---- > [!theorem] Theorem. ([[singular value soft thresholding]]) > Consider the low-[[rank]] [[matrix]] approximation problem $\hat{X}= \arg \min_{X \in \mathbb{F}^{M \times N}} \frac{1}{2} \|Y-X\|_{F}^{2} + \beta \|X\|_{*},$ > where $\|\cdot\|_{*}$ is the [[nuclear norm]], chosen as a [[convex function|convex]] relaxation of $\rank(\cdot)$. > Because the [[Frobenius norm]] and [[nuclear norm]] are [[unitarily invariant norm of matrices|unitarily invariant]], we can apply the [[characterization of unitarily invariant unconstrained regularized low-rank approximations]] to ultimately obtain the solution $\begin{align} \hat{X} = & \sum_{k=1}^{r} h_{\text{soft}}(\sigma_{k}; \beta) u_{k}v_{k}' \\ h_{\text{soft}}(\sigma_{}; \beta) := & [\sigma - \beta]_{+} = \max(\sigma - \beta, 0). \end{align},$ where $U_{r}\Sigma_{r}V_{r}'=\sum_{k=1}^{r} \sigma_{k}u_{k}v_{k}'$ is a [[compact svd]] of the observed data $Y$. ---- #### ----- #### References > [!backlink] > ```dataview TABLE rows.file.link as "Further Reading" FROM [[]] FLATTEN file.tags GROUP BY file.tags as Tag > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```