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Random ‘Clouds’ on Matrix Lie Groups

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Motivations Notation in Lie group theory Random clouds Open questions Random ‘Clouds’ on Matrix Lie Groups Simone Fiori Dipartimento di Ingegneria dell’Informazione Universit`a Politecnica delle Marche (Ancona, Italy) GSI2013 - Geometric Science of Information MINES ParisTech – August 2013 Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Outline 1 Motivations 2 Notation in Lie group theory 3 Random clouds 4 Open questions Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Outline 1 Motivations 2 Notation in Lie group theory 3 Random clouds 4 Open questions Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations Methods for solving inverse problems rely high-dimensional signal/data processing techniques: Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations Methods for solving inverse problems rely high-dimensional signal/data processing techniques: Tensor factorization in blind source separation [T.M. Rutkowski, R. Zdunek, A. Cichocki, International Congress Series, Elsevier, 2007] High resolution Doppler imagery modeling [F. Barbaresco, Emerging Trends in Visual Computing, Lecture Notes in Computer Science, 2009] Bayesian methods for signal/image processing [A. Mohammad-Djafari, EURASIP Journal on Advances in Signal Processing, 2012] Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations In the development of information processing algorithms, testing is a fundamental step: Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations In the development of information processing algorithms, testing is a fundamental step: Testing is necessary to prove the correctness of an algorithm Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations In the development of information processing algorithms, testing is a fundamental step: Testing is necessary to prove the correctness of an algorithm Testing consists in applying a set of inputs to an algorithm Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations In the development of information processing algorithms, testing is a fundamental step: Testing is necessary to prove the correctness of an algorithm Testing consists in applying a set of inputs to an algorithm If the output does not match the expected output, the algorithm is flawed Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations In the development of information processing algorithms, testing is a fundamental step: Testing is necessary to prove the correctness of an algorithm Testing consists in applying a set of inputs to an algorithm If the output does not match the expected output, the algorithm is flawed Testing (and test design) is part of quality assurance for an algorithm Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations Several algorithms use Lie-group-valued data: Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Motivations Several algorithms use Lie-group-valued data: Study of plate tectonics [M.J. Prentice, The Journal of the Royal Statistical Society Series C (Applied Statistics), 1987] Modeling of DNA chains [K.A. Hoffman, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2004] Tuning of kernel machines [H. Xiong et al., IEEE Transactions on Neural Networks, 2005] Averaging over matrix-type Lie groups [S. Fiori and T. Tanaka, IEEE Transactions on Signal Processing, 2009] Automatic object pose estimation [Q. Rentmeesters et al., Proceedings of the 2010 IEEE International Conference on Acoustics Speech and Signal Processing, 2010] Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Example Testing of averaging algorithms over Lie groups −0.6 −0.4 −0.2 0 0.2 −1 −0.8 −0.6 −0.4 −0.2 −0.9 −0.8 −0.7 −0.6 −0.5 Entry (X11 k ) Distribution: Gaussian, Manifold: SO(3), Variance = 0.114 Entry (X21 k ) Entry(X31 k) Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Outline 1 Motivations 2 Notation in Lie group theory 3 Random clouds 4 Open questions Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Lie group = Algebraic group + Differentiable manifold An algebraic group structure (G, µ, ι, e) is made of a set G endowed with: Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Lie group = Algebraic group + Differentiable manifold An algebraic group structure (G, µ, ι, e) is made of a set G endowed with: A multiplication operation µ Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Lie group = Algebraic group + Differentiable manifold An algebraic group structure (G, µ, ι, e) is made of a set G endowed with: A multiplication operation µ An inversion operation ι Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Lie group = Algebraic group + Differentiable manifold An algebraic group structure (G, µ, ι, e) is made of a set G endowed with: A multiplication operation µ An inversion operation ι An identity element e. Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Lie group = Algebraic group + Differentiable manifold Associated with the Lie group G is a Lie algebra g def =TeG. Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Basic Lie-group operators: Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Basic Lie-group operators: Left translation about a point g ∈ G, denoted by ℓg : G → G, defined by ℓg (h) def =µ(ι(g), h) for h ∈ G Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Basic Lie-group operators: Left translation about a point g ∈ G, denoted by ℓg : G → G, defined by ℓg (h) def =µ(ι(g), h) for h ∈ G Inverse left translation is defined by ℓ−1 g (h) def =µ(g, h) Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Basic Lie-group operators: Left translation about a point g ∈ G, denoted by ℓg : G → G, defined by ℓg (h) def =µ(ι(g), h) for h ∈ G Inverse left translation is defined by ℓ−1 g (h) def =µ(g, h) Lie group automorphism is denoted by ϕ ∈ Aut(G) Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Definition and properties of Lie groups Basic Lie-group operators: Left translation about a point g ∈ G, denoted by ℓg : G → G, defined by ℓg (h) def =µ(ι(g), h) for h ∈ G Inverse left translation is defined by ℓ−1 g (h) def =µ(g, h) Lie group automorphism is denoted by ϕ ∈ Aut(G) Inner automorphism ϕh ∈ Aut(G), with h ∈ G, defined as ϕh(g) def =µ(µ(h, g), ι(h)) Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Outline 1 Motivations 2 Notation in Lie group theory 3 Random clouds 4 Open questions Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Rule for generating random clouds The following rule is proposed to generate random clouds of points on a Lie group: Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Rule for generating random clouds The following rule is proposed to generate random clouds of points on a Lie group: h = (ϕ ◦ ℓ−1 g ◦ expe)(ω) where: expe : g → G denotes Lie-group exponential Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Rule for generating random clouds The following rule is proposed to generate random clouds of points on a Lie group: h = (ϕ ◦ ℓ−1 g ◦ expe)(ω) where: expe : g → G denotes Lie-group exponential ω is a randomly-picked element of g Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Rule for generating random clouds The following rule is proposed to generate random clouds of points on a Lie group: h = (ϕ ◦ ℓ−1 g ◦ expe)(ω) where: expe : g → G denotes Lie-group exponential ω is a randomly-picked element of g h is the random element generated in G Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Examples Numerical examples presented with G = SO(3): SO(n) def ={X ∈ Rn×n|XT X = I, det(X) = 1} so(n) = {Ω ∈ Rn×n|ΩT = −Ω} ϕ(X) = X3 or ϕ(X) = X4 Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Examples −1 −0.9 −0.8 −0.7 −0.6 −0.5 0 0.5 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Entry (Y 11 k ) Manifold: SO(3), Transform: Inner Entry (Y 21 k ) Entry(Y31 k) −0.5 0 0.5 1 −0.5 0 0.5 1 −1 −0.9 −0.8 −0.7 −0.6 −0.5 −0.4 −0.3 Entry (Z11 k ) Distribution: Gaussian, Transform: Quartic Entry (Z21 k ) Entry(Z31 k) Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Examples −1 −0.5 0 0.5 1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 ℜ{Y 11 k } Distribution on the Lie algebra: Uniform. Variance = 0.325 ℑ{Y 11 k } ℜ{Y12 k} Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Examples −1 −0.9 −0.8 −0.7 −0.6 −0.5 −1 −0.5 0 0.5 1 −1 0 1 ℜ{Y 11 k } Distribution on the Lie algebra: Gaussian. Variance = 0.316 ℑ{Y 11 k } ℜ{Y12 k} −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 0 1 ℜ{Z11 k }ℑ{Z11 k } ℜ{Z12 k} Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Outline 1 Motivations 2 Notation in Lie group theory 3 Random clouds 4 Open questions Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Open questions Measure the shape of the generated distributions Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Open questions Measure the shape of the generated distributions Up to now: Distribution of the distances d(h, g) (see paper) Simone Fiori Random ‘Clouds’ on Matrix Lie Groups Motivations Notation in Lie group theory Random clouds Open questions Open questions Measure the shape of the generated distributions Up to now: Distribution of the distances d(h, g) (see paper) Introduce directional/density descriptors on the tangent bundle Simone Fiori Random ‘Clouds’ on Matrix Lie Groups