现代采样理论中的凸优化技术及其应用

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Optimization Techniques in Modern Sampling Theory Convex Optimization in Signal Processing and Communications 作为一个 IT 行业的大师，我将为您生成相关的知识点，总结如下： **知识点1：凸优化在信号处理方面的应用** 凸优化是一种非常重要的数学工具，在信号处理方面有着广泛的应用。本文将详细介绍凸优化在信号处理方面的应用，包括信号重建、信号优化、信号压缩等方面。 **知识点2：信号优化的重要性** 信号优化是信号处理的核心问题之一，旨在从信号中提取有用的信息。凸优化可以用于解决信号优化问题，例如信号去噪、信号压缩、信号重建等。 **知识点3：现代采样理论** 现代采样理论是信号处理的基础理论之一，旨在研究信号的采样和重建问题。凸优化可以用于解决现代采样理论中的优化问题，例如信号的采样和重建、信号的压缩和重建等。 **知识点4：信号的采样和重建** 信号的采样和重建是信号处理的核心问题之一，旨在从信号中提取有用的信息。凸优化可以用于解决信号的采样和重建问题，例如信号的采样、信号的重建、信号的压缩等。 **知识点5：优化方法** 优化方法是解决信号优化问题的关键，旨在找到最优的信号优化解决方案。凸优化可以用于解决优化问题，例如最小二乘法、最小极值法、罚函数法等。 **知识点6：子空间优先** 子空间优先是一种常用的优化方法，旨在找到最优的信号优化解决方案。凸优化可以用于解决子空间优先问题，例如子空间优先的最小二乘法、子空间优先的最小极值法等。 **知识点7：平滑性优先** 平滑性优先是一种常用的优化方法，旨在找到最优的信号优化解决方案。凸优化可以用于解决平滑性优先问题，例如平滑性优先的最小二乘法、平滑性优先的最小极值法等。 **知识点8：约束优化** 约束优化是一种常用的优化方法，旨在找到最优的信号优化解决方案。凸优化可以用于解决约束优化问题，例如约束优化的最小二乘法、约束优化的最小极值法等。 **知识点9：无约束优化** 无约束优化是一种常用的优化方法，旨在找到最优的信号优化解决方案。凸优化可以用于解决无约束优化问题，例如无约束优化的最小二乘法、无约束优化的最小极值法等。凸优化在信号处理方面的应用非常广泛，包括信号优化、信号压缩、信号重建等方面。凸优化可以用于解决信号优化问题，例如信号的采样和重建、信号的压缩和重建等。

Optimization Techniques in Modern Sampling Theory

Table 1.1: Diﬀerent scenarios treated in this chapter

Unconstrained Reconstruction Constrained Reconstruction

Least-Squares Minimax Least-Squares Minimax

Subspace Priors

Noise-Free Samples

Section 1.5.1.1 Section 1.5.1.2 Section 1.5.2.1 Section 1.5.2.2

Smoothness Priors

Noise-Free Samples

Section 1.6.1.1 Section 1.6.1.2 Section 1.6.2.1 Section 1.6.2.2

Smoothness Priors

Noisy Samples

Section 1.8 Section 1.8 Section 1.8 Section 1.8

comparisons between the various recovery algorithms. Finally, in Section 1.8 we

treat the case in which the samples are noisy. There, we focus our attention on

smoothness priors and on the minimax objective. We use semi-deﬁnite relaxation

to tackle the resulting non-convex quadratic programs. This section also includes

a summary of recent results on semi-deﬁnite relaxation of non-convex quadratic

programs, which is needed for our derivations.

1.2 Notation and mathematical preliminaries

The exposition in this chapter is based on a Hilbert-space interpretation of sam-

pling techniques, and relies on the concepts of frames and projections. In this

section we introduce some notations and mathematical preliminaries, which form

the basis for the derivations in the sections to follow.

1.2.1 Notation

We denote vectors in an arbitrary Hilb ert space H by lowercase letters, and the

elements of a sequence c ∈ `

by c[n]. Traditional sampling theories deal with

signals, which are deﬁned over the real line. In this case the Hilbert space H of

signals of interest is the space L

of square integrable functions, i.e., every vector

x ∈ H is a function x(t), t ∈ R. We use the notations x and x( t) interchangeably

according to the context. The continuous-time Fourier transform (CTFT) of a

signal x(t) is denoted by X(ω) and is deﬁned by

X(ω) =

∞

−∞

x(t)e

jωt

dt. (1.1)

Similarly, the discrete-time Fourier transform (DTFT) of a sequence c[n] is

denoted by C(e

jω

) and is deﬁned as

C( e

jω

) =

∞

n=−∞

c[n]e

jωn

. (1.2)

The inner product between vectors x, y ∈ H is denoted hx, yi, and is linear

in the second argument; kxk

= hx, xi is the squared norm of x. The `

norm

6 Chapter 1. Optimization Techniques in Modern Sampling Theory

of a sequence is kck

|c[n]|

. The orthogonal complement of a subspace

A is denoted by A

⊥

. A direct sum between two closed subspaces W and S is

written as W ⊕ S, and is the sum set {w + s; w ∈ W, s ∈ S} with the prop erty

W ∩ S = {0}. Given an operator T , T

∗

is its adjoint, and N(T) and R(T ) are

its null space and range space, respectively.

1.2.2 Projections

A projection E in a Hilbert space H is a linear operator from H onto itself that

satisﬁes the property

= E. (1.3)

A projection operator maps the entire space H onto the range R(E), and leaves

vectors in this subspace unchanged. Property (1.3) implies that every vector in

H can be uniquely decomposed into a vector in R(E) and a vector in N(E),

that is, we have the direct sum decomposition H = R(E) ⊕ N(E). Therefore, a

projection is completely determined by its range space and null space.

An orthogonal projection P is a Hermitian projection operator. In this case

the range space R(P ) and null space N(P) are orthogonal, and consequently

P is completely determined by its range. We use the notation P

to denote

an orthogonal projection with range V = R(P

). An important property of an

orthogonal projection onto a closed subspace V is that it maps every vector in

H to the vector in V which is closest to it:

y = arg min

x∈V

ky − xk. (1.4)

This property will be useful in a variety of diﬀerent sampling scenarios.

An oblique projection is a projection operator that is not necessarily Hermi-

tian. The notation E

⊥

denotes an oblique projection with range space A and

null space S

⊥

. If A = S, then E

⊥

= P

[16]. The oblique projection onto A

along S

⊥

is the unique operator satisfying

⊥

a = a for any a ∈ A;

⊥

s = 0 for any s ∈ S

⊥

. (1.5)

Projections can be used to characterize the pseudo-inverse of a given operator.

Speciﬁcally, let T be a bounded operator with closed range. The Moore-Penrose

pseudo inverse of T , denoted T

†

, is the unique operator satisfying [17]:

N(T

†

) = R(T )

⊥

R(T

†

) = N(T )

⊥

T T

†

x = x ∀x ∈ R(T ). (1.6)

The following is a set of properties of the pseudo-inverse op erator, which will b e

used extensively throughout the chapter [17].

Optimization Techniques in Modern Sampling Theory

Lemma 1.1. Let T be a bounded operator with closed range. Then:

1. P

R(T )

= T

†

T .

2. P

N(T )

⊥

= T T

†

3. T

∗

has closed range and (T

∗

)

†

= (T

†

)

∗

1.2.3 Frames

As we will see in the sequel, sampling can be viewed as the process of taking

inner products of a signal x with a sequence of vectors {a

}. To simplify the

derivations associated with such sequences, we use the notion of set transforms.

Deﬁnition 1.1. A set transformation A : `

→ H corresponding to vectors {a

}

is deﬁned by Ab =

b[n]a

for all b ∈ `

. From the deﬁnition of the adjoint, if

c = A

∗

x, then c[n] = ha

, xi.

Note that a set transform A corresponding to vectors {a

}

n=1

in R

is simply

an M × N matrix whose columns are {a

}

n=1

To guarantee stability of the sampling theorems we develop, we concentrate

on vector sets that generate frames [18, 19].

Deﬁnition 1.2. A family of vectors {a

} in a Hilbert space H is called a frame

for a subspace A ⊆ H if there exist constants α > 0 and β < ∞ such that the

associated set transform A satisﬁes

αkxk

≤ kA

∗

≤ βkxk

, ∀x ∈ A. (1.7)

The norm in the middle term is the `

norm of sequences.

The lower bound in (1.7) ensures that the vectors {a

} span A. Therefore, the

number of frame elements, which we denote by N, must be at least as large as

the dimension of A. If N < ∞, then the right hand inequality of (1.7) is always

satisﬁed with β =

. Consequently, any ﬁnite set of vectors that spans

A is a frame for A. For an inﬁnite set of frame vectors {a

}, condition (1.7)

ensures that the sum x =

b[n]a

converges for any sequence b ∈ `

and that

a small change in the expansion coeﬃcients b results in a small change in x [20].

Similarly, a slight perturbation of x will entail only a small change in the inner

products with the frame elements.

1.3 Sampling and reconstruction setup

We are now ready to introduce the sampling and reconstruction setup that will

be studied in this chapter. Before we elaborate on the abstract Hilbert space

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