Sparse approximation by modified Prony method Gerlind Plonka and Vlada Pototskaia Institut f¨ ur Numerische und Angewandte Mathematik Georg-August-Universit¨ at G¨ ottingen Alba di Canazei, September 19, 2016 Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 1 / 29
Outline Sparse approximation problem for exponential sums 1 Prony’s method 2 The AAK theorem for samples of exponential sums 3 Method for sparse approximation of exponential sums 4 Numerical example 5 Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 2 / 29
Outline Sparse approximation problem for exponential sums 1 Prony’s method 2 The AAK theorem for samples of exponential sums 3 Method for sparse approximation of exponential sums 4 Numerical example 5 Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 3 / 29
Sparse approximation of exponential sums Consider a function of the form N � a j z x f ( x ) = with | z j | < 1 , j j =1 where a j , z j ∈ C . Goal: Find a function n ˜ � z x f ( x ) = ˜ a j ˜ with | ˜ z j | < 1 , j j =1 such that n < N and � f − ˜ f � ≤ ε Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 4 / 29
Discrete sparse approximation problem Consider a sequence of samples f := ( f k ) ∞ k =0 given by N � a j z k f k := f ( k ) = with | z j | < 1 , j j =1 where a j , z j ∈ C . Find a sequence ˜ f := ( ˜ f k ) ∞ Goal: k =0 of the form n ˜ � z k f k = ˜ a j ˜ with | ˜ z j | < 1 , j j =1 such that n < N and � f − ˜ f � ℓ 2 ≤ ε Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 5 / 29
Possible applications We consider here a structured low-rank approximation problem for model reduction. Problem : Low-rank approximation using the SVD destroys the Hankel structure. [Markovsky, 2008] Applications Approximation of special functions by exponential sums, e.g. Bessel functions, or x − 1 / 2 to avoid quadrature methods for Schr¨ odinger equations. [Beylkin, Monzon, 2005], [Hackbusch, 2005] Signal compression by sparse representation of the (discrete) Fourier transform. Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 6 / 29
Our approach (1) Given a sufficiently large number of samples f k , reconstruct z j and a j such that N � a j z k f k = with | z j | < 1 j j =1 using a Prony-like method. Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 7 / 29
Our approach (1) Given a sufficiently large number of samples f k , reconstruct z j and a j such that N � a j z k f k = with | z j | < 1 j j =1 using a Prony-like method. (2) Given the representation (1), find ˜ z j and ˜ a j such that for n ˜ � z k f k = a j ˜ ˜ with | ˜ z j | < 1 j j =1 and n < N we have � f − ˜ f � ℓ 2 ≤ ε using the AAK Theorem [Adamjan, Arov, Krein], (1971). Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 7 / 29
Outline Sparse approximation problem for exponential sums 1 Prony’s method 2 The AAK theorem for samples of exponential sums 3 Method for sparse approximation of exponential sums 4 Numerical example 5 Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 8 / 29
Classical Prony’s Method (1795) Assume N � a j z k with z j := e T j f ( k ) = j j =1 Given: N and f k := f ( k ) for k = 0 , . . . , 2 N − 1 Wanted: z j ∈ C , a j ∈ C Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 9 / 29
Classical Prony’s Method (1795) Assume N � a j z k with z j := e T j f ( k ) = j j =1 Given: N and f k := f ( k ) for k = 0 , . . . , 2 N − 1 Wanted: z j ∈ C , a j ∈ C Consider the Prony polynomial N N � � p k x k , P ( x ) := ( x − z j ) = with p N = 1 . j =1 k =0 Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 9 / 29
Classical Prony’s Method (1795) Assume N � a j z k with z j := e T j f ( k ) = j j =1 Given: N and f k := f ( k ) for k = 0 , . . . , 2 N − 1 Wanted: z j ∈ C , a j ∈ C Consider the Prony polynomial N N � � p k x k , P ( x ) := ( x − z j ) = with p N = 1 . j =1 k =0 We have for l = 0 , . . . , N − 1 N N N N N a j z ( l + k ) � � � � a j z l � p k z k p k f l + k = p k = j = 0 j j k =0 k =0 j =1 j =1 k =0 Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 9 / 29
Classical Prony method (1795) Since p N = 1 , the last equation can be written as N N − 1 N − 1 � � � p k f l + k = p k f l + k + f l + N = 0 ⇔ p k f l + k = − f l + N k =0 k =0 k =0 and defines a homogeneous difference equation of order N . Matrix-vector-representation: f 0 f 1 · · · f N − 1 p 0 f N f 1 f 2 · · · f N p 1 f N +1 . . . . = − . . ... . . . . . . . . . . f N − 1 f N · · · f 2 N − 2 p N − 1 f 2 N − 1 Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 10 / 29
Literature [Prony] (1795) Reconstruction of a difference equation [Schmidt] (1979) MUSIC (Multiple Signal Classification) [Roy,Kailath] (1989) ESPRIT (Estimation of signal parameters via rotational invariance techniques) [Hua,Sakar] (1990) Matrix-Pencil method [Stoica,Moses] (2000) Annihilating filters [Potts,Tasche] (2010,2011) Approximate Prony method [Kunis et al.], [Sauer] (2015) Multivariate Prony’s method Golub, Milanfar, Varah (’99); Vetterli, Marziliano, Blu (’02); Maravi´ c, Vetterli (’04); Elad, Milanfar, Golub (’04); Beylkin, Monz´ on (’05,’10); Batenkov, Yomdin (’12,’13); Filbir et al. (’12); Potts, Tasche (’11,’12,’13); Plonka, Wischerhoff (’13, ’16); Peter, Plonka (’13); Cuyt, Lee, Tsai (’16); Diederichs, Iske (’16) .... Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 11 / 29
Outline Sparse approximation problem for exponential sums 1 Prony’s method 2 The AAK theorem for samples of exponential sums 3 Method for sparse approximation of exponential sums 4 Numerical example 5 Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 12 / 29
AAK Theorem for Samples of Exponential Sums Consider the sequence f := ( f k ) ∞ k =0 given by samples N � a j z k f k = f ( k ) = with 0 < | z j | < 1 j j =1 and let D := { z ∈ C : 0 < | z | < 1 } . We define the infinite Hankel matrix · · · f 0 f 1 f 2 · · · f 1 f 2 f 3 = ( f k + j ) ∞ Γ f := · · · f 2 f 3 f 4 k,j =0 . . . ... . . . . . . with respect to f . Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 13 / 29
AAK theorem for samples of exponential sums Then Γ f has the following properties: Γ f defines a compact operator on ℓ 2 = ℓ 2 ( N ) . Γ f has finite rank N . The singular values of Γ f are of the form σ 0 ≥ σ 1 ≥ . . . ≥ σ N − 1 > σ N = . . . = σ ∞ = 0 . Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 14 / 29
AAK theorem for samples of exponential sums Then Γ f has the following properties: Γ f defines a compact operator on ℓ 2 = ℓ 2 ( N ) . Γ f has finite rank N . The singular values of Γ f are of the form σ 0 ≥ σ 1 ≥ . . . ≥ σ N − 1 > σ N = . . . = σ ∞ = 0 . [Young] (1988) An Introduction to Hilbert Space Discrete H ∞ optimization [Chui, Chen] (1992) [Peller] (2000) Hankel Operators and Their Applications [Beylkin,Monz´ on] (2005) On approximation of functions by exponential sums [Andersson et al.] (2011) Sparse approximation of functions using sums of exponentials and AAK theory Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 14 / 29
The AAK theorem (Adamjan, Arov, Krein, 1971) Let f := ( f ( k )) ∞ k =0 be given as before. Let ( σ n , u n ) be a fixed singular pair of Γ f with σ n / ∈ { σ k } k � = n and σ n � = 0 . Then ∞ � u n ( k ) x k P u n ( x ) := k =0 has exactly n zeros ˜ z 1 , . . . , ˜ z n in D , repeated according to multiplicity. Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 15 / 29
The AAK theorem (Adamjan, Arov, Krein, 1971) Let f := ( f ( k )) ∞ k =0 be given as before. Let ( σ n , u n ) be a fixed singular pair of Γ f with σ n / ∈ { σ k } k � = n and σ n � = 0 . Then ∞ � u n ( k ) x k P u n ( x ) := k =0 has exactly n zeros ˜ z 1 , . . . , ˜ z n in D , repeated according to multiplicity. If the ˜ z k are pairwise different, then there are ˜ a 1 , . . . , ˜ a n ∈ C such that for � n � ∞ f = ( ˜ ˜ � f j ) ∞ z kj j =0 = ˜ a k ˜ k =1 j =0 it holds that f � ℓ 2 → ℓ 2 = σ n . � Γ f − Γ ˜ Gerlind Plonka, Vlada Pototskaia Prony method for sparse approximation Canazei 2016 15 / 29
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