Andrei P Kirilyuk
email: andrei.kirilyuk@gmail.com

UNIFIED COMPLEX-DYNAMICAL THEORY OF FINANCIAL, ECONOMIC, AND SOCIAL RISKS AND THEIR EFFICIENT MANAGEMENT: REASON-BASED GOVERNANCE FOR SUSTAINABLE DEVELOPMENT

An extended analysis compared to observations shows that modern "globalised" world civilisation has passed through the invisible "complexity threshold", after which usual "spontaneous", empirically driven kind of development ("invisible hand" etc.) cannot continue any more without major destructive tendencies. A much deeper, non-simplified understanding of real interaction complexity is necessary in order to cope with such globalised world development problems. Here we introduce the universal definition, fundamental origin, and dynamic equations for a major related quantity of (systemic) risk characterising real complex system development tendencies at any level of dynamics. Practically important conclusions are derived, opening further detailed applications in economy, finance and development practice.

P a g e | 2

In this time of deep and rapid global changes in all aspects of life and activities of the world civilisation, the growing problems of global development criticality, its further directions and related risks acquire the unprecedented and vital importance, with the special, encompassing role of financial risks and economic uncertainty (see e. g. [START_REF]Challenges of a Changing Earth[END_REF][START_REF] Rees | Our final hour: A scientist's warning: How terror, error, and environmental disaster threaten humankind's future in this century-on earth and beyond[END_REF][START_REF]The Future of Life and the Future of Our Civilisation[END_REF][START_REF] Szombatfalvy | The Greatest Challenges of Our Time[END_REF][START_REF]Global Catastrophic Risks[END_REF][START_REF] Toffler | Future shock[END_REF][START_REF] Vinge | The Coming Technological Singularity: How to Survive in the Post-Human Era[END_REF][START_REF] Battiston | Complexity theory and financial regulation[END_REF][START_REF] Sergueiva | Systemic Risk Identification, Modelling, Analysis, and Monitoring: An Integrated Approach[END_REF][START_REF] Caccioli | Network models of financial systemic risk: a review[END_REF]). Despite huge efforts applied to understanding and efficient management of respective critical development problems, the dominating unitary, dynamically single-valued science paradigm (including its complexity imitations) fails to provide any efficient approach and problem solutions, which should be expected in view of complexity correspondence and complex-dynamical control principles of our universal science of complexity [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | Sustainable Reason-Based Governance after the Globalisation Complexity Threshold[END_REF], implying that the planetary system of extremely high unreduced dynamic complexity cannot be efficiently simulated in the framework of over-simplified zero-complexity projection of unitary science, irrespective of computer powers applied. The announced crucial goal of sustainable development (e. g. [START_REF]Challenges of a Changing Earth[END_REF]) becomes thus seriously compromised.

In particular, usual theory fails to provide the universal and consistent definition itself of risk or stability, even in separate fields of financial and economic development, let alone the related efficient strategy to cope with the emerging critical phenomena and the intrinsically unified, globalised character of culminating changes and critical risks. In the absence of the necessary genuine understanding of respective system dynamics, the dominating unitary approach tries the obvious and empirically motivated protective kind of strategy, with growing mechanical restrictions on the emerging changes and their assumed causes, thus inevitably limiting progressive development (turning then into degradation), but also introducing unpredictable and unexpected changes of any (but most probably negative) kind. The accepted unitary strategy of risk mitigation leads thus inevitably to growing, maybe even more dangerous risks, and in general to the growth of the most dangerous risk of advancing global degradation.

P a g e | 3

We arrive here at the fundamental origin of unitary approach deficiency, which fails to see the omnipresent dynamic instability of any real, dynamically multivalued interaction process and system evolution [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF]. In this sense the dominating unitary, dynamically single-valued theory and approach cannot provide the correct, causal understanding of progress either, replacing it with inconsistent, empirically driven and vague ideas about "good" (progressive) and "bad" (destructive) changes. Our unreduced interaction analysis shows that the desired "stability" of any real system opposed to "crisis" and other "risks" is nothing but its permanent progressive changes of entropycomplexity growth (or sustainability thus rigorously defined) based on the intrinsic dynamic instability of real, multivalued interaction dynamics and related dynamically random changes. Therefore our universal principle of complexdynamical control [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF] emphasizes the optimal growth of entropycomplexity realised by the dynamically random changes, instead of the false unitary control strategy of mechanistic restrictions in order to preserve the desired (external and illusive) regularity and status quo.

Based on our universal complexity development paradigm [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF], with the discrete transformation of latent action-complexity ( , )

xt (dynamic information) to the explicit form of entropy-complexity ( , ) S x t (dynamic entropy), ( , ) ( , ) 0 S x t x t     , we can propose the universal definition of "undesired", negative instability kind referred to as risk in the form of increased (maximum) probability of destructive entropy-complexity growth ("the death branch" on the universal complexity evolution curve [START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF]), with the quantitative risk magnitude, R, equal (eventually up to a coefficient) to the reciprocal entropy-complexity growth rate, or reciprocal generalised energy E:

11 const const 0 1 xx S R t t E                       ; , (1)
with the opposite sustainability magnitude,

1 R  S
, coinciding (again up to a coefficient) with the total (development) energy E:

0 const const 1 xx S E R t t           ; S , (2)
where 0 and  are the characteristic values of action-complexity magnitude (or variation) and change period respectively. P a g e | 4

Since the generalised total energy characterises the temporal rate of action-complexity transformation to entropy-complexity, i. e. actually the rate of progress, it is natural that the positive, progress-bringing result of universally unstable multivalued dynamics, or sustainability S , is proportional to the rate of progress, while the negative result of the same omnipresent instability, in the form of degradation probability, or risk R, is inversely proportional to the same progress rate E, so that 1 R  S . The risk magnitude R is high (on any scale) during stages of weak entropy-complexity growth (between its stepwise transition jumps) and especially during and after the fatal transition to the death branch of slow, destructive entropy growth, while it is minimal during rapid system transitions to higher complexity sublevels. Correspondingly, all risks will grow with slowing down (higher  ) of transformation of smaller amounts of action-complexity 0 , eq. ( 1). It is important that the unified dynamic complexity and its growth rate of the above risk definition emerge from the unreduced, multivalued interaction dynamics [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF] taking the form of permanent probabilistic change of incompatible system realisations and thus including the totality of occurring, dynamically random events. The obtained unified risk magnitude definition of eq. [START_REF]Challenges of a Changing Earth[END_REF] shows that the truly reliable (and actually universal) way of risk reduction can only be based on sustainable, intrinsically progressive complexity development liberated from inevitable crises and impasses of unitary kind of organisation. While the definite transition to that genuine sustainability regime of intrinsically low risks occurs only as a transition to the superior complexity level of Harmonical System [11,14,15,18.19], the proposed universal risk definition and criterion can be useful also at lower complexity levels.

Whereas the main way of risk reduction (in particular in economy and finance) is the search for further progressive growth of unreduced entropycomplexity, in accord with the underlying universal symmetry of complexity, ( , ) ( , ) 0 S x t x t     , the unreduced complex dynamics analysis at a given complexity sublevel (for example during phases of slow complexity growth) also provides universal understanding of the origin of risky events and ways of their probability reduction. We can see, in particular, that any "stable", lowrisk system operation mode corresponds to the unified SOC regime of internal-P a g e | 5 ly chaotic, but externally quasi-regular complex dynamics [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF], while the opposite limiting regime of global chaos implies maximum risk values. Recalling the unified criterion of global (or partial) chaos regime in terms of major frequency resonances between system operation modes, we arrive at the rigorously specified conclusion that in order to reduce the risk probability, one should avoid frequency, or temporal, resonances between major repeated operations involving essential quantities of money or other exchange matter, i. e. one should avoid the condition

Q mn   
, where m, n are small integers and ,   are major system operation frequencies involving essential exchanges.

While in a number of cases respective rules are practically implemented based on intuitively felt empirical considerations and experience, we provide the universal and rigorously substantiated criterion in terms of any unreduced interaction dynamics.

Moreover, for situations with relatively smooth complexity transformation we can apply our unified Hamilton-Schrödinger formalism [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF] in order to describe the inhomogeneous, highly nonlinear and chaotic dynamics, distribution and evolution of the introduced risk magnitude R (eq. ( 1)). Consider, for example, an often suitable form of the generalised Hamilton-Jacobi equation describing the unreduced dynamics of action-complexity ( , )

Xt:

      2 1 1, 1 , , , 0 2 NN i i ij i j ii i j j i U x t V x x X t t m x                       , (3)
where 12 ( , ,..., )

n X x x x 
is the vector of all relevant agent coordinates i x (con- figuration measures), ( , ) ii U x t is the generally time-dependent external influence potential acting on the i-th agent with the generalised mass i m , ( , )

ij i j
V x x is the potential of interaction between the i-th and j-th agents, and N is the total number of agents (maybe hierarchically organised). Using our unreduced interaction analysis with the help of the generalised EP formalism [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF], we can find its complete, dynamically multivalued, or suitable approximate solution for action-complexity ( , ) Xt and then using eq. ( 1) find the corresponding, in general also multivalued, risk distribution and evolution in time and space of relevant (financial, economic, technological, social, or political) variables,

1 ( , ) ( ) R X t t     
. This approach opens absolutely new, mathematical-P a g e | 6

ly exact and causally substantiated possibilities for the totally objective analysis of risk dynamics and evolution in arbitrary real system at any level of its (growing) complexity.

As shown in refs. [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF][START_REF] Kirilyuk | Sustainable Reason-Based Governance after the Globalisation Complexity Threshold[END_REF], further progressive entropy-complexity growth of the global civilisation system towards lower-risk dynamics is actually possible today only by a step-like transition to the superior complexity level corresponding to vanishing risk values, with the single alternative of destructive, death-branch complexity evolution realising the opposite tendency of essentially growing risks. Today we have therefore the nontrivial, historically significant bifurcation point leading to that qualitatively big and always growing difference between the near-future low-risk and high-risk levels and evolution tendencies. Using our universal complexity laws realising the unified symmetry of complexity [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF], we can specify major features of the desired superior-complexity level of development also referred to as the Harmonical System, or genuine sustainability [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF]. Based on the clear, explicit understanding of the unreduced complex dynamics of the entire planet system, those features include the new, emergent kind of social structure and corresponding intrinsically progressive, reason-based governance, creative (complexity-increasing) production ways and processes, new kind of settlement, infrastructure and lifestyle, and certainly new organisation, content and role of the intrinsically complete knowledge of (growing) unreduced dynamic complexity.

The qualitatively new, reason-based (global) governance system [START_REF] Kirilyuk | Sustainable Reason-Based Governance after the Globalisation Complexity Threshold[END_REF] deserves a special mention here as the concrete and indispensable element to be introduced in addition to (and then increasingly instead of) traditional, unitary governance. In accord with the universal complexity correspondence principle, it appears in this higher-complexity tendency as a superior governance structure oriented to explicit, rigorously substantiated and clearly presented problem solutions, which need not immediately replace the existing, unitary governance structures preserving their full decision power. Now, however, these traditional governments and populations they govern are provided with the consistently substantiated and openly presented guidelines of those real problem solutions in the spirit of intrinsically progressive complexity growth. And although they are not formally obligatory for realisation, they clearly P a g e | 7

demonstrate the extended possibilities of intrinsically progressive complexity development.

We obtain thus the superior-complexity level of (unified) social conscious intelligence of the planetary "organism" (including all its interactive and omnipresent global networks), which is absent in the known traditional social structures relying only on the empirical, "animal" and very short-sighted kind of general intelligence. However, that traditional "spontaneous", "invisiblehand" kind of development of social structure is now totally exhausted at the attained high level of "globalised" civilisation complexity, which can further progress only with the help of explicit social consciousness layer, providing also the advanced, superior-complexity version of traditional unitary democracy (usually erroneously considered to be the best possible one, within its own paradigm). This superior-complexity, explicitly conscious democratic system is not limited in its intrinsically progressive, low-risk development and provides its ever growing, creative liberties due to the qualitatively extended, causally complete kind of knowledge and the new system of science of unreduced dynamic complexity [START_REF] Kirilyuk | Universal Concept of Complexity by the Dynamic Redundance Paradigm: Causal Randomness, Complete Wave Mechanics, and the Ultimate Unification of Knowledge[END_REF][START_REF] Kirilyuk | Universal Science of Complexity: Consistent Understanding of Ecological, Living and Intelligent System Dynamics[END_REF][START_REF] Kirilyuk | Complex Dynamics of Real Quantum, Classical and Hybrid Micro-Machines[END_REF][START_REF] Kirilyuk | Complex-Dynamic Origin of Consciousness and Sustainability Transition[END_REF][START_REF] Kirilyuk | Theory of Everything, Ultimate Reality and the End of Humanity: Extended Sustainability by the Universal Science of Complexity[END_REF][START_REF] Kirilyuk | Unreduced Dynamic Complexity: Towards the Unified Science of Intelligent Communication Networks and Software[END_REF][START_REF] Kirilyuk | New Mathematics of Complexity and Its Biomedical Applications[END_REF][START_REF] Kirilyuk | Towards Sustainable Future by Transition to the Next Level Civilisation[END_REF][START_REF] Kirilyuk | Dynamic Origin of Evolution and Social Transformation[END_REF]. P a g e | 8


Mayeul Arminjon

Lorentz-invariant second-order tensors and an irreducible set of matrices

We prove that, up to multiplication by a scalar, the Minkowski metric tensor is the only second-order tensor that is Lorentz-invariant. To prove this, we show that a specific set of three 4 × 4 matrices, made of two rotation matrices plus a Lorentz boost, is irreducible.

Introduction

It is a basic result of special relativity that the Minkowski metric tensor is invariant under the Lorentz group. The main aim of this paper is to prove that, up to a scalar, this property characterizes the Minkowski metric:

Theorem. Let (M, γ 0 ) be the four-dimensional Minkowski spacetime. Any (0 2) tensor on M that is invariant under the Lorentz group is a scalar multiple of the Minkowski metric tensor γ 0 .

(See Note 1 for the extension to a Lorentzian spacetime.) This result is not very surprising and seems to be heuristically known. For instance, after having introduced the classical totally antisymmetric fourth-order tensor, Maggiore [START_REF] Maggiore | A Modern Introduction to Quantum Field Theory[END_REF] states: "The only other invariant tensor of the Lorentz group is η µν [the Minkowski metric, M.A.]; its invariance follows from the defining property of the Lorentz group, eq. (2.13)." (The latter equation is equivalent to Eq. (3) below.) Nevertheless, we saw neither a precise statement of the above Theorem nor a correct proof in the literature that we could find. The proof that we present here appeals to Schur's lemma (Section 2). However, to identify a relevant irreducible set of 4 × 4 matrices in order to use Schur's lemma was not completely obvious. To prove the irreducibility of that set S, we had to study in detail which are the invariant subspaces of each of the matrices that constitute S (Sect. 3): although it is often easy to check that some subspace is invariant under some endomorphism (or some matrix), it is less trivial to identify the complete list of the invariant subspaces. To do that in the case at hand, we state and prove a result (Lemma 2) about the invariant subspaces of a specific class of endomorphisms (Sect. 4).

Proof of the Theorem

A (0 2) second-order tensor T at some point X ∈ M is Lorentz-invariant iff, in any Cartesian coordinates x µ on M, and for any 4 × 4 real matrix L = (L µ ν ) belonging to the (real) Lorentz group O(1, 3), we have1

T µν := L ρ µ T ρσ L σ ν = T µν , (1)
or (T = (T µν ) being the component matrix of T at X in the coordinates x µ )

L T T L = T. (2)
In particular, the Minkowski metric tensor γ 0 , with component matrix η := diag(1, -1, -1, -1) in any Cartesian coordinates, is of course a Lorentzinvariant (0 2) second-order tensor on M, since by definition a matrix L belongs to the Lorentz group O(1, 3) iff

L T η L = η. (3)
Because the matrix η is its own inverse:

η η = 1 4 := diag(1, 1, 1, 1), we deduce from (3) that L -1 η (L T ) -1 = η. (4)
Multiplying on the right each side of (2) by the corresponding side of (4), we get:

L T T η (L T ) -1 = T η, (5)
or

L T (T η) = (T η) L T . (6)
That is, the matrix M := T η commutes with the transpose of any matrix L in the Lorentz group. The Theorem results immediately from this and from the following two statements.

Lemma 1 (Schur's lemma) (e.g. [START_REF] Godement | Cours d'Algèbre[END_REF][START_REF] Richtmyer | Principles of Advanced Mathematical Physics[END_REF]). Let M be a k × k complex matrix which commutes with any matrix in an irreducible set S of k × k matrices, i.e., in a set S of k × k complex matrices such that no nontrivial subspace of C k is invariant under all mappings x → N x, N ∈ S. Then M is a complex multiple of the identity matrix 1 k .

Proposition 1. The set of the matrices L T , L ∈ O(1, 3), is an irreducible set of 4 × 4 complex matrices.

Proof of Proposition 1. In view of the sentence right after Eq. (9) below, Proposition 1 is an immediate consequence of the following stronger result.

Proposition 2. Let the complex vector space C 4 be endowed with its canonical basis (e µ ) (µ = 0, ..., 3), with e µ = (δ ν µ ) ν=0,...,3 , and identify an endomorphism of C 4 with its matrix in that basis. The set S made by the two spatial rotations L i having axis e i , i = 1, 2, each having a given angle θ i with 0 < θ i < π, plus the Lorentz boost L 1 in the direction e 1 with a given coefficient β v := v c , 0 < β v < 1, is an irreducible set of three 4×4 complex matrices.

Proof of Proposition 2

The rotation matrix having axis e 1 and angle θ (with θ = θ 1 in the sequel) is

L 1 =     1 0 0 0 0 1 0 0 0 0 cos θ -sin θ 0 0 sin θ cos θ     . (7)
The rotation matrices with axes e 2 and e 3 and angle θ are deduced from (7) by the permutations (1 2 3) → (2 3 1) and (1 2 3) → (3 1 2) of the indices, respectively. The boost matrix in direction e 1 and with coefficient β v is

L 1 =     γ v -γ v β v 0 0 -γ v β v γ v 0 0 0 0 1 0 0 0 0 1     , γ v := 1 1 -β 2 v . (8)
Note that all matrices L in the set

{L 1 , L 2 , L 3 , L 1 } have real coefficients.
Therefore, an endomorphism of either the real space R 4 or the complex space C 4 can be given by such a matrix L in the same canonical basis (e µ ), which is also a basis of R 4 . The difference lies in the real or complex coefficients z µ below:

z = z µ e µ → Lz = L µ ν z ν e µ . (9)
Any matrix L in the set

{L 1 , L 2 , L 3 , L 1 } has the form L T , L ∈ O(1,
3), of course, because L T i is the rotation matrix having axis e i and angle -θ i , which belongs to (the real group) O(1, 3), and because L 1 is symmetric and also belongs to O(1, 3).

Henceforth, we consider only the complex vector space C 4 and its complex subspaces, as well as their complex endomorphisms, including those defined by the matrices L ∈ S. Clearly, the complex vector plane Span{e 2 , e 3 } or (in a shorter notation) [e 2 , e 3 ] is invariant under the rotation L 1 . It is well known and easy to check that the restriction of L 1 to [e 2 , e 3 ] has complex eigenvalues λ ± = exp (±iθ 1 ) (which here are distinct from one another and from 1 since 0 < θ 1 < π), with corresponding eigenvectors a ± = e 2 ∓ ie 3 . The complex eigenvalues of the rotation matrices with axes e 2 and e 3 are exp (±iθ i ) (i = 2, 3), and the corresponding eigenvectors are deduced from a ± by the permutations (1 2 3) → (2 3 1) and (1 2 3) → (3 1 2) of the indices, respectively. We will now show that the set S is an irreducible set of complex matrices, by using these facts about the eigenvectors of the L i 's, plus the lemma below -whose proof is deferred to Sect. 4 for convenience.

Lemma 2. Assume the finite-dimensional vector space E is the direct sum of two subspaces F and G: E = F ⊕ G, each of which being invariant by the endomorphism T of E, with, moreover, T |F = λ Id F , and with T |G admitting a basis (v j ) (j = 1, ..., n) of eigenvectors corresponding with pairwise distinct eigenvalues λ j such that, in addition, λ j = λ (j = 1, ..., n).

Then, each invariant subspace W of E by T has the form

W = [(u i ) i∈I ; (v j ) j∈J ] := Span{(u i ) i∈I ; (v j ) j∈J }, (10)
where (u i ) i∈I (0 ≤ p := Card(I) ≤ dim F) is a family of linearly independent vectors of F (p = 0 meaning that the family is empty), and where (v j ) j∈J (0 ≤ q := Card(J) ≤ n = dim G) is a family of eigenvectors extracted from (v j ) j=1,...,n (q = 0 meaning that the family is empty).

Lemma 2 applies to the endomorphism T = L 1 of E := C 4 given by the matrix (7), with F := [e 0 , e 1 ] and G := [e 2 , e 3 ], the latter being stable by L 1 and admitting the basis of eigenvectors (a + , a -). This allows us to easily write the complete list of the non-trivial vector subspaces of C 4 which are invariant under the endomorphism L 1 , Eq. (7):

• The (complex) "lines" [a] := Ca, where the vector a ∈ C 4 is either:

a linear combination a = λe 0 + µe 1 , λ, µ ∈ C, λ = 0 or µ = 0; or a + = e 2 -ie 3 ;
or a -= e 2 + ie 3 .

• The following "planes" (2D complex subspaces): The list of the non-trivial vector subspaces of C 4 which are invariant under the endomorphism L i (i = 2 or i = 3) obtains by applying the permutation (1 2 3) → (2 3 1) or respectively (1 2 3) → (3 1 2) to the indices in the list above. Therefore, it is clear that the only line which is invariant under the rotations L 1 and L 2 (or under L 1 and L 3 , or under L 2 and L 3 ) is [e 0 ]. However, that line is obviously not invariant under the boost (8). Thus no complex line is invariant under the set S. As to the "planes": it is clear also that none of the invariant planes by L 1 numbered (i) and (ii) in the list above is invariant under L 2 (nor by L 3 , in fact). Still clear is the fact that an invariant plane by L 1 , of the form (iii) or (iv): [λe 0 + µe 1 , e 2 + ie 3 ] ( = ±1), cannot coincide with either [e 0 , e 2 ] or [e 3 , e 1 ], which are invariant planes by L 2 . The only remaining possibility to have an invariant plane by L 1 and by L 2 is if an invariant plane by L 1 , of the form (iii) or (iv): [λe 0 + µe 1 , e 2 + ie 3 ] ( = ±1), can coincide with an invariant plane by L 2 , of one of the corresponding forms: [λ e 0 + µ e 2 , e 3 + ie 1 ] ( = ±1, not necessarily = ). Thus the question is whether, for any α, β ∈ C, one can find α , β ∈ C such that u(α, β) := α(λe 0 + µe 1 ) + β(e 2 + ie 3 ) = α (λ e 0 + µ e 2 ) + β (e 3 + ie 1 ), ( 11) that is, such that Let us finally look if there can be a 3-space invariant under the set S, beginning with asking: which are, if any, the 3-spaces invariant under both L 1 and L 2 ? 1) We start the latter question by searching if a 3-space invariant under L 1 , of the form (a) above, can coincide with a 3-space invariant under L 2 , of the corresponding form. That is: can we have

αλ = α λ , αµ = i β , β = α µ , i β = β . (12)
i.e.,

αλ = α λ , αµ = γ , β = α µ , γ = β . (15)
-If λ = 0, we have from (15) 1 : α = αλ/λ . Then if µ = 0, we get from (15) 3 : α = β/µ , hence β/µ = αλ/λ . If instead µ = 0, (15) 3 gives us β = 0. In either case, v in Eq. ( 14) is assigned to depend at most on two parameters, hence (14) cannot hold on the whole 3-space [λe 0 + µe 1 , e 2 , e 3 ].

-If λ = 0, we have from (15) 1 : αλ = 0. Then if λ = 0, this gives α = 0, so again (14) cannot hold on the whole 3-space [λe 0 + µe 1 , e 2 , e 3 ] . If instead λ = 0, then since we are considering the case λ = 0, necessarily µ = 0 and µ = 0 to have indeed a 3-space on both sides of (13), thus this is the case that both of them coincide with the 3-space [e 1 , e 2 , e 3 ], which is indeed invariant under L 1 and under L 2 .

2) Then we have to see if a 3-space invariant under L 1 , of the form (a) above, can coincide with a 3-space invariant under L 2 and corresponding with the cases (b) or (c), though of course after the relevant permutation (1 2 3) → (2 3 1). Thus, we ask if we can have

[λe 0 + µe 1 , e 2 , e 3 ] = [e 0 , e 2 , e 3 + i e 1 ] ( = ±1), (16)
i.e., we ask if, for any α, β, γ ∈ C, we can find α , β , γ ∈ C such that α(λe 0 + µe 1 ) + βe 2 + γe 3 = α e 0 + β e 2 + γ (e 3 + i e 1 ),

or αλ = α , αµ = i γ , β = β , γ = γ . (17)
Thus we must have γ = γ = -i αµ, so once more the relevant equality, here (17), cannot hold on a whole 3-space.

3) The remaining possibility is the equality

The same trivial discussion as for the case (16) leads here (with the same notations) to β = i αµ, so (19) cannot happen.

Thus there is just one 3-space that is invariant under both L 1 and L 2 , namely the "spatial 3-space" [e 1 , e 2 , e 3 ]. (That 3-space is invariant under L 3 as well.) But it clearly is not invariant under the boost L 1 , Eq. (8). We conclude that no proper subspace of C 4 is invariant under the set S, which is therefore an irreducible set of matrices. This proves Proposition 2.

It is clear that e.g. the set S := {L 2 , L 3 , L 2 }, or the set S := {L 3 , L 1 , L 3 } (with L i the boost in the direction e i ), are irreducible also.

Proof of Lemma 2

Let the endomorphism T of E = F ⊕ G be as in the statement of Lemma 2, and let W be a vector subspace of E that is invariant under T . Suppose first that W ⊂ G. Then the fact that W has indeed the form (10) claimed by Lemma 2 is a direct application of the following known result (e.g. [START_REF] Gohberg | Invariant Subspaces of Matrices with Applications[END_REF]): Lemma 3. Assume the endomorphism T of the vector space G admits a basis of eigenvectors (v j ) (j = 1, ..., n) corresponding with pairwise distinct eigenvalues. Then any subspace of G that is invariant under T has the form

W = [(v j ) j∈J ] := Span{v j ; j ∈ J}, ( 20
)
where J is some subset of {1, ..., n}.

If instead W ⊂ G, define W and W as the projection space of W onto F or G, respectively, in the decomposition E = F ⊕ G. This definition can be written explicitly as:

W = {y ∈ F; ∃z ∈ G : y + z ∈ W}, (21)
and the like for W . The subspace W is not reduced to zero, for otherwise we would have W ⊂ G. Indeed, E = F ⊕ G implies that for any x ∈ W there is a unique pair y ∈ F, z ∈ G such that x = y + z; if W = {0} we must have y = 0 by (21). So let (u i ) i=1,...,p (1 ≤ p ≤ dim F) be a basis of W . Since each of the u i 's belongs to W , by the definition (21) there is for each of them a vector z ∈ G such that the vector x := u i + z belongs to W. Let us decompose z on the basis (v j ) of G, thus getting numbers z j such that z = n j=1 z j v j . Thus we have

x = u i + n j=1 v j ∈ W, (22)
where, by the assumption of Lemma 2, v j := z j v j is still an eigenvector of T for the eigenvalue λ j (even though possibly v j = 0), i.e., T v j = λ j v j , while u i is an eigenvector of T for the eigenvalue λ = λ j (j = 1, ..., n). Thus by Lemma 4 below, we have u i ∈ W.

Lemma 4 (e.g. [START_REF]Unnamed contributor to the forum math.stackexchange.com "All Invariant Subspaces of a Linear Transformation[END_REF]). Assume that v 1 , ..., v m are eigenvectors of the endomorphism T of the vector space E corresponding with pairwise distinct eigenvalues λ j . If W is an invariant subspace of E under T such that v 1 + ... + v m ∈ W, then for each j = 1, ..., m we have v j ∈ W.

End of the proof of Lemma 2. Suppose first that W := {z ∈ G; ∃y ∈ F : y + z ∈ W} ( 23) is reduced to zero. Then, because E = F ⊕ G, we have W ⊂ F, just like, as we showed after (21), we have W ⊂ G in the symmetric case W = {0}. Therefore, it is immediate to check that W = W , so (u i ) i=1,...,p is a basis of W, hence W has indeed the form (10).

If instead W = {0}, we build a basis (v j ) j∈J of W extracted from the basis (v j ) j=1,...,n of G (made of eigenvectors of T ), as follows. For any x ∈ W, there is a unique pair y(x) = P F (x) ∈ F, z(x) = P G (x) ∈ G such that x = y(x) + z(x).

(24)

Since (v j ) is a basis of G, we can decompose z(x) uniquely on the basis (v j ) j=1,...,n , so

z(x) = j∈J(x)
z j (x) v j , z j (x) = 0 for j ∈ J(x).

(Note that J(x) can be empty, which occurs iff z(x) = 0.) By a very similar argument, also using Lemma 4, to that developed around Eq. ( 22), we see that (24) and (25) imply that v j ∈ W if j ∈ J(x). Hence v j ∈ W if j ∈ J(x), since W ∩ G ⊂ W from (23). Then define J := x∈W J(x).

(26)

  (i) [e 0 , e 1 ]; (ii) [e 2 , e 3 ] = [e 2 -ie 3 , e 2 + ie 3 ]; (iii) [λe 0 + µe 1 , e 2 -ie 3 ], λ, µ ∈ C, λ = 0 or µ = 0; (iv) [λe 0 + µe 1 , e 2 + ie 3 ], λ, µ ∈ C, λ = 0 or µ = 0. • The following 3-spaces: (a) [λe 0 + µe 1 , e 2 , e 3 ], λ, µ ∈ C, λ = 0 or µ = 0; (b) [e 0 , e 1 , e 2 -ie 3 ]; (c) [e 0 , e 1 , e 2 + ie 3 ].

If = 1 ,

 1 we thus have β = iβ by (12) 4 , whence αµ = -β by (12) 2 . Then if µ = 0, we must have β = 0. If instead µ = 0, we must have α = -β/µ. In either case, (12) can apply only when either α or β is determined by the other number, thus it cannot occur on the whole complex plane [λe 0 +µe 1 , e 2 + ie 3 ]. The case = -1 gives rise to the same discussion. Thus no 2-D subspace of C 4 is invariant under both L 1 and L 2 , a fortiori none is invariant under the set S.

[λe 0

 0 + µe 1 , e 2 , e 3 ] = [λ e 0 + µ e 2 , e 3 , e 1 ]? (13) This is true iff, for any α, β, γ ∈ C, one can find α , β , γ ∈ C such that v := α(λe 0 + µe 1 ) + βe 2 + γe 3 = α (λ e 0 + µ e 2 ) + β e 3 + γ e 1 ,

[λe 0 +

 0 µe 2 , e 3 , e 1 ] = [e 0 , e 1 , e 2 + i e 3 ] ( = ±1).

This definition and the Theorem extend immediately to any Lorentzian spacetime (V, γ), by considering, instead of Cartesian coordinates, coordinates that are Cartesian at the given point X ∈ V, i.e., such that γ µν (X) = η µν .

Note that we have by construction J ⊂ {1, ...n}. We claim that the finite family (v j ) j∈J is a basis of W . Indeed, consider any z ∈ W . From (23), there is some y ∈ F such that

Hence we have y = y(x) and z = z(x) from the uniqueness of the decomposition (24), so that from (25):

Thus the family (v j ) j∈J does generate W , and being a free family as an extracted family from the basis (v j ) j=1,...,n , it is indeed a basis of W . Since we showed that u i ∈ W (i = 1, ..., p) and that v j ∈ W (j ∈ J), it is clear that

Conversely, note that in the decomposition (24) of any x ∈ W, we have y(x) ∈ W from (21) and z(x) ∈ W from (23). Hence the reverse inclusion follows from the fact that (u i ) i∈{1,...,p} is a basis of W and that (v j ) j∈J is a basis of W . This completes the proof of Lemma 2.


S Barrat
email: barrat@loria.fr

S Tabbone
email: tabbone@loria.fr

P Nourrissier

A Bayesian classifier for symbol recognition

Keywords: Symbol recognition, data analysis, probabilistic graphical models, Bayesian networks, variable selection

We present in this paper an original adaptation of Bayesian networks to symbol recognition problem. More precisely, a descriptor combination method, which enables to improve significantly the recognition rate compared to the recognition rates obtained by each descriptor, is presented. In this perspective, we use a simple Bayesian classifier, called naive Bayes. In fact, probabilistic graphical models, more specifically Bayesian networks, are a simple and intuitive way of probability distribution representation. In order to solve the dimensionality problem, we use a variable selection method. Experimental results, obtained in a supervised learning context and tested on GREC symbol database, are very promising.

Introduction

Classification is a basic task in data analysis and pattern recognition. This task requires a classifier, i.e a function that assigns a class label to instances described by a set of features. The induction of classifiers from data sets of labelled data (we speak about supervised learning) is a central problem in machine learning. In fact, in numerous applications, the aim is to assign a feature vector f = {f 1 , f 2 , ..., f n } to a class c i among k classes, designed by the vector c = {c 1 , c 2 , ..., c k }. Some approaches to this problem are based on various functional representations such as decision trees, neural networks, decision graphs [START_REF] Bishop | Pattern Recognition and Machine Learning[END_REF], associated to decision rules. Probabilistic approaches also play a central role in classification. A way to reach the previous goal, by using probabilities, is to compute the conditional probability distribution P (c i |f ), ∀i ∈ {1, 2, ..., k} and assign the instance f to the class c i for which this probability is maximal. We could formulate and solve complicated probabilistic models purely by algebric manipulations. However, we shall find it highly advantageous to improve the analysis using diagrammatic representations of probability distributions, called probabilistic graphical models. In fact they provide a simple way to visualize the structure and the properties (including conditional independence properties) of a probabilistic model, but they especially enable to perform complex computations like inference and learning, using graphical manipulations.

Thus we propose, in this paper, an original method of descriptor combination applied to symbol classification. We have adapted probabilistic graphical model theory to symbol recognition problem. The originality of our approach also relies on the use of the variable selection method LASSO [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF], proposed to overcome the dimensionality problem related to the size of feature vectors and the inherent network complexity.

Bayesian networks as classifiers

Definitions and notations

We remind that our aim is to assign a feature vector f = {f 1 , f 2 , ..., f n } to a class c i among k classes. Recent works in supervised learning [START_REF] Domingos | Beyond independence: Conditions for the optimality of the simple bayesian classifier[END_REF][START_REF] Friedman | Bayesian network classifiers[END_REF][START_REF] Friedman | On bias, variance, 0/1loss, and the curse-of-dimensionality[END_REF] have shown that the naive Bayesian classifier has a surprisingly good performance. This classifier learns from training data (labelled data) the conditional probability of each feature F j , ∀j ∈ {1, 2, ...n, } given the class label c i , ∀i ∈ {1, 2, ..., k}. Classification is then done by applying Bayes rule to compute the probability of c i given the particular instance of F 1 , F 2 , ..., F n , ∀i ∈ {1, 2, ..., k} and then predicting the class with the highest posterior probability. This computation is rendered feasible by making a strong independence assumption: all the features F j are conditionally independent given the value of the class C. Thus we obtain this formula:

P (c i |f 1 , f 2 , ..., f n ) = P (f 1 , f 2 , ..., f n , c i ) P (f 1 , f 2 , ..., f n ) = P (f 1 , f 2 , ..., f n |c i ) × P (c i ) P (f 1 , f 2 , ..., f n )
where

P (f 1 , f 2 , ..., f n ) = k i=1 P (f 1 , f 2 , ..., f n , c i ) = k i=1 P (f 1 , f 2 , ..., f n |c i ) × P (c i )
In order to represent and manipulate independence assertions, we need an appropriate language and efficient machinery. Both are provided by Bayesian networks [START_REF] Pearl | Probabilistic reasoning in intelligent systems: networks of plausible inference[END_REF]. These networks are directed acyclic graphs that provide an efficient and effective representation of the joint probability distribution over a set of random variables. Each vertex in the graph represents a random variable, and edges represent dependence relations between the variables. More precisely, the network encodes the following conditional independence statement: each variable is independent of its nondescendants in the graph given the state of its parents. These independencies are then exploited to reduce the number of parameters needed to characterize a probability distribution, and to efficiently compute posterior probabilities given an evidence. If we represent a naive Bayesian classifier as a Bayesian network, it has the simple structure depicted in Fig. 1. This network encodes the main assumption behind the naive Bayesian classifier, namely, that each feature (each leaf in the graph) is independent from the rest of the features, given the state of the class variable (the root of the graph).

Parameter learning

The parameters of our model, i.e the probabilities P (c i |f 1 , f 2 , ..., f n ) and P (c i ), ∀i ∈ {1, 2, ..., k}, are obtained by Maximum Likelihood Parameter Estimation method, because it is one of the most robust parameter estimation techniques, from a statistical point of view. The idea of this method is to obtain the most likely values of the parameters, for a given distribution, that will best describe the data. But when some instances are not observed in the training set, this method will evaluate the corresponding parameters by null values. If the resulted network is used to infer on these instances, the network will predict that these instances have a null probability. Thus this method can generate overfitting. To overcome this problem, we introduce additional pseudo counts at every instance in order to ensure that they are all 'virtually' represented in the training set. It doesn't mean that we have to construct a new training set, but that we will use Dirichlet priors. Using maximum likelihood estimator, with Dirichlet priors, implies that every instance, even if it is not represented in the training set, will have a not null probability. This method is explained more precisely in [START_REF] Robert | A decision-Theoretic Motivation[END_REF].

Inference

A query image, f j , that we want to classify, characterized by its features f j1 , f j2 , ..., f jn , can be considered as an evidence which would be represented by:

P (f j ) = P (f j1 , f j2 , ..., f jn ) = 1
when the network will be evaluated. Then the inference process consists in computing posterior probability distributions of one or several other subsets of nodes. In the case of classification, we infer the class node. According to our Bayesian network topology, the inference process propagates the values from the image feature level, represented by the n nodes F 1 , F 2 , ..., F n , to the class node C. Currently we count several methods of exact or approximate inference techniques. Although the exact inference is generally a NP-hard problem [START_REF] Cooper | The computational complexity of probabilistic inference using bayesian belief networks[END_REF], it is still efficient for some classes of Bayesian networks. Moreover the complexity in time of exact inference methods is early computable. When the result exceeds a reasonable threshold, we will prefer to use an approximate method [START_REF] Jaakkola | Variational probabilistic inference and the QMR-DT network[END_REF][START_REF] Jordan | An introduction to variational methods for graphical models[END_REF]. In our problem, the size of the network is not large, because we reduce the feature dimensionality with a variable selection method (see subsection 3.2), so we can use an exact inference method. One of the most robust methods of exact inference is the JLO algorithm, called junction tree algorithm too, developped in [START_REF] Jensen | Bayesian updating in recursive graphical models by local computations[END_REF][START_REF] Lauritzen | Local computations with probabilities on graphical structures and their application to expert systems[END_REF]. This algorithm transforms the network into a junction tree: the variables are merged into clusters in order to delete all loops in the network and to make the clusters as small as possible. After converting the network to a tree, a message passing algorithm [START_REF] Kim | A computational model for combined causal and diagnostic reasoning in inference systems[END_REF] is applied to the network. In this technique, each node is associated to a processor, which can send some messages to its neighbours, in an asynchronous way, until it reachs a stability. Then the probabilities of each node are updated in function of the evidence instance. After the belief propagation, we know the posterior probability P (c i |f j1 , f j2 , ..., f jn ), ∀i ∈ {1, 2, ..., k}. The query f j is assigned to the class c i which maximizes this probability.

Symbol recognition

This section explains how we have adapted the theoretical method before-mentioned to the symbol recognition problem. We present the used features and the method we have had to use in order to overcome large dimensionality problem.

Features

We have to classify black and white images, so we have choosen these three shape descriptors: Generic Fourier Descriptor (GFD), Zernike descriptor and the R-signature 1D. We briefly present each descriptor below.

Generic Fourier Descriptor Generic Fourier Descriptor is based on Fourier transform [START_REF] Zhang | Shape-based image retrieval using General Fourier Descriptor[END_REF]. It is a pixel descriptor. The rotation invariance is achieved by using the modified polar Fourier transform (MPFT) proposed by Dengsheng Zhang and Guojun Lu in [START_REF] Zhang | Shape-based image retrieval using General Fourier Descriptor[END_REF]. Then scaling invariance is achieved after normalization.

Zernike descriptor Zernike descriptor [START_REF] Kim | A modified Zernike moment shape descriptor invariant to translation, rotation and scale for similarity-based image retrieval[END_REF] is a pixel descriptor based on Zernike moments. Zernike moments of a given shape are calculated as correlation values of the shape with Zernike basis functions, in that all the pixels of the shape, independently of their position, contribute with the same weight to the Zernike moments. These moments are rotation invariant. To make the Zernike moments of the shape descriptor invariant also to translation and scaling, a given shape is scaling and translation normalized, by obtaining the smallest circle centered at the center of mass, covering all the shape pixels. Then the obtained circle is adjusted to match the radius of Zernike moment basis functions. The Zernike shape descriptor consists of low order magnitudes of Zernike moments.

R-signature 1D

The R-signature 1D [START_REF] Tabbone | Technical Symbols Recognition Using the Twodimensional Radon Transform[END_REF] uses Radon transform to represent an image. The Radon transform is the projection of an image in a particular plan. This projection has interesting properties which make it a good descriptor. According to these geometrical properties, a 1D signature of the transform is created. This signature checks the properties of invariance to some geometrical transformations, such as the translation and the scaling (after normalization). The rotation invariance is achieved by a cyclic permutation of the signature, or directly from its Fourier transform.

Dimensionality reduction

Once the n, n ∈ {1, 2, 3} descriptors we want to combine are computed on each image, we dispose of n signatures per image. The concatenation of these signatures provides us a new feature vector per image. Each vector component is considered as a discrete random variable and corresponds to a node from the feature level of the network. The large dimensions of the initial signatures imply a large dimension for the feature vector and too many variables in the network compared to the size of training set: in fact existing algorithms on Bayesian networks are efficient until around 1000 variables, only if we have quite a few training data. To overcome this problem, we have to use a dimensionality reduction method, which enables to extract from the feature vectors, just the most relevant and discriminating features, with a minimal information loss. We have choosen the regression method LASSO for its stability and implementation facility, but especially because this method enables to select variables (when some regression coefficients are null) and is competitive in complex situations, contrary to the Principal Component Analysis (PCA) or the traditional methods like subset selection or ridge regression, for example.

Least Absolute Shrinkage and Selection Operator: LASSO The LASSO is an approach of variable selection and coefficient shrinkage introduced by R. Tibshirani [START_REF] Tibshirani | Regression shrinkage and selection via the lasso[END_REF]. This method shrinks the regression coefficients by imposing a penalty on their size. The LASSO coefficients minimize a penalized residual sum of squares:

β lasso = arg min β N i=1 (y i -β 0 - p j=1 x ij β j ) 2 subject to p j=1 |β j | ≤ s (1)
The LASSO uses a L1 penalty: p j=1 |β j |. This constraint implies that for smaller values of s (s ≥ 0), some of the coefficients β will be null. So choosing s is like choosing the number of predictors in a regression model. Then the selection criterion is simple: we will select the variables corresponding to the coefficients different from zero.

The computation of the lasso solutions is a quadratic programming problem, and can be tackled by standard numerical analysis algorithms. One of the most adapted is the Least Angle Regression (LAR) procedure. In fact it exploits the special structure of the lasso problem, and provides an efficient way to compute the solutions simultaneously for all values of s. This algorithm is presented in detail in [START_REF] Efron | Least angle regression[END_REF]. We have applied this method to our training data: in this case y represents our class variable C, and x j = {x j1 , x j2 , ..., x jp } the p features of the instance j.

From continuous to discrete data

The descriptors before-mentioned provide us signatures of non integer values. So, once we have selected variables, we dispose of a reduced set of continuous variables. But the naive Bayesian classifier requires discrete variables. So it is necessary to discretize our variables. We have used the method presented in [18], because it enables an important data reduction, by passing from continuous to discrete data, without any loss of information. Moreover it is easy to implement. Concretely this method consists in approximating probability laws by histograms. These histograms must optimally approximate, in the sense of the maximum likelihood and a mean squares cost, the unknown law of a random process with a single n-sample. In order to obtain the bin number defining the histogram and the distribution of these bins, the information criterion of Akaike (AIC), habitually used for model order selection, has been generalized to this problem. More precisely, an m-bins histogram is built from the sample. Then, the idea to reduce the histogram size is to merge the two adjacent bins, for which the difference between the AIC before merging and after merging is maximal. This process is repeated until this difference would be negative.

We apply this method for all variables on the training set. We dispose of a n-sample per variable and the method provides us one histogram per variable. Then considerate one histogram: each bin corresponds to a possible discrete value for the variable corresponding to this histogram, and we can obtain the probability of each possible value thanks to the number of elements in each bin. Thus we can estimate the probability distribution for each variable.

Experimental results

We have used the symbols of GREC database [START_REF]Graphics Recognition, Recent Advances and Perspectives, 5th InternationalWorkshop[END_REF] to test our method. We have selected a subset of 50 models and generated a database of 3600 symbols, composed of 72 different images per model. Each image corresponds to a model, with some degradations and linear transformations (rotation and scaling) of different intensities.

On this database we have defined some learning tests: we have defined various training and test sets of different sizes. We have tested the method by using the signatures issued from one, two or three descriptors and executing all possible combinations. The tables below present the recognition rates obtained by the different tests. The variable selection method LASSO has enabled us to select 13 variables among 225 initial variables for GFD, 15 among 34 for Zernike and 13 among 180 for the R-signature. Concerning the combinations, we have obtained 4 variables issued from GFD and 6 issued from Zernike for the combination of GFD and Zernike, 7 variables issued from GFD and 7 from the R-signature for the combination of Zernike and the R-signature, and 6 from Zernike and 6 form the R-signature for the combination of Zernike and the R-signature. Finally, for the combination of the 3 descriptors, we have 2 variables from GFD and the R-signature and 3 from Zernike. The table 1 shows the means of the recognition rates obtained with some random selections of the same numbers of variables than the ones obtained with LASSO. The table 2 presents the results obtained after the automatic selection, with LASSO method, of a subset of relevant variables. In the both tables, G means that the GFD descriptor is used, Z that the Zernike descriptor is used and R for the R-signature. Finally the '+' operator indicates that we combine the descriptors.

The experimental results show the interest of our method: the LASSO method enables us to choose a suitable number of variables for each descriptor, for our Bayesian classifier. Moreover, in the great majority of the examples, selecting the variables with the LASSO is more efficient than a random selection of the same number of variables. We also observe that, after applying the LASSO method, whatever the recognition rates obtained by one descriptor, we improve these rates by combining 2 or 3 descriptors. In the same way, whatever the recognition rates obtained by combining 2 descriptors, we improve these recognition rates by combining 3 descriptors.

Besides, even if we obtain a high recognition rate with Zernike descriptor, this rate will not decrease when we combine this descriptor with one or two other descriptors, whatever these descriptors and even if the added descriptors have a low rate (it is the case with the R-signature), i.e the bad behaviour of a descriptor doesn't impede the other descriptor behaviours.

Conclusion and future work

In this paper we have proposed an original adaptation of Bayesian theory. Moreover, the LASSO method has been proposed to solve the dimensionality problem of feature vectors. The experimental results are promising and show that our method enables to improve the recognition rate by combining descriptors.

In our future work, we want to integrate a relevance feedback approach. More precisely, it is a matter of taking into account some informations given by the user, on some queries, in order to modify the network parameters after inferred the class of a query image. Besides we want to use the information given by keywords associated to a subset of training data. It's possible because Bayesian networks enable to manage, in a same network, different kinds of information (in this case different media), and thanks to their ability to manage incomplete data.

Fig. 1 .

 1 Fig. 1. Naive Bayes

Table 1 .

 1 Recognition rates without automatic variable selection Specifications G Z R G+Z G+R Z+R G+Z+R training test ims 1-54 ims 55-72 88.2 92.3 60.2 93.8 94.7 95.7 97.1 ims 19-72 ims 1-18 83.3 88.2 50 89.2 87.8 91.3 94 ims 1-36 ims 37-72 83.8 89.8 50.8 90 86.7 92.9 95.4 ims 37-72 ims 1-36 81 90.5 48.5 91.4 86 93.2 93.8 ims 1-18 ims 19-72 75.8 86.9 44.2 89.3 81.2 92.2 90.6 ims 55-72 ims 1-54 66.2 79.1 36.9 81.3 73.4 84.6 85.4

	Specifications training test	G Z R G+Z G+R Z+R G+Z+R
	ims 1-54 ims 55-72 87.2 90.9 70.4 92.4 89.7 93.3	94.7
	ims 19-72 ims 1-18 81.8 86.4 61.2 85.3 85.1 89.8	90.1
	ims 1-36 ims 37-72 83 88.5 63 86.9 86.5 92	90.9
	ims 37-72 ims 1-36 81.2 89.2 63.2 86.4 86.3 92.5	90.8
	ims 1-18 ims 19-72 80.6 83.8 54.2 84.4 83.2 90.9	88.2
	ims 55-72 ims 1-54 64.3 79.4 49.5 76.8 70.9 85.2	82.7

Table 2 .

 2 Recognition rates with the variable selection method LASSO