Blind equalization and sequence estimation using Markov chain Monte Carlo methods

This paper discusses the problem of restoring a digital input signal that has been degraded by an unknown FIR filter in noise, using the Gibbs sampler. A method for drawing a random sample of a sequence of bits is presented; this is shown to have faster convergence than a scheme by Chen and Li, which draws bits independently. ©1998 IEEE.

State estimation schemes for independent component coupled hidden Markov models

Conventional Hidden Markov models generally consist of a Markov chain observed through a linear map corrupted by additive noise. This general class of model has enjoyed a huge and diverse range of applications, for example, speech processing, biomedical signal processing and more recently quantitative finance. However, a lesser known extension of this general class of model is the so-called Factorial Hidden Markov Model (FHMM). FHMMs also have diverse applications, notably in machine learning, artificial intelligence and speech recognition [13, 17]. FHMMs extend the usual class of HMMs, by supposing the partially observed state process is a finite collection of distinct Markov chains, either statistically independent or dependent. There is also considerable current activity in applying collections of partially observed Markov chains to complex action recognition problems, see, for example, [6]. In this article we consider the Maximum Likelihood (ML) parameter estimation problem for FHMMs. Much of the extant literature concerning this problem presents parameter estimation schemes based on full data log-likelihood EM algorithms. This approach can be slow to converge and often imposes heavy demands on computer memory. The latter point is particularly relevant for the class of FHMMs where state space dimensions are relatively large. The contribution in this article is to develop new recursive formulae for a filter-based EM algorithm that can be implemented online. Our new formulae are equivalent ML estimators, however, these formulae are purely recursive and so, significantly reduce numerical complexity and memory requirements. A computer simulation is included to demonstrate the performance of our results. © Taylor & Francis Group, LLC.

Improved bounds for subband-adaptive iterative shrinkage/thresholding algorithms.

This paper presents new methods for computing the step sizes of the subband-adaptive iterative shrinkage-thresholding algorithms proposed by Bayram & Selesnick and Vonesch & Unser. The method yields tighter wavelet-domain bounds of the system matrix, thus leading to improved convergence speeds. It is directly applicable to non-redundant wavelet bases, and we also adapt it for cases of redundant frames. It turns out that the simplest and most intuitive setting for the step sizes that ignores subband aliasing is often satisfactory in practice. We show that our methods can be used to advantage with reweighted least squares penalty functions as well as L1 penalties. We emphasize that the algorithms presented here are suitable for performing inverse filtering on very large datasets, including 3D data, since inversions are applied only to diagonal matrices and fast transforms are used to achieve all matrix-vector products.

Robust noise reduction for speech and audio signals

Statistical model-based methods are presented for the reconstruction of autocorrelated signals in impulsive plus continuous noise environments. Signals are modelled as autoregressive and noise sources as discrete and continuous mixtures of Gaussians, allowing for robustness in highly impulsive and non-Gaussian environments. Markov Chain Monte Carlo methods are used for reconstruction of the corrupted waveforms within a Bayesian probabilistic framework and results are presented for contaminated voice and audio signals.

Rigorous finite-time guarantees for optimization on continuous domains by simulated annealing

Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete combinatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated annealing which allows one to guarantee finite-time performance in the optimization of functions of continuous variables. The results hold universally for any optimization problem on a bounded domain and establish a connection between simulated annealing and up-to-date theory of convergence of Markov chain Monte Carlo methods on continuous domains. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory.

Tree-Structured Stick Breaking Processes for Hierarchical Data

Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeable. One can view our model as providing infinite mixtures where the components have a dependency structure corresponding to an evolutionary diffusion down a tree. By using a stick-breaking approach, we can apply Markov chain Monte Carlo methods based on slice sampling to perform Bayesian inference and simulate from the posterior distribution on trees. We apply our method to hierarchical clustering of images and topic modeling of text data.

Coupled foundation-superstructure analysis and influence of building stiffness on foundation response

This paper proposes a simple method to include superstructure stiffness in foundation analyses. The method involves extracting a small "condensed structural matrix" from finite element models of the superstructure, which can then be incorporated into pile group or piled raft analyses using common approaches such as elastic continuum or load transfer methods. The matrix condensation method directly couples structural and geotechnical analyses, and eliminates the need for iterative analyses between structural and geotechnical engineers. Effectiveness of the approach is illustrated through analyses of several buildings designed with a typical floor plan but with varying heights. The parametric study illustrates that superstructure stiffness can have a significant influence on foundation settlement estimates, and the stiffening effects are dominated by the lower stories of the superstructure. The proposed method aims to bridge the gap between structural and geotechnical analyses. Also, being a computationally simple and accurate approach, it is applicable to parametric or optimization studies that would otherwise involve large amounts of analyses. © 2010 ASCE.

Simulated Annealing: Rigorous finite-time guarantees for optimization on continuous domains

Simulated annealing is a popular method for approaching the solution of a global optimization problem. Existing results on its performance apply to discrete combinatorial optimization where the optimization variables can assume only a finite set of possible values. We introduce a new general formulation of simulated annealing which allows one to guarantee finite-time performance in the optimization of functions of continuous variables. The results hold universally for any optimization problem on a bounded domain and establish a connection between simulated annealing and up-to-date theory of convergence of Markov chain Monte Carlo methods on continuous domains. This work is inspired by the concept of finite-time learning with known accuracy and confidence developed in statistical learning theory.

A reversible infinite HMM using normalised random measures

Copyright 2014 by the author(s). We present a nonparametric prior over reversible Markov chains. We use completely random measures, specifically gamma processes, to construct a countably infinite graph with weighted edges. By enforcing symmetry to make the edges undirected we define a prior over random walks on graphs that results in a reversible Markov chain. The resulting prior over infinite transition matrices is closely related to the hierarchical Dirichlet process but enforces reversibility. A reinforcement scheme has recently been proposed with similar properties, but the de Finetti measure is not well characterised. We take the alternative approach of explicitly constructing the mixing measure, which allows more straightforward and efficient inference at the cost of no longer having a closed form predictive distribution. We use our process to construct a reversible infinite HMM which we apply to two real datasets, one from epigenomics and one ion channel recording.

Sequential Monte Carlo computation of the score and observed information matrix in state-space models with application to parameter estimation

Sequential Monte Carlo (SMC) methods are popular computational tools for Bayesian inference in non-linear non-Gaussian state-space models. For this class of models, we propose SMC algorithms to compute the score vector and observed information matrix recursively in time. We propose two different SMC implementations, one with computational complexity $\mathcal{O}(N)$ and the other with complexity $\mathcal{O}(N^{2})$ where $N$ is the number of importance sampling draws. Although cheaper, the performance of the $\mathcal{O}(N)$ method degrades quickly in time as it inherently relies on the SMC approximation of a sequence of probability distributions whose dimension is increasing linearly with time. In particular, even under strong \textit{mixing} assumptions, the variance of the estimates computed with the $\mathcal{O}(N)$ method increases at least quadratically in time. The $\mathcal{O}(N^{2})$ is a non-standard SMC implementation that does not suffer from this rapid degrade. We then show how both methods can be used to perform batch and recursive parameter estimation.