Sparsity Problem Involving Rational Basis Functions

Uncertainty principle and sparse reconstruction in pairs of orthonormal rational function bases

Sparse gammatone signal model optimized for English speech does not match the human auditory filters

1 - Variables and Functions

Iterative learning control approaches often suffer from poor extrapolability with respect to exogenous signals, including setpoint variations. The aim of this paper is to introduce rational basis functions in ILC. Such rational basis function have the potential to both increase performance and enhance extrapolability. The key caveat that is associated with these rational basis function lies in a significantly more complex optimization problem when compared to using polynomial basis functions. In this paper, a novel iterative optimization procedure is proposed that enables the use of rational basis functions in ILC. A simulation example confirms (1) the advantages of rational basis functions compared to pre-existing results, and (2) the efficacy of the proposed iterative algorithm.

The importance of sparse signal structures has been recognized in a plethora of applications ranging from medical imaging to group disease testing to radar technology. It has been shown in practice that various signals of interest may be (approximately) sparsely modeled, and that sparse modeling is often beneficial, or even indispensable to signal recovery. Alongside an increase in applications, a rich theory of sparse and compressible signal recovery has recently been developed under the names compressed sensing (CS) and sparse approximation (SA). This revolutionary research has demonstrated that many signals can be recovered from severely undersampled measurements by taking advantage of their inherent low-dimensional structure. More recently, an offshoot of CS and SA has been a focus of research on other low-dimensional signal structures such as matrices of low rank. Low-rank matrix recovery (LRMR) is demonstrating a rapidly growing array of important applications such as quantum state tomography, triangulation from incomplete distance measurements, recommender systems (e.g., the Netflix problem), and system identification and control. In this dissertation, we examine CS, SA, and LRMR from a theoretical perspective. We consider a variety of different measurement and signal models, both random and deterministic, and mainly ask two questions. How many measurements are necessary? How large is the recovery error? We give theoretical lower bounds for both of these questions, including oracle and minimax lower bounds for the error. However, the main emphasis of the thesis is to demonstrate the efficacy of convex optimization---in particular l1 and nuclear-norm minimization based programs---in CS, SA, and LRMR. We derive upper bounds for the number of measurements required and the error derived by convex optimization, which in many cases match the lower bounds up to constant or logarithmic factors. The majority of these results do not require the restricted isometry property (RIP), a ubiquitous condition in the literature.

CHAPTER 1 - INTRODUCTION

Analytic approximate eigenvalues by a new technique. Application to sextic anharmonic potentials

Integration in finite terms with elementary functions and dilogarithms

A Galerkin method using rational basis functions

Iterative learning control yields accurate feedforward input by utilizing experimental data from past iterations. However, typically there exists a tradeoff between task flexibility and tracking performance. This study aims to develop a learning framework with both high task-flexibility and high tracking-performance by integrating rational basis functions with frequency-domain learning. Rational basis functions enable the learning of system zeros, enhancing system representation compared to polynomial basis functions. The developed framework is validated through a two-mass motion system, showing high tracking-performance with high task-flexibility, enhanced by the rational basis functions effectively learning the flexible dynamics.

Compressive (or namely compressed) sensing (CS) exhibits superior performance in sparse spatial spectrum estimation from a predefined vandermonde based dictionary. The CS theory requires that dictionary is as incoherent (orthogonal) as possible since the number of atoms is generally much more than observation vectors, and, namely, the dictionary is over-complete or redundant. Previous researches focus on designing sensing matrix to reduce the mutual coherence of dictionary. However, according to Grassmannian frames, it is still a problem that the coherence of a given dictionary is hard to break through an equiangular tight frame (ETF). To address the problem, we proposed and proved a KR-KSVD method to break through the original lower bound of mutual coherence. In the Khatri-Rao subspace, measurement matrix is designed by minimizing the cost function between the Gram matrix of the equivalent dictionary and an identity matrix with the KSVD method. Simulations demonstrate that the method can produce a better performance in terms of mutual coherence property and sparse recovery accuracy.Compressive (or namely compressed) sensing (CS) exhibits superior performance in sparse spatial spectrum estimation from a predefined vandermonde based dictionary. The CS theory requires that dictionary is as incoherent (orthogonal) as possible since the number of atoms is generally much more than observation vectors, and, namely, the dictionary is over-complete or redundant. Previous researches focus on designing sensing matrix to reduce the mutual coherence of dictionary. However, according to Grassmannian frames, it is still a problem that the coherence of a given dictionary is hard to break through an equiangular tight frame (ETF). To address the problem, we proposed and proved a KR-KSVD method to break through the original lower bound of mutual coherence. In the Khatri-Rao subspace, measurement matrix is designed by minimizing the cost function between the Gram matrix of the equivalent dictionary and an identity matrix with the KSVD method. Simulations demonstrate that the method can produce a better...

Error-correcting decoding is generalized to multivariate sparse rational function recovery from evaluations that can be numerically inaccurate and where several evaluations can have severe errors (outliers). The generalization of the Berlekamp-Welch decoder to exact Cauchy interpolation of univariate rational functions from values with faults is by Kaltofen and Pernet in 2012. We give a different univariate solution based on structured linear algebra that yields a stable decoder with floating point arithmetic. Our multivariate polynomial and rational function interpolation algorithm combines Zippel's symbolic sparse polynomial interpolation technique [Ph.D. Thesis MIT 1979] with the numeric algorithm by Kaltofen, Yang, and Zhi [Proc. SNC 2007], and removes outliers (cleans up data) through techniques from error correcting codes. Our multivariate algorithm can build a sparse model from a number of evaluations that is linear in the sparsity of the model.

Effective coefficient asymptotics of multivariate rational functions via semi-numerical algorithms for polynomial systems

This paper proposes a new adaptive algorithm for identification Hammerstein systems by using the rational orthogonal basis functions. By adopting an adaptive decomposition algorithm for the Hardy space functions, identification of the linear part can be achieved with the sampling input and output data by selecting poles for the rational orthogonal basis functions in order to get optimal approximations. After that the nonlinear part can be estimated with general least-squares method. Example shows the proposed algorithm is efficient.

Heartbeat classification using electrocardiogram (ECG) data is a vital assistive technology for wearable health solutions. We propose heartbeat feature classification based on a novel sparse representation using time-frequency joint distribution of ECG. Fundamental to this is a multi-layer perceptron, which incorporates these signatures to detect cardiac arrhythmia. This approach is validated with ECG data from MIT-BIH arrhythmia database. Results show that our approach has an average 95.7% accuracy, an improvement of 22% over state-of-the-art approaches. Additionally, ECG sparse distributed representations generates only 3.7% false negatives, reduction of 89% with respect to existing ECG signal classification techniques.