K-SVD
Rachel Zhang
1. k-SVD introduction
1. K-SVD usage:
Design/Learn a dictionary adaptively to betterfit the model and achieve sparse signal representations.
2. Main Problem:
Y = DX
Where Y∈R(n*N), D∈R(n*K), X∈R(k*N), X is a sparse matrix.
N is # of samples;
n is measurement dimension;
K is the length of a coefficient.
2. Derivation from K-Means
3. K-Means:
1) The sparse representationproblem can be viewed as generalization of the VQ objective. K-SVD can be viewed as generalization of K-Means.
2) K-Means algorithm for vectorquantization:
Dictionary of VQ codewords is typically trained using K-Means algorithm.
When Dictionary D is given, each signal is represented as its closestcodeword (under l2-norm distance). I.e.
Yi = Dxi
Where xi = ej is a vector from the trivial basis,with all zero entries except a one in the j-th position.
3) VQ的字典训练:
K-Means被视作一个sparse coding的特例,在系数x中只有一个非零元,MSE定义为:
所以VQ的问题是:
4) K-Means 算法实现的迭代步骤:
1) 求X的系数编码
2) 更新字典
3. K-SVD,generalizing the K-Means
4. Objective function
5. K-SVD的求解
Iterative solution: 求X的系数编码(MP/OMP/BP/FOCUSS),更新字典(Regression).
K-SVD优化:也是K-SVD与MOD的不同之处,字典的逐列更新:
假设系数X和字典D都是固定的,要更新字典的第k列dk,领稀疏矩阵X中与dk相乘的第k行记做
上式中,DX被分解为K个秩为1的矩阵的和,假设其中K-1项都是固定的,剩下的1列就是要处理更新的第k个。矩阵Ek表示去掉原子dk的成分在所有N个样本中造成的误差。
6. 提取稀疏项
如果在5.中这一步就用SVD更新dk和
7. 总结
K-SVD总可以保证误差单调下降或不变,但需要合理设置字典大小和稀疏度。
在我的实验中,随着字典的增大,K-SVD有整体效果提升的趋势,但不一定随着稀疏度的增大使得整体误差下降。
8. Reference:
1) K-SVD: An algorithm fordesigning overcomplete dictionaries for sparse representation (IEEE Trans. OnSignal Processing 2006)
2) From sparse solutions of systemsof equations to sparse modeling of signals and images (SIAM Review 2009 240')
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