U N I V E R S I T Y O F C O P E N H A G E NU N I V . - DIKU

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UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENWelcome to Copenhagen!Schedule:8MondayRegistration andwelcomeTuesdayWednesdayCrash course onDifferential andRiemannianGeometry 1.1(Feragen)Crash course onDifferential andRiemannianGeometry 3(Lauze)Introduction toInformation Geometry3.1(Amari)Crash course onDifferential andRiemannianGeometry 1.2(Feragen)Tutorial on numericsfor Riemanniangeometry 1.1(Sommer)Introduction toInformation Geometry3.2(Amari)Crash course onDifferential andRiemannianGeometry 1.3(Feragen)Tutorial on numericsfor Riemanniangeometry 1.2(Sommer)Introduction toInformation Geometry3.3(Amari)ThursdayFriday9Information Geometry & Information Geometry &Stochastic Optimization Stochastic Optimization1.1in Discrete Domains 1.1(Hansen)(Màlago)10Information Geometry & Information Geometry &Stochastic Optimization Stochastic Optimization1.2in Discrete Domains 1.2(Hansen)(Màlago)11121314LunchLunchCrash course onDifferential andRiemannianGeometry 2.1(Lauze)Introduction toInformation Geometry2.1(Amari)Crash course onDifferential andRiemannianGeometry 2.2(Lauze)Introduction toInformation Geometry2.2(Amari)Introduction toInformation Geometry1.1(Amari)Introduction toInformation Geometry2.3(Amari)Introduction toInformation Geometry1.2(Amari)Introduction toInformation Geometry2.4(Amari)LunchInformation Geometry & Information Geometry &Stochastic Optimization Stochastic Optimization1.3in Discrete Domains 1.3(Hansen)(Màlago)LunchLunchInformation Geometry & Information Geometry &Information Geometry &Reinforcement Learning Stochastic OptimizationCognitive Systems 1.11.11.4(Ay)(Peters)(Hansen)Information Geometry & Information Geometry &Information Geometry &Reinforcement Learning Stochastic OptimizationCognitive Systems 1.21.21.5(Ay)(Peters)(Hansen)1516Information Geometry & Stochastic OptimizationInformation Geometry &Reinforcement Learningin Practice 1.1Cognitive Systems 1.31.3(Hansen)(Ay)(Peters)Social activity/Networking eventStochastic OptimizationInformation Geometry &in Practice 1.2Cognitive Systems 1.3(Hansen)(Ay)Coffee breaks at 10:00 and 14:45 (no afternoon break onWednesday)Slide 1/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENWelcome to Copenhagen!Social Programme! Today: Pizza and walking tour! 17:15 Pizza dinner in lecture hall 18:00 Departure from lecture hall (with Metro – we have tickets) 19:00 Walking tour of old university Wednesday: Boat tour, Danish beer and dinner 15:20 Bus from KUA to Nyhavn 16:00-17:00 Boat tour 17:20 Bus from Nyhavn to NÃ rrebro bryghus (NB, brewery) 18:00 Guided tour of NB 19:00 Dinner at NBSlide 1/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENWelcome to Copenhagen! Lunch on your own – canteens and coffee on campus Internet connection Eduroam Alternative will be set up ASAP Emergency? Call Aasa: 4526220498 Questions?Slide 1/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENFaculty of ScienceA Very Brief Introduction to Differentialand Riemannian GeometryAasa Feragen and François LauzeDepartment of Computer ScienceUniversity of CopenhagenPhD course on Information geometry, Copenhagen 2014Slide 2/57

UNIVERSITY OF COPENHAGENOutline1MotivationNonlinearityRecall: Calculus in Rn2Differential GeometrySmooth manifoldsBuilding ManifoldsTangent SpaceVector fieldsDifferential of smooth map3Riemannian metricsIntroduction to Riemannian metricsRecall: Inner ProductsRiemannian metricsInvariance of the Fisher information metricA first take on the geodesic distance metricA first take on curvatureSlide 3/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22UNIVERSITY OF COPENHAGEN

UNIVERSITY OF COPENHAGENOutline1MotivationNonlinearityRecall: Calculus in Rn2Differential GeometrySmooth manifoldsBuilding ManifoldsTangent SpaceVector fieldsDifferential of smooth map3Riemannian metricsIntroduction to Riemannian metricsRecall: Inner ProductsRiemannian metricsInvariance of the Fisher information metricA first take on the geodesic distance metricA first take on curvatureSlide 4/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22UNIVERSITY OF COPENHAGEN

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENWhy do we care about nonlinearity? Nonlinear relations between data objects True distances not reflected by linear representation”Topographic map example”. Licensed under Public domain via Wikimedia Commons http://commons.wikimedia.org/wiki/File:Topographic map example.png#mediaviewer/File:Topographic map example.pngSlide 5/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENMildly nonlinear: Nonlinear transformationsbetween different linear representations Kernels! Feature map nonlinear transformation of (linear?) data spaceX into linear feature space H Learning problem is (usually) linear in H, not in X .Slide 6/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENMildly nonlinear: Nonlinearly embeddedsubspaces whose intrinsic metric is linear Manifold learning! Find intrinsic dataset distances Find an Rd embedding that minimally distorts those distancesSlide 7/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENMildly nonlinear: Nonlinearly embeddedsubspaces whose intrinsic metric is linear Manifold learning! Find intrinsic dataset distances Find an Rd embedding that minimally distorts those distances Searches for the folded-up Euclidean space that best fits thedata the embedding of the data in feature space is nonlinear the recovered intrinsic distance structure is linearSlide 7/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENMore nonlinear: Data spaces which areintrinsically nonlinear Distances distorted in nonlinear way, varying spatially We shall see: the distances cannot always be linearized”Topographic map example”. Licensed under Public domain via Wikimedia Commons http://commons.wikimedia.org/wiki/File:Topographic map example.png#mediaviewer/File:Topographic map example.pngSlide 8/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENIntrinsically nonlinear data spaces:Smooth manifoldsDefinitionA manifold is a set M with an associated one-to-one map ϕ : U Mfrom an open subset U Rm called a global chart or a globalcoordinate system for M.Slide 9/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENIntrinsically nonlinear data spaces:Smooth manifoldsDefinitionA manifold is a set M with an associated one-to-one map ϕ : U Mfrom an open subset U Rm called a global chart or a globalcoordinate system for M. Open set U Rm set that does not contain its boundary Manifold M gets its topology ( definition of open sets) from Uvia ϕSlide 9/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENIntrinsically nonlinear data spaces:Smooth manifoldsDefinitionA manifold is a set M with an associated one-to-one map ϕ : U Mfrom an open subset U Rm called a global chart or a globalcoordinate system for M. Open set U Rm set that does not contain its boundary Manifold M gets its topology ( definition of open sets) from Uvia ϕ What are the implications of getting the topology from U?Slide 9/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENIntrinsically nonlinear data spaces:Smooth manifoldsDefinitionA smooth manifold is a pair (M, A) where M is a set A is a family of one-to-one global charts ϕ : U M from someopen subset U Uϕ Rm for M, for any two charts ϕ : U Rm and ψ : V Rm in A, theircorresponding change of variables is a smooth diffeomorphismψ 1 ϕ : U V Rm .Slide 9/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENOutline1MotivationNonlinearityRecall: Calculus in Rn2Differential GeometrySmooth manifoldsBuilding ManifoldsTangent SpaceVector fieldsDifferential of smooth map3Riemannian metricsIntroduction to Riemannian metricsRecall: Inner ProductsRiemannian metricsInvariance of the Fisher information metricA first take on the geodesic distance metricA first take on curvatureSlide 10/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22UNIVERSITY OF COPENHAGEN

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDifferentiable and smooth functions f : U open Rn Rq continuous: write(y1 , . . . , yq ) f (x1 , . . . , xn )Slide 11/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDifferentiable and smooth functions f : U open Rn Rq continuous: write(y1 , . . . , yq ) f (x1 , . . . , xn ) f is of class C r if f has continuous partial derivatives r1 ··· rn yk x1r1 . . . xnrnk 1 . . . q, r1 . . . rn r .Slide 11/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDifferentiable and smooth functions f : U open Rn Rq continuous: write(y1 , . . . , yq ) f (x1 , . . . , xn ) f is of class C r if f has continuous partial derivatives r1 ··· rn yk x1r1 . . . xnrnk 1 . . . q, r1 . . . rn r . When r , f is smooth. Our focus.Slide 11/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDifferential, Jacobian Matrix Differential of f in x: unique linear map (if exists) dx f : Rn Rqs.t.f (x h) f (x) dx f (h) o(h).Slide 12/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDifferential, Jacobian Matrix Differential of f in x: unique linear map (if exists) dx f : Rn Rqs.t.f (x h) f (x) dx f (h) o(h). Jacobian matrix of f : matrix q n of partial derivatives of f : y1 x1 (x) Jx f Slide 12/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22. yq x1 (x) y1 xn (x). yq xn (x)

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDifferential, Jacobian Matrix Differential of f in x: unique linear map (if exists) dx f : Rn Rqs.t.f (x h) f (x) dx f (h) o(h). Jacobian matrix of f : matrix q n of partial derivatives of f : y1 x1 (x) Jx f . yq x1 (x) y1 xn (x). yq xn (x) What is the meaning of the Jacobian? The differential? How dothey differ?Slide 12/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDiffeomorphism When n q: If f is 1-1, f and f 1 both C r f is a C r -diffeomorphism. Smooth diffeomorphisms are simply referred to as adiffeomorphisms.Slide 13/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDiffeomorphism When n q: If f is 1-1, f and f 1 both C r f is a C r -diffeomorphism. Smooth diffeomorphisms are simply referred to as adiffeomorphisms. Inverse Function Theorem: f diffeomorphism det(Jx f ) 6 0. det(Jx f ) 6 0 f local diffeomorphism in a neighborhood of x.Slide 13/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDiffeomorphism When n q: If f is 1-1, f and f 1 both C r f is a C r -diffeomorphism. Smooth diffeomorphisms are simply referred to as adiffeomorphisms. Inverse Function Theorem: f diffeomorphism det(Jx f ) 6 0. det(Jx f ) 6 0 f local diffeomorphism in a neighborhood of x. What is the meaning of Jx f ? Of det(Jx f ) 6 0?Slide 13/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDiffeomorphism f may be a local diffeomorphism everywhere but fail to be aglobal diffeomorphism. Examples: Complex exponential:f : R2 \0 R2 ,(x, y ) (ex cos(y ), ex sin(y )).Recall its inverse (the complex log) has infinitely many branches.”Complex log” by Jan Homann; Color encoding image comment author Hal Lane, September 28, 2009 - Own work. Thismathematical image was created with Mathematica. Licensed under Public domain via Wikimedia Commons http://commons.wikimedia.org/wiki/File:Complex log.jpg#mediaviewer/File:Complex log.jpgSlide 14/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDiffeomorphism f may be a local diffeomorphism everywhere but fail to be aglobal diffeomorphism. Examples: Complex exponential:f : R2 \0 R2 ,(x, y ) (ex cos(y ), ex sin(y )).Recall its inverse (the complex log) has infinitely many branches. If f is 1-1 and a local diffeomorphism everywhere, it is a globaldiffeomorphism.”Complex log” by Jan Homann; Color encoding image comment author Hal Lane, September 28, 2009 - Own work. Thismathematical image was created with Mathematica. Licensed under Public domain via Wikimedia Commons http://commons.wikimedia.org/wiki/File:Complex log.jpg#mediaviewer/File:Complex log.jpgSlide 14/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENDiffeomorphism f may be a local diffeomorphism everywhere but fail to be aglobal diffeomorphism. Examples: Complex exponential:f : R2 \0 R2 ,(x, y ) (ex cos(y ), ex sin(y )).Recall its inverse (the complex log) has infinitely many branches. If f is 1-1 and a local diffeomorphism everywhere, it is a globaldiffeomorphism. What is the intuitive meaning of a diffeomorphism?”Complex log” by Jan Homann; Color encoding image comment author Hal Lane, September 28, 2009 - Own work. Thismathematical image was created with Mathematica. Licensed under Public domain via Wikimedia Commons http://commons.wikimedia.org/wiki/File:Complex log.jpg#mediaviewer/File:Complex log.jpgSlide 14/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENOutline1MotivationNonlinearityRecall: Calculus in Rn2Differential GeometrySmooth manifoldsBuilding ManifoldsTangent SpaceVector fieldsDifferential of smooth map3Riemannian metricsIntroduction to Riemannian metricsRecall: Inner ProductsRiemannian metricsInvariance of the Fisher information metricA first take on the geodesic distance metricA first take on curvatureSlide 15/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22UNIVERSITY OF COPENHAGEN

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENBack to smooth manifoldsDefinitionA smooth manifold is a pair (M, A) where M is a set A is a family of one-to-one global charts ϕ : U M from someopen subset U Uϕ Rm for M, for any two charts ϕ : U Rm and ψ : V Rm , theircorresponding change of variables is a smooth diffeomorphismψ 1 ϕ : U V Rm .Slide 16/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENBack to smooth manifoldsDefinitionA smooth manifold is a pair (M, A) where M is a set A is a family of one-to-one global charts ϕ : U M from someopen subset U Uϕ Rm for M, for any two charts ϕ : U Rm and ψ : V Rm , theircorresponding change of variables is a smooth diffeomorphismψ 1 ϕ : U V Rm . What are the implications of inheriting structure through A?Slide 16/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENBack to smooth manifolds ϕ and ψ are parametrizations of MSlide 16/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22UNIVERSITY OF COPENHAGEN

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENBack to smooth manifolds ϕ and ψ are parametrizations of M Set ϕj (P) (y 1 (P), . . . , y n (P)), thenϕj ϕ 1i (x1 , . . . , xm ) (y1 , . . . , ym ) k and the m m Jacobian matrices yare invertible. x hk ,hSlide 16/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENExample: Euclidean space The Euclidean space Rn is a manifold: take ϕ Id as globalcoordinate system!Slide 17/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENExample: Smooth surfaces Smooth surfaces in Rn that are the image of a smooth mapf : R2 Rn . A global coordinate system given by fSlide 18/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENExample: Symmetric Positive Definite Matrices P(n) GLn consists of all symmetric n n matrices A thatsatisfyxAx T 0 for any x Rn ,(positive definite – PD – matrices) P(n) the set of covariance matrices on Rn P(3) the set of (diffusion) tensors on R32 Global chart: P(n) is an open, convex subset of R(n n)/2 A, B P(n) aA bB P(n) for all a, b 0 so P(n) is a convex2cone in R(n n)/2 .Middle figure from Fillard et al., A Riemannian Framework for the Processing of Tensor-Valued Images, LNCS 3753, 2005, pp112-123. Rightmost figure from Fletcher, Joshi, Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors,CVAMIA04Slide 19/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENExample: Space of Gaussian distributions The space of n-dimensional Gaussian distributions is a smoothmanifold Global chart: (µ, Σ) Rn P(n).Slide 20/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENExample: Space of 1-dimensional Gaussiandistributions The space of 1-dimensional Gaussian distributions isparametrized by (µ, σ) R R , mean µ, standard deviation σ Also parametrized by (µ, σ 2 ) R R , mean µ, variance σ 2 Smooth reparametrization ψ 1 ϕSlide 21/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENIn general: Manifolds requiring multiple chartsThe sphere S 2 {(x, y , z), x 2 y 2 z 2 1}For instance the projection from North Pole, given, for a pointP (x, y , z) 6 N of the sphere, by xyϕN (P) ,1 z 1 zis a (large) local coordinate system (around the south pole).Slide 22/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENIn general: Manifolds requiring multiple chartsThe sphere S 2 {(x, y , z), x 2 y 2 z 2 1}For instance the projection from North Pole, given, for a pointP (x, y , z) 6 N of the sphere, by xyϕN (P) ,1 z 1 zis a (large) local coordinate system (around the south pole).In these cases, we also require the charts to overlap ”nicely”Slide 22/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENIn general: Manifolds requiring multiple chartsThe Moebius strip1 1u [0, 2π], v [ , ]2 2 cos(u) 1 21 v cos( u2 ) sin(u) 1 1 v cos( u ) 221u2 v sin( 2 )Slide 22/57 — Aasa Feragen and François Lauze — Differential Geometry — September 22The 2D-torus(u, v ) [0, 2π]2 , R r 0 cos(u) (R r cos(v )) sin(u) (R r cos(v )) r sin(v )

UNIVERSITY OF COPENHAGENUNIVERSITY OF COPENHAGENSmooth maps between manifolds f : M N is smooth if

Differential and Riemannian Geometry 1.1 (Feragen) Crash course on Differential and Riemannian Geometry 3 (Lauze) Introduction to Information Geometry 3.1 (Amari) Information Geometry & Stochastic Optimization 1.1 (Hansen) Information Geometry & Stochastic Optimization in Discrete Domains 1.1 (M lago) 10 Cra

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