The Euclidean Distance Degree of an Algebraic Variety Bernd - PowerPoint PPT Presentation

The Euclidean Distance Degree of an Algebraic Variety Bernd Sturmfels UC Berkeley and MPI Bonn joint work with Jan Draisma, Emil Horobet ¸, Giorgio Ottaviani, and Rekha Thomas 1 / 26

Getting Close to Varieties Many models in the sciences and engineering are the real solutions to systems of polynomial equations in several unknowns. Such a set is an algebraic variety X ⊂ R n . Given X , consider the following optimization problem: for any data point u ∈ R n , find x ∈ X that minimizes the squared Euclidean distance d u ( x ) = � n i =1 ( u i − x i ) 2 . 2 / 26

Getting Close to Varieties Many models in the sciences and engineering are the real solutions to systems of polynomial equations in several unknowns. Such a set is an algebraic variety X ⊂ R n . Given X , consider the following optimization problem: for any data point u ∈ R n , find x ∈ X that minimizes the squared Euclidean distance d u ( x ) = � n i =1 ( u i − x i ) 2 . What can be said about the algebraic function u �→ x ( u ) from the data to the optimal solution? Its branches are given by the complex critical points for generic u . Their number is the Euclidean distance degree, or short, the ED degree, of the variety X . 3 / 26

Logo 4 / 26

Plane Curves Fix a polynomial f ( x , y ) of degree d and consider the curve ( x , y ) ∈ R 2 : f ( x , y ) = 0 � � X = . Given a data point ( u , v ) we wish to find ( x , y ) on X such that ( u − x , v − y ) is parallel to the gradient of f . Must solve two equations of degree d in two unknowns: � � u − x v − y f ( x , y ) = det = 0 ∂ f /∂ x ∂ f /∂ y ezout’s Theorem, we expect d 2 complex solutions ( x , y ). By B´ Proposition A general plane curve X of degree d has EDdegree ( X ) = d 2 . 5 / 26

The Cardioid The cardioid is a special curve of degree 4. Its ED degree equals 3 . ( x , y ) ∈ R 2 : ( x 2 + y 2 + x ) 2 = x 2 + y 2 � � X = . The inner cardioid is the evolute or ED discriminant . It is given by 27 u 4 + 54 u 2 v 2 + 27 v 4 + 54 u 3 + 54 uv 2 + 36 u 2 + 9 v 2 + 8 u = 0 . 6 / 26

Linear Regression If X is a linear subspace of R n then EDdegree ( X ) = 1 . Which non-linear varieties do arise in applications? ◮ Control Theory ◮ Geometric Modeling ◮ Computer Vision ◮ Tensor Decomposition ◮ Structured Low Rank Approximation ◮ ..... In many cases, X is given by homogeneous polynomials, so X is a cone. View it as a projective variety in P n − 1 . 7 / 26

Ideals Let I X = � f 1 , . . . , f s � ⊂ R [ x 1 , . . . , x n ] be the ideal of X and J ( f ) its s × n Jacobian matrix. The singular locus X sing is defined by � � I X sing = I X + c × c -minors of J ( f ) , where c = codim ( X ) . The critical ideal for u ∈ R n is � � � u − x ��� � ∞ � I X + ( c +1) × ( c +1)-minors of : I X sing J ( f ) Lemma For generic u ∈ R n , the function d u has finitely many critical points on the manifold X \ X sing , namely the zeros of the critical ideal. − → EDdegree ( X ) 8 / 26

Ideals Let I X = � f 1 , . . . , f s � ⊂ R [ x 1 , . . . , x n ] be the ideal of X and J ( f ) its s × n Jacobian matrix. The singular locus X sing is defined by � � I X sing = I X + c × c -minors of J ( f ) , where c = codim ( X ) . The critical ideal for u ∈ R n is � � � u − x ��� � ∞ � I X + ( c +1) × ( c +1)-minors of : I X sing J ( f ) Lemma For generic u ∈ R n , the function d u has finitely many critical points on the manifold X \ X sing , namely the zeros of the critical ideal. − → EDdegree ( X ) If f 1 , . . . , f s are homogeneous, so that X ⊂ P n − 1 , we use instead   u � � �� � ∞ I X sing ·� x 2 1 + · · · + x 2 � I X + ( c +2) × ( c +2)-minors of x : n �   J ( f ) 9 / 26

Bounds Proposition Let X ⊂ R n be defined by polynomials f 1 , f 2 , . . . , f c , . . . of degrees d 1 ≥ d 2 ≥ · · · ≥ d c ≥ · · · . If codim ( X ) = c then EDdegree ( X ) ≤ ( d 1 − 1) i 1 ( d 2 − 1) i 2 · · · ( d c − 1) i c . � d 1 d 2 · · · d c · i 1 + i 2 + ··· + i c ≤ n − c Equality holds when f 1 , f 2 , . . . , f c are generic. Example If X is cut out by c quadratic polynomials in R n then its ED degree is at most 2 c � n � . c Similar bounds are available for projective varieties X ⊂ P n − 1 . 10 / 26

Singular Value Decomposition Fix positive integers r ≤ s ≤ t and n = st . Given an arbitrary s × t -matrix U , we seek a matrix of rank r that is closest to U . Here X is the determinantal variety of s × t -matrices of rank ≤ r . Proposition � s � EDdegree ( X ) = . r Proof. Compute the singular value decomposition U = T 1 · diag ( σ 1 , σ 2 , . . . , σ s ) · T 2 . with σ 1 ≥ σ 2 ≥ · · · ≥ σ s . By the Eckart-Young Theorem, U ∗ = T 1 · diag ( σ 1 , . . . , σ r , 0 , . . . , 0) · T 2 is closest rank r matrix to U . All critical points are given by r -element subsets of { σ 1 , . . . , σ s } . 11 / 26

Closest Symmetric Matrix For symmetric U = ( U ij ), consider two unconstrained formulations: s s r r � 2 � 2 . � � � � � � � Min t U ij − Min t U ij − t ik t kj or t ik t kj i =1 j =1 k =1 1 ≤ i ≤ j ≤ s k =1 Eckart-Young applies only in the first case: � s � � s � EDdegree ( X ) = or EDdegree ( X ) ≫ . r r Here X is the variety of symmetric s × s -matrices of rank ≤ r . 12 / 26

Closest Symmetric Matrix For symmetric U = ( U ij ), consider two unconstrained formulations: s s r r � 2 � 2 . � � � � � � � Min t U ij − Min t U ij − t ik t kj or t ik t kj i =1 j =1 k =1 1 ≤ i ≤ j ≤ s k =1 Eckart-Young applies only in the first case: � s � � s � EDdegree ( X ) = or EDdegree ( X ) ≫ . r r Here X is the variety of symmetric s × s -matrices of rank ≤ r . For 3 × 3-matrices with r = 1 , 2 we have EDdegree ( X ) = 3 EDdegree ( X ) = 13 . or Fixing the Euclidean metric on R 6 , put rank constraints on either √  2 x 11 x 12 x 13    x 11 x 12 x 13 √ x 12 2 x 22 x 23 or x 12 x 22 x 23  √    x 13 x 23 2 x 33 x 13 x 23 x 33 13 / 26

Critical Formations on the Line d’apr` es [Anderson-Helmke 2013] Let X denote the variety in R ( p 2 ) with parametric representation d ij = ( z i − z j ) 2 for 1 ≤ i ≤ j ≤ p . The points in X record the squared distances among p interacting agents with coordinates z 1 , z 2 , . . . , z p on the real line. The ideal I X is generated by the 2 × 2-minors of the Cayley-Menger matrix 2 d 1 p d 1 p + d 2 p − d 12 d 1 p + d 3 p − d 13 d 1 p + d p − 1 , p − d 1 , p − 1   · · · d 1 p + d 2 p − d 12 2 d 2 p d 2 p + d 3 p − d 23 d 2 p + d p − 1 , p − d 2 , p − 1 · · ·   d 1 p + d 3 p − d 13 d 2 p + d 3 p − d 23 2 d 3 p d 3 p + d p − 1 , p − d 3 , p − 1   · · ·     . . . . ...   . . . .   . . . .   d 1 p + d p − 1 , p − d 1 , p − 1 d 2 p + d p − 1 , p − d 2 , p − 1 d 3 p + d p − 1 , p − d 3 , p − 1 2 d p − 1 , p · · · Theorem The ED degree of the Cayley-Menger variety X equals � 3 p − 1 − 1 if p ≡ 1 , 2 mod 3 2 EMdegree ( X ) = 3 p − 1 − 1 p ! − if p ≡ 0 mod 3 2 3(( p / 3)!) 3 14 / 26

Hurwitz Stability A univariate polynomial with real coefficients, x ( t ) = x 0 t n + x 1 t n − 1 + x 2 t n − 2 + · · · + x n − 1 t + x n , is stable if each of its n complex zeros has negative real part. Can express this using Hurwitz determinants  x 1 x 3 x 5 0 0  0 0 x 0 x 2 x 4   1 ¯   Γ 5 = · det 0 x 1 x 3 x 5 0 .   x 5   0 0 x 0 x 2 x 4   0 0 x 1 x 3 x 5 Theorem The ED degrees of the Hurwitz determinants are EDdegree (¯ EDdegree (Γ n ) Γ n ) n = 2 m + 1 8 m − 3 4 m − 2 n = 2 m 4 m − 3 8 m − 6 Here Γ n = ¯ Γ n | x 0 =1 15 / 26

Average ED Degree E X π 2 R n # π − 1 1 3 5 3 1 2 ( u ) Equip data space R n with a probability measure ω . Taking the standard Gaussian centered at 0 is natural when X is a cone: 1 (2 π ) n / 2 e −|| x || 2 / 2 dx 1 ∧ · · · ∧ dx n . ω = The expected number of critical points of d u is � aEDdegree ( X , ω ) := R n # { real critical points of d u on X } · | ω | . Can compute this integral in some interesting cases. 16 / 26

Tables of Numbers Hurwitz Determinants: EDdegree (¯ aEDdegree (¯ n EDdegree (Γ n ) Γ n ) aEDdegree (Γ n ) Γ n ) 3 5 2 1.162... 2 4 5 10 1.883... 2.068... 5 13 6 2.142... 3.052... 6 9 18 2.416... 3.53... 7 21 10 2.66... 3.742... ED degree can go up or down when replacing an affine variety by its projective closure. Our theory explains this .... Important Application: Tensors of Rank One Format aEDdegree EDdegree 2 × 2 × 2 4.2891... 6 2 × 2 × 2 × 2 11.0647... 24 2 × 2 × n , n ≥ 3 5.6038... 8 2 × 3 × 3 8.8402... 15 2 × 3 × n , n ≥ 4 10.3725... 18 3 × 3 × 3 16.0196... 37 3 × 3 × 4 21.2651... 55 3 × 3 × n , n ≥ 5 23.0552... 61 17 / 26

Duality X x 1 u x 2 u − x 2 X ∗ u − x 1 Figure: Bijection between critical points on X and critical points on X ∗ . 18 / 26