Talking about the compression perception (15): The spark constant of the measurement matrix

In the sense of compression, there are some indicators used to evaluate the measurement matrix, such as common rip, and spark constants can be used to measure the suitability of a measurement matrix in addition to rip.

1, 0 space conditions null spaces Condition

Before introducing Spark, consider the 0 space of the measurement matrix.

Here we consider the conditions to be satisfied with the measurement matrix from the 0 space of the matrix: for K sparse signal X, when and only if the 0 space of the measuring matrix does not intersect with the linear space of the 2K base vector spanned, or the vector in 0 space is not in the linear space of the 2K base vector spanned.

The nature of the above description seems a bit difficult to understand, then the equivalent of the expression is the spark constant.

2. The Spark constant definition

In simple terms, the minimum number of linear correlation vector groups in a matrix column

3. Spark Evaluation theorem

When and only if Spark (Φ) >2k, it is possible to get the exact approximation of K-sparse signal x by the minimum 0 norm optimization problem.

4. Linear Correlation Definition:

Theorem:

Properties:

5. Proof of theorem

Contradiction

The first step proves that, for any vector y, there is at most one k sparse signal X, so.

Proof: Assume that the linear correlation column defined by the measurement matrix is less than or equal to 2K, starting from a linear-related definition,

There is a vector, that is, so and H is not equal to 0.

Because, the H can be expressed as:, so

thereby getting.

But our condition shows that at most there is only one K sparse signal X, so it is inconsistent with the original condition, so the hypothesis is not established, the original proposition is established.

The second step proves that, for any vector y, the K sparse signal x satisfies and has at most one.

Proof: Suppose K sparse signal X has at least two, set to, then.

Because, by definition, the linear correlation column of the measurement matrix is greater than 2K, starting from the linear correlation definition,

There is a vector, so, and H is not equal to 0. and the measurement matrix of 0 space should be greater than 2K, and assuming that the H space is less than or equal to 2K dimensions, to meet, when and only when h=0, that is, with the original hypothesis contradiction, so the assumption is not established, the original proposition is established.

Talking about the compression perception (15): The spark constant of the measurement matrix