Path-complete LMI automaton ( Ahmadi et al. ) Solve family of LMIs: � Q v ≻ 0 , ∀ v ( P ρ ) ρ 2 Q w � A T i Q v A i , ∀ w = τ d ( v, i ) Bisection: ρ d := smallest ρ such that ( P ρ ) is feasible. Theorem ( Ahmadi et al. - SICON 2014 ) An optimal solution ( Q v ) v provides a norm v ( x T Q v x ) 1 / 2 ν ( x ) = max such that 1 ρ d � ρ ( A ) � 2( d +1) ρ d 1 n (asymptotically exact as d → ∞ ). Proof based on the Loewner-John theorem: the Barabanov norm can be approximated by an Euclidean norm up to a √ n multiplicative factor.
Before... Figure: Computation time (s) vs dimension: red Ahmadi et al. , ,
...Now Figure: Computation time (s) vs dimension: red Ahmadi et al. , blue “quantum” dynamic programming (this talk),
...Now Figure: Computation time (s) vs dimension: red Ahmadi et al. , blue “quantum” dynamic programming (this talk), green specialization to nonnegative matrices (this talk - MCRF, 2020)
How do we get there ? A closer look at simplified LMIs ρ 2 Q � A T Q ≻ 0 i QA i , ∀ i ∈ [ m ] .
How do we get there ? A closer look at simplified LMIs ρ 2 Q � A T Q ≻ 0 i QA i , ∀ i ∈ [ m ] . Solving a wrong equation We would like to write: “ ρ 2 Q � sup A T i QA i ” . i ∈ [ m ]
How do we get there ? A closer look at simplified LMIs ρ 2 Q � A T Q ≻ 0 i QA i , ∀ i ∈ [ m ] . Solving a wrong equation We would like to write: “ ρ 2 Q � sup A T i QA i ” . i ∈ [ m ] The supremum of several quadratic forms does not exist ! ⇒ will replace supremum by a minimal upper bound
How do we get there ? A closer look at simplified LMIs ρ 2 Q � A T Q ≻ 0 i QA i , ∀ i ∈ [ m ] . Solving a wrong equation We would like to write: “ ρ 2 Q � sup A T i QA i ” . i ∈ [ m ] The supremum of several quadratic forms does not exist ! ⇒ will replace supremum by a minimal upper bound Fast computational scheme Interior point methods are relatively slow → Replace optimization by a fixed point approach. For nonnegative matrices, reduces to a risk-sensitive eigenproblem.
Minimal upper bounds x is a minimal upper bound of the set A iff � � A � x and A � y � x = ⇒ y = x . The set of minimal upper bounds: � A .
Minimal upper bounds x is a minimal upper bound of the set A iff � � A � x and A � y � x = ⇒ y = x . The set of minimal upper bounds: � A . Theorem (Krein-Rutman - 1948) A cone induces a lattice structure iff it is simplicial ( ∼ = R + n ).
Minimal upper bounds x is a minimal upper bound of the set A iff � � A � x and A � y � x = ⇒ y = x . The set of minimal upper bounds: � A . Theorem (Krein-Rutman - 1948) A cone induces a lattice structure iff it is simplicial ( ∼ = R + n ). Theorem (Kadison - 1951) The L¨ owner order induces an anti-lattice structure: two symmetric matrices A, B have a supremum if and only if A � B or B � A .
Introduction Minimal upper bounds Noncommutative Dynamic Programming Risk sensitive eigenproblem Concluding remarks The inertia of the symmetric matrix M is the tuple ( p, q, r ) , where • p : number of positive eigenvalues of M , • q : number of negative eigenvalues of M , • r : number of zero eigenvalues of M . Definition (Indefinite orthogonal group) O ( p, q ) is the group of matrices S preserving the quadratic form x 1 1 + · · · + x 2 p − x 2 p +1 − · · · − x 2 p + q : � � S T = � � I p I p S =: J p,q − I q − I q 12 / 38
Introduction Minimal upper bounds Noncommutative Dynamic Programming Risk sensitive eigenproblem Concluding remarks The inertia of the symmetric matrix M is the tuple ( p, q, r ) , where • p : number of positive eigenvalues of M , • q : number of negative eigenvalues of M , • r : number of zero eigenvalues of M . Definition (Indefinite orthogonal group) O ( p, q ) is the group of matrices S preserving the quadratic form x 1 1 + · · · + x 2 p − x 2 p +1 − · · · − x 2 p + q : � � S T = � � I p I p S =: J p,q − I q − I q � � ǫ 1 ch t ǫ 2 sh t O (1 , 1) is the group of hyperbolic isometries , ǫ 1 sh t ǫ 2 ch t where ǫ 1 , ǫ 2 ∈ {− 1 , 1 } 12 / 38
Introduction Minimal upper bounds Noncommutative Dynamic Programming Risk sensitive eigenproblem Concluding remarks The inertia of the symmetric matrix M is the tuple ( p, q, r ) , where • p : number of positive eigenvalues of M , • q : number of negative eigenvalues of M , • r : number of zero eigenvalues of M . Definition (Indefinite orthogonal group) O ( p, q ) is the group of matrices S preserving the quadratic form x 1 1 + · · · + x 2 p − x 2 p +1 − · · · − x 2 p + q : � � S T = � � I p I p S =: J p,q − I q − I q � � ǫ 1 ch t ǫ 2 sh t O (1 , 1) is the group of hyperbolic isometries , ǫ 1 sh t ǫ 2 ch t where ǫ 1 , ǫ 2 ∈ {− 1 , 1 } O ( p ) × O ( q ) is a maximal compact subgroup of O ( p, q ) . 12 / 38
Theorem ( Stott - Proc AMS 2018 , Quantitative version of Kadison theorem) If the inertia of A − B is ( p, q, 0) , then � R pq . ∼ ∼ � O ( p, q ) { A , B } = = O ( p ) × O ( q )
Introduction Minimal upper bounds Noncommutative Dynamic Programming Risk sensitive eigenproblem Concluding remarks Example p = q = 1 . � O (1 , 1) O (1) × O (1) is the group of hyperbolic rotations: �� ch t sh t � � | t ∈ R sh t ch t 14 / 38
Canonical selection of a minimal upper bound Ellipsoid: E ( M ) = { x | x T M − 1 x � 1 } , where M is symmetric pos. def. Theorem ( L¨ owner - John ) There is a unique minimum volume ellipsoid containing a convex body C .
Canonical selection of a minimal upper bound Ellipsoid: E ( M ) = { x | x T M − 1 x � 1 } , where M is symmetric pos. def. Theorem ( L¨ owner - John ) There is a unique minimum volume ellipsoid containing a convex body C . Definition-Proposition ( Allamigeon, SG, Goubault, Putot, NS , ACM TECS 2016) Let A = { A i } i ⊂ S ++ and C = ∪ i E ( A i ) . We define ⊔A so that E ( ⊔A ) n is the L¨ owner ellipsoid of ∪ A ∈A E ( A ) , i.e., ( ⊔A ) − 1 = argmax X { log det X | X � A − 1 i , i ∈ [ m ] , X ≻ 0 } .
Canonical selection of a minimal upper bound Ellipsoid: E ( M ) = { x | x T M − 1 x � 1 } , where M is symmetric pos. def. Theorem ( L¨ owner - John ) There is a unique minimum volume ellipsoid containing a convex body C . Definition-Proposition ( Allamigeon, SG, Goubault, Putot, NS , ACM TECS 2016) Let A = { A i } i ⊂ S ++ and C = ∪ i E ( A i ) . We define ⊔A so that E ( ⊔A ) n is the L¨ owner ellipsoid of ∪ A ∈A E ( A ) , i.e., ( ⊔A ) − 1 = argmax X { log det X | X � A − 1 i , i ∈ [ m ] , X ≻ 0 } . Then, ⊔A is a minimal upper bound of A , and ⊔ is the only selection that commutes with the action of invertible congruences: L ( ⊔A ) L T = ⊔ ( L A L T ) ,
Theorem ( Allamigeon, SG, Goubault, Putot, NS , ACM TECS 2016) Computating X ⊔ Y reduces to a square root (i.e., SDP-free!). X ⊔ I = 1 2 ( X + I ) + 1 Suppose Y = I : 2 | X − I | . General case reduces to it by congruence: add 1 Cholesky decomposition + 1 triangular inversion. Complexity: O ( n 3 ) .
Theorem ( Allamigeon, SG, Goubault, Putot, NS , ACM TECS 2016) Computating X ⊔ Y reduces to a square root (i.e., SDP-free!). X ⊔ I = 1 2 ( X + I ) + 1 Suppose Y = I : 2 | X − I | . General case reduces to it by congruence: add 1 Cholesky decomposition + 1 triangular inversion. Complexity: O ( n 3 ) . � The Loewner selection ⊔ is
Theorem ( Allamigeon, SG, Goubault, Putot, NS , ACM TECS 2016) Computating X ⊔ Y reduces to a square root (i.e., SDP-free!). X ⊔ I = 1 2 ( X + I ) + 1 Suppose Y = I : 2 | X − I | . General case reduces to it by congruence: add 1 Cholesky decomposition + 1 triangular inversion. Complexity: O ( n 3 ) . � The Loewner selection ⊔ is • continuous on S ++ × S ++ but does not extend continuously to the n n closed cone,
Theorem ( Allamigeon, SG, Goubault, Putot, NS , ACM TECS 2016) Computating X ⊔ Y reduces to a square root (i.e., SDP-free!). X ⊔ I = 1 2 ( X + I ) + 1 Suppose Y = I : 2 | X − I | . General case reduces to it by congruence: add 1 Cholesky decomposition + 1 triangular inversion. Complexity: O ( n 3 ) . � The Loewner selection ⊔ is • continuous on S ++ × S ++ but does not extend continuously to the n n closed cone, • not order-preserving,
Theorem ( Allamigeon, SG, Goubault, Putot, NS , ACM TECS 2016) Computating X ⊔ Y reduces to a square root (i.e., SDP-free!). X ⊔ I = 1 2 ( X + I ) + 1 Suppose Y = I : 2 | X − I | . General case reduces to it by congruence: add 1 Cholesky decomposition + 1 triangular inversion. Complexity: O ( n 3 ) . � The Loewner selection ⊔ is • continuous on S ++ × S ++ but does not extend continuously to the n n closed cone, • not order-preserving, • not associative.
Introduction Minimal upper bounds Noncommutative Dynamic Programming Risk sensitive eigenproblem Concluding remarks Reducing the search of a joint quadratic Lyapunov function to an eigenproblem Goal x T Qx such that max i ∈ [ m ] ν ( A i x ) � ρν ( x ) . � Compute norm ν ( x ) = Computation: single quadratic form Corresponding LMI: ρ 2 Q � A T i QA i , ∀ i . Eigenvalue problem for a multivalued map ρ 2 Q ∈ � A T i QA i . i 17 / 38
Quantum dynamic programming operators Quantum channels (0-player games) Completely positive trace perserving operators: � � A i XA ∗ A ∗ K ( X ) = i , i A i = I n . i i
Quantum dynamic programming operators Quantum channels (0-player games) Completely positive trace perserving operators: � � A i XA ∗ A ∗ K ( X ) = i , i A i = I n . i i Propagation of ”non-commutative probability measures” (analogue of Fokker-Planck). Quantum dynamic programming operator (1-player game) � A T T ( X ) = i XA i i with � the set of least upper bounds in L¨ owner order (multivalued map).
Quantum dynamic programming operators Quantum channels (0-player games) Completely positive trace perserving operators: � � A i XA ∗ A ∗ K ( X ) = i , i A i = I n . i i Propagation of ”non-commutative probability measures” (analogue of Fokker-Planck). Quantum dynamic programming operator (1-player game) � A T T ( X ) = i XA i i with � the set of least upper bounds in L¨ owner order (multivalued map). Propagation of norms (backward equation).
Quantum dynamic programming operator associated with an automaton τ d transition map of the De Bruijn automaton on d letters: n ) ( m d ) X ∈ ( S + T d � A T and w ( X ) := i X v A i w = τ d ( v,i ) Reduces to the earlier d = 1 case by a block diagonal construction. Theorem Suppose that ρ 2 X ∈ T d ( X ) with ρ > 0 and X positive definite. Then, ρ ( A ) � ρ .
Theorem Suppose that A is irreducible. Then there exists ρ > 0 and X such that � v X v is positive definite and ρ 2 X = T d ⊔ ( X ) ∈ T d ( X ) where � [ T d A T ⊔ ( X )] w := i X v A i . w = τ d ( v,i )
Exercise: find the mistake in the following proof We want to show that the following eigenproblem is solvable: � [ T d A T i X v A i = ρ 2 X w ⊔ ( X )] w := w = τ d ( v,i ) 1. suppose, w.l.g., d = 0 .
Exercise: find the mistake in the following proof We want to show that the following eigenproblem is solvable: � [ T d A T i X v A i = ρ 2 X w ⊔ ( X )] w := w = τ d ( v,i ) 1. suppose, w.l.g., d = 0 . 2. Consider the noncommutative simplex, ∆ := { X � 0: trace X = 1 } . This set is compact and convex.
Exercise: find the mistake in the following proof We want to show that the following eigenproblem is solvable: � [ T d A T i X v A i = ρ 2 X w ⊔ ( X )] w := w = τ d ( v,i ) 1. suppose, w.l.g., d = 0 . 2. Consider the noncommutative simplex, ∆ := { X � 0: trace X = 1 } . This set is compact and convex. 3. Consider the normalized map ˜ T d ⊔ ( X ) = (trace T d ⊔ ( X )) − 1 T d ⊔ ( X ) . It sends ∆ to ∆
Exercise: find the mistake in the following proof We want to show that the following eigenproblem is solvable: � [ T d A T i X v A i = ρ 2 X w ⊔ ( X )] w := w = τ d ( v,i ) 1. suppose, w.l.g., d = 0 . 2. Consider the noncommutative simplex, ∆ := { X � 0: trace X = 1 } . This set is compact and convex. 3. Consider the normalized map ˜ T d ⊔ ( X ) = (trace T d ⊔ ( X )) − 1 T d ⊔ ( X ) . It sends ∆ to ∆ 4. By Brouwer fixed point theorem, it has a fixed point
Exercise: find the mistake in the following proof We want to show that the following eigenproblem is solvable: � [ T d A T i X v A i = ρ 2 X w ⊔ ( X )] w := w = τ d ( v,i ) 1. suppose, w.l.g., d = 0 . 2. Consider the noncommutative simplex, ∆ := { X � 0: trace X = 1 } . This set is compact and convex. 3. Consider the normalized map ˜ T d ⊔ ( X ) = (trace T d ⊔ ( X )) − 1 T d ⊔ ( X ) . It sends ∆ to ∆ 4. By Brouwer fixed point theorem, it has a fixed point 5. This fixed point is an eigenvector of T d
Exercise: find the mistake in the following proof We want to show that the following eigenproblem is solvable: � [ T d A T i X v A i = ρ 2 X w ⊔ ( X )] w := w = τ d ( v,i ) 1. suppose, w.l.g., d = 0 . 2. Consider the noncommutative simplex, ∆ := { X � 0: trace X = 1 } . This set is compact and convex. 3. Consider the normalized map ˜ T d ⊔ ( X ) = (trace T d ⊔ ( X )) − 1 T d ⊔ ( X ) . It sends ∆ to ∆ 4. By Brouwer fixed point theorem, it has a fixed point 5. This fixed point is an eigenvector of T d
Exercise: find the mistake in the following proof We want to show that the following eigenproblem is solvable: � [ T d A T i X v A i = ρ 2 X w ⊔ ( X )] w := w = τ d ( v,i ) 1. suppose, w.l.g., d = 0 . 2. Consider the noncommutative simplex, ∆ := { X � 0: trace X = 1 } . This set is compact and convex. 3. Consider the normalized map ˜ T d ⊔ ( X ) = (trace T d ⊔ ( X )) − 1 T d ⊔ ( X ) . It sends ∆ to ∆ 4. By Brouwer fixed point theorem, it has a fixed point 5. This fixed point is an eigenvector of T d � ⊔ is continuous in int S + n × int S + n , but not on its closure.
Exercise: find the mistake in the following proof We want to show that the following eigenproblem is solvable: � [ T d A T i X v A i = ρ 2 X w ⊔ ( X )] w := w = τ d ( v,i ) 1. suppose, w.l.g., d = 0 . 2. Consider the noncommutative simplex, ∆ := { X � 0: trace X = 1 } . This set is compact and convex. 3. Consider the normalized map ˜ T d ⊔ ( X ) = (trace T d ⊔ ( X )) − 1 T d ⊔ ( X ) . It sends ∆ to ∆ 4. By Brouwer fixed point theorem, it has a fixed point 5. This fixed point is an eigenvector of T d � ⊔ is continuous in int S + n × int S + n , but not on its closure. → cannot apply naively Brouwer.
Fixing the proof of existence of eigenvectors Lemma For Y i ≻ 0 , we have m m m 1 � � � Y i � Y i � Y i m i =1 i =1 i =1 Corollary For all X ∈ S + n , we have 1 mK d ( X ) � T d ⊔ ( X ) � K d ( X ) , with K d � A T T d � A T w ( X ) = i X v A i ⊔ ,w ( X ) = i X v A i . w = τ d ( v,i ) w = τ d ( v,i )
Proof i A T Reduction to K : X �→ � i XA i strictly positive: X � 0 = ⇒ K ( X ) ≻ 0 .
Proof i A T Reduction to K : X �→ � i XA i strictly positive: X � 0 = ⇒ K ( X ) ≻ 0 . Let X ∈ ∆ := { X � 0: trace X = 1 } . By compactness: αI � K ( X ) � βI , with α > 0 .
Proof i A T Reduction to K : X �→ � i XA i strictly positive: X � 0 = ⇒ K ( X ) ≻ 0 . Let X ∈ ∆ := { X � 0: trace X = 1 } . By compactness: αI � K ( X ) � βI , with α > 0 . Then α mI � T ⊔ ( X ) � βI , so T ⊔ (∆) ⊂ compact subset of int ∆ .
Proof i A T Reduction to K : X �→ � i XA i strictly positive: X � 0 = ⇒ K ( X ) ≻ 0 . Let X ∈ ∆ := { X � 0: trace X = 1 } . By compactness: αI � K ( X ) � βI , with α > 0 . Then α mI � T ⊔ ( X ) � βI , so T ⊔ (∆) ⊂ compact subset of int ∆ . Conclude by Brouwer’s fixed point theorem.
Computing an eigenvector We introduce a damping parameter γ : T γ � � A T � ⊔ ( X ) = i XA i + γ (trace X ) I n . i Theorem The iteration T γ ⊔ ( X ) X k +1 = trace T γ ⊔ ( X ) converges for a large damping: γ > nm (3 d +1) / 2 Conjecture The iteration converges if γ > m 1 / 2 n − 1 / 2 . Experimentally: γ ∼ 10 − 2 is enough! Huge gap between conservative theoretical estimates and practice. How theoretical estimates are obtained?
Lipschitz estimations Riemann and Thompson metrics Two standard metrics on the cone S ++ n d R ( A, B ) := � log spec( A − 1 B ) � 2 . d T ( A, B ) := � log spec( A − 1 B ) � ∞ . They are invariant under the action of congruences: d ( LAL T , LBL T ) = d ( A, B ) for invertible L . d M ( X 1 ⊔ X 2 ,Y 1 ⊔ Y 2 ) Lipschitz constant: Lip M ⊔ := sup d M ( X 1 ⊕ X 2 ,Y 1 ⊕ Y 2 ) . X 1 ,X 2 ,Y 1 ,Y 2 ≻ 0
Lipschitz estimations Riemann and Thompson metrics Two standard metrics on the cone S ++ n d R ( A, B ) := � log spec( A − 1 B ) � 2 . d T ( A, B ) := � log spec( A − 1 B ) � ∞ . They are invariant under the action of congruences: d ( LAL T , LBL T ) = d ( A, B ) for invertible L . d M ( X 1 ⊔ X 2 ,Y 1 ⊔ Y 2 ) Lipschitz constant: Lip M ⊔ := sup d M ( X 1 ⊕ X 2 ,Y 1 ⊕ Y 2 ) . X 1 ,X 2 ,Y 1 ,Y 2 ≻ 0 Theorem Lip T ⊔ = Θ(log n ) Lip R ⊔ = 1 Proof. d T , d R are Riemann/Finsler metrics → work locally + Schur multiplier estimation (Mathias).
Scalability: dimension Table: big-LMI vs Tropical Kraus Dimension CPU time CPU time Error vs LMI n (tropical) (LMI) 5 0 . 9 s 3 . 1 s 0 . 1 % 10 1 . 5 s 4 . 2 s 1 . 4 % 20 3 . 5 s 31 s 0 . 4 % 30 7 . 9 s 3 min 0 . 2 % 40 13 . 7 s 18 min 0 . 05 % 45 18 . 1 s − − 50 25 . 2 s − − 100 1 min − − 500 8 min − −
Introduction Minimal upper bounds Noncommutative Dynamic Programming Risk sensitive eigenproblem Concluding remarks Figure: Computation time vs dimension 27 / 38
Introduction Minimal upper bounds Noncommutative Dynamic Programming Risk sensitive eigenproblem Concluding remarks Scalability: graph size − 1 1 − 1 − 1 1 − 1 A 1 = − 1 − 1 1 A 2 = − 1 − 1 0 0 1 1 1 1 1 Table: big-LMI vs Tropical Kraus: 30 − 60 times faster. Order d 2 4 6 8 10 Size of graph 8 32 128 512 2048 CPU time 0 . 03 s 0 . 07 s 0 . 4 s 2 . 0 s 9 . 0 s (tropical) CPU time 1 . 9 s 4 . 0 s 24 s 1 min 10 min (LMI) Accuracy 1 . 1 % 1 . 3 % 0 . 4 % 0 . 4 % 0 . 6 % 28 / 38
Special case of nonnegative matrices Suppose A i ∈ R n × n , replace the quantum dynamic programming + operator n ) ( m d ) X ∈ ( S + T d � A T and w ( X ) := i X v A i w = τ d ( v,i )
Special case of nonnegative matrices Suppose A i ∈ R n × n , replace the quantum dynamic programming + operator n ) ( m d ) X ∈ ( S + T d � A T and w ( X ) := i X v A i w = τ d ( v,i ) by the classical dynamic programming operator + ) ( m d ) x ∈ ( R n T d A T and w ( x ) := sup i x v w = τ d ( v,i )
Special case of nonnegative matrices Suppose A i ∈ R n × n , replace the quantum dynamic programming + operator n ) ( m d ) X ∈ ( S + T d � A T and w ( X ) := i X v A i w = τ d ( v,i ) by the classical dynamic programming operator + ) ( m d ) x ∈ ( R n T d A T and w ( x ) := sup i x v w = τ d ( v,i ) Operators of this type arise in risk-sensitive control Anantharam, Borkar, also in games of topological entropy Asarin, Cervelle, Degorre, Dima, Horn, Kozyakin, Akian, SG, Grand-Cl´ ement, Guillaud.
Special case of nonnegative matrices Suppose A i ∈ R n × n , replace the quantum dynamic programming + operator n ) ( m d ) X ∈ ( S + T d � A T and w ( X ) := i X v A i w = τ d ( v,i ) by the classical dynamic programming operator + ) ( m d ) x ∈ ( R n T d A T and w ( x ) := sup i x v w = τ d ( v,i ) Operators of this type arise in risk-sensitive control Anantharam, Borkar, also in games of topological entropy Asarin, Cervelle, Degorre, Dima, Horn, Kozyakin, Akian, SG, Grand-Cl´ ement, Guillaud. Theorem Suppose the set of nonnegative matrices A is positively irreducible. Then, there exists u ∈ ( R + ) ( m d ) \ { 0 } such that T d ( u ) = λ d u .
Special case of nonnegative matrices Suppose A i ∈ R n × n , replace the quantum dynamic programming + operator n ) ( m d ) X ∈ ( S + T d � A T and w ( X ) := i X v A i w = τ d ( v,i ) by the classical dynamic programming operator + ) ( m d ) x ∈ ( R n T d A T and w ( x ) := sup i x v w = τ d ( v,i ) Operators of this type arise in risk-sensitive control Anantharam, Borkar, also in games of topological entropy Asarin, Cervelle, Degorre, Dima, Horn, Kozyakin, Akian, SG, Grand-Cl´ ement, Guillaud. Theorem Suppose the set of nonnegative matrices A is positively irreducible. Then, there exists u ∈ ( R + ) ( m d ) \ { 0 } such that T d ( u ) = λ d u . Follows from SG and Gunawardena, TAMS 2004.
A monotone hemi-norm is a map ν ( x ) := max v ∈ V � u v , x � with u v � 0 such that x �→ ν ( x ) ∨ ν ( − x ) is a norm. Theorem (Coro. of Guglielmi and Protasov ) If A ⊂ R n × n is positively irreducible, there is a monotone hemi-norm ν + such that ∀ x ∈ R n i ∈ [ m ] ν ( A i x ) = ρ ( A ) ν ( x ) , max + Theorem (Polyhedral monotone hemi-norms) If A ⊂ R n × n is positively irreducible, if T d ( u ) = λ d u , and + + ) ( m d ) \ { 0 } , then u ∈ ( R n � x � u := max v ∈ [ m d ] � u v , x � is a polyhedral monotone hemi-norm and i ∈ [ m ] � A i x � u � λ d � x � u . max
A monotone hemi-norm is a map ν ( x ) := max v ∈ V � u v , x � with u v � 0 such that x �→ ν ( x ) ∨ ν ( − x ) is a norm. Theorem (Coro. of Guglielmi and Protasov ) If A ⊂ R n × n is positively irreducible, there is a monotone hemi-norm ν + such that ∀ x ∈ R n i ∈ [ m ] ν ( A i x ) = ρ ( A ) ν ( x ) , max + Theorem (Polyhedral monotone hemi-norms) If A ⊂ R n × n is positively irreducible, if T d ( u ) = λ d u , and + + ) ( m d ) \ { 0 } , then u ∈ ( R n � x � u := max v ∈ [ m d ] � u v , x � is a polyhedral monotone hemi-norm and i ∈ [ m ] � A i x � u � λ d � x � u . max Moreover, ρ ( A ) � λ d � n 1 / ( d +1) ρ ( A ) , in particular λ d → λ as d → ∞ .
How to compute λ such that T d ( u ) = λu for some u � = 0 , u � = 0
How to compute λ such that T d ( u ) = λu for some u � = 0 , u � = 0 • Policy iteration: Rothblum
How to compute λ such that T d ( u ) = λu for some u � = 0 , u � = 0 • Policy iteration: Rothblum • Spectral simplex: Protasov
How to compute λ such that T d ( u ) = λu for some u � = 0 , u � = 0 • Policy iteration: Rothblum • Spectral simplex: Protasov • non-linear Collatz-Wielandt theorem + convex programming = ⇒ polytime : Akian, SG, Grand-Cl´ ement, Guillaud (ACM TOCS 2019)
How to compute λ such that T d ( u ) = λu for some u � = 0 , u � = 0 • Policy iteration: Rothblum • Spectral simplex: Protasov • non-linear Collatz-Wielandt theorem + convex programming = ⇒ polytime : Akian, SG, Grand-Cl´ ement, Guillaud (ACM TOCS 2019)
How to compute λ such that T d ( u ) = λu for some u � = 0 , u � = 0 • Policy iteration: Rothblum • Spectral simplex: Protasov • non-linear Collatz-Wielandt theorem + convex programming = ⇒ polytime : Akian, SG, Grand-Cl´ ement, Guillaud (ACM TOCS 2019) policy iteration/spectral simplex requires computing eigenvalues (demanding), need to work with huge scale instances (dimension N = n × m d )
Krasnoselski-Mann iteration x k +1 = 1 2( x k + F ( x k )) applies to a nonexpansive map F : � F ( x ) − F ( y ) � � � x − y � .
Krasnoselski-Mann iteration x k +1 = 1 2( x k + F ( x k )) applies to a nonexpansive map F : � F ( x ) − F ( y ) � � � x − y � . Theorem ( Ishikawa ) Let D be a closed convex subset of a Banach space X , let F be a nonexpansive mapping sending D to a compact subset of D . Then, for any initial point x 0 ∈ D , the sequence x k converges to a fixed point of F .
Krasnoselski-Mann iteration x k +1 = 1 2( x k + F ( x k )) applies to a nonexpansive map F : � F ( x ) − F ( y ) � � � x − y � . Theorem ( Ishikawa ) Let D be a closed convex subset of a Banach space X , let F be a nonexpansive mapping sending D to a compact subset of D . Then, for any initial point x 0 ∈ D , the sequence x k converges to a fixed point of F . Theorem ( Baillon, Bruck ) � F ( x k ) − x k � � 2 diam( D ) √ , πk
Definition (Projective Krasnoselskii-Mann iteration) Suppose f : R N + → R N + is order preserving and positively homogeneous of degree 1 . Choose any v 0 ∈ R N i ∈ [ N ] v 0 > 0 such that � i = 1 , � 1 / 2 � f ( v k ) v k +1 = � ◦ v k , (1) � G f ( v k ) where x ◦ y := ( x i y i ) and G ( x ) = ( x 1 · · · x N ) 1 /N .
Definition (Projective Krasnoselskii-Mann iteration) Suppose f : R N + → R N + is order preserving and positively homogeneous of degree 1 . Choose any v 0 ∈ R N i ∈ [ N ] v 0 > 0 such that � i = 1 , � 1 / 2 � f ( v k ) v k +1 = � ◦ v k , (1) � G f ( v k ) where x ◦ y := ( x i y i ) and G ( x ) = ( x 1 · · · x N ) 1 /N . Theorem Suppose in addition that f has a positive eigenvector. Then, the projective Krasnoselskii-Mann iteration initialized at any positive vector v 0 ∈ R N i ∈ [ N ] v 0 + such that � i = 1 , converges towards an eigenvector of f , and G ( f ( v k )) converges to the maximal eigenvalue of f .
Definition (Projective Krasnoselskii-Mann iteration) Suppose f : R N + → R N + is order preserving and positively homogeneous of degree 1 . Choose any v 0 ∈ R N i ∈ [ N ] v 0 > 0 such that � i = 1 , � 1 / 2 � f ( v k ) v k +1 = � ◦ v k , (1) � G f ( v k ) where x ◦ y := ( x i y i ) and G ( x ) = ( x 1 · · · x N ) 1 /N . Theorem Suppose in addition that f has a positive eigenvector. Then, the projective Krasnoselskii-Mann iteration initialized at any positive vector v 0 ∈ R N i ∈ [ N ] v 0 + such that � i = 1 , converges towards an eigenvector of f , and G ( f ( v k )) converges to the maximal eigenvalue of f . Proof idea. This is Krasnoselski iteration applied to F := log ◦ f ◦ exp acting in the quotient of the normed space ( R n , � · � ∞ ) by the one dimensional subspace R 1 N .
Corollary Take f := T d , the risk-sensitive dynamic programming operator, and let i ∈ [ N ] ( f ( v k )) i /v k β k := max i . Then, 4 d H ( v 0 , u ) + log n log ρ ( A ) � log β k � log ρ ( A ) + √ d + 1 πk where d H is Hilbert’s projective metric.
Level d CPU Time (s) Eigenvalue λ d Relative error 1 0 . 01 2 . 165 6 . 8 % 2 0 . 01 2 . 102 3 . 7 % 3 0 . 01 2 . 086 2 . 9 % 4 0 . 01 2 . 059 1 . 6 % 5 0 . 02 2 . 041 0 . 7 % 6 0 . 05 2 . 030 0 . 1 % 7 0 . 7 2 . 027 0 . 0 % 8 0 . 32 2 . 027 0 . 0 % 9 1 . 12 2 . 027 0 . 0 % Table: Convergence of the hierarchy on an instance with 5 × 5 matrices and a maximizing cyclic product of length 6
Dimension n Level d Eigenvalue λ d CPU Time 10 2 4 . 287 0 . 01 s 3 4 . 286 0 . 03 s 20 2 8 . 582 0 . 01 s 3 8 . 576 0 . 03 s 50 2 22 . 34 0 . 04 s 3 22 . 33 0 . 16 s 100 2 44 . 45 0 . 17 s 3 44 . 45 0 . 53 s 200 2 89 . 77 0 . 71 s 3 89 . 76 2 . 46 s 500 2 224 . 88 5 . 45 s 3 224 . 88 19 . 7 s 1000 2 449 . 87 44 . 0 s 3 449 . 87 2 . 7 min 2000 2 889 . 96 4 . 6 min 3 889 . 96 19 . 2 min 5000 2 2249 . 69 51 . 9 min 3 2249 . 57 3 . 3 h Table: Computation time for large matrices
MEGA The Minimal Ellipsoid Geometric Analyzer, Stott - available from http://www.cmap.polytechnique.fr/~stott/ • implements the quantum dynamic programming approach • 1700 lines of OCaml and 800 lines of Matlab • uses BLAS/LAPACK via LACAML for linear algebra • uses OSDP/CSDP for some semidefinite programming • uses Matlab for other semidefinite programming
Concluding remarks • Reduced the approximation of the joint spectral radius to solving non-linear eigenproblems
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