Uniformly smooth space

In applied mathematics, K-SVD is an algorithm of deciding dictionary of sparse representation in a singular value decomposition approach. K-SVD is a generalization of the k-means clustering method, iteratively alternates between sparse coding coefficient of the example based on current dictionary and atoms in the dictionary. The update of dictionary enables K-SVD algorithm to better fit the data. K-SVD can be found widely use in applications such as image processing, audio processing, biology, and document analysis.

Sparsity description

Given an overcomplete dictionary matrix $D\in \mathbb {R} ^{n\times K}$ that contains $K$ signal-atoms for each columns, a signal $y\in \mathbb {R} ^{n}$ can be represented as a linear combination of these atoms. To represent $y$ , the sparse representation $x$ should satisfy the exact condition $y=Dx$ , or approximate condition $y\approx Dx$ , or $\|y-Dx\|_{p}\leq \epsilon$ . The vector $x\in \mathbb {R} ^{K}$ contains the representation coefficients of the signal $y$ . Typically, norm $p$ is selected as $1,2,{\text{ or }}\infty$ .

If $n<K$ and D is a full-rank matrix, an infinite number of solutions are available for the representation problem, Hence, constraints should be set on the solution. Also, to ensure sparsity, the solution with the fewest number of nonzero coefficients is preferred. Thus, the sparsity representation is the solution of either

(P_{0})\quad \min \limits _{x}\|x\|_{0}\qquad {\text{subject to }}y=Dx

or

(P_{0,\epsilon })\quad \min \limits _{x}\|x\|_{0}\qquad {\text{subject to }}\|y-Dx\|_{2}\leq \epsilon

where the $L_{0}$ norm counts the nonzero entries of a vector. Please see Matrix norm.

Choice of dictionary

Generalize the K-means:

The k-means clustering can be also regarded as a method of sparse representation. That is, finding the best possible codebook to represent the data samples $\{y_{i}\}_{i=1}^{M}$ by nearest neighbor, by solving

\quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\forall i,x_{i}=e_{k}{\text{ for some }}k.

or can be written as

\quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\quad \forall i,\|x_{i}\|_{0}=1.

The sparse representation term $x_{i}=e_{k}$ enforces K-means algorithm to use only one atom (column) in dictionary D. To relax this constraint, target of the K-SVD algorithm is to represent signal as a linear combination of atoms in D.

The K-SVD algorithm follows the construction flow of K-means algorithm. However, In contrary to K-means, in order to achieve linear combination of atoms in D, sparsity term of the constrain is relaxed so that nonzero entries of each column $x_{i}$ can be more than 1, but less than a number $L_{0}$ .

So, the objective function becomes

\quad \min \limits _{D,X}\{\|Y-DX\|_{F}^{2}\}\qquad {\text{subject to }}\quad \forall i\;,\|x_{i}\|_{0}\leq T_{0}.

or in another objective form

\quad \min \limits _{D,X}\sum _{i}\|x_{i}\|_{0}\qquad {\text{subject to }}\quad \forall i\;,\|Y-DX\|_{F}^{2}\leq \epsilon .

In the K-SVD algorithm, the $D$ is first to be fixed and the best coefficient matrix $X$ . As finding the truly optimal $X$ is impossible, we use an approximation pursuit method. Any such algorithm as OMP, the orthogonal matching pursuit in can be used for the calculation of the coefficients, as long as it can supply a solution with a fixed and predetermined number of nonzero entries $T_{0}$ .

After the sparse coding task, the next is to search for a better dictionary $D$ . However, finding the whole dictionary all at a time is impossible, so the process then update only one column of the dictionary $D$ each time while fix $X$ . The update of $k-th$ is done by rewriting the penalty term as

\|Y-DX\|_{F}^{2}=\left|Y-\sum _{j=1}^{K}d_{j}x_{T}^{j}\right|_{F}^{2}=\left|\left(Y-\sum _{j\neq k}d_{j}x_{T}^{j}\right)-d_{k}x_{T}^{k})\right|_{F}^{2}=\|E_{k}-d_{k}x_{T}^{k}\|_{F}^{2}

where $x_{T}^{k}$ denotes the k-th row of X.

By decomposing the multiplication $DX$ into sum of $K$ rank 1 matrices, we can assume the other $K-1$ terms are assumed fixed, and the $k-th$ remains unknown. After this step, we can solve the minimization problem by approximate the $E_{k}$ term with a $rank-1$ matrix using singular value decomposition, then update $d_{k}$ with it. However, the new solution of vector $x_{T}^{k}$ is very likely to be filled, because the sparsity constrain is not enforced.

To cure this problem, Define $\omega _{k}$ as

\omega _{k}=\{i\mid 1\leq i\leq N,x_{T}^{k}(i)\neq 0\}.

Which points to examples $\{y_{i}\}$ that use atom $d_{k}$ (also the entries of $x_{i}$ that is nonzero). Then, define $\Omega _{k}$ as a matrix of size $N\times |\omega _{k}|$ , with ones on the $(i{\text{-th}},\omega _{k}(i))$ entries and zeros otherwise. When multiplying $x_{R}^{k}=x_{T}^{k}\Omega _{k}$ , this shrinks the row vector $x_{T}^{k}$ by discarding the nonzero entries. Similarly, the multiplication $Y_{k}^{R}=Y\Omega _{k}$ is the subset of the examples that are current using the $d_{k}$ atom. The same effect can be seen on $E_{k}^{R}=E_{k}\Omega _{k}$ .

So the minimization problem as mentioned before becomes

\|E_{k}\Omega _{k}-d_{k}x_{T}^{k}\Omega _{k}\|_{F}^{2}=\|E_{k}^{R}-d_{k}x_{R}^{k}\|_{F}^{2}

and can be done by directly using SVD. SVD decomposes $E_{k}^{R}$ into $U\Delta V^{T}$ . The solution for $d_{k}$ is the first column of U, the coefficient vector $x_{R}^{k}$ as the first column of $V\times \Delta (1,1)$ . After updated the whole dictionary, the process then turns to iteratively solve X, then iteratively solve D.