K-SVD算法：设计适应稀疏表示的字典
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收藏 1.68MB PDF 举报 KSVD（K-Singular Value Decomposition）算法是一种在2006年提出的重要技术，它与小波变换类似，但在信号处理领域展现出了独特的优势。该方法的核心是利用一个过度完备的字典（overcomplete dictionary），即包含多个原型信号原子的矩阵，来表示信号。信号被表示为这些原子的稀疏线性组合，这意味着信号可以通过少数几个关键特征或原子来近似，这对于压缩、逆问题的正则化、特征提取等应用具有显著作用。
在近年来的研究中，主要关注点在于如何设计出更适合于稀疏表示模型的字典。KSVD算法在此背景下脱颖而出。与预先设定一组线性变换（如傅立叶变换或小波变换）相比，KSVD算法更侧重于通过适应性方法来调整字典，以更好地适应训练信号。这种方法允许字典在实际应用中自我学习和优化，以达到更高效的信号表示。
KSVD算法的具体流程包括以下几个步骤：
1. **信号重构**：首先，给定一个训练信号集，对每个信号进行表示，目标是找到最接近原始信号且表示系数尽可能稀疏的字典表示。
2. **字典更新**：通过迭代过程，每次迭代都会更新字典中的原子，使得它们更符合训练信号的稀疏表示需求。这通常涉及到奇异值分解（SVD）和最小化重构误差。
3. **稀疏编码**：对于新的未知信号，使用优化过的字典执行稀疏编码，找出最接近信号的稀疏表示。
4. **在线学习与扩展**：对于实时或在线应用， KSVD算法可以采用增量学习的方式，不断更新字典以适应新的信号数据。
尽管已有了一些设计字典的方法，如基于固定变换的选择和基于训练信号的自适应调整，但KSVD算法的独特之处在于其迭代优化过程，能够自动调整字典结构，从而提升信号表示的质量。这种灵活性使得KSVD在众多领域中展现出强大的潜力，特别是在那些信号复杂、需要高效压缩和分析的场景中。随着技术的发展，KSVD及其变种算法将继续为未来的信号处理和机器学习提供强大工具。
AHARON et al.: -SVD: ALGORITHM FOR DESIGNING OVERCOMPLETE DICTIONARIES 4313
quantization (VQ) coding method, called gain-shape VQ, where
this coefficient is allowed to vary [39]. In contrast, in sparse
representations as discussed in this paper, each example is rep-
resented as a linear combination of several vectors
.
Thus, sparse representations can be referred to as a generaliza-
tion of the clustering problem.
Since the
-means algorithm (also known as the generalized
Lloyd algorithm—GLA [39]) is the most commonly used pro-
cedure for training in the vector quantization setting, it is nat-
ural to consider generalizations of this algorithm when turning
to the problem of dictionary training. The clustering problem
and its
-means solution will be discussed in more detail in
Section IV-A, since our work approaches the dictionary training
problem by generalizing the
-means. Here we shall briefly
mention that the
-means process applies two steps per each it-
eration: i) given
, assign the training examples to their
nearest neighbor; and ii) given that assignment, update
to better fit the examples.
The approaches to dictionary design that have been tried so
far are very much in line with the two-step process described
above. The first step finds the coefficients given the dictio-
nary—a step we shall refer to as “sparse coding.” Then, the
dictionary is updated assuming known and fixed coefficients.
The differences between the various algorithms that have been
proposed are in the method used for the calculation of coeffi-
cients and in the procedure used for modifying the dictionary.
B. Maximum Likelihood Methods
The methods reported in [22]–[25] use probabilistic rea-
soning in the construction of
. The proposed model suggests
that for every example
the relation
(3)
holds true with a sparse representation
and Gaussian white
residual vector
with variance . Given the examples
, these works consider the likelihood function
and seek the dictionary that maximizes it. Two assumptions are
required in order to proceed: the first is that the measurements
are drawn independently, readily providing
(4)
The second assumption is critical and refers to the “hidden vari-
able”
. The ingredients of the likelihood function are computed
using the relation
(5)
Returning to the initial assumption in (3), we have
(6)
The prior distribution of the representation vector
is assumed
to be such that the entries of
are zero-mean i.i.d., with Cauchy
[24] or Laplace distributions [22], [23]. Assuming for example
a Laplace distribution, we get
(7)
This integration over
is difficult to evaluate, and indeed, Ol-
shausen and Field [23] handled this by replacing it with the ex-
tremal value of
. The overall problem turns into
(8)
This problem does not penalize the entries of
as it does for
those of
. Thus, the solution will tend to increase the dictio-
nary entries’ values, in order to allow the coefficients to become
closer to zero. This difficulty has been handled by constraining
the
-norm of each basis element, so that the output variance
of the coefficients is kept at an appropriate level [24].
An iterative method was suggested for solving (8). It includes
two main steps in each iteration: i) calculate the coefficients
using a simple gradient descent procedure and then ii) update
the dictionary using [24]
(9)
This idea of iterative refinement, mentioned before as a general-
ization of the
-means algorithm, was later used again by other
researchers, with some variations [36], [37], [40]–[42].
A different approach to handle the integration in (7) was sug-
gested by Lewicki and Sejnowski [25]. They approximated the
posterior as a Gaussian, enabling an analytic solution of the inte-
gration. This allows an objective comparison of different image
models (basis or priors). It also removes the need for the ad-
ditional rescaling that enforces the norm constraint. However,
this model may be too limited in describing the true behaviors
expected. This technique and closely related ones have been re-
ferred to as approximated ML techniques [37].
There is an interesting relation between the above method and
the independent component analysis (ICA) algorithm [43]. The
latter handles the case of a complete dictionary (the number of
elements equals the dimensionality) without assuming additive
noise. The above method is then similar to ICA in that the algo-
rithm can be interpreted as trying to maximize the mutual infor-
mation between the inputs (samples) and the outputs (the coef-
ficients) [24], [22], [25].
C. The MOD Method
An appealing dictionary training algorithm, named method
of optimal directions (MOD), is presented by Engan et al. [36],
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