PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-31

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Truncated SVD and Tikhonov regularisation act as hard and smooth spectral filters on the singular components

Statement

Let $F \in {R, C}$ , let $A \in M_{m \times n} (F)$ have singular value decomposition $A = U Σ V^{*}$ , let $b \in F^{m}$ , let $τ \geq 0$ , and let $λ > 0$ . In singular-value coordinates, applying the two filters to $b$ gives the following coefficients: truncated SVD multiplies the $i$ th singular component by

$g_{τ} (σ_{i}) = {\begin{cases} σ_{i}^{- 1}, & σ_{i} > τ, \\ 0, & σ_{i} \leq τ, \end{cases}$

whereas Tikhonov regularisation multiplies it by $g_{λ} (σ_{i}) = σ_{i} / (σ_{i}^{2} + λ)$ . Thus truncated SVD is a hard spectral filter and Tikhonov regularisation is a smooth spectral filter.

Facts & Assumptions

Given: A scalar field $F \in {R, C}$ , a singular value decomposition $A = U Σ V^{*}$ , a right-hand side $b \in F^{m}$ , a threshold $τ \geq 0$ , and a parameter $λ > 0$ .

[L1]

The truncated pseudoinverse replaces retained singular values by their reciprocals and discards the rest (The truncated singular-value pseudoinverse obtained by discarding singular values below a declared numerical-rank threshold).

[L2]

Tikhonov regularisation scales the $i$ th singular direction by $σ_{i} / (σ_{i}^{2} + λ)$ (Tikhonov regularisation scales each singular component by the filter factor $σ / (σ^{2} + λ)$ ).

Proof

technique · direct

1.1L1algebra

By [L1], applying $A_{τ}^{+}$ to $b = \sum_{i} ⟨ b, u_{i} ⟩ u_{i}$ gives $A_{τ}^{+} b = \sum_{σ_{i} > τ} \frac{1}{σ_{i}} ⟨ b, u_{i} ⟩ v_{i} .$ Thus the filter is $g_{τ} (σ_{i})$ with a hard cutoff at $τ$ .

1.2L2algebra

By [L2], the Tikhonov solution is $x_{λ} = \sum_{σ_{i} > 0} \frac{σ_{i}}{σ_{i}^{2} + λ} ⟨ b, u_{i} ⟩ v_{i} .$ Hence every nonzero singular direction is retained but damped smoothly according to $g_{λ} (σ_{i})$ , while zero singular directions have filter value $g_{λ} (0) = 0$ .

2.1step 1.1step 1.2∎

Step 1.1 is the truncated-SVD filter and step 1.2 is the Tikhonov filter, so the two methods are hard and smooth spectral filters respectively.

Depends on

Used by

Truncated SVD denoising on a diagonal problem keeps the declared large singular directions and discards the noisy ones Example

Dependency tree · two levels

5 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

Gene H. Golub, Least squares, singular values and matrix approximations (standard reference, not scraped)
Stanford CS205L, Unit 12: Regularization (standard reference, not scraped)