Approximating Covering and Minimum Enclosing Balls in Hyperbolic Geometry Frank Nielsen 1 etan Hadjeres 2 Ga¨ ´ Ecole Polytechnique 1 Sony Computer Science Laboratories, Inc 1 , 2 Conference on Geometric Science of Information � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 1
The Minimum Enclosing Ball problem Finding the Minimum Enclosing Ball (or the 1-center ) of a finite point set P = { p 1 , . . . , p n } in the metric space ( X , d X ( ., . )) consists in finding c ∈ X such that p ∈ P d X ( c ′ , p ) c = argmin c ′ ∈ X max Figure : A finite point set P and its minimum enclosing ball MEB ( P ) � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 2
The approximating minimum enclosing ball problem In a euclidean setting, this problem is ◮ well-defined: uniqueness of the center c ∗ and radius R ∗ of the MEB ◮ computationally intractable in high dimensions. We fix an ǫ > 0 and focus on the Approximate Minimum Enclosing Ball problem of finding an ǫ -approximation c ∈ X of MEB ( P ) such that d X ( c , p ) ≤ (1 + ǫ ) R ∗ ∀ p ∈ P . � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 3
The approximating minimum enclosing ball problem: prior work Approximate solution in the euclidean case are given by Badoiu and Clarkson’s algorithm [Badoiu and Clarkson, 2008]: ◮ Initialize center c 1 ∈ P ◮ Repeat ⌊ 1 /ǫ 2 ⌋ times the following update: c i +1 = c i + f i − c i i + 1 where f i ∈ P is the farthest point from c i . How to deal with point sets whose underlying geometry is not euclidean ? � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 4
The approximating minimum enclosing ball problem: prior work This algorithm has been generalized to ◮ dually flat manifolds [Nock and Nielsen, 2005] ◮ Riemannian manifolds [Arnaudon and Nielsen, 2013] Applying these results to hyperbolic geometry give the existence and uniqueness of MEB ( P ), but ◮ give no explicit bounds on the number of iterations ◮ assume that we are able to precisely cut geodesics. � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 5
The approximating minimum enclosing ball problem: our contribution We analyze the case of point sets whose underlying geometry is hyperbolic. Using a closed-form formula to compute geodesic α -midpoints, we obtain ◮ a intrinsic (1 + ǫ )-approximation algorithm to the approximate minimum enclosing ball problem ◮ a O (1 /ǫ 2 ) convergence time guarantee ◮ a one-class clustering algorithm for specific subfamilies of normal distributions using their Fisher information metric � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 6
Model of d -dimensional hyperbolic geometry: The Poincar´ e ball model e ball model ( B d , ρ ( ., . )) consists in the open unit ball The Poincar´ B d = { x ∈ R d : � x � < 1 } together with the hyperbolic distance 2 � p − q � 2 � � ∀ p , q ∈ B d . ρ ( p , q ) = arcosh 1 + , (1 − � p � 2 ) (1 − � q � 2 ) This distance induces on the metric space ( B d , ρ ) a Riemannian structure. � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 7
Geodesics in the Poincar´ e ball model Shorter paths between two points (geodesics) are exactly ◮ straight (euclidean) lines passing through the origin ◮ circle arcs orthogonal to the unit sphere Figure : “Straight” lines in the Poincar´ e ball model � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 8
Circles in the Poincar´ e ball model Circles in the Poincar´ e ball model ◮ look like euclidean circles ◮ but with different center Figure : Difference between euclidean MEB (in blue) and hyperbolic MEB (in red) for the set of blue points in hyperbolic Poincar´ e disk (in black). The red cross is the hyperbolic center of the red circle while the pink one is its euclidean center. � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 9
Translations in the Poincar´ e ball model � x � 2 + 2 � x , p � + 1 1 − � p � 2 � � � � x + p T p ( x ) = � p � 2 � x � 2 + 2 � x , p � + 1 Figure : Tiling of the hyperbolic plane by squares � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 10
Closed-form formula for computing α -midpoints A point m is the α -midpoint p # α q of two points p , q for α ∈ [0 , 1] if ◮ m belongs to the geodesic joining the two points p , q ◮ m verifies ρ ( p , m α ) = αρ ( p , q ) . � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 11
Closed-form formula for computing α -midpoints A point m is the α -midpoint p # α q of two points p , q for α ∈ [0 , 1] if ◮ m belongs to the geodesic joining the two points p , q ◮ m verifies ρ ( p , m α ) = αρ ( p , q ) . For the special case p = (0 , . . . , 0) , q = ( x q , 0 , . . . , 0), we have p # α q := ( x α , 0 , . . . , 0) with � α � x α = c α, q − 1 � � 1 + x q c α, q := e αρ ( p , q ) c α, q + 1 , = . where 1 − x q � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 11
Closed-form formula for computing α -midpoints Noting that ∀ p , q ∈ B d p # α q = T p ( T − p ( p ) # α T − p ( q )) we obtain ◮ a closed-form formula for computing p # α q ◮ how to compute p # α q in linear time O ( d ) ◮ that these transformations are exact. � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 12
(1+ ǫ )-approximation of an hyperbolic enclosing ball of fixed radius For a fixed radius r > R ∗ , we can find c ∈ B d such that ρ ( c , P ) ≤ (1 + ǫ ) r ∀ p ∈ P with Algorithm 1: (1 + ǫ )-approximation of EHB ( P , r ) 1: c 0 := p 1 2: t := 0 3: while ∃ p ∈ P such that p / ∈ B ( c t , (1 + ǫ ) r ) do let p ∈ P be such a point 4: α := ρ ( c t , p ) − r 5: ρ ( c t , p ) c t +1 := c t # α p 6: t := t+1 7: 8: end while 9: return c t � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 13
Idea of the proof p t By the hyperbolic law of r ′ ≤ r r cosines : ch ( ρ t ) ≥ ch ( h ) ch ( ρ t +1 ) ρ t +1 θ c t +1 c ∗ ch ( ρ 1 ) ≥ ch ( h ) T ≥ ch ( ǫ r ) T . θ ′ h > ǫ r ρ t c t Figure : Update of c t � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 14
(1+ ǫ )-approximation of an hyperbolic enclosing ball of fixed radius The EHB ( P , r ) algorithm is a O (1 /ǫ 2 )-time algorithm which returns ◮ the center of a hyperbolic enclosing ball with radius (1 + ǫ ) r ◮ in less than 4 /ǫ 2 iterations. � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 15
(1+ ǫ )-approximation of an hyperbolic enclosing ball of fixed radius The EHB ( P , r ) algorithm is a O (1 /ǫ 2 )-time algorithm which returns ◮ the center of a hyperbolic enclosing ball with radius (1 + ǫ ) r ◮ in less than 4 /ǫ 2 iterations. Our error with the true MEHB center c ∗ verifies � ch ((1 + ǫ ) r ) � ρ ( c , c ∗ ) ≤ arcosh ch ( R ∗ ) � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 15
(1 + ǫ + ǫ 2 / 4)-approximation of MEHB ( P ) In fact, as R ∗ is unknown in general, the EHB algorithm returns for any r : ◮ an (1 + ǫ )-approximation of EHB ( P ) if r ≥ R ∗ ◮ the fact that r < R ∗ if the result obtained after more than 4 /ǫ 2 iterations is not good enough. � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 16
(1 + ǫ + ǫ 2 / 4)-approximation of MEHB ( P ) In fact, as R ∗ is unknown in general, the EHB algorithm returns for any r : ◮ an (1 + ǫ )-approximation of EHB ( P ) if r ≥ R ∗ ◮ the fact that r < R ∗ if the result obtained after more than 4 /ǫ 2 iterations is not good enough. This suggests to implement a dichotomic search in order to compute an approximation of the minimal hyperbolic enclosing ball . We obtain ◮ a O (1 + ǫ + ǫ 2 / 4)-approximation of MEHB ( P ) � N � 1 ◮ in O �� ǫ 2 log iterations. ǫ � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 16
(1 + ǫ + ǫ 2 / 4)-approximation of MEHB ( P ) algorithm Algorithm 2: (1 + ǫ )-approximation of MEHB ( P ) 1: c := p 1 2: r max := ρ ( c , P ); r min = r max 2 ; t max := + ∞ 3: r := r max ; 4: repeat P , r , ǫ � � c temp := Alg1 , interrupt if t > t max in Alg1 5: 2 if call of Alg1 has been interrupted then 6: r min := r 7: else 8: r max := r ; c := c temp 9: end if 10: dr := r max − r min ; r := r min + dr ; 11: 2 t max := log(ch(1+ ǫ/ 2) r ) − log(ch( r min )) log(ch( r ǫ/ 2)) 12: until 2 dr < r min ǫ 2 13: return c � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 17
Experimental results ◮ The number of iterations does not depend on d . Figure : Number of α -midpoint calculations as a function of ǫ in logarithmic scale for different values of d . � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 18
Experimental results ◮ The running time is approximately O ( dn ǫ 2 ) (vertical translation in logarithmic scale). Figure : execution time as a function of ǫ in logarithmic scale for different values of d . � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 19
Applications Hyperbolic geometry arises when considering certain subfamilies of multivariate normal distributions. For instance, the following subfamilies µ, σ 2 I n ◮ N � � of n -variate normal distributions with scalar covariance matrix ( I n is the n × n identity matrix), σ 2 1 , . . . , σ 2 ◮ N � � �� µ, diag of n -variate normal distributions with n diagonal covariance matrix ◮ N ( µ 0 , Σ) of d -variate normal distributions with fixed mean µ 0 and arbitrary positive definite covariance matrix Σ are statistical manifolds whose Fisher information metric is hyperbolic. � 2015 Frank Nielsen - Ga¨ c etan Hadjeres 20
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