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Tadashi Wadayama

Publications and source records attributed to Tadashi Wadayama.

81 records · Page 5Linked to original sources
arXiv2010

Statistical Mechanical Analysis of a Typical Reconstruction Limit of Compressed Sensing

We use the replica method of statistical mechanics to examine a typical performance of correctly reconstructing $N$-dimensional sparse vector $bx=(x_i)$ from its linear transformation $by=bF bx$ of $P$ dimensions on the basis of minimization of the $L_p$-norm $||bx||_p= lim_{epsilon to +0} sum_{i=1}^N |x_i|^{p+epsilon}$. We characterize the reconstruction performance by the critical relation of the successful reconstruction between the ratio $alpha=P/N$ and the density $rho$ of non-zero elements in $bx$ in the limit $P,,N to infty$ while keeping $alpha sim O(1)$ and allowing asymptotically negligible reconstruction errors. We show that the critical relation $alpha_c(rho)$ holds universally as long as $bF^{rm T}bF$ can be characterized asymptotically by a rotationally invariant random matrix ensemble and $bF bF^{rm T}$ is typically of full rank. This supports the universality of the critical relation observed by Donoho and Tanner ({em Phil. Trans. R. Soc. A}, vol.~367, pp.~4273--4293, 2009; arXiv: 0807.3590) for various ensembles of compression matrices.

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arXiv2008

A Cutting Plane Method based on Redundant Rows for Improving Fractional Distance

In this paper, an idea of the cutting plane method is employed to improve the fractional distance of a given binary parity check matrix. The fractional distance is the minimum weight (with respect to l1-distance) of vertices of the fundamental polytope. The cutting polytope is defined based on redundant rows of the parity check matrix and it plays a key role to eliminate unnecessary fractional vertices in the fundamental polytope. We propose a greedy algorithm and its efficient implementation for improving the fractional distance based on the cutting plane method.

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arXiv2008

On Undetected Error Probability of Binary Matrix Ensembles

In this paper, an analysis of the undetected error probability of ensembles of binary matrices is presented. The ensemble called the Bernoulli ensemble whose members are considered as matrices generated from i.i.d. Bernoulli source is mainly considered here. The main contributions of this work are (i) derivation of the error exponent of the average undetected error probability and (ii) closed form expressions for the variance of the undetected error probability. It is shown that the behavior of the exponent for a sparse ensemble is somewhat different from that for a dense ensemble. Furthermore, as a byproduct of the proof of the variance formula, simple covariance formula of the weight distribution is derived.

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arXiv2008

Gradient Descent Bit Flipping Algorithms for Decoding LDPC Codes

A novel class of bit-flipping (BF) algorithms for decoding low-density parity-check (LDPC) codes is presented. The proposed algorithms, which are called gradient descent bit flipping (GDBF) algorithms, can be regarded as simplified gradient descent algorithms. Based on gradient descent formulation, the proposed algorithms are naturally derived from a simple non-linear objective function.

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arXiv2008

Interior Point Decoding for Linear Vector Channels

In this paper, a novel decoding algorithm for low-density parity-check (LDPC) codes based on convex optimization is presented. The decoding algorithm, called interior point decoding, is designed for linear vector channels. The linear vector channels include many practically important channels such as inter symbol interference channels and partial response channels. It is shown that the maximum likelihood decoding (MLD) rule for a linear vector channel can be relaxed to a convex optimization problem, which is called a relaxed MLD problem. The proposed decoding algorithm is based on a numerical optimization technique so called interior point method with barrier function. Approximate variations of the gradient descent and the Newton methods are used to solve the convex optimization problem. In a decoding process of the proposed algorithm, a search point always lies in the fundamental polytope defined based on a low-density parity-check matrix. Compared with a convectional joint message passing decoder, the proposed decoding algorithm achieves better BER performance with less complexity in the case of partial response channels in many cases.

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arXiv2008

Asymptotic Concentration Behaviors of Linear Combinations of Weight Distributions on Random Linear Code Ensemble

Asymptotic concentration behaviors of linear combinations of weight distributions on the random linear code ensemble are presented. Many important properties of a binary linear code can be expressed as the form of a linear combination of weight distributions such as number of codewords, undetected error probability and upper bound on the maximum likelihood error probability. The key in this analysis is the covariance formula of weight distributions of the random linear code ensemble, which reveals the second-order statistics of a linear function of the weight distributions. Based on the covariance formula, several expressions of the asymptotic concentration rate, which indicate the speed of convergence to the average, are derived.

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arXiv2007

Average Stopping Set Weight Distribution of Redundant Random Matrix Ensembles

In this paper, redundant random matrix ensembles (abbreviated as redundant random ensembles) are defined and their stopping set (SS) weight distributions are analyzed. A redundant random ensemble consists of a set of binary matrices with linearly dependent rows. These linearly dependent rows (redundant rows) significantly reduce the number of stopping sets of small size. An upper and lower bound on the average SS weight distribution of the redundant random ensembles are shown. From these bounds, the trade-off between the number of redundant rows (corresponding to decoding complexity of BP on BEC) and the critical exponent of the asymptotic growth rate of SS weight distribution (corresponding to decoding performance) can be derived. It is shown that, in some cases, a dense matrix with linearly dependent rows yields asymptotically (i.e., in the regime of small erasure probability) better performance than regular LDPC matrices with comparable parameters.

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arXiv2005

Average Coset Weight Distribution of Combined LDPC Matrix Ensemble

In this paper, the average coset weight distribution (ACWD) of structured ensembles of LDPC (Low-density Parity-Check) matrix, which is called combined ensembles, is discussed. A combined ensemble is composed of a set of simpler ensembles such as a regular bipartite ensemble. Two classes of combined ensembles have prime importance; a stacked ensemble and a concatenated ensemble, which consists of set of stacked matrices and concatenated matrices, respectively. The ACWD formulas of these ensembles is shown in this paper. Such formulas are key tools to evaluate the ACWD of a complex combined ensemble. From the ACWD of an ensemble, we can obtain some detailed properties of a code (e.g., weight of coset leaders) which is not available from an average weight distribution. Moreover, it is shown that the analysis based on the ACWD is indispensable to evaluate the average weight distribution of some classes of combined ensembles.

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