压缩感知的优化学习

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"Learning Compressed Sensing - 探索优化线性投影在压缩感知中的应用" 压缩感知（Compressed Sensing，简称CS）是一种数学理论，它表明稀疏信号可以通过少量的线性测量实现精确重构。这一领域的研究在近年来引起了广泛的关注，因为它挑战了传统的信号处理理念，即需要获取信号的完整采样才能进行有效的恢复。论文作者Yair Weiss、Hyun Sung Chang和William T. Freeman深入探讨了这一前沿主题。在理想情况下，对于具有高稀疏度的信号且无测量噪声时，随机测量可以确保完美重构。然而，当面对实际应用中的信号和噪声分布时，基于主成分分析（PCA）或独立成分分析（ICA）的测量方法在某些情况下可能表现出更好的性能，能够在较少的测量次数下实现重构。该论文的核心问题是：在我们希望测量的信号类型训练集条件下，什么样的线性投影是最优的压缩感知测量方式？作者发现，最优的投影通常既不是数据的主成分，也不是其独立分量。相反，它们是一组新颖的投影，这些投影能够捕获数据中仍然存在的不确定性，即那些未被充分解析的部分。为了找到这些最优投影，论文可能会涉及一系列优化算法和统计学习方法，包括但不限于凸优化、机器学习模型以及非负矩阵分解等技术。通过学习和适应训练集的特性，这些方法能够构建出一组投影矩阵，该矩阵能够有效地捕获信号的稀疏结构，即使在存在噪声的情况下也能实现高效的重构。此外，论文可能会讨论如何评估这些投影的性能，比如通过重构误差、信噪比（SNR）或者计算复杂度等指标。同时，可能会探讨这些结果在实际应用中的意义，如医学成像、无线通信、图像处理等领域，以及如何将这些理论成果转化为实际的硬件系统设计。 "Learning Compressed Sensing"这篇论文不仅揭示了压缩感知理论的新见解，还提出了寻找最优线性投影的策略，这对于提高信号处理效率和降低数据采集成本具有重要的科学价值和实际应用前景。

Learning Compressed Sensing

Yair Weiss

1,2

Hebrew University of Jerusalem

yweiss@cs.huji.ac.il

Hyun Sung Chang

and William T. Freeman

MIT CSAIL

{hyunsung,billf} @mit.edu

Abstract— Compressed sensing [7], [6] is a recent set of

mathematical results showing that sparse signals can be exactly

reconstructed from a small number of linear measurements.

Interestingly, for ideal sparse signals with no measurement

noise, random measurements allow perfect reconstruction while

measurements based on principal component analysis (PCA)

or independent component analysis (ICA) do not. At the same

time, for other signal and noise distributions, PCA and ICA can

signiﬁcantly outperform random projections in terms of enabling

reconstruction from a small number of measurements.

In this paper we ask: given a training set typical of the signals

we wish to measure, what are the optimal set of linear projections

for compressed sensing ? We show that the optimal projections

are in general not the principal components nor the independent

components of the data, but rather a seemingly novel set of

projections that capture what is still uncertain about the signal,

given the training set. We also show that the projections onto

the learned uncertain components may far outperform random

projections. This is particularly true in the case of natural images,

where random projections have vanishingly small signal to noise

ratio as the number of pixels becomes large.

I. INTRODUCTION

Compressed sensing [7], [6] is a set of recent mathematical

results on a classic question: given a signal x ∈ R

and a

set of p linear measurements y ∈ R

, y = W x, how many

measurements are required to allow reconstruction of x ?

Obviously, if we knew nothing at all about x, i.e. x can be

any n dimensional vector, we would need n measurements. Al-

ternatively, if we know our signal x lies in a low-dimensional

linear subspace, say of dimension k, then k measurements are

enough. But what if we know that x lies in a low-dimensional

nonlinear manifold ? Can we still get away with fewer than

n measurements ?

To motivate this question, consider the space of natural

images. An image with n pixels can be thought of as a vector

in R

but natural images occupy a tiny fraction of the set of

all signals in this space. If there was a way to exploit this fact,

we could build cameras with a small number of sensors that

would still enable us perfect, high resolution, reconstructions

for natural images.

The basic mathematical results in compressed sensing deal

with signals that are k sparse. These are signals that can

be represented with a small number, k of active (non-zero)

basis elements. For such signals, it was shown in [7], [5],

that ck log n generic linear measurements are sufﬁcient to

recover the signal exactly (with c a constant). Furthermore,

the recovery can be done by a simple convex optimization or

by a greedy optimization procedure [8].

These results have generated a tremendous amount of ex-

citement in both the theoretical and practical communities. On

the theoretical side, the performance of compressed sensing

with random projections has been analyzed when the signals

are not exactly k sparse, but rather compressible (i.e. can

be well approximated with a small number of active basis

elements) [7], [5] as well as when the measurements are

contaminated with noise [11], [19]. On the practical side,

applications of compressed sensing have been explored in

building “single-pixel” cameras [20], medical imaging [14]

and geophysical data analysis [12].

Perhaps the most surprising result in compressed sensing is

that perfect recovery is possible with random projections. This

is surprising given the large amount of literature in machine

learning and statistics devoted to ﬁnding projections that are

optimal in some sense (e.g. [4]). In fact, as we review in

the next section, for ideal sparse signals with no measure-

ment noise, random measurements signiﬁcantly outperform

measurements based on principal component analysis (PCA)

or independent component analysis (ICA). At the same time,

for other signal and noise distributions, PCA and ICA can

signiﬁcantly outperform random projections.

In this paper we ask: given a training set typical of the

signals we wish to measure, what are the optimal set of linear

projections for compressed sensing ? We show that the optimal

projections are in general not the principal components nor the

independent components of the data, but rather a seemingly

novel set of projections that capture what is still uncertain

about the signal, given the training set. We also show that

the projections onto the learned uncertain components may

far outperform random projections. This is particularly true

in the case of natural images, where random projections have

vanishingly small signal to noise ratio as the number of pixels

becomes large.

II. RANDOM PROJECTIONS VERSUS PCA AND ICA

To compare random projections to PCA and ICA, con-

sider the sparse signals illustrated by the image patches in

ﬁgure 1. In this dataset, each signal x has exactly one non-

zero component, and this non-zero component is uniformly

distributed in the range [−U

, U

]. We assume that all indices

i have approximately the same range, i.e. U

≈ 1, but to break

symmetries we set U

= 1 + /i.

We are interested in the probability of correct reconstruc-

tion, from a projected signal:

y = Wx (1)

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