There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

On robust estimation of the location parameter

Digital processing of speech signals

In this paper we consider sparsity on a tensor level, as given by the n-rank of a tensor. In an important sparse-vector approximation problem (compressed sensing) and the low-rank matrix recovery problem, using a convex relaxation technique proved to be a valuable solution strategy. Here, we will adapt these techniques to the tensor setting. We use the n-rank of a tensor as a sparsity measure and consider the low-n-rank tensor recovery problem, i.e. the problem of finding the tensor of the lowest n-rank that fulfills some linear constraints. We introduce a tractable convex relaxation of the n-rank and propose efficient algorithms to solve the low-n-rank tensor recovery problem numerically. The algorithms are based on the Douglas–Rachford splitting technique and its dual variant, the alternating direction method of multipliers.

/pdf/tensor-completion-and-low-n-rank-tensor-recovery-via-convex-5g3xjgvj70.pdf

Tensor completion and low-n-rank tensor recovery via convex optimization

Scattered Data Approximation

This paper exemplifies that the use of multiple kernels leads to efficient adaptive filtering for nonlinear systems. Two types of multikernel adaptive filtering algorithms are proposed. One is a simple generalization of the kernel normalized least mean square (KNLMS) algorithm [2], adopting a coherence criterion for dictionary designing. The other is derived by applying the adaptive proximal forward-backward splitting method to a certain squared distance function plus a weighted block l1 norm penalty, encouraging the sparsity of an adaptive filter at the block level for efficiency. The proposed multikernel approach enjoys a higher degree of freedom than those approaches which design a kernel as a convex combination of multiple kernels. Numerical examples show that the proposed approach achieves significant gains particularly for nonstationary data as well as insensitivity to the choice of some design-parameters.

/pdf/multikernel-adaptive-filtering-yhq04km1xt.pdf

Multikernel Adaptive Filtering

In this paper, we propose a novel adaptive filtering algorithm based on an iterative use of (i) the proximity operator and (ii) the parallel variable-metric projection. Our time-varying cost function is a weighted sum of squared distances (in a variable-metric sense) plus a possibly nonsmooth penalty term, and the proposed algorithm is derived along the idea of proximal forward-backward splitting in convex analysis. For application to sparse-system identification problems, we employ the (weighted) l 1 norm as the penalty term, leading to a time-varying soft-thresholding operator. As the simple example of the proposed algorithm, we present the variable-metric affine projection algorithm composed with the time-varying soft-thresholding operator. Numerical examples demonstrate that the proposed algorithms notably outperform their counterparts without soft-thresholding both in convergence speed and steady-state mismatch, while the extra computational complexity due to the additional soft-thresholding is negligibly low.

A sparse adaptive filtering using time-varying soft-thresholding techniques

The first aim of this paper is to present a useful toolbox of quasi-nonexpansive mappings for convex optimization from the viewpoint of using their fixed point sets as constraints. Many convex optimization problems have been solved through elegant translations into fixed point problems. The underlying principle is to operate a certain quasi-nonexpansive mapping T iteratively and generate a convergent sequence to its fixed point. However, such a mapping often has infinitely many fixed points, meaning that a selection from the fixed point set Fix(T) should be of great importance. Nevertheless, most fixed point methods can only return an “unspecified” point from the fixed point set, which requires many iterations. Therefore, based on common sense, it seems unrealistic to wish for an “optimal” one from the fixed point set. Fortunately, considering the collection of quasi-nonexpansive mappings as a toolbox, we can accomplish this challenging mission simply by the hybrid steepest descent method, provided that the cost function is smooth and its derivative is Lipschitz continuous. A question arises: how can we deal with “nonsmooth” cost functions? The second aim is to propose a nontrivial integration of the ideas of the hybrid steepest descent method and the Moreau–Yosida regularization, yielding a useful approach to the challenging problem of nonsmooth convex optimization over Fix(T). The key is the use of smoothing of the original nonsmooth cost function by its Moreau–Yosida regularization whose the derivative is always Lipschitz continuous. The field of application of hybrid steepest descent method can be extended to the minimization of the ideal smooth approximation Fix(T). We present the mathematical ideas of the proposed approach together with its application to a combinatorial optimization problem: the minimal antenna-subset selection problem under a highly nonlinear capacity-constraint for efficient multiple input multiple output (MIMO) communication systems.

Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings

This paper proposes a robust reduced-rank scheme for adaptive beamforming based on joint iterative optimization (JIO) of adaptive filters. The novel scheme is designed according to the constant modulus (CM) criterion subject to different constraints. The proposed scheme consists of a bank of full-rank adaptive filters that forms the transformation matrix, and an adaptive reduced-rank filter that operates at the output of the bank of filters to estimate the desired signal. We describe the proposed scheme for both the direct-form processor (DFP) and the generalized sidelobe canceller (GSC) structures. For each structure, we derive stochastic gradient (SG) and recursive least squares (RLS) algorithms for its adaptive implementation. The Gram-Schmidt (GS) technique is applied to the adaptive algorithms for reformulating the transformation matrix and improving the performance. An automatic rank selection technique is developed and employed to determine the most adequate rank for the derived algorithms. A detailed complexity study and a convexity analysis are carried out. Simulation results show that the proposed algorithms outperform the existing full-rank and reduced-rank methods in convergence and tracking performance.

/pdf/adaptive-reduced-rank-constrained-constant-modulus-1blpfetkxo.pdf

Adaptive Reduced-Rank Constrained Constant Modulus Algorithms Based on Joint Iterative Optimization of Filters for Beamforming

This paper presents a new approach to efficient acoustic echo cancellation (AEC) based on reduced-rank adaptive filtering equipped with selective-decimation and adaptive interpolation. We propose a novel structure of an AEC scheme that jointly optimizes an interpolation filter, a decimation unit, and a reduced-rank filter. With a practical choice of parameters in AEC, the total computational complexity of the proposed reduced-rank scheme with the normalized least mean square (NLMS) algorithm is approximately half of that of the full-rank NLMS algorithm. We discuss the convergence properties of the proposed scheme and present a convergence condition. First, we examine the performance of the proposed scheme in a single-talk situation with an error-minimization criterion adopted in the decimation selection. Second, we investigate the potential of the proposed scheme in a double-talk situation by employing an ideal decimation selection. In addition to mean squared error (MSE) and power spectrum analysis of the echo estimation error, subjective assessments based on absolute category rating are performed, and the results demonstrate that the proposed structure provides significant improvements compared to the full-rank NLMS algorithm.

/pdf/efficient-acoustic-echo-cancellation-with-reduced-rank-28cundivcg.pdf

Masahiro Yukawa

Papers

Multikernel Adaptive Filtering

A sparse adaptive filtering using time-varying soft-thresholding techniques

Minimizing the Moreau Envelope of Nonsmooth Convex Functions over the Fixed Point Set of Certain Quasi-Nonexpansive Mappings

Adaptive Reduced-Rank Constrained Constant Modulus Algorithms Based on Joint Iterative Optimization of Filters for Beamforming

Efficient Acoustic Echo Cancellation With Reduced-Rank Adaptive Filtering Based on Selective Decimation and Adaptive Interpolation