Testing membership in varieties, algebraic natural proofs, and - PowerPoint PPT Presentation

Testing membership in varieties, algebraic natural proofs, and geometric complexity theory Markus Bl¨ aser Saarland University with Christian Ikenmeyer, Gorav Jindal, Vladimir Lysikov, Anurag Pandey, and Frank-Olaf Schreyer

Membership testing in varieties Orbit problems in computer science The minrank problem

Variety membership problem Variety membership problem ◮ “Given” a variety V and ◮ given a point x in the ambient space ◮ decide whether x ∈ V ! What is the complexity of this problem? − → depends on the encoding of V

Varieties given by circuits Theorem If V is given by a list of arithmetic circuits, then the membership problem is in coRP . Proof: ◮ Let C 1 , . . . , C t computing f 1 , . . . , f t such that V = V ( f 1 , . . . , f t ) . ◮ Test whether f 1 ( x ) = · · · = f t ( x ) = 0 by evaluating C τ at x . (Polynomial Identity Testing) Remark Can be realized as a many-one reduction to PIT.

PIT reduces to PIT for constant polynomials Lemma There is a many-one reduction from general PIT to PIT for constant polynomials. Proof: ◮ Let C be a circuit of size s computing f ( X 1 , . . . , X n ) . ◮ The degree and the bit size of the coefficients are exponentially bounded in s . ◮ f is not identically zero iff f ( 2 2 s2 , . . . , 2 2 ns2 ) � = 0 . Remark The proof yields a many-one reduction from PIT to hypersurface membership testing when the surface is given as a circuit.

Further ways to specify varieties ◮ Explicitely in the problem: Let V = ( V n ) and consider V -membership ◮ As an orbit closure: Let G = ( G n ) be a sequence of groups acting on an n -dimensional ambient space. Given ( x, v ) decide whether x ∈ G n v ! ( Orbit containment problem ) ◮ By a dense subset: Given circuits computing a polynomial map, decide whether x lies in the closure of the image.

Membership testing in varieties Orbit problems in computer science The minrank problem

Tensor rank and matrix multiplication Definition u ⊗ v ⊗ w ∈ U ⊗ V ⊗ W is called a rank-one tensor. Definition (Rank) R ( t ) is the smallest r such that there are rank-one tensors t 1 , . . . , t r with t = t 1 + · · · + t r . Lemma Let t ∈ U ⊗ V ⊗ W and t ′ ∈ U ′ ⊗ V ′ ⊗ W ′ . ◮ R ( t ⊕ t ′ ) ≤ R ( t ) + R ( t ′ ) ◮ R ( t ⊗ t ′ ) ≤ R ( t ) R ( t ′ )

Strassen’s algorithm and tensors Observation: Tensor product ∼ = Recursion Strassen’s algorithm: ◮ � 2, 2, 2 � ⊗ s = � 2 s , 2 s , 2 s � ◮ R ( � 2, 2, 2 � ⊗ s ) ≤ 7 s Definition (Exponent of matrix multiplication) ω = inf { τ | R ( � n, n, n � ) = O ( n τ ) } Strassen: ω ≤ log 2 7 ≤ 2.81 Lemma log r If R ( � k, m, n � ) ≤ r , then ω ≤ 3 · log kmn .

Restrictions Definition Let A : U → U ′ , B : V → V ′ , C : W → W ′ be homomorphism. ◮ ( A ⊗ B ⊗ C )( u ⊗ v ⊗ w ) = A ( u ) ⊗ B ( v ) ⊗ C ( w ) ◮ ( A ⊗ B ⊗ C ) t = � r i = 1 A ( u i ) ⊗ B ( v i ) ⊗ C ( w i ) for t = � r i = 1 u i ⊗ v i ⊗ w i . ◮ t ′ ≤ t if there are A, B, C such that t ′ = ( A ⊗ B ⊗ C ) t . (“restriction”). Lemma ◮ If t ′ ≤ t , then R ( t ′ ) ≤ R ( t ) ◮ R ( t ) ≤ r iff t ≤ � r � . ( � r � “diagonal” of size r .)

Orbit problems Let ( A, B, C ) ∈ End ( U ) × End ( V ) × End ( W ) act on U ⊗ V ⊗ W by ( A, B, C ) u ⊗ v ⊗ w = A ( u ) ⊗ B ( v ) ⊗ C ( w ) . and linearity. We can interpret t ∈ U ′ ⊗ V ′ ⊗ W ′ as an element of U ⊗ V ⊗ W by embedding U ′ into U , V ′ into V , and W ′ into W . Lemma R ( t ) ≤ r iff t ∈ ( End ( U ) × End ( U ) × End ( U )) � r � .

Border rank and orbit problems ◮ S r be the set of all tensors of rank r . ◮ X r := S r is the set of tensors of border rank ≤ r . Lemma log r If R ( � k, m, n � ) ≤ r , then ω ≤ 3 · log kmn . Lemma R ( t ) ≤ r iff t ∈ ( GL r × GL r × GL r ) � r � .

Identity testing Lemma (Valiant) If a polynomial f ∈ k [ X 1 , . . . , X n ] can be computed by a formula of size s , then there is a matrix pencil of size m × m A := A 0 + X 1 A 1 + · · · + X n A n such that f = det ( A ) . We have m = O ( s ) . Observation f is identically zero iff A does not have full rank. SL m × SL m acts on ( A 0 , . . . , A n ) by ( S, T )( A 0 , . . . , A n ) := ( SA 0 T, . . . , SA n T ) .

Noncommutative identity testing Definition Let G act on V . The null cone are all vectors v such that 0 ∈ Gv . One can define a noncommutative version of the rank of a matrix pencil. Theorem A does not have full noncommutative rank iff A is in the null cone of the left-right-SL-action. Theorem (Garg–Gurvits–Oliviera–Wigderson) This null-cone problem can be solved deterministically in polynomial time.

Valiant’s world ◮ Let X = X 1 , X 2 , . . . be indeterminates. ◮ A function p : N → N is p-bounded , if there is some polynomial q such that p ( n ) ≤ q ( n ) for all n . Definition A sequence of polynomials ( f n ) ∈ K [ X ] is called a p-family if for all n , 1. f n ∈ K [ X 1 , . . . , X p ( n ) ] for some polynomially bounded function p and 2. deg f n ≤ q ( n ) for some polynomially bounded function q . Definition The class VP consists of all p-families ( f n ) such that L ( f n ) is polynomially bounded.

Projections as orbit problems Definition 1. f ∈ K [ X ] is a projection of g ∈ K [ X ] if there is a substitution r : X → X ∪ K such that f = r ( g ) . “ f ≤ g ” 2. A p-family ( f n ) is a p-projection of another p-family ( g n ) if there is a p-bounded q such that f n ≤ g q ( n ) . “ ( f n ) ≤ p ( g n ) ” ◮ End n acts on k [ X 1 , . . . , X n ] by ( gh )( x ) = h ( g t x ) for g ∈ End n , h ∈ k [ X 1 , . . . , X n ] , x ∈ k n . ◮ If f ∈ End n h and h is homogeneous of degree d , then f is homogeneous of degree d ◮ If f ≤ h , then deg f can be smaller than deg h . ◮ Padding: Replace f by X deg h − deg f f . 1 ◮ If f ≤ h , then X deg h − deg f f ∈ End n h 1 ◮ VP and VP ws are closed under End n .

Valiant’s conjecture Conjecture (Valiant) VP � = VNP ◮ the weaker conjecture VP ws � = VNP is equivalent to per �≤ p det . Conjecture (Mulmuley & Sohoni) VNP �⊆ VP ws ◮ equivalent to X n − m per m / ∈ GL n 2 det n for any n = poly ( m ) . 11

Orbit closure containment problem ◮ We want to understand the complexity of deciding x ∈ Gv ? ◮ We will focus on tensors. ◮ Tensor rank is NP-hard (Hastad). ◮ Very little is known about closures. ◮ In partcular, we do not know any hardness results for border rank.

Membership testing in varieties Orbit problems in computer science The minrank problem

The minrank problem Definition Let A 1 , . . . , A k ∈ K m × n . The min-rank of A 1 , . . . , A k is the minimum number r such that there are scalars λ 1 , . . . , λ m , not all being 0 , with rk ( λ 1 A 1 + · · · + λ k A k ) ≤ r. We denote the min-rank by minR ( A 1 , . . . A k ) . ◮ Can also be phrased in terms of a matrix pencil X 1 A 1 + · · · + X k A k . ◮ Can be phrased in terms of tensors by stacking the matrices on top of each other.

Geometric description Theorem Let U , V , W be vector spaces over an algebraically closed field F . The set of all tensors T ∈ U ⊗ V ⊗ W with minrank at most r is Zariski closed. Definition We call the projective variety P M U ⊗ V ⊗ W,r = { [ T ] ∈ P ( U ⊗ V ⊗ W ) | ∃ x � = 0 : rk ( Tx ) ≤ r } the projective minrank variety , and the corresponding affine cone M U ⊗ V ⊗ W,r = { T ∈ U ⊗ V ⊗ W | ∃ x � = 0 : rk ( Tx ) ≤ r } the affine minrank variety , or just the minrank variety .

Simple properties Lemma Let V ′ and W ′ be subspaces of V and W respectively. Then M U ⊗ V ′ ⊗ W ′ ,r = M U ⊗ V ⊗ W,r ∩ ( U ⊗ V ′ ⊗ W ′ ) . Lemma Let dim U = k , dim V = n and dim W > s = n ( k − 1 ) + r . Then � M U ⊗ V ⊗ W,r = M U ⊗ V ⊗ W ′ ,r . W ′ ⊂ W dim W ′ = s Lemma The variety M U ⊗ V ⊗ W,r is invariant under the standard action of GL ( U ) × GL ( V ) × GL ( W ) on U ⊗ V ⊗ W .

Orbit problem ◮ Let L = ( F n ) ⊕ ( k − 1 ) ⊕ F r , dim L = s := n ( k − 1 ) + r . ◮ Let L i be the i -th summand with standard basis e ij , 1 ≤ j ≤ dim L i . ◮ Let U = F k with standard basis e i . r k n � � � T k,n,r = e 1 ⊗ ( e 1j ⊗ e 1j ) + e i ⊗ ( e ij ⊗ e ij ) , j = 1 i = 2 j = 1 ◮ The group GL ( U ) × GL ( L ) × GL ( L ) acts on U ⊗ L ⊗ L . Theorem Suppose V and W are subspaces of L . Then M U ⊗ V ⊗ W,r = ( GL ( U ) × GL ( L ) × GL ( L )) T k,n,r ∩ ( U ⊗ V ⊗ W ) .

Symmetries Theorem If r < n , then the stabilizer of T k,n,r in GL k × GL s × GL s is isomorphic to ( GL r × GL 1 ) × ( GL n × GL 1 ) k − 1 ⋊ S k − 1 . ( Z 1 , z 1 , . . . , Z k , z k ) ∈ ( GL r × GL 1 ) × ( GL n × GL 1 ) k − 1 is embedded into GL k × GL s × GL s via ( diag ( z 1 , . . . , z k ) , diag ( Z 1 , . . . , Z k ) , diag (( z 1 Z 1 ) − T , . . . , ( z k Z k ) − T )) and S k − 1 permutes the last k − 1 coordinates of U and the last k − 1 summands of L simultaneously. Theorem If stab T = stab T k,n,r , then T lies in ( GL k × GL s × GL s ) T k,n,r . If stab T ⊃ stab T k,n,r , then T ∈ ( GL k × GL s × GL s ) T k,n,r