A continuous analogue of the tensor-train decomposition

We develop new approximation algorithms and data structures for representing and computing with multivariate functions using the functional tensor-train (FT), a continuous extension of the tensor-train (TT) decomposition. The FT represents functions using a tensor-train ansatz by replacing the three-dimensional TT cores with univariate matrix-valued functions. The main contribution of this paper is a framework to compute the FT that employs adaptive approximations of univariate fibers, and that is not tied to any tensorized discretization. The algorithm can be coupled with any univariate linear or nonlinear approximation procedure. We demonstrate that this approach can generate multivariate function approximations that are several orders of magnitude more accurate, for the same cost, than those based on the conventional approach of compressing the coefficient tensor of a tensor-product basis. Our approach is in the spirit of other continuous computation packages such as Chebfun, and yields an algorithm which requires the computation of "continuous" matrix factorizations such as the LU and QR decompositions of vector-valued functions. To support these developments, we describe continuous versions of an approximate maximum-volume cross approximation algorithm and of a rounding algorithm that re-approximates an FT by one of lower ranks. We demonstrate that our technique improves accuracy and robustness, compared to TT and quantics-TT approaches with fixed parameterizations, of high-dimensional integration, differentiation, and approximation of functions with local features such as discontinuities and other nonlinearities

Structure of Disk Dominated Galaxies I. Bulge/Disk Parameters, Simulations, and Secular Evolution

(Abridged) A robust analysis of galaxy structural parameters, based on the modeling of bulge and disk brightnesses in the BVRH bandpasses, is presented for 121 face-on and moderately inclined late-type spirals. Each surface brightness (SB) profile is decomposed into a sum of a generalized Sersic bulge and an exponential disk. The reliability and limitations of our bulge-to-disk (B/D) decompositions are tested with extensive simulations of galaxy brightness profiles (1D) and images (2D). Galaxy types are divided into 3 classes according to their SB profile shapes; Freeman Type-I and Type-II, and a third ``Transition'' class for galaxies whose profiles change from Type-II in the optical to Type-I in the infrared. We discuss possible interpretations of Freeman Type-II profiles. The Sersic bulge shape parameter for nearby Type-I late-type spirals shows a range between n=0.1-2 but, on average, the underlying surface density profile for the bulge and disk of these galaxies is adequately described by a double-exponential distribution. We confirm a coupling between the bulge and disk with a scale length ratio r_e/h=0.22+/-0.09, or h_bulge/h_disk=0.13+/-0.06 for late-type spirals, in agreement with recent N-body simulations of disk formation and models of secular evolution. This ratio increases from ~0.20 for late-type spirals to ~0.24 for earlier types. The similar scaling relations for early and late-type spirals suggest comparable formation and/or evolution scenarios for disk galaxies of all Hubble types.Comment: 78 pages with 23 embedded color figures + tables of galaxy structural parameters. Accepted for publication in the Astrophysical Journal. The interested reader is strongly encouraged to ignore some of the low res figures within; instead, download the high resolution version from http://www.astro.ubc.ca/people/courteau/public/macarthur02_disks.ps.g

The wavelet-NARMAX representation : a hybrid model structure combining polynomial models with multiresolution wavelet decompositions

A new hybrid model structure combing polynomial models with multiresolution wavelet decompositions is introduced for nonlinear system identification. Polynomial models play an important role in approximation theory, and have been extensively used in linear and nonlinear system identification. Wavelet decompositions, in which the basis functions have the property of localization in both time and frequency, outperform many other approximation schemes and offer a flexible solution for approximating arbitrary functions. Although wavelet representations can approximate even severe nonlinearities in a given signal very well, the advantage of these representations can be lost when wavelets are used to capture linear or low-order nonlinear behaviour in a signal. In order to sufficiently utilise the global property of polynomials and the local property of wavelet representations simultaneously, in this study polynomial models and wavelet decompositions are combined together in a parallel structure to represent nonlinear input-output systems. As a special form of the NARMAX model, this hybrid model structure will be referred to as the WAvelet-NARMAX model, or simply WANARMAX. Generally, such a WANARMAX representation for an input-output system might involve a large number of basis functions and therefore a great number of model terms. Experience reveals that only a small number of these model terms are significant to the system output. A new fast orthogonal least squares algorithm, called the matching pursuit orthogonal least squares (MPOLS) algorithm, is also introduced in this study to determine which terms should be included in the final model

Tensor Analysis and Fusion of Multimodal Brain Images

Current high-throughput data acquisition technologies probe dynamical systems with different imaging modalities, generating massive data sets at different spatial and temporal resolutions posing challenging problems in multimodal data fusion. A case in point is the attempt to parse out the brain structures and networks that underpin human cognitive processes by analysis of different neuroimaging modalities (functional MRI, EEG, NIRS etc.). We emphasize that the multimodal, multi-scale nature of neuroimaging data is well reflected by a multi-way (tensor) structure where the underlying processes can be summarized by a relatively small number of components or "atoms". We introduce Markov-Penrose diagrams - an integration of Bayesian DAG and tensor network notation in order to analyze these models. These diagrams not only clarify matrix and tensor EEG and fMRI time/frequency analysis and inverse problems, but also help understand multimodal fusion via Multiway Partial Least Squares and Coupled Matrix-Tensor Factorization. We show here, for the first time, that Granger causal analysis of brain networks is a tensor regression problem, thus allowing the atomic decomposition of brain networks. Analysis of EEG and fMRI recordings shows the potential of the methods and suggests their use in other scientific domains.Comment: 23 pages, 15 figures, submitted to Proceedings of the IEE

Distribution Regression with Sample Selection, with an Application to Wage Decompositions in the UK

We develop a distribution regression model under endogenous sample selection. This model is a semiparametric generalization of the Heckman selection model that accommodates much richer patterns of heterogeneity in the selection process and effect of the covariates. The model applies to continuous, discrete and mixed outcomes. We study the identification of the model, and develop a computationally attractive two-step method to estimate the model parameters, where the first step is a probit regression for the selection equation and the second step consists of multiple distribution regressions with selection corrections for the outcome equation. We construct estimators of functionals of interest such as actual and counterfactual distributions of latent and observed outcomes via plug-in rule. We derive functional central limit theorems for all the estimators and show the validity of multiplier bootstrap to carry out functional inference. We apply the methods to wage decompositions in the UK using new data. Here we decompose the difference between the male and female wage distributions into four effects: composition, wage structure, selection structure and selection sorting. After controlling for endogenous employment selection, we still find substantial gender wage gap -- ranging from 21% to 40% throughout the (latent) offered wage distribution that is not explained by observable labor market characteristics. We also uncover positive sorting for single men and negative sorting for married women that accounts for a substantive fraction of the gender wage gap at the top of the distribution. These findings can be interpreted as evidence of assortative matching in the marriage market and glass-ceiling in the labor market.Comment: 72 pages, 4 tables, 39 figures, includes supplement with additional empirical result

Modularity and the spread of perturbations in complex dynamical systems

We propose a method to decompose dynamical systems based on the idea that modules constrain the spread of perturbations. We find partitions of system variables that maximize 'perturbation modularity', defined as the autocovariance of coarse-grained perturbed trajectories. The measure effectively separates the fast intramodular from the slow intermodular dynamics of perturbation spreading (in this respect, it is a generalization of the 'Markov stability' method of network community detection). Our approach captures variation of modular organization across different system states, time scales, and in response to different kinds of perturbations: aspects of modularity which are all relevant to real-world dynamical systems. It offers a principled alternative to detecting communities in networks of statistical dependencies between system variables (e.g., 'relevance networks' or 'functional networks'). Using coupled logistic maps, we demonstrate that the method uncovers hierarchical modular organization planted in a system's coupling matrix. Additionally, in homogeneously-coupled map lattices, it identifies the presence of self-organized modularity that depends on the initial state, dynamical parameters, and type of perturbations. Our approach offers a powerful tool for exploring the modular organization of complex dynamical systems