Lecture II. The reach of a manifold Algebraic Geometry with a view - PowerPoint PPT Presentation

KTH ROYAL INSTITUTE OF TECHNOLOGY Lecture II. The reach of a manifold Algebraic Geometry with a view towards applications Sandra Di Rocco, ICTP Trieste

Plan for this course ◮ Lecture I: Algebraic modelling (Kinematics) ◮ Lecture II: Sampling algebraic varieties: the reach. ◮ Lecture III: Projective embeddings and Polar classes (classical theory) ◮ Lecture IV: The Euclidian Distance Degree (closest point) ◮ Lecture V: Bottle Neck degree from classical geometry (back to sampling) 2/24

Main goals ◮ Definition of reach. ◮ Sampling. ◮ Recovering the topological signature. 3/24

The reach of a manifold References: ◮ Estimating the reach of a Manifold. E. Aamari, J. Kim, F . Chazal, B. Michel, A. Rinaldo, et al.. Electronic journal of Statistics, Vol. 13, 2019 1359-1399. ◮ Computing the homology of basic semialgebraic sets in weak exponential time P . Bürgisser, F . Cucker and P . Lairez. Journal of the ACM 66(1), 2019 . ◮ Learning algebraic varieties from samples . P . Breiding, S. Kalisnik, B. Sturmfels and M. Weinstein. Revista Matemática Complutense. Vol. 31, 3, 2018 , pp 545-593. ◮ Sampling Algebraic Varieties . DR, D. Eklund, O. Gävfert. In progress. 4/24

(Real) Sampling 5/24

(Real) Sampling 5/24

(Real) Sampling X ⊂ R N , I X ⇒ Cloud data on X ⇒ (CW) Complex ⇒ Invariants of X 5/24

(Real) Sampling X ⊂ R N , I X ⇒ Cloud data on X ⇒ (CW) Complex ⇒ Invariants of X Two important steps 5/24

(Real) Sampling X ⊂ R N , I X ⇒ Cloud data on X ⇒ (CW) Complex ⇒ Invariants of X Two important steps ◮ density 5/24

(Real) Sampling X ⊂ R N , I X ⇒ Cloud data on X ⇒ (CW) Complex ⇒ Invariants of X Two important steps ◮ density ◮ complex 5/24

Application: Variety Sampling ◮ Consider a compact submanifold M ⊆ R n . 6/24

Application: Variety Sampling ◮ Consider a compact submanifold M ⊆ R n . ◮ And a point cloud E ⊂ M . 6/24

Application: Variety Sampling ◮ Consider a compact submanifold M ⊆ R n . ◮ And a point cloud E ⊂ M . ◮ Put growing balls around the points E . 6/24

Application: Variety Sampling ◮ Consider a compact submanifold M ⊆ R n . ◮ And a point cloud E ⊂ M . ◮ Put growing balls around the points E . 6/24

Application: Variety Sampling ◮ Consider a compact submanifold M ⊆ R n . ◮ And a point cloud E ⊂ M . ◮ Put growing balls around the points E . ◮ Geometry of M captured by union of balls for ball sizes in certain interval. 6/24

Application: Variety Sampling ◮ Consider a compact submanifold M ⊆ R n . ◮ And a point cloud E ⊂ M . ◮ Put growing balls around the points E . ◮ Geometry of M captured by union of balls for ball sizes in certain interval. 6/24

Application: Variety Sampling ◮ Consider a compact submanifold M ⊆ R n . ◮ And a point cloud E ⊂ M . ◮ Put growing balls around the points E . ◮ Geometry of M captured by union of balls for ball sizes in certain interval. 6/24

Application: Variety Sampling ◮ Consider a compact submanifold M ⊆ R n . ◮ And a point cloud E ⊂ M . ◮ Put growing balls around the points E . ◮ Geometry of M captured by union of balls for ball sizes in certain interval. 6/24

Application: Variety Sampling 7/24

Application: Variety Sampling ◮ Consider a growing tubular neighborhood of M . 7/24

Application: Variety Sampling ◮ Consider a growing tubular neighborhood of M . ◮ At some point it becomes singular ( M � = { p } ). 7/24

Application: Variety Sampling ◮ Consider a growing tubular neighborhood of M . ◮ At some point it becomes singular ( M � = { p } ). ◮ Half the black distance is called the reach of M . ◮ And the line is a bottleneck . 7/24

Set up Consider R n , endowed with the euclidean inner product < x , y > = � x i y i . Let X ⊂ R N be a smooth compact algebraic variety defined by an ideal I = ( f 1 , ..., f k ) ⊂ R [ x 1 , ..., x n ] . Let p ∈ X and consider the Jacobian matrix J ( f 1 , ..., f k ) = ( ∂ f i ) i , j ∂ x j evaluated at p . 8/24

◮ Since X is smooth, the rows of the Jacobian matrix span an ( n − d ) -dimensional linear subspace of R n , where d is the local dimension of X at p . ◮ This subspace translated to the point p is called the normal space of X at p , and is denoted N p ( X ) (fibers of the Normal Bundle ). 9/24

10/24

For p ∈ R n , Consider the distance function d X ( p ) = min x ∈ X || x − p || and π X ( p ) = { x ∈ X | || x − p || = d X ( p ) } 10/24

For p ∈ R n , Consider the distance function d X ( p ) = min x ∈ X || x − p || and π X ( p ) = { x ∈ X | || x − p || = d X ( p ) } For r � 0 let X r the tubular neighborhood of X of radius r X r = { p ∈ R n | d X ( p ) < r } 10/24

For p ∈ R n , Consider the distance function d X ( p ) = min x ∈ X || x − p || and π X ( p ) = { x ∈ X | || x − p || = d X ( p ) } For r � 0 let X r the tubular neighborhood of X of radius r X r = { p ∈ R n | d X ( p ) < r } ∆ X = { p ∈ R n : π X ( p ) > 1 } , the medial axis M X = ∆ X 10/24

11/24

∆ X = { p ∈ R n : π X ( p ) > 1 } , the medial axis M X = ∆ X Picture: International Conference on Cyberworlds (CW’07). Henning Naß, Franz-Erich Wolter and Hannes Thielhelm.DOI:10.1109/CW.2007.55 11/24

The Reach 12/24

The Reach Define the reach of X as: τ X = inf x ∈ X , y ∈ ∆ X { d ( x , y ) } 12/24

The Reach Define the reach of X as: τ X = inf x ∈ X , y ∈ ∆ X { d ( x , y ) } Notice that for each point x ∈ X r where r < τ | π X ( p ) | = 1 . This point is called the closest point of p on X . 12/24

The Reach Define the reach of X as: τ X = inf x ∈ X , y ∈ ∆ X { d ( x , y ) } Notice that for each point x ∈ X r where r < τ | π X ( p ) | = 1 . This point is called the closest point of p on X . Facts: ◮ X is compact, thus τ X > 0 . ◮ dim X > 0 (not convex) τ X < ∞ . ◮ τ X is a combination of local and global estimates: 12/24

13/24

◮ Locally : ρ X is the minimal radius of curvature on X ( radius of curvature at x ∈ X is the reciprocal of the maximal curvature of a geodesic passing through x .) 13/24

◮ Locally : ρ X is the minimal radius of curvature on X ( radius of curvature at x ∈ X is the reciprocal of the maximal curvature of a geodesic passing through x .) ◮ Globally : bottlenecks : η X = 1 2 inf {|| x − y || : ( x , y ) ∈ X × X , x � = y , y ∈ N x X , x ∈ N y X } . 13/24

◮ Locally : ρ X is the minimal radius of curvature on X ( radius of curvature at x ∈ X is the reciprocal of the maximal curvature of a geodesic passing through x .) ◮ Globally : bottlenecks : η X = 1 2 inf {|| x − y || : ( x , y ) ∈ X × X , x � = y , y ∈ N x X , x ∈ N y X } . τ X = min { ρ X , η X } 13/24

From:Estimating the reach of a Manifold 14/24

From:Estimating the reach of a Manifold Picture: Estimating the reach of a Manifold. E. Aamari, J. Kim, F . Chazal, B. Michel, A. Rinaldo, et al. 14/24

Sample 15/24

Sample X ⊂ R N smooth, compact variety. Definition Let ε > 0 . A finite set of points E ⊂ X is called an ε - sample of X if for every x ∈ X there is a point e ∈ E such that IIx − eII < ε. 15/24

Sample X ⊂ R N smooth, compact variety. Definition Let ε > 0 . A finite set of points E ⊂ X is called an ε - sample of X if for every x ∈ X there is a point e ∈ E such that IIx − eII < ε. Definition ◮ Let q ∈ R N and let c ( q , ε ) ⊂ R N denote the closed ball centered at q and of radius ε. ◮ Consider � C ( ε, E ) = c ( q , ε ) . q ∈ E 15/24

Sample gives homology 16/24

Sample gives homology Theorem 2 sample of X . If ε < 1 Let ε > 0 and consider E an ε 2 τ for all p ∈ E then X ֒ → C ( E , ε ) is a homotopy equivalence. 16/24

Sample gives homology Theorem 2 sample of X . If ε < 1 Let ε > 0 and consider E an ε 2 τ for all p ∈ E then X ֒ → C ( E , ε ) is a homotopy equivalence. proof: ◮ π X : C ( E , ε ) → X is continuous. ◮ C ( E , ε ) × [ 0 , 1 ] → C ( E , ε ) defines as ( x , t ) �→ t π X ( x ) + ( 1 − t ) x is a deformation retract. To prove: t π X ( x ) + ( 1 − t ) x ∈ C ( E , ε ) . 16/24

One way to sample via Numerical AG 17/24

One way to sample via Numerical AG Let X ∈ R N be a variety of dimension d . Let T d ⊂ { 1 , 2 , . . . , N } be the set of unordered d -tuples. Let e 1 , . . . , e N be the standard basis. For t = ( t 1 , . . . , t d ) ∈ T d we let V t = Span ( e t i , . . . , e t d ) . For δ > 0 consider the grid G t ( δ ) = δ Z ∩ V t and the projection π t : R N → V t . Then � X ∩ π − 1 E δ = ( g ) t t ∈ T d , g ∈ G t is a finite sample (up to random perturbation). 17/24

One way to sample via Numerical AG 18/24

One way to sample via Numerical AG Theorem ε If 0 < δ < N and E δ � = ∅ , then E δ is ε sample of X . √ 18/24

One way to sample via Numerical AG Theorem ε If 0 < δ < N and E δ � = ∅ , then E δ is ε sample of X . √ Use numerical methods to construct sample: 18/24