Numerical computation of the homology of basic semialgebraic sets - - PowerPoint PPT Presentation

Numerical computation

• f the homology of basic semialgebraic sets

Pierre Lairez joint work with Peter Bürgisser and Felipe Cucker

Inria Saclay

TAGS 2018

Linking Topology to Algebraic Geometry and Statistics 22 February 2018, Leipzig

Basic semialgebraic sets

definition A basic semi algebraic set is the solution set of finitely many polynomial equation and inequations.

Picture : https://de.wikipedia.org/wiki/Steinmetz-Körper

1

Semialgebraic sets in applications

motion planning

• J. T. Schwartz, M. Sharir, “On the piano movers problem”

2

Complexity bounds in real algebraic geometry

symbolic algorithms

(basic) semialgebraic set defined by equations or inequalities of degree . polynomial time algorithm membership Decide if single exponential time algorithm — emptyness Decide if (Grigoriev, Vorobjov, Renegar) dimension Compute (Koiran) CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if single exponential time algorithm — emptyness Decide if (Grigoriev, Vorobjov, Renegar) dimension Compute (Koiran) CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if single exponential time algorithm — emptyness Decide if (Grigoriev, Vorobjov, Renegar) dimension Compute (Koiran) CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — emptyness Decide if (Grigoriev, Vorobjov, Renegar) dimension Compute (Koiran) CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if (Grigoriev, Vorobjov, Renegar) dimension Compute (Koiran) CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute (Koiran) CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dimW (Koiran) CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dimW (Koiran) #CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dimW (Koiran) #CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) b0,b1,b2,... Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dimW (Koiran) #CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) b0,b1,b2,... Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dimW (Koiran) #CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) b0,b1,b2,... Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — (sD)2O(n) homology Compute the homology groups of CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dimW (Koiran) #CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) b0,b1,b2,... Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — (sD)2O(n) homology Compute the homology groups of W CAD Compute the cylindrical algebraic decompositon (Collins)

3

Complexity bounds in real algebraic geometry

symbolic algorithms

W ⊆ Rn (basic) semialgebraic set defined by s equations or inequalities of degree D. polynomial time algorithm membership Decide if x ∈ W single exponential time algorithm — (sD)nO(1) emptyness Decide if W = ∅ (Grigoriev, Vorobjov, Renegar) dimension Compute dimW (Koiran) #CC Compute the number of connected components (Canny, Grigoriev, Vorobjov) b0,b1,b2,... Compute the first few Betti numbers (Basu) Euler Compute the Euler characteristic (Basu) double exponential algorithms — (sD)2O(n) homology Compute the homology groups of W CAD Compute the cylindrical algebraic decompositon (Collins)

3

Approaching sets by union of balls

numerical algorithms

• Homotopically equivalent to its nerve
• Combinatorial computation of the homology
• Tricky choice of the parameters:
• sufgiciently many points
• radius not too small
• radius not too large
• How to quantify “sufgiciently many”, “too small”

and “too large” in an algebraic setting?

• Can we derive algebraic complexity bounds for the

computation of the homology of semialgebraic sets?

4

Approaching sets by union of balls

numerical algorithms

• Homotopically equivalent to its nerve
• Combinatorial computation of the homology
• Tricky choice of the parameters:
• sufgiciently many points
• radius not too small
• radius not too large
• How to quantify “sufgiciently many”, “too small”

and “too large” in an algebraic setting?

• Can we derive algebraic complexity bounds for the

computation of the homology of semialgebraic sets?

4

Approaching sets by union of balls

numerical algorithms

• Homotopically equivalent to its nerve
• Combinatorial computation of the homology
• Tricky choice of the parameters:
• sufgiciently many points
• radius not too small
• radius not too large
• How to quantify “sufgiciently many”, “too small”

and “too large” in an algebraic setting?

• Can we derive algebraic complexity bounds for the

computation of the homology of semialgebraic sets?

4

Approaching sets by union of balls

numerical algorithms

• Homotopically equivalent to its nerve
• Combinatorial computation of the homology
• Tricky choice of the parameters:
• sufgiciently many points
• radius not too small
• radius not too large
• How to quantify “sufgiciently many”, “too small”

and “too large” in an algebraic setting?

• Can we derive algebraic complexity bounds for the

computation of the homology of semialgebraic sets?

4

Approaching sets by union of balls

numerical algorithms

• Homotopically equivalent to its nerve
• Combinatorial computation of the homology
• Tricky choice of the parameters:
• sufgiciently many points
• radius not too small
• radius not too large
• How to quantify “sufgiciently many”, “too small”

and “too large” in an algebraic setting?

• Can we derive algebraic complexity bounds for the

computation of the homology of semialgebraic sets?

4

Approaching sets by union of balls

numerical algorithms

• Homotopically equivalent to its nerve
• Combinatorial computation of the homology
• Tricky choice of the parameters:
• sufgiciently many points
• radius not too small
• radius not too large
• How to quantify “sufgiciently many”, “too small”

and “too large” in an algebraic setting?

• Can we derive algebraic complexity bounds for the

computation of the homology of semialgebraic sets?

4

Approaching sets by union of balls

numerical algorithms

• Homotopically equivalent to its nerve
• Combinatorial computation of the homology
• Tricky choice of the parameters:
• sufgiciently many points
• radius not too small
• radius not too large
• How to quantify “sufgiciently many”, “too small”

and “too large” in an algebraic setting?

• Can we derive algebraic complexity bounds for the

computation of the homology of semialgebraic sets?

4

Approaching sets by union of balls

numerical algorithms

• Homotopically equivalent to its nerve
• Combinatorial computation of the homology
• Tricky choice of the parameters:
• sufgiciently many points
• radius not too small
• radius not too large
• How to quantify “sufgiciently many”, “too small”

and “too large” in an algebraic setting?

• Can we derive algebraic complexity bounds for the

computation of the homology of semialgebraic sets?

4

A numerical algorithm for homology

arxiv:1706.07473

input W = { x ∈ Rn f1(x) = ··· = fq(x) = 0,g1(x) 0,...,gs(x) 0 } input space tuples of polynomial equations/inequalities

• f degree at most

. input size dimension of this space. condition number (to be defined later) main result One can compute with

• perations

probability measure Gaussian probability distribution probabilistic analysis cost with probabiliy cost with probabiliy . grid methods Initiated by Cucker, Krick, Malajovich, Shub, Smale, Wschebor

5

A numerical algorithm for homology

arxiv:1706.07473

input W = { x ∈ Rn f1(x) = ··· = fq(x) = 0,g1(x) 0,...,gs(x) 0 } input space H = tuples of s + q polynomial equations/inequalities

• f degree at most D.

input size dimension of this space. condition number (to be defined later) main result One can compute with

• perations

probability measure Gaussian probability distribution probabilistic analysis cost with probabiliy cost with probabiliy . grid methods Initiated by Cucker, Krick, Malajovich, Shub, Smale, Wschebor

5

A numerical algorithm for homology

arxiv:1706.07473

input W = { x ∈ Rn f1(x) = ··· = fq(x) = 0,g1(x) 0,...,gs(x) 0 } input space H = tuples of s + q polynomial equations/inequalities

• f degree at most D.

input size N = dimension of this space. condition number (to be defined later) main result One can compute with

• perations

probability measure Gaussian probability distribution probabilistic analysis cost with probabiliy cost with probabiliy . grid methods Initiated by Cucker, Krick, Malajovich, Shub, Smale, Wschebor

5

A numerical algorithm for homology

arxiv:1706.07473

input W = { x ∈ Rn f1(x) = ··· = fq(x) = 0,g1(x) 0,...,gs(x) 0 } input space H = tuples of s + q polynomial equations/inequalities

• f degree at most D.

input size N = dimension of this space. condition number κ∗ (to be defined later) main result One can compute with

• perations

probability measure Gaussian probability distribution probabilistic analysis cost with probabiliy cost with probabiliy . grid methods Initiated by Cucker, Krick, Malajovich, Shub, Smale, Wschebor

5

A numerical algorithm for homology

arxiv:1706.07473

input W = { x ∈ Rn f1(x) = ··· = fq(x) = 0,g1(x) 0,...,gs(x) 0 } input space H = tuples of s + q polynomial equations/inequalities

• f degree at most D.

input size N = dimension of this space. condition number κ∗ (to be defined later) main result One can compute H∗(W ) with (sDκ∗)n2+o(1) operations probability measure Gaussian probability distribution probabilistic analysis cost with probabiliy cost with probabiliy . grid methods Initiated by Cucker, Krick, Malajovich, Shub, Smale, Wschebor

5

A numerical algorithm for homology

arxiv:1706.07473

input W = { x ∈ Rn f1(x) = ··· = fq(x) = 0,g1(x) 0,...,gs(x) 0 } input space H = tuples of s + q polynomial equations/inequalities

• f degree at most D.

input size N = dimension of this space. condition number κ∗ (to be defined later) main result One can compute H∗(W ) with (sDκ∗)n2+o(1) operations probability measure Gaussian probability distribution probabilistic analysis cost with probabiliy cost with probabiliy . grid methods Initiated by Cucker, Krick, Malajovich, Shub, Smale, Wschebor

5

A numerical algorithm for homology

arxiv:1706.07473

input W = { x ∈ Rn f1(x) = ··· = fq(x) = 0,g1(x) 0,...,gs(x) 0 } input space H = tuples of s + q polynomial equations/inequalities

• f degree at most D.

input size N = dimension of this space. condition number κ∗ (to be defined later) main result One can compute H∗(W ) with (sDκ∗)n2+o(1) operations probability measure Gaussian probability distribution probabilistic analysis cost (sD)n3+o(1) with probabiliy 1−(sD)−n cost 2O(N 2) with probabiliy 1−2−N. grid methods Initiated by Cucker, Krick, Malajovich, Shub, Smale, Wschebor

5

A numerical algorithm for homology

arxiv:1706.07473

input W = { x ∈ Rn f1(x) = ··· = fq(x) = 0,g1(x) 0,...,gs(x) 0 } input space H = tuples of s + q polynomial equations/inequalities

• f degree at most D.

input size N = dimension of this space. condition number κ∗ (to be defined later) main result One can compute H∗(W ) with (sDκ∗)n2+o(1) operations probability measure Gaussian probability distribution probabilistic analysis cost (sD)n3+o(1) with probabiliy 1−(sD)−n cost 2O(N 2) with probabiliy 1−2−N. grid methods Initiated by Cucker, Krick, Malajovich, Shub, Smale, Wschebor

5

Condition number

Condition number for linear systems

problem How much the solution of a linear system Ax = b is afgected by a pertubation of b ? (Goldstine, von Neuman, Turing) distance to ill-posed set singular matrices (Eckart, Young, Mirsky) many analogues [e.g. Demmel] Is there a considition number for closed sets?

6

Condition number for linear systems

problem How much the solution of a linear system Ax = b is afgected by a pertubation of b ? ∥δx∥/∥δb∥ κ(A) = ∥A∥∥A−1∥ (Goldstine, von Neuman, Turing) distance to ill-posed set singular matrices (Eckart, Young, Mirsky) many analogues [e.g. Demmel] Is there a considition number for closed sets?

6

Condition number for linear systems

problem How much the solution of a linear system Ax = b is afgected by a pertubation of b ? ∥δx∥/∥δb∥ κ(A) = ∥A∥∥A−1∥ (Goldstine, von Neuman, Turing) distance to ill-posed set κ(A) = ∥A∥/dist(A,singular matrices) (Eckart, Young, Mirsky) many analogues [e.g. Demmel] Is there a considition number for closed sets?

6

Condition number for linear systems

problem How much the solution of a linear system Ax = b is afgected by a pertubation of b ? ∥δx∥/∥δb∥ κ(A) = ∥A∥∥A−1∥ (Goldstine, von Neuman, Turing) distance to ill-posed set κ(A) = ∥A∥/dist(A,singular matrices) (Eckart, Young, Mirsky) many analogues [e.g. Demmel] Is there a considition number for closed sets?

6

Condition number for linear systems

problem How much the solution of a linear system Ax = b is afgected by a pertubation of b ? ∥δx∥/∥δb∥ κ(A) = ∥A∥∥A−1∥ (Goldstine, von Neuman, Turing) distance to ill-posed set κ(A) = ∥A∥/dist(A,singular matrices) (Eckart, Young, Mirsky) many analogues [e.g. Demmel] Is there a considition number for closed sets?

6

Reach of a closed set

The reach of a set is its minimal distance to its medial axis.

https://en.wikipedia.org/wiki/Local_feature_size

7

Reach of a closed set

W a closed subset of Rn the reach is the largest real number such that (Federer)

8

Reach of a closed set

W a closed subset of Rn the reach τ(W ) is the largest real number such that d(x,W ) < τ(W ) ⇒ ∃!y ∈ W : d(x,W ) = ∥x − y∥. (Federer)

8

Reach of a closed set

W a closed subset of Rn the reach τ(W ) is the largest real number such that d(x,W ) < τ(W ) ⇒ ∃!y ∈ W : d(x,W ) = ∥x − y∥. (Federer)

8

Reach of a closed set

W a closed subset of Rn the reach τ(W ) is the largest real number such that d(x,W ) < τ(W ) ⇒ ∃!y ∈ W : d(x,W ) = ∥x − y∥. (Federer) τ(W ) = ∞ ∞ > τ(W ) > 0 τ(W ) = 0

8

The Niyogi-Smale-Weinberger theorem

W ⊆ Rn closed finite assumption

Hausdorfg

conclusion For any

Hausdorfg

,

9

The Niyogi-Smale-Weinberger theorem

W ⊆ Rn closed X ⊂ Rn finite assumption

Hausdorfg

conclusion For any

Hausdorfg

,

9

The Niyogi-Smale-Weinberger theorem

W ⊆ Rn closed X ⊂ Rn finite assumption 6distHausdorfg(X ,W ) < τ(W ) conclusion For any

Hausdorfg

,

9

The Niyogi-Smale-Weinberger theorem

W ⊆ Rn closed X ⊂ Rn finite assumption 6distHausdorfg(X ,W ) < τ(W ) conclusion For any δ ∈ ( 3distHausdorfg(X ,W ), 1

2τ(W )

) , ∪

x∈X

Bδ(x) ∼ = W.

9

The Niyogi-Smale-Weinberger theorem

W ⊆ Rn closed X ⊂ Rn finite assumption 6distHausdorfg(X ,W ) < τ(W ) conclusion For any δ ∈ ( 3distHausdorfg(X ,W ), 1

2τ(W )

) , ∪

x∈X

Bδ(x) ∼ = W.

9

An algebraic condition number

ill-posed problems

Non-transversal intersection of the boundaries Singularity in the boundary

10

An algebraic condition number

ill-posed problems

Non-transversal intersection of the boundaries Singularity in the boundary

10

An algebraic condition number

real spherical varieties

homogeneous setting X ⊂ Sn defined by homogeneous polynomial equations f1 = 0,..., fq = 0 (denoted F = 0) of degree at most D. singular solution is a singular solution if the jacobian matrix is not full-rank. ill-posed problems The system is ill-posed if it has a singular solution. condition number { ill-posed problems } .

11

An algebraic condition number

real spherical varieties

homogeneous setting X ⊂ Sn defined by homogeneous polynomial equations f1 = 0,..., fq = 0 (denoted F = 0) of degree at most D. singular solution x ∈ X is a singular solution if the jacobian matrix ( ∂fi/∂x j )

i,j is

not full-rank. ill-posed problems The system is ill-posed if it has a singular solution. condition number { ill-posed problems } .

11

An algebraic condition number

real spherical varieties

homogeneous setting X ⊂ Sn defined by homogeneous polynomial equations f1 = 0,..., fq = 0 (denoted F = 0) of degree at most D. singular solution x ∈ X is a singular solution if the jacobian matrix ( ∂fi/∂x j )

i,j is

not full-rank. ill-posed problems The system F = 0 is ill-posed if it has a singular solution. condition number { ill-posed problems } .

11

An algebraic condition number

real spherical varieties

homogeneous setting X ⊂ Sn defined by homogeneous polynomial equations f1 = 0,..., fq = 0 (denoted F = 0) of degree at most D. singular solution x ∈ X is a singular solution if the jacobian matrix ( ∂fi/∂x j )

i,j is

not full-rank. ill-posed problems The system F = 0 is ill-posed if it has a singular solution. condition number κ(F) = ∥F∥/dist(F,{ ill-posed problems }).

11

Geometry of ill-posedness

dimension, degree

What is the geometry of {ill-posed problem} ⊂ H ? codimension 1 degree example A cubic plane curve: . and the ill-posed set is given by the following degree 12 polynomial with 2040 monomials

12

Geometry of ill-posedness

dimension, degree

What is the geometry of {ill-posed problem} ⊂ H ? codimension 1 degree example A cubic plane curve: . and the ill-posed set is given by the following degree 12 polynomial with 2040 monomials

12

Geometry of ill-posedness

dimension, degree

What is the geometry of {ill-posed problem} ⊂ H ? codimension 1 degree n2nDn example A cubic plane curve: . and the ill-posed set is given by the following degree 12 polynomial with 2040 monomials

12

Geometry of ill-posedness

dimension, degree

What is the geometry of {ill-posed problem} ⊂ H ? codimension 1 degree n2nDn example A cubic plane curve: a0x3+a1x2+a2xy2+a3y3+a4x2+a5xy+a6y2+a7x+a8y+a9 = 0. dimH = 9 and the ill-posed set is given by the following degree 12 polynomial with 2040 monomials −19683a4

0a4 3a4 9+26244a4 0a3 3a6a8a3 9−5832a4 0a3 3a3 8a2 9−5832a4 0a2 3a3 6a3 9−7290a4 0a2 3a2 6a2 8a2 9+388

−1836a4

0a3a3 6a3 8a9+216a4 0a3a2 6a5 8−432a4 0a6 6a2 9+216a4 0a5 6a2 8a9−27a4 0a4 6a4 8+26244a3 0a1a2a3 3

+3888a3

0a1a2a3a3 6a3 9+4860a3 0a1a2a3a2 6a2 8a2 9−2592a3 0a1a2a3a6a4 8a9+288a3 0a1a2a3a6 8−129

−8748a3

0a1a3 3a5a8a3 9−8748a3 0a1a3 3a6a7a3 9+5832a3 0a1a3 3a7a2 8a2 9+5832a3 0a1a2 3a5a2 6a3 9+4860

+4860a3

0a1a2 3a2 6a7a8a2 9−5184a3 0a1a2 3a6a7a3 8a9+864a3 0a1a2 3a7a5 8−5184a3 0a1a3a5a3 6a8a2 9+183

+1836a3

0a1a3a3 6a7a2 8a9−360a3 0a1a3a2 6a7a4 8+864a3 0a1a5a5 6a2 9−360a3 0a1a5a4 6a2 8a9+36a3 0a1a 12

Distance to ill-posedness

theorem dist(F, {ill-posed}) ≃ min

x∈Sn

( 1 ∥dxF †∥2 +∥F(x)∥2 ) 1

2

• vanisihes at a singular root

(Cucker) is easily approximable.

13

Distance to ill-posedness

theorem dist(F, {ill-posed}) ≃ min

x∈Sn

( 1 ∥dxF †∥2 +∥F(x)∥2 ) 1

2

• vanisihes at a singular root

(Cucker)

⇝ κ(F) is easily approximable.

13

An algebraic condition number

spherical semialgebraic sets

homogeneous setting W ⊂ Sn defined by homogeneous polynomial equations F = 0 and inequalities G 0 of degree at most D. afgine spherical Homogenize and constrain . ill-posed problems is ill-posed some subsystem , with , is ill-posed. condition number . theorem { ill-posed problems } .

14

An algebraic condition number

spherical semialgebraic sets

homogeneous setting W ⊂ Sn defined by homogeneous polynomial equations F = 0 and inequalities G 0 of degree at most D. afgine → spherical Homogenize and constrain x0 > 0. ill-posed problems is ill-posed some subsystem , with , is ill-posed. condition number . theorem { ill-posed problems } .

14

An algebraic condition number

spherical semialgebraic sets

homogeneous setting W ⊂ Sn defined by homogeneous polynomial equations F = 0 and inequalities G 0 of degree at most D. afgine → spherical Homogenize and constrain x0 > 0. ill-posed problems W is ill-posed some subsystem F ∪ H, with H ⊆ G, is ill-posed. condition number . theorem { ill-posed problems } .

14

An algebraic condition number

spherical semialgebraic sets

homogeneous setting W ⊂ Sn defined by homogeneous polynomial equations F = 0 and inequalities G 0 of degree at most D. afgine → spherical Homogenize and constrain x0 > 0. ill-posed problems W is ill-posed some subsystem F ∪ H, with H ⊆ G, is ill-posed. condition number κ∗(F,G) = maxL⊆G κ(F ∪L). theorem { ill-posed problems } .

14

An algebraic condition number

spherical semialgebraic sets

homogeneous setting W ⊂ Sn defined by homogeneous polynomial equations F = 0 and inequalities G 0 of degree at most D. afgine → spherical Homogenize and constrain x0 > 0. ill-posed problems W is ill-posed some subsystem F ∪ H, with H ⊆ G, is ill-posed. condition number κ∗(F,G) = maxL⊆G κ(F ∪L). theorem κ∗(F,G) ∥F,G∥/dist((F,G),{ ill-posed problems }).

14

Reach and condition number

homogeneous setting W ⊂ Sn defined by homogeneous polynomial equations F = 0 and inequalities G 0 of degree at most D. theorem corollary finite. For any

Hausdorfg

,

15

Reach and condition number

homogeneous setting W ⊂ Sn defined by homogeneous polynomial equations F = 0 and inequalities G 0 of degree at most D. theorem D

3 2 τ(W )κ∗(F,G) 1

7

corollary finite. For any

Hausdorfg

,

15

Reach and condition number

homogeneous setting W ⊂ Sn defined by homogeneous polynomial equations F = 0 and inequalities G 0 of degree at most D. theorem D

3 2 τ(W )κ∗(F,G) 1

7

corollary X ⊂ Sn finite. For any δ ∈ ( 3distHausdorfg(X ,W ), ( 14D

3 2 κ∗(F,G)

)−1) , ∪

x∈X

Bδ(x) ∼ = W.

15

Sampling and thickening

Tentative algorithm

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } 1 Compute 2 Pick a

• grid
• n

. (That is, any point of is

• close to

.) 3 Compute

• utput The homology of

. correctness Niyogi-Smale-Weinberger theorem + estimate of . efgiciency How to check ?

16

Tentative algorithm

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } 1 Compute δ = ( 14D

3 2 κ∗(F,G)

)−1 2 Pick a

• grid
• n

. (That is, any point of is

• close to

.) 3 Compute

• utput The homology of

. correctness Niyogi-Smale-Weinberger theorem + estimate of . efgiciency How to check ?

16

Tentative algorithm

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } 1 Compute δ = ( 14D

3 2 κ∗(F,G)

)−1 2 Pick a 1

3δ-grid G on Sn.

(That is, any point of Sn is 1

3δ-close to G.)

3 Compute

• utput The homology of

. correctness Niyogi-Smale-Weinberger theorem + estimate of . efgiciency How to check ?

16

Tentative algorithm

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } 1 Compute δ = ( 14D

3 2 κ∗(F,G)

)−1 2 Pick a 1

3δ-grid G on Sn.

(That is, any point of Sn is 1

3δ-close to G.)

3 Compute X = { x ∈ G

• dist(x,W ) 1

3δ

}

• utput The homology of

. correctness Niyogi-Smale-Weinberger theorem + estimate of . efgiciency How to check ?

16

Tentative algorithm

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } 1 Compute δ = ( 14D

3 2 κ∗(F,G)

)−1 2 Pick a 1

3δ-grid G on Sn.

(That is, any point of Sn is 1

3δ-close to G.)

3 Compute X = { x ∈ G

• dist(x,W ) 1

3δ

}

• utput The homology of Bδ(X ).

correctness Niyogi-Smale-Weinberger theorem + estimate of . efgiciency How to check ?

16

Tentative algorithm

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } 1 Compute δ = ( 14D

3 2 κ∗(F,G)

)−1 2 Pick a 1

3δ-grid G on Sn.

(That is, any point of Sn is 1

3δ-close to G.)

3 Compute X = { x ∈ G

• dist(x,W ) 1

3δ

}

• utput The homology of Bδ(X ).

correctness Niyogi-Smale-Weinberger theorem + κ∗ estimate of τ(W ). efgiciency How to check ?

16

Tentative algorithm

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } 1 Compute δ = ( 14D

3 2 κ∗(F,G)

)−1 2 Pick a 1

3δ-grid G on Sn.

(That is, any point of Sn is 1

3δ-close to G.)

3 Compute X = { x ∈ G

• dist(x,W ) 1

3δ

}

• utput The homology of Bδ(X ).

correctness Niyogi-Smale-Weinberger theorem + κ∗ estimate of τ(W ). efgiciency How to check dist(x,W ) 1

3δ? 16

Easier sampling

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } , κ∗ = κ∗(F,G) thickening . theorem If then

interesting!

remark remark bounds the variations of under small pertubations of the equations: it is a genuine condition number idea Replace by (for a suitable ).

17

Easier sampling

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } , κ∗ = κ∗(F,G) thickening W (r) = { x ∈ Sn |fi(x)| r∥fi∥,g j(x) −r∥g j∥ } ⊇ W . theorem If then

interesting!

remark remark bounds the variations of under small pertubations of the equations: it is a genuine condition number idea Replace by (for a suitable ).

17

Easier sampling

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } , κ∗ = κ∗(F,G) thickening W (r) = { x ∈ Sn |fi(x)| r∥fi∥,g j(x) −r∥g j∥ } ⊇ W . theorem If r ( 13D

3 2 κ2

∗

) then Tube(W,D−1/2r) ⊂ W (r) ⊂ Tube(W,3κ∗r)

• interesting!

remark remark bounds the variations of under small pertubations of the equations: it is a genuine condition number idea Replace by (for a suitable ).

17

Easier sampling

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } , κ∗ = κ∗(F,G) thickening W (r) = { x ∈ Sn |fi(x)| r∥fi∥,g j(x) −r∥g j∥ } ⊇ W . theorem If r ( 13D

3 2 κ2

∗

) then Tube(W,D−1/2r) ⊂ W (r) ⊂ Tube(W,3κ∗r)

• interesting!

remark W (r) ̸= ∅ ⇒ W ̸= ∅ remark bounds the variations of under small pertubations of the equations: it is a genuine condition number idea Replace by (for a suitable ).

17

Easier sampling

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } , κ∗ = κ∗(F,G) thickening W (r) = { x ∈ Sn |fi(x)| r∥fi∥,g j(x) −r∥g j∥ } ⊇ W . theorem If r ( 13D

3 2 κ2

∗

) then Tube(W,D−1/2r) ⊂ W (r) ⊂ Tube(W,3κ∗r)

• interesting!

remark W (r) ̸= ∅ ⇒ W ̸= ∅ remark κ∗ bounds the variations of W under small pertubations of the equations: it is a genuine condition number idea Replace by (for a suitable ).

17

Easier sampling

input W = { x ∈ Sn | F(x) = 0,G(x) 0 } , κ∗ = κ∗(F,G) thickening W (r) = { x ∈ Sn |fi(x)| r∥fi∥,g j(x) −r∥g j∥ } ⊇ W . theorem If r ( 13D

3 2 κ2

∗

) then Tube(W,D−1/2r) ⊂ W (r) ⊂ Tube(W,3κ∗r)

• interesting!

remark W (r) ̸= ∅ ⇒ W ̸= ∅ remark κ∗ bounds the variations of W under small pertubations of the equations: it is a genuine condition number idea Replace dist(x,W ) 1

3δ by x ∈ W (r) (for a suitable r). 17

Covering algorithm

input A spherical semialgebraic set W = { x ∈ Sn | F(x) = 0,G(x) 0 } assumption κ∗(F,G) is finite.

• utput A finite set X ⊂ Sn and an ε > 0 such that Bε(X ) ∼

= W . algorithm function COVERING(F, G) r ← 1 repeat r ← r/2 Compute a r-grid Gr in Sn k∗ ← max{κ(F ∪L,x) | x ∈ Gr and L ⊆ G} until 71D

5 2 k2

∗r < 1

return the set X = Gr ∩W (D

1 2 r) and the real number ε = 5Dk∗r

end function

18

Complexity analysis

Condition-based analysis

computation of the covering (sDκ∗)n1+o(1) computation of the homology How big is ? worst case complexity unbounded average complexity unbounded ?!

19

Condition-based analysis

computation of the covering (sDκ∗)n1+o(1) computation of the homology #X O(n) = (sDκ∗)n2+o(1) How big is ? worst case complexity unbounded average complexity unbounded ?!

19

Condition-based analysis

computation of the covering (sDκ∗)n1+o(1) computation of the homology #X O(n) = (sDκ∗)n2+o(1) How big is κ∗? worst case complexity unbounded average complexity unbounded ?!

19

Condition-based analysis

computation of the covering (sDκ∗)n1+o(1) computation of the homology #X O(n) = (sDκ∗)n2+o(1) How big is κ∗? worst case complexity unbounded average complexity unbounded ?!

19

Condition-based analysis

computation of the covering (sDκ∗)n1+o(1) computation of the homology #X O(n) = (sDκ∗)n2+o(1) How big is κ∗? worst case complexity unbounded average complexity unbounded ?!

19

Weak complexity bounds

If the average case is unbounded, is the algorithm slow? example The power method for computing the dominant eigenpair of a real symmetric matrix (compute for large ). Unbounded average case (Kostlan). Used in practice with success. weak complexity cost with probability . (Amelunxen, Lotz)

20

Weak complexity bounds

If the average case is unbounded, is the algorithm slow? example The power method for computing the dominant eigenpair of a real d ×d symmetric matrix (compute Mnx for large n). Unbounded average case (Kostlan). Used in practice with success. weak complexity cost with probability . (Amelunxen, Lotz)

20

Weak complexity bounds

If the average case is unbounded, is the algorithm slow? example The power method for computing the dominant eigenpair of a real d ×d symmetric matrix (compute Mnx for large n). Unbounded average case (Kostlan). Used in practice with success. weak complexity cost poly(d) with probability 1−exp(−d). (Amelunxen, Lotz)

20

Probabilistic analysis

general bound If Σ ⊂ H is an homogeneous algebraic hypersurface, and if X ∈ H is a Gaussian isotropic random variable, P ( ∥X ∥ dist(X ,Σ) t ) 11dimH degΣ t . degree bound { ill-posed problems } corollary 1 cost with probabiliy corollary 2 cost with probabiliy .

21

Probabilistic analysis

general bound If Σ ⊂ H is an homogeneous algebraic hypersurface, and if X ∈ H is a Gaussian isotropic random variable, P ( ∥X ∥ dist(X ,Σ) t ) 11dimH degΣ t . degree bound deg{ ill-posed problems } n2n(s +1)n+1Dn corollary 1 cost with probabiliy corollary 2 cost with probabiliy .

21

Probabilistic analysis

general bound If Σ ⊂ H is an homogeneous algebraic hypersurface, and if X ∈ H is a Gaussian isotropic random variable, P ( ∥X ∥ dist(X ,Σ) t ) 11dimH degΣ t . degree bound deg{ ill-posed problems } n2n(s +1)n+1Dn corollary 1 cost (sD)n3+o(1) with probabiliy 1−(sD)−n corollary 2 cost with probabiliy .

21

Probabilistic analysis

general bound If Σ ⊂ H is an homogeneous algebraic hypersurface, and if X ∈ H is a Gaussian isotropic random variable, P ( ∥X ∥ dist(X ,Σ) t ) 11dimH degΣ t . degree bound deg{ ill-posed problems } n2n(s +1)n+1Dn corollary 1 cost (sD)n3+o(1) with probabiliy 1−(sD)−n corollary 2 cost 2O(N 2) with probabiliy 1−2−N.

21

Perspectives

Ill-posedness is relative to a data representation example Given by a rational parametrization, the lemniscate is well-conditionned next goal Given , compute the homology of any set obtain from the sets and by union, intersection and complementation, assuming . Work in progress by Josué Tonelli Cueto.

Thank you!

22

Perspectives

Ill-posedness is relative to a data representation example Given by a rational parametrization, the lemniscate is well-conditionned next goal Given , compute the homology of any set obtain from the sets and by union, intersection and complementation, assuming . Work in progress by Josué Tonelli Cueto.

Thank you!

22

Perspectives

Ill-posedness is relative to a data representation example Given by a rational parametrization, the lemniscate is well-conditionned next goal Given F = (f1,..., fs), compute the homology of any set obtain from the sets { fi 0 } and { fi 0 } by union, intersection and complementation, assuming κ∗(F) < ∞. Work in progress by Josué Tonelli Cueto.

Thank you!

22

Perspectives

Ill-posedness is relative to a data representation example Given by a rational parametrization, the lemniscate is well-conditionned next goal Given F = (f1,..., fs), compute the homology of any set obtain from the sets { fi 0 } and { fi 0 } by union, intersection and complementation, assuming κ∗(F) < ∞. Work in progress by Josué Tonelli Cueto.

Thank you!

22