﻿ 广义测不准原理中的数学问题研究 Study on the Mathematical Problems of Generalized Uncertainty Principles

Vol.05 No.03(2016), Article ID:18469,24 pages
10.12677/AAM.2016.53064

Study on the Mathematical Problems of Generalized Uncertainty Principles

Guanlei Xu1, Xiaotong Wang2, Lijia Zhou1, Limin Shao1, Yonglu Liu1, Xiaogang Xu2

1Ocean Department of Dalian Navy Academy, Dalian Liaoning

Received: Aug. 11th, 2016; accepted: Aug. 25th, 2016; published: Aug. 31st, 2016

ABSTRACT

The uncertainty principle is the elementary rule in the crossed fields of mathematics, information and physics and so on, which plays an important role in scientific sense and engineering value. This paper discussed the mathematical problems in the research of widely studied generalized uncertainty principles (i.e., the generalized uncertainty principles on time-frequency analysis and the generalized uncertainty principles on sparse representation), including the extension of the traditional inequalities to the generalized domains, the optimization of various p-norms, the optimal matrix factorization and so on. The review of these mathematical problems is the focus in this paper, and the disadvantages and the future work of these mathematical problems are discussed as well.

Keywords:Generalized Uncertainty Principle, Sparse Representation, Time-Frequency Analysis, Resolution Analysis, Norm, Entropy, Matrix Factorization

1海军大连舰艇学院军事海洋系，辽宁 大连

2海军大连舰艇学院航海系，辽宁 大连

1. 引言

2. 时频分析广义测不准原理中的数学问题

2.1. LCT基本定义和数学特性及应用

, (1)

1) 叠加特性：, (2)

2) 可逆性：, (3)

3) 时移性：, (4)

4) 尺度特性：, (5)

5) 乘积特性：,(6)

6) 广义Parseval准则：. (7)

Parseval准则/定理的物理意义是能量守恒，时域能量等于频域能量，不会因为变换而发生改变。而广义Parseval准则讨论了在广义域内(分数阶Fourier变换域和线性正则变换域内)的能量守恒问题，即时域能量等于广义频域能量。

, (8)

. (9)

.

.

.

Cauchy-Schwartz不等式有另一形式，还可以用范数(见论文第三部分)的写法表示：

.

2.2. Minkowski不等式

Minkowski不等式在广义测不准原理的推导中也颇为重要，首先简要回顾下Minkowski不等式的推导过程。我们考虑连续函数形式的p次幂：

.

.

.

,

.

.

2.3. 广义Hausdorff-Young不等式

,

,

.

.

.

，即得

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.

.

,

.

.

,

.

2.4. 广义Pitt不等式

,

，可得：

,

.

.

.

，并将其代入上式，可得：

.

，则

.

.

.

,

.

,

.

.

2.5. 不同广义测不准原理证明过程中的数学问题

3. 信号稀疏表示广义测不准原理中的数学理论及方法

Table 1. Mathematical problems of generalized Heisenberg uncertainty principles

Table 2. Mathematical problems of generalized Shannon entropic uncertainty principles

Table 3. Mathematical problems of generalized Rényi entropic uncertainty principles

Table 4. Mathematical problems of generalized windowed uncertainty principles

Table 5. Mathematical problems of generalized logarithmic uncertainty principles

Table 6. Generalized Hausdorff-Young and Pitt Inequalities

3.1. p范数

p范数(p-norm)可以看成2范数的扩展，但是：p的范围是[1, inf)。p在(0,1)范围内定义的并不是范数

Table 7. State of the generalized uncertainty principles for signal sparse representation

Table 8. Mathematical problems of generalized uncertainty principles for signal sparse representation

(但是，我们有时也笼统地称之为0-范数、1/2-范数等)，因为违反了三角不等式。在p范数下定义的单位球(unit ball)都是凸集(convex set，简单地说，若集合A中任意两点的连线段上的点也在集合A中，则A是凸集)，但是当0 < p < 1时，在该定义下的unit ball并不是凸集(注意：我们没说在该范数定义下，因为如前所述，0 < p < 1时，并不是范数)。下图展示了p取不同值时单位圆(因为p取2时为标准的单位圆，故以单位圆为标准比对对象)的形状，见图1

0-范数是稀疏表示中常用的范数，其物理意义就是求非零数据的个数。由于信号严格稀疏表示采用最小0-范数来进行量化和界定，但是最小0-范数的求解是个NP问题(即数学上需要把所有的情况都穷举完才能找到最优的解)，所以Denoho等很多学者又给出了信号稀疏表示的最小0-范数和最小1-范数等价的广义测不准原理边界条件，然后用1-范数代替0-范数进行问题的求解。

3.2. 数学优化问题

P0问题：

. (10)

P1问题：

. (11)

Pe问题：

, (12)

1) 基追踪算法

Chen等人 [77] 提出了一种极小化1-范数的稀疏求解思路(P1问题)。实际上，需要特别说明的是：基追踪算法并非基于一个最优化原则，其原理本质是给定一些限制条件后，通过极小化1-范数可以获得最稀疏的解。主要是通过单纯形法、内点法或对数障碍发来进行求解。它需要最少的测度，但其高算法复杂性会影响到实际大规模应用。假设线性系统个非零的稀疏解，也就是，。而且，假设。匹配追踪或者基础追踪可以成功的恢复稀疏解吗？显而易见的，这样的成功并不是对于所有的矩阵A的所有的都可以的，因为这个可能和一般情况下的已知的NP问题发生冲突。然而，如果这个等式有一个充分稀疏的解，那么这些算法在寻址原始的目标(P0)的成功性就有所保证了。

Figure 1. The different shapes of unit ball for different p

, (13)

. (14)

(15)

, (16)

, (17)

, (18)

, (19)

. (20)

2) 贪婪算法

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3) 熵优化方法

and.

. (21)

.(22)

4) RIP条件

(23)

. (24)

. (25)

3.3. 矩阵稀疏及秩最小化问题

. (26)

， (27)

. (28)

. (29)

， (30)

4. 结论

Study on the Mathematical Problems of Generalized Uncertainty Principles[J]. 应用数学进展, 2016, 05(03): 536-559. http://dx.doi.org/10.12677/AAM.2016.53064

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