} b .$

Facts & Assumptions

Given: A matrix $A \in M_{m \times n} (F)$ , a vector $b \in F^{m}$ , and a real parameter $λ > 0$ , where $F \in {R, C}$ .

[L1]

The regularised objective is $J_{λ} (x) = ∥ A x - b ∥_{2}^{2} + λ ∥ x ∥_{2}^{2}$ (The Tikhonov regularised least-squares objective $∥ A x - b ∥_{2}^{2} + λ ∥ x ∥_{2}^{2}$ for $λ > 0$ ).

[L2]

$A$ admits a singular value decomposition $A = U Σ V^{*}$ (Every linear map between finite-dimensional real or complex inner product spaces admits a singular value decomposition).

Proof

technique · direct

1.1L1L2algebra

By [L2], write $A = U Σ V^{*}$ and set $x = V y$ and $\hat{b} : = U^{*} b$ . Since $U$ and $V$ are unitary, $J_{λ} (V y) = ∥ Σ y - \hat{b} ∥_{2}^{2} + λ ∥ y ∥_{2}^{2} .$

2.1step 1.1algebra

If $σ_{1}, \dots, σ_{r}$ are the nonzero singular values, then step 1.1 becomes $J_{λ} (V y) = \sum_{i = 1}^{r} ∣ σ_{i} y_{i} - {\hat{b}}_{i} ∣^{2} + \sum_{i = r + 1}^{m} ∣ {\hat{b}}_{i} ∣^{2} + λ \sum_{i = 1}^{n} ∣ y_{i} ∣^{2} .$ Each variable $y_{i}$ appears in a one-variable quadratic with positive coefficient $σ_{i}^{2} + λ$ or $λ$ , so the objective is strictly convex.

3.1step 2.1algebra

Minimising coordinatewise gives $y_{i} = \frac{σ_{i}}{σ_{i}^{2} + λ} {\hat{b}}_{i} (1 \leq i \leq r), y_{i} = 0 (r < i \leq n) .$ Equivalently, $y = (Σ^{*} Σ + λ I)^{- 1} Σ^{*} \hat{b} .$

4.1L2step 3.1algebra

Returning to $x = V y$ and using $A^{*} A = V Σ^{*} Σ V^{*}$ , one gets $x_{λ} = V (Σ^{*} Σ + λ I)^{- 1} Σ^{*} U^{*} b = (A^{*} A + λ I)^{- 1} A^{*} b .$

5.1step 2.1step 4.1∎

Step 2.1 proves strict convexity and step 4.1 gives the unique minimiser.

Depends on

Used by

Tikhonov regularisation scales each singular component by the filter factor σ/(σ²+λ) Proposition

Dependency tree · two levels

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Sources

Stanford CS205L, Unit 12: Regularization (standard reference, not scraped)
Andrew Stuart and Jochen Voss, Matrix Analysis and Algorithms (standard reference, not scraped)

For every λ>0λ>0, the Tikhonov objective is strictly convex and has the unique minimiser (A∗A+λI)−1A∗b(A∗A+λI)−1A∗b