compressed Sensing Reconstruction via Belief Propagation
Introduction
One of the key theorem in digital signal processing is Shannon/Nyquist theorem. This theorem specifies the conditions on which a band limited signal can be reconstructed uniquely from its discrete samples. This property of band limited signals made signal processing viable on natural analog signals. However, in some applications even the sampled signal lays in a extremely high dimensional vector space. To make signal processing algorithms computationally tractable, many researchers work on compressiblity of signals. Here, it is assumed that most of the information content of a signal lays in a few samples with large magnitude. This lead to study and investigation on a class of signals, known as compressible signals.
Compressible signals can be stored efficiently by ignoring small value coefficients. Naturally it means that during sampling procedure loosing those samples is unimportant. This lead to a natural questions: Can we sample compressible signals in a compressed way? Is there any method to sense only those large value coefficients? In parallel works by Donoho <ref name="R1"> D. Donoho, “Compressed Sensing,” in IEEE Trans. on Info. theory, vol 52, no 4,pp. 1289–1306, Apr. 2006.</ref> and Candes et. al <ref name="R2"> E. Candes, J. Romberg, J.; T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” in IEEE Trans. on Info. theory, vol 52, no 2,pp. 489–509, Feb. 2006.</ref> this question was answered. They showed that this procedure can be done by measuring a small number of random linear projection of the source signal. They also provided a decoding scheme for reconstructing the source signal using those measurements and provided theoretical proofs for different aspects of this topic.
Tow important questions in CS theory is: 1. How many measurement do we need to be able to reconstruct a specific signal in stable and robust way? 2. What is the decoding algorithm complexity? These are vital questions which must be answered in order to make CS theory an alternative procedure for ordinary sampling. These questions have been answered by various researchers. Possibly the most important limitation of CS theory is the complexity of the current decoding algorithms. Those algorithms with strong theoretical background which can be applied to different applications are computationally expensive and can not be used for large scale problems. For this reason one of the active research topics in this field, is to develop tractable CS decoding algorithms. A recently published paper <ref name="R3"> D. Baron, S. Sarvotham, and R.G. Baraniuk, “Compressive Sensing Via Belief Propagation ,” in IEEE Trans. on Signal Processing, vol 58, no 1,pp. 269–280, Jan. 2010.</ref> tries to provide a decoding algorithms via belief propagation (BP). Authors connect CS to graphical models in an interesting approach and then use BP on the corresponding graph to provide a decoding algorithm. In the current paper summary we will follow their approach which indicates the importance of graphical models and the wide range of problems that these models can be used on. It is shown that this approach leads to a decoding algorithm with much less computational load.
Compressed Sensing
Compressive sensing is a method for sampling and representing compressible signals at below the Nyquist rate measurement rates. Let [math]\displaystyle{ \ x }[/math] is a [math]\displaystyle{ N \times 1 }[/math] column vector in [math]\displaystyle{ \mathbb{R}^N }[/math] that only has [math]\displaystyle{ K \le N }[/math] non-zero entries. To measure this source signal we measure a small number of linear combinations of its elements, [math]\displaystyle{ \ M }[/math], as follows
where [math]\displaystyle{ \ y \in \mathbb{R}^M }[/math] is the vector containing the measurements and [math]\displaystyle{ \psi \in \mathbb{R}^{M \times N} }[/math] is a matrix that carries the coordinate basis we use in our application. The goal of CS theory is, given [math]\displaystyle{ \psi }[/math] and [math]\displaystyle{ \ y }[/math],to recover the source signal in a stable and robust approach.
It is obvious that this problem is ill-posed and infinitely many solutions satisfy (1). Since we assume that our source signal is sparse we can take advantage of this knowledge and assert sparsity as prior to limit the feasible region of solutions. Interestingly it has been proven that under this priority if the source signal is sparse enough then it can be recovered as the solution of optimization problem <ref name="R1"/> and <ref name="R2"/>:
where [math]\displaystyle{ \ ||.||_0 }[/math] denotes [math]\displaystyle{ \ l_0 }[/math] norm that measures the number of non-zero entries. Unfortunately [math]\displaystyle{ \ l_0 }[/math] optimization is NP hard and needs combinatorial enumeration and thus is intractable. Interestingly, it has been proven that if the sensing matrix satisfies restricted isometry property (RIP), then (2) can be equivalently solves as:
Here we have replaced [math]\displaystyle{ \ l_0 }[/math] norm by [math]\displaystyle{ \ l_1 }[/math] norm which is convex and robust towards additive noise. This optimization can be solved efficiently using linear programming and reconstruction complexity is [math]\displaystyle{ \ O(N^3) }[/math] . It has been shown that matrices with independently identical distributed Gaussian or Bernoulli entries satisfy RIP condition and can be used effectively as sensing matrix. In such cases the linear programming decoder requires [math]\displaystyle{ \ K log(1+N/K) }[/math] measurements for signal reconstruction.
Compressed Sensing Reconstruction Algorithms
Linear programming is the core approach for designing CS decoding algorithms. This approach is tractable with complexity of [math]\displaystyle{ \ O(N^3) }[/math], however this complexity is intractable for large scale problems. For example a [math]\displaystyle{ \ 256\times 256 }[/math] image will need [math]\displaystyle{ \ 2^{48} }[/math] operations for decoding! One the other hand the encoding stage will need [math]\displaystyle{ \ O(NM) }[/math] computations for a dense sensing matrix. For this reason one important direction in this filed is to design and simplify the encoding and decoding algorithms.
One alternative decoding approach is iterative greedy algorithm. This approach needs slightly more measurements but it decreases computational complexity significantly. Perhaps the most well know algorithms in this category are Orthogonal Matching Pursuit <ref name="R4"> J. Tropp and A. C. Gilbert, “Signal recovery from partial information via orthogonal matching pursuit,” Apr. 2005, Preprint.</ref> and Matching Pursuit (MP) <ref name="R5"> EM. F. Duarte, M. B. Wakin, and R. G. Baraniuk, “Fast reconstruction of piece-wise smooth signals from random projections ,” in Proc. SPARS05, France, Nov. 2005.</ref>. OMP requires [math]\displaystyle{ \ 2K ln(N) }[/math] measurements and the coputational complexity is [math]\displaystyle{ \ O(KN^2) }[/math]. Even though it is better but it still is intractable for large [math]\displaystyle{ \ N }[/math] and [math]\displaystyle{ \ K }[/math]. More advanced versions like StOMP can recover the signal in [math]\displaystyle{ \ Nlog(N) }[/math] complexity. It is much faster and can be used for larger class of signals.
The common assumption in these approaches is that sensing matrix is assumed to be dense. Recently some researchers are trying to use sparse sensing matrix and use it to simplify the decoding algorithms. For example chaining pursuit uses this assumption and recovers the signal with complexity [math]\displaystyle{ \ K log^2(N) log^2(K) }[/math]. Table. 1 summaries CS schemes including their complexity as well as required measurements for each method. These results indicates that wecan reduce the complexity only at expense of having more measurements. The last row indicates the algorithm of the current paper. This shows superiority of this algorithm. Later we will see that this superiority comes from using BP, though some simplifying assumption has been made and the method is not general proposed. (Click on the image for table enlargement)
Lessons from LDPC codes applied to CS
At first glance it may seem impossible or at lease complicated to connect CS and a graphical model. The main idea for this procedure comes from the world of error control coding. Sparse error control coding schemes such as LDPC strongly suggests the use of sparse compressed sensing matrices. Can we use sparse CS matrices to encode sparse signals efficiently? Can we take advantage of the sparse structure to reduce the encoding and decoding complexity? To answer these questions we will have a brief review on LDPC code which will clarify how we can assign a graphical model to a CS problem.
An LDPC code is a blco code that has a sparse parity check matrix [math]\displaystyle{ \ H }[/math]. The parity check matrix of an LDPC code can be represented by a biparitite graph (see Fig. 1).
There are two class of nodes in the graph. The first class represents the transmitted data and the second class represents the check bits. A check bit is resulted as the linear combination of transmitted data. An edge exists in the graph if a transmitted bit appears in linear combination which produces an arbitrary check bit. This graph representation is not a special characteristics of LDPC codes. What makes it special is that for these codes the graph is sparse.: the number of edges in the graph grows linearly with [math]\displaystyle{ \ N }[/math] rather than [math]\displaystyle{ \ N^2 }[/math]. Since the graph is sparse decoding can be done efficiently using BP algorithm on the corresponding graph. This the key idea that suggests that CS decoding algorithms may develop using BP. Though our graph can have cycles but experiments show that even for cyclic graphs BP provides an acceptable approximation of the real answer.
Connecting CS decoding to graph decoding algorithms
Similar to error codes, a sensing matrix can be represented by a biparitie graph.The neighbors of a given measurement node is the set of coefficient nodes that are used to generate that measurement. In order to follow the LDPC scheme we need to choose sensing matrix to be sparse. As the first step in using graphical models in CS we assume the sensing matrix to be very simple: we assume the entries are only belonging to the set [math]\displaystyle{ \ \{0,1,-1\} }[/math]. Hence the encoding procedure is very simple and only involves summation and difference. This assumption make the corresponding graph to be sparse and hence we can use message passing algorithms for inference (note message passing for a general graph, possibly not sparse, is NP hard problem). We assume that each row of [math]\displaystyle{ \ \psi }[/math] contains exactly [math]\displaystyle{ \ L }[/math] non zero elements. This number is chosen based on properties of the source signal.
Encoding complexity
If the row weight of [math]\displaystyle{ \ \psi }[/math] is equal to [math]\displaystyle{ \ L }[/math] and we have [math]\displaystyle{ \ M }[/math] measurements then the encoding complexity will be [math]\displaystyle{ \ O(ML) }[/math]. Choosing [math]\displaystyle{ \ M=Nlog(N)/L }[/math] experimental seems to be appropriate and thus [math]\displaystyle{ \ O(ML)= O(Nlog(N)) }[/math]
CS-LDPC decoding of sparse signals
In this section we focus on approximately sparse signals. Similar to other inference problems which use BP we need to know probability distributions. Here we will assume a prior on source signal and then use it in BP procedure.
Source model
Since we want to use graphical model techniques it is necessary to assert a prior on the source signal. Here we will model the source signal by mixture of Gaussians model. This model is simple and captures prior knowledge about the source sparsity.
Let [math]\displaystyle{ \ X=[X(1),...,X(N)] }[/math] be a random vector in [math]\displaystyle{ \mathbb{R}^N }[/math] and [math]\displaystyle{ \ x=[x(1),...,x(N)] }[/math] be an instance of [math]\displaystyle{ \ X }[/math]. An approximately sparse signal has a few large magnitude coefficients and a lot of small magnitude coefficients. One can assume that we have two states for coefficients. One state is "high" which corresponds to large magnitude coefficients and the other state is "low" which corresponds to small magnitude coefficients. We denote the state of the j'th coefficient by [math]\displaystyle{ \ q(j) }[/math], where it is either equal to 1 (high) or 0 (low). The vector [math]\displaystyle{ \ q=[q(1),...,q(N) }[/math] which contains these state variables determines the source state configuration. We assume that this vector is an outcome of the state random variable [math]\displaystyle{ \ Q=[Q(1),...,Q(N)] }[/math] which is simply assumed to have multinomial distribution. For each [math]\displaystyle{ \ x(j) }[/math] we associate for the "high" state an outcome of a high variance Gaussian distribution and for the "low" state an outcome of a low variance Gaussian distribution to model the coefficients. Thus:
We assume that coefficients are distributed independently. To have about [math]\displaystyle{ \ K }[/math] large magnitude coefficients we choose the Bernoulli parameter of the [math]\displaystyle{ \ Q }[/math] equal to [math]\displaystyle{ \ S=K/N }[/math]. Fig. 2 illustrates the mixture of Gaussian model PDF and Fig. 3 is an outcome of such a distribution for [math]\displaystyle{ \ N=50, K=5, \sigma_1^2=30, \sigma_0^2=1 }[/math].
Decoding via statistical inference
The framework that was set transforms the original CS problem to a Bayesian inference problem. As we have seen in the course we can used message passing BP algorithm to solve these problems over factor graphs.
As we know (1) is undetermined linear system and assuming [math]\displaystyle{ \ \phi }[/math] to be full rank the solution lie in a hyperplane of dimension [math]\displaystyle{ \ N-M }[/math]. We use MAP and MMSE estimations to for the CS reconstruction. Note that MMSE and MAP estimates are identical for multivariate Gaussian rv's but we need both of them to derive the solution.
First we derive the exact MMSE estimate of the source in our framework. Unfortunately it has exponential complexity and is intractable for large scale systems. Due to sparse structure assumption of [math]\displaystyle{ \phi }[/math] we can use message passing scheme to estimate [math]\displaystyle{ \ \phi }[/math] approximately but in a computationally robust and stable approach.
Exact solution to CS statistical inference
To obtain MAP and MMSE estimates first we will assume [math]\displaystyle{ \ q }[/math] is known and will derive the MMSE estimate given [math]\displaystyle{ \ q }[/math]. Later we use this estimate to derive MMSE estimate. Let covariance matrix of [math]\displaystyle{ \ X }[/math] conditioned on [math]\displaystyle{ \ q }[/math] is denoted by [math]\displaystyle{ \ \Sigma_q }[/math]. It is a diagonal matrix with the j'th diagonal entry equal to [math]\displaystyle{ \ \sigma_0^2 }[/math] if the state is low and [math]\displaystyle{ \ \sigma_1^2 }[/math] if the state is high.
Theorem 1
Given [math]\displaystyle{ \ y=\psi x }[/math], then the MMSE and the MAP estimate of [math]\displaystyle{ \ x }[/math] conditioned on knowing the state vector is given:
Proof: If [math]\displaystyle{ \ q }[/math] is known then we have:
We do change of variable [math]\displaystyle{ \ \theta=\Sigma_q^{-1/2} x' }[/math]. Then we have [math]\displaystyle{ \ \theta_{MAP}=argmin_{\theta}{\theta^T\theta} }[/math] such that [math]\displaystyle{ \ y=\Phi \Sigma_q^{1/2}\theta }[/math]. This is a simple least square problem and the solution can be computed using pseudo inverse: [math]\displaystyle{ \ |theta_{MAP}=\Sigma_q^{1/2} \Phi^T (\Phi\Sigma_q \Phi^T)^{-1}y }[/math]. Inverse change of the variable gives the result. Finally as mentioned MMSE and MAP estimates are the same for multivariate Gaussian random variables.
Now we use this conditional solution to result in the general solution.
Theorem 2
The MMSE estimate of [math]\displaystyle{ \ x }[/math] when the state configuration is unknown is given by
The expected value of MMSE error obeys the equalities:
[math]\displaystyle{ \ =\Sigma_{q\in[0,1]^N}P[Q=q|Y=y]\hat{x}_{MMSE,q}=\Sigma_{q\in[0,1]^N}P[Y=y|Q=q]P[Q=q]\hat{x}_{MMSE,q} }[/math] which completes the proof.
Approximate solution to CS statistical inference via message passing
Numerical results
Conclusion
References
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