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The Tikhonov regularised least-squares objective $∥ A x - b ∥_{2}^{2} + λ ∥ x ∥_{2}^{2}$ for $λ > 0$

Definition

Let $F \in {R, C}$ , let $A \in M_{m \times n} (F)$ , let $b \in F^{m}$ , and let $λ \in R$ satisfy $λ > 0$ . The Tikhonov regularised least-squares objective is the map

$J_{λ} (x) : = ∥ A x - b ∥_{2}^{2} + λ ∥ x ∥_{2}^{2} (x \in F^{n}),$

where $∥ \cdot ∥_{2}$ is the Euclidean norm induced by the standard inner product (The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product).

The parameter $λ$ is part of the problem data. The second term penalises large norms of $x$ , so the regularised problem is not the same optimisation problem as the unregularised least-squares problem.

Depends on

The norm $\lVert v\rVert=\sqrt{\langle v,v\rangle}$ induced by a real or complex inner product

Used by

For every λ>0, the Tikhonov objective is strictly convex and has the unique minimiser (A^*A+λ I)⁻¹A^*b Theorem

Dependency tree · two levels

3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Stanford CS205L, Unit 12: Regularization (standard reference, not scraped)

The Tikhonov regularised least-squares objective ∥Ax−b∥22+λ∥x∥22∥Ax−b∥22​+λ∥x∥22​ for λ>0λ>0