Section: Overall Objectives
Overall Objectives

Glossary
 Singularity exponent
A measure of the unpredictability around a point in a complex signal. Based on local reconstruction around a point, singularity exponents can be evaluated in different ways and in different contexts (e.g. nonlocalized, through the consideration of moments and structure fonctions, trough the computation of singularity spectra). In GEOSTAT we study approaches corresponding to far from equilibrium hypothesis (e.g. microcanonical) leading to geometrically localized singularity exponents.
 Singularity spectrum
The mapping from scaling exponents to Hausdorff dimensions. The singularity spectrum quantifies the degree of nonlinearity in a signal or process, and is used to characterize globally the complexity of a signal.
 Most Singular Manifold
The set of most unpredictable points in a signal, identified to the set of strongest transitions as defined by the singularity exponents. From that set the whole signal can be reconstructed.
 Fully Developed Turbulence (FDT)
Turbulence at very high Reynolds numbers; systems in FDT are beyond deterministic chaos, and symmetries are restored in a statistical sense only.
 Compact Representation
Reduced representation of a complex signal (dimensionality reduction) from which the whole signal can be reconstructed. The reduced representation can correspond to points randomly chosen, such as in Compressive Sensing, or to geometric localization related to statistical information content (framework of reconstructible systems).
 Sparse representation
The representation of a signal as a linear combination of elements taken in a dictionary (frame or basis), with the aim of finding as less as possible nonzerio coefficients for a large class of signals.
 Universality class
In theoretical physics, the observation of the coincidence of the critical exponents (behaviour near a second order phase transition) in different phenomena and systems is called universality. Universality is explained by the theory of the renormalization group, allowing for determination of the changes a physical system undergoes under different distance scales. As a consequence, different macroscopic phenomena displaying a multiscale structure (and their acquisition in the form of complex signals) can be grouped into different sets of universality classes.
Every signal conveys, as a measure experiment, information on the physical system whose signal is an acquisition. As a consequence, it seems therefore natural that signal analysis or compression makes use of physical modelling of phenomena: the goal is to find new methodologies in signal processing that goes beyond the simple problem of interpretation. Physics of disordonned systems, and specifically physics of spin glasses is putting forward new algorithmic resolution methods in various domains such as optimization, compressive sensing etc. with significant success notably for NP hard problems.
Physics of turbulence also introduces phenomelogical approaches based on singularity exponents. Energy cascades are indeed closely related to singular geometrical sets defined randomly. At these structures' scales, information of the process is lost by dissipation. However, all the cascade is encoded in the singular sets. How do these structures organize in space and time, in other words, how do the entropy cascade itself ? To unify these two notions, a description in term of free energy of a generic physical model is possible, such as an elastic interface model in a random nonlinear energy landscape: this is for instance the correspondance between compressible stochastic Burgers equation and directed polymers in a disordonned medium. Typical of such systems, dekrieging transition indicates that each singularity can be understood as a transition between two metastable states. Each of these transitions marks large fluctuations of the system which visits randomly the minima of the free energy. A signal which is an acquisition of such systems displays statistical properties characteristic of different classes, and the nature of noise is determinant. In particular, the dynamics enters a socalled Griffiths phase if for example noise gets structured like a hierarchical network, connected on long range distances, locally recursive, and randomly sparse. In such a context, phenomenologies related to cascades and multiaffine intermittence are present. As a typical example, in the study of cardiac dynamics (a subject that gets interest among statistical physicists) an effective model belonging to a similar category can be contemplated. In such model, an explicitely broken symetry is restaured spontaneously. This is a consequence from the existence of an abelian symetry of topological solutions, where these topological solutions are fronts excited by limiting conditions. A statistical jauge is the signature of a nonlinear intrinsic disorder which emerges for certain regions of parameter space. Such areas have a vitrous nature (Griffiths). The jauge is responsible for jumps between different metastable states and allows to recover singularity exponents of acquired signals. Conversely, the exploration of the phase space in such models can lead to a possible "testing" of the measured system through the computation of singularity exponents and the determination of the nature of intrinsic noise. From these considerations and in a heuristical framework, one sees that the recovering of a semantics in a measured signal can be contemplated.
GEOSTAT is a research project in nonlinear signal processing which develops on these considerations: it considers the signals as the realizations of complex dynamic systems. The driving approach is to understand the relations between complexity (or information content) and the geometric organization of information in a signal. For instance, for signals which are acquisitions of turbulent fluids, the organization of information is related to the effective presence of a multiscale hierarchy, of multifractal nature, which is strongly related to intermittency and multiplicative cascade phenomena; the determination of this geometric organization unlocks key nonlinear parameters and features associated to these signals; it helps understanding their dynamical properties and, as a consequence, their analysis. We use this approach to derive novel solution methods for superresolution and data fusion in Universe Sciences acquisitions [10]. Another example can be found heartbeat signal analysis, where singularity exponents help understand the distribution of activation points in a signal during episodes of atrial fibrilation. Specific advances are obtained in GEOSTAT in using this type of statistical/geometric approach to get validated dynamical information of signals acquired in Universe Sciences, e.g. Oceanography or Astronomy. The research in GEOSTAT encompasses nonlinear signal processing and the study of emergence in complex systems, with a strong emphasis on geometric approaches to complexity. Consequently, research in GEOSTAT is oriented towards the determination, in real signals, of quantities or phenomena, usually unattainable through linear methods, that are known to play an important role both in the evolution of dynamical systems whose acquisitions are the signals under study, and in the compact representations of the signals themselves. Research in GEOSTAT is structured in two parts:

Applicative aspects which encompass biomedical data (heartbeat signal analysis, biomedical applications in speech signal analysis) and the study of universe science datasets.
The theoretical objectives are:

multiscale description in terms of multiplicative cascade (essential in the characterization of turbulent systems).

Excitable systems (cardiac electrophysiology): study of intermittency phenomena.
The methodological tools used in reaching these objectives place GEOSTAT at the forefront of nonlinear signal processing and complex systems. We cite: singularity exponents [48], [7] [11], how these exponents can be related to sparse representations with reconstruction formulae [13] [49], [5] and superresolution in Oceanography and Earth Observation [10], [2], comparison with embedding techniques, such as the one provided by the classical theorem of Takens [46], [38], the use of Lyapunov exponents [34], how they are related to intermittency, large deviations and singularity exponents, various forms of entropies, persistence along the scales, optimal wavelets [6], comparison with other approaches such as sparse representations and compressive sensing [https://hal.inria.fr/tel01239958], and, above all, the ways that lead to effective numerical and high precision determination of nonlinear characteristics in real signals. Presently GEOSTAT explores new methods for analyzing and understanding complex signals in different applicative domains [47]. Derived from ideas in Statistical Physics, the methods developed in GEOSTAT provide new ways to relate and evaluate quantitatively the local irregularity in complex signals and systems, the statistical concepts of information content and most informative subset. As a result, GEOSTAT is aimed at providing radically new approaches to the study of signals acquired from different complex systems (their analysis, their classification, the study of their dynamical properties etc.). A common characteristic of these signals, which is related to universality classes [41] [42] [39], being the existence of a multiscale organization of the systems. For instance, the classical notion of edge or border, which is of multiscale nature, and whose importance is well known in Computer Vision and Image Processing, receives profound and rigorous new definitions, in relation with the more physical notion of transition and fits adequately to the case of chaotic data. The description is analogous to the modeling of states far from equilibrium, that is to say, there is no stationarity assumption. From this formalism we derive methods able to determine geometrically the most informative part in a signal, which also defines its global properties and allows for compact representation in the wake of known problematics addressed, for instance, in timefrequency analysis. It appears that the notion of transition front in a signal is much more complex than previously expected and, most importantly, related to multiscale notions encountered in the study of nonlinearity [44]. For instance, we give new insights to the computation of dynamical properties in complex signals, in particular in signals for which the classical tools for analyzing dynamics give poor results (such as, for example, correlation methods or optical flow for determining motion in turbulent datasets).