On the Smallest Enclosing Riemannian Balls — On Approximating the Riemannian 1-Center — http://www.sonycsl.co.jp/person/nielsen/infogeo/RiemannMinimax/ Marc Arnaudon 1 Frank Nielsen 2 1 Universit´ e de Bordeaux, France 2 ´ Ecole Polytechnique & Sony CSL e-mail: Frank.Nielsen@acm.org Computational Geometry 46(1): 93-104 (2013) arXiv 1101.4718 � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 1/39
Introduction: Euclidean Smallest Enclosing Balls Given d -dimensional P = { p 1 , ..., p n } , find the “smallest” (with respect to the volume ≡ radius ≡ inclusion) ball B = Ball ( c , r ) fully covering P : c ∗ = min n max i =1 � c − p i � . c ∈ R d ◮ unique Euclidean circumcenter c ∗ , SEB [19]. ◮ optimization problem non-differentiable [10] c ∗ lie on the farthest Voronoi diagram � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 2/39
Euclidean smallest enclosing balls (SEBs) ◮ 1857 : d = 2, Smallest Enclosing Ball? of P = { p 1 , ..., p n } (Sylvester [16]) ◮ Randomized expected linear time algorithm [19, 5] in fixed dimension (but hidden constant exponential in d ) ◮ Core-set [3] approximation: (1 + ǫ )-approximation in O ( dn ǫ 2 )-time in arbitrary dimension, O ( dn ǫ 4 . 5 log 1 1 ǫ + ǫ ) [7] ◮ Many other algorithms and heuristics [14, 9, 17], etc. SEB also known as Minimum Enclosing Ball (MEB), minimax center, 1-center, bounding (hyper)sphere, etc. → Applications in computer graphics (collision detection with ball cover proxies [15]), in machine learning (Core Vector Machines [18]), etc. � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 3/39
Optimization and core-sets [3] Let c ( P ) denote the circumcenter of the SEB and r ( P ) its radius Given ǫ > 0, ǫ -core-set C ⊂ P , such that P ⊆ Ball ( c ( C ) , (1 + ǫ ) r ( C )) ⇔ Expanding SEB ( C ) by 1 + ǫ fully covers P Core-set of optimal size ⌈ 1 ǫ ⌉ , independent of the dimension d , and n . Note that combinatorial basis for SEB is from 2 to d + 1 [19]. → Core-sets find many applications for problems in large-dimensions. � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 4/39
Euclidean SEBs from core-sets [2] B˘ adoiu-Clarkson algorithm based on core-sets [2, 3]: BCA : ◮ Initialize the center c 1 ∈ P = { p 1 , ..., p n } , and ◮ Iteratively update the current center using the rule c i +1 ← c i + f i − c i i + 1 where f i denotes the farthest point of P to c i : s = arg max n f i = p s , j =1 � c i − p j � ⇒ gradient-descent method ⇒ (1 + ǫ )-approximation after ⌈ 1 ǫ 2 ⌉ iterations: O ( dn ǫ 2 ) time ⇒ Core-set: f 1 , ..., f l with l = ⌈ 1 ǫ 2 ⌉ � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 5/39
Euclidean SEBs from core-sets: Rewriting with # a # t b : point (1 − t ) a + tb = a + t ( b − a ) on the line segment [ ab ]. D ( x , y ) = � x − y � 2 , D ( x , P ) = min y ∈P D ( x , y ) Algorithm 1: BCA ( P , l ). c 1 ← choose randomly a point in P ; for i = 2 to l − 1 do // farthest point from c i s i ← arg max n j =1 D ( c i , p j ); walk on the segment [ c i , p s i ] // update the center: c i +1 ← c i # 1 i +1 p s i ; end // Return the SEB approximation return Ball ( c l , r 2 l = D ( c l , P )) ; ⇒ (1 + ǫ )-approximation after l = ⌈ 1 ǫ 2 ⌉ iterations. � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 6/39
Bregman divergences (incl. squared Euclidean distance) SEB extended to Bregman divergences B F ( · : · ) [13] B F ( c : x ) = F ( c ) − F ( x ) − � c − x , ∇ F ( x ) � , B F ( c : X ) = min x ∈ X B F ( c : x ) F H ′ q ˆ p B F ( p, q ) = H q − H ′ q H q ˆ q p q ⇒ Bregman divergence = remainder of a first order Taylor expansion. � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 7/39
Smallest enclosing Bregman ball [13] F ∗ = convex conjugate of F with ( ∇ F ) − 1 = ∇ F ∗ Algorithm 2: MBC ( P , l ). // Create the gradient point set ( η -coordinates) P ′ ← {∇ F ( p ) : p ∈ P} ; g ← BCA ( P ′ , l ); return Ball ( c l = ∇ F − 1 ( c ( g )) , r l = B F ( c l : P )) ; Guaranteed approximation algorithm with approximation factor 1 depending on min x ∈X �∇ 2 F ( x ) � , ... but poor in practice (1 + ǫ ) 2 r ′∗ ∀ s , S F ( x ; ∇ F − 1 ( c ( g ))) ≤ min x ∈X �∇ 2 F ( x ) � with S F ( c ; x ) = B F ( c : x ) + B F ( x : c ) � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 8/39
Smallest enclosing Bregman ball [13] A better approximation in practice ... Algorithm 3: BBCA ( P , l ). c 1 ← choose randomly a point in P ; for i = 2 to l − 1 do // farthest point from c i wrt. B F s i ← arg max n j =1 B F ( c i : p j ); walk on the η -segment // update the center: [ c i , p s i ] η c i +1 ← ∇ F − 1 ( ∇ F ( c i )# 1 i +1 ∇ F ( p s i )) ; end // Return the SEBB approximation return Ball ( c l , r l = B F ( c l : X )) ; θ -, η -geodesic segments in dually flat geometry. � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 9/39
Basics of Riemannian geometry ◮ ( M , g ): Riemannian manifold ◮ �· , ·� , Riemannian metric tensor g : definite positive bilinear form on each tangent space T x M (depends smoothly on x ) ◮ � · � x : � u � = � u , u � 1 / 2 : Associated norm in T x M ◮ ρ ( x , y ): metric distance between two points on the manifold M (length space) �� 1 � ϕ ∈ C 1 ([0 , 1] , M ) , ρ ( x , y ) = inf � ˙ ϕ ( t ) � d t , ϕ (0) = x , ϕ (1) = y 0 Parallel transport wrt. Levi-Civita metric connection ∇ : ∇ g = 0. � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 10/39
Basics of Riemannian geometry: Exponential map ◮ Local map from the tangent space T x M to the manifold defined with geodesics (wrt ∇ ). ∀ x ∈ M , D ( x ) ⊂ T x M : D ( x ) = { v ∈ T x M : γ v (1) is defined } with γ v maximal (i.e., largest domain) geodesic with γ v (0) = x and γ ′ v (0) = v . ◮ Exponential map : exp x ( · ) : D ( x ) ⊆ T x M → M exp x ( v ) = γ v (1) D is star-shaped . � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 11/39
Basics of Riemannian geometry: Geodesics ◮ Geodesic : smooth path which locally minimizes the distance between two points. (In general such a curve does not minimize it globally.) ◮ Given a vector v ∈ T x M with base point x , there is a unique geodesic started at x with speed v at time 0: t �→ exp x ( tv ) or t �→ γ t ( v ). ◮ Geodesic on [ a , b ] is minimal if its length is less or equal to others. For complete M (i.e., exp x ( v )), taking x , y ∈ M , there exists a minimal geodesic from x to y in time 1. γ · ( x , y ) : [0 , 1] → M , t �→ γ t ( x , y ) with the conditions γ 0 ( x , y ) = x and γ 1 ( x , y ) = y . ◮ U ⊆ M is convex if for any x , y ∈ U , there exists a unique minimal geodesic γ · ( x , y ) in M from x to y . Geodesic fully lies in U and depends smoothly on x , y , t . � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 12/39
Basics of Riemannian geometry: Geodesics ◮ Geodesic γ ( x , y ): locally minimizing curves linking x to y ◮ Speed vector γ ′ ( t ) parallel along γ : D γ ′ ( t ) = ∇ γ ′ ( t ) γ ′ ( t ) = 0 d t ◮ When manifold M embedded in R d , acceleration is normal to tangent plane: γ ′′ ( t ) ⊥ T γ ( t ) M ◮ � γ ′ ( t ) � = c , a constant (say, unit). ⇒ Parameterization of curves with constant speed... � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 13/39
Basics of Riemannian geometry: Geodesics Constant speed geodesic γ ( t ) so that γ (0) = x and γ ( ρ ( x , y )) = y (constant speed 1, the unit of length). x # t y = m = γ ( t ) : ρ ( x , m ) = t × ρ ( x , y ) For example, in the Euclidean space: x # t y = (1 − t ) x + ty = x + t ( y − x ) = m ρ E ( x , m ) = � t ( y − x ) � = t � y − x � = t × ρ ( x , y ) , t ∈ [0 , 1] ⇒ m interpreted as a mean (barycenter) between x and y . � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 14/39
Basics of Riemannian geometry: Injectivity radius Diffeomorphism from the tangent space to the manifold ◮ Injectivity radius inj ( M ): largest r > 0 such that for all x ∈ M , the map exp x ( · ) restricted to the open ball in T x M with radius r is an embedding. ◮ Global injectivity radius : infimum of the injectivity radius over all points of the manifold. � 2013-14 Frank Nielsen, ´ c Ecole Polytechnique & Sony Computer Science Laboratories 15/39
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