# K SVD for Dictionary Learning

In applied mathematics, K-SVD is a dictionary learning algorithm for creating a dictionary for sparse representations, via a singular value decomposition approach. K-SVD is a generalization of the k-means clustering method, and it works by iteratively alternating between sparse coding the input data based on the current dictionary, and updating the atoms in the dictionary to better fit the data. It is structurally related to the expectation maximization (EM) algorithm.[1][2] K-SVD can be found widely in use in applications such as image processing, audio processing, biology, and document analysis.

## 从字典学习的角度看 K-means¶

$\min _{D, X}\left\{\|Y-D X\|_F^2\right\} \quad \text { subject to } \forall i, x_i=e_k \text { for some } k$

$\min _{D, X}\left\{\|Y-D X\|_F^2\right\} \quad \text { subject to } \forall i, \|x_i\|_0 = 1$

$\min _{D, X}\left\{\|Y-D X\|_F^2\right\} \quad \text { subject to } \forall i, \|x_i\|_0 = T_0$

$\underset{D,X}{\min} \sum_i \| x_i \|_0 \quad \text{subject to }\forall i, \| Y-DX \|_F^2 \le \epsilon.$