深入浅出SPARSE之【Chapter 2】 Uniqueness and Uncertainty（如何证明一个解是不是全局最优的稀疏解）

我要解方程。但是有两个shortcoming（1）The equality requirement b = Ax is too strict，我们找个近似的条件，（2）The sparsity measure is too sensitive to very small entries in x,

但是还有两个questions：Q1: When can uniqueness of the sparsest solution be claimed?Q2: Can a candidate solution be tested to verify its (global) optimality?

This section addresses these questions,, we first consider special matrices A for which the analysis seems to be easier, and then extend our answers to the general A.

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  2.1 Treating the Two-Ortho Case（本段目的，在正交矩阵下，证明稀疏解是全局最优的，即唯一性。采用倒序看更好。。。一个不确定准则和定义，3打定理！）

    A is the concatenation of two orthogonal matrices,  

2.1.1 An Uncertainty Principle（不确定准则）

a signal cannot be sparsely represented both in time and in frequency（why 芥末说呢？

高斯函数的傅里叶变换

 f(x)=(a>0)

举个特例：

For an arbitrary pair of bases            ;an interesting phenomenon occurs: either  a can be sparse, or b can be sparse, but not both，就是说a，b不可能同时稀疏，

dependent on the distance between ，

所以回答了：为什么要定义互相干性？？？？？？？？？？？？？？？？？

有以下性质（左边的结论易得）因为

要proof这个非常容易的。

首先由定义可以得到：1=b'*b<=u(||alpha||1*||beta||1)

并且当||alpha||2=1时，有||alpha||1<=||alpha||0，

故1/u<=||alpha||1*||beta||1<=(||alpha||0+||beta||0)/2

Uncertainty of Redundant Solutions(冗余解的不确定性)

要proof这个非常容易的。

||X1||0+||X2||0>=||X1-X2||0=||e||0=||e1||0+||e2||0>=2/u(Theorem 2.1)

From Uncertainty to Uniqueness（Theorem 2.2直接可以得到）

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2.2 Uniqueness Analysis for the General Case

Uniqueness via the Spark(The spark is at least as difficult to evaluate as solving (P0).)

Definition 2.2. : The spark of a given matrix A is the smallest number of columns from A that are linearly-dependent

有以上定义我们可以看出：spark的本质：spark（A）=min||x||0,  s.t. Ax=0,描述一个A的稀疏性！无关的向量越多越稀疏！||x||0>=spark（A）

mpute, and as such, it allows us to lower-bound the spark, which is often hard to compute）

Definition 2.3. : The mutual-coherence of a given matrix A is the largest absolute  normalized inner product between different columns from A. Denoting the k-th column

in A by ak, the mutual-coherence is given by

描述的是两列的夹角，描述列之间的依赖性， Mutual-Coherence的作用是用来估计spark的范围的，如下公式

为什么呢？我没有证明粗来，直观上来所，u=1时候，spark(A)=2.

SparkPKMutual-Coherence

        Spark


Mutual-Coherence

range [2,n+1]range [1/sqrt(N),1],1/u+1属于[2,sqrt(N)+1]功能更加强大，上述更tight，对许素判断得更加的紧密，只要小于spark(A)/2就是最稀疏的了.
但是不好计算，比L0更大的计算量，NP问题功能更加小
但是好计算

Uniqueness via the Babel Function（不以讨论）

怎么班呢？Mutual-Coherence计算简单，但是约束不够， Spark约束很好，但是不好计算，我们采用了以下方法。换一种角度定义spack（采用近似的办法）

（可用LP来做）

Numerical experiments show that this bound tends to be quite tight, and close to the true spark.