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Guideline Introduction String Functions Solution of the Identifiability Problem A Simple and Efficient Solution of the Identifiability Problem for Hidden Markov Models and Quantum Random Walks Alexander Schönhuth Pacific Institute for the Mathematical Sciences School of Computing Science Simon Fraser University February 2009 Alexander Schönhuth Identifiability Problem

Guideline Introduction String Functions Solution of the Identifiability Problem Guideline Introduction 1 Identifiability Problem Hidden Markov Processes (HMPs) Quantum Random Walks (QRWs) String Functions 2 Stochastic Processes as String Functions Hankel Matrices and Dimension of String Functions Observable Operators Dimension of HMPs and QRWs Minimal Representations Solution of the Identifiability Problem 3 Computational Bottleneck Key Insight Algorithm Alexander Schönhuth Identifiability Problem

Guideline Identifiability Problem Introduction Hidden Markov Processes (HMPs) String Functions Quantum Random Walks (QRWs) Solution of the Identifiability Problem Identifiability Problem Situation : Φ : P → S where P is a set of parameterizations and S is the corresponding set of stochastic processes. Definition A stochastic process Φ( P ) as induced by the parameterization P is said to be identifiable iff Φ − 1 (Φ( P )) = { P } (1) Alexander Schönhuth Identifiability Problem

Guideline Identifiability Problem Introduction Hidden Markov Processes (HMPs) String Functions Quantum Random Walks (QRWs) Solution of the Identifiability Problem Hidden Markov Processes (HMPs) 0.5 0.45 0.25 0.3 0.25 0.25 Initial probabilities π = ( 0 . 8 , 0 . 2 ) T a b c a b c Transition probabilities M = ( m ij := P ( i → j )) i , j = 1 , 2 0.5 � 0 . 3 � 0 . 7 1 2 = 0.5 0.3 0 . 5 0 . 5 0.7 Emission probabilities, 0.8 0.2 e.g. e 1 b = 0 . 5 , e 2 c = 0 . 45. START Alexander Schönhuth Identifiability Problem

Guideline Identifiability Problem Introduction Hidden Markov Processes (HMPs) String Functions Quantum Random Walks (QRWs) Solution of the Identifiability Problem Hidden Markov Processes (HMPs) 0.5 0.45 0.25 0.3 0.25 0.25 Initial probabilities π = ( 0 . 8 , 0 . 2 ) T a b c a b c Transition probabilities M = ( m ij := P ( i → j )) i , j = 1 , 2 0.5 � 0 . 3 � 0 . 7 1 2 = 0.5 0.3 0 . 5 0 . 5 0.7 Emission probabilities, 0.8 0.2 e.g. e 1 b = 0 . 5 , e 2 c = 0 . 45. START Random source ( X t ) with values in Σ = { a , b , c } : e.g.: P X ( X 1 = a , X 2 = b ) = π 1 e 1 a ( a 11 e 1 b + a 12 e 2 b ) + π 2 e 2 a ( a 21 e 1 b + a 22 e 2 b ) Alexander Schönhuth Identifiability Problem

Guideline Identifiability Problem Introduction Hidden Markov Processes (HMPs) String Functions Quantum Random Walks (QRWs) Solution of the Identifiability Problem Quantum Random Walks (QRWs) A QRW Q = ( G , U , ψ 0 ) consists of a directed graph G = ( V , E ) , a unitary operator U : C | E | → C | E | and a wave function ψ 0 ∈ C | E | Alexander Schönhuth Identifiability Problem

Guideline Identifiability Problem Introduction Hidden Markov Processes (HMPs) String Functions Quantum Random Walks (QRWs) Solution of the Identifiability Problem Quantum Random Walks (QRWs) A QRW Q = ( G , U , ψ 0 ) consists of a directed graph G = ( V , E ) , a unitary operator U : C | E | → C | E | and a wave function ψ 0 ∈ C | E | Classical random source associated with QRW Q = ( G , U , ψ o ) : Sequences of symbols v 0 ... v t v t + 1 ... from V Underlying sequences of states ψ o ...ψ t ψ t + 1 ... from C | E | Alexander Schönhuth Identifiability Problem

Guideline Identifiability Problem Introduction Hidden Markov Processes (HMPs) String Functions Quantum Random Walks (QRWs) Solution of the Identifiability Problem Quantum Random Walks (QRWs) A QRW Q = ( G , U , ψ 0 ) consists of a directed graph G = ( V , E ) , a unitary operator U : C | E | → C | E | and a wave function ψ 0 ∈ C | E | Classical random source associated with QRW Q = ( G , U , ψ o ) : Sequences of symbols v 0 ... v t v t + 1 ... from V Underlying sequences of states ψ o ...ψ t ψ t + 1 ... from C | E | Mechanism: e ∈ E , e =( v t , u ) | ( U ψ t ) e | 2 . Generate symbol v t ∈ V with probability � e ∈ E , e =( v t , x ) | ( U ψ ) e | 2 ) · � ∈ C | E | ψ t + 1 = ( 1 / � e ∈ E , e =( v , u ) ( U ψ ) e Return to first step. Alexander Schönhuth Identifiability Problem

Guideline Identifiability Problem Introduction Hidden Markov Processes (HMPs) String Functions Quantum Random Walks (QRWs) Solution of the Identifiability Problem Identifiability Problem Identifiability Problem Given the parameterizations of two HMPs M 1 , M 2 or two QRWs Q 1 , Q 2 , decide whether the associated random processes p 1 , p 2 are equivalent. Input : Two parameterizations of two HMPs M 1 , M 2 or two QRWs Q 1 , Q 2 . Output: Yes, if p 1 = p 2 , no else. Solution for HMPs: Ito, Amari and Kobayashi, IEEE Tr. Inf. Th., 1992. Algorithm is exponential in the number of hidden states. No solution for QRWs known! Alexander Schönhuth Identifiability Problem

Stochastic Processes as String Functions Guideline Hankel Matrices and Dimension of String Functions Introduction Observable Operators String Functions Dimension of HMPs and QRWs Solution of the Identifiability Problem Minimal Representations String Functions Let Σ ∗ := ∪ t ≥ 0 Σ t be the set of all strings of finite length over an alphabet Σ . Treat random processes ( X t ) with values in Σ as string functions p X : Σ ∗ → R by p X ( v = v 0 v 1 ... v t ) := P ( X 0 = v o , X 1 = v 1 , ..., X t = v t ) . By standard arguments: ∀ v ∈ Σ ∗ : ( X t ) = ( Y t ) ⇔ p X ( v ) = p Y ( v ) . Alexander Schönhuth Identifiability Problem

Stochastic Processes as String Functions Guideline Hankel Matrices and Dimension of String Functions Introduction Observable Operators String Functions Dimension of HMPs and QRWs Solution of the Identifiability Problem Minimal Representations Dimension of String Functions The Hankel Matrix Let wv = w 1 ... w m v 1 ... v n ∈ Σ m + n be the concatenation of two strings w = w 1 ... w m ∈ Σ s , v = v 1 ... v n ∈ Σ t . Consider the (infinite-dimensional) Hankel matrix P p := [ p ( wv )] v , w ∈ Σ ∗ ∈ R Σ ∗ × Σ ∗ ∼ = R N × N . for a string function p : Σ ∗ → R . Alexander Schönhuth Identifiability Problem

Stochastic Processes as String Functions Guideline Hankel Matrices and Dimension of String Functions Introduction Observable Operators String Functions Dimension of HMPs and QRWs Solution of the Identifiability Problem Minimal Representations Dimension of String Functions The Hankel Matrix Let wv = w 1 ... w m v 1 ... v n ∈ Σ m + n Example : Let Σ = { 0 , 1 } . be the concatenation of two p ( � ) p ( 0 ) p ( 1 ) strings w = w 1 ... w m ∈ Σ s , v =  . . .  p ( 0 ) p ( 00 ) p ( 10 ) v 1 ... v n ∈ Σ t . . . .    p ( 1 ) p ( 01 ) p ( 11 )  . . .   Consider the (infinite-dimensional) P p =  p ( 00 ) p ( 000 ) p ( 100 )  . . .   Hankel matrix  p ( 01 ) p ( 001 ) p ( 101 )  . . .   P p := [ p ( wv )] v , w ∈ Σ ∗ ∈ R Σ ∗ × Σ ∗ ∼  . . .  = R N × N . ... . . . . . . for a string function p : Σ ∗ → R . Alexander Schönhuth Identifiability Problem

Stochastic Processes as String Functions Guideline Hankel Matrices and Dimension of String Functions Introduction Observable Operators String Functions Dimension of HMPs and QRWs Solution of the Identifiability Problem Minimal Representations Dimension of String Functions The Hankel Matrix Let wv = w 1 ... w m v 1 ... v n ∈ Σ m + n Example : Let Σ = { 0 , 1 } . be the concatenation of two p ( � ) p ( 0 ) p ( 1 ) strings w = w 1 ... w m ∈ Σ s , v =  . . .  p ( 0 ) p ( 00 ) p ( 10 ) v 1 ... v n ∈ Σ t . . . .    p ( 1 ) p ( 01 ) p ( 11 )  . . .   Consider the (infinite-dimensional) P p =  p ( 00 ) p ( 000 ) p ( 100 )  . . .   Hankel matrix  p ( 01 ) p ( 001 ) p ( 101 )  . . .   P p := [ p ( wv )] v , w ∈ Σ ∗ ∈ R Σ ∗ × Σ ∗ ∼  . . .  = R N × N . ... . . . . . . for a string function p : Σ ∗ → R . We define the dimension of p to be dim p := rk P p ∈ N ∪ {∞} . Alexander Schönhuth Identifiability Problem

Stochastic Processes as String Functions Guideline Hankel Matrices and Dimension of String Functions Introduction Observable Operators String Functions Dimension of HMPs and QRWs Solution of the Identifiability Problem Minimal Representations Observable Operators Let p v resp. p w be the row resp. column vector of P p referring to strings v resp. w. Alexander Schönhuth Identifiability Problem

Stochastic Processes as String Functions Guideline Hankel Matrices and Dimension of String Functions Introduction Observable Operators String Functions Dimension of HMPs and QRWs Solution of the Identifiability Problem Minimal Representations Observable Operators Let p v resp. p w be the row resp. column vector of P p referring to strings v resp. w. Definition The linear operators R Σ ∗ R Σ ∗ ρ v , τ w : − → p v , p w p �→ for v , w ∈ Σ ∗ are called observable operators. Alexander Schönhuth Identifiability Problem