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Signal processing makes extensive use of point estimators and accompanying error bounds. These work well up until the likelihood function has two or more high peaks. When it is important for an estimator to remain reliable, it becomes necessary to consider alternatives, such as set estimators. An obvious first choice might be confidence intervals or confidence regions, but there can be difficulties in computing and interpreting them (and sometimes they might still be blind to multiple peaks in the likelihood). Bayesians seize on this to argue for replacing confidence regions with credible regions. Yet Bayesian statistics require a prior, which is not always a natural part of the problem formulation. This paper demonstrates how a re-interpretation of the prior as a weighting function makes an otherwise Bayesian estimator meaningful in the frequentist context. The weighting function interpretation also serves as a reminder that an estimator should always be designed in the context of its intended application; unlike a prior which ostensibly depends on prior knowledge, a weighting function depends on the intended application. This paper uses the time-of-arrival (TOA) problem to illustrate all these points. It also derives a basic theory of region-based estimators distinct from confidence regions., Comment: Accepted for presentation at IEEE Conference on Acoustics, Speech and Signal Processing (ICASSP) 2025

This paper addresses the problem of nonconvex nonsmooth decentralised optimisation in multi-agent networks with undirected connected communication graphs. Our contribution lies in introducing an algorithmic framework designed for the distributed minimisation of the sum of a smooth (possibly nonconvex and non-separable) function and a convex (possibly nonsmooth and non-separable) regulariser. The proposed algorithm can be seen as a modified version of the ADMM algorithm where, at each step, an "inner loop" needs to be iterated for a number of iterations. The role of the inner loop is to aggregate and disseminate information across the network. We observe that a naive decentralised approach (one iteration of the inner loop) may not converge. We establish the asymptotic convergence of the proposed algorithm to the set of stationary points of the nonconvex problem where the number of iterations of the inner loop increases logarithmically with the step count of the ADMM algorithm. We present numerical results demonstrating the proposed method's correctness and performance., Comment: 6 pages, 1 Figure

Recursive filters are treated as linear time-invariant (LTI) systems but they are not: uninitialised, they have an infinite number of outputs for any given input, while if initialised, they are not time-invariant. This short tutorial article explains how and why they can be treated as LTI systems, thereby allowing tools such as Fourier analysis to be applied. It also explains the origin of the z-transform, why the region of convergence is important, and why the z-transform fails to find an infinite number of solutions.

Approximate message passing (AMP) algorithms break a (high-dimensional) statistical problem into parts then repeatedly solve each part in turn, akin to alternating projections. A distinguishing feature is their asymptotic behaviours can be accurately predicted via their associated state evolution equations. Orthogonal AMP (OAMP) was recently developed to avoid the need for computing the so-called Onsager term in traditional AMP algorithms, providing two clear benefits: the derivation of an OAMP algorithm is both straightforward and more broadly applicable. OAMP was originally demonstrated for statistical problems with a single measurement vector and single transform. This paper extends OAMP to statistical problems with multiple measurement vectors (MMVs) and multiple transforms (MTs). We name the resulting algorithms as OAMP-MMV and OAMP-MT respectively, and their combination as augmented OAMP (A-OAMP). Whereas the extension of traditional AMP algorithms to such problems would be challenging, the orthogonal principle underpinning OAMP makes these extensions straightforward. The MMV and MT models are widely applicable to signal processing and communications. We present an example of MIMO relay system with correlated source data and signal clipping, which can be modelled as a joint MMV-MT system. While existing methods meet with difficulties in this example, OAMP offers an efficient solution with excellent performance.

The generalization error of a learning algorithm refers to the discrepancy between the loss of a learning algorithm on training data and that on unseen testing data. Various information-theoretic bounds on the generalization error have been derived in the literature, where the mutual information between the training data and the hypothesis (the output of the learning algorithm) plays an important role. Focusing on the individual sample mutual information bound by Bu et al., which itself is a tightened version of the first bound on the topic by Russo et al. and Xu et al., this paper investigates the tightness of these bounds, in terms of the dependence of their convergence rates on the sample size $n$. It has been recognized that these bounds are in general not tight, readily verified for the exemplary quadratic Gaussian mean estimation problem, where the individual sample mutual information bound scales as $O(\sqrt{1/n})$ while the true generalization error scales as $O(1/n)$. The first contribution of this paper is to show that the same bound can in fact be asymptotically tight if an appropriate assumption is made. In particular, we show that the fast rate can be recovered when the assumption is made on the excess risk instead of the loss function, which was usually done in existing literature. A theoretical justification is given for this choice. The second contribution of the paper is a new set of generalization error bounds based on the $(\eta, c)$-central condition, a condition relatively easy to verify and has the property that the mutual information term directly determines the convergence rate of the bound. Several analytical and numerical examples are given to show the effectiveness of these bounds., Comment: 32 pages, 1 figure. arXiv admin note: substantial text overlap with arXiv:2205.03131

The success of deep learning over the past decade mainly relies on gradient-based optimisation and backpropagation. This paper focuses on analysing the performance of first-order gradient-based optimisation algorithms, gradient descent and proximal gradient, with time-varying non-convex cost function under (proximal) Polyak-{\L}ojasiewicz condition. Specifically, we focus on using the forward mode of automatic differentiation to compute gradients in the fast-changing problems where calculating gradients using the backpropagation algorithm is either impossible or inefficient. Upper bounds for tracking and asymptotic errors are derived for various cases, showing the linear convergence to a solution or a neighbourhood of an optimal solution, where the convergence rate decreases with the increase in the dimension of the problem. We show that for a solver with constraints on computing resources, the number of forward gradient iterations at each step can be a design parameter that trades off between the tracking performance and computing constraints., Comment: 18 pages, 1 figure

Transfer learning, or domain adaptation, is concerned with machine learning problems in which training and testing data come from possibly different probability distributions. In this work, we give an information-theoretic analysis of the generalization error and excess risk of transfer learning algorithms. Our results suggest, perhaps as expected, that the Kullback-Leibler (KL) divergence $D(\mu\|\mu')$ plays an important role in the characterizations where $\mu$ and $\mu'$ denote the distribution of the training data and the testing data, respectively. Specifically, we provide generalization error and excess risk upper bounds for learning algorithms where data from both distributions are available in the training phase. Recognizing that the bounds could be sub-optimal in general, we provide improved excess risk upper bounds for a certain class of algorithms, including the empirical risk minimization (ERM) algorithm, by making stronger assumptions through the \textit{central condition}. To demonstrate the usefulness of the bounds, we further extend the analysis to the Gibbs algorithm and the noisy stochastic gradient descent method. We then generalize the mutual information bound with other divergences such as $\phi$-divergence and Wasserstein distance, which may lead to tighter bounds and can handle the case when $\mu$ is not absolutely continuous with respect to $\mu'$. Several numerical results are provided to demonstrate our theoretical findings. Lastly, to address the problem that the bounds are often not directly applicable in practice due to the absence of the distributional knowledge of the data, we develop an algorithm (called InfoBoost) that dynamically adjusts the importance weights for both source and target data based on certain information measures. The empirical results show the effectiveness of the proposed algorithm.

Recent advancements in unsupervised domain adaptation (UDA) and semi-supervised learning (SSL), particularly incorporating causality, have led to significant methodological improvements in these learning problems. However, a formal theory that explains the role of causality in the generalization performance of UDA/SSL is still lacking. In this paper, we consider the UDA/SSL scenarios where we access $m$ labelled source data and $n$ unlabelled target data as training instances under different causal settings with a parametric probabilistic model. We study the learning performance (e.g., excess risk) of prediction in the target domain from an information-theoretic perspective. Specifically, we distinguish two scenarios: the learning problem is called causal learning if the feature is the cause and the label is the effect, and is called anti-causal learning otherwise. We show that in causal learning, the excess risk depends on the size of the source sample at a rate of $O(\frac{1}{m})$ only if the labelling distribution between the source and target domains remains unchanged. In anti-causal learning, we show that the unlabelled data dominate the performance at a rate of typically $O(\frac{1}{n})$. These results bring out the relationship between the data sample size and the hardness of the learning problem with different causal mechanisms., Comment: 56 pages Including Appendix

A recent line of works, initiated by Russo and Xu, has shown that the generalization error of a learning algorithm can be upper bounded by information measures. In most of the relevant works, the convergence rate of the expected generalization error is in the form of O(sqrt{lambda/n}) where lambda is some information-theoretic quantities such as the mutual information between the data sample and the learned hypothesis. However, such a learning rate is typically considered to be "slow", compared to a "fast rate" of O(1/n) in many learning scenarios. In this work, we first show that the square root does not necessarily imply a slow rate, and a fast rate (O(1/n)) result can still be obtained using this bound under appropriate assumptions. Furthermore, we identify the key conditions needed for the fast rate generalization error, which we call the (eta,c)-central condition. Under this condition, we give information-theoretic bounds on the generalization error and excess risk, with a convergence rate of O(\lambda/{n}) for specific learning algorithms such as empirical risk minimization. Finally, analytical examples are given to show the effectiveness of the bounds., Comment: 15 pages, 1 figure

Approximate Message Passing (AMP) is an efficient iterative parameter-estimation technique for certain high-dimensional linear systems with non-Gaussian distributions, such as sparse systems. In AMP, a so-called Onsager term is added to keep estimation errors approximately Gaussian. Orthogonal AMP (OAMP) does not require this Onsager term, relying instead on an orthogonalization procedure to keep the current errors uncorrelated with (i.e., orthogonal to) past errors. \LL{In this paper, we show the generality and significance of the orthogonality in ensuring that errors are "asymptotically independently and identically distributed Gaussian" (AIIDG).} This AIIDG property, which is essential for the attractive performance of OAMP, holds for separable functions. \LL{We present a simple and versatile procedure to establish the orthogonality through Gram-Schmidt (GS) orthogonalization, which is applicable to any prototype. We show that different AMP-type algorithms, such as expectation propagation (EP), turbo, AMP and OAMP, can be unified under the orthogonal principle.} The simplicity and generality of OAMP provide efficient solutions for estimation problems beyond the classical linear models. \LL{As an example, we study the optimization of OAMP via the GS model and GS orthogonalization.} More related applications will be discussed in a companion paper where new algorithms are developed for problems with multiple constraints and multiple measurement variables., Comment: 15 pages, 2 figure

Transfer learning is a machine learning paradigm where knowledge from one problem is utilized to solve a new but related problem. While conceivable that knowledge from one task could be useful for solving a related task, if not executed properly, transfer learning algorithms can impair the learning performance instead of improving it -- commonly known as negative transfer. In this paper, we study transfer learning from a Bayesian perspective, where a parametric statistical model is used. Specifically, we study three variants of transfer learning problems, instantaneous, online, and time-variant transfer learning. For each problem, we define an appropriate objective function, and provide either exact expressions or upper bounds on the learning performance using information-theoretic quantities, which allow simple and explicit characterizations when the sample size becomes large. Furthermore, examples show that the derived bounds are accurate even for small sample sizes. The obtained bounds give valuable insights into the effect of prior knowledge for transfer learning, at least with respect to our Bayesian formulation of the transfer learning problem. In particular, we formally characterize the conditions under which negative transfer occurs. Lastly, we devise two (online) transfer learning algorithms that are amenable to practical implementations, one of which does not require the parametric assumption. We demonstrate the effectiveness of our algorithms with real data sets, focusing primarily on when the source and target data have strong similarities., Comment: 45 pages, 12 figures

Transfer learning is a machine learning paradigm where the knowledge from one task is utilized to resolve the problem in a related task. On the one hand, it is conceivable that knowledge from one task could be useful for solving a related problem. On the other hand, it is also recognized that if not executed properly, transfer learning algorithms could in fact impair the learning performance instead of improving it - commonly known as "negative transfer". In this paper, we study the online transfer learning problems where the source samples are given in an offline way while the target samples arrive sequentially. We define the expected regret of the online transfer learning problem and provide upper bounds on the regret using information-theoretic quantities. We also obtain exact expressions for the bounds when the sample size becomes large. Examples show that the derived bounds are accurate even for small sample sizes. Furthermore, the obtained bounds give valuable insight on the effect of prior knowledge for transfer learning in our formulation. In particular, we formally characterize the conditions under which negative transfer occurs., Comment: Paper accepted to ISIT2021

Hidden Markov chain, or Markov field, models, with observations in a Euclidean space, play a major role across signal and image processing. The present work provides a statistical framework which can be used to extend these models, along with related, popular algorithms (such as the Baum-Welch algorithm), to the case where the observations lie in a Riemannian manifold. It is motivated by the potential use of hidden Markov chains and fields, with observations in Riemannian manifolds, as models for complex signals and images., Comment: accepted for publication at MTNS 2020

Hopfield neural networks are a possible basis for modelling associative memory in living organisms. After summarising previous studies in the field, we take a new look at learning rules, exhibiting them as descent-type algorithms for various cost functions. We also propose several new cost functions suitable for learning. We discuss the role of biases (the external inputs) in the learning process in Hopfield networks. Furthermore, we apply Newtons method for learning memories, and experimentally compare the performances of various learning rules. Finally, to add to the debate whether allowing connections of a neuron to itself enhances memory capacity, we numerically investigate the effects of self coupling. Keywords: Hopfield Networks, associative memory, content addressable memory, learning rules, gradient descent, attractor networks, Comment: 8 pages, IEEE-Xplore, 2020 International Joint Conference on Neural Networks (IJCNN), Glasgow

We study numerical optimisation algorithms that use zeroth-order information to minimise time-varying geodesically-convex cost functions on Riemannian manifolds. In the Euclidean setting, zeroth-order algorithms have received a lot of attention in both the time-varying and time-invariant cases. However, the extension to Riemannian manifolds is much less developed. We focus on Hadamard manifolds, which are a special class of Riemannian manifolds with global nonpositive curvature that offer convenient grounds for the generalisation of convexity notions. Specifically, we derive bounds on the expected instantaneous tracking error, and we provide algorithm parameter values that minimise the algorithm's performance. Our results illustrate how the manifold geometry in terms of the sectional curvature affects these bounds. Additionally, we provide dynamic regret bounds for this online optimisation setting. To the best of our knowledge, these are the first regret bounds even for the Euclidean version of the problem. Lastly, via numerical simulations, we demonstrate the applicability of our algorithm on an online Karcher mean problem., Comment: 27 pages, 2 figures

In this paper, two novel algorithms to estimate a Gaussian Vector Autoregressive (VAR) model from 1-bit measurements are introduced. They are based on the Yule-Walker scheme modified to account for quantisation. The scalar case has been studied before. The main difficulty when going from the scalar to the vector case is how to estimate the ratios of the variances of pairwise components of the VAR model. The first method overcomes this difficulty by requiring the quantisation to be non-symmetric: each component of the VAR model output is replaced by a binary "zero" or a binary "one" depending on whether its value is greater than a strictly positive threshold. Different components of the VAR model can have different thresholds. As the choice of these thresholds has a strong influence on the performance, this first method is best suited for applications where the variance of each time series is approximately known prior to choosing the corresponding threshold. The second method relies instead on symmetric quantisations of not only each component of the VAR model but also on the pairwise differences of the components. These additional measurements are equivalent to a ranking of the instantaneous VAR model output, from the smallest component to the largest component. This avoids the need for choosing thresholds but requires additional hardware for quantising the components in pairs. Numerical simulations show the efficiency of both schemes.

Transfer learning, or domain adaptation, is concerned with machine learning problems in which training and testing data come from possibly different distributions (denoted as $\mu$ and $\mu'$, respectively). In this work, we give an information-theoretic analysis on the generalization error and the excess risk of transfer learning algorithms, following a line of work initiated by Russo and Zhou. Our results suggest, perhaps as expected, that the Kullback-Leibler (KL) divergence $D(mu||mu')$ plays an important role in characterizing the generalization error in the settings of domain adaptation. Specifically, we provide generalization error upper bounds for general transfer learning algorithms and extend the results to a specific empirical risk minimization (ERM) algorithm where data from both distributions are available in the training phase. We further apply the method to iterative, noisy gradient descent algorithms, and obtain upper bounds which can be easily calculated, only using parameters from the learning algorithms. A few illustrative examples are provided to demonstrate the usefulness of the results. In particular, our bound is tighter in specific classification problems than the bound derived using Rademacher complexity., Comment: Accepted paper in 2020 IEEE International Symposium on Information Theory (ISIT)

Impropriety testing for complex-valued vector has been considered lately due to potential applications ranging from digital communications to complex media imaging. This paper provides new results for such tests in the asymptotic regime, i.e. when the vector dimension and sample size grow commensurately to infinity. The studied tests are based on invariant statistics named impropriety coefficients. Limiting distributions for these statistics are derived, together with those of the Generalized Likelihood Ratio Test (GLRT) and Roy's test, in the Gaussian case. This characterization in the asymptotic regime allows also to identify a phase transition in Roy's test with potential application in detection of complex-valued low-rank subspace corrupted by proper noise in large datasets. Simulations illustrate the accuracy of the proposed asymptotic approximations., Comment: 11 pages, 8 figures, submitted to IEEE TSP

Let $M$ be a simply-connected compact Riemannian symmetric space, and $U$ a twice-differentiable function on $M$, with unique global minimum at $x^* \in M$. The idea of the present work is to replace the problem of searching for the global minimum of $U$, by the problem of finding the Riemannian barycentre of the Gibbs distribution $P_{\scriptscriptstyle{T}} \propto \exp(-U/T)$. In other words, instead of minimising the function $U$ itself, to minimise $\mathcal{E}_{\scriptscriptstyle{T}}(x) = \frac{1}{2}\int d^{\scriptscriptstyle 2}(x,z)P_{\scriptscriptstyle{T}}(dz)$, where $d(\cdot,\cdot)$ denotes Riemannian distance. The following original result is proved : if $U$ is invariant by geodesic symmetry about $x^*$, then for each $\delta < \frac{1}{2} r_{\scriptscriptstyle cx}$ ($r_{\scriptscriptstyle cx}$ the convexity radius of $M$), there exists $T_{\scriptscriptstyle \delta}$ such that $T \leq T_{\scriptscriptstyle \delta}$ implies $\mathcal{E}_{\scriptscriptstyle{T}}$ is strongly convex on the geodesic ball $B(x^*,\delta)\,$, and $x^*$ is the unique global minimum of $\mathcal{E}_{\scriptscriptstyle{T\,}}$. Moreover, this $T_{\scriptscriptstyle \delta}$ can be computed explicitly. This result gives rise to a general algorithm for black-box optimisation, which is briefly described, and will be further explored in future work., Comment: V1 ; extended version of conference paper ; GSI 2019

Computational models of the respiratory central pattern generator (rCPG) are usually based on biologically-plausible Hodgkin Huxley neuron models. Such models require numerous parameters and thus are prone to overfitting. The HH approach is motivated by the assumption that the biophysical properties of neurons determine the network dynamics. Here, we implement the rCPG using simpler Izhikevich resonate-and-fire neurons. Our rCPG model generates a 3-phase respiratory motor pattern based on established connectivities and can reproduce previous experimental and theoretical observations. Further, we demonstrate the flexibility of the model by testing whether intrinsic bursting properties are necessary for rhythmogenesis. Our simulations demonstrate that replacing predicted mandatory bursting properties of pre-inspiratory neurons with spike adapting properties yields a model that generates comparable respiratory activity patterns. The latter supports our view that the importance of the exact modeling parameters of specific respiratory neurons is overestimated., Comment: 12 Pages, ICONIP 2018 conference

This paper considers the problem of localising a stationary signal source using a team of mobile agents which only take binary measurements. Background false detection rates and missed detection probabilities are incorporated into the framework. A Bayesian estimation algorithm that discretises the search environment is employed, and analytical convergence and consistency results for this are derived. Fisher Information is then used as a metric for the design of optimal agent geometries. Knowledge of the probability of detection as a function of the source and agent locations is assumed in the analysis, with special attention given to range-dependent functions. The behaviour of the algorithm under inexact knowledge of the probability of detection is also analysed. Finally, simulation results are presented to demonstrate the effectiveness of the algorithm., Comment: 17 pages, 4 Figures, Submitted to IEEE Transactions on Signal Processing

Recently, Riemannian Gaussian distributions were defined on spaces of positive-definite real and complex matrices. The present paper extends this definition to the space of positive-definite quaternion matrices. In order to do so, it develops the Riemannian geometry of the space of positive-definite quaternion matrices, which is shown to be a Riemannian symmetric space of non-positive curvature. The paper gives original formulae for the Riemannian metric of this space, its geodesics, and distance function. Then, it develops the theory of Riemannian Gaussian distributions, including the exact expression of their probability density, their sampling algorithm and statistical inference., Comment: 8 pages, submitted to GSI 2017

In 3D single particle imaging with X-ray free-electron lasers, particle orientation is not recorded during measurement but is instead recovered as a necessary step in the reconstruction of a 3D image from the diffraction data. Here we use harmonic analysis on the sphere to cleanly separate the angu- lar and radial degrees of freedom of this problem, providing new opportunities to efficiently use data and computational resources. We develop the Expansion-Maximization-Compression algorithm into a shell-by-shell approach and implement an angular bandwidth limit that can be gradually raised during the reconstruction. We study the minimum number of patterns and minimum rotation sampling required for a desired angular and radial resolution. These extensions provide new av- enues to improve computational efficiency and speed of convergence, which are critically important considering the very large datasets expected from experiment.

This paper considers the problem of estimating probability density functions on the rotation group $SO(3)$. Two distinct approaches are proposed, one based on characteristic functions and the other on wavelets using the heat kernel. Expressions are derived for their Mean Integrated Squared Errors. The performance of the estimators is studied numerically and compared with the performance of an existing technique using the De La Vall\'ee Poussin kernel estimator. The heat-kernel wavelet approach appears to offer the best convergence, with faster convergence to the optimal bound and guaranteed positivity of the estimated probability density function.

This paper presents several results about isotropic random walks and multiple scattering processes on hyperspheres ${\mathbb S}^{p-1}$. It allows one to derive the Fourier expansions on ${\mathbb S}^{p-1}$ of these processes. A result of unimodality for the multiconvolution of symmetrical probability density functions (pdf) on ${\mathbb S}^{p-1}$ is also introduced. Such processes are then studied in the case where the scattering distribution is von Mises Fisher (vMF). Asymptotic distributions for the multiconvolution of vMFs on ${\mathbb S}^{p-1}$ are obtained. Both Fourier expansion and asymptotic approximation allows us to compute estimation bounds for the parameters of Compound Cox Processes (CCP) on ${\mathbb S}^{p-1}$., Comment: 16 pages, 4 figures

Reproducing kernel Hilbert spaces are elucidated without assuming prior familiarity with Hilbert spaces. Compared with extant pedagogic material, greater care is placed on motivating the definition of reproducing kernel Hilbert spaces and explaining when and why these spaces are efficacious. The novel viewpoint is that reproducing kernel Hilbert space theory studies extrinsic geometry, associating with each geometric configuration a canonical overdetermined coordinate system. This coordinate system varies continuously with changing geometric configurations, making it well-suited for studying problems whose solutions also vary continuously with changing geometry. This primer can also serve as an introduction to infinite-dimensional linear algebra because reproducing kernel Hilbert spaces have more properties in common with Euclidean spaces than do more general Hilbert spaces., Comment: Revised version submitted to Foundations and Trends in Signal Processing

This primer explains how continuous-time stochastic processes (precisely, Brownian motion and other Ito diffusions) can be defined and studied on manifolds. No knowledge is assumed of either differential geometry or continuous-time processes. The arguably dry approach is avoided of first introducing differential geometry and only then introducing stochastic processes; both areas are motivated and developed jointly., Comment: 19 pages

This article demonstrates how an understanding of the geometry of a family of cost functions can be used to develop efficient numerical algorithms for real-time optimisation. Crucially, it is not the geometry of the individual functions which is studied, but the geometry of the family as a whole. In some respects, this challenges the conventional divide between convex and non-convex optimisation problems because none of the cost functions in a family need be convex in order for efficient numerical algorithms to exist for optimising in real-time any function belonging to the family. The title "Optimisation Geometry" comes by analogy from the study of the geometry of a family of probability distributions being called information geometry., Comment: 14 pages

An efficient coordinate-free notation is elucidated for differentiating matrix expressions and other functions between higher-dimensional vector spaces. This method of differentiation is known, but not explained well, in the literature. Teaching it early in the curriculum would avoid the tedium of element-wise differentiation and provide a better footing for understanding more advanced applications of calculus. Additionally, it is shown to lead naturally to tensor products, a topic previously considered too difficult to motivate quickly in elementary ways., Comment: 9 pages

The Newton iteration is a popular method for minimising a cost function on Euclidean space. Various generalisations to cost functions defined on manifolds appear in the literature. In each case, the convergence rate of the generalised Newton iteration needed establishing from first principles. The present paper presents a framework for generalising iterative methods from Euclidean space to manifolds that ensures local convergence rates are preserved. It applies to any (memoryless) iterative method computing a coordinate independent property of a function (such as a zero or a local minimum). All possible Newton methods on manifolds are believed to come under this framework. Changes of coordinates, and not any Riemannian structure, are shown to play a natural role in lifting the Newton method to a manifold. The framework also gives new insight into the design of Newton methods in general., Comment: 36 pages

This paper introduces several fundamental concepts in information theory from the perspective of their origins in engineering. Understanding such concepts is important in neuroscience for two reasons. Simply applying formulae from information theory without understanding the assumptions behind their definitions can lead to erroneous results and conclusions. Furthermore, this century will see a convergence of information theory and neuroscience; information theory will expand its foundations to incorporate more comprehensively biological processes thereby helping reveal how neuronal networks achieve their remarkable information processing abilities., Comment: 18 pages, 7 figures, to appear in Biological Cybernetics

Noncommutative harmonic analysis is used to solve a nonparametric estimation problem stated in terms of compound Poisson processes on compact Lie groups. This problem of decompounding is a generalization of a similar classical problem. The proposed solution is based on a char- acteristic function method. The treated problem is important to recent models of the physical inverse problem of multiple scattering., Comment: 26 pages, 3 figures, 25 references