P-recursive moment sequences of piecewise D-finite functions and Prony-type algebraic systems Dmitry Batenkov Gal Binyamini Yosef Yomdin Weizmann Institute of Science, Israel 18th International Conference on Difference Equations and Applications July 23-27, 2012, Barcelona D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 1 / 27
1 Prony-type systems D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 2 / 27
Linear recurrences with constant coefficients Definition k = 0 ∈ C ω is C -recurrent if ∃ A 0 ,..., A d ∈ C such The sequence { m k } ∞ that ∀ k ∈ N : A 0 m k + A 1 m k + 1 + ··· + A d m k + d = 0 . D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 3 / 27
Linear recurrences with constant coefficients Definition k = 0 ∈ C ω is C -recurrent if ∃ A 0 ,..., A d ∈ C such The sequence { m k } ∞ that ∀ k ∈ N : A 0 m k + A 1 m k + 1 + ··· + A d m k + d = 0 . General form of solution Exponential polynomials (Binet’s formula) K P i ( k ) ξ k ∑ m k = i i = 1 where { ξ i } are the roots of the characteristic polynomial A 0 + A 1 x + ··· + A d x d . D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 3 / 27
Prony system K P i ( k ) ξ k ∑ m k = i i = 1 Reconstruction problem Given few initial terms m 0 ,..., m N , reconstruct { ξ i , P i } . D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 4 / 27
Prony system K P i ( k ) ξ k ∑ m k = i i = 1 Reconstruction problem Given few initial terms m 0 ,..., m N , reconstruct { ξ i , P i } . Examples Padé approximation: { m k } are Taylor coefficients of a � � ξ − 1 rational function with poles at i D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 4 / 27
Prony system K P i ( k ) ξ k ∑ m k = i i = 1 Reconstruction problem Given few initial terms m 0 ,..., m N , reconstruct { ξ i , P i } . Examples Padé approximation: { m k } are Taylor coefficients of a � � ξ − 1 rational function with poles at i High resolution methods in Signal Processing D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 4 / 27
Example: finite rate of innovation Problem: recovering a signal which has been sampled below Nyquist rate D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 5 / 27
Example: finite rate of innovation Problem: recovering a signal which has been sampled below Nyquist rate Assumption: the signal is finite-parametric. For example: K ∑ x ( t ) = a j δ ( t − ξ j ) j = 0 D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 5 / 27
Example: finite rate of innovation Problem: recovering a signal which has been sampled below Nyquist rate Assumption: the signal is finite-parametric. For example: K ∑ x ( t ) = a j δ ( t − ξ j ) j = 0 Method: choose a sampling kernel h ( t ) with certain algebraic properties s.t. K a j e − ı ξ j n ∑ y n = � h ( t − n ) , x ( t ) � = j = 0 D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 5 / 27
Example: finite rate of innovation Problem: recovering a signal which has been sampled below Nyquist rate Assumption: the signal is finite-parametric. For example: K ∑ x ( t ) = a j δ ( t − ξ j ) j = 0 Method: choose a sampling kernel h ( t ) with certain algebraic properties s.t. K a j e − ı ξ j n ∑ y n = � h ( t − n ) , x ( t ) � = j = 0 Generalized to piecewise polynomials D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 5 / 27
Prony solution method K K ∑ P i ( k ) ξ k ∑ m k = deg P i = C i ; i = 1 i = 1 D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 6 / 27
Prony solution method K K ∑ P i ( k ) ξ k ∑ m k = deg P i = C i ; i = 1 i = 1 1 Solve Hankel-type system ··· m 0 m 1 m C − 1 A 0 m C ··· m 1 m 2 m C A 1 m C + 1 × = − . . . . . . . . . . . . . . . . . . ··· m C − 1 m d + 1 m 2 C − 1 A C − 1 m 2 C � �� � def = M D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 6 / 27
Prony solution method K K ∑ P i ( k ) ξ k ∑ m k = deg P i = C i ; i = 1 i = 1 1 Solve Hankel-type system ··· m 0 m 1 m C − 1 A 0 m C ··· m 1 m 2 m C A 1 m C + 1 × = − . . . . . . . . . . . . . . . . . . ··· m C − 1 m d + 1 m 2 C − 1 A C − 1 m 2 C � �� � def = M 2 � � are the roots of x d + A d − 1 x d − 1 + ··· + A 1 x + A 0 = 0 . ξ j D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 6 / 27
Prony solution method K K ∑ P i ( k ) ξ k ∑ m k = deg P i = C i ; i = 1 i = 1 1 Solve Hankel-type system ··· m 0 m 1 m C − 1 A 0 m C ··· m 1 m 2 m C A 1 m C + 1 × = − . . . . . . . . . . . . . . . . . . ··· m C − 1 m d + 1 m 2 C − 1 A C − 1 m 2 C � �� � def = M 2 � � are the roots of x d + A d − 1 x d − 1 + ··· + A 1 x + A 0 = 0 . ξ j 3 Coefficients of { P i } are found by solving a Vandermonde-type linear system. D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 6 / 27
Subspace methods Observations M = V T BV , with V -confluent Vandermonde. The range of M and V are the same. V has the rotational invariance property : V ↑ = V ↓ J � � where J is the block Jordan matrix with eigenvalues ξ j . D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 7 / 27
Subspace methods Observations M = V T BV , with V -confluent Vandermonde. The range of M and V are the same. V has the rotational invariance property : V ↑ = V ↓ J � � where J is the block Jordan matrix with eigenvalues ξ j . ESPRIT method 1 Compute the SVD M = W Σ V T . ↓ W ↑ . 2 Calculate Φ = W # 3 Set { ξ i } to be the eigenvalues of Φ with appropriate multiplicities. D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 7 / 27
Prony systems - solvability l j − 1 K K ξ k − i ∑ ∑ ∑ m k = a i , j k ( k − 1 ) ····· ( k − i + 1 ) l j = C ; k = 0 , 1 ,..., 2 C − 1 ; j � �� � j = 1 i = 0 j = 1 def = ( k ) i Theorem The Prony system has a solution if and only if the sequence ( m 0 ,..., m 2 C − 1 ) is C -recurrent of length at most C . D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 8 / 27
Prony systems - solvability l j − 1 K K ξ k − i ∑ ∑ ∑ m k = a i , j k ( k − 1 ) ····· ( k − i + 1 ) l j = C ; k = 0 , 1 ,..., 2 C − 1 ; j � �� � j = 1 i = 0 j = 1 def = ( k ) i Theorem The Prony system has a solution if and only if the sequence ( m 0 ,..., m 2 C − 1 ) is C -recurrent of length at most C . Theorem � � The parameters a i , j , ξ j can be uniquely recovered from the first 2 C measurements if and only if 1) ξ i � = ξ j for i � = j , and 2) a l j − 1 , j � = 0 for all j = 1 ,..., K . D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 8 / 27
Prony systems - local stability Theorem (DB,YY 2010) Assume that max k < C | ∆ m k | ≤ ε for sufficiently small ε . Then there exists a positive constant C 1 depending only on the nodes ξ 1 ,..., ξ K and the multiplicities l 1 ,..., l K such that for all i = 1 , 2 ,..., K : C 1 ε j = 0 � � | ∆ a ij | ≤ 1 + | a i , j − 1 | C 1 ε 1 ≤ j ≤ l i − 1 | a i , li − 1 | 1 | ∆ ξ i | ≤ C 1 ε | a i , l i − 1 | This behaviour is observed in experiments D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 9 / 27
Prony systems - local stability Theorem (DB,YY 2010) Assume that max k < C | ∆ m k | ≤ ε for sufficiently small ε . Then there exists a positive constant C 1 depending only on the nodes ξ 1 ,..., ξ K and the multiplicities l 1 ,..., l K such that for all i = 1 , 2 ,..., K : C 1 ε j = 0 � � | ∆ a ij | ≤ 1 + | a i , j − 1 | C 1 ε 1 ≤ j ≤ l i − 1 | a i , li − 1 | 1 | ∆ ξ i | ≤ C 1 ε | a i , l i − 1 | This behaviour is observed in experiments Prony method fails to separate the parameters, worst performance D.Batenkov,G.Binyamini,Y.Yomdin (WIS) Moments of piecewise functions ICDEA 2012 9 / 27
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