Error-Correcting Sparse Interpolation in the Chebyshev Basis Andrew - PowerPoint PPT Presentation

Error-Correcting Sparse Interpolation in the Chebyshev Basis Andrew Arnold* Erich Kaltofen University of Waterloo North Carolina State University a4arnold@uwaterloo.ca kaltofen@math.ncsu.edu AndrewArnold.ca kaltofen.us ISSAC ’15, Bath, UK 2015-07-08

Introduction Let T n ( x ) be the n th Chebyshev polynomial of the first kind: T 0 = 1 , T 1 = x , T n ( x ) = 2 xT n − 1 ( x ) − T n − 2 ( x ) , for n ≥ 2 T 4 = 8 x 4 − 8 x 2 + 1 , E.g. T 2 = 2 x 2 − 1 , T 3 = 4 x 3 − 3 x , . . . Useful Properties: ▶ T n ( T m ( x )) = T mn ( x ) ▶ T m ( x ) T n ( x ) = 1 2 ( T m + n ( x ) + T | m − n | ( x )) 2 ( x n + x − n ) ▶ T n ( x + x − 1 ) = 1 2 ▶ Over R , for | ξ | > 1, T n ( ξ ) ̸ = 0 We are interested in the interpolation and error-correcting properties of sparse Chebyshev polynomials (i.e. polynomials that are sparse in the Chebyshev basis). 2

Contributions Covered in this talk An error-correcting black-box interpolation algorithm for sparse Chebyshev polynomials. Not covered in this talk An alternative sparse Chebyshev interpolation algorithm for f ∈ K[ x ], char (K) ̸ = 2, that reduces the problem to sparse interpolation in the power basis, i.e. 1 , x , x 2 , x 3 , . . . . ▶ Allows for early termination (Kaltofen, Lee; 2003), such that we can (probabilistically) interpolate f with t terms, with cost sensitive to t even when bounds for t are not supplied as input. 3

The Problem ▶ Suppose f ∈ R [ x ] is of the form t ∑ f = c i T δ i ( x ) � �� � i =1 c i ̸ =0 and with δ 1 >δ 2 >...>δ t ▶ We are given a black box ■ for f . For j = 0 , 1 , . . . , L − 1, we query ■ with x j and get back a j , where either a j = f ( x j ) or a j is an erroneous evaluation. f ? x j a j = ■ ( x j ) = f ( x j ) ▶ Problem: reconstruct f and identify errors, given ■ and bounds B ≥ t , D ≥ deg( f ) , E ≥ # { errors } while minimizing, L the number of queries to ■ . 4

Interpolating with Errors - Example E.g. deg( f ) ≤ D = 19, ≤ B = 3 terms, ≤ E = 5 errors 5

Interpolating with Errors - Example E.g. deg( f ) ≤ D = 19, ≤ B = 3 terms, ≤ E = 5 errors ▶ Minimizing ℓ 2 -error gives a dense approximation for the model, 0 . 786462 T 19 − 0 . 253808 T 19 − 0 . 270838 T 18 + 0 . 101009 T 16 + 0 . 206344 T 15 − 0 . 135857 T 15 − 0 . 076361 T 14 + 0 . 051550 T 12 − 0 . 699793 T 12 + 0 . 003612 T 10 − 0 . 473865 T 10 + 0 . 352537 T 8 − 0 . 307681 T 8 − 1 . 054240 T 7 + 0 . 753950 T 5 − 0 . 112232 T 5 − 1 . 388821 T 4 + 1 . 025795 T 2 + 1 . 364547 T 1 + 3 . 325460 T 0 5

Interpolating with Errors - Example E.g. deg( f ) ≤ D = 19, ≤ B = 3 terms, ≤ E = 5 errors ▶ Minimizing ℓ 2 -error gives a dense approximation for the model, 0 . 786462 T 19 − 0 . 253808 T 19 − 0 . 270838 T 18 + 0 . 101009 T 16 + 0 . 206344 T 15 − 0 . 135857 T 15 − 0 . 076361 T 14 + 0 . 051550 T 12 − 0 . 699793 T 12 + 0 . 003612 T 10 − 0 . 473865 T 10 + 0 . 352537 T 8 − 0 . 307681 T 8 − 1 . 054240 T 7 + 0 . 753950 T 5 − 0 . 112232 T 5 − 1 . 388821 T 4 + 1 . 025795 T 2 + 1 . 364547 T 1 + 3 . 325460 T 0 ▶ But if we identify 3 errors... 5

Interpolating with Errors - Example E.g. deg( f ) ≤ D = 19, ≤ B = 3 terms, ≤ E = 5 errors ▶ Minimizing ℓ 2 -error gives a dense approximation for the model, 0 . 786462 T 19 − 0 . 253808 T 19 − 0 . 270838 T 18 + 0 . 101009 T 16 + 0 . 206344 T 15 − 0 . 135857 T 15 − 0 . 076361 T 14 + 0 . 051550 T 12 − 0 . 699793 T 12 + 0 . 003612 T 10 − 0 . 473865 T 10 + 0 . 352537 T 8 − 0 . 307681 T 8 − 1 . 054240 T 7 + 0 . 753950 T 5 − 0 . 112232 T 5 − 1 . 388821 T 4 + 1 . 025795 T 2 + 1 . 364547 T 1 + 3 . 325460 T 0 ▶ But if we identify 3 errors...we get a sparse fit T 15 − 2 T 11 + T 2 5

Interpolating with Errors - Example E.g. deg( f ) ≤ D = 19, ≤ B = 3 terms, ≤ E = 5 errors ▶ Minimizing ℓ 2 -error gives a dense approximation for the model, 0 . 786462 T 19 − 0 . 253808 T 19 − 0 . 270838 T 18 + 0 . 101009 T 16 + 0 . 206344 T 15 − 0 . 135857 T 15 − 0 . 076361 T 14 + 0 . 051550 T 12 − 0 . 699793 T 12 + 0 . 003612 T 10 − 0 . 473865 T 10 + 0 . 352537 T 8 − 0 . 307681 T 8 − 1 . 054240 T 7 + 0 . 753950 T 5 − 0 . 112232 T 5 − 1 . 388821 T 4 + 1 . 025795 T 2 + 1 . 364547 T 1 + 3 . 325460 T 0 ▶ But if we identify 3 errors...we get a sparse fit T 15 − 2 T 11 + T 2 5

Related Work - Sparse Interpolation in Power Basis Theorem (Prony; 1795) f ( x ) = ∑ t i =1 c i x e i ∈ K[ x ] , Let K be a field, ω ∈ K , and t − 1 t ∏ ∑ ( y − ω e i ) = y t + ϕ i y i . Φ( y ) = i =1 i =0 Then for a j = f ( ω j ) ,       a 0 a 1 a t − 1 φ 0 a t · · · a 1 a 2 a t φ 1 a t +1   · · ·        =  . . . .  .   .  ...   . . . . .   . . . . .   a t − 1 a t a 2 t − 2 φ t − 1 a 2 t − 1 · · · � �� � = H t ▶ i.e., exponents of f are encoded in a solution to a Hankel system. ▶ Corollary: H t ′ is singular for t ′ > t . 6

Prony’s Method for Interpolation in the Power Basis 1 Inputs: Error-free ■ for t -sparse f ∈ K[ x ], bounds B ≥ t , D ≥ deg( f ) 1. Choose ω ∈ K of order > D . Let x j = ω j . Evaluate a j = ■ ( x j ) for j = 0 , 1 , . . . , 2 B − 1. 1 See also Ben-Or–Tiwari; 1988 7

Prony’s Method for Interpolation in the Power Basis 1 Inputs: Error-free ■ for t -sparse f ∈ K[ x ], bounds B ≥ t , D ≥ deg( f ) 1. Choose ω ∈ K of order > D . Let x j = ω j . Evaluate a j = ■ ( x j ) for j = 0 , 1 , . . . , 2 B − 1. 2. Determine t ≤ B as the largest value such that H t is nonsingular. 1 See also Ben-Or–Tiwari; 1988 7

Prony’s Method for Interpolation in the Power Basis 1 Inputs: Error-free ■ for t -sparse f ∈ K[ x ], bounds B ≥ t , D ≥ deg( f ) 1. Choose ω ∈ K of order > D . Let x j = ω j . Evaluate a j = ■ ( x j ) for j = 0 , 1 , . . . , 2 B − 1. 2. Determine t ≤ B as the largest value such that H t is nonsingular. 3. Solve size- t Hankel system to get Φ( y ) with roots ω e i , 1 ≤ i ≤ t . 1 See also Ben-Or–Tiwari; 1988 7

Prony’s Method for Interpolation in the Power Basis 1 Inputs: Error-free ■ for t -sparse f ∈ K[ x ], bounds B ≥ t , D ≥ deg( f ) 1. Choose ω ∈ K of order > D . Let x j = ω j . Evaluate a j = ■ ( x j ) for j = 0 , 1 , . . . , 2 B − 1. 2. Determine t ≤ B as the largest value such that H t is nonsingular. 3. Solve size- t Hankel system to get Φ( y ) with roots ω e i , 1 ≤ i ≤ t . 4. Factor Φ to get ω e i . Determine e i from ω e i for 1 ≤ i ≤ t . 1 See also Ben-Or–Tiwari; 1988 7

Prony’s Method for Interpolation in the Power Basis 1 Inputs: Error-free ■ for t -sparse f ∈ K[ x ], bounds B ≥ t , D ≥ deg( f ) 1. Choose ω ∈ K of order > D . Let x j = ω j . Evaluate a j = ■ ( x j ) for j = 0 , 1 , . . . , 2 B − 1. 2. Determine t ≤ B as the largest value such that H t is nonsingular. 3. Solve size- t Hankel system to get Φ( y ) with roots ω e i , 1 ≤ i ≤ t . 4. Factor Φ to get ω e i . Determine e i from ω e i for 1 ≤ i ≤ t . 5. Obtain the coefficients c i as the solution to the Vandermonde system       x e 1 x e 2 x e t . . . c 1 a 0 0 0 0 x e 1 x e 2 x e t . . . c 2 a 1       1 1 1        =  , . . . . . ...       . . . . . . . . . .     x e 1 x e 2 x e t . . . c t a t − 1 t − 1 t − 1 t − 1 and output f = ∑ t i =1 c i x e i . 1 See also Ben-Or–Tiwari; 1988 7

Related Work - Sparse Chebyshev Interpolation Theorem (Lakshman, Saunders; 1995) Let f ( x ) = ∑ t i =1 c i T δ i ( x ) ∈ R [ x ] , ξ > 1 , and t − 1 t ∏ ∑ ( y − T δ i ( ξ )) = y t + ϕ i y i . Φ( y ) = i =1 i =0 Then for a j = f ( T j ( ξ )) , ([ a i + j ) [ ϕ i ] t − 1 [ a | i − j | ] t − 1 ] t [ a t + i + a t − i ] t − 1 + i =0 = i =0 . i , j =0 i , j =0 � �� � � �� � H t T t ▶ i.e., indices δ i are encoded in solution to a Hankel+Toeplitz system. ▶ One can show H t ′ + T t ′ is nonsingular for t ′ > t. 8

Lakshman–Saunders Sparse Chebyshev Interpolation Inputs: Error-free ■ for t -sparse f ∈ R [ x ]; bounds B ≥ t , D ≥ deg( f ) 1. Choose ξ > 1. Let x j = T j ( ξ ). Evaluate a j = ■ ( x j ) for j = 0 , . . . , 2 B − 1. 9

Lakshman–Saunders Sparse Chebyshev Interpolation Inputs: Error-free ■ for t -sparse f ∈ R [ x ]; bounds B ≥ t , D ≥ deg( f ) 1. Choose ξ > 1. Let x j = T j ( ξ ). Evaluate a j = ■ ( x j ) for j = 0 , . . . , 2 B − 1. 2. Determine t ≤ B as the largest value such that H t + T t is nonsingular. 9

Lakshman–Saunders Sparse Chebyshev Interpolation Inputs: Error-free ■ for t -sparse f ∈ R [ x ]; bounds B ≥ t , D ≥ deg( f ) 1. Choose ξ > 1. Let x j = T j ( ξ ). Evaluate a j = ■ ( x j ) for j = 0 , . . . , 2 B − 1. 2. Determine t ≤ B as the largest value such that H t + T t is nonsingular. 3. Solve Hankel+Toeplitz system to obtain Φ( y ) with roots T δ j ( ξ ). 9