基于贝塔-伯努利过程的鲁棒贝叶斯稀疏表示

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在近年来的研究中，稀疏表示因其在图像处理、计算机视觉、信号处理等领域中的广泛应用而备受关注。它在图像去噪、图像修复（inpainting）、盲源分离（BSS）以及压缩感知等任务中发挥着核心作用。然而，一个重要的挑战是训练数据中的异常值（outliers），这些异常值可能导致估计结果的不准确性和鲁棒性降低。本文提出了一种基于非参数贝叶斯框架的鲁棒贝叶斯稀疏表示方法。传统的稀疏表示模型假设噪声可以被简化为一个高斯噪声项，但这种假设在存在异常值的情况下并不适用。为此，作者将噪声分解为一个高斯噪声和一个稀疏的异常噪声项，旨在更精确地捕捉数据中的复杂性。作者引入了贝塔-伯努利过程作为稀疏解的先验分布。贝塔-伯努利过程是一种概率模型，它能够自适应地学习数据的潜在结构，对于处理稀疏性和不确定性具有优势。相比于固定的先验分布，非参数贝叶斯方法能够更好地应对数据的异质性，包括异常值的情况。关键词：异常值检测、非参数贝叶斯方法、贝塔-伯努利过程、稀疏表示。这种方法的主要目标是通过利用贝塔-伯努利过程的特性来提高模型的鲁棒性，从而在面对训练数据中的异常值时，依然能获得更加稳健和可靠的稀疏表示估计。在实际应用中，该模型可能通过迭代优化算法求解，如 Expectation-Maximization (EM) 或 Variational Inference (VI)，以找到最优的贝塔-伯努利过程参数和稀疏系数。通过这种方式，模型能够在保持整体稀疏性的同时，有效地抑制异常值的影响，提升整个系统的性能。总结来说，这篇研究论文创新地结合了非参数贝叶斯统计与贝塔-伯努利过程，为稀疏表示问题提供了一种新的解决策略，特别适用于那些包含异常值的复杂数据集。其潜在应用领域包括但不限于图像处理中的去噪和复原，信号处理中的信号恢复，以及机器学习中的特征选择，有望显著提高这些领域的算法性能和鲁棒性。

Robust Bayesian Sparse Representation Based on

beta-Bernoulli Process Prior

Zengyuan Mi , Qin Lin , Yue Huang , Xinghao Ding

School of Information Science and Technology, Xiamen University, Xiamen China, mzy19890220@163.com

Abstract

—There has been a significant growing interest in

the study of sparse representation recent years. Although

many algorithms have been developed, outliers in the

training data make the estimation unreliable. In the

paper, we present a model under non-parametric

Bayesian framework to solve the problem. The noise term

in the sparse representation is decomposed into a

Gaussian noise term and an outlier noise term, which we

assume to be sparse. The beta–Bernoulli process is

employed as a prior for finding sparse solutions.

Keywords-outliers; non-parametric Bayesian; beta-Bernoulli

process; sparse representation

NTRODUCTION

There has been a significant growing interest in the study

of sparse representation recent years. It has found applications

in a wide range of diverse fields. These include image

denoising, inpainting, blind source separation (BSS) and

compressive sensing [1]. The sparse representation of a signal

is modeled by [2]:

αε

(1)

where Y represents an

MN×

signal matrix, D is an

MK×

matrix with K atoms

{, , , }

dd d⋅⋅⋅

called the

dictionary,

is a

KN×

coefficient matrix which is sparse

and

is an

MN×

noise matrix.

The representation of a signal with an overcomplete

dictionary has several advantages，including the fact that it

encourages a simple model and therefore over-training is

often avoided [3].

Many algorithms have been proposed to find sparse

coefficient when the dictionary matrix is given. The simplest

one is the relevance vector machine (RVM) [4]. Kernel

function matrix is chosen as the dictionary and Student’s t-

distribution is employed to enhance the sparsity of the

coefficients. Designing dictionaries to better fit the above

model can be done by either selecting one from a pre

specified set of linear transforms or adapting the dictionary to

a set of training signals. In [5] Michal Aharon and Michael

Elad proposed a novel algorithm for dictionary training

(KSVD), this algorithm could be interpreted as a

generalization of the K-Means clustering process. In [6] non-

parametric Bayesian techniques are considered for learning

dictionaries for sparse image representations (BPFA), which

gives a more intuitive constraint on the sparsity than RVM.

All the above is done with the assumption that the noise

obeys a Gaussian distribution. However, outliers in the

training data (such as salt and pepper noise in the images) will

make the assumption unreliable [7].

In the paper we extend the BPFA algorithm to efficiently

find out outliers. A robust model under non-parametric

Bayesian framework is proposed to decompose the noise term

in the BPFA model into a Gaussian noise term and an outlier

noise term, which we assume to be sparse. The beta-Bernoulli

prior [8] is employed for finding sparse solutions to the image

representation.

The remainder of the paper is organized as follows. We

review beta-Bernoulli process and present the proposed model

in the next section with inference discussed in the Section III.

Experiment results on image denoising are showed in Section

IV. Then we make conclusions in Section V.

II. MODEL

CONSTRUCTION

A. Beta-Bernoulli process

The two-parameter beta-Bernoulli process, which is a

non-parametric prior with three inputs:

0a >

and

0b >

, and

base measure

is represented as:

~(,,)HBPabH

.The base

measure

is denoted as:

∑

, where

is the

column of D. Provided that K is reasonably small, the

discrete beta-Bernoulli process,

∑

, requires the

generation of the vector

, where

~()

~(/,(1)/)

z Bernoulli

Beta a K b K K

−

(2)

B. Model construction

The noise term in the model (1) described in BPFA is

often assumed to be independent Gaussian samples with zero

mean. However, in the presence of outliers, Gaussian noise is

not an appropriate assumption for

. We considered splitting

The project is supported by the National Natural Science Foundation of

China (No.30900328, 61172179), the Fundamental Research Funds for the

Central Universities (No. 2011121051), the Natural Science Foundation of

Fujian Province of China(No. 2012J05160)

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