Definability and stability of multiscale decompositions for manifold-valued data

We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold-analogues of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midpoint-interpolating. For definable multiscale decompositions we obtain a stability result

A global approach to the refinement of manifold data

A refinement of manifold data is a computational process, which produces a denser set of discrete data from a given one. Such refinements are closely related to multiresolution representations of manifold data by pyramid transforms, and approximation of manifold-valued functions by repeated refinements schemes. Most refinement methods compute each refined element separately, independently of the computations of the other elements. Here we propose a global method which computes all the refined elements simultaneously, using geodesic averages. We analyse repeated refinements schemes based on this global approach, and derive conditions guaranteeing strong convergence.Comment: arXiv admin note: text overlap with arXiv:1407.836

Curvature corrected tangent space-based approximation of manifold-valued data

When generalizing schemes for real-valued data approximation or decomposition to data living in Riemannian manifolds, tangent space-based schemes are very attractive for the simple reason that these spaces are linear. An open challenge is to do this in such a way that the generalized scheme is applicable to general Riemannian manifolds, is global-geometry aware and is computationally feasible. Existing schemes have been unable to account for all three of these key factors at the same time. In this work, we take a systematic approach to developing a framework that is able to account for all three factors. First, we will restrict ourselves to the -- still general -- class of symmetric Riemannian manifolds and show how curvature affects general manifold-valued tensor approximation schemes. Next, we show how the latter observations can be used in a general strategy for developing approximation schemes that are also global-geometry aware. Finally, having general applicability and global-geometry awareness taken into account we restrict ourselves once more in a case study on low-rank approximation. Here we show how computational feasibility can be achieved and propose the curvature-corrected truncated higher-order singular value decomposition (CC-tHOSVD), whose performance is subsequently tested in numerical experiments with both synthetic and real data living in symmetric Riemannian manifolds with both positive and negative curvature

Level-dependent interpolatory Hermite subdivision schemes and wavelets

We study many properties of level-dependent Hermite subdivision, focusing on schemes preserving polynomial and exponential data. We specifically consider interpolatory schemes, which give rise to level-dependent multiresolution analyses through a prediction-correction approach. A result on the decay of the associated multiwavelet coefficients, corresponding to a uniformly continuous and differentiable function, is derived. It makes use of the approximation of any such function with a generalized Taylor formula expressed in terms of polynomials and exponentials

Interview with Endre Szemerédi

Endre Szemerédi is the recipient of the 2012 Abel Prize of the Norwegian Academy of Science and Letters. This interview was conducted in Oslo in May 2012 in conjuction with the Abel Prize celebration

Essays on the nonlinear and nonstochastic nature of stock market data

The nature and structure of stock-market price dynamics is an area of ongoing and rigourous scientific debate. For almost three decades, most emphasis has been given on upholding the concepts of Market Efficiency and rational investment behaviour. Such an approach has favoured the development of numerous linear and nonlinear models mainly of stochastic foundations. Advances in mathematics have shown that nonlinear deterministic processes i.e. "chaos" can produce sequences that appear random to linear statistical techniques. Till recently, investment finance has been a science based on linearity and stochasticity. Hence it is important that studies of Market Efficiency include investigations of chaotic determinism and power laws. As far as chaos is concerned, there are rather mixed or inconclusive research results, prone with controversy. This inconclusiveness is attributed to two things: the nature of stock market time series, which are highly volatile and contaminated with a substantial amount of noise of largely unknown structure, and the lack of appropriate robust statistical testing procedures. In order to overcome such difficulties, within this thesis it is shown empirically and for the first time how one can combine novel techniques from recent chaotic and signal analysis literature, under a univariate time series analysis framework. Three basic methodologies are investigated: Recurrence analysis, Surrogate Data and Wavelet transforms. Recurrence Analysis is used to reveal qualitative and quantitative evidence of nonlinearity and nonstochasticity for a number of stock markets. It is then demonstrated how Surrogate Data, under a statistical hypothesis testing framework, can be simulated to provide similar evidence. Finally, it is shown how wavelet transforms can be applied in order to reveal various salient features of the market data and provide a platform for nonparametric regression and denoising. The results indicate that without the invocation of any parametric model-based assumptions, one can easily deduce that there is more to linearity and stochastic randomness in the data. Moreover, substantial evidence of recurrent patterns and aperiodicities is discovered which can be attributed to chaotic dynamics. These results are therefore very consistent with existing research indicating some types of nonlinear dependence in financial data. Concluding, the value of this thesis lies in its contribution to the overall evidence on Market Efficiency and chaotic determinism in financial markets. The main implication here is that the theory of equilibrium pricing in financial markets may need reconsideration in order to accommodate for the structures revealed